1. GEOLOGY 1B: CLASTIC SEDIMENTS
26 Fluid flow Fluid flow
27 Sediment transport Sediment transport
28 Bedform dynamics Bedforms and cross bedding
Reading:
P.A. Allen: Earth Surface Processes. Blackwell Science, 1998.
J.R.L. Allen: Principles of Physical Sedimentology. Allen & Unwin, 1985.
M. Leeder: Sedimentology and Sedimentary Systems. Blackwell Science, 1999.
G.V. Middleton and J.B Southard: Mechanics of Sediment Movement.
SEPM Short Course 3, 1984.
Contact:

2. 1B Clastic Sediments
Lecture 26
FLUID MECHANICS NH

3. RHEOLOGY Elastic: Strain linearly proportional
to stress; strain recoverable.
Earth’s crust
Plastic:
Above yield stress, material
deforms permanently (by flow),
With no additional increase of stress.
Ice sheet
Viscous:
to stress; strain permanent.
Flow velocity ~ stress.
Water

4. CLEAR FLUID UNDERGOING SHEAR
Linear velocity gradient U/L ~ F force applied to move upper plate
At any point in the viscous fluid: t = m du/dy
shear stress velocity gradient
viscosity of the fluid
Laminar flow is dominated by molecular viscosity.

5. LAMINAR FLOW PAST CYLINDER

6. DRAG Fluid approaching grain is decelerated from free
stream velocity u. Loss of kinetic energy.
Volume of fluid undergoing deceleration: uA
Mass of this volume: rfuA
Kinetic energy: mu2/2
Loss of kinetic energy: rfu3A/2
Conservation of energy:
power = loss of kinetic energy
Power = Fu
Drag Force FD = ru2/2 A
Particle shape affects fluid motion near grain:
FD = CD ru2/2 A
drag coefficient CD = FD/ru2D2
A ~ D2 D

7. Flow lines bend around grain:
DRAG
Flow lines bend around grain:
Viscosity should be included in treatment
Drag Force FD = ru2/2 A
Particle shape affects fluid motion near grain:
FD = CD ru2/2 A
drag coefficient CD = FD/ru2D2 D

8. DIMENSIONAL ANALYSIS Identify all parameters relevant to problem.
Group parameters to obtain dimensionless products.
Problem with N parameters and n dimensions: (N – n) dimensionless products
Dimensions in Mechanics: Mass M
Length L
Time T
Choose three repeating parameters with independent dimensions:
No two can be combined to produce dimensions of third.
Do not use key variables as repeating parameters.
Combine the three repeating parameters with each of the remaining parameters
to make them dimensionless.

9. DIMENSIONAL ANALYSIS: DRAG ON GRAIN
Variable: Dimension:
Velocity of fluid, u LT-1
Viscosity of fluid, m ML-1T-1
Density of fluid, r ML-3
Size of particle, D L
Drag force, FD MLT-2
Repeating variables: r, u, and D
To make drag force [ML-1T-2] dimensionless:
eliminate [M] by dividing by r [ML-3]
eliminate [T] by dividing by u2 [LT-1]2
eliminate [L] by dividing by D2 [L]2
To make viscosity [ML-1T-1] dimensionless:
eliminate [T] by dividing by u [LT-1]
eliminate [L] by dividing by D [L]
FD/ru2D2 = CD
m/ruD

10. DIMENSIONAL ANALYSIS: DRAG ON GRAIN
FD/ru2D2 = CD
m/ruD Re = ruD/m inertia/viscous force
often very small Reynold’s number

11. SETTLING GRAIN Stoke’s Law: u = D2g’/18m
Settling velocity of grain with diameter D and density rs through
a still fluid with density rf:
FD = pD3g’/6
Drag force
submersed weight of grain
g’ = (rs – rf)g submersed specific weight
Fluid is static: ignore rf
Remaining variables: FD, u, m, and D
Dimensionless product: FD/muD = 3p
Stoke’s Law:
u = D2g’/18m
Only when flow is laminar: small Reynolds number.

12. DIMENSIONAL ANALYSIS: DRAG ON GRAIN
Stoke’s Law only applies in laminar flow

13. LAMINAR FLOW PAST CYLINDER

14. Energy cannot be lost from system, but may change form.
BERNOULLI’S THEOREM
Energy cannot be lost from system, but may change form.
Energy in flow:
Kinetic energy (rfu2/2)
Potential energy (rfgh)
Pressure energy (p)
Frictional heat loss: small
For constant potential energy,
an increase in flow velocity
results in a decrease in pressure.
How much work can stream do?

15. Stream Power is the rate at which a flow does work on its bed.
Work: rate of conversion of potential energy into kinetic energy.
Principal control on sediment transport and formation of bedforms.
Rate of loss of gravitational potential energy per unit area of stream bed:
rgSdu S is channel bed slope, d is flow depth.
rgSd is downslope component of gravity force
acting on unit water column.
Opposed by an equal shear stress t0
exerted by unit bed area.
Stream Power w = t0u
Need to know velocity profile in stream

16. VELOCITY PROFILE IN LAMINAR FLOW
At channel bed: t0 = rgSd
At height y: ty = rgS(d-y)
ty = t0(1-y/d)
Shear stress varies linearly from
maximum at bed to zero at surface.
Using t = m(du/dy),
du/dy = rgS(d-y)/m
Integrate to obtain velocity at any
point above bed, assuming that
fluid density and viscosity are constant:
u = (rgS)/m (yd – y2) + C
If C = 0, then velocity profile is parabolic.

17. TURBULENT FLOW Re = ruD/m > 500 In turbulent flow, fluid
particles take part in rapidly
varying 3-D motion in
turbulent eddies. In these
eddies, local accelerations
are very important; viscosity
plays a minor role.
Re = ruD/m > 500
Turbulent flows are well
mixed.

18. DIMENSIONAL ANALYSIS: DRAG ON GRAIN
Stoke’s Law only applies in laminar flow

19. BURSTS AND SWEEPS Flow streaks in wall region. Spacing of streaks, l
depends on flow properties:
Re* = ru*l/m = 100
Re* is boundary Reynolds no.
u* = √t0/r is shear velocity.
Burst-sweep process is
main creator of turbulence.
Inrush of high-velocity sweeps
may locally exceed threshold
of sediment motion.

20. BOUNDARY LAYER In boundary layer: Total stress =
Viscous stress m(du/dy) +
Turbulent stress -r(uv).
Turbulent stress:
= (m + h)du/dy
is eddy viscosity,
>>m
hdu/dy = -r(uv)
h/r is kinematic eddy
viscosity, e
The origin of turbulence is linked with presence of
a boundary. The effects of the boundary are felt in
motion of fluid over certain distance away from boundary:
boundary layer.
Hydraulically smooth boundary: roughness elements contained within viscous sublayer

21. VELOCITY PROFILE IN TURBULENT FLOWS
Within turbulent boundary layer there is a viscous sublayer. In this layer,
flow is laminar, with a high velocity gradient.
In outer part of boundary layer, where the kinematic eddy viscosity is large,
transfer of momentum is efficient, and the fluid is well mixed with a
small gradient of average velocity.
Velocity u in viscous sublayer is f(t0, m, and y)
One dimensionless product: mu/t0y, which is constant, roughly unity.
If the shear velocity is the shear stress at the boundary expressed in
dimensions of velocity:
u*2 = t0/r ,
then the velocity u at any height y within the viscous sublayer can be
found from
mu/ru*2y = Thickness of viscous sublayer < 1 mm.

22. VELOCITY PROFILE IN TURBULENT FLOWS
In the core of the boundary layer, the velocity gradient only depends on
the shear stress at the boundary (or shear velocity).
There are three parameters: velocity gradient (du/dy), shear velocity (u*),
and height above the boundary (y).
One dimensionless product:
u*/(y du/dy) = k ≈ 0.4 k is von Karman’s constant.
It can be shown that
u/u* = 1/k ln(y/y0), the law of the wall
where the roughness length y0 is the height above the bed at which the
flow velocity appears to be zero.
The velocity profile in a turbulent flow has a logarithmic form.

23. FLOW SEPARATION Flow separation occurs where a positive
pressure gradient is set up in the flow,
i.e., a downstream increase in pressure,
causing the boundary layer to separate
from the solid boundary by a region of
slow, upstream moving fluid.
This is an important cause of turbulence,
and a principal factor in the dynamics
of bedforms.