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

M. HÖLLING, H. HERWIG: Computation of turbulent natural convection at vertical
walls using new wall functions
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Computation of turbulent natural convection at vertical walls using new wall
functions
M. Hölling, H. Herwig
Technical Thermodynamics
Hamburg University of Technology
Denickestraße 15, 21073 Hamburg, Germany
m.hoelling@tu-harburg.de
h.herwig@tu-harburg.de
Turbulent natural convection at vertical walls has been under investigation in the last decades
but is still not sufficiently understood. The viscous sublayer, i.e. the region very close to the wall,
can be described properly, see e.g. Tsuji and Nagano [1], since there the turbulent fluctuations
are damped by the wall and the governing equations can be solved directly. But in the more
interesting region in which turbulence dominates, the flow cannot be described properly. George
and Capp [2] offer analytical temperature and velocity profiles which have become a kind of
standard for natural convection. But, it was shown by Versteegh and Nieuwstadt [3] and by
Henkes and Hoogendoorn [4] that at least the velocity profile is erroneous. We employ a different
approach presented in Hölling and Herwig [5] to describe the turbulence affected region of the
flow field.
The starting point for the analysis is a channel with a hot and a cold wall of infinite extent
as used by Versteegh and Nieuwstadt [3] for their DNS study. The governing equations then
reduce to:
0 = ∂
�
∂y
0 = ∂
∂y
∂y − v′ T ′
�
a ∂T
�
ν ∂u
∂y − u′ v ′
�
+ gβ � �
T − T0
It is found that the temperature field consists of a viscosity influenced wall layer and a fully
turbulent outer layer. The temperature profile is obtained by matching of gradients between
these layers that reveals a logarithmic profile. It is in good agreement with DNS as well as
experimental temperature profiles from various studies.
For the velocity profile a different approach is chosen; the profile is not obtained by matching
of gradients. Instead the momentum equation (2) is rewritten in such a way that the temperature
profile and the Reynolds stresses are expressed as a function of the wall distance. The Reynolds
stresses are modelled using the eddy viscosity approach. A constant turbulent Prandtl number
is assumed as can be concluded from DNS data. Then the eddy viscosity is directly linked to the
turbulent thermal diffusivity and therefore is a linear function of wall distance. Once all terms
are expressed as a function of wall distance the momentum equation can be integrated and a
velocity profile emerges. This profile is in good agreement with DNS and experimental data.
Straightforward numerical solutions without adequate near wall treatment, like with FLU-
ENT 6.2, to reproduce the DNS data of Versteegh and Nieuwstadt [3] for Ra = 5.0 · 10 6 show
that even with fine grids also in the viscous sublayer only poor agreement can be achieved.
Thus, we conclude that it would be diserable to have a new approach and improved near wall
treatment.
Therefore, we apply the new universal profiles as wall functions for CFD calculations. They
are implemented in the two dimensional CAFFA code of Ferziger and Peric [6] that uses the kω-turbulence
model and the Boussinesq-approximation. The standard k-ω model is used inspite
Speaker: HÖLLING, M. 91 BAIL 2006
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