The exergy concept and compressible turbulence

Abstract

Turbulence models facilitated by Kolmogorov’s theory play an important role for compressible flows. Typically the basis of these models is the power spectrum of the velocity ${\mathbf {u}}$ or of the density-weighted velocity ${\mathbf {w}}\equiv \rho ^{1/3}{\mathbf {u}}$. While for incompressible flow the quantity turbulent kinetic energy characterises turbulent motions, from the thermodynamic point of view, due to fluctuations of the density and the temperature other kinds of energies play a role at the different scales in compressible turbulence. We generalise the power spectrum of the velocity ${\mathbf {u}}$ from incompressible flows to compressible flows by introducing the exergy spectrum as an application of the exergy concept. Furthermore, we discuss the application of the concept of turbulent exergy to turbulence modelling and demonstrate this approach with a direct numerical simulation and a Large-Eddy-Simulation of homogeneous isotropic turbulence. The advantage of turbulence modelling based on turbulent exergy is shown on the example of the Approximate Deconvolution Model (ADM) where, at smallest scales for its newly introduced entropy formulation, more available energy is extracted from the flow, and this occurs in a more physical way than for the classical equation set of the model using the total energy.

Appendices

Relations of exergy expressions

From Eq. (1), the averaged exergy per unit volume is expressed as

$$\begin{aligned} {\bar{\rho }}{\tilde{\epsilon }}={\bar{\rho }}{\tilde{e}}-{\bar{\rho }}e_r +p_r\left( 1-\frac{{\bar{\rho }}}{\rho _r}\right) -T_r({\bar{\rho }}{\tilde{s}} -{\bar{\rho }}s_r)+\frac{1}{2}{\bar{\rho }} {\widetilde{(u_j-u_{j,r})(u_j-u_{j,r})}}. \end{aligned}$$

(31)

Together with the special reference state a (Eq. 4), we obtain the turbulent exergy per unit volume

$$\begin{aligned} {\bar{\rho }}\epsilon _t={\bar{\rho }}{\tilde{\epsilon }}(r=a) ={\bar{\rho }}{\tilde{e}}-{\bar{\rho }}e_a+\frac{1}{2}{\bar{\rho }} {\widetilde{(u_j-u_{j,a}) (u_j-u_{j,a})}}. \end{aligned}$$

(32)

If we subtract Eq. (32) from Eq. (31)

$$\begin{aligned}&{\bar{\rho }}{\tilde{\epsilon }}-{\bar{\rho }}\epsilon _t =\underbrace{{\bar{\rho }}{\tilde{e}}-{\bar{\rho }}e_r+p_r \left( 1-\frac{{\bar{\rho }}}{\rho _r}\right) -T_r({\bar{\rho }}{\tilde{s}} -{\bar{\rho }}s_r)+\frac{1}{2}{\bar{\rho }} {\widetilde{(u_j-u_{j,r})(u_j-u_{j,r})}}}_{{\bar{\rho }} {\tilde{\epsilon }}}\nonumber \\&\quad -\underbrace{\left[ {\bar{\rho }}{\tilde{e}} -{\bar{\rho }}e_a+\frac{1}{2}{\bar{\rho }} {\widetilde{(u_j-u_{j,a})(u_j-u_{j,a})}}\right] }_{{\bar{\rho }} \epsilon _t}~, \end{aligned}$$

(33)

we obtain the exergy per unit volume for the special reference state a

$$\begin{aligned} ={\bar{\rho }}e_a-{\bar{\rho }}e_r+p_r\frac{{\bar{\rho }}}{\rho _a} \left( 1-\frac{\rho _a}{\rho _r}\right) -T_r({\bar{\rho }}s_a-{\bar{\rho }}s_r) +\frac{1}{2}{\bar{\rho }}(u_{j,a}-u_{j,r})(u_{j,a}-u_{j,r})~, \end{aligned}$$

(34)

which shows that the exergy is the sum of the turbulent exergy and the exergy of the special reference state a (Eq. 7).

Acoustic energy density

The exergy of compressible flow is related to the acoustic energy density as summarised subsequently: For small disturbances and a perfect gas, the equation for the exergy (Eq. 1) multiplied with $\rho$ and expanded with a Taylor series around the reference state r becomes after Reynolds averaging the equation of the disturbance energy per unit volume

$$\begin{aligned} \epsilon _d=\frac{1}{2}\left( \frac{T_r}{\rho _r\gamma M^2}\overline{\rho '^2}+\frac{\rho _r}{\gamma (\gamma -1)T_rM^2}\overline{T'^2} +\rho _r(\overline{u_i'u_i'})\right) ~, \end{aligned}$$

(35)

introduced in Ref. [14] as an integral formulation, where $f'=f-f_r$ represents the deviation from the average reference or base state $f_r={\bar{f}}$. This expression is equivalent to

$$\begin{aligned} \epsilon _d=\frac{1}{2}\left( \frac{1}{\gamma p_r}\overline{p'^2}+(\gamma -1)\gamma M^4p_r\overline{s'^2}+\rho _r(\overline{u_i'u_i'})\right) ~, \end{aligned}$$

(36)

(see Ref. [35] and references therein) and is as the exergy, positive definite. For isentropic fluctuations ($s'=0$), Eq. (36) reduces to the well-known formula for the acoustic energy density with the contributions of potential and kinetic energy. A more general expression of disturbance energy which is closely related to the exergy has been derived in Ref. [32].

Fourier filter

In Sect. 3, the filtering procedure was introduced based on cancelling out Fourier modes in spectral space. One can obtain the same result with a real space Fourier filter which might be more illustrative, when compared to our incompressible flow analogy (Fig. 2). Note that in Sect. 3 the Fourier modes were in the range $0\le k\le N/2$, and the complex conjugate Fourier coefficients were not stored. The zero mode and odd-ball mode N/2 coefficients were real. Here we use the notation that the range of Fourier modes is $0\le k\le N-1$ (conjugate complex coefficients are represented in contrast to the definition of Sect. 3 and Fourier modes are transformed to positive k for simplicity).

For the one-dimensional case, the Fourier filter is constructed from the discrete Fourier transform forward and backward matrices, where the entries of the forward transformation matrix ${\mathbf {W}}$ are

$$\begin{aligned} w_{jl}=\frac{e^{(-2\pi i/N)jl}}{\sqrt{N}}\qquad 0\le j,l<N-1~, \end{aligned}$$

(37)

and the backward transformation matrix ${\mathbf {W}}^{-1}$ is

$$\begin{aligned} {\mathbf {W}}^{-1}={\mathbf {W}}^*~, \end{aligned}$$

(38)

where here $()^*$ denotes conjugate complex numbers. Together with the matrix ${\mathbf {G}}_k$, which cancels out Fourier modes k and $N+1-k$ in spectral space

$$\begin{aligned} g_{jl,k}= {\left\{ \begin{array}{ll} 1 &{}\quad j=l\ne k \wedge j=l\ne N+1-k~,\\ 0 &{}\quad \text {otherwise}~, \end{array}\right. } \end{aligned}$$

(39)

the filter matrix is

$$\begin{aligned} {\mathbf {F}}_k={\mathbf {W}}^{-1}{\mathbf {G}}_k{\mathbf {W}}~. \end{aligned}$$

(40)

We filter the vector

$$\begin{aligned} {\varvec{\varTheta }}= \begin{bmatrix} {\varvec{\sigma }}_0\\ {\varvec{\sigma }}_1\\ \vdots \\ {\varvec{\sigma }}_{N-1} \end{bmatrix}~, \end{aligned}$$

(41)

of the control volumes $0,\ldots ,N-1$ of the state vector ${\varvec{\sigma }}$ mode by mode. The stepwise filtering procedure can be expressed recursively as

$$\begin{aligned} {\varvec{\varPhi }}_k= {\left\{ \begin{array}{ll} {\varvec{\varTheta }} &{}\quad k=k_{\max }~,\\ {\mathbf {F}}_{k+1}{\varvec{\varPhi }}_{k+1} &{}\quad 0\le k<k_{\max }~, \end{array}\right. } k_{\max }=N/2~, \end{aligned}$$

(42)

for the filter steps $k,k-1,\ldots ,$0. The computation of the exergy of mode k with respect to mode $k-1$

$$\begin{aligned} \begin{aligned} \epsilon _k=\sum _{l=0}^{N-1}\sum _{j=0}^{N-1}\epsilon \Big (&{\varvec{\sigma }} =\begin{bmatrix}\phi _{j1}&\phi _{j2}&\cdots&\phi _{j5}\end{bmatrix}_k, \\&{\varvec{\sigma }}_r=\begin{bmatrix}\phi _{j1}&\phi _{j2}&\cdots&\phi _{j5}\end{bmatrix}_{k-1}\Big )f_{jl, k}~,~k>0~, \end{aligned} \end{aligned}$$

(43)

is done with the filter coefficients $f_{jl, k}$. In order to exploit the favourable computational complexity of the FFT algorithm scaling with $N\log N$, one would typically compute $\epsilon _k$ as the difference between the exergy of the unfiltered and filtered solution with respect to one set of modes using FFTs for the filtering step (Sect. 3). The extension of the filtering procedure to multiple dimensions is straightforward.