Compressible flow (gas dynamics)

modified2026-07-21statusfinished

Virginia Tech AOE 3114 compressible-flow relation set — isentropic flow, normal shock, oblique shock (θ-β-M), Fanno friction flow, and Rayleigh heat-addition flow, in one reference.

Index

Isentropic flow relations
Normal shock wave relations
Oblique shock wave relations (θ-β-M)
Conical shock wave relations (Taylor-Maccoll)
Fanno flow with friction
Rayleigh flow with heat transfer
Air-breathing propulsion cycle

Each relation set below includes a Voici¹1Voici compiles a notebook to a static, serverless dashboard — mocked here as a static page, since this pipeline has no Jupyter kernel to run against. dashboard: a matplotlib-styled chart with the live calculator input marked as a red point on the curve.

2.1 · Isentropic flow relations

[2.1] Isentropic flow relations

$$\frac{T_0}{T}=1+\frac{\gamma-1}{2}M^2,\quad \frac{p_0}{p}=\left(\frac{T_0}{T}\right)^{\frac{\gamma}{\gamma-1}},\quad \frac{A}{A^*}=\frac{1}{M}\left[\frac{2}{\gamma+1}\left(1+\frac{\gamma-1}{2}M^2\right)\right]^{\frac{\gamma+1}{2(\gamma-1)}}$$

Mach angle $\mu=\arcsin(1/M)$ and Prandtl-Meyer angle $\nu(M)$ defined for $M>1$. Source: Devenport, VT AOE 3114.

FIG. 01 — converging-diverging (Laval) nozzle, sonic throat A*, subsonic in / supersonic out

Mach number M Specific heat ratio γ

T₀/T	—
p₀/p	—
ρ₀/ρ	—
A/A*	—
Mach angle μ	—
Prandtl-Meyer ν	—

import numpy as np, matplotlib.pyplot as plt
M = np.linspace(0.01, 3, 300)
T0_T = 1 + (gamma-1)/2*M**2
plt.plot(M, T0_T); plt.scatter([M_input],[T0_T_input], color='red', zorder=3)

FIG. 02 — T₀/T vs. Mach number, current M marked in red

2.2 · Normal shock wave relations

[2.2] Normal shock wave relations

$$M_2^2=\frac{2+(\gamma-1)M_1^2}{2\gamma M_1^2-(\gamma-1)},\quad \frac{p_2}{p_1}=1+\frac{2\gamma}{\gamma+1}(M_1^2-1)$$

$$\frac{p_{02}}{p_{01}}=\left(\frac{(\gamma+1)M_1^2}{(\gamma-1)M_1^2+2}\right)^{\frac{\gamma}{\gamma-1}}\left(\frac{\gamma+1}{2\gamma M_1^2-(\gamma-1)}\right)^{\frac{1}{\gamma-1}}$$

Source: Devenport, VT AOE 3114 Compressible Aerodynamics.

FIG. 02 — stationary normal shock: upstream supersonic, downstream subsonic, denser flow lines after the jump

Upstream Mach M₁ (>1) Specific heat ratio γ

M₂	—
p₂/p₁	—
ρ₂/ρ₁	—
T₂/T₁	—
p₀₂/p₀₁	—

FIG. 03 — M₂ and property ratios vs. upstream Mach M₁ (1 ≤ M₁ ≤ 6), current M₁ marked in red

2.3 · Oblique shock wave relations (θ-β-M)

[2.3] Oblique shock wave relations (θ-β-M)

$$\tan\theta=2\cot\beta\left[\frac{M_1^2\sin^2\beta-1}{M_1^2(\gamma+\cos2\beta)+2}\right]$$

Solved numerically for weak/strong shock angle β given M₁, γ, deflection θ. Source: Devenport, VT AOE 3114.

FIG. 04 — supersonic wedge: flow turn angle θ, oblique shock angle β, both measured from the upstream flow direction

Upstream Mach M₁ (>1) Flow turn angle θ (deg) Specific heat ratio γ

β weak	—
β strong	—
M₂ (weak)	—
p₂/p₁ (weak)	—

import numpy as np, matplotlib.pyplot as plt
theta = np.linspace(0.1, theta_max, 200)
beta_weak = solve_theta_beta_M(M1, theta, gamma)
plt.plot(theta, beta_weak); plt.scatter([theta_input],[beta_input], color='red', zorder=3)

FIG. 05 — weak shock angle β vs. deflection θ at fixed M₁, current θ marked in red

2.4 · Conical shock wave relations (Taylor-Maccoll)

[2.4] Conical shock wave relations (Taylor-Maccoll)

$$\frac{\gamma-1}{2}\left(1-V_r'^2-V_\theta'^2\right)\left(2V_r'+V_\theta'\cot\theta+\frac{dV_\theta'}{d\theta}\right)+V_\theta'\left(V_r'V_\theta'+\frac{dV_\theta'}{d\theta}\right)=0,\quad V_\theta'=\frac{dV_r'}{d\theta}$$

Axisymmetric supersonic flow past a rigid cone: unlike a 2D wedge, streamlines behind the (conical, attached) shock keep turning as they approach the cone surface — the shock angle β, cone half-angle θ_c, and surface Mach M_c are related through the Taylor-Maccoll ODE, not a closed-form expression. Solved here by shooting: guess β, integrate the ODE pair (V′_r, V′_θ) from the post-shock state at θ=β inward, and bisect on β until V′_θ=0 lands exactly at θ=θ_c (flow tangent to the cone surface). Source: Virginia Tech AOE 3114 (Devenport); NACA Report 1135 tables for cross-check.

FIG. 04b — axisymmetric cone: shock envelope at angle β, cone surface at half-angle θ_c < β, streamlines between them curve smoothly (no discrete second turn, unlike a 2D wedge's uniform post-shock state)

Upstream Mach M₁ (>1) Cone half-angle θ_c (deg) Specific heat ratio γ

Shock angle β	—
Surface Mach M_c	—
Surface pressure ratio p_c/p₁	—
Max attached θ_c at this M₁	—

from scipy.integrate import solve_ivp
from scipy.optimize import brentq
sol = solve_ivp(taylor_maccoll_rhs, [beta, theta_c], [Vr0, Vth0])
beta_solution = brentq(lambda b: theta_at_Vtheta_zero(b) - theta_c_target, mu, beta_max)

FIG. 04c — shock angle β vs. cone half-angle θ_c (weak-shock branch) at fixed M₁, current θ_c marked in red; curve terminates at θ_c,max (shock detachment)

2.5 · Fanno flow with friction

[2.5] Fanno flow with friction

$$\frac{4fL^*}{D}=\frac{1-M^2}{\gamma M^2}+\frac{\gamma+1}{2\gamma}\ln\left(\frac{(\gamma+1)M^2}{2+(\gamma-1)M^2}\right)$$

Constant-area adiabatic duct with wall friction; * denotes the sonic (choked) reference state. Source: Devenport, VT AOE 3114.

FIG. 05 — constant-area duct with wall friction: Mach number rises toward the choked length L*

Mach number M Friction factor f Duct diameter D (m) Specific heat ratio γ

4fL*/D	—
L* max length	—
p/p*	—
T/T*	—
p₀/p₀*	—

import numpy as np, matplotlib.pyplot as plt
M = np.linspace(0.05, 1, 300)
fLD = (1-M**2)/(gamma*M**2) + (gamma+1)/(2*gamma)*np.log((gamma+1)*M**2/(2+(gamma-1)*M**2))
plt.plot(M, fLD); plt.scatter([M_input],[fLD_input], color='red', zorder=3)

FIG. 06 — 4fL*/D vs. Mach number, current M marked in red

2.6 · Rayleigh flow with heat transfer

[2.6] Rayleigh flow with heat transfer

$$\frac{T_0}{T_0^*}=\frac{2(\gamma+1)M^2\left(1+\frac{\gamma-1}{2}M^2\right)}{(1+\gamma M^2)^2},\quad \frac{p}{p^*}=\frac{\gamma+1}{1+\gamma M^2}$$

Constant-area frictionless duct with heat addition/rejection; * denotes the thermally choked (M=1) state. Source: Devenport, VT AOE 3114.

FIG. 06 — constant-area frictionless duct: heat addition drives flow toward thermal choking

Mach number M Specific heat ratio γ

T₀/T₀*	—
T/T*	—
p/p*	—
p₀/p₀*	—

import numpy as np, matplotlib.pyplot as plt
M = np.linspace(0.05, 3, 300)
T_Tstar = M**2 * ((gamma+1)/(1+gamma*M**2))**2
plt.plot(M, T_Tstar); plt.scatter([M_input],[T_Tstar_input], color='red', zorder=3)

FIG. 07 — T/T* vs. Mach number, current M marked in red (peaks near M=1/\u221a\u03b3, falls off toward thermal choke)

2.7 · Air-breathing propulsion cycle

[2.7] Air-breathing propulsion cycle (ideal turbojet/ramjet Brayton cycle)

$$F_{\text{spec}}=\sqrt{2c_pT_a\left(\tau_\lambda-\tau_r(\tau_c-1)-\frac{\tau_\lambda}{\tau_r\tau_c}\right)}-V_0,\quad \tau_r=1+\frac{\gamma-1}{2}M_0^2,\quad \tau_c=\pi_c^{\frac{\gamma-1}{\gamma}},\quad \tau_\lambda=\frac{T_{04}}{T_a}$$

Ideal (no losses) turbojet/ramjet Brayton cycle: diffuser and compressor bring station 0 (freestream) to station 3 isentropically, the burner adds heat at constant stagnation pressure to T₀₄, and the turbine/nozzle expand back to ambient. Setting π_c=1 reduces this to the ramjet case (no compressor, no turbine). T_a is the ambient static temperature at the flight altitude (ISA troposphere/lower-stratosphere approximation below). Source: Virginia Tech AOE 3114 air-breathing propulsion sub-calculator (Devenport); ideal-cycle formulation per Hill & Peterson, Mechanics and Thermodynamics of Propulsion.

FIG. 07 — turbojet station numbering (0 freestream, 2 diffuser exit, 3 compressor exit, 4 burner exit/turbine inlet, 5 turbine exit, 9 nozzle exit); π_c=1 collapses stations 2-3 to a bare duct (ramjet)

Flight Mach M₀ Altitude (km) Compressor pressure ratio π_c (=1 for ramjet) Turbine inlet temp T₀₄ (K) Specific heat ratio γ

Ambient T_a (ISA)	—
Specific thrust F_spec	—
TSFC	—
Thermal efficiency η_th	—
Propulsive efficiency η_p	—
Overall efficiency η_o	—

import numpy as np, matplotlib.pyplot as plt
pi_c = np.linspace(1, 40, 200)
Fspec, TSFC = ideal_turbojet_cycle(M0, altitude, pi_c, T04, gamma)
fig, (ax1, ax2) = plt.subplots(1, 2)
ax1.plot(pi_c, Fspec); ax2.plot(pi_c, TSFC*1e6)

FIG. 08 — specific thrust F_spec and TSFC vs. compressor pressure ratio π_c at fixed M₀, altitude, T₀₄; current π_c marked in red