Convex integration and dissipative anomaly in compressible turbulence

基本方面的普遍性，在小尺度上的统计特性在高雷诺数流体湍流主要是研究不可压缩的情况下，类似的努力，完全可压缩制度最近才开始。这种情况的一个原因是，除了（泰勒微尺度）雷诺数R_lambda之外，还必须包括其他无量纲参数。最近对直接数值模拟（DNS）数据的综合记录进行的比较表明，不仅必须包括R_lambda附近的湍流马赫数M_t，而且还必须包括湍流速度与螺线管速度均方根（rms）之比δ。另一方面，一个规模的能量平衡，类似于工作的Duchon和罗伯特的不可压缩的情况下，揭示了两个不同的可能的级联机制，称为变形和正压密度的工作，这可能是负责可压缩湍流的耗散异常。这一建议的基本前提是，可压缩湍流的进一步发展需要详细研究这些机制及其对M_t和δ的依赖性。虽然变形的工作是密切相关的不可压缩的情况下，昂萨格的经典猜想，预计占主导地位的滞弹性制度，膨胀为主的制度是迄今为止完全未探索。本项目旨在将现有的Onsager弱解理论统一地推广到整个M_t-delta参数平面上的可压缩Euler方程组。数学方面的重要里程碑问题包括容许弱解的一般非唯一性、Onsager临界Besov空间中弱解的构造以及极端耗散事件的几何理论的数学基础。在流体力学方面的关键里程碑问题是尺度相关的数值分析的变形和气压密度功与能量耗散事件在充分发展的可压缩湍流。达到这些科学目标，因此需要共同努力的数学和基本的流体力学调查，汇集了分析计算，建设性的证明技术的凸积分，直接数值模拟，可压缩湍流的统计和几何分析。我们的项目特别涉及优先计划的研究领域A，重点是弱熵解的分析和数值研究及其与可压缩Navier-Stokes湍流中小尺度不稳定性的相关性。

Fundamental aspects of the universality of statistical properties at the small scales in high-Reynolds-number fluid turbulence have been mostly studied for the incompressible case and similar efforts for the fully compressible regime have been started only recently. One reason for this circumstance is that further dimensionless parameters have to be included next to the (Taylor microscale) Reynolds number R_lambda. A recent comparison of a comprehensive record of direct numerical simulation (DNS) data suggests that not only the turbulent Mach number M_t next to R_lambda, but also the ratio of dilatational to solenoidal root mean square (rms) velocities, delta, have to be included. On the other hand, a scale-by-scale energy balance, analogous to work of Duchon and Robert for the incompressible case, reveals two distinct possible cascade mechanisms called deformation and baropycnal work, which may be responsible for the dissipative anomaly in compressible turbulence. The basic premise of this proposal is that further progress in compressible turbulence requires the study of these mechanisms in detail and their dependencies on M_t and delta. Whilst deformation work is closely related to the incompressible case and the classical conjecture of Onsager and is expected to dominate in the anelastic regime, the dilatation-dominated regime is so far completely unexplored. The present project aims at extending the currently available Onsager-theory of weak solutions to the compressible Euler equation system in the entire M_t—delta parameter plane in a unifying manner. Key milestone problems on the mathematical side include the generic non-uniqueness of admissible weak solutions, the construction of weak solutions in Onsager-critical Besov spaces as well as the mathematical foundations of a geometric theory of extreme dissipation events. The key milestone problem on the fluid mechanical side is the scale-dependent numerical analysis of deformation and baropycnal work in relation with energy dissipation events in fully developed compressible turbulence. Reaching these scientific objectives requires consequently a joint effort of mathematical and basic fluid mechanical investigations that brings together analytical calculations, constructive proof techniques of convex integration, direct numerical simulations, and statistical and geometrical analysis for compressible turbulence. Our project addresses in particular the research area A of the Priority Programme, focusing on the analytical and numerical study of weak entropy solutions and their relevance for the small-scale intermittency in compressible Navier-Stokes turbulence.