T. Vaithianathan

Numerical approach to simulating turbulent flow of a viscoelastic polymer solution

In this paper, we present two new numerical algorithms for updating the equations of motion for a viscoelastic fluid that can be described by the finite extensible nonlinear elastic polymer model with the closure proposed by Peterlin (so called FENE-P model) in a transient calculation. In particular, our algorithms address two difficulties found in earlier formulations. First, the polymer extension, represented by the trace of the conformation tensor, can numerically exceed the finite extensible length causing the restoring spring force to change sign and the calculation to rapidly diverge. In our formulations, we have redefined the conformation tensor so that this possibility no longer exists. Secondly, the conformation tensor must remain symmetric and positive definite at all times for the calculation to remain stable. The accumulation of numerical errors can cause loss of this property, leading to the growth of Hadamard instabilities [J. Non-Newtonian Fluid Mech. 60 (1995) 53]. We present two matrix decompositions that enable us to construct the conformational tensor in a manner that ensures positive definiteness. Numerical tests of the new algorithms show significant departures from other approaches that rely on filtering to remove the instabilities.

A spectral study of differential diffusion of passive scalars in isotropic turbulence

In a companion paper, we applied the eddy-damped quasi-normal Markovian (EDQNM) turbulence theory to the mixing of two inert passive scalars with different diffusivities in stationary isotropic turbulence. This paper showed that a rigorous application of the EDQNM approximation leads to a scalar covariance spectrum that violates the Cauchy-Schwartz inequality over a range of wavenumbers. The violation results from the improper functionality of the inverse diffusive time scales that arise from the Markovianization of the time evolution of the triple correlations. The modified inverse time scale they proposed eliminates this problem and allows meaningful predictions of the scalar covariance spectrum both with and without a uniform mean gradient. This study uses the modified EDQNM model to investigate the spectral dynamics of differential diffusion. Consistent with recent DNS results by Yeung, we observe that whereas spectral transfer is predominantly from low to high wavenumbers, spectral incoherence, being of molecular origin, originates at high wavenumbers and is transferred in the opposite direction by the advective terms

Comparison between a spectral and probability density function model for turbulent reacting flows

This study compares the performance of a newly developed spectral model based on the eddy damped quasi-normal Markovian (EDQNM) theory with a standard probability density function (PDF) model for the case of two initially unmixed reactants undergoing a finite-rate bimolecular reaction. The two models were chosen because they involve complementary treatments of the nonlinearities and mixing terms. That is, nonlinearities are exactly treated in the PDF and mixing is modeled, whereas the opposite is true for EDQNM. The predictions of the two models are compared to direct numerical simulations. The results show that the PDF model is capable of describing the mixing of the major species reasonably well, but fails to describe the correlations between the reactants and the products even qualitatively. This suggests that the mixing model in the PDF is adequate for describing mixing between major species, but is incapable of describing mixing of the more spatially segregated product species. The EDQNM model does a slightly better job of describing the mixing of reactant species and a much better job of describing mixing of the product species. Presumably the improvement is associated with the more accurate description of the interscale dynamics that are especially important for the segregated products. The implication is that a model that combines the strengths of the EDQNM for describing mixing and the PDF for describing the nonlinearities would yield the best of both worlds.