Adaptively Refined Large-Eddy Simulations of Galaxy Clusters 3 Filter formalism 5 LES and SGS model

Chapter 4 Favre-filtered equations of fluid dynamics

4.1 Basic equations

Using the Favre-Germano formalism developed in 3.3 to filter the equations of compressible selfgravitating fluid dynamics (2.1)-(2.3) leads to

$\displaystyle\frac{\partial}{\partial t}\langle\rho\rangle+\frac{\partial}{% \partial r_{j}}\hat{v}_{j}\langle\rho\rangle=$	$\displaystyle\ 0$		(4.1)
$\displaystyle\frac{\partial}{\partial t}\langle\rho\rangle\hat{v}_{i}+\frac{% \partial}{\partial r_{j}}\hat{v}_{j}\langle\rho\rangle\hat{v}_{i}=$	$\displaystyle-\frac{\partial}{\partial r_{i}}\langle p\rangle+\frac{\partial}{% \partial r_{j}}\langle\sigma^{\prime}_{ij}\rangle+\langle\rho\rangle\hat{g}_{i% }-\frac{\partial}{\partial r_{j}}\hat{\tau}(v_{i},v_{j})$		(4.2)
$\displaystyle\begin{split}\displaystyle\frac{\partial}{\partial t}\langle\rho% \rangle\hat{e}+\frac{\partial}{\partial r_{j}}\langle\rho\rangle\hat{v}_{j}% \hat{e}=&\displaystyle-\frac{\partial}{\partial r_{j}}\langle v_{j}p\rangle+% \frac{\partial}{\partial r_{j}}\langle v_{i}\sigma_{ij}\rangle+\langle\rho% \rangle\hat{v}_{i}\hat{g}_{i}\\ &\displaystyle+\hat{\tau}(v_{i},g_{i})-\frac{\partial}{\partial r_{j}}\hat{% \tau}(v_{j},e)\end{split}$		(4.3)

with

\displaystyle\hat{e}=\hat{e}_{int}+\frac{1}{2}\hat{v}_{i}\hat{v}_{i}+\frac{1}{% 2}\frac{\hat{\tau}(v_{i},v_{i})}{\langle\rho\rangle}

(4.4)

and

\displaystyle\hat{\tau}(v_{j},e)=\hat{\tau}(v_{j},e_{int})+\frac{1}{2}\hat{% \tau}(v_{j},v_{i},v_{i})+\hat{v}_{i}\hat{\tau}(v_{j},v_{i})

(4.5)

Filtering the equation for the kinetic energy and the internal energy alone we get:

	$\displaystyle\begin{split}\displaystyle\frac{\partial}{\partial t}\langle\rho% \rangle\hat{e}_{k}+\frac{\partial}{\partial r_{j}}\hat{v}_{j}\langle\rho% \rangle\hat{e}_{k})=&\displaystyle-\langle v_{i}\frac{\partial}{\partial r_{i}% }p\rangle+\langle v_{i}\frac{\partial}{\partial r_{j}}\sigma^{\prime}_{ij}% \rangle+\langle\rho\rangle\hat{v}_{i}\hat{g}_{i}\\ &\displaystyle+\hat{\tau}(v_{i},g_{i})-\frac{\partial}{\partial r_{j}}\hat{% \tau}(v_{j},e_{k})\end{split}$		(4.6)
	$\displaystyle\begin{split}\displaystyle\frac{\partial}{\partial t}\langle\rho% \rangle\hat{e}_{int}+\frac{\partial}{\partial r_{j}}\hat{v}_{j}\langle\rho% \rangle\hat{e}_{int}=&\displaystyle-\langle p\frac{\partial}{\partial r_{j}}v_% {j}\rangle+\langle\sigma^{\prime}_{ij}\frac{\partial}{\partial r_{j}}v_{i}% \rangle-\frac{\partial}{\partial r_{j}}\hat{\tau}(v_{j},e_{int})\end{split}$		(4.7)

with

\displaystyle\hat{e}_{k}=\frac{1}{2}\hat{v}_{i}\hat{v}_{i}+\frac{1}{2}\frac{% \hat{\tau}(v_{i},v_{i})}{\langle\rho\rangle}

(4.8)

and

\displaystyle\hat{\tau}(v_{j},e_{k})=\frac{1}{2}\hat{\tau}(v_{j},v_{i},v_{i})+% \hat{v}_{i}\hat{\tau}(v_{j},v_{i})

(4.9)

4.2 Resolved energy and turbulent energy equations

Multiplying the filtered equation for the momentum (4.2) with the Favre-filtered velocity $\hat{v}_{i}$ yields the balance equation for the resolved kinetic energy:

\displaystyle\begin{split}\displaystyle\frac{\partial}{\partial t}\langle\rho% \rangle\frac{1}{2}\hat{v}_{i}\hat{v}_{i}+\frac{\partial}{\partial r_{j}}\hat{v% }_{j}\langle\rho\rangle\frac{1}{2}\hat{v}_{i}\hat{v}_{i}=&\displaystyle-\hat{v% }_{i}\frac{\partial}{\partial r_{i}}\langle p\rangle+\hat{v_{i}}\frac{\partial% }{\partial r_{j}}\langle\sigma^{\prime}_{ij}\rangle+\langle\rho\rangle\hat{v}_% {i}\hat{g}_{i}\\ &\displaystyle-\hat{v}_{i}\frac{\partial}{\partial r_{j}}\hat{\tau}(v_{i},v_{j% })\end{split}

(4.10)

Adding the equation for the resolved kinetic energy (4.10) to the equation for the filtered internal energy (4.7) one gets the equation for the total resolved energy $e_{res}=\hat{e}_{int}+\frac{1}{2}\hat{v_{i}}\hat{v_{i}}$ :

\displaystyle\begin{split}\displaystyle\frac{\partial}{\partial t}\langle\rho% \rangle e_{res}+\frac{\partial}{\partial r_{j}}\hat{v}_{j}\langle\rho\rangle e% _{res}=&\displaystyle-\hat{v}_{i}\frac{\partial}{\partial r_{i}}\langle p% \rangle+\hat{v}_{i}\frac{\partial}{\partial r_{j}}\langle\sigma^{\prime}_{ij}% \rangle+\langle\rho\rangle\hat{v}_{i}\hat{g}_{i}\\ &\displaystyle-\langle p\frac{\partial}{\partial r_{i}}v_{i}\rangle+\langle% \sigma^{\prime}_{ij}\frac{\partial}{\partial r_{j}}v_{i}\rangle-\hat{v}_{i}% \frac{\partial}{\partial r_{j}}\hat{\tau}(v_{i},v_{j})\\ &\displaystyle-\frac{\partial}{\partial r_{j}}\hat{\tau}(v_{j},e_{int})\end{split}

(4.11)

The arising four terms $\langle p\frac{\partial}{\partial r_{i}}v_{i}\rangle,\langle\sigma^{\prime}_{% ij}\frac{\partial}{\partial r_{j}}v_{i}\rangle,\hat{\tau}(v_{i},v_{j}),\hat{% \tau}(v_{j},e_{int})$ in the total resolved energy represent the coupling of the unresolved fluctuations to the filtered resolved flow. We could now try to find equations based on quantities of the resolved flow, to model each of these terms independent of each other. Nevertheless we will see that the first three of these four terms can be connected by an equation for another quantity, called the turbulent energy $\varepsilon_{t}$ . From solving the equation for this quantity we get the three terms $\langle p\frac{\partial}{\partial r_{i}}v_{i}\rangle,\langle\sigma^{\prime}_{% ij}\frac{\partial}{\partial r_{j}}v_{i}\rangle,\hat{\tau}(v_{i},v_{j})$ . Only the fourth term $\hat{\tau}(v_{j},e_{int})$ is not connected with the turbulent energy and has to be modeled separately.

We get the balance equation for the turbulent energy¹⁹¹⁹Interpreting the quantity $\hat{\tau}(v_{i},v_{i})=\hat{\tau}_{ii}$ as an energy is only possible, if $\hat{\tau}_{ii}\geq 0$ . This is only guaranteed, if the filter convolution kernel is a semi-positive function in position space (Vreman et al., 1994; Sagaut, 2006). $\varepsilon_{t}=\langle\rho\rangle e_{t}=\frac{1}{2}\hat{\tau}(v_{i},v_{i})$ by subtracting the balance equation for the resolved kinetic energy (4.10) from the balance equation of the filtered kinetic energy (4.6) :

\displaystyle\begin{split}\displaystyle\frac{\partial}{\partial t}\langle\rho% \rangle e_{t}+\frac{\partial}{\partial r_{j}}\hat{v}_{j}\langle\rho\rangle e_{% t}=&\displaystyle-{\left[\langle v_{i}\frac{\partial}{\partial r_{i}}p\rangle-% \hat{v}_{i}\frac{\partial}{\partial r_{i}}\langle p\rangle\right]}\\ &\displaystyle+{\left[\langle v_{i}\frac{\partial}{\partial r_{j}}\sigma^{% \prime}_{ij}\rangle-\hat{v}_{i}\frac{\partial}{\partial r_{j}}\langle\sigma^{% \prime}_{ij}\rangle\right]}\\ &\displaystyle+\hat{\tau}(v_{i},g_{i})-\frac{1}{2}\frac{\partial}{\partial r_{% j}}\hat{\tau}(v_{j},v_{i},v_{i})-\hat{\tau}(v_{j},v_{i})\frac{\partial}{% \partial r_{j}}\hat{v_{i}}\end{split}

(4.12)

For a better comparison with Schmidt et al. (2006b) we will transform the following terms like

$\displaystyle\langle v_{i}\frac{\partial}{\partial r_{i}}p\rangle$	$\displaystyle=\frac{\partial}{\partial r_{i}}\langle v_{i}p\rangle-\langle p% \frac{\partial}{\partial r_{i}}v_{i}\rangle$	(4.13)
$\displaystyle\hat{v}_{i}\frac{\partial}{\partial r_{i}}\langle p\rangle$	$\displaystyle=\frac{\partial}{\partial r_{i}}\hat{v}_{i}\langle p\rangle-% \langle p\rangle\frac{\partial}{\partial r_{i}}\hat{v}_{i}$	(4.14)
$\displaystyle\langle v_{i}\frac{\partial}{\partial r_{j}}\sigma^{\prime}_{ij}\rangle$	$\displaystyle=\frac{\partial}{\partial r_{j}}\langle v_{i}\sigma^{\prime}_{ij}% \rangle-\langle\sigma^{\prime}_{ij}\frac{\partial}{\partial r_{j}}v_{i}\rangle$	(4.15)
$\displaystyle\hat{v}_{i}\frac{\partial}{\partial r_{j}}\langle\sigma^{\prime}_% {ij}\rangle$	$\displaystyle=\frac{\partial}{\partial r_{j}}\hat{v}_{i}\langle\sigma^{\prime}% _{ij}\rangle-\langle\sigma^{\prime}_{ij}\rangle\frac{\partial}{\partial r_{j}}% \hat{v}_{i}$	(4.16)

and rewrite the balance equation for the turbulent energy:

\displaystyle\begin{split}\displaystyle\frac{\partial}{\partial t}\langle\rho% \rangle e_{t}+\frac{\partial}{\partial r_{j}}\hat{v}_{j}\langle\rho\rangle e_{% t}=&\displaystyle-\frac{\partial}{\partial r_{j}}{\left[\frac{1}{2}\hat{\tau}(% v_{j},v_{i},v_{i})+\langle v_{i}p\rangle-\hat{v}_{i}\langle p\rangle-\langle v% _{i}\sigma^{\prime}_{ij}\rangle+\hat{v}_{i}\langle\sigma^{\prime}_{ij}\rangle% \right]}\\ &\displaystyle-{\left[\langle p\rangle\frac{\partial}{\partial r_{i}}\hat{v}_{% i}-\langle p\frac{\partial}{\partial x_{i}}v_{i}\rangle\right]}+{\left[\langle% \sigma^{\prime}_{ij}\rangle\frac{\partial}{\partial r_{j}}\hat{v}_{i}-\langle% \sigma^{\prime}_{ij}\frac{\partial}{\partial r_{j}}v_{i}\rangle\right]}\\ &\displaystyle+\hat{\tau}(v_{i},g_{i})-\hat{\tau}(v_{j},v_{i})\frac{\partial}{% \partial x_{j}}\hat{v_{i}}\end{split}

(4.17)

If we introduce now in analogy to Schmidt et al. (2006b) the following definitions

$\displaystyle-\mu$	$\displaystyle=\langle v_{i}p\rangle-\hat{v}_{i}\langle p\rangle$	(4.18)
$\displaystyle-\kappa$	$\displaystyle=\langle v_{i}\sigma^{\prime}_{ij}\rangle-\hat{v}_{i}\langle% \sigma^{\prime}_{ij}\rangle$	(4.19)
$\displaystyle\mathbb{D}$	$\displaystyle=-\frac{\partial}{\partial r_{j}}{\left[\frac{1}{2}\hat{\tau}(v_{% j},v_{i},v_{i})-\mu+\kappa\right]}$	(4.20)
$\displaystyle\langle\rho\rangle\lambda$	$\displaystyle={\left[\langle p\rangle\frac{\partial}{\partial r_{i}}\hat{v}_{i% }-\langle p\frac{\partial}{\partial r_{i}}v_{i}\rangle\right]}$	(4.21)
$\displaystyle\langle\rho\rangle\epsilon$	$\displaystyle=-{\left[\langle\sigma^{\prime}_{ij}\rangle\frac{\partial}{% \partial r_{j}}\hat{v}_{i}-\langle\sigma^{\prime}_{ij}\frac{\partial}{\partial r% _{j}}v_{i}\rangle\right]}$	(4.22)
$\displaystyle\Gamma$	$\displaystyle=\hat{\tau}(v_{i},g_{i})$	(4.23)

we can write the balance equation for the turbulence energy like

\displaystyle\frac{\partial}{\partial t}\langle\rho\rangle e_{t}+\frac{% \partial}{\partial r_{j}}\hat{v}_{j}\langle\rho\rangle e_{t}=\mathbb{D}+\Gamma% -\langle\rho\rangle(\lambda+\epsilon)-\hat{\tau}(v_{j},v_{i})\frac{\partial}{% \partial r_{j}}\hat{v_{i}}.

(4.24)

With the help of equations (4.13) to (4.16) and the definitions (4.21) and (4.22) we can also rewrite the equation (4.11) for the total resolved energy:

\displaystyle\begin{split}\displaystyle\frac{\partial}{\partial t}\langle\rho% \rangle e_{res}+\frac{\partial}{\partial r_{j}}\hat{v}_{j}\langle\rho\rangle e% _{res}=&\displaystyle-\frac{\partial}{\partial r_{i}}\hat{v}_{i}\langle p% \rangle+\frac{\partial}{\partial r_{j}}\hat{v}_{i}\langle\sigma^{\prime}_{ij}% \rangle+\langle\rho\rangle\hat{v}_{i}\hat{g}_{i}\\ &\displaystyle+\langle\rho\rangle(\lambda+\epsilon)-\hat{v}_{i}\frac{\partial}% {\partial r_{j}}\hat{\tau}(v_{i},v_{j})-\frac{\partial}{\partial r_{j}}\hat{% \tau}(v_{j},e_{int}).\end{split}

(4.25)

4.3 Summary

The last two equations (4.24) and (4.25) together with equation (4.1) and (4.2) (and additionally the Poisson equation for the gravity term and the equation of state) form a complete system of partial differential equations for fluid dynamics

$\displaystyle\frac{\partial}{\partial t}\langle\rho\rangle+\frac{\partial}{% \partial r_{j}}\hat{v}_{j}\langle\rho\rangle=$	$\displaystyle\ 0,$		(4.26)
$\displaystyle\frac{\partial}{\partial t}\langle\rho\rangle\hat{v}_{i}+\frac{% \partial}{\partial r_{j}}\hat{v}_{j}\langle\rho\rangle\hat{v}_{i}=$	$\displaystyle-\frac{\partial}{\partial r_{i}}\langle p\rangle+\frac{\partial}{% \partial r_{j}}\langle\sigma^{\prime}_{ij}\rangle+\langle\rho\rangle\hat{g}_{i% }-\frac{\partial}{\partial r_{j}}\hat{\tau}(v_{i},v_{j}),$		(4.27)
$\displaystyle\begin{split}\displaystyle\frac{\partial}{\partial t}\langle\rho% \rangle e_{res}+\frac{\partial}{\partial r_{j}}\hat{v}_{j}\langle\rho\rangle e% _{res}=&\displaystyle-\frac{\partial}{\partial r_{i}}\hat{v}_{i}\langle p% \rangle+\frac{\partial}{\partial r_{j}}\hat{v}_{i}\langle\sigma^{\prime}_{ij}% \rangle+\langle\rho\rangle\hat{v}_{i}\hat{g}_{i}\\ &\displaystyle+\langle\rho\rangle(\lambda+\epsilon)-\hat{v}_{i}\frac{\partial}% {\partial r_{j}}\hat{\tau}(v_{i},v_{j})-\frac{\partial}{\partial r_{j}}\hat{% \tau}(v_{j},e_{int}),\end{split}$		(4.28)
$\displaystyle\frac{\partial}{\partial t}\langle\rho\rangle e_{t}+\frac{% \partial}{\partial r_{j}}\hat{v}_{j}\langle\rho\rangle e_{t}=$	$\displaystyle\ \mathbb{D}+\Gamma-\langle\rho\rangle(\lambda+\epsilon)-\hat{% \tau}(v_{j},v_{i})\frac{\partial}{\partial r_{j}}\hat{v_{i}},$		(4.29)

where it is often useful to split the equation for resolved energy into an equation for the resolved kinetic energy and internal energy respectively

	$\displaystyle\begin{split}\displaystyle\frac{\partial}{\partial t}\langle\rho% \rangle\hat{e}_{k}+\frac{\partial}{\partial r_{j}}\hat{v}_{j}\langle\rho% \rangle\hat{e}_{k})=&\displaystyle-\hat{v}_{i}\frac{\partial}{\partial r_{i}}% \langle p\rangle+\hat{v}_{i}\frac{\partial}{\partial r_{j}}\langle\sigma^{% \prime}_{ij}\rangle+\langle\rho\rangle\hat{v}_{i}\hat{g}_{i}\\ &\displaystyle-\hat{v}_{i}\frac{\partial}{\partial r_{j}}\hat{\tau}(v_{i},v_{j% }),\end{split}$		(4.30)
	$\displaystyle\begin{split}\displaystyle\frac{\partial}{\partial t}\langle\rho% \rangle\hat{e}_{int}+\frac{\partial}{\partial r_{j}}\hat{v}_{j}\langle\rho% \rangle\hat{e}_{int}=&\displaystyle-\langle p\rangle\frac{\partial}{\partial r% _{j}}\hat{v}_{j}+\langle\sigma^{\prime}_{ij}\rangle\frac{\partial}{\partial r_% {j}}\hat{v}_{i}+\langle\rho\rangle(\lambda+\epsilon)\\ &\displaystyle-\frac{\partial}{\partial r_{j}}\hat{\tau}(v_{j},e_{int}).\end{split}$		(4.31)

The explicit forms of the quantities $\mathbb{D},\lambda,\epsilon,\Gamma$ and $\hat{\tau}(v_{i},v_{j})$ are unknown and have to be modeled in terms of the turbulence energy $e_{t}$ . The term $\hat{\tau}(v_{j},e_{int})$ has to be modeled independently of $e_{t}$ . The models for all these terms represent our turbulence or subgrid model.