# Supersonic Flow in Diverging Nozzle

Clearly, at low speeds the velocity of a fluid increases when the area through which it is travelling decreases. I am curious as to why a fluid travelling faster than the speed of sound increases its velocity when its area is increased. Thank you

Im not 100% sure why, however the reason it works going from big to small is because of the pressure differential. (P1-P2)=(rho/2(V2^2-V1^2)), as you can see the density times velocity needs to equal the pressure difference. so at a higher speed the pressure differential going from small to large may be larger thus needing a larger V2 to keep the ratio equal to the pressure differential. My best guess!!

Gold Member
At low speeds the fluid is treated as incompressible, so when the nozzle contracts, the only way for the molecules to get out of the way of one another and conserve mass is by speeding up.

In a compressible flow, that same fluid can change in density, meaning it isn't required to follow the same rules as incompressible flows.

I may be wrong here, as I haven't ever really considered this question aside from what the equations say, but when the area increases, the density decreases and the temperature drops. This is a net loss of energy that is balanced by the resulting increase in kinetic energy.

Mech_Engineer
Gold Member
From my fuzzy recollection of intermediate thermodynamics (and looking it up):

$\frac{dA}{A}=-\frac{dV}{V}(1-M^{2})$

So in the design of a nozzle's cross-sectional area using mass and energy balances, the rate of change in area of the nozzle at any point is related to the area, velocity, change in velocity, and Mach number.

Putting it another way, compressed fluids going through a converging nozzle can only pass through the nozzle at up to Mach 1 (speed of sound in the fluid). This limitation is due to back pressure and "choked flow," meaning the maximum mass flow rate through an orifice is limited to Mach 1 through that orifice. To increase velocity after a throat (minimum area) requires a diverging (supersonic) nozzle which allows the fluid's pressure to drop, reducing back pressure and accelerating the flow. This is of course not taking into account things like normal shockwaves and the like...

http://en.wikipedia.org/wiki/De_laval_nozzle

Not to intrude, but I have a question for Mech. Engineer. How sensative are these to different levels of temperature of the working fluid? In my case, an ICE.