A spectral model for homogeneous shear-driven anisotropic turbulence in terms of spherically averaged descriptors

Abstract

International audienceA nonlinear spectral model in terms of spherically averaged descriptors is derived for the prediction of homogeneous turbulence dynamics in the presence of arbitrary mean-velocity gradients. The governing equations for the tensor $\hat{R}_{ij}(\mathbf k,t)$, the Fourier transform of the two-point second-order correlation tensor, are first closed by an anisotropic eddy-damped quasinormal Markovian procedure. This closure is restricted to turbulent flows where linear effects induced by mean-flow gradients have no essential qualitative effects on the dynamics of triple correlations compared with the induced production effects in the equations for second-order correlations. Truncation at the first relevant order of spectral angular dependence allows us to derive from these equations in vector $\mathbf k$ our final model equations in terms of the wavenumber modulus $k$ only. Analytical spherical integration results in a significant decrease in computational cost. Besides, the model remains consistent with the decomposition in terms of directional anisotropy and polarization anisotropy, with a spherically averaged anisotropic spectral tensor for each contribution. Restriction of anisotropy to spherically averaged descriptors, however, entails a loss of information, and realizability conditions are considered to quantify the upper boundary of anisotropy that can be investigated with the proposed model. Several flow configurations are considered to assess the validity of the present model. Satisfactory agreement with experiments on grid-generated turbulence subjected to successive plane strains is observed, which confirms the capability of the model to account for production of anisotropy by mean-flow gradients. The nonlinear transfer terms of the model are further tested by considering the return to isotropy (RTI) of different turbulent flows. Different RTI rates for directional anisotropy and polarization anisotropy allow us to correctly predict the apparent delayed RTI shown after axisymmetric expansion. The last test case deals with homogeneous turbulence subjected to a constant pure plane shear. The interplay between linear and nonlinear effects is reproduced, yielding the eventual exponential growth of the turbulent kinetic energy

Spectral modelling for passive scalar dynamics in homogeneous anisotropic turbulence

“…In a second step, a tractable model is obtained in terms of spherically-averaged descriptors, so that the state vector depends only on the wavenumber modulus k: the integration on spherical shells is performed analytically. This procedure in two steps generalizes the pioneering work of Cambon et al (1981) for the velocity field, and consistently extends the recent model by Mons et al (2016) to the scalar and velocity-scalar correlations modelling.…”

“…As a conclusion, the dynamics of a passive scalar field in homogeneous anisotropic turbulence is driven by six spherically-averaged compact equations within this anisotropic EDQNM modelling: three for the velocity field first given in Mons et al (2016) 20) two for the passive scalar ((3.7) and (3.8)), and one for the scalar flux (3.9). The last three ones are original results of the present work.…”

“…In this part, the main theoretical aspects of Mons et al (2016) are recalled for the sake of clarity and to introduce important notations. The spectral velocity second-order correlation -or spectral Reynolds stress tensor -is defined aŝ…”

“…where (e (1) , e (2) , e (3) ) is the Craya-Herring orthonormal right-handed frame of reference associated with a privileged direction n. The first step of the derivation proposed by Mons et al (2016) consists in using EDQNM to close the generalized transfer terms in the evolution equations of E and Z, equivalent to (2.2).…”

“…Hereafter, an EDQNM closure adapted to homogeneous anisotropic flows is derived similarly to Mons et al (2016). The complete anisotropic form of the second-order spectral tensors is kept in the quasi-normal (QN) formulation, and is therefore expressed in terms of the k-vector descriptors of Table 1.…”