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High-speed disperse multiphase flows are present in numerous environmental and engineering applications with complex interactions between turbulence, shock waves, and particles. Compared with its incompressible counterpart, compressible two-phase flows introduce new scales of motion that challenge simulations and experiments. This review focuses on gas–particle interactions spanning subsonic to supersonic flow conditions. An overview of existing Mach-number-dependent drag laws is presented, with origins from eighteenth-century cannon firings and new insights from particle-resolved numerical simulations. The equations of motion and phenomenology for a single particle are first reviewed. Multiparticle systems spanning dusty gases to dense suspensions are then discussed from numerical and experimental perspectives. [ABSTRACT FROM AUTHOR]

We perform particle-resolved simulations of subsonic and transonic flows past random arrays of spherical particles. The particle Reynolds number is held at R e ≈ 300 to ensure the flow remains in the continuum regime. At low volume fractions, the drag force increases sharply near a critical Mach number due to the formation of shock waves and reaches a maximum value when the bulk flow is supersonic. Neighbor-induced aerodynamic interactions reduce the critical Mach number at higher volume fractions. An effective Mach number is introduced to capture the increase in compressibility effects on drag. A new drag correlation is proposed valid for subsonic and weakly supersonic flow from dilute to moderately dense suspensions. • Performed PR-DNS of subsonic and transonic flows past random arrays of particles. • Drag increases sharply near a critical Mach number due to the formation of shocks. • Fluid-mediated interactions reduce the critical Mach at higher volume fractions. • An effective Mach number is introduced to capture increase in compressibility on drag. • A new drag correlation is proposed valid at finite Mach number and volume fraction. [ABSTRACT FROM AUTHOR]

Models for quasi-steady drag, quasi-steady drag variation, and transverse forces for compressible flows through random, fixed, particle suspensions are presented. Correlations are formulated using drag force measurements obtained from particle-resolved simulation data spanning subsonic to supersonic flow conditions. The newly proposed drag models are extensions of existing models for incompressible dense gas-solid flows to finite Mach numbers, while the transverse particle force model is new and incorporates the covariation between streamwise and transverse forces. • Compressible particle-resolved simulations of flow through suspensions are conducted. • Simulation data from multiple sources are used to develop force correlations. • Models for quasi-steady drag, drag variation, and transverse forces are developed. • The drag models extend existing dense gas-solid flow models to finite Mach numbers. • The transverse force model incorporates covariation effects between lift and drag. [ABSTRACT FROM AUTHOR]

Coarse grid simulations of circulating fluidized bed (CFB) reactors require an accurate estimation of the particle drag force. Conventional homogeneous drag laws are known to be insufficient for such configurations owing to the tendency of particles to cluster. The Energy Minimization Multi-Scale (EMMS) model has received much attention in recent years for capturing heterogeneity in simulations of CFB reactors. In this paper, we critically review the EMMS model by revisiting the original formulation, underlying assumptions, key modifications, and improvements. Some capabilities as well as discrepancies of the model are explained along with areas for future improvements. Leveraging highly resolved Euler–Lagrange (EL) simulations, we present a critical assessment of EMMS and its fundamental assumptions. A fully-developed homogeneous flow of solid particles settling under gravity is considered as a test case. Strong inter-phase coupling results in the spontaneous generation of dense clusters that generate and sustain underlying turbulence. This setup is analogous to an individual computational cell within a coarse grid two-fluid model or EL simulation for which the EMMS model is typically applied. A structure tracking algorithm is applied to the EL data to identify clusters and measure their characteristics. Significant variation is observed in the two-phase flow parameters, in contrast with the key assumptions in the EMMS model. Variance-based sensitivity of the model parameters is performed to identify individual terms that contribute most to the EMMS prediction. This work provides a framework for incorporating the observed variation and improving the standard EMMS model. [Display omitted] • A critical review of EMMS and its underlying assumptions is presented. • The EMMS model is assessed using highly resolved Euler–Lagrange simulations. • Cluster-induced turbulence generated by particle settling is used as a test case. • EMMS is unable to capture statistical variations in many of the statistics. • Variance-based sensitivity of the EMMS model is performed. [ABSTRACT FROM AUTHOR]