Why Prandlt Mixing Length Theory works at all?

[FluidStu]

There are several unreasonable assumptions in the formulation of the Prandlt Mixing Theory. However, it works reasonably well for simple 1-D flow. An attempt to explain why is given by Davidson in his book 'Turbulence', on page 122 - 124, section 4.1.4.

It's stated that it still works because the mixing length theory is really just an extension of vortex dynamics. How? Actually there are two specific questions:

1) In the case of a planar x-y flow, the mean vorticity (that of the mean flow) points in the z-direction, while that of the turbulent vorticity (that of the fluctuating flow) is random. How does this mean that the vorticity of the large eddies is the same order of magnitude as that of the mean flow? Doesn't that assume that the fluctuating velocity is relatively insignificant?

2) We might define the vorticity as:

ω_z ~ ∂ū_x/∂y,

where ū_x is the mean velocity. How then, can we extend this to say that:

ω_z ~ ∂ū_x/∂y ~ u/l,

where u is a typical measure of u' (fluctuating velocity) and l is typical size of the large eddies? I don't understand the physical meaning of this last statement, particularly since in (1) it seems like we said that the fluctuating component of the velocity is relatively insignificant? Why should the vorticity then be defined on the scale of these fluctuating velocities?

Thanks

 

[boneh3ad]

1) In the case of a planar x-y flow, the mean vorticity (that of the mean flow) points in the z-direction, while that of the turbulent vorticity (that of the fluctuating flow) is random. How does this mean that the vorticity of the large eddies is the same order of magnitude as that of the mean flow? Doesn't that assume that the fluctuating velocity is relatively insignificant?

Provided that the large eddy length scale, $\ell$, is significantly smaller than the boundary layer thickness (or other such inertial length scale), then yes, this implies that the fluctuating velocities are small. However, this isn't how Davidson is trying to argue that the orders of magnitude are the same. Instead, he uses more of heuristic explanation about how the turbulent fluctuations tend to tease out the initially-straight mean vortex lines into three dimensional structures (e.g. the hairpin vortices typical of boundary layers)., and that these structures tend to organize on the scale of the large eddies but retain roughly the same vorticity as they did originally due to conservation of vorticity.

2) We might define the vorticity as:

ωz ~ ∂ūx/∂y,

where ūx is the mean velocity. How then, can we extend this to say that:

ωz ~ ∂ūx/∂y ~ u/l,

where u is a typical measure of u' (fluctuating velocity) and l is typical size of the large eddies? I don't understand the physical meaning of this last statement, particularly since in (1) it seems like we said that the fluctuating component of the velocity is relatively insignificant? Why should the vorticity then be defined on the scale of these fluctuating velocities?

It doesn't really matter if the velocity fluctuations themselves are small, because if the large eddy length scale, $\ell$, is also small, then the gradient can still be large and of the same order of magnitude as the mean flow vorticity (which has a larger numerator and denominator). In fact, assuming that both $u$ and $\ell$ are small means that $u/\ell$ more accurately approximates a derivative, which is how they justify using that to represent the turbulent vorticity.

Of course, the main thing to take away here is that turbulence models are inherently hand-wavy, but as long as the scaling arguments hold for a given physical situation, the model will perform reasonably well.