Normal Shock Relations Simulator — Compressible Flow
Visualize the state across a normal shock in a supersonic stream with the Rankine-Hugoniot relations. Adjust the upstream Mach number, specific-heat ratio, temperature and pressure to see how the downstream Mach, pressure, temperature and density ratios change.
Parameters
Upstream Mach M_1
—
Specific-heat ratio γ
—
Upstream static temperature T_1
K
Upstream static pressure P_1
kPa
Perfect gas, adiabatic, 1-D, steady flow is assumed. The shock is physical only for M_1 ≥ 1.
Results
—
Downstream Mach M_2
—
Downstream static pressure P_2
—
Downstream static temperature T_2
—
Density ratio ρ_2/ρ_1
Shock Front and Upstream/Downstream Flow
Left = upstream (cool color, long arrows = high speed) / center vertical line = shock / right = downstream (warm color, short arrows = low speed)
Ratios vs M_1 (P_2/P_1, T_2/T_1, M_2)
Red = P_2/P_1 (left axis) / orange = T_2/T_1 (left axis) / blue = M_2 (right axis) / yellow vertical line = current M_1
Theory & Key Formulas
For a 1-D steady normal shock in a perfect gas, the Rankine-Hugoniot relations follow from conservation of mass, momentum and energy. The upstream Mach M_1 and the specific-heat ratio γ determine the entire downstream state.
Downstream Mach M_2 (asymptotes to 1/√((γ−1)/(2γ)) as M_1 grows):
The shock vanishes at M_1 = 1 and grows stronger as M_1 increases. Physically meaningful solutions exist only in the supersonic regime M_1 \gt 1.
What is the Normal Shock Relations Simulator
🙋
In photos of supersonic fighter jets you sometimes see those white cone-shaped clouds around the airframe. Are those shock waves?
🎓
Yes — those are condensation clouds tied to shocks. The cone is an oblique shock, but the underlying physics is the same as for the normal shock relations here. Whenever a supersonic gas has to slow down to subsonic, it does so across a thin sheet where pressure, temperature and density jump up and the Mach number drops. With M_1 = 2 in the simulator, the downstream Mach is 0.577 — try it and watch the value appear in the M_2 stat card.
🙋
So the flow downstream of a shock is always subsonic?
🎓
For a normal shock, always. From $M_2^2 = (1 + \tfrac{\gamma-1}{2}M_1^2)/(\gamma M_1^2 - \tfrac{\gamma-1}{2})$, the larger M_1 gets, the smaller M_2 becomes — but for γ = 1.4 it asymptotes to 1/√7 ≈ 0.378 even as M_1 → ∞. Push M_1 to 5 in the simulator and you will see M_2 hovering around 0.4.
🙋
Looking at the curves, the pressure ratio shoots up forever but the density ratio seems to flatten out. Why?
🎓
Sharp eye. The density ratio ρ_2/ρ_1 has a hard ceiling of (γ+1)/(γ−1), which for γ = 1.4 is exactly 6. The pressure ratio, on the other hand, grows like M_1² without bound. So a strong shock has "huge pressure but only moderately denser gas", which means the temperature must skyrocket. That is exactly why a re-entry vehicle's nose becomes a glowing plasma sheath.
🙋
And why does γ go from 1.20 to 1.67 in the slider?
🎓
Different gases have different γ. Monatomic gases (helium, argon) are around 1.67, air is 1.40, and triatomic gases like CO₂ sit near 1.30. At very high temperatures, vibrational modes and dissociation push the effective γ down to about 1.20. Re-entry CFD and combustion-shock studies routinely vary γ, so the simulator lets you explore it. Reduce γ and you will see the density-ratio ceiling rise — at γ = 1.20 the limit jumps to 11.
Frequently Asked Questions
At M_1 = 1 the shock degenerates into a Mach wave with zero strength: substituting M_1 = 1 in the relations gives P_2/P_1 = 1, ρ_2/ρ_1 = 1, T_2/T_1 = 1 and M_2 = 1, so there is effectively no shock. Physically meaningful shock solutions exist only for M_1 \gt 1, which is why the simulator's slider starts at 1.05.
No — the second law of thermodynamics forbids it. If you assume a shock at M_1 \lt 1, the Rankine-Hugoniot relations give a solution in which entropy decreases, which is impossible in an adiabatic system. So a normal shock can only run from supersonic to subsonic. The reverse "expansion shock" that would accelerate a flow does not exist either; supersonic acceleration must instead happen via a continuous Prandtl-Meyer expansion.
A converging-diverging (Laval) nozzle smoothly accelerates the flow past the throat to supersonic when run at its design pressure. If the back pressure is higher than the design value, the supersonic exit flow has to be brought down to subsonic to match the downstream pressure — and this happens through a normal shock somewhere in the diverging section. As the back pressure drops, the shock moves toward the exit and eventually disappears.
The entropy change is Δs/R = ln[(P_2/P_1)·(ρ_1/ρ_2)^γ], and it grows rapidly with shock strength (M_1). At M_1 = 2, γ = 1.4 it is about 0.33, giving a stagnation-pressure ratio P_02/P_01 = exp(−Δs/R) ≈ 0.72. By M_1 = 3, Δs/R ≈ 1.04 and P_02/P_01 ≈ 0.35 — a single shock has lost more than half of the available stagnation pressure.
Real-World Applications
Supersonic aircraft and rocket nozzles: In fighter jets and rockets, hot high-pressure gas from the combustion chamber is accelerated to supersonic speeds in a Laval nozzle. Operating off-design causes a normal shock to stand inside the diverging section, causing stagnation-pressure loss and reduced thrust. The "shock diamonds" seen in ground tests are alternating oblique shocks and expansions caused by exit/ambient pressure mismatch. Both the normal-shock and oblique-shock (θ-β-M) relations are core design tools.
Atmospheric re-entry thermal protection: Capsule-type re-entry vehicles enter the atmosphere at altitudes of 50-80 km with M_1 ≈ 20-30 hypersonic flow. A strong detached bow shock forms in front of the nose. Applying the normal-shock relations along the stagnation streamline gives stagnation temperatures of several thousand to ten thousand kelvin, hot enough to dissociate and partially ionize the air. Designing the thermal protection system (TPS) — ablators or high-temperature ceramic tiles — depends critically on accurate post-shock temperature and pressure predictions.
Supersonic intakes (air-breathers): Compressors of jet engines only work subsonic, so on supersonic aircraft the intake must decelerate the flow to subsonic. The simplest pitot intake does this with a single normal shock, but the stagnation-pressure loss is large. The SR-71 and Concorde use multi-shock intakes that decelerate the flow through several oblique shocks first and finish with a weak normal shock — minimizing the total-pressure loss is what gives those intakes their characteristic appearance.
Shock tubes and explosion propagation: The shock tube — widely used in defense, aerospace and basic-physics research — generates a normal shock inside a duct by suddenly rupturing a diaphragm separating high- and low-pressure sections. The Rankine-Hugoniot relations let designers choose diaphragm pressure ratio and gas composition to hit any target M_1. They are used for hypersonic combustion studies, dynamic material testing and reduced-scale models of interstellar shocks.
Common Misconceptions and Cautions
The most common error is to think that the total (stagnation) temperature rises across a shock. A normal shock is adiabatic, so by energy conservation the total temperature T_0 is exactly preserved across it. What rises is the static temperature T, because the velocity drops sharply and kinetic energy is converted into internal energy. When the simulator shows T_2 = 489 K, that is the static temperature; the stagnation temperature stays at T_0 = T_1(1+(γ−1)/2·M_1²) = 290·1.8 = 522 K on both sides.
The next pitfall is to overlook that the density ratio has a hard upper bound. The relations show that ρ_2/ρ_1 cannot exceed (γ+1)/(γ−1), which equals exactly 6 for γ = 1.4 no matter how large M_1 becomes. The pressure ratio P_2/P_1, by contrast, grows without bound as M_1². This "pressure unbounded, density saturated" mismatch is what drives the post-shock temperature so high in strong shocks. The simulator's curve clearly shows T_2/T_1 and P_2/P_1 separating dramatically as M_1 approaches 5.
One more caution is the thermodynamic restriction that the relations only run in the supersonic-to-subsonic direction. Mathematically the Rankine-Hugoniot equations also produce solutions for M_1 \lt 1, but those solutions decrease entropy and are physically forbidden. There is no such thing as a "shock travelling from downstream" or an "expansion shock" that boosts subsonic flow to supersonic. The simulator's M_1 slider starts at 1.05 specifically because only the supersonic regime is meaningful.