## 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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