A stochastic model for the relative motion of high Stokes number particles in isotropic turbulence

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).