Flow fields near wall boundaries are dominated by viscous forces
where flow velocity changes from zero on the wall to the mean free
stream velocity away from the wall surface. In fluid mechanics, the
boundary layer is the zone where the flow velocity changes from zero to
the %99 of the free stream velocity. In turbulent flows, wall boundaries
are the main source of turbulence, while, at the same time the
dissipation rate is the highest near the wall surfaces due to high
viscous forces. For low Re numbers, boundary layers may be fully
laminar. For high Re number turbulent flows, laminar zone in the
boundary layer becomes thinner. In this respect, turbulent boundary
layers are divided into two regions; outer layer and inner layer. In the
outer boundary layer, the flow is assumed to be fully turbulent while
the inner layer is further divided into three sub-zones; log layer,
buffer layer, and viscous zone. In the log layer, flow is equally
dominated by viscous and turbulent effects. In the viscous layer, flow
is assumed to be fully laminar while the buffer layer is the transition
zone between viscous and log layers.
Each layer requires a different numerical approach if a direct
numerical simulation (DNS) mesh is not used for these zones. Viscous
zones may be easily resolved for many problems, however, for other zones
analytical/empirical approaches are required.
For low Re number flows, some turbulence models such as models can be solved to the wall
surface with some damping functions accounting for the vorticity
vorticity-blocking effect of the solid surfaces. In this case, still, a
mesh resolution of is still
needed with finely resolved boundary layer mesh. In the case of high Re
number flows, a hybrid turbulence model may be used such as SST model, in which, model is used in high Re
free flow zones while is
used for low Re number near wall zones. This type of approach resolving
boundary layer with a proper turbulence model is called
Near-Wall Turbulence Modelling (NWTM).
Another approach is to provide semi-empirical boundary conditions for
all flow variables effected by the wall boundary. This approach is
called Wall Treatment (WT) or Wall
Functions method. Wall functions impose values of transported
values on the cell centers of the near wall cells. In NeuralFlow, both VWTM
and wall functions are used in NeuralFlow for SST method. For only wall function methods
must be used.
16.1 Wall Treatment
Wall functions are some type of semi-empirical formulations providing
boundary condition-like values for near-wall cell center values. In
literature there are mainly two types of wall functions;
Standard Wall Functions and Blended Wall
Functions (or y+ independent wall functions).
Standard wall functions are only available for the viscous sub-layer
and the log layer. Since the viscous sub-layer is assumed to exist for
, these functions are
generally used for
boundary cell resolutions where the log layer starts. It is generally
assumed that a buffer layer exists for .
Normalized mean velocity profile in a turbulent boundary
layer in semi-log coordinates (Schultz et al. 2010)
independent wall functions
cover all three sublayers up to and enable the use of various
size boundary layer meshes with very high or low resolution. These
functions automatically adjust if a high-resolution boundary layer
solution is required such as at .
Wall functions are generally defined in terms of some non-dimensional
quantities which are described in the following sections.
16.1.1 Non-dimensional Wall
Distance
Non-dimensional wall distance or simply y plus is defined as; (16.1) where is the friction velocity (velocity
scale) at the near wall cell described in Section 16.1.5 and is the distance to the nearest
wall.
16.1.2 Wall Tangential
Non-dimensional Velocity
Dimensionless wall tangential component of the velocity is given by; (16.2)
16.1.3 Dimensionless
Temperature
(16.3) where is the wall heat flux and
is the wall surface
temperature.
16.1.4 Dimensionless Turbulence
Values
Turbul eddy viscosity; (16.4) Turbulent kinetic energy; (16.5) Production of turbulent kinetic energy;
(16.6) Dissipation rate; (16.7) Specific
dissipation rate; (16.8)
16.1.5 Velocity Scale
Friction velocity represents the value of the velocity component
tangent to the wall at near wall cell center. The formulation for this
value depends on the used wall function type and value. Remember that in the case of
standard wall functions values for buffer zone are not available.
For standard wall functions; (16.9) where is a model coefficient.
For independent wall
treatment; (16.10) where is a blending function; (16.11) where
is the wall distance
Reynolds number; (16.12)
Wall tangential velocity is calculated as;
(16.13) where and
indicate near cell center and
wall surface values, and is
wall face normal unity vector.
16.1.6 Velocity Wall Functions
Velocity wall functions provide the distribution of in a turbulent boundary layer.
For standard wall treatment; (16.14) where is the von Karman constant
is the modified log law
offset defined as; (16.15) where is the log law offset and is the roughness function which is 1.0
for smooth surfaces.
Velocity wall function for independent treatment based
on the paper (Reichardt
1951) is described by; (16.16) where; (16.17)(16.18)(16.19)
Normalized mean velocity profile in a turbulent boundary and
wall function fit of Reichardt (Reichardt 1951)
16.1.7 Temperature Wall
Functions
The temperature wall functions describe the distribution of
dimensionless temperature in
the turbulent boundary sublayers.
The standard temperature wall function is defined
as; (16.20) where Pr is the Prandtl number; (16.21) where is the thermal conductivity.
is the turbulent
Prandtl number. is
the von Karman constant.
is defined in Eq. (16.15). is a dimensionless parameter known as
the -function, which accounts for
the resistance to heat transfer across the viscous sublayer (Malin 1987). Several sublayer resistance
functions are proposed in the literature. In NeuralFlow, the function is taken
from (Jayatilleke
1966) and defined as; (16.22) where .
independent wall
function for is
described as; This approach is taken from Kader’s study (Kader 1981)(16.23) where; (16.24)(16.25) where is the roughness function
which is 1 for smooth surfaces.
16.1.8 Modification for
Compressible and Separated Boundary Layers
16.1.9 Turbulent Dissipation Wall
Function
Dimensionless turbulent dissipation wall function at the center of a near
wall center is calculated by the following wall functions;
Standard wall function for is described as;
(16.26) where is the von Karma
constant.
independent wall
function for (16.27) where is given in Eq. (16.11).
16.1.10 Specific Turbulent
Dissipation Wall Function
Standard wall function for is described as; (16.28) where is given in Eq. (14.19) and is a model constant.
independent wall
function for is
described as; (16.29)
16.1.11 Near Wall Production Of
Turbulent Kinetic Energy
There are no wall functions available for turbulent kinetic energy
, however, their dimensionless
near wall production can be
described with wall functions. Standard wall function for is given by; (16.30) where
since for in the case of standard velocity
wall function (see Eq.(16.14)). is the von Karman constant.
independent wall
function for is
given by; (16.31)
16.2 Wall Treatment for Momentum
Transfer
Wall shear stress near a wall is always higher for turbulent boundary
layers due to higher mixing. Wall shear stress is calculated as: (16.32) where is the wall friction velocity and
described as; (16.33)
where is calculated as given in
Sec.16.1.5 and is calculated as in Sec.16.1.6.
16.3 Wall Treatment of Heat
Flux
For turbulent boundary layers, the wall heat flux is calculated as:
(16.34) where is calculated as given in Sec.16.1.5 and is calculated as in Sec.16.1.7.
