Adaptively Refined Large-Eddy Simulations of Galaxy Clusters 2 Turbulence 4 Favre-filtered equations of fluid dynamics

Chapter 3 Filter formalism

Next we present a very general filter formalism, which is useful when dealing with the compressible equations of fluid dynamics. We collect the most important rules¹²¹²These rules have already been used implicitely by Canuto (1997); Schmidt et al. (2006b), but our work is the first to summarize them explicitly., which are necessary to filter the compressible equations of fluid dynamics.

3.1 Reynolds filter

In general filtering means splitting some quantity $a$ in a mean value $\langle a\rangle$ generated by the filter procedure and some deviation $a^{\prime}$ from the mean value. If a filter fulfills the following relations,

$\displaystyle\langle A+B\rangle$	$\displaystyle=\langle A\rangle+\langle B\rangle,$	(3.1)
$\displaystyle\langle C\rangle$	$\displaystyle=C,\ \text{if $C=$\ const.},$	(3.2)
$\displaystyle\langle\langle A\rangle B\rangle$	$\displaystyle=\langle A\rangle\langle B\rangle,$	(3.3)

it is called a Reynolds filter or Reynolds operator. From equation (3.2) and (3.3) we see that for $B=C=const.$

\displaystyle\langle\langle A\rangle C\rangle=\langle A\rangle C.

(3.4)

From this relation (3.4) it follows for $C=1$

\displaystyle\langle\langle A\rangle\rangle=\langle A\rangle.

(3.5)

From the last equation ((3.5)) and (3.3) we get for $B=\langle D\rangle$

\displaystyle\langle\langle A\rangle\langle D\rangle\rangle=\langle A\rangle% \langle\langle D\rangle\rangle=\langle A\rangle\langle D\rangle.

(3.6)

If we split a quantity $a$ in a sum of some kind of mean value $\langle a\rangle$ (computed by a filter that satisfies the Reynolds criteria (3.1)-(3.3)) and some deviations $a^{\prime}$

\displaystyle a=\langle a\rangle+a^{\prime}

(3.7)

it follows from equation (3.5) for the mean of the deviations

\displaystyle\langle a^{\prime}\rangle=\langle a-\langle a\rangle\rangle=% \langle a\rangle-\langle\langle a\rangle\rangle=\langle a\rangle-\langle a% \rangle=0.

(3.8)

One can show that the so called central moments of this quantity ( $\langle a^{\prime}b^{\prime}\rangle$ , $\langle a^{\prime}b^{\prime}c^{\prime}\rangle$ , $\langle a^{\prime}b^{\prime}c^{\prime}d^{\prime}\rangle$ , $\ldots$ ) can be expressed in terms of the classical moments ( $\langle ab\rangle$ , $\langle abc\rangle$ , $\langle abcd\rangle$ , $\ldots$ ) like¹³¹³See for example (Monin and Yaglom, 1971).

$\displaystyle\langle a^{\prime}b^{\prime}\rangle=$	$\displaystyle\ \langle ab\rangle-\langle a\rangle\langle b\rangle$		(3.9)
$\displaystyle\begin{split}\displaystyle\langle a^{\prime}b^{\prime}c^{\prime}% \rangle=&\displaystyle\ \langle abc\rangle-\langle a\rangle\langle b\rangle% \langle c\rangle\\ &\displaystyle-\langle a\rangle\langle b^{\prime}c^{\prime}\rangle-\langle b% \rangle\langle a^{\prime}c^{\prime}\rangle-\langle c\rangle\langle a^{\prime}b% ^{\prime}\rangle\end{split}$		(3.10)
$\displaystyle\begin{split}\displaystyle\langle a^{\prime}b^{\prime}c^{\prime}d% ^{\prime}\rangle=&\displaystyle\ \langle abcd\rangle-\langle a\rangle\langle b% \rangle\langle c\rangle\langle d\rangle\\ &\displaystyle-\langle a\rangle\langle b^{\prime}c^{\prime}d^{\prime}\rangle-% \langle b\rangle\langle a^{\prime}c^{\prime}d^{\prime}\rangle-\langle c\rangle% \langle a^{\prime}b^{\prime}d^{\prime}\rangle-\langle d\rangle\langle a^{% \prime}b^{\prime}c^{\prime}\rangle\\ &\displaystyle-\langle a\rangle\langle b\rangle\langle c^{\prime}d^{\prime}% \rangle-\langle a\rangle\langle c\rangle\langle b^{\prime}d^{\prime}\rangle-% \langle a\rangle\langle d\rangle\langle b^{\prime}c^{\prime}\rangle\\ &\displaystyle-\langle b\rangle\langle c\rangle\langle a^{\prime}d^{\prime}% \rangle-\langle b\rangle\langle d\rangle\langle a^{\prime}c^{\prime}\rangle-% \langle c\rangle\langle d\rangle\langle a^{\prime}b^{\prime}\rangle\end{split}$		(3.11)
$\displaystyle\langle a^{\prime}b^{\prime}c^{\prime}d^{\prime}e^{\prime}\rangle=$	$\displaystyle\ \ldots$		(3.12)

3.2 Germano formalism

Germano (1992) postulates that the relations between moments and central moments for non-Reynolds operators¹⁴¹⁴Non-Reynolds operators do not fulfill 3.8, so their mean of the deviations is unequal zero. are of similar form as for Reynolds operators. Therefore he introduces the so called generalized central moments $\tau(a,b),\tau(a,b,c),\ldots$ for non-Reynolds operators. These should fulfill in analogy to equation (3.9)-(3.12) the following relations¹⁵¹⁵It seems to be very difficult to prove these relations even in case of very simple non-Reynolds operators.

$\displaystyle\tau(a,b)=$	$\displaystyle\ \langle ab\rangle-\langle a\rangle\langle b\rangle$		(3.13)
$\displaystyle\begin{split}\displaystyle\tau(a,b,c)=&\displaystyle\ \langle abc% \rangle-\langle a\rangle\langle b\rangle\langle c\rangle\\ &\displaystyle-\langle a\rangle\tau(b,c)-\langle b\rangle\tau(a,c)-\langle c% \rangle\tau(a,b)\end{split}$		(3.14)
$\displaystyle\begin{split}\displaystyle\tau(a,b,c,d)=&\displaystyle\ \langle abcd% \rangle-\langle a\rangle\langle b\rangle\langle c\rangle\langle d\rangle\\ &\displaystyle-\langle a\rangle\tau(b,c,d)-\langle b\rangle\tau(a,c,d)\\ &\displaystyle-\langle c\rangle\tau(a,b,d)-\langle d\rangle\tau(a,b,c)\\ &\displaystyle-\langle a\rangle\langle b\rangle\tau(c,d)-\langle a\rangle% \langle c\rangle\tau(b,d)-\langle a\rangle\langle d\rangle\tau(b,c)\\ &\displaystyle-\langle b\rangle\langle c\rangle\tau(a,d)-\langle b\rangle% \langle d\rangle\tau(a,c)-\langle c\rangle\langle d\rangle\tau(a,b)\end{split}$		(3.15)
$\displaystyle\langle a^{\prime}b^{\prime}c^{\prime}d^{\prime}e^{\prime}\rangle=$	$\displaystyle\ \ldots$		(3.16)

For the generalized central moments the following rules apply:

1.

They are symmetric in their arguments

$\displaystyle\tau(a,b)=\tau(b,a);\tau(a,b,c)=\tau(b,a,c),\ldots$ (3.17)
2.

The generalized central moment of a constant is zero

$\displaystyle\tau(a,c)=0,\tau(a,b,c)=0,\ \text{if $c=$ const.}$ (3.18)
3.

In case of a static (time independent) filter operator it permutes with the time derivative and the chain rule applies

$\displaystyle\frac{\partial}{\partial t}\tau(a,b)=\tau{\left(\frac{\partial}{% \partial t}a,b\right)}+\tau{\left(a,\frac{\partial}{\partial t}b\right)}$ (3.19)
4.

If the filter operator is isotropic (independent of position in space) then it applies

$\displaystyle\frac{\partial}{\partial x_{i}}\tau(a,b)=\tau{\left(\frac{% \partial}{\partial x_{i}}a,b\right)}+\tau{\left(a,\frac{\partial}{\partial x_{% i}}b\right)}$ (3.20)
5.

Additionally the following relation can be proved

$\displaystyle\tau(a,Cb)$ $\displaystyle=C\cdot\tau(a,b),\ \text{if $C=$ const.}$ (3.21)

$\displaystyle\tau(a,b+c)$ $\displaystyle=\tau(a,b)+\tau(a,c)$ (3.22)

$\displaystyle\tau(a,bc)$ $\displaystyle=\tau(a,b,c)+\langle b\rangle\tau(a,c)+\langle c\rangle\tau(a,b)$ (3.23)

3.3 Favre-Germano formalism

In the case of compressible fluid dynamics, the moments appearing in the filtered equations are one order higher than in non-compressible fluid dynamics (eg. $\langle\rho u_{i}u_{j}\rangle$ instead of $\langle u_{i}u_{j}\rangle$ ). If we would adopt the Germano relations (3.13) to (3.16) in this case, we would get many terms which are difficult to interpret physically. But if we use density weighted quantities¹⁶¹⁶For a modern review of this procedure see Veynante and Vervisch (2002). similar to Favre (1969) and develop relations in analogy to the Germano relations for these density weighted quantities, we can write the filtered compressible equations of fluid dynamics in a much simpler way (Canuto, 1997; Schmidt et al., 2006b).

We define density weighted quantities according to Favre like

\displaystyle\langle\rho a\rangle=\langle\rho\rangle\hat{a}\Rightarrow\hat{a}=% \frac{\langle\rho a\rangle}{\langle\rho\rangle}.

(3.24)

In analogy to Germano we postulate the following relations:

$\displaystyle\hat{\tau}(a,b)=$	$\displaystyle\ \langle\rho ab\rangle-\langle\rho\rangle\hat{a}\hat{b}$		(3.25)
$\displaystyle\begin{split}\displaystyle\hat{\tau}(a,b,c)=&\displaystyle\ % \langle\rho abc\rangle-\langle\rho\rangle\hat{a}\hat{b}\hat{c}\\ &\displaystyle-\hat{a}\hat{\tau}(b,c)-\hat{b}\hat{\tau}(a,c)-\hat{c}\hat{\tau}% (a,b)\end{split}$		(3.26)
$\displaystyle\hat{\tau}(a,b,c,d)=$	$\displaystyle\ \ldots$		(3.27)

For the quantities $\hat{\tau}(\ldots)$ the same rules apply as for the generalized central moments $\tau(\ldots)$ introduced by Germano:

1.

$\hat{\tau}(a,b)=\hat{\tau}(b,a);\hat{\tau}(a,b,c)=\hat{\tau}(b,a,c),\ldots$
2.

$\hat{\tau}(a,c)=0,\hat{\tau}(a,b,c)=0$ , if $c=$ const.
3.

$\frac{\partial}{\partial t}\hat{\tau}(a,b)=\hat{\tau}{\left(\frac{\partial}{% \partial t}a,b\right)}+\hat{\tau}{\left(a,\frac{\partial}{\partial t}b\right)}$ for static filter.
4.

$\frac{\partial}{\partial x_{i}}\hat{\tau}(a,b)=\hat{\tau}{\left(\frac{\partial% }{\partial x_{i}}a,b\right)}+\hat{\tau}{\left(a,\frac{\partial}{\partial x_{i}% }b\right)}$ for isotropic filter.
5.
1. (a)
  
  $\hat{\tau}(a,Cb)=C\cdot\hat{\tau}(a,b)$ , if $C=$ const.
2. (b)
  
  $\hat{\tau}(a,b+c)=\hat{\tau}(a,b)+\hat{\tau}(a,c)$
3. (c)
  
  $\hat{\tau}(a,bc)=\hat{\tau}(a,b,c)+\langle b\rangle\hat{\tau}(a,c)+\langle c% \rangle\hat{\tau}(a,b)$

If we compare the Favre relations to the Germano relations we see:

$\displaystyle\text{Germano:}\ \langle\rho a\rangle=$	$\displaystyle\ \langle\rho\rangle\langle a\rangle+\tau(\rho,a)$	(3.28)
$\displaystyle\text{Favre:}\ \langle\rho a\rangle=$	$\displaystyle\ \langle\rho\rangle\hat{a}$	(3.29)
$\displaystyle\Rightarrow\hat{a}=$	$\displaystyle\ \langle a\rangle+\frac{\tau(\rho,a)}{\langle\rho\rangle}$	(3.30)
$\displaystyle\Rightarrow\hat{a}=$	$\displaystyle\ \langle a\rangle,\ \text{if $\rho=$ const.}$	(3.31)

This means in the case of a constant density, the formalism with Favre density weighted quantities is equivalent to the Germano formalism. ¹⁷¹⁷This can also be proved for the higher moments.

3.4 Explicit filtering

The rules for filtering described in the last sections do not depend on an explicit form of a filter procedure. However we now want to introduce a commonly used representation of a filter procedure, namely the convolution filter. Using a convolution filter the mean value $\langle a\rangle$ of some quantity $a(x)$ is defined as

\displaystyle\langle a(x)\rangle=\int_{-\infty}^{\infty}G(x-x^{\prime})a(x^{% \prime})dx^{\prime},

(3.32)

where $G(x-x^{\prime})$ is called the convolution kernel, and is associated with some cutoff length $l_{\Delta}$ . The deviation of the mean value is then defined as

\displaystyle a^{\prime}=a(x)-\langle a(x)\rangle=a(x)-\int_{-\infty}^{\infty}% G(x-x^{\prime})a(x^{\prime})dx^{\prime}.

(3.33)

The importance of the convolution filter stems from the fact that it can be used to generalise discrete operators, e.g. we can write the well-known second-order central difference formula for the derivative of a continuous variable like¹⁸¹⁸See S and Moin,P (1984).

	$\displaystyle\frac{a(x+h)-a(x-h)}{2h}$	$\displaystyle=\frac{d}{dx}{\left(\frac{1}{2h}\int_{x-h}^{x+h}a(x^{\prime})dx^{% \prime}\right)}$
		$\displaystyle=\frac{d}{dx}\int_{-\infty}^{\infty}G(x-x^{\prime})a(x^{\prime})% dx^{\prime}$
		$\displaystyle=\frac{d}{dx}\langle a\rangle$

with

\displaystyle G(x-x^{\prime})=\begin{cases}\frac{1}{2h}&\text{if }\left\lvert x% -x^{\prime}\right\rvert\leq h\\ 0&\text{otherwise}\end{cases}.

(3.34)

The convolution kernel (3.34) is also called a box or top-hat filter and is most often used for performing explicit spatial scale separation.

$\displaystyle\tau(a,Cb)$	$\displaystyle=C\cdot\tau(a,b),\ \text{if $C=$ const.}$	(3.21)
$\displaystyle\tau(a,b+c)$	$\displaystyle=\tau(a,b)+\tau(a,c)$	(3.22)
$\displaystyle\tau(a,bc)$	$\displaystyle=\tau(a,b,c)+\langle b\rangle\tau(a,c)+\langle c\rangle\tau(a,b)$	(3.23)