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Astron. Astrophys. 337, 311-320 (1998)

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3. Numerical results

3.1. Calculated cases

The cases which we have calculated are summarized in Table 2. Our main interest is in the thin accretion column of supersonic flows, thus Mach number [FORMULA]=4 was chosen as the standard case. The results obtained with the LMC scheme exhibited more mesh dependency than those of the AMC scheme, therefore we show mainly the AMC results


[TABLE]

Table 2. Parameters and nondimensional values of each case. [FORMULA] is the root mean square of fluctuations from the averaged value of [FORMULA]. [FORMULA] and [FORMULA] are the average of mass and momentum accretion rate respectively, and [FORMULA] is the root mean square of [FORMULA].


3.2. Mesh dependency of the LMC and AMC schemes

Table 2 summarizes how for both the LMC and AMC scheme the mass accretion rate, [FORMULA], and the RMS value of the angular momentum accretion rate, [FORMULA] depend on the details of the models. The mass accretion rate changes from 0.420 to 0.817 when the size of the central hole increases from 0.005 [FORMULA] to 0.02 [FORMULA] in the LMC calculations (case LM040__005, LM040__01, LM040__02). On the other hand, the mass accretion rates for the AMC cases are in the range 0.838 to 0.951.

For the LMC scheme, as the central hole becomes smaller, mass and angular momentum accretion also become smaller with the standard grid. When the central hole is large, a temporary accretion disk is formed and the averaged mass accretion rate is close to the Hoyle & Lyttleton value. But for a small central hole an almost permanent accretion disk is formed near the hole which blocks further accretion. However in the fine grid case this decrease of the mass accretion is not found and the existence of the accretion disk is temporary. On the other hand for the AMC scheme, accretion disks are always transient and accretion rates are similar.

This fact indicates that the conservation of angular momentum is very important for the accretion problem. Since the results based on the AMC scheme have less mesh dependency, they seem to be more reliable. Thus only the results from the AMC scheme will be shown in the rest of this section.

3.3. Occurrence of oscillations and formation of accretion disks

The accretion column of the supersonic accretion flow is narrow and large oscillations are found. In Fig. 1, the typical sequence of the swinging of the accretion column of case AM040005 is shown. The oscillating accretion column swings over 180 degrees. This oscillation is similar to that shown in the computation of Boffin and Anzer (1994), but there is also an accretion disk in our computation. The radius of the accretion disk is about 0.1 [FORMULA].

[FIGURE] Fig. 1. Density contour for the AM040__005 case, at times T=22.1, 22.3, 22.5, 22.7 and 22.9. The scale is logarithmic from 0 to 3, in steps of 0.3 .

Enlarged views are shown in Fig. 2. The formation and destruction of the accretion disk is clearly seen in these figures. An anti-clockwise rotating accretion disk is formed at T=22. The accreting matter falls from the upper-left direction through the accretion column, thus this matter has anti-clockwise angular momentum and it accelerates the disk. Then the column is pushed backward by accreting matter from upstream. When the column moves behind the object, the accreting matter has clockwise angular momentum (T=22.2). This matter collides with the disk and destroys the disk (T=22.4). Then a clockwise rotating disk is formed (T=22.6 and following). As shown in Fig. 3, mass and angular momentum is accreted when the disk collapses.

[FIGURE] Fig. 2. Density contour for the AM040__005 case, at times T=22., 22.2, 22.4, 22.6 and 22.8. The scale is logarithmic from 0 to 3, in steps of 0.6, and we have zoomed close to the star. The arrows visualize the velocity field

[FIGURE] Fig. 3. Time history of the mass accretion rate (left) and angular momentum accretion rate (right) for the AM040__005 case at times corresponding to Fig. 2, AMC scheme

The averaged mass accretion rate of this case is 0.855, but the instantaneous rate is over 10 times larger than the Hoyle & Lyttleton estimate. The averaged momentum accretion is almost zero. However, a fair amount of instantaneous angular momentum accretion with altering direction is also found. This can be understood as follows: the symmetric accretion column cannot have angular momentum, but the portions of an asymmetric accretion column can have angular momentum. When the positive angular momentum portion falls downward, a positive accretion disk is formed. This accretion disk loses its angular momentum and accretes onto the object, when it collides with the following portion which has negative angular momentum.

3.4. History of mass accretion rate and angular momentum accretion

The histories of mass accretion and angular momentum accretion of representative cases from the AMC scheme are shown in Fig. 4 and 5. Those from the LMC scheme are shown in Fig. 6.

[FIGURE] Fig. 4. Time history of the mass accretion rate. [FORMULA]=4.0; [FORMULA]=0.02, .01, .005 and .01 fine grid, AMC scheme

[FIGURE] Fig. 5. Time history of the angular momentum accretion rate. [FORMULA]=4.0; [FORMULA]=0.02, .01, .005 and .01 fine grid, AMC scheme

[FIGURE] Fig. 6. Time history of the mass accretion rate (upper frames) and angular momentum accretion rate (lower frames). [FORMULA]=4.0 [FORMULA]. The results from the standard grid (left) and from the fine grid (right) are shown, LMC scheme

3.4.1. Effects of [FORMULA] and mesh size

The histories of accretion rates for [FORMULA]=4.0 from the AMC scheme with different [FORMULA] and mesh size are shown in Figs. 4 and 5. As seen in those figures, the fluctuations of the accretion rate become bigger with smaller [FORMULA].

We can see from Table 2 and Fig. 4 that the mean values of the mass accretion rate are quite similar for different values of [FORMULA], whereas the peak amplitudes of [FORMULA] increase with decreasing [FORMULA]. This then implies that the accretion bursts are correspondingly shorter for smaller values of [FORMULA]. Although the results for the fine grid (Fig. 4 & 5) have fluctuations of higher frequency and are more chaotic, they are generally similar to those for the standard grid. The similarity is also clearly seen in Table 2. All values except [FORMULA] are similar for different [FORMULA]. The sequences of the oscillations are also similar. The difference in [FORMULA] can be explained by the mechanism of the accretion disk destruction. The orbits of the anti-clockwise rotating accretion disk in Fig. 2 are almost circular, but become elliptical when the accreting matter which has clockwise angular momentum interacts with matter of the anti-clockwise rotating disk (T=22.2 in Fig. 2). The matter is taken away from the computational space when its elliptic orbit touches the central hole. Thus, the bigger central hole absorbs the matter over a longer period than the smaller hole. Therefore the fluctuation of mass accretion becomes larger for a smaller hole.

On the other hand, the LMC scheme shows a different [FORMULA] dependence. For the standard grid, the mass accretion rate becomes smaller as [FORMULA] decreases, as can be seen from the average values of [FORMULA] which are given in Table 2. Because an almost permanent accretion disk is formed it blocks further accretion. But this permanent accretion disk is not found for the fine grid. In this case the accretion rates show large fluctuations. The conservation of angular momentum is only achieved to second order accuraey in the LMC scheme. Thus, we believe that the standard grid is not fine enough for the LMC scheme to capture the accretion disk formation.

3.4.2. Effects of the Mach number

If the Mach number of the uniform flow is 1, the flow is perfectly steady, but for larger Mach number it is always non-steady. Note that the case of [FORMULA]=1.4 is also non-steady and shows small fluctuations from the beginning. However, non-steadiness with large amplitudes occurs only after 40 time units. The Mach number dependence of the averaged mass accretion rate, the fluctuation of the mass accretion rate and the fluctuation of the angular momentum accretion rate are summarized in Fig. 7. These are the results from the AMC scheme with the standard grid.

[FIGURE] Fig. 7. Mach number dependency of characteristic values for the AMC case with standard grid

As shown in Fig. 7 the amplitude of the oscillations is largest when the Mach number is 2, and the fluctuations become smaller as the Mach number becomes greater than 4. This is also shown in the history of the mass and angular momentum accretion of the Mach 8 case (case AM080 01) in Fig. 8.

[FIGURE] Fig. 8. Time history of the mass accretion rate (left) and of the angular momentum accretion rate (right). [FORMULA]=8.0 [FORMULA]

As the Mach number increases, the amplitudes of the accretion column oscillation become smaller. This behavior is clearly shown in the time averaged density contours of Fig. 9. In the time averaged solution, the wiggles of the accretion column are smeared out and the high density regions around the accreting object are seen. The high density regions are concentrated in a narrow cone in the high Mach number cases (see the [FORMULA]=8 case in Fig. 9). It is interesting that the transient accretion disks are also seen as high density regions in the averaged solutions and that averaged accretion columns look like bow shocks.

[FIGURE] Fig. 9. Time averaged density contour. Rmin=0.01, [FORMULA]=2,4 and 8. The contours are logarithmically spaced between 0 and 3, in steps of 0.3

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© European Southern Observatory (ESO) 1998

Online publication: August 6, 1998
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