Subsonic models

The subsonic models that we have tested are UC0 and UB0 (see Table 8.1), with a Mach number of M= 0.6. These models differ primarily in their adiabatic indices, although the fluid velocity has to be changed to allow for the maximum velocity permitted bycs<√

Γ−1. We run the simulations to time t= 400M, and the results are shown in Figures 8.3 and 8.4.

For both models we see that the density contours near the hole are nearly circular, but offset slightly in the downstream direction relative to the centre of the hole so that the density at the horizon is higher on the downstream side. Model UC0, with the higher adiabatic index, gives a lower fluid density close to the horizon than does model UB0, since the fluid is less compressible for higher adiabatic index.

The fluid velocity contour plots are smooth, showing the features we would expect, namely, that the fluid falls in with increasing velocity, and that there is a stagnation point downstream. The velocity at a given distance upstream of the singularity is higher for model UC0, and the velocity at a given distance

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

Figure 8.4: Contour plots of density and velocity for a uniform subsonic flow past a non-spinning black hole with flow parameters given by model UC0. The plots are evaluated at timet= 400M, and the axes are in units of M.

downstream of the hole is higher for model UB0, the stagnation point for which lies further downstream than for model UC0.

In Figures 8.5 and 8.7 we show the effect on these models of including spin a = 0.9 and density perturbationǫρ= 0.2. The flow is still in the equatorial plane of the black hole, however. In both cases the flow is wrapped around the horizon significantly, in the direction of the spin. The density contours now have slight indentations, showing that there is a line of slightly lower density fluid. This is more evident for model UB0. Examination of cross-sections for other values ofz, such asz= 1 (see Figure 8.6), shows that the indentation is more prominent to either side of thez= 0 plane.

The velocity contours show the development oftworegions of lower velocity flow. From the form of the ellipse-shaped contours extending into the lower-left of the plots, it seems that the main flow morphology has been twisted around the horizon by about 45^◦, although there are now no stagnation points, due to the added effect of the fluid rotating around the hole before falling in. Examining iso-surfaces of the velocity shows that neither are there are any stagnation points away from thez= 0 plane.

These flow features arise because the spin of the black hole tends to pull fluid in the lower half of the plot around in the anti-clockwise direction, while fluid in the upper half of the plot is decelerated in the same (angular) direction. The fluid from the lower half therefore moves further anti-clockwise around the black hole than it would otherwise have done before it is decelerated sufficiently to fall into the black

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

Figure 8.5: Contour plots of density and velocity for a subsonic flow with density perturbed by ǫρ = 0.2 past a black hole with spin a= 0.9 and flow parameters given by model UB0. The plots are evaluated at timet= 400M, and the axes are in units ofM.

Figure 8.6: Density as in Figure 8.5, now evaluated onz= 1 plane.

hole. This accounts for the change in the locations of the regions of low velocity.

The density perturbation gives a higher density towards the bottom of the plot (on the left of the figure, upstream of the black hole). The effect of the perturbation is similar to that caused by the spin in

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

Figure 8.7: Contour plots of density and velocity for a subsonic flow with density perturbed by ǫρ = 0.2 past a black hole with spin a= 0.9 and flow parameters given by model UC0. The plots are evaluated at timet= 400M, and the axes are in units ofM.

−15 −10 −5 0 5 10 15

−15

−10

−5 0 5 10 15

(a)Model UB0

−15 −10 −5 0 5 10 15

−15

−10

−5 0 5 10 15

(b)Model UC0

Figure 8.8: Accretion regions for all four subsonic models. The solid line shows the accretion region for a= 0,ǫρ= 0, and the dashed line shows the accretion region fora= 0.9,ǫρ= 0.2.

that, due to the lower momentum of the fluid at the top of the plot (being less dense), the fluid coming from the lower part of the plot (being more dense) can travel further around the black hole before being

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4 4.5 5 5.5 6 6.5

t/M M˙

(a)Model UB0

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15 20 25 30 35 40 45

t/M M˙

(b)Model UC0

Figure 8.9: Variation of mass accretion rate with time for two subsonic 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. The accretion rates are evaluated on a sphere of radius 5M.

decelerated sufficiently to fall into it.

The cross-sections of the accretion regions upstream of the black hole are shown in Figure 8.8. As expected by symmetry, the accretion regions for the unperturbed, non-spinning cases are circular. The plots also show that the combined effect of the spin and density perturbation is to deflect fluid from the upper half of the domain away from the black hole so that it does not accrete onto it. We also see that the accretion regions for the higher adiabatic index (model UC0) are somewhat smaller than those for model UB0.

We note that the tilted form of the flow and the development of a line of lower density are features that also appear in Ruffert’s simulations involving a density perturbed fluid onto a compact object [95].

This suggests that the major changes to the flow features are the result of the density perturbation, rather than the spin of the black hole, as such an effect was not included in Ruffert’s simulations, not being relevant in the Newtonian regime.

In Figure 8.9 we show the mass accretion rates for the four subsonic models we have presented. We see that although all the rates have reached a nearly constant value, not showing any short-period oscillations, the rates for model UB0 are slowly decaying. However, the fact that the rates are very nearly constant shows that we have attained steady-state flow, thus validating our numerical methods.

In both cases we see that the joint effect of the spinning black hole and the density perturbation is to lower the rate at which mass is accreted onto the black hole. This is as suggested by the reduction in the size of the accretion regions. The reasons for this will be discussed in the full parametric study in§8.4.

Note that the mass accretion rates for the two models cannot be compared directly, as they have already been scaled by the factor given in equation (8.1).

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4.5 5 5.5 6 6.5 7 7.5 8

t/M x

/M

UB0

UC0

Figure 8.10: Evolution of stagnation point locations with time for the subsonic models UB0 (+signs) and UC0 (circles).

In Figure 8.10 we show the evolution of the stagnation point for the two non-spinning, non-perturbed models we have presented. We see that they both reach a constant state by the end of the evolution.

These plots are derived from interpolating the total velocity along thex-axis (the direction of the wind in this case) at equally spaced points, and finding the location of the minimum. This accounts for the discrete nature of the points at which the stagnation point is located.

The streamlines for the perturbed model UB0 with spinning black hole are shown in Figure 8.11.

Overall, these show the flow being diverted from its original direction by the presence of the black hole.

However, at the base of the main outgoing tracer pattern, we see a concave feature, which suggests that the fluid is forming two vortices. We shall see this feature developed more strongly in supersonic flows.

The streamlines for the perturbed and spinning UC0 model are shown in Figure 8.12. These essentially show only that the flow has been diverted from its original direction by the black hole. However, there is a small feature at the bottom of the main circle of tracers. This may well be an undeveloped version of the vortex structure previously referred to.

Figure 8.11: Plot of streamlines for a perturbed flow withǫρ= 0.2, model UB0, parallel to the equatorial plane, onto a Kerr black hole with spina= 0.9.

Figure 8.12: Plot of streamlines for a perturbed flow withǫρ= 0.2, model UC0, parallel to the equatorial plane, onto a Kerr black hole with spina= 0.9.

(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.