Hydrodynamic and thermal boundary layers

Hydrodynamic (velocity) boundary layer

When a fluid of free-stream velocity $u_\infty$ flows over a plate, the fluid in contact with the surface is at rest (no-slip) and velocity rises across a thin region to $0.99\,u_\infty$. This region, of thickness $\delta$, is the velocity boundary layer. All the viscous shear is confined to it; outside it the flow is practically inviscid.

Thermal boundary layer

If the plate temperature $T_s$ differs from the free-stream temperature $T_\infty$, the fluid temperature changes from $T_s$ at the wall to $T_\infty$. The thickness $\delta_t$ at which $(T-T_s)/(T_\infty - T_s) = 0.99$ is the thermal boundary layer. All convective heat transfer resistance lies within it.

Development

 u_inf --> . . . . . . . . . . . . . . . . .
                          _________ turbulent
                  _______/  ~~~~~~~  (buffer,
         _______/  transition        viscous
   _____/ laminar                    sublayer)
 -+--------------+-------------+------------> x
 leading edge    x_cr

• Laminar region near the leading edge: smooth layers, $\delta \propto \sqrt{x}$.
• Transition region: disturbances grow and the flow becomes unstable.
• Turbulent region: eddies mix the fluid, the layer grows faster ($\delta\propto x^{4/5}$), with a thin laminar sublayer at the wall. Both $\tau_w$ and $h$ increase sharply.

The transition is fixed by the critical Reynolds number

$$Re_{x,cr} = \frac{u_\infty x_{cr}}{\nu}\approx 5\times10^5$$

(it ranges from $10^5$ to $3\times10^6$ with roughness and turbulence level).

Effect of Prandtl number

$Pr = \nu/\alpha = \mu c_p/k$ compares momentum diffusion with heat diffusion. For laminar flow:

$$\frac{\delta}{\delta_t}\approx Pr^{1/3}$$