Standard Wall Functions

In document ANSYS Fluent Theory Guide.pdf (Page 156-160)

Using This Manual

Chapter 3: Flows Using Sliding and Dynamic Meshes

4.16. Near-Wall Treatments for Wall-Bounded Turbulent Flows

4.16.2. Standard Wall Functions

The standard wall functions in ANSYS Fluent are based on the work of Launder and Spalding [246] (p. 772), and have been most widely used in industrial flows. They are provided as a default option in ANSYS

Fluent.

4.16.2.1. Momentum

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

(4.302) where

(4.303)

is the dimensionless velocity.

(4.304)

is the dimensionless distance from the wall.

and

= von Kármán constant (= 0.4187) = empirical constant (= 9.793)

= mean velocity of the fluid at the wall-adjacent cell centroid, = turbulence kinetic energy at the wall-adjacent cell centroid, = distance from the centroid of the wall-adjacent cell to the wall, Turbulence

= dynamic viscosity of the fluid

The range of values for which wall functions are suitable depend on the overall Reynolds number of the flow. The lower limit always lies in the order of ~15. Below this limit, wall functions will typically deteriorate and the accuracy of the solutions cannot be maintained (for exceptions, see Scalable Wall Functions (p. 122)). The upper limit depends strongly on the Reynolds number. For very high Reynolds numbers (for example, ships, airplanes), the logarithmic layer can extend to values as high as several thousand, whereas for low Reynolds number flows (for example, turbine blades, and so on.) the upper limit can be as small as 100. For these low Reynolds number flows, the entire boundary layer is frequently only of the order of a few hundred units. The application of wall functions for such flows should therefore be avoided as they limit the overall number of nodes one can sensibly place in the boundary layer. In general, it is more important to ensure that the boundary layer is covered with a sufficient number of (structured) cells than to ensure a certain value.

In ANSYS Fluent, the log-law is employed when . When the mesh is such that at the wall-adjacent cells, ANSYS Fluent applies the laminar stress-strain relationship that can be written as

(4.305) It should be noted that, in ANSYS Fluent, the law-of-the-wall for mean velocity and temperature are

based on the wall unit, , rather than ( ). These quantities are approximately equal in equilibrium turbulent boundary layers.

4.16.2.2. 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 (mo-mentum) viscous sublayer, and changes from fluid to fluid. For example, the thickness of the thermal sublayer for a high-Prandtl-number fluid (for example, oil) is much less than its momentum sublayer thickness. For fluids of low Prandtl numbers (for example, liquid metal), on the contrary, it is much larger than the momentum sublayer thickness.

(4.306) 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 [489] (p. 786).

The law-of-the-wall is implemented in ANSYS Fluent for the non-dimensional temperature using the wall scaling:

(4.307)

Here the convective-conductive part and viscous heating part are modeled using the following composite forms:

Near-Wall Treatments for Wall-Bounded Turbulent Flows

(4.308)

(4.309)

where is computed by using the formula given by Jayatilleke [201] (p. 770):

(4.310)

and

= turbulent kinetic energy at the wall-adjacent cell centroid, = density of fluid

= specific heat of fluid = wall heat flux

= temperature at the wall-adjacent cell centroid, = 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 pressure-based solver, the terms

and

will be included in Equation 4.307 (p. 119) only for compressible flow calculations.

The non-dimensional thermal sublayer thickness, , in Equation 4.307 (p. 119) 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 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, , is computed from the intersection of the linear and logarithmic profiles, and stored.

During the iteration, depending on the value at the near-wall cell, either the linear or the logarithmic profile in Equation 4.307 (p. 119) is applied to compute the wall temperature or heat flux (depending on the type of the thermal boundary conditions).

Turbulence

The function for given by Equation 4.310 (p. 120) is relevant for the smooth walls. For the rough walls, however, this function is modified as follows:

(4.311)

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 6.99 in Wall Roughness Effects in Turbulent Wall-Bounded Flows in the User's Guide.

4.16.2.3. Species

When using wall functions for species transport, ANSYS Fluent assumes that species transport behaves analogously to heat transfer. Similarly to Equation 4.307 (p. 119), the law-of-the-wall for species can be expressed for constant property flow with no viscous dissipation as

(4.312)

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 correspond-ing Schmidt numbers.

4.16.2.4. 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 wall-adjacent cells. The

boundary condition for imposed at the wall is

(4.313) where is the local coordinate normal to the wall.

The production of kinetic energy, , and its dissipation rate, , at the wall-adjacent 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 wall-adjacent control volume.

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

(4.314)

and is computed from

(4.315)

The equation is not solved at the wall-adjacent cells, but instead is computed using Equa-tion 4.315 (p. 121). and Reynolds stress equations are solved as detailed in Wall Boundary Condi-tions (p. 65) and Wall Boundary CondiCondi-tions (p. 89), respectively.

Near-Wall Treatments for Wall-Bounded Turbulent Flows

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 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 are offered as an additional option, which can potentially improve the results in such situations.

Important

Standard wall functions are available with the following viscous models:

• - models

• Reynolds Stress models

In document ANSYS Fluent Theory Guide.pdf (Page 156-160)