Stump the Scientist: Fluid dymanics
We’ve got a new Stump the Scientist responding to today’s question. Show him some love and send any additional fluid dynamics questions in for him!
Question posted by Edison’s Desk reader Alex Webster on 11/22/10, 1:41 PM:
My question is concerning fluid dynamics, and my be a little specific, but has always bothered me: a DeLaval Nozzle (converging-diverging nozzle) increases the velocity of airflow from subsonic to supersonic with a decreasing, then increasing, cross-section area. This is because subsonic airflow will increase in velocity as the nozzle’s cross-section decreases in area, while supersonic acts the exact opposite.
My question is: why does this work? What happens at the throat of the nozzle that allows the airflow to become supersonic, rather than falling back to subsonic airflow beyond the throat when the cross-sectional area increases?
There is one key equation. The mass flow in a duct is given by:
Mass flow = density x velocity x area
If the density does not change – or changes very little (i.e. either we have a liquid like water, or a gas flowing at very low speeds), then the velocity and area are inversely related. Thus, if the duct is a converging / diverging nozzle, the flow accelerates in the converging section ahead of the throat (the position of minimum area) and decelerates again downstream of the throat, where the area increases.
But the density does change: how does it change? When the fluid is accelerating, there must be a force causing it to accelerate. This force is provided the pressure, and if the fluid is to accelerate, the pressure must fall in the direction of the flow. Similarly, if the fluid decelerates, the pressure must rise in the direction of flow. As the pressure decreases or increases, the density decreases or increases. Factoid: for air the absolute pressure is proportional to the density if the temperature is constant, or proportional to the density raised to the power 1.4 if the pressure change involves no total energy exchange with the environment.
Going back to our one equation, as the velocity increases in the converging part of the nozzle, the pressure decreases, and thus the density decreases, which tends to cause the velocity to increase some more. Is it possible to imagine a situation in which the rate of decrease in density is so great that it could overwhelm any rate of increase in area, such that the fluid continues to accelerate even in the diverging part of the nozzle, downstream of the throat?
This is exactly what happens. What I cannot show you, with this vague hand-waving, is (a) that the density-decrease beats area-increase situation happens when the flow is supersonic – i.e. faster than the local sound speed in the gas, and (b) that the cross-over from subsonic to supersonic flow can only happen at the throat. That requires Newton’s second law, thermodynamics, a general formula for the speed of sound in a gas, and equations. I’d be happy to supply all of that if you wish!