Recursive one-way Navier-Stokes equations: accurate, low-cost spatial marching

We introduce a new recursive formulation of the one-way Navier-Stokes (OWNS) equations that reduces cost to a level similar to the parabolized stability equations (PSE) while retaining superior accuracy. Many free-shear and wall-bounded flows contain a slowly varying direction that can be leveraged to efficiently study their linear and nonlinear stability, transition, and turbulent dynamics using spatial marching methods. The widely used parabolized stability equations (PSE) have severe limitations, due to regularization required to overcome the ill-posedness of the spatial march, for flows involving multiply instability mechanisms, transient growth, and acoustics. One-way Navier-Stokes (OWNS) equations overcome these limitations by formally removing the upstream-traveling waves responsible for ill-posedness, but doing so increases CPU and memory cost by one to two orders-of-magnitude relative to PSE. Our new formulation avoids this increased cost by modifying the OWNS equations to avoid large systems of equations in favor of a recursive series of smaller ones, reducing computational complexity. The new formulation is also easier to implement and comes with a priori error estimates. The efficiency of the method is demonstrated using a hypersonic boundary layer and a turbulent jet.