Update
23 March 2024:
For the readers of this and other similar articles about ramjet
propulsion, be aware that GW’s ramjet
book is finally available as a self-published item. Its title is “A Practical Guide to Ramjet
Propulsion”. Right now, contact GW at gwj5886@gmail.com to buy your copy.
He will, upon receipt of payment by surface mail or Western
Union (or similar), manually email the
book to you as pdf files. This will take
place as 9 emails, each with 3 files
attached, for a total of 27 files (1 for
the up-front stuff, 1 each for 22
chapters, and 1 each for 4
appendices). The base price is
$100, to which $6.25 of Texas sales tax
must be added, for an invoice total of
$106.25.
This
procedure will get replaced with a secure automated web site, that can take credit cards, and automatically send the book as
files. However, that option is not yet available. Watch this space for the announcement when it
is.
GW is working
on a second edition. No projections yet
for when that will become available.
--------------
Update 11-21-2020: The readership on this article has been extraordinary the last few days. I hope you found it interesting and useful. Feel free to comment. If I can help you, please contact me. This topic was a prerequisite to my becoming a ramjet expert, among several other things that I did.
Update 4-25-2022: This article is about inlets used for gas turbine engines and for subsonic-combustion ramjets. These all feature subsonic flow in a divergent channel, at the final delivery point for the air. Inlets for supersonic-combustion ramjets are quite different, despite using similar external compression features: having no min area point, and delivering a still-supersonic flow in an essentially-constant area duct termed an isolator. Those are out of scope here.
-----------------------------
The purpose of this article is more to explain how inlets
work in terms of physics, than it is to
inform how exactly to make the calculations. The reader is presumed conversant with
compressible flow, shock waves, and boundary layers. Inlets are useful for both airbreathing
engine propulsion streams, and for flows
of cooling air for all sorts of equipment.
The focus here is on airbreathing propulsion.
Scope
There are three common types of inlets seen on a variety of
flight vehicles. All turn the kinetic
energy of an approaching airstream into a pressure rise after capture. Some do
this better than others, depending upon
the application. There are the NACA
flush inlet, the pitot/normal shock
inlet, and the “supersonic” inlet
featuring external shock wave compression devices (two forms).
The NACA flush inlet has low external drag, being flush to the surface, but it has poor pressure recovery and mass
ingestion-for-the-size characteristics.
It has not been used in a propulsion application since its use for the
Douglas D-558-2 Skyrocket experimental plane of the early 1950’s, and so is left out of consideration
here. It is often still used for
cooling air streams, however.
Two Kinds of Inlet for the Various Propulsion Applications
The two inlet types suitable for modern propulsion
applications are the pitot/normal shock inlet,
and the “supersonic” inlet. See Figure 1. Note that (1) there is no such thing as a “low-subsonic
ramjet”, and (2) that the very same
components are used quite differently by gas turbine engines versus ramjet
engines. This shows up in the different
flow fields depicted. Shock waves are
shown in red, ingested air blue, and decelerated-but-spilled air green. “Spilled” means diverted around outside the
inlet.
Figure 1 – Modern Propulsion Inlet Types and Typical
Applications
Of these, the pitot/normal
shock inlet is the older. It has been used
in jet aircraft and ramjet vehicles for subsonic and supersonic flight since
the 1940’s, up to a max practical speed
of about Mach 2 beginning in the 1960’s. In supersonic flight, it always ingests air that has gone through a
single normal (perpendicular) shock wave,
the strongest type with the greatest pressure loss, and an always-subsonic flow speed
downstream. The higher the Mach number, the larger these effects. See Figure 2.
Figure 2 – Pitot/Normal Shock Inlet Variations and
Applications
The supersonic inlet has a ramp or spike-shaped surface
protruding ahead of the cowl lip that generates one or more oblique shock
waves, prior to a “terminal normal shock
wave” at (or downstream of) the inlet minimum flow area. Flow is still supersonic, but at a lower Mach number, after each wave, until the final normal shock, which always produces subsonic flow behind
itself.
Oblique shock waves are less strong with less pressure loss
across them, but also with a lower but
still-supersonic Mach number downstream.
The effects are larger with higher local approach Mach number, and with a larger compression surface flow
deflection away from free stream. As it
turns out, the sum of the pressure
losses across multiple oblique shocks,
plus a final normal shock at a low approach Mach, is less than the pressure loss across a
single normal shock at full flight speed,
if you size and locate the deflections correctly. How to actually do that is beyond scope here.
The typical supersonic inlet has either planar ramp surfaces
that deflect flow away from freestream,
generating the oblique shock waves as planar surfaces, or it has a conical spike that generates a
conically-symmetric deflection and conical oblique shock waves. The planar-wave form has two-dimensional flow
characteristics, and we call these “2-D”
or “rectangular” inlets. The conical
spike is associated with an axisymmetric cylindrical flow geometry. We call these “axisymmetric” inlets.
In either geometry,
there are one or two oblique shock waves generated by the external
compression surface prior to flow passing the cowl lip and becoming a contained
internal flow. This change from an
external to an internal flow we call “capture”,
and the swept area at which it occurs we call the “capture
area”, denoted as Ac. The minimum flow area of the inlet may be
located at “capture”, or it may be
located further downstream. The final
portion is always a divergent subsonic diffuser passage.
If there is no contracting supersonic flow channel, we call this “(all-) external (shock)
compression”. Any supersonic contracting
channel has a pattern of reflecting oblique shock waves inside it. There is such an internal compression scheme when
the min area is downstream of capture. We
call that “mixed compression”, because
there is both external and internal compression. (There are no practical inlets that
use only a confined contracting channel, for all-internal compression.)
At design conditions,
the terminal normal shock wave is located at the min area. It can be located further downstream in a
diverging channel, but never upstream in
a contracting channel. That is
mathematically unstable, and also never, ever happens in nature! The shock system is instead
disgorged (unswallowed) to a position out in front of the cowl lip, including the terminal normal shock.
All-external compression inlets will “start” spontaneously
(meaning swallow the shock system spontaneously at design speed and theoretically
design backpressure, or in practical
reality at design speed but with a somewhat-lower-than-design backpressure. Mixed compression inlets will not start
at design, needing to be significantly
oversped (and with lower backpressure) to start, unless starting bleed cutbacks and slots are
added. These making starting
easier, but also reduce the efficiency
of mass ingestion. See Figure 3. Cutbacks are in the cheekwalls of 2-D
inlets, bleed slots can be used with
both 2-D and axisymmetric inlets.
Often, both must be used to
achieve tolerable start characteristics.
Figure 3 – The Two Types of Supersonic Inlets (All-External
and Mixed Compression)
At flight speeds between about Mach 1 and Mach
1.5-to-2, there’s not a lot of difference
in the pressure recovery characteristics of the pitot/normal shock vs
supersonic inlets, but above Mach
1.5-to-2, the difference is quite
substantial, and grows ever larger with
increasing speed. This explains why most
of the early supersonic jets used the simpler pitot/normal shock inlet. They could not reach Mach 2 in the first
place. The complications were not worth
the trouble to implement.
Once combat dash speeds got to about Mach 2, you saw the supersonic inlet increasingly employed
on jet fighter aircraft, despite its
inferior mass ingestion characteristics below about Mach 1.5 due to its
detached shock system. Oversizing the
inlet, and employing blow-in air bleed
doors on the subsonic passage, are the
means by which the inferior mass ingestion characteristics got overcome for the
jet aircraft applications. Ramjets
employing supersonic inlet technology rarely fly slower than about Mach
1.8-1.9, if that slow, which eliminates that particular
consideration for the ramjet application.
The pitot/normal shock inlet will also start
spontaneously, at any supersonic flight
speed, as long as the backpressure is
low enough. The common thread here
is all these inlets will have unswallowed shock systems (inlets not
started) if the backpressure demand is too high! The oblique shocks of the supersonic types
complicate that overall picture only a little bit further.
Behaviors of Pitot/Normal Shock Types
The behavior of the pitot/normal shock inlet is simpler to
understand, and so it will be discussed
first. From a speed standpoint, there is only subsonic and supersonic
behavior, and even these are similar in
most ways. The main variation for both
regimes is air ingestion versus demanded backpressure. See Figure 4.
Figure 4 – Behavior of Pitot/Normal Shock Inlets Vs Speed
and Backpressure Demand
In subsonic flight,
there is a simple tradeoff of captured streamtube size versus the final delivered pressure (demanded
backpressure). Higher pressure demand
reduces mass ingested (shown in blue in the figure). This shows up in the reduced size of the
captured streamtube. Thus there is
subsonic streamtube divergence ahead of capture, which is a sort of external compression
(diffusion) that adds to the internal compression (diffusion) afforded by the subsonic
divergent channel shape. That is, in point of fact, how the higher pressure demand gets met. But air that theoretically might get ingested,
instead gets decelerated and then
spilled around the outside of the inlet (shown in green in the figure). There is a drag for this, which must be accounted-for, in some way.
The upper limit on delivered pressure is called “stagnation”
or “total” pressure, when you stop-up
the outlet, at zero massflow throughput. That is also called the “pitot” or “ram”
pressure. Its value depends upon the oncoming
flow speed: higher values at higher
speeds, shocked or not.
In supersonic flight,
the normal shock wave is the complicating factor. At design conditions, termed “critical”, it resides in the min area at the capture
location. Flow is flight-velocity-supersonic
right up the wave, and then subsonic
downstream of it. There is no spillage, as is illustrated bottom center of the
figure.
If you reduce the demanded backpressure, the shock migrates further downstream into
the divergent channel, and takes place
at a higher-than-flight Mach number.
There is a larger shock loss, and
less of the divergent channel to diffuse the subsonic flow. That is how you achieve the lowered
backpressure. But again, there is no spillage, as shown bottom right of the figure.
If you raise the demanded backpressure above design, the shock suddenly jumps out in front of the
inlet, so that there is subsonic
diffusion compression in the divergent streamtube just prior to capture. There is also less air massflow
ingested, and thus there is inherent spillage,
which causes more drag.
About Backpressure
Backpressure demand is just not something to do with
the inlet! It is instead generated
by whatever the inlet is connected to.
This could be any of a variety of devices, all of which have their own pressure vs
massflow characteristics. For this
article, we are interested in the gas
turbine-based jet engine, and in the
ramjet engine. They are quite different. Quite fundamentally so.
The simplest way to think about this is to compare a
pitot/normal shock inlet coupled to an air pump, to a pitot/normal shock inlet coupled to a
simple flow resistance (an outlet nozzle or orifice). The air pump literally sets the massflow
passing through the system, while the
outlet resistance does not set the massflow,
but instead sets the pressure just ahead of itself. This comparison is shown in Figure 5.
The gas turbine engine is very nearly a constant-volume flow
rate device at any given core rotation speed,
with higher volume flow rate at higher rotation rate. This would apply to the (low bypass)
“turbojet”, and to the (high bypass)
“fanjet”, and to all the variants in
between those extremes. At any one altitude,
the air density makes the volume flow rate a massflow rate. Thus the gas turbine engine is very much an
air pump. Its rotation speed at any one
altitude sets the massflow, which the
inlet must then provide.
At very low speeds (such as during ground runs), the engine-demanded massflow can actually
exceed what the inlet could otherwise sweep out with its capture area. That leads to severe suction conditions
inside the duct, in order to pull in
more air than could be swept-out otherwise.
Under this scenario, adding
spring-loaded blow-in doors to the duct, can augment the total ingested air, without making the inlet so very large
otherwise.
The ramjet is very much like the simple outlet restriction
in its behavior, which is also a good
model for most cooling air installations.
How rich a mixture, and how much
burning you actually get in the combustor, will act to modify this behavior, but it essentially is very much like the simple
outlet restriction on a duct. The restriction sets the pressure just ahead of
itself, and the inlet adjusts its
massflow ingestion to provide that demanded backpressure. In this scenario, there is nothing to create suction conditions
inside the duct, so that the max
possible ingestible airflow is only that which could be swept out by
the capture area Ac.
Figure 5 – Behavior of Inlet Coupled to an Air Pump vs an Outlet
Restriction
The fundamental notion here is that the gas turbine behaves
like an air pump, while the ramjet
behaves like a simple outlet restriction.
The two are fundamentally different,
and thus there are fundamentally-different responses of the inlet to
whatever device it is coupled-to. And
that is exactly why the very same inlet components get used so
differently with the two types of engine.
Behaviors of Supersonic (External Compression) Types
This is similar to,
but somewhat more complicated than,
the behaviors of the pitot/normal shock inlet. That is because of the oblique waves shed
from the external compression surface,
and where they fall in relation to the cowl lip.
It is perhaps easiest to first understand the full design
speed range-of-behavior of a supersonic inlet.
The significant variables are flight speed and backpressure demand. Flight speed is significant, relative to the design speed where the oblique
shocks are focused right on the cowl lip,
but must also exceed the speed Matt at which the oblique
shock system detaches from the compression surface, and moves out in front as a normal shock bow
wave.
Therefore, we must
consider three speeds, and three
demanded backpressure conditions. The
three speeds are below design Mach (but above Matt), design Mach,
and above-design Mach. The three
demanded backpressure conditions are above what can be supplied at the
speed, on the max design value for the
speed, and below the max design value
for the speed. These influence where the
shocks fall, the flow pattern, and the spillage. See Figure 6.
Figure 6 – Behavior of Supersonic Inlets vs Speed and
Backpressure Demand, Attached Bow Shock
Of the nine sketches in the figure, the center one is the design point, at which the oblique shocks fall on the cowl
lip, there is no spilled air, and the backpressure demand is
“critical”, meaning at its max design
value. Accordingly, the terminal normal shock falls at the min
area within the inlet passage. The
sketches show all-external compression,
but the same overall behavior obtains with mixed compression. The terminal shock is just at the min area, which is further downstream in mixed
compression.
If you slow down but maintain a just-critical
backpressure, you get the top center
sketch. The oblique shocks fall ahead of
the cowl lip, but supersonic flow is
maintained through capture to the terminal normal shock. However,
the captured streamtube (blue) is inherently smaller than what could
have been swept out by the capture area. Thus, there is air that is shock-decelerated, yet spilled (diverted around the outside of
the inlet). There is a drag for that, which must be accounted-for.
If you speed up past design,
you get the bottom center sketch.
Now the oblique shocks fall inside the cowl lip. Part of the ingested air sees the oblique
shocks and weaker terminal normal shock.
Part of the air sees only a strong normal shock, as indicated in the figure. The recovered pressure is thus inherently
lower at higher flight speeds because of this.
But, no air is spilled.
If you raise the demanded backpressure above critical
(called “subcritical inlet”), at the
design speed, you get the center left
sketch. The inlet “unstarts”, meaning its shock system gets
“unswallowed”. The oblique shock(s) is(are)
still there, but it(they) is(are) followed
by a normal shock, partway up the
compression surface. From
there, flow is subsonic all the way into
(and on through) the inlet. The
delivered pressure will be quite close to the critical value, but the air ingestion is sharply reduced by
lots of spillage, as shown.
The same things happen if you demand too much backpressure
at a speed less than design (top left sketch) or at a speed greater than design
(bottom left sketch). The inlet
unstarts, with the normal shock out on
the external compression surface,
spillage, and a delivered
pressure pretty close to design critical.
Only the position of the normal shock,
and the associated amount of spillage,
changes.
The pattern with lower-than-critical backpressure (called
“supercritical inlet”) is even simpler.
At design speed, you get the
center right sketch. The only difference
between it and the design sketch in the center,
is the position of the terminal shock deeper down the divergent diffuser
passage. Neither spills air. The supersonic expansion associated with that
diverged -larger area makes the terminal wave stronger, and its pressure loss larger. That is exactly how the inlet adjusts to
deliver a lower demanded backpressure.
Below design speed (top right sketch), at lower-than-critical backpressure, all the external flow field is the same as
the top center sketch, including the
inherent spillage. Only the terminal
shock is deeper down the passage,
leading to lower delivered backpressure,
same as at design speed. Above
design speed (lower right sketch), the
external pattern is the same as at critical operation (bottom center
sketch), with no spillage. Again,
the terminal shock is deeper down the passage, leading to lowered delivered backpressure.
It is customary to “book-keep” the spillages
differently, depending upon whether they
are backpressure-induced, or are caused
by shocks falling ahead of the cowl lip. When there is backpressure-induced spillage
(as in the center left sketch), there
is subsonic flow at capture, and we
call that “spillage”, and book-keep its
effect as “spillage drag”. When there is
reduced capture due to shock position, but
with still-supersonic flow at capture (as in the top center sketch), we don’t call it spillage, but we do lump its effects into “additive
drag”, along with some other effects (as
applicable).
Using these two models simultaneously is a pretty good
model for what happens in the top left sketch. There is an underspeed component of additive
drag associated with top center and top right sketches. And,
there is a subcritical spillage drag associated with the center left and
bottom left sketches. There is both
an underspeed additive drag component and a spillage drag associated
with the top left sketch. They add to
the total effect, if both expressed as
drags or as spillages, which they are
customarily not.
There is a speed below which the oblique shock shed by the
tip of the external compression surface can no longer stay attached. This happens usually in the general vicinity
of Mach 1.5, but every design is
different in detail! The shock
suddenly jumps out in front of the compression surface as a locally-normal-shock
bow wave. From there, behavior resembles that of the pitot/normal
shock inlet, as shown in Figure 7. The difference is that, even for the exact same swept-out capture
area Ac, the air ingestion of the supersonic inlet is lower than that of the
pitot-normal shock inlet. This is caused
by the blockage presented to the oncoming locally-subsonic flow, of the external compression surfaces.
The effects between Mach 1 and the shock-detachment speed Matt
are shown in the three lower sketches of the figure. The air ingested and the shock position are
what varies from sketch to sketch. Flow
from the bow wave to capture is subsonic,
just like with the pitot/normal shock inlet. At higher backpressure, the shock is further ahead, and less air is ingested (bottom left
sketch). At lower backpressure, the shock is closer in, and more air is ingested (bottom right
sketch.
In subsonic flight (top 3 sketches in the figure), the pattern and physics depicted are the
same, except that there is no bow shock.
Again, this pattern is less ingestion at higher
backpressure, and more ingestion at
lower backpressure, very much like the
pitot/normal shock inlet.
There is one extra sketch in the top right of the
figure. That shows what happens at
trivial oncoming velocity in a ground run.
This can only happen with a gas turbine engine trying to pump the air
through the inlet! All through the
inlet, one sees suction below
atmospheric, with the greatest near the
min passage area. That is where
spring-loaded blow-in doors can let in enough air for the engine to run at that
power setting.
Figure 7 – Behavior of Supersonic Inlets Below the Attached
Bow Wave Speed
Other Additive Drag Effects
These occur with side-mounted inlets, for which the captured stream(s) are in
contact with the vehicle forebody. There
is, of course, skin friction associated with that
contact. But, there are also vehicle forebody flow field
effects, primarily the bow shock from
the nose, which causes a bit of loss in total pressure.
This is effectively a drag force acting upon the captured
streamtube, long before it is ever
captured. These are also additive
drag effects. Lumped together
with the underspeed spillage, they
comprise what is usually book-kept as an additive drag table.
How the Data Are Usually Book-Kept
Inlet delivered pressure data are usually reported as the
ratio of critical-inlet (design) delivered total (stagnation) pressure Pt2crit
to free-stream total pressure Ptoo.
That would be PRcrit = Pt2crit/Ptoo. The main variable affecting these values is
free-stream Mach number, with vehicle
angle of attack a weaker secondary parameter,
and vehicle roll angle sometimes a weaker-still tertiary parameter.
Inlet mass ingestion is usually reported as a ratio of area
ratios. The area ratio in question is
the streamtube area ratio from free-stream to the capture point AR = Aoo/AC. It has a critical-inlet value ARcrit. Both are functions of primarily Mach
number, but also angle-of-attack (AOA), and maybe roll angle. So the data are usually book-kept as ARcrit
vs M, parametric on AOA and roll.
Additive drag is usually book-kept as a drag coefficient
increment CDadd whose reference area is the swept-out inlet
capture area Ac. Coefficient
times reference area times free-stream dynamic pressure is the additive drag
force. This is primarily a
function of flight Mach number, with AOA
and roll as weaker secondary and weaker-still tertiary parameters.
The supercritical pressure margin is PM = 1 –
Pt2/Pt2crit. This is
zero at critical operation, and a number
between zero and one for supercritical operation. It is considered to be zero if subcritical.
The subcritical inlet spillage margin SM = 1 –
AR/ARcrit. This is zero at
critical, and a number between zero and one
during subcritical operation. It is
considered to be zero during supercritical operation.
PM and SM cannot both be nonzero at the same time! The inlet is either supercritical or it is
subcritically spilling! Both parameters can
be exactly zero at the same time, but
only if the inlet is operating at exactly critical conditions.
Sometimes you will see an “inlet margin”
reported, that is negative at -SM for
subcritical operation, zero at
critical, and positive at PM for
supercritical operation. This is
rather commonly done.
The format and appearance of typical data are shown in Figure 8. How a typical inlet operates is shown in Figure 9.
Figure 8 – What Typical Inlet Data Look Like
Figure 9 – How a Typical Inlet Operates
The way this data gets used is as follows:
For a given point in flight,
Mach number, AOA, and roll are all known. These are used, usually as part of a table look-up computer
routine, to determine the values of PRcrit, ARcrit, and CDadd. The flight speed and altitude, for any given model of a day (hot, cold,
or other), determine the values
of Ptoo and dynamic pressure qoo. The flight condition total pressure thus allows
one to compute Pt2crit. You
will also need the oncoming free-stream total temperature, which is conserved throughout the inlet.
This is usually done within some larger computer analysis program
that determines the engine-inlet match.
One always starts at critical (PM = SM = 0), and iterates the calculation until it
balances within the program. One
must initially find out if the demanded Pt2 exceeds Pt2crit
or not. If it does, you are on the subcritical spillage
path, and you stay there looking
for the right value of SM until the analysis converges. If not,
you are on the supercritical path,
and you stay there looking for the right value of PM until the
analysis converges. You do not switch
from one path to the other, while
analyzing any one flight point!
AR is ARcrit * (1-SM), so that Aoo = Ac * AR, and Pt2 is Pt2crit
(1-PM), subject to the constraint
already mentioned that PM and SM cannot both be nonzero. Once these values are converged and fully
known, you can compute ingested massflow
and the state variables at every station in the inlet. You can also compute additive drag and spillage
drag (if any), and the ram drag of the
ingested airstream, which is RD = wair
Voo / gc. What you
do with those drags depends upon the thrust-drag accounting system you are
using (net jet or installed). That
choice is beyond scope here.
I don’t really recommend attempting all this iteration as
purely hand calculations! You need to be
using proper computer programs for this!
I have used several, and written
several of my own.
What works well for gas turbine propulsion is a performance
estimating model based on “typical pressure ratios”. Looking again at Figure 9,
and realizing that all gas turbine inlets are operating in spillage
mode, where the pressure ratio at any
Mach is just about constant, plus the
compressor pressure rise dominates over inlet pressure rise by far, one can justify using such a simplified model!
That kind of model just doesn’t work for ramjet, which has no compressor, so that the only pressure rise item is the
inlet. Looking again at Figure 9, we see that this value varies wildly, since nearly all ramjet inlets are operating
in supercritical mode. What that means
is any constant input “typical pressure ratio” is really going to be wildly
wrong! So, you must run a real cycle analysis based on
actual calculations of state variables,
at every station throughout the inlet,
engine, and nozzle, for a ramjet.
Design Estimates for Pitot/Normal Shock
Pitot/normal shock inlets are the only type for which you
can make realistic performance estimates from first principles. They won’t be “right”, but they really are somewhere in the
ballpark.
You need a guess for the subsonic diffuser “efficiency” as a
ratio of delivered-to-freestream total pressure that you can use up to Mach
1. It’ll be crudely around 0.98. Above Mach 1,
you multiply this by the normal shock total pressure ratio, from a table such as that in ref. 1.
The guess for critical streamtube area ratio is just a
constant from subsonic through supersonic.
Again, a crude guess would be
something like 0.98.
As long as the pitot/normal shock inlet is a nose
inlet, there is no additive drag to
worry about. If it is a side inlet, you have to use some sophisticated methods to
estimate the friction and bow wave forces acting on the entering
streamtube. Ref. 2 addresses that.
A properly sized pitot/normal shock inlet will always
operate on the subcritical spillage path when flying subsonic. If that choice fails to obtain in your cycle
analysis, you have mis-sized the
inlet! Subsonically, there is no way to reduce the total pressure
in the diffuser for supercritical operation (other than gross flow
separation, which utterly defeats the
purpose of the inlet), because there
cannot be a shock wave. Thus, it must always operate in subcritical
spillage mode.
Design of Supersonic (External Compression) Types
This is just not something you can do from first
principles! In the old days, this was done cut-and-try with models tested
in supersonic wind tunnels. It was a
very expensive and time-consuming process.
Today, the inlet design
process is quite different, and makes extensive
use of computational fluid dynamics (CFD) programs. To do supersonic inlet work, these codes must have adequate turbulence
models, the ability to do shock
waves, and the ability to handle separated
zones of flow. Otherwise, shock-boundary layer interactions will not
be adequately modeled, and the answer will
simply be wrong, converged or not.
Further, there must
be a way to determine reliably when the calculation has converged to its final
form of the answer. There are many CFD
codes widely available today that are employed for this work. However,
few of them have all the necessary attributes. In my experience, the most common failure with many of them is actually
recognizing a reliable convergence to an answer. The second most common failure mode is
converging to a wrong answer, because
not all the necessary fluid mechanics attributes are included.
In any event, no
matter what CFD code you use, you still
must still experimentally-verify the inlet performance in a wind tunnel. This is because of the still-high risk that
the code predictions are just plain wrong.
That experimental wind tunnel testing is still expensive and
time-consuming, just not nearly as
extensive in scope as it once was. But
you still have to do it, to fly
successfully!
I have some old empirical correlations that I use to create
ballpark-realistic guesses for the three inlet performance curves, in terms of a user-selected design
shock-on-lip Mach number. There is a set
for 2-D inlets, and a set for
axisymmetric inlets. They are for
near-zero AOA and roll angles only. I put
these in my yet-to-be-published ramjet book,
listed as ref. 3, but not yet publicly available as of this
writing.
What Are the Real-World Flaws in Inlet Performance?
There are five types of problems: (1) shock-boundary layer interactions causing
flow separation problems, whether
located in a divergent channel or elsewhere,
(2) flow separation problems in divergent diffuser channels, whether a shock is present or not, (3) ingestion of low-energy boundary layer or
separation-zone air that reduces pressure and streamtube recovery, (4) flow distortions and separations caused
by turn elbows, and (5) solving inlet starting
problems, especially with
mixed-compression inlets.
How to Approach Fixing the Real-World Flaws
Here are the best solution approaches that I know-of, for the 5 problem areas listed above. I give no numbers, because each has to be tailored for each
application. This is done with a
combination of CFD and wind tunnel test.
You simply cannot avoid the testing needed to verify your CFD
answers!
shock-boundary layer interactions causing flow
separation problems, whether located in
a divergent channel or elsewhere
These can occur on the external compression surfaces
themselves, on the confined-channel
surfaces leading to the inlet min area in an all-external compression inlet (or
to just downstream of it in a mixed-compression inlet), or anywhere else in the divergent diffuser
channel. The terminal shock can quite fall deep in the divergent diffuser, if operating at large supercritical pressure
margin.
On the external compression ramps, typically for each segment, there is a constant pressure with increasing
distance downstream, until you reach the
next shock. These are short
distances, with the start of boundary layer
formation at the surface tip, so the boundary layer is rather thin compared to
the dimension of the streamtube to be captured,
at least through the first oblique shock.
But, if there is a
problem with too much low-energy boundary-layer air, the surface can be made porous, and the air that sinks through the porosity
collected and sent to an overboard dump channel built into the structure. Because of its momentum parallel to the surface, the higher energy air won’t make the turn
through the porosity holes, only the
lower energy air. Thus the method is
rather selective that way. See Figure 10 for the
concept.
Figure 10 – Surface Porosity “Fix” for External Compression
Surfaces
Where the next shock is located along the compression
surfaces, the surface takes another
deflection turn toward the cowl lip.
“Toward” helps keep the flow attached to the surface, although the boundary layer of lower-energy
air gets suddenly thicker where it passes through the adverse pressure “jump”
of the shock wave. This effect is far
more pronounced with the far stronger normal shock waves, than it is with oblique shock waves. That is why you rarely see surface porosity
applied to external compression surfaces,
and when you do, it is only after
the second wave in the train. Between
that second wave and the entrance at the cowl lip is where you might get a
shock-thickened boundary layer that would need treatment with the surface
porosity “fix”, which is where it is
shown in the figure.
With all-external compression, at critical flow, the normal shock is at the min area, which is located at the cowl lip. Flow is subsonic behind that normal
shock. The whole capture pattern is that
of a turn away from the preceding deflection turn directions, back toward (and a little past) the
freestream direction. Actually, the same reversing turn effect is true of
mixed-compression inlets, but the
terminal normal shock is located deeper in, at the min area, in critical operation.
The “outside of the turn” surface behind the cowl lip is
curved back toward the flow direction,
so you usually do not have a normal shock boundary layer interaction
causing separation there. The “outside
of the turn” effect helps prevent that.
It is on the “inside of the turn” at the “lump” of the min area surface,
that shock-separation will likely
occur, as shown in Figure 11.
What has to be done to correct the separation is a more
drastic “fix” than surface porosity leading to an overboard dump. You need an actual slot from the surface
right behind the normal shock, leading
to a channel that dumps directly overboard, where the pressure is much lower. The thicker the boundary layer and separated
zone, the higher into the inlet flow
channel the rear edge of the slot must be,
in order to “slice off” most of the low energy air and divert it
overboard. This slot extends across the
inlet from cheek wall to cheek wall.
That is what the figure illustrates.
Figure 11 –Bleed Slot “Fix” at Inlet Min Area (Inlet Throat)
In supercritical operation,
the min area location is supersonic,
with the terminal shock located in the divergent diffuser channel
downstream. In accelerating supersonic
flow, pressure falls towards
downstream, and there is no normal shock
at the min area, so boundary layer
separation there is not a problem. The
layer is thinner, so that more of the
channel has higher-energy air, which
is why the height of the rear edge of the bleed slot is a tradeoff: you don’t want it too high during
supercritical operation, if you are
using the inlet for a ramjet engine.
With gas turbine, you are always
subcritical, so this issue does not
affect your bleed slot design.
In subcritical operation,
the shock system is unswallowed out in front of the cowl lip, and flow is everywhere subsonic through the
capture location and min area location.
Subsonic air has lower momentum and can make a turn into the bleed slot
easier. The min area slot will also act
to make the shock system easier to swallow and start the inlet. This is especially helpful with
mixed-compression inlets, to avoid the
need for overspeed, to enable
starting. And again, this applies more to the ramjet
application, which requires supercritical
operation, when gas turbine applications
do not.
flow separation problems in divergent diffuser
channels, whether a shock is present or
not
This is a phenomenon common to both all-external and
to mixed compression inlets, and also
to the pitot/normal shock inlets. All
have a divergent channel in which subsonic diffusion occurs, and in which the terminal normal shock
resides, when operating
supercritically. That occurs all the
time, if the inlet is used with a
ramjet, because ramjets require
supercritical operation for best results.
It is rare in inlet installations for gas turbines, because those almost always require
subcritical operation for best results.
The terminal normal shock interacts with an inherently-thick
boundary layer in a divergent channel,
which already favors flow separation because of the adverse pressure
gradient behind the shock in the subsonic flow.
Flow ahead of the shock is supersonic,
with a favorable pressure gradient,
that does not favor separation. Yet
these things interact, with the massive
flow separation actually moving the shock slightly upstream of where you would
think it might be, based on simple
compressible flow calculations. This
massive flow separation essentially destroys the subsonic diffusion you would
get out of the subsonic portion of the diffuser.
The bleed slot “fix” won’t work for the ramjet
situation, because the position of the
normal shock can be anywhere in the divergent channel, depending upon the exact value of the
operating supercritical pressure margin.
This position inherently varies “all over the map”. There is no shock subcritically, for gas turbine.
The empirical “fix” that works the best (shock-separation or
no) is a downstream flow-straightening grid,
whose drag exerts a slight extra backpressure upon what is happening in
the diffuser channel. This concept is shown in Figure 12. That slight effect usually reduces greatly
the sizes of the separation zones in the duct.
Not much can be done about the presence of lower-energy boundary layer
(“shear layer”) zones. But at least
there is some of the theoretical diffusion taking place, and the distribution of flow is somewhat more
even across the flow channel.
Figure 12 – The Flow-Straightening Grid “Fix” for Diffusers
Containing Shock Waves
It would take a sophisticated CFD code indeed, to properly predict all these phenomena, which is exactly why experimental
verification in the wind tunnel is still absolutely required for proper
inlet development.
Experience has shown that the grid shape has to be very
streamlined, in order to minimize the flow
separations behind it downstream. Actual
grids made of streamline aircraft tubing were the earliest implementations of
this concept. The later implementations
in ramjet applications have trended toward a plate with holes cut through
it, each contoured as a venturi
passage. If not truncated too soon, these passages lead to a uniform distribution
of very small wake zones downstream,
which close fairly quickly.
Another potential advantage of the venturi-passage
plate form of the grid is that when supercritical margins are very high, the total of the venturi throat areas can be
sized to choke, which then acts as a
shock-position limiter in the divergent diffuser channel. From that point, the rest of the supercritical shockdown
pressure loss takes place in the venturi exit passages, which distributes the flow disturbances as smaller items spread evenly across the
flow channel downstream.
ingestion of low-energy boundary layer or
separation-zone air that reduces pressure and streamtube recovery
This is a problem with side-mounted inlets well aft on a
vehicle, whether lateral side inlets, an inlet on the belly, or a top-mounted inlet on a dorsal
surface. Nose inlets and chin inlets do
not suffer this problem, because they
are located at the nose, without a
significant path length for boundary layer growth. For the side inlets, the long path available offers the
opportunity for a very significant boundary layer thickness, by the time the flow reaches the inlet(s). This would be as true for side-mounted
pitot/normal shock inlets, as it is for
side-mounted supersonic inlets (of either type).
In the case of a dorsal inlet on a vehicle at significant
positive angle of attack, the thickness
of the boundary layer can be greatly magnified by the flow separation, converting it into a shear layer. In the extreme case, the entire captured inlet airstream can be
low-energy (or even no-energy) air.
The “fix” for this is to stand the inlet off of the
surface, out in the higher-energy air
outside the boundary layer (and any separated wakes). The air between the vehicle surface and the
inlet has to be diverted around whatever stand-off structure is used, quite often because there is a turn elbow
needed to take the captured air inside the vehicle. This diversion structure is termed a
“boundary layer diverter”. The concept
is shown in Figure 13.
Figure 13 – The Inlet on Standoff with Boundary Layer
Diverter “Fix”
flow distortions and separations caused by turn elbows
That brings up flow separations and turbulent losses in flow-turning
elbows. These are necessary with
side-mounted inlets to bring the captured air into the vehicle. A milder form is the S-duct entry for the
center engine of a three-engine jet aircraft with tail-mounted engines (such as
the Boeing 727 or the Lockheed L-1011).
The S-duct is actually just two mild turns in sequence. The common feature here is that all elbow
turns must contain all-subsonic flow.
You cannot have a contained supersonic turn without a strong
shock wave that will essentially choke off the flow from going through the
elbow!
The older and more common “fix” is a set of turning vanes in
the corner of the turn elbow. This
derives from early subsonic wind tunnel practice for closed-circuit wind
tunnels. It was developed very early in
the 20th century. You may not
need very many corner vanes, if
eliminating the flow separation is more important than near-uniformity of flow
exiting the elbow. Ramjet elbows
typically might have only one such vane.
Such elbows have inner and outer surfaces that must follow gentle
curves.
You can actually make a flow-straightening grid do double
duty as a corner turning vane. You do
this by mounting it at the proper angle, in a miter-cut turn elbow. The outlet streams from the passages in the
grid are already inherently uniformly distributed, at the proper mounting angle, and are already pointed in exactly the right
direction. In addition, the miter-cut turn is much easier to
fabricate.
These concepts are shown in Figure 14. Note that in a very gentle S-duct, no vanes may be needed, but some boundary layer anti-separation
control can be afforded by strategically-placed turbulence generators. There is less total turbulence in a round
duct than a rectangular duct, because
there are no corners in which to “house” it.
Energy of circulation in turbulence is not the bulk energy of flow that
contributes to effective pressure rise.
You want to suppress such turbulence to the extent possible.
Figure 14 – The Turning Elbow Problem with Corner Vane “Fix”
variations
solving starting problems, especially with mixed-compression inlets
The first thing is lower demanded backpressure on the
inlet. This is equally true for
pitot/normal shock inlets, external
compression supersonic inlets, and
mixed-compression supersonic inlets.
In the ramjet application,
this is fairly easy to achieve,
because just before combustor ignition,
the air-only pressure in the engine chamber is substantially lower than
its burning pressure. Despite the
effects of combustor and air entry pressure drops, the backpressure demanded of the inlet is
lower air-only, than it is burning.
Usually this issue does not even arise in gas turbine
applications, because operation is always
subcritical. But if it does arise, blow-out doors near the inlet throat can
relieve the flow mismatch.
For the ramjet application,
the more-or-less standard “fixes” are cheekwall cutbacks to the inlet
throat, and direct overboard bleed slots
at the inlet throat. These are depicted in Figure 15.
Figure 15 – Inlet-Starting “Fixes” for Mixed Compression
What causes the problem is that the sonic max flow rate at
the min throat area is less massflow through the inlet, than it will scoop in critical operation. The only way to force the ingestion, without significant overspeed, is to bleed off the excess massflow that
the throat will not pass if only sonic. Cheekwall cutbacks and throat bleed
slots offer the means to do this. Once
the shock system is swallowed, and flow
adjacent to these overboard bleed passages is high-momentum supersonic, the air largely will not make the turn and
bleed overboard. Some small portion always
will make the turn, which is why the
cost you pay in mixed compression for easier start capability is reduced
pressure and area recovery.
The all-external compression inlet does not suffer from
this, because the capture point is the
min area. There is no supersonic
contracting channel in which a shock is unstable. But,
the cowl lip drag and basic inlet fairing drag are higher, because of the steeper cowl angle, and the larger-dimension inlet fairing.
So, the designer’s
choice is either a higher-drag inlet that is easy to start without overspeed, versus a lower-drag inlet that needs some
rather sophisticated, empirically
developed and verified “band-aids” to avoid overspeed starting. This matters more than you might think with
ramjet propulsion systems: it is very
easy to drive an inlet unstarted by a brief excursion to too-rich a mixture
that demands too-high an inlet delivered backpressure! Without the “band-aids”, the inlet never restarts, you lose most of your thrust to
spillage drag and low air ingestion effects,
and your vehicle “falls out of the sky”.
The Cooling Air Problem
This is an easy problem if very subsonic (incompressible)
flow is presumed. There is a pitot inlet
feeding a subsonic diffuser that in turn feeds air to, and through, a cooling radiator, such as for an engine. This could be an oil cooler, or the radiator for a liquid cooled
engine. Incompressible flow is presumed, but there is a significant temperature
difference in the flow through the radiator.
See Figure 16.
Therefore, the
density “after” is lower than the density “before”, due to its hotter temperature after passage
through the radiator. After the
radiator, the passage converges to an
exit area. The duct area Ad, exit area Ae,
and the before and after temperatures Tcool and Thot
are known to start the problem, as is
the oncoming air velocity Voo and incoming density.
The objective is to find the air velocity at the radiator
inlet face V, and the oncoming
streamtube area Aoo, so as to
determine limits on the size of the Ac that is appropriate for that particular
Ae. This has to be done for all the
possible values of Voo, so
that the largest Ac is the design choice.
Plus, the throughput massflow
must be compared to that for the heat transfer design of the radiator, to confirm or deny the appropriateness of the
value of Ae (which can be made variable if necessary).
Figure 16 – The Radiator Cooling Duct Problem
Note that the diffuser model here is different from that of
compressible and supersonic inlets. I
have expressed its pressure loss as a KD factor times the oncoming
freestream dynamic pressure. KD
would be a rather small fraction. The
radiator pressure loss has been expressed as a KR factor times the
dynamic pressure approaching the radiator inside the duct at Ad. KR
would be some modest-to-larger fraction,
to at most a number approaching unity. Both of these must come from
experimental test values.
The massflow continuity statement in the figure reflects the
change in density behind the radiator.
The Bernoulli equation is cast in terms of total pressure being static
plus dynamic. The pressure losses
subtract from total pressure. The
contraction from duct to exit is analyzed as lossless.
The “trick” was to recast the velocity at station d in terms
of the exit velocity, using the
continuity relation. Then one gathers
all the terms in Voo on one side,
and the terms in Ve on the other side,
of the total pressure relationship.
Factoring the velocities out and solving for exit velocity gets the
relation shown in the figure.
Then the mass continuity relationships define both duct
velocity V and oncoming streamtube area Aoo for you. The selected inlet area Ac must lie somewhere
between the streamtube area Aoo and the duct area Ad. Smaller is a lower-drag installation, because of spillage. But it must be largest of the various values
obtained across the flight envelope.
Sizing Inlets For Ramjets
The details of exactly how to size ramjet engine geometries
are out of scope here. That topic is
well-covered in ref. 3.
This discussion addresses more how the inlet characteristics influence those
choices.
Per the discussions above, for a high-speed design using a supersonic inlet, you want the oblique shocks to always fall on,
or inside, the cowl lip,
in order to maximize ingested airflow all across the vehicle flight
envelope. That means the inlet
design shock-on-lip speed must equal or exceed the very minimum flight
Mach number of your flight envelope.
(This shock-on-lip speed will never fall below about Mach 1.8 or
so, due to shock detachment limitations
associated with the external compression surfaces!) Some designs violate this min-Mach-at-shock-on-lip
constraint, but only very slightly, driven there by trading very inefficient use
of ramjet fuel for reduced required booster rocket size.
Best overall results across the flight envelope are usually
obtained if you size the ramjet geometry at its shock-on-lip Mach somewhere low
in the stratosphere, where the inlet air
total temperature is coldest. The inlet
operating PM will be minimum there, and usually
will only need to cover design dimensional tolerances and other statistical
variations (such as controlled fuel flow levels), at PM ~ 2-3%.
The best tradeoff
between higher thrust at higher throughput massflow, and still feasible max (and min) inlet
velocities approaching the flame stabilizer all across the flight envelope, will be near a maximum throat/combustor area ratio
not to exceed 65%, and an inlet
area/combustor area ratio in the 40-50% range.
The exact details depend upon the details of your flight envelope, as much as anything.
If you are sizing a low speed design with a pitot-normal shock inlet, the massflow-spillage tradeoff has a sort of an
inflection point where the ramjet nozzle throat first chokes, which is usually near Mach 1.1 in the real
world of actual component efficiencies.
That is the point at which combusted Mach number and inlet Mach number
maximize, posing the greatest flame
blowout risks.
Size at that Mach 1.1 speed for a high-subsonic-capable
design, or else size at your max Mach in
your flight envelope (for a design with a max speed well into in the low
supersonic range). The best tradeoff between higher thrust at higher
throughput massflow, and still-feasible
max inlet velocities approaching the flame stabilizer, will be near a nozzle throat/combustor area
ratio not to exceed 65% (as long as the throat is choked), and inlet area/combustor area ratio
40-50%. Higher thrusts in the low
supersonic range occur with higher sizepoint Mach than ~1.1, at the cost of slightly-lowered subsonic
thrusts.
Sizing Inlets For Gas Turbines
There are basically two types of engines and two types of
inlets to consider, but there are only 3
possible combinations of these. The two
types of engines are (1) subsonic non-afterburning gas turbine engines for
transport aircraft, with a bypass ratio
between 0 and 6, and (2) supersonic afterburning
gas turbine engines for supersonic fighters and supersonic-capable
bombers, with bypass ratios between 0
and not quite 1, and a max flight Mach
number under about 2.5.
The two types of inlets are (1) pitot/normal shock for max
flight speeds under about Mach 1.4, and
(2) supersonic inlets with external compression features for flight speeds up
to about Mach 2.5. There were a very few aircraft indeed, that have flown up to and a little past Mach
3, and they were very specialized, unique designs, both in their engine designs, and in their inlet designs. That is out of scope for this article.
The three combinations to consider are:
(1) subsonic/non-afterburning engines with pitot inlets in
only subsonic flight,
(2) supersonic afterburning engines with pitot/normal shock
inlets up to at most Mach 1.4, and
(3) supersonic afterburning engines with supersonic inlets
for flight up to about Mach 2.5.
For all these combinations,
the engine sets the air massflow through the inlet, and the inlet must adapt to supply the
demanded amount. The inlet must
be sized to successfully deliver the maximum air massflow that is demanded
anywhere in the aircraft flight envelope (at max engine thrust setting), and must spill the excess air at all the
other flight conditions.
In Ref.
4, Raymer has some crude but
useful empirical models for estimating the max engine air massflow demand as a
function of the design maximum flight Mach number and the engine diameter. These models are in his Chapter 10, as his Figure 10.13, plus some very useful modelling equations for
aircraft design. The variation of max airflow versus flight Mach number is
quite small subsonic, and quite strong
in supersonic flight.
Subsonic Non-Afterburning Engines / Pitot Inlets
Estimate the engine air massflow using the equation in
Raymer figure 10.13 (wa = 0.183 Di2), where wa is air flow in lbm/sec
and Di is the inlet face diameter in inches. Adjust this by adding the various air bleed
demands for nonpropulsive use, per
Raymer table 10.2, bearing in mind that
many aircraft have separate flush inlets for various cooling air flows, but also that cabin air-conditioning air is
often compressor bleed air. Then at the aircraft
max design flight Mach number M, use the
plot of ref. 4 Raymer
figure 10.13 to determine the appropriate value of wa/Ac
(lbm/sec-in2). Divide the
total demanded max air massflow by the wa/Ac value to
determine your rough (initial) estimate of the inlet capture area Ac, sq.in. This is not good enough for an
actual aircraft layout drawing.
At the max flight speed design point, Raymer suggests letting half the deceleration
be done inside the subsonic diffuser,
and half externally in the approaching all-subsonic streamtube. That puts the capture station Mach number at
the arithmetic average of the max flight Mach number (usually near 0.8 to 0.9)
and the max allowable engine inlet face diameter Mach number (usually in the
0.3 to 0.4 range). Raymer’s equations
10.16 and 10.17 are useful for this process.
His equation 10.16 is the ratio of area ratios (to the sonic
area A*) AC/Aengine = (AC/A*)/(Aengine/A*), where each area ratio is determined with the
compressible streamtube relation,
expressed in 10.17 as evaluated at air’s specific heat ratio of 1.4: A/A* = (1/M)[(1 + 0.2 M2)/1.2]3.
The max design absolute demanded massflow then sets the
capture area via wa = gc ρ∞ V∞
A∞
= gc ρengine Vengine Aengine =
gc ρC VC AC, remembering that it is compressible flow! The
units Raymer uses are lbm/sec for massflow,
lbm/cu.ft for density, ft/sec for
velocity, and sq.ft for area, with gc = 32.174 ft-lbm/lb-sec2. This more complicated calculation of AC
supersedes the initial estimate described in the previous paragraph. It
is just a better estimate.
One needs to determine inlet massflow capability at climb
and approach speeds relative to full thrust (max airflow) conditions. The engine air massflow demand needs to fall
within the inlet capability (capture station Mach < or at most = flight
Mach) at those speeds. If it does
not, one should first try reducing the sizing
value of the capture station Mach number (at max flight speed design) below the
average of that max speed and the engine inlet face speed. Reduce it until full thrust can be used at
climb and approach speeds. That puts
more of the deceleration on the external streamtube, and less on the diffuser internal divergence.
At zero speed before starting the takeoff roll, the huge shortfall in massflow capability
must be made up by spring-loaded blow-in doors on the inlet diffuser duct.
Capture station massflow capability is limited to significantly-subsonic flow
(something in the Mach 0.8 to 0.9 range at most). Being the min area in these
circumstances, it is the natural
chokepoint for flow.
Supersonic Afterburning Engines / Pitot/Normal Shock
Inlets
This is done just about the same way, using the same tools, as the subsonic non-afterburning pitot
case, except for the max flight Mach design
point value used in figuring cowl entry Mach. Instead of using the max flight design Mach
number, one uses the subsonic Mach
number behind the normal shock, as evaluated
at the flight design Mach, using a
normal shock table such as that in ref. 1. This change
applies at all supersonic speeds, and is
irrelevant at all subsonic speeds.
Supersonic Afterburning Engines / Supersonic Inlets
A number of flight conditions must be investigated to
determine which of them actually sizes the swept-out inlet capture area AC. This might be the max supersonic flight Mach
number, the subsonic cruise Mach
condition, or even the low Mach number at
takeoff climb speed. There is a need for
max thrust at all of these conditions.
Cruise thrust is actually far lower,
but what do you do if you suddenly need to accelerate? You go to max thrust setting! The inlet AC must be large
enough to supply the worst of these conditions,
and it will have to spill the excess air at all the other flight
conditions.
The inlet shock-on-lip Mach number is usually set 0.1 to
0.2 Mach higher than the actual max flight Mach number. This is to cover random variations and
overspeed events, and also to allow room
for the unswallowed terminal normal shock wave, that is just barely in front of the cowl lip, for negligible spillage operation, if this really is the max airflow delivery
condition.
These supersonic inlet/supersonic afterburning engine applications
typically have larger bleed air requirements by far, so be sure to add these flows to the engine
propulsive airflow demand at max thrust.
The inlet must supply the total of all of these together.
These designs will usually have inlet bleed (or “inlet
dump”) doors on the diffuser section before the engine inlet face. These blow open and dump inlet-captured air
massflow that the engine cannot use, for
a somewhat lower-drag option than simple massive subcritical spillage at the
cowl lip. These are distinct from the
similarly-located blow-in doors that let the engine suck in extra air for
takeoff. They are also quite distinct
from the engine bleeds, which affect
thrust as well as drag.
What you will need to evaluate these air massflow
trades are (1) a model for engine air massflow demand, (2) a model for inlet critical total pressure
recovery all the way from subsonic to a max supersonic speed, (3) a model for inlet critical streamtube
area recovery all the way from subsonic to a max supersonic speed, and (4) a model of inlet spillage drag.
For the engine airflow model, I’d use Raymer’s model at max thrust, and ratio that by the rpm for reduced thrust
settings. You will also need to ratio it
by the increased density at the engine face,
as provided by the inlet. The
inlet critical pressure recovery and spillage margin tell you what the total
pressure is at the engine inlet face.
The total temperature is that of the freestream. Use the inlet face Mach number to reduce those
total values to static values. Then
compute the density at the inlet face from these, using the ideal gas equation of state. The ratio of that, to ambient air density (at that altitude), is a multiplier that increases the air
massflow demanded by the engine at that thrust setting.
The inlet critical pressure ratio, and its critical streamtube area ratio are
just empirical data versus primarily Mach number. Spillage drag is based on fluid flow
fundamentals (momentum of the stream).
The critical pressure recovery will be very high from
subsonic to Mach 1, and still fairly high
up to the speed at which the oblique shocks first attach to the initial
compression surface. That is in the
vicinity of Mach 1.5. Then they trend
down increasingly rapidly, as Mach
number increases into the supersonic range,
usually with a slope break at shock-on-lip. For an inlet that
spills, actual pressure recovery will
always be at the critical value (or perhaps a small percentage better in some
cases).
The inlet critical streamtube recovery ratio (A∞/AC)critical
will be quite low from subsonic up to around the Mach number at which the
oblique shock attaches to the initial compression surface. From there it more-or-less linearly increases
to a high value at the design shock-on-lip Mach number. Above that,
it is usually approximately constant at that high value. The actual streamtube ratio with spillage is
less than the critical value by the factor (1-SM). Inlet ingested air massflow is:
wa = ρ∞
V∞
AC (A∞/AC) = ρ∞
V∞
AC (A∞/AC)critical (1 – SM)
So, as long as engine
demand is under critical inlet ingestion capability (SM = 0), your design is adequate. If not,
you need a bigger AC,
or you need blow-in doors. Or
both.
The spillage drag model is actually fairly
simple. Twice the spillage margin SM multiplied
by (A∞/AC)critical
is the spillage drag coefficient. This coefficient
multiplied by AC, and by the
freestream dynamic pressure, is the
spillage drag. Freestream dynamic
pressure can be calculated in either of two equivalent ways, whichever is more convenient: q = 0.5 gc ρ V2 = 0.7 P M2. For q in lb/sq.ft, use gc = 32.174, ρ in lbm/cu.ft, and V in ft/sec. Or use P in lb/sq.ft = 144 times P in
lb/sq.in in the Mach form.
Some fraction of the ram drag of airflow dumped from an
inlet bleed door is a good estimate for that kind of bleed drag. Figure ram drag as w V / gc for
massflow w in lbm/sec, V in ft/sec, and using gc = 32.174.
For takeoff from a standing start, you will simply have to use enough blow-in
doors on the diffuser to allow-in the required air massflow to enable max
thrust. There is no way around this
requirement, any more than there was
with a pitot inlet on a subsonic non-afterburning engine. These are quite separate from the inlet bleed
doors that let air massflow out.
If you get the impression that this is not an easy set of
calculations to make, then you got the
right impression. This is not a job for
amateurs!
References
#1. National Advisory Council for Aeronautics, “Report
1135 Equations, Tables, and Charts for Compressible Flow”, Ames Research Staff, 1953.
#2. J. Seddon and E. L. Goldsmith, “Intake Aerodynamics”, AIAA Education Series, 1985
#3. G. W. Johnson, “A
Practical Guide to Ramjet Propulsion”,
yet to be published, copyrighted
2017.
#4. D. P. Raymer,
“Aircraft Design: A Conceptual
Approach”, AIAA Education Series, 1989.
Final Comments
This article provides a sense of the basic physics that
governs inlet operation, whether for gas
turbine or for ramjet applications, or
even incompressible cooling flow designs.
But, I have not given the
detailed analysis techniques necessary to do real design analysis, in this article!
There are three chapters in my as-yet unpublished
ramjet book (ref. 3) that also deal with these physics, just more-or-less restricted to the ramjet
application only. The book includes some
rough estimating tools for “typical” supersonic inlet data curves, but it also cautions (just as I do here) that
these data are empirical, and must be
generated, or at least verified, by actual supersonic wind tunnel testing.
If the reader has no background in compressible flow
analysis, then he has no business
trying to turn what I say in this article into his own inlet designs! Such instruction in compressible flow
analysis I considered to be out-of-scope here.
It fills whole textbooks.
There are many in the literature.
Nothing I have put into this article (or the ramjet book)
applies to flight speeds high enough to cause significant air dissociation in
the captured subsonic flow, or in the
engine combustor. That happens at about Mach
6 to 7 in the stratosphere, and closer
to Mach 5 on the surface, or higher
up. You cannot use standard compressible
flow analysis at those conditions, because
it really isn’t air anymore, it
is plasma. The ideal gas assumption
underlying compressible flow analysis has totally broken down!
It should be obvious that I know a lot about aerodynamics
and compressible flow analysis. I do
consult professionally in these topics.
I may be retired, but I still
know the stuff. I just no longer sign
drawings.
-----------------------------
Update 11-12-2020:
Oops, I forgot to
clarify exactly how all those inlet characteristic data are really used.
The max possible delivered total pressure is Pt2crit, obtained from the freestream total pressure Ptoo
as:
Pt2crit
= Ptoo (Pt2/Ptoo)crit = delivered Pt2
if subcritical
The actual delivered total pressure Pt2 really is
this critical value if in spillage mode,
or it is less than this critical value if in supercritical mode:
Pt2
= Pt2crit (1 - PM) if supercritical
The max possible ingestible air massflow is the critical
massflow, computed from the critical
ratio, inlet capture area, and freestream conditions as:
wacrit
= ρoo
Voo Ac (Aoo/Ac)crit =
delivered wa if supercritical
The actual captured air massflow is this value if
supercritical, or else it is less than
this value if subcritical:
wa = ρoo
Voo Ac (Aoo/Ac)crit (1 -
SM) if subcritical
The additive drag is calculated from a coefficient whose
area basis is Ac. It can
include bow wave effects and skin friction forces on the entering airstream (if
any), and it can include the drag
effects of inherently-spilled air (even supercritically) for operation below
shock-on-lip speed, or both.
Dadd
= CDadd qoo Ac
same whether supercritical or subcritical
In supersonic flight,
the spillage drag applies only when operation is subcritical and
the shock system is not swallowed. In
subsonic flight with a pitot inlet,
spillage is inherent, being the
only way the inlet can adjust to backpressure demand. That spillage drag is basically the ram drag
of the spilled air:
Dspil
= CDspil qoo Ac nonzero only if subcritical
where CDspil
= 2 SM (Aoo/Ac)crit nonzero only if subcritical
A reminder: SM and PM
cannot both be nonzero. The inlet must
either operate supercritically or subcritically. They can both be zero, but only when the inlet is operating exactly
at critical conditions.
What you do with the additive and spillage drags depends
upon what thrust-drag accounting system you use. The basic net jet thrust of the inlet/engine/nozzle
system is the nozzle thrust force minus the ram drag of the captured airstream:
Fnet
jet = Fnozzle – wa Voo/gc
In the “net jet” accounting system, the thrust is the net jet thrust, and both the additive and spillage drags must
be added to the basic airframe drag.
Thrust =
Fnet jet
Drag =
airframe drag + additive drag + spillage drag
In the “installed thrust” accounting system, the thrust is the net jet thrust minus both
the additive and spillage drags, and the
drag is just the basic airframe drag.
Thrust =
Fnet jet – additive drag – spillage drag
Drag =
airframe drag
Because the fuel flow rates are what they are, regardless of the accounting system, you will get different specific impulse or
thrust specific fuel consumption values because the thrust values depend
upon the accounting system.
Propulsion specialists often like to work in “net jet”
accounting, because it is often easier
to deal with, in that particular area of
specialty. Most everybody else prefers “installed
thrust” accounting all the time. It is
usually the default among performance and trajectory specialists.
-----------------------------
Update 2-15-2022: Here is a brief explanation of why the very same supersonic inlet components work entirely differently, when used with a gas turbine engine than they do with a ramjet. The gas turbine engine is basically an air pump that sets the mass flow through the inlet. The ramjet does no such thing: whatever is swept out by the inlet cowl lip is all the massflow that can ever go through the inlet.
The gas turbine engine acting as a pump can pull the inlet diffuser down to a suction, below local atmospheric. The designer can use that suction to open blow-in doors to increase the airflow reaching the outlet of the diffuser duct at the engine compressor face. That effect is precisely how one can put supersonic inlet features on the inlets of a jet aircraft, and still get good performance at low transonic and subsonic speeds! Without that extra air, the engine could not make adequate thrust, because it otherwise sucks all the pressures, throughout the engine and nozzle, down too low.
The ramjet just simply cannot do any of that (it is not a pump)! If you operate a supersonic inlet attached to a ramjet at too low a speed, the shock wave system not only unswallows, it actually detaches from the compression ramp or spike, and jumps out in front as a bow wave (that also happens with the gas turbine, just to a lesser extent).
The flow from cowl lip all through the inlet is subsonic, for both engine types, when that happens. That subsonic flow sees the compression ramp features as an obstruction, and simply diverts around the cowl lip outside the inlet entirely! Very little air goes through the inlet at all. Without inlet-scooped air, first and foremost, there is no thrust at all, with a ramjet. But there is also absolutely no pumping effect to use for diffuser suction, to open blow-in doors, to get any extra air.
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