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.
Growth of laminar boundary layer
Can be laminar or turbulent.
Once turbulent, it thickens, and nearbed stress increases
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
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
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
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.
Make figure for laminar and turbulent from data we already have
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
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
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