When the accuracy provided by wall functions in the k-ε model is not enough, a so called
low Reynolds number model can be used. “Low Reynolds number” refers to the region close to the wall where viscous effects dominate.
Most low Reynolds number k-ε models adapt the turbulence transport equations by introducing damping functions. This module includes the AKN model (after the inventors Abe, Kondoh, and Nagano;
Ref. 12). The AKN
k-
ε model for compressible flows reads (
Ref. 8 and
Ref. 12):
lw is the distance to the closest wall.
Realizability Constraints are applied to the low Reynolds number
k-
ε model.
The wall distance variable, lw, is provided by a mathematical Wall Distance interface that is included when using the low Reynolds number
k-
ε model. The solution to the wall distance equation is controlled using the parameter
lref. The distance to objects larger than
lref is represented accurately, while objects smaller than
lref are effectively diminished by appearing to be farther away than they actually are. This is a desirable feature in turbulence modeling since small objects would get too large an impact on the solution if the wall distance were measured exactly.
δw is the distance to the closest wall. The boundary conditions for the momentum equations are a no-penetration condition
u ⋅ n = 0 and a shear stress condition
Here, κv, is the von Kármán constant (default value 0.41),
U|| is the velocity parallel to the wall, and
B is a constant that by default is set to 5.2.
These expression can be combined with the lift-off concept shown in Figure 3-7 which gives
δw = hw/2. The
k-equation formally fulfills
both at the wall and in the log-layer, so this condition is applied for all
δw+.
The conditions for the turbulent dissipation, ε, is given by the Wolfshtein model, which is commonly employed in two-layer
k-
ε implementations (
Ref. 13):
The resulting wall resolution, δw+, is available as the postprocessing variable.
Delta_wPlus.
The damping terms in the equations for k and
ε allow for a no slip condition to be applied to the velocity, that is
u = 0 which is the case when
Wall Treatment is set to
Low Re.
where n is the wall normal direction. This condition is however numerically very unstable. Therefore,
ε is not solved for in the cells adjacent to a solid wall and instead the analytical relation
is prescribed in those cells (using the variable εw, which only exists in those cells).
Equation 3-152 can be derived as the first term in a series expansion of
is the distance, measured in viscous units, from the wall to the center of the wall adjacent cell. The boundary variable Distance to cell center in viscous units,
lplus_cc, is available to ensure that the mesh is fine enough. Observe that it is unlikely that a solution is obtained at all if
The guidelines given in Inlet Values for the Turbulence Length Scale and Turbulent Intensity for selecting the turbulence length scale,
LT, and the turbulence intensity,
IT, apply also to the low-Reynolds number
k-
ε model.
The low-Reynolds number k-
ε model has the same default initial guess as the standard
k-
ε model (see
Initial Values) but with
replaced by
lref.
In some cases, especially for stationary solutions, a fast way to convergence is to first solve the model using the ordinary k-
ε model and then to use that solution as an initial guess for the low-Reynolds number
k-
ε model. The procedure is then as follows:
The low-Reynolds number k-
ε model applies absolute scales of the same type as the
k-
ε model (see
Scaling for Time-Dependent Simulations).