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March 1981
sus
1.54
SPRING ICE JAMS IN STREAM CHANNELS
PHYSICAL PRINCIPlES AND QUANTITATIVE ANAlYSIS
lu.A. Deev and A.F . Popov
UNIVERSI TY O F ALASKA
ARCTIC f'" , ... ,.. .... -" ... • '-I FORMA 110M
I ' • \ ( , -R
ic:· ":;;O>HT
~NO-!.."-"'~. AlA~ 99501
UNITED STATES ARMY
CORPS OF ENGINEERS
COLD REGIONS RESEARCH AND ENGINEERING LABORATORY
HANOVER. NEW HAMPSHIRE .. U.S.A .
O.sttibulioot llmllrd to U S Cowmn"'"l ... .,.;., only,
Otlwr r~UHII f01 ihls document m •lll M ..t.....t to USACIIIUL .
NOTICE
The .::on tents of thil publication have been translated as presented in the original text. No
attempt has been made to verify the accuracy of any statement contained herein . Tbil
translation is publilbed without copy editing or gapbics preparation in order to expedite
the disaemination of information ..
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Draft Trantlation 7S9 .. TITLil (-..,de) t. TY"'Il 01' ltiE."'O"T I ll'll ... OO COVIItllD
SPRING ICE JAMS IN STREAM CHANNElS Translation
Pb)'lical Principles and Quanti tali~ Aaalylis •• ltlllti'OftMING 0110. "E"'DIIT NUMel:lt
'7. AUTHO'WII) •• CONTitACT Olt OltA.NT NUM81llt(e)
lu.A. Deev and A.F. Popov
t . "'lllti'OitiiiNCI OltOANIZATION HAMil AND ADOitllU 10. "'ltOG._AM ELlEMENT. "'IIOJIECT, TASK
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1'7. DISTiti8UTION STATIEIIIlMT (ef lite ... .,._, onleft4 In •r•dr 20, It.,,__,,_ lt .. tt)
II. IU~DIDTAIIY MOTU
Trlllllation by lntenutional TI'IUlations, West Peabody, MaaachUJetts .
CRREL Btbiiotraphy no. 33-2832 .
lt. KIEY WOitOS (C..fl-on ,..._ .. ef• II fleceeeary -l*"fll:r ., •roclr ,_,._)
Ana!ysil (mathemat!o;s) Ice conditions
Experimentation Ice floes
Icebound riten Ice jams
Ice breakup .. A-TitACT rc-----... II ., ... ...,,, ., .... ~)
l.:e ,;an. u nawral, reauiar formatiooa with aped fie, but not identical forma, sizes and properties were examined. This
permitted 1 ~riptioo of the theoretical principles of the investipted proces~a of jam formation in rellti~ly total
form. The material presented includes a) the estabUIIunent of initial tbe()retical coo.cepts of a model of ice m~men.t
and ice jams; b) an in'fttiptioo of tbe causes and formation .mechanilm of ice jams and their special features as a func·
tion of the concrete conditions of the external en.Uonment ; c) a quantitati.¥e analysis of the elements or jam formation
in the form uted for solms applied problems; and d) theoretical bales an dl practical collliderations for methodJ of
combat tins ice ,jams and controWns them.
Unclallified
• •
•
Iu. A. Deev, A. F. Popov
SPRING ICE JAMS IN CHANNEL
FLOWS
Physical Principles and
Quantitative Analysis
/seal/ Leningrad Gid.rometeoizdat,
1978
/109/
•
[Tt-is pa.ge ie at the end of the book in the Russian. J
TABLE OF CONTENTS
Foreword
Russian page
3
Conventional Designations
[List of symbols used}
Chapter 1. Physical Premises and
Theoretical Models for Studying
Ice Jams
1.1. Properties and special
features of the investigated
medium
1.2. Testing and conditions of
the theoretical inveatiga.tion
of the investigated. object
1. 3 .• Rheological diagram of the
formation of 1.ce jams. A system
of initial equations and supple-
mentary relationships
1. 4. Physical modellinq of ice
jams
Chapter 2. Jam-Less Regime: of Ice
Movement and Losses of Ice-
Block Stability
2.1. Channel throughput in
jam-less ice floating
2.2. General conditions of ice-
block stability loss. Hummocking
and plunging1 of ice blocks in
collisions with obstacles
2.3. Tightening of blocks i n
collision with an obstacle
2.4. The stressed state and losses
of st.ability of a single-layer
ice-block a .ccretion
Chapter 3. Emergence and Formation
of Ice Jams
3.1. Causes and focal points of
ice jams in field observations
3.2. General conditions of jam
formo11tion . Jam classificatione
5
7
7
12
15
24
27
27
32
36
42
45
45
49
English page
1
5
7
7
16
22
37
43
43
51
58
67
72
72
79
/110/
Russian page
3 . 3 •. Formation of jmna under
stationary cor.ditions. Emer-
gence and. consolidation of jams
3. 4. For11ation of L':le j,UI core
3.5. Formation and packing of
the jam body
Chapter 4. Equilibrium Jam State
4.1. Regularities of ice distri-
bution and the equilibrium state
53
57
63
69
of the jam body 69
4 .2. Conditions of static
equilibrium of jaas 72
4.3. Level head and other char-
acteristics of jams under
stationary conditions 75
4.4. Jams with a limited forma-
tion and jams under nonstationary
conditions 79
Chapter 5. Practical Assumptionf for
Calculation and. Regulation of Jam
Formation 83
5 .1. Bydromechanical principles
of counte.racting jams and methods
for jam control 83
5. 2. Some practical conside·rati ons
for the calculat ion and control
of jams 89
5.3. Longitudinal dimensions and
block form 96
Conclwsion 102
References 103
.c.nglish page
86
93
101
113
113
119
123
131
136
136
146
159
158
170
('l'he last page of the book contains printing informati on]
/3/*
Fo.reword
Ice jama, a reqular and widespread phenomenon in the freez-
ing and thawing of the many rivers of the USSR, can cause con-
siderable loss as a conse.quence of the. floods that they cause and
the ice damage t .o water resources equipn.ent, ships, etc. The
losses are o .ften. aggravated as a consequence of the sharply expressed
elementary nature of the phenomenon, the inten.se dynamics o .f the
process of jam formation a .nd the inconsistency from one year to
the next of the size and location of these jams. In the total
complex of water resources practice, the formation of jams, includ-
ing artificial ones, can have a positive eff.ect, for example, for
purposes of improving the irrigation of .flood plains, reducing
th.e intensity of ice moveMent, and preventing ice jams in low-lying
sections of channel flows.
Under the conditions of an intensive water .resources organiza-
tion, the problem of mana.ging ice movement includi·ng its regulation,
combatting ice obstacles, predictions,and a quantitative ana-
lysis of ice· jams should be classified as an important hy~raulic
*Numbers in slashes in the lef.·t-hand margin indicate the original
Russian page.
2.
eJ19in .. ring problem. At the present time there is a large accumula-
tion of •teriala investigating the causes and conditions of the
for .. tion of spring ice jaaa. These investigations are to a great
extent lillit:ed to a qualitative st1.~dy of the phenomenon, although
in recent years a nUIIber of i11portant investigations have appeared
devoted to a quantitative analysis of the individual problems of
jam fo~tion.
A cardinal problem still remains with respect to the develop-
-nt of a general theory of ice j.... The present study is de-
voted to solving some questions .related to this problem. Con-
side.rable apace is devoted to an analysis (based on physics
wit.h the dra.wing in of special fields of mechanics) of the
recJularity of ice jaas which arise in the process of tha..,ing and
ice JDOvement in channel flows. Investigations i .n this area, bequn
in 1948 by Iu. A. Deev, include the theoretical study of ice jams
cQCDbined with field. obse.rvations, experiments, and lite.ra.ture
sources.
We examined ice jams as regular natural fo·rmations with
specific, but not identical forms, sizes, and properties. The
individual properties of particles of a larqe-uni t systen' required
uainq metru>ds of investigation somewhat different from those
usually used for the study of accumulations of •finely crushed
ice•, •qranulated blocks•, etc. (see. Section 1. 2); these were considered
here in apri.ng ice. j ama comprised of the accumula.tion of. rela-
tively large ice blocks. This permitted a description of the
3.
/4/ theoret.ical prin.ciples of the investigated processes of jam forma-
tion in relatively total form. Ind.ividual questions relative to
the general theory of ice moveme.nt are included in the examination.
The material presented includes:
(a) The establishment of initial theoret.ical concepts o .f
a IDOd.el of ice movement and ice jams;
(b) An, investigation of· the causes and formation mechanism
of ice jams and their speciul features as a function of the con-
crete c .ondition.s of the 3Xternal environmen.t;
(c) A quantita.tive analysis of the elements of jam forma-
tion in the form used for solving applied. problems:
(d) Theoretical banes and practical considerations for
methods of combatting ice jams and controlling them.
Th.e very comprehensive set of questions and the large volume
of investigations tha.t enter into the problem of ice jams did not
permit exami.ning all sides of the. investigated phenomenon with the
same detai.l within the limits of t .he present st.udy. l'his ·pre-
determined the necessity of future supplements, rev h. ad editions
and corrected versions of some of. the provisions set forth in the.
book . Separate. aspects can be perceived, such as dlscussions that
fu l,ly conform to the modern stage of investigation of t h is phenomena .
The princ:iples of the established regularities correspond to
th.e results of fiel d observations and agree wi.th experimental
studies related to the quantitative aspects of t .he process; they
can be used in solving practical and ap·pli.ed problems as well as
4.
in .baaJ.c re•ea.rch.
Part of the eata.b1Uibe5 retJUlari.t:i• and iSependences ob-
vioud.y 1UY be ~U•d to antu.n ice jil!lla, but the spet:ial
featu,:-e.e of tbeae latter -must be taken into in•e•tiqation. In-
cluded. here are: jamlesa paasaqe anc'S holdi.nq back aecumulations
of ice blocks. losses of st~i:lity o'f ice· blocks in collisions,
the atx:a.i.ned IJ'tate .of the ice accretion, tbe f~~tion of jams, and
tbe incre .. e in ice thickne:aa in theaa., etc. However, autumn. ice jams
~s a whole require special inveatiqat~ans and ar~ not examined
in tbe preaent study~
It. shoulc'S be noted that durinq the time the' stu.dy was
conCSucted, i.ts i:ndividual -.apeC't11 ~e studied 'by other research-
era; however, the authors introduce all of their conclusions into
the book, since thi.• helps to maintain the integrity of the dis-
cuarion and points out d.if'ferences in tbe initial poai tiona of
·the .analytd.cal method itself. :some general determinations and
aspeet.a that touch upon ice ~t and ice jams . .are 'included
in the study in order to s .iJUplify its utili-zation by a broad ci.rcle
of water resourcea specialists.
The authors express their sincere thanks to Profe•sor B. B.
BoqOiilovakii, to s. N. Buatov, and F. I. Bydin, Ph. D., 'Techni-
cal s .cienoe:;; .R-v. Donebenko, and A. !f. Chizhov·, Ca:Dd. Technical
Seiencea; L-JL Marqol.in and A. Ya. Rybkina, Cand. Geoqrapbical
Sci-encea f.or their va~uable observation• and reC'OIIImendations, raade
by reading and criticizinq the ~~anuscript-: tbeee-were taken into
considera·tion in preparing the study for publication.
/5/
s.
Conventional Designations
Q, q • B, H, 8, L, Bo -discharge, specific discharge, depth,
width, len.gth of a flow section, "effe.ctive"
flow width
u, u 0 = ru, uk and up -flow rate: surface, average, critical, and
guaranteed tightening of t.he ice. blocks (see
1, b, h -
section 2.1)
uv -wind velocity
r -transfer coefficient from surface velocity
to the central
head jam level in the given line, of
direction (flowing), maximum level and
level corresponding to the stable state
of a single layer of ice block a .ccreti.on
length, width, and thickness of the ice block
thickness of the immersed section of the
ice block accretion in the given line
(flow line) and the maximum thickness
v, vk, v 1 , v 2 , and v 3 -velocity of ice movement: total, critical,
and guaranteeing hummock formation, sub-
v 0 = u-v -
s and SP -
merging and piling up of the ice blocks;
relative velocity of ice movement
ice discharge, total and probable amount
(according to quantity and size of the ice
blocks)
I
E -
'·
-ice blodt t.lrroggb-put (drift) of the flow
and jam (limiting) aecti.ona
specific gravit}' and den•it}' of the water and ice
.blocks (and their accretion)
-porosity coefficient o£ the ice ~ccretion
-populati.on coefficient of ice movement
3, T, 3p' a .nd Up
A1 , A2 , A3 , At, A1 ,
As' Ad, ~
P1• P2' P3• P4• Ps·
P6' Po
-total and specific discharge and co-
efficient of water filtration through
the jam
-kinetic renergy of ·the ice bloc~, pressure
energy of the ice a ·ccretion (including
the water plus ice total!.),. the kinetic
flow en~rgy and the potential flow energy;
-critical values of the work of force• act-
ing on the ice block, which, if •urpa••ed,
and relcrtive to t .he indices, l:eads to a break-
down in the stability of the floating
blocks, their plunging (tightening)
piling up, pushing against each
other, movement under the ice, the sur-
mounting of coupling forces bet.ween the
block.&, between the blocks and the Ftream
bed, a breakdown of blocks;
-force-; of hydrodynamic pressure (pro-
file), water friction against the ice sur-
a, T, ak'
a i, and a 'P
(bend)
6a.
fa,ce, th.e component of ice· weight, wind
pressure, filtration pressure, centri-
fugal, shore resistance, relative to
a su.rface ·unit (see section l. 3). The
additional index y denotes the trans-
verse component of these forces .
Tk' as, -normal and tangential stresses, their
(comp.)
critical values, breakdown stresses
in compression and bending, and the
strength of the jam obstacle.
-critical values of normal stresses at
which there arises a packing of the
accretion, ice hummocking, jam pileup
of ice, the breakdown of ~he jam
F = 1 lll h2
1 and F 2 = -active and passive pressure (resistance)
2 · ll2hl of the ice accretion (see section l. 3)
ko, kl, k2, k and k. v l. -resistance (pressure} coefficients of
the water flow to the block movement;
general, form, friction, wind pressure,
and proportionality coefficient
slopes of t he bottom, the water surfa~e (longi-
tudinal and crosswise} and in the ice
accretion jam
~ , 41 1 , ~ 2 , n , r;; and B -coefficients of internal friction in
the block accretion underwater, f.riction
C and n
e
6b.
~ainat the ahore a.nd tb~ bed., ice
againat ice, lateral preaaure, non-
pri ... tic fora of the channel, and
the •arch effect• (aee section• 1.1,
2.1)
elope angles. of th.e botta. slope
of the jaaa and blocks in the jaa (aee
Fig. 6)
-the Chezy coefficient and the rough-
ness coefficient
-time an.d temperature:
-probability coef.ficient (reliability)
of the given calculated value
-angle of internal friction of ice accre-
tion
-line corresponding to the stable state
of a one-layer block accretion
7.
Chapter 1
Physical Premis.es and Theoretical Models for Studying Ice Jams
1. 1. Propertie.s and Special Features of the Investigated
Medium
Ice: janlB are accumulations of ice blocks which cause a
crowding of the usetu:.. channel cross se.ction a.nd a rise in the
water level associated with this; they form in the case when--
for a specific quantity, strength, and pressure energy of the
blocks--the throughput of the channel is insufficient for their
tram~portion. The re.sults of jam studies* show that their emer-
gence, formation me.chanism, distribution of ice in the jam, the· value,
of level rise, and other characte~istics are determined by many
factors and the.ir various combinations. Thes·e factors may be
cla.ssified in three groups:
(1) Hydrometeorological, including the velocity and direc-
tion of the water current , the mechanical and geometric characte·r-
istics of the blocks that participate in the ice movement, the
nature and sequence of the river opening and the devel.opment. of
high water, as well a .s the preceding ice regime of the river,
weather conditions of the autumn-winter and spring pet:iods, etc.;
(2) Geomorphological, which determine the nature and
special features of the channel structure in longitudinal,cross-
wise, and planar re·lationships, the nature of the flood plain,
trough , etc. ~
(3) Factors. of human activity, i.e ., various water resources
equipment crowding the channel, as well as measureme:nts for control-
*Some of the publi.shed studies on this problem are given in the
refe.rences. A more complete bibliography may be f.ound in t .he
Studies of coordina.ted meetings on hydrotechnology, the Stud-
ies of the GGI,and other special publications.
7a.
ling the discharge, channel, and intensity of ice move.ent,
etc.
So.e of the enumerated factors are interdependent, mutually
dependent, complex (sum.ed up) phenomena, the investigation of
which ia possible only after study of the regularity of action
on them of separate •ela.entary• factors, which determine these
phenome,na.
/8/
8 .
Thu•, ice j ... are a coaplex, aultifaceted. natural fo .naa-
tion. In o.rder to inve•tigate them it is nec.es•ary to f .irst
e•tabU.•h the physical nature of the phenQaenon and. the tora of
.. terial motion corre•ponding to it. This in turn peraits
deteTai ning claaaification• and pbysic~l law• necea•ary for
the investigation and drawing up a diagram of the phamonenon
with delineation of its lllo!Un. interconnections (external and in-
terna.l). The baaic properties of the object subject to the in-
ve•tigation and th.e method of: its analysis are established on the,
basis of the above prerequisites.
It follows from the determination of an ice jam that this
pheno~~enon is in ita physical nature a process of retardin.g
(building up) ice blocks entering a certain flew segment during
ice movement or during the breaking open of the ice cover. There-
f .ore, the flow section on which the process of ice block retarda-
tion a ·riaes and develops under c ·ertain conditions can be called
a jam section.
Below, we will use· the term ice blocks to infer monolithic
polycrystalli.ne solid bodies of finite dimensions, prod.ucts of the
breaking up of the ice cover, capable of travelLing in the flow
channel. under the action of hydrodynamic and. other forces, due
to the abeence of rigid bo.nds with the surrounding enviroruMnt and
due to their size. The breakdown of the ice cover into very large,
less mobile units, "ice fields", whose longitud.inal dimension• can
exceed the width of the channel, usually precedes the eme.rgence o .f
9.
ice blocks. Desp.i te the usual te11p0rary existence of these fo·.l"1U
inte~diata between the ice cover and the ice blocks, they may
have a substantial influence on t .he opening process of the chan-
ne·l and are the iaaediate cause of ice jams.
Ice· 'blocks, inheriting properties from their natural pa.rent,
the ice cover, acquire a number of' special features, v.hich, al-
though they vary as a function o .f the concrete condi.tions of
their fon~Ation and exi.stence, possess, however, a specific gen-
eral character, whi.ch was examin.ed earlier in /19/.
In their phys.ical properties, ice blocks are deformed solid
bodies of nonuniform and anisotropic structure, of nonprismatic
fo.ra and nonuniform. size. In the ""'vement of ice, these char-
acteristics, including. the relative density of distribution of
the ice blocks in the channel are nonanalytical, discontinuous.
(or piecewise continuous) functions of the. coordinates and
time. The noted circumstance is justified both with respect
to a fixed. channe.l section as well as to the mobile system of
ice-block accretion. This is due to the fact that factors which
determine the emergence, characteristics, and behavior of ice
blocks in the flow are themselves variables along coordinates
and in t .ime.
/9/ Various forms of ice-block movement can arise and develop
next t:o each other in this process. In order to make an analy-
sis, it is expedient to classify t~hese as:
(1) The free JDOvement of ice, blocks, including perio<Uc
cont.act wi.th soli.d bodies:
10.
(2) The .a•••nt of ic:e bloc:Jta in the for. of on-layer
(plane or ba..ock) accretion:
(.3) Move1Mnt of ice bl;)Cka w:itb the foraation (or break-
down) of multilayer jaa acc:r:etion of ice .blocks.
In their 110ve.ent, ice blocks are subjected to nonuniforw
external actions and exhibit differett rheological properties
ot elasticitity, plasticity, brittleness, etc. Therefore, in
a rapid •dynaaic• application of external forces,tbe elastic
properties O·f the ice step into f .irat place, while in slow appli-
cation, plastic deformations, fluidity of the ice, et.c. are
iaporta.nt.
In ice movement it is possible to determine two types of
force transfer in the accretio,n of ice blocks e~ined as a
discontinQOQs ayst-, consisting of unbroken (continuous) deforaable
bodies of finite dilaensions.
(1) Continual force trana.fer -in3ide the ice block•;
(2) Discrete transfer -between the ice blocks with non.-
atationary (rheonomic) connections and force transfer of the
contact type. In a single-layer accretion there are. unidirec-
tional conne.ctions, which cre.!lte only compressive stresea (in
the absence of freezing of the ice blocks), while in aultilayer
accretions, by remaining rheono•ic, they possess in part a
ci.al ch«racter du.e to tbe pre.s•ure of tbe ice blocks against each
other, of r89elation, and of filling the spaces between the ioe
blocks in the process of plastic deformation (fluidity) of tbe ice.
/10/
11.
The· compx:ession reaistanceof the accretion of ice blocks
accumulates froiD those types of counteractions which a deforme·d
system can be subject to: The internal resistance of the ice
blocks to compression and breakdown (to cleavage, shearing),
resistance to slipping against each other and resistance t .o
expulsion (p.ressing out) of: the ice blocks. From the viewpoint
of meclla.nics, the formation of a jam is a process of particle
displacements of a system from a state of less stable equilibrium
to a more stable state, which can be accompanied by a change :l.n
the fox:m and size of these particles (a breakdown of the ice blocks)
as a consequence of their deformation.
By taking a general rheological stance towax:d tha properties
of the investigated medium, the evolution of a single-layer accre-
tion of ice blocks during an increase in comp.ressi ve fox:ce can be
p.resented with a known schematization of the phenomenon as follows.
When the compressive force has reache.d a certain critical value
a 2 , sufficient to overcome the force of resistance of the ice
blocks toward slipping along contact sections, a proce.ss of coming
together occurs, a consolidation of the ice blocks to a more
dense packing, which is accompanied by a scattering of t 'he contact-
ing side surfaces of the ice blocks, a shattering of fine i.ce
chi.ps, etc.
A fl·rther increase in the stress to a 3 leads to the forma-
tion of slipping (displacement) areas within the ice blocks with
shearing along their contacting edges (usually less strong than
the inside sect.ion of the ice blocks /19/), at a certain angle to
12.
the direction of action a 3 • A portion of the pressu.re on these
areas is trans.ferred to a plane normal to a 3 , and if its value
is high enough, a c .rHping and under-r..aing*of the ice blocks
againat each other occurs (the pheno .. non of ice h~cking).
This process h made easier by the oblique shearing of edges in
the direction of large tangential forces and .by th.e bending of
not very •rigid• ice blocks (.see Section 1. 3).
Fig. l. Breakdown of the longitudinal stability (swelling) o.f
the ice cover in a sandbank.
Koweve.r, losses in t .he longitudinal stability with wave-form
~. rupture of the blocks and the subsequent hummocking
of their fragments can occur even ea.rlier for large size ice
blocks--with a smaller compressive force--as follows from the
theory of t.he resistance o .f material·s. Such a breakdown is
especially characteristic for ice fields and the ice cover in the
sandbanks :>f many rivers, where it arises unde.r the pressure of
the increasing head of the water after the formation of border
zones and transverse cracks separating the ice on the sandbank from
the above-situated reacheS. Such a phenomena., was clearly observed
in the _,del in~estigation (Fig. 1).
*or plunging--Translator
/11/
13.
Jams can arise also directly in the collision of floating ice
blocks with an obstacle (the ice field, the edge of the unbroken
ice cover, etc.), omitting the intermediate stage mentioned above.
This happens in the case when the kinetic energy of the ice block
is sufficient £or their packing or under-plunging. A similar
phenomenon may take place directly in the brea.king open of the
i ce cover under conditions of considerable compressive force.
With a very rapid growth of compre.ssive pres.sure in the ice
block accretions (or with a . large. store of kinetic energy of
the blocks knocking against. each other) , the creeping process,
which requires a known time, cannot successfully develop and
af.ter a .tta.ining the dynamic. stability limit, the ice bloc.Jts will
break down. With a s.low increase in the load, which does not
exceed the rate of the dislocation shifts within the blocks,
plastic deformation--fluidity with a bending and warping of the
ice blocks will prevail. At a temperature close to 2.ero, the
examined phenomenon, as experimental investigations show /19/,
will be accompanied by a thawing of the ice. Under conditions
of ice movement. in rivers, this process usua.lly doe.s not develop,
but in the defo,rmation and breakdown of ice packings in jams, it
can play a significant role.
After a jam has emerged, two processes occur in it that
are opposite in their action direction. On the one hand, under
the effects of ice pressure and the water mass, a consolidation
of the ice mass of the jam occurs due to packing, the phenomenon
14.
of regelation, and fillinq the apaces between tbe ice blocks dur-
.i:aq their plastic deforaation . At teeperatur• below free.zing,
this process is suppleaented by a freezing of the ice blocks
which leads to an. increaae in t .he elastic connections in the ice
JMsa. On the other hand, the piling-up atrenqt.h is· weakened due
to the aechanical and beat action of the water filte.ri.ng and flow-
ing under the ice as well as by the insulation, the conyective
heat suppl.Y f'roaa the ataospbere, et:.c.
An. increase in the longit.udinal. dimensions of the. jaa and
in the head. of the level in the: continuing ice movement can lead
t.o the fact that compressive stresses exceed the value of internal
resistance of the ice-block accretion. In this case., the accretion
thickness should increase (ice shifts in the jam), and with
an insufficient. quantity of ice, it brea..ks down by a shift of the
ice block.s . with respect to each other or of the entire mass of ice
along the flov bed. It is obvious that the nature of breakdown
depends on the ratio of the va l ues for the coefficients of interna~
friction in the. accretion of ice blocks and their coupling with
banks, and. also depends on the intens.i ty of the growth of pressure
and level head. and on the nature of the obstacle to the jam itself.
The packing density of the ice-block accretion is analogous
to the process of coming togther und.er pressure of skeletal
earth part.ic.les (cal led compre:ssive contraction in soil mechanics),
but has a more complex c.ha.racter. This character is explained by
deformation, nonuniform dimensions, and by t .he s ·tress of the ice blocks,
the phenomenon of regelation and recrystallization of the ice
15.
under pressure, as well as by the scatterin.g and cleaving of
the contacting edges. In a single-layer accretion of ice blocks,
the compressive contractions correspond to plane deforaation,
and in multi-layer accretion, to volumetric deformation.
With an increase in compression o to a limiting value
equal to the internal reaction of the plane ice-block accretion,
the porosity coefficient e: .reaches a minimum value. A f·urther
increase in stress causes a loss of stability of the ice blocks,
their hummmocking, and piling up with new compressive contraction,
deformation, or breakdown of the blocks.
It follows fran the exanined properties of the investigated medium
that the overall dependence e: = f(o ) , is a multifactor piecewise
/12/ continuous function and cannot be expressed in simple analytical form.
The investigation of this phenomenon is insufficient to construct
a functional dependen::e; therefore an evaluation of the limiting
porosity values can now be based only on empirical data.
For an ideal free-flowing body, the coefficient of inter-
action with the walls is determined by a constant value for the
product of thrust and friction coefficients. In the block
accretion, this value is a variable dependent on the nature of
the equilibrium state, the nature and speed of movement of the
accretion, the channel structure, etc. Thus, according to the
investigative data of the Central Scientific Research Institute
of Ice Melting /57/, the interaction coefficient of the flowing
material with the banks decreases according to a nonlinear law
16.
vi.th an increaae in the Froude number and with a videninq of the
c .bannel, while it increaaea during • contraction.
A. ~·on, R. Hauaaer, E •. Pariset: /15/ and D. F. Panfilov
/58/ introduced a couplinq coefficient. al.a into the condition
of interaction vitb the banks. Hovever, accordinq to the· a.ta
of the Canadian investiqators, it haa a substantial value only
in the case of a . thin ice accretion. They note also that little
study baa been made of the internal friction coe·fficient of cUi accre-
tion, and exaaine it as a variable. The. results of oar inveatiqa-
tiona also· show that the breakdown in the stability of the blocks
witb relatively large lonqitudinal di..JDenaions ia not deterained
by tllis coefficient, but by other factors (aee Section 2.4).
In addition, due to the relative mobility of· the blocJta with-
in the accretion, the so-called "arch effect" can arise reqularly,
which is examined in the theory of free-flowing bodies /25/.
It ia an additional reCJl.stance to lonqitudinal coapresai.on and
to the movement of the ice accretion. This resistance can be considered
by introducing into the friction f .orce of the shore some coeffi-
cient s ~ 1, which is. also a variable c!epending in our inveatiqa-
t.ion on ·tbe accretion atructure, the relative size of tbe ice
block, et.c. (see sect ion 2. 1. ) .
1.2. Testing and Conditions of the Theoretical Investigation of
the rn.veatigated Object
In the study of ice jams, the object of the investigations
:..s a regular .formation consistinq of interacting individualized
and de.formable solid bodies--ice blocks: the primary external
/13/
17.
interconnections o .f these fol"lMtions a .re exaain.ed by way of a
aecbanical interaction with the surrounding 1\ediua. Even in the
study o .f such processes as, for exaaple, the external and i n ternal
thawing of the ice, the formation of •temperature cracks•, the
thermody naaical and ph:ysica.l-cheaical effects in the models of
calcu.lation taken for the investigation\ are diaregard.ed, since
they are val:ues of a lesser magnit ude.
The construction of an ove·ralll theoretical model which
takes. in·to consideration all of the pro,perties and effects of'
the object noted above, is difficult due to both mathematical
complexity as well as the insufficient development of a theory
o .f d.iscontinuous deformable bodies and the extent of study of the
object itself. In addition, an attempt at including in one dia-
gram all of the basic proper t ies o f the i n vestigated medium would
lead ·to a complicated model little suited for investigation. This
forces us to proceed by applying simplified diagrams designed to
solve individual problems /70/.
There is already available in ice technology a new ·test
for applying similar systems. The model of a linear elastic
body with the introduction o .f certain additional conditions and
assumptions is wide l y used for the •dynamic interaction• of ice
blocks /39, 61/.
Model s o f a nonlinear elastic body, an elastic-plastic
body /14/, etc. are examined f .or describing ice behavior beyond
the elasticity limits . With a small rate of deformation, plast.ic
18.
deforaat1ona of ice are a.xlelled by a viecoua liquid diagram.
B. v. P.roskuryakov and v. P. Berdennikov /65/, for example,
applied such a diagram to the investigation of teaape_rature de-
foraations in aJl ice mass.
Ice accretions form a compact maaa with specific properties
and regularities in their movement and evolutio·n under the action
of eoapressive forces. However, they can only partially be
described by existing models of continuous or discrete media.
Thus, a viscous liquid mode-l ia used to study the established
.oveaent and equilibrium state of an accretion. of small-size
particles (slush ice) /23/. With larger particles -•crushed ice•,
"crushed blocks" --as,pects of discontinuous body theory are applied.
Here, models of an "ideal free-flowing body" /51/, a free-f'lowing
body with interparticle coupling and variable coefficients of
inte:rnal friction and interaction with the shore /7, 57/, and also
tClking into consideration the shore-coupling coefficient /15, 57/
w·ere employed.
The force theory of K. Yansen (A. Gan'on, v. P. Berdennikov,
etc.) and A. Kako (B. Mishel', I. Ulle) and the general aspects
of the free-flowing medium theory (D. F. Panfilov, et al) were
used for an evaluation of the pressure increase in an ice accre-
tion and its equilibrium state.
The limiting value of stress which determines the jam strength
is evaluated either by the limit of ice strength or by the limiting
value of internal (passive) resistance to slipping which is obvious-
/14/
19.
ly better established /15, 50, 58/. The attractive force* of
water and air f .low is determined by diffe·rent methods:
(1) According to the usual method of st:i'eamline flow
of soli.d bodies in a viscous liquid /39, 61/:
(2) According to the v. M. Makkaveyev diagram with
the breakdown of the current along the vertical into two sections
and by applying to it the Chezy formula /5, 61, 74/;
(3) According to the Strikler diagram /15/;
(4) According to a calculated structure proceeding from
the ·A. Kako diagram where the tangential stresses from the flow
and the component of ice gravity are considered simultaneously
/SO, 86/.
In the general dependence obtained by Berdennikov /6/ for
determining the moving force of ice jam masses, the forces of
flow friction on the ice and on the river bed and the component
'>f ice weight and the water column underneath the ice are
simultaneously considered.
In solving dynamic problems (nonuniform movement, change
in the kinetic energy of the moving ice, etc.), ordinary Euler
equations are used along with the law of kinetic energy change
/36/, including taking into consideration deforn:ation forces
'19/ and the impulsive nature of ice block accretion stoppage
at the obstacle /60/, etc.
A quantitative analysis of jam formation includes hydraulic
calculations of the flow below the ice. The basic theoretical
*This concept is given in more detail in Section 1. 3.
20.
foundations and principles o ,f t .heae calculations are established
in the works of N. N. Pavlovskii, A. N. Rakhma.nov, P. R. Belokon'
and other researchers. In the developaent of these principles
up· t .o the present t:.ime, a number of working diagrasu have been
developed for determining these hydraulic characteristics of
the. sub-i.ce flow. The method f .or determining the Chezy coe.f-
ficient vas examined in a similar case by A. N. Marchuk /46/.
Calculation diagrams· f 'or determining t .'he coefficient of rough-
ness have been set forth by A. A. Sabaneyev, D. F. Panfilov,
V. I. Sinot.in /58, 71, 78/, etc.
A less studied problem is the filtration of water through
the ice accretion jam. There are still no sufficiently caaplete
theore.tical de.velop.ents and calculat.ions along this line.
Closest to the given problem is the question of filtration
through a stone brid.ge studied by s. v. IZbash /28/ et al.
We took the calculation diagram calculated for this case as
the bas.is for examin~tion of filtration through a. jam (see
section 4 .1).
It must be noted that the use of a mathematical apparatus
in the st.udy of ice accretions is complicated by the fact that
the in.t .ernal proceases occurring in them are expressed, as
has bee.n noted, by nonanalytical functions. Theref.ore, an
analysis is usually carried out according to average character-
istics and without taking into consideration the individualized
properties of the particles: of the system. T:he latter may be
21.
considered sufficiently valid for accretions of small particles
(slush, ice dams, crushed ice, etc.) for which the given models
are predominantly utilized. However, for accretions of ~tive
ly large (spring) ice blocks, the consideration of some indi-
vidual properties is necessary for solving single problems.
/15/ Some characteristics can be expressed here only by the law of
distribution of random values. More developed models based on
a continual or "ideally free-flowing" concept of the medium are
insufficient for describing specific properties and features of
the object. A more.rigid investigation based on a discrete
structure of accretion construction and based on principles of
statistical mechanics is made difficult by the insufficient
theoretical development.
Therefore, taking into consideration all of the properties
of interest of the object within th~ framework of classical or
statistical mechanics is practically unachievable. Here, it is
expedient to apply both the continuousand discrete methods of
investigations, as well as elementar~ methods of mechanics, in-
specting and comparing the results with natural and experimental
data in order to solve the various problems. In specific cases
it may be necessary to apply a combined method using different
rheological models, the introduction of supplementary conditions,
probability characteristics, etc. A rheological diagram of jam
formation is created from these considerations and a system of
basic initial equations and additional relationships is estab-
22 •
. U .shed with the use of necesnry "finishing" tr-t:llent of the
existing .odels.
1. 3. Rheoloqical Diagraa of the Fonaation of Ice .Jaaa. A
Syst-o·f Initial Equations and Suppl_.n,tary Jte_latiom-
ships
A rheoloqical diagram constructed on the basis of the requ-
lari ti.es of ice accreti.on evolution examined in sec.t ion 1.1. is
taken fox the initial IDOdel. Here it is considered ·that deforma-
tion diaplace.ent.a in the process o •f ice-jaa formation in the
geneial case cause five basic forma of coapr·esaabil .ity:
(1) Elastic, viscous-plastic--vi thin individual blocks
with thei.r destruc·tion at a > a 1 , s;
(2) Compressive -In ice-block accretion with packing
consolidation due to a change in the blocks' 111utual disposition f . rom
' • •o to ~ • ~ .. :
(3) "Hu.DOcking• -In ice block acc.retion with con-
solida·tion by humaock: forta&tion an<l the loaa of stability
of the ic-e block•;
(4) "Pilings" -In multi-layer (jam) accretion of ice
blocks during formation and breakdown;
(5) Viscous-plastic -In ice-block accretion with a
filling of the spaces between the blocks due to the ice
fl.uidity.
Displacementsof the compressability forms 1, 2, a.nd 5
de·termine for the most part a volumetric deformation of the
for .. , while 3 and 4 - a fonn change, and forma 2-4 correspond
/16/
23.
to finite displacements of large-size particles (ice blocks).
Therefore, the application to this case of mechanics equations
for a continuous medium has a formally conditioned character and
is allowed mainly only to describe critical (limiting) stresses
of states,with the introduction of additional parameters that
take into consideration the special features of the processes
examined. Taking into consideration the noted features of the
behavior of a system, the degree of its study, and experience in
applying simplified diagrams, a rheological model of ice move-
ment is established for its development in stages (Fig. 2) with
increasing compressive stress -from the floating of solitary
ice blocks to the formation and destruction of the ice jam. Models
applied in rheo~ and designations of internal relationships are
used for this /54, 66/ •
., ,.,
•• ,,,, I
~ ·~
~~ji:iJ,:
L~·· _. ---+'~~s __ ._._ t._ !-'t.:..s ----+:.__----lt---P.-,-
Fig. 2. Rheological model of internal connections of ice movement
and ice jams.
Ice movement stages: I -floating of individual ice blocks;
II -single-layer plane accretion of ice blocks with compressive
contractions; Ilia and IIIb -stability losses (hummocking) of
ice blocks: IV -formation of multi-layer ice-block accretion
jams; V -consolidation of the jams; VI -breakdown of the jam
24.
clue to atabilit.y loaaJ 1 -riqid oonaectioll; 2 -elaetie;
3 -Yiaooua; 4 -aNence of reaiat.ance: S and 6 -• internal
friction• reaiat.ance and brittle resiatance of the ice~loclt
accretion.
The fi,rst step is 1110delled with the assuaption that ideal
solid bodies participate in the· p ·rocess. In the second staqe,
in model Ila, ice blocks are a .ssw.ed in the form of elastic-vis-
coua relaxinq bodies, as in study /59/; in Model lib, they are
assumed in the .form of brittle-plastic bodies· with connections
correspond.ing to compressive contraction, while in Mod.el .lie,
where ice-block accretion/shore coupling forces are taJt.en into
consideration -in the form of brittle-plastic bodles possessing
/17/ internal friction . Fo.r the case of the loss of stability of
the ice blocks, the model is made correspondinq to the resulta
of investigations (see Chapter 3) according to the diagraas:
I .Ila -as.suaing the participation of elastic-plastic brittle
bodies in the process, in IIIb -ela~tic-plastic bodies with
internal friction. The same models are used with respect to
stages IV and VI of the formation and breakdown of the jal'R. In
stage v, elastic-viscous-plastic bodies with increasing amounts
of elastic couplings are taken for the ice blocks: this cor-
responds to the case of ice co·nsolidation in the jaa under
conditions of regelation and ice-block freezing that prevail.
A diagram similar to the latt.er, with the additional inclusion
of the Newton cell was developed by T. Tabat /91/ for studying
25.
the fluidity of ocean ice.
The corresponding rheological equations for the considered
stages can be written as follows:
Step IIa -form chal"ge:
D -1c1D -k,D + k1 b;
~ 1'\
Stage Ilb -voluaetr·ic deformation:
'1,-1/vT'"-(1-_G)tri
Stages III, IV, and VIa
Step IIIb
St.age V
( 1.1)
(1. 2)
(1. 3)
(1. 4)
(1. 5)
where On and. D are stress and deformat.ion de:viators; T0 and Tn a1 ·e
spherical tensors, while D, on, T0 , Tn are their rates: k 1 , k 2 ,
k 3 , v an.d G arc rheological coefficients that characterize the
relaxation of stresses, elastic and viscous form change, volumetric
compressibility, and viscosity /4., 54, 66/.
The examined rheological diagram can be taken to solve var-
ioua problema, usings its individual eleme:nts or their combina-
tiona. Thus, the free floating of ice is modelled by sta.ge I,
if the stress created by the water pressure is disregarded, where-
as if the latter is considered, the diagram of combined stages I
and II are used. In the investigation of the interaction of ice
bloc'kB with solid bodie!l, the diagram II-IIlb is taken, which in
a dynamic load stage can be simplified by excluding the viscous
/18/
26.
component of dia.gram II; the value a 3 will correspond here to
the limit of ice strength. In order to study the jaaless pas-
sa.ge of ice blocks and the break.down of ice fields, dia.gram IIa-
IIb-IIh, may be used; while taking into consideration shore
couplin9, a.odel IIa-IIb-II.Ib is used or even a simpler, less
pre.cisc model, without. considering the viscous component in
IIa or just one unit may be used directly: Inb. The process
of breaking open of the ice cove·r may be described by the model
of units II-IIla. Under these conditions, the critical value
a 3 will correspond to the. loss of sta.bility of the ice hlocJts,
the, limit of ice stre.ngth vs. shearing (in the passage o: the
ice blocks through the channel construction) an.d to the limit
of bending streng·th. The model can include stages .II'-II £-IV,
et.c. for the case of .formation o .f the jam direcly in the break-
ing up of· the ice cover.
Stages IIa, III, and IV, which are described by models
that are used in the investigation of discontinuous (free.-flow-
ing) bodies /25/ a.re of basic significance to ·the present in-
vestigation. The theory of free-flowing bodies is developed
more fully in the field of statistics. Problems of dynamics
are solved primarily by the method of limiting equilibri~m -
by the fixation of the state of the sy.stem in times that cor-
respond to the emergence of the stationary regime with specific
values. of the primary parameters.
General conditions for the equi ll ibrium of a free-flowing
body take the form:
/19/
27.
(1. 6)
and the •tability condition:
Drrl(a). (1. 7)
For models II.a, IIIb, and IV, condition (1. 7) can lead to
the requirements:
(a) ln contact sections between the ice blocks:
(1. 8)
(b) On the surfa.ce adjacent to the water:
(1. 9)
(G)
(1.10)
(d) On a free surface:
(1.11)
and .for models IIIa and VI -to the requirement:
(1.12)
where ox z and cr . z are stress components on the coordinates; ,y, x,y,
a and T -stresses acting on the surface of the free-flowing
body; and a~ and T~ are their critical values; F and "' "' x,y,z
Mx,y,z are components of volumetric forces and moments acting
with respect to any pole; o'p is the strength limit of the
obstacle which contributes the compressive .forces in the lee
blocks. Here and .below it wi 11 be assumed that the x axis is
direct.ed along the surface of the flow, the y axis is at right
angles to the flow, and z is down·ward.
Taking into consideration the inter-particle coupling and
the coupling between the ice and the banks (for diagram IIb) a
28.
-the •cou.pU.nq coefficient" which determin .. t.he value of the
initi£1 reaht.ance to diaplaceaent. Eqa. (1. 9-l.ll) det.erllline
the Uaiting c:onditiona, whi.le the equality aigna in Eqs. (l. 7-1.12)
correapon~ to the case of liait.!ng equilibrium. We note here that
the given condition• do not conaider the iaotropy of the ice-block
structure an~ do not 110~el thei.r breakdown.
In order to obtain a clo.ed equation syetem, whi~h permit•
eolving the problen.e eet forth in the etudy, F.qe. (1. 6-1.1 2) are
added aa further dependence• that reflect the epecia.l character
of the behavior of a eyet.elll comprised of deforsable particle• of
finite diaeneione and moving· in a water-air medium under condi-
tiona of occurrence of filtration, f 'reezing, etc. phenOIMna.
1. The balanced equati on that deecribee a change in tbe
quantity of ice in time dt in a :flow eection bounded by eur-
facee Fi and with volume V ie taken as follow•:
(1.13)
while wi th finite, difference• for time lit:
• AS •. ~ s,-:-&.c. .. , ( 1.14 > ·-·
where the totah in the right hand sectio ~• of equations (1.13) and
(1.14) determine the entrance (exit) of the ice through the upper and
lower boundaries of the section and through the cMnne.l of right
and left tributariea; 11 s 0 i• the change in the quantity of ice
in the examined eection due to f.reezing and thawing ; vn is the
c011ponent of ice-block movement speed norma.l to the surf:ace F 1 -P' 4 •
/20/
29.
Equations (1.13) and (1.14) permit classifying ice move-
ment (ice transit) according to the direction of its motion:
(1) Longitudinally-forward and backward (upstream);
(2) Crosswise -through the tributary channel;
(3) Longitudinally-transversely.
Different directional combinations yield 15 types of jams
which theoretically exhaust all possible cases. The longitudinal-
crosswise type movement is observed at the mouth of the tributary,
and the countercurrent flow downstream of large rivers in a wind
surge.
2. The dynamic equations which more fully correspond to
the object's properties are equations of the quantity and moment
of movement of deformable bodies /30/. Their application to
the objectives examined is difficult and it is hardly expedient
at the present stage; however in principle it is not promising
if we proceed from the increasing possibilities of computer
application and development of the theory of ice movement itself.
3. The law regarding a change in the ice-block energy fl ~
can be used in the D. Bernoulli equation form or in another form
which has been applied for this purpose by P. A. Kuznetsov /61/
and by Iu. A. Deev /20/:
• 48 .,.9 +~A, (1.15) . I
where ~ is the elastic energy of the ice block; r A is the
sum of the work of all forces including ~ consumed in plastic
deformation, local crumpling, thawing, etc.
30.
1'lae foraatton of a j .. ia accc:.panied by an. iDCr••e i.n
the apecific enerqy (preaaue head) of the flow froa }0 to \
in order to aurmoamt tbe .additional hydraulic resiatancea h'
which ariae in the useful cross section in ice-block accretions.
With a stationary vater/.ice IDIOvement recJi•e (d/dt. ., 0) •ccord-
ill9 to the law of conservation of enerqy:
.9, .. il.+9h+Ht.+llt .. _ (1.16)
where Bt ia the change in the apecific enerfiY of turbu~lent acatter-
ing; Bf is the energy loas in the filtration of water through the
ice block accretion.
Thia equation, which we examined earlier in /21/, pe~its
G~btaininq additional eharacteristj.ca and cd.teria in the atudy
of jaa fonaation. The particular caae of Eq. (1.16) ia elCAiained
by B. Mishel•/U/ in his criterion of bydrodynaaic stability of
the. ice-cover edge.
4. The forces of interact ion. between the ice blocks and
the aurrounding medium -their caaposition and physical character
-have been exaJIIined in detail earlier /20/. Here there is intro-
duced interaction .forces of ice blocks (and their accretion) with
a water/air .edila and with a.olid bodies contacting the ice blocks,
forcea caused by the earth • s mass and its rotation, interacti.ve
forces between the .blocks thanselves, which arise when they col-
lide, disperse, etc. The action of the_se forces. is not unifora
in its specific contribution to the ja111 formation process. With-
in the framework of the problems presented in this study, the
following forces are taken into consideration.
/21/
31.
(a) The force interaction of water or air currents
(wind pressure) on a solid body can, as is known, lead to the pri-
mary vector P • /J! + ~~ + P: and to the primary moaent. The ..
vector Px corresponds to the force of the so-called head pres-
sure (resistance) of the current, which can be presented as a
...
sum of the force (Pl)x of the hydrodynamic pressure (profile and
...
wave) on the edge of the ice block and the force (P 2 )x caused by
the friction of the current against the ice surface.
In investigating the interaction between the current and
ice blocks, the force of head pressure Px is usually called the
"attractive force" of the current. We will use this term in the
.. -9
following. For a determination of forces (P 1 )x and (P 2 )x in the
study, the general aerodynamic method is taken, in which these
forces that refer to the action area unit are expressed by the
formulas:
(1.17a)
and
( 1. 17b)
(For convenience we w~ll omit writing the index •x• here and below.)
For a rectangular form of ice blocks and their accretion, as
has been suggested by K. N. Korzhavin /39/, this can be written:
(1.18a)
where
(1.18b)
If the transverse component of the current s~ appears,
for example in the bending of the current, the corresponding trans-
32.
verse fore• compone~t (pl, z> y" wttdlctt, i .a dlet.,•aaiaect. wdinq to
foanrJaa.. analogaua to (1.17) aDd (1.1!1) -rqea... 'l'Q 1101 .. the
incli.v-idual probl ... , 1 t is convenient 1!o expresa ( ~) y by tba
depth and radius of the c:uzzeat taJ::Ia.::
wbaz:e k.' =
...... ..,.,,. ,..-.. ...:.--..-.
w.2 • (C, y); IC :::: 1. 3; the:; function
(l.I9)
daotermined accord.inq to A. V. Karauahav'a table /33/1.
The values of c:oe.fficiaa:ta kl' 11 2 , and 11 0 ¢'dill b¥ various
author• a.re diiferent .. In regulati.als fo.r determination of ice
loads ( Sll1 7 6-66J tbe.ae values are: t&ken eqU&l to 1c.1 • 0. 0 5 tt • • 2 I m 4
and k 2 • 0-005 t•a 2 /m4 • Hawever, these data obYioualy relate only
to tlnl ca.ae of a hori:a:onital disposition of the :ilce bloda. If
they are distributed in t.he jam at an angle to the cu.rreat s.ur-
fac«, thus to the x ax.is , the calculations for the. det.er1aill&.Uon
o.f the e'lCalllined coefficients becoae lllUCh -=>re campllcated ( ...
section 2 •. 2) •
~
The vec.to.r P z in the unbalan~ed (or s..U-aul:Derqed') rtat•
of the solid body can have positive and negative values, co~
responding to t~e total tor the three forcea (Arc~dea',
•hydradynamic head", "hydrodynamic inflow• -of the dip of the
body into the current.
The hydrodynamic head force aria:es in the presence o.f tbe
"a.ttac.k. angle", determined by the disposition of the body r.ela-
tive to the direction of cu.rrents flowing around it. Tha hydro-
dynamic inflow force has a complex c .haracter. In the mav~nt
33.
of a solid body at a speed different from the speed of the cur-
rent it appears and is caused by a crowding of the useful cross
section, the flowing of currents under it with the formation of
/22/ whirlpool zones, the subsequent loss in energy, etc. These phenomena
reduce the pressure (evacuation) under the body, which expands
some distance from its other boundary where the flow of the cur-
rent has arisen. An evaluation of this force can be made only
approximately in the solution of applied problems.
(b) The component of ice weight, which refers to a unit
of surface, is expressed, depending on the problem to be
solved, by the angle or average velocity of the current:
•• .._ ""'y.AI = Y1h _C!_ • ,.., • r•C~It ' ( l. 20)
(1.21)
where C • l/n R11 6 • Here, for accretions of spring ice blocks,
which are characterized by a relatively great roughness and by a wide
channel with R • 0.5 H, where H is the current depth under the
ice /46, 67, 73/. The value of the coefficient r of passage from
the below-ice to the average speed of the current is a function
of the nature of the ice and hydraulic characteristics. Accord-
ing to the data of some field observations /36, 39/ in the absence
of slush under the ice,r = 0.8-0.9. The values of the coefficient
k" '"" f(C) are given by A. v. Kar:.aushev /33/.
(c) The force of sb:>re resistance, which corresponds to
the equilibrium level (1.9) can be expressed according to the
relatively we~l confirmed diagran\ of K. Yansen /5, 15/. However,
/23/
34.
with reapect. t .o the peculiaritie.a of the interaction between the
ice-:Plock accretion and the shore, .examined in Section 1.1, we
also introduced the coefficients t and B , which take into con-
aideration the effect of the nonprismatic fonn of the shore
/51/ and tbe emergence of the "arch effect", respectively.
The force .rela.ting, t .o the unit of aide surfa.ces of the accre-
tion is then dete.nuined by:
(1.22)
t is the coupling coefficient with
the shore, whi.ch is taken into consideration if the accretion
is not very thick /15/: n changes from \J I ( 1-\J ) to tan,2 ( 11 /4-e /2)
-in the limited stress state; here \J is Poisson's ratio.
In addition to the forces examined above, the following are
considered in solving individual problems: centrifugal force of
inertia, which. arises in a change in the directior1 of movement
of the accretion, the force which is consume-d in the deformation
of the ice on impact, friction bet een the ice blocks, etc.
5. The limiting conditions of the jam. strength (1.12)
taking into consideration the de.formability of the blocks should be
aupple.anted by the fo l lowing requirements;
(1) The· nondestruction of the jam and ice blocks:
o,.<!IJI <a.: 1 l.,S
(1.23)
(2) Nondestruction of the jam in the crushing of the
ice blocks:
G > 1'/l > 0 • v 1 l, s ( 1. 24)
•
35.
6. Active and passive pressure (internal resistance) of
the ice accretion ia determined accordinq to the known /15/
dependences:
tan ,.,1 '·-,..1'1-p· c1 -p"Jr> tr c:c'"-e12> 2 1: ( 1. 25)
'·,.. .. s~a· =-p" c1 -Jflr> r;,rn,1 + 9/2) ·~\ 1 -
(1. 26)
7. We divided ice blocks into three qroups accordinq to deqree
of riqidity in bendinq upon collision for convenience in this
inveatiqation. The idea for this classification is borrowed
from the theory of beams on an elastic base /75/ with a certain
chanqe in the calculation method and substitutinq plate riqidity for
cylindrical beam stability.
Short ice blocks in which the bucklinq due to bendinq is
neqliqibly small in comparison to the saqqinq of the base. These
are "riqid" ice blocks, the lenqth of which complies to the
condition:
(k _ "' 1-.<0,6(E/fk)''•· -sJ.ort, .. (1.27)
where I = bh7'12 -the smallest main moment of inertia: E -is
the elasticity modulus of the ice, and k = yb -the coefficient
of the foundation bed; at y = 0.001 kq/cm 3 :
'· < 2,1i (l::hl)'l•. (1. 27a)
For E = 1·10 5 kq/cm2 and h = 100 em, lk .. 14.5 m, while
with h • 20 em, lk < 4.0 m.
Lonq ice blocks comply to the condition a k < a., where
1
ak is the critical compressive stress, which causes a loss in
the lonqitudinal stability of the block. By applyinq the formula
/24/
36.
fo:r co.ap:reaaion of a :rec.tangular freely supported plate for deter-
mining thia value, we find that the length of the ice block
should coaaply to the condition:
where a • l/b.
ld > 0,92 (I+ "'1)A (E/"t~,
d•long
(1.28)
With a i • 10 kg/cm2 , m • 0. 5, and for h • 100 CM, 1 11 >
115 m, while at h • 20 ca, > 23 m.
lee blocks of average· size should. obviously satiafy the
equation:
I <I <I
... k llV •'""' (1.29)
Aa was noted, some characteristics and parameters of the
investigated system and medium are nonanalytical and discontinu-
ous functions diat.ributed a .ccording to the law of random values.
This requires introducing avera<Je values and applying pro))able
charac·teristics and values of the investigated phenoaaena in the
analysis. In >;his connection, a sectional averaginC] of hydraulic
elements ia taken for morphometrically uniform aegment.s of the
current.. Speeds, slopea, population of ice movement., etc. are aver-
aged acco·rding to width and lengt.h of the segment.s. For a Jnathe-
matical description o .f the .form of the jam "body" t .he step char-
acter of piling is approxima.ted by primary curves. A probability
evaluation of the object's characteristics in view of its inauf-
ficient study is made by introducing the transfer coefficient
from one confidence level to another.
37.
1.4 Physical Modelling of Ice Jams
For the purpose of confirming and refining the mechanism
established theoretically for jam formation a~d its individual
re.gularities, several model investigations we·re made in 1968-
1974. It was learned that under the conditions of multifaceted
phe.nomena similar to what we are exa.min ing, it is possible only
to approximate.ly reproduce likenesses to simplif.ied models de-
signed for solving the individual problems, and also that
physical modelling as applied to ice jams asswnes the fo·llow-
ing indispensable conditions:
(1) Modelling by concrete models that allow the re-
prod\\ction of specific aspects of the investigated objects
should satisfy the initial theoretical .diagram (see
Section 1.2) and the results of field investigations~
(2) Only the primary active forces and deformations
which determine the nature of the examined process in -:he
given problem are considered in the modelling:
(3) The possibility of reproducing on the models
formations of multi-layer accretions with a large number o .f
ice bloc.ks, while keeping in them the necessary indivi·iuality
as particles of the system~
(4) The possibility of determining intermediate (interconnectin~)
steps of the character of· ice jam formation under very dynamic
conditions and with process instability .
The practical fulfillment of these requirements presented
considerable difficulty since a modelling theory for ice jams
/25/
38.
has hardly been developed. A brief preeentation of aome modelling
results f'o:t .i.n.dividual theoretical problema is published in
/15, 64, 72, 82, 87/.
In the present s .tudy the available experience for model
investigations in jams and the general aspects of the theory of
physical 1110delling were utilized. The f 'ollowing procedures,
scales, criteria, and etru.cture of the modelling were taken
in this connection and taking into account practical possi-
bilitiee.
(1) Only the mechanical ; nteraction was .modelled without
considering the anisotropy and the nonunifot:m structure of the
ice blocks using an ice substitute . A rigid paraffin was used
for the ice substitute which, as is noted by N. Roien, •has a
kncM:l siJnil.ax ity to ice, but possesses a greater viscosi ty• and a
smaller strength. The physical-mechanical properties of the paraf-
fin used are determined experimentally.
(2) For purposes of excludin.g the uncertainty of form and
size of natural ice blocks, these blocks were simulated by square
pla·t•s of one or severa~ sizes, which are distributed according
to the usual distribu·tion law for random values with finite limits.
(3) For the bas1c model, ice blocks of' an average length
were taken which for rigid paraffin, in agreement with Eq. (1. 29),
complied with the condition BOh > 1 > lOh. Some of' the experi-
ments were conducted with short and long blocks.
(4) The investigation. of jam formation was conducted
for. condition (1.25) with nondestruction of the blocks. The
39.
losses in block stability were studied both under the nondestruc-
tive condition (suberging and tightening of short and average size
blocks) and under the breakdown state -long blocks model wice
fieldaw and the breaking open (loss of longitudinal stability)_
ol the ice cover on sandbanks.
(5) The channel model was a hydraulic trough with a vary-
ing bottom slope (rigid channel of rectangular prismatic form) with a
depth scale distortion of 5-15 times that usually observed in rivers.
Various obstacles, which reduce the channel throughput were modelled
in the trough (see section 3.2).
(6) The dynamic pattern of the current was modelled by two
basic forces: gravity and water pressur~observing the equality
and the compatability of the Froude and Euler numbers. The fric-
tional force was considered in separate problems observing the
compatability of the Reynolds' numbers.
(7) The stressed state of the blocks under a load was
assumed planar described by the known equation of"· Levi /4/,
while the connection between stresses and deformations was taken
according to the rheological diagrams II-IV-VI (see Fig. 2).
(8) A dependence is deteDmined from the Levi equation for
converting the critical stres~es ak, which determine the step-
wise development of the jam formation (according to diagrams II-IV:
t,.n;t;• ( 1 .. )'a (1. 30) "··· ~ *1ft,.. -;;-.,,,,
where lm and ln are the linear dimensions of the model and the field.
The critical stresses are expressed by the strength limits
with respect to ak • ki a i' where ki < 1 is the proportionality
/26/
40.
coefficient.
It should be noted that an investigation of the qualitative
mechanism of jam formation genera1ly speaking did not require special
conversions· of the mode·l to the field. This is touched upon · and
confirmed by a number of formulas cf a physical nat'UFe taken for
the model and for the field assuming the constancy of the respec-
tive physical constants a:nd parameters characterizing the mate-
rial used, the hydrodynamic. conditior."~, etc.
The model investigations of ice jams were conducted along the
following basic lines.
(1) The study of the ice jam formation mechanism:
(a) On an unopened section of the river and on "blocked-up"
ice fields, tal..i.ng into consideration jams only of transit ice and
rigid part-icipation of local ice wh .ich forms in the breaking up
of a pOrtion of the ice cover:
(b) For a0 < b (see Section 3.2), i.e. with insufficient
throughput of the current along the width of the channel (divi-
sion of the river into branches, stone sections, bridge forma-
tions, etc.);
(c) For H < h, i.e. with an insufficient channel depth
(shallow rifts, sandbar!, etc.). More than 70 tests were
conducted for various wat.er flow rates, initial depths, cha·n-
nel slopes, etc.
(2) The study of the mechanism of loss of ice-block sta-
bility (see Section 3.2) when they collide with obstacles (dynamic
form) and when the longitudinal comp.reasion increases in the. ice-
41.
block accretion to a critical value (static form). More than
60 teats were conducted under various hydraulic conditions, ice-
block forms and dimensions.
(3) The stu~y of the process of ice-cover buckling and
breakdown with an increased water flow rate,keeping it in a
stretch of water*(see Pig. 1); 15 tests were conducted.
(4) A study of the breakdown of ice jams of various
types with water influx; 15 tests were conducted.
In the investigation of jams on a ~~ ~ ~.
the Lottom stretch of the trough was raised on the sandbank and had
a large slope. The models of the ice cover and ice fields were
made of paraffin, while the stylized models of obstacles for
a0 < b and H < h were made of modelling clay or solid materials.
In order to established the coupling moments of the examined proc-
esses, scale photographs were taken, which were then used in the
treatment of the results.
The conducting of model investigations has a kn~n limit-
ing character with respect to possibilities, volume, and detail,
which permits only clarifying the qualitative nature of the
investigated phenomena and processes. However, their results
permitted confirming the earlier formulated basic theoretical
aspects /18-20/, and also defining somewhat more accurately and
detailing the mechanism of the jam formation. In particular,
we were successful in establishing a gradation of jams according
to the extent of their development (see Section 3.1) clarifying
the form of the jam body and the regularity of ice-block distribu-
*the "reaches"--Trans.
4 2.
tion (see Sections 3.2 and 3.3), specifying and generalizing the
mechan-ism of ice-block st!ability loss, it wae also possible·
to determine some· quantitati-ve dependences.
•
43.
Chapter 2
.Jam-Lese Regime of Ice Movement and Losse.s of Ice-Block Stability
2.1. Channel Throughput in .Jam-Less Ice Floating
The ice d .ischarge t .hrough a channel!. section normal at each
point to vector ; , expressed by t .he mass per second, i.e. , the n
mass passing through surface F per unit of time, is ex·pressed by
the equation:
( 2 .1)
.In the general case in the current section L, the. value .
S • f (x, y, z, t) has a large sm with a maximum integr and for
Eq. (2.1). In a stationary regime, S • f(x) will correspond to m
the discharge paasing through the "elementary section .. line with
the s.mal!.lest limiting throughput s 0 in the given sect.ion L. The
"element.ary section" is the current section. the length of which is
equal to the ice-block length, and bounded below by
the investiga.ted line. A change in the ice transit regime in
this line ia evidently immedia.tely and. automatically distributed
upward over the entire length o .f the section.
By averaging ~ according to F and assuming h and p 1 constant,
for the examined case we find tha.t the floating capacity of the cur-
rent section L with the bounda.ry line s 0 is determined by:
(2.2)
where B0 is the "effective wid.th" -the current width with depth
H0 ~ 0. 9h for b < B0 (in the case b > B0 and T ~ Ap' the
value s 0 • 0).
/28/
44.
The index min Eq. (2.2) indicates the naaximum value of
the. product 1"tli s depen«le:n~e corresponds
to the jam-less IDOVeatent of the ice blocks in the cur.rent
IUtction with the boundary line s 0 •
The jaa-less floatinq of ice bLocks can occur in three
(1) A sinq-le-layer plane ice•-.bloc:k accretion:
(2) A single-layer hUDilOck accretlon without block
breakdown:
(3) A ainqla-layer hw.ock accretion with block break-
down (ahearing -in contraction; rupture -on curves of de-
crease1 etc.). Floating is also found along the free channel,
including ita bottlenecks, t .urns, etc., and in the presence of
bead re•istance of· natural or artificial origin (ice · daJu', bric!qes,
embankaent cresta, etc.).
rce passage through hydraulic structures is examined. in a
number of: special investi.gations /16, 17, 34-39, 46-.tS/. The.
condition of jam-less advance of finely crushed ice accretion
subject to the basic la.w of mechanics of a free-flowin4} aedi.um,
for the case of a wide rectangular and straight channel, was
formulated by D. P. Pan.filov /58/. The jam-leas floating of
a packed, single-layer ice-block accreti.on taking into account the
blodts' individual properties, pat·ticularly the condition h < 1
and the presence of transverse caaponen.ts of force act,ing on the
a.c_cretion ia examined. below.
45.
In this case, some laws of free-flowing med.ium mechanics
are insufficient to desc.ribe tbe system behavior and Eq. ( 2. 2)
requires the consideration of some of its peculiarities, inherent
to:
(1) The discontinuous (free-flowing) body: (a) for
condition (1.6) and (1.9) -(1.11) or (1.12): (b) fo·r
where t a are tangential forces. on the surface of contact with
the shore, created by.the •arch effect•J
( 2) The system O·f solid bodies: (a) for plane-paral-
lel movement of the blocks (vz • 0), which generally speaking is
already guaranteed by Eqs. (1. 9) -(1.12): (b) observing the
requirement a 0 > b: (c) in agreement with (1.14) of the condi-
tion ~s • o for s > O:
( 3) The system of deformable bodi.es -ratio A2 3 4 < , , , p
" " t At' where. t At is the total work. of all forces acting on the
' l
ice blocks.
Therefore, for investigating jam-less (and in general the
equilibrium) regime! and the s 0 determination, we have a clos.ed
system of equations:
(1) The balance ~ s • 0:
(2) The equilibrium equation (1.6):
(3) The rheological dependences (1.8)-(1.12) for diagram.a
lila and Illb, w.hich determine the boundary conditions, while for
--
the s 0 calculation, Eqs. (1.8)-(1.12) will correspond to the
limiting equilibrium condition;
46.
(4) The requirements s 0 ~ b and T k < T • a
The possibility of the emerqence o .f the arch ef.fect
is considered by the latter condition -the transfer o .f ·forces in
/29/ the ice blocks at an angle to their movement with the force
distribut i .on in tbe form of an a .rch pushinq 49ainst the shore,
so that ta > 0 can cause a retardation, while for
there. is a wedging of blocks. According to R. L. Zenkov•s
investigations /25/, tlle emergence of arches (domes) in free-
flowing bodies is possible with a compressive stress that qua.r-
antees dome formation. This phenomenon is particularly cha.r-
acteristic of constricted seqments and channel turns, but takes
place also in rectangular sections of prismatic channels.
This phenomenon has not been sufficiently studied, and
a possibility o .f considering it obviously lies within linrits
of the direct use of t .he experimental data. The braking action
of unstable arc.hes of sn1all blocks (for e0 » b) in the estiu-
tion of v can be Lpproximately considered by introducing the
empi.rical correction factor B • f (80 /b) > 1 into the formula
for determining the shore int.eraction force. The appearance of'
stable arches can be es.timated by a probability method using
the results of field tests. V. V. Piotrovich /62/ examined
t.he case of formation of stable arches of' three blocks in a river
turn.
The results of the investigation (see Section 2.4) show
that the arch effect appears bot.h in the ice-block movement, and
47.
in an immobile accretion found in the stressed state. This
phenomenon is particularly characteristic for "narrow rivers",
where the ice-block dimensions are relatively large. Thus,
in modelling ice-jam formations forb • (0.10 -0.25) a 0 in the
rear section of jams, due to the formation of arches upstream,
sections free of ice blocks emerge. These break down with an
increase in pressure or a change in the level of discharge [of
water). It may be assumed that the ·paces free of ice observed in
the Dniesber River within the jam limits are associated with
this phenomenon /31/.
The block accretion/shore interaction coefficient ( n, ~1 >
is a variable, as noted above, dependent on the velocity of ice
movement, the nonprismatic form of the shore, etc. /57/. In the
current level of study of this phenomenon, it can also be approximately
considered by introducing into the shore interaction force Po the
channel nonprismatic coefficient established empirically,
(see Section 1.3).
> r; <
The general expression of ice throughput through a river
section was obtained earlier /20/ proceeding from the following
reasons. The moving accretion of contacting ice blocks bounded
1.0
by the locking lines can be examined with known approximation as
a material system of mass M and all forces may be related to its
center of mass. Taking Eq. (1.14) for this case and the theorem
of a change in kinetic energy of a material system, we obtain:
( 2. 3)
48.
/30/ where ,1 • •an; a is the portion of the river width occupied
" by the ice movement; n is the number of block layers 1 r A. is
' l.
the work o .f all forces in the block movesaent plane relative to
mass M and. pa.th L; c 1 = c 2 • M applies to a rectangular river
section, while c1 _ = .H and c 2 • I/R2 at a . turn; I is the moment
of inertia of the system; R is the turn radius; v 0 is the velo-
city of motion of the system at the section origin.
The application of Eq. (2.3) for practical puz:poses
presents known difficulties in the general ca.se. However, it
provides relatively complete qualitative information on the dynamic
conditions and. possible regimes of ice movement including condi-
tions that d.etermine the "jam risk" of a . flow section /20/.
The dependence for the determinat.'ion of ice movement velo-
city v in the s 0 calculation according to Eq. (2.2) for the first
type of jam-less regime, more convenient for practical use, is
obtained by applying the equilibri.um equation (1.6) to the case
which establishes th~ motion of a .. ingle-layer block accretion in
a wide channel with a turn radius R for a 0 > 0 and o.
Here we assume that with s .ufficient dista.nce fram the frontal edge
of the a .ccretion and under the action of external stresses, the
longitudinal stresses in the packed. block accretion (rheologic.al
diagram IV in Fig. 2) reach a critical value in the boundary line
and do not change further. The fo·rces p 2 , p 3 , p 4 , p 0 , and the
transverse forces resulting from these directed along the y axis:
p 6 , P 2y' P)y' and :!:. p 4 y act on the t.ransver•e strip .taken
here with a length equal to unity. The first two of these are
/31/
49.
directed to the concave shore, the third to the convex, and the
fourth can have either direction.
The equilibrium conditio:t can be written as:
P2 + P'l ± Pa-ff& (p,.+ flo.-no:!:: /'&w)-P~ = 0, ( 2. 4)
where Po • Po h/B, while in parentheses is given the difference
of· cross forces in absolu.te value.
l!!q. (2.4) holds both for the free channel, and when a front
obstacle is present as long as this resistance prevails with a
stress not ex.ceed::.ng ak given above. By expanding the force
value p 2 and expre.ssing t .he values p 3 , p 2 y' and p 3 Y by the cross
c.urrent velocity according to Eqs. (1.19) -(1.21) and consider-
ing the energy loss for impacting, friction, and deformation of
the ice blocks, by means of the coefficient A0 = o.9o-o.ss,
we find the desired ice movement velocity i.n the form:
The value v is determined by trial and error. a k enter-
ing into p 0 may be takP.n according to Eq. (2.20), while for
crushed ice, the product nokh will correspond to the active ice
pressure oothe shore calculated according to Eq. (1. 25). In pre-
'lious calculations, the a coef.ficient which. considers the arch
effect .for relatively large ice blocks B < (4-10) u may be
taken equal ~o 1.2-1.3, while for small blocks -1.0 -1.10.
/32/
so.
These coefficient values lllllst be taken as orienting values
subject to further precision by special investigations.
To determine the thrtJU9hput s 1 • s 0 'for a known v, it i s
.necessary to find the smallest value (for the length of the
section) of product (B 0-;)m for each line for the greatest value of
ice move.ent population ljl • 0. 80 -0. 85 &nd to put the value
obtained in Eq. (2. 2). In the gener-al caae, the analytical
expression s 0; • f (x) is not assigned and the probelm is solved
by trial and error or by the method of control lines in char-
acteriatic river sections: constrictions, turns, a decrease in
the f1ow velocity, etc.
r· -... ~ .. ,. ..... -.. ...... ~"'"=·-.:~ :-· .. -· r .,. a •-•• 0 ·~-~~~ -·:~-.....,..,l: _ • ...---r-.,;JI"':;;;:»=--l
I
I
I
I
.· ·, /
. '•
•· . . ........
.:··~--;.· ..
. .. . . ,. .. .,.-
.' .. ~...--···-.
r·• ... -"~"'"'~ :
.. .... •"
.. ·.. ~ ·• :, . ' \ '~·;1 ..::~···\Jl
.. .
I .
··.>1 ,·.:·· .. J1
• '\, '·.~ .................... ··--:.··.~· ,-. t, I 0 a ) L-..:....~ 4!+ I t'.te.D ....:...-~..._'at. ·• ._ .. ·• ·• -. · -:..,;;,_,t.llit?-;;.,'trrlllllo.llit!.llo;nn..ird..,·o.ar.rl":-r"~)ololi't.,_,.,. • .,.iooiWolii'iooi'IO:'~~...:~
Fig. 3. Jam f1,·rmation at the ice-cover edge.
By putting v • 0 into Eq. (2.5), we obtain the V{llue of
the flow velocity for which there results the stand.atill of the
blocJt acc.retion:
(2.6)
51.
and for a rectangular river se.ction without considering wind.
pressure:
(2.7a)
or
Fr ----=--:;.'=-eA~:-::-~ •ll(ta+ ~~~ ) • (2.7b)
where Fr is the .Froude nUlllber.
A series of curves Fr • f(h/H) for var.ious values h may be
constru.cted according to Eq. (2. 7b) for B = const. Eqs. (2.6) -
(2.7b) may be used for solving applied problems, for example,
for the ca:lculation of ice-retaining barriers, etc. Eqs. (2.5)
-{2. 7} were subjected to a certain confi.rmation on paraffin.
block models in a straight channel section and in a 90° turn
for R • 38 (Fig. 3). The mean-squa.re deviation of the calculated
and experimental data was 17\. Here the reasoning resulting
from an examination of the equations is confirmed by the fact
that with a block movement velocity close to the current velo-
city, when. p 6 + p 2y > p 3y, the accretion experiences an addi-
tional co~striction (piling up) into a concave shore, while at
low velocities, on the other hand, a convex shore.
2. 2. General Condi.tions of Ice Block Stability Loss. Hummock.ing
and Plunging of Ice Blocks in Collisions with Obstacles
Under a nonstationary regime, S > s 0 may appear
both with an increase in ice discharge S, and with a decrease
in the channel throughput s 0 • Here, a portion of the block energy
1\
equal to l1~ =-~ -fA> 0 (or all energy at s 0 = 0), is converted
to the er.ergy of interaction between the ice blocks and both the ob-
/33/
52.
stacle and current. This causes a .t A) > A1 a breakdown
of block stability, while for A) < A 1 , the joining of blocks
floating one und.er the other.
In the latter case, as the length of the block accretion
increases upstream, their pressure against each other a will
increase (see Section 1. 4), so that at a > a k, this also leads
to a loss of their stability. Thus, block stability losses may
arise both in their collision with an obstacle ("dynamic dila-
gram" of the phenomenon), as well as due to thei r compression
in the accretion ("static diagram").
The stability bre.akdovn phenomenon for ice blocks under
·field and laboratory conditions has been studied. by many re-
searchers: F. I. Bydin, Ye. l. loganson, L. G. Latyshenkov,
G. F. Kennedy, E. Pariset, G. s. Shadrin, L. G. Shulyakovakii
and others, including the a•lthors of the pre!lent study. .Ana-
lyzing and gene.ralizing these data lead to the conclusion that
a breakdown in ice-block stabilit.y upon collision with an
obstacle can be manifest·ed. in the following forms :
(1) A huiiiiDOcking of the ice, i.e., partial submerging
of the blocks in water at an angle to the sur.face and. lock-
ing them into this position by other blocks:
(2) A plunging of th~ front-edge blocks under the
obstacle:
(3) A tighten i ng of the blocks under the obstacle by
the lowering of the rear edge:
53.
(4) A packing (piling up) of the ice blocks on the
obstacle and on each other,
(5) A sinking (flatwise) of the blocks in the water due
to their loss of buoyancy.
The individual forms of block stability loss have already
been subjected to a special analysis and theoretical study:
The tightening of blocks-L. G. Latyshenkov /42/, V. K. Troinin
/76/, G. F. Kennedy and M. s. Uzuner /77/J the plunging -v. I.
Sinotin /12n buoyancy losses -E. Pariset, R. Hausser /89/: the
plunging and creeping of blocks -Iu. A. Deev /19/, and others.
Establishing qualitative dependences of the various forma of
stability loss is done on the basis of assumptions in the
present study for the initial conditions and models given in
Chapter 1. Analyzing the examined phenomenon permitted us to
draw the following conclusions:
(1) The br~ form of block stability in collision
with an obstacle is basically a function of:
(a) The relative kinetic energy of the blocks 3,
determined by the block approach speed and block dimensions;
(b) The block form;
(c) The nature of the obstacle;
(d) The st4te of the current.
(2) Three cases for the breakdown mechanism (form) of
block stability are established:
(1) The kinetic energy of the blocks 3 is directly con-
verted to the dnergy of mechanical pressure on obstacle T; hum-
/34/
54.
mockinCJ, plungi.nq, and pilinq of the blocks (Model IIIb) occur
dependinq on the value o .f ~ and the nature of the obstacle;
(2) The ratio between kinetic and potential enerqy of
the current uncleir the block changes when it stops at the
obstacle, which causes a tightening of the bloc!kS (Model IIIa);
(3) The total pressure of block accretion increases to
o > ok, upon which there occurs a conversion of the system
(according to diagram IIIa) to a state of more stable equilibrium
(hummocking, formation of a two-layer accretion, etc.).*
In the first case, the stabi lity loss** is caused by the
fo~tion O!f torque primari.ly as a consequence of t .he cent·ral
i•pact of' the block with pla.ne obstacles (the edqe of the ice
cover, a block, a barrier, etc.). The process may be described
in the form of two steps;
(1) Initial -The advance of the front (impacting) edge
of the block with its partial breakdown und.er the obstacle,
(or on it) which causes the phenomenon of hummock formation;
(2) Plunging per se (creep) of the block under the
obsta.cle (on it) with the' surmounting of the frict:ion force.
A general expression was obtained earlier for the approach
velocity of the block ~l' which guarantees the first step -the
hu~cking of t .he ice,, by solving a s ystem of differential equa-
tiona of motio·n and a change in the kinetic energy of the block
/20/. Analogous results are given by a simpler method. The cal-
*The mechanism of vertical block sinking is not examined. here;
we refer t.hoae i.nterested to the stud.ies by E. Pariset /89/
and V. K. Troinin /76/.
**See Sections 2. 3 and 2.4 for the second and third case>s.
----------
/35/
55.
culation for the phenomenon, the rea.eoning behind it and assumptions
for this method may be pre3ented as follows.
1. When the block collides with the edge (or with another
block) and there is a sufficient approach velocity, the edge of
the impacting ice moves out to the border (or under it) as a
function of the direction of torque, which arises due to the
nonuniform thickness of the blocks /19/, the nonuniform strength
and form of the edge of the blocks along the vertical, the tan-
gential stress due to water friction against the lower surface
of the ice, the drop in pressure under the rear portion of the
block; the curve or decline of the movement in water under the
ice-cover edge.
2. In the block's advance, its center of mass effects a
forward motion, while the impacting edge effects a plane-paral-
lel rotation in the xoz plane. The kinetic energy of the block
is consumed for this work, i.e.
,.·a ... •'lp 1-., v (2.8a)
..
where n• is the coefficient that considers the additional pres-
sure of ice masses approaching from above; Ap -the work expended
in the forward motion of the center of block mass; Av-the same
for rotation.
3. It is assumed that a projection of the block hits at
angle a 2 against the plane face of the cover without slipping,
but with the partial breakdown of the contacting sections. This
process, on the one hand, is similar to U .e phenomenon of fric-
tion with dispersal, and on the other hand, it is analogous to
56.
a cleavinq <•hearii'lq) of the ice, which takes place in the reqion
of IIUIXimum t .anqent.ial stress. Por conditions of linear cornpres-
sion by lonqitud.inal force, P can be. wr.itten.:
(2.8b)
where n 2 is a coefficient. which accounts f ·or a veakeninq of the
block edqe: w1 is the area of the sheared secti,on, normal to
tbe direction of action of horizontal force.
Under t .he assumed conditions, the shearinq depth is equal
to h with a drop (rise) of the block e.dge to 0. 5 h, while the
block lowering (rise) angle. will be presented reapectively by
a • 2h/l.
4. We will express the work. A and A by tbe average force p v
·value p • ps, which correspon~.e to the average value w1 :
A +A.-=Illl+e."2A/l, p . v.. ss
where. rs is the average value of the force
assumed equal to h at h << 1.
(2.8c)
arm p , which can be s
The dependence ~ on path x takes lhe form:
tan 4ola u 2(h-:) Z l(u1 • (2.8d)
. By integra.ting Eq. (2. B.d) in limit.s of x • 0 to x • h and
dividing the. result by h, we find. Ill • s Substituting this value
in Eq. (2.8c) we find that:
t• :...O,If~finpJ,::~Jl,f_ 8 8 • •
By evaluating 3 in Eq. (2.8a) and solving it jointly with
Eqa. (2.8c) and (2.8d) accounting for the probable nature of
the· values n 2 and tan a 2 , we obtain:
(2.9)
Pa= ~-' h ( .. ,., ) ...
\ ,.,,.. .
57.
Al is a coefficient which
considers the probable nature of the value n and tan a 2 •
In guaranteeing p < 10%, the value Al • 0.5-0.6; for
P • SOt, the value
> 90%, the value
Al = 1 (n 2 = 0. B, tan a 2 = 1), and with P
Al = 1.5-1.6. The coefficient n1 = (l/n•)2
with sparse ice movement is taken equal to 1.0, for average move-
ment-O.B, and for dense movement-0.6-0.7.
The approach veloc:ties of the block for which there occurs
its plunging <v 2 ) or creep (v 3 ) are obtained from considering
the values of work loss on these processes:
·Ps-= (u~ + 0,6h)0
'
1
;
~-(u~ + 10h )"·'.
(2.10)
(2.11)
/36/ A confirmation of these equations under field con~itions
in the Prut River (5 surveys) and on models (30 tests) gave
satisfactory results. The mean-square deviation of experimental
values from those calculated was 12-15%.
Assuming a connection between the pressure force p and the
values h, a
8
, which correspond to the strength of an elastic
plate, in the form p = 3.5 aa (h/1)2 , the value v 1 (for P = 0.094
t•s 2;m 4 and lb 5 1 2 ) is attained in the form:
(2.12)
Eq. (2.12) is similar to the dependence of Pariset /89/
obtained by another method:
~-t,25k Jfh. (2.12a)
58.
where k • 0. 7-1.2 ia the fora coefficient. Bovever, it is
obvious that Eq. (2.12) bas a -.ore. general character.
I. Ya. Liser /43/ obtained for Siberian rivera v 1 • 1.2-1.5 m/s
according 'tO Eq. (2.12a) fork • 1.2 an.d b • 0.6-1.0 m, but it is
noted that this value should .be· saaller taking into account the
pressure of other blocks. Eq. (2.12) under the same data and for
>.• 1.0, n 1 • 0.8 gives v 1 •· 0. 7-0.8 11/s, and v 2 c 1.0-1.1 11/s.
2. 3. Tightening o .f the Blocks in the Collision with an Obstacle
The second of the stability loss fonus enumerated in
Section 2. 2 is the tigh.tening of the blocks in the impact with
a plane obstacle, such as the border of the edge cover, a block
whlch has stopped earlier, etc. A generalizing of the results
of' an experimental-theoretical study of' the tightening tMChanism
f .or the blocks in the case o .f their sudden stoppage at an
obs.tacle (considering the results of investigations of the
authors given in section 2.2) shows that it is caused by:
(1) a decrease in the kinetic ~ and an increase in the p
potent • '\l Up energy of the current on the rear f :ace of the block
as a consequence of the braking of the jets; and (2) an increase
in )p and a decrease in Up under the block due to the crowding
of ita useful sectio n.
The consequence of the first phenomenon is the rise (in-
crease) in the wave pressure force . This force is introduced
into the general equation of .hydrodynamic pressure which, in the
given case, acting on the rear edge of the block, will have a
component directed downward Poz· The consequence of the second
/31/
59.
phenomenon is the appearance of the hydrodynamic inflow force
Ppz' also directed downward. This force is determdned by a
decrease in pressure under the blJck in its back section for
a certain length 1 1 •
In addition to this, a wave pressure force causes a braking
of the jets along the length of the current (next to the block)
with a change in the form of free water surface in the wave form
with a corresponding decrease in 3p and UP. The force which
arises as a consequence of a formation of whirlpools along the
side surfaces of the block is related to the same category. And
finally, in the case of an increase in the block thickness toward
the front surface, which creates an "attack angle" (see Section 1.3),
there arises a force Phz of the hydrodynamic head directed upward.
If a sudden stopping (collision) of the block does not
cause a complete or partial destruction of the obstacle or the
edge of the block (which causes its plunging or creeping), and
also a slipping of the block in the vertical plane (loss of bouyancy)
then the front edge of the blocks under the pressure of the current links
up with the obstacle by propping against it. In this case, the
forces named above as well as the force of gravity Pgv and Archi-
medea' force Paz form, relative to the aupports,the total for
force momenta acting in the vertical plane. If the total of force
moments directed upward is more than the main resistance force
moment, than the ice block begins to sink by its rear edge under
water with increasing speed as a consequence of the increase
60 •
.in forces Pp2 and Pvz· Here, the block is tightened under
the obat:acle or fox; R < 1 it is wedged between the, stream bottom and.
t:b.e obstacle. Mo.ents of tightening and wedg,ing o·f block
JIOdle1s in a hydraulic trough are given in the photog.raphs
(Fig. 4). In the presence of a block accretion, the tighten-
inq proeesa can encompass two or :110re blocks at one time du·~
to the eaergenee of a total to•rque for them. I :t should be
noted that the tota,l o ·f force .al!Mlnts directed downward in-
creases with a decrease in the current depth as a consequence
of an i .nc:rease in the hydrodynamic inflow force (see Eq. 2.13b).
The establishment of a critical current velocity, an in-
c .rea.ae o ·f which gives rise to the tightening of blocks of
given ciU..naioDS, is carried out for the exa.ple o ·f a sud.den
stopping O·f a block with rectangular shape of constant thick-
ness unde:r conditions ot aplane problea, taking into account the:
probable fora of contact . !'ur this case, the equilibriua equa-
tion of force 801Mnt relative to the y axis lyinC} in the inter-
section of the front vertical face and the lo~er block surface,
after excluding second-order values of smallness takes, tbe fona:
Uft .
~Jt.+G.SI&cu,P .. -!-(1+0,5/J P,.-0,511' •• -0.rJtP •• =-0, (2.13)
where 1 1 is t .be length of the distribution of decreased pressure
under the block (in the detachment zone of the jet).
'rbe deteraination of forces whicb enter into Eq. l2.13),
excluding f .orce P pz' do not cause difficulties. The latter
aay be evaluated only approximately. v. K. Troinin /76/ intro-
61.
duces in Eq. (1.17a) for this coefficient, which considers the
/38/ vacuum gauge pressu.re, the coefficient Jc 1 which is similar in
struct.ure •
/39/
.. ·' rr~"!:~y----'"14o:"·'~. ~.~~ ..... "!"!.'~----.. ~·-.. -___..~ .
I
.. •r:· •• P..,•'":~·•' .. ·--• • :•"$•J·;4t• ,. •' ":.·:··,.:::~~·~~~·.-:::.::1;:~"·~::'.~:~·· ~_:·.:.;~!""'!~ ... • . ~-· .... ., ... -· .. ..... ,.. ~ •.; .. -..
. ::· ..
,.
.....
l'ig. 4. Tightening of blocks upon collision with an obs.tacle.
a -Initial step of block tiqhtening under t :he edge of t.he ice
cover -submersion of it!l rear edge in water; b -Wedging of
th.e tight.ened block for 1 > h propped between the ice cover
edge and the bottom of the stream.
The expression of the unknown .fo·rce is presented more
genera.lly by the pressure drop under the block 4h, which con-
siders both the current . contraction with a disruption in the j ·et
as well as local head losses, i.e.:
(2.13a)
The value t.h is determined from Bernoulli • s equatior.
w.hich consists of two sections: in front of the rear face of
62.
the block and und.er 1 t:
where H8 ,. H/ (H -h) ; a is, the kinetic energy correction in
the Bernoulli .equa.tion; r; is tbe coef'ficient of local m
losses in the inflow of jets under the block.
The joint solution of t .he given equations with respect
0 to flow velocity for a 2 • 45, a= l.1and Cm • 0.5 with
conslderation of the various block for.'ls leads to the equation:
u' ... A n,r,if',ltl •PI
p 1 Ah (l-l.t2t)-lr1• -Al1 -... ' (2.14)
A-0.5~r (t,al/:-0,511.-1);
(2 .Ua)
1 2 is the coefficient of the probable contact form which is taken
with 50\ probability equal to 0.9; for P • 25\ the value 1 2 •
0.85-0.80 and for P • 75\, 1 2 • 1.1-1.2.
M. s. Uzuner and G. F. Kennedy /77/ in concluding a similar
dependenc.e also used the Bernoulli equation, but in another
stricter form. However, the value c , which they introduced •
into the formula and which depends on pI p 1 , h/ 1, h/H and bas
a considerable effect on u, was established by thea only in an
approxiaate expe.rimental manner.
Eq. (2.14) was confirmed under laboratory conditions and
partly in the f .ield. For the natu.ral river ice blocks, there
was obtained a good agrenaent. between the field te.sta and the
calculated data with probability liaita of S0-75\. Coaparative
/40/
63.
calculations according to Kennedy's and Pariset•s equation
show that Eq. (2.14) provides a close similarity of results,
but the Trainin formula gives higher velocities. we note that
due to the introduction of thepnXability coefficient A 2 ,
Eq. (2.14) is more flexible for practical application.
The theoretical analysis and the experiment show that
the form of stability loss of ice blocks upon collision with an
obstacle is a function of the kinetic energy and the velocity of
block approach, as well as the relative depths h/H and dimensions
h/1. With large } and v values and a small h/H value, there is
observed (or there prevails) a plunging and creep of the blocks.
The tightening process here either does not arise or it develops
with a retardation with respect to plunging. With small energy
values, insufficient for a partial destruction (~leaving) of
the edge of the blocks, and also with small relative depths,
the tightening of the blocks prevails.
The phenomenon of flat dropping under the water is char-
acteristic of ice blocks with small h/1 values (according to
Kennedy's data, for ice blocks with h/1 < 0.1). The process
of plunging or tightening (see Fig. 4) can arise, as was shown
by the experiment, with a relatively high approach velocity even
in these cases. In ge~era~ the principle of minimum energy con-
sumption can be followed in the calculations and the loss of
stability can be taken in the form which requires the least
energy consumption (smallest u).
64.
'l'Nue is an intereat in ut&blishinq the crlt.i~al current
velocity, the exceeding of vhlch causu al.iwiNJ \.1 OC' ax.-• tionaJ,-forvard .ation u v,p of the bloc.lua under t.be ice
ac.cretion.. ln ot:der to derive the r-pective forDllaa, the
same method is used as for obtaining Eq. (2.14). Kovever, a
.are general case is exaai.ned here, when the lower surface of
the accretion and conaeque.ntly the ice blocks pressing dovnvard
against it are distributed a .t sa.e angle a to the cur.rent
direction. This ca.se is characteristic in the for .. tion of
"developed" jaas (see Section 3.5). Here, so.e values, vhich
enter into t .he equations for deterai.ning the force of hydro-
dynamic pressure P1 ,2 and. of hydrodynamic inflow Ppz' a.re variables.
A tera ia introduced into the formula for deteraininq the hydto-
dynaaic inflow force, which u.kea into conaideratio.n the addi,tion-
al coapresaion of the cur.rent under the block as a con•equence of
ita inclination. Taking this fact ot: into consideration, the
equation• for de.teraining the indicated forces take the follow-
ing form:
Pa.1 -k ... 1 l'b-lkt +k• ..!!..) u'aJJ-(loa••+ t.-.) u1& -kwaa't; . t ·~
(2.15)
(2.16)
where a ... 1 cos 9 + h sin e an.d a5 -h cos e + 1 sin 8
A1 and A2 are detertlline:1 according to Eq. (2.14a) with the sub-
stitution there for the value H 8 :
8.1 -Jj 11 --~P. -II (2.16a) • Ac.»~llllD""n. ll-hira8 •
Then we obtain t.he desired dependence for obtaining
\.1. from the conditions of equilibrium of· the active forces and
65.
for -from the conditions of equilibrium of the
force moments with respect to the axis lying at the intersection
between the front edge surface and the lower ice ·~tUmsur-
face: u•-~''""'''''"' a ..... +(AJilcos*'+A~a) ••
ut -0,5 a.,.,,.,•.ls
•· a A. (}+0.5&1:8 }+.tala (t +t' .. CIS 8+Ae (0.~+ ~ 1lll8) '• • (2 • 17 )
where
(2.18)
a, -lfD 9 -t .,,cos.:; c 1 -COS 8 -•• sfD 8; 1.11 -sfu • 9 +
+ •• cos 8: a, -1 + rsiD e .
P'"-. t: .. . · ... · .....
~: ..
a'•,
. -~~-·--~~-·-·~-~--·~-~]
., .· .. •' '•'" :: · .... :.. . . .. . . . . .. ~ .: ·-······, . .. ,;' :1
I, .... :~' ,l .. ....
.......
i ·t
·!
. '
::."~
.... !'I
~;
. . I . -· L . J. ... : _.. I ....... ·-----~·--_--_:.:::.::.. •. -•.... ~ .......•• ~· ·.:: ___ :.:~~:.-. ___ .. ___ ..:.~::.: ::.:._:~. '.l
Fig. S. Formation of a jam ~-
Eqs. (2.17) and (2.18) may be simplified to the form:
"~~AI --;:----•'\.:..,!'~,_.1--r--· 8 ,:~ ·H·,II (-i;-+t}+Al • (2.17a)
u• " A.. &p~llt
v,p 2At, (1-~i )+211(t1 +!·1 ) •
(2.1Ba)
For an examination of Eqs. (2.17) and (2.18), it can be
seen that the form of block motion under the ice, i.e., the slip-
/
/
66.
ping or ~ot&ting of the blocka dependa prt.ari.ly on the valuea
+ 1 , e an4 on the degree of c~reasion. of the· current by the
block, deterained by the h/8 ratio an.d to a lea .. r ext.-tt on
the h/1 value. For: e • 0° and h/H < 0.1-o .. os, Eq. (2.18)
has smaller values for the c .rltical velocity of the cu1.rent than
does Eq. (2.17) for any ratlo h/1 only for .1 > 0.8 -0.9.
For h/H,. 0.1, the limiting value •1 is reduced to 0.7-0.8; for
h/H • 0. 3 to o .• 6, etc. It follows from this that for 6 a 0° and
small values of. ~, for exaaple, under the co.ntinuous ice cover,
/42/ the slipping of blocks will prevail, while with large values,
their rotation will occur under the jam accretion of ice. The
paraaeters h/H and h/1 will effect the deqree of developaent
of this or t "hat form. In the general case for jaa fortaation,
the values • , h/H, h/1 and 8 are functions of spatial coo.rdinate
and. time. Thus, for different sections of width and lenqth of
the cu.rrent and in differen-t phases of jam foi'lMtion, thi• o ·r
that form of ice-bloc'k movement under the ice will arise. This
reqularity was cc -.firraed qualitatively by the results of JDOdel
~ nveatigations. The correlatioo of data from 30 experiments with
calculations according to Eqs. (2.17)and (2.18) gave an average
convergence on the order of 15'. The phen0111ena of turning a .nd
slipping of the ice blocks und.er the ice a .ccretion is shown in the
photoqraphs for the .model investigation and can be discerned in
Fig. 5.
67.
2.4. The Stressed State and Losses of Stability of a Single-Layer
Ice-Block Accretion
The analysis of the stressed state of an immobile current-
racked plane-layer accretion of ice blocks is conducted according
to the diagram of K. Yansen, which has already been applied for
this purpose (see Section 1.2). As in the derivation of Eq. (2.5),
transverse components of attractive current force and gravity are
introduced into the number of active forces, while coefficients
which consider the arch effect and the non-prismatic form of
the channel are included in the •side thrust• in agreement with
Bq. (1.22). The equilibrium equation is set up tor the elementary
transverse strip of the accretion with length dx. The integration
of this equation i~ made from x = 0 to x = L.
We will assume, as proposed by A. Gan'on et al. /15/ that for
x • O, the normal stress is equal to the hydrodynamic pressure on
the rear edge of the accretion. Solving the obtained equation
with respect to the longitudinal stress a , we find:
(2.19)
where
The stress in Eq. (2.19) reaches a limiting value:
(2.19a)
for L • aB 0 , where a • 4-10 is a coefficient taken as a function
of the formula used for p 0 /6, 15, 42, 88/.
It is seen from Eq. (2.19) that for p 1 • B0p/o0 the second
term is converted to 0 and the pressure a in the given line te-
/43/
68.
aa.i.na c:oua,tut with an inc:reaae in the length of the ice: ac·cre-
ti.on upstream. For p 1 > s 0 p;p0 , the IIWl.Ximum valu.e fo·r pressure
wil-l be. for L • 0: for L > 0 the preaaU!'e dec·rease~P., which cor-
responds to tbe concept of the '"narrow river" proposed by A. Gan 'on,
R. Hausae.r, and E. Pariset /15/. As a con.sequence of a . more com:?lete
accounting for the. forc.ea acting on the ice-block accretion, Eq. (2.19)
permits establishing not. only the limiting width of a narrow river,
but also th,e corresponding ratios between the hydraulic character-
istics and the ice-block sizes. This feature is o .f interest.
in establishing conditions for possible fonnation of jams
and predictinq them, which is examined in Section 4.2.
A brea..-down in the stability of a one-layer ice-block accre-
tion, in ag·reement with the rheological model Ilia. (see .Fig. 2) and
Eq. (1.12) is det.ermined by the ratio~ o m > ok, at which the:re
arises a shift in the particles along the slip sections. For a
free.-flowing body, the value ok ca.rrespands to it·a int.ernal
(passive) resistance determined according to Eq. (1.26). As cthe
result of published investigations show, this method is also
acceptable far studying an a .ccretia·n of fine crush,ed ice /5, 6, 15,
51, 58/.
At the same time, the investigations that we conducted of
the beha~vior of the ice-blac.k ac,cretion far 1 >> b showed that.
losses in the stability of a . single-layer plane accretion arise
beginnin.g approximately at 1 > (3-4) h with. langitudina.l campres-
sian which exceeds the value o k, obtained according to the theory
of free-flow.ing bodies, but is cansiclerably leas than the strength
69.
limits of the .ice. The indicated phenomenon is associated with
the fact tha.t the mechanical system comprised of a packed plane
ice-block accretion is more. stable than the accretion of lumped
particles.
The breakdown in the stability of the ex.am.ined. system is
ca.used by the presence of a number of regularly occurring phe-
nomena (factors):
( 1) Off.-center contraction with a nonunif·orm value and
fom of the ice blocks;
(2) The phenomenon of inertial forces during ice move-
ment, which is caused by the crushing of ice chips, and the over-
thrust of the ice blocks on the shore;
(3) The formation of ice-block inclinations in the vertical
plane as a consequence of these movements, etc.;
(4) The presence of oblique cuts and projections on the
contacting edges of the ice blocks and a decrease in their
strength /19/, which causes the appearance of vertical components
of compressive force, the chipplng off of the ed.ges of the ice
blocks and their slipping against each other at an angle to the
water surface.
The finding of a gen.eral form for the dependence of system
stability breakdown on these factors is made difficult by their
complex probable manifesta.tion and the insufficient development
of the respective fields of mechanics. Therefore, the re,-
sulta of an experimental verification of the various mechanisms
of the phenomenon are taken as a base and the following operating
•
/44/
70.
diagraa is proposed.
For a 0 » b and 1 » hr the instability of one-sided
conn.ecti,ons of the syst,em under the taction o·f tbe factors
enumerated above reqularly leads to the emergence of the inclina-
tion o .f individual ice blocks in the vertical plane with their
bracing against each other and t .o the phenomenon of the respective
Jk>ments of active forces .. For a = a k, this .slope reaches a
critical value, the exceeding of which causes a breakdown in ice-
block stability in the system due t .o the proqressive increase in
the. torque of the force.s. The corresponding de.pendence for a k
takes the form:
(2.20)
11 -l -~(-I + ~) . •• •r '
A 4 is the probability co-
efficient for the occurrence of the phenomenon: for P = 50\, the
value of A4 = 1.1; for P ~ 10\ the valu~· of A 4 • 0.3-0.5,
and for P > 90\, the value of. A4 = 2-3. The values presented
for the probability coefficient for P ~ 10\ and P > 90\ have
an orienting natur:e. A confirmation of Eq. (2.20) for A 4 • 1.0
on ice-block models with h/1 • 0.1 gave a COIIlparatively sa.tie-
factory corr espondence for 25 experimental and calculated values
of ok (mean-square deviation 11\).
It should be noted that in establishing the dependence f ·or
determining the critical value of compressive stress, 5 work-
inq diagrams were examined, including that of D. F. Panfilov and
the diagram of the Canadian res.,archers A. Gan'on and B. Michel',
in which the stability is determined by the internal resistance, of
71.
the ice,like the resistance of a free-flowing body. This
resistance depends on h 1 and ~ , but does not depend on 1.
For a single-layer accretion o& spring ice blocks with 1 >> h
the bask factor determining the conditions of stability loss
is the longitudinal dimension of the blocks.
72.
/4 s I Chapter 3
Eaerqenoe and ~tion of IDe J ...
3.1. Cauaea •nd Focal Po'.iDta _of .I:ee Jams in Field Obs~ations
The general conditione, cause11 a .nd places o ·f ic.e jam forma-
tion in ella:nne:l currents have already been determined relative-
ly c~ompletely /1, 2, 7, 8, 10-12, 24-29, 43-45, 48, 52, 56,
68-6"9, 77, 80-85/. Certain investigation·• along these 'lines
will be reca1..led below. The resQ1te of a stuay of ice jams
un~ "fi.eld condition• leads to the follOYing conclusions of a
qualitative nat.ure.
Ice jams may form: (1) At the beginning of the ice move-
ment or in th.e breaking apart of the ice cove.r; (2) In the
period of ice mc·,ement or as a consequence of retarding the
opening of the :d ver in individual sections. In this caae
they are us.ually formed in ~he rise of flood waters and in rare
cases in their fall, with a utationary water flow (dQ/dt) = 0
or c1os:e to this.
Sections with a limited throughput (of ice transit) are
focal points of jams, as has been noted, under the condit.ion
o.f sufficient quantities of incoming ice and the energy pressure
value determined by condition } or T >
In the breaking up of the ice cover, "hummocking jams"
according to B. V. P.ros.kuryakov and V. P. Be·rdennikov /69/ are
fonned in the brceakdown of the ice cover as a whole, when hum-
mocking , pu11h.ing under and piling up of ice blocks against each
/46/
73.
other and against the shore create accretions of ice masses
in the channel. These same researchers classified two types
of hummocking jams: (1) Jams formed when couplings with tha
shore were absen.t; and (2) In the presence of ice mass coup-
lings with the· shore, when a side thrust is included in the
active forces. Hummocking jams are observed in both small and
large rivers (the Se-verna,ya Dvina, Yenisei, Lena, etc.). Jams
are relatively widely distributed in holding back the opening
in individual sections of. the river /2, 32, 44, 68, 90/. A
delay in the vpening may be caused by the structure of the chan-
nel, the hydraulic flow or ice conditions -factors that are
interrelated to a certain extent. Places of sharp discontinuity
in the general profile (with a decrease in the inclinations and
velocities of· the current), channel b ffurcations, etc. are char-
acteristic in the examined situation /11, 83/. Similar jams
' are observed in many rivers of Siberia /2, 43, 48/, the central
V·olga /34/, Central Asia /24, 84/, and the Carpathians /18/. A
frequent cause of jam formation is the opening delay in reaches,
particularly confined to sharp turns in the river /2, 44, 68/.
Such jams are characteristic of rivers flowing northward, but may
also be observed. in rivers of another current direction. '!he cause
of opening delays providing risk of jams is the stronger and thick-
er ice in sections: (1) In autumn-winter jams, in rivers con-
taining slush, including regions with a relatively cold winter,
for example, in the Northeast region of the USSR /48/: (2) The
74.
fonaation of large ice layers observed in Siberian rivers /29/;
(3) Inc~lete winter openinq with ice hU!IaOCking in reg i ons
with unstable winters, such as for examp l e, in the Dti.ester River
/ll-32/, in the lsrael River and the Wight, River-USA /77/ .and
some Carpathian rivers (the jam on the Chernaya Tisa river of
March 23, 1964, observed. by· IU. A. Oeev).
Th.e later opening of lakes, rese.rvolrs or primary ·river·s
with respect. to t .heir .inflow is also rarely the cau.s :e of jam
emergence at. the edge of the retained ice cover. A sect ion of
jam underwater may be the site o.f . jam formation, when the ice
cover is broken down in it. by the influx of water and ice blocks
accumulate downstream /1/.
Other focal points of ice jams when ice movement orig-
ina.tes. or is developed may be, as noted. by L. G. Sh.ulyakovskii
/83/, a diff.erent type of obstacle to ice field pa.ssage or large
transit i c e flux in the ice movement. These include channel
constrictions and bends, canals, islands, sandbanks, manmade
structures (bridge abutments, piers, etc.). Here both the
d.irect. holding back of blocks in the case of (B 0 < b) .as well as
their wedging in a turn may occu r. All of the enumerated jams
are formed a .s a rule for dQ/dt > 0.
Ice jams during ice movement may be formed. in those places
where the flow throughput is insufficient for any reason for
the f .ree transit of ice blocks. T'his ma.y b t! caused by an in-
crease in the ice flow rate or block size as well as by a de-
crea.se in the ice transit throughput. The increase in ice flow
/47/
75.
rate ~aay i ·n turn be cauaed by the development of ice movement
on the upper section of the river or the eruption of a jam there
/83/, and aLto by a concurrence of intense ic.e movement i ·n the
main river and ita tributat:y. Such ice movement has been ob-
served, for exaap,le, by F. N. Bydin /10/ in the Sviri River and
by Iu. A. Deev in the Nieman River below the Vili River junction
in 1950. In siJnilar case.s, jams are formed in low-lying limited
sections or in lines. (see Section 2 .1) confined to regions of
a decrease in current inclination and velocity, increased cha.n-
nel resistance in sharp turna, abrupt constrictions, etc.
v. s. Antonov /2/ notes that in places of sharp funnel-shaped
channel constriction, ice accumula.tes very readily and two types
of jams are formed: (1) Due to the wedging of large ice blocks~
and (2) Due to the insufficient floating capacity fo.r the passage
of all the ic.e.
We note that the appea.rance of large ice blocks and their
wedging is more o .ften observed at the beginning of ic.e movement,
but may also occur during it. The appearance of large ice blocks
during ice movement .may be associated with different cal:."'es , for
example, with the detachment of lar.ge blocks from underneath
islands -"tailings•, which Deev observed on the Angara .R.
Jams associated with the upward thrust of very thick blocks may
form in river heads, outflows from lakes and reservoirs, where
the ice thic.kness is usually thick.er than in the river. Lake ice
blocks in narrow river sections (h > H) will be held back
76.
and can cauae a jaa. Such a j-was obeerved in 1948 in the
Anqaza R. in a shallow nnd bank at tJae Patronr this arose
due to the inflow of Lake Baikal ice. It aay be as8\mll8d that
the j-on the Neva R. described by R. A. Nezbikhovskii /52/
are of this type. This is indicated by the sharp decrease in
the river f1.ow rate below the jaas, which is evidently caused
by the settling (landing) of Lad09ian blocks. at the bottom.
The decrease .in ice-transit channel throughput during i.ce
mo~nt aay be caused by a decrease in the wa.ter flow rate and.
a redu_ction in the water level or even a wind surge. In turn,
the decrease in flow rate and reduction in water level are caused by
a change in weather conditions or a delay of· the runof'f in upper-
lying illpOund st.x:uctures.. Jams associated with t .he occu.rrence
of negative tewaperatures are charact.eristic, for example, of.
northeastern rivers of the USSR /48/ where the beginning ice move-
' ment i8 retarded by routine cooling of.f. This leads to a . decrease
in the transport capacity of the current and the ice sinks. Such
jams are evidently formed under conditions of. a decrease in the
value of or a change in the sign of the derivative dQ/dt to the
negative. Jams which are. associated with a w•ind surge of the ice
IIWlY ar.ise· in the .mouth regions of large low-land rivers, such as,
for e .xample, in the lower Ob R.
The stability and the relative risk of jams in the examined
focal sites is nonunifo·rm. In regulations for counteracting ja_ms
/4 7 I, t .w:o types of permanent places of jam formation are clas-
sified: (1) Places with a discontinuity in the general profile
/48/
77.
ca.usioq a decrease in the slopes and flow velocity;
(2) Sharp turns of the river (greater than 110-115°). Here
it is necessary to include a_lso tapering sections of reservoir
hea(l curves, and in a numbtlr of cases places of manmade
channel constriction. Sections with a delayed opening gi.ve a
grea·ter risk of jam formation with respect to t .he ice conditions.
Gen.erally speaking, the other focal points o .f jam formation listed
above are less stable and permanent. Th.ei.r risk of jam forma-
tion in a number of cases is a £:unction. of the weather conditions
of autumn-winter-spring and concrete morphological, hydraulic,
and other special featu~es of the river or its section.
Places with an insufficient throughput, which represent
jam focal points, in agreement with Eq. (2.2) are characterized .. by a decrease there (up to zero) of a 0v - a multifactor func-
tion., derived fromEqs. (2.5), (2.15), and (2.16) which depends .
on spatial coordina.tes and time,. i.e.:
R.,. =I (u, n. 11, i 0 , b. h. ~).
The specific combination of these factors, which causes
a decrease in a 01, depends on the natural features of the flow,
numan ac.tivities, or both together. The classification of the
rivers of the USSR according to these indicators was not an
objective of the present study. Similar classifications for
specific purposes have been carried out in the investigat.ions
mentioned above. However, for a more complete analysis here o.f
... the cause of a reduction in s 0v, it is expedient to make the
71.
(1) lce-tbe~l factor• vllich cause a reducUon in a 0 4nd
-t v u a oon•~ence of the 110v-.ent reaiatance of ice blockcs of
it-.obile ice f~tiona (flow sections with unbroken ice eover,
slow-movin.q sections, incl~dtnq blocked-up iee fields, ice
laye.ra, etc. ) ;
(l) Morpholoqical factors, where a 0~ deere•••• with a de-
creue in the. flaw rate with a reduction in the ch•nnel inclina-
tion or wate.r aurf•ce (for the river outflow tro• mountains, in-
flux into rese~rvoira, raises in va.ter level, and in other cases
ot discontinuity of~ lonqitudinal profile);
(3) Mechanical factors which liait B·o due to a dee~rease
in tb.e depth (sandbanks, ab4llo;.r waters, etc.) and the flow
vidtb (constriction and branchinq of the channel, brid.ge abut-
~nt, channel dams, etc.);
(4) Aerohydrodynam'ic factora, which cause a chanqe in
the value or the direction of vecto.r ~ with respect to the flow
direction due to centrifugal foreea, circulatinq currents, wind
pressure, etc.;
( 5) Combined -) factors, where the decrease in a 0 and v
ia caused by several of the causes nam9d above.
After. analyzing ·t}\e literature. sources, it seems that most
larqft jams ar• related to the last group in the above cla,ssifica-
tion; for example, jams with a simultaneous decrease in th4:. cur-
rent velocity and division of the channel into canals while re-
/49/
79.
taining the ice cover there, etc.
3.2. General Conditions of Jam Formation. Jam Classifications.
According to the causes enumerated in 3.1, a reduction in
• B0v for various types of ice transit can be classified into
more than 50 types of jamA. Most of these were found in rivers
of the USSR, USA, Canada and other countries. Disregarding the
many factors involved in the many types of ice jams, their forma-
tion is subject to certain general conditions. An analysis of
these conditions permits establishing general regularities and
concrete features of the mechanism of jam formation and the
classification of their basic fo~.
Jams arise in segments with an insufficient ice transit
throughput (see sections 1.2-1.4, 3.1) in the case of a loss
of block stability there upon collision (dynamic formation
form), and with an incr~ase in the longitudinal compression
in the accretion of blocks to a critical value ak (static
form).
The possibility of jam formation on a flow section with
ice movement sp with a certain probability (with respect to
quantity and dimensions of ice blocks) is determined by the
general criterion for jam formation:
(3.1)
For k < 0, the jam cannot form. The relationship
0 < k < 1 corresponds to a "jam-risk8 section (line).
The smallest value k • 1 for sp > 0 will be in the case where
s 0 • 0 and the jam may form only directly at line s 0 • Criterion
/50/
80.
(3.1) cietermi·n• only one condition of j-formation -the in-
su_fficient thro\&9hput of a flow aection.
The neceaaary and au.fficiemt condition• of jaa foraation
are determined from Eqa. (1.8), (1.12), (2.16), (2.19), (2.20)
and (3.1) and reault in the requirement•:
0 < k-.:;1 with s,. > 0;
.11,-4 ..;;3 { or T) ~A • ,.
(3.2a)
(3.2b)
(3.2c)
Requirement (3.2b) corre»ponds to the conditione
v > v 2 , 3 (u > u ) and p a • f (L) ~ a • It is necesaa:ry that
the ·condition oS/a L < 0 be fulfilled for the extendinq of
the j • formation o .f a certain aection downward from a line
< The dimensi011s and maqnitude of the jam in-
crease with s 0;sp • o• (SP > o >; akta L > o, and T • T•
< Ap. By varyinq the dependences k, v, and o on factors
controlling their paraJHters a• a function of x and t, it is
possible to establish the possibility and place of jam forma-
tion. This question is examined in Seet.ion 5. 1.
8y Eq. (3.2) we predict the entrance of the ice into the
foraillq j .. through some initia.l line c 0 in the quantity SOn'
equal to the throughput of this line, which applies to
S > s 0 > 0. The value SP is a piecewise continuous function of time and P -n
bounded by the throughput of the flow section above the jaa.
The value SOn ia determined by the value (and, generally speak·-
inc;J, the sigfi) of the resulting work of' all the forces actin9
*sic
I
81.
on the ice blocks.
In the general case, forces act on the ice blocks in
the formed jam that have a geometrical sum equal to:
( 3. 3)
where iG and iYG are components of ice weight, di.rected along .. .. inclines i and iy' respectively: P1 and P 2 are forces of hydro-
dynamic and aerodynamic pressure (resistance) on the blocks:
F 1 and F 2 are internal forces of elastic and non-elastic de-
formation of the ice; N1 and N2 are the side and vertical
components of elastic compressio1. and the weight of the ice
caused by ice friction against ice, the bed, and the shore;~
-+ Pm is the force expended on thermodynamic and physical chemical ... processes: Pa is Archimedes• force: I is inertia.
The ~~ of L~ forces can be indicated by the dif-
terence in the energy of pressure acting in the direction of
' ice-block movement in the forming jam, and the work of forces
opposing this movement, i.e.; T-A2 _4 = f (x, y, z, t). The
value and sign of this difference are determined by parameters
that characterize the hydraulic and ice movement regime of
the current, which has been examined in detail earlier /20/.
Then, bearing in mind that T - A2_4 > 0 and Sp > 0, the
value s 1 _! 0, with the given inclination i 0 , the following
can be written:
(3.4)
where Hn is the value of the variable head, which arises due
to the inflows of tributaries or of impounds and jams situated below,
*Including the movement resistant force fron transverse obstacles
(bridge supports, wedged ice fields, etc.).
82.
etc; kf b the filtut.t.on coeffic.i6Jlt through the jam; R ia the
radius of curv•ture for the turn (bend) of the river.
The analysis of Eq. (3.4) facilitates the investigation
of the con<!itions and peculiaritiea of jp formation for dif-
ferent water/ice .-oveJHnt, type• of ice-block tranait, and the
structure O'f the flow channel (see Chapter 4) •
Two basic mechanisiU o·t jam formation are found as a
function of the. direction of the resulting forces enumerated in
Eq. (3. 3) with reape.ct to the nov direction. (1) With the in-
put of ice into the jaa aection in fonerd and forward longitudinal-
si<Se block transport; and (2) For rever•• transit. Jams of. the
firat ·type wi.ll be examined below.
A theoretical and experimental atudy shove that in the
general caae t:here are five stages of jam formation. /18/:
(1) Formation of the baae; (2) Formation of the core; (3) For1114-
/51/ tion of the rear aection -the jam body wi.th i .ce packing in the
ja• and the format .. ion of the head; (4) The fonu.t .ion of the
ta.il aection of t he jam: (5) T.he slow consolidation of the jam
mae• (or ita development state) during which ita breakdown be-
gina.
While the.re are general regularitiea in jal!l foraation,
there are considerable differences caused by concrete cond,i-
tiona in which the jams are formed. Theae relate to: (1) The
type of obstacle which createa the ratio k > 0; (2) The hydro-
dynamic and ice conditione (nature and regime of ice moveJMnt,
phyaical-mechanical characteristics of the ice block, etc.);
83.
( 3) The structure O·f the channel and flood plain o·n jam
sections of the river and adj,acent places. The resultinq
action of these factors determines a number O·f speci<!ll features
in the emerge.nce and mechanism of jam formation, as well as in
their f .orm, dime.nsions, filtration properties, and certain other
characteristics.
The experimental-theoretical investigations conducted by
the a.ut-hors ~ined with the results of fie.ld investigations
published in the litera.ture for ice jams permit the following
classi.fication of jam types.
1. The cause of jam formation may be subdivided into
those that form: (a) according to a dynamic diagram due to
loss of ice-block stability upon collision under the conditions
~ > A2-4' i.e. for v > v 2 3 or u > u ; {b) accordinq , p
to a a.tatic diagram, as a consequence of block stability loss .
upon collision with an increase in t .he longitudinal compressive
pressure T > A 2 _4 , i .e. for >
2. According to the formation mechanism and body shape,
jams are classi.fied as: (a) undeveloped, where the jam body
formation proceed.s under the conditions .) and T <
i.e. for 30'/aL < 0 and • 01 (b) developechri th
the formation of the body for ~ and T > A2 , 4 , 30'/i}L > 0
and a hf/ a L > 0; (c) the transition type, which is
formed for A3 ,4 > ~and T) > A1 ,2 • The formation mechanism
of these jams is examined in Sections3.3-3.5, and the criteria
/52/
84 •
. for estAblishing th.e type of ja. are given ih Section 5.1. The
diftereuce• i:n the atructare and for. of the body of the given
j-, and alao th.e distribution of the ice black.a in thea .. Y be
ex.ained in Pig" 6.
3. A.ccorclinC} to the na.ture of stress in the blocka, jams
that ewe:rge under the prevailing conditions are clasaified as:
(a) tangential stress (type T >a), which occurs in sections
with H < h (sandbanks, shallow water, divisio·n of the chan-
nel into -.11 canals) and for a closed channel (ice cover,
wedged ice fields, etc.); (b) not"'llal co~reasive stresses
< a > T under conditions of a 0 < b as a consequence o .f
retardi.ng the blocks at transverse obstacles (constriction•
and divi8'ions of the charrnel (bridge suppOrts, etc.): (c) tan-
gential an~ norJaal stresses (t, a) for s 0 < b and. H < h or
i .n a closed channel (bridge supports with unbroke.n ice cover
between the., etc.).
4. According ·to the filtration properties and special
features of the forJUtio.n of a !P.AXi•wn head level, jallls are classified
as: (a) freely filtering jaas with ice accretion poroaity, which
assures an average filtration velocity that is sufficient for pas-
sage of the entire water flow even with a COiftplete c .logging
of the channel by ice blocks; (b) slightly filtering; and
{c) nonfiltering jams. Filtering jaas may be formed when ice-
block accretions consist priJDarily ot large, strong blocks,
while nonfiltering jams fora with accretions of predoainantly
85.
small blocka of varying size (crushed or ground ice) or blocks
witb a small atrength A2 4 < A • - - p
S. According to head section form, jams .are classifi.ed as
t:hoee: (a) w.itb bottom alope (aee Pig. 6a, b, c): (b) without
bottom slope (aee P'ig. 6d), which are formed under conditions
of b < B0 at transverae obstacles in sharp constrictions of
the channel, etc •.
6. According to condi.tions of jam head shift (floating)
along the. ve·rtical, which depends on the nature o .f the obstacle,
jau are divided into: (a) those with free shifting: (b) with
a ~inched" base and with stopping at the obstacle, which
corresponds to the condition • 0 (jams at the
edge of a st.rong ice cover, at transverse obstacles (see
Pig. 6d).
7. Accord.ing to the morphological features of the jam .
section, the strength of the obstacle, and the ice probabi.lity,
jams are examined: (a) with a formation not limited\ to the
given causes; (b) with a formction limited to these· caus~s.
which is examined in more detail in £ection 4.4.
B. According to the ,.,ater discharge regime and ice move-
ment in the formation of the jam, jams are classified as form-
ing: (a) under stationary conditions, i.e. at dO/dt • 0 and
dS/dt .. 0; {b) under non-stationary conditions with
acce.lerated, retarded, and contimlous development,
examined in Section 4.4.
/54/
86.
In add.ition, jaaa should .be mentioned tha.t form ·in t.he
absence of shore binding and in the presence of this bond /64/.
Jaaa of transit (coaing t ,o the top) blocks, of local blocks
which are foraed in the breaking apart of the i .ce cover, and
frOID blockB of lliscellaneoua origi.n JUY also be distinguished.
The characteristics (forma) of jaas which are given under
1-3, 5 and 6 above are the results of investigations conducted
in the present study, while those given in 4, 7, and 8 represent
a clarification and gen.eralization of those data on .janaa knotrn
from the. literature. The jaa formation mechanisa and it11
peculiad ties, the .form of the jana body, the equil.ibriWII state,
and their other characteristics will be examined below.
3. 3. Formation of Jams Under Stationary Conditions. Eaergence.
and Consolidation of Jams
The forJUtion of jama under etat.ionary water/ice move~Dent
' conditione and in the absence o! growth liaits both in flow
length, as well as in height, i.e. for L 3 > L1 , Lv
and Hb > Hp' where L 3 and Lv ere the lengths of the jam and
rectangular top bordering flow sections (see L0 , L1 , and L 2 in
Fig. 6). The developed jam with a "dyna111ic form" of .formation
is taken as the basis of analysis. The formation mechanism of
this t,ype ot jam haa been examined earlier /18/, and ia further
refined with the following experiment.al-theoretical study. The
features of ja.-.s of the second type are established according
to a stepwise analysis of their formation mechanis111.
/52/
87.
----------------------------
o) -t -. - -..!J_:::u. •
-----lit
C) --...---
;pl
·r·
.
~
·'
Fig. 6. Schematic longitudinal jam section. (a) Undeveloped
freely-filtering jam with a bottom slope; (b) A
developed freely-filtering jam with a bottom •lope:
(c) A developed non-filtering jam with a bottom slope:
(d) A developed freely filtering jam without a .bottom
slope (at a transverse obstacle with b less ban B0 :
L1 and L 2 -head; L 0 -rear ( •body•) and L 3 -tail
(lingle-layer) part of the jam: iJ-channel slope:
a-angle of block inclination in the jam body:
a 1 -angle of inclination of the bottom slope
of t:he j1am.
88.
Tbe emergence of jams •• vell as tb_e formation of
their core along vi.th general regularitiea alao have dif-
ferences that are determined by: (1) The initial v;alue of
the bLock energy (tand T) vith respect to the work necessary
f .or plunging, piling, and pushing the blocJe (A2 _4 ): (2) The
nature and features of the jam obstac:le in the channel. we
examined the emergence of jams on sections where the. condition
k < 0 arises as a consequence of: (a) Insufficient depth;
(b) Inst.ff.icient. vidth; (c) The presence of the unbroken ice
cover (ice fields) in the path of 110vement of the transit blocks;
(d) Non-simultaneous opening of the specified flow ael~tion;
(e) The presence of obstacles opposing t .he formation of ice
movement. in the opening of the river.
The emergence am.1 consolidation of the jam under condi-
tions of } > .A 3 ,4 in the section with H < h begin with the
' formation of the lower subjacent block layer -the jam base.
Approaching the jam section, the blocks push deep into it
under the action of energy 3 and pressure T of returning
blocks (for ~ or T > < A3 ). With an increase in
the length of the pushed-through layer of blocks, the work
A4 increases and at T < A4 the formation of the base is
termin.ated and the core formation begins.
With an initial energy value of less than A4 , the blocks
remain immediately in front of the jam section, where the core
begin& to form. An analogous phenomenon take& place in the
/55/
89.
presence of. transvers.e obstacles for 1 and b > B, including
also the wedging of large blocks (jams without the bottom
slope, see Fig. 6).
The emergence of a jam on a section with an unbroken ice
cover, in the approach of the blocks to its edge, requires the
fulfi.llment of the additional condi ti.on:
( 3. 5)
where s 1 and s 2 are thequanti.ties of ice carried off under the ice
cover and along its surface.
The bltx::ks can move under the ice cover at a flow rate of
u > us or uv, as established in Section 2.3, and the ice
flow rate can be approximated according to the equation:
(3.6)
The mod.el in.vestiga.tion showed that for ~ > A2 , the
block stability losses at the edge of the ice cove.r caused
both their plungin9 and their tightening with the wedging of
blocks for 1 > H braced between the bottom and the edge (see
Figs. 4a, b). These blocks are used as a springboa.rd for the
creeping of other blocks to the edge and at the same time
create an obstacle to the plunging of new blocks under the edge.
The respective decrease in s 1 causes the accretion of blocks
at the edge and a raise in the water level with its discharge
onto the ice cover, so that blocks striking there can be car-
ried away along it in quantity s 2 which depends on the values
Hp' s 0 , hand the ice roughness (hummocking).
90'.
In fulfilling Eq. (3. 5), the first laye.r of ice blocks
on the ice cover foraa the jam base. In the qeneral case,
Eq. (3.5) can reault from an inc.reaae in SP' as well as in
a decrease in 5 1 or 5 2 (for exaJIIple, according to the clogging
of the current under the ice cover by blocks, etc.). Further,
the formation mechanism of .both types of jams (w·ith H < h)
is siiRila..r also on sections v·ith the ice cover.
In the delayed openinq of a rela•tively larqe section of
the .river, for example on a sandb~nk. while ke.e .pin.q intact the
ic.e . cover in the lov·er-lyinq reaches, large blocks may form
the jam base, and these blocks are formed in the breakinq apart
of the ic:e cover and their penetra.tion downward under the edge.
of the ice· of the unopened sect ion.
Model investiqa·tions permitted the rather clear tracinq
of the entire process of a similar non-simultaneous openinq·
and emergence of the jam on a conventional model of the geo-
morpholoqical pair sandbank / reaches (at t .he channel turn)
with a thicker ice cove.r on the reaches. The process is
developed in the follovinq manner. With an increase in the
wa·ter flow rate, the ice cover in the section vi th the siNlll-
est at.renqth and the weakest external couplings (sandbank.) lost
its couplinq to the shore, rose up, protruded, lost its longi-
tudinal stability and then broke apart into larqe units thrust
over the edqe cf the stronqe.r ice cover (reaches). Depending
on the current ve l ocity and the increase in the va.ter flow rate,
/56/
91.
two mechani.sms of the. examined phenomenon are discerned.
With a high current velocity and an intensive rise
in the water level, the deformation and breakdown of the ice
cover corresponds to the behavior of elastic freely supported
plates, examined in the theory of materials' resistance. In our
tests, under such conditions, the• ice cover buckle.d, forming
bulging waves and then broke into large units (according to the
number of pathways of the bulging).
With a relatively low current velocity and a slow rise
in the water level, the ice. cover at the. sandbank (which is not
connected with the upper reaches) floats up, forming at its
place of rein.forcement with the lower-lying ill'lllobile cover only
one bend. With a rise in the water level to a value 2-4 times
the initial depth there is observed a breakdown and shifting of
the ice to the reach·es with its subsequent breakdown as a con-
sequence of collision with the edge of the strong ice in the
reaches (Fig. 7).
/57/
92.
lee blocks formed in this 111ay partially creeplng or
t.ighteninq undex:· the edge 111ere wedged toge.ther r ·ather than
creating the jam base. In individual cases, the 111edging of
particularly large blocks 111ith the moving out of their front
edge. under the edge of the ice cover created a clogging up
of a single-branch channel and a raising of the water level
like in a type of spillway dam with a wall inclined
along the current. In front of such "dams" there are observed
curves of fall and an increase. in the current velocity, so
that the su.bsequent smaller (transit) blocks freely pass through
the jam cre.st and the further development of the jam is termi-
nated. This phenome.non can obviously be used in some cases for
preventing the formation of large jams.
Therefore, in the examined type of jam, the base and low-
er portion of the core are comprised of large blocks of "local .
origin". The uppe:r section is formed of small "transit" blocks .
A similar distribution of the ice in a jam has bee-n established
also, for example, on the Lena R. by A. s. Rudne'iT /68/.
The consolidation of the "hwnmocking jams" .is ana.logous in
its general nature, with only the difference that in thia case,
jams are confined not only to the, edge of the ice cover, but
also to places of reduced channel throughput. After consolidating
the jam (the formation of its base) the core is formed, and then
the jam body. The mechanism of these force.s, according to the
resu:•.ts of experimental-theoretical and field tests, are exam-
93.
ined in Section 3.4 and 3.5.
3.4. Formation of the Jam Core
After the jam is consolidated, its core is formed, which
is a relatively compact piling up of ice directly on that
section where the blocks are stopped or their movement is
slowed down as a consequence of insufficient throughput
(Pig. 5, Ba, 9a).
With a large value for the initial pressure energy of
the blocks () and T > A3 ,4 >, which is characteristic for
developed jams, the core formation proceeds as a consequence
of the regularly alternating processes of accumulation (piling
up), submerging, tightening and pushing through of the blocks
into the depths of the jam section. This alternation is caused
by a change in the values of ~ 1 T, A2 _4 and the ratio between
the' and the growth of ,the water level rise and dimensions of
the core.
A certain level head caused by the formation of the base
reduces the work value for the accumulation of blocks A3 and
at 3 > A3 < A4 , the blocks that have not pushed through
earlier 1 upon stopping, pile against them. This piling in tut·n
increases the crowding of the channel and the rise in the water
level, decreasing the frictional force between the blocks (and
consequently also work A4 ) and again gufranteeing the process
of the pushing forward deep into the section of the blocks
piled up on the base, which occurs at T > Thus a
second layer of blocks forms for the specific length of which
94.
there again arises the ratio ) > A3 < A4 and the pushing
throuqh alternates with the piling up of a third layer.
Theref.ore, tbe formation of the core in this stage of its
development proceeds,as it. were,by the subsequent accumula-
tion of block layers. However, due to the nonuni.form size of
the blocks, and also the values v, u, and H , this process p
occurs irregularly, accompanied by hummocking and upsetting of
blocks.
A further increase in the water level with an increase in
the size of the core .reduces the flow velocity and consequently
the block pressure ene.rgy. Consequently, the length L1 of each
ice bloc.k subsequent in the height of the layer decreases and
the bottom inclination of the co·re acquires a stepw.ise acute
angled form (with respect to i 0 ). The upper (upstream) frontal
slope of the core during this time has an angle close to a
straight line (see Figs. 5, Ba and 9a), which corresponds to
the essence of the examined process.
With a furt.her decrease in the flow rate, only large
blocks still possess energy of -~ > Al' At the same ti~e, condi-
/60/ tiona of block tiqhtening arise at the core and are facilitated
dut! to an increase in the depth and the emergence of the ratio
>> < The tightened ice blocks take root flatwise
against the front side of the core (Figs. 9a and lOa), as a
consequence of which the filt·ation coefficient decreases and
consequently so does the filtration flow rate, while the water
/58/
95.
level increases. Because of this, the va.lue o .f work A4 is
reduced <1ccording to the pushing in of the float.ing blocks
along the core surface.
Fig. 8. The formation of the
edge of the cover.
develo.ped filtering jam at the
(a) The core formation due to creeping and tighten-
i n g
(c)
of the blodts; (b) The .formation of the jam body;
The fo·rmed jam after shifting.
/59/
Fig. 9.
96.
.•
. rz..~ ....... ...,., t:* r:e= • • . ,:J,...-...-:rt~ ~~~-~.;.~ ... ,_.,,_.;t .. .._, • ..-:.,,,.,.,,. • .._; .1 .:0-a~,...._.alfli...,,..,.J~I .J..-a.~~,l;j
• ' • ·.., • : • •• --•·· I ., ..., . •• .' -· -•-. :-· ...... -..-,._ ..
,.._.,._., _.,,, 0 ·I 0 o 'I ' ,\ ', • ..... -· .·
The formation o .f the developed filtering jMl at the edge
of the ice cover for a turbulent flow atate.
(a) Moment of core formation due to block tightening:
(b) Foraing of the head section
formed jam after shifting.
and jaa body: (c) The
/60/
97.
Fig. 10. Formation of an undevel oped jam at the edge of. the
ice cover.
(a) Core formation; (b) The formed jam.
At this second sta.ge of core forming, the process of tighten-
ing and packing of the blocks in the f .ront part of the core, which
a.lternates with the pushing through of the blc cks a.long the sur-
face of ita lower bank acquires, along with a growth ir the wate.r
level,a very large specific gravity, and at the en.d of core forma-
tion (for A3 » ~ ~ A2 ) it becomes the only process. 'lbe pl~.mging blocks
t .ightening aga.inat the core are at f irs·t distributed verticall.Y,
and then as the core CJXOG,they acquire an inverse slope (see F'ig. 9a).
Thus, in the formed core, the ice block.s are distributed more or
less fan-shaped: in its lower section -horizontall y, in its mid-
. 1-
/61/
I' • •• -.. ':"' ••• •• 7 ~ . , .. _.... ....... -.-..:.....·-· ----· ·--~-
98.
.... ..:· ( ~ ..
·-J--.-..1111. ·-
dle section -vertically, and in its front end -at an obtuse
angle to i 0 (see Fig. s, 9b).
The pressure T may cauoe shifts of the ice in the core,
which increa.se .its dimensions and the angle of the bottom slope
The avera.ged value of this angle (olong the line which envelopes
the slope proflle) lies within limits which are deter111ined from
the conditions of block stability toward slipping an.d shif~ing
along the current bed, i.e.:
tan
•~tca.c; .. ·, (3.7)
whe.ce · •• -m,tp 1 + 0,7~1'1tt 11/. -m 11,; IRa -0.4-0,7--. -· t» . . -. -- .
is the coef-
fi~ient which considers the activity of the filtration current
through the jam. For + < ~' • in Eq. (3. 7) instead of+
1 (?) I
we uee • •
~ith a further decrease in th.e flow velocity due to the
rise in the water level to a value of v < v 2 in front of
the core, only a hummocking of the blocks will occur, and with
u • uk (i.e., for u < u land v p < v 1 , their joining to-
gether,and in this way the core formation is terminated.
The water level head at the formed, freely filterinq jam
and the angle of t .he water &urface, which reaches a minimua
value a.t .1is time, is determined. from the equations:
H-....-"-· . '"• .
•••• ~ .... ----0,4rJII:·t· •
(3.8)
(3.9)
In non-filtering jams, the level head will be greate·r than
H by the value. 1
ing Eq. (4 .19b).
A H • H1 -H' 0 , where. H '0 is deterained accord-
An explanatlon of the cause of this phenomenon
------------~__... __ ··-.
99.
is given in Section 4.3.
The core formation for a turbulent current cond~tion·with
an initial depth of H < Hk (where Hk is the critical depth)
ia characterized by the emergence of an hydraulic"jum~ above
the core, after the level head reaches the value Hp > Hk
with an increase in the ground current velocity and turbulent
flux. As a model investigation has shown, this leads to a more
intense packing of the blocks in the submerged portion of the
front core inclination. Consequently, qf is reduced, Hp in-
c::.reas5, and so ciJespreasure T on the core, which is experiencing
in this case greater shifts and the packing of the ice with
an increase in tan a up to a limiting value of 41 (see Fig. 9b).
The core formation under the initial condition
A3 ,4 > ) > A2 (which is characteristic of undeveloped jams)
begins with the stopping (or hummocking) of the first blocks
in front of the jam section. The subsequent blocks push under
them, creating a rise in the head and decreasing the work
value A3 , 4 . For T > A4 , the upper layer of blocks pushes
into the jam section, forming a core base. Then the base
forma by the repeating pattern: submerging (tightening), head [water 1~]
/62/ increase, and pushing of the blocks along the core su~face into
the depth of the jam section. In thla case the core dimensions
and the bottom slope angle haveamaller values (Fig. lOa and 11),
than in developed jams. The core is formed analcgoualy in jams
which form according to the static diagram with am > a k"
'·
· .. ~ . ... · .. . ,.
100.
, ........ __ .. ····-: ---~ r·-•· .. -··· ..... _....-
... . .. :· ·~·r,.·"'(~·-·~"'! .;.·~·• ~"'~~ .. -..... ··: :··.·:' , :· ....... :
.·
Fig. 11. Transition-type jam .
The examined core formatio·n mechanism relates fully to
the case of fre.ely filtering ice-jam accretions, wh.e·re there is
a more or lus complete piling up by the ice of the useful cross
aect:ion. In sl.ightl :~r filtering and non-filtering jams, there
remaina a space under the ice accretion sufficient for passage
of the water flow under conditions of a closed channel under
the ice (see Fig. 6c). A number of investigaton: have studied
the par .. eters of such a flow in the established regime /5, 15,
SO, 58/. A theoretical dependence between hydraulic chara.cter-
iati.ca of open and closed flows of continuous ice accretions
eatabli8hed by Berdenni'kov showed that the, hydraulic angl e of
the sub-ice flow increases considerably.
The basic feature of dightly filter i ng ~tnd non-fi.ltering
jams under the condition of a possible .free ahift.ing of the blocks
in the vertica: section (floating) is the fo·rmation of .the core
allliOst canpletely (excluding the initial stage) as a consequence
101.
of the blocks pushing under those that have stopped earlier.
Jams with a pinched base and supported against an obstacle
form primarily due to the piling up of blocks one on top of
the other. The pushing under of blocks here can develop sub-
sequently if the upper layers of the block pilings are not
choked. In the contrary case, a continuously increasing water
level head will cause the rise of the core, which creates condi-
/63/ tiona for the piling up of new blocks and their movements against
the obstacle. The formation of the head section will proceed
continuously until the ice movement terminates.
3.5. Formatior and Packing of the Jam Body
In the formed core, the slope, as noted, has a minimum
value corresponding to the head curve type a 1 , while for
H < Hk' it corresponds to type a 2 • In front of the
jam core, with u = uk, there is formed a single-layer accretion
of blocks, which in developed jams and transitional-type jams
is further transformed into a multi-layer accretion -the jam
body.
It follows from the determination of the developed jam
that these are formed in flows with relatively large channel
slopes. Under these conditions, the level head decreases rela-
tively rapidly with distance upstream from the core, and the
current speed increases, and also at a certain line c 1 with
u > uk, the tightening and submerging of the blocks increases
with their movement toward the core under the upper layer of
102.
t!he idnobile ioe. 'l!his movemen.t: is. assured for 3 > A:1 ,
which is mad.e possible by a decrease in the hydrostatic pres-
sure foli each subsequent block layer du.e to dll-increase in Hs
and the alternation of block slippin:q with their .Uatwise rota-
tion., and also an increase in the head and consequently of the
slope of the water surface (in the section, the line c 1 is the
co-re ,. due to the cracking of ice blocks there). 'I!he head
curve with minimum slope in this case pushes away from the. core
to line c 1 ~ a .s a consequenc.e of which, the. current velocity de-
creases there and the block submerging terminates, replaced by
their abutting at u a uk.
the block movement toward the core under the ice accretion
still contin·ues during this time, since the current speed. is
greater there due to the ice crowding o .f the useful section.
At first t>locks that have reached the nucleus come to a atop
and press against it along the entire plane or at a certai.n angle
to the direction of vector~ (?] (Fig. 9b), since here the work
A1 increases, and part of the block energy is consumed in
packi.ng. The growth of the ice accretion at the core again de-
creases filtration through the core, incre.asing the head le.•el
and decreasing slope i, which slows down the movem.ent of the other
blocks in the section:·. ' 11 ne c 1 -core. The wor.k of pushing the
blocks under the ice accretion also increases due to the incre.ase
in its thicknesil. The stoppi.ng of the block movement over the
entire section (and consequently an increase in the. level head
/64/
103.
and a decrease in slope) begins after the ratio ) < Al
is established under the ice, which corre.sponds to the equilibrium
condition (1.10).
In the final analysis in the line c 1 -core section, the
blocks are established obliquely with a decrease in the thick-
ness h 1 of the accretion as a function of distance from the core
(see Figs. 8b and 9b). Upstream from line c 1 at a certain line c 2 ,
as a consequen.ce in the decrease of the level hea.d, the process is
repeated, and ~ ari~ in a subsequent upper line Ci, etc.
Thus the increase in the rear section of the
jam consists of a subsequent regressive -upstream -formation
(at 3 > A1 ) and an automatic braking (at ~ < A1 ) of the
processes of block plunging a.nd piling up in the channel (see
Figs. 8b and 9c). As the line c. moves a.way from the core, the
1
thickness of the block accretion h 1 and the quantity of ice pene-
trating the nucleus decreases as a consequen.ce of the general
level increase, the lengtheni.ng of the block movement path, and
the increase in hi in the direction toward the core. The de-
gree of subsequent piling up of blocks at the core will be pro-
portional to i 0 and q; for high values, the increase in the rear
section upstream will continue until the ice movement is terminated
or until a new -upstream -jam forms, or until the present one is
broken down.
As the jam increases in dimensions, the slope of its sur-
face i 3 and the pressure of ice and water on the core increase.
With a pressure value of T > As, d '· i.e. when it exceeds
104.
the int.ernal resistance of the accretion F 2 (passive pressure),
there arises a shift of the entire ice mass to t .he core side,
increasing the thickness and density of the accretion there.
The increased and deformed core forms a head s!!ction of
the jam (see• .Figs. Bb and 9b, c) with a corresponding increase
in the level head there and a decrease in the filtration dis-
charge uf and angle i 3 • The nucleus packing in freely filter-
ing jams in turn causes the additional rise of the head, neces-
sary for increasing the hydraulic slope, which guarantees the
ratio qf = q, in maintaining jam stability. The increase in
the level head and the de.crease in the slope reduce pre.s.sure T
on the nucleus and the packing process upstream, creating an
overall increase in 1 over the ent.ire jam section. This process
3
again can cause a lengthening (for u > u ) and a thicke.ning p
(for ~ > A1 > of the rear section of the jam with a subsequent
increase in pressure up to T > A d, and a new ice movement s,
at a higher head level with a subsequent new increa 3e in the
jam head.
During these shifts, the block slope at the water su:rface
acquires in the base section of the jam body an angle close to
the angl!.e of internal friction, which corresponds to t 'he medium
conditions in whic.h the block accretion is found (for the paraf-
fin modeLs, this angle was 30-40°).
The stopping of the growth of the head level and block accre-
tion thickness below the specific line (including the jam heod)
/65/
105.
corresponds to the simultaneous fulfilling (emergence) of
the following conditions:
blocks of T < A d: s,
t .he block accretion T
free surface of Tv <
<
(1) the current pressure on the
(2) the water current pressure under
T vk:
(3) the wind pressure on the
(4) the specific water dis-
charge of q = qf + q 1 , where q 1 is the discharge under the block
accretion. According to the rheological diagram IV, the first
three conditions result in the fulfilling of requirement.s (1.8)
and (1.11). The fourth condition is det.ermined by the filtra-
tion processes of the jam block accretion, which is examined in
mo .re detail in Chapter 4.
As the jam is formed, line ci moves away upstream with the
simultan.eous increase in h 1 • ~•t some line c 0 , where condition
(1.12) is fulfilled in agreement with diagram Ilia, the multi-layer
accretion of blocks is converted to a single layer. Here the
level .head will be determined by the value:
n. -__.!._+•"+ "-'••· r ... (3.10)
where E = 0.15 -0.20: h . is the smallest block thickness in r.u.n
the given ice movement.
The two last terms in Eq. (3.10) present a "correction"
fo.r the head, c.rea.ted by the single-layer block accretion above
line c 0 and the additional raising of the wa.ter from smaller
size blocks, which will advance by pushing under the blocks of cal-
culated dimensions which have stopped earlier. The value of
the second correct.ion is determined from a. diagram which allows
106.
relating the phenomenon to the determinat.ion of the sum of two
infinitely decreasing geometrical prOCJressions.
Eq. ( 3.10) corresponds to the stabUi·ty condition of both
the separate blocks that are pushing under, as we~l as to their
single-layer accretion. Thus, for the calculated value of uk
in Eq. (3.10) the smallest value obta.ined from Eqs. (2. 9) and
(2.14) should be taken . A checking of Eq. (3.10) on model and
field. t .ests (the Dliester and Prut rivers) gave relatively satis-
factory results. The mean-square deviation of the calcul ated
va.lues from the experi•ental was 16\.
Below line c 0 towa::d the jam head on section L0 (see
Fig. 6b), the block accretion thickness h 1 increases according
to a nonlinear law. The lower envelope of the longitudinal
profi~e of the block accretio·n in a prismatic-formed. channel
approximated by a flat curve is described by an exponential-type
curve, as an analysis has sn ... .,wn (see Section 4.1) and as experi-
ments have confirmed (see Figs. 7, 8b, 9c).
Therefore, the developed jam whic:h has formed with a . bo.tto•
slope has three characteristic regions in the longitudinal direc-
tion (see Fig. 6b, c): (1) the head reqions, where the ice fills
all or most of the channel with two sections: the lower section -
triangular form with a bottom slope of length L1 and the top
section - a rectangular section of length L 2 : (2) the rear
(jaa body) region of length L0 with decreasing thickness of the
accretion h 1 upstream: (3) the tail reqion consisting, of a
/66/
107.
single-layer accretion of blocks of length L3"
Local changes in h 1 , which deviate from the exponential
law may arise in natural channels in certain sections from change.s of
B, H, 1 0 and n in the jam body. The tail o .r rear S1ection of one
jam may come into direct contact with the. head of another, upper
jam. It should be noted that if the pressure o -f a single-layer
block accretion does not permit establishing equilibrium as deter-
mined acco.rding to Eq. (3 .10) then the jam will con.tinue to form
due to the upstream movemen.t of section L0 and consequently there is
increase in the length of section L 2 (for a further explanation,
see section 4.2).
The maximum level head Hm in the formed jam is recorded at
the upper boundary of section L2 . Downstream, the hydraulic
resistance creates a depression curve, while upstream, the water
surface in the filtering jam is found under conditions of water
rise, with slope 13" In line c 0 , this slope corresponds to the
value 1 1 for head dimensions determined according to Eq. (3.10).
The slope of the water surface 1 3 a f (L) in section L0 in the
general case has a nonlinear character.
The body of developed non-filtering jams forms almost like
that of the filtering jams. The basic difference consists of the
fact that under the ice accretion of the non-filtering jam, a
space rema.ins sufficient for the passage of the incoming water
discha.rge. The minimum sub-ice. dept.h H' 0 is determined at the
place of the s!"allest ice accreti.on thickness, i.e., at the
108.
boundary of sections L0 and L2 (see Fig. 6c). The average value
of this depth in a prismatic-form
Eq. (4.19b).
channel is determined by
Slightly filtering developed jams represent a transitional
type between freely filtering and non-filtering jams. With re-
spect. to their filtration properties, which determine the dis-
chArge qf, there remains a free space under the ice a .ccretion
that is sufficient for the discharge passage q 1 = q -qf. Sub-
stituting the valu.e q 1 for q in Eq. (4.19b) it is possible to
determine the respective averaged depth H0.
Depending on the formation conditions, undeveloped jams
can be classified into two types:
(1) Undeveloped jUul, which consist of the head L1 (o•r
L2 = 0) and the tail section L 3 adjacent to it -a single-layer
b~. accretion. The rear section (body) is absent in these
jams, i.e. L0 "' 0 (see Figs. 6a, lOb):
(2) Jams of a transitional type, where L2 • 0 also, but
which have a jam body, comprised of several layers of blocks
with a predominantly horizontal disposition and decreas·ing up-
stream of value h 1 (see Fig. 11).
Jams of the first t,ype are formed in the. case when the
minimum pressure of block accretion on the formed nucleus
does not n.xceed a cri,tical value, determined according to Eq. (2.20),
i. •• I < and u <
Jams of the second type form when the formation of the jam
body occurs under conditions of crm ~ crk' but T < A8 d. In
/67/
109.
SJCh a case, the formation of the jam body proceeds without
shifting and packing of the ice. More complete criteria for
the determination of undeveloped jams and transitional-type jams
are given in Chapter 4.
The question of a limiting value of ice filling of the use-
ful cross section in jams has not been fully clarified as yet:
certain theoretical treatments /43, 50, 73/, confirmed by a
n~r of !ield test.s /43, 44,74/ lead to the conclusion that
a complete cl~gging of the c hannel with ice is impossible, but
cases of channe~. clogging of rivers down to the very bottom ha.ve
been noted, f.or EXample in the Lena River /2/, the .Israel R.
in the USA /18/, etc.
It should be noted that in this connection, first of a.ll,
the measurement of the ice thickness of jams in the field (par-
ticularly at their head section where it is smallest) is coupled
with a number of difficulties and often tbt thickness is given as an
approximate value according to .!ndirect indicators. Secondly,
the theoretical treatments mentioned here cela.te to the formati on
of non-filtering (or slightly filtering) jams of crushed ice and
are fully justified for such a case.
Howeve.r, in jams of large, strong blocks, which assu·re suf-
ficien t filtration through the jam head (Of • Q) a complete ice
clogging of the u.seful cross section can oc:cur in the head
section of the jam, which is confirmed by the model investiga-
tion (see Figs. 7-9). A clogging of the channel up to the very
bottom is also possible with crushed ice in the formation of a jam
llO.
under the ice cove.r or under a chok.e .c! ice field. Such a
phenomenon is observec! in the settlinq to the bottom of the
already formed jam~ accre,tions in the decline of hiqb water and
i~1 the retaining of jam masses in shallow· water after jam
eruption in the preeence in both ca.ses of an outlet. path .for
the water to go into "channels", along reaches or rifts. With
free filtration of water through the jam, the' flow can. be directed
along the out.let path (the branch, reaches, etc.) or it can break
through "pressure tunnels" in the ice accretion. The fillinq up
of the channel with ice down to the bottom also takes place in
jams at large ice layers /2/.
Therefore, the extent of filling of a single-br.anch (with-
out outlet path) cha.nnel with ice at the jam head naay be dif-
ferent and depends primarily on the filtration propertiea o .f the
ice a .ccretion determined by the dit~ensions and stre.ncJth of ice
blocks and the degree of uniformity of these oharacteriatics,
and also on the type of j.~m. i .ts vigor, the channel peculiaritiea,
the flow hydraulics, etc. We will not continue. further with non-
filtering jams, which have already been subjected to a detailed
investigation elsewhere, but we will note briefly so.a character-
istics of filtering jams.
In freely filtering jams, which ariae wi't.h R > h, as the
thickness of t .he .ice accre.tion grows at the jam head, it aettles
and canes into contact with more elevated bed aectiona. The flow
of water under the i.ce decreases in si.ze and 11ay be divided into
separate pressure currents passing along the deprraa.a bottOift
111.
sections. Having a high hydrodynamic pressure, these flows
/68/ may change their direction,breaking open a path in the direction
of least resistance to the movement, including bursting open of
the jam surface. Ye. I. Ioganson described such a phenomenon
on the Volkhov R. /29/.
Deev observed pressure currents within a pile-up of
ice and slush covered at the top by a continuous ice cover in
the Angara R. in 1947-48. Their direction was clearly followed
by the strong noise of the blocks carried along by the water and
striking against each other. The breakthrough of the current
onto the surface found here had an explosive nature; blocks with
an area of several square meters and a thickness of more than 0.5 m
were heaved up and thrown several tens of meters from the place of
breakthrough. According to calculations, the maximum current
velocity in the breakthrough was on the order of 10-12 m/s. With
a further growth of the jam "head", continuous sub-ice flows,
shrinking and dividing, can be converted to filtration flows.
However, an increase in water discharge with an intensity high-
er than the critical intensity, i.e. at q > qf, which causes a
jam rise, may lead to a partial floating of the ice and formation
of new sub-ice currents.
Sub-ice flows are not characteristic of freely filtering
jams which arise for H < h, i.e. due to the mechanical contact
between the blocks and the bottom. They are formed only in
longitudinal recesses of the bed, and also in the rise and break-
through of ~ater in some section of the core ("head") of the jam
112.
as a consequence of ita insufficient atrenqth, i.e., under the
condition T > A d' wtlich corresponds to the breakdown of B,
Eq. (1.12). With an increase in the jam head sue, the possibility
of fo·I'1l'linq such flows decreases. An increase in the size of
the jam head during its .formation leads to a decrease in the
concentrated sub-ice flows due to the gradual filling of the chan-
nel depressions with ice. Therefore, for large freely-filtering
jaJU, the filling of the entire channel with ice in t.he jam head
section is characteristic. The large.r dimensions and density
of the ice packing can lead here in individual cases to a . con-
siderable decrease in filtration through the jam and to a reduc-
tion in the water discharge and level in the river below the jam,
i. e., a freely-filtering ,jam is converted into a slightly .filter-
in.g one. such a pattern evidently occurred in the jam formation
in the single-branched channel of the Suhon R. at t 'he Opock D.
/14/.
/69/
113.
Chapter 4
Equilibrium Jam State
4 .1. Regularities of Ice Distribution and the Equilibrium Stat,e
of the Jam Body
The static equilibrimn of non-filtering jam accretions of
consta.nt thickness of crushed ice wa& examined by a number of
authors /5, 15, 43, SO, 51, 60,73/. Applying the concepts of
the theory of a free-flowing .medium for this case, they estab-
lish dependences which associat.e the hydraulic characteristics of the
current with the thickness of the ac·cretion, the level head,
etc. Essentially it. is not. jama that have been considered in
thes.e investigations, but the equilibrium state of masses of
constant thickness of crushed ice, packed by a current. The
finite dimensions, the special features of the structure and the
form of the jams and regular natural formations taking into con-
sideration the individual pro.pert.ies of the blocks that compose
the jam are not considered nor investigated. From this point of
view, an analysis' is given below which allows establishing
gene.ral conditions of equilibrium and certain special features
of various types of jams.
The developed fi 1 ter ing jam is tak.en as the general case
with consideration of the fact that other types of jams (section
3.2) may be derived from its particular cases.• The body of t .he
developed jam is characterized by an increase in section L0 of
the accretion thickness h 1 of the ice from h in its tail section to
*Such classification of jams into general and special cases is
taken only for the convenience of mathema~ical analysis and
should not be confused with the concept of relative fre-
quency of jams of dif'f.erent types observed in nature.
/10/
114.
hm in ita head section accord.i ·ng to a nonlinear law.
To establish the type of dependence h 1 • f(x) on the
jam aection L 0 , a crossviae strip of length dx is tak.en and
an equilibriwa equat.ion (1.6) ia established for it under the
following assumptions and conditions:
(1) For a wide rectangular channel with a constant bot-
t.om slope. and with straight current e ·ections L greater than pr
the jam head L > L 0 + L2 , the jam ice accretion is found in pr -
a bounded streseed state. In agree~~ent with form IV of the
rheological model (see Fig. 2), it corresponds to so111e critical
value of stress under the action of which a separate. compartment is
found. Under these conditions, the external force should be
equal to the maximum internal react.ance of the accretion: t .he
passive resistance i n the longitudinal di .rection and the active resistance
in the crosswise direction, i.e., on the shore;
(2) A gradual increase in the i ce accretion thickness along
the length of section L0 is approximated by a flat curve, wherein
the. acc.retion thickness in.creaees to a value dhl along the length
of the elementary segment dx;
(3) In the general case, the accretion consists of blocks
with h < 1, inclined towards the direction of current at an
angle equal to the angle of' internal friction · (rupture) of the
accretion under the water (see Fig. 6b)7
(4) Under equilibrium conditions of the jam body, the
current velocity unde,r the ice accret.ion is equal to a c .ritical
value uk, at wh i ch there is still g'l.aranteed the i.-.obility (ata-
--· ........----·--~---.---.-----------
115.
bility) of the blocks in the jam body. In section L 0 , this
velocity~ • u is determined by Bqs. (2.17) and (2.18), -K 8 1 V
and at the upper boundary of the section for L0 • 0, where the
multi-layer accretion is transformed into a single layer U 1 p
is calculated according to Bq. (2.14). The average depth H0 is
established by the latter condition on the upper boundary, as
it follows from Bq. (3.10)7
(5) Forces are considered:p1 , p 2 , p 3 , p 4 , p 5 , p 0 , which
act in the direction of the x axis, and the resultant of the
cross~ise forces p 2 y' p 3y' p 4y, which create an additional
resistance to friction against the ~. The force p 0 , directed
to the negative side of the x axis is taken according to
Bq. (1.22) without considering coupling, while forces p 1 and p 2
under the ice accretion are taken according to Eq. (2.15). In
addition to this, the relatively small value of the slopes per-
mits taking tan a =sin a •
Taking into consideration the above assumptions and condi-
tions, the equilibrium equation for the strip dx takes the form:
( 4 .1)
or
(4 .la)
where At-p,-p;B-2q;a~~"~-YaB (I-q~_.~t,); .t1 -Bp/Af; p.,.
Pa. 1 + P•:!: p~; -
a'kis the transverse force, determined by the active pressure of
the accretion on the bank1 ii • 4 • (1 -• 1 ) p 4 ; Ps is the filtra-
tion pressure approximately determined according to the depend-
1--.. --
.•
J.-...
/71/
~.., "
.. ·-··-.. _ ....... ---------· -----··-
116.
-ence p 5 • 2
£ ko uf; uf is the average f .iltration velocity
throuqh the jam bod:y.
By inteqrat ing Eq. (4.la) from x • 0 (for h 1 • h) to
x • L, we obtain;
( 4. 2)
Eq. (4. 2) also describe.& the reqularit:y of the growth of
thickness of the aul:lmerged section of the bloc·k accretion along
the length L 0 of the jam body .
By ana:.lyzing Eqe . (4.la) and (4.2) for M ~ 0 and A ~ h,
it ia possible to determine the following .
1. For M > 0 and A > 0, the derivative of (4.la) is
positive and consequently, the accretion thickness increases
with an increa.se in L from the smallest. value h 1 • h at the· upper
boundary of the section L 0 (i.e . at L0 = 0) to the maximum
h 1 • ~ at the lower boundary. With an increase in the Y&lue L,
the exponential function exp-(ML/akB l in Eq. (4.2) tends toward
0, wh i le hm ~ A. Such a limit is practically achieved in
a now section L0 r ·ela.t .ively limi.ted in length. With the condi-
tion that the expontn.eial function is equal to zero, the lengt.b
of this section is found by the expression :
(4. 3)
~ the coe.f.ficient m3 may be taken equal to 5-6.
Eq . (4.3) obviou3ly determines the limitinq value. of the.
length of the body of the developed jam with a maximum thick-
ness of the ice accretion:
(4.4)
. /72/
117.
Two cases result under the conditions that M > 0 and
A > h:
(1) hm > H1 and i 3 < il' where H1 and i 1 are the level
head and slope for the formed jam core, which are determined
according to Eqs. (3.8)and (3.9). In this case, the jam body
is formed with an increase in the thickness of the ice accre-
tion in the·core and a continuous decrease of slope therein,
which takes place due to shifts, ramming, and packing of the ice
mass in the body and head of the jam. In this case, the jum
section L2 is formed (see Figs. 6b, 7, Be);
(2) hua? ~ H1 . The formation of the jam takes place smooth-
ly without packing and shifting of the ice.
The first case, in agreement with the classification used
in this study corresponds to the formation of the developed jam,
while the second is of the transitional type.
Therefore section L2 is formed in the developed jam after
the thickness of the ice accretion in front of the core exceeds
its height and brings about a packing of the ice under the jam
body, a compressing of the ice mass, and shifting to the core
side. The respective increase in the level head at the core for
some value ~ in turn creates a supplemental volume of ice
accretion within the limits of the jam body: ~hB.
Since the thickness of the ice accretion h 1 decreases upstream,
only a portion of this volume goes into the formation of the
jam body, while the remaining portion in the volume w2 --
obtained on the basis of very simple geometrical considerations,
118.
goes into the. formation of the jam section L 2 • w2 • kw1 ,
where k < 1 may be written, and then:
L.-kL8 (1-Il111&r..) (4. 5)
According to the results of sun.e experimental investiga-
tions, the coefficient k is on the order of 0.5.
2. For M > 0 and A < h, the derivative dh1 /dx ~ 0,
so that the highest value h 1 = h occurs at L = 0. we conclude
from thi.s that a single-layer ice accretio·n is formed abov-e the
jam nucleus a .nd this has the smallest longitudinal compression
om < a k a .nd a . constant minimum thickness of the ice accre-
tion equal to· the block thickness h 1 = h. This case corresponds
to the formation of the undeveloped jam .
3. For M < 0, the derivative in (4.la) is positive, and
the value h, in agreement with Eq. (4. 2) monotonically increases
with an increase in L from h to infinity. A.t the same time,
the value L 0 is negative, as seen from Eq. (4.3). For such a
case, static equilibrium is impossible due to the continuous
in~rease in the ice. thickness at the jam .head. However, this
process is a .ccompanied by a decrease .in the slope at the. coxe,
which, under co:1ditions of an unlimited height of the jam
may be reduced to i 3 < p 0 /y B. In this case, the value M
becaaes positive and the further formation of the. jam will proceed
as a function of the ratio between A and h, according to the
first or second of the examined cases.
/13/
119.
4.2. Conditions of Static Equilibrium of Jams
The relationships A > h, i.e., B > p 0 - p 3B) h/p, and
A < h in Eq. (4.2) together with the inequalities obtained
from Eq. (2.19), correspond to the concepts of a wide and a
narrow river, respectively, introduced by Canadian researchers
/15/, but are more general, determining the possibility of an
increase ~long the length of the current of both stress and
ice thickness, and also the type of jam, if in parameter A the
value i is determined according to Eq. (3.9).
The equilibrium cunditiona of non-filtering ice accretions
of a constant thickness of the front edge and at a cistance
from it have already been subjected to analysis /5, 15, 49, SO/.
The equilibrium condition of jams as uniform formations of
specific, size, structure, and properties comprised of indi-
vidual units -blocks -is examined below. In this connection,
the equilibrium conditions of a free-flowing body (1.6) -(1.12)
are supplemented by regularities and special features of the
properties ot the investigated object, reflected in Eqs. (2.9) -
(2.11), (2.14), (2.19), (4.2), etc. Inthiscase, thecondi-.
tiona of static equilibrium of the jam will include the follow-
ing requirements:
1. In each cross section of the jam, in agreement with
Eq. (2.4), the condition M > 0 is fulfilled, where the pres-
sure on the shore is determined according to the dependences:
(a) a k 21
for undeveloped jams~ so that for iy • 0, we have the expressions:
_,
120.
8•, < I'Jl.: .
Bla < a 111 ,
a and 1 1 are dete~ined according to Eq. (2.20).
( 4. 6a)
(4.6b)
The average slope of the water surface in the developed
jam, obtained from the geometric structures, i_s equal to
'·-··-i;-(,~ .. -A..)+t.. (4. 7)
and for the undeveloped jam, it is de.termined according to
Eq. (3. 9) with the substitution there of the: average depth
value according to Eq. ( 3 .10).
2. On the lowe.r surface uf the jam body COftling into
contact with the curren·t below the ice, the current speed is
,, -~" :au -. .41 v ( 4. 8)
where u v is de.termined a.ccording to Eqs. (2.17), (2.18), for s,
tan e = ~ •
3. At the upper boundary of the jam, where a single-
layer block accretion is formed, the stability conditions are
described by the following equations:
(a) The static state:
"'· r:; h: .
n., <If,;
.k
(4.9a)
(4. 9b)
(b) The dynamic state (in the collision of blocks float-
ing up to the edge of the jam):
(4.10a)
(4.10b)
"':•
I .
J. __ .:..
/14/
-;;··-·· -----·--r----·-----------·· ------··.
'J
121.
where
a k is de.termined according to Eq. (2.20) and u
8
according to
Eq. (2.6).
We note that requiremen.t ( 4. 10) is already provided by
condition (4.9), since in the derivation of Eq. (4.12) on the
upper boundary of: th.e jam, the depth a 0 at which u = uk was
fixed. Eqs. (4.9) and (4.10) are examined in developed form
in Section 4 .4.
4. The jam s .tability (as a . single body) vis-a-vis shift-
in.g along the current bed is guaranteed:
(a) By con.dition (1.12) for jams at transverse obstacles;
(b) By the relationship ~l ~ ~, while for •1 < •
by equation:
(4.11)
for d.eveloped and freely filt.ering jams, where LP • o.sm1 a 2 L1 +
m2 L 2 ; m1 • 0. 4 -0. 7; m2 • 0. 7 -0. 9 are coef·ficients which con-
sider the action of the filtration cur.rent through the. jam;
a 2 • 0.9-1.2 is the coefficient of the form of the bottom
slope of the jam. L2 • 0 in Eq. ( 4 .11) for undeveloped j a ·ms.
It may happen that the equilibrium condition (4.10a, b)
ie not obs'lrved if the pressure of the a.ingle-layer block acc·re-
tion adjacent to the jam increases to am > a k (with a cur-
rent velocity of up to u > uk) due to the large slope in
this section o .r for other reasons. This causes a thickening
of the ice at the upper jaa section, as well as an additional
I
I
-··· .4
···--.. ... -......_. ... ....._ .. --· . ·-·-
122.
rise in the water level, d.eter111ined traa the equilibrium condi-
ti.on by the equation:
~-...!!!..-L ... ( 4 .12)
The process of .ice tbickn.esa increase by plungin.g, piling up,
and shifting o .f the· ice is gradually spread downstream -to the
jam head. A new equilibrium position is establisned for the
altered jam parameters. The nature and size of tb~se changes
depends on ~he jam fo.r. and the value Ah. In thi s case, the
undeveloped jam acquires a transitional chara.cte·r, while the
latter becanes deve l oped. The pa.cking o .f ice in the head section
of a developed jam first leads to upstream shifting of the
section L0 and a respective lengthening section of L2 by the
value.:
(4.12a)
For ll h > i 0 L0 , the thickness of the. ice accre.tion in-
creases at the jam head, causing an addi t~onal level rise there:
(4 .12b)
At the same. time, the slope of the water surface decreases
on section L0 which ia found in a new position -shifted upstream.
The value of thi8 slope is determined according to Eq. (4. 7) for
h' • h + Ah and R'o for m m m ak and ~, calculated accordinq
to the equations examined in Chapter 2.
When th.e slope is reduced to a critical value:
(4.13)
the process o f ice thick.nesa increase in section L 2 terminate•,
and only an increase in the length of this secti.on can take place
/75/
123.
af.ter this as well as. a corresponding removal upstream of the
section L0 • The noted phenomena of conversion of one type of
jam to another, the growth of section L2 , and of the ice thick-
ness a.t the head of the jam were recorded in model investiga-
tions.
4. 3. Level Head and Othe.r Characteristics of Jams Under Sta-
tionary Conditions
It is not hard to see from the examined regularities of
jam formation that the maximum level head Hm occurs in the head
section of jams, and over a period of time coincides with the
termination of their formation, i.e., with the occurrence of the
equil~brium state. · Special features in the form, structure,
.filtration properties, and equilibrium conditions of various
types of jams evidently predetermine the necessity of applying
a nonuniform procedure for evaluating the level head. For this
is taken the data. Q, i 0 , p, 4>• 41 1 , the dimensions and form of
the channel, the dependence B = f (H), and also the calculated
block characteristics: 1 , h, h . and a . . • m1n 1,s
In undeveloped jams, where the L0 and L 2 sections are
absent, the equilibrium conditions relate directly to the formed
nucleus and the maximum level head will be determined by the
water depth at the core. This depth is expressed by Eq. (3.10)
0 for 0 = 0 , and the level head:
1/., ... Hn ···ll ·.·~+l''l ·.L 11. b ,.... " .., ,..,,.-.... (4.14)
where Hb is the actual depth up to the jcl.~l fo·.rmation.
124.
The slope of the water surface at the jam and the
longitudinal dimensions of the core may be determined from
the dependences given in 2.4 and the equilibrium conditions
(4.11).
Freely filtering, developed jams under equilibrium condi-
tions comply with Eqs. (4.6) -(4.10). The maximum value of
the level head here will correspond to the maximum thickness h m
of the submerged section of the ice accretion in the jam, which is
situated at the boundary of sections L0 and L2 • For determining
this value, we have the basic equation (4.2) in agreement with
which:
(4.15)
and also Eq. (4.7), and with a breakdown in the equilibrium condi-
tions (4.10) -the relationship (4.13). For purposes of ob-
taining a closed system of equations for given hydraulic and
ice-movement characteri~tics, it is necessary to have an equa-
tion for determining the filtration pressure. However, due to
the insufficient study of this phenomenon, its. value can be
established only empirically or semi-empirically. The results
of our tests show that in a first approximation, the filtration
pressure can be evaluated according to the equation p 5 = k'ouf 2 '
where it is ass~ued that the resistance coefficient k'o = k 0 is
determined according to Eq. (1.18), and the average filtration
Jl
/76/
125.
The dimensions of the jam sections L0 and L2 are deter-
mined according to Eqs. (4.3), (4.5), and (4.7), established
for the case of prismatic channels with constant bottom slopes
and roughness. In natural channels, changes in the morphological
and hydraulic characteristics along the length of the river c~used
corresponding deviations of the dependence h 1 = f(L) from an
exponential type curve described by Eq. (4.2), creating local
increases and decreases of the accretion. These latter may be
called secondary cores. Such deviations were clearly traced in
the investigation of jam formation in channel models with a
variable depth, slope, and roughness.
In jam sections L1 and L2 , the hydraulic slope 1 increases,
forming a depression curve. The value of this slope in freely
filtering jams should assure surmounting hydraulic resistance
to the passage of the filtration discharge Qf = Q. With an in-
crease in the length of the L2 section, the filtration path
increases and evidently there exists a limiting length of this
section which corresponds to the given values hm and L1 , at
which a sufficient filtration through the jam is still assured.
The precise determination of this value is difficult for a
number of reasons, even for jams with constant hydraulic and
filtration characteristics. The approximate limiting value of
section L2 can be shown by the equation:
L~ .• =~-11~-l •. itJ T/., •It ( 4. 16)
where I 0 is the averaged hydraulic slope.
126.
A further increase in the L2 ]._ength ca.n lead to a de-
crease in the porosity coefficient, and consequently the
filtration as a consequ,ence of the packing of the ice accre-
tion .which arises under the condition T >
'l'he determination of the porosity coefficient and conse-
quently kf under field conditions is one of the basic diffi-
culties that arise in the jam calculation, since this value
depends on a number of ambiguously and nonunifCi>rmly acting
factors. Thus, unde.r water and ice pressure in a jam, as a
consequence of ice settling and fluidity, the coefficient £
decreases, while under the thermal and mechanical action of
the filtering water, it increases, but decreases in the freezing
of the blocks under conditions of negative air temperature, re-
gelation, etc. For the short-term equilibrium existence o.f the
jam wi·th tno -freezing of the blocke, we may take in a first
approximation E = 0.20 -0.25 = canst, as a consequence of
--------
some interbalancing of the given phenomena. However, this ques-
tion requires further investigation. A decrease in the filtra-
tion coefficient may lead to a conversion of the freely filter-
ing jam to a slightly filtering one with the formation of sub-ice
flows under the ice accretion (or within it) (see Section 3.5).
With slightly filtering jams, i.e., under the conditions
Q > Qf > 0, the ·value of the level head should be greater than
the thickness of the block accretion jam determined according to
1111 Eg. (4.15). For determining the level head in this case, as an
initial position, which has been confirmed by experiments, it is
127.
aaswmed that in the head section of the jam, the blocks are
distributed at a density that quauantees the filtration
of the entire water discharge for a given accretion thickness
h'm and section length L1 + L2 • Then the value of the level
head is determined from the equation:
(4.17)
The value h' is based on the following arguments. The ID
head section of the jam L1 + L2 may be examined from the view-
point of hydraulic engineering as a filtering dam with a vertical
upper slope and an acute-.angled bottom slope -with angle a of
slope to the bed. Filtration through the jam ~has a turbulent
nature and Chezy's formula may be taken in the following generalized
form for determining uf with some degree of approximation, as is
found in the caiculation of filtration through a stone talus:
"f .... c~ y. 1,;,,1 , (4 .17a)
where C0 b is the generalized Chezy coefficient which is a function
of the dimensions of the block, the form, structure and density
of their packing; I is the hydraulic gradient.
By comparing Eq. (4.17a) with the linear filtration formula
uf • kfi' by introducing the conditional concept of a "nonlinear
filtration coefficient"
(4.17b)
it is possible to formally arrive at the Darcy law in the form
uf • k'fi for a quasi-constant value kf. By substituting
Eq. (4.17b) for kf in the formula for determing qf through an
earth dam with a vertical front slope /79/, we obtain the
/78/
128.
dependence for estimating h'm in the form:
~ . ., z. (.l.a + Lt-6,.ff/'At ]',_...,..-r.t "-.~--~-~~ .. . . ~~ . ~ . (4.18)
where ha is the depth in the tapering of the depression curve on
the bottom slope, which is determined according to known
hydraulic methods /79/.
In a first approximation it is possible to take ha ~
' (1.2 -1.5) H'b' where H b is the actual depth below the jam.
The coefficient c0 b in the assumed structure k'f and hm > 0.2 m
depends primarily on the block formsand their packing method
/28, 79/.
We obtained a value of C
0
b from 1.5 for free filtration
to 0.4-0.5 for slight filtrationwith paraffin plates. Such
values should be somewhat higher for the field tests.
In developed non-filtering jams, the sub-ice flows (or
flow) axe retained, as has been mentioned, sufficient for passage of
the entire water discharge at the current velocity under the
ice accre~ion uk' which assures the condition of its strength
(4.8) ~·which assures the condition of its strength (4.8).
With a wide rectangular ~. the sub-ice flow in the jam
section L0 is characterized hydraulically as pressure head movement.
The depth of the current downstream is reduced as a consequence
of the increase in tht• ice accretion thickness in section L0 ,
while the velocity increases (see Fig. 6c). The possibility
of an increase in the velocity uk • u while observing Eq. (4.8) s,v
is assured in agreement with Eqs. (2.17a) and (2.18a) by an in-
crease in the slope angle e of the blocks and the density of
129.
their packing.
To sum up, on the upper boundary of a non-filtering jam,
there is established a maximum sub-ice average flow depth H0 ,
determined according to Eq. (2.10) for e ~ 0°, as in the case
of filtering jams. The minimum value of this depth H'o is
established in the line with h 1 = hm' i.e., on the lower
boundary of section L0 • It is possible to distinguish two
cases here:
(1) In transitional-type jams, formed without shifts
and packing down, i.e., for hm < H1 , this depth is deter-
mined according to Eq. (3.8) at a critical current velocity,
calculated for the block slope angle equal to the angle of
internal friction (rupture) of the ice accretion under water,
i.e. for tan e -~
(2) In the formation of developed jams as a consequence
of the shifting and packing of the ice mass, the blocks in the
head section of the jam can assume almost a vertical position,
densely pushing against each other. The strength of the accre-
tion considerably increases and may exceed the internal resist-
ance determined according to the law of free-flowing bodies
(1.26), and the stability of the lower surface of the accretion
increases and may oppose current velocities considerably
larger than the critical one for tan e = ~ . Corresponding-
ly, the depth of the sub-ice flow decreases. According to
theoretical investigations /43, 50, 73/, the maxim'lm degree of
filling with ice of the useful cross s•~ction is diverse: from
• .JI • ,I • ·., .-·.· •• ~-\, .~ .~;.: ·~
.... ~..:,;. ~ ..... _ ....... __ .,,.l....c; __ .,._...,_,.,..:..,.~ ..... ------............ = ....... =--~----·-·
/19/
130.
o.~· to-·0.8.· However, fi-ltntion properties of the. iame were not
considered or wex<e eva-luated only. very approximatelY...
The fltablishJI\ent of a r -eliable method for deterai.ni.ng
the-mlnimum H' 0 in the exallliried case requires. a SReaiaiJ., expe~i
menta-1-theoret.ical· stiudy. T-he maximum crt:tical ve~ocity ~
ahd the minilllum depth H '0 at the jam . head may be estimated
approximate1.y from the equality of the initial. and final
specific ene·rgies O·f see-tion L 0 taking into consideration
Eq. (1.16) acco.rding to the fol.lowing equations:
. 11;, .... -!l..r • .·u.
(4.19a)
(4 . 19b)
whe.re uk is the critical vel ocity at th'e upper jam boundary at
the deptti a0 •
It follows from Eq . (4.19) that in developed n~n-filtering
jams, there b a co·nnect.ion between the values hm' L 0 , and AH =
Ho -H'o· Elementary geometric structures per.it it to be
expressed in the form :
( 4. 20)
A combined sol ution of Eqs. (4.2), (4.7), and (4.20) per-
mit obtaining the value h and the head level: m.
1/~.,.. h,.+ 11~-1/t:i (4. 2 1)
It i .s possible to note that in the case H • 0 < Hk (where
Hk is the critical depth), the sub-ice flow h converted to a
, turbulent state. In this case, d i rectly below the jam or in
section L 1 there arises a h ydraulic jump, which joins t he sub-ice
flow with the wa t er level in the jam under water.
'·
131.
4.4. Jama with a Limited Formation and Jams under Nonstationary
ConcUtions
The jam characteristics examined above refer to jams with
a ca.plete (unlimited) formation (see Section 3.2) under sta-
tionary conditions of the water/ice-movement reqime. Under
natural co.1ditions, the jam formation may terminate earlier
than would follow fro~ the examined regularities, for a number
of reasons; these include:
1. The termination of ice movement or in the general
case the reduction of its intensity to sp < s 0 , as a conse-
quence of which, the jam development may terminate at any inter-
mediate stage. For the given function Spa f(t), the value hm
may be determined according to the formulas given above, from
the ice volume going into the jam format.i.on.
2. The insufficient length L'1 of the jam section, when
L'1 < L1 , where L1 corresponds to conditions of a jam unlimited
in length. Such jams can arise in ahallow sandbanks, in ice fields,
etc. The value hm for jams with a bottom slope in this case is
determined by the equation:
(4.22)
where a 2 is the same as in Eq. (4.11).
3. The insufficient length of the flow section for posi-
tioning the developed jam body. For example, this length may be
found in the presence of large channel turns directly above the
jam section with sp > s 0 • In such a case, the maximum possible
jam body length (L 0 + L2 ) will be equal to the length of the rec-
.... ]. -r ·-r: ... -..
. . ._ ·-···ol'· ' ... -.......... , .. ., .,........ ........... --.~.,..... ..
. i . I.. __ . -··-------· ~ .... --~.:-... _ ..... ---~-----· .. _ ___ _._ ____ -·-·
1
' ~ .
/80/
132.
tanqular section of the river above the jam core.
4. The insufficient heiqht. of th.e obetac:le (bank•) in
th.e jam sect ion when an outlet path for water and ice. ia formed
at. some head level H'm· Such a case was recorded, for example,
in A j aat O"'l the Ni.eman R. /53/.
5. The inaufficient st.renqth of th.e obstacle which. crowds
the channel when Eq. (1.12) breaks down at some head level
8" m < hm (jam in st.ru::tures of reduced outlay, in A section with
_,/
the i ce cover, etc.). The maximum value of the bead in this
case will correspond to the moment of the appearance of the dis-
ruptive preaaure on the obstacle a'p• caused by 'the weiqht of
the ice jam cr the water and ice pressure. Por f .reely filter-
ing jams unlimited in lenqth, the maximum thickness h18 is
approxim&tely de.termined according to the value of the maximum
active pressure:
(4.23)
Special formulas that determine the value. of ice pressure
on a stru.cture provide a more exact solution /6 1 25, 36 1 39, 51/.
For jams which form on the ice cover, not restinq against the.
bottom, the dimensions of section L1 and L2 (and also the value
H18 ) should correspond to the condition
o.~•: 0,5L, +La----.;-• (4.24)
where h 0 is the thickness of the ice. cover.
Eq. (4.24) is correc.:.ed after establishing the physical
constants in the. mk (force.) s system. It is investigated from
conditions of stability of a loaded plate and holds true for
\
'·
•.
. -~-~ . J
i
j_)_ -
/81/
133.
au.ch a speed of load bui.ldup on the. ice cover Dh (see Eqs • 1. 1 ,
l. 5), whe.n au ia not a function of this factor -accordi.ng
to Dzhellinak • • da.ta /59 I -for Dh > 0. 5 kg/ (cm2 ·a) •
6. T.he breakdown of the condition of stability (4.11)
of the jam head upon dislocation along the flow bed as a conse-
quence. o ·f: (a) An increase in t wi. th an increase in hm (jams
in bends, in channel constrictions, on ice layers., etc. ) ;
(b) A decrease in •a as a consequence o.f the rise of the jam
from low-lying dams and other j .ams, or due to a decrease in
filtration through the jams because of freezing and deforma-
tion of the blocks. P'or a preliminary evaluation of hm in a
channel constriction of length 1s with a sho.re convergence angle
as' the approximat.e equation, which we suggest, may be used in
the mk (force) s system:
(4.25)
and for other types of jams in the case of .their l .imited equi-
librium-Eq. 4.11.
7. A change in the throughput of the jam section as a
consequence of: (a) A change in the water discharge of the jam
with a rise and fall of the high water; (b ) The emergence or
a change in the force of wind pressure on the blocks (piling up
and f .loating. of ice, including wind surges of the jam with re-
verse ice transit, observed, f;or example, in the Ob R. lower
section).
The force of wind pressure may be tak.en into considera-
tion by including it in the com.position of active forces P in
(
0. : i J.. ....
i -~ -;
'· -~------------,. __ .....,. __ ~. -,--r ·· ___ ..... ~ __ _....._ __ .....__ .. --·--
134.
the equilibriua equation (4 .1). The formation o ·f this jam with
a variable ice diecharqe te related to the case of a non-sta-
tio.nary re9i ... exaaine<S below.. It should be noted that the
qualitativ·e aepect of the influence of' moat ot the causes examined
above waa noted earlier by F. I. Bydin /12/.
The apecial feature• of jam formation under the conditions
cSq/dt ~ 0 are clari!ied by an analysis of Eq. (3. 4), which will
take the following fora with constant valuea for sp, n, '\,, kf ,
and "n?• 0:
;._<; ·~··-liS.,' •--'!L ell/ ·r ~/· "'"' ~ ....
(4.26)
As shown in Chapter 3, with an increase in '\>~ the quan-
tity of ice· entering into the jam s 0 n decreases, reaching zero when
its formation is terminated, due to a reduction in the energy of
preesure T and an increa!"e in the work of reaistant forces A2 3 4 ,
f I
i.e., in lq. (4. 26), the deriva.tive 3, s 0 _/(J H < O, while the n p
derivativet 3Son/ aq > 0, since with an increase in discharge,
the energy of pres.sure T increases and work A3 , 4
Slmning up, it can be said that:
l. At dq/dt > 0, three cases can occur:
dec.reases .
(1) as0 n/ (JHP > SOn/a q -the sloW' formation of the
jam at higher levels Hp and a larger volu.11e of ice in it, w, than
in the case q • c~nst.;
(2) ason/a q -unlimited (continuous)
development of the iam at constant intensity;
(3) as0 n/a HP < as0 n/ aq -the breakdown of an earlier
formed jam as a consequence of the increase in throughput of
/82/
135.
the jam section according to the condition SP < s 0 •
2. For dq/dt < 0 an acceleration in the jam development
will occur with smaller level rise and ice volume in the jam.
The effect of the following conditions:
tlu.jtlt V: 0~ dktdt ~ 0:. ·dS rJdt .P 0.
on the jam formation can be examined in a similar way.
With a wind surge (or intensifying) downstream, the
energy of block pressure T increases and the jam is formed for
large values of level rise and ice volume, while with a reversal
of ita direction -slowly with smaller values of HP and W, and
with a large uv value, the sign of T can change to the negative,
which causes the breakdown of the jam with the emergence of a
reverse ice transit -upstream.
With a decrease in the filtration coefficient as a conse-
quence of freezing and fluidity of the ice, the level rise in-
creases at the jam, the work A2 ,3 ,4 is reduced, and the through-
put of the jam increases or its rise increases. For s 0 > sp
or T > A2 ,3 ,4 , the jam breaks down. An increase in the filtra-
tion coefficient due to ice thawing and other phenomena causes a
reduction in Hp' which increases A2 ,3 ,4 a1.d leads to an accelerated
formation of the jam with small level rise and ice volume (see also
section 4.3).
--------·--
136.
Chapter 5
Practical Assumptions for Calculation and Regulation of Jam
Formation
5.1. Hydromechanical Principles of Couneeracting Jams and
Methods for Jam Control
The question to be examined is a part of the total problem
of controlling the ice movement for wat.er management purposes.
The struggle against jams to prevent the harmful effects that
accompany them. is of basic importance·; such phenomena are:
(1) The rise in the· wat.er level with the flooding of
the territory around the. bank;
(2) The static and dynamic ice pressure against the shore
and hydraulic structures:
(3) The formation of a ~idal wave" saturated with the
ice mass to the extent of a mud-laden torrent during a rapid
breakdown of the jam;
(4) A reduction in the wa.ter discharge and level oelow
the place of jam formation, which may cause interruptions in the
work of water management equipment in certain cases.
At the same time, in specific circumstances it is expedient
to form artificial jams in certain sect i ons of the flow or to
limit the jams by certain dimensions for the purpose of inundating
flood plains, maintaining or reducing the intensity of ice move-
ment, etc . Such measures belong to the field of jam regulation
and encounter additional difficulties., since their theory arii
p17actice necessitates considerable development.
/84/
137.
The measures used for controlling jam formation and for
counteracting the harmful consequences of various types of jams
under diverse conditions cannot be identical. The drawback to
this position is that unsatisfactory and even contradictory re-
sults tend to appear, such as has been shown, for example, by
F. I. Bydin /12/ applicable to the case of poor htmm intervention of
the ice cover at bridges without taking into consideration the
action on the dynamics of ice movement under the actual condi-
tiona of the given object.
A haphazard, not reasonably thought out action directly
on the jam using explosives and bombing may not cause a break-
down,but rather a strengthening of the jam due to packing and
subsequent ice filling, an increase in the total area of crushed
ice coupling with the bed, and a decrease in filtration through
the jam. This in turn may cause an additional level rise, which
decreases work A 2 3 4 and facillitates the piling up of new blocks. , ,
To sum up, the strength and dimensions of the jam increase accord-
ing to Eq. (1. 9) •
Correctly placed explosives, primarily at the jam head
may be very effective under certain conditions /3, 12, 68/. For
example, if there is a sharp decrease in filtration through the
jam with the use of explosives, then it can break down as a conse-
quence of the rise and loss of stability of the bottom slope f.or
tan a > 41 or of its entire head section in the breakdown of
Eq. (4 .11) .
--------------~
138.
The general principle for combatting jam• and controlling
them, as can be co·ncluded from the regulari tie• examined above
and the mechanism of jam formation, consists o·f a . fully controlled
ch~nge in ·.the ratio between the factors ( f.orces and parameters) ,
which cause the emergence and development of jams on the one hand,
and the jamle•s ice movement regime on the other. The question
of which parameters, where and when , to what/degree and what form
of action to take, should be solved independently in each case.
This principle O·f the struggle against jams in essence was
exp.reued by Bydin in 1961 /1 2/.
For general purposes and for overall control, the measures
for combatting jams may be classified into three groups:
(1) Preventive -banning jam formati.on;
(2) Liquidating measures -breakdown of formed (or form-
ing) jams:
(3) Control measures -The limited development of a jam
by allowing a certain level rise or ice pressure to exist,
changing the rate of formation {or breakdown) and the place of
jam formation, as well as an artificial jam formation at a
specified place.
The determination of a more effective method of applying
these measure11 should be based o·n an analysis of the ca.uses
and mechanism of formation of the examined jam, the flow
hydraulics, the chann.el structure, the ice movement conditions,
etc. , and. a .lso the expected damage from the jam or its bene-
ficial effect taking into consideration technical-economic pos-
/85/
139.
sibilities.
There is now available a relatively large number of pub-
lications concerning questions with regard to testing, effective-
ness, and principles of various engineering measures for counter-
acting jams.
The established quantitative regularities of jam formation
allow some generalizations to be made and hydromechanical
principles for the examined measures to be given as a part of
the theory for managing ice jams. The problem is solved by work-
ing out a system of hydromechanical criteria that determine the
conditions of the emergence, development, and termination of
jam formation properties. The fully controlled change in the
ratio between values entering into the criteria may permit
controlling the process in the desired direction.
Three groups of criteria have been developed, which deter-
mine the conditions of emergence (I), the degree of development
(II) and the jam size (III), as well as the conditions of their
!imitation and breakdown.
I. The emergence (prevention) of jams in accord with the
rheological equations (1.1) -(1.2) and ·equilibrium conditions
(1.2) -(1.12), is determined by the ratios between the values
entering into Eqs. (3.2a, b, c). They are expressed in developed
form in the following three criteria subgroups.
(1) Criteria of flow section throughput with respect to
Eqs. ( 1.12), (2. 12), ( 3. 2) and the requirEments given in
Chapters 2 and 3:
:I. ,.,
' I .
·i:J
.......... ~-·-. ..,-. ........... .6--4 ..... :. ...... '"'"' ......... ·~------~::.&~ .......... . ... ,_-.,. __ .. . -----. --. ..... .. . -...
/86/
140.
,'t ••.•• fl ..
d,. -l'fii&'J, ~~!;
" 7i; ~..;I;
(5.1)
(S.la)
(S.lb)
(S.lc)
vhere a • is the maximwj longitudinal stress (pressure) of the
blocks; o' is the strength limit of the jam obstacle. p
The value s 0 can be determined by the theoretical
formula {2.3); in a number of cases other methods are also suit-
able. Thus in the passage of ice through hydraulic equipment
including breakdown of blocks, the. throughput may be determined
by the methods examined in special investigations /16, 3?, 37, 47,
63/. For ice floating according to the first or second regime of
ice movement (see Sect ion 2. 1) , i.e. for a ..,/ a 8 < 1, in an open
channel, Eq. (2.2) may be taken with the determination of v
according to Eq. (2.5). In a closed channel (movement of blocks
under the ice cover, through ice bridges, the blocked up ice field,
etc.), the value s 0 may be approxima.tely determined according
to Eq. ( 3 • 6) •
The different combinations of values of the different
criteria correspond in a given section to free (floating) ice,
t~ partial and complete retardation (stopping) of the blocks.
(2) Criteria of emergence (prevention of formation) of a jam aOOJrd-
ing to the static diagram (due to the pressure of block accre-
tion) :
(a) In the presence of a connection between the blocks
and the shore according to Eqa. (2.6) an~ (2.19) or (4.9a)
. l
f I . .. .. ~ .....
Ll-
Ul •
(5.2a)
(b) In the absence of a connection with the shore,
with the investigation of the equilibrium conditions for
Po • 0 an4 p 4 • 0: t
(5.2b)
or in asimpler approximate form, the calculation according to
which for the Dni~R. gave a satisfactory correspondence
with field data: A+0,02l ill! j { + o·'·+ALp) '• • ( 5. 2c)
where Lpr is the length of the straight section of the river
above the place of jam forma~ion.
(3) The criteria of emergence (prevention) of a jam under
dynamic conditions (du& to losses in block stability in collision)
for the cases:
(a) Plunging and piling up of blocks -according to
Bqs. (2.9) and (2.10):
(5.3a)
(b) The tightening of the blocks-according to Eq. (2.14):
(5.3b)
Therefore, for the emergence of a jam with the arresting of
blocks that is expressed by certain ratios according to criterion
(5.1), it is necessary that one of the criteria (5.2) or (5.3)
be lese than unity. With the corresponding values of some of
these criteria, the jam is formed according to that diagram which
corresponds to the least value for the criterion.
----y
!
I
. ' .,__......,._ ~ ..... ..,:M~Wo~a ... ·-.. ... -............... _..__....,_..__._.._ .......... __ ....,... __ -··-.,.,_ -----1-........ .
142.
'Instead of slope, the rate of flow or the discbar·qe may be
introduced into Bqs. (5.2a, b, c) and (5.3a, b), uaing the
Chezy formula for condi.tiona of a useful cross secU.on.
~ly, these crit.eri a aaay be expressed by the Froude number
P'r, which in certain cases may prove expedient. Thus, criterion
(5. 2a) in such form will be:
II. The type of jam accordinq to the extent of its develop-
ment with l';'espect to the .rheoloqical equation (1. 3) and the equi-
/87/ librium condition (1.8) is established by specific ratios between
A and .h in Eq . (4.1a) or (4.9a) with a substitution there of the
slope value i according to Eq . (3.8) and the depth -acco·rding
to Eq. (3.10). 'l'h.e corresponding criterion, which may be ob-
·tained also directly from Eq. (2.6) takes t:he form:
(5.4)
When the criterion is less t'nan unity, the undeveloped
jam forms with the limiting value om < o k·
III. The jam dimensions, the limit to its developaent
according to the given value hm and the condition of ita break-
down, in agreement with the initial equations(l.3) -(1.12) are
established according to Eqs . (2.5), (2.19), (4.2), and certain
additional relationships examined in sectiona4.1 -4.4. They
permit obtaining obtaining a system of the following total and
partial criteria:
(1) The criterion of jam throughput with respect to
Eqs. (5.1) and (2. 5) with the substitution th.ere of variables
I
• 1r-~ . .
L . .L-
143.
.
correspondin~ to the jam regime under limitin~ or given (cal-
culated) dimensions for ita
4;-• ,:;~~ [ u-( ':--.t•' )"'1 $it, (5~·5)
where s 0 z is the jam· throughput (Soz > .. o-~·£or an incomplete
retardation of the blocks in the jam, while the index "z"
corresponds to the characteristics of the forming jam. For
SOz/Sp > 1, the jam dimensions decrease; SOz/Sp • 1 cor-
responds to the equilibrium state of the jam, while for s 0 z/Sp < 1,
the jam may increase, grow upst:ream, or remain unchan~ed. For
sp -0, the jam formation terminates.
(2) The criterion of hydromechanical stability (equi-
librium) of the upper boundaries -"the tail section" -of the
jam with respect to Eq. (4.9):
(5.6)
for a and 1 1 -see Eq. (2.20).
(3) The criterion of jam body stability in each cross-
section obtained from Eq. (4.6a):
I'··~ --1 ,:;-,, -:-;: . (5.7)
For values of Eqs. (5.la), (5.5), (5.7) of leas than
unity, the stability breaks dowr• and the jam force increases.
In this case, an undeveloped jam may acquire the character
of a transitional jam, and a transitional jam, that of a
developed one. If Eq. (5.5) is greater than unity, a pack-
ing of ice will occur (shift) with a decrease in the length of
the jam body, the piling up of ice in the head section and the
/88/ possible breakdown of the jam for the corresponding values of
I '
:.l +-1-
! 1
' ' ~ -i. -;
L.
,. .. · .. ~ '!" ."":-~---;-~ -~ -;~ -;~:--,...........-:----.--...-------... ~..;. _______ r.._..~.-----·-·-·---------·....,_ ...... __ _
144.
crl1utri• that deterain.e the stability and st.renqth of the jam
head (see below).
.. ., .....
Under the coruU tiona whe.n Bqs. ( S • la) and ( S • 7) are greater than 1.ni ty,
the j .. formation discontinues or there is a decrease ln the
head due to ioe settling. The case when Eq. •(5. 7) is equal
to unity corresponds to the equilibrium condition of the jam
body for any ice accretion t:hic!(ness (breakdown in continuity
Eq. (4.2)).
4. The criterion of hydroaechanical st&bi~ity of the jam
hea.d is established according to Eq. (4.2) in the form:
pwl:,--r.,,&... ~ 1·. (5.8)
Up
where I; -(I -.... t,),.
When this criterion i.s grea.ter than or equal to unity,
the ice accret-ion in the head section of the jam is statically
•tabl:.e, .while if it is less than 1, it is unstable. In this
aaae, the jam power may increase o .r it breaks down, depending
on the tyJ?e of jam and jam obstacle, conditions of ice mo~rement,
etc.
(5) The criterion of jam head strength with respect to
Eqs. (1. 5), (1.12), and (1. 25):
"• """ 1. ...... r;e ' ( 5 • 9 )
where o i8 the limit of block stre·ngth o r jam obstacle
3
strength. When this criterion is leas than one, the jam breaks
(6) The partial criteri.:m of jam stability with a rapid
increase in water diac::h.arge ·in the form of a straiqht packing
wave with. respect to Eq. ( 4. 6a)
,, .~
•,
. ! .
! I
L .J
L.l.
/89/
us.
-. . ~ ..
(5.10)
where di h the additional -slope of the water surface created
by the water discharge admission. If this criterion is less
than. unity, the equilibrium jam bod.y is broken down, which
may lead to an increase in the jam power or its breakdown, as
is noted in the examination of criterion (5.8).
(7) The partial criterion of stability against head
shif.ting ot a freely filtering jam. with a bottom slope, resting
on 'the flow bed conforming to Eq. (4.11)
Lr(aT,+2~et .. l ..c:: f ,
·, 1'1"-d ;:::o; •
(5.11)
Th.e stabili.ty of the jam against. shifting is a .ssu.red. if
this criterion is greater than or equ.al to unity, while if i t is
l ess than 1 , the jam breaks down.
(8) The partial criterion of limiting the possible jam
height with respect to Eqs. (4. 5), (4. 22) -(4. 25) due to the
ca.uses:
(a) Ins.ufficient height of the jam obsta.cle:
(b) Its insufficient length
~r.~··· .... tS (:
(5.12a)
(S.l2b)
(c) The insufficient streng·th of the base for jams at
the edge of th.e ice. cover
' ( "· )+ ~ ~ •B t n.~r., 1 ---c--;;or J ,s f :
II ·~ h' . i .
(5.12c)
(d) The insufficient s t rength of the base for jams
in a constriction .
1 . • ........... _.,,.. .. .~.... . ..,. ..... .... . -· t ; • . . . ,. . ·~;
.. ... · .. _ ~ ...... , ... ~ ........... _.._. ........... . ., ................ -... -.......
·. _!,·l--1-.. -... _ ····~-·· ---~-----··_.... __ .....,_....., ____ _
146. ···-. . ·-
(5.12dl
where 1\. ia determined as a function of the type of jam accord-
ing to the equations give.n in section 4. 3: H1 -according to
Bq. (3.8).
Criteria (5.12c, d) are qiven in the mk (f·orce.) s system.
The he.ight of the jam can increase only to the criteria values
equal to unity, and the jam breaks down with larger values for
crit.er.ia. (5.12c), an.d (5.12d).
In addi:tio·n tot:he partial criteria examined here, i f neces-
sary, addit.ional criteria, which limit the dimensions, strength ,
and stability of jams for various other reasons may also be estab-
l.iahed by a similar method (see section 4.4).
5. S. So&e Practical Considerations for the Calculation aBl
Control of Jams
A quant.itative analysis of jam formation in the general
case .may int;lude the determination of the place, time, emergence,
and dimensions of the jam . For given morphometric, hydrau l ic,
a n d ice movement cha·cacteristics, such a calculation with
the use of the quantitative dependences determined above will
include th~ fc:llowing sequence of operation :
{1) The determination of the location of the jam section
and the moment of emergence of the jam regime for a g i ven function
SP • f(t) according to the relat i onships given in section 2.1, or
a .ccording to values for criteria {5.1) -(S.lc), which correspond
to the cases of holding back or stopping the ice blocks1
. .
/90/
147.
(2) The determination of ·the possibility of jam emer-
gence in the jam sec·tion according to dynamic or static dia-
grams, according to the data in sections 4.3, 4.4, or accord-
ing to the. respe.ctive values o .f criteria ( 5. 2a, b, c) and ( 5. 3a, b):
(3) The determination of hydraulic characteristics in the
jam section for the period of jam formation including H • f (L, B):
(4) The establishment of a . jam type according to the
degree of its development. on the basis of data given in section 4.1
or according to the value of criterion (5.4);
(5) The determination of the maximum block accretion thick-
ness hm and the height of the level head Hm under conditions of
a.n unlimited jam deve·lopme,nt according to the equations given
in section 4.3 or taking int.o consideration criterion (5.8) with its
value equal to one:
(6) The establishment of the stability and st.rengt.h of
the jam according to the data of section 4.2 or criteria (5.6)-
(5.9):
(7) The determination of the possibility of lim1.ting the
jam formation to causes examined in section 4.4 by means of an
investigation of the values for criteria (5.10) -(5.12a, b, c, d):
( 8) The establ.ishment of limited jam dimensions correspond-
ing to values equal to unity, those of criteria (5.6) -(5.12),
according to which the instability of a jam is establi~hed or
its development is limited:
/91/
--------------------------~--------
148.
(9) The deter:111ination of the calculated value of the
head level an.d other characteristics (which are established
in the presence of some limi.tationsl , as a minimum value
of those obta.ined by calculation.
For a. nonstationary regime, the establishment of the level
head and other jam cha.racte.:-istics requires introducing and
analyzing the respective functions Q • t 1 (t), uk ~ t 2 (t) with the
detenninatioo of dependenc.e Hm • f ( Q, uk) , etc.
The regularities examined in. Chapters 3 and 4 also permit
giving a theoretica.l basis to the established field observations,
cases, causes, and "factors" o .f large jam formation /2, 11, 29, 44,
68/. In the general case, jams are larger, the greater the energy
and strength of the blocks in the approach to the section with an
insuf.ficient floating capacity (jam section) and the smaller this
capacity, the longer the jam section, and the higher and steeper
the. bank.
The regularities of jam formation obtained in the present study,
whicn characterizes them according to physical illdicators and
a developed system of hydromechanical criteria, as well as ave--
all experienced gained in ice technology permit making some
general 'zat.ions with regard to measures for counteracting jams.
The regularities examined in Chapter 2 and expressed in
the form of the system of criteria (5 .1) -(5.3b) can serve as
a theoretical basis for preventive measures. It follows from the
above that a jam does not arise in a given place with certain
combinations of values of the given criteria, which guarantee the
149.
free passage. of blocks or their holdup (stoppage) with the
formation of a single.-layer a .cc.retion.
A free jam-les.s ice passage, in a .greement with criterion
(5.1), requires the fulfill~ent of condition s 0 > SP which
is supplemented f ·or b/B0 > 1 by the requirement of sufficient
pressure eneryy at the block .. for the breakdown of the obstacle
or shearing of the blocks in constricted areas. The ice pa.ssage
in a closed channel (ice bridges, etc.) requiteS the fulf.illment
of the addi tiona! cor . .:ii t ions.: u .> u
8
, v (see section 2. 3) and
1 < H, which t ... mts the ._,edging of tightened bl.ocks (see
section 3.3). •ne holding back (stopping) of blocks in a given
section for s 0 < SP does not cause the formation of a jam for
values of criteria (5.2) and (5.3) not greater than one, and also
with the current velocity corresponding to Eqs. ( 2.10) and ( 2. 14) .
Therefore, the prevention of a jam (for Sp > 0) may be
achieved with a decrease in SP or an increase in the throu.ghput
of the section, s 0 = f (u, H, B, R, uv, h, n, •1 , n , S ,~
by me·ans of influencing the values on which it dependa, and also
by changing, if necessary, the respective criteria.
functions of the hydraulic characteristics of the current,
dimensions, and strength of the blocks, the obstacle, etc. (see
Eqs. ( 2. 6) , ( 2. 9) , 12.14) , ( 2. 20) , ( 5. 1) , ·etc.) •
Att.ention should be dra.wn to the fact that the same measures .
may under diverse conditions lead to opposite results. Thus,
an increase in the current velocity may assure t .he passage of ict!
/92/
150.
in constrictions due to an increase· in the pressure energy,
but wi.th sufficient strength of the obstacle, it may lead to
the formation of a jam due to a dec:rease in criteria (5.2a, b)
or ( 5. 3a) to vaLues leas than one. It follows frODL this tha.t
a careful analysis of the concrete conditions and the special
featurea of the jam formation in the given section should precede
the carrying o~t of preventive meas ure s.
Some of the means for preventing j•ama, derived from the
examined overall aspects are given below.
1. A decrease in the intensity of ice movement to SP < s 0
may be fulfilled by holding back the ice in traps, semi-dams,
e.tc., including artificial jams. created upstream in low-risk
places.
2. An increase in the throughput of the section to
s 0 ~ Sp by·: 1) Increases in the ice thrust front a 0 , the
current speed, the hydraulic drop at hydraulic stiuctures, by means
of corrective works, admissions f.rom above-situated reservoirs,
etc.; 2) Dec.reases in the resistance of the blocks ·to movement
by means of correcting the banks, and also decrea.ses in the angle
of their convergence and increasl!s in the radi.us of channel turns
in artificial currents, etc;.; 3) Decreases in block size -by
their direct mechanical breakdown, injury, or indirectly -by hold-
ing back the growth of block thickness in the winter, reducing
the strength of. the ice cover by using radiational heat, applying
chemicals, etc. /47h by the breakdown or reduction to a /a • > 1 m p-
of the strer..gth of the jam obstacle (for example, ice bridges, ice
' '
151.
layers, etc.).
3. A reduction in the bloc.k p.ressure energy
and an increase in the: value of critical pressure
and cr m
a k fo·r the
formation of a single-la.yer block accretion on the jam sect.ion,
including preventing the jam Jirectly a ·t the broken ice cover
by means of(l) Re.ductions in the current speed and inclination
at the approach to the jam section by means of correcting struc-
tures, reduction in the water discharge, formation of a level
t-.ead from lower-lying head equipment, the creating of artificial
obstacles, etc.; (2) Increases in the blocks' resistance to
movement due to an increase in the degree of non-prismatic
form and decrease in the radius of channel turn in artificial
channels, etc.; ( 3) Inct·eases in the longitudinal dimensions
and strength of the blocks by means of. a marunade premature
breaking through of the ice cover, etc.
The principles for measures with regard to breakdown of
the formed or still forming jams are established from the regu-
larities examined in sections 4.1-5.1 and criteria (5.5) -(5.12d).
It follows from an analysis of these regularities that jams
will break down in the case when the·re is:
( 1) An increase in the jam throughput in ag·reement. with
(5 .5) to s 0 z > SP, including its being due to a thawing of
the ice i~ the jam;
(2) A bre~~down in the stability of the jam head accord-
ing to Eq. (5.8) with the subsequent loss of ice strength in
the jam or jam obstacle, determined by criteria (5.9), (5 .12b),
/93/
152.
and (5.12d), or losses in stability to shifting according to
Eq. (5.11) -for jams with a bottom slope. The loss of s .ta-
bility of the jam head itself ma.y be caused by a loss in
stability of the jam body due to an increase in pressure
forces, as well as an inc.rease in water discharge according
to Eq. (5.10);
( ).) Losses in the strength of the jam head caused by
other factors: manmade breakdown of the jam obstacle, col-
lapse of ice in the bottom slope of the jam, etc.:
(4) A ri·se in the jam due to an additional influx of
water, producing a rise in the water level under the jam.
The.refore the jam can be bro~en down by action on it as
on a single solid body (stability against shifting, etc.) or
on a block accretion (free-flowing body). In the latter case,
the stability and strength of the jam, as derived from Eqs. (5.5)
and (5.12) depends analogously on the characteristics presented
in the examination of jam prevention.
A break.down of undeveloped jams and of the core of form-
ing developed jams requires a smaller energy consumption than
the breakdown of fully developed jams. However, in the dynamic
method of breakdown with the movement of ice along the jam,
for example, in the water passage, the developed jam breaks down,
as has be,~.n shown by models, with a lower energy consumption.
This is associated with the fact that the large ice mass in
. body of the developed jam, having o,btained some a .ccele.ration
and inertia, causes ·the jam breakdown due to an avalanche-like
•
153.
movement of the ice in the direction of its head.
Of the developed jams, those possessing the greatest
stability are those formed at H < h and dq/dt ~ 0 as a conse-
quence of the expanse of section L2 (see Fig. 6) over the chan-
nel clogged with ice and the stability of ~he bottom inclina-
tion, which has a slope of tan a < ~ • On the other hand,
jams that form at dq/dt > 0, ) and T > A have less stability due to a 2,3,4
shorter L2 section and the limiting value tan a ~ •
With a fall in the head level, caused by a decrease in
the discharge under water or an increase in the filtration
coefficient kf' jams, by settling at the bottom, possess a
high stability. Correspondingly, an increase in the water level
above the jam, due to a change in the water discharge and filtra-
tion decreases its stability, which at 350 /3 H n p <
leads to its breakdown. The same causes may provoke the breakdown
of jam stability due to ice floating or an increase in the jam
throughput.
The nonuniform distribution of the current speed and depth,
as well as the special features of the bed relief und other
causes create a nonuniform distribution of pressure and the de-
gree of stability, and, in a number of cases the cross section~l
jam strength. Therefore, in the head section of jams found
completely under conditions of stable equilibrium, there are
often local sections with a stressed 1:tate that is limiting or
close to it where, in agreement with Eqs. (1.8) -(1.12),
a = 2 Tk = k 0 u k or (J =
154.
Here, the applic.ation of a re.latively small force causes a
local breakdown in equilibrium, which may be converted to a
total breakdown for a jam in such a case. when it would develop
further due to a. natura.l pr essure increase t n rough the flow of
water and ic·e . Such key sections sometimes are called "jam
locks". The searching out of such a section and destroying it
correctly in order to evoke a "chain ·reaction" in the jam is
the most effe.ctive and economical means of jam breakdown, con-
firmed. by corresponding studies /48/.
Therefore, the jam may ~e broken down by destroying its
stability and strength on. the whole or on key sections by var-
ious methods derived from the analysis that ha.s been made. We
will now note some of these methods.
I. "Jam rinsing" by passa.ge of ~Water fr•om an upstream
reservoir as a function of the type and special features of the
jam may bring about its destruction due to: (1) a breakdown in
jam stability as a single body according to Eq. (5.10); (2) the
rising of and floating of all the jam ice vith the loss of sta-
/94/ bility against displacement according to Eq. (5.11); (3) a.
reduction in the internal resistance of the block accretion jam aoa)rding
to the condition Tk > ~a, due to block floating and a decrease
in the joining of the blocks, ~hich corresponds to the condition
dq/dt > 0 for <
The model investi9ati ons showed that this method is more
effective for breaking down developed jams formed in sections
with insufficient depth (shallow water, sandbanks). For jams at
..
. . .
, I
,i__j_ ..
155 •
t .he edge of the ice cover ~rinsing" is expedient o·nly after a
breakdown in the ice cover below the jam. There is
established an inversely. proportional and approximately linear ·relatialShip
between the water 4ischarge speed buildup rate and the maximum
discharge. value at which the jam breaks down.
II. A rise in the jam due to a rise in the water level
underwater AHb' wh i ch acts on jam stability in the same direc-
tion as the "rinsing", but creates a certain reduction in T.
The necessary va.lue for AHb for jams with a bottom incline
may be established from the conditions of breakdown of static
jam st.ability. In a simpler approximate form, the respective
equation is in the form:
Ml -~11 (~-t) b m t::o ' tan (5.13)
According to the data of experimental investigations, the
examined method it: more effective for jams of the type H < h
in sections with s .nall bottom slopes or with a small L2 section
le11 gth.
III. The maintenance (regulation) of ice movement above
the jam, which forms with a rise in high water for the purpose
of raising the jam and breaking down its stability at
The method .may be effective for jams which form
for H < h and at the edge of the ice co·ver with a sufficiently
intens~ increase in the water discharge.
IV. An increase in the hydraulic drop to guarantee the
condition T > AP for jams of the type b > 8 0 chie~ly at arti-
ficial constrictions in the channel by the respective technological
/95/
156.
measures.
V. Breakdown of· blocks in jams of the same type to
dimensions of b < B0 by mechanical, impact (explosives, bomb-
ing, etc.), acoustical, and hydraulic Dethods.
VI. Disruption of strength (breakdown) of the jam
obstacle according to Eq. (5. 9) by one of the methods named
above .for jams at the ice cover, in layers of ice, in channel
constrictions, canal.s, etc.
VII. A reduction in filtration through the jam for an
increase in H to .fulfill: the conditions examined in Paragraph I, p
in jams of the type H < h, in a closed chann.el, etc., that
also may be brought about by the methods given in Par. V.
VIII. A breakdown in the stability of the bottom slope
according to Eq. (5.11) by one of the methods examined. tn
Par. V. This method will obviously be more effect1.ve, the
closer tan a is to ~ and the shorter the L2 jam section is.
IX. Breakdown in jam stability according to the same
equation by a .ction on key sections at the ja_m head or the
forna:ation of longitudinal channels in the jaa, the direction
of. which correspond to the special features of the channel _,
topography and coincid-e with the direction of vector Tm. Such
a condition may be fulfilled by the method named above. For
example, a method of burning through channels by thermite loads
was suc.cessfully applied by Kh. Barnes /3/ on the St. Lawrence R.
The arrangement of the channels ·may be made a~iately under
157.
the conditions of separate stable jams of the type. H h.
The following are particularly important in controlling
the processes of jam formation: (1) Artificial formation of the
jam in a specif:ic flow section, and (2) A limi ti.ng of jam
development to specified dimensions. There may also be an
int.erest i .n controlling the time of the beginning of formation,
the cont.inuance o.f formation, and the existence of: the jam, etc.
The theoretica.l basis for measures of artificial jam
format ion are dependen.ces that de.termine the measures for
predicting jams, .but in an inverse numerical ratio of
criteria (5.1)-(5.3). Correspondingly, the me·thods of
action on the process of jam format.ion in the given case will
also be directly the opposite when compared with conditions
of jam prediction.
The basic measures for 1 imiting the development. of jams
will be dependences examined on the basis of methods of jam
breakdown; criteria (5.5) -(5.12),which determine possible
methods of both b .reakdown and limiting of jan development, also
apply here. The following ca.n be attributed tc these methods:
(1) A decrease in the intensity of ice movement to
SP < s 0 by holding t .he ice above the. jam il'l semi-jams, arti-
ficial jams, etc. This method is applied in principle to jams
of all types;
(2) A limitation of the length of the jam obstacle L1
for jams with a bottom slope by corrective, explosive, and other
measures to a magnitude determinable from the condition cor-
/96/
·--···---~----···---···-...... , --..
158.
responding to th.e form of the bottom section of the jam:
(5.14)
(3) The arrangement of an outlet path for water and
ice if the head reaches a value of Hm by clearing the banks
and flood plains, the making of an ou.tlet canal with
Scan ~ SP -s 0 • With an ap(.ropriate flood plain struc-
ture, this method may be sufficiently effective and relia.ble.
Thus, results of the investigation .of jam formation at the
mouth ·region o .f the Nieman R., published by I. Ya. Nechai /53/
showed tha.t rises in the water level caused by the formation
of ice jams on the basic bra.nch of the Nie.me.n are insignificant
due to the. disperal of water from above the jam along the. un-
dammed up riqht-eank flood plain;
( 4) A reduction in the water discharge by "t\olding the.
discharge to a valu.e(corresponding to Eq. (5.8) of less than unity,
for which an increase in the level head terminates. However,
in th.ie case., the jam may continue to grow in length -upetream.
This method is in principle applicable to dams of all types;
(S) The crushing of blocks that ernter into th-e jam to
a value of b < a0 and ) < A3 with the simultaneous increase
in q and a decrease in kf or without this for the purpose of
block pas.sage along the surfac.e of the ice accretion jam. Other
methoda examined above in the measures for jam breakdow·n are
also fully possible.
It is evident that for purposes of controlling, predict-
ing, and breaking down jams, a combination of the examined
159.
methods and measures may be applied.
5 .3. Longitudinal Dimensions and Block Form
Calculating the dimensions and form of the blocks is
necessary for· solving a nwnber of problems of jam formation.
For example, the block dimensions ent.er into· formulas of
block and jam stability loss (2.9) -(2.18), (2.20), and
(5.6), whi.le ·the block form is implicitly considered by the
probability coefficients which determine the block contact
conditions (the l coefficients, etc.).
At the same time t .here ace as yet no analytical methods
for calculating the dimension and form of the blocks: this
results from ·the insufficient study of the phenomenon. The
data f :rom field tests, generalized in the form of probable or
approximate analytical dependences should lie at the basis
of methods for evalua·ting the exar:ined values. The results
o .f some treatments along these lines are given below.
In examining spring ice blocks as a product of the break-
down of the ice cover (or ice fields), the .following should be
noted:
1. In the general case, the ice cover on :;:-ivers is a
complex,polycrystalline aggregate of stratified structure of
nonuniform texture and strength. Four different types of layers
are characteris.tic. The uppermost layer, the ice-snow layer, is
cha.racterized by the disorderly array of crystals and the low
resistance to opening /9/. Therefore, it is of little impo.rtance
/97/
160.
in the block mechanics, although its thickness may reach 30\
or more of the total ice thickness.
The second layer, the •primary congealed ice" (according
to P. A. Shumskii • s te.rminology /85/), also of an unorderly
st.ructure, is the primary autumn formation (slush, caked ice,
etc.) of small thickness, comprising 10-1!..• of the total. Be-
low this is the third layer of the secondary (orthotropic) ice
formed by the cryst.allization of water under the primary ice
in its transport through the river. The forced crystallization
and geomet.ric separating here lead to the formation of an
ordered columnar structure. However, in the pre.sence of slush
under the primary ice layer, this orderliness breaks down. The
thickness of the given layer can reach one-half or more of the
total.
The fourth and lower layer, formed by direct crystalliza-
tion of the river water, is a polycrystalline body of columnar
structure. The third and fourth layers have the g .reatest strength
that is almost the same in both cases /9, 19/. The ice cover in
various rivers and river sections and in different winters may
have fewer or more :ayers -alternate combinations of both
examined above (for example, in sections of autumn ice jams),
depending on hydromet.eorologic<tl and other conditions. A
high contamination of the liquid with various i~~~purities causes
many defects in river ice crystals, which facilitates the
development of dislocation movements.
..
... ;_ . J 161.
2. In the spring, before opening, the ice cover is
subjected to an internal and external thawing an.d to an in-
c.=eaaed water pressure, which is reflected on its texture a.nd
strength. At this time, lhe ice prepares for its new form of
existence as blocks. There tak.es place a partial breakdown of
the intercrystalline layers in the lower ice layer due to processes
of di ~•olution and absorption by solar radiation, which leads
to a decrease in th.e specifi.c surface energy. Here, heat energy
alone suffices .for the break.down of a · solid body along a de-
feet network. Therefore, the lower surface of the. ice cover
and the blocks as well as the lateral surface is covered with
developing c~vities.
3. The thickness ·of the ice cover i .n rive.rs regularly
decreases from the bank·s t .o the dyna:.mic current axis and from
the reaches to sandbanks. This is associated primarily with
the nonunifo·rm amount of heat energy, which evolves in the
process of hydraulic-resistance energy di·ssipation. In other
words, there is an inverse relationship between the thickness of
the ice and the current speed, as a measure of the energy of
hydraulic resistances. The ratio of the.ice cover thickness on
sections of the stream with different flow velocities ma.y be
expressed according to the follow~_ng equation /41/:
. ' s, +. -t·•·rt •:
:.: e ----''I
:J .• --·-" : r 4'••t·~ •: _, '!
(5.15)
where ) 1 and ~ 2 is the heat transfer of the bott.om; c1 and c 2
are the Che.zy coefficients. Eq. (5.15) may be used for evaluating
162.
possible fluctatio.ns in h.
/98/ 4. The appearanc:e of compressive, tensile, and bending
stresses in the ice cover is associated wit.h the ac·tion of
forces that are variable with respect to time and place.
These include:
(1) The attractive fo1rc.e (1.18), which creates a tensile
stress on the upper boundary of a certain ice cover section
(striving to tear the cover from the above-lying section), while
it creates a compressive stress at the lower boundary;
(2) The static ice pressure, which arises due to thermal
expansion -contractions with daily fluct•Jations in tempErature
and those which create a uniform compression -elongation in
the longitudinal and transverse directions, and the f.ormation
of thermal cracks in the limiting case ;
(3) The vertical water pressure force (from bottom to top)
on the ice cover, which arises with a raising up of the ice
(Archimedes' force) and with an increase in the hydrostatic head,
due to an increase in the water influx into the river and the
.formation of a temporary head. The latter leads to the appear-
ance of bending moments with their maximum value at the shore
in places where the ice covl!r is fastened to it;
(4) The wind pressure force, which creates tensile and
compressive stresses in the longitudinal and transverse direc-·
tion s;
(5) The force of dynamic and static ice (.block) pres-
sure on the uppe.r edge. of the ice ice cover in the investigated
•
163.
section.
Under the action of the given forces, the ice cover may be
broken up due to : (a) its rupture or compression in the longi-
tudinal and crosswise direction: (b) bending in the crosswise
direction, including 4~der the action of the gravity in the
•sag• of the ice cover; (c) losses of longitudinal stability.
The separate sections which form in the breakdown of the ice
cover (ice fields, primary blocks) acquire a known mobility,
which lea.ds to the appearance of two new forces: (1) Inertia,
and (2) Collision forces (dynamic pressure.) o ·f blocks during
their collision. The further formation and reformation o .f dimen-
sions and forms of the blocks, which continues up to t .he end of
ice movemen~ occur under the action of these forces.
The dimensions of the individual blocks at this time are
determined by the joint action of a large number of variable
factors and may be examined as random. The average . · irnensions
of the blocks under given hydrodynamic, conditions are a more
definite value·, yielding to a _known degree of quantitative
analysis. For sufficiently long periodBof time under conditions
of the developed dynamics of the process, i.e., when the given
forces play a leading role, it is not difficu.lt to establish
the existence of a close, almost functional, connection o .f the ... type 1 = f (v, h, C1 • ) • l.,s
The presence of such a relationship, confirmed by field
observations, permits expressing the horizontal dimensions of
blocks as a function of thickness and strengt·h, which are assumed
' .; __ ~
/99/
·----·-··-·-··.
164.
to be known. The principles of the theory of materials' resist-
ance as well as special investigations, on 'the basis of whic ...
is deterroin.ed the length of the blocks, precedi·ag from the
condition. of their not bre.akin.g down upon collision with each
other a·n1 with the banks, are used for this purpose . It is
assumed that part o ·f . the impact force F 1 is consumed in the par-
tial breakdown. of the block edges according to the diagram
examined in section 2.2. As a result of this, conditions of
the creeping (plun.ging) of one block on another are facilitated.
The bloc.ks experience both bending and compressive stress in
whic}'l there is consumed a seco!ld basic portion of impact strength
F • / F~ + F~, where F 2 is the bendi.ng force, and F 3 is the
simple compressive force. Other forces will be excluded .from
the examination, due to their known insignificance /61/, and
then we can write:
(5 .16)
where m is the block mass.
Using a method similar to that for deriving Eq. (2. 9),
it is established that for average forces, with a reliability
of · SOl, the impact force F1 = h 2 o 8 The value F 2 is found
from the relation known fran naterials' resistance:
F2 ... alUa:Jr,l.
•t:::::ording to investigations of block impact against an
inclined. surface, conducted by B. V. Zylev /27/, the results of
which nre applied to the examined case, the maximum. compressive
and bending forces may be associated with the relationship
...
-\.
•
165.
F 311 • 1.1F211tan 8. Substituting in Eq . (5.16) the values of
mass and forces F1 , F2 , F 3 with 8,. 45°, 1 • b, and solving
the equation with respect to block length, we obtain:
1-~ [h (" -f-0.2'.o ,r~-~ (Ita )'1.5,
AIY 8 i ...... i
(5.17)
where A1 -see Eq. ( 2. 9) •
A check of !:q. (5.17) showed that it gives the most satis-
factory results for current speeds of 0 .6-0 .8 to 1.3 -1.5 m/s
and for ice movement of average intensity. 'For A1 = 1 and
average values of h and a, Eq. (5.17) gives average block
dimensions. As field experimental investigations have shown
(Fig. 12), the quantity of such block.s within deviation limits
of + 15-20\ may comprise 30-60\ of the total quantity of blocks
in a given ice· movement. Substituting in Eq. (5.17) the extreme
values for A1 , h, an.d a, the limiting dimensions of blocks
may be obtained. How·ever, one must bear in mind that the
physically limiting block dimensions will obviously be: the
maximum 1 • &0 and the minimum 1 i = h. In addition, as max m n
shown by a statistical treatment of photographs of ice movement,
/100/ the distributio·n of block sizes is close to asymmetric curves
for distribu.tion of random values (with left asymmetry), while
the form of the curves, l i ke the block dimensions, changes with
the passage of time.
The fo·rm of t.he individual blocks and their size may be examined
aa random values. In averaging these values, a known regula.rity,
which has a t .heoretical basis, may also be examined. As. has.
already been mentioned, the thickness of t.he ice cover in rivers
166.
usually decreases from the banks to the middle of the stream
and from the reaches to the ~andbanks. By applyin9 in this
case the tenets of the theory of plate defo.rmation, it can be
shown that with pressure on the ice cover, ch:>ked ice fields
and, in general, formations of low mobility on the lower sur-
face and upstream of the edge have the most probable breakdown:
(a) parallel to the banks and (b) at an acute angle to it, along
the diagonal. It follows from this that the most probable form of
the blocks in a plane is close to an irregular tr~jectory with
more acute angles for large gradients of ice thickness varia-
tion along the width of the stream. The nonuniform ioe thick-
ness along the length of rivers and various local abnormalities
finally complicate the picture, but the general regular nature
is evidently maintained.
Along I-I
Along II-II
Fig. 12. Stylized dia9ram of the. formation of bloc.ks with a
rapid (dynamic) breakdown of the ioe cover due to
longitudinal compression and crosswise bending. ·
..
•
/101/
•
167.
Thus, the results of our investigations of more than
1500 average-size and large, blocks (according to photographs)
at the be.ginning of ice movement on rivers with flow velocities
of 0. 7 -0. 3 m/s sh.owed the following:
(1) 50-75\ of the blocks have a form in the plain that
is close to an irregular trajectory, 15-25' are triangular,
and 10-20' are rectangular, polyhedral, etc.;
(2) The acute angles of the blocks in 75-80\ of the
cases lie within limits of 40-60°; 10-15' have angles of 20-40°;
and 5-10\, greater than 60°. For obtuse an.gles, 75\ of the blocks
have 90-120°, 20\ have angles of 120-140°, and 10\ -greater than
140°.
The results of field investigations (observations) ag.ree
with the· data of certain experimental i ·nvestigations on the
breakdown of the ice cover of natural ice in a long hydraulic
trough. The noted features are examined in Fig. 12, where a
ttylized diagram of the distribution of form and relative block
dimensions is shown; these are forms in. the trough with a suf-
ficiently rapid dynamic breakdown of the ice cover with the
simultaneous action of compressive and bending forces.
The given. data may be utilized for probability evalua-
tion of block forma. It should be borne in mind that these
relate to "young" blocks (at the beginning of ice movement).
As time goes by, the dimensions and form of the blocks change.
As a rule, blocks acquire a more curvilinear profile, gradually
approximatin.g circular and ellipse forms with curvilinear angles
168.
of 90° and 1110re. We can conclude that it is in principle pos-
sible to construct in future the· ca.lculation depen.den.ces
for determining the dimensions and form of blocks for specific
mechanical properties of the ice and conditions of block forma-
tion. Such a structure may be conducted along lines of elastic
and plastic theory, introducing probability characteristics.
Such constructions require a rather large number of special
field-experimental and theoretical investigations.
/102/ CONCLUSION
Inve.stigations of ice jams, based on prin-
ciples of. general physical laws of mechanics and rheology of
both continuous and discrete media, indicates on the one hand
the fruitfulness and the many possibilities of this method, which
allows the. solutio·n of a .number of important problems examined
here., while on the other hand, indicates the urgent need of
studying many new complex problems derived from actually comhat-
ting jams and contro·lling them.
While t:he regul:arities established here permit conducting
a more or less reliable state of analysis of jam formation. under
stationary water conditions and known characteristics o f ice·
mo\·ement, the question regarding calculations o·f jams under condi-
tions of sharply expressed unstable conditions remains to a great
extent unsolved. The same may be said of det.eraining quantita-
tive characteris.tics of ice movements, the conditions' of its
development, and changes over time, etc.
...
•
•
•
169 .
In order to· solve S\lCh problems, it is necessary to con-
solidate the methods and laws of mechanics of continuous and
discrete media with the methods and procedures of hydrology,
meteorology, and also• special divisions of the sciences, such
as physical chemistry, ice technology, etc. Thi.s complex
approach to methods of investigating ice jams should also
apply to so•lving problems of controllir.g them.
It is obvious that future studies should be directed to
improving and detailing mathematical models of ice jams, field
and e.x.perimental methods of inve.stigation, the treatment of
methods for solving various applied problems, as well as to
checking the accuracy of cert.ain hypotheses and calcula-
tion models.
In conclusion, we think it necessary to again· state that
the present study is not exhaustin.. The author• viewed their
task. o.nly with respect to the. possibility of characterizing
the most important questions of the examined problem from the
point of view o :f general physical laws, mechanics, and rheology
of discrete media, comprised of individual particles -ice
blocks. We note also that in following through with this
concept, the particular bia.se!s and op · nions of the authors
have appeared to a certain extent .
'
I
170.
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Gauthier -Uillln. Pari<~, t.;;;.;,. ;;l}:i il·
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R. I. Keno~y, Joloutr.·.a!, 1:1~. 20:· ;p. •
58. !II i c he 1 B. W;n:.:r rcgiOJlo 11r rive:~ :.nd lakos. Cc;ltl P.egior .. Science
end l::~!iiDHri,n:. Mono~er:.pl: Ill -Diu, U. ii. Arn•y Corr-s of Enai~~Mn. f30 p •
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N t, p. 89-100. . .
90. c ..... a . .n,·:;.o~:·: pll:o:::!.l r:a p. J~~·:;n. -•X::Apo.llor•ll • M<r.e-
OpG.'lC:.rliRt, Co4;:tll , f9(i$, loi 1, C. 4£--11.
lit. T a ll a ~ a T. A rr.r.hur..·r.~cl'L c! vi..cr~l~"c of · sea ic:e: J. Oceano.. •
s:r:~. Soc, Jepeu, ts~s. v, :!, :;. ~. :p, 28-:9. .