Langley Research Center

Turbulence Modeling Resource

The Langtry-Menter 4-equation Transitional SST Model

This web page gives detailed information on the equations for various forms of the Langtry-Menter transitional shear stress transport turbulence model. All forms of the model given on this page are linear eddy viscosity models. Linear models use the Boussinesq assumption for the constitutive relation:

$\tau_{ij} = 2 \mu_t \left(S_{ij} - \frac{1}{3} \frac{\partial u_k}{\partial x_k} \delta_{ij} \right) - \frac{2}{3} \rho k \delta_{ij}$

Unless otherwise stated, for compressible flow with heat transfer this model is implemented as described on the page Implementing Turbulence Models into the Compressible RANS Equations, with perfect gas assumed and Pr = 0.72, Pr_t = 0.90, and Sutherland's law for dynamic viscosity.

Return to: Turbulence Modeling Resource Home Page

Langtry-Menter 4-equation Transitional SST Model (SST-2003-LM2009)

This model is also sometimes known as the "gamma-Retheta-SST" model, because it makes use of equations for "gamma" and "Retheta" in addition to SST's "k" and "omega" equations. The primary reference for the standard implementation of the Langtry-Menter 4-equation Transitional SST model is:

Langtry, R. B. and Menter, F. R., "Correlation-Based Transition Modeling for Unstructured Parallelized Computational Fluid Dynamics Codes," AIAA Journal, Vol. 47, No. 12, December 2009, pp. 2894-2906, https://doi.org/10.2514/1.42362.

Note that this reference is missing a key functional definition ( $F_{turb}$ ). The missing definition for the "standard" implementation is instead taken from:

Menter, F. R., Langtry, R., and Volker, S., "Transition Modelling for General Purpose CFD Codes," Flow, Turbulence and Combustion, Vol. 77, No. 1, Nov. 2006, pp. 277-303, https://doi.org/10.1007/s10494-006-9047-1.

In the interest of clarity and to remove ambiguity in the implementation of this model, some nomenclature and the presentations of some functional definitions have been altered from the original reference.

Also note that there is some minor confusion in the literature for this model. The papers mostly reference the original (standard) Menter SST model, but the PhD thesis from which the model was derived as well as Flow, Turbulence, and Combustion 77(1):277-303, 2006 both make use of the SST-2003 base model. Thus, the default four-equation model here is based on the two-equation SST-2003 model, augmented by two additional equations to describe the laminar-turbulent transition process. (See ending paragraph for naming conventions when basing this transition model off of different SST base models.) The model is given by the following:

$\frac{\partial \left( \rho k \right)}{\partial t} + \frac{\partial \left( \rho u_j k \right)}{\partial x_j} = \hat P_k - \hat D_k + \frac{\partial}{\partial x_j} \left[ \left( \mu + \sigma_k \mu_t \right) \frac{\partial k}{\partial x_j} \right]$

$\frac{\partial \left( \rho \omega \right)}{\partial t} + \frac{\partial \left( \rho u_j \omega \right)}{\partial x_j} = P_{\omega} - D_{\omega} + \frac{\partial}{\partial x_j} \left[ \left( \mu + \sigma_{\omega} \mu_t \right) \frac{\partial \omega}{\partial x_j} \right] + 2 \left( 1 - F_1 \right) \frac{\rho \sigma_{\omega 2}}{\omega} \frac{\partial k}{\partial x_j} \frac{ \partial \omega}{\partial x_j}$

$\frac{\partial \left( \rho \gamma \right)}{\partial t} + \frac{\partial \left( \rho u_j \gamma \right)}{\partial x_j} = P_\gamma - E_\gamma + \frac{\partial}{\partial x_j} \left[ \left( \mu + \frac{\mu_t}{\sigma_f} \right) \frac{\partial \gamma}{\partial x_j} \right]$

$\frac{\partial \left( \rho\hat Re_{\theta t} \right)}{\partial t} + \frac{\partial \left( \rho u_j \hat Re_{\theta t} \right)}{\partial x_j} = P_{\theta t} + \frac{\partial}{\partial x_j} \left[ \sigma_{\theta t} \left( \mu + \mu_t \right) \frac{\partial \hat Re_{\theta t}}{\partial x_j} \right]$

The equations have been written above to be in proper conservation form, consistent with, e.g., Wilcox (in Turbulence Modeling for CFD, DCW Industries, Inc., La Canada, CA, 2006), Menter et al (in Turbulence, Heat and Mass Transfer 4, 2003, pp. 625-632), and Menter (in NASA TM 103975, 1992, https://ntrs.nasa.gov/citations/19930013620).

The source terms for the $\gamma$ equation are defined as:

$P_\gamma = F_{length} c_{a1} \rho S \left[ \gamma F_{onset} \right] ^{0.5} \left( 1 - c_{e1} \gamma \right)$

$E_\gamma =c_{a2} \rho \Omega \gamma F_{turb} \left( c_{e2} \gamma - 1 \right)$

where

$F_{onset1} = \frac{Re_V}{2.193 Re_{\theta c}}$

$Re_V = \frac{\rho S d^2}{\mu}$

$F_{onset2} = min \left( max \left( F_{onset1},F_{onset1}^4 \right), 2.0 \right)$

$R_T = \frac{\rho k}{\mu \omega}$

$F_{onset3} = max \left[ 1 - \left( \frac{R_T}{2.5} \right)^3, 0 \right]$

$F_{onset} = max \left(F_{onset2} - F_{onset3}, 0 \right)$

$F_{turb} = exp \left[ - \left( \frac{R_T}{4} \right) ^4 \right]$

$F_{length} = F_{length,1} \left( 1 - F_{sublayer} \right) + 40.0 F_{sublayer}$

$F_{length,1} = \left\{ \begin{array}{ll} 39.8189 + (-119.270 \cdot 10^{-4}) \hat Re_{\theta t} + (-132.567 \cdot 10^{-6}) \hat Re_{\theta t}^2, & \hat Re_{\theta t} < 400; \\ 263.404 + (-123.939 \cdot 10^{-2}) \hat Re_{\theta t} + (194.548 \cdot 10^{-5}) \hat Re_{\theta t}^2 + (-101.695 \cdot 10^{-8}) \hat Re_{\theta t}^3, & 400 \leq \hat Re_{\theta t} < 596 ;\\ 0.5 - (3.0\cdot 10^{-4})(\hat Re_{\theta t} - 596.0), & 596 \leq \hat Re_{\theta t} < 1200; \\ 0.3188 & 1200 \leq \hat Re_{\theta t} \end{array} \right.$

$F_{sublayer} = exp \left[ - \left( \frac{Re_\omega}{200} \right) ^2 \right]$

$Re_{\omega} = \frac{ \rho \omega d^2}{\mu}$

$Re_{\theta c} = \left\{ \begin{array}{ll} (-396.035 \cdot 10^{-2}) + (10120.656 \cdot 10^{-4}) \hat Re_{\theta t} + (-868.230 \cdot 10^{-6}) \hat Re_{\theta t}^2 + (696.506 \cdot 10^{-9}) \hat Re_{\theta t}^3 + (-174.105 \cdot 10^{-12}) \hat Re_{\theta t}^4, & \hat Re_{\theta t} \leq 1870; \\ \hat Re_{\theta t} - \left( 593.11 + 0.482 \left( \hat Re_{\theta t} - 1870.0 \right) \right) & 1870 < \hat Re_{\theta t} \end{array} \right.$

(Note that this last expression looks slightly different from Eq. (16) of AIAA Journal 47(12):2894-2906, 2009 because two of the terms have been combined: (10120.656x10^-4) $\hat Re_{\theta t}$ = $\hat Re_{\theta t}$ + 120.656x10^-4 $\hat Re_{\theta t}$ .) In the above, $\rho$ is the density, $\mu$ is the molecular dynamic viscosity, $d$ is the distance from the field point to the nearest wall, $S = \sqrt{2S_{ij}S_{ij}}$ is the strain rate magnitude, and $\Omega = \sqrt{2W_{ij}W_{ij}}$ is the vorticity magnitude, with

$S_{ij}=\frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right)$

$W_{ij}=\frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} \right)$

The source term of the $\hat Re_{\theta t}$ equation is defined as:

$P_{\theta t} = c_{\theta t} \frac{\rho}{T} \left( Re_{\theta t}^{eq} - \hat Re_{\theta t} \right) \left( 1.0 - F_{\theta t} \right)$

for which

$T = \frac{500 \mu}{\rho U^2}$

$U = \sqrt{ u_k u_k}$

$F_{\theta t} = min \left[ max \left( F_{wake} exp \left(- \left( \frac{d}{\delta} \right) ^4 \right) , 1.0 - \left( \frac{c_{e2}\gamma - 1}{c_{e2} - 1}\right) ^2 \right), 1.0 \right]$

$\delta = \frac{375 \Omega \mu \hat Re_{\theta t} d}{\rho U^2}$

$F_{wake} = exp \left[ - \left( \frac{Re_{\omega}}{1\cdot 10^{5}} \right) ^2 \right]$

$\lambda_{\theta} = \frac{\rho \theta_t ^2}{\mu} \frac{dU}{ds}$

$Tu = 100 \frac{\sqrt{2k/3}}{U}$

$\frac{dU}{ds} = \frac{u_m u_n}{U^2} \frac{\partial u_m}{\partial x_n}$

$Re_{\theta t} ^{eq} = \left\{ \begin{array}{ll} \left( 1173.51 - 589.428 Tu + 0.2196 Tu^{-2} \right) F \left( \lambda_{\theta} \right), & Tu \leq 1.3; \\ 331.50 \left(Tu - 0.5658 \right) ^{-0.671} F \left( \lambda_{\theta} \right) & Tu > 1.3 \end{array} \right.$

$F \left( \lambda_{\theta} \right) = \left\{ \begin{array}{ll} 1 + \left[ 12.986 \lambda_{\theta} + 123.66 \lambda_{\theta} ^2 + 405.689 \lambda_{\theta} ^3 \right] exp \left( -\left( \frac{Tu}{1.5} \right)^{1.5} \right), & \lambda_{\theta} \leq 0; \\ 1 + 0.275 \left[1 - exp \left( -35.0 \lambda_{\theta} \right) \right] exp \left( - \frac{Tu}{0.5} \right) & \lambda_{\theta} > 0 \end{array} \right.$

IMPORTANT: The expression for $Re_{\theta t}^{eq}$ is an implicit function of $\theta_t$ through the presence of $\lambda_{\theta}$ since

$Re_{\theta t} ^{eq} = \frac{\rho U \theta_t }{\mu}$

(the equations for $Re_{\theta t}^{eq}$ are typically solved by iterating on the value of $\theta_t$ ). Note that in the nomenclature of AIAA Journal 47(12):2894-2906, 2009, the expression for $Re_{\theta t}^{eq}$ uses a "local freestream velocity" for U, which is actually intended to be the velocity at the edge of the boundary layer. But in the functionality of the model, this velocity needs to be the local velocity. Although $Re_{\theta t}^{eq}$ is small for small velocities (i.e. near the wall in a boundary layer), this effect is accounted for in the model with the F_theta_t term. Outside of the boundary layer, the transported $\hat Re_{\theta t}$ is "attracted" to the equilibrium value ( $Re_{\theta t}^{eq}$ ) and is physically correct at the edge of the boundary layer. Inside the boundary layer, the attraction is suppressed and the value at the edge is diffused into the boundary layer.

The calibration constants for the Langtry-Menter model are:

$\begin{array}{l} c_{a1} = 2.0 \\ c_{a2} = 0.06 \\ c_{e1} = 1.0 \\ c_{e2} = 50 \\ c_{\theta t} = 0.03 \\ s_1 = 2 \\ \sigma_{f} = 1.0 \\ \sigma_{\theta t} = 2.0 \end{array}$

The boundary conditions for $\gamma$ and $\hat Re_{\theta t}$ are:

$\frac{\partial \gamma}{\partial n} \vert _{wall} = 0$

$\gamma_{farfield} = 1$

$\frac{\partial \hat Re_{\theta t}}{\partial n} \vert _{wall} = 0$

$\hat Re_{\theta t, farfield} = \left\{ \begin{array}{ll} \left( 1173.51 - 589.428 Tu _{\infty} + 0.2196 Tu_{\infty}^{-2} \right), & Tu_{\infty} \leq 1.3; \\ 331.50 \left(Tu_{\infty} - 0.5658 \right) ^{-0.671} & Tu_{\infty} > 1.3 \end{array} \right.$

The effects of laminar-turbulent transition are introduced to the underlying SST model by modifying the turbulent-kinetic-energy source terms as:

$\hat P_k = \gamma_{eff} P_{k,SST}$

$\hat D_k = min \left( max \left( \gamma_{eff},0.1 \right), 1.0 \right) D_{k,SST}$

$\gamma_{eff} = max \left( \gamma, \gamma_{sep} \right)$

$\gamma_{sep} = min \left( s_1 max \left[0, \left( \frac{Re_V}{3.235 Re_{\theta c}} \right) - 1 \right] F_{reattach},2 \right) F_{\theta t}$

$F_{reattach} = exp \left[ - \left( \frac{R_T}{20} \right) ^4 \right]$

where the subscript 'SST' refers to the functional definitions of the base SST model being used. The form of the specific dissipation equation is unaltered.

A modification to the SST $F_1$ blending function is required with the Langtry-Menter model:

$R_y = \frac{\rho d \sqrt{k}}{\mu}$

$F_3 = exp \left[ - \left( \frac{R_y}{120} \right)^8 \right]$

$F_1 = max \left( F_{1,SST},F_3 \right)$

For numerical robustness, the following three limits are enforced:

$-0.1 \leq \lambda_{\theta} \leq 0.1$

$Tu \geq 0.027$

$Re_{\theta t} ^{eq} \geq 20$ (missing "eq" added 8/16/2021)

Unless stated otherwise above, the functional definitions and calibration constants of the underlying SST-2003 turbulence model should not be altered when used with the Langtry-Menter transition model. This includes the definition of the turbulent eddy viscosity, which should be calculated in an identical manner to that of the SST-2003 model.

If the Langtry-Menter model is applied to a different base SST model, the naming of the model should reflect it. As mentioned above, the SST-2003-LM2009 model is based off of the SST-2003 model. If based off of the "standard" SST model instead, then the transition model should be referred to as SST-LM2009. Similarly, if based off of SST-2003m, it would be SST-2003m-LM2009. Or, if adding Langtry-Menter transition to SST-V, the model should be referred to as SST-V-LM2009, etc.

This model is not Galilean invariant, due to its explicit use of the velocity vector.

Langtry-Menter 4-equation Transitional SST Model with Stationary Crossflow Extension (SST-2003-LM2015)

The SST-2003-LM2015 version represents an extension to the SST-2003-LM2009 model. It incorporates transition due to stationary crossflow (SCF) instability. The reference for this version of the model is:

Langtry, R. B., Sengupta, K., Yeh, D. T., and Dorgan, A. J., "Extending the Gamma-Rethetat Local Correlation Based Transition Model for Crossflow Effects," AIAA Paper 2015-2474, June 2015, https://doi.org/10.2514/6.2015-2474.

As discussed in AIAA-2020-1034, the SST-2003-LM2009 and SST-2003-LM2015 models are supposed to be implemented with SST-2003 as their underlying turbulence model. However, the calibration of SST-2003-LM2015 reported in AIAA-2015-2474 was likely done based on an implementation with the original (SST) model version as its underlying turbulence model. Therefore, the calibration may need to be revisited.

The model is the same as SST-2003-LM2009, except that it adds the term D_SCF to the right-hand-side of the transport equation for $\rho\hat Re_{\theta t}$ . D_SCF is a new sink term that accounts for the SCF effects. It takes the form:

$D_{SCF} = c_{\theta t} \frac{\rho}{T} c_{CF} min \left[(Re_{\theta t})_{SCF} - \hat Re_{\theta t}, 0 \right] (F_{\theta t 2})$

where $c_{CF} = 0.6$ ; and $c_{\theta t}$ is the same as in the SST-2003-LM2009 model. However, the timescale T has been limited here to improve robustness for some high unit Reynolds number flows:

$T = min \left( \frac{500 \mu}{\rho U^2} , \frac{\rho L^2}{(\mu + \mu_t)} \right)$ (added 10/13/2021)

with L being the local grid length. The quantity $(Re_{\theta t})_{SCF}$ represents the transition onset momentum thickness Reynolds number corresponding to SCF induced transition. Details pertaining to the evaluation of this quantity are given below. The last term in the equation for D_SCF, $F_{\theta t 2}$ , has been included to ensure that the sink term is only active within the laminar parts of the boundary layer. It is expressed as:

$F_{\theta t 2} = min \left(F_{wake} e^{-(y/\delta)^4}, 1 \right)$

where F_wake takes on the same value as that used in the formulation of the production term in the original SST-2003-LM2009 model.

Since a dominant source for the excitation of the stationary crossflow instability is believed to be the surface roughness, $(Re_{\theta t})_{SCF}$ is correlated as a logarithmic function of the nondimensional rms amplitude of the surface roughness:

$(Re_{\theta t})_{SCF} = \frac{\rho \theta_t \left(U/0.82 \right)}{\mu} = -35.088 ln \left( \frac{h_{roughness}}{\theta_t} \right) + 319.51 + f(\Delta H_{CF+}) - f(\Delta H_{CF-})$

The h_roughness is a required input to this model. It should be noted that the above equation involves $\theta_t$ on both sides, so it must be solved iteratively by using a procedure such as the Newton-Raphson or the shooting method. Note that this correlation was derived on the basis of transition measurements for the NLF(2)-0415 45 degree swept aifoil, together with additional data derived from the stability computations of the same NLF(2)-0415 airfoil at various sweep angles. The last two terms on the right hand side of the above correlation represent the shifts needed to account for the changes in the crossflow strength relative to that in the NLF(2)-0415 experiment. They are defined as:

$f(\Delta H_{CF+}) = 6200(\Delta H_{CF+}) + 50000(\Delta H_{CF+})^2$

$f(\Delta H_{CF-}) = 75.0 tanh \left( \frac{\Delta H_{CF-}}{0.0125} \right)$

where

$\Delta H_{CF+} = max(0.1066 - \Delta H_{CF}, 0)$

$\Delta H_{CF-} = max[-1.0(0.1066 - \Delta H_{CF}), 0]$

$\Delta H_{CF} = H_{CF} \left[1.0 + min \left(\frac{\mu_t}{\mu}, 0.4 \right) \right]$

$H_{CF} = \frac{\Omega_{streamwise} d}{U}$

$\Omega_{streamwise} = | \vec U \cdot \vec \Omega |$

$\vec U = \left( \frac{u}{\sqrt{u^2+v^2+w^2}}, \frac{v}{\sqrt{u^2+v^2+w^2}}, \frac{w}{\sqrt{u^2+v^2+w^2}} \right)$

$\vec \Omega = \left[ \left(\frac{\partial w}{\partial y}-\frac{\partial v}{\partial z} \right), \left(\frac{\partial u}{\partial z}-\frac{\partial w}{\partial x} \right), \left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y} \right) \right]$

This model is not Galilean invariant, due to its explicit use of the velocity vector.

Wing-like Crossflow-Extension to the Langtry-Menter 4-equation Transitional SST Model (SST-2003-LM2009-CFC1)

The SST-2003-LM2009 model was extended to predict transition due to crossflow instability in three-dimensional boundary layers. This model is identical to the SST-2003-LM2009 model except for the source term in the $\gamma$ equation. To account for the additional transition mechanism it is changed to:

$P_{\gamma} = \left(F_{length} [\gamma F_{onset}]^{0.5} + F_{length,CF} [\gamma F_{onset,CF}]^{0.5}\right) c_{a1} \rho S \left(1 - c_{e1} \gamma\right)$

The primary reference of the Crossflow-Extension is:

Grabe, C., Nie, S. and Krumbein, A., "Transport Modeling for the Prediction of Crossflow Transition," AIAA Journal, Vol. 56, No. 8, 2018, pp. 3167-3178, https://doi.org/10.2514/1.J056200.

This model is one of two different approaches described in the above reference. While this approach (SST-2003-LM2009-CFC1) strongly relies on the infinite yawed wedge flow at zero angle of attack and the C1-criterion, the other approach (SST-2003-LM2009-CFHE, described below) is based on the local helicity in the flow. Both approaches are not Galilean invariant due to the explicit use of the velocity vector.

For both approaches a function $F_{length,CF}$ is introduced:

$F_{length,CF} = 5$

The application of the SST-2003-LM2009-CFC1 approach is restricted to wing-like geometries. Applying it to arbitrary geometries is beyond the model's scope.

The $F_{onset,CF}$ functions are defined by:

$F_{onset,CF} = \max\left(F_{onset2,CF} - F_{onset3,CF}, 0.0\right)$

$F_{onset1,CF} = \frac{Re_{\delta 2}^{\star}}{C\ Re_{\delta 2t}^{\star}}$

$F_{onset2,CF} = \min\left(\max\left(F_{onset1,CF}, F_{onset1,CF}^4\right), 2.0\right)$

$F_{onset3,CF} = \max\left(1 - \left(\frac{R_T}{1.5}\right)^3, 0.0\right)$

where $C = 0.75$ . The local crosswise displacement thickness Reynolds number $Re_{\delta 2}^{\star}$ in the numerator of $F_{onset1,CF}$ is defined by:

$Re_{\delta 2}^{\star} = \frac{Re_{\nu,max} \sin \vartheta^{\star}} {f_{corr} (H_{12}^{\star}, \vartheta^{\star})}$

$f_{corr} (H_{12}^{\star}, \vartheta^{\star}) = 4.3 \times 10^{-7} \exp(f_1 + f_2 + f_3 + f_4)$

$f_1 = 6.6588 \, H_{12}^{\star}$

$f_2 = \frac{1}{2} \sin^2 \vartheta^{\star}$

$f_3 = 0.12 \cos\left(2 \vartheta^{\star}\right)$

$f_4 = -1.8 \sin\left(2 \vartheta^{\star}\right) \left(H_{12}^{\star} - 2.3\right)$

$\cos \vartheta^{\star} = \frac{\vec{u}}{\left|\vec{u}\right|} \cdot \left.\frac{dp}{d\vec{x}}\right|_{pr} / \left|\frac{dp}{d\vec{x}}\right|_{pr}$

$\left.\frac{dp}{dx}\right|_{pr} = \frac{dp}{dx} - \left(\frac{dp}{dx} n_{x} + \frac{dp}{dy} n_{y} + \frac{dp}{dz} n_{z}\right) \frac{n_{x}}{\left|\vec{n}\right|^2}$

In the denominator of $F_{onset1,CF}$ , the local crosswise displacement thickness Reynolds number at transition onset $Re_{\delta 2t}^{\star}$ is determined by the C1 criterion:

$Re^{\star}_{\delta 2t} = \frac{300}{\pi} \arctan \left[ \frac{0.106}{\left(H_{12}^{\star}-2.3\right)^{2.052}} \right],$ $2.3 < H^{\star}_{12} \le 2.7$

$Re^{\star}_{\delta 2t} = 150.0,$ $H^{\star}_{12} \leq 2.3$

The approximated local shape factor is defined by:

$H_{12}^{\star} = \max\left(-7.349 \lambda^{\star} + 2.5916, H_{12,min}^{\star}\right),$ $\lambda^{\star} > 0$

$H_{12}^{\star} = \min\left(-6.3215 \lambda^{\star} + 2.5916, 4.0\right),$ $\lambda^{\star} \leq 0$

$H_{12,min}^{\star} = \max\left(0.0085 \vartheta^{\star} + 1.818, 2.35\right)$

$\lambda^{\star} = \frac{\rho \theta^{\star 2}}{\mu} \frac{d\left|\vec{u}_{e}\right|}{d\vec{s}}$

$\theta^{\star} = \frac{\left.d^2 \Omega\right|_{max}}{2.193 \left|\vec{u}_{e}\right|}$

$\left|\vec{u}_{e}\right| = \sqrt{u_\infty^2 + \frac{2 \tilde\gamma}{\tilde\gamma - 1} \left[1 - \left(\frac{p}{p_{\infty}}\right)^{1-\frac{1}{\tilde\gamma}}\right] \frac{p_\infty}{\rho_{\infty}}}$ <- typo fixed 11/11/2018

$\frac{d\left|\vec{u}_{e}\right|}{d\vec{s}} = \frac{u}{\left|\vec{u}\right|} \frac{d\left|\vec{u}_{e}\right|}{dx} + \frac{v}{\left|\vec{u}\right|} \frac{d\left|\vec{u}_{e}\right|}{dy} + \frac{w}{\left|\vec{u}\right|} \frac{d\left|\vec{u}_{e}\right|}{dz}$

$\frac{d \left| \vec{u}_e \right|}{d x_i} = -\frac{1}{\left| \vec{u}_e \right| \rho_{\infty}} \left( \frac{p}{p_{\infty}} \right)^{-1/ \tilde \gamma} \frac{dp}{dx_i}$

where $\tilde \gamma$ is the heat capacity ratio.

General Crossflow-Extension to the Langtry-Menter 4-equation Transitional SST Model (SST-2003-LM2009-CFHE)

$P_{\gamma} = \left(F_{length} [\gamma F_{onset}]^{0.5} + F_{length,CF} [\gamma F_{onset,CF}]^{0.5}\right) c_{a1} \rho S \left(1 - c_{e1} \gamma\right)$

The primary reference of the Crossflow-Extension is:

Grabe, C., Nie, S. and Krumbein, A., "Transition Transport Modeling for the Prediction of Crossflow Transition," 34th AIAA Applied Aerodynamics Conference, AIAA Aviation Forum (AIAA 2016-3572), June 2016, https://doi.org/10.2514/6.2016-3572.

This model is one of two different approaches described in the above reference. This approach (SST-2003-LM2009-CFHE) is based on the local helicity in the flow. The other approach (SST-2003-LM2009-CFC1, described above) strongly relies on the infinite yawed wedge flow at zero angle of attack and the C1-criterion. Both approaches are not Galilean invariant due to the explicit use of the velocity vector.

For both approaches a function $F_{length,CF}$ is introduced:

$F_{length,CF} = 5$

There are no restrictions concerning the application of the SST-2003-LM2009-CFHE approach to arbitrary geometries.

$F_{onset,CF} = \max\left(F_{onset2,CF} - F_{onset3,CF}, 0.0\right)$

$F_{onset1,CF} = \frac{Re_{He}}{C\ Re_{He,t}^{+}}$

$F_{onset2,CF} = \min\left(\max\left(F_{onset1,CF}, F_{onset1,CF}^4\right), 2.0\right)$

$F_{onset3,CF} = \max\left(1 - \left(\frac{R_T}{2}\right)^3, 0.0\right)$

where $C = 0.7$ . Note that this constant also appears in the SST-2003-LM2009-CFC1 model, but with a different numerical value. The local Helicity Reynolds number $Re_{He}$ is defined by:

$Re_{He} = \frac{\rho d^2}{\mu} \frac{He}{\left|\vec{u}\right|}$

$He = \vec{u} \cdot \left( \nabla \times \vec{u}\right)$

The Helicitiy Reynolds number at transition onset $Re_{He,t}^{+}$ is given by an empirical criterion:

$Re_{He,t}^{+} = \max\left(-456.83\, H_{12}^{+} + 1332.7, 150.0\right)$

${H}_{12}^{+} = 4.02923 - \sqrt{-8838.4 \lambda^{+4} + 1105.1 \lambda^{+3} - 67.962 \lambda^{+2} + 17.574 \lambda^{+} + 2.0593}$

$\lambda^{+} = \frac{\rho l^{2}}{\mu} \frac{d\left|\vec{u}_{e}\right|}{d\vec{s}}$

$l = \frac{1}{C_{He,max}} \frac{2}{15} d$

$C_{He,max} = 0.6944$

See above SST-2003-LM2009-CFC1 section (or the original reference) for details on how to compute the $\frac{d \left| \vec{u}_e \right|}{d \vec{s}}$ term.

Return to: Turbulence Modeling Resource Home Page

Jim Coder of Penn State is acknowledged for putting together the SST-2003-LM2009 portion of this webpage, and Cornelia Grabe of DLR is acknowledged for putting together the SST-2003-LM2009-CFC1 and SST-2003-LM2009-CFHE portions of this webpage. Balaji Venkatachari of NIA is acknowledged for his help with SST-2003-LM2015.

Recent significant updates:
10/13/2021 - inserted missing equation for limited timescale in SST-2003-LM2015
08/16/2021 - added clarification about equation for Re_theta_c; fixed typo in limit term
04/06/2021 - added description of SST-2003-LM2015
02/25/2021 - added a few clarifying equations regarding the computation of d|u_e|/ds
11/11/2018 - fixed typo in abs(u_e) equation for SST-2003-LM2009-CFC1 (removed "d")
09/14/2018 - added notes describing apparent differences between SST-2003-LM2009 and the original AIAA Journal
11/09/2017 - added SST-2003-LM2009-CFC1 and SST-2003-LM2009-CFHE models
05/24/2017 - fixed typo in E_gamma equation (it was missing a gamma)

Privacy Act Statement

Accessibility Statement

Responsible NASA Official: Ethan Vogel
Page Curator: Clark Pederson
Last Updated: 03/10/2022