This study looks at numerical modeling
of laminar flow around a circular cylinder.

In this case , viscosity model is
laminar with the steady formulation.

For Re inferior to 1, our results
are easily similars to the reality, as you can see with the picture below
:

Streamlines of the flow , Re = 0.1

In the boundary layer around the
circular cylinder, the static pressure is a maximum at the stagnation
point and gradually decreases along the front half of the cylinder.

The flow stays attached in this
favorable pressure region as expected.

However, the pressure starts to
increase in the rear half of the cylinder and the particle now experiences
an adverse pressure gradient.

Consequently, the flow separates
from the surface and creating a highly turbulent region behind the cylinder
called the wake.

The pressure inside the wake region
remains low as the flow separates and a net pressure force (pressure drag)
is produced.

Dynamic Pressure around the
cylinder, Re = 25

Recirculation flow behind the cylinder, Re = 25

Vortex shedding :

The pictures below show how vortices
appear with Re number equal to 200 (i.e. u = 2.4 m/s)

Time step is 0.005 s .

t = 0.8 s t = 0.85 s

t = 0.90 s
t = 0.95 s

As a fluid particle flows toward the leading edge of a cylinder, the pressure in the fluid particle rises from the free stream pressure to the stagnation pressure.

The high fluid static pressure near
the leading edge impels flow about the cylinder as boundary layers develop
about both sides.

On the other hand, the high pressure
is not sufficient to force the flow about the back of the cylinder at high
Reynolds numbers.

The boundary layers separate from
each side of the cylinder surface and form two shear layers that trail
aft in the flow and bound the wake.

Dynamic Pressure, Re = 200

Velocity vectors, Re = 200

Streamlines, Re = 200

Since the innermost portion of the
shear layers, which is in contact with the cylinder, moves much more slowly
than the outermost portion of the shear layers, which is in contact with
the free flow, the shear layers roll into the near wake, where they fold
on each other and coalesce into discrete swirling vortices.

Instability emerges as the shear
layer vortices shed from both the top and bottom surfaces interact with
one another.

They shed alternatively from the
cylinder and generates a regular vortex pattern (the Karman vortex street)
in the wake .

Dynamic Pressure, Re = 200

Velocity vectors, Re = 200

\

Streamlines, Re = 200

vortices periodicities in time step,
Re = 200

A vortex periodicity, Re = 200

The vortex shedding occurs at a
discrete frequency and is a function of the Reynolds number.

The dimensionless frequency of
the vortex shedding, the shedding Strouhal number, St = f D/V, (f=frequency;
D=characteristic diameter; V = characteristic velocity) is
constant, approximately equal to 0.20 .

In theory , we must find a frequency
: f = (0.2*2.4)/0.1 = 4.8 Hz which corresponds to a
period : Tth = 1/4.8 = 0.21 s

In our case, time step is 0.01 s . And we can see that a period is equal to 22 time steps. So the numerical results give us a period T = 0.22 s .

We rationally can think that ours
results are similar to the reality.

IV.4 High Reynolds
number flow

In this part, k-epsilon model was
used .

As we did not found realistics
results, we decided not to display them .