US ARMY-MATERIEL

COMMAND

MEMORANDUM REPORT BRL-MR-3523

"DRAG PREDICTIONS FOR PROJECTILES ATTRANSONIC AND SUPERSONIC SPEEDS

Jubaraj Sahu

June 1986

C-,S0~JUL. 1 8 1986L•:

US ARMY BALLISTIC RESEARCH LABORATORYABERDEEN PROVING GROUND, MARYLAND

7 ci

8 ¢., ?' J.'-Jo •,0

Destroy this report when it is no longer needed.

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S.... . ...-22161. - - .. .

The findings in this report are not to be construed as an officialDepartment of the Army position, unless so designated by otherauthorized documents.

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Memorandum Report nRL-MR-3523 j_____-________________"_-A. TITLE (And Subtitle) S. TYPE OF REPORT & PERIO0 COVEREO

DRAG PREDICTION'S FOR PROJECTILES AT TRANSONIC FINALAND SUPERSONIC SPEEDS 4. PERFORMING ORO. REPORT NUMBER

7. AUTHOR(a) S. C.)NTRACT OR GRANT NUMmER(s)

Jubaraj Sahu

S. PERFORMING ORGANIZATION NAIM AND ADORESS 10, PROGRAMd ELEMENT. PROJECT, TASK

US Army Ballistic Research Laboratory AREA A WORýZ UNIT NUMBERS

ATTN: SLCBk-LF RDTE 11 162618AH80Aberdeen Proving Ground, _M 21005-5066

SII. CONTROLLING OFFICE NAME AND ADORESS I2. REPORT DATE

US Army Ballistic Research Laboratory June 1986ATTN: SLCBR-DD-T ,3 NUMBEROFPAGESAberdeen Proving Ground, Marvland 21005-5066 46

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Approved for public release, distribution unlimited

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IS. SUPPLEMENTARY NOTES

IS. KEY WORDS (Continue on ttn s, lde .• f neci.psom Mad Idmeltly by block numa ber)

Na~vier-Stokes Computations Supersonic FlowDrag Components Pressure DragTransonic Flow Viscous DragDesign Codes Base Drag

26. ASUAC1 rcetim.am iii *"arw a-b m Inuemvesso mid~ 57rb block ,Zimbar)

The breakdown of the total drag into its individual components (pressuredrag, viscous drag, and base drag) is important in the preliminary design stageof shell. Design codes are available to predict the individual drag componentsand thus, the total drag. Typically, the totdl drag predicted by these designcodes agrees well with the total drag measured from flight tests. But, howwoi Zldo these codes predict the individual drag components. Experimental verifi-cation of the prediction of the drag components is an extremely difficult tasl,.

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SECURITY CLASSIFICATION OF THIS PAGE (Wn Data Entered)

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20. ABSTRACT (Continued)

Thus, a NavieriStokes computational procedure is used in this report to predictthe individual drag components and test the accuracy of the predictions of thedesign codes. A thin'-layer Navier4Stokes code has been used to compute theentire flow field over projectiles including the base region. Numerical calcu-lations have been made for various Mach numbers in the transonic and supersonicregimes. Pressure drag, skin friction drag, base drag 3r, thus, the total dragare obtained from the computed results. Comparison of dI'ag has been made withavailable experimental data and also with predictions from design codesemploying semi-empirical techniques.

O.

TABLE OF CONTENTS

LIST OF ILLUSTRATIONS ........ s .................................. 5I. INTRODUCTION ................ 7

II. COMPUTATIONAL TECHNIQUE ........................................... 7

III. nESIGN CODES ............................................... I....... 10

A. NSWCAP ............. as.......................................... 10

B. MCDRAG .................... . ........ ...................... 11

IV. RESULTS ........................................................... 13

A. SOC Configuration ............................................. 13

S. SOCBT Configuration ........................................... 14

C. M549 Projectile ............................. 15

V. CONCLUDING REMARKS .... ....... .*so* #................................ 16

REFERENCES ........................................................ 43

DISTRIBUTION LIST ............... ......................... 45

6. o

QU1ALIry

INSP~ Ten ~CodeS

3r

LISf OF ILLUSTRATIONS

Figure e

1 Model Geometry (SOC) ..................................... 17

2 Expanded View of the Grid Near the SOC Projectile .............. 18

3 Expanded View of the Grid in the Base Region ................... 19

4 Variation of Pressure Drag Coefficient with Mach Number,a = 0, SOC .................................................. ?0

5 Variation of Viscous Drag Coefficient with Mach Number,= , SOC .................................................. 21

6 Variation of Base Drag Coefficient with Mach Number, a = 0,SOC ..................... ... ............... 22

7 Base Drag vs Mach Number at Transonic Speeds, a = 0 ........... 23

8 Variation of Total Drag Coefficient with Mach Number, a = 0,SOC..... ... ....................... ........................... 24

9 Model Geometry (SOCBT) ....................................... 25

10 Expanded View of the Grid for the SOCBT Projectile ............ 266

11 Expanded View of the Grid in the Base Region ................... 27

12 Variation of Pressure Drag Coefficient with Mach Number,a = 0, SOCBT, M = 0.8 to 3.0 ................................ 28

13 Variation of Pressure Drag Coefficient with Mach Number,a = 0, SOCBT, M = 0.9 to 1.2 ................................ 29

14 Variation of Viscous Drag Coefficient with Mach Number,0= 0, SOCBT .............. .................... #...... 30

15 Variation of Base Drag Coefficient with Mach Number, a = 0,SOCBT, M a 0.1 to 3.0 ....................................... 31

16 Variation of Base Drag Coefficient with Mach Number,a = 0, SOCBT, M - 0.9 to 1.2 .......................... 32

17 Base Drag Comparison for SOC and SOCBT, a = 0 .................. 33

18 Variation of Total Drag Coefficient with Mach Number, a = 0,SOCBT ......................... .. ............................ 34

19 M549 Projectile ................................................ 35

20 Computational Model ............................................ 36

5

LIST OF ILLUSTRATIONS (Continued)

Figure Page

21 Physical Grid for M549 Projectile ...... ..................... 37

22 Velocity Vectors, M. - .9, 0, M549 ......................... 38

23 Pressure Drag r'koefficient vs Mach Number, a 0 0, M549 .......... 39

24 Viscous Drag Coefficient vs Mach Number, a 0, M549 ........... 4025 Base 'rag Coefficient vs Mach Number, a 0, M549 .............. 41

26 Total Drag Coefficient vs Mach Number, a = 0, M549 ....... ..... 42

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I. INTRODUCTION

One of the most important aerodynamic performance characteristics forshell is the total drag. The total drag for projectiles can be divided intothree components: (i) pressure drag (excluding the base), (ii) viscous (skinfriction) drag, and (iii) base drag. The base drag is a major contributor tothe total drag, particularly at transonic speeds. Thus, the determination ofbase pressure is essential in predicting the total drag for projectiles. Thebreakdown of the total drag into various components is important in the pre-liminary design stage of shell. This information can aid the designer to findpotential areas for drag redurtion and achieve a desired increase in rangeand/or terminal velocity of projectiles.

Design codes are available that can predict the individual drag compo-nents and thus, the total aerodynamic drag. Typically, the total dragpredicted by these design codes agrees well with the total drag measured fromflight tests. But, how well do these codes predict the individual drag com~po-nents for a projectile? It is difficult to measure the individual drag compo-

nents experimentally (especially the base drag and skin friction drag) andverify the accuracy of these predictions. Recently developed Navier-Stokescomputational procedures are capable of predicting all the individual drag

components for projectiles and can be used to determine how well the designcodes predict the individual drag components. This report describes such aNavier-Stokes computational study to predict the individual drag components ofprojectiles and determine the accuracy of the predictions from availabledesign codes.

II. COMPUTATIONAL TECHNIQUE

The Azimuthal Invariant (or Generalized Axisymmetric) thin-layer Navier-Stokes equations for general spatial coordinates F,, n, 4 can he written as'

a 3 G + H =Re-" (1)

where F = 4(x,y,z,t) is the longitudinal coordinate

n -- n(y,z,t) Is the circumferential coordinate

S= ;(x,y,z,t) is the near normal coordinate

T = t is the time

and

i I. C.J.Ni'etubicz, r.H. PuZliam, and J.L. Steger, "VureriCa7 SoZittion f h

AzimuthaLoInvayr-ant Navier-Stokes Equations," US Arrmj Ba"7itict -7esear'ch

Laboratory, Aberdeen Proving Ground, 4aryLand, AR1RL-TR-1222-, k1arch1980. (AD A085716) (Also see AlAA -Jour-nal, Vol. Z3, No. 72., Dece.-ber

. 1980, p . 1411-1412)

I. - - . - . . . . " . • •7

pU pW

Ou puU+ Xp puW+ xP

q J" PV E= J pvU+4yp G =J" pvW+4yp

Pw owU++zp pwW+czp

e (e+p)U-&tp (e+p)W-ýtp

01U 0 (

H = J" 0V[R(U - + R(W -t

-pVR(V - nt)- p/R

0

h'. 2(4x + + ;2)u4 + (p/3)(;u U + v4 + W )r

x y z) x y (4/)~u z, xp(;x2 + ;z + 4x)v• + (P/3)(;xU{ + yV¢ + zW)y

S = U(;2 + C+ ÷ 2 )w + (/3)(xU + yv + w)cx y z C x y4 C Z CZ

((;2 + ;2 + ;[ 2 + W2 )) +pr-V(2 2 +)-'(a 9 )x y z

+ (1,/3)( xu + 4yV + .zw)(;xuc + y VC + w

The velocities

U =t + x u + &y V + •zw

V =n +nU nyv + nzW (2)

W + ; U +4 v + ;zWt x y z

represent the contravariant velocity components.The Cartesian velocity components (u, v, w) are nondimensionalized with

respect to a. (free stream speed of sound). The density (P) is referenced

8

I

to P. and total energy (e) to Pa.2. The local pressure is determined using

the equation of state,

P - (Y - 1)[e - 0.Sp(u2 + v 2 ÷ w2)] (3)

where y is the ratio of specific heats.

In Equation (1), axisymmetrlc flow assumptions have been made which

result in the source term, H. The details o, how this is obtained can befound in Reference 1 and are not discussed hp. Equation (1) contains onlytwo spatial derivatives. However, it re! -ins all three momentum equations andallows a degree of generality over the standard axisymmetric equations. Inparticular, the circumferential velocity is -.v assumed to be zero, thusallowing computations for spinning projectilEs tru ,e accomplished.

The numerical algorithm used is the Beam-Warming fully implicit, approxi-mately factored finite difference scheme. The algorithm can be first orsecond order accurate in time and second or fourth order accurate in space.Since the interest is only in the steady-state solution, Equation (1) issolved in a time asymptotic fashion and first order accurate time differencingis used. The spatial accuracy is fourth order. Oetails of the algorithm areincluded in References 2-4.

-4

For the computation of turbulent flows, a turbulence model must be sup-plied. In the present calculations, the two layer algebraic eddy viscositymodel developed by Baldwin and Lomaxs is used. In their two-layer model, theinner region follows the Prandtl-Van Driest formulation. Their outerformulation can be used in wakes as %ell as in attached and separated boundarylayers. In both the inner and outer formulations, the distribution ofvorticity is used to determine length scales, thereby avoiding the necessityof finding the outer edge of the boundary layer (or wake). The magnitude ofthe local vorticity for the axisymmetric formulation is given by:

2. J.L. Steger, "Implicit Finite Difference Simulation of Flow AboutArbitrar,i Geometries with Application to Airfoils," AIAA Journal, Vol. 16,-No. 7, JuLy 1978, pp. 679-686.

3. T.H. Pu7.Zi-n and J.1L. Steger, "On Implicit Finite-Difference Simulationsof Three-Dimensional Flow," AIAA Journal, Vol. 18, No. 2, February 1980,pp. 159-167.

4. R. Beam and R.F. War ing, "An Implicit Factored Sche-ie fOr the Jompres-sible Navier-Stokes Equations," AIAA Journal, Vol. 16, No. 4, April .1978',pp. 393-402.

S. B.S. Baldwin and H. Lomax, "Thiz-Layer Approzination and Algebraic k!odele

for Separated Turbulent F ows," AZAA Piper No. 78-257, i976.

9 U

Iu 2 + (3V aW)2 +aw au(.. O = T Tz Ty (7 + (4)

In determining the outer layer length scale a functions

F(y) y ~wJ [I exp(-y'/A4 )] (5)

was used where y+ and A+ are boundary layer parameters.

The thin-layer Navier-Stokes computational technique described above hasbeen used in conjunction with a unique flow field segmentation proce-dure 6 , 7 which allows the entire flow field over a projectile including thebase region flow to be computed An important advantage of this segmentationprocedure lies in the preservation of the sharp corner at the hase; in otherwords, no approximation or rounding of the actual sharp corner at the base ismade. The details of this procedure can be found in References 6 and 7.Since the entire flow field over the projectile is calculated, all theindividual drag components can be computed and thus, the total aerodynamic"drag can be determined.

"Ill. DESIGN CODES

A. 1N1SWCAP

The NSWCAP code 8 , 9 is a semi-empirical/analytical technique whichprovides fast predictions of static and dynamic coefficients of shell attransonic and low to moderate supersonic velocities. This code is relativelysimple to use and is formulated to be a 'design tool'.

6. J. Sahu, C.J. Nietubica, and T.L. Steger, "Numerical Computation of EaseFLow for a Projectile at Transonic Speeds," US Arnmy Ballistic ResearchLaboratory, Aberdeen Prooing Ground, Maryland, ARBRL-TR-02495, June 1983.(AD A130293) (Also see AIAA Paper No. 82-1358, August 1982)

7. J. Sahu, C.J. Nietubica, and J.L. Steger, "Nahier-Stokes Computations ofProjectile Base Flow with and without Base Injection," US Army BaLlisticResearch Laboratory, Aberdeen Proving Ground, Maryland, ARBRL-TR-02532,November 1983. (AD A135738) (Also see AIAA Journal, Vol. 23, No. 9,Septenber 19856, pp. 1348-1355)

8. F.G. Moore and R.C. Swanson, "Aerodynamics of Tactical Weapons to VachNAmber 3 and A 1e- of-Attack 150, Part I - Theor-j and Appication,"NSWC/DL TR-3584, February 197?.

9. F.X. Woore and R.C. Swanson, "Aerodynamics of Tactical Weapons to "!achNwmber 3 and Angle-of-Attack 15r, Part II - Co.-puter Pro-ro- and "Isaae,"NSWC/DL TR 3600, March 1?97?.

10

The total drag is obtained by a linear superposition of pressure drag,skin friction drag and base drag. This code also has the ahility to estimatethe drag due to a rotating band if it is present.

The pressure drag consists of contributions from the nose and the boat-tail. The nose drag is based on empirical as well as experimental data attransonic speeds. The boattail pressure drag is based upon a small distur-bance potential solution. The estimation of nose drag and boattail drag atsupersonic speeds is based on the Van Dyke second-order theory'°.

The skin friction component of the total drag is computed using the modelof Van Driest 1 1 The base drag prediction is empirical. It is assumed thatthe boattail is located after a relatively long afterbody so that theapproaching external flow is at free stream conditions. The base drag isdetermined by the expression

CAB = -CP (M.)(RB/Rref).BA

where Cp (M.) is the base pressure coefficient for a long afterbody with no

boattail. RB is the base radius and Rref is the reference body radius

(usually the radius of the cylindrical section). CPA is based on the dataBA

for a long cylindrical afterbody and a fully turbulent boundary layer ahead ofthe base. The effect of base bleed or rocket exhaust can not be accounted forin this approach.

B. MCDRAG

This is another program1 2 based on semi-empirical technique whichprovides a oick response and is very easy to use. This code is used forestimating the drag of a projectile in the Mach number range of 1.5 to 5.0l.The total drag takes the form

C9 Cp + Cv + CBD0 P 0V 0B

10. M.J. Van Dykes, "The 5imiLarity RuLes for Second-Ortser Subsonic andSupersonic F2.w," NACA Tech Vote 3875, October 1958.

11. E.R. Van Driest, "Turbulent Boundary Layers in :ompressibl.e Fi'i8,".ucurnaZl of the Aeronauticr! SenceS, Vo0. 28, ;Vo. 3, 52, pp. 2145-260.

,2. R.L. vc1oll y, - A Computer Program for E'stimatinqi the Dr--a-Coefficient of Projectiles," US Ar'ny 3allistic Research Laborator.j,Aberdeen Proving Ground, Mary'.nd, ARRK-TR-O2293 r..b.a-' :S61. (AA0981:0)

11

where a total drag coefficient at zero angle of attack

a pressure drag or wave drag coefficient'DP

ai - viscous or skin friction drag coefficient'DV

C base drag coefficient.

The pressure drag consists of drag due to the projectile nose, boattail andthe rotating band (if any). Prediction of the projectile nose drag is basedon analytical theories at supersonic speeds while the transonic nose drag isbased on correlations with experimental data. This is similar to the pro-ced'.re used in the NSWCAP code. The effect of leading edge bluntness isaccounted for in estimating the nose drag. As for the boattail drag, secondorder thecry is used to correlate with experimental data to estimate thiscomponent of drag at supersonic speeds and a similar procedure is used attransonic speeds as well. The drag due to the rotating band is usually verysmall (of the order a few percent of the total drag) and is based oncorrelations with few experimental tests.

The skin friction drag coefficient, C0 v, is given by,

Cv -4CF SW

where CF = skin friction coefficient for a smooth flat plate

SW - projectile wetted surface area (excluding the base)

CF is estimated analytically depeoding on whether the bounc;ary layer islaminar or turbulent on the nose. The flow over the afterbody is assumed tobe turbulent always.

The base drag coefficient, Cr is estimated empirically. The approach

taken here differs from the procedure used in the NSWCAP code. The pressuredrag and skin friction drag are estimated as described previously. These con-tributions are then subtracted out from the measured total drag coefficientswhich are available from free flight data of various projectiles. An averagebase pressure is then inferred from the derived base drag coefficient. Theestimate of base drag coefficient is obtained from the relation,

12

I.

2d 2 -

where P/P.= base pressure obtained from a least square fit of the data which

Includes Mach number and boattail effects

and dB * diameter at the base.

IV. RESULTS

Computations of various drag components, and thus the total drag havebeen made for three projectiles: SOC; SOCBT; and the M549. For the Navier-Stokes computations, solutions were marched in time until the steady stateresults were achieved. Results are now presented for the three casesconsidered.

A. SOC Configuration

This projectile has a 3 caliber (1 caliber 1 1 reference body diameter)secant-ogive nose and a 3 caliber cylindrical afterbody as shown in Figure1. One of the first steps before performing Navier-Stokes calculations is thedetermination of a computational grid. An expanded view of the grid near theprojectile is shown in Figure 2. The grid consists of 114 points in thelongitudinal direction (including 30 points in the base region) and 50 pointsin the normal direction. The grid points in the normal direction arestretched away from the surface exponentially. The clustering near the bodysurface is required to resolve the boundary layer. Additional grid clusteringis used in the longitudinal direction near the nose-cylinder junction and thebase where appreciable changes in the flow field variables are expected.Figure 3 shows an expanded view nf the grid in the base region. Fifty gridpoints are used in the base region from the base corner down to the centerline of symmetry. This is done to provide adequate grid resolution along thebase of the projectile.

The individual drag components (pressure drag, viscous drag and basedrag) are obtained from the computed solutions and their variations with Machnumber are presented next. These results are compared with the predictionsfrom the design codes. As shown in Figure 4, the pressure drag predicted bythe Navier-Stokes code agrees well with the MCDRAG and NSWCAP predictionsexcept at M - 1.2 where a small discrepancy is found. Comparison .)f viscousdrag is shown in Figure 5. Here the viscous drag predicted by the Navier-Stokes code is In very good agreement with the MCDRAG prediction. The NSWCAPcode underpredicts this drag by about 15% in the transonic speed regime. Thiscontribution to the tocal drag is, however, small.

Figure 6 shows the variation of base drag as a function of Mach number.As expected, the base drag decreases as Mach number is increased from 1.2 to3. For this range of speeds the Navier-Stokes code result agrees very wellwith the NSWCAP prediction. MCDRAG overpredicts these results by about 12%.

13

Comparison of the base drag Is shown more clearly in Figure 7. The Navier-Stokes results are Indicated by circles, experimental results 1 3 by triangles,data base 14 results by diamonds and MCDRAG results by squares. The resultsfrom the data base are based on correlation of base pressures obtained from anumber of experi.,ents and other analytical techniques. The increase in basedrag with increase in Mach number in the transonic regime is predicted by alltechniques. However, large scatter in the predictions of base drag exists.The Navier-Stokes result agrees well with the result from the data base exceptat M = 1.1 where it overpredicts the base drag by 12%. The MCDRAG predictionof bose drag is consistently higher (up to 10%.) than the data base result.

The total aerodynamic drag is obtained from the individual components

and is plotted as a function of Mach number in Figure 8. The Navier-Stokesresult compares very well with NSWCAP prediction at all speeds (transonic andsupersonic). The MCDRAG prediction is in reasonable agreement with otherresults at higher Mach n" ibers (M > 1); however, discrepancy up to 20% isfound at low Mach numbers (.9 < M < 1.0). The total drag, as expected,increases as Mach number increases from 0.9 to 1.2 in the transonic speed

.. regime.

B. SOCBT Configuration

The model geometry for this secant-ogive-cylinder-boattail (SOCBT) con-figuration Is shown in Figure 9. It has a 3 caliber secant-ogive nose, a 2caliber cylinder and a 1 caliber, 70 boattail. A computational grid was ob-tained for this configuration and an expanded view of the grid near the pro-jectile is shown in Figure 10. This grid consists of 114 points in thestreamwise direction and 50 points in the normal direction. Grid clusteringin the streamwise direction has been used near the ogive-cylinder andcylinder-boattail junctions as well as the base. Figure 11 shows the expandedview of the grid in the base region and again shows the large number of gridpoints used in this region.

The drag components (pressure drag, viscous drag and base drag) for this

shape are shown in Figures 12 through 18. Figure 12 shows the pressure dragcomparison. At higher supersonic Mach numbers .all predictions are inreasonable agreement. The pressure drag at transonic speeds is under pre.dicted by both the design codes. As shown in Figure 13 the pressure drag

13. L.D. Kayder, "Base Pressure Measurements on a Projeotile Shape at lachNumbers for 0.91 to 1.20," US Army Ballistic Research Laboratoyj,Aberdeen Prooving Ground, Maryland, ARBRL-MR-03353, April 1984. (ADA141341)

14. P.R. Payne and R.M Hartley, "Afterbody Drag, Volume I - Drag of Conicaland Circular Arc Afterbodies without Jet Flow," Final Report,DTNSRDC/ASED-80/10, Bethesda, Maryland, May 1980.

14

obtained from the Navier-Stokes results is compared with experiment"5 and isin excellent agreement with experiment. The viscous drag is compared inFigure 14. The Navier-Stokes result is in good agreement with the MC0RA(Gprediction at all Mach numbers from .9 to 3.0. The NSWCAP code prediction isin reasonable agreement at supersonic Mach numbers and is underpredicted atall transonic Mach numbers. Comparison of the base drag is shown in Figure15. As shown in this figure, the Navier-Stokes code prediction is in goodagreement with the predictions from the design codes at higher Mach numbers (M> 1.5). Large discrepancies are seen between the predictions at transonicspeeds. Both design codes MCDRAG and NSWCAP grossly overpredict the base dragin this speed regime. The Navier-Stokes result is compared with experiment 13

and the MCDRAG result in Figure 16. Although there is a small discrepancybetween the numerical result and the experimental data, the trend i.e.,decrease and increase of base drag with Mach number seen experimentally isclearly observed in the computational results. MCDRAG, on the other hand,does not show the correct trend and grossly overpredicts the base drag exceptat M - 1.2. NSWCAP, predicts even higher base drag at transonic speeds andthus, is in worse agreement with the Navier-Stokes result and the experiment.The computational result and the experiment show a negative base drag (orthrust) at Mach numbers 0.94, 0.96 and 0.98 which is not predicted by thedesign codes.

Figure 17 shows the variation of base drag with Mach number for both SOCand SOCBT configurations. These are the computational results and the reduc-tion in base drag due to the boattail is clearly seen. Figure 18 shows thetotal drag as a function of Mach number. The Navier-Stokes result is comparedwith the design code predictions. The agreement is good at all Mach numbersexcept between 1.1 and 1.2 where about 15-20% discrepancy is found between thedesign code predictions and the Navier-Stokes result.

C. M549 Projectile

The R549 Is a modern, low drag, Army artillery shell. The geometry ofthis shell is shown in Figure 19. It has a 3 caliber ogive nose, an approxi-mately 2 caliber cylindrical section, and a .59 caliber, 7 1/20 boattail.Certain simplifications have been made on this shape. The flat nose wasmodeled as a hemisphere cap and the rotating band was eliminated. As aresult, a modified configur'ation, shown in Figure 20, was obtained and usedfor the numerical computations.

An expAnded view of the computational grid near the projectile is shownin Figure 21. This grid has 114 points in the streamwise direction and 40points in the normal direction. Grid points are clustered near the surfaceand also in the base region where large changes In the flow variables areexpected. Figure 22 shows the velocity vectors in the base region obtainedfrom the Navier-Stokes calculations. The recirculatory base flow is clearlyevi dent.

15. L.D. Xajneer, "Surface Pressure Measuremente on a Boattail.ed ProjectileShape at Traneonic Speede," US Army Ballietic Research Laboratory,Aberdeen Proving Ground, Maryland, ARBRL-MR-03161, March 1982. (ADA213620)

15

The individual drag components (pressure drag, viscous drag, base drag)and the total drag are shown in Figures 23, 24, 25 and 26, respectively. Thedrag components obtained from Navier-Stokes solutions are compared with theNSWCAP and MCDRAG predictions. The pressure drag predicted by the Navier-Stokes code, NSWCAP code and MCDRAG is in good agreement at low Mach numbers(.9 < M < 1.0). MCDRAG predicts slightly higher pressure drag at higher Machnumbers compared to the Navier-Stokes and NSWCAP predlctions. The viscousdrag predicted by all codes is in good agreement. As for the base drag, theNavier-Stokes result agrees very well with the NSWCAP prediction whereasMCDRAG underpredicts the base drag by as much as 20 to 25%. The total dragcomparison is shown In Figure 26. Again, the Navier-Stokes result is comparedwith predictions from NSWCAP and MCDRAG codes. In addition, these predictionsare compared with the LCWSL 16 data base and are all in good agreement.

IV. CONCLUDING REMARKS

A thin-layer Navier-Stokes code has been used to compute the flow fieldover projectiles including the base region. Numerical computations have been

made for three projectiles (SOC, SOCBT and M549) for various Mach numbers anda 0 -. The individual drag components (pressure drag, viscous drag, and basedrag) and total drag were obtained from computed sclutions. In addition, twodesign codes (NSWCAP and MCDRAG) that employ semi-empirical techniques wereused to predict the drag for projectiles. Predictions from the Navier-Stokescode and the design codes were compared with experiment and/or data baseresults where available.

The viscous drag predicted by the design codes is generally in goodagreement with the prediction from the Navier-Stokes code. However, signifi-cant discrepancies in the prediction of pressure drag and base drag arefound. When pressure drag is underpredicted, the base drag is overpredictedand vice versa. Thus, when the individual drag components are added up toobtain the total drag, the discrepancy is usually small and reasonably goodagreement is found between the predictions of total drag from the Navier-Stokes code and the design codes.

I

16. A. Loeb, "Private Covvunioation," Large Caliber Weapons SystemsLaboratory, ARX•, AMCCOM, Dover, Ne Jersey.

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REFERENCES

1. C.J. Nietublcz, T.H. Pulliam and J.L. Steger, "Numerical Solution of theAzimuthal-Invariant Navier-Stokes Equations," US Army Ballistic ResearchLaboratory, Aberdeen Proving Ground, Maryland, ARBRL-TR-fl2227, March1980. (AD A085716) (Also see AIAA Journal., Vol. 18, No. 12, December1980, pp. 1411-1412)

2. J.L. Steger, "Implicit Finite Difference Simulation of Flow AboutArbitrary Geometries with Application to Airfoils," AIAA Journal, Vol. 16,No. 7, July 1978, pp. 679-686.

3. T.H. Pulliam and J.L. Steger, "On Implicit Finite-Difference Simulationsof Three-Dimensional Flow," AIAA Journal, Vol. 18, No. 2, February 1980,pp. 159-167.

4. R. Beam and RF. Warming, "An Implicit Factored Scheme for the Compres-sible Navier-Stokes Equations," AIAA Journal, Vol. 16, No. 4, April 1978,pp. 393-402.

5. B.S. Baldwin and H. Lomax, "Thin-Layer Approximation and Algebraic Modelfor Separated Turbulent Flows," AIAA Paper No. 78-257, 1978.

6. J. Sahu, C.J. Nietubicz and J.L. Steger, "Numerical Computation of BaseFlow for a Projectile at Transonic Speeds," US Army Ballistic ResearchLaboratory, Aberdeen Proving Ground, Maryland, ARBRL-TR-O2492 , June1983. (AD A13n293) (Also see AIAA Paper No. 82-1358, August 1982)

7. J. Sahu, C.J. Nietubicz and J.L. St.ager, "Navier-Stokes Computations ofProjectile Base Flow with and without Base Injection," US Army BallisticResearch Laboratory, Aberdeen Proving Ground, Maryland, ARBRL-TR-02532,November 1983. (AD A135738) (Also see AIMA Journal, Vol. 23, No. 9,September 1985, pp. 1348-1355)

S. F.G. Moore and R.C. Swanson, "Aerodynamics of Tactical Weapnns to MachNiumber 3 and Angle-of-Attack 150, Part I - Theory and Application,"

INSWC/DL TR-3584, February 1977.

9. F.G. Moore and R.C. Swanson, "Aerodynamics of Tactical Weapons to MachNumber 3 dnd Angle-of-Attack 150, Part II - Computer Program and Usage,"NSWC/DL TR-3600, March 1977.

10. M.J. Van flykes, "The Similarity Rules for Second-Order Subsonic andSupersonic Flow," NACA Tech Note 3875, October 1956.

11. F.R. Van Driest, "Turbulent Boundary Layers in Compressible Fluids,"Journal of the Aeronautical Sciences, Vol. 18, No. 3, 1951, pp. 145-160.

12. R.L. Mc.oy, "Mcrg - A Computer Program for Estimating the DragCoefficient of rTectiles," US Army Ballistic Research Laboratory,Aberdeen Proving Ground, Maryland, ARBRL-TR-02293, February 1981. (ADA098110)

43

REFERENCES (Continued)

13. L.O. Kayser, "Base Pressure Measurements on a Projectile Sahpe at MachNumbers from O.9lto 1.20," US Army Ballistic Research Laboratory, AberdeenProving Ground, Maryland, ARRRL-MR-03353, April 1984. (AD A141341)

14. P.R. Payne and R.M. Hartley, "Afterbody Drag, Volume 1 - Drag of Conicaland Circular Arc Afterbodles without Jet Flow," Final Report,nTNSRDC/ASED-80-10, Bethesda, Moryland, May 1980.

15. LA. Kayser, "Surface Pressure Measurements on a Boattailed ProjectileSahpe h.t Transonic Speeds," US Army Ballistic Research Lahoratory,Aberdeen Proving Ground, Maryland, ARBRL-MR-03161, March 1982. (AnA113520)

16. A. Loeb, "Private Communication," Large Caliber Weapons SystemsLaboratory, ARr)C, AMCCOM, Dover, New Jersey.

I]

IJ

44

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