Compressible & Sonic Flow可壓縮與音速流

What you'll be able to do本章學習成果

• Define the speed of sound and the Mach number, and identify subsonic, sonic, and supersonic flow.
• Explain how area change affects velocity and pressure differently in subsonic vs supersonic flow.
• Describe the effect of back pressure on a nozzle, including choking and the maximum mass flow rate.
• Explain the occurrence of normal shocks and analyze isentropic flow of ideal gases with constant specific heats.

Key equations重要公式

The speed of sound音速

A sound wave is a small pressure disturbance that travels through a medium at a speed $c$ set by the medium's properties. Across the wave the process is very nearly isentropic, which gives $c^2 = -v^2(\partial p/\partial v)_s$. For an ideal gas with constant specific heats ($pv^k = $ const), this becomes:

So sound travels faster in a hotter gas — about 347 m/s in air at 300 K, rising to ~506 m/s at 650 K. The speed of sound is an intensive property: it depends on the local state of the flow.

Mach number & the stagnation state馬赫數與滯止狀態

The Mach number is the ratio of the local flow velocity to the local speed of sound:

$M<1$ is subsonic, $M=1$ is sonic, $M>1$ is supersonic. Every state in a flow has a reference stagnation state — the state it would reach if brought to rest isentropically. Stagnation temperature $T_0$ and pressure $p_0$ are the anchors for the isentropic-flow relations below.

Area change & the flow截面積變化與流動

For steady isentropic flow, velocity and pressure always move oppositely — a rising velocity means falling pressure ($dV>0 \Rightarrow dp<0$). But how area change affects velocity flips at $M=1$. There are four cases:

The counterintuitive result: to keep accelerating a supersonic flow you must widen the duct.

Converging–diverging ducts收縮–擴張管道

Chaining a converging section to a diverging one makes a converging–diverging nozzle: subsonic flow accelerates to $M=1$ at the minimum-area throat, then continues to accelerate supersonically in the diverging section. Run in reverse it's a supersonic diffuser. The crucial rule: $M=1$ can occur only at the throat (the minimum area) — though it need not occur there.

Isentropic-flow explorer等熵流動探索器

Sweep the Mach number and read the isentropic-flow functions — the area ratio $A/A^\*$ and the temperature, pressure, and density relative to stagnation — plus the local sound speed and velocity. Watch $A/A^\*$ bottom out at 1 exactly at $M=1$: the same area ratio above 1 corresponds to two Mach numbers, one subsonic and one supersonic.

Ideal gas, $k=1.4$, $R=287$ J/kg·K. $A^\*$ is the throat area for sonic flow at the same stagnation state and mass flow.

Back pressure & choking背壓與壅塞

Feed a converging nozzle from a fixed stagnation state and lower the back pressure $p_B$ downstream. At first the mass flow rate rises and the exit pressure tracks $p_B$. But once the exit reaches $M=1$ — at the critical pressure $p^\*$ — the nozzle is choked: it passes the maximum possible mass flow, and further lowering $p_B$ changes nothing inside.

Normal shocks正衝擊波

A normal shock is a near-discontinuous, irreversible jump from supersonic to subsonic flow, with an abrupt rise in pressure. Across it the mass, energy, and momentum balances must all hold simultaneously — their solutions are the intersections of the Fanno line (mass + energy) and the Rayleigh line (mass + momentum) on an $h$–$s$ diagram. Because the shock generates entropy ($s_y > s_x$):

• The upstream state is supersonic; the downstream state is subsonic.
• Stagnation enthalpy is unchanged across the shock (adiabatic), but stagnation pressure drops — the signature of the loss.

For ideal gases with constant specific heats, the temperature, pressure, Mach-number, and stagnation-pressure ratios across the shock are tabulated functions of the upstream Mach number (Table 9.3).

Sound speed & Mach number音速與馬赫數