Mathematical Formulation
The wall -function approach in ANSYS CFX is an extension of the method of Launder and
Spalding [13]. In the log-law region, the near wall tangential velocity is related to the wall -shear -stress, , by means of a logarithmic relation.
In the wall -function approach, the viscosity affected sublayer region is bridged by employing empirical formulas to provide near-wall boundary conditions for the mean flow and turbulence transport
equations. The formulas connect the wall conditions (e.g., the wall -shear -stress) to the dependent variables at the near-wall mesh node which is presumed to lie in the fully-turbulent region of the
boundary layer.
The logarithmic relation for the near wall velocity is given by: where:
u + is the near wall
velocity, is the friction velocity, U t is the known velocity tangent to the wall
at a distance of from the wall , y
+ is the dimensionless distance from the wall ,
is the
wall shear stress, is the von Karman constant and C is a log-layer constant depending on wall
roughness (natural logarithms are ud).
A definition of
in the different wall
formulations is available. For details, e Solver Yplus and
Yplus. Scalable Wall Functions Equation 164 has the problem that it becomes singular at paration points where the near wall velocity, U t , approaches zero. In the logarithmic region, an alternative velocity scale, u* can be ud instead of u +:
This scale has the uful property that it does not go to zero if U t goes to zero (in turbulent flow k is never completely zero). Bad on this definition, the following explicit equation for
can be
obtained: The absolute value of the wall shear stress , is then obtained from:
where: and u * is as defined earlier.
Modeling Flow Near the Wall
Prev Up / Home Wall Distance Formulation Next
Equation 164.
Equation 165.
Equation 166.
Equation 167.
Equation 168.
Equation 169.
Equation 170.
ANSYS CFX-Solver Theory Guide | Turbulence and Wall Function Theory | Modeling Flow Near the Wall |
One of the major drawbacks of the wall -function approach is that the predictions depend on the location of the point nearest to the wall and are nsitive to the near-wall meshing; refining the mesh does not necessarily give a unique solution of increasing accuracy (Grotjans and Menter [10]). The problem of inconsistencies in the wall -function, in the ca of fine meshes, can be overcome with the u of the Scalable Wall Function formulation developed by ANSYS CFX. It can be applied on arbitrarily fine meshes and allows you to perform a consistent mesh refinement independent of the Reynolds number of the application.
The basic idea behind the scalable wall -function approach is to limit the y * value ud in the logarithmic formulation by a lower value of . 11.06 is the interction between the
logarithmic and the linear near wall profile. The computed is therefore not allowed to fall below this limit. Therefore, all mesh points are outside the viscous sublayer and all fine mesh inconsistencies are avoided.
It is important to note the following points:
●To fully resolve the boundary layer, you should put at least 10 nodes into the boundary layer.
●Do not u Standard Wall Functions unless required for backwards compatibility.
●The upper limit for y + is a function of the device Reynolds number. For example, a large ship may have a Reynolds number of 109 and y + can safely go to values much greater than 1000. For lower
Reynolds numbers (e.g., a small pump), the entire boundary layer might only extend to around
y + = 300. In this ca, a fine near wall spacing is required to ensure a sufficient number of nodes in the boundary layer.
If the results deviate greatly from the ranges, the mesh at the designated Wall boundaries will require modification, unless wall shear stress and heat transfer are not important in the simulation.
Solver Yplus and Yplus
In the solver output, there are two arrays for the near wall spacing. The definition for the Yplus
variable that appears in the post processor is given by the standard definition of
generally ud in CFD:
where is the distance between the first and cond grid points off the wall .
In addition, a cond variable, Solver Yplus, is available which contains the ud in the logarithmic profile by the solver. It depends on the type of wall treatment ud, which can be one of three different treatments in ANSYS CFX. They are bad on different distance definitions and velocity scales. This has partly historic reasons, but is mainly motivated by the desire to achieve an optimum performance in terms of accuracy and robustness:
●
Standard wall function (bad on ) ●
Scalable wall function (bad on ) ●Automatic wall treatment (bad on )
The scalable wall function y + is defined as:
and is therefore bad on 1/4 of the near wall grid spacing.
Note that both the scalable wall function and the automatic wall treatment can be run on arbitrarily fine meshes.
Equation 171.
Equation 172.
Automatic Near-Wall Treatment for Omega-Bad Models
While the wall -functions prented above allow for a consistent mesh refinement, they are bad on physical assumptions which are problematic, especially in flows at lower Reynolds numbers (Re<105), as the sublayer portion of the boundary layer is neglected in the mass and momentum bal
ance. For flows at low Reynolds numbers, this can cau an error in the displacement thickness of up to 25%. It is therefore desirable to offer a formulation which will automatically switch from wall -functions to a low-Re near wall formulation as the mesh is refined. The
model of Wilcox has the advantage that an analytical expression is known for in the viscous sublayer, which can be exploited to achieve this goal. The main idea behind the prent formulation is to blend the wall value for between the
logarithmic and the near wall formulation. The flux for the k -equation is artificially kept to be zero and the flux in the momentum equation is computed from the velocity profile. The equations are as follows: Flux for the momentum equation, F U :
with:
Flux for the
k -equation: In the -equation, an algebraic expression is specified instead of an added flux. It is a blend between
the analytical expression for
in the logarithmic region: and the corresponding expression in the sublayer:
with being the distance between the first and the cond mesh point. In order to achieve a smooth
blending and to avoid cyclic convergence behavior, the following formulation is lected: While in the wall -function formulation, the first point is treated as being outside the edge of the viscous sublayer, the location of the first mesh point is now virtually moved down through the viscous sublayer as the mesh is refined in the low-
Re mode. It is to be emphasized, that the physical location of the first mesh point is always at the wall (y = 0
). The error in the wall -function formulation results from this virtual shift, which amounts to a reduction in displacement thickness. This error is always prent in the wall -function mode, but is reduced to zero as the method shifts to the low-Re model. The shift is bad on the distance between the first and the cond mesh point y = y 2 - y 1 with y being the wall normal distance. Treatment of Rough Walls
The above wall function equations are appropriate when the walls can be considered as hydraulically smooth. For rough walls , the logarithmic profile still exists, but moves clor to the wall . Roughness
Equation 173.
Equation 174.
Equation 175.
Equation 176.
Equation 177.
Equation 178.
Equation 179.
effects are accounted for by modifying the expression for u + as follows:
where:
and y R is the equivalent sand grain roughness [14].
You are advid to exerci care in the u of the rough wall option together with the standard wall -function approach. Inaccuracies can ari if an equivalent sand grain roughness is of the same order, or
larger than the distance from the wall to the first node.
The first element off the
wall should not be much thinner than the equivalent sand grain roughness. If this element becomes too thin, then negative values of u +
can be calculated, resulting in the ANSYS CFX-Solver failing. For details, e Wall Roughness.
Heat Flux in the Near-Wall Region
The thermal boundary layer is modeled using the thermal law-of-the-wall
function of B.A. Kader [15]. Heat flux at the wall can be modeled using a wall function approach or the automatic wall treatment. Using similar assumptions as tho above, the non-dimensional near-wall temperature profile follows a universal profile through the viscous sublayer and the logarithmic region. The non-dimensional
temperature, T +, is defined as:
where T w is the temperature at the wall , T f the near-wall fluid temperature, c p the fluid heat capacity and q w the heat flux at the wall . The non-dimensional temperature distribution is then modeled as:
where: Pr is the fluid Prandtl number, given by:
where is the fluid thermal conductivity. Combining the equations leads to a simple form for the wall heat flux model:
Turbulent fluid flow and heat transfer problems without conjugate heat transfer objects require the specification of the wall heat flux, q w , or the wall temperature, T w . The energy balance for each
boundary control volume is completed by multiplying the wall heat flux by the surface area and adding to the corresponding boundary control volume energy equation. If the wall temperature is specified, the wall heat flux is computed from the equation above, multiplied by the surface area and added to the boundary energy control volume equation.
Equation 180.
Equation 181.
Equation 182.
Equation 183.
Equation 184.
Equation 185.
Equation 186.
Equation 187.
Additional Variables
The treatment of additional scalar variables in the near wall region is similar to that for heat flux.
Treatment of Compressibility Effects
With increasing Mach number (Ma > 3), the accuracy of the wall -functions approach degrades, which can result in substantial errors in predicted shear stress, wall heat transfer and wall temperature for supersonic flows.
It has been found that the incompressible law-of-the-wall is also applicable to compressible flows if the velocity profile is transformed using a so-called ”Van Driest transformation” [16]. The logarithmic velocity profile is given by:
where , , = 0.41 and C = 5.2. The subscript w refers to wall conditions, and the subscript “comp” refers to a velocity defined by the following equation
(transformation):
Near a solid wall , the integrated near-wall momentum equation reduces to:
while the energy equation reduces to: Expressions for shear stress and heat flux applicable to the boundary layer region are:
and If Equation 190, Equation 192 and Equation 193 are substituted into Equation 191 and integrated, the resulting equation is:
which provides a relationship between the temperature and velocity profiles. Using the perfect gas law, and the fact that the pressure is constant across a boundary layer, Equation 194 replaces the density ratio found in Equation 189. Performing the integration yields the following equation for the “compressible” velocity:
where:
Equation 188.
Equation 189.
Equation 190.
Equation 191.
Equation 192.
Equation 193.
Equation 194.
Equation 195.
Equation 196.
Equation 197.Equation 198.
