Instability analysis of compressible flows over open cavities by a Jacobian-free numerical method - Mathias, Marlon Sproesser (2017)
The influence of the Mach number and the boundary layer thickness over the stability of Rossiter-like modes in a subsonic compressible flow over an open rectangular cavity is studied. This work describes the implementation and use of the relevant computational methods. The most straight-forward way of accessing the stability of a flow is to build the Jacobian matrix of its governing equations and to compute its eigenvalues and eigenvectors. The so called matrix forming methods explicitly compute this matrix and use numerical algorithms to solve its eigenproblem. When the flow grows more complex, the Jacobian matrix may become unfeasibly large. The algorithm implemented here is of the Jacobianfree type, which means that this matrix is not explicitly needed. Therefore, the Arnoldi iteration method is used as all it needs is the ability of multiplying the Jacobian by an arbitrary vector. The algorithm is built in a way that a call to a flow solver is equivalent to this multiplication. The development of this solver is also covered by this work, it is a DNS (Direct Numerical Solver) for the compressible Navier-Stokes equations, which means that no turbulence models are used. High numerical precision is an important requirement as small disturbances, many orders of magnitude smaller than the base flow, must be well resolved. High order spectral like differentiation methods are employed. A validation work is performed for both the DNS and the instability analysis algorithm. Finally, this code becomes a tool to access the effect of a cavity on a boundary-layer flow. Two-dimensional cases are run for various incoming boundary layer thicknesses and Mach numbers. This work focuses on Rossiter modes and the physical phenomena that cause them to be either stable or unstable. Three types of phenomena are checked for their influence in the Rossiter modes: resonance with standing waves; spatial amplification at the mixing layer; and transfer from the flow disturbances to acoustic energy. Finally, the linear stability results are compared to DNS runs, which include non-linear effects. In the current parametric space, it was concluded that the instability at the mixing layer has an important role in selecting the Rossiter modes, while the increased flow to acoustic energy transfer caused by higher Mach numbers influence the mode amplification rate. The resonance with standing waves only plays a small role in this case.