Ivey, H Reese

Klunker, E Bernard

English

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GOVf. DOC, 
NATIONAL ADVISORY COMMITTEE 
FOR AERONAUTICS 
TECHNICAL NOTE
No. 1371 
CONSIDERATIONS OF ThE TOTAL DRAG OF 
SUPERSONIC AIRFOIL SECTIONS 
By H. Reese Ivey and E. Bernard Kiunker 
Langley Memorial Aeronautical Laboratory
Langley Field, Va. 
Washington 
July 1947
& (E 1t .	 c'I 
JUL 21 147
https://ntrs.nasa.gov/search.jsp?R=19930081999 2020-06-17T21:15:22+00:00Z
NATIONAL ADVISORY COMtEfl FOR AONAtJTICS 
dc:.No	 o. 1371 
CONSIJiERATIONS OF 'I TOTAL ]EAG OF•

SUPSONIC AflDIL SECTIONS 
By H. Reese Ivey and E. Berna'd Kiunker 
SUMMARY 
The, results of calculations of the viscous and pressure drags 
of some two-dimensIonal superson±c : alrfoils at zero lift are presented. 
The results Indicate that inclusion of viscous drag alters many 
previous, results regarding the desirability of certain airfoil shapes 
for securing low drags at supersonic spedo.' At' certain Reynolds and 
Mach numbers, for instance, a circular-arc airfoil may theoretIcally 
have lees drag than the' prevIously advocated. syimnetrical wedge-shape 
profile; although under different conditIons, the circular-arc 
airfoil may have the higher drag. 
Drag calculatIons for 6-percent-thick symnetrIcal circular-arc 
and double-wedge airfoils are presented for Mach numbers o 1.35 
and 1.6 and Reynolds numbers ±'rcm
	 to io8. Unseparated. flows are 
considered and approximate corrections for boundary-leyer and shock-
wave interaction are applied only to the momentum thicithess at the 
trailing-edge shockwave. The theory of viscous supersonic flows 
will have to be extended before. a more exact ana1y1s of the drag is 
possible.
INODUCTION 
Recent experiments indicate that airfoil shapes heretofore 
considered good. for supersonic .speeds may in f.ct be in.ferior to 
pl'ofiles having higher pressure drags. In order to understand better 
the behavior of airfoils at supersonic speeds, It is desirable to 
eliminate arid explain apparent contradictions between e±perlment and 
theory. The drag of thin airfoils may be considered. as the sum of the 
pressure and viscous drags. Although a great deal of work has been 
done on the. calculation of the pressure drag, lIttle theoretical work 
has been attempted on the calculation of the viscous drag. The purpose 
of this paper is to consider the separate effects of viecos arid 
pressure drag on the total drag for two thin airfoil sections.
2
	
NAOA	 No. 1371 
• The pressure drag as deteiytiie approximate linearized 
relations is a minimum for a
	 trica wedge -shape airfoil of a 
given thicimeas ratio with maxim th1ciés located at the 
50 -percent. chord (reference 1). .. Thmoré exac methods of refer-
ences 2 and. 3 show that other loca1 jojis of. the aximum thIcIiees for 
the syetrical wedge -sha pe pro±iie iave. lower .pre6ure drag. On all 
these flat-side profiles, however, the velocity gradient is zero over 
the flat leading and trailing sections of the airfoils and., hence, 
boundary-layer transition and viscoü& drag may be somewhat similar to 
that of a flat plate. Viscous. drags lower than those usually obtained 
for a flat. plate can be obtained, by using a falling-pressure gradient 
to increase the extent of the laminar flow. It appears possible then 
that a curved airfoil, such as a doi.ible .cIic1ar arc, which has a 
favorable pressure gradient may also have more laminar flow than a 
f1at-id,e airfoil and therefore loss vicOuu drag over a certain range 
of Mach aM Reynolds numbers. Although the theoretical pressure drag 
of the circular-arc profile Is higher. than that of the syimietrIca1 
ved.ge'ehäpes, the total drag (viscous aM pressure) of the circular 
arc might possibly be the lower f or certain conditions. 
In order to. demonstrate the effects of viscous and prossure'drag 
on the total drag, calculations were made for two, 6-percent-thick 
airfoil shapes - a double circular arc and asynmietrlcal wedge with 
maximum thicithess at the 50-percent chord. The calculations were 
made for Mach numbers f 1.35 aM 1.6 and, covered a range of Reynolds 
numbers from 10 to 10 
The present paper Is Intended to serve only as a preliminary 
study of the total drag of a two-dimensional airfoil and. neglects 
many faOtors. A more complete analysis would requIre consideration 
of the effects of separation, interaction between the boundary layer 
and. the shock waves, and. angle of attack. Existing theories need 
considerable development before all these factors can be included. 
DISCUSSION 
The drag Of a two-dimensional airfoil at zero lift at suDersonic 
speeds is assumed to be the sum of the pressure drag .aid the drag due 
to viscosity. • The pressure distribution Is calculated by assuming 
the absence of the boundary layer, and, the shock drag Is then readily 
determined from the pressure d,lsti'ibutIon . The boundary 1ayei 
momentum thic1Qiese corresponding...to..thiO.. pressure distribution Is 
calculated.,and. the viscous drag Is 't1en deterxthied from the momentum 
thIc1ie5s at the trailing edge of the aIrfOil. •
NACA	 No. 1371	 3 
The method of reference 1 was used. to determine the shock drag 
as weli as the local values of Mach number, velocity, density, and. 
other conditions along the airfoil. This method requires the 
existence of attached, shock waves and, is therefore restricted to 
sharp-nose airfoils. In the absence of a boundary layer and, flow 
seDaration the calculations are accurate for the wedge airfoil and 
are a close approximation for the circular-arc airfoIl. In a more 
complete analysis the pressure distributIon should be adjusted. for 
changes in the flowpattern caused by boundary-layer thickness end.. 
by sudden changes In slope of the boundary-layer displacement 
thickness due to transition or separation.
	
0 
The calculated pressure distribution Is used. to compute the 
boundary-layer momentum thickness along the airfoil by the method 
of reference 5 . Reference 5 assumes the following: The skin-
friction coefficient is Independent of Mach number and pressure 
gradient; a fixed; velocity profile independent of pressure gradient 
may be used; the Prandtl number is 1; and. no heat conduction occurs. 
Both laminar and. turbulent boundary layers may be
.
 computed. approxi-
matelyby this method with the use of the, appropriate constants 
given in the reference. For the present calculations, transition 
from laminar to turbulent flow 'vas considered to occur suddenly. 
Therefore, in order to àompute the momentum thickness along the surface 
of the airfoil, parameters corresponding to laminar flow were used 
from the leading edge to the point of transition,. and parameters 
corresponding to turbulent flow were used In the equations from the 
point of transition back to the trailing edge. 
Transition from laminar to turbulent flow is dependent on such, 
factors as Mach number, Reynolds number, pressure gradient, stream 
turbulence, and. surface roughness. Determination of transition for 
a given body and flow conditions is therefore difficult. Lees (refer-
ence 6) end Schlichting (reference 7) have Investigated the effect of 
Mach number and velocity gradient, respectively, on the stability of 
the laminar boundary layer by assuming vanishingly small disturbances. 
In the absence of a stability theory which accounts for.both velocity 
gradients arid compressibility, it was necessary to combine the work 
of these references. The boundary-layer thickness for neutral 
stability was considered to be the value for a flat plate in Incom-
pressible flow multiplied, by factors to correct for Mach number arid 
pressure gradient. The criterion used to estimate transition in the 
present investigation was consideration of the neutral stability of 
the laminar boundary layer'. The notation N =. 1 •is used to denote 
transition occurring when the boundary-layer momentum thickness 
reaches the value for neutral stability. If the airfoil. is well 
falred and in a stream of low initial turbulence, transition need not
lj.	 NACA TN No. 1371 
occur until the boundary J.ayer is thicker. For example,tbe transition 
curve N 2 corresponds to transition at a point where the boundary-
layer momentum thicess Is twice that for neutral stability. 
Before a more extended treatment is possible,the stability 
theory must be developed to Include the combIned effects of compressi-
bilIty end pressure gradient. A more complete analysis wIll require 
an Investigation to determine the existence of tranit1on regions, or 
shock waves (end hence changes in the presure distribution) due to 
transition or separation. In the, present paper values of. the momentum 
thickness of the laminar boundary lager wex'e- calculated along the 
surface of. the airfoil and compared with the corresponding values of 
momentum thickness for neutral 'stability. The intersection of the 
curves through these points gIva apcsib1e pbint of instability 
for a given Mach number axid 'Reynolds number. The stability criterions 
indicate that the critical part of the boundary layer from consideration 
of.'possible transition. is well forward. on the curved airfoil. 
The momentum thickness increases in. passing through the trailing 
shock. Approximate corrections for this effect have been supplied to 
the authors by	 Neal Tetervin of , the Physical Research Division. 
The assumptions are that the boundary -layer zacmentum equation applies 
and. that the length over which the pressure rise takes place on the 
- surface. is so short that the skin frIction can be neglected. The 
correction becomes:
- 
(\T\0av+2P 
sv2 ,	 p2 
where 0 is the momentum thickness, V 'is the velocity tangeitia1 
to the boundary layer, U 0	 is the average valixe.for compressible 
av	 .	 . 
flow of the ratio of displacement thickness to momentum thickness 
across the shock, and 'p is the density at the edge: of the.. boundary 
layer. The subscripts 1 and 2 refer to conditions before and after 
the shock,respectively.	 . .	 . 
In he actual' Oase, the interaction of the 'hock wavq, and 
boundary layer results In en 'increase iii the momentum thickne8s, an 
increase in the displacement thIcknes,and a change in the velocity 
profile and may cause flow separation. The effect of separation is 
to reduce the pressure drag and increase the viscous drag. At'low 
Reynolds numbers there may' be considerable separation resulting in a 
total drag less than the theoretical shock drag (reference 8). The 
method used was chosen for lack of amorO exact method. '
NACA TN No. 1371	 5 
PRETATION OF FIGURES 
Figure 1(a) gives the viscous-drag coefficient of aflat plate 
as a function of Reynolds number based 'on chord. The transition 
curves for a Mach number M0 of 1.35 are numbered according to the 
ratio of boundary-layer momentum thic1ese at transition to the 
bounary-1ayer thickness for neutral stability (N 2 to io). 
Pigvres . l(b) and 1(c) give the corresponding curves of viscous drag 
at Mo = I35 for a 6-percent-thick circular-aró airfoil and. 6-percent-
thick double-wedge airfoil, respectively. The transition occurs at 
a very much higher Reynolds. number for the curved airfoil then for 
the flat plate or wedge,
 airfoil. As the Reynolds number Is Increased, 
an abrupt rise In the drag for the circular-arc airfoil Is noticed. 
The reason for'the sudden drag rise is hat the ratio of local 
boundary-layer thickness to the thickness for local neutral stability 
(as given by Schlich-ting 's theory) reaches a maximum well forward on 
this. airfoil so that the critical part of the airfoil (as regards 
stability or transition) is also well forward. A complet wing would 
not necessarIly experience a sharp drag rise as the Reynolds number 
Is increased, since transition may occur at different Reynolds numbers 
over d.Iffereit sections. Parts of the transition curves for the 
wedge are dashed to 1 indlcatê possible theoretical errors introduced 
by the sudden expansion at the niidchord,. The actual curves may be 
somewhat to the left of the dashed curves (closer to flat-plate 
conditions). 
Figures 2(a) and. 2(b) give the vIscous-drag coefficient at 
= 1.6 for the 6-percent-thick circular-arc and wedge airf oils, 
respectively. These curves show again that transition occurs at. a 
much higher Reynolds number on the circular arc than on the wedge; 
however, transition on both types of airfoil occurs at a lower 
Reynolds number at M0
 = 1.6 than at .M0
 = 1.35. (See fig. 1.) 
FIgures 3(a) and 3(b) compare the viscous-drag coefficients of 
the airfoils at M0
 = 1.35 and, at M0 = i.6, respectively, for a 
constant transition number N 5 . These figures indIcate that the 
circular-arc airfoil may have a much lower viscous-th'agcoefficint 
than the wedge over a crtain range of Reynolds numbers. 
The pressure-drag coefficients are determined to be as follows 
(from reference
NACA TN No. 1371 
Airfoil Pressure.drag coefficient 
(6 percent thiàk) M0 = 1 . 35	 M0=f0 
Cjràular arc 
Double wedge
0.0218 
.0160
0.0158 
.0116
Whenthese pressure-drag coefficients are ad,d.edto the viscous-
drag coefficients of figures 3(a) and 3(b) (or in general to figs. 1 
and 2), curves are obtained for the total -drag coefficIents. See 
figures li. (a) and )i. (b). For N = 5 the circular-arc airfoil has at 
least slightly more drag than the wedge for all Reynolds numbers 
investigated at M0 = 1.35; however, the analjais shows that the 
wedge may have the more drag over a certaIn Reynolds number range 
at M0
 = 1.6. The actual comparison obtained d.opend.s on the value 
of. N used for the analysis. Since . N is a function of many 
paremeters,.such as surface finish and streamturbulence, it is 
difficult to assii N a proper value for a given profile. Some 
Information concerning the effect of stream turbulence on transition 
at low speeds is given in reference 9 .	 . 
This paper demonstrates the need, for including both Mach number 
and Reynolds number, as well as auth factors as stream turbulence 
and. surface finish whore possible, In papers of experimental work 
since these factors may influence the interpretation of the data for 
full-scale application. 	 .. 
• It must be kept in mind, that the present paper has compared the 
drags of two airfoil shapes of the same thickness ratio. A cbmparison 
of airfoils giving the same structural etronth or stiffness would be 
more favorable to the curved aIrfoil. 
C0NCLUDDG REMARKS 
An analysIs has been résented. which serves as a preliminary 
study of the total (viscous and pressure) drag of supersonic airfoil 
sections at zero lift. . Within the limitations of the present paper 
certain conclusions have been drawn: 
The relative pressure drags of airfoils at supersonic speeds is 
In general different from the relative total drags even at zero lift. 
The airfoil shape for minimum drag varies with Reynolds number, Mach 
NACA TN No. 1371	 7 
number, turbulehce, surface finish, and other factors, and is not 
necessarIly the shape that would give ni.trimi theoretical pressure 
drag. 
Langley Memorial Aeronautical Laboratory 
National Advisory Conittee for Aercnautics 
Langley Field, Va., May 9, 1947 
IE]TCES 
1. Taylor, G-. I.: Applications to-Aeronautics of Ackeret's Theory 
of Aerofoils Moving at Speeã.s Greater Than That of Sound. 
R. & M. No. 111.67, British A.R.C., 1932. 
2. Busemann, A.: Aerodynamic Lift at Supersonic Speeds. 2814., .Ae. 
Techl. 1201, BrItish A.R.C., Feb. 3, 1937. (irom 
Luftfahtforschun, Bd.. 12, Ni'. 6, Oct. 3, 1935, pp. 210-220.) 
3. Ivey, H. Reese: Notes on the Theoretical Characteristics of 
Two-Dimensional Suersonic Airfoils. NACA TN No. 1179, 1911.7. 
4. Ivey, H. Reese, Stickle, Goore W., and Schuettler, Alberta: 
Charts for Determining the Characteristics of Sharp-Iose 
Airfoils in Two-DiDiensional Flow at Supersonic Speeds. 
NACA TN No. 1111.3, 1947. 
5. Tetervin, Neal: Approximate Formulas or the Computation of 
Turbulent Boundary-Layer Momentum Thicbiesses in Compressible 
F1o•rs. MACA ACR No. L6A22, 1911.6. 
6. Lees )
 Lester: The Stability of the Laminar Boundary Layer in a 
Compressible Fluid. MACA TN No. 1360, 194'. 
7. Schlichting, H.: Der Urns chiag laminar/turbulent und. d.as  
WiderstandsprobleDl des Tra;flçe1s. Forschuncsbericht Mr. 111.10, 
Deutsche Luftfahrforschun (Bravnachweig), 19)41. 
8. Fern, Antonio: Experimental Results with Airfoils Tested in the 
High-Speed Tunnel at Guidonia. MACA TM No. 9 11.6, 191.0. 
9. Schubauer, G. B., and S1amstad., H. K.: Laminar -Boundary -Layer 
Oscillations and Transition on a Flat Plate. MACA ACR, 
April 1943.
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Considerations of the total drag of supersonic airfoil sections

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