## 2. Numerical procedure and initial conditionsSince the numerical procedures and initial conditions are mostly identical to what has already been described and used in previous papers (cf. Ruffert 1997, R1, and references therein) I will refrain from repeating every detail, but only give a brief summary. ## 2.1. Numerical procedureThe distribution of matter is discretised on multiply nested equidistant Cartesian grids (e.g. Berger & Colella, 1989) with zone size and is evolved using the "Piecewise Parabolic Method" (PPM) of Colella & Woodward (1984). The equation of state is that of a perfect gas with a specific heat ratio (see Table 1). The model of the maximally accreting, vacuum sphere in a softened gravitational potential is summarised in Ruffert & Arnett (1994) and Ruffert & Anzer (1995).
A gravitating, totally absorbing "sphere" moves relative to a medium that has a distribution of density and velocity far upstream (at ) given by with the redefined accretion radius (note slight difference to Eq. 1) In this paper I only investigate models with gradients of the
The function "" is introduced in
Eq. (3) and (4) to serve as a cutoff at large distances I'll assume both and for the following estimates. Accreting all mass within the accretion cylinder, and taking the matter to have the density and velocity distributions as given by Eqs. (3) and (4) one obtains the mass accretion rate to lowest order in and to be an equation very similar to Eq. (2). Further assuming that all angular momentum within the deformed accretion cylinder is accreted too, the specific angular momentum of the accreted matter follows to be (Ruffert & Anzer 1995; Shapiro & Lightman 1976; again to lowest order in ) For positive the density is lower
on the positive side of the The values of the specific angular momentum obtained from the
numerical simulations will be compared to the values that follow from
this Eq. (7) to conclude which of the above mentioned views - low or
high specific angular momentum of the accreted material - is most more
appropriate. As was stated further above, R1 found that a sizable
amount (between 7% and 70%) is accreted, when velocity gradients are
present. I will implicitly assume the component
when discussing properties like
fluctuations, magnitudes, etc. From the symmetry of the boundary
conditions the average of the Apart from serving as cut-off, the tanh-dependencies in Eqs. (3) and (4) have a gradient that is slightly less steep than simply linear. Thus less specific angular momentum is present at a given radius from the accretor and smaller values in the magnitude of the accreted quantity result. One can numerically approximate the integrals of the mass flux and
angular momentum over the cross section of the accretion cylinder, to
obtain the coefficients Here, I will only consider the effect of a density gradient. The
unitless functions
## 2.2. ModelsThe combination of parameters that I varied, together with some results are summarised in Table 1. The first letter in the model designation indicates the strength of the gradient: M, P, and T have , while N, Q, and U have and V is . The second letter specifies the relative wind flow speeds, F (fast), S (slow) and V stand for Mach numbers of 10, 3 and 1.4, respectively. I basically simulated models with all possible combinations of the two higher flow speeds (Mach numbers of 3 and 10), the two gradients (3% and 20%) and varying the adiabatic index between 5/3, 4/3 and 1.01. Model VS with the largest gradient of 100% facilitates a comparison to a previous two-dimensional simulation by Fryxell & Taam (1988). The grids are nested to a depth such that the radius of the accretor spans several zones on the finest grid. As far as computer resources permitted, I aimed at evolving the
models for at least as long as it takes the flow to move from the
boundary to the position of the accretor which is at the centre
(crossing time scale). This time is given by
and ranges from about 1 to about 10
time units. The actual time that the
model is run can be found in Table 1, as well as the parameters
of the grids ( When modeling a BHL flow with a density gradient, one has to pay attention to the fact that matter parcels with possibly very different densities (which initially are separated upstream) will be focussed to find themselves close to each other along the accretion axis (U. Anzer, personal communication). This might render the flow additionally unstable. Although true in principle, this problem does not affect, in practice, the models I will present: a closer inspection of Eq. (3) reveals that the density jump from one end of the accretion radius to the other is only a factor 1.5 for the case with large gradient (). This does not seem to have an additional influence as compared to models with constant density presented in paper R1. The calculations are performed on a Cray-YMP 4/64 and a Cray J90 8/512. They need about 12-16 MWords of main memory and take approximately 160 CPU-hours per simulated time unit. © European Southern Observatory (ESO) 1999 Online publication: June 17, 1999 |