Tuesday, June 28, 2011

MrsDrPoe: "Control, Control! You must learn control [volumes]!"

Happy Thesis Tuesday to you all! Last week, we briefly mentioned control volumes; this concept is crucial to correctly applying fluid mechanics equations to a particular problem. Thus, borrowing a line from Yoda, "You must learn control [volumes]!"

Tips for CV Selection
Selection of a control volume is easy; however, selection of the best control volume can be a bit more difficult and often takes practice. Here are some helpful tips:

1) make sure the points in the flow that you have information for (velocity, pressure, temperature, etc.) are located on the control surface

2) making the control surface perpendicular to the flow often makes problems easier to solve

3) if you are looking for some support force of a beam or stand, the control volume should intersect the beam or stand

4) for internal flow, the pipe that fluid is flowing through is often not included in the control volume when the "no-slip" condition is applied to the pipe surface

Moving or Deforming CVs
We know that fluid is allowed to flow across the control surface (i.e. into and out of the control volume), but in many cases (looking at jet engines, deflating balloons, etc.) employing a control volume that moves or deforms relieves some of the difficulty in finding a solution.

Today we will focus primarily on moving CVs. First, it is important to note that the shape, size and orientation do not change for a moving control volume- the volume is simply linearly or angularly translated with some velocity Vcv. One of the major differences between moving and stationary control volumes is that the the velocity with which the fluid crosses the control surface is a relative velocity (W) for a moving CV and an absolute velocity (V) for a stationary one.
(The relative velocity is what we will use later in the continuity equation involving a moving control volume.) To obtain the absolute velocity of the fluid at the control surface, the relative velocity must be added to the velocity of the control volume:

V = Vcv + W

As a quick example to get these differences in velocity cemented in your minds, let's consider a man, John, entering a train car at 0.1 m/s while the train itself is traveling at 160 m/s. If we draw the control volume around the train car that John is moving through, the velocity of the CV is 160 m/s. The 0.1 m/s velocity is John's relative velocity; it is such because that is what a person sitting in the train car would measure his velocity as. A person sitting on a station platform outside the train car would measure his velocity as his absolute velocity, which would be 160.1 m/s.

Control volumes can be a bit daunting, but the more you deal with fluid mechanics problems, the easier it is to determine what type of CV you should use and where you should place it. Never fear! We'll soon get into some interesting applications of these concepts. Until next week...may the force be with you!


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