BOUNDARY LAYERS Zone of flow immediately in vicinity of boundary Motion of fluid is retarded by frictional resistance Boundary layer extends away from boundary until unaffected by frictional resistance and flow is same velocity as free stream Growth of laminar boundary layer

BOUNDARY LAYERS Can be laminar or turbulent. Once turbulent, it thickens, and nearbed stress increases

BOUNDARY LAYERS Effect of turbulence is to transport things such as heat, suspended sediment, and momentum Momentum gets diffused towards the boundary, thing like sediment tend to diffuse away

BOUNDARY LAYERS Given an equation that describes this motion in the x-direction as: Assumes no velocity variation in x-direction (uniform), flow solely in x-direction

BOUNDARY LAYERS Assume hydrostatic pressure distribution inside the boundary layer. Further, if the shear stresses vanish away from the boundary (velocity gradients go to zero), then Euler’s equation arises and we have Where the subscript infinity signifies far away from boundary substituting

BOUNDARY LAYERS Previous equation assumed laminar flow with molecular eddy viscosity Should really be Where the eddy viscosity cannot be moved outside the integral because it likely depends on the elevation The subscript on the second eddy viscosity denotes turbulent There are a bunch of ways to determine the turbulent eddy viscosity: assume a certain shape (linear, parabolic etc), use a turbulence closure scheme.

WHAT DOES SINUSOIDAL MOTION OVER PLATE LOOK LIKE Make figure for laminar and turbulent from data we already have

BOUNDARY LAYERS What is really typically wanted is no just the turbulent eddy viscosity but the bed stress because this is what is used to help estimate transport. The first part after equal sign is the turbulent Reynolds stress and it depends on correlations between the horizontal and vertical velocity fluctuations These are estimated using the last part of equation

BOUNDARY LAYERS Define something called friction velocity (a pseudo-velocity) as Prandtl, developed a mixing length hypothesis for the turbulent eddy viscosity. Basically it gets bigger with distance from bed. (linear) as Not exactly correct because would suggest infinitely large eddy far from boundary

BOUNDARY LAYERS Leads to something called the “Law of the Wall” Gives us u as a function of elevation. Trouble is, how do we know u *. We normally don’t. So we estimate the shear stress from a quadratic drag law as Where f is a friction factor. Then if we wanted to we could rearrange to use in the Law of the Wall