Example of Velocity Field with Zero Constant Vorticity.

We first consider a simple velocity field which satisfies Laplace's equation and has zero vorticity. The velocity field is


\begin{displaymath}V_{x} = y
\end{displaymath}

and


\begin{displaymath}V_{y} = x.
\end{displaymath}

This velocity field satisfies Euler's equation if the pressure gradient is given by the equations:


\begin{displaymath}\frac{\partial p}{\partial x} = -\rho x
\end{displaymath}

and


\begin{displaymath}\frac{\partial p}{\partial y} = -\rho(y+g)
\end{displaymath}

where $g$ is the gravity acceleration constant. From the two equations above, the pressure is given by the equation


\begin{displaymath}p(x,y) = -\frac{\rho}{2}(x^{2}+y^{2}+gy) +C.
\end{displaymath}

where $C$ is an integration constant defined by the boundary conditions. Then, this velocity field satisfies the hypothesis in Kelvin's theorem, so the circulation on any closed material contour should be constant, and in this special case the value of the circulation is zero. One way to check this affirmation is just to compute the circulation on a closed material contour, and the reader can use the vector field manipulation application. Another way is to take a look on the vorticity of this velocity field. The vorticity for this case is given by the equation


\begin{displaymath}\vec{\omega} = \left(\frac{\partial V_{y}}{\partial x}-\frac{\partial V_{x}}{\partial y}\right) \hat{k} = 0 \hat{k}.
\end{displaymath}

The circulation around any material contour is related to the vorticity flux across a surface limited by the integration contour according to Stoke's theorem. Therefore, for this particular velocity field the circulation on any closed material contour is zero. The reader can use the vector field manipulation application to check this by doing the following exercise:

1.
The velocity field above is the default vector field in the vector field manipulation application. The user should chose the option Draw Contour from the drop down box in the application panel, and then click the button Draw from the application panel.
2.
By mouse clicking in the display area the user can input a contour made of straight lines. For example, the user may input a square contour.
3.
The user can know click the circulation button for the application to evaluate the circulation on the inputed closed contour. It should be zero.
4.
Next, the user should change the value of time in the Input t text box. After that, the user should click the Draw button, and the new shape and position of the contour is displayed. To have a better representation of the convected contour, the user should input a large value for the number of points per side of the contour to advance in time in the text box named # of points per side to advance in time. Since the velocity field satisfies Laplace's equation, the area enclosed by the contour remains the same.
5.
Now, the user should click the circulation button to see the new value of the circulation around the new contour. It should be zero.



Karl P Burr
2003-07-07