In order to study Strouhal instability we simulated the two-dimensional flow past a circular cylinder with Computational Fluid Dynamics software and observed the influence of the parameters. The method used is described in this page.

### I - Modeling and meshing (Gambit)II - Computation (Fluent)III - Results

Results Sheet (.html - 500 Ko)

## I - Modeling and meshing (Gambit)

A rectangular plane domain was designed around the circle representing the cylinder. The cylinder was 1cm in diameter. The sides of the rectangle had to be far enough not to interfere with computation. So they were placed 10cm upstream, 50cm downstream and 20cm both sides of the cylinder.

We chose to create an unstructured mesh because of the cylinder. Indeed unstructured meshes are more efficient and very handy when meshing curves. In a two-dimension problem, the unstructured mesh is made of triangles.

In preliminary study, a simple mesh was created in order to check the validity of the domain choice. It was refined on the cylinder and large away. The cells were 1mm long on the cylinder and 2cm long on the sides of the rectangle.

Preliminary study

First simulations based on this mesh show that Von Karman Vortex Street disappeared when cells were larger. So we decided to refine the mesh in the cylinder wake. Refining the outlet was insufficient because only the part of the mesh close to the outlet was refined. The only solution we found was to create constraints lines in the domain. Two lines were added along x-axis direction between the cylinder and the inlet and the outlet. But it created problems when defining boundary conditions because outlet mass flow ratio of each new outlet evolved with time and remained unknown before computation. The solution was to keep only one outlet and creating constraints lines between the cylinder and the corners of the domain.

New domain with constraints lines

On the edges of this domain, the nodes were placed with the repartition below:

 inlet: 20 nodes - regular outlet: 50 nodes - biexponential, ratio 0.4 sides: 30 nodes - regular constraints lines: 100 nodes - successive ratio, ratio 1.025 front half-cylinder: 10 nodes - regular back half-cylinder: 15 nodes - regular

Nodes repartition

Nodes repartition on the cylinder

Then the two surfaces of the domain were meshed separately. The constraints lines allows a good link between the meshes of the two surfaces and a better repartition in the domain. This mesh is made with about 14.000 cells.

Mesh with constraints lines

The last step with the pre-processor was the definition of the boundary conditions. The constraints lines had to be non-material. We defined it as "interior" so that the fluid go through this artificial lines. The others boundary conditions are classical. The sides were defined as "symmetry" (i. e. the flow is null) and it is estimated that all the flow exhausts only through the back of the domain. Indeed, computation is easier and real outflow through the sides is hard to estimate. Considering the sides far enough, the approximation of no flow is correct. The boundary conditions are defined below:

 inlet: velocity inlet outlet: outflow - flow rate: 1 sides: symmetry constraints lines: interior front half-cylinder: wall back half-cylinder: wall

## II - Computation (Fluent)

We worked with the solver Fluent5. The computations were done with water, considering a laminar model and using the boundary conditions presented before. In order to initialize the computations, the inlet velocity was chosen to obtain a Reynolds about 40 because according to experimental data, it is the critical Reynolds, i.e. up to this Reynolds, the flow becomes instable and Von Karman Vortex Street appears.

However with this preliminary study we did not obtain some oscillations. Then the simple order unsteady model was changed to a second order model in order to decrease excessive stability due to computational methods. This was not enough. Finally a disturbance was created to cause a vortex release. A 5 seconds disturbance created 3 vortices but the flow re-stabilized. So it was decided to increase Reynolds number to 100. Von Karman Vortex Street appeared without any disturbance with such a Reynolds number.

Then this results seemed insufficient because vortices disappeared where the cells of the mesh became larger. That's why the second mesh presented above was created. With the new mesh, the results were better and Von Karman Vortex Street developed in a longer distance.

Computations with preliminary mesh

### Comment about computational method and Strouhal instability

The computations were done in 2 stages: first they were done with a steady model to find a steady solution. Like this, the wake with re-circulations developed behind the cylinder in the whole domain. After the steady computations converged, new computations were done with an unsteady model but from the steady solution. Then Von Karman Vortex Street was shaped.

There is a physical interpretation to this. Indeed, all instabilities are a combination of the two basic instabilities: saddle-node and pitchfork. This instabilities show that up to a critical value of a control parameter (Reynolds number considering Strouhal instability), the number of fluid balances increases from 0 to 2 (saddle-node) or from 1 to 3 (pitchfork). This change is called a fork. However some balances are stable and others are instable. This instable balances do not exist in nature. But they are a mathematical solution that can be computed. That is the reason why with steady computations, we observe an instable solution of Strouhal instability that do not exist in the nature. Indeed a disturbance will always create an instability and will drive the fluid to a stable balance.