Page 287 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

SOME EXAMPLES OF FLUID FLOW AND HEAT TRANSFER PROBLEMS

(a) Finite element surface mesh (b) Instantaneous u 1 velocity contours 279

Figure 9.17 Isothermal flow past a circular cylinder. Three-dimensional finite element
mesh and an instantaneous u 1 velocity contour, Re = 100

surfaces were assumed to have no flow in the direction normal to the surfaces. Since the
two-dimensional problem was solved in three dimensions by introducing a third dimension,
the width of the domain in the third dimension is arbitrary. The smaller the size of the
domain in the third dimension, the smaller will be the number of elements in the mesh.
For the three-dimensional computations carried out here, the length in the third dimension
wasassumedtobeequalto 0.5D.
The three-dimensional surface mesh is shown in Figure 9.17(a). The volume mesh
used within the domain was generated using linear tetrahedral elements. A total number
of approximately 600,000 elements were used in the calculations. As may be observed,
the mesh is very fine behind the cylinder, along the expected von Karman vortex street.
This is essential in order to accurately predict the flow. A mesh convergence study in three
dimensions is time-consuming and difficult, and it is advisable to analyse many meshes
in order to prove the convergence of the results. Alternatively, if the problem has existing
results, then a comparison with these will give confidence about the results generated. Here,
we chose the alternative approach and compared our results with the existing data.
The calculation was carried out using the fully explicit form of the CBS scheme
(Nithiarasu 2003). The initial values of u 1 and u 2 were assumed to be equal to unity
and zero respectively. Note that these values are non-dimensional. All the velocity values
are non-dimensionalized using the reference inlet velocity value (see Chapter 7 for details).
Similarly, the distances are scaled with respect to the diameter of the cylinder. These scal-
ings result in a non-dimensional inlet velocity value of unity and a cylinder diameter of
unity in the non-dimensional space. The initial values of pressure were assumed to be zero
everywhere in the domain.
As mentioned previously, the solution to this problem is known to be periodic with
respect to time. Once the solution reaches a steady periodic state, the periodic vortex shed-
ding continues indefinitely. This process consists of vortex formation behind the cylinder
and shedding.