Where the unknown boundary is the known one

Every boundary-value problem in this collection so far has had the same comfortable structure: a region that is known, and a function to be found in it. A free surface breaks that structure. The jet leaving a hole, the cavity behind a plate, the surface of a wave — in each case the shape of the boundary is part of what is being solved for, so the region the equation is posed in is not known until the equation is solved.

That is not a hard problem. It is a differently shaped problem, and the difference is worth a rung.

Two conditions on a curve nobody knows #

A free surface carries two boundary conditions rather than one, and that is the reason it is tractable at all.

It is a streamline. Nothing crosses it, because on one side is moving fluid and on the other is still fluid or vapour, and a surface with no thickness cannot pass anything. So $\psi$ is constant on it.

The speed on it is known. The pressure just outside the surface must match the pressure just inside, which is a constant; Bernoulli along the surface then fixes the speed, and fixes it to be constant too.

An ordinary wall gives one condition on a known curve. A free surface gives two conditions on an unknown one, and the arithmetic of that trade is exactly even — one extra condition pays for one extra unknown. The problem is well posed and it is not posed in a way any ordinary method can attack, because every method for Laplace’s equation begins by discretising a region.

The change of variable #

Here is the move, and it is one sentence: stop plotting the flow in the plane of positions and plot it in the plane of velocities.

Write $\zeta = (u - iv)/U$, the complex velocity divided by the free stream. Every point of the fluid has one, and the map from the physical plane to the $\zeta$ plane is the hodograph — a word from Hamilton, who used it for the same construction in planetary orbits.

What happens to the boundaries is the whole point:

• the free streamline had unknown shape and known speed, so it becomes an arc of the unit circle, $|\zeta| = 1$, whose position is known exactly and in advance;
• the plate had known shape and unknown speed, so it becomes a segment of the imaginary axis with an unknown endpoint;
• the axis of symmetry, where the flow is along $x$, becomes a segment of the real axis.

Why it works, and where the linearity went #

The hodograph is not a trick that could be tried anywhere. It works because of a fact about two-dimensional potential flow that this collection has already leaned on: the whole flow is one analytic function, $w(z) = \phi + i\psi$, and its derivative $dw/dz = u - iv$ is analytic too.

So $\zeta$ is an analytic function of $z$, and the map is conformal wherever $\zeta' \ne 0$. Analytic maps carry harmonic functions to harmonic functions, so a problem for Laplace’s equation in the physical plane is a problem for Laplace’s equation in the hodograph plane, with the boundaries in different places.

Nothing has been linearised and nothing has been approximated. The nonlinearity in a free-surface problem is not in the equation — Laplace’s equation is as linear as an equation gets — it is in the domain, and a change of variable that fixes the domain removes it entirely. That is a much rarer piece of luck than it sounds, and it is what makes free-streamline theory exact where almost every other treatment of a free surface is a perturbation series.

What it looks like when nothing is free #

The transformation is not reserved for free surfaces, and looking at an ordinary flow through it makes clear what the free surface is contributing.

Take the cylinder. Its hodograph image is bounded by the curve traced by the surface velocity, which runs from zero at the front stagnation point up to $2U$ at the shoulder, back to zero at the rear stagnation point, and round again — a closed curve through the origin twice. Nothing about it is simple, and nothing about it is needed, because the body’s shape was known from the start and the physical plane was a perfectly good place to work.

A hodograph earns its keep exactly when a boundary’s speed is simpler than its shape. On a wall the shape is simple and the speed is whatever it turns out to be; on a free surface the reverse. The transformation is a bet on which of the two is easier, and free-streamline problems are the ones where the bet pays.

Solving in the strip #

The quarter disc is still not the easiest place to work, so one more map is taken. Writing $\Omega = -\ln\zeta$ turns it into a semi-infinite strip: the unit circle becomes the line $\operatorname{Re}\Omega = 0$, and the two straight boundaries become the strip’s edges.

A strip is a polygon with two vertices at infinity, so Schwarz–Christoffel maps a half-plane onto it in closed form, and the half-plane is where the flow already is — because the whole boundary is one streamline, so $\psi = 0$ everywhere on it and the $w$-plane image is the upper half plane with no work required.

Composing the two gives the single expression the previous rung quoted:

$$\frac{dz}{dw} = \frac{\sqrt{w-b}+i\sqrt b}{U\sqrt w}.$$

The check that the composition was right #

Three maps composed by hand is three chances to drop a factor, and a dropped factor produces a perfectly smooth wrong flow. So the expression is checked rather than trusted, and the check is that every boundary falls out of it by inspection.

For $w < 0$ both square roots are imaginary and the ratio is real — motion along the axis. For $0 < w < b$ the numerator is imaginary and the denominator real — motion straight up the plate. For $w > b$ the modulus is exactly $1/U$ — constant speed, the free streamline.

And the one that would catch a scaling error: integrating $|dz/dw|$ up the plate from the stagnation point to the edge has to return the plate’s own half-width. It does, to five figures, and that is what fixes $b$.

What it costs #

The method is exact and it is narrow, and the narrowness is structural rather than a matter of effort.

The body must be made of straight segments, or the strip is not a polygon and Schwarz–Christoffel does not apply. A flat plate, a wedge, a slot in a wall, a pair of plates — all fine. A circular cylinder is not, and Levi-Civita’s 1907 extension handles curved bodies at the cost of an integral equation, which is to say at the cost of the closed form.

The flow must be two-dimensional, because there is no analytic function of two complex variables doing this job. Every result in free-streamline theory is a plane result, and the axisymmetric cavity behind a disc — the case a real cavitating body is closest to — has no hodograph solution at all.

And the hodograph map need not be one-to-one. Two different points in the flow can have the same velocity, and when they do the hodograph image folds over itself. For the Kirchhoff plate it does not, which is why the picture above is a simple quarter disc; for a plate at incidence it is more complicated, and for a body with a stagnation point in the middle of a free surface the method breaks down entirely.

The plate at incidence, where it gets harder #

Tilt the plate and every step above still applies, and every step above becomes more work.

The two free streamlines now leave edges at which the flow arrives at different speeds and different angles, so the hodograph image is no longer symmetric and the Schwarz–Christoffel map acquires an extra parameter. Worse, the stagnation point moves off the centre of the plate, so the plate’s image in the hodograph is two segments rather than one and the strip becomes a polygon with a slit in it.

The solution exists — Rayleigh published it in 1876 — and it produces the normal-force coefficient

$$C_N = \frac{2\pi\sin\alpha}{4+\pi\sin\alpha},$$

which at ninety degrees returns the plate result of the previous rung and at small incidence gives a lift-curve slope of $\pi/2$ per radian. That last number is a quarter of the attached theory’s 2π, and the discrepancy is the whole difference between a wing and a flat plate with a wake behind it: a section that separates at its leading edge produces a quarter of the lift of one that does not.

The other problem it was invented for #

Free-streamline theory arrives in most courses as an attempt to explain drag, which is the application it does worst. It was invented for jets, which is the application it does best, and where the constant pressure on the free surface is not an assumption but the ambient atmosphere.

A jet leaving a sharp-edged slot in a plane wall contracts, and the contraction ratio is not measured but computed: $\pi/(\pi+2) = 0.611$, from a hodograph map in which the free surface’s shape is part of the answer. This collection prices that elsewhere, and the number is one of the very few in the subject that is exact, dimensionless and non-obvious.

The same construction gives the deflection of a jet striking a plate, the force on a Pelton bucket, the shape of a two-dimensional nappe leaving a weir, and the profile of a Borda mouthpiece — where the answer is exactly one half and can be had by momentum alone, which makes it the one case where the hodograph can be checked against a completely independent argument.

The compressible version, and the theorem that ended a design programme #

The history above ends with Chaplygin’s observation that the same change of variable linearises a genuinely nonlinear equation, and that observation had a consequence in aircraft design worth following, because it is the one case where a hodograph result closed a question rather than opening one.

In the physical plane the transonic potential equation is nonlinear: the density depends on the speed, so the coefficients depend on the unknown. In the hodograph plane the speed and the flow direction are the independent variables, the coefficients become known functions of them, and the equation is linear — exactly, with no small parameter anywhere. That is a far stronger gain than the free-surface case, where the equation was linear all along and only the region was awkward.

The gain was used. Through the 1960s and 70s, aerofoils with shock-free transonic flow were designed by solving in the hodograph plane and mapping back — sections carrying a supersonic pocket over the upper surface that is decelerated smoothly back through the speed of sound, with no shock and therefore none of the total-pressure loss a shock costs. They exist, they were built, and they were measured to work at their design point.

And then Morawetz proved that they cannot survive. A shock-free transonic flow is isolated: an arbitrarily small change to the profile, or to the free-stream Mach number, or to the incidence, destroys the shock-free character and a shock appears. The solutions are exact points in a design space with no neighbourhood — which means no manufacturing tolerance, no off-design condition and no gust may be admitted.

That is an unusual kind of result to get from a fluid-mechanical theorem, and it settled a practical argument outright. The design goal changed from no shock to a weak shock, deliberately placed and deliberately mild, with the recovery behind it shaped so that the boundary layer survives it. Every supercritical wing in service is that compromise, and it is a compromise because a theorem said the alternative was not available rather than because nobody could build it.

The hodograph has one more warning to give in the same setting, and it is visible in the mapping rather than in the flow. Where the transformation’s Jacobian vanishes the map folds, and the physical-plane solution constructed through it becomes multivalued — a limit line, at which the solution stops being a function of position. It is not a shock and it is not physical; it is the method announcing that the flow it is describing cannot exist. A change of variable that makes a hard problem easy will generally tell the truth about the problem’s own limits, and the limit line is the compressible hodograph doing exactly that.

What the picture cannot show #

The hodograph plane has no geometry in it. Distance in that picture is a difference of velocities, not of positions, so nothing about the shape of the quarter disc says anything about the shape of the plate. Two flows past very different bodies can have similar hodographs and two similar bodies can have very different ones.

The mapping’s inverse is an integral. Getting from the hodograph back to a picture of the flow means integrating $dz/dw$ along a path, and the constant of integration has to be pinned down somewhere — here by requiring the stagnation point to be at the origin. A figure drawn from a hodograph solution is always one quadrature away from the solution itself.

And the interior of the cavity is not in the hodograph at all. The still fluid has zero velocity, so all of it maps to a single point. Everything the model declines to say about the wake is, in that plane, a single dot.

Who found it, and when #

Hamilton introduced the hodograph in 1846 for orbital mechanics, where the velocity of a body in an inverse-square field traces a circle — a fact that is nearly invisible in the orbit and obvious in the hodograph, which is exactly the property this whole method exploits. Helmholtz and Kirchhoff brought it into fluid mechanics in 1868 and 1869. Levi-Civita generalised it to curved boundaries in 1907, and Chaplygin used it in 1902 for compressible flow, where the hodograph plane linearises an equation that is genuinely nonlinear and not merely posed on an awkward region.

The surprising connection is that one: in a compressible flow the hodograph transformation makes a nonlinear equation linear, exactly, with no approximation. The gas-dynamic equations in the physical plane are nonlinear because the density depends on the speed; in the hodograph plane the roles of dependent and independent variables swap and the nonlinearity moves into the coefficients, where it is harmless. The transonic small-disturbance solutions of the 1940s were built that way. It is the same change of variable, applied to a different difficulty, and it works for the same reason: the trouble was never in the equation, it was in which variables were being treated as known.

There is one more reason to know the method beyond the flows it solves. It is the clearest case in the collection of a difficulty that is not in the physics at all. Laplace’s equation is linear, the boundary conditions are linear, and the problem is nonetheless hard — because a region that depends on its own solution is a nonlinearity hiding in the geometry. Recognising that shape is worth more than the particular map: it turns up in the shape of a soap film, in the position of a free surface in a cavitating flow, and in asking for a pressure distribution and finding out what shape has it, which is the same trade of a known region for a known boundary value run in the opposite direction.

Where the ladder goes next #

Below this rung is the flow it solves and the analytic function that makes the change of variable legitimate.

Beside it is the jet it was really invented for, where the constant pressure on the free surface is the atmosphere and nothing has to be assumed.

And above it, the general lesson these rungs keep finding: a problem that cannot be solved as posed is often a problem posed in the wrong variables, and the missing information — here, the shape of a boundary — is sometimes not missing at all but merely written down somewhere else.