Lagrangian gradient transport for variable-density turbulence

We apply Lagrangian analysis to materially conserved scalars for turbulent transport in variable-density flows. The consequences of the material conservation of density leads to meaningfully different expressions for the turbulent transport. Formal Lagrangian analysis produces gradient transport expressions substantially different from those obtained by the physically intuitive “argument by analogy” method used in computational models. These intuitive arguments, in Favre and Reynolds averaged settings, are contrasted to the formal Lagrangian results. It is shown that the basis functions of gradient transport theory are different when there is an additional materially conserved variable. In variable density flows, the turbulent fluxes now depend on the mean density gradient indicating the possibility of counter gradient transport from first principles as seen in some laboratory experiments. Using expressions from the formal analysis, we derive consistent gradient transport expressions for the turbulent transport terms that appear in the first- and second-order Favre moment equations. It is shown that arguments by analogy from constant density transport for the Favre fluxes are not consistent with the Lagrangian results. The analysis is limited to transport by high Reynolds number turbulent fluctuations in the presence of mean gradients and negligible dilatation of the fluctuating velocity. The results are applicable to turbulent combustion and to stellar convection problems in which the density fluctuations are on the order of the mean density.