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.

中文翻译：

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中等雷诺数下可压缩流体动力湍流中动能的尺度依赖性和跨尺度转移

我们通过两个具有初始马赫数的衰减湍流的直接数值模拟，研究了可压缩三维水动力湍流中动能的尺度依赖性和跨尺度转移的性质。 $\mathrm{中号}=\mathrm{1个}/3$ 和1，并且具有中等的雷诺数， ${\mathrm{[R}}_{\lambda }〜100$。使用动态频谱传输（ST）和Kármán-Howarth-Monin（KHM）方程的可压缩和不可压缩版本来分析湍流动力学。我们发现非线性耦合导致动能的通量在小范围内消散。同时，可逆的压力-膨胀机制导致动能与内能之间发生振荡交换，平均净净能量传递为零。虽然不可压缩的KHM和ST方程在模拟中通常无效，但它们的可压缩对等方程令人满意，并以定量相似的方式描述了动能的大尺度衰减，跨尺度能量传递/级联，压力膨胀和耗散。$ķ$ 和空间间隔长度 $升$ 作为 $ķ升\simeq \sqrt{3}$。在给定的时间内，KHM方法中的耗散和压力膨胀项在大范围内很强，而ST项在小范围内占主导。这是由于这两种方法的互补累积行为。当在其整个振荡周期内进行平均时，压力膨胀的影响很弱，并且可能导致动能从大尺度转移到小尺度，而动能和内能之间没有净交换。我们的结果表明，对于足够大的系统，动能级联存在一个惯性范围。这种传递部分是由于经典的，非线性对流驱动的级联，部分是由于压力膨胀引起的能量传递。我们还使用ST和KHM方法研究内部能量的性质。在模拟中很好地满足了内部能量的动态ST和KHM方程，但在粘性耗散方面的行为却大不相同。我们得出结论，ST和KHM方法最好分别用于动能和内能。