On High Speed Aerodynamics and Heat Transfer

These topics are complicated,  interconnected,  and difficult to master well enough to enable doing the work.  Yet they can be intuitively understood easier than most people suspect.  This is really just an “understanding” article,  not a “how-to” article.

For atmospheric flight from high subsonic to high supersonic,  and even low hypersonic,  the standard compressible flow analyses apply,  which are based upon the ideal gas assumption.  Primarily,  that means all the kinetic energy of motion goes into internal energy,  which raises temperature.  None goes into ionization.  And,  the usual equation of state PV = nRT can be used.

This energy deposition effect shows up quite directly as the “stagnation” or “total” temperature trend with Mach number,  for any given static (thermodynamic) temperature,  and for any given gas specific heat ratio ϒ.  That equation is quite simple:

 T_tot = T_static (1 + c1 M²)  where c1 = 0.5*(ϒ – 1) 

That same ideal-gas fluid mechanics model underlies much of the heat transfer correlations that apply to various geometries and situations.  At low speeds,  even fairly high subsonic,  there is not enough viscous dissipation due to velocity shear in the boundary layer,  to raise its cross-layer temperature profile.  Thus,  under those conditions,  the static gas temperature is usually taken as the driving temperature for heat transfer to the adjacent surface. 

Once supersonic,  that boundary layer temperature profile shows a temperature peak termed “recovery temperature”,  which is only a little less than the total temperature.  Because that hotter layer of gas is so closely adjacent to the surface (being down in the boundary layer),  it effectively drives the heat transfer to that surface.  In that case,  the driving temperature is the recovery temperature.

The equation for calculating recovery temperature is also fairly simple,  but requires knowledge of the gas Prandtl number PR and whether or not the flow is laminar or turbulent:

 T_rec = r*(T_tot-T_static) + T_static  where r = PRⁿ  and n = 0.5 laminar,  0.333 turbulent

In the supersonic speed ranges where those correlations actually apply,  there are heat transfer models for stagnation zones that use total temperature,  and there are heat transfer models for lateral surfaces that use recovery temperature.  Those lateral surface models can be the simpler static temperature models,  but only if the speed is slow enough that no viscous dissipation occurs (essentially subsonic).

The “division” between “supersonic flow” and “hypersonic flow” has to do with how rapidly the overall shock wave geometry changes with increasing speed.  That rate of change is quite rapid at low supersonic speeds,  but decreases to near-zero at hypersonic speeds.  The changeover point depends upon the bluntness of the flying object.  Very blunt objects such as space capsules traveling heat-shield-forward are hypersonic at about Mach 3,  while “pointy” objects like aircraft at near-zero angle-of-attack,  or missiles,  are hypersonic at about Mach 5.

The limit to this type of compressible flow analysis occurs at hypersonic speeds when the gas begins to ionize,  throwing the underlying ideal gas-based compressible fluid mechanics model into error.  That in turn throws the heat transfer correlations into error.  Once the error becomes significant,  one needs to be using non-ideal gas assumptions,  where the equation of state PV = nRT no longer applies.  And,  one needs to be using heat transfer correlations actually developed for that regime.

There is a rule-of-thumb for the effective gas temperature that drives heat transfer,  in such hypersonic flight that ionization is significant.  That rule-of-thumb is only approximate,  but quite simple:

Temperature in deg K is numerically equal to velocity in meters/second

What Figure 1 shows is the trends of ideal-gas compressible flow-based total temperature,  and the entry rule-of-thumb temperature,  versus flight Mach number from 0 to 10,  for air at 15 C.

 Figure 1 – Calculated Driving Temperature Trends Mach 0 to 10

In the figure,  the blunt and “pointy” speeds for hypersonic flight are noted.  The two temperature estimates cross at Mach 5,  and are still close at Mach 6,  but not so much at Mach 7.  Your vehicle aerodynamic coefficients (such as drag and lift coefficients) become more-or-less constant at the hypersonic transition points indicated. 

Once you are supersonic,  the heat transfer correlations that use total or recovery temperature apply.  These and the compressible fluid mechanics techniques are the models to use,  up to about Mach 5 for sure,  maybe to Mach 6,  but probably not Mach 7.  The errors of not allowing for ionization are getting too large for any sort of design analysis that you could trust,  past about Mach 6.

This not only applies to aerodynamics and heat transfer,  but also to any sort of propulsion cycle analysis that uses ideal-gas compressible fluid mechanics!    Or to anything else that uses those same compressible fluid mechanics modeling techniques.

Figure 2 shows just how rapidly the modeling error increases as speeds increase to those in the entry range.  This is the same sort of plot of the same variables as in Figure 1,  just extended out to Mach 30.  Typical of a re-entering warhead would be Mach 15. 

At Mach 15,  the entry rule-of-thumb is that heat transfer driving temperature for the heat shield stagnation point will be near 5000 K,  and actual non-ideal gas models would get something in that same ballpark.  But classical ideal-gas compressible flow models would estimate that temperature to be about 13,000 K.  The error disparity is immense!  The difference is due to kinetic energy going into ionization instead of raising thermodynamic temperature.  It’s no longer the same gas.  

 Figure 2 – Calculated Driving Temperature Trends Mach 0 to 30

Recommendations:

If you are very “pointy” in shape,  the supersonic/hypersonic transition is about Mach 5;  if blunt,  it is just about Mach 3. 

“Pointy” objects at very high angles of attack (~20-30 degrees or more) are going to behave more like blunt objects,  because the “pointy” nose simply does not face directly into the oncoming slipstream.

You may use ideal-gas based compressible fluid mechanics up to about Mach 5,  to maybe Mach 6,  without significant error.  For flight in air,  that corresponds to roughly 2500 K total temperatures,  just about where air dissociation into ionized plasma becomes significant.

Beyond about Mach 6,  you need to be using a non-ideal gas based fluid mechanics model,  or else just real experimental data. None of the compressible-fluids-based heat transfer correlations apply.  

Once there is supersonic flow,  your heat transfer models for lateral surfaces should include viscous dissipation effects:  they should be formulated in terms of recovery temperature as the driving temperature.  It is the hot sublayer in the boundary layer whose temperature drives heat transfer.

Compressibility itself is modeled differently for heat transfer:  typically as a reference temperature for gas properties termed T* that is distinct from the average film temperature.  One needs to be doing that,  even at high subsonic speeds. 

Beyond about Mach 6,  your stagnation point heat transfer model should switch to an entry-range correlation;  below that,  you can use one based on classical compressible flow total temperatures.

If you are flying fast enough that ideal-gas compressible fluid mechanics is not applicable,  it is very likely that any propulsion cycle analysis (other than a simple rocket) is also inapplicable,  as these are almost universally based on ideal-gas compressible fluid flow.

Heat Protection Applications:

For entry hypersonics,  the situations depicted in Figure 3 apply for a blunt capsule traveling above Mach 3,  at zero or low angle-of-attack (AOA).  There is a stagnation point near the center of the heat shield.  That is the location of highest applied pressure,  and highest heat transfer coefficient “h”. 

Because of the bluntness,  subsonic flow behind the bow shock prevails over the heat shield,  with the sonic line near its periphery,  as illustrated.  Most such shapes have a “tumble-home” angle exceeding 20 degrees,  so that the flow separation point is also at the periphery of the heat shield.  

 Figure 3 – Typical Flow Field Characteristics for a Blunt Shape in Entry Hypersonics

The same tumble-home angle limits the achievable AOA to no more than the tumble-home angle,  because otherwise,  the flow would re-attach on the more-windward side,  leading directly to very much higher heat transfer coefficient “h”,  due to the scrubbing action. 

Otherwise,  the lateral sides of the capsule are within the wake zone,  where velocities relative to the surface are quite low,  and so there is little scrubbing action.  That means heat transfer coefficients “h” are low,  relative to those on the other surfaces.  These lateral sides therefore receive the least heating rates,  far below any of those seen on the heat shield surface.  They could cool by re-radiation,  if the local equilibrium material temperature is low enough to be tolerable for the material. 

 

The situation for “pointy” shapes at high angle-of-attack is actually rather similar,  as illustrated in Figure 4.  If dead broadside,  there is a stagnation line along the belly.  If otherwise high AOA,  there is a stagnation point near the nose end of the belly,  with near-stagnation conditions along what otherwise would have been the stagnation line along the belly. 

Either way,  it is the crossflow picture that is informative.  While the effective cross section shape varies with AOA,  the basic conditions are otherwise similar,  as illustrated in the figure.  The sonic line delimiting the subsonic region is much closer to the stagnation point or line,  with supersonic flow over much of the windward side.  The resulting heat transfer coefficients are higher than one might otherwise expect,  due to the larger supersonic scrubbing action,  in spite of the lower pressures. 

Figure 4 – Typical Flow Field Characteristics for a “Pointy” Shape at High AOA in Entry Hypersonics

The point at which flow separation typically occurs in turbulent flow is at (or just past) the maximum cross section width,  as shown.  Conditions in the separated wake zone are very low velocity,  with low scrubbing action,  and low heat transfer coefficients. 

If flow were instead laminar,  the separation point would occur somewhat upstream of the location of maximum width.  However,  that would obtain only on very small objects.  Otherwise,  the conditions are pretty much similar to what is pictured in the figure. 

Bear in mind that hypersonic heating is both convective and radiative.  The convective heating varies with velocity cubed,  and dominates the picture at speeds below 10 km/s in air.   Radiative heating comes from the incandescent plasma that is the shocked flow surrounding the object.  This effect varies as velocity to the 6^th power,  and dominates the picture above 10 km/s in air.

Even if the atmospheric gas is not Earthly air,  there is not a lot of difference,  qualitatively.   Specific heat ratio and Prandtl number are different,  so the numbers are a little different.  

On the windward side,  there must be some sort of ablative or refractive heat shield material,  in direct contact with a significant mass of structure within the object.  That structure is the heat sink,  into which the heat conducted inward must be deposited.  The whole process is a short transient,  only minutes in duration.  And THAT is why entry heat protection is fundamentally a transient heat-sinking process quite distinct from sustained hypersonic atmospheric flight!

Because of the far lower heating,  the lateral-side heat protection problem is quite different,  as illustrated in Figure 5.  If the entry situation is not too demanding,  and the lateral wall material can survive the high equilibrium temperatures,  then bare metal sides can be used,  as in the old Mercury and Gemini capsules returning from low Earth orbit at about 8 km/s speeds.

If the situation is a little more demanding,  similar to Apollo returning from the moon at just under 11 km/s,  there must be some heat shield ablative or refractory over the structure of the lateral sides.  This is also shown in the figure.  That substructure is the heat sink for the lateral heat shield material. 

This kind of “backshell” construction (whether it needs heat shielding or not) is required in order to protect cargo within that is “delicate”,  in the sense that its temperature may not be allowed to go high enough to re-radiate.  Such is typically in the vicinity of 1000+ F = 811+ K for steels and other alloys.

On the other hand,  the cargo might be “tough”,  in the sense that it can withstand getting hot enough to re-radiate effectively all by itself.  In that case,  there is no need for any backshell at all,  only a windward-side heat shield is required.

For re-radiation to work as the only cooling,  then the material temperature and spectrally-averaged emissivity must be high enough for that re-radiation to effectively balance all of the convective and radiative heat inputs at their peak values,  and to do this at an acceptable material temperature.

For ablative heat shields,  the heat balance for design purposes is effectively that the convective and radiative inputs must balance the re-radiative output plus the conduction inward plus the energy consumed to pyrolize.  The surface temperature (in part governing conduction inward) is effectively that at which the material has just fully charred. 

Figure 5 – Technical Solutions For Heat Protection During Entry

For refractory heat shields,  the surface temperature may freely “float” to equilibrium.  Only practical material service temperatures limit this.  Otherwise,  the balance is the same as for ablative heat shields,  except that there is no energy consumed by pyrolysis. 

Minimum conduction inward occurs when the heat sink temperature has maximized.  The temperature rise of the heat sink,  its specific heat capacity,  and  the mass of the heat-sink substrate,  must together be enough to contain all of the heat conducted into it,  during the total entry heating transient.

Steady Low-Hypersonic Flight:

For analyzing steady cruise up to about Mach 6 in the atmosphere,  the heat protection problem tends to become a steady-state issue,  not a transient heat-sink issue.  This is because the time scale is so much longer:  several minutes to a few hours. 

The same basic ideal-gas compressible-flow analyses apply,  except that your heat sink “gets full” quicker than you can reach your destination.  That means there are two,  and only two,  practical means to deal steady-state with the heat energy you continuously absorb:

(1)    Re-radiate the energy away to the environment

(2)    Put the absorbed energy into the propulsive fuel,  so that it ultimately exits the tailpipe

To radiate away the energy means the radiating surface must get hot,  it must have good spectrally-averaged thermal emissivity to be efficient,  and it must have a direct view of the environment.  The Earthly environment has a warm temperature,  which reduces the energy radiation rate somewhat.  A form of Boltzmann’s Law applies to the re-radiated energy rate:

Q/A,  BTU/hr-ft² = e σ (T⁴ – T_E⁴) for T’s in deg R and σ = 0.1714 x 10^-8 BTU/hr-ft²-R⁴

where T is the material temperature,  T_E is the Earthly environment temperature (near 540 R = 300 K),  e is the spectrally-averaged emissivity (a number between 0 and 1),  and σ is Boltzmann’s constant.

If instead you “dump” the heat into the fuel as it is used,  it will get very hot,  and you must prevent that fuel from boiling,  which produces vapor lock,  that stops the propulsion.  That requires very high fuel delivery pressures,  especially in the passages where it is the coolant for the aeroheated part.  

Depending upon the nature of the fuel,  there is also the risk of “coking” deposits,  which will plug up small passages very quickly,  even if boiling is successfully avoided.

Above Mach 6,  the compressible flow and heat transfer relations have to be replaced by their non-ideal gas equivalents,  but Boltzmann’s Law still applies for re-radiation. It’s still the same steady-state heat balance for each and every part of the vehicle.

Airbreathing Engine Cycle Analysis Applications:

Gas turbine engines of multiple types can be modeled fairly accurately with simple “pressure ratio” models,  because the compressor pressure-rise and turbine pressure-drop dominate the cycle pressure picture by far,  over all the other effects.  Those “others” would be the inlet pressure rise,  and the pressure losses associated with all the other components.  These are usually input as fixed ratios,  because the variations with operating conditions are quite small compared to compressor and turbine pressure-rise and pressure-drop effects. 

That last becomes increasingly inaccurate as flight speeds get hypersonic,  because the variations of component performance with operating conditions are going to get far more important as conditions get more extreme.  Offsetting that is this simple fact:  so far,  there are no turbine engines that have ever been operational,  which flew any faster than about Mach 3.5.

Those same pressure ratio models can be used to model subsonic combustion ramjet engines,  but this is far too imprecise to be a useful design analysis!  There are no compressor pressure-rise and turbine pressure-drop phenomena in a ramjet.  The only pressure-rise item is the inlet,  and its variation with conditions and its interplay with the pressure drops of all the other components,  dominates ramjet behavior.  These things simply cannot accurately be modeled as “typical constant ratios”. 

At the other end of the spectrum is finite-difference computer fluid dynamics modeling.  This is accurate to the extent that the turbulence model,  the combustion models,  the flow separation models,  and the ionization/non-ideal gas models are all accurate (not all codes have such).  Such analyses require great effort to set up and to interpret the results.  One analysis for one situation in one design means the investigator must run a lot of them,  for just the one design.  However,  this is about the only realistic way to model supersonic combustion ramjets (scramjets),  especially at speeds well above Mach 6.

In between these two modeling extremes are the quasi-1-D ideal-gas compressible flow-based models of subsonic combustion ramjet cycle analysis.  These are quite accurate,  up to the Mach 6 point where the ideal gas model fails,  and they require a lot less effort to set up,  and very little to interpret. 

These can be tailored to provide sizing or performance,  and for repeated performance runs across the flight envelope.  They provide fluid flow state information at every modeled location within the engine.  Those are all very distinct advantages.  But they inherently do not provide accurate results past about Mach 6. 

Recommended Simple Heat Transfer Models for Back-of-the-Envelope Stuff:

These divide into what applies in when compressible fluid mechanics is “good”,  and what applies when it is “not good”.  That changeover is about Mach 6.

               Supersonic to About Mach 6

These take the forms of lateral surfaces,  stagnation zones,  and internal duct flow (for propulsion items such as combustors and subsonic ducts).  You look up gas properties from standard tables,  or else estimate them with empirical equations. 

Turbulent flow with compressibility and viscous dissipation on a flat plate parallel to freestream;  applicable to exposed skins in high speed flight at speed V

Need:                   total T_t,  and static T,  plus fluid Prandtl number Pr and plate surface T_s

                              plus density rho and velocity V at edge of boundary layer,  and length L

Calculations:      recovery factor r = Pr^1/3

                              recovery temperature T_r = r (T_t – T) + T

                              ref temp. T* = 0.5(T + T_s) + 0.22(T_r – T)   this models compressibility

                              evaluate properties at T*

                              ReL* = rho V L / mu   (properties at T*,  V = freestream/edge of boundary layer)

                              average NuL* = .036 ReL*^0.8 Pr^1/3

                              average h = NuL* k/L   (k at T*)

                              Q/A = h(T_r – T_s)   this models the viscous dissipation, positive to surface for T_r>T_s

Source:                Chapman (ref. 2) eqn. 8.41,  attributed therein to Eckert

Stagnation-Point heating in very high speed flow (supersonic and hypersonic)

Need:                   stagnation T_t2,  P_t2,  k_t2, mu_t2,  Pr_t2,  and rho_t2 behind the shock wave,  surface T_s,  freestream V and rho;  diameter D = 2 R_n,  where R_n is the nose radius

Calculations:      Rpt = P_t2/P_t1 = (N1/D1)^E1*(N2/D2)^E2 where

                              N1 = (γ + 1)M²,   D1 = (γ – 1)M² + 2,  and E1 = γ/(γ – 1)

                              N2 = γ + 1,    D2 = 2 γ M² – (γ – 1),  and E2 = 1/(γ – 1)

P_t2 = P_t1 Rpt  (this procedure is total pressure ratio across a normal shock,  any γ)

ReD = rho_t2 V D / mu_t2

                              Cylinder:  NuD = 0.95 ReD^1/2 Pr_t2^0.4 (rho/rho_t2)^1/4  applies to leading edges

                              Sphere:    NuD = 1.28 ReD^1/2 Pr_t2^0.4 (rho/rho_t2)^1/4  applies to nose tips

                              h = NuD k_t2 / D

                              Q/A = h(T_t1 – T_s)    where T_t1 = total ahead of wave = T_t2 behind wave

Source:                Chapman (ref. 2) eqn. 8.45,  see also ref. 1 for total pressure ratio across shock wave

Turbulent flow inside a pipe or duct or tube,  with a nontrivial difference between fluid T and surface T_s;  applicable to combustor and subsonic inlet air duct inside film coefficients

Need:                   flow rate w,  duct diameter D,  fluid static T,  and surface T_s

Calculations:      evaluate all properties at bulk T fluid, plus a second mu_s at T_s

                              ReD = rho V D / mu = 4 w / pi D mu             (second form very convenient!)

                              NuD = 0.027 ReD^0.8 Pr^1/3 (mu/mu_s)^0.14       (“Seider and Tate”)

                              h = NuD k/ D

                              Q/A = h(T – T_s)                  positive to surface for T > T_s

Source:                Chapman (ref. 2) eqn. 8.16,  attributed therein to Seider and Tate

               Above About Mach 6

This takes the form only of stagnation heating.  The lower values farther away from the stagnation zone must come from experimental data for each shape and situation.  However,  for a blunt capsule heat shield,  a first-cut over-design is to size the heat shield to the stagnation zone requirements.  It must be “that thick” there,  and can be about half that thick near the sonic line.

The stagnation point heating model is proportional to density/nose radius to the 0.5 power,  and proportional to velocity to the 3.0 power.  The equation used here is H. Julian Allen’s simplest empirical model from the early 1950’s,  converted to metric units.  It is:

q, W/sq.cm = 1.75 E-08 (rho/rn)^0.5 (1000*V)^3.0,  where rho is kg/cu.m,  rn is m,  and V is km/s.  The 1000 factor converts velocity to m/s. 

The source for this (that I have in my possession) is a US customary units form of the same equation,  obtained from ref. 3.

Getting Gas Properties

If you cannot obtain more accurate data from tabulations in various references,  there are some empirical estimating relations that will get you “into the ballpark”.  These are from the “grab bag” chapter of my yet-to-be-published book on ramjet propulsion,  ref. 4.

It is always preferable to look up the properties of real gases and liquids in standard references.  For hot combustion gases,  this is often problematical.  However,  it is possible to approximate the properties from simple inputs.  I obtained these correlations informally from a colleague,  James M. Cunningham,  who was the head thermal analyst at Rocketdyne/Hercules in McGregor,  Texas,  about 2 to 4 decades ago.  These also work reasonably well for air itself,  although real data tables for air are better.  Choose a gas MW and γ appropriate to the temperature range you are working in.  Those and the temperature are all you need to estimate realistic properties for heat transfer purposes.  Ideal gas behavior is inherently assumed.

Input items as constants (not very dependent upon temperature at all):

Gas molecular weight MW           (can be summed up from the chemical balance equation)

Gas specific heat ratio γ                (usually in the vicinity of 1.2 for real combustion-product gases)

Functions of inputs but not very dependent upon temperature:

Specific heat at constant pressure c_p = 1.987 γ / [(γ – 1) MW]    units will be BTU/lbm-R

Prandtl number Pr = 4 γ / (9γ – 5)                                                          dimensionless

Gas constant   R, ft-lb/lbm-R  =  Ru/MW               where Ru = 1545.4 ft-lb/lbmole-R

Functions of inputs and strongly-dependent upon temperature:

Viscosity  µ, lbm/in-sec = 46.6 x 10^-10 MW^0.5 (T, R)^0.6  or µ, lbm/ft-sec = 5.592 x 10^-8 MW^0.5 (T, R)^0.6

Thermal conductivity k, BTU/hr-ft-R = [1 x 10^-4 (9γ – 5) (T, R)^0.6] / [(γ – 1) MW^0.5]

General Remarks:

None of these above are considered to be significantly pressure-dependent at all.  The property that is strongly pressure-dependent is density.  The ideal-gas relation to define density is:

               rho,  lbm/ft³  =  (P, psfa) / [(R, ft-lb/lbm-R)*(T, R)]             (“psfa” is lb/ft²,  absolute)

If your situation is not quite amenable to the ideal gas model,  you can still use P = Z rho R T,  where Z is an empirical function of P, T,  and the critical P_crit and T_crit for your gas.  Reference 5 is an old thermodynamic textbook that contains a universal approximation chart for Z.  Any modern text should contain the same chart.  This is more applicable to cold conditions approaching the triple point.

It should be noted that the usual Mach number formulation of compressible flow analysis does in fact assume both adiabatic and ideal-gas behavior!

References:

There are many references for classical compressible fluid mechanics.  But fundamentally,  virtually all of the basics in any of them,  trace back to the famous old NACA report number 1135:

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

There are many books available on heat transfer,  covering a variety of situations with empirical correlations for each.  My reference is an old college textbook on the subject:

               #2. Alan J. Chapman,  “Heat Transfer” second edition,  MacMillan,  1967.

There is a variety of good information on very many topics in the following reference.  I cite it for the simple entry stagnation heating equation attributable to H. Julian Allen,  equation 4B-4 page 520.  In the reference,  this is given in US customary units.  I converted this to metric,  specifically for entry calculations using speed in km/s and heating rates in W/cm². 

#3. “SAE Aerospace Applied Thermodynamics Manual”,  second edition,  Society of Automotive Engineers,  1969.

The next reference is not yet available,  but soon should be.  AIAA has chosen not to offer it,  so that I must self-publish it.

#4. Gary W. Johnson,  “A Practical Guide to Ramjet Propulsion”,  yet to be published  but copyrighted as of 2017.

This one is just another old textbook on classical (not statistical) thermodynamics.  Any modern classical thermodynamics textbook should have the same information in it.

#5. Gordon J. Van Wylen and Richard E. Sonntag,  “Fundamentals of Classical Thermodynamics”,  John Wiley and Sons,  1965.