Gravity flow past a cylinder moving beneath a free surface

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HYDRONAUTICS, Incorporated

TEC}ICAL REPORT 117-16

/

GRAVI'IY FLOW PAST A CYLIND MOVING BENEATh A FREE-SURFACE

By

G. Dagan

June 1970

This document has been approved for public

release and sale; its distribution is unlimited

Prepared for

Office of Naval Research Department of the Navy

Contract Nonr-3321-9(O0)

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-1-TABLE OF CONTENTS

Page

ABSTRACT 1 INTRODUCTION 2

DISCUSSION OF THE.DEEP SUBMERGENCE SOLUTION 5 FIRST -ORD INNER AND OUTER (PANS IONS IN THE

PHYSICAL PLANE 7

k.

THE ZERO-ORDER INNER SOLUTION 10

THE FIRST-ORDER OUT _SOLUTION 16

MATCHING OF THE INNER AND OUTER EXPANSIONS 19

THE LIFT AND THE DRAG 20

SU4ARY AND DISCUSSION OF RESULTS 22

APPROXIMATE SOLUTION FOR MODERATE FROTJDE NUTERS

AND MODERATE TO DEEP SUBMERGENCE 27

SUrVIIVIARY AND CONCLUSIONS

31

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LIST OF FIGUR

Figure 1 - Steady Free-Surface Flow Past a Submerged Cylinder Figure 2 - Three Models of Free-Gravity Flow Past a Cylinder Figure

3 -

The Flow Domain in the Plane

Figure 14. _{- The Shapes of the Free-Surface and of the Closed}

Body Derived with the Aid of the Inner Solution

Figure.

5 -

The Relationship Between Different Inner and Outer

Paramet ers

Figure 6

-

The Lift and the Drag as Functions of x

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NOTATI ON

a' - The cylinder radius

d,e - Parameters of the inner solution f' - The complex potential

f = f'g/1P3 - Dimensionless outer potential f = f'/a'U' - E.imensionless inner potential

Fr = U'/(ga') - Froude number based on cylinderts radius g - Gravity acceleration

h' - Depth of immersion beneath the unperturbed upstream level

h = h'g/U'2 - as above, dimensionless, outer H = h'/a' - as above, dimensionless, ihner m - Parameter of the inner solution

= Rx'_iRy' - The complex force acting on the cylinder

U' - Body constant velocity -- Complex velocity

w .= = w'/U' - Dimensionless complex velocity z' = x + ly' - Complex variable

z .= z'g/U'2 - Complex variable., dimersionless, outer

= z'/a' - Complex variable, dimensionless, inner

a - Square root of the doublet strength in the

C plane.

y - Et1ler constant

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iv

-Parameter related to the location of the doublet beneath the free-surface in the plane

cp','Jj' - Potential and stream functions

- Free-surface elevation

= _{- Free-surface elevation, dimensionless, outer} - Free-surface elevation, dimensionless, inner - Inner variables

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-1-ABS TRACT

The gravity flow past a cylinder moving close to a free-surface at high Fr is investigated by the method of matbhed asymptotic expansions. In contrast with the linearized solu-tion in which the dimensionless depth of immersion h = h'g/U'2 is kept constant, in the present analysis h 0 as Fr

The inner flow model is that of a nonseparated non-linear gravity-free flow past a doublet. The non-linear effects are

strong when the depth of immersion H = h'/a' is small. The

lift and the wave-drag are different from those given by the linearized solution. When the depth of immersion is mode'ate the linearized solution is valid provided that the depth of immersion is replaced by an effective depth larger than the actual one. This result agrees with the second-order computa-tions of the lift and drag carried out by Tuck

_(1965).

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-2-1.

INTROJDTJCTION

We consider the problem of a steady potential flow of a

heavy liquid prast a submerged cylinder. The exact equations of flow, given here for convenience of reference, are

(Fig-ure 1) 2 dw? w? igw?) _{= 0}

= 0

= const. + y'2 = a? w? = TJ? ,;

= h

The nonlinearity of quaticn

_Li]

has defied any attempt to solve the problem analytically, or even numerically.

Approximate solutions have been sought by perturbation methods, I.e. by expanding the solution in a small parameter

series. _{With physical parameters of the problem TV, g, h'}

and a?, there are only two possible dimensiohless sthall

parameters, e.g., c = a?g/TP2 and h = h?g/TP2. _{To proceed in} an orderly way, let us consider e as the basic parameter of: the problem. Two types of asymptotic solutions may then be considered:

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so far. The limit 1/c order approximation is y' = h', replacing the have been investigated

1969)

in relation with a ship bow. Under the unchanged and the drag case is not considered

(i) The small Fr number flow, i.e. the limit

U!2/arg _i/c 0. This case has not been studied in detail = 0 represents rest, while the first-that of a flow beneath a rigid waii at free-surface. Second-order effects in a .previous work (Dagan and Tulin, the stability of the free-surface near

limit process the body shape remains is zero at any order. The small Fr again in the present work.

(ii) High Fr flow, i.e. £ 0. This case has been investigated in much detail. The unperturbed flow (e = 0) is uniform. The free-surface condition (Equation

_[11)

is lin-nearized and transferred to y' = h' and gravitywaves appear far behind the body. The perturbation expansion achieves the simultaneous linearization of the free-surface condition and of the body condition, the body degenerating into a singularity.

The details of the solution depend on the relationship between h and c. The solutions obtained so far pertain to

the case. h = 0(1). The frst.-order solution, in which the body is replaced bya doublet (Lamb, _1932). is quite old. and

wilibe analyzed in detail in the following section. Higher-order corrections have been derived by Havelock

_(1936),

the

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-k

-(Wehausen and Laitone 1960, p.

572.1.).

In these higher-order

approximations only the flow in' the vicinity of the.body has

been correcLed, while the free-surface condition has been

kept in its first-order version.

This procedure is not

con-sistent in principle, since the body and the free-surface

contributions are of the same order.

Moreover, detailed

consistent computations carried out by Tuck (1965) have shown

that the free-surface second order effect is more important

than that of the body.

In fact, the solution based on the. asymptotic process

in which c -

0, h = 0(1) is one in which the relative

sub-mergence H = h'/a' tends to infinity like 1/c when c

0. An atterript to apply

ft

to a cylinder located not tOo far

from the free-surface (H = 2) by Tuck (1965) has shown

in-deed that the second-order effects are Very irportat in this

case.

The validity of this type of solution in the. case of

bodies moving close tO the free-surface and extrapolation of

the results to floating bodies is, therefore, questionable..

In both applications and theory it is important to

determine the flow pattern in the case of submerged bodies

moving close to the free.-surface, which is also the case in

which the wave drag component is significant.

This is the

motivation of thepresent work in which we let h = h(.c)

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-5-body submergence Froude number tends to infinity as the Froude number based on the cylinder radius tends to

in-finity. This type of solution is, hopefully, helpful in understanding non-linear ship-wave effects as well as other related problems.

2,. DISCUSSION OF ThE DEEP SUEMERGCE SOLUTION (e 0, h = 0(1)) Lamb's first-order solutiOn (given in 1913) for a doublet my be written in the following form:

= + 2

-

z-21h +

U

where Ei(iu) dx and the variables are outer ones

-(see Notation).

If we examine the solution in the vicinity of the doublet (z -, a) and in the vicinity of the free-surface (e.g. z = ih.)

we find that thebod.y and the free-surface interact weakly at first order, i.e. f tends uniformly to the potential ofa uniform flow past a doublet in an infinite medium and of an unperturbed uniform flow, respectively, as e 0.

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When h = h(e) = 0(1) a consistent expansion of Equation [5] yields

f = z + 2ie2eiZEi(iz) 2hC2 - + 21e_iZEi(iz)] + o 3

y

[6]

f of Equation

[6]

olonger'has the properties

enumerated above. The flow near z = 0 is not

a

uniform flow past a doublet and for

2 =

ih the third tern.in Equatiofl

[6]

becomes of larger order than the second one,. The lirt acting

on the cylinder (divided by the buoyancy) is, according to Equation [5] proportional to e2/h3. It becomes infinite If htends to zero fastel' thaPi

In fact, the untform convergence of the solution near the origin is no longer ensured. This conclusion is also supported by the examination of the second-order quantities computed by Tuck

_(1965).

The deterioration of the solution expressed by Equation

[5]

near the origin is due to the fact that the near-body and near-free-surface flows interact non-linearly even at first

order.

In the following section a uniform solution for the case

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3.

FIRST-ORDER INNER AND OUT

PANSIONS IN ThE PIrZSICPLPLNE

The outer expansion of w and r, the outer Vriabies being referred to U' and. U'2/g (see Notation), is as follows:

w =1

_± 1(e)w1 + 2(e)w2 +

Ti = h ± :

The substitution of w and Ti fl the exact equations [i]

-[k] leads, at first-order, tO the well-known linearized equations:

= -i

0

and similarly for higher orders.

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(3r = 0)

iw1

=0

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-8-At the limit 0 the body approaches the unperturbed free-surface at the origin. For this reason the free-surface conditions

[9]

and [Ia] apply toy = o ard w1 and i]. are

singular near the origin. Their behavior there, as well as (c), have to be determined from the inner expansion.

With the inner variable , f, and refer-red to U?

and a? (see Notation) and an thner expansion of the type:

+ A1(eW1 +

+

1(e)1 +

we obtain, at zero order, after substituting. in Equations

[1] -

[3]:

It = I 0

=0

= const (37 = (37 =

(2 + 372

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Hence, the zero-order inner approximation is that of a gravity-free flow past a cylinder. The velocity on the

free-surface (Equation [15]) has been selected such that the match-ing with the zero-order outer velocity is ensured. The location of the cylinder beneath the free-surface has to be determined by further matching of r1 and ?i.

Now, there are various models of free-gravity flow past a body, three o.f them being suggested in. Figure 2. Any mOdel

is plausible if it produces a solution which can be matched with an appropriate outer solution. We select in the present work the model T1a'T in Figure 2 of a smooth free-surface and nonseparated flow, since we are primarily interested in the non-linear effect of the free-surface as the body approaches it from deep submergence.

If we assume that all the suggested models, as well as other conceivable ones, are plausible, we are left with a certain degree of arbitrariness. On the other hand the order of h(c) cannot be assigned apriori; it depends on the assumed

model. As.long as we c&m.not o1Ve the non-linear problem

exactly, we have to rely partially on intuition and experirents. In addition, since the solution of free-gravity flow past a

cylinder is not iple, we replace it by a doublet. Although the closed body generated in this way (see Figure

k) is

slightly different from a cylinder, the simplicity of the solution makes the approximation worthwhile.

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-10--k.

THE ZERO-ORDER IN SOLUTION

The complex plane f() is a two-sh'eet surface.

We map

it on to the auxiliary one-sheet

=

+ i

plane by:

-

_-

_a2

f

C-2ix

such that

0 on

_{= x (Figure}

3).

The constants a and x'

which are related tO the strength and the lOcatIon of the

doublet in the

plahe, reSpectiveiy, will be determined

subsequently.

d2

dl

The function

= =

is regular In the lower

dCWo

half-pan

< x and has a known real part along

_{= x given}

by Equations [15] and [18]:

Re w = 2n

l-2a2

- x2

_[19]

(2

_{+ x2)2,}

With a new variable

the detePiinatiOn of

becomes a Dirichle.t problem

or the

lower half-plane ThiX = 0.

The problem is solved by the

Cauchy integral along IniX = 0, withproper care to the cuts

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-11-along the branch lines of w. The fLiiction 1-2

(2

₁

obtained from Equion [19] by subst1tutig Equation [20]

has two zeros and a double pole in the lower half plane

- (X-d-ie) (X+d-ie)

-

(x-i)2

the logarithm being taken at its principal value, such that for X ± the argument is zero.

Wi-th the aid of Equations [18] and, [25] we immediately find

d and e being positive f or a/ > 1/2.

The result of the Cauchy integral is readily found as

[25] X = - iè [21] [22] Where d = 2/x2±l)1/2 ₊ 2

-

l]2

[23] 2 e (ko2/x2 -

a4/4

1/2 [2)] 2d

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HYDRONAIJTICS, Incorporated 1. -12-a 1

1_.x4

-

_-

C2

(-2ix)2J

C [26]

0 -

d! - -+ i(l-+e)1 j ± d - i'(1+e)12 L C

JI

_C

I

and by further integration

z=

(ki(ln

-2i

+ {d2 - 3(le)2](

-2i

-iI

22

2ij' 2 3 21 I I X .i

-

2iX)a

II

The inspection of: Equations [26] and [27] shows that has its only singulariy, of thedoublet type, at = 0,

while 2.is regular in the half plane _{< X and vanishes at}

C = 0. All the singularities of are located at

= 2i,

the image of the doublet across ₌

. The logarithmic term

in Equation [27] has zero argument along _{= 0,} _{< 2x.} The next step is the derivat±on of the relationship

betweena

(the doublet strength) and

_x

(its location in the

C plane). Two possibilities have been considered:

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-13-The doüblet .treigth iti the plane equal to

utiity. This is tañtmount to defining aT (and ccordingiy) strictly as the doib1et strength. Th ecpanion of Equations

26j

and 27]in the vicinity cf 0 gives for the

cOeffi-dent of -l/ in the

.

expressiOn

16a2

d2 + (i+e)2j2 - 1

it turns out, however, that the closed boy generated by the doublet, a1thouh close in its shape to a cylinder has a

DadiusdiffOrent fromunity small X.

The radiuC of the closed body approximately equal to unity.

he stanätiOn poit of OoQriate

st Xd + i(l-e)j 129J

has been selected aC the represeittive pont of the body ad

itC abscissa in the plane has been made equal to unity. By

using Equations [27J and

[9J

the ±ollOwth relatiOnship is easily found:

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J

)4(i-e)

d: d2 + 3(l-e)2

= xd\1

- d arctg 1 + e + 2 d2 + (l+e)2

)-1.(1-e) d2.+ (1-e)2] (1+e)

[d2 + (l+e)22

± £d

-3d

(1+e)2J =

work.

In Figure k we give the shapes of the free-surface and of the closed body for a few values of a. and . The computations

have been carried out numerically in the following manner: different arbitrary values have been assigned to a2/x2;

d,e,a and have been computed with the aid of Equations

23], 2k] and [30]; (iii) the value of

+

This latter conditioi has been aopted in the present

- a2 a2

-ix

L

C

C-2i

has been computed at the stagnation point and (iv) the corresponding streamline, of equatiOn

='

as well as the shape of the free-surface, have been computed by iising. Equations [31] and [27], respectively.

[30]

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-15-The flow.betng symmetrical, only half of the field is reproduced. In Figure 5 we depict graphically the dependence

of. a, x and

d2+(l±e)2 (the square 'oot of the coefficient of

in

,

quatoi [28]) on a2/2.

Fina11y, and have the following asymptotic expansion .f or (Equations [26] and [22])

= 1 '4.i.(1-e) + 1. [32]

+ ki(l-e)2z + ixm.+ 0 . [33]

where

m = -k(l-e)2n2 - d2 - 3(l-e.,)2 + (l-e) [d2 + (I_e)2]

- . [d2 + (l_e)2]2

The potential, and the free-surface equation, are singular at infinity like .

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5.

THEPIRST.ORDER OUTER SOLUTION

It is useful to operate in the plane rather than z, since the inner epansion is defined parametrically as

_a

function of (the procedure is similar to that Of Tulin

(1963)

and Wu

(1967)).

Taking = c as an outer variable in the plare, we transfer the exact equat±os L1 arid J frOm z to with

the aId of Equation

i8]

in hich f = ef. Hence

dw form:

r

=1

(C-21ex)2 + 1w

=0

(p=ex)

_35i

± ....) (1 2e2 ...) d

[38]

w = 1 ; Im z = h

(c -+iex)

[36]

The outer o1ution is now expanded ti the _plane

w() = 1+ 1(c)w(ç) ±

2(c)w2(g) *,...

_[37]

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Wi = 0

-17-With the requirement that at upstream infinity

(C + ieX). the free-surface elevation is Imz h, we get from Equation [38]:

z = + ih - ie -

()

f

w1(c)dc + 0(e2)

[39]

Substituting the expansion of Equation [31] into

Equation

[35]

we arrive, to first-order, at

(ImC=0)

[ko]

equations sirnilar.to Equation [9]. w1(C) is singular at = 0, which is the body location when c 0. Its behavior there. is determined by the matching With the olutior.

The outer limit of the inner solution is readily formed from Equation [32] by substituting ₌ as

= 1 - )-l.i(1-e)

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-18-Hence, the matching requires that

=

At = 87ra'X(l-e) . [k6J

Wi 1i(l-e)

C

i.e. the singularity of w1 is of a vortex type. By integration of Equation LkO] in the whole _C plane (Wu,

1967) w1(C)

becomes

+ e Ei(iC)]

[3]

z is foind from Equations

39]

and

_k3]

as follows

z=

+ ih - - kX(l_e)eie1C _{Ei(iC) +}

O(2)

For x - ( -. ), far behind the body, we get fo' the

free-surface profile

= h(c) - - 8irc(l-e)sin +

= h() - ex - 8ire(1-e)sin x + O(e2) [L.5]

The amplitude of the free-waves behind the body is, therefore

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-19-6.

MATCHING OF THE INNER AND 0UTJFL EXPANSIONS

The matching is carried out in general by an intermediate expansion (Cole,

_1968),

but in the present case it may be achieved by the simple rule: the outer limit of the inner solution is

identical with the inner limit of the outer solution (Van Dyke,

1967).

The inner limit of the outer solution relies on the well known expansi.on of Ei(iC).

Ei(iç)

yn(-i) +

_L

n(n)

n= 1

where y =

0.5772...

By substituting _C = c andz = c in Equations

[13J

and

[kui] and expanding with c -' 0, = 0(1), we get

w=

4iX(l-e)

- )-L(1-e)c2nc + 0(c) [k8_J

C

= -kix(1-:e)nc + + i - i)( _'X(1-e)iy k(l-e)nC + 0(c)

(id

0)

[7J

The inner limit of w (Equation [k81) obviously matches the outer limit of (Equation

32].

Moreover, we find that the orders of the higher order terms in the inner expansion are Ai(c) = A2(c) = c,

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-20-The matching of (Equations k9] and

_E33)

requires the additional relationship

h = X(l-e)eQftc + ck(i-e)y +

or

H = (l-e)2nc + 4X(l-e)y + 5o

I

Equation 5O], which is one of the central results of the present analysis, establishes the order of magnitude of h, as well as its dependence on the inner parameters a and x.

Uniform expansions of w and z may be written by adding the inner and outer expansions and subtracting their common

part.

7. THE LIFT AND THE DRAG

The Tforce acting on the body may be determined by the Blasius theorem applied to the inner solution

R=R -iR, =

x y

2J

5 1]

where R = R?/patU?2.

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-21-2(c)

a.

dC

the integral enclosing the origin inthe plane.

By substituting Equations E26] and [27j into Equation

[2.1.9] and by applying the residue theorem, ,R is readily found

as a2 a2:.\

_I

2(1+e') R0 -- ± (l±e)]2 -1 +

f[l

.d2' + '(l+e)2

[531

As one would expect from the. ymrnetry of theinner solution, R is a negative lift force. ts dependence on

2/2is represented in Figure

6.

The drag appears in the inner solution in higher order

terms. But we may compute the wave-drag from the outer solu-tion, since, it remains bounded under the limit h 0. Using the well known formula based on wave energy radiation, the drag is found from Equation [1.6] as

)

RT

= i.6ir2pga2(i-e)2

[5k]

or, by referring the drag to buoyancy

R!/1rpga?2.= l&ir2(1-e)2. _[55-i

[52]

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-22-Equation _E55J is also represented graphiclly in Figure 6.

8. SUrVUVIARY AND DISCUSSION OF RUL

It is worthwhile at this point to analyze the results of the preceding sections before proceeding to further

develop-ments.

The uniform solutions Of w and are completely determined if two independent parameters of the problem, e.g., H and

are given.

The examination of the inner flow picture (Figures 1 and

5) reveals a fewinteresting points:

(i) The approximation of the cylinder by the closedbody generated by a doublet is surrisingiy accurate

(Figure 24.). _{Excepting the slight deviations near the crest,}

the body!s shape coincides with a circle of unit radius, pro-vided that the doublet strength is suitably adjusted. This

result suggests that the approximation of a body by a singu-larity distribution may be quiVe accurate, even close to the free-surface.

(ii)' The free-surface is convex near the body and for this reason the lift is downwards and increases in mag-nitude. as x decreases (Figure

6).

For relatively small x the curvature of the free-surface is pronounced and it may

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23

-become unstable. The free surface may breakupstream in the zone Of convexity because of the negative pressure gradient inside the body-water or it may detach in the rear of the

body.' In both cases the model of inner flow adopted here is no longer adequate. For this reaso we have not considered values of x < 0.9 although the accurate computatioti of the range of validity of the model is a question for further

study.

If is sufficiently large, -he free-surface is flat. In

this case a simpler solution of the inner flow may be derived by linearizing the inner free-surface condition. This type of solution is presented in the next section.

(iii) Far from the body the free-surface rises logarithmically (Equation [33]). This singular behavior is common to free-gravity flows and is indeed corrected,by the outer expression. I.f x is large, however, the logarithmic termbecomes insignificant (see next section) and the far behavior is regular.

Turning now to the outer so'lution, the following observations seem appropriate:

(i) Although the solution obtained here is an asymptotic one, valid for c - 0 (Fr - ), in applications

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-2k

-values of Fr. For this reason we have considered in Figures 5, .6 and 7 only values of Fr smaller than ten and we limit the discussion to this range alone. This is done with an optim-istic belief in the sufficiently rapid convergence of the expansions.

(ii) Let us consider first Equation

ko]

and Figure 5 which represent the relationship between H (immersion below the far upstream level), _x (proportional roughly to the free-surface rise above the cylinder's crest, Figure

k)

and the Froude number.

For very large. H, say } >

k,

is also large, independent

of Fr, arid H

_x.

This estimate results from Equation [koj in which we neglect terms of order l/. d (Equation _23]),

e (Equatin

_E2k])

and m (Equation

3kj)

have the following

values for large

_x:

d

l/, e

l-l/S2 and m l-l/2.

In this approximation the free-surface in the inner solution is extremely flat and coincides with the level at infinity. The terms of orders

_i/C

and 1/C2 in the expansion of

(Equation [32]) vanish, and the same is true for the outer solutIon (Equation

_[k3]).

In fact, as it is shpwn in the next section, the linearized deep submergence solution

(Equa-tion [5]) is valid.iri this range and coincides with the solu-tion presented here.

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-25-'For moderate to large H (2 < H < k), _x is still insensitive to the Fr variation (Figure 5), but the relationship between H and

_x

is better approximated by H

x -

_l/.

The free-surface near the body is still flat, and the inner solution may be

simplified. This regime is discussed in detail in the following

section. In this range the first term of '(Equation,

32fl

and the corresponding first-order outer solution are still negligible, and the linearized deep submergence solution is approximately valid provided that the depth of immersion is taken equal to

_x.

The non-linear effect of the free-surface manifests itself in the relationship between

_x

and H, solely.

For small H (H < 2),

_x

and H differ sensibly, and even when the cflnderts crest approaches the upstreamlevel (H 0), X is still of the order of the unity. For a givet H, _x

in-creases with Fr,

i.e.

the water piles up aboVe the body as the speed increases, but the growth is very slow. In.this range,

x'

H and Fr are interrelated non-linearly in a complex manner through Equation [.F01; the coefficient of the first-order solution (Equation _[1+31) also depends non-linearly on for

intermediate values of . ' In this range the first-order outer

term is significant and the body singularity is of a vortex type, rather th'an a doublet type.

(iii) The variation of the lift with _x is given

'in Figure

6.

The relationship between the lift, (re±erred to buoyancy),'and Fr for a fewvalues of H is represented in

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-26-Figure 7. The lift is strongly dependent on the near field and for this reason the result of the present computà.tions is markedly different from that based on the linearized

solu-tion. In the linearized solution the lift behaves for small H like -l/2eH3, being eventually infinite for H -. 0, while

in Figure 7 we find finite values even when H is equal or less than zero.

Generalizing, we may conclude that for all the flow features related to the near-body field the present non-linear analysis yields results which are sensibly different of those of the deep submergence solution, for small H.

(iv) The wave drag at first-order (represented in Figures 6 and

₇₎

dropsrapidly as

_x

increases. For sufficiently large

x (x

>> 1) Equation

[55]

gives R'/irpga'2 which may be much smaller than the first-order value of the drag based on the linearized solution (Equation [5])

R'ATpa?2

These estimates suggest that the wave drag component found here is sigriificant only for small H, otherwise the higher-order components may be more important for moderate Froude numbers. In other words, a realistic evaluation of the wave drag requires the computation of a few more terms in the outer solution. We plan to extend the analysis to such terms in the future.

(32)

-27-Now, we examine the approximate solution pertinent to moderate H.

9. APPROXIMATE SOLUTION FOR MODERATE FROUDE NIJMEERS AND MODERATE TO DEEP SUBMGENCE

As stated .before, the first orcer term of 'the outer solution is of little value for constant Fr and high _x

(H > 2, _{x >3).}

The next term of the outer expansion,

_52(.c)w2(C),

is of order 2 times a function of . It originates from a forcing

term in the free-surface condition, which comes from the first-order solution, and from a singular term at the origin, which comes from the matching with the inner solution. For suffi-ciently large x the forcing term becomes neglig.ible.(being of order l/X6) and the second-order outer solution is determined mainly by the singular term. This observation helps to cir-cumvent the computation of the cothplete secod-order solution,

which is tedious because of the forcing term.

We can proceed with the computations on a sounder physical basis, which lends itself to extension to three-dimensional flows, if we observe that the omission of the term 4ix(l-e)/c in the inner solution (Equation 32]) is equivalent to the

linearization of Equation [15], i.e. assuming that = 1

along the free-surface, rather than = 1. This assuthption

(33)

HYDFLONAUTICS, Incorporated

with a = 1.

(çj, regular in the lower half-plane and vanishing at = 0, with the real part along

_{= x}

given by Equation [56], is immediately found by the aid of the Cauchy integral as

follows

d2 df

= Re

d

-2

We linearize accordingly the inner problem and replace Equation [19] by ± 1

-l

- (-i)

The asymptotic expressions of. arid , obtained from

Equations [57] and

_[58],

for large , are

(

= x)

[56]

while = becomes, according to Equations [18] and [51], ..[57]

[s8]

= 1 + + 0 C 1

[59]

(34)

Now, it is easy to ascertain that the above inner solution may be matched with the familiar Lamb solution (Equation [5])

for a doublet in the plane, provided that h= c. Differ-entiating Equation [5], replacing z and h by _C and cH, res-pectively, and expanding inc we arrive at

w = 1 +

c2[t

+

2eEi(iC)]

₊ + kiç

±

2+XeTi.i(iC)]

± 0(c4)

[61]

Taking the inner limit, with ç = c, - 0, = 0(1),

we get from Equation [l

= 1 + 0(c)

[62]

which matches the outer limit of the inner solution (Equation

[59 ).

The inner limit of z is determined by a similar pro-cedure from Equations

_[38]

and [61] in the form

[63.)

C _C

-29-XC

C C

(35)

-:o-and the two solutions (Equations [603--:o-and [63]: match if

[6k]

Since z ₌ + Q(e2), the outer first-order solution of w

.ma be rewritten in terms of z by just replacing by z, The

results now have a simple physical interpretation: the non-linear effect of the free-surface does not change the form of the classical deep submergence solution (Equation [5]), but changes h to _x depending non-linearly on H through Equa-tion [6k], which tnay be rewritten as

H+/H2 +k

2

In other words, the deptb of itnmesion beneath the unperturbed level His .epiacedby an..effebtiveand.iarger depth

_x.

For la'gê H the two dépthe c.dincide, i.e. y. H,

and we obtain the deep submergence solution exactly.

Tuck (1965)

has obtained a simi1at result by computing the second-order correction in the linearized theory with

= 0(1). The curves of drag and lift, for instance, com-puted from first- and second-order aOximations, are. sim-ilar, but shifted by an approximately cohstant amount.

(36)

-3

1-10. SUMMARY AND CONCLUSIONS

A uniform first-order solution of the flow past a

cylinder moving close to a freesurface at high Fr has been obtained by the method of matched asymptotic expansions

The solution is based on a model of nonseparated gravity-free flow near the body. The non-linearity of this model

in-fluences significantly the solution for small depths of immersion H. The near-body flow is sensibly different, in this case, from that given by the deep submergence linearized solution. The far downstream free-waves are also different fiom thcse of the linearized solution: they ar.e related to a vortex singularity replacing the body rather than a doublet. The amplitude of thee first order waves., decays very fast as the depth of immersion increases, however; and the same is true for the associated wave drag.

Ou±' main purpose is the study of non-linear free-surface effects near blunt bow ships. The present analysis suggests that these effects are important in the near body field and it supports the approach of or previous work (Dagan and Tulin,

1969).

A fe extensions of the present work seem to be of interest, for instance: evaluation of higher-order effects for a cylinder, the solution of flow past a hydrofoil and the flow past a sphere or other three-dimensional body.

(37)

-32-R JF E-32-R C ES

Cole, D. J., "Perturbation Methods in Applied Mathematics," BlaisdeilPubi. Comp.,

260 p., 1968.1

Dagan, G., and Tulin, M. P., "Bow WavesBefore Blunt Ships," HYDROMAUTICS, Iticorporated Technical Report

117_ill.,

1+5

p.,

1969.

Havelock, T. H., "The Methodof Images in Some Problems of Surface Waves," Proc. Roy. Soc., A. 115, p.

268, 1926.

Lamb, H., "Hydrodynamics," 6th ed., Cambridge Univ. Press.,

132.

5. Tulin, M. P., "Supercavitatirig Flows Small-Perturbation Theory," Proc.

mt.

Symp. on Appi. of Theorof Functions, Tbilisi USSR, Vol.

2,

pp.

_{1#03_1+39, 1963.}

Van Dyke, M. D., "PerEirbation Methods in Fluid Mechanics," Academic Press, 229 p.,

_1967..

Wu, T. Y. T., "A Singular Perturbation Theory for Non-Linear Free-Surface Flow Problems," Int. Shipbuilding,

(38)

HYDRONAUrICS, INCORPORATED

U'

-V

FIGURE] _{STEADY FREE-SURFACE FLOW PAST A SUBMERGED CYLINDER}

DOUBLET

FIGURE 3 - THE FLOW DOMAIN IN THE PLANE

LI

PLANE

CONTINUOUS b ) JET FLOW _{(c ) BREAKING WAVE}

FREE-SURFACE

FIGURE 2 - THREE MODELS OF FREE-GRAVITY FLOW PAST A CYLINDER

(39)

HYDRONAUTICS, INCORPORATED 3 2 DOUBLET

'V

-.

N.

-FREE-SU RFACE

CYLINDER OF UNIT RADIUS

x = 0.85, a = 0.85 (a2 /x2 1)

=1.00, a = 0.86 (a2 /x2= 0.75)

FIGURE 4 - THE SHAPES OF THE FREE-SURFACE AND THE CLOSED BODY DERIVED WITH THE AID OF THE INNER SOLUTION

(40)

H V DR ON A UT I CS, INCORPORATED

3.0

2.0

1.0

0.0

X( LOCATION OF DOUBLET IN PLANE)

(STRENGTH OF DOUBLET IN 'PLANE)

---4a / ( d2

+ (14- e2)) (STRENGTH OF DOUBLETS in ZPLANE) H - h' /a' (IMMERSION/RADIUS)

--H=X-1/X

U' /(ga')

= 1

FIGURE 5 - THE RELATIONSHIP BETWEEN DIFFERENT INNER AND OUTER PARAMETERS

-j

(41)

HYDRONAUTICS, INCORPORATED 0.1 A

0'

0.0

/

- - R'y/p'a'U'2 (LIFT)

_R'/tTp'ga'2 (DRAG)

1.0 X

I.

0.5 1.0

FIGURE 6 - THE LIFT AND THE DRAG AS FUNCTIONS OF )(

/

0.3 0.2

/

3.0 2.0

(42)

HYDRONAUTICS, INCORPORATED

____-R'/ITpga'2 (LIFT/BUOYANCY)

R'/itp ga'2 (DRAG/BUOYANCY)

/

r

/

.1

_/

/

I

II

I

/

_/

/

_/

/

I

/,

I

//

,

/ ',

H=il

0 1 2 3 4 5

U' /(g a')

I

/

H=0

H = 0.5 H = 1 ,O

=0

H =0.5

FIGURE 7 - THE LIFT AND THE DRAG AS FUNCTIONS OF Fr FOR GIVEN H

0.5

0.4

0.1

(43)

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3, REPORT TITLE -

-GRAVITY FLOW PAST A CYLINDER MOVING BENEATH A FREE-SURFACE

4.DESC RIP TI yE NOTES (Type of report andirrclusive dates)

-Technical Report

5. AIJTHOR(SI (First name, middle initial, last name)

G., Dagan

6 REPORT DATE

June ₁₉₇₀

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(3. ABSTRACT - - .

-The gravity flow past a cylinder moving close to a free-surface. at

high Fr is investigated by the method of matched asymptotic expansions. In contrast with the linearized solution in which the dimensionless

depth of immersion h h'g/U'2 is kept constant, in the present analysis

0 as Fr

The inner fl.ow.model is that of a nonseparated non-linear gravity-free flow past a doublet. The non-linear effects are strong when the depth of immersion H =. h'/a' is small. The lift and the wave-drag are

different from those given by the linearized solution. Wl-iefi the depth of immersion is moderate the linearized solution is valid provided that the depth of immersion is replaced by an effective depth larger than the

actual one. This result agrees with the second order computations of