Re: how does volume reduction effect velocity

Dear Ben,

Fluid Dynamics isn't my strongest subject.  I survived it during school 
with a combination of basic understanding and LOTS of help from classmates! 
 So I can sort of tell you what is going on with fluids but certainly can't 
lecture on it.  Therefore I am going to deal with your question in the most 
simplest form and then just mention some of the other issues involved that 
can greatly complicate the problem.

Let's first define the control volume for your experiment as your pipe.  
That way we are just looking at the flow in the pipe itself.  If we 
consider the air to be incompressible, we can define a simple solution for 
this problem.  As you know, gas is compressible, but these effects are 
negligible for flow speeds under about 1/3 of the fluids Mach number (speed 
of sound).  So unless you are going to be doing your experiment at 200 mph 
I think we are OK with this assumption!  So then our density doesn't change 
inside our control volume.  If the inlet and outlet are simple in shape and 
perpendicular to the flow (as yours will be) we have the following 
equation, which is the incompressible version of the "equation of 
continuity":

Sum(ViAi)out = Sum(ViAi)in

This is if you have more than one inlet and outlet.  If you have one of 
each, it becomes:

VinAin = VoutAout

Where:
Vin = Velocity at inlet
Ain = Area of inlet
Vout = Velocity at outlet
Aout = Area of outlet

So for you problem we know that Vin = 30 mph = 44 ft/s,  Ain = 2in^2*pi = 
12.56 in^2 = .087 ft^2 
(remember to keep your units the same!) and Aout = 3.14in^2 = .021ft^2.  
Plugging into our equation and solving for Vout we get:

Vout = 44 ft/s*.087 ft^2 / .021 ft^2 = 182.3 ft/s = 124 mph!

This velocity is the average velocity of the air, not every particle will 
be going this speed, but for a low viscosity fluid like air at this speed 
this is a good approximation.  We are also ignoring effects of friction at 
the walls (pretty minimal in your case) and we assume a gentle transition 
from 4" to 2" (like you stated).  If it were a sharp transition the speeds 
would be less.  I could attempt to describe boundary layer and Reynolds 
number effects, tube smoothness effects, and some others but I think I'd 
give myself a headache and I'm hoping the above equation is all you will 
need.  Hope this helps.
BK

P.S.  One thing to note, if you speed up to 65 mph your exit velocity will 
be close to 300 mph which means we are getting into the compressible range 
for air which may change the output.

Reference:  "Fluid Mechanics: 2nd Ed." By Frank White and "Physics for 
Scientists and Engineers: 2nd Ed" by Douglas Giancoli.