Wind Energy

66 C. Wagner
12.3 Governing Equations
In (12.1) the dimensionless incompressible Navier–Stokes equations which contain
the Reynolds number Re are presented in cartesian coordinates using
Einstein’s summation convention
∂ui
=0,
∂xi
∂ui
∂t
+ ∂uiuj
∂xj
= − ∂p
+
∂xi
1
Re
�
∂ ∂ui
+
∂xj ∂xj
∂uj
�
. (12.1)
∂xi
� �� �
Almost any known analytical solution of (12.1) is valid only in the laminar
regime, i.e. for low Reynolds numbers. For higher Reynolds numbers, disturbances
which are introduced for example by imperfect or rough walls tend to
grow into three-dimensional vortical structures. Due to stretching and tilting
of the associated vortex lines an initially simple flow pattern changes into
complicated turbulence, which is characterized by irregularity in space and
time, increased dissipation, mixing and nonlinearity.
Often, one is not interested in all details of the time-dependent threedimensional
velocity and pressure field, but in the behaviour of the statistical
mean or of the large scales. To reduce the level of detail of the solution any
turbulent variable f is split into a filtered value 〈f〉 and its fluctuation f −〈f〉.
The corresponding filtering of (12.1) leads to the filtered Navier–Stokes
equations which contains the extra nonlinear term τij in comparison to (12.1).
∂〈ui〉
=0,
∂xi
∂〈ui〉 ∂〈ui〉〈uj〉
+ +
∂t ∂xj
∂τij
= −
∂xj
∂〈p〉
+
∂xi
1
�
∂ ∂〈ui〉
+
Re ∂xj ∂xj
∂〈uj〉
�
.
∂xi
� �� �
2〈Sij〉
(12.2)
Using statistical averaging as a filter function one obtains the so-called
Reynolds stress tensor τij = 〈u ′′
i u′′ j 〉, where u′′ i and u′′ j denote the statistical
velocity fluctuations. Then (12.2) represents the RANS equations. The statistical
approach is associated with the highest loss of information and with
a closure problem which is not satisfactorily solved. Spectral information is
completely lost, since any statistical quantity is an average over all turbulent
scales. The obtained flow field describes the mean flow, which is enough for
many applied problems, while the turbulent information is described with the
Reynolds stress tensor.
The LES technique resembles a compromise between RANS and DNS since
it allows to predict the dynamics of the large turbulent scales while the effect
of the fine scales are modeled with a subgrid-scale model. The governing
equations which have to be solved in a LES are also derived applying a filter
function on the Navier–Stokes equations (12.1) to formally remove scales with
a wavelength smaller than the grid mesh. To distinguish top-hat filtering from
statistical averaging in (12.2) 〈ui〉 and 〈p〉 are substituted by ui and p and
the subgrid scale tensor τij = u ′
iu′ j where u′ i denotes the subgrid scale velocity
fluctuations.
2Sij