As pioneered by Donzis and Maqui ({J. Fluid Mech.}, vol. 797, 2016, pp. 181-200) and Khurshid and Donzis ({Phys. Fluids}, vol. 31, 2019, 015103), the compressible isotropic turbulence in thermal non-equilibrium is drawing attention in the fluid dynamics community. In the present study, the vibrational rate and the dissipation/production of vibrational energy fluctuation of compressible isotropic turbulence with solenoidal forcing in vibrational non-equilibrium are investigated using numerical simulations. The turbulent Mach number ($ M_t $)issettobe0.44and1.09,andtheTaylorReynoldsnumber($ Re_$)equalsto98.58approximately.Thefocusisontheeffectofthenormalizedvibrationalrelaxationtime($ K_$)andcharacteristictemperature($_v $) on statistical features of the vibrational rate and the dissipation/production components of vibrational energy fluctuation. { When $K_{\tau }$ is small $(\lesssim 1.0)$, the average of normalized vibrational rate conditioned on the normalized dilatation is positive in the compression region and negative in the expansion region; enhances with increasing of compression and expansion level, respectively.} It indicates that, on average, the energy transfers from translational-rotational to vibrational modes in the compression region and in the inverse direction in the expansion region. The conditional average of the normalized vibrational rate reduced with increasing of $K_{\tau }$, and is insensitive to ${\theta }_v$. The joint PDFs and conditional PDFs of the normalized vibrational rate are also analysed, to reveal the effect of compression and expansion motions on the internal energy exchange between the translational-rotational and vibrational modes. The dissipation/production of vibrational energy fluctuation can result from effects of dilatation, thermal diffusion and vibrational relaxation. The dissipation component due to thermal diffusion always weakens the vibrational energy fluctuation in both compression and expansion regions for the weakly and highly compressible turbulence; but its effect is insignificant as compare to other two components. For the weakly compressible turbulence, the dissipation/production of vibrational energy fluctuation mainly comes from effects of dilatation and vibrational relaxation when $K_{\tau }$ is small $(\lesssim 1.0)$; and the vibrational relaxation component loses its significance with increasing of $K_{\tau }$. For the highly compressible turbulence, both the dilatation and vibrational relaxation effects play an important role in the dissipation and production of vibrational energy fluctuation, and their effects are significantly different from that of the weakly compressible turbulence. The conditional average, joint PDFs and conditional PDFs of each dissipation/production component are calculated and analysed in detail to reveal their effects on the vibrational energy fluctuation in different compression and expansion regions. { Furthermore, the bulk viscosity effect on the compression/expansion motion in flow field, the vibrational rate, and the dissipation/production components of vibrational energy fluctuation is discussed briefly.

中文翻译：

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具有热非平衡的可压缩各向同性湍流中的振动弛豫

正如Donzis和Maqui（{J.Fluid Mech。}，第797卷，2016年，第181-200页）和Khurshid and Donzis（{Phys。Fluids}，第31卷，2019年，015103）的开创者一样，可压缩各向同性热非平衡中的湍流引起了流体动力学界的关注。在本研究中，利用数值模拟研究了在非平衡状态下利用螺线管强迫的可压缩各向同性湍流的振动速率和振动能量波动的耗散/产生。湍流马赫数（$ M_t$）\mathrm{一世}ssËŤŤØbË0.44\mathrm{一种}ñd1.09，\mathrm{一种}ñdŤHËŤ\mathrm{一种}ÿ升Ø\mathrm{[R}\mathrm{[R}ËÿñØ升dsñü米bË\mathrm{[R}（$ 回覆_$）Ëqü\mathrm{一种}升sŤØ98.58\mathrm{一种}pp\mathrm{[R}ØX\mathrm{一世}米\mathrm{一种}ŤË升ÿ。ŤHËFØCüs\mathrm{一世}sØñŤHËËFFËCŤØFŤHËñØ\mathrm{[R}米\mathrm{一种}升\mathrm{一世}žËdv\mathrm{一世}b\mathrm{[R}\mathrm{一种}Ť\mathrm{一世}Øñ\mathrm{一种}升\mathrm{[R}Ë升\mathrm{一种}X\mathrm{一种}Ť\mathrm{一世}ØñŤ\mathrm{一世}米Ë（$ K_$）\mathrm{一种}ñdCH\mathrm{一种}\mathrm{[R}\mathrm{一种}CŤË\mathrm{[R}\mathrm{一世}sŤ\mathrm{一世}CŤË米pË\mathrm{[R}\mathrm{一种}Ťü\mathrm{[R}Ë（$振动速率和振动能量波动的耗散/产生分量的统计特征。{ 什么时候${ķ}_{\tau }$ 是小 $（\lesssim 1.0）$，以归一化膨胀为条件的归一化振动率的平均值在压缩区域为正，在膨胀区域为负。}表示平均而言，能量在压缩区域中从平移-旋转模式传递到振动模式，并且在膨胀区域中沿反方向传递。归一化振动率的条件平均值随着${ķ}_{\tau }$，并且对 ${\theta }_v$。还分析了归一化振动速率的联合PDF和条件PDF，以揭示压缩和膨胀运动对平移-旋转和振动模式之间的内部能量交换的影响。振动能量波动的耗散/产生可能是由于膨胀，热扩散和振动松弛的影响。由于热扩散而产生的耗散分量始终会减弱弱和高度可压缩湍流在压缩和膨胀区域的振动能量波动；但与其他两个组件相比，其作用微不足道。对于弱可压缩湍流，振动能量波动的耗散/产生主要来自膨胀和振动松弛的影响。${ķ}_{\tau }$ 是小 $（\lesssim 1.0）$; 振动松弛分量随其增加而失去其重要性。${ķ}_{\tau }$。对于高度可压缩的湍流，膨胀和振动松弛效应在振动能量波动的散发和产生中都起着重要作用，并且它们的影响与弱可压缩湍流的影响显着不同。计算并详细分析了每个耗散/生产组件的条件平均值，联合PDF和条件PDF，以揭示它们对不同压缩和膨胀区域中振动能量波动的影响。{此外，简要讨论了体粘度对流场中压缩/膨胀运动，振动速率以及振动能量波动的耗散/产生分量的影响。