A nonlinear model for the subgrid timescale experienced by heavy particles in large eddy simulation of isotropic turbulence with a stochastic differential equation

Abstract

The effects of subgrid scale (SGS) motions on the dispersion of heavy particles raise a challenge to the large-eddy method of simulation (LES). As a necessary first step, we propose the use of a stochastic differential equation (SDE) to represent the SGS contributions to the relative dispersions of heavy particles in LES of isotropic turbulence. The main difficulty is in closing the SGS-SDE model whilst accounting for the effects of particle inertia, filter width and gravity. The physics of the interaction between heavy particles and SGS turbulence is explored using the filtered direct numerical simulation method. It is found in the present work that (i) the ratio of the SGS Lagrangian and Eulerian timescales is different from that of the full-scale Lagrangian and Eulerian timescales. The ratios are also dependent on filter widths. (ii) In the absence of gravity, the SGS timescale seen by heavy particles non-monotonically changes with particle Stokes number and has a maximum at particle Stokes number (St = τ p /δT E ) near 0.5. (iii) In the presence of gravity, a similarity law exists between the SGS Lagrangian correlation function seen by a heavy particle within a timedelay τ and the SGS spatial correlation function with the displacement w τ , where w is the average settling velocity of a heavy particle. The joint effects of particle inertia and gravity are accounted for using the elliptic model for pair fluctuations to the dispersion of particles [9,10]. Chibbaro and Minier [10] addressed the importance of introducing specific features related to the near-wall coherent structure in pipe flows based on the Langevin probability density function (PDF) method. In the SDE, the Lagrangian integral timescale of fluid velocity seen by heavy particles (the fluid-particle interaction timescale), T Lp , is one of the key parameters and it determines the dispersion rates of heavy particles [11][12][13][14]. Direct numerical simulation (DNS) shows that T Lp does not vary monotonically with particle Stokes number St K from T L to T E , but has an 'N' shape with a maximum near St K = 1 due to the relative motion of particles near rotational vortical structures [15][16][17], where St K ≡ τ p /τ K , τ p is the particle Stokes response time and τ p = ρ p d 2 p /18µ, d p is the particle diameter, µ the dynamical viscosity, ρ p particle density and τ K the turbulent Kolmogorov timescale.The RANS method finds wide application in engineering calculations [9] and the DNS method has manifested its power in fundamental research [17][18][19][20]. However, DNS is still limited to flows at relatively low or moderate Reynolds numbers. RANS does not predict well the unsteady structures in turbulent flows which are of importance for turbulent dispersion of heavy particles. Motivated by the limitations of RANS and DNS, large-eddy simulation (LES) is becoming a potential method for future engineering calculations. In LES, large-scale velocitiesū(x, t) are resolved explicitly using the filtered Navier-Stokes equations. The smallscale velocities...

Large-eddy simulation of turbulent dispersed flows: a review of modelling approaches

“…Particles follow the motion of the large eddies but slip predominantly on the small ones: This dynamics is characterized by particle-to-fluid relative velocities much smaller than u l and hence by small particle Reynolds numbers. In the following, we will focus on the models that have been developed in this regime (see [31,55,62,117] among others). In the opposite limit (St SG S >> 1), particles become SGS-inertial with respect to subgrid eddies [125], and (in principle) no particle SGS model is necessary: The dominant eddies controlling the relative velocity are resolved in LES, and the corresponding velocity u is enough to solve Eq.…”

Focus on dynamics of particles in turbulence

“…New methods in numerical simulations. On the numerical aspects, new strategies of largeeddy-simulations (LES) are proposed, with novel sub-grid scale models based on a stochastic differential equation to account for particles inertia [1] and coupling hybrid Eulerian-Lagrangian approaches [2], improving the capacities of LES to handle pair separation and collisions for point-like particles. A long standing limitation of simulations of particles in turbulence was their insufficient capability to address the effects of finite particle sizes, as usual models for particle motion are based on the Maxey-Riley-Gatignol equation that was derived for particles with vanishingly small sizes [3,4] 3 .…”

“…The case of small, heavy particles, whose dynamics can be reasonably approximated using the linear Stokes equation (with the Stokes number as the only parameter characterizing particle inertia), has been extensively investigated numerically in recent decades using high resolution DNS in homogeneous isotropic conditions. The knowledge accumulated from this canonical situation offers a solid ground for the development of new numerical strategies in more realistic flow configurations using LES, as mentioned above [1,2], and for addressing more complex situations where collective effects can arise (see section 5).…”

Scaling law of resolved-scale isotropic turbulence and its application in large-eddy simulation

“…Besides, in the simulation of heavy particles, it is also found that the choice of subgrid model is related to the computational accuracy of particles [30][31][32][33][34]. The aim of the research on resolved-scale scaling law is to provide a better tool for the subgrid modeling in LES.…”