Intermittency Transport Equation

The transport equation for the intermittency $γ$ is formulated as:

Figure 1. EQUATION_DISPLAY

\frac{d}{d t} (ρ γ) + \nabla\cdot ρ γ \bar{v} = \nabla\cdot [(μ + \frac{μ_{t}}{σ_{f}}) \nabla γ] + P_{γ} - E_{γ}

(1500)

where:

Production Term

The transition production term is defined as:

Figure 2. EQUATION_DISPLAY

P_{γ} = F_{l e n g t h} C_{a 1} ρ S {[γ F_{o n s e t}]}^{1 / 2} (1 - C_{e 1} γ)

(1501)

where $F_{o n s e t}$ is a trigger function that describes the initiation of intermittency for the different modes of transition:

Figure 3. EQUATION_DISPLAY

F_{o n s e t} = max (F_{o n s e t 2} - F_{o n s e t 3}, 0)

(1502)

with:

Figure 4. EQUATION_DISPLAY

F_{o n s e t 2} = min [max (({F_{o n s e t 1}^{4}, F}_{o n s e t 1}), 2)]

(1503)

Figure 5. EQUATION_DISPLAY

F_{o n s e t 3} = max [1 - {({R e}_{t} / 2.5)}^{3}, 0]

(1504)

Figure 6. EQUATION_DISPLAY

F_{o n s e t 1} = \frac{{Re}_{ν}}{C_{onset 1} {Re}_{θ c}}

(1505)

and:

$F_{l e n g t h}$ is given by Eqn. (1549)
$C_{a 1}$ and $C_{e 1}$ are Model Coefficients.
$S$ is the modulus of the mean strain rate tensor, see Eqn. (1129).
${R e}_{t}$ is the turbulent Reynolds number given by Eqn. (1136).
${Re}_{ν}$ is the strain-rate Reynolds number given by Eqn. (1139).
$C_{onset 1}$ is a Model Coefficient.
${Re}_{θ c}$ is the critical momentum thickness Reynolds number given by Eqn. (1548).

The destruction/relaminarization term is defined as:

Figure 7. EQUATION_DISPLAY

E_{γ} = C_{a 2} ρ W γ F_{t u r b} (C_{e 2} γ - 1)

(1506)

where:

F_{t u r b} = exp [- {(\frac{{R e}_{t}}{4})}^{4}]

(1507)

and:

The modification for separation-induced transition is:

Figure 8. EQUATION_DISPLAY

γ_{s e p} = min [s_{1} max (0, \frac{{Re}_{ν}}{3.235 {Re}_{θ c}} - 1) F_{r e a t t a c h}, 2] F_{θ t}

(1508)

where:

F_{r e a t t a c h} = exp [{- (\frac{{Re}_{t}}{20})}^{4}]

(1509)

where ${Re}_{t}$ is given by Eqn. (1136).

The effective intermittency that enters the turbulence transport equations is defined as:

Figure 9. EQUATION_DISPLAY

γ_{e f f} = max (γ, γ_{s e p})

(1510)

For details on how $γ_{e f f}$ enters the turbulence transport equations, see Coupling with SST (Menter) K-Omega Model.


$σ_{f}$	$C_{e 1}$	$C_{a 1}$	$C_{e 2}$	$C_{a 2}$	$C_{onset 1}$	$s_{1}$
1	1	2	50	0.06	2.193	2