naca conical boatail pressure affects

7/29/2019 NACA Conical Boatail Pressure Affects

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TECHNICAL NOTE 2972

THEORETICAL PRESSURE DISTRIBUTIONS AND WAVE DMGS

FOR CONICAL BOATTAILSBy John R. Jack

Lewis Flight Propulsion LaboratoryCleveland, Ohio

WashingtonJuly 1953

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TECHUBRARY tC/WB, NMlM16MlMlllo

1s13CIL5952maw fmmom COMMITIEE~ MROIWUZ!ICS

TKXKKML NOTE 2972

TEIEOREIIICALRESSURE DISZKIBUITONS AND MNll IWWSFoR CONICALomTAm3By John R. Jack

Afterbody pressure distributions and wave drags were calculatedusing a second-order theory for a variety of conical boattails at zerosugle of attack. Results are presented for Mch numbers from 1.5 to4.5, area ratios from 0.200 to 0.800, and boattail angles from 3 to II.

The results indicate that for a given boattail angle, the wave dragdecreases with increasing Mach nuriber~d area ratio. The wave drag,for a constant srea ratio, increases with increasingboattail angle.For a specific l&ch number, area ratio, and fineness ratio, a comparisonof the wave-drag coefficientsfor conical, tangent-parabolic,and secant-parabolic boatta~ showed the conical boattail to have the smallestwave drag.

INIROIXXTIONOne of the major components of a missile configurateon is theafterbody section known as the boattail. There are, however, littledata available to serve either as a guide in the design of boattails

for supersonicbodies or as a basis for esthating the aerodynamic loadsand wave drags associated with boattails. It has generally been assumedthat boundary @er effects render the potential flow computationstoward the rear of the missile meaningless. However, avafible qyzri-mental data indicate that potential theory does predict the boattailchsxacteristics adequately for most design purposes. An investigationwas therefore undertaken at the IIMR Lewis Laboratory to study systemat-ically the variations of pressure distributions and wave drags of ~conical.boattaila with Mach number, area ratio, and boattail angle .

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1)

After completion of the work presented h this report, a report of sim-ibr content came to the attention of the author (ref. 1). However,since reference 1 does not present say pressure dlstributions, whichare quite valuable for structural desiw and for esttit ing base pres-sues, ptilication of the present report was considered warranted.

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Rressure distributions and wave-dragnumbers fran 1.5 to 4.5, area ratiosangles from 3 to U_.

NACA TN 2972b

coefficientsare-presented for &hfrom 0.200 to 0.800, and boattail

MEI!HODOF COMWIMTIONPressure distributions for each boattafi were calculated using thesecond-ofiertheory developed in reference 2. It was assumed that theboattails are preceded by a cylindrical.section of sufficient length togive uniform flow at the free-stresm lhch number at the beginning ofeach boattail. The calculating procedure followed was that presentedti reference 3, in which the approximate boundary condition at the sur-face of the body is used to obtain the perturbation velocities, and theexact isentropic pressure relation is used for evaluating the pressurecoefficient at each point on the bdy. ti each case the solution wascarried downstream to the petit at whSch the radius of the I&ch cone from

the beginnhg of the boattail has grown to ten times the local radiusof the bcattail. As indicated in reference 2, the second-order solutioncould be carried beyond this petit by extending the tables used in thecomputations;however, the area ratios of practical interest correspondto boattai.1.engths which, in general, are within this Mmitat ion. !l?heprocedure presented in reference 3 proves to be an expedient means ofobtatiing both a first- and a second-ordersolution. The average timefor calculating the combined first- and second-order solutions wasapprccdmately 11 cmput er hours. .

Wave-tig coefficientsfor each boattail were obtained by graph-ically integrating the pressure distributio~ over the boattail surfaces.The wave-drag coefficient was based on the maximum area of the boattailand is deftied by

where Cp is the exact isentropic pressure coefficient, r is the localboattail radius, and ~ and R are the maxhum and minimum boattailradii, respectively.

I?XSQECSAND DEXUSSIONPressure distributions and wave drags. - The variations of pressure

coefficient in the axial direction for the conical boattaiM me pre-sented in figure 1 for selected values of boattail angle and free-stresm


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NACA TN 2972

&ch nmiber. In each case, the pressuxethe Prandtl-Meyer value at the beginning

coefficient hasof the boattail

3

a~roxhmt el.ybecause of the

~ans ive turning through the boattail mgl.e, which is fold.owedby aco~tinuous compression along the boattail surf~.ce. The pressurecoefficient level increases with increasingMach nuniberfor a givenboattail angle and, for a given I&h nuniber,decreases as the boattailangle increases.

The dependence of the pressure distributionsupon fineness ratioand area ratio msy be found by correlating boattafl angle with finenessratio and area ratio. The variation of fineness ratio and area ratiowith boattail angle is given in figure 2.

Graphical integration of the pressure distributions presented infigure 1 over the boattail surfaces yields the variation of wave-dragcoefficient with area ratio and with boattail.angle (figs. 3 and 4,respectively). The dashed lines presented in figme 3 represent thelimit@g area ratio to which the theory of reference 3 can be appliedwithout extending the tables used in the computations. The wave dxagcoefficientspresented in figure 4 were obtained by extrapolating someof the data of figures 3(a) and 3(b). For a given lkch number, thecoefficient of wave drag increases with ticreasingboattail angle anddecreases with increasing area ratio.

To extend the I&ch number ~nge investigated, the wave-drag coef-ficient has been plotted as a function of the reciprocal of the I&chnuiber (fig. 5). The results of the present calculationswere extra-pcilded to 1~~ = O by using the concepts of Newtonian flow theorywhich predict a zero drag coefficient for all boattail angles as% +- (ref. 4). For a specific boattail angle, the coefficient ofwave drag decreases as the Mach nuniberis ticreased.

Body contour effect. - To obtain the effect of body contour onpressure distributions and wave drags, a tangent- and a secant-parabolicboattail contour each having the same len@h and area ratio have beeninvesti~ted and the results compared with those obtatied from the cor-respon~g inscribed conical boattail. Design parameters for theseboattails were arbitrarily chosen and are:l%chnmber, ~........ . . . . . . . . . . . . . . ..O .2.5Ftieness ratio, L/~...... . . . . . . . . . . . . . ..2255Arearatio, A/~ . . . ...08. ....~~oo-2ooSecant-parabolicboattail leading-edge angle, deg . . . . . . . . . 3.5Conical boattail angle, e,deg . . . . . . . . . . . . . . . . . ..7The contours considered for the comparison are slike inasmuch as theyme members of the psrabolic fsmily, with each having a cliffrent expan-sion [email protected] at the leading edge of the boattail. Defining equations forthese boattail contours axe as follows:

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4

Tangent-parabolic:

NACA TN 2972

2ii=l ()0.02719 +%Secant-parabolic:

2~=1 ()

- 0.06116 = - 0.01364 ~% mconical:

t=l - 0.1228 ~%Pressure distributions for the tangent- and secant-parabolictails are ccmpared with the pressure distribution for the conicaltail h figure 6. Graphical.integration of these pressure distributions

to obtain the respective wave drags shows the conical.boatt-ail.o havethe least wave drag (~ = 0.0465). The tangent-parabolicboattail hma drag 1.27 times the conical boattail drag; while the secant-parabolicboattail &rag is 1.02 times as large.

SUMMAlwaEAfterbody pressure distributions

l.atedusing a second-ordertheory for and wave drags have been calcu-various conical boattails at zeroangle of a%ck. The following conclusions have been rbached aftermm the res~ts for a ~ch ntier range from 1.5 to 4.5 and forboattail angles from 3 to KLO:

1. For a given boattail angle, the pressure coefficient levelticreased with increasing Mach number and for a given &h numberdecreased as the boattail angle increased.

2. The wave-drag coefficient for a given boattail angle decreasedwith ficreasing I&h number and area ratio. For a given area ratioand increasing boattail.angle, the boattail wave-drag coefficientticreased.

3. For a I&ch nunber of 2.5, an area ratio of 0.200, and a finenessratio of 2.25, a comparison of the wave drags for a conical, a tangent-parabolic, and a secant-parabolicboattail.shows the conical boattailto have the smallest wave drag.Lewis Flight Propulsion LaboratoryNational Advisory Committee for AeronauticsCleveland, Ohio, April.28, 1953

boat-boat-


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WA TN 2972 5

REFERENCES1. Huth, J. H., and Dye, H. M.: Axial and Normal Force Coefficients forPointed Bodies of Revolution at Super- and l&personic Speeds.Psxtu - Boattails. RM-905, U. S. Air Force Proj. RAND, The RAND

Corp., Aug. 5, 1952.2. Van Dyke, Milton D.: First- and Second-Order Theory of SupersonicFlow Past Bodies of Revolution. Joux. Aero. Sci., VO1. 18, no. 3,

Mar. 1951, PP. 161-178.3. Van Dyke, Milton D.: Practicsl Calculation of Second-Order Super-

sonic Flow Past Nonlifting Bcxliesof Revolution. N21CATN 2744, 1952.4. Grinmd.nger,G., Williams, E. P., and Young, G. B. W.: Lift on

hclined Bdies of Revolution in Hypersonic Flow. Jour. Aero. Sci.,vol. 17, no. M_, Nov. 1950, pp. 675-690.

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NACA TN 2972

-. 3

-. 2

-.1

0 1(a) lb - 1.5.

0(c) l% - 3.5.

-.06

-.04

-.02

=*v J . z 5 4 5 6

.7

\

L I 1 I I 1 1 1 1- 1

Axial station, *(d) l% - 4.5.

Fl@re 1. - Variation of pressure coefficient in axial directionfor conical boatt.ails.

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NACA TN 2972 7.

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1.0

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0

Ie/ -L!iIDJ,D

1+----L .+

\ .$=(l.2&my

\ G,deg\ \

3 .\

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2 4 6 8 10Fineness ratio, L/~

Figure 2. - Relations between boattail angle, area ratio, andness ratio.fine-

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.08

.04,

0(a) Mach nuniber,1.5.

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.08 1--~I

\ A

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0 .2 .4 .6 .8 1.0Area ratio, A/k

(b) Mach number, 2.5.Figure 3. - Variation of wave-drag coefficient with area ratio. (Dashedlines represent limiting area ratio for theory of ref. 3.)

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.06

.04

.02

0

.06

.04

.02

0

(c) Mach number, 3.5.

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10 NACA TN 2972

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0 2 4 6 8 10 12E@attall angle, f3,deg(b) Mach nuuiber,2.5.

Figure 4. - Variation of wave-drag coefficient with boattall angle.

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NACA TN 2972 11

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.06

.04

#

.02

0(c) Mach number, 3.5.00, .06

qgs

.04

.02 T

o 2 4 6 8 10 12Wattail angle, 9, deg(d) Mach muiber, 4.5.

Figure 4. - Concluded. Variation of wave-drag coefficient with boattdlangle.

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NAM TN 2972

c!?-.

.16

.12

.08

.04

0(a) Area ratio, 0.2~.

.16 q

U..12

9.08 / /

7/ A

/ /5.04-/ - 3

I0 .2 .4 .6 .8Mach number reciprocal, l/~

(b) Area ratio, O .4WI.Figure 5. - Variation of wave-drag coefficient withreciprocal of Mach nuuiber.

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.12e,degI-1.

.08- A

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.04

0(c) Area ratio, 0.600.

.0611

/9.047

{5

.02r

o

Figure 5.ficfent

.2 .4 .6 .8Mach number reciprocal, l/~

(d) Area ratio, 0.800.- Concluded. Variation of wave-drag coef-with reciprocal of Mach number.


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-.10

-.08

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0

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o +f

Tangent parabolic8ecant parabolic

-Conical

=M[ (Not to scale) I I I

\ / y \~ Tangent-parabola

\

// parabola

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.2 .4 .6 .8 1.0Mal station, xiL

Figure 6. - Pressure distributions over three boattail contours.Free-stream I.kchnumber, 2.5; area ratio, 0.200.

NACA-h@q -7-6-53- 1004

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naca conical boatail pressure affects

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