wiki:u/johannjc/injection

Version 6 (modified by Jonathan, 7 years ago) ( diff )

When calculating injection from outflows, we generally have some prescription for the mass, momentum, and possibly energy injection as a function of position within the outflow kernel. However, we would like the numerical injection to satisfy certain requirements.

  1. The total mass injected should equal
  2. The center of mass of the injected mass should equal
  3. The total momentum injected should be (bipolar)
  4. The scalar momentum injected should equal
  5. The angular momentum injected should equal

First, let's consider a simpler example. Let's assume we want to inject a certain amount of mass around a point (in 2D) with a kernel given by

where our normalized kernel

for .

Here is the distance from a sink particle.

For a grid with 10 points in x and y over the interval x=[-1,1] y=[-1,1] - and for a sink particle at [.05,.05] the result is

Note the center of mass of the injected material is not .05,.05 and the injected mass is not 1.0. This is because of discretization error.

We would like to find a solution for that is close to this, but subject to the constraints involving the center of mass and the total mass injected.

If we consider each zone inside the kernel, we can write the constraints as the following matrix equation

This can be expressed as and we can solve this underconstrained problem using the pseudo-inverse of A

While this solves our constraints - it looks nothing like our desired profile. What we would like to do is to solve , in a way that does not minimize , but that minimizes where is our vector of initial given by our kernel.

We can do this using Matlab's lsqlin function - or we can use the prescription given in https://see.stanford.edu/materials/lsoeldsee263/08-min-norm.pdf (page 8-13 through 8-15)

This gives

Now for the more complicated case, we have terms in involving momentum, and we could simultaneously solve for and , though it is probably more important to have be close to the desired values. Errors in and could lead to unphysical velocities. We cannot solve for and at the same time, since there are non-linear terms that involve the product of and .

However, if we separate out the solution for using the constraints on total mass and mass moment, we can treat those as constants when solving for . This then gives us the matrix equation

where

Taking into account particle motion and orientation

If we are explicit about our constraints, we have

Change in mass of particle

Change in mass moment of particle

Change in momentum of particle (due to loss of mass)

Change in internal angular momentum of particle

Amount of outward momentum in jet

which can be reduced to

An alternative approach is to try to match the amount of outward momentum in the jet for each direction independently.

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