A variational principle associated with a localized numerical solution of unsteady free-surface flows

DTNSRDC-82/077

ONAL PRINCIPLE ASSOCIATED WITH A LOCALIZED NUMERICAL SOLUTION

ADY FREE-SURFACE FLOWS

DAVID W. TAYLOR NAVAL SHIP
RESEARCH AND DEVELOPMENT CENTER

Bethesda, Maryland 20084

A VARIATIONAL PRINCIPLE ASSOCIATED WITH A
LOCALIZED NUMERICAL SOLUTION OF UN-
STEADY FREE-SURFACE FLOWS

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SHIP PERFORMANCE DEPARTMENT
RESEARCH AND DEVELOPMENT REPORT

September 1982 DTNSRDG-82/077

MAJOR DTNSRDC ORGANIZATIONAL COMPONENTS

DTNSRDC
COMMANDER 00
TECHNICAL DI athe hy

OFFICER-IN-CHARGE
CARDEROCK

OFFICER-IN-CHARGE

ANNAPOLIS

SYSTEMS
DEVELOPMENT
DEPARTMENT

AVIATION AND
SURFACE EFFECTS
DEPARTMENT

SHIP PERFORMANCE
DEPARTMENT

15

COMPUTATION,
MATHEMATICS AND
LOGISTICS SERA es

STRUCTURES

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PROPULSION AND

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eal DEPARTMENT
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A VARIATIONAL PRINCIPLE ASSOCIATED WITH A ae
LOCALIZED NUMERICAL SOLUTION OF UNSTEADY are,

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David W. Taylor Naval Ship Research Program Element 61153N
and Development Center Task Area RRO140302

Bethesda, Maryland 20084 Work Unit 1542-018

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September 1982

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Variational Principle Numerical Ship Hydrodynamics
Finite Element Technique Nonlinear Theory

Free Surface Convolution

Functional

. ABSTRACT (Continue on reverse side if necessary and identify by block number)

In this report, a variational principle for unsteady body wave problems
is treated both with and without a convolution integral and with both linear
and nonlinear free-surface conditions. Functionals are obtained for the
numerical computation of unsteady flow fields near a body that moves on or
beneath the free surface. This formulation can be applied to ship hydro-
dynamic performance problems of water entry and body slamming, as well as to
arbitrary body motion.

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TABLE OF CONTENTS

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NOTATION

D Fluid domain

f; Function defined in Equation (23)
g Acceleration of gravity

By Function defined in Equation (24)
h Body or free surface defined in Equation @))
J Functional

m Source distribution

Pp Pressure distribution

Sp Free surface

Sy Interface of near and far fields
S. Body surface

t Time

Viet Rectangular Cartesian coordinates
0) Variation

0 Water density

T Time

a) Potential

w Function defined in Equation (21)
w Frequency defined in Equation (21)
Subscript

i The ith order

n Normal derivative toward fluid

fe) Projection on the z = 0 surface

t The time derivative

iv

The z derivative
Near field, the first order

Far field, the second order

ABSTRACT

In this report, a variational principle for unsteady body
wave problems is treated both with and without a convolution
integral, and with both linear and nonlinear free surface con-
ditions. Functionals are obtained for the numerical computation
of unsteady flow fields near a body that moves on or beneath the
free surface. This formulation can be applied to ship hydro-
dynamic performance problems of water entry and body slamming,
as well as to arbitrary body motion.

ADMINISTRATIVE INFORMATION
The work reported herein has been supported by the Numerical Naval Hydrodynamics
Program at the David W. Taylor Naval Ship Research and Development Center. This

program is jointly sponsored by the Office of Naval Research and DINSRDC under Task
Area RRO140302, Work Unit 1542-018.

INTRODUCTION

In the early 1970's the David W. Taylor Naval Ship Research and Development
Center (DTNSRDC) recognized the demand for advanced numerical methods to predict the
hydrodynamic performance characteristics of naval ships, particularly when classical
methods proved inadequate.

Thus, in 1974 the Numerical Naval Ship Hydrodynamics Program was begun at
DTNSRDC. Under this program the author previously investigated the steady ship-—wave
problem using a variational principle associated with a localized finite-element
Pochndaeee This method is useful particularly to analyze the flow field near the
ship in detail; in the far field, the Michell approximation can be used. This report
extends the problem to the unsteady case.

For both the steady and unsteady problems, the simple calculation is for a
linear free surface condition with exact body boundary conditions. An iterative
method is needed for a nonlinear free surface condition. However, in the unsteady
problem, the variational principle requires an integration with respect to time using
the initial conditions; in this instance, a convolution integral is useful. The
variational principle for the unsteady body wave problem with exact body boundary

conditions is treated both with and without convolution, and with both linear and

*A complete listing of references is given on page 13.

ai

nonlinear free surface conditions. For the linear free surface problem, a func-
tional for the variational principle is obtained with a convolution rather than a
general integral. The convolution cannot be applied to a nonlinear free surface
problem; the condition is required for large values of time. A nonlinear solution
derived using an iteration scheme having the linear convolution form is also dis-
cussed. The time integration can be eliminated if the motion is sinusoidal.

This formulation can be applied to problems of water entry and body slamming,

as well as to arbitrary body motion.

NONLINEAR PROBLEM
Since problems dealt with here can be generalized easily to three-dimensions,
for simplicity we first consider a two-dimensional problem in the rectangular Car-
tesian (x,z) coordinate* plane. When a body whose surface is represented by

S. [z=h(x,t)] | @)

enters the water surface Sa (Ze= 10), bes a = h(x,t), t > 0) at time t = O, or when
a semisubmerged or fully submerged body starts to move at time t = O and either exits
the water or stops moving at t = th» then the boundary conditions for a velocity

potential $¢ are as follows:

¢ = >, = 0 for t < 0 everywhere

+ (1b)? - 6, + gh=0 oS (2)

h - VoV (h-z) = 0
h_ - ® eo = 0
t n x

Here, S(t) is the submerged body surface varying with time t, and n is the normal

direction into the fluid. We consider potentials 1 in the domain Dy and >, in the

*Definition of notations are given on page iv.

domain Dos where dD, is the near field including 85> and Dy is outside of D,- Then
at the interface Sy of D, and Das we need to have

(4)
¢,--¢

n 2n
The outer potential 5 in Ds is assumed to satisfy the linear free surface condition

ate t SPon © o

For such 5 we know the time-dependent Green fundetor

Now we will construct a Lagrangian for the previously described problem, con-

sidering the Lagrangian that feateee used

ic

‘ 2
1 re x
i -| \j (5 Ma % 1-41, ) azexet +f | g > dxdt a | (0,- 5 by) bo. dzdt (6)

a
0 Dd, 0 Sip 0 Sy

where P> %, and h vary with time, and t is a sufficiently large time after the body
has either exited from or come to rest in the water so that we can safely assume that
the variation bo), Sb, aue 5b, vanishes at t = tT. It will be shown later that the
use of a convolution integral necessitates only the initial condition without the
Condition at t = T.

Since

hiGcat ye ~niGc, t)
ma bdz = ods + $(z=h) a (7)

we have

re h(x,t) h(x,t)

{ fx] >. dzdt I aE al odz — o(z=h) hy dxdt
0

heat) T AE
“| ee | | dt (z=h) he dx
0
or
Ag cr 5
{| ), dzdxdt = S| | h, ¢dxdt +f bdxdz (8)
Dis Ce) Q ss. D, (t=t)

Now we take a variation of J in Equation (6), and use the Green theorem, and the

He eye aL
condition at large t = T obtaining

T
1
6J -{ | € Vo, Vy ,t2h ) Oh dxdt

0 Sie

au 40
“| il vo, 89, dzdxdt I { (19/65 *,) 6h, dxdt
0 0 3, ,Us.

Di 1F

i ic
=| | (>, +45,) 66, dade = | (¢,-$5) 665, dadt

0 Ss; 0 Sy

i
1
+ “| J (50. P5409 56, ) dzdt = 0 (9)
0 Ss

However trom the identity,

ff (9° 865-50" (66,)} ards -| {$, 66,-$5(8b,,)} ds = 0
S

7) Sor

and from the linear free surface condition on Sop Equation (5), and the condition

at t = T, we have

€ i
| J (9 55,,-$5565,) dadt = =| A (8$505,-¢5505,,) dxdt
0 Sj 0 Se
(10)
iG
. Al SUT he STP ores a | (8$.565,7$95¢5,) = dx = 0
ope w Sop

so that the last integral of Equation (9) vanishes. Since 5by5 5b, Sbo0> and 6h

are arbitrary, we obtain from Equations (9) and (10) the corresponding time-
dependent, free surface boundary value problem represented by Equations (2)-(4)
together with the Laplace equation for ¢. Therefore, solving the variational problem
with the functional J of Equation (6) is equivalent to solving the Laplace equation
with the boundary conditions as set forth in Equations (2)-(4), provided we assume

that

6b, = 5b, = Ob, = 0 at t=T

LINEAR PROBLEM
If we assume a linear free surface condition in both Sip and Sop and keep the

exact boundary condition [Equation (3)] on the body surface,

Paper ie! nH ad

then we may use

18
a {fa Vo,V$, dzdxdt - ak [ $191 ep dxdt
OD

iL

{Jom f]

where S is the projection of S, on z= 0. When we take the variation of J, we have

Fo
S13 AD.
Sace Veoh edule dsdese Oe ee deat
me Tele eee pa | UA aes, lee) oe
anny SN

1
(i> 2&2) $,,dsdt (12)

ie
( o,, ve -,) dxdt

T c
-| j ($4 -%5) bo) dzdt -| j (6) 5,257 dzdt (12a)

0 Sy 0 Sy

Here, in addition to Equation (10), the following equation holds

J js Care dxdt -f J Tee dxdt f i ee dxdt

als

Z 2) a +f 9750 OO ter OX 2
0 “Ss 0

Fo ene ae (13)

Thus, as in the previous section, we can easily derive the corresponding linear free
surface boundary value problem using Equation (5) with the exact body boundary con-

dition in Equation (11).

USE OF CONVOLUTION
Equations (6) and (12) are Lagrangians in a time-dependent, two-dimensional

space with nonlinear and linear free surface conditions, respectively. They could

be localized in space but not in time. Namely, we had to specify the conditions on
Se at t = 0 and t = Tt with a sufficiently large T. On the other hand, we did not
require the initial condition un = 0 at t = 0. In addition, such time T when 1 = 0
on Sp may be too fleeces for practical use. For linear free surface boundary condi-
tions, we can treat our variational problem in the same way as those who have treated

variation principles for linear initial value problems using convolutions defined

by
i
* = oe
oT 5 :{ >, (x,z,t) (85257 t) dt
0
(14)
ab ad ao ao
= 1 2 1 D
15 * VO5 ~ Ox z ox us Oz ; Oz
We change Equation (12) to
Ta ye. eo deol] Caren ecee
2 al 1 22 il det
an SG
5 ak ;
| he x %) dx | (0° 7 +, * Pon dz (GLE)
ne Si

If we use the identity relation

and the initial condition %) =o = 0 at t = 0, instead of the conditions Ot = 0 on

lt

Sp at t = 0 and t = Tt with large T used in the previous section, we obtain, for any

time, T

T 1G

s| 1 * ee dx -| 5o, (x, t) Prep Hott) dxdt | | >, (x,t) 5b pp He T-t)dxdt
0s 0s

Fo Fo Fo

T
= 2 ( 6, (x,t) dy pp So Tot dxdt
05
Fo

+f {($, (x,t) S$, , (x, t-1)-$,, (x, t) 84, (x, t-t)} 14 dx
S
Fo

il
ho
o>

T
| dp, (x,t) by pp Ho Tt) dxdt
S

Fo

(16)

HI
N
n>
OQ
a
ke
>
rR
ct
Gr
a.
*

where T need not be large, and x represents a point on the free surface z = 0. Thus,

we obtain as in Equation (12a)

2 x :
D
i

Sao

f2.
- | 6b, * (1, hy+1 - hy] dx

S
Ss

|| (5-95) * 86, dz - i (b, 45.) * 60,42 (17)
eT aT

To obtain Equation (17), we used a convolution expression of Equation (16), where we

can use the initial condition 4 = =1 0;

Ve

Since Soy > Soo, and Soo, are arbitrary, we obtain the corresponding equations
for a time-dependent linear free surface boundary value problem that has a unique
solution.

If we lift out Sy so that D, occupies the entire fluid domain, then the last

integral of Equation (15) ieee The resulting equation appears much simpler
than that obtained by Mapeae! due to a simple difference in the treatment of the
free surface condition. Equation (15) does not give the wave height as a natural
boundary condition, whereas Murray's corresponding equation does. However, from h =
$,/8 the wave height can be obtained.

If we consider eigen solutions that satisfy only the Laplace equation; the
linear free surface condition, Equation (5); and the radiation condition in dD, U Dy»

then >, which satisfies the body boundary condition in Equation (11), can be derived

from Equations (15) and (17) by using

1 Vid.
= —_— — *
J | (3 Oe hyt1 ny] odx (18)
s
s
When we know a functional whose minimum value is attained by the solution, we
can find the solution numerically by such methods as the finite-element eechatiques
F P 9
or singularity method.

For example
b= D7 a9) (19)

where o, is the Green function, which is available for this problem for a source
distribution on the body surface. The source distribution m; will be obtained from

soltuion of the simultaneous equations,

SO . fa1'2,°.2N (20)

SINUSOIDAL MOTION

If we consider a sinusoidal ship oscillation such as h =f Bawe for S, of
Equation (1), we substitute
i iwt
into Equation (12), integrate with respect to t, and obtain
-iwt Al, Ne 2.
e J= Jy = > Vb, dzdx - oe vy dx
Si Sp
+) fb, dx-] (vw -tu,)u, 4 (22)
i ices ( P22 5) Dass © ji

This is exactly the same Lagrangian that Bai and Yeung. used. Working from Equation

(18), we can apply eigen solutions to the whole field by using

i =| (5 ¥,-F) ds

where
h,/ vno+1 ape

A similar functional was used by Sao et Fale? to solve the problem of a heaving

oscillation of a dock.

ITERATIVE SCHEME
For problems with the linear free surface condition, we can completely localize
the numerical scheme in DS with 0 < t < Tt for any T with the help of the convolution
form. However, for nonlinear problems, the finite-element technique has to rely on

: P “ : saa oe
an iteration. We may thus use an iterative free surface condition on z = 0

10

OVS gel f=: 0 (23)
gh, - bat +g, = 0 (24)

where, for the first order perturbation solution, fy = 0, 8) = QO and, for the nth

order, ae and g, are known functions of 5 of the (n-1)th or the lower order

solutions.

Then the Lagrangian for each f. is

1 i
= —- * —_——
oe {J Paes eae | oie uae &
Dy (z=0)ND,
ae * *
| $,, *f, det fo, 4, ax
(z=0)9D,

oa
oJ, = 0 (25)
where the solutions for 1= 1, 2,...n - 1 should be used to determine the solution

when i = n. Equation (24) gives the wave height h for each i.
If we specify a time-dependent, free surface pressure distribution p on the

projection oan of S, to z = O instead of hy in S. we may use

£, = 8) — p/P OD ee

in Equations (23) and (24) for the first order, where p is the water density.

lel!

Although we have discussed the two-dimensional time-dependent problem, Equations

(19) through (24) can be extended to the three-dimensional time-dependent problem.

| al il
= = *k Sek
D

That is

l (2=0)ND,
- \| , * fs dxdy (fr, * 4 dxdy
(z=0)ND,
alt
-{{ (¢)- 2 b,)* 6, de
=F

and soon.

CONCLUDING REMARKS

We have formed functionals with both linear and nonlinear free surface boundary
conditions. For the former but not the latter case, we could apply a convolution
integral. However, the body boundary condition is satisfied exactly in both cases.
In many cases, the flow field near an arbitrary body is of interest, and eigen solu-
tions with linear free surface conditions are known. Even in any large unsteady
motion such as ship slamming, the free surface condition for a short period in the
beginning may be linear, then the convolution may be applied in the early time
period. Especially in the slamming problem, the peak pressure is known to be reached
early in the beginning and estimation of the early pressure distribution on the
slamming body is required. With this functional, we can find the solution for an
arbitrary body numerically by such methods as the finite element technique or singu-

larity method. Thus, a wide application of such functionals can be expected.

12

REFERENCES
1. Yim, B., "A Variational Principle Associated with a Localized Finite-
Element Technique for Steady Ship-Wave and Cavity Problems," Proceedings of the First
International Conference on Numerical Ship Hydrodynamics, David Taylor Naval Ship

R&D Center (1976).

2. Wehausen, J.V. and E.V. Laitone, "Surface Waves," Encyclopedia of Physics,

Springer-Verlag, Berlin, Vol. IX, pp. 446-778 (1960).

3. Stoker, J.J., "Water Waves," Interscience Publishers, Inc., New York (1957),
pp. 187-196.

4, Yim, B., "Investigation of Gravity and Ventilation Effects in Water Entry
of Thin Foils," Proceedings of the International Union of Theoretical and Applied
Mechanics Symposium held in Leningrad (1971), NAUK Publishing House, Moscow, pp.
471-475 (1973).

5. Luke, J.C., "A Variational Principle for a Fluid with a Free Surface,"

Journal of Fluid Mechanics, Vol. 27, Part 2, pp. 395-397 (1967).

6. Gurtin, M.E., "Variational Principles for Linear Initial-Value Problems,"

Quarterly Journal of Applied Mathematics, Vol. 22, pp. 252-256 (1964).

7. Murray, J.C., "A Note on Some Variational Principles for a Class of Linear
Initial-Boundary Value Problems," Journal of the Institute of Mathematics and

Applications, Vol. 12, pp. 119-123 (1973).

8. Bai, K.J. and R. Yeung, "Numerical Solutions of Free-Surface Flow Problems,"

10th Symposium on Naval Hydrodynamics, Office of Naval Research (1974).

9. Sao, K. et al., "On the Heaving Oscillation of a Circular Dock," Journal of

the Society of Naval Architects of Japan, Vol. 130, pp. 121—=13F~ GVA”:

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ANS) 1540 B. Yim

i 1542 T.T. Huang

Al 1542 J. Bai

i 1560 D. Cieslowski
iL 1560 Division Head
1 1561 GAG Cox

1 1561 S.L. Bales

1 1562 D.D. Moran

il 1562 Es. Zarniek
il 1562 Y.S. Hong

16

Copies
aL
AL
1

10

Code
1563
1564
1564
Byala Db
522.1
D226 2

Name

W.E. Smith

J.P. Feldman

R.M. Curphey

Reports Distribution
Library (C)

Library (A)

Ny

DTNSRDC ISSUES THREE TYPES OF REPORTS

1. DTNSRDC REPORTS, A FORMAL SERIES, CONTAIN INFORMATION OF PERMANENT TECH-
NICAL VALUE. THEY CARRY A CONSECUTIVE NUMERICAL IDENTIFICATION REGARDLESS OF
THEIR CLASSIFICATION OR THE ORIGINATING DEPARTMENT.

2. DEPARTMENTAL REPORTS, A SEMIFORMAL SERIES, CONTAIN INFORMATION OF A PRELIM-
INARY, TEMPORARY, OR PROPRIETARY NATURE OR OF LIMITED INTEREST OR SIGNIFICANCE.
THEY CARRY A DEPARTMENTAL ALPHANUMERICAL !DENTIFICATION.

3. TECHNICAL MEMURANDA, AN INFORMAL SERIES, CONTAIN TECHNICAL DOCUMENTATION
OF LIMITED USE AND INTEREST. THEY ARE PRIMARILY WORKING PAPERS INTENDED FOR IN-
TERNAL USE. THEY CARRY AN IDENTIFYING NUMBER WHICH INDICATES THEIR TYPE AND THE
NUMERICAL CODE OF THE ORIGINATING DEPARTMENT. ANY DISTRIBUTION OUTSIDE DTNSRDC
MUST BE APPROVED BY THE HEAD OF THE ORIGINATING DEPARTMENT ON A CASE-BY-CASE
BASIS.