Approximate Analyses Interrelating Pressure Distribution and Axisymmetric Body Form

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U niv e rs ity of lows

LI 8 K a H Y

A dissertation submitted in partial fulfillment of the require­ ments for the degree of Doctor of Philosophy, in the Department of Mechanics and Hydraulics, in the Graduate College of the State University of Iowa February, 19-50

ProQuest Number: 10991959

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T v ^ o co^, '2L

This dissertation is hereby approved as a credit­ able report on an engineering project or research carried out and presented in a manner which warrants its acceptance as a prerequisite for the degree for which it is submitted. It is to be understood, however, that neither the Department of Mechanics and Hydraulics nor the dissertation advisor is responsible for the statements mad© or for the opinions expressed.

Dissertation Advisor

ACKNOVDIDGMTS Grateful acknowledgments are due to Dr, John S* LfcNown for his continuous advice and helpful suggestions and to Dr. Hunter House* Director of the Iowa Institute of Hydraulic Hesearch for his general encouragement and support.

Gratitude is also expressed to Hr. Emmett VL

Laursen for the special assistance which he so generously provided* to Ifessrs. Wen Heieh and Yu Ch h Soong for their assistance in carrying out the relaxation routines# and to Miss Leona Amelon for her untiring efforts in the preparation of this dissertation. Hie investigation was sponsored by the Office of Kaval Research under Contract H8onr-S00.





THEORETICAL ANALYSIS ............................. Preliminary Considerations ....................... Relaxation Method ....................... Mathematical Analysis ..........................

4 4 7 14

RESULTS AND DISCUSSIONS............................ A Symmetrical Profiles............... G. UnaynaoBtrlcal Profiles............... C Cavitation Pocket F o r m ...................... D. Profile with a linear Pressure Gradient . . . . . . E. Applications..............................

29 29 SI S3 36 37










Cylindrical coordinate system



Characteristics of relaxation elements. (a) General case, (b) Specific case encountered . . . . . .


S. Relaxation pattern for a regular net . . . . . . . . . . . .




Evaluation of velocity at boundary, (a) Velocity cDisponente. (b) StreaEKfunetlon gradient ............... 11


Sketch showing velocity relationship for a slender body ............................... 22

8. Characteristics of assumed linear velocity distribution

. . 24

7. Symmetrical profiles with velocity distribution

a * c *0


8. Symmetrical profiles with velocity distribution

a * c * b/Z


9. Symmetrical profiles with velocity distribution

a * c »b


10. Symmetrical profiles with velocity distribution 11. Summary plot for symmetrical profiles

a • c * 1.2b 44

.................. 45

12. Unsymmetrical profiles with 1 » -1.0.................... 48 13. Unsyimaetrical profiles with X * -0.4.................... 47 14. UnsyiMetrical profiles with X» -0,2

........... 48

15. Unsymetricai profiles with X « 0

. 49

16. Cmyranetrical profiles with X » +0.2

. 50

17. Summary plot for unaymmetrical profiles


18. Coarse net for the cavitation pocket around a hemispherical head K « 0.2


51 62



19. Final net for the cavitation pocket around a hemispherical head K » 0.2 20. 21.

Silhouette photograph for the vapor pocket around a hemispherical head K « 0.2

63 64

Comparison of th© vapor pockets and pressure distributions for hemispherical head K * 0.2 (Relaxation and experiment) ........................


22. Comparison of the vapor pockets and pressure distributions for hemispherical head K ® 0.2 (Mathematical and experiment) ................


23. 24.

Comparison of the vapor pockets and pressure distributions for 2*1 ellipsoidal head Z » 0.2


Profile with an assumed linear pressure gradient......



LIST CF TABLES Table I. Values of

Pane ft*X 1 Pn(Ml) •**« '




1 INTRODUCTION Usually the main interest in the design of an immersed body form moving in a fluid medium is the reduction of the resistances encountered* in effect, the reduction of the drag coefficient.


portion of the total resistance known as form drag can be varied since It is dependent on the pressure distribution which is a function of the body shape.

Separation of the flow from the boundary will result in a

low pressure on the rear of the body and therefore will have a major influence on the form drag. Although no satisfactory methods have been developed for the prediction of the point of separation in either the laminar or turbulent boundary layer, various pressure distributions can be compared qualitatively for separation effect.

Two methods are pro­

posed herein that allow the design of a body form to a predetermined pressure distribution and thereby control of the probable separation point. Similarly cavitation around a body can be prevented by design­ ing the shape for a pressure distribution having a maximum negative pressure less than the vapor pressure.

The same methods may be used in

the solution of this problem. The past analyses have had in common the approach in which the body form is assumed and the pressure distribution determined. A basic assumption which is always made is that the Reynolds number of the flow is so large that the flow is essentially irrotational. For a given axisysBaetric body* von Karman developed a method in which a series of

2 line sources and sinks was assumed# the strengths of 'which were deter­ mined by the condition that the body is necessarily a stream surface in a parallel field of flow [1# 2], Kaplan [3] presented a more rigorous mathematical analysis of the potential flow around a body of revolution. A series solution of the velocity potential function was obtained by expressing the profile of the body in a power series in the elliptical coordinate system. Recently Southwell developed a numerical method of solving differential equations* known as the relaxation method, which may be used to solve the Laplacian equation for a given boundary condition. However* this method does not give the general function of the Laplacian solution, but furnishes only numerical values of either the velocity potential or the stream function* at a large number of points throughout the region. It is a seraigraphical method and has its major achievement in solving the difficult problem of free stream lines [4, 5]. In this dissertation Southwell's relaxation method is applied in finding the shape of a cavitation pocket having a known pressure distribution by trial and error.

In a recent publication Birkhoff has

also suggested this method [6]. Although the method proved successful* it was not considered wholely satisfactory because of the number of times the boundary had to be modified, and the time required to liquidate all the residues in each trial in the entire region. A mathematical analysis lias also been developed for irrotational flow around slender bodies in which either the body form or the pressure distribution can be specified.

This analysis consists of

3 expressing the general velocity function derived from a solution of the Laplacian equation* in terms of Legendre’s polynominal vdth unknown coefficients, and simplification and integration of the velocity func­ tion by assuming for a slender body a velocity (or pressure) distribu­ tion.

Once the unknown coefficients are evaluated from the preassigned

conditions* the body profile can be determined. Y/hile this method is simpler for long slender bodies, the relaxation method may be used for a wider variety of shapes. A series of profiles both symmetrical and unsymraetrical (fore and aft)* of different slenderness ratio have been calculated according to this mathematical analysis.

The cavitation pocket form for the cavi­

tation index K » 0.3 around a hemispherical head form with a parallel afterbody has been evaluated* based upon the relaxation method and also the mathematical analysis. The results agree closely with the silhou­ ette photograph taken In the water tunnel. An exact analysis based on Kaplan’s method for the calculated pocket form was also made* the result of which agrees closely with that of the mathematical analysis.


Flow past a well-streamlined body at high Reynolds number approaches the ideal limit of irrotational motion. Therefore the rota­ tional components at every point in the fluid must be zero which may be expressed as c)w Qy


__ dz * dz

3v _ 3u dx ’


in a rectangular coordinate system (x , y , z) where u , v , and w represent the velocity components in the x , y , and z direction, respectively.

Under these conditions there will exist a velocity poten­

tial $ throughout the flow, the space derivatives of which in any given direction will equal the velocity component in the same direction: u = _ | i .

v = _ 9 i



= - M




The fluid is assumed to be incompressible and hence the continuity equa­ tion, 9“ + 2 X. + §* » o c>x dy dz must be satisfied.

(2 )

If Eq. (2) is expressed in terms of the velocity

potential utilizing Eq. (1), it follows that v 2cj> =






(3 )


which isgenerallyknown as theLaplacian

equation. Forany flow prob­

lem thequestion is then to solve Eq. (3) for agivenboundary


6 In some cases expression of the Laplacian equation in a curvi­ linear orthogonal coordinate system is preferable to the rectangular system used in Eqs. (1) and (5). In the analysis of flow past a sphere or an ellipsoid, the use of a spherical or elliptical coordinate system introduces considerable simplicity.

The equations of transformation of

a curvilinear orthogonal coordinate system X = f i( x , , x 2)x3) , y = f 2(xl;X£,x3) , Z = f3(x,,x2,x3)

where x^ , x2 , and

(4 )

x3 are variables In a curvilear orthogonal

system. The Laplacianequation, in general,

L- _L(JL£M\+i f Ml

E,E2E3|9Xi\ Et 3xJ

m y then be expressed as

d fE 'Ez = 0 3x2\ Ez 3Xz/ 3x3\ E3 Sx3j


and dsq * dsg , and de3 are the arc lengths along the axis of 2tq , xg , and xg » respectively.

The Laplacian equation In any orthogonal

coordinate system may be immediately obtained by using Eqs. (4), (5) and (8). In solving axisymmetric flow problems it Is sometimes more

convenient to use the stream function 4

than the velocity potential

function ^ . For such flow it is also expedient to us© the cylindri­ cal coordinates (x , r , 0 ) usher© x denotes the distance along the axis of symmetry, r the perpendicular distance from the x-axis, and 0 the angle between any meridian plan© and the x-y plane (see Fig. 1). The continuity equation in cylindrical coordinates for incom­ pressible flow must again be satisfied, and, hence, Stokes' stream

Fig. 1. Cylindrical coordinate system. function ^ exists, such that u.-lli r dr

y= i M . r dx


where u and v are the velocity components parallel along the axis of symmetry and in the radial direction, respectively.

Since the flow

is irrotational, the rotational component expressed in cylindrical coordinates must also be aero, i.e.,

7 By substituting Eq* (7) Into Eq. (8), one obtains


r or


c) r

It may be remarked that# unlike the two-dimensional case where both the stream function and the velocity potential function satisfy the Lapla­ cian equation# Stokes* stream function does not satisfy the Laplacian equation* Any flow problem requires the solution of either Eq. (5) or Eq, (9) for a given boundary condition. However, the solution is often complex and may even be impossible in terms of elementary functions satisfying all the conditions. An approximate solution may# in general# be obtained either by the use of a numerical technique or by making some simplifying assumptions. Relaxation Ifethod The relaxation method is a numerical method of solving differ­ ential equations based on the elementary theory of finite difference. The differential equation, written in its corresponding finite-difference form# can be applied successively to a large nuntoer of points which are the points of intersection, or node points# In an arbitrarily drawn net­ work. This network subdivides the region of flew into a number of squares (partial squares along the boundary). At each node point a value of the desired quantity


) is assumed with reference

to a crude flow net and the relaxation technique is then used which leads to a correction to be applied to each assumed value, such that the

8 flnite-difference equation is satisfied throughout the considered region*

Once the large errors have been eliminated in a coarse net*

a finer net may be drawn. The intermediate values on the finer net may be estimated closely by the use of the corrected values on the pre­ vious net. The accuracy of the desired values will be increased as successively finer nets are drawn. In solving flow problems by relaxation, it is more convenient to use the stream function, since the boundary is one of the stream­ lines along which a|i

is constant. Equation (9) may be written in its

corresponding finite-difference form as follows: ■f—— i-r~■ k,+k3\


i-- -*r


*i— t■— — — t-— \ti^— —■-i .. . ( i\ .i= —i;_ ju/ ,A oi kj Ik.kj 2r(kAi)% 4j

where a is the side length of the regular net, and


is the dis­

tance between points 0 and 1 to the length a , ^ l i£ the streamfunction value at point 1 , similarly for kg * kg , k^ , and ■^3 , \|>4 > as shown in Fig. 2a.

3 k*a %

k,a l i.-r 1



In the region where the net is regular.

T r

Fig. 2. Characteristics of relaxation elements, (a) General case, (b) Specific case encountered.

9 that is, where kx • kg * kg * k4 , Eq. (10) reduces to

+ i + 4* + 4 3+ V - 4 +o~ ^ ^ ' + 4) = 0

For the solution of problems with free streamlines by the relaxation method, the form of the free boundary is assumed and then the entire region is subdivided into squares, except for irregular poly­ gons along the boundaxy. Estimated values of ^

are assigned to each

node point, and it is usually convenient to assume the valises at the boundaxy to be zero, as shown in Fig. 2b. Either Eq. (10) or Eq. (11) is then applied to each node point, depending upon whether the net is regular or irregular.

In general, the assumed values will not satisfy

the difference equation. The algebraic summations of the left-hand side of Eq. (10), called residues, must then be eliminated -throughout the region. The systematic process of residue elimination is known as the relaxation method.

If S'0 is increased by one unit, it can be seen

from Eq, (11) that the residue at point 0 ia decreased by four units and the residues at points 1 and 3 are increased by one unit respec­ tively. Similarly, the residues at points 2 and 4 are increased by 1 ♦ (a/2r) and 1 - (a/Sr) when Eq. (10) is applied at these points. The pattern of relaxation is shown in Fig. 3. The same principle may be applied to the irregular star near the boundaxy.

If all the points

in the region are successively and repeatedly relaxed, the residues can be decreased to a negligible magnitude.

It is highly advisable to

check the residues after a long process of relaxation, since only in


r 2r

Fig. S. Relaxation pattern for a regular net.

this way can a numerical error be eliminated. For the more detailed procedure and the systematic bookkeeping process, reference can be made to Southwell’s original publication [4]. An advance to a finer net is mad© in regions where the

assum p­

tion of linear variation of the function is not satisfied. A close estimate of the values of the stream function for the finer net may be obtained by using curvilinear interpolation. The previous procedure is repeated, i.e., computing and liquidating the residues on the finer net, until the residues have been decreased to a negligible magnitude.


general, each advance of the net is confined to a smaller region where linear variation of the stream-function values ceases to hold on a previous net, but the amount of vrork required increases considerably with each advance, since the number of node points varies according to the square of the number of divisions. Equations (10) and (11) are (satisfied after liquidating all

11 the residues on the final net. Another boundaxy condition which must be satisfied in the free-streamline problem is that the velocity along the free boundaxy is constant. The velocity components at any point can b© evaluated from Eq. (7) and the resultant velocity q at any point on the boundary, as may be seen from Fig. 4, is equal to a




cos p


— 5in p


^ boundary



Fig. 4, Evaluation of velocity at boundary. (a) Velocity components, (b) Streamfunction gradient. where |3 is the angle between the normal to the boundary and a verti­ cal line. The first partial derivative

H/c>x or d^/c)r in Eq. (7)

can be evaluated by dividing the stream function values by the dis­ tances

Ax or

At near the boundary.

It is evident that the gra­

dient evaluated is not on the boundary, but it is at the middle point of Ax or Ar . However, if the differentiated Lagrange formula is

12 used, u better approximation can be obtained. By actual calculation it has been found that the differentiated Lagrange formula of the third degree is sufficient for the purpose. Differentiating ■ftx. _ ( x - a z X x - Q ^ - ( x - a n ) ( a r d 2 ) ( 0 r q 3) - ( a , - a n )


(x -a ,)(x -q 9 )--(x -a r 0 /




y If ^


0 £ ^ £ 2-rr

are eliminated in turn from Eq. (16), families of con-

focal hyperbolas and ellipses are obtained for constant values of r|

16 and ^

, the distance between foci being 4a. For any point P(x, y* z) in space, the equation of trans­

formation corresponding to Eq. (4) in the elliptical coordinate system ( h , ^ , G ) is

X = Za\ ja = r co50 = 2a(Az-l) ^Cl-ju^cos© z — r sin0 = 2a (A2-1),/2(|- ju2)'/25in6


where A = cosh %



X varies from 1 to 00 , and ju varies from -1 to +1. If Eq. (IS) is substituted into Eq. (5), expression of the Laplacian equation in the elliptical coordinate system results c)

55 = ae2




For axisymmetric flow, the velocity potential is independent of © and hence is a function of \

and p

only. Equation (17) reduces to

it SX






which is the Laplacian equation for the case of axisymmetric flow and is to be solved for given boundary conditions. Equation (18) is a homogenous and linear partial differential equation of second order and may be solved by the method of separation of variables.

Hence, if the velocity potential

has a solution of

10 the form (19)

4 = Ux)

It follows that

J- A . L(x) dx


M(|^) cty




The above expression represents two ordinary differential equations where o is an arbitrary constant.

If c ■ n (n + 1) • then the

above equation may be reduced to dU\V

+ n(n + 0L(M = 0



_d_ dju

dMtfO'4 n(n+ OM(ju) ~0



which are in the form of Legendre’s differential equation with the solution Pn(\ ) and Pn(fA) , respectively. Hence, the general solu­ tion of Eq. (18) is 4 =2aU 2


AnPnCM P n W

where An’s are constant coefficients, U is a constant velocity (factoring out 2aU for the simplifications of later development), and Pn(x) is represented as p,,,. idf Pn(x) ~ 2nn! dxh It is noted that Pn( \ ) is a polynominal in x becomes infinite as X

of the nth degree and

oo , Since the region outside the surface

including the region at infinity, is to be considered, another solution

17 for L( X ) is required.

The solution, linearly independent of Pn( X ),

is the second kind of Legendre polynorainal and is denoted by Qn(X } where QrM = jPnlM In y q - ~ k nU) ■where Kn(x ) is a polynondnal of (n-l)^1 degree and is expressed as

K n W = T T ? p"-'w

+ I S ) p^ w + '’’

Qn( X ) vanishes at X « oo. The general solution may now be expressed as 4 = 2qU L W^.6 AnQ„WR,(|j) 1


which represents the velocity potential function for an axisymmetric body moving with a velocity U in the negative x-dlrection. The body may conveniently be supposed at rest and the fluid moving with a veloc­ ity U in the positive direction of the axis of symmetry. The velocity potential of such a parallel field of flow is 4 = - 2a UX pi Hence, the general solution of 4 ^or 2111 axisymmetric body in a steady state of motion is 00

4= >2aUZoAnQ„WPn(^-2aU


The above equation can be used to evaluate any relationship, such as velocity distribution or pressure distribution, in the field of flow or along the boundary.

(k). EvalmtlQH



In order to find a body form with a preassigned pressure distribution, the expression for the velocity can be derived from Eq, (22) and related to an assumed velocity distribution along a body.

If the body is

assumed to be slender, the normal velocity components along the boundary may be neglected. Thereby the resulting velocity expression will be simplified %nd can be integrated with an assumed velocity distribution. Since (A , y. ) is an orthogonal coordinate system, It fol­ lows that U)v = _ ^ £ where u^ and ux

“u= ~ ^ £

are the tangential and normal velocity components

at any point along the curve of \

* constant and ju * constant,

respectively. From Eq. (8) the arc lengths along such curves may be expressed as


19 Differentiating Eq. (22) and substituting into Eq. (23), one obtains

( $ 2=


,+ * T)


where q is the resultant velocity at any point in the region and equals

>A\2 + u^,2 . In order to express the velocity along the boundary as a known

function, a simplification of Eq. (24) must be accomplished.

Since the

body is approximately elliptical In shape, the profile of the body niay be eaq>ressed by a functional relationship of X and \a

. If the body

Is assumed slender and represented approximately by X » X i , which is close to unity, the velocity components u \

may be neglected. Then


-t X, or



is close to unity, the left-hand

further simplified by setting


of Eq, (26) may be

\ i ■ 1 • There is obtained


i r = - l ^ Q where


A Q. represents the difference of the velocity along the boundary

and the velocity of the oncoming undisturbed flow, and p i « (xi/2a) ,

20 In order to determine the constants An » Eq, (26) mast be integrated. Integrating Eq. (28) between the limits of 1 and |>}_ yields


= -?



[P n ^ .) - Pn( 0 j

or (27) since Pn(i) « 1 . Equation (27) may be integrated again by using the orthogonality property of the Legendre polynominal, i.e., jVfOP*'


0 20rl

ISultiplying Eq. (27) by Pm(^)


IT]1* n


m= n


and integrating between the limits of

-1 and +1 , it follows that >i r>i. ^2)d|A Pm ((U.)dh(,=-Z A hQ„(X,^ Pm C^[P,C^- l]dp

By the orthogonality property as expressed in Eq. (28), the above equa­ tion may be reduced to +1 Mi *"AnQy,(\i)




Pn(^ 3 ) dju^ * 0 , for n £ 1 . The expression J ■i-1 in the above equation can be simplified since

I, (1$ V

- L lir) V = Ftp.^ - k


(Aq/u) d^q’


21 where k * constant. Substituting Eq. (SO) into Eq. (29) yields f*i “ AnQnCM 2n+Y ” J '-1 or r+\

AnQnW= - • ^ 1 J FV0P„(p,)dp,


since J ^ k P^C pi) dpi « 0 , for n » 1 . The above relationship, Eq. (31), can b© used to determine the unknown values of An for an assumed velocity distribution along the boundary. (c) Relationship between the coeffioients An and the coordinates of the body profile.

Hie combination of a uniform flow and

a distribution of sources and sinks will produce a certain shape of boundary. Hence, for a certain shape of boundary, there must exist a distribution function to give such a boundary,

this unknown distribu­

tion function can be found by equating the velocity potential due to the unknown distribution and the general Laplacian solution. If one considers a continuous distribution of sources and sinks along the axis of symmetry between x * -2a and x * +2a , and denotes the strength of the distribution per unit length at the point xi by I(xi) , the velocity potential, at any point (x , r) on the meridian plane, due to such distribution is r+2oi

>, -L

4 Y

4 v

IM4* (u-Xi)2+ r*yz



from elementary hydrodynamics. If the above equation is identical to

zz the general solution of Laplacian, Eq. (21)* an integral equation is obtained with the unknown function I(xi) . The unknown distribution function will produce the specified body* form and as shown by Kaplan [5] equals I(2ap,)= 4-n-aU ^ A nPn(|Ui)


where f*^ « (x^/Sa) • The coordinates of the body profile and the coefficients An can be related as follows: If the body is again assumed to be slender and the velocity u due to the distribution of sources and sinks is small compared with the main stream velocity U * as shown in Fig. (5), then the following relationship will hold v U



dro dX|


is the radial coordinates of the profile. Since the body is

assumed to be slender, the distribution function will vary gradually along the axis of symmetry. And if the differential element of the distribution function is assumed to be responsible for the looal distor-

U Fig. 6. Sketch showing velocity relationship for a slender body.

23 ticm of the flow pattern, the following expression will follow (34) Eliminating v from Eqs* (33) and (34) yields IWH2-nr.U$j = U ^ ( W )


The above equation shows the fact that the strength of the distribution is directly proportional to the gradient of r$2 in the direction of the axis of symmetry. Th© slope of the r-squared curve for the front portion is positive and hence the distribution is a continuous source; the slope of the r-squared curve for the rear portion is negative and hence the distribution is a continuous sink. Elindnating I(xq) be­ tween Eqs. (32) and (35), there is obtained

4-naUZ AnRilfM” U^ ^ r» 2) Since

• 2api * the above equation reduces to (S3)

Integrating Eq. (3®) between the limits of -1 and pi , it follows that (37) which represents the desired profile with th© preassigned velocity distribution.

24 id) Relationship to velocity (llgtributlon. The results ffrom the preceding analysis may be applied by assuming a velocity distribution along a profile with X » \\ , the function F( pi)

in Eq. (SO) nay

then be determined. By substituting th© function F(pi) the coefficients kn are obtained.

into Eq. (31),

Once th© coefficients are known,

the body profile can be evaluated from Eq. (37). It may be seen from Eq. (SO) that th© function F( pi) depends upon the type of velocity function along the boundary assumed.


order to delay the separation, the assumed velocity should be such that the pressure distribution curve resulting has either a zero or a small gradient along the body profile. Equation (31) may be easily integrated if the assumed velocity function is a polynominal. example,

( Aq/j) is assumed

varies linearly from pi * -1to to b and from

pi * X to


to have a linear variation with pi . It pi “ I with the magnitude of


p i » +1 with the magnitude of b to c ,

as shown in Fig. 8. The equations representing such a distribution are

Fig. 6. Characteristics of assumed linear velocity distribution.

25 as follows: for L| , Af


+ i = a (fJ|+0


and for L2 ,

Acf, , , — = c + b-c — r(p,Ix )


The functions F(

“ ■ X-.

In Eq* (SO) are then expressed as fHir

F^p.) =

^ ^ I a-v ~~"r(u,+ 0 dw»

J0 L

A+ I