Local Collocation Approach for Solving Turbulent Combined Forced and Natural Convection Problems (Advances in Applied Mathematics and Mechanics, 2021)
An application of the meshless Local Radial Basis Function CollocationMethod (LRBFCM) [22, 30–33] in solution of incompressible turbulent combinedforced 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 withLaunder and Sharma [23] and Abe et al. [1] closure coefficients. The involved temperature,velocity, pressure, turbulent kinetic energy and dissipation fields are representedon overlapping 5-noded sub-domains through the collocation by usingmultiquadrics Radial Basis Functions (RBF). The involved first and second orderpartial derivatives of the fields are calculated from the respective derivatives of theRBF's. The involved equations are solved through the explicit time stepping. Thepressure-velocity coupling is based on Chorin's fractional step method [11]. Theadaptive upwinding technique, proposed by Lin and Atluri [27], is used becauseof the convection dominated situation. The solution procedure is represented fora 2D upward channel flow with differentially heated walls. The results have beenassessed by achieving a reasonable agreement with the direct numerical simulationof Kasagi and Nishimura [20] for Reynolds number 4494, based on the channelwidth, and Grashof number 9.6 × 105. The advantages of the represented mesh-freeapproach are its simplicity, accuracy, similar coding in 2D and 3D, and straightforwardapplicability in non-uniform node arrangements.