Volume 3, Issue 3
Local Collocation Approach for Solving Turbulent Combined Forced and Natural Convection Problems

Adv. Appl. Math. Mech., 3 (2011), pp. 259-279.

Published online: 2011-06

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• Abstract

An application of the meshless Local Radial Basis Function Collocation Method (LRBFCM) [22, 30–33] in solution of incompressible turbulent combined forced and natural convection is for the first time explored in the present paper. The turbulent flow equations are described by the low-Re number $k − ε$ model with Launder and Sharma [23] and Abe et al. [1] closure coefficients. The involved temperature, velocity, pressure, turbulent kinetic energy and dissipation fields are represented on overlapping 5-noded sub-domains through the collocation by using multiquadrics Radial Basis Functions (RBF). The involved first and second order partial derivatives of the fields are calculated from the respective derivatives of the RBF's. The involved equations are solved through the explicit time stepping. The pressure-velocity coupling is based on Chorin's fractional step method [11]. The adaptive upwinding technique, proposed by Lin and Atluri [27], is used because of the convection dominated situation. The solution procedure is represented for a 2D upward channel flow with differentially heated walls. The results have been assessed by achieving a reasonable agreement with the direct numerical simulation of Kasagi and Nishimura [20] for Reynolds number 4494, based on the channel width, and Grashof number 9.6 × 105. The advantages of the represented mesh-free approach are its simplicity, accuracy, similar coding in 2D and 3D, and straightforward applicability in non-uniform node arrangements.

76F60, 76M25, 76R05, 76R10, 65D05, 65M22 and 65M70

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@Article{AAMM-3-259, author = {Vertnik , Robert and Šarler , Božidar}, title = {Local Collocation Approach for Solving Turbulent Combined Forced and Natural Convection Problems}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2011}, volume = {3}, number = {3}, pages = {259--279}, abstract = {

An application of the meshless Local Radial Basis Function Collocation Method (LRBFCM) [22, 30–33] in solution of incompressible turbulent combined forced and natural convection is for the first time explored in the present paper. The turbulent flow equations are described by the low-Re number $k − ε$ model with Launder and Sharma [23] and Abe et al. [1] closure coefficients. The involved temperature, velocity, pressure, turbulent kinetic energy and dissipation fields are represented on overlapping 5-noded sub-domains through the collocation by using multiquadrics Radial Basis Functions (RBF). The involved first and second order partial derivatives of the fields are calculated from the respective derivatives of the RBF's. The involved equations are solved through the explicit time stepping. The pressure-velocity coupling is based on Chorin's fractional step method [11]. The adaptive upwinding technique, proposed by Lin and Atluri [27], is used because of the convection dominated situation. The solution procedure is represented for a 2D upward channel flow with differentially heated walls. The results have been assessed by achieving a reasonable agreement with the direct numerical simulation of Kasagi and Nishimura [20] for Reynolds number 4494, based on the channel width, and Grashof number 9.6 × 105. The advantages of the represented mesh-free approach are its simplicity, accuracy, similar coding in 2D and 3D, and straightforward applicability in non-uniform node arrangements.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.10-10s2-01}, url = {http://global-sci.org/intro/article_detail/aamm/168.html} }
TY - JOUR T1 - Local Collocation Approach for Solving Turbulent Combined Forced and Natural Convection Problems AU - Vertnik , Robert AU - Šarler , Božidar JO - Advances in Applied Mathematics and Mechanics VL - 3 SP - 259 EP - 279 PY - 2011 DA - 2011/06 SN - 3 DO - http://doi.org/10.4208/aamm.10-10s2-01 UR - https://global-sci.org/intro/article_detail/aamm/168.html KW - Turbulent combined convection, two-equation turbulence model, radial basis function, collocation, meshless method, upward channel flow. AB -

An application of the meshless Local Radial Basis Function Collocation Method (LRBFCM) [22, 30–33] in solution of incompressible turbulent combined forced and natural convection is for the first time explored in the present paper. The turbulent flow equations are described by the low-Re number $k − ε$ model with Launder and Sharma [23] and Abe et al. [1] closure coefficients. The involved temperature, velocity, pressure, turbulent kinetic energy and dissipation fields are represented on overlapping 5-noded sub-domains through the collocation by using multiquadrics Radial Basis Functions (RBF). The involved first and second order partial derivatives of the fields are calculated from the respective derivatives of the RBF's. The involved equations are solved through the explicit time stepping. The pressure-velocity coupling is based on Chorin's fractional step method [11]. The adaptive upwinding technique, proposed by Lin and Atluri [27], is used because of the convection dominated situation. The solution procedure is represented for a 2D upward channel flow with differentially heated walls. The results have been assessed by achieving a reasonable agreement with the direct numerical simulation of Kasagi and Nishimura [20] for Reynolds number 4494, based on the channel width, and Grashof number 9.6 × 105. The advantages of the represented mesh-free approach are its simplicity, accuracy, similar coding in 2D and 3D, and straightforward applicability in non-uniform node arrangements.

Robert Vertnik & Božidar Šarler. (1970). Local Collocation Approach for Solving Turbulent Combined Forced and Natural Convection Problems. Advances in Applied Mathematics and Mechanics. 3 (3). 259-279. doi:10.4208/aamm.10-10s2-01
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