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12.4.5 Effects of Buoyancy on Turbulence in the $k$- $\epsilon$ Models

When a non-zero gravity field and temperature gradient are present simultaneously, the $k$- $\epsilon$ models in FLUENT account for the generation of $k$ due to buoyancy ( $G_b$ in Equations  12.4-1, 12.4-4, and 12.4-15), and the corresponding contribution to the production of $\epsilon$ in Equations  12.4-2, 12.4-5, and 12.4-16.

The generation of turbulence due to buoyancy is given by

 G_b = \beta g_i \frac{\mu_t}{{\rm Pr}_t} \frac{\partial T}{\partial x_i} (12.4-23)

where Pr $_t$ is the turbulent Prandtl number for energy and $g_i$ is the component of the gravitational vector in the $i$th direction. For the standard and realizable $k$- $\epsilon$ models, the default value of Pr $_t$ is 0.85. In the case of the RNG $k$- $\epsilon$ model, Pr $_t$ = $1/\alpha$, where $\alpha$ is given by Equation  12.4-9, but with $\alpha_0 = 1/{\rm Pr} = k/\mu c_p$. The coefficient of thermal expansion, $\beta$, is defined as
 \beta = - \frac{1}{\rho} \left(\frac{\partial \rho}{\partial T}\right)_p (12.4-24)

For ideal gases, Equation  12.4-23 reduces to
 G_b = - g_i \frac{\mu_t}{\rho {\rm Pr}_t} \frac{\partial \rho}{\partial x_i} (12.4-25)

It can be seen from the transport equations for $k$ (Equations  12.4-1, 12.4-4, and 12.4-15) that turbulence kinetic energy tends to be augmented ( $G_b > 0$) in unstable stratification. For stable stratification, buoyancy tends to suppress the turbulence ( $G_b < 0$). In FLUENT, the effects of buoyancy on the generation of $k$ are always included when you have both a non-zero gravity field and a non-zero temperature (or density) gradient.

While the buoyancy effects on the generation of $k$ are relatively well understood, the effect on $\epsilon$ is less clear. In FLUENT, by default, the buoyancy effects on $\epsilon$ are neglected simply by setting $G_b$ to zero in the transport equation for $\epsilon$ (Equation  12.4-2, 12.4-5, or 12.4-16).

However, you can include the buoyancy effects on $\epsilon$ in the Viscous Model panel. In this case, the value of $G_b$ given by Equation  12.4-25 is used in the transport equation for $\epsilon$ (Equation  12.4-2, 12.4-5, or 12.4-16).

The degree to which $\epsilon$ is affected by the buoyancy is determined by the constant $C_{3 \epsilon}$. In FLUENT, $C_{3 \epsilon}$ is not specified, but is instead calculated according to the following relation [ 140]:
 C_{3\epsilon} = {\rm tanh} \left\vert \frac{v}{u} \right\vert (12.4-26)

where $v$ is the component of the flow velocity parallel to the gravitational vector and $u$ is the component of the flow velocity perpendicular to the gravitational vector. In this way, $C_{3 \epsilon}$ will become 1 for buoyant shear layers for which the main flow direction is aligned with the direction of gravity. For buoyant shear layers that are perpendicular to the gravitational vector, $C_{3 \epsilon}$ will become zero.


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