Sensitivity Analysis of a Plane Mixing Layer using the Sensitivity Equation Method

Sensitivity field evolution of the incompressible, two-dimensional mixing layer to perturbations in both Reynolds number, $Re_{\delta_0}$, and Prandtl number, $Pr$, has been examined using the sensitivity equation method. In this method, the sensitivity coefficients (i.e., the partial derivative of vorticity and temperature with respect to $Re_{\delta_0}$ and $Pr$) are obtained from direct numerical simulation of the sensitivity equations coupled with the governing equations of the fluid motion. This is achieved using an unsteady finite volume based fractional step algorithm. Coherent structures in the sensitivity field depict mechanisms responsible for enhanced vortex growth and scalar mixing with increasing $Re_{\delta_0}$ and $Pr$, respectively. Two distinct configurations were found in the sensitivity field of vorticity. The first configuration represents an increasing growth in the mixing layer as $Re_{\delta_0}$ increases, while the second configuration depicts the saturation state for the vorticity field. The sensitivity of the temperature field to changes in $Pr$ exhibits a third configuration describing enhanced scalar with increasing $Pr$. These interpretations are confirmed with calculations of integral quantities, namely the rate of growth of the mixing layer with $Re_{\delta_0}$ and the evolution of the probability density function of the scalar field with $Pr$.