Advection Diffusion Equation with Star CD

By Benoit Seille & Ugo Schuck

Advection-diffusion problem is one of the suitable problem to check the efficiency of a computational fluid dynamic software. During this study, we will try
to check the efficiency of STARCD on a advection and diffusion problem in specific conditions. Therfore we will first introduce the problem we want to solve with the
advection-diffusion equation, the geometry and boundary conditions. Then we will see some interesting results to finnaly try to conclude on the interest of STARCD.

• The Starting Problem
-The Advection Diffusion Equation

The problem, we want to study is based on the advection-diffusion equation which is in steady and two-dimensional conditions:

The left terms represent advection which is controled by the velocity, The right ones represent diffusion controled by conductivity.

-The Geometry of the Problem

The geometry of the problem is quite simple, this is a rectangle of about 1m*2m.

-Condition of the Problem

the fluid choosen during the study is water. The conditions are summarized is the next figure:

The boundary conditions are:   *four inlet velocity Ux=Uy=u where u depends on tests.
*two inlet conditions of temperatures T=293 K and concentrtion c=1 (inlet1)
*two inlet conditions of temperatures T=273 K and concentrtion c=0 (inlet2)
*two outlet with pressure conditions P=0 Pa

• Running this Case with Star CD
-The Mesh

We have choosen to make our study on a 50*100 cells mesh. The next figure represent this mesh with velocity inlet conditions:

-Initialization of the Problem

The initialization is made with a velocity Ux=Uy=u, T=273 K and c=0 everywhere.

The concentration does not exist as a parameter of STARCD therefore we have had to generate a passive additional scalar representing concentration.
This operation is easy to do with STARCD.

-Differencing Schemes

In fact we use only one scheme, the MARS scheme.

-Number of Iteration

• Results and Comparison with Theory
-A Reminder of Theory

To validate the study we will use approximations of  diffusion time (Td) and convection time (Tc) defined by Td=L*L/a and Tc=L/u.

We will now check this estimate time on reults we have obtain with STARCD.

-The different Scalars

In this study, there is no differences between results obtained with convection and diffusion of a temperature or a concentration because both are considered
as passive scalar. They have no influence on the flow.

-Results

*Convection leads the flow:

The first test we have made was with an inlet velocity u=1 m/s and a diffusivity a=6.32 E-9 m*m/s. We have obtained the following figure:

Temperature evolution for u=1 m/s and a=6.32.10^9 m*m/s

We can see that the flow is quite controled by the advection. By comparing the diffusion time and the advection time we have found that the diffusion time 5 millions bigger
than advection time. Therefore we can consider that the convection leads the flow. However the diffusion seems to have a little effect on the flow but it can also be numerical
diffusion.

*diffusion leads the flow:

The second test we have made was with a null inlet velocity and a diffusion coefficient a=10 E-5 m*m/s. We have obtained the following result:

Temperature evolution without advection.

We can see on the previous figure that it has worked because there is no advection.

• Conclusion
During this study, we have spent more time in succeed in the use of STARCD than in obtaining results. In fact, STARCD remains difficult to use.
We have just succeed in obtaining some results but it appears clear that our study is not enought efficient to conclude on the resolution of a advection-diffusion problem
with STARCD.
However, this study permited to discover a new computational fluid dynamic software, although it remains difficult to understand well such a complex software in such a short time.