Fanno Flow

Fanno Flow in a Constant-Area Adiabatic Duct (aerodynamics/high-speed/fanno-flow)

Use when the task is the Fanno flow of a steady adiabatic perfect-gas flow in a constant-area duct with wall friction: the Fanno-line integral fL*/D, the friction-duct choking length, the downstream Mach number recovered on the inlet branch for a given friction parameter, the friction required to choke a duct of given length and diameter, and the total-pressure and static ratios along the duct to its sonic reference state. This leaf implements the standard Fanno relations (the same compressible-flow framework that NACA TR-824 tabulates) in pure Python, stdlib only, with a deterministic 1-D bisection for the branch inversion. It pairs with aerodynamics/high-speed/isentropic-flow-relations as the frictionless complement (that leaf's area-Mach and choked mass flow have no wall-friction term and no Fanno line), with aerodynamics/high-speed/normal-shock for shock-jump stagnation loss at a station, and with aerodynamics/high-speed/shock-tube, which supplies the deterministic-bisection precedent. It does NOT do Rayleigh heat-addition flow of a frictionless duct (rayleigh-flow, same wave; friction with heat addition is out of scope), boundary-layer skin friction and heating on surfaces (flat-plate-skin-friction-heating), or the unsteady shock-tube wave system (shock-tube).

Domain quick reference

• Fanno-line integral: phi(M) = fL*/D = (1 - M^2)/(gamma M^2) + ((gamma + 1)/(2 gamma)) ln((gamma + 1) M^2 / (2 + (gamma - 1) M^2)). phi is the friction parameter that chokes the duct from the Mach number M: its only zero and minimum sit at M = 1, it falls from infinity at M = 0 to 0 at M = 1 and rises toward the plateau ((gamma + 1)/(2 gamma)) ln((gamma + 1)/(gamma - 1)) - 1/gamma (0.8215 at gamma 1.4).
• Segment rule: a segment of friction parameter fL/D strictly below phi(M1) moves the Mach number toward 1 on the same branch, consuming fL/D = phi(M1) - phi(M2) of the choke margin. Subsonic friction accelerates the flow (M2 in (M1, 1)); supersonic friction decelerates it (M2 in (1, M1)). At fL/D = phi(M1) the exit is exactly sonic and beyond it the duct chokes internally (out of scope, the module raises).
• Choking length: L* = (D/f) fL*/D from the inlet Mach and Fanning factor; the friction factor that makes a duct of length L exactly choked is f_req = (D/L) fL*/D(M1).
• Starred total-pressure ratio: p0/p0* = (1/M) (2 (1 + (gamma - 1) M^2 / 2)/(gamma + 1))^((gamma + 1)/(2 (gamma - 1))), equal to 1 at M = 1 and growing on both branches away from it.
• Starred static ratios: T/T* = (gamma + 1)/(2 + (gamma - 1) M^2), p/p* = (1/M) sqrt((gamma + 1)/(2 + (gamma - 1) M^2)), rho/rho* = (1/M) sqrt((2 + (gamma - 1) M^2)/(gamma + 1)), all 1.0 at M = 1 and consistent with p/p* = (rho/rho*)(T/T*).
• Station ratios across a segment (one duct, one sonic reference): p02/p01 = (p0/p0*)2/(p0/p0*)1 below 1 on both branches, and for X in p, T, rho: X2/X1 = (X/X*)2/(X/X*)1. Entropy rise ds/R = -ln(p02/p01)

  0; the duct is adiabatic so T02/T01 = 1.

• Conventions: f is the Fanning (skin-friction) factor f = tau_w / (rho u^2 / 2); Darcy-factor tables tabulate (4 f) L/D, so divide f_D by 4 before calling. D is the hydraulic diameter. gamma = 1.4 air by default and a parameter elsewhere.

Workflow

1. Fix the duct state: inlet Mach number M1, Fanning friction factor f, hydraulic diameter D and gas gamma (fanno_flow_logic constants and guards).
2. Evaluate the Fanno-line integral with fanno_function(M1): the friction parameter fL*/D that would choke the duct from the inlet Mach.
3. Compute the friction-duct choking length with choke_length(M1, f, D): the duct run that brings the exit exactly to M = 1 (0.0 when the inlet is already sonic).
4. Form the segment friction parameter fL/D = f L / D for the duct length L and recover the downstream Mach number with downstream_mach(M1, fL/D) on the inlet branch: subsonic flow accelerates toward M = 1, supersonic flow decelerates toward M = 1.
5. Read the starred reference ratios at each station with total_pressure_ratio(M) and static_ratios(M): p0/p0*, T/T*, p/p* and rho/rho* to the sonic state of the same duct.
6. Assemble the station-to-station duct state with duct_state(M1, fL/D): the p02/p01 total-pressure loss and the p2/p1, T2/T1, rho2/rho1 quotients, then verify the entropy rise -ln(p02/p01) > 0 and the adiabatic identity T02/T01 = 1.
7. Find the friction required to choke with friction_to_choke(M1, L, D): the Fanning factor f_req that makes the given duct exactly choked, and compare it with the actual f to read the choke margin.
8. Confirm the deterministic checks with the contract test scripts/test_fanno_flow.py.

Worked example

Representative duct: D = 0.1 m, Fanning f = 0.005 (friction parameter gradient f/D = 0.05 per metre), gamma = 1.4. All values are real module outputs.

• Fanno-line integral: fL*/D(0.3) = 5.299253105091 and fL*/D(2.0) = 0.304996502581, so the 0.005-friction duct chokes from M = 0.3 in L* = 105.985062101823 m and from M = 2.0 in L* = 6.099930051630 m: the supersonic duct chokes about 17.4 times sooner at the same friction.
• Starred references: p0/p0* = 2.035065262346 at M = 0.3, 1.6875 at M = 2.0 and 1.0 at M = 1. At M = 0.3: T/T* = 1.178781925344, p/p* = 3.619057466836, rho/rho* = 3.070167084366. At M = 2.0: T/T* = 0.666666666667, p/p* = 0.408248290464, rho/rho* = 0.612372435696, the standard Fanno table row.
• Subsonic branch: M1 = 0.3 through L = 50 m gives fL/D = 0.005 * 50 / 0.1 = 2.5, below the choke value 5.299253105091. downstream_mach(0.3, 2.5) = 0.375749134774, the subsonic flow accelerated toward M = 1; the round-trip residual phi(M2) + fL/D - phi(M1) is 2.498e-12 and the margin left to sonic is phi(0.3) - 2.5 = 2.799253105091 = phi(M2). duct_state(0.3, 2.5): p02/p01 = 0.822735477531 (17.7% total-pressure loss), p2/p1 = 0.794420493240, T2/T1 = 0.990043659533, rho2/rho1 = 0.802409555973; the static pressure falls most and the static temperature almost stays (T02/T01 = 1.000000000000). Exit reference p0/p0* = 1.674320390423; ds/R = -ln(p02/p01) = 0.195120542447. friction_to_choke(0.3, 50, 0.1) = 0.010598506210, about double the actual 0.005, so the duct is not choked. downstream_mach(0.3, 0.0) = 0.300000000000 in the zero-friction limit.
• Supersonic branch: M1 = 2.0 through L = 3 m gives fL/D = 0.15, below the choke value 0.304996502581. downstream_mach(2.0, 0.15) = 1.552005392004, the supersonic flow decelerated toward M = 1; residual -8.171e-14, margin phi(2.0) - 0.15 = 0.154996502581. duct_state(2.0, 0.15): p02/p01 = 0.718851057841 (28.1% loss), p2/p1 = 1.420320685669, T2/T1 = 1.214784619331, rho2/rho1 = 1.169195479649: the static pressure, temperature and density all rise as supersonic flow slows, the mirror image of the subsonic branch. Exit p0/p0* = 1.213061160106; ds/R = 0.330101094541. friction_to_choke(2.0, 3, 0.1) = 0.010166550086.
• Read-off: friction pulls both branches toward M = 1 and spends total pressure doing it. A Mach-0.3 duct run must stay below fL/D 5.299 to avoid choking, a Mach-2.0 run below fL/D 0.305, and a duct above the choke parameter at its inlet is physically over-length for that inlet state (the module raises ValueError).

Verification

• Confirm fanno_function returns the Fanno table anchors: 5.299253105091 at M = 0.3, 1.069060312718 at 0.5, 0.304996502581 at 2.0 and 0.786830832907 at 10.0, with the gamma parameter honored (fanno_function(2.0, 1.3) = 0.357277365682).
• Confirm p0/p0* is minimal at M = 1 and the starred static ratios all equal 1.0 there, with p/p* = (rho/rho*)(T/T*) at every Mach.
• Confirm downstream_mach round-trips: phi(M2) + fL/D - phi(M1) is below 1e-9 on both branches and the zero-friction limit returns the inlet Mach within 1e-9.
• Confirm every non-physical input raises ValueError: Mach at 0 and negative, a sonic inlet, gamma at 1.0, friction factor, diameter and duct length at 0 and negative, and fL/D negative or at/above phi(M1) (the duct chokes at or before the exit; the module never returns a fake Mach below the branch floor).
• Confirm determinism: identical inputs give identical bits, no RNG, no imports beyond math.
• Run the contract test offline: python3 scripts/test_fanno_flow.py (41 tests, deterministic).

Related leaves

• aerodynamics/high-speed/isentropic-flow-relations: the frictionless complement of this leaf, the isentropic total-to-static ratios, area-Mach inversion and choked mass flow at a geometric throat.
• aerodynamics/high-speed/normal-shock: shock-jump stagnation loss at a station, the discontinuous counterpart of the integrated friction loss here.
• aerodynamics/high-speed/shock-tube: the unsteady diaphragm wave system and the deterministic-bisection precedent reused by the branch inversion.
• aerodynamics/high-speed/flat-plate-skin-friction-heating: boundary-layer skin friction, drag and heating on surfaces, a different quantity from the duct-averaged Fanning factor of a confined channel.

Pitfalls

• Mixing Fanning and Darcy friction factors: f here is the Fanning factor; tables that tabulate the Darcy factor f_D = 4 f give (4 f) L/D, so a caller converting table values must divide f_D by 4 before calling or the computed lengths come out four times too short.
• Confusing fL/D with fL*/D: fL*/D(M1) is the total choke parameter from the inlet, while fL/D is what a finite duct of length L actually consumes; a segment is only legal while fL/D stays strictly below fL*/D(M1), and the margin left to sonic is phi(M2) = phi(M1) - fL/D.
• Feeding an over-length duct: at fL/D = phi(M1) the exit is exactly sonic and beyond that the duct chokes internally, which has no steady-state solution on the Fanno line; the module raises ValueError instead of returning a fake Mach below the branch floor.
• Expecting friction to always slow the flow: in a constant-area adiabatic duct friction accelerates the subsonic branch (M2 above M1) and decelerates the supersonic branch (M2 below M1); both move toward M = 1, and the static temperature falls on the subsonic side but rises on the supersonic side.
• Reading p0/p0* as a loss along the duct: p0/p0* is the ratio to the sonic reference and grows away from M = 1 on both branches; the loss along the duct is the quotient p02/p01 = (p0/p0*)2/(p0/p0*)1, always below 1.
• Inverting a sonic inlet: downstream_mach raises at M1 = 1 because a sonic inlet is already choked; there is no duct inversion to perform.
• Adding heat: this leaf is adiabatic by claim (T0 constant). Friction with heat addition belongs to neither this leaf nor its rayleigh-flow sibling; it needs a combined Fanno-Rayleigh treatment that is out of scope.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_fanno_flow.py

The test covers the Fanno table anchors of the fanno_function Fanno-line integral on both branches and the gamma parameter, branch monotonicity, the p0/p0* total-pressure-ratio anchors and its minimum at the sonic state, the starred static-ratio rows with the p/p* = (rho/rho*)(T/T*) identity, the friction-duct choking lengths and their closed-form identity, the branch-preserving downstream Mach inversion with round-trip residuals, the zero-friction limit and deterministic bits, the friction required to choke and its consistency, the duct_state loss ratios with entropy rise and the adiabatic T02/T01 check on both branches, and ValueError rejection of every non-physical input (the 12 anchor cases: Mach at 0 and negative, a sonic inlet, gamma at 1.0, zero friction factor, diameter and duct length, fL/D negative and at/above the choke value at M1 = 0.3 and M1 = 2.0).

Compliance

• Standards referenced, not reproduced: NACA TR-824 (Equations, Tables and Charts for Compressible Flow) frames the compressible-flow context; the Fanno relations above are standard engineering methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.