**Numerical simulations**

The difficulty in CFD is to have correct simulation results. When a new Navier Stokes code is used, it has to be tested on some academical cases. Usually, the turbulent codes (RANS) are tested on the axysymmetric impicting flow and the laminar cases are tested on the boundary layer cases.

As it has been described before, the main point here is to give reliable thermal boundary layer thickness in order to estimate the evaporation fluxes at the bottom of the geometry. The Navier Stokes solver adapted for such a goal is Icofoam: a transient solver for incompressible, laminar flow of Newtonian fluids. As there is no equation of temperature in this solver, it has been hanmade implemented.

Once the solver is chosen, it has to be validated on a test case: the boundary layer case.

**Numerical simulations **

** 1- Problem's geometry**

In this part, we simulate the boundary layer over a flat-plate. For this, we consider a flat plate of length L. This flat-plate is situated in a uniform flow with a uniform velocity U.

*Figure: Flat-plate in a uniform flow*

** Flat-plate's mesh**

We use OpenFoam tools to mesh the domain. The mesh is particularly refined above the plate in order to describe better the evolution of the boundary layer over the flat-plate. The global domain is divided to (100x200) meshes.

The picture below represents the meshed domain viewed by Paraview.

* Figure: Domain mesh *

** 3- "MyIcoFoamT" solver**

OpenFoam doesn't offer any standard solver that can resolve both thermic and dynamic equations. That's why, we create our own solver "**MyIcoFoamT**" in order to simulate the thermic boundary layer. In fact, we modify the standard solver "IcoFoam" that solves the incompressible laminar Navier - Stokes equations using the PISO algorithm, by adding temperature equations.

The next screenshots are representation of modifications that we have done in the source code:

* Figure: Screenshot of source code*

** 4- Test case**

We study the boundary layer over a flat-plate under the following circumstances:

- The flat-plate is situated in a uniform flow of air with a uniform velocity U=0.025m/s. We consider that the air kinematic viscosity is equal to 0.55 10
^{-6}m^{2}/s. Therefore, the Reynolds number is approximately 327.15. Thus, the test case is a laminar flow. - The plate's temperature is equal to T
_{p}=90°C while the air temperature is T_{0}=60°C.

The next figure shows the domain at initial instant of simulation

* Figure: the domain at initial instant of simulation*

**Numerical simulations **

** ****Estimation of the boundary layer thicknessâ€‹**

Once we have created the geometry and then generated the refined mesh of the domain above the flat-plate, we can launch the simulation to visualize the boundary layer over the flat-plate. We will now expose the results obtained.

We will first study the dynamic boundary layer, then the thermic boundary layer. The aim of this project's part is to estimate the evolution of the boundary layer thickness and to compare it with theoretical profiles corresponding to Blasius solutions.

**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.

**Numerical simulations **

** Thermal boundary layer**

The thermal study is very important for the further evaporation in the container which represents the main work of this BEI.

We suppose that the incident air flow is at a uniform temperature, and that the surface is maintained in a temperature T_{p} also uniform but different from the air temperature T_{0}.

** a- Temperature field**

We consider now the temperature equations while simulating the problem via OpenFoam. For this, we add the equations in the source code of our own solver as we have already explained.

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

- By exploring the temperature field perpendicularly to the plate, we shall observe a progressive variation of T
_{p}to the air temperature T_{0}; in fact this variation is at first fast then it becomes slower and slower as we penetrate into the incident air flow. - The region in which T varies in a significant way corresponds the the thermal boundary layer. It is noticeable that the thermal boundary layer gets thicker when we go away from the leading edge: it has the same altitude as the dynamic boundary layer.
- At the origin, the temperature profile is uniform, then the influence of T
_{p}appears gradually in the fluid. As a consequence, a temperature gradient emerged as well as a heat flux directed from the plate towards the fluid.

So, as well as the dynamic boundary layer is the expression of momentum diffusion, the thermal boundary layer results from the thermal diffusion in the fluid in movement.

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

** b- Boundary layer thickness**

The aim of this part is to estimate the thermal boundary layer thickness values over the flat plate.

The region in which T varies in a significant way corresponds the the thermal boundary layer. However this definition is too vague. The problem was moreover the same with the dynamic boundary layer, and will be approached on the same spirit, namely a conventional definition of the thermal boundary layer thickness.

We define this thickness as following $$\frac{T(x,\delta_T(x)) - T_p (x)}{T_0 - T_p (x)}= 0.99 $$ .

**Blasius model for a flat plate**

Source:http://help.solidworks.com/2013/french/SolidWorks/cworks/c_Convection_Heat_Coefficient.htm?format=P

The Blasius solution estimates the thermal thickness in such case as follows:

$$ \frac {\delta_T}{x} = 4.64 \sqrt {\frac{a}{U_0 x}}$$ where a is the thermal diffusivity.

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

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

* Figure: the thermal thickness profiles*

As is the case of dynamic boundary layer, the thermal boundary layer thickness has the same profile as the Blasius solution.

** d- Conclusion:**

The results correspond perfectly to Blasius solution. Therefore, the study of the boundary layer has enabled us to validate our model in order to use it in the evaporation study of the fuel container.