Enhanced stability of flows through contraction channels: Combining shape optimization and linear stability analysis

Abstract

The first flow bifurcation, in channels with a sudden geometry contraction, is controlled through shape optimisation to delay the onset of asymmetry. First, we confirm the existence of a pitchfork type bifurcation instability, already reported in similar geometries. The global mode responsible for this bifurcation leads to asymmetric flow for Reynolds numbers beyond a critical value. Second, we propose a global shape optimisation methodology to introduce small modifications in the channel geometry that lead to flows with enhanced stability. Our results include three contraction ratios C = 2, 4 and 8, where C is the ratio of upstream to downstream channel widths.The shape optimisation aims at minimising the growth rate of the unstable mode responsible for asymmetry. Sensitivity analysis is used to find an appropriate geometry parametrisation, which is defined through super-elliptic curves, and limited to small deformations. Additionally, a dynamically updated surrogate model (based on Radial Basis Functions) is developed to drive the optimisation. This substitutes expensive function evaluations, each requiring the solution of a steady Navier-Stokes base flow computation and a solution of an eigenvalue problem (linear stability analysis). Finally, a mode tracking algorithm identifies the eigenmode responsible for the onset of asymmetry during the optimisation.The optimised geometries show rounded corners and are stable for Reynolds numbers well beyond the original values. For all contraction ratios, the critical Reynolds number increases by at least 7.9 times with respect to the original values. three-dimensional simulations confirm that the optimised geometry is more stable than the original when periodic boundaries are used on the side walls. When comparing the drag of the optimised geometry to the original, we obtain a reduction of at least 64%.

Sensitivity gradients of surface geometry modifications based on stability analysis of compressible flows

“…To solve the linear systems associated with the iterative generation of the Krylov subspace, a full lower-upper (LU) decomposition of the sparse matrix A(q, X) is calculated in parallel and distributed memory, using the MUMPS package [32]. This methodology has been validated in different scenarios to recover the physical eigenvalues; see, for instance, [3,11,[33][34][35]. In this work, the calculation of eigenvalues involves a Krylov-Schur technique, implemented in the public numerical library SLEPc [36].…”

“…This approach has, however, the drawback of the memory limitations related to the matrix storage, as the scaling of the full LU decomposition behaves as (N v × N ) 3 [38]. However, domain reduction strategies [39] and efficient matrix distribution [40] reduce memory demands and allows one to perform stability analyses of three-dimensional configurations [3].…”

“…In the search for improved aerodynamic configurations, new tools that couple flow control, stability analysis, and optimization loops are emerging [1][2][3]. Recent studies [4][5][6] have developed flow control methodologies using global stability analysis as a source for gradient calculations.…”

“…More recently, on a planar X-junction, Lashgari et al [16] optimized the distribution of wall suction and blowing for the first flow bifurcation, also giving a physical explanation for the second bifurcation using the results of the sensitivity analysis. As an example of an eigenvalue-based and gradient-free optimization problem, Wang et al [3] stabilized the global instability, present on channel contraction, using a genetic-algorithm optimization approach combined with structural sensitivity analysis, which enable one to delay the onset of the instability measured in terms of critical Re number, by 750 percent. This list probably omits many relevant works on flow control based on global stability, so the reader is encouraged to follow the reviews of Luchini [17] and Sipp [18] on adjoint analysis and global stability analysis of open flow configurations.…”

Domain reduction strategy for the stability analysis of complex aerodynamic flows

“…Under certain circumstances, the purged flow generates nonsymmetrical configurations due to the pressure imbalance provoked by the injected flow. Martinez‐Cava et al and Wang et al combined RANS simulations and global stability analysis to investigate the physical reasons behind this flow bifurcation. An antisymmetric global mode was ultimately identified as the responsible of triggering the main mechanism forcing the changes in the flow topology.…”