Parabolized stability equations (PSE) have opened new avenues to the analysis of the streamwise growth of linear and nonlinear disturbances in slowly varying shear flows such as boundary layers, jets, and far wakes. Growth mechanisms include both algebraic transient growth and exponential growth through primary and higher instabilities. In contrast to the eigensolutions of traditional linear stability equations, PSE solutions incorporate inhomogeneous initial and boundary conditions as do numerical solutions of the Navier-Stokes equations, but they can be obtained at modest computational expense. PSE codes have developed into a convenient tool to analyze basic mechanisms in boundary-layer flows. The most important area of application, however, is the use of the PSE approach for transition analysis in aerodynamic design. Together with the adjoint linear problem, PSE methods promise improved design capabilities for laminar flow control systems.

Parabolized stability equations

A nonlocal, linear instability theory is used to model convective amplification of waves with weakly divergent or curved wave-rays and wave-fronts, propagating in a weakly nonuniform flow. The compressible, nonlocal stability equations, which represent an eighth order system of parabolic differential equations in generalized coordinates, are derived from a consistent scaling method. Amplification rates from spatial DNS and nonlocal instability analysis of unstable first mode disturbances in hypersonic flow past a flat plate are shown to be in excellent agreement. Conical Divergence is shown to be stabilizing on first mode disturbances in nonuniform, nonparallel hypersonic flow past a pointed cone. Convex wall camber is found to have a strongly stabilizing effect, concave wall camber a strongly destabilizing effect on first and second mode disturbances. Due to concave curvature a third mode unstable region develops merged with the second mode.

Nonlocal and Local Instability Analysis of Hypersonic Flows

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Nonlocal and Local Instability Analysis of Hypersonic Flows