Coupling time-stepping numerical methods and standard aerodynamics codes for instabilitiy analysis of flows in complex geometries

Abstract

The development of a global instability analysis code coupling a time-stepping approach, as applied to the solution of BiGlobal and TriGlobal instability analysis 1, 2 and finite-volume-based spatial discretization, as used in standard aerodynamics codes is presented. The key advantage of the time-stepping method over matrix-formulation approaches is that the former provides a solution to the computer-storage issues associated with the latter methodology. To-date both approaches are successfully in use to analyze instability in complex geometries, although their relative advantages have never been quantified. The ultimate goal of the present work is to address this issue in the context of spatial discretization schemes typically used in industry. The time-stepping approach of Chiba 3 has been implemented in conjunction with two direct numerical simulation algorithms, one based on the typically-used in this context high-order method and another based on low-order methods representative of those in common use in industry. The two codes have been validated with solutions of the BiGlobal EVP and it has been showed that small errors in the base flow do not have affect significantly the results. As a result, a three-dimensional compressible unsteady second-order code for global linear stability has been successfully developed based on finite-volume spatial discretization and time-stepping method with the ability to study complex geometries by means of unstructured and hybrid meshes.

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“…35 In a different vein, application of Jacobian-Free Newton-Krylov methods to standard aerodynamic codes with the aim to study compressible flow instability over objects of arbitrary shape has commenced. 120 Details may respectively be found in the related contributions to this Conference.…”

“…Although this situation is slowly changing in recent literature, 22,[117][118][119] global instability analysis is yet to penetrate into the world of classic CFD schemes, the latter typically being based on variants of low-order finite-volume methods. Gómez et al 120 present a Jacobian-Free Newton-Krylov based time-stepping algorithm of the class introduced into global instability analysis by Eriksson and Rizzi 121 and successfully employed to analyze BiGlobal 81,122 and TriGlobal 8 EVPs combined with a standard finite-volume solver for compressible flow capable of addressing compressible flows over complex geometries. Validation results, one of which is shown in figure 2, have been obtained by reference to the classic lid-driven cavity flow at low Reynolds numbers, as well as flow over a stalled airfoil; details may be found in Ref.…”

“…Validation results, one of which is shown in figure 2, have been obtained by reference to the classic lid-driven cavity flow at low Reynolds numbers, as well as flow over a stalled airfoil; details may be found in Ref. 120.…”