How Propulsion Nozzles Work

Update 4-8-2024:  Should any readers want to learn how to do what I do (estimating performance of launch rockets or other space vehicles),   be aware that I have created a series of short courses in how to go about these analyses,  complete with effective tools for actually carrying it out.  These course materials are available for free from a drop box that can be accessed from the Mars Society’s “New Mars” forums,  located at http://newmars.com/forums/,  in the “Acheron labs” section,  “interplanetary transportation” topic,  and conversation thread titled “orbital mechanics class traditional”.  You may have scroll down past all the “sticky notes”. 

The first posting in that thread has a list of the classes available,  and these go far beyond just the two-body elementary orbital mechanics of ellipses.  There are the empirical corrections for losses to be covered,  approaches to use for estimating entry descent and landing on bodies with atmospheres,  and spreadsheet-based tools for estimating the performance of rocket engines and rocket vehicles.  The same thread has links to all the materials in the drop box. 

The New Mars forums would also welcome your participation.  Send an email to newmarsmember@gmail.com to find out how to join up.

A lot of the same information from those short courses is available scattered among the postings here.  There is a sort of “technical catalog” article that I try to main current.  It is titled “Lists of Some Articles by Topic Area”,  posted 21 October 2021.  There are categories for ramjet and closely-related,  aerothermodynamics and heat transfer,  rocket ballistics and rocket vehicle performance articles (of specific interest here),  asteroid defense articles,  space suits and atmospheres articles,  radiation hazard articles,  pulsejet articles,  articles about ethanol and ethanol blends in vehicles,  automotive care articles,  articles related to cactus eradication,  and articles related to towed decoys.  All of these are things that I really did. 

To access quickly any article on this site,  use the blog archive tool on the left.  All you need is the posting date and the title.  Click on the year,  then click on the month,  then click on the title if need be (such as if multiple articles were posted that month).  Visit the catalog article and just jot down those you want to go see.

Within any article,  you can see the figures enlarged,  by the expedient of just clicking on a figure.  You can scroll through all the figures at greatest resolution in an article that way,  although the figure numbers and titles are lacking.  There is an “X-out” top right that takes you right back to the article itself. 

----------     

Update 23 March 2024:  For the readers of this and other similar articles about ramjet propulsion,  be aware that GW’s ramjet book is finally available as a self-published item.  Its title is “A Practical Guide to Ramjet Propulsion”.  Right now,  contact GW at gwj5886@gmail.com to buy your copy. 

He will,  upon receipt of payment by surface mail or Western Union (or similar),  manually email the book to you as pdf files.  This will take place as 9 emails,  each with 3 files attached,  for a total of 27 files (1 for the up-front stuff,  1 each for 22 chapters,  and 1 each for 4 appendices).  The base price is $100,  to which $6.25 of Texas sales tax must be added,  for an invoice total of $106.25. 

This procedure will get replaced with a secure automated web site,  that can take credit cards,  and automatically send the book as files.  However,  that option is not yet available.  Watch this space for the announcement when it is.  

GW is working on a second edition.  No projections yet for when that will become available.

--------------

This article applies to anything that thrusts from a subsonic chamber through (at least) a sonic throat.  It is intended to give readers a means to compute accurate and realistic thrusts.  This plus a knowledge of chamber characteristic velocity c* is sufficient to do very elementary rocket ballistics.

Update 11-16-18:  see stuff added at the very end,  below,  past the original figures.  

Update 1-26-19:  see stuff added at the very end below,  past the first update.

Update 1-31-19:  see stuff added at the very,  very end below,  past the second update.

Update 10-1-19:  see plume spread stuff added at the very end

Update 2-9-23:  this article uses expansion area ratio and assumed geometry to estimate aerospike and other free expansion designs.  I have since posted a new article "Rocket Nozzle Types",  posted 4 Feb 23,  that revisits the aerospikes using actual Prandtl-Meyer expansion-corner analysis.  It gets about the same answers ultimately,  but allows determination that the gas generators benefit from some bell-confined expansion followed by the free expansion on the spike.  

Most (but not all) nozzles that have a sonic throat also have a supersonic expansion bell.  Scramjet is excluded as being without a sonic throat:  the feed to the nozzle inlet is already supersonic,  and there is no contraction in flow area to a throat. 

Rockets of any type are typically high pressure ratio PR chamber-to-exit,  and high area ratio AR exit-to-throat.  These can be ablatively cooled,  or actively liquid-cooled.

Gas turbine engine nozzles are typically low pressure ratio PR chamber-to-exit,  and low area ratio AR exit-to-throat.  These are usually air-cooled,  and variable geometry:  anything from convergent-only to a mild supersonic expansion bell.  Lower turbine inlet temperatures require lean mixtures and cooler flames,  making air cooling possible,  as long as the air itself isn’t too hot.  That high speed air heat effect limits the flight speeds achievable with gas turbine engines.

Ramjet engine nozzles are typically low pressure ratio PR chamber-to-exit,  and low area ratio AR exit-to-throat.  Modern missile designs are usually ablative.  Some of the oldest designs were air-cooled,  similar to gas turbines,  but this approach is severely limiting in a modern ramjet design,  which can run far richer,  and at far higher flight speeds,  where the air itself is far hotter.   

Fundamentals

Conservation of mass:  the same massflow exists throughout the nozzle (any air cooling bleed effects or other injections or leaks are ignored,  if any exist at all).

Conservation of momentum:  a control volume drawn about the rocket engine is pierced by the exit stream exactly at its exit area,  and the momentum of the propellant feeds are either inconsequential,  or perpendicular to the thrust axis,  or they come from tanks inside the control volume.  This could be any combination of those situations,  or even all three.  Balancing stream momentum and the pressure forces against a restraining force,  leads to evaluating the thrust.

Conservation of energy:  the drop (from chamber to exit) in enthalpy,  as measured by the drop in static temperature,  equals the increase in kinetic energy of the stream,  with essentially zero kinetic energy inside the chamber.  There is an accompanying drop in static pressure,  in an amount defined by the ideal gas assumptions and the corresponding equation of state.  See Figure 1.  All figures are at the end.

We use enthalpy “h” instead of internal energy “u”,  because it includes the effects of pressure change upon energy content,  and internal energy does not.  Enthalpy difference Δh is essentially the temperature difference ΔT,  multiplied by the specific heat at constant pressure c_p.  (Internal energy change uses the specific heat at constant volume c_v.)

Book-keeping:  this is done the easiest way in Mach number-pressure-temperature variables,  instead of the primitive variables,  as long as the ideal gas assumption applies.  That last means we may use as our equation of state P = ρ R T,  and we may use as the change in enthalpy Δh = c_p ΔT. 

In this book-keeping scheme,  we make good use of total (or stagnation) pressures P_t and temperatures T_t.  Assuming no appreciable friction losses,  flow is “isentropic”,  meaning total pressure and total temperature are constant through the nozzle,  a very good assumption in almost every conceivable case. 

The ratio of specific heats γ = c_p/c_v becomes a very useful value to relate totals to statics.  At a location where the Mach number is M,  the total/static temperature ratio TR = 1 + 0.5*(γ – 1) M²,   and the total/static pressure ratio is PR = TR^exp,  where exp = γ / (γ – 1).    

The streamtube area model is more complicated than the simple mass conservation-derived relation in incompressible flow,  and is based off of sonic conditions at the throat area A_t.  If you know the Mach number M at another station where the area is A,  you can find that area ratio AR = A/A_t as easily as the total/static ratios TR and PR.  If you know instead the area ratio A/A_t,  finding the Mach number M is inherently a transcendental (iterative) solution: 

                A/A_t = (1 / M) [TR / const1]^const2

where TR is defined as above,  const1 = 0.5 (γ + 1),  and const2 = 0.5 (γ + 1) / (γ – 1)

Heat transfer:  this is driven not by static temperature but by recovery temperature!   We must do this because the supersonic flow in the nozzle is both highly compressible,  and highly dissipative.  At any given Mach number M,  recovery temperature T_r is very nearly the same as total temperature T_t.  How it varies depends upon laminar versus turbulent flow,  and the gas property Prandtl number P_r:

                T_r = T + r (T_t – T)  where the recovery factor r = P_r^0.5 laminar,  P_r^0.33 turbulent

Only the heat transfer film coefficient h responds significantly to the varying Mach number,  pressure,  and temperature down the nozzle profile.  It does this in a very empirical way.  Multiple models exist for this,  not covered here.  The local heat flux at any station is of the form:

                Q/A = h (T_r – T_s) where T_s is the material surface temperature

For heat transfer purposes,  failing real data,  you can estimate Prandtl number P_r = 4 γ /(9γ – 5).

Conventional Nozzle Thrust Coefficient C_F

Your ideal gas model of the gas flowing through the nozzle comprises its constant specific heat ratio γ,  and its constant molecular weight MW.  These can come from thermochemistry calculations,  and need to reflect the high temperatures involved. 

The thrust F of an idealized nozzle evaluated at its exit plane is the momentum of the exiting gas,  plus the exit area A_e times the difference in pressure between the exiting stream static pressure P_e and the ambient backpressure of the surroundings P_b.  Ideally,  all the streamlines are parallel to the thrust axis.  

In the real world,  they are not.   This streamlines-off-angle effect is modeled with the nozzle kinetic energy efficiency factor η_ke.  It applies to the momentum term in thrust,  but not the pressure term,  as long as the exit plane is perpendicular to the axis.

                F = η_ke m V_e + (P_e – P_b) A_e where m = mass flow rate and V_e = exit velocity

To convert this to compressible flow variables,  we make use of the m = ρ_e A_e V_e massflow definition,  the ideal gas equation of state P_e = ρ_e R T_e with R = R_univ/MW,  and the exit plane speed of sound for an ideal gas c_e = (γ g_c R T_e)^0.5.  The variable g_c is the “gravity constant” to make the equation consistent with inconsistent mass and force units.  If those units are consistent,  g_c will be 1.

                F = η_ke (P_e / R T_e) Ve² Ae + (P_e – P_b)A_e using massflow,  then equation of state

                F = γ η_ke P_e A_e M_e² + (P_e – P_b)A_e  using speed of sound

Note that the first term in the equation just above is the momentum term,  and the second term is the static pressure difference term.  Distribute the A_e so that there are 3 separate terms,  and regroup. 

                F = P_e A_e [1 + γ η_ke M_e²] – P_b A_e  recombining terms such that P_e A_e factors out

Here,  inside the bracket,  the 1 now corresponds to the exit static pressure term with P_e A_e factored out,  and the γ η_ke M_e² corresponds to the momentum term with P_e A_e factored out.  The backpressure effect is still a separate force term,  with the recombined bracket-containing force term really just being thrust into vacuum.

Now we introduce the definition of thrust coefficient C_F = F / P_c A_t with the understanding that the P_c is the total (stagnation) pressure feeding the nozzle.  If the contraction from chamber to throat area is large enough,  there is no measurable difference between total and static pressure at the nozzle entrance.

                C_F = F / P_c A_t = (P_e A_e / P_c A_t)[1 + γ η_ke M_e²] – (P_b A_e / P_c A_t)

                CF = (P_e/P_c)(A_e/A_t)[1 + γ η_ke M_e²] – (P_b / P_c)(A_e / A_t)    regrouping P’s and A’s together

                CF = (AR / PR) [1 + γ η_ke M_e²] – AR / PR_op     (the thrust coefficient equation)

    with PR_op = P_c/P_b using the actual design P_c and P_b

                with PR = P_c/P_e = (1 + 0.5 (γ – 1) M_e²)^exp = TR^exp   where exp = γ / (γ – 1)

                and AR = A_e/A_t = (1/M_e)[ TR/const1]^const2

    with const1 = 0.5 (γ + 1) and const2 = 0.5 (γ + 1) / (γ – 1)

This last formulation is particularly convenient when one wants a certain exit Mach number M_e,  because AR = A_e/A_t and PR = P_c/P_e are easily calculated from M_e using the ideal gas γ.  Otherwise,  if conditions at a certain AR are desired,  one iteratively tries M_e values until the desired AR obtains,  then computes PR.  Essentially,  M_e and P_c/P_e are “locked in” by the AR value regardless of the value of P_c,  although they are not most conveniently figured in that order. 

The “operating pressure ratio” PR_op = P_c/P_b depends directly upon your design choices for P_c and P_b.  One had to choose a P_c to do the thermochemistry,  and P_b is set by the altitude,  or else 0 if vacuum.

Once γ,  M_e,  PR,  AR,  and PR_op are all known,  evaluating C_F is easy,  per the above equation.  If you have used a value of c* to size a throat A_t elsewhere in your fundamental ballistics,  then the nozzle thrust is easily obtained as F = C_F P_c A_t.  From ballistics,  choked nozzle massflow w = P_c C_D A_t g_c / c*,   see Figure 2 below.  C_D is the nozzle throat’s discharge coefficient (or efficiency).

If all the hot gas generated in the engine workings upstream of the nozzle entrance goes through the nozzle,  then I_sp = C_F c* / g_c.  If not,  you must ratio down your calculated I_sp,  F,  and A_t by 1 + f,  where f is the fraction of generated hot gas massflow that does not go through the nozzle.

Example Problem:  Conventional Nozzle,  Sea Level and 20 kft Designs

I automated these calculations into a spreadsheet,  and verified the numbers with hand calculations.  An image of the spreadsheet for the sea level design is given in Figure 3.  In the spreadsheet,  items highlighted yellow are the user inputs,  and items highlighted blue are the principal outputs from the sizing calculations.  These are used to generate the performance table versus altitude,  which is not highlighted. 

For this example,  I assumed Pc = 1800 psia,  and a conical nozzle of 15 degree half angle.  I used specific heat ratio γ = 1.20,  and a c* = 5900 ft/sec so that specific impulse would be near 300 sec,  similar to LOX-RP1.  I used A_t = 1.0 square inch,  with a nozzle C_D = 0.99 to size flow rate.  The resulting design is a nominal 3000 lb thrust design,  completely immune to backpressure-induced separation,  since it is never over-expanded.  How the nozzle kinetic energy efficiency is calculated from half angle is discussed below.

Keeping all the data the same except for the design backpressure,  I ran the spreadsheet again for perfect expansion at 20 kft instead of sea level.  The effect is to increase the expansion ratio for a higher momentum term,  and then accept the negative pressure difference term reducing thrust below 20 kft altitude.  The gas generating chamber and throat are exactly the same.  An image of the 20 kft design spreadsheet is given as Figure 4.  The spreadsheet includes a separation backpressure estimate (see that discussion just below),  which shows the risk starts at pressures about 9 psi larger than sea level air pressure.  So,  this design is also very likely immune to backpressure-induced separation risks.

Flow Separation Risks

These can only be estimated empirically.  There are many correlations.  My preferred one uses the inverse of PR = P_c/P_e.  P_sep is the estimated backpressure,  at and above which nozzle flow separation is to be expected.  It is empirical,  and it is a rough estimate.  The designer should allow significant margin. 

                P_sep / P_c = (1.5 P_e/P_c)^0.8333

For the 20 kft design example just above,  P_c/P_e = 266.3,  so that P_e/P_c = 0.003755.  Thus P_sep/P_c = 0.013355,  and for P_c = 1800 psia,  the expected P_sep = 24.04 psia,  quite a margin above sea level pressure.  We can conclude that there is no risk of separation in the example nozzle,  all the way down to sea level,  where P_b is only 14.7 psia.  The risky backpressure is even higher at about 45 psia for the sea level design.

KE-Efficiency Correlations

Most conventional nozzles are axisymmetric.  Those streamlines near the axis are aligned along that axis,  so that the cosine factor for off-angle alignment is cos(0^o) = 1.00.  Those near the nozzle wall are aligned at the angle of that wall off the axis.  For a conical nozzle,  this is the half angle of the cone.  The cosine factor for off-angle alignment is cos(a) where “a” is the half angle of the cone.  See Figure 1 again.

Thus,  there is a distribution of local off-axis alignments for the streamlines across the exit plane.  While the “correct” way to determine the effective cosine factor for the distribution would be to integrate them for an average,  there is an easier model that is just as good.  Simply compute the arithmetic average of the centerline cosine factor value (1.00) and the wall cosine factor value cos(a),  and call that the nozzle kinetic efficiency factor:

                η_ke = 0.5 [1 + cos(a)]  where “a” is the effective average half-angle of the nozzle wall

For a conical nozzle,  “a” is the cone’s geometric half-angle.  At 15 degrees,  η_ke = 0.983.  For a curved bell,  there is a local “a” near the throat,  and a smaller local “a” at the exit lip.  One simply averages the two local a’s,  and uses that average as “a” in the kinetic energy efficiency formula.  For most practical curved bell designs,  that average “a” won’t be far from 15 degrees.  See again Figure 1.

Free-Expansion Designs By “Last Point of Contact = Perpendicular Exit Plane Model”

There are multiple techniques and geometries by which a nozzle can be made self-compensating for perfect expansion at any altitude backpressure.  They all share two features:  (1) a free streamtube surface unconfined by a physical shell before the “exit plane”,  and (2) a point of last contact with physical structure that is wetted by the propulsion stream that locates the exit plane.   We want the components of the actual distribution of exhaust velocities,  that are aligned with the engine axis. 

Most,  if not all,  these free-expansion designs can be analyzed for expected performance using the very same ideal gas compressible flow techniques used for conventional nozzles.  It is just that the order in which things need to be done is revised.  Note that the very same gas-generating chamber and throat area serves as the feed to the free-expansion “nozzle” at all values of P_b. 

Conceptually,  we are interested in an effective planar exit area located at the “point of last contact” (just as the exit lip is the “last point of contact” with conventional bell nozzles), and oriented perpendicular to the engine axis.  This is shown in Figure 5. 

Unlike conventional nozzles,  these are always perfectly expanded,  so that P_e = P_b,  as long as P_b is not exactly zero!  Once a P_b is known,  then PR = P_c/P_e = P_c/P_b is known.  One solves the PR equation for M_e at this value of PR,  which is not a transcendental iteration,  just a simple direct solution:

                M_e =  { 2/(γ – 1) [PR^(γ-1)/γ – 1]}^0.5

With M_e now known,  find the area ratio from the streamtube relation,  and use it with the throat area to find the effective value of the exit area A_e:

                TR = 1+ 0.5 (γ – 1) M_e²

                const1 = 0.5 (γ + 1)

                const2 = 0.5 (γ + 1) / (γ – 1)

                AR = (1 / M_e) [TR / const1]^const2

                A_e = AR A_t

Referring again to Figure 5,  there is obviously a distribution of streamline directions at the exit plane,  which is different for each backpressure.  Each geometry is different,  but the idea is to find the largest half-angle off of axial and use it as “a”.  This goes into the correlation for kinetic energy efficiency.  That correlation is generally for “a” < 30 degrees,  so we are misusing this here!  But,  it is the best I have at this time to offer.  Any such “a”-dependent model,  even if flawed,  is better than no model at all!

                η_ke = 0.5 [1 + cos(a)]

Instead of a thrust coefficient,  we estimate thrust directly from the calculated exit plane conditions,  remembering that P_e = P_b,  and from that thrust,  the thrust coefficient (to use with c* for I_sp):

                F = η_ke γ P_e A_e M_e²

                C_F = F / P_c A_t

One should note that neither A_e nor η_ke are constants here,  as P_b changes.  At high backpressures (low altitudes),  “a” is small,  η_ke is high,  and A_e and M_e are lower.  At low backpressures (high altitudes),  “a” is quite large,  η_ke is lower,  and A_e and M_e are high.  Exactly how “a” varies is quite geometry-dependent. 

If P_b = 0 (vacuum of space),  PR = infinite,  leading to infinite M_e and A_e.  There can be no planar exit plane,  and Prandtl-Meyer expansion says “a” > 90 degrees by a small amount.  There is no point trying to use this compressible flow analysis technique on a free-expansion nozzle in vacuum,  quite unlike a conventional nozzle!  (Which means this free-expansion design approach is inappropriate in vacuum!)

However,  for an axisymmetric center-spike design (aerospike nozzle),  one could estimate a = tan^-1[(R_e-R_t)/L_spike].  For this,  R_e = (A_e/pi)^0.5,  and R_t = approximately (A_t/pi)^0.5.  L_spike is the distance from throat plane to exit plane.  Longer is lower effective “a”,  but higher weight,  and a tougher cooling design.

I made another worksheet in the spreadsheet for axisymmetric aerospike nozzles,  embodying the above calculation techniques,  and I verified it with hand calculations.  It lays out differently,  since the sequence is different,  and more items vary with altitude.  The same grouping of design point data vs altitude performance is maintained,  and the same color-coding for highlighted items.   However,  the volume of data is larger,  requiring two figures (vs one) to display herein.

Example Axisymmetric Aerospike Problem

The fairest way to compare this type of nozzle design with any conventional nozzle design is to size both with the same P_c,  A_t,  and γ.  If thrust is the issue,  and it usually is for launch vehicles,  then the preferred performance variable to examine is thrust. 

For the example problem,  we use P_c = 1800 psia,  A_t = 1.0 in²,  and γ = 1.20,  same as the conventional nozzle examples earlier.  The same c* and nozzle throat C_D are used.  In effect,  this engine shares the very same gas generator as the two conventional examples.  The same altitude backpressures are also used,  so that this design can be compared directly to the earlier examples,  except that vacuum performance cannot be included.

The spreadsheet results are given in Figures 6 and 7 below.  The two figures together provide the image of the spreadsheet.  I have repeated the altitude data in Figure 7 for convenience. 

Comparisons Among the Example Nozzle Designs

How these designs compare,  especially as regards altitude performance,  does not “jump off the page” from tabular data.  That takes plots,  something this spreadsheet software offers.  I used the same altitudes and air pressure data for all 3 examples.  Copying selected data from each worksheet into yet another worksheet provides a way to directly plot performance from all 3 nozzles on the same page.  I did this for thrust,  specific impulse,  thrust coefficient,  and nozzle kinetic energy efficiency. 

Bear in mind that all three share the same gas generator at P_c = 1800 psia,  A_t = 1 square inch,  γ = 1.20,  chamber c* = 5900 ft/sec,  and nozzle throat discharge coefficient C_D = 0.99.  All three are roughly the same 3000 lb thrust at their design points,  within a percentage point or three. 

The thrust comparison is given in Figure 8 below.  The conventional sea level design has slightly better thrust at sea level ( by about 82 lb out of a nominal 3000 lb) than the conventional 20 kft design.  This reflects the effects of the negative pressure difference term at sea level,  for the slightly-overexpanded 20 kft design. 

The 20 kft design has about a 107 lb thrust advantage,  above 100 kft,  over the sea level design.  This reflects the larger expansion ratio of the 20 kft design,  and the fact that the exit momentum term dominates by far over the pressure difference term,  in thrust.

The axisymmetric aerospike design is “right in there” with the other two,  up to about 50 kft or 60 kft altitude.  Then its performance drops dramatically with increasing altitude,  something the free expansion is supposed to compensate!  It is a little better than the conventional sea level design at sea level,  and it remains superior all the way up to about 55 kft.  It is equivalent or very slightly better to the conventional 20 kft design at sea level,  and remains essentially equivalent to about 20 kft.  Its downturn in thrust performance is quite dramatic,  and starts at about 40 kft or 50 kft. 

It should not surprise anyone that the specific impulse trends in Figure 9 tell the same tale as the thrust in Figure 8,  since all three share the same gas generator with the same propellant massflow.  Nor should it surprise anyone that the thrust coefficient trends in Figure 10 also tell exactly the same tale,  since all 3 designs share the same gas generator operating at the same chamber pressure. 

The reason for the dropoff in aerospike performance,  versus the conventional designs,  traces directly to the trends of nozzle kinetic energy efficiency,  something that in turn depends upon the effective average half-angle of the propulsion stream bondary.  This is really nothing but the cosine factors of streamlines that are aligned off-axis.  Kinetic energy efficiency trends are given in Figure 11.

Remember,  for the conventional designs,  half-angle is locked-in by the physical bell,  right up to the exit plane.  Downstream of the exit lip,  gas expands laterally into the vacuum,  but this happens downstream of the “last point of contact”,  where thrust is actually calculated.  This is implied by how we draw the control volume about the engine and nozzle,  something shown in the lower right corner of Figure 1,  touching at that last point of contact.

For the axisymmetric aerospike free-expansion design,  the last point of contact is the tip of the spike.  The free expansion surface of the plume is inside the control volume,   as is the bell of the conventional nozzle.  At high altitudes where the air pressure is low,  the plume boundary must expand quite far laterally,  between the throat,  and the “exit plane” at the last point of contact.  This is precisely how large AR and M_e are achieved,  in order to match P_e = P_b.  Since the length of the free-expansion zone is fixed,  the boundary half-angle must be quite large at high AR.  That reduces kinetic energy efficiency. 

The two conventional designs share a constant kinetic energy efficiency of 98.3%,  as shown.  The aerospike starts out slightly better at 99.1% (due to the choice of L_spike used),  but drops below conventional at about 20 kft,  and falls ever more rapidly to only about 77.7% at 100 kft.  This traces directly to the effective half-angle of the plume boundary between the throat,  and the exit plane at last point of contact. 

That is why I included a plot of the axisymmetric aerospike half-angle vs altitude as Figure 12.  Looking at this,  please remember that half-angle is constant-with-altitude at 15 degrees for the two conventional designs.  At 100 kft,  cos(56.335^o) = 0.5543.  Averaging that with 1 inherently produces η_ke = 77.7%.

Conclusion

I don’t see any significant advantage to the free-expansion nozzle approach.  The small performance improvement is restricted to the lower atmosphere,  and this design approach is entirely inappropriate for use in vacuum!  The complications with cooling the spike outweigh any tangible performance benefits,  which are low (unless you cheat by not accounting for the streamline divergence effects).  

Update 11-16-18:  This conclusion is correct as far as it goes,  but it is also incomplete.  See the update just below the original 12 figures.  I explored the free-expansion nozzle approach a lot further,  in two different forms.  It can be made to work better at high altitudes,  but NOT in vacuum,  and not really significantly better than a conventional nozzle sized at the highest altitude consistent with not flow-separating at sea level.

 Figure 1 – Nozzle Fundamentals

 Figure 2 – Modeling Nozzles with Compressible Flow

 Figure 3 – Spreadsheet Image for 15 Degree Conical Nozzle As Sea Level Design

 Figure 4 – Spreadsheet Image for 15 Degree Conical Nozzle As 20 Kft Design

Figure 5 – Analogous Procedure for Free-Expansion Designs

 Figure 6 – Example Axisymmetric Aerospike Nozzle Results,  Part A

 Figure 7 – Example Axisymmetric Aerospike Nozzle Results,  Part B

Figure 8 – Thrust Comparison Among the 3 Designs vs Altitude

Figure 9 – Specific Impulse Comparison Among the 3 Designs vs Altitude

Figure 10 – Thrust Coefficient Comparison Among the 3 Designs vs Altitude

Figure 11 – Nozzle Kinetic Energy Efficiency Comparison Among the 3 Designs vs Altitude

Figure 12 – Trend of Effective Half-Angle “a” for Axisymmetric Aerospike Design

Update 11-16-18:  

                                                                         

When I did the original article,  I made the axisymmetric aerospike design at the same spike length as the nozzle bell length for the 20 kft conventional nozzle design.  This made its performance comparable to the 20 kft conventional design up to around 40 kft altitude,  then above that,  the ever-increasing streamline divergence angles “killed” its nozzle kinetic energy efficiency,  reducing its performance below conventional,  at the higher altitudes.  

This design approach essentially determined 20 kft as the altitude at which the effective average boundary half-angle of the circular cross-section free-expansion streamtube was 15 degrees,  same as the 15 degree conical conventional nozzles.  The fixed input here is the aerospike expansion surface’s length.  The area ratio (and expanded diameter) are determined by expanding to local pressure.  The difference between exit and throat diameters (ignoring the spike),  divided by the spike length,  is the tangent of the streamtube half-angle,  presuming a conical shape. 

I did not originally check separation pressures for the conventional nozzles,  but added that later.  As it turns out,  for this study’s 1800 psia chamber pressure,  you do not want to attempt conventional designs above about 30 kft.  This is because the margin between expected separation backpressure and sea level pressure gets to be too small to trust.

To the three designs in the original article,  I added a 30 kft conventional design,  and a revised axisymmetric aerospike with a much longer spike that has a 15 degree half angle at 100 kft altitude.  These five designs (are depicted in Figure 13 (all figures at the end of this update).

There is a different idea about the free-expansion aerospike geometry that limits expanded half-angle better:  I call it the “twin aerospike”.  Instead of being axisymmetric with the spike immersed along the plume centerline,  in the twin aerospike,  the nozzle bell is conceptually cut away,  from throat to exit,  top and bottom,  leaving two symmetrically-placed spikes along each side,  at the physical bell half angle.  This is shown in Figure 14.  

I have never before seen a proposal like this;  therefore, it is my idea.  Please give me credit for it,  if you pursue it. 

This alternate approach leads to an elliptical plume cross section,  with one diameter fixed by the position of the twin spikes.  The other diameter is smaller at low altitudes,  and larger at high altitudes.  The effective average boundary half angle is the arithmetic average of the fixed spike angle,  and the variable angle produced by plume diameter,  throat diameter,  and spike length.  The plume cross section is circular at the design point,  where the free expansion half angle matches the physical spike half-angle. 

I ran this twin spike design approach at a spike half-angle same as the conventional bell (15 degrees),  and two spike lengths,  one set by a design altitude the same as the 30 kft conventional nozzle,  the other 100 kft,  same as the added axisymmetric aerospike design.  That makes a total of 7 designs to compare,  as given in the following table:

It is very important to understand what happens to effective average boundary half-angle for these various designs,  in order to understand what happens to nozzle kinetic energy efficiency.  That in turn governs the thrust and impulse performances that can be achieved. 

Accordingly,  the first comparison plot (Figure 15) is effective average boundary half-angle vs altitude for the 7 designs.  Its cosine averaged with 1 is the nozzle kinetic energy efficiency.  That item is plotted vs altitude in the second comparison plot (Figure 16) for all 7 designs.   The other performance measures are thrust,  specific impulse,  and thrust coefficient,  as given in Figures 17, 18,  and 19. 

Looking at Figure 15,  note that all three conventional designs share exactly the same 15 degree half-angle.  They fall right on top of each other,  so only the 30k design is visible in the plot,  being the last group plotted.

Both of the lower-altitude free-expansion designs show similar upward trends of half-angle to about 70-80 kft,  with only the twin aerospike 30k design trend “bending over” with increasing altitude.  This is the beneficial effect of the elliptical cross-section shape,  with one longitudinal section fixed at 15 degrees,  and averaged with the other that has a variable boundary half angle,  to meet exit area.

The two high-altitude free-expansion designs have the same design altitudes of 100 kft,  and very similar trends of effective average boundary half-angle with increasing altitude.  The differences attribute to the averaging of effective boundary half-angle that takes place in the twin aerospike,  but not in the axisymmetric aerospike. 

I did not have available atmosphere data higher than 200 kft;  indeed such is questionable,  as we are heading into something more like free-molecule flow,  than continuum flow,  at such altitudes.  Had there been atmospheric pressure data available between 200 kft and 300 kft,  the twin 100k design would have “flattened out” the way the twin 30k design did.  Neither axisymmetric design can do that.

What those half-angle data represent is the effective average boundary values whose cosines average with 1 in the nozzle kinetic energy efficiency correlation equation.  Those data for the 7 designs are given in Figure 16.  Note that all 3 conventional designs share a constant half angle of 15 degrees,  and thus a constant nozzle efficiency of 98.3%.  There is not much room above that for the free-expansion designs to do “better” at low altitudes,  although they certainly do. 

However,  above about 30-40 kft,  the two low-altitude free-expansion designs (“axi a 20k” and “twin 30k”) have efficiencies that drop below conventional.  These drop to around 80% efficiency,  although at two different altitudes.  That difference is an artifact of the two design altitudes,  and (more importantly) how the plume cross section forces the averaging (or not) of the effective half angle.  The “twin 30k” design sort-of “bottoms out” around 80% efficient,  while the “axi a 20k” design does not.  We could “eyeball-extrapolate” the “twin 30k” design to an efficiency near 80% at 300 kft (effectively out in space);  the “axi a 20k” design’s efficiency evidently continues to drop.

Designing the free expansion nozzles at higher altitude (getting far longer spikes) is quite evidently a better deal from a fluid mechanics standpoint.  The nozzle efficiencies in Figure 16 show both those designs with similar trends to 200 kft altitude.  An “eyeball guess” says the twin 100k design might “bottom out” somewhere near 70% efficiency at 300 kft,  while the “axi a 100k” design’s efficiency will probably just continue to fall precipitously. 

Free expansion nozzle efficiencies are equal to,  or very slightly better than,  conventional,  all the way up to the design altitude;  and backpressure-induced separation is by definition no risk at all,  all the way down to sea level.  Then,  at altitudes above the design point,  the nozzle efficiencies must inherently fall off,  according to one or the other behavior,  depending upon whether an axisymmetric aerospike or a twin aerospike.  There is an altitude above which free-expansion efficiency is less than conventional.

These half-angle and nozzle efficiency behaviors combine with the fluid mechanics of expansion,  to produce the thrust vs altitude curves of Figure 17.  These show quite clearly that we do not want the lower-altitude free-expansion designs,  since by about 40 kft,  the conventional nozzles outperform them,  just as I said in the original article.  The 30k twin aerospike design does better than the 20k axisymmetric aerospike design,  mainly because of the half-angle averaging,  but neither maintains equal or better performance than conventional,  past 50 kft “for sure”. 

The two high-altitude free-expansion designs do indeed show equal or slightly superior performance to the conventional designs,  but only up to about 170-180 kft altitudes.  They “peak” at their design altitudes of 100 kft.  The “axi a 100k” design’s performance will fall precipitously past 200 kft,  while the “twin 100k” design’s performance may (or may not) “bottom out” well below conventional performance levels,  at 300 kft. 

This outcome also shows in the delivered specific impulse and thrust coefficient trends of Figures 18 and 19.  That is because “all else really is equal”,  in particular chamber pressure,  throat area,  c* velocity,  and thus nozzle mass flow rate. 

Could we “push” the free-expansion advantage to higher altitudes still?  Probably,  by making the effective average boundary half angle = 15 degrees at higher altitudes than 100 kft.  The cost,  as indicated in Figures 13 and 14,  is very long spikes indeed!  There is no fluid mechanical optimum here! 

But,  increasingly-long spikes are increasingly infeasible from both the constructional,  and the thermo-structural,  viewpoints.  It is that trade-off,  not fluid mechanics,  which determines whether either of the free-expansion design approaches is “better” than a conventional nozzle design.  And,  as shown in Figures 17-19,  the performance advantage of the free-expansion design over a 30 kft conventional design is never very large at all!  Just as the original article conclusion indicated.

Updated Conclusions

The conclusions in the original article are correct,  but a bit incomplete.  Free-expansion designs can NEVER outperform conventional designs in vacuum,  just as originally concluded.  The trends are just wrong to support such a conclusion.   

Designed at sufficiently high altitude,  the performance of free-expansion designs can be made to equal,  or to slightly exceed,  conventional designs up to some fairly-useful altitude in the 200-300 kft range.  But,  there will always be a performance penalty to pay in full vacuum!

This performance gain at some altitudes with free-expansion designs comes at a cost:  an ever-longer expansion spike (or spikes).  These can easily be infeasible for constructional,  or thermo-structural reasons.  Fluid mechanics does not limit this!

Of the two free-expansion designs considered here,  the twin aerospike approach offers somewhat better performance potential than the axisymmetric aerospike.  This is because of the way the fundamentally-elliptic cross-section of the twin aerospike forces the average boundary half-angle to behave,  relative to the fundamentally-circular cross-section of the axisymmetric aerospike. 

Two-dimensional linear aerospike geometries were not included in this study.  However,  their performance characteristics should fall within the bounds of the axisymmetric aerospike and the twin aerospike.  There will be no “breakthroughs” with the 2-D linear form,  relative to the other two. 

Conventional nozzle technology is well-established and has been flying for over a century.  Free-expansion nozzle designs (of any kind) have never been flown up to this time.  They thus cannot be considered a well-established technology,  the failed X-33 program notwithstanding. 

So,  for the time being,  my personal recommendation is just continue with conventional nozzles,  designed to (at most) about 30 kft perfect expansion,  as long as the somewhat lower sea level thrust is tolerable,  relative to a sea level-expanded design.  If not,  reduce the design altitude,  and accept the penalty at high altitudes.  Simple.  Effective.  Well-proven.  “KISS”,  which means “Keep It Simple,  Stupid”.  

 Figure 13 – Pertinent Dimensions and Conditions for Conventional and Axisymmetric Aerospike Designs

 Figure 14 – Pertinent Dimensions and Conditions for Twin Aerospike Designs

Figure 15 -- Comparison of Effective Half-Angle vs Altitude for the 7 Designs

 Figure 16 -- Comparison of Nozzle Kinetic Energy Efficiency vs Altitude for the 7 Designs

 Figure 17 – Comparison of Thrust vs Altitude for the 7 Designs

 Figure 18 -- Comparison of Specific Impulse vs Altitude for the 7 Designs

Figure 19 -- Comparison of Thrust Coefficient vs Altitude for the 7 Designs

Update 1-26-19: 

In response to questions from a friend,  I revisited the nozzle performance comparisons with one more free-expansion design:  an axisymmetric aerospike designed at 150 kft altitude.  I already had an axisymmetric 100 kft design,  and a twin aerospike 100 kft design,  both of which performed very much better at very high altitudes,  than the corresponding free-expansions designed at lower altitudes. 

These numbers in this article and its updates were run for a US 1962 standard day atmosphere table that I happened to have,  that only extends to 200 kft.  It’s not quite a hard vacuum at 300 kft,  but it is getting fairly close.  The difference is essentially irrelevant for a conventional bell nozzle,  but not for the free expansion designs.  So,  I added the axisymmetric 150 kft design to see what happened to it at 200 kft.  This verifies the trends.  A 200 kft design is possible,  but will have an even longer spike (see below).

I eliminated all the conventional designs except the 30 kft bell,  for better clarity in the plots.  That design expands perfectly at 30 kft on a standard day,  while still being unseparated at sea level.  It has the best average performance,  and with a very modest-length bell of normal expansion ratio.  Because of the negative backpressure term,  its sea level thrust is lower than for a bell optimized there,  but is quite comparable to the sea level performance for all the free expansion designs. 

The way all of these rocket engines were analyzed is at constant chamber pressure Pc = 1800 psia,  with a constant chamber c* = 5900 ft/sec,  a fixed throat area A_t = 1 square inch,  at a fixed throat discharge coefficient C_D = 0.99.  Whether conventional bell or free-expansion,  the streamtube area ratio at design is such that expanded pressure equals ambient atmospheric pressure at the design altitude. 

The conventional bell is conical at fixed 15 degree half angle.  That geometry sets its length,  which is quite short in comparison to the spike lengths of the high-altitude free-expansion designs.  The free-expansion designs had their spike lengths set such that the effective average half-angle of the free-expansion plume boundary was 15 degrees at design.  The details of exactly how that geometry was estimated is in the original article above.  For these high-altitude free-expansion designs,  those spikes are very long indeed.  Whether such long designs are practically achievable thermally and structurally is NOT addressed here.  Only the fluid mechanics is considered.

The axisymmetric aerospikes feature a circular cross section geometry,  while the twin aerospike has an elliptical cross section geometry;  thus they are different in geometry analysis and altitude behavior.  Because the twin is such a novel geometry,  I didn’t do a 150 kft version of it.  Most folks’ perceptions are more accustomed to the axisymmetric version,  or its very close cousin,  the 2-D linear aerospike. However,  they are all more-or-less comparable in overall performance.

Figure A shows a direct comparison of the conventional and free-expansion engines. All share EXACTLY the SAME chamber and throat areas.  Note the 10 inch bell versus the 45 to 108-inch long spike geometries.  For the record, the original lower-altitude axisymmetric aerospike design featured an 8-inch spike at about 20 kft.  The original lower-altitude twin spike design featured a 10.44 inch spike at 30 kft.  These expansions compare quite closely with 20 kft and 30 kft design conventional bells,  actually. 

Figure B shows the results of the expanded plume geometries in terms of effective average half-angle “a” that goes into the kinetic energy efficiency as nke = 0.5*(1 + cos(“a”)).  The conventional bell is,  of course,  the constant one,  at 15 degrees.  The two 100 kft free expansion designs show slightly-different but very comparable trends that rise rather rapidly to near 50 degrees at 200 kft.  The higher-altitude / longer spike 150 kft axisymmetric design lowers that to nearer 30 degrees at 200 kft,  but it is still obviously rising rapidly.

Figure C shows the resulting trends of nozzle kinetic energy efficiency nke,  which is the multiplier upon the exit momentum term in thrust.  It does NOT multiply the pressure-difference term in the conventional bell,  while the free-expansion designs do not have a pressure-difference term in thrust.

Figure D shows thrust versus altitude for all these four designs (conventional 30 kft design,  axisymmetric aerospike 100 kft design,  twin aerospike 100 kft design,  and axisymmetric aerospike 150 kft design).  All are quite comparable near sea level and up into the stratosphere.  The free-expansion designs are very slightly “better” than the conventional design in that range of altitudes. 

Above about 60 kft and onward to around 200-250 kft,  the free-expansion designs are significantly better than the conventional design.  Bear in mind that if the very long spikes are not practically achievable,  then this advantage cannot be realized! 

Note however that as the altitude heads toward 300 kft = 90 km = “space”,  that the free expansion trends are increasingly downward.  Those trends are quite clear:  out in hard vacuum,  the conventional bell gets better thrust performance with much shorter (and more practical) expansion structures.   

Figure E shows trends of specific impulse (Isp) versus altitude for all four designs.  These tell exactly the same story as the thrust data.  There is a significant benefit to be had with free-expansion nozzles between somewhere near 60 kft and somewhere around 200-250 kft.  This advantage is ONLY attainable IF you can practically achieve the VERY LONG expansion spike lengths!  No free expansion designs ever tested so far have ever had anything but very short expansion spikes!

Figure F shows the trends of effective thrust coefficient vs altitude for all four designs.  These curves tell the very same story as thrust and specific impulse.

My overall conclusions remain the same as they have been,  in both the original article and the first update. 

There is an advantage of free-expansion spike nozzles over conventional bell nozzles in the higher stratosphere,  but ONLY if the required very long spikes are really practically-achievable designs. 

If only short spikes are feasible,  that advantage goes away,  as the 20 kft and 30 kft designs in the original article so very clearly show. 

And,  out in hard vacuum,  the conventional bell is just inherently superior.  That is because its exit streamline directions are well-collimated,  while those of the free-expansion designs inherently cannot be.

 Figure A – Geometry and Ballistic Analysis Assumptions

Figure B – Effective Average Half-Angle Trends vs Altitude

 Figure C – Effective Nozzle Kinetic Energy Efficiency Trends vs Altitude

 Figure D – Thrust vs Altitude Trends

 Figure E – Specific Impulse vs Altitude Trends

Figure F – Thrust Coefficient vs Altitude Trends

Update 1-31-19:                                                                                                              

I updated my spreadsheet file and its user manual to a second version.  These are rocket nozzle2.xlsx and user2.docx.  In the spreadsheet file,  there are fewer worksheets,  one each for the three nozzle types,  and one for comparison plotting.  The twin aerospike nozzle model was revised to input half angle at design,  and use that to solve for the tip spike spacing Dspike. 

With both the twin and axisymmetric aerospikes,  the input for Lspike is the analog to design backpressure with the bell.  One varies spike length to obtain a certain half angle at the desired design altitude. 

With all three models,  a generic study would begin with At = 1 sq.in.  Vary that to enforce exactly the sea level thrust that you desire.  If you do adjust At,  then for the aerospikes,  go back and readjust Lspike to be compatible.  It varies proportional to throat diameter.

Yet Another Study:  “Best Long” Designs

Making use of the previous results,  generic nozzles were sized starting from At = 1 sq.in,  with Pc = 1800 psia and chamber c* = 5900 ft/sec.  The conventional 15 degree half-angle conical bell was designed at 30 kft to get best compromise performance from sea level to hard vacuum.  Both aerospikes were designed for effective average half-angle 15 degrees at 100 kft expansions,  to get as good a very high-altitude performance as seemed practical.  These produce quite long expansion spikes compared to the conventional bell length,  hence the terminology “long”.  All three had their throat areas At resized slightly,  so as to provide exactly 3000 lb thrust at sea level.   See Figure I below.

These results gave effective average half-angles versus altitude as shown in Figure II.  Of course,  that trend is constant with the conventional bell.  With the two aerospikes,  it varies quite strongly,  reflecting the perfectly-expanded condition,  although with slightly-different curve shapes.  These differences reflect the circular plume shape for the axisymmetric design,  versus the elliptical plume shape for the twin design.  Those half-angle trends produce the nozzle kinetic energy efficiency trends shown in Figure III.  Again,  the aerospike designs show varying efficiencies,  versus the constant efficiency for the conventional bell. 

The trends with altitude for thrust,  specific impulse,  and thrust coefficient are given in Figures IV, V, and VI.  They all tell exactly the same story.  Not shown is expanded Mach number.  This is fixed for the conventional bell,  and very variable for both aerospike designs. 

Conclusions:

The conventional bell produces slightly better thrust below the “long” aerospike design altitude of 100 kft,  with the aerospike thrusts increasingly worse than the bell above their design altitude.  

The “long” aerospike designs confer a specific impulse advantage up to about 160 kft.  It is slight at low altitudes,  and more significant near design,  then drops:  more advantageous for TSTO than SSTO.   

 Figure I – Geometries for the Three Designs,  with “Long” Aerospikes

 Figure II – Effective Average Half-Angle Trends With Altitude, “Long”

 Figure III – Nozzle Kinetic Energy Efficiency Trends With Altitude, “Long”

 Figure IV – Thrust Trends With Altitude, “Long”

 Figure V – Specific Impulse Trends With Altitude, “Long”

Figure VI – Thrust Coefficient Trends With Altitude, “Long”

Still Another Study:  “Short” Designs                                                                    

For the “long” designs,  the expansion spikes were approximately 4 times longer than the conical conventional bell.  Spikes that long may not be practical from a thermal or structural standpoint,  or even a practical geometric envelope standpoint. 

Accordingly,  I took exactly the same analyses and put in the short spikes for a nominal 30 kft design altitude.  These spikes are pretty close to the same length,  but no longer than,  a conventional bell.  These are the “short” designs.

Again,  I resized the generic At = 1.0 sq.in slightly to produce exactly 3000 lb of sea level thrust.  The axisymmetric and twin aerospike designs were then compared to exactly the same 30 kft design conventional bell as before.  See Figure VII below.

These results gave effective average half-angles versus altitude as shown in Figure VIII.  Of course,  that trend is constant with the conventional bell,  as before.  With the two aerospikes,  it varies quite strongly,  reflecting the perfectly-expanded condition,  although with slightly-different curve shapes.  These differences reflect the circular plume shape for the axisymmetric design,  versus the elliptical plume shape for the twin design.  They are larger,  because the spikes are very much shorter,  for the same expanded exit area.

Bear in mind that effective half angles above about 60 degrees are essentially “junk”,  as the conical plume expansion presumption breaks down (the geometry just gets ridiculous).  Those half-angle trends produce the nozzle kinetic energy efficiency trends shown in Figure IX.  Again,  the aerospike designs show varying efficiencies,  versus the constant efficiency for the conventional bell,  just not quite as good as the “long” designs,  especially at very high altitudes.    

The corresponding trends with altitude for thrust,  specific impulse,  and thrust coefficient are given in Figures X, XI, and XII.  They all tell exactly the same story.  Not shown is expanded exit-plane Mach number.  This is fixed for the conventional bell,  and very variable for both aerospike designs. 

Conclusions:

The conventional bell produces significantly better thrust at every altitude above sea level than either “short” aerospike design.   At very high altitudes,  the twin does better than the axisymmetric.

The “short” aerospike designs confer a slight specific impulse advantage up to about 40 kft.  It then drops,  the axisymmetric strongly,  the twin less so,  but still below the bell.

Final Observations:

The aerospike designs are better-suited to first stages of TSTO designs.  The “long” designs are better,  but the “short” designs are still almost competitive.  The bell nozzle is far superior for the second stage,  because its performance is superior in hard vacuum.  This same effect says the bell is better for SSTO.

 Figure VII – Geometries for the Three Designs,  with “Short” Aerospikes

Figure VIII – Effective Average Half-Angle Trends With Altitude, “Short”

 Figure IX – Nozzle Kinetic Energy Efficiency Trends With Altitude, “Short”

 Figure X – Thrust Trends With Altitude, “Short”

 Figure XI – Specific Impulse Trends With Altitude, “Short”

Figure XII – Thrust Coefficient Trends With Altitude, “Short”

Update 10-1-19:  Jet Plume Spreading Stuff

This topic is important for the plume impact forces that can blow around surface materials and excavate blast craters during retropropulsive landings.

The main point is the our intuitive feel for the effects of jet plume impact here on Earth is entirely inappropriate for jet plumes in vacuum,  or near-vacuum,  conditions.

As the Figure U-1 indicates,  the difference in fundamental behavior is quite substantial.  In evaluating this figure  bear in mind that exit Mach number Me varies from around 4.2 at area ratio 40 to around 5.4 at area ratio 200,  which pretty well covers the range of practical rocket engines.

Figure U-1 -- Jet Plume Behaviors in Dense Atmospheres and in Vacuum

Jet plumes from rocket nozzles in Earth's atmosphere have expansion pressures at the bell exit not largely different from the ambient atmospheric backpressure.  After leaving the exit,  these plumes stay relatively well-collimated,  affected only by mixing and dilution with the surrounding atmosphere.

Accordingly,  the jet blast wind pressure is essentially axially directed,  and equal to the nozzle thrust force,  at least up close where no significant mixing has yet occurred.  The mixing and dilution effect starts getting to be barely significant about 10 plume diameters downstream of the exit,  and has become very significant somewhere around 100 (or more) diameters downstream.

Jet plumes in vacuum (or near-vacuum) behave very differently!  There is a short core that remains collimated axially,  but its extent is of very limited proportion,  being at most about 10 exit diameters long.  This region is bounded roughly by the similar right triangle of exit velocity axial,  and speed-of-sound radial.  In other words,  the core cone length is about twice the exit Mach number times the exit diameter.

The rest of the plume outside that core cone immediately and drastically "fans out" radially.  This is the effect of a compressible flow physics phenomenon called "Prandtl-Meyer expansion".  This happens so as to take the finite exit-plane expanded pressure,  and immediately reduce it to the surrounding vacuum or near-vacuum pressure by sharply turning.  In vacuum,  this sharp turn is slightly more than 90 degrees.

The resulting jet blast effect is reduced to low force values very quickly by the pressure drop (wind pressure is proportional to gas pressure times gas velocity squared),  and rapidly becomes mostly radial,  not at all axial.  Only the very-limited-extent core is strong enough to disturb anything!  

The net result of this is that a rocket jet blast which on Earth would dig a big blast crater and fling copious amounts of rocks and dirt,  will have very little effect upon a plume-impacted surface out in vacuum.  This is exactly why there was only a little blowing dust and a couple of pebbles seen during the Apollo moon landings, and no big excavated craters were seen under the lunar modules after landing.

There are conspiracy theorists out there who claim we never actually landed on the moon,  and one of the things they point to is the lack of a blast crater under the lunar modules in the photos.  I just told you why they are wrong to make that claim!