[PDF] an engineering aerodynamic heating method for hypersonic flow

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[PDF] an engineering aerodynamic heating method for hypersonic flow

The thermal design of hypersonic vehicles involves accurately and reliably predicting the convective heat- ing over the surface of the vehicle Such results may

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3807_3aiaa_92_0499.pdf AN ENGINEERING AERODYNAMIC HEATING METHODFOR HYPERSONIC FLOWChristopher J? Riley ?NASA Langley Research Center? Hampton? VAFred R? DeJarnette yNorth Carolina State University? Raleigh? NCAbstract

A capability to calculate surface heating rates hasb een incorp orated in an approximate three?dimensionalinviscid technique?Surface streamlines are calculatedfrom the inviscid solution? and the axisymmetric analogis then used along with a set of approximate convective?heating equations to compute the surface heat trans?fer? The metho d is applied to blunted axisymmetric andthree?dimensional ellipsoidal cones at angle of attack forthe laminar ?ow of a p erfect gas? The metho d is also ap?plicable to turbulent and equilibrium?air conditions? Thepresent technique predicts surface heating rates that com?pare favorably with exp erimental ?ground?test and ?ight?data and numerical solutions of the Navier?Stokes ?NS?and viscous sho ck?layer ?VSL? equations? The new tech?nique represents a signi?cant improvementover currentengineering aerothermal metho ds with only a mo dest in?crease in computational e?ort?Nomenclature

A?B?D?Jgeometric factorse

?s ?e?t tangential unit vectors on b o dy surfacee x ?er ?e? unit vectors of cylindrical co ordinate systeme ? ?e? ?en unit vectors of sho ck?oriented co ordinatesysteme ?? ?e?? ?e?n unit vectors of streamline co ordinate systemfsho ck radius? fb o dy radiush ? ?h? scale factors of sho ck?oriented co ordinatesystemh ?? ?h ?? scale factors of streamline co ordinate system ?Aerospace Technologist? Aerothermo dynamics Branch? SpaceSystems Division? Memb er AIAA?y

Professor? Mechanical and Aerospace Engineering Department?Asso ciate Fellow AIAA?Copyrightc?1992 by the American Institute of Aeronautics andAstronautics? Inc? No copyright is asserted in the United Statesunder Title 17? U?S? Co de? The U?S? Governmenthasaroyalty?freelicense to exercise all rights under the copyright claimed herein forGovernment purp oses? All other rights are reserved by the copyrightowner?

MMachnumbernco ordinate normal to sho ck?nco ordinate normal to b o dypstatic pressureqheat?transfer rateRradius of curvatureu? v ? wvelo city comp onents of sho ck?orientedco ordinate systemVvelo city magnitudeVvelo cityvectorx? r??cylindrical co ordinate systemx? y ? zCartesian co ordinate system?angle of attack?sho ck angle relative to freestream velo city?

?b o dy angle relative to freestream velo city? ? sho ck angle in circumferential direction? ? ? b o dy angle in circumferential direction?stream function ratio? ???s? ?inclination angle of surface streamlines? ? ??? sho ck curvatures?? ?sho ck co ordinates? ?? ??streamline co ordinates?density?sho ck angle???????b o dy angle??? ?? ????stream functionsSubscripts bbodyssho ckwwall1freestream conditionsIntro duction

The thermal design of hyp ersonic vehicles involvesaccurately and reliably predicting the convective heat?ing over the surface of the vehicle?Such results mayb e obtained bynumerically solving the Navier?Stokes?NS? equations?

1or one of their various subsets suchas1

the parab olized Navier?Stokes ?PNS?

2and viscous sho ck?layer ?VSL? equations

3?4for the ?ow?eld surrounding thevehicle?However? due to the excessive computer stor?age requirements and run times of these detailed ap?proaches? they are impractical for the preliminary designenvironment where a range of geometries and ?ow pa?rameters are to b e studied? On the other hand? engineer?ing inviscid?viscous metho ds

5?8have b een demonstratedto adequately predict the heating over a wide range ofgeometries and aerothermal environments? Various ap?proximations in the inviscid and b oundary?layer regionsreduce the computer time needed to generate a solution?This reduction in computer time makes the engineeringaerothermal metho ds ideal for parametric studies?Two of the simpler engineering aero dynamic heatingmetho ds that are currently used are AEROHEAT

5?6andINCHES?

7Both use the axisymmetric analog concept

9which allows axisymmetric b oundary?layer techniques tob e applied to three?dimensional ?3?D? ?ows provided thesurface streamlines are known? AEROHEAT calculatesapproximate surface streamlines based solely on the b o dygeometry? INCHES uses an approximate expression forthe scale factor in the windward and leeward planes whichdescrib es the spreading of surface streamlines? Circum?ferential heating rates are then generated by an empiricalrelation? Another area of approximation is the surfacepressure distribution employed by the engineering meth?o ds? AEROHEAT assumes mo di?ed Newtonian theorywhich is inaccurate for slender b o dies? while INCHESuses an axisymmetric Maslen technique?

10The de?cien?cies and limitations of these approximations to the surfacestreamlines and pressures in the engineering aerothermalmetho ds? along with their corresp onding e?ects on thesurface heat transfer? have b een do cumented in Refs? 11to 13?An approximate 3?D inviscid metho d

14?15has b eendevelop ed that is more accurate than mo di?ed Newto?nian theory and has a wider range of applicability thanthe axisymmetric Maslen technique? The inviscid tech?nique uses two stream functions that approximate theactual stream surfaces in the sho cklayer and a mo di?edform of the Maslen second?order pressure equation?

16Themetho d has b een shown to calculate the inviscid ?ow?eldab out 3?D blunted noses as well as 3?D afterb o dies reason?ably accurately and much faster than numerical solutionsof the inviscid ?Euler? equations?

14In this pap er? the approximate inviscid technique em?ploys the axisymmetric analog to predict laminar andturbulent surface heating rates using the approximateconvective?heating equations of Zoby et al?

17Both p erfectgas and equilibrium?air ?ows are considered? Improvedsurface streamlines are calculated based on b oth the b o dygeometry and surface pressure distribution? Surface heat?ing rates are presented for spherically?blunted and asym?metric ellipsoidal cones at angle of attack? Comparisonsare made b etween results of the present technique? VSLand NS solutions? and available exp erimental data to

x G y V ¥ e n

Figure 1?Sho ckwave geometry: side view?demonstrate the accuracy and capability of the presentengineering technique?Analysis

This section describ es the 3?D inviscid technique? thepro cedure for computing inviscid surface streamlines? andthe application of the axisymmetric analog? Approxima?tions and coupling issues are also discussed?Inviscid Metho d

Since a detailed description of the approximate 3?Dinviscid metho d has b een presented previously?

14?15onlya brief outline of the inviscid metho d is given here?Co ordinate Systems

The three?dimensional sho ck surface can b e repre?sented byr s

?f?x? ???1?where ?x? r??? are wind?oriented cylindrical co ordinateswith corresp onding unit vectors ?ex

?er ?e?

?? Thex?axis isaligned with the freestream velo cityvector and is normalto the sho ck surface at the origin? Two angles???

?x? ??and ??x? ??? describ e the sho ckwave shap e and are de??ned astan?? ? 1 f ?f@ tan ? ? ?f@x cos?? ?2?An additional angle is given by?????? ? All anglesare shown in Figs? 1 and 2? For the sp ecial case of ax?isymmetric ?ow?rs ?f?x?? ? ? ??x????

?0?and????Next? a sho ck?oriented curvilinear co ordinate system??? ?? n? is de?ned where?and?represent co ordinates ofa p oint on the sho ck surface andnis the inward distancenormal to the sho ck? Di?erential arc lengths along eachco ordinate direction at the sho ck areh?

d??h? d?? and2 f y z s d f e f r e r

Figure 2?Sho ckwave geometry: rear view?dnwhereh?

andh?

are scale factors for the corresp ond?ing co ordinates? This co ordinate system is well?suited forhyp ersonic ?ow?M1

?1? and thin sho cklayers?The unit vectors?e? ande? ? are tangent to the sho cksurface and are chosen such thate? is in the direction ofthe tangential velo city just inside the sho ck surface? Theunit vectore? is then de?ned to b e p erp endicular toe?anden ? In cylindrical co ordinates? the unit vectors of thecurvilinear co ordinate system are given bye ? ?cos?ex ? sin? ?cos?? er ?sin?? e? ?e ? ?sin?? er ? cos?? e? ?3?e n ?sin?ex ?cos ? ?cos?? er ?sin?? e? ?Although this curvilinear co ordinate system is orthogo?

nal at the sho ck surface? it is nonorthogonal within thesho cklayer for a general three?dimensional sho ck? How?ever? for thin sho cklayers? orthogonalitymay b e assumedeverywhere?The velo city is de?ned in terms of the unit vectorsat the sho ckasV?ue?

?ven ?we? ?4?From the de?nition ofe? ande? ? the cross?owvelo citycomp onent at the sho ck?ws

? is equal to zero?Governing EquationsThe governing equations for 3?D inviscid ?ow aresimpli?ed by assuming that the velo city comp onentwis equal to zero not only at the sho ck but throughout thesho cklayer? This yields two stream functions? ? ?whichis equal to?here? and ?? which approximate the actualstream surfaces in the sho cklayer? The stream function? is analogous to the Stokes stream function for axisym?metric ?ow?Approximate expressions for the pressure and nor?mal velo city comp onent are then obtained by transform?ing the normal momentum and continuity equations tostreamline co ordinates and evaluating the ?owvariables

at the sho ck?Along a line normal to the sho ck? theseexpressions arep????ps ?p1 ???1? ?p2 ??

2?1??5?v????vs

?v1 ???1??6?wherep 1 ? ? s us ?? h?p 2 ?? ? s vs tan? 2h? ??? ??? ?v 1 ? ? s vs h? cos? ??? ??? ?and ?? ?

sDe?ning ? ? 0 to b e the b o dy surface gives??1onthe sho ck and?? 0 on the b o dy?Note that Eq? ?5?reduces to Maslen?s second?order pressure equation

16foraxisymmetric ?ow if the scale factorh?

is equal to thesho ck radiusrs ?Metho d of Solution

Since the inviscid metho d is an inverse one? the sho ckshap e must b e varied until the correct b o dy shap e is pro?duced? The resulting iteration pro cedure is handled dif?ferently in each region of the ?ow?In the stagnation region of a blunt body travelingat hyp ersonic sp eeds? the ?ow is subsonic and the sho ckshap e for the entire subsonic?transonic region must b edetermined globally?A 3?D sho ck given by longitudi?nal conic sections blended in the circumferential directionwith an ellipse is assumed? The parameters describing thesho ck are iterated until the b o dy shap e ?? ? 0? generatedby the approximate inviscid metho d matches the actualb o dy shap e at several discrete p oints? In this study? sixsho ck parameters are varied until the calculated b o dy ismatched to the actual b o dy at six lo cations?Once past the transonic region? the inviscid ?owistotally sup ersonic and a marching scheme is well p osed?The sho ck surface from the transonic region forms a start?ing solution for the marching pro cedure? The sho ckvari?ables are extrap olated in?along a numb er of constant?lines which circle the sho ck? On each line? the sho ckcurvature??

is lo cally iterated until the calculated b o dyshap e matches the correct b o dy? The sho ckvariables arethen advanced downstream to the next??lo cation and thepro cess rep eated?Axisymmetric Analog

The 3?D b oundary?layer analysis is simpli?ed by us?ing the axisymmetric analog

9as is done in most engi?neering aerothermal metho ds? The 3?D b oundary?layer3

equations are ?rst written in a streamline co ordinate sys?tem? The cross?owvelo city comp onent tangenttothesurface but normal to the streamline is then assumed tob e zero? This simpli?cation reduces the 3?D b oundary?layer equations to the axisymmetric form provided thedistance along the streamline is substituted for the surfacedistance and the scale factor describing the divergence ofthe streamlines is interpreted as the axisymmetric b o dyradius? Axisymmetric b oundary?layer metho ds can thenb e employed in the existing 3?D inviscid technique?Inviscid Surface StreamlinesBefore applying the axisymmetric analog? inviscidsurface streamlines are computed from the approximateinviscid solution?Inviscid surface streamlines maybecalculated from the surface pressure distribution

5orfrom the velo city comp onents?

8The approximate inviscidmetho d

14?15used here predicts accurate surface pressures?but the direction of the velo city on the surface is not ac?curate? Therefore? in the present metho d? streamlines arecalculated from the surface pressures?A streamline co ordinate system

5? ??? ????n? is de?nedwhere ??and

??are co ordinates of a p oint on the b o dysurface and ?nis the distance normal to the b o dy? Thebars indicate the variables apply to the b o dy and notthe sho ck? Di?erential arc lengths along each co ordinatedirection at the b o dy areh??

d ???h?? d ??? andd?nwhereh??andh?? are scale factors for the corresp onding co ordinates?If the b o dy surface is represented byrb ?

?f?x? ?? in windaxes with the axial co ordinate parallel to the freestreamvelo city and passing through the stagnation p oint? theunit vector normal ?outward? to the b o dy surface is givenbye

?n ??sin ??ex ? cos ?? ?cos ?? ? er ?sin ?? ? e? ??7?The b o dy angles are de?ned in the same fashion as the sho ck angles and aretan ?? ? ? 1 f ? ?f @ tan ??? ? ?f@x cos ?? ? ?8?The tangential unit vectors at the surface?e?? ande?? ? aresimilar to the tangential unit vectors at the sho ck? FromRef? 5? they are given ase ?? ?cos ??e?s ? sin ??e?t ?9?e ?? ??sin ??e?s ? cos ??e?t ?10?wheree ?s ?cos ??ex ? sin ?? ?cos ?? ? er ?sin ?? ? e? ??11?e ?t ?sin ?? ? er ? cos ?? ? e? ?12?and the angle ??represents the orientation of the surfacestreamlines? Note that the vectors?e?s ande?t ? are identi?cal in form to the unit vectors?e? ande? ? de?ned at thesho ck? The orientation of the inviscid surface streamlines?given by

??? is found by applying the momentum equationsalong the b o dy surface using the pressure distributiongenerated by the inviscid solution? By writing the mo?mentum equations in streamline co ordinates? taking thescalar pro duct withe??

? and substituting the unit vectors?Eqs? ?9? and ?10?? this may b e expressed as1 h?? ? ??@ ?? ?? sin ??h?? ???@ ?? ? 1?b V 2b 1h?? ?p b? ?? ?13?where ????? ?? ? ? The scale factorh?? can b e determinedby noting that for an orthogonal curvilinear co ordinatesystem ? @ ?? ? h ?? e ?? ? ? ?@ ?? ? h ?? e ?? ?Taking the scalar pro duct of this equation withe?? andagain substituting the unit vectors? Eqs? ?9? and?10??yields 1 h?? ?lnh??? ?? ? 1h?? ? ??@ ?? ? sin ??h?? ???@ ??

?14?Equations ?13? and ?14? maybe integrated along asurface streamline to obtain the streamline direction

??and the scale factorh??

?Although the surface stream?lines can b e determined after the inviscid solution hasalready b een calculated? it was found to b e more con?venient to compute the inviscid solution and the surfacestreamlines simultaneously? Before applying these equa?tions along sho ck co ordinates? transformation op eratorsrelating derivatives with resp ect to the the streamline co?ordinates ?

???

??? to derivatives with resp ect to the sho ckco ordinates ??? ?? are needed? In the approximate invis?cid metho d? the curvilinear co ordinate system is assumedto b e orthogonal throughout the sho cklayer? This as?sumption simpli?es the analysis but do es not change theform of the approximate pressure and velo city relations?Eqs? ?5? and ?6?? since the ?ow?eld variables are evaluatedat the sho ck where the co ordinate system is orthogonal?However? at the b o dy surface? the correct co ordinate di?rections need to b e considered? Following the approachof Ref? 15 and using the nonorthogonal directions at thesurface? the transformation op erators areJ

h?? ?@ ?? ? ?Be?? ?e? ?De?? ?e? ? 1h? ?@? ??De?? ?e? ?Ae?? ?e? ? 1 h? ?@ ?15?and J h?? ?@ ?? ? ?Be?? ?e? ?De?? ?e? ? 1h? ?@? ??De?? ?e? ?Ae?? ?e? ? 1 h? ?@ ?16?where

A?1?nb

??4

B?1?nb

??D? n b h? ????J?AB ? D

2These op erators can b e used to calculate the pressure

derivative in Eq? ?13? as well as allow Eqs? ?13? and ?14?to b e integrated with resp ect to the sho ck co ordinate??Boundary?Layer Metho dThe axisymmetric analog allows any axisymmetricb oundary?layer metho d to b e applied along an inviscidsurface streamline? In this study? a set of approximateconvective?heating equations develop ed by Zoby et al?

17is used for the b oundary?layer solution? Laminar and tur?bulent heating rates may b e calculated from these rela?tions for b oth p erfect gas and equilibrium?air ?ows? Ap?proximate expressions for the b oundary?layer thickness atb oth laminar and turbulent conditions are also given inRef? 17? Results using this technique have b een shown tocompare favorably with more detailed metho ds for windtunnel and ?ight conditions?

18?20Boundary?layer edgeconditions are found byinterp olating in the approximateinviscid solution a distance away from the wall equal tothe b oundary?layer thickness?This approach has b eendemonstrated to approximately account for the e?ects ofentropy?layer swallowing?Results and Discussion

Surface heating rates are presented at p erfect gas andlaminar conditions over spherically?blunted and 3?D ellip?soidal cones at angle of attack in order to demonstrate thecapability and accuracy of the present technique? A com?parison with ?ight data obtained at laminar and turbulent?ow conditions is also presented based on equilibrium?aircalculations?Spherically?Blunted Cones

Computed laminar surface heating rates are pre?sented in Figs? 3 and 4 for the windward plane of a 15deg spherically?blunted cone at angles of attackof5and10 deg?The freestream Machnumb er is 10?6 and thenose radius is 1?1 inches? Results of the present metho dare compared with results of an engineering aerothermalmetho d AEROHEAT

5?6and exp erimental data?

21Goodagreement ?within 10 p ercent? b etween the results of thepresent metho d and the exp erimental data is shown inFigs? 3 and 4? The AEROHEAT results fail to predictthe correct magnitude of the surface heating as well asthe lo cal maximum in the heating? These discrepanciescan b e attributed to the approximate pressure distribu?tion and streamlines used in AEROHEAT? Circumferen?tial heating rates are presented in Figs? 5 and 6 at two

Alt = 150 kft

M ¥ = 15 R b = 0.125 ft T w = 2260 deg R

Figure 7?Comparison of surface heating rates for 5 degsphere?cone at?? 3 deg?axial lo cations on the blunted cone for angles of attackof 5 and 10 deg? The windward plane is lo cated at??0 deg and the side plane is at?? 90 deg? The compar?ison of the exp erimental and predicted heating rates isseen to b e go o d at b oth axial stations of 4?86 and 10?13nose radii? This comparison illustrates that the presenttechnique is capable of computing heating rates o? thewindward plane of symmetry?In order to demonstrate the signi?cant improvementof the present metho d over current engineering aero dy?namic heating metho ds? the surface heating rates in thewindward plane of symmetry are calculated for a 5 degspherically?blunted cone at an angle of attack of 3 deg?The freestream Machnumb er is 15 and the freestreamconditions corresp ond to an altitude of 150?000 ft? Thewall temp erature is 2260 deg R and the nose radius is0?125 ft? Heating rates are computed using the presenttechnique? AEROHEAT? INCHES?

7and a detailed VSLmetho d?

11The resulting surface heating rates are pre?sented in Fig? 7? The surface heating rates generated byAEROHEAT and INCHES di?er byas muchas 40per?cent from the more accurate VSL solution? On the other

0.02.55.07.510.012.5

10 1 10 2 10 3 q w , B T U / f t 2 - s

Present method

x, ft

Laminar

Turbulent

Alt = 80 kft

M ¥ = 19.97 R b = 0.01167 ft

EQLB Air

Reentry F (Ref. 22)

Figure 8?Comparison of surface heating rates withReentry F ?ight data ?5 deg sphere?cone at?? 0?14deg??hand? the solution of the present metho d shows much b et?ter agreement ?within 15 p ercent? with the VSL resultsand also predicts the correct trend in the surface heatingrate levels?The surface heating rates over a 5 deg spherically?blunted cone at equilibrium?air and turbulent conditionsare examined next in Fig? 8? Results from the presentmetho d are compared with heat?transfer data obtainedfrom the ?ight exp eriment Reentry F?

22The Reentry Fvehicle was a 5 deg spherically?blunted cone with a lengthof 13 ft and an initial nose radius of 0?1 inches?Thedata shown in Fig? 8 corresp ond to a tra jectory p ointat 80?000 ft? The freestream Machnumb er is approxi?mately 20 and the angle of attack is 0?14 deg? The resultsdepicted corresp ond to the leeward plane of the vehicle?In the present technique? equilibrium air prop erties areobtained from Hansen?

23while transition is assumed tob egin at the rep orted distance?

22The calculated heatingrates in the transition region are based on the Dhawanand Narasimha

24mo del? Excellent comparison b etweenthe results from the present technique and the ?ight lam?inar and turbulent data is noted?Ellipsoidal Cones

The p erfect gas? laminar solution over a blunted 2:1ellipsoidal cone is examined next at angles of attackof0and 15 deg? The cone angles in the windward and sideplanes are 5 and 9?93 deg? resp ectively? The freestreamMachnumb er is 10?19 and the nose radius in the sideplane is 1?0 inch? Surface heating rates from the presenttechnique are compared with results from a NS metho d?LAURA?

1and exp erimental data?

25The LAURA metho dis chosen for comparison purp oses b ecause of its abilityto compute the ?ow?eld ab out a 3?D nose? In addition?there is an apparent lack of heat?transfer data availablein the op en literature on 3?D nose shap es? Thirty?sevenstreamlines are used to obtain the solution around the6

0.02.55.07.510.0

10 -1 10 0 10 1 10 2 q w , B T U / f t 2 - s

Present method

x /R b f = 90 deg LAURA f = 0 deg M ¥ = 10.19 a = 0 deg

Figure 9?Comparison of surface heating rates for 2:1ellipsoidal cone?ellipsoidal cone in the present technique? A grid of 64 cellsin the axial direction? 30 cells around the circumferenceof the b o dy? and 64 cells in the normal direction is usedto obtain the LAURA solution? The present techniquerequires approximately 200 CPU sec on a Sun workstationto obtain a solution? while the LAURA solution requiresapproximately 4 CPU hrs on a CRAY?2 sup ercomputer?No e?ort was made to optimize the LAURA calculations?Axial surface heating rates are depicted in Fig? 9 for

the windward ??? 0 deg? and side ??? 90 deg? planesat an angle of attack of 0? Go o d agreement is noted nearthe nose and in the side plane downstream?However?in the windward plane downstream? the results from thepresent technique overestimate the results generated byLAURA by 25 p ercent? For the ellipsoidal cone? the sur?face streamlines diverge rapidly from the side plane andconverge towards the windward plane? Unfortunately?inthis in?ow region near the windward plane? it app earsthat the approximate surface pressures are not accurateenough to predict reasonable streamline paths? For thisreason? the solution over the ellipsoidal cone at 0 deg angleof attack is computed using simpli?ed surface streamlinesby setting the streamline angle

??equal to zero? Account?ing for the in?ow correctly downstream would reduce theheating rates near the windward plane? However? at an?gle of attack? the streamlines are again computed usingthe surface pressures since the in?ow is reduced?Circumferential heating rates for the ellipsoidal coneat 0 deg angle of attack are depicted in Figs? 10 ? 13 atfour axial lo cations on the b o dy? The ?rst is on the 3?Dnose? while the remaining three are downstream on the3?D afterb o dy? Excellent agreement ?within 10 p ercent?is seen atx?Rb

?0?4 on the 3?D nose? Atx?Rb

M ¥ = 10.19 a = 15 deg x/R b = 9.7

Figure 18?Comparison of circumferential surface heat?ing rates for 2:1 ellipsoidal cone?over present engineering metho ds? but the applications to3?D b o dies signi?cantly enhance current capabilities?Concluding Remarks

A rapid but reliable engineering aero dynamic heatingmetho d has b een develop ed by coupling an approximate3?D inviscid technique with the axisymmetric analog anda set of approximate convective?heating equations? Sur?face streamlines are calculated using b oth the b o dy ge?ometry and surface pressure distribution?The metho dis applied to the solution over spherically?blunted conesand 3?D ellipsoidal cones at angle of attack for the laminarand turbulent?ow of a p erfect gas and equilibrium air?The present technique predicts surface heating rates thatcompare favorably with exp erimental data? equilibrium?air ?ight data? and numerical solutions of the NS andVSL equations? It also represents a signi?cant improve?mentover current engineering aerothermal metho ds withonly a mo dest increase in computational e?ort?Acknowledgements

Research p erformed by the second author was sup?p orted by Co op erative Agreement NCC1?100 with theAerothermo dynamics Branch of the Space Systems Di?vision at NASA Langley Research Center? The authorswish to thank Mr? E? Vincent Zoby? Dr? Peter A? Gno?o?and Mr? H? Harris Hamilton of the Aerothermo dynamicsBranch and Dr? F? McNeil Cheatwo o d of North CarolinaState University for their guidance and assistance?References

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Cleary? J? W?? ?E?ects of Angle of Attack andBluntness on Laminar Heating?Rate Distributions of a15 oCone at a Mach Numb er of 10?6?? NASA TN D?5450?Oct? 1969? 22

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