About

This is a growing collection of illustrated essays about how air and water actually move. Each one takes a single idea and draws it until the argument is visible — and every flow in every picture was solved and then checked before it was allowed to appear.

Why the checking matters more here than usual

A wrong flow field is beautiful. Streamlines come out smooth and plausible whatever nonsense produced them, and neither a reader nor an author can tell a solution of the equations from a function that merely looks like one. It is the characteristic failure of this subject, and it is the reason every figure here is generated by a solver that asserts.

• Mass is conserved. The velocity field is divergence-free to better than one part in ten million, checked over the whole domain.
• Nothing flows through a wall. The velocity is tangent to every body surface, which is the condition a plausible-looking wrong field fails first.
• Lift is computed twice. Once from the circulation by Kutta–Joukowski, once by integrating pressure over the surface — an unrelated calculation that never mentions circulation. They must agree, in magnitude and in sign.
• d'Alembert's zero drag comes out as zero. Not "small": the ideal-flow drag on a cylinder is computed as about one part in ten thousand million million, which is arithmetic noise and nothing else.

A figure whose field fails those checks throws, and the page is never made. That is not a formality. Five real errors were caught this way before a single essay existed, and not one of them looked wrong on screen — including a circulation whose sign was reversed, so the aerofoil was generating lift downwards while every other check passed.

The wrong explanations are tested here, not ignored

This subject is taught worse than almost any other in physics. The story that air parting at the leading edge must meet up again at the trailing edge is in a great many textbooks, and it is simply false. Bernoulli's equation is routinely applied across streamlines, through fans and inside boundary layers, where it does not hold. d'Alembert's paradox — that the exact theory predicts no drag at all — tends to be skipped.

Rather than ignore all this, the site states each explanation fairly and then tests it against a solved flow. The rule is that a refutation must be a measurement: it is not enough to say equal transit time is wrong, the figure has to compute the two quantities the claim equates and show they differ.

Vector, not raster

Continuous fields — pressure, vorticity, speed — are drawn as contour bands built by marching squares, not as bitmaps. A bitmap cannot recolour for the dark theme, cannot be checked, is opaque to the layout gates, and is far too expensive to sweep when a reader drags a parameter. Bands cost a few dozen filled paths instead of forty thousand pixels, and they are better explanatory practice besides: a smooth ramp hides the values that bands let a reader read off.

There is an escape hatch for genuine texture — the kind of image where the pattern itself is the point — and one rule that goes with it: raster carries texture, never information a reader must read.

Where the models stop

Three places, and they fail differently.

Ideal flow is exact, closed-form and predicts that an aeroplane needs no engines. Its failure is the most useful thing in the subject rather than an embarrassment, and several essays are organised around it. Viscous flow is solved on a grid here, and that grid is coarse: it resolves attached flow becoming separated flow, and it does not resolve turbulence. The unsteady wake is beyond it at present — the solver does not shed a vortex street, and nothing on this site claims one until it does.

Where a figure depends on the grid being fine enough, the essay says so. Nothing here quotes a number the code did not produce.

And one kind of figure that is not a flow at all

Turbulence cannot be solved at any Reynolds number worth the name — resolving the flow over a wing needs about 10¹⁷ grid points — so the essays about it argue from things that can be established exactly: the arithmetic of averages, the consequences of a stated closure, and the statistics of a field built to order from a spectrum somebody chose. None of those is a solution of the equations of motion, and each figure drawn from one says which of the three it is, in the same words every time.

The distinction matters most where a picture is persuasive. A field synthesised from the Kolmogorov spectrum looks exactly like turbulence and contains no cascade whatever, which is the subject of one of the essays; a velocity profile from a mixing-length closure looks exactly like a measurement and is a consequence of an assumption. A figure that is not a solution is labelled as one, and the model note under it is where to look.

On being wrong

Corrections are welcome and will be made. An elegant wrong answer is worse than none in this subject, because it is indistinguishable from a right one until somebody checks — which is the whole argument for checking.