Time in seconds, for both Reynolds numbers and the two methods described for the liddriven cavity problem.
Abstract
In this work, we discuss the numerical solution of the Taylor vortex and the liddriven cavity problems. Both problems are solved using the Stream functionvorticity formulation of the NavierStokes equations in 2D. Results are obtained using a fixed point iterative method and working with matrixes A and B resulting from the discretization of the Laplacian and the advective term, respectively. We solved both problems with Reynolds numbers in the range of 3200 ≤ Re ≤ 7500. Results are also obtained using the velocityvorticity formulation of the NavierStokes equations. In this case, we are using only the fixed point iterative method. We present results for the liddriven cavity problem and for the Stream functionvorticity formulation with Reynolds numbers in the range of 3200 ≤ Re ≤ 7500. As the Reynolds number increases, the time and the space step size have to be refined. We show results for 3200 ≤ Re ≤ 20,000. The numerical scheme with the velocityvorticity formulation uses a smaller step size for both time and space. Results are not as good as with the Stream functionvorticity formulation, although the way the scheme behaves gives us another point of view on the behavior of fluids under different numerical schemes and different formulation.
Keywords
 NavierStokes equations
 velocityvorticity formulation
 Stream functionvorticity formulation
 Reynolds number
 fixed point iterative process
1. Introduction
The fixed point iterative method has already been used for solving the NavierStokes equations and the Boussinesq system under different formulations, see [1, 2, 3, 4].
The idea behind this iterative method was to work with a symmetric and positive definite matrix. The scheme worked very well, as shown in [1, 2, 3, 4], but the processing time was, in general, very large especially for high Reynolds numbers. Working with matrixes A and B, we are dealing with a nonsymetric matrix, but it can be proved that it is strictly diagonally dominant for Δt sufficiently small. The processing time with the second method was reduced in approximately 30 or 35%.
Additionally, in order to show that the fixed point iterative method works well for moderate and high Reynolds numbers, we report results for the liddriven cavity problem and Reynolds numbers in the range of 3200 ≤
Results, in both formulations, are obtained using the fixed point iterative method reported in [5], which is applied to a nonlinear elliptic system resulting after time discretization. The method has shown to be robust enough to handle moderate and high Reynolds numbers, which is not an easy task, see [1, 2].
As the Reynolds number increases, the mesh has to be refined and a smaller time step has to be used, in order to capture the fast dynamics of the flow and, numerically, because of stability reasons, as mentioned in [6], although, with the velocityvorticity formulation [7, 8], a finer mesh has to be used, both in time and in space.
The computing time is, in general, very large with this numerical scheme and for both formulations, so that is why we are looking forward to reduce computing time working with both matrixes A and B resulting from the discretization of the Laplacian and the advective term, respectively, instead of working just with matrix A, which is symmetric and positive definite.
With the Stream functionvorticity formulation, and for moderate and high Reynolds numbers, the second scheme was faster than the fixed point iterative method (see [9, 10]). With respect to the velocityvorticity formulation, we are just showing results using the fixed point iterative method, and for lower Reynolds numbers, but we are looking forward to modify the scheme also in order to reduce computing time.
2. Mathematical model
Let
These are the NavierStokes equations in primitive variables. This system must be provided with appropriate initial conditions
When working in a twodimensional region
followed by (Eq. (1b)), with ψ the Stream function,
The following system of equations is then obtained:
where ω is the vorticity given by ω =
This system represents the NavierStokes equations in the Stream functionvorticity formulation. The incompressibility condition (Eq. (1b)), because of (Eq. (2)), is automatically satisfied and the pressure does not appear any more.
3. The velocityvorticity formulation
Taking the curl in
and using the identity
Two Poisson type equations for the velocity are obtained, which together with the equation for the vorticity give us:
These are the NavierStokes equations in the velocityvorticity formulation.
4. Numerical method for the Stream functionvorticity formulation
The following secondorder approximation for the time derivative is used:
where
The resulting discretization system reads:
where
At each time level, the following nonlinear system has to be solved.
To obtain
For solving this system of equations, two strategies were used in this work: First, we used the fixed point iterative method described in [5]:
Denoting
our system is equivalent to
Then, at each time level, the following the fixed point iterative process (see [5]) is used:
Given
and then take
To reduce the computing time, we worked in solving the system by the following method at each time step:
where A and B are the matrixes resulting from the discretization of the Laplacian and the advective term respectively, and (Eq. (13b)) is solved using GaussSeidel method.
5. Numerical method for the velocityvorticity formulation
The secondorder approximation (Eq. (7)) for the time derivative is used and the following nonlinear system is obtained in Ω
where
Using again the fixed point iterative method previously described, we have:
Given
and then take
6. Numerical experiments
With respect to the liddriven cavity problem and using the Stream functionvorticity formulation
where
In
Figures 1
and
2
, we show results for the liddriven cavity problem with
For the Taylor vortex problem, results are shown in
Figures 3
and
4
for
The exact Stream function and the vorticity are also shown in
Figure 5
, for
In the primitive variables formulation, we have as initial conditions:
For the Stream function, the boundary conditions are:
For the vorticity, the boundary conditions are:
In Tables 1 and 2 , we show computing times for the abovementioned problems with both the methods; the Fixed Point Iterative Method and working with matrixes A and B.

Fixed point iterative method (s)  Working with A and B (s) 

5000  153  120 
7500  801  610.25 

Fixed point iterative method (s)  Working with A and B (s) 

5000  15.5  12.75 
7500  15.5  12.75 
In
Figure 6
, we show the streamlines and isovorticity contours for
Then, in
Figure 8
, we show just the isovorticity contours for
In the case of the velocityvorticity formulation and the liddriven cavity problem, the boundary condition for u is given by
Not all the results were obtained using the secondorder discretization. In some cases, a fourthorder discretization has to be used, using the fourthorder option of Fishpack [12] (used in this work for solving the elliptic problems appearing).
In
Figure 9
, we show the streamlines and the isovorticity contours for
In
Figure 10
, we show the isovorticity contours for
As can be noticed with the Stream functionvorticity formulation, we are using a value of h half of the size of the one used with the velocityvorticity formulation. We assume the results obtained with the firstmentioned formulation are more reliable. Computing time for the velocityvorticity formulation was much larger. We think there are still some numerical problems with this formulation and for very high Reynolds numbers.
7. Conclusions
For the liddriven cavity problem results agree very well with those reported in the literature [1, 2, 3, 4, 13, 14], and by working with matrixes A and B it was possible to reduce computing time between a 30 and 35%.
As can be seen in Figures 1 and 2 , numerical oscillations occurred, given the high Reynolds numbers used in such a way that it is necessary to use smaller values of h [6], numerically because of stability of the method and physically in order to capture the fast dynamics of the flow.
For high Reynolds numbers and small values of h the computational work takes a lot of time, so reducing computing time becomes a very important fact. For the Taylor Vortex Problem [8, 15], processing time was also reduced between 30 and 35%.
With the velocityvorticity formulation, as already mentioned, we only show results using the Fixed Point Iterative Method, and we are looking forward working with both matrixes A and B, in order to reduce computing time also with this formulation. This is the reason why we only show results till
In conclusion, the numerical scheme applied with the stream functionvorticity formulation is not as good with the velocityvorticity formulation, although, the way it behaves with some values of the parameters and the order of discretization, gives us another point of view about the behavior of the fluids under different numerical methods and different formulations.
We must also say that our code has not been parallelized since it is difficult to do this. It must be taken into account that the equations, in both formulations, are coupled. We are looking forward to use a solver for the system of linear equations that can be parallelized. This can be viewed as a future work.
Acknowledgments
The authors would like to acknowledge the support received by the National Laboratory of Supercomputing from of the southeast of Mexico BUAPINAOEUDLAP (Laboratorio Nacional de Supercómputo del Sureste de México).
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