Lagrangian analysis for turbulent transport in variable-density turbulence

Lagrangian analysis of materially conserved scalars is applied to the problem of turbulent transport in variable-density flows. The consequences of an additional material conserved quantity, the density, is generally not acknowledged and leads to significant and meaningfully different expressions for turbulent transport in the moment equations. The formal Lagrangian analysis produces gradient transport expressions substantially different from those obtained by the physically intuitive “argument by analogy” method used in computational models. Various intuitive arguments, in Favre and Reynolds averaged settings, are contrasted to the formal Lagrangian results. Using expressions from the formal analysis, we derive consistent gradient transport closures for the turbulent transport terms that appear in the first- and second-order Favre moment equations. Results for coupled multispecies turbulent transport are given. The analysis is limited to variable-density turbulence in which the dilatation of the fluctuating velocity is small. 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.

Figure 1

Illustration of a Lagrangian trajectory in stationary homogeneous isotropic turbulence: the Lagrangian displacement vector ${\xi }_i$, Lagrangian velocity $v_i$, and the Lagrangian fluctuation $c^L$ (introduced in the next section) are labeled.

Figure 2

Case 1: Scalar slab diffusing with time (left) and the instantaneous velocity component field of the stationary homogeneous isotropic turbulence (right).

Figure 3

(a) Third moment magnitude, mean, and variance gradient as a function of time and (b) normalized profiles in $x_2$, averaged between $4\le t/t_{\mathrm{LE}}\le 15$.

Figure 4

Case 2. Buoyancy-driven turbulent mixing layers: (left) instantaneous density fields for three Atwood numbers and (right) peak fluxes in the fully developed mixing zones. Arrows indicate direction of acceleration.