Application of Mo ving Adaptiv e Meshes in - ccas.ru fileving mesh is applied to generate mo ving...
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Transcript of Application of Mo ving Adaptiv e Meshes in - ccas.ru fileving mesh is applied to generate mo ving...
Application of Moving Adaptive Meshes in
Hyperbolic Problems of Gas Dynamics �
Boris N. Azarenoky
Abstract
Coupled algorithm of using the Godunov's type solver and adap-
tive moving mesh is applied to generate moving adaptive mesh when
modeling a 2D explosion with complicated wave structure.
1 Introduction
Coupled algorithm of using the Godunov's type solver and adaptive movingmesh has been o�ered in [1, 2]. In this approach after every mesh iterationthe �nite-volume ow solver updates the ow parameters at the new timelevel directly on the curvilinear moving grid without interpolation from onemesh to another [3, 4]. Thus, we eliminate the errors caused by interpola-tion procedure which smears the discontinuities. This scheme on one handutilizes the idea of the Godunov's scheme on the deforming meshes [5] andon the other hand is of the second-order accuracy in time and space.
Method of adaptive grid generation has been suggested in [6]. Methodis variational, i.e. we consider the problem of minimizing a �nite-di�erencefunction approximating the Dirichlet's functional written for surfaces. Thediscrete functional has an in�nite barrier at the boundary of the set of gridswith all convex quadrilateral cells and this guarantees unfolded grid genera-tion during computations both in any simply connected, including noncon-vex, and multiply connected 2D domains. This folding-resistant propertyis very important since if any of the cells becomes folded we have to stopcalculation of the ow problem and use special procedures to continue mod-eling. The barrier property is also of particularly importance in the vicinity
�This work was supported by the Russian Fund of Fundamental Research (project
code 02-01-00236)yComputing Center of Russian Academy of Sciences, Vavilov str. 40, GSP-1, Moscow,
Russia. Email: [email protected]
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of shock waves where the cells are very narrow and maximal aspect ratioachieves 50 and larger [7].
When modeling 2D hyperbolic problems with discontinuous solution onthe moving adaptive mesh it is possible to reduce the errors, caused byshocks smearing over the cells, by many factors of ten and, therefore, todecrease signi�cantly, by several times [7], the overall error.
2 Modeling of explosion
We consider a spherical explosion between two parallel walls at z=0 andz=1 [8]. Initially the gas is at rest with parameters (p; �)out=(1; 1) eve-where except in a sphere centered at (0; 0; 0:4) with radius 0:2. Inside thesphere (p; �)in=(5; 1), the ratio of speci�c heats =1:4. The initial jumpin pressure results in an outward moving shock and a contact discontinuityand an inward moving rarefaction wave. There occurs interactions betweenthese waves and between waves and the walls. The ow by t=0:7 consistsof several shocks and strong tangential discontinuity surrounding the low
density region near the center.First, until the initial outward shock reaches the wall z=0, the solution is
spherically symmetric. Further, it remains cylindrically symmetric and weuse the 2D axisymmetric formulation of the problem. In half-plane �=const:
(where � is an angle in cylindrical coordinates) we compute the domain(x; z)2[0; 1:4]�[0; 1] and on the axis x=0 de�ne the symmetric boundaryconditions. On the walls z=0; 1 we use the condition of re ection and atline x=1:4 undisturbed values (u; v; p; �)out are de�ned. The �xed mesh isrectangular. The pressure contours at t=0:7 computed on the �xed mesh100�140 are presented in Fig.1.
We perform adaptation by minimizing the Diriclet's functional [1, 2].On the walls and axis of symmetry we apply constrained minimization toprovide consistent redistribution of the grid nodes inside the domain andon its boundary [7]. Up to t=0:45 we calculate on the rectangular meshand then the adaptive procedure is switched on. Since strong grid linescompression results in decreasing the admissible time step (value of 4t fallsproportionally to the increase of the cell aspect ratio) we use the followingstrategy. Main part of the time interval we compute on not very condensedmesh and at some moment t
0=0:7�Æt we attain a maximum compressionof grid lines in the vicinity of discontinuities. Following this methodologywithin the time interval (0:45; 0:65) we set rather a small coeÆcient of adap-tation [1, 2] ca=0:07 to 0:1. Here it is suÆcient to perform 3 mesh iterations
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at every time step. The admissible time step 4t is decreased by 5 to 10times in comparison with 4t on the rectangular mesh.
Further, within the �nal interval (0:65; 0:7), the coeÆcient ca is increasedup to 0:18 to 0:4 and then time step falls down to 0:05 to 0:005 of 4t on therectangular mesh. It is not a constant value during computation becausethe mesh constantly "breaths" and accordingly the value of 4t periodically�rst decreases then increases, etc.. At this stage it is suÆcient to perform1 to 2 mesh iterations at every time step.
In the �rst series of computations with adaptation the pressure p is usedas a control function f [1, 2]. In Figs.2,3 the pressure contours and adaptedmesh are shown, respectively. We see that the shocks thickness is decreasedsigni�cantly in comparison with the �xed mesh solution presented in Fig.1.However the grid lines can be strongly condensed to the outer shock and twomore ones adjoining to it. This is due to insuÆcient number of grid points.In the next calculations we use the mesh 200�250, see Fig.4. Here boththe intensive shocks and compression waves are depicted by the condensedgrid lines rather �ne. Pressure contours, computed on the rectangular andadapted meshes, are presented in Figs.5,6.
Since p is continuous across the tangential discontinuity the mesh doesnot feel it. In the next calculations the density � it is used as a controlfunction f . Here the mesh 100�140 depicts the tangential discontinuityvery clear, see Fig.7. Density contours computed on the quasiuniform andadaptive meshes are presented in Figs.8,9. Comparing the grids with thecontrol function f=p, Fig.3, and f=�, Fig.7, one can note that in spite ofthe fact that in the second case ca is twice larger, in the �rst case the shocksare indicated by the grid lines a bit more clear. This occurs due to in uenceof the contact discontinuity on the mesh in the second case.
References
[1] B. N. Azarenok and S.A. Ivanenko, Application of adaptive grids in numer-
ical analysis of time-dependent problems in gas dynamics, Comput. Maths.
Math. Phys., 40(2000), No. 9, pp. 1330{1349.
[2] S.A. Ivanenko and B.N. Azarenok, Application of moving adaptive grids for
numerical solution of nonstationary problems in gas dynamics, Int. J. for
Numer. Meth. in Fluids, 39(2002), No. 1, pp. 1-22 (See also
http://www.math.ntnu.no/conservation/2001/043.html).
[3] B.N. Azarenok, Realization of a second-order Godunov's scheme, Comp.
Meth. in Appl. Mech. and Engin., 189 (2000), pp. 1031{1052.
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Figure 1: Pressure contours from 0:775 to 1:5 with increment 0:0125 in the x�z
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Figure 2: Pressure contours computed on the adapted mesh 100�140.
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Figure 3: Adapted mesh 100�140, p is a control function, ca=0:21.
Figure 4: Adapted mesh 200�250, p is a control function, ca=0:18.
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Figure 5: Pressure contours computed on the rectangular mesh 200�250.
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Figure 6: Pressure contours computed on the adapted mesh 200�250.
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Figure 7: Adapted mesh 100�140, � is a control function, ca=0:4.
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Figure 8: Density contours from 0:2 to 1:3 with increment 0:0125 computed on
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Figure 9: Density contours computed on the adapted mesh 100�140.
[4] B.N. Azarenok, S.A.Ivanenko and T.Tang, Godunov's scheme and moving
adaptive grids, Report on applied mathematics,
http://www.math.ntnu.no/conservation/2002/016.html
[5] S.K. Godunov (Ed.), et al., Numerical Solution of Multi-Dimensional Prob-
lems in Gas Dynamics, Nauka press, Moscow, (1976), (in Russian).
[6] S.A.Ivanenko, Adaptive grids and grids on surfaces. Comp. Maths. Math.
Phys, 33(1993), No. 9, pp. 1179{1193.
[7] B.N. Azarenok, Variational barrier method of adaptive grid generation in
hyperbolic problems of gas dynamics, SIAM J. Num. Anal., to appear. (See
also http://www.math.ntnu.no/conservation/2001/042.html)
[8] J.O. Langseth and R.J. LeVeque, A wave propagation method for 3D hy-
perbolic conservation laws, J. Comp. Phys. 165(2000), pp. 126{166.
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