Compressible Aerodynamics: Isentropic Flow, Shocks, Expansions, Nozzles and Duct Flows

Small pressure disturbances travel at the speed of sound a = √(γRT) — 340.3 m/s at sea-level ISA and 295.1 m/s at 216.65 K — and the Mach number M = V/a measures how far the flow is from being able to "warn" the fluid ahead. For steady adiabatic flow the energy equation is c_pT + V²/2 = c_pT0, which gives the stagnation relations: T0/T = 1 + ((γ − 1)/2)M²; for isentropic deceleration p0/p = (T0/T)^(γ/(γ−1)) and ρ0/ρ = (T0/T)^(1/(γ−1)). At M = 1 these give the sonic (starred) ratios T/T0 = 2/(γ + 1) = 0.8333, p/p0 = 0.5283 and ρ*/ρ0 = 0.6339. The stagnation temperature is conserved in any adiabatic flow, shocks included; the stagnation pressure is conserved only if the flow is also reversible.

Combining continuity, momentum and the speed of sound gives the area-velocity relation dA/A = (M² − 1)dV/V. In subsonic flow (M < 1) a converging duct accelerates the flow; in supersonic flow (M > 1) a diverging duct does; and M = 1 can occur only at a throat, where dA = 0. So a flow is accelerated from rest to supersonic speed only by a converging-diverging (de Laval) nozzle with sonic flow at the throat. The isentropic area ratio is A/A* = (1/M)[(2/(γ + 1))(1 + ((γ − 1)/2)M²)]^((γ+1)/(2(γ−1))), which is 1 at M = 1, larger on either side, and 1.6875 at M = 2. Once the throat is sonic the mass flow is fixed by the reservoir: ṁ = (p0A/√T0)·√(γ/R)·(2/(γ + 1))^((γ+1)/(2(γ−1))) = 0.0404 p0A/√T0 for air in SI units — the nozzle is choked, and lowering the back pressure further cannot draw more flow.

• Back pressure slightly below p0: subsonic everywhere, a venturi; mass flow rises as p_b falls.
• First critical back pressure: the throat just reaches M = 1 and the divergent part decelerates the flow subsonically — the nozzle is now choked.
• Lower still: supersonic flow after the throat ends in a normal shock inside the divergent section, which moves towards the exit as p_b falls.
• Over-expanded: the flow is supersonic to the exit but p_e < p_b, so oblique shocks form outside to raise the pressure.
• Design condition: p_e = p_b, a clean supersonic jet.
• Under-expanded: p_e > p_b, and expansion fans outside the lip complete the expansion.

A diffuser does the reverse job, decelerating the flow and recovering pressure. A subsonic diffuser simply diverges gently enough to avoid separation. An ideal supersonic diffuser would be a converging-diverging passage that slows the flow isentropically to M = 1 at its throat and then subsonically, but it cannot be started: while the flow is being established a normal shock stands ahead of it, and swallowing that shock needs a throat larger than the isentropic one (or overspeeding, or variable geometry). Real supersonic diffusers and intakes therefore use a system of oblique shocks followed by a weak normal shock.

A normal shock is a thin, irreversible jump from supersonic to subsonic flow. Mass, momentum and energy across it give, for upstream Mach number M1: M2² = (1 + ((γ − 1)/2)M1²)/(γM1² − (γ − 1)/2); p2/p1 = 1 + (2γ/(γ + 1))(M1² − 1); ρ2/ρ1 = V1/V2 = (γ + 1)M1²/(2 + (γ − 1)M1²); and T2/T1 = (p2/p1)/(ρ2/ρ1). The stagnation temperature is unchanged, the entropy rises and the stagnation pressure falls, s2 − s1 = −R ln(p02/p01). The Prandtl relation V1V2 = a*² is a compact check. At M1 = 2: M2 = 0.5774, p2/p1 = 4.5, ρ2/ρ1 = 2.667, T2/T1 = 1.6875 and p02/p01 = 0.7209.

When supersonic flow is turned into itself by a wedge or ramp of angle θ, an oblique shock forms at wave angle β. The tangential velocity is unchanged, and the normal component M1n = M1 sin β obeys the normal-shock relations, M2 = M2n/sin(β − θ). The θ-β-M relation tan θ = 2 cot β (M1² sin²β − 1)/(M1²(γ + cos 2β) + 2) has two solutions for each θ below a maximum θ_max: the weak shock, which is what normally occurs and usually leaves the flow supersonic, and the strong shock, which leaves it subsonic. Beyond θ_max no attached solution exists and a detached bow shock stands ahead of the body. As the deflection tends to zero the oblique shock weakens to a Mach wave at the Mach angle μ = sin⁻¹(1/M).

When supersonic flow turns away from itself round a convex corner it expands through a fan of Mach waves — a centred Prandtl-Meyer expansion. The process is isentropic, so p0 and T0 are constant, the Mach number rises and the pressure falls. The Prandtl-Meyer function ν(M) = √((γ + 1)/(γ − 1)) tan⁻¹√((γ − 1)(M² − 1)/(γ + 1)) − tan⁻¹√(M² − 1) is zero at M = 1, 26.38° at M = 2, and tends to 130.45° as M → ∞; a turn of θ takes the flow from ν(M1) to ν(M2) = ν(M1) + θ. Oblique shocks and expansion fans together give shock-expansion theory for supersonic aerofoils; for a thin flat plate at small α, linear theory gives c_l = 4α/√(M∞² − 1).

Flow over the upper surface of an aerofoil is faster than the free stream, so the local Mach number reaches 1 somewhere before the free stream does. The critical Mach number M_cr is the free-stream Mach number at which sonic flow first appears on the surface. It is found by intersecting two curves: the minimum pressure coefficient corrected for compressibility, which by the Prandtl-Glauert rule is C_p = C_p0/√(1 − M∞²), and the pressure coefficient at which the local flow is sonic, C_p,cr = (2/(γM∞²))[((2 + (γ − 1)M∞²)/(γ + 1))^(γ/(γ−1)) − 1]. Slightly above M_cr a pocket of supersonic flow terminated by a shock forms; as it strengthens the shock thickens the boundary layer and can separate it, and the drag coefficient rises steeply at the drag-divergence Mach number.

Two constant-area duct flows show what friction and heating do to a compressible stream. Fanno flow is adiabatic flow with wall friction: the stagnation temperature is constant, entropy rises along the duct, and friction always drives the Mach number towards 1 — a subsonic stream accelerates and a supersonic one decelerates. The maximum length that can be sustained before the flow chokes at the exit is given by 4f̄L/D = (1 − M²)/(γM²) + ((γ + 1)/(2γ)) ln[(γ + 1)M²/(2 + (γ − 1)M²)]. If the duct is longer than L, a subsonic flow reduces its mass flow and a supersonic flow forms a shock inside the duct.

Rayleigh flow is frictionless flow with heat addition or removal; p + ρV² is constant. Heating raises the stagnation temperature and drives the Mach number towards 1 from either side, just as friction does: subsonic flow accelerates and its static pressure falls, supersonic flow decelerates and its pressure rises. Cooling does the opposite. The maximum stagnation temperature occurs at M = 1, so there is a limit to the heat a given inlet state can accept — thermal choking. Curiously, in subsonic heating the static temperature peaks at M = 1/√γ ≈ 0.845 and then falls even though heat is still being added, the extra energy going into kinetic energy.