Gas Dynamics · Compressible Flow
U.S. Standard Atmosphere, isentropic, normal shock, oblique shock, Fanno, Rayleigh, and isothermal relations. Pick any supported input variable and the remaining quantities are calculated.
Valid from −2.0 to 84.852 km geopotential altitude. The model uses dry air, hydrostatic equilibrium, and the 1976 standard temperature layers.
Total (stagnation) conditions are constant along an isentropic streamline; ratios shown are static-to-total.
Valid for a calorically perfect gas; M₁ ≥ 1 is required for a physically realizable (entropy-increasing) shock. Here ΔV/a₁ = (V₁ − V₂)/a₁, so the reported velocity decrease is positive.
The θ-β-M relation is solved numerically; results use the normal-shock relations on the wave-normal Mach number M₁sinβ.
Adiabatic flow with friction in a constant-area duct. Starred (*) values are the properties at the point where M = 1 would be reached by friction alone. The parameter 4f̄Lmax/D uses the Fanning factor f̄ and is numerically equal to fDLmax/D when fD is the Darcy factor.
Frictionless flow with heat addition/rejection in a constant-area duct. Starred (*) values are the sonic (M = 1) reference state.
Steady, constant-area perfect-gas flow with friction and heat transfer maintaining constant static temperature (T/T* = 1). Starred (*) properties refer to isothermal choking at Mc = 1/√γ, rather than M = 1. The remaining friction parameter 4fFLmax/D equals fDLmax/D; fF is the Fanning factor and fD is the Darcy factor. For varying friction, this parameter represents the integral of 4fF dx/D. Stagnation properties are defined by local isentropic deceleration.
Ideal, steady, adiabatic quasi-one-dimensional flow in a converging–diverging nozzle, with at most one internal normal shock. No boundary-layer separation model. Reservoir = state 1; throat = state 2; immediately before/after a shock = states 3/4; nozzle exit = state 5. Shock location is reported as A₃/A₂; an axial location requires a nozzle contour. For a shock at the exit, state 5 denotes the inner exit plane before the shock and state 4 the external post-shock state. External jet adjustments are classified but not solved.
Constant-area perfect-gas flow. Friction parameter = 4fF L/D = fD L/D (Fanning/Darcy). Fanno and isothermal friction must be nonnegative and moves Mach toward the process critical value. Rayleigh heat addition/removal is restricted to the same side of M = 1; q is positive for heat added to the gas. Rayleigh parameter input is q/(cp T₀₁), with cp = γR/(γ−1). Isothermal choking occurs at M = 1/√γ. The two-state calculation does not insert a shock or continue through choking.
Centered, isentropic expansion of a calorically perfect gas around a convex corner. Turning angle is the nonnegative expansion magnitude. Compression turns require an oblique-shock solution. Stagnation pressure and temperature are conserved. Angles are in degrees. A finite downstream solution requires θ < ν∞ − ν₁.
Ideal aligned probe. For M ≤ 1 the probe reads free-stream stagnation pressure; for M > 1 it reads stagnation pressure behind a normal shock. Impact pressure q꜀ = pprobe − p₁ differs from dynamic pressure q = ½ρV². Equivalent airspeed uses ρref = 1.225 kg/m³. Calibrated airspeed uses the ideal pitot relation at standard sea-level air conditions (γref = 1.4, Rref = 287.05287 J/(kg·K), Tref = 288.15 K, pref = 101325 Pa), including the supersonic branch; these reference constants remain standard air when the working-gas γ changes. Instrument/position corrections are not modeled.