Accurate solution for natural convection around single and tandem circular cylinders inside a square enclosure using SEM

In this article, the spectral element method is applied to produce and serve as accurate solutions of benchmark quality for the natural convection around single and tandem circular cylinders inside a square enclosure at Rayleigh number Ra = 10(4), 10(5), 10(6), and Prandtl number Pr = 0.71. To capture the cylindrical boundary better, two cases of affine map with clear geometric meaning are introduced such that the collocation grids are well located on the physical boundary. An improved time-splitting method is applied to deal with the coupled pressure and velocity in the momentum equation, and the computational framework is further promoted in this work when solving the convection term such that no components in the coefficient matrices of the algebraic equations need to be updated as the computational time is advanced. Numerical solutions of high quality are presented by the average Nusselt number, maximum value and contours of stream function, isotherms, and the quantitative dimensionless temperature along the horizontal midline, which could be a reference result for this classical problem.