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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References
Åkervik, E., Brandt, L., Henningson, D.S., Hoepffner, J., Marxen, O., Schlatter, P.: Steady solutions of the Navier-Stokes equations by selective frequency damping. Phys. Fluids 18(6), 068102 (2006)
Åkervik, E., Hœpffner, J., Ehrenstein, U., Henningson, D.S.: Optimal growth, model reduction and control in a separated boundary-layer flow using global eigenmodes. J. Fluid Mech. 579, 305–314 (2007)
Alves, L.S.B.: Preconditioned implicit Runge–Kutta schemes for unsteady simulations of low mach number compressible flows. In: Idelsohn, S., Sonzogni, V., Coutinho, A., Cruchaga, M., Lew, A., Cerrolaza M. (eds.) 1st Pan-American Congress on Computational Mechanics (2015)
Alves, L.S.B., Kelly, R.E., Karagozian, A.R.: Transverse jet shear layer instabilities. Part II: linear analysis for large jet-to-crossflow velocity ratios. J. Fluid Mech. 602, 383–401 (2008)
Bagheri, S., Schlatter, P., Schmid, P.J., Henningson, D.S.: Global stability of a jet in crossflow. J. Fluid Mech. 624, 33–44 (2009)
Barletta, A., Alves, L.S.B.: Transition to absolute instability for (not so) dummies (2014). arXiv:1403.5794
Barone, M.F., Lele, S.K.: Receptivity of the compressible mixing layer. J. Fluid Mech. 540, 301–335 (2005)
Bijl, H., Carpenter, M.H., Vatsa, V.N., Kennedy, C.A.: Implicit time integration schemes for the unsteady compressible Navier–Stokes equations: laminar flow. J. Comput. Phys. 179, 313–329 (2002)
Blaschak, J.G., Kriegsmann, G.A.: A comparative study of absorbing boundary conditions. J. Comput. Phys. 77, 109–139 (1988)
Bodony, D.J.: An analysis of sponge zones for computational fluid mechanics. J. Comput. Phys. 212(2), 681–702 (2006)
Brevdo, L., Laure, P., Dias, F., Bridges, T.J.: Linear pulse structure and signalling in a film flow on an inclined plane. J. Fluid Mech. 396, 37–71 (1999)
Chomaz, J.M.: Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357–392 (2005)
Collis, S.S., Lele, S.K.: Receptivity to surface roughness near a swept leading edge. J. Fluid Mech. 380, 141–168 (1999)
Colonius, T.: Modelling artificial boundary conditions for compressible flow. Annu. Rev. Fluid Mech. 136, 315–345 (2004)
Colonius, T., Lele, S.K.: Computational aeroacoustics: progress on nonlinear problems of sound generation. Prog. Aerosp. Sci. 40, 345–416 (2004)
Cunha, G., Passaggia, P.Y., Lazareff, M.: Optimization of the selective frequency damping parameters using model reduction. Phys. Fluids 27(094103), 1–22 (2015)
Dahlquist, G.: A special stability problem for linear multistep methods. BIT Numer. Math. 3, 27–43 (1963)
Falcao, C.E.G., Medeiros, F.E.L., Alves, L.S.B.: Implicit Runge–Kutta physical-time marching in low mach preconditioned density-based methods. In: 7th AIAA Theoretical Fluid Mechanics Conference, AIAA 2014-3085. AIAA Aviation (2014)
Germanos, R.A.C., de Souza, L.F., de Medeiros, M.A.F.: Numerical investigation of the three-dimensional secondary instabilities in the time-developing compressible mixing layer. J. Braz. Soc. Mech. Sci. Eng. 31(2), 125–136 (2009)
Jones, L.E., Sandberg, R.D., Sandham, N.D.: Stability and receptivity characteristics of a laminar separation bubble on an aerofoil. J. Fluid Mech. 648, 257–296 (2010)
Jordi, B.E., Cotter, C.J., Sherwin, S.J.: Encapsulated formulation of the selective frequency damping method. Phys. Fluids 26(034101), 1–10 (2014)
Jordi, B.E., Cotter, C.J., Sherwin, S.J.: An adaptive selective frequency damping method. Phys. Fluids 27(094104), 1–8 (2015)
Kelly, R.E., Alves, L.S.B.: A uniformly valid asymptotic solution for the transverse jet and its linear stability analysis. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Sci. 366, 2729–2744 (2008)
Knoll, D.A., Keyes, D.E.: Jacobian-free Newton–Krylov method: a survey of approaches and applications. J. Comput. Phys. 193, 357–397 (2004)
Lardjane, N., Fedioun, I., Gokalp, I.: Accurate initial conditions for the direct numerical simulation of temporal compressible binary shear layers with high density ratio. Comput. Fluids 33, 549–576 (2004)
Lax, P.D., Richtmyer, R.D.: Survey of the stability of linear finite difference equations. Commun. Pure Appl. Math. IX, 267–293 (1956)
Loiseau, J.C., Robinet, J.C., Cherubini, S., Leriche, E.: Investigation of the roughness-induced transition: global stability analyses and direct numerical simulations. J. Fluid Mech. 760, 175–211 (2014)
Lomax, H., Pulliam, T.H., Zingg, D.W.: Fundamentals of Computational Fluid Dynamics. Scientific Computation. Springer, Berlin (2001)
Megerian, S., Davitian, J., Alves, L.S.B., Karagozian, A.R.: Transverse jet shear layer instabilities. Part I: experimental studies. J. Fluid Mech. 593, 93–129 (2007)
Michalke, A.: Survey on jet instability theory. Prog. Aerosp. Sci. 21, 159–199 (1984)
Pier, B.: Local and global instabilities in the wake of a sphere. J. Fluid Mech. 603, 39–61 (2008)
Pulliam, T.H., Steger, J.L.: Implicit finite-difference simulations of three-dimensional compressible flow. AIAA J. 18(2), 159–167 (1980)
Saric, W.S., Reed, H.L., White, E.B.: Stability and transition of three-dimensional boundary-layers. Annu. Rev. Fluid Dyn. 35, 413–440 (2003)
Schmid, P.J.: Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162 (2007)
Sparrow, C.: The Lorenz Equations: Bifurcations, Chaos and Strange Attractors, Applied Mathematical Sciences, vol. 41. Springer, New York (1982)
Steinberg, S., Roache, P.J.: Symbolic manipulation and computational fluid dynamics. J. Comput. Phys. 57, 251–284 (1985)
Teixeira, R.S., Alves, L.S.B.: Modeling far field entrainment in compressible flows. Int. J. Comput. Fluid Dyn. 26, 67–78 (2012)
Theofilis, V.: Global linear instability. Annu. Rev. Fluid Mech. 43, 319–352 (2011)
Wang, L., Mavriplis, D.J.: Implicit solution of the unsteady Euler equations for high-order accurate discontinuous Galerkin discretizations. J. Comput. Phys. 225(2), 1994–2015 (2007)
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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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DOI: https://doi.org/10.1007/s00162-017-0426-0