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Flow whose mass flux is limited by a sonic condition in a controlling section, for fixed upstream thermodynamic conditions. In a smooth isentropic converging-diverging nozzle, the throat reaches Mach number one. Lowering downstream pressure further cannot indefinitely increase that fixed-upstream throat mass flux; shocks and downstream structure can still change.

For a steady isentropic flow with $p=K{\rho }^{\gamma }$, $K>0$ and $\gamma >1$, write the Bernoulli function asAt a fixed position, hold $C_B$, $K$ and the external potential $\Phi$ fixed and assume $H=C_B-\Phi >0$. The positive scalar mass flux is $j=\rho u$ for $u>0$. Differentiating the Bernoulli constraint gives $d\rho /du=-\rho u/c_s^2$, henceThe maximum occurs at $u=c_s=c_{\mathrm{cr}}$. Substituting this equality back into the energy constraint givesBoth the small-speed endpoint and the zero-enthalpy endpoint have vanishing $j$, so this interior stationary point is the maximum. The familiar formula with $C_B$ alone assumes a zero potential at the chosen section, or defines $C_B$ to exclude that fixed contribution.

For a perfect gas with no potential change, the result is equivalent to $T_{\mathrm{cr}}=2T_{\mathrm{stag}}/(\gamma +1)$. The local maximum explains choked flow; a complete nozzle solution must also connect the prescribed upstream state to the controlling section.^[ref]

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.

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

The Mach number is the ratio of fluid flow speed to local sound speed.

More precisely, the Mach number is the dimensionless quantitywhere $U$ is the magnitude of the body's relative velocity through the surrounding gas, and $c>0$ is the local sound speed measured in the gas rest reference frame. For a fluid flow past a stationary body, the same Mach number uses the local fluid speed in the body's rest reference frame. A Mach number below, equal to, or above one means subsonic flow, sonic flow, or supersonic flow, respectively. The denominator is a property of the local gas: its sound speed depends on temperature and composition.^[ref]

The relative velocity matters even when the body is at rest in a wind tunnel. A uniform change of reference frame subtracts the same velocity from body and gas, so their relative velocity and the Mach number stay unchanged under a Galilean transformation. Ground speed alone cannot supply $U$ when there is wind. For an original collinear example, a body moving at $680\mathrm{m}{\mathrm{s}}^{-1}$ over the ground through gas moving in the same direction at $170\mathrm{m}{\mathrm{s}}^{-1}$ has $U=510\mathrm{m}{\mathrm{s}}^{-1}$. With a stipulated sound speed $c=340\mathrm{m}{\mathrm{s}}^{-1}$, its Mach number is $1.5$, rather than $2$. A moving observer changes apparent acoustic wave arrival times, as in the radial Mach number, without changing the body's Mach number relative to the gas.

Why is the threshold one so useful? Acoustic waves carry small pressure disturbances at the sound speed. In subsonic flow, such information can travel upstream relative to the body; in a uniform supersonic flow, it is swept downstream. The Mach cone turns this difference in communication into visible geometry. Matching Mach number between a wind tunnel and flight is therefore essential to matching compressible flow effects.^[ref]

An actual supersonic wind tunnel experiment makes the distinction between wavefronts and shock waves concrete. In NASA Glenn's 1-by-1-foot wind tunnel, interchangeable converging-diverging nozzle blocks set the Mach number. A strut-mounted exhaust-nozzle model had a nosecone, an air plenum and transparent sidewalls for schlieren photography. Before installing the model, the researchers photographed the empty test section: two shock wave patterns originated at a wall joint after the nozzle block. This control guided model placement to keep reflected shock waves away from the plume-study region. The rig included internal total pressure and static pressure sensing and a traversing external static pressure probe. A baseline nozzle and previous pressure data supplied a planned validation comparison; the report distinguishes this plan from the observed empty-tunnel control. This establishes an apparatus origin for those visible shock waves, not a measurement of the Mach cone from an infinitesimal source. The reported finite shock wave angle cannot simply be inserted into the Mach angle formula to calibrate $M$: shock wave strength and geometry also matter.^[ref]

In conventional schlieren photography, a bright slit source and two concave mirrors send parallel light rays through the wind tunnel and focus them at a knife edge before an optical camera. Spatial changes in mass density cause refraction; the knife edge converts the deflection into image contrast, making shock waves visible. This is the general optical principle, rather than a specification of every component in the preceding rig. A photograph projects a three-dimensional fluid flow into two dimensions; contrast alone does not provide a calibrated pressure jump or acoustic wave amplitude.^[ref]

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.

For a calorically perfect gas on a steady isentropic nozzle branch,Here $A_{\ast }$ 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.

Treat the gas as a smooth quasi-one-dimensional flow through a rigid passage, neglecting axial body forces and heat input. It is the confining medium that is incompressible; the flowing gas must remain compressible. With area $\mathcal{A}(x)$ and axial speed $u$, continuity and inviscid momentum balance giveCombining their logarithmic differentials yields the nozzle area-velocity relationA smooth crossing of $\mathcal{M}=1$ requires an area extremum. The given radius is positive and has derivative $r^{\prime }(x)=16r_{0}x\exp [-16x^2/L^2]/L^2$, which vanishes only at $x=0$ within the slab and changes from negative to positive there. Thus this station is the unique throat, with $r_s=r_0/2$ and ${\mathcal{A}}_s=\pi r_0^2/4$. The specified subsonic inlet and supersonic outlet select the smooth accelerating branch, soA throat by itself does not guarantee transonic flow for arbitrary reservoir and back pressures; that branch is specified here. Before the throat a converging section accelerates subsonic flow; beyond it a diverging section accelerates supersonic flow.^[ref]

For the temperature task, evaluate the mass flux and sound speed at the throat using $p=K{\rho }^{\gamma }$:The ideal-gas relation then givesThis expresses the sonic temperature in terms of the entropy constant, geometry, mass flux and composition; no numerical value of $K$ or $\dot{M}$ is provided. Equivalently $c_s^{\gamma +1}=\gamma K(\dot{M}/{\mathcal{A}}_s{)}^{\gamma -1}$ and $T_s=\mu c_s^2/(\gamma R^{\ast })$. The slab length drops out because the sonic thermodynamic state depends on the minimum area, not on the axial distance over which a lossless smooth nozzle contracts. As a check, conserved stagnation enthalpy gives $T_s=2T_{\mathrm{stag}}/(\gamma +1)$.^[ref]

The result assumes cross-sectional thermodynamic uniformity and negligible transverse kinetic energy. Exact multidimensional flow through a sharply varying passage can have a curved sonic surface; the radius profile alone does not solve that boundary-value problem. The original numerical plot below computes the standard quasi-one-dimensional branch, rather than reproducing the supplied passage sketch.

A model representing a passage by its varying cross-sectional area and approximately uniform section-averaged flow variables. It neglects detailed transverse velocity and thermodynamic structure. In a smooth lossless nozzle, conservation laws then connect area to Mach number; this model does not by itself determine the curved sonic surface of an arbitrary multidimensional passage.

A supersonic wind tunnel produces controlled supersonic flow in its test section. Its nozzle geometry and imposed pressure conditions set the test Mach number.