Improved Parabolization of the Euler Equations

We present a new method for stability and modal analysis of shear flows and their acoustic radiation. The Euler equations are modified and solved as a spatial initial value problem in which initial perturbations are specified at the flow inlet and propagated downstream by integration of the equations. The modified equations, which we call one-way Euler equations, differ from the usual Euler equations in that they do not support upstream acoustic waves. It is necessary to remove these modes from the Euler operator because, if retained, they cause instability in the spatial marching procedure. These modes are removed using a two-step process. First, the upstream modes are partially decoupled from the downstream modes using a linear similarity transformation. Second, the error in the first step is eliminated using a convergent recursive filtering technique. A previous spatial marching method called the parabolized stability equations uses numerical damping to stabilize the march, but this has the unintended consequence of heavily damping the downstream acoustic waves. Therefore, the one-way Euler equation could be used to obtain improved accuracy over the parabolized stability equations as a low-order model for noise simulation of mixing layers and jets.