A block's front face is pulled upstream by its own lip

A step’s face is pushed at its foot and pulled at its lip, equally mapped ideal flow up a forward-facing step with one Schwarz–Christoffel factor and found something exact. The stagnation at the concave foot pushes the face downstream; the suction at the convex lip, where the flow turns the corner at infinite speed, pulls it upstream; and the two are equal — $2/\pi$ of the dynamic pressure times the height each — at every ramp angle, because one reflection of the map exchanges the two corners. The face, on balance, feels nothing.

The step never comes down. A body has a back as well as a front, and a block on a floor has four corners: two concave where it meets the floor, two convex at its top edges. The essay ended on the question that adding the back raises — whether each face still balances its own push and pull, or whether a short block’s two lips borrow from each other — and on how much of a real block’s drag separation at each lip explains. The first answer is that the balance is lost, in a definite direction, and the direction changes what the second question means.

Four corners in one map #

A Schwarz–Christoffel map takes the upper half of an auxiliary plane onto any region bounded by straight lines, with one factor for each corner: a factor $(w - w_k)^{\alpha_k/\pi - 1}$ for a corner of interior angle $\alpha_k$ on the fluid’s side. A block standing on a floor, with the fluid above, has two feet where the fluid sees a right angle, $\alpha = \pi/2$, and two lips where it sees three right angles, $\alpha = 3\pi/2$. Placing the four corners symmetrically on the real axis — front foot at $-a$, front lip at $-b$, rear lip at $b$, rear foot at $a$ — gives

$$\frac{dz}{dw} = \sqrt{\frac{w^2 - b^2}{w^2 - a^2}},$$

which tends to one far away, so the stream there is uniform and the complex potential is simply $Uw$. The speed on the wall is then the modulus of $dw/dz$: zero at each foot, infinite at each lip, as the flow near any corner requires. The two constants $a$ and $b$ are fixed by the block’s height and length, which are integrals of $|dz/dw|$ along the faces and the top; their ratio sets the block’s proportions and their scale its size. Computed with the endpoint singularities removed by substitution, the map draws the block asked for to a part in $10^{12}$.

The figure shows the pressure the map gives along the whole wall. Ahead of the block the floor’s pressure rises towards stagnation at the front foot; up the front face it falls, crosses zero and plunges to minus infinity at the lip; along the top it recovers partway, and on a long block nearly to ambient; down the back the pattern repeats in mirror image. On a block as long as it is tall the top never recovers: its middle is at a pressure coefficient of −1.89, which is to say the flow there is moving 1.7 times as fast as the stream.

The face is pulled #

Split the front face’s force into the part where the pressure is above ambient — the push, near the foot — and the part below — the pull, near the lip. On a long block both tend to the step’s $2/\pi = 0.637$ and cancel. On a short block they do not. The push hardly changes: 0.63 on a block ten heights long, 0.59 on one as long as it is tall, 0.57 on one a quarter as long. The pull grows: 0.68, 0.89, 1.21. The face’s net force is a pull upstream, of 0.052 of the dynamic pressure times the height on the ten-height block, 0.300 on the cube-like one and 0.648 on the short one. A block a tenth of a height long — a fence — has its front face pulled upstream by nearly the whole dynamic pressure times its height, 0.96.

The reason is the lip. The suction at a convex corner is set by how fast the flow is when it arrives there, and at the front lip of a short block the flow is already fast, because it is in the region of fast flow over the top that the rear lip is also making: on a short block the two lips are close enough that each sits in the other’s acceleration. That is the borrowing the earlier essay asked about, and it is one-way: the feet, which are stagnations, are not made more stagnant by each other’s presence, so the push stays where it was while the pull grows. The pressure up the front face crosses zero at 0.816 of the height on a step — exactly $\tfrac12 + 1/\pi$, as the step essay found — at 0.778 on the cube-like block and 0.748 on the short one: the low pressure of the top reaches down the front face.

The back face, by the map’s fore-and-aft symmetry, is the mirror image, and the mirror reverses what each pressure does to the block. Its foot’s stagnation presses on a face whose outward normal points downstream, so it pushes the block upstream; its lip’s suction pulls the block downstream; and again the lip wins. The back face’s net force is 0.300 downstream for the cube-like block, exactly cancelling the front’s. That is d’Alembert’s paradox again, a zero total drag, but distributed in a way the usual picture reverses: each face of the ideal block is pulled by its own lip, the front upstream and the back downstream.

Why the usual picture is upside down #

The usual picture of a bluff body’s drag — a stagnation pressure pushing on the front, a low pressure sucking on the back — is the separated flow’s picture, and it is the reverse of the ideal flow’s. In ideal flow the front face is pulled upstream by its lip and the back pulled downstream by its own; the total is zero. In a separated flow the lips do not get their suction: the flow leaves each lip as a free shear layer instead of turning the corner, the corner’s infinite speed never happens, and the pressure near the lip stays close to that of the separated region beside it. The front face then keeps its push and loses its pull, and its net force changes sign — from 0.3 upstream to about 0.6 downstream for the cube-like block. The back face sits in the separated wake at a base pressure below ambient and is pulled downstream by it.

So separation does not add a drag to an ideal flow that had none; it removes a distribution of forces that happened to sum to zero and keeps some of them. The drag that results is roughly the front face’s push, 0.55 to 0.63 of the dynamic pressure times the height whatever the block’s length, plus whatever the wake’s suction on the back is worth. That is the second half of the earlier essay’s question in a form a measurement can answer: the push the ideal flow computes is a lower bound on what separation at the front lip leaves on the front face, and it hardly depends on the block’s proportions. What depends on the proportions, through the wake’s reattachment on the top and the base pressure behind, is the rest.

The top is lifted #

The fast flow over the top sucks the block off the floor. On the cube-like block the top’s lift is four times the dynamic pressure times the height, and it falls only slowly as the block shortens — a quarter as long, the lift is still 3.5 — because the suction near two close lips is intense: per unit of the top’s own length the short block’s top is under a suction of fourteen dynamic pressures on average. A long block’s top, whose middle has recovered nearly to ambient, carries its lift near its two ends.

That lift is the force that real flows over roofs, low buildings and shipping containers on a deck exert most reliably, because a top is usually long enough for the separated flow from the front lip to reattach, and the region just behind the front lip is under strong suction in separated flow too. Ideal flow overstates the lift, since it gives each lip an infinite suction; but it puts the suction where real flows put it, at the edges, and that is why roof sheeting fails first at the eaves.

The floor feels it ahead #

The block’s stagnation reaches upstream along the floor. Ahead of a cube-like block the floor’s pressure coefficient is above a tenth for 3.8 heights, and a longer block, which turns more of the stream, reaches further — 5.7 heights for one twenty heights long. That upstream pressure rise is what a boundary layer approaching the block meets, and an adverse gradient of that length is what makes the layer separate ahead of the foot and roll into the horseshoe vortex that wraps round every obstacle on a wall. The ideal flow says how far ahead the layer is warned; how much uphill a layer can take says whether it survives the warning.

A fence, the block with no length #

Shorten the block until it is a fence, a plate standing on the floor, and two limits appear, one infinite and one exact. The front face’s pull grows without bound: 1.5 of the dynamic pressure times the height for a block three hundredths of a height long, 2.0 at a hundredth, 2.5 at three thousandths — half a unit for every threefold shortening, a logarithm. The two lips have merged into one sharp edge, round which ideal flow turns through a full half-turn, and its speed there grows as the inverse square root of the distance rather than the inverse cube root of a right-angled lip; integrated up the face, the suction diverges logarithmically. The back face’s pull grows in exactly the same way, and on a plate of no thickness the two faces are one object, so the fence feels their difference, which is zero.

The top’s lift, meanwhile, tends to a finite number: 3.15 of the dynamic pressure times the height at a hundredth of a height long, 3.146 at a thousandth, 3.143 at three ten-thousandths — converging on $\pi$. A finite upward force on a top of vanishing length is an edge force, the same kind of object as the finite force an infinite speed makes at the leading edge of a thin plate at incidence: a suction of unbounded intensity over a length that shrinks as fast, leaving a definite pull. For the fence it pulls the plate upward, along itself, with a force of $\pi$ times the dynamic pressure times its height, and a fence anchored to a floor carries that tension in its own structure — in ideal flow. A real fence’s flow separates from its edge and the force goes, which is the leading-edge suction’s fate too.

In numbers, a container on a deck #

The figures scale with the dynamic pressure, and a concrete case shows their size. A shipping container 2.6 metres high, standing broadside to a wind of twenty metres a second, sees a dynamic pressure of 245 pascals. If its flow stayed attached — which it does not — the front face would be pulled upstream by 0.3 of that over its height, 190 newtons per metre of the container’s length, the back pulled downstream by as much, and the top, for a container as long in the wind’s direction as it is tall, lifted by four times the dynamic pressure times the height, 2.5 kilonewtons per metre. The real container’s flow separates at its front lip; its front face is pushed rather than pulled, by about 0.6 of the dynamic pressure times its height, 380 newtons per metre; and its top, in the separated region behind the lip, is still sucked up, by an amount the reattachment decides. The ideal flow gets the sign of the front face’s force wrong and the location of the top’s suction right, which is the general rule for bluff bodies: ideal flow is a poor guide to drag and a fair one to where the low pressures are.

Where the two-dimensional block lives #

Every number here is for a block infinitely long across the stream, in a uniform stream above a floor that is a streamline. The floor is also a plane of symmetry: the flow over a block on a floor is the upper half of the flow round a rectangular cylinder of twice the block’s height in an open stream, with the floor as its image. So the forces computed here are also the forces on the front and back faces of a free rectangular cylinder, per half — a free cylinder’s front face is pulled upstream by twice these numbers, in ideal flow, and its sides lifted outward.

Real blocks on real floors stand in boundary layers, not uniform streams; the oncoming flow is slower near the floor, so the stagnation at the foot is weaker than ideal flow’s and the push smaller. Real blocks are also finite across the stream, and air flows round their ends as well as over their tops, which lowers every pressure difference. Neither changes the structure: each face pulled by its own lip in ideal flow, a pushed front and a wake-sucked back once the lips separate.

The convention the numbers rest on #

Forces are per unit span across the stream, over $\tfrac12\rho U^2 h$ with $h$ the block’s height and $U$ the uniform stream’s speed; the top’s lift is also given over $\tfrac12\rho U^2\ell$. Positive streamwise force is downstream. The pressure coefficient is $1 - q^2/U^2$. Lengths are in block heights, and the block’s length $\ell$ is that of its top. The lips’ suctions are infinite and integrable, so every force is finite; the figures draw the suction cut off at −6.

Who mapped it #

Schwarz and Christoffel’s maps are of the 1860s; their use for flows over steps, fences and blocks on walls is in Milne-Thomson’s text, and the step’s equal push and pull is the earlier essay’s. Flows over two-dimensional blocks, ribs and fences have been measured since the 1930s for building aerodynamics and heat-exchanger ribs, and the separated picture — a stagnation-pushed front and a wake-sucked back — is the one those measurements established. The ideal flow’s reversed distribution, and the push that survives separation, are what the map computes here.

How it was checked #

For blocks a quarter, one and four heights long the map’s face height comes out at one to $2 \times 10^{-12}$ and its top at the asked length to $3 \times 10^{-15}$. As the block lengthens the front face’s net force falls towards the step’s zero — 0.052 at ten heights, 0.015 at forty — and its push towards $2/\pi$, 0.627 at forty, which are the step essay’s results recovered as a limit rather than assumed. The pressure on the face crosses zero at 0.816 of the height on the forty-height block, against the step’s exact $\tfrac12 + 1/\pi = 0.818$. The model refuses a block of no length, a negative one and one longer than its quadratures can resolve.

Still open: the block whose front lip separates #

The ideal block keeps every lip’s suction, and a real one keeps none at its front lip: the flow leaves as a free shear layer, and on a block long enough it reattaches on the top. The next calculation would let the front lip separate as a free streamline at a stated cavity pressure, in the way a stalled plate’s flow is computed, and ask how the front face’s force, the top’s lift and the reattachment length depend on the block’s length and the cavity pressure — and whether the push that ideal flow computes, which separation is supposed to leave untouched, is in fact untouched once the lip’s pressure is the cavity’s rather than minus infinity.