PHAK Ch. 4 · graduate stack · learning aid

Lift Theory Lab

From pilot Bernoulli notes to circulation & Kutta–Joukowski. Study home · Bernoulli note · Performance Lab · Quiz

Before you play: layers of the same lift

Lift is the aerodynamic force perpendicular to the freestream. It comes from pressure and shear on the wing surface. Different “theories” are usually different descriptions of one flow—not competing religions.

Pilot (PHAK)

Faster upper flow → lower static pressure (Bernoulli). Wing also imparts downwash (Newton). PHAK: both operate. Symmetrical airfoils still lift at AOA.

Engineering

\(C_p = 1 - (V/V_\infty)^2\). Section \(C_L \approx 2\pi\alpha\) (thin airfoil, rad). 2-D: \(L' = \rho V_\infty\Gamma\) (Kutta–Joukowski). Finite wing: induced drag & downwash.

Why viscosity matters

Ideal inviscid flow doesn’t pick a unique circulation. A sharp trailing edge + viscosity enforces the Kutta condition, selecting \(\Gamma\) so flow leaves the TE smoothly.

  1. Bernoulli — venturi & static vs dynamic pressure
  2. Airfoil — AOA, camber, schematic \(C_p\) and surface speed
  3. Newton — momentum / downwash picture
  4. Circulation — \(\Gamma\), Kutta–Joukowski, thin-airfoil \(C_L\)
  5. Synthesis — myths vs complete stack
  6. Guide — story + quiz

Suggested path: read the Bernoulli note, then walk tabs 1→5, finish with Guide.

Skip to interactive lab ↓

Venturi & Bernoulli

Continuity: mass flow conserved. Speed up at the throat → static pressure falls.

p + ½ρV² = H   (steady, inviscid, incompressible, along streamline)
V throat
—
Δp (in − throat)
—
Cp at throat
—
Dynamic ½ρV² (in)
—
PHAK link: “As the velocity of a moving fluid increases, the pressure within the fluid decreases.” Same idea on the upper surface of a wing.

Schematic venturi

Arrow length ∝ speed; color cool = lower static pressure.

Schematic

Airfoil pressure field

AOA and camber set a schematic surface-speed / Cp distribution (not a panel code).

Thin-airfoil CL
—
Dist. ΔCp proxy
—
Lift (S=15 m² toy)
—
Cp = 1 − (V/V∞)²
local
Try: camber = 0, AOA = 0 → CL ≈ 0 (symmetrical zero-lift). Raise AOA with camber still 0 — lift without a “longer path.”

Section + Cp sketch

Cyan = upper suction tendency; orange = lower. TE speeds converge (Kutta-ish).

Schematic Cp

Momentum / downwash

Control-volume view: the wing gives the air downward momentum; reaction is lift.

F = ṁ Δv   →   L ≈ ṁ · vdown   (order-of-magnitude picture)
Lift ≈ ṁ vd
—
Power into wake (toy)
—
ε downwash (from CL, AR)
—
CD,i ≈ CL²/(πeAR)
—
PHAK: downwash meets flow from below at the TE; Newton’s third law → upward force on the airfoil. Finite span: tip vortices sustain the downwash field → induced drag.

Streamtube deflection

Streamlines bend downward aft of the wing (schematic).

Schematic

Circulation & Kutta–Joukowski

2-D inviscid airfoil: lift per unit span is fixed by freestream and circulation.

L′ = ρ∞ V∞ Γ   ·   Γ = ½ c V∞ CL   ·   CL ≈ 2π α
CL ≈ 2πα
—
Γ circulation
—
L′ = ρVΓ
—
L′ via ½ρV²cCL
—
Kutta condition: real flow leaves a sharp trailing edge smoothly; that selects \(\Gamma\). Ideal potential flow without Kutta cannot decide how much lift the airfoil carries.

Bound vortex sketch

Circulation around the section; freestream + vortex → net streamline pattern (cartoon).

2-D idealization

Complete stack (one flow, many views)

Use this as the “graduate” summary card.

  1. Definition: Lift from integrated surface pressure & shear.
  2. Bernoulli / Cp: maps surface speed to pressure.
  3. Newton / momentum: far-field downwash & force balance on a control volume.
  4. Circulation: L′ = ρVΓ with Γ fixed by Kutta (2-D).
  5. 3-D: tip vortices, induced drag, span efficiency.
  6. Reality: viscosity, separation, compressibility, unsteady flow limit every simple model.
Myth: Equal transit time — air above and below must rejoin after equal times.
Truth: Upper flow is typically faster; parcels do not take equal time. Lift does not require that myth.
Myth: Bernoulli or Newton — pick one.
Truth: Complementary. PHAK: both operate when lift is produced.
Myth: Only the bottom “pushes up.”
Truth: Full pressure field; upper-surface suction is often large.

Live consistency check

Thin-airfoil CL from current airfoil sliders; KJ lift matches ½ρV²cCL form.

α / camber
—
CL
—
Γ
—
L′ match?
—
Linked to Airfoil tab
Change AOA on the Airfoil tab, then return here — synthesis numbers follow the same state.

Guided story

1 / 5

—

—

Check yourself

About / equations / limits

  • Bernoulli form shown is incompressible, steady, inviscid, along a streamline.
  • Thin-airfoil \(C_L \approx 2\pi\alpha\) (α in radians) is small-AOA theory for thin sections; stalls are cartooned.
  • Kutta–Joukowski is 2-D; real wings need finite-span corrections.
  • Canvas drawings are pedagogical—not wind-tunnel data.
  • Sources: FAA PHAK Ch. 4; standard undergrad aero (Anderson / Kuethe-style curriculum topics).