Turbulent Transition Models

All of the previously described models are incapable of predicting boundary layer transition. To include the effects of transition additional equations are necessary.

Transitional flows can be found in many industrial application cases including gas turbine blades, airplane wings and wind turbines. It is known that conventional turbulence models over predict the wall shear stress for transitional flows. Thus, transition models can be used to improve the accuracy of CFD solutions when flows encounter turbulent transition of the boundary layer.

γ-Reθ Model

Langtry and Menter (2009) proposed one of the most commonly used transition models in industrial CFD applications. The $γ - R e_{θ}$ $γ - R e_{θ}$ model is a correlation based intermittency model that predicts natural, bypass and separation induced transition mechanisms.

The $γ - R e_{θ}$ $γ - R e_{θ}$ model is coupled with the Shear Stress Transport (SST) turbulence model and requires two additional transport equations for the turbulence intermittency ( $γ$ $γ$ ) and transition momentum thickness Reynolds number ( $R e_{θ}$ $R e_{θ}$ ). The turbulence intermittency is a measure of the flow regime and is defined as $γ = \frac{t_{turbulent}}{t_{laminar} + t_{turbulent}}$ $γ = \frac{t_{turbulent}}{t_{laminar} + t_{turbulent}}$ where $t_{turbulent}$ $t_{turbulent}$ represents the time that the flow is turbulent at a given location, while $t_{laminar}$ $t_{laminar}$ represents the time that the flow is laminar.

For example, while the intermittency value is zero, the flow is considered to be laminar. A value of one is for fully turbulent flow. The transition momentum thickness Reynolds number is responsible for capturing the non local effects of turbulence intensity and is defined as $R e_{θ} = \frac{ρ ¯ u θ}{μ}$ $R e_{θ} = \frac{ρ \bar{u} θ}{μ}$ where the momentum thickness is $θ = δ \int 0 \frac{u}{¯ u} (1 - \frac{u}{¯ u}) d y$ $θ = \int_{0}^{δ} \frac{u}{\bar{u}} (1 - \frac{u}{\bar{u}}) d y$ .

To couple with the SST model, Langtry and Menter (2009) modified the production term and dissipation term of the turbulent kinetic energy to account for the changes in the flow intermittency.

Transport Equations

Turbulent Kinetic Energy k

(1)

\frac{\partial (ρ k)}{\partial t} + \frac{\partial (ρ ¯ ¯¯ ¯ u_{j} k)}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [(μ + σ_{k} μ_{t}) \frac{\partial k}{\partial x_{j}}] + {˜ P}_{k} + D_{k}

$\frac{\partial (ρ k)}{\partial t} + \frac{\partial (ρ \bar{u_{j}} k)}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [(μ + σ_{k} μ_{t}) \frac{\partial k}{\partial x_{j}}] + {\tilde{P}}_{k} + D_{k}$

Eddy Frequency (Specific Dissipation Rate) ω

(2)

\frac{\partial (ρ ω)}{\partial t} + \frac{\partial (ρ ¯ ¯¯ ¯ u_{j} ω)}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [(μ + σ_{ω} μ_{t}) \frac{\partial ω}{\partial x_{j}}] + P_{ω} + D_{ω} + 2 (1 - F_{1}) \frac{ρ σ_{ω 2}}{ω} \frac{\partial k}{\partial x_{j}} \frac{\partial ω}{\partial x_{j}}

$\frac{\partial (ρ ω)}{\partial t} + \frac{\partial (ρ \bar{u_{j}} ω)}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [(μ + σ_{ω} μ_{t}) \frac{\partial ω}{\partial x_{j}}] + P_{ω} + D_{ω} + 2 (1 - F_{1}) \frac{ρ σ_{ω 2}}{ω} \frac{\partial k}{\partial x_{j}} \frac{\partial ω}{\partial x_{j}}$

Turbulent Intermittency

γ

$γ$

(3)

\frac{\partial (ρ γ)}{\partial t} + \frac{\partial (ρ ¯ ¯¯ ¯ u_{j} γ)}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [(μ + \frac{μ_{t}}{σ_{f}}) \frac{\partial γ}{\partial x_{j}}] + P_{γ} + D_{γ}

$\frac{\partial (ρ γ)}{\partial t} + \frac{\partial (ρ \bar{u_{j}} γ)}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [(μ + \frac{μ_{t}}{σ_{f}}) \frac{\partial γ}{\partial x_{j}}] + P_{γ} + D_{γ}$

Transition Momentum Thickness Reynolds Number

{Re}_{θ}

${Re}_{θ}$

(4)

\frac{\partial (ρ {Re}_{θ})}{\partial t} + \frac{\partial (ρ ¯ ¯¯ ¯ u_{j} {Re}_{θ})}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [σ_{f} (μ + μ_{t}) \frac{\partial {Re}_{θ}}{\partial x_{j}}] + P_{θ}

$\frac{\partial (ρ {Re}_{θ})}{\partial t} + \frac{\partial (ρ \bar{u_{j}} {Re}_{θ})}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [σ_{f} (μ + μ_{t}) \frac{\partial {Re}_{θ}}{\partial x_{j}}] + P_{θ}$

Production Modeling

Turbulent Kinetic Energy k

(5)

{˜ P}_{k} = γ_{e f f} min (P_{k}, 10 ρ β^{*} k ω) \frac{\partial ¯ ¯ ¯ u_{i}}{\partial x_{j}}

${\tilde{P}}_{k} = γ_{e f f} min (P_{k}, 10 ρ β^{*} k ω) \frac{\partial \bar{u_{i}}}{\partial x_{j}}$

where

Production $P_{k} = μ_{t} S^{2}$ $P_{k} = μ_{t} S^{2}$
Strain rate magnitude $S = \sqrt{2 S_{i j} S_{i j}}$ $S = \sqrt{2 S_{i j} S_{i j}}$
Strain rate tensor $S_{i j} = \frac{1}{2} (\frac{\partial ¯ ¯ ¯ u_{i}}{\partial x_{j}} + \frac{\partial ¯ ¯¯ ¯ u_{j}}{\partial x_{i}})$ $S_{i j} = \frac{1}{2} (\frac{\partial \bar{u_{i}}}{\partial x_{j}} + \frac{\partial \bar{u_{j}}}{\partial x_{i}})$
Constant $β^{*}$ $β^{*}$

Turbulent Dissipation Rate ε

(6)

P_{ε} = α ρ S^{2}

$P_{ε} = α ρ S^{2}$

where

Blending function for specifying constants $α = α_{1} F_{1} + α_{2} (1 - F_{1})$ $α = α_{1} F_{1} + α_{2} (1 - F_{1})$
$α_{1} = \frac{5}{9}$ $α_{1} = \frac{5}{9}$ for the inner layer
$α_{2}$ $α_{2}$ = 0.44 for the outer layer.
Turbulent Intermittency $γ$ $γ$

(7)

P_{γ} = F_{l e n g t h} C_{a 1} S \sqrt{(γ F_{o n s e t})} (1 - γ C C_{e 1})

$P_{γ} = F_{l e n g t h} C_{a 1} S \sqrt{(γ F_{o n s e t})} (1 - γ C C_{e 1})$

where $F_{o n s e t} = m a x {m i n (m a x [\frac{R e_{ν}}{2.193 R e_{θ 0}}, {(\frac{R e_{ν}}{2.193 R e_{θ 0}})}^{4}], 2) - m a x (1 - {(\frac{R_{T}}{2.5})}^{3}, 0)}$ $F_{o n s e t} = m a x {m i n (m a x [\frac{R e_{ν}}{2.193 R e_{θ 0}}, {(\frac{R e_{ν}}{2.193 R e_{θ 0}})}^{4}], 2) - m a x (1 - {(\frac{R_{T}}{2.5})}^{3}, 0)}$ $R e_{ν} = \frac{S d^{2}}{ν}$ $R e_{ν} = \frac{S d^{2}}{ν}$ , $R_{T} = \frac{k ν}{ω}$ $R_{T} = \frac{k ν}{ω}$

Transition Momentum Thickness Reynolds Number ${Re}_{θ}$ ${Re}_{θ}$

(8)

P_{θ} = C_{θ t} \frac{1}{t} ({Re}_{θ} - ¯ ¯¯¯¯ ¯ {Re}_{θ}) (1 - F_{θ})

$P_{θ} = C_{θ t} \frac{1}{t} ({Re}_{θ} - \bar{{Re}_{θ}}) (1 - F_{θ})$

where $F_{θ} = min (max [F_{w a k e} e^{- {(\frac{d}{δ})}^{4}}, 1 - {(\frac{γ - 1 / C_{e 2}}{1 - 1 / C_{e 2}})}^{2}], 1)$ $F_{θ} = \min (\max [F_{w a k e} e^{- {(\frac{d}{δ})}^{4}}, 1 - {(\frac{γ - 1 / C_{e 2}}{1 - 1 / C_{e 2}})}^{2}], 1)$ , $F_{w a k e} = e^{- {(\frac{R e_{ω}}{1 E + 5})}^{2}}$ $F_{w a k e} = e^{- {(\frac{R e_{ω}}{1 E + 5})}^{2}}$ , $R e_{ω} = \frac{ω d^{2}}{ν}$ $R e_{ω} = \frac{ω d^{2}}{ν}$ , $δ = \frac{50 Ω d}{¯ u} δ_{B L}$ $δ = \frac{50 Ω d}{\bar{u}} δ_{B L}$ , $δ_{B L} = 7.5 θ_{B L}$ $δ_{B L} = 7.5 θ_{B L}$ , $θ_{B L} = \frac{¯ ¯¯¯¯ ¯ R e_{θ} ν}{¯ u}$ $θ_{B L} = \frac{\bar{R e_{θ}} ν}{\bar{u}}$

Dissipation Modeling

Turbulent Kinetic Energy k

(9)

D_{k} = - min (max [γ_{e f f}, 0.1], 1.0) ρ β^{*} k ω

$D_{k} = - \min (\max [γ_{e f f}, 0.1], 1.0) ρ β^{*} k ω$

Turbulent Dissipation Rate ω

(10)

D_{ω} = - ρ β ω^{2}

$D_{ω} = - ρ β ω^{2}$

Turbulent Intermittency

γ

$γ$

(11)

D_{γ} = - C_{a 2} F_{turbulent} Ω γ (C_{e 2} γ - 1)

$D_{γ} = - C_{a 2} F_{turbulent} Ω γ (C_{e 2} γ - 1)$

where $F_{turbulent} = e^{{(- \frac{R_{T}}{4})}^{4}}$

Modeling of Turbulent Viscosity $μ_{t}$

(12)

$μ_{t} = \frac{ρ a_{1} k}{max (a_{1} ω, S F_{2})}$

where

The second blending function $F_{2} = t a n h {(m a x (2 \frac{\sqrt{k}}{β^{*} ω d}, \frac{500 ν}{d^{2} ω}))}^{2}$ ,
Fluid kinematic viscosity $ν$ ,
Constant $β^{*}$ .

Correlations

(13)

$R e_{θ} = (1173.51 - 589.428 T u + \frac{0.2196}{T u^{2}}) F (λ_{θ})$

(14)

$R e_{θ t} = {\begin{matrix} (1173.51 - 589.428 T u + \frac{0.2196}{T u^{2}}) F (λ_{θ}) f o r T u \leq 1.3 \\ 331.5 {(T u - 0.5658)}^{- 0.671} F (λ_{θ}), f o r T u > 1.3 \end{matrix}$

(15)

$F (λ_{θ}) {\begin{matrix} 1 - (- 12.986 λ_{θ} - 123.66 λ_{θ}^{2} - 405.689 λ_{θ}^{3}) e^{- {(\frac{T u}{1.5})}^{5}} f o r λ_{θ} \leq 0 \\ 1 + 0.275 {(1 - e^{- 35 λ_{θ}})}^{- 0.671} e^{- (\frac{T u}{1.5})}, f o r λ_{θ} > 0 \end{matrix}$

where $λ_{θ} = \frac{θ^{2}}{ν} \frac{d \bar{u}}{d s}$ is the pressure gradient.

Model Coefficients

$β^{*}$ = 0.09.

The following constants for SST are computed by a blending function $ϕ = ϕ_{1} F_{1} + ϕ_{2} (1 - F_{1})$ : $σ_{ω 1}$ = 0.5, $σ_{ω 2}$ = 0.856, $σ_{k 1}$ = 0.85, $σ_{k 2}$ = 1.00, $β_{1} = \frac{3}{40}$ , $β_{2}$ = 0.0828. $C_{e 1}$ = 1.0, $C_{e 2}$ = 50, $C_{a 1}$ = 2.0, $C_{a 2}$ = 0.06, $C_{θ 1}$ = 1.0, $C_{θ 2}$ = 2.0, $σ_{f}$ = 1.0.

Turbulent Transition Models

γ-Reθ Model

Transport Equations

Production Modeling

Dissipation Modeling

Modeling of Turbulent Viscosity μt<math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msub> <mi>μ</mi> <mi>t</mi> </msub> </mrow> </math>

Correlations

Model Coefficients

Modeling of Turbulent Viscosity $μ_{t}$