Tuesday, September 27, 2011

MrsDrPoe: Potential Flow, Part III

Once again Thesis Tuesday is upon us!  Today we'll be concluding our investigation into Potential Flow by examining combinations of the flows we've looked at so far in both Cartesian and cylindrical coordinates.  This is possible because potential flows obey the superposition principle.


A doublet is the combination of a source and a sink of equal strenghts (+/- m) located 2a apart, as we can see in the figure below:

To find the potential function and streamfunction for the doublet (in relation to point P), we simply add the two potential functions and the two stream functions respectively:

PHI = PHIsource + PHIsink = 0.5*(m/pi)*ln(r1) - 0.5*(m/pi)*ln(r2)

PHI = 0.5*(m/pi)*ln(r1/r2)

PSI = PSIsource + PSIsink = 0.5*(m/pi)*theta1 - 0.5*(m/pi)*theta2

PSI = 0.5*(m/pi)*(theta1 - theta2)

If we allow the distance between the source and sink shrink (a goes to 0, theta1 goes to theta 2, r1 goes to r2), we arrive at:

PHI = ((m*a)/(pi*r))*cos(theta)

PSI = -((m*a)/(pi*r))*sin(theta)

Flow Over a Cylinder

To examine flow over a cylinder, we will combine a doublet and uniform flow.  Before we can combine these flows, however, we must first translate uniform flow into cylindrical coordinates (remember- you can never mix coordinate systems) using:

x = r*cos(theta), y = r*sin(theta)

Next we simply combine the potential function and streamfunction from each flow to find these functions for the combination:

PHI = PHIdoublet + PHIuniform = ((m*a)/(pi*r))*cos(theta) - U*r*cos(theta)

PSI = PSIdoublet + PSIuniform = -((m*a)/(pi*r))*sin(theta) + U*r*sin(theta)

If we let (m*a)/pi = U*R*R where R is the radius of a cylinder we're interested in:

PHI = -U*cos(theta)*(r - (R*R)/r)

PSI = U*sin(theta)*(r - (R*R)/r)

Note the streamline where PSI = 0, r = R is taken as the surface of a cylinder of radius R.  We can do this because streamlines run parallel to the flow, meaning no flow crosses the streamline, just like no fluid can cross a solid (non-permeable) boundary; therefore, any streamline in an invisicd flow field can be considered a solid boundary.  All streamlines for PSI > 0 give the inviscid flowfield over the cylinder.  

Streamlines exist inside the cylinder (from the doublet) as seen below, but we neglect these.  We can also determine the stagnation point for the flow by finding the velocity from the potential function and streamfunction and noting where it equals zero.  By combining other potential flows in a similar manner, we can examine inviscid flow around other objects.

And that's potential flow.  Tune in next week when we'll look into the third major governing equation of fluid mechanics - the energy equation.


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