A swept wall makes vorticity along its isobars, not across its flow

Where vorticity comes from traced the vorticity in a flow past a body back to the wall: a fluid that starts irrotational acquires vorticity only where it touches a surface it cannot slip over, and the rate at which a still wall supplies it is set by the pressure gradient along the wall. A wall puts in exactly its own speed followed that source through moving and spun-up walls and found its total over a start-up equal to the wall’s own change of speed. In every case drawn there the source pointed one way — parallel to the surface and across the flow — and the essay ended by naming what three dimensions require: a surface on which the source has a direction, and part of that direction is along the flow.

The source is a vector #

At a still no-slip wall the vorticity that diffuses out into the fluid per unit area and time is, in Lighthill’s form,

$$\boldsymbol\sigma = -\nu\,\frac{\partial\boldsymbol\omega}{\partial n} = -\frac{1}{\rho}\,\mathbf{n}\times\nabla p,$$

with $\mathbf n$ the normal into the fluid. The cross product puts $\boldsymbol\sigma$ in the plane of the wall and at right angles to the pressure gradient: the source lies along the isobars. That is not a convention. The fluid at the wall is at rest, so the momentum equation there reduces to the pressure gradient balancing the viscous stress’s divergence, $\nabla p = \mu\nabla^2\mathbf u = -\mu\nabla\times \boldsymbol\omega$, and the vorticity that viscosity must carry away from the wall to keep that balance is whatever the pressure gradient’s direction requires.

On a two-dimensional body the pressure gradient is along the flow, so the isobars run across it, and the source points across the flow, which is the direction a two-dimensional boundary layer’s vorticity has. The two directions coincide, and the vector character of the source has nothing to show. On a swept body they part.

Where they part #

The cleanest swept body is an infinitely long circular cylinder yawed to the stream. The wind a swept wing feels established the independence principle that makes it simple: the chordwise flow round it is the flow round an unswept cylinder in the stream’s chordwise component, $U_e = 2U_\infty\cos\Lambda\sin\theta$ outside the layer, and the spanwise component $W = U_\infty\sin\Lambda$ rides along unchanged. Nothing varies along the span, so the pressure gradient is chordwise everywhere and the isobars, and with them the source, run exactly along the span.

The flow outside the layer does not. At the attachment line its chordwise speed is zero and it runs purely along the span; round the cylinder the chordwise speed grows, peaks at the suction peak, and the flow turns towards the chord. The share of the source that points along the local flow is $W/\sqrt{U_e^2 + W^2}$, the cosine of the angle between the span and the flow. It is one at the attachment line, whatever the sweep. At the suction peak it is 0.088 for 10° of sweep, 0.18 for 20°, 0.33 for 35° and 0.51 for 50°. On an unswept cylinder it is zero everywhere.

The part of the source that is not along the flow is the ordinary across-the-flow vorticity of a boundary layer. The part that is along the flow is streamwise vorticity, created at the wall as streamwise vorticity — not made later by tilting spanwise vortex lines, which is the account usually given of where a three-dimensional layer’s streamwise vorticity comes from.

How much, and where #

The share alone overstates the attachment line’s importance, because the source there is small: the pressure gradient divided by the density is $U_e\,dU_e/ds$, which vanishes where $U_e$ does. Weighted by its size, the source on a cylinder swept 35° peaks forty-five degrees round from the attachment line, where 44 per cent of it points along the flow, and falls to zero at the suction peak; beyond the peak the flow decelerates, the gradient reverses and the source changes sign, putting in vorticity of the opposite sense — the vorticity that in two dimensions eventually separates the layer.

Integrated over the accelerating front, from the attachment line to the suction peak, the share of the new vorticity that is streamwise is 0.30 at 20° of sweep, 0.50 at 35° and 0.68 at 50°. A typical transport wing is swept between 25° and 35° at its leading edge, so between a third and a half of the vorticity its leading-edge region generates points along the flow from the moment it is made. That is not a small three-dimensional correction to a two-dimensional layer; at those sweeps it is a comparable share of the whole.

The integral in closed form #

The integrated share has a closed form, and it says what the source is adding up. From the attachment line to the suction peak the whole source integrates to $\int U_e\,dU_e = \tfrac12 U_{e,\max}^2$: the chordwise kinetic energy the flow gains round the front, which is Bernoulli’s equation read as a statement about vorticity added up. The part along the flow integrates to $\int U_e\,dU_e\,W/\sqrt{U_e^2+W^2} = W\big(\sqrt{U_{e,\max}^2 + W^2} - W\big)$. With $U_{e,\max} = 2U_\infty\cos\Lambda$ and $W = U_\infty\sin\Lambda$, the share is

$$\frac{2\sin\Lambda\big(\sqrt{4\cos^2\Lambda + \sin^2\Lambda} - \sin\Lambda\big)}{4\cos^2\Lambda},$$

0.497 at 35°, which is the figure’s number without a quadrature. It grows like $\Lambda$ at small sweep, because the streamwise part is the spanwise speed times the change in total speed, and reaches one only as the sweep approaches ninety degrees, where the chordwise flow and the source both vanish.

The flow curves across the isobars #

Unrolling the cylinder makes the geometry visible. The isobars, along which the source lies, are straight lines along the span. The streamlines of the outer flow start near the attachment line running almost spanwise — the chordwise speed is small there — and curve over towards the chord as the flow accelerates round the cylinder. At 50° of sweep they travel several radii along the span before turning; at 20° they turn within a radius. The angle between a streamline and the isobar it is crossing is the angle between the flow and the source, and the figure is the previous ones drawn as a map.

Seen this way the result is almost a tautology, and its consequence is not. A boundary layer’s vorticity is its shear: vorticity across the flow is a velocity profile that grows away from the wall in the flow’s direction, vorticity along the flow is a profile that turns with height — a crossflow. The source along the isobars therefore means that a swept layer is born skewed. Its velocity near the wall points more along the isobars’ normal than the outer flow does, because the pressure gradient that drives the near-wall fluid is chordwise while the outer flow’s momentum is partly spanwise.

A crossflow with no spanwise pressure gradient #

The usual explanation of a swept wing’s crossflow is pressure: the slow fluid near the wall is pushed sideways by a pressure gradient the fast fluid outside can resist. On an infinite swept cylinder there is no spanwise pressure gradient at all, so the explanation has to be read carefully, and the source makes it precise. The only gradient is chordwise. It acts on the whole layer, but the fluid near the wall, slowed by friction, has less chordwise momentum to resist it with and is turned further towards the chord than the outer flow; its spanwise momentum, which no gradient acts on, is only diffused. The near-wall flow is therefore turned towards the chord, relative to the outer flow, while the layer is accelerating, and away from it once the gradient reverses past the suction peak.

The vorticity statement and the momentum statement are the same statement. The source along the span is the chordwise pressure gradient expressed as vorticity, and the part of it along the flow is the turning of the near-wall fluid that the momentum argument describes. Neither needs a spanwise pressure gradient, and the commonly drawn picture — boundary-layer fluid draining spanwise under a spanwise gradient — belongs to a tapered or finite wing, where that gradient exists, and adds to the crossflow computed here rather than causing it.

The turn, in an exact solution #

Near the attachment line the boundary layer has an exact solution of the full Navier–Stokes equations, Hiemenz’s stagnation flow with Cooke’s spanwise flow added: the chordwise velocity is $ax\,f'(\eta)$ and the spanwise $W\,g(\eta)$, with $f''' + ff'' + 1 - f'^2 = 0$ and $g'' + fg' = 0$. The two profiles have different shapes. The chordwise one is driven by the pressure gradient and rises steeply from the wall, its gradient there 1.2326; the spanwise one feels no pressure gradient, only viscosity dragging it towards the outer flow, and rises more slowly, 0.5705. At the station where the outer flow runs at 45°, the flow near the wall therefore runs at 25° — turned towards the chord — and swings round to 45° through the layer.

That turn is the streamwise vorticity the source put in, read off the velocity rather than the pressure. The same solution gives the source both ways: differentiating the computed profiles at the wall, the flux of vorticity out of it has a spanwise part equal to the pressure gradient divided by the density, to two parts in a million, and a chordwise part that is zero to the accuracy of the differencing. The source points along the span, as the isobars do, in an exact solution of the equations rather than by the formula.

The attachment line of a real wing #

The exact solution has a practical reading at a wing’s leading edge, where the attachment line is the first place the layer forms. Its thickness scale is $\sqrt{\nu/a}$, with $a$ the chordwise strain rate of the outer flow there, and the spanwise flow along it is $W$, so the layer’s own Reynolds number is $\bar R = W/\sqrt{\nu a}$. For a cylinder of radius $r$, $a = 2U_\infty\cos\Lambda/r$. A transport wing at 230 metres a second at cruise altitude, swept 30°, with a leading-edge radius of two centimetres, has $a$ near twenty thousand per second and $\bar R$ about 130. Experiments put the value above which turbulence carried along the attachment line from the wing root survives near 250, and the value at which the laminar attachment line itself becomes unstable near 580; like every transition number, both depend on the disturbances present. The leading edge of this wing is laminar, and the layer that leaves it carries the skew computed above. Because $\bar R$ grows as the square root of the leading-edge radius, it is largest inboard, where the chord and the radius are biggest, and that is where turbulence from the fuselage junction threatens to run out along the attachment line — one reason the root is treated separately in a swept wing’s design.

What the crossflow becomes #

The turned layer is not merely a curiosity of the profile. A velocity profile that turns through the layer has an inflection in the direction normal to the outer flow, and an inflected profile can be unstable to disturbances that do not need viscosity to grow, the kind whose growing waves all lie inside one circle. On a swept wing that is the crossflow instability: co-rotating streamwise vortices, aligned within a few degrees of the outer flow, that appear close behind the leading edge and are among the main routes to turbulence on swept wings. Its growth is set by the size of the crossflow, which the source analysis says is a large share of the layer’s vorticity at practical sweeps from the very start. The instability’s existence follows from the vector character of a scalar-looking statement: the wall makes vorticity where the pressure changes, along the isobars, and the isobars on a swept wing are not where the flow is going.

How it was checked #

The swept-attachment-line equations were solved by shooting with a fourth-order Runge–Kutta march to eight boundary-layer thicknesses, the wall gradient of $f$ found by bisection so that $f'$ reaches one and $g$ scaled so that it reaches one. They return 1.23259 and 0.57047, the tabulated 1.2326 and 0.5705. The vorticity flux at the wall was then computed from the velocity field — the derivatives of the vorticity taken by one-sided differences of the computed profiles, not from the equations at the wall — and compared with the pressure gradient: the spanwise part agrees to $2\cdot10^{-6}$ and the chordwise part is $7\cdot10^{-7}$ of the spanwise. The potential-flow share calculation has no free parameters; its integrals are midpoint sums of four thousand points.

The convention: the source’s sign, and the share as a cosine #

The source is the flux of vorticity out of the wall into the fluid, $-\nu\,\partial\boldsymbol\omega/\partial n$, which with the normal into the fluid is $-(\mathbf n\times\nabla p)/\rho$ at a still wall; its sign changes where the pressure gradient does. The share along the flow is the cosine of the angle between the source and the outer flow’s direction, unsigned. The cylinder has unit radius, the stream unit speed, and angles round it are measured from the attachment line.

What the picture cannot show #

The outer flow here is potential flow round a cylinder, which a real cylinder departs from beyond about 80° when its layer separates; the accelerating front, where the integrated shares are taken, is where the potential flow is good. The analysis says where vorticity is made and in which direction, not where it goes: once made it is carried, tilted and stretched by the layer’s own flow, and the crossflow the source starts is then amplified or reduced by the turning of the outer streamlines, which the share does not compute. Definitions of the flux differ on a curved wall — Lyman’s adds a term involving the wall’s curvature and its vorticity — and the difference, small for a thin layer, is left out here. And a moving wall adds its own acceleration, the term the previous essay was about. The flow is also incompressible. On a transonic swept wing the suction peak is far sharper than a cylinder’s, concentrated within a few per cent of the chord behind the leading edge, so the source is concentrated there too, in the region where the outer flow is still turning fastest — which on these arguments raises the streamwise share rather than lowering it, but is not computed here.

Who found it, and when #

Lighthill set out the wall as the source of vorticity, with the flux equal to the pressure gradient’s tangential part, in his 1963 chapter on boundary layers; Morton in 1984 and Lyman in 1990 gave the general vector forms for moving and curved walls. Hiemenz’s stagnation flow is from 1911 and Cooke’s extension to the swept attachment line from 1950. The crossflow instability was identified on swept wings by Gray in flight tests in 1952 and explained by Owen and Randall the same year as an inflectional instability of the skewed profile. Reading the skew as the source pointing along the isobars is the connection this essay draws between them. The two are usually told in different places: the vorticity-source papers are about how a wall makes vorticity in general, and the crossflow papers about one instability on one kind of wing, though the second is the first’s clearest consequence.

Still open: the source on a wall curved two ways #

A cylinder is curved one way and swept, and its isobars are straight. A body curved in two directions — a wing’s leading edge near the root, a ship’s bow, a fuselage nose — has isobars that curve across the surface, and its source turns with them while the flow turns differently. The next calculation takes potential flow over an ellipsoid at incidence, whose surface pressure is known in closed form, maps the isobars and the surface streamlines, and integrates the streamwise share of the source over the region ahead of the pressure minimum, asking how much streamwise vorticity a nose makes without any sweep at all — which would say how much of the vorticity that rolls up into a body’s lee vortices at incidence was born streamwise at the wall.