Laminar and Turbulent Boundary Layers

A boundary layer may be laminar or turbulent. A laminar boundary layer is one where the flow takes place in layers, i.e., each layer slides past the adjacent layers. This is in contrast to Turbulent Boundary Layers shown in Fig.6.2 where there is an intense agitation.

In a laminar boundary layer any exchange of mass or momentum takes place only between adjacent layers on a microscopic scale which is not visible to the eye. Consequently molecular viscosity $ \mu$ is able predict the shear stress associated. Laminar boundary layers are found only when the Reynolds numbers are small.

Figure 6.2: Typical velocity profiles for laminar and turbulent boundary layers

A turbulent boundary layer on the other hand is marked by mixing across several layers of it. The mixing is now on a macroscopic scale. Packets of fluid may be seen moving across. Thus there is an exchange of mass, momentum and energy on a much bigger scale compared to a laminar boundary layer. A turbulent boundary layer forms only at larger Reynolds numbers. The scale of mixing cannot be handled by molecular viscosity alone. Those calculating turbulent flow rely on what is called Turbulence Viscosity or Eddy Viscosity, which has no exact expression. It has to be modelled. Several models have been developed for the purpose.

Figure 6.3: Typical velocity profiles for laminar and turbulent boundary layers

As a consequence of intense mixing a turbulent boundary layer has a steep gradient of velocity at the wall and therefore a large shear stress. In addition heat transfer rates are also high. Typical laminar and turbulent boundary layer profiles are shown in fig.6.3. Typical velocity profiles for laminar and turbulent boundary layers Growth Rate (the rate at which the boundary layer thickness $ \delta$ of a laminar boundary layer is small. For a flat plate it is given by

$\displaystyle {\delta \over x}~=~{5.0 \over \sqrt{Re_x}}$ (6.1)

where Rex is the Reynolds Number based on the length of the plate. For a turbulent flow it is given by

$\displaystyle {\delta \over x}~=~{0.385 \over {Re_x}^{0.2}}$ (6.2)

Wall shear stress is another parameter of interest in boundary layers. It is usually expressed as Skin friction defined as

$\displaystyle C_f~=~{\tau_w \over {1/2 \rho U_\infty ^2} }$ (6.3)

where $ \tau_w$ is the wall shear stress given by

$\displaystyle \tau_w~=~\mu \left({{\partial u} \over {\partial y}}\right)_{y=0}$ (6.4)

and $ U_\infty$ is the free stream speed.

Skin friction for laminar and turbulent flows are given by

$\displaystyle C_f~$ $\displaystyle =~{0.664 \over {\sqrt{Re_x}}}, \texttt{Laminar Flow}$, Laminar    
$\displaystyle C_f~$ $\displaystyle =~{0.0594 \over {{Re_x}^{0.2}}}, \texttt{Turbulent Flow}$,Turbulent (6.5)

(c) Aerospace, Mechanical & Mechatronic Engg. 2005
University of Sydney