Scale dependence and cross-scale transfer of kinetic energy in compressible hydrodynamic turbulence at moderate Reynolds numbers

We investigate the properties of the scale dependence and cross-scale transfer of kinetic energy in compressible three-dimensional hydrodynamic turbulence by means of two direct numerical simulations of decaying turbulence with initial Mach numbers $M=1/3$ and 1, and with moderate Reynolds numbers, $R_{\lambda }\sim 100$. The turbulent dynamics is analyzed using compressible and incompressible versions of the dynamic spectral transfer (ST) and the Kármán-Howarth-Monin (KHM) equations. We find that the nonlinear coupling leads to a flux of the kinetic energy to small scales where it is dissipated; at the same time, the reversible pressure-dilatation mechanism causes oscillatory exchanges between the kinetic and internal energies with an average zero net energy transfer. While the incompressible KHM and ST equations are not generally valid in the simulations, their compressible counterparts are well satisfied and describe, in a quantitatively similar way, the decay of the kinetic energy on large scales, the cross-scale energy transfer/cascade, the pressure dilatation, and the dissipation. There exists a simple relationship between the KHM and ST results through the inverse proportionality between the wave vector $k$ and the spatial separation length $l$ as $kl\simeq \sqrt{3}$. For a given time, the dissipation and pressure-dilatation terms are strong on large scales in the KHM approach, whereas the ST terms become dominant on small scales; this is due to the complementary cumulative behavior of the two methods. The effect of pressure dilatation is weak when averaged over a period of its oscillations and may lead to a transfer of the kinetic energy from large to small scales without a net exchange between the kinetic and internal energies. Our results suggest that for large-enough systems, there exists an inertial range for the kinetic energy cascade. This transfer is partly due to the classical, nonlinear advection-driven cascade and partly due to the pressure dilatation-induced energy transfer. We also use the ST and KHM approaches to investigate the properties of the internal energy. The dynamic ST and KHM equations for the internal energy are well satisfied in the simulations but behave very differently with respect to the viscous dissipation. We conclude that ST and KHM approaches would better be used for the kinetic and internal energies separately.

Figure 1

Evolution in run 1: (a) the relative changes in the kinetic energy $\mathrm{\Delta }{E}_{\mathrm{k}}$ (solid line), the total energy $\mathrm{\Delta }{E}_{\mathrm{t}}$ (dotted line), and the internal energy $\mathrm{\Delta }{E}_{\mathrm{i}}$ (dashed); (b) rms of the vorticity ${\omega }_{\text{rms}}$; (c) Mach number $M$; (d) rms of the density fluctuations ${\rho }_{\text{rms}}$; and (e) the dissipation rate $\langle \mathit{\tau }:\mathit{\Sigma }\rangle$ (solid line), the pressure-dilatation rate $\langle p\theta \rangle$ (dashed line), and the compressible dissipation rate $4\mu \langle {\theta }^2\rangle /3$ (dotted line) as functions of time.

Figure 2

Evolution in run 2: (a) the relative changes in the kinetic energy $\mathrm{\Delta }{E}_{\mathrm{k}}$ (solid line), the total energy $\mathrm{\Delta }{E}_{\mathrm{t}}$ (dotted line), and the internal energy $\mathrm{\Delta }{E}_{\mathrm{i}}$ (dashed); (b) rms of the vorticity ${\omega }_{\text{rms}}$; (c) Mach number $M$; (d) rms of the density fluctuations ${\rho }_{\text{rms}}$; and (e) the dissipation rate $\langle \mathit{\tau }:\mathit{\Sigma }\rangle$ (solid line), the pressure-dilatation rate $\langle p\theta \rangle$ (dashed line), and the compressible dissipation rate $4\mu \langle {\theta }^2\rangle /3$ (dotted line) as functions of time.

Figure 3

Power spectral density of $\mathit{u}$, compensated by $k^{5/3}$ as a function of the wave vector $k$ in run 1 (solid line) and run 2 (dashed line).

Figure 4

Spectral transfer in run 1. (a) The validity test $O_k$ of Eq. (24) (black solid line) as a function of $k$ along with the different contributions (solid lines): blue, the losses/decay term $-\partial E_{\mathrm{k}k}/\partial t$; green, the transfer term $-S_k$; orange, the pressure dilatation term ${\mathrm{\Psi }}_k$; and red, the dissipation term $-D_k$. Dashed lines show the incompressible equivalent $O_k^{\left(i\right)}$ [Eq. (25)] and its contributions. (b) Time-averaged contributing terms (solid lines) with their minimum and maximum values (dotted lines) for blue, the decay $-{\langle \partial E_{\mathrm{k}k}/\partial t\rangle }_t$; green, the transfer $-{\langle S_k\rangle }_t$; orange, the pressure-dilatation term ${\langle {\mathrm{\Psi }}_k\rangle }_t$; and red, the dissipation term $-{\langle D_k\rangle }_t$. All the quantities are normalized with respect to $Q_{\mu }$.

Figure 5

Spectral transfer in run 2 in the same format as in Fig. 4.

Figure 6

KHM equation for run 1: (a) The validity test $O$ of Eq. (34) (black solid line) as a function of the separation scale $l$ along with the different contributions: blue, $-\partial \mathcal{S}/\partial t/4$; green, $\mathcal{K}$; orange, ${\mathrm{\Psi }}^{\prime }$; and red, $-D^{\prime }$. Dashed lines show the incompressible equivalents. (b) Time-averaged contributing terms (solid lines) with their minimum and maximum values (dotted lines) for blue, $-{\langle \partial \mathcal{S}/\partial t\rangle }_t/4$; green, ${\langle \mathcal{K}\rangle }_t/4$; orange, ${\langle {\mathrm{\Psi }}^{\prime }\rangle }_t$; and red, $-{\langle {\mathcal{D}}^{\prime }\rangle }_t$. All the quantities are normalized with respect to $Q_{\mu }$.

Figure 7

KHM equation for run 2 in the same format as in Fig. 6.

Figure 8

Direct comparison between ST and KHM methods for (a) run 1 and (b) run 2: Solid lines denote the time-averaged ST terms (see Figs. 4 and 5): blue, the losses/decay ${\langle \partial E_{\mathrm{k}k}/\partial t\rangle }_t$; green, the transfer ${\langle S_k\rangle }_t$; orange, the pressure-dilatation term $-{\langle {\mathrm{\Psi }}_k\rangle }_t$; and red, the dissipation term ${\langle D_k\rangle }_t-Q_{\mu }$ (i.e., shifted by $Q_{\mu }$) as functions of $k$. Dashed lines denote the corresponding time-averaged KHM contributions (see Figs. 6 and 7): blue, $-{\langle \partial \mathcal{S}/\partial t\rangle }_t/4-Q_{\mu }$ (i.e., shifted by $Q_{\mu }$); green, $\mathcal{K}$; orange, ${\langle {\mathrm{\Psi }}^{\prime }\rangle }_t$; and red, $-{\langle {\mathcal{D}}^{\prime }\rangle }_t$ as functions of $l=\sqrt{3}/k$. All the quantities are normalized with respect to $Q_{\mu }$.

Figure 9

Spectral transfer of the internal energy in run 2: (a) The validity test $O_{\mathrm{i}k}$ of Eq. (44) (black line) as a function of $k$ along with the different contributions: blue, the time variation $-\partial E_{\mathrm{i}k}/\partial t$; green, the energy transfer term $-S_{\mathrm{i}k}$; orange, the pressure-dilatation term ${\mathrm{\Psi }}_{\mathrm{i}k}$; magenta, the diffusion ${\mathrm{\Phi }}_{\mathrm{i}k}$; and red, the dissipation term $D_{\mathrm{i}k}$. (b) Time-averaged contributing terms (solid lines) with their minimum and maximum values (dotted lines) for blue, the decay $-{\langle \partial E_{\mathrm{i}k}/\partial t\rangle }_t$; green, the transfer $-{\langle S_{\mathrm{i}k}\rangle }_t$; orange, the pressure-dilatation term ${\langle {\mathrm{\Psi }}_{\mathrm{i}k}\rangle }_t$; magenta, the diffusion ${\langle {\mathrm{\Phi }}_{\mathrm{i}k}\rangle }_t$; and red, the dissipation term ${\langle D_{\mathrm{i}k}\rangle }_t$. All the quantities are normalized with respect to $Q_{\mu }$.

Figure 10

KHM equation for the internal energy in run 2: (a) The validity test $O_{\mathrm{i}}$ of Eq. (49) (black line) as a function of the separation scale $l$ along with the different contributions: blue, $-\partial {\mathcal{S}}_{\mathrm{i}}/\partial t/2$; green, ${\mathcal{K}}_{\mathrm{i}}$; orange, $-{\mathrm{\Psi }}_{\mathrm{i}}$; red, ${\mathcal{D}}_{\mathrm{i}}$; and magenta, ${\mathrm{\Phi }}_{\mathrm{i}}$. (b) Time-averaged contributing terms (solid lines) with their minimum and maximum values (dotted lines) for blue, $-{\langle \partial {\mathcal{S}}_{\mathrm{i}}/\partial t\rangle }_t/2$; green, ${\langle {\mathcal{K}}_{\mathrm{i}}\rangle }_t$; orange, $-{\langle {\mathrm{\Psi }}_{\mathrm{i}}\rangle }_t$; red, ${\langle {\mathcal{D}}_{\mathrm{i}}\rangle }_t$; and magenta, ${\langle {\mathrm{\Phi }}_{\mathrm{i}}\rangle }_t$. All the quantities are normalized with respect to $Q_{\mu }$.