The governing equations are the continuity and Navier-Stokes equations. These equations are written in a steady rotating frame of reference. This has the advantage of making our simulation not require a moving mesh.
This form of the Navier Stokes equations has additional terms, namely the centripetal acceleration term and the Coriolis term. The equations that we will use look as follows:
Conservation of mass:
{latex}
\begin{equation*}
\frac{\partial \rho}{\partial t}+\nabla \cdot \rho \vec{v}^{\,}_r =0
\end{equation*}
{latex} |
Conservation of Momentum (Navier-Stokes):
{latex}
\begin{equation*}
\nabla \cdot (\rho \vec{v}^{\,}_r \vec{v}^{\,}_r)+\rho(2 \vec{\omega}^{\,} \times \vec{v}^{\,}_r+\vec{\omega}^{\,} \times \vec{\omega}^{\,} \times \vec{r}^{\,})=-\nabla p +\nabla \cdot \overline{\overline{\tau}}_r
\end{equation*}
{latex} |
Where
{latex}$vec{v}^{\,}_r${latex} |
is the relative velocity (the velocity viewed from the moving frame) and
{latex}$vec{\omega}^{\,}${latex} |
is the angular velocity.
In Fluent, we'll turn on the additional terms for a moving frame of reference and input
{latex}$vec{\omega}^{\,}=\omega \mathbf{\hat{k}}${latex} |
For more information about flows in a moving frame of reference, visit ANSYS Help View > Fluent > Theory Guide > 2. Flow in a Moving Frame of Reference and ANSYS Help Viewer > Fluent > User's Guide > 9. Modeling Flows with Moving Reference Frames.
Important: We use the Reynolds Averaged form of continuity and momentum and use the SST k-omega turbulence model to close the equation set.
We model only 1/3 of the full domain using periodicity assumptions:
{latex}
\begin{equation*}
\vec{v}^{\,}(r_1,\theta_1) = \vec{v}^{\,}(r_1,\theta_1 - 120)
\end{equation*}
{latex} |
Inlet: Velocity of 12 m/s with turbulent viscosity of 5% and turbulent viscosity ratio of 10
Outlet: Pressure of 1 atm
Blade: No-slip
Side Boundaries: Periodic
Under Construction |
FLUENT converts these differential equations into a set of algebraic equations. Inverting these algebraic equations gives the value of (u, v, omega, p) at the cell centers. Everything else is derived from the cell centers values (post-processing). In our mesh, we'll have around 400,000 cells. The total number of unknowns and hence algebraic equations is:
400,000 * 4 = 1.6 million.
This huge set of algebraic equations is inverted through an iterative process. The matrix to be inverted is huge but sparse.
Solver: Pressure-based
Under Construction |
One simple hand-calculation that we can do now before even starting our simulation is to find theoretical wind velocity at the tip. We can then later compare our answer with what we get from our simulation to verify that they agree.
The velocity, v, on the blade should follow the formula
{latex}
\begin{equation*}
v=r \times \omega_{}
\end{equation*}
{latex} |
Plugging in our angular velocity of -2.22 rad/s and using the blade length of 43.2 meters plus 1 meter to account for the distance from the root to the hub, we get
{latex}
\begin{equation*}
v=-2.22\ \mathrm{rad/s}\ \mathbf{\hat{k}} \times -44.2\ \mathrm{m}\ \mathbf{\hat{i}}
\end{equation*}
\begin{equation*}
v=98.1\ \mathrm{m/s}\ \mathbf{\hat{j}}
\end{equation*}
{latex} |
*Insert video
Under Construction |
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