A scalar subgrid model with flow structure for large-eddy simulations of scalar variances

Abstract

A new model to simulate passive scalar fields in large-eddy simulations of turbulence is presented. The scalar field is described by clouds of tracer particles and the subgrid contribution of the tracer displacement is modelled by a kinematic model which obeys Kolmogorov's inertial-range scaling, is incompressible and incorporates turbulent-like flow structure of the turbulent small scales. This makes it possible to study the scalar variance field with inertial-range effects explicitly resolved by the kinematic subgrid field while the LES determines the value of the Lagrangian integral time scale TL. In this way, the modelling approach does not rely on unknown Lagrangian input parameters which determine the absolute value of the scalar variance.The mean separation of particle pairs displays a well-defined Richardson scaling in the inertial range, and we find that the Richardson constant GΔ ≈ 0.07 which is small compared to the value obtained from stochastic models with the same TL. The probability density function of the separation of particle pairs is found to be highly non-Gaussian in the inertial range of times and for long times becomes Gaussian. We compute the scalar variance field for an instantaneous line source and find good agreement with experimental data.

Turbulent pair separation due to multiscale stagnation point structure and its time asymmetry in two-dimensional turbulence

“…For simplicity of notation in the backward case, the minus sign shall be dropped henceforth, and it is understood that time in the backward case always refers to times smaller than the initial flow time t = 0. For flows whose dynamics are time reversible, like Gaussian flows or kinematic simulations ͑KSs͒, one could expect that the mean square separation of the forward and backward cases coincides and indeed, this has been shown for Gaussian flows and KS, [4][5][6] ͗⌬ 2 ͑t͉͒⌬ 0 ͘ fwd = ͗⌬ 2 ͑t͉͒⌬ 0 ͘ bwd . ͑1͒…”

“…Therefore, the underlying quantity of mixing processes is the so-called backward dispersion, that is, the distribution of particle separations ͗⌬ 2 ͑t͒͘ at a time t for a prescribed separation ⌬ 0 at a later time t 0 Ͼ t. [3][4][5][6][7] Let us denote the forward mean square separation at time ͑the time refers to the lapsed time of the underlying flow of the separation process; t = 0 denotes the time when the initial/finial separation ⌬ 0 is fixed͒ t with an initial separation ⌬ 0 at t =0 by ͗⌬ 2 ͑t͉͒⌬ 0 ͘ fwd . Equivalently, the backward mean square separation shall be denoted by ͗⌬ 2 ͑−t͉͒⌬ 0 ͘ bwd .…”

Presence of a Richardson’s regime in kinematic simulations

“…The coefficient vectors a n and b n are chosen randomly and independently in the plane normal to k n , a n · k n = b n · k n = 0 (7) to ensure that the random field is incompressible. In order to impose an energy spectrum, E(k) upon the field, the magnitudes of the coefficients are chosen as follows…”

“…where X 1 (t) is the position of the first particle and X 2 (t) the position of the second particle at time t. The first quantity of interest is the mean-square separation between the two particles ∆ 2 (t) as a function of time which has received much research attention since the pioneering work of [1] (see for example [2][3][4][5][6][7][8][9][10][11][12][13][14]). …”

The Relevance of a Back-Scatter Model for Depth-Averaged Flow Simulation

“…a passive tracer, the dynamics on the subgrid scale will contribute to the mixing in a way that can not be modelled as diffusion since the real process is advection by the small scales. This has consequences for the statistics of two-particle dispersion, chemical reactions and other processes where the small scale kinetics needs to be separated from diffusion [10].…”