zbMATH — the first resource for mathematics

Extending the meshless local Petrov-Galerkin method to solve stabilized turbulent fluid flow problems. (English) Zbl 1404.76165
Summary: The aim of this paper is to extend the meshless local Petrov-Galerkin method to solve stabilized turbulent fluid flow problems. For the unsteady incompressible turbulent fluid flow problems, the Spalart-Allmaras model is used to stabilize the governing equations, and the meshless local Petrov-Galerkin method is extended based on the vorticity-stream function to solve the turbulent flow problems. In this study, the moving least squares scheme interpolates the field variables. The proposed method solves three standard test cases of the turbulent flow over a flat plate, turbulent flow through a channel, and turbulent flow over a backward-facing step for evaluation of the method’s capability, accuracy, and validity purposes. Based on the comparison of the three test cases results with those of the experimental and conventional numerical works available in the literature, the proposed method shows to be accurate and quite implemental. The new extended method in this study together with the previously published works of the authors (on extending the meshless local Petrov-Galerkin method to solve laminar flow problems) now, for the first time, empower the meshless method to solve both laminar and turbulent flow problems.
Reviewer: Reviewer (Berlin)

76M10 Finite element methods applied to problems in fluid mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76F65 Direct numerical and large eddy simulation of turbulence
Full Text: DOI
[1] Arefmanesh, A.; Mahmoodi, M.; Nikfar, M., Effect of position of a square- shaped heat source on the Buoyancy-driven heat transfer in a square cavity filled with nanofluid, Sci. Iran., 21, 1129-1142, (2014)
[2] Arefmanesh, A.; Najafi, M.; Musavi, S. H., Buoyancy-driven fluid flow and heat transfer in a square cavity with a wavy baffle-Meshless numerical analysis, Eng. Anal. Bound. Elem., 37, 366-382, (2013) · Zbl 1351.76208
[3] Arefmanesh, A.; Najafi, M.; Nikfar, M., Meshless local Petrov-Galerkin simulation of Buoyancy-Driven fluid flow and heat transfer in a cavity with wavy side walls, Comput. Model. Eng. Sci., 62, 113-149, (2010) · Zbl 1231.76133
[4] Arefmanesh, A.; Najafi, M.; Nikfar, M., MLPG application of nanofluid flow mixed convection heat transfer in a wavy wall cavity, Comput. Model. Eng. Sci., 69, 91-117, (2010) · Zbl 1231.76302
[5] Arefmanesh, A.; Nikfar, M., Analysis of natural convection in a nanofluid-filled triangular enclosure induced by cold and hot sources on the walls using stabilized MLPG method, Can. J. chem. Eng., 91, 1711-1728, (2013)
[6] Atluri, S. N.; Zhu, T., A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Comput. Mech., 22, 117-127, (1998) · Zbl 0932.76067
[7] Barik, N. B.; Sekhar, T. V. S., An efficient local RBF meshless scheme for steady convection-diffusion problems, Int. J. Comput. Meth., 14, 1750064, (2017) · Zbl 1404.65288
[8] Cai, Y.; Han, L.; Tian, L.; Zhang, L., Meshless method based on Shepard function and partition of unity for two-dimensional crack problems, Eng. Anal. Bound. Elem., 65, 126-135, (2016) · Zbl 1403.74082
[9] Dehghan, M.; Abbaszadeh, M.; Mohebbi, A., Analysis of a meshless method for the time fractional diffusion-wave equation, Numer. Algorithm, 73, 445-476, (2016) · Zbl 1352.65298
[10] Dehghan, M.; Haghjoo-Saniji, M., The local radial point interpolation meshless method for solving Maxwell equations, Eng. Comput., 33, 897-918, (2017)
[11] Dehghan, M.; Mohammadi, V., A numerical scheme based on radial basis function finite difference (RBF-FD) technique for solving the high-dimensional nonlinear Schrödinger equations using an explicit time discretization: Runge-Kutta method, Comput. Phys. Commun., 217, 23-34, (2017)
[12] Enjilela, V.; Arefmanesh, A., Two-step Taylor-characteristic-based MLPG method for fluid flow and heat transfer applications, Eng. Anal. Bound. Elem., 51, 174-190, (2015) · Zbl 1403.76043
[13] Enjilela, V.; Salimi, D.; Tavasoli, A.; Lotfi, M., Stabilized MLPG-VF-based method with CBS scheme for laminar flow at high Reynolds and Rayleigh numbers, Int. J. Mod. Phys. C, 27, 1650081, (2016)
[14] Hidayat, M. I. P.; Ariwahjoedi, B.; Parman, S.; Irawan, S., A mesh free approach for transient heat conduction analysis of nonlinear functionally graded materials, Int. J. Comput. Meth., 15, 1850007, (2018) · Zbl 1404.74202
[15] Hoffmann, K. L.; Chiang, S. T., Computational Fluid Dynamics, (2000), Engineering Education System: Engineering Education System, USA
[16] Hosseini, V. R.; Shivanian, E.; Chen, W., Local radial point interpolation (MLRPI) method for solving time fractional diffusion-wave equation with damping, J. Comput. Phys., 307, 307-332, (2016) · Zbl 1352.65348
[17] Islam, S.; Singh, V., A local meshless method for steady state convection dominated flows, Int. J. Comput. Meth., 14, 1750067, (2017) · Zbl 1404.76195
[18] Jafari, N.; Azhari, M., Buckling of moderately thick arbitrarily shaped plates with intermediate point supports using a simple hp-cloud method, Appl. Math. Comput., 313, 196-208, (2017)
[19] Ji, Y.; Huang, T.; Huang, W.; Rong, L., Meshfree method in geophysical electromagnetic prospecting: The 2D magnetotelluric example, Int. J. Comput. Meth., 14, 1750084, (2017) · Zbl 1404.86033
[20] Jovic, S. and Driver, D. M. [1994] “Backward-facing step measurements at low Reynolds number, \(\text{Re}_H = 5 0 0 0\),” NASA technical memorandum 108807, pp. 21-24, https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19940028784.pdf.
[21] Latin, R. M.; Bowersox, R. D. W., Flow properties of a supersonic turbulent boundary layer with wall roughness, AIAA J., 38, 1804-1821, (2000)
[22] Laufer, J. [1951] “Investigation of turbulent flow in a two-dimensional channel,” NACA Report1053.
[23] Le, H.; Moin, P.; Kim, J., Dirct numerical simulation of turbulent flow over a backward-facing step, J. Fluid Mech., 330, 349-374, (1997) · Zbl 0900.76367
[24] Lei, Z. X.; Zhang, L. W.; Liew, K. M., Meshless modeling of geometrically nonlinear behavior of CNT-reinforced functionally graded composite laminated plates, Appl. Math. Comput., 295, 24-46, (2017)
[25] Lin, H.; Atluri, S. N., The meshless local Petrov-Galerkin (MLPG) method for solving incompressible Navier-Stokes equations, Comput. Model. Eng. Sci., 2, 117-142, (2001)
[26] Liu, Ch. Sh.; Younga, D. L., A multiple-scale Pascal polynomial for 2D Stokes and inverse Cauchy-Stokes problems, J. Comput. Phys., 312, 1-13, (2016) · Zbl 1351.76219
[27] Loukopoulos, V. C.; Bourantas, G. C., MLPG6 for the solution of incompressible flow equations, Comput. Model. Eng. Sci., 88, 531-558, (2012) · Zbl 1356.76059
[28] Mardani, A.; Hooshmandasl, M. R.; Hosseini, M. M.; Heydari, M. H., Moving least squares (MLS) method for the nonlinear hyperbolic telegraph equation with variable coefficients, Int. J. Comput. Meth., 14, 1750026, (2017) · Zbl 1404.65199
[29] Mohammadi, M. H., Stabilized meshless local Petrov-Galerkin (MLPG) method for incompressible viscous fluid flows, Comput. Model. Eng. Sci., 29, 75-94, (2008) · Zbl 1232.76028
[30] Najafi, M.; Arefmanesh, A.; Enjilela, V., Meshless local Petrov-Galerkin method-higher Reynolds numbers fluid flow applications, Eng. Anal. Bound. Elem., 36, 1671-1685, (2012) · Zbl 1351.76073
[31] Najafi, M.; Arefmanesh, A.; Enjilela, V., Extending MLPG primitive variable-based method for implementation in fluid flow and natural, forced and mixed convection heat transfer, Eng. Anal. Bound. Elem., 37, 1285-1299, (2013) · Zbl 1287.65081
[32] Najafi, M.; Enjilela, V., Natural convection heat transfer at high Rayleigh numbers — Extended meshless local Petrov-Galerkin (MLPG) primitive variable method, Eng. Anal. Bound. Elem., 44, 170-184, (2014) · Zbl 1297.80003
[33] Najafi, M.; Nikfar, M.; Arefmanesh, A., Inclination angle implications for fluid flow convection in complex geometry enclosure numerical analyses, J. Theor. Appl. Mech., 53, 519-530, (2015)
[34] Nikfar, M.; Mahmoodi, M., Meshless local Petrov-Galerkin analysis of free convection on nanofluid in a cavity with wavy side walls, Eng. Anal. Bound. Elem., 36, 433-445, (2012) · Zbl 1245.76141
[35] Nithiarasu, P.; Liu, C. B., An artificial compressibility based characteristic based split (CBS) scheme for steady and unsteady turbulent incompressible flows, Comput. Meth. Appl. Mech. Eng., 195, 2961-2982, (2006) · Zbl 1176.76086
[36] Pope, S. B., Turbulent Flows, (2000), Cambridge University Press: Cambridge University Press, New York · Zbl 0966.76002
[37] Rumsey, C. [2012]. NASA Langley turbulence modeling resource, http://turbmodels.larc.nasa.gov.
[38] Sharif, M. A. R.; Guo, G., Computational analysis of supersonic turbulent boundary layers over rough surfaces using the k-x and the stress-x models, Appl. Math. Model., 31, 2655-2667, (2007) · Zbl 1225.76180
[39] Skouras, E. D.; Bourantas, G. C.; Loukopoulos, V. C.; Nikiforidis, G. C., Truly meshless localized type techniques for the steady-state heat conduction problems for isotropic and functionally graded materials, Eng. Anal. Bound. Elem., 35, 452-464, (2011) · Zbl 1259.80029
[40] Stevens, D.; Power, H., The radial basis function finite collocation approach for capturing sharp fronts in time dependent advection problems, J. Comput. Phys., 298, 423-445, (2015) · Zbl 1349.65526
[41] Vertnik, R.; Šarler, B., Solution of incompressible turbulent flow by a mesh-free method, Comput. Model. Eng. Sci., 44, 65-95, (2009) · Zbl 1357.76030
[42] Wu, X. H.; Tao, W. Q.; Shen, Sh. P.; Zhu, X. W., A stabilized MLPG method for steady state incompressible fluid flow simulation, J. Comput. Phys., 229, 8564-8577, (2010) · Zbl 1381.76272
[43] Wu, Y. L.; Liu, G. R.; Gu, Y. T., Application of meshless local Petrov-Galerkin (MLPG) approach to simulation of incompressible flow, Numer. Heat Transf. B, 48, 459-475, (2005)
[44] Xiao, L.; Yang, J.; Peng, T.; Tao, L., A free surface interpolation approach for rapid simulation of short waves in meshless numerical wave tank based on the radial basis function, J. Comput. Phys., 307, 203-224, (2016) · Zbl 1351.76228
[45] Yang, J.; Hu, H.; Koutsawa, Y.; Ferry, M. P., Taylor meshless method for solving non-linear partial differential equations, J. Comput. Phys., 348, 385-400, (2017) · Zbl 1380.65393
[46] Zheng, H.; Zhang, Ch.; Wang, Y.; Sladek, J.; Sladek, V., A meshfree local RBF collocation method for anti-plane transverse elastic wave propagation analysis in 2D phononic crystals, J. Comput. Phys., 305, 997-1014, (2016) · Zbl 1349.74380
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.