Compressible flow and shock waves

In short

Compressible flow is gas flow in which density changes are large enough to matter — conventionally above about Mach 0.3 — so the continuity, momentum and energy equations must be solved together with the equation of state. It explains why a convergent-divergent nozzle is needed for supersonic exhaust, why drag rises sharply near Mach 1, and why shock waves form ahead of blunt supersonic bodies. The topic is heavily examined in GATE Aerospace (gas dynamics section), ISRO/DRDO/NAL and HAL recruitment tests, and in IIT/IISc M.Tech interviews. It is applied daily in ISRO launch-vehicle nozzle design, gas-turbine and ramjet work at GTRE and HAL, supersonic tunnel testing at NAL Bengaluru, and high-speed CFD validation.

Explained

Speed of sound, Mach number and flow regimes

• A small pressure disturbance travels isentropically at the speed of sound a = √(γRT): in air (γ = 1.4, R = 287 J/kg·K) a ≈ 340 m/s at 288 K, ≈ 295 m/s at 216.65 K (11 km, ISA).
• a depends only on temperature for a perfect gas — not on pressure or density separately, since p/ρ = RT.
• Mach number M = V/a; its square M² is proportional to the ratio of flow kinetic energy to internal (thermal) energy, and M is the single governing parameter of compressibility.
• Regimes: incompressible (M < 0.3, density change < 5%), subsonic (0.3–0.8), transonic (0.8–1.2, mixed subsonic/supersonic pockets with shocks), supersonic (1.2–5), hypersonic (M > 5, with real-gas dissociation, viscous interaction and thin shock layers).
• Physical meaning of M = 1: disturbances can no longer travel upstream. Subsonic flow "knows" about a downstream obstacle; supersonic flow does not — information is confined to a Mach cone of half-angle μ = sin⁻¹(1/M).
• Density ratio check: ρ₀/ρ = (1 + 0.2M²)^2.5 → at M = 0.3 this is 1.046, i.e. 4.6% — the origin of the M = 0.3 rule.

Isentropic and stagnation relations

• Stagnation (total) properties are those reached on isentropically, adiabatically decelerating the flow to rest: T₀ fixed by energy (adiabatic) alone; p₀ and ρ₀ require isentropy too.
• Energy equation for an adiabatic perfect-gas flow: h + V²/2 = h₀, i.e. cpT + V²/2 = cpT₀ → T₀/T = 1 + (γ−1)M²/2.
• With p/ρ^γ = const: p₀/p = (T₀/T)^{γ/(γ−1)}, ρ₀/ρ = (T₀/T)^{1/(γ−1)}.
• Key distinction: T₀ is conserved across a shock and along any adiabatic duct (no work, no heat); p₀ *falls* wherever entropy rises (shocks, friction). Exam answers hinge on this.
• Critical (sonic, starred) state at M = 1: T*/T₀ = 2/(γ+1) = 0.8333; p*/p₀ = 0.5283; ρ*/ρ₀ = 0.6339 for γ = 1.4. The 0.528 value is the critical pressure ratio of a nozzle.
• Maximum (limiting) speed on expanding to vacuum: Vmax = √(2cpT₀); characteristic speed a* = √(2γRT₀/(γ+1)).

Variable-area flow, nozzles and choking

• Differential area-velocity relation: dA/A = (M² − 1)·dV/V. Hence for M < 1, area must *decrease* to accelerate; for M > 1 it must *increase*; M = 1 can occur only where dA = 0, i.e. at the throat.
• Area-Mach relation: A/A* = (1/M)[(2/(γ+1))(1 + (γ−1)M²/2)]^{(γ+1)/2(γ−1)} — double-valued, giving one subsonic and one supersonic solution for each A/A*.
• Convergent nozzle: flow accelerates until p_back/p₀ = 0.528; below that it is choked — exit stays at M = 1 and mass flow is fixed; further expansion happens outside as an underexpanded jet.
• Choked mass flow: ṁ = ρ*A*a* = 0.6847 p₀A*/√(RT₀) for γ = 1.4 — depends only on p₀, T₀ and throat area. This is why a choked orifice is a reliable flow meter and why rocket thrust is set by chamber conditions.
• Convergent-divergent (de Laval) nozzle operating modes as back pressure falls: (1) venturi (fully subsonic), (2) choked with subsonic divergent flow, (3) normal shock inside the divergent portion, (4) shock at exit plane, (5) overexpanded — oblique shocks outside, (6) design/correct expansion, (7) underexpanded — expansion fans outside.
• Diffusers reverse the logic: a supersonic diffuser is convergent-divergent, which is why supersonic intakes (and supersonic tunnels) need a second throat and have a starting problem.

Normal shock waves

• A shock is an extremely thin (order of a few mean free paths, ~10⁻⁷ m) irreversible adiabatic compression across which p, T, ρ rise and V, M, p₀ fall. It exists only when the upstream flow is supersonic: M₁ > 1 and always M₂ < 1.
• Governing equations: ρ₁V₁ = ρ₂V₂; p₁ + ρ₁V₁² = p₂ + ρ₂V₂²; h₁ + V₁²/2 = h₂ + V₂²/2 (so T₀₁ = T₀₂).
• Rankine-Hugoniot relation links density and pressure ratios without velocity; unlike isentropic compression it predicts a *finite* density ratio limit of (γ+1)/(γ−1) = 6 as p₂/p₁ → ∞.
• Prandtl relation: V₁V₂ = a*² ⇒ M₁*M₂* = 1 — the neat proof that a shock must take supersonic flow to subsonic.
• Entropy rise Δs = cp ln(T₂/T₁) − R ln(p₂/p₁) > 0 only for M₁ > 1; a hypothetical expansion shock would give Δs < 0 and violates the second law — this is the standard "why no expansion shock" answer.
• Stagnation pressure loss p₀₂/p₀₁ (= A₁*/A₂*) is mild near M = 1 (0.99 at M = 1.2) but severe at high Mach (0.72 at M = 2, 0.33 at M = 3) — the reason multi-shock intakes are preferred over a single normal shock.
• Moving shocks and the shock tube: a shock of speed Cs into still gas is handled by shifting to shock-fixed coordinates, M₁ = Cs/a₁. A shock tube generates a shock, contact surface and expansion fan from a bursting diaphragm — the standard lab device for short-duration high-enthalpy testing.

Oblique shocks and expansion waves

• A supersonic flow turning into itself (a concave corner, a wedge, a compression ramp) produces an oblique shock at wave angle β to the flow, deflecting it by θ.
• Only the normal component M_{n1} = M₁ sin β obeys the normal-shock relations; the tangential component is unchanged. Hence M_{n1} > 1 but M₂ may still be supersonic (the usual case, called a *weak* shock).
• θ–β–M relation: tan θ = 2 cot β (M₁²sin²β − 1)/(M₁²(γ + cos 2β) + 2). For each M₁ and θ < θmax there are two β roots — the weak (observed) and strong (subsonic downstream) solutions; above θmax no straight attached shock is possible and the shock detaches into a curved bow shock.
• As M₁ → ∞, θmax → about 45.5° for γ = 1.4; at M = 2, θmax ≈ 23°. Sharp slender leading edges are therefore essential for attached shocks.
• A Mach wave is the limiting zero-strength oblique shock, β → μ = sin⁻¹(1/M).
• Prandtl-Meyer expansion: flow turning away from itself at a convex corner expands isentropically through a continuous fan; ν(M₂) = ν(M₁) + θ, where ν(M) = √((γ+1)/(γ−1)) tan⁻¹√((γ−1)(M²−1)/(γ+1)) − tan⁻¹√(M²−1). νmax = 130.45° for γ = 1.4.
• Shock reflections and interactions: a shock reflecting off a solid wall reflects as a shock; off a free (constant-pressure) boundary it reflects as an expansion — the mechanism of the diamond pattern in an underexpanded jet.
• Shock-expansion theory gives flat-plate and diamond-aerofoil supersonic lift and drag directly from these two relations, and explains wave drag, absent in subsonic flow.

Flow with friction and heat addition (1-D duct flows)

• Fanno flow: adiabatic flow with friction in a constant-area duct. h₀ constant; entropy rises, so M moves *towards* 1 from either side; friction cannot carry flow through M = 1. Governing length: 4fL*/D (or fL*/D depending on text convention). Over-long ducts cause choking — subsonic flow reduces its mass flow; supersonic flow adjusts via a shock.
• Rayleigh flow: frictionless constant-area flow with heat transfer. Heating drives M towards 1 (accelerating subsonic flow, decelerating supersonic flow); cooling moves it away. Maximum heat addition reaches M = 1 — thermal choking, the real limit in combustors, ramjets and afterburners.
• Both are solved with tabulated starred ratios (p/p*, T/T*, p₀/p₀*) and the superposition rule: for two stations, 4fL/D = (4fL*/D)₁ − (4fL*/D)₂.
• Contrast: in Fanno flow T₀ is constant and p₀ drops; in Rayleigh flow T₀ changes with heat and p₀ also drops. Across a shock both p₀ drops and T₀ is constant — a shock is the jump linking the supersonic and subsonic branches of the same Rayleigh/Fanno line.

Measurement, standards and Indian practice

• Subsonic compressible pitot: p₀/p = (1 + 0.2M²)^3.5; using incompressible Bernoulli overestimates speed above M ≈ 0.3.
• Supersonic pitot (Rayleigh pitot-tube formula): the probe sits behind a normal shock, so the measured ratio is p₀₂/p₁, requiring the shock relations; the airspeed indicator logic in a supersonic aircraft must account for this.
• Facilities and codes: blowdown and continuous supersonic tunnels at NAL Bengaluru and the 1.2 m trisonic tunnel; hypersonic shock tunnels at IISc (HST series) and IIT Madras; ISRO's VSSC and LPSC nozzle test stands. Airworthiness in India is regulated by DGCA under the Bharatiya Vayuyan Adhiniyam, 2024 and its rules (this Act replaced the Aircraft Act, 1934); military airworthiness is certified by CEMILAC.
• Standard atmosphere data follow ISO 2533 (ICAO standard atmosphere), used for all Mach-altitude conversions; SI units throughout — p in Pa or bar, T in K.
• Gas-property values quoted here are for air as a calorically perfect gas with γ = 1.4; above about 1,000 K cp varies and above roughly 2,000 K dissociation makes γ and R effective values — real-gas or equilibrium models are then required, and quoted tables no longer apply.

Key points

• M = V/a — governing parameter; a = √(γRT) depends on temperature only.
• M = 0.3 threshold — density error ~5%; below it incompressible analysis is acceptable.
• T₀ — constant in any adiabatic flow including across a shock; p₀ falls whenever entropy rises.
• Critical pressure ratio p*/p₀ = 0.528 (γ = 1.4) — the choking condition for a convergent nozzle.
• Choked mass flow — fixed by p₀, T₀ and A*; lowering back pressure further changes nothing inside.
• dA/A = (M²−1) dV/V — the reason a supersonic nozzle must be convergent-divergent.
• A/A\* — double-valued; one subsonic and one supersonic Mach number per area ratio.
• Normal shock — M₁ > 1 always gives M₂ < 1; p, T, ρ, s rise; V, M, p₀ fall.
• Prandtl relation V₁V₂ = a\*² — compact proof of the subsonic downstream result.
• Rankine-Hugoniot limit — density ratio cannot exceed (γ+1)/(γ−1) = 6 for air.
• No expansion shock — it would decrease entropy, violating the second law.
• Oblique shock — only the normal velocity component shocks; M₂ usually stays supersonic.
• θ–β–M — weak and strong solutions below θmax; beyond θmax the shock detaches as a bow shock.
• Mach angle μ = sin⁻¹(1/M) — zero-strength limit of an oblique shock.
• Prandtl-Meyer expansion — isentropic turning fan; ν₂ = ν₁ + θ, νmax = 130.45°.
• Fanno flow — adiabatic + friction; M → 1 from both sides, p₀ falls, T₀ constant.
• Rayleigh flow — heat addition drives M → 1; thermal choking limits combustor heat release.
• Wave drag — pressure drag from shock losses, with no subsonic counterpart; drives the transonic drag rise and area-ruled fuselages.

Formulas and examples

Example 1 — Stagnation conditions on a cruising aircraft. An aircraft flies at M = 0.85 at 11 km, where T = 216.65 K and p = 22.63 kPa.

• a = √(1.4 × 287 × 216.65) = √(87,046) ≈ 295.0 m/s; V = 0.85 × 295.0 ≈ 250.8 m/s (≈ 903 km/h).
• T₀/T = 1 + 0.2 × 0.7225 = 1.1445 → T₀ = 216.65 × 1.1445 ≈ 247.9 K.
• p₀/p = 1.1445^3.5. ln 1.1445 = 0.13497; × 3.5 = 0.47240; e^0.47240 = 1.6040 → p₀ ≈ 22.63 × 1.6040 ≈ 36.3 kPa.
• Incompressible Bernoulli would give q = ½ρV² with ρ = p/RT = 22,630/(287 × 216.65) = 0.3639 kg/m³ → q = 0.5 × 0.3639 × 250.8² ≈ 11.44 kPa, i.e. p₀ ≈ 34.1 kPa — about 6% low, showing why compressibility correction is needed.

Example 2 — Choked convergent-divergent nozzle (rocket-style). Chamber: p₀ = 50 bar, T₀ = 3,000 K, γ = 1.4, R = 287 J/kg·K (air values for the exam arithmetic), throat A* = 0.01 m².

• Throat conditions: T* = 3,000 × 0.8333 = 2,500 K; p* = 50 × 0.5283 = 26.4 bar.
• ṁ = 0.6847 × (50 × 10⁵ × 0.01)/√(287 × 3,000) = 0.6847 × 50,000/√(861,000) = 34,235/927.9 ≈ 36.9 kg/s.
• For an exit area ratio A/A* = 4, the supersonic root of the area-Mach relation gives M_e ≈ 2.94.
• Then T₀/T_e = 1 + 0.2 × 2.94² = 1 + 0.2 × 8.6436 = 2.7287 → T_e ≈ 3,000/2.7287 ≈ 1,099 K; a_e = √(1.4 × 287 × 1,099) = √(441,568) ≈ 664.5 m/s → V_e ≈ 2.94 × 664.5 ≈ 1,954 m/s.
• Ideal momentum thrust = ṁV_e ≈ 36.9 × 1,954 ≈ 72.1 kN (pressure-thrust term extra).

Example 3 — Normal shock in a supersonic intake. M₁ = 2.0, p₁ = 40 kPa, T₁ = 250 K, γ = 1.4.

• M₂² = (1 + 0.2 × 4)/(1.4 × 4 − 0.2) = 1.8/5.4 = 0.3333 → M₂ ≈ 0.577.
• p₂/p₁ = 1 + (2.8/2.4)(4 − 1) = 1 + 1.16667 × 3 = 4.50 → p₂ = 180 kPa.
• ρ₂/ρ₁ = (2.4 × 4)/(0.4 × 4 + 2) = 9.6/3.6 = 2.667.
• T₂/T₁ = 4.50/2.667 = 1.6875 → T₂ = 421.9 K.
• Δs = 1,005 ln(1.6875) − 287 ln(4.50) = 1,005 × 0.52325 − 287 × 1.50408 = 525.9 − 431.7 ≈ +94.2 J/kg·K (positive, as the second law demands).
• p₀₁ = 40 × 1.8^3.5 = 40 × 7.824 ≈ 313.0 kPa; p₀₂ = 180 × (1 + 0.2 × 0.3333)^3.5 = 180 × 1.06667^3.5 = 180 × 1.2534 ≈ 225.6 kPa → p₀₂/p₀₁ ≈ 0.721, i.e. 28% total-pressure loss across one normal shock at M = 2.

Example 4 — Oblique shock on a wedge. M₁ = 3.0 air flows over a wedge half-angle θ = 20°; the weak-solution wave angle is β ≈ 37.8°.

• M_{n1} = 3.0 × sin 37.8° = 3.0 × 0.6130 = 1.839.
• p₂/p₁ = 1 + (2.8/2.4)(1.839² − 1) = 1 + 1.16667 × 2.382 = 3.78.
• M_{n2}² = (1 + 0.2 × 3.382)/(1.4 × 3.382 − 0.2) = 1.6764/4.5348 = 0.3697 → M_{n2} = 0.608.
• M₂ = M_{n2}/sin(β − θ) = 0.608/sin 17.8° = 0.608/0.3057 ≈ 1.99 — still supersonic, the signature of a weak oblique shock.

Example 5 — Prandtl-Meyer expansion. M₁ = 1.5 air turns 15° around a convex corner. ν(1.5) = 11.91°.

• ν₂ = 11.91 + 15 = 26.91° → from Prandtl-Meyer tables, M₂ ≈ 2.0.
• Isentropic, so p₀ and T₀ are unchanged: p₂/p₁ = (p₂/p₀)/(p₁/p₀) = 0.1278/0.2724 ≈ 0.469 — pressure falls to under half.

Where it is used

• Launch vehicles and propulsion: ISRO (VSSC, LPSC) nozzle contour and area-ratio design, altitude-compensation and overexpansion studies for PSLV, GSLV and LVM3 stages; semi-cryogenic and cryogenic engine testing at IPRC Mahendragiri.
• Gas turbines and military engines: HAL and GTRE (Kaveri/dry Kaveri) work on transonic compressor stages, choked turbine nozzle guide vanes, afterburner thermal choking, and supersonic intake design for Tejas-class aircraft.
• Wind-tunnel and test engineering: supersonic and hypersonic tunnel operation at NAL Bengaluru, IISc and IIT Madras shock tunnels — nozzle blocks, second throats, blowdown run-time and starting-load calculations.
• CFD and design roles: high-speed aerodynamics engineers at Airbus/Boeing India, Honeywell, Collins Aerospace, Safran and Tata Advanced Systems validating shock-capturing solvers against shock-tube and tunnel data.
• Process and energy industries: choked flow through safety relief valves, orifice plates and control valves; critical-flow nozzles for gas flow calibration; steam-turbine nozzle blocks at BHEL.
• Defence systems: missile aerodynamics at DRDL/RCI (BrahMos, Akash, Astra) — bow shocks, wave drag, nose-cone shaping, and ramjet/scramjet intake shock structures.

Interview questions

• Q: Why is a convergent-divergent nozzle needed for supersonic flow? A: Because dA/A = (M²−1)dV/V — subsonic flow accelerates in a contraction and supersonic flow only in an expansion, so M = 1 must occur at a throat with the divergent part continuing the acceleration.
• Q: What does choking mean physically? A: The throat has reached M = 1, so downstream pressure signals cannot travel upstream; mass flow is fixed by p₀, T₀ and throat area regardless of how far back pressure drops.
• Q: Which stagnation property is conserved across a shock, and which is not? A: Stagnation temperature is conserved (the shock is adiabatic with no work); stagnation pressure drops because entropy increases.
• Q: Why can an expansion shock not exist? A: The shock equations for a supersonic-to-higher-Mach jump give a negative entropy change, violating the second law; expansions must occur gradually through an isentropic Prandtl-Meyer fan.
• Q: Can the flow behind an oblique shock be supersonic? A: Yes — only the normal velocity component is reduced below sonic, and the unchanged tangential component usually keeps the resultant supersonic in the weak-shock solution.
• Q: When does a shock detach from a wedge? A: When the required turning angle exceeds θmax for that Mach number (about 23° at M = 2); the shock then stands off as a curved bow shock with a locally normal portion.
• Q: Why do supersonic intakes use multiple oblique shocks rather than one normal shock? A: Several weak oblique shocks followed by a weak terminal normal shock recover far more stagnation pressure — a single normal shock at M = 3 loses about 67%.
• Q: What is the difference between Fanno and Rayleigh flow? A: Fanno is adiabatic flow with friction (T₀ constant); Rayleigh is frictionless flow with heat transfer (T₀ changes) — in both, the Mach number is driven towards 1.
• Q: Why does a pitot tube read incorrectly at supersonic speed if treated conventionally? A: A normal shock stands ahead of the probe, so it senses the post-shock stagnation pressure; the Rayleigh pitot formula relating p₀₂/p₁ must be used.
• Q: What causes the transonic drag rise? A: Local supersonic pockets on the surface terminate in shocks, producing wave drag and shock-induced boundary-layer separation; area ruling and swept or supercritical aerofoils delay it.
• Q: What is the maximum density ratio across a shock in air and why is it finite? A: (γ+1)/(γ−1) = 6, from the Rankine-Hugoniot relation — temperature rises so steeply with pressure that density saturates, unlike isentropic compression.
• Q: An underexpanded nozzle exhaust shows a diamond pattern — why? A: Expansion fans at the lip reflect off the constant-pressure jet boundary as compressions that coalesce into shocks, which reflect back as expansions, repeating periodically.

Common mistakes

• Using incompressible Bernoulli or ½ρV² dynamic pressure above M ≈ 0.3, or applying Bernoulli across a shock.
• Assuming stagnation pressure is conserved across a shock; only stagnation temperature is.
• Taking the full upstream Mach number in the normal-shock formulae for an oblique shock, instead of M₁ sin β, or forgetting to divide by sin(β − θ) to recover M₂.
• Picking the wrong root of the area-Mach relation — reading the subsonic value when the nozzle is running supersonic.
• Believing a convergent nozzle can give supersonic exit flow by dropping back pressure further.
• Mixing temperature units (using °C in √(γRT)) or pressure units (bar with Pa) in gas-dynamic formulae.
• Assuming γ = 1.4 and constant cp at combustion or re-entry temperatures, where dissociation and variable specific heats make the perfect-gas tables invalid.

Related topics

Fundamentals of aerodynamics · Airfoils and wings · Fluid flow: Bernoulli and flow measurement · Gas turbines and jet propulsion · High-speed and hypersonic aerodynamics · Computational fluid dynamics