Fundamentals of Inlets

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.

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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.  

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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 M_att 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 M_att),  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 M_att 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 P_t2crit to free-stream total pressure P_too.  That would be PR_crit = P_t2crit/P_too.  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 = A_oo/A_C.  It has a critical-inlet value AR_crit.  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 AR_crit vs M,  parametric on AOA and roll.

Additive drag is usually book-kept as a drag coefficient increment C_Dadd 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 – P_t2/P_t2crit.  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/AR_crit.  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 PR_crit,  AR_crit,  and C_Dadd.  The flight speed and altitude,  for any given model of a day (hot,  cold,  or other),  determine the values of P_too and dynamic pressure q_oo.  The flight condition total pressure thus allows one to compute P_t2crit.  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 P_t2 exceeds P_t2crit 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 AR_crit * (1-SM),  so that A_oo = Ac * AR,  and P_t2 is P_t2crit (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 = w_air V_oo / g_c.  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 20^th 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 T_cool and T_hot are known to start the problem,  as is the oncoming air velocity V_oo and incoming density.

The objective is to find the air velocity at the radiator inlet face V,  and the oncoming streamtube area A_oo,  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 V_oo,  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 K_D factor times the oncoming freestream dynamic pressure.  K_D would be a rather small fraction.  The radiator pressure loss has been expressed as a K_R factor times the dynamic pressure approaching the radiator inside the duct at Ad. K_R 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 V_oo 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 A_oo for you.  The selected inlet area Ac must lie somewhere between the streamtube area A_oo 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 (w_a = 0.183 D_i²),  where w_a is air flow in lbm/sec and D_i 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 w_a/A_c (lbm/sec-in²).  Divide the total demanded max air massflow by the w_a/A_c value to determine your rough (initial) estimate of the inlet capture area A_c,  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*) A_C/A_engine = (A_C/A*)/(A_engine/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 w_a = g_c ρ_∞ V_∞ A_∞ = g_c ρ_engine V_engine A_engine = g_c ρ_C V_C A_C,  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 g_c = 32.174 ft-lbm/lb-sec².  This more complicated calculation of A_C 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 A_C.  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 A_C 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_∞/A_C)_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:   

w_a   =   ρ_∞ V_∞ A_C (A_∞/A_C)   =   ρ_∞ V_∞ A_C  (A_∞/A_C)_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 A_C,   or you need blow-in doors.  Or both.

The spillage drag model is actually fairly simple.  Twice the spillage margin SM multiplied by (A_∞/A_C)critical is the spillage drag coefficient.  This coefficient multiplied by A_C,  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 g_c ρ V² = 0.7 P M².  For q in lb/sq.ft,  use g_c = 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 / g_c for massflow w in lbm/sec,  V in ft/sec,  and using g_c = 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.

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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 P_t2crit,  obtained from the freestream total pressure P_too as:

               P_t2crit = Pt_oo (P_t2/P_too)_crit = delivered P_t2 if subcritical

The actual delivered total pressure P_t2 really is this critical value if in spillage mode,  or it is less than this critical value if in supercritical mode:

               P_t2 = P_t2crit (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:

               wa_crit = ρ_oo V_oo A_c (A_oo/A_c)_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 = ρ_oo V_oo A_c (A_oo/A_c)_crit (1 - SM) if subcritical

The additive drag is calculated from a coefficient whose area basis is A_c.  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.

               D_add = C_Dadd q_oo A_c  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:

               D_spil = C_Dspil q_oo A_c  nonzero only if subcritical

               where C_Dspil = 2 SM (A_oo/A_c)_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:

               F_{net jet} = F_nozzle – wa V_oo/g_c 

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 = F_{net 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 = F_{net 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.

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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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