Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-22T18:21:45.581Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Computations of Gas-Solid Mixture Two-Phase Flow

Published online by Cambridge University Press:  03 June 2015

D. Zeidan  and
R. Touma
D. Zeidan*
Affiliation:
Department of Mathematics, Al-Balqa Applied University, Al-Salt, Jordan
R. Touma
Affiliation:
Department of Computer Science &Mathematics, Lebanese American University, Beirut, Lebanon
*
*Corresponding author. Email: [email protected][email protected]
Get access

Abstract

This paper presents high-resolution computations of a two-phase gas-solid mixture using a well-defined mathematical model. The HLL Riemann solver is applied to solve the Riemann problem for the model equations. This solution is then employed in the construction of upwind Godunov methods to solve the general initial-boundary value problem for the two-phase gas-solid mixture. Several representative test cases have been carried out and numerical solutions are provided in comparison with existing numerical results. To demonstrate the robustness, effectiveness and capability of these methods, the model results are compared with reference solutions. In addition to that, these results are compared with the results of other simulations carried out for the same set of test cases using other numerical methods available in the literature. The diverse comparisons demonstrate that both the model equations and the numerical methods are clear in mathematical and physical concepts for two-phase fluid flow problems.

Keywords

76T1576N1035L6565M08Hyperbolic conservative equationstwo-phase flowscompressible gas-solidmixture conservation lawsapproximate Riemann solverupwind Godunov methodsnumerical simulation
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 6 , Issue 1 , February 2014 , pp. 49 - 74
Copyright
Copyright © Global-Science Press 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Baer, M. and Nunziato, J., A two-phase mixture theory for the deflagration-to-detonation tran-sition (DDT) in reactive granular materials, In T. J. Multiphase Flow, 12 (1986), pp. 861–889.Google Scholar
[2]Banks, J. W. et al., A high-resolution Godunov method for compressible multi-material flow on overlapping grids, J. Comput. Phys., 223 (2007), pp. 262–297.Google Scholar
[3]Deledicque, V. and Papalexandris, M. V., An exact Riemann solver for compressible two-phase flow models containing non-conservative products, J. Comput. Phys., 222 (2007), pp. 217–245.Google Scholar
[4]Drew, D. and Passman, S., Theory of Multicomponent Fluids, (Applied Mathematical Sciences, Vol. 135), New York, Springer-Verlag, 1998.Google Scholar
[5]Enwald, H., Peirano, E. and Almstedt, A.-E., Eulerian two-phase flow theory applied to fluidization, Int. Multiphase Flow, 22 (1996), pp. 21–66.Google Scholar
[6]Friis, H. A., Evje, S. and Flatten, T., A numerical study of characteristic slow-transient behavior of a compressible 2D gas-liquid two-fluid model, Adv. Appl. Math. Mech., l (2009), pp. 166–200.Google Scholar
[7]Godunov, S. K. and Romenski, E., Elements of Continuum Mechanics and Conservation Laws, Kluwer Academic/Plenum Publishers, 2003.Google Scholar
[8]Guillard, H. and Duval, F., A Darcy law for the drift velocity in a two-phase flow model, J. Comput. Phys., 224 (2007), pp. 288–313.Google Scholar
[9]Harten, A., Lax, P. D. and Leer, B. van, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25 (1983), pp. 35–61.Google Scholar
[10]Hu, X. Y. and Khoo, B. C., An interface interaction method for compressible multifluids, J. Comput. Phys., 198 (2004), pp. 35–64.CrossRefGoogle Scholar
[11]Ishii, M., Thermo-Fluid Dynamic Theory of Two-Phase Flow, Paris, Eyrolles, 1975.Google Scholar
[12]Johnsen, E. and Colonius, T., Implementation ofWENO schemes in compressible multicomponent flow problems, J. Comput. Phys., 219 (2006), pp. 715–732.CrossRefGoogle Scholar
[13]Kapila, A. K. et al., Two-phase modeling of deflagrationto-detonation transition in granular materials: Reduced equations, Phys. Fluids, 13 (2001), pp. 3002–3024.Google Scholar
[14]Kataoka, I., Local instant formulation of two-phase flow, Int. J. Multiphase Flow, 12 (1986), pp. 745–758.CrossRefGoogle Scholar
[15]Kreeft, J. J. and Koren, B., A new formulation of Kapila’s five-equation model for compressible two-fluid flow, and its numerical treatment, J. Comput. Phys., 229 (2010), pp. 6220–6242.Google Scholar
[16]Liang, Q. and Marche, F., Numerical resolution of well-balanced shallow water equations with complex source terms, Adv. Water Res., 32 (2009), pp. 873–884.Google Scholar
[17]Luo, H., Baum, J. D. AND LööHner, R., On the computation of multi-material flows using ALE formulation, J. Comput. Phys., 194 (2004), pp. 304–328.CrossRefGoogle Scholar
[18]Luke, E. A. and Cinnella, P., Numerical simulations of mixtures of fluids using upwind algo-rithms, Comput. Fluids, 36 (2007), pp. 1547–1566.Google Scholar
[19]Markatos, N. C. and Kirkcaldy, D., Analysis and computation of three-dimensional, transient flow and combustion through granulated propellants, Int. J. Heat Mass Trans., 26 (1983), pp. 1037–1053.Google Scholar
[20]Nessyahu, H. AND Tadmor, E., Non-oscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys., 87 (1990), pp. 408–463.Google Scholar
[21]Raviart, P. A. and Sainsaulieu, L., A nonconservative hyperbolic system modeling spray dynamics, Part I: solution of Riemann problem, Math. Model. Methods Appl. Sci., 5 (1995), pp. 297–333.Google Scholar
[22]Romate, J. E., An approximate Riemann solver for a two-phase flow model with numerically given slip relation, Comput. Fluids, 27 (1998), pp. 455–477.Google Scholar
[23]Romenski, E. and Toro, E. F., Compressible two-phase flow models: two-pressure models and numerical methods, Comput. Fluid Dyn. J., 13 (2004), pp. 403–416.Google Scholar
[24]Romenski, E. AND Drikakis, D., Compressible two-phase flow modelling based on thermodynamically compatible systems of hyperbolic conservation laws, Int. J. Numer. Methods Fluids, 56 (2007), pp. 1473–1479.CrossRefGoogle Scholar
[25]Resnyanskya, A. D. and Bourne, N. K., Shock-wave compression of a porous material, J. Appl. Phys., 95 (2004), pp. 1760–1769.Google Scholar
[26]Sainsaulieu, L., Finite-volume approximation of two phase-fluid flows based on an approximate Roe-type Riemann solver, J. Comput. Phys., 121 (1995), pp. 1–28.Google Scholar
[27]Saurel, R. and Abgrall, R., A multiphase Godunov method for compressible multifluid and multiphase flows, J. Comput. Phys., 150 (1999), pp. 425–467.Google Scholar
[28]Schwendeman, D. W., Wahle, C. W. and Kapila, A. K., The Riemann problem and a highresolution Godunov method for a model of compressible two-phase flow, J. Comput. Phys., 212 (2006), pp. 490–526.Google Scholar
[29]Shukla, R. K., Pantano, C. and Freund, J. B., An interface capturing method for the simulation of multi-phase compressible flows, J. Comput. Phys., 229 (2010), pp. 7411–7439.CrossRefGoogle Scholar
[30]StäDtke, H., Gasdynamic Aspects of Two-Phase Flow: Hyperbolicity, Wave Propagation Phenomena, and Related Numerical Methods, Weinheim, Wiley-VCH, 2006.Google Scholar
[31]Stewart, H. B. and Wendroff, B., Two-phase flow: models and methods, J. Comput. Phys., 56 (1984), pp. 363–409.Google Scholar
[32]Toro, E. F., Riemann-problem based techniques for computing reactive two-phase flows, In: Dervieux, Larrouturrou, editors, Lecture Notes in Physics, Numerical Combustion, 351 (1989), pp. 472–481, Springer-Verlag.Google Scholar
[33]Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics, A practical introduction, Berlin, Heidelberg, Springer-Verlag, 2009.Google Scholar
[34]Tokareva, S. A. and Toro, E. F., HLLC-type Riemann solver for the Baer-Nunziato equations of compressible two-phase flow, J. Comput. Phys., 229 (2010), pp. 3573–3604.Google Scholar
[35]Touma, R., Central unstaggered finite volume schemes for hyperbolic systems: applications to unsteady shallow water equations, Appl. Math. Comput., 213 (2009), pp. 47–59.Google Scholar
[36]Toumi, I., and Kumbaro, A., An approximate linearized Riemann solver for two-fluid model, J. Comput. Phys., 124 (1996), pp. 286–300.Google Scholar
[37]Valero, E., De, J. Vicente and Alonso, G., The application of compact residual distribution schemes to two-phase flow problems, Comput. Fluids, 38 (2009), pp. 1950–1968.CrossRefGoogle Scholar
[38]Leer, B. van, Towards the ultimate conservative difference scheme V, a second-order sequel to Godunov’s method, J. Comput. Phys., 32 (1979), pp. 101–136.CrossRefGoogle Scholar
[39]Wackers, J. and Koren, B., A fully conservative model for compressible two-fluid flow, Int. J. Numer. Methods Fluids, 47 (2005), pp. 1337–1343.CrossRefGoogle Scholar
[40]Wang, B. AND Xu, H., A method based on Riemann problem in tracking multi-material interface on unstructured moving grids, Eng. Appl. Comput. Fluid Mech., 1 (2007), pp. 325–336.Google Scholar
[41]Yeom, G. S. and Chang, K. S., Numerical simulation of two-fluid two-phase flows by HLL scheme using an approximate Jacobian matrix, Numer. Heat Trans. B, 49 (2006), pp. 155–177.Google Scholar
[42]Zeidan, D., Slaouti, A.Romenski, E. AND Toro, E. F., Numerical solution for hyperbolic conservative two-phase flow equations, Int. J. Comput. Methods, 4 (2007), pp. 299–333.Google Scholar
[43]Zeidan, D., Romenski, E., Slaouti, A. and Toro, E. F., Numerical study of wave propagation in compressible two-phase flow, Int. J. Numer. Methods Fluids, 54 (2007), pp. 393–417.Google Scholar
[44]Zeidan, D. and Slaouti, A., Validation of hyperbolic model for two-phase flow in conservative form, Int. J. Comput. Fluid Dyn., 23 (2009), pp. 623–641.Google Scholar
[45]Zeidan, D., Applying upwind Godunov methods to calculate two-phase mixture conservation laws, AIP Conference Proceedings, 1281 (2010), pp. 155–158.Google Scholar
[46]Zeidan, D., Numerical resolution for a compressible two-phase flow model based on the theory of thermodynamically compatible systems, Appl. Math. Comput., 217 (2011), pp. 5023–5040.Google Scholar
[47]Zheng, H. W., Shu, C. and Chew, Y. T., An object-oriented and quadrilateral-mesh based solution adaptive algorithm for compressible multi-fluid flows, J. Comput. Phys., 227 (2008), pp. 6895–6921.Google Scholar