LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOWThe basic equations of incompressible Newtonian fluid mechanics are the incompressible forms of the Navier-Stokes equations and the continuity equation:These equations specify four equations (continuity is a scalar equation, Navier-Stokes is a vector equation) in four unknowns ui (i = 1..3) and p.

LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOWThe physical meaning of the terms in the Navier-Stokes equations can be interpreted as follows. Multiplying by and using continuity, the equations can be rewritten asTerm A ~ time rate of change of momentum

Term B ~ pressure force

Term C ~ net convective inflow rate of momentum ~ inertial force

Term D ~ viscous force ~ net diffusive inflow rate of momentum

Term E ~ gravitational force A B C D E

LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOWWe make the transformations (u1, u2, u3) = (u, v, w) and (g1, g2, g3) = (gx, gy, gz). Expanding out the equations we then obtain the following forms for the Navier-Stokes equations:and the following form for continuity:

LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOWThe simplest flow we can consider is constant rectilinear flow. For example, consider a flow with constant velocity U in the x direction and vanishing velocity in the other directions, i.e. (u, v, w) = (U, 0, 0). This flow is an exact solution of the Navier-Stokes equations and continuity.Thus for any constant rectilinear flow, all that needs to be satisfied is the hydrostatic pressure distribution (even though there is flow):or

LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOWFor plane Couette flow we make the following assumptions: the flow is steady (/t = 0) and directed in the x direction, so that the only velocity component that is nonzero is u (v = w = 0); the flow is uniform in the x direction and the z direction (out of the page), so that /x = /z = 0; the z direction is upward vertical; the plate at y = 0 is fixed; and the plate at y = H is moving with constant speed U

For such a flow the only component of the viscous stress tensor isH

LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOWThat is, the components of the viscous stress tensor areyxumoving with velocity UfixedfluidHere we abbreviateH

LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOWThus u = u(y) only, and v = w = 0. This result automatically satisfies continuity:Momentum balance in the x, y and z directions (z is upward vertical)

LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOWMomentum balance in the z direction (out of the page):That is, the pressure distribution is hydrostatic. Recall that the general relation for a pressure distribution ph obeying the hydrostatic relation is:H

LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOWMomentum balance in the x (streamwise) direction:The no-slip boundary conditions of a viscous fluid apply:the tangential component of fluid velocity at a boundary = the velocity of the boundary (fluid sticks to boundary) H

LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOWIntegrate once:Thus the shear stress must be constant on the domain.HIntegrate again:Apply the boundary conditions to obtainC2 = 0, C1 = U/H and thus

LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOWFor open-channel flow in a wide channel we make the following assumptions: the channel has streamwise slope angle ; x denotes a streamwise (not horizontal) coordinate, z denotes an upward normal (not vertical) coordinate and y denotes a cross-stream horizontal coordinate; the flow is steady (/t = 0) and directed in the x direction, so that the only velocity component that is nonzero is u (v = w = 0); the flow is uniform in the x direction and the y direction (out of the page), so that /x = /y = 0; the bottom of the channel at z = 0 is fixed; there is no applied stress at the free surface where z = H.

LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOWThe channel width is denoted as B. It is assumed that the channel is sufficiently wide (B/H

LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOWContinuity is satisfied if u = u(z) and v = w = 0.The equations of conservation of streamwise and upward normal momentum reduce to:

The equations thus reduce to:LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOWSinceThe first equation can thus be rewritten aswhere is an abbreviation for 13 = 31.

Assuming that a) pressure is given in gage pressure (i.e. relative to atmospheric pressure) and there is no wind blowing at the liquid surface, the boundary conditions on LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOWareviscous fluid sticks to immobile bedno applied shear stress as free surfacegage pressure at free surface = 0 (surface pressure = atmospheric)

Now the condition LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOWstates that the hydrostatic relation prevails perpendicular to the streamlines (which are in the x direction). Integrating the relation with the aid of the boundary conditionyields a pressure distribution that varys linearly in z:

LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOWThe equation subject tosimilarly yields a linear distribution for shear stress in the z direction:Note that the bed shear stress b at z = 0 is given as

LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOWThus subject toIntegrates to give the following parabolic profile for u in z:

LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOWThe maximum velocity Us is reached at the free surface, where z = H and = 1); ThusDepth-averaged flow velocity U is given asThus

LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOWA dimensionless bed friction coefficient Cf can be defined asHere Cf = f/8 where f denotes the Darcy-Weisbach friction coefficient. Between the above relation and the relations belowit can be shown thatHere Re denotes the dimensionless Reynolds No. of the flow, which scales the ratio of inertial forces to viscous forces.

LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOWNow suppose that there is a wind blowing upstream at the free surface, exerting shear stress w in the x direction. The governing equations of the free surface flow remain the same as in Slide 15, but one of the boundary conditions changes toThe corresponding solution to the problem iswhere r is the dimensionless ratio of the wind shear stress pushing the flow upstream to the force of gravity per unit bed area pulling the flow downstream:

LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOWThe solution for velocity with the case of wind can be rewritten aswhere und is a dimensionless velocity equal to 2u/(gsinH2).A plot is given below of und versus for the cases r = 0. 0.25, 0.5, 1 and 1.5.r = 0(no wind)r = 0.25r = 0.5r = 1r = 1.5