Flow velocity: Prandtl Mixing and the Law of the Wall
Rigorous derivation of turbulent mixing (Prandtl) and the boundary-layer fluid-velocity profile.
Learning Goals
- Understand how turbulent mixing exchanges momentum (and therefore, velocity) up and down through the water column
- Learn how this turbulent exchange creates a different velocity profile than would be expected for a laminar flow
- Relate stress and strain in both laminar and turbulent flows
- Define the bed roughness, $z_0$
The Law of the Wall: The Equation
For quick reference; see course notes and videos for derivation:
\[u(z) = \frac{u_\tau}{\kappa} \ln\left(\frac{z}{z_0}\right)\]Course notes
Prandtl Turbulent Mixing
Covers both laminar (i.e., viscous) and turbulent transfer of momentum through a flow, and defines the eddy viscosity (though I only note that exact term in passing at the end).
Law of the Wall
Steps to derive the turbulent velocity profile based on Prandtl turbulence theory. This also includes the formal definition of the shear velocity.
The roughness length, \(z_0\)
Estimating \(z_0\) from grain size and the thickness of the laminar sublayer.
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