shaping of axisymmetric bodies for minimum drag in incompressible flow

8/10/2019 Shaping of Axisymmetric Bodies for Minimum Drag in Incompressible Flow

1/8

100 J. HYDRONAUTICS VOL.8, NO. 3

Shaping of Axisymmetric Bodies for Minimum Drag

in Incompressible Flow

Jerome S. Parsons* and Raymond E. GoodsontPur due University, Lafayette, Ind.

andFabio R. Goldschmiedt

Westinghouse Electric Corporation, Pittsburgh, Pa.

A method is presented for the automatic synthesis of minimum drag hull shapes for axisymmetricvehicles of specified enclosed volume and constant speed submerged in incompressible, nonseparat-ing, noncavitating flow at zero incidence. The computer-oriented optimization procedure does notconsider propulsion or maneuvering; drag reduction is accomplished solely through manipulation ofthe vehicle hull shape. The selected optimization formulation is a nongradient algorithm in a finite,constrained parameter space. This study considers an eight-parameter class of rounded-nose tail-boom bodies constrained to be well behaved as determined by previous hydrodynamic experience.The drag model, for nonseparating flows at zero incidence, is based on classical hydrodynamics andconsists of computer programs from work available in the literature; the model is representative of

state-of-the-art drag prediction methods. By exploiting laminar flow while avoiding turbulent sepa-ration, a body with a drag coefficient one-third below the best existing laminar design has been ob-tained. The evidence suggests that the method produces realistic hull shapes useful in engineeringdesign. The optimization procedure is independent of any particular drag model so that the effectson minimum drag shapes due to alternate drag models can be studied.

Nomenclature

a t = component of parameter set at, i = 1, . . ., N.a(Li), a(Ut) - for Complex Method, the temporary lower and upper

parameter boundaries, respectively, used duringthe initial complex generation.C D = drag coefficient de fined as Drag/(l/2 pt/^V 2 / 3 ).C f = skin friction coefficient defined as r w / l/2 p U 2 ).C p - pressure coefficient d efined as p - pm )/(l/2 pll2}.D = vehicle drag. Also maxi mum diameter of body

Y X m ).fr = fineness ratio defined as L/D where D is the maxi-

mum diameter.H = shape factor defined as d*/6.K - number o f vertices in N-dimensional complex figure

of the Complex Method. Normally K = 27V.KI = body curvature at X m defined as d 2Y X m )/dX 2 .f e i = nondimensional curvature at X m defined as

-2X m2 /D)Ki = -2x m

2 fr }KiL where D is themaximum diameter.

f e ( lm) = nondimensional curvature at X m in terms of mid-body coordinates. Defined in Eq. ( 4 k ) .

L - over-all body length.N = number of independent parameters. Equivalently,

the dimensionali ty of the search space.NN = total number of body profile points used in drag

computation.

PF = performance function. Fo r drag minimization PF isth CD

P O ORtR n

Rs

RvRe

r0SSi

s( ia)

Tt

free-stream static pressure.profile radius at X / defined as Y(X/) .radius of curvature at nose defined as l/ d 2X/dY 2 )

atX = 0.

running surf ace -length Reynolds number defined asS u e /v.body volume Reynolds number defined as V^^U^/v.momentum thickness Reynolds number defined as

radial coordinate of a point in the boundary layerdefined as r 0 + y cos a .

nondimensional profile radius at X / defined as2Rt/D = 2 fr Rt/L.

nondimensional radius of curvature at nose definedas 4X m /D

2 )R n = 4x mfr2)Rn/L.

body radius. Same as Y.surface length. Same as x.profile slope at X defined as dY X t)/dX.total surface arc -length.nondimensional profile slope at Xj defined as

[- Xt - X m}/ D/2 - Ri)]Si = [-2fr(x i - x m )/ I ~ r t ) ] St .

nondimensional profile slope at Xj in terms of tail-boom aftbody coordinates. D efined in Eq. ( 4 q ) .

terminal profile radius defined as Y ( L ) .nondimensional terminal profile radius defined as

2 T / D = 2frT/L.tangential velocity component at a point (x,y) in

h b d l


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the same as CD the boundary layer

JULY 1974 SHAPING O F AXISYMMETRIC BOD IES FOR MINIMUM D R A G 101

a = angle between the surface tangent and the body cen-terline in a meridional plane. Search strategy ex-

pansion factor in Eq. (8 ) .' < = boundary-layer thickness defined as the y-distance

fo r which u/u e = 0.995.5* = displacement thickness associated with momentumintegral equation. D efined as

JY>/r 0 (l ~ u/Ue)dyfo r incompressible flow.

e = eddy viscosity.^ 2 , ^ 3 = relative tolerances used in search method0 = momentum thickness associated with momentum

integral equation and Young's f o r m u l a for c o m -puting drag. Defined as

for/r 0 u/u e (1 - u/u e}dyfo r incompressible flow.

= fluid dynamic viscosity.v = fluid kinematic viscosity.p = fluid density.TW = wall shear stress.

I. Introduction

A FUNDAMENTAL interest in the field of hydrody-namics is the reduction of submerged vehicle power re-quirements. This is accomplished, in general, by some

combination of 1) vehicle hull shaping, 2) boundary-layercontrol; e.g., polymer injection 1 or slot suction, 3) effi-cient propulsion; e.g., a wake propeller or a suction slotwith a stern jet, 2 and 4) efficient steering control consis-tent with hydrodynamic stability.

Items 1 and 2 attempt to reduce skin friction and pres-sure drag while item 3 attempts to extract energy lost tothe fluid surrounding the vehicle. A complete systems de-sign simultaneously takes into account hull shaping,boundary-layer control, propulsion, and steering control. 2

But the overwhelmingly complicated nature of the com-plete problem has prohibited the analytical systems de-sign approach. Hydrodynamicists are forced to rely heavi-ly on experimentally based methods to reduce submergedvehicle power requirements.

The present study form al izes a portion of the completesystems problem. Herein is developed a computer-orient-ed optimization procedure for synthesizing vehicle hullshapes producing minimum drag for a specified vehiclevolume and a constant vehicle speed in incompressible,nonseparating, noncavitating flow at zero incidence. Sincepropulsion is not considered, drag rather than power isminimized. The results should guide the hydrodynamicdesigner in selecting a vehicle shape to be combined withthe mechanisms for boundary layer control, propulsion,and steering. Typical applications are those deeply sub-merged vehicles, such as torpedoes and submarines, whichcarry a payload in an enclosed volume at a nominallyfixed design speed.

The hydrodynamic problem considered here is madem o r e tractable by restricting the analysis to the class ofaxisymmetric vehicle shapes without appendages im-

boundary-layer separation region, and in the wake regionfollowing separation. All of these phenomena are of pri-mary importance in drag prediction. In addition, there areother factors which complicate the prediction of drag inpractice. 3 Examples are the effects of ambient turbulence,body surface waviness, and body vibration on the bounda-ry-layer development. With the lack of reliable drag mod-els, all published work to date directed toward the designof low-drag axisymmetric bodies in incompressible flowhas been mainly experimental in nature. Two studies arenoted here.

An experimental study of drag for a systematic series ofaxisymmetric bodies was reported by Gertler in I960. 4

The series, known as Series 58, is systematic in thatfive parameters characterizing the rounded-nose, pointed-tail shapes are perturbed one at a time about a selected parent model. Twenty-four designs were tested by tow-ing models through water at Reynolds numbers up to Rv= 5 X 10 6 (7 ft3 at 18 knots). Not considered in the studywere bodies designed to reduce drag by exploiting laminarflow. It was believed that such flow could not be used ef-fectively at the submarine Reynolds numbers Rv ^ 3 X10 8) to which the designs were to be extrapolated. Thestudy concludes that the resistance of submerged stream-

lined bodies can be reduced by proper geometrical shap-ing, and presents a minimum resistance submarine hullshape for all turbulent flow.

A laminar flow shape fo r torpedo-type Reynolds num-bers Rv ^ 10 7) was developed by North American Avia-tion, Inc., and was reported by Carmichael in 1966. 3 Thepurpose of the study was to determine if significant dragreduction was possible through shape manipulation alone.A fo rm al shape-synthesizing procedure was not developed;rather, one body was designed based on N A C A low drag

airfoil shapes. The one m odel , dubbed the Dolphin, wastested extensively by gravity-powered drop tests in thePacific Ocean. A significant drag reduction was noted, theDolphin having half the drag of a conventional torpedo atsimilar Reynolds numbers. The low drag was achievedprimarily by the Dolphin's ability to maintain a long runof laminar boundary layer. This fact in itself is importantsince prior to the Dolphin testing it was generally accept-ed that no significant amount of laminar boundary layerflow could be maintained at such high Reynolds numbers.While the Dolphin is not a minimum drag shape, it cer-tainly demonstrates the practicality of exploiting laminarboundary layers and body shaping in general to produceefficient low-drag designs, at least in the tested Reynoldsnumber range.

This paper proposes and tests a drag minimization pro-cedure for synthesizing low-drag hull shapes useful to thehydrodynamic designer. The problem divides naturallyinto three parts: the optimization formulation, the mathe-matical description of the vehicle hull shapes, and thedrag model. The method is demonstrated by com par ing adesign with the best experimentally verified design avail-


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102 PARSONS, G O O D S O N , A ND G O L D S C H M I ED J. HYDRONAUTICS

Fig. 1 Example of the eight-parameter class of rounded-nosetailboom bodies with fixed boundary conditions used in thepresent study. Fixed boundary conditions: Y o) 0, dY o)/dx

= -9dY(L)/dX = 0; d 2Y L)/dX 2 = 0.

plex computer algorithm meant that an analytical optimi-zation procedure such as Calculus of Variations wouldhave to have strong positive benefits to justify its beingused. After careful study it was decided that a parametrictreatment provided the best balance of all the consider-ations .

As such, the general function Y(X) is assumed to satisfy

Y(X) = Y 2)

where Ft(X) are a sequence of polynomials, G(x) is chosento satisfy boundary conditions, and a t are parameters tobe determined. For a parameterically defined body thedrag coefficient depends on the number N, the nature ofthe functions Ft(X), and the multiplicative constants a t, i= 1, . . . ,N. This may be expressed as

C _ _ cD[N,a-,F-(X)f i = 1,. , J V ] ( 3)

Once N and the Ft(X)y i = 1, . .. ,N, are fixed, it is possibleto obtain minimum drag shapes for the restricted class

defined by Eq. (2) by manipulating the set of constants a t,i= 1, . .., , inEq. (3).The formulation implied in Eqs. (2) and (3) requires so-

lution by a parametric optimization method, whetherusing classical or other methods. By assuming the solutionform to within a set of variable parameters, i.e., Eq. ( 2 ) ,the optimization results lose a degree of generality thatthe classical variational calculus solution retains. How-ever, the parametric formulation has been adopted as themost feasible procedure for the drag minimization prob-

lem, particularly since the variational calculus approachappears to require an external iterative method to obtainan explicit solution. It is felt that the loss of generality ofthe parametric approach is not significant so long as thenumber of parameters a t, i = 1, . . . ,N, is not unreasonablysmall.

III. Mathematical Description of the Vehicle HullShape

In order to make a direct comparison between the pres-ent method and the best low-drag vehicle design available

mial. The zero slope and curvature at X = L allows a cy-lindrical tailboom extension to be added without loss ofprofile continuity through the second derivative. The bodyprofile, open at X = L, is essentially a truncation of thecomplete body. For this body class, the short tailboomlength seems to be an adequate approximation to the fulltailboom for the purpose of computing the drag coefficientCo since the computed CD approaches a constant valuenearX = L(seeSec.VI).

The nine parameters shown in Fig. 1 are reduced to theeight nondimensional, nonnegative parameters r n , fr, x m ,ki, Xi, n, Si , and t. The analytical expressions for this di-mensionless eight-parameter body class are given below:For 0 < X < X m (forebody):

Y(X)/L =

where

FI(X) = -2*(* - I) 3

F 2 (*).= -x2 (x-l) 2

G(x) = x 2 3x 2 -8* 6)

ForX m < X < X t ( m i db o dy ) :Y(X)/L = (l/2/ r )(r f + (1 -

wherex = X i-X}/ X i-X m )

F I X ) = - 1/2 x\x - I)2

F2(x ) = x - x*(3x2 - 8* + 6)

G(x) = x3(6x2 -15*+ 10)

(4a)

(4b)

(4c )

(4d)

(4e )

(4f )

(4g)

(4h)

( 4 i )

46)

For Xi < X < L (tailboom af tbody):

Y(X}/L = ( r f /2 / r )(l + [ t / r{ ) - l]

wherex=(L

FI(X) = 1 - - 15* + 10)

7*+ 4)

(4n )

(4o)

(4p)(4q)

A number of limiting type constraints, explicit in thedimensionless parameters, are placed on the eight-param-eter class of rounded-nose, tailboom bodies. The con-straints arise from physical limits as well as previous hy-drodynamic experience. The role of the constraints is sig-nificant; they prevent the optimization strategy fromfinding low drag designs which are unrealistic or impracti-cal in an engineering sense. These geometric constraints

i d b l


4/8

JULY 1974 SHAPING O F AXISYMMETRIC BOD IES FO R MINIMUM D R A G 103

constraint boundaries in s i a ) t/n) space 6 (see Fig.2 ) .

The inflection constraints (7-9) due to Granville, 5 are in-

troduced primarily to maintain a smooth vehicle shapewith a well-behaved stream wise pressure distribution.An advantage of Granville's procedure of dividing the

body into sections is now apparent; the two-dimensionalconstraints give geometric insight into the problem. Theconstrained body class is well behaved in that the bodyprofiles are hydrodynamically smooth, i.e., continuousthrough the second derivative and without unwanted in-flections. Other constraints are introduced by the numeri-cal drag model discussed in Sec. IV.

IV. Numerical Drag Model

The complicated physics involved in drag predictioncan be divided into five parts: mainstream inviscid flow,laminar boundary layer flow, boundary layer transition,turbulent boundary layer flow, and turbulent boundary-layer separation. Methods for predicting mainstream in-viscid flow and laminar boundary layers are well estab-lished. But there is less confidence in methods for predict-ing transition, turbulent boundary layers, and separationdue to an incomplete understanding of the fundamentalnature of the physics involved. Even so, reasonably goodmethods relying on some empirical correlations have ap-peared in the literature. We have used the method due toSmith, Cebeci, and their co-workers 7-11 ; however, early inthe project it was decided to develop the optimizationstrategy independent of any particular drag model so thatfuture improvements in drag prediction methods could beincorporated into the minimization program if desired.

Following Cebeci, 7 the flow field and basic notation areshown in Fig. 3. The mainstream flow outside the bound-ary layer is accurately modeled by inviscid potential flowtheory. The boundary layer is laminar from the leadingedge to the transition point A and is assumed fully turbu-lent from point A to the trailing edge. Turbulent separa-tion is not allowed; this requirement is a constraint on theoptimization problem since the drag model is not valid forseparated boundary layer flow. Turbulent boundary-layerseparation is assumed to occur when the computed skinfriction becomes negative causing the iterative solution to

diverge. In addition, to help avoid separating flow, wehave constrained the velocity ratio u e x)/U a>not to exceed1 . 2 i n v a l u e f o r O < X < L.

The mainstream inviscid velocity distribution along thebody surface is computed using the Douglas-Neumannmethod 8 '9 ; viscous displacement effects on the inviscidstreamlines are not considered. The computed inviscidflow is assumed to be a reasonable approximation to thereal mainstream flow so long as turbulent boundary-layerseparation does not occur.

Using the standard no-slip and no-mass-transfer bound-ary conditions the incompressible axisymmetric bound-

NO INFLECTIONON FOREBODY

Fig. 2 Feasible regions for no-inflection constraints on theeight-parameter class of rounded-nose tailboom bodies used in

the present study.

contains an unspecified arbitrary constant.The drag coefficient is computed using Young's f o rm u -

la 13 given as

C n Drag 477

5)

where Co is the drag coefficient, p is the fluid density, U mis the reference velocity, V is the body volume, r 0 is thebody radius, 6 is the momentum thickness, u e is the ve-locity at the edge of the boundary layer, H = 5*/6 is theshape factor, and 6* is the displacement thickness. Thesubscript T.E. denotes quantities at the trailing edge ofthe body. Since r 0 , 6, u e , and H vary along the body, thedrag coefficient may be treated as Co(X). Such a treat-ment is not implied in Young's derivation, but the mono-tonic behavior of Co(X) is of interest (see Sec.

V I ).The validity of the d rag m odel described here is be-


5/8

104 PARSONS, GOO D SON, AND G O L D S C H M IE D J. HYD RONAUTICS

INPUT: N, K 2 < 3 INVISCID VELOCITY RAT/0(.2

Body X-35S E T j = I

R N D O M L Y GENERATE VERTEX j IN N-SPACE O U N D E D BY L . AN D ay. , i = I .... N

C H E C K S T O P P I N G C O N D I T I O N : A R EF I V E B E S T P F V A L U E S W I T H I N < 3

N E I Q H B O R H O O D O F B E S T P F V A L U E ?

F i g . 4 Flow chart of the modified Box-Guin Complex Methodused in the present study.

A discussion of the drag model is given in Ref. 7. Experi-mental drag data are compared with numerical predictionsfor the Dolphin 3 in Sec. VI.

V O ti i ti St t g A M difi ti f B

AXIAL COORDINATE, X

F i g . 5 Comparison of profiles and velocity distributions forthe low-drag laminar body X 3 5 and the Dolphin.

ic gradients dCo/dat, i = 1, . . . ,N , are available; hence, anongradient (direct) search method is used since it is be-lieved that nongradient methods are more efficient with nongradient problems than finite-differencing used withgradient methods. 14

For this project we have employed a constrained directsearch method rather than on of the unconstrainedmethods 15 18 used with a penalty function. 14 '19 An earlymethod for inequality constrained problems is due to Ro-senbrock. 20 In this study, the straightforward Box-GuinC o m p l e x Method 21 '22 with a m odif ied starting procedureis used. Powell's Method 16 with a penalty function hasbeen used as well as the search strategy described here.

Shown in F i g . 4 is a flow chart of the search strategy.No initial guess is needed. The procedure starts by r a n -domly generating K vertices (body shapes) in the c o n -

strained N- dimensional space. That is, vertex; has N r a n -domly generated components

7)j = (rn)[aUf -a L

where i = 1, . . . , 7V, j = 1, . . . ,K, (rn) is a random numberon the interval [0,1], and a(Lt) and a(Ut) a re , respectively,temporary lower and upper parameter boundaries. Thevertices are un i fo rm ly distributed over the N- dimensionalrectangular v o lu m e defined by the boundaries a(Lt) anda(Ui), i = 1, . . . ,N. Normally a portion of the rectangular

volume is outside the feasible region defined by the ex-plicit and implicit constraints (geometric and hydrody-namic); when a randomly generated vertex falls outsidethe feasible region, it is simply thrown out and regenerat-ed. This starting procedure is a m odification of Box'sComplex Method which calls for an unfeasible vertex tobe retracted halfway to the centroid of the partially c o m -pleted complex f igure. The modification was necessary toobtain a well dispersed initial set of K vertices in a n o n -rectangular feasible search space.

Once the initial complex is form ed, the search proceedsby generating the trial vertex at by


6/8

JULY 1974 SHAPING OF AXISYMMETRIC BODIES FO R MINIMUM DRAG 105

Table 1 Geometry and flow data for body X-35 at R v = 107

fr = 4.848805xm = 0.588774

fc i = 0.171086r n = 0.757355

n = 0.6472 98Si = 2.286662

X i = 0.785317t = 0.173127 L/FVa = 3.714341Wetted surface area/F 2 / 3 = (> . 45 1445

X/L Y/L e /L x io3 H Cf X IO4 Rs X 10- R 9 X 10- 4

0.00019 88

0.00223 418

0.006610.01269 289

5 38

0.06329 9414

0.112430.132910.155510.178510.201590.236360.271250.306250.34132

0.376450.411610.44681 482 2

0.51724

55247

0.58770 62292

0.64639 66979

0.69303

0.00155 467

772

0.010660.01347 1878

0.028630.038180.04301 529

5798

0.06315 683

73 4

0.077330.083000.087820.091860.09516

9780.099810.10127 1 224

0.10280 1 3 6

0.10312

0.102910.10210 1 23

9687

0.115550.32147 48467

6 8 7

69484

0.801880.90073 948 6

0.96456 99 92

1.002111.01241

1.021931.030081.036961.045311.05162

1.056161.05916

1.060821.061391.061251 6 94

1.061451 648

1.076921.110661.136601.153881.15319

788

0.00548 6 2

0.00681 766

944

0.01320

1696

0.01882

2261

2456

2657

2864

3 63

0.032560.035380.03818 41

0.04389

4686 4993

0.05308 5624

5924

6167

0.06234 5971

583

5927

6456

2.214532.27812

2.350992.382342.390932.415482.470612 497 4

2 5 424

2.512702.515232.517232.518402 52 23

2.522552.527042.532992.540582.54986

2 56 462.571602.581192 58484

2.573252.527222.398612 23787

2 2 649

2.25221

2 448 6

0.00441488

0.103040.152680.19184 23872

0.27173 28698

0.29350 3 546

0.310690.315120.318800.320380.320190.316720.309720.299590.28683

272220.256710.24183 22994

0.22538 23772

0.30109 4 482

0.41915

35837

0.20100

0.00066 564

0.014540.02623 396

6977

0.14195 23224

0.285660.41272 488 4

57248

0.665850.761010.856551.00021

1 143851.286991.42922

1.570261.709991.848611.986902.126742 27277

2.439532.661312.822602 96616

3.06499

0.003380.006540.010840.015380.019770.028120.044150.05971 6744

0.083210.09142 9993

0.108720.117210.125410.137370.14913 16 84

0.17265

18464 19686

0.20925 22164

0.233570.243910.24935 24631

0.24613 254 4

0.27651

Transition

7 454

0.71595 72726

0.749530.78222 81472

0.847590.881150.915500.95045

0.973910.99576

0.094560.09182 8866

0.081200.06806 5445

0.041770.031080.023370.01913

0.018060.01785

1 144171.12914

1 108491.053950.960070.891520.849250.83255 84268

0.87511

9 4540.93457

0.08787 1 862

0.13570

21 97

0.39712

0.682011.086021.591282.061132 25892

2.178242.01739

1.573301.407411.375731.375561.399041.414811.416291.393031.345641.29339

1.263691.24245

2.449323.276783

08453

2.290891.32631

0.822190.565090.430630.346690.30584

0.313370.33866

3.090923.099553.09121

3.031052.886692.797232.775722 83 6

2 97478

3 2 374

3.390383.58447

37344

45554

0.55870 8259

1.416162.258433.425754 92 82

6.451387.34251

7.318377.00301

the m ethod to start with random ly an d globally distribut-ed inform ation (body shapes) so that it has some chanceof finding a global opt im um in the feasible region, al-though there is no guarantee that such is found. The com-plex figure's ability to roll along boundaries helps toprevent prem ature convergence on a boundary. Since them ethod uses large-scale features of the function surface, itis not sensitive to local irregularities and sm all errors

constraints, when properly used, prevent unrealistic opti-m al solutions. Optim al solutions are no better than them odels used in the optim ization; those solutions restingon constraint boundaries are no better than the assum p-tions behind the constraints. The optim ization strategyused here is independent of both m odel and constraints sothat the effect on the m inim um drag shapes due to changesin either the m odel or constraints can be studied.


7/8

106 PARSONS, G O O D S O N, A ND GOLDSCHMIED J . H Y D R O N A U T I C S

3.6

3.4CD

3.0

2.8

Laminar Separation/Turbulent Reattachment

LAMINAR

DBody X-35

O Dolph in

6.6 7.0 7.4

Log R Q7.8

Fig. 6 Comparison of transition predictions for body X-35and the Dolphin at Rv = 1 7 using the Michel-e 9 correlation.

of a conventional torpedo. 3 The results are subdividedinto hydrodynamic aspects and optimization aspects.

Hydrodynamic Aspects

The design Rv of 10 7 for body X-35 corresponds to theDolphin vo lum e of 5.6 cubic feet at 40 knots in water.Body X-35 with its inviscid velocity distribution and

Co(X) are shown in Fig. 5 along with the Dolphin forcomparison. The C D fo r body X-35 is about one-thirdlower than that of the Dolphin at Rv = 10 7. Geometry andflow data at R v = 10

7 for body X-35 are given in Table 1.The long run of near-zero velocity gradient suggests thatthe forebody is similar to a Reichardt section. The regionof locally accelerated flow at X/L = 0.65, caused by con-tinuous but rapidly changing curvature, suppresses transi-tion until X/L = 0.7. The turbulent boundary layer sur-vives the region of decelerated flow without separatingand enters the terminal accelerated region in which thevelocity u e asymptotically approaches the freestreamvalue. The CD(X) computed by Young's form ula, Eq. (5) ,increases monotonically to the trailing edge value of CD =0.0051.

Transition predictions at R v = 107 for body X-35 and

the Dolphin are shown in Fig. 6 with the Michel-e 9 curve.For body X-35, the laminar Re RS trajectory never en-ters the indicated 10% band; transition is predicted bylaminar separation and assumed turbulent reattachment.By contrast, the Dolphin laminar Re - RS trajectorymonotonically approaches the curve and crosses it atabout X/L = 0.5.

Predicted drag coefficients CD for body X-35 and theDolphin, as well as experimental data for the Dolphin, 3over a range of Reynolds number RV are shown in Fig. 7.The reasonably good agreement between predicted and

roBoundary Layer

Tripped atX L S

O olphin Data[3] Dolphin Prediction01 Body X-35 Prediction

o

8 iI ' -ll i .

a

JT PA R A M E T E R S PERTURBED ONE-AT-A-T1MEV 0t P~ WHILE HOLDING OTHERS AT DESIGN VALUES

Q

.95 1.0 1.05

NORMALIZED PARAMETER PERTURBATION

Fig. 8 Parameter-perturbation studies on the eight-parame-ter body X-35 at its design Reynolds number Rv = 10 7.

experimental values of CD for the Dolphin indicates valid-ity of the drag model at least over the test Rv range. Theexperimental an d predicted C D values for the trippedlaminar boundary layer are in very good agreement andestablish that the Dolphin's low drag is due to extensivelaminar flow. The fact that extensive laminar flow can

exist in the ocean environment at these high Reynoldsnumbers is in itself significant. The predicted CD valuesfor body X-35 are one-third lower than those for the Dol-phin in this Rv range. The Dolphin's demonstration oflaminar flow exploitation combined with the inferred(from Fig. 6) superior laminar stability of body X-35suggests that body X-35 is a realistic improvement overthe Dolphin body.

Optimization Aspects

For the sample run given here, the input values were N= 8,K= 16, e2 = 3 = 1%, Rv = 10

7, and NN = 30. Thesearch strategy found body X-35 at the 35th CD evalua-tion. However, the method was unable to generate a newfavorable search direction and the search was terminatedwithout fo rm al convergence after 80 CD evaluations (80minutes CPU time on CDC 6500 computer). Such perfor-mance suggests that the m odified Box-Guin C o m p le xMethod is a good starting procedure that may be ade-quate for design purposes, but that another procedureshould b e used fo r fo rm al con vergence if it is required.

Perturbation studies on body X-35 at its design Rv -10 7 are shown in Fig. 8. A plus and minus five percentperturbation is made on each dimensionless parameterwhile holding the other seven parameters at their designvalues given in Table 1. The perturbations reveal thatbody X-35 is suboptimal, a consequence of stopping thesearch after 80 CD evaluations without form al conver-gence. The perturbations also reveal that CD is locallyquite sensitive to the maximum diameter location x m and

the inflection point slope St . CD is locally insensitive tothe fineness ratio /> the nose radius r the maximum di


8/8

JULY 1974 SHAPING OF AXISYM METRIC BODIES FO R MINIMUM D RAG 107

proved drag models, different design constraints, or alter-native shape characterizations may be utilized in the de-sign of the minimum drag bodies. Experience with themethod has generated the following specific conclusionsand recommendations:

1) Low-drag axisymmetric body designs exploiting ap-preciable runs of laminar flow have resulted from the op-timization technique and particular drag model describedin this paper.

2) Comparison of the Dolphin 3 experimental bodydata and design and a minimum drag design at the sameReynolds number indicates that the optimization tech-niques generates realistic candidates for low drag bodyshapes with significantly less drag (% less drag for a 5.6ft

3 body at 40 knots.)3) Further work examining body shapes as a function of

Reynolds' number indicates that the optimization tech-nique provides an effective low drag body design mecha-nism for a wide range of parameters.

4) The optimization program has been designed to ac-cept improved drag computations and different design orhydrodynamic constraints.

5) For nonseparating flow on axisymmetric bodies, thedrag model used for this work agrees reasonably well with

data at moderate Reynolds numbers (R ~ 107

).6) Development of more generally applicable drag mod-els is needed.

7 ) Development of better drag models and experimentalverification of the low-drag designs should be jointly un-dertaken. Particular attention should be given to the tran-sition criterion from laminar to turbulent flow and thestability of any given long, laminar run.

8) Further work on the optimization scheme could re-sult in more efficient convergence.

References

1 Granville, P. S., Progress in Frictional D r ag Reduction, Sum-m er 1968 to Summer 1969, Hydromechanics Lab. TN-143, Au-gust 1969, Naval Ship Research and Development Center, Wash-ington, D.C.

2Goldschmied, F. R., Integrated Hull D esign, Boundary-LayerControl, an d Propulsion of Submerged Bodies, Journal of Hy-dronautics, V ol. 1, No. 1, July 1967, pp. 2-11.

3

Carmichael, B. H., Underwater V ehicle Drag ReductionThrough Choice of Shape, AIAA Paper 66-657, June 1966.4Gertler, M., Resistance Experiments on a Systematic Series

of Streamlined Bodies of RevolutionFor Application to the D esignof High-Speed Submarines, DTMB Rept. C-297 , April 1950, NavalShip Research and Development Center, Washington, D.C.

5Granville, P. S., Geometrical Characteristics of StreamlinedShapes, NSRDC Rept. 2962, March 1969, Naval Ship Researchand Development Center, Washington, D.C.

6Parsons, J. S., The Optimum Shaping of Axisymmetric Bod-ies for Minimum Drag in Incompressible Flow, Ph.D. thesis,June 1972, School of Mechanical Engineering, Purdue Univ., La-fayette, Ind.

7

Cebeci, T. and Mosinskis, G. J., Calculation of Viscous Dragand Turbulent Boundary-Layer Separation of Two-Dimensionaland Axisymmetric Bodies in Incompressible Flows, Rept.MDC-J0973-01, Nov. 1970, Douglas Aircraft Co., Long Beach,Calif.

8 Smith, A. M. O. and Pierce, J., Exact Solution of the Neu-mann Problem. Calculation of Non-Circulatory Plane and AxiallySymmetric Flows about or within Arbitrary Boundaries, Rept.ES26988, April 1958, Douglas Aircraft C o. , Long Beach, Calif.

9 Faulkner, S., Hess, J. L., and Giesing, J. P., Comparison ofExperimental Pressure Distributions with Those Calculated bythe Douglas Neuman Program, Rept. LB31831, Dec. 1964, D o u g -las Aircraft Co., Long Beach, Calif.

10 Cebeci, T., Smith, A. M. 0., and Wang, L. C., A Finite-D if-ference Method for Calculating Compressible Laminar and Tur-bulent Boundary Layers, Rept. DAC-67131, March 1969, D o u g -las Aircraft Co., Long Beach, Calif.

Smith, A. M. O. and Gamberoni, N., Transition, PressureGradient, and Stability Theory, Rept. ES26388, Au g. 1956,Douglas Aircraft Co., Long Beach, Calif.

12 Schlichting, H., Boundary Layer Theory, 6th ed., McGraw-Hill, New York, 1968.

13 Young, A. D., The Calculation of the Total and Skin Fric-tion Drags of Bodies of Revolution at Zero Incidence, R. and M.No. 1874, April 1939, Aeronautical Research Council, London,England.

14 Beveridge, G. S. G. and Schechter, R. S., Optimization:Theory and Practice, McGraw-Hill, New York, 1970, Chap. 8.

15 Hooke, R. and Jeeves, T. A. , Direct Search Solutions of Nu-merical an d Statistical Problems, Journal of the Association f orComputing Machinery, Vol . 8, No. 2, April 1961, pp. 212-229.

16 Powell, M. J. D., An Efficient Method for Finding the Mini-mum of a Function of Several Variables without Calculating De-rivatives, Th e Computer Journal, Vol. 7 , No. 2 , July 1964, pp .155-162.17 Wozny, M. J. and Heydt, G. T., A Directed RandomSearch, ASME Paper 70-WA/Aut-7, 1970. (Also available underthe same title as a Ph.D. thesis by G. T. Heydt, Aug. 1970, School ofElectrical Engineering, Purdue University, Lafayette, Ind.)

18 Masters, C. O. and Drucker, H., Observations on DirectSearch Procedures, IEEE Transactions on Systems, Man, andCybernetics, Vol . SMC-1, No. 2, April 1971, pp. 182-184.

19 Fiacco, A. V. and McCormick, G. P., Nonlinear Program-ming: Sequential Unconstrained Minimization Techniques,Wiley, New York, 1968.

20

Rosenbrock, H. H., An Automatic Method for Finding theGreatest or Least Va l u e of a Function, The Computer Journal,Vol. 3, No. 3, Oct. 1970, pp. 175-184.

21 Box, M. J., A New Method of Constrained Optimizationand a Comparison with Other Methods, The Computer Journal,Vol . 8, No. 1, April 1965, pp. 42-52.

22 Guin, J. A., Modification of the C om plex Method of Con-strained Optimization, The Computer Journal, Vol. 10, No. 4,February 1968, pp. 416-417.

shaping of axisymmetric bodies for minimum drag in incompressible flow

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