A stochastic model for the relative motion of high Stokes number particles in isotropic turbulence
A stochastic model for the relative motion of high Stokes number particles in isotropic turbulence
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DOI：
10.1017/jfm.2014.461
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发表时间：
2014-09
影响因子：
3.7
通讯作者：
S. Rani;R. Dhariwal;D. Koch
中科院分区：
文献类型：
--
作者：
S. Rani;R. Dhariwal;D. Koch
摘要描述湍流中惯性粒子对相对运动的概率密度函数（PDF）动力学方程需要封闭相空间扩散电流。提出了一种适用于具有Stokes数的高惯性粒子对$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}{\mathit{St}}_r \gg 1$的扩散电流解析闭包。这里${\mathit{St}}_r$是基于漩涡的时间尺度$\tau _r$的斯托克斯数，漩涡的大小随对分离$r$而变化。在${\mathit{St}}_r \gg 1$的渐近极限下，对PDF动力学方程化为Fokker-Planck形式的方程。表征Fokker-Planck方程中扩散电流的扩散张量等于$1/\tau _v^2$乘以沿粒子对轨迹流体相对速度的拉格朗日相关的时间积分。其中$\tau _v$为粒子粘性弛豫时间。扩散张量的闭合是通过将流体相对速度的拉格朗日关联转换成欧拉流体速度关联来实现的，这种关联在$O(\tau _r)$的时间尺度上基本保持不变；然而，这对质心不是静止的，它对涡旋的响应时间尺度相当于或小于$\tau _v$。对于各向同性湍流，欧拉流体-速度相关性可以表示为速度谱张量的傅里叶变换，使我们能够推导出扩散张量的封闭形式表达式。该闭包的一个显著特征是，它具有跨越整个湍流尺度谱的对分离的单一独特形式，而不像以前的闭包涉及积分、惯性子范围和kolmogorov尺度分离的不同形式的速度结构函数。利用这一闭包，我们求解了统计上等同于福克-普朗克方程的朗之万方程，以求得静止各向同性湍流中粒子对相对速度和分离的演化。Langevin方程方法能够模拟成对相对运动的完整PDF，而不是像基于力矩的方法那样只模拟PDF的前几个力矩。因此，计算并给出了不同分离$r$的PDF $\varOmega (U|r)$和$\varOmega (U_r|r)$，其中前者是相对速度的PDF $U$，后者是相对速度径向分量的PDF $U_r$，两者都以分离$r$为条件。与Sundaram ＆ Collins （J. Fluid Mech.）的直接数值模拟（DNS）研究相一致。， vol. 335, 1997, pp. 75-109)， Langevin模拟捕获了$\varOmega (U|r)$从积分尺度分离的高斯到kolmogorov尺度分离的指数PDF的转变。从这些模拟中计算的径向分布函数（RDFs）也显示出与fsamvrier， Simonin和Legendre（第四届多相流国际会议论文集，新奥尔良，2001）的DNS研究结果在数量上的合理一致。
Abstract The probability density function (PDF) kinetic equation describing the relative motion of inertial particle pairs in a turbulent flow requires closure of the phase-space diffusion current. A novel analytical closure for the diffusion current is presented that is applicable to high-inertia particle pairs with Stokes numbers $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}{\mathit{St}}_r \gg 1$ . Here ${\mathit{St}}_r$ is a Stokes number based on the time scale $\tau _r$ of eddies whose size scales with pair separation $r$ . In the asymptotic limit of ${\mathit{St}}_r \gg 1$ , the pair PDF kinetic equation reduces to an equation of Fokker–Planck form. The diffusion tensor characterizing the diffusion current in the Fokker–Planck equation is equal to $1/\tau _v^2$ multiplied by the time integral of the Lagrangian correlation of fluid relative velocities along particle-pair trajectories. Here, $\tau _v$ is the particle viscous relaxation time. Closure of the diffusion tensor is achieved by converting the Lagrangian correlations of fluid relative velocities ‘seen’ by pairs into Eulerian fluid-velocity correlations at pair separations that remain essentially constant during time scales of $O(\tau _r)$ ; the pair centre of mass, however, is not stationary and responds to eddies with time scales comparable to or smaller than $\tau _v$ . For isotropic turbulence, Eulerian fluid-velocity correlations may be expressed as Fourier transforms of the velocity spectrum tensor, enabling us to derive a closed-form expression for the diffusion tensor. A salient feature of this closure is that it has a single, unique form for pair separations spanning the entire spectrum of turbulence scales, unlike previous closures that involve velocity structure functions with different forms for the integral, inertial subrange, and Kolmogorov-scale separations. Using this closure, Langevin equations, which are statistically equivalent to the Fokker–Planck equation, were solved to evolve particle-pair relative velocities and separations in stationary isotropic turbulence. The Langevin equation approach enables the simulation of the full PDF of pair relative motion, instead of only the first few moments of the PDF as is the case in a moments-based approach. Accordingly, PDFs $\varOmega (U|r)$ and $\varOmega (U_r|r)$ are computed and presented for various separations $r$ , where the former is the PDF of relative velocity $U$ and the latter is the PDF of the radial component of relative velocity $U_r$ , both conditioned upon the separation $r$ . Consistent with the direct numerical simulation (DNS) study of Sundaram & Collins (J. Fluid Mech., vol. 335, 1997, pp. 75–109), the Langevin simulations capture the transition of $\varOmega (U|r)$ from being Gaussian at integral-scale separations to an exponential PDF at Kolmogorov-scale separations. The radial distribution functions (RDFs) computed from these simulations also show reasonable quantitative agreement with those from the DNS study of Février, Simonin & Legendre (Proceedings of the Fourth International Conference on Multiphase Flow, New Orleans, 2001).