Theory of optimum shapes in free surface flows. Part 2: Minimum drag profiles in finite cavity flow

J. Fluid Mech. (1972), vol. 55, part 3, pp. 457-472 Printed in Great Britain

Theory of optimum shapes in free-surface flows.

Part 2. Minimum drag proffles in infinite cavity flow

By ARTHUR K. WHITNEYt

California Institute of Technology, Pasadena, California

(Received 8 September 1971 and in revised form 6 July 1972)

7

The problem considered here is to determine the shape of a symmetric

two-dimensional plate so that the drag of this plate in infinite cavity flow is a

mini-mum. With the flow assumed steady and irrotational, and the effects due to

gravity ignored, the drag of the plate is minimized under the constraints that the

frontal width and wetted arc-length of the plate are fixed. The extremization

process yields, by analogy with the classical Euler differential equation, a pair of coupled nonlinear singular integral equations. Although analytical and numerical

attempts to solve these equations prove to be unsuccessful, it is shown that

the optimal plate shapes must have blunt noses. This problem is next formulated

by a method using finite Fourier series expansions, and optimal shapes are ob-tained for various ratios of plate arc-length to plate width.

1. Introduction

We consider the two-dimensional cavity flow of an incompressible fluid past

a symmetric plate of arbitrary shape. The flow far upstream is uniform with

velocity U, pressure p, and density p. The pressure p inside the cavity is

assumed to be a constant, so that by Bernoulli's law the fluid velocity at the cavity wall is a constant V, where

pV2±p = pU2+p.

The cavity flow may be characterized by the cavitation number

= (PuPc)/PU2.

As the cavitation number decreases, the length and width of the cavity grow

indefinitely and the flow approaches Helmholtz flow', in which the cavity is

infinitely long and the cavity pressure equals the free-stream pressure (o-= O).

The specific problem considered here is to find the shape of a symmetric plate (see figure 1) of given wetted arc-length s and given frontal width y0, so that the drag of this plate in infinite cavity flow (o = O) is a minimum. A precise definition of the class of plates under consideration will be given in § 2. The solution of this problem has obvious applications in the design of struts or other two-dimensional non-lifting surfaces which may operate in the super-cavitating range. For flows

t Present address: Lockheed Palo Alto Research Laboratory, Lockheed Missiles and

Space Co., Palo Alto, California. 2 APR. 1973

..ah. y. Scheepsbouwkwd

ARCHIEF

Technische Hogeschool

Deilt

458 A. K. Whitney

I

z plane

X

FIGURE 1. The physical plane. Note that SA and S'A' are parts of the body boundary that are exposed to the same constant pressure p, as on the cavity boundary.

at large Reynolds number, the viscous effects may be ignored as a first approxima-tion; however, corrections due to viscous drag can be calculated once the potential

flow is known. Although optimal shapes are sought for the case o- = 0, these

shapes should remain approximately optimal for o-> 0, since the rule CD(o-) = CD(0)(l+o)

is known to relate approximately the drag coefficient CD at o' = O to the drag coefficient of the same body for O < o 1 (see e.g. Gilbarg 1960). Thus, to

minimize CD at a given o'> 0, we could just as well minimize CD(0). It should be

noted that this rule appears to be a good approximation only for blunt bodies,

so the above argument may be less accurate when the arc-length becomes much

larger than the width of the plate.

Lavrentieff (1938) gave the solution to a related minimum drag problem (see e.g. Gilbarg 1960), under the constraining condition that the plate is confined to lie within a rectangle which circumscribes the nose and ends of the plate. If the nose of the plate is at the origin and the plate ends are at (x0, ± the solution for the optimal profile (see figure 2) was found by Lavrentieff to consist of a fiat nose section, of breadth h and the free streamlines ensuing from the ends of this

section, as a continuation of the body, to pass through (x0, ± y0), the flat nose

length h, being uniquely determined for given x0 and y0. The pressure difference

across the flat portion of the plate is the only contribution to drag, since the

fluid pressure equals the cavity pressure (p = P) on the free streamlined sections

of the plate. This solution was obtained by the use of several comparison and

monotonicity theorems which follow from the maximum principle for harmonic functions.

The present work was originally conceived as an extension to Lavrentieff's problem; but it was hoped that the general problem with varied constraining

conditions could be solved by using the variational calculus method introduced

earlier by Wu & Whitney (1971). It may be noted that if the condition that the

plate be confined to lie within the rectangle is dropped, one can easily construct a sequence of plate shapes which, in the limit, have zero drag (neglecting viscous

U

Theory of optimum shapes in free-surface flows. Part 2 459

-vo

FIGURE 2. The Lavrontieff profile.

X0

X

X

FIGUItE 3. A sequence of plate shapes tending towards one with vanishing drag.

effects). Such a sequence is ifiustrated in figure 3. A typical plate consists ofan

inverted cap of width h1 and length h2 plus the free streamlines which issue from

the ends of the cap and go on to pass through the corners of the rectangle. All

other plate shapes in this sequence are found by decreasing h1 and increasing h2

so that the free streamlines always pass through (x0, ± As h1

-

O, the flow

inside the cap becomes a dead water region with stagnation pressure

= pU2+pn,

so that the drag of the plate is just p U2h1, which can be made arbitrarily small by proper choice of h1. Note that the pressure difference across the back face of

the cap is the only contribution to drag.

This observation led the author to consider the problem described earlier. It

was thought that by fixing the arc-length of the plate, shapes such as those

described above (with vanishing drag) would be eliminated.

f=AU2

I' A'

-c

1

0

Fiouxx 4. Transformation from the complex potential plane to the plane.

2. The problem of the symmetric cavitating plate

The class of flows under consideration is limited to those flows with an infinite cavity past a plate (see figure 1) of frontal width Yo' total arc-length s, having a continuous slope except at the nose where the vertex angle is 2, with O

a

It is further possible to have a rear stretch of the plate (SA and S'A' say) at the

cavity pressure p = PC while it remains everywhere wetted; in other words, SA

and S'A' are free streamlines. In addition, we assume that the pressure p on the

plate satisfies p e PC.

The last condition p PC is an obvious statement of the fact that the vapour

pressure (assumed to equal p) is the minimum pressure in the flow. The sections SA and S'A' are included since free streamlines have already been shown to make up part of Lavrentieff's profiles, and similar results are expected for the present problem. It is more convenient than not accounting for this expectation from the beginning. Note, also, that the assumption imposes no undue restrictions on the problem, since the actual locations of S and S' are not known apriori, but must be

found as part of the optimization requirement.

By proper choice of origin and magnification, the complex potential plane

f = ç + ir is mapped to the upper half = + i

plane (see figure 4) by

o A. K. TVhitn,ei 11 S A S' .4' +1 +C f plane Ç plane 0

where ç is the velocity potential, Lr the stream function, and A a real, positive

constant which is chosen so that S'OS maps to 1. The sections SA and S'A'

map to i

o, where o i (equality holds only if S = A, S' = A'); the

re-maining sections of the free streamlines lie on o s ¡ cc.

We shall adopt the same notation as used by Wu & Whitney (1972) in part i

S .4 I

The hodograph variable w() may be split into two parts, w()=w0(') + w1(),

where w0 accounts for the singular behaviour of w at the stagnation point =0, and w1( F1 + i,81, forj j o 1) is analytic in the entire upper half plane including

the real axis. It can be shown, in fact, that

2a

w()

= log {{(2 - i ) + i]/} + w(),

where (- 1)1 is defined in the

plane cut along the real axis 1 and is

positive for

= >

1; the logarithm function is defined to be that branch which is real for a real, positive argument, with a cut along the negative real axis of the

argument. Letting

- +

iO, J 4J 1, in (9) and comparing this with (6), we have 2a

F()

=

_{__log{[1+(1_2)i]/4T}+F1(),}

(lOb)

so that the proper logarithmic singularity of F

= Re w = log (U/q) and the proper jump in flow angle, fl(0 + ) - fl(0 ) =

2, are exhibited at

= 0. From

the conditions required of w on the free surface and at infinity, and from our

choice for w0, it readily follows that

= Re{w1(+iO)} = O

(jj >

1); (11)

as

jj-a.

(12)

Finally, F in (lOa) must be chosen so that inequality (7) is satisfied. Also, (8) and (lOa) imply

F1(±1)=0.

(13)

(10 a) Theory of optimum shapes in free-surface flows. Part

of this paper (hereafter referred to as part 1). In particular, the complex w and the logarithmic hodograph w are

w= df/dz=

u-iv

=qe10,

w()

=log(U/w) = log(U/q)+iO

The pressure p is given by Bernoulli's law

= p(U2q2) =

pU2(le-2).

On the free streamlines SI and S'I', p p; therefore, by (4),

r(,0+)

= O

(Ii

1).

If the boundary value of w on S'OS is denoted by

(

then, sincep is the minimum pressure in the flow, the inequality

2 461 velocity (2) (3) (4) (5) 1), (6) (7) (8)

F()

O _(I 1),

follows by (4); and, since the pressure is continuous at S and S',

['(±l)=O.

462 A. K. Whitney

The function w is best determined for the so-called inverse problem by re-garding either F1() or

_fl1()

as a known function of .This information, together

with (11) and (12), determines w1 uniquely. We now suppose that F1 =Rew1

is given which satisfies (13) and is Hölder continuous (see part 1 for the definition)

onj

1. The Dirichlet problem for w1,

w(i) =

J1

F(t)dt

(20) lT

Re{w1(+ iO)) fF1() for igi (O for

jj>1, J

(14)

together with (12), has the solution

(ù()

_f1F1(t)dt

(15)

which may be verified by letting

_{- +iO(II}

1) in (15) and using Plemelj's formula (e.g. see Muskhelishvili 1953, §17). The imaginary part of w on S'OS

is found to be

1 1 F1(t)dt

(I1),

(16)

whereÇdenotes the Cauchy principal valuo. The Hölder continuity of_fi1follows from the assumptions already made on F1. (Note that if F1 does not satisfy (13),

but approaches a non-zero value at an endpoint, then ,8, as given by (16), will have a logarithmic singularity at that endpoint.) By (2) and (3), the plate shape and the cavity boundary are given parametrically by

z()

=

x()+iy()

=

Af ed;

(17) and, since the plate is symmetric, its frontal width is given by

Yo =

ImAfe(dc

(18)

It is convenient for subsequent analysis to convert this expression for y0 to an integral from - i to + 1. To do this, we first continue w() into the lower half

plane by w()

= - w().

Next, we consider the function which appears in the integrand in (18). This function is uniquely determined by the jump in its

value (due to the discontinuity of w) across the cut 1 and by its expansion for large j (see Muskhelishvili 1953,§78). In fact, it can be shown that

=

J'

d+fl

F(t) dt + , (19)

in which the first integral exhibits the correct discontinuity across the cut

i

and the last two terms are required by the expansion of the left side of

(19) for 3. 1 (see Whitney 1969). In this expansion, we use the integral

Theory of optimum shapes in free-surface flows. Part 2 463

which is found by exactly the same procedure as that used in determining tù1()

(see (14) and (15)). By substituting (19) into (18), and noting that/J(

-

₌

-we obtain

Yo =

{f'

tsinh I'(t) [sinfl(t)+ cosß(t) log

(L)]

dt+

2cf' r(t)dt}

(21) An element of plate arc-length ds is found from (17), (5), and (6) to be

IAe'jJdT for

1,

ds = IdzJ =

IAId

for ¡ > 1. Thus, the total arc-length of the plate is given by

so

fc ds = A [(c2_1)+f1e1'Itjdt].

(22) The complex force acting on a plate element dz is given by

dF = (pp)(idz),

or, by (4), (5), (6), and (17),

(II e 1),

and dF = O for J > 1. By integrating this expression, noting again the asym-metry of fl(), we obtain for the drag D

D = ,oU2Af1tsinh

F(t)sin/3(t)dt, (23) and the lift vanishes, as we should expect for a symmetric shape at zero angle of incidence. An alternative expression for the integral which appears in (23), and also in the expression for the width (21), may be obtained by substituting (20) in

the left side of (19) and expanding this equation for large _.By matching the

coefficients of ' on both sides of this expansion,we obtain the identity

r'

t sinh F(t) sin ß(t) dt =

_{- (}

F(t) dt) . (24)

J-i

/

Since p and U are kept constant in the minimization of the drag, it is convenient

to set D* = D/pU2 so that D* has the dimension of length. By (23) and (24)

we have

D" = 4(Ç['(t)dt)2.

(25)

3. The minimum drag problem

In § 2 we showed that the problem of minimizing the drag of a symmetric profile of given width and arc-length reduces to finding functions F) and fl,() and constants A, e and ,such that D* in (25) is a minimum subject to the

con-straints (21) and (22), in whichYo and are given fixed quantities. The functions

464 A. K. Whitney

angle of the plate by (10); furthermore, fi,() is related to F,() by (16), in which

I', satisfies the end conditions (13) and is chosen so that F in (lOa) is positive.

Equivalently, we may state this problem in terms of F(), fi(e), A, and c, and omit

further reference to F1(), ß1(), and ; however, the discontinuous behaviour of

I' and fi, as exhibited in (10), should still be recognized. By letting

-- +i0

(II

1) in (20), wo obtain the identity

_' F(t)dt

_-

_{H[I'(t)] (26)}

which may also be verified by (10) and (16). In the above, the symbol LI denotes the finite Hilbert transform. Finally, in (21), (22), and (25), the factor A is a real, positive constant and the parameter e, which determines the location of the

end-points of the plate in the plane, satisfies i c < cx.

This problem is equivalent to finding a pair of extremal arcs, F() and fi(e),

which satisfy (26) and minimize the functional I{F(), ¡3(e); A, c] = D*

-

- 2irA,y0

ri

A

= A f(F(t), ¡3(t), t; A1, A,, e) dt + -(1_A,) _I ['(t) dt) , (27 a)

J-1

\J-1

/

where, by (21), (22), (24) and (25),

f(F(), ß(),

; A,, A,, e) = - A,[(c' - 1) + eT()II]

- 2A,[sinh F() cosfi() log {(c- )/(e + )}+ 2cF()].

(27 b)

In the above, A,, A, are Lagrange multipliers, and the integral identity (24) has

been used in the expression (21) for

Yo-The general variational problem of this type has been investigated earlier

by Wu & Whitney (1971, 1972). For the present problem the method of solution

will follow the same approach with some slight modifications. Let the set

{F(), fi(e), A, e} denote the optimal solution and let {F(), /), A, } be an arbi-trary neighbouring set which also satisfy (21), (22), and (26). The differences

6F() = F-F,8ß() = g-/?,8A = A-A,8c = -c,formasetofsmallvariations;

6A, 8e,, and either 8F or 8/3 are independent of one another since by (26), 8F and

8/3 are related by

_{fi() = - I![8F(t)].}

(28)

The variation of the functional 1, about its optimal value, due to the variations

{8F, 8/3, 8A, 8c} is given by

= I[F+ÔF,fi+8fi;A +8A,c+&c]-1{F,fi;A,c]

=ô1+82I+...,

where, by (27), the first variation 81 and second variation 821 are

81 = IA8A

+J6c+Af {[í+i _A2)J] ÔF(t) +ffl8fi(t)}dt,

(29)

8'I = 21 AC8A

8C+I(8C)2+28Af {[fr+(1 _A2)J]or+f8/3}dt

/'l 2A

+ A {frr(8fl' + 2frfl8F&/3 +ffiß(8fi)'} dt + (1_A,) (8J)', (30) .1 -1

Theory of optimum shap&s in free-surface flows. Part 2 465

where the subscripts denote partial differentiation, and

(1

J

_{= I}

F(t)dt, _{8J =} 61'(t)dt. (31)

J-1

J-1

For I to be a minimum, we must have 61 = O and 821> 0 for arbitrary 6A, 6c,

and 8F(). Therefore, the three terms in the expression (29) for 81 must vanish

separately to ensure 61 = O. The first of these requires 'A = = I/A = O, or

f1 f(F(t),fl(t), t; A1, A2, c)

dt4

(1 À2) J2 = 0, (32a)

or, by (27a) and (31), D* = A1s0+2A2y0. (32b)

The second equation comes from eI/.c = O, or by (27 a, b),

f'fcd1t =

Aic_2Azf'

= 0. (33)

Finally, for the last integral of (29), we substitute (28) for 6fl, then change the

order of integration, giving

f1

fF+(1_A2)J+1][fß]}6F(t)dt

= 0. (34)

Now, since 6F() is arbitrary, we obtain the nonlinear singular integral equation A1,À9,c)+&[f(F,fl,t;A1,A2,cfl = - (1 _A2)fF(t)d.t, (35a)

where, by (27 b),

fr = - A1 e')jj - 2A2 k cosh F() cosfl() log (--) +

2c],

f = 2A2 sinh F() sin fl() log

(

-This integral equation, which contains À1, A2, and e as parameters, is to be solved together with the linear integral equation (26) for the extremal arcs F() and j3(). The Lagrange multipliers A1, À2 can be determined from (32) and (33). The para-meter e is most conveniently determined from the relation

s0/y0 = k (36)

say, where k is a specified quantity, and y0, s are given by (20), (21). Finally, the optimal drag coefficient (based on plate width) is given by

CD = D/pU2y0 = D*/y0, (37)

in which D* assumes the form (25). (Note that actual calculation of the

para-meter A is curtailed as A drops out of the expressions for CD and o/Y0. It is neces-sary, however, to consider the variation 6A in order to obtain (32), which relates 5o/Yo and CD.)

As explained previously, this optimal drag coefficient is a minimum if the

second variation 621 > 0. Upon using (24) to derive an alternative expression for

the second variation of J2, the last two terms on the right-hand side of (30)

30 _{FLM 55}

466 A. K. Whitney

become of the same form and can be readily combined. Further noting that I = O implies automatically _'Ac= 0, since I =

= IJA, the inequality

2J> O now reduces to

A(8c)2JÇdt+AJ{grr(&F)2+2grfiF8fl+gßfi(8fi)2}dt> _{0, (38)}

where g

f+ 2A(1 - A2)sinh F() sinj9(). Since A > 0, this inequality holds

if both integrals are positive. The first term is positive when, by (27),

f'fccd1t

= A1+2A J'

t2sinhF(t)cosß(t)dt 0 (39)

1

(C2 -t2)2

It has been shown by Wu & Whitney (1971, 1972) that a necessary condition for the second integral in (38) to be positive is that

grr+gßß = À,e"(I > o

( < 1),

or simply À, < 0, (40)

since F is real. Once an optimal solution is found, conditions (39) and (40) may be checked to determine if the solution is actually a minimum.

The singular integral equation (35) can be reduced to an integral equation with

a regular kernel by using the identity

cosh F () cosß() = fl[tsinh F(t) sinß(t)] + ,

(41) which can be obtained by letting

-

± i0( 1) in (19). Finally, substitution of cosh F cos/3 given by (41) into (35) yields

2À2J' t smb ['(t) sinfl(t) K(t,

; e) dt À,e')I

= 2À9(2c+1og{(c)/(c+)}--(1

_À0)f

F(t)dt, (42a)

where

K(t,; e) =

1log{(ct) (c+)/(c+t) (cg)}/(t).

(42b)

We note that the kernel K(t, ; e) is regular even at t =

It is now possible to show that, if an optimal shape exists, it must have a fiat

nose, i.e. = w. To show this, we first notethat by (42) e"(jI possesses a regular

series expansion about = O of the form

r()jj =c0+c12+...,

(43) where the explicit forms of e0, e,, . . .,etc., are easily found but are unnecessary.

On the other hand, from (lOa) we can deduce that

er(Il =

= (2)24 1-2cthi eri( + 0(1 2(1_/r))

as

-- 0, since F,() is

assumed to be continuous, at = 0, in particular. This

expansion agrees with (43) if and only if cz = , i.e.when the nose of the plate is fiat.

Since there are no known analytical methods for solving the system of integral

Theory of optimum shapes in free-surface flows. Part 2 467

attempts have not been successful; however, in order to illustrate some of the many difficulties which beset such procedures, we briefly mention one of the

schemes that has been tried. First, the integrals in (26), (32), (33) and (42) were

approximated by numerical quadratures which involved the values of F() and ,8() at N points {} from - i to + 1. An initial guess of {F()} was made and the

set {fi()} was then calculated by (26). Next the Lagrange multipliers À1, À2 were

found from (32) and (33), in which the current values of F and fi were used. Finally,

new values of{F(1)}were calculated by solving for F() in (42), and the process was

repeated, with the hope that the iterations would converge. These calculations

were done for arbitrary values of c ? 1, with different c's corresponding to

different ratios s0/y0.

As mentioned above, this method and several similar ones did not work.

Among the more disagreeable features encountered are the following: (i) the

iterations do not always converge; (ii) the values of F(), as given by the solution of (42) for F(), are not always positive, so that (7) is violated; and (iii) the values

of F at = ± lis not zero, so that fi in (26) becomes large as -- ± i and

integra-tion of terms involving sin fi and cos fi by numerical quadrature fails. Corrective steps were taken, such as enforcing F( ± 1) = O at each iteration step, and by in-creasing N; these measures did not help, however.

Note that fi(e) in (26) has a logarithmic singularity at

= ± i if

F( ± 1) = const. + O,

and fi has a higher-order (branch-type) singularity if F itself is singular at the

endpoints (e.g. see Muskhe]ishvili 1953, §29). By studying the linear case,

in which the functionals are quadratic in F and fi (see Wu & Whitney 1971, 1972), it was deduced that the endpoint condition (7) cannot be satisfied in general. If this conclusion is also true for the present nonlinear case, it is likely that a solution to (26) and (42) will have no direct physical relevance since F and fi will be singular at = ± 1. Nevertheless, such a solution (if one exists)

would provide a mathematical bound for the drag which could then be used in judging the 'degree of optimality' of results obtained by other (approximate)

methods, such as that presented in § 4.

4. Solution by expansions in finite Fourier series

We now investigate a method to obtain an approximate optimal solution by

the expanding F1(g) and in Fourier series in which the constant coefficients are chosen so that the drag is minimized, subject to the same isoperimetric con-straints as before. Let the expansion for F1 be given by

N

asin(2n-1)O,

(44)

n= i

where = cos 0(0 O n). This F1 satisfies (13) and the required symmetry

property. From the identity

sin mçb sin ç! dç

= rcosmO,

o

cosç5cos0

468 A. K. Whitney wo see that (16) is satisfied, term by term, if

N

acos(2n-1)O.

(45)

n1

Upon setting = iT in (9) (as dictated by the result of § 3), we obtain = log{[(2

- i ) + i]J} +

Therefore, by (17), the frontal width of the plate is given by

Yo = -2AImJ Cew1()[(2_

1)+i]d.

(46)

This integral is most easily evaluated by the change of variables

= 4(v+v'),

(47)

which maps the upper half plane into the half circle _i' 1, Im y O (see figures 4 and 5) and maps the endpoints of the plate = ± c to y = K, where

K=c(c2-1).

(48)

It is readily verified that w1 as a function of y is given by

N

w(v) = i

a1v2' - i(v).

(49) n=1

From (46) to (49), the expression for the width now becomes

Yo = Im

_J

°()(V+ 2i - 2v-1 - 2iv_2 + v3) dv,

which is evaluated by taking the path of integration L6 shown in figure 5. In the limit e 0, it can be shown (see Whitney 1969) that

Yo

(Io

[(2t

{2- í'(t)}2\

Ç) + (4+ 1U)

cosc(t)] cit

= t

+ ir(2 - a1)2 +

_

cos ç -- sin (K)1 (50)

K K2 _J'

where '(t) = dc/dt, í"(t) = d22/dt2. By (lOa), (44), and (48), the expression for the arc-length (22) becomes

I ¡'ir N

i

so=AJi(K+K_1)2_1+J exp -asin(2n-i)O(i+sinO)sinOdOj. (51)

L .10 n=1

j

Finally, from (10 a), (25), and (44), the drag is found to depend only on the first of the Fourier coefficients,

= Air(2 -a1)2.

(52)

The optimization problem reduces to minimizing D* in (52), subject to the

constraints (50) and (51), over the (N + 2)-dimensional space (A, K, a1, a9, ..., aN).

For general values of N> 1, this problem must be done numerically; however,

if N = i the integrals in (50) and (51) may be evaluated in terms of special

Theory of optimum shapes in free-surface flows. Part 2 469

Fiotms 5. The path of integration L6 in the y plane from V = i to y = K.

= (v+l/v)

functions. Note that by (lOb) and (45), ¡3(e) =

sgn a1 cos O, for the case

N = 1, so the plate section S'OS is convex (or concave) when viewed from the approaching flow as a1 is positive (or negative). This section is a flat plate,

corresponding to the Lavrentieff profile (see § 1) when a1 = 0.

where Si(x) is the sine integral, L(x) and I(x) are the modified Struve and Bessel functions, respectively (see e.g. Abramowitz & Stegun 1964).

The problem of finding the optimal plate shape from the class of plates with N = i is now equivalent to extremizing

I(A, K, a1) = D*(A, a1) - A1s0(A, K, a1) 2nA2y0(A,K, a1),

where, as before, A1 and A2 are unknown Lagrange multipliers and D*, _{and Yo'}

are given by (52), (54) and (53), respectively. If I is extremal, the three partial derivatives 'A' I, and_'a1must vanish. This gives three relations among the quan-tities A, K, a1, A and A2. By eliminating A1 and A2 from these three equations we

obtain _D*1

_{i D"}

0

55

A L50a1Yoa - 50k-YOaii - a1L5OA Yo, 50,cYOAJ

-where OK as0/aK, etc. Let the solution of (55) be denoted by a1 = f(K). For

K i (e 1) it can be shown, by expansions of(52), (53), (54) and (55), that

a1 =f(K) = {8/(3ir+ 16)}(1 K)2{241r/(31T+ 16)2}(l K)3+O(1 K)4. (56)

With N = lin (49)(51),

(t) = a1t, so that (50) and (51) can be reduced to

Yo = +A{(2ai-2 + 4ai-' - K2) sin (a1K) + (4K' - a1 K1 - 2Kaj1) cos (a1 K)

+ (2 a1)2 {-7T - Si(a1K)]}, (53)

= A[(K+K_1)2_

i

+fe_ais0(l+sinO)sinOdO]

470 A. K. Whitneg

=fi)

01 02 03 04 05 06 07 OES 09 1 0

K

FIaiiaE 6. The curve a1 = f(K) satisfying equation (55).

The general solution, as plotted in figure 6, if found by fixingKat various values between O and i and numerically solving for a1 from (55). AsK -i- O (c -- cc), a1

is found to be the root of the transcendental equation

a(3-2a)

a1{L0(a1)

- I(a)} -

(a + 2) {L1(a1) - I1(al)}

_{= n4}

- a)

This root is given by a1 0.1020, which provides an upper bound for a1; the

corresponding optimal shapes are only slightly curved over S'OS.

This one relation, a1 = f(K),is all that is needed to complete the solution, since

the factor A drops out of the expressions for the drag coefficient and the ratio

So/Yo of arc-length to chord. Thus, by (36) and (37)

CD = CD(al,K) = D*/y0, (57)

k k(a1,K) =s0/y0. (58)

Evaluation of (57) and (58) (in which D*, Yo' and s are given by (52), (53),

and (54)) for a1 = f(K) gives a parametric representation of CD against k. This is plotted in figure 7, where CD = 2ir/(ir +4) is the drag coefficient of a fiat plate in infinity cavity flow (see e.g. Lamb 1932). As k - cc, it can be shown from (57) and

(58)thatCD (4+n/(7T+8k).

The minimum drag proffles for various values of k are obtained by numerically

integrating dz in (17) and, as shown in figure 8, are seen to be quite similar to

Lavrentieff's profiles discussed earlier in § 1. However, by expanding Yo in (53) and s0in (54) for small a1, it can be shown that for (k 1) 1

CD=

{ly(k-1)+O(k-1)},

(59)

where y = 4{2(ir+4)} F0584 (60)

for the Lavrentieff profiles (a1 = 0), and

y = 4{(91r+64)/(1T+4)(37T+16)}l F1641 (61) 010 008 006 004 002

Theory of optimum shapes in free-surface flows. Part 2 471 k = 103 150 A - 05 10

FIGURE 8. Some optimum plate profiles for the case N = 1.

10 I1 12 13

k =

Fioui 7. Minimum drag coefficient against k for the case N = 1.

CD/CD (4+n/(ir+8k) as k-4. Ø

I:

10 08 06 04 02

472 A. K. Whitney

for the proffles in figure 7. Therefore, for k close to unity, the drag coefficients of the proffles in figure 7 are slightly less than those for Lavrentieff's profiles. This is

not too surprising, however, since the constraining conditions in the present

problem differ from Lavrentieff's.

The cases N = 2, 3, ..., could, in principle, be carried out along similar lines

and should result in improved drag coefficients for a given k = .s0/y0. The numeri-cal examples given by Wu & Whitney (1971), in which the exact solutions to the

variational problems are known, indicate that expansion in Fourier series is a very effective method, at least for the case of quadratic functionals. Whether the same holds true for the present problem, in which the functional is of a

dif-ferent type, remains to be seen.

This paper is based on part of the author's doctoral research which was sup-ported by the National Science Foundation and carried out at the California

Institute of Teclmology under Professor T. Y. Wu, whose interest and encourage-ment is gratefully acknowledged. The present work was sponsored by the Naval

Ship System Command General Hydrodynamics Research and Development

Center and the Office of Naval Research, under contract 220(5 1).

REFERENCES

XBRAMOWITZ, M. & STEGUN, I. 1964 Handbook of Mathematical Functions. National Bureau of Standards.

GILBARG, D. 1960 Jets and Cavities. Handbuch der Physik, vol. 9. Springer.

LIB, H. 1932 Hydrodynamics (6th edn.). Cambridge University Press.

LAVEENTIEFF, M. 1938 Sur certaines properiétés des fonctions univalentes et leurs applica-tions à la théories des sillages. Mat. Sbornik, 46, 391.

MuS1rrELISuVILI, N. 1953 Singular Integral Equations. Groningen, Holland: Norcihoff.

WB1TY, A. K. 1969 Minimum drag profiles in infinite cavity flows. Ph.D. thesis, Cali-fornia Institute of Technology, Pasadena, CaliCali-fornia.

Wu, T. Y. & WHImrv, A. K. 1971 Theory of optimum shapes in free-surface flows.

Calif. Inst. Tech. Rep. E 132 F. 1.

Wu, T. Y. & WHITNEY, A. K. 1972 Theory of optimum shapes in free-surface flows. Part 1. Optimum profile of sprayless planing surface. J. Fluid Mech., 55, 439.