Fluid Mechanics · Chapter 12

Compressible Flow

When a gas moves fast enough, its density can no longer be treated as constant, and thermodynamics enters the analysis alongside fluid dynamics. This chapter develops stagnation properties, the speed of sound and Mach number, isentropic nozzle flow and choking, normal and oblique shocks, and the duct flows driven by heat (Rayleigh) and friction (Fanno).

Chapter Objectives

Distinguish static, stagnation, critical, and post-shock states with absolute pressure.
Classify compressible regimes using local Mach number and thermodynamic property data.
Apply mass, momentum, and energy balances to isentropic nozzles and normal shocks.
Recognize choking and select nozzle, Rayleigh, or Fanno models from physical cues.
Verify state calculations through conservation, entropy direction, units, and limiting behavior.

Section 12–1

Stagnation Properties

Earlier chapters assumed density was constant. Here we lift that restriction and study flows where density changes are significant — high-speed gas flows in nozzles, diffusers, and turbines, where fluid dynamics and thermodynamics become inseparable.

Stagnation (Total) Properties

The state a flowing fluid would reach if brought to rest adiabatically. Stagnation enthalpy combines static enthalpy and kinetic energy; for an ideal gas this defines a stagnation temperature. When the deceleration is also reversible (isentropic), the corresponding pressure and density are the stagnation pressure and density.

Stagnation Enthalpy & Temperature

\[ h_0 = h + \frac{V^2}{2}, \qquad T_0 = T + \frac{V^2}{2c_p} \]

The term \(V^2/2c_p\) is the dynamic temperature — the temperature rise on bringing the flow to rest. At low speeds it is negligible and \(T_0\approx T\); at high speeds it is large, which is why high-speed probes read elevated temperatures.

Physical meaning

Static properties describe a fluid element moving with the stream. Stagnation temperature follows from an adiabatic energy balance and remains constant even across an adiabatic shock; stagnation pressure additionally assumes reversible deceleration and therefore falls when entropy is produced. Compressible-flow pressures must be absolute in property ratios.

Problem-solving strategy

Draw a control volume and label static, stagnation, throat, and downstream states without mixing them.
Choose downstream-positive coordinates and declare pressure as absolute, temperature basis, gas model, and signs of heat and shaft work.
Use continuity plus steady-flow energy and momentum, then add isentropic, shock, Rayleigh, or Fanno closure only when justified.
Determine local Mach number and check for choking before iterating pressure or area relations.
Audit mass and total-energy conservation, entropy direction, units, state ordering, and sensitivity to gas-property assumptions.

Section 12–2

Speed of Sound & Mach Number

Speed of Sound

The rate at which an infinitesimal pressure disturbance travels through a medium. For an ideal gas it depends only on temperature, through the specific-heat ratio \(k\) and gas constant \(R\).

Speed of Sound (ideal gas)

\[ c = \sqrt{\left(\frac{\partial P}{\partial \rho}\right)_s} = \sqrt{kRT} \]

For gases, density changes are often negligible below about \(\mathrm{Ma}=0.3\) when heating is also weak, but Mach number alone cannot excuse strong heat addition or large pressure changes. Reynolds number still governs viscous behavior, and Froude number matters when gravity competes with inertia.

EXAMPLE 1Static and stagnation state of a flight stream

Determine local Mach number, stagnation temperature, and stagnation pressure for an air stream entering a research probe at a known static state.

Recognition strategy

Known static temperature and velocity give sound speed and Mach number; adiabatic reversible deceleration then connects static and stagnation states.

Given information

Static \(T=285\ \text{K}\), \(P=82.0\ \text{kPa}\) absolute, \(V=220\ \text{m/s}\); \(+x\) follows the flow.
Supplied calorically perfect air properties on the \(285\)–\(310\ \text{K}\) basis: \(k=1.40\), \(R=287\ \text{J/(kg}\!\cdot\!\text{K)}\), \(c_p=1005\ \text{J/(kg}\!\cdot\!\text{K)}\).

Model and interpretation

A streamline control volume is steady and adiabatic with no shaft work or elevation change. Isentropic stagnation pressure is an approximate ideal-gas result; stagnation temperature follows the exact model energy balance.

Step-by-step solution

1. Find acoustic speed. \(c=\sqrt{kRT}=\sqrt{(1.40)(287)(285)}=338.4\ \text{m/s}\).

2. Classify the flow. \(\mathrm{Ma}=V/c=220/338.4=0.650\), so compressibility cannot be neglected.

3. Apply total enthalpy. \(T_0=T+V^2/(2c_p)=285+220^2/[2(1005)]=309.1\ \text{K}\).

4. Recover total pressure. \(P_0=P(T_0/T)^{k/(k-1)}=(82.0\ \text{kPa})(309.1/285)^{3.5}=108.9\ \text{kPa}\) absolute.

Verification

Units: \(kRT\) produces \(\text{m}^2/\text{s}^2\), kinetic energy divided by \(c_p\) produces kelvins, and the pressure ratio is dimensionless.

Signs and direction: Deceleration from positive \(V\) raises both \(T_0\) and \(P_0\); neither can be below its static value for isentropic stagnation.

Plausibility: \(T_0/T=1.0845\), also equal to \(1+0.2\mathrm{Ma}^2\), independently confirming the speed, energy, and Mach calculations.

Final result: \(\mathrm{Ma}=0.650\), \(T_0=309.1\ \text{K}\), and \(P_0=108.9\ \text{kPa}\) absolute.

Engineering interpretation

A real probe may not recover the full isentropic pressure because of viscous loss, shock formation, alignment error, and thermal lag; calibration must quantify those uncertainties.

Mach Number

The ratio of flow speed to local speed of sound — the single most important parameter in compressible flow. It sorts flows into regimes: subsonic (\(\mathrm{Ma}<1\)), sonic (\(=1\)), supersonic (\(>1\)), and hypersonic (\(\gg1\)).

Mach Number

\[ \mathrm{Ma} = \frac{V}{c} \]

Section 12–3

One-Dimensional Isentropic Flow

Property Ratios via Mach Number

For isentropic flow of an ideal gas, every stagnation-to-static property ratio is a function of Mach number and \(k\) alone — the working relations tabulated in every compressible-flow text.

Temperature, Pressure & Density Ratios

\[ \frac{T_0}{T} = 1 + \frac{k-1}{2}\mathrm{Ma}^2, \quad \frac{P_0}{P} = \left(\frac{T_0}{T}\right)^{\!\frac{k}{k-1}}, \quad \frac{\rho_0}{\rho} = \left(\frac{T_0}{T}\right)^{\!\frac{1}{k-1}} \]

Critical (Sonic) Conditions

Properties evaluated where \(\mathrm{Ma}=1\) are critical conditions, marked with an asterisk. They serve as the reference for choked-flow calculations. For air (\(k=1.4\)), the critical pressure ratio \(P^*/P_0\approx0.528\).

Section 12–4

Isentropic Flow through Nozzles

Area–Velocity Relation

How area change affects velocity flips sign with the Mach number. In subsonic flow a converging passage accelerates the gas; in supersonic flow it decelerates it. To accelerate gas past Mach 1 the passage must first converge to a throat, then diverge — the converging–diverging (de Laval) nozzle.

Area–Mach Relation

\[ \frac{dA}{A} = \frac{dV}{V}\left(\mathrm{Ma}^2 - 1\right) \]

Choked Flow

Once the throat reaches \(\mathrm{Ma}=1\), the mass flow rate hits a maximum and cannot increase by lowering the back pressure further — the flow is choked. This caps the throughput of nozzles, valves, and orifices in high-speed gas systems.

Figure 12–1. A converging–diverging nozzle accelerates gas to subsonic speed in the converging part, sonic at the throat, and supersonic in the diverging part. (Original schematic.)

EXAMPLE 2Ideal choked purge-nozzle capacity

An ideal converging purge nozzle is fed by a regulated air reservoir; determine whether it chokes and calculate its maximum ideal mass flow.

Recognition strategy

A reservoir, converging minimum area, and low back pressure signal a choking test followed by the sonic ideal-gas mass-flux relation.

Given information

\(P_0=520\ \text{kPa}\) absolute, \(T_0=310\ \text{K}\), throat area \(A^*=1.80\times10^{-4}\ \text{m}^2\), back pressure \(P_b=240\ \text{kPa}\) absolute.
Supplied constant air properties at this temperature basis: \(k=1.40\), \(R=287\ \text{J/(kg}\!\cdot\!\text{K)}\); \(+x\) is downstream.

Model and interpretation

The fixed nozzle control volume is steady, adiabatic, one-dimensional, and isentropic to the throat with negligible inlet velocity. The equation is exact for this ideal model and approximate for a real nozzle.

Step-by-step solution

1. Compute the critical ratio. \(P^*/P_0=[2/(k+1)]^{k/(k-1)}=0.5283\).

2. Test choking. \(P_b/P_0=240/520=0.4615<0.5283\), so the throat reaches \(\mathrm{Ma}=1\) and mass flow is capped.

3. Form sonic mass flux. \(\dot m/A^*=P_0\sqrt{k/(RT_0)}[2/(k+1)]^{(k+1)/[2(k-1)]}\).

4. Substitute absolute states. The ideal maximum is \(\dot m=(1.80\times10^{-4})(520000)\sqrt{1.4/(287\!\cdot\!310)}(2/2.4)^3=0.2149\ \text{kg/s}\).

Verification

Units: \(AP_0\sqrt{1/(RT_0)}\) reduces from \(\text{m}^2\!\cdot\!\text{Pa}\!\cdot\!\text{s/m}\) to \(\text{kg/s}\), as required.

Signs and direction: The positive mass flow is downstream; lowering back pressure further cannot reverse or increase the ideal choked throat flux.

Plausibility: The back-pressure ratio is safely below critical, and mass flow scales linearly with area and reservoir absolute pressure while decreasing with \(\sqrt{T_0}\).

Final result: Choked operation gives an ideal maximum mass flow of \(\dot m_{\max}\approx0.215\ \text{kg/s}\).

Engineering interpretation

Actual capacity needs a measured discharge coefficient and real-gas or variable-property review; neither a coefficient nor a vendor limit is assumed in this ideal preliminary estimate.

Section 12–5

Shock Waves & Expansion Waves

Normal Shock

In supersonic flow, a sudden, nearly discontinuous compression can occur across a thin normal shock. The flow drops abruptly from supersonic to subsonic; pressure, temperature, and density jump up, while velocity and stagnation pressure drop. The process is adiabatic but strongly irreversible — entropy increases.

Across a Normal Shock

\[ T_{0,2} = T_{0,1}\ \ (\text{adiabatic}), \qquad P_{0,2} < P_{0,1}\ \ (\text{entropy rises}) \]

Normal-shock static ratios

\[ \mathrm{Ma}_2^2=\frac{1+\tfrac{k-1}{2}\mathrm{Ma}_1^2}{k\mathrm{Ma}_1^2-\tfrac{k-1}{2}}, \qquad \frac{P_2}{P_1}=1+\frac{2k}{k+1}\left(\mathrm{Ma}_1^2-1\right) \]

Oblique Shocks & Expansion Fans

When supersonic flow is turned into itself it forms an angled oblique shock (compression); when turned away from itself it passes through a smooth, isentropic Prandtl–Meyer expansion fan. These govern the flow around supersonic wedges, inlets, and airfoils.

EXAMPLE 3Normal shock in an inlet test section

A normal shock stands in a constant-area air-flow test section; calculate the downstream Mach number and static-state jumps, then quantify total-pressure loss.

Recognition strategy

A stationary discontinuity normal to a supersonic stream requires normal-shock balances, not isentropic relations across the shock.

Given information

Upstream \(\mathrm{Ma}_1=2.20\), \(P_1=36.0\ \text{kPa}\) absolute, \(T_1=245\ \text{K}\); \(+x\) is downstream.
Supplied calorically perfect air value \(k=1.40\) over the computed temperature range; the duct is adiabatic and has no shaft work.

Model and interpretation

A thin stationary control volume encloses the shock. One-dimensional mass, momentum, and total-energy balances are exact within the perfect-gas model; viscosity acts internally and makes the process irreversible.

Step-by-step solution

1. Find downstream Mach number. Substitution into the normal-shock relation gives \(\mathrm{Ma}_2=\sqrt{(1+0.2(2.2)^2)/(1.4(2.2)^2-0.2)}=0.547\).

2. Calculate pressure jump. \(P_2/P_1=1+[2(1.4)/2.4][(2.2)^2-1]=5.480\), hence \(P_2=197.3\ \text{kPa}\) absolute.

3. Use mass conservation. \(\rho_2/\rho_1=(k+1)\mathrm{Ma}_1^2/[2+(k-1)\mathrm{Ma}_1^2]=2.951\), so \(T_2/T_1=(P_2/P_1)/(\rho_2/\rho_1)=1.8569\) and \(T_2=455.0\ \text{K}\).

4. Quantify irreversibility. The perfect-gas normal-shock relation gives \(P_{0,2}/P_{0,1}=0.628\), while adiabatic energy conservation requires \(T_{0,2}=T_{0,1}\).

Verification

Units: Shock functions are dimensionless; multiplying ratios by upstream kilopascals absolute and kelvins returns valid downstream state units.

Signs and direction: Flow remains downstream but decelerates to subsonic speed; static pressure, density, and temperature rise while total pressure falls.

Plausibility: Continuity is honored through \(\rho_1V_1=\rho_2V_2\); constant stagnation temperature and reduced stagnation pressure confirm adiabatic yet irreversible behavior.

Final result: \(\mathrm{Ma}_2=0.547\), \(P_2=197\ \text{kPa}\) absolute, \(T_2=455\ \text{K}\), and \(P_{0,2}/P_{0,1}=0.628\).

Engineering interpretation

The shock destroys about \(37.2\%\) of upstream stagnation pressure, directly reducing recoverable inlet performance; shock position, boundary-layer interaction, and variable properties require validated analysis or test.

Section 12–6

Duct Flow with Heat Transfer & Negligible Friction (Rayleigh Flow)

Rayleigh Flow

Compressible flow in a constant-area duct where heat transfer is the driving effect and friction is neglected — a model for combustors and heat exchangers. Adding heat drives a subsonic flow toward \(\mathrm{Ma}=1\) and a supersonic flow toward it as well; the sonic point is the thermal limit beyond which more heating chokes the flow.

Counterintuitive result: heating a supersonic Rayleigh flow lowers its temperature over part of the range, because the accompanying changes in velocity and pressure outweigh the direct heating effect.

Section 12–7

Adiabatic Duct Flow with Friction (Fanno Flow)

Fanno Flow

Compressible flow in a constant-area insulated duct where wall friction is the driving effect and heat transfer is neglected — a model for long gas pipelines. Friction drives the Mach number toward unity from either side: subsonic flow accelerates toward \(\mathrm{Ma}=1\), supersonic flow decelerates toward it. There is a maximum duct length for given inlet conditions, beyond which the flow chokes.

Instructor Note

Compressible flow lives and dies by the Mach number — get it first, every time. Memorize the one stagnation-temperature relation; the pressure and density ratios follow from it by the isentropic exponents. And hold onto the central picture: a converging–diverging nozzle is the only way to take a gas smoothly from subsonic to supersonic, with \(\mathrm{Ma}=1\) pinned at the throat.

Chapter recap

Mach number compares flow speed with information-propagation speed and controls area response.
Stagnation temperature follows adiabatic energy conservation; stagnation pressure records irreversibility.
Choking fixes sonic mass flux, while shocks conserve mass, momentum, and total enthalpy but raise entropy.
Rayleigh and Fanno models isolate heat transfer and friction, respectively, and both drive flow toward sonic conditions.
Before moving on, be able to label states, select a model, use absolute properties, and verify conservation and regime limits.