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• Wind Turbine Blade FSI (Part 1) - Pre-Analysis & Start-Up

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Conservation of mass:

Latex

\begin{equation*}
\frac{\partial \rho}{\partial t}+\nabla \cdot \rho \vec{v}^{\,}_r =0
\end{equation*}


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*}


Where

Latex
$\vec{v}^{\,}_r$

is the relative velocity (the velocity viewed from the moving frame) and

Latex
$\vec{\omega}^{\,}$

is the angular velocity.

Note the additional terms for the Coriolis force (

Latex
$2 \vec{\omega}^{\,} \times \vec{v}^{\,}_r$

) and the centripetal acceleration (

Latex
$\vec{\omega}^{\,} \times \vec{\omega}^{\,} \times \vec{r}^{\,}$

) in the Navier-Stokes equations. In Fluent, we'll turn on the additional terms for a moving frame of reference and input

Latex
$\vec{\omega}^{\,}= -2.22 \mathbf{\hat{k}}$

.

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

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We model only 1/3 of the full domain using periodicity assumptions:

Latex

\begin{equation*}
\vec{v}^{\,}(r_1,\theta) = \vec{v}^{\,}(r_1,\theta_1 - 120n)
\end{equation*}


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This therefore proves that the velocity distribution at theta of 0 and 120 degrees are the same. If we denote theta_1 to represent one of the periodic boundaries for the 1/3 domain and theta_2 being the other boundary, then

Latex
$\vec{v}^{\,}(r_i,\theta_1)=\vec{v}^{\,}(r_i,\theta_2)$

.

The boundary conditions on the fluid domain are as follow:

...

Latex

\begin{equation*}
v=r \times \omega_{}
\end{equation*}


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


$$v=-2.22\ \mathrm{rad/s}\ \mathbf{\hat{k}} \times -44.2\ \mathrm{m}\ \mathbf{\hat{i}}$$
$$v=98.1012\ \mathrm{m/s}\ \mathbf{\hat{j}}$$



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Thus, at rated wind speed,

Latex

\begin{eqnarray*}
C_p = \frac{P_{rated}}{P_{wind}}
= \frac{P_{rated}}{0.5\rho A V_{rated}^3}
= \frac{P_{rated}}{0.5(1.225\frac{kg}{m^3})(\frac{\pi(82.5m)^2}{4})(11.5\frac{m}{s})^3}
= 0.30
\end{eqnarray*}



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