II. GENERAL KNOWNLEDGE

II.1 Experimentals Observations

A cylinder with a diameter d, along the z axe is placed in a flow with a velocity u in the direction of the x axe .
We can see the differents following flow patterns, function of the Reynolds number value Re.

• Experimental evidence shows that with a uniform freestream, flow about a circular cylinder remains uniform until a Reynolds number based on cylinder diameter of about 4, when vortices  which rotate in opposite directions appear, attached to the trailing surface of the cylinder.
• At a critical Reynolds number of about 47, the cylinder wake becomes unstable, i.e. fluid velocity depends on time, and the characteristic Karman vortex street begins to appear. The attached eddies finally shed at Reynolds numbers of about 60, and the alternate shedding pattern continues.
• Starting at Re =150 the vortex street becomes turbulent further downstream, and at Re =400, the vortices themselves become turbulent from the point of generation.
• For higher values of Reynolds number, vortices lose their regular shape and coherency, making visualization difficult. Then for extremely large  Reynolds numbers (Re > 3.10e6), it has been recently etablished that a regular street  appears, when the Strouhal number assumes S =0.27.

II.2 Scientific Theory

Flow Separation

The presence of the fluid viscosity slows down the fluid particles very close to the solid surface and forms a thin slow-moving fluid layer called a boundary layer.
The flow velocity is zero at the surface to satisfy the no-slip boundary condition.  Inside the boundary layer, flow momentum is quite low since it experiences a strong viscous flow resistance.   Therefore, the boundary layer flow is sensitive to the external pressure gradient (as the form of a pressure force acting upon fluid particles).
If the pressure decreases in the direction of the flow, the pressure gradient is said to be favorable.   In this case, the pressure force can assist the fluid movement and there is no flow retardation.  However, if the pressure is increasing in the direction of the flow, an adverse pressure gradient condition exists.  In addition to the presence of a strong viscous force, the fluid particles now have to move against the increasing pressure force.  Therefore, the fluid particles could be stopped or reversed, causing the neighboring particles to move away from the surface.  This phenomenon is called the boundary layer separation.

Aerodynamic Forces

According to the Newton's second law, time rate change of the linear momentum is equal to the sum of all external forces acting on a system.
Therefore, an integration of the linear momentum inside a control volume surrounding the circular cylinder can provide information of the aerodynamic forces (lift and drag) acting on the cylinder. There are alternative upward and downward flows in the wake as the result of vortex shedding.
Consequently, there must be also an oscillatory up and down force acting periodically on the cylinder.
This periodic forcing exerting on the cylinder body is responsible for the  vortex-induced vibrations as described earlier.

Momentum Balance

As started earlier, the external force acting on an object can be determined using the momentum balance concept.   In general, there is a momentum deficit in the wake profile along the streamwise direction as relative to the incoming momentum upstream of the object. So, there is net force acting on the object.  This net force along the flow direction is called the drag.  Averaged velocity profiles of the flow past a circular cylinder is provided as a general representation of the wake flow field.
Near the cylinder, flow separates from the surface.   Immediately behind the cylinder, a recirculation region exists with a strong reversing flow.  The region between the cylinder and the end of the recirculation region is called the vortex formation region.   The centerline velocity becomes zero at the end of the vortex formation region.
Further downstream, the two separating shear layers merge and the velocity profile presents a typical wake profile.  It is clear that there is a deficit in the center of the wake.
This deficit in the momentum flow is the direct result of drag force acting on the cylinder.