report 1123 a study of inviscid flow about airfoils at high supersonic speeds 1

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REPORT 1123
A STUDY OF INVISCID FLOW ABOUT AIRFOILS AT HIGH SUPERSONIC SPEEDS 1
ByA.J.Emma, Jr.,CLMUJNCEA. SYVERTSONa,ndSAMUEKLIWJS
. .
SUMMARY dieting the pressures (and velocities) at the surface of an
airfoi.-in steady motion at low supersonic speeds. Thus the
8tdy ji’owabmdcurvedaiq%ih ai highsupersonicsped b
lineartheory of Ackeret (ref. 1) has proven particularly use-
investigated andytidly. W& therwumption i%aiairbehava
a8an ideal diai!omicgm, d ia-foundthd thes?wck-expana-ii.m ful in studying the flow about relatively thin, sharp-nosed
methodmaybeu-d topredicttlujlowaboutmmoedairfoib up to airfoils at small angles of attack, while the second+rder
arbitrarily high .MaG?Lnumbem, provided the $OWdejlztion theory of Busemann (ref. 2) has found application when
thicker airfoils at larger angles of attack were under consid-
anglesare not too dose to t?w8ecorre$pordingto 8hock&t.ach-
eration. Athighfree-streamMach numberstherangeofappli-
ment, Thi8r& appli~ not only to thecikterminutionof the
eurfaixpressurediw%hiim, butabo tothedeterminationofthe cabili@-of any potential theory isserioudy limited, however,
wholejlowfield abouian airfoil. Verijicu.hh of thti ob8erviz- due to theproduction of strong shocksby even therelatively
smallflow deflections caused by thin airfoils. The aswmp-
tion ix obtaimdwiththeaid of& methodof chartierbtics by
tion of potential flow is invalidated, of course, by the pro-
extensixecalcw?uiiu?Mof % pre.wuregradient and 8hock+vave
nounced entropy risesoccurring through th~e shock.
ourmture ai the leading edge, and by cu.bw?dti ofthepnwure
disti”butionona 10-percen$-thickbiconvexairfoil ai0° ang~ of This limitation on potential th~ries was early recognized
and led to the adoption (seeref. 3) of what isnow commonly
attuck.
called the shock-expansion method. The latter method de-
An approximation to the 8hock-ezpan&onmethodfor thin
a~oi18 athighMach numlwr8I%alsoinve@@ed andh found rives its advantage over potential theories, principally, by
accounting for the entropy rise through the oblique shock
to yield presmmesin error by l.emthun 10 per& a$M_z.ch
emanating from the leading edge of a sharp-nosed airfoil.
number8above3 and @w deji%ctionanglee up to $6°. Thti
Consequently, so long as the disturbed air behaves essen-
slender-airfm”lmethodis relalixly simpleinform andthm may
tially like an ideal gas, and so long aa entropy gradients
proveu8ejdjor someengineeringpwrpo8es.
normal to the streamlines (due to curvature of the surface)
.E$ecte of ca.brie imperfedti of air m.ma~txtin,di8turbed
do not significantly influence flow at the surface, the shock-
jlow jkllh at high Mach number8are inv~tigatd, pari%ukr
expansion theory should predict the pressuresat the surface
ai!i%ntionbeing given to the redua%-n qf the ratw of 8peci@
of an airfoil with good accuracy-it is tacitly assumed, of
heats, So longm thisraiio do.anotdecreaeeappreciablybelow
course, that the flow velocity is everywhere supersonic, and
1.3, it ;Sindim.tedthaith.s8hock-expaw”on method,generaliad
that the Reynolds number of the flow is sufhoientlylarge to
i!oincludethee$ec&?of th+%eimperfect%, 8h0u?dbe sub8tivn-
minimize viscous effects on surfacepreswres.
t~y asaccurateafor ideal-gin@we. !f’htiob8erzzz&nis veri-
The of the behavior of airfrom that of an ideal
jiedun”ththeaidof agenera.?imd8hock-ezpaM”onmethodanda
gas at the temperatures encountered in flight at l@h super-
generalimdmethodof cbracten%tia employedinform appli-
sonic speeds has been the subject of some investigation in
cable-for1oM-Jah i%nperaiurtxup ti about6,000° Rankine.
the caseof flows through oblique shock waves. In reference
The [email protected] methodis modijiedto employ an axwage
4, the effects of thermal and caloric imperfections on the
value of the ratw of 8petifi Lm#.8for a particukx @w ji.ek?.
prasmre riseacrossan oblique shook-wavewereinvestigated
Thi8 vimplifid nw!h.odha8 @8entiu/.lythe 8ameaccurq for
at sea-level Mach numbers of 10 and 20, and it was found
imperfectqa.sjhni.wash counterpartha8fio.dreul-gasjlowe.
that these eflects decreaaed the rise by less than 5 percent
An apjroxima.tej?owan.dyti b madeatm%mely highMach
for maximum temperatures up to 3,000° R. (corresponding
numberswhereitie indicgtedthuth ratioof specij?cheaikmay
toflow deflectionanglesup to 240). This decreasewasfound
approach clo8eto 1. In thiz7me, it h found thatthe 8h0ck-
to be due ahnost entirelyto caloricimpixfections, or changes
cxpansion nw!k.odmay be in cmwidawbieemor; however,.the
in vibrational heat capaciti~ of the airpassingththroeugh
Bu.wmannmethodfor the limii of in$nite free-8tream Mach
shock wave. The changesin temperatureand density of the
number and specific-ht ratio of 1 appear8 to apply w-i%
air-passingthrough the wave wereaffected to a considerably
reasonableaccurq. .
greater extent. Subsequently, an inw+gation was carried
INTRODUCI’ION
out by Ivey and Cline up to lMaoh numbers ashigh as 100
SmWdisturbrmce potentiql-flow theories have been (ermef-. 5), using the results for normal shock waves obtained
ployed widely, and for the most part successfully, for bpyre-Bethe and Teller considering effects of dissociation (ref.
1Supermh NAOA TN 21341‘T$nviscidF1OIVAboutAMcIIIYatHlgb SrI~ Sfkd& byA. J.Eggew Jr.,andOlarencnA. Syvce-txm,19~ andNAOA TN ‘ZiZ9“An Analysk of
SuperwIIo Flowh tbeR@m oftheLcmifnRgdgoOfO- Mm IIIdU~g CM fmDe@.~g ~m~ QradfonatndShook-WavOeorvatmrab”y SamualKraus,1!25!2.
339
_.— __—
340 REPORT 1123—NATIONALADVIBORY COMMITTED FOR AERONAUTICS
6). As would be expected, the pressureswere found to be
pressurecoefficient, ~
affected to a somewhat grhter ‘&tent at the higher Mach
numbers. speciiicheat at constant pressure
The extent i%which flow in the region of theleading edge specificheat at constant volume
of an airfoil departs from the simple W-mdtLMeyer type curvature
hasalsobeen investigated at high supemcmicairspeeds. If Mach number (ratioof local velocity tolocal spood
the surface is curved, for example, to give an expanding of sound)
flow downstream of the leading edge, expansion waves from
pr~e ratio, ~
the surface will interact with the nose shock wave, thereby
curving it and yielding a nonisentropic flow field. This static pressure
flow field may be characterized not only by disturbances dynamic pressure “. ‘
emanating from the surface but alsoby disturbance reflect- gas constant
ing to some extent from the shock wave back toward the rectangular coordinates (in strearnlim diroo~ion
surface. The manner in which these phenomeqa dictate and normal to streamline direction, respec-
shock-wave curvature and surface pressure gradient in tively)
ideal-gas flows at the leading edge has been tieated by temperature, ‘R.
Crocco (ref. 7) andmore recently by Schaefer (ref. 8), lMunk time
and Prim (ref. 9), Thomas (ref. 10], and others. In the raultant velocity
cams considered by Munk and Prim, it was found that sur- distance measured from leading edge along airfoiI
face pressure gradients were less (ii absolute value) than surface
those obtained assuming Prandtl-Meyer flow at the higher rectangnhu coordinates
lMachnumbers (i. e., Mach numbers greater than about 3) angle of attack, radians unlessotherwise spocificd
although, generally, by no more than about 10 percent.
Mach angle, arc sin —~1 , radians
Since curved airfoilsarelikely to be of fundamental interest ()
athighflightspeeds (see,e.g.,ref. 11), the ef%ctsof reflect+xl
mtiOOfSpCCifiCheats,~
disturbances would appear to merit further investigation,
both at the leading edge and as regards their influence on (Average value of Yis-rC.)
the whole flow field. In addition, it would appear desirable flow deflection angle, rndi&s unless othorwim
to considereilectsof gaseousimperfections through thefield. spec$ed
Such an investigation has therefore been undertaken in angle between shock wave and flow direction just
the present report, using the method of characteristics to downstream of shock wave, radians
obtain accurately flow fields and as a basisfor obtaining the molecular vibrational energy constant, ‘R. (6,600°
more approximate methods of analysis. The method is ‘ R. for air)
employed in a generalized form which allows caloric imper- ratio of shock-wave curvature to thatgiven by tlm
fections, as well as entropy gradients, in the flow to be con- shock-expansion method
sideredattemperaturesup to theorderof 5,000°R.—thermal massdensity
imperfections areneglected (seeref. 4. A 10-percent-thick + shock-wave angle,radians .
biconvti airfoil is treated at Mach numbers from 3.5 to ratio of surfacepre9suregradient to that given by
hdin.ity, and the results are compared with the predictions the shock-expansion method
of the shock+xpansion method, includ~ a simplifiedform ray anglefor Prandtl-lMeyerflow, radinna
of the method applicable to slender airfoils at high Mach
SUBSCBWP8
numbers and a generalized form of the method including
effects of cnloric imperfections. In addition, flow”in the o . free-streamconditions
region of the leading edge of curved airfoilsis considered in A,B,CJI conditions at dii7erentpoints in flow field
some detail. Values of the surface.pressure gradient and ‘i ideal-gas quantitiw
shock-wave curvature are presented for a wide range of iv .conditions at the L&ling edge immediately clown-
Mach numbers and flow deflection angles. streamof the shock wave
s conditions on streamline
SYMROLS w conditions along airfoilsurface
10Cd Spied Ofsound u- conditions along shock wave
chord
SUPERSCRIPT
&mteris(& Wordinate (G positively inclined
and G negatively inclined with respect to ‘the — - vector quantitie9
local velocity vector) . DEVELOPMENTOFMETHODSOFANALYSIS
section drag coefficient
section lift coefficient GENERAL METHODS
section moment coefficient (moment taken about ~Method of charaateristios.—Two-dimensional rotation(d
leading edge) supersonicflowshave been treatedby numerousauthorawith
A STUDY OF RWISCID FLOW’ ABOUT AIRFOILS AT HfKK SUPERSONIC SPEEDS 341
tlmaid of themethod of characteristicajand varioti adapta- Furthermore, it readily follows fim the differential energy
tions of the method have been found which are especially equation and these expressions that the speed of sound is
suited for studying particular types of such flows. In the given by the simplerelation
cme.of steadyflowsinwhich atmosphericairdoesnot behave
as an ideal diatomic gas, a very familiar and simpleform of
thecompatibility equationsmay be employed. To illustrate,
Combi@ug equations (8) and (10) and noting that
considor theEuler equation
sin /3=a/Vthere is then obtained
(1) N-=-2&
(11)
the continuity equation ,
Hence, on combining this equation with equations (6) and
(7), it isapparent that the familiar compatibility equations
div (P~=O (2)
and the equation for the speed of sound (evaluated at con- (12)
stant entropy)
~, dp and
(3)’
‘&
(13)
Rewriting equations (1) and (2) in the form of partial dif-
ferentialequationsandtransformingtheresultingexpressions
alsohold for the more general type of flow under considera-
to the characteristic, or C’l,Cz,coordinate system, there is
tion. These equations are basic, of coume, to two-dimen-
obtained, upon combination with equation (3), the following
siomd characteristics theory, and, as will be shown later, -
relationsfor steady flow:
form a convenient starting point for developing simpler
theories of two-dimensional supersonic flow.
(4)
In order to apply equations (12) and (13), it is evident
that the manner in which -j and /3.or M are co~ected to p
or 3must be known. Relations implicitly connecting these
variablesat temperaturesup to the order of 5000° R maybe
readily obtained from the results of reference 4 by simply
A simple addition or subtraction of equations (4) and (5) eliminating the terms therein accounting for thermalimper-
then yields the compatibility equations (see, e. g., ref. 12) fections. Thus we have as a functiori of the local st&ic
temperature and free-stream conditions
(6)
l+KW’L+L 1
l’=’Yi 82 ~err (14)
1+(’yf——v ji (em- 1)2
[ ()
and
Now, in reference 4 both caloric and thermal imperfections
of air wero considered, and it was found that the latter
imperfections 2have a negligible effect on shock processesin
atmosphericair, It may easilybeshownthatthisconclusion (15)
also applies to espansion processes and, for this reason,
For isentiopic flow along astreamline,thepressureisrelated
caloric imperfections only are considered in detail in the
to the temperatureby the expression
present paper. These imperfections become significant in
air at temperatures greater than about 800° R. and first
p _A(TJ
manifest themselves as changes in the vibrational heat (16)
p. A(T)
capacitieswith temperature. Thus, thespecfic heatsc, and
where
c, and theirratioy for thegasalsochange., The equation of
state remains,however,
(17)
p=pRT (8)
and thespeciiicheats arestillrelated to the gas constant by If thereis a shock wave in the flow,3in particular, a nose or
the expression leading-edge shock, then the folknving additional relations
CV—C,=R (9) obtained withequations (8), (10), and (15) andtheconditiom
fThwnml hnpmfectfonaOWWYappearfntheformoffntermokular formsandmolecular- $Ifthmaranoshock!vave&k the.mWrfpt afxreqnation(16)can,ofcourse,beraplaeed
slracl?ecisandmaybeaccountedforwithaddltkmaltermsfntheeqnatkmofstate. Wftbtheantmcrlpto.
—
342 REPORT 1128—NATIONti” ADVISORY COMMITTEE FORAERONAUTICS
for continuity of flow and conservation of momentum along re901vedinto the streamlinedirection and combined, noting
a streamlinethrough the shock arealsorequired: that
ap
(21)
G=&(%+%)
i
and
(22)
~
(19) there is then obtained the relation
and [—1—aslac,
tan6=— 1 1 (20) m_ a6/ac2 27P as (23)
tan u 70M02 ati c1 -G
‘— I+&
(pa/pQ)–l-l ‘
defining the gradient of p alongs. If flow along stremnlirw
By use of thelocal static txxnperatureas aparameter, the
downstream of thenoseisof thesimplePrandtl-Moyer typo,
term 2Yp/sin 2P in equations (12) and (13) may now be
evaluated with equations (14) through (17). Equations (18) however, wehave
through (20) define the initial conditions downstream of a
leading edge or other shock wave in the flow field. Thus, (;4)
equations (12) through (20) provide all the information
necessary to calculate the flow about an airfoil by,means of Hence, it isevident that therequirementfor thistype flow is
the method of characteristics. As described in detail in
Appendix A, the calculations are of three general types:
(26)
(1) determination of conditions at a point in the flow field
between the shock and the surface; (2) determination of
conditions at a point on the surface; and (3) determination Equation (25) iq of co,urse,simply anapprosinmte stmtamont
of condition’sat apoint just downstreamof theshock. Case of a well-known property of Prandtl-Meyer flows; namely,
(1) entails the use of both compatibility equations, whiIe that flow inclination angles are essentially constant along
cnse (2) entails the use of the compatibility equation for a fit-family Mach lines. It follows from equation (12) that
second-family characteristic line in combination with the if equation (25) holds, then the pressureswill alsobe essen-
equation of the airfoil surface, and case (3) involves the tially constant along these lines. 1{ does not follow, how-
compatibility equation for a first-family line in combination ever, that the.Mach number will be constant or, for that
with theoblique-shock equations.. With the aid of the three matter, that the first-family characteristic lines will be
general types of calculations, the entireflow field about an straight (aais the case for isentropic expansion flows about
airfoil can be built up numerically using a computing pro- a corner). In fact, it may easily be shown th& tho Mach
cedureworking from theleading edgedownstream. In cases number gradient along Clisproportional to thelocnlentropy
where changes in the vibmtional heat capacities with tem- gqdient normal to the streamlinesand that the G linesme
perature are neglected, the calculations are, of course, curved according to the change in M. Thus we sm thnt
simplifkd since-Yof the gas can be considered wnstant, and there is really only one basic assumption underlying the
temperature, pressure, and density ratios are simply the shock-espansion method; namely, disturbances incident on
ided-gas functions of Mach number. the nose shock (or, for tha~ matter, any other shock) ILm
‘Shock-expan&on method.-This method of calculating consumed almost entirely in changing the direction of the
supersonicflow of an idenlgas at the surface of _Wairfoil is shock. In this regard, it is interesting to note that tlm
well known, entailing simply the calculation of flow at the assumption of Thomas (ref. 13), that pressureis a function
nosewith the oblique-shock equations and flow downstream only of flow deflection angle and entropy, is equivalent to
of thenose with the PrandtLMeyer equations. Determina- this wmmption. It follows, of course, that the most
tion of airfoil characteristic in this manner requires only a general solution obtainable with Thomas’ seriesroprmenta-
smallamount of time, of course, compared to that involved tion of the pressure is that given by the shock-qmpsion
when the method of characteristic is used, hence, the method.
advantage of the former method. The qu~tions arise, With the assumptionthat alldisturbancesincident on tho
however, as to exactly what the simplifying assumptions shock wave are consumed, it is evident that the shock-
underl@g the shock+xpansion method are, and what form expamion method provides a relatively simple means for
the method takes (for calculative purposes) when the gas calculating the whole flow field about an airfoil, including
displays varying vibrational heat capacities. the eiTectsof shock-wave curvature. The details of such
The matter of simplifying assumptions may perhaps best calculations are presented in Appendix B. In general, of
be considered by employing equations (12) and (13), the course, the vali@ty of this assumption and the ticcumcy of
basic compatibility equations. If these expressions are the shock-expansion method can only be checked by com-
A STUDY OF INWSCIJI FLOW ABOUT AIRFOILS AT HIGH SUPERSONIC SPEEDS 343
parison of calculations using this method with those using thereisobtained from theoblique-shock equations,eimplifled
the method of characteristics. to conform with thisanalysis,.
The shock-expansion method for a calorically imperfect,
Mc?q=– I
diotomic gas is rend~y deduced from the equations previ- f(M&)= (33)
ously obtained. For example,flow conditions at theleading 2
M&2+-
. edge of an airfoil can be evaluated with the,obl.ique-shock- 4( .=-l )( %~’””’-1)
wuve expressions(eqs. (18) through (20)) and the expression and
27aMO%N2-(7=—1) “
for conservation of energy (eq. (15)). The variation of flow g(MOaN)= (34)
inclination angle with pressure along the surface is then ‘Y.+ 1
obtahied by graphically integrating equation (24); namely, where
s
~st3in2P ~p (35)
8&!-i3N= (26)
p* %p
With equations (3o) through (35), the pressureson the snr-
where the variables y, p, and 13are evaluated using equa-
face of an airfoil may easilybe obtained. In terms of pres-
tions (14) through (17), employing the static temperature
surecoefficientmehave
as a parameter. When extreme accuracy is not essential,
this rather tedious calculation can be avoided, and a rela~
(36)
tively simple algebraic solution of the flow downstream of “%ii7[(%9-11
tlm nose can be employed.4 The details of this solution or
me presented in .Appendix C. In the special case of flo~
~t high supersonic speeds about slender airfoils, the whole ‘3n
c.=a{g(~o’”)[’-’(u”)(l(:)lal’}’}
crdcukttionbecomes particularly simpleand warrantsspecial
attention. -The advantage of these slender-airfoil expressions lies, of
If it is assumed that the local surface slopes are small course, in their relative simplicity and, thus, the ease of
compared to 1 and, in addition,.that the free-stream Mach calculation which is inherent to them. It maybe noted in
number is large compared to 1, it follows that u and f?are thisregard that the functionsf(MoL$N)and g(M&) can‘be
everywhere smallcompmed to 1. In thiscase,equation (24) calculatedonce andfor allwith equations (33), (34), and (35),
takeson the approximate form provided the variation of Y. with &f& is lmown. This
calculation has been carried out for a constxmtvalue of -y
I
dp “
~ ~=~pM (27) equal to 1.4 and for average values of y, assuming TO=5000
R.s The resultsarepresentedin tableI.
It should alsobe noted that the slender-airfoilexpressiomq
Furthermore,ifitisassumedthatYisconstant atanaverage
of the shock-expansion method satidy the hypersonic simi-
vrdue~=for a particular flow field (this assumption appears
larity law for airfoils first-deduced by Tsien (ref. 15).0 A
reasonable since, in the temperature range up to 5,000° R,
necessary condition for the validity of these expressionsis
thechangeinYislessthanabout 15percent), thentheMach
thus satiefied;however, the accuracy of the shock-expansion
number andpressuremay berelatedby thesimpleexpression
method, whether for slenderairfoilsor otherwise,remainsto
T.-1 be investigated.
~=u~ @ K
(28)
()P METEODS FOR CALCULATING THE FLOW lN THE REGION OF TEE
.LRAD~Q EDGE
Equations (27). and (28) combine to yield the differential As was pointed out previously, pressure disturbances
equation
emanating from tho surface of a curved airfoilint~act with,
and thereby curve, the leading-edge shock wave. The
*N&)- d(zJ=d, (29)
geometry of thk phenomenon near the leading edge of a
convex airfoil is illustrated in figure 1. The pressure dis-
whichreadilyintegrates (betweenN andS) to theform turbances horn the airfoil (expapsion waves for a convex
airfoil) travel along &et-family Mach lines Cl. In addition
to changing theinclination of theshockwave, theinteraction .
Ps-[1-(qhi;)(l-:)]= (30)
between these disturbances and the shock produces another
ET
system of disturbances which travel along second-family
nowdenoting Mach lines Czhorn the shock wave to the surface.
-l’a—1 Method of characteristics.-An exact solutionfor thesur-
~ MNtiN=j@do6J (31)
facepres.suregradientandahock-wave curvatureattheleading
‘ and
5Fora@vanvolneofT%Tx, totbe ac?nraoyof MS a- istie Meal-gasfnnctkm
~=g(Mo6N) ‘ , (32) afMi.3x. Thn8,knowfng TN, -rHma k dotormined. The,avemgovalueofy nmdlY
Y-=.@mO=(’rN+Ym.
6ThisfactwasemployedbyLlnnell (ref.16)toobtainanexpresdonforp~ cwfikkmt
{Thetabnlotcdm2uft2OfI+oyes(ref.14)my okmPr0V9m?fdfntbfncamforMach cqnlvakmttoequation(37)fortbecamofconstantTandtoobtainexplloitmlutfonafortlm
numbcmupto3. lffb drag,andpitcMnganomentcmfklenk ofs@vanfafrfcihatbyperwnfusp+ids.
.
344 REPORT 112S—NATIONAIJ AINTSORY COMMITt’EE FOR AERONAUTICS
A procedure for evaluating equations (38) and (39) for n
calorically imperfect, &atomic gaa, as well as for an ideal
gas, is presented in Appendix D. Siice they m-eexact for
two-dimensional, steady, inviscid flow, values of the surfrum
pressure gradient,and shock-wave curv&me at the leading
edge, determined using these equations, may be put to two
uses. 13’imt,the accuracy of approximate methods of cal-
culating theflow fieldabout acurved airfoilcanbe evaluated
at the leading edge by comparing the values of the surface
pressure gradient and shock-wave curvature predicted by
these approximate methods to the values obtained using
equations (38) and (39). Second, the pressuregradient ond
shock-wave curvature can be used to calculate tlm initial
points of acharacteristicsolutionfor theflowabout anairfoil,
Shook-expansion method,-One approximate solution for
the surface preswqe gradient and shock-wave curvaturo has
alreadybeenindicatedintheprevious discussionoftheshock-
—.— expansion method. The requirement for the application of
the shock-expansion method is given by equation (26),
Henee, it is apparent that under this condition the oxpros-
sion for the surface pressure gradienL(eq. (38)) reduces to
(41)
\
FIGUREl.-Schematio _ of supersonic flow past a curved
Similarly, the.expressionfor the shock-wave curvaturo (eq.
dwp-nosa airfoil.
(39)) reducm to
edgemaybe determinedinthefollowingmanner. It isclear,
(42)
referring to figure 1, that the change in flow anglebetween
points AandCgivenby thecompatibility equations (seeeqs. \(m/ .
(12) and (13)) along the path ABC must equal the chinge
It shoul~ be realized that the flow field is detwmined by
determined by the airfoil surface horn A to C.. Similarly,
the basic flow equations in conjunction with the shock wavo
the diilerence in pressurebetween points B and D given by
and airfoil surface asboundary conditions. Thus, the addit-
the compatibility equations along the path BCD must equaI
ional requirement for this shock-expaneion method of zero
that determined by the change-in shock-wave inclination
pressure gr&lient along first-family Mach lima means that
between B and D. In reference 9, these conditions were
one of theflow relationscannot be satisfiedexactly (i. e., tlm
employed at theled.ing edge to obtain equations, in simple
flow field is overdetermined). . Equations (41) and (42)
parametric form, for determining the surface pressure
satisfy the shock relations and the airfoil surface M boun-
gradientsand shock-wave curvatur~. These equations can
dary condition’; however, the compatibility equations me
be written in the form
only approximately satisfied. (SeeAppendix B.)
The error in surface pressure gradient associated with
neglecting the reflected disturbances might be expected to
be largestin theregion of theleading edge of a curved airfoil
due to the closeproximity of theshock wave andthesurfrtco.
The magnitude of the error in this region may be deducocl,
for the surfacepressuregradient, and of course, from the ratios of values of surface prwsum
gradient and shock-wave curvature given by the character-
(t?-u+a)+(w J- istic method to thosegiven by theshock-expansion method.
sin (l+wx9;~:”-’)’
‘= The surface-pressure-gradient ratio and shock-wavo-curvn-
‘3’)
tureratio can be written (using eqs. (38) and (41))
amo,
1——aipz
,
for theshock-wave curvature, where *= (43)
amcl
-1= [=%llpm-.+o] and (using eqs. (39) and (42))
,40,
atipes
Sin(bu+a)a+t(i=pcl
[i%+(%ll ‘ho+”-’) ti(B+PL$)
)
(44)
It should be pointed out that equation (38) is, of course, ‘=
equivalent to equation (23). ‘ , (’+w)ti@-”+’)
.
A STUDY OF INVISCID FLOW ABOUT AIRFOIZS AT HI(3H SUPERSONIC SPEEDS 345
rwpectively, A procedure for evaluating equations (43) and airfoils; and second, an investigation of the complete flow
(44) for the flow of a calorically imperfect gas.,aswell asfor field about an esample airfoil. Each of these sections is
an ideal gas, ispresented inAppendix D. (The application further subdivided into a consideration of the effects of
of eq. (43) for anided gashas already been given inref. 9.) Mach number, assuming air behaves as an ideal gas, and
Aswasstatedprevkdy, thepressuregradient andshock- into aconsideration of the combined eflectsof Mach number
wave curvature can be used to calculate the initialpoints of and gaseous imperfections. In the latter regard, the prin-
a chrmcteristic solution for the flow about an airfoil. It is cipal emphasis is placed on the caloric imperfections pre-
apparent that the initial points for a shock-expansion solu- viously discussed.
tion can-befound in a similar manner. With either type of
FLOWINTEEREGIONOFT= LBADR?EGDGEOFCURvl?NllRFO~
solution, flow conditions at an initial point on the surface
downstream of the leading edge can be calculated with the Ideal-gas flow.-The results of the calculations (using
aidof the appropriate value of the surface pressuregradient. eq. @4)) of the surface pressure gradient are presented in
Similarly,theflow conditions at aninitialpoint on theshock table H andiigure2.7 The valuespresentedinthe table are
for arange of Mach numbem from 1.5to y and for leading-
wave can be obtained with the aid of the corresponding
valuo of the shock-wave curvature. Additional points can edge deflection angles horn 0° to 45°. Where no value
be obtained between these two by linear interpolation. If appearsin the table, the flow behind the shock wave issub-
the two initial points arechosen asthe endsof a iirst-family sonic.” Corresponding resultsof the calctiations of surface-
Mach line, there is sufficientinformation available to deter-. pressure-gradientratio arepresentedintableIII andiigure3.
J?romthese resultsit is seenthat except near shock detach-
minethecurvature of thisMach line. Therefore, if detailed
knowledge of the flowintheregion of the leading edgeis re ment, the surface-pressure-gradient ratio variea only from
quired, the surface, shock wave, and &at-family Mach line 0.98 to 1.02for lMachnumberslessthan4. Therefore, very
little error will result from the use of the shock-espansion
crmbe appro.simatedby circular arcs. (Seeref. 17.)
method for the surface pressure gradiant at these lower
INVESTIGATIONOFTHEFLOVVABOUTAIRFOILSAND Mach numbers. For Mach numbers greater than 4 (even
DISCUSSIONOFlZESULTS
f0hart9 wei-eok prtsantedforsurfmepresmregradfen~mrfaa+~t rfwo,
This study isdivided into two sections: first, aninvestiga- endshookwavecnrvalumforldesl+!asflowsfnrekmm %however,theremfts@vaninthe
pi-rmntreportaremrnewhotmoreexknsfve. Tbesa@dts me ah presmted fa cmss-
tion of the flow in the region of the leading edge of curved
plottedformslareferemm17.
175(
150(
125C
Iooc
75C
50C
25C
c )
Leading-edge deflection angle, 3, degrees
(a) M% 1.5 to 6 ‘ (b) M%6t020
FIGURE 2.—Variationof surfamPrwmre gradientwithleading+dgedeflectionanglesforvm”ousfres-stremMachnum&,
,
321a064&28
. .
—
.
346 .REPORT 1123—NATIONM ADV180RY COMMI’?XEE FOR AERONAUTICS
.
(eq. (D5)) arepresentedintableIV andfigure4.’ Similarly,
L5
theresultsof the calculations of shock-wave-curvature ratio
(eq. (44)) arepresentedintableV andfigure6. Except near
detachment, the curvature ratio varies from 0.92 to 1.08for
all Mach numbers including OY.1° Thus, only small errors
1.4
would result horn using the value of the shock-wave curvrL-
ture given by the shock-expansion method for all flow con-
ditions except near shock deta&ment.
Calorically-imperfeot-gas flows,-With increasing Mach
1,3
number andleading-edge slope, the temperatureratio across
> an oblique shock wave increwwaas shown in figure 6. As
=0- the temperature behind the shock wave increases, the be-
k havior of air diverges from that of an.ideal gas as discussed
=z 1.2 . previously. Below 800° R., the’divergence isnot significant,
u
o
and the equationsfor ideal-gaaflow can be appliedwith only
s
w. minute errorsrew.dting. Above 800° R., the energy of the
“,two vib~tional degreesof freedom of the gasmolecules isappre-
ii
:1.1 ciable and becomes greater with increasing tempemtum.
&or l?or these conditions, the specific heats and theirratio vary,
0
% significantly with temperature. The equations developed
$
previously consider theseeffects and permit the extension of
LO the solution for surface pressure gradient and shock-tirme
curvature to the case of calorically imperfect gases. These
. .
.9
.80
10 20
“
Le;ding-edge deflection ang e, &degrees”
FIGURE3.—Variation ofsurfaw-pressure-~tient ratiowith deflection
angle forvarious freetim Mach nurnbem
including o),* the ratio only vmies between about’ O.9”and
1.1 (again except near detachment) and, therefore, only small 3 4!
error might be expected from the use of the shock-expansion
method. Near detachment, at the higher Mach numbers,
the surface-pr-ure-gradient ratio attains a very large range
of vrdues and the maximum value increases with Mach num- /
ber. I?or these conditions, then, the use of the shock-ex-
pansion method for calculating the surface pressure gradient
at the leading edge would result in appreciable. error.
L--l
The flow along the surface is isentropic. Hence, it can be
shown that ~, the ratio of surface pressuregradient to that
given by thegeneralizedshock+ucpansionmethod, isalsothe
velocity-grdent ratio, theMach number gradient ratio, the
@
Mach angle gradient ratio, the densi~-gradient ratio, and
.5
thetemperature-gradientratio. hy of these~dients may .:
be found, then, by calculating the gradient, using the shock-
.:
expmsion method and applying the appropriate value of ~. -2
Thisproperty of theratio #makes itusefulintheapplication 1.5
0 10 2 ) o
of the method of characteristicswith any of the coordinate Leoding-edge d flection am e, 8, degrees
systemscommonly employed in the compatibility equations.
FIGURE4.-Variation of shook-wave ourvatun with lmclhwedaro
The results of the calculations of shock-wave curvature deflection anglefor variousfree-stnwn Maah numbom “ -
$Forthepmtfcnk cassoflmlnitefree-mreamlfa~ nnmlmrendKZIIdeiktfm mgle,tbe *Asfnthemaof#,E./K. hesthedeublevalws ofOand
pmsmm-gmdfentratfekdonblevalad. FromWInatkm(De), itkaprarmt that* bonlty Z$%f”’N’””,’-”.
7’+1
orm dsEwifcm. YeL at fmlnfte Mach nmnbsr.#apprmdm 10M fnthemseof#endK./K.,xhasthedoublev@Jueaof1end for
+le=~e Z(~7-1)[1-~~Y]
deileden sngleapprmclmsu?im. MO_=., a-o.
/
1
*
A STUDY OF R?VISCID FLOW ABOUT AIRFOHA5 AT HKIK SUPERSOMC SPEEDS 347
equations are valid for temperatures up to the order of 5,0000
R. I’or afree~tream temperatureof 500” R., therefore, the
shaded area between lines of constant temperature ratio of
-1.5and 10,in figure 6,representstherange of conditions for
which the method developed in thisreport for the flow of a
&atomic, calorically imperfect gaswould apply.
The excitation of thevibrational degreesof freedom of the
gas molecules requires nfinite number of collisions, causing
the well-known heat-capacity lag discussed in references
5 and 6. The flow distance (i. e., along the streamline)
required to establishequilibriumisusuallysmallin denseair
and willbe considered infinitesimalin thisreport. Also, the
dissociation of air (seeref. 6) will not be considered here.
Since the free-stream static temperature is an additional
parameter in calculations of flow of imperfect gases, only n
limited number of calculations of (1/KJ (dP/dW), ~, .Q.K,
,Yul 1 au,,, I and K were made. The purpose of these calculations is to
o 10 20 30 40 50
compare the variations of these quantities with the values
LeadJng-edge deflection angle, 8, degrees
FIGURE5.—Variationofshook-wave-curvaturraetiowithIeading+dge as given by the ideal-gas-flow computations. The calcula-
deflmtionangle for varioue free-stream Mach numbers. tion foIIowed the procedure described in Appendix D. A
free-stream static temperature of 5000 R. was used. The
50 results of these calculations are presented in table VI for
I
various Mach numbers and leading-edge deflection angles.
The surface pressuregradients for an ideal gas and for a
calorically imperfect, diatonic gas are compared in figure 7.
In all cases calculated, the gradient is smaller for the im-
perfect gas and diverges gradually, with increasing free-
stream AMachnumber and deflection angIe, from the value
of thegradientfor anidealgas. This&vergence isconsist~t
with the increasing effects of the caloric imperfections due
to the increasingtemperaturebehind the shock wave.
The surface-pressure-gradient ratio for the imperfect gas
is compared infigure 8with the ratio for anideal gas. The
surface-prwsure-gradient ratio is smaller for $npwfect-gas
flows than for ideal-gas flows indicating that the effects of
shock-wave and expansion-wave interaction are greater.
This result is attributed, in pait, to the fact that the angle
between the shock wave and the airfoilsurfaceissmallerfor
—
Ideal gas
4
<*
—— -- ‘Colarically imperfect
gas, & =500° R
-E 750
. I
z, Free-stream
.U~-uu. Mach number,MO
: 500
)
0I 4 8 1I2 1I6 210 2I4 Leoding-edge deflection angle, 8, degrees
Free -streom Mach number, 1440 Fmmm 7.—Compariaon ‘of the “variation of surfam presmre gradient
FIGURD&-Varfation ofleading+dge deflectionanglewithfree-stream With leading-edge deflection angle for varioue free-stream Mauh
Mach numberforvariousvalueaof thetempemture ratio. numk-a foranidealw andfor aaalorieally imperfeat, diatomia ga...
REPORT 1123—NATIONAL ADV160RY COMldITI’E13 FOE AERONAUTIC%
imperfecbgas flows than for ideal-gas flows. The difference
between the imperfect- and ideal-gas calculations increases
with increasingMach number and deflection angle, asmight
be expected. In therangeof Mach numbersandflow deflec-
tion angles investigated, however, the extremevalues of *
differonly by about 5percent (seefig. 8). Thus, itisappar-
ent that w-bilethe shock~ansion method willnot bo quite
asaccuratefor calorically-imperfect-gas flowsasfor idenl-gaa
flows, the method willnot be expected to be invalidated.
A divergence with Mach number and deflection anglo is
alsoapparent infigure 9inwhich theshock-wwvecurvatures
for anideal gas and a calorically imperfect, diatonic gas are
I I I J compared. This divergence is compatible with the change
‘
.81
o 10 20 30 40 50 in surface pressuregradient due to the caloric imperbc.tions
Leading-edge deflection ongle, 8, degrees of the gas. The shock-wave-curvature ratio for acalorically
FIGURE8.—Compariacq of the variation of surface-pre~ure-gwlient imperfect, diatomic gas and this ratio for an ideal gas me
ratiowithleading-edgedeflectionangleforvariousfrea+trwwnMach showninfigure 10. Again, itisseenthat theeffect of caloric
numbersforanidealgasandforacaloricallyimperfect, diatomicgas. imperfection is to increas.6the effects of shock-wawe and
expansion-wave interaction.
2.0, I I Ill 1 COMPLETE PLOW FIELDS
— Idealgas I Ideal-gas flows.-The effectsof Mach number of primary
——— Calorically imperfect
interesthereare,of course,‘thosewhich resultfrom theinter-
gas, ~= 500° R
action between the leading-edge (or other) shock wave and
b ~ 1.5 small disturb~ces originating on the surface of an airfoil,
k~
Although the reflected disturbances that are the product of
ai-
2 this interaction willhave the largeat effect on the flow nom
~
the leading edge, their influence on the complete flow field
: Lo- Free-stream about an airfoil also -warrantainvestigation. Further in-
o Moth number,IWO
$ 20-, sight into these effects can be obtained in the region just
downstream of the shock wave without regard for the shapo
$
0 of the airfoil producing the shock. To this end, it is con-
6 .5 .. a8/acl
—=—w (seeeq. (26))
‘timt h ‘wider‘e ‘t10a8/ac,
which may be termed “the disturbance strengthratio” since,
in theregion under consideration, it isammaureof therutio
0 10 20 30 40 50 of strengths of disturbances reflected from the shock wam
Leoding-edge deflection angle, 8, degrees to ,disturbances incident on the wave. This ratio moy be
FIGURE9.-Comparison of the variation of the shock-wave curvature evaluated with equation (4o). This calculation has been
with the leading-edge deflection anglefor variousfree-stream Mach carried out for Mach numbers from 3.5 to co (7=1.4) ml
numbersforanidealgasandforacaloricallyimperfect, diatomicgas.
flow deflection anglea approaching those corresponding to
shock detachment (i. e., Ma = 1), and the results me pre-
sented in figure 11. It is evident that except near ill,= 1,
1.10
I I
Free -slreom
Y Mach number, ---- ------- .
Iwo /H /--’ -—---- ----
=0- / ‘ -—. ___ -.
0
: L05
20
2
z
>
5
u
1
: Loo
3 \
A Ideal gas 1
0
.2 -–--— Cabfically imperfect
m gas, ~ = 500” R
I I
.950 10 20 30 40 50
Leading-edge deflecticm angle, 8, degrees Deflection angle, 8, degrees
I?mmm 10.—Comparison of the variation of shock-wave-curvature FImJREIIl.—Variation withdeflectionangleofthedisturbancestrength
ratio with leading+d~ deflection anglefor VW’OUSfree-strcm Mach ratio behind an oblique shock wave for various free-stnmm Mnch
numbers foranideal gasandforacalorically imperfect, diatmnic gae. numlxm (7=1.4).