progress in aerospace sciencespop.h-cdn.co/assets/cm/15/06/54d151700212d_-_biplane_jpass.pdf ·...

Supersonic biplane—A review

Kazuhiro Kusunose a, Kisa Matsushima b,n, Daigo Maruyama c

a JAXA, Japanb University of Toyama, Japanc ONERA, France

a r t i c l e i n f o

Available online 19 November 2010

a b s t r a c t

One of the fundamental problems preventing commercial transport aircraft from supersonic flight is the

generation of strong sonic booms. Sonic booms are the ground-level manifestation of shock waves created by

airplanes flying at supersonic speeds. The strength of the shock waves generated by an aircraft flying at

supersonic speed is a direct function of both the aircraft’s weight and its occupying volume; it has been very

difficult to sufficiently reduce the shock waves generated by the heavier and larger conventional supersonic

transport (SST) configuration to meet acceptable at-ground sonic-boom levels. It is our dream to develop a

quiet SST aircraft that can carry more than 100 passengers while meeting acceptable at-ground sonic-boom

levels. We have started a supersonic-biplane project at Tohoku University since 2004. We meet the challenge

of quiet SST flight by extending the classic two-dimensional (2-D) Busemann biplane concept to a 3-D

supersonic-biplane wing that effectively reduces the shock waves generated by the aircraft. A lifted airfoil at

supersonic speeds, in general, generates shock waves (therefore, wave drag) through two fundamentally

different mechanisms. One is due to the airfoil’s lift, and the other is due to its thickness. Multi-airfoil

configurations can reduce wave drag by redistributing the system’s total lift among the individual airfoil

elements, knowing that wave drag of an airfoil element is proportional to the square of its lift. Likewise, the

wave drag due to airfoil thickness can also be nearly eliminated using the Busemann biplane concept, which

promotes favorable wave interactions between two neighboring airfoil elements. One of the main objectives

of our supersonic-biplane study is, with the help of modern computational fluid dynamics (CFD) tools, to find

biplane configurations that simultaneously exhibit both traits. We first re-analyzed using CFD tools, the classic

Busemann biplane configurations to understand its basic wave-cancellation concept. We then designed a 2-D

supersonic biplane that exhibits both wave-reduction and cancellation effects simultaneously, utilizing an

inverse-design method. The designed supersonic biplane not only showed the desired aerodynamic

characteristics at its design condition but also outperformed a zero-thickness flat-plate airfoil. (Zero-thickness

flat-plate airfoils are known as the most efficient monoplane airfoil at supersonic speeds.) Also discussed in

this paper is how to design 2-D biplanes, not only at their design Mach numbers but also at off-design

conditions. Supersonic biplanes have unacceptable characteristics at their off-design conditions such as flow

choking and its related hysteresis problems. Flow choking causes rapid increase of wave drag and it continues

to be kept up to theMach numbers greater the cruise (design) Mach numbers due to its hysteresis. Somewing

devices such as slats and flaps, which could be used at take-off and landing conditions as high-lift devices,

were utilized to overcome these off-design problems. Then supersonic-biplane airfoils were extended to 3-D

wings. Because that rectangular-shaped 3-D biplanewings showed undesirable aerodynamic characteristics at

their wingtips, a tapered-wing planformwas chosen for the study. A 3-D biplane wing having a taper ratio and

aspect ratio of 0.25 and 5.12, respectively, was designed utilizing the inverse-design method. Aerodynamic

characteristics of the designed biplane wing were further improved by using winglets at its wingtips. Flow

choking and its hysteresis problems, however, occurred at their off-design conditions. It was shown that these

off-design problems could also be resolved by utilizing slats and flaps. Finally, a study on the aerodynamic

characteristics of wing–body configurations was conducted using the tapered biplane wing. In this study a

bodywas chosen in order to generate strong shockwaves at its nose region. Preliminary parametric studies on

the interference effects between the body and the tapered biplane wing were performed by choosing several

different wing locations on the body. From this study, it can be concluded that the aerodynamic characteristics

of the tapered biplane wing are minimally affected by the disturbances generated from the body, and that the

biplane wing shows promise for quiet commercial supersonic transport.

& 2010 Elsevier Ltd. All rights reserved.

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/paerosci

Progress in Aerospace Sciences

0376-0421/$ - see front matter & 2010 Elsevier Ltd. All rights reserved.

doi:10.1016/j.paerosci.2010.09.003

n Corresponding author. Tel./fax: +81 76 445 6796.

E-mail addresses: [email protected], [email protected] (K. Matsushima).

Progress in Aerospace Sciences 47 (2011) 53–87

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2. History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.1. Origin of the Busemann biplane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.2. Continuing research on the Busemann biplane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.3. Descendants and derivatives of the biplane concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3. 2-D theory and construction of biplane airfoils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.1. Basic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.1.1. Wave-reduction effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.1.2. Wave-cancellation effect (Busemann biplane) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.1.3. An ideal biplane configuration, the Licher biplane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.2. CFD analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.2.1. Analyses using unstructured grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.2.2. Hysteresis analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4. Design for biplane airfoil of better performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.1. Inverse design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.1.1. Pressure and geometry relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.1.2. Basic equation and procedure for the inverse-problem design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2. Inverse design for biplane airfoil at a Mach number of 1.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2.1. Design process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2.2. General features of the designed airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3. Drag polar diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5. 3-D extension of biplane airfoils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.1. Busemann biplane wing with rectangular planform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.2. Planform parametric study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.2.1. Effect of changing sweep angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.2.2. Effect by changing taper ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.3. Introduction of winglet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.4. Design for better-performance biplane wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.4.1. Application of the 2-D designed airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.4.2. 3-D inverse design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6. Wing–fuselage interference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.1. Wing–fuselage configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.2. Aerodynamic performance at cruise condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.3. Application of designed wing to wing–fuselage combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

7. Experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

8. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

A.1. Aerodynamic center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

A.1.1. Biplane and diamond airfoils. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

A.1.2. Relation between an A.C. location and load distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

1. Introduction

Beginning with the first flight achieved by the Wright brothers in1903, the past 100 years of aviation history have been full ofremarkable milestones. In 1947, the Bell X-1 experimental aircraftbroke the sound barrier, ushering in the era of supersonic flight. Inmilitary aviation, airplane speed records have been continuouslybroken and topped; one of NASA’s experimental aircraft recently flewat Mach-10 speeds. Commercial aviation, however, has not enjoyedas many advances in supersonic flight; Concorde (1969–2003) wasthe first but also the last supersonic transport aircraft ever built.Concorde’s supersonic flights were, unfortunately, terminated, due toits poor fuel efficiency and unacceptable at-ground noise level.

A fundamental problem preventing commercial transportaircraft from supersonic flight is the creation of strong shockwaves, whose effects are felt on the ground in the form of sonicbooms. Because the strength of the shock waves generated by anaircraft flying at supersonic speed is a direct function of both theaircraft’s weight and its occupying volume, it has been deemednearly impossible to sufficiently reduce the shock waves gener-ated by the heavier and larger conventional commercial aircraft tomeet acceptable at-ground sonic-boom levels. In this paper, wefocus on supersonic biplanes proposed by a group at Tohoku

University [1–6]. They tried to extend the classic Busemannbiplane concept [7–10] to develop a practical supersonic-biplanewing that will generate sufficient lift at supersonic flights withoutincreasing a severe wave-drag penalty.

In general, a lifted airfoil generates shock waves through twofundamentally different mechanisms: one is due to its lift and theother is due to its thickness. The wave drag due to lift cannot beeliminated completely; it can only be reduced through multi-airfoilconfigurations. Based on the 2-D supersonic thin-airfoil theory [8]wave drag of an airfoil is proportional to the square of its lift. Multi-airfoil configurations redistribute the system’s total lift among theindividual airfoil elements, reducing the lift of each of the individualelements and, therefore, the total wave drag of the system. This willbe referred to as the ‘‘wave-reduction effect’’ in the rest of this paper[6]. Likewise, wave drag due to airfoil thickness can also be nearlyeliminated using the Busemann biplane concept, which promotesfavorable wave interactions between the two neighboring airfoilelements. By choosing their geometries and their relative locationsstrategically, the waves generated by the two elements cancel eachother (the ‘‘wave-cancellation effect’’ [6]). However, it is important toremember that skin friction of the biplane system will increasebecause of the increased surface area. One of the main objectives ofour supersonic-biplane study is, with the help of modern CFD tools,

K. Kusunose et al. / Progress in Aerospace Sciences 47 (2011) 53–8754

to obtain 2-D and 3-D biplane configurations that simultaneouslyexhibit both traits.

Kusunose and his group at Tohoku University in 2004 beganto study supersonic biplanes for the next generation supersonictransport by utilizing CFD tools (Navier–Stokes codes, but mainlyin their inviscid Euler modes) [11–13]. In this paper we introducea brief history of the classic Busemann biplane and its relatedresearches, followed by a discussion on the fundamental theory ofsupersonic biplane. In general, supersonic biplanes show superioraerodynamic characteristics at their design Mach numbers.However, they have poor performances at their off-designconditions. Flow choking occurs at high subsonic speeds [9,14],and continues to Mach numbers greater than the designMach number in the acceleration stage due to flow hysteresis[14,15]. Since the internal flow of a biplane is identical to thatof an intake diffuser, the characteristics of choking and hysteresisof supersonic biplanes can be analyzed based on suchcharacteristics of a supersonic diffuser under start and un-startconditions [16].

We then proceed with an overview of important progress madeby Kusunose’s group on 2-D and 3-D supersonic biplanes. It will bediscussed how to design supersonic biplanes, not only at theirdesign Mach numbers but also at their off-design conditions. At theirdesign Mach numbers, 2-D and 3-D supersonic biplanes that exhibitboth wave-reduction and wave-cancellation effects simultaneouslyare designed using an inverse-design method [3,6,17,18]. At off-design conditions hinged slats and flaps are applied to avoid flowchoking and accompanying hysteresis problems [6,17–22]. Theseslats and flaps can be used as high-lift devices at take-off andlanding conditions. Several studies on sonic-boom propagationmechanisms through the atmosphere are given in Ref. [23,24].Aerodynamic studies on wing planforms and wing–body configura-tions are also conducted using a tapered biplane wing [25–30].Preliminary parametric studies on the interference effect betweenthe body and biplane wing are performed by choosing severaldifferent wing locations on the body [31,32]. We finally make a briefreview on recent experimental investigations closely related tocurrent biplane developments [33–39].

Nomenclature

Symbols

A cross sectional areaAn cross sectional area at the throat of nozzleA1, Ai section area of inletA2, At section area of throatAin cross sectional area at the inlet of nozzleAR aspect ratiob semi-span lengthc, C chord lengthcmid position of the mid-chord at the wingtip of the three-

dimensional wingcref reference chord lengthcroot chord at the wing root of the three-dimensional wingctip chord at the wingtip of the three-dimensional wingc1, c2 Busemann coefficientsCd wave-drag coefficient of airfoil (wing section)Cdp pressure drag coefficient of the two-dimensional

airfoil in Navier–Stokes simulationsCdf friction drag coefficient of the two-dimensional airfoil

in Navier–Stokes simulationsCdfric friction drag coefficientCdtotal total drag coefficient in Navier–Stokes simulationsCD drag coefficientCl lift coefficient (wing section)CL lift coefficientCp pressure coefficientD wave dragf, g functionf(x) airfoil geometryF pressure force acting on airfoilFU, FL pressure forces acting on upper and lower elements of

biplaneG, h gap between biplane elementsh gap between two elements of the biplanel surface length defined along the airfoil surfaceL liftl/d lift-to-drag ratio of the two-dimensional airfoil (wing

section)L/D lift-to-drag ratio

M Mach numberMsw swallowing Mach numberMt Mach number behind oblique shock waveMN free-stream Mach numberMN inlet Mach numberP static pressureP0, PN total and free-stream pressuresDP pressure jump due to local flow inclinationq dynamic pressurer, y, xm parameters of skewed cylindrical coordinate systemR gas constantRe Reynolds numberDs entropy productionS wing reference areat airfoil thicknessUN free-stream velocityw span width of biplanewlt winglet (wingtip panel)x, y, z parameters of the Cartesian coordinate systemx streamwise coordinatey span-wise coordinatez vertical coordinateX, Y, Z body coordinate systemxi incident point from the leading edgexw the location of wing for wing–fuselage configurationsz gap between biplane elementsa angle of attackb, b0 shock-wave angleg ratio of specific heatsd margine wedge angley df/dx-ay reflection anglem Mach angler fluid density

Subscripts

i inlett throatN free-stream

K. Kusunose et al. / Progress in Aerospace Sciences 47 (2011) 53–87 55

2. History

2.1. Origin of the Busemann biplane

Today, one can learn about the Busemann biplane only in atraditional textbook. For example, in Liepmann and Roshko [8], it isintroduced as one of the wave-drag reduction methods utilizinginterference between its two element wings. The interference issimply explained in Fig. 1, which is extracted from Ref. [8]. Thefirst proposal of the supersonic-biplane concept was made byDr. Adolf Busemann at the Fifth Volta Congress in Rome in 1935.Fig. 2 shows a picture of the congress where many famousaerodynamicists, including A. Busemann himself, are identified[40]. This 1935 Volta Congress is regarded as the threshold ofmodern high-speed aerodynamics. The main topic was ‘‘HighVelocity in Aviation,’’ with the President being G.A. Crocco. Here,Busemann presented a paper titled ‘‘Aerodynamic lift at supersonicspeed’’ [7] in which he mostly discussed the concept of the sweptwing. In the supplement of the conference proceedings, Busemannindicated how to cancel wave drag to make a wing equal to aflat plane by means of biplane configuration. The article was

written in German. Its title is ‘‘Auftribe des Doppeldeckersbei Uebershallgeschwindigkeit’’, which means ‘‘Investigation ofBiplanes in Supersonic flows’’. Later, in 1955 in Ref. [41], hepresented the figure of the 1935 biplane again, which is shown inFig. 3. He introduced it as the biplane of zero wave drag and noisethat he had demonstrated at the Volta Congress in 1935.

2.2. Continuing research on the Busemann biplane

After the Fifth Volta Congress, several researchers werededicated to the study of a supersonic biplane. Until 1958, studieson Busemann-type biplanes appeared in many papers. Almost allof the studies were two dimensional because of the lack ofcomputational power and refined experimental strategy whencompared with the technology of the present day.

V.O. Walcher carried out a parametric study regarding theleading and trailing edge (L.E. and T.E.) angles of each upper andlower element of a biplane [42].

In 1944, M.J. Lighthill discussed the advantages and disadvan-tages of the Busemann biplane. As a disadvantage, he pointed outthat the biplane had a greater wing area than a monoplane, whichincreased skin-friction drag, and that the passage between thetwo elements might ‘‘choke’’ [43]. In the same year, wind-tunnelexperiments were conducted by A. Ferri in Italy [14]. These weredone for both non-lifting and lifting cases. Ferri measuredaerodynamic forces and took Schlieren photographs to observethe viscous effect of the boundary layer and shock movementphenomena. He observed undesirable ‘‘choke’’ and ‘‘hysteresis’’occurrences, both of which will be discussed later in this paper. Inspite of the disadvantages, Ferri concluded that the biplane waspromising to obtain better efficiency than that of an isolated wing.

In a 1947 NACA paper published in the United States,W.E. Moeckel performed a detailed parametric study on thespacing of the gap and the L.E. and T.E. angles of the two elementsof the biplane for lifting cases. One of his results was to find thatunsymmetrical biplanes whose lower elements were thickerthan upper elements would have higher L/D than symmetricalbiplanes [44].

Based on Moeckel’s conclusion, R.M. Licher made an analysis todesign the optimal L/D unsymmetrical biplane with the given Cl asa constraint by using the supersonic linear theory in 1955 [10].This is an interesting paper, which would lead us to a practicalsupersonic-biplane transport. This will also be discussed in thenext chapter. After the publication of Ref. [10], however, nofurther publication could be found in the literature except forRef. [45]. It appears that something had discouraged researchersfrom further studying the Busemann biplane.

Wave-cancellation

Busemann biplane

Pressure distribution on inside surface

Off-design

Wave reflection

Busemann biplane

Fig. 1. Simple description for mutual cancellation of waves on a biplane (Ref. [8]).

A. Busemann C.Wieselsberger

G.A.Crocco L.Prandtl

Fig. 2. The Fifth Volta Congress (Ref. [40]). By the courtesy of http://www.dglr.de/

literatur/publikationen/pfeilfluegel/Kapitel1.pdf

Figure 1 of Ref. [41] titled “Biplane of zero drag and noise”.

HIGH-SPEED AERONAUTICS

Fig. 3. Busemann biplane proposed in (Ref. [41]).


2.3. Descendants and derivatives of the biplane concept

Inspired by the Busemann biplane concept, a lot of researchwas inspired to consider the configuration wherein shock wavesinteract with other multiple elements to produce favorable waveinteraction and reduce wave drag. Recently, in 2004, suchresearch was summarized in Ref. [46]. The author of Ref. [46],Dr. D.M. Bushnell of NASA Langley Research Center, called theconcept the ‘‘favorable shock-wave-interference approach’’ fordrag reduction. He concluded that the issues of the concept can beincreasingly addressed through the progress of ‘‘smart material’’and ‘‘flow control’’ technologies. Some of the issues such as thederivatives of the biplane concept can be found in Refs. [47–49].

3. 2-D theory and construction of biplane airfoils

3.1. Basic theory

3.1.1. Wave-reduction effect

In this section we demonstrate the wave-reduction effect of2-D biplanes. We compare the wave drag of two different airfoils,a single flat-plate airfoil and a parallel flat-plate biplane, at asupersonic flow condition. The conditions are such that bothairfoils generate the same amount of lift (the ‘‘constant-liftcondition’’). Using the 2-D (inviscid) supersonic thin-airfoil theory[8], the local pressure jump can be expressed as a function of the(small) local flow inclination angle a, measured from theoncoming flow direction, as sketched in Fig. 4

DPP1

¼ p�p1P1

¼ gM21ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

M21�1

q a ð1Þ

where g and M1 represent the ratio of specific heats and the Machnumber of the oncoming flow, respectively. Symbols P and P1 arethe local and oncoming flow pressures.

It can be shown that the lift and wave drag of a single flat-plateairfoil with a small angle of attack aS as sketched in Fig. 5 are

given by

LS ¼4aSffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM21�1

p q1C ð2Þ

DS ¼4a2

SffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM21�1

p q1C ð3Þ

where the symbols qN and C represent the free-stream dynamicpressure defined by

q1 � 1

2r1U2

1 ¼ gM21

2P1

and the chord length of the flat-plate airfoil, respectively [6,8].Next, we calculate lift and drag of a biplane airfoil. The biplane,

as shown in Fig. 6, is constructed of two parallel flat plates havingthe same chord length as that of the single flat plate C at an angleof attack ab. Assuming that the compression wave (or expansionwaves) generated from the leading edge (L.E.) of one element doesnot interact with the neighboring element, lift and drag of thebiplane can be calculated in the same way as they were calculatedfor the single flat-plate airfoil [6].

The lift and drag of the biplane airfoil can be calculated asfollows:

Lb ¼8abffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM21�1

p q1C ð4Þ

Db ¼8a2

bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM21�1

p q1C ð5Þ

We then adjust the biplane’s angle of attack ab such that theconstant-lift condition is met. Equating the lift of the biplane withthe lift produced by the single flat-plate airfoil (Eqs. (2) and (4)),the two incidence angles ab and aS have the followingrelationship:

ab ¼aS

2ð6Þ

Under the constant-lift condition, the wave drag of the biplanereduces to

Db ¼8ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

M21�1p aS

2

� �2q1C ¼ 1

2

4a2Sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

M21�1p q1C

!ð7Þ

Compared with Eq. (3), it is clear that the wave drag of the biplanereduces to 1/2 of the original single-plate airfoil under theconstant-lift condition [6].

Fig. 4. Pressure jump through a compression or an expansion wave.

Fig. 5. Single flat-plate airfoil with angle of attack aS. Fig. 6. Biplane airfoil with angle of attack ab.


Similarly, it can be easily shown that the wave drag of ann-plate system reduces to 1/n of the original single-plate airfoilunder a specified lift condition, as long as the compression wave(or expansion waves) generated from the L.E. of each element ofthe n-plate biplane system does not interact with the neighboringelements. We should remember, however, that skin-friction dragof the n-plate airfoil will increase to n-times that of the single flat-plate airfoil because of the increased surface area. The increase inskin-friction drag is an inevitable byproduct of our wave-dragreduction process using multi-element airfoil systems.

The skin friction of a single flat-plate airfoil can be estimatedusing the following incompressible turbulent boundary-layerformula [50]:

Cdfric ¼ 2Cf ð8Þ

where

Cf ¼0:027

ðReÞ1=7ð9Þ

In this paper lift and drag coefficients of both single-plate andbiplane airfoils are defined by Cl�L/qNC, Cd�D/qNC based on thechord length of single-plate airfoil C. The symbol Re denotes theReynolds number based on the flat-plate chord length, C. For theconversion from incompressible to compressible flows, a relation-ship between the skin-friction coefficient (for incompressibleflows) and Mach number plotted in Fig. 13.10 of Ref. [8] canbe used.

3.1.2. Wave-cancellation effect (Busemann biplane)

The biplane configuration can also significantly reduce wavedrag due to airfoil thickness. Within a supersonic thin-airfoilapproximation, Busemann [7] showed that the wave drag of azero-lifted diamond airfoil can be completely eliminated bysimply splitting the diamond airfoil into two elements andpositioning them in a way such that the waves generated bythose elements cancel each other [8,9], as sketched in Fig. 7. Thewedge angles of the diamond airfoil and the Busemann biplaneare 2e and e, respectively. Remember that pressures acting on thefront half and rear half of its inner surfaces, P1 and P2, respectively(see Fig. 8), are identical. Total lift and wave drag acting on theBusemann biplane are, therefore, both zero under the zero-liftcondition.

Within a supersonic thin-airfoil approximation, lift and dragacting on the Busemann biplane are identical to those acting on aflat-plate airfoil at a small angle of attack [6,9]. However, in theactual case wave drag is always larger than that of the flat-plateairfoil, because of the entropy produced due to the shock wavesexisting between the biplane elements.

Generally, in supersonic speeds, wave drag due to airfoilthickness is large relative to that due to its lift. Supersonic aircraftare therefore severely limited in their wing thickness. If the

wave-cancellation effect is used effectively, the strong restrictioncurrently imposed on the wing thickness of supersonic aircraftmay be relaxed considerably. Remember that lift and wavedrag of a diamond airfoil (see Fig. 9) with a small incidenceangle a can be expressed, using the same supersonic thin-airfoilapproximation, as ([8])

LD ¼ 4affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM21�1

p q1Cþ � � � ð10Þ

DD ¼ 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM21�1

p a2þ t

C

� �2" #

q1CþUUU ð11Þ

Symbol t/C represents the thickness–chord ratio of the diamondairfoil. In Fig. 10 wave-drag components due to lift and due tothickness are calculated for a lifted diamond airfoil for twodifferent airfoil thicknesses t/C¼0.5 and 0.10 at constant-liftcondition Cl¼0.10 and flow condition MN¼1.7. Skin-frictioncoefficients are calculated using Eq. (8), which is based on a flat-plate airfoil with chord length C, with the help of the Machnumber correction factor given in Ref. 8.

3.1.3. An ideal biplane configuration, the Licher biplane

The previously discussed wave-reduction model (flat-platebiplane model shown in Fig. 6), however, cannot be combined toBusemann’s wave-cancellation concept directly, because waveinteractions that are required for the Busemann biplane donot occur. We, therefore, need to seek a biplane configurationthat has those two desirable characteristics simultaneously inorder to attain a significant wave-drag reduction. An unsymmetricalFig. 7. Wave-cancellation effect of Busemann biplane.

Fig. 8. Pressure distribution on Busemann biplane.

Fig. 9. Diamond airfoil at zero angle of attack.

Fig. 10. Wave-drag components of a diamond airfoil.


biplane configuration discussed by R. Licher in 1955 [10](sketched in Fig. 11) exhibits both of the desirablecharacteristics: the wave-reduction effect and the wave-cancellation effect. By promoting favorable wave interactionsbetween the upper and lower elements, the wave drag due tolift can be reduced to 2/3 of that of a single flat plate underthe identical lift condition. Additionally, the Busemann wave-cancellation concept can be applied to the system to eliminatewave drag due to airfoil thickness. Wave-reduction effect of theLicher biplane is discussed next.

Utilizing the supersonic thin-airfoil theory [8], the Licherbiplane can be split into its lift component and thicknesscomponents, as shown in Fig. 11. Analysis of wave drag due tothickness for the Licher biplane is identical to that of theBusemann biplane, and has therefore been discussed in theprevious section. We, therefore, focus on lift and wave drag due toits lift component. The symbol a denotes the flow inclinationangle of the upper (flat-plate) element; it also denotes the flowinclination angle of the lower surface of the lower element(see Fig. 12). The wedge angle of the lower element (having a half-diamond shape) has been also chosen to be a. This particularshape and location of the lower element cause the compressionwave generated from the leading edge of the upper element andthe expansion waves generated from the throat of the lowerelement to cancel each other. This can be shown througharguments similar to the wave-interaction analysis of theBusemann biplane. Finally, we can show that the pressure alongthe entire upper surface of the lower element is uniform, and thatits value is identical to that of the free-stream. It is, therefore,clear that the upper element of the system acts like single flat-plate airfoil with the incidence angle of a, and pressure actingon the lower surface of the lower element contributes to thesystem as an additional aerodynamic force. Detailed discussionsare given in [6].

Lift acting on the Licher biplane system shown in Fig. 12 can becalculated as

L¼ 4affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM21�1

p q1Cþ 2affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM21�1

p q1C ¼ 3

2

4affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM21�1

p q1C

!ð12Þ

Comparing Eq. (12) with Eq. (2), it is clear that the Licher biplanegenerates 1.5 times the lift of a single flat-plate airfoil with thesame angle of attack. We now adjust the angle of attack a of theLicher biplane so that it generates the same amount of lift as asingle flat-plate airfoil with an angle of attack aS (constant-liftcondition). Since both the single and Licher airfoils have the samereference chord length C, a and aS have the following relationship:

a¼ 2

3aS ð13Þ

Because of the constant-lift condition

L¼ 3

2

4affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM21�1

p q1C

!¼ 4aSffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

M21�1p q1C ¼ LSingle ð14Þ

Similar to the lift analysis, wave drag of the Licher biplane systemcan be expressed as ([6])

D¼ 4a2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM21�1

p q1Cþ 2a2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM21�1

p q1C ¼ 3

2

4a2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM21�1

p q1C

!ð15Þ

When the constant-lift condition (L¼LSingle) is considered, thedrag Eq. (15) reduces (with the help of Eq. (13)) to

D¼ 3

2

4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM21�1

p 2

3aS

� �2

q1C

!¼ 2

3

4a2Sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

M21�1p q1C

!¼ 2

3DSingle

ð16Þ

Eq. (16) shows that the wave drag due to lift of the Licherbiplane reduces to 2/3 that of a single flat-plate airfoil underconstant-lift condition [6,10].

3.2. CFD analysis

3.2.1. Analyses using unstructured grid

In this section, analysis results from CFD simulations using anunstructured grid approach are mentioned [18–20]. For thepurpose of examining the characteristics of Busemann biplanesat off-design conditions with zero incidence angle, the thickness–chord ratio (t/c) of the Busemann biplane and a diamond airfoilwere selected as 0.10 (its equivalent wedge angle, e, being 5.711)(shown in Fig. 13). The gap of the biplane has been adjusted(z/c¼0.5) to obtain minimum drag at free-stream Mach number,MN¼1.7, which will be referred as the design Mach numberhereinafter in this paper. Inviscid flow analyses were performedusing the TAS code (Ref. [51,52]). The grids around the diamondairfoil and the Busemann biplane are shown in Fig. 14. Wave-dragcoefficient (Cd) calculated from CFD analyses and theory based onthe supersonic thin-airfoil approximation are shown in Table 1.They are in good agreement. The reliability of the CFD code was

Fig. 11. Decomposition of Licher Biplane into its lift and thickness components.

Fig. 12. Geometry of lift component of Licher Biplane.

Busemann biplane Diamond airfoil

z

c

t

c

t1=t2

t2

t1

z

c

(z/c=0.5, t/c=0.10)

t

c

(t/c=0.10)

Fig. 13. Baseline models at zero-lift conditions (MN¼1.7).


discussed and validated in Ref. (6). It is clear that wave drag of theBusemann biplane is almost completely eliminated being lessthan 1/10 of that of the diamond airfoil.

Because the biplane configuration is similar to the supersonicconverging and diverging nozzle, supersonic biplanes may havethe disadvantage of having choked flow at a wide range oftransonic flow regions. The choked-flow phenomenon generatessignificant wave drag at the biplane’s off-design Mach numbers.Fig. 15 shows the detailed wave-drag characteristics of theBusemann biplane over a range of flight Mach numbers(0.3oMNo3.0), including its design Mach number MN¼1.7.Flow-hysteresis problem that caused by the continuous change inthe free-stream Mach number will not be considered. CFDanalyses, including the flow-hysteresis problem, will bediscussed in the next section. Flow choking and its concomitanthysteresis problem were also observed in experiments [14,34,35].

Let us now return to Fig. 15 again. We can observe a range ofMach numbers (1.64oMNo2.0) where wave drag remainsnearly at its minimum value. The existence of this low wave-drag range is critical for the development of actual airplanes in thefuture. From Fig. 15, however, we also observe a high wave-dragrange of Mach numbers in the transonic flow region where wavedrag is even greater than that of the baseline diamond airfoil.

Some of the strategies to counter against choking (high drag)are airfoil morphing and the adaptation of Fowler motion. Figs. 16and 17 show simple diagrams of morphing and Fowler motionused in this study. Morphing alters the area ratio of the inlet areato the throat. With the morphing strategy, reduction in wave dragis expected because of the change in airfoil thickness. Thethickness–chord ratio (t/c) is changed from 0.10 to 0.06 on eachelement as shown in Fig. 16.

Fig. 18 shows wave-drag characteristics at various Machnumbers of the Busemann biplane with morphing and withFowler motion. It can be observed that much lower wave dragthan that of the diamond airfoil is achieved over a wide range offree-stream Mach numbers [6,18]. However, it is obvious that thebiplane with Fowler motion has higher friction drag than otherbiplanes because of the increase in surface area.

3.2.2. Hysteresis analysis

In real flight, airplanes accelerate from take-off to cruise Machnumber continuously. During the acceleration stage, choking will

occur and it may cause flow-hysteresis problems. Therefore it isnecessary to simulate the actual processes by changing theairplane speed continuously.

3.2.2.1. One-dimensional theory on intake diffuser. It is necessary todiscover methods that are applicable to Busemann-type biplanesin order to avoid the choked-flow and flow-hysteresis problems atoff-design conditions. Before we examine how these problems canbe overcome, it may be useful to discuss the start/un-start char-acteristics of a supersonic inlets diffuser (see Fig. 19), for they

Fig. 14. Mesh visualizations of a Busemann biplane baseline model (two-

dimensional unstructured grid, grid numbers are 0.20 million).

Table 1Wave-drag coefficients (Cd) of a diamond airfoil and a Busemann biplane

(zero lift).

CFD Theory Error (%)

Diamond airfoil 0.0291 0.0292 0.34

Busemann biplane 0.00189 – –

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.0 0.5 1.0 1.5 2.0 2.5 3.0M

Cd

Busemann Biplane Diamond Airfoil

c

zt(z/c=0.5, t/c=0.05)

(t/c=0. 10)ct

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.0 0.5 1.0 1.5 2.0 2.5 3.0M

Cd

Busemann Biplane Diamond Airfoil

c

zt(z/c=0.5, t/c=0.05)

(t/c=0. 10)ct

c

ztc

zt(z/c=0.5, t/c=0.05)

(t/c=0. 10)ct

ct

Zero-Lift conditions

Fig. 15. Wave-drag characteristics of the diamond airfoil and the Busemann

biplane (zero lift).

Fig. 16. A simple diagram of the Busemann biplane with morphing.

Fig. 17. A simple diagram of the Busemann biplane with Fowler motion.

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

M

Cd

Busemann BiplaneDiamond AirfoilMorphingFowler

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Fig. 18. Wave-drag characteristics of the biplane with morphing (zero lift).


share many similar characteristics with the Busemann biplane. InFig. 19, the line in red shows the Kantrowitz limit [16,53] (givenby Eq. (17)), which is the Mach number that the flow speed ofinlet diffusers must exceed before the inlets can go from the un-start to start condition, once a bow shock is generated in front ofits inlet.

At

Ai¼ ðg�1ÞM2

1þ2

ðgþ1ÞM21

� �1=22gM2

1�ðg�1Þðgþ1ÞM21

� �1=ðg�1Þð17Þ

where Ai is the area of inlet and At is the area of throat. Also, theline in blue refers to the isentropic contraction limit [53], wherethe Mach number is MN¼1.0 at the throat of supersonic inlets.The isentropic contraction limit is calculated by Eq. (18).

At

Ai¼M1

ðg�1ÞM21þ2

gþ1

� ��ðgþ1Þ=2ðg�1Þð18Þ

It is reasonable to assume that this rule can be applied to theBusemann biplane in order to avoid the choked-flow and flow-hysteresis of the Busemann biplane. In fact, the results from CFDanalyses are in good agreement with the values that arecalculated using Eqs. (17) and (18). Here the At/Ai of theBusemann biplane is 0.8, as shown by the solid line in Fig. 19.The predicted Mach number of the starting and unstarting are2.154 and 1.600, respectively.

3.2.2.2. Quasi-unsteady CFD simulations. Inviscid CFD analyses(Euler simulation) of the Busemann biplane were conducted totrace the hysteresis. For the simulation we use the followingstrategy: a quasi-unsteady simulation. We divide the accelerationprocess from 0.6 to 2.18 of free-stream Mach numbers into smallintervals of 0.1 or so. We conducted a series of simulations as MN

was raised discretely along the intervals using the previoussimulation result imposed as the initial condition.

Figs. 20 and 21 show Cp color maps at each Mach number inacceleration and deceleration stages. Fig. 22 shows a Cd–MN graph.We observe that a detached shock wave is generated at an upstreamlocation of the airfoils and it gets closer to the leading edge witheach increment of a free-stream Mach number. Once a certain Machnumber is reached, the detached shock wave attaches to the leadingedge and is swallowed to a downstream location of the throat. As aconsequence, choking disappears (in this case at MN¼2.18).This phenomenon is similar to that of the starting process of theintake diffuser. The value of the starting Mach number givenfrom CFD agrees with that predicted from the 1-D flow equation(Eq. (17)).

3.2.2.3. How to minimize hysteresis problem. From Fig. 19 it isevident that as the area ratio At/Ai increases, the biplane becomesmore and more startable at lower Mach numbers. The biplaneequipped with hinged slats and flaps, shown in Fig. 23, was usedfor the study. With the hinged slat the sectional area ratio of theinlet to the throat (At/Ai) increases to 0.909. According to the

Fig. 20. Cp color maps around the Busemann biplane on acceleration (from 0.6 to 2.18 of free-stream Mach number per about 0.1) (0.6rMNr1.1); (1.2rMNr1.7) and

(1.8rMNr2.18). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

At

Kantrowitz limitIsentropic Contraction limit

Ai

Fig. 19. Start/un-start characteristics of supersonic inlet diffuser.


equations of the intake diffuser, the biplane equipped with hingedslats and flaps can start atMN¼1.55. Choking will no longer occurat the cruise Mach number (MN¼1.7).

CFD analyses were conducted on the Busemann biplaneequipped with slats and flaps using a quasi-unsteady simulation

[6,18,21]. Fig. 24 shows Cp color maps near the starting Machnumber. Fig. 25 shows a Cd–MN graph. We can see the startingoccurs at MN¼1.56 and that choking disappears. Note that basedon the one-dimensional theory from Eq. (17), the predictedstarting Mach number is 1.55. For the Busemann biplane withmorphing discussed in the previous section (see Fig. 16), thestarting condition is MN¼1.41. The ratio of the section area of theinlet to that of the throat (At/Ai) is 0.936 (With the theoreticalstarting Mach number being 1.40).

Fig. 21. Cp color maps around the Busemann biplane on deceleration (from 2.18 to

1.7 of free-stream Mach number per about 0.1) (2.18ZMNZ1.7). (For

interpretation of the references to color in this figure legend, the reader is referred

to the web version of this article).

0.08

M

Impalsive start Analyses

Acceleration

Deceleration

HysteresisHysteresis

MM =2.18=2.18MM =1.64=1.64

Design Mach number Design Mach number MM =1.7=1.7

0.00

0.02

0.04

0.06

0.10

0.12

0.14

0.16

M

Cd

Impalsive start Analyses

Acceleration

Deceleration


MM =2.18=2.18MM =1.64=1.64


0.0 0.5 1.0 1.5 2.0 2.5 3.0

Fig. 22. Wave-drag characteristics of Busemann biplane on quasi-unsteady flow

(zero lift).

Fig. 23. A simple diagram of the Busemann biplane equipped with hinged slats

and flaps.

Fig. 24. Cp color maps of the biplane equipped with hinged slats and flaps on

acceleration (zero lift). (For interpretation of the references to color in this figure

legend, the reader is referred to the web version of this article).

M

Busemann Deceleration

Busemann Acceleration

Slat and Flap Deceleration

Slat and Flap Acceleration


MM =1.56=1.56

MM =2.18=2.18


0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

M

Cd

Busemann Deceleration

Busemann Acceleration

Slat and Flap Deceleration

Slat and Flap Acceleration


MM =1.56=1.56

MM =2.18=2.18


2.52.01.51.00.5

Fig. 25. Wave-drag characteristics of the Busemann biplane and the one equipped

with hinged slats and flaps on quasi-unsteady flow (zero lift).


3.2.2.4. High-lift condition. In this section, take-off and landingconditions (MN¼0.2) are discussed [6,18]. The Navier–Stokesequation was used for the analyses, with a Reynolds number of 30million. A one-equation turbulence model by Spalart–Allmaras[54] is adopted to treat turbulent boundary layers for viscous flowcomputations. The same hinged slats and flaps mentioned beforeare used as a high-lift-device. The geometry used for the analysisis shown in Fig. 26. The positions of the hinges are the same asthose of the biplane shown in Fig. 23 (30% chord length). Fig. 27shows a drag polar diagram (where, Cdtotal refers to the total dragcoefficient: wave drag plus friction drag) and Fig. 28 shows Cp andeddy viscosity color maps at an angle of attack of 41. Thecalculated lift and drag are given in Table 2. It can be observedthat sufficient lift (Cl42.0) is generated by utilizing hinged slatsand flaps.

Viscous effects at off-design conditions are also discussedbriefly. Fig. 29 shows Cdtotal characteristics near its start and un-start Mach numbers. Here, results from both Navier–Stokes andEuler simulations are compared. It shows that there are no drasticchanges in flow characteristics caused by viscous effects, with theexception of the starting Mach (reduced from 2.18 to 2.14).

4. Design for biplane airfoil of better performance

4.1. Inverse design

4.1.1. Pressure and geometry relation

As discussed in the previous section, biplane airfoils can begood candidates for the wings of next generation SST. We,therefore employed inverse designing to optimize the biplaneairfoil at its lifted condition. The inverse-design process calculates

the airfoil geometry, necessary for the specified target pressuredistribution to occur along its surface [6,17–19]. In order todetermine the desired geometry, the relationship between the

Fig. 26. A simple diagram of the Busemann biplane equipped with hinged slats

and flaps utilized as a high-lift-device.

0

12

3

4

5

01

23 4 5 6 7 8

-1

Busemann biplane

Busemann biplane with HLD

every 1degree plotted

0

12

3

4

5

01

23 4 5 6 7 8

-1

0

0.5

1

1.5

2

2.5

0

Cdtotal

Cl

Busemann biplane

Busemann biplane with HLD

0.05 0.1 0.15 0.2 0.25

Fig. 27. Drag polar diagram of the Busemann biplane and the one equipped with

hinged slats and flaps utilized as a high-lift-device angle of attack 41.

Eddy viscosity

Busemann biplane

Cp

Cp

Eddy viscosity

Busemann biplane equipped with slats (10deg) and flaps (15deg)

Fig. 28. Cp and eddy viscosity color maps of the Busemann biplane and the one

equipped with slats and flaps as a high-lift-device (A.o.A¼41). (For interpretationof the references to color in this figure legend, the reader is referred to the web

version of this article).

Table 2CFD analysis results of both two biplanes at an angle of attack 41.

Cl Cdtotal

Busemann biplane 0.612 0.0485

HLD 2.025 0.0955

M

Cd t

otal

Euler AccelerationEuler DecelerationNS AccelerationNS Deceleration

MM =2.18=2.18

MM =1.64=1.64


MM =2.14=2.14

0.00

0.02

0.04

0.06

0.08

0.10

0.12

M

Euler AccelerationEuler DecelerationNS AccelerationNS Deceleration

MM =2.18=2.18

MM =1.64=1.64


MM =2.14=2.14

1.2 1.7 2.2

Fig. 29. Cd characteristics of the Busemann biplane on quasi-unsteady simulation

over a range of free-stream Mach numbers (Euler and Navier–Stokes analyses).


surface pressure distribution and geometry is required. Here, weuse an algebraic relationship derived from the oblique shockrelations [8].

An airfoil’s geometry f(x) is related to its pressure distribution,as follows:

Cp ¼ c1yþc2y2 ð19Þ

where y represents the local flow deflection angle (df/dx�a) alongthe airfoil surface. The symbols c1, c2, are the Busemanncoefficients given as

c1 ¼2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

M12�1

q , c2 ¼ðM1

2�2Þ2þgM14

2ðM12�1Þ2

ð20Þ

a is the angle of attack of the airfoil. Also, x represents the airfoil-chord direction (see Fig. 30) and g is the ratio of specific heats.MN

(41.0) is the free-stream Mach number.Eq. (19) is the second-order equation with respect to change in

geometry. Splitting the airfoil geometry into the upper and lowersurfaces, Eq. (19) yields

Cpþ ¼ c1dfþ ðxÞdx

�a� �

þc2dfþ ðxÞdx

�a� �2

Cp� ¼�c1df�ðxÞdx

�a� �

þc2df�ðxÞdx

�a� �2

ð21Þ

where subscripts + and � denote the upper and lower surfaces,respectively.

4.1.2. Basic equation and procedure for the inverse-problem design

Eq. (21) can be used for design problems. In Ref. [55], Ogoshiand Shima performed the wing-section design of an SST usingEq. (21). However, the interaction effects between the two biplaneelements must be considered. Then, we must adopt theperturbation form (i.e. D-form) of Eq. (21) as the basic equationfor our inverse-design procedures [56]. Taking the smallperturbation form (Cp-Cp+DCp and f-f+Df) of Eq. (21), weobtain D-form equations [57]:

DCpþ ¼ c1dDfþ ðxÞ

dx

� �þ2c2

dfþ ðxÞdx

�a� �

dDfþ ðxÞdx

� �þc2

dDfþ ðxÞdx

� �2

ð22Þ

DCp� ¼�c1dDf�ðxÞ

dx

� �þ2c2

df�ðxÞdx

�a� �

dDf�ðxÞdx

� �þc2

dDfþ ðxÞdx

� �2

ð23Þwhere + and � indicate upper and lower surfaces, respectively(see Fig. 31).

With the above D-form equations, an iterative design methodis constructed. Fig. 32 illustrates the iterating process. First, theflow field around the initial configuration is analyzed to obtainthe ‘‘initial’’ pressure distribution of the airfoil. At this time,the target pressure distribution should be specified. Next, aninverse-problem solver is employed to calculate the x-derivativeof the correction value for the airfoil geometry, dDf7/dx; this

x-derivative is related to the difference between the target andthe current pressure distributions, denoted as DCp (Cp-residual). Inparticularly, the geometry correction term Df (see Fig. 31 again) isdetermined from DCp. Solving Eqs. (22) and (23) for dDf+/dx anddDf�/dx, the airfoil geometry is updated:

f update7 ðxÞ ¼ f7 ðxÞþZ x

0

dDf7dx

ðxÞdx ð24Þ

where the symbol 0 indicates the x coordinate of the airfoilleading edge.

In this approach, however, it is evident that there is noguarantee in obtaining an airfoil that has a closed trailing edge.We, therefore, may need to make further (but minor) modifica-tions to the obtained geometry with the closed trailing edge [6].We, then, calculate the flow field around the updated airfoilgeometry for the next cycle. An optimal airfoil design can beobtained through repeating this process until DCp (Cp-residual)becomes negligible.

4.2. Inverse design for biplane airfoil at a Mach number of 1.7

4.2.1. Design process

For the inverse design, a Licher-type biplane was selected asthe initial biplane configuration (a¼1.01, Cl¼0.0812, Cd¼0.00449)and from this the geometries of upper and lower elements wouldbe designed [18,19]. The design procedure is shown in Fig. 33.As a design condition, free-stream Mach number MN¼1.7 andangle of attack a¼11 were selected (here, a represents theangle of the lower surface of the lower element against the free-stream direction). A flow solver called TAS code, usingunstructured grids, was used to calculate the flow fields aroundthe biplane.

Both the target and initial pressure distributions for both theupper and lower elements used for the biplane design are shownin Fig. 34. Target Cp distributions are constructed in such ways togenerate more lift on the upper surface of the upper element andalso to create additional lift but generating lower drag onthe lower surface of the upper element (especially near theFig. 30. Airfoil and flow direction.

x

z f (x)

FlowA.O.A =α

f(x)+Δ f(x)

Fig. 31. Airfoil geometries based on the current and target pressure distributions.

Initial Airfoil

Grid Generation

Target Cp

Flow Solver(Euler)

Inverse Design

Grid GenerationCp<

Designed Airfoil

YESNO

Cp Target Cp Current Cp

f f + f

Grid Generation

Target Cp

Flow Solver(Euler)

Inverse Design

Grid GenerationCp<


Current Cp

f f + f

f : Geometry Correction

Fig. 32. The iterative design method of the inverse-problem design.


trailing edge). The obtained Cp distributions (after 14 timesiterations) of the upper and lower elements, along with the targetvalues, are plotted as shown in Fig. 35. The initial and designedgeometries are compared in Fig. 36. The gain of the angle of attack(of the lower surface on the lower element) is approximately0.191 at its design point, a¼1.01 (compared with the initial Licher-type biplane). The total maximum thickness–chord ratio (t/c) ofthe designed biplane is 0.102. Its lift and wave drag are Cl¼0.115,Cd¼0.00531 (L/D¼21.72). It is clear that a biplane having betteraerodynamic performances was designed compared with that ofthe Licher biplane. Detailed aerodynamic performances of those

biplanes are given in Table 3. A Cp contour map at this designpoint is shown in Fig. 37. It was confirmed that this inverse-designmethod would work well for the 3-D biplane configurations todetermine its wing-section geometries [30,58].

4.2.2. General features of the designed airfoil

Observing the geometry of the designed biplane, the trailingedge of the upper element of the designed biplane configurationwas modified so that its curvature was aligned to the free-streamdirection, creating additional lift. It should also be noted that the

Initial Airfoil

Grid Generation

Target CpofUpper Wing

Flow Solver(Euler)

Inverse Design Grid Generation

Cp<

Designed Airfoil

YES

NO

f f + f

f : Geometry Correction

Grid Generation

f f + f

Inverse Design

Cp<

Designed Airfoil

YES

NOFlow Solver

(Euler)

Target CpofLower Wing

Initial Airfoil

Grid Generation

Target Cp of Upper Wing

Flow Solver(Euler)


Cp<

Designed Airfoil

Current Cp of Upper Wing

f f + f

Grid Generation

Current Cp of Lower Wing

f f + f



Inverse Design

Cp<

Designed Airfoil

NOFlow Solver

(Euler)

Target Cp of Lower Wing

Fig. 33. Design cycle.

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

(x/c)

Cp

Cp

INITIAL TARGET

Lift on the upper surfaceof the upper element

An additional liftbut lower drag

Remove

Upper Element

INITIAL TARGET

Lift on the upper surfaceof the upper element

An additional liftbut lower drag

Remove

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

INITIAL TARGET

Remove

A cause ofreflection shock wave

Lower Element

INITIAL TARGET

Remove

A cause ofreflection shock wave

-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

(x/c)-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

Fig. 34. Target Cp distributions.

Upper Element

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

Cp

TARGET Designed

Lower Element

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

(x/c)

Cp

TARGET Designed

-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

(x/c)

-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

Fig. 35. Cp distributions of the designed biplane configuration.


compression waves from the leading edge of the biplane elementsand the expansion waves generated from its throats nearlycancelled each other out, eliminating the initially observedpressure peaks at the throats.

Navier–Stokes analyses were also performed on the designedbiplane. The Reynolds number is 32 million. Flow conditions areidentical to those discussed in the previous section. Fig. 38 showsthe pressure distributions of the designed biplane obtained from

Navier–Stokes simulations. We observed that pressure peaks arisea short distance in front of the throats due to the boundary layereffect. Cp distributions obtained from Euler simulations are very

Table 3The aerodynamic performance in Euler simulations.

a (deg.) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Busemann

Cl 0.0000 0.0284 0.0571 0.0858 0.1146 0.1435 0.1727 0.2021

Cd 0.00218 0.00245 0.00325 0.00458 0.00647 0.00891 0.01192 0.01551

L/D 0.00 11.61 17.59 18.72 17.72 16.11 14.49 13.03

Diamond

Cl 0.0000 0.0257 0.0515 0.0773 0.1031 0.1290 0.1550 0.1810

Cd 0.02891 0.02914 0.02983 0.03100 0.03264 0.03475 0.03734 0.04041

L/D 0.00 0.88 1.73 2.49 3.16 3.71 4.15 4.48

Licher

Cl 0.0231 0.0521 0.0812 0.1102 0.1394 0.1687 0.1982 0.2279

Cd 0.00345 0.00370 0.00449 0.00586 0.00780 0.01031 0.01346 0.01725

L/D 6.71 14.10 18.06 18.80 17.88 16.36 14.73 13.21

Designed

Cl 0.0580 0.0867 0.1154 0.1442 0.1730 0.2018 0.2307 0.2598

Cd 0.00336 0.00414 0.00531 0.00701 0.00925 0.01202 0.01534 0.0192

L/D 16.38 20.93 21.72 20.55 18.70 16.79 15.04 13.51

Flat plate (theory)

Cl 0.0000 0.0254 0.0508 0.0762 0.1016 0.1270 0.1523 0.1777

Cd 0.00000 0.00022 0.00089 0.00199 0.00355 0.00554 0.00798 0.01086

L/D – 114.59 57.30 38.20 28.65 22.92 19.10 16.37

Upper Element

0.15

0.20

0.25

0.30

(z/c

)

Initial

Designed Biplane

Lower Element

-0.30

-0.25

-0.20

-0.15

(x/c)

(z/c

)

Initial

Designed Biplane

-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

(x/c)

-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

Fig. 36. Section airfoil geometries of the designed biplane (t/c¼0.102).

Fig. 37. Cp contour map of the designed biplane at MN¼1.7 (affi11).

LicherAB1.5 NS Cp distributions

-0.1

0.0

0.1

0.2

0.3

0.4

(x/c)

Cp

Euler Upper Euler Lower

NS Upper NS Lower

-0.1 0.1 0.3 0.5 0.7 0.9 1.1

Fig. 38. Cp distributions of the designed biplane configuration in Navier–Stokes

simulations.


similar to those of Navier–Stokes simulations. In Fig. 39 the totalCd (shown as Cdtotal: wave drag plus friction drag) at the identicallift conditions (Clffi0.11) are compared among the diamondairfoil, the Busemann biplane and the designed biplane. Dragcomponents are tabulated in Table 4. It was observed that thedesigned biplane had nearly the same friction drag as the originalBusemann biplane but with lower wave drag reducing the totaldrag coefficient from 142 counts to 127 counts (here, 1countmeans 10�4).

4.3. Drag polar diagrams

Fig. 40 shows the drag polar curve (from the inviscid analysis)of the designed biplane at the cruise Mach number (MN¼1.7)compared against the curves of other airfoils. The figure includes azero-thickness single flat-plate airfoil, a Busemann biplane, aLicher-type biplane and the designed biplane. Numerical data aregiven in Table 3. The characteristics of the single flat-plate airfoil

are obtained by the supersonic thin-airfoil theory whose equationis given from Eq. (3), as,

Cd ¼4a2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM21�1

p ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM21�1

p4

C2l ð25Þ

Note that the single flat-plate airfoil of zero thickness has onlywave drag due to its lift. Therefore, during supersonic flightsthe single flat-plate airfoil has the lowest wave drag amongmonoplanes.

The designed biplane has lower wave drag than the Licher-type biplane over a wide range of Cl. Particularly when Cl40.14,the total wave drag of the designed biplane becomes less thanthat of the zero-thickness single flat-plate airfoil. It is surprising tofind a biplane configuration that can have a lower wave drag thanthat of a flat-plate airfoil. From Table 3, one can see that at a rangeof sufficient lift of 0.1rClr0.2, the designed biplane exhibits a12 to 35 count reduction in Cd, compared to that of the Busemannbiplane.

Fig. 41 shows a Cl-a graph on the above-mentioned airfoils. Thelower the a of an airfoil is, the weaker the shock waves generatedfrom the lower surface are. It can be observed from Fig. 41 that thedesigned biplane has the highest Cl among those with the same a.In other words, the designed biplane emits the weakest shockwaves toward the ground at a given Cl condition. Tabulatedgeometry of this 2-D designed biplane is given in Ref. [6,28].

5. 3-D extension of biplane airfoils

5.1. Busemann biplane wing with rectangular planform

The 2-D Busemann biplane was then extended to a 3-Drectangular wing. The wing-section geometry of this rectangularwing was identical to that of the 2-D Busemann biplane. Thereference wing area (based on one element of the wing) of this wingwas chosen to be 1.0 and the semi-span length and aspect ratio ofthe wing were 1 and 2, respectively. Inviscid flow analysis aroundthe biplane wing was conducted at the free-streamMach number of1.7. The number of nodes used in the CFD analysis wasapproximately 1.10 million. Typical meshes are shown in Fig. 42.It also shows Cp contours of the rectangular biplane wing. It is clearthat adequate interference between shock waves and expansionwaves does not occur at wingtip regions. Figs. 43 and 44 show Cpdistributions at wing-span stations and span-wise Cd distributions ofthe rectangular biplane wing. Pressure leaks are observed inside ofthe Mach cone generated at the leading edge of the wingtip. As aresult of those pressure leaks, the two-dimensionality of the flow is

Cd t

otal

0

0.01

0.02

0.03

0.04

Diamond Designed

Cd wave (Cd)

Cd friction (Cdf)

Busemann

Fig. 39. Evaluation of Cdtotal at the same lift conditions (Clffi0.11) (diamond airfoil,

Busemann biplane, designed biplane).

Table 4The aerodynamic performance in Navier–Stokes simulations.

a (deg.) Cl Cd Cdfric Cdtotal Cl (in Euler) Cd (in Euler)

Diamond

0 0 0.0292 0.00394 0.0332 0 0.0289

2 0.104 0.0329 0.00414 0.0370 0.103 0.0326

Busemann

0 0 0.00185 0.00777 0.00962 0 0.00218

2 0.116 0.00639 0.00780 0.0142 0.115 0.00647

Designed

1.19 0.116 0.00479 0.00794 0.0127 0.115 0.00531

Fig. 40. Wave-drag polar diagrams of the designed biplane.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

-1 0 1 2 3 4

� [deg]

Cl

Flat Plate

Busemann

Licher

Designed

Fig. 41. Cl-a characteristics of various airfoils.


lost at the wingtip area. The wave-drag coefficient of the biplanewing increased to 0.00685, knowing that Cd of the 2-D Busemannbiplane is 0.00218 [30–32].

5.2. Planform parametric study

In order to reduce the undesirable pressure leaks occurring atthe wingtips, tapered wings were considered. Fig. 45 shows thesurface Cp contours of a tapered Busemann biplane wing. Thewing reference (based on one element of the wing) area is also setto 1 and its taper ratio is 0.25. Therefore, the semi-span length andthe aspect ratio will be 1.6 and 5.12, respectively. It is clear thatthe area affected by the Mach cones generated from the wingtipsis significantly reduced. The CD of this tapered wing is 0.00300,which is significantly lower than that of the above-mentioned

Fig. 46. Span-wise Cd distributions of a rectangular and a tapered Busemann

biplane wings (reference area 1, zero-lift conditions).

Free stream

x

z

y

Span

direction

Cp visualizationsCp visualizations

Effects of Mach cone

x

z

y

Fig. 42. Surface Cp and mesh visualizations of a rectangular Busemann biplane

wing (reference area 1).

Fig. 43. Cp distributions at each 10% span stations of a rectangular Busemann

biplane wing (reference area 1).

2D

Fig. 44. Span-wise Cd distributions of a rectangular Busemann biplane wing

(reference area 1).

Front view

Side view

y

y x

z

mid-chord line

Top view

y

x

z

flow

croot

Fig. 45. Orthographic drawing, and mesh and surface Cp visualization of a tapered

Busemann biplane wing (reference area 1).


rectangular wing. Fig. 46 shows span-wise wave-drag distri-butions along the rectangular and the tapered biplane wings. Italso shows Cd of the 2-D Busemann biplane is included as areference condition. This tapered wing also indicates anotherfavorable effect. The Cd distributions in the mid-wing areas arelower than that of the 2-D case (see Fig. 46). In several of thefollowing sections, the reasons for this favorable effect will beinvestigated.

5.2.1. Effect of changing sweep angles

First of all, we checked the sweep angle effect on a taperedwing, with fixed taper ratio of 0.25. Here, the sweep angle isdefined as the angle of the wing’s leading edge to the free-stream.Fig. 47 shows the CD characteristics relative to sweep angle. In thisstudy the sweep angle effect was evaluated by choosing aparameter defined by the mid-chord apex at the wingtip (Cmid/Croot) (see Fig. 47 for the definition of Cmid/Croot). It can be seen thatthe tapered wing with Cmid/Croot of around 0.5 achieved the lowestwave drag.

Fig. 48 illustrates simple diagrams of two different wingstermed Case 1 and Case 2, and their span-wise Cd distributions.Case 1 has no sweep (Cmid/Croot¼0.125). Case 2 has a sweep angleand its mid-chord line is normal to the free-stream direction(Cmid/Croot¼0.5). As shown in Fig. 47, Case 2 has a lower CD thanthat of Case 1. Fig. 49 shows Cp contours of the inner surfaces ofCase 1 and Case 2 with the help of Cp distributions of those twowings at 50% semi-span stations. It also shows the Cp distributionsof the 2-D case are included as a reference. It can be observed thatCp distributions of Case 1 and Case 2 wings differ from that of thetwo-dimensional result due to the sweep effect of the wing. It isclear that the Case 2 wing performs better than the Case 1 wing.

5.2.2. Effect by changing taper ratios

The characteristics of CD were examined by changing the taperratio of wing. The mid-chord of the wing was fixed to be normalto the free-stream direction (Case 2, Cmid/Croot¼0.5 in Fig. 47).

Fig. 50 shows CD characteristics due to the change of aspect ratio.The taper ratio of the wing and aspect ratio are uniquely relatedbecause that the reference wing areas of the wings are fixed to 1.The smaller the taper ratio is, the lower the CD is. The aspect ratioalso increases with decreasing taper ratio. However, CD does notdecrease significantly when the taper ratio is less than 0.25.Therefore, a taper ratio of 0.25 was selected. The aspect ratio ofthis wing becomes 5.12. In the next section, the design of a three-dimensional biplane wing, with this selected planform (theparameters of the planform are shown in Table 5), will bediscussed using the inverse-design approach. Remember that thewave-drag coefficient of the Busemann biplane wing (whose wingsection has Busemann biplane geometry) in this planform hasvalues of CD¼0.00300 at zero-lift conditions [30].

In Ref. [26], various biplane-wing planforms were systema-tically investigated in terms of drag reduction.

5.3. Introduction of winglet

During the extension of a supersonic-biplane airfoil to a three-dimensional wing unfavorable pressure leaks, that would destroy

x

y

crootctip

croot/ctip= 0.25

Reference area = 1

flow

cmid

x

y

2D2D

Fig. 47. CD characteristics with changes of the sweep angle (reference area 1,

zero-lift conditions).

Fig. 48. Simple diagram of interaction of the shock waves and the expansion

waves and span-wise Cd distributions of Case 1 and Case 2 (zero-lift conditions).


the desirable wave interactions were observed at the wingtipregions (see Figs. 42 and 43). In order to eliminate those pressureleaks at the wingtips, a winglet was introduced to the supersonicbiplanes. Fig. 51 shows Cp and mesh contours of Busemannbiplane wings with and without winglets. Table 6 shows dragcoefficients of these biplane wings. Note that the wing planformsare identical to those discussed in the previous sections (seeTable 5).

It is clear that the unfavorable wingtip effects can be nearlyeliminated and the aerodynamic performance improved by usingthe winglets. Furthermore, the winglets may be a necessity from astructural point of view. As can be seen in Fig. 51, the pressures ofthe inner surfaces of the biplane wing are higher than those of theouter surfaces. Therefore, winglets may play a critical role inminimizing flutter and bending problems.

5.4. Design for better-performance biplane wing

5.4.1. Application of the 2-D designed airfoil

The practical design of a 3-D biplane wing for high L/D atsufficient lift conditions (CL40.1) is discussed in this section[30,32]. The inverse-problem method used for 2-D biplanedesigns in the previous section was also used. In this section,we discuss how to design both the upper and lower elements ofthe wing utilizing the inverse method at each span-station. Thepreviously discussed 2-D designed airfoil is introduced in thiscase as the initial geometry of the wing section. The finallydesigned wing, therefore, will have different wing-sectiongeometries at each span-station. Note that the planform of thebiplane wing is fixed in this study.

5.4.2. 3-D inverse design

5.4.2.1. Design condition. For the inverse-design approach, aninitial wing model is required. For the planform of the initial wing,the tapered wing shown in Table 5 was chosen. The referencewing area and its taper ratio are 1 and 0.25, respectively. Thesemi-span length and the aspect ratio of the wing are 1.6 and5.12, respectively. For the geometries of wing sections of theinitial wing, the 2-D designed biplane geometry (defined as‘‘Designed Airfoil’’ in Section 4.2) was used. The number of nodesused for the simulations is approximately 1.10 million. Note thatthis wing, which we call the ‘‘Initial Wing’’, has values ofCL¼0.111, CD¼0.00621 and L/D¼17.9. As a reference, theaerodynamic performances of the ‘‘Designed Airfoil’’ and ‘‘InitialWing’’ are tabulated in Table 7.

5.4.2.2. Design process. The inverse-design method discussed inSection 4 had been confirmed to work well for the 3-D design[58]. We then applied the method at ten span stations (every 10%of wing span from 0 to 90% of the wing). The design procedureshown in Fig. 52 was carried out using the specified (target)pressure distributions along both the upper and lower elements.First, design iterations were performed on the upper element untilthe obtained Cp distributions converged to the target (upper) Cpdistributions, while the lower element wing configuration waskept fixed. Then, the lower element was designed, while theconfiguration of the newly designed upper element remainedfixed. If the aerodynamic performance of the newly designedwing section exhibits no improvement over that of the initialwing section, geometry of the initial wing section will be usedinstead.

5.4.2.3. Analysis of the initial model. The ‘‘Initial Wing’’ was used asthe initial model of the inverse-design process. Target pressuredistributions required for this wing design will be discussed in the

croot ctip

Reference area = 1

flow

Fig. 50. CD and aspect ratio characteristics with changes of taper ratio (zero-lift

conditions).

Fig. 49. Cp Visualizations of the inner surfaces and Cp distributions at 50% semi-

span stations (not affected by Mach cones) of Case 1 and Case 2.

Table 5Geometric parameters of the selected wing planform as a baseline model.

Parameters Conditions

Taper ratio 0.25

Aspect ratio 5.12

Semi-span length 1.6

Reference area 1

Mid-chord line Normal to the free-stream


next section. Fig. 53 shows Cp visualizations of the inner surfacesand Cp distributions of the ‘‘Initial Wing’’. Let us focus on the Cpdistributions along the inner surfaces (Cp distributions along thelower surface of the upper element and along the upper surface ofthe lower element). Compared to the Cp distributions of the‘‘Designed Airfoil’’ (see Fig. 35 shown in Section 4), large pressurepeaks near throats were recognized on both the upper andlower elements. Additionally, the values of the pressurecoefficients in the areas not affected by the wing root (y/b of50% to 80%) were greater than those in the areas that wereaffected (y/b of 0 to 40%). Note that the region affected by the

wing root is confined inside the Mach cones generated from thewing root.

Fig. 54 shows span-wise Cl, Cd and l/d distributions of the‘‘Initial Wing’’. It can be observed that the areas affected by theMach cones from the wing root and the wingtip are characterizedby poor aerodynamic performance. However, those regions whichare not affected by the Mach cones have a higher lift-to-dragcoefficient than those of 2-D designed airfoil. These are typicalcharacteristics of the 3-D tapered biplane wings as mentionedbefore. The goal of the design here is to generate a biplane winggeometry, which has better aerodynamic performance than theinitial wing by specifying desirable Cp distributions (as targetpressures) at each wing-span station.

5.4.2.4. Guideline for setting target pressure distributions. In orderto prescribe target pressure distributions for our three-dimen-sional wing design, we employed the following two strategies.First we tried to remove the unnecessary pressure peaks occurredaround the mid-chord (near the throat) areas in the baseline‘‘Initial Wing’’. Second, we extend the ‘‘good’’ Cp distributions thatare obtained from the mid-wing region of the ‘‘Initial Wing’’, intothe regions near the wingtip and wing root. Target pressure dis-tributions for the upper and lower elements used for the three-dimensional wing design are shown in Figs. 55 and 56,respectively.

5.4.2.5. Design results. Let us now focus on the target Cp dis-tributions and the obtained Cp distributions from the designedwings. As mentioned earlier, the upper element was first designed.

Busemann biplane without winglet

Spanwise Cd distributions

Flow Flow

Busemann biplane with winglet

Fig. 51. Cp and mesh visualizations and Cd distributions of the tapered Busemann biplane wing with and without a winglet.

Table 6Drag coefficients of Busemann biplanes without winglet and with winglet at zero-

lift conditions.

CD

Without winglet 0.00300With winglet 0.00258

Table 7Aerodynamic performance of 2-D ‘‘Designed Airfoil’’ and 3-D ‘‘Initial Wing’’.

Lift coefficient Drag coefficient Lift-to-drag ratio

Designed airfoil 0.115 0.00531 21.7

Initial wing 0.111 0.00621 17.9


Details of the target Cp distributions of the upper element areshown in Fig. 55. Fig. 57 shows the obtained Cp distributions of thedesigned wing (after 14 iterations of the inverse-design cycle) andthe Cp contours of the inner surfaces of the upper wing. Thisdesigned wing was termed the ‘‘Upper Designed Wing’’. Theobtained Cp distributions successfully converged with the targetdistributions. CL, CD and L/D are 0.120, 0.00662 and 18.1,respectively. L/D was improved with the increase of CL. Fig. 58shows the designed section geometries of the ‘‘Upper DesignedWing’’. Fig. 59 shows its span-wise Cl, Cd and l/d distributions. Thegeometries around the wing symmetry section were altered tohave greater angles of attack. Span-wise Cl in the areas affected by

Mach cones from the wing root increased without reductions of thespan-wise l/d.

Next, the lower element was designed. The Cp distributions ofthe lower element obtained after 14 iterations of the previouslydiscussed upper-element design cycle, were used as the initialdistributions on the lower-element wing. The initial and target Cpdistributions of the lower element are shown in Fig. 56. Asmentioned in the previous section, the purpose of the discrepancybetween the initial and the target Cp distributions is to remove thepressure peaks at the mid-chords. Fig. 60 shows the obtained Cpdistributions after 14 iterations and the Cp visualization of theinner surfaces of the designed biplane wing. The newly designed

Upper element

010%20%30%40%50%60%70%80%90%

Lower element

010%20%30%40%50%60%70%80%90%

M

M

Fig. 53. Cp visualizations of the inner surfaces and Cp distributions of the ‘‘Initial Wing’’.

Initial Wing

Grid Generation

Target Cp ofOne Element

Flow Solver(Euler)


Cp <

NO


f f + f

Designed Wing

YES

start

Design cycle of each element

Initial Wing

Grid Generation

Target Cp of One Element

Flow Solver(Euler)


Cp <

NO


f f + f

Designed Wing

YES

goal

at each span station

Design cycle of each element

Fig. 52. Design flow chart of one element of a three-dimensional biplane wing.


wing was termed ‘‘Upper and Lower Designed Wing’’. It isclear that the pressure peaks were successfully removed.Unfortunately, the Cp distribution at the wing root did notconverge to its target distributions. The designed wing has CL,CD and L/D of 0.122, 0.00664 and 18.3, respectively. Fig. 61 showsdesigned section geometries of the ‘‘Upper and Lower DesignedWing’’. Fig. 62 shows its span-wise Cl, Cd and l/d distributions.Although improvements were seen throughout most of the areas,the span-wise Cl and l/d at the wing root section showed noimprovement over those of the ‘‘Initial Wing’’. We, therefore,conducted a further modification.

The geometry at the wing root section of the lower element ofthe ‘‘Upper and Lowe Designed Wing’’ was replaced with that ofthe ‘‘Initial Wing’’. This wing was termed the ‘‘Final Wing’’. Fig. 63shows span-wise Cl, Cd and l/d distributions, and Fig. 64 shows Cpvisualizations of the inner surfaces of the ‘‘Final Wing’’. This wingmaintained higher l/d than the ‘‘Initial Wing’’ at all span stations,and more lift was created in the areas affected by Mach conesoriginating from the wing root. The ‘‘Final Wing’’ has values ofCL¼0.125, CD¼0.00678 and L/D¼18.4. The 3-D designing wasterminated at this point.

5.4.2.6. Comparison with other three-dimensional supersonic-biplane

wings. Table 8 shows the aerodynamic performances of variousthree-dimensional tapered biplane wings. The designed biplanewings show the highest aerodynamic performance among theBusemann and Licher biplane wings. Fig. 65 shows a drag polardiagram of the biplane wings. Fig. 65 also includes the drag polardiagram of the 2-D zero-thickness single flat-plate airfoil. Notethat the 2-D flat-plate airfoil is identical to the 3-D flat-plate wingwith an infinite span that has only wave drag due to its lift. It isnotable that the ‘‘Final Wing’’ having a sufficient thicknessachieves lower wave drag than the 3-D zero-thickness flat-platewing with an infinite span at CL40.17 at its design Machnumber 1.7.

5.4.2.7. Final wing with winglet. The winglet discussed in theprevious sections was applied to the ‘‘Final Wing’’. The lift-to-dragratio was significantly improved from 18.4 to 19.6 at the identicallift coefficients of 0.125. Fig. 66 shows Cp contours of the innersurfaces of the ‘‘Final Wing with Winglet’’. Fig. 67 shows a dragpolar curve of the ‘‘Final Wing with Winglet’’. The wing achievedlower drag than the single flat-plate airfoil at CL40.16.

6. Wing–fuselage interference

As a first step in working towards the designing a practical SSTwith biplane wings, the interference effects between the biplanewing and body (fuselage) were investigated. In this chapter,aerodynamic characteristics of wing–body configurations aresimulated utilizing CFD and the aerodynamic performance ofthe biplane wings affected by those wing–body configurations isdiscussed [27–32].

6.1. Wing–fuselage configuration

A wing–body configuration shown in Fig. 68 (Refs. [27,28,31])was adopted to investigate wave-interference effects from thebody. The body has a conical configuration at the nose and arectangular parallelepiped configuration in the rear. This bodyshape was chosen in order to generate strong shock waves andexpansion waves. A biplane wing is attached to the body in a waythat the disturbances from the body can influence the entire wing.The baseline wing configuration used in the study is a Busemann

Fig. 54. Span-wise Cl, Cd and l/d distributions of the ‘‘Initial Wing’’.

010%20%30%40%50%60%70%80%90%

Fig. 55. Target Cp distributions of the upper element.

010%20%30%40%50%60%70%80%90%0 Target10% Target20% Target30% Target40% Target50% Target60% Target70% Target80% Target90% Target

Fig. 56. Target Cp distributions of the lower element, and Cp distributions of the

lower element of the ‘‘Upper Designed Wing’’ as initial Cp distributions.


biplane wing with a tapered planform. Details of the geometrieswill later be presented in Fig. 70. Fig. 69 shows typical bodyeffects on the biplane wing.

Two types of supersonic-biplane wings are used for the wing–body configurations in this study. The configurations of these twobiplane wings are identical except for the existence of winglets.The wing has the section shape of a Busemann biplane airfoilwhose total thickness–chord ratio and gap-to-chord ratiobetween its two elements are 0.10 and 0.505, respectively. Thewing has the tapered planform (discussed in the previous section)with a taper ratio of 0.25. The wing reference area is 1, and theaspect ratio is 5.12 with the mid-chord line being normal to thefree-stream direction. Details of the wing parameters are shownin Fig. 70 and given in Table 9. In the study, four different winglocations relative to the body were investigated. Totally, thereare eight types of wing–body configurations (without winglet

(w/o wlt) and with winglet (w/wlt)). The overview of those fourdifferent wing locations is shown in Fig. 71.

6.2. Aerodynamic performance at cruise condition

Fig. 72 shows a mesh visualization used for the inviscid flowanalysis at the cruise Mach number of 1.7. The number of nodes ofeach wing–body configuration is about 2.0 million. Fig. 73 showsCp contours of the body symmetry plane z¼0 for each wing–bodyconfiguration. The cases of ‘‘w/wlt’’ are described in order to showshock and expansion waves generated from body. When theleading edge of the wing root is positioned at xw¼3 or 3.5, thewings are strongly affected by both compression and expansionwaves from the body. On the other hand, when xw¼4 or 4.5, thewings are exposed only to the expansion waves. The wings w/owlt are equally affected by those waves from the body. Fig. 74

Fig. 57. Cp visualization of the inner surface, and target and obtained Cp distributions of the upper element of the ‘‘Upper Designed Wing’’.

Fig. 58. Designed section geometries of the ‘‘Upper Designed Wing’’.Fig. 59. Span-wise Cl, Cd and l/d distributions of the ‘‘Upper Designed Wing’’.


shows drag polars calculated from the wings themselves of thewing–body configurations (w/o wlt and w/wlt) and of the wing-alone (simple wing) cases. The details of the aerodynamicperformance are tabulated in Tables 10 and 11. All the wing–body configuration cases with the exception of the case wherexw¼3 have better aerodynamic performance than the wing-alone cases.

Fig. 75 shows the surface Cp distributions at zero lift conditions(zero angle of attack) of the wing–body configurations w/o wltand w/wlt. The Cp contours of those are along the inner surfaces ofthe biplane wings. For the instances where xw¼4 and 4.5, inwhich the wings are affected only by the expansion waves, thereis very little difference in aerodynamic characteristics betweenthe w/o wlt and w/wlt cases. However, there are noticeable

differences when xw¼3 and 3.5, in which wings are affected bynot only the expansion waves but also the compression waves.Flow choking occurs on the wing–body configuration withwinglet of xw¼3, where the wingtip region is widely exposed tothe compression waves, and then thus causing high wave drag. Onthe other hand, for the configuration without a winglet of xw¼3.5,the wing is less affected by those compression waves, and thehigh pressure regions near the wing tip causes the reduction ofthe wave drag of the wing. These phenomena can be explained byobserving the reflection mechanism of the compression wavesfrom the winglet. For the cases of without winglets, flow chokingdoes not occur because the pressure increase caused by thereflection of the compression waves from the winglet does notexist. Drag reduction effects nar the wingtip are also not detected.

010%20%30%40%50%60%70%80%90%0 Target10% Target20% Target30% Target40% Target50% Target60% Target70% Target80% Target90% Target

M

Fig. 60. Cp visualization of the inner surface, and target and obtained Cp distributions of the lower element of the ‘‘Upper and Lower Designed Wing’’.

Fig. 61. Designed section geometries of the ‘‘Upper and Lower Designed Wing’’.

Fig. 62. Span-wise Cl, Cd and l/d distributions of the ‘‘Upper and Lower Designed

Wing’’.


In the cases of both w/wlt and w/o wlt at xw¼4 and 4.5, at whichpoint the wings are affected by only the expansion waves, flowchokings are not observed.

Fig. 76 shows span-wise Cd distributions along the wing forboth w/o wlt and w/wlt cases. In the cases w/o wlt at xw¼3 and3.5, we notice several high Cd regions, which are affected by thecompression waves from the body, and low Cd regions, which are

caused by the expansion waves. Similar trends are observed forthe case w/wlt, with the exception of one unique characteristic forxw¼3.5. The reflection of the compression waves from the wingletcreates a thrust force (low drag) near the winglet. These samereflection waves are what cause the flow choking in the xw¼3case. In summary, for both cases w/o wlt and w/wlt, the areasaffected by the compression waves will generate greater wavedrag than the wing-alone cases, when there is reflection of thecompression waves from the winglet. On the other hand, the areasaffected by the expansion waves have better aerodynamicperformance in all cases. Therefore, the configurations of xw¼4

Fig. 64. Cp visualizations of the inner surfaces of the ‘‘Final Wing’’.

Fig. 63. Span-wise Cl, Cd and l/d distributions of the ‘‘Final Wing’’.

Table 8Aerodynamic performance of three-dimensional biplane wings.

Three-dimensional biplane wing CL CD L/D

Busemann biplane wing (a¼21) 0.110 0.00706 15.6

Licher biplane wing (a¼1.51) 0.107 0.00642 16.7

Initial wing (affi11) 0.111 0.00621 17.9

Final wing (affi11) 0.125 0.00678 18.4

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014

Flat Plate Airfoil (Theory) Busemann Biplane WingLicher Biplane Wing 2D Designed Wing3D Designed Wing

Lower Drag

Required CLfor Cruise

every 0.5deg plotted0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22

CL

CD

Flat Plate Airfoil (Theory) Busemann Biplane WingLicher Biplane Wing 2D Designed Wing3D Designed Wing

Lower Drag

Required CLfor Cruise

every 0.5deg plotted

Fig. 65. Drag polar curves of three-dimensional biplane wings and single flat-plate

wing with an infinite span.

Fig. 66. Cp visualizations of the inner surfaces of the ‘‘Final Wing with Winglet’’.

0.000.020.040.060.080.100.120.140.160.180.200.22

0.002

Cd

Cl

Flat Plate (Theory) (t/c=0) Final Wing with Winglet

0.000 0.004 0.006 0.008 0.010 0.012 0.014

Fig. 67. Drag polar diagram of ‘‘Final Wing with Winglet’’.


and xw¼4.5 are recommended. Fig. 77 shows span-wise Cl, Cd andl/d distributions at an angle of attack of 21. It can be concluded thebiplane wing is reasonably robust against disturbances generatedby fuselage. Moreover, when the wing from the wing–bodyconfiguration is exposed to expansion waves, aerodynamiccharacteristics of the wing itself can be improved compared tothose of the wing without the body (simple wing).

6.3. Application of designed wing to wing–fuselage combination

In order to design practical flight carriers, the designed biplanewing discussed in the previous section was applied to a wing–body configuration. From the view point not only of aerodynamicperformances at the cruise condition but also of avoiding the un-start problem near the cruise Mach number, the wing–bodyconfiguration w/wlt of xw¼4 was selected. Under these conditionsthe designed biplane wing will be exposed only to expansionwaves from the body. This wing–body configuration with thedesigned wing, defined as the ‘‘Final Wing with Winglet’’ in theprevious section, will be referred to in this study as the ‘‘Finalwing–body configuration’’.

x

y

z

Flow

Symmetry Plane

0.505C(root)

1.02C(root)

0.5C(root)

1.98C(root)

2.5C(root)

0.2525C(root)

C(root)

Side view of Nose region

n=tan-1(0.125/0.5) 14.0[deg]

s=tan-1(0 1275/1 02) 7 1[deg]

ε

ε

Fig. 68. Wing–body configuration proposed by Odaka and Kusunose [27,28,31].

Fig. 69. Cp visualization of a wing–body configuration proposed by Odaka and

Kusunose at z¼0 [27,28,31].

ctip

H(tip)

b/2

Flow

S(ref)H(root)

y

zx

C(root)

Fig. 70. Configuration of tapered Busemann biplane wing.

Table 9Geometric parameters of wing planform.

Parameters Conditions

Taper ratio 0.25

Aspect ratio 5.12

Semi-span length 1.6

Reference area 1

Mid-chord line Normal to the free-stream

Fig. 71. Positions at which a wing is attached.

Fig. 72. Mesh visualization for analysis at cruise condition.


At an angle of attack of 1.191, CL and CD were improved from0.126 to 0.131 and from 0.00647 to 0.00631, respectively. L/Dincreased from 19.4 to 20.8. This value is very close to that of the2-D designed biplane discussed in Section 4. Note that the lift-to-wave-drag ratio of the 2-D designed biplane is 21.7. Fig. 78 shows

Fig. 73. Cp visualizations of the bodies with the winglets at z¼0 (zero angle of attacks).

w/o wlt

0.000.020.040.060.080.100.120.140.160.180.200.22

0.014

CD

CL

simple wing w/o wlt x=3 w/o wlt x=3.5 w/o wltx=4 w/o wlt x=4.5 w/o wlt

w/ wlt

0.000.020.040.060.080.100.120.140.160.180.200.22

CD

CL

simple wing w/ wlt x=3.5 w/ wlt x=4 w/ wlt x=4.5 w/ wlt

0.000 0.002 0.004 0.006 0.008 0.010 0.012

0.0140.000 0.002 0.004 0.006 0.008 0.010 0.012

Fig. 74. Drag polar diagrams.

Table 10Aerodynamic performance of wings of the body without winglet.

A.o.A 0 1 2 3

xw¼3

CL �0.0001 0.0603 0.1208 0.1817

CD 0.00397 0.00511 0.00852 0.01426

L/D 0.0 11.8 14.2 12.7

xw¼3.5

CL 0.0000 0.0587 0.1173 0.1758

CD 0.00338 0.00446 0.00769 0.01308

L/D 0.0 13.2 15.3 13.4

xw¼4

CL 0.0000 0.0586 0.1170 0.1753

CD 0.00313 0.00421 0.00743 0.01280

L/D 0.0 13.9 15.7 13.7

xw¼4.5

CL 0.0000 0.0582 0.1163 0.1747

CD 0.00307 0.00414 0.00736 0.01274

L/D 0.0 14.0 15.8 13.7

Simple wing

CL 0.0001 0.0544 0.1090 0.1643

CD 0.00326 0.00425 0.00725 0.01229

L/D 0.0 12.8 15.0 13.4


the surface Cp contours of the ‘‘Final wing–body configuration’’.Fig. 79 shows drag polar diagrams and aerodynamic performance ofthe wing itself from ‘‘Final wing–body configuration’’ (designed X¼4w/wlt) and the ‘‘Final Wing’’ without body (simple designed wing).The values of the aerodynamic performance of the wings aretabulated in Table 12. The wing from the ‘‘Final wing–bodyconfiguration’’ achieved the highest performance. In Fig. 79, a dragpolar curve of the two-dimensional flat-plate airfoil obtained fromthe linear theory [8] is also included. In its aerodynamicperformance, the wing of the ‘‘Final wing–body configuration’’achieves the lowest drag at CL40.14. Fig. 80 shows span-wise Cl, Cdand l/d distributions of the designed wing of the wing–bodyconfiguration and the designed wing itself (simple designed wing).The general trends in their aerodynamic characteristics are almostthe same as those of the Busemann biplane wing shown in Fig. 76. Itcan be seen that the flow disturbances generated by a body will notdeteriorate the performance of the biplane wing if the winglocations are carefully selected.

Finally, the hinged slats and flaps were installed into the ‘‘Finalwing–body configuration’’ to overcome the choked-flow problemat its off-design conditions [32]. Wing forms as well as its CL–MN,CD–MN curves are shown in Fig. 81. The hinged slats and flaps wereequipped at the leading edge and trailing edge, respectively, along

Table 11Aerodynamic performance of wings of the body with winglet.

A.o.A 0 1 2 3

xw¼3

CL �0.0002 0.0630 0.1263 0.1904

CD 0.02199 0.02329 0.02727 0.03417

L/D 0.0 2.7 4.6 5.6

xw¼3.5

CL 0.0001 0.0591 0.1180 0.1769

CD 0.00279 0.00387 0.00714 0.01267

L/D 0.0 15.3 16.5 14.0

xw¼4

CL �0.0001 0.0584 0.1167 0.1750

CD 0.00271 0.00378 0.00700 0.01235

L/D 0.0 15.4 16.7 14.2

xw¼4.5

CL 0.0001 0.0582 0.1162 0.1745

CD 0.00274 0.00381 0.00702 0.01238

L/D 0.0 15.3 16.6 14.1

Simple wing

CL 0.0001 0.0543 0.1089 0.1642

CD 0.00283 0.00382 0.00681 0.01184

L/D 0.0 14.2 16.0 13.9

Fig. 75. (a) Surface Cp distributions of wing–body configurations and simple wing w/o wlt at zero lift conditions (zero angle of attacks). (b) Surface Cp distributions of wing–

body configurations and simple wing w/wlt at zero lift conditions (zero angle of attacks).


the wing span from 0.1 to 0.9. Fig. 82 shows surface Cp contoursof the wing–body configurations equipped with the hinged slatsnear the starting Mach number [16,53]. It can be observed thatwhen the Mach number is at 1.57 the detached shock waves aroundthe wingtip are not swallowed due to the winglet effect. However, atthe other span-wise locations the detached shock waves are alreadyswallowed. It is clear that by installing hinged slats and flaps on tothe wing–body configurations, choked-flow condition can beavoided at its design Mach number of 1.7.

7. Experiment

Shortly after A. Busemann proposed his supersonic-biplaneconcept in 1935 [7], experimental investigations began. In theearly 1940s, fundamental experiments were already conducted byFerri in order to measure aerodynamic forces acting on theBusemann biplanes near its design Mach number [14]. Based onhis experiments he concluded that the supersonic biplane may, aspredicted by theory, lead to notable advantages from theperspective of both reducing drag and increasing efficiency ofthe wing unit. He also observed flow choking phenomena andtheir related hysteresis problems by varying the wing gap. Thisoperation was carried out by moving one of the wing elementsparallel to itself.

In this review paper we focus on the recent experimentalinvestigations that are closely related to the current biplanedevelopments.

Supersonic and transonic flow fields around the Busemannbiplane were examined by Kuratani et al. [34]. The purpose of thisstudy was to demonstrate the shock-wave-interference and

cancellation effects between the wing elements of the biplane.Wind-tunnel testing in supersonic and transonic flow regions wasperformed along with CFD analysis to investigate the flowcharacteristics around the 2-D supersonic-biplane model.Schlieren images and CFD analysis clarified the fundamentalflow characteristics around the biplane at its design Mach numberof 1.7. However, it was also observed in the transonic flow regionthat the supersonic biplane acted like a subsonic nozzle that wasaccompanied by choked flows.

The experimental model for the ballistic range was designedand tested by Toyoda et al. [37] to examine the low-boomcharacteristic of the supersonic biplane. In their study wingletswere attached to the ballistic-range models. Those winglets weredesigned with the help of CFD analysis in order to avoid the

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6y/ croot

Cd

Cd

x=3x=3.5x=4x=4.5

region affected by compression waves

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

simplex=3x=3.5x=4x=4.5

region affected by expansion waves

- 0.004

- 0.002

0

0.002

0.004

0.006

0.008

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

simple wingx=3.5x=4x=4.5

- 0.004

- 0.002

0

0.002

0.004

0.006

0.008

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

x=3.5x=4x=4.5

reflection effect of winglet

y/ croot

w/o wlt

w/ wlt

Fig. 76. Span-wise Cd distributions at zero-lift conditions (zero angle of attacks).

w/o wlt Cl , Cd

w/o wlt l / d

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6y/croot

y/croot

y/croot

C l

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

C d

Cl simple Cl x=3Cl x=3.5 Cl x=4Cl=4.5 Cd simpleCd x=3 Cd x=3.5Cd x=4 Cd x=4.5

0

5

10

15

20

25

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6l/

d

l / d simplel /d x=3l /d x=3.5l/d x=4l/d x=4.5

w wlt Cl, Cd

w wlt l / d

0

0.02

0.04

0.06

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.002

0.004

0.006

0.08

0.1

0.12

0.14

0.008

0.01

0.012

0.014

Cl simple wing Cl x=3.5Cl x= 4 Cl x=4.5Cd simple Cd x=3.5Cd x=4 Cd x=4.5

0

10

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

20

30

40

50

60

l/d

l/d simple wingl/d x=3.5l/d x= 4l/d x=4.5

y/croot

C l C d

Fig. 77. Span-wise Cl, Cd and l/d distributions at angle of attacks of 21.


un-start (choked flow) phenomena. As the CFD simulationsindicated the designed experimental model was successfullylaunched without encountering the un-start problem. In theirexperiment, supersonic-biplane models were flown at a Machnumber of 1.7 with zero angle of attack. The Schlieren imagesrevealed presence of the shock-wave-cancellation mechanisms asexpected from the CFD simulations.

Using Pressure Sensitive Paint (PSP), Nagai et al. [36] measuredthe pressure distributions along the surfaces of the Busemannbiplane in a small supersonic indraft wind tunnel. Due to the

effects of the boundary layers developed along the wind-tunnelwalls, the observed pressure distributions were considerablydifferent from the values predicted through the simple 2-Dsupersonic thin-airfoil theory. They concluded that a further studywas necessary to completely understand the interaction-flowmechanism of the 2-D Busemann biplane by carefully isolatingthe wind-tunnel wall effects.

The low-speed aerodynamic characteristics of a baseline Buse-mann biplane (without high-lift devices) were investigated usingexperimental and CFD approaches by Kuratani et al. [35]. Thepurpose of the study was to analyze the fundamental low-speedaerodynamic and flow characteristics of the Busemann biplane. Inthis study, biplane stall characteristics were clarified. At the free-stream velocity of 30 m/s, the Busemann airfoil stalled at an angle ofattack of approximately 201. During the study, end plates wereattached to a rectangular-shaped biplane wing to simulate 2-Dflows. At high incidence angles, flow around the upper element ofthe biplane tended to separate earlier than that of the lower one. Thestall of the lower element was suppressed due to the presence of theupper element and the total lift of the biplane systemwas, therefore,primarily generated by the lower element of the biplane. It wasconcluded that the fundamental aerodynamic characteristics of theBusemann biplane obtained through the wind-tunnel testing werein good agreement with those obtained from CFD analysis.

At take-off and landing conditions, flow visualizations around theBusemann biplane airfoil equipped with leading-edge and tailing-edge flaps (or slats and flaps) were performed by Kashitani et al.[38,39] in a low-speed smoke wind tunnel. The lift coefficient of thebiplane airfoil was estimated by utilizing a method based on smoke-line pattern measurements. With the help of hinged slats and flaps,the maximum lift coefficients of the Busemann biplane could reachto roughly 2.0 when the lift coefficient was normalized by using thebaseline chord length of the wing element of the biplane. The dragcoefficient was estimated by measuring the velocity defects in the

0.000.020.04

60.00.080.100.120.140.160.180.200.22

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014CD

CL

Flat Plate (Theory) simple designed wing designed x=4 w/ wltsimple wing w/ wlt x=4 w/ wlt

Fig. 79. Drag polar curves of ‘‘Final Wing with Winglet’’ and the wing of ‘‘Final

wing–body configuration’’.

Table 12Aerodynamic performance of the wings of ‘‘Final wing–body configuration’’ and

‘‘Final Wing with Winglet’’.

A.o.A 0.19 0.69 1.19 1.69 2.19 2.69

Wing of ‘‘Final wing–body configuration’’

CL 0.0721 0.1017 0.1312 0.1607 0.1900 0.2192

CD 0.00434 0.00506 0.00631 0.00810 0.01044 0.01330

L/D 16.6 20.1 20.8 19.8 18.2 16.5

‘‘Final wing with Winglet’’

CL 0.0710 0.0983 0.1257 0.1530 0.1803 0.2077

CD 0.00448 0.00523 0.00647 0.00821 0.01046 0.01323

L/D 15.8 18.8 19.4 18.6 17.2 15.7

Fig. 80. Span-wise Cl, Cd and l/d distributions of the wing of ‘‘Final wing–body

configuration’’ and ‘‘Final Wing with Winglet’’.

Fig. 78. Surface Cp visualization of ‘‘Final wing–body configuration’’ at an angle of

attack of 1.191.


wake at a downstream station of the airfoil. The experimental datafor the lift and drag coefficients at low speeds were in goodagreement with the reference CFD data [6,32]. It was confirmed thatthe aerodynamic characteristics of the Busemann biplane equippedwith slat and flap were similar to those of a conventional airfoil withhigh-lift devices.

8. Conclusions

This paper reviewed the progresses on supersonic biplanesmade by a group at Tohoku University since 2004 [1–6]. Theyhave extended the classic Busemann biplane concept [7] todevelop a practical supersonic biplane, that will generatesufficient lift at supersonic flights while making its shock-wavestrength minimum.

As a fundamental characterization the classic 2-D Busemannbiplane and its extended biplane configurations such as the Licherbiplane [10] were analyzed with CFD. The thickness–chord ratio ofeach element of those biplanes was approximately 0.05. The liftcoefficient was set from 0.10 to 0.20 to generate sufficient lift attheir cruise (design) Mach number of 1.7. It was confirmed that theBusemann biplane could eliminate almost all the wave drag causedby its airfoil thickness. At zero-lift condition wave drag generatedwas more than ninety percent smaller compared to that of adiamond airfoil, with the same thickness–chord ratio of 0.10 [6]. At asmall angle of attack, the lift and wave drag of the Busemannbiplane are identical to those of a flat-plate airfoil except for a smallwave-drag penalty. This penalty is due to the entropy produced byshock waves, which exist between the biplane elements. The Licherbiplane was able to further reduce lift-related wave drag by two-thirds from that of the Busemann biplane.

We then studied the aerodynamic characteristics of theBusemann biplane at off-design conditions. In the study flowchoking and its concomitant hysteresis problems were found tooccur at a wide range of free-stream Mach numbers, from 0.50 to2.18, producing an unacceptable level of wave drag. As acountermeasure, hinged slats and flaps, which can also be usedas high-lift devices during take-off and landing conditions, wereutilized to control the area ratios of the inlet and the throat ofbiplanes. By applying these slats and flaps, the biplane was able toachieve the same level of wave drag (coefficient) as that of thediamond airfoil with equal thickness–chord ratio of 0.10, over awide range of free-stream Mach numbers [20,22].

The Licher biplane was further improved by utilizing aninverse-design method with specified pressure distributionsalong the biplane surfaces [17,18]. The aerodynamic perfor-mance of the original Licher biplane was of Cl¼0.0812,Cd¼0.00449 with lift-to-(wave) drag ratio of 18.1 at an angle ofattack of 11, based on Euler (inviscid) simulations. The designedbiplane had a Cl of 0.115, Cd of 0.00531 and a lift-to-drag ratio of21.7 at the same angle of attack. The designed biplane generated

2.5 1.19deg2.5

Slat & FlapSlat

slat & flap slat original

Slat & FlapSlat

Original

2.5

1.71.00.8M

1.57

Hysteresis of choking disappears

4deg 1.19deg

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

CL

Slat & Flap

Slat

Sufficient CL

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.5 1 1.5 2 0.5 1 1.5 2MM

CD

Slat & Flap

SlatSimple Busemann (zero lift)

Original

Fig. 81. Wing form and CL–MN, CD–MN graphs.

M∞ =1.56 M∞ =1. 57

M∞ =1.60

Fig. 82. Surface Cp visualizations of ‘‘Final wing–body configuration’’ equipped

with hinged slats near the starting Mach number.


less wave drag than the flat-plate airfoil at Cl40.14 [18]. It isimportant to remember that the flat-plate airfoil has the lowestwave drag among the entire monoplane airfoils. The designedairfoil with slat and flap devices was also analyzed at flowconditions from take-off to cruise. It was confirmed that thedesigned airfoil with hinged slats and flaps could generate enoughlift for its entire flying speeds, while nearly elminating flowchoking and its related hysteresis problems [6,18].

To extend the previously discussed 2-D biplane to 3-D biplanewings, an additional study was conducted. For this study thetapered-wing planform with taper ratio of 0.25 and aspect ratio of5.12 was selected [26,30]. Utilizing the previously discussed 2-Ddesigned biplane configuration as the (initial) wing-sectiongeometry of the tapered wing, the inverse design wassimulated. The lift coefficient and lift-to-(wave) drag ratio of thedesigned wing were 0.125 and 18.4 (without viscous effect) at theangle of attack of approximately 11 [30,31]. In order to minimizepressure leaks at its wingtip regions, a winglet was introduced.When the winglet was applied to the designed wing, the lift-to-(wave) drag ratio further improved to 19.6. For viscous flowanalyses, the designed wing with the winglet achieved lift-to-(total) drag ratios of 7.0–9.5 at a range of lift coefficients between0.10 and 0.20 [32]. Flow choking and hysteresis also occurred forthis designed wing. However the flow choking was milder thanthat of the 2-D designed biplane itself, and they were nearlyeliminated, similar to the 2-D cases, with the use of slat and flapdevices [30,32].

Finally, several wing–body configurations were simulated toinvestigate the wave-interference effects of the body on thetapered biplane wings [31,32]. In the study a body geometrythat would generate strong shock waves at its nose region wasselected to investigate the sensitivity of the biplane wing to non-uniform upstream flows. Preliminary parametric studies on theinterference effects between the body and the supersonic-biplanewings were performed by choosing several different winglocations on the body. When the biplane wings were affected bycompression waves from the body, the wings with wingletsperformed better than those without. It was, however, easier forflow choking to occur even with a small change in upstream flowcondition when the biplane had winglets. When the wings wereaffected by expansion waves, the Mach number range at whichflow choking would occur was reduced compared to that of thewing-alone (isolated wing) case. In general, supersonic-biplanewings performed better when they were exposed to expansionwaves. The previously discussed designed wing with winglet wasthen applied to the wing–body configurations. Lift coefficient andlift-to-(wave) drag ratio of the wing were found to be 0.131 and20.8, when the biplane wings were positioned in a region wherethey were exposed in the region of expansion waves. Theaerodynamic performances of the designed biplane wing withbody was able to approach those of the 2-D designed biplane bytaking advantage of the interference effect of the body [32].

In summary, a 2-D designed supersonic biplane with approxi-mately ten percent thickness–chord ratio can achieve lower wavedrag better than a zero-thickness flat-plate airfoil at lift coefficientgreater than 0.14. For both 2-D and 3-D biplane configurations,slats and flaps can be used to counteract flow choking that ocursat off-design conditions. For wing–body configurations, whenbiplane wings are located in expansion-wave regions, the biplanewings perform better than the wing-alone configurations both attheir design and at off-design conditions. These results indicateaerodynamic feasibility of supersonic biplanes for future super-sonic transports. It is, however, clear that further fundamentalstudies incorporating viscous flow analysis to determine bothbiplane-wing and optimized fuselage geometries of these futuretransports are necessary.

Acknowledgments

The authors are grateful for the help, information inputsand contributions provided by Professor S. Obayashi ofTohoku University, Professor A. Sasoh of Nagoya University,Dr. H. Yamashita, a post doctoral fellow of Tohoku University,Dr. M. Yonezawa of Honda R&D Co. Ltd, who finished the Ph.Dcourse of Tohoku University, and Mr. Y. Utsumi, a Master coursestudent of Tohoku University. We also would like to appreciateMs. S. Tuchikado at University of Toyama for her efforts ofsupporting us to prepare the manuscript and Mr. J. Kusunose atthe University of California at Davis for proofreading ourmanuscript.

Appendix A

A.1. Aerodynamic center

Shock waves generated from a wing flying at supersonicspeeds result in both of high wave drag and sonic-booms, whichare disadvantageous for supersonic flight. In addition to theseproblems, issues concerning the shift in the wing’s aerodynamiccenter (A.C.) should be resolved for the practicality of supersonicflight. The authors would like to show some interesting resultsrelated to the A.C. of the biplane in this appendix.

For considering a monoplane wing, the A.C. is at the 25% chordduring subsonic flights (i.e.MNo1.0) and at the 50% chord duringsupersonic flights (i.e. MN41.0). Therefore, the trim control ofthe supersonic airplanes becomes complex and difficult. In thecase of the Concorde, eleven fuel tanks were equipped so that fuelcould be transferred from tank to tank mid-flight. The fuelmovement would accommodate for the change of theA.C. location due to change in speed. Thus, the shift of the A.C.location of a biplane should be investigated for a wide range offlight speeds. In this appendix, a two-dimensional analysis will bediscussed [59,60].

A.1.1. Biplane and diamond airfoils

To identify the A.C. location, CFD computation is applied to themodels shown in Fig. 13 (Section 3.2). In order to comparethe biplane’s A.C. with the monoplane’s, both the Busemannbiplane and the diamond airfoil were numerically analyzed.Fig. A1 shows both airfoils and their coordinate axes. The chordlength is 1.0, and the leading and trailing edge positions are 0.0and 1.0, respectively. The airfoil thickness is 0.1; for the biplane,the thicknesses of both of upper and lower elements are 0.05, for atotal thickness of 0.1. The distance between the upper and lowerelements of the biplane is 0.505. This value was chosen topromote the favorable shock and expansion-wave interaction atthe designed Mach number of 1.7 [60].

CFD flow simulations were conducted to obtain lift andpitching moment coefficients for the 27 cases for each of airfoil.Utilizing those lifts and pitching moments A.C. locations are

Fig. A1. Airfoil models.


determined [59,60]. The 27 cases contain a subsonic range ofMach numbers from 0.2 to 0.6 and a supersonic range from 1.7 to2.0 with angles of attack of 01, 11 and 21. To determine the A.C.location, we assumed that the A.C. would be located along thex-axis as shown in Fig. A1. From the study, it was found that theA.C. of biplanes remained within 25–27% chord at both insubsonic and in supersonic ranges. On the other hand, the A.C.location of diamond airfoils was 28% chord for flow speeds ofMach 0.2–0.6, and 44–45% chord for Mach 1.7–2.0.

A.1.2. Relation between an A.C. location and load distributions

Further investigation was conducted to relate the A.C. locationand the load distribution (lift) along a chord axis. The surface Cpdistributions are displayed with the airfoil geometry in Fig. A2.The figure includes four simulation cases: the diamond airfoil atthe speed of MN¼0.5 (a), the case of the biplane airfoil at thespeed of MN¼0.5 (b), the case of the diamond airfoil at the speedof MN¼1.7 (c) and the case of the biplane airfoil at the speed ofMN¼1.7 (d). Each case has three different Cp distributions for the

Fig. A2. Surface Cp distributions along the chord. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of

this article).


angle of attack of 01, 11 and 21, respectively. In the diagrams (a)and (c) of diamond airfoils the pink lines indicate the upper-surface Cps while the blue lines represents the lower surface Cps.In (b) and (d) of biplanes case the pink lines indicate the upper-element Cps while the blue ones indicate the lower element Cps.For the biplane cases, it can be seen that in subsonic flows, thelower element generates the lift whereas in supersonic flows, theupper element generates the lift.

The load distributions along the chord were next studied.Using the Cp distributions, the load distribution graphs are

produced and graphed in Fig. A3. Like Fig. A2, four differentcases are compared here. Each case consists of their loaddistributions along the chord at angles of attack of 1.01 and 2.01as well as their load variation distributions when the angle ofattack increases from 0.01 to 1.01 and from 1.01 to 2.01. Focusingon the load variations, we note that in case (c) of diamond airfoil,the rear half of the airfoil generates lift, causing the A.C. locationto shift to near 50% chord. However, in the other cases, the rearhalf of the airfoil does not generate lift, resulting in the A.C.location around 25% chord.

Fig. A3. Load distributions along the chord.


Therefore, the A.C. of Busemann biplanes designed forMN¼1.7 remains fairly constant, at 25% chord, both in subsonicflows of MN from 0.2 to 0.6 and in supersonic flows of MN from1.7 to 2.0.

References

[1] Kusunose K, New A. Concept in the development of boomless supersonictransport. In: Proceedings of the first international conference on flowdynamics, Sendai, Japan, November 2004, p. 46–7.

[2] Kusunose K, Matsushima K, Goto Y, Yamashita H, Yonezawa M, Maruyama D.et al. A fundamental study for the development of boomless supersonictransport aircraft. AIAA Paper, AIAA-2006-0654, January 2006.

[3] Maruyama D, Matsushima K, Kusunose K, Nakahashi K. Aerodynamic designof biplane airfoils for low wave drag supersonic flight. AIAA Paper, AIAA-2006-3323, June 2006.

[4] Kusunose K, Matsushima KM, Maruyama D, Yamashita H, Yonezawa M.A study of 2-D supersonic biplanes for the development of nearly boomless-supersonic transport. In: Proceedings of the 44th aircraft symposium, Omiya,Japan, October 2006 [in Japanese].

[5] Kusunose K, Matsushima K, Goto YM, Maruyama D, Yamashita H, YonezawaM. A study in the supersonic biplane utilizing its shock wave cancellationeffect. Journal of The Japan Society for Aeronautical and Space Sciences2007;55(636):1–7. [in Japanese].

[6] Kusunose K, Matsushima K, Obayashi S, Furukawa T, Kuratani N, Goto Y, et al.Aerodynamic design of supersonic biplane, cutting edge and related topics.The 21st century COE Program. International COE of flow dynamics lectureseries, vol. 5. Sendai: Tohoku University Press; 2007.

[7] Busemann A. Aerodynamic lift at supersonic speeds. Luftfahrtforschung, Ed.12, No. 6, October 3, 1935, p. 210–20.

[8] Liepmann HW, Roshko A. Elements of gas dynamics. New York: John Wiley &Sons, Inc.; 1957 p. 107–123, 389.

[9] Ferri A. Elements of aerodynamics of supersonic flows. New York: TheMacmillan Company; 1949 (Also published from Dover Phoenix Edition,Dover Publications, Inc., New York, 2005), p. 154–160.

[10] Licher RM. Optimum two-dimensional multiplanes in supersonic flow.Report No. SM-18688, Douglass Aircraft Co., 1955.

[11] Japan Aerospace eXploration Agency (JAXA). Introduction to UPACS /http://www.ista.jaxa.jp/res/c02/upacs/index.htmlS.

[12] Nakahashi K, Ito Y, Togashi F. Some challenge of realistic flow simulations byunstructured grid CFD. International Journal for Numerical Methods in Fluids2003;43:769–83.

[13] Ito Y, Nakahashi K. Surface triangulation for polygonal models based on CFDdata. International Journal for Numerical Methods in Fluids 2002;39(1):75–96.

[14] Ferri A. Experiments at supersonic speed on a biplane of the Busemann type.Translated by M. Flint, British R.T.P. Translation, No. 1407, Ministry of AircraftProduction, 1944, p. 1–40.

[15] Furukawa T, Kumagai N, Oshiba S, Ogawa T, Saito K, Sasoh A. Measurement ofpressure field near Busemann’s biplane in supersonic flow. In: Proceedings ofthe annual meeting and the seventh symposium on propulsion system forreusable launch vehicles, Northern Section of the Japan Society forAeronautical and Space Sciences, vol. G-17, 2006 [in Japanese].

[16] Matsuo K. Compressive fluid dynamics. Rikogakusha, Tokyo, 1994, p. 70–96,142–71 [in Japanese].

[17] Matsushima K, Kusunose K, Maruyama D, Matsuzawa T. Numerical designand assessment of a biplane as future supersonic transport. In: Proceedings ofthe 25th ICAS congress, ICAS 2006-3.7.1, 2006, p. 1–10.

[18] Maruyama D, Kusunose K, Matsushima K. Aerodynamic characteristics ofa two-dimensional supersonic biplane, covering its take-off to cruiseconditions. International Journal on Shock Waves 2009;18(6):437–50.

[19] Maruyama D, Matsushima K, Kusunose K, Nakashi K. Aerodynamic design ofbiplane airfoils for low wave dreg supersonic flight. In: Proceedings of the24th AIAA applied aerodynamics conference, AIAA Paper, AIAA-2006-3323,2006.

[20] Maruyama D, Matsuzawa T, Kusunose K, Matsushima K, Nakahashi K.Consideration at off-design conditions of supersonic flows around biplaneairfoils. In: Proceedings of the 45th AIAA aerospace sciences meeting andexhibit, AIAA Paper, AIAA-2007-0687, 2007.

[21] Yamashita H, Yonezawa M, Obayashi S, Kusunose, K. A study of Busemann-type biplane for avoiding choked flow. AIAA Paper, AIAA 2007-0688, 2007.

[22] Yamashita H, Obayashi S, Kusunose K. Reduction of drag penalty by means ofplain flaps in boomless Busemann biplane. International Journal of Emerging.Multidisciplinary Fluid Sciences 2009;1(2):141–64.

[23] Yamashita H, Obayashi S. Global variation of sonic boom intensity to seasonalatmospheric gradients. AIAA Paper, AIAA 2010-1389, 2010.

[24] Yamashita H, Obayashi S. Sonic boom variability due to homogeneousatmospheric turbulence. Journal of Aircraft 2009;46(6):1886–93.

[25] Yonezawa M, Yamashita H, Obayashi S, Kusunose K. Investigation ofsupersonic wing shape using Busemann airfoil. AIAA Paper, AIAA 2007-0688, 2007.

[26] Yonezawa M, Obayashi S. Reducing drag penalty in the three-dimensionalsupersonic biplane. Journal of Aerospace Engineering 2009;223(7):891–9.

[27] Odaka Y, Kusunose K. Interference effect of body on supersonic biplanewings. In: Proceedings of the 40th fluid dynamics conference/aerospacenumerical simulation symposium, 2008, Paper 1B10, June 2008 [in Japanese].

[28] Odaka Y, Kusunose K. Effects of a bulky fuselage on aerodynamiccharacteristics of supersonic biplane wings. In: Proceedings of the 40thJSASS annual meeting, April 2009, p. 68–73 [in Japanese].

[29] Maruyama D, Nakahashi K, Kusunose K, Matsushima K. Aerodynamicdesign of supersonic biplane—toward efficient supersonic transport. In:Proceedings of the CEAS 2009, European Air and Space Conference,Manchester, UK, 2009.

[30] Maruyama D, Matsushima K, Kusunose K, Nakahashi K. Three-dimensionalaerodynamic design of low wave-drag supersonic biplane using inverseproblem method. Journal of Aircraft 2009;46(6):1906–18.

[31] Odaka Y, Kusunose K. A fundamental study for aerodynamic characteristics ofsupersonic biplane wing and wing–body configurations. Journal of JapanSociety of Aeronautical Space Sciences 2009;57(664):217–24. [in Japanese].

[32] Maruyama D. Aerodynamic feasibility study of supersonic biplane. Disserta-tion for the degree of Doctor of Philosophy (Engineering), AerospaceEngineering, Tohoku University, July 2009.

[33] Furukawa T, Kumagai N, Oshiba S, Ogawa T, Saito K, Sasoh A. Measurement ofpressure field near Busemann’s biplane in supersonic flow. In: Proceedings ofthe annual meeting and the seventh symposium on propulsion system forreusable launch vehicles, Northern Section of the Japan Society forAeronautical and Space Sciences, Sendai, Japan, March, 2006 [in Japanese].

[34] Kuratani N, Ogawa T, Yamashita H, Yonezawa M, Obayashi S. Experimentaland computational fluid dynamics around supersonic biplane for sonic-boomreduction. AIAA Paper, AIAA 2007-3674, 2007.

[35] Kuratani N, Ozaki S, Obayashi S, Orawa T, Matsuno T, Kawazoe H. Experimentaland computational studies of low-speed aerodynamic performance and flowcharacteristics around a supersonic biplane. Transactions of the Japan Society forAeronautical and Space Sciences 2009;52(176):89–97.

[36] Nagai H, Oyama S, Ogawa T, Kuratani N, Asai K. Experimental study oninterference flow of a supersonic biplane using pressure-sensitive painttechnique. In: Proceedings of the ICAS 2008, 26th international congress of theaeronautical sciences ICAS/ATIO 2008, Anchorage, Alaska, USA, 2008, p. 1–9.

[37] Toyoda A, Okubo M, Obayashi S, Shimizu K, Matsuda A, Sasoh A. Ballisticrange experiment on the low sonic boom characteristics of supersonicbiplane. AIAA Paper, AIAA 2010-0873, 2010.

[38] Kashitani M, Yamaguchi Y, Kai Y Hirata K, Kusunose K. Preliminary study onlift coefficient of biplane airfoil in smoke wind tunnel. AIAA Paper, AIAA2008-0349, 2008.

[39] Kashitani M, Yamaguchi Y, Kai Y, Hirata K, Kusunose K. A study of theBusemann airfoil in smoke wind tunnel. Transactions of the Japan Society forAeronautical and Space Sciences 2010;52(178):213–9.

[40] Ferrari C. Recalling the Vth VOLTA congress: high speed in aviation. AnnualReview of Fluid Mechanics 1996;28:1–9.

[41] Busemann A. The relation between minimized drag and noise at supersonicspeed. In: Proceedings of the Conference on high-speed aeronautics,Polytechnic Institute of Brooklyn, January 20–22, 1955, p. 133–44.

[42] Walcher VO. Discussion on aerodynamic drag of wings at supersonic speedusing biplane concept. Luftfahrtforshung 1937;14:55–62. [in German].

[43] Lighthill MJ. A note on supersonic biplane. British A.R.C. R.&M. No. 2002,1944, p. 294–8.

[44] Moeckel WE. Theoretical aerodynamic coefficients of two-dimensionalsupersonic biplane. NACA T.N. No. 1316, 1947.

[45] Lomax H, Sluder L. A method for the calculation of wave drag on supersonic-edge wing and biplane. NACA-TN no. 4232, 1958.

[46] Bushnell DM. Shock wave drag reduction. Annual Review of Fluid Mechanics2004;36:81–96.

[47] Kulfan RM. Application of favorable aerodynamic interference to supersoinicairplane design. SAE Technical Paper Series 901988, 1990, p. 1–19.

[48] Taylor RM. Variable wing position supersonic biplane. US Patent 4,405,102,US Navy, 1983.

[49] Gerhardt HAA. Lifting shock wave cancellation module. US Patent 4,582,276,Northrop Corp., 1986.

[50] White FM. Viscous fluid flow. 2nd ed. New York: McGraw-Hill, Inc.; 1991p. 429–30.

[51] Nakahashi K, Ito Y, Togashi F. Some challenge of realistic flow simulations byunstructured grid CFD. International Journal for Numerical Methods in Fluids2003;43:769–83.

[52] Ito Y, Nakahashi K. Surface triangulation for polygonal models based on CFDdata. International Journal for Numerical Methods in Fluids 2002;39(1):75–96.

[53] Van Wie DM, Kwok FT, Walsh RF. Starting characteristics of supersonic inlets.In: 32nd AIAA/ASME/SAE/ASEE joint propulsion conference, AIAA 96-2914,July 1996.

[54] Spalart PR, Allmaras SRA. One-equation turbulent model for aerodynamicflows. AIAA Paper, AIAA 92-0439, January 1992.

[55] Ogoshi H, Shima E. Role of CFD in aeronautical engineering (15). SpecialPublication of National Aerospace Laboratory, SP-37. In: Proceedings of the15th NAL symposium, Tokyo, Japan, June 1997, p. 81–6.

[56] Matsushima K, Takanashi S. In: Fujii K, Dulikravich GS, editors. An inversedesign method for wings using integral equations and its recent progress,Notes on numerical fluid mechanics, vol. 68. Wiesbaden: Vieweg & Sohn;1999. p. 179–209.


[57] Matsushima K, Maruyama D, Nakano T, Nakahashi K. Aerodynamic design oflow boom and low drag supersonic transport using favorable waveinterference. In: Proceeding of the 36th JSASS annual meeting, April 2005,p. 130–3 [in Japanese].

[58] Matsushima K, Maruyama D, Matsuzawa T. Numerical modeling for super-sonic flow analysis and inverse design. In: Proceedings of the lectures andworkshop international—recent advances in multidisciplinary technologyand modeling, JAXA-SP-07-008E, 2008, p. 185–94.

[59] Noguchi R, Maruyama D, Matsushima K, Nakahashi K. Study of the aerodynamiccenter and aerodynamic characteristics of supersonic biplanes in a wide Machnumber range. In: Proceedings of the 22nd CFD symposium by the Japan Societyof Fluid Mechanics, vol. B6-1, December, 2008, p. 1–7 [in Japanese].

[60] Noguchi R. Study of aerodynamic center and low boom for realization of thenext generation supersonic transport. Thesis for the degree of Master ofSciences (Aerospace Sciences), Department of Aerospace Engineering, TohokuUniversity, January 2010.


progress in aerospace sciencespop.h-cdn.co/assets/cm/15/06/54d151700212d_-_biplane_jpass.pdf ·...

Documents