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Numerical computation of high Rayleigh number natural convection and prediction of hot radiator induced room air motion

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Abstract

The two-dimensional stationary turbulent buoyant flow and heat transfer in a cavity at high Rayleigh numbers was computed numerically. The k−ε turbulence model was used. The time-averaged equations for momentum, energy and continuity, which are coupled to the turbulence equations, were solved using a finite difference formulation. In order to validate the computer code, a comparison exercise was carried out. The test results are in good agreement with the internationally accepted benchmark solution. Grid-refinement shows the necessity of a very fine grid at high Rayleigh numbers with especially small grid-distances in the near-wall region. The computed boundary layer velocity profiles are in excellent agreement with available experimental data. The local heat transfer in the turbulent part of the boundary layers is predicted 20% too high. Computations were carried out for the natural convective flow in a room induced by a hot radiator and a cold window. Various radiator configurations and types of thermal boundary conditions were applied including thermal radiation interaction between surfaces.

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Abbreviations

a :

thermal diffusivity (m2/s)

C μ :

constant in ν t expression

D :

cavity dimensions (m)

g :

acceleration of gravity (m/s2)

G k :

production/destruction of k by buoyancy (kg/ms3)

h :

enthalpy (J/kg)

IX:

index of grid point

k :

turbulent kinetic energy (m2/s2)

m :

dimensionless stratification parameter

Nu:

overall Nusselt number

Nu y :

local Nusselt number

NX:

total number of grid points

p :

pressure (N/m2)

P k :

production of k by shear stress (kg/ms3)

Q :

heat flux through wall (W/m)

Ra:

overall Rayleigh number

Ra y :

local Rayleigh number

Re t :

turbulent Reynolds number

S ε :

source term in ε-equation (kg/ms4)

S φ :

source term for φ

T c, T h :

temperatures of cold and hot walls (K)

T s (y) :

stratification temperature on vertical mid-line (K)

T 0 :

mean cavity temperature (K)

u, v :

horizontal and vertical velocity components (m/s)

u 0 :

Brunt-Vaisälä velocity scale (m/s)

x, y :

horizontal and vertical coordinates (m)

α :

non-linearity parameter for grid

β :

coefficient of thermal expansion (l/K)

γ :

jet angle (°)

Гφ :

diffusivity for φ

S ε :

dissipation rate for turbulent kinetic energy (m2/s3)

φ :

variable to be solved

λ :

thermal conductivity (W/mK)

ν, ν t :

kinematic and eddy viscosities (m2/s)

ψ :

stream function (kg/ms)

ρ :

density (kg/m3)

σ k, σ ε, σ t :

constants in k−ε model

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Authors and Affiliations

  1. Faculty of Applied Physics, Heat Transfer Section, Delft University of Technology, P.O. Box 5046, 2600 GA, Delft, The Netherlands

    A. M. Lankhorst & C. J. Hoogendoorn

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  1. A. M. Lankhorst
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  2. C. J. Hoogendoorn
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Lankhorst, A.M., Hoogendoorn, C.J. Numerical computation of high Rayleigh number natural convection and prediction of hot radiator induced room air motion. Applied Scientific Research 47, 301–322 (1990). https://doi.org/10.1007/BF00386241

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