zbMATH — the first resource for mathematics

Techniques for improving monotonicity in a fourth-order finite-volume algorithm solving shocks and detonations. (English) Zbl 1440.76099
Summary: Techniques are proposed to reduce numerical oscillations in a fourth-order, finite-volume algorithm for solving thermally-perfect, reacting fluid flows with strong discontinuities, such as shock or detonation waves. These additional mechanisms have proven necessary for multispecies flows solved at fourth-order accuracy, and contribute towards bounding the variation of the solution in the vicinity of strong discontinuities. There, oscillations can form due to strong gradients in the flow and may be further intensified by numerical procedures introduced to treat the thermally-perfect thermodynamic system and physical constraints on species mass fractions. The new techniques are designed to respect the conservative property of the base algorithm, retain fourth-order accuracy of the solution in regions of smooth flow, and cooperate with the high-order piecewise parabolic method limiter. Extensive numerical tests, ranging from multispecies mixing flows to reacting flows with detonations, are performed to verify that the new techniques meet the design criteria while effectively suppressing oscillations. The proposed techniques are applied to solve the Shu-Osher and double Mach reflection problems and a set of oblique detonation wave problems. The results demonstrate the effectiveness and robustness of the algorithm.
76M12 Finite volume methods applied to problems in fluid mechanics
76V05 Reaction effects in flows
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
76T30 Three or more component flows
76L05 Shock waves and blast waves in fluid mechanics
Full Text: DOI
[1] Wang, Z. J.; Fidkowski, K.; Abgrall, R.; Bassi, F.; Caraeni, D.; Cary, A.; Deconinck, H.; Hartmann, R.; Hillewaert, K.; Huynh, H. T.; Kroll, N.; May, G.; Persson, P.-O.; van Leer, B.; Visbal, M., High-order CFD methods: current status and perspective, Int. J. Numer. Methods Fluids, 72, 8, 811-845 (2013)
[2] Wang, Z. J., High-order computational fluid dynamics tools for aircraft design, Philos. Trans. R. Soc. A, Math. Phys. Eng. Sci., 372, 2022 (2014)
[3] (Jan 2018), 5th international workshop on high order CFD methods
[4] Colella, P.; Sekora, M., A limiter for PPM that preserves accuracy at smooth extrema, J. Comput. Phys., 227, 15, 7069-7076 (2008) · Zbl 1152.65090
[5] McCorquodale, P.; Colella, P., A high-order finite-volume method for conservation laws on locally refined grids, Commun. Appl. Math. Comput. Sci., 6, 1, 1-25 (2011) · Zbl 1252.65163
[6] Henrick, A. K.; Aslam, T. D.; Powers, J. M., Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points, J. Comput. Phys., 207, 2, 542-567 (2005) · Zbl 1072.65114
[7] Henrick, A. K.; Aslam, T. D.; Powers, J. M., Simulations of pulsating one-dimensional detonations with true fifth order accuracy, J. Comput. Phys., 213, 1, 311-329 (2006) · Zbl 1146.76669
[8] Yoon, S.-H.; Kim, C.; Kim, K.-H., Multi-dimensional limiting process for three-dimensional flow physics analyses, J. Comput. Phys., 227, 12, 6001-6043 (2008) · Zbl 1388.76218
[9] Park, J. S.; Kim, C., Multi-dimensional limiting process for finite volume methods on unstructured grids, Comput. Fluids, 65, 8-24 (2012) · Zbl 1365.76167
[10] Houim, R. W.; Kuo, K. K., A low-dissipation and time-accurate method for compressible multi-component flow with variable specific heat ratios, J. Comput. Phys., 230, 23, 8527-8553 (2011) · Zbl 1337.76054
[11] Karni, S., Hybrid multifluid algorithms, SIAM J. Sci. Comput., 17, 5, 1019-1039 (1996) · Zbl 0860.76056
[12] Abgrall, R.; Karni, S., Computations of compressible multifluids, J. Comput. Phys., 169, 2, 594-623 (2001) · Zbl 1033.76029
[13] Ma, P. C.; Lv, Y.; Ihme, M., An entropy-stable hybrid scheme for simulations of transcritical real-fluid flows, J. Comput. Phys., 340, 330-357 (2017) · Zbl 1376.76033
[14] Billet, G.; Abgrall, R., An adaptive shock-capturing algorithm for solving unsteady reactive flows, Comput. Fluids, 32, 10, 1473-1495 (2003) · Zbl 1033.76031
[15] Kim, K. H.; Kim, C., Accurate, efficient and monotonic numerical methods for multi-dimensional compressible flows: Part II: Multi-dimensional limiting process, J. Comput. Phys., 208, 2, 570-615 (2005) · Zbl 1329.76265
[16] Taylor, E. M.; Wu, M.; Martin, M. P., Optimization of nonlinear error for weighted essentially non-oscillatory methods in direct numerical simulations of compressible turbulence, J. Comput. Phys., 223, 1, 384-397 (2007) · Zbl 1165.76350
[17] Thornber, B.; Mosedale, A.; Drikakis, D.; Youngs, D.; Williams, R. J.R., An improved reconstruction method for compressible flows with low mach number features, J. Comput. Phys., 227, 10, 4873-4894 (2008) · Zbl 1388.76188
[18] Kang, H.-M.; Kim, K. H.; Lee, D.-H., A new approach of a limiting process for multi-dimensional flows, J. Comput. Phys., 229, 19, 7102-7128 (2010) · Zbl 1425.76185
[19] Schmitt, T.; Selle, L.; Ruiz, A.; Cuenot, B., Large-eddy simulation of supercritical-pressure round jets, AIAA J., 48, 9, 2133-2144 (2010)
[20] Terashima, H.; Koshi, M., Approach for simulating gas-liquid-like flows under supercritical pressures using a high-order central differencing scheme, J. Comput. Phys., 231, 20, 6907-6923 (2012) · Zbl 1351.76291
[21] Guzik, S. M.; Gao, X.; Owen, L. D.; McCorquodale, P.; Colella, P., A freestream-preserving fourth-order finite-volume method in mapped coordinates with adaptive-mesh refinement, Comput. Fluids, 123, 202-217 (2015) · Zbl 1390.65091
[22] Guzik, S. M.; Gao, X.; Olschanowsky, C., A high-performance finite-volume algorithm for solving partial differential equations governing compressible viscous flows on structured grids, Comput. Math. Appl., 72, 2098-2118 (2016) · Zbl 1368.76040
[23] Gao, X.; Owen, L. D.; Guzik, S. M.J., A parallel adaptive numerical method with generalized curvilinear coordinate transformation for compressible Navier-Stokes equations, Int. J. Numer. Methods Fluids, 82, 664-688 (2016)
[24] Owen, L. D.; Guzik, S. M.; Gao, X., A high-order adaptive algorithm for multispecies gaseous flows on mapped domains, Comput. Fluids, 170, 249-260 (2018) · Zbl 1410.76256
[25] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, II, (Upwind and High-Resolution Schemes (1989), Springer), 328-374
[26] Gordon, S.; McBride, B. J., Computer Program for Calculation of Complex Chemical Equilibrium Compositions and Applications I. Analysis, Reference Publication, vol. 1311 (1994), NASA
[27] McBride, B. J.; Gordon, S., Computer Program for Calculation of Complex Chemical Equilibrium Compositions and Applications II. Users Manual and Program Description, Reference Publication, vol. 1311 (1996), NASA
[28] Adams, M.; Colella, P.; Graves, D. T.; Johnson, J. N.; Johansen, H. S.; Keen, N. D.; Ligocki, T. J.; Martin, D. F.; McCorquodale, P. W.; Modiano, D.; Schwartz, P. O.; Sternberg, T. D.; Van Straalen, B. (2014), Lawrence Berkeley National Laboratory, Chombo Software Package for AMR Applications - Design Document, Tech. Rep. LBNL-6616E
[29] Lomax, H.; Pulliam, T. H.; Zingg, D. W., Fundamentals of Computational Fluid Dynamics, Scientific Computation (2001), Springer-Verlag Berlin Heidelberg · Zbl 0970.76002
[30] Hirsch, C., Numerical Computation of Internal and External Flows: The Fundamentals of Computational Fluid Dynamics (2007), Elsevier
[31] Riviere, B., Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation (2008), SIAM · Zbl 1153.65112
[32] Pulliam, T. H.; Zingg, D. W., Fundamental Algorithms in Computational Fluid Dynamics, Scientific Computation (2014), Springer International Publishing: Springer International Publishing New Delhi, India · Zbl 1302.76004
[33] Lambert, J. D., Numerical Methods for Ordinary Differential Systems (1991), John Wiley & Sons: John Wiley & Sons New York · Zbl 0745.65049
[34] Chaplin, C.; Colella, P., A single-stage flux-corrected transport algorithm for high-order finite-volume methods, Commun. Appl. Math. Comput. Sci., 12, 1, 1-24 (2017) · Zbl 1362.65091
[35] Woodward, P. R.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54, 1, 115-173 (1984) · Zbl 0573.76057
[36] Colella, P., Multidimensional upwind methods for hyperbolic conservation laws, J. Comput. Phys., 87, 1, 171-200 (1990) · Zbl 0694.65041
[37] Brent, R. P., An algorithm with guaranteed convergence for finding a zero of a function, Comput. J., 14, 4, 422-425 (1971) · Zbl 0231.65046
[38] Owen, L. D.; Guzik, S. M.; Gao, X., A fourth-order finite-volume algorithm for compressible flow with chemical reactions on mapped grids, (23rd AIAA Computational Fluid Dynamics Conference. 23rd AIAA Computational Fluid Dynamics Conference, AIAA 2017-4498, AIAA AVIATION Forum (2017))
[39] Colella, P.; Woodward, P. R., The piecewise parabolic method (PPM) for gas-dynamical simulations, J. Comput. Phys., 54, 1, 174-201 (1984) · Zbl 0531.76082
[40] Berger, M. J.; Colella, P., Local adaptive mesh refinement for shock hydrodynamics, J. Comput. Phys., 82, 1, 64-84 (1989) · Zbl 0665.76070
[41] PeleC, version 00 (2017)
[42] Motheau, E.; Wakefield, J., Investigation of finite-volume methods to capture shocks and turbulence spectra in compressible flows, Commun. Appl. Math. Comput. Sci. (2020), forthcoming, preprint version at
[43] Peer, A. A.I.; Gopaul, A.; Dauhoo, M. Z.; Bhuruth, M., A new fourth-order non-oscillatory central scheme for hyperbolic conservation laws, Appl. Numer. Math., 58, 5, 674-688 (2008) · Zbl 1138.65075
[44] Guzik, S. M.; McCorquodale, P.; Colella, P., A freestream-preserving high-order finite-volume method for mapped grids with adaptive-mesh refinement, (50th AIAA Aerospace Sciences Meeting. 50th AIAA Aerospace Sciences Meeting, AIAA 2012-0574 (2012), AIAA)
[45] Figueria da Silva, L. F.; Deshaies, B., Stabilization of an oblique detonation wave by a wedge: a parametric numerical study, Combust. Flame, 121, 1, 152-166 (2000)
[46] Billet, G., Improvement of convective concentration fluxes in a one step reactive flow solver, J. Comput. Phys., 204, 1, 319-352 (2005) · Zbl 1143.76474
[47] Woodward, P. R., Piecewise-Parabolic Methods for Astrophysical Fluid Dynamics, 245-326 (1986), Springer: Springer Netherlands, Dordrecht
[48] Colella, P.; Dorr, M. R.; Hittinger, J. A.F.; Martin, D. F., High-order finite-volume methods in mapped coordinates, J. Comput. Phys., 230, 2952-2976 (2011) · Zbl 1218.65119
[49] Li, C.; Kailasanath, K.; Oran, E. S., Detonation structures behind oblique shocks, Phys. Fluids, 6, 4, 1600-1611 (1994) · Zbl 0825.76396
[50] Thaker, A. A.; Chelliah, H. K., Numerical prediction of oblique detonation wave structures using detailed and reduced reaction mechanisms, Combust. Theory Model., 1, 4, 347-376 (1997) · Zbl 1140.80410
[51] Wang, T.; Zhang, Y.; Teng, H.; Jiang, Z.; Ng, H. D., Numerical study of oblique detonation wave initiation in a stoichiometric hydrogen-air mixture, Phys. Fluids, 27, 9, Article 096101 pp. (2015)
[52] Kee, R. J.; Ruply, R. M.; Meeks, E.; Miller, J. A., Chemkin-III: a FORTRAN chemical kinetics package for the analysis of gas-phase chemical and plasma kinetics (1996), Sandia National Laboratories: Sandia National Laboratories Livermore, Technical Report SAND96-8216
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.