4.12.2 Standard Wall Functions

The standard wall functions in ANSYS FLUENT are based on the work of Launder and Spalding [ 181], and have been most widely used in industrial flows. They are provided as a default option in ANSYS FLUENT.

Momentum

The law-of-the-wall for mean velocity yields

where

is the dimensionless velocity.

is the dimensionless distance from the wall.

and	$\kappa$	=	von Kármán constant (= 0.4187)
	$E$	=	empirical constant (= 9.793)
	$U_P$	=	mean velocity of the fluid at the near-wall node $P$
	$k_P$	=	turbulence kinetic energy at the near-wall node $P$
	$y_P$	=	distance from point $P$ to the wall
	$\mu$	=	dynamic viscosity of the fluid

The logarithmic law for mean velocity is known to be valid for $30 < y^* < 300$. In ANSYS FLUENT, the log-law is employed when $y^* > 11.225$.

When the mesh is such that $y^* < 11.225$ at the wall-adjacent cells, ANSYS FLUENT applies the laminar stress-strain relationship that can be written as

It should be noted that, in ANSYS FLUENT, the laws-of-the-wall for mean velocity and temperature are based on the wall unit, $y^*$, rather than $y^+$ ( $\equiv \rho u_{\tau} y/\mu$). 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 law-of-the-wall for mean velocity, the law-of-the-wall for temperature employed in ANSYS FLUENT comprises the following two different laws:

• linear law for the thermal conduction sublayer, or thermal viscous 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 high-Prandtl-number 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 near-wall region can be significantly different from that of low subsonic flows, due to the heating by viscous dissipation. In ANSYS FLUENT, the temperature wall functions include the contribution from the viscous heating [ 357].

The law-of-the-wall implemented in ANSYS FLUENT has the following composite form:

where $P$ is computed by using the formula given by Jayatilleke [ 150]:

and

	$k_P$	=	turbulent kinetic energy at the first near-wall node P
	$\rho$	=	density of fluid
	$c_p$	=	specific heat of fluid
	$\dot{q}$	=	wall heat flux
	$T_P$	=	temperature at the first near-wall node P
	$T_w$	=	temperature at the wall
	${\rm Pr}$	=	molecular Prandtl number ( $\mu c_p/k_f$)
	${\rm Pr}_t$	=	turbulent Prandtl number (0.85 at the wall)
	$A$	=	Van Driest constant (= 26)
	$U_c$	=	mean velocity magnitude at $y^{*}=y_T^{*}$

Note that, for the pressure-based solver, the terms

and

will be included in Equation  4.12-6 only for compressible flow calculations.

The non-dimensional thermal sublayer thickness, $y^*_T$, in Equation  4.12-6 is computed as the $y^*$ 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 law-of-the-wall 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, $y^*_T$, is computed from the intersection of the linear and logarithmic profiles, and stored.

During the iteration, depending on the $y^*$ value at the near-wall cell, either the linear or the logarithmic profile in Equation  4.12-6 is applied to compute the wall temperature $T_w$ or heat flux $\dot{q}$ (depending on the type of the thermal boundary conditions).

The function for $P$ given by Equation  4.12-7 is relevant for the smooth walls. For the rough walls, however, this function is modified as follows:

where $E'$ is the wall function constant modified for the rough walls, defined by $E' = E/f_r$. To find a description of the roughness function $f_r$, you may refer to this equation in this section in the separate User's Guide.

Species

When using wall functions for species transport, ANSYS FLUENT assumes that species transport behaves analogously to heat transfer. Similarly to Equation  4.12-6, the law-of-the-wall for species can be expressed for constant property flow with no viscous dissipation as

where $Y_{i}$ is the local species mass fraction, ${\rm Sc}$ and ${\rm Sc}_t$ are molecular and turbulent Schmidt numbers, and $J_{i,w}$ is the diffusion flux of species $i$ at the wall. Note that $P_c$ and $y^*_c$ are calculated in a similar way as $P$ and $y^*_T$, with the difference being that the Prandtl numbers are always replaced by the corresponding Schmidt numbers.

Turbulence

In the $k$- $\epsilon$ models and in the RSM (if the option to obtain wall boundary conditions from the $k$ equation is enabled), the $k$ equation is solved in the whole domain including the wall-adjacent cells. The boundary condition for $k$ imposed at the wall is

where $n$ is the local coordinate normal to the wall.

The production of kinetic energy, $G_k$, and its dissipation rate, $\epsilon$, at the wall-adjacent cells, which are the source terms in the $k$ equation, are computed on the basis of the local equilibrium hypothesis. Under this assumption, the production of $k$ and its dissipation rate are assumed to be equal in the wall-adjacent control volume.

Thus, the production of $k$ is based on the logarithmic law and is computed from

and $\epsilon$ is computed from

The $\epsilon$ equation is not solved at the wall-adjacent cells, but instead is computed using Equation  4.12-12. $\omega$ and Reynolds stress equations are solved as detailed in Sections  4.5.3 and 4.9.9, respectively.

Note that, as shown here, the wall boundary conditions for the solution variables, including mean velocity, temperature, species concentration, $k$, and $\epsilon$, 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 ANSYS FLUENT. The standard wall functions work reasonably well for a broad range of wall-bounded flows. However, they tend to become less reliable when the flow situations depart from the ideal conditions that are assumed in their derivation. Among others, the constant-shear and local equilibrium assumptions are the ones that most restrict the universality of the standard wall functions. Accordingly, when the near-wall flows are subjected to severe pressure gradients, and when the flows are in strong non-equilibrium, the quality of the predictions is likely to be compromised.

The non-equilibrium wall functions offered as an additional option that can potentially improve the results in such situations.

Standard wall functions are available with the following viscous models: k- $\epsilon$ models Reynolds Stress models

Scalable Wall Functions

The purpose of scalable wall functions is to force the usage of the log law in conjunction with the standard wall functions approach. This is achieved by introducing a limiter in the y* calculations such that

where $y*_{limit} = 11.225$. The use of Equation  4.12-13 in the context of the scalable wall functions concept is straightforward, i.e. the y* formulation used for any standard wall function formula is replaced by ${y*}{\widetilde{}}$.

Scalable wall functions can be enabled only when the standard wall functions are used and only via the following text command:

define/models/viscous/near-wall-treatment/scalable-wall-functions?