BL zones sketch and displacement thickness

Characteristics of the flow

A flat plate held parallel to a free stream develops a boundary layer, in which viscosity slows the fluid near the surface. Velocity is zero at the wall and equals the free-stream velocity $U$ at the edge of the layer ($y = \delta$). The thickness $\delta$ grows with distance $x$. The flow is laminar near the leading edge, changes through a transition region and becomes turbulent downstream, with the change at about $Re_x = Ux/\nu \approx 5\times10^5$. Outside the layer the flow is practically inviscid.

Sketch

 U ->
 ---------------------------------------------------
                                      Turbulent BL
          Laminar BL    Transition  _________________
 U -> ----------____________......___________________
                                      ..... laminar sub-layer
 ========================================= plate
 leading edge      x ------------------------>
   (i) laminar  (iii) transition  (ii) turbulent
   (iv) laminar sub-layer: thin film at the wall

• (i) Laminar boundary layer: smooth flow, $\delta = 5x/\sqrt{Re_x}$.
• (iii) Transition zone: flow changes from laminar to turbulent, $Re_x \approx 3\times10^5$ to $5\times10^5$ or more.
• (ii) Turbulent boundary layer: eddy mixing, thicker, fuller velocity profile.
• (iv) Laminar sub-layer: very thin viscous layer next to the wall, even under the turbulent layer.

Displacement thickness

It is the distance by which the outer streamlines are displaced away from the plate because of the reduction of flow in the boundary layer:

$$\delta^* = \int_0^{\delta}\left(1 - \frac uU\right)dy$$