Dimensionless Space and Time

[WebPilot]

Space and Time

These are the two variables with which you have to deal when solving
1D compressible flow problems.

I found this problem in a text book:

Isentropic Back Flow. A duct of constant cross section is
filled with air ( gamma = 1.4) that is isentropically compressed from
atmospheric pressure, and the right end of the duct is suddenly
opened to the atmosphere. What fraction of the initial mass
in the duct is discharged, and how does the pressure at the
closed end vary with time? (Note: assume p/p0 is 2.83)

This reads like "Kadenacy" to me.

The fact that a sudden discharge of air or gas from a previously
pressurized container left a depression within the container has been known
since before the turn of the century and was often referred to as the
'Kadenacy effect'.

The author solved it using pencil, paper, triangles and a calculator (a graphical
method). I am altering his method somewhat by using a little bit of pencil, paper
and calculator and a lot of a spreadsheet and a CAD program. You learn by doing.

Ideally, consider a thin membrane covering the open end of a half open tube. This
membrane is ruptured at time, t=0.

A centered expansion wave (a.k.a. fan) exists at the now open (right) end.
It begins to propagate toward the left approaching the closed end of the
tube. The head is shown in the diagram as being reflected as an
expansion (or rarefaction) wave when it reaches the closed end.
The rest will follow suit in time, but they must first traverse a region
occupied with other waves.



Of course, this is just the beginning to the solution.

-fde

[WebPilot]

I'll give a little decryption later today.

-fde

[WebPilot]

I changed the point label color. Compression waves are now medium blue.

Disturbances go further than I have drawn, but their slopes change dramatically
sometimes, so I have gotten in the habit of "trimming" them.

Note the right travelling expansion wave from 1-6 has been reflected
at the open end as a compression wave, 6-11. The velocity of
the outflow was checked and it is indeed, subsonic.

Remember, the y-axis is tau (pronounced as the "t" in let) and
represents dimensionless time. It's still time, whether it has
dimensions of "secs" or not. The x-axis, xi (pronounced as the "x" in
fox), represents the station location along the tube, expressed as a
fraction of the overall tube length.

The dotted line, beginning at (0.15,0) is a particle path
line for that location in space. It remains at rest (vertical
line) until the left moving expansion wave through 1 arrives at
its location and pulls it to the right. It continues along this
course until the time comes when it is pulled by the expansion wave
through 2. As it crosses the characteristic curve through 2-3, a
right moving expansion wave pulls it to the left. As it
crosses the characteristic curve through 3-8, it is again pulled to the
right by a left moving expansion wave.

Presently, it is unknown whether this particle will be expelled out the right
end of the tube, or not. This process of tugging and as
we shall later see, pushing, will be continued as the construction of
the diagram proceeds.

-fde

[Mike Everman]

Yeah, nice.

[WebPilot]

Thanks.



A discussion follows, shortly.

-fde

[WebPilot]

I finished the reflection of expansion wave from the open end to point
7, at point 12. Note that the reflection at point 7 is a right moving
expansion wave, but the reflection at point 12, the open end, is a
compression wave that travels to the left (compression waves
are denoted in a medium blue).

Also note, that the compression waves through 6-11-16 and 12-7 appear
to be converging.

You haven't seen the table I had to produce to do this drawing, but
the velocity (actually speed) of the outflow at point 18 has become
0. If the next wave that gets reflected at the open end has inflow at
the open end, then point 18 can be imagined as the interface which
differentiates between what's left ot the original gas in the tube, and the
fresh gas now entering the tube.

Assuming this to be true for now (I'll prove it, later), if I backward
construct a pathline from point 18, then I can answer the first
question:

What fraction of the initial mass in the duct is discharged?

When I drew the pathline starting @ (0.15,0), the slopes of the
pathlines at every end point on a characteristic line were previously
computed. As the pathline crossed a characteristic, it is a simple
matter to interpolate for the required value from those at the
endpoint - at least working in the forward direction (I knew the
intersection coordinate values).

Working backwards, as at 18, I neither know the slope, nor the
intersection coordinates. Hmmm, what to do?

It turns out I discovered a graphical technique to ease the chore.
I love CAD programs!

Working backwards, I got a value of (0.288,0) or 28.8% of the mass
remains in tube after the first cycle, or 71.2% is discharged!

The author computed about 72% of the initial mass is
discharged. I am within 1.111% of his number.

I am content with this.

-fde

[Mike Everman]

I figured out recently that being content for me is excelling at things meaningless to 99% of humanity.

[pezman]

Forrest,

Nice presentation. I was working on getting some flow simulation codes working a while back, and this might be enough of an impetus to actually get me back to it.

Thanks again for publishing this.

[WebPilot]

Mike: to have a + effect on 1% of humanity is quite an achievement.
------------
pezman: thanks, good luck with your codes and you're welcome.
------------
Ben: yes, this is MOC.
------------

A partial MOC data table can be seen here, for this problem.


-fde

[WebPilot]

We are now in a position to provide a partial answer to the 2nd
question:

... how does the pressure at the closed end vary with time?

From the CAD diagram, go to Xi=0, look up the vertical axis and see what points
cross there. Write the numbers down. Now go to the data table, for each of
those points go to the ninth column and write that number down beside
it. This is y, for the plot, which is the pressure ratio.

You still need x for the plot, which will be dimensionless time. This
can be taken again from the CAD drawing. Since I have made the
drawing, I can let it compute exactly what the distances involved are
and then calculate its respective Tau value. You will have to approximate.

A plot was made with the spreadsheet. I am not happy with its capabilities,
but it will do for now.



-fde

PS Happy Thanksgiving to those of you who celebrate it.

Weather conditions here are: Snow 28°F(-2°C)

[Mike Everman]

Happy Thanksgiving, Forrest, best to you and yours. You may know this, but:
If you right click on the graph and change the graph type to XY scatter, subtype "connected by lines", then right click the graphed line and select "add trend line", then you can specify an nth order polynomial fit curve, as well as have it display a descriptive equation of the fit curve. Very handy, though it looks as if your points fall on a sinusoid, as one would expect, no?

[WebPilot]

Mike,

Thanks for the tip.

The top of the graph must be flat. Why? Because it takes a finite amount
of time for a disturbance (the head of the expansion fan) to reach station
0. Thus, the pressure remains at the initial pressure even though the
frangible diaphragm has been ruptured.

I wouldn't mind a smooth curve through the rest of the points, but it
doesn't seem to allow me to choose which points to do this.

So, it was a choice of just dots, or line segments.

Thanks to your input, I now prefer line segments - perhaps, it is less confusing. (?)

-fde

[WebPilot]

The left travelling compression waves, 11-16 and 12-17 have converged
to form a weak shock wave at points 21, 22. Its technical name
is a "left-facing, left-travelling (or Q), weak shock wave". I have
denoted it in the drawing using this color.

There are several waves that are going to cross it, and it will be reflected
twice in the analysis.

-fde

[pezman]

I found a pretty accessible tutorial on computational fluid dymanics here:

http://www.astro.uu.se/~bf/course/numhd ... tents.html

[WebPilot]

The weak shock is a discontinuity - parameters change abruptly as the
characteristics cross it (e.g. pressures jump). Gas property values both "in
front of" and "in back of" the shock have to be addressed - thus, two points
are labelled at each crossing.

The slopes of the characteristics and that of the shock are altered by mutual
interaction.


Paperback Writer