×

zbMATH — the first resource for mathematics

A versatile embedded boundary adaptive mesh method for compressible flow in complex geometry. (English) Zbl 1415.76495
Summary: We present an embedded ghost fluid method for numerical solutions of the compressible Navier Stokes (CNS) equations in arbitrary complex domains. A PDE multidimensional extrapolation approach is used to reconstruct the solution in the ghost fluid regions and imposing boundary conditions on the fluid-solid interface, coupled with a multi-dimensional algebraic interpolation for freshly cleared cells. The CNS equations are numerically solved by the second order multidimensional upwind method. Block-structured adaptive mesh refinement, implemented with the Chombo framework, is utilized to reduce the computational cost while keeping high resolution mesh around the embedded boundary and regions of high gradient solutions. The versatility of the method is demonstrated via several numerical examples, in both static and moving geometry, ranging from low Mach number nearly incompressible flows to supersonic flows. Our simulation results are extensively verified against other numerical results and validated against available experimental results where applicable. The significance and advantages of our implementation, which revolve around balancing between the solution accuracy and implementation difficulties, are briefly discussed as well.

MSC:
76M25 Other numerical methods (fluid mechanics) (MSC2010)
76N15 Gas dynamics, general
76J20 Supersonic flows
65D05 Numerical interpolation
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
Software:
Chombo
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Berthelsen, P.; Faltinsen, O., A local directional ghost cell approach for incompressible viscous flow problems with irregular boundaries, J. Comput. Phys., 227, 9, 4354-4397, (2008) · Zbl 1388.76199
[2] Verzicco, R.; Mohd-Yusof, J.; Orlandi, P.; Haworth, D., Large-eddy simulation in complex geometric configurations using boundary body forces, AIAA J., 38, 3, 427-433, (2000)
[3] Peskin, C., Flow patterns around heart valves: a numerical method, J. Comput. Phys., 10, 2, 252-271, (1972) · Zbl 0244.92002
[4] Mittal, R.; Iaccarino, G., Immersed boundary methods, Annu. Rev. Fluid Mech., 37, 239-261, (2005) · Zbl 1117.76049
[5] Peskin, C., Numerical analysis of blood flow in the heart, J. Comput. Phys., 25, 3, 220-252, (1977) · Zbl 0403.76100
[6] Peskin, C., Fluid dynamics of heart valves: experimental, theoretical, and computational methods, Annu. Rev. Fluid Mech., 14, 235-259, (1982)
[7] Goldstein, D.; Handler, R.; Sirovich, L., Modeling a no-slip flow boundary with an external force field, J. Comput. Phys., 105, 2, 354-366, (1993) · Zbl 0768.76049
[8] Saiki, E.; Biringen, S., Numerical simulation of a cylinder in uniform flow: application of a virtual boundary method, J. Comput. Phys., 123, 2, 450-465, (1996) · Zbl 0848.76052
[9] Lai, M.-C.; Peskin, C., An immersed boundary method with formal second-order accuracy and reduced numerical viscosity, J. Comput. Phys., 160, 2, 705-719, (2000) · Zbl 0954.76066
[10] Zou, J.-F.; Ren, A.-L.; Deng, J., Study on flow past two spheres in tandem arrangement using a local mesh refinement virtual boundary method, Int. J. Numer. Methods Fluids, 49, 5, 465-488, (2005) · Zbl 1079.76061
[11] Su, S.-W.; Lai, M.-C.; Lin, C.-A., An immersed boundary technique for simulating complex flows with rigid boundary, Comput. Fluids, 36, 2, 313-324, (2007) · Zbl 1177.76299
[12] Pan, D., An immersed boundary method for incompressible flows using volume of body function, Int. J. Numer. Methods Fluids, 50, 6, 733-750, (2006) · Zbl 1086.76055
[13] Mohd-Yusof, J., Combined immersed-boundary/B-spline methods for simulations of flow in complex geometries, (Annual Research Briefs, (1997), Center for Turbulence Research)
[14] Fadlun, E.; Verzicco, R.; Orlandi, P.; Mohd-Yusof, J., Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations, J. Comput. Phys., 161, 1, 35-60, (2000) · Zbl 0972.76073
[15] Verzicco, R.; Fatica, M.; Iaccarino, G.; Orlandi, P., Flow in an impeller-stirred tank using an immersed-boundary method, AIChE J., 50, 6, 1109-1118, (2004)
[16] Balaras, E., Modeling complex boundaries using an external force field on fixed Cartesian grids in large-eddy simulations, Comput. Fluids, 33, 3, 375-404, (2004) · Zbl 1088.76018
[17] Gilmanov, A.; Sotiropoulos, F.; Balaras, E., A general reconstruction algorithm for simulating flows with complex 3D immersed boundaries on Cartesian grids, J. Comput. Phys., 191, 2, 660-669, (2003) · Zbl 1134.76406
[18] Gilmanov, A.; Sotiropoulos, F., A hybrid Cartesian/immersed boundary method for simulating flows with 3D, geometrically complex, moving bodies, J. Comput. Phys., 207, 2, 457-492, (2005) · Zbl 1213.76135
[19] Yang, J.; Balaras, E., An embedded-boundary formulation for large-eddy simulation of turbulent flows interacting with moving boundaries, J. Comput. Phys., 215, 1, 12-40, (2006) · Zbl 1140.76355
[20] Majumdar, S.; Iaccarino, G.; Durbin, P., RANS solvers with adaptive structured boundary non-conforming grids, (Annual Research Briefs, (2001), Center for Turbulence Research, Stanford University), 353-466
[21] Tseng, Y.-H.; Ferziger, J., A ghost-cell immersed boundary method for flow in complex geometry, J. Comput. Phys., 192, 2, 593-623, (2003) · Zbl 1047.76575
[22] Franke, R., Scattered data interpolation: tests of some methods, Math. Comput., 38, 157, 181-200, (1982) · Zbl 0476.65005
[23] Gao, T.; Tseng, Y.-H.; Lu, X.-Y., An improved hybrid Cartesian/immersed boundary method for fluid-solid flows, Int. J. Numer. Methods Fluids, 55, 12, 1189-1211, (2007) · Zbl 1127.76045
[24] Iaccarino, G.; Verzicco, R., Immersed boundary technique for turbulent flow simulations, Appl. Mech. Rev., 56, 3, 331-347, (2003)
[25] Lundquist, K.; Chow, F.; Lundquist, J., An immersed boundary method enabling large-eddy simulations of flow over complex terrain in the WRF model, Mon. Weather Rev., 140, 12, 3936-3955, (2012)
[26] Aslam, T., A partial differential equation approach to multidimensional extrapolation, J. Comput. Phys., 193, 1, 349-355, (2004) · Zbl 1036.65002
[27] Ng, Y.; Min, C.; Gibou, F., An efficient fluid solid coupling algorithm for single-phase flows, J. Comput. Phys., 228, 23, 8807-8829, (2009) · Zbl 1245.76019
[28] Min, C.; Gibou, F., A second order accurate level set method on non-graded adaptive Cartesian grids, J. Comput. Phys., 225, 1, 300-321, (2007) · Zbl 1122.65077
[29] Chiu, P.; Lin, R.; Sheu, T. W., A differentially interpolated direct forcing immersed boundary method for predicting incompressible Navier Stokes equations in time-varying complex geometries, J. Comput. Phys., 229, 12, 4476-4500, (2010) · Zbl 1305.76073
[30] Uddin, H.; Kramer, R.; Pantano, C., A Cartesian-based embedded geometry technique with adaptive high-order finite differences for compressible flow around complex geometries, J. Comput. Phys., 262, 379-407, (2014) · Zbl 1349.76542
[31] Cummings, J.; Aivazis, M.; Samtaney, R.; Radovitzky, R.; Mauch, S.; Meiron, D., A virtual test facility for the simulation of dynamic response in materials, J. Supercomput., 23, 1, 39-50, (2002) · Zbl 0994.68536
[32] Arienti, M.; Hung, P.; Morano, E.; Shepherd, J., A level set approach to eulerian-Lagrangian coupling, J. Comput. Phys., 185, 1, 213-251, (2003) · Zbl 1047.76567
[33] Colella, P., Multidimensional upwind methods for hyperbolic conservation laws, J. Comput. Phys., 87, 1, 171-200, (1990) · Zbl 0694.65041
[34] Saltzman, J., An unsplit 3D upwind method for hyperbolic conservation laws, J. Comput. Phys., 115, 1, 153-168, (1994) · Zbl 0813.65111
[35] Colella, P.; Graves, D.; Ligocki, T.; Martin, D.; Van Straalen, B., AMR Godunov unsplit algorithm and implementation, (2008), Applied Numerical Algorithms Group, NERSC Division, Lawrence Berkeley National Laboratory, Tech. rep.
[36] Sethian, J., Level-set method and fast marching methods, (1996), Cambridge · Zbl 0852.65055
[37] Gibou, F.; Fedkiw, R., A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem, J. Comput. Phys., 202, 577-601, (2005) · Zbl 1061.65079
[38] M. Adams, P. Colella, D. Graves, J. Johnson, N. Keen, T. Ligocki, D. Martin, P. McCorquodale, D. Modiano, P. Schwartz, T. Sternberg, B.V. Straalen, Chombo software package for AMR applications design document.
[39] Berger, M.; Oliger, J., Adaptive mesh refinement for hyperbolic partial differential equations, J. Comput. Phys., 53, 3, 484-512, (1984) · Zbl 0536.65071
[40] Berger, M.; Colella, P., Local adaptive mesh refinement for shock hydrodynamics, J. Comput. Phys., 82, 1, 64-84, (1989) · Zbl 0665.76070
[41] Rendleman, C. A.; Beckner, V. E.; Lijewski, M.; Crutchfield, W.; Bell, J. B., Parallelization of structured, hierarchical adaptive mesh refinement algorithms, Comput. Vis. Sci., 3, 3, 147-157, (2000) · Zbl 0971.65089
[42] Al-Marouf, M.; Rapaka, N. R.; Samtaney, R., An embedded-boundary AMR method for compressible flow in moving complex geometry, (28th International Conference on Parallel Computational Fluid Dynamics, Kobe, Japan, May 9-12, (2016)), submitted for publication
[43] Choi, H.; Moin, P., Grid-point requirements for large-eddy simulation: Chapman’s estimates revisited, Phys. Fluids, 24, (2012)
[44] Sagaut, P., Large-eddy simulation for incompressible flows: an introduction, (2006), Springer Science & Business Media · Zbl 1091.76001
[45] Williamson, C., Vortex dynamics in the cylinder wake, Annu. Rev. Fluid Mech., 28, 1, 477-539, (1996)
[46] Hartmann, D.; Meinke, M.; Schröder, W., A strictly conservative Cartesian cut-cell method for compressible viscous flows on adaptive grids, Comput. Methods Appl. Mech. Eng., 200, 9-12, 1038-1052, (2011) · Zbl 1225.76211
[47] Provansal, M.; Mathis, C.; Boyer, L., Bénard-von Kármán instability: transient and forced regimes, J. Fluid Mech., 182, 1-22, (1987) · Zbl 0641.76046
[48] Tritton, D., Experiments on the flow past a circular cylinder at low Reynolds numbers, J. Fluid Mech., 6, 4, 547-567, (1959) · Zbl 0092.19502
[49] Le, D.; Khoo, B.; Peraire, J., An immersed interface method for viscous incompressible flows involving rigid and flexible boundaries, J. Comput. Phys., 220, 1, 109-138, (2006) · Zbl 1158.74349
[50] Dennis, S.; Chang, G.-Z., Numerical solutions for steady flow past a circular cylinder at Reynolds numbers up to 100, J. Fluid Mech., 42, 471-489, (1970) · Zbl 0193.26202
[51] Coutanceau, M.; Bouard, R., Experimental determination of the main features of the viscous flow in the wake of a circular cylinder in uniform translation. I. steady flow, J. Fluid Mech., 79, 231-256, (1977)
[52] Gautier, R.; Biau, D.; Lamballais, E., A reference solution of the flow over a circular cylinder at \(R e = 40\), Comput. Fluids, 75, 103-111, (2013) · Zbl 1277.76067
[53] Taira, K.; Colonius, T., The immersed boundary method: a projection approach, J. Comput. Phys., 225, 2, 2118-2137, (2007) · Zbl 1343.76027
[54] Brehm, C.; Hader, C.; Fasel, H., A locally stabilized immersed boundary method for the compressible Navier Stokes equations, J. Comput. Phys., 295, 475-504, (2015) · Zbl 1349.76437
[55] Park, J.; Kwon, K.; Choi, H., Numerical solutions of flow past a circular cylinder at Reynolds numbers up to 160, KSME Int. J., 12, 6, 1200-1205, (1998)
[56] Stålberg, E.; Brüger, A.; Lötstedt, P.; Johansson, A.; Henningson, D., High order accurate solution of flow past a circular cylinder, J. Sci. Comput., 27, 1-3, 431-441, (2006) · Zbl 1115.76058
[57] Berger, E.; Wille, R., Periodic flow phenomena, Annu. Rev. Fluid Mech., 4, 1, 313-340, (1972)
[58] White, F., Viscous fluid flow, (2006), McGraw-Hill New York
[59] Russell, D.; Wang, Z., A Cartesian grid method for modeling multiple moving objects in 2D incompressible viscous flow, J. Comput. Phys., 191, 1, 177-205, (2003) · Zbl 1160.76389
[60] Liu, C.; Zheng, X.; Sung, C., Preconditioned multigrid methods for unsteady incompressible flows, J. Comput. Phys., 139, 1, 35-57, (1998) · Zbl 0908.76064
[61] Kunz, P.; Kroo, I., Analysis, design and testing of airfoils for use at ultra-low Reynolds numbers, (Proceedings of a Workshop on Fixed and Flapping Flight at Low Reynolds Numbers, Notre Dame, (2000))
[62] Mittal, R.; Dong, H.; Bozkurttas, M.; Najjar, F. M.; Vargas, A.; von Loebbecke, A., A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries, J. Comput. Phys., 227, 10, 4825-4852, (2008) · Zbl 1388.76263
[63] Zhang, W.; Samtaney, R., A direct numerical simulation investigation of the synthetic jet frequency effects on separation control of low-re flow past an airfoil, Phys. Fluids, 27, 5, (2015)
[64] Ghias, R.; Mittal, R.; Dong, H., A sharp interface immersed boundary method for compressible viscous flows, J. Comput. Phys., 225, 1, 528-553, (2007) · Zbl 1343.76043
[65] Mittal, S.; Balachandar, R., On the inclusion of three dimensional effects in simulations of two-dimensional bluff body wake flows, (1997)
[66] Rajani, B.; Kandasamy, A.; Majumdar, S., Numerical simulation of laminar flow past a circular cylinder, Appl. Math. Model., 33, 3, 1228-1247, (2009) · Zbl 1168.76305
[67] Kravchenko, A.; Moin, P.; Shariff, K., B-spline method and zonal grids for simulations of complex turbulent flows, J. Comput. Phys., 151, 2, 757-789, (1999) · Zbl 0942.76058
[68] Nam, J.; Lien, F., A ghost-cell immersed boundary method for large-eddy simulations of compressible turbulent flows, Int. J. Comput. Fluid Dyn., 28, 1-2, 41-55, (2014)
[69] Johnson, T.; Patel, V., Flow past a sphere up to a Reynolds number of 300, J. Fluid Mech., 378, 19-70, (1999)
[70] Mittal, R., A Fourier-Chebyshev spectral collocation method for simulating flow past spheres and spheroids, Int. J. Numer. Methods Fluids, 30, 7, 921-937, (1999) · Zbl 0957.76060
[71] Kim, J.; Kim, D.; Choi, H., An immersed-boundary finite-volume method for simulations of flow in complex geometries, J. Comput. Phys., 171, 1, 132-150, (2001) · Zbl 1057.76039
[72] Dütsch, H.; Durst, F.; Becker, S.; Lienhart, H., Low-Reynolds-number flow around an oscillating circular cylinder at low Keulegan-Carpenter numbers, J. Fluid Mech., 360, 249-271, (1998) · Zbl 0922.76024
[73] Guilmineau, E.; Queutey, P., A numerical simulation of vortex shedding from an oscillating circular cylinder, J. Fluids Struct., 16, 6, 773-794, (2002)
[74] Koochesfahani, M., Vortical patterns in the wake of an oscillating airfoil, AIAA J., 27, 9, 1200-1205, (1989)
[75] Ramamurti, R.; Sandberg, W., Simulation of flow about flapping airfoils using finite element incompressible flow solver, AIAA J., 39, 2, 253-260, (2001)
[76] Young, J.; Lai, J., Oscillation frequency and amplitude effects on the wake of a plunging airfoil, AIAA J., 42, 10, 2042-2052, (2004)
[77] Lai, J.; Platzer, M., Jet characteristics of a plunging airfoil, AIAA J., 37, 12, 1529-1537, (1999)
[78] LeVeque, R., Finite volume methods for hyperbolic problems, vol. 31, (2002), Cambridge University Press · Zbl 1010.65040
[79] Glaz, H.; Colella, P.; Glass, I.; Deschambault, R., A numerical study of oblique shock-wave reflections with experimental comparisons, (Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 398, (1985), The Royal Society), 117-140
[80] Graves, D.; Colella, P.; Modiano, D.; Johnson, J.; Sjogreen, B.; Gao, X., A Cartesian grid embedded boundary method for the compressible Navier Stokes equations, Commun. Appl. Math. Comput. Sci., 8, 1, 99-122, (2013) · Zbl 1282.76006
[81] de Tullio, M.; Palma, P.; Iaccarino, G.; Pascazio, G.; Napolitano, M., An immersed boundary method for compressible flows using local grid refinement, J. Comput. Phys., 225, 2, 2098-2117, (2007) · Zbl 1118.76043
[82] Bashkin, V.; Vaganov, A.; Egorov, I.; Ivanov, D.; Ignatova, G., Comparison of calculated and experimental data on supersonic flow past a circular cylinder, Fluid Dyn., 37, 3, 473-483, (2002) · Zbl 1161.76321
[83] Nam, J.; Lien, F., Assessment of ghost-cell based cut-cell method for large-eddy simulations of compressible flows at high Reynolds number, Int. J. Heat Fluid Flow, 53, 1-14, (2015)
[84] Bailey, A.; Hiatt, J., Sphere drag coefficients for a broad range of Mach and Reynolds numbers, AIAA J., 10, 11, 1436-1440, (1972)
[85] Grétarsson, J.; Kwatra, N.; Fedkiw, R., Numerically stable fluid-structure interactions between compressible flow and solid structures, J. Comput. Phys., 230, 8, 3062-3084, (2011) · Zbl 1316.76070
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.