A step's face is pushed at its foot and pulled at its lip, equally

Nothing turns a sharp corner found the flow near a corner to be fixed by the corner’s angle and nothing else. Close to a corner of interior angle $\alpha$ the speed goes as $r^{\pi/\alpha - 1}$: in a concave corner, where $\alpha < \pi$, it falls to zero at the apex, and at a convex one it grows without limit, as $r^{-1/2}$ round a sharp edge. That is a statement about a single corner in isolation. A real body has its corners in company — the simplest has two, a concave one and a convex one, and the flow between them, and the force they make, depend on both at once.

The simplest such body is a step: a wall that runs along the floor, rises, and runs on at a new height. The stream approaching it must climb. In the corner at the foot of the face it slows to rest; over the lip it turns a convex corner and speeds up without limit. Both are local corner flows, and the face of the step lies between them. Ideal flow up a step can be solved exactly, and the answer to the obvious question — how hard does the stream push on the face — is that it does not push at all.

The map that solves it #

A wall made of straight pieces is the image of a straight line under a Schwarz–Christoffel map, and the step needs only one factor per corner. Take the upper half of a complex plane $w$, and map it by

$$\frac{dz}{dw} = A\left(\frac{w - 1}{w + 1}\right)^{k}, \qquad \theta = k\pi.$$

The real axis of $w$ becomes the wall: the part $w < -1$ is the floor, $-1 < w < 1$ is the ramp’s face rising at angle $\theta$, and $w > 1$ is the top. At $w = -1$ the map turns the wall up through $\theta$, making the concave corner at the foot; at $w = 1$ it turns it back down, making the convex corner at the lip. The constant $A$ sets the size: the face’s length is $A$ times $\int_{-1}^{1}((1-w)/(1+w))^k\,dw$, which is $2\pi k/\sin\pi k$ in closed form, and its height is that times $\sin\theta$. In the $w$-plane the flow is a uniform stream, $f = UAw$, and since $dz/dw \to A$ far away the stream in the physical plane is $U$ there.

The speed on the wall is then

$$q = U\left|\frac{w + 1}{w - 1}\right|^{k},$$

zero at the foot and infinite at the lip. Close to each corner it is the local corner flow of the single-corner theory: up the face from the foot the speed grows as the distance to the power $\theta/(\pi - \theta)$, linearly for a right-angled step; along the top from the lip it falls as the distance to the power $-\theta/(\pi + \theta)$, the minus one-third power for a right angle. The map knows nothing about corners beyond the factor it has for each, and reproduces both exponents.

Pushed and pulled #

The face is the only part of the wall that faces upstream, so the streamwise force on the step is the pressure on the face times its projected height, $\int p\,dy$. Over the lower part of the face the pressure is above ambient — the stream is slowing into the foot’s corner — and it pushes the step back. Over the upper part the stream is already accelerating towards the lip and the pressure is below ambient, and near the lip it falls without limit, as the speed’s square, which goes as the distance to the minus two-thirds: infinite, but integrable. That suction pulls the face forward.

For a right-angled step the arithmetic can be done by hand. With $w = -\cos t$ the face’s height element is $(2h/\pi)\cos^2(t/2)\,dt$ and the pressure coefficient $1 - \tan^2(t/2)$, so the force is

$$\frac{D}{\tfrac12\rho U^2 h} = \frac{2}{\pi}\int_0^{\pi}\big(\cos^2\tfrac t2 - \sin^2\tfrac t2\big)\,dt = \frac{2}{\pi}\int_0^\pi \cos t\,dt = 0.$$

The pressure crosses ambient where $\cos t$ does, at $t = \pi/2$, which is a height $h(\tfrac12 + \tfrac1\pi) = 0.818\,h$. Below it the face is pushed by $\tfrac{2}{\pi}\int_0^{\pi/2}\cos t\,dt = 2/\pi$ of a dynamic pressure times the height; above it, in the top 18 per cent, it is pulled by the same $2/\pi$. A step’s face carries a push of 0.64 of a dynamic pressure on its height, and an equal and opposite pull squeezed into the fifth nearest its lip.

How tightly the pull is squeezed can be read off the same integral. The height up the face is $h(t + \sin t)/\pi$ and the pull accumulated above a station $t$ is $\sin t$ of the total, so the top tenth of the face carries 95 per cent of the pull, the top twentieth 84 per cent, the top hundredth 55 per cent and the top thousandth still 26 per cent. A quarter of the force that balances the whole face lives within a thousandth of the step’s height of its lip — a millimetre on a metre-high step. That is the sense in which the suction is concentrated: not at a point, as a sharp edge’s is, but within a distance that shrinks without limit while still carrying a finite share of the force, which is what an $r^{-2/3}$ singularity in pressure does. Any real fluid that cannot follow the lip over that last millimetre loses a quarter of the balance at once.

Why they cancel at every angle #

The cancellation is not special to the right angle, and the map says why in one line. For a ramp at any angle the face force is

$$D = \tfrac12\rho U^2 A\sin\theta\int_{-1}^{1}\left[\left(\frac{1-w}{1+w}\right)^{k} - \left(\frac{1+w}{1-w}\right)^{k}\right]dw,$$

and the integrand is odd under $w \to -w$: the reflection that swaps the two ends of the face swaps the factor that makes the stagnation with the factor that makes the infinity. Whatever the ramp’s angle, the foot’s push and the lip’s pull are images of each other in the mapped plane, and they cancel.

The push grows with the angle — a gentle ramp barely slows the stream, a vertical face brings it to rest — from 0.23 at 30 degrees to 0.45 at 60 and $2/\pi$ at 90, and the pull grows with it exactly. The point where the pressure crosses ambient climbs the face as the ramp steepens, from 0.61 of the height at 30 degrees to 0.82 at 90: the steeper the face, the more of it is pushed and the more tightly the pull is packed into the lip.

This is d’Alembert’s paradox in a form that shows its mechanism. The paradox says an ideal flow exerts no drag on a body, and the usual demonstration is a symmetry — the cylinder’s pressure is the same fore and aft. A step has no fore-and-aft symmetry at all: its front is a wall, its back is a plateau that runs on for ever. The zero force survives anyway, and it survives by a cancellation between two corners of opposite kinds, one pushing with a stagnation pressure spread over most of the face and one pulling with an infinite suction concentrated at a point.

The force that lives in the lip #

That second half is the structure a finite force from an infinite speed found at the leading edge of a flat plate. There, the plate’s pressure can only act normal to it, the lift has to be perpendicular to the stream, and the difference is made up by a suction force concentrated at the sharp leading edge, where the speed goes as $r^{-1/2}$. Here, the suction at the lip goes as $r^{-2/3}$ in pressure and is spread over a short distance rather than concentrated at a point, but its role is the same: it is the force that keeps an ideal flow from having drag, and it lives exactly where the speed is infinite.

And it is lost the same way. A real stream cannot turn the lip at infinite speed; its boundary layer, which has just climbed the face against the rising pressure of the foot’s corner and then been flung round a convex corner into a falling pressure, separates at the lip. The suction never forms, the face’s push is no longer balanced, and the step has drag: very nearly the push, 2/π of a dynamic pressure times its height, less whatever suction the separated bubble on top recovers. The drag a real step has is, in this reading, the ideal theory’s push with the ideal theory’s pull taken away — a drag made of a force that ideal flow said was there and balanced.

What the foot’s corner does to the floor #

The stagnation at the foot reaches upstream. Along the floor ahead of the face the pressure rises from ambient far off to the full stagnation value at the corner: 0.65 of a dynamic pressure a step height ahead of the face, 0.38 two heights ahead. A boundary layer on the floor meets that rising pressure — how much uphill a layer can take is a question of exactly this kind — and near the foot the adverse gradient grows without limit as the corner is approached, so a real layer separates ahead of the step and a standing vortex sits in the corner. The real step thus loses both of the ideal flow’s features: the stagnation at the foot becomes a separated corner vortex and the suction at the lip becomes a separated bubble. What survives is the push on the face between them. Round an obstacle of finite width the same foot vortex wraps the obstacle’s sides and trails downstream as a horseshoe: the vortex that digs scour holes at the feet of bridge piers in a river and that sets the heat transfer round the base of a turbine blade on its hub. Its origin is the ideal flow’s stagnation in the foot’s corner, reached by a boundary layer that cannot climb the pressure rise into it.

The same step, run backwards #

Reverse the stream and the step faces downstream: the flow runs along the high level, turns down round the lip — now the first corner it meets, still convex and still at infinite speed — and slows into the foot’s concave corner on the low level. The map is the same, with the stream’s direction reversed, and the speed on the wall, which depends only on $|w \pm 1|$, is unchanged: the face of a backward-facing step is pulled at its lip and pushed at its foot by the same amounts, now in the opposite sense, and still cancels. Ideal flow does not distinguish a step up from a step down. The exact theory is reversible in exactly this sense: reverse every velocity and every pressure is unchanged.

Real flow distinguishes them completely. A stream reaching a backward-facing step’s lip separates there at once, because it is asked to turn a convex corner into a region it cannot fill, and the layer it sheds reattaches several step heights downstream. The face then sits in the dead air of the separated region at a pressure below ambient, and the step’s drag comes from that base suction rather than from a lost lip suction. The forward step separates because of its lip too, but its drag is the push on its face that the lip can no longer balance; the backward step’s drag is a suction on its face that the ideal flow never put there. Where the flow lets go decides both, and neither is in the ideal solution.

Why a map rather than mirrors #

Corners of special angles can be solved by images: a right-angled corner is the flow of a source and its reflections in two walls, and the corners that can be done with mirrors are exactly those whose angle is π over a whole number, where the reflections close up. A step has two corners of different kinds a finite distance apart, and no arrangement of images reproduces both walls and the face between them. The Schwarz–Christoffel map does it for any angles at once, by building the wall’s direction out of one factor per corner: each factor turns the real axis through the corner’s exterior angle and leaves it straight elsewhere. That is why the map’s exponents and the local corner theory’s agree — the map is the local solutions multiplied together, with the scale between them fixed by the step’s height.

The same construction gives the lip suction’s resemblance to a sharp trailing edge’s singularity. At a sharp edge the Kutta condition removes the infinite speed by choosing the circulation; at a step’s lip there is no circulation to choose, the infinity is forced by the geometry, and the only thing that can remove it is a real fluid’s separation.

The size of the push #

For a wall two metres high in a ten-metre-a-second wind, a dynamic pressure is about 61 pascals, and the push on its face is $2/\pi$ of that times its height: 78 newtons on every metre of its length, 7.8 kilograms’ weight. In ideal flow an equal pull at its top edge cancels it; in air, the flow separates at the top edge, the pull is lost, and something close to the push is the load the wall must carry — plus the base suction behind it if the wall is thin, since a thin wall is a forward step and a backward step back to back. The ideal solution does not give the real load, but it gives its structure: which part of the face pushes, which part would have pulled, and where on the face the load changes sign.

The convention, and what the picture leaves out #

The step is two-dimensional, with a floor and a top that run to infinity in each direction, and the stream far away is uniform at $U$ over a semi-infinite depth. Forces are per unit span, in units of $\tfrac12\rho U^2 h$; the face force counts only the face, since the horizontal floor and top carry no streamwise pressure force. The flow is ideal and attached: every effect of viscosity — the separation at the lip, the corner vortex at the foot — is outside it, and so is the drag they produce. The vertical force is not computed: the suction along the top and the stagnation pressure along the floor each integrate logarithmically to infinity over a wall that runs on for ever, and only a body of finite length has a finite lift.

Who mapped the corners #

Schwarz and Christoffel found the polygon map independently in the late 1860s, and its use for flows over steps and into corners is in every treatment of free-streamline and ideal flow since, from Lamb to Milne-Thomson. That the face force of a step vanishes is d’Alembert’s theorem, 1752, applied to a body whose symmetry is in its map rather than in its shape. Measurements of forward-facing steps in boundary layers, and their separation bubbles at the lip, are a staple of the twentieth-century experimental literature on surface roughness and on the flow over buildings.

How it was checked #

The face’s length, integrated numerically along the mapped wall with a substitution that removes the endpoint singularities, reaches the step’s height to two parts in a hundred million at 30°, three in ten million at 60° and rounding at 90°. The face force, integrated over two hundred thousand intervals, is zero to $4 \times 10^{-16}$, $2 \times 10^{-13}$ and $1.3 \times 10^{-11}$ at those angles, and the right angle’s push is $2/\pi$ to nine figures. Fitted over five decades of distance, the speed near the foot and the lip follows the local corner flow’s exponents, 1 and −1/3, to $10^{-4}$, and the pressure crosses ambient at $h(\tfrac12 + \tfrac1\pi)$ to a part in a thousand. And the ramp refuses no angle, an overhanging face and a negative height.

Still open: the step that has a back #

The step here never comes down. A body has a back as well as a front, and a rectangular block in a stream has four corners: two concave where it meets the floor and two convex at its top edges. The next calculation maps a block of given height and length with a Schwarz–Christoffel map of four factors, computes the force on its front and back faces separately, and asks how the front’s push and pull and the back’s are shared — whether the front’s lip suction still balances its own face, or whether a short block’s two lips borrow from each other — and how much of the block’s real drag separation at each lip explains.