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High-accuracy bench mark solutions to natural convection in a square cavity

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A fourth-order high-accuracy finite difference method is presented for the bouyancy-driven flow in a square cavity with differentially heated vertical walls. The two bench mark solutions against which other solutions can be compared were obtained. The present solution is seemed to be accurate up to fifth decimal. The proposed scheme is stable and convergent for high Rayleigh number, and will be applicable to general problems involving flow and heat transfer, especially in three dimensions.

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a :

thermal diffusivity

g :

gravitational acceleration

h :

mesh length in the x-direction

k :

time step

L :

side length of cavity

Pr :

Prandtl number

Ra :

Rayleigh number = gβΔTL 3a

t :


T c, T h :

surface temperatures (see Fig. 1a)

T :


ΔT :

temperature difference = T h - T c

u, v :

velocity in the x and y directions, respectively

x, y :


Δx, Δy :

mesh lengths in the x- and y-directions, respectively


volumetric expansion coefficient


kinematic viscosity


stream function




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Saitoh, T., Hirose, K. High-accuracy bench mark solutions to natural convection in a square cavity. Computational Mechanics 4, 417–427 (1989). https://doi.org/10.1007/BF00293047

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  • Heat Transfer
  • Convection
  • Information Theory
  • Finite Difference
  • General Problem