c~ / HYDRAULICS AND MECHANICS OF RIVER ICE JAMS
by
Mehmet Secil Uzuner and John F. Kennedy
Sponsored by Rock Island District U.S. Army Corps of Engineers Contract No. DAC25-73-C-0032 and National Science Foundation Grant No. GK-35918X1
TC IIHR Report No. 161 24 .164 Iowa Institute of Hydraulic Research no.161 The University of Iowa 1974 Iowa City, Iowa May, 1974
f LIBRA "V JAN 375
Bureau or Ree.amafien Denver, Coioracfa
I BUREAU OF RECLAMATION DENVER LIBRARY 92072357 ^ / V v vX ( y >0\A 92072357 ? 7 7HYDRAULICS AND MECHANICS OF RIVER ICE JAMS
b by
Mehmet Secil Uzuner and John F. K e n n e d y /
Sponsored by Rock Island District U.S. Army Corps of Engineers Contract No. DAC25-73-C-0032 and National Science Foundation Grant No. GK-35918X1
n Report 16U?
Iowa Institute of HydrauHclydrauHc ]Research */) The Umvorcityu»lovÎowaC/i ( Iowa City, Iowa
^May, 1974 1 Abstract
An analytical model is developed for evolution of embacles in rivers. The differential equations underlying the analysis are the static force equilibrium relation for the fragmented ice cover; the unsteady, nonuniform momentum relation for the liquid flow beneath the ice; and the unsteady continuity equations for the solid and liquid phases. The analysis is limited to the case of uniform, rectangular channels. It yields predictions for the streamwise distribution of the thickness of the ice cover and the depth of flow beneath it.; the flow depth and cover thickness over the uniform downstream reach where the jam thickness and flow depth are constant; the flow depth at the upstream end of the jam; and the velocity of propagation of the jam front. Esti mates are also obtained for the rate of growth of a jam during early stages of its evolution, just following initiation. The formulation of the force equilibrium for the cover includes relations for the compres sive and shear strengths of floating fragmented ice, which are based on a failure analysis of the material. Experiments were conducted to deter mine the compressive strength, and Merino’s experimental results served as the basis for the determination of the shear strength. A computer program for the simultaneous, numerical solution of the governing equations is presented, and a specific example is solved in some detail to illustrate application of the mathematical model to the prediction of the development of embacles. The mode of thickening of jams was determine by analyzing the incip ient motion of ice blocks resting beneath an embacle. The critical thick ness, densimetric Froude number for transport was compared with that re quired for submergence of a floe at the upstream end of the cover. It was concluded that jams thicken primarily by internal collapse rather than by transport of floes beneath the floating cover.
l AC KNOWLEDGEMENTS
The writers wish to express their sincere gratitude to Dr.
Joachim Schwarz for his help and suggestions during the experiments.
A debt of gratitude is owed to Mr* Dale Karris and his shop personnel
for their timely help in the course of experiments, and many thanks to
Mrs. Ann Janes for typing the manuscript.
ii TABLE OF CONTENTS
Page
LIST OF FIGURES v LIST OF TABLES ix LIST OF SYMBOLS x
Chapter I INTRODUCTION 1
A. Introductory Remarks 1
B. The Genesis and Evolution of Ice Jams 8
C. Literature Review l6
Chapter II THEORETICAL MODEL 39
A. Force Equilibrium in a Floating Fragmented Ice Cover 39
B. Effective Stresses In Floating, Fragmented Covers and the Nondimensional Force-Equilibrium Relation k 2
C. Gradually Varied, Unsteady Flow Under a Floating Ice Cover k 6
D. Determination of V , Reduced Water Discharge Under a Floating Ice Cover, Equilibrium Values of h and t and Boundary Condition on t 53
E. Presentation and Discussion of the Solution of Governing Differential Equations 58
F. Transport of Floes as Cover Load Beneath an Ice Jam 71
Chapter III THE EVOLUTION STAGE OF ICE JAMMING 79
A. Introductory Remarks 79
B. Estimate of Time Required for an Embacle to Reach the Quasi-Steady State 80
iii Page
C. Evolution of Length, Downstream Thickness, and Propagation Velocity of Evolving Embacles 82
Chapter IV EXPERIMENTAL DETERMINATION OF k , C AND C x 0 i 89 A. Introductory Remarks 89
B. Apparatus Test Materials and Experimental Procedure 89
C. Experimental Results and Determination of k 97 X D. The formation of Cohesive Bonds, and Dimensional Analysis for k 110 X E. Merinofs Determination of C and C. o 1 117 Chapter V SUMMARY AND CONCLUSIONS 122
REFERENCES I27
Appendix A SOLUTION TECHNIQUE FOR THE DIFFERENTIAL EQUATION dh (-rr2-) = f(t ,h ), AND CALCULATION OF COVER- az o o o THICKNESS PROFILES OVER THE NONUNIFORM REACH OF A JAM 130
Appendix B COMPUTER PROGRAM FOR IBM-360 FOR SOLVING THE GOVERNING DIFFERENTIAL EQUATIONS AND CALCULATING ICE COVER THICKNESS PROFILES AND DEPTH OF FLOW AT THE NONUNIFORM FLOW REGION 138
Appendix C A METHOD FOR DETERMINING THE HYDRAULIC ROUGHNESS OF ICE JAMS FORM FIELD DATA 153
iv LIST OF. FIGURES
Figure Page
1 Oblique aerial photograph of an ice jam on the Rock River just upstream, from its confluence the Mississippi River (March, 1972). h
2 Close-up views of the surface of the jammed ice on the Rock River (1973)* 5
3 Aerial photograph of an ice jam on the Mississippi River (February, 1966). 6
k View of the jam that formed on the Israel River at Lancaster, New Hampshire (1967). 6
5 Schematic representation of development of Jams of fragmented ice. 10
6 Accumulation of blocks beneath a floating frag mented cover in a laboratory flume experiment on ice-jam formation. (Block dimensions: 1-75 in.x 1.75 in.x 0.50 in., Specific gravity = O.85). 15
7 Photo of deposition of ice on overbank regions of the Iowa River. 17
8 Michel’s (1957) test results on the form coefficient, kQ , for right parallelepied blocks. 22
9 Definition sketch for Berdennikov’s (196k) and Pariset et al. (1966) analysis of stresses in fragmented ice covers. 25
10 Definition sketch for Michel’s (1965) analysis of stresses in ice jams. 28
11 Definition sketch for floe submergence. 36
12 Definition sketch for the force equilibrium of an elemental control volume taken from a frag mented ice cover. ho
V Figure Page
13 Definition sketch for the upstream propaga tion of an ice jam on the nonuniform flow region. 1+7
lA Normalized ice thickness profiles calculated for selected values of flow, ice, and channel properties. 60
15 Variation of equilibrium thickness and velocity of propagation with C for selected, values of o flew, ice, and channel properties. Note that t and V are independent of k . e i w X 6 16 Variation of ice cover length in the nonuniform reach with Cq for selected values of flow, ice, and channel properties. 62
17 Relation between equilibrium depth of flow and Cq for selected values of flow, ice, and channel properties. Note that h is independent of k . e x 63 18 Variation of depth of flow in front of an ice cover with CQ for selected values of flow, ice, and channel properties. 6h
19 Variation of total volume of fragmented ice in the nonuniform flow region with Cq for selected values of flow, ice, and channel properties. 65 20 Schematic depiction of a fragmented ice-cover profile. 67 q. C 21 Variation of and with C for selected q. C o m cn values of flow, ice, and channel properties. 69 (a) C = 0.50 cn (8) C = 0.75 cn 22 Schematic representation of an ice block at rest beneath a fragmented ice cover. 73
23 Variation of T and T with C for selected e eo o values of ice, flow, and channel properties. 83 vi Figure Page
2k Distribution of ice volume along nonuniform reach of an ice jam for selected values of flow, ice, and channel properties, 86
25 Normalized volume of ice in an equilibrium jam upstream from section with normalized cover thickness, tQ , for selected values of flow, ice, and channel properties. 87
26 Schematic rendering of apparatus used in k experiments. 90
27 Photo of apparatus used in k^ experiments. 91
28 Dynamometer and force measuring components 93
29 Force transducer circuit block diagram. 9^
30 Typical voltage (force)-time record 98 (a) Compressive test with parallelepiped ice blocks. (b) Compressive test with ice blocks with random plan form.
31 Experimentally determined longitudinal stress coefficient, k^, for parallelepiped ice blocks. 105
32 Experimentally determined longitudinal stress coefficient, k , for ice blocks with random plan form. X 108
33 Effect of strain rate on k for random plan- form ice blocks. X 109
3U Phase diagram for water. Ill
35 Mohr circle for a fragmented, floating ice cover. Ilk
36 Unit shear strength versus cover thickness H 9 (a) Ice cubes (b) Phndom plan form ice blocks
37 Shear stress predictor 121
vii Figure Eage
38 Photo of an ice jam that formed on the Mississippi River, Pool 22, in January 1969- 123
39 Photo of blasting of an ice jam on the Iowa River just upstream from its confluence with the Mississippi River (February 1973). 12U
A .1 Block diagram of the computer program. 136
C.l Schematic depiction of flow under an ice jam. 157
C .2 Composite roughness as a function of ice and bed roughness as calculated by Larsen (1969). 158
viii LIST OF TABLES
Page
Summary of investigations on floating ice block stability 20
Summary of experimental results from compression tests to determine k for parallelpiped ice blocks 103
Summary of experimental results from compression tests to determine k for blocks with random plan-form x 106 LIST OF SYMBOLS
A o = total flow area
A1 = area which is dominated by the bed
area which is dominated by the ice cover
b = length of ice block perpendicular to largest plan dimension
C = Chezy coefficient
C cn = surface concentration of ice blocks at normal depth C cu = surface concentration of ice blocks at x = 0 C. 1 = cohesion intercept C o = shear stress coefficient C s square of the ratio of the surface velocity to the mean flow velocity of the approaching stream
d = largest plan dimension of an ice block
d e = effective block diameter F cr - //¡F - critical Froude number of approach flow for block submergence
= reaction of the banks per unit length
hydrodynamic force exerted by the flow on the upstream end of the cover
= lateral force per unit length
w component of the weight of the ice in the flow direction per unit cover surface area
force per unit width due to the normal stress in the ice cover in x direction f1 = Darcy-Weisbach friction factor for bed
x = -Darcy-Weisbach friction factor factor for ice cover
g = gravitational acceleration
H = depth of approach flow
Hq = H/hn = normalized depth of flow in front of the jam
h = depth of flow under a -uniform ice cover or a block
h(x) = depth of flow under an ice cover in nonuniform flow reach
he = h^/h^ = normalized equilibrium depth of flow
h = eauilibrium depth of flow
h^ - normal depth of flow
= h(x)/h = normalized depth of flow under an ice cover in nonuniform flow reach
K = permeability of jammed ice
k = Rankine’s active pressure coefficient (Michel’s analysis)
k. = stress coefficient for C. and a* i lx ' k. = roughness for each boundary (i.e., j = 1 for bed, j = 2 for J ice cover)
k^ = streamwise stress coefficient
k^^ = streamwise stress coefficient for individual runs
k = Michel’s form coefficient o k^ = coefficient equal or smaller than unity depending on loading state
L = length of ice cover tested in the Ice Force tank
1 = length of parallelepiped ice blocks used in stability analysis
Mg = moment caused by the buoyancy of the floe
xi - moment due to the normal pressures exerted on the “ d upstream and downstream faces of the block
moment of the local pressure reduction under the block near its upstream end
moment resulting from the shear force exerted on the bottom of the block
n ~ composite Manning's roughness coefficient
nn = Manning's roughness coefficient for toed
= Manning's roughness coefficient for ice cover
= wetted perimeter for bed section
— wetted perimeter for ice cover section
P — porosity of the fragmented ice cover
Q = water discharge at normal depth
a = water discharge, per unit channel width
iu ice discharge, per unit channel width, in fixed coordinates at x = 0
m = ice discharge, per unit channel width, at normal depth in fixed coordinates
= water discharge, per unit channel width, S through the jammed ice
9 * -water discharge, per unit channel width, in moving coordinate system
- ice discharge in moving coordinates, per unit channel width
R = hydraulic radius of the flow section
= net resistance from the banks in Michel's (1965,1966a) analysis
S o = sin 0 = channel slope
xii ~s energyv gradient sf * : normalized standard deviation of the values of k T : time required for the jam to reach equilibrium thickness e at its downstream end T = : normalized time required to reach equilibrium eo 2 T(V ) =1 time required for a volume ¥ h , of ice to accumulate o ,, ^ o n ? m the jam t = fragmented ice cover thickness t : normalized equilibrium thickness of fragmented ice cover e -P 1! t . = thickness of ice block 1 -P II O normalized ice block thickness t = t/h .?= normalized ice cover thickness O n normalized leading edge thickness tle -P 1! OJ t = submerged ice cover thickness P u carriage velocity c U*. shear velocity for sections effected by bed (j = l) or ice cover (j = 2) V critical velocity of approach flow, at which the blocks cr are just swept under the cover II < mean velocity of the flow under the jam V mean velocity of flow under equilibrium thickness e mean velocity for each section (i*e., j = 1 for bed, v j ■ j = 2 for ice cover) = mean velocity at normal depth n = actual percolation velocity through the jam Y = surface velocity of the stream in front of the ice cover su = mean velocity of the stream in front of the jam U V = velocity of upstream propagation of leading edge of the jam w Vf = V + = flow velocity in moving coordinate system Y = normalized volume of fragmented ice cover in the nonuniform flow reach Y = nondimensional volume of ice contained between the leading edge and x = x (V ) o x o v.(y) = velocity at a distance y. from the boundary J J (v ). = maximum velocity max j W = width of channel x = distance from the leading edge of the cover x = normalized cover length in the nonuniform flow reach x = x/h = normalized distance from the leading edge o n x f = distance from the leading edge in moving coordinates Y. = distance from the two boundaries to the point of ■ maximum velocity z = distance measured positive upwards from the phreatic line d - angle between the tangent to the phreatic line in the control volume taken from the fragmented ice cover g = velocity head coefficient of the flow under the separation zone beneath a floating ice block Y = specific weight of water Ye = equivalent weight of the cover xiv specific weight of ice coefficient of friction for the interface between the ice cover and the banks (Berdennikov, 196^') angle between the channel bed and horizontal Berdennikovf s proportionality constant between a and a y x density of water density of ice streamwise normal force per un it plan area of the stream (M ichel 1965, 1966a) weight of the fragmented ice per unit plan area streamwise normal, effective stress averaged over the fragmented ice cover applied horizontal, compressive, effective normal stress a t a p o in t lateral, normal, effective stress averaged over the fragmented ice cover vertical effective stress at a point shear stress at the banks shear stress due to flow (j = 1 for bed, j = 2 for ice cover bottom effective shear stress averaged over the ice cover maximum shear stress, t , at the banks xy shear stress exerted on the bottom of ice cover by th e flo w angle of internal friction Chapter I INTRODUCTION A. Introductory Remarks Ice jams pose many nettlesome problems to river engineers, including flooding caused by blockage of channels, damage to structures interference with navigation, obstruction of water diversion intakes, etc. As the demands on Earthfs water resources steadily increase and require more complete utilization, and as river flood plains become more extensively developed, river engineers are faced with more and more challenges arising from jamming. Ice jamming is such a complex phenomenon, or to be more precise, ensemble of phenomena, that it is difficult even to frame a concise definition of an ice jam. Uzuner and Kennedy (1972) proposed a definition which, with slight modification, reads as follows: An ice jam is an accumulation of ice on a stream which produces extensive blockage of the flow section; the accumulation is initiated by a channel obstruction, which may consist of other ice, a change in channel alignment or cross-section, or natural or man made obstacles in the stream. According to this definition, there are two primary requirements for the formation of an ice jam: a large discharge of frazil or fragmented solid ice, and an obstacle which impedes the downstream passage of the ice. The usual sources of ice include: i. Production of excessive frazil ice, which adheres to channel 2 boundaries and/or structures and blocks the channel. ii. Sudden release of large quantities of lake ice into a river, usually following wind-or seiching-induced collapse of a ' natural ice-arch just above the outlet, iii. A sudden rise in air temperature, or injection of a large amount of heat (e.g., from a power plant) into a stream at an upstream location, causing weakening and break-up of sheet ice, which then moves downstream. iv. Break-up of an ice cover by increased river discharge. The most generally encountered channel obstacles are: i. Channel constrictions, such as sharp bends, bridge piers, protruding abutments, flow regulating structures, and sand bars. ii. Shorefast ice, bottomfast ice, or both which are produced by extensive freezing. iii. Lake-or sea-ice driven to a river mouth by high wind, where •it is grounded in the relatively shallow river channel. iv. Floes grounded in a river mouth during receding tide. v. An abrupt streamwise decrease in slope along a channel, accompanied by a decrease in velocity and an increase in the width or depth of the channel. Pariset, Hausser, and Gagnon (1966) gave the following short description of the life of an ice jam: "As winter begins, large quantities of ice floes form on rivers flowing in cold regions. The floes are carried downstream by the current until they reach an artificial obstacle such as bridge pier in a low velocity zone, or a river section where slower surface currents and topography cause the floes to wedge, and result in the eventual build-up of a continuous ice bridge. As new ice floes arrive and come to rest on the ice bridge, the front of the cover proceeds upstream at a rate depending on the quantity of ice coming from upstream.... In some sections where the velocity is slower, a relatively thin ice cover progresses quickly and smoothly, whereas in other sections considerable local thickening - termed ice jamming - always occurs before the progression of the cover can proceed upstream.... Even more damaging ice jams can form during the late winter when the warmer rays of the sun and first increase of flow combine to dislocate the ice cover. A large volume of ice is thus released and can jam downstream under an ice cover which is not yet weakened. This is typical in some rivers flowing north to colder regions” , (p.2). Figure 1 is an oblique aerial photograph of an ice jam that formed on the Rock River, near Rock Island, Illinois. The obstruction in this case is the 1-80 bridge. The ice was formed upstream on the Rock River, which frequently produces large quantities of ice that flow into the Mississippi River and cause ice jamming there and extreme flooding (Army Corps of Engineers, 1967). Figure 2 shows a close-up view of the surface of the jammed ice on the Rock River, just upstream from the site shown in figure 1. Figure 3 is an aerial photograph of the jam that formed on the Mississippi River in the vicinity of Rock Island, Illinois, just downstream from the confluence with the Rock River. This jam caused in large measure by ice produced on the Rock River that flowed into the Mississippi River. Figure 4 shows a view of the jam that formed on the Israel River at Lancaster, New Hampshire in 1967. December, 1967, after an extended period of mild weather, Figure 1 Oblique aerial photograph of an ice jam on the Rock River just upstream from its con fluence -with the Mississippi River (March, 1972). (b) Figure 2 Close-up views of the surface of the jammed ice on the Rock River (1973). Figure 3 Aerial photograph of an ice jam on the Mississippi River (February, 1966). Figure b View of the jam that formed, on the Israel River at Lancaster, New Hampshire (1967). the ice on Israel River broke up and formed a jam near Lancaster -with out serious flooding. Subsequent cold weather produced more frazil ice and solidified this jam. Finally rapid thawing in March, 1968, produced a new jam which caused catastrophic flooding and damage amounting to one-half million dollars. A detailed report of this jam is given by Frankenstein and Assur (1970). The goal of the analytical and experimental study reported herein is to develop a theoretical framework for prediction of and for presentation of data on the distributions of thickness of accumulation, streamwise normal force and stress, lateral normal stress, and shear stress in a two-dimensional jam of fragmented ice on a stream flowing in a uniform, rectangular channel. The model also predicts the stream- wise distribution of flow depth beneath the jam as well as in the reach upstream from the leading edge.. The differential equations underlying the analysis are the static force equilibrium relation for the ice cover, the unsteady, nonuniform momentum relation for the flow beneath and approaching the ice jam, and the unsteady continuity .equation for the liquid and frozen water. These derivations are presented in Chapter II. Relations between cover thickness, longitudinal and lateral normal stresses, and shear stress are then derived from hypotheses concern ing the nature of the force interaction between ice fragments. These are discussed in Chapter II, Sections A,B,C,D and E. The stress relations are introduced into the governing differential equations which are then solved simultaneously by means of a high speed computer, in order to obtain the thickness and flow-depth profiles along the jam. The details of the integration procedure are presented in Appendix A. For determining the mode of thickening (i.e. either by blodk transport or internal failure) an analysis of incipient motion of an ice block resting on the bottom of a fragmented cover is developed, in Chapter II Section F, The critical Froude number for transport is compared with that for the submergence of a floe at the upstream end of the cover. In Chapter III the calculation of ice cover thickness and length during the evolution of an ice jam is presented. Finally, the experi mental data on streamwise stress coefficient are presented and discussed, in Chapter IV. B. The Genesis and Evolution of Ice Jams An ice jam is the composite result of several constituent phenomena. In order to gain better understanding of these and further insight into the mechanics of ice jams, a rather complete description of genesis and evolution of a typical ice jam (if there is such a thing!) will be presented here. For simplicity, the discussion will be limited to a uniform, rectangular, channel of very great depth (the large depth being included to permit the depth of flow plus the ice thickness to increase indefinitely without the added complication of overbank flow). As noted above, a jam is initiated when the down stream passage of floating ice being transported by a stream or a 9 river is blocked by a bridge, change in channel alignment, shorefast ice, etc. Thereby, an obstacle to passage of subsequent floes arriving at the section is produced and a jam is initiated. The modes of the subsequent evolution of fragmented-ice jams are depicted in figure 5, which is based on observations made in flume experiments using wooden and paraffin blocks to simulate ice, and on re ports of ice jam behavior. The floes arriving at the leading edge of the stationary cover may have sufficient momentum that they are submerged immediately (Case A in figure 5). In this situation, the upstream boundary condition for the jam states that the leading edge -of the jab thickens until its strength is great enough to withstand the momentum of the arriving floes. Alternately, the arriving ice blocks may come to rest at the leading edge of the cover (Cases B1 and B2, figure 5). If the Froude number of the flow is sufficiently small, the floes will accumulate in a single layer (Case Bl). At large Froude numbers, however, the ice blocks will be submerged and transported beneath the cover (Case B2). The stability of arriving blocks against submergence depends on the flow conditions and ice geometry; the submergence process is described and analyzed in detail by Uzuner and Kennedy (1972), and in the review of literature on ice jamming presented in Section C of this Chapter. The blocks swept under the cover may come to rest near the up stream end of the cover to form a "hanging dam", of the type shown in figure 6 and the second step of Case B2 in figure 5, or be transported further downstream, either coming to rest at some location or possibly (but not usually) being swept clear under the jammed ice, past the ICE BLOCKS MOVING DOWNSTREAM A HIGH VELOCITY LOW VELOCITY B ■ * Momentum of arriving floes produces F loes first come to immediate submergence and a multi rest at surface at layered leading edge. leading edge of -----f=---- — floating cover. Blocks arrested at leading edge Blocks submerged and swept under the F lov deepens and velocity is reduced of cover, which extends as a one cover, thereby forming hanging dam by obstruction produced by jam until layered jam until forces applied a n d c o n s e q u e n t l y m u l t i - l a y e r e d covet*. arriving floes are not submerged by cause collapse at some point. ------“ ------momentum. May be submerged, as in ------"T ------:------J 1 C a s e C. ______ The water in front of the hanging Aam T he cover progresses in upstream backs up due to the added energy loss direction by the blocks coming to caused by the constriction, and later rest in front of the fragmented lay arriving floes come to rest in front C over lengthens and stress in jam er. In order to resist the increas of the cover. increases until internal collapse ing streamvise forces, cover thickens 1 occurs, to provide strength required and finally reaches equilibrium thick* ------:--- for force equilibrium. Jam then ---develoDs----- as in Cases^ ------B and C. Figure 5 Schematic representation of development of jams of fragmented ice. channel obstruction, and continuing downstream. The presence of the ice cover produces an additional local energy loss by lengthening the wetted perimeter of the channel section along the reach occupied by the arrested ice. As the jam lengthens and thickens, the magnitude of the flow obstruction increases, and in order for the stream*s water discharge to pass beneath the cover, the cross-sectional area available to the flow is increased by the ice fTfloating upTf to deepen the flow section and thereby reduce the flow velocity. Moreover, the e n e rg y gradients along the reaches just downstream and upstream from the evolving jam are reduced due to the increased depth (compared to the normal depth) along these "backwater" reaches. Thus the added energy dissipated by the flow because of the presence of the jam is made available by the reduced energy gradients upstream and downstream from the cover, and by the dissipation rate of the flow beneath the cover being reduced (compared to what it would be if the flow depth did not increase) by the arrested ice cover 11 floating upTT. It should be noted that, just as in the case of an isolated bend in a long channel, there is no net additional energy dissipation due to the presence of the jam, but rather only a redistribution of the rate of energy dissipation along the channel. As the depth of flow at the upstream end of the jam increases and the velocity there decreases, the Froude number at that section is decreased and hence so also are the ability of the flow to submerge ice blocks that are first arrested after reaching the jam (see section C of this Chapter, and figure 11)and the momentum of the 2.2 arriving floes,«Finally, the flow depth becomes so large and the velocity so small that the floes are no longer submerged, but instead come to rest at the upstream edge of the jam, causing it to lengthen in the upstream direction. This development of the backwater profile upstream from the arrested cover results in storage of a significant volume of water and reduces the liquid discharge beneath the jam. At this point it is worthwhile to consider the balance of forces that comes into play in the stationary ensemble of ice blocks that constitutes the jam. The principal external forces applied to the ice field are the streamwise component of the weight of the ice and pore water it contains, the shear force applied by the flowing water to the bottom of the cover and the impact force of the arriving floes. In some cases wind may also exert a significant shear force on the top surface of the ice. These forces are balanced by the longitu dinal gradient of the internal effective normal force acting on planes oriented perpendicular to the flow direction, and the shear forces exerted on the ice by the banks; this force balance is formulated and analyzed in Chapter II, section A, It appears self-evident that both the shear strength and the compressive strength of the ice cover increase as its thickness increases. Now as the length of the jam is increased by the arrest of the. newly arrived floes near its upstream end, the total streamwise external force applied to the jam upstream from any section will cause the jam to thicken by failure or "collapse” of the ice cover (generally not fracture of individual blocks)., until its shear and compressive strengths are great enough to balance the 13 applied forces (see, e.g., Steps 2 and 3, Case Bl, figure 5). At large distances downstream from the upstream end of the jam, the cover will become sufficiently thick that just the bank shear will be equal to the applied external forces, both per unit length of channel, and no streamwise gradient in the compressive force, nor therefore in the thickness either, will be required. The jam will then have reached its equilibrium thickness at this section (final Step of Cases A and B). Note that it may still be growing at its upstream end, due to arrival of more ice, and thickening to equilibrium over its upstream reaches. Over the reach where the jam has attained equilibrium, the slope of the energy grade line likely will be close or equal to that of the channel. Once the jam has attained its equilibrium thickness at some point, its subsequent development is quasi-steady; i.e. when viewed in a coordinate system moving upstream with a velocity equal to the velocity of the leading edge, both the jam profile and the flow field appear steady. Several other aspects of ice jam formation are deserving of description. During early stages of jam formation, while the depth is still relatively small and the velocity and Froude number of the approach flow are large so that arriving floes are being submerged, the ice cover may thicken by deposition of submerged blocks. That is, the flow may be able to submerge the floes but unable to transport them beneath the cover (Step 2, Case B2). This is an especially complex problem near the upstream end of the jam, where the boundary layer under the ice cover is developing. Here the relatively high velocity’ of flow near the cover may be able to transport floes along quite steep slopes of the under side of the cover, resulting in formation of a hanging dam, illustrated in figure 6. Further along under the ice, where the thickening of the boundary layer has reduced the shear stress applied to the cover and the velocity near the ice boundary, the ice- transport capacity of the flow will be greatly diminished. Thus ice blocks may be transported some distance under the cover and then be deposited, thereby thickening the jam locally; this represents an intermediate case. At the two extremes ice may be transported under a jam that has attained its equilibrium thickness, or it may be deposited immediately downstream from the leading edge of the jam, depending on the depth and velocity of flow, size of fragments, channel slope, etc. This transport of ice as "cover load” (in analogy with the term "bed load” used in sediment transport mechanics) is examined in Section F of Chapter II. The evolution of ice jams was examined above without consider ation of limitations on its development. Generally, ice will continue to accumulate in a jam until the supply from upstream is exhausted. Thus if one knows the volume of ice moving into the jammed reach and can predict the thickness of the jammed ice cover along its length, he can.predict the streamwise extent of the jam and the increase in river stage it produces. In nature, river channels generally do not have vertical walls and very great depth; the complex geometry of rivers and their flood plains adds further complication on the analysis of ice jams. As the Figure 6 Accumulation of blocks beneath a floating fragmented cover in a laboratory flume experiment on ice-jam formation. (Block dimen sions: 1.75 in. x 1.75 in. x 0,50 in., Specific gravity = 0 ,85). 16 flow increases in depth, it occupies a progressively wider channel, due to the sloping of the banks. This gives the flow system an added degree of freedom, and significantly complicates the problem. In an extensive ly ice-jammed river, significant portions of the water and ice discharge may occur out of the main channels, in the overbank area. Figure 7 presents a photo of deposition of ice on overbank regions of the Iowa River. It is the ice blocks transported in the overbank areas that often are responsible for extreme damage to houses, engineering struc tures, trees, etc. In a complete analysis of a specific jam, one must take into account the cross-sections of the individual river and valley under consideration, and the movement and storage of ice and water that occur in the overbank areas. A further complication is also added thereby to the formulation of the force balance in the jam, since the lateral constraint condition at the banks is not well defined in the case of sloping banks. C. Literature Review The hightened concern for Earth's water, resources which has emerged in the past few years has been accompanied by steadily increas ing levels of interest in and research on the genesis and character istics of ice jams. However, only limited progress has been made towards an adequate understanding and formulation of ice jam mechanics. The analyses that have been developed are limited generally to one or another aspect of ice jamming, such as submergence of floes, backwater Figure T Photo of deposition of ice on overhank regions of the Iowa River. 18 effects, forces exerted by jams on structures, etc., and little attention has been directed toward the interaction of the constituent phenomena to obtain a comprehensive understanding and formulation of the mechanics of ice jamming. The ensuing literature review is presented in two parts. In the first the criteria for the stability of floating ice blocks and the upstream propagation of the leading edge of fragmented ice covers on rivers will be reviewed. The second part recounts the earlier theoret ical studies that have been conducted to develop analytical models for calculating the stresses in and thicknesses of ice jams. 1. Criteria for stability of floating ice blocks, and upstream propagation of the leading edge of fragmented covers. In figure 5 three conditions are depicted at the upstream end of an ice jam. In Case A, the upstream end of the jam thickens until it has sufficient strength to withstand the momentum of the arriving floes. This balance is used as the upstream boundary condition for the theoretical model of ice jams developed in Chapters II and III. In Case B2 in figured, the floes may come to rest, and then be submerged by underturning or vertical sinking. According to Ashton (197*0, the criterion for submergence is only very slightly affected by the motion of the block; i.e., the critical condition for submergence of initially stationary block and moving blocks are virtually identical. The orderly sub mergence of the blocks is amenable to analytical treatment, and has been the subject of several studies. Early investigators of this phenomenon observed and sought to quantify the critical conditions a t , which, arriving blocks are just submerged and carried under a floating obstruction (e.g., a fragmented or continuous ice cover); this condition has been termed the criterion for the progression of an ice cover (Bolsegna, 1968, p.5). The criterion was given either as the observed critical velocity at which the approaching blocks are just swept under the cover, or cast in terms of a nondimensional parameter such as the critical Froude number of the approach flow or a Froude number based on the approach flow velocity and block length or thickness. The theoretical investigations of floe stability have utilized the one-dimensional continuity and Bernoulli equations (with energy losses disregarded) between sections jpst upstream and down stream from the leading edge of the arrested cover. In these analyses incipient submergence has been assumed to occur when the "no spill" condition is reached; i.e., when the water surface elevation at the stagnation point at the upstream end of the floe just equals the elevation of the upper edge of the block, which is reduced by the flow acceleration and reduced pressure under the floe. This condition, which is one of the most questionable aspects of the analytical models of floe submergence would not have to be utilized if the formulation of the hydrodynamic moment exerted on the floe were made complete by inclusion of a more precise expression for the contribution of the pressures over the region of flow separation and on the upstream face and top. However, this is difficult to achieve. The criteria suggested by previous investigators and the results of the theoretical analyses are summarized in table 1. Most of Table 1 Summary of investigations on floating ice block stability. INVESTIGATOR CRITERION REMARKS Determined from the data on the St. Lawrence River. This criterion is based on observation that: "....ice covers may thicken and progress at velocities up to 2 . 2 5 ft/sec without floes pass McLachlan, D. W. ing underneath". Estiveef (1958) gave V as (1926) Vcr= 1 .2 5 it/sec between 2 . 3 to 2 . 6 ft/sec. The value given by McLachlan was later confirmed by Cosineau (196I) on the basis of many years of observations on the St. Lawrence River. He noted that this value is the upper limit which can be reached only under ideal conditions. Based on moment equilibrium of a single block Michel, B. . arrested in front of an obstruction. The form (1 9 5 7 ) coefficient, k , was determined in flume tests with right parallelepiped blocks, with the results presented in figure 8. Sinotin, V. I. Obtained from experiments in laboratory using and Vcr = (o .0 3 5 < jd 2 paraffin blocks. G-uenkin, Z. A. (1970) Based on his observations, Kivisild noted that: ?!After a section with a Froude number 0 .0 8 is Kivisild, H. R. reached it is estimated that ice will be carried (1959) under the ice cover.....velocity might be a FcW s h ' 0 0 8 better criterion than F . Tentatively velocity could be assumed to be Setermining factor with a critical velocity about 2 .0 ft/sec". Later, F i—;------!__ q l __ (Table 1 continued) INVESTIGATOR CRITERION REMARKS Kivisild (cont’d) vas given "by Cartier (1959) as 0.13, by Mathieu (1968) as 0 .1 1 , and by Oudshoorn (1970) as between 0.06 to 0.09. Michel (1965) notes that for square blocks with the thickness is 0.25 of the length, the leading edge Is stable when F <0.12. cr Pariset, E., Based on one-dimensional Bernoulli and continuity Hausser, R., equations between sections with and without ice, and with the no-spill condition met when the upper Gagnon, A. leading edge of the ice blocks are at the same (1966) elevation as the water surface. Utilizes same principles as model of Pariset et Michel, B. al., except that porosity, p, of the fragmented (1966) cover is introduced into the analysis. Based on one-dimensional hydrodynamic analysis of Uzuner, M. S. the flow passing beneath the upstream end of a and floating cover, and force and moment equilibrium Kennedy, J. F. j & ' - p of the block arrested in front of a stationary « (1972) r 1 cover. Same no-spill condition as Pariset et al. V C 5 4 i -Li . was utilized. Valid for tyicO.l. U 7T J Makes use of a simplified moment equilibrium analysis with the no-spill condition as the Ashton, G. D. F 20 hs ) ' criterion for incipient submergence. (197*0 ro H 22 0 0A 0.8 1.2 1.6 20 l /w Figure 8 Michel’s (1957) test results on the form coefficient, kQ , for right parallel piped blocks. 23 ■the references included m this table are cited by Bolsegna (1.968) in his comprehensive review of the literature on river ice jams. The notation used in this table is as follows: Y=-critical velocity of approach flow, at which the blocks are just swept under the cover; ^cr~ ^cr^1^®^5 critical Froude number of the approach flow; g = gravita tional acceleration; p f = ice density; p = water density; t^ = thickness of the ice blocks; p - porosity of the fragmented ice cover; 1 = length of the blocks; C^= square of the ratio of the surface velocity to the mean flow velocity of the approaching stream; g = a function of the geometrical and flow variables which represents the effects of the velocity increase just outside the separation zone (g = 0* for long blocks i.e. , t^/1 < 0.1). In the analyses represented in table 1, only the criterion for block submergence was investigated and the question of whether the blocks will come to rest under the cover or be transported downstream b^ the flow after being submerged was not treated, The submerged transport of floes was the subject of one of the phase of the present study, and will be discussed in detail in Section F of Chapter II. 2. Normal compressive stresses in fragmented ice covers and prediction of leading edge thickness. As was noted in Section B of this chapter, ice jams thicken until their compressive and shear strengths are adequate to withstand the streamwise forces applied upstream from any section. Accordingly, the strength-thickness relation for floating, fragmented covers is an integral part of analyt ical models of ice jams. Michel (1966a) recognized this when he noted 2k that; investigations- of this problem have shown that the forces exerted on structures by fragmented ice fields do not increase indefinitely with the upstream length of the cover, but reach a limiting value which, if exceeded, will cause the ice field to fail. Indeed, many of the studies of forces in fragmented covers have been prompted by interest in the forces they exert on structures, and by the need to predict required strengths of ice booms. According to Michel (1966a), studies of ice forces date back to 1935. The damage resulting from ice forces can be very extensive. Frankenstein (1971)* for example, reported that total damage caused by a moving fragmented ice field on White River below Windsor, Vermont, in 196k was about $2.5 million. The two analyses of the maximum stress in a fragmented cover, that are summarized in the following paragraphs were developed by Berdennikov (196U) and Michel (1965). Berdennikov1s (196k) analytical model for calculating the normal compressive stresses in a broken ice field begins with the equation for the equilibrium of forces acting upon an elemental transverse strip of length dx taken from the ice cover. The stresses acting on this element are: the streamwise normal stress, a^, shear stress, x^, exerted on the bottom of the ice cover by the flow; and the shear stress, x^, applied to the cover by the banks. For static equilibrium, the algebraic sum of these forces must be zero; (U„ + d c r ^ w t - - TjWd fc + 2 ^4 d * m 0 (1) Figure 9 Definition sketch for Berdennikov’s (196k) and Pariset et al, (-19-66) analysis of stresses in fragmented ice covers. ro \n 2 6 where t is the cover thickness and W is the width of "the river9 a,s depicted in figure 9. It is assumed in his analysis that the lateral stress, a , is proportional to streamwise stress, a , y ’ x ’ ^ Ç G*. (2) and that is proportional to a , (3) hence W where ç is the coefficient of friction for the interface between the ice cover and banks. Substituting (1+) into (l), rearranging terms, and integrating the resulting differential equation with the boundary condition = 0 at x = 0 yields (5) where x is positive streamwise direction, as shown in figure 9. At- large distances downstream, where x >> W, (5) becomes fl w (<0 (6) «MC Equation 6 indicates that for x >> W, the streamwise forces 27 are balanced by the bank shear. The field data ayailable to Berdennikov led him to recommend ££ = 0,5. The x component of the weight of the ice cover was not explicitly introduced into his analysis, and therefore must be included in cr . x An obvious shortcoming of this analysis is the treatment of cover thickness as constant along the channel in the integration of the force balance equation. This consideration indicates that the analytical model is applicable only to the reaches where equilibrium thickness has been reached. Accordingly (5) is somewhat meaningless. Michel (1965, 1966a) envisioned the streamwise forces exerted on an ice jam as being supported by arching or bridging of floes across the channel, as occurs, for example, in the blockage of grain elevators. Accordingly, he analyzed the force equilibrium of a control volume with the form of a catenary subject to a uniform lateral load, as shown in figure 10. The forces applied to the ice are then balanced by pure compression along the arches. The development of his analysis proceeds as follows: The streamwise force per unit plan area of the stream, , acting on the cover is given by a a =* CTW sin9 + T2 (7 ) where af^ is the weight of the fragmented ice per unit plan area, is the shear stress exerted on the bottom of the cover by the flowing water, and sin 0 is the slope of the stream. The quantities, a and t9 are given by 28 Figure .10 Definition sketch for Michel's (1965) analysis of stresses in ice jams. 29 Cw = x (' - p)-£- -t and tz = y V~ sin 9 (9) in which p and p f are the densities of the liquid water and ice, respectively, p is the porosity of the cover, f = pg is the specific weight of liquid water, and R is the hydraulic radius of the flow section, From the Darcy-Weisbach relation for wide channels, sin 0 and R may be expressed as sin9 = ------;______(10) 4 3 (h-i^W5 and « -g- (h (i d where t^ and h are as depicted in figure 10, Q is the discharge, and ? 2 is the friction factor for the underside of the cover. Note that equilibrium of the buoyant and gravity forces acting on the floating ice requires tg = t . *. Introducing (10) and (ll) into (9)» then (8) and (9) into (7) yield * Michel's paper is not consistent in the definition of the hydraulic radius. In some steps, (ll) is used, while in others, (ll) with tp replaced by t is adopted. Actually, K = h/2 appears to be preferable . to either of these. 30 fix[h-h 12 (1 -2-frÿj Q- (12) B (h-l^W2 According to Michel, the internal stresses s^ and s^, depicted in figure 10, can he expressed as s, - it(l-O Y '(H > ') (13) and (1U) where k is Rankine's active pressure coefficient for granular materials (Lambe and Whitman, 1969) and is given by (15) in which 0 is the internal friction angle of the fragmented material., The net resistance, r, from the hanks can he written as r = t2s2 +i£t-Os>z-4r ^1 (1 6 ) 2 ^ % % 2. In (13), (lU)5 and (16) it is seen that the lateral strength of the ice cover is related to the effective normal stress; this concept is ex panded upon in Section B of Chapter II. Substituting (15) into (13) and (lU) and the result into (l6) yields for the total force transmit-* ted to the hanks 31 r - Xil=*l £ dr) 2 \ 4 2 / y ' H For conditions of static equilibrium, the sum of the forces due to the weight of the cover and shear force under the cover must be balanced by the streamwise component of the force transmitted to the banks: 2 r 5 'm ^ = W ^ (18) Introducing from (12) and r from (17) into (l8) and letting o cl 1 4 8 ("(-p) Wn'Ci-^Cy.jf') (19) results in a ! G f (h -U U V ___ (20) [h + l20-2p)] ~ w for reaches where the cover has attained its equilibrium thickness. Introducing the nondimensional variables t f and h f, defined as 1 w * i ' = (21) and h w % h ' = ( 22.) a, q v* 32 into (20) leads to ( 23) which can he solved for given values of p and hf. For p = 0 . 1+ and t / h = 0 . 1+, (23 ) has a minimum o f h f = 2 .3 6 , a t sm a lle r v a lu e s o f h f , according to Michel, the jam w ill thicken. Although Michel’s analysis represented a significant forward stride at the time it was developed, it contains several deficiencies. F irst, the streamwise component of the weight of the pore water con tained in the jam is not included in (8). This force is not negligible compared to the others, and must be taken into account in the force balance. Second, the analysis does not consider the effects of non uniformity of the ice-cover thickness on the force balance, and hence can give no information about distribution of thickness along the upstream, nonuniform reach of the jam. Finally, it appears unlikely that the arching type of support could be very effective in wide rivers with sloping banks. In channels with other than steep banks, the lateral thrust would merely force the ice onto the banks. The present ly available information suggests that the cohesive shear strength of the ice is an important factor in supporting the applied loading; this is discussed in Section C of this Chapter. Pariset, Hausser and Gagnon (.1966). analyzed ice-jam formation in straight rivers flowing with constant discharge in uniform channels. Underlying their analysis is the assumption that the mechanics of 33 jamming are independent of the rheological properties of ice. Their analytical model includes the following forces acting on a control vol ume of streamwise length dx and extending across the full width of the channel: i. The hydrodynamic force per unit width exerted by. the current on the upstream end of the cover, f . o ii. The shear stress applied to the bottom of the cover by the flowing water, x^. iii. The component of the weight of the ice in the flow direc tion per unit cover-surface area, f . w iv. The shear stress exerted by the wind on the top of the cover. According to Pariset et al. , this generally is very small and was not retained in their analysis. v. The reaction of the banks per unit length along the chan nel, consisting of a cohesive contribution and an ice-over—ice friction term, f . vi. The force per unit width due to the normal stress in the ice cover in the x-direction, f . ’a Static equilibrium among the forces acting on a control volume dx long and extending over the full width of the channel and thickness of the jam is then expressed as wdf,, 4 = (X+OWck (2k) where W is the channel width (see figure 9 ) and df is the resultant a differential force per unit width due to the x gradient of the ice thickness, with f presumably given by (25) and the bank force is given by (26) in which Ch is the cohesive intercept, f is the lateral force per unit length, and 0 is the angle of internal friction of the ice. The later al force, f , is assumed in their analysis to be proportional to streamwise force, f , per unit width 7 cr ( 27 ) where k^ is a coefficient equal or smaller than unity, depending on the loading state (active or neutral) of the granular mass of ice. Intro ducing (27) into (26) and the result into ( 2i+) and integrating with the boundary conditions f =?..fQ-at x = 0 y ie l d s NX/ Ci t (28) f a + Q - a Acip 2 k, ^ 0 k, W» in w hich Ç W (29) 2 k4 4 As was the case with Berdennikow’s (l96k) analysis, the integration leading to (29) is inconsistent because t is treated as a constant. Similarly, the quantities which are dependent on t, notably x0 and f also are treated as independent of t. Thus, according to (25), the variation in along the channel results from unexplained changes in V On the basis of the result given by (28) and (29), Pariset et al. distinguished two different types of jams; a. Narrow Rivers: If a^ given by (29) is negative, the force f^ in the cover and presumably also t (although t was treated as con stant in the integration) have their maxima at the upstream end of the jam* In this case, the forces applied to segment of the ice cover are balanced entirely by bank shear, with the streamwise gradient in f a actually adding to the net force in the x direction. Thus, they argue that the thickness of the ice coyer is governed by conditions at its upstream end and will not be altered during the upstream propagation of the jam. Pariset et al. then formulated equilibrium thickness for this case at the upstream edge on the basis of the one-dimensional continuity and Bernouli equations (disregarding energy losses) between sections just upstream from the ice cover, section 1 in figure 11, and immediately downstream from the leading edge, section 2. The mechanism analyzed by these investigators by which the leading edge ice blocks become unstable and swept under the cover involves the following fea tures. The flow is accelerated as it passes under the cover, and consequently the pressure supporting the blocks is reduced; as a result Figure 11 Definition sketch for floe submergence* Go ON 37 "the elevation of the blocks is lowered below that corresponding to quiescent conditions. At the stagnation point at the leading edge of the ice cover, the local water-surface elevation is increased by the amount of the velocity head. According to their concept, an ice block becomes unstable and is swept under the cover when the water-surface elevation at the leading edge becomes equal to that of the top of the lowered block. This criterion for submergence of ice blocks is re ferred to as no-spill condition which was discussed in the preceding section of this chapter. The resulting equation for the thickness at the leading edge as a function of approach-flow Froude number is given ^7 (30) A detailed derivation and discussion of this relation is given by Uzuner (1971). Other analyses of the leading edge block stability are summarized in table 1. b. Wide River: This is the case when a^ given by (29) is positive, in which case ‘f and presumably also t increase with x. The equilibrium thickness of the jam is obtained by letting x ■* 00 in (28) and making the following substitutions: t (31) 38 (32) V p3 c1 in which is the mean velocity of the flow under the jam and C is the Chezy coefficient; and (33) The resulting equation for the equilibrium thickness of the cover is M /|., fltV M iL (3U) I^ H T f R-j pi P/Hl in which k^ = k^k^ tan 0. A final objection to be lodged against the analysis of Pariset et al. is that it does not take an adequate accounting of the hydraulics of the flow. The presence of the ice jam increases the depth and de creases the velocity of the approaching flow at the upstream end of the cover, and thereby affects the equilibrium thickness, as the inclusion of and H in (3b) demonstrates. However, their model provides no means for determining the effects of the jam on the river hydraulics. * Note that these investigations take Rankinefs passive pressure coef ficient as the limiting value for the wide river case, whereas Michel uses the active value to arrive at (25.) • 39 Chapter II THEORETICAL MODEL A, Force Equilibrium in a Floating Fragmented Ice Cover Consider an element of rectangular plan-form taken from an ice cover floating on a flowing stream, as depicted in figure 12. The co ordinate axes and the notation adopted for the various quantities arising in the analysis are defined in this figure. Note in particular that the x axis is parallel to, and the lateral faces of the control volume are taken normal to, the phreatic surface in the cover. The a normal and shear stresses in the ice, x9y 9z and xy , are effective values averaged over the ice cover thickness. The internal surface forces actually are transmitted by particle-to-particle contact, but the stresses are calculated on the basis of the transmitted force divided by the entire surface area (and not Just the interparticle contact area). It is assumed that at every point the z component of the weight of the cover is supported by buoyancy; in other words, there are no bending stresses in the cover. The force balance in the y- direction is of no particular interest here except as it affects the normal stress on x-z planes, a , which in turn plays a role in deter- y mining t , as discussed later. The x component of the weight of the ice and the pore water it contains, per unit area in the x-y plane, is given by G*tdy Figure 12 Definition sketch for the force equilibrium of an elemental control volume taken from a fragmented ice cover. 41 ^ f ^ (35) where (0 + a) is the streamwise slope of the phreatic line. For the case in which the z component of the weight of the ice is supported by- displacement of liquid water, the summation of forces normal to the phreatic surface gives i (I f^ c**(e> + d%dy » c^(e +1) 1 % (36) whence to - P t (37) Substituting (37) into (35) yields ^ = f'$ 1 stv\ (s> 401s) (38) Note that according to (37), the slope of the water surface and that of Q ^ the bottom of the ice cover differ only by — -?r- J ^ p ox Summation of the forces acting in the x direction on the con trol volume depicted in figure 12 and given by (38) leads to (39) 1 = ) - 11 " ° Note that the x component of the hydrostatic forces acting on the surface of the control volume are themselves in balance. k2 If the analysis is limited to situation in which a « 0 and cos (£». = i (39) reduces in the limit to p ax ~ ^ ^ - f 'c ^ W n 0 t = O (k0) for a laterally uniform jam; Ii.e., for the case *~(t, t 0) = 0], d j id Equation ^-0 is the governing relation for the force equilibrium of the ice cover. Before it can be integrated, however, to obtain t as a function of x and time, it is necessary to express a^, x^, and x ^ as functions of t. Chapter IY presents the results of experiments con ducted to determine a , and a discussion of the dependency of a and x x x on t and the strain rate. These relations are introduced into (Uo) in the following sections. B. Effective Stresses in Floating, Fragmented Covers, and the Non- dimensional Force-Equilibrium Relation. Consider the element depicted in figure 12 taken from a float ing, fragmented ice cover. As the external loads applied through x^ and F are increased, a point will be reached beyond which the strength of the cover, reflected in the critical values of a and x , is inade- x xy quate to support the external loading. The cover will then fail, and rearrangement of the fragments and accompanying thickening of the cover will occur until the internal surface forces are adequate to support the externally applied loads. It should he h o m e in mind that the cover will not thicken hy this process until its strength is exceeded; that is t will not he U3 dependent on the loading until the onset of failure. Moreover, the failure results from displacement between fragments, and not in general from rupture of the pieces of ice. Now the vertical thickening of the cover will be resisted by the submerged weight of the ice fragments, reflected in , and by the cohesive strength of the interparticle bonds az which are also dependent on as well as on several other features. az Accordingly, it is of interest to examine a z . Consider the vertical equilibrium of the floating, fragmented ice cover. For z > 0, with z measured positive upwards from the phreatic line O>S(0+Od') (bl) while for z < 0, (V )* <0 = t+2) c~('e+°0 (1,2) For z > 0, az supports the weight of the overlying ice, while over the range z < 0 it resists the upward force due to the buoyancy of the •underlying fragments. The average value of over the whole thickness, az a , is given by z ^ = i <**3) where v d - ^ 0 — §-Y O - K ) e'% <*%9 (^) is an equivalent weight of the cover, and again the restriction a << 0 has been introduced and will he retained in subsequent steps hereinafter. It will now be assumed that when ice field is undergoing fail ure by displacement of the fragments relative to each other, 5*=k,A <1,5) and where k is a stress coefficient which is somewhat similar to the x Rankine passive stress coefficient, Cq is a shear stress coefficient, Ck is the cohesive intercept (Larnbe and Whitman, 1969)9 an(i the failure value of t . Further justification for adoption of (45) xy and (46) is given in Chapter 4 and by Merino (1974). Suffice here to note that k and C , and likely also C., for ice are strongly strain- x o 1 rate dependent, but they will be treated as constant in the ensuing analysis. In the case of a straight prismatic channel, x^ is uniformly distributed under the ice cover. The internal stress, x , however, xy varies linearly across the stream, in the y direction. This can be seen from (40) when it is remembered that t, a , and xp all are independent X 9x of y. Therefore ■ --SL is constant at a fixed location along the stream, 3y Tx y = ' 0 a "t y = 0 •(■'the midplane of the channel) because of symmetry, and hence is given by Tl*M - - TvT ^ ("T ^ Ji CVr) * 3 “ w x j ;k where ¥ is the width of the channel. Substitution of (¡+3), (>5), and (hj) into (40) and carrying out the indicated differentiation yields 21 . 2t (48) where S - sin 0. Equation W will be rendered nondimensional by introduction of 1C i - i _ (49) l o = h ° ~ h *r where h is the normal depth for the open channel flow, given by Z~\ hn 5 I lI (50) Introduction of x and t into (48) gives o o ■toft = a + + c-t, (51) in which 1+6 and , _ r'% -2dc>/w) (53) and c Co ih (5U) It should be noted here that in (52) is a function of depth of flow beneath the cover, h. The momentum equation for unsteady, nonuniform flow under the cover, which is required to calculate h(x ) is derived o and discussed in Section C of this Chapter. C. Gradually Varied, Unsteady Flow Under a Floating Ice Cover A complete mathematical model of ice jams includes the distri bution of flow depth and mean velocity under the arrested ice and upstream from its leading edge. This information is also necessary to determine the shear stress exerted on the bottom of the ice jam. In this section the unsteady momentum equation for nonuniform flow under an ice cover is derived. Then the time dependence is eliminated from the unsteady equation by transforming the fixed system of coordinates (i.e., one that is stationary relative to the stream bed) to a system moving with the speed of the leading edge of the jams. Consider the ice-conveying flow, depicted in figure 13 Figure 13 Definition sketch for the upstream propagation of an ice jam in the nonuniform flow region. kQ passing along an open reach of uniform channel from which the liquid phase flows under a floating fragmented cover arrested in front of an obstacle. The transported ice accumulates along the upstream portion of the jam, thereby causing the jam to extend upstream. The continuity equation for the liquid flow through and beneath the ice cover between sections 1 and 2 shown in figure 13 i s si =• o ( 55) d% dJ + P 3T dx where q and q^ are, respectively, the water discharges per unit channel width under the cover and through the jammed ice, and T is time. The quantity of q can be written V V P IP f ( 56) in which is the actual percolation velocity of the water through the jam and is given by ( 57) where K is the permeability and p is the porosity of the fragmented ice in the cover. Note that Dupuit’s assumptions (Harr, 1962) have been used as the basis for (56) and (57). The momentum equation per unit width for the control volume bounded by the stream bed and the bottom of the cover can be written b9 p)_P2-('r,-iT2«s.«i.)d^+^ ^ 0cl*-= ^¡.ievf>)cl^+ |^ O v2l-iW +ev{§? + F"p fyV*- (5f where and are the forces by the hydrostatic pressure distribu tions at sections 1 and 2 , respectively, and the last term on the left-hand side of (58) is the x component of the force exerted on the control volume by the weight of the ice and its pore water. Note that in (58), the x axis has been treated as parallel to the plane stream bed, in keeping with the restriction a << 0 introduced at the end of Section A of this chapter. The last term on the right-hand side of (58) is the change of momentum due to the flow passing through the upper boundary of the control volume shown in figure 13 (i.e., the bottom of the ice cover). The quantities P^ and P^ in (58) can be written p4 =('21^3 * e (59) and (60) Substituting (59) and (60) into (58), dividing by yhdx, and rearranging terms results in _L 2_1 f a*- 3 b 50 3 d 1 5 O (61) g|i (JU'V3^ + r f 9T j where ?, (62) f -- Y h a«nd the approximations cos-0 = 1 and cosa = 1 have been adopted. Multi- plying the unsteady continuity equation, (55), by q / h g, subtracting the result from (6l), and using the identity 2 1 (Vh)-h^ +V lb (63) 3 * 3>T 3 T yields lit + iiy_ (6|f) Sh2 Equation 6U is the unsteady momentum equation in fixed coordinates for flow under a floating ice cover* Due to the floes arriving to the leading edge, the jam length ens in the upstream direction and thickens in order to resist the increased streamwise forces applied to it* According to the hypotheses made in Section B of Chapter I, a developing jam will continue to thicken at any section until it finally reaches a thickness at which the applied streamwise forces per unit length are balanced only by bank 51 shear. Thereafter the jam does not thicken any further, and is said to have achieved its equilibrium thickness at that section. After the cover reaches equilibrium thickness at its downstream end, the leading edge will propagate upstream with constant velocity. This is apparent when it is recalled that thickness of the jam at any distance from the leading edge will remain steady if x^ at that point is steady; this is demonstrated by the fact that time does not appear explicitly in the force balance relation, (Uo). Therefore, the volume of ice contained in the jam upstream from the point at which equilibrium thickness is reached is constant, and the arriving floes can be envisioned as in creasing the length of the equilibrium reach, and therefore the front of the jam will move upstream with constant velocity if the ice discharge reaching the jam is constant. The jam and flow then appear steady in coordinate system moving upstream with speed V . For this case the unsteady terms in (55) and (6^) can be replaced by O _ \ / ^ (65) 9T " w Soc' where x f is the new streamwise coordinate and is given by V!* K + \V T (66) Introducing (65) and the identity J L 15. - ( V i ' l - v l S b 4 . (67) g in 1 9 * 3 52 into i6k) yields V d v ' din , dl -V o ( 68) dxy f d x/ 3 d *' where v 7® v +vw (69) Kote that (68) is now independent of time and hence, the partial deriva tives are replaced hy ordinary derivatives. Introduction of (65) into (55) yields for the continuity equation in moving coordinates (v'W) 4 =. O (70) d * ' dot' Substituting from (70) into (68) and neglecting the term involving q^ as small in comparison to ^ f ( V oh) results in v^\dVi -t- S f = 0 (71) %h Jdx.' Equation 71 is the momentum equation for the flow beneath the ice cover in a coordinate system moving upstream with a velocity V . Note that, w as could have been anticipated, it is identical to the equation derived as if the flow were steady. Moreover, for x-g = 0 and ~ = 0, (7l) reduces to the classical equation of nonuniform flow. The simultaneous solution of the equations expressing the force balance in the jam, (51), and the momentum equation for the flow, 53 (71)» yield h and t as a function of x* for a jam that has reached the quasi-steady state (i.e., a jam that has reached equilibrium thickness at its downstream end). The procedure used in the numerical integration of these equations is presented in Appendix A, and results are given in Section E of this chapter. D, D eterm in atio n o f V , Reduced Water Discharge Under a Floating Ice ______W______■______■_____ ■ C over, Equilibrium Values 'of h and t and Boundary Condition on t . The flow obstruction presented by the jam causes formation of a backwater reach upstream from the leading edge of the jam. As the jam front propagates upstream with constant velocity, V after equilibrium conditions are reached at its downstream end, a significant quantity of water is stored in the backwater profile, which moves as a monotonic wave, and under the lengthening jam, and the actual discharge (measured in fixed coordinates) under the ice and downstream from the jam is reduced. For example, Bajorunas (1963) found that ice jams in the St. Clair and Detroit Rivers reduced the discharge up to 80,000 c f s under extreme conditions. Larsen (1969) also studied this problem on two Swedish power channels and found that the head losses in the river when ice covered is up to 62 percent greater than under open channel conditions. The reduced discharge must be used in calculating the shear stress, x^, appearing in the governing equations, ( 5l ) and ( 7l ) . (a) Determination of V . The speed of propagation of the w leading edge of an ice jam, V , can be determined from the conservation w of the volume of ice floes moved downstream by the flow: 5 ^ = ^lln + Vw (72) in which q^n is the ice volume discharge per unit width far upstream where the flow is uniform and has normal depth, q^ is the ice discharge through a fixed section at the leading edge of the jam, C is the sur face concentration of the floating ice blocks, ^ is the floe thickness, and the subscript n denotes values at normal flow conditions. After eqxiilibrium thickness has been achieved, q.^ may be expressed (73) where t is the equilibrium thickness and p is the porosity of the 0CL fragmented cover. Introducing q ^ from (73) into (72) and solving for V yields (7*0 where t and t. are the normalized cover and block thicknesses, e 10 * respectively, which are defined as i - hv, 1 _ ^ (75) (b) Determination of q. In the moving coordinate system the unit liquid discharge, q f, under the ice across any section is constant, according to (TO), if the discharge through the ice is neglected, and is given by 55 (76) and the corresponding Telocity is v'= ^ - (V„+Y*) (77) TT The discharge in fixed coordinates, q, is obtained from (76) as 1 = (V'-Vw)h- ■!„ Ww(h„-b) (78) (c) Determination of t^. A fragmented ice cover is said to have reached its equilibrium thickness over these reaches where t is constant. In terms of dynamic quantities, this is the case when the bank shear is equal to the externally applied streamwise forces, both per unit length along the channel. Then, S- 2 is zero and the force- o X O equilibrium equation for the cover, (71), reduces to = a 4- \>L h- c (79) Winch is a quadratic equation for which, after replacing t by t o e* the physically meaningful solution is given by J — h - J lo2- 4 O.C x.e * ------(80) 0-C where a, b, and c are as defined by (52), (53), and (5*0. The shear stress x2 , in (52) can be written T ~ ^ P -2=- (81) U " T ? h 56 where fg is the friction factor for the bottom of the ice cover and q is given by (78). .The corresponding expression for the energy gradient is given by (82) Over the equilibrium reach, where g and g are zero, the energy gradient must be close or equal to that of channel slope. The momentum equation, (71), after substitution of (82), reduces to (83) In the equilibrium reach, h should be replaced by h . In (7*0, (78), (80), and (83), V^, q, tg and h are the unknowns. Using (8l) and (8 3) in (80) yields an equation for t as a function of e h. Then, substituting this expression for t^ into (7^0 produces an equation for as a function of h, which,when substituted into (78 ), gives q(h). Finally, introducing q(h) into (83), substituting h for h e results in (8U) Vi e ~ m where -b-A>*-2ce,he vn = (Cc.i-X (85) 2a 57 and (86) ç + ft y6 k*. and (87) h~ Equation 8U is a nonlinear relation with as the only unknown. This equation was solved numerically using Wegstein’s iter ation technique, as described in Appendices A and B. The determination of q, t , and are then straightforward calculations from (78), (80), and (7*0, respectively. (d) Upstream boundary condition on t. Ice blocks reaching the upstream end of an ice jam move downstream with the surface veloc ity of the stream in front of the ice cover, V . The leading edge thickness of the jam must be sufficient to withstand the momentum of the arriving floes. The equation expressing the dynamic condition at the upstream end of the fragmented cover reads (88) where t0= t at x0= 0 and, from (78) (89) where 58 = K + - f - ( 90 ) Note that a discontinuity in the flow depth at the upstream end of the cover is introduced into the analysis; at x q = 0,'the flow depth is treated as changing abruptly from Hq to hQ (o). Thus the integration of the governing differential equation for the force balance in the fragmented ice cover must extend from t = t at x = 0 to t = t at O le o o e Xo ~ Xe ^w^ere xe normalized length of the cover from the leading edge to the beginning of equilibrium reach). jjL*. Presentation and Discussion of the Solution of Governing Differential Equations In the analytical model developed in Sections A, B, C, and D it was argued that a fragmented ice cover thickens at each section until its strength is adequate to support the forces applied upstream. The longitudinal ice-thickness profiles, calculated as described in Appendix A for jams that have achieved equilibrium thickness, are then such that force equilibrium is satisfied at every section, and the ice floats up until the depth and velocity beneath the cover produce an energy grade line over the equilibrium reach that has slope equal to that of the channel bed. The analysis considers only the ice-flow- system; the effects of complex channel cross-section and overbank topography were eliminated by considering only a deep channel with vertical walls. Typical results obtained from numerical solution of the governing differential equation, as described in Appendix A, are 59 presented in figures lA to 21, Figure ik shows typical fragmented ice-thickness profiles for different values of k^ and C . The plotted profiles are shown to the point where t = 0.99t . Note that t is independent of k and C , as o e e ^ x cn can also be seen in figure 15; this is a consequence, of course, of the externally applied streamwise forces in the equilibrium reach being balanced by the bank shear through the shear stress, t . The xy shear strength of the ice cover is, in turn, dependent on t , C^, and Cq . The length of the nonuniform reach, x^, on the other hand, depends on k , Ccn, and Cq , as can be seen in figures ik and 16 , because in this region both internal normal and shear forces act to balance the applied forces. Figure 15 also shows V calculated from (7^)* As C increases w cn with the flow conditions held constant, the rate at which ice reaches the upstream end of the cover and accumulates in the jam also will increase; hence the dependence of on C . The equilibrium flow depth, h^, shown in figure 17» varies with both Ccn and CQ . These dependencies arise from the fact that the discharge (measured in fixed coordinates) under the ice decreases as increases with increasing Ccn. The reduced discharge produces smaller shear stress, x^, on the bottom of the ice and hence smaller t . However, V also increases with reduced e ’ w t (see (80)). Note, however, that h is only very weakly dependent on e e C and C for a given set of flow conditions, cn o Figure 18 gives H as a function of k , C , and C . From the o x cn o way in which Hq was calculated, by integrating in the upstream direction from x q = x^, and consideration of the energy equation for the flow, it I t =0.05 Q = 39,500 cfs t./l = 0 . 1 f = 0. 02 i 1 Figure ik Normalized ice-thickness profiles calculated for selected values of flow, ice, and channel properties. ON o 6i 0 12 3 U 5 6 7 8 9 Co Figure 15 Variation of equilibrium thickness and velocity of propagation with C for selected values of flow, ice, and channel properties. Note that t and V are independent of k . e w x 3000 x e 1000 700 500 c .= 1 V 300 f2= p = 0.1+ t. = 0.05 10 t./l = 0.1 1 W = 1000 ft s = o.oooU o Q = 39,500 cfs 100 \ ■ i___ 10 5c* 8 Lgure 16 Variation of ice cover length in the nonuniform, reach with Cq for selected values of flow, ice, and channel properties. 63 Figure 17 Relation between equilibrium depth of flow and C for selected values of flow, ice, and o channel properties* Note that h is independent of k . 6 x 6k 01 2 3 4 5 _ 6 7 8 9 Co Figure 18 Variation of depth of flow in front of an ice coyer with C for selected yalues of flow- ice, o . ’ and channel properties. C^= 2.0 .psf % -— Ccn = 0.50 t = 0.02 ----Ccn = 0.75 f2= 0,1 p = 0.4 t.10 =0.05 1000 500 300 100 gure 19 Variation of total volume of fragmented ice in the nonuniform flow region with C for selected o values of flow, ice, and channel properties. 66 is clear that depends on h^ and the flow-depth profile in the non- uniform reach; hence the dependence on k , C , and C . x ? cn’ o Finally, figure 19 gives the volume of ice contained in the jam upstream from x^, calculated by numerical integration of the pro files given in figure 1^. Consider a schematic profile as depicted in figure 21. From trapezoidal rule the incremental volume A¥., between J any two sections can be expressed as AM = -¿-¿i 6-t-j + tj« (9; where t^., t^ +1, and Ax ^ are as shown in figure 21, and p is the porosity of the fragmented ice cover. The limiting values for t. are J the leading edge thickness at the upstream end, determined by the momentum of the floes moving downstream by the stream, and 0.991 at e the downstream end of the nonuniform reach. The total volume of the fragmented ice within these lim its is the sum of incremental volumes given by (91) and can be written V.-Z (92) He where is the nondimensional volume of ice per unit channel width contained in the nonuniform reach. Use will be made of V in the approx imate analysis of the initial evolution of ice jams, presented in the next chapter. In fixed coordinates, the ice concentration, C , on the surface cn Xo i-cover profile. 68 of the stream and ice discharge, q^, differ along the backwater reach from the normal values, due to the change in depth of flow and the velocity in this reach. The ice discharge in moving coordinates can be written f ■ = Cc M V W -v TT ^ (93> where V and q are as given by and (78), respectively, and C is w (7*0 c the surface concentration in the backwater reach. The ice discharge in fixed coordinates, is given by \ = "C i9h) Solving for from (93), after introducing (73) for q f^ and then substituting the resulting expression for into (9^-) yields a relation for q .: i V ^ W ' - O O - t t V-") (95) The ice concentration and ice discharge in fixed coordinates at x = 0 will be denoted by C and q^u , respectively. These quantities, normal ized by Ccn(ice concentration at normal depth) and q^n, are depicted <11U- in figures 21a and 21b. It is seen in figures 21 that — < 1, as qin expected due to ice storage that occurs in the unsteady backwater reach. On the other hand, C in the backwater reach is larger than C , for 5 c cn 69 01 2 3 4 5 6 7 8 co (a) C = 0.50 cn -L u. C U Figure 21 Variation of --- and with C for q. C o in cn selected values of flow, ice, and channel properties. 70 (Figure 21 continued) 71 the following reason; along the nonuniform flow reach, the flow depth increases, and hence the flow velocity must decrease faster than the decrease in water discharge. Now the ice is transported at roughly the mean velocity of the flow. Although the ice discharge also decreases in the x direction the ice-floe velocity must decrease more rapidly requiring an increase in C^, This can occur only by floes ”shelving”, or overriding one another. Perhaps in reality increases only to about unity, and the *■ ^suiting surface blockage modifies the backwater profile in such a way that does not exceed unity. This point remains to be clarified. F. Transport of Floes as Cover Load Beneath an Ice Jam In the foregoing analysis of ice jams, thickening by internal collapse or failure of the cover was the only mode of ice transport along the jam considered. However, there remains the possibility of floe transport beneath the cover by sliding, rolling, and saltating; this mode of transport will be referred to as cover load, in analogy to sediment transport along stream bottoms as bed load. This transport process was included in the description of ice jam evolution presented in Section B of Chapter I. One might reason that if a floe is set into motion as cover load near the upstream end of the jam, it will continue to be transported beneath the cover, since the flow velocity and hence presumably also the transport potential of the flow increase along the nonuniform reach of the jam. There does remain;however, the possibility 12 of locally intensified transport potential under just the upstream end of the cover, where the boundary layer is developing and the fluid velocity near the cover and the shear stress on it are large. This may account for the formation of "hanging dams", illustrated in figure 6 and discussed in Chapter I, Section B. The question of stability of floes beneath a cover clearly is closely akin to that of the initiation of motion of prisms resting on a stream bed. This latter problem has been studied by several investi gators since about 19^+2, with the goal of determining the size and weight of riprap material which may safely be employed for bed and bank pro tection. Of particular interest here are the experimental and analyti cal model of Allen (19^2); aspects of these studies will be incorpora ted into this analysis developed below. Experimental observations made in the course of the present investigation, and also reported by others (Allen, I9I+2 ) have revealed that the blocks generally move by rotation or "underturning" (rotation similar to but opposite in direction from overturning), rather than by sliding or saltation. Hence it is in order to examine the balance of moments exerted on the floe. The moments per unit width acting on the block shown in figure 22 may be summarized as follows: a. The moment due to the normal pressures exerted on the upstream and downstream faces of the block may be expressed Mb =*. ^ V2 -k^ (96) 73 / Figure 22 Schematic representation of an ice block at rest beneath a fragmented ice cover. where C, is a moment coefficient, t. is block thickness, and V is the mean flow velocity under the cover. b. The moment resulting from the shear force exerted on the bottom of the floe is M 5 = Cjf ^ V i (97) where Cf is the friction coefficient for the flow along the underside of the block and 1 is the length of the block. c. The moment of the local pressure reduction under the block near its upstream end, due mainly to flow separation, is of the form (98) where is the moment coefficient. d. The moment caused by the buoyancy of the floe is Mb. (V-vOU-l (99) where y and y' are the specific weights of the liquid and solid phases of water, respectively. For static equilibrium, the overturning moments, ?L, M„, and M^, should be balanced by the restoring moment M^. After equating the sum of (96)., (97), and (98) to (99) and rearranging terms, there results, V L_ (100) /Y-y'n 1 ' *■ c bu t; 75 in which -V cbL Cl o i) The relation developed analytically and quantified experimentally by Novak (19^8) for incipient motion of a block resting on the bottom of an ice cover can be cast in nondimensional form with the result (102) /I?-,------1 v Y The form of (lOO) and (102) are identical if the friction and lift terms are small enough to be- neglected ' or are of the form of the C term d In the derivation of (102), both lift and drag were accounted for by introducing a single coefficient. The shear force on the underside of the block, however, was not taken into consideration. In (lOO) and (101), on the other hand each moment contribution was accounted for by a separate coefficient. To examine the possibility of cover-load transport occurring, it suffices to determine if ice that becomes submerged at the upstream end of the jam will be tranported along the equilibrium reach where, as discussed earlier, the flow has the greatest cover-load transport capac ity. The following question will nowbe examined: If a floe is submerged by underturning at the upstream end of the cover, will it be transported as cover load? To this end, relations for the incipient 7 6 motion of cover load and submergence of floes will be compared. In a coordinate system moving with the velocity of the upstream end of cover that has attained its equilibrium thickness, the continu ity equation for the liquid is H ^ (.103) where the terms are as depicted in figure 22. Solving for V from e (103) and substituting- into (102), in which Y and h are replaced by Y^ and h respectively, yields Vw (loll) where H , t . , and h are the normalized values of H, t . , and h o ’ 1 0 ’ e 9 i 5 , respectively, and the it-subscript denotes incipient transport. TJzuner and Kennedy (1972) have developed a theory for and reported data on the submergence of blocks at the leading edge of an ice jam for different t^/l ratios and various specific gravities. Since, for most natural floes,t^/l is small, the present comparison will be limited to their formulation for t./r << 1. However, the same com- parison can be made for all other sizes of blocks by using a proper 77 equation with the appropriate coefficients. The critical Fronde number for submergence of long blocks (i.e., for t.,/1 < % 0.1) is given by Uzuner and Kennedy (1972) as V 2 [**' q),' Ly' X ò ^ - is where the is-subscrij' refers to incipient submergence, and C , the S coefficient that accounts for the surface velocity being greater than the mean velocity , was taken as 1.3 on the basis of float-velocity measurements in a flume and consideration of the logarithmic velocity distribution for wide open channels. Uzuner (1971) presents a detailed discussion of the determination of Cg . Note that the flow conditions appropriate to incipient submergence , included in (104), are those at the upstream end of the jam. Equations■10U and 105 do not admit to easy comparison, primarily because the several variables contained in (l0l+) are rather compli cated functions of other variables. As one part of the computer program presented in Appendix B, the Froude numbers appearing on the left of (101+) and (105) were compared. For all of the cases examined, it was found that a larger value of is needed to transport the floes than to submerge them, indicating that motion as cover load is not an important aspect of jam behaviour. One observation should be added, however. The analytical model of Uzuner and Kennedy (1972) examines floe sub- mergence occurring beneath a downstream cover of ice that is only one 78 floe thick, wiille the upstream end of jams are treated herein as being sufficiently thick to support the momentum transferred to the jam by arriving floes. This boundary condition is expressed in (88), It would appear that a larger is required to submerge floes in the latter case than in the former, so the foregoing condition likely remains valid. For example, for the flow and ice jam where properties are summarized in figures Ik to 21, the critical density-thickness Froude number for transport, given by (lOU), varied from 8.76 to 10.05, while the submergence Froude number, given by (105), was only 1.22. 79 Chapter III THE EVOLUTION'STATE OP ICE JAMMING A» Introductory Remarks In Chapter II the governing d ifferen tial equations for the force equilibrium of a two-dimensional jam subjected to streamwise nor mal, shear, and body forces and for nonuniform flow beneath a fragmented ice cover were derived, and a fa ir ly complete numerical solution was constructed for the quasi-steady case, in which the jam has achieved its equilibrium thickness at its downstream end and propagates upstream at constant velocity. In th is case, the mathematical formulation could be rendered formally steady, and the downstream boundary conditions, t = t given by (80), h = h determined from (81+), and q = q obtained ®q 0 from (78) for h = h , were readily established. The present chapter w ill he given over to consideration of evolution of ice jams during the early stages of their formation, before equilibrium values of t and h have been reached at the downstream end of the accumulation. During this stagey the behavior of the jam is s t i l l governed by the force balance relation for the ice , (5d.), the nonuniform,unsteady momentum equation for the flow, (7-1) 9 the equation of continuity for the liquid phase, (55)» and continuity and momentum relations for the solid phase at the upstream end of the jam. However, the set of equations cannot be readily or concisely solved in this case, for several reasons. First, the partial d ifferen tia l equations can not be reduced to ordinary, d ifferen tial 80 equations 'by' a simple transformation, as was done in the preceding chapter, because the upstream end of the jam does not propagate with constant velocity'. Since time dependency can not be removed from ths simultaneous equations, it would be necessary to solve them for discrete time steps, employing an iterative process, and using the temporal changes in the dependent variables between steps to estimate the time derivatives. Finally, it is not altogether clear what should be used for the initial condition. One might start the calculation with a short uniform ice accumulation of a small thickness. But the formulation is not applicable until the jam acts as a continuum ;i.e., until it is several floes thick. In any event, it is doubtful that the required effort would be justified, since the evolutionary stage usually is of relatively short duration, arid considering that the validity of this formulation is subject to some question until the jam has thickened somewhat. Instead, this chapter will be concerned with development of approximate estimates for the time required for the jam to achieve its equilibrium thickness, the downstream thickness and length of the jam at any time, and its rate of upstream propagation. B, Estimate of Time Required for an Embâcle to Reach thé Quasi-Steady State Puring early stages of ice jamming the unsteady term in the governing differential equations may be significant. When the first ice floes become arrested and initiate jamming, the ice discharge (in fixed 8l coordinates) is nearly constant along the upstream reach. However, as the jam extends upstream and thickens, the flow deepens, due to the added resistance presented by the embâcle, and the ice cover floats up. Hence the ice discharge is reduced as a result of the reduced surface velocity of the flow in the backwater reach upstream from the jam. An exact determination of the ice discharge reaching the jam and the veloc ity of.upstream propagation of the leading -edge of the embâcle would require solution of the full set of simultaneous partial differential ■ equations, as discussed above, to determine the nonuniform, unsteady flow characteristics and ice-concentration distribution in the backwater reach. As an alternate, in order to formulate an estimate for T , the e time required for the jam to reach equilibrium thickness at its down stream end, it will be assumed that the ice discharge at the upstream end of the jam varies linearly with time from its initial value, q to its quasi-steady value, q.^ The.volume of solid ice in the jam when it reaches equilibrium, will be equal to the ice discharged across the section reached by the leading edge when the jam has just achieved quasi-steady conditions, plus the volume of ice initially in the reach of length x^ j Ms = + 'K-c hn C cn "Li© (106) where qiu is given by (72), Note that in (l06), If and x are nondimen- G 0 sional quantities. Solving Tg from (106) and expressing the normalized time, Teo , as 82 (107) yields Teo=_£jk_ (V.-^CnV) (108.) («!*+■■ «1..Y where V and can he obtained from figures 16 and 19 > respectively, for the indicated flow and ice conditions. The variation of T and T eo e with k^, C^, and C^n for the jam analyzed in Chapter II and described in figures Ik to 21 is given in figure 23. The quantity q^n was calculated from In (109) The fact that surface velocity, and hence that of the floes, is larger than the mean flow velocity can be accounted for by using a value of Ccn that is, say, 10 to 15 percent greater than the actual value. C. Evolution of Length, Downstream Thickness, and Propagation Velocity of Evolving Embacles. For calculation of estimates of the length and downstream thickness of evolving jams and the propagation velocity during evolution, a major assumption will be employed. Specifically, it will be assumed that the 83 Tp(hrs) Figure 23 Variation of T and T with C for e eo o selected values of ice, flow, and channel properties. 81+ profile of an evolving embâcle (the one that has not reached equilibrium thickness at its downstream end)» containing a volume of h^V of ice is n o the same as the profile of a quasi^steady jam between its upstream end, Xo = an<^ station x^(yq )h^ upstream from which it contains volume 2 of ice. This assumption is at best only approximately correct, for the streamwise distributions of shear stress on the bottom of the jam and weight of ice and pore water, and the momentum transferred to the upstream end of the jam by the arriving floes, all of which determine the jam profile, are somewhat different between evolving and equilibrium embâcles. Nevertheless, the assumption appears to be adequate for the purpose at hand. Further, it will be assumed, as in the preceding section, that the ice discharge (in fixed coordinates) at the upstream end of the embâcle varies linearly with time. Using these assumptions the time, T, required for a volume 2 ¥ h of ice to accumulate in the jam can be obtained from o n , ¡.T itl v X .p * 2^ r ‘ - ) + (no) in which V is the nondimensional volume of ice contained in the jam o 0 between x = 0 and x = x (¥ ). Solving for Ï yields o o o o ( 1 1 1 ) To apply (.in), it is necessary to know the relation betwsen Tf and h . o o This obtained by integrating the thickness profile of the equilibrium embacle from the leading edge to x . This calculation is a part of the computer program given in Appendix B and illustrated in the block dia gram in Appendix A. Examples of these relations are given in figure 2h for two of the profiles shown in figure 1^. The estimates of jam length and downstream thickness as func tions of time are made as follows: ¥ 1. Selected values — and determine the corresponding values of x o V-e ~ from graphs of the type given in figure 2h, but calculated for the e jam under consideration. 2. Determine ¥ and x from the known values of ¥ and x . o o e e 3. Calculate T(¥ ) from (ill) ' o Determine the thickness at the downstream end of the jam ¥ from the graphical relation between -2 and t (e.g.r figure 25), deter- e mined from the ice thickness profile and the graphical relation between ¥ x — and ~ (e.g., figure 2k)\ *e e The velocity of propagation of an evolving jam, V (T), can be estimated from the relation between x and T (V ) obtained from steps 2 o o * and 3. It can also be estimated from the downstream thickness, the ice discharge, and surface concentration as TOO vv(rK(T)h„(i-^ = -V Vv Ccn ( 112) xe Figure 2k Distribution of ice volume along nonuniform reach of an ice jam for selected values of flow, ice, and channel properties. 87 Figure 25 Normalized volume of ice in an equilibrium jam upstream from section with normalized cover thickness, t , for o ’ selected values of flow, ice, and channel properties. 88 from which, there cam he obtained Q . O liSl ^lf> * '»'u i (113) to {.T ^ (i—^ —Ccn 89 Chapter IY EXPERIMENTAL DETERMINATION OF k , C AND C x* O i A, Introductory Remarks The analytical model of* ice jams developed and presented in Chapters II and III contained three coefficients, k , C and C . rela- x 7 o i* ting the strength ana thickness of floating fragmented ice covers. For the calculation of ice-jam profiles, propagation velocities, equilibrium thickness, the time of development, etc. reported in the preceding two chapters, ranges of values of these coefficients were considered. To quantify k^, a series of laboratory experiments was performed; these are summarized in the following sections. Additionally, the analytical model developed to provide a framework for presentation of the experimental results is described. Finally the. results of a recent experimental study conducted at the Institute of Hydraulic Research to determine C and C o i are summarized. The findings of these two experimental investigation were used to determine the values of k , C , and C. used in the afore- x o i mentioned calculations. B. Apparatus,:Test Materials, and Experimental Procedure 1, Experimental Unit, The experiments to quantify k^ were conducted in the Institute’s Ice Force-Facility, depicted schematically in figure 26 and photographically in figure 27« The principal component of this test unit is a 2-foot deep, 3—foot wide, 19-foot long insulated Figure 26 Schematic rendering of apparatus used in'k experiments. VO o Figure 27 Photo of apparatus used in kx experiments. 92 tank located in temperature controlled room which, houses the Institute’s Low Temperature Flow Facility , The compression apparatus consisted of a driving plate attached to the motor driven carriage through a dynamometer equipped with a strain-gage force transducer. The .carriage was supported by four ball-bushings which rode on one-inch diameter rails. It was driven by a variable speed, one-horsepower DC motor through a cone-pulley system, geared speed reducer, two pinion gears at the end of the drive shaft that extended across the tank, and two racks, one affixed to the top of each of the long walls of the flume, The motor speed was remotely regulated through an SCR drive control; the carriage velocity could be varied from 0.0(A cm/sec to 2,k cm/sec, The driving plate was attached to the carriage through the moment insensitive dynamometer illustrated in figure 28. The force sensing element of the dynamometer was Statham Universal Transducing Cell, Model UC3, with a load accessory Model TJL^, hereinafter referred to collectively as the transducer head. The transducer head had a built-in foil spring with a load limit of 200 pounds. A block diagram of the transducer circuit is shown in figure 29. The output of the strain-gage bridge was amplified by Dana 2850 amplifier and recorded with a Beckman Type RS Qrnagraplu The force measuring system was cali brated by applying known horizontal loads to the driving plate. The system was found to be linear with a calibration constant of 80.0 lb/volt. Periodic checks of the calibration revealed no drift in the system. It was also found out that the calibration constant was independent of the position of load application on the driving plate. Figure 28 Dynamometer and force measuring components. 9h TRANSDUCER Figure 29 Force transducer circuit block diagram. 95 2. Test Materials, Two different types of fragmented ice were used in the tests, in order to investigate shape and size effects on k , « x The ice blocks used in the first series of experiments w'ere right paral- lelepipeds prepared in the cold room in tin pans with movable partition vails. The block dimensions and other properties are reported in table 2. The second series of experiments to quantify kx was carried out with ice blocks of random plan form and uniform thickness, t_^. The ice fragments for these experiments were prepared by fracturing ice frozen in the tank to the desired thickness. The largest plan dimension, d, and the dimen sion perpendicular to it, b, of each fragment of a sample of 100 or more pieces were measured in order to obtain a measure of the size' distribu tion and effective block diameter, d , 9 e 5 (ni») For each series of runs the block dimensions of this series are presented in table 3. Porosity measurements for the two test materials were made in a specially built lucite box for the parallelepiped materials, and in large pails for the random blocks. The material whose porosity was to be measured was placed in the box (with a screen lid on top to prevent float ing) and filled with water. The weight of the water required to fill the interstiaes between the particles in the box was then used to calculate the porosity. In neither the first series nor the second series of kx experi ments were the ice fragments used for a large number of tests or after 96 they had been left in the cold room for an extended period of time, since their quality deteriorated with time. 3* Experimental Procedure; The procedure followed in each experimental run was as follows. The compression apparatus, depicted in figures 26 and 27 was installed in the ice-force tank, which then was filled with water which had been chilled to about 0°C. The cold-room thermostat was set to 0°C and the water and the surroundings allowed to come to this temperature. The quantity of ice blocks required to give the desired cover thickness then was placed in the tank in front of the driving plate, and the floating ice cover was gently agitated with a rod to produce uniform thickness of the cover and homogenous orientation of the fragments. The cover thickness was measured by means of a spe cially designed thickness gage, made of an aluminum staff fitted with a measuring tape affixed to one surface and a 3 in. by 5 in. plate attached perpendicular to the graduated staff. The gage was inserted through the ice and raised until its foot came into contact with the underside of the floating cover. Then after adjucting the SCR drive control to give the de sired velocity, the carriage was set into motion. The velocity of the carriage was at first kept low in order to minimize inertial effects on the force record. After the strong effect of velocity on the compressive strength was discovered experiments were conducted using higher carriage veloci ties. The length and thickness of the cover tested were such that the change in the cover thickness at time of failure was small compared to the initial thickness. Considerable variation in the failure force was observed between 97 -nominally identical experiments. These seeming inconsistencies were attributed to variations in the orientation and arrangement of the frag ments from run to run. In some instances the ice blocks would become arranged in such a way that relatively strong ice ,fcolumns" formed between the driving plate and the end of the tank, or on ice arch formed across the tank. In other cases, these stronger arrays were not formed. Because of this variability it was judged necessary to repeat each exper iment several times. The average of the failure force obtained from ten or more tests for each condition was used in the calculation of the reported values of k . C. Experimental Results and Determination of k ______;______x In the tests it was observed that the force applied to the cover increased gradually with time until failure was reached, whereupon it diminished abruptly. If the carriage motion was continued, the applied force would again increase with time until a second peak was reached, and so on. Typical voltage (force)-time records for random floes and for ice cubes are shown in figures 30a and 30b, respectively. The scale gradation is shown on each figure separately; the calibration constant was 80.0 lb/volt. The maximum force, measured just before the sudden drop, was used in the calculation of k . These points are indicated on the force records shown in figures 30. These two experiments were run with the same carriage velocity. Note the different type of failure for each type of block. The reason for the larger force required for failure in VOLTAGE (a) Compressive test with parallelepiped ice blocks Figure 30 Typical voltage (force) - time record VO OO VOLTAGE Cb) Compressive test with ice "blocks with random plan form (Figure 30 continued) vo 100 the case of ice cubes is believed to be the smaller porosity arid result- ting larger contact area and hence greater internal cohesive strength of these covers. Additionally, these regular shapes probably could become arranged in face-to-face patterns that could support larger forces before a 1 buckling" type of failure occurred. The failure force, F, can be expressed (see figure 25) us F = i (115) where is the longitudinal 5 effective, normal stress, W is the width of the tank, and t is the thickness of the fragmented cover. Intro ducing cxx from (U5) and (1+3) into (115) and solving for k yields k = _____ (n6) where p 1 and p are the densities of solid and liquid phases of water, respectively. Further justification for adopting this definition of k^ is given in Section D of this chapter. The results of tests conducted using right parallelepiped ice blocks and random plan-form floes are summarized in tables 2 and 3, respectively, and are presented graphically in figures 31 and 32, re- t. spectively. The number by each point gives the value of r-*, and the t lines are drawn through the average values of k^ for the different values In the calculation of k from (ll6), the specific gravity of ice was assumed to toe 0.92 and W was taken as the inside width of the tank, 3.0 ft. The quantity (S3)) appearing in tables 2 and 3 is the 101 normalized standard deviation of the values of k determined from the x individual runs for each condition, (st^ kx (117) where k , represents the individual determinations of k for a particu- la/r condition* Other quantities appearing in tables 2 and 3 are defined in footnotes, to the tables* The nondimensional parameters used in the graphical presentations are derived in Section D. Finally, figure 33 gives the results of some tests conducted with random plan-form floes to investigate the effects of carriage velocity (i.e., deformation rate) on k . Note that^in this figureythe individual x determinations of k . is indicated for each velocity. xi Several interesting observations can be made from figures 31, 32, and 33. First, it is seen in figure 31 that k is quite sensitive to X t. the geometry of the fragments, as represented by — , generally increasing e as the floes become thicker relative to their lateral dimensions. The scatter about the mean line is much smaller for the regular ice blocks than for the floes with random plan form. It is believed that the rela tively high degree of scatter in the latter case results in large measure from the variability in the ice dimensions between batches of ice, and also between individual floes in the case of tests using the same tests material. There is no discernible pattern in the dependence of k^ on t . the variation of the former with the latter apparently is so slight . h that it is masked by the experimental scatter. For given values of — e 102 and 1 , k. ip somewhat larger for the geometrically regular blocks than t X for the irregular ones, This is not surprising, since the parallelpiped floes can become aligned end to end so as to support relatively large compressive loads-,Probably even greater importance in this regard is that the larger interparticle contact area per unit volume in the case of the regular blocks makes possible the development of larger cohesive bonds, which are discussed in the following section. For the constant carriage velocities represented in figures 31 and 32, k^ initially increases with cover length, then decreases after at taining a maximum at I — of 30 to U0. The initial increase of k^ with — is due, no doubt, to the lower strain rates occurring in the longer t covers. As Is discussed in the following section, the strength of the cohesive bonds diminishes as the strain rate is increased. This also seems to be the case in figure 33, where k^ diminishes monotonically with increasing carriage velocity and hence with strain rate. In the experiments with the longer covers, it was observed that the part of the cover far away from the driving plate was largely inactive, and did not become stressed during loading of the cover. Instead, the load applied by the driving plate was transferred through shear to the walls of the tank. Therefore, the strain rate in the cover did not decrease continu ously with increasing cover length after a certain point, but instead reached an asymptotic value. This consideration explains the decrease of k with increasing S- for the longer covers. It also points up the fact x . t that the normal stress was not constant throughout the cover, but de creased with distance from the driving plate. The failures in the cover 103 Table 2 Summary of experimental results from compression tests to determine k^ for parallelepiped ice blocks. Expt . L/t k t . / d ti/t t t X P i e i b d L Wo. (cms) (cms) P-1 60 63 0,27 0.75 0,29 0,25 8 2 7 7 o P-2 50 61+ 0.21 0.75 0.29 0.22 9 2 7 7 o P-3 h3 90 0.37 0.75 0.29 0.18 11 2 7 7 A P-7 73 81 0.29 0.75 0.29 0.18 11 2 7 7 A 30 o .7 i P-5 95 0.75 0.29 0.17 12 2 7 7 O P-6 30 97 0,51 0.75 0.29 0.22 9 2 7 7 O P-7 30 85 0.77 0.75 0.29 0.33 6 2 7 7 O P-8 1+7 10 0 .1 7 ’ 0 .7 l 0.29 0.22 9 2 7 7 □ H H P-9 10 38 O 0 .7 l 0.29 0.17 12 2 7 7 □ P-10 10 72 0.16 o. 7i 0.29 0.13 15 2 7 7 □ p-ii 10 7 l 0.17 0 .7 l 0.29 0.11 18 2 7 7 □ P-12 20 68 0.18 0.72 0.29 0.22 9 2 7 7 0 P-13 20 62 0.71 0.72 0.29 0.17 12 2 7 7 0 P-17 20 58 0.29 0.72 0.29 0.13 15 2 7 7 0 P-15 10 12 0.21 0.75 0.17 0.22 9 2 17 17 n P -l6 10 12 0.17 0.75 0.17 0.17 12 2 17 17 m P-17 10 17 0.27 0.75 0.17 0.13 15 2 lb 17 m P-18 10 15 0.21 0.75 0.17 0.11 1.8 2 lb 17 ■ P-19 10 - 19 0.32 0.75 0 .l7 0.10 21 2 17 17 ■ P-20 30 18 0.17 0.75 o.i7 0.13 15 2 17 17 • P-21 30 26 0.2 6 0.75 0.17 0.17 12 2 17 17 • XOl+ (Table 2. continued) P-22 30 2 k 0,26 0,1+5 0,ll+ 0.22 9 2 ll+ 11+ • P-23 10 10 U 0.13 0.37 1.00 0.39 '9 3.5 3.5 3.5 9 ro 0 P-2U 10 129 O 0.37 1.00 0,29 12 3.5 3.5 3.5 9 P-25 10 130 0.20 0.37 1.00 0.23 15 3.5 3.5 3.5 9 P-26 10 119 0.11 0.37 1,00 0.19 18 3.5 3.5 3.5 9 P-27 10 101 0.17 0.37 1.00 0.17 21 3.5 3.5 3.5 9 P-2 8 30 179 0.20 0.37 1.00 0.23 15 3.5 3.5 3.5 A P-29 30 196 0.19 0.37 1.00 0.29 12 3.5 3.5 3.5 A P-30 30 182 0.21 0.37 1.00 0.39 9 3.5 3.5 3.5 A Figure 31 Experimentally determined longitudinal stress coefficient, k^, for parallelepiped ice "blocks. ■106- Table 3 Summary of experimental results from compression tests to determine k for blocks with random plan-form, x L/t k t /d t i/t t Expt. X (SD!te P i' e No. (cms) R-l 10 1 5 0,17 0,1*9 0,20 0.21 9 O R-2 10 ll+ o.i6 0,1+9 0.20 0.16 12 O R-3 10 18 0.11 0.1+9 0.20 0.13 15 O t . 2.0 cm R-1+ 10 18 0.09 0.1+9 0.20 0.11 18 O 1 f> = R-5 10 2k 0.17 0.1+9 0.20 0.09 21 O = 11.1* cm R-6 10 22 0.18 0,1+9 0.20 0.08 2l+ 3.0 cm O < R-7 30 3k 0.30 0.1+9 0.20 0.13 15 # a == 15.9 cm * R-8 30 38 0.20 = 1 0.25 0,1+9 0.16 12 # ad 1+.3 cm 30 O.lH 0.20 0.21 d = 9.7 cm R-9 51 0.1+9 9 # e R-10 20 26 0.20 0,1+9 0.20 0.11 18 0 R-ll 20 27 0.21+ 0.1+9 0.20 0.13 15 © R-12 20 37 0,21 0.1+9 0,20 0,16 12 e R-13 20 37 ' o. 31+ 0.1+9 0.20 0,21 9 0 R-ll+ 20 50 0.11+ 0.1+9 0.25 0.10 2l+ a R-15 20 6l 0.28 0.1+9 0,25 0.12 21 a R-l6 20 30 . + 0.1+9 0,25 0.11+ l8 a = 2.5 cm 0 31 *i R-17 20 26 0.29 0.1+9 0.25 0,17 15 a b =■■ 12.3 cm R-18 20 20 0,21 12 =3.3 cm 0.30 0.1+9 0,25 a 0b R-19 20 36 0.30 0.1+9 0.25 0.28 9 a d == 18.6 cm R-20 30 0.21 = 1+.8 cm 60 0,1+9 0.25 0.17 15 0 °d R-21 30 50 0.21 12 d = 10.2 cm 0.27 0.1+9 0.25 0 A 107 (Table 3. continued) R-22 30 37 0,23 0,1+9 0,25 0.28 9 a R-23 i+o k b 0.1+7 0.1+9 0,25 0.21 12 is R-21+ 1+0 b 6 0.25 0,1+9 0.28 9 0.25 is \f R-25 20 15 0.22 0.1+9 0.20 0.11 2l+ V R- 2 6 2 0 13 0 . 2 0 0 .U8 0.20 0.12 21 V R-27 2 0 17 0.13 0 ,1+8 0.20 0.11+ 18 V R-2.8 2 0 2 0 0 . 1 6 0 .1+8 0.20 t. = 2 .6 cm 0.17 15 V l R-29 20 20 . 0 . 2 0 0 .1+8 0.20 0.22 12 V b = 11+.9 cm R-30 30 2 2 O . l H 0 .1+8 0,20 0 cm 0.17 15 ▼ CTb = 1+‘ R-31 30 2 8 0.09 0 .1+8 0.20 0.22 12 ▼ d = 20.6 cm R-32 30 32 0 . 1 8 0 .1+8 0.20 0.29 9 0 cm T ad = 5* 1+0 30 0 . 2 0 0 .1+8 0.20 0.22 12 d = 13; . 0 cm R-33 W e R-31+ ko 29 0 .2 1+ 0 .1+8 0.20 0.29 9 V 1 Standard deviation of d Standard deviation of b 108 Figure 32 Experimentally determined longitudinal stress coefficient, kx , for ice blocks with random plan-form. --- ,-----!-----,-----1----- r---- 1---- (-----1---- t——|-----1---- 140 O _L Symb. L * Mean 130 • 4.88 m 12.2cm 1.5cm 40 ------' O 3.66 m 18.3 cm 1,5cm 20 — — 120 no . b -10. k cm t d = 16.5 cm ¡00 [ dp =8. 5 cm 1 ■ 90 I ! All other kx experiments were run 80 70 60 50 4 0 3 0 20 10 0 i i i i i i i i______j ______i — 0.02 0.04 0.06 0.08 0.10 Carriage Velocity (cm/sec) ■e 33 Effect of strain rate on for random plan-form ice blocks 110 were observed generally to occur at sections close to the driving plate, and hence the force measured with the dynamometer appears to be the correct one to use in calculating the compressive strength of the covers. D. The Formation of Cohesive Bonds, and Dimensional Analysis for fc x From the results of the compression tests conducted on floating, fragmented ice covers it was concluded that the rate of deformation is one of the dominant :.:tors influencing the compressive strength of embacles. This phenomenon is attributed to cohesion and the manner in which it develops within the ice cover. According to Merino (197^-) * cohesion between ice fragments occurs when the water film which surrounds the particles freezes, thus forming a natural weld between ice fragments. The nature of the weld is directly related to crystallisation of the water film, and can be best explained, following Merino (197^)* hy use of the phase diagram (see figure 3^) defined by two independent para meters: pressure and temperature. Let the initial conditions of the contact surface between two fragments be a pressure of one atmosphere and a temperature very slightly below 0°C. Note that there will be a temperature gradient in the cubes, from about 0 C at the surface to some slightly lower temperature at their interiors. In figure let the points s^ and correspond to the contact surface and the interiors of the cubes, respectively, before the cover is loaded and the fragments are pressed against each other. Application of a compressive stress, say of , across the contact surface will produce a Pressure Figure 3^ Phase diagram for water. 112 othermal changes at the interface, and hence the point s will move c t0 Sca- with some ice being melted. However, the heat of fusion taken up m this melting process reduces the temperature of the liquid pro duced, causing the phase point to shift to or The liquid water is then supercooled and consequently will refreeze with the passage of tune, and the phase condition will change from that of point o± to that the ice line, s_p. The energy released by the temperature drop from Sca fc° sf ls fca; er 1JP by the heat of fusion of the water which remains liquid at the interface. At the interior points of the ice fragments, on the other hand, the phase diagram point moves from s to s for i ia ’ example, and no melting occurs. The important point to note is that the application of pressure can produce melting and refreezing at particle interfaces, with the accompanying production of cohesive welds. Lower rates of defoliation allow more time for refreezing to develop. This explains the increase of compressive strength with decreasing rate of deformation illustrated in figure 33. At higher rates of deformation, on the other hand, the cohesive bonds do not have time to develop as completely and therefore the cover strength is lower. It seems reason able to expect that there is a characteristic time, Tc , associated with formation those cohesive bonds. From the foregoing explanation one might expect that larger applied pressures would produce weaker bonds, since with increasing interfacial stress the temperature drop from s ^ to sf increases, leaving more residual liquid water and hence presumably a weaker bond. However, in a floating, fragmented cover the ensemble of particles will be compacted 113 as the applied effective compressive stress is increased, resulting in a greater area of particle—to—particle contact, and hence a stronger cover. This notion "will be developed further. In the following part of this section, the state of stress in the cover will be examined using Mohr’s circle, and a dimensional analysis will be developed to justify the format used in figures 31 and 32 for the presentation of results. The Mohr circle for a floating, fragmen ted ice cover is shown In figure 35* The vertical,effective stress at any point is a ^ 9 and a ^ is the applied horizontal^compressive,effective stress at that point. The failure line is shown as C-D; the mechanical friction angle of the material is 0, and Ch indicates the cohesive strength resulting from freezing together of the particles. It is assumed that passive failure at a point will occur when a T is increased x until the circle becomes tangent to the failure line. Similarly active failure occurs if a^ is decreased until another tangent point, denoted by E occurs. The goal of the ensuing calculation is to express the pas sive value of as a function of the other variables appearing in figure 35* The vertical distance AB can be written either from the geometry of the circle as (118) 2 X Figure 35 Mohr circle for a fragmented, floating ice cover. -p- 115 or from the equation of the failure line, CD, and the circle geometry as (119 ) Equating the expression for AB given by (ll8) and (119) and then solving for o' yields X (120) which is the desired relation for a !. Now when the cover is undergoing x compressive failure, (120) will be fulfilled at every point on the vertical plane being loaded. Therefore, af and a may be replaced by x z their average values, a and a , respectively, over the cover thickness. x z In the discussion of the development of cohesive bonds it was noted that the cohesive strength can be expected to increase with applied compres sive stress,.a , because of the ice the accompanying increase in interparticle contact area. It will be assumed therefore that (121) Substituting (l2l) into (120) and replacing o' and 0 by their average X z values, 0 and 0 , respectively, results in X z 116 -4 r n jj 1 -4, (122) ) - 1 _ •stwQi) where is given by (^3). Since for a given granular medium and az degree of compaction, 0 may be regarded as constant, (122) takes the form (123) where -* r I (12k) l — Whitman, 1969 ) but modified in order to account for the cohesion between blocks. The quantity k_^ will depend on the rate of deformation, the geometry of the test setup, and the properties of the test material. An expression for the applied stress at active failure, corresponding to the intersection at point E in figure 35 * can be obtained from a similar analysis. However, this type of failure is not of particular interest here. Dimensional analysis now will be used to express k as a function of those quantities upon which it depends. The general functional form adopted for k is x ^ ^ t > h, Uc ) ^ (1 2 5 ) 11.7 vhere is the characteristic time for cohesive bond formation and the other quantities are as used previously. Equation 125 can be written nondimensionally as which is given quantitative expression in figures 31 and 32. The third quantity on the right of (126) is not included in the figures, because U U T T is unknown; instead the value of —— [observe that — ~—— is a linear comb— c d. d ^ e e ination of the other quantities in (126)] is noted for each set of data. E. Merino’s Determination of C and C. o l Merino (197*0 investigated the shear strength of floating fragmented ice covers in the Institute’s Ice Force Facility described in detail in the previous section. The experimental unit installed in the Ice Force Facility for his tests consisted of a stationary and a moving part. The former was made up of two rectangular half-compartments which contained the test specimen, and a channel to receive the forward plate of the lat ter . The I-shaped moving part, which was connected directly to the car riage, had a flange which separated the two identical test compartments. Two walls following the driving plate prevented ice fragments from es caping from the test compartments after some displacements of the moving part had occurred. The force on the driving plate was measured with the dynamometer used in the investigation of k and illustrated in figure 28. The test materials consisted of wooden blocks boiled in oil to reduce 118 their water uptake, and two different freshwater ice materials: ice cubes and random plan-form, constant thickness floes. Only the results of the experiments conducted with ice are considered here. Figures 36 show the measurement variation of shear strength, t , with cover thickness, t, and shearing velocity. It is seen that t increases more or less linearly with t for each carriage velocity, 'U , and decreases with increasing U ■ until a minimum value of about 2 psf is reached. This suggests that Ch is about 2 psf for the ice tested. In figure 37 9 it is seen that at small deformation rates, the normalized shear strength, T ! 3-i-nearly proportional to de/ Uc . Superposition of the con stant t at high shear velocities with the linearly decreasing relation observed at low displacement rates and shown in figure 37 5 leads to V = 4 - 2 . 0 (ft - lb - sec) (127a) Uc. which may be written T ~ - . C.L - 4- c (ft - lb - sec) (127b) Vet where = O.Jd^/U^ and Ch = 2.0 psf. Equation 127a is depicted in figure 37. For Merinofs experiments, y ranged from 1.20 to 1.1*7» and t e C. was varied from 0.3 ft to 0.9 ft. Thus for C. = 2.0 psf, — 7— - was 1.51 1 * y t e to 5.56. Figure 37 and (127b) then suggest that Cq varies from about 2 to 100 for the conditions in Merino’s tests. In the numerical examples of Chapters II and III, however, Cq was varied only from 1 to 8 . 119 L i f t ) V *10 3 L (f t ) 2 0 ( it/sec) 1.2 o 0.82 # i t 1.64 3.28 A ▲ 5 36 yf t - - V 9.84 « i______I .... I v l______I_____ I______L____ _ 0 *2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 t (ft) (a)' Ice cutes Figure 3 6 Unit shear strength versus cover thickness 120 (psf ) (b) Random plan form ice blocks (Figure 36 continued) 7 igure 37 Shear stress predictor 122 Chapter V SIMILARY AND CONCLUSIONS The mathematical model developed in the preceding chapters has , of necessity * treated ice jamming as a well ordered phenomenon. Thus it seems in order to recall at this point, as the thesis draws to a closer that natural jams are, at best, untidy, disordered phenomena, as figure 38 attests, and the usults of any theoretical analysis of embacles can only be expected to predict the gross or general features of their be havior. However, this often is the type of information required by the engineers. One is, of course, well advised to solve ice jam problems whenever possible by preventing initiation and formation of the jam. Once an embacle has formed, the river engineer often must resort to heroic measures, such as blasting (see figure 39 ) 5 ice breaking, dusting, etc. The state of art for preventing and combatting ice jams has been reviewed by Moor and Watson (l97l) and Sinotin (1973). In reality, when confronted with a massive jam and the high river stages it produces, the engineer often has little recourse but to spend his days raising the levees and his nights praying for a warm, early spring. The analytical model developed herein is based on four types of relations: the static force equilibrium of floating, fragmented ice; continuity relations for the liquid and frozen water; the unsteady, non- uniform momentum relation for the flow beneath the ice; and stress relations concerning the compressive and shear strengths of floating, Figure 38 Photo of an ice jam that formed on the Mississippi River, Pool 22, in January 1969. H ro LO Figure 39 Photo of blasting of an ice jam on the Iowa River just upstream from its confluence with the Mississippi River (February 1973). ro -P" 125 fragmented ice, and the shear stress applied to the coyer by.the'flow beneath it. A computer program was developed which integrates the governing equations for the case in which the embâcle has achieved quasi- steady conditions (i.e., has achieved equilibrium conditions at its downstream and is propagating upstream with constant velocity) to obtain a relatively complete description of the behaviour of jams for this con dition. Estimates were also obtained for the rate of jam evolution during the early stages of its development (before it reaches quasi steady state). Ihe principal uncertainties in the analysis surround the coeffi cients k^, CL, Cq , and f^, arising in the strength and stress relations. These should be regarded at present as free parameters which may be adjusted in order to bring theoretically predicted and observed embâcle behaviours into conformity. Field observations should also be used to quantify these coefficients further. The next research steps in the study of ice jams should be directed toward elucidation of the nature of the shear and compressive strengths of ice jams, and quantification of the aforementioned coefficients. Further studies of laboratory ice jams should be carried out to verify the analysis and quantify the strength/stress coefficients. It should be pointed out that the heavy dependence of ice-jam behavior on the strength properties of ice cover, especially on its ability to form cohesive bonds between fragments, suggests that model studies of ice jams should be conducted with real ice, since artificial materials, such as wax, polyethylene, or wood, generally do not exhibit the cohesive behavio 126 to the extent ice does. Finally, it must be emphasized again that ice jams pose very complicated engineering problems 'which do not lend themselves to simple, all encompassing explanations of their behavior. 127 references Allen, J, (A91+2). 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(1972) "Stability of Floating Ice Blocks", Journal of the Hydraulics Division, ASCE, Vol. 93, No. HY12, Proc. Paper 9^-18, pp. 2117-2133 Uzuner, M. S. (197*0 "The Composite Roughness of Ice-Covered Streams", (Submitted to Intern. Assn, for Hydr. Res. for publication) 130 Appendix A <3h SOLUTION TECHNIQUE FOR THE DIFFERENTIAL EQUATION (-rr^) = f(t ,h ), CLo o O O AND CALCULATION OF COVER-THICKNESS PROFILES OVER THE NONUNIFORM REACH OF A JAM Determination of the cover-thickness and flow-depth profiles along an ice jam requires simultaneous solution of the force-balance equation for the jam, (5l), and the momentum equation for the flow under the ice cover, (Tl). After introducing the bottom shear stress, t ^, given by (8l) into the force balance equation, (51), and rearranging terms, there results