GEOLOGY 1B: CLASTIC SEDIMENTS 26Fluid flowFluid flow 27Sediment transportSediment transport...

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Transcript of GEOLOGY 1B: CLASTIC SEDIMENTS 26Fluid flowFluid flow 27Sediment transportSediment transport...
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:[email protected]
1B Clastic SedimentsLecture 26
FLUID MECHANICS
NH 012007
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 LT1
Viscosity of fluid, ML1T1 Density of fluid, ML3
Size of particle, D LDrag force, FD MLT2
Repeating variables: , u, and D
To make drag force [ML1T2] dimensionless:eliminate [M] by dividing by [ML3]eliminate [T] by dividing by u2 [LT1]2
eliminate [L] by dividing by D2 [L]2
To make viscosity [ML1T1] dimensionless:eliminate [M] by dividing by [ML3]eliminate [T] by dividing by u [LT1]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(dy)
y = 0(1y/d)
Shear stress varies linearly from maximum at bed to zero at surface.
Using (du/dy),
du/dy = gS(dy)/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 3D 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.
Burstsweep process ismain creator of turbulence.
Inrush of highvelocity 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.