Cross-Flow Term

The cross-flow term allows you to account for cross-flow effects in three dimensional boundary layer transition.

Activating the cross-flow term modifies the transition trigger function $F_{o n s e t}$ , which activates the production term in the intermittency transport equation given by Eqn. (1550).

The modified transition trigger function is given by:

图 1. EQUATION_DISPLAY

F_{o n s e t} = \max (F_{o n s e t}, F_{o n s e t, S C F})

(1562)

where $F_{o n s e t}$ is given by Eqn. (1552) and $F_{o n s e t, S C F}$ is defined as:

图 2. EQUATION_DISPLAY

F_{o n s e t, S C F} = \min [\max (0, f_{S C F} - 1), 1]

(1563)

Depending on the model variant, the cross-flow function $f_{S C F}$ is defined as:


Model Variant	$f_{S C F}$
Gamma Transition	图 3. EQUATION_DISPLAY $\frac{C_{S C F} C_{r} Δ H_{S C F} {Re}_{θ c}}{{({Re}_{δ 2 t}^{*})}_{t r}}$ (1564)
SA Gamma Transition	$\frac{C_{S C F} C_{r} Δ H_{S C F} {Re}_{ν}}{{({Re}_{δ 2 t}^{*})}_{t r}}$ (1565)

where:

${({Re}_{δ 2 t}^{*})}_{t r}$ is the Critical Cross-flow Displacement Thickness Reynolds Number.
$C_{r}$ is the Surface Roughness Constant.
$R e_{θ c}$ is the critical momentum thickness Reynolds number given by Eqn. (1558).
${Re}_{ν}$ is the strain-rate Reynolds number given by Eqn. (1139).
$C_{S C F}$ is a Model Coefficient.

Critical Cross-flow Displacement Thickness Reynolds Number

According to Arnal's C1-criterion [373], cross-flow triggers transition as long as the following condition is satisfied:

图 4. EQUATION_DISPLAY

\frac{{Re}_{δ 2 t}^{*}}{{({Re}_{δ 2 t}^{*})}_{t r}} \geq 1

(1566)

where:

${Re}_{δ 2 t}^{*}$ is the cross-flow displacement thickness Reynolds number.
${({Re}_{δ 2 t}^{*})}_{t r}$ is the critical cross-flow displacement thickness Reynolds number, beyond which the cross-flow triggers transition in the boundary layer.

The critical cross-flow displacement thickness Reynolds number is defined as a function of a boundary layer shape factor as:

图 5. EQUATION_DISPLAY

{({Re}_{δ 2 t}^{*})}_{t r} = {\begin{matrix} - \frac{300}{π} \arctan [\frac{0.106}{{(H - 2.3)}^{2.05}}] & i f 2.3 \leq H < 2.7 \\ 150 & i f H < 2.3 \end{matrix}

(1567)

where the shape factor $H$ is derived from the pressure gradient parameter $λ_{θ L}$ given by Eqn. (1560) as [387]:

图 6. EQUATION_DISPLAY

H = 2 + 4.14 l - 83.5 l^{2} + 854 l^{3} - 3337 l^{4} + 4576 l^{5}

(1568)

with:

图 7. EQUATION_DISPLAY

l = 0.25 - λ_{θ L}

(1569)

The cross-flow displacement thickness Reynolds number is a non-local quantity that needs integration across the boundary layer. This term is proportional to the three-dimensionality of the boundary layer and approximated using helicity. Helicity as a cross-flow measure was reported by Langtry et al. [379] and Mueller et al. [382].

The streamwise helicity $Ω_{s}$ is defined as:

图 8. EQUATION_DISPLAY

Ω_{s} = | \frac{\bar{v}}{| \bar{v} |} \cdot \bar{w} |

(1570)

where:

$\bar{v}$ is the mean velocity vector.
$\bar{w} = \nabla\times \bar{v}$ is the mean vorticity vector.

Following Langtry et al. [379], a dimensionless cross-flow strength is defined from the streamwise helicity as:

图 9. EQUATION_DISPLAY

H_{S C F} = \frac{Ω_{s} d}{| \bar{v} |}

(1571)

where:

$d$ is the wall distance.
$| \bar{v} |$ is the magnitude of the mean velocity vector.

The cross-flow displacement thickness Reynolds number is finally approximated as:

图 10. EQUATION_DISPLAY

{Re}_{δ 2 t}^{*} \approx C_{S C F} Δ H_{S C F} {Re}_{θ c}

(1572)

where:

图 11. EQUATION_DISPLAY

Δ H_{S C F} = H_{S C F} [1 + \min (\frac{μ_{t}}{μ}, 0.4)]

(1573)

and $\frac{μ_{t}}{μ}$ is the turbulent viscosity ratio.

Surface Roughness Constant

Langtry et al. [379] reported a log dependence between the stationary cross-flow Reynolds number and the surface roughness and supplemented the Gamma Transition model to include this correlation. A similar approach is used to mimic the roughness effect into the cross-flow model by adding a roughness constant $C_{r}$ . Depending on the model variant, $C_{r}$ is defined as:


Model Variant	$C_{r}$
Gamma Transition	图 12. EQUATION_DISPLAY $2 - {0.5}^{\frac{h}{0.25 \times 10^{- 6}}}$ (1574)
SA Gamma Transition	$1.03 - \frac{0.4}{2^{h / (0.25 \cdot 10^{- 6} m)}}$ (1575)

where $h$ is the cross-flow inducing roughness height (in meters) and a Model Coefficient.

Model Coefficients


$C_{S C F}$	$h$
1	$3 \times 10^{- 6}$