The theory behind an LES model adapted for both high and low Reynolds number flows is explained here

A general LES model

Before attempting to develop any kind of LES model one MUST do a DNS.

The reason for this is that only DNS can demonstrate without any doubt that the numerical method used is capable of resolving local turbulent oscillations. We can define a DNS as a simulation in which the turbulence kinetic energy of the flow is conserved without the introduction of any kind of turbulence modelling ( i.e. the turbulence kinetic energy statistics balance). Typically a CFD code will have been written to conserve momentum explicitly and so a DNS is the only real test that the system is truly capable of capturing the local spatial and time oscillations associated with "natural" turbulence.

It is not important to study different Reynolds numbers as a single DNS is enough to demonstrate the viability of the numerical scheme.

The objective of an LES

Viewed in a different way a DNS shows us how much oscillation the numerical scheme is capable of dealing with. This means that if a given cell encounters an oscillation that is stronger it must be reduced or "smoothed" to a level that the numercial scheme can cope with. This "smoothing" process is what the LES model must do.

One of the key problems with LES modelling (or indeed any type of turbulence modelling) is that the behaviour of the "smoothing" process changes significantly with the amount of work it has to do.

For example a tiny grid cell very close to a wall doesn't require the same sort of modelling as a large grid cell that covers an entire boundary layer.

Some features necessary in an LES model

There are several features that are important for the turbulent subgrid viscosity eddy viscosity $\nu_t$ of an LES model:

  • $\nu_t=0$ on a no slip surface

  • $\nu_t=0$ in the centre of a vortex

It should be noted that there is no physical reason for the turbulent subgrid viscosity to always be positive. It is common practice to prevent negative subgrid viscosity from occuring to ensure numerical stability but this has no physical justification.

In order to gain a better understanding of how LES models work it is useful to look at the classic Smagorinsky model.

How much smoothing do we have to do in an LES?

The answer is: it depends on the flow !!

Next to walls however there is a relatively good non-dimensional property which can be used to quite accurately describe the flow behaviour.

This quantity is $y^+ = { y u^* \over \nu}$ where $u^*=\sqrt{\nu {\partial \bar u \over \partial y} \right|_{wall}} $

For example near the wall $u^+=y^+$ where $u^+={\bar u \over u^*}$

and the popular empiric profile of Reichardt can also be used.

The main problem with $y^+$ is that it is inherently related to the distance from the wall $y$ whereas LES (and indeed any kind of turbulence modelling) should be only related to the local flow behaviour.

A local version can be given by

$\Delta^+={ \Delta \sqrt{ \nu \sqrt{T} } \over \nu}$ with $T= {\partial u_i \over \partial x_j} {\partial u_i \over \partial x_j}$ in other words $u^*$ has become local $u^*=\sqrt{ \nu \sqrt{T} } $

so

$\Delta^+={ \Delta \sqrt[4]T \over \sqrt\nu}$ a local version of $y^+$

The $\Delta^+$ is a useful parameter to analyse LES turbulent flow field.

N.B. For $k-\epsilon$ models $\Delta^+ = { \Delta \sqrt{ \nu \sqrt{   {\partial \overline u_i \over \partial x_j} {\partial \overline u_i \over \partial x_j} + {\epsilon \over \nu}   } } \over \nu} = { \Delta \left( {\partial \overline u_i \over \partial x_j} {\partial \overline u_i \over \partial x_j} + {\epsilon \over \nu}  \right)^{1/4} \over \sqrt{\nu}}$

The old version was developed to make use of this parameter. The main difficulty with the usage of this parameter within a model seems to be its Reynolds number dependence.

Another quantity that has a far more subtle Reynolds number dependence is related to the Smagorinsky turbulent eddy viscosity.

$\Delta^t={ \Delta^2 \sqrt T \over u^* L }$ where the reference unity values of the friction velocity and and length scale are used to non dimensionalise the parameter.

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: r19 - 2009-12-28 - 16:26:19 - DominiqueLaurence
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