On the Formulation of the Resolvent Operator for Compressible Flows

This paper investigates the non-uniqueness of resolvent analysis in the context of compressible fluid flows. Specifically, we compare two mathematically equivalent formulations of the compressible Navier-Stokes equations (NSEs) in two sets of flow variables related via a nonlinear transformation, which we refer to as the ‘Cubic+’ and ‘Quadratic+’ formulations. The Cubic+ formulation is based on the conventional representation of compressible NSEs using the primitive variables of density, velocity components and temperature, whereas the Quadratic+ formulation utilizes a representation where density and temperature are replaced by specific volume and pressure, respectively. These formulations are implemented on a compressible plane Couette flow for a broad range of Mach numbers. Although the Quadratic+ formulation generally predicts higher amplifications than the Cubic+ formulation when the Chu energy is utilized for the compressible inner product, this trend reverses if kinetic energy is instead used to define the inner product. Furthermore, for some combinations of the streamwise and spanwise wavenumber pairs, the two formulations predict substantially different temporal behaviors. These range from differences in estimating the temporal frequencies for the largest gains to instances wherein one formulation predicts a single global peak when the other predicts multiple local peaks. In addition, the Quadratic+ results feature localized regions of wavenumber space associated with high resolvent gains that are absent in the Cubic+ results, but are qualitatively comparable to one another otherwise. The inconsistencies observed in resolvent analysis based on the two formulations considered here suggest that any insights drawn from resolvent analysis should be accompanied by computational and/or experimental studies to corroborate findings and interpretations of the underlying physics.