The standard wall functions in
FLUENT are based on the proposal of Launder
and Spalding [
197], and have been most widely used for
industrial flows. They are provided as a default option in
FLUENT.
Momentum
The lawofthewall for mean velocity yields
(12.101)
where
(12.102)
(12.103)
and
=
von Kármán constant (= 0.4187)
=
empirical constant (= 9.793)
=
mean velocity of the fluid at point
=
turbulence kinetic energy at point
=
distance from point
to the wall
=
dynamic viscosity of the fluid
The logarithmic law for mean velocity is known to be valid for
.
In
FLUENT, the loglaw is employed when
.
When the mesh is such that
at the walladjacent cells,
FLUENT
applies the laminar stressstrain relationship that can be written as
(12.104)
It should be noted that, in
FLUENT, the lawsofthewall for mean velocity
and temperature are based on the wall unit,
, rather than
(
).
These quantities are approximately equal in equilibrium
turbulent boundary layers.
Energy
Reynolds' analogy between momentum and energy transport gives a similar
logarithmic law for mean temperature. As in the lawofthewall for mean
velocity, the lawofthewall for temperature employed
in
FLUENT comprises the following two different laws:
linear law for the thermal conduction sublayer where conduction is
important
logarithmic law for the turbulent region where effects of turbulence
dominate conduction
The thickness of the thermal conduction layer is, in general, different from the
thickness of the (momentum) viscous sublayer, and changes from fluid to fluid.
For example, the thickness of the thermal sublayer for a highPrandtlnumber
fluid (e.g., oil) is much less than its momentum sublayer thickness. For fluids
of low Prandtl numbers (e.g., liquid metal), on the contrary, it is much larger
than the momentum sublayer thickness.
In highly compressible flows, the temperature distribution in the nearwall
region can be significantly different from that of low subsonic flows, due to
the heating by viscous dissipation. In
FLUENT, the temperature wall functions
include the contribution from the viscous heating [
382].
The lawofthewall implemented in
FLUENT has the following composite form:
(12.105)
where
is computed by using the formula given by Jayatilleke [
165]:
(12.106)
and
=
turbulent kinetic energy at point P
=
density of fluid
=
specific heat of fluid
=
wall heat flux
=
temperature at the cell adjacent to wall
=
temperature at the wall
=
molecular Prandtl number (
)
=
turbulent Prandtl number (0.85 at the wall)
=
Van Driest constant (= 26)
=
mean velocity magnitude at
Note that, for the pressurebased solver, the terms
and
will be included in Equation
12.105 only for compressible flow calculations.
The nondimensional thermal sublayer thickness,
, in Equation
12.105
is computed as the
value at which the linear law and the logarithmic law
intersect, given the molecular Prandtl number of the fluid being modeled.
The procedure of applying the lawofthewall for temperature is as follows.
Once the physical properties of the fluid being modeled are specified, its
molecular Prandtl number is computed. Then, given the molecular Prandtl
number, the thermal sublayer thickness,
, is computed from the
intersection of the linear and logarithmic profiles, and stored.
During the iteration, depending on the
value at the nearwall cell, either
the linear or the logarithmic profile in Equation
12.105 is applied to
compute the wall temperature
or heat flux
(depending on the type
of the thermal boundary conditions).
The function for
given by equation Equation
12.106 is relevant for the smooth
walls. For the rough walls, however, this function is modified as follows:
(12.107)
where
is the wall function constant modified for the rough walls,
defined by
. To find a description of the roughness function
, you may refer to Equation
7.133 in Section
7.13.1.
Species
When using wall functions for species transport,
FLUENT assumes that
species transport behaves analogously to heat transfer. Similarly to
Equation
12.105, the lawofthewall for species can be expressed for
constant property flow with no viscous dissipation as
(12.108)
where
is the local species mass fraction,
and
are molecular and turbulent Schmidt numbers, and
is the diffusion flux
of species
at the wall. Note that
and
are calculated in a
similar way as
and
, with the difference being that the Prandtl
numbers are always replaced by the corresponding Schmidt numbers.
Turbulence
In the

models and in the RSM (if the option to obtain wall boundary
conditions from the
equation is enabled), the
equation is solved in the
whole domain including the walladjacent cells. The boundary condition for
imposed at the wall is
(12.109)
where
is the local coordinate normal to the wall.
The production of kinetic energy,
, and its dissipation rate,
,
at the walladjacent cells, which are the source terms in the
equation, are
computed on the basis of the local equilibrium hypothesis. Under this
assumption, the production of
and its dissipation rate are assumed to be
equal in the walladjacent control volume.
Thus, the production of
is computed from
(12.1010)
and
is computed from
(12.1011)
The
equation is not solved at the walladjacent cells, but instead is
computed using Equation
12.1011.
and Reynolds stress equations are solved as detailed in Sections
12.5.3 and
12.7.9, respectively.
Note that, as shown here, the wall boundary conditions for the solution
variables, including mean velocity, temperature, species concentration,
, and
, are all taken care of by the wall functions. Therefore, you do not
need to be concerned about the boundary conditions at the walls.
The standard wall functions described so far are provided as a default option in
FLUENT. The standard wall functions
work reasonably well for a broad range of wallbounded flows. However, they
tend to become less reliable when the flow situations depart too much from the
ideal conditions that are assumed in their derivation. Among others, the
constantshear and local equilibrium hypotheses are the ones that most restrict
the universality of the standard wall functions. Accordingly, when the
nearwall flows are subjected to severe pressure gradients, and when the flows
are in strong nonequilibrium, the quality of the predictions is likely to be
compromised.
The nonequilibrium wall functions offered as an additional option can improve
the results in such situations.
Standard wall functions are available with the following viscous models: