Supersonic Inlets for Ramjets

This is a more detailed version of an earlier posting on this topic.  That one,  "Inlets Capable of Supersonic Flight Use" dated 21 August 2026 below,  presumed the reader was an amateur.  This one presumes the reader actually knows something about compressible flow with shock waves,  and about how inlet performance data are usually book-kept and used.  

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Supersonic Inlets for Ramjets                                    

Supersonic inlets have spikes or ramps as external compression features that act before the airstream is captured at the cowl lip,  going to internal flow from there.  Spikes go with axisymmetric inlets that have circular cross sections,  ramps with two-dimensional (2-D) inlets that have a rectangular cross section.  The chin inlet is a variation on the nose-mounted axisymmetric spike inlet,  being just a sector of the full circle,  and using part of the vehicle nose as its external compression spike surface.   And,  there are a variety of side-mounted inlets,  both round or rectangular in cross section,  or even half-round.

The external compression surfaces create oblique shock waves with a flow deflection off of the axial direction.  The flow behind an oblique shock wave is still supersonic,  and the total pressure loss across the wave is quite modest.  There is at least one,  and most often,  two (or more),  such oblique waves in a typical supersonic inlet.  The last (“terminal”) shock wave is a normal shock,  perpendicular to the flow,  creating subsonic flow behind the wave,  and with a much larger total pressure loss across the wave,  when compared to an oblique shock at the same Mach number.  See Ref.1 for equations,  tables,  and charts.

By choosing the deflection angles, their geometric locations,  and the number of waves correctly,  about the same total pressure loss ratio across each wave can be had.  The sum of these losses is less than the loss across a single normal shock at the full flight Mach number (or even more if supercritical),  as in a pitot/normal shock inlet.

All of these inlets have a minimum cross section area called the “inlet throat”,  located at,  or just downstream of,  the inlet capture location into internal flow.  They all have divergent diffuser passages downstream of that throat,  where subsonic flow is finally achieved.  With the throat at the cowl lip,  there is no internal flow contraction,  these are called “all-external compression” inlets.  With the throat downstream of the cowl lip,  there is internal flow contraction for more compression,  these are called “mixed-compression” inlets.  The local Mach number at this inlet throat is always still somewhat supersonic.  There are no feasible all-internal compression inlets,  not with any technology known today!  In-line diffusers,  yes.   Inlets,  no. 

All of these inlets share similar-looking characteristic curve shapes for what is called the critical total pressure recovery ratio PRCR,  the critical captured streamtube area ratio ARCR,  and the additive drag coefficient C_Dadd.  The item of greatest influence on the shapes of these curves is the shock-on-lip Mach number,  that being the speed (measured as flight Mach number) at which the oblique shock waves created by the external compression surfaces are focused to fall exactly upon the cowl lip,  by deliberate design.

There is a modest slope change from below shock-on-lip to faster than shock-on-lip in the PRCR curve,  yet it still decreases from high values to low values as flight speed increases.  There is a sharp slope change at shock-on-lip speed in the ARCR curve.  Below shock-on-lip,  the recovered streamtube is much smaller than what is theoretically swept out by the cowl lip.  Above shock on lip,  the recovered streamtube is very close to that theoretical swept-out value.  What that really means is that the inlet can ingest lots of air flow above shock-on-lip speed,  and very little air flow below shock-on-lip speed.  The first priority by far for ramjet application is maximum possible ingested air flow rate!

There is also a sharp slope change in the C_Dadd curve at shock -on-lip speed.  C_Dadd is defined as the drag coefficient of “additive drag”,  based on the swept-out inlet capture area A_C.  The higher values at the lower speeds are the effect of minimally-decelerated still-supersonic air diverted past the cowl lip by the streamtube geometry below shock-on-lip.  This is because the local streamlines behind the oblique shock waves must run parallel to the external compression surfaces that created the waves.  Above shock-on-lip speed,  the C_Dadd values are low,  and in the case of nose-mounted inlets,  often zero.

All of these behaviors and characteristics are illustrated in Figure 1.

Figure 1 – Basic Characteristics of Supersonic Inlets with External Compression

These components must be used differently with turbojets versus ramjets,  because of the quite-different characteristics of those engines.  The turbojet is almost like a positive-displacement pump,  in that,  at any given rotation speed and air density,  there is only one engine air massflow!  Rotation speed is determined by the throttle setting.  The turbojet inlet must deliver the exact air flow rate that the engine demands,  regardless of the flight speed and the inlet ARCR recovery characteristic curve!  To do that,  it must operate “subcritically”,  with the terminal shock “unswallowed”,  and thus located on the external compression surface in front of the cowl lip,  as shown in Figure 2.  There is a sometimes-large spillage of subsonic air around the cowl lip,  in order to deliver the “right” ingested air massflow rate to the turbojet engine.

In contrast,  the ramjet acts like a simple obstruction,  producing a pressure at the entrance to the min obstruction area that depends only on the total air and fuel massflow rates fed to the engine,  and the combusted temperature and properties corresponding to them.  The thrust of the ramjet depends most strongly on the amount of air flow ingested!  That means the inlet must be operated supercritically (with a swallowed terminal shock),  for ingesting the maximum airflow!  The more air,  the higher the combustor pressure,  and the higher the combustor pressure,  the greater that nozzle thrust force!  This is also shown in Figure 2.

Figure 2 – The Same Components Must Be Used Differently for Turbojets and Ramjets

Exactly how the ramjet engine-inlet balance works is shown in Figure 3,  for which the most important behavior is supercritical versus subcritical operation.  The inlet characteristics are represented by the cross plot of PRCR versus ARCR at any one flight Mach number,  very similar to the one shown in Figure 1,  just with some extra details added here.  The corresponding depictions of the inlet flow field and the distribution of conditions through the engine are shown in the figure,  one each for subcritical operation,  operation exactly at the critical inlet point,  and for supercritical operation.  As already discussed,  best ramjet performance obtains when ingested air flow rate is maximized (supercritical inlet). 

Note that the ramjet engine determines the total pressure at the nozzle entrance P_t4 from the flow rates,  and it determines the pressure drops across hardware items and flame zones,  so that it determines the actual operating total pressure P_t2 at the inlet diffuser outlet!  The inlet determines only the maximum-possible diffuser total pressure P_t2CR! 

Only at critical are these equal!  Supercritically,  P_t2CR > P_t2,  the difference measured by the supercritical pressure margin PM.  If the engine-demanded P_t2 is too high,  the inlet will instead operate subcritically,  and spill just enough air to make P_t2 = P_t2CR.  The amount of this air spillage is measured by the subcritical spillage margin SM.

Figure 3 – Exactly How the Engine-Inlet Balance Works with Ramjet

So,  for a supersonic inlet,  there are basically 3 modes related to shock-on-lip speed,  and 3 modes related to inlet criticality.  That’s a total of 9 possible modes and flow patterns,  as depicted in Figure 4.  The critical-inlet condition right at shock-on-lip speed is the design point for the inlet (but not necessarily for the engine-inlet system in the ramjet).  All supersonic inlets,  regardless of the application,  share these 9 behaviors.  Only some of them are appropriate for ramjet,  and only some of them for turbojet.  In the upper left illustration,  both the streamtube geometry is reduced,  and there is subcritical spillage.

Figure 4 – The Nine Behaviors Common to All Supersonic Inlets

The differences between all-external compression inlets and mixed-compression inlets are shown in Figure 5.  The mixed-compression inlet has both external compression by oblique shock waves,  and internal compression by both passage contraction and another oblique shock wave.  The all-external compression inlet has no internal compression,  only an always-divergent internal passage!  It only has the external oblique shock waves,  and its min-area throat is right at the cowl lip station.  At inlet design Mach,  both types of inlet have a terminal normal shock very close to the min-area throat, wherever it is.

The all-external compression inlet has a higher cowl lip drag because of its strongly-curved shape toward the spike or ramp.  But it has no internal contraction,  and so is easy to start (swallow the shock system),  because that geometry entirely avoids the Kantrowitz-Donaldson conundrum (see Ref. 2) peculiar to internal supersonic contractions!  It helps if the demanded backpressure at start is also low,  at values more like large supercritical margin. 

Without any throat bleed means,  the initially-subcritical mixed compression inlet throat area simply cannot subsonically pass the full swept-out supersonic massflow rate (at its design speed),  thus requiring very serious overspeed above design Mach,  in order to start!  Even so,  in order to start at overspeed,  the demanded backpressure must also be low,  corresponding to levels seen at high supercritical pressure margin. 

This difficulty with this approach is that once the inlet is started,  if you slow down to,  or below,  design speed,  and also happen to experience inlet unstart for any reason whatsoever,  you cannot re-start the inlet!  There is no restart,  unless you have some other means to re-accelerate the vehicle back up to the feasible starting overspeed.

Throat bleeds can be through the lateral cowl cheekwalls,  as cutback slots in the 2-D variants,  or by slots cut in the axisymmetric cowl wall tube from ahead of,  to just past,  the throat.  It is also possible to put a throat bleed slot in the centerbody spike or ramp,  leading overboard directly in the 2-D variant,  and leading outward through the centerbody support struts,  for the axisymmetric variant.

Figure 5 – Characteristics for All-External Vs Mixed Compression Inlets

The supersonic inlet has a swept-out capture area defined by the cowl lip and the bottom or centerline of the ramp or spike external compression surface.  When viewed from ahead,  it is obvious to even-the-untrained-eye that the ramp or spike features present an obstruction to the oncoming flow.  If the oblique shock waves are formed properly by the external compression surfaces,  this blockage is not a problem,  as those waves divert the flow parallel to the external compression surface,  and compress it to a smaller size that actually will fit through the available passage area at the inlet throat station!  This is illustrated in the right-hand image of Figure 6.  This is still true,  even if there is some air not ingestible due to the streamtube and shockwave geometry,  while traveling at somewhat lower than the supersonic shock-on-lip speed.

However,  as speed decreases further,  there comes a point where the external compression surfaces cannot maintain the existence of oblique shock waves that are attached to the surface.  The shock system detaches,  and becomes a curved bow wave,  out in front of the external compression features at some standoff distance,  with locally-normal shock characteristics.  This is shown in the middle illustration in Figure 6.  A nominal speed at which this occurs is shown in the figure,  but the actual value depends upon the angles off axial of the compression surfaces.  The reader is referred to the ramp and cone charts for oblique shock waves in NACA 1135 for further details. 

This behavior persists at some bow shock standoff distance that varies,  until flight speed is subsonic,  and the bow shock disappears entirely.  Flow ingestion is still very poor due to the presence of the obstruction that the subsonic approaching flow can “sense”.  This is shown in the left illustration in Figure 6.

Figure 6 – Low-Speed Behaviors Seen Only in Turbojet Installations

The only “fix” by means of which to increase air ingestion is the spring-loaded blow-in door,  shown in the bottom illustration of Figure 6.  This only works if a suction can be pulled in the inlet duct,  and only a gas turbine can do that.  Ramjet cannot.

Various inlet integrations are possible.  For modern high-speed combustor layouts,  some options are shown in Figure 7.  The old stovepipe-engine designs with V-gutter flameholders,  used in the 1950’s,  were suitable only as engine nacelles added to the basic vehicle,  as in the IM-99 Bomarc interceptor missiles.  In the stovepipe designs,  there is no place inside the engine structure,  in which to package anything,  even the fuel.  And these were limited by materials survivability issues,  to flight speeds under about Mach 3. 

The nose inlet offers the lowest vehicle drag,  with the air transfer duct either coaxial along the centerline as illustrated,  or alternatively,  as an annular passage about a centerbody containing the equipment and fuel. 

The chin inlet is sort-of like a circumferential segment of a nose inlet,  using part of the vehicle nose as its external compression spike.  But,  to achieve shock-on-lip at a reasonable flight Mach number,  the nose half-angle may have to be steeper (more drag) than would otherwise be desired for low drag but high-volume packaging.  Even so,  it still offers a very low-drag vehicle design,  plus it is well-adaptable to noncircular vehicle cross sections that have the flat bottoms by which to be “wave riders” for higher lift.  ASALM-PTV was one of these,  and reached Mach 6,  although that was by accident.  It was designed for cruise at Mach 4,  limited more by materials survival than by thrust-over-drag.

The side entry inlets offer packaging in the front of the vehicle without any geometric interference from the air transfer ducting.  They do increase the vehicle drag,  so that top speeds usually fall in the Mach 3 to 4 range,  limited by thrust-over-drag.  These can take a variety of forms,  leading to a variety of different sudden-dump flameholding geometries. 

The single side inlet offers no added lift capacity,  only just the inlet fairing drags,  which are cowl lip,  skin drag,  aft boat-tail pressure drag,  and aft face base drag.  All of these are traditionally charged against airframe drag,  no matter the thrust-drag accounting scheme.  The inlet needs to go on the belly,  so that at lifting angles of attack,  the inlet flow is not blocked by the vehicle body.  Remember,  ramjets cannot suck air like a pump,  only gas turbines can!

The two-inlet configurations are either symmetrically placed on the two sides,  or asymmetrically placed on those two sides,  but at or below the horizontal.  Both of these geometries offer added lift from the inlet fairings,  which can replace the presence of the forward wing-fins of missiles.   You really do not want inlets anywhere around the dorsal side,  as they have a hard time ingesting air:  the missile body blocks the oncoming stream to one extent or another.

Three inlets are possible,  but that configuration has not been used.  It always suffers from one or two inlets suffering the “dorsal inlet” problem,  to one extent or another. 

Four inlets might do better in the X-configuration,  but in some cases having the fins in the X-configuration might be more important to the vehicle designer.  There are missiles out there with X-configuration inlets,  like the “Sunburn” anti-ship missile.  There was another with the fins in X,  and the inlets in cross-configuration,  as shown in Figure 7.  That was the SA-6 “Gainful” surface to air missile.  Both work.

Any of the illustrated geometries are able to serve as flameholders for LFRJ engines,  but not the GG-fed RJ’s that do not use hypergolic magnesium fuels!  The asymmetric two-inlet configuration is the most successful for GG-fed RJ using fuels that do require flameholding.

There is a preferred location zone for the capture features of any of the side inlet configurations.  This is about 4 to 6 diameters aft of the nose.  Too far forward,  and the flow about the body has not yet re-stabilized from the disturbances presented by that nose.  That puts unrepeatability into the inlet capture characteristics.  Too far aft,  and you get into inlet and fin interference,  plus increased additive drag along the greater wetted length.

Figure 7 – Various Inlet Integrations Are Possible

The SFRJ to date has shown a preference for nose or chin inlet designs,  although side inlets could also feed the same air plenum just ahead of the fuel chamber.  The real design restriction would be,  not being forced to put the side inlet capture features too far forward. 

A very important aspect of the design of ramjets or gas turbine-powered vehicles is the flight envelope.  This is a region in altitude-speed space in which the inlet/engine combination is expected to operate,  and is not expected to operate outside of it.  This kind of thing is illustrated in Figure 8.  Bear in mind that the minimum ramjet flight speed is an appropriately-high ramjet takeover speed,  while the minimum gas turbine speed is literally zero on the surface up to some elevation,  and below vehicle stall speed above it.

As already discussed above,  the minimum takeover speed of a ramjet is usually where you set the design shock-on-lip Mach number for its inlet.  Somewhere along that low-speed boundary in the flight envelope,  is your ramjet design point!  At design,  and at the max mixture ratio expected,  you need enough supercritical pressure margin PM at engine design,  to maintain a PM > 0 all along and near that inner boundary.  PM will increase as you speed up across the envelope toward its outer boundary,  and it will also increase as you lean back mixture ratio in the ramjet engine.  Ambient air temperature also affects it.

Most gas turbine installations do not use actual supersonic inlets,  they use the simpler pitot/normal shock inlets,  because they never fly more than about Mach 2.5.  Pitot/normal shock inlets are a separate topic,  not discussed here.  They operate down to very subsonic speeds.

But for a really high-speed gas turbine installation,  not only are the same inlet components used differently,  but the design approach itself is entirely different!  Right up to the maximum possible speed,  you want the oblique shocks to fall ahead of,  or at worst on,  the cowl lip.  (That is utterly different than what you do designing for ramjet!)

That high-speed-side shock-on-lip gives you full freedom to spill air with subcritical spillage margin all across the flight envelope,  in response to changes in engine throttle setting (and engine core rotation speed).  So,  your inlet design shock-on-lip Mach number is the highest speed you ever expect to fly with gas turbine.  To change that,  requires a translating-spike variable inlet geometry technology,  something done with the SR-71.

With gas turbine installations,  both the nozzle throat and nozzle exit areas are variable.  Their maxima get set by the max thrust available at max throttle setting at low altitudes. 

The selection of the inlet capture area A_C is a compromise between the large values needed at low altitude (and ground running),  versus all the rest of the flight envelope.  This gets compromised further by how many blow-in doors are included. 

If you really have a supersonic inlet with a gas turbine installation,  you may have to seriously consider including variable inlet geometry,  in the form of relaxing the compression surfaces back to “flat” relative to the oncoming flow.  That is how you get the removal of the obstruction effect,  that you otherwise inherently have with supersonic inlet technology,  at low or subsonic speeds,  or even up to around Mach 1.5.

Figure 8 – Ramjet and Gas Turbine Use Different Design Approaches

There are a lot of internal flow problems in all of these different inlets,  which can be “fatal” to a given design.  That is also beyond scope here,  being a separate topic entirely,  as are the pitot/normal shock inlets themselves. 

There is also a variation on the mixed-compression inlet that starts easily without overspeed or bleed slots.  It is called the “shock-follower inlet”,  and it compromises its pressure recovery with suboptimal internal shock strengths and locations to accomplish this!  The air ingestion was not compromised,  as this was for a ramjet engine application on the old Russian surface-to-air missile known to NATO as the SA-6 “Gainful”,  which had 4 axisymmetric side-mounted mixed-compression side inlets.

The means to accomplish this was undersizing the centerbody,  in order to oversize the inlet throat area enough to pass the swept-out airflow subsonically.  This is often done in supersonic diffuser work,  per Ref. 2,  although many diffusers are adjustable to a smaller,  more-efficient throat after the shock system is swallowed.  In the SA-6,  throat oversize was done,  and left that way at fixed geometry,  in order to swallow the shock system easily in a very simple design.  That pushed the terminal shock at critical away from the throat,  and closer to the full duct area,  at the rear of the centerbody.  Interference with the waves shed by the centerbody struts does complicate this shock field,  but we will ignore that here.

Subcritically,  all of this unswallows out in front of the cowl lip as with all the other supersonic inlets.  The SA-6 being a ramjet,  operating subcritically was avoided by design,  even at the takeover points.  Supercritically,  there is a complicated train of oblique shocks trailing off the centerbody down into the constant-area ducts.  The terminus of this was complicated,  but effectively it is like a weak normal shock somewhere down in that duct.  Having long ducts to accommodate large supercritical margins is essential.  See Figure 9.

References:

#1.  NACA Technical Report 1135 “Equations,  Tables,  and Charts for Compressible Flow”,  Ames Research Staff,  1953.

#2.  Kantrowitz, A., and Donalson, C., “Preliminary Investigation of Supersonic Diffusers”,  NACA TR L5D20,  1945.

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Inlets Capable of Supersonic Flight Use

This posting is a primer about supersonic-capable air inlets for people who know very little about them.  I had intended to put t up in September,  but ended up posting it late in August.   I have another one coming soon that goes into more details about how these things behave and how the data are "book-kept".  Together,  they make a pretty good "get started" lesson in supersonic inlets.  This topic is crucial to the success of anything that flies fast with any kind of airbreathing propulsion.  

The posting follows:

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There are two kinds of inlets that could possibly be used in supersonic flight,  especially in their simplest forms for ramjet-propelled vehicles.  One is the so-called “pitot/normal shock” inlet,  useful from subsonic speeds all the way up to about Mach 1.5,  maybe even 2.0.  The other is the “supersonic inlet” that features external compression surfaces ahead of its cowl lip capture station.  These are only useful in their simplest forms from roughly Mach 1.5-to-2,  up to high supersonic,  even low hypersonic speeds.  That would be Mach 6+.

That brings up the definition of “capture area”.  It is defined as the area that is swept-out by the inlet structure as it moves at flight speed.  That varies with inlet type,  because of the varying requirements for external compression structures and cowl lip shapes.  However,  Figure 0 may help the reader to better understand how these things are defined.

Figure 0 – Definitions of Capture Area for the Various Inlets

Shock waves are inherent once supersonic flight speed is attained.  Shock waves come in two basic forms:  the “normal shock” and the “oblique shock”.  They have different characteristics and effects.  This is illustrated in Figure 1.  The normal shock occurs when a supersonic streamtube must go straight into a region of sufficiently-higher pressure.  The oblique shock forms when an angled surface forces the streamtube to divert to a new direction.  The math behind this is complex,  and given in the indicated source (Ref. 1).  The normal shock is the “stronger”,  in that flow behind it is subsonic,  and at a lower total (stagnation) pressure.  Flow behind an oblique shock is still supersonic,  with less loss of total pressure.  Total is larger than the static pressure,  reflecting high speed flow energy. 

Figure 1 – Basic Characteristics of the Two Kinds of Shock Waves

The pitot/normal shock inlet is just a pitot inlet flying supersonic.  A bow wave shock begins to form out in front (“subcritical” operation) if the outlet is obstructed.  If the pressure at the outlet of the duct reduces to just the right pressure,  the shock moves to the min area right at the cowl lip.  This is the “critical” operation point.  If the pressure at the outlet (“the backpressure”) is reduced further,  the shock will move downstream further into the divergent passage of the inlet duct (“supercritical” operation).  The pitot/normal shock inlet has no “shock-on-lip” design point,  it simply “is”.  This is illustrated to the left side in Figure 2 at critical operation.

The supersonic inlet has a compression surface exposed in front of the cowl lip capture station.  This surface is a spike if circular in cross section,  or is a flat ramp if rectangular in cross section.  It works by diverting the air stream’s direction,  thus creating oblique shocks.  At the design “shock-on-lip” speed,  these shock waves converge upon the cowl lip,  as illustrated for the two slightly-different designs on the right side of Figure 2.   

There is always a minimum channel area in a supersonic inlet,  as noted in the figure,  termed the “inlet throat”.  If the inlet throat is downstream of the cowl lip capture station,  there is also internal compression in the contracting channel.  That is the “mixed compression” design,  bottom of the two illustrations.  The cowl lip can be close to parallel with the oncoming flow for lowest drag,  and sheds an internal oblique shock,  which at design should be focused upon the inlet throat station.   At critical operation,  the terminal normal shock is also located at that inlet throat,  which is still somewhat supersonic.

Figure 2 – How the Various Inlets Incorporate Shock Waves in Their Compression Process

If the cowl lip is angled more parallel to the external compression surface,  the inlet throat can be located right at the cowl capture station.  The cowl lip does cause higher cowl lip drag,  oriented that way.  There is no internal compression by any contracting channel,  and thus no internal oblique shock shed by the cowl.  This is the “all-external compression” inlet design.  At critical operation,  the final terminal normal shock is right there at the inlet throat,  which is the min flow area at the cowl capture point.

In either supersonic design,  the sequence of oblique shocks slows the flow speed in steps,  at each oblique shock shed by a surface feature.  The diversion angles generate the oblique shock waves that the flow encounters.  Done “right” (that math is also very complex),  there is about the same total pressure ratio (a number less than 1) across each shock,  which can also be made about equal to the total pressure ratio of the terminal normal shock.  That last is because the terminal shock occurs at a quite low supersonic flow speed,  instead of full flight speed. 

The product of these total pressure ratios is essentially (but not entirely,  because of friction in its various forms) the “critical pressure recovery ratio” of the inlet (a number less than 1).  That product of ratios is higher than the total pressure recovery ratio across the single normal shock of the pitot/normal shock inlet.  That higher total pressure recovery ratio is why these supersonic inlet types are much preferred beyond Mach 1.5 to 2 flight speeds.

There is a classic problem in fluid mechanics texts,  dealing with using a converging-diverging passage as a supersonic inlet diffuser to slow a flow down,  instead of as a nozzle to speed it up.  At the “design” speed,  flow slows in the contraction to a speed just above Mach 1,  and then tries to expand again supersonically just as the channel starts to diverge,  but then shocks down subsonic with a normal shock,  for (ideally) final subsonic diffusion. 

When one attempts this experimentally,  the shock forms out front as a bow shock,  and the subsonic flow behind it tries to pass through the min area at flow speeds up to Mach 1.  But the subsonic massflow passable at Mach 1 is less than the supersonic massflow that is being swept out!  One has to speed-up the oncoming flow to considerably higher speed than the “design” speed,  in order to “swallow” the bow shock,   in order for it to become that terminal normal shock. 

This is the Kantrowitz-Donaldson problem (Ref. 2),  and mixed-compression inlets inherently suffer from it,  because such a converging-diverging passage is a part of the inlet’s design.  The only way to get the shock system swallowed is either (1) by very significant overspeed,  or (2) by adding very significant throat bleed slots that lead overboard,  in order to pass the requisite supersonic massflow rate.  The subsonic flow can make the turn to go out through the bleed slots easily!  Once supersonic,  the flow cannot easily make the turn,  so the slot bleed flow is very much lower,  once supersonic internally. 

The all-external compression inlet does not have this “starting” (shock-swallowing) problem!  That is because there is only a diverging channel,  once the flow is inside the cowl.  These inlets start easily without need of overspeed,  or any need of significant throat bleed slots.  They simply have no contracting channel with which to suffer the Kantrowitz-Donaldson problem.  That leads to the list of pros and cons for the two types of supersonic inlets in Table 1.  The pitot inlet is included for completeness.

All these inlets share the behaviors that we call subcritical,  critical,  and supercritical flow.  This is driven by the “backpressure” the inlet “sees” at its subsonic outlet.  In effect,  this backpressure is a resistance to swallowing the shock system and keeping it swallowed.  In both critical and supercritical flow,  the shock system is swallowed.  At critical,  the terminal normal shock is right at the inlet throat.  Supercritical,  it is located further down in the divergent diffuser and is thus stronger,  taking place at a higher flow speed,  and thus creating a lower total pressure in the subsonic flow downstream of it. 

In subcritical flow the entire shock system is unswallowed,  located out on the external compression surfaces.  Its position along that surface determines how much air is spilled around the cowl lip and not ingested at all,  even after passing through the oblique shocks and terminal normal shock and getting the max possible final total pressure.  This behavior is shown in Figure 3 just below for all three inlets,  at their design speeds (excepting pitot,  which is simply shown in supersonic flow).  Subsonically,  the pitot always effectively operates “subcritically” by spilling air,  it’s just that there is no shock wave at all.

Only the supersonic inlets have a design speed,  the pitot does not.  There is the possibility of operating below design speed,  at design speed,  and above design speed,  quite independently of the possibilities of operating sub- or supercritical.  At or above design speed,  the ingested streamtube is its full swept-out size.  Below it,  the ingested streamtube is smaller for geometric reasons,  because the local flow behind each oblique shock must be parallel to the surfaces of the external compression features.  This behavior is shown only at critical,  for simplicity,  in Figure 4.  The shock and spillage patterns are more complex supercritical,  and simply different subcritical. 

Figure 3 – Subcritical,  Critical,  and Supercritical Operation at Design Speed

Figure 4 – Below vs At or Above Design Speed with Supersonic Inlets at Critical

The net effect of all this is (1) subcritical spillage reduces air ingestion while maintaining full pressure recovery,  while supercritical operation ingests full air flow while reducing pressure recovery.  The point where both ingestion and recovery are their max possible values is the critical point.  This has the effect that at any one flight speed Mach number,  the operation of the inlet is per Figure 5 below.  In that figure,  the notation PRCR is the critical total pressure recovery,  which is that maximum recovery.  The notation ARCR is the critical area ratio of the captured streamtube,  to that which could be swept out by the inlet’s capture area.  It is maximum at critical.

That “any one flight Mach number” could be below,  at,  or above the design shock-on-lip Mach number for the supersonic-type inlets.  There is no such design shock-on-lip Mach number for a pitot/normal shock inlet.  Below design,  the possible ARCR ratio of the supersonic-type inlets is reduced by the geometries involved,   while above it,  ARCR is not reduced.   The pitot has no such effect.  

Figure 5 – How All Supersonic Inlets Operate at Any One Flight Mach Number

Pressure recovery is aways reduced at higher speeds,  but it reduces faster above design than below it,  with the supersonic-type inlets.  Pitot has no slope break at design because it has no design speed,  but it always has lower pressure recovery than the supersonic types,  just because of the stronger single normal shock.  That gives rise to the recovery curve shapes shown generically in Figure 6.  

Figure 6 – Generic Representation of Typical Inlet Curve Shapes with Mach number

There are drags associated with the air that is either spilled subcritically,  or not ingested for being below design speed,  among many other reasons.  These vary with the details of the inlet designs,  and where they are positioned on the vehicle.  That is a very complicated item,  beyond scope here.

The use of these inlets on turbojet and turbofan aircraft is another very large separate topic.  These inlets,  especially the supersonic types,  require a myriad of “band-aids” like spring-loaded blow-in doors admitting extra air,  with the duct sucked down below local atmospheric by the turbine engine,  in order to function well when below design speed.  The pitot inlet may need some of these “band-aids” for low subsonic and ground-run operation.  Even so,  the very same components get used quite differently for turbojet or turbofan,  as compared to ramjet!  That is also beyond scope here,  except to say that ramjet needs supercritical operation,  and turbojet/turbofan needs subcritical operation.

References

#1.  NACA Technical Report 1135 “Equations,  Tables,  and Charts for Compressible Flow”,  Ames Research Staff,  1953.

#2.  Kantrowitz, A., and Donaldson, C.,  “Preliminary Investigation of Supersonic Diffusers”,  NACA TR L5D20,  1945.

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search code          21082026

search keywords  aerothermo, ramjet

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What Nozzle Designs to Use Where, and Why

I see proponents of aerospike (altitude-compensating) nozzles claiming that the improved efficiencies seen vs altitude down in the atmosphere,  extend out into vacuum,  making such nozzles “perfect” for single-stage to orbit vehicle designs.   But,  such aerospike nozzles CANNOT perform well in vacuum!  Compressible flow physics says so!  Read on.

Aerospike is one type of a “free-expansion” nozzle.  All these expand the flow partly against a suitably-shaped surface,  and partly as an unbounded free-surface,  both along the expanding stream.  As indicated in Figure 1 below (all figures are located at the end of this article),  at the design point (at some altitude),  the free surface of the stream pretty much moves straight aft,  getting its expanding stream cross-section area from the shape of the spike or ramp.  That design point operation is the top right image in Figure 1.

All the momentum vectors at this design condition point more-or-less axially aft,  at the last point of contact with the spike or ramp,  which is also where the expanded stream pressure matches the surrounding atmospheric pressure.  Only the cosine components of these momentum vectors apply to axial thrust,  but because the angles are nearly axial,  those cosine factors are high,  very nearly 1!  Their averaged value is the “nozzle kinetic energy efficiency” η_KE.  Thrust is F = m V_exp η_KE with no pressure term (P_exp – P_atm)*A_exp,  since P_exp = P_amb by design and the term is zero,  where m is the mass flow rate through the nozzle.

Below the design altitude,  less area expansion is required to reach the higher ambient atmospheric pressure,  and so the final size of the stream at the last point of contact is smaller,  and the expanded velocity V_exp is less,  although still high.  The momentum vectors are all still rather closely aligned with the axial direction,  so η_KE is still high.  One is still getting very close to the theoretically-possible thrust.  This condition is the top left image in Figure 1.

Above design altitude,  more area expansion is required to reach P_exp = P_amb by about the last point of contact with the spike or ramp.  The expanded velocity V_exp will be higher than design there.  But since the spike or ramp geometry is fixed,  the only way to get the extra expanded stream area is for the free surface to angle outward,  off the axis!  This is the bottom left image in Figure 1. 

Note that many of the momentum vectors now have significant angles off of axial,  reducing their cosine-factored axial components.  The average of all those cosine factors is now lower,  so that η_KE is lower,  meaning thrust is reduced significantly below what could be theoretically had.  The lesson here is that free expansion designs have high efficiencies at and below their design altitudes,  but those efficiencies drop above design altitude!

The extremized case is taking the free-expansion nozzle out into vacuum,  which is the lower right image in Figure 1.  The stream needs to expand to an “infinite” area in order to achieve an expanded pressure P_exp that is zero,  but it cannot!  It can deflect only to some angle off the axis,  if the free expansion starts at the Mach 1 throat area of the nozzle.  This is the limiting angle value as determined by a part of compressible flow analysis called “Prandtl-Meyer flow”.  V_exp will be as high as it can be,  but probably somewhere on a curved control volume surface intersecting the axis somewhat aft of the last point of contact. 

Look again at the bottom right image in Figure 1.  Note that most of the momentum vectors are significantly angled away from axial,  out past the last point of contact with the spike or ramp!  That geometry will not change,  even if the pressure reaches its minimum somewhere aft of that last point of contact.  Some of those vectors are right out to the side,  where they contribute nothing to axial thrust,  or even past that,  subtracting from thrust! 

The average of all those cosine factors η_KE is so very clearly sharply reduced below 1,  and so the deliverable thrust is far below what is theoretically possible!  The lesson here is painfully obvious:  you do NOT want to use an aerospike (or by extension any free-expansion nozzle design) out in vacuum!

There is a way to limit the vacuum fan-out angles to something less.  This is shown in Figure 2 below,  where the design has a partial fixed bell to a point where the Mach number is greater than 1,  followed by a free expansion,  as before.  The simplest form of this notion is the “scarf-cut bell”,  but there can be aerospikes fed by supersonics.   

The outer free boundary angle of the stream in vacuum is driven by the Prandtl-Meyer flow limitation starting from a supersonic Mach number,  which is a lower final angle than when starting from Mach 1.  But the expanded velocity is less because the expanded stream area is less at the last point of contact,  reducing the theoretically-obtainable thrust.   The spread of momentum angles is less,  leading to a higher (but still reduced) η_KE. 

As Figure 2 indicates,  the further supersonic you start the free boundary expansion,  the more you can limit the fan-out angle.  But you are losing expanded speed doing this!  It’s a trade-off!  And it does not change the fundamental lesson of not using a free expansion nozzle in vacuum!  Only the detailed numbers are different.

Conventional fixed-bell nozzles as illustrated in Figure 3 below,  do not face this free-surface fan-out problem,  as their stream is confined by the bell surface right up to the last point of contact at the exit plane!  THAT is where you figure thrust,  and thus you DO NOT CARE that the plume fans out suddenly just downstream of the exit plane,  by the same mechanisms as described above.  This free-surface fan-out expansion,  just aft of the exit plane,  has been observed on every launch vehicle leaving the sensible atmosphere since the space age began.  It is quite real! 

The momentum vector angles at the exit plane of a fixed bell are constrained by the physical bell walls,  right up to the exit plane where the area is A_exp.  The shapes and angles of those walls determine the exiting angle into the free surface edge of the plume.  The vector on the axis of the bell is axially-directed at the exit.  Thus all the cosine factors are close to 1,  and η_KE is a high value,  fixed by the bell geometry,  and independent of the ambient atmospheric pressure!   The differences from all the free expansion designs are (1)  that there is now a backpressure correction term on thrust:   Fth = m V_exp η_KE + (P_exp – P_amb)*A_exp,  and (2) that η_KE is unaffected by altitude ambient atmospheric pressure. 

Figure 3 also shows that a curved bell and a conical bell can have exactly the same high η_KE,  it is just that the curved bell can be a little shorter,  but is more difficult to design.

Because of the backpressure term on thrust,  and how backpressure-induced separation limits design options,  for a true “sea level” nozzle design,  you set P_exp = P_amb = P_SL,  and accept the lower area expansion ratio and V_exp.  That will be all the thrust you can get at sea level,  and while it increases as you ascend,  it does not increase by very much,  because the largest effect on thrust is area expansion ratio to reach V_exp,  not the backpressure term.

The so-called “vacuum-optimized” nozzle bell is nothing of the sort!  It is entirely constraint-driven:  you simply go to the biggest nozzle exit area that will actually fit behind the stage or vehicle!  It’s the same constraint,  whether one engine or several.  Being able to thrust-vector-control the vehicle with engine gimbal angles usually reduces that max nozzle exit size a little.  This vacuum design approach is also shown in Figure 3.

The thrust equation above for a fixed bell can be converted with compressible flow to a very convenient thrust coefficient form,  as shown in Figure 3.  This allows one to determine the expansion ratio-related data without knowing the actual dimensions first!  That is also shown in Figure 3.  For the thrust coefficient equations shown in the figure,  there are 3 items related to the expansion ratio selected.   They are the area expansion ratio Ae/A* itself,  the exit plane Mach number Me,  and the ratio Pe/Pc of the exit plane expanded pressure and the chamber stagnation pressure Pc,  that feeds the nozzle.  If you know any one of them,  basic compressible flow quickly finds the other two for you. 

Thrust coefficient has two parts,  as indicated in Figure 3.  Those are the vacuum thrust coefficient C_Fvac,  and a backpressure correction term that varies with altitude ambient pressure Pa and the chamber pressure Pc.  The actual thrust coefficient down in the atmosphere C_F = C_Fvac – (Pa/Pc)*(Ae/A*).  And thrust F_th = C_F Pc A*,  by definition.  All of this presupposes no separated flow in the bell (dealing with that is beyond scope here).

Now,  Figure 4 below shows two altitude-range sketches for launch and ascent toward low circular Earth orbit (LEO).  The left one is marked up to show what happens when using a free-expansion nozzle like an aerospike for a single-stage to orbit (SSTO) vehicle,  as aerospike proponents want.  Most of the time (and propellant) is spent down low,  because the vehicle has yet to accelerate to high speeds.  As it leaves the sensible atmosphere,  the trajectory has bent over.  Most of the so-called “delta-vee to orbit” is obtained essentially out in vacuum!  For an aerospike,  this is where η_KE is seriously degraded!  It has good η_KE only early-on,  down in the sensible atmosphere,  as discussed thoroughly above.

The plot on the right in Figure 4 is marked for a two-stage to orbit (TSTO) vehicle,  using sea level fixed-bell engines in the first stage,  and vacuum fixed-bell engines in the second.  Again,  most of the “delta-vee to orbit” is obtained essentially in vacuum by the second stage,  with engines that are operating at high efficiency.  In contrast,  the aerospike SSTO is getting most of its “delta-vee to orbit” out in vacuum at low efficiency!

The only other point to make about Figure 4 is also in the plot on the right.  Down low in the atmosphere during the TSTO first stage burn,  both the fixed-bell sea level nozzle and the aerospike nozzle would be operating at high efficiency!  The aerospike might even outperform the fixed bell in that role,  if the staging altitude is not too high! 

Figure 5 below shows 2 sketches.  The left sketch is thrust vs altitude for a fixed-bell sea level nozzle,  plus an alternate ascent design option for that same role.  The plot on the right is an altitude-range plot marked for a TSTO vehicle,  showing where staging occurs as you leave the atmosphere at only-supersonic speeds.  (The entry interface altitude is the edge of the sensible atmosphere only when moving at truly orbital-class speeds.)

The sea level nozzle on the left shows an increase in thrust at higher altitudes,  because the P_amb in the backpressure term is going down.  You can design for expansion to ambient at a higher altitude,  say in the 5 – 10 km range.  That thrust trend will have a larger expansion ratio and a higher thrust going into vacuum,  but at the cost of lower sea level thrust when the vehicle is heaviest!  It can actually average a slightly higher thrust over the ascent to staging,  than a true sea level nozzle.  It is an engineering design trade available to you!

Bear in mind that an aerospike (or other free-expansion design) designed at high altitude will have an increasing V_exp and a still-high η_KE,  almost all the way to staging,  which is essentially just outside the sensible atmosphere.  That makes such a nozzle a real competitor in the first stage role of a TSTO!

The second stage of the TSTO just needs to use a fixed-bell vacuum design.  That will be efficient,  because all of that burn is outside the sensible atmosphere,  essentially in vacuum.  And as thoroughly discussed above,  a free-expansion design will be quite inefficient at such altitudes,  due to the excessive stream-spreading angles of its free edge.

Conclusions

(#1) I have no doubt at all,  that free expansion nozzle designs perform quite well throughout the sensible atmosphere,  and probably better than sea level fixed bells,  for ascent to staging altitudes.

(#2)  Conclusion (#1) extends to even the compromise fixed-bells designed at altitudes modestly above sea level. 

(#3)  A fixed vacuum bell will far out-perform any free-expansion design,  when operating out in vacuum!  That is because free-expansion efficiency starts falling somewhere above design altitude,  and it has fallen drastically upon going into vacuum!

(#4)  The best role for free-expansion nozzle designs is in the first stage of a TSTO.  They will do better in vehicles that have lower staging altitudes,  because the vacuum free-surface spreading effects upon overall impulse delivery get reduced.

Figure 1 – Free-Expansion Plume Behaviors (Aerospike Shown)

Figure 2 – Limiting the Fanning-Out In Vacuum Is a Tradeoff

Figure 3 – Fixed-Bell Nozzles Can Be Either Atmospheric or Vacuum But Not Both

Figure 4 – SSTO to LEO With Aerospike vs TSTO to LEO With SL and Vacuum Bells

Figure 5 – The Best Application of Aerospike Is First Stage of a TSTO

Followup:  Actual Numbers Estimated

I went ahead and ran a study to see how bad the fan-out and fan-in angles might be,  and what effect they might really have,  on the ideal expansion numbers.  I ran it for an axisymmetric aerospike designed for “perfect” expansion at 60,000 feet altitude (18.29 km),  with an arbitrary spike radius at the throat station of 1.000 feet,  and a spike cone half-angle of 13 degrees.  I presumed liquid oxygen (LOX) and liquid hydrogen (LH2) as propellants,  in a full-flow cycle with no dumped bleed,  and a Pc of 4400 psia.

The basic calculations are depicted in Figure 6.  At design,  the streamtube boundary goes straight back:  all the area expansion ratio is provided by the spike and lip geometry.  The last point of contact with the spike tip is where the control volume about the engine must be located,  in order to exclude from thrust calculations the effects of the conical oblique shock that turns the spike-angled surface flow back to straight axial. 

Below design altitude,  the flow must “fan-in” from the throat lip by some angle C in order to “hit” the right area ratio at lower altitudes.  Above design altitude,  the flow must fan-out by some angle B in order to “hit” the right area ratio at the higher altitudes.  The angle B may not exceed the turning angle “v” from Prandtl-Meyer flow (P-M)!  If it does,  the control volume must move aft of the tip,  and you must account for the oblique shock losses,  too.  

Figure 6 – How the Numbers Were Estimated

At design,  a decent estimate of the averaged cosine factor for correcting all streamline momentum vectors back to axial,  would be the average of cosine(0) = 1 for the plume edge,  and cosine(A) for the axis.  That averaged cosine factor is the value of the nozzle kinetic energy efficiency η_KE that applies to both thrust coefficient C_F and specific impulse Isp,  as well as to thrust,  which I did not explicitly figure for this study.  (F = C_F Pc At.)

Skipping thrust,  one can go directly from C_F to Isp as Isp = C_F c* (1 – BF) / (gc C_D),  where BF is the dumped bleed fraction,  and C_D is the effective-area factor applied to the geometric throat area,  for computing massflow.  In this study,  BF = 0,  and I used a nice high C_D = 0.995.

The design sizing numbers are shown in Figure 7.  Sizing at 60,000 feet is why Re is only slightly larger than Rt.  This has to be set iteratively,  to hit the “right” Ae.

Figure 7 – Design Values for This Study

The ideal expansion numbers are very easy to compute.  One needs the ambient pressure Pa at that altitude,  and sets the expanded pressure Pe equal to it.  Thus,  at any altitude,  Pe/Pc is known,  and can solved for the expanded Mach number Me,  at the last point of contact with the spike tip.  From that,  the area ratio Ae/At is easily found.  The thrust would be massflow x velocity x averaged cosine factor  m Ve η_KE,  or in compressible flow variables F = γ η_KE Pe Ae Me²,  since the (Pe-Pa)Ae term is zero.  Dividing by Pc and At,  we have the ideal free-expansion thrust coefficient: 

C_F = (Pe/Pc) (Ae/At) γ η_KE Me²  for which ideally η_KE = 1

The Isp is then Isp = Fth/w = Pc C_F At / (Pc C_D At gc /c*(1 – BF)) = C_F c* (1-B_F)/(gc C_D).  For this study BF = 0.0 and C_D = 0.995. 

I used a power-function c* correlation to estimate c* at Pc from some old data for LOX-LH2 at 1000 psia:     c* = 7950 ft/s (Pc/1000 psia)^0.006051

In these units,  one uses gc = 32.174.

All of these ideal numbers figured at η_KE = 1 are totally uncorrected for the fan-in and fan-out angles that inherently occur!  Those angles affect the average of the cosine factors that must be applied to the streamline momentum at every point on the exit surface of the control volume.  Not looking at the angle effects on η_KE  is the mistake that proponents of aerospike (and other free-expansion nozzle designs) make,  unless they are adept at modeling real-world effects in compressible flow. Such are NOT easy to calculate!

For ideal flow at η_KE = 1,  the performance looks like that depicted in Figure 8.  This “looks great” all the way from the surface into vacuum!  And that is what non-adept proponents of aerospikes use to make their claims,  including that these nozzles are “perfect” for single-stage to orbit usage.  But,  η_KE is NOT 1 all the way to vacuum;  it CANNOT BE!

Figure 8 – Ideal Expansion Performance Looks Deceptively Great

You have to account as best you can for the exit area-averaged cosine factors of all the streamline momenta.  In other words,  η_KE is a function of the fan-in angle C below design,  the fan-out angle B above design,  and the spike half angle A.  I showed some well-inside-the-ballpark approximations for those dependencies in Figure 6 above.

This angle-dependent η_KE has little effect below design altitude,  but the effect increases rapidly above design altitude,  becoming quite severe,  as indicated in Figure 9.  The fan-in angle C is shown vs altitude top left,  the fan-out angle B is shown vs altitude top right,  and the values of η_KE and C_F vs altitude that result,  are shown bottom left.  The resulting Isp for this study that results from this properly-corrected C_F trend,  is shown bottom right.  

Figure 9 – Actual Angle-Corrected Performance Falls Off at High Altitudes

Two things should “jump off the page at you” immediately,  looking at Figure 9 just above:

(#1) These aerospike- (and by extension any free-expansion-) design nozzles are NOT good vacuum engines!  They start losing performance in a serious way somewhere above design altitude,  which for this example is in the vicinity of 150,000 feet altitude (45.72 km).  Such is actually below typical staging altitudes for two-stage-to-orbit (TSTO) vehicles,  which usually fall in the 50-70 km range,  and it is far below any proper orbital altitude (150+ km).

(#2) Given (#1),  the “best application” for the aerospike- (or any other free-expansion-) nozzle design,  is in the first stage of a TSTO vehicle,  NOT in an SSTO vehicle!

And (#2) confirms what I said,  in the non-numerical article above this “Followup” section.

Final Remarks

Otherwise,  I may not have run the “right” aerospike design.  My throat area is very small,  to get from a very high Pc down to a very low Pa at the high altitude of 60,000 feet.  Both a lower Pc and a lower design altitude would act to increase throat area,  showing up as a larger difference between Rt and Re in Figure 7 above. 

However,  that does not matter to the basic trends outcome here! 

The trends identified with fan-in and fan-out angles would remain,  with only the numbers shifting a little.  The trend here that is important,  is the free-surface fan-out angle that increases with increasing altitude above design,  no matter what else is done!  That WILL ALWAYS happen!  And it will always dominate performance somewhere above design!   

Which in turn means aerospike nozzle designs (and by extension any free-expansion designs) WILL ALWAYS show performance degradation,  far below what is theoretically possible,  while leaving the sensible atmosphere at only-supersonic speed.  That will inherently happen somewhere near 45 km altitude,  FAR BELOW orbital altitudes,  just because the air is so thin up there!

It is very easy to estimate the theoretical performance of free-expansion-type nozzles.  It is not an easy task to estimate that plume fan-out angle,  unless you are very well-versed in such aspects of compressible flow.  Further,  you must understand where and how (and why “there”) to draw the control volumes that let you properly calculate thrust. 

What I have repeatedly seen is that most proponents of free-expansion designs as vacuum-capable,  do not have those compressible flow skills,  beyond the very elementary theoretical expansion calculation!  Further, they simply do not understand that the fanning-out of the free surface of their plume strongly lowers thrust efficiency,  by the cosine component effects upon the momentum vectors of every streamline.  Or that this effect is “real” precisely because it happens INSIDE the control volume that one MUST draw!  

And,  usually at least some of them will really dislike being shown the error of their claims!  To them I can only say “sorry,  but the truth will prevail”.

Meanwhile,  the aerospike nozzle has great application down in the atmosphere,  both in rockets and in jet propulsion! 

In point of fact,  it has been flying in some turbojet engines for decades.  The ones with the spike sticking out of the nozzle “turkey feathers”,  those are axisymmetric aerospikes,  when the “turkey feather” outlet is converging and choked!  It works great,  down in the atmosphere!

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search code:     02082026

search keywords:  aerothermo,  space program

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