Realizable high-order finite-volume schemes for quadrature-based moment methods. (English) Zbl 1419.76465
Summary: Dilute gas-particle flows can be described by a kinetic equation containing terms for spatial transport, gravity, fluid drag and particle-particle collisions. However, direct numerical solution of kinetic equations is often infeasible because of the large number of independent variables. An alternative is to reformulate the problem in terms of the moments of the velocity distribution. Recently, a quadrature-based moment method was derived for approximating solutions to kinetic equations. The success of the new method is based on a moment-inversion algorithm that is used to calculate non-negative weights and abscissas from the moments. The moment-inversion algorithm does not work if the moments are non-realizable, which might lead to negative weights. In [ibid. 227, No. 4, 2514–2539 (2008; Zbl 1261.76027)], O. Desjardins et al. showed that realizability is guaranteed only with the 1st-order finite-volume scheme that has an inherent problem of excessive numerical diffusion. The use of high-order finite-volume schemes may lead to non-realizable moments. In the present work, realizability of the finite-volume schemes in both space and time is discussed for the 1st time. A generalized idea for developing realizable high-order finite-volume schemes for quadrature-based moment methods is presented. These finite-volume schemes give remarkable improvement in the solutions for a certain class of problems. It is also shown that the standard Runge-Kutta time-integration schemes do not guarantee realizability. However, realizability can be guaranteed if strong stability-preserving (SSP) Runge-Kutta schemes are used. Numerical results are presented on both Cartesian and triangular meshes.
MSC:
| 76M12 | Finite volume methods applied to problems in fluid mechanics |
| 76T15 | Dusty-gas two-phase flows |
Keywords:
kinetic equation; quadrature method of moments; number density function; realizablility; finite-volume scheme; unstructured gridsCitations:
Zbl 1261.76027References:
| [1] | T.J. Barth, D.C. Jespersen, The design and application of upwind schemes on unstructured meshes, AIAA 89-0366, 1989.; T.J. Barth, D.C. Jespersen, The design and application of upwind schemes on unstructured meshes, AIAA 89-0366, 1989. |
| [2] | T.J. Barth, P.O. Frederickson, High-order solution of the Euler equations on unstructured grids using quadratic reconstruction, AIAA 90-0013, 1990.; T.J. Barth, P.O. Frederickson, High-order solution of the Euler equations on unstructured grids using quadratic reconstruction, AIAA 90-0013, 1990. |
| [3] | Benamou, J.-D., Big ray tracing: multivalued travel time field computation using viscosity solutions of the Eikonal equation, Journal of Computational Physics, 128, 463-474 (1996) · Zbl 0860.65052 |
| [4] | Beylich, A. E., Solving the kinetic equation for all Knudsen numbers, Physics of Fluids, 12, 444-465 (2000) · Zbl 1149.76318 |
| [5] | Bhatnagar, P. L.; Gross, E. P.; Krook, M., A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Physical Reviews, 94, 511-525 (1954) · Zbl 0055.23609 |
| [6] | Bird, G. A., Molecular Gas Dynamics and the Direct Simulation of Gas Flows (1994), Clarendon Press: Clarendon Press Oxford · Zbl 0709.76511 |
| [7] | Bouchut, F.; Jin, S.; Li, X. T., Numerical approximations of pressureless gas and isothermal gas dynamics, SIAM Journal of Numerical Analysis, 41, 135-158 (2003) · Zbl 1060.76080 |
| [8] | Carrillo, J. A.; Majorana, A.; Vecil, F., A semi-Lagrangian deterministic solver for the semiconductor Boltzmann-Poisson system, Communications in Computational Physics, 2, 1027-1054 (2007) · Zbl 1164.82326 |
| [9] | Cercignani, C., The Boltzmann Equation and Its Applications (1988), Springer: Springer New York · Zbl 0646.76001 |
| [10] | Chapman, S.; Cowling, T. G., The Mathematical Theory of Nonuniform Gases (1970), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0098.39702 |
| [11] | Chu, C. K., Kinetic-theoretic description of the formation of a shock wave, Physics of Fluids, 8, 12-22 (1965) |
| [12] | Collela, P., The piecewise parabolic method for gas-dynamical simulations, Journal of Computational Physics, 54, 174-201 (1984) · Zbl 0531.76082 |
| [13] | S. Deshpande, A second order accurate, kinetic theory based, method for inviscid compressible flows, Technical Report NASA Langley 2613, 1986.; S. Deshpande, A second order accurate, kinetic theory based, method for inviscid compressible flows, Technical Report NASA Langley 2613, 1986. |
| [14] | Desjardins, O.; Fox, R. O.; Villedieu, P., A quadrature-based moment method for dilute fluid-particle flows, Journal of Computational Physics, 227, 2514-2539 (2008) · Zbl 1261.76027 |
| [15] | Engquist, B.; Runborg, O., Multiphase computations in geometrical optics, Journal of Computational and Applied Mathematics, 74, 175-192 (1996) · Zbl 0947.78001 |
| [16] | Enwald, H.; Peirano, E.; Almstedt, A. E., Eulerian two-phase flow theory applied to fluidization, International Journal of Multiphase Flow, 22, 21-66 (1996) · Zbl 1135.76409 |
| [17] | Fox, R. O., A quadrature-based third-order moment method for dilute gas-particle flows, Journal of Computational Physics, 227, 6313-6350 (2008) · Zbl 1388.76247 |
| [18] | Fox, R. O., Higher-order quadrature-based moment methods for kinetic equations, Journal of Computational Physics, 228, 7771-7791 (2009) · Zbl 1391.82049 |
| [19] | Fox, R. O., Optimal moment sets for multivariate direct quadrature method of moments, Industrial and Engineering Chemistry Research, 48, 9686-9696 (2009) |
| [20] | Godunov, S. K., Finite difference methods for the computation of discontinuous solutions of the equations of fluid dynamics, Math. Sbornik, 47, 271-306 (1959) · Zbl 0171.46204 |
| [21] | Gosse, L., Using K-branch entropy solutions for multivalued geometric optics computations, Journal of Computational Physics, 180, 155-182 (2002) · Zbl 0999.78003 |
| [22] | Gosse, L.; Jin, S.; Li, X. T., On two moment systems for computing multiphase semiclassical limits of the Schrödinger equation, Mathematical Models and Methods in Applied Science, 13, 1689-1723 (2003) · Zbl 1055.81013 |
| [23] | Gottlieb, S.; Shu, C. W.; Tadmor, E., Strong stability-preserving high-order time discretization methods, SIAM Review, 43, 89-112 (2001) · Zbl 0967.65098 |
| [24] | Grad, H., On the kinetic theory of rarefied gases, Communications on Pure and Applied Mathematics, 2, 331-407 (1949) · Zbl 0037.13104 |
| [25] | Jin, S.; Li, X. T., Multi-phase computations of the semiclassical limit of the Schrödinger equation and related problems: Whitham vs Wigner, Physica D, 182, 46-85 (2003) · Zbl 1073.81041 |
| [26] | Jin, S.; Liu, H.; Osher, S.; Tsai, R., Computing multi-valued physical observables for the semiclassical limit of the Schrödinger equation, Journal of Computational Physics, 205, 222-241 (2005) · Zbl 1072.65132 |
| [27] | Knight, D. D., Elements of Numerical Methods for Compressible Flows (2006), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1103.76003 |
| [28] | Leveque, R., Finite Volume Methods for Hyperbolic Problems (2002), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1010.65040 |
| [29] | Li, X. T.; Wöhlbier, J. G.; Jin, S.; Booske, J. H., Eulerian method for computing multivalued solutions of the Euler-Poisson equations and applications to wave breaking in klystrons, Physical Review E, 70, 016502 (2004) |
| [30] | N.L. Lostec, R.O. Fox, O. Simonin, P. Villedieu, Numerical description of dilute particle-laden flows by a quadrature-based moment method, in: Proceedings of the Summer Program 2008, Center for Turbulence Research, Stanford University, 2008, pp. 209-221.; N.L. Lostec, R.O. Fox, O. Simonin, P. Villedieu, Numerical description of dilute particle-laden flows by a quadrature-based moment method, in: Proceedings of the Summer Program 2008, Center for Turbulence Research, Stanford University, 2008, pp. 209-221. |
| [31] | Marchisio, D. L.; Fox, R. O., Solution of population balance equations using the direct quadrature method of moments, Journal of Aerosol Science, 36, 43-73 (2005) |
| [32] | McGraw, R., Description of aerosol dynamics by the quadrature method of moments, Aerosol Science and Technology, 27, 255-265 (1997) |
| [33] | R. McGraw, Correcting moment sequences for errors associated with advective transport, 2006. <http://www.ecd.bnl.gov/pubs/momentcorrection_mcgraw2006.pdf>; R. McGraw, Correcting moment sequences for errors associated with advective transport, 2006. <http://www.ecd.bnl.gov/pubs/momentcorrection_mcgraw2006.pdf> |
| [34] | McGraw, R., Numerical advection of correlated tracers: Preserving particle size/composition moment sequences during transport of aerosol mixtures, Journal of Physics: Conference Series, 78, 012045 (2007) |
| [35] | Nicodin, I.; Gatignol, R., Unsteady half-space evaporation and condensation problems on the basis of the discrete kinetic theory, Physics of Fluids, 18, 127105 (2006) · Zbl 1146.76493 |
| [36] | Ogata, Y.; Im, H.-N.; Yabe, T., Numerical method for Boltzmann equation with Soroban-grid CIP method, Communications in Computational Physics, 2, 760-782 (2007) |
| [37] | Passalacqua, A.; Fox, R. O.; Garg, R.; Subramaniam, S., A fully coupled quadrature-based moment method for dilute to moderately dilute fluid-particle flows, Chemical Engineering Science, 65, 2267-2283 (2010) |
| [38] | Perthame, B., Boltzmann type schemes for compressible Euler equations in one and two space dimensions, SIAM Journal of Numerical Analysis, 29, 1-19 (1990) · Zbl 0744.76088 |
| [39] | Prather, M. J., Numerical advection by conservation of second-order moments, Journal of Geophysical Research, 91, 6671-6681 (1986) |
| [40] | Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P., Numerical Recipes in Fortran 77: The Art of Scientific Computing (1992), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0778.65002 |
| [41] | Pullin, D. I., Direct simulation methods for compressible inviscid ideal gas-flows, Journal of Computational Physics, 34, 53-66 (1980) · Zbl 0419.76049 |
| [42] | Runborg, O., Some new results in multiphase geometrical optics, Mathematical Modelling and Numerical Analysis, 34, 1203-1231 (2000) · Zbl 0972.78001 |
| [43] | Runborg, O., Mathematical models and numerical methods for high frequency waves, Communications in Computational Physics, 2, 827-880 (2007) · Zbl 1164.78300 |
| [44] | M. Sakiz, O. Simonin, Numerical experiments and modelling of non-equilibrium effects in dilute granular flows, in: Proceedings of the 21st International Symposium on Rarefied Gas Dynamics, Cépaduès-Éditions, Toulouse, France, 1998.; M. Sakiz, O. Simonin, Numerical experiments and modelling of non-equilibrium effects in dilute granular flows, in: Proceedings of the 21st International Symposium on Rarefied Gas Dynamics, Cépaduès-Éditions, Toulouse, France, 1998. |
| [45] | Schiller, L.; Naumann, A., A drag coefficient correlation, V.D.I. Zeitung, 77, 318-320 (1935) |
| [46] | Struchtrup, H., Macroscopic Transport Equations for Rarefied Gas Flows (2005), Springer: Springer New York · Zbl 1119.76002 |
| [47] | Torrilhon, M.; Struchtrup, H., Regularized 13-moment equations: shock structure calculations and comparison to Burnett models, Journal of Fluid Mechanics, 513, 171-198 (2004) · Zbl 1107.76069 |
| [48] | Vance, M. W.; Squires, K. D.; Simonin, O., Properties of the particle velocity field in gas-solid turbulent channel flow, Physics of Fluids, 18, 063302 (2006) |
| [49] | van Leer, B., Towards the ultimate conservative difference schemes V. A second order sequel to Godunov’s method, Journal of Computational Physics, 135, 229-248 (1997) · Zbl 0939.76063 |
| [50] | Wang, Z. J., A quadtree-based adaptive Cartesian/quad grid flow solver for Navier-Stokes equations, Computers and Fluids, 27, 529-549 (1998) · Zbl 0968.76053 |
| [51] | Wang, Z. J., Spectral (finite) volume method for conservation laws on unstructured grids: basic formulation, Journal of Computational Physics, 178, 210-251 (2002) · Zbl 0997.65115 |
| [52] | Wang, Z. J., High order methods for Euler and Navier-Stokes equations on unstructured grids, Journal of Progress in Aerospace Sciences, 43, 1-47 (2007) |
| [53] | Wheeler, J. C., Modified moments and Gaussian quadratures, Rocky Mountain Journal of Mathematics, 4, 287-296 (1974) · Zbl 0309.65009 |
| [54] | Williams, F. A., Spray combustion and atomization, Physics of Fluids, 1, 541-545 (1958) · Zbl 0086.41102 |
| [55] | Wöhlbier, J. G.; Jin, S.; Sengele, S., Eulerian calculations of wave breaking and multivalued solutions in a traveling wave tube, Physics of Plasmas, 12, 023106 (2005) |
| [56] | Wright, D. L., Numerical advection of moments of the particle size distribution in Eulerian models, Journal of Aerosol Science, 38, 352-369 (2007) |
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