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Planar potential flow on Cartesian grids

Published online by Cambridge University Press:  27 April 2022

Diederik Beckers Open the ORCID record for Diederik Beckers [Opens in a new window]  and
Jeff D. Eldredge Open the ORCID record for Jeff D. Eldredge [Opens in a new window]
Diederik Beckers
Affiliation:
Mechanical and Aerospace Engineering, University of California, Los Angeles, CA 90095-1597, USA
Jeff D. Eldredge*
Affiliation:
Mechanical and Aerospace Engineering, University of California, Los Angeles, CA 90095-1597, USA
*
†Email address for correspondence: [email protected]
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Abstract

Potential flow has many applications, including the modelling of unsteady flows in aerodynamics. For these models to work efficiently, it is best to avoid Biot–Savart interactions. This work presents a grid-based treatment of potential flows in two dimensions and its use in a vortex model for simulating unsteady aerodynamic flows. For flows consisting of vortex elements, the treatment follows the vortex-in-cell approach and solves the streamfunction–vorticity Poisson equation on a Cartesian grid after transferring the circulation from the vortices onto the grid. For sources and sinks, an analogous approach can be followed using the scalar potential. The combined velocity field due to vortices, sinks and sources can then be obtained using the Helmholtz decomposition. In this work, we use several key tools that ensure the approach works on arbitrary geometries, with and without sharp edges. First, the immersed boundary projection method is used to account for bodies in the flow and the resulting body-forcing Lagrange multiplier is identified as the bound vortex sheet strength. Second, sharp edges are treated by decomposing the vortex sheet strength into a singular and non-singular part. To enforce the Kutta condition, the non-singular part can then be constrained to remove the singularity introduced by the sharp edge. These constraints and the Poisson equation are formulated as a saddle-point system and solved using the Schur complement method. The lattice Green's function is used to efficiently solve the discrete Poisson equation with unbounded boundary conditions. The method and its accuracy are demonstrated for several problems.

JFM classification

Mathematical Foundations: Computational methods Vortex Flows: Vortex dynamics Wakes/Jets: Separated flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 941 , 25 June 2022 , A19
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

REFERENCES

Baker, G.R. 1979 The ‘cloud in cell’ technique applied to the roll up of vortex sheets. J. Comput. Phys. 31 (1), 76–95.CrossRefGoogle Scholar
Beckers, D. & Eldredge, J.D. 2021 JuliaIBPM/GridPotentialFlow.jl v0.3.2. Available at: https://github.com/JuliaIBPM/GridPotentialFlow.jl/tree/v0.3.2.Google Scholar
Benzi, M., Golub, G.H. & Liesen, J. 2005 Numerical solution of saddle point problems. Acta Numerica 1, 1–137.CrossRefGoogle Scholar
Chatelain, P., Curioni, A., Bergdorf, M., Rossinelli, D., Andreoni, W. & Koumoutsakos, P. 2008 Billion vortex particle direct numerical simulations of aircraft wakes. Comput. Meth. Appl. Mech. Engng 197 (13–16), 1296–1304.CrossRefGoogle Scholar
Chatelin, R. & Poncet, P. 2014 Hybrid grid-particle methods and penalization: a Sherman–Morrison– Woodbury approach to compute 3D viscous flows using FFT. J. Comput. Phys. 269, 314–328.CrossRefGoogle Scholar
Chen, S.S. 1975 Vibration of nuclear fuel bundles. Nucl. Engng Des. 35 (3), 399–422.CrossRefGoogle Scholar
Chorin, A.J. & Bernard, P.S. 1973 Discretization of a vortex sheet, with an example of roll-up. J. Comput. Phys. 13 (3), 423–429.CrossRefGoogle Scholar
Christiansen, J.P. 1973 Numerical simulation of hydrodynamics by the method of point vortices. J. Comput. Phys. 13 (3), 363–379.CrossRefGoogle Scholar
Colonius, T. & Taira, K. 2008 A fast immersed boundary method using a nullspace approach and multi-domain far-field boundary conditions. Comput. Meth. Appl. Mech. Engng 197 (25–28), 2131–2146.CrossRefGoogle Scholar
Coquerelle, M. & Cottet, G.-H. 2008 A vortex level set method for the two-way coupling of an incompressible fluid with colliding rigid bodies. J. Comput. Phys. 227 (21), 9121–9137.CrossRefGoogle Scholar
Cottet, G.-H. & Koumoutsakos, P. 2000 Vortex Methods: Theory and Practice. Cambridge University Press.CrossRefGoogle Scholar
Cottet, G.-H. & Poncet, P. 2004 Advances in direct numerical simulations of 3D wall-bounded flows by vortex-in-cell methods. J. Comput. Phys. 193 (1), 136–158.CrossRefGoogle Scholar
Couët, B., Buneman, O. & Leonard, A. 1981 Simulation of three-dimensional incompressible flows with a vortex-in-cell method. J. Comput. Phys. 39 (2), 305–328.CrossRefGoogle Scholar
Cserti, J. 2000 Application of the lattice green's function for calculating the resistance of an infinite network of resistors. Am. J. Phys. 68, 896–906.CrossRefGoogle Scholar
Darakananda, D. & Eldredge, J.D. 2019 A versatile taxonomy of low-dimensional vortex models for unsteady aerodynamics. J. Fluid Mech. 858, 917–948.CrossRefGoogle Scholar
Ebiana, A.B. & Bartholomew, R.W. 1996 Design considerations for numerical filters used in vortex-in-cell algorithms. Comput. Fluids 25 (1), 61–75.CrossRefGoogle Scholar
Eldredge, J.D. 2019 Mathematical Modeling of Unsteady Inviscid Flows, Interdisciplinary Applied Mathematics, vol. 50. Springer.CrossRefGoogle Scholar
Eldredge, J.D. 2022 A method of immersed layers on Cartesian grids, with application to incompressible flows. J. Comput. Phys. 448, 110716.CrossRefGoogle Scholar
Gazzola, M., Chatelain, P., van Rees, W.M. & Koumoutsakos, P. 2011 Simulations of single and multiple swimmers with non-divergence free deforming geometries. J. Comput. Phys. 230 (19), 7093–7114.CrossRefGoogle Scholar
Gillis, T., Marichal, Y., Winckelmans, G. & Chatelain, P. 2019 A 2D immersed interface vortex particle-mesh method. J. Comput. Phys. 394, 700–718.CrossRefGoogle Scholar
Gillis, T., Winckelmans, G. & Chatelain, P. 2017 An efficient iterative penalization method using recycled Krylov subspaces and its application to impulsively started flows. J. Comput. Phys. 347, 490–505.CrossRefGoogle Scholar
Gillis, T., Winckelmans, G. & Chatelain, P. 2018 Fast immersed interface Poisson solver for 3D unbounded problems around arbitrary geometries. J. Comput. Phys. 354, 403–416.CrossRefGoogle Scholar
Goza, A., Liska, S., Morley, B. & Colonius, T. 2016 Accurate computation of surface stresses and forces with immersed boundary methods. J. Comput. Phys. 321, 860–873.CrossRefGoogle Scholar
Hejlesen, M.M., Koumoutsakos, P., Leonard, A. & Walther, J.H. 2015 Iterative Brinkman penalization for remeshed vortex methods. J. Comput. Phys. 280, 547–562.CrossRefGoogle Scholar
Jones, M.A. 2003 The separated flow of an inviscid fluid around a moving flat plate. J. Fluid Mech. 496 (496), 405–441.CrossRefGoogle Scholar
Katsura, S. & Inawashiro, S. 1971 Lattice Green's functions for the rectangular and the square lattices at arbitrary points. J. Math. Phys. (Journal of Mathematical Physics) 12, 1622–1630.Google Scholar
LeVeque, R.J. & Li, Z. 1994 The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM. J. Numer. Anal. 31 (4), 1019–1044.CrossRefGoogle Scholar
Liska, S. & Colonius, T. 2014 A parallel fast multipole method for elliptic difference equations. J. Comput. Phys. 278, 76–91.CrossRefGoogle Scholar
Marichal, Y., Chatelain, P. & Winckelmans, G. 2014 An immersed interface solver for the 2-D unbounded Poisson equation and its application to potential flow. Comput. Fluids 96, 76–86.CrossRefGoogle Scholar
Meng, J.C.S. & Thomson, J.A.L. 1978 Numerical studies of some nonlinear hydrodynamic problems by discrete vortex element methods. J. Fluid Mech. 84 (3), 433–453.CrossRefGoogle Scholar
Monaghan, J.J. 1985 Extrapolating B splines for interpolation. J. Comput. Phys. 60 (2), 253–262.CrossRefGoogle Scholar
Poncet, P. 2009 Analysis of an immersed boundary method for three-dimensional flows in vorticity formulation. J. Comput. Phys. 228 (19), 7268–7288.CrossRefGoogle Scholar
Ramesh, K., Gopalarathnam, A., Granlund, K., OL, M.V. & Edwards, J.R. 2014 Discrete-vortex method with novel shedding criterion for unsteady aerofoil flows with intermittent leading-edge vortex shedding. J. Fluid Mech. 751, 500–548.CrossRefGoogle Scholar
Rasmussen, J.T., Cottet, G.-H. & Walther, J.H. 2011 A multiresolution remeshed vortex-in-cell algorithm using patches. J. Comput. Phys. 230 (17), 6742–6755.CrossRefGoogle Scholar
Rossinelli, D., Bergdorf, M., Cottet, G.-H. & Koumoutsakos, P. 2010 GPU accelerated simulations of bluff body flows using vortex particle methods. J. Comput. Phys. 229 (9), 3316–3333.CrossRefGoogle Scholar
Saffman, P.G. 1993 Vortex Dynamics, Cambridge Monographs on Mechanics, vol. 87. Cambridge University Press.CrossRefGoogle Scholar
Spietz, H.J., Hejlesen, M.M. & Walther, J.H. 2017 Iterative Brinkman penalization for simulation of impulsively started flow past a sphere and a circular disc. J. Comput. Phys. 336, 261–274.CrossRefGoogle Scholar
Wiegmann, A. & Bube, K.P. 2000 The explicit-jump immersed interface method: finite difference methods for PDEs with piecewise smooth solutions. SIAM J. Numer. Anal. 37 (3), 827–862.CrossRefGoogle Scholar
Yang, X., Zhang, X., Li, Z. & He, G.W. 2009 A smoothing technique for discrete delta functions with application to immersed boundary method in moving boundary simulations. J. Comput. Phys. 228 (20), 7821–7836.CrossRefGoogle Scholar