A nose at incidence makes its vorticity along the flow

A swept wall makes vorticity along its isobars, not across its flow took the source of vorticity at a no-slip wall — Lighthill’s flux, which at a stationary wall equals the pressure gradient turned through a right angle in the surface — and showed that it lies along the isobars. On a swept cylinder the isobars run along the span and the flow runs at an angle to them, so part of the source points along the flow: vorticity born streamwise, the kind that rolls up into the vortices a swept wing sheds. It ended on a body curved in two directions, whose isobars curve across its surface, and asked how much streamwise vorticity such a body makes without any sweep at all.

The answer, for the simplest such body, is most of it. A slender nose at a few degrees of incidence makes the larger share of its wall vorticity pointing along the flow, and the reason is not curvature of the isobars but which way the pressure changes fastest.

A body whose surface flow is one vector #

A prolate spheroid has a remarkable potential flow. On any ellipsoid in a uniform stream, the velocity at the surface is the component tangential to the surface of a single constant vector,

$$\mathbf q = \mathbf V - (\mathbf V\cdot\mathbf n)\,\mathbf n, \qquad \mathbf V = \big((1+k_1)U\cos\alpha,\; (1+k_2)U\sin\alpha,\; 0\big),$$

where $\alpha$ is the incidence and $k_1$, $k_2$ are Lamb’s added-mass coefficients for motion along and across the axis. For a slender body $k_1$ is small and $k_2$ near one, so the crossflow component is nearly doubled and the axial one barely changed. The surface speed, the pressure and its gradient are therefore closed form, and so is the source, $\boldsymbol\sigma = \mathbf n \times \nabla_s(q^2/2)$, which lies in the surface and along the isobars. Checked numerically: the surface velocity is tangent to $3\times10^{-16}$, the source perpendicular to the pressure gradient to $2\times10^{-17}$, and a stagnation point at zero incidence has a pressure coefficient of one to $2\times10^{-11}$.

Most of the source points along the flow #

The measure is the cosine of the angle between the source and the local surface flow, weighted by the source’s strength and integrated over the region where the flow is accelerating along itself — ahead of the pressure minimum on every surface streamline, where the source is the vorticity the boundary layer will carry downstream. At zero incidence it is exactly zero: the flow runs along the meridians, the isobars are circles round the axis, and the source is everywhere across the flow. A sphere at any incidence also makes none, because its two coefficients are equal, the vector $\mathbf V$ is parallel to the stream, and its flow is axisymmetric about the stream’s direction.

The sphere’s zero has a reading in terms of how the flow is built. Any body’s potential flow far away is the stream plus a doublet, and flows add up is the statement that the two can be superposed. A sphere’s doublet always points along the stream, whatever the incidence, because a sphere has no direction of its own. A spheroid’s does not: its doublet strengths along and across its axis differ by the ratio of $1 + k_2$ to $1 + k_1$, so at incidence the effective stream $\mathbf V$ that shapes its surface flow is tilted further from the axis than the real stream is — by about twice as much for a slender body. That tilt is the crossflow, and the streamwise source is its signature at the wall. No elongation, no tilt, no streamwise source.

Give a slender body a little incidence and the share climbs fast. At 10° a body twice as long as it is wide makes 20 per cent of its source streamwise, one four times as long 55 per cent, one eight times as long 81 per cent and one twelve times 89 per cent. At 5° the fineness-8 body is already at 63 per cent, and at 2° at 34. On a fuselage at an ordinary angle of attack, most of the vorticity the wall makes is born pointing along the flow.

Why: the pressure changes round the body faster than along it #

The pressure on a slender body at incidence is the sum of two parts with different shapes. The axial part — the body moving along its own axis — varies slowly along the length, from a stagnation point at the nose to a shallow suction along the middle and back. The crossflow part — the body moving across its own axis — varies round each cross-section like the flow round a circular cylinder, from stagnation on the windward line to a strong suction on the sides, and its coefficient is nearly twice the crossflow’s own because $k_2$ is near one. At the middle of a fineness-4 body at 10°, the windward line sits at a pressure coefficient of −0.13 and the side line at −0.24, against −0.17 all round at zero incidence.

The source is the pressure gradient turned through a right angle. A gradient that runs round the section — the crossflow’s — gives a source that runs along the body, and along the body is nearly the direction of the flow, which is mainly axial. A gradient that runs along the body — the axial flow’s — gives a source round the section, across the flow. On a slender body at incidence the crossflow’s gradient is the larger over most of the surface, because it changes over the body’s small radius while the axial part changes over its long length, and so most of the source is streamwise. A sphere has no such disparity of lengths, and its share is zero; a longer body has more, and its share rises towards one.

Where on the body it is made #

Along the windward line, the plane of symmetry, the share is exactly zero: the flow runs along the line and the pressure changes only along it. On the flanks — between the windward line and the side, and between the side and the leeward line — the share rises to 0.88 at mid-length, where the crossflow round the body is accelerating or decelerating and its gradient runs round the section. On the side line itself it falls to zero at mid-length, where the crossflow is fastest and its gradient vanishes, and is large ahead of and behind that point. The streamwise source is made on the flanks, either side of the crossflow’s suction peak.

A pair, from the start #

The streamwise source has a sign, and it is opposite on the two sides of the plane of incidence: the crossflow runs from windward round each side in opposite senses, and the source it makes points forward on one flank and aft on the other. Integrated over one side, it grows nearly in proportion to the incidence at small angles — 0.021 at 2° and 0.132 at 10°, in units of the stream speed squared times the body’s half-length, for fineness 4 — and faster than that beyond.

That is the beginning of the pair of lee vortices a body at incidence carries. Their vorticity is streamwise and of opposite signs on the two sides, and it is usually described as appearing where the crossflow boundary layer separates from the leeward flanks. The calculation here says that the streamwise vorticity is already there before anything separates: the wall makes it, on the flanks, in attached flow, and separation only decides where it leaves the wall. Lift out of a failure is the same process on a delta wing, where a sharp edge fixes the separation line; on a smooth body the line is set by the crossflow boundary layer, and the vorticity it releases was supplied upstream.

The longer the body, the more streamwise #

At a fixed incidence the share rises steadily with fineness — zero, 20, 55, 81, 89 per cent at fineness 1, 2, 4, 8, 12 — and it is a slender body’s property in exactly the way a body with no lift, and a moment anyway is: slender-body theory treats each cross-section as a two-dimensional cylinder in crossflow, and a cylinder in crossflow makes its source along its own axis. The longer the body, the more exactly each section is such a cylinder, and the more of its source is axial. The fineness of an airliner fuselage, around ten, puts it near the top of the curve.

An airliner’s fuselage in the climb #

Put numbers on a fuselage of fineness ten — a common airliner proportion — at the few degrees of incidence a climb asks of it. At 3° the streamwise share of its wall source over the accelerating region is 55 per cent; at 5°, 72. These are angles at which nothing separates on a smooth fuselage, at which the lee vortices are weak or absent, and at which an account of the vorticity in terms of separation would say there is none of the streamwise kind. There is, and it is the larger part. It is carried in the attached boundary layer along the flanks and up towards the top of the fuselage, where it meets the opposite-signed vorticity from the other flank; how much of it cancels there and how much rolls up is the question attached flow cannot answer.

The same flanks are where the wing root sits. A fuselage lowers its wing root’s critical Mach number found the fuselage’s overspeed adding to the wing’s at the root; the streamwise vorticity the fuselage’s flank makes is delivered to the same place, into the root’s boundary layer, where it is one of the sources of the junction’s own horseshoe vortex.

What the vorticity amounts to #

The source is a flux, the rate at which vorticity enters the fluid per unit area of wall, and its integral over a region of the surface is the rate at which that region adds to the fluid’s total vorticity, a vorticity integrated over volume. The streamwise part integrated over one side of a fineness-4 body at 10° is 0.132 in units of $U^2$ times the half-length $a$. Over the time the stream takes to pass the body, $2a/U$, that supplies a volume integral of streamwise vorticity of $0.264\,U a^2$, and spread along the body’s length $2a$ it is the vorticity of a line vortex of circulation $0.13\,Ua$ running the length of each flank. For a body of half-length one metre in a stream of fifty metres a second that is about six and a half square metres per second on each side, of opposite signs. Circulation is vorticity, added up is the identity that turns the wall’s supply into that number.

It is an upper bound, not a prediction. Some of the streamwise vorticity made on one flank meets the opposite-signed vorticity of the other near the top of the body and cancels; some is carried into the wake without rolling up. What the potential-flow calculation does say is that the supply is large enough, in attached flow, to account for lee vortices of a substantial fraction of $Ua$ — without appealing to anything that happens at separation, which is a different statement from the usual one.

How this changes the swept-wing picture #

The swept cylinder of the earlier essay made streamwise vorticity because its isobars were straight and its flow was not along them. The nose makes it for a different reason: its isobars curve round the body, and its flow, mainly along the body, crosses them at an angle that depends on how the crossflow’s pressure compares with the axial flow’s. Both are the same statement, that a source along the isobars has a streamwise part whenever the flow is not perpendicular to the isobars, and both are statements about attached flow. The sweep is one way to arrange the angle and incidence on a slender body another; the swept wing’s share at 40° of sweep and the nose’s at 10° of incidence are of the same order, and on an aircraft both operate at once, at the wing root where the fuselage’s flank meets the wing’s leading edge.

Two exact zeros the source must return #

The source is computed by central differences of $q^2/2$ in the surface’s orthogonal coordinates — the meridional angle and the azimuth — and its integrals on a grid of a hundred by a hundred points. The two exact zeros test the whole chain: at zero incidence the streamwise share is $5\times10^{-11}$, and on a sphere at 25° it is $10^{-11}$, both the differencing’s error. Lamb’s coefficients are the existing closed forms, with the series used near the sphere where the closed form cancels. A wall puts in exactly its own speed checked the source’s total against the wall’s speed on a flat plate; this body has no such total, because its source has two components and the streamwise one integrates to zero over the whole surface by symmetry.

The convention: the source along the isobars #

The source is $\boldsymbol\sigma = -(\mathbf n \times \nabla p)/\rho$ at a stationary wall, with $\mathbf n$ the outward normal, which with Bernoulli’s pressure is $\mathbf n \times \nabla_s(q^2/2)$. The streamwise share is the absolute cosine of the angle between the source and the local surface velocity, weighted by the source’s magnitude; the accelerating region is where the surface velocity points up the gradient of $q^2$. Incidence is in the body’s plane of symmetry, and the fineness ratio is the length over the diameter. Integrals are in units of the stream speed squared times the half-length, and positions along the body are measured from the upstream nose as a fraction of the length. The share is a weighted mean of a cosine, so it is one when every bit of source points along the flow and zero when every bit points across it, regardless of how strong the source is in total.

What the inviscid picture cannot show #

The boundary layer’s own response. The source is what the wall supplies — the thin layer at the wall carries it; how the vorticity is carried, diffused and eventually released depends on the boundary layer, whose crossflow on a body at incidence is strong and three-dimensional. Separation. Past a crossflow separation line the potential flow is no longer the outer flow, and on the leeward side of a body at more than a few degrees it is not. The nose’s exact shape. A real nose is not an ellipsoid; a blunter or sharper one changes the axial pressure gradient near the tip, where the share is smallest. Unsteadiness at high incidence, where the lee vortices become asymmetric, is outside a steady potential flow altogether.

Lighthill’s source and slender bodies #

Lighthill wrote the wall’s vorticity flux in terms of the pressure gradient in 1963. That the flow over any ellipsoid is the tangential part of a constant vector goes back to the nineteenth-century theory of the ellipsoid in a uniform field; Lamb gave the coefficients. Crossflow on slender bodies at incidence is the basis of Allen’s and Jorgensen’s methods of the 1950s for missile and fuselage loads, in which the lee vortices are attributed to crossflow separation; the streamwise share of the attached wall’s source, and its dependence on fineness and incidence, follow from putting the two classical pieces together.

Still open: where the streamwise vorticity leaves the wall #

The source says where streamwise vorticity is made, not where it goes. The next calculation solves the crossflow boundary layer on the spheroid at incidence along its surface streamlines, carries the vorticity the source supplies, and asks where the crossflow layer separates on the leeward flanks, how much of the streamwise vorticity made upstream of that line is shed into the lee vortices rather than cancelled by the opposite-signed vorticity of the leeward side, and whether the vortices’ strength at a given incidence follows the integrated streamwise source computed here — which would make the attached-flow source a predictor of the separated flow’s vortices.