Example of Velocity Field with Non-constant Vorticity.

Next, we consider a velocity field which satisfies Laplace's equation and which has variable vorticity. It is given by


\begin{displaymath}V_{x} = y^{2}/2
\end{displaymath}

and


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

If this velocity field satisfies Euler's equation, the pressure gradient should satisfies the equations:


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

and


\begin{displaymath}\frac{\partial p}{\partial y} = -\rho(\frac{y^{2}}{2}+g).
\end{displaymath}

There is no pressure function $p(x,y)$ which has the gradient given by the two equations above. Therefore, this velocity field does not satisfy Euler's equation, and so it does not satisfy Kelvin's theorem, and the circulation on a closed material contour may change with time. This can be confirmed by evaluating the vorticity of this velocity field. The vorticity in this case is given by


\begin{displaymath}\vec{\omega} = -y \hat{k},
\end{displaymath}

and the circulation on a closed contour in this case is not the vorticity times the value of the area defined by the closed contour. In terms of the vorticity the circulation is given by the integral of the vorticity over the area defined by the closed contour. What we want to illustrate in this case is that, as the closed material contour is being convected with the fluid, the circulation on it changes. To see this, the reader can use the vector field manipulation application and do the following exercise:

1.
The velocity field above can be typed in the appropriate text boxes of the vector field manipulation application. The user should also change the default values in the text boxes Input x0, Input xN, Input y0 and Input yN. The user should type:

After that, 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. It should be a contour with simple geometry, like a square for example. It should be inside the area $-0.4 < x < 0.4$ and $-0.4 < y < 0.4$.
3.
The user can know click the circulation button for the application to evaluate the circulation on the inputed closed contour. Keep the circulation value given by the application.
4.
Next, the user should change the value of time in the Input t text box from $0$ to $1$. 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 has a different value and it should have decrease, since most of the area inside the contour is transported by the fluid to the first quadrant and by the fact that the vorticity is negative for positive values of the $y$ variable.
6.
The reader can repeat the process from the third item down, but with a different value of time. Be careful for the convected contour do not fall outside the display area. Try a value of time $t < 2$.



Karl P Burr
2003-07-07