Linear Stability Analysis of Shear Flows with Thermal Effects

The study of shear flows with thermal effects finds several applications in weather prediction, study of ocean currents, heat exchangers, electronics cooling, electroplating, gas turbines, chemical vapor deposition (CVD), among others. In some of these applications, for example in CVD, one needs to ensure a laminar flow to keep the quality of the process. To investigate the thermal influence in shear flows, a linear stability approach was adopted in this work. This analysis allows the investigation of such flows including their response to infinitesimal disturbances. The incompressible form of the Navier-Stokes equations is employed together with the Oberbeck-Boussinesq approximation to model the present flows and, therefore, include buoyancy effects. Two different flow setups are studied: the Poiseuille-Rayleigh-Bénard (PRB) problem, which is a channel flow with an unstably stratified temperature gradient, and the Blasius boundary layer heated from below, here called mixed convection boundary layer. To assess the stability properties of these flow configurations with respect to critical parameters, modal and non-modal analyses are employed. The former consists in the eigenvalue solution to obtain the neutral curves while the latter computes the transient growth and the resolvent/input-output analysis. The PRB flow analysis started with the validation of the formulation and the numerical methods by comparing the obtained results with the literature. Then, the resolvent and input-output analyses were performed, providing similar features compared to those obtained for the transient growth studies available in the literature. The present mixed convection boundary layer results are novel. Hence, an initial validation of the formulation was performed for the isothermal Blasius boundary layer flow. Then, a mesh independence study was performed to ensure convergence of the eigenvalue solution for the mixed convection case. Then, results in terms of the neutral curves, transient growth and resolvent/input-output maps are presented. For the mixed convection boundary layer, the temperature fluctuations introduce a new energy amplification mechanism which is dominant for low wavenumbers. In the input-output analysis of the PRB flow, it was found that the maximum amplification occurs for the streamwise velocity. However, for the mixed convection boundary layer, the maximum amplification occurs for the wall-normal velocity.