Compressibility

Finite volume method

Turbulence

Eulerian path

Mathematical physics

Shock (circulatory)

Mach number

Lagrangian

Particle-laden flows

Classical mechanics

Statistical physics

Geology

Oceanography

Thermodynamics

Physics

Two-phase flow

Knudsen number

Flow (mathematics)

Compressible flow

Internal medicine

Particle (ecology)

Medicine

Mechanics

Volume (thermodynamics)

International Journal of Multiphase Flow

In this work, we present a rigorous derivation of the volume-filtered viscous compressible Navier–Stokes equations for disperse two-phase flows. Compared to incompressible flows, many new unclosed terms appear. These terms are quantified via a posteriori filtering of two-dimensional direct simulations of shock-particle interactions. We demonstrate that the pseudo-turbulent kinetic energy (PTKE) systematically acts to reduce the local gas-phase pressure and consequently increase the local Mach number. Its magnitude varies with volume fraction and filter size, which can be characterized using a Knudsen number based on the filter size and inter-particle spacing. A transport equation for PTKE is derived and closure models are proposed to accurately capture its evolution. The resulting set of volume-filtered equations are implemented within a high-order Eulerian–Lagrangian framework. An interphase coupling strategy consistent with the volume filtered formulation is employed to ensure grid convergence. Finally PTKE obtained from the volume-filtered Eulerian–Lagrangian simulations are compared to a series of two- and three-dimensional direct simulations of shocks passing through stationary particles.

A volume-filtered description of compressible particle-laden flows