Incompressible Aerodynamics: Fluid Basics, Viscous Flow, Potential Flow, Aerofoils and Wings

A flow can be described by following particles (Lagrangian) or by watching fixed points (Eulerian); the link is the material derivative D/Dt = ∂/∂t + V·∇. Three kinds of line are drawn: a streamline is everywhere tangent to the velocity at one instant; a pathline is the track of one particle over time; a streakline joins all particles that have passed through one point, which is what a dye or smoke filament shows. In steady flow all three coincide; in unsteady flow they generally differ. Vorticity ω = ∇ × V measures local rotation, and a flow with ω = 0 everywhere is irrotational.

The conservation laws are written for a control volume (integral form) or at a point (differential form). Mass: ∂/∂t∫ρ dV + ∮ρV·dA = 0, or ∂ρ/∂t + ∇·(ρV) = 0. Momentum: the rate of change of momentum in the volume plus its net outflow equals pressure, body and viscous forces — at a point, ρDV/Dt = −∇p + ρf + viscous terms, which for a Newtonian fluid are the Navier-Stokes equations and without viscosity the Euler equations. Energy: the first law for a flowing fluid, which in steady adiabatic flow with no work reduces to constant stagnation enthalpy h + V²/2. A Newtonian fluid is one whose shear stress is linear in the rate of strain, τ = μ du/dy in simple shear, with μ independent of the strain rate; air and water are Newtonian.

Buckingham’s π theorem says a relation among n variables involving k independent dimensions can be written among n − k dimensionless groups. For the force on a body, F = f(ρ, V, l, μ, a) has six variables and three dimensions, so C_F = F/(½ρV²l²) depends on two groups only: the Reynolds number Re = ρVl/μ and the Mach number M = V/a (plus the shape and the angle of attack). Two flows are dynamically similar when they are geometrically similar and every relevant π group is equal; then all the force and pressure coefficients are equal too. This is why a wind-tunnel model result can be scaled to flight — and why it is hard: a 1/10 scale model in air at the same temperature needs ten times the speed to match Re, which destroys the Mach match. Pressurised or cryogenic tunnels raise Re by raising density or lowering viscosity instead.

Two exact solutions of the Navier-Stokes equations show viscosity at work. In plane Couette flow a fluid of depth h lies between a fixed plate and one moving at U with no pressure gradient; the velocity is linear, u = Uy/h, and the shear stress is the same everywhere, τ = μU/h. In Hagen-Poiseuille flow a pressure gradient drives fully developed laminar flow along a pipe of radius R: the profile is the paraboloid u = (−dp/dx)(R² − r²)/(4μ), the centre-line speed is twice the mean, the volume flow is Q = πR⁴Δp/(8μL), and the Darcy friction factor is f = 64/Re. Between parallel plates the same analysis gives a parabola whose maximum is 1.5 times the mean.

At high Reynolds number viscosity matters only in a thin boundary layer next to the surface, where the velocity rises from zero (no-slip) to the outer inviscid value. Its thickness δ is conventionally where u = 0.99U. Two integral thicknesses are more useful: the displacement thickness δ* = ∫(1 − u/U)dy, the distance the outer flow is pushed out, and the momentum thickness θ = ∫(u/U)(1 − u/U)dy, a measure of the momentum lost to skin friction. For the laminar flat plate (Blasius) δ ≈ 5.0x/√Re_x, δ* = 1.72x/√Re_x, θ = 0.664x/√Re_x and the local skin-friction coefficient is c_f = 0.664/√Re_x, so δ grows as √x. A turbulent layer grows faster, roughly δ ≈ 0.37x/Re_x^(1/5). Transition on a smooth flat plate is usually taken near Re_x ≈ 5 × 10⁵. An adverse pressure gradient (pressure rising downstream) slows the near-wall fluid until the wall shear falls to zero and the layer separates.

In irrotational flow the velocity is the gradient of a potential, V = ∇φ, and in incompressible flow continuity makes φ satisfy Laplace’s equation ∇²φ = 0; in two dimensions the stream function ψ (u = ∂ψ/∂y, v = −∂ψ/∂x) satisfies it too, and lines of constant ψ are streamlines. Because Laplace’s equation is linear, elementary solutions can be added. Bernoulli’s equation p + ½ρV² = constant holds along a streamline in steady, inviscid, incompressible flow, and everywhere in the flow field when the flow is also irrotational — which is what turns a potential-flow velocity field into a pressure distribution.

• Uniform stream + source = Rankine half-body. The stagnation point sits Λ/(2πV∞) upstream of the source, and far downstream the body is Λ/V∞ thick.
• Uniform stream + source + equal sink = Rankine oval, a closed body.
• Uniform stream + doublet = non-lifting circular cylinder of radius R = √(κ/(2πV∞)). The surface speed is 2V∞ sin θ, so C_p = 1 − 4 sin²θ: +1 at the stagnation points, −3 at the shoulders, and the symmetric pressure gives zero drag — d’Alembert’s paradox.
• Add a vortex = lifting cylinder. The lift per unit span is ρ∞V∞Γ. As Γ grows the two stagnation points move together towards the bottom of the cylinder, meet there when Γ = 4πV∞R, and leave the surface above that.

An aerofoil is defined by its chord line (leading to trailing edge, length c), its mean camber line (midway between upper and lower surfaces), the maximum camber and its position, the thickness distribution and maximum thickness, and the leading-edge radius. In the NACA four-digit series, NACA 2412 has a maximum camber of 2% of the chord located at 40% chord and a maximum thickness of 12% of the chord; NACA 0012 is symmetric and 12% thick. The section coefficients are c_l = L′/(q∞c), c_d = D′/(q∞c) and c_m = M′/(q∞c²), the primes denoting per unit span. The centre of pressure is where the resultant acts, and it moves with incidence; the aerodynamic centre is the point about which c_m does not change with α, and it is the more useful reference.

The Kutta-Joukowski theorem says that for any two-dimensional body in steady, inviscid, incompressible flow the lift per unit span is L′ = ρ∞V∞Γ, where Γ is the circulation round the body, and the drag is zero. What fixes Γ for an aerofoil is the Kutta condition: the flow leaves a sharp trailing edge smoothly, with finite velocity there — in vortex-sheet language, γ(TE) = 0. Viscosity is what enforces it in reality. Kelvin’s circulation theorem (circulation round a closed fluid curve is constant in inviscid, barotropic flow) then explains how the circulation appears: as the aerofoil starts moving, a starting vortex of strength −Γ is shed from the trailing edge and left behind, so the total circulation of the fluid stays zero while the aerofoil carries a bound vortex +Γ.

Thin-aerofoil theory replaces the aerofoil by a vortex sheet on its camber line and imposes flow tangency and the Kutta condition. For a symmetric aerofoil it gives c_l = 2πα (α in radians), c_m about the quarter chord equal to zero, the centre of pressure and the aerodynamic centre both at c/4, and c_m,LE = −c_l/4. For a cambered aerofoil c_l = 2π(α − α_L0), with α_L0 negative for positive camber, and c_m,c/4 = constant (negative for positive camber) — so the aerodynamic centre is still at the quarter chord, but the centre of pressure now moves aft as c_l falls. The lift slope 2π per radian (about 0.11 per degree) is the result to remember.

On a finite wing the higher pressure below spills round the tips, rolling the trailing vortex sheet into two tip vortices. By Helmholtz’s theorems a vortex filament cannot end in the fluid, so the bound vortex turns at the tips and trails downstream — the horseshoe-vortex model. The trailing vortices induce a downwash w at the wing, which tilts the local relative wind down by the induced angle α_i ≈ w/V∞. The wing section therefore sees a smaller effective angle α_eff = α − α_i, and the local lift, perpendicular to the tilted wind, has a rearward component: the induced drag, drag due to lift that exists even in inviscid flow.

Prandtl’s lifting-line theory models the wing as a bound vortex of varying strength Γ(y) along the span, shedding trailing vorticity dΓ/dy; the downwash at each station follows from the Biot-Savart law and the fundamental equation matches it to the section lift. Its central result is the elliptic lift distribution, Γ(y) = Γ0√(1 − (2y/b)²): the downwash is constant along the span, α_i = C_L/(πAR), and C_D,i = C_L²/(πAR), the least induced drag possible for a given lift and span. Any other planform loading gives C_D,i = C_L²/(πeAR) with span efficiency e < 1. The finite wing’s lift slope falls below the section value a0: a = a0/(1 + a0/(πeAR)), so a wing of aspect ratio 6 with a0 = 2π has a = 4.71 per radian. Here AR = b²/S; an untapered, untwisted elliptic planform produces the elliptic loading.

A U-tube manometer measures a pressure difference by the height of a liquid column: with manometric liquid of density ρ_m under a fluid of density ρ, p1 − p2 = (ρ_m − ρ)gh, and for air over water or mercury the ρ term is usually negligible. Inclining the tube at angle θ to the horizontal spreads the same vertical height over a length L = h/sin θ, raising the sensitivity. A Pitot tube facing the flow brings the fluid to rest at its mouth and senses the stagnation (total) pressure p0; a static port on the side senses the static pressure p; a Pitot-static tube combines both. In incompressible flow Bernoulli gives V = √(2(p0 − p)/ρ). In subsonic compressible flow the isentropic relation p0/p = (1 + (γ − 1)M²/2)^(γ/(γ−1)) must be used instead; in supersonic flow a bow shock stands ahead of the probe, and the Rayleigh Pitot formula relates the measured total pressure behind the shock to the free-stream Mach number.