Wind Energy

12 Turbulence Modelling and Numerical Flow Simulation 67
12.4 Direct Numerical Simulation
Reliably solving (12.1) without any additional turbulence and/or transition
model in a DNS requires accurate numerical methods, which do not inhibit
significant artificial viscosity. Spectral methods, which provide very accurate
spatial discretization have been used successfully in the past mainly for simple
geometries. Besides spectral methods, finite difference methods based on
second- or fourth-order accurate central difference schemes are widely used
for DNS of turbulent flows. They are in general easily applicable in complex
computational domains. Further, as pointed out by Choi et al. [1] it is possible
to obtain “spectral” accuracy if the first and second order moments of
the velocity fluctuations are considered for the same number of grid points.
12.5 Statistical Turbulence Modelling
The unknown Reynolds stress tensor 〈u ′′
i u′′ j 〉 in (12.2) is a symmetric tensor,
which, for statistically three-dimensional flows, contains six different elements.
In most applied flow problems 〈u ′′
i u′′ j 〉 is approximated using one- or two equation
turbulence models. One popular two equation turbulence model is the
so-called (k, ω)-model by Wilcox [9] (ω does not correspond to the vorticity).
This model which allows to predict seperated flows better than the well-known
(k, ω)-model reads:
−〈u ′′
i u ′′
j 〉 = νt2〈Sij〉− 2
3 kδij, νt = k 〈u′′ i
, k =
ω u′′
2
i 〉
, ω = ε
cµk
(12.3)
with the unknown properties k and ω. Closure is obtained solving the following
transport equation for these two unknowns and using a set of empirical
constants (cµ, σk,cω1,cω2, σω).
〈ui〉 ∂k k
= cµ
∂xi
2
ε 2〈Sij〉〈Sij〉 + 1
Reτ
〈ui〉 ∂ω
= cω12〈Sij〉〈Sij〉−cω2ω
∂xi
2 + 1
Reτ
∂(1 + νt ∂k
σkν ) ∂xi
∂xi
∂(1 + νt ∂ω
σων ) ∂xi
∂xi
− ε, (12.4)
. (12.5)
In Fig. 12.1 the mean axial velocity component and the mean pressure computed
by Frahnert and Dallmann [2] in a RANS calculation of the fully
developed turbulent channel flow with the (k, ω)-model are compared to the
DNS results of Kim et al. [4] for a Reynolds number Reτ = uτH/ν = 360
(uτ denotes the friction velocity and H the channel height). While the profiles
of the mean axial velocity component 〈u〉 in Fig. 12.1 compare very well, the
mean pressure distributions in Fig. 12.1b reveal strong discrepancies. Considering
the comparison of the turbulence intensities in Fig. 12.2, which are the
four different components of the Reynolds stress tensor, good agreement is
obtained for the Reynolds stress 〈u ′
v ′
〉, but the normal stresses 〈u ′ ′
2 〉, 〈v 2〉
and 〈w ′ 2 〉 compare rather poorly.