Dimensionless Space and Time

[WebPilot]

Hey Sam,

Care to reveal a little more about your code? What numerical scheme are you using? What
boundary conditions are you including? (open and closed ends by the sounds of it). I assume
its 1D - Does it allow for non-uniform cross-sections?

Sure.

Hmmm. I asked the same question of you, here, but received no answer. No matter, there
are many ways to solve these partial DiffEQ's. I am sure you are well aware of this
fact.

Yes, open and closed BC's exist.

ODUsF stands for One Dimensional Unsteady Flow.

'Variable cross section' is allowed.

I've got a fair bit of experience in CFD codes and I think at times it can be one of the most
frustrating things I've ever undertaken. Feeling the pain yet?

I can appreciate that. I've written other 'sticky' codes for structural analyses (static and
vibration) and non-linear vibratory and there is no better 'rush' than when they 'work' as they
should. They always do what I tell them to do, they just don't always do what I 'want' them
to do ! ha ha

I don't feel pain anymore. I just develop a higher level of patience, perseverance
and diligence - which, I believe, a lot of the people up here, have in common.
When I get 'burnt out' working on it, I just 'walk away' from it and come
back at a later time.

The code that was most 'unforgiving' was the system dynamical one I
wrote for a pulse combustor. ODUsF wasn't (isn't) easy, either, but I
haven't developed 'dark circles' under my eyes from it (yet) .

Laters,

-fde

PS Why do my 'personal messages' to you go undelivered ?

[WebPilot]

This is/was one of my better posts. It's almost a year old.

During my absence over the past 6 mos. the links to the
images have been broken. So, I've restored for now the
missing images so those of you who missed it, and wish to
read it, can.

Remember, this is a tedious calculation, but imagine the work of the
pioneers in this field, who did it using drafting instruments for drawing,
slide rules for multiplication and pencil and paper for addition and
subtraction (I would have learned how to use an abacus, myself).

I hope you enjoy it and helps you understand a little better
about the gas dynamics in a tube. I had a lot of fun with this
thread.

[Graham C. Williams]

Dear FDE.
I have much enjoyed rereading this thread. As you say it's one of the best.

thanks
Graham.

[Mike Everman]

Yeah, good stuff.

[WebPilot]

Over a year ago, I wrote this thread.

I have something to add, namely a point 50 on the wave diagram
I developed previously. It occurs at approximately tau=4.4, p/p0=1.585 .
This pressure ratio is slightly higher than the 1.475 obtained for
point 40, occurring at an earlier time.



To refresh your memory, this graph is one of the pressure
at the closed end of the tube as a function of time, in this
case dimensionless. NOTE: P/P0=1, indicates atmospheric
pressure.

Now, tau or dimensionless time is (a0/L0)*t . a0 is the
speed of sound at some reference temperature, let's say
room temperature. L0 is the total length of the tube and t
is time. If you work out the units, the dimension of tau
is [1], thus the term is unit- or dimensionless.

Let's look more closely at this pressure pulse. It begins at 2.83,
overexpands to below atmospheric (.295) and recompresses to
1.585. If we assume this to be where the second maximum occurs,
then the elapsed tau is the period of a cycle, and thus we can
compute its corresponding frequency, f, in cycles per second.

4.4 = tau = (a0/L0)*t
period (secs) = t = 4.4 * L0 / a0

f = 1/period= a0/(4.4*L0)

The reader should realize:

- This is a one dimensional flow analysis of a gas pressure
pulse of finite amplitude (2.83), not an acoustic wave of infinitesimal
amplitude.

- Infinitesimal amplitude acoustic theory predicts pressure peaks
and troughs of equal but opposite (gauge) amplitude, not the larger +tive
pressure area with a corresponding smaller -tive one as depicted here.
This disparity in area determines how much "thrust" is produced per
cycle.

- Infinitesimal amplitude acoustic theory predicts a fundamental
frequency of f = a0/(4*L0). The 4 in the denominator is close to the
4.4 I've obtained here, but the similarity between the two theories
ends there.

[WebPilot]

Using NUDiS again, I increased the time of solution for my model
and changed the number of cells from 10 to 20.

Notice:

- the "jumps" are beginning to "steepen" from my 10 cell model to
those depicted in my wave diagram results (I attribute this to the
increase in the number of cells)

Look at the prominent "fishhook"!

- a somewhat "double peak" occurs during recompression. The
"mean secondary pressure maximum" occurs at approximately
tau = 5.1

In otherwords, the fundamental frequency is

f = 1/period= a0/(5.1*L) , Again, this is NOT 4!

As I posted before, I realize there is a bit of a timing discrepancy
between my wave diagram results and those predicted by NUDiS.

However, they are both making predictions in the same direction.

[WebPilot]

I have not drawn (yet) the wave diagram performed by F. Shultz-Grunow.
He published it in "Gasdynamische Untersuchungen am
Verpuffungstrahlrohr", in August 1944.

We know it as NACA TM-1131, (english translation) February 1947.

F. Shultz-Grunow developed a pulse jet model by incorporating some
combustion information provided by none other than Paul Schmidt.

I had to read "in between the lines" to figure out the time scale,
tau, since said scale is missing on the tau-xi plot.

From this I was able to plot the pressure ratio vs. tau just behind
the valves at startup:



NOTES:

1. The initial max pressure ratio is very high, as reported by
Paul Schmidt. The max pressure ratio of the next cycle is smaller
due to a motion process that now exists due to the 1st cycle.

2. I too, have plotted the characteristic "flat" spot due to the "stiffness"
of the valves not allowing them to open immediately even though a
-tive pressure differential exists across them.

3. The pressure spends more time +tive than -tive (or more time above
than below p/p0=1). This results in thrust.

4. The dimensionless period of the 1st cycle is tau = 2.66

frequency = 1/period (expressed in secs)
= ( a0 / (2.66 * L) ) <= 2.66 NOT 4.0!

Now, as a check let's compute this frequency since the well read
"pulsejeteer" knows this frequency is about 40 Hz for a V-1.

a0 = speed of sound @ 293 °K (or 20 °C or 68 °F)
= 343.1 [m/sec]

L = length of V-1 clone = 3.335 m

frequency = 343.1 [m/sec] / (2.66 * 3.335 m) = 38.7 Hz

I am amazed. Aren't you?

In 1944, F. Shultz-Grunow had to have been considered a GENIUS.

I consider him to be one still, today.

[WebPilot]

Additions and corrections to this graph are forthcoming.
...Stay tuned....

[WebPilot]

[WebPilot]

This is the gas velocity plot which is the companion
to the prior pressure plot.

Remember:

1. This is for a cylindrical pulse jet.
2. This is the initial or startup cycle.

Discussion:

1. For over a little more than a cycle, there is no flow
(0-a), outflow (a-d), no flow (d-e) and inflow (e-h). +tive
and -tive signs indicate outflow and inflow respectively.

2. There are two components of outflow. Initially,
it is sonic or above (b-c) and then subsonic (c-d).

3. Tau = 2.57 is the end of the first cycle and
combustion begins. Notice inflow is still continuing
from this time on into part of the next cycle!

This wraps it up for the 1st cycle. Realize we are
dealing with a transient response and not the steady
state solution. It should converge over time (meaning
more cycles need to be inspected)

[Irvine.J]

Very interesting...

[WebPilot]

I am pleased to read you found it so, James.

Thanks to Mr. F. Shultz-Grunow, pulse jet theory has
been on some firm mathematical ground since the
early 40's.

Franco Marcenaro took a very little bit of it when
writing his article on Pulsejet Theory. Mysteriously, he did not
reference F. Shultz-Grunow, "Gas-Dynamic Investigations
of the Pulse-Jet Tube", (translated) 1947.

[larry cottrill]

Again, beautiful work, Forrest. Even without following the details of the math, the principle of thrust development shines through. Nicely presented!

L Cottrill

[WebPilot]

Thank-you, Larry.

F. Shultz-Grunow writes:

This phenomenum of inward flow at the exhaust end
of the tube was experimentally observed by Paul Schmidt
and termed by him "intrusion of air".

[WebPilot]

Note: the above tau-velocity plot is my doing.
For some reason, Mr. F. Shultz-Grunow did not
include it in his treatise. It should have been. He
did include a pressure-tau plot.

Note:
Dimensionless parameters are merely an engineering
"shorthand" in order to include as much information
as possible in a minimal amount of space resulting in
a minimal amount of cal- or recalculation.

Definitions:

The open end is where station xi/L=1.0 is on the
wave diagram.

Dimensionless time, tau = a0/L * t ,
where t is time
a0 is the reference speed of sound and
L is the length

Exhaust velocity ratio, M = u/a0
where u is the exhaust velocity of the effluence
and a0 is the reference speed of sound