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12.9.4 Inlet Boundary Conditions for the LES Model

This section describes the three algorithms available in FLUENT to model the fluctuating velocity at velocity inlet boundaries.



No Perturbations


The stochastic components of the flow at the velocity-specified inlet boundaries are neglected if the No Perturbations option is used. In such cases, individual instantaneous velocity components are simply set equal to their mean velocity counterparts. This option is suitable only when the level of turbulence at the inflow boundaries is negligible or does not play a major role in the accuracy of the overall solution.



Vortex Method


To generate a time-dependent inlet condition, a random 2D vortex method is considered. With this approach, a perturbation is added on a specified mean velocity profile via a fluctuating vorticity field (i.e. two-dimensional in the plane normal to the streamwise direction). The vortex method is based on the Lagrangian form of the 2D evolution equation of the vorticity and the Biot-Savart law. A particle discretization is used to solve this equation. These particles, or "vortex points'' are convected randomly and carry information about the vorticity field. If $N$ is the number of vortex points and $A$ is the area of the inlet section, the amount of vorticity carried by a given particle $i$ is represented by the circulation $\Gamma_i$ and an assumed spatial distribution $\eta$:

$\displaystyle \Gamma_i(x,y)$ $\textstyle =$ $\displaystyle 4 \sqrt{\frac{\pi A k(x,y)}{3N [2 \ln(3) - 3\ln(2)]}}$ (12.9-24)
       
$\displaystyle \eta({\vec x})$ $\textstyle =$ $\displaystyle \frac{1}{2\pi \sigma^2} \left(2e^{-\vert x\vert^2/2\sigma^2} -1 \right) 2e^{-\vert x\vert^2/2\sigma^2}$ (12.9-25)

where $k$ is the turbulence kinetic energy. The parameter $\sigma$ provides control over the size of a vortex particle. The resulting discretization for the velocity field is given by
 {\vec u}({\vec x}) = \frac{1}{2\pi} \sum_{i=1}^N \Gamma_i \... ...'\vert^2/2\sigma^2} ) }{\vert{\vec x} - {\vec x}'_i\vert^2} (12.9-26)

Where $\vec z$ is the unit vector in the streamwise direction. Originally [ 327], the size of the vortex was fixed by an ad hoc value of $\sigma$. To make the vortex method generally applicable, a local vortex size is specified through a turbulent mixing length hypothesis. $\sigma$ is calculated from a known profile of mean turbulence kinetic energy and mean dissipation rate at the inlet according to the following:
 \sigma = \frac{c k^{3/2}}{2\epsilon} (12.9-27)

where $c=0.16$. To ensure that the vortex will always belong to resolved scales, the minimum value of $\sigma$ in Equation  12.9-27 is bounded by the local grid size. The sign of the circulation of each vortex is changed randomly each characteristic time scale $\tau$. In the general implementation of the vortex method, this time scale represents the time necessary for a 2D vortex convected by the bulk velocity in the boundary normal direction to travel along $n$ times its mean characteristic 2D size ( $\sigma_m$), where $n$ is fixed equal to 100 from numerical testing. The vortex method considers only velocity fluctuations in the plane normal to the streamwise direction.

In FLUENT however, a simplified linear kinematic model (LKM) for the streamwise velocity fluctuations is used [ 231]. It is derived from a linear model that mimics the influence of the two-dimensional vortex in the streamwise mean velocity field. If the mean streamwise velocity $U$ is considered as a passive scalar, the fluctuation $u'$ resulting from the transport of $U$ by the planar fluctuating velocity field $v'$ is modeled by
 u' = -{\vec v'} \cdot \vec{g} (12.9-28)

where $\vec{g}$ is the unit vector aligned with the mean velocity gradient $\vec{\nabla U}$. When this mean velocity gradient is equal to zero, a random perturbation can be considered instead.

figure   

Since the vortex method theory is based on the modification of the velocity field normal to the streamwise direction, it is imperative that the user creates an inlet plane normal (or as close as possible) to the streamwise velocity direction.



Spectral Synthesizer


The spectral synthesizer provides an alternative method of generating fluctuating velocity components. It is based on the random flow generation technique originally proposed by Kraichnan [ 186] and modified by Smirnov et al. [ 338]. In this method, fluctuating velocity components are computed by synthesizing a divergence-free velocity-vector field from the summation of Fourier harmonics. In the implementation in FLUENT, the number of Fourier harmonics is fixed to 100.


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