Kinetic and internal energy transfer in implicit large-eddy simulations of forced compressible turbulence

We revisit the problem of how energy transfer through the turbulent cascade operates in compressible hydrodynamic turbulence. In general, there is no conservative compressible cascade since the kinetic and internal energy reservoirs can exchange energy through pressure dilatation. Moreover, statistically stationary turbulence at high Mach number can only be maintained in nearly isothermal gas, i.e., if excess heat produced by shock compression and kinetic energy dissipation is continuously removed from the system. We mimic this process by a linear cooling term in numerical simulations of turbulence driven by stochastic forcing. This allows us to investigate turbulence statistics for a broad range of Mach numbers. We compute the rate of change of kinetic and internal energy in wave-number shells caused by advective, compressive, and pressure dilatation effects and constrain power-law fits to compressible turbulence energy spectra to a range of wave numbers in which the total energy transfer is close to zero. The resulting scaling exponents are significantly affected by the forcing. Depending on the root mean square Mach number, we find a nearly constant advective component of the cross-scale flux of kinetic energy at intermediate wave numbers for particular mixtures of solenoidal and compressive modes in the forcing. This suggests the existence of a natural, Mach number dependent mixture of forcing modes. Our findings also support an advection-dominated regime at high Mach numbers with specific scaling exponents (Burgers scaling for the pure velocity fluctuation $u$ and Kolmogorov scaling for the mass-weighted variable $v={\rho }^{1/3}u$).

Figure 1

Schematic view of energy transfer in the $\mathrm{Q}-\mathrm{K}$ plane for a local energy cascade. Shell-to-shell transfers are shown as diagonal bands above and below the median (long dashed line), with red and green (online version only) indicating positive and negative transfers, respectively. The long dashed line for which $\mathrm{Q}=\mathrm{K}$ is the median line. The transfer terms on the right-hand side of the shell energy Eqs. (29) and (30) are given by summation over all $\mathrm{Q}$ for given $\mathrm{K}$ [Eq. (31)], corresponding to the horizontal dashed lines. Cross-scale fluxes are obtained by summing over rectangular regions $\mathrm{Q}\le k$ and $\mathrm{K}>k$, where $k$ is an arbitrary but fixed wave number. If the local energy transfers vanish sufficiently fast with distance from the median line (locality of energy transfer), the cross-scale fluxes for the gray-shaded regions are approximately constant.

Figure 2

Time evolution of the mean internal energy minus initial energy (code units) and rms Mach number for different cooling coefficients $\alpha$ in simulations with forcing parameters $V=1.0$ and $\zeta =2/3$ (see also Table 1).

Figure 3

Phase diagrams for cooling coefficients $\alpha =1000$ (left) and 1 (right) at the end of simulations with $V=1.0$ and $\zeta =2/3$. The temperature $T$ and density $\rho$ are normalized to their initial values.

Figure 4

Slices of the vorticity $\omega$ (a), (d), divergence $d$ (b), (e), and $q$ (c), (f) defined by Eq. (8) for cooling coefficients $\alpha =1000$ (a)–(c) and 1 (d)–(f) at the end of simulations with $V=1.0$ and $\zeta =2/3$ (the corresponding phase diagrams are shown in Fig. 3).

Figure 5

Time evolution of the mean internal energy minus initial energy (a, in code units) and rms Mach number (b) for different forcing amplitudes in combination with $\zeta =2/3$ and $\alpha =1000$ in the statistically stationary phase starting at $t=t_1$.

Figure 6

Time-averaged PDFs of the logarithmic mass density $s=log(\rho /{\rho }_0)$, where ${\rho }_0=1$, for different Mach numbers and fixed $\zeta =2/3$ (a) and for varying $\zeta$ (b). The thin solid lines indicate log-normal fits defined by (42) in the range $-3{\sigma }_s\le s-s_0\le 3{\sigma }_s$.

Figure 7

Time-averaged transfers ${\mathcal{T}}_{\mathrm{XY}}^{\mathrm{K}}$ into shell $\mathrm{K}$ [see Eq. (31)] for energy reservoirs X and Y indicated by the legends in the right column. Forcing with $V=1.0$ and $\zeta =2/3$ was applied for the transfer functions in plots (a), (e), (i), (m) and (b), (f), (j), (n), while $V=4.0$ and $\zeta =1/4$ for (c), (g), (k), (o) and (d), (h), (l), (p). In both cases the cooling coefficient was varied: $\alpha =1000$ for (a), (e), (i), (m) and (c), (g), (k), (o), and $\alpha =10$ for (b), (f), (j), (n) and (d), (h), (l), (p). See also Table 1 for an overview. All transfers are scaled by $V^3$. The range between one standard deviation above and below the time average is filled (except for the top panels). The estimated inertial range on the basis of near zero transfer is indicated by gray shaded wave-number ranges.

Figure 8

Time-averaged cross-scale fluxes of kinetic and internal energy in nearly isothermal turbulence ($\alpha =1000$) for increasing forcing magnitude $[V=0.5$ for (a), (e), (i), $V=1.0$ for (b), (f), (j), $V=2.0$ for (c), (g), (k), and $V=4.0$ for (d), (h), (l)] and a constant mixing ratio $\zeta =2/3$. All fluxes are normalized by $V^3$.

Figure 9

Cross-scale fluxes for supersonic turbulence ($V=4.0$) with decreasing fraction of solenoidal forcing modes $[\zeta =2/3$ for (a), (e), (i), $\zeta =1/2$ for (b), (f), (j), $\zeta =1/4$ for (c), (g), (k), and $\zeta =1/8$ for (d), (h), (l)] as in Fig. 8.

Figure 10

Normalized turbulence energy spectra for the three velocity variables $\mathbf{u}$ (a), (d), $\mathbf{v}={\rho }^{1/3}\mathbf{u}$ (b), (e), and $\mathbf{w}=\sqrt{\rho }\mathbf{u}$ (c), (e) compensated with $k^{5/3}$. The mixture of solenoidal and dilatational forcing modes is varied for integral velocity scales $V=1.0$ (a)–(c) and $V=4.0$ (d)–(f). The power-law fits listed in Table 3 are shown as thin black line segments for $0.5k_{\min }<k<2k_{\max }$. The vertical gray lines in the left plots indicate the sonic wave numbers $k_{\mathrm{s}}$ defined by Eq. (41).

Figure 11

Normalized compensated turbulence energy spectra as in Fig. 10 for different Mach numbers and $\zeta =2/3$.

Figure 12

Time evolution of the mean internal energy minus initial energy (a) and kinetic energy (b) for different numerical resolutions in the case $V=1.0$ and $\zeta =2/3$.

Figure 13

Time-averaged transfers as in Fig. 7 for numerical resolutions ${1024}^3$ (a), (d), (g), (j), ${512}^3$ (b), (e), (h), (k), and ${256}^3$ (c), (f), (i), (l) in the case $V=1.0$ and $\zeta =2/3$. In panels (a)–(c) also the transfer term associated with the forcing, ${\mathcal{F}}^{\mathrm{K}}\equiv {\mathcal{T}}_{\mathrm{FU}}^{\mathrm{K}}$, is shown for comparison.

Figure 14

Normalized compensated turbulence energy spectra as in Fig. 11 for different numerical resolutions in the case $V=1.0$ and $\zeta =2/3$.