LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW The 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 FLOW The 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 as A B C D E Term 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

LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW We 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 FLOW The 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 FLOW For 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 is  y x u moving with velocity U fixed fluid H

LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW That is, the components of the viscous stress tensor are Here we abbreviate  moving with velocity U u fluid H y x  fixed

LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW Thus 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 FLOW Momentum 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:  y x u moving with velocity U fixed fluid H

LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW Momentum 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)  y x u moving with velocity U fixed fluid H

LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW Integrate once: Thus the shear stress  must be constant on the domain. Integrate again: Apply the boundary conditions to obtain C2 = 0, C1 = U/H and thus  y x u moving with velocity U fixed fluid H

LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW For 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 FLOW The channel width is denoted as B. It is assumed that the channel is sufficiently wide (B/H << 1) so that sidewall effects can be ignored. Thus streamwise velocity u is a function of upward normal distance z alone, i.e. u = u(z). H B The vector of gravitational acceleration is (gx, gy, gz) = (gsin, 0, -gcos)

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

LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW The equations thus reduce to: Since The first equation can thus be rewritten as where  is an abbreviation for 13 = 31.

LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW 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 are viscous fluid sticks to immobile bed no applied shear stress as free surface gage pressure at free surface = 0 (surface pressure = atmospheric)

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

LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW The equation subject to similarly 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 FLOW Thus subject to Integrates to give the following parabolic profile for u in z:

LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW The maximum velocity Us is reached at the free surface, where z = H and  = 1); Thus Depth-averaged flow velocity U is given as Thus

LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW A dimensionless bed friction coefficient Cf can be defined as Here Cf = f/8 where f denotes the D’arcy-Weisbach friction coefficient. Between the above relation and the relations below it can be shown that Here 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 FLOW Now 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 to The corresponding solution to the problem is where 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 FLOW The solution for velocity with the case of wind can be rewritten as where und is a dimensionless velocity equal to 2u/(gsinH2). 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 = 0.5 r = 0.25 r = 1.5 r = 1