01 / MACH NUMBER
Moving towards sonic flow
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The dotted horizontal line is M = 1. A smooth branch ends there; it does not cross to the other side.
THE SOUND BARRIER, WITHOUT A THROAT
Compressible Ducts
Two different processes. One sonic limit.
ANALYTICAL 1D MODELConstant area · fixed inlet · perfect gas
WALL FRICTION
Friction can accelerate a gas.
A subsonic stream gains speed as friction lowers its pressure. A supersonic stream slows. Both move towards Mach 1.
MASS FLOW · CONSTANTTOTAL TEMPERATURE · CONSTANTENTROPY · NON-DECREASING
CONSTANT-AREA SECTIONGeometry shown schematically
Mach number
01
Solving…
Exit Mach—At x / L = 1
Total-pressure change—Inlet to exit
Total-temperature change—No heat added
Mass flow—Fixed by inlet and diameter
Preparing the visual time scale…
FOLLOW THE SAME STATION THROUGH EVERY CURVE
One flow, several signatures.
B · liveA · frozenSolid: static / dashed: total
02 / STATIC AND TOTAL PRESSURE
A pressure rise can hide a loss
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Static pressure can rise in a supersonic duct while total pressure falls. Each pressure is normalised by its own inlet value.
03 / STATIC AND TOTAL TEMPERATURE
Temperature is not total energy
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Friction redistributes static enthalpy and kinetic energy. The adiabatic duct retains its total temperature.
04 / ENTROPY
The approach to a limiting state
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Entropy is measured from each case’s inlet. The maximum on this branch occurs at M = 1.
Checking mass and energy consistency…
Watch the inlet branch.
Compare the subsonic and supersonic presets. Friction always opposes motion, but the compressible pressure and density response determines whether speed rises or falls.
Find the boundary.
Choose “Sonic at exit”, then “Beyond the limit”. The hatched section has no continuation on this fixed-inlet branch. The solver does not silently change the inlet or insert a shock.
Separate energy from pressure.
Freeze a case as A, then change the loading in B. Total temperature stays fixed in Fanno flow, while irreversible friction reduces total pressure. “Energy conserved” does not mean “isentropic”.
READ THE BOUNDARY OF THE MODEL
Equations, assumptions and evidence.
These are steady one-dimensional analytical states. Moving tracers illustrate an already solved flow; they are not a time-dependent CFD simulation.
Mass and energy
A fixed inlet determines mass flux. A subsonic or supersonic branch is selected explicitly.
Fanno · adiabatic wall friction
F(M) is the remaining friction exposure to sonic flow. At the limit, fDL*/D = F(M₁). The Darcy factor is four times the Fanning factor.
Rayleigh · frictionless heat addition
Heat is prescribed uniformly per unit duct length. The maximum allowable heat on a fixed-inlet branch is q* = cₚ(T₀* − T₀₁), where * denotes M = 1.
Static and stagnation properties
Stagnation pressure is the pressure reached by a hypothetical local isentropic deceleration. It is not the duct’s static pressure.
What this model includes
Steady flow of a calorically perfect gas; circular constant-area duct; cross-section averages; no shaft work; one effect at a time. Air constants are γ = 1.4, R = 287.05 J kg⁻¹ K⁻¹ and cₚ = 1004.675 J kg⁻¹ K⁻¹.
The limit applies to the specified inlet, not to every possible flow through the same pipe. A real system may change inlet mass flow, develop shocks or become unsteady.
What remains outside it
Friction and heat are not combined. Boundary layers, entrance losses, variable heat capacity, chemistry, heat-transfer coefficients, wall temperature and shock location are omitted. A high-temperature warning marks where the constant-air-property approximation becomes less representative.
Analytical and numerical consistency checks do not establish experimental accuracy or suitability for engineering design.
Sources, implementation and verification record
The equations were independently implemented from conservation laws and checked against university teaching references. All diagrams are original. No third-party software or figure has been bundled.
- Kyle Niemeyer, Oregon State University · Fanno flow — analytical friction-distance and sonic-state relationships.
- Kyle Niemeyer, Oregon State University · Rayleigh flow — mass, momentum and heat-addition relationships.
- NPTEL · Fanno line and fixed-inlet choking — interpretation of the branch limit.
- NASA / Alabama A&M · compressible-flow verification study (2007) — context for analytical Fanno and Rayleigh benchmarks; this lab does not run GFSSP.
Model ledger and equations · Source scope and attribution · Recorded analytical checks · JSON
Recorded checks include 29,160 sampled states, mass and energy closure, Rayleigh momentum, Fanno wall-shear momentum, an independent Rayleigh quadratic, and Fanno differential-equation refinement. See the record for tolerances and scope.