1. HYDRAULICS AND SEDIMENT TRANSPORT: RIVERS AND TURBIDITY CURRENTS
CEE 598, GEOL 593
TURBIDITY CURRENTS: MORPHODYNAMICS AND DEPOSITS
LECTURE 6
HYDRAULICS AND SEDIMENT TRANSPORT:
RIVERS AND TURBIDITY CURRENTS
Head of a turbidity current in the laboratory
From PhD thesis of M. H. Garcia

2. STREAMWISE VELOCITY AND CONCENTRATION PROFILES: RIVER AND TURBIDITY CURRENT
u = local streamwise flow velocity averaged over turbulence
c = local streamwise volume suspended sediment concentration averaged over turbulence
z = upward normal direction (nearly vertical in most cases of interest) z z

3. VELOCITY AND CONCENTRATION PROFILES BEFORE AND AFTER A HYDRAULIC JUMP
The jump is caused by a break in slope
Garcia and Parker (1989)

4. VOLUME FLUX OF FLOWING FLUID AND SUSPENDED SEDIMENT
The flux of any quantity is the rate at which it crosses a section per unit time per unit area.
So flux = discharge/area x
ut A u
utA
The fluid volume that crosses the section in time t is Aut
The suspended sediment volume that crosses is cAut
The streamwise momentum that crosses is wuAut
The fluid volume flux = u
The suspended sediment volume flux = uc
The streamwise momentum flux = wu2

5. LAYER-AVERAGED QUANTITIES: RIVER
In the case of a river, layer = depth
H = flow depth
U = layer-averaged flow velocity
C = layer-averaged volume suspended sediment concentration
(based on flux)
Now let
qw = fluid volume discharge per unit width (normal to flow)
qs = suspended sediment discharge per unit width (normal to flow)
discharge/width = integral of flux in upward normal direction

6. FOR A RIVER:
Flux-based average values U and C
Or thus z

7. LAYER-AVERAGED QUANTITIES: TURBIDITY CURRENT
The upper interface is diffuse!
So how do we define U, C, H?

8. USE THREE INTEGRALS, NOT TWO
Let
qw = fluid volume discharge per unit width
qs = suspended sediment discharge per unit width
qm = forward momentum discharge per unit width
Integrate in z to “infinity.” z

9. FOR A TURBIDITY CURRENT
Three equations determine three unknowns U, C, H, which can be computed from u(z) and c(z).

10. BED SHEAR STRESS AND SHEAR VELOCITY
Consider a river or turbidity current channel that is wide and can be approximated as rectangular.
The bed shear stress b is the force per unit area with which the flow pulls the bed downstream (bed pulls the flow upstream) [ML-1T-2]
The bed shear stress is related to the flow velocity through a dimensionless bed resistance coefficient (bed friction coefficient) Cf, where
The bed shear velocity u [L/T] is defined as
Between the above two equations,
where Cz = dimensionless Chezy resistance coefficient

11. SOME DIMENSIONLESS PARAMETERS
D = grain size [L]
 = kinematic viscosity of water [L2/T], ~ 1x10-6 m2/s
g = gravitational acceleration [L/T2]
R = submerged specific gravity of sediment [1]
Froude number ~ (inertial force)/(gravitational force)
Flow Reynolds number ~ (inertial force)/viscous force): must be >~ 500 for
turbulent flow
Particle Reynolds number ~ (dimensionless particle size)3/2

12. SOME DIMENSIONLESS PARAMETERS contd.
Shields number ~ (impelling force on bed particle/ resistive force on bed
particle): characterizes sediment mobility
Now let c denote the “critical” Shields number at the threshold of motion of a particle of size D and submerged specific gravity R. Modified Shields relation:

13. SHIELDS DIAGRAM motion no motion
The silt-sand and sand-gravel borders correspond to the values of Rep computed with R = 1.65,  = 0.01 cm2/s and D = mm and 2 mm, respectively.
motion
no motion

14. CRITERION FOR SIGNIFICANT SUSPENSION
But recall
where
and
Thus the condition
and the relation of Dietrich (1982):
specifies a unique curve
defining the threshold for significant suspension.

15. SHIELDS DIAGRAM WITH CRITERION FOR SIGNIFICANT SUSPENSION
Suspension is significant when u/vs >~ 1

16. NORMAL OPEN-CHANNEL FLOW IN A WIDE CHANNEL
Normal flow is an equilibrium state defined by a perfect balance between the downstream gravitational impelling force and resistive bed force. The resulting flow is constant in time and in the downstream, or x direction.
Parameters:
x = downstream coordinate [L]
H = flow depth [L]
U = flow velocity [L/T]
qw = water discharge per unit width [L2T-1]
B = width [L]
Qw = qwB = water discharge [L3/T]
g = acceleration of gravity [L/T2]
= bed angle [1]
tb = bed boundary shear stress [M/L/T2]
S = tan = streamwise bed slope [1]
(cos   1; sin   tan   S)
w = water density [M/L3]
The bed slope angle  of the great majority of alluvial rivers is sufficiently small to allow the approximations

17. THE DEPTH-SLOPE RELATION FOR NORMAL OPEN-CHANNEL FLOW
Conservation of water mass (= conservation of water volume as water can be treated as incompressible):
Conservation of downstream momentum:
Impelling force (downstream component of weight of water) =
resistive force
Reduce to obtain depth-slope product rule for normal flow:

18. THE CONCEPT OF BANKFULL DISCHARGE IN RIVERS
Let  denote river stage (water surface elevation) [L] and Q denote volume water discharge [L3/T]. In the case of rivers with floodplains,  tends to increase rapidly with increasing Q when all the flow is confined to the channel, but much less rapidly when the flow spills significantly onto the floodplain. The rollover in the curve defines bankfull discharge Qbf.
Bankfull flow ~ channel-forming flow???
Minnesota River and floodplain, USA, during the record flood of 1965

19. PARAMETERS USED TO CHARACTERIZE BANKFULL CHANNEL GEOMETRY OF RIVERS
In addition to a bankfull discharge, a reach of an alluvial river with a floodplain also has a characteristic average bankfull channel width and average bankfull channel depth. The following parameters are used to characterize this geometry.
Definitions:
Qbf = bankfull discharge [L3/T]
Bbf = bankfull width [L]
Hbf = bankfull depth [L]
S = bed slope [1]
Ds50 = median surface grain size [L]
n = kinematic viscosity of water [L2/T]
R = (rs/r – 1) = sediment submerged specific gravity (~ 1.65 for natural
sediment) [1]
g = gravitational acceleration [L/T2]

20. SETS OF DATA USED TO CHARACTERIZE RIVERS
Sand-bed rivers D  0.5 mm
Sand-bed rivers D > 0.5 mm
Large tropical sand-bed rivers
Gravel-bed rivers
Rivers from Japan (gravel and sand)

21. SHIELDS DIAGRAM AT BANKFULL FLOW
Compared to rivers, turbidity currents have to be biased toward this region to be suspension-driven!

22. FROUDE NUMBER AT BANKFULL FLOW
Turbidity currents?

23. DIMENSIONLESS CHEZY RESISTANCE COEFFICIENT AT BANKFULL FLOW
Turbidity currents?

24. DIMENSIONLESS WIDTH-DEPTH RATIO AT BANKFULL FLOW
Turbidity currents?

25. THE DEPTH-SLOPE RELATION FOR BED SHEAR STRESS DOES NOT NECESSARILY WORK FOR TURBIDITY CURRENTS!
In a river, there is frictional resistance not only at the bed, but also at the water-air interface. Thus if I denotes the interfacial shear stress, the normal flow relation generalizes to:
But in a wide variety of cases of interest, I at an air-water interface is so small compared to b that it can be neglected.

26. A TURBIDITY CURRENT CAN HAVE SIGNIFICANT FRICTION ASSOCIATED WITH ITS INTERFACE
If a turbidity current were to attain normal flow conditions,
where
and Cf denotes a bed friction coefficient and Cfi denotes an interfacial frictional coefficient.
But turbidity currents do not easily attain normal flow conditions!

27. REFERENCES
Garcia and Parker (1989)