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## Practical Turbulence Modelling in Fluent

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Reynolds Averaging

- Most engineering CFD problems solve the RANS
- These are derived by splitting the instantaneous velocity into a mean and a fluctuating component:
- This decompsition is then substituted into the equations of motion for all velocities:

Reynolds Averaging

- The equations are then averaged over all scales of the turbulence
- In general:
- but:
- The mean of the fluctuating product is non zero and ui’& uj’are said to be correlated

Reynolds Averaging

- The result is the Reynolds Averaged Navier Stokes Equations

Reynolds Stresses

- Averaging lead to an additional tensor of stresses in the momentum equation. These stresses are called the Reynolds Stresses
- The Reynolds Stresses
- Are six additional terms in the momentum equations (The Tensor is symmetric)
- They characterise the transfer of momentum by turbulence
- These are unknowns and must be modelled

The Averaging Domain

- Historically this has been interpreted as a time interval which is long relative to the fluctuations and short relative to any mean dynamics of the flow.
- However it can be interpreted as a spatial domain or an ensemble of measurements. The Ergodic hypothesis postulates that these three averages are equal (for stationary homogeneous turbulence). See Wilcox p31
- There are a number of other averaging techniques including phase averaging and Favre averaging

Modelling the Reynolds Stresses

- A good model must characterise the transfer of momentum by the turbulence but can only do so in terms of the mean flow and mean values of the Reynolds Stresses
- This is because averaging removes all information about the turbulent energy cascade and the scales of the turbulence
- We can still account for anisotrophy to some extent because we still have a tensor
- But the reality still is that after averaging we have lost a huge amount of information about a complex process and we must expect that this will limit the applicability of the model
- Algebraic, One and two equation models make the Boussinesq approximation. RSM models do not

The Bousinessq Approximation

- The Bousinessq approximation proposes that the transport of momentum by turbulence is a diffusive process and thus that the Reynolds stresses can be modelled using an eddy viscosity which is analagous to molecular viscosity
- What does this mean?
- It assumes that the turbulence is isotropic (Why?)
- It assumes a local equilibrium between stress and strain (Why?)
- It assumes that eddies behave like molecules and this is the original basis for the approximation.

Do eddies behave like molecules?

- The first obvious flaw is that molecules are very small but eddies are comparable in size to the scale of the flow
- The derivation of the molecular viscosity involves truncation of a Taylor series to a first order term. This is valid for moeculaes because the mean free path is much shorter than the mean scale of the flow. But eddies have a mean free path with is comparable to the scale of the flow
- Molecules participate in a lot of collisions because the mean free path is short. This implies that they have a lot of opportunities to reach local equilibrium wr to momentum. Again not the case for eddies.

Mixing Length Models

- These are algebraic models for turbulence and proposes that the eddy viscosity is a function of a mixing length.
- The mixing length of the eddy is analogous to the mean free path of the molecule
- Wilcox discusses the mixing length model because it highlights the flaws in the Bousinessq approximation.
- Nevertheless mixing length models are useful because they describe relatively simple shear flows quite well
- Typical applications include turbulent boundary layers, mixing layers and free shear flows such as far wakes,mixing layers and jets

Mixing Length Models

- The earliest algebraic model is the Prandtl mixing length model, but other authors refined the model for particular applications
- Typically ymight be the distance from the wall as in a turbulent boundary layer.

Law of the Wall

- All turbulent boundary layers display this behaviour when uand yare made non-dimensional with respect to the friction velocity u

Law of the Wall

- Fig 1.5 p 15 wilcox or fig 3.7 p 70 Wilcox

Law of the Wall

- This is a very powerful relationship because
- It is observed in all fully developed boundary layers
- Accurate calculation of the velocity profile in the boundary layer is easy to determine provided we know the shear stress at the wall
- From a CFD perspective it means that we may not have to mesh into the boundary layer because we have an algebraic solution to the equations of motion in this region (Fluent’s standard Wall functions!!)
- If you apply a mixing length model to the equations of motion for a fully developed turbulent boundary layer you will derive the Law of the Wall

One Equation and Two Equation Models

- The aim of these models is to incorporate non-local and flow history effects on the eddy viscosity.
- They solve an extra transport equation for the turbulent kinetic energy. Two equation models also solve a transport equation for the disippation
- The turbulent kinetic energy is defined as the trace of the Reynolds Stress tensor
- A rigorous transport equation for the turbulent kinetic energy can be derived by the taking the trace of the rigorous Reynolds Stress Equations

The turbulent kinetic energy equation

- The turbulent kinetic energy equation as modeled has a number of simplifications from the rigorous equation. The modeled equation is:
- The first term on the RHS is the production of k, the second term () is the specific dissipation per unit mass. The last terms describe the transport of k by molecular and turbulent diffusion
- Virtually all one and two equation turbulence models solve this equation

The turbulent kinetic energy equation

- To close the k equation we need to calculate e and we still have to calculate the eddy viscosity t.
- From dimensional considerations:
- and
- the closure problem in this instance is to calculate the length scale l

One equation models

- One equation models prescribe the length scale l algebraically
- There is a whole class of one equation models described in the literature. See Wilcox
- Fluent has implemented the Spalart-Allmaras model. This solves a transport equation for the eddy viscosity rather than the turbulent kinetic energy. It has been tuned for airfoil and wing applications but gives poor results for jet spreading rates.
- One equation models probably suffer the same limitations as a mixing length model for basically the same reasons (They prescribe the length scale)

Two - Equation models (k-e and k- w)

- Two equation models are the simplest “complete” turbulence models and use the Bousinessq approximation
- They solve two additional transport equations - one for the turbulent kinetic energy and the other is for the turbulent disippation rate. Example is the standard k-e model
- They have been the basis of most turbulence modeling reasearch
- The eddy viscosity is calculated from dimensional considerations

The k-e model and its variants

- An exact transport equation for e can derived by taking the following moment about the Navier Stokes equations. (The procedure is basically the same as the derivation of rigorous Reynolds Stress Transport equation - see Wilcox p123)
- The problems are
- The rigorous equation contains more unknowns. The data to determine approximations for these are sparse
- The end result is that the modeled equation is drastically simplified
- Wilcox suggests that the modeled equation is just an empirical equation for the rate of energy transfer from the large eddies

The Standard k-e model

- This standard k-e model is the default turbulence model in Fluent. Rather than solving for a length scale it solves a second transport equation for the disippation rate.

The Standard k-e model

- This model was derived and tuned for Flows with high reynolds numbers
- This implies
- It is suited for flows where the turbulence is nearly iso-tropic
- And is suited to flows where the energy cascade proceeds in local equilibrium with respect to generation
- Fluent also has the RNG and Realizable k-e models

The RNG k-e model

- The RNG k-e model solves the same transport equations are the standard model but the constants are derived analytically (?) from RNG methods and the turbulent viscosity is calculated differently
- This improves the prediction at low Reynolds numbers and for near wall flows
- The model also includes a swirl constant which improves the prediction for swirling flows. However this constant assumes different values depending on how strong the swirl is. (Ie an empirical constant)
- In cyclone studies it has been shown to give poor results and increasing the swirl constant creates numerical instability

The Realizable k-e model

- The equation for the compoent of the Reynolds stress tensor that is normal:
- This must be > 0 by definition (it is a square). However this equation implies that if the strain is sufficiently large it will go negative
- The realizable k-e uses a variable Cso that this will never happen.
- It is better suited to flows where the strain is large. This includes flows with strong streamline curvature vorticies and rotation

k-w models

- These are also two equation models but formulate the disippation differently
- w is the specific disippation rate
- Are in Fluent 6
- Wilcox is one of the developers of this type of model. He claims the model is better for some applications

Reynolds Stress Models

- Models which make the Boussinesq approximation are inaccurate for flows with sudden changes in the mean strain rate.
- Similarly the Boussinesq approximation fails for flows which experience extra rates of strain caused by rapid dilation, out-of plane straining and significant streamline curvature
- The failure arises because the normal reynolds stresses are not equal ii jj kk
- Flow history effects on ij persist for long distances in turbulent flows. This arises because the eddies exchange momentum with other eddies relatively slowly
- The Boussinesq approximation assumes that eddies behave like molecules and exchange momentum with other molecules quickly. ( Bear in mind the difference in scales) - it is not surprising then that eddy viscosity models may be flawed

Reynolds Stress Models

- Reynolds stress models attempt to solve transport equations for the individual Reynolds stresses
- In principle this is a better approach but the problem is providing closure approximations to the extra 22 unknowns (correlations) that arise in the derivation of the rigorous or exact equations. Unfortunately closure is provided by empiricism
- The model derived by Launder Reece and Rodi is the best known model and is the RSM implemented in Fluent
- This model still solves a transport equation for the dissipation and a transport equation for the turbulent kinetic energy
- The Fluent RSM model solves 7 additional transport equation compared to two for a k-e model. This makes it numerically more intensive and it tends to be less stable.

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