MrsDrPoe: Characteristics of Flow Past an Object

Happy Thesis Tuesday to you all!  Today we'll be starting a series on external flow by examining some of this flow's general characteristics.

Flows in which an object is completely submerged in a fluid are termed external flows; however, flows such as those around buildings are also considered external flows, even though buildings aren't completely submerged.  We can consider cases where the object or body is stationary and fluid is flowing around it, where the object is moving through a stationary fluid, or some combination of the two.  For all of these situations, if we fix our coordinate system with the body, we can analyze these scenarios as if fluid were flowing over a stationary body.  We will consider the velocity upstream of the body (Uinfinity) to be constant with respect to time and space.  The bodies in the flow can be classified using one of two systems:

1a) 2D object extending infinitely in a third direction
1b) axi-symmetric bodies formed by rotating a cross-section about an axis of symmetry
1c) 3D bodies

2a) streamlined bodies
2b) blunt bodies

As you can imagine, flow past an object is influenced by both the fluid properties and the size and shape of the object.  These characteristics are typically grouped in dimensionless parameters; those used most often for external flows are Reynolds number, Mach number, and the Froude number.

We can examine some general differences in flows for a range of Reynolds number values by taking a closer look at flows over a flat plate.  Here we consider a plate of length, l, in the same fluid with the same viscosity and density, but we will continue to increase the upstream velocity, Uinfinity, so that Re = 0.1, 10, and 10^7.  As the Reynolds number increases, the region around the plate where the viscous forces/stresses are important shrinks considerably, causing outer streamlines to be deflected from the plate less and less.  The wake region behind the plate also shrinks as the Reynolds number increases.  It should be noted that flows with Re < 1 are dominated by viscous effects while flows with Re > 1 are dominated by inertia.

If we look at flow over a cylinder, we can make some more generalizations about how flow over blunt bodies is affected by changes in Reynolds number.  For low Reynolds number flows (0.1), again we see large deflection of the streamlines far away from the body.  These streamlines appear to be symmetric about the center of the cylinder (the stagnation streamline) as we saw when we investigated the velocity potential function.  As the Reynolds number increases (50, 10^5), we see this area affected by the viscous stresses again shrinking.  We can also see that the flow separates from the surface of the clyinder, creating a recirculation bubble or wake behind the cylinder.

Next week we'll continue looking at external flows, focusing on the region affected by the viscous forces.  Until then, happy studying!

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.

Doublet

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.