Application of the minimum rate of energy dissipation theory to gradually varied flow in gravel-bed streams by Paul Arthur Janke A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Civil Engineering Montana State University © Copyright by Paul Arthur Janke (1986) Abstract: Predictability of the magnitude of stream changes resulting from man-made works can be improved. This study develops the MINRED method. The MINRED method utilizes (a) the minimum rate of energy dissipation theory, (b) a combination of Shield's diagram and Raudkivi's graphical relation, and (c) the direct step method of determining the gradually varied flow profile. This method calculates equilibrium bankfull values of the dependent variables (stream depth, width, and sediment size) in reaches of gravel-bed alluvial streams having gradually varied flow and low sediment discharge. The MINRED method's accuracy is evaluated through application of the method to two Alaskan streams, Copper River tributary and Brooks Creek tributary. Results of this application indicate poor reliability when compared with measured values and trends in values predicted in the literature. This study supplies a rationale for the low reliability and offers suggestions for future work.  APPLICATION OF THE MINIMUM RATE OF ENERGY DISSIPATION THEORY TO GRADUALLY VARIED FLOW IN GRAVEL-BED STREAMS by Paul Arthur Janke A thesis submitted in partial fulfillment of the requirements for the degree O t Doctor of Philosophy in Civil Engineering MONTANA STATE UNIVERSITY Bozeman, Montana December 1986 AS?# a sus# 11 APPROVAL of a thesis submitted by Paul Arthur Janke This thesis has been read by each member of the thesis committee and has been found to be satisfactory regarding content, English usage, format, citations, bibliographic style, and consistency, and is ready for submission to the College of Graduate Studies. - ( - Q %3, Date Approved for the Major Department Date Head, Major Department Approved for the College of Graduate Studies Graduatfe DeanDate ill STATEMENT OF PERMISSION TO USE In presenting this thesis in partial fulfillment of the requirements for a doctoral degree at Montana State University, I agree that the Library shall make it available to borrowers under rules of the Library. I further agree that copying of this thesis is allowable only for scholarly purposes, consistent with "fair use" as prescribed in the U.S. Copyright Law. Requests for extensive copying or reproduction of this thesis should be referred to University Mocrofilms International, 300 North Zeeb Road, Ann Arbor, Michigan 48106, to whom I have granted "the exclusive right to reproduce and distribute copies of the dissertation in and from microfilm and the right to reproduce and distribute by abstract in any format." Signature Date____ I £> — 2 { — $ /p VACKNOWLEDGMENTS I am deeply indebted to my committee chairman, Mr. Theodore Williams, for his guidance and assistance at critical times during the development of this document. Thanks is also due to the other engineering committee members for their comments and suggestions. These are Dr. Richard Brustkern, Dr. Al Cunningham, and Dr. William Hunt. I want to thank Greg Shepherd and Dan Burkwit for assisting with the data collection, Bruce Arndt for providing surveying equipment, and Dale Walberg for offering support during the development of the computer program. I also want to thank my parents, Paul and Shirley Janke, for their support during this investigation. I especially want to thank my wife, Jill Real Janke, for assisting with the data collection, typing the manuscript and providing invaluable support during the entire process. vi TABLE OF CONTENTS Page LIST OF TABLES..... X LIST OF FIGURES..................... xii ABSTRACT.......... ......... ................ xiii 1. INTRODUCTION.................... ....... I Establishment of the Problem........... I Minimum Rate of Energy Dissipation . . Theory...... ........... .......... . 3 Objectives...................... 4 2. LITERATURE REVIEW........................ 6 Adjustment Time Periods Independent Variables.. Water Discharge.... Sediment Discharge.................. 10 Dependent Variables.................... 12 Functional Relationships.......... 13 Cross Sectional Shape............... 15 Depth, Width, and Velocity.......... 15 Parameter Value Trends............... . . 16 Minimum Rate of Energy Dissipation Theory................................ 16 Statement for Alluvial Streams....... 18 Application. . . ..................... 20 Summary........................ 22 3. FIELD INVESTIGATION AND MEASURED VALUES... 25 Data Collection........................ 27 Independent Variables . . . . ..... 27 Sediment Density................. 28 Water Discharge.............. .•••• 28 Man-Made Works . .... 28 Dependent Variables.............. ; • 29 Method and Equipment.. .......... 29 . Mean Sediment Size........ 30 to t- CO via Stream Depth.................... 32 Stream Width..................... 32 Thalweg Slope.................. . . 32 Presentation and Discussion of Measured Values........................ 33 Average Measured Values............. 33 Subreaches Not Altered by the Culvert........................ 36 Subreaches Altered by the Culvert........... 37 Pre-Culvert...................... 37 Post-CulvSrt. . . ......... 37 Conclusion................ 39 Gradually Varied Flow.............. . 39 Upstream. . ................. 39 Downstream....................... 40 4. MINRED METHOD................ 43 Development............................ 43 General............ 45 Preliminary........... 46 Direct Step Method........ 47 Shields Diagram and Raudkivi1s Graphical Relation................. 49 Shields Diagram.................. 50 Raudkivi1s Graphical Relation..... 54 Minimum Rate of Energy Dissipation Theory.................. 59 Description............................ 63 D Loop................. 65 DS Loop............ 66 Simultaneous Solution of D Loop and DS Loop....................... 66 B Loop.............................. 67 Summary.................. 68 5. CALCULATED VALUES. . ..................... 71 Application................. 71 Presentation and Discussion of Calculated Values . ..................... 72 Slope of the Energy Grade Line. . . . ......... 72 Average Calculated Values........... 73 Summary....... 75 TABLE OF CONTENTS— Continued Page viii 6 . RESULTS AND DISCUSSION.................. 76 Parameter Changes Between Subreaches. . . . ......................... 77 Measured Value Changes. ................ 78 Calculated Value Changes............... 78 Literature Predictions.............. 80 Results of Comparisons . ................ 80 Parameter Changes At Subreaches........... 81 Global Minimum Calculated.......... . 81 Literature Predictions.............. 81 Ideal Ratios.................... 82 Actual Ratios....................... 82 Results of Comparisons............ 84 Global Minimum Not Calculated........ 84 Ideal Ratios.................. 85 Actual Ratios.................... 85 Results of Comparisons.............. 86 Summary........................ 86 7. SUMMARY, CONCLUSIONS, AND SUGGESTIONS FOR FUTURE WORK....................... 88 Summary and Conclusions................... 88 Assumptions and Approximations......... 89 Assumptions.................... 89 Approximations................. 90 Suggestions for Future Work............ 92 REFERENCES CITED. . ....................... 94 APPENDICES......... 98 Appendix A - References Consulted but not Cited....... 99 Appendix B - Definition of Symbols...................... 103 Appendix C - Development of the Minimum Rate of Energy Dissipation Theory for Alluvial Streams.............. 109 Equation of Continuity. . .............. Ill Stress Tensor................. . .... Ill Equation of Motion. . . ................. 113 Rate of Energy Dissipation.......... . 115 Velocity Vector Satisfying the Equation of Motion............. 115 TABLE OF CONTENTS— Continued Page ix TABLE OF CONTENTS — Continued Velocity Vector Not Satisfying the Equation of Motion........................ 120 Velocity Variation Vector........................ 122 Hypothetical Velocity Vector................. 123 Statement of the Theory.......... 129 Appendix D - Bankfull Discharge........ 132 Appendix E - Measured Values................ 138 Appendix F - Flow Conditions Near Culverts......... 151 Appendix G - Summary of the MINRED Method................................. 158 Preliminary...................... ... 159 D Loop....................... 159 DS Loop................... 161 B Loop................. 162 Appendix H - Equations for Shields Diagram and Raudkivi1s Graphical Relation........................ 163 Appendix I - Calculated Values. . ....... 169 Page XLIST OF TABLES Page 1. Functional relationships for dependent variables...................... 14 2. Average measured data.................... 35 3. Bankfull elevations at cross section OU............................... 38 4. Average calculated data. . ................ 74 5. Direction of average parameter value changes from unaltered to altered subreaches................*.......... ... 79 6 . Ratios of average measured values to average calculated values for subreaches altered by the culvert........ 83 7. Summary of comparisons that agree........ 87 8 . Annual maximum discharges for Copper River tributary.......................... 135 9. Annual maximum discharges for Brooks Creek tributary.................. 136 10. Bankfull discharge calculation........... 137 11. Culvert data............................ 139 12. Copper River tributary, downstream,. measured data............ 140 13. Copper River tributary, upstream, measured data............................ 141 14. Brooks Creek tributary, downstream, measured data.................. 142 15. Brooks Creek tributary, upstream, measured data...................... 143 Continued 16. Copper River tributary, bankfulI Froude number............................ 148 17. Brooks Creek tributary, bankfulI Froude number............................ 149 18. Cross sections locating break in measured data................ 150 19. Estimated pre-culvert bank elevation, cross section OU......................... 153 20. Culvert control location at bankfulI conditions............................... 154 21. Estimated post-culvert bank elevation, cross section OU......................... 157 22. Copper River tributary, downstream, calculated data.......... 170 23. Copper River tributary, upstream, calculated data.......................... .171 24. Brooks Creek tributary, downstream, calculated data.... . ..................... 172 25. Brooks Creek tributary, upstream, calculated data...... ......... 173 xi LIST OF TABLES — Page xii LIST OF FIGURES Page 1. Location map........................ 26 2. Typical stream with culvert creating gradually varied flow upstream...... 41 3. Typical stream with culvert creating gradually varied flow downstream.... 42 4. Shields diagram........ 51 5. Raudkivi' s graphical relation....... 56 6 . Flow chart of the MINRED method..... 64 7. Notation for stresses............... 115 8 . Forces per unit area on a typical water volume between two cross sections in an alluvial stream...... 117 9. Copper River tributary, downstream, measured data....................... 144 10. Copper River tributary, upstream, measured data....................... 145 11. Brooks Creek tributary, downstream, measured data.................. 146 12. Brooks Creek tributary, upstream, measured data................... 147 13. Schematic location of downstream cross sections...................... . 170 14. Schematic location of upstream cross sections.......... 171 TABLE OF CONTENTS — Continued Velocity Vector Not Satisfying the Equation of Motion......... 119 Velocity Variation Vector........................ 121 Hypothetical Velocity Vector........................ 122 Statement of the Theory.. .... 128 Appendix D - Bankfull Discharge........ 132 Appendix E - Measured Values........... 138 . Appendix F - Flow Conditions Near Culverts.............. . . ........... 151 Appendix G - Summary of the MINRED Method........................... 158 Preliminary............ 159 D Loop................... 159 DS Loop. .................... 161 B Loop............ 162 Appendix H - Equations for Shields Diagram and Raudkivi's Graphical Relation............................... 163 Appendix I - Calculated Values......... 169 ix Page XLIST OF TABLES Page 1. Functional Relationships for Dependent Variables...................... 14 2. Average Measured Data.................... 35 3. Bankfull Elevations at Cross Section OU..................... 38 4. Average Calculated Data.............. 74 5. Direction of Average Parameter Value Changes from Unaltered To Altered Subreaches........ 79 6. Ratios of Average Measured Values to Average Calculated Values for Subreaches Altered by the Culvert...... 83 7. Summary of Comparisons That Agree........ 87 8 . Annual Maximum Discharges for Copper River Tributary.......................... 135 9. Annual Maximum Discharges for Brooks Creek Tributary........................ 136 10. Bankfull Discharge Calculation........... 137 11. Culvert Data........ 139 12. Copper River Tributary, Downstream, Measured Data........... 140 13. Copper River Tributary, Upstream, Measured Data.............. 141 14. Brooks Creek Tributary, Downstream, Measured Data................... 142 15. Brooks Creek Tributary, Upstream, Measured Data................. *......... 143 16. Copper River Tributary, Bankfull Froude Number......................... . . 148 17. Brooks Creek Tributary, Bankfull Froude Number...................... 149 18. Cross Sections Locating Break in Measured Data. . ................... 150 19. Estimated Pre-Culvert Bank Elevation, Cross Section OU... . ..................... 153 20. Culvert Control Location at Bankfull Conditions. .............................. 154 21. Estimated Post-Culvert Bank Elevation, Cross Section OU................... 157 22. Copper River Tributary, Downstream, Calculated Data.......................... 170 23. Copper River Tributary, Upstream, Calculated Data.................. 171 24. Brooks Creek Tributary, Downstream, Calculated Data.......................... 172 25. Brooks Creek Tributary, Upstream, Calculated Data....................... 173 xi LIST OF TABLES — Continued Page xii LIST OF FIGURES Page 1. Location map.............. 26 2. Typical stream with culvert creating gradually varied flow upstream...... 41 3. Typical stream with culvert creating gradually varied flow downstream.... 42 4. Shields Diagram...... 51 5. Raudkivi 1 s Graphical Relation....... 56 6 . Flow chart of the MINRED method..... 64 7. Notation for stresses........ 114 8 . Forces per unit area on a typical water volume between two cross sections in an alluvial stream...... 116 9. Copper River tributary, downstream, measured data............ 144 10. Copper River tributary, upstream, measured data....................... 145 11. Brooks Creek tributary, downstream, measured data....................... 146 12. Brooks Creek tributary, upstream, measured data................... 147 13. Schematic location of downstream cross sections................... ... 170 14. Schematic location of upstream cross sections............. 171 xiii ABSTRACT Predictability of the magnitude of stream changes resulting from man-made works can be improved. This Study develops the MINRED method. The MINRED method utilizes (a) the minimum rate of energy dissipation theory, (b) a combination of Shield's diagram and Raudkivi’s graphical relation, and (c) the direct step method of determining the gradually varied flow profile. This method calculates equilibrium bankfulI values of the dependent variables (stream depth, width, and sediment size) in reaches of gravel-bed alluvial streams having gradually varied flow and low sediment discharge. The MINRED method's accuracy is evaluated through application of the method to two Alaskan streams. Copper River tributary and Brooks Creek tributary. Results of this application indicate poor reliability when compared with measured values and trends in values predicted in the literature. This study supplies a rationale for the low reliability and offers suggestions for future work. — — ' - I I i ,r I - - - - - - - - - - - - 1 m u — • ! _ I ! , 1 I I » ■ i s [ I I CHAPTER I INTRODUCTION Alluvial streams are self-adjusting. Through the processes of scour and deposition, the physical con­ figuration and flow characteristics in a reach of an alluvial stream can be modified by changes in conditions imposed at a particular location. These imposed conditions are the independent variables. A stream's physical configuration and flow characteristics are its. dependent variables. When values of the independent variables change, it is desirable to estimate values of the dependent variables in equilibrium with the modified values of the independent variables. Establishment of the Problem l Hey (1975b) considers it important to know what impacts imposed human activity will have on.alluvial channels. With this information, appropriate measures can be taken to sustain channel stability, to prevent destruction of riparian land, to maintain channel flood flow capacity, and to retain recreational appeal. Hyra (1978) expounds that instream recreational activities have specific water velocity and depth requirements. Further, Bovee (1978) 2emphasizes that some fish species have specific instream habitat requirements for water velocity, depth, and mean sediment size. Regardless of the degree of equilibrium attained by a stream, localized human activity may produce major changes in values of a stream's dependent variables both near the activity and throughout an entire reach (Simons, 1979). Conversely, alterations in values of a stream's dependent variables may impact human activities. Knowing that a specific human activity will be imposed on a stream, it is desirable to estimate values of the dependent variables that will be in equilibrium with the modified values of the independent variables. These estimated values can be compared to the existing values to predict impacts from the human activity on (a) channel flood flow capacity; (b) riparian land use; and (c) various instream activities such as aquatic habitat, recreation, and commerce. With this information, appropriate measures could be taken to minimize negative impacts. For example, estimates could indicate that a specific human activity would modify the stream mean sediment size so as to make it unusable to an important fish species during spawning. This information could be used to modify the human activity to minimize this negative impact. Trends in values of a stream's dependent variables due to a specific human activity can be predicted with a high degree of reliability. However, methods currently available 3to water resources engineers are inadequate to predict the magnitude of these changes. Minimum Rate of Energy Dissipation Theory By determining the values of a stream's dependent variables before and after a human activity is imposed, the magnitude of changes in these values can be predicted. Values of a stream's dependent variables can be calculated by the simultaneous solution of relationships which link the independent and dependent variables. This solution method requires that one such relationship is used for each dependent variable. However, alluvial streams have more dependent variables than relationships available to link them to independent variables. Also, many human activities imposed on streams create a considerable length of gradually varied flow. The availability of another relationship usable in alluvial streams with gradually varied flow increases the ability of engineers to predict the magnitude of stream changes resulting from human activity. The minimum rate of energy dissipation theory has recently been proposed for use with alluvial streams. This theory states that, provided certain conditions are met, equilibrium values of the dependent variables are adjusted so that alluvial streams flow with the minimum rate of energy dissipation that is compatible with the independent variable values. 4Alluvial streams with no imposed human activity experience conditions approaching uniform fIowi Although not correct, this.study considers such streams to have uniform flow conditions. The terms gradually varied flow and rapidly varied flow are used in the traditional sense. Recently, investigators have used the minimum rate of energy dissipation theory as a relationship to link the independent and dependent variables of alluvial streams with no imposed human activity. Results of the simultaneous solution of this with other relationships indicate values of the calculated dependent variables agree with published data measured in flumes and streams which have low sediment discharges and uniform flow. Objectives The first objective is to develop a method that uses the minimum rate of energy dissipation theory to calculate , equilibrium bankfulI values of the dependent variables in reaches of gravel-bed alluvial streams having gradually varied flow. The second objective is to answer the question: Given a gravel-bed alluvial stream which satisfies certain conditions, with what reliability can the above method be used to calculate equilibrium bankfulI values of the dependent variables in reaches having gradually varied flow imposed by a highway culvert? The first objective will be addressed by developing a 5method which solves three relationships to calculate equilibrium bankfulI values of the dependent variables in reaches of gravel-bed alluvial streams having gradually varied flow. The three relationships used by this method are the direct step method of determining the gradually varied flow profile, a combination of Shields diagram and Raudkivi1s graphical relation, and the minimum rate of energy dissipation theory. The dependent variables of gravel-bed alluvial streams are the stream width, stream depth, and mean size of the sediment that is stationary on the bed. Because this method uses the minimum rate of ' energy dissipation theory, it is named the MINRED method. Two graveI-bed streams are studied. Each has rapidly and gradually varied flow imposed by a highway culvert at the bankfulI water discharge. The second objective will be addressed by evaluating, the reliability with which the MINRED method can be used to calculate equilibrium bankfulI values of the dependent variables in the gradually varied flow reaches of the two streams studied. 6CHAPTER 2 LITERATURE REVIEW This literature review provides the background for developing the MINRED method and for evaluating the reliability with which it can be applied to reaches of gravel-bed alluvial streams having gradually varied flow. The literature review is divided into five sections discussing: the time period required for equilibrium to be developed between the independent and dependent variables, the independent variables of graveI-bed streams for the required time period, the dependent variables of graveI-bed alluvial streams for the required time period, the trends in values of the dependent variables expected after values of the independent variables are changed by the installation of a culvert, and the minimum rate of energy dissipation theory as applied to alluvial streams, including its use to estimate equilibrium values of the dependent variables. Adjustment Time Periods Independent and dependent variables of a stream are considered to be in equilibrium if values of the dependent variables are adjusted in response to values of the independent variables. Such a stream may be defined as I"graded" (Mackin, 1948), "in regime" (Blench, 1969), or "poised" (Simons, 1979). In this study, the term equilibrium is used exclusively. The time periods which can be considered in a study of stream response to independent variables are short term, long term, and very long term (Vanoni, 1975). Short term time periods are days, weeks, or months. Streams seldom if ever achieve equilibrium in the short term. This is because after values of the independent variables change, the time required for the dependent variables to change to the new equilibrium values is greater than the length of the short term time period. Therefore, the short term is too brief for studying stream equilibrium. The long term is required for an equilibrium condition to be created between the independent and dependent variables (Vanoni, 1975). Alluvial streams with 10 to 100 years to react to a change in an imposed condition can be considered to be adjusting toward equilibrium (Santos-Cayade and Simons, 1973). The very long term is a geologic time period (Vanoni, 1975). The long term time period is considered in this study. Independent Variables For the long term time period, the independent variables of alluvial streams are sediment density, fluid molecular viscosity and density, acceleration due to gravity, water discharge, and sediment discharge (Vanoni, 81975). "Man-made works" are independent variables in the very long term (Vanoni, 1975). In this study, however, man­ made works are considered to be independent variables in the long term time period. This is because man-made works are known to cause major impacts on streams in the short term and long term time periods and are believed to have only a negligible influence on stream morphology in the very long term. The number of independent variables to be considered herein is reduced below that given by Vanoni (1975). Simons and Senturk (1976) and Stelczer (1981) indicate that when low density particles of volcanic origin can be. neglected, the sediment specific gravity can be considered equal to a constant 2.65. The measured data indicate that this is true for the study streams. Therefore, the sediment density is constant. Daily and Harleman (1966) and Olson (1966) indicate the water molecular viscosity and density depend on the temperature. In this study, a constant water tempera­ ture is approximated to represent the temperature fluctua­ tions in streams. Therefore, the water molecular viscosity and density are constant. Water Discharge According to Prins and de Vries (1971) and Bray (1982b), river studies frequently assume that dependent variables are adjusted in response to a constant discharge 9rather than to the discharge fluctuations. This constant, channel forming flow rate is called the dominant discharge. Wolman and Leopold (1957), Wolman and Miller (1960), and Hey (1975a) studied natural channels and laboratory flumes. These studies conclude that compared to any other single discharge, the bankfulI flow rate transports the most sediment in the long term and hence is the dominant discharge. Hey (1975b) concludes that for alluvial channels in equilibrium, the average bankfulI dimensions that are formed in response to a constant bankfulI discharge are similar to those formed in response to discharge fluctuations. Ackers (1982) studied meander wavelength in laboratory flumes at a constant discharge concluding that the bankfulI flow rate provided the best correlation with the results than any other discharge. Therefore, the dominant discharge is the bankfulI flow rate. Consider two streams in equilibrium differing only in the imposed water discharge. The first stream has a constant water discharge and the second stream does not. The average bankfull dimensions of these two streams are similar if the constant discharge imposed on the first stream equals the bankfulI discharge of the second stream. Wolman and Leopold (1957), Wolman and Miller (1960), Hey (1975a), Williams (1978), and others have studied streams in the United States, United Kingdom, and India. These studies show the recurrence interval of the bankfulI 10 discharge to be one to two years for many streams flowing in diverse climatic and physiographic environments. These studies also show that a recurrence interval of 1.5 years on the annual maximum series is a good average, although some localities diverge greatly from this value. Dunne and Leopold (1978) indicate that for streams with limited discharge information, the best approximation of the bankfulI flow rate is the discharge with a recurrence interval of 1.5 years on the annual maximum series. This recurrence interval of the bankfulI discharge is widely accepted in the literature. Sediment Discharge The sediment discharge can be divided into the wash load and the bed material load. The wash load is that part of the sediment discharge which remains in suspension at any water flow rate and is composed of particle sizes finer than those found on the stream bed. Therefore, the wash load is determined by the upsIope supply rate and can not be predicted by the transport capacity of the stream (Shen, 1971). Wash load can be imposed on streams by glacial ( runoff, human activity, or erosion from the drainage basin. The study streams presented in Chapter 3 are not fed by glacial runoff. There is no human activity nor is there any exposed soil in the drainage basins upstream from the highway culverts in question. The latter because vegetation 11 ranging from grass to spruce and willow completely cover the ground. For these reasons, there is a limited source of wash load to the sites of interest in the study streams and hence the wash load is not considered further. Lane (1947) indicates the bed material load is that part of the sediment discharge transported as either bed load or suspended load and is composed of particle sizes which can be found on the stream bed. Shen (1971) indicates the bed material load equals the sediment transport capability of the flow. According to Leopold, Wolman, and Miller (1964), the bed load is that part of the bed material load which moves on or very near the bed. Dunne and Leopold (1978) indicate that the suspended load is that part of the bed material load which is supported by the fluid column. They also indicate that the sediment grains of the suspended load are nearly always less than 0.5 mm in diameter. Lane (1947) indicates that gravel-bed streams have a mean grain diameter larger than 2 mm. For the graveI-bed study streams, therefore, suspended load is low; hence, most sediment transported must move as bed load. Bray (1982a) indicates that initiation of motion of a gravel-bed occurs near bankfulI stage. This infers that the sediment discharge and the bed load is low in gravel-bed streams at the bankfulI water flow rate. Also, Hey (1982) indicates that tracer experiments at a variety of gravel-bed rivers in the United Kingdom found bed loads low at the 12 bankfull flow rate. Vanoni (1975) indicates that bed load samplers collect between roughly 45% and 70% of the material transported as bed load. He also compares various analytical sediment discharge relationships and concludes that predictions vary by roughly two orders of magnitude. Therefore, accurate sediment discharge information can not be determined by either direct measurements or calculations. Sediment discharge is low and can not be accurately determined in graveI-bed streams at the bankfull water flow rate. For simplicity and possibly for accuracy, the sediment discharge is assumed to be constant at the bankfull discharge for the gravel-bed study streams. Dependent Variables Values of the dependent variables are determined in response to values of the independent variables. According to Yang and Song (1979), in general, the dependent variables describe the physical configuration and flow characteristics of streams. There is some disagreement in the literature regarding which parameters are the dependent variables of alluvial streams. Vanoni (1975) indicates much of the disagreement stems from not making adequate distinctions between the behavior of streams in the short term, long term, and very long term. 13 For the long term, the dependent variables of alluvial streams are: b = stream width, d = stream depth, ^s = mean size of sediment stationary on the bed, f = Darcy-Weisbach friction factor, Pf = planform, R = hydraulic radius, S = slope of the energy grade line, V = mean velocity of flow, Wf = fall velocity of the bed material, and a = geometric standard deviation of the bed material size distribution (Vanoni, 1975). Some parameters not on Vanoni's list are considered to be dependent variables by other investigators, for example: sinuosity, wetted perimeter, and bed form height. These parameters are less important than those given by Vanoni because, for gravel-bed alluvial streams at the bankfulI discharge, these parameters are either constant, negligible, or functions of the dependent variables given by Vanoni (1975) . Functional Relationships Table I presents functional relationships between the independent and dependent variables and the references from which these were determined. These relationships reduce the 14 Table I. Functional relationships for dependent variables. Variable Function of References O Constant c,h ds Q c,h,m d Q 7 dg b,i,k,l b Q » dg b, i , k S Q/ ds, d, b b, e, f , h, I Pf Q, ds , d, b a, j R Q, ds , d d,g,h,o V Q, ds, d, b h,i,k,l Wf ds c,o f Q 7 dg, d, b n Note. References are as follows: (a) Ackers (1982), (b) Ackers and Charlton (1970), (c) Bathurst (1982), (d) Bray (1982a), (e) Bray (1982b), (f) Fahnestock (1963), (g) Hey (1978), (h) Hey (1982) , (i) Leopold and Maddock (1953), (j) Leopold and Wolman (1957), (k) Leopold et al. (1964), (I) Schumm and Khan (1972), (m) Simons (1971), (n) Simons and Richardson (1966), and (o) Vanoni (1975). 15 dependent variables of gravel-bed alluvial streams to functions of the mean sediment size that is stationary on the bed, stream depth, and stream width. Independent variableswere omitted from these relationships if their impacts on the dependent variables are constant or can not be predicted in general. Therefore, the water discharge, Q, was the only independent variable considered. Cross Sectional Shape Vanoni (1975) indicates that there have been many successful studies that assume channel cross sections are wide and rectangular. Presumably this means the width is at least several times the depth. Bray (1982b) adds that this generally results in an error of less than 3% in the value of the hydraulic radius, even for small channels near bankfulI stages. Further, Leopold et al. (1964) show that the assumption of a rectangular cross section is accurate for a wide variety of streams in the United States and Europe. Therefore, channels in which the width is at least several times the depth can be considered rectangular, and the cross section defined by the width and depth. Depth, Width, and Velocity Leopold and Maddpck (1953), Leopold et al. (1964), and others indicate stream depth, stream width, and mean velocity of flow are exponentially related to the water 16 discharge. These investigators also show the discharge exponent for each of these variables is constant for flumes, canals, and streams that are in equilibrium, with no imposed human activity, and flowing in temperate climates. Therefore, the equilibrium condition of a stream satisfying these criteria can be evaluated by comparing its discharge exponents to those given in the literature. Parameter Value Trends The independent variables of gravel-bed streams are the bankfulI water discharge and man-made works. A culvert placed in a stream is a man-made work. Values of the dependent variables respond to changes in the independent variable values. Therefore, after the’'installation of a culvert, bankfulI values of the dependent variables change until equilibrium is established. A culvert placed in a stream creates a gradually varied flow condition upstream when its discharge capacity is less than that of the stream. Schumm (1971) and Simons (1979) indicate that this causes an increase in the depth and a decrease in the slope of the energy grade line and sediment discharge. Because of this reduced sediment discharge, subreaches upstream of the culvert with gradually varied flow can not transport the larger particles and thus ex­ perience sediment deposition. The moving sediment particles are smaller than the stationary bed sediment. Thus, the 17 mean sediment size of the sediment deposited in the gradually varied flow reach during a certain water discharge is smaller than the mean sediment size of the pre-culvert bed material which was stationary during the same water discharge. Consequently, the mean sediment size on the stream bed upstream of the culvert decreases as a result of the culvert. The culvert reduces sediment discharge and the mean sediment size transported into the reach downstream of the culvert to below the pre-culvert condition. Degradation occurs downstream of the culvert increasing the sediment discharge and compensating for the reduction in the imposed sediment discharge (Simons, 1979). As a result of this degradation, stream depth increases downstream of the culvert reducing the velocity head and the rate of head loss. Therefore, the slope of the energy grade line decreases (Schumm, 1971 and Simons, 1979). This increased stream depth and reduced slope of the energy grade line decreases the sediment discharge of the stream. In time this condition will equal the imposed sediment discharge. Similar to upstream of the culvert, the mean sediment size on the stream bed downstream of the culvert decreases as a result of the reduced sediment discharge (Simons, 1979). For the above conditions, channel width will decrease if the stream transports a significant amount of fine material (Santos-Cayade and Simons, 1973). This does not I18 occur in gravel-bed streams, hence the width remains unchanged as a result of the culvert. A culvert placed in a graveI-bed stream may impose gradually varied flow reducing its sediment discharge, This causes no change in the width, an increase in the depth, and a decrease in the slope of the energy grade line and the mean sediment size stationary on the bed. This occurs both upstream and downstream of the culvert. Minimum Rate of Energy Dissipation Theory Discussion of the minimum rate of energy dissipation theory is divided into two subsections. The first subsection states the theory and the conditions where it can be applied to alluvial streams. The second subsection discusses a method which uses the theory to estimate equilibrium values of the dependent variables for alluvial streams with uniform flow. Statement for Alluvial Streams A development of the minimum rate of energy dissipation theory is summarized in Appendix C. The development considers three velocity vectors. Provided certain conditions are met, the flow described by a velocity vector satisfying the equation of motion necessarily results in a lesser rate of energy dissipation than the flow described by a velocity vector not satisfying the equation of motion. 19 Fluid flows must satisfy the equation of motion. Thus, the flow with the least rate of energy dissipation necessarily satisfies the equation of motion and the real flow situation. The energy grade line is a line connecting values of the total energy in the flow at various locations. The difference in the energy grade line at two locations is the difference in total energy (Chow, 1959). Assuming no external energy is added to or removed from the fluid, this energy difference is the amount of energy dissipated by the flow. The average rate with which energy is dissipated by the flow is determined by dividing the change of total energy by the distance between the two locations. This is equivalent to the slope of the energy grade line. Therefore, to flow with the least rate of energy dissipation is to necessarily flow with the least slope of the energy grade line. Thus, the minimum rate of energy dissipation theory, as it applies to alluvial streams, is as stated below. Provided certain conditions are met, equilibrium values of the dependent variables are adjusted so alluvial streams flow with the minimum slope of the energy grade line that is compatible with the imposed independent variable values (Chang, 1979). The above conditions require alluvial streams to have a constant water temperature with steady uniform or gradually varied flow and negligible sediment discharge. 20 Application Chang (1979, 1980a, 1980b), presented a method which solves three relationships to calculate the equilibrium values of the stream width, stream depth, and slope of the energy grade line for alluvial streams with uniform flow. This method uses a specific water discharge, sediment discharge, mean sediment size, water temperature, and trapezoidal cross sectional shape. Chang's method uses three relationships: an empirical sediment transport equation, an empirical flow resistance equation, and the minimum rate of ehergy dissipation theory represented by a minimum slope of the energy grade line. The steps used by Chang's method are: 1. Input data. 2. Assign a value to the stream width. 3. Assign a value to the stream depth. 4. Calculate the slope of the energy grade line using the sediment transport equation. 5. Calculate the water discharge using the flow resistance equation. 6. If the calculated water discharge does not equal the input water discharge, modify the stream depth and go to item 4. 7. If the calculated slope of the energy grade line does not equal the global minimum slope, modify the stream width and go to item 3. 21 8. Output stream width, stream depth, and slope of the energy grade line. Chang's 1979 study used a mean sediment size of medium sand and the method given. Calculated values of the minimum energy grade line slope and of the associated stream width and stream depth were found for six different sediment discharge values. With a low sediment discharge, one minimum value of the slope was found. For a higher sediment discharge two minimum values of the slope were found; each at different values of the width and depth. One minimum slope was found in the lower flow regime and the other in the upper flow regime. The slope in the lower flow regime had the lower value, the global minimum. Therefore, Chang (1979) determined a minimum value of the slope can be found. He did not explain, however, why the physical processes produce this minimum slope. These results were compared to published river and canal data; conclusions related planform to the water discharge and slope. For low sediment discharges, both of Chang's 1980 studies yield calculations similar to his 1979 work. In one study (1980a) he used a mean sediment size of medium sand and compared the calculated values to published data describing the equilibrium values of alluvial sand- and gravel-bed canals with uniform flow. In his subsequent study, Chang (1980b) used a mean sediment size of coarse gravel to large cobbles and compared the calculated values Ito published data describing the equilibrium values of alluvial gravel- and cobble-bed streams with uniform flow. The values of the width arid depth compared were those associated with a sediment discharge which produced a minimum slope equal to the measured slope. The calculated values of the width and depth in both studies compared ■ favorably to the published data. Chang concludes that a method using the minimum rate of energy dissipation theory in combination with other relationships can accurately calculate equilibrium values of the dependent variables for alluvial streams with uniform flow and low sediment discharge. The mean sediment size is a dependent variable. In the above studies, however, Chang assigns this variable a , constant value for each stream studied. This practice is acceptable for stream reaches with uniform flow because values of the dependent variables are relatively constant with location. For stream reaches with gradually varied flow, however, values of the dependent variables change with location. Hence, for the gradually varied flow reaches under study herein, values of the mean sediment size can not be considered constant for each study stream. Summary Alluvial streams with 10 to 100 years (a "long-term period") reaction time to a change in the value of an 22 23 independent variable are considered to be adjusting toward equilibrium. The independent variables of gravel-bed streams are the bankfulI water discharge and man-made works. The dependent variables of gravel-bed alluvial streams are the mean size of the sediment that is stationary on the bed, stream depth, and stream width. After a culvert is placed in a gravel-bed stream, the stream width does not change, the stream depth increases, and the slope and mean sediment size decrease from the pre-culvert condition. The minimum rate of energy dissipation theory states that provided certain conditions are met, equilibrium values of the dependent variables adjust so alluvial streams flow with the minimum slope of the energy grade line compatible with the imposed independent variable values. These conditions require alluvial streams to have a constant water temperature with steady uniform or gradually varied flow and negligible sediment discharge. A method using the minimum rate of energy dissipation theory in combination with other relationships accurately calculates equilibrium values of the dependent variables for alluvial streams with uniform flow and low sediment discharge. Information presented in this chapter provides the background for developing the MINRED method and for evaluating the reliability with which it can be applied to reaches of graveI-bed alluvial streams having gradually varied flow. Information in Chapter 3 describes the data 24 collection and presents measured data from two gravel^bed streams having gradually varied flow imposed by a highway culvert. The MINRED method is presented in Chapter 4. The application of the MINRED method to the study streams is discussed in Chapter 5 along with the resulting calculated data. The results of applying the MINRED method to the study streams is presented in Chapter 6. 25 CHAPTER 3 FIELD INVESTIGATION AND MEASURED VALUES Preliminary study sites were chosen where long term U .S. Geological Survey (U.S.G.S.) annual maximum water discharge records are available and where gradually varied flow has been imposed by a single highway culvert. Eleven streams located in the southcentral and the interior portions of Alaska meet these criteria. Each stream was visually inspected prior to continuing the study; nine were discarded. To simplify the analysis, study of four streams was discontinued because man-made works other than the culvert appeared to have significantly altered the dependent variable values. The MINRED method utilizes the minimum rate of energy dissipation theory which is not valid for rapidly varied flow. Study of three streams was discontinued because the bed was composed of boulders equal to or greater than the bankfulI depth, creating rapidly varied flow at the bankfulI discharge. Study of two streams was discontinued because the mean sediment size oh the bed was significantly smaller than gravel. The two streams selected for study are located in Alaska as shown in Figure I. The 9.5 foot diameter culvert Barrow Figure I. Location map. Brooks Creek Tributary ~ d ) !Fairbanks ^jCopper River [Tributary ■2/ Anchorage 27 in the Copper River tributary is located in southcentral Alaska at mile 51.8 of the Glenn Highway, or about nine miles west of Slana. U.S.G.S. water resources publications indicate this has a drainage area of 4.32 square miles. The five foot diameter culvert in the Brooks Creek tributary is located in the Alaskan interior at mile 84 of the Elliott Highway, or about 15 miles southwest of Livengood. U.S.G.S. water resources publications indicate this has a drainage area of 7.81 square miles. These culverts were in­ stalled in the mid 1950s. Both streams flow through forests composed mainly of spruce and willow. The banks are held nearly vertical, by roots of the adjacent vegetation. This gives further justification for using the rectangular cross section discussed in Chapter 2. Most of the spruce trees near the streams are small suggesting much of the ground surrounding these trees is frozen all year. Both streams flow in soils of discontinuous permafrost (Hartman & Johnson, 1978). Data Collection Independent Variables Provided the sediment density is constant, the independent variables of graveI-bed streams are the bankfulI water discharge and man-made works. 28 Sediment Density There is no evidence of volcanic activity hear the sites studied. For such streams, the sediment specific gravity can be considered equal to 2.65. Therefore, the sediment density for the study streams.is constant. Water Discharge Each study stream has a small drainage area at the culvert of interest and hence the time of concentration for each is small. Both study streams are remotely located. The travel time to each site from any population center is much greater than the time of concentration. Therefore, direct measurement of the bankfulI water discharge is impractical and was not done. The bankfulI water discharge can be approximated as the discharge having a 1.5 year recurrence interval on the annual maximum series. The study streams have long term U.S.G.S. annual maximum water discharge records at the culverts of interest. The bankfulI water discharge was obtained from, these records. The method and the resulting discharges are presented in Appendix D . Man-Made Works See Appendix E, Table 11 for the measured data describing the culvert that is imposing gradually varied flow on each study stream. 29 Dependent Variables The dependent variables of gravel-bed alluvial streams are the mean size of the sediment that is stationary on the stream bed, stream depth, and stream width. As discussed in Chapter 2, channels in which the width is at least several times the depth can be considered rectangular and the cross section defined by the width and depth. A preliminary inspection of the study streams found that the width was approximately five to ten times the depth and the cross sectional shape was approximately rectangular. Therefore, values of the width and depth are the only data that need be collected to describe the cross sections of the study streams. The discharge of interest is the bankfulI value. Therefore, the bankfulI values of the dependent variables are those of interest. However, because data were collected during low flow rates, some estimation was required to obtain values representing the dependent variables at the bankfulI discharge. Method and Equipment The study streams were divided into subreaches. Cross section lines define the subreach boundaries. The cross section located in the stream, adjacent to the upstream end of the culvert is designated cross section CU. Moving upstream, cross sections are designated IU, 2U, 3U, and so 30 forth. Similarly, the cross section located in the stream, adjacent to the downstream end of the culvert is designated cross section GD. Moving downstream, the cross sections are designated ID, 2D, 3D, and so forth. Measured values were determined with a Hilger and Watts SL 9060-2 automatic level, level rod, and Lietz-Eslon 165 foot tape. There were no bench marks near the streams. A temporary bench mark of elevation 100.00 feet was arbitrarily assigned to each upstream culvert invert. Mean Sediment Size The sediment discharge is low at the bankfulI water discharge. However, some of the smaller particles stationary on the bed at low flow rates are transported at the bankfulI discharge. GraveI-bed' rivers usually have a marked sorting of bed materials With sections composed of the largest sizes spaced about every five to seven channel widths (Mercer, 1971). This was found in the study streams. For these two reasons, at a low water flow rate it is difficult to determine the mean sediment size that is stationary on the bed at the bankfulI discharge. The bed sediment at each cross section was visually examined. A location was selected such that the mean sediment size at that location is representative of the cross section. At this location, the finer material on the stream bed was assumed to be in transport during bankfulI 31 flow. The infrequently occurring smaller and larger sizes were excluded and the mean of the remaining sizes was measured. This is the value of the mean sediment size recorded. Although the above method to determine the mean sediment size at the bankfulI discharge is crude, other methods are prohibitively expensive and/or would not yield results of higher quality. For example, sophisticated bed load samplers are very expensive and do not collect representative samples. Also, a shovel or similar device could be used to collect samples for later, analysis but the water velocity would not allow the smaller sizes to be retained. Further, unless these samples were collected at bankfulI flow, the results would have to be adjusted for this anyway. Finally, to collect samples at bankfulI flow would create both logistical and operational problems. Therefore, the measured values describing the mean sediment size are as representative of the bed as values which could have been determined by other, more sophisticated methods. Lane (1947) standardizes the terms for a particle diameter as follows: gravel: 64 mm - 2 mm (0.21 ft - 0.0067 ft), sand: 2.0 mm - 0.250 mm (0.0067 ft - 0.0008 ft), and fine sand: 0.250 mm - 0.125 mm (0.0008 ft - 0.0004 ft). Nearly all of the study cross sections have a mean sediment size in the range of gravel. 32 Stream Depth For streams in equilibrium, the difference between the elevations of the bank and the thalweg is equal to the stream depth at the bankfulI discharge. Dunne and Leopold (1978) and others provide guidelines to help field locate the stream bank at low flow rates. Although these guidelines were used, field determination of the stream bank was difficult. For this reason, some variation in measured values of the stream depth is anticipated. Stream Width . The stream width at the bankfulI discharge was measured. Because field location of the stream bank was difficult, some variation in measured values of the stream width is expected. Thalweg Slope For both uniform and gradually varied flow in prismatic channels, the slopes of the energy grade line, stream bed, and hydraulic grade line are similar. For an open channel, the hydraulic grade line coincides with the water surface. For streams in equilibrium, the water surface at the bankfulI discharge coincides with the stream bank. By approximating the study streams to be prismatic, the measured slope of the stream bed and/or stream bank represents the slope of the energy grade line during the 33 bankfulI discharge. Because field determination of the stream bank was difficult, the measured thalweg slope is used. Collecting data to determine the thalweg slope was no simple task. Mercer (1971) indicates gravel-bed streams form a series of riffles and pools; this was found in the study streams. Hence, the thalweg slope fluctuates. The MINRED method does not account for pools and riffles and therefore calculates the average slope of the energy grade line. To allow for valid comparisons between the measured and calculated data, measurements were not collected near either the top of riffles or the bottom of pools. Thus, attempts were made to collect data representing the average thalweg slope. Because this was difficult to do by eye, some variance of the thalweg slope is expected. Presentation and Discussion of Measured Values Measured data are presented in Appendix E. These describe each culvert and study stream. Average Measured Values Because streams rarely flow in homogeneous soils, it is not surprising that the measured values fluctuate somewhat with location. In spite of this fluctuation, the measured data presented in Tables 12 through 15 and Figures 9 through 12 indicate, for most parameters, the values in the Isubreaches near the culvert are different from those in the subreaches farther away. Values of the dependent variables in the subreaches near the culvert are changed as a result of the culvert installation. Therefore, subreaches near the culvert are altered by the culvert and subreaches, farther away are unaltered. Cross sections approximating the break in the measured data are given in Table 18. Subreaches between the culvert and these cross sections are considered to have been altered by the culvert. Subreaches further from the culvert than these cross sections are considered to have been unaltered by the culvert. The minimum rate of energy dissipation theory is valid for uniform and gradually varied flow. However, rapidly varied flow occurs in the subreaches adjacent the culvert at the bankfulI discharge. For valid comparisons between the measured and calculated data, values measured in the subreaches adjacent the culvert are not considered further. The measured values fluctuate with location. For simplicity, average measured values from the subreaches altered by the culvert are used rather than using measured values from each altered subreach. Similarly, average measured values from the subreaches unaltered by the culvert are used. These average measured values are presented in Table 2. For the portion of the Copper River tributary not altered by the culvert, the data presented in Table 2 are 34 'i 1 1 I l I 35 Table 2. Average measured data. Parameter Copper River tributary Brooks Creek tributary Not altered Altered Not altered Altered Cross Upstm 5U-14U 1U-4U 6U-14U 1U-5U sections Dnstm 7D-12D 3D-6D 5D-12D 3D-4D Width, Upstm 7.10 15.88 7.80 8.48 ft Dnstm 11.08 23.25 8.19 12.15 Depth, Upstm 1.66 1.96 0.84 1.42 ft Dnstm 1.60 3.70 I . 29 1.37, MSS, Upstm 0.0312 0.0048 0.0290 0.0419 ft Dnstm 0.0660 0.0417 0.0451 0.0208 Thalweg Upstm 0.0267 0.0115 0.0184 0.0095 slope Dnstm 0.0378 0.0476 0.0234 0.0278 Note.’ MSS = mean sediment size. I' averages from 10 cross sections upstream and six cross sections downstream. For the portion of the Copper River tributary altered by the culvert, the data presented are averages from four cross sections upstream and four cross sections downstream. For the portion of the Brooks Creek tributary not altered by the culvert, the data presented in Table 2 are averages from nine cross sections upstream and eight cross sections downstream. For the portion of the Brooks Creek tributary altered by the culvert, the data presented are averages from five cross sections upstream and two cross sections downstream. Subreaches Not Altered by the Culvert The bankfulI water discharge at each stream has probably been constant for many years. Also, the culvert is the only man-made, work having any significant impact on the study streams. Therefore> the independent variable values imposed on the study streams were probably constant for many years prior to the culvert installation. For this reason, the study streams are presumed to have attained equilibrium before the culvert was installed. For the subreaches unaltered by the culvert, this equilibrium has not been changed; hence, the measured values describe the pre-culvert equilibrium. ---------------------- - - " 1— '■— — -------------------------- ---------------------- -'.VnJ— _______________IU-I— _________ u ____________________ 111 n r v . .. I . 1 . 37 Subreaches Altered by the Culvert The culverts in the study streams, were installed approximately 30 years ago. These streams should be adjusting toward equilibrium. Thus, values of the dependent variables in subreaches altered by the culvert should be adjusting toward equilibrium with post-culvert values of the independent variables. This trend is examined by comparing the measured bank elevation just upstream of the culvert to the elevations at the same location estimated for both the pre-culvert and post-culvert equilibrium conditions. Pre-Culvert Subreaches unaltered by the culvert have attained equilibrium unchanged by culvert installation. For these subreaches, measured values describe the pre-culvert equilibrium condition; the pre-culvert equilibrium slope is equal to the measured slope. The pre-culvert equilibrium bank elevation at cross section OU is approximated by extending this measured slope. For determination of the pre-culvert equilibrium bank elevation, see Appendix F, j Table 19. The results are given in Table 3. Post-Culvert For determination of the post-culvert water surface elevation at the bankfulI discharge and at stream cross section OU, see Appendix F1 Tables 20 and 21. This Ii Table 3. Bankfull 38 elevations at cross section OU. Condition Copper River tributary Brooks Creek tributary Equilibrium pre-culvert 99.27 100.25 Measured 101.18 101.32 Equilibrium post-culvert 102.17 102.65 Note. Elevations are in feet.calculated water surface elevation is the estimated post-culvert equilibrium bank elevation. These elevations for cross section OU are presented in Table 3. •r 39 calculated water surface elevation is the estimated post- culvert equilibrium bank elevation. These elevations for cross section OU are presented in Table 3. Conclusion . The measured bank elevation at cross section OU is between the estimated pre-culvert and the estimated post­ culvert elevation. For this reason, the bahkfull water surface elevations are probably adjusting toward their post­ culvert equilibrium values. It is assumed that all values of the dependent variables in subreaches altered by the culvert have not attained equilibrium with the post-culvert values of the independent variables, but are adjusting toward this condition. Gradually Varied Flow Upstream Values in Table 2 show that upstream of each culvert, the average measured depth in the unaltered subreaches is less than the average measured depth in the altered subreaches. Also, values in Tables 2 and 21 show that the depth at cross section OU required to pass the bankfulI discharge through the culvert is greater than the depth in the unaltered subreaches upstream of the culvert. At the bankfulI discharge, an Ml gradually varied flow profile is created in those subreaches with subcritical flow that have 40 been altered by the culvert. This is shown in Figure 2. Downstream Values in Table 2 show that downstream of each culvert, the average measured depth in the altered subreaches is greater than the average measured depth in the unaltered subreaches. Further, values in Tables 12 and 14 show that a scour hole is created downstream of both culverts. At the bankfulI discharge, an A2 gradually varied flow profile is created at the downstream end of the scour hole. This is shown in Figure 3. T = Thalweg ---- Pre-culvert WS = Water Surface - - - - Existing EGL = Energy Grade Line - - - - Post-culvert Cross Section Culvert Subreach Length, X Longitudinal SectionDetail A Figure 2. Typical stream with culvert creating gradually varied flow upstream. VJS EGL T WS Water Surface Energy Grade Line • Thalweg :Water Surface -- Existing — Post-culvert Pre-culvert Culvert Cross Section Detail A Detail A Pl g „ r e 3 . Typical s t r e a m w i t h culvert v a r i e d flow downstream. creating gradually 43 . CHAPTER 4 MINRED METHOD The MINRED method was developed for this study. This method uses the minimum rate of energy dissipation theory and two other relationships. Results should describe the equilibrium bankfulI values of the dependent variables in reaches of gravel-bed alluvial streams having gradually varied flow and low sediment discharge. Development The MINRED method uses measured values describing the independent and dependent variables for the subreach and the downstream cross section. The independent variables of the gravel-bed study streams are the bankfulI water discharge and the culvert (man-made works). The dependent variables of the gravel-bed study streams are the mean sediment size that is stationary on the bed, stream depth, and stream width. The downstream end of the stream subreach is designated the downstream cross section. The upstream end of this subreach is designated the upstream cross section. The MINRED method uses a trial and error solution of three relationships linking the independent and dependent variables: (a) the direct step method of determining the 44 gradually varied flow profile, (b) a combination of Shields diagram and Raudkivi1s graphical relation, and (c) the minimum rate of energy dissipation theory. These relationships have not been previously coupled in the literature. Results of the MINRED method are values of the dependent variables at the upstream cross section. The MINRED method uses measured values from the subreach and the downstream cross section to calculate values at the upstream cross section. Because stream subreaches flowing subcritical have downstream control, calculations must start downstream and work upstream. Conversely, calculations for stream subreaches flowing supercritical must work downstream. Therefore, the MINRED method can only be used for stream subreaches with subcritical flow. Figures 2 and 3 are control volume diagrams showing a typical stream with a culvert creating gradually varied flow. Parameters shown in these figures are the stream width, stream depth, mean sediment size, and subreach length (distance between the upstream and downstream cross sections). Some parameters appearing in equations used by the MINRED method are not shown in Figures 2 and 3 because these parameters do not have a demonstrable physical significance. These figures do, however, show all the dependent variables. The parameters not shown can best be understood as mathematical combinations of, or functions of. 45 the dependent variables and the water discharge as given in the following equations. Recall that the cross section of the study streams can be considered rectangular and thus defined by the width and depth. General The MINRED method assumes many values for the stream width at the upstream cross section. For each assumed width, a value is calculated fqr both the stream depth at the upstream cross section and the mean size of sediment that remains stationary on the stream bed at the upstream cross section. With this information, the slope of the energy grade line at the upstream cross section is also determined for each assumed width. Therefore, the MINRED method determines many sets of values for the stream width, depth, mean sediment size stationary on the bed, and energy grade line slope compatible with the independent variable values. The minimum rate of energy dissipation theory states that of all these sets of dependent variable values, the one set that describes the stream equilibrium condition is the set associated with the smallest slope of the energy grade line. As discussed in Chapter 2, Chang (1979, 1980a, 1980b) studied streams with uniform flow and concluded that a minimum slope satisfying the independent variable values can be found. These studies, however, did not state any 46 reason why this should occur. Chang also concludes that the dependent variable values associated with this minimum slope describe the stream's equilibrium condition. The MINRED method determines the one set of dependent variable values (width, depth, and mean sediment size) associated with the smallest calculated slope of the energy grade line. If the MINRED method finds the minimum slope of the energy grade line that satisfies the independent variable values, the results should be the dependent variable values describing the post-culvert equilibrium condition. Details of the MINRED method are given in this chapter and in Appendix G. Preliminary The MINRED method uses the bankfulI water discharge, the measured distance between the upstream and downstream cross sections, and the measured thalweg slope between the upstream and downstream cross sections. The MINRED method also uses measured values of the stream depth, width, and mean sediment size at the downstream cross section. The latter value is used to determine, the Manning roughness coefficient. Chow (1959) indicates the Manning roughness coefficient can be calculated by: n = (nQ + U 1 + n 2 + n 3 + n 4) Ht5. (4-1) 47 The above symbols are as follows: n = Manning roughness coefficient; n g = the value for a straight, uniform, smooth channel in the natural materials and thus is due to the effect of grain roughness; n ^ = correction for"the effect of surface irregularities; n g = correction for the effect of variations in channel cross section size and shape; n„ = correction for the effect of obstructions; 3 . n^ = correction for the effect of vegetation; and ntj. = correction for the effect of channel meandering. The determination of the Manning roughness coefficient for the study streams is given in Appendix G, item 6. Direct Step Method The MINRED method uses the direct step method to determine the depth of flow at the Upstream cross section. The depth in a stream varies with location when gradually varied flow is imposed. Chow (1959) gives several methods for calculating the stream depth in gradually varied flow. One of these is the direct step method. This method divides the gradually varied flow reach into subreaches and considers each to have uniform flow. The stream cross sectional shape is considered rectangular, see Chapter 2. Therefore: JL A = b d, R = bd/(b + 2d), and V = Q/A. 48 (4-2) (4-3) (4 — 4 ) The symbols are as follows: A = cross sectional area of flow, b = stream width, d = stream depth, R = hydraulic radius, V = mean velocity of flow, and Q = water discharge. The. specific energy, E, and slope of the energy grade line, S, are calculated for the downstream cross section. The equations used are: E = d + V2/2g and (4-5) S = [Vn/(1.49R0-67 )]2. (4-6) The symbol g represents the acceleration due to gravity. Equation 4-5 is the definition of specific energy and equation 4-6 is derived from the Manning uniform flow equation. 49 The MINRED method temporarily assumes values for the stream width, depth, and mean sediment size at the upstream cross section. Equations 4-1 through 4-6 are recalculated for the upstream cross section. The distance between the upstream and downstream cross sections, X, is calculated by: X = (E2 - E1)/{S0 - [(S1 + S20/2]}. (4-7) The subscripts I and 2 represent the upstream and downstream cross sections, respectively. The symbol AP represents the measured thalweg slope between the upstream and downstream cross sections. Equation 4-7 is given in Chow (1959) and is derived from the energy equation. The MINRED method uses a trial and error procedure that modifies the value of the depth at the upstream cross section until the calculated distance between the two cross sections equals the measured distance. Details of how the MINRED method uses the above equations are given later in this chapter and in Appendix G . Shields Diagram and Raudkivi's Graphical Relation The MINRED method uses a combination of Shields diagram and Raudkivi's graphical relation to determine the mean size of the sediment that remains stationary on the stream bed at the upstream cross section. This is the mean sediment size associated with both the calculated depth and assumed width I IL 50 at the upstream cross section from the direct step method. Shields Diagram Shields diagram is an empirical curve proposed by Rouse from data obtained by Shields and other workers from experiments in flumes with fully developed turbulent flows of noncohesive sediments. This diagram is given by Vanoni (1975) and is shown in Figure 4. The left portion of the curve is the laminar flow region, the right portion of the curve is the turbulent flow region, and the central portion of the curve is the transition between the two. Critical conditions are said to exist when the hydrodynamic force acting on a sediment grain has reached its maximum value for which no sediment movement occurs: (Vanoni, 1975). . The critical value of a variable is defined as that occurring at the critical conditions. These terms are used below. I ■'! Shields diagram is used by the MINRED method to determine the critical value of one parameter. Knowing the boundary Reynolds number, Shields diagram is used to determine the critical value of the dimensionless shear stress. This value is used by the MINRED method to determine the critical value of other parameters, including the mean sediment size that is stationary on the stream bed during bankfulI conditions. The original work to develop Shields diagram was Cr it ic al Va lu e of th e Di me ns io nl es s Sh ea r S t r e s s , 2. ? I Ul>- * H U*H Boundary Reynolds Number, Reit = U^d^/v UlH Figure 4. Shields diagram 52 published in 1936. Some workers since then have reported slightly different results for portions of the curve. Shields diagram is the standard by which these new results are evaluated. Therefore, Shields diagram is the most widely accepted means for determining the critical conditions. The results from Shields diagram are approximate. The reason for this is described below. Shields diagram uses mean parameter values to determine the critical conditions. For turbulent flow, however, there is no single mean parameter value for which sediment motion begins or ends suddenly. Rather, near critical conditions, the motion of sediment grains occurs randomly in both time and space. The incidence of such movement increases as the mean shear stress increases. For this reason, to describe the initiation of sediment movement by mean parameter values requires a range of such values corresponding to a low to high incidence of sediment movement. Vanoni (1975) indicates that one researcher found the results of Shields diagram correspond to a high incidence of sediment movement for turbulent flow. The. MINRED method assumes values for the stream width and mean sediment size at the upstream cross section. With these, the direct step method calculates values for the stream depth and energy grade line slope at the upstream cross section. The MINRED method approximates a water 53 temperature and uses these above values to determine values for the shear velocity, U*, and the boundary Reynolds number, Re* at the upstream cross section. The equations used to make these calculations are given by Vanoni (1975) as follows: U* = (gRS) 0J5 and (4-8) Re* = U t ds/v . (4-9) The symbol v represents the kinematic molecular viscosity. The sediment specific gravity for the study streams equals 2.65. Using this value, the value for the boundary Reynolds number just calculated, and equations for the curve in Shields diagram, the MINRED method calculates the critical value of the dimensionless shear stress,LIC , at the upstream cross section. The equations used to make this calculation are discussed in the next paragraph. The MINRED method uses this value and the equation given by Shields diagram to calculate the critical value of the bed shear stress,Lc, at the upstream cross section. This is done as follows: Y)ds (4-10) Previously undefined symbols are: Jl 54 Yg = sediment specific weight and Y = fluid specific weight. Shields diagram is empirical and the literature does not give equations for the curve. Equations were developed by using program 1415 G from the Hewlett-Packard Users Library. This program, run on a Hewlett-Packard 4ICV calculator, uses data points from the curve and the method of least squares to determine second order through ninth order equations of the form: T O NP + A^x + A2x 2 + ... Agx9. {4-11) For each equation, this program calculates the coefficient of determination, r2. The lowest order equation with the largest coefficient of determination is used by the MINRED method. Curve equations for Shields diagram and the coefficients of determination are given in Appendix H. Raudkivi's Graphical Relation Two relationships given in the literature can be used to correlate the depth and mean sediment size of gravel-bed streams. These are given by Raudkivi (1967, 1971) and Chang (1980b). Both are empirical. Empirical relationships have limited applicability; therefore, care should be exercised to ensure they are applied only to the conditions for which they were 55 developed. Both the Raudkivi and the Chang relationship were developed from information similar to but not completely representative of all the measured data. Raudkivi's relationship is used in this study because it was developed from data more closely resembling the study streams than the data used to develop Chang's relationship. The relationship developed by Raudkivi is known in the literature as Raudkivi's graphical relation. Raudkivi's graphical relation is a set of empirical curves that relate the mean velocity of flow to the shear velocity and mean sediment size. This relation is presented in Figure 5. Raudkivi developed this relation by using data from his own flume experiments and data reported by others from large rivers to laboratory flumes. The data cover a sediment size from 0.0003 ft to 0.016 ft (very fine sand to very fine gravel). The left portion of the curves in Raudkivi's graphical relation corresponds to the initiation of sediment movement and the development of dunes on gravel-bed streams. Vanoni (1975) states that left of this portion of the curve corresponds to flows with velocities just above the critical value for the initiation of motion. It is this portion of the curves that is used by the MINRED method. The central portion of the curves corresponds to a moderate sediment discharge, erosion of the dunes, and the development of a flat bed. The fight portion of the curves corresponds to a n)/A 100 \ T A X 4 _ Fine Sand Upper ' \ \ / I Flow Regi ne X\ Gravel — Sand - \ \\Z Fine San I, Dn> Transit! > ■ mentioned in this paragraph are presented in Table 6. Actual Ratios Table 6 presents the actual ratios of average measured values to average calculated values for subreaches altered 83 Table 6. Ratios 1 Dnstm 0.67 1.35 =1 >1 Depth Upstm 1.07 1.48 <1 <1 Dnstm 1.75 1.21 <1 <1 MSS Upstm 6.86 6.65 >1 -I Dnstm 9.48 0.19 >1 -I Slope Upstm 230. 27.14 >1 -I Dnstm 432 . I . 88 >1 -I Note. MSS = mean sediment size. Ideal, minimum = ideal ratio assuming global minimum slope was calculated. Ideal, no minimum = ideal ratio assuming global minimum slope was not calculated. 84 by the culvert. These ratios are calculated from values in Tables 2 and 4. Results of Comparisons Measured values used in this study are approximations of the actual stream conditions. Therefore, ratios of average measured values to average calculated values must also be considered approximations. Actual ratios between 0.25 and 1.00 are considered to be somewhat less than unity. Also, actual ratios between 1.00 and 4.00 are considered to be somewhat greater than unity and actual ratios between 0.67 and 1.50 are considered to be equal to unity. For the Copper River tributary, the actual ratios and the ideal ratios agree for one of the eight comparisons. This is the downstream width. For the Brooks Creek tributary, the two ratios agree for two of the eight comparisons. These are the downstream width and downstream slope. Global Minimum Not Calculated The following is a summary of information presented in the previous subsection. For subreaches altered by the culvert, measured values of the stream depth should be smaller and measured values of the mean sediment size and slope should be larger than the post-culvert equilibrium values. Also for these subreaches, measured values of the stream width should equal the post-culvert equilibrium values. The MINRED method does not calculate values describing the equilibrium condition if the global minimum slope of the energy grade line is not found. The following is an extension of information presented in Chapter 5. When the minimum slope is not determined, calculated values of the stream width should be smaller and calculated values of the stream depth, mean sediment size, and slope should be larger than the post-culvert equilibrium values. Ideal Ratios The information in the previous two paragraphs is combined to yield conclusions valid for subreaches altered by the culvert when the minimum slope is not found. Ratios of measured to calculated values should be somewhat greater than unity for the stream width, somewhat less than unity for the stream depth, and approximately equal to unity for the mean sediment size and slope. These ideal ratios are given in Table 6. Actual Ratios Actual ratios of average measured values to average calculated values for subreaches altered by the culvert are given in Table 6. These ratios are calculated from values 86 in Tables 2 and 4. Results of Comparisons Measured values are approximations of the actual stream conditions. For this reason, actual ratios between 1.00 and 4.00 are considered to be somewhat greater than unity and actual ratios between 0.25 and 1.00 are considered to be somewhat less than unity. Also, actual ratios between 0.50 and 2.00 are considered to be approximately equal to unity. For the Copper River tributary, the actual ratios and the ideal ratios do not agree for any of the eight comparisons. For the Brooks Creek tributary, the two ratios agree for two of the eight comparisons. These are the downstream width and downstream slope. Summary This chapter compares values calculated by the MINRED method to measured values and to changes of these values predicted in the literature. A summary of these comparisons is given in Table 7. For the Copper River tributary, 14 of 32 or 44% of the comparisons agree. For the Brooks Creek tributary, 10 of 32 or 31% of the comparisons agree. Assuming the minimum slope is found, 3 of 16 or 19% of the comparisons agree. Assuming the minimum slope is not found, 2 of 16 of 13% of the comparisons agree. 87 Table 7. Summary of comparisons that agree. Item Conditions Copper Brooks River Creek trib. trib. Parameter changes Calc to meas 6 of 8 3 of 8 between subreaches Calc to lit 7 Of 8 3 of 8 Parameter changes Minimum I of 8 2 of 8 at subreaches No minimum 0 of 8 2 of 8 Note. Calc to meas = comparisons of changes in average calculated values to changes in average measured values. Calc to lit = comparisons of changes in average calculated values to changes predicted by the literature. Minimum = comparisons assuming global minimum slope was calculated. No minimum = comparisons assuming global minimum slope was not calculated. 88 CHAPTER 7 SUMMARY, CONCLUSIONS, AND SUGGESTIONS FOR FUTURE WORK Summary and Conclusions 1. This study develops a method (referred to herein as the MINRED method) to calculate equilibrium bankfull values of the dependent variables in reaches of gravel-bed alluvial streams which have gradually varied flow. 2. The MINRED method uses three relationships. These are (a) the direct step method of determining the gradually varied flow profile, (b) a combination of Shields diagram and Raudkivi1s graphical relation, and (c) the minimum rate of energy dissipation theory. 3. Data from two gravel-bed alluvial streams which have gradually varied flow imposed by a highway culvert are used to evaluate the reliability of the MINRED method. 4. Values of the dependent variables calculated by the MINRED method are compared to measured values and to changes of these values resulting from the culvert installation, as predicted in the literature. The results of these comparisons are similar whether or not the MINRED method calculates the global minimum slope of the energy grade line. Therefore, conclusions do not depend on whether the global minimum slope is found. H I 5. Values calculated by the MINRED method compare poorly to both measured values and to changes of these values resulting from the culvert installation, as predicted in the literature. Thus, for the data available, the MINRED method calculates equilibrium bankfulI values with a low degree of reliability. 6. The MINRED method needs further refinement and/or field testing before it can be recommended for use by practicing engineers. I ' • Assumptions and Approximations There is no evidence to indicate that the MINRED method is not valid for use in gradually varied flow. Therefore, the low reliability of the MINRED method must be due to the assumptions and approximations which are required to apply this method to the gravel-bed study streams. Assumptions Following is a summary of the assumptions made in applying the MINRED method to the graveI-bed study streams. I. Sediment discharge is low and can not be accurately determined, in graveI-bed streams at the bankfulI water flow rate. For simplicity and possibly for accuracy, the sediment discharge is assumed to be constant at the bankfulI water discharge for the gravel-bed study streams. Streams with over ten years to react to a change in 89 2 . 90 an imposed condition can be considered to be adjusting toward equilibrium. The study streams have had approximately 30 years to react to the culvert. Just upstream of the culvert in each study stream, the bankfulI water surface elevation is adjusting toward post-culvert equilibrium. For these reasons, all stream locations altered by the culvert are assumed to have not attained post-culvert equilibrium, but are assumed to be adjusting toward this condition. Approximations Following is a summary of the approximations made in applying the MINRED method to the gravel-bed study streams. 1. The discharge of interest is the bankfulI value. Therefore, the bankfulI values of the dependent variables are those of interest. For practical reasons however, data were collected during low flow rates. For this reason, measured values representing the dependent variables at the bankfulI discharge are approximate. The procedures used to obtain data representative of the bankfull conditions are discussed in Chapter 3. 2. Subreaches near the culvert are altered by the culvert and subreaches farther away are unaltered. Measured data are used to determine which subreaches are altered and which are unaltered by the culvert. However, because the measured data are approximate and fluctuate somewhat with 91 location, this determination is approximate. 3. As described in Chapter 4, values determined by Shields diagram can only approximate the critical conditions for turbulent flow. . Therefore, results obtained from Shields diagram must be considered approximate. 4. As discussed in Chapter 4, results from Raudkivi's graphical relation must be considered approximate for the following three reasons. First, there is some scatter of the data points from which this relation was developed. Second, the measured values of this study are not identical to the data from which this empirical relation was develop­ ed . Third, the portion of the curve in this relation that is used has a moderately steep slope, compounding any error in the input data. 5. There are no water temperature data for the study streams. The drainage area for both streams is in discontinuous permafrost and therefore the temperature of the groundwater influent to these streams is just above freezing. Data in Appendix D show that the annual maximum discharge for the Copper River tributary occurs primarily during June, July, and August and thus results from summer rainstorms. Data in Appendix D also show that the annual maximum discharge for the Brooks Creek tributary occurs primarily in May, June, and August. Althpugh early May is typically snow melt season at this stream, these events are mostly a result of summer rainstorms. The temperature of 92 the falling rain from these storms is significantly above freezing. The bankfull water discharge has a recurrence interval of 1.5 years on the annual maximum series and thus occurs at a similar time of year as the annual maximum discharge. From above, the water temperature during a typical bankfull discharge event is not known exactly, but must be somewhat above freezing. This water temperature is approximated to be 38 degrees F . Suggestions for Future Work Reliability of the MINRED method could be improved by: 1. Developing a calculation method or measurement technique which can accurately determine the sediment discharge in graveI-bed streams at the bankfull discharge. This would eliminate the need to assume a constant sediment discharge at the bankfull condition. This would also ensure that the minimum rate of energy dissipation theory and the MINRED method are applied only to streams with a negligible to low sediment discharge. 2. Either waiting much longer after culvert installation to study the streams or collecting data over many years to determine changes in the variables. This would eliminate the need to assume that all the dependent variables in the study streams are adjusting toward equilibrium with the post-culvert independent variables. 3. Determining exponents of the stream width, depth, 93 and mean velocity that relate these variables to the discharge, for streams in equilibrium with no imposed human activity, and flowing in soils with discontinuous permafrost. Such exponents are available for the above conditions in temperate climates only. Determining these exponents for streams flowing in discontinuous permafrost would allow the altered and unaltered subreaches in the study streams to be determined, rather than approximated. 4. Improving Shields diagram or developing some other method which more accurately determines the critical conditions for turbulent flow. This would allow a better estimate of the mean sediment size. 5. Improving Raudkivi1s graphical relation or some other method which more accurately applies to the gravel-bed streams under study. This would allow a better estimate Of the mean sediment size. 6. Developing a method which calculates the one water temperature having the same influence on the dependent variables as the combined effects of all the water temperatures actually occurring. This would eliminate the need to approximate such a water temperature and hence result in a more accurate determination of the mean sediment size. REFERENCES CITED 95 Ackers, P. (1982). Meandering Channels and the Influence of Bed Material. In R.D. Hey, J.C. Bathurst, & C.R. Thorne (Eds.), Gravel-Bed Rivers (PP. 389-421). Chichester: John Wiley & Sons. Ackers, P., & Charlton, F.G. (1970, March). Dimensional Analysis of Alluvial Channels with Special Reference to Meander Length. Journal of Hydraulic Research, 8(3),287-316. Albertson, M.L., & Simons, D.B. (1964). Fluid Mechanics. In V.T. Chow (Ed.), Handbook of Applied Hydrology (pp. 7-1 to 7-49) . New York: McGraw- Hill Book Co. Bathurst, J.C. (1982). Theoretical Aspects of Flow Resistance. In R.D. Hey, J.C. Bathurst, & C.R. Thorne (Eds.), Gravel-Bed Rivers (pp. 83-108). Chichester: John Wiley & Sons. Bird, R.B., Stewart, W.E., & Lightfoot, E.N. (1960). Transport Phenomena. New York: John Wiley & Sons. Blench, T. (1969, November). Coordination in Mobile-Bed Hydraulics. Journal of the Hydraulics Division, 95, 1871-1898. Bovee, K.D. (1978, January). Probability-of-Use Criteria for the Family Salmohidae. (Instream Flow Information Paper No. 4). Fort Collins, Colorado: Cooperative Instream Flow Service Group. Bray, D .I. (1982a). Flow Resistance in Gravel-Bed Rivers. In R.D. Hey, J.C. Bathurst, & C.R. Thorne (Eds.), Gravel-Bed Rivers (pp. 109-137). Chichester: John Wiley & Sons. Bray, D . I. (1982b). Regime Equations for GraveI-Bed Rivers. In R.D. Hey, J.C. Bathurst, & C.R. Thorne (Eds.), Gravel-Bed Rivers (pp. 517-552). Chichester: John Wiley & Sons. Chang, H .H. (1979, May). Minimum Stream Power and River Channel Patterns. Journal of Hydrology, 41 (3/4), 303-327. Chang, H.H. (1980a, May). Stable Alluvial Canal Design. Journal of the Hydraulics Division, 106, 873-891. Chang, H.H. (1980b, September). Geometry of Gravel Streams. Journal of the Hydraulics Division, 106, 1443-1456. Chow, V.T. (1959). Open-Channel Hvdraulics. New York: McGraw-Hill Book Company. Daily, J.W., & Harleman, D .R . (1966). Fluid Dynamics. Reading, Massachusetts: Addison-Wesley Publishing Co., Inc. Dunne, T., & Leopold, L.B. (1978). Water in Environmental Planning. San Francisco: W.H. Freeman & Co. 96 Fahnestock, R .K . (1963), Morphology and Hydrology of a Glacial Stream-White River, Mount Rainier Washington. (U.S.G.S. Professional Paper No. 422-A), Washington, D.C.: U.S. Government Printing Office. Fox, R.W., & McDonald, A.T. (1978). Introduction to Fluid Mechanics (2nd ed.). New York: John Wiley & Sons. Hartman, C.W. & Johnson, P.R. (1978). Environmental Atlas of Alaska (2nd ed.). Fairbanks, Alaska: University of Alaska. Hey, R.D . (1975a). Design Discharge for Natural Channels. In R.D. Hey, & T.D. Davies (Eds.), Science, Technology and Environmental Management (pp. 73-88). Westmead, England: Saxon House, D-C. Heath Ltd. Hey, R.D. (1975b, December 12-16). Response of Alluvial Channels to River Regulation. Proceedings of the Second World Congress on Water Resources, 5^ 183-188. Hey, R.D. (1978, June). Determinate Hydraulic Geometry of River Channels. Journal of the Hydraulics Division, 104, 869-885. Hey, R.D. (1982). Design Equations for Mobile Gravel-Bed Rivers. In R.D. Hey, J.C. Bathurst, & C.R . Thorn (Eds.), Gravel-Bed Rivers (pp. 553-580). Chichester: John Wiley & Sons. Hyra, R . (1978, June). Methods of Accessing Instream Flows for Recreation. (Instream Flow Information Paper No. 6). Fort Collins, Colorado: Cooperative Instream Flow Service Group. Lane, E.W. (1947, December). Report of the Sub­ committee on Sediment Terminology. Transactions, American Geophysical Union, 28 (6), 936-938. Langbein, W.B. (1949, December). Annual Floods and the Partial-Duration Flood Series. Transactions, American Geophysical Union, 30, 879-881. Leopold, L.B., & Maddock Jr., T . (1953). The Hydraulic Geometry of Stream Channels and Some Physiographic Implications. (U.S.G.S. Professional Paper No. 252). Washington, D.C.: U.S. Government Printing Office. Leopold, L.B., & Wolman, M .G . (1957). River Channel Patterns: Braided, Meandering and Straight. (U.S.G.S. Professional Paper No. 282-B). Washington, D.C.: U.S. Government Printing Office. Leopold, L.B., Wolman, M.G., & Miller, J.P. (1964). Fluvial Processes in Geomorphology. San Francisco: W.H. Freeman & Co. Mackin, J.H. (1948, May). Concept of the Graded River. Geological Society of America Bulletin, 59 (5), 463-512. Mercer, A.G . (1971). Analysis of Alluvial Bed Forms. In H.W . Shen (Ed.) River Mechanics, Volume I (pp. 10-1 to 10-26). Fort Collins, Colorado: Hsieh Wen Shen, Pub. 97 Morel-Seytoux, H.J. (1979). Forecasting of Flows-Flood Frequency Analysis. In H.W. Shen (Ed.), Modeling of Rivers (pp. 3-1 to 3-46). New York: John Wiley & Sons. Olson, R.M. (1966). Engineering Fluid Mechanics (2nd ed.). Scranton, Pennsylvania: International Textbook Company. Prins, A., & de Vries, M. (1971, August 29 to September 3). On Dominant Discharge Concepts for Rivers. International Association for Hydraulic Research 14th Congress, 3, 161-170. Raudkivi, A.J. (1967, September). Analysis of Resistance in Fluvial Channels. Journal of the Hydraulics Division, 93, 73-84. Raudkivi, A.J. (1971, December). Discussion of: Sediment Transportation Mechanics: F . Hydraulic Relations for Alluvial Streams by the Task Committee for Preparation of the Sedimentation Manual, January 1971. Journal of the Hydraulics Division, 97, 2089-2093. Raudkivi, A.J. (1976). Loose Boundary Hydraulics (2nd ed.). Oxford: Pergamon Press. Santos-Cayade, J., & Simons, D.B. (1973). River Response. In H.W. Shen (Ed.), Environmental Impact on Rivers (pp. 1-1 to 1-25). Fort Collins, Colorado: Hsieh Wen Shen, Pub. Schumm, S.A. (1971). Fluvial Geomorphology: Channel Adjustment and River Metamorphosis. In H.W. Shen (Ed.), River Mechanics, Volume I (pp. 5-1 to 5-22). Fort Collins, Colorado: Hsieh Wen Shen, Pub. Schumm1 S.A., & Khan, H.R. (1972, June). Experimental Study of Channel Patterns. Geological Society of America Bulletin, 83 (6), 1755-1770. Shen, H.W. (1971). Wash Load and Bed Load. In H.W. Shen (Ed.), River Mechanics, Volume I (pp. 11-1 to 11-30). Fort Collins, Colorado: Hsieh Wen Shen, Pub. Simons, D.B. (1971). River and Canal Morphology. In H.W. Shen (Ed.), River Mechanics, Volume II (pp. 20-1 to 20-60). Fort Collins, Colorado: Hsieh Wen Shen, Pub. Simons, D.B. (1979). River and Canal Morphology. In H.W. Shen (Ed.), Modeling of Rivers (pp. 5-1 to 5-81). New York: John Wiley & Sons. Simons, D .B ., & Richardson, E.V. (1966). Resistance to Flow in Alluvial Channels. (U.S.G.S. Professional Paper No. 422-J). Washington, D.C.: U .S. Government Printing Office. Simons, D .B., & Senturk, F. (1976). Sediment Transport Technology. Fort Collins, Colorado: Water Resources Publications. 98 Song, C.C.S., & Yang, C.T. (1980, September). Minimum Stream Power: Theory. Journal of the Hydraulics Division, 106, 1477-1487. Stelczer, K . (1981). Bed-Load Transport, Theory and Practice. Littleton, Colorado: Water Resources Publications. Straub, W.O. (1978, December). A Quick and Easy Way to Calculate Critical and Conjugate Depths in Circular Open Channels. Civil Engineering, 48, 70-71. Vanoni, V.A. (Ed.). (1975). Sedimentation Engineering. New York: American Society of Civil Engineers, Publishers. Williams, G.P. (1978, December). Bank-Full Discharge of Rivers. Water Resources Research, 14 (6), 1141-1154. Wolman, M.G., & Leopold, L.B. (1957). River Flood Plains: Some Observations on their Formation. (U.S.G.S. Professional Paper No. 282-C). Washington, D.C.: U.S. Government Printing Office. Wolman, M.G., & Miller, J.P. (1960, January). Magnitude and Frequency of Forces in Geomorphic Processes. Journal of Geology, 68 (I), 54-74. Yang, C.T., & Song, C.C.S. (1379, July). Theory of Minimum Rate of Energy Dissipation. Journal of the Hydraulics Division, 105, 769-784. 99 APPENDICES 100 APPENDIX A REFERENCES CONSULTED BUT NOT CITED 101 Ackers, P., & Charlton, F.G. (1970, September). Meander Geometry Arising from Varying Flows. Journal of Hydrology, XI (3), 230-252. Andrews, E.D. (1980, April). Effective and Bankfull Discharges of Streams in the Yampa River Basin, . Colorado and Wyoming. Journal of Hydrology, 46 (3/4), 311-330. Bagnold, R.A. (1956, December). The Flow of Cohesionless Grains in Fluids. Philosophical Transactions of the Royal Society of London, Series A, 249 (964), 235-297.. Brownlie, W.R. (1980, March). Discussion of: Velocity Profiles and Minimum Stream Power by C.C.S. Song and C.T. Yang, August 1979. Journal of the Hydraulics Division, 106, 473-474. Carlston, C.W. (1965, December). The Relation of Free Meander Geometry to Stream Discharge and its Geomorphic Implications. American Journal of Science, 263, 864-885. Chang, H.Y., Simons, D.B., & Woolhiser, D.A. (1971, February). Flume Experiments on Alternate Bar Formation. Journal of the Waterways, Harbors and Coastal Engineering Division, 97, 155-165. Church, M., & Jones, D . (1982). Channel Bars in Gravel-Bed Rivers. In R.D. Hey, J.C. Bathurst, & C.R. Thorne (Eds.), Gravel-Bed Rivers (pp. 291-338). Chichester: John Wiley & Sons. Falkner, F.H. (1935, January). Studies of River Bed Materials and Their Movement, with Special Reference to the Lower Mississippi River. (U.S. Waterways Experiment Station Paper No. 17). Vicksburg, Mississippi: War Department, U.S. Army Corps of Engineers. Henderson, F.M. (1966). Open Channel Flow. New York: Macmillan Publishing Co. Hey, R.D. (1982). Gravel-Bed Rivers: Form and Processes. In R.D. Hey, J.C. Bathurst, & C.R. Thorne (Eds.), Gravel-Bed Rivers (pp. 5-13). Chichester: John Wiley & Sons. Ikeda, S. (1984, April). Prediction of Alternate Bar Wavelength and Height. Journal of Hydraulic Engineering, H O , 371-386. Kazemipour, A.K., & Apelt, C.J. (1983, October). Energy Losses in Irregular Channels. Journal of Hydraulic Engineering, 109, 1374-1379. Lamb, H . (1932). Hydrodynamics (6th ed.). New York: Cambridge University Press. Lane, E.W. (1957). A Study of the Shape of Channels Formed by Natural Streams Flowing in Erodible Material. (M.R.D. Sediment Series No. 9). Omaha: U.S. Army Engineer Division, Missouri River, Corps of Engineers. 102 Langbein, W.B. (1942, November). Hydraulic Criteria for Sand-Waves. Transactions, American Geophysical Union. Part II, 615-618. Laushey, L.M., Kappus,.U., & Ofwona, M.P. (1967, September 11-14). Magnitude and Rate of Erosion at Culvert Outlets. Proceedings of the Twelfth Congress of the International Association for Hydraulic Research, 3, 338-345. Leopold, L.B., & Wolman, M.G. (1960, June). River Meanders. Geological Society of America Bulletin, 71 (6), 769-793. Li, R.M., & Simons, D.B. (1982). Dynamic Modelling of Channel Responses. In R.D. Hey, J.C. Bathurst, & C .R . Thorne (Eds.), Gravel-Bed Rivers (pp. 469-485). Chichester: John Wiley & Sons. Maddock Jr., T . (1969). The Behavior of Straight Open Channels with Movable Beds. (U.S.G.S. Professional Paper No. 622-A). Washington, D.C.: U.S. Government Printing Office. McNown, J.S., & Malaika, J. (1950, February). Effects of Particle Shape on Settling Velocity at Low Reynolds Numbers. Transactions, American Geophysical Union, 31 (I),,74-82. Nixon, M . (1959, February). A Study of the Bank-Full Discharges of Rivers in England and Wales. Pro­ ceedings, Institution of Civil Engineers, 12, 157-174. Shen, H.W. (1971). Stability of Alluvial Channels. In H.W. Shen (Ed.), River Mechanics, Volume I (pp. 16-1 to 16-33). Fort Collins, Colorado: Hsieh Wen Shen, Pub. Shen, H.W. (1979). Additional Remarks on Extremal Floods, Basic Equations, River Channel Patterns, Modeling Techniques and Research Needs. In H.W. Shen (Ed.), Modeling of Rivers (pp. 20-1 to 20-21). New York: John Wiley & Sons. Simons, D.B. (1967, September 11-14). River Hydraulics. Proceedings of the Twelfth Congress of the Inter­ national Association for Hydraulic Research , 5^, 376-398. Song, C.C.S., & Yang, C.T. (1980, October). Closure to: Velocity Profiles and Minimum Stream Power, August 1979. Journal of the Hydraulics Division, 106, 1696-1697. Task Committee for Preparation of Sediment Manual. (1971, April). Sediment Transportation Mechanics: H . Sediment Discharge Formulas. Journal of the Hydraulics Division, 97, 523-567. Task Force on Bed Forms in Alluvial Channels. (1966, May). Nomenclature for Bed Forms in Alluvial Channels. Journal of the Hydraulics Division, 92, 51-64. White, W .R., Bettess, R., & Paris, E. (1982, October). Analytical Approach to River Regime. Journal of the Hydraulics Division, 108, 1179-1193. 103 Williams, G.P. (1978). Hydraulic Geometry of River Cross Sections^-Theory of Minimum Variance. (U.S.G.S. Professional Paper No. 1029). Washington, D.C.: U .S. Government Printing Office. Yang, C.T. (1972, October). Unit Stream Power and Sediment Transport. Journal of the Hydraulics Division, 98, 1805-1826. Yang, C.T. (1973, October). Incipient Motion and Sediment Transport. Journal of the Hydraulics Division, 99, 1679-1704. Yang, C.T. (1976, July). Minimum Unit Stream Power and Fluvial Hydraulics. Journal of the Hydraulics Division, 102, 919-934. Yang, C.T. (1984, December). Unit Stream Power Equation for Gravel. Journal of Hydraulic Engineering, H O , 1783-1797. Yang, C.T., & Stall, J.B. (1976, May). Applicability of Unit Stream Power Equation. Journal of the Hydraulics Division, 102, 559-568. APPENDIX B DEFINITION OF SYMBOLS cross sectional area of flow; stream width; exponent in equation D-2; stream depth; mean sediment size; total energy at a location; specific energy at the upstream cross section; specific energy at the downstream cross section; Darcy-Weisbach friction factor; Froude number; acceleration due to gravity; vertical direction; head loss at culvert entrance; unit vector in the x-direction; unit vector in the y-direction; unit vector in the z-direction; correction of the Manning roughness coefficient for the effect of channel meandering; Manning roughness coefficient; Manning roughness coefficient due to the effect of grain roughness; correction of the Manning roughness coefficient for the effect of surface irregularities; correction of the Manning roughness coefficient for the effect of variations in the channel cross section size and shape; 106 P Pf P($) P (V + V) Pe Pne Pw Q Q R Re* S S0 SI S2 = correction of the Manning roughness coefficient for the effect of obstructions; = correction of the Manning roughness coefficient for the effect of vegetation; = fluid pressure; = planform; = rate of energy dissipation due to the original velocity vector; •- rate of energy dissipation due to the hypothetical velocity vector; = probability of exceedance; = probability of nonexceedance; = wetted perimeter; = water discharge; = mean water discharge; = bankfulI water discharge; = coefficient of determination; - hydraulic radius; = boundary Reynolds number; = slope of the energy grade line; = measured thalweg slope between the upstream and downstream cross sections; = slope of the energy grade line at the upstream cross section; = slope of the energy grade line at the downstream cross section; 107 t T U U U* U*c v V V + V W W X X = time; = recurrence interval; = component of the original velocity vector in the x-direction; = component of the velocity variation vector in the x-direction; = shear velocity; = critical value of the shear velocity; = component of the original velocity vector in the y-direction; = component of the velocity variation vector in the y-direction; = magnitude of the mean velocity of flow; = velocity in the culvert at critical flow; = critical value of the mean velocity; = original velocity vector representing the mean velocity vector at a point; = hypothetical velocity vector; = velocity variation vector; = component of the original velocity vector in the z-direction; = component of the velocity variation vector in the z-direction; = fall velocity of the bed material; = axis in the longitudinal stream direction; = distance between the upstream and downstream 108 y Yc Yn z V V YS e n y V P O T T TC 4) (V) -» /< (V) vector; and = rate of energy dissipation per unit volume of fluid due to the velocity variation vector. 109 APPENDIX C DEVELOPMENT OF THE MINIMUM RATE OF ENERGY DISSIPATION THEORY FOR ALLUVIAL STREAMS Ill A development of the minimum rate of energy dissipation theory applicable to alluvial streams is given in condensed form by Yang and Song (1979) and Song and Yang (1980). This development is summarized and expanded in this appendix. The development shows that the theory is obtained from the equation of motion. Therefore, provided the conditions of the development are met, the minimum rate of energy dissipation theory is an alternate way of stating the equation of motion. The discussion in this appendix is divided into four sections. The first three sections present the equation of continuity, equations describing the stress tensor, and equation of motion. The last section develops the theory from these equations and determines the conditions for which it is applicable to alluvial streams. In order to develop the theory from these equations, three velocity vectors are considered: the original velocity vector, the hypothetical velocity vector, and the velocity variation vector. The equations to be developed by the first three sections consider the original velocity vector only. The hypothetical velocity vector and the velocity variation vector are discussed in the last section. The minimum rate of energy dissipation theory is developed from equations written in rectangular co­ ordinates. The x, y, and z axes represent the longitudinal, lateral and vertical stream directions, respectively. The symbols u, v, and w represent the components of the time 112 averaged or mean velocity vector at a point in the x, y, and z directions, respectively. Therefore, ^ ^ V = ui + vj + w k ( C - I ) is the mean velocity vector at a point. This is named the original velocity vector. The symbols i, jf, and k represent unit vectors in the x, y , and z directions, respectively. An arrow superscript indicates a vector. Equation of Continuity Daily and Harleman (1966) indicate that water can be considered an incompressible fluid. For an incompressible fluid, Bird, Stewart, and Lightfoot (1960) indicate the equation of continuity is: 9u . 3v . 9w 97 + 97 + 97 VV = O (C-2) Stress Tensor Fox and McDonald (1978) state that water can be considered a Newtonian fluid. Equations given by Bird et al. (1960) show for laminar flow of Newtonian fluids, the constant of proportionality between the fluid stress components and the velocity gradient is the dynamic molecu­ lar viscosity. Bird et al. (1960) and Daily and Harleman (1966) indicate equations describing the fluid stress \ 113 components in turbulent flow are similar to those in laminar flow except the constant of proportionality is the total fluid viscosity. They further indicate the total fluid vis­ cosity is the sum of the dynamic molecular viscosity and the dynamic eddy viscosity. Equation C-2 and the total fluid viscosity can be substituted into the equations describing the fluid stress components for laminar flow of Newtonian fluids. This yields the fluid stress components for laminar or turbulent flow of incompressible Newtonian fluids. These are: tXX = ‘ 2 P ( V + £)I7 (C-3) (C-4) (C-5) (C-6) ( C - 7 ) (C- 8) The above symbols are as follows: T = fluid stress component, P = fluid density, v = y/p = kinematic molecular viscosity,. 114 E = n/P = kinematic eddy viscosity, y = dynamic molecular viscosity, and tI = dynamic eddy viscosity. The sum of all the fluid stress components acting on a water volume equals the stress tensor, L U The water temperature is considered constant, hence the molecular viscosity is also constant. The eddy viscosity depends on the state of the turbulent motion, thus varies with location in the flow. The subscript convention for the stress components in ' equations C-3 to C-8 can be better understood by considering the cubic fluid volume shown in Figure 7. The first subscript indicates the direction of the normal to the face of the fluid volume on which the stress acts. The second subscript indicates the direction in which the stress acts. A normal stress has identical subscripts and a shear stress does not. A stress tensor is symmetric if the shear stress components with subscripts differing only in their order are equal in magnitude (Bird et al., 1960). The shear stress components given in equations C-6 to C-8 are symmetric. Equation of Motion Bird et al. (1960) and Daily and Harleman (1966) indicate the equation of motion for steady flow with 115 Jfc xz Figure 7. Notation for stresses. 116 gravitational body forces is: ,atf p ( u a7 + Vg- + w x f ) + 7P + YVh 'V0T (C- 9) Previously undefined symbols are as follows: p = fluid pressure, Y = fluid specific weight, and h = vertical direction measured positive upward. Bird et al. (1960) indicate the term on the far left of equation C-9 represents the rate of change of momentum per unit volume of fluid. They further indicate from left to right, the other terms in equation C-9 represent pressure surface force per unit volume of fluid, gravitational body force per unit volume of fluid, and viscous surface force per unit volume of fluid. For forces per unit area applied on a typical water volume between two cross sections in an alluvial stream, see Figure 8. Rate of Energy Dissipation This section states the minimum rate of energy dissipation theory and the conditions for which it is applicable,to alluvial streams. Velocity Vector Satisfying the Equation of Motion The substantial time derivative of a variable describes its rate of change with respect to both time and location 117 Z Profile Plan Section A - A Figure 8. Forces per unit area on a typical water volume between two cross sections in an alluvial stream. 118 for a path following the fluid motion (Bird et al., i960). For fluid at a constant temperature under steady flow with gravitational body forces, the substantial time derivative of the total energy per unit volume of fluid is: S t ( ^ ) = + v |~ + WM ) + VP + YVh ] (C- IO) This is deduced from Bird et al. (1960). Previously undefined symbols are as follows: t = time B = total energy at a location, and V = fluid volume. The right side of equation C-IO equals the dot product of the original velocity vector (given in equation C-I) and the left side of equation C-9. Combining equations C-9 and C-IO and rewriting yields: U t ^ ) = z 9 u .3 v9w ( C - 99 2 . )566w Equation C-Il describes the rate of change of energy per unit volume of fluid for a path following the fluid motion. This equation is valid for fluid at a constant temperature under steady flow with gravitational body forces. Bird et al. (1960) indicate the first term on the right of equation C-Il describes an energy change due to viscous surface forces which is reversible. Therefore, this term 119 does not describe the rate at which energy is lost from the fluid. Dunne and Leopold (1978) posit most energy lost from a stream is converted into internal energy and dissipated as heat and a small part of energy lost from a stream is used to transport sediment. Bird et al. (1960) indicate the second term on the right of equation C-Il describes the rate of irreversible conversion of kinetic energy to internal energy by viscous dissipation, per unit volume of fluid. Bird et al. (1960) and Daily and Harleman (1966) show flow in open channels does not cause appreciable temperature changes in the water, hence all the energy converted into internal energy is lost. Thus, when no energy is used to transport sediment, the second term on the right of equation C-Il represents the rate at which energy is lost from a stream (Bird et al., 1960). Energy dissipation is a loss of energy from a stream, hence a negative change of energy. Therefore, the rate of energy dissipation per unit volume of fluid equals the negative of the second term on the right of equation C-Il. Bird et al. (1960) expand the negative of the second term on the right of equation C-Il for symmetrical stress tensors. Combining this with equations C-3 to C-8 yields: 120 (C-1 2 ) The symbol represents the rate of energy dissipation per unit volume of fluid. Equation C-12 is valid for incompressible Newtonian fluids at a constant temperature in steady, laminar or turbulent flow conditions under * gravitational body forces with negligible sediment transport. The (^ ) on the left side indicates that this equation is written considering the original velocity vector given in equation C-I. This velocity vector satisfies the equation of motion. Similar notation is also used with other velocity vectors. Velocity Vector Not Satisfying the Equation of Motion The original velocity vector satisfies the equation of continuity and the equation of motion. Also, this velocity vector has a certain kinematic eddy viscosity and certain . boundary conditions. Yang and Song (1979) consider a hypothetical velocity vector that is assigned the following characteristics. This vector does not satisfy the equation of motion but does satisfy the equation of continuity and has both the same kinematic eddy viscosity and the same boundary conditions as the original velocity vector. The 121 equation of action as given in equation C-9 does not place any specific requirements on either the kinematic eddy viscosity or the boundary conditions. Therefore, it is possible for the hypothetical velocity vector to not satisfy the equation of motion but have the same kinematic eddy viscosity and boundary conditions as the original velocity vector. A velocity vector describing flow in an alluvial stream must necessarily satisfy the equation of motion. Therefore, the hypothetical velocity vector does not describe a real flow situation. This vector is considered merely for the purposes of argument to develop the minimum rate of energy dissipation theory for alluvial streams. This development will show that the flow described by a velocity vector satisfying the equation of motion results in a lesser rate of energy dissipation than the flow described by a velocity vector not satisfying the equation of motion. The hypothetical velocity vector is denoted by: -f" A A ^ A -> A ^ V + v = ( u + u ) i + (v + v ) j + (w + w) k . (C-13) The velocity variation vector is the difference between the original velocity vector and the hypothetical velocity vector. The velocity variation vector is denoted by: 122 "i V = ui + vj + w k . ( C - 1 4) Velocity Variation Vector Albertson and Simons (1964) and Daily and Harleman (1966) indicate the kinematic eddy viscosity is proportional to the average intensity of the turbulence. Because the original and hypothetical velocity vectors given in equations C-I and C-I3 have the same kinematic eddy viscosity, they must also have the same turbulent characteristics. The sum of the original and velocity variation vectors given in equations C-I and C-14 equals the hypothetical velocity vector given in equation C-13. Because the velocity vectors given in equations C-I and C-13 have the same turbulent characteristics, the velocity variation vector given in equation C-14 necessarily has no turbulent characteristics, hence represents laminar flow. Therefore, the kinematic eddy viscosity of the velocity variation vector is zero. Thus, similar to equation C-12, the rate of energy dissipation per unit volume of fluid due to the velocity variation vector is: 4>(V) = pv{2[(-|^-) + RGL Y D3y ♦ ♦ < ! = ( I f ) i O - ( H + I f ) 2 3u 1 + f f > I 123 Hypothetical Velocity Vector The rate of energy dissipation per unit volume of fluid due to the hypothetical velocity vector is developed belowi Similar to equation C-12, 4>(V + V) = p ( v + E ) [ 2 . + f > I 4 + u ) I ( C - 1 6 ) + [-|y(v + v)] + —'-I i, r , 1m u + [|y( U + u) + -|^ (v + v) ] r —•-I i V + V) + |y( W + W) ] + + w) + | y ( u + u ) ] ] . Expanding equation C-16 yields: ( V + V) = p(v + e){2[(|-^ + L L Y D D 3 u 2 3x 3 v 3y 3v 2 3y ( € - 17 ) + ( H + I f ) 2 ] + + I r + I t + + B ^ + l? + Bu . 3 u \ 2 i B i + 3 l } 1 Expanding and rearranging equation C-17 yields: 1 2 4 4>( V + V ) = p ( v + e ) { 2 [ ( | ^ ) + ( | ^ ) + ( 3* ( C + p(v + e){2[2( M # ) + 2 ( ^ % + 2 ( % # ) : 3 x 3 x 3y 3y‘ + 2 ( M % + 2 ( M # ) + 2 ( ^ ^ ) + 2(|l%) 3y S y 7 ' fc' 3 y 3 x 7 ' " x 9 y + 2 ( | | I ^ ) + 2 < l r l f >3z 3z 3 z 3 y + 2 ( ^ l f ) + * < & & > + 2 ^ H ) + 2 < H H 1 W * # " ] + ( f f > .2 + p ( v + e By + 2(is + # ' + 2t|| % ,3y 9 x 2 Equation C-18 can be simplified to: -18) 125 t ( V + V ) = P ( v + e ) { 2 [ ( | ^ ) + ( | j ) + ( f f ) ] ( C - 1 9 ) + + ' ( I f » T?) * ( I f + I f ) ^ » 2p(v + «){«(|a|i) + ( % % ) + (ff ff)]3 v 3 v 3 w 3 w + (If If) + + ( I f + I f ) 2 J ' Note that: p (v + e) = p\) + Pe PV( I + 77) (C-20) Substituting equations C-12, C-I5, and C-20 into equation C-19 gives: 126 4*(V + V) = *(V) + 2p(v + e ) {2[ ("l^ - (C-21 ) + + ( ^ 1 # ) ] + ( ^ 1 ? ) + 3 v 3 v lSy ^yj 3 u 3 u ‘17 3 ? + » (15 %) fi> + 'il 37>3 v 3 v 3 v 3 v 3 v 3 w ♦ < U $ > ♦ ( % i f ) M i f i f ) * ( i f i f ) M i f i f M (if if)} + ( i + f)*(v) Equation C-21 can be simplified to: ( MV + V ) = 4>( V ) + 2 p ( v + e ) { 2 [ ( - § j f | £ ) (C-22) M | f i f ) » ( I f i f ) ] + ( i f + I f x i i3 i I T ' 3x'' 3y + (if + ^xli +f ) + ' f +Ifx-H +If) I3u w 3w + (1 + -5) M V ) . Substituting equations C-3 to C-8 into equation C-22 yields: 127 SVRF D FY O SVRFY W - B L b b + Tyy (C-23) + T z z i l + \ y + I ? ) + V M + + Tz x (I 7 + i i ) ] + (1 + "#)*(%) . Integration by parts is described in many calculus textbooks. This procedure is: /u d v = u v - / vdu ( C - 2 4 ) The original velocity vector and the hypothetical velocity vector have the same boundary conditions. Therefore, the velocity variation vector is zero at the boundaries. Integrating the middle term on the right side of equation C-23 by parts as given above yields: U yLU 4>(V + V) =, (V) + 2 [ u ( ^ U yL U + v (■ •9y (C-25 ) ^ t i ) ] + (I + (V) Substituting equations C-3 to C-8 into equation C-25 yields 128 4>(V + V) = (V) - 2p(v + e){u[2g|(|^) (C-26) + ^ + 1 # ) ] + ^ » f e ) ♦ * , | ( U + + ♦ ! * > ♦ , ± < 1 * + |s > ♦ 2 4 ( i ? ) ] j + (I + ~)( V + V) = ( V ) - 2p(v + E ) { u [ -—- + - —- 3x2 3y2 (C-27) + 0 + ^ + # + H)] + ![ill + ill + ill + ^ifli ♦ |1 + If)] 3yz 3z 3 v 3w. 2 " 3yv 3x ' 3y 3z + ;cS + 0 + 0 + ^ + ^ + l?)]1 + (I + |)*(V) . Substituting equation C-2 into equation C-27 yields: 129 <}>( V + V ) = ) (C-28) + ( 7 p + T F n + (1 + • Statement of the Theory Equations C-I2 and C-I5 indicate that the first and last terms on the right of equation C-28 are the sum of the The first derivative of the velocity can also be called th6 velocity gradient or acceleration. It represents the change in the velocity with location. The middle term on the right velocity components. The second derivative of the velocity can also be called the first derivative of the velocity gradient, the first derivative of the acceleration, or the acceleration gradient. It represents the change in the acceleration with location. i'or rapidly varied flow, the velocity, acceleration, and acceleration gradient can all change significantly with ! ■ location. Therefore, the acceleration in the first and/or the last term in equation C-28 may be comparable to the acceleration gradient in the middle term. For this reason, equation C-28 can not be reduced when considering rapidly squares of the first derivative of the velocity components. of equation C-28 is the sum of the second derivative of the 130 varied flow. For uniform and gradually varied flow, however, the velocity changes little with location. For this type of flow, the acceleration is small and the acceleration gradient is very small. Therefore, the acceleration gradient in the middle term of equation C-28 is much less than the acceleration in the first and last term. For this reason, the middle term on the right of equation C-28 is negligible for uniform and gradually varied flow. Under these conditions, equation C-28 can be reduced to: Equation C-15 indicates the second term on the right of equation C-30 is the sum of squared terms, hence is always positive. Equation C-30 can be rewritten: Integrating equation C-31 over the total volume of fluid yields: zx -*• ^ * *(v + v) = XV) + (I + £)(V) < 4>(V + V) . (C-31 ) P(V) < P(V + V) (C-32) where P represents the rate of energy dissipation. Equation C-32 mathematically states the minimum rate of energy dissipation theory. Equation C-32 is valid for incompressible Newtonian fluids at a constant temperature in steady, laminar or turbulent flow conditions, uniform or gradually varied flow under gravitational body forces with negligible sediment discharge. Water is an incompressible Newtonian fluid. Alluvial streams flow turbulent under gravitational body forces. Thus, for alluvial streams, these requirements are met. Therefore, equation C-32 is valid for alluvial streams having a constant water temperature with steady uniform or gradually varied flow and negligible sediment discharge. The original velocity vector satisfies the equations of continuity and motion. The hypothetical velocity vector does not satisfy the equation of motion, but satisfies the equation of continuity and has the same kinematic eddy viscosity and boundary conditions as the original velocity vector. Therefore, the minimum rate of energy dissipation theory as represented by equation C-32 states: provided certain conditions are met, the flow described by a velocity vector satisfying the equation of motion necessarily results in a lesser rate of energy dissipation than the flow described by a velocity vector not satisfying the equation 131 of motion. APPENDIX D aNulrthh JMnCkesHi 133 Langbein (1949) and others define the annual maximum discharge as the highest instantaneous value in a given water year. Dunne and Leopold (1978) indicate a compilation of these discharges is the annual maximum series. The recurrence interval from this series is the average interval between the occurrence of annual maximum discharges that equal or exceed a given value. The bankfulI discharge has a recurrence interval of approximately 1.5 years On the annual maximum series. Therefore, on the average once in 1.5 years, or two years out of three, the highest discharge for the year will be equal to or will exceed the bankfulI capacity of the channel. Dunne and Leopold (1978) and Morel-Seytoux (1979) indicate: P = I - P = 1 - 1/T = I - 2/3 = .3333. (D-l)ne e * i The above symbols are as follows: P n e = probability the bankfulI discharge will not be exceeded; P0 = probability the bankfulI discharge will be exceeded; and T = recurrence interval, in years, between events that will equal or exceed the bankfulI discharge. Morel-Seytoux (1979) indicates using the Gumbel distribution, the probability of non-exceedance of any 134 discharge is: P = e ne -e-b (D-2) He further indicates that: mdppfoWVRP W q D dWSogY (D-3) The above symbols are as follows: o = standard deviation of the discharges, Q = any water discharge, and Q = mean water discharge. Combining equations D-2 and D-3 gives: Q = Q - 0 . 45a - 0 . 78a1n [ - I n ( P pe ) ] . (D“ 4 > Combining equations D-I and D-4 yields: Qb = Q - 0 . 52a . (D-5) The symbol represents the bankfulI water discharge. The annual maximum series for the streams under study, is presented in Tables 8 and 9. The bankfulI discharges can be found by combining the information in these tables and equation D-5. These discharges are given in Table 10. 135 Table 8. Annual maximum discharges for Copper River tributary. Water year Month Cf S Discharge, Water year Month Discharge, Cf S 1963 Aug. 13 1973 June 98 1964 July 173 1974 June 57 1965 July 135 1975 July 97 1966 June 28 1976 Aug. 40 1967 N . R. 7 1977 Aug. 20 1968 June 29 1978 July H O 1969 Aug. 15 1979 June 45 1970 July 26 1980 June 205 1971 Aug. 19 1981 Aug. .62 1972 July 48 1982 Aug. 115 Note. These data are from U.S.G.S . crest stage gage station no, 15199000. N .R . not recorded. 136 Table 9. Annual maximum discharges for Brooks Creek tributary. Water year Month Discharge, Cf S Water year Month Discharge, cfs 1964 Aug. 137 1974 June 2 1965 Sept. 85 1975 May 168 1966 May 90 1976 N.R. 96 1967 Aug. 43 1977 May 43 1968 N . R . 98 1978 June 47 1969 June 2 1979 May 98 1970 June 42 1980 May 40 1971 Aug. 46 1981 Aug. 9 1972 N . R . 67 1982 May 45 1973 May 32 Note. These data are from U.S.G.S. crest stage gage, station no. 15519200. N. R . not recorded. 137 Table 10. Bankfull discharge calculation. Stream name Mean discharge, Q, Cf S Standard deviation, O Bankfull discharge, Q b , Cf S Copper River 67.10 56.66 37.64 tributary Brooks Creek tributary 62.63 44.24 39.63 APPENDIX E MEASURED VALUES 139 Table Tl. Culvert data. Parameter Copper River Brooks Creek tributary tributary Diameter, ft 9.5 5 Length, ft 164 86 Elevation of upstream invert, ft 100.00 100.00 Elevation of downstream invert, ft 91.67 94.90 Slope 0.0508 0.0593 Material corrugated corrugated metal metal 140 Table 12. Copper River tributary, downstream, measured data. Cross section Thalweg elev, ft Bank elev, ft Bankfull width, ft MSS, ft Distance, ft OD 88.63 91.62 21.5 - 3 ID 88.43 91.93 40 - 60 2D 88.52 91.58 18 - 35 3D 88.16 93.04 41 0.0417 52 4D 85.97 89.41 20 0.0417 33 5D 84.44 87.49 12 0.0417 34 6D 82.49 85.91 20 0.0417 36 7D 82.03 83.40 20 0.0625 31 8D 80.37 81.38 7.5 0.0625 131 9D 74.81 76.16 10 0.0625 162 IOD 68.77 70.32 6.5 0.0625 262 IlD 59.19 61.21 13 0.0833 207 12D 52.04 54.33 9.5 0.0625 + Note. Elev = elevation. MSS = mean sediment size. Unreported data indicated by a -. Unapplicable data indicated by a +. Distance is that to next downstream cross section. 141 Table 13. Copper River tributary, upstream, measured data. Cross section Thalweg elev, ft Bank elev, ft Bankfull width ft MSS, ft Distance, ft 14U 124.93 126.78 7.5 0.0276 + 13U 120.99 122.47 3.5 0.0276 260 12U 113.69 114.51 6 0.0208 236 IlU 111.15 112.92 4 0.0833 103 IOU 107.69 109.10 5.5 0.0417 111 9U 105.57 107.28 8 0.0104 75 8U 103.72 105.49 12.5 0.0276 42 7U 102.68 104.30 11 0.0417 . 44 6U 101.38 103.70 6.5 0.0104 13 5U 100.09 101.91 6.5 0.0208 48 4U 99.98 101.85 9 0.0130 8 3U 99.90 101.51 13.5 0.0052 32 2U 99.52 102.14 21 0.0005 8 IU 99.30 101.05 20 0.0005 19 OU 99.60 101.18 27 0.0005 28 Note. Elev = elevation. MSS = mean sediment size. Unapplicable data indicated by a +. Distance is that to next upstream cross section. 142 Table 14. Brooks Creek tributary, downstream, measured • data. Cross section Thalweg elev, ft Bank elev, ft Bankfull width, ft MSS, ft Distance, ft OD 94.71 97.16 7.5 - 10 ID 93.42 96.55 21 - 19 2D 95.38 97.40 19.4 - 23 3D 95.32 96.57 15.3 0.0208 32 4D 94.43 95.91 9 0.0208 28 SE) 94.22 95.08 10.7 0.0208 40 GD 93.09 94.76 7.3 0.0272 63 7D 90.71 91.90 7 0.0208 131 8D 86.61 88.04 6 0.0208 115 9D 84.34 85.82 6.5 0.1042 170 IOD 81.28 82.60 8 0.1250 172 IlD 77.90 79.09 12 0.0208 213 12D 73.09 74.29 8 0.0208 + Note. Elev = elevation. MSS = mean sediment size. Unrepdrted data indicated by a -. Unapplicable data indicated by a +.. Distance is that to next downstream cross section. 143 Table 15. Brooks Creek tributary, upstream, measured data. Cross section Thalweg elev, ft Bank elev, ft Bankfull width, ft MSS, ft Distance, ft 14U 118.30 118.99 7 0.0104 + 13U 113.25 113.95 10 0.0208 212 12U 111.06 111.46 7.5 0.0313 167 IlU 108.37 109.09 7 0.0417 137 IOU 105.62 106.71 8.5 0.0313 151 9U 104.17 104.88 7 0.0313 108 8U 102.00 102.95 7.5 0.0058 103 7U 101.25 102.29 t-CO 0.0449 43 6U 101.04 102.31 7 0.0438 17 5U 100.74 103.21 6.7 0.0428 16 4U 100.71 102.03 6.2 0.0417 19 3U 101.05 101.69 7.5 0.0417 11 2U 100.79 102.21 10 0.0417 14 IU 100.20 101.45 12 0.0417 13 OU 100.08 101.32 7.2 0.0417 10 Note. Elev = elevation. MSS = mean sediment size. Unapplicable data indicated by a +. Distance is that to next upstream cross section. Sl op e Wi dt h, Ft De pt h, Ft Me an Se di me nt Si ze , Ft 144 OD 2D 4D 6D BD IOD 0.0833 0.0625 0.0417 5 3* I' 40. 30. 20 10 0.07 0.05 0.03. 0.01 U < < I I ► , I I I i I I > I > I. I I C O O O O O O O O I 500 12D I 750 I 1000 Distance from culvert. Ft Figure 9. Copper River tributary, downstream, meas u r e d d a t a . Sl op e W i d t h , Ft De pt h, Ft Me an Se di me nt Si ze , Ft 145 OU 4U 8U IOU 12U 14U 0.08 0.04. > 0.00. 2.5- 2 . 0 - 1.5- 1.0. 30. 20. 10 . I 0 - 0.12. 0.08 0.04. 0.00 0 250 500 750 1000 Distance from Culvert, Ft Figure 10. Copper River tributary, upstream, measured data. si oP e Wi dt h. Ft De pt h, Ft Me an Se di me nt Si ze , Ft 1 4 6 OD 4D 8D IOD 0 . 1 4 _ 4J E £ o . i o . (► "S , L I I C U Cl * 2 0 )x : * ’ u a S 1 0 C < > > < \ C i I> ( < O I> 2 0 „! 4J 1 5 ) (\ £ f I ) •H 3 ( I < I C < ' « I > ) Cl 0 . 1 2 - Cla O w 0 . 0 4 p O O O U O O O O O 0 . 0 0 *■ 0 250 500 750 1000 Distance from Culvert, Ft ■Figure 11. B r o o k s C r e e k t r i b u t a r y , d o w n s t r e a m , m e a s u r e d data. Sl op e Wi dt h, Ft De pt h, Ft Me an Se di me nt Si ze , Ft 147 Ou 4u 8u 0.05 0.03 0.01 0.04 0.02 0.00 1000 F i g u r e 12 Distance from Culvert, Ft B r o o k s C r e e k t r i b u t a r y , u p s t r e a m , m e a s u r e d d a t a 148 Table 16. Copper River tributary. bankfulI Froude number. Cross section Frbude number Cross section Froude number OD 0.06 OU 0.12 ID 0.03 IU 0.14 2D 0.07 2U 0.07 3D 0.02 3U 0.24 4D 0.05 4U 0.29 5D 0.10 5U 0.42 6D 0.05 6U 0.29 7D 0.21 7 U 0.29 8D 0.87 8U 0.23 9D 0.42 9U 0.37 IOD 0.53 IOU 0.72 IlD 0.18 IlU 0.70 12D 0.20 12U 1.49 13U 1.05 14U 0.35 Table 17. Brooks Creek 149 tributary. bankfulI Froude number. Cross Froude Cross Froude section number section number OD 0.24 OU 0.70 ID 0.06 IU . 0.42 2D 0.13 2U 0.41 3D 0.33 3U 1.82 4D 0.43 4U 0.74 5D 0.82 5U 0.27 6D 0.44 6U . 0.70 7D 0.77 7U 0.76 8D 0.68 8U 1.00 9D 0.60 9U 1.67 IOD 0.58 IOU 0.72 IlD 0.45 IlU 1.63 12D 0.66 12U 3.68 13U 1.19 14U 1.74 150 Table 18. Cross sections locating break in measured data. Parameter Copper River tributary Brooks Creek tributary Upstream Down­ stream Upstream Down­ stream Width 3-4 7-8 2-3 5—6 Depth NB 6-7 6-7 2-3 MSS 4—5 6-7 7-8 NB Slope 4 — 5 NB NB 4—5 Average 4—5 6-7 5—6 4-5 Note. MSS = mean sediment size. NB = no break in measured data. 151 APPENDIX F FLOW CONDITIONS NEAR CULVERTS This appendix estimates the pre-culvert equilibrium bank elevations and the post-culvert equilibrium bank elevations at stream cross section OU. These elevations are used in Table 3 to determine whether the dependent variables in subreaches altered by the culvert are adjusting toward equilibrium. Table 19 estimates the pre-culvert equilibrium bank elevations at stream cross section OU. The measured bank elevation from the unaltered cross section nearest the culvert is extended to Cross section OU by using the average measured thalweg slope from the subreaches unaltered by the culvert. Table 20 shows that the normal depth in both culverts is less than critical depth at the bankfulI discharge. For the Brooks Creek tributary, however, at bankfulI discharge the tai!water depth above the culvert invert is greater than the critical depth. Does this force the culvert to flow subcritical? Table 20 also shows this stream flows sub- critical downstream of the culvert. Further, it shows that, if at least part of the culvert flows supercritical, the sequent depth to the culvert normal depth, that is, the depth in the culvert after a hydraulic jump, is greater than the imposed tai!water depth. When the sequent depth is greater than the tai!water depth, the hydraulic jump occurs in the mild channel below the culvert (Chow, 1959). Table 20 shows for the Copper River tributary, the tailwater 152 J fl I ■ /■ I 153 Table 19. Estimated pre-culvert bank elevation, cross section OU. Item Copper River Brooks Creek tributary tributary Upstm Dnstm Upstm Dnstm Closest unaltered 5U 7D 6U 5D cross section (a) Bank elevation 101.91 83.40 102.31 95.08 in (a), ft (b) Distance from (a) 95 417 83 198 to OU, ft (c) Slope in un- 0.0267 0.0378 0.0184 0.0234 altered subreaches (d) Estimated bank elev. 99.37 99.16 100.78 99.71 at OU, ft (e) Average of ■(e), ft 99.27 100.25 154 Table 20. Culvert control location at bankfulI conditions. Item Copper R. tributary Brooks Creek tributary Critical depth, yc fta 1.43 1.74 Manning's coeff., n a 0.024 0.024 Normal depth, y , fta 0.96 1.14 Stream WS, culvert exit, ft 91.62 97.16 Dnstm culvert inv. el.. ft 91.67 94.90 Tailwater depth above Below 2.26 culvert invert, ft invert Stream Fr dnstm culvert - 0.24 Culvert Fr at y • n 2.23 2.27 Sequent depth, ftb — 2.46 Note. Unapplicable data indicated by a -. WS = water surface elevation, inv. = invert, and Fr = Froude number. Critical depth in culvert = y . Normal depth in the culvert = y^. ® Calculated from Chow (1959). ^ Calculated from Straub (1978). 155 elevation is below the culvert invert at the bankfulI discharge. Therefore, both of the study culverts flow supercritical at the bankfulI discharge. Hence, the discharge through each culvert is controlled at the culvert entrance provided the following two conditions are met. First, flow must pass through critical at the culvert en­ trance. This requires subcritical flow in the stream just upstream of the culvert. Second, the culvert entrance must not be submerged. This requires the headwater depth over the top of the culvert to not exceed about half the culvert diameter. The steady flow energy equation written between a cross section just upstream and just downstream of the culvert entrance is: Zs + a .+ V2/2g = Zc + yc + Vc2/2g + (F-l) The above symbols are as follows: Z s = thalweg elevation in the stream at cross section OUf d = depth in the stream at cross section OU, V = velocity in the stream at cross section OU, Z = culvert invert elevation, c y = critical depth in the culvert, V = velocity in the culvert at critical flow, and c = 0.5 Vj2 /2g = head loss at culvert entrance. Using equation F-l, Table 21 presents the depth and the 156 water surface elevation in each stream just upstream of the culvert. Table 21 shows for each depth found, the assump­ tions required for inlet control are true. That is, flow does pass through critical at the culvert entrance and the culvert entrance is unsubmerged. 157 Table 21. Estimated post-culvert bank elevation, cross section OU- Item Copper River tributary Brooks Creek tributary Discharge, cfs . (a) 37.6 39.6 Culvert dia., ft (b) 9.5 5.0 Z , ft C (c) 100.00 100.00 y C- ft (d) 1.43 1.74 Vc, fps (e) 5.64 6.47 z s, ft (f) 99.60 100.08 V, fps (g) 1.39/d 5.50/d Depth, d, ft (h) 2.57 2.57 WS at OU, ft (f) + (h) 102.17 102.65 Elev. at top of upstm 109.50 105.00 end of culvert, ft Stream Froude no. upstm culvert 0.06 0.24 Note. Culvert invert elevation = Z . Critical depth in --- c culvert - y . Velocity in culvert at critical flow = Vc. c Thalweg elevation at OU = Z^. Velocity at OU = V. Depth at OU = d. WS = water surface elevation. 158 APPENDIX G SUMMARY OF THE MINRED METHOD I159 Preliminary 1. Input Q, the water discharge with a 1.5 year recurrence interval on the annual maximum series. 2. Input Sg, the measured thalweg slope between the upstream and downstream cross sections. 3. Input the measured distance between the upstream and downstream cross sections. 4. Input the stream width, b, measured at the downstream cross section. 5. Input the mean sediment size, d , measured at the s downstream cross section. 6. Calculate the Manning roughness coefficient. Chow (1959) indicates that nQ = 0.0342d . For the streams under study, it is estimated from Chow (1959) that (a) n^ = 0.010 (b) n2 = 0.005, (c) n 3 = 0.000, (d) n^ = 0.005, and (e) m =1.15. Therefore, from equation 4-1, the Manning 5 1/6 roughness Coefficient, n = (0.0342ds + 0.02) (1.15). 7. If D loop has never been executed, go to item 10. 8. If the existing value of n is similar to the value of n at last exit from D loop, go to item 40. 9. Go to item 20. 10. Input the stream depth, d, measured at the down­ stream cross section. D Loop Calculate the cross sectional area of flow, A=bd.11. 12 . Calculate the wetted perimeter, E^=b+2d. 13. Calculate the hydraulic radius, R=AyrPw . 14. Calculate the magnitude of the mean velocity of flow, V=Q/A. 15. If this is not the first time through D loop, go to item 20. 16. Calculate the specific 2 cross section, E2=d+V /(2g). 17. Calculate the slope of the downstream cross section, S2 18. Modify d. 19. Go to item 11. 20. Calculate the specific 2 cross section, E1=d+V ,/(2g). 21. Calculate the slope of the upstream cross section, S 1 = 22. Calculate the distance downstream cross sections, X = (E2 - E1) /{S0 - [(S1 + S12)/2]} . 23. If the measured distance between the upstream and downstream cross sections is not similar to the calculated distance, go to item 18. 24. If DS loop has never been executed, go to item 28. 25. If the current value of d is different from the value of d at last exit from DS loop, go to item 28. 26. If B loop has never been executed, go to item 40. 160 energy at the downstream the energy grade line at = [Vn/ (1.49R°*67 ) ]2 . energy at the upstream the energy grade line at 0.67 2 tVn/(1.49R ) ] . between the upstream and 161 27. If the current value of b is the same as the value of b at last exit from DS loop, go to item 40. DS Loop 28. Calculate the shear velocity, u* = CgRS1)0"5. 29. Calculate the boundary Reynolds number, Re* = U*ds/v. 30. Enter Shields diagram with the value of Re* to determine the critical value of the dimensionless shear stress,LI*c 31. Calculate the critical value of the bed shear stress, LI O LIC Rcn W 32. Calculate the critical value of the shear velocity, U*c O RvPz,Yd1U 33. Calculate the dimensionless shear stress, T*= U*2p/[(YS - Y) 500.0 3 a+ V g2gxgg (H-5) Raudkivi1s Graphical Relation I. Gravel: a. 0 .0325 I L I < 0.081 (H-6) 3 5 V = (1 .8125 x 10 - I .4854 x 10 x* cr + 4.9529 x 10 T* - . 8 . 2 2 7 0 x 10 x* + 6 .7710 x IO8X* + 2.2050 x 1 0% * ) ( U* - U*^) = . 9948 166 b. 0.081 ! _ » _< 0.14 (H-7) 2 Vcr = (53.1301 - 9 .7323 x 10 _ » + 7.8340 x I O N * - 2.1152 x 1 0% * ) r 2 = . 9944 Vcr = (13 .8092 - 2 0 . 9980 ( V = ppvE)[2uV C Epv)2 - -( V1 I 6 V - U*c ) * r 2 = . 9964 2. Sand a . 0 .0325 < _ » < 0.081 V = (31.1591 - 3 .1520 x 10 _ » cr * + 1 . 5535 x 1 0% * - 2.6422 x 1 0% * ) ^ 0 .14 < T* <_ 0.51 (H-B) Equation H-6 b. 0.081 < _ » _< 0.20 (H - 9 ) r2 .9982 c. 0 .20 < T* j: 0 .325 (H- IO) Vcr = (15 .2180 - 50 .9079 r * + 1.1585 x 10%* W9pdUdd8pLIYRtI - U*c)= r 2 = .9916 d. 0 .325 m LI m 0 .70 ( H - I l ) 2 Vcy, = (7 .9830 + 8.00 64 T * - 36 .7524?* = E [ vp ) -2 ( V 1 IU * - U * c ) = r 2 = .9973 e. 0 .70 < ( V H ] v-r I t C ] e1 2 2 Vcr = ( - 1 . 3 042 x 10 + 4 .4408 x 10 ( V 2 2 2 3 - 4.3875 x 10 L I + 1.4198 x 10 ?* ) 2 2 I (U* - U*c): r 2 = .9775 167 3. F ine Sand, Lower Flow Regime a. 0 .0325 < LI _< 0.081 Equation H-6 168 b. 0.081 < Cf P g2Yg E q u a tio n H-9 c . 0 .20 < LI _< 0 .95 (H -13) 2 V = ( 11 .6343 - 18.8550 LI + 28.5969 %* 5 6h2hSYt 3 a w^.= a 5 = a |wa r 2 = .9667 F in e Sand, Upper F low Regime (H -14) 0 .575 <. %* ^ 3 .70 Vc r = (25 .0635 + 9.5466%* - 28.6237%* +18.2024%* - 4.6962%* + 0.43.56%*) ( U * - U * c ) = .9964 169 APPENDIX I CALCULATED VALUES I 170 Table 22. data. Copper River tributary, downstream. calculated Cross section Width, ft Depth, ft MSS, ft EGL slope 3D 26.93 2.40 0.0048 0.0000880 4D 26.61 2.33 0.0042 0.0000999 5D 27.08 2.62 0.0041 0.0000663 GD 58.91 1.14 0.0043 0.0001880 7D - - - - GD 13.08 0.64 0.3845 0.0606194 9D 6.18 I . 12 0.4531 0.057848 IOD 11.21 0.72 0.3305 0.054898 IlD 9.85 0.87 0.3600 0.041153 Note. MSS = mean sediment size. EGL = energy grade line. No calculated data indicated by a Culvert Schematic location of downstream crossFigure 13 sections 171 Table 23. data. Copper River tributary. upstream, calculated Cross section Width, ft Depth, ft MSS, ft EGL slope IlU 6.77 1.27 0.2082 0.0284891 IOU 68.97 0.96 0.0047 0.0002418 9U 73.37 1.00 0.0039 0.0001855 8U 56.80 1.13 0.0048 0.0002119 7U 41.30 1.62 0.0039 0.0001237 6U 53.37 1.27 0.0041 0.0001611 5U - - - - 4U 55.88 1.59 0.0008 0.0000567 3U 33.50 2.36 0.0007 0.0000459 2U 61.85 1.58 0.0007 0.0000473 Note. MSS = mean sediment size. EGL = energy grade line. No calculated data indicated by a -. Figure 14. Schematic location of upstream cross sections. 172 Table 24. data. Brooks Creek tributary, downstream, calculated Cross section Width, ft Depth, ft MSS, ft EGL slope 3D 9.00 1.13 0.1082 0.0147986 4D - - - - SD 46.39 1.16 0.0068 0.0003389 GD 8.62 1.02 0.2895 0.0373274 7D 7.83 I . 14 0.2669 0.0322400 8D 8.90 I . 38 0.1099 0.0109072 9D 9.40 I . 22 0.1315 0.0152865 IOD 14.77 0.77 0.1765 0.0239626 IlD 10.19 1.01 0.2104 . 0.0240062 Note. MSS = mean sediment size. EGL = energy grade line. No calculated data indicated by a -. 173 Table 25. data. Brooks Creek tributary, upstream, calculated Cross Width, Depth, MSS, EGL section ft ft ft slope 8U 48.63 1.23 0.0056 0.0002491 7U - - - - 6U - - - - 5U - - - — 4U - - - 3U - - - - 2U 61.96 0.96 0.0063 0.0003537 Note. MSS = mean sediment size. EGL = energy grade line. No calculated data indicated by a -. MONTANA STATE UNIVERSITY LIBRARIES 3 1762 1001 5247 7