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 M^{2})
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 = PR^{n} 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-ft^{2} = e σ
(T^{4} – T_{E}^{4}) for T’s in deg R and σ =
0.1714 x 10^{-8} BTU/hr-ft^{2}-R^{4}

^{}

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)M^{2}, D1 = (γ – 1)M^{2}
+ 2, and E1 = γ/(γ – 1)

N2
= γ + 1, D2 = 2 γ M^{2} – (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/ft^{3} = (P,
psfa) / [(R, ft-lb/lbm-R)*(T, R)] (“psfa”
is lb/ft^{2}, 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/cm^{2}.

#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.