Department of Mechanical Engineering, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel.
Phys Rev E. 2019 Aug;100(2-1):023111. doi: 10.1103/PhysRevE.100.023111.
We study the statistics of fluid (gas) density and concentration of passive tracer particles (dust) in compressible turbulence. As Ma increases from small or moderate values, the density and the concentration in the inertial range go through a phase transition from a finite continuous smooth distribution to a singular multifractal spatial distribution. Multifractality is associated with scaling, which would not hold if the solenoidal and the potential components of the flow scaled differently, producing transport which is not self-similar. Thus, we propose that the transition occurs when the difference of the scaling exponents of the components, decreasing with Ma, becomes small. Under the smallness assumption, the particles' volumes obey a power-law evolution. That, by the use of conservation of the total volume of the flow, entails the volumes' shrinking to zero with probability 1 and formation of a singular distribution. We discuss various concepts of multifractality and propose a way to calculate fractal dimensions from numerical or experimental data. We derive a simple expression for the spectrum of fractal dimensions of isothermal turbulence and describe limitations of lognormality. The expression depends on a single parameter: the scaling exponent of the density spectrum. We demonstrate that the pair-correlation function of the tracer concentration has the Markov property. This implies applicability of the compressible version of the Kraichnan turbulence model. We use the model to derive an explicit expression for the tracer pair correlation that demonstrates their smooth transition to multifractality and confirms the transition's mechanism. The obtained fractal dimension explains previous numerical observations. Our results have potentially important implications for astrophysical problems such as star formation as well as technological applications such as supersonic combustion. As an example, we demonstrate the strong increase of planetesimal formation rate at the transition. We prove that finiteness of internal energy implies vanishing of the sum of Lyapunov exponents in the dissipation range. Our study leads to the question of whether the fluid density which is an active field that reacts back on the transporting flow and the passive concentration of tracers must coincide in the steady state. This is demonstrated to be crucial both theoretically and experimentally. The fields' coincidence is provable at small Mach numbers; however, at finite Mach numbers, the assumption of mixing is needed, which we demonstrate to be not self-evident because of the possibility of self-organization.
我们研究了可压缩湍流中流体（气体）密度和被动示踪粒子（尘埃）浓度的统计特性。当马从较小或中等值增加时，惯性范围内的密度和浓度经历了从有限连续光滑分布到奇异多重分形空间分布的相变。多重分形性与标度有关，如果流的旋度和位势分量以不同的方式标度，产生非自相似的输运，则标度将不成立。因此，我们提出当分量的标度指数差随着马的减小而变小时，相变发生。在微小性假设下，粒子的体积服从幂律演化。这意味着，通过使用流动的总体积守恒，必然会导致体积以概率 1 收缩到零，并形成奇异分布。我们讨论了各种多重分形概念，并提出了一种从数值或实验数据计算分形维数的方法。我们推导出了等温和湍流的分形维数谱的简单表达式，并描述了对数正态性的局限性。该表达式取决于单个参数：密度谱的标度指数。我们证明示踪剂浓度的自相关函数具有马尔可夫性质。这意味着可压缩克努森湍流模型的适用性。我们使用该模型推导出示踪剂对关联的显式表达式，该表达式证明了它们从平滑过渡到多重分形的过程，并证实了相变的机制。得到的分形维数解释了以前的数值观测结果。我们的研究结果对于天体物理问题（如恒星形成）以及技术应用（如超音速燃烧）具有潜在的重要意义。例如，我们证明了在相变时行星形成率的急剧增加。我们证明了内部能量的有限性意味着耗散范围内的 Lyapunov 指数之和为零。我们的研究提出了这样一个问题，即作为对输运流做出反应的主动场的流体密度和被动示踪剂浓度的浓度是否必须在稳定状态下重合。这在理论和实验上都是至关重要的。在小马赫数下可以证明场的重合；然而，在有限的马赫数下，需要假设混合，我们证明这不是不言而喻的，因为存在自组织的可能性。
