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An efficient semi-implicit finite volume method for axially symmetric compressible flows in compliant tubes. (English) Zbl 1326.76070
Summary: A new efficient semi-implicit finite volume method for the simulation of weakly compressible, axially symmetric flows in compliant tubes is presented. The fluid is assumed to be barotropic and a simple cavitation model is also included in the equation of state in order to model phase transition when the fluid pressure drops below the vapor pressure. The discretized flow equations lead to a mildly nonlinear system of equations that is efficiently solved with a nested Newton technique. The new numerical method has to obey only a mild CFL condition based on the flow velocity and not on the sound speed, leading to large time steps that can be used. The scheme behaves well in the presence of shock waves and phase transition, as well as in the incompressible limit. In the present approach, the radial velocity profiles and therefore the wall friction coefficient are directly computed from first principles. In the compressible regime, the new method is carefully validated against quasi-exact solutions of the Riemann problem, while it is validated against the exact solution found by Womersley for an oscillatory flow in a rigid tube in the incompressible regime.

76M12 Finite volume methods applied to problems in fluid mechanics
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
76N15 Gas dynamics (general theory)
Full Text: DOI
[1] Abgrall, R.; Karni, S., A comment on the computation of non-conservative products, J. Comput. Phys., 229, 2759-2763, (2010) · Zbl 1188.65134
[2] Andrianov, N.; Warnecke, G., On the solution to the Riemann problem for the compressible duct flow, SIAM J. Appl. Math., 64, 878-901, (2004) · Zbl 1065.35191
[3] Beck, M.; Iben, U.; Mittwollen, N.; Iben, H.-K.; Munz, C.-D., On the solution of conservation equations in cavitated hydraulic pipelines, (3rd International Symposium on Computational Technologies for Fluid/Thermal/Chemical Systems with Industrial Applications, 2001 ASME Pressure Vessels and Piping Conference, July 22-26, Hyatt Regency, Atlanta, Georgia, USA, (2001), ASME), 87-98
[4] Berg, A.; Iben, U.; Meister, A.; Schmidt, J., Modeling and simulation of cavitation in hydraulic pipelines based on the thermodynamic and caloric properties of liquid and steam, Shock Waves, 14, 111-121, (2005) · Zbl 1178.76350
[5] Berry, R.; Saurel, R.; LeMetayer, O., The discrete equation method (DEM) for fully compressible, two-phase flows in ducts of spatially varying cross-section, Nucl. Eng. Des., 240, 3797-3818, (2010)
[6] Brugnano, L.; Casulli, V., Iterative solution of piecewise linear systems, SIAM J. Sci. Comput., 30, 463-472, (2007) · Zbl 1155.90457
[7] Brugnano, L.; Casulli, V., Iterative solution of piecewise linear systems and applications to flows in porous media, SIAM J. Sci. Comput., 31, 1858-1873, (2009) · Zbl 1190.90233
[8] Brugnano, L.; Sestini, A., Iterative solution of piecewise linear systems for the numerical solution of obstacle problems, J. Numer. Anal. Ind. Appl. Math., 6, 67-82, (2012) · Zbl 1432.65079
[9] Castro, M. J.; Gallardo, J. M.; Parés, C., High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. applications to shallow-water systems, Math. Comput., 75, 1103-1134, (2006) · Zbl 1096.65082
[10] Castro, M. J.; LeFloch, P. G.; Muñoz-Ruiz, M. L.; Parés, C., Why many theories of shock waves are necessary: convergence error in formally path-consistent schemes, J. Comput. Phys., 227, 8107-8129, (2008) · Zbl 1176.76084
[11] Casulli, V., A high-resolution wetting and drying algorithm for free-surface hydrodynamics, Int. J. Numer. Methods Fluids, 60, 391-408, (2009) · Zbl 1161.76034
[12] Casulli, V., A semi-implicit numerical method for the free-surface Navier-Stokes equations, Int. J. Numer. Methods Fluids, 74, 605-622, (2014)
[13] Casulli, V.; Cattani, E., Stability, accuracy and efficiency of a semi-implicit method for three-dimensional shallow water flow, Comput. Math. Appl., 27, 99-112, (1994) · Zbl 0796.76052
[14] Casulli, V.; Dumbser, M.; Toro, E. F., Semi-implicit numerical modeling of axially symmetric flows in compliant arterial systems, Int. J. Numer. Methods Biomed. Eng., 28, 257-272, (2012) · Zbl 1242.92019
[15] Casulli, V.; Stelling, G., A semi-implicit numerical model for urban drainage systems, Int. J. Numer. Methods Fluids, 73, 600-614, (2013)
[16] Casulli, V.; Zanolli, P., A nested Newton-type algorithm for finite volume methods solving Richards’ equation in mixed form, SIAM J. Sci. Comput., 32, 2255-2273, (2009) · Zbl 1410.76209
[17] Casulli, V.; Zanolli, P., Iterative solutions of mildly nonlinear systems, J. Comput. Appl. Math., 236, 3937-3947, (2012) · Zbl 1251.65073
[18] Clain, S.; Rochette, D., First- and second-order finite volume methods for the one-dimensional nonconservative Euler system, J. Comput. Phys., 228, 8214-8248, (2009) · Zbl 1422.76171
[19] Correia, J.; LeFloch, P.; Thanh, M., Hyperbolic conservation laws with Lipschitz continuous flux-functions. the Riemann problem, Bol. Soc. Bras. Mat., 32, 271-301, (2001) · Zbl 1009.35053
[20] Dumbser, M.; Castro, M.; Parés, C.; Toro, E., ADER schemes on unstructured meshes for non-conservative hyperbolic systems: applications to geophysical flows, Comput. Fluids, 38, 1731-1748, (2009) · Zbl 1177.76222
[21] Dumbser, M.; Casulli, V., A staggered semi-implicit spectral discontinuous Galerkin scheme for the shallow water equations, Appl. Math. Comput., 219, 15, 8057-8077, (2013) · Zbl 1366.76050
[22] Dumbser, M.; Iben, U.; Munz, C., Efficient implementation of high order unstructured WENO schemes for cavitating flows, Comput. Fluids, 86, 141-168, (2013) · Zbl 1290.76098
[23] Dumbser, M.; Toro, E. F., A simple extension of the osher Riemann solver to non-conservative hyperbolic systems, J. Sci. Comput., 48, 70-88, (2011) · Zbl 1220.65110
[24] Fambri, F.; Dumbser, M.; Casulli, V., An efficient semi-implicit method for three-dimensional non-hydrostatic flows in compliant arterial vessels, Int. J. Numer. Methods Biomed. Eng., 30, 1170-1198, (2014)
[25] Fung, Y., Biomechanics: circulation, (2010), Springer New York, Berlin, Heidelberg
[26] Glaister, P., Conservative upwind difference schemes for compressible flows in a duct, Comput. Math. Appl., 56, 1787-1796, (2008) · Zbl 1152.76457
[27] Han, E.; Hantke, M.; Warnecke, G., Exact Riemann solutions to compressible Euler equations in ducts with discontinuous cross-section, J. Hyperbolic Differ. Equ., 9, 403-449, (2012) · Zbl 1263.35178
[28] Helluy, P.; Hérard, J.; Mathis, H., A well-balanced approximate Riemann solver for compressible flows in variable cross-section ducts, J. Comput. Appl. Math., 236, 1976-1992, (2012) · Zbl 1427.76204
[29] Iben, U.; Morozov, A.; Winklhofer, E.; Wolf, F., Laser pulse interferometry applied to high-pressure fluid flow in micro channels, Exp. Fluids, 50, 597-611, (2011)
[30] Iben, U.; Wrona, F.; Munz, C.; Beck, M., Cavitation in hydraulic tools based on thermodynamic properties of liquid and gas, J. Fluids Eng., 124, 1011-1017, (2002)
[31] Liu, T., Transonic gas flow in a duct of varying area, Arch. Ration. Mech. Anal., 80, 1-18, (1982) · Zbl 0503.76076
[32] Maso, G. D.; LeFloch, P. G.; Murat, F., Definition and weak stability of nonconservative products, J. Math. Pures Appl., 74, 483-548, (1995) · Zbl 0853.35068
[33] Müller, S.; Voss, A., The Riemann problem for the Euler equations with nonconvex and nonsmooth equation of state: construction of wave curves, SIAM J. Sci. Comput., 28, 651-681, (2006) · Zbl 1114.35127
[34] Parés, C., Numerical methods for nonconservative hyperbolic systems: a theoretical framework, SIAM J. Numer. Anal., 44, 300-321, (2006) · Zbl 1130.65089
[35] Rochette, D.; Clain, S.; Bussière, W., Unsteady compressible flow in ducts with varying cross-section: comparison between the nonconservative Euler system and the axisymmetric flow model, Comput. Fluids, 53, 53-78, (2012) · Zbl 1271.76294
[36] Sherwin, S.; Formaggia, L.; Peirò, J.; Franke, V., Computational modelling of 1D blood flow with variable mechanical properties and its application to the simulation of wave propagation in the human arterial system, Int. J. Numer. Methods Fluids, 43, 673-700, (2003) · Zbl 1032.76729
[37] Tavelli, M.; Dumbser, M., A high order semi-implicit discontinuous Galerkin method for the two dimensional shallow water equations on staggered unstructured meshes, Appl. Math. Comput., 234, 623-644, (2014) · Zbl 1298.76120
[38] Tavelli, M.; Dumbser, M.; Casulli, V., High resolution methods for scalar transport problems in compliant systems of arteries, Appl. Numer. Math., 74, 62-82, (2013) · Zbl 1302.76134
[39] Thanh, M., The Riemann problem for a nonisentropic fluid in a nozzle with discontinuous cross-sectional area, SIAM J. Appl. Math., 69, 1501-1519, (2009) · Zbl 1181.35146
[40] Toro, E. F., Riemann solvers and numerical methods for fluid dynamics, (2009), Springer · Zbl 1227.76006
[41] Womersley, J., Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known, Am. J. Physiol., 127, 553-563, (1955)
[42] Wrona, F., Simulation von kavitierenden strömungen in hochdrucksystemen, (2005), Universität Stuttgart, Institut für Aerodynamik und Gasdynamik, Ph.D. thesis
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