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.