Boundary layer turbulence as a paradigm for inhomogeneous turbulence

DNS data can be used to plot $\overline{\nu_t}=\overline{{u_i}' {u_j}'} / {\partial \bar u_i \over \partial x_j}$ and compare it with different powers of $y$

From this we can get an idea of how the $y$ dependence of $\nu_t$ should change with $y^+$.

For example

$y^+=0$, $\nu_t\sim y^3$

$y^+\approx20$, $\nu_t\sim y^2$

$y^+\approx40$, $\nu_t\sim y$

$y^+\approx70$, $\nu_t\sim y^{1/2}$

This sort of way at looking at $\nu_t$ suggests that we require at least 5 different models for $\nu_t$ or in another sense the sensitivity of the turbulent viscosity is different at different distances from the wall.

In order to obtain the y dependence we can make use of $Q={\partial u_i \over \partial x_j} {\partial u_j \over \partial x_i}$ and the approximate assumption that $Q \sim y^2$. In addition we can convert the wall associated $y^+$ dependence to a local flow $\Delta^+$ dependence

$\Delta^+=0$, $\nu_t\sim Q^{3/2}$

$\Delta^+\approx20$, $\nu_t\sim Q$

$\Delta^+\approx40$, $\nu_t\sim Q^{1/2}$

$\Delta^+\approx70$, $\nu_t\sim Q^{1/4}$

Since the dimensions of $T={\partial u_i \over \partial x_j} {\partial u_j \over \partial x_i}$ are the same as $Q$: $T\sim {1\over s^2}$ and $Q \sim {1\over s^2}$ and also $T$ has no $y$ dependence, we can use $T$ with $\Delta $ to provide the necessary dimensions for $\nu_t$

$\Delta^+=0$, $\nu_t=c_1 \Delta^2 {Q^{3/2} \over T}$

$\Delta^+\approx20$, $\nu_t=c_2 \Delta^2{Q\over  T^{1/2}}$

$\Delta^+\approx40$, $\nu_t=c_3 \Delta^2 Q^{1/2}$

$\Delta^+\approx70$, $\nu_t=c_4 \Delta^2 (Q T)^{1/4}$

For larger values of $\Delta^+$a turbulent viscosity associated with high Reynolds numbers needs to be developed.

High Reynolds number LES

The most common way of dealing with the wall shear stress in high Reynolds number wall flows is to use a power law or log law.

We must remember that we are trying to model part of the momentum equation for the near wall grid

$ {\partial \over \partial x_j} \left[ \left( \nu + \nu_t\right) {\partial u_i\over \partial x_j} \right] $

and the model is represented as

$ {\partial \over \partial x_j} \left[ {u_*}^2 \right] $

In effect $\nu_t {\partial u_i\over \partial x_j} = {u_*}^2$ at high Reynolds numbers because the the viscosity contribution becomes negligeable.

The friction velocity $u_*$ can either be obtained from a power law e.g.

$u^+=A (y^+)^B$

which can be written as

${u \over u_*}=A ( {y u_*\over \nu} )^B$ (Werner Wengle, $A=8.3$, $B=1/7$)

or a log law

$u^+={1 \over K} ln (y^+) + C$

which can be written as

$ {u \over u_*}= {1 \over K} ln ( {y u_*\over \nu}) + C$ ($K=0.41$, $C=5.1$)

In both cases the value of the friction velocity can be obtained as a function of the local grid velocity, wall distance and the molecular viscosity:

$u_*=f(u, y, \nu)$

and this is the value that is included in the momentum equation.

For the power law shown above the relation is

$u_* =\left[ {\nu \over y} \left({u \over 8.3}\right)^7\right]^{1/8}$

so

${u_*}^2\sim \left( {\nu u^7 \over y} \right)^{1/4}$

This suggests that at high Reynolds numbers the turbulent viscosity should be modelled as

$\nu_t {\partial u_i\over \partial x_j} \sim \left( {\nu u^7 \over y} \right)^{1/4}$

and

$\nu_t  \sim \left( {\nu u^3  y^3} \right)^{1/4}$

Modelling high Reynolds number oscillations

The wall laws and power laws discussed above demonstrate that the turbulent viscosity needs to have some molecular viscosity dependence at high Reynolds numbers.

$\Delta^+\approx100$, $\nu_t\sim y^{1/2} \nu ^{1/4}$

$\Delta^+\rightarrow \infty $, $\nu_t\sim \left( {\nu \over y} \right)^{1/4}$


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Topic revision: r3 - 2010-01-22 - 10:28:14 - YacineKahil
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26 Sep 2018

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