The internal structure of shock waves is well known for shocks with a Prandtl number of the order of unity, but it is still poorly understood for shocks with extremely large and small Prandtl numbers, and the latter case is of particular interest given the non-existence of smooth solutions for strong inviscid shocks. The literature has suggested unequal contributions of viscosity and heat conduction to shock compression, specifically that viscosity is of more significance, but the observations therein do not provide further insight into the exact role of viscosity in shock waves, i.e., how viscosity affects the shock structure such that a smooth transition is maintained. To elucidate it, this research numerically investigates the internal structure of shock waves, with emphasis on the asymptotic behavior and features of shock profiles in the inviscid (zero-Prandtl-number) limit. The asymptotic analysis demonstrates a curious fact: the distinction between strong shock solutions with positive and zero Prandtl numbers does not vanish as the former's Prandtl number tends to zero. The only way to avoid this gap is to allow the emergence of an unphysical discontinuity (e.g., an isothermal jump) in inviscid solutions, which clearly indicates the essentiality of viscosity. For a better understanding of the underlying mechanisms, features of small-Prandtl-number shocks are further studied and compared with those of ordinary shocks. The result shows a peak in viscous local entropy generation rates within small-Prandtl-number shocks, and reducing the viscosity paradoxically makes it higher. Through this peak, the effect of lowered viscosity is neutralized, and the isothermal jump is smoothed, hence the critical role of viscosity in shock transition.

中文翻译：

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激波的内部结构：无粘极限下的渐近行为和小普朗特数下的特征


冲击波的内部结构对于具有单位阶普朗特数的冲击是众所周知的，但对于具有极大和极小普朗特数的冲击仍然知之甚少，并且由于不存在后一种情况而特别令人感兴趣强无粘冲击的平滑解决方案。文献提出粘度和热传导对冲击压缩的贡献不等，特别是粘度更重要，但其中的观察结果并没有进一步深入了解粘度在冲击波中的确切作用，即粘度如何影响冲击结构从而保持平稳过渡。为了阐明这一点，本研究对激波的内部结构进行了数值研究，重点研究了无粘性（零普朗特数）极限下激波剖面的渐近行为和特征。渐近分析证明了一个奇怪的事实：正普朗特数和零普朗特数的强激波解之间的区别并没有随着前者的普朗特数趋于零而消失。避免这种差距的唯一方法是允许在无粘性溶液中出现非物理不连续性（例如等温跳跃），这清楚地表明了粘度的重要性。为了更好地理解潜在机制，进一步研究了小普朗特数冲击的特征，并将其与普通冲击进行了比较。结果表明，小普朗特数激波内的粘性局部熵生成率达到峰值，而降低粘性反而会使其更高。通过这个峰值，粘度降低的影响被中和，等温跳跃变得平滑，因此粘度在冲击转变中起着关键作用。