Transition Momentum Thickness Reynolds Number Transport Equation

The transport equation for the transition momentum thickness Reynolds number $\bar{{R e}_{θ t}}$ is formulated as:

Figure 1. EQUATION_DISPLAY

\begin{matrix} \frac{d}{d t} (ρ \bar{{R e}_{θ t}}) + \nabla\cdot (ρ \bar{{R e}_{θ t}} \bar{v}) = \nabla\cdot [σ_{θ t} (μ + μ_{t}) \nabla \bar{{R e}_{θ t}}] + P_{θ t} + D_{S C F} \end{matrix}

(1511)

where:

$ρ$ is the density.
$\bar{v}$ is the mean velocity vector.
$σ_{θ t}$ is a Model Coefficient.
$μ$ is the dynamic viscosity.
$μ_{t}$ is the turbulent eddy viscosity.
$P_{θ t}$ is the Production Term.
$D_{S C F}$ is the Cross-Flow Term.

Production Term

The production term is defined as:

Figure 2. EQUATION_DISPLAY

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

(1512)

where $C_{θ t}$ is a Model Coefficient and ${Re}_{θ t}$ is the transition onset momentum thickness Reynolds number, which can be approximated by different correlations. See Correlations for ReTheta_t.

The time-scale $t$ is given by:

Figure 3. EQUATION_DISPLAY

t = \frac{500 μ}{{ρ U}^{2}}

(1513)

with $U = | \bar{v} |$ .

Figure 4. EQUATION_DISPLAY

F_{θ t} = min {max [F_{w a k e} \exp ({- (\frac{{ρ U}^{2}}{375 W μ \bar{{Re}_{θ t}}})}^{4}), 1 - {(\frac{C_{e 2} γ - 1}{C_{e 2} - 1})}^{2}], 1}

(1514)

where:

$W$ is the modulus of the mean vorticity tensor, see Eqn. (1131).
$C_{e 2}$ is a Model Coefficient.

$F_{w a k e}$ is defined as:

F_{w a k e} = exp (- {(R e_{ω} / 10^{5})}^{2})

(1515)

where:

Figure 5. EQUATION_DISPLAY

{Re}_{ω} = \frac{ρ ω d^{2}}{μ}

(1516)

and $d$ is the distance to the nearest wall.

Cross-Flow Term

The cross-flow term is defined according to [379] as:

Figure 6. EQUATION_DISPLAY

D_{S C F} = C_{θ t} \frac{ρ}{t'} C_{S C F} \min ({Re}_{S C F} - \bar{{Re}_{θ t}}, 0) F_{θ t 2}

(1517)

where $C_{S C F}$ is a Model Coefficient and:

Figure 7. EQUATION_DISPLAY

t' = \min (t, \frac{ρ L^{2}}{(μ + μ_{t})})

(1518)

Figure 8. EQUATION_DISPLAY

F_{θ t 2} = \min (F_{w a k e} e^{- {(\frac{{ρ U}^{2}}{375 W μ \bar{{Re}_{θ t}}})}^{4}} - \bar{{Re}_{θ t}}, 1)

(1519)

The empirical correlation for stationary cross-flow transition is defined as:

Figure 9. EQUATION_DISPLAY

{Re}_{S C F} = \frac{θ_{t} ρ (\frac{U}{0.82})}{μ} = - 35.088 l n (\frac{h}{θ_{t}}) + 319.51 + f ({Δ H}_{{S C F}^{+}}) - f ({Δ H}_{{S C F}^{-}})

(1520)

where $h$ is the cross-flow inducing roughness height (in meters) and a Model Coefficient. The cross-flow strength shift terms are given by:

f ({Δ H}_{{S C F}^{+}}) = 6200 ({Δ H}_{{S C F}^{+}}) + 50000 {({Δ H}_{{S C F}^{+}})}^{2}

(1521)

f ({Δ H}_{{S C F}^{-}}) = 75 \tanh (\frac{{Δ H}_{{S C F}^{-}}}{0.0125})

(1522)

using:

{Δ H}_{{S C F}^{+}} = \max [0.1066 - {Δ H}_{S C F}, 0]

(1523)

{Δ H}_{{S C F}^{-}} = \max [- (0.1066 - {Δ H}_{S C F}), 0]

(1524)

and:

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

(1525)

The non-dimensional cross-flow strength is defined as:

H_{S C F} = \frac{d W_{S t r e a m w i s e}}{U}

(1526)

where $W_{S t r e a m w i s e}$ is the stream-wise vorticity given by:

W_{S t r e a m w i s e} = | u \cdot w |

(1527)

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

The correlation for stationary cross-flow Eqn. (1520) is an implicit function that is solved iteratively using a Newton-Raphson method to derive the momentum thickness $θ_{t}$ .

Model Coefficients


$C_{θ t}$	$C_{S C F}$	$h$	$σ_{θ t}$
0.03	0.6	$3 \times 10^{- 6}$	2