Good morning, and welcome to another Thesis Tuesday on the blog! Today we'll be continuing our discussion of potential flow, looking at some problems better examined in cylindrical coordinates.
Although cylindrical coordinates are useful, transforming equations from Cartesian to cylindrical coordinates often results in some 'hairy' expressions. For our purposes today, we'll need the definitions for the velocity potential function and the streamfunction in this new coordinate system:
Velocity Potential: vr = d/dr(PHI), vtheta = d/dtheta(PHI/r), vz = d/dz(PHI)
Streamfunction: vr = d/dtheta(PSI/r), vtheta = -d/dr(PSI)
Next, we'll apply these relations to a few potential flow scenarios.
Source and Sink
A source/sink is a flow radially outward/inward from/to a point at some volumetric flowrate, m. We'll define the velocity components for this type of flow as:
vr = m/(2*pi*r) and vtheta = 0
For a source type of flow, m > 0; for a sink, m < 0. We should also note that at r = 0, the velocity becomes infinite, which is physically impossible. "Wait," you may be asking yourself, "If this flow is physically impossible, why do we care?" The answer to this question is that you'll have to wait till next week to find out. (Don't you just love cliff-hangers?) Let's figure out the velocity potential and stream functions that govern a source (realizing that if we simply make m negative, we'll also have the functions that govern a sink):
Velocity Potential:
d/dr(PHI) = m/(2*pi*r)
int(1)dPHI = int(m/(2*pi*r))dr
PHI = 0.5*(m/pi)*ln(r) + c
(We can see that if we take the derivative of this function with respect to theta, we get zero; therefore, this expression also matches the second constraint below.)
d/dtheta(PHI/r) = 0
Streamfunction:
d/dtheta(PSI/r) = m/(2*pi*r)
int(1)dPSI = int(m/(2*pi))dtheta
PSI = 0.5*(m/pi)*theta + c
(We
can see that if we take the derivative of this function with respect to r, we get zero; therefore, this expression also matches the second
constraint below.)
-d/dr(PSI) = 0
Vortex
It seems odd that we could consider a vortex to be irrotational; however, we must remember that rotation refers to the orientation of a fluid element and not the path followed by the element. An irrotational vortex is called a free vortex, while a rotational vortex is called a forced vortex. (Recall that the velocity potential function is only valid for irrotational flows.) A combined vortex is one with a forced vortex at its core and a free vortex outside its core.
The property of circulation is often associated with vortex motion. Circulation, GAMMA, is defined as the line integral of the tangential component of the velocity taken around a closed curve in the flow field. The velocity potential and streamfunctions for free vortices are often written in terms of the circulation of the flow*.
For a free vortex, we will define the following velocity components:
vr = 0 and vtheta = K/r
where K is the magnitude of the velocity.
Velocity Potential:
d/dtheta(PHI/r) = K/r
int(1)dPHI = int(K)dtheta
PHI = K*theta + c
(Again the derivative of this function with respect to r matches the second matching condition.)
d/dr(PHI) = 0
Streamfunction:
d/dr(PSI) = K/r
int(1)dPSI = -int(K/r)dr
PSI = -K*ln(r) + c
(The derivative of this function with respect to theta matches the second matching condition.)
d/dtheta(PSI/r) = 0
Cool, huh? Remember that if we plot constant values of PHI and PSI, we will obtain a flow net that helps us visualize the flow. Next week, we'll do some fun things with the results from today and last week.
*PHI = (0.5*GAMMA*theta)/pi and PSI = -(0.5*GAMMA*ln(r))/pi
