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 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

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.

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: