3.1.5 · D5Compressible Flow & Aerodynamics
Question bank — Area-velocity relation — dA - A = (M² − 1)(dV - V) — derivation (explains de Laval nozzle)
True or false — justify
Squeezing any duct always speeds the gas up.
False. Only when (subsonic): then , so forces . When the sign flips and squeezing slows the gas down.
If area increases, the flow always slows down.
False. That is the subsonic habit. In supersonic flow , so goes with — a widening duct accelerates the gas.
At the throat of a de Laval nozzle the flow is guaranteed to be sonic ().
False. at the throat only permits ; it also permits a subsonic maximum-speed point. Sonic conditions require a large enough pressure ratio to "choke" the flow — see De Laval Nozzle and Choked Flow.
The area–velocity relation was derived using only mass conservation.
False. It needs three laws: continuity, Euler's momentum equation, and the isentropic sound-speed relation .
For an incompressible liquid the same equation applies with replaced by zero.
Partly true, but read it right. Setting gives , i.e. const with constant — the familiar "narrow pipe = faster" rule. The full relation reduces to it only because incompressible flow is the limit.
To go supersonic you keep converging the duct harder and harder.
False. At the factor , so ; converting further () would force in supersonic flow, forbidding acceleration. You must switch to diverging past the throat.
A gas can accelerate without any drop in pressure.
False. Euler gives , so demands . A gas buys speed only by spending pressure.
The equation says nothing about density, so density is irrelevant here.
False. Density is hidden inside it. The whole sign-flip comes from : supersonically density drops faster than speed rises, and that is why the area must grow.
Spot the error
"Since is constant, and we squeeze down, must always rise to compensate."
The error: it assumes stays fixed. In compressible flow also changes. Supersonically falls so steeply that can fall while stays constant.
"At the throat , and the equation gives , so must be zero."
The error: when the bracket is also zero, making it — indeterminate. is unconstrained, which is exactly how the flow slides smoothly from subsonic to supersonic.
"A converging–diverging nozzle diverges because the gas needs room to expand thermally."
The error: mixes up cause. The divergence is demanded by continuity + the sign of in supersonic flow, not by thermal expansion. The geometry follows the equation, not a heating argument.
"For the term is negative, so the flow can never accelerate."
The error: a negative factor doesn't forbid acceleration; it just pairs with . Subsonic flow accelerates fine — you converge the duct.
" means the speed of sound depends on how fast pressure changes with time."
The error: it is a derivative with respect to density, not time, at constant entropy. It measures how stiffly pressure responds to being compressed — see Speed of Sound in a Gas.
"We can drop the isentropic assumption; the relation still holds for any flow."
The error: Step 3 uses , valid only for isentropic changes. With friction or heat addition, relates to differently and this exact form breaks.
Why questions
Why do we log-differentiate const instead of differentiating directly?
A product's derivative is messy, but hands us the fractional changes directly — the language the whole topic speaks.
Why does act like a "switch" in this equation?
Because the sign of flips at : negative below, positive above. That single sign change reverses whether "accelerate" means converge or diverge.
Why must sonic flow occur only at a throat?
makes , forcing . A point where is an area extremum — in an accelerating nozzle that is the minimum, the throat.
Why is supersonic acceleration "expensive" in area?
With , , so a small needs a larger . At , gaining speed needs more area — hence rocket bells flare so wide.
Why does the density term carry a minus sign in ?
Because faster flow means lower pressure (Euler), and lower pressure means lower density (sound-speed relation). Speed up, density down — opposite signs.
Why is there no gravity term in the Euler equation used here?
The nozzle is treated as horizontal and the gas is light, so the pressure force per unit mass utterly dominates any weight term; it is dropped as negligible.
Why can't a purely converging nozzle ever produce supersonic exhaust?
The most it achieves is at its exit throat. To exceed you need , which a converging-only duct never provides — you must add a diverging section.
Edge cases
What does the relation say at (stagnant limit)?
: pure incompressible behaviour. Density plays no role and the flow follows simple pipe continuity.
What happens as (hypersonic)?
, so — enormous area increases yield only tiny speed gains. Nozzles become impractically large; other physics (real-gas, dissociation) also intervenes.
Exactly at , what is the value of ?
Zero. regardless of . The area is momentarily stationary — a genuine extremum.
If the flow decelerates () in a subsonic duct, what shape is it?
gives ; with we get — a diverging duct (a subsonic diffuser). Everyday nozzles run backwards here.
Can occur somewhere that is not a throat and not sonic?
Yes — if locally (flow momentarily neither speeding up nor slowing) the equation is satisfied for any . signals "no acceleration here," which sonic flow is only one way to cause.
What if a diverging duct carries subsonic flow?
and force : the gas decelerates and pressure rises. This is a subsonic diffuser, recovering stagnation pressure — the opposite of a nozzle.
Recall One-sentence summary
The single sign of is the entire story: it decides whether "speed up" means squeeze or spread, forces sonic flow to live only at a throat, and dictates the hourglass shape of the de Laval nozzle.