ically. Theregion i

k leadingnts. Results

Theoretical aerothermal concepts for configuration designof hypersonic vehicles

Shripad P. Mahulikar∗,1

Department of Aerospace Engg, Indian Institute of Technology, Bombay, P.O. IIT Powai, Mumbai 400076, India

Abstract

Convection coefficients and heat fluxes due to aerodynamic heating on critical surfaces of hypersonic vehicle are obtained analytapplicability of recovery temperature for stagnation regions is discussed. Convection coefficient for the bicurvature forward stagnationsobtained directly from 2-D stagnation region correlation, using the two principal radii of curvatures. Convective heat flux to swept-bacedge (SBLE) surface is obtained from the 2-D stagnation region and flat plate heat fluxes, using the respective velocity vector componereveal the concepts of temperature-minimised-sweepback, and the thermally-benign sharp SBLE effect at high sweepback angles.

Keywords: Aerodynamic heating; Aerothermal; Hypersonic; Sweepback

1. Introduction 2. Theoretical analysis of convection to stagnation region

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The aerodynamic heat flux on a flat plate wall at 0◦-

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Aerodynamic heating increases as (V∞) , but aerodynamicdrag increases only as (V∞)2 [1]. The ascent peak stagnatiopoint and wing leading-edge equilibrium wall temperatureshypersonic trans-atmospheric vehicles (TAVs) are about 4and 3000 K, respectively [13]. Aerothermal environmentsthe design of X-34 were generated from conservative approbased solely on engineering methods applied to criticaleas [4,15]. Approximate analyses were used to estimatestagnation-point heat flux for TAVs [6]. Study of the inflence of nose bluntness on heating levels for simple geomecorroborated that blunting reduces laminar heat transfer tohypersonic vehicle [16]. An overview revealed that to compthe surface temperature accurately, it is necessary to desthe convection mechanisms [3]. Based on the survey, it is fothat convection correlations are available for axisymmetric2-D stagnation regions [1,14], but not for bicurvature forwstagnation and SBLE regions.

f0

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incidence (denoted by subscript ‘w, fp’), is defined as,

q ′′w,fp = hfp(Tr − Tw). (1)

The recovery temperature for calorically perfect air is givenTr = Tl{1 + rl[(γ − 1)/2](Ml)

2}; where,Tl is local static tem-perature,Ml is local Mach number,rl is recovery factor baseon local parameters (= Prn

1), andγ is the ratio of specific heatsThe exponent of Prandtl number,n = 0.5 for laminar flow, and1/3 for turbulent flow [1,14]. There is confusion in literatuon the applicability of Eq. (1) for convection to stagnationgions. However, Eq. (1) is for flat plate at 0◦-incidence, andusesTr as the basis. Truitt [14] proposes the use ofTr; butAnderson Jr. [1] has also suggested the use ofT0∞, in addi-tion to the use of the adiabatic wall temperature,Taw. However,the stagnation region velocities are so low that the heaterated due to viscous dissipation is insignificant. Hence,heat source term in the boundary layer energy equationnot exist. The mechanism of temperature rise in the station region flow is due to reversible conversion of flow-woto heat. Hence, convection occurs betweenTw and the statictemperature at the low velocity boundary layer edge,T0∞. As

682

Nomenclature

Cp specific heat of air at constant pressure J kg−1 K−1

h convection coefficient . . . . . . . . . . . . . . W m−2 K−1

q ′′w convective heat flux at wall/surface of hypersonic

vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W m−2

T temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . KV velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m s−1

x distance from stagnation point/leading edge offlat-plate measured along flow . . . . . . . . . . . . . . . . m

Subscripts

stag stagnation region0 stagnation value∞ free-stream value

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four temperatures:Taw, Tr, T0∞, andT∞, are used in literature, the following clarification illustrates their applicability: (In the stagnation region,Taw = T0∞ �= Tr; (ii) For flat plate at0◦-incidence,Taw = Tr �= T0∞; and (iii) AsM∞ → 0 (low sub-sonic flow),Taw = T∞ = T0∞ = Tr.

The h is based on the Reynolds analogy, which holdsV∞ < 3 km/s [7]. Stagnation regions are characterised by hh, because the full impact of air-flow promotes thermal dfusion. Since a bow shock stands in front of the stagnaregion, the flow in its vicinity is low subsonic and laminar; fwhich, the Stanton number is,Stl = f/(Re1x)

1/2 [2]. The Re1x

is Reynolds number, and subscript ‘x ’ denotes the dimensiofor estimatingRe. Thus,h = f Cpl(ρlVlµl/x)1/2, where,ρ isdensity of air,µ is dynamic viscosity of air; and since,Vl = βx,

h = f Cpl(ρlβµl)1/2. (2)

For cylinders, the velocity gradient along 2-D stagnationgion is,β = 4V∞/D; and for spheres,β = 3V∞/D; whereD

is diameter. The 2-D and axisymmetric laminar stagnation pmeterf in Eq. (2), are respectively given as,f = 0.57(Prl)

−0.6,andf = 0.763(Prl)

−0.6. The lowerf for 2-D stagnation is dueto lower flow relieving. Hence,

h2-D,stag= 1.14(Prl)−0.6Cpl(ρlµlV∞/D)1/2, (2.1)

and

haxy,stag= 0.763(3)1/2(Prl)−0.6Cpl(ρlµlV∞/D)1/2. (2.2)

Thus,h ∝ D−0.5, and (h2-D,stag/haxy,stag) < 1 (≈ 0.863).Fig. 1 shows schematic views of the forward stagnation

typical hypersonic vehicle. The bicurvature forward stagnais described as: “a cylinder of radiusr that is bent in a plane pependicular tor by radiusR(generallyr R)”. The h is linkedto the boundary layer thickness (by Reynolds analogy), wdepends on the extent of flow relieving. The flow relieving retive to geometries for which convection coefficient correlatioare known, is the physical basis for estimatinghbicurv,stag. Basedon flow relieving by the geometries: bicurvature, axisymmetand 2-D stagnations, the following inequalities must be satisby hbicurv,stag:

hbicurv,stag> h2-D,stag(r), (3)

and

hbicurv,stag> h2-D,stag(R). (4)

The effect of bicurvature is to reduce the 2-D thermal retances due tor and R (because flow is more relieved than

r

n

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-

a

h

,d

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(a)

(b)

Fig. 1. Schematic sketch of lifting body of typical air-breathing hypersonichicle. (a) Plan view. (b) Side view.

2-D flow). Axisymmetric flow has three directions to moveup, down, and sideways; in which it encounters the sameresistance. Hence, the flow is most relieved; consequently,

hbicurv,stag< haxy,stag(r). (5)

Thehbicurv,stag must satisfy inequalities (3)–(5), which are tphysical bases. It ishypothesised thathbicurv,stag is,

hbicurv,stag={[

h2-D,stag(R)]2 + [

h2-D,stag(r)]2}1/2; (6)

which resembles in form the approach in [8] (Eq. (8)). Eq.satisfies inequalities (3) and (4); but from Eqs. (2.1) and (2inequality (5) is satisfied if,

R > 2.908r. (6.1)

Eq. (6) suggests that:

hbicurv,stag= h2-D,stag(Rbicurv,stag). (7)

Eq. (7) resembles the Hypersonic Equivalence Principlesince,hbicurv,stag (for 3-D geometry) is obtained fromh2-D,stag.From Eqs. (2.1) and (7),hbicurv,stag= 1.14(Prl)

−0.6Cpl{ρlµl ×V∞[(1/R) + (1/r)]/2}1/2; hence, [1/Rbicurv,stag(R, r)] =(1/R) + (1/r), where,Rbicurv,stag(R, r) is the equivalent radius. Eq. (7) is used when inequality (6.1) is satisfied; e

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hbicurv,stag= haxy,stag(r). For (R/r) < 2.908, bicurvature stagnation behaves as axisymmetric stagnation of radiusr , leadingto higherh and conservative results. This approach is supeto the use of currently available correlations for 2-D andisymmetric stagnations, which respectively under-predictover-predicthbicurv,stag. Alternative hypotheses forhbicurv,stagare possible that satisfy inequalities (3) and (4), and satisfacof inequality (5) can expand the range of applicability relatto Eq. (6.1). However, the proposed hypothesis enables estion of hbicurv,stag directly fromh2-D,stag, usingRbicurv,stag; andis reasonably conservative.

3. Convection model for swept-back leading edge (SBLE)region

The wing leading edge heating was computed using swcylinder theory as [8],

q ′′w,SBLE = (

0.5q ′′2w,2-D,stagcos2 Λ + q ′′2

w,fp sin2 Λ)1/2

. (8)

For Λ = 90◦, q ′′w,SBLE = q ′′

w,fp; however, whenΛ = 0◦,

q ′′w,SBLE = q ′′

w,2-D,stag/21/2 �= q ′′w,2-D,stag. For SBLE of fin [5],

hSBLE = h2-D,stagcos0.24Λ. (9)

For sweepback angleΛ = 0◦, hSBLE = h2-D,stag; but whenΛ =90◦, hSBLE = 0 �= hfp. For a turbulent flat plate at an angleφ tothe freestream (forV∞ < 3962 m/s) [13],

q ′′w(W/cm2) = 3.89× 10−8(ρ∞)0.8(V∞)3.37cos1.78(φ)

×sin1.6(φ)(xT)−1/5(Tw/556)

×[1− 1.11STw/(CpT0∞)

]. (10)

Theφ is angle of incidence with respect to freestream (= 90◦ −Λ◦), and xT is measured along the flow from the pointcommencement of fully-turbulent boundary layer. In the limφ → 0◦ andφ → 90◦, q ′′

w → 0, which is incorrect. An engineering formulation for heat flux based on the swept-infincylinder theory is given as [12],

q ′′w = 1.29× 10−4(2ρ∞/D)0.5(V∞)3(1− 8 sin2 Λ)cosΛ

×{1− STw/

[CpT∞ + 0.5(V∞)2(1− 0.18 sin2 Λ)

]}, (11)

where, S is the specific heat of wall material. WhenΛ =90◦, q ′′

w = 0 �= q ′′w,fp; and whenΛ = 0◦, q ′′

w = 1.29× 10−4 ×(2ρ∞/D)0.5(V∞)3{1− STw/[CpT∞ + 0.5(V∞)2} �= q ′′

w,2-D,stag.Thus, Eqs. (8)–(11) fail to captureq ′′

w-variation and convectionmechanisms, over the entire range ofΛ.

A methodology is now derived that describes the convtion mechanisms at SBLE. It is assumed that there is nopingement of bow-shock, which is true for typical configrations (unlessΛ again decreases along the span). AsΛ in-creases from 0◦–90◦, the convection mechanism must gradally change from 2-D stagnation to flat plate at 0◦-incidence.Assuming the oblique shock to be weak,V∞-vector compo-nent along SBLE surface isV∞ sinΛ, and perpendicular tothis surface isV∞ cosΛ (Fig. 1(a)). ForV∞ cosΛ componentperpendicular to the surface,h2-D,stag-correlation (Eq. (2.1))

r-d

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is used; and forV∞ sinΛ component along the surface,hfp-correlation is used. Thehfp-correlation is based on turbuleflow assumption throughout SBLE; and is given as,hfp =0.0292(ρlVl)

0.8(Cpl)1/3(kl)

2/3/[(µl)7/15x0.2] [11]; where,k is

thermal-conductivity of air. As viscous dissipation andversible compression mechanisms co-exist, their thermal rtances are in parallel; and the surface area for these mechais identical. Hence,h2-D,stagandhfp (which are normal to SBLEsurface) should be added, if these convections occur ovesame temperature difference. Addition of the twoh’s results inaddition of their respectiveq ′′

w’s; and in this case, vice-versalso holds. But in hypersonic flow, the temperature differefor heat transfer corresponding toh2-D,stagis (T0 −Tw), and cor-responding tohfp is (Tr − Tw) (Section 2). Hence,h2-D,stagandhfp cannot be directly added; but the twoq ′′

w’s (also normal toSBLE surface) can be added as,

q ′′w,SBLE = q ′′

w,2-D,stag+ q ′′w,fp. (12)

The heat fluxes in Eq. (12) are:

q ′′w,2-D,stag= h2-D,stag(V∞ cosΛ)(T0 − Tw), (12.1)

where, for calorically perfect air,T0 = T∞+(V∞ cosΛ)2/(2Cp)

and

q ′′w,fp = hfp(V∞ sinΛ)(Tr − Tw), (12.2)

where,Tr = T∞ + rl · (V∞ sinΛ)2/(2Cp). The splitting ofV∞-vector is also the basis for reducing wave drag experiencethe vehicle; because for highΛ,V∞-component perpendiculato the surface is reduced. For estimating the total drag on aclined wedge also,V∞-vector is split in to components, normand tangential to the inclined wedge surface [10]. These prdures based on superposition ofq ′′

w’s assume dominant linearitin the superposed mechanisms. Though all real processenon-linear, the additional non-linearity does not alter the qutative characteristics. However, estimated parameters do chin quantity; which does not affect the objective of this invegation. Hence, this procedure is reasonable to reveal additconcepts in configuration design; which are based on qualittrends of parameters. At low Mach numbers,T0 ≈ Tr ≈ T∞;hence, Eq. (12) reduces to,q ′′

w,SBLE = hSBLE(Tw −T∞), where,hSBLE = h2-D,stag(V∞ cosΛ) + hfp(V∞ sinΛ). From Eqs. (12)–(12.2), asΛ → 0◦, q ′′

w,SBLE → q ′′w,2-D,stag; and asΛ → 90◦,

q ′′w,SBLE → q ′′

w,fp. For turbulent hypersonic flow, thermophyical properties are evaluated at the local Eckert’s referetemperature (Tref) [2,9]. The reference density is obtainedρref = Pstat/(RgTref); where,Rg is gas constant for air.

To obtain the local static pressurePstat, on SBLE surfacewithout solving the Euler equation numerically, an analytiprocedure is derived here. Since SBLE of hypersonic vehiis not sharp, the normal component ofV∞ experiences a normashock (if it is supersonic). The corresponding static presafter the normal shock (P2) depends only on the normal component of the freestream Mach number:M∞ cos(Λ). Hence,P2 = P∞{[2γ /(γ + 1)](M∞ cosΛ)2 − (γ − 1)/(γ + 1)} [1];where, subscript ‘2’ refers to station downstream of norm

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Fig. 2. Variation ofq′′w vs. x along SBLE surface (Λ = 74◦).

shock. The normal component of Mach number (denotedsubscript ‘n’) is,

Mn2 =√

(M∞ cosΛ)2 + [2/(γ − 1)][2γ /(γ − 1)](M∞ cosΛ)2 − 1

.

The corresponding static temperature is

T2 = T∞(P2/P∞){[

(γ − 1)(M∞ cosΛ)2 + 2]

/[(γ + 1)(M∞ cosΛ)2]};

and the velocity of sound is,a2 = (γRgT2)1/2. Since,Vn2 =

Mn2a2, the resultant velocity is,V2 = [(Vn2)2+(V∞ sinΛ)2]1/2,

because the parallel component of velocity is unaffected bynormal shock; thus,M2 = V2/a2. The static temperature in thSBLE region is,TSBLE = T0∞ − (V∞ sinΛ)2/(2Cp); where,T0∞ = T∞{1 + [(γ − 1)/2](M∞)2}, and the Mach number isMSBLE = V∞ sinΛ/(γRgTSBLE)1/2. ThePstat is obtained usingisentropic relation between the region immediately downstrof the normal shock and the SBLE region as

Pstat= P2{[

1+ (γ − 1)(M2)2/2

]/[1+ (γ − 1)(MSBLE)2/2

]}γ /(γ−1).

The SBLE of lengthL is discretised in to isothermal finite segments. The temperature of each segment is obtainbalancing net convection input (q ′′

w,SBLE) to radiation loss tosky (q ′′

w,rad), under steady-state. The inter element conducis assumed negligible, which gives a conservative maximtemperature. The nodal temperature of segments is obtaby the Newton–Raphson method. The lifting body of a tycal hypersonic vehicle has an SBLE region of varying thickn

y

e

by

n

ed

s

Fig. 3. Variation ofTw,SBLE vs. x for variousΛ.

(Fig. 1(b)). Fig. 2 givesq ′′w,2-D,stag, q ′′

w,fp, andq ′′w,SBLE, along the

length (L) of SBLE; at cruise altitudeH = 35 km andM∞ = 7.The dimensionless distancex (= x/L), is measured from thforward stagnation point. The componentq ′′

w,2-D,stagis negative,i.e. it ‘virtually’ cools SBLE, heated byq ′′

w,fp; since,Tw,SBLE >

T0. Hence, stagnation region with smaller diameter isvirtuallycooled more effectively; butq ′′

w,SBLE is positive. Temperaturvariations along SBLE are in Fig. 3, for lifting body with typicdimensions (scaled) shown in Fig. 1. The temperatures areest whenΛ = 0◦, and monotonically decrease up to aboutΛ =80◦ (dashed curve). These results demonstrate the temperminimised-sweepback, which conceptually differs fromdrag-minimised-sweepback. The drag-minimised-Λ results inminimum heat generation and the temperature-minimiseΛ

results in minimum heat transfer to the vehicle. Beyondtemperature-minimised-Λ, the temperatures again increaseto Λ = 90◦ (shown by dashed curve forΛ = 89◦). The exper-imental data that supports the temperature-minimised-Λ is in[5]. The temperatures forΛ = 89◦ exceed those forΛ = 80◦,which is not predicted by presently available correlations (E(8)–(11)). The temperature-minimised-Λ is especially beneficial during re-entry; since, the additional drag is benign.powered flight, optimumΛ should be based on a trade-off btween low drag and temperature.

Table 1 givesTw,SBLE,max for different Λ for two cases:Case 1 is for the actual radius of SBLE, and Case 2 is forthe actual radius of SBLE. ForΛ = 0◦, the smaller radius leading edge surface has a higher temperature (as expected

685

Table 1Illustration of thermally benign sharp SBLE effect

Λ◦ → 0 30 50 Thermally-benign sharp SBLE effect

60 70 80 89

Tw,SBLE,max Case 1 1382.7 872.0 675.0 598.5 552.7 546.0 600.4(◦C) Case 2 1489.3 955.8 705.6 597.6 526.7 512.8 590.0�Tw,SBLE,max (◦C) 106.6 83.8 30.6 −0.9 −26.0 −33.2 −10.4

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Eq. (2.1)). However, forΛ > 60◦, the smaller radius SBLE iat a lower temperature; which is contrary to popular belief [that blunting reduces the temperature. This effect is obsefor Λ > 60◦; because the normal component ofV∞ reduces,thereby reducing the corresponding virtual stagnation tempture. Hence, this component of heat flux (q ′′

w,2-D,stag) is negative,and virtually cools SBLE. ForΛ → 90◦, the normal component ofV∞ reduces, thereby reducingh2-D,stag(V∞ cosΛ) and|q ′′

w,2-D,stag|. Hence, the difference between the temperatuof blunt and sharp SBLEs reduces. For typicalΛ (∼ 74◦), asharper leading edge radius is thermally-benign. This unventional observation is termed as the ‘thermally-benign sharpSBLE effect’. The variation of this effect is also in Tablewhich gives the difference inTw,SBLE,max between Cases 2 an1 (�Tw,SBLE,max). At Λ ≈ 75◦, this effect produces best resufor a sharper SBLE.

4. Summary and conclusions

(i) It is wrong to use the recovery temperature for aeronamic heat input to stagnation region.

(ii) The convection coefficient for bicurvature stagnation cbe obtained as,hbicurv,stag = h2-D,stag(Rbicurv); where,h2-D,stagis the coefficient for 2-D stagnation, andRbicurv =Rr/(R + r).

(iii) The convective heat flux to swept-back leading ed(SBLE) surface can be obtained by splitting the velity vector along and normal to the surface. This heat flis given as a sum of heat fluxes due to flat plate ‘q ′′

w,fp’ and2-D stagnation ‘q ′′

w,2-D,stag’, respectively.(iv) For large sweepback angles (Λs), the velocity vector com

ponent normal to SBLE, ‘virtually’ cools the surface; anheating is due to the component of velocity along the sface. This ‘virtual cooling’ increases as the leading edradius is reduced. Thethermally-benign sharp SBLE ef-fect, which is captured for largeΛs, is the reduction inleading edge surface temperature for a smaller radiusthe case presented, this benign effect begins atΛ ≈ 60◦,peaks atΛ ≈ 75◦, and thereafter reduces asΛ → 90◦.

(v) There existsΛ for which the temperatures of SBLE athe least. Thistemperature-minimised-sweepback concep-tually differs from the drag-minimised-sweepback. Ttemperature-minimised-Λ results in minimum heat transfer to SBLE, and drag-minimised-Λ results in minimumheat generation.

(vi) The temperature-minimised-sweepback and thermally-benign sharp SBLE effect should be considered in th

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configuration design of hypersonic vehicles, for optimdesign.

Acknowledgements

The author thanks the Defence Research and DevelopLaboratory, Hyderabad, India; for the support. The authorknowledges the improvement in 3rd paragraph of Section 3Mr. S. Jayakumar of his Department. The author is gratefuMr. A. Gujarathi of his Department; for assistance in editiThe author also thanks the A. von Humboldt Foundation – Gmany, for the rich exposure to research.

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