Order 104 speedup in global linear instability analysis using matrix formation

Abstract

A unified solution framework is presented for one-, two-or three-dimensional complex non-symmetric eigenvalue problems, respectively governing linear modal instability of incompressible fluid flows in rectangular domains having two, one or no homogeneous spatial directions. The solution algorithm is based on subspace iteration in which the spatial discretization matrix is formed, stored and inverted serially. Results delivered by spectral collocation based on the Chebyshev-Gauss-Lobatto (CGL) points and a suite of high-order finite-difference methods comprising the previously employed for this type of work Dispersion-Relation-Preserving (DRP) and Pade finite-difference schemes, as well as the Summationby-parts (SBP) and the new high-order finite-difference scheme of order q (FD-q) have been compared from the point of view of accuracy and efficiency in standard validation cases of temporal local and BiGlobal linear instability. The FD-q method has been found to significantly outperform all other finite difference schemes in solving classic linear local, BiGlobal, and TriGlobal eigenvalue problems, as regards both memory and CPU time requirements. Results shown in the present study disprove the paradigm that spectral methods are superior to finite difference methods in terms of computational cost, at equal accuracy, FD-q spatial discretization delivering a speedup of 0(10''). Consequently, accurate solutions of the three-dimensional (TriGlobal) eigenvalue problems may be solved on typical desktop computers with modest computational effort.

Investigation on the wake flow instability behind isolated roughness elements on the forebody of a blunt generic reentry capsule

“…The chosen nonuniform node distribution minimizes the error of the piecewise polynomial interpolation and in case of q = I, the resulting ¦nite di¨erence schemes are equivalent to the Chebyshev collocation methods. Paredes et al [22] showed that this ¦nite di¨erence method outperforms spectral collocation methods for stability analysis calculations and is the best compromise in terms of accuracy and computational e©ciency, while recovering spectral-like accuracy. In the present study, the eighth-order scheme, FD-q8, has been chosen and grid convergence was checked by increasing the number of discretization nodes in spanwise and wall-normal direction, respectively.…”

The linearized pressure Poisson equation for global instability analysis of incompressible flows

“…First, using a modified version of the nek5000 internal routine int_tp, the base flow is interpolated spectrally from the spectral element grid onto a cubic domain discretized by 101 mapped Chebyshev Gauss Lobatto points in each spatial direction. Subsequently, the interpolated base flow is again interpolated spectrally onto the mesh used for the eigenvalue problem solutions, the latter built using 4th-and 6th-order FD-q methods [44] and comprising from 31 3 to 61 3 discretization nodes. The system (7-10) is then solved subject to homogeneous Dirichlet boundary conditions for the disturbance velocity components on all boundaries and pressure perturbation boundary conditions provided by the three-dimensional LPPE (13).…”

“…The two-dimensional (β = 0) temporal BiGlobal eigenvalue problem is solved in a rectangular domain defined by x ∈ [100, 650]× y ∈ [1,36], corresponding to a Reynolds number Re = 2400 at the beginning of the domain. The matrix-forming approach discussed by Paredes et al [44] is used, in which equations (2)(3)(4)(5) are discretized by a 16th-order FD-q finite-difference method using N x = 601 and N y = 101 nodes along the x− and y−directions, respectively. The eigenspectra delivered by the LPPE and the PC closures are shown in Fig.…”

Spatial linear global instability analysis of the HIFiRE-5 elliptic cone model flow

“…26 These libraries exploit the high level of sparsity pattern offered by the finite-difference spatial differentiation, improving substantially on numerical efficiency while keeping accuracy; see 27 for more details. The (η, ζ) directions are discretized in a coupled manner using the stable high-order finite-differences numerical schemes of order q (FD-q) developed in.…”