How Isentropic Flow Relations Work
Isentropic flow describes a gas moving through a duct, nozzle, or over a body with no heat added or removed and no friction or shock losses, so entropy stays constant along each streamline. Under that assumption, the local (static) properties of the gas — temperature T, pressure p, and density ρ — can be related to the stagnation (or total) properties T₀, p₀, and ρ₀ that the same gas would have if it were brought to rest isentropically, purely as a function of the local Mach number M and the gas's specific heat ratio γ = cp/cv. These relations are the backbone of nozzle design, wind-tunnel calibration, pitot-static airspeed measurement, and compressible-flow textbook tables.
Deriving the stagnation temperature, pressure, and density ratios
Starting from the steady-flow energy equation for an adiabatic, calorically perfect gas, cpT + V²/2 = cpT₀ leads directly to the temperature ratio:
- Temperature: T₀/T = 1 + [(γ-1)/2]·M², where M = V/a is the local velocity divided by the local speed of sound.
- Pressure: because an isentropic process for an ideal gas obeys p/ρ^γ = constant (equivalently p ∝ T^(γ/(γ-1))), the pressure ratio is the temperature ratio raised to the γ/(γ-1) power: p₀/p = (T₀/T)^(γ/(γ-1)).
- Density: similarly, ρ ∝ T^(1/(γ-1)), so ρ₀/ρ = (T₀/T)^(1/(γ-1)).
All three ratios equal 1 at M = 0 (static equals stagnation when the gas is already at rest) and increase monotonically as M grows — a faster-moving gas has more kinetic energy to "give back" as it is decelerated to rest.
The area-Mach number relation and practical limits
Combining the isentropic relations with mass continuity (ρAV = constant) through a variable-area duct gives the area ratio A/A*, where A* is the cross-sectional area at which the local flow would be exactly sonic (M = 1) — the throat of a converging-diverging nozzle:
A/A* = (1/M) · [ (2/(γ+1)) · (1 + (γ-1)/2·M²) ] ^ [(γ+1) / (2(γ-1))]
Because this expression depends on M², the same A/A* value corresponds to both a subsonic and a supersonic Mach number; you need to know which branch of the nozzle you're in (converging vs. diverging past the throat) to pick the physically correct root. These relations assume a calorically perfect gas with constant γ, no chemical reactions or dissociation, and flow free of shock waves or strong viscous losses — good approximations for air below roughly Mach 5, but real-gas and vibrational-excitation effects require corrected γ values at hypersonic speeds or high temperatures.