BL growth along thin plate, examples

Concept of boundary layer

The boundary layer is the thin region next to a solid surface where the velocity changes from zero at the surface (no-slip) to the free-stream value $U$, because of viscosity. Outside it the flow is effectively inviscid.

Growth along a thin plate

 U ->   ------------------------------------------
        laminar         transition      turbulent
 ------------___________.........______________________
   delta ~ sqrt(x)                        delta ~ x^0.8
 ========================================= plate -->
 leading edge

Laminar flow (near the leading edge, $Re_x < 5\times10^5$)

• The layer starts with zero thickness at the leading edge and grows slowly as viscous effects spread outward.
• The fluid moves in smooth layers. Blasius's result: $\delta = \dfrac{5x}{\sqrt{Re_x}}$, so $\delta \propto \sqrt{x}$.
• The velocity profile is nearly parabolic, with a gentle gradient at the wall.

Turbulent flow (downstream, $Re_x > 5\times10^5$)

• After a transition region the flow becomes turbulent and eddies mix the fluid strongly.
• The layer is thicker and grows faster: $\delta = \dfrac{0.37x}{Re_x^{1/5}}$, $\delta \propto x^{0.8}$.
• The profile is fuller ($u/U = (y/\delta)^{1/7}$), the wall shear stress is larger, and a thin laminar sub-layer stays at the wall.

Here $Re_x = Ux/\nu$.

Two examples of use

1. Finding the skin-friction drag on ship hulls, aircraft wings and flat surfaces.
2. Designing streamlined shapes to delay separation and reduce drag, and wind tunnel sections using displacement thickness.