On constructing realizable, conservative mixed scalar equations using the eddy-damped quasi-normal Markovian theory

Abstract

The eddy-damped quasi-normal Markovian (EDQNM) turbulence theory has been applied to the covariance spectrum of two passive isotropic scalars with different diffusivities in stationary isotropic turbulence. A rigorous application of EDQNM, which introduces no new modelling assumptions or constants, is shown to yield a covariance spectrum that violates the Cauchy–Schwartz inequality over some of the wavenumbers. One approach to this problem is to derive a model based on a stochastic differential equation, as its presence guarantees realizability. For an isotropic scalar, it is possible to construct a Langevin equation for the Fourier transform of the scalar concentrations that is consistent with each EDQNM scalar autocorrelation spectrum. The Langevin equations can then be used to construct a model for the covariance spectrum that is realizable. However, the resulting covariance transfer term does not properly conserve the scalar covariance, and so the model is still not satisfactory. The problem can be traced to the Markovianization step, which leads to the presence of the scalar diffusivities in the transfer functions in an unphysical fashion. A simple fix is described which reconciles the two approaches and gives conservative, realizable results for all time.Next, we apply the EDQNM theory to a more general system involving the mixing of anisotropic scalars. Anisotropy in this case results from a uniform mean gradient of the two scalar concentrations in one direction. As with the isotropic scalars, direct application of the EDQNM closure results in a covariance spectrum that violates the Cauchy–Schwartz inequality; however, in this case it is not as simple to construct a Langevin model that reproduces all of the spectral interactions that result from the EDQNM procedure. Nevertheless, we show that the same modification of the inverse time scale as is done for the isotropic scalar results in an anisotropic scalar covariance spectrum that is realizable for all times.