GEOLOGY 1B: CLASTIC SEDIMENTS

26 Fluid flow Fluid flow27 Sediment transport Sediment transport28 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:nhovius@esc.cam.ac.uk

1B Clastic SedimentsLecture 26

FLUID MECHANICS

NH 01-2007

RHEOLOGY

Elastic:Strain linearly proportional to stress; strain recoverable.Earth’s crust

Plastic:Above yield stress, materialdeforms permanently (by flow),With no additional increase of stress.Ice sheet

Viscous:Strain linearly proportionalto stress; strain permanent.Flow velocity ~ stress.Water

CLEAR FLUID UNDERGOING SHEAR

Linear velocity gradient U/L ~ F force applied to move upper plate

At any point in the viscous fluid: = du/dy

shear stress velocity gradientviscosity of the fluid

Laminar flow is dominated by molecular viscosity.

LAMINAR FLOW PAST CYLINDER

DRAG

Fluid approaching grain is decelerated from freestream velocity u. Loss of kinetic energy.

Volume of fluid undergoing deceleration: uAMass of this volume: fuA

Kinetic energy: mu2/2

Loss of kinetic energy: fu3A/2

Conservation of energy: power = loss of kinetic energy

Power = Fu

Drag Force FD = u2/2 A

Particle shape affects fluid motion near grain: FD = CD u2/2 A

drag coefficient CD = FD/u2D2

D

A ~ D2

DRAG

Drag Force FD = u2/2 A

Particle shape affects fluid motion near grain: FD = CD u2/2 A

drag coefficient CD = FD/u2D2

D

Flow lines bend around grain: Viscosity should be included in treatment

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 MLength LTime 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 parametersto make them dimensionless.

DIMENSIONAL ANALYSIS: DRAG ON GRAIN

Variable: Dimension:Velocity of fluid, u LT-1

Viscosity of fluid, ML-1T-1 Density of fluid, ML-3

Size of particle, D LDrag force, FD MLT-2

Repeating variables: , u, and D

To make drag force [ML-1T-2] dimensionless:eliminate [M] by dividing by [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 [M] by dividing by [ML-3]eliminate [T] by dividing by u [LT-1]eliminate [L] by dividing by D [L]

FD/u2D2 = CD

/uD

DIMENSIONAL ANALYSIS: DRAG ON GRAIN

FD/u2D2 = CD

/uD Re = uD/inertia/viscous forceoften very small Reynold’s number

SETTLING GRAIN

Settling velocity of grain with diameter D and density s througha still fluid with density f:

FD = D3’/6

Drag forcesubmersed weight of grain’ = (s – f)g submersed specific weight

Fluid is static: ignore f

Remaining variables: FD, u, , and D

Dimensionless product: FD/uD = 3

Stoke’s Law:

u = D2’/18

Only when flow is laminar: small Reynolds number.

DIMENSIONAL ANALYSIS: DRAG ON GRAIN

Stoke’s Law only applies in laminar flow

LAMINAR FLOW PAST CYLINDER

P

0

BERNOULLI’S THEOREM

Energy cannot be lost from system, but may change form.

Energy in flow: Kinetic energy (fu2/2) Potential energy (fgh) 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?

STREAM POWER

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:

gSdu S is channel bed slope, d is flow depth.gSd is downslope component of gravity force

acting on unit water column.

Opposed by an equal shear stress 0 exerted by unit bed area.

Stream Power = 0uNeed to know velocity profile in stream

VELOCITY PROFILE IN LAMINAR FLOW

At channel bed: 0 = gSd

At height y: y = gS(d-y)

y = 0(1-y/d)

Shear stress varies linearly from maximum at bed to zero at surface.

Using (du/dy),

du/dy = gS(d-y)/Integrate to obtain velocity at any point above bed, assuming that fluid density and viscosity are constant:

u = (gS)/ (yd – y2) + CIf C = 0, then velocity profile is parabolic.

TURBULENT FLOW

In turbulent flow, fluidparticles take part in rapidlyvarying 3-D motion inturbulent eddies. In theseeddies, local accelerationsare very important; viscosityplays a minor role.

Re = uD/ > 500

Turbulent flows are wellmixed.

DIMENSIONAL ANALYSIS: DRAG ON GRAIN

Stoke’s Law only applies in laminar flow

BURSTS AND SWEEPS

Flow streaks in wall region.

Spacing of streaks, depends on flow properties:

Re* = u*/ = 100

Re* is boundary Reynolds no.u* = √0/ is shear velocity.

Burst-sweep process ismain creator of turbulence.

Inrush of high-velocity sweepsmay locally exceed thresholdof sediment motion.

BOUNDARY LAYER

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

In boundary layer:Total stress = Viscous stress (du/dy) +Turbulent stress -(uv).

Turbulent stress:= (+ )du/dy

is eddy viscosity,>>

du/dy = -(uv)

/ is kinematic eddy viscosity,

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(0, , and y)One dimensionless product: u/0y, which is constant, roughly unity.

If the shear velocity is the shear stress at the boundary expressed in dimensions of velocity:

u*2 = 0/ ,

then the velocity u at any height y within the viscous sublayer can be found from

u/u*2y = 1 Thickness of viscous sublayer < 1 mm.

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.

FLOW SEPARATION

Flow separation occurs where a positivepressure 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 dynamicsof bedforms.