Bounded stochastic shell mixing model for turbulent mixing of multiple scalars with arbitrary diffusivities

Abstract

In the paper by Xia et al. [Flow, Turbul. Combust. 85, 689 (2010)], we derived a stochastic shell mixing model (SSMM) for turbulent mixing of multiple scalars with arbitrary diffusivities. The framework uses a Monte Carlo scheme to advance notional particles that move with the fluid and has scalar concentrations that are subdivided into “shells” that loosely represent a spectral decomposition of the scalar fluctuations. The variance within each shell is equivalent to the scalar spectrum evaluated at the corresponding wavenumber. Small-scale phenomena such as differential diffusion and the dependence of mixing on the scalar length scale are captured by the inherently spectral nature of the model. Furthermore, the sum of the scalar over shells for each notional particle simultaneously provides a fine-grained representation of the joint scalar probability density function (PDF). This aspect of the model was introduced, but not fully explored in the earlier paper, which focused on scalar fluctuations arising from a uniform mean gradient. At the level of this model, the scalar PDF reduces to a joint Gaussian. In this study, we explore more complex mixing scenarios, including the initial double delta function, which arises when two streams are brought into sudden contact. This introduces two important complexities that are the focus of this paper. The first is the shape changes to the PDF that result as the scalar PDF relaxes into a Gaussian form. The second is due to the bounds that the scalar must satisfy. Monte Carlo schemes inherently violate bounds; we have modified the model so as to ensure the bounds are satisfied by nearly all of the particles (bounded SSMM or BSSMM). Extensive comparisons between the BSSMM predictions and direct numerical simulations (DNS) are made for two decaying scalars (with different diffusivities) in forced, isotropic turbulence. Overall the agreement is very good. Additionally, we test a conjecture made in deriving the BSSMM model, namely the rate of mixing of a scalar is assumed to be a function of its spectrum, but not its PDF. Using DNS, we contrast the mixing rate of two scalars with identical PDFs, but different spectra against two scalars with identical spectra, but different PDFs. We conclude from this comparison that the mixing rate is much more sensitive to the scalar spectrum than to its PDF, in support of the assumption used to derive the new mixing model.