ISEGORIA / MATH ENCYCLOPEDIA
Active flow control: steering with jets of air
A wing changes its lift by changing its circulation, and for a century that has meant moving a flap. Active flow control does it with air instead: a thin jet blown from a slot near the trailing edge. Blown over a rounded edge, the jet clings to the curve, the Coanda effect, because the pressure inside a curved jet sheet must fall to hold it on its curved path, and it drags the point where the flow leaves the wing round with it. That moves the rear stagnation point, which a sharp edge would pin by the Kutta condition, and so changes the circulation and the lift with no moving parts. The jet’s strength is measured by its momentum coefficient Cμ, and a little goes a long way: the lift gained is many times the jet’s own thrust. BAE Systems and the University of Manchester flew the MAGMA UAV with circulation control on its wing and fluidic thrust vectoring in 2017, after DEMON in 2010. DARPA’s CRANE program is building the Aurora Flight Sciences X-65, an uncrewed demonstrator of about Mach 0.7 with fourteen such effectors across its flying surfaces; it keeps conventional control surfaces too, for testing and for expanding its flight envelope, and is expected to fly in 2027. For a stealthy aircraft the attraction is a skin with no hinge lines or gaps, fewer moving parts and less weight. These experiments start from jet-flap theory, build a circulation-control wing on the Joukowski map from the aerodynamics topic, and end with the allocation problem every such aircraft must solve: which jets to blow to get the moment the pilot asks for.
Before you begin: Aerodynamics (circulation, the Kutta condition, the Joukowski map), complex numbers, linear algebra
Predict, manipulate, then test yourself: every experiment ends with practice questions that are generated fresh at three levels. Graphs illustrate the mathematics; they do not replace a proof.
1. The jet flap: lift from a sheet of air
Blow a thin, fast sheet of air out of the trailing edge, turned down at an angle \(\tau\) to the chord. The sheet carries momentum, and it can only curve if the pressure differs across it, so it behaves like a flexible extension of the wing that carries load behind it. Spence solved the thin-aerofoil problem in 1956: the lift grows with the momentum coefficient \(C_\mu\), the jet’s momentum flux divided by the dynamic pressure and the wing area, at first like \(\sqrt{C_\mu}\). The jet’s own thrust, turned down, supplies only \(C_\mu\sin(\tau+\alpha)\); the rest is supercirculation, extra circulation that the jet induces round the whole wing. Drag the operating point, turn the jet, and compare it with a hinged flap.
Worked example. Blow \(C_\mu=0.05\) at \(\tau=30^\circ\) on a wing at \(\alpha=4^\circ\). Then \(\partial C_L/\partial\tau=0.809\) and \(\partial C_L/\partial\alpha=6.564\) per radian, so \(C_L=0.5236\times0.809+0.0698\times6.564=0.882\), against \(2\pi\times0.0698=0.439\) unblown. The jet adds \(\Delta C_L=0.443\), an augmentation ratio \(\Delta C_L/C_\mu\) of 8.9, and its own reaction, \(0.05\sin34^\circ=0.028\), is only 6% of that. A plain flap of 25% chord, with a slope of 3.83 per radian, would have to turn 6.6° to do the same. On a 10 m² wing at 50 m/s (\(q_\infty=1531\) Pa) with a 200 m/s jet, the mass flow is 3.83 kg/s and the blowing power \(\tfrac12\dot mV_j^2\) is 76.6 kW, for 6.78 kN of extra lift.
Watch out. Spence’s theory is linear: small \(\tau\) and \(\alpha\), an infinitely thin jet leaving a sharp edge, attached, incompressible, two-dimensional flow. The interpolation formulas of Williams, Butler and Wood fit his computed values at \(C_\mu=1\) and 4 and as \(C_\mu\to0\). On a wing of finite span some of the gain is lost to downwash. The jet’s shape in the diagram is schematic; the curves and the arrows are computed.
Why is the lift gained so much larger than the jet’s thrust?
The jet sheet can only curve if there is a pressure difference across it, so it carries load itself and acts like a flexible flap trailing behind the wing. For the flow to leave the trailing edge in the jet’s direction, the flow round the whole aerofoil must take on extra circulation, and the Kutta–Joukowski lift \(\rho U\Gamma\) of that supercirculation is many times the jet’s momentum: at small \(C_\mu\), \(\Delta C_L/C_\mu\approx2\tau\sqrt{\pi/C_\mu}\).
2. The Coanda wing: circulation without a sharp edge
Round off the trailing edge and the Kutta condition loses its grip: with no sharp edge there is no special point where the flow must leave, and the circulation is free. A circle that clears \(\zeta=1\) by a margin \(\varepsilon\) maps under \(z=\zeta+1/\zeta\) to a section with a rounded trailing edge. Now blow a thin jet tangentially from a slot just ahead of the edge. Its sheet is curved, and the pressure inside it must fall by about \(\rho u_j^2h/R\) to keep it on the curve, so it clings to the surface, the Coanda effect, and drags the rear stagnation point of the outer flow round to the underside. That is circulation control: where the jet lets go sets \(\Gamma\), and the lift follows from Kutta–Joukowski as before. Drag the separation point round the edge.
Worked example. Take \(\mu=0.04i\) and \(\varepsilon=0.2\), so \(a=1.2008\): the section is 18% thick, with an edge radius of 1.6% of the chord. The old sharp-edge direction is \(\theta=-2.29^\circ\). At \(\alpha=4^\circ\) a stagnation point there gives \(\Gamma=4\pi\times1.2008\times\sin6.29^\circ=1.653\) and, with chord 4.067, \(C_L=0.813\). Blow until the jet separates 12° further round, at \(\theta_s=-14.29^\circ\): \(\Gamma=4\pi\times1.2008\times\sin18.29^\circ=4.736\) and \(C_L=2.329\), almost three times the lift with nothing moved. Integrating the pressure round the surface gives the same 2.329 and zero drag. The flow now leaves the edge 53.6° below the chord, and jet-flap theory puts the blowing at about \(C_\mu=0.18\). Through a 0.8 mm slot on a 0.5 m chord at 30 m/s that is a 225 m/s jet, holding a pressure drop of \(C_\mu q_\infty c/R=11\,q_\infty\) across itself.
Watch out. This is potential flow with the stagnation point placed by hand: it says what circulation a given separation point implies, not where the jet separates, which depends on \(C_\mu\), \(h/R\) and the boundary layers and is found by experiment. Real jets entrain air and spread, and a jet that is too thick for its curve, or supersonic and underexpanded, detaches. The link to jet-flap theory matches only the direction in which the flow leaves the edge; it is a rough estimate, and measured circulation-control sections usually need less blowing.
Why does circulation control need a rounded trailing edge?
A sharp edge fixes the circulation through the Kutta condition: the flow must leave there whatever the jet does, so a jet can only act by turning the flow behind the edge, as a jet flap does. A rounded edge has no such point. The jet can carry the stagnation point round it, and every degree it moves changes \(\Gamma\) continuously, through \(\Gamma=4\pi Ua\sin(\alpha-\theta_s)\).
3. Fourteen jets, three moments: control allocation
An aircraft with no ailerons, elevators or rudder still needs roll, pitch and yaw. With fluidic effectors it gets them by blowing: a slot on the lower trailing edge turns the flow down and adds lift there, a slot on the upper surface spoils it, and blowing at a wingtip adds drag that yaws the nose. Each effector’s moment is its force times its lever arm, so the moments are linear in the blowing, \(m=Bu\): three equations in fourteen unknowns. Of the infinitely many solutions the controller wants the cheapest, the one with the least \(\|u\|^2\), subject to one hard fact: a jet can blow but cannot suck, so \(0\le u\le u_{\max}\). Fly the stick and watch the allocation; push it far enough and the effectors saturate.
Worked example. Yaw alone is the simplest case. Only the two tip effectors make drag, each with a lever arm \(y/b=0.475\) and \(\Delta C_D=3u\). A nose-right command \(C_n=0.002\) is met by the right tip alone blowing \(u=0.002/(0.475\times3)=1.40\times10^{-3}\); blowing both tips would only cancel. With the stick at 45% right roll, 30% nose up and 20% right yaw, and limits of \(2\times10^{-3}\), the allocator blows the three right upper slots (0.55, 0.77 and 0.87, in units of \(10^{-3}\)) to drop that wing, the left outboard lower slot (0.53, with 0.09 from its neighbour) to raise the other, the left inboard upper slot (0.25) to help lift the nose, and the right tip (0.42) for yaw: \(3.48\times10^{-3}\) in all, about 3.8 kg/s of bleed air at the reference condition.
Watch out. The effector strengths \(k_i\), their positions and the limits are schematic, chosen to show the mathematics: real effectors are nonlinear, interact with one another, depend on angle of attack and Mach number, and respond with a lag. Here lift effectors make no drag and drag effectors no lift. The change in total lift is left to the angle of attack. The bleed figure assumes a 40 m² wing at 200 m/s at about 10 km, with 300 m/s jets. The aircraft is a generic planform, not any real type.
Why not simply invert B?
\(B\) is 3 by 14, so it has no inverse: a whole eleven-dimensional family of blowing patterns gives the same moments. The pseudo-inverse picks the least \(\|u\|^2\), but it happily returns negative entries, which would mean sucking air into a blowing slot. Adding \(0\le u\le u_{\max}\) turns the problem into a small quadratic programme, and when the command lies outside what the bounded effectors can reach, the best that can be done is the nearest attainable moment.