Title: A mass invariant in a compressible turbulent medium Show Chinese title

Title: 可压缩湍流介质中的质量不变量

Abstract: Predicting the measurable statistical properties of density fluctuations in a supersonic compressible turbulent flow is a major challenge in physics. In 1951, Chandrasekhar derived an invariant under the assumption of the statistical homogeneity and isotropy of the turbulent density field and stationarity of the background density. Recently, Jaupart & Chabrier (2021) extended this invariant to non-isotropic flows in a time-evolving background and showed that it has the dimension of a mass. This invariant $M_{\rm inv}$ is defined by $M_{\rm inv} = \mathbb{E}(\rho)\text{Var}\left(\frac{\rho}{\mathbb{E}(\rho)}\right)(l_{\rm c}^\rho)^3$ where $\rho$ is the density field and $l_{\rm c}^\rho$ is the correlation length. In this article, we perform numerical simulations of homogeneous and isotropic compressible turbulence to test the validity of this invariant in a medium subject to isotropic decaying turbulence. We study several input configurations, namely different Mach numbers, injection lengths of turbulence and equations of state. We confirm that $M_{\rm inv}$ remains constant during the decaying phase of turbulence. Furthermore, we develop a theoretical model of the density field statistics which predicts without any free parameter the evolution of the correlation length with the variance of the logdensity field beyond the assumption of the gaussian field for the logdensity. Noting that $M_{\rm inv}$ is independent of the Mach number, we show that this invariant can be used to relate the non-gaussian evolution of the logdensity probability distribution function to its variance with no free parameters. Show Chinese abstract

Abstract: 预测超音速可压缩湍流中密度涨落的可测量统计特性是物理学中的一个重大挑战。 在1951年，钱德拉塞卡在假设湍流密度场的统计同性和各向同性以及背景密度平稳性的前提下推导出一个不变量。 最近，Jaupart & Chabrier（2021）将这个不变量扩展到时间演化背景中的非各向同性流动，并表明它的量纲为质量。 这个不变量$M_{\rm inv}$由$M_{\rm inv} = \mathbb{E}(\rho)\text{Var}\left(\frac{\rho}{\mathbb{E}(\rho)}\right)(l_{\rm c}^\rho)^3$定义，其中$\rho$是密度场，$l_{\rm c}^\rho$是相关长度。 在本文中，我们对均匀且各向同性的可压缩湍流进行数值模拟，以测试该不变量在受到各向同性衰减湍流影响的介质中的有效性。 我们研究了几种输入配置，即不同的马赫数、湍流注入长度和状态方程。 我们确认在湍流衰减阶段，$M_{\rm inv}$保持恒定。 此外，我们建立了一个密度场统计的理论模型，该模型无需任何自由参数即可预测对数密度场方差之外的关联长度演变，超越了对数密度的高斯场假设。 注意到$M_{\rm inv}$与马赫数无关，我们证明该不变量可以用来将对数密度概率分布函数的非高斯演化与其方差联系起来，而无需任何自由参数。