Instability and Perturbation Evolution in High Speed Boundary Layers: Flow-Thermodynamic Interactions

Abstract

In high speed flows the nature of pressure changes. Pressure evolves from a Lagrange multiplier in low speed flows to a thermodynamic variable at high speeds. The compressible velocity field develops a finite dilatational component allowing for net pressure-work which is absent in low speed flows. Consequently, pressure action triggers significant flow-thermodynamic inter-actions and transfers energy between the velocity and thermal fields. The emergence of flow-thermodynamic interactions has a marked impact on the stability and overall nature of perturbation evolution in high speed flows. 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 primary motivation of this work is to understand the flow-thermodynamic interactions during instability and perturbation evolution in high speed boundary layer flows. These interactions manifest in the flow in two ways: (i) thermal-transport effects and (ii) pressure-velocity interactions. In this work we use linear stability analysis (LSA) and direct numerical simulations (DNS) to study both aspects in the linear regime of the high speed boundary layer transition process. In the first study, thermal transport effects on the boundary layer stability are established by examining the influence of Prandtl number on instability and perturbation evolution. It is shown that increasing Prandtl number has a destabilizing effect. The behavior of production, pressure-strain correlation and pressure-dilatation as functions of Mach and Prandtl numbers is characterized. First and second instability modes exhibit similar stability trends but the underlying flow physics is shown to be diametrically opposite.

In study 2, Helmholtz decomposition is used in conjunction with linear stability analysis to examine the pressure-velocity interactions for the boundary layer instability. A corresponding de-composition of the pressure field is also proposed. The contributions of perturbation solenoidal kinetic, dilatational kinetic and internal energy to the various instability modes are examined as a function of Mach number (M). As expected, dilatational and pressure field effects play insignificant part in the first-mode behavior at all Mach numbers. The second (Mack) mode however is dominantly dilatational in nature and perturbation internal energy is significant compared to perturbation kinetic energy. The observed behavior is explicated by examining the key processes of production and pressure-dilatation. Production of the second mode dilatational kinetic energy is mostly due to the solenoidal-dilatational covariance stress tensor interacting with the mean (back-ground) velocity gradient. This cross production component also inhibits first mode. The dilatational pressure facilitates energy transfer from the kinetic to the internal field in the near wall region, whereas the energy transfer away from the wall is mostly due to the solenoidal pressure work.

Finally, the flow-thermodynamic interactions in the transient linear regime of high speed boundary layers is studied using direct numerical simulations (DNS) at different Mach numbers M. We observe that for M ≥ 3 random pressure perturbations evolve to their asymptotic state in three distinct stages. In the initial two stages, pressure-dilatation is the dominant flow-process and exhibits disparate behaviour during the two stages. In stage 1, strong action by pressure-dilatation facilitates a rapid one way transfer of energy from internal to kinetic energy. The rapid transfer of energy in the first stage brings about equipartition between the dilatational kinetic and internal energy. On the other hand in stage 2, pressure action leads to a reversible coupling wherein energy is transferred back and forth between pressure and velocity perturbations. Eventually, in stage 3 modal behaviour commences and the evolution is characterized by the most unstable eigenmode. At high Mach numbers (M > 4), the second mode is dominant and as a result dilatational fluctuations and the ensuing flow-thermodynamic interactions are significant in the final stage. In contrast at lower M, the solenoidally dominated first mode leads to negligible flow-thermodynamic interactions.