3.1.8 · D5Compressible Flow & Aerodynamics
Question bank — Choked flow — condition M = 1 at throat, maximum mass flow
True or false — justify
Choking always means the flow is supersonic somewhere in the nozzle.
False. In a purely converging nozzle, choking means at the throat and subsonic everywhere upstream — nothing is supersonic inside the nozzle.
Once a converging nozzle is choked, lowering the back pressure still lowers the throat pressure.
False. The throat pressure is locked at (for air). Signals of the lower can't travel upstream past the sonic throat, so the throat "can't hear" the change.
A hotter reservoir (higher ) pushes more mass through a choked throat.
False. : hotter gas is less dense, and density loss beats the rise in sound speed, so drops.
Doubling the reservoir pressure doubles the choked mass flow.
True. linearly (through ), with fixed.
At the choking condition, maximizes the mass flow per unit throat area.
True. Setting (with fixed) yields ; that stationary point is the maximum of the flux function .
The critical pressure ratio is a universal constant for all gases.
False. It's , which depends on . is specifically air (); monatomic gases () give a different value.
When choked and exhausting to a very low , the exit jet is under-expanded.
True. Exit pressure stays at , so the gas finishes expanding outside the nozzle, forming an under-expanded jet with shock diamonds.
A choked flow is not isentropic because it's "stuck."
False. The flow inside the nozzle up to the throat remains isentropic; choking is a condition, not a loss mechanism. Irreversibility (shocks) may appear downstream, not from choking itself.
Spot the error
"Since the exit is open to atmosphere, the throat pressure must equal atmospheric pressure."
Error: true only for subsonic (unchoked) jets. When choked, throat pressure sticks at , which can exceed ambient; the mismatch resolves outside the nozzle.
"To get supersonic flow, just keep dropping the back pressure on a converging nozzle."
Error: a converging nozzle tops out at . Supersonic flow requires a Converging–Diverging (de Laval) Nozzle — diverging area is what accelerates a supersonic stream.
"The mass flow keeps rising because a bigger pressure difference always drives more flow."
Error: that's incompressible/subsonic intuition. Past the throat is deaf to , so saturates regardless of how large becomes.
"At the flow velocity is maximal, so mass flow is maximal."
Error: velocity keeps rising past in a diverging section, but density falls faster. Mass flux (not alone) peaks at .
"In a de Laval nozzle, area increasing downstream of the throat makes the flow slow down."
Error: for supersonic flow, increasing area accelerates it (from the Area–Mach Relation (A/A*)). The subsonic intuition flips sign once .
"Choked mass flow depends on the exit area of the converging nozzle."
Error: it depends on the throat (minimum) area . In a plain converging nozzle the exit is the throat, but the controlling area is always the sonic minimum.
"Because sound speed rises with , hotter reservoirs give higher ."
Error: rises only as , but density falls as . Combined, — hotter means less flow.
Why questions
Why can't downstream pressure signals travel back into a sonic throat?
Pressure disturbances travel at the local sound speed . At the flow moves at downstream, so an upstream-going signal has zero net upstream speed — it's frozen at the throat and never reaches the reservoir.
Why does the maximum mass flow occur exactly at and not some other Mach number?
The mass-flux function has its only stationary point at ; that's where per unit area is largest, balancing rising velocity against falling density.
Why does the throat pressure "lock" once the nozzle chokes?
Because the throat is isolated from (no upstream signal), its state is fixed by the reservoir alone: , independent of anything downstream.
Why is a diverging section needed after the throat for supersonic flow?
Once , the Area–Mach Relation (A/A*) shows that further acceleration () requires increasing area. Only a diverging duct can supply this, so de Laval nozzles are converging–diverging.
Why is proportional to but the ratio independent of ?
The critical ratio comes from the isentropic relation evaluated at , which involves only . Absolute scales with reservoir density , so raising raises flow linearly while leaving the ratio untouched.
Why doesn't the under-expanded jet's extra expansion increase the mass flow?
That expansion happens outside the nozzle, downstream of the sonic throat. Mass conservation fixes at the throat; nothing downstream of can feed back to change it.
Edge cases
What happens to exactly at the critical back-pressure ratio (air)?
This is the onset of choking: the throat just reaches and just reaches its maximum. It's the boundary — subsonic-everywhere on one side, choked-and-frozen on the other.
If is slightly above , is the nozzle choked?
No. The flow is subsonic throughout, the exit pressure matches , and still responds to (increases as you lower) the back pressure.
What is the mass flow in the degenerate limit (no pressure difference)?
Zero. With no pressure drop there's no driving force, everywhere, and — the far-left end of the flux curve, opposite to the choked maximum.
In the limit (perfect vacuum downstream) for a converging nozzle, what is ?
Still exactly — the same as at . The nozzle chokes at and stays frozen; a vacuum only makes the external jet more under-expanded, not the flow larger.
For a monatomic gas () versus air (), does choking still occur at ?
Yes — the result comes from maximizing for any . Only the critical ratios (, ) change with , not the sonic condition.
In a de Laval nozzle running fully supersonic, is the throat still the choke point?
Yes. The throat still sits at and still sets . The diverging section changes the exit Mach number, but the mass flow is fixed by the sonic throat.
What if the "throat" area is not the true minimum (e.g. an even narrower spot exists upstream)?
The actual minimum area is the throat that chokes first; occurs there, and that section — not the geometric label — controls .
Recall Self-test summary
Choked ::: at the throat; at its maximum for given . Throat pressure when choked (air) ::: locked at , independent of . Effect of hotter reservoir ::: less mass flow, since . Supersonic flow requires ::: a converging–diverging (de Laval) nozzle, not a plain converging one.