Derivation of and Simulations with BiGlobal Stability Equations

Laminar to turbulent transition has an important role in the aerospace domain in view of its impact on aerodynamic drag and, regarding the high velocity regime, heat transfer. State of the art computational methods, like DNS, LES and RANS are found to be too expensive or rely on case dependent turbulence models to be used for obtaining information regarding the transition phenomenon. Transition is typically initiated by the onset of instability of the laminar flow. Linear stability theory describes the eigenmode growth mechanism. Although this yields a restriction, because additional mechanisms play a role too, the eigenmode growth phase establishes an important base in many practical situations. However, the linearization provides a considerable step in the simplification of the analysis, while the stability theory can be adapted according to the structure of the given mean flow. At the Von Karman Institute (VKI), the VKI Extensible Stability and Transition Analysis (VESTA) toolkit has been developed, which mainly involves methods based on the linear stability theory. In the current project, the main goal was to extend the already present tools to incorporate the BiGlobal stability equations, which, together with appropriate boundary conditions, form an eigenvalue problem. This particular problem is solved for perturbations inhomogeneous in two spatial directions and their complex growth rate and frequency. This extension involved a new version of the tool for the derivation of the BiGlobal stability equations, a tool for their automatic implementation in Matlab via the spectral collocation method and a simulation tool to apply boundary conditions and execute the analysis corresponding to a prescribed mean flow. The derivation of the BiGlobal equations and their verification formed the first part of the project. Both incompressible and compressible versions are derived for different kinds of coordinate systems (e.g. Cartesian and cylindrical) and formulations in the compressible case (e.g. involving temperature and pressure and the energy equation based on static enthalpy). This allowed the verification of the tool with a large number of previously published references. All references, to the knowledge of the current author, that have thus far reported the compressible equations were found to contain errors and had to be cross-verified to yield the ultimate positive outcome. It is hence deemed that the present treatment is the first to report the full compressible BiGlobal stability equations in primitive variable formulation correctly. The second part of the project involved the verification of the performance of the combination of the derivation, implementation and simulation tools. This was done by considering three test cases (mean flows). In all cases, the eigenvalue problem was solved using the QZ algorithm. In cases that required high resolution, the Arnoldi algorithm was used in addition, because of its lean performance with respect to required memory. The first test case was the parallel Blasius boundary layer. Because of its one-dimensional nature, this flow has been intensively analysed in the past by means of the classic local stability analysis type (LST). This allowed the BiGlobal analysis of this mean flow to be thoroughly verified in both the incompressible and supersonic regime. The second case involved the developing incompressible Blasius boundary layer. This flow was chosen because of its better affinity with the actual Blasius boundary layer flow, which has an intrinsic developing nature. The BiGlobal approach involved artificial in- and outflow boundary conditions. Analyses were performed on a domain with a small and large streamwise extent to focus on a flow that is weakly and strongly developing, respectively. The former analyses were again compared to LST simulations to yield an internal verification and consistency check. The results of the analyses on the larger domain could be compared to the literature and were found to agree well in a qualitative sense. The Tollmien-Schlichting branch obtained in this study was found to lie too high with respect to the one reported in the literature. Although the exact reason for this could not yet be established, the most likely cause is a (small) difference in the prescribed mean flow. It is expected that the test case will yield identical results when exactly the same mean flow will be used, as some key differences can be identified in the literature in this regard. It was found that the artificial boundary conditions caused an odd/even effect with respect to the continuous eigenmode branches in the spectrum when the number of points in the streamwise direction was taken to be either odd or even. A similar behaviour was observed when consulting the literature, although the effect was never elaborated on explicitly. Lastly, the incompressible complex lamellar bidirectional vortex was considered. This mean flow is defined on a cylindrical coordinate system and is highly inhomogeneous in at least two spatial directions. Therefore, this case requires the BiGlobal approach and all power of the newly developed tools could be tested. A test case handled in the literature was very precisely reconstructed. Although it was found that no part of the spectrum was converged, the results were nearly identically retrieved. The solutions to all three test cases have been obtained successfully and compare reasonably well with the literature. It is therefore concluded that all capabilities of the newly developed tools have been tested successfully and the tools can be considered to be verified.