The computation of turbulent natural convection flows - Turbulence ...

Numerical Implementation 128
In the solution sequence, in order to prevent steep changes in the vari-
ables because of the strong non-linearities of the transport equations, under-
relaxation is used to slow down the rate of the solution process. The under-
relaxation can be implemented as:
φP = αφ new
P +(1−α)φ old
P
(5.34)
where “old” and “new” superscripts refer to old and current iterations and the
under-relaxation factor α has a value 0 < α < 1.
This under-relaxation can be introduced directly to the discretized equa-
tions of momentum, temperature, k, ε, etc. as:
a φ
P
α φP = i a φ
iφi +S φ + (1−α)
α
a φ
Pφold P
(5.35)
In the pressure correction equation, it is implemented by adding only a fraction
ofP ′
to P∗ as:
P = P ∗ +αP ′
(5.36)
The under-relaxation α is set to 0.1 in all of the transport equations and
pressure-correction for all of the 2D simulations. For 3D steady-state simula-
tions of inclined cavities, 0.1 is used as under-relaxation factor. The value of
under-relaxation is set to 0.2 for the transport equations and 0.1 for pressure-
correction in 3D time-dependent simulations of inclined cavities. For the 3D
simulation of the horizontal circular cavity, 0.2 is the under-relaxation factor
for the transport equations while 0.1 is set for under-relaxation of pressure-
correction. The under-relaxation factors needed to be as low as the values
mentioned above, at least during some stages of the calculations, since com-
putations carried out with larger under-relaxation factors became unstable and
led to divergence.