Compressible Flow Tables: Method

The Report 1135 Relations, the Inverse Solutions and the Fanno and Rayleigh Lines

Scope

The tool is the working form of NACA Report 1135, Equations, Tables, and Charts for Compressible Flow (Ames Research Staff, 1953), the compilation that every gas-dynamics table since has reprinted. It evaluates, for a calorically perfect gas of any specific-heat ratio γ, the relations the report prints for continuous (isentropic) one-dimensional flow, normal and oblique shock waves and the Prandtl-Meyer expansion, and adds the two constant-area problems the report leaves out, the Fanno line (adiabatic flow with friction) and the Rayleigh line (frictionless flow with heat addition), in the forms of Shapiro's Dynamics and Thermodynamics of Compressible Fluid Flow. Each mode takes any of the quantities the report tabulates and solves for the Mach number first, prints the whole row of ratios with the report's equation numbers and the substituted numbers, lists the neighbouring Mach numbers in the report's table layout, draws the chart of the computed case, and, when a static pressure and temperature are entered, gives the dimensional state. The report's imperfect-gas corrections (its Charts 9 to 24, for air above about 1,000 K where the vibrational heat capacity matters) are not implemented: the tool is a perfect-gas table, exact in that model and valid for air to about Mach 5 at ordinary total temperatures.

Notation

The report's symbols are kept throughout, because the equation numbers quoted on the page refer to them: subscript t marks total (stagnation) conditions, an asterisk the sonic state (M = 1), β = √|M² − 1|, μ = sin⁻¹(1/M) the Mach angle, ν the Prandtl-Meyer angle, δ the flow-deflection angle across an oblique shock (the wedge half-angle) and θ the shock-wave angle measured from the upstream flow direction. Most textbooks after Liepmann and Roshko write θ for the deflection and β for the shock angle, so their "θ-β-M diagram" is the report's Chart 2 with the axis labels exchanged; the chart drawn here is labelled the report's way, δ across and θ up. V_m is the maximum speed, reached by expanding to zero absolute temperature, and a_t, a* and V_m are related by eq. 33: (a_t/a*)² = (γ+1)/2, (V_m/a*)² = (γ+1)/(γ−1).

Isentropic flow

For adiabatic flow of a perfect gas the energy equation c_pT + V²/2 = c_pT_t gives T/T_t = [1 + (γ−1)/2 M²]⁻¹ (eq. 43); the isentropic relation p/ρ^γ = constant then gives p/p_t = [1 + (γ−1)/2 M²]^{−γ/(γ−1)} (eq. 44) and ρ/ρ_t = [1 + (γ−1)/2 M²]^{−1/(γ−1)} (eq. 45). The dynamic pressure is q = γpM²/2 (eq. 47), so q/p_t = (γM²/2)(p/p_t) (eq. 48), a quantity that rises from zero to a maximum of 0.4312 at M = √2 (for every γ) and falls again, which is why its inverse needs a subsonic and a supersonic branch. V/a* (eq. 50), often written M*, is the speed referred to the critical speed of sound; unlike M it stays finite, tending to √[(γ+1)/(γ−1)] = 2.4495 for air. The stream-tube area ratio comes from continuity ρVA = ρ*a*A* (eq. 79) as A*/A = [(γ+1)/2]^{(γ+1)/(2(γ−1))} M [1 + (γ−1)/2 M²]^{−(γ+1)/(2(γ−1))} (eq. 80); the tool prints its reciprocal A/A*, which is 1 at the throat and has a subsonic and a supersonic Mach number for every value above 1. The ratios to total conditions invert in closed form (M = √{2/(γ−1) [(p/p_t)^{−(γ−1)/γ} − 1]} and the like); A/A*, q/p_t and every inverse of the later sections are solved by bisection, 200 halvings on the branch chosen, which places the Mach number to machine precision.

Normal shock waves

Conservation of mass, momentum and energy across a stationary normal shock (eqs. 84 to 86), with the entropy condition that rules out expansion shocks, gives Prandtl's relation u₁u₂ = a*² (eq. 89), the Rankine-Hugoniot relations between the pressure and density ratios (eqs. 90 to 92), and the working forms in the upstream Mach number: p₂/p₁ = [2γM₁² − (γ−1)]/(γ+1) (eq. 93), ρ₂/ρ₁ = u₁/u₂ = (γ+1)M₁²/[(γ−1)M₁² + 2] (eq. 94), T₂/T₁ as their quotient (eq. 95), M₂² = [(γ−1)M₁² + 2]/[2γM₁² − (γ−1)] (eq. 96), and the total-pressure ratio p_t2/p_t1 (eq. 99), whose logarithm is the entropy rise, Δs/R = −ln(p_t2/p_t1) (eq. 101). The total temperature does not change (eq. 88). Equation 100, p_t2/p₁, is the Rayleigh supersonic pitot formula: a pitot tube in a supersonic stream sits behind its own bow shock and reads the total pressure after the shock, so the Mach number of a supersonic wind tunnel or aircraft is found from the ratio of pitot to static pressure through this equation, which the tool inverts. The report also writes every ratio in the static-pressure ratio ξ = p₂/p₁ (eqs. 105 to 112); eq. 105, M₁² = [(γ+1)ξ + (γ−1)]/(2γ), is the closed-form inverse the tool uses when the pressure ratio is given, and eq. 96 inverts directly when M₂ is given. Density ratios approach (γ+1)/(γ−1) = 6 and M₂ approaches √[(γ−1)/(2γ)] = 0.378 as M₁ grows without bound, so a density ratio of 6 or more, a downstream Mach number outside 0.378 to 1, or any ratio below 1 is refused as impossible.

Oblique shock waves

An oblique shock is a normal shock to the velocity component normal to it, the tangential component being unchanged (eqs. 115 to 124), so with M₁ sin θ in place of M₁ the normal-shock ratios give p₂/p₁, ρ₂/ρ₁, T₂/T₁ and p_t2/p_t1 directly (eqs. 128 to 130, 142), and M₂ sin(θ − δ) is the downstream normal Mach number (eq. 131). The deflection follows from the kinematics as cot δ = tan θ [(γ+1)M₁²/(2(M₁² sin²θ − 1)) − 1] (eq. 138). Given the deflection instead of the shock angle, the report notes that "no convenient explicit relations exist" and offers the cubic in sin²θ of eq. 150; the tool solves eq. 138 for θ by bisection on each branch, between the Mach angle and the angle of maximum deflection for the weak shock, and between that angle and 90° for the strong one. The maximum deflection δ_max (22.97° at M₁ = 2 in air) is found by a golden-section search of eq. 138; a wedge steeper than δ_max cannot hold an attached shock and the page says so instead of computing. The weak shock is the one observed on an attached wedge or cone in a free stream; the strong shock occurs only when the downstream pressure is imposed high enough, as behind a blunt body near its axis. The line M₂ = 1, the sonic limit, lies slightly on the strong side of the maximum-deflection locus, so a small range of weak-branch shocks has subsonic flow behind it. The page draws Chart 2 of the report, θ against δ for the report's set of Mach numbers with the given one in black and the computed shock marked, and the shock polar of eq. 164a (Busemann's hodograph, the report's Figure 3): the locus of the downstream velocity (u₂/V₁, v₂/V₁) over all shock angles, from the normal shock on the axis to the Mach wave at u₂ = V₁, with the sonic circle V = a* separating weak from strong solutions. For a cone the shock is weaker than for a wedge of the same half-angle; the report's Charts 5 to 8 give the cone solution (Taylor-Maccoll), which the tool does not reproduce.

Prandtl-Meyer expansion

A supersonic stream turning around a convex corner expands isentropically through a fan of Mach waves. Integrating the turning from M = 1 gives the Prandtl-Meyer angle ν = √[(γ+1)/(γ−1)] tan⁻¹√[(γ−1)/(γ+1)(M² − 1)] − tan⁻¹√(M² − 1) (eq. 171c), with the maximum ν_max = (√[(γ+1)/(γ−1)] − 1) × 90° = 130.45° for air (eq. 172) reached only at infinite Mach number and zero pressure. A stream at M₁ turning through Δν reaches the Mach number whose ν equals ν(M₁) + Δν, read back from the same function, which the tool inverts by bisection; the static ratios then follow from the isentropic relations, since the expansion is isentropic (eq. 173 writes p/p_t in ν directly). A compression through a small angle may be treated the same way with the sign reversed; the report observes that this gives better accuracy than the weak-shock series in δ² and is nearly as good as the series through δ³.

Fanno line

Steady adiabatic flow in a constant-area duct with wall friction has constant total temperature and mass flux but rising entropy; the states it can pass through lie on the Fanno line of the T-s plane, whose entropy maximum is at M = 1. Friction therefore drives a subsonic stream faster and a supersonic one slower, both toward the sonic point, and a duct long enough to reach it is choked. Referring every property to the sonic state the stream would reach, T/T* = (γ+1)/[2 + (γ−1)M²] from the energy equation, V/V* = M√(T/T*) and ρ/ρ* its reciprocal from continuity, p/p* from the gas law, and p_t/p_t* = (1/M){[2 + (γ−1)M²]/(γ+1)}^{(γ+1)/(2(γ−1))}, the same function of M as the isentropic A/A*. Integrating the momentum equation gives the duct length to choking, fL*/D = (1 − M²)/(γM²) + (γ+1)/(2γ) ln{(γ+1)M²/[2 + (γ−1)M²]}, written here with the Darcy friction factor f (the Moody-chart value, four times the Fanning factor used in some tables, whose column is 4fL*/D). The length between two sections is the difference of their fL*/D values. On the supersonic branch fL*/D tends to a finite limit as M grows, (γ+1)/(2γ) ln[(γ+1)/(γ−1)] − 1/γ = 0.8215 for air; a longer supersonic duct cannot stay shock-free, and a normal shock forms inside it. The tool returns the Mach number for a given fL*/D on either branch, and with a static state entered prints the choking section's pressure, temperature and velocity.

Rayleigh line

Frictionless constant-area flow with heat addition keeps the mass flux and the impulse p + ρV² constant; its states lie on the Rayleigh line, which also has its entropy maximum at M = 1, so heating drives both subsonic and supersonic streams toward sonic conditions and cooling drives them away. The momentum equation alone gives p/p* = (1 + γ)/(1 + γM²); the gas law and continuity then give T/T* = M²(p/p*)², ρ/ρ* = (1 + γM²)/[(1 + γ)M²], and the total conditions follow with the isentropic ratios, T_t/T_t* = 2(γ+1)M²[1 + (γ−1)/2 M²]/(1 + γM²)². The total-temperature ratio measures the heat: q = c_p(T_t2 − T_t1) between two sections, and c_p(T_t* − T_t) is the heat that chokes the flow from the given section. Two features are worth remembering: the static temperature reaches its maximum (γ+1)²/(4γ) = 1.0286 T* at M = 1/√γ = 0.845 in air, so heating a subsonic stream between that Mach number and 1 lowers its temperature while raising its total temperature; and T_t/T_t* tends to (γ² − 1)/γ² = 0.4898 as M grows, so a supersonic stream can absorb at most about twice its total enthalpy before choking. The inverse of T_t/T_t* is solved on the branch chosen.

Dimensional state

With the static pressure and temperature of the stream at the given Mach number, the density follows from ρ = p/(RT), the speed of sound from a = √(γRT) (eq. 29b), the velocity from V = Ma, the dynamic pressure from q = γpM²/2 (eq. 31b), and the total conditions from the isentropic ratios. Across a shock the downstream static state is the upstream one times the ratios, and the downstream velocity is M₂a₂. The gas constant of air (53.35 ft·lbf/(lbm·°R) = 287.05 J/(kg·K)) and helium (386.0 = 2,077) are built in; for another gas enter R in the unit of the selected system. Temperatures are plain numbers in °F or °C and are turned into absolute temperatures inside; they are not changed by the unit switch. The pressure goes through the site's unit dictionary.

Accuracy and sources

The library reproduces every column of Report 1135 Table I (subsonic) and Table II (supersonic) at the 41 Mach numbers read from the report's pages, within one unit of the last printed digit (five of the 447 cells differ by that one unit, the report's values having been computed on desk calculators in 1947 to 1953), nine points digitized from Chart 2 to within 0.5°, and the closed-form limits of every relation. An independent numpy and scipy check takes different routes to the same numbers: the isentropic ratios from the integrated energy equation, the normal shock from the three conservation laws solved numerically, the oblique shock from the kinematics of the tangential velocity with a root finder, the Prandtl-Meyer angle and the Fanno length from quadrature of their differential relations, and the Rayleigh sonic state from the conservation of mass flux and impulse; agreement is to the sixth figure. The relations hold for any γ between 1 and 5/3; the report's own tables are for γ = 7/5 only.