BL development along thin flat plate

When a uniform stream of velocity $U$ flows parallel to a thin flat plate, the fluid at the surface sticks to it (no-slip). Viscous shear slows the layers above, and a region of velocity gradient grows along the plate. This is the boundary layer. Its thickness $\delta$ increases with distance $x$ from the leading edge.

Development

 U ->  free stream (inviscid)
 ---------------------------------------------------
        .---- laminar ----.transition.---- turbulent ---
 U -> ----------___________......_____________________
                   boundary layer   (laminar sub-layer
 =================================   at wall)
 leading edge         x_cr        flat plate -->

1. Laminar zone: Starting at the leading edge, the layer is thin and the flow is orderly, in smooth layers. Thickness grows as $\delta \propto \sqrt{x}$ (Blasius, $\delta = 5x/\sqrt{Re_x}$).
2. Transition zone: At a critical Reynolds number $Re_{x,cr} \approx 5\times10^5$ (for a smooth plate and low free-stream turbulence), disturbances amplify and the flow starts to change to turbulent.
3. Turbulent zone: Beyond transition, mixing by eddies makes the layer thicker ($\delta \propto x^{0.8}$) with a fuller velocity profile. A very thin laminar sub-layer remains next to the wall.

where $Re_x = Ux/\nu$.

Salient features

• Velocity is zero at the wall and reaches $0.99U$ at $y = \delta$.
• $\delta$ increases downstream; the pressure gradient along a flat plate is zero.
• Wall shear stress is highest near the leading edge in the laminar region, drops, then jumps at transition and is higher in the turbulent region than for laminar flow at the same $x$.
• A larger viscosity or lower velocity (lower $Re$) gives a thicker layer.
• Pressure across the layer is nearly constant, $\partial p/\partial y \approx 0$.
• The effect of the plate on the flow outside the boundary layer is represented by the displacement thickness $\delta^*$.