A Ford 302, modelled from first principles in C++23, for no reason.
make && ./windsor spec
No dependencies. Nothing to configure. No website. It is an engine, and it runs.
A cross-plane V8 burbles. A flat-plane V8 shrieks. Everyone knows this and almost nobody can say why, and the reason is not in the exhaust, the camshaft, the displacement or the fuel. It is four numbers on a crankshaft.
A 90° V8 fires evenly on either crank — a power stroke every 90° of rotation, eight per cycle, perfectly spaced. At the flywheel the two engines are identical in rhythm. There is no measurement you can take at the output shaft that tells them apart.
But an engine does not exhale through its flywheel. It exhales through two exhaust manifolds, and a manifold is not connected to an engine — it is connected to four cylinders, and it only ever hears those four:
| each bank hears | |
|---|---|
| cross-plane (throws at 0, 90, 180, 270) | 90 - 180 - 270 - 180 |
| flat-plane (throws at 0, 180, 0, 180) | 180 - 180 - 180 - 180 |
That is the entire difference. A cross-plane bank coughs twice in quick succession, waits three quarters of a turn, and coughs again — a limping, syncopated pulse train, beating against the other bank's, never resolving. That is the sound of a Mustang idling outside a diner. A flat-plane bank exhales in perfect thirds of a revolution, both banks lock in phase, and the harmonics stack cleanly at twice the frequency. That is the sound of a 458 leaving a tunnel.
So: if the model is honest, changing one part should change the sound and nothing else.
./windsor record # the factory cross-plane crank
./windsor record --flat # a billet flat crank, same block, same everything
A firing order is not a design input. It is a result. You forge a crank, you hang eight rods on four journals, you grind a cam that decides which cylinder on each pin fires on the first revolution and which on the second — and the firing order is whatever falls out.
So windsor.hpp specifies the forging, and nothing else:
Crankshaft{
90.0_deg,
{ 45.0_deg, 315.0_deg, 135.0_deg, 225.0_deg },
{{
// cylinder journal bank revolution
/* 1 */ { 0, Bank::right, 0 },
/* 2 */ { 1, Bank::right, 0 },
/* 3 */ { 2, Bank::right, 1 },
...and ./windsor spec reports what comes out of it:
---- everything above was specified. everything below came out. ----
CRANKSHAFT
forging cross-plane
firing order 1-5-4-2-6-3-7-8
the engine fires every 90 deg, eight times per cycle
WHAT EACH BANK HEARS
right bank cyl 1 4 2 3 fires at 180 - 90 - 180 - 270
left bank cyl 5 6 7 8 fires at 270 - 180 - 90 - 180
LOPSIDED. Each bank coughs twice in quick succession, waits three
quarters of a turn, and coughs again.
This is the burble.
1-5-4-2-6-3-7-8 is what is cast into a 1968 302 intake manifold. It was
never typed in.
Pass --ho and you get the 1982 5.0 H.O. camshaft on the identical
forging — which is exactly what Ford did — and the order becomes
1-3-7-2-6-5-4-8, the 351W order. Same crank, different cam, different order,
identical sound, because the sound was never in the cam.
./windsor verify
Sixty-three checks in a couple of seconds, every one against something outside the project: a derivative against finite differences, a burn rate against its own integral, a cylinder head against a flow bench, a firing order against the casting, a shock tube against its exact solution, and the thesis against a Fourier transform. Nothing in the model was fitted to any of it.
The cylinder head, against a flow bench. Curtain area, a discharge coefficient, a throat cap. How fast that coefficient sags past its peak is the only number in the port model fitted to anything, and this is what it is fitted to:
| lift | model | published, stock C8OE-F |
|---|---|---|
| 0.400 in | 153.5 cfm | ~150 cfm |
| 0.426 in (the cam's peak) | 149.9 cfm | — |
The engine, against Ford.
| model | Ford, 1968 302-2V (gross) | |
|---|---|---|
| peak torque | 306 lb-ft @ 1985 | 295 lb-ft @ 2400 |
| peak power | 206 hp @ 4994 | 210 hp @ 4400 |
Within four percent and two, with nothing fitted. The magnitudes are close and the curve is too broad — torque peaks 400 rpm low, power 600 high — which is the intake runner tuning still missing. It was ten percent optimistic with the power peak a thousand rpm high until the carburettor's venturis went in: a two-barrel engine breathes through a hole that cannot be opened.
The indicator card. ./windsor card draws Watt's diagram, wide open and
throttled, to the same scale. At 3000 rpm the trace peaks at 64 bar, 9.5° after
top dead centre — late enough to use the crank's leverage, early enough that the
end gas has not gone off by itself. Throttled:
| at 2000 rpm | gross | pumping | net |
|---|---|---|---|
| wide open | 11.94 bar | −0.07 bar | 11.87 bar |
| throttled | 0.98 bar | −0.93 bar | 0.05 bar |
Ninety-five percent of everything it makes, spent on suffocating itself.
The fuel, which is what stops the engine. Livengood–Wu over Douaud–Eyzat:
the end gas spends dt/τ of its patience each instant and goes off when the
account reaches one. It reports an index and not a verdict — the threshold
belongs to the CFR engine the correlation was fitted on — but the comparisons
are what compression ratio and advance are actually chosen by, and they all come
out right. Knock worsens at low rpm, with advance, with compression, with
cheaper fuel, and eases on a rich mixture. Which makes 9.5:1, 34° of advance and
what was in the tank in 1968 one decision instead of three.
The balance, which is why the crank exists. The same rod table, asked a different question:
| at 3000 rpm | cross-plane | flat-plane |
|---|---|---|
| primary force | 0 N | 0 N |
| secondary force | 0 N | 5000 N |
| what it traces | a circle — counterweights take it | a line — nothing can |
Half a tonne, a hundred times a second, and nothing bolted to a shaft turning at ω can oppose a force that goes at 2ω. The flat crank sounds better and shakes. That is the bill for the noise, and why almost nobody pays it.
The sound, against itself. Half-order energy against whole-order energy at idle, half-orders being the signature of a train that repeats every two revolutions instead of one:
| cross-plane | flat-plane | ||
|---|---|---|---|
| one bank alone | 0.70 | 0.0009 | 742× |
Ninety-six percent of the energy sits below 90 Hz, where the burble lives, and nothing measurable above 1.4 kHz.
./windsor record writes stereo, because a V8 does not have an exhaust — it has
two, down opposite sides of the car, permanently out of step on a cross-plane
crank. Two microphones 0.6 m apart, 1.5 m behind two tailpipes 1.0 m apart, each
hearing both pipes: quieter by the extra distance, later by the time sound takes
to cross the gap. Nothing is widened or panned; the image is the geometry, and
moving the microphones moves it. Summing to mono recombines the banks toward an
even train and cancels much of the unevenness — real, and why people have argued
about H-pipes for sixty years, but it throws the evidence away.
include/engine/
si.hpp inside the engine, everything is SI. one rule, no exceptions
geometry.hpp the crank-slider, which is the entire machine
crankshaft.hpp where the sound comes from
balance.hpp why the cross-plane crank exists at all
charge.hpp the working fluid, and what is dissolved in it
wiebe.hpp how fast the fire spreads
knock.hpp the constraint that decides everything
woschni.hpp heat going where it does no good
camshaft.hpp the only part of the engine that decides anything
port.hpp the bottleneck
cylinder.hpp one cylinder, as an open system
friction.hpp the tax
induction.hpp the throttle, and the vacuum behind it
ignition.hpp deciding when, with no way of knowing
riemann.hpp what two bodies of gas do to each other
exhaust.hpp four pipes, a collector, and the noise
engine.hpp the assembly
windsor.hpp the engine itself, as built
apps/
spec.cpp the shop manual page
dyno.cpp a water brake on the flywheel
run.cpp a gauge cluster, wired to the sensors
card.cpp let it draw its own picture, as Watt did
record.cpp three feet behind the tailpipe
verify.cpp the inspection sheet
The engine is a sealed mechanism that turns and gets hot and knows nothing about being observed. The six apps are instruments bolted to it afterward, and none of them may reach into the physics to make its own job easier.
Every header opens with prose explaining why the part exists and what it is arguing with. They are meant to be read in roughly the order above.
Each cylinder is an open thermodynamic system, integrated in crank angle, five terms of the first law per radian:
m·cᵥ·dT/dθ = −p·dV/dθ the piston, taking or giving work
+ dQ_burn/dθ the fire (Wiebe)
− dQ_wall/dθ the coolant (Woschni)
+ Σ (dmᵢ/dθ)·hᵢ gas arriving (compressible orifice flow)
− u·dm/dθ the same, as mass
Open and not closed, because everything anyone cares about happens in the half of the cycle that the air-standard Otto cycle deletes — the half where the valves are open and the mass is a variable.
The exhaust is a digital waveguide: a pipe is two delay lines, one per direction, which is d'Alembert's 1747 result sampled at 176 kHz rather than in the continuum. Junctions scatter by continuity of pressure and conservation of volume flow. The loop is closed — each cylinder's exhaust boundary is the pressure standing in its own primary pipe, so a header has a length that matters and not merely a note. Only by about two percent here, for reasons under nonlinear gas dynamics below.
γ varies with temperature, falling from 1.400 in a cold intake charge to 1.246 in combustion products. A cycle computed at a constant 1.4 will promise you a fifth more thermal efficiency than any engine can deliver.
Stated plainly, because an unstated simplification is a lie and a stated one is a design decision. Each is named in the file where it bites.
-
Wrist-pin offset. Real pistons carry 0.5–1.5 mm toward the thrust side. It buys a rattle you cannot hear and costs a closed-form solution.
-
Intake runner tuning. Built three times, shipped none. The third gave the eight runners one shared junction — the same scattering node the exhaust uses for its collector — which was the hypothesis the second attempt ended on. It was only half right. Sharing redistributes energy between runners; it does not dissipate any, and eight low-loss pipes passing a wave around still have it.
What mattered more was length. At 0.30 m the ram gain came out at 34% where a stock log manifold is worth a few percent, and that is what made the engine unholdable — because a torque-controlled brake on a rising torque curve is an unstable equilibrium, and ram tuning is exactly what makes a curve locally rise. Shortened to 0.18 m the gain falls to 4% and the curve comes out smooth.
It still could not be held between 1000 and 1500 rpm, at any controller gain, and I do not know why. That is written down in
induction.hppas the next thing to find out rather than explained away. -
Nonlinear gas dynamics.
riemann.hppis a second-order finite-volume Euler solver, verified against Sod's shock tube to one part in 10⁵ and conserving mass and energy to machine precision. Wired in place of the delay lines it removed the Mach 1 clamp and made header length worth 14% of torque — and collapsed the difference between the two crankshafts, which is the one measurement this project exists to make. So it was reverted. A delay line has no numerical dissipation: it is the exact solution to the linear problem, not an approximation to it, and for a problem that is mostly linear propagation the cruder-looking model is the more faithful one.What survives is a measurement.
./windsor verifyclocks a real blowdown front at 1409 m/s — Mach 2.4, against 586 for sound in the gas ahead of it. It is a shock. The shipped model cannot make one, and now says so with a number instead of an apology. -
Blow-by, oil temperature, crankshaft torsion, dissociation above 2000 K.
None of these would change the shape of the project.
make
./windsor spec what was built, and what falls out of it
./windsor dyno put it on a water brake and sweep it
./windsor run watch it idle (0-9 throttle, SPACE wide open, Q quit)
./windsor card let it draw its own indicator diagram
./windsor record stand behind it with two microphones
./windsor verify check every number this project quotes
Flags: --flat fits the billet flat-plane crankshaft, --ho the 1982 H.O.
camshaft.
A C++23 compiler and make. That is the whole of it, and it is deliberate:
every dependency is a bet that something else will still build in fifteen
years. The WAV writer is 44 bytes of header and some samples, written here.
No reason. That is the point. It is an engine that turns over in a terminal and cannot move anything, burning fuel that does not exist, wasting a third of it out of a pipe into nowhere, and it does all of that correctly.
MIT.