Effect of variable mass density on the kinematics of scalar gradient

Abstract

Analysis of the equations for the orientation and norm of the gradient of a passive scalar shows that the kinematics of the scalar gradient may be deeply altered by non-solenoidal effects felt through the velocity gradient. While these effects are explicitly expressed in the equation for the gradient norm, the orientation equations are unaltered as compared to the incompressible case. In two-dimensional flows the behavior of the scalar gradient is governed by both the strain persistence parameter, r, and the ratio, δ/σ, of velocity divergence to the norm of the deviatoric part of the strain tensor. In particular, in the dilatational case, while large dilatation (δ/σ > 1) unconditionally lessens the scalar gradient, moderate dilatation (0 <δ/σ < 1) needs effective rotation (r ≠ 0) to drive the scalar gradient to decrease. In three-dimensional flows the analytic study needs the assumption that vorticity is aligned with a strain principal axis and leads to a more complex picture based on the strain persistence parameter, the velocity divergence and the strain eigenvalue corresponding to the strain direction parallel to vorticity. In this case, too, for moderate non-solenoidal dilatational effects, rotation appears to oppose the rise of the scalar gradient norm. Possible inferences of the analysis relevant to reacting flows with heat release are briefly discussed