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Astron. Astrophys. 343, 641-649 (1999)

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4. Axi-symmetrical flow over a sphere

In this section we present numerical simulation results for an axi-symmetrical bow shock flow over a perfectly conducting sphere, for parameter values of [FORMULA] and [FORMULA] which are situated in the switch-on domain (Fig. 3). These are the same parameters as for the bow shock flow around a cylinder which was studied in De Sterck et al. (1998b). We will investigate if the axi-symmetrical bow shock flow over a sphere in the switch-on regime exhibits a complex bow shock topology similar to the topology of a flow around a cylinder in that parameter regime. In Fig. 8a we show a global view of the converged axi-symmetrical bow shock solution. The horizontal x-axis (coinciding with the stagnation streamline) is an axis of rotational symmetry. The leading shock front shows a clear dimple, and there seem to be additional discontinuities behind the leading shock front. The shock front is much closer to the object than in the case of the flow around a cylinder with the same inflow parameters, which is shown in Fig. 8b for comparison.

[FIGURE] Fig. 8. a  Steady axi-symmetrical solution of the flow over a perfectly conducting sphere, with [FORMULA] and [FORMULA] ([FORMULA] grid). Density contours pile up in shocks, and streamlines come in horizontally from the left. b  For comparison, the steady bow shock solution for the flow around a cylinder (Fig. 2), with the same inflow parameters. In the flow over a sphere, the shock fronts are much closer to the object than in the cylinder case, because a sphere obstructs the flow much less than a cylinder.

In Fig. 9 we show a detailed representation of the central part of the axi-symmetrical bow shock solution near the stagnation streamline. This plot is to be compared to its cylinder flow equivalent shown in Fig. 2 (where only the upper part of the symmetrical flow is plotted). The flow clearly exhibits a topology which is very similar to the topology of the flow around a cylinder. Inspection of the way in which the field lines are refracted when the shocks are passed, reveals that the shocks in all the flows are of the same type as the shocks in the model flow of Fig. 2 which were discussed in Sect. 2.1, and detailed analysis of upstream and downstream Mach numbers, along the lines of the detailed analysis in De Sterck et al. (1998b), confirms this conclusion.

[FIGURE] Fig. 9. Detail of the steady axi-symmetrical solution of the flow over a perfectly conducting sphere ([FORMULA] grid). Density contours pile up in shocks, and streamlines come in horizontally from the left. For the symmetrical flow over a sphere, parameters in the switch-on domain lead to a complicated topology which is very similar to the topology of the flow around a cylinder.

We can thus conclude that in the switch-on regime the axi-symmetrical flow over a sphere exhibits a complex bow shock topology very similar to the topology of a bow shock flow around a cylinder in that parameter regime.

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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999
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