Intuition The ONE core ideaAir flowing past a wing obeys equations that are too hard to solve by hand, so we chop the problem into many small pieces and let a computer solve a giant set of simple equations. Everything in the parent note — panels, sources, influence coefficients, meshes — is just a different way of doing that chopping.
This page assumes nothing . If the parent note used a symbol, a word, or a picture, we build it here first. Read top to bottom; each block earns the next.
Before any symbol, hold this image: at every point in the air there is a little arrow saying how fast and which way the air moves there. A carpet of arrows filling space is called a vector field . The whole subject is: find that carpet of arrows around the wing.
Definition Velocity fieldV V direction , not just a size". In 2-D it has two numbers: how fast rightward (V x V y Picture: the carpet of arrows above.
Why we need it: lift and drag come from how these arrows bend around the body.
x , y P x y point in space is written P ( x , y ) Picture: the usual graph paper — right and up.
Definition Freestream speedV ∞ α V ∞ far away from the wing, where the body hasn't disturbed it yet. The little ∞ α angle of attack : the tilt between that incoming wind and the wing.
Picture: straight parallel arrows coming in from the left, tilted by α Why: this is the "input" wind; the wing's job is to bend it.
The parent said "incompressible ⇒ ∇ ⋅ V = 0 ⇒ V = ∇ ϕ ∇ ⋅
Definition The nabla symbol∇ ∇ not a number . It is a recipe meaning "take slopes in each direction". Think of it as a tiny machine that measures how a quantity changes as you step in x y
∇ ⋅ V out minus arrows coming in . If more leaves than enters, the box is a source of fluid — divergence is positive.
∇ ⋅ V = ∂ x ∂ V x + ∂ y ∂ V y Incompressible means the fluid can't be squashed, so in equals out for every box: ∇ ⋅ V = 0 Picture: left box in the figure — balanced arrows.
Definition The partial-slope symbol∂ ∂ x ∂ V x V x x y ∂ "slope while pretending the other variables are frozen."
Why we need it: velocity changes from point to point; the slope tells us the change.
Definition Irrotational / curlDrop a tiny paddle-wheel in the flow. If it spins , the flow has rotation (curl) there. If it merely drifts without spinning, the flow is irrotational .
Picture: right box in the figure — the wheel drifts but does not turn.
Why: irrotational flow lets us replace the whole arrow-carpet with a single scalar "height map" — the potential ϕ
This is the master trick. Instead of storing two numbers (V x , V y one number ϕ slopes of that height.
ϕ V = ∇ ϕ ϕ scalar (one number) at each point, like the height of a hill.
V = ∇ ϕ downhill-steepest on that height map, and its length is the steepness.
Picture: contour lines of a hill; arrows cross them at right angles, longest where lines crowd.
Why: one number per point is far easier than two — and it automatically guarantees the flow is irrotational (a hill has no built-in swirl).
Intuition Why "potential" earns its keepA single height map ϕ ϕ ϕ
Combine the two facts from §2:
incompressible: ∇ ⋅ V = 0
irrotational: V = ∇ ϕ
Substitute the second into the first:
∇ ⋅ ( ∇ ϕ ) = ∇ 2 ϕ = 0
∇ 2 ϕ ∇ 2 ϕ = ∂ x 2 ∂ 2 ϕ + ∂ y 2 ∂ 2 ϕ above or below the average of its neighbours on the height map. Setting it to zero says: no point may be a bump or dip — the surface is as smooth as a stretched drumhead.
Picture: a soap film pulled flat across a wire loop; every point is the average of those around it.
Why: see Laplace's Equation & Potential Flow . This is the governing equation of ideal flow.
Linear adding two valid solutions gives another valid solution.
Because we may add solutions, we keep a box of simple ones. Each is a valid ϕ
r θ r θ Picture: rings (r θ
Definition Source strengthλ λ total volume of fluid pumped out per unit time , spread equally in all directions.
Picture: a garden sprinkler at the centre; bigger λ Why: on a wing surface we tune each panel's λ skims the wall and never flows through it.
Γ Γ Picture: a whirlpool; Γ Why: lift needs swirl. By Kutta–Joukowski Theorem & Circulation , lift per length L ′ = ρ V ∞ Γ Γ = 0
ln r 2 π ln r e r 2 π 36 0 ∘ Why this tool: ln r unique radially-symmetric solution of Laplace's equation in 2-D — no other simple function both spreads equally and satisfies ∇ 2 ϕ = 0
Definition Panel and control pointA panel = a short straight segment approximating a piece of the curved surface. Its control point = its midpoint, where we enforce the rule.
Picture: a smooth airfoil traced by N Why: we can't find a smooth strength λ ( s ) N λ 1 … λ N
n ^ n ^ out of the panel, perpendicular to it. The tangent points along it.
Picture: on each toothpick, one arrow sticking outward, one lying flat.
Why: the wall condition is about flow through the wall, i.e. the normal direction.
V ⋅ n ^ V ⋅ n ^ = ∣ V ∣ cos θ V n ^ Why this tool: we need only the through-the-wall component, and the dot product is exactly the "how much of one arrow points along another" machine.
β i I ij β i i I ij at panel i by a unit source on panel j at i from j ".
Why: these numbers fill the matrix that the computer inverts.
Definition Pressure coefficientC p C p = 1 − ( V ∞ V t ) 2 V t tangential (sliding-along) speed at a panel. C p dimensionless pressure: where air speeds up (V t > V ∞ C p Bernoulli's Equation (fast air = low pressure).
Why: integrate C p
λ Feels right: strength sounds like power. Catch: lift lives in Γ λ Fix: recall L ′ = ρ V ∞ Γ
Definition Navier–Stokes, viscosityμ ρ stickiness (viscosity μ squeezed (density ρ Navier–Stokes Equations , which are nonlinear — you cannot add solutions anymore, so the whole brick trick dies and we need Finite Volume Method on a volume mesh . See also Boundary Layer Theory & Skin Friction Drag , Turbulence Modelling — RANS, LES, DNS , and why inviscid theory predicts zero drag: d'Alembert's Paradox .
Nonlinear solutions may not be added; superposition fails.
divergence = 0 incompressible
linear so we may add solutions
elementary bricks source vortex stream
wall condition V dot n = 0
panels and control points
linear system A lambda = b
add viscosity and density
Navier Stokes then CFD mesh
What does the little arrow on V it has direction and size — it is a vector, one arrow per point in space.
What does ∇ ⋅ V = 0 as much fluid leaves every tiny box as enters it — incompressible.
What does irrotational mean with the paddle-wheel picture? a tiny wheel drifts but never spins.
What is ϕ V = ∇ ϕ a one-number height map; velocity is its steepest slope.
Why must ∇ 2 ϕ = 0 no point may bump above its neighbours — a smooth drumhead surface.
Why is linearity the whole secret? valid solutions may be added, so we stack simple bricks into complex flows.
What does source strength λ volume of fluid pumped out per unit time, spread evenly.
What does circulation Γ λ swirl, hence lift, via L ′ = ρ V ∞ Γ
What single boundary condition sets the panel strengths? zero normal velocity,
V ⋅ n ^ = 0 .
What does n ^ straight out of the wall; the dot picks the through-the-wall part to kill.
What is C p dimensionless pressure 1 − ( V t / V ∞ ) 2
What breaks when we add viscosity and density? the equations turn nonlinear — no adding solutions — so we need CFD.
