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: nhovius@esc.cam.ac.uk

1B Clastic Sediments Lecture 26 FLUID MECHANICS NH 01-2007

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

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.

LAMINAR FLOW PAST CYLINDER

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

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

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.

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

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

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.

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

LAMINAR FLOW PAST CYLINDER

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?

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

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.

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.

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

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.

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

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 = 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 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.