Blade-Vortex Interaction in Potential Flow
 

When studying blade-vortex interaction in potential flow, a flat plate can be used to simulate a thin airfoil for small angles of attack.  A flat plate can be mathematically constructed using a vortex sheet, which satisfies the no-penetration condition.  It is of interest to see how the strength and the configuration of free vortices affect the lift on the plate, and to observe the wake that is created for different configurations.

For this problem, a Fortran program used in an advanced fluid mechanics class, and provided by M. Zeiger, was used to generate the data.  The program calculates the wake for two scenarios: the impulsively started solution, and one where free vorticity already exists.  When free vortices exist, these vortices are convected across the plate at the free stream velocity.  For this to model a real flow, the velocities of the upper and lower surfaces must be equal at the trailing edge.  This phenomenon is known as the Kutta Condition, which is satisfied in this problem by shedding vortices from the trailing edge.

The Fortran program generates data in the form of a text file, which will have to be transferred somewhere for viewing.  Want to show the positions of the vortices at each time step relative to the plate, the sign of the circulation of the vortices, the coefficients of pressure along the plate, and the lift at each time step.  In order to extract information from data, it is useful to see all of the information at the same time.  The best way to show the positions of the vortices is to represent them as dots on the screen.  The dots are redrawn relative to a fixed origin for each time step.

 Depending on the difference in vorticity, the shed vortex can have a positive or negative direction.  For this example, vortices with a positive sign were colored green, and those with a negative sign were colored red.  It is interesting to see how the sign of the shed vortices changes for increasing or decreasing lift.  The coloring of the vortices makes it easy see when the time step is too large, because the vorticity at the trailing edge will oscillate with a large amplitde.  This is a good tool for illustrating the limitations of code, as it shows where the code breaks down.

In this problem the vortices are shed in the x-direction, therefore, in order to view all of the data at once, the graph becomes very long in the x-direction, and very short in the y-direction.  Obviously, everything on the screen would have to be scaled so that as the height of the window gets smaller, the coordinate system also gets smaller.  This works very well, because when the window is tall end of animation is cut off, however, it is easy to see what is happening over the plate.  When the window is long, it is easy to see the overall wake of the vortices.