Boundary Layer Theory, Drag & Lift Notes

Boundary Layer Theory, Drag & Lift

Boundary layer concept, laminar and turbulent boundary layers on a flat plate, laminar sub-layer; boundary layer, displacement, momentum and energy thicknesses and shape factor; von Kármán momentum integral equation; Blasius solution and approximate profiles; drag on a flat plate; separation and its control; drag and lift on immersed bodies — friction and pressure drag, streamlined and bluff bodies, drag on sphere and cylinder, Stokes' law, terminal velocity, Magnus effect and circulation, aerofoils — with solved numericals.

Last reviewed 16 Sept 2026 · 7 min read

The boundary layer concept

Prandtl (1904) showed that when a real fluid flows past a solid surface, the effects of viscosity are confined to a thin layer next to the surface — the boundary layer — where velocity rises from zero (no-slip) to the free-stream value $U$ . Outside it the flow may be treated as ideal.

On a flat plate held parallel to the flow:

Laminar boundary layer forms at the leading edge and thickens with distance.
Transition occurs at a local Reynolds number $R e_{x} = U x / ν$ of about 5 × 10⁵ (the value depends on free-stream turbulence and surface roughness).
Turbulent boundary layer follows, growing faster; beneath it lies a thin laminar (viscous) sub-layer.

Boundary layer thicknesses

FormulaThickness definitions (

u

= velocity in the layer)

Boundary layer thickness $δ$ — distance from the surface where $u = 0.99 U$ .
Displacement thickness — distance the surface would have to be displaced outward to give the same flow rate in ideal flow:

δ^{*} = \int_{0}^{δ} (1 - \frac{u}{U}) d y

Momentum thickness — loss of momentum flux:

θ = \int_{0}^{δ} \frac{u}{U} (1 - \frac{u}{U}) d y

Energy thickness — loss of kinetic energy flux:

δ^{**} = \int_{0}^{δ} \frac{u}{U} (1 - \frac{u ^{2}}{U ^{2}}) d y

Shape factor $H = δ^{*} / θ$ (always > 1; 2.59 for the Blasius laminar profile, about 1.3–1.4 for turbulent layers).

Order: $δ > δ^{*} > θ$ (and $δ^{**} > θ$ ).

Von Kármán momentum integral equation

For a flat plate with zero pressure gradient:

\frac{τ _{0}}{ρ U ^{2}} = \frac{d θ}{d x}

Assuming a velocity profile, this gives $δ$ , $τ_{0}$ and drag.

Laminar boundary layer on a flat plate

FormulaBlasius (exact) solution

δ \approx \frac{5.0 x}{R e _{x}}, δ^{*} = \frac{1.72 x}{R e _{x}}, θ = \frac{0.664 x}{R e _{x}}

Local skin-friction coefficient: $c_{f} = \frac{τ _{0}}{\frac{1}{2} ρ U ^{2}} = \frac{0.664}{R e _{x}}$ Average drag coefficient for plate length $L$ : $C_{D} = \frac{1.328}{R e _{L}}$ Drag force (one side, width $b$ ): $F_{D} = C_{D} \frac{1}{2} ρ U^{2} b L$

$δ \propto x$ in laminar flow; $τ_{0} \propto 1/ x$ (highest near the leading edge).

Approximate profiles (momentum integral):

Velocity profile $u / U$ ( $η = y / δ$ )	$δ R e_{x} / x$	$C_{D} R e_{L}$
Linear: $η$	3.46	1.155
Cubic: $\frac{3}{2} η - \frac{1}{2} η^{3}$	4.64	1.292
Sinusoidal: $sin (π η /2)$	4.79	1.31
Blasius (exact)	5.0 (4.91 at 99%)	1.328

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