Compound Matrix Method for Linear Stability Analysis of Flow Over a Flat Plate

Flow transition from laminar to turbulent regime is marked by instability characteristics of the former. The question of transition evolves into how stable the incoming flow is. Linear stability analysis takes in nonlinear governing equations and converts them into the corresponding perturbation equation, that can be studied for solutions seeking the most (un)stable modes and study their propagation. This study uses the Compound Matrix Method (CMM) for solving stability of compressible boundary layer flows. The problem involves flow of air over an adiabatic flat plate. This boundary conditions can be found in surfaces exposed to high-speed flows, such as outer walls of aircraft and spacecrafts. The solution to the problem comes in the form of determining stable and unstable pairs of \(\left( {{\text{Re}} ,\omega ,\alpha_{i} } \right)\) , with \(\alpha_{i} = 0\) being the neutral curve. The initial and boundary conditions appropriate to the problem are converted appropriately. The use of free stream boundary conditions helps to simplify the system of 6 equations by retaining only 3 decaying modes in the free stream. The region on \({\text{Re}} - \omega\) with \(\alpha_{i} > 0\) depicts the stable region and \(\alpha_{i} < 0\) depicts unstable. The spatial stability analysis for adiabatic flow over a flat plate at two Mach numbers using CMM to give the critical Reynolds number as \(523.7\) and \(567.7\) for \(M = 0.1,0.6\) , and results are compared against the incompressible Orr-Sommerfeld equation (OSE).