Supersonic models

(a)Equally spaced contours oflogρ. (b)Velocity contours evaluated in Boyer-Lindquist coordi- nates. The stagnation point is marked.

Figure 8.13: Contour plots of density and velocity for a uniform supersonic flow past a non-spinning black hole with flow parameters given by model UA1. The plots are evaluated at timet= 300M.

(a)Equally spaced contours oflogρ. (b)Velocity contours evaluated in Boyer-Lindquist coordi- nates. The stagnation point is marked.

Figure 8.14: Contour plots of density and velocity for a uniform supersonic flow past a non-spinning black hole with flow parameters given by model UB1. The plots are evaluated at timet= 300M, and the axes are in units of M.

−15 −10 −5 0 5 10 15

−15

−10

−5 0 5 10 15

(a)Model UA1

−15 −10 −5 0 5 10 15

−15

−10

−5 0 5 10 15

(b)Model UB1

Figure 8.15: Accretion regions for all four supersonic models. The solid line shows the accretion region fora= 0,ǫρ= 0, and the dashed line shows the accretion region for a= 0.9,ǫρ= 0.2.

Upstream of the black hole, the velocity contours are almost circular and show the fluid increasing in velocity as it approaches the horizon. As the fluid passes through the shock, it is decelerated significantly.

(a)Equally spaced contours oflogρ. (b)Velocity contours evaluated in Boyer-Lindquist coordi- nates.

Figure 8.16: Contour plots of density and velocity for a supersonic flow with density perturbed byǫρ= 0.2 past a black hole with spin a= 0.9and flow parameters given by model UA1. The plots are evaluated at timet= 400M, and the axes are in units ofM.

Close to the horizon inside the shock-cone, the contours are once again circular arcs, but upstream the contours distort to surround the stagnation point. The flow for model UB1 has its stagnation point closer to the black hole than that for model UA1 but upstream, outside the shock-cone, the flow velocity is lower for model UB1. Since both models have the same Mach number, M = 1.5, we see that both shock-cones have approximately the same opening-angles.

In Figures 8.16 and 8.18 we show the effect on these models of including a spin of a = 0.9 and a density perturbation ofǫρ = 0.2. The flow is still in the equatorial plane of the black hole, however. We see that the shock-cone is pulled round the hole in the positive (anti-clockwise) direction. This effect is so significant near to the black hole that the shock-cone attaches to the upstream side of the horizon and far from the horizon the cone is still at an angle to the horizontal. We also see that there is a line of significantly lower density dividing the shock-cone into two sections. This is a more extreme case of the effect we noticed in the subsonic regime. Examining iso-surfaces of the density reveals that there are two regions of lower density either side of the z = 0 plane that extend down towards the horizon.

Unsurprisingly, model UA1 still has a higher density near to the horizon than model UB1.

The velocity contour plots show similar twisting and distortion to the subsonic case. The velocity drops significantly across the shock-cone, and outside the cone, the contours retain their circular appearance.

There are then two regions of low velocity. It is not clear from the numerical solution whether either of

Figure 8.17: Plot of streamlines for a perturbed flow withǫρ= 0.2, model UA1, parallel to the equatorial plane, onto a Kerr black hole with spina= 0.9.

these contain stagnation points.

It is more instructive, however, to examine the streamlines (shown in Figure 8.19). For the tracers that we choose to examine for model UB1, we see that all of those approaching the black hole where it is rotating against the flow direction fall into it, whereas none of those approaching it on the co-rotating side do so. The flow is compressed in from the sides, and the tracers that escape the domain downstream of the black hole form a tear-drop shape.

For model UA1, where the streamlines are shown in Figure 8.17 we note that the tracers were started on a larger radius circle, but we still see a qualitatively fairly similar flow. However, at the top of the

(a)Equally spaced contours oflogρ. (b)Velocity contours evaluated in Boyer-Lindquist coordi- nates.

Figure 8.18: Contour plots of density and velocity for a supersonic flow with density perturbed byǫρ= 0.2 past a black hole with spin a= 0.9 and flow parameters given by model UB1. The plots are evaluated at timet= 300M, and the axes are in units ofM.

tear-drop, we see that two vortices have developed in the flow. These correspond to the two low density regions mentioned above.

The cross-sections of the accretion regions upstream of the black hole are shown in Figure 8.15. Again the accretion regions corresponding to the unperturbed, non-spinning cases are circular. We also see that the combined effect of the spin and perturbation is to move the accretion region towards the top of the domain, and to cause it to decrease in size. We also see that the accretion regions for the higher adiabatic index (model UB1) are somewhat smaller than those for model UA1.

Again, the gross flow features are borne out by an examination of Ruffert’s simulations in a density perturbed flow [95]. We also note that the angle by which the shock-cone is tilted is approximately the same in both models.

In Figure 8.20 we show the mass accretion rates for the four supersonic models we have presented.

The rates for model UA1 have not quite reached a constant value at timet= 400M, but those for model UB1 have already attained their limit at t= 250M. Again we see that the effect of spinning the black hole and introducing a density perturbation is to lower the mass accretion rate, as suggested by the decrease in area of the accretion regions.

In Figure 8.21 we show the evolution of the stagnation point for the two non-spinning, non-perturbed supersonic models we have presented. We see that they both reach a constant state by the end of the

Figure 8.19: Plot of streamlines for a perturbed flow withǫρ= 0.2, model UB1, parallel to the equatorial plane, onto a Kerr black hole with spina= 0.9.

evolution.

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t/M M˙

(a) Model UA1

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t/M M˙

(b)Model UB1

Figure 8.20: Variation of mass accretion rate with time for two supersonic models. In both plots the + signs correspond to the uniform, non-spinning case, and the circles correspond to the perturbed,ǫρ= 0.2, spinning,a= 0.9 case.

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t/M x

/M

UB1 UA1

Figure 8.21: Evolution of stagnation point locations with time for the supersonic models UA1 (+signs) and UB1 (circles). A loss of data, for technical reasons, caused model UB1 to stop earlier, rather than any numerical effects.