Erosion Rate is a Function of Erodibility and Excess Shear Stress = k ( o - c ) From Relation between Shear Stress and Erosion We Calculate c and

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Equilibrium, Shear Stress, Stream Power and Trends of Vertical Adjustment Andrew Simon USDA-ARS, Oxford, MS asimon@msa-oxford.ars.usda.

Cohesive: silts and clays Resistance is derived from electro-chemical interparticle forces under zero normal stress Shields Diagram Shields Diagram by Particle Diameter Cohesive

1 Equilibrium, Shear Stress, Stream Power and Trends of Vertical Adjustment Andrew Simon USDA-ARS, Oxford, MS Non-Cohesive versus Cohesive Materials Non-cohesive: sands and gravels etc. Resistance is due solely to particle size, weight, shape and hiding. Cohesive: silts and clays Resistance is derived from electro-chemical interparticle forces under zero normal stress Shields Diagram Shields Diagram by Particle Diameter Cohesive Materials Denotes uncertainty Excludes cohesives Heterogeneous Beds Erosion of Cohesives by Hydraulic Shear Need for a means to determine critical shear stress ( c ) and the erodibility coefficient (k) in-situ for soils and sediments. k s = 3* D 84 1

2 Erosion Rate is a Function of Erodibility and Excess Shear Stress = k ( o - c ) = erosion rate (m/s) Obtained from jet-test k = erodibility coefficient (m 3 /N-s) device o = boundary shear stress (Pa) c = critical shear stress (Pa) ( o - c ) = excess shear stress Critical shear stress is the stress required to initiate erosion. Impinging Jet Applies Shear Stress to Bed Jet Nozzle Impinging Jet Applies Shear Stress to Bed Jet Nozzle From Relation between Shear Stress and Erosion We Calculate c and As scour hole depth increases, shear stress decreases. 3 /sec Erosion Rate, cm 3 k (cm 3 /Pa/sec) c Shear Stress, Pa General Relation for Erodibility and Critical Shear Stress Erodibility, m 3 /N-s k = 0.1 c -0.5 Where; c = critical shear stress (Pa), x, y = empirical constants ERODIBILITY COEFFICIENT (k), IN cm 3 /N N-s k = 0.09 c ERODIBILITY COEFFICIENT (k) Revised Erodibility Relation y = x R 2 = E E E E E E E E E E E+03 CRITICAL SHEAR STRESS (Pa) CRITICAL SHEAR STRESS, IN Pa 2

Distributions: Critical Shear Stress PERCENTI LE

Kalamazoo River James Creek Shades Creek 40

Nebraska N Fork Broad River 20 Tualatin River

Erodibility Coefficient PERCEN TILE 100 Yalobusha

Creek 80 Missouri River Upper Truckee River 70 W.

Nebraska N Fork Broad River Tualatin River System

Data 40 30 20 10 0 0.001 0.010 0.100 1.000 10.

000 ERODIBILITY COEFFICIENT (k) Mapping Critical

3 Distributions: Critical Shear Stress PERCENTI LE Yalobusha River System 50 Kalamazoo River James Creek Shades Creek 40 Missouri River Upper Truckee River 30 W. Iowa, E. Nebraska N Fork Broad River 20 Tualatin River System Tombigbee River 10 S Branch Buffalo River All Data CRITICAL SHEAR STRESS (Pa) Distributions: Erodibility Coefficient PERCEN TILE 100 Yalobusha River System Kalamazoo River 90 James Creek Shades Creek 80 Missouri River Upper Truckee River 70 W. Iowa, E. Nebraska N Fork Broad River Tualatin River System 60 Tombigbee River SB Branch hbuffalo River 50 All Data ERODIBILITY COEFFICIENT (k) Mapping Critical Shear Stress: Yalobusha River Basin, Mississippi VS Idealized Adjustment Trends For a given discharge (Q) S e n c d Adjustment: Boundary Shear Stress Adjustment: Increasing Resistance 3

4 Adjustment: Increasing Resistance Adjustment: (Excess Shear Stress) Degrading Reach Boundary Shear Stress: Range of Flows Adjustment: Excess Shear Stress Degrading Reach Shear stress, in N/m 2 Excess shear stress Adjustment: (Excess Shear Stress) Aggrading Reach Adjustment of Force and Resistance 4

5 Results of Adjustment Decreasing Sediment Loads with Time Experimental Results Toutle River System Total and Unit Stream Power = w y V S = Q S = total stream power per unit length of channel = specific weight of water w = water-surface width y = hydraulic depth v = mean flow velocity Q = water discharge S = energy slope Adjustment: Unit Stream Power w = / ( w y) = V S where w = stream power per unit weight of water Flow Energy Total Mechanical Energy H = z + y + ( v2 / 2 g) where H = total mechanical energy (head) z = mean channel-bed elevation (datum head) = coefficient for non-uniform distribution velocity y = hydraulic depth h( (pressure head) g = acceleration of gravity Head Loss over a reach due to Friction hf = [z1 + y1 + ( 1 v12 / 2g)]- [z2 + y2 + ( 2 v22 / 2g)] Head, Relative to channel bed Es = y + ( v2 / 2g) = y + [ Q2 / (2 g w2 y2)] Adjustment: Total Mechanical Energy As a working hypothesis we assume that a fluvial system has been disturbed in a manner such that the energy available to the system (potential and kinetic) has been increased. We further assume that with time, the system will adjust such that the energy at a point (head) and the energy dissipated over a reach (head loss), is decreased. Now, for a given discharge, consider how different fluvial processes will change (increase or decrease) the different variables in the energy equations. 5

6 Adjustment: Energy Dissipation Trends of Vertical Adjustment and Determining Equilibrium Minimization of energy dissipation Determining Equilibrium Recall definition A stream in equilibrium is one in which over a period of years, slope is adjusted such that there is no net aggradation or degradation on the channel bed (or widening or narrowing) OR There is a balance between energy conditions at the reach in question with energy and materials being delivered from upstream Causes of Channel Incision Trends of Incision: Channelization Trends of Incision: Below Dams 6

7 Bed-level Trends Along a Reach Bed-level Trends Along a Reach Empirical Functions to Describe Incision E = at b E = elevation of the channel bed a = coefficient; approximately, the pre-disturbance elevation t = time (years), since year before start of adjustment b = dimensionless exponent indicating rate of change on the bed (+) for aggradation, (-) for degradation E/ E o = a + b e -kt E = elevation of the channel bed E o = initial elevation of the channel bed a = dimensionless coefficient, = the dimensionless elevation a > 1 = aggradation, a < 1 = degradation b = dimensionless coefficient, = total change of elevation b > 0 = degradation, b < 0 = aggradation k = coefficient indicating decreasing rate of change on the bed Empirical Model of Bed-level Response Comparison of the Two Bed-level Functions A Natural Disturbance (Toutle River System) 7

8 Bed-Level Response Bed Response: Toutle River System Upstream disturbance, addition of potential energy, sub-alpine environment Comparison with Coastal Plain Adjustment Model of Long-Term Bed Adjustment Downstream disturbance, increase in gradient, coastal plain environment 8

PART 2:! FLUVIAL HYDRAULICS" HYDROEUROPE

PART 2:! FLUVIAL HYDRAULICS HYDROEUROPE PART 2:! FLUVIAL HYDRAULICS" HYDROEUROPE 2009 1 HYDROEUROPE 2009 2 About shear stress!! Extremely complex concept, can not be measured directly!! Computation is based on very primitive hypotheses that