**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 Re_{x} 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.