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Order-of-Magnitude Speedup for Steady States and Traveling Waves via Stokes Preconditioning in Channelflow and Openpipeflow

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Book cover Computational Modelling of Bifurcations and Instabilities in Fluid Dynamics

Part of the book series: Computational Methods in Applied Sciences ((COMPUTMETHODS,volume 50))

Abstract

Steady states and traveling waves play a fundamental role in understanding hydrodynamic problems. Even when unstable, these states provide the bifurcation-theoretic explanation for the origin of the observed states. In turbulent wall-bounded shear flows, these states have been hypothesized to be saddle points organizing the trajectories within a chaotic attractor. These states must be computed with Newton’s method or one of its generalizations, since time-integration cannot converge to unstable equilibria. The bottleneck is the solution of linear systems involving the Jacobian of the Navier–Stokes or Boussinesq equations. Originally such computations were carried out by constructing and directly inverting the Jacobian, but this is unfeasible for the matrices arising from three-dimensional hydrodynamic configurations in large domains. A popular method is to seek states that are invariant under numerical time integration. Surprisingly, equilibria may also be found by seeking flows that are invariant under a single very large Backwards-Euler Forwards-Euler timestep. We show that this method, called Stokes preconditioning, is 10–50 times faster at computing steady states in plane Couette flow and traveling waves in pipe flow. Moreover, it can be carried out using Channelflow (by Gibson) and Openpipeflow (by Willis) without any changes to these popular spectral codes. We explain the convergence rate as a function of the integration period and Reynolds number by computing the full spectra of the operators corresponding to the Jacobians of both methods.

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Acknowledgements

We thank Dwight Barkley and John Gibson for their contributions. We acknowledge the support of TRANSFLOW, provided by the Agence Nationale de la Recherche (ANR).

Author information

Authors and Affiliations

  1. Laboratoire de Physique et Mécanique des Milieux Hétérogènes (PMMH), CNRS, ESPCI Paris, PSL Research University, Sorbonne Université, Univ. Paris Diderot, Paris, France

    Laurette S. Tuckerman

  2. School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK

    Jacob Langham

  3. School of Mathematics and Statistics, University of Sheffield, Sheffield, S3 7RH, UK

    Ashley Willis

Authors
  1. Laurette S. Tuckerman
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  2. Jacob Langham
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  3. Ashley Willis
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Correspondence to Laurette S. Tuckerman .

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  1. School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv, Israel

    Alexander Gelfgat

Appendix

Appendix

This Appendix presents samples in Figs. 9, 10, 11 and 12 of previous computations carried out by Stokes preconditioning.

Fig. 9
figure 9

From Borońska and Tuckerman [17]

Rayleigh–Bénard convection in a cylinder of aspect ratio with radius / height \(=2\), \(Pr=6.7\) and insulating lateral boundaries. Left: Bifurcation diagram shows 17 branches of steady states, with azimuthal symmetries \(m=2\) (pizza, four-roll), \(m=0\) (two-tori, torus), \(m=3\) (marigold, Mitsubishi, cloverleaf, Mercedes), \(m=1\) (dipole, three-roll, tiger, asymmetric three-roll). Right: Partial schematic diagram showing branches with \(m=3\) symmetry. Transition from conductive state to marigold and then Mitsubishi branches occur via circle and ordinary pitchfork bifurcations, respectively, and to cloverleaf and Mercedes branches via two successive saddle-node bifurcations. The only stable states are on a portion of the Mercedes branch, shown by the thick curve. The results are from a pseudospectral simulation with \((M_r,M_\theta ,M_z)=(60,130,30)\).

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Fig. 10
figure 10

From Mercader, Batiste, Alonso and Knobloch [55]

Binary fluid convection in a domain with aspect ratio width / height \(= 14\), Neumann boundary conditions, Prandtl and Lewis numbers \(Pr=7\), \(Le=0.01\) and separation ratio \(S=-0.1\). Left: Partial bifurcation diagram showing two-pulse point-symmetric states based on 15 rolls. Right: Temperature and concentration fields for the three solutions indicated as dots on the bifurcation diagram.

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Fig. 11
figure 11

From LoJacono, Bergeon and Knobloch [61]

Three-dimensional binary fluid convection in a porous medium of size \(6\times 6 \times 1\). Left: bifurcation diagram. Right: vertical velocity at mid-layer. The transition from a four-armed structure with arms oriented along the diagonals (panels c and d, black curve) to an eight-armed structure with arms oriented along both the diagonals and the principal axes of the domain (panel a, orange curve) via a target pattern (panel b). Simulations use a spectral element method with 6 elements in the quarter domain, each with (23, 23, 17) points.

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Fig. 12
figure 12

From Beaume, Chini, Julien and Knobloch [97, 109]

Steady states of a streamwise-independent reduced model for plane Couette flow. Bifurcation diagram on left. Representative states from lower (middle) and upper (right) branches at \(Re\approx 1000\). Colored contours show the streamfunction of streamwise rolls, while the black curves show contours of the streamwise velocity.

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Tuckerman, L.S., Langham, J., Willis, A. (2019). Order-of-Magnitude Speedup for Steady States and Traveling Waves via Stokes Preconditioning in Channelflow and Openpipeflow. In: Gelfgat, A. (eds) Computational Modelling of Bifurcations and Instabilities in Fluid Dynamics. Computational Methods in Applied Sciences, vol 50. Springer, Cham. https://doi.org/10.1007/978-3-319-91494-7_1

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