ANSYS CFX Transition Model Formulation

The full transition model is based on two transport equations, one for the intermittency and one for the transition onset criteria in terms of momentum thickness Reynolds number. It is called ‘Gamma Theta Model’ and is the recommended transition model for general-purpose applications. It uses a new empirical correlation (Langtry and Menter) which has been developed to cover standard bypass transition as well as flows in low free-stream turbulence environments. This built-in correlation has been extensively validated together with the SST turbulence model ([101 (p. 286)], [102 (p. 286)], [103 (p. 286)]) for a wide range of transitional flows. The transition model can also be used with the BSL or SAS-SST turbulence models as well.

It should be noted that a few changes have been made to this model compared to original version (that is, the CFX-5.7 formulation, [101 (p. 286)],[102 (p. 286)]) in order to improve the transition predictions. These include:

1. An improved transition onset correlation that results in improved predictions for both natural and bypass transition.

2. A modification to the separation induced transition modification that prevents it from causing early transition near the separation point.

3. Some adjustments of the model coefficients in order to better account for flow history effects on the transition onset location.

You should use the new formulation [103 (p. 286)], although the original version of the transition model (that is, the CFX-5.7 formulation) can be recovered by specifying the optional parameter “Transition Model Revision = 0” in the CCL in the following way:

FLUID MODELS:

…

TURBULENCE MODEL:

Option = SST

TRANSITIONAL TURBULENCE:

Option = Gamma Theta Model Transition Model Revision = 0 TRANSITION ONSET CORRELATION:

Option = Langtry Menter END

END END … END

The transport equation for the intermittency, γ, reads:

(Eq. 2.142)

+ = − + − + ⎡

⎣⎢⎛

⎝⎜ + ⎞

⎠⎟ ⎤

⎦⎥

∂

∂

∂

∂

∂

∂

∂

P E P E μ ∂

ρ γ t

ρU γ

x γ γ γ γ x

μ σ

γ x

( )

( )

1 1 2 2

j

j j

t

γ j

The transition sources are defined as follows:

(Eq. 2.143)

= ⎡⎣ ⎤⎦ =

P_γ1 2F_length ρS γF_onset ^cγ3 ;E_γ1 P_γ1γ

where S is the strain rate magnitude. F_length is an empirical correlation that controls the length of the transition region. The destruction/relaminarization sources are defined as follows:

(Eq. 2.144)

= =

P^γ2

(

²c^γ1

)

ρ Ω γ F^{turb ;}E^γ2 c^γ2P^γ2γ

where Ω is the magnitude of vorticity rate. The transition onset is controlled by the following functions:

(Eq. 2.145)

= =

Re_ν ^{ρy S}_μ² ;R_{T μ ω}^ρk

ANSYS CFX Transition Model Formulation

(Eq. 2.146)

=

= F ⋅

F ^{min max}

( (

F ^,F

)

^{, 2.0}

)

Re onset 1 2.193 Re

onset 2 onset 1 onset 14 ν

θc

(Eq. 2.147)

= ⎛

⎝⎜ − ⎞

⎠⎟

= −

= ⁻ F

F F F

F e

( )

( )

max 1 , 0

max , 0

R

( )

onset 3 2.5

3

onset onset 2 onset 3 turb

T

RT4 4

Re_θc is the critical Reynolds number where the intermittency first starts to increase in the boundary layer. This occurs upstream of the transition Reynolds number, Re^∼ _θt, and the difference between the two must be obtained from an empirical correlation. Both the F_length and Re_θc correlations are functions of Re^∼ _θt.

The constants for the intermittency equation are:

(Eq. 2.148)

=

=

=

= c

c c σ

0.03 50 0.5 1.0

y y y y 1 2 3

The modification for separation-induced transition is:

(Eq. 2.149)

= ⎛

⎝⎜ ⋅ ⎡

⎣⎢⎛

⎝⎜

⎞

⎠⎟− ⎤

⎦⎥ ⎞

⎠⎟

=

=

−

γ F F

F e

γ

(

γ γ

)

min 2 max 1, 0 , 2

max ,

Re

Re θt

eff

( )

sep 3.235 reattach

reattach

sep

ν θc RT20

4

The model constants in Equation 2.149 (p. 76) have been adjusted from those of Menter et al. [101 (p. 286)] in order to improve the predictions of separated flow transition. The main difference is that the constant that controls the relation between Re_ν and Re_θc was changed from 2.193, its value for a Blasius boundary layer, to 3.235, the value at a separation point where the shape factor is 3.5 (see, for example Figure 2 in Menter et al. [101 (p. 286)]).

The boundary condition for γ at a wall is zero normal flux while for an inlet γ is equal to 1.0.

The transport equation for the transition momentum thickness Reynolds number, Re^∼ _θt, reads:

(Eq. 2.150)

+ = + ⎡

⎣⎢

⎢

⎛

⎝⎜ + ⎞

⎠⎟ ⎤

⎦⎥

⎥

∂

∂

∂

∂

∂

∂

∂

∂

∼ ∼ ∼

P σ

(

μ μ

)

ρR e t

ρU R e

x θt x θt t

R e x

(

^θt

) (

^{j θt}

)

j j

θt j

The source term is defined as follows:

(Eq. 2.151)

= −^∼ − =

Pθt cθt ρ

(

Re Re

) (

^1.0 F

)

^;t

t θt θt θt μ

ρU 500 2

(Eq. 2.152)

= ⎛

⎝⎜⎜ ⎛

⎝⎜ ⋅ − ⎞

⎠⎟ ⎞

⎠⎟⎟

− −

F^{θt min max} F^wake e

^{( )}

^{, 1.0}

(

1.0 1⁄50^γ −^1⁄50

)

^{, 1.0}

y 2 δ

4

(Eq. 2.153)

= = = ⋅

∼

θ_BL ^{R e}_ρU^θtμ ;δ_BL ¹⁵₂ θ_BL ;δ ⁵⁰_U^Ωy δ_BL

ANSYS CFX Transition Model Formulation

(Eq. 2.154)

= = ^{− ⎛⎝⎜ × ⎞⎠⎟} Re_ω ^{ρ ωy}_μ ;F_wake e

Reω 2

1 105 2

The model constants for the Re^∼ _θt equation are:

(Eq. 2.155)

= =

c_θt 0.03 ;σ_θt 2.0

The boundary condition for Re^∼ _θt at a wall is zero flux. The boundary condition for Re^∼ _θt at an inlet should be calculated from the empirical correlation based on the inlet turbulence intensity.

The model contains three empirical correlations. Re_θt is the transition onset as observed in experiments. This has been modified from Menter et al. [101 (p. 286)] in order to improve the predictions for natural transition. It is used in Equation 2.151 (p. 76). F_length is the length of the transition zone and goes into Equation 2.143 (p. 75). Re_θc is the point where the model is activated in order to match both Re_θt and F_length; it goes into Equation 2.146 (p. 76).

At present, these empirical correlations are proprietary and are not given in this manual.

(Eq. 2.156)

= = ^∼ = ^∼

Reθt f Tu^{( , ) ;}λ Flength f Re

( )

θt ^,Reθc f Re

( )

θt

The first empirical correlation is a function of the local turbulence intensity, T u, and the Thwaites pressure gradient coefficient λ_θ defined as:

(Eq. 2.157)

= ⎛⎝⎜

⎞ λ_θ ^θ_v²⎠⎟ ^dU_ds

where dU ds⁄ is the acceleration in the streamwise direction.

The transition model interacts with the SST turbulence model, as follows:

(Eq. 2.158)

+ = − + ⎛

⎝⎜ + ⎞

⎠⎟

∂

∂

∂

∂

∼ ∼ ∂

∂

∂

ρk

(

ρu k

)

P D

⁽

μ σ μ

⁾

∂

( )

t x j k k x k t

k

j j xj

(Eq. 2.159)

= =

∼ ∼

P_k γ_effP_k ^;D_k ^{min max}

( (

γ_eff, 0.1 , 1.0

) )

D_k

(Eq. 2.160)

= = ^{− ⎛⎝⎜ ⎞⎠⎟} =

R^{y ρy k}_μ ^;F³ e ^{120R y ;}F¹ max F

(

^{1 orig,}F³

)

8

where P_k and D_k are the original production and destruction terms for the SST model and F_{1 orig} is the original SST blending function. Note that the production term in the ω-equation is not modified. The rationale behind the above model formulation is given in detail in Menter et al. [101 (p. 286)].

In order to capture the laminar and transitional boundary layers correctly, the grid must have a y⁺ of approximately one. If the y⁺ is too large (that is, > 5) then the transition onset location moves upstream with increasing y⁺. You should use the High Resolution discretization scheme (which is a bounded second-order upwind biased discretization) for the mean flow, turbulence and transition equations.

Note

The default production limiter for the turbulence equations is the ‘Kato-Launder’ formulation when the transition model is used.

As outlined inCFX Transition Model (p. 101) in the ANSYS CFX-Solver Modeling Guide, two reduced variants of the transition model are available beside the two-equation Gamma Theta transition model. A zero equation model, where the user can prescribe the intermittency directly as a CEL expression Specified Intermittency, and a one equation model, which solves only the intermittency equation using a user specified value of the transition onset Reynolds number Gamma Model.

ANSYS CFX Transition Model Formulation