Treatment of Particle Collisions in Direct Numerical Simulations of High Speed Compressible Flows

Direct numerical simulations of heavy particles suspended in a turbulent fluid are performed to study the rate of inter-particle collisions as a function of the turbulence parameters and particle properties. The particle volume fractions are kept small (∼10−4) so that the system is well within the dilute limit. The fluid velocities are updated using a pseudo-spectral algorithm while the particle forces are approximated by Stokes drag. One unique aspect of the present simulations is that the particles have finite volumes (as opposed to point masses) and therefore particle collisions must be accounted for. The collision frequency is monitored over several eddy turnover times. It is found that particles with small Stokes numbers behave similarly to the prediction of Saffman & Turner (1956). On the other hand, particles with very large Stokes numbers have collision frequencies similar to kinetic theory (Abrahamson 1975). For intermediate Stokes numbers, the behaviour is complicated by two effects: (i) particles tend to collect in regions of low vorticity (high strain) due to a centrifugal effect (preferential concentration); (ii) particle pairs are less strongly correlated with each other, resulting in an increase in their relative velocity. Both effects tend to increase collision rates, however the scalings of the two effects are different, leading to the observed complex behaviour. An explanation for the entire range of Stokes numbers can be found by considering the relationship between the collision frequency and two statistical properties of the particle phase: the radial distribution function and the relative velocity probability density function. Statistical analysis of the data, in the context of this relationship, confirms the relationship and provides a quantitative description of how preferential concentration and particle decorrelation ultimately affect the collision frequency.

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Direct Numerical Simulations (DNS) of Natural Transition in High-Speed Boundary Layers Using a Broadband Random Forcing Approach

IUTAM Laminar-Turbulent Transition - IUTAM Bookseries ◽

10.1007/978-3-030-67902-6_49 ◽

2021 ◽

pp. 565-574

Author(s):

Christoph Hader ◽

Hermann F. Fasel

Keyword(s):

Numerical Simulations ◽

Boundary Layers ◽

High Speed ◽

Direct Numerical Simulations ◽

Random Forcing

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Evolution of enstrophy in shock/homogeneous turbulence interaction

Journal of Fluid Mechanics ◽

10.1017/jfm.2012.265 ◽

2012 ◽

Vol 707 ◽

pp. 74-110 ◽

Cited By ~ 24

Author(s):

Krishnendu Sinha

Keyword(s):

Shock Wave ◽

Numerical Simulations ◽

High Speed ◽

Isotropic Turbulence ◽

Stokes Equations ◽

Interaction Analysis ◽

Direct Numerical Simulations ◽

Homogeneous Isotropic Turbulence ◽

Mean Flow ◽

Linear Interaction

AbstractInteraction of turbulent fluctuations with a shock wave plays an important role in many high-speed flow applications. This paper studies the amplification of enstrophy, defined as mean-square fluctuating vorticity, in homogeneous isotropic turbulence passing through a normal shock. Linearized Navier–Stokes equations written in a frame of reference attached to the unsteady shock wave are used to derive transport equations for the vorticity components. These are combined to obtain an equation that describes the evolution of enstrophy across a time-averaged shock wave. A budget of the enstrophy equation computed using results from linear interaction analysis and data from direct numerical simulations identifies the dominant physical mechanisms in the flow. Production due to mean flow compression and baroclinic torques are found to be the major contributors to the enstrophy amplification. Closure approximations are proposed for the unclosed correlations in the production and baroclinic source terms. The resulting model equation is integrated to obtain the enstrophy jump across a shock for a range of upstream Mach numbers. The model predictions are compared with linear theory results for varying levels of vortical and entropic fluctuations in the upstream flow. The enstrophy model is then cast in the form of$k$–$\epsilon $equations and used to compute the interaction of homogeneous isotropic turbulence with normal shocks. The results are compared with available data from direct numerical simulations. The equations are further used to propose a model for the amplification of turbulent viscosity across a shock, which is then applied to a canonical shock–boundary layer interaction. It is shown that the current model is a significant improvement over existing models, both for homogeneous isotropic turbulence and in the case of complex high-speed flows with shock waves.

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