Starship/Superheavy Hot Staging

This is not insider knowledge.  It is based only upon what SpaceX has publicly released. 

There is some kind of geometrically-shaped shield plate that protects the front end of the Superheavy booster where the forward dome of the forward propellant tank is otherwise exposed.  This shield was shed after hot staging with Version 2,  but is now integral to the booster stage in Version 3.  It deflects the plumes of the Starship upper stage engines,  from striking and damaging or destroying that forward tank end,  during hot staging.

This shield structure is coupled with an interstage ring between the stages,  part of the Superheavy,  that is “porous”,  in the sense that there are openings through which the deflected engine plumes can exit laterally,  in a more-or-less radial direction,  during the transient hot-staging event.  The force to turn a plume gets exerted upon that shield.

The Superheavy booster had four grid fins spaced 90 degrees apart,  in Version 2.  Plume blast damage had been noted to the grid fin rotated into the way of plumes,  during the booster flip maneuver,  in Version 2.  In Version 3,  there are only 3 grid fins,  still spaced 90 degrees  apart,  but somewhat larger than the Version 2 grid fins.  That leaves a space where there is no grid fin,  which is clocked to align with the dorsal side of the upper stage Starship,  as shown in Figure 1. 

Figure 1 – Version 3 Starship/Superheavy Arrangement and Intended Booster Flip Direction

The design direction of the booster flip is a plane aligned with the dorsal-ventral plane of Starship,  and the front of the booster is supposed to flip in the direction where it moves “down” as shown in the figure.  Thus,  no grid fin is exposed to plume blast damage anymore,  in Version 3.  The side of the booster is less vulnerable to plume blast damage,  being cooled by the cryogenic propellant vapors inside.  

In Version 2,  the torque to flip the booster came from a strong gimbal angle of the 3 booster engines firing during the staging event to keep the booster propellants settled into the aft ends of their tanks,  where the suction-feeds to the engines are located.   This is still the case in Version 3,  with some control of the deflected plume force directions added,  by one or both of two means:   (1) the geometric shaping of the shield and of the “porosity” of the interstage ring,  can favor a plume deflection force component in the intended flip direction,  and (2)  the startup sequencing of the Starship engines can also produce a temporary plume deflection force component favoring the intended flip direction. 

There are two separate effects going on during the hot-staging event,  which must both be successful,  but which are also linked to each other in ways that make the successful “solution space” rather small!  These are indicated in Figure 2 below,  which depicts the conditions as the hot staging begins,  but before the flip starts.  (Drag was ignored.)

First,  the acceleration of the upper stage Starship has to be larger than the acceleration of the Superheavy booster (before the flip starts),  in order not to risk a collision. 

Second,  the axial acceleration of the Superheavy booster must be a positive number large enough to keep the propellants settled,  all during the staging event,  and all during the flip maneuver. 

That last is also complicated by the flip motion possibly flinging the propellants forward,  in only the forward tank,  if the flip is too fast!

The variables available for controlling this event are the thrust levels used in the booster and upper stage engines,  the shaping of the shield and interstage ring porosity geometries,  and the sequence of ignition of the upper stage engines. 

To simplify this first look,  the author chose only a full thrust setting in the 3 booster engines,  and lighting all 6 upper stage engines at once,   but a some reduced thrust.  Guesses were made for plume deflection force component angles and booster engine gimbal angles.   All of this is indicated in the figure.

The author has no actual data for the engine thrust levels at the staging altitude,  or for the actual masses of either stage at the time of staging.  He made reasonable guesses for these,  as well,  as indicated in the figure.  So the results are inherently approximate!

Figure 2 – Finding A Thrust Setting For Starship That Keeps Superheavy Accelerating

It would not be possible to get full thrust in an engine at the moment of ignition!  It is more practical to ignite at a partial-thrust setting propellant flow rate,  stabilize after a split second,  and then throttle up quickly a split second after that,  as desired. 

The author chose to look at full and half thrust initially,  and then added a 40% thrust setting,  after getting negative booster acceleration at full upper stage thrust,  and zero booster acceleration at half thrust.  The 0.1 gee booster axial at 40% Starship thrust is in the rough ballpark of other ullage thrust applications.  The 40% thrust setting is within the capability of the Raptor-3 engine design,  so this point really is a feasible thing to do!

As the figure indicates,  the Starship upper stage has the acceleration to leave the booster behind,  at 40% thrust on 6 engines.  It cannot throttle up until the flip has proceeded enough to direct the front end of Superheavy out of the way of the Starship engine plumes.

Also indicated in the figure,  the Superheavy axial acceleration is zero at half Starship thrust,  and only 0.10 gees at 40% Starship thrust.  Starship cannot throttle down much more than that,  and still leave the vicinity at about 0.4 gees!  Which is exactly why this author says the solution space here is quite narrow!  Screw this up,  and the propellants will unsettle,  the booster engines will suck vapor,  and their turbopumps will explode!

The next step was to look at starting the booster flip.  This is illustrated in Figure 3,  which presumes the 40% thrust solution just obtained above.  Two cases were examined:  a booster engine gimbal angle of 10 degrees,  and a comparable lateral deflection force generated by shield geometry and/or interstage porosity distribution.  The booster mass moment of inertia was approximated by the solid bar formula for rotation about its center of gravity (cg).  That cg was simply presumed to be halfway along the booster.  

Figure 3 – Starting the Booster Flip Maneuver During Hot Staging

The results for time-to-clear shown lower right of figure are amazingly close to what is in the publicly-released videos of these flight tests (about a second or two to clear)!  Considering how crude all the assumed data is,  that outcome is quite remarkable!  Letting deflected upper stage plumes help the gimballed booster engines to start the flip maneuver is definitely beneficial,  but it is not an overwhelming effect.  That would seem to be the lesson to be learned here.

That brings up the “spin gravity” effect of the flipping booster upon the settling of propellants in its forward tanks by engine-induced acceleration.  As depicted in Figure 4,  this depends upon which side of the booster cg falls the free surface in the forward tank.  One must take into account relative tank sizes,  and how full those tanks are.  A very crude estimate is shown in the figure,  indicating that spin will help settle the propellants.

Figure 4 – Evaluating Whether Spin of Flip Affects Propellant Settling

Note that had the LOX tank been forward in the design instead of aft,  this outcome would not be true!

The magnitude of the “spin gravity” effect depends upon the rotation rate ω at the time the booster clears the plumes,  which is crudely angular acceleration times the clearance time,  for the two cases shown in Figure 3 above. Those would be 0.139 rad/s for the gimbal only case,  and 0.186 rad/s for both effects together. 

The acceleration in gees at the free surface would be L ω²,  where L is the distance between the cg and the free surface.  As measured from the left as shown in Figure 4,  as long as the free surface is left of the cg,  the spin gravity gees adds to the axial thrust gees.  If the free surface falls to right of the cg as shown in the figure,  then the spin gees would subtract from the axial gees!  There is a coupling between stage layout and “spin gravity” risks.

For the dimensions indicated in Figure 4,  the LCH4 free surface is 10 m left of the cg. So,  if only gimballing,  one would add about 0.020 gees to the axial 0.1 gees.  If faster for both effects together,  one would add about 0.035 gees.  The additions would be a bit over 3 times larger for the LOX tank,  as the L to its free surface is larger at something like 33.7 m.

Much more accurate dimensions masses,  and propellant-remaining percentages would be needed to get reliable results!  But these results obtained here do seem to crudely indicate what is going on during Starship/Superheavy hot staging.

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A Question

My wife found this,  and it is just too good not to post!  But it does raise some other serious questions,  beyond just the one in the image. 

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A few other questions might be:

#1.  “Why is everything that is not good always said by Trump to be either Biden’s or Obama’s fault?”  Not all those things claimed by Trump to be their fault,  can actually be their fault!  A lot of them have been found to actually be Trump’s fault,  which just means he is lying to deflect blame from himself.  Many others are no one’s fault,  just bad things that happened.  Like Covid-19 and the resulting economic depression caused by containing it.

#2. Virtually everything that Trump has ever claimed,  which have actually been fact-checked,  have turned out to be lies.  That’s not opinion,  that’s real data,  right in front of you,  me,  God,  and everybody!  Many of these were really bad,  “pants-on-fire” lies!  Given that track record,  the second question is “Why would anybody believe anything Trump says,  at all?”

#3. Trump is a draft dodger (his dad bought him the “bone spur” draft deferment),  a convicted financial felon (New York state case),  and a civilly-adjudged sex offender (the E. Jean Carrol case).  Those are also not opinion,  but fact,  real data in front of us all.  The question:  “Why would any American ever want a US Navy ship to be named after such a person?”  Re-naming the under-construction USS Doris Miller for Trump is truly a travesty!  Dorrie Miller was a true combat hero aboard the battleship West Virginia during Pearl Harbor,  and he died serving aboard the USS Liscombe Bay during that war!

You can probably think up more for yourself!

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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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World’s Longest Stuff

I found this image in someone’s post on LinkedIn.  It is too funny (and too true) not to post here!  Enjoy.

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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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Kudos to SpaceX for Flight 13

After a last-second automatic abort on the 16^th,  and a bad weather delay Thursday,  SpaceX was able to launch its Starship/Superheavy vehicle successfully on Friday (24 July 2026).  This was Flight 13,  called by some “Lucky 13”,  the second flight for the new version 3 vehicle,  which is bigger,  and uses a new,  upgraded version of the rocket engines. 

Unlike the previous test flight,  nearly everything about this flight appeared to work perfectly.  Kudos to SpaceX for a job very well done!  Very well indeed!

The upper stage Starship vehicle flew perfectly,  with all its engines working,  and re-lighting just fine.  It deployed some 20 examples of the latest version Starlink satellites,  performed the in-space re-light of an engine (which proved not possible on the previous Flight 12),  re-entered the lower atmosphere without any problems over the Indian Ocean near Australia,  descended to low altitude,  re-ignited its engines,  made its flip to nose-up,  and a very soft splashdown landing in the sea.  

It even survived toppling over into the water without exploding!  So,  it might even be towed to Australia for direct inspection,  a real bonus for evaluating the heat shield performance!

The Superheavy booster flew most of its intended flight plan quite well,  excepting the final touchdown burn into the sea,  just off the SpaceX launch site near Brownsville,  Texas.  The “hot staging” seemed to work fine,  with 3 of the 33 booster engines keeping the propellants settled in free fall,  inside the tanks during its flip-around,  so that 13 engines could be ignited to perform the boost-back burn. 

This failed on Flight 12,  due to multiple lost engines,  but it worked just fine this time on “Lucky 13”.  There was something wrong about that Flight 12 flip-around maneuver,  which may have in some way induced those failures-to-ignite.  It occurred in the wrong plane,  for one thing,  according to reports.

The coasting booster then arced over,  back toward the launch site,  and fell into the lower atmosphere,  stabilized tail-first by its grid fins,  toward a low altitude touch-down burn event.  This last burn is where the trouble happened!  Instead of igniting the intended 13 engines to slow down its very supersonic speed,  a handful of engines failed to ignite!  The booster descended stably,  but did not decelerate adequately on too-few engines,  and so it struck the sea at something still near the speed of sound!  That intended soft booster splashdown was the only failed objective in this entire test flight,  as near as I can tell.

We did not see quite so many engine ignition failures in the version 2 Starship/Superheavy test flights that happened before Flight 12 (which was the very first version-3 flight).  Something is going on with version 3 that will have to be found and fixed,  before SpaceX can risk attempting to catch the booster with the arms on the launch tower.  They did that catch  successfully more than once,  with version 2 in the earlier flights.  A truly spectacular thing to watch,  too!

But a failure to decelerate to a slow hover would end up destroying both the booster and the launch-and-catch tower!  And right now,  there is only one of those facilities in existence that is compatible with the version 3 booster!  They have to solve this engine ignition problem with version 3,  before they risk attempting another booster catch.

The upper stage Starship is another story entirely,  now after its complete success during “Lucky 13”,  and near-complete success during Flight 12!  It would be a reasonable bet that they might attempt a ship catch,  on the next flight.  Musk himself even hints at that,  on social media. If they do attempt this feat,  that would be the very first recovery of a Starship that has been into space at near-orbital speeds.  Such would be invaluable,  both for heat shield evaluations and improvements,  and for eventually demonstrating the readiness of this vehicle to actually do real orbital missions.

There will be tension within SpaceX over this.  Some will want to see another Starship upper stage water landing success,  before risking the catch facility.  Others will want to forge ahead as fast as possible.  That is normal.  We will soon hear what they decide to do next. 

I wish them success,  especially since doing spaceflight successfully,  really is a very,  very hard thing for any outfit to do.  And once again,  kudos to SpaceX for a flight test that was almost entirely successful! 

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Note:  this same article was submitted to the Waco “Tribune-Herald” 7-25-2026 as a possible column for the op-ed page.  I am on their board of contributors. 

Meanwhile,  here are the search items for this article:

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Update 8-1-2026:  The Waco "Tribune-Herald" ran the article as a column on the op-ed page in the Saturday paper today (8-1-26).  They ran it pretty much as submitted,  and posted here.  They did not use the illustration that I drew of the booster flip event.  

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Call to Action

I received this image from HOTNIR,  which is the Heart of Texas Network for Immigrant Rights,  located in and around Waco,  Texas.  The image is self-explanatory:  we so very clearly have government officers behaving like a secret police force intended to mistreat and “disappear” the demonized few (immigrants),  and intimidate the rest of us into submission.  

I am sick and tired of seeing government agencies weaponized and abused into the tools of dictatorship!  On this,  our nation’s birthday,  I call upon all Americans to come together,  and rise up to depose our wannabee-king and all his enabling minions!  Your last chance to do this peacefully at the ballot box is this November! 

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