Predicting the nonlinear amplification of disturbances using the Spatial Perturbation Equations in a viscous boundary layer*

The Spatial Perturbation Equations (SPE) are a well-posed streamwise marching formulation that utilizes a series of downstream traveling solutions to stabilize the marching procedure. This technique avoids the inconsistencies of the Parabolized Stability Equations (PSE) which require ad-hoc remedies to stabilize the inherently ill-posed marching procedure.  

A consistent use of streamwise non-parallel terms, viscous terms, and nonlinear terms are introduced and used to march the disturbances and remove undesirable upstream-traveling solutions that arise due to numerical error.

Our novel framework does not rely on modal wave behavior and enables the use of an explicit advancement scheme.

Additionally, it incorporates a robust treatment of nonlinear interactions of harmonics which allows the accurate capturing of high-amplitude perturbations.   

Comparisons of the evolution of linear and nonlinear disturbances in a boundary-layer flow show excellent agreement with direct numerical simulations (DNS) for both incompressible and compressible flow at a fraction of the computational cost.