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Bernoulli
Equation
J-2X Progress: The Next Phase For E10001
William Greene
In January, the Chinese
people celebrated their traditional New Year and formally initiated the year of
the Dragon.
I was born in the year of the
Dragon (it comes up every twelve years) and I started thinking about previous
Dragon years and where I was when they occurred.
My first year of the Dragon
after my birth happened to be the 200th birthday of our great country and I was
starting sixth grade.
My second year of the Dragon
was the year that I got married so that was kind of important to me on a personal
level.
My third year of the Dragon
was the year that I started working for NASA after spending a decade working
for defense and space industry contractors.
It is interesting looking at
one’s life in such a series of widely separated snapshots. Things move
on.
The same is true for
J-2X. Last year was momentous for our project. We assembled and
tested our first development engine, E10001.
We celebrated and received
well-deserved (if I do say so myself) kudos and pats on the back. But now
things move on and the life of our good friend E10001 enters its next
phase.
And the next phase for E10001
involves changes to its nozzle configuration. So, before I tell you
specifically what we’re doing to E10001, we need to discuss how a supersonic
nozzle works.
Below is a schematic of what, on a rocket engine, would be
called the thrust chamber assembly or the main injector plus main combustion
chamber plus the nozzle.
Within the realm of
compressible flow this is known as a convergent-divergent nozzle, or as a “de
Laval nozzle” after a late 19th-century Swedish engineer, Gustaf de Laval, who
pioneered using such shapes as part of steam engines […and you woke up this
morning not realizing that you’d learn something historical today!].
How it works is simple.
Fluid flows from high pressure at the head end on the left towards the low
pressure at the exhaust on the right.
In between, the flow area of
the “pipe” in which the fluid flows is manipulated to accelerate the
fluid.
The most narrow point in the
flow is called the throat. Fluid flow to the left, upstream, of the
throat is subsonic, i.e., traveling at less than the speed of sound.
If the ratio of “high” to
“low” pressure at the two ends is large enough, then fluid flow to the right,
downstream, of the throat is supersonic, i.e., traveling at greater than the
speed of sound.
Under such conditions, the
velocity at the throat itself is exactly that of the speed of sound.
In other words, the fluid is
traveling at “Mach 1” at the throat [the term named for Ernst Mach, an Austrian
scientist and philosopher also from the late 19th century].
Oh, and all of this only
works if your “fluid” is compressible, or in other words a gas like air or, in
a rocket, combustion products.
How and why this happens gets
a little heavy on the thermodynamics, so please just trust me for now.
But the really neato thing
that Mr. De Laval learned when playing with convergent-divergent nozzles like
this is that:
(1)
for subsonic flow, as the flow area gets smaller, the flow velocity goes up,
(2)
for supersonic flow, as the flow area gets larger, the flow velocity goes
up.
In other words, they act the
opposite of each other.
For a rocket, this is
absolutely fantastic since the whole idea of a rocket is to fling stuff out the
back end at very, very high velocity and this cool device accomplishes that
with just a little bit of creative geometry.
Okay, with me so far?
Then, here’s another thing to think about regarding supersonic
flow: You can’t shout upstream.
Sound is nothing more than
pressure waves traveling through a fluid.
A gas has a characteristic
speed at which pressure waves are conveyed within it. That, then, is the
speed of sound.
So, if the gas is traveling
at greater than the speed of sound, then pressure waves cannot travel
upstream.
Think of it this way: imagine
yourself to be a gas molecule. Normally, when traveling less than the
speed of sound, you can receive signals from all directions.
Your motion can be impacted
by pressure waves both upstream and downstream of where you sit at any given
time.
However, now imagine that you
are that gas molecule hurtling along in a supersonic flow.
Now, because you’re traveling
faster than the ability of pressure waves to get back upstream, you can have no
idea what’s going on downstream. You’re flying along blindly.
Thus, the bottom line is that once the ratio of high and low
pressures are sufficient to cause this situation of supersonic flow in the
divergent portion of the nozzle (a term that we use is that the throat is
“choked”), then the nozzle flow is the nozzle flow.
In other words, it is largely
independent of what happens beyond the exit plane. Largely, but not
entirely. I’ll explain below. Hold on.
Next, we’re going to talk about the Bernoulli Equation [developed
by an 18th-century father and son team of Swiss professors Johann and Daniel
Bernoulli].
No, we’re not going to do any
math. All that we have to do is understand the concept of the Bernoulli
Equation and how it relates to the flow in the divergent portion of our
nozzle.
Here it is: Absent other
factors, when fluid is accelerated, its pressure drops. You can think of
this in terms of energy.
Pressure is like stored
energy, as in electrical energy in a battery.
Velocity is active energy, as
in electrical energy spinning a fan. Absent any other input or output,
when you show more active energy (velocity), you then have less stored energy
(pressure).
Just for fun, here are some pictures of the men I’ve mentioned
so far. Oh, and I tossed in a friend of Daniel Bernoulli’s named Leonhard
Euler.
Anyone who knows anything
about mathematics or fluid dynamics knows all about Mr. Euler. He was
truly a genius on par with Sir Issac Newton. (BTW, I kinda like the white,
powdered wig thing the Bernoulli guys had going there. Maybe I’ll adopt
it myself…)
Back to the
topic at hand. Where do we stand once we combine compressible fluid flow
through the divergent portion of a de Laval nozzle, traveling at speed greater
than Mach 1 (meaning that pressure waves cannot travel upstream), and with the
application of the Bernoulli Equation and the effect on pressure?
I will
attempt to show you in a picture… So, if I make my nozzle
longer and longer and longer, with a larger and larger exit size, my exhausting
gas goes faster and faster and faster.
Again, that’s why rocket
engines have big divergent nozzles. Ta-da!
But, there are limits. There
always are. Nothing is free.
The first limit is weight. As your nozzle gets bigger and
bigger, your nozzle structure gets heavier and heavier.
As some point, any gain in
engine performance is offset by the loss of vehicle performance because your
engine is too heavy to lift.
The second limit is due to
what’s on the other side of the exit plane. What’s outside the nozzle is,
well, the ambient environment.
If you’re sitting at the NASA
Kennedy Space Center in Florida, where we usually launch our rockets, the
ambient conditions are known as “sea level” conditions, meaning that the
atmospheric pressure averages about 14.7 pounds per square inch.
On the other hand, if you’re
floating around in space and in orbit around the earth, then your ambient
conditions are, to a pretty good approximation, a vacuum, meaning 0.0 pounds
per square inch pressure.
What happens if you’re that gas molecule hurtling along in the
flow at supersonic velocity down the nozzle and then you’re suddenly flung into
ambient conditions?
Well, if you’re in the main
part of the flow, not much. You eventually slow down through a series of
oblique shocks external to the nozzle.
As I said above, if you’re
moving supersonically within the nozzle, then you’re not affected by what’s
downstream.
But what if you’re not in the
main flow but instead along the wall? Here’s a secret: The flow along the
wall is slower than the main, core flow.
Indeed, exactly at the wall,
in the limit, the velocity is zero. That changes things.
So, exactly at the wall, the velocity is zero, and just
fractions of an inch into the flow the velocity is supersonic.
This transition zone is known
as the “boundary layer” and the fluid dynamics complexity here can be nearly
mind boggling and it has to do with viscous friction between the fluid and the
wall.
But the important point is
that there is a thin layer that is not supersonic. Below is a typical
textbook-like representation of boundary layer flow.
Remember when I said that
what happens beyond the exit plane largely doesn’t affect the fluid flow in the
nozzle? The boundary layer is the exception.
Because the flow here is
subsonic, pressure conditions downstream can influence things upstream. And
here is the source of the other limit on your nozzle size.
If the ambient pressure is much, much higher than the pressure
of the nozzle flow, then this pressure can slow up the subsonic portion along
the wall.
If you slow it up enough, you
can make the boundary layer thicker and thicker until it’s no longer just
fractions of an inch thick.
Having a thick boundary layer
means that your nozzle is not flowing “full.” The flow can become
“detached” from the wall and such a situation is inherently unstable.
All around the nozzle, in
local pockets, the boundary can grow and collapse and grow again causing
localized pressure variations. Shock waves start bouncing around.
Then the nozzle structure
itself, usually not built very stiff so that it doesn’t weigh too much, starts
to respond to these local pressure variations and shock waves and it wobbles
and ripples and buckles.
To put is more succinctly, if
your nozzle expands the rocket exhaust flow too much for the ambient
conditions, you have an “over-expanded” condition and this can literally tear
the nozzle apart.
Below is a picture that tells
the story of the impact of ambient pressure on nozzle flow.
Now, finally, we’ll get back
to J-2X E10001.
For all of the tests conducted to date, the nozzle that we’ve
tested on E10001 has had an expansion ratio of 35 to 1, meaning that the area
of the exit plane is thirty-five time larger than the area of the throat.
With this kind of expansion
ratio for this engine, the nozzle flow is not over expanded.
The nozzle “flows full” at
sea level conditions like those seen at the NASA Stennis Space Center (SSC)
where we test the engines and all is good.
But the J-2X is intended to
be an upper stage engine in flight, meaning that when it fires during the
mission, it will be at over 100,000 feet in the altitude where the ambient
pressure is much less than sea level conditions.
Because of that, we designed
the engine to use a larger nozzle, get more performance from greater exit
velocity, and not over expand the exhaust flow at THOSE conditions way up in
the upper atmosphere, practically in space.
But then how do we test it? If we have a nozzle that flows
full at altitude, but does not flow full (i.e., it’s over expanded) at sea
level, then how do we perform a test showing that the nozzle works?
We can’t exactly build a test
stand at 100,000 feet in the sky. Instead, we make the test stand simulate
these high-altitude conditions.
Below is a picture of NASA
SSC test stand A-2. What you see there in the middle, the big tube several
stories tall surrounded by structures, is the passive diffuser.
The diffuser, combined with a
clam-shell enclosure structure around the bottom portion of the engine, uses
Bernoulli effects (see, they come into play again!) such that when the engine
is firing, it does so into an ambient environment that “appears” to be like
that at high altitude.
By doing this, for the next
phase of J-2X E10001 development, we will be able to do testing with a nozzle
extended to an expansion ratio of 59 to 1.
That is one step closer to
the ultimate flight configuration for the J-2X as part of the exploration
mission and therefore one step closer to fulfilling that mission.
It takes a bit of explaining
to understand why all this is necessary, but the bottom line truly is that we
are getting closer and closer to our exploration goals.
So, enjoy come on along with us to celebrate the Year of the
Dragon with the generation of lots of smoke and fire from the J-2X. It’s
going to be fun. But first, maybe a few traditional Chinese New Year’s
treats…
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