This work is devoted to the study of the influence of variation of the controlled dissipative properties on the accuracy of the QGDFoam solver and the visual representation of this influence. The work continues a series of studies on the comparative assessment of the accuracy of various numerical methods and solvers built on their basis. To carry out a comparative assessment, a generalized computational experiment for classes of problems with a reference solution is constructed and implemented. A generalized computational experiment based on the synthesis of solutions of mathematical modeling problems, parallel technologies and visual analysis tools makes it possible to obtain solutions not only for individual problems, but for whole classes of problems determined by the specified ranges of key parameters. Accordingly, a comparative assessment of the accuracy of numerical methods is also carried out for a class of problems. Earlier, a similar computational experiment was carried out for a comparative assessment of the accuracy for solvers of the OpenFOAM open source software package on the well-known classical problem of an oblique shock wave formation. One of the solvers participating in the calculations, namely the QGDFoam solver, was the only one of all to have controlled dissipative properties. New generalized computational experiment was implemented to study the effect of variation of the parameter that controls the dissipative properties. The target was to reduce the error in comparison with the reference solution. The research results are presented in this work.
assessment of the accuracy
computational experiment, visualization, QGDFoam solver, comparative
This work develops an approach to constructing a generalized computational experiment for
comparative analysis of the accuracy of numerical methods and solvers implemented on their basis. The
work continues a series of studies devoted to the comparative assessment of the accuracy for solvers of
the open source software package OpenFOAM [
The general essence of the research carried out can be described as follows. Problems are selected
that have a reference solution in certain ranges of key parameters. The region of the key parameter
space is meshed. At each point of the grid partition, the problem under consideration is solved using
several solvers and compared with the exact solution. The target functional here is the error of each
solver - a deviation from the reference solution. The data obtained are investigated using visual analysis
and give a sufficient idea of the comparative accuracy of the considered solvers in a specific class of
problems [
Previously, a similar study was carried out for 4 solvers of the OpenFOAM open source software package for the problem of oblique shock wave formation. A shock wave was formed when a supersonic flow fell on a plate at an angle of attack. Results have been obtained that allow a comparative assessment of the accuracy [6, 7]. However, one solver from the comparison, namely the QGDFoam solver, has
2021 Copyright for this paper by its authors. unique properties. This solver has a parameter that allows one to control the dissipative properties of the implemented numerical method, that is, to regulate the artificial viscosity. Therefore, we had an idea to investigate the issue of improving the accuracy of this solver by varying this parameter. These studies are summarized in this paper.
This work is based primarily on the use of a generalized computational experiment. A generalized computational experiment is a computational technology based on the synthesis of solutions to problems of mathematical modeling, parallel technologies and visual analysis tools. This technology makes it possible to obtain solutions not only for individual problems, but for entire classes of problems, determined by the specified ranges of changes in key parameters. As a rule, the purpose of constructing a generalized computational experiment is to study the dependence of a certain target functional on changes in the defining parameters of the problem under consideration. The result of such an experiment is multidimensional data, the study of which requires the use of visualization and visual analytics methods.
This work continues the research cycle. Previous papers contain a description of the main approaches
to the construction of a generalized computational experiment and the visualization problems arising in
this case. Papers [
The system of quasi-gas dynamic (QGD) equations was created in the eighties by a group of scientists from the
B.N. Chetverushkin [9, 10]. Later, quasi-gas dynamic equations were presented in the form of conservation laws, investigated in detail, and theoretically substantiated [11-15]. Monographs dedicated to the derivation of the equations and their applications to numerical modeling problems were also written [16, 17]. A fundamental and essential difference of the QGD approach from the Navier-Stokes theory was the use of a space-time averaging procedure to determine the basic gasdynamic quantities. Thus, additional terms appear. More precisely, the mass flux density vector, the viscous stress tensor and the heat flux vector are represented as a sum of the corresponding quantities in the Navier-Stokes form and small additions of a significantly nonlinear form, proportional to a small parameter having the time dimension.
Many calculations have been carried out based on the QGD system of gas dynamics equations. However, all of them were performed using individual programs. In order to extend the application of the QGD approach to a wider range of problems, the OpenFOAM solver [18-20] was developed under the guidance of T.G. Elizarova at the Keldysh Institute of Applied Mathematics of the Russian Academy of Sciences.
One of the most important properties of this approach is the possibility to control the dissipative properties. In the framework of the Euler equations the dissipative coefficient can be written out as: where α is a dimensionless parameter, ℎ is the spatial grid step, is the sound speed. The presence of a controllable parameter with dissipative terms allows one to successfully suppress undesirable oscillations in numerical simulations of problems with discontinuities.
= ℎ
During numerical experiments, the problem formulation fully corresponded to that described in [6, 7]. We considered a two-dimensional problem of oblique shock wave formation. A supersonic flow of inviscid gas falls on a half-plate at the angle of attack. An oblique shock wave is formed at the end of the plate. The problem was considered with a variation of the defining parameters, where the Mach number varied from 2 to 4 in 0.5 step, and the angle of attack varied from 6° to 20°. The deviation from the known exact solution in different norms, L1 and L2, was calculated. Using the obtained data, the error surfaces were plotted for all solvers involved in the calculations.
In the past studies [6, 7] the parameter α was taken equal to 0.1 for the solver QGDFoam. But we need to find the optimal values of this parameter to minimize the error in the norms L1 and L2. Taking into account that a single uniform grid is used for all calculations, we have to solve the inverse problem. We need to find such a value of the parameter α at which the value of the error is minimal in both norms. Then the problem of finding such values can be formulated as follows:
Argmin ErrL1(α) , ErrL1(α) = ∑ | ( )− | for norm L1
∑ | | Argmin ErrL2(α) , ErrL2(α) =
for norm L2 √∑ ( ( )− √∑ ( )2 )2 (2) (3)
The search for optimal values of α was carried out for fixed values of the defining parameters M=2, β=6°. This variant was chosen as a basic one in [6, 7]. To begin with, the minimum possible value of the parameter was found, at which the solver worked. For this problem the value was α=0.031. Further a numerical calculation with variation of parameter α was carried out and an error for the pressure field in norms L1 and L2 was found. The minimum values of the norm are highlighted in bold. The results are presented in Table 1.
The values of the parameter α that provide the minimum error in the norms L1 and L2 were found. These values differ from the previously used value α = 0.1. For norm L1 the optimal value is α = 0.125, for norm L2 – α = 0.1075. It was decided to carry out a generalized computational experiment for each norm separately with a fixed value of α at variations in Mach number and angle of attack. It would be natural to solve such an inverse problem for each point of the grid partitioning of the area of the defining parameter space, but this is too computationally expensive. Numerical solutions with parameter α computed from the point for the base case should provide insight into the effect of variation in parameter α on the accuracy of the solver under consideration. The results are presented in Tables 2-11.
rCF
rCF
rCF
rCF
rCF
pCF
pCF
pCF
pCF
pCF
rCF
rCF
rCF
rCF
pCF
pCF
pCF
pCF
It should be noted that during the calculations the authors were interested in the high error that appeared at Mach number M = 2 and angle β = 20° for the solvers rhoCentralFoam and pisoCentralFoam in norm L1. A similar phenomenon was observed in the calculations at the same Mach number and angles β = 15°, 20° for the L2 norm. When solving these solvers numerically, oscillations perpendicular to the shock wave front appear (Figure 1). The QGDFoam solver has a parameter that affects the numerical dissipation and, with the optimal values of the parameter used, the oscillations are not visually distinguishable (Figure 2). This is serious evidence of the advantages of the QGDFoam solver, which has the ability to limit undesirable oscillations due to the presence of controlled dissipative properties.
During numerical experiments, the problem formulation fully corresponded to that described in [6, 7]. We considered a two-dimensional problem of oblique shock wave formation. A supersonic flow of inviscid gas falls on a half-plate at the angle of attack. An oblique shock wave is formed at the end of the plate. The problem was considered with a variation of the defining parameters, where the Mach number was varied from 2 to 4 in 0.5 step, and the angle of attack was varied from 6° to 20°. The deviation from the known exact solution in different norms, L1 and L2, was calculated. Based on the data obtained, error surfaces were plotted for all solvers involved in the calculations.
Figure 3 shows the error surfaces in the L2 norm for all four solvers that took part in the comparison. The calculations for the QGDFoam solver were performed with a single chosen value of α = 0.1.
The Figure 4 shows the results for the same solvers in the L1 norm, but added surface, denoted as QGDF*, calculated at α = 0.125. It can be seen that the variation of the parameter α reduced the error. The surface QGDF* lies substantially below the surface QGDF.
The following Figure 5 presents in the L1 surface norm for the solver QGDFoam when choosing α = 0.1 (QGDF) and α = 0.125 (QGDF*).
The figure shows a significant reduction in the error when the previously found value of the parameter α = 0.125 is chosen. The maximum error reduction is 12.6% with respect to the values obtained by choosing α = 0.1.
Next, we consider similar results for the norm L2. Here, to plot the surface corresponding to the error for the QGDFoam solver, the parameter α = 0.1075 was chosen according to Table 1. Figure 6 shows the results for the solvers rhoCentralFoam, pisoCentralFoam, sonicFoam, and QGDFoam when α = 0.1 (QGDF) and α = 0.125 (QGDF*) were chosen.
The overall picture is about the same as in Figure 4. However, for the norm L2 the error reduction is significantly smaller. The maximum error reduction here is 2.3% with respect to the values obtained by choosing α = 0.1. The following Figure 7 presents in the L2 norm a close-up of the error surface for the QGDFoam solver when α = 0.1 (QGDF) and α = 0.1075 (QGDF*) are chosen.
Thus, the implemented generalized computational experiment allows us to assert that variations of the parameter regulating the dissipative properties of the QGDFoam solver can significantly reduce the error in comparison with the reference solution and increase the accuracy of calculations. This should be considered as a clear advantage of this solver and a great potential in its use for solving practical problems of mathematical modeling.
In this paper, we studied the effect of variation in controllable dissipative properties on the accuracy of the QGDFoam solver and a visual representation of this effect. This work continues a series of studies on the comparative evaluation of various numerical methods accuracy and solvers built on their basis. The study is based on the construction of a generalized computational experiment for a comparative analysis of the accuracy of numerical methods and solvers based on them. The realization of the generalized computational experiment has shown that variations of the parameter regulating dissipative properties of the solver QGDFoam can significantly reduce the error in comparison with the reference solution and increase the accuracy of calculations.
The study was supported by a grant from the Russian Science Foundation № 19-11-00169, https://rscf.ru/project/19-11-00169/
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