Two-Dimensional Airflow Around an Aerofoil - ATPL 081 — Principles of Flight

The previous lesson summed the pressure pattern into one resultant force and left the pattern itself as a promise. This lesson keeps it: the detailed anatomy of the flow around a two-dimensional aerofoil: where the air rises to meet the wing, where it stagnates, where the suction really lives, why the force's grip point wanders while another point stays fixed, and where drag is born. Almost every later argument in this subject (the stall, the moment balance, the drag polar) is read off this map.

Reading the streamline pattern

The first lesson established that streamline spacing is a pressure map; here is the rule stated once, exam-tight:

Converging streamlines — the stream tube is narrowing, so the flow accelerates and its static pressure falls.
Diverging streamlines — a widening tube: deceleration, and rising static pressure.

Around a lifting aerofoil the pattern is unmistakable: over the curved upper surface the streamlines crowd hard together (fast flow, deep suction) while under the lower surface at positive angle of attack they spread slightly apart (slowed flow, mild overpressure). One glance at the spacing tells you which surface is doing what.

The pattern also extends beyond the wing, and the exam expects both halves. Ahead of the leading edge the flow begins rising before it arrives: the upwash, because the pressure field announces the wing into the air upstream of it. Behind the trailing edge the flow leaves with a downward slant: the downwash. Downwash is not exhaust; it is the receipt for lift. The previous lesson's third-law argument lives exactly here: the wing continuously deflects a mass of air downward, and the reaction to that deflection is the lift. Both angles return later with consequences: the three-dimensional lesson will show the tip vortices strengthening the downwash (and charging induced drag for it), and the control module will find the tailplane flying permanently inside this downwash field.

The stagnation point and its travels

Where the flow divides into the upper and lower streams sits the stagnation point: the one point where the oncoming free stream itself divides and stops, with its full total pressure intact, unlike the air that friction later brings to rest in the surface layer, where the full total pressure of the first lesson acts, and where the dividing streamline meets the surface. At a positive angle of attack it does not sit on the nose's tip but just below and behind it, on the lower surface: the air aiming slightly up at the wing splits under the chin, not at the point.

Now raise the angle of attack and watch it move: the stagnation point slides further aft along the lower surface. That migration is quietly powerful. The air destined for the upper surface must first travel briefly forward (against the general stream, toward the leading edge) and around an ever larger stretch of nose, through ever tighter curvature, so it accelerates ever more violently, and the upper surface's suction deepens. And the migration is measurable: the stall-warning vane or tab on many leading edges works precisely by sensing it: when the stagnation point has slid far enough aft, the local flow direction at the sensor flips, and the horn sounds. A moving point on a diagram is, in the aircraft, the thing that warns you before the wing lets go.

The pressure distribution

Put numbers on every point of the surface and the pattern of the previous lesson gets its true shape. On the upper surface the suction is not spread evenly: it rises abruptly from the stagnation region, reaches its extreme (the suction peak) just aft of the leading edge, where the flow is locally fastest, then decays gradually toward the trailing edge as the streamlines relax apart. On the lower surface a modest overpressure sits near the stagnation point and fades to near-ambient further aft. The upper surface's share dominates: roughly two thirds of the lift, as the previous lesson flagged.

The map responds to both of its controls:

Angle of attack — raising α deepens the suction peak, pulls it further forward, and strengthens the lower-surface overpressure: the whole force grows.
Camber — a cambered section's curvature squeezes the upper stream tubes even at zero angle of attack, so it shows suction (and lift) with the chord dead-level into the flow; a symmetrical section at zero α carries mirror-image patterns above and below, which cancel to zero lift.

That is the pressure-map reading of facts stated drily in earlier lessons: camber lifts at α = 0, symmetry does not.

Centre of pressure and aerodynamic centre

The previous lesson placed the resultant at the centre of pressure (CP) and admitted it moves. Now watch the movement itself, and what it does to the aircraft. Raise the angle of attack and the suction peak grows near the leading edge, dragging the force's balance point forward. Lower the lift toward zero on a cambered section and the CP does something stranger: it slides aft without limit. At exactly zero lift a cambered section still carries its nose-down twisting moment (a pure couple with no net force), and a couple cannot be represented by any single point of action. A reference that wanders with every control input, and occasionally leaves the aircraft entirely, is a miserable thing to build any calculation on.

Why any of this matters: the point where the lift acts is the lever arm the wing uses on the rest of the aircraft. When the CP moves, the wing's nose-up or nose-down leverage about the centre of gravity changes (with every gust, every control input, every speed change) so the tailplane must continuously re-balance a load that never sits still, and the pilot feels it as changing trim. For the engineer it is worse still: every balance or stability calculation anchored to the CP must be redone at every angle of attack, against a reference that sometimes is not even on the aircraft. That is the problem the second point exists to solve.

So aerodynamics keeps a second point: the aerodynamic centre (AC), at roughly quarter chord in subsonic flight. Its defining property: the pitching moment about the AC does not change with angle of attack. As α rises, the lift grows while its arm about the AC shrinks, and the two effects cancel, so the aerodynamicist writes the wing as one lift force acting at a FIXED point plus one CONSTANT moment, and never chases the CP again. This bookkeeping is how aircraft are actually designed and loaded: the stability module balances the wing (lift at the AC plus its constant moment) against the tailplane, and the loading limits of subject 031 quote the centre of gravity in % MAC precisely so it can be placed against the AC. Load an aircraft outside those limits and the fixed point the whole balance relies on sits in the wrong place relative to the weight: CP and AC stop being abstractions the day you fill in a loadsheet. The symmetrical section closes the circle from two lessons ago: it has no moment, so its CP has nothing to wander for: it sits on the AC and stays there.

Where drag is born

Two-dimensional drag comes from exactly two physical phenomena: friction between the air and the surface, and a pressure imbalance between the front and the rear of the body. Every drag name the later lessons introduce is one of these two wearing different clothes.

Skin friction drag exists because air is viscous. The layer of air in contact with the skin sticks to it completely (its speed is zero) and each layer above shears past the one below, so the wing permanently drags a thin sheet of air along with it. The rearward tug of all that shearing, summed over every square metre of wetted surface (the total area the air touches), is skin friction. It grows with wetted area, with speed, and with roughness, which is why performance aircraft are polished, flush-riveted and gap-sealed: every rivet head is a tiny drag tax collected for the whole flight.

Form drag (pressure drag) appears when the flow cannot follow the body's shape. Behind a blunt shape (or an aerofoil at too high an angle of attack) the flow separates: it lets go of the surface, and the rear of the body sits in a churning wake where the static pressure never recovers to match the overpressure at the nose. High pressure pushing on the front, missing pressure behind: the difference is a net rearward force. Streamlining is nothing more than shaping a body so the flow can decelerate gradually and the rear pressure does recover: the teardrop section of a wing is the shape of that idea.

The wake also explains where the drag energy goes. Air that was calm before the aircraft passed is left behind slowed, deflected and swirling, and that kinetic energy was paid for by thrust. In momentum terms: the wake's air has lost forward momentum relative to the stream, and the rate of that momentum loss is the drag force. Wind tunnels exploit this directly: survey the momentum missing in the wake behind a model, and you have measured its drag without touching it.

Two naming notes complete the picture. Taken together, the friction drag and form drag of a section are called its profile drag: the drag of the shape itself, a name the total-drag lesson will reuse when it assembles the whole aircraft's drag budget. And one famous family member is missing here on purpose: induced drag, the drag that is the price of lift itself, does not exist in two-dimensional flow at all: it is born from the spanwise flow around a real wing's tips (spanwise: along the span, from root to tip), and it waits for the three-dimensional lesson.

What the angle of attack does to lift

Assemble the whole lesson and the master variable's influence follows as one chain. Raising α slides the stagnation point aft under the nose; the upper flow rounds a tighter nose through more violent acceleration; the suction peak deepens and advances; the downwash behind the wing steepens. Every link adds lift, and through the normal working range they add it almost in proportion: each degree of α buys a similar extra slice of lift. The proportionality is not free-running. It holds only while the upper-surface flow can still follow the surface; as α grows, the pressure recovery toward the trailing edge steepens until the flow begins to separate, the growth bends over, and beyond the critical angle it collapses: the stall. Drawing that whole story as one curve, with a coefficient that strips away speed and size, is precisely the next lesson's job: the $C_L$ –α graph. The previews below already wear that proper axis label: read $C_L$ simply as 'lift' for now; the next lesson defines the coefficient, so the curve is an old friend before the mathematics arrives.

One more preview, because the curve's shape is a design variable, not a law of nature. Lowering full flap shifts the whole curve upward (more lift at every angle of attack and a higher maximum) at the price of a slightly earlier stall; how flaps manage that trick is the maximum-lift lesson's subject, where the symbol $C_{L\,max}$ gets its proper introduction. And the wing's aspect ratio (the slenderness number from the geometry lesson) sets the curve's steepness: a low-aspect-ratio wing gains less lift per degree of angle of attack, so its curve is shallower, and it keeps flying to a visibly higher angle before stalling, which is why short-winged deltas land nose-high. The three-dimensional lesson explains the mechanism.

What the exam asks

This lesson is the anatomy the examiner probes with 'describe' and 'explain'. The overview follows the learning objectives; expect to:

Describe the streamline pattern around a lifting aerofoil, and state the converging/diverging rule with its speed and static-pressure consequences

Describe upwash and downwash, and connect downwash to lift as the momentum it pays

Describe the stagnation point (flow at rest, full total pressure, on the lower surface just behind the nose) and its AFT migration along the lower surface as the angle of attack increases

Describe the pressure distribution and local speeds: where the suction peak sits, how the map changes with angle of attack, and how camber gives lift at zero alpha while symmetry gives none

Explain both reference points: the wandering centre of pressure (forward with α, aft without limit at zero lift) versus the fixed aerodynamic centre at quarter chord with its constant moment

List the two physical phenomena behind drag, describe skin friction (viscous shear over the wetted area) and form drag (separation, wake, unrecovered pressure), and explain the wake as lost energy and momentum

Explain the influence of the angle of attack on lift as a chain (stagnation shift, deeper suction, steeper downwash) nearly proportional until separation ends it