Prandtl number effects on the hydrodynamic stability of compressible boundary layers: flow–thermodynamics interactions

Abstract

Hydrodynamic stability of compressible boundary layers is strongly influenced by the Mach number ( $M$ ), Prandtl number ( $Pr$ ) and thermal wall boundary condition. These effects manifest on the flow stability via the flow–thermodynamics interactions. Comprehensive understanding of stability flow physics is of fundamental interest and important for developing predictive tools and closure models for integrated transition-to-turbulence computations. The flow–thermodynamics interactions are examined using linear analysis and direct numerical simulations in the following parameter regime: $0.5 \leq M \leq 8$ ; and $0.5 \leq Pr \leq 1.3$ . For the adiabatic wall boundary condition, increasing Prandtl number has a destabilizing effect. In this work, we characterize the behaviour of production, pressure–strain correlation and pressure dilatation as functions of the Mach and Prandtl numbers. First and second instability modes exhibit similar stability trends but the underlying flow physics is shown to be diametrically opposite. The Prandtl number influence on instability is explicated in terms of the base flow profile with respect to the different perturbation mode shapes.

Effect of flow–thermodynamics interactions on the stability of compressible boundary layers: insights from Helmholtz decomposition

“…The modal form of perturbations is substituted into the linearized perturbation equation (2.6) to formulate the eigenvalue problem Here, are the eigenmode shapes corresponding to the eigenvalue . The elements of the fifth-order coefficient matrices and are listed in Sharma & Girimaji (2022). No-slip and zero thermal perturbation boundary conditions are used for velocity and temperature, while a Neumann boundary condition for pressure is obtained by solving the wall-normal momentum equation (2.6 b ).…”

“…It is worth noting that the second mode is stable for . The solenoidal and dilatational energy levels of an eigenmode are quantified by considering global averages as defined in Sharma & Girimaji (2022). The global-averaged kinetic () and internal energy () are obtained by integrating the amplitude of velocity and pressure perturbations in the wall-normal direction: where denotes the complex conjugate of .…”

“…The elements of the fifth-order coefficient matrices A and B are listed in Sharma & Girimaji (2022). No-slip and zero thermal perturbation boundary conditions are used for velocity and temperature, while a Neumann boundary condition for pressure is obtained by solving the wall-normal momentum equation (2.6b).…”

“…In particular, the kinetic energy extracted from the mean flow by the production mechanism can be diverted away from the velocity field to the thermodynamic field via the pressure-dilatation mechanism. Recently, Sharma & Girimaji (2022) investigated the kinetic-internal energy exchange brought about by pressure-dilatation for both first and second modes. Their findings suggest that pressure-dilatation is negligible compared to production for the first mode, whereas it plays an important role in the second-mode dynamics.…”

“…Red dashed lines correspond to the (a) divergence and (b) vorticity of the solenoidal velocity u s . Blue dash-dotted lines represent the (a) divergence and (b) vorticity of the dilatational velocity u d .The solenoidal and dilatational energy levels of an eigenmode are quantified by considering global averages as defined inSharma & Girimaji (2022). The global-averaged kinetic (k g ) and internal energy (e g ) are obtained by integrating the amplitude of velocity and pressure perturbations in the wall-normal direction:k g = 1 l y l y…”