Shooting method for linear inviscid bi-global stability analysis of non-axisymmetric jets

Abstract

The shooting method is commonly used to solve the linear parallel-flow stability problem for axisymmetric jets, i.e., a flow having one inhomogeneous direction. The present extension to two inhomogeneous directions – i.e., a bi-global stability problem – is motivated by inviscid non-axisymmetric jets. The azimuthal direction is Fourier transformed to obtain a set of coupled one-dimensional shooting problems that are solved by two-way integration from both radial boundaries – centreline and far field. The overall problem is formulated as one of iterative root-finding to match the solutions from the two integrations. The approach is validated against results from the well-established matrix method that discretizes the domain to obtain a matrix eigenvalue problem. We demonstrate very good agreement in two jet problems – an offset dual-stream jet, and a jet exiting from a nozzle with chevrons. A disadvantage of the shooting method is its sensitivity to the initial guess of the solution; however, this becomes an advantage when the need arises to track an eigensolution in a sweep over a problem parameter – say with increasing offset in the dual-stream jet, or with downstream distance from the nozzle exit. We demonstrate the performance of the shooting method in such tracking tasks.