The vortex generator model is implemented to simulate the effects caused by vortex generators without the need to resolve them in the computational mesh, which would lead to a very fine mesh. Thus, some computational cells close to the wall will simulate the effect of the vortex generator.

The model used to simulate roughness elements independently from their shape is based on the vortex generator model by Chima.

Usually vortex generators are used to introduce vortices into the flow field and create as little drag as possible. In the present case, the vortex generator model will be used perpendicularly to the flow in order to introduce the resistance of the flow. Then, the effect of the vortex generator should be the same as the effect of the roughness element described previously.

The effect of the vortex generator on the flow will be taken into account via an additional source term into the Navier Stokes equations. The Navier Stokes equations for a compressible flow can be written for a generalized coordinates system as:

\begin{equation} \frac {\partial {\hat u}} {\partial t} + \frac {\partial \hat E}{\partial \xi} + \frac {\partial \hat F}{\partial \eta} + \frac {\partial \hat G}{\partial \zeta} = \hat K + \hat B \end{equation}

Where

\begin{equation} \hat u = \begin{pmatrix} \rho \\ \rho u \\ \rho v \\ \rho w \\ e \end{pmatrix} \end{equation}

E, F and G are the fluxes according to the different coordinate directions. K is the "classical" source term and B is the "Vortex Generator" source term. So, the modifications of the flow due to the vortex generator are computed in every computational cell where the vortex generator should be present.

The additional source term B is a function of the lift and drag forces:

\begin{equation} \hat B = \frac 1 J \begin{pmatrix} 0 \\ {(\vec {F_L} + \vec {F_D})}_x \\ {(\vec {F_L} + \vec {F_D})}_y \\ {(\vec {F_L} + \vec {F_D})}_z \\ r \Omega \vec {F_\theta} \end{pmatrix} \end{equation}

r is the radius and \(\Omega{}\) is the vorticity.

\begin{equation} \vec{F_L} = c_L \frac \rho 2 { \| \vec v \|}^2 S \end{equation}

(Lift Force)

\begin{equation} \vec{F_D} = c_D \frac \rho 2 { \| \vec v \|}^2 S \end{equation}

(Drag Force)

S is the surface, J the Jacobian Matrix

with:

\begin{equation} {\| \vec v \|}^2 = u^2 + v^2 + w^2 \end{equation}

The lift and drag coefficients are computed as:

\begin{equation} c_L = c_{L \alpha} \cdot \alpha \end{equation}

\begin{equation} c_D = c_{D0} + k_2 \cdot c_L^2 \end{equation}

where \(c_\text{l\(\alpha{}\)}\), \(c_\text{D0}\) and \(k_\text{2}\) are constants and

\begin{equation} \alpha \end{equation}

is the angle of attack.

The vortex generator model is controlled via the control file. Specifically, "vortex generators" is an attribute that can be given to a solid boundary panel. Detailed instructions how to apply the vortex generator model can be found in ChgPanelAtt -VORTGEN.