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Minimal gain marching schemes: searching for unstable steady-states with unsteady solvers

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Abstract

Reference solutions are important in several applications. They are used as base states in linear stability analyses as well as initial conditions and reference states for sponge zones in numerical simulations, just to name a few examples. Their accuracy is also paramount in both fields, leading to more reliable analyses and efficient simulations, respectively. Hence, steady-states usually make the best reference solutions. Unfortunately, standard marching schemes utilized for accurate unsteady simulations almost never reach steady-states of unstable flows. Steady governing equations could be solved instead, by employing Newton-type methods often coupled with continuation techniques. However, such iterative approaches do require large computational resources and very good initial guesses to converge. These difficulties motivated the development of a technique known as selective frequency damping (SFD) (Åkervik et al. in Phys Fluids 18(6):068102, 2006). It adds a source term to the unsteady governing equations that filters out the unstable frequencies, allowing a steady-state to be reached. This approach does not require a good initial condition and works well for self-excited flows, where a single nonzero excitation frequency is selected by either absolute or global instability mechanisms. On the other hand, it seems unable to damp stationary disturbances. Furthermore, flows with a broad unstable frequency spectrum might require the use of multiple filters, which delays convergence significantly. Both scenarios appear in convectively, absolutely or globally unstable flows. An alternative approach is proposed in the present paper. It modifies the coefficients of a marching scheme in such a way that makes the absolute value of its linear gain smaller than one within the required unstable frequency spectra, allowing the respective disturbance amplitudes to decay given enough time. These ideas are applied here to implicit multi-step schemes. A few chosen test cases shows that they enable convergence toward solutions that are unstable to stationary and oscillatory disturbances, with either a single or multiple frequency content. Finally, comparisons with SFD are also performed, showing significant reduction in computer cost for complex flows by using the implicit multi-step MGM schemes.

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Authors and Affiliations

  1. Programa de Pós-Graduação em Engenharia de Defesa, Instituto Militar de Engenharia, Rio de Janeiro, RJ, 22290-270, Brazil

    Renan de S. Teixeira

  2. Laboratório de Mecânica Teórica e Aplicada, Departamento de Engenharia Mecânica, Universidade Federal Fluminense, Niterói, RJ, 24210-240, Brazil

    Leonardo S. de B. Alves

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  1. Renan de S. Teixeira
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  2. Leonardo S. de B. Alves
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Correspondence to Leonardo S. de B. Alves.

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Communicated by Vassilios Theofilis.

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de S. Teixeira, R., S. de B. Alves, L. Minimal gain marching schemes: searching for unstable steady-states with unsteady solvers. Theor. Comput. Fluid Dyn. 31, 607–621 (2017). https://doi.org/10.1007/s00162-017-0426-0

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