engineering & technology publications
ISSN 1759-3433
PROCEEDINGS OF THE THIRD INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY
Numerical Simulation of the Flow Induced by a Circular Cylinder Subject to Forced Oscillations
+Department of Mathematics and Statistics, Memorial University, St. John's, Newfoundland, Canada
*Department of Mechanical Engineering, Menoufia University, shebin El-kom, Egypt
with respect to the oncoming uniform flow is investigated
numerically for the first time. Not only do these oscillations
have practical consequences relating to the design of engineering
structures, but from a fundamental standpoint the forced
oscillations of cylinders at an arbitrary angle with respect to
the oncoming uniform flow form an important and relatively
unexplored class of oscillatory flows. The flow is incompressible
and two-dimensional, and the cylinder oscillations are harmonic.
From the standpoint of controlling laminar two-dimensional vortex
shedding from a circular cylinder by using "active control" of
the nature of the vortex shedding process can be significantly
altered by cylinder oscillation. Generally speaking, the control
of flow physics and near-wake structure of a bluff body may take
the form of global control, where the entire body is subjected to
prescribed motion, or local control, involving localized
application of unsteady blowing/suction or heating at specified
positions on the surface of the stationary body. Attention here is
focused on the case of global control where the time-dependent
dimensionless recti-linear oscillatory velocity of a circular
cylinder is represented by
where
is the dimensionless amplitude of oscillatory
velocity, 
is the forced Strouhal number and 
represents
the dimensionless displacement amplitude of recti-linear
oscillation. In this study the forced frequency 
is normalized
with the free-stream flow speed 
and the cylinder diameter 
.
The dimensionless time 
is related to the dimensional time 
through 
where 
is the cylinder radius.
For a cylinder, placed in a uniform free-stream, subject to recti-linear
oscillations the flow field depends mainly on three dimensionless
parameters. The first is the Reynolds number, defined as 
,
where 
is the coefficient of kinematic viscosity of the fluid. The
second is the forced Strouhal number 
which characterizes the
forced oscillation frequency. The third is the dimensionless displacement
amplitude of recti-linear oscillation . If a fluid is in relative
motion past a bluff cylinder which is forced to vibrate over a range
of frequencies near the Kármán vortex-shedding frequency,
then resonant flow-induced oscillations of the cylinder occurs
(i.e the lock-on or synchronization phenomenon,
,
where 
is the Kármán vortex-shedding frequency for the
flow past the cylinder without oscillation). It is noted that
is normalized with the constant speed of the cylinder translation
and the cylinder radius ; it varies with 
and remains
practically constant (namely, at the value of 
) for 
. Results of this study are examined for a Reynolds number of
and 
and a fixed motion amplitude of 
at two
values of angle of inclination
and 
of cylinder
oscillation with respect to free stream. The study concentrates on
a domain of oscillations frequencies
.
The investigation is based on the solution of unsteady Navier-Stokes equations together with the mass conservation equation in the case of viscous fluids. The method of solution is based on the use of truncated Fourier series representations for the stream function and vorticity in the angular polar coordinate. A non-inertial coordinate transformation is used so that the grid mesh remains fixed relative to the accelerating cylinder. The Navier-Stokes equations are reduced to ordinary differential equations in the spatial variable and these sets of equations are solved by using finite difference methods, but with the boundary vorticity calculated using integral conditions rather than local finite-difference approximations.
The cylinder motion starts suddenly from rest at time 
.
Immediately following the start of the cylinder motion, a very
thin boundary-layer develops over the cylinder surface and grows
with time. Accordingly, we divide the solution time into two
distinct zones. First zone begins following the start of fluid
motion and continues until the boundary-layer becomes thick enough
to use physical coordinates. In this zone, we use boundary-layer
coordinates, which are appropriate to the flow field structure, in
order to obtain an accurate numerical solution. The spacing of the
grid points used in this zone are such that they are spaced closer
together near the surface and further apart at large distances. In
addition, the adopted grid mesh continually grows in time to
properly accommodate the vortex shedding process and
boundary-layer development of the flow. The second zone starts
following the first one and continues until the termination of
calculations. The change from boundary layer coordinates to
physical coordinates is made when boundary layer thickens which
also ensures that the same grid points can be used in
boundary-layer and actual physical space. In this way the
numerical solution procedure can be started with good accuracy and
continued with comparable accuracy until a periodic
vortex-shedding regime is established.
The numerical method is verified for small times by comparison
with the analytical results of a perturbation series solution and
an excellent agreement is found. The time variation of the in-line
and transverse force coefficients are first presented for three
values of the forced frequency and the flow field
development is also first presented for the case 
when
in the form of streamline patterns. The results of
this study are consistent with previous experimental predictions.
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