turbulence and near wall theory

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CFX-5 Solver Theory

Turbulence and Wall Function Theory

• Turbulence Models (p. 60)

• Eddy Viscosity Turbulence Models (p. 63)

• Reynolds Stress Turbulence Models (p. 75)

• Large Eddy Simulation Theory (p. 82)

• Detached Eddy Simulation Theory (p. 86)

• Modelling Flow Near the Wall (p. 88)

• Wall Distance Formulation (p. 99)

Page 59 CFX-5.7.1

CFX-5 Solver TheoryTurbulence Theory

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Turbulence Models

Turbulence consists of fluctuations in the flow field in time and space. It is a complex process, mainly because it is three dimensional, unsteady and consists of many scales. It can have a significant effect on the characteristics of the flow. Turbulence occurs when the inertia forces in the fluid become significant compared to viscous forces, and is characterised by a high Reynolds Number.

In principle, the Navier-Stokes equations describe both laminar and turbulent flows without the need for additional information. However, turbulent flows at realistic Reynolds numbers span a large range of turbulent length and time scales and would generally involve length scales much smaller than the smallest finite volume mesh which can be practically used in a numerical analysis. The Direct Numerical Simulation (DNS) of these flows would require computing power which is many orders of magnitude higher than available in the foreseeable future.

To enable the effects of turbulence to be predicted, a large amount of CFD research has concentrated on methods which make use of turbulence models. Turbulence models have been specifically developed to account for the effects of turbulence without recourse to a prohibitively fine mesh and Direct Numerical Simulation. Most turbulence models are statistical turbulence model, as described below. The two exceptions to this in CFX-5 are the Large Eddy Simulation model (Large Eddy Simulation Theory (p. 82)) and the Detached Eddy Simulation model (Detached Eddy Simulation Theory (p. 86)).

Statistical Turbulence Models and the Closure Problem

When looking at time scales much larger than the time scales of turbulent fluctuations, turbulent flow could be said to exhibit average characteristics, with an additional time-varying, fluctuating component. For example, a velocity component may be divided into an average component, and a time varying component.

In general, turbulence models seek to modify the original unsteady Navier-Stokes equations by the introduction of averaged and fluctuating quantities to produce the Reynolds Averaged Navier-Stokes (RANS) equations. These equations represent the mean flow quantities only, while modelling turbulence effects without a need for the resolution of the turbulent fluctuations. All scales of the turbulence field are being modelled. Turbulence models based on the RANS equations are known as Statistical Turbulence Models due to the statistical averaging procedure employed to obtain the equations.

Simulation of the RANS equations greatly reduces the computational effort compared to a Direct Numerical Simulation and is generally adopted for practical engineering calculations. However, the averaging procedure introduces additional unknown terms containing products of the fluctuating quantities, which act like additional stresses in the fluid. These terms, called ‘turbulent’ or ‘Reynolds’ stresses, are difficult to determine directly and so become further unknowns.

Page 60 Turbulence Models CFX-5.7.1


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The Reynolds (turbulent) stresses need to be modelled by additional equations of known quantities in order to achieve “closure”. Closure implies that there is a sufficient number of equations for all the unknowns, including the Reynolds-Stress tensor resulting from the averaging procedure. The equations used to close the system define the type of turbulence model.

Reynolds Averaged Navier Stokes (RANS) Equations

As described above, turbulence models seek to solve a modified set of transport equations by introducing averaged and fluctuating components. For example, a velocity U may be divided into an average component, , and a time varying component, .

The averaged component is given by:

where Dt is a time scale that is large relative to the turbulent fluctuations, but small relative to the time scale to which the equations are solved.

Substituting the time averaged quantities into the original transport equations (see Transport Equations (p. 21)) results in the Reynolds-averaged equations given below. In the following equations, the bar is dropped for time-averaged quantities, except for products of fluctuating quantities.

(Eqn. 23)

(Eqn. 24)

where τ is the molecular stress tensor.

(Eqn. 25)

The continuity equation has not been altered but the momentum and scalar transport equations contain turbulent flux terms additional to the molecular diffusive fluxes. These are the Reynolds stress, , and the Reynolds flux, . These terms arise from the non-linear convective term in the un-averaged equations. They reflect the fact that convective transport due to turbulent velocity fluctuations will act to enhance mixing over and above

U u

U U u+=

U1∆t----- U td

t

t ∆t+

∫=

ρ∂t∂

------ ρU( )∇•+ 0=

ρU∂t∂

----------- ρU U⊗ ∇•+ τ ρ– u u⊗ SM+∇•=

ρφ∂t∂

--------- ρUφ( )∇•+ Γ∇φ ρ– uφ( ) SE+∇•=

ρu u⊗ ρuφ



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that caused by thermal fluctuations at the molecular level. At high Reynolds numbers, turbulent velocity fluctuations occur over a length scale much larger than the mean free path of thermal fluctuations, so that the turbulent fluxes are much larger than the molecular fluxes.

The Reynolds-averaged energy equation is:

(Eqn. 26)

where the mean Total Enthalpy is given by

In addition to the mean flow kinetic energy, the Total Enthalpy now contains a contribution from the turbulent kinetic energy, k, given by

Turbulence models close the Reynolds-averaged equations by providing models for the computation of the Reynolds stresses and Reynolds fluxes. CFX-5 models can be broadly divided into two classes: eddy viscosity models and Reynolds stress models.

ρhtot∂t∂

--------------- ρUhtot ρuh λ∇T–+( )∇•+ p∂t∂

------=

htot h12--U

2k+ +=

k12--u

2=



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Eddy Viscosity Turbulence Models

One proposal suggests that turbulence consists of small eddies which are continuously forming and dissipating, and in which the Reynolds stresses are assumed to be proportional to mean velocity gradients. This defines an ‘eddy viscosity model’.

The eddy viscosity hypothesis assumes that the Reynolds stresses can be related to the mean velocity gradients and Eddy (turbulent) Viscosity by the gradient diffusion hypothesis, in a manner analogous to the relationship between the stress and strain tensors in laminar Newtonian flow:

(Eqn. 27)

Here, µt is the Eddy Viscosity or Turbulent Viscosity. This has to be prescribed.

Analogous to the eddy viscosity hypothesis is the eddy diffusivity hypothesis, which states that the Reynolds fluxes of a scalar are linearly related to the mean scalar gradient:

(Eqn. 28)

Here, Γt is the Eddy Diffusivity, and this has to be prescribed. The Eddy Diffusivity can be written:

(Eqn. 29)

where Prt is the turbulent Prandtl number. Eddy diffusivities are then prescribed using the turbulent Prandtl number.

The above equations can only express the turbulent fluctuation terms of functions of the mean variables if the turbulent viscosity, µt, is known. Both the k-ε and k-ω two-equation turbulence models provide this variable.

Subject to these hypotheses, the Reynolds averaged momentum and scalar transport equations become:

(Eqn. 30)

(Eqn. 31)

where B is the sum of the body forces, µeff is the Effective Viscosity, and Γeff is the Effective Diffusivity, defined by,

ρu u⊗–23--ρkδ–

23--µt Uδ µt ∇U ∇U( )T

+( )+∇•–=

ρuφ– Γt∇φ=

Γt

µt

Prt--------=

ρU∂t∂

----------- ρU U⊗( )∇•+ B ∇ p'– µeff ∇U ∇U( )T+( )( )∇•+=

ρφ∂t∂

--------- ρUφ Γeff ∇φ–( )∇•+ S=

µeff µ µt+=

Page 63 Eddy Viscosity Turbulence Models CFX-5.7.1


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and,

and is a modified pressure, defined by:

where ζ is the bulk viscosity.

The Reynolds averaged energy equation becomes:

(Eqn. 32)

Note that although the transformation of the molecular diffusion term may be inexact if enthalpy depends on variables other than temperature, the turbulent diffusion term is correct, subject to the eddy diffusivity hypothesis. Moreover, as turbulent diffusion is usually much larger than molecular diffusion, small errors in the latter can be ignored.

The Reynolds averaged transport equation for Additional Variables (non-reacting scalars) becomes:

(Eqn. 33)

Eddy viscosity models are distinguished by the manner in which they prescribe the eddy viscosity and eddy diffusivity.

The Zero Equation Model in CFX-5Very simple eddy viscosity models compute a global value for µt from the mean velocity and a geometric length scale using an empirical formula. Because no additional transport equations are solved, these models are termed ‘zero equation’.

The zero equation model in CFX-5 uses an algebraic equation to calculate the viscous contribution from turbulent eddies. A constant turbulent eddy viscosity is calculated for the entire flow domain.

The turbulence viscosity is modelled as the product of a turbulent velocity scale, Ut, and a turbulence length scale, lt, as proposed by Prandtl and Kolmogorov,

(Eqn. 34)

where is a proportionality constant. The velocity scale is taken to be the maximum velocity in the fluid domain. The length scale is derived using the formula:

Γeff Γ Γt+=

p'

p' p23--ρk U∇• 2

3--µeff ζ–⎝ ⎠

⎛ ⎞+ +=

ρhtot( )∂t∂

-------------------- P∂t∂

------ ρUhtot( )∇•+– λ Tµt

Prt-------- h∇+∇

⎝ ⎠⎜ ⎟⎛ ⎞

SE+∇•=

ρΦ∂t∂

----------- ρUΦ( )∇•+ ΓΦµt

σΦ-------+

⎝ ⎠⎜ ⎟⎛ ⎞

∇Φ∇• S+=

µt ρ f µUtlt=

f µ

4 Eddy Viscosity Turbulence Models CFX-5.7.1


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(Eqn. 35)

where VD is the fluid domain volume. This model has little physical foundation and is not recommended.

Two Equation Turbulence ModelsTwo-equation turbulence models are very widely used, as they offer a good compromise between numerical effort and computational accuracy. Two-equation models are much more sophisticated than the zero equation models. Both the velocity and length scale are solved using separate transport equations (hence the term ‘two-equation’).

The k-ε and k-ω two-equation models use the gradient diffusion hypothesis to relate the Reynolds stresses to the mean velocity gradients and the turbulent viscosity. The turbulent viscosity is modelled as the product of a turbulent velocity and turbulent length scale.

In two-equation models the turbulence velocity scale is computed from the turbulent kinetic energy, which is provided from the solution of its transport equation. The turbulent length scale is estimated from two properties of the turbulence field, usually the turbulent kinetic energy and its dissipation rate. The dissipation rate of the turbulent kinetic energy is provided from the solution of its transport equation.

The k-ε model in CFX-5

k is the turbulence kinetic energy and is defined as the variance of the fluctuations in velocity. It has dimensions of (L2 T-2), e.g. m2/s2. ε is the turbulence eddy dissipation (the rate at which the velocity fluctuations dissipate) and has dimensions of k per unit time (L2 T-3), e.g. m2/s3.

The k-ε model introduces two new variables into the system of equations. The continuity equation is then:

(Eqn. 36)

and the momentum equation becomes:

(Eqn. 37)

where B is the sum of body forces, µeff is the effective viscosity accounting for turbulence, and is the modified pressure given by

(Eqn. 38)

lt V D

13--

⎝ ⎠⎜ ⎟⎛ ⎞

7⁄=

ρ∂t∂

------ ρU( )∇•+ 0=

ρU∂t∂

----------- ρU U⊗( ) µeff U∇( )∇•–∇•+ p'∇ µeff U∇( )T∇• B+ +=

p'

p' p23--ρk+=



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The k-ε model, like the zero equation model, is based on the eddy viscosity concept, so that

where µt is the turbulence viscosity. The k-ε model assumes that the turbulence viscosity is linked to the turbulence kinetic energy and dissipation via the relation

where Cµ is a constant. Refer to List of Symbols (p. 2) for its value.

The values of k and ε come directly from the differential transport equations for the turbulence kinetic energy and turbulence dissipation rate:

(Eqn. 39)

(Eqn. 40)

where Cε1 , Cε2 , σk and σe are constants. Refer to List of Symbols (p. 2) for their values.

Pk is the turbulence production due to viscous and buoyancy forces, which is modelled using:

(Eqn. 41)

For incompressible flow, is small and the second term on the right side of (Eqn. 41) does not contribute significantly to the production. For compressible flow, is only large in regions with high velocity divergence, such as at shocks.

The term 3µt in (Eqn. 41) is based on the “frozen stress” assumption [54]. This prevents the values of k and ε becoming too large through shocks, a situation that becomes progressively worse as the mesh is refined at shocks. The parameter Compressible Production (accessible on the Advanced Model Control part of the Turbulence Model section in CFX-Pre) can be used to set the value of the factor in front of µt, the default value is 3, as shown. A value of 1 will provide the same treatment as CFX-4.

Buoyancy Turbulence

If the full buoyancy model is being used, the buoyancy production term Pkb is modelled as:

(Eqn. 42)

µeff µ µt+=

µt Cµρk2

ε-----=

ρk( )∂t∂

-------------- ρUk( )∇•+ µµt

σk------+

⎝ ⎠⎜ ⎟⎛ ⎞

k∇∇• Pk ρε–+=

ρε( )∂t∂

-------------- ρUε( )∇•+ µµt

σε------+

⎝ ⎠⎜ ⎟⎛ ⎞

ε∇∇• εk-- Cε1Pk Cε2ρε–( )+=

Pk µt U U∇ U∇ T+( )• 2

3-- U 3µt U∇• ρk+( )∇•–∇ Pkb+=

U∇•U∇•

Pkb

µt

ρPrt-----------– g ρ∇•=



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and if the Boussinesq buoyancy model is being used, it is:

(Eqn. 43)

This buoyancy production term is included in the k equation if the Buoyancy Turbulence option in CFX-Pre is set to Production. It is also included in the ε equation if the option is set to Production and Dissipation expert parameter is set and if Pkb is positive:

If the directional option is enabled, then is modified modified by a factor accounting for the angle α between velocity and gravity vectors:

Default model constants are given by:

Sct = 0.9 for Boussinesq buoyancy, Sct = 1 for full buoyancy model

C3 = 1

Directional Dissipation = Off

For omega based turbulence models the buoyancy turbulence terms for the ω equation are derived from and according to the transformation .

The RNG k-ε Model in CFX-5

The RNG k-ε model is based on renormalisation group analysis of the Navier-Stokes equations. The transport equations for turbulence generation and dissipation are the same as those for the standard k-ε model, but the model constants differ, and the constant Ce1 is replaced by the function Ce1RNG.

The transport equation for turbulence dissipation becomes:

(Eqn. 44)

where,

and,

Pkb

µt

ρPrt-----------ρβg T∇•=

Pεb C3 max 0 Pkb,( )⋅=

Pεb

Pεb C3 max 0 Pkb,( ) αsin⋅⋅=

Pkb Pεb ε β'ωk=

ρε( )∂t∂

-------------- ρUε( )∇•+ µµt

σεRNG-----------------+

⎝ ⎠⎜ ⎟⎛ ⎞

ε∇∇• εk-- Cε1RNGPk Cε2RNGρε–( )+=

Cε1RNG 1.42 f η–=



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Refer to List of Symbols (p. 2) for the values of constants.

The k-ω Model in CFX-5

One of the advantages of the k-ω formulation is the near wall treatment for low-Reynolds number computations. The model does not involve the complex non-linear damping functions required for the k-ε model and is therefore more accurate and more robust. A low-Reynolds k-ε model would typically require a near wall resolution of y+ < 0.2, while a low-Reynolds number k-ω model would require at least y+ < 2. In industrial flows, even y+ < 2 cannot be guaranteed in most applications and for this reason, a new near wall treatment was developed for the k-ω models. It allows for smooth shift from a low-Reynolds number form to a wall function formulation.

The k-ω models assumes that the turbulence viscosity is linked to the turbulence kinetic energy and turbulent frequency via the relation

The Wilcox k-ω ModelThe starting point of the present formulation is the k-ω model developed by Wilcox [11]. It solves two transport equations, one for the turbulent kinetic energy, k, and one for the turbulent frequency, ω. The stress tensor is computed from the eddy-viscosity concept.

k-equation:

(Eqn. 45)

ω-equation:

(Eqn. 46)

f η

η 1 η4.38----------–⎝ ⎠

⎛ ⎞

1 βRNGη3+( )

-----------------------------------

ηPk

ρCµRNGε-------------------------=

=

µt ρ kω----=

ρk( )∂t∂

-------------- ρUk( )∇•+ µµt

σk------+

⎝ ⎠⎜ ⎟⎛ ⎞

∇k∇• Pk β′ρkω–+=

ρω( )∂t∂

--------------- ρUω( )∇•+ µµt

σω-------+

⎝ ⎠⎜ ⎟⎛ ⎞

∇ω∇• αωk----Pk βρω2

–+=



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In addition to the independent variables, the density, ρ, and the velocity vector, U, are treated as known quantities from the Navier-Stokes method. Pk is the production rate of turbulence, which is calculated as in the k-ε model (see (Eqn. 41)).

The model constants are given by:

The unknown Reynolds stress tensor, τ, is calculated from:

(Eqn. 47)

In order to avoid the build-up of turbulent kinetic energy in stagnation regions, a limiter to the production term is introduced into the equations according to Menter [9]:

with clim = 10 for ω based models. This limiter does not affect the shear layer performance of the model, but has consistently avoided the stagnation point build-up in aerodynamic simulations. This limiter is also available for the k-ε model and is controlled through the parameter Production Clip Factor (accessible on the Advanced Model Control part of the Turbulence Model section in CFX-Pre).

If Pkb is positive, the buoyancy production term is included in the k equation if the Buoyancy Turbulence option in CFX-Pre is set to Production. It is also included in the ω equation if the the if the Option is set to Production and Dissipation. See Buoyancy Turbulence (p. 66) for more details.

The Baseline (BSL) k-ω Model

The main problem with the Wilcox model is its well known strong sensitivity to freestream conditions (Menter [12]). Depending on the value specified for ω at the inlet, a significant variation in the results of the model can be obtained. This is undesirable and in order to solve the problem, a blending between the k-ω model near the surface and the k-ε model in the outer region was developed by Menter [9]. It consists of a transformation of the k-ε model to a k-ω formulation and a subsequent addition of the corresponding equations. The Wilcox model is thereby multiplied by a blending function F1 and the transformed k-ε model by a

β′ 0.09=

α 5 9⁄=

β 0.075=

σk 2=

σω 2=

µt2 ρ23--δk–=τ S

Pk min Pk climε( , )=



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function 1-F1. F1 is equal to one near the surface and switches over to zero inside the boundary layer (i.e. a function of the wall distance. See Wall Distance Formulation (p. 99)). At the boundary layer edge and outside the boundary layer, the standard k-ε model is therefore recovered.

Wilcox model:

(Eqn. 48)

(Eqn. 49)

Transformed k-ε model:

(Eqn. 50)

(Eqn. 51)

Now the equations of the Wilcox model are multiplied by function F1, the transformed k-ε equations by a function 1-F1 and the corresponding k- and ω- equations are added to give the BSL model:

(Eqn. 52)

(Eqn. 53)

The coefficients of the new model are a linear combination of the corresponding coefficients of the underlying models:

All coefficients are listed again for completeness:

ρk( )∂t∂

-------------- ρUk( )∇•+ µµt

σk1--------+

⎝ ⎠⎜ ⎟⎛ ⎞


ρω( )∂t∂

--------------- ρUω( )∇•+ µµt

σω1----------+

⎝ ⎠⎜ ⎟⎛ ⎞

∇ω∇• α1ωk----Pk β1ρω2

–+=

ρk( )∂t∂

-------------- ρUk( )∇•+ µµt

σk2--------+

⎝ ⎠⎜ ⎟⎛ ⎞


ρω( )∂t∂

--------------- ρUω( )∇•+ µµt

σω2----------+

⎝ ⎠⎜ ⎟⎛ ⎞

∇ω∇• 2ρ 1σω2 ω--------------- k ω∇∇ α2

ωk----Pk β2ρω2

–+ +=

ρk( )∂t∂

-------------- ρUk( )∇•+ µµt

σk3--------+

⎝ ⎠⎜ ⎟⎛ ⎞


ρω( )∂t∂

--------------- ρUω( )∇•+ µµt

σω3----------+

⎝ ⎠⎜ ⎟⎛ ⎞

∇ω∇• 1 F1–( )2ρ 1σω2 ω--------------- k ω∇∇ α3

ωk----Pk β3ρω2

–+ +=

Φ3 F1Φ1 1 F1–( )Φ2+=

β′ 0.09=



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The Shear Stress Transport (SST) k-ω Based Model

The k-ω based SST model accounts for the transport of the turbulent shear stress and gives highly accurate predictions of the onset and the amount of flow separation under adverse pressure gradients.

The BSL model combines the advantages of the Wilcox and the k-ε model, but still fails to properly predict the onset and amount of flow separation from smooth surfaces. The reasons for this deficiency are given in detail in Menter [9]. The main reason is that both models do not account for the transport of the turbulent shear stress. This results in an overprediction of the eddy-viscosity. The proper transport behaviour can be obtained by a limiter to the formulation of the eddy-viscosity:

where

Again F2 is a blending function similar to F1, which restricts the limiter to the wall boundary layer, as the underlying assumptions are not correct for free shear flows. S is an invariant measure of the strain rate.

Blending FunctionsThe blending functions are critical to the success of the method. Their formulation is based on the distance to the nearest surface and on the flow variables.

α1 5 9⁄=

β1 0.075=

σk1 2=

σω1 2=

α2 0.44=

β2 0.0828=

σk2 1=

σω2 1 0.856⁄=

vt

a1k

max a1ω SF2,( )---------------------------------------=

vt µt ρ⁄=

F1 arg14( )tanh=



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with:

where y is the distance to the nearest wall and v is the kinematic viscosity and:

with:

The Wall Scale EquationDuring the solution of a simulation using the SST or BSL model, you will see a plot in the CFX-Solver Manager for “Wall Scale”. These models require the distance of a node to the nearest wall for performing the blending between k-ε and k-ω. The wall scale equation is the equation solved to get the wall distance, simply:

where φ is the value of the wall scale. The wall distance can be calculated from the wall scale through:

The (k-ε)1E Eddy Viscosity Transport Model

A very simple one-equation model has been developed by Menter [32] [33]. It is derived directly from the k-ε model and is therefore named the (k-ε)1E model:

where is the kinematic eddy viscosity, is the turbulent kinematic eddy viscosity and σ is a model constant.

The model contains a destruction term, which accounts for the structure of turbulence and is based on the von Karman length scale:

arg1 min maxk

β′ωy------------ 500v

y2ω

-----------,⎝ ⎠⎜ ⎟⎛ ⎞ 4ρ k

CDkwσω2 y2

--------------------------------,⎝ ⎠⎜ ⎟⎛ ⎞

=

CDkω max 2ρ 1σω2ω-------------- k ω 1.0 10 10–×,∇∇⎝ ⎠

⎛ ⎞=

F2 arg22( )tanh=

arg2 max2 kβ′ωy------------ 500v

y2ω

-----------,⎝ ⎠⎜ ⎟⎛ ⎞

=

∇2φ 1–=

Wall Distance ∇φ 2 2φ+( ) ∇φ–=

∂ρvt

∂t-----------

∂ρU jvt

∂x j------------------+ c1ρvt S c2ρ

vt

LvK---------

⎝ ⎠⎜ ⎟⎛ ⎞ 2

µρvt

σ--------+⎝ ⎠

⎛ ⎞ ∂v∂x j--------+–=

v vt



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where S is the shear strain rate tensor. The eddy viscosity is computed from:

In order to prevent the a singularity of the formulation as the von Karman length scale goes to zero, the destruction term is reformulated as follows:

The coefficients are:

By default the model is solved in combination with the scalable wall functions described in Scalable Wall-Functions (p. 88).

Coefficient Value

c1 0.144

c2 1.86

c3 7.0

A+ 13.5

κ 0.41

σ 1.0

LvK( )2 S2

∂S∂x j-------- ∂S

∂x j--------

-----------------=

µt ρvt=

Ek ε–

vt

LvK---------

⎝ ⎠⎜ ⎟⎛ ⎞ 2

=

EBB

∂vt

∂x j--------

∂vt

∂x j--------=

E1e c3EBB

Ek ε–c3EBB----------------

⎝ ⎠⎜ ⎟⎛ ⎞

tanh=

∂ρvt

∂t-----------

∂ρU jvt

∂x j------------------+ c1D1ρvtS c2ρE1e µ

ρvt

σ--------+⎝ ⎠

⎛ ⎞ ∂vt

∂x j--------+–=



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Low Reynolds Number Formulation

Low Reynolds formulation of the model is obtained by including damping functions. Near wall damping functions have been developed to allow integration to the surface:

where D2 is required to compute the eddy-viscosity which goes into the momentum equations:

The low Reynolds formulation of the (k-ε)1E model requires a near wall mesh resolution of y+ < 1.

D1vt v+

vt v+-------------=

D2 1vt

A+κv

-------------⎝ ⎠⎜ ⎟⎛ ⎞ 2

–exp–=

µt ρD2vt=



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Reynolds Stress Turbulence Models

These models are based on transport equations for all components of the Reynolds stress tensor and the dissipation rate. These models do not use the eddy viscosity hypothesis, but solve an equation for the transport of Reynolds stresses in the fluid. The Reynolds stress model transport equations are solved for the individual stress components.

Algebraic Reynolds stress models solve algebraic equations for the Reynolds stresses, whereas differential Reynolds stress models solve differential transport equations individually for each Reynolds stress component.

The exact production term and the inherent modelling of stress anisotropies theoretically make Reynolds Stress models more suited to complex flows, however practice shows that they are often not superior to two-equation models.

The Reynolds averaged momentum equations for the mean velocity are

(Eqn. 54)

where is a modified pressure, B is the sum of body forces and the fluctuating Reynolds stress contribution is . Unlike eddy viscosity models, the modified pressure has no turbulence contribution and is related to the static (thermodynamic) pressure by:

(Eqn. 55)

In the differential stress model, is made to satisfy a transport equation. A separate transport equation must be solved for each of the six Reynolds stress components of

. The differential equation Reynolds stress transport is:

(Eqn. 56)

where P and G are shear and buoyancy turbulence production terms of the Reynolds stresses respectively, φ is the pressure-strain tensor, and C is a constant.

Buoyancy turbulence terms are controlled in the same way as for the k-ε and k-ω model. See the text below (Eqn. 43) (p. 67).

The Reynolds Stress ModelThe standard Reynolds Stress model in CFX-5 is based on the ε-equation. The CFX-Solver solves the following equations for the transport of the Reynolds stresses:

ρU∂t∂

----------- ρU U⊗( ) µ U∇( )∇•–∇•+ p''∇– ρu u⊗( )∇•– B+=

p''ρu u⊗

p'' p U∇• 23--µ ζ–⎝ ⎠

⎛ ⎞+=

u u⊗

ρu u⊗

ρu u⊗∂t∂

-------------------- ρu u⊗ U⊗( ) ρCkε--u u⊗ u u⊗∇( )

T

⎝ ⎠⎛ ⎞∇•–∇•+ P G

23--δρε–+= + φ

Page 75 Reynolds Stress Turbulence Models CFX-5.7.1


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(Eqn. 57)

which can be written in index notation as

(Eqn. 58)

where φij is the pressure-strain correlation, and P, the exact production term, is given by:

(Eqn. 59)

As the turbulence dissipation appears in the individual stress equations, an equation for ε is still required. This now has the form:

(Eqn. 60)

In these equations, the anisotropic diffusion coefficients of the original models are replaced by an isotropic formulation, which increases the robustness of the Reynolds stress model.

The Reynolds Stress model is also available with anisotropic diffusion coefficients. In this case, the CFX-Solver solves the following equations for the transport of the Reynolds stresses:

(Eqn. 61)

where φij is the pressure-strain correlation, and P, the exact production term, is given by (Eqn. 59).

The equation for ε is:

(Eqn. 62)

The model constants are listed below for each model.

ρu u⊗∂t∂

-------------------- ρU u u⊗⊗( )∇•+ P µ 23--csρk

2

ε-----+⎝ ⎠

⎛ ⎞ u u⊗∇⎝ ⎠⎛ ⎞∇• 2

3--δρε–+= φ +

t∂∂ ρuiu j( )

xk∂∂

Ukρuiu j( )+ Pij φij xk∂∂ µ 2

3--csρk

2

ε-----+⎝ ⎠

⎛ ⎞ uiu j∂xk∂

------------- 23--δijρε–+ +=

P ρ u u⊗ U∇( )TU∇( )u u⊗+( )–=

ρε( )∂t∂

--------------xk∂∂ ρUkε( )+

εk-- cε1P cε2ρε–( )

xk∂∂ µ

µt

σε------+

⎝ ⎠⎜ ⎟⎛ ⎞

xk∂∂ε

+=

t∂∂ ρuiu j( )

xk∂∂

Ukρuiu j( )+ Pij φij xk∂∂ µδkl csρk

ε--ukul+⎝ ⎠

⎛ ⎞ uiu j∂xl∂

------------- 23--δijρε–+ +=

ρε( )∂t∂

--------------xk∂∂ ρUkε( )+

εk-- cε1P cε2ρε–( )

xk∂∂ µδkl cερk

ε--ukul+⎝ ⎠

⎛ ⎞xl∂

∂ε+=



Contents Index Help
Pressure-Strain Terms

One of the most important terms in Reynolds stress models is the pressure-strain correlation, φij.

The pressure strain correlations can be expressed in the general form

where

and

In this formulation, a is the anisotropy tensor, S is the strain rate and W is the vorticity. This general form can be used to model linear and quadratic correlations by using appropriate values for the constants. The model constants are listed below.

Alternative Reynolds Stress Constants

There are three varieties of the standard Reynolds stress models available in CFX-5. These are known as LRR-IP, LRR-QI and SSG. Each model has different model constants.

The LRR-IP and LRR-QI models were developed by Launder, Reece and Rodi [4]. “IP” stands for Isotropisation of Production, and “QI” stands for Quasi-Isotropic. In these models, the pressure-strain correlation is linear.

The SSG model was developed by Speziale, Sarkar and Gatski [5]. This model uses a quadratic relation for the pressure-strain correlation.

The table below shows the values of the constants for each model.

φij φij1 φij2+=

φij1 ρε Cs1a Cs2 aa13--a aδ•–⎝ ⎠

⎛ ⎞+⎝ ⎠⎛ ⎞–=

φij2 Cr1Pa Cr2ρkS Cr3ρkS a a•–

Cr4ρk aST

SaT 2

3--a–+ S• δ⎝ ⎠

⎛ ⎞

Cr5ρk aWT

WaT

+( )

+

+

+

–=

a u u⊗k

--------------23--δ

S

–

12-- U∇ U∇( )T

+( )

W12-- U∇ U∇( )T

–( )

=

=

=



Contents Index Help
Selection of the appropriate model is carried out on the Fluid Models panel of the Domains form in CFX-Pre. The following options correspond to the types of models listed above:

Reynolds Stress Model - LRR-IP SSG Reynolds Stress Model - SSG QI Reynolds Stress Model - LRR-IQ

The Reynolds Stress-ω ModelsCFX-5 provides two Reynolds Stress-ω models: the Omega Reynolds Stress and Baseline (BSL) Reynolds Stress models. The two models relate to each other in the same way as the two equation k-ω and BSL models (see The Baseline (BSL) k-w Model (p. 69)).

The Reynolds Stress-ω turbulence model, or SMC-ω model, is a Reynolds Stress model based on the ω-equation. The advantage of the ω-equation is that it allows for a more accurate near wall treatment with an automatic switch from a wall function to a low-Reynolds number formulation based on the grid spacing.

The modelled equations for the Reynolds stresses can be written as follows:

(Eqn. 63)

ω for the Omega Reynolds Stress Model

The Omega Reynolds Stress Model uses the following equation for ω:

(Eqn. 64)

Model CµRS seRS cs cε1 ce2

LRR-IP 0.1152 1.10 0.22 1.45 1.9

LRR-QI 0.1152 1.10 0.22 1.45 1.9

SSG 0.1 1.36 0.22 1.45 1.83

Model Cs1 Cs2 Cr1 Cr2 Cr3 Cr4 Cr5

LRR-IP 1.8 0.0 0.0 0.8 0.0 0.6 0.6

LRR-QI 1.8 0.0 0.0 0.8 0.0 0.873 0.655

SSG 1.7 -1.05 0.9 0.8 0.65 0.625 0.2

∂ ρτij( )∂t

-----------------∂ Ukρτij( )

∂xk-------------------------+ ρPij–

23--β′ρωkδij ρΠij–

∂∂xk-------- µ

µt

σ*------+⎝ ⎠

⎛ ⎞ ∂τij

∂xk---------

⎝ ⎠⎜ ⎟⎛ ⎞

+ +=

∂ ρω( )∂t

---------------∂ Ukρω( )

∂xk-----------------------+ αρω

k----Pk βρω2

–∂

∂xk-------- µ

µt

σ-----+⎝ ⎠

⎛ ⎞ ∂ω∂xk--------

⎝ ⎠⎜ ⎟⎛ ⎞

+=



Contents Index Help
The following coefficients apply:

ω for the BSL Reynolds Stress Model

The coefficients α and β of the ω-equation, as well as both the turbulent Prandtl numbers σ∗ and σ, are blended between values from the two sets of constants, corresponding to the ω-based model constants and the ε-based model constants transformed to an ω-formulation:

(Eqn. 65)

• Set 1 (SMC-ω zone):

The value of β here corresponds to the k-ω model, described in The Wilcox k-w Model (p. 68). The von Karman constant κ has a commonly used value of 0.41.

• Set 2 (SMC-ε zone):

The blending of coefficients is done by smooth linear interpolation with the same weight function F as the one used in a cross-diffusion term of the ω-equation (Eqn. 65):

(Eqn. 66)

where with

and

σ∗ 2= σ 2= β 0.075= α ββ′----- κ2

σ β′( )0.5--------------------– 5

9--= =

ρω( )∂t∂

--------------- ∂ Ukρω( )+ α3ωk----Pk β3ρω2

– ∂∂xk--------+ µ

µt

σω3----------+

⎝ ⎠⎜ ⎟⎛ ⎞ ∂ω

∂xk-------- 1 F1–( )2ρ 1

σ2ω---------- ∂k

∂xk-------- ∂ω

∂xk--------+=

σ∗1 2= σ1 2= β1 0.075= α1ββ′----- κ2

σ β′( )0.5--------------------– 0.553= =

σ∗2 1.0= σ2 0.856= β2 0.0828=

α2ββ′----- κ2

σ β′( )0.5--------------------– 0.44= =

φ3 F φ1⋅ 1 F–( )φ2+=

F arg4( )tanh=

arg min max kβ′ωy------------ , 500υ

y2ω

------------⎝ ⎠⎜ ⎟⎛ ⎞

, 4ρ k

CDkωσk ε– y2

-------------------------------------⎩ ⎭⎨ ⎬⎧ ⎫

=



Contents Index Help
Pressure-strain correlation

The constitutive relation for the pressure-strain correlation is given by:

(Eqn. 67)

The production tensor of Reynolds stresses is given by:

(Eqn. 68)

The tensor Dij, participating in the pressure-strain model (Eqn. 67), differs from the production tensor in the dot-product indices:

(Eqn. 69)

The turbulent viscosity in the diffusion terms of the balance equations (Eqn. 63) and (Eqn. 64) is calculated in the same way as in the Wilcox k-ω model, described in The Wilcox k-w Model (p. 68).

(Eqn. 70)

The coefficients for the model are:

0.09

C1 1.8

C2 0.52

CDkω max 2ρ 1σk ε– ω------------------- ∂k

∂x j-------- ∂ω

∂x j-------- , 10 10–

⎝ ⎠⎛ ⎞=

Πij β′C1ω τij23--kδij+⎝ ⎠

⎛ ⎞ α Pij23--Pδij–⎝ ⎠

⎛ ⎞– β Dij23--Pδij–⎝ ⎠

⎛ ⎞– γ k Sij13--Skkδij–⎝ ⎠

⎛ ⎞–=

Pij τik

∂U j

∂xk---------- τ jk

∂Ui

∂xk--------- ;+= P

12--Pkk=

Dij τik

∂Uk

∂x j---------- τ jk

∂Uk

∂xi----------+=

µT ρ kω----=

β′

α 8 C2+( ) 11⁄

β 8C2 2–( ) 11⁄

γ 60C2 4–( ) 55⁄



Contents Index Help
Wall Boundary Condition

The SMC-ω model is used in combination with the automatic wall treatment developed for the k-ω based models (k-ω, BSL and SST). The formulation has been recalibrated to ensure a constant wall shear stress and heat transfer coefficient under variable near wall resolution.

Rotating Frame of Reference for Reynolds Stress ModelsOne of the advantages of Reynolds stress transport models, when compared to k-ε and k-ω models, is their ability to simulate the additional anisotropy of the Reynolds stresses due to the Coriolis forces appearing in the rotating frame of reference.

The necessary additional source terms to account for the Coriolis forces have been implemented into CFX-5 for use with any of the available Reynolds stress transport models. These terms are described in a book by Wilcox [30], and in more detail in a paper by Launder, Tselepidakis and Younis [31].

If the flow equations are written in the frame of the coordinate system, which rotates relative to the steady inertial frame with the vector angular velocity , then one new source term Gij has to be added directly to the right hand side of the transport equation for τij, (Eqn. 63):

(Eqn. 71)

where εijk is a Levi-Chivita factor, equal to 1 if its indices i,j,k form an even permutation of 1,2,3, equal to -1 for an odd permutation of 1,2,3, and equal to 0 if any two indices are equal.

Besides, the absolute velocity gradient tensor, participating in the production tensor (Eqn. 68) and in the model equation for the pressure-strain correlation (Eqn. 67), is written in the rotating frame as a sum of the strain rate tensor Sij :

and the rotation tensor :

This representation of the velocity gradient results in an apparent additional source term Gij (Eqn. 71), coming from the production term (-ρPij). That is why in reference [31] the Coriolis source term Gij differs from (Eqn. 71) by an additional factor of 2.

Ω

Gij ρ Ωk τ jmεikm τimε jkm+( )⋅ ⋅=

Sij 0.5∂Ui

∂x j---------

∂U j

∂xi----------+

⎝ ⎠⎜ ⎟⎛ ⎞

=

Ωij

Ωij 0.5∂Ui

∂x j---------

∂U j

∂xi----------–

⎝ ⎠⎜ ⎟⎛ ⎞

εijk Ωk⋅+=



Contents Index Help
Large Eddy Simulation Theory

This section outlines the theoretical details of the LES model in CFX-5. For details on setting up an LES simulation and modelling advice, see The Large Eddy Simulation Model (p. 107 in CFX-5 Solver Modelling).

The non filtered Navier Stokes equations are:

Large Eddy Simulation (LES) is about filtering of the equations of movement and decomposition of the flow variables into a large scale (resolved) and a small scale (unresolved) parts. Any flow variable f can be written such as:

where , the large scale part, is defined through volume averaging as:

where is the filter function (called the hat filter or Gaussian filter).

After performing the volume averaging and neglecting density fluctuations, the filtered Navier Stokes equations become:

The non linear transport term in the filtered equation can be developed as:

In time averaging the terms (2) and (3) vanish, but when using volume averaging this is no longer true.

Introducing the sub-grid scale (SGS) stresses, τij, as:

we can rewrite the filtered Navier Stokes equations as:

∂ ρUi( )∂t

------------------∂ ρUiU j( )

∂x j-------------------------+ ∂p

∂xi-------– µ

∂2Ui

∂x j∂x j-----------------+=

f f f '+=

f

f xi t,( ) G xi xi'–( ) f xi' t,( ) xi'd

Vol∫=

G xi xi'–( )

∂ ρUi( )∂t

------------------∂ ρuiu j( )

∂x j----------------------+ ∂ p

∂xi-------– µ

∂2Ui

∂x j∂x j-----------------+=

uiu j Ui ui'+( ) U j u j'+( )=

UiU j Uiu j' U jui' ui'u j'+ + +=

(1) (2) (3) (4)

τij uiu j UiU j–=

Page 82 Large Eddy Simulation Theory CFX-5.7.1


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with

Smagorinsky ModelThe Smagorinsky model [34] can be thought of as combining the Reynolds averaging assumptions given by Lij + Cij = 0 with a mixing-length based eddy viscosity model for the Reynolds SGS tensor. It is thereby assumed that the SGS stresses are proportional to the modulus of the strain rate tensor, ,of the filtered large-scale flow:

To close the equation, we need a model for the SGS viscosity vSGS. Based on dimensional analysis the SGS viscosity can be expressed as:

where l is the length scale of the unresolved motion (usually the grid size ∆ = (Vol)1/3) and qSGS is the velocity of the unresolved motion.

In the Smagorinsky model, based on an analogy to the Prandtl mixing length model, the velocity scale is related to the gradients of the filtered velocity:

∂ ρUi( )∂t

------------------∂ ρτij ρUiU j+( )

∂x j-----------------------------------------+ ∂ p

∂xi-------– µ

∂2Ui

∂x j∂x j-----------------+=

∂ ρUi( )∂t

------------------∂ ρUiU j( )

∂x j-------------------------+ ∂ p

∂xi-------– µ

∂2Ui

∂x j∂x j-----------------

∂ ρτij( )∂x j

-----------------–+=

τij uiu j UiU j–=

UiU j Uiu j' U ju' ui'u j' UiU j–+ + +=

Lij Cij Rij+ +=

Lij UiU j UiU j– Leonard Stresses= =

Cij Uiu j' U jui'+ Cross Terms= =

Rij ui'u j' SGS Reynolds Stresses= =

Sij

τij13--– τkk 2 vSGS Sij⋅ ⋅– vSGS

∂Ui

∂x j---------

∂U j

∂xi----------+

⎝ ⎠⎜ ⎟⎛ ⎞

⋅= = =

vSGS l qSGS∝



Contents Index Help
This yields the Smagorinsky model [34] for the SGS viscosity:

with CS the Smagorinsky constant. The value of the Smagorinsky constant for isotropic turbulence with inertial range spectrum

is:

For practical calculations the value of CS is changed depending on the type of flow and mesh resolution. Its value is found to vary between a value of 0.065 (channel flows) and 0.25. Often a value of 0.1 is used.

Wall DampingClose to walls, the turbulent viscosity can be damped using a combination of a mixing length minimum function, and a viscosity damping function fµ:

with

CS and κ are constants which you can set; their default values are 0.1 and 0.4 respectively.

By default, the damping function fµ is 1.0. A Van Driest and a Piomelli like damping can be specified by the user. For the Van Driest case, the damping function is:

with A = 25. For the Piomelli case it is:

with A = 25. The normalised wall distance

qSGS ∆ S= where S 2SijSij( )1 2⁄

=

vSGS CS∆( )2S=

E k( ) Ckε2 3⁄k

5 3⁄–=

CS1π--- 2

3Ck---------⎝ ⎠

⎛ ⎞ 3 4⁄0.18= =

µT ρ min lmix f µCS∆,( )2 2SijSij⋅=

lmix κ ywall⋅=

f µ 1 y A⁄–( )exp–=

f µ 1 y A⁄–( )3[ ]exp–=

y y u⋅( ) v⁄=



Contents Index Help
is defined as a function of the calculated wall distance y, kinematic viscosity v and local velocity scale .

The Van Driest or Piomelli wall damping can be switched on when the LES turbulence model is selected. The damping factor A is defaulted to 25.0.

u



Contents Index Help
Detached Eddy Simulation Theory

This section outlines the theoretical details of the DES model in CFX-5. For details on setting up a DES simulation and modelling advice, see The Detached Eddy Simulation Model The Detached Eddy Simulation Model (p. 113 in CFX-5 Solver Modelling).

Experience has shown that the use of LES in boundary layer flows at high Re numbers is prohibitively expensive [57] and therefore not useful for many industrial flow simulations. On the other hand, turbulent structures can be resolved in massively separated regions, where the large turbulent scales are of the same dimension as the geometrical structure generating them (airfoil flaps, buildings, blunt trailing edges on turbine blades). DES is an attempt to combine elements of RANS and LES formulations in order to arrive at a hybrid formulation, where RANS is used inside attached and mildly separated boundary layers and LES is applied in massively separated regions. While this approach offers many advantages, it is clearly not without problems, as the model has to identify the different regions automatically. DES models require a detailed understanding of the method and the grid generation requirements and should not be used as “black-box”. You are advised to read the CFX technical report [55] on the subject, which explains all details on grid generation requirements, zonal formulation and boundary conditions.

The CFX version of the DES model is based on the SST formulation. The advantage of this combination is that the accurate prediction of turbulent boundary layers up to separation and in mildly separated regions carries over from the SST model. In addition, the SST model supports the formulation of a zonal DES formulation [56], which is less sensitive to grid resolution restrictions than the standard DES formulation, as proposed by Strelets [58]. Refer to the CFX technical report [55] for details.

SST-DES Formulation Strelets et al.

The idea behind the DES model of Strelets [58] is to switch from the SST-RANS model to an LES model in regions where the turbulent length, Lt, predicted by the RANS model is larger than the local grid spacing. In this case, the length scale used in the computation of the dissipation rate in the equation for the turbulent kinetic energy is replaced by the local grid spacing, ∆.

(Eqn. 72)

The practical reason for choosing the maximum edge length in the DES formulation is that the model should return the RANS formulation in attached boundary layers. The maximum edge length is therefore the safest estimate to ensure that demand.

The DES modification of Strelets can be formulated as a multiplier to the destruction term in the k-equation:

ε β∗kω k

32--

Lt⁄ k

32--

CDES∆( )⁄ for CDES∆ Lt<( ) ∆→ max ∆i( ) ; Lt k( ) β⁄ ∗ω= = = =

Page 86 Detached Eddy Simulation Theory CFX-5.7.1


Contents Index Help
(Eqn. 73)

with CDES equal to 0.61, as the limiter should only be active in the model region. The numerical formulation is also switched between an upwind biased and a central difference scheme in the RANS and DES regions respectively.

Zonal SST-DES formulation in CFX

The main practical problem with the DES formulation (both for the Spalart Allmaras and the standard SST-DES model) is that there is no mechanism for preventing the limiter from becoming active in the attached portion of the boundary layer. This will happen in regions where the local surface grid spacing is less than the boundary layer thickness with c of the order of one. In this case the flow can separate as a result of the grid spacing (grid-induced separation), which is undesirable. In order to reduce this risk, CFX offers a zonal formulation of the DES formulation, based on the blending functions (which are functions of the wall distance. See Wall Distance Formulation (p. 99).) of the SST model [56]:

In case FSST is set to 0, the Strelets model is recovered. The default option is , which offers the highest level of protection against grid-induced separation, but might also be less responsive in LES regions. For details, refer to the CFX technical report [55].

Numerical Treatment of DES Model

Following Strelets [58], the zonal DES model switches from a second-order upwind scheme in the RANS region to a central difference scheme in the LES region. The details of the switching mechanism are beyond the current description and can also be found in the CFX technical report [55] on the subject.

Boundary Conditions

For LES simulations, unsteady fluctuations have to be specified in most cases at the inlet boundaries. This greatly complicates the use of LES in industrial flows, as the details of these fluctuations are generally not known. In DES simulations, it is in most cases possible to specify the same boundary conditions as for the SST model. This is particularly true if the default setting of the zonal model are used (for details, refer to [55]).

ε β∗kω β∗kω FDES with FDES maxLt

CDES∆------------------ 1( , )=⋅→=

k ε–

∆s cδ>

FDES CFX– maxLt

CDES∆------------------ 1 FSST–( ) 1( , ) with FSST 0 F1 F2, ,==

FSST F2=

Page 87 Detached Eddy Simulation Theory CFX-5.7.1


Contents Index Help
Modelling Flow Near the Wall

This section presents the mathematical details of how flow near to a no-slip wall is modelled in CFX-5. An introduction to near-wall flow, modelling details and guidelines on using wall functions are presented in Modelling Flow Near the Wall (p. 119 in CFX-5 Solver Modelling).

Mathematical FormulationThe wall-function approach in CFX-5 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, τω, 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. These 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:

(Eqn. 74)

where:

u+ is the near wall velocity, uτ is the friction velocity, Ut is the known velocity tangent to the wall at a distance of ∆y from the wall (for a definition of ∆y in the different wall formulations, see Solver Yplus and Yplus (p. 90)), y+ is the dimensionless distance from the wall, τω is the wall shear stress, κ is the von Karman constant and C is a log-layer constant depending on wall roughness (natural logarithms are used).

Scalable Wall-Functions

(Eqn. 74) has the problem that it becomes singular at separation points where the near wall velocity, Ut, approaches zero. In the logarithmic region, an alternative velocity scale, u* can be used instead of u+:

u+ Ut

uτ------

1κ--- y

+( )ln C+= =

y+ ρ∆yuτ

µ----------------=

uττωρ------⎝ ⎠

⎛ ⎞1 2⁄

=

u* Cµ1 4⁄

k1 2⁄

=

Page 88 Modelling Flow Near the Wall CFX-5.7.1


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This scale has the useful property that it does not go to zero if Ut goes to zero (in turbulent flow k is never completely zero). Based on this definition, the following explicit equation for uτ can be obtained:

(Eqn. 75)

The absolute value of the wall shear stress τω, is then obtained from:

(Eqn. 76)

where:

and u* is as defined earlier.

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 sensitive 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 case of fine meshes can be overcome with the use of the Scalable Wall Function formulation developed by 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 used in the logarithmic formulation by a lower value of . 11.06 is the intersection 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 use 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 case a fine near wall spacing is required to ensure a sufficient number of nodes in the boundary layer.

If the results deviate greatly from these ranges the mesh at the designated Wall boundaries will require modification, unless wall shear stress and heat transfer are not important in the simulation (see Creating a Surface Plot of y+ (p. 146 in CFX-5 Tutorials)).

uτUt

1κ--- y*( )ln C+------------------------------=

τω ρu*uτ=

y* ρu*∆y( ) µ⁄=

y* max y* 11.06( , )=y*



Contents Index Help
Solver Yplus and Yplus

In the solver output, there are two arrays for the near wall y+ spacing. The definition for the YPLUS variable that appears in the post processor is given by the standard definition of y+ generally used in CFD:

where ∆n is the distance between the first and second grid points off the wall.

In addition, a second array (SOLVER YPLUS) is available which contains the y+ used in the logarithmic profile by the solver. It depends on the type of wall treatment used, which can be one of three different treatments in CFX-5. They are based 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 (based on ∆y = ∆n / 4)

• Scalable wall function (based on ∆y = ∆n / 4)

• Automatic wall treatment (based on ∆y = ∆n)

The scalable wall function y+ is defined as:

and is therefore based 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.

Automatic Near-Wall Treatment for k-ω Based Models

While the wall-functions presented above allow for a consistent mesh refinement, they are based 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 balance. For flows at low Reynolds numbers, this can cause 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 k-ω model of Wilcox has the advantage that an analytical expression is known for ω in the viscous sublayer, which can be exploited to achieve this goal. The main idea behind the present formulation is to blend the wall value for ω 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, FU:

y+ τω ρ⁄ ∆n⋅

v------------------------------=

y* max y* 11.06( , )= y* u*∆n 4⁄v

--------------------=



Contents Index Help
with:

Flux for the k-equation:

In the ω-equation, an algebraic expression is specified instead of an added flux. It is a blend between the analytical expression for ω in the logarithmic region:

and the corresponding expression in the sublayer:

with ∆y being the distance between the first and the second mesh point. In order to achieve a smooth blending and to avoid cyclic convergence behaviour, the following formulation is selected:

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 emphasised, 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 present in the wall-function mode, but is reduced to zero as the method shifts to the low-Re model. The shift is based on the distance between the first and the second mesh point ∆y = y2 - y1 with y being the wall normal distance.

FU ρuτ u*–=

uτ v ∆U∆y--------=

u* max a1k uτ( , )=

Fk 0=

ωlu*

a1κy------------

1a1κv------------ u*2

y+

--------= =

ωs6v

β ∆y( )2-----------------=

ωω ωs 1ωl

ωs------

⎝ ⎠⎜ ⎟⎛ ⎞ 2

+=



Contents Index Help
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 closer to the wall. Roughness effects are accounted for by modifying the expression for u+ as follows:

(Eqn. 77)

where:

and yR is the equivalent sand grain roughness [14].

You are advised to exercise care in the use of the rough wall option together with the standard wall-function approach. Inaccuracies can arise 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 CFX-Solver failing. See Wall Roughness (p. 72).

Heat Flux in the Near-Wall Region

The thermal boundary layer is modelled using the thermal law-of-the-wall function of B.A. Kader [15].

Heat flux at the wall can be modelled using a wall function approach or the automatic wall treatment. Using similar assumptions as those 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 Tw is the temperature at the wall, Tf the fluid temperature that is being non-dimensionalised, cp the fluid heat capacity and qw the heat flux at the wall. The non-dimensional temperature distribution is then modelled as:

where:

u+ 1κ--- y*

1 0.3k++----------------------⎝ ⎠

⎛ ⎞ln C+=

k+ yRρµ---u*=

T+ ρC pu* Tw T–

f( )

qw------------------------------------------=

T+

Pr y*eΓ–( ) 2.12 y*( )ln β+[ ]e

1– Γ⁄( )+=

β 3.85Pr1 3⁄ 1.3–( )

22.12 Pr( )ln+=



Contents Index Help
Pr is the fluid Prandtl number, given by

where λ is the fluid thermal conductivity. Combining these equations leads to a simple form for the wall heat flux model:

(Eqn. 78)

Turbulent fluid flow and heat transfer problems without conjugate heat transfer objects require the specification of the wall heat flux, qw, or the wall temperature, Tw. 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.

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:

(Eqn. 79)

where uτ = (τw / ρw)1/2, y+ = uτ y / vw, κ = 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):

Γ 0.01 Pr y*( )4

1 5Pr3y*+

------------------------------=

PrµC p

λ-----------=

qw

ρC pu*

T+

----------------- Tw T f–( )=

Ucomp

uτ----------------

1κ--- u* y

v--------⎝ ⎠

⎛ ⎞ln C+=



Contents Index Help
(Eqn. 80)

Near a solid wall the integrated near-wall momentum equation reduces to:

(Eqn. 81)

while the energy equation reduces to:

(Eqn. 82)

Expressions for shear stress and heat flux applicable to the boundary layer region are:

(Eqn. 83)

and

(Eqn. 84)

If (Eqn. 81), (Eqn. 83) and (Eqn. 84) are substituted into (Eqn. 82) and integrated, the resulting equation is:

(Eqn. 85)

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, (Eqn. 85) replaces the density ratio found in (Eqn. 80). Performing the integration yields the following equation for the “compressible” velocity:

(Eqn. 86)

where:

Ucompρ

ρw------ Ud∫=

τ τw=

q qw U τw+=

τ µtU∂y∂

-------=

qµtC p

Prt------------

⎝ ⎠⎜ ⎟⎛ ⎞ T∂

y∂------⎝ ⎠

⎛ ⎞–=

T Tw

Prt qwU

C p τw---------------------–

Prt U2

2C p----------------–=

Ucomp BA U+

D--------------⎝ ⎠

⎛ ⎞ AD----⎝ ⎠

⎛ ⎞asin–asin=

A qw τw⁄=

B 2C pTw( ) Prt( )⁄=

D A2

B+=



Contents Index Help
The above derivation provides most of the equations necessary for the implementation of wall-functions which are applicable to compressible flows. These wall-functions are primarily implemented in two locations in CFX-5: hydrodynamics and energy. First, consider the implementation in the hydrodynamics section. The equation for the wall-shear-stress is obtained after a slight rearrangement of (Eqn. 79):

(Eqn. 87)

This is similar to the low speed wall-function, except that Ucomp now replaces U. The Van Driest transformation given by (Eqn. 86), must now be performed on the near wall velocity. In the implementation (Eqn. 86) is re-written in the following form:

(Eqn. 88)

where:

This completes the wall-function modifications for the hydrodynamics.

τw

ρ------

u*1k-- u* y

v--------ln C+

----------------------------Ucomp=

Ec( )comp1 2⁄ 2

Prt--------

Tw

T-------

1 2⁄B* Ec+

D*-------------------⎝ ⎠

⎛ ⎞ B*D*-------⎝ ⎠

⎛ ⎞asin–asin=

u* Cµ1 4⁄

k1 2⁄

=

τ*w ρ u*( )2=

EcU

2

C p T------------=

B*qw

C p T τ*w--------------------U=

D* B*2 2Prt--------

Tw

T-------Ec+

⎝ ⎠⎜ ⎟⎛ ⎞ 1 2⁄

=

Ucomp C pT Ec( )comp( )1 2⁄=

C 5.2=

Prt 0.9=

Tw

T------- 1 Prt B*

Prt

2------- Ec+ +=



Contents Index Help
The relationship between wall heat transfer, near wall velocity and near wall temperature is given by (Eqn. 85) (and also in rearranged form after (Eqn. 88): the equation for Tw / T). A dimensionless variable, θ+, can be defined as:

(Eqn. 89)

and hence:

(Eqn. 90)

From (Eqn. 90) it is clear that knowing θ+, the near wall velocity, the near wall temperature and the wall heat transfer, the wall temperature can be calculated (if instead the wall temperature is known, then the wall heat transfer is calculated). Before proceeding, it is instructive to consider the equation that is used in CFX-5 for low Mach number heat transfer. The equation is subsequently modified for use at higher Mach numbers. This unmodified equation is given by:

(Eqn. 91)

where:

It can be seen that this equation blends between the linear and logarithmic near wall regions, and hence is more general than just using the logarithmic profile that is implied by (Eqn. 89). (Eqn. 91) has been extended for use in the compressible flow regime by interpreting the linear and logarithmic velocity profiles given above as “compressible” or Van Driest transformed velocities:

θ+Prt

Uu*-----=

θ+Prt

Uu*------

ρC p u*qw

------------------ Tw T–Prt U

2

2C p----------------–

⎝ ⎠⎜ ⎟⎛ ⎞

= =

θ+Pr U1

+( ) eΓ–

Prt U2+( ) e

1 Γ⁄–+=

U1+

y+

=

U2+ 1

Prt-------- 2.12 y

+( )ln β+( )=

Γ f 1 Pr y+( )=

β f 2 Pr( )=

Ucomp1+

y+

=

Ucomp2+ 1

Prt-------- 2.12 y

+( )ln β+( )=



Contents Index Help
This change is consistent with (Eqn. 79). The “untransformed” velocities required by (Eqn. 91) can be obtained by applying the inverse Van Driest velocity transformation to these “compressible” velocities. The inverse transformation is given by:

(Eqn. 92)

where:

and A and B are defined following (Eqn. 86) above. The reverse transformed velocities obtained from (Eqn. 92) are substituted into (Eqn. 91) to obtain the value of θ +. Either the wall heat transfer or the wall temperature can then be obtained from (Eqn. 90).

Guidelines for Mesh Generation The mesh generation has to be performed before any information on wall shear stresses or boundary layer thickness is available. It is therefore necessary to provide estimates for the required mesh density based on correlations for flat plate boundary layers. The problem is to determine the required near wall mesh spacing in terms of Reynolds number and running length for a given y+ target value.

The estimates will be based on correlations for a flat plate with a Reynolds number of:

with the characteristic velocity and the length of the plate, L.

The correlation for the wall-shear-stress coefficient, cf, is given by:

(Eqn. 93)

The definition of ∆y+ for this estimate is:

(Eqn. 94)

with ∆y being the mesh spacing between the wall and the first node away from the wall. With the definition:

(Eqn. 95)

U+ 1

R--- RUcomp

+( )sin H 1 RUcomp+( )cos–( )–=

R u*( ) B( )⁄=

H A u*( )⁄=

ReL

ρU∞L

µ---------------=

U∞

c f 0.025Rex1– 7⁄

=

∆y+ ∆yuτ

v------------=

c f 2ρuτ

2

ρU∞2

------------ 2uτ

U∞--------

⎝ ⎠⎜ ⎟⎛ ⎞ 2

= =



Contents Index Help
Eliminating uτ between (Eqn. 94) and (Eqn. 95) gives:

Replacing cf with the help of (Eqn. 93) gives:

This equation allows us to set the proper y+ value at a given location Rex, for example, a fraction C of . As , except for very small Rex, the result is:

A second criterion for a good mesh is that there should be a minimum of mesh points inside the boundary layer in order for the turbulence model to work properly. The authors’ best estimate is to use at least 10 points for a wall-function method and at least 15 for a sublayer computation. The distance of the selected mesh point ∆yδ can then be computed from the correlation:

to be:

The boundary layer for a blunt body does not start with zero thickness at the stagnation point for Rex. It is therefore safe to use some fraction of ReL, say 25%. With this assumption you get:

You would therefore select a point, say the fifteenth off the surface and check if:

In a true geometry, assuming grid lines normal to the surface, you must compute the distance as:

These are simple guidelines for the generation of meshes which satisfy the minimal requirements for accurate boundary layer computations.

∆y ∆y+ 2

c f-----

vU∞--------=

∆y L ∆y+ 80 Rex

1 14⁄ 1ReL---------=

Rex C ReL= C1 14⁄ 1≈

∆y L ∆y+ 80ReL

13– 14⁄=

Reδ 0.14Rex6 7⁄

=

δ 0.14 L Rex6 7⁄ 1

ReL---------=

δ 0.035 L ReL1 7⁄–

=

n 15( ) n 1( )– δ≤

∆yδ ∆x2 ∆y

2 ∆z2

+ +=



Contents Index Help
Wall Distance Formulation

The wall distance is used in various functions that control the transition between near-wall and freestream models. Its derivation is described here.

1-D Illustration of Concept

Consider the 1-D case of a horizontal surface, with the y direction normal to the surface:

Consider the following Poisson equation for a variable :

(Eqn. 96)

with a boundary condition of at the wall.

Integrate once:

(Eqn. 97)

Integrate again:

(Eqn. 98)

Since at , we deduce that . We also know from (Eqn. 97) that:

y

wally=o

φ

y2

2

d

d φ 1–=

φ 0=

yddφ

y– C1+=

φ 0.5y2

– C1y C2+ +=

φ 0= y 0= C2 0=

C1 yddφ

y+=

Page 99 Wall Distance Formulation CFX-5.7.1


Contents Index Help
Substituting into (Eqn. 98) for and , we are left with a quadratic equation for in terms of and :

Rearranging this gives:

where , , and .

Solving this quadratic and choosing the positive root gives:

(Eqn. 99)

Concept Generalized to 3-D

(Eqn. 96) becomes a diffusion-only transport equation with a uniform source term of unity:

We solve this equation with wall boundary conditions of .

We then evaluate (Eqn. 99), replacing with , and interpret as the distance to the

nearest wall, or “wall distance”. Since and are always positive, the wall distance is also

always positive.

C1 C2 yφ

yddφ

φ 0.5y2

–yd

dφy+⎝ ⎠

⎛ ⎞ y+=

ay2

by c+ + 0=

a 0.5= byd

dφ= c φ–=

yyd

dφ–yd

dφ⎝ ⎠⎛ ⎞ 2

2φ++=

φ∇2 1–=

φ 0=

yddφ φ∇ y

φyd

dφ

Page 100 Wall Distance Formulation CFX-5.7.1

turbulence and near wall theory

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