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

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 = tau(p)/delta T-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 tau and the SGS spatial correlation function with the displacement < w > tau, 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 correlation of SGS velocity seen by heavy particles. The SGS timescale seen by heavy particles is extracted from the elliptic model and used to close the SGS-SDE model. The validations of the model against direct numerical simulation show that the SGS-SDE model can improve the performance of LES on relative dispersions especially when their initial separations are in the inertial subrange. Furthermore, we assess the performance of the SGS-SDE model by comparing the results with the approximate deconvolution method. The results show that the SGS-SDE model is more suitable for particles with small Stokes numbers, St(K) < 2. The model developed here provides a basis for the development of a more advanced SGS model for particles in non-homogeneous and anisotropic turbulent flows in pipes or channels.