Introduction to Turbulence Modeling

Background• The need for turbulence modeling

– In order to predict turbulent flows by numerical solutions to the Reynolds equations, it becomes necessary to make closing assumptions about the apparent turbulent stress.

• What is a turbulence model ?– A turbulence model is a computational procedure to close the the

system of mean flow equations so that a more or less wide variety of flow problems can be calculated.

• The purpose of this section – is to introduce the methodology commonly used in turbulence

modeling.

Categories of Turbulence Models• Turbulent viscosity models (Category I)

– Models using the Boussinsq assumption

– Most models currently used in engineering calculations are of this type.

• Zero equation models

• One equation models

• Two equation models

• Reynolds stress or stress-equation models (Category II)– Models that affect closure to the Reynolds Equations without

the Boussinsq assumption.

• Large-eddy simulations (Category III)– Not based entirely on the Reynolds equations

Boussinsq assumption• The apparent turbulent shearing stress might be related to the rate

mean strain through an apparent scalar turbulent or “eddy” viscosity.

• For the general Reynolds stress tensor, the Boussinsq assumption gives

' ' 23

ji ii j t t ij

j i i

uu uu u kx x x

ρ µ ρ µ δ⎛ ⎞∂ ⎛ ⎞∂ ∂

− = + − +⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠⎝ ⎠

where µt is the turbulent viscosity,

ui is the mean velocity component,

δij is “Kronecker delta” : if i=j, δij=1; otherwise δij =0

k is the kinetic energy of turbulence

( )' '

'2 '2 '212 2i iu uk u v w= = + +

The turbulent viscosity model

• The turbulent or “eddy” viscosity model is to relate the turbulent viscosity, µt to the mean properties.

• According to the number of differential equations used in determining µt , the eddy viscosity models are classified into the followings:– Zero equation models

– One equation models

– Two equation models

Zero Equation Models• Zero equation models are also called mixing length or simple

algebraic models.

• It relate the turbulent viscosity, µt to the mean properties by simple algebraic formulae, not by differential equations.

• One of the most successful of this type of model was suggested by Prandtl in the 1920s:

2t m

uly

µ ∂=

∂

• The apparent turbulent shearing stress is given by

' ' 2i j m

u uu u ly y

ρ ρ ∂ ∂− =

∂ ∂where lm2 is a “mixing length”, determined by the algebraic formulae or experiments

Comments on Zero Equation Models• There exist some other mixing length models that are the most widely

used turbulence models in external aerodynamic calculations in the aerospace industry.– Baldwin and Lomax (1978)

– Cebeci and Smith (1974)

• Advantages of mixing length models – Easy to implement and cheap in terms of computing resource

– Good predictions for thin shear

– Well established

• Disadvantages of mixing length models – Completely incapable of describing flows with separation and recirculation

– Only calculates mean flow properties and turbulent shear stress

One Equation Models

• Motivations

– Shortcomings of algebraic models• The turbulent viscosity, µt and mixing length lm2 are related

to the gradient of the mean velocity only.

• The µt and lm2 account for local effects (the gradient of the mean velocity), not for the convection and diffusion properties.

– It is suggested that one more transport equation is needed for the kinetic energy of turbulence k.

– There should be a relation between k and µt .

One Equation Models

• The modeled form of the kinetic energy of turbulence equation

3/ 2( )( ) ji t i it D

i j k j j i j

uku u uk k kCt x x x x x x l

ρ µρ µ µ ρσ

⎡ ⎤ ⎛ ⎞∂⎛ ⎞∂ ∂ ∂∂ ∂ ∂+ = + + + −⎜ ⎟⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦ ⎝ ⎠

Particle rate ofincrease of k

Diffusion rate for k

Generation rate for k

Dissipation rate for k

• Based on the suggestion of Prandtl and Kolmogoroc, the turbulent viscosity can be evaluated as

t C l kµµ ρ=

where σk=1, Cµ = 0.09, CD = 0.08 ~ 0.38

l is a length parameter, determined by the algebraic formulae or experiments

Comments on One Equation Models

• The one equation models appear to provide a definite improvement over algebraic models.

• The recent one equation models have provided better agreement with experiment data for some separated flows than has generallybeen possible with algebraic models.– Baldwin-Barth model

– Spalart-Allmaras model

• On the whole, however, the performance of most one equation models has not offer a significant improvement over that of the algebraic models.

• It is still difficult to evaluate the length parameter l.

Two Equation Models• What is the difference between one equation models

and two equation models.– In two equation models, one more equation, called turbulent

dissipation equation, is added to Reynolds Averaged N-S Equations in addition to the turbulent kinetic energy equation.

• The typical two equation models is k-ε models.

• In k-ε models, the turbulent dissipation rate, ε is defined as

' 'i i

k k

u ux x

µερ⎛ ⎞⎛ ⎞∂ ∂

= ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠

The Standard k-ε model

• The turbulent viscosity µ can be expressed in terms of k and ε :

2

tkCµµ ρε

=

• Where Cµ is an ad hoc constant.

• In the standard k-ε model, k and ε are unknown variables. There are two equations for k and ε:

1 3 2

( )( )

( )( ) ( )

i tk b M k

i j k j

i tk b

i j k j

kuk k G G Y St x x x

u C G C G C St x x x kε ε ε ε

ρ µρ µ ρεσ

ρε µρε ε εµ ρεσ

⎡ ⎤⎛ ⎞∂∂ ∂ ∂+ = + + + − − +⎢ ⎥⎜ ⎟∂ ∂ ∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞∂∂ ∂ ∂+ = + + + − +⎢ ⎥⎜ ⎟∂ ∂ ∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦

The Standard k-ε model

• where Gk is the generation rate of turbulent kinetic energy due to mean velocity

ji ik t

j i j

uu uGx x x

µ⎛ ⎞∂∂ ∂

= +⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠

• Gb is the generation rate of turbulent kinetic energy by pressure• YM is the generation of k by Reynolds stress• C1ε, C2ε, C3ε, Cµ, σK and σs are ad hoc adjustment constants.

1 2

3 3

1.44, 1.92, 0.09, 1.0,1.3, 1 0

k

s

C C CC or C

ε ε µ

ε ε

σ

σ

= = = =

= = =

Shortcomings of The Standard k-ε model

• The standard k-ε model is not appropriate for use in the near-wall region because the damping effect solid boundary has not been included in the model.

• Not appropriate for use in the flows with the low Reynolds numbers.

• Poor performance in some cases such as– Curved boundary layer

– Swirling flows

Modifications to The Standard k-ε model

• The model with modification by the addition of damping terms to extend applicability to the near-wall region.

• The low Reynolds number k-ε model (Re < 150)

• The RNG k-ε model:– RNG stands for Re-Normalization Group (重整化群k-ε模型)

– Can be used in curved boundary layer

• Realizable k-ε model:– Can be used in swirling flows

Comments on The Standard k-ε model

• Advantages– Excellent performance for many industrially relevant flows

– Well established; most widely validated turbulence model

• Disadvantage– More expensive to implement than mixing length model and one

equation model ( two extra PDEs)

– Poor performance in some cases• Some separated flows

Reynolds Stress Equation Models

• What is the differences between the Turbulent viscosity models and Reynolds Stress Equation Models ?– Reynolds Stress Equation Models do not assume that the

turbulent shearing stress is proportional to the rate of mean strain (Boussinsq assumption ).

– Reynolds Stress Equation Models need Reynolds stress transport equations, i.e. a set equations about i ju u

Reynolds Stress Equation Models

• Advantages– Potentially the most general of all classical turbulence models

– Only initial and/or boundary conditions need to be supplied

– Very accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets, asymmetric channel and non-circular duct flows and curved flows.

• Disadvantages– Very large computing costs (seven extra PDEs)

– Not widely validated as the mixing length and k-ε models

– Performs just as poorly as the k-ε models in some flows

Algebraic Stress Equation Models

• What is the differences between the Algebraic Stress equation and Reynolds Stress Equation Models ?– The algebraic stress model (ASM) is an economical way of

accounting for the anisotropy of Reynolds stress without going to the full length of solving the Reynolds stress transport equations.

– The Reynolds stress equations reduce to a set of algebraic equations.

Algebraic Stress Equation Models• Advantages

– Cheap method to account for Reynolds stress anisotropy

– Potentially combines the generality of approach of the RSM with the economy of the k-ε model

– If convection and diffusion terms are negligible the ASM performs as well as the RSM

• Disadvantages– Only slight more expensive than the k-ε model ( two PDEs and a system

of algebraic equations)

– Not widely validated as the mixing length and k-ε models

– Model is severely restricted in flows where the transport assumption for convective and diffusive effects do not apply.

Large Eddy Simulation

• In Large Eddy Simulation (LES), the large-scale structure of the turbulent flow is computed directly and only the effects of the smallest (subgrid-scale) and more nearly isotropic eddies are modeled.

• It is accomplished by “ filtering” the Navier-Stokes equations to obtain a set of equations that govern the “resolved” flow.

• The computational effort required for LES is less than that of DNS by approximately a factor of 10 using present-day methods.

Summary on the Turbulence Models

• Turbulent viscosity models (Category I)– Zero equation models

– One equation models

– Two equation models

• Reynolds stress or stress-equation models (Category II)– The Reynolds stress equation model (RSM)

– The algebraic stress equation model (ASM)

• Large-eddy simulations (Category III)– Subgrid Scale Models

Final Remarks• The mixing length model and the k-ε models are most

widely applied models.

• The standard k-ε model still comes highly recommended for general purpose CFD computations.

• Large eddy simulation models required large computing resources and are not yet of use as general purpose tools.

• Different turbulence models may give you different results.

• The turbulence models don’t yet perform very well for use in severely separated flows.