Time‐dependent natural convection in a square cavity: Application of a new finite volume method

Abstract

SUMMARYA new finite volume (FV) approach with adaptive upwind convection is used to predict the two-dimensional unsteady flow in a square cavity. The fluid is air and natural convection is induced by differentially heated vertical walls. The formulation is made in terms of the vorticity and the integral velocity (induction) law. Biquadratic interpolation formulae are used to approximate the temperature and vorticity fields over the finite volumes, to which the conservation laws are applied in integral form. Image vorticity is used to enforce the zero-penetration condition at the cavity walls. Unsteady predictions are carried sufficiently forward in time to reach a steady state. Results are presented for a Prandtl number (Pi-) of 0 7 1 and Rayleigh numbers equal to to3, lo4 and lo5. Both 11 x 11 and 21 x 21 meshes are used. The steady state predictions are compared with published results obtained using a finite difference (FD) scheme for the same values of Pr and Ra and the same meshes, as well as a numerical bench-mark solution. For the most part the FV predictions are closer to the bench-mark solution than are the FD predictions.

Rayleigh–Bénard convection in tall rectangular enclosures

“…In particular, the average Nusselt numbers throughout the cavity as well as the maximum horizontal and vertical velocity components, respectively on the vertical and the horizontal midplane of the enclosure, are within 1% of the benchmark data, as indicated in Table 1, where other reference solutions obtained by other authors through finite-volume methods are also reported (i.e., the results by Mahdi and Kinney [20], for Ra = 10 3 , and those by Hortmann et al [21], for Ra = 10 4 to 10 6 ). It seems worth noticing that our dimensionless velocity results have been multiplied by the Prandtl number before being inserted in Table 1, so as to account for the choice of the ratio between kinematic viscosity and characteristic length of the cavity as scale factor for the velocity, instead of the ratio between thermal diffusivity and characteristic length, used by de Vahl Davis in Ref.…”

Buoyant heat transport in fluids across tilted square cavities discretely heated at one side

“…The following E = 0.1, ⏐ψ⏐ ⏐ ψ⏐ ⏐ ψ⏐ max = 9.35 E = 0.5, max = 11. 86 E = 0.9, max = 13.36 additional benchmark solutions are also reported for further comparison: (a) the results obtained through finite-volume methods by Mahdi and Kinney [29], for Ra ¼ 10 3 , and by Hortman et al [30], for Ra ¼ 10 4 to 10 6 , are listed in column BM2; (b) the results obtained through a finite-element method by Wan et al [31] are listed in column BM3; and (c) the results obtained through a discrete singular convolution algorithm by Wan et al [31] are listed in column BM4. It is worth noticing that our dimensionless velocity results have been multiplied by the Prandtl number before being inserted in Table 3, so as to account for the choice of the ratio between kinematic viscosity of the fluid and characteristic length as scale factor for the velocity, instead of the ratio between thermal diffusivity of the fluid and characteristic length, used in refs.…”

Enhanced natural convection heat transfer of nanofluids in enclosures with two adjacent walls heated and the two opposite walls cooled

“…As far as the time step setting is concerned, its effects on the solutions relevant to configuration AA are displayed in Table 4 for Finally, with the scope to validate the numerical code used for the present study, three different tests have been carried out. In the first test, the steady-state solutions obtained for an air-filled square enclosure differentially heated at sides, assuming m av = 0 and constant physical properties, have been compared with the benchmark solutions of de Vahl Davis [35], Mahdi and Kinney [36], Hortman et al [37], and Wan et al [38]. In the second test, the steady-state average Nusselt numbers computed numerically for a Prandtl number Pr = 7 (corresponding to water at T av = 293 K) and Rayleigh numbers Ra = 10 3 5 Â 10 7 (calculated using a fixed DT = 20 K), again assuming m av = 0, have been compared with the usually recommended BerkovskyPolevikov correlation based on experimental and numerical data of laminar natural convection in vertical rectangular cavities heated and cooled from the side with an aspect ratio near unity, see, e.g., Bejan [39] and Incropera et al [40].…”