//
Final
Modeling
Technical
to: Monitor:
NASA
Ames
Raymond
Research
Grant:
Submitted
by:
Group
NAG2708
Department
Cambridge, Investigator:
Effects
Center
Aerodynamics
of Aeronautics
Massachusetts
Principal
! l
Hicks
Computational NASA
for:
of HeavyGas Airfoil Flows
on
Submitted
Report
Mark
MA
Astronautics
of Technology
02139
Drela
Associate MIT
Institute
and
Professor
Aeronautics
and
11 May
Astronautics
92
Nq22755 (NASAC_190_7) EFFECTS _N
&TRFO[L
MODELING FLO_S
_F _inJ1
H_AVYGAS Report
(MII}
G_/uz
Uric as 0091531
1
Summary
A nonideal element
gas model
airfoil
has
program.
tunnels
employing
(SF6),
although
been
The
heavy most
developed
specific
gases. heavy
and
applications
The particular gases
retrofitted
could
into
targeted
the
are
gas modeled
viscous/inviscid
compressible
in this
be implemented
MSES
work
airfoil
has
if adequate
been
state
flows
sulfur
and
multiin wind
hexMtuoride
caloric
data
were
available. Numerical airfoil
predictions
behavior
tunnels.
in transonic
The
dominant
in supersonic smaller)
zones
effect
somewhat
with
flows, effect
are
is that
more
MSES
especially
is that
lower,
and
for a given
resistant
indicate
that
the
at the
higher
for a given
shocks
edge
are
Mach
to an adverse
nonideality total
freestream
pressures Mach
correspondingly
number,
pressure
of SF6
envisioned
number,
weakened.
a boundary
gradient
due
significantly
layer
for pressurized
local
Mach
Another
(but
in a heavy
to reduced
influences
gas
adiabatic
numbers apparently
is theoretically
heating
near
the
is valid
for
wall. As
pointed
nonideal
out
gases.
by
Wagner
Similarity
and
Schrnidt
between
two
[1], transonic
flows
smalldisturbance
can be obtained
if the
theory
transonic
similarity
parameter
1  M 2
K
[M2(7'+1)]2/3 is matched,
and
if the
pressure
coefficients
are
scaled
by
the
factor
M 2 A so that
the
above
are
quantity deft_ned
B for the
flows.
rameter the
for and
addition
of the
viscous Re
number
on
boundary
layer
that
the
parameters
M,
flows. and
the
of %, is not
effect
1 atm,
and
for
of a fixed
M.
Figure
and
SF6
Re)
Cp vs z/c at
7',
curves
good
for the
for air).
2 makes the
Clearly,
the
comparison matching
2822
airfoil
comparison at M*
a fixed
an
A
therefore
into
air
and
[2] at M
= 0.735
K
and
or K is more
M*
ACL
appropriate
scales viscous
flows. similarity
for air, = 0.765
It is rule
K,
ACL,
heavygas
parameters.
a fixed
pa
and
matching
three
at
tran
affects
one
con
additional
number,
that
between other
in the
in compressible
indicate
correspondence
RAE
and
for viscous
that
all be combined
as the
K
in Appendix
is rigorous
Mach It
gradients
experiments
as significant
local
thickness.
7_ can
numerical
Figure makes
and
flows
C shows and
to pressure
is derived
be formulated
Appendix
properties
parameters
in MSES.
cannot
displacement
Re,
gives
nearly
3 atm.
3 in turn
CL = 0.743
still
the
gas
response
Fortunately,
M °, CL,
1 compares
= 0.735
Mach
on the
The
7', which
transonic
rule
Re,
flows.
heats
inviscid
number
depends
two
employed
a similarity
Reynolds
and
Figure
M
the
the
of specific
nonideal
theory,
This
local
transonic (or
ratio
and
effects
unlikely
Apparently,
at
to
between
formulation
ideal
smalldisturbance
In
displacement highly
between
(7'+1)1__2
be matched
of an "equivalent"
7_ is introduced.
effect
also
smalldisturbance
similarity
of transonic
sonic
must
in terms
secondorder
Although text
ACL
=
flows.
for
SF_
instead
(corresponding for evaluating
to
RAE
....
Figure
1:
atm).
Cp distributions
CL
= 0.743,
transonic
flow
and
(3 atm)
SF6
comparisons, the K
the
ACL,
effects
the
unscaled.
However,
the
total
good
that match
slatted
Figure number reasonable.
= Voo/a_
incompressible
type
clear
drag
matching
a fairly
of M.
how
5.200.106
ALFA
:
2.q83
CL
=
0.7q30
CD
:
0.01367
CM
=
0.0872
L/D
=
5q.36
="
9:00
= 0.735
......
for
air,
and
be scaled
ACL
(1 atm),
SF6
and
SF6
(3
described
strong
shock
It should
be
where
by
on
stressed
any
rise
much
A,
while
these
drag
the
are
results
drag
for air,
As
smaller not
coefficient friction
SF6
expected
were
CO
drag
components
theory
flows.
(1 atm),
from if M*
the
over
should
at
this
Cp
is used
performed
as
fixed
sweep.
In
be
left
perhaps
in an experiment,
is obviously
matching
Figure
6 shows
in reference
[3].
the
a somewhat
slat on
to M again
C/; gives
5.
sweep
profile
(or alternatively
Cj, distributions
unscaled
the
4 and
drag
Mach
behavior
since
only
survey.
heavygas
airfoil
dragdivergence
The
to scale
a wake
(corresponding
flows
M
on transonic
to separate
= 0.3257
and
:
in Figures
smalldisturbance
air and
the
M*
of gas
should
from
K
between
at
further,
and
in lieu
configurations,
7 compares M*
M
it is impossible
twoelement
air produces
of the
is obtained
For highlift indicate
versus
it is not
pressure
drag
0.735
RE
NCRIY
airfoil
To illustrate
parameter
since
2822
=
= 6.2 million.
is shown
compressibility and
RAE
characteristics.
principle,
M
Re
for
2822
MACH
the
slat
= 0.30
that
a very
and
poor
gas nonideality
M*
and
the
inviscid
A freestream
for the
for air)
simply
invalid,
three
and
matching
match
in all
CL)
the
usual except
gives
number
gas cases = 2.85.
cases,
still
shock
CL
numerical
studies
a reasonably
Cp distributions
Mach
weaker
but
on at
The
of M the
main
a fixed
of course
= 0.30
Mach
a in
element.
sonic
comparison
freestream
over
Mach is quite number
in effectively
is irrelevant.
3
RAE  2.0 I"sE_
sF63 _T._ I "l'M
5F6
v 2.1
1,5 Cp
A IR _7 L
_'_I
'
:
0.735
.E ALFA
:
_2.q83 2o0,_o'
CL
:
0.7q30
co 0.01367 _. :oo872 _,o . s. 35
111 III
1.0
2822
MACH
,
T = 9.00
O.S
0.0
O.S i
1.0
j
f
Figure
2:
arm).
CL
Cp distributions = 0.743,
2.0
Re
for RAE
2822
airfoil
at
M*
= 0.765
for
air,
SF6
(1 atm),
and
SF6
(3
= 6.2 million.
RAE WSE5
1.5
Cp 1.0
2822
MACH
:
0.735
RE
=
6.200"!06
ALFA
:
2.483
CL
=
O, 7q30
C0
=
0.01367
CM
=
0.
0872
L/O
=
5q.
36
Nc_Ir
=
9.00
0.5
0°0
¸
0.5
1.0 f
Figure atm).
3: Cp distributions ACL
= 2.095,
Re
for ttAE
2822
airfoil
at
K
= 0.3867
for air,
SFe
(1 arm),
and
SFe
(3
= 6.2 million.
#
0.030
.TOTRL I
RIRFOIL /
R2822
RIR eI
R2822
SF6
1 RTM
R2822
SF6
3
RTM
/
0.020
,VlSC II
_D
.
0.010
WRVE
0.000 O. 55
O. 70
0.80
F1 Figure
4: RAE
C/: = 0.743,
2822 dragdivergence
behavior
versus
M for air, SF8 (I atm),
and
SF6
(3 atm).
Re = 6.2 million.
0.030 RIRFOIL '
R2822
/
RIR
0.020
CI]
0.010
0.000 O. 55
O. 70
O. 75
O. 80
H* Figure
5: RAE
CL = 0.743,
2822 dragdivergence
Re = 6.2 million.
behavior
versus
M ° for air, SF8 (1 atm),
and
SFs
(3 atm).
]0.0
NASA A
MSES V 2.11
9.0 8.0
=
ALFA EL
= [9.q67 = 2.8500
CO
= 0.00582
0.300
= 0.1098 =q89.49
CM L/D
7.0
SLAT 2
MACH
6.0
C ,o
i
q.O
I
J
3.0
I i
2.0
I I I
1.0 0.0 1.0
Figure
6: Cp distributions
SF6
3
5FS
i
for slatted
airfoil
in air.
¢ITM _TM
#]R
NASA
IO.OTwsES
_9.0j v2" 8,01
1 7.0i S.O
'
i
:
i
::i
:
::::::i:i!_z
SLAT =
ALF_
=
]9._66
CL
=
0.8601
CD CM
I
A
MACH
0
2
{EL
I)
0.300
= 0.00000 = 0.2355 :=:O;OO
:
I
4.0 3.0 2.0 i .0 0.0:
i
1.0
Figure
7: Cp distributions
(3 atm).
over slat at M* = 0.3257
j and CL = 2.85 for air, SF6 (1 atm),
and SF6
The bulk of the heavygas model development and applicationto transonic, inviscidflows is documented in the SM
Thesis of Marc Schafer,which isattached as Appendix A. As mentioned
previously,Appendix B derivesthe farfield behaviorof a nonidealairfoil flow. This was required for implementation of new outer boundary conditionsforthe MSES
code. Appendix C derives
the shape parameter compressibility correctionforan adiabaticboundary layerin nonidealflow. This was requiredto implement new heavygas correlations forthe MSES
integralhoundary layer
formulation.
References
[11B. Wagner and W. Schmidt. Theoreticalinvestigations of realgas effectsin cryogenic wind tunnels.
AIAA
[2] P. H. Cook,
Journal,
16(6),
M. A. McDonald,
and boundary
Jun 1978. and M. C. P. Firmin.
layer and wake measurements.
Assessment,
AR138.
AGARD,
Aerofoil
In Ezperimental
RAE
2822 pressure
distributions
Data Base for Computer
1979.
[3]E. Omar, T. Zierten,and A. Mahal. Twodimensional wind tunneltestsof a NASA airfoil with varioushighlift systems.Contractor Report 2214, NASA, [4] M. Drela.
TwoDimensional
Equations. Report
PhD thesis,
Aerodynamic
Design
Dec 1985. Also, MIT Gas Turbine
and
supercritical
Apr 1973. Analysis
& Plasma
Using
Dynamics
the
Euler
Laboratory
No. 187, Feb 1986.
[5] J.D. Cole and L.P. Cook. Mathematics [6] M. B. Giles Journal,
Transonic
MIT,
Program
Aerodynamics,
and Mechanics.
NewHolland,
and
Twodimensional
M. Drela.
25(9),
Sep 1987.
[7] D. L. Whitfield.
Analytical
AIAA781158,
Transonic
description
volume
Amsterdam,
New York,
transonic
of the complete
30 of NewHolland
in Applied
1986.
aerodynamic
turbulent
Series
boundary
design
method.
layer velocity
AIAA
profile.
1978.
6
N
!
Appendix
A
Modeling
of Heavy
Gas
Effects
on
Airfoil
Flows
by Marc
Submitted
Alan
to the Department
Schafer
of Aeronautics and Astronautics
on May
3, 1992
in partial fulfillmentof the requirements for the degree of Master
Thermodynamic
of Science in Aeronautics and Astronautics
models
were constructed for a caloricallyimperfect gas and
for
a nonideal gas. These were incorporated into a quasi one dimensional flow solver to develop an understanding the perfect gas model.
of the differencesin flow behavior between the new models and
The
models
were also incorporated into a two dimensional flow
solver to investigate their effectson transonic airfoilflows. Specifically,the calculations simulated results but those
airfoil indicated
that
matching
testing that
in a proposed the
high Reynolds
nonidealities
caused
of an appropriate
number
significant
nondimensional
heavygas differences
parameter
led
test
in the to flows
in air.
Thesis
Supervisor:
Mark
Drels,
Associate
Professor
of Aeronautics
and
facility.
Astronautics
flow
The field,
similar
to
4
Acknowledgments
I would liketo expressmy thanks to allthose who made thisthesispossible.First, to Mark Drela whose brilliance and ingenuityhave served as an inspirationin allof my studies.Also, to Harold 'Guppy' Youngren whose leadershipduring the Daedalus projecthelped me to reMize what itreallymeans to be an engineer.
I would also liketo thank my parents and the rest of my family. Without your support,I never would have made itas faras I have.
My
appreciationalso goes to the NASA
Ames
research centerand the NDSEG
fellowshipprogram without whose financial support thisthesiswould never have happened.
PRECED:NG
PAGE
BLANK
5
NOT
FILMED
Contents
Abstract
3
Acknowledgments
1
Introduction
10
2
Real
11
3
Gases
2.1
Calorically
2.2
Nonldeal Gases
Solving
the
Imperfect
Euler
Gases
...............................
Equations
3.1 Calorically Imperfect Gas
3.2
4
NonIdeal
.........................
11
12
15
..........................
Gas .................................
16
17
20
Results
4,1
One Dimensional
Duct Flow .........................
21
4,2
Two Dimensional
Res_flts
23
5
Conclusions
A
Curve
pRECAE_.r2.!NGPAGE
Fit
..........................
28
For SFs
State
2T
Equation
..... FILMED _:,_;._.._ _w:3T ,.
7
11
B
MSES
Bibliography
Subroutine
for NonIdeal
Gas
Model
28
43
jJ7
List
4.1
Stagnation and
Pressure
SF8 at latin
of Figures
Ratio(Strength)
vs.
Upstream
Mach
No.
for Air
and 3atm
.........................
2O
Duct Flow
.........................
21
4.2
One dimensional
4.3
Shock Strength
and
Location
vs. _ .....................
22
4.4
Shock
and
Location
vs. Z0
22
4.5
Comparison
of Air and SF6 at Fixed M and
4.6
Comparison
of SFs at latm
and
3atm to Air, M* = .740, CL = .9
4.7
Comparison
of SFs at latin
and
3stm to Air, M" = .732, CL = .75
4.8
Comparison
of SFe at latm
and 3atm to Air, _¢ = .439, ACL
Strength
9
....................
Cr,
.............
24
= 2.18
. . .
24
. .
25
. .
25
Chapter
I
Introduction
In the past few decades, the design and development of large transport aircrafthas reLiedon wind tunnel data taken at significantly lower Reynolds numbers than those found in operation. The drawbacks of thissubscaledata become apparent when one considersphenomena
such as attachment linetransition or similaraspectsof boundary
layerbehavior at high Reynolds numbers.
The need for accurate wind tunnel data clearlymandates the constructionof a suitablehigh Reynolds number testfacility. However, the cost of buildinga largeatmospheric tunnel and largetunnelmodels isprohibitive. Higher Reynolds numbers are oftenachievedby pressurizing tunnelsto effectively increasethe densityof the air.This alternative ispracticalonly up to a point.
A potentialsolutionfollowingthe same basicidea reliesupon the use of gases with significantly higher molecular weights than air. Candidate gases include Freon12 or SuLfur Hexaflouride(SFs), but the use of nonbreatlutblegases clearlycauses some problems. These problems willlikelybe insignificant to the cost and operationaladvantagesof such a facility. Combining heavy gases with pressurization would allow test Reynolds numbers comparable to those on largetransportsin flight[1].
One complicationisthat Freon and SFs have si_plificantly different thermodynamic propertiesthan air,especial]y at elevatedpressures.Heavy gases do not followthe ideal equationof stateP  pRT nearlyas wellas airdoes,nor do they maintain a constant ratioof specific heats _  c_/c_ over any significant temperature range. The following discussionwillattempt to quant_y the potentialimportance of these effects through a computational study.
Chapter
Real
The thermodynamic
relations
2
Gases
specifically
subject
to real gas effects are the state
equa
tion
p : p_T
(2._)
and the caloric equation, h = /cpdT
(_.2)
= cpT
theseparticularforms only being validfora perfectgas. Real gas effects may be divided into two cate$ories:
I. Calorically imperfectgases forwhich cp depends on temperature, but which still satisfyequation (2.I). 2. Nonideal gases for which cp depends on both pressure and temperature, and equation (2.1)no longerholds.
The firsteffectresultsfrom the introductionof multiplevibrationalmodes for polyatomic molecules which become more important at higher temperatures. The second effectdepends on intermolecular fozceswhich become strongeras a gas moves towards liquefaction, ie.higher pressuresand lower temperatures.
2.1
Calorically
Imperfect
Gases
The only difference between a perfectand an imperfectgas stems from the dependence of c_ on temperature
in the imperfect
case.
A cursory _T_mln_tion
for $F6 shows that,
in the range of temperatures
11
of experimental
data
likely to be found in a wind tunnel
test,thisdependence islinearin temperature.
(2.3)
%(T) = a + bT Therefore,
equation
(2.2) becomes bT 2
(2.4)
h(T) = _T + 7which may be easily inverted
to find T(h).
 _+
2.2
NonIdeal
(2.5)
+T
Gases
The stateequationfora perfectgas (2.1)derivesfrom a kineticmodel of gas molecules which assumes that themoleculesarepointmasses and that they do not exertany forces on one another except instantaneouslyduring collisions.Clearly these assumptions become lessaccurateas the molecular weight of the gas increases.Van der Wsals's equation
(p + p2 )I1 contains molecular
two correction attraction,
to equation and _ corrects
(2.1):
= pRT
a corrects
for the volume
the pressure of the molecules
/26) to account
for inter
themselves.
Using a nonidealstateequationlikeVan der Waals's causes many seriouscomplicationsas enthalpy,%, % etc.now depend on pressureas wellas temperature. Despite these complir2ttions, enthalpyand entropy must remain stateva_'iables regardlessof the form of the state equation. That is,localentropy and enthalpy must depend only on the localpressureand temperature and not on the upstream conditions(ie.the gas
history).
Liepmann and Roshko [2] equatethisconditionwith therequirementthat a canonical equationof statemust have one of thesefour forms:
e = e(_,p)
(2.T)
h
g Here e  h  p/p
is the usual
internal
=
h(s,p)
(2.8)
=
f(T,p)
(2.9)
=
g(T,p)
(2.10)
energy,
f
 e  Ts
is the free energy,
and
g _ h  Ts is the free enthalpy.
For a conventional specifying readily
flow solver,
the enthalpy
defintion
(2.8) appears
the state in this specific form is not convenient
available
to the flow solver.
Liepmann
because
and Rosh.ko propose
best; however,
the entropy
s is not
a more suitable
_LP = z(p,r)
(2.11)
pRT
which requires
T(p, h) to have a form which makes
For a Van der Waals's
form
h a state
variable.
gas
Z =
1 1  _p
ap fit
(2.12)
which clearlyapproaches the ideal stateequation fora, _9 _
O. For typically
small
valuesof a and/3
Z
_
1 + p
where the second approximation Roshko
write equation
(2.13)

_ 1 +
_
(2.13)
is made to make Z = Z(p, T) explicitly.
Liepmann
and
in more general form as
Pc with Pc and Tc being the critical pressureand temperature of the gas,and _bevidently being a universalfunctionwhich they tabulatefor gases other than airbut with approximatdy
the same molecular
curve to experimental
data
weight.
For heavier
gases such as SFs it is best to fit a
as explained
in Appendix
A. For SFs, s good curve fit takes
the form
= c= It is now necessary function
to determine
h(p, T) can be obtained.
+
the specific heat c_pa_ity Liepmann
13
and
(2.15)
+
Roshko
c_(p, r) *o that the enthalpy combine
two forms of the
equation
of state
h(p, T) and
s(p, T) into the fundamental
reciprocity
relation
between
h(p,r) _d p(p, r) Oh Op which
is valid for any gas.
Oh Since Oh/Op = 5(r) the pressure
_
Combining
TO(1/p) OT
I p
this with the state
RT 2 (OZ)p
only depends
(2.16) equation
RT¢ _,(_)
on the temperature,
_ both
(2.11)
gives
9v(r )
(2.17)
h and cp must
be linear
in
as follows.
h(p,r)
=
/cp(r)dT
+ pF(r)
(2.18)
Oh
cp(p,r) = aT =
(2.10)
d_" cp(T) + p__
= cp(r)  R _p
As in the case of the calorically
imperfect
6p(T)
Substituting
(2.20) T_ _b" (_)
(2.21)
gas, cp(r) has the form
= a + bT
(2.22)
thisinto the enthalpy equationgives
bT _ + pRTe h(p,r) = aT + V Pc ¢'( T/ ) It is also possible
to determine
the caloric
equation
(_) as e(p, r) [3].
14
by expressing
(2.23) the internal
energy
Chapter
Solving
These
gas models
the integral
may
the
be readily
form of the steady
3
Euler
integrated
Eater
into an existing
(pff . h ff + p_) dA ho = h +
These equations
p to the enthalpy
used to capture
conditions
where () denotes
0
(3.1)
=
0
(3.2)
constant
(3.3)
lu__2 = 2
of shock
the local Mach number
the equations,
quantitiy
and (),el
p =
_/_._
Y
_'/Y,d
=
h =
and
the upwinding
while the boundary
losses require the local stagnation
to nondimensionalize the dimensional
with a state equation
h and the density p. In addition,
the shocks requires
and evaluation
It is desirable
=
are exact for any fluid flow, but must be supplied
to relate the pressure scheme
flow solver which solves
equations:
j_._dA f
Equations
conditions.
the following
denotes
a reference
scheme
is used
quantity:
h __P_
Furthermore, nondimensional
%, _,
and R are nondimensionalized
using
R resulting
in several
new
parameters.
a
=
a/R bT,,_ 2a
15
]q
For the resultspresented here, the referenceconditionsare chosen to be stagnation conditions.
3.1
Calorically
Imperfect
Gas
The nondimensional form of the caloricequation which governs the behavior of the imperfectgas is:
h(r)
=
/ %dT
(3.4)
=
aT + a_T 2
(3.5)
which may be invertedto give T as a functionof h. T(h)
With
T obtained
specified speed
from
value of p.
=
h, p may
I
+ y_l +4nh/a 2j9
be determined
The local Mach
number
(3.6)
using the ideal
comes from
gas law (2.1)
the familiar
defintion
and a of the
of sound:
0p a2 =
_.
=7T
(3.7)
The localvalue of 7 may be found from equation (2.3). 7 =
The last remaining pressure,
density,
conditions
•
difficulty
a + 2a_T 1  a  2a_T
is the determination
and temperature.
from flow conditions.
_oo =
% =
These relations The familiar
of the isentropic are necessary
perfect
relations
to calculate
between
stagnation
gas relations _A_
1
1+
(3.8)
M 2
do not hold for a calorically
imperfect
The proper forms are obtained
gas.
from the formal statement, dh = T d# + dp P
16
(3.9)
_'_
and for an isentropic
process
ds = O:
dh = _
(3.zo)
P From the definition of enthalpydh = cpdT, and foran idealgas p/p = T, so equation (3.10)becomes cp(T)dT
= dp 
T
(3.1_)
p
Integratingthisequationgives
po = exp(_logT + 2_(1  r))
(3.12)
P and the isentropicdensityrelationthen followsdirectlyfrom the stateequation. p T(ho) Po T(h)
p = Po
(3.13)
Strictlyspeaking,solutionof the Euler equationsrequiresnothing else.However, if a NewtonIL_phson techniqueisused, allof the necessaryequationsmust be linearized for the Jacobian matrix. In the case of the calorically imperfectgas, the equations axe slightlymore complicated than for a perfectgas, but they may still allbe written explicitly. Therefore the linearizations axe easilydone by differentiating the relevant equations.
3.2
NonIdeal
Gas
The nondimensional equationsdescribingthe nonidealgas are the stateequation
p
1 + p,_(,_)
pT
(3.14)
Zo
and the caloric equation.
=
.q,,.
z ,l 1
(3.15)
Z0 isanother parameter which may be describedin terms of Iand r.
= 1 + _(
zo = po o
1
)
(3.16)
17 ,,4"
The nonidealgas presentssome difficulty as the enthalpydepends on the temperature and the pressure.Therefore,from equations (3.14)and (3.15),p and T may be found using a NewtonRaphson
system to drivethe followingresidualsto zero. p
I + _(_)
R1(p,r)= pT
(3.17)
Zo
(3.1s)
_r + _T2 + p ¢'(_.r )
R2(p,r)= h
The localMach number depends on the speed of sound which must be found from the definition: a2 :
_p,
(3.19)
This iscalculatedas follows:
dp but dh = dp/p
for an isentropic
=
_P Op h dp +
process,
°2_
OP IIp _
dh
(3.20)
and hence
(3.21)
_
The local7 reallyhas no me_nlng and need not be calculated.
The extra complexity of the nonidealgas appears in the calculationof the sensitivities. Since p and T axe found by an iterative processthey must be found by perturbing the Jacobian matrix of the converged NewtonRaphson and p is related remain
to a perturbation
system. A perturbationin h
in p and T by the condition
that the R(p, r, h, p) must
zero. 6RI
Numerically

inverting
"
_
this system
oa
Jr
gives the required
derivatives.
(3.23)
The second derivatives are found in a similarfashionstartinginsteadwith !_ a_ the residuals.Using a subscriptnotationforthe derivatives (_
Jr
18
and _p
_=Ph):
(3.24)
_
A similar system with Rzp invertinggives _
and R_p as residualsisalsoformed. As above, numerically
: _, _
: __p _ etc. These manipulationsare implemented in
the sottrce code in Appendix B.
The lastremaining task is calculationof the stagnationconditionsand, again,it is not possibleto find an analyticexpression. Another NewtonRaphson
system is
constructedwhere the first residualcomes from equation(3.15):
RI : ho h(p,T)
(3.25)
The second residualisderivedby rearrangingequation(3.9) dh
ds
Integrating
T
+
dp
pT
(3.26)
=
_dT
+ d(p_) 7"

_dT
+ d(lx/rl_
dpz p e) 
(3.27) wd(p_)
dp P
(3.28)
gives:

(3.29)
The second residualmay then be formed
R2 = _1. siP, T)
(3.30)
where sz is the entropy of the staticconditions.
Driving these two residualsto zero gives the stagnation conditionsPo, To. The dezivatives_p,_z, etc,needed forthe NewtonRsphson
solvermay then be found by
perturbing the converged Jscobian matrix and relatingthe resultingderivatives to the staticconditionsthxongh the chain rulesad eqtmtions (3.15)sad (3.29).This process is identical to the one used above to findp sad T and theirderivatives.
19
Chapter
4
Results
After developingthe models forthe calorically imperfectand nonidealgases,the next step was to evaluatethe differences these changes caused in inviscidflows.The primary quantities of interest are thelocationof shocks and theirstrengthwhich isdefinedas the ratioof of stagnationpressuresacrossthe shock. For a perfectgas,the shock strength may be expressedas a functionof the upstream Mach number M1.
po_.__: = [1+ y+ 27 1_''' _,,_I)]I/('I)[ m_ However,
for the nonideal
(7+I)M_ [(#T)M_'_ 7  2'],/(,I)
gas, this relation
must be calculated
1.000
""'.._ _.._
0.9511
,..
_
P er£ect
......
Po: Po:
",
%" ",
0.9049
N.
"',
%'% ",.
%'%" %.
•, "',
,
1._
Istm 3arm
"_%
.
1.2oo
numerically.
, """
Strength
0.8S0
(4.1)
,
,
%%"
1._
,
M1
Figure
4.1: Stagnation
SFs at latm
Pressure
Ratio(Strength)
and 3atm
20
vs. Upstream
Mach
No. for Air and
2.00 __......
Perfect rmperfect NonIdead
Math 1.20
I; t_
0.40
!
I
I
o.2s
0.00
O.SO X
'
0.TS
'
1.o0
Figure 4.2: One dimensionalDuct Flow
4.1
One
The first
Dimensional
comparison
ing/diverging
Duct
of the different
Flow
gas models
was a study
of the flow in a converg
nozzleusing a quasi one dimensionalEuler solver.This flowischaracter
izedby sonicflow at the throat with a shock downstream to match the specifiedexit pressureas shown in figuxe(4.2).
As a basis dimensional
for comparison
reference
of the
enthalpy
(hopo/Po)
ho =
With
ho held constant,
also held tittle
constant.
7 therefore
Under
different
these
was made
models
in a duct
equal for all three
flow,
the
non
cases.
3' 71
(4.2)
=
a(1 + ]3)
(4.3)
=
a(1 + a) + x_'(_)
(4.4)
zo
depends
on a, _, _r, and _'. The exit presure ratio is
conditions,
or no effect on shock strength
gas
the slope of the c_ versus T curve 09) had
or position
a e(4.s).
21
relative
to the perfect
gas as shown
in
0.9440
__ ___
0.860
Peffect Imperfect
__ Perfect ___/mperfe_
0.84G
0.9360 Location
Strength
0.82C
0.9280
0.9200
I
I
0.00
I
!
1.00
0.80_
!
2.00
o.oo
3.00
1;0
Figure
For
the
combined
nonideal into
functions
Zo.
of Z0 and
stagnation
enthalpy
the
effects
of the
gas.
The
difference
becomes
less
'
2;o
'
and
may
3_00
Beta
Beta
gas,
4.3:
Shock
_ and
Figure(4.4)
r are not shows
the corresponding as above.
the
plots
by changing
in shock
strength
and
and
really
Location
vs.
independent
variation
perfect
These
nonideality
Strength
in shock
gas results clearly
show
7 as in the position
parameters strength
with
7 adjusted
that case
becomes
and
position
to preserve
it is not possible
be as the
to mimic
of the
calorically
imperfect
larger
and
as the gas
larger
ideal.
0.9400
__ ___
0.860
Perfect NonIde,d
__ ___
0.S40 /
Strength
ss
Location
/
.s J
0.9300 ////_'
0.820
// // //
0.9200 0.60
0._)
'
O.SO00.60
1.00
'
Figure
4.4:
Shock
Strength
22
and
Location
0.80
Zo
ZO vs.
Zo
'
1.00
Perfect NonIde_
The Last test conducted effects
of the various
numerical
scheme.
with the one dhneusional
gas models
flow model
on the upwindlng
The flow solver
drives
scheme
the momentum
was to determine
needed
equation
for stability residual
(4.5)
2
speed
is defined
qi
and p_ is nonzero
as
:
qi
only if M_ is greater
_(M,(q,))
Initially,
the exact
earizations converged
and used in the upwinding
value of 7 even though perturbations.
_'/(qi
(4.6)
 q_l)
than Me.
1
(4.?)
M?J
at each node along scheme.
analysis
with all the necessary
Under these conditions,
the upwindlng
the stability
Using a constant

= 7
7 was calculated
with Mc ___1. However,
of the
to zero,
R1  p_qiA_(¢_  q;_) + p_A_  p_IA___ +/_ + P_I (A_  A__I) where the upwinded
the
is relatively
used to derive
value of 7 had absolutely
l_
the flow solver
insensitive
to the exact
equation(4.7)
ignored
7
no effect on the viable range
for Mc or the rate of convergence.
4.2
Two
Dimensional
Results
The subroutine
which appears
element
of the two dimensional
version
Numerical
experiments
clearly demonstrate Mach distributions
in Appendix
transonic
Airfoils
tests
airfoil design/mmlysis
the effect of the new gas model.
inviscid
Figure(4.5)
freestream
and 3atm and
in heavy
a stagnation
the airfoil
code ISES [4]. cases to more
conditions
Mach number and lift coefficient. The SFs is characterized temperature
performance
and Note
by stagnation
of 310K.
gases will be much more worthwile
be found so that the tests reflect
the multi
shows am overlay of the
for a test airfoil run in 5Fs at two different stagnation
that they are not at the same angle of attack. of latin
into MSES,
carried out were limited to singleelement
in air. All three cases axe at matched
pressures
B was incorporated
if some relationship
in air.
may
The only parameters
2.0
RRE
_.us ,Y
2822
IEL
1)
i_,l
J L
I.Sl
1,0
//_/SPII
(161m)
" t
......
11.51 I
0.0! I
0.51 l.n
Figure
which
4.5: Comparison
may be adjusted
ditions,
and angle
M and
C_ constant:
in a wind tunnel
of atttack
experimentatatJon,
of Air and SF6 at Fixed
or CL.
clearly,
test are the Math
Fignre(4.5)
shows
the best match
sonic conditions.
Figure(4.6)
(latin
at the same M*.
3atm)
2.0
number,
an attempted
this is not an eft'ective technique. was achieved
After
velocity
stagnation match
_RE
2022
con
keeping
a good
deal of
gases at the
to the speed of sound at
shows the case in air from fig_ze(4.5)
mill
C_
by running the different
same M* which is defined as the ratio of freestream
and
M and
tEL
compared
with SF6
l)
V_T,I
t.5 _%///
C_.
r_ l (l,.m)
s_'e(3,o)
[.0
0.5
o.o: O.S
1.0
Figure
4.6: Comparison
of SFe at latin
24
and 3atm to Air, M* = .740, Cr_ = .9
2822
Rfi[
((I.
tl
vl.I
1.51
/_ •_z
,s,lJr
_
SPe (3.*=)
I ......
0.5
l
0.0_
11.5[ 1.0
Figure
4.7: Comparison
of SFs at latin
and 3arm to Air, Mr* = .732,
A case with s weaker shock, figure(4.7) The match
is slightly
was used to further
worse, but this is to be expected
more sensitive
to small changes
M*, Anderson
[5] proposes
in M than a strong
mateh/ng
because
CL  .75
verify this relationship. a weak shock
one. As an alternative
the small disturbance
similarity
is much
to matching
parameter
_ and
ACL where
(,M_(7, A
+ 1))2/s
(4.8)
' + z) z
(4.9) _g[
211:P1 ((/
I!
Vi.I
l.S
Cp l.O
___
s_
illllt,lte_lll
13.m)
istTl_VlVlllllelllgl_lllll
0.5
0.0
O.S
1.0
Figure 4.8: Comparison
of 5Fs at latm
and 3arm to Air, x = .439, ACe
25
= 2.18
Chapter
5
Conclusions
The models derivedabove adequately describethe thermodynamic behaviorof nonideal and calorically imperfectgases.Despite some minor complicationsin l£nearizing these models, they were implemented in routinessuitableforincorporationintoexistingflow solversbased on Newton's method. First,a quasi onedimensionalflow solverwas used to examine the in_uence of the variousnondlmensionalparameters which govern the behavior of the different gases.
Transonicairfoil testcasesfor air and SFs were then used to study the influenceof parameters which may be controlledin a wind tunnelexperiment: stagnationpressure, freestreamMach number, and angle of attack. The goal of thisstudy was determine the conditionsunder which a wind tunnel testin a heavy gas would produce results comparable to those found in air. Matching M*
and CL or _ and ACL
were both
effective for the test cases presented here. Further study is necessary to determine which isbest formultielementcases.
The resultsare encouragingin that they definitely hint at the possibility of directly relating mentally SOWS.
heavy
gas test data to performance
the mode]
for SFs,
and to invest_ate
in air. It is first necessary
to verify experi
the effects of nonideal
gases on viscous
Appendix
Curve
A curve
Fit
For
A
SF6
fit may be found for the function
State
$ (_)
Equation
for any gas given experimental
state
data. With the density(p) measured at a number of different pressures(p) and temperatures(T), a vectoris definedcontainingthe difference between the realgas and a perfectgas at each data point. at_t
 1
Z=
(A.1)
_I Defaxing0 = _, the matrix A containsthe stateinformation.
1
•
:
.oo
:
:
:
(A.2)
/_.e!
The goal isto find a stateequationagreeingcloselywith the experimental data in but of the simple form: e,,
Z(p,T)=l+
p
f  C,_ L
_1 (A.3)
C._I ... Co] 1
Therefore _ A_ and _ is found
by the technique
of linear
regression:
= (ATA)ZATZ The results presented data for SF6.
in this thesis were based on a quadratic
The required
(A.4)
data may be found in [6].
27
(A.S) fit for _bfrom approximate
Appendix
MSES
Subroutine
Jubroutine
for
B
hgpare(alfl,btal,
NonIdeal
Gas
ccO,ccl,cc2,
taul,
Model
hO)
C
nonideal derived
Initializes Formulation
Input: alfl
Constants
gas routines. Schafer SM thesis.
in
for
Cp(T)
in
caloric
equation:
Cp
= a(1
+ bT)
betl
taul
Constant
ccO ccl
Constants
cc2
in
phi(T)
in
defining
phi
= cO
nonideality
phi(T)
+ cl(tau/T)
factor
in
polyuonial
+
c2(tau/T)**2
Z(p,T)
fore:
Ousput: hO
Enthalpy
Tnternal
at
reference
conditions
pO,
TO
output:
zO
Nonideality
factor
Z(pO,TO)
at
reference
conditions
C .......
real*4
implicit common
/nongss/
coamon
all, /nonfit/
k k
c2,
(ah,m,oz)
bta, cl,
pi,
tau,
zO
cO
c pu$
input
all
=
bta
= btal
t_
•
paraaetera
into
comaon
blocks
all1
taul
cO =ccO cl
:ccl
c2
= cc2
pi
:
c 1.0
c calculate zO
=
hO :
reference
1.0
+ pie(c2/tause2
(airs(1.
+ bta)
nonideality
factor
+ cl/tau
+
+ pi/tausphid(1./tau))
28
and
enthalpy
cO) /
zO
_/
subroutine
nideal(hO,r,q,
p
t
,p_r
,p_q,
msq,msq_r,msq_q)
C ....
c
Calculates
pressure
c
stagnation
enthalpy,
and
Mach
density,
number and
for speed.
c c
Input
c
hO
: stagnat
c
r
density
c
q
speed
ion
enthalpy
c c
:
Output
c
p
pressure
c
p_r
dp/dr
c
p_q
dp/dq
c
msq
square
c
msq_r
dM" 2/d.r
msq_q
d]4" 2/dq
c C ....
of
Math
number
N'2
"
implicit
set
static
h
= hO
h_q
=
subroutine
real*4
(ah,n,oz)
enthalpy 
0.Seqe*2 q
ngasp$(h,r,p,p_r,p_h,p_rr,p_hh,p_rh,
29
specified
•
_,t_r,__h,t_rr,t
C
Calculates
C
specified
pressure
and
static
temperature
enthalpy
and
hh,t_rh)
for
density.
C C
Input:
C
h
enthalpy
C
r
density
C C
Output
:
C
p
preHure
C
p_r
dp/dr
C
p_h
dp/dh
C
p_rr
d'2p/dr*2
C
p_hh
d'2p/dh'2
C
p_rh
d*2p/drdh
C
t t_r
t emp er aSur • dr/dr ... etc.
C
C ........................................................
implicit
real*4
(ah,m,oz)
dimension
a(2,2),
b(2,2),
bh(2,2),
couon
ai(2,2),
aih(2,2),
air(2,2),
br(2,2)
/nongu/ all,
bta,
pi,
tau,
zO
C
c ....
Newton data
convergence eps
tolerance
/5.0E6/
C
c ....
initial
guess
from
if(bta.eq.O.O) t
imperfect
ideal
gas
then
= h/all
else t
=
(1.0
+
sqrt(1.O
+ 4.0ebtaeh/alf))
to
converge
on
/
endif p = ret
Newton itcon do
=
100
set
loop 16 iter=l,
and
correct
p,t
itcon
lineazize
nonideality
ttc
=
ttc_t
= l./(taue_ee2)
factor
1./(taue_)
z = I. z_p = z_t =
÷ p_pi*phi(tt¢) pi*phi(ttc) p_pi*phld(t¢c)*ttc_t
C
residual
I:
eta_e
p/(r*t)
equation
reel
=
rl_p
=
l./(r*_)

 z
/zO
rl_t
:
p/(rstse2)
 z_t/zO
z_p/zO
3o
Z(p,t)
(2.0ebta)
tml
=
tnl_p
= O.
(alf*t
+
tml_t
=
tm2
= p*pi/tau*phid(ttc)
/
zO
tm2_p
=
/
zO
tm2_t
= p*pi/tau*ph/dd(ttc)*ttc
/
zO
(all
all*bract**2)
+ 2.*alf*bta*t
)
2: caloric
zO
equation
= h  (tml
+ tm2)
r2_p
=
 (tml_p
+ tm2_p)
r2_t
=
 (tml_t
+ tm2_t)
Jacobian
matrix
a(1,1)
= rl_t
a(1,2)
= rl_p
a(2,1)
= r2_t
a(2,2)
= r2_p Jacobianmatrix
find
inverse
dstinv
=
1.0
ai(1,1)
=
a(2,2).detinv
ai(2,2)
=
ai(1,2)
=
a(1,2)*dstinv
ai(2,1)
=
a(2,1)edstinv
Newton
/

(a(1,1)*a(2,2)
a(1,2)*a(2,1))
a(1,1)*dstlnv
changes
dt
: (ai(l,l)*reml
+ ai(l,2)*re82)
dp
= (ai(2,1)*resl
+
rlx
/
t
res2
sot
zO
pi/tau*phid(ttc)
residual
set
/
al(2,2)sras2)
= 1.0
if(rlx*dp
.gt.
2.Sap)
rlx
= 2.5*p/dp
if(rlxsdp
.It.
.8*p)
rlx
= .8*p/dp
if(rlxsdt
.gt.
2.5*t)
rlx
= 2.Set/dr
if(rlxsdt
.it.
.Set)
rlx
= .8*_/dt
updat o varlablo8 t = t + rlxsdt p
= p
+ rlx*dp
convorgenca if
check
(abs(dplp)
.le.
spa
.and.
abe(dr/t)
.le.
epm)
C
IO0
coal_ians
C
grtte(s,*)
'IGISPT:
rrite(e,e)
'dp
write(.,*)
'p
Convergence
dT T h
r:',
p,
failed.'
dt
:', dp, t,
h,
r
C
3
continue
C
set
residual
derivatives
r1_r
= p/(ree2et)
rl_h
= O.
er_
input
z,h
31
variables
So:o
S
r2_r=O. r2_h=l.
b(l,l)
= rl_r
b(l,2)
= rl_h
b(2,1)
= r2_r
b(2,2)
= r2_h
C C ....
set t_r
p,t derivatives wrt r,h = (ai(1,1)*b(1,1) + ai(1,2)sb(2,1))
t_h
= (ai(1,1)*b(1,2)
+ ai(1,2)*b(2,2))
p_r
= (ai(2,1)*b(1,1)
+ ai(2,2)sb(2,1))
p_h
= (ai(2,1)*b(1,2)
+ ai(2,2)mb(2,2))
set
second
C C C ....
residual
derivatives
tic
=
ttc_t
= 1./(tau*t**2)
tic_it
=
wrt
r,h
i,l(tau*t) 2./(tau*t**3)
z
= 1.
z_p
=
+ p*pi*phi(ttc) pi*phi
z_pt
=
pi*phid(ttc)*ttc_t
z_pp
= O.
z_t
=
pspiSphid(ttc)*ttc_t
z_tt
=
p*pi*
rl
=
rl_p
=
rl_pt
=
rl_pp
=
rl_t
=
rl_tt
='2.*p/(ret**3)
(tic)
(phidd(ttc)*ttc_t**2
p/(r*t)

z
/zO
l./(r*t)

z_p
/zO
l./(r*t**2)

z_pt/zO

z_pp/zO

z_t

z_tt/zO
p/(r*t**2)
r1_r
=
rl_h
= O.
rl_hp
= O.
rl_ht
= O.
rl_rp
=
r1_rt
=
r1_rr
=
+ phid(ttc)
/zO
p/(r.*2et)
l./(r*s2st) p/(rss2st*s2) 2.*p/(r**S*t)
tal
=
(alf*t
+
tal_t
=
(all
+
/
zO
2.*alf*bta*_
alfebtaet**2) )
/
tal_tt
=
(
zO
2.*alf*bta
)
/
Cml_p¢
= O.
zO
Cml_p
= O.
tml_pp
= O.
tm2
= pspl/tau*phid(ttc)
/
zO
ta2_p
=
pl/tauSphid(ttc)
/
zO
ta2_p¢
=
pl/tau*ph/dd(ttc)sttc__
/
zO
ta2_pp
=
tm2_t
= p*pi/tau*
tB2_tt
= pspl/taue(ph/ddd(t¢c)*ttc_t**2
O. ph/dd(ttc)*ttc_t
/ +
32
zO
*tic_it)
&
ph/dd(ttc)*t_c_tt) r2
= h

(tml
+ tm2)
r2_p
=

(tml_p
+ tm2_p)
r2_t
=

(tm1_t
+ tm2_t)
r2h
=
/
zO
I.
c c c ....
set
and
ph
= p_h
linearize
th
= t_h
new
+ rl_t
*th
rlh
rlh
= rl_p
rlh_ph
= r1_p
rlh_th
=
rlh_p
= r1_pp*ph
+ rl_pt*th
+ rl_hp
rlh_t
= rl_pt*ph
+ rl_tt*_h
+ rl_ht
rlh_h
= O.
rlh_r
*ph
residuals:
= drl/dh
= O,
r2h
= dr2/dh
= 0
+ rl_h
rl_t
= ph/(r**2*t)

+ th*p/(r**2*t**2)
r2h
= I.
r2h_ph
=
tml_t*th
r2h_th
=

tal__
r2h_p
=

tml_pt*th

r2h_t
=

tml_tt*th

r2h_h
= O.
r2h_r
= O.
a(1,1)
= rlh_th
a(1,2)
= rlh_ph
a(2,1) a(2,2)
= rlh_th = r2h_ph
detinv
:
1.0
aih(l,l)
:
a(2,2)edetinv
/
aih(2,2)
=
aih(l,2)
=
a(l,2)*dotinv
aih(2,1)
=
a(2,1)*do_Inv

tml_p*ph

tml_p

tm2_t*th

tm2_p*ph

tm2_p

t12_t
tm1_pp*ph

ti2_pt*th

tm2_pp*ph
tml_pt_ph

tm2__*th

tm2_p_eph
(a(l,l)ea(2,2)

a(l,2)*a(2,1))
a(l,l)edetinv
dth
= (a:Lh(1,1)*rlh
+ aih(1,2)*r2h)
dph
= (aih(2,1)*rth
+ alh(2,2)*r2h)
c C
_
l
c
_h
= th
_l_
+
I_h
+ dth
c c c ....
sot
end
pr
= p_r
lt.no_tzÙ
tr
= t_r
rlr
= rl_p
rlr_pr
= rl_p
rlr__r
:
nor
*pr
roetduL].s:
+ rl_t
*tr
rlr
+ rl_r
rl_t
33
= drl/dr
= O,
r2r
= dr2/dr
= 0
rl=_p rlr__
= zl_ppspr
+ rl_pt*_r
+ r1_rp
= r1_pt*pr
+ rl_tt*tr
+ rl_rZ
rlr_r
= r1_rp*pr
+ r1_rt*_r
+ r1_rr
rlr_h
=
r2r
= 
r2r_pr
=
r2r_tr
= 
tml.t
r2r_p
= 
tml_pt*tr

r2r_t
= 
tml_tt*tz

r2r_h

O.
r2r_r
=
O.
O.
tml_t
a(1,1)
= rlr_tr
a(1,2)
= rlr_pr
a(2,1)
= r2r_tr
a(2,2)
= r2r_pr
*tr

tml_p

tml_p
*pr

t=2_t
*tr

tm2_p

tm2_p
*pr

tm2_t
tml_pp*pt

tm2_pt*_r

tm2_pp*pr
tml_pt*pr

tm2_tt*tr

tm2_pt*pr

a(1,2)*a(2,1))
C
detinv
=
1.0
air(l,1)
=
a(2,2)*detinv
/
(a(1,1)*a(2,2)
air(2,2)
=
a(1,1)*detinv
air(l,2)
= a(1,2)*detinv
air(2,1)
=
a(2,1)*detinv
dtr
=
(air(1,1)*rlr
+ air(1,2)*r2r)
dpr
=
(air(2,1)*rlr
+ air(2,2)*r2r)
C
c
pr
c
tr =
= pr
+
dpr
tr + dtr
C C
c ....
calculate
responses
in
dZ/dh
and
dxlh
= rlh_h
+ rlh_p*ph
+ rlh_¢*¢h
dr2h
= r2h_h
+ r2h_p*ph
+ r2h_t*¢h
dxlr
= rlr_h
+ rlr_psph
+ rlr__s_h
dr2r
= r2r_h
+ r2r_psph
+ r2r_tsth
dth
=
(alh(1,1)*drlh
unit
h
pertuxba_ion
_o
unig
r perturba¢ion
+ alh(2,2)*dr2h)
*.hh
= dth
d_h
•
(alr(1,1)*drlr
+ alz(1,2)*dr2r)
dph
= (alr(2,1)*drlz
+ alr(2,2)*dz2r)
_hr
= dth
phr
= dph
responses
in
dt/dh
and
dzlh
ffi rlh_r
+ rlh_p*pr
+ rlh_tstr
dr2h
=
+
+ r2h_t*tr
r2h_r
to
+ aih(1,2)*dr2h)
dl_ = (alh(2,1)*d=_h
calculate
dp/dh
r2h_p*pr
dp/dh
34
drlz
= rlr_r
÷ rlz_pepz
+ rlz_t*tr
dr2z
= r2r_r
+ r2r_pepr
+ r2r_t*tr
dth
= (ath(1,1)*dxlh
+ aih(1,2)*dx2h)
dpb
= (aih(2,1)*dxlh
+ aih(2,2)*dx2h)
trh
= dth
prh
= dph
d_h
= (air(1,1)sdxlr
+ air(1,2)*dx2r)
dph tx_r
= (air(2,1)*dxlr = d_h
+ air(2,2)*dx2r)
prr
= dph
set
final
first
p_r
= pr
t_r
=
p_h
= ph
t_h
=
th
p_hh
=
phh
t_hh
= thh
p_rr __rr
= prr = trr
p_rh
=
.5*(prh+phz)
t_rh
=
.S*(trh+thr)
and
second
derivatives
wrt
(r,h)
tr
return end
subroutine
nonstag(hO,rho,q,
pO,pO_r,pO_q,
•
rO,rO_r,rO_q
C
Calculates
C
specified
s_agna_ton
presses
and
:_a_attenauthalpy,
C
Input:
C
hO
stagnat
C
rho
denJlty
C
q
.pe_
£on
enChalpy
C C
OUtlm_:
¢
pO
station
C C
po_r pO_q
dpO/dr epO/dq
C
rO
8ta6na¢
C
rO_r
drO/dz
C
rO_q
drO/dq
lapllclt dimension reals4
reals4 a(2,2),
pzeesure
ton
density
(ah,a,oz) aI(2,2),
d_usi_y
denslSy,
C
b(2,2)
h_p,h_t
35
)
for and
speed.
coon
/non,
u/ all,
comaon
bta,
pi,
tau,
zO
/non.fit/ c2,
cl,
cO
C
data
eps
/S.OE6/
C
CCC
z(pp,tt)
=
z_p(pp,tt)
=
z_t(pp,tt)
=
h
= hO 
h_q
=
h_hO
=
1.
+ pp*pi*phi
(1./(tau*tt))
pi*phi
(1./(tau*tt))
pp*pi*phid(l./(taustt))
/
(tau*tt**2)
.5.q*.2

q
1.0
C
r
= rho
set
input
call
preliure
and
temperature
and
derivatives
ngaspt(h,r,p,p_r,p_h,p_rz,p_hh,p_rh, t,t_r,t_h,t_rr,t_hh,t_rh)
C
set ttc
entropy =
• and 1./(tau*t)
ttc_t
= l./(tau*t**2)
tic_it
=
derivatives
wrt
p,t
2./(tau*tee3)
ph
= phi(tic)
phd
= phid(ttc)
phdd
= phidd(ttc)
phddd
= phiddd(ttc)
C
ph_t
= phd
•
ttc_t
phd_t
= phdd
*
ttc_t
phdd_t
= phddd
* ttc_t
•
= alfelog(t) 
i_p • _t
+ 2.0*alfebtaet
p'p•*(
_*phd
=  pi*( = alf/t
t*phd

p*pi*(
*ttc_t *ttc_t + 2.0*alfebta
phd
•
*ttc_t
+ ph
)

log(p)
÷ ph
)

1.0/p
+ ph_t
+ t_hd_tettc_t + t*phd
)
*ttc_tt
c c ....
initial
gues
cc cc
if(bta.eq.O.O) tO = hO/al£
cc
elle
¢c cc
tO audif
cc
pO
= = p
(1.0
for
pO,tO
from
imperfect
+
•qrt(1.O
• exp(alfelog(t)
+ 4.0*btaehO/alf)) + alf*2.0ebtae(1.Ot))
C
tO
gas
then
= t
pO = p
36
/
(2.0*bta)
lerton itcon
C ....
do
=
100
loop IS
to
converge
iter=l,
correcZ
pO,tO
i$con
tic
=
tic_tO
= l./(tau*tO**2)
tic_fro
=
ph
on
l./(tau*tO)
2./(tau*tO**3)
= phi(tic)
phd
= phid(ttc)
phdd
= phidd(ttc)
phddd
= phiddd(ttc)
ph_tO
= phd
* ttc_tO
phd_tO
= phdd
* ttc_tO
phdd_tO
= phddd
* tic_tO
enthalpy
residual
reJl
=
(alf*(tO
rl_pO
=
(
rlotO
=
(alf*(l.O+
+ b_a*tO**2)
+ pO*pi/tau*phd
)/zO
pi/tau*phd bta*tO*2.)
)/zO
+ pO*pi/tau*phd_tO)/zO
C
residual
entropy reg2
= all*log(tO)
&
+ 2.0*alf*bta*tO
pO*pi*(
r2_pO
=

r2_tO
=
all/tO
k
pie(
tO*phd
*tic_tO
+ ph
)

log(pO)
tO*phd
*ttc_tO
+ ph
)

l.O/pO
+
pO*pi*(
k
2.0*alf*bta
phd
*ttc_tO
+ ph_tO
+ tOSphd_tOsttc_tO + tOaphd
_ttc_ttO
)
C
getup
and
invert
a(1,1)
= rl_tO
a(1,2)
= rl_pO
a(2,1)
: r2_tO
a(2,2)
= r2_pO
Jacobian
detlnv
=
1.0
ai(1,1)
=
a(2,2)edetlnv
/
ai(2,2)
=
a(1,1)edatlnv
ai(1,2)
= m(1,2)sd.tlnv
ai(2,1)
= a(2,1)edetlnv
matrix
(aCi,1)em(2,2)

a(1,2)ea(2,1))
C
sa_
Ionon
vtrimbles
dt
•
d_
= (_L(2,1)e=esl
r].z
(ai(1,1)eresl
:
+ 8i(1,2)eres2) + ai(2,2)eres2)
1.0
if(r].xsdt:
.IF.
2.S,pO)
rlx
: 2.S*pO/dp
if(rlx*_p
.it.
.8,I_0)
rlx
:
£f(r].xedt if(rlxedt
._t. .it.
2.5stO) o.SStO)
rlx rlx
= 2.5etO/d_ = .8.tO/dr
update
Q.e_pOl_p
variables
37
hO
pO = pO + rlx*dp _0 = tO + rlx*dt
convergence
check
if(abm(dp/pO)
.le.
eps
.and.
abs(dt/tO)
.le.
apt)
go to
C
lO0
continue
C
write(*,*)
'NONSTiG:
Convergence
write(*,*)
'dp
dT
:',dp,
write(*,*)
'po
To
h r:',pO,tO,h,r
failure.'
dt
C
2
continue
C C ....
set rl
residual s
=
derivatives
wrt
(s,hO)
O.
r2_s
= I.0
rl_h
=
r2_h
= O.
1.0
b(1,1)
= rl_s
b(1,2)
= rl_h
b(2,1)
= r2_s
b(2,2)
= r2_h
C C ....
CCC
CCC
set
(tO,pO)
derivatives
_rt
(s,hO)
tO_s
= (ai(1,1)eb(1,1)
+ ai(1,2)*b(2,1)) + al(l,2)*b(2,2))
tO_hO
:
pO_s
= (ai(2,1)sb(1,1)
(ai(1,1)*b(1,2)
+ ai(2,2)*b(2,1))
pO_hO
= (at(2,1)*b(1,2)
+ ai(2,2)*b(2,2))
C C ....
conver_
derivatives
tO_t
= tO_s*s_t
tO_p
=
pO_t
= pO_seS_t
pO_p
= pO_s*s_p
(s,hO)
er_
to
er_
(p,t,hO)
tO_leS_p
C C Cw
set
stqnation
zz
= z(pO,tO)
zz_p
= z_p(pO,tO)
zz__
= z_t(pO,_O)
rO
=
rO_z
= zO/zze*2
zO/zz
rO
and
darivatlves
•
= rO_z*zz_p
+ zO/(zzezO)
= rO_z*zz_t

zO*pO/(zzetO*e2)
derivatives
from
rO_pOepO_p
+
rO_tOetO_p
= rO_pO_pO_t
+
rO_tOetO_t
= rO_pO*pO_hO
+
rOtOstO_hO
=
rO_t rO_hO
(pO,tO)
pO/tO
tO_tO
rO_p
_
* pO/tO
rO_pO
convert
CCC
density
m
(pO,tO)
C
38
to
wr$
(p,t,hO)
2
c ....
convor$ rO r = rO
derivatives rO_pep_r
q =
pO_r
=
pO_q
=
from grz + rO_t*t_r
(rO_pip_h
+
pO_p*p_r
rO_t*t
(p,t)
to
erz
(r,q,hO)
h)*h_q
+ pO_t*t_r
(pO_p*p_h
+ pO_t*t_h)*h_q
C
c¢¢
rO_hO
=
(rO_p*p_h
+ rO_t*t_h)*h_hO
+
rO_hO
¢cc
pO_hO
=
(pO_p*p_h
+ pO_t*t_h)*h_hO
+
pO_hO
C
return end
real*4
function
implicit
phi(ttc)
reale4(ah,m,oz)
C ........
c c
Returns function Z = 1 + pi*phi(ttc)
phi
used
in
nonideality
C ........
common
/nonfit/
l
c2, phi
=
cl,
c2*ttc**2
cO + cl*ttc
+ cO
return end
reali4 inplicit common
function /nonfit/ c2,
i phld
phid(ttc)
reali4(ah,m,oz) cO
cl,
= 2.*c2ettc
+ cl
return end
reale4 intlli¢tt comion
functlonphldd(ttc) reali4(ah,n,oz) /noniit/ c2,
i plLtmkl
cl,
cO
= 2. ec2
retu.rn end
reil*4 iliplicii coumon •
functlonphlddd(ttc) rule4(ah,n,oz) /no.it/ c2,
cl,
cO
C
39
pazameter
phlddd
= O.
C
return end
subroutine
hgent(hO,r,q,
Returns
entropy
con:on
/nongas/
common
all, /nonfit/
t
s
= hO 
set
input
from
bta,
c2,
h
s)
input
pi,
cl,
vaziables
tau,
hO,r,q
zO
cO
.5.q*.2
C
c ....
call
pressure
and
temperature
and
derivatives
ngupt(h,r,p,p_r,p_h,p_rr,p_hh,p_rh, t,t_r,t_h,t_rr,t_hh,t_rh)
tic
=
1./(tau*t)
+tc_t
=
1./(tau*t**2)
ph
= phi(tic)
phd
= phid(ttc)
s
= all*log(t)
+
2.0*alf*bta*t
 p*pi*(t*phd*ttc_t
+ ph)

log(p)
return end
subroutine
c
non_v(hO,r,q,
Returns
gam,gma_r,gal_q)
"equlvLlen_"
co,,on
/nongu/
common
[nonfJ.$/
k
Llf,
k
ganma
bta,
c2,
cl,
pi,
for
tau,
BL density
prof£1o
zO
cO
C
c ....
see
e¢a¢ic
snChalpy
h
:
0.$*q**2
h_q
=
set
pressure
hO 
q
C
c ....
call
and
ngaspt
(h,
temperatuzo
and
dorivm$ives
r,p,p_r,p_h,p_rr,p_hh,p_rh,
k
t, t_r,
t_h,
t_rr,
t_hh,
t_rh)
C
c ....
set asq
speed = p_r
of
sound
squared:
/

(I.
p_h/r)
a'2
 alp/dr
(at
¢onmtant
s)
asq,r = p_rr
/
(I,

p_h/r)

/
(I.

p_h/r)**2
= p_rh
/
(I.

p_h/r)
+ p_r
/
(I.

p_h/r)**2
asq_h •
p_r
t_c
=
_Cc_t
= l./(Cau*t**2)
tic_it
=
= phi(tic) = phld(t_c)
phdd
= phldd(_tc)
phddd
= phlddd(t_c) = =
z
t
cp
*p_hh/r
2,/(_au*t**3)
phd
z_p
 p_rh/r)
l./(tau*t)
ph
z
*(p_h/r**2
I.
+ p*pi*ph pi*ph
=
p*pi*phd*tZc_t
=
( alf*(1.0
phdds_tc_t
)
/
zO
phddst_c_t
)
/
zO
cp_p
=
(
cp__
=
( all*(
•
+ 2.0*bta*t)
+ p*pi/_au* pi/tau*
2.0*bta
)
+ pspi/taus(phddd*ttc_t*82
h/(cp*t)*(l.O
+
phddettc__t)
)
zet
=
 p*pi/(C*Cau)*phd/z)
* zO
zet_h
= l.O/(cp*C)*(1.0

p*pi/(t*_au)*phd/z)
* zO
zet_p
=

pi/(t*_au)*phd/z
h/(cp*_)*(
&
/
zO

p*pl/(t*Zau)*phd/zs(z_p/z))
* zO

pepi/(t*$au)Sphd/z*(z_t/z

l.O/t)

pmpi/(ts_au)*phdd*t$c_t/z
)
* zO
(ze_/cp)*cp_p zet_t
=
h/(cp*$)*(
& &
(zet/cp)*cp_t

_e/m
=
gsmr
=
gam_h
=
uq/(hszet)e(zet_h/zet
gaa_p
=
aJq/(heze_)e(zet_p/zet)
gu_t
= uq/(hezst)s(ze__t/zet)
gaa.q
uq/(h*zet)
+
(ze_/t)
1.0 alq_r/(hezez) 
l.O/h)
gam_pep_h
+ gma_t*t_h
+ gam_h
gaa_p*p_r
+ gam_t*t_r
+ gam_r
= gaa_h*h_q
subrout4_e
sonAc(hO,pO,rO,
c
calculates
sonic
c
from
specified
q,p,r)
quan_i_ies sonic
quantities
q,p,r hO,pO,rO
41
+ aiq_h/(heze_)
C"
implicit
real
data
/ l.Oe5
/
_ith
perfect
epm
initialize
(m)
gam
= rOshO
/ (rO*hO
gml
= gam 
1.0
gas
 pO)
q = sqr_(2.0*hO/(2.0/gml
:
trat p
1.0
+ 1.0))
+ O.5*gml
= pO*trat**(gam/gml)
r =
rO*trat**(1.O/gml)
converge do
on
nonideal
I0 liars=l, call
values
by
forcing
N'2
nideal(hO,r,q,
p
,p_r
call
nonstag(hO,r,q,
resl
= msq
all
= msq_r
a12
= msq_q
rea2
= pstag
a21
= pstag_r
a22
= pstag_q

) )
1.0

pO
1.0/(all*a22
dr
=
(resl*a22
dq
=
(all
dp
= p_r*dr
rlz
and
pstag,patag_r,pstag_q, rstag,rstag_r,rstag_q
=
I,
,p_q,
msq,mJq_r,msq_q
detinv
=
15
*ros2

a12*a21)

a12
*res2)*detinv

reel*a21
)*detinv
+ p_q*dq
1.0
:
= 1.Set/dr
if(rlx*dz
.gt.
1.5*z)
zlx
if(rlx*dz
.i_.
.6*r)
rlx
= .6*r/dz
if(rlx*dq
.gt.
1.6.q)
rlx
=
1.6*q/dq
If(rlx*dq
.I_.
.e,q)
rlx
=
.e*q/dq
c r = r + rlxedr q
= q
+ rlxedq
p

+ rlz.dp
p
c dlax
= aaaxl(
abs(dz)/r
,
abs(dq)/q
)
c if(_
.it.
spa)
So
$o
11
c 10
continue 'sonic:
write(*,*) 11
conversence
fa/led.
continue
C
return end
!
son/c
42
,daax
patag
= pO
Bibliography
[1] J. H. McMasters, transport
airplane
[2] H. W. Liepmann
W. H. Roberts,
[4] M. Drela and number
and A. Roshko.
[5] W. Anderson. wind tunnels. [6] O. Coufal. 298.15
AIAA
AIAA
16(6),
of Gasdynamics.
airfreon
tests
of s
1988. Wiley, New York,
1957.
of real gas effects in cryogenic
June 1978.
25(10),
analysis
of transonic
and low Reynolds
Oct 1987.
study on the use of sulfur hexaflouride
AIAA901421,
Thermodynamic
Recent
AIAA882034,
investigations
Viscousinviscid
Journal,
A numerical
Elements
Theoretical
Journal,
M. B. Giles.
airfoils.
F. M. Payee.
in high ].ift configurations.
[3] B. Wagner and W. Schinidt. wind tunnels.
and
as a test gas for
1990. properties
 30,000 K and pressure
of sulphur hexaflouride
range .101325  2 MPa.
1986.
43
ACTA
in temperature Teehnica
range
CSAV, 31,
Appendix HighOrder
Airfoil
for Ideal
The entropy
steady
flow around
and total
is an ideal
enthalpy,
or a nonideal
the perturbation potential relations are obtained.
The
¢.
Boundary
NonIdeal
away
and hence gas.
Farfield
and
an airfoil
B
from
Gas
flow can then
These
V__/
viscous
regions
properties
still be decribed
the freestream
4' =
Flows
shock wakes and
is also irrotational.
Assuming
Conditions
is aligned
hold whether
by the with
has
velocity the
zaxis,
constant the fluid
potential the
@ or
following
q_,(z + ¢)
(1)
=
q_[(1+¢_)i
+ CuJ]
(2)
q2 = iq]2 =
q2 [(l+¢x)
2 + ¢2u]
(3)
1 _V(q 2) = qVq
2 qoo[ (¢x_+¢_
=
¢_+¢uCxu)i
+ (¢_u + ¢_ ¢_ + ¢_¢_)j] The governing
flow equation
(4)
is:
v.(pv_) v_
or
=
o
(5)
=
V!.w
(6)
P In isentropic
flow (s = constant),
p = p(p), so
dp s Vp _p
Vp = and
where
P a2 qVq

(7)
hence
a is the speed
to the speed:
of sound.
a = a(q).
W2_
=
1 _qVq.
a 2V2dp
=
qVq.[(l+¢z)i
In isentropic,
For a perfect
and/or
adiabatic
(8) + _j]
(9)
flow, the speed of sound
is uniquely
related
gas, a(q) is given by
as while for an imperfect
[email protected]
=
nonideal
2 a_
7 1 2 (q2_q_)
gas it is necessary
(10)
here to linearize
a(q) about the freestream
conditions.
a2
,
d(a_)
_ a® + It is convenient
to define
an "equivalent"
7'
d(ql )
[q2ql][
ratio of specific
=
d(a_) 2 d(q_)
1 8
heats
,
(il)
'_ 7' for the nonideal
gas as
(12)
so that the a(q) relation for the nonideal gas can be compactly
a2 For to
a perfect note
that
invariably
gas, 7'
this
can
reverts
easily
7 > 1 for perfect
Substituting
._
to the
be
less
2 a°°
_exact
than
written as
(13)
7'1( 2 q2 _ q _) form
unity
(10)
since
for heavy
in this
gases
case
such
as
7' = 7. sulfur
It is interesting
hexafluoride,
while
gases.
for a s, qS, and
qVq
in equation
_ q_ (2¢. + _ + ¢
(9),
[¢= + ¢_]
we obtain
=
q_ [(1 + ¢_)(¢=
+ ¢_ ¢= + ¢_¢_)
+ ¢_ (¢_ + ¢_ ¢_ + ¢_¢_)1 F
l 7,I 2 M_2 2¢=J [¢z=+¢_1
[1
'
+ ¢_
is the
freestream
 M®)¢== where
M=
= q=/a_
Equation nonideal as
(16)
2D second
compressible
derived
the
is the
potential
in references
"equivalent"
firstorder
version
In terms
of the
:
M s [(7 +1)¢.¢_= Mach
(16)
P randtlIt has
PrandtlGlauert
equation
same
that
the (12).
7' in lieu
form
usual
the
which
as the ratio
Wagner
and
+ 2_b_C=u]
+
()
CO ¢3
(16)
governs
equation
of specific Schmidt
smallperturbation
for a perfect heats [1] have
ideal
gas
7 is replaced
by
considered
the
of 7
z//_
(17)
# = y
= _k_,
(15)
coordinates
=
where/3
O(¢3)
+ (7'l)_bz¢_
Glauert
the
equation
using
+
number.
[5], except by
Cz=+2¢u¢=_l
'
order
7' defined
of equation
M2[¢=+2¢=
flows.
[4] and
value
:
(14)
general
(is)
_s
:
_2 + _2
0
=
arctan_
solution
to equation
:
_r °
r_
+
D=cos0 2_r f
+
+
\
2_r )
(19) (20)
"V
(16)
is
D_sin8 27r
kl'_'ks
(21)
where
kl = _1(/'+1 __ + Terms
of order
1/_ s and
above
have
(22)
._) been
discarded.
In a flow solver,the circulation r can be determined eitherdirectlyfrom the liftper unit span L p (Euleror NavierStokescode), Ll r
p_q_
(23)
or indirectlyby specifyinga Kutta condition(potentialsolveror MSES). The source strengthZ can be determined from the totalprofile drag per unit span D _,or from the asymptotic mass defect behind the airfoil includingthe shock wake. £)i 
p®q_
(24)
In the case of a potentialsolver, D _ should not includethe wave drag sincethere isno shock wake (unlessan entropy correctionscheme is employed). Note that r and Z here have units of length since_bin (1) correspondsto a unitfreestreamspeed. Cole and Cook [5]give explicit expressionsforthe doublet coefficients Dz and D_ in terms of fieldintegralsover the domain. Unfortunately,these expressionsare unwieldy and for a nonideal gas would be ratherexpensive.A simplerand economical approach isto iteratively update D_ and D u by minimizing the mismatch between V6
and the velocity_olution from the flow solveron the
outer boundary. The approach taken in reference[4], forexample, isto minimize the integral 1
I=_
/ IVq_× _olutionl _ dz
(25)
taken over the outermost streamlines.The doublet terms in the farfield expansion (21) decay fasterthan the others,and so can be neglectedforsufficiently distantouter boundaries.However, retainingthem greatlyreduces thesensitivity of the solutionto domain size,especially fortransonic
flows[6]. With itsterm coefficients defined,equation (21) gives a very accurate representationof the perturbationpotential_ away from the airfoil.The gradientof equation (21) accuratelygives the totalvelocity_ via relation(2).Either _b,_, or an appropriatederived quantitymay then be imposed at the outer domain boundary as a highorderboundary condition. A potentialsolver would typicallyimpose _ or OctOn, whereas an Euler or NavierStokessolverwould typically impose the flow angle at the inflowand pressureat the outflow,both being determined from V6.
10
Appendix Shape for Ideal
The profile,
major which
influence alters
flows, this effect is mostly shape
parameter
V(y)
and
to pressure
and R(y)
by the correlation
are the velocity
of the
the velocity density
profile
profile
from its definition
U(y)
R(y)
(26).
near
Ha
is a nonuniform
In an integral shape
scheme
parameter
parameters
H, the kinematic are defined
as
(26)
profiles. R = p Pe
Ha are only weakly
the vonKarman
density
for adiabatic
f(1  U) dy f(1  U)Vdy

the wall due to adiabatic
In turn,
behavior
Me. The shape
and density
and hence
layer
between
V = u ue Since
Flows
gradients.
f(1  RU) dy f(1  U)RUdy

Gas
on boundary
response
captured
Relations
NonIdeal
Ha, and the edge Mach number
H where
Parameter
of compressibility
the layer's
C
integral
(27)
affected
heating
by compressibility, will increase
momentum
reduction
H as can be seen
equation
dO dz shows that adverse ISES
an increase
pressure
C1 2
in H will increase
gradient.
[4]) employs
gas model
_
The
a correlation
(2 + gthe momentum
integral
boundary
of the form Hk(H,
in Appendix
A, the state
P
the corresponding dh the
the caloric
thickness
Me) for air.
(28) growth
layer formulation
equation
pT_T
where
_._du....._ u, dz rate
in MSES
This is rederived
d_/dz (and
for a given its precursor
for the nonideal
as follows.
As developed
while
M_)
caloric equation
= _p(r) dT
approximation equation
across
the boundary
factor
Z for most
+ p._"(r)
dp __ 0 across
%(TT,)
: 1= "_
The nonideality
(29)
dT
= cp(p, T) dT a boundary
layer.
(30) Linearizing
layer we have
hh,
•
that
as
form is
__ en(r) dT
on the basis
gas can be written
 Z(p, T)
in differential
+ d[pSr(T)]
is made
of a nonideal
nonideal
(31)
1
c_
I
%,T_
(32)
gases has the form
Z(p, T) = 1 + P¢(rc/r) Pc 11
(33)
where
Pc and Tc are the criticalpressure and temperature.
This can likewise be linearized about
the edge conditions as follows.
p_
Z Combining
this
with
equation
(32),
the
equation
of state
(29),
= 1
1
"Z
(35)
%.T_
Z_ .
(36)
p_T,
we have
_Using
T
=
1
the
 1
Pc T_
density
profile
R
p p_
is then

related
to the
T and
Z profiles
as
T_Zep T Z p_
T_ Z, with
the
the
density
usual
boundary
profile
layer
can
R
R
approximation
be written
 1

1+
_
1+(__1)(
of the enthalpy
_
(__1)
(37)
Z
p __ pe being
in terms
1+
=
T
1
Using
profile
P'Tc¢:_ peT, Z,/
relations
(32)
and
(36),
alone.
+
(38)
c ,T, z,J

T,
(1
h.
made.
O[(Lh,/h
(39)
1 ,)] 2
(40)
where %,T,
CFor
turbulent
enthalpy
across
unity.
Since
in a gas the
the
the
the
boundary
layer,
turbulent
(convection
assumption
enthalpy,
adiabatic
by
velocity
and
this
diffusion
mechanisms
static
h.
true
enthalpy
numbers
h
for
are
are
then
ho  u_/2 ho _ U2/2
heat
typically With
related
a constant
a turbulent
and
is reasonable.
profiles
_
only
to assume
of momentum
Prandtl
enthalpy
(41)
Z,]
it is reasonable
is strictly
turbulent
stagnation
pcT,
flows,
although
eddies),
of constant
layer
1
h0
are
stagnation
Prandtl
number
essentially
the
close
to unity.
denoting
the
of same
Hence,
stagnation
by
1  i_u2 U2 1  __
(42)
u 2
h_ce_1

h
_
(U21) u2 U2 1_2_:_
(43)
and the density and velocity profilesare then related by u 2
R
=
1 +
_
12 u2_ 12
(U 2U 2
1)C.
(44)
Since
u_/ho
and
( are both functions
implicitly
defines
Hk in terms
to assume
a smalldefect
of the edge Mach
of H and
Me. To obtain
the density
this relation
Me,
the
in closed
density form,
profile
(44)
it is necessary
profile U = 1e
so that
number
profile
;
can be approximated
e