3 RHEOLOGY Elastic: Strain linearly proportional to stress; strain recoverable.Earth’s crustPlastic:Above yield stress, materialdeforms permanently (by flow),With no additional increase of stress.Ice sheetViscous:to stress; strain permanent.Flow velocity ~ stress.Water
4 CLEAR FLUID UNDERGOING SHEAR Linear velocity gradient U/L ~ F force applied to move upper plateAt any point in the viscous fluid: t = m du/dyshear stress velocity gradientviscosity of the fluidLaminar flow is dominated by molecular viscosity.
6 DRAG Fluid approaching grain is decelerated from free stream velocity u. Loss of kinetic energy.Volume of fluid undergoing deceleration: uAMass of this volume: rfuAKinetic energy: mu2/2Loss of kinetic energy: rfu3A/2Conservation of energy:power = loss of kinetic energyPower = FuDrag Force FD = ru2/2 AParticle shape affects fluid motion near grain:FD = CD ru2/2 Adrag coefficient CD = FD/ru2D2A ~ D2D
7 Flow lines bend around grain: DRAGFlow lines bend around grain:Viscosity should be included in treatmentDrag Force FD = ru2/2 AParticle shape affects fluid motion near grain:FD = CD ru2/2 Adrag coefficient CD = FD/ru2D2D
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 productsDimensions in Mechanics: Mass MLength LTime TChoose 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.
9 DIMENSIONAL ANALYSIS: DRAG ON GRAIN Variable: Dimension:Velocity of fluid, u LT-1Viscosity of fluid, m ML-1T-1Density of fluid, r ML-3Size of particle, D LDrag force, FD MLT-2Repeating variables: r, u, and DTo make drag force [ML-1T-2] dimensionless:eliminate [M] by dividing by r [ML-3]eliminate [T] by dividing by u2 [LT-1]2eliminate [L] by dividing by D2 [L]2To make viscosity [ML-1T-1] dimensionless:eliminate [T] by dividing by u [LT-1]eliminate [L] by dividing by D [L]FD/ru2D2 = CDm/ruD
10 DIMENSIONAL ANALYSIS: DRAG ON GRAIN FD/ru2D2 = CDm/ruD Re = ruD/m inertia/viscous forceoften very small Reynold’s number
11 SETTLING GRAIN Stoke’s Law: u = D2g’/18m Settling velocity of grain with diameter D and density rs througha still fluid with density rf:FD = pD3g’/6Drag forcesubmersed weight of graing’ = (rs – rf)g submersed specific weightFluid is static: ignore rfRemaining variables: FD, u, m, and DDimensionless product: FD/muD = 3pStoke’s Law:u = D2g’/18mOnly when flow is laminar: small Reynolds number.
12 DIMENSIONAL ANALYSIS: DRAG ON GRAIN Stoke’s Law only applies in laminar flow
14 Energy cannot be lost from system, but may change form. BERNOULLI’S THEOREMEnergy 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: smallFor constant potential energy,an increase in flow velocityresults 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 forceacting on unit water column.Opposed by an equal shear stress t0exerted by unit bed area.Stream Power w = t0uNeed to know velocity profile in stream
16 VELOCITY PROFILE IN LAMINAR FLOW At channel bed: t0 = rgSdAt height y: ty = rgS(d-y)ty = t0(1-y/d)Shear stress varies linearly frommaximum at bed to zero at surface.Using t = m(du/dy),du/dy = rgS(d-y)/mIntegrate to obtain velocity at anypoint above bed, assuming thatfluid density and viscosity are constant:u = (rgS)/m (yd – y2) + CIf C = 0, then velocity profile is parabolic.
17 TURBULENT FLOW Re = ruD/m > 500 In turbulent flow, fluid particles take part in rapidlyvarying 3-D motion inturbulent eddies. In theseeddies, local accelerationsare very important; viscosityplays a minor role.Re = ruD/m > 500Turbulent flows are wellmixed.
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 = 100Re* is boundary Reynolds no.u* = √t0/r is shear velocity.Burst-sweep process ismain creator of turbulence.Inrush of high-velocity sweepsmay locally exceed thresholdof sediment motion.
20 BOUNDARY LAYER In boundary layer: Total stress = Viscous stress m(du/dy) +Turbulent stress -r(uv).Turbulent stress:= (m + h)du/dyis eddy viscosity,>>mhdu/dy = -r(uv)h/r is kinematic eddyviscosity, eThe origin of turbulence is linked with presence ofa boundary. The effects of the boundary are felt inmotion 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 asmall 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 indimensions of velocity:u*2 = t0/r ,then the velocity u at any height y within the viscous sublayer can befound frommu/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 onthe 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 thatu/u* = 1/k ln(y/y0), the law of the wallwhere the roughness length y0 is the height above the bed at which theflow 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 separatefrom the solid boundary by a region ofslow, upstream moving fluid.This is an important cause of turbulence,and a principal factor in the dynamicsof bedforms.