Nothing in the present picks the flow

The constant a hole leaves behind establishes the mathematics: in a region without holes, Laplace’s equation and the boundary values have exactly one solution, and cutting a hole gives a one-parameter family. The count of free constants is the number of holes, and nothing in the mathematics chooses between the members.

This essay is about what does choose, and the answer is the only thing left: what happened before.

That makes the circulation round a body the purest memory in this collection. It is not a residue of a history, or a correlation, or a quantity that decays. It is a number the flow was given at some earlier time and has carried unchanged ever since, and the present state of the problem contains no trace of where it came from.

Six flows, one boundary condition #

The family is the classical one: uniform flow past a circular cylinder, with a line vortex of arbitrary strength at its centre. Six members are computed, at circulations from zero to twelve pi.

Every one of them satisfies the wall condition. That is not an approximation and it is not a tolerance: the added vortex has no radial velocity anywhere, so it cannot push fluid through a circle centred on it, and the largest normal velocity found anywhere on any of the six surfaces is exactly zero in double precision.

And they are completely different flows. The fastest point on the surface runs from twice the free stream to eight times it. The peak suction runs from −3 to −63, a factor of twenty-one. The pressure distribution changes from symmetric about the horizontal to entirely one-sided.

The lift column is the circulation column, digit for digit, and the drag column is machine zero at every member — so the two theorems the family is usually quoted for fall out of the surface integral rather than being assumed by it.

The lift, from the body’s own pressure #

The lift is obtained here by integrating each cylinder’s own surface pressure rather than by quoting the theorem, which is worth doing because the agreement is the check that the family has been constructed correctly.

It comes out as rho U Gamma to nine figures for every member — which is one formula, and it does not ask what the shape is arriving as a by-product — and the drag comes out at 10⁻¹⁶, which is the exact theory says nothing has any drag arriving as another.

So the six flows differ by a factor that a force balance would measure immediately, and by nothing at all that the statement of the problem contains.

Where the flow stops, and the circulation that has no stopping point #

There is one more visible difference and it is the one a flow visualisation would show first.

At zero circulation the flow stops at the front and the back of the cylinder — at 0 and π radians, and the computation reads 0.000000000 and 3.14159. Raising the circulation moves the two stopping points round towards one another, both towards the same side, and they meet at exactly four pi U a, which is 12.5664. Above that there is no stopping point on the body at all: the circulation is strong enough that the surface is moving everywhere, and the stopping point has left the body and sits out in the fluid.

That critical value is not a parameter of the model. It is where the added swirl at the surface, Γ/2πa, equals the largest speed the free stream can produce there, 2U — so it is a comparison between two known quantities and it comes out as a clean multiple of pi.

What actually fixes the constant #

Kelvin’s theorem is the answer, and it is worth stating exactly what it does and does not say.

The circulation round any material circuit is constant for ever in an ideal barotropic flow with conservative body forces — which is what survives being wound up, and which the drift was the instrument shows is exact rather than approximate. Take a body at rest in a fluid at rest, and a large material circuit surrounding it. Its circulation is zero. Now accelerate the body however violently, along any path, for as long as desired: the circuit is still material, so its circulation is still zero.

Since no vorticity has been shed — a cylinder has no sharp edge — the circulation round the body is also zero, and the flow that results is the member of the family with Γ = 0. Not because it is the simplest or has the least energy, but because that is the one the initial condition permits.

Now take the same body and spin it first, while it is stationary. Spinning requires a no-slip surface, which requires viscosity for some interval; the fluid near the body is dragged round and acquires circulation. Then stop the spin, switch the viscosity off, and accelerate the body forward to exactly the same speed as before.

At the end the two bodies are the same shape, at the same place, moving at the same speed, in the same fluid, obeying the same equation with the same boundary condition. One has a lift of −6.3·10⁻¹⁶ and the other 18.8496, which is 6π to six figures; both have a drag below 10⁻¹⁴. The difference between them is 18.8496 and nothing in the present state of either flow says where it came from.

Why “the least-energy member” is not an answer #

There is a tempting way out of the non-uniqueness and it is worth closing off, because it is the first thing most readers reach for.

The member with zero circulation has the least kinetic energy of the family — the energy of the circulation and the energy of the stream add without a cross term, which the constant a hole leaves behind computes. So one might hope that nature picks the least-energy member and that the constant is determined after all.

It does not, and the reason is that there is no mechanism by which it could. Kinetic energy is conserved in an ideal flow, so a flow with circulation in it has no way to shed the energy that circulation carries, and no principle in the equations drives a system towards its minimum energy in the absence of dissipation.

The minimum-energy statement is real and is a different theorem: it says that among all flows satisfying the same boundary condition, the irrotational one has the least energy, which is the flow with the least energy in it. It picks the member that would be reached if energy could be shed. Nothing here can shed any.

A rotating cylinder in a real fluid does eventually settle, and it settles at a circulation set by the balance between what the spinning surface puts in and what viscosity takes out — which is a statement about dissipation and not about a variational principle.

Why the Kutta condition has to exist #

This is the light in which the Kutta condition is best understood, and it is the reason the sharp edge decides is a separate essay from what actually holds a wing up.

A wing in steady flight has a definite circulation, and something has to have chosen it. For a cylinder the choice is made by the history and stays made for ever. For a wing the choice is made by the history too — but the history now contains a mechanism for changing the circulation, because a sharp edge sheds vorticity, and the shedding continues until the flow leaves the edge smoothly.

So the Kutta condition is not an extra equation added to close the system. It is a statement about where the shedding stops, and it works because the process it describes is self-limiting: any circulation other than the Kutta value leaves the flow turning a corner at the trailing edge, which sheds more vorticity, which changes the circulation. The circulation on a wing is fixed by history too; the wing just has a fast way of forgetting the wrong ones.

A cylinder has no such mechanism, which is exactly why a rotating cylinder can carry any circulation it is given and a wing cannot.

What the solver computed, and how it was checked #

The family is the closed-form potential flow, so nothing here is approximated and the checks are aimed at the claims.

That every member satisfies the wall, by differencing the stream function on the surface and requiring the radial velocity to vanish. It reads 0.000000000 on all six members, and the check is written to fail on anything above 10⁻⁶ — which would catch an algebra error in the vortex term.

That the family is genuinely spread, by requiring the peak suction to vary by at least a factor of five across it. It varies by twenty-one — −3 at the bottom of the family and −63 at the top. A demonstration that six nearly identical flows are hard to tell apart would not be a demonstration.

That the lift is rho U Gamma and the drag is zero, both from the integrated surface pressure at twenty thousand points rather than from the formula: the lifts come back as 0, 6.2832, 12.566, 18.850, 25.133 and 37.699 against circulations of the same six numbers, and the largest drag anywhere in the family is 7.7·10⁻¹⁵.

And that the stopping points merge at the critical circulation, with the position at half the critical value checked against the closed form and the case above it checked to have no stopping point on the surface at all.

One thing was got wrong and is recorded because it is the classic error in this subject. The first version of the lift check compared against minus rho U Gamma and failed on every member. The sign convention of the stream function used here makes the swirl clockwise for a positive constant, so the counterclockwise circulation is the negative of it. A sign error in a circulation makes a wing fly downwards while every other check passes, which is why this collection’s own gate lists it among the four errors it was built to catch.

How much of the family a real fluid can reach #

Since the theory permits the whole family, it is worth asking which members a fluid can actually be put into, because the answer is a much smaller set and it is set by viscosity twice over.

Putting circulation in. The only mechanisms are a moving surface, a shed vortex, or a body force that is not a gradient. The first two need viscosity — a no-slip surface to drag the fluid round, or a sharp edge at which a boundary layer leaves — so the family is reached through exactly the ingredient the model excludes.

Keeping it there. A circulation once established diffuses outward: the vortex spreads, its core grows as the square root of the elapsed time times the viscosity, and the circulation round a fixed circuit far away is unchanged while the circulation round the body’s own surface falls. So the constant is only constant on the time scale over which diffusion has not reached the circuit being measured.

And the high members are unreachable in practice. At a circulation of twelve pi — three times the critical value of 4πUa — the suction peak is sixty-three dynamic heads against the zero-circulation flow’s three, which no real flow sustains: it would cavitate in water and separate in air long before. A rotating cylinder’s measured lift saturates at a few times its diameter’s worth of circulation, far below what this theory allows.

What this means for an experiment #

The practical form is a warning about repeatability, and it is not hypothetical.

Two runs of the same experiment can measure different flows, if the start-up differs. A model that was disturbed while the tunnel was coming up to speed can carry a circulation the next run does not, and the difference is a lift rather than a scatter in a lift.

Which is why start-up procedures are specified. The convention of running a tunnel up in a fixed way, and of discarding the first minutes, exists because the flow’s constant is set then and not afterwards.

And why a rotating-cylinder experiment is delicate. The circulation there is being set continuously by the no-slip surface, so it depends on the spin rate, on the surface finish, and on how long the cylinder has been spinning — and the measured lift depends on all three in a way that the inviscid theory cannot describe at all.

The same non-uniqueness, elsewhere in this collection #

The shape of the problem — an exact theory that determines everything except one number, which the history supplies — turns up three more times here and is worth collecting.

The vorticity on a closed streamline. The vorticity nothing decides finds a whole function’s worth of freedom rather than one constant: a streamline that comes from upstream carries its vorticity with it, and a closed one comes from nowhere, so nothing in the steady problem says what it carries.

The separation point on a free streamline. Drag in the theory that forbids it needs a separation point that the inviscid theory cannot supply, and it is supplied by the boundary layer’s history upstream.

And the profile behind an exact total. Exact in the total, free in the profile is the general statement of how much an exact constraint leaves open, and this essay is one of its sharper instances: one condition on a whole function space.

In every case the practical rule is the same. When an exact theory leaves a constant free, look for the mechanism that sets it, and expect the mechanism to be dissipative — a shed vortex, a boundary layer, a viscous spin-up. That is not a coincidence: the free constant exists because the ideal theory has no memory, and the only things that give a fluid a memory are the things the ideal theory left out.

What the picture cannot show #

The six flows are drawn as surface distributions rather than as fields, and the reason is that the fields differ most in a place the drawing would not show: far from the body, where the circulation’s own velocity falls only as one over the distance while the cylinder’s disturbance falls as one over the square.

So the members of this family are most different far away and least different near the body, which is the opposite of what a picture of the near field suggests. It is also why the circulation is the quantity the far field keeps — what the far field remembers — and why a measurement made a long way off is the one that can determine it.

Who found it, and when #

The non-uniqueness in a multiply-connected region is Helmholtz’s and Kelvin’s, and its resolution by the initial condition is Kelvin’s 1869 theorem. The cylinder-with-circulation solution is older than either and is a standard exercise by the 1860s.

What took longer was the recognition that the same argument settles the wing, which is Prandtl’s school in the 1900s and 1910s: the circulation is set by the start-up, the starting vortex carries away its opposite, and the Kutta condition is the statement of where the shedding stops. The cylinder is the case with no shedding mechanism, and it is therefore the clean demonstration that the constant is historical.

Limits recorded rather than smoothed over #

Two dimensions. A hole in the plane leaves a constant behind; a body in three dimensions does not disconnect the region, so there is no free constant of this kind. The three-dimensional statement is about the vorticity in the wake instead.

No viscosity, except during the spin. The two-history demonstration cheats openly: putting circulation into the fluid requires a no-slip surface for some interval, so the spun case is not an ideal flow throughout. That is not a flaw in the argument — it is the argument, since the only ways to change a circulation are the ways Kelvin’s theorem excludes.

The critical circulation is the cylinder’s. Four pi U a is a property of a circle, and a body of another shape has its own value at which the stopping point leaves the surface.

The circulation is taken as a point vortex at the centre. Any distribution of vorticity inside the body would give the same exterior flow, since the exterior only sees the total, so the family is parameterised by one number however the vorticity is arranged. That is the same statement the inside a flow does not decide makes about singularities generally, and it is why the constant is a single number rather than a function.

And no member of this family is stable. The high-circulation members have enormous suction peaks and would separate immediately in any real fluid, which is why a Flettner rotor’s measured lift falls far short of what this theory allows.