(1)
[1]Jing Chunbo, Liang Dongfang, Wang Guangqian
Dept. Hydraulic Eng., Tsinghua University, Beijing 100084
E-mail:
jcb@mail.tsinghua.edu.cn
Abstract:
The incompressible flows are solved by using the
fractional step finite volume method. As the weakly compressible model is used,
it is possible to solve the velocity components and pressure explicitly, so high
computational efficiency can be achieved. The benchmark examples of
incompressible flows, i.e., the cavity flow, sudden expansion flows and flow
around a circular cylinder are simulated. The results show its numerical
advantages.
Keywords: finite volume scheme, weakly compressible model, incompressible flows
In part1, the proposed finite volume formulations are given in very detail, and the stability analysis is carried out. In this paper, this new scheme is applied to solve the incompressible flows. In numerical solutions, the incompressibility of the fluid makes the problem more difficult[1-3], because the continuity equation for incompressible flows is independent of the derivative of pressure, no explicit numerical integration scheme can be used. In order to overcome this difficulty, the penalty function method has been proposed to solve steady and unsteady flow problems of high Reynolds number. Their computations are based on the implicit integration in time, so the computational scale is limited. The ideas of the penalty function method are close to that of the artificial compressibility method.
A fractional step FEM has been proposed[4], it has good accuracy and stability properties and is computationally efficient. Based on this finite element method, a new finite volume formulation is proposed to solve weakly compressible equations using unstructured grid in present investigation. The two dimensional incompressible flows, such as cavity flows and sudden expansion flows, are computed, the results are compared with that of the former computation[5,6]. The results show that the present FVM is computational efficient and can be applied to solve problems with complicated boundary conditions.
Considering the following compressible Navier-Stokes equations (the details of the equations can be found in Part1), its dimensionless form can be expressed as
(1)
(2)
where,
is the velocity component, p is the pressure, c is the speed of the sound,
is
coordinate, t is time,
. Re is Reynolds number defined as
, L and U are the reference length and reference velocity respectively.
The finite volume formulations of cell-centered form of equations (1) and (2) can be obtained as(for the details, please see Part1)
(3)
where
represents
or
;
represents
or
(the details of the equations can be found in Part1). As equation(3) possesses diagonal
dominated property, it can be solved by the Jacobian iteration
(4)
where
is the lumping matrix and k is
iteration step. In general, two times of iteration is enough in the present
computations. From above processes, we can find that it is unnecessary to form
the global coefficient matrix and only cell level matrix is needed in the
computations. When this formulation is combined with the 3-step time
integrations, much more stable and accurate solution can be obtained for
convection dominated flows. The three-step time integrations can be written as
(5)
Two-dimensional Cavity flow at Reynolds number
400 is calculated by present FVM. The 41
41 mesh, as shown in Fig.1 (a) is used. The time step is limited by the CFL
condition, which requires the grid Courant number
(
), where u is the maximum velocity
value in the computational domain, h is the minimum grid side length. The velocity boundary conditions
are : u=1, v=0, on the top side; the non-slip velocity boundary
conditions are introduced on the other three boundary sides. The vector field
and streamline distributions are shown in Fig.1 (b), (c), besides the large main
recirculation region, the secondary recirculation region at the bottom Connor
can be obtained clearly. The velocity profiles along the vertical and horizontal
center lines are plotted in Fig.1 (d), the results obtained by Ghia at al.
(1982) which using very fine grid (129
129) and multi-grid implicit solution method are also shown in the same figure.
It can be seen that the present results are agree well with Ghia’s solution,
even if the relatively coarse grid is used in the present FVM computations.
(a) Computational grid
(b) Velocity
(c) Stream lines
(d) Velocity profiles (◆:
Ghia et al., 1982)
Fig. 1
Two-dimensional cavity flow (Re=400)
Another benchmark example to check the numerical scheme
is the sudden expansion flow through a channel. The boundary conditions are:
horizontal velocity component u is parabolic distributed and the maximum value
, v=0, on the inlet (left side); u=v=0, on the solid wall; p=0
and
on outlet boundary (right side).
The Reynolds number is 60, which is based on the maximum inlet velocity and the
width of the inlet boundary. The results are given in the dimensionless form,
the velocity vector and the pressure contour lines are shown in Fig.2 (a) and
(b), the recirculation region can be obtained in the present computation. The
computed pressure along the line y=0.0, as shown in Fig.2 (c), is compared with
the result calculated by Kawahara and Hirano (1983) which use weakly
compressible flow model and the Lax-Wendroof time integration. It can be seen
that the present simulation is in agreement with the former results.
(a) Velocity field
(b) Pressure contour
(c) Pressure on y=0.0
Fig. 2 Two-dimensional sudden expansion flows (Re=60)
The vortex evolution behind a circular cylinder is computed by the present finite volume method, as the Karman vortex is produced in the wake region, the flow field is unsteady. This flow is more complicated than that of the cavity flows and the sudden expansion flows. The numerical scheme should simulate the separation point on the circular cylinder face and the vortex shedding properties accurately. The diameter of the cylinder is taken as 1,
(a)
Computational mesh
t=183s
t=189s
(b) Stream lines near the cylinder
Fig. 3 Streamline patterns around a circular cylinder
The computational domain is about 45 in the flow direction, 20 in the direction perpendicular to the main flow direction, which can insure the flow properties in the wake region do not affected by the outside boundary effect. The boundary conditions are: u=1, v=0 on the inlet side (left); v=0 and the shear stress equals zero on the upper and down sides; v=0, p=0 and shear stress equals zero on the outlet boundary (right); non-slip velocity boundary condition is given on the cylinder face. The Reynolds number is 100 in the present simulation, the computational grid shown in Fig. 3(a), there are 2815 nodal points and 5424 triangular control cells. In order to simulate the separation point, the very fine grid is used near by the cylinder face, where the minimum grid side length is 0.05. The time step is limited by this small grid size. As the present numerical model uses the explicit scheme both for velocity components and pressure, it is computation efficient compared with the implicit scheme or using the Poission equation for the pressure. The streamlines at time 183 and 189 seconds are shown in Figs.3 (b). In order to seen the flow properties in the wake region, only the streamlines near by the cylinder and in the wake region are plotted. It can be seen that the Karman vortex can be obtained, the vortex-shedding period can be described by the Strouhal number (St). From the streamline pattern obtained by the present computation, it can be found that St is equal to 6, which is in agreement with that reported by Williamson (1996).
A new finite volume formulation, which using
fractional step time integration and consistent volume integration matrix, has
been proposed to solve the incompressible flows. The weakly compressible model
is used in the simulations, which need less computer storage and CPU time. The
cavity flow and flows around circular cylinder are simulated by using the
present method, the results are in agreement with the literature.
Acknowledgements
This research was supported by the State key 973-Project for fundamental Research and Development Program(Grant No. G1999043607 ) and the National Science Foundation (Grant No. 59979013).
References
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