Bondi Flow/Accretion Subgrid Model Questions

The equations for Bondi flow admit 4 classes of solution only two of which are physical

  1. A solution with u(r) = 0 at r = infinity. (AB'C' in Bondi 1954 Fig 2)
  1. A solution with u(r) = 0 at r = 0. (A'BC in Bondi 1954 Fig 2)

Note both are of type ii in Bondi's formulation.

From first principles we expect the global solution (not just what happens in the kernel to be solution 1. We have a spherically collapsing cloud which has .

That solution however must be matched as cleanly as possible with what happenes within the kernel (whose radius is ) with the kernel values being given as

So the question becomes are we expecting that within the kernel we may switch from solution type 1 to solution type 2?

Physically this can only occur if a shock has formed at the "surface" of the accreator which has then expanded such that .

So given the values of velocity and location, we can calcualte x0 and y0, which give us lambda and lambda_c. This then divides the x0-y0 space into super/subsonic and super/subcritical regions.

  • In the yellow region, you can integrate inward (or outward) without ever becoming subsonic.
  • In the cyan region you can integrate inward (or outward) without ever becoming supersonic.
  • In the left supercritical region,s you should be ok - as you should be able to integrate inward - though not outward
  • The right supercritical region presents a problem - as the integral inward will diverge. It is these regions which have no steady state solution and require something like the 'shock assumption'

And for reference, here's the same figure ignoring Gamma

Not the line dividing sub/super critical follows x02*y0*z0=lambda=lambda_c - as z0 = 1 by definition.

And here's the same plot but for a range of .

Attachments (5)

Download all attachments as: .zip

Comments

1. Adam Frank -- 7 years ago

I was thinking about the Bondi problem earlier this week - or in general about the solutions to steady state, adiabatic, flows with 1/r potentials.

Since it is adiabatic, we can throw out the energy evolution equation. The density and momentum equations, can then be expressed as either differential equations (integrated from the boundary inward) - or recast as full differentials leading to analytic equations for density and velocity

To arrive at Equation 10 in the paper, the density equation is written as a full differential leading to a conserved quantity

rho v r2 = \Lambda

the solution of v is then substituted into the momentum differential equation followed by some simplifications to arrive at a single differential equation for \rho with auxilliary equations for c (from the EOS and the boundary terms) and v (from the continuity equation)

Bondi, takes the integral form of both equations - and ends up with two equations for two unknowns (rho, v) as a function of position and the boundary terms (rho_inf, c_inf, and v_inf). v_inf is effectively built into lambda - though the ram pressure (½ v2) is assumed to be negligible at infinity - along with the potential.

In our case, we have rho, c, and v at the boundary, but we can't ignore the potential or the ram pressure terms. But we can modify Bondi's approach to include this extra term. Much of the analysis is unchanged - though the critical value for lambda depends on the boundary ram pressure and potential. I've attempted this in the Paper in the section entitled Bondi V3

I also have a matlab notebook where I'm investigating the behavior of the different types of solutions (super/sub sonic and super/sub critical) and will keep you posted.