LIBRARY

RESEARCH REPORTS DIVISION

NAVAL POSTGI SCHOOL

MONTEREY, CALIFORNIA 93940

NPS69-79-007PR

rNAVAL POSTGRADUATE SCHOOL

Monterey, California

EFFECTS OF CAVITATION ON UNDERWATER

SHOCK LOADING -- PLANE PROBLEM, PART I

R. E. NEWTONJuly 1979

Progress Rep<art for Period ApriJune

1 1979 -

1979

Approved for Public Release: Distribution Unlimited

Prepared for: Defense Nuclear AgencyWashington, DC 20 30 5

FEDDOCSD 208.14/2:

NPS-69-79-007PR

Naval Postgraduate SchoolMonterey, California

Gh DUATE SCHOOLDNTEREY, CA 93943-5101

Rear Admiral T. F. Dedman Jack R. Borstinj

Superintendent Provost

The work reported herein was supported by the Defense Nuclear

Agency SPSS.

Reproduction of all or part of this report is authorized.

This report was prepared by:

SECURITY CLASSIFICATION OF THIS PACE (Whan Data Entarad)

REPORT DOCUMENTATION PAGEI. REPORT NUMBER

NPS69-79-007PR2. GOVT ACCESSION NO.

READ INSTRUCTIONSBEFORE COMPLETING FORM

1. RECIPIENT'S CATALOG NUM8E*

4. TITLE (and Subtitle)

EFFECTS OF CAVITATION ON UNDERWATER SHOCKLOADING - PLANE PROBLEM, PART 1

S. TYPE OF REPORT a PERIOD COVERED

Progress April-June1979

* PERFORMING ORG. REPORT NUMBERNPS69-79-007PR

7. AUTHORf»J

R. E. Newton

1. CONTRACT OR GRANT NUMSEftfJ

MIPR 79-608

» PERFORMING ORGANIZATION NAME AND ADDRESS

Naval Postgraduate SchoolMonterey, CA 93940

10. PROGRAM ELEMENT. PROJECT, TASKAREA * WORK UNIT NUMBERS

Y99QAXSF501Work Unit 16

H. CONTROLLING OFFICE NAME AND AOORESS

Defense Nuclear Agency SPSSWashington, D. C. 20305

12. REPORT DATE

Julv 1979IS. NUMBER OF PAGES

IT MONITORING AGENCY NAME a AOORESS/f/ dlttarant Imm Coniralllnd Olllea) IS. SECURITY CLASS, (el thta report;

UNCLASSIFIEDISa. DECLASSIFICATION/ DOWNGRADING

SCHEDULE

16. DISTRIBUTION STATEMENT (el thla Hapott)

Approved for Public Release: Distribution Unlimited

17. DISTRIBUTION STATEMENT (el tha a*at.-act antarad In Block 30, II diltarant tram Maport)

It. SUPPLEMENTARY NOTES

I*. KEY WORDS (Continue an rawaraa alda it p«c<«c«? and Idmnlltr fry aiock nummar)

Underwater shock, cavitation, finite elements

20. ABSTRACT (Centlnum an ravaraa alda II naeaaaawy and Identity Or kiec* mamoat)

Development and testing of a finite element program for shock-induced cavitation is described. A displacement potential function is _the dependent variable used. Graphical output is pre-sented for an encounter which induces repeated cavitation.

DO,:°:

M7, 1473

(Page 1)

EDITION OF I NOV •• IS OBSOLETES/N 0102-014- 4601

I

UNCLASSIFIEDSECURITY CLASSIFICATION OF THIS PAGE (Whan Data Xniarad)

DUDLE1.

• w^VAI

= SCHOOEFFECTS OF CAVITATION ON UNDERWATER SHOCK

LOADING - PLANE PROBLEM, PART I

1. Introduction

In Ref. 1 possible problem formulations for predicting

the effects of cavitation on underwater shock propagation were

examined. It was concluded that using either the displacement

potential (a scalar) or the particle displacement (a vector)

as the dependent variable could produce useful results.

Ref. 2 gave a summary account of an unsuccessful effort to

use ADINA for applications with axisymmetric geometry.

The present report is a summary of progress to date in

developing a capability for modeling a two-dimensional fluid

region in which a structure is submerged. Examination of

available codes disclosed no candidate allowing implementation

of the displacement potential formulation, with cavitation,

for the fluid and simultaneously permitting an appropriately

coupled structural model. Accordingly, an ad hoc code for

the fluid has been developed. A compatible structural code

is being constructed. All codes referred to in this report

are written in FORTRAN IV.

2. Program Development

2. 1 Mesh Generator Q1GNEW)

For a plane problem with a cylindric structure whose

length is normal to the plane, the fluid region is the portion

of the plane exterior to a circle (radius A) . Taking advantage

of symmetry and truncating gives the rectangular region of

Fig. 1. The mesh is generated from a set of super-elements

as shown in Fig. 2. The transitional super-elements (those

adjacent to the structure) each have two curved sides. The

remaining elements are square. Note that, if each quadrant

of the structural "hole" is treated as an element, the region

is topologically equivalent to MU layers, each containing

ML + MR squares. In the region of Fig. 2 the entry face is

at x = MR*(0.9*A) and the exit face at x = ML*(0.9*A). The

top face is at y = MU*(0.9*A).

In the finite element grid each super-element is sub-

divided N*N. The final elements are four-noded quadrilaterals

using linear shape functions. For the transitional elements

the additional nodes are found by using quadratic isoparametric

shape functions for mapping.* The square grid on which the

element mesh is based includes N*(2*N+1) nodes which are

inside the structure. It is advantageous to retain these nodes

in the numbering scheme.

All of the nodal locations within the transition region

can be derived from those in the sector from to 45 by

successive reflections. It follows that there are only

(3*N+l)*N/2 distinct transition elements for which fluid "mass"

and "stiffness" matrices need to be calculated. The remaining

elements are identical squares and only a single additional

pair of fluid matrices is required.

A program (MGNEW) has been written to generate this

element grid. Required input data are the structure radius A

^Mapping with isoparametric shape functions is described inRef. 3.

ExitFace

y

Top

Plane of Symmetry

Fig. 1. Fluid region nomenclature

EntryFace

ML spaces —»"* MR spaces

Fig. 2. Super-element grid

and the mesh descriptors ML, MR, MU, and N. Output data in-

clude nodal coordinates, numbers of nodes and elements, and

global node numbers for each element.

2. 2 Fluid Program (DPLPOT)

A program for tracking shock wave propagation, including

cavitation, has been developed. Equations for the displacement

potential formulation used are given in Ref. 1.

2.21 Initial conditions

A subroutine of the program determines initial values

of the displacement potential and its first temporal derivative

.

These correspond to a uniform hydrostatic pressure plus a

plane shock wave with step rise and exponential decay. The

shock wave is travelling in the negative x direction with

front at x = A at time t = 0.

2.22 Boundary conditions

The governing equation requires that values of the

normal derivative of the displacement function be specified at

boundary nodes. Subroutines are provided to impose a radiation

boundary condition on the entry and exit faces. At the entry

face explicit provision is made for the continuing entry of

the incident shock wave. The analytic basis for these boundary

conditions is given in Ref. 1. A radiation boundary condition

is also imposed at the top face, again including explicit

provision for the passage of the incident shock wave. In the

absence of specific intervention the solution scheme enforces

a zero value of the normal derivative at boundary points

.

Accordingly, no boundary inputs are required for the plane

of symmetry y = 0.

2.23 Temporary structure

Initial testing of the fluid program DPLPOT has been

done with a simplified boundary condition at the fluid-

structure interface. This condition is that the boundary

pressure remains equal to the initial hydrostatic pressure.

Physically this corresponds to substituting an inflated

cylindric bag for the structure. Such a "hole" in the fluid

can readily be made to induce cavitation.

2.24 Coefficient matrices

Coefficient matrices for the fluid are calculated at

the element level. The fluid "mass" matrix is lumped (diagonal)

rather than consistent. Total storage requirements for element

matrices are quite modest because of the limited number of

different elements. The diagonal system mass matrix is

assembled in a single vector whose length is equal to the

number of nodes. No system "stiffness" matrix is assembled.

2.25 Time integration

Numerical integration in time is based on central

difference formulas. The calculation is organized in such a

way that the nonlinearity introduced by cavitation does not

affect the fluid mass or stiffness matrices. Because the

integration algorithm is explicit and the mass matrix is

diagonal, no matrix decomposition is required. As a result,

the central processor time is directly proportional to the

number of fluid nodes. Processing time is a linear function

of the number of time steps. "Overhead" time requirements

for mesh generation, coefficient matrix calculation and

establishment of initial values are approximately equivalent

to 9 time steps.

2.

3

Fluid Program (VELPOT)

The initial form of the program DPLPOT, described above,

required a computationally expensive damping calculation.

To avoid this, the alternative of using the velocity potential

as the dependent variable was explored. Pertinent equations

are given in Ref. 1. For this purpose it was necessary to

revise four of the subroutines of DPLPOT. The alternate

version, called VELPOT, reduced the central processor time by

35%. Although VELPOT results closely resembled those from

DPLPOT, they were not identical. Efforts to account for the

discrepancies led to the development of two one-dimensional

programs, DIS and VEE, which were made to yield identical

results

.

Subsequent development of DPLPOT replaced the inefficien-

damping calculation. This improvement eliminated the computa-

tional advantage of VELPOT and VELPOT was abandoned.

2.4 Structural Program (STRUK2)

A structural program has been under development by

Jack T. Waller, a thesis student of mine. This program

utilizes trigonometric series representations of the loading

and displacements of an orthotropic shell in a state of plane

strain. It is now being tested separately, but has not yet

been coupled with the fluid.

3. Results

Results obtained thus far are relevant to determining

when cavitation will occur, how big the cavity will become,

and how long it will persist. For this purpose, various

combinations of shock pressure, hydrostatic pressure, and

shock decay length have been tested. Effects of region size

and shape, element size, length of time step, damping parameter,

and total integration time have also been examined.

5 . 1 Line Printer Plot

The program provides for both tabular and graphical

output, but the latter has been the more useful in this phase

of the investigation. It is a line printer plot of fluid

nodal pressures, repeated at selected time intervals. For

the purpose of this plot the nodes of the structural and

transition regions are mapped into the topologically equivalent

rectangular array.

Fig. 3 shows the arrangement of the line printer plot.

The left-hand column is the plane of symmetry (x axis of Fig. 1)

and the right-hand column is the "top" face. The entry face

is the first row and the exit face is the last row. Along the

left-hand edge is a rectangle filled with X's, 4 columns wide

and 7 rows high. The X's denote dummy nodes inside the

structure. The rows immediately above and below, and the

column to the right (all marked H) contain the 17 structural

nodes

.

The pressure map of Fig. 3 is for hydrostatic pressure

0.2 MPa and shock pressure 2.0 Mpa with a decay length of 15 m.

The pressure ranges for the mapping characters are shown at

the right. Since the map is for t = there are no pressures

below hvdrostatic.

MESH DATA

ft 3.909 rtTERS

N 3 M.-3 MU -13 MR - 9 MP « 43

ICCflY LENGTH - 13.9 M ACOUSTIC CCLCCITY - 1309. hVS

HYBPOSTflTIC PRESSURE - 9.2939 KPfl SHOCK PRESSURE - 2.999 KPfl

TirC STEP - 9.239 US TCAX * 9.9 ttS TttflP - 9.9 MS

ETR 2.039B-92

PRINT CODE: 121111111 1 112 1

T 9.9 MS

11 1 1111 8 3 931imn»H!;n!'HHlUHiHiin i SW1 1 a n

;

; i : )i : h i ! ; n ; i i j j i M : j ! j i < h j j ] i m i n i ; >»<H>frlHHHHKHHH H H I J H HI 8 3 H J H i 1 J 1 i i l 1 3 3 ii i H a 3 I 3 H 1 1

l f w " m y i v *. 4 -i . * 3 . j " n j i ^ ! j j ji Mill rTmrrmrm iij j /i

riTfriPffi.c!Fift:^f^ft^

BBBEBBEBBBBEBB33BEBBSBBE2EE3BEEBBBBBBBBBBS333:BBBBBBBBBBBBB3B33BBSB3BBSEB3EBBBSBBBBBBBBBB3333

ccccccccccccc:cccccccccc:cccrxccccccccrxrxcrxccctccccc^BDDDDDBDLBnDIErDIiDriiDrrQEELLDSrDEDIiDrEDEEEBLEDDDBIIIIIIBriBEBriDEBDI'liliDDIiDDr'IiEIiDBDBBIlIiIiBBLIiBBBBBa

DICIiDbI'BI'BDLBBI:IiIiprDIiIiIi3BI'I:I'riBI'r BDIi3BSr'BI'DI!3DD

>CC'>HK->»iHH '1 g i) i H 3 M a 1 i » 8 iK J H 3 H ii 8 3 ? 8 iH-q-8-JH->W

?CC>tHHHH> 'I 3 i * i H i i 'HHHW-HH-ii ! g w 3 } IN a C 3 3 i ) I g iHHH-34>o<*hhw-h»-s 4? vi 9 a i j ii a 3 r> 3 n h * -h n 1 ? n *

»

3 .)m 3 3 1 w a a a 1

>C<>M a 9 S 3 .1 4 » 8 3 J-H"hH4HHH-ibH> i>» l 'HHH 3 8 ii 9 3 i !HHHHHHHI H-S^HK-^^r^HH^^J-fr^H) H-tHHHHH iHSr! ) H-HW<HHHHHWHrM-fri«iHH-frg i l> » 9 v -H~>-i-H>-§-HHH a- 3 3-fr»S .1 iHHHHHHHHH-M-iH 'M3^ja> iHHHHHHHHHHHH < l il 'J a >>^HHHH-H-->>HHHH » ->y*-g BHHH »»WKHKH) j IH i) M-iHHHhH^^HHHHHHHH

HHHHH JWH-3-3^-3 <>S-* IHHHHHHJ » » » *H-^-tf-a a 4 i? 3 3 »vTH-i-*HWHHHHHHHK^-i) 3 3 a i 8 a J a IHKHH i tf »d»0 3 8 H-H-fr* i d-B-i-B-a ii jfrH

PRESSURE CODE

Symbol

X

blanki

3

6

3

H

A

B

C

E

F

Pressure

StructureCavi tated

- .04

.04 - .09

.09 - .14

.14 - .18

.18 - .43

.43 - .77

.77 - 1.101 .10 - 1.431 .43 - 1.771 .77 - 2.10over 2.10

Fig. 3. Initial pressure map

3.2 Cavitation Example

Fig. 4 exhibits a sequence of pressure maps at 16 ms

intervals based on the initial conditions of Fig. 3. Note

that the cavity develops rapidly in the first two frames,

shrinks in the next two, and is completely closed by 30 ms

.

The succeeding frames in which the cavity opens again represent

the first instance in which this behavior was encountered.

It has been established by a longer duration run that the

cavity does not open again after 144 ms

.

2.001

: u 1 1'. ii 1 11 1 1 1 1 1 1 1 ! 1 1 - -

I

I

u 1 1 1 n 1 1 1 1 1 : 1 1 1 n :

:

niiuiiuiiuuiiiiiiuisisuiiiiimiiiiiiiiuiiiilixUUllllllllllllultllllll -

I I I i i : i iiuuimiuu:mi uui:

1111

-EEjEESrEEEESEEEEESEEEEEiEEfeeeeeeseeeeeeseeeeeeeeeeeejees-eeeeeeeeeeeeeiseeeeee:e e e £ e i e • e e e e e s e e e e e e s e • i e •

!

minium: 1 1 1 1 1 1 1u 1

1

Illlllllllll1111111111111111U1UU11 1m 1 1 1 : 1 1

1

uiuuuui.uuiiiiiii

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1

. : . : : :-.-.,-:-

11 1 11 1 1 1 1 1 1 1 1 1 1

1

inn ..::::--;1 1 1 1 1 1 1 1 1 1 11 1 1 1

1

mm:::1111111111111111 u::i::::::::::v^tllllllllllllll] i : 1 1 ; : : : .

.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1

1111111 . : 1 1

.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1

iiiiiiiiuiii!!'.::lUllllllUlllll 11111111111111113:1 1 1 1 1 1 1 1 1 1 1 1 1 11

1

l ! 1 1 1 1 ! l : 1 1 1 1 : i . i

.

1111111111111 uiiiimuuuui111111111 iiniiiiuiuiiii!nun llllllllllllllllll

llUllllllUlllllluuimuim mi

1 1 11

' 1 1 1 1 ! 1 1

1

: 1

1

1 1 1 1 1 1 1 1 1 1 1 1 1 1

1

IIUUUIUUI!niiiiiiiiiini

i : -. i

:

iiuuuninn: i : i : : 1 1 : : i ! : 1 ! :U U 1 11 U 1 11

lllll: :lllillllllin: . iiiiiu

11 1 1 1 11". '

: :;:.::..

..::..:.:.::..': ... :

miiiiuunmmmuiilii ..:.:...:

: i n 1 1 1 n n i n : 1 1 1 1 1 1 n 1 1

:

niuiiiiuiuiiiiiuui::iiiiiiinuiuuiiuiu':;:.iiuinuiiuiiiiiiii::::

iiiiiiiimi

.luiiiuiiillllllll!

iiiinu::::: -r--.

mini::.:: .::

limiiiiii: :::::nmumiuuc:'niimmiiiii

llllllllllllll: um 1 1

!

n i u n l

mil '. i : i

:

muuumniimumuuuiuimnuimn

' m : mumu-.:::..m l u i u

:85838838886e6e6eocoj ::::::!; : : 3&£<»<£«eeeebe68S

Fig. 4. Cavitation example

10

REFERENCES

1. Newton, R. E., "Effects of Cavitation on Underwater ShockLoading - Part 1," NPS-69- 78-013 , Naval PostgraduateSchool, Monterey, CA, July 1978.

2. Newton, R. E., "Effects of Cavitation on Underwater ShockLoading - Axisymmetric Geometry," NPS-69- 78-017PR,Naval Postgraduate School, Monterey, CA, November 197

3. Zienkiewicz, 0. C, The Finite Element Method , 3rd ed.,Chapter 8, McGraw-Hill, London, 1977.

11

Initial Distribution List

Copies

1. Research Administration, Code 012A 1

Naval Postgraduate School

Monterey, California 93940

2. Professor R. E. Newton 10

Mechanical Engineering Department

Code 69NeNaval Postgraduate SchoolMonterey, California 93940

3. Weidlinger AssociatesConsulting Engineers110 East 59th StreetNew York, NY 10022ATTN: Melvin Baron

4. Lockheed Missiles and Space Company, Inc. 1

3251 Hanover StreetPalo Alto, California 94304

ATTN: Tom Geers D/52-33Building 205

5. DirectorDefense Nuclear AgencyWashington, D. C. 20305

ATTN: SPSS

26. Superintendent

Naval Postgraduate SchoolMonterey, California 93940ATTN: Code 2124 Tech. Rpts. Lib.

12

U 188307

IS KM 6*l 3 8 2 3

U188307

DUDLEY KNOX LIBRARY - RESEARCH REPORTS

5 6853 01062453 9

111