Massively Parallel Solution of the BiGlobal Eigenvalue Problem Using Dense Linear Algebra

Abstract

Linear instability of complex flows may be analyzed by numerical solutions of partial-derivative-based eigenvalue problems; the concepts are, respectively, referred to as BiGlobal or TriGlobal instability, depending on whether two or three spatial directions are resolved simultaneously. Numerical solutions of the BiGlobal eigenvalue problems in flows of engineering significance, such as the laminar separation bubble in which global eigenmodes have been identified, reveal that recovery of (two-dimensional) amplitude functions of globally stable but convectively unstable flows (i.e., flows which sustain spatially amplifying disturbances in a local instability analysis context) requires resolutions well beyond the capabilities of serial, in-core solutions of the BiGlobal eigenvalue problems. The present contribution presents a methodology capable of overcoming this bottleneck via massive parallel solution of the problem at hand; the approach discussed is especially useful when a large window of the eigenspectrum is sought. Two separated flow applications, one in the boundary-layer on a flat plate and one in the wake of a stalled airfoil, are briefly discussed as demonstrators of the class of problems in which the present enabling technology permits the study of global instability in an accurate manner.= memory required for one floating-point complex number mem proc = memory available for computations on each processor N arrays = number of arrays to be stored N var = number of variables and equations N x , N y = Chebyshev-Gauss-Lobatto collocation points in the x and y directions p = number of processors Re = Reynolds number t = time t AR;it = time required for one Arnoldi iteration t EIG = time required for the eigenvalues and eigenvectors calculation t EVP = time required for the eigenvalue problems creation t LU = time required for the matrix shifting and lower-upper decomposition u, v, w = basic flow streamwise, wall-normal, and spanwise velocity components x, y, z = streamwise, wall-normal, and spanwise spatial coordinates

BiGlobal linear stability analysis on low-Re flow past an airfoil at high angle of attack

“…[30][31][32][33] The objective of the present study is to investigate the stability characteristics of incompressible flow past an airfoil in the low-Re regime Re = 400-1000. The Reynolds number in this study is comparably higher than the previous studies 15,28,29 and the flow is unsteady, thus the results reveal the flow instabilities beyond the first Hopf bifurcation point. The numerical methods employed are free of any assumption or turbulence model, thus both the base flow and the perturbation characteristics are obtained following the original (linearized) Navier-Stokes equations.…”

“…The physical domain size is 80C in the x-direction with a wake length of 49C, and 60C in the y-direction. This large domain size is chosen based on similar studies by Kitsios et al 15 and Rodríguez and Theofilis 29 to minimize the effect of the artificially imposed boundary conditions on the flow around the airfoil. The domain is discretized by a 1024 × 128 C-type grid.…”

“…There appears a branch of weakly unstable modes with medium frequencies as shown in the inset of the figure, which is also observed in the works of Kitsios et al 15 and Rodríguez and Theofilis. 29 These weakly unstable modes correspond to the far wake instability biased towards to the outflow boundary which makes the flow less regularized compared with the time-periodic state in the near wake region (see the curve of P5 in Figure 3) and are possibly caused by the truncation of the computational domain. 47 To examine the dependency on mesh resolution, a similar analysis with the same domain size (80C × 60C) but a finer 2048 × 256 mesh is carried out at Re = 1000.…”

“…18,35 Since the near wake flow of a bluff body is accelerated by the flow instability, 36 stability analysis based on the time-averaged flow leads to dramatically different predictions regarding both the growth rate and frequency of the perturbation compared with the stability analysis of a steady-state base flow. It is concluded from the above discussion that presently the global stability analysis on flow past an isolated airfoil is only performed for low-Re incompressible flow where the base flow can be obtained by directly integrating the governing equations, 15,28,29 and for high-Re compressible flow either in the RANS framework or using the time-averaged flow field as the base flow. [30][31][32][33] The objective of the present study is to investigate the stability characteristics of incompressible flow past an airfoil in the low-Re regime Re = 400-1000.…”

On the birth of stall cells on airfoils

“…A massively parallel implementation of the Arnoldi algorithm is used in order to recover a window of the eigenspectrum which contains the physically-interesting most unstable/least stable eigenvalues. Implementation details of the parallel solution algorithm may be found in [19].…”