Re: How to calculate the volume of a streamline bundle?

Thanks.

I can simplify the question like this. There are 50 curves in 3D space.
Each curve has a limited length (not necessarily equal). Since these
curves are quite close to each other, it looks like a tube or a fiber
bundle. I want to know the volume of this bundle.

Because the surface or the boundary of a bundle is unknown, I use
convex hull to find the extreme points (the edge points), then use the
edge points as bundle boundaries to compute the volume. However, the
convex hull algorithm did not work well in curved tubes. For example,
it could compute a straight tube, but not a U-shape tube.

Ben



Lee Rudolph wrote:
"Ben" <laserbin@xxxxxxxxx> writes:

Hi,

I asked a question several days ago and got the answer that I thought
it should be enough to solve my problem. However, the problem is more
complicated than I expected. I decided to start a new topic instead.

My question is how to calculate the column from streamlines. I have
about 50 streamlines integrated by Runge-Kutta method. Each streamline
is a series of points in (x,y,z). These streamlines are close to each
other (look like a bundle of fibers) by visual inspection under Matlab.
I want to calculate the volume the streamlines cover in space.
...

What is it that is "streaming"? If it is (or can be considered to be)
an incompressible fluid, then (unless I'm very confused and talking
through my hat, which is perfectly possible) then the 3-dimensional
volume swept out over your elapsed time interval by *all* the infinitely
many streamlines that start in your initial 2-dimensional configuration
and end in the terminal 2-dimensional configuration dictated by your
problem is *exactly* the integral, over that initial surface, of the
function that associates to each initial value the length of its streamline.
For large values of 50, and making other assumptions that may or may not
be justified, the volume is then well approximated by the sum of the
lengths of your 50 flowlines, multiplied by some factor like the
surface area of the initial surface divided by 50. ... If you are
faced with compressibility, then instead of integrating the lengths
of all the streamlines (or, to an approximation, adding up the lengths
of your 50 samples), you would need to integrate over the surface a
function which is itself an integral over each flowline of something
or other that quantifies the compressibility. I can be vaguer if
need be.

Lee Rudolph

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