Looking for gas expansion at nozzle exit

[Rvanpelt]

I am trying to characterize the expansion of air as it leaves a nozzle into the ambient environment. Let's assume a round cross section. I know that the air flow will initially have a cross section resembling the nozzle outlet with a slighty smaller diameter equivalent to the vena contracta which will be a slight distance beyond the outlet. I know that the cross section of the air flow will expand outward as you travel further from the nozzle outlet.

I am looking for a way to determine what the rate of change of the cross sectional area will be with respect to distance travelled from the vena contracta (or nozzle outlet since that can be approximated).

I know that the air pressure prior to the outlet may have an effect. The ambient pressure will most definitely have an effect. The temperature of the air stream and the ambient will have an effect. The flow rate and consequent velocity may also have an effect.

Can someone provide me with a direction to follow to derive a mathematical analysis and what parameters I will need to know in order to solve it?

Also I would like to note that this does not have to be exact or highly detailed. If there is a way to approximate this relationship without a lengthy computer program to run through pages of partial differential equations that would be acceptable.

 

[Latexman]

What you are describing is commonly called "free jet behavior". This is discussed and there are several references in Perry's Chemical Engineers' Handbook (7th Ed.) on page 6-20. It gives longitudal and radial velocity distribution relationships you can use.

Good luck,
Latexman