Converging-diverging (de Laval) nozzle — subsonic, supersonic flow: Question bank

Reminder of symbols before we start (so no line uses an unearned term):

$A$ = cross-sectional area of the tube at a point; $d A$ = tiny change in it as you move downstream.
$V$ = flow speed; $d V$ = tiny change in speed.
$ρ$ = density; $a$ = local speed of sound; $T$ = local temperature.
$γ$ = ratio of specific heats $c_{p} / c_{v}$ of the gas (about $1.4$ for air); it appears in $a = γ R T$ and in the pressure/temperature ratios ( $R$ = specific gas constant).
$M = V / a$ = Mach number, the flow speed measured in units of the local sound speed $a$ .
$\overset{m}{˙} = ρ A V$ = mass flowing per second. Constant along the tube by continuity.
Throat = the narrowest point (area minimum, $d A = 0$ ).
Starred quantities ( $A^{*}, ρ^{*}, a^{*}, p^{*}, T^{*}$ ) = the values at the sonic condition $M = 1$ , i.e. the throat when it is choked. So $A^{*}$ is the throat (sonic) area, $p^{*}$ the throat pressure, and so on.
$A_{e}$ = the exit area of the nozzle (its opening at the far end of the diverging section); the ratio $A_{e} / A^{*}$ compares exit to throat.
$p_{0}$ = stagnation (reservoir) pressure = the pressure in the tank before the gas moves, where $V = 0$ . Likewise $T_{0}$ is the stagnation temperature. As the gas accelerates its static pressure drops below $p_{0}$ .
$p_{b}$ = back pressure = the pressure of the environment at the nozzle exit.

Because this bank leans on the sign of $(M^{2} - 1)$ and on the hourglass geometry, three quick sketches anchor everything before the reveals:

Figure — Converging-diverging (de Laval) nozzle — subsonic, supersonic flow

True or false — justify

True or false: Narrowing a tube always speeds a gas up.

False. Narrowing speeds up only subsonic flow (

M < 1

). For

M > 1

the factor

(M^{2} - 1)

is positive, so accelerating needs

d A > 0

— you must widen the tube.

True or false: In a properly running de Laval nozzle, the fastest the gas ever goes is at the throat.

False. At the throat

M = 1

exactly. The gas keeps accelerating through the diverging section, so the highest speed is at the exit.

True or false: Sonic flow (

M = 1

) can occur anywhere in the nozzle if you set the pressure right.

False. The master equation forces

d A = 0

when

M = 1

, so

M = 1

can only sit at a local area minimum — the throat.

True or false: Once the throat is choked, sucking harder (lower

p_{b}

) pulls more mass through.

False. At

M = 1

the throat blocks pressure signals from travelling upstream, so

\overset{m}{˙}

is locked at its maximum

\overset{m}{˙}_{ma x} = ρ^{*} A^{*} a^{*}

(the density, area and sound speed all evaluated at the sonic throat). See Choked Flow & Mass Flow Limit.

True or false: A converging-only nozzle can produce supersonic exhaust if the pressure drop is big enough.

False. Its smallest area is the exit, where at best

M = 1

. Passing

M > 1

needs a diverging section (

d A > 0

), which it doesn't have.

True or false: For a given diverging area ratio

A / A^{*} > 1

there is exactly one possible flow.

False. There are two solutions of the area-ratio relation: one subsonic (decelerating) and one supersonic (accelerating). The back pressure decides which Nature picks.

True or false: The area–velocity relation was derived assuming friction and heat loss are small.

True. It assumes steady, 1-D, isentropic flow — no friction, no heat transfer. That's why it uses the Euler momentum equation, not a viscous one.

True or false: A normal shock inside the diverging section makes the flow supersonic.

False. A normal shock is a sudden jump from supersonic down to subsonic; it decelerates the flow abruptly, it never accelerates it.

Spot the error

Error: "

\frac{d A}{A} = (M^{2} - 1) \frac{d V}{V}

, so at

M = 1

the velocity can't change."

The error is confusing

d V

with

d A

. At

M = 1

the equation gives

d A = 0

(area is stationary), but

d V

can still be nonzero — the gas keeps accelerating right through the throat.

Error: "To go supersonic, keep the tube narrowing so it never lets up."

Wrong sign for

M > 1

. Continued narrowing (

d A < 0

) with

M > 1

would decelerate the gas. Beyond the throat the tube must diverge.

Error: "Mass flow

\overset{m}{˙} = ρ A V

grows without limit as we drop

p_{b}

Once the throat reaches

M = 1

the flow is choked and

\overset{m}{˙}

is fixed. The formula is right; the claim that it keeps growing ignores the sonic gate at the throat.

Error: "Density is essentially constant, so shrinking the area is what speeds the gas up everywhere."

Only true for low

M

. For

M > 1

density drops faster than area grows in a diverging section; that density collapse is exactly what forces

V

up to keep

ρ A V

constant.

Error: "The throat pressure equals the reservoir pressure

p_{0}

No —

p_{0}

is the stagnation (zero-speed) pressure. At a choked throat the pressure has fallen to

p^{*} \approx 0.528 p_{0}

(for

γ = 1.4

), because the gas has picked up speed and its static pressure dropped.

Error: "At the exit of a design-matched nozzle the exit pressure must equal the reservoir pressure."

At the design condition exit pressure equals the back pressure

p_{b}

, which is far below

p_{0}

. Matching to

p_{0}

would mean the gas never accelerated.

Error: "Because

a = γ R T

and

T

falls downstream, the sound speed rises, so

M

automatically drops."

As the gas accelerates and cools,

a

does fall — but

V

rises much faster, so

M = V / a

increases downstream in the diverging section.

Why questions

Why does the sign of

(M^{2} - 1)

flip the whole behaviour?

It multiplies

\frac{d V}{V}

: for

M < 1

it's negative (accelerating needs shrinking area), for

M > 1

it's positive (accelerating needs growing area). The single sign change is the entire subsonic/supersonic reversal.

Why must

M = 1

live precisely at the throat and nowhere else?

M = 1

the factor

(M^{2} - 1) = 0

, forcing

d A = 0

. A point where

d A = 0

and area then increases is an area minimum — that's the definition of the throat.

Why can't a pressure signal from downstream reach the reservoir once the throat is choked?

Pressure signals travel at the local sound speed

a

relative to the gas. At the throat the gas moves at

a

(

M = 1

), so a signal trying to go upstream stands still — it can't get past. See Speed of Sound in a Gas.

Why do rockets need the diverging bell, not just a converging cone?

A converging cone can only reach

M = 1

at its exit. Thrust demands high exit velocity, so you need

M ≫ 1

, which requires the diverging section to keep

d A > 0

for supersonic acceleration. See Rocket Propulsion & Thrust.

Why does lowering the back pressure below choking not change what happens upstream of the throat?

The converging section and throat are already at their choked state; the sonic throat isolates them. Only the flow downstream of the throat (shocks, expansion) responds to further

p_{b}

drops.

Why is

A / A^{*}

never less than 1?

A^{*}

is the sonic (minimum) area. The relation has its minimum value of exactly 1 at

M = 1

; any real area is

\geq

the throat, so the ratio is

\geq 1

for both branches.

Edge cases

Edge case: What flow occurs if

p_{b}

is set exactly equal to

p_{0}

No pressure difference means no driving force — the gas doesn't move at all,

V = 0

\overset{m}{˙} = 0

. The nozzle is idle.

Edge case: What is the highest

M

a converging-only nozzle can produce at its exit?

Exactly

M = 1

(sonic), reached only when it is choked. It physically cannot exceed 1 because there is no diverging area to satisfy

d A > 0

Edge case: If

p_{b}

is just barely below the choking value, where does the supersonic flow end?

The gas goes briefly supersonic just past the throat, then a normal shock stands in the diverging part, snapping it back to subsonic before the exit. Lowering

p_{b}

pushes the shock farther downstream.

Edge case: What happens to

\overset{m}{˙}

p_{b}

drops from the design point to well below it (under-expanded)?

Nothing —

\overset{m}{˙}

stays fixed at

\overset{m}{˙}_{ma x}

because the throat is still choked. Only the plume outside the nozzle changes (expansion waves appear).

Edge case: In the limiting case

M \to 0

(nearly stationary gas), what does the area–velocity relation reduce to?

(M^{2} - 1) \to - 1

, giving

\frac{d A}{A} \approx - \frac{d V}{V}

: to speed up you shrink the area proportionally — the plain incompressible venturi behaviour, as classic nozzle theory predicts.

Edge case: If the diverging section is far too long (huge

A_{e} / A^{*}

) for the given

p_{b}

, what appears?

The flow over-expands: pressure falls below

p_{b}

, so the environment shoves back and a shock (or shock system) forms inside or at the exit to raise the pressure to match

p_{b}

Recall One-line memory hook for the whole bank

Sign of $(M^{2} - 1)$ rules everything ::: minus below Mach 1 (squeeze to speed up), plus above Mach 1 (widen to speed up), zero only at the throat.