BoundaryLayer

A boundary layer is a thin region of fluid near a solid boundary where viscous forces are important, causing the velocity to transition from zero at the wall (no-slip) to the free-stream value away from the surface. In high-Reynolds-number external flows, most drag and heat transfer originate within the boundary layer; outside it the flow can often be treated as inviscid.

Two main regimes exist: laminar and turbulent boundary layers. The laminar layer is described by Prandtl's boundary-layer equations, a reduced form of the Navier–Stokes equations that apply to a thin layer with relatively small pressure variation across it. For steady, incompressible flow with constant properties, the equations are: continuity ∂u/∂x + ∂v/∂y = 0; momentum u ∂u/∂x + v ∂u/∂y = - (1/ρ) ∂p/∂x + ν ∂^2 u/∂y^2, with ∂p/∂y ≈ 0 and boundary conditions u=0, v=0 at the wall and u → U∞ as y → ∞. The pressure field is determined by the outer inviscid flow.

On a flat plate with zero pressure gradient, the laminar solution is the Blasius solution. The laminar boundary-layer thickness scales as δ ~ 5 x / sqrt(Re_x), where Re_x = U∞ x / ν. Turbulent boundary layers thickens more rapidly, with estimates such as δ ~ 0.37 x Re_x^{1/5}.

Boundary layers may separate when the pressure gradient along the surface is adverse enough to reverse near-wall flow, leading to flow detachment and increased drag. Turbulent layers are more resistant to separation but can still detach under strong gradients or geometry changes.

Boundary-layer concepts extend to atmospheric, industrial, and marine contexts, where the layer influences heat, mass transfer, and surface interactions.