Dynamic boundary layer

Numerical simulations

1- Dynamic boundary layer

a- Velocity field

In this part, we consider that the fluid and the plate have the same temperature. The physical problem is then reduced to a dynamic study.

The following figure shows the evolution of the dynamic boundary layer over the flat plate.

Figure: The evolution of the dynamic boundary layer over the flat plate

We notice that the boundary layer is quickly established trough the area over the plate. The velocity values increase as we go away from the plate.

Moreover, it is noticeable that:

• All fluid particles in touch with the wall are immobile relatively  to the plate.
• The flow near the plate is slowed down.
• The normal gradient of velocity near to the wall is important due to the viscosity effect.
• The viscosity, in the boundary layer, is small but it has an important impact on the shear stress on the plate $$\tau_p = \mu \frac{\partial U}{\partial y} |_{paroi}$$  which could have high values.

Velocity profile

In the figure below we plot the velocity profile versus the altitude above the plate in x= 0.9 cm.

Figure:  the velocity profile versus the altitude above the plate in x= 0.9 cm

b- Boundary layer thickness

The most important in this study is to estimate the boundary layer thickness values over the flat plate. We define this thickness as following  $$U[x,\delta(x)] = 0.99 U_0$$ .

Blasius model for a flat plate

source: "Mécanique des fluides" Patrick Chassing

In the case of a flat plate disposed in a uniform parallel flow of a viscous fluid, the thickness of the boundary layer is announced by Blasius as:

$$\delta (x) = 4.92 \frac{x}{\sqrt {Re_x}}$$

such as  Rex is the local Reynolds number $$Re_x= \frac {x U(x)}{\nu}$$

c- Comparison

We  compare the numerical boundary layer thickness with the Blasius solution.

The following figure shows the dynamic thickness profile versus Blasius solution:

Figure: the dynamic thickness profiles

It can be seen that numerical curve of thickness is similar to Blasius solution. They both have the same order of magnitude ~10-3 m.