Isentropic nozzle flow

Smooth adiabatic nozzle flow without entropy production. In a steady quasi-one-dimensional ideal-gas model,

ρ u A

and

u^{2} /2 + c_{p} T

are constant along the passage. A converging-diverging passage can connect subsonic and supersonic branches through a sonic throat when reservoir and back-pressure conditions select that branch.

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Adiabatic nozzle terminal speed ( $u_{\infty} = c_{*} (γ + 1) / (γ - 1)$ )

 0  0

For a lossless polytropic flow with

γ > 1

, negligible gravity and a sonic throat, the Bernoulli function is

u^{2} /2 + c_{s}^{2} / (γ - 1) = (γ + 1) c_{*}^{2} / [2 (γ - 1)]

. On an expanding supersonic branch for which

ρ \to 0

, the specific enthalpy vanishes and the limiting velocity is the displayed value. If area grows as

C r^{2}

, mass conservation further gives

ρ \sim \dot{M} / (C u_{\infty} r^{2})

. A subsonic pressure-confined branch need not have this asymptotic behavior.

Sonic density of a polytropic nozzle ( $ρ_{} = [\dot{M} / (A_{} γ K)]^{2/ (γ + 1)}$ )

 0  0

For steady polytropic flow with

p = K ρ^{γ}

K > 0

γ > 1

, a sonic section has

u_{*} = c_{*} = (γ K ρ_{*}^{γ - 1})^{1/2}

. Substitution in mass conservation

\dot{M} = A_{*} ρ_{*} u_{*}

gives the displayed density. These conditions fix the sonic state for a prescribed flux and throat area, subject to compatible reservoir and downstream boundary conditions.

de Laval nozzle

 1  0

A converging-diverging nozzle that can accelerate a compressible flow from subsonic to supersonic speed through a sonic throat. The nozzle area-velocity relation requires acceleration in a converging subsonic section and a diverging supersonic section. Reservoir and downstream pressures determine whether this transonic branch operates; the shape alone does not ensure choked flow.

Nozzle area-Mach number relation ( $A / A_{*}$ )

 0  0

For a calorically perfect gas on a steady isentropic nozzle branch,

\frac{A}{A _{*}} = \frac{1}{M} [\frac{2}{γ + 1} (1 + \frac{γ - 1}{2} M^{2})]^{(γ + 1) / (2 (γ - 1))} .

(1)

Here

A_{*}

is the sonic area. An area ratio greater than one generally admits a subsonic and a supersonic root; boundary conditions choose the physical branch. Numerical inversion must retain that branch information.

Nozzle area-velocity relation ( $\frac{d A}{A} = (M^{2} - 1) \frac{d u}{u}$ )

 0  0

Steady mass conservation, inviscid momentum balance and

d p = c_{s}^{2} d ρ

give

d ln A = (M^{2} - 1) d ln u

, where

M = u / c_{s}

. A smooth sonic crossing needs an area extremum; a minimum-area throat supports an accelerating subsonic-to-supersonic branch. A throat is a necessary geometric condition, not a sufficient specification of operating pressures.

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Isentropic nozzle flow refers to the flow of a compressible fluid (such as a gas) through a nozzle under idealized conditions where the process is isentropic. An isentropic process is one that is both adiabatic (no heat transfer occurs with the surroundings) and reversible (no entropy is generated). In simpler terms, it is an idealized process that assumes no friction and no heat loss, making it highly efficient.

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