Nozzle

A nozzle shapes a passage to control the speed and direction of a fluid flow. In compressible flow, converging-diverging passages can accelerate a gas through a sonic flow throat into supersonic 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.

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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A nozzle is a device designed to control the direction or characteristics of a fluid flow as it exits an enclosed chamber or pipe. Nozzles are commonly used in various applications, including: 1. **Aerospace and Aviation**: In jet engines, nozzles help to accelerate exhaust gases, providing thrust. In rocket engines, they are critical for directing high-speed gases to create lift-off.

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