Pre-Analysis & Start-Up

Pre-Analysis

In the Pre-Analysis step, we'll review the following:

• Mathematical model: We'll look at the governing equations + boundary conditions and the assumptions contained within the mathematical model.
• Numerical solution procedure in ANSYS: We'll briefly overview the solution strategy used by ANSYS and contrast it to the hand calculation approach.
• Hand-calculations of expected results: We'll use an analytical solution of the mathematical model to predict the expected particle velocity at the blade tip. We'll pay close attention to additional assumptions that have to be made in order to obtain an analytical solution.

Mathematical Model 

Governing Equations

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 looks as follow:

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}

Numerical Solution Procedure in ANSYS

Hand-Calculations of Expected Results

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}

{latex}
\begin{equation*}
\mathbf{m \ddot{x} + k x =0}
\end{equation*}
{latex}

{latex}
\begin{equation*}
m \ddot{x} + k x =0
\end{equation*}
{latex}

{latex}
\begin{equation*}
\boldsymbol{m \ddot{x} + k x =0}
\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}

Start-Up

*Insert video

 It will cover:

-Opening ANSYS

-Setting up the project schematic for the geometry and the Fluid flow, naming them. 

Go to Step 2: Geometry

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