Skip to main content Accessibility help
Hostname: page-component-99c86f546-45s75 Total loading time: 0.277 Render date: 2021-11-29T15:57:59.893Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

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

Published online by Cambridge University Press:  03 June 2015

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


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.

Research Article
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.)


[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. 861889.CrossRefGoogle 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. 262297.CrossRefGoogle 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. 217245.CrossRefGoogle 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. 2166.CrossRefGoogle 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. 166200.Google Scholar
[7]Godunov, S. K. and Romenski, E., Elements of Continuum Mechanics and Conservation Laws, Kluwer Academic/Plenum Publishers, 2003.CrossRefGoogle 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. 288313.CrossRefGoogle 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. 3561.CrossRefGoogle Scholar
[10]Hu, X. Y. and Khoo, B. C., An interface interaction method for compressible multifluids, J. Comput. Phys., 198 (2004), pp. 3564.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. 715732.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. 30023024.CrossRefGoogle Scholar
[14]Kataoka, I., Local instant formulation of two-phase flow, Int. J. Multiphase Flow, 12 (1986), pp. 745758.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. 62206242.CrossRefGoogle 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. 873884.CrossRefGoogle 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. 304328.CrossRefGoogle Scholar
[18]Luke, E. A. and Cinnella, P., Numerical simulations of mixtures of fluids using upwind algo-rithms, Comput. Fluids, 36 (2007), pp. 15471566.CrossRefGoogle 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. 10371053.CrossRefGoogle Scholar
[20]Nessyahu, H. AND Tadmor, E., Non-oscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys., 87 (1990), pp. 408463.CrossRefGoogle 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. 297333.CrossRefGoogle 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. 455477.CrossRefGoogle 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. 403416.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. 14731479.CrossRefGoogle Scholar
[25]Resnyanskya, A. D. and Bourne, N. K., Shock-wave compression of a porous material, J. Appl. Phys., 95 (2004), pp. 17601769.CrossRefGoogle 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. 128.CrossRefGoogle Scholar
[27]Saurel, R. and Abgrall, R., A multiphase Godunov method for compressible multifluid and multiphase flows, J. Comput. Phys., 150 (1999), pp. 425467.CrossRefGoogle 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. 490526.CrossRefGoogle 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. 74117439.CrossRefGoogle Scholar
[30]StäDtke, H., Gasdynamic Aspects of Two-Phase Flow: Hyperbolicity, Wave Propagation Phenomena, and Related Numerical Methods, Weinheim, Wiley-VCH, 2006.CrossRefGoogle Scholar
[31]Stewart, H. B. and Wendroff, B., Two-phase flow: models and methods, J. Comput. Phys., 56 (1984), pp. 363409.CrossRefGoogle 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. 472481, Springer-Verlag.Google Scholar
[33]Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics, A practical introduction, Berlin, Heidelberg, Springer-Verlag, 2009.CrossRefGoogle 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. 35733604.CrossRefGoogle Scholar
[35]Touma, R., Central unstaggered finite volume schemes for hyperbolic systems: applications to unsteady shallow water equations, Appl. Math. Comput., 213 (2009), pp. 4759.Google Scholar
[36]Toumi, I., and Kumbaro, A., An approximate linearized Riemann solver for two-fluid model, J. Comput. Phys., 124 (1996), pp. 286300.CrossRefGoogle 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. 19501968.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. 101136.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. 13371343.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. 325336.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. 155177.CrossRefGoogle 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. 299333.CrossRefGoogle 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. 393417.CrossRefGoogle 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. 623641.CrossRefGoogle Scholar
[45]Zeidan, D., Applying upwind Godunov methods to calculate two-phase mixture conservation laws, AIP Conference Proceedings, 1281 (2010), pp. 155158.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. 50235040.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. 68956921.CrossRefGoogle Scholar

Send article to Kindle

To send this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the or variations. ‘’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

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

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

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

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

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

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *