BAIL 2006 Book of Abstracts - Institut für Numerische und ...

O. SHISHKINA, C. WAGNER: Boundary and Interior Layers in Turbulent Thermal
Convection
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Boundary and Interior Layers in Turbulent Thermal Convection ∗
1. Introduction
O. Shishkina & C. Wagner
DLR - Institute for Aerodynamics and Flow Technology,
Bunsenstrasse 10, 37073 Göttingen, Germany
Olga.Shishkina@dlr.de, Claus.Wagner@dlr.de
Numerous scientifical problems and industrial applications require solutions of the Rayleigh-
Bénard problem, i.e. turbulent convection of fluids heated from below and cooled from above,
with the Rayleigh number (Ra) from 10 5 up to 10 20 . For a review on this classical problem and
for the references to earlier literature we refer to the paper Ahlers, Grossmann and Lohse [1].
Since the diffusion coefficient in the Navier-Stokes equation, which is inversely proportional to
the square root of Ra, is very small, the solution - both the temperature and the velocity fields -
have very thin boundary layers near the horizontal walls. For moderate Rayleigh numbers (from
10 5 up to 10 8 ) interior layers, i.e. thermal plumes, are also observed. The boundary layers, the
thermal plumes and the turbulent background are indicated, respectively, by high, moderate and
small values of the temperature gradient norm and, hence, by large, moderate and small values
of the thermal dissipation rate. By means of direct numerical simulations (DNS) we invesigate
boundary and interior layers which take place in turbulent Rayleigh-Bénard convection.
2. Governing equations and the numerical method
The governing dimensionless equations for the Rayleigh-Bénard problem in Boussinesq approximation
can be written in cylindrical coordinates (z, r, ϕ) as follows
ut + u · ∇u + ∇p = Γ −3/2 Ra −1/2 P r 1/2 ∇ 2 u + T z, (1)
Tt + u · ∇T = Γ −3/2 Ra −1/2 P r −1/2 ∇ 2 T, (2)
∇ · u = 0, (3)
where u is the velocity vector, T the temperature, ut and Tt their time derivatives, p the
pressure. Here Ra = αgH 3 ∆T/(κν) denotes the Rayleigh number, P r = ν/κ the Prandtl
number, Γ = D/H the aspect ratio with H the height and D the diameter of the cylindrical
container. Further, α is the thermal expansion coefficient, g the gravitational acceleration, ∆T
the temperature difference between the bottom and the top plates, ν the kinematic viscosity and
κ the thermal diffusivity. The dimensionless temperature varies between +0.5 at the bottom
plate to −0.5 at the top plate. An adiabatic lateral wall is prescribed by ∂T/∂r = 0. Finally, on
the solid walls the velocity field vanishes according to the impermeability and no-slip conditions.
To investigate turbulent Rayleigh-Bénard convection in cylindrical containers of the aspect
ratios Γ = 10 and Γ = 5 for Ra from 10 5 to 10 7 and P r = 0.7 we conducted DNS. The simulations
were performed with the fourth order accurate finite volume method developed for solving (1)
– (3) in cylindrical coordinates on staggered non-equidistant grids. For the numerical method
used in the DNS we refer to [2].
∗ The work is supported by Deutsche Forschungsgemeinschaft (DFG) under the contract WA 1510-1
Speaker: SHISHKINA, O. 138 BAIL 2006
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