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In this study, the velocity-vorticity formulation, in
combination with boundary element method and Euler-Lagrangian
method is utilized to solve the unsteady two-dimensional
incompressible Navier-Stokes equations. The correctness of
present model is verified by a simple driven cavity flow by
comparing with existing literature. The contents of present
study are divided into three parts. In the first part, the
computation is simplified to a steady Stokes flow, in order to
fully grasp the influence of vorticity boundary conditions on
the flow field. Four different kind of boundary conditions are
specified to analyze numerical results. The moving top lid and
the vorticity distribution of singular points at the
intersection where two side walls meet, are found to have
decisive effect on the flow field. After refining the grid
points near the singular points, the flow circulation renders a
value of -1.001. This magnitude approaches the analytic value -1
with a relative error less then 0.2%. In the meantime the
continuity equation reaches the accuracy of O(-14), which is
very closed to the theoretical value 0.The vorticity of vortex
center located at (0.50,0.760) has a value of -3.12, and stream
function value -0.00995, which are all very consistent with
previous studies. In the second part, two-dimensional unsteady
viscous incompressible flow, whose Reynolds number ranging from
0.001 to 1000 is under consideration. Several important flow
characteristics, such as velocity distribution of central line,
center of vortex, stream function , and size of corner eddies,
are all compared with the results from Burggraf, Ghia, Pan,
Young, etc. Good validation is obtained for the present model
through careful comparison. In the third part, the dynamic
characteristics of flow field is investigated. In most
conventional studies, only one monitor point is taken to
determine the dynamic behavior of flow field. In this study, 25
monitor points are put to record the flow domain. It is found
that even under the same Reynolds number, the dynamic phenomena
vary greatly at different positions. In addition, the flow is
found to take periodic as well as quasi-periodic motions at
Reynolds number 1 and 10. A steady solution converging to O(-11)
at Reynolds 0.001 is found.. In addition, when Reynolds number
is 1, this model finds that the steady state and periodic
solutions will coexist at different locations. It is interesting
to observe that a steady state solution is reached for a
particular position from a macroscopic scale point of view.
However an oscillatory motion is revealed from a microscopic
scale measurement. Near the boundary, if observed by a
microscopic scale, the flow has simple periodic solution.
However , the periodic solution will contain more than 10
harmonics around the cavity center. When the Reynolds number is
increased to 10, the chaotic characteristics will also change
with the location. The amplitude of the oscillation is smaller
than Reynolds number 1. Subharmonics still exist but the number
of frequency drops to 6 approximately. The distribution of
different kinds of solutions will be symmetric about the
vertical center line, and non-symmetric with the horizontal
center line, due to velocity distribution.
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