Next, we consider a velocity field which satisfies Laplace's equation but has a constant vorticity. It is given by the equations:
and
This velocity field satisfies Euler's equation if the pressure gradient is given by the equations:
and
where
is the gravity acceleration constant. From the two equations above, the pressure
is
and 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 equal to the value of the area enclosed by the contour. 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 in this case is
Since the vorticity for this velocity field is constant, for a closed contour which encircles an area of value
,
the circulation
in this case is just the vorticity times the area, then
The reader can use the vector field manipulation application to check this by doing 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 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 which encloses an area with value easily evaluated. A good choice is 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 equal to the value of the area encircled by the contour.
- 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 the same as before.
Karl P Burr
2003-07-07