Hydrodynamic impact simulated by impacting rigid, right circular cylinder on smooth water surface, and computer progra

Kettleborough, C. F.

English

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CASE FILE 
C O P Y  
TEXAS ENGINEERING EXPERIMENT STATION 
T E X A S  A B M  U N I V E R S I T Y  
C O L L E G E  S T A T I O N  T E X A S  7 7 8 4 3  
. . .  < I _  . -  
, ,  
https://ntrs.nasa.gov/search.jsp?R=19700002490 2020-03-12T04:50:04+00:00Z
STATUS REPORT TO NATIONAL AERJNAUTICS 
AND SPACE ADMINISTRATION 
Apollo Water Impact 
NASA Research Grant No .  NGR-44-001-071 
Report  f o r  Per iod 
A p r i l  1, 1969 t o  October 1, 1969 
Prepared by 
Research Assoc ia t e  
October 1969 
Submitted by 
Texas Engineering Experiment S t a t i o n  
Texas A&M Unive r s i ty  
Col lege S ta t ion ,  Texas 
Abs t rac t 
T h i s  r e p o r t  d e s c r i b e s  (a) the s t a t u s  of t h e  computer program be ing  
developed t o  s imula t e  the  i m p a c t  of a r i g h t  c i r c u l a r  c y l i n d e r  on a smooth 
water s u r f a c e  and (b) t h e  experimental  equipment and methods. 
APOLLO WATER IMPACT 
In t roduc t ion  
Hydrodynamic impact h a s  been s imulated by impacting a s t a t i o n a r y ,  
r i g i d ,  r i g h t  c i r c u l a r  p r o j e c t i l e  wi th  a v iscous  incompressible  f l u i d  i n i t i a l l y  
a t  v e l o c i t y  V . 
pac t  v e l o c i t y  of t h e  f a l l i n g  p r o j e c t i l e .  
cal  model p r i o r  t o  t h e  i n s t a n t  of impact is shown i n  F igure  1. 
The i n i t i a l .  v e l o c i t y  of t h e  f l u i d  corresponded t o  t h e  i m -  
0 
A diagram of t h e  s imulated physi-  
This  problem h a s  been a t t acked  numerical ly  us ing  t h e  o r i g i n a l  Marker- 
And-Cell (MAC) code wi th  t h e  a d d i t i o n  of a moving coord ina te  system. The 
MAC method w a s  developed by Harlow and Welch (1, 2 ,  3) of Los Alamos Sci- 
e n t i f i c  Laboratory f o r  s tudy ing  t r a n s i e n t  f lu id- f low problems involv ing  f r e e  
su r faces .  The l a r g e  number of d e t a i l s  concerning i t s  use  can be found i n  
t h e  l i t e r a t u r e  c i t e d .  
form of t h e  complete Navier-Stokes equa t ions  t o  t h e  moving f l u i d  and keeps 
t r a c k  of t h e  f r e e  s u r f a c e  by moving about  massless f l u i d  markers i n  ac- 
B a s i c a l l y ,  t h e  MAC method a p p l i e s  t h e  f i n i t e  d i f f e r e n c e  
-cordance  wi th  average  l o c a l  v e l o c i t i e s .  P re s su res  are obta ined  from v e l o c i t y  
i n i t i a l  cond i t ions  us ing  a n  i m p l i c i t  technique.  
v e l o c i t i e s ,  t h e  momentum equat ions  are i n t e g r a t e d  e x p l i c i t l y  y i e l d i n g  new 
v e l o c i t i e s  which are used t o  move t h e  markers and t o  c a l c u l a t e  p re s su res  
f o r  t he  next  t i m e  event .  Results of t h e  MAC s imula t ion  con ta in  p r e s s u r e  
and v e l o c i t y  d i s t r i b u t i o n s  beneath t h e  p r o j e c t i l e  as w e l l  as a g r a p h i c a l  
r e p r e s e n t a t i o n  of t h e  deformed f r e e  su r face .  
Using t h e s e  p re s su res  and 
Appl ica t ion  t o  a Right  C i r c u l a r  Cyl inder  
The computat ional  g r i d  system is  shown i n  F igure  1. Many problems were 
-2- 
encountered wi th  t h e  o r i g i n a l  MAC method dur ing  t h e  course  of t h i s  per iod .  
Computer r e s u l t s  ob ta ined  us ing  t h i s  method w e r e  more o f t e n  than  not  
p h y s i c a l l y  impossible ,  such as, nega t ive  p r e s s u r e  f i e l d s ,  u n r e a l i s t i c  v e l o c i t y  
d i s t r i b u t i o n s ,  and a l i n e a r  dependence upon t i m e  s t e p .  These poor r e s u l t s  
are brought  about  by t h e  f a c t  t h a t  t h e  o r i g i n a l  MAC method does not  r e l i a b l y  
main ta in  con t inu i ty .  
Severa l  modi f ica t ions  have been made t o  the o r i g i n a l  MAC method of 
s o l u t i o n  i n  o rde r  t o  o b t a i n  r e s u l t s  which are of t h e  same o r d e r  as those  
obta ined  experimental ly .  
ensur ing  t h a t  c o n t i n u i t y  i s  being maintained throughout t he  f l u i d .  An i m -  
The most important  mod i f i ca t ion  is  c o n s i s t e n t l y  
p l i c i t  method of so lv ing  f o r  v e l o c i t i e s  has  been inc luded  t o  make s u r e  
t h a t  t h e  v e l o c i t y  f i e l d  is  compatible wi th  t h e  p re s su re  f i e l d  and a t  t h e  
same t i m e  ensure  con t inu i ty .  A b r i e f  gene ra l  f low diagram o u t l i n i n g  t h e  
computat ional  procedure i s  shown i n  F igure  2. This  procedure can be termed 
a n  i m p l i c i t - i m p l i c i t  procedure due t o  t h e  i m p l i c i t  c a l c u l a t i o n  of bo th  
p re s su res  and v l e o c i t i e s .  
r' 
With t h i s  scheme, p re s su res  and v e l o c i t i e s  have been obta ined  and a 
v a l i d  range of t i m e  s t e p s  e s t ab l i shed .  
dV 
t i o n  ve r sus  t i m e  f o r  var ious  t i m e  s t e p s  are shown i n  F igure  3. 
Typ ica l  p l o t s  of p r o j e c t i l e  dece lera-  
d t  
T i m e  s t e p s  l a r g e r  t han  those  shown cause t h e  i m p l i c i t - i m p l i c i t  scheme t o  
d iverge .  I f  t h e  t i m e  s t e p  i s  much smaller than  t h a t  shown i n  F igure  3, no 
advantage i s  gained because of the  l eng th  of computer t i m e  r equ i r ed  t o  ob- 
t a i n  a s o l u t i o n .  Any t i m e  s t e p  chosen w i t h i n  t h e  accep tab le  range i s  too  
s m a l l  t o  a l low t h e  markers t o  b e  moved a v i s i b l e  d i s t a n c e  w i t h i n  a reason- 
a b l e  amount of computer t i m e .  
S e v e r a l  ex tens ions  t o  t h e  prev ious ly  descr ibed  i m p l i c i t  method of 
s o l u t i o n  are p r e s e n t l y  be ing  considered. 
implemented a t  t h i s  t i m e .  These ex tens ions  are: 
The f i r s t  of t hese  is  being 
1) C a l c u l a t e  p re s su res  and v e l o c i t i e s  i m p l i c i t l y  f o r  t h e  f i r s t  
several t i m e  cyc le s  u n t i l  t h e  pressure  and v e l o c i t y  f i e l d s  are s u f f i c i e n t l y  
developed. 
from t h e  i m p l i c i t  method of c a l c u l a t i n g  v e l o c i t i e s  t o  an e x p l i c i t  method. 
Experiments w i th  t h e  MAC method have shown t h a t  t h e  e x p l i c i t  method of 
s o l u t i o n  a l lows  t h e  use  of l a r g e r  t i m e  s t e p s  and thus  l a r g e r  marker d i s -  
placements.  
more economically f e a s i b l e .  
The exac t  number of t i m e  cyc le s  is y e t  t o  be determined. Change 
The a b i l i t y  t o  use  l a r g e r  t i m e  s t e p s  makes l a r g e r  t i m e  events  
2) Inc lude  a i r  i n  the  computational reg ion  t o  t h e  s i d e  of and several 
r a d i i  beneath t h e  r i g h t  c i r c u l a r  cy l inde r .  
t h e  f a l l i n g  c y l i n d e r  and t h e  quiescent  w a t e r  s u r f a c e  depends upon t h e  
c h a r a c t e r i s t i c s  of t h e  v e l o c i t y  and p res su re  f i e l d s  and is y e t  t o  be  de t e r -  
mined. 
v e l o c i t i e s  s t u d i e d ,  t he  i m p l i c i t - e x p l i c i t  scheme descr ibed  above w i l l  b e  
used t o  e v a l u a t e  p re s su res  and v e l o c i t i e s  f o r  a i r  and water as s e p a r a t e  
f l u i d s .  The a d d i t i o n  of a i r  al lows t h e  f r e e  s u r f a c e  t o  deform be fo re  
impact.  
e f f e c t "  o f t e n  noted i n  p re s su re  d i s t r i b u t i o n s  measured exper imenta l ly .  
The i n i t i a l  d i s t a n c e  between 
Assuming t h e  a i r  t o  be  incompressible  f o r  t h e  ve ry  low e n t r y  
The deformed f r e e  su r face  w i l l  t h e o r e t i c a l l y  v e r i f y  t h e  l tcol l iseum 
3) Follow t h e  same procedure as i n  ex tens ion  (2), except  assume a i r  
t o  b e  compressible  even f o r  low impact  v e l o c i t i e s .  With t h i s  assumption 
d e n s i t y  becomes a v a r i a b l e  which must b e  evaluated dur ing  t h e  i m p l i c i t  
s o l u t i o n  of t h e  p re s su re  f i e l d .  I f  t h e  procedure o u t l i n e d  i n  ex tens ion  ( 2 )  
-4- 
. 
y i e l d s  r e s u l t s  wi th  engineer ing to l e rances ,  ex tens ion  (3)  w i l l  no t  be 
necessary .  
Development of a P a r a l l e l  Program i n  Car t e s i an  Coordinates  
I n  June 1969, development was i n i t i a t e d  on a computer program which 
a p p l i e s  t h e  Marker and C e l l  technique i n  Car t e s i an  coord ina te s .  The 
a n a l y t i c a l  theory  and numerical  technique involved are  e s s e n t i a l l y  t h e  
same as i n  t h e  c y l i n d r i c a l  case, the  major except ion  being t h a t  a s t a t i o n a r y  
r a t h e r  than moving coord ina te  system is  used. 
program are: 
The purpose of t h i s  p a r a l l e l  
1) To s imula t e  t h e  dropping of a "block" of water on a smooth f r e e  
s u r f a c e  i n  o r d e r  t o  gene ra t e  t r a v e l i n g  s u r f a c e  waves. 
2) To s e r v e  as a b a s i c  program which can be modified t o  s imula t e  
ob l ique  impact of a v e h i c l e  on a f r e e  s u r f a c e  wave. 
3)  To s e r v e  as a b a s i c  program which can be modified r e a d i l y  t o  
experiment w i th  d i f f e r e n t  v a r i a t i o n s  of t he  Marker and C e l l  technique.  
I n i t i a l  va lues  of p re s su res  and v e l o c i t i e s  must be  known through- 
o u t  t h e  f l u i d  body b e f o r e  impact i n  o r d e r  t o  c a r r y  o u t  t h e  p r e s e n t  impact 
s t u d i e s .  Since such parameters are n o t  known f o r  s u r f a c e  waves, i t  i s  
necessary  t o  gene ra t e  t h e  s u r f a c e  waves a r t i f i c i a l l y  and hence, compute 
t h e  r equ i r ed  parameters  by following through t h e  process  wi th  t h i s  computer 
program. 
The assumption of a x i a l  symmetry i n  t h e  program dea l ing  w i t h  c y l i n d r i -  
ca l  coord ina te s  prec ludes  t h e  i n v e s t i g a t i o n  of ob l ique  impact ,  whereas t h e  
u s e  of two-diemnsional Ca r t e s i an  coord ina tes  makes such i n v e s t i g a t i o n s  
i 
i 
-5- 
poss ib l e  wi thout  undue d i f f i c u l t i e s .  
The non-constant c o n t r o l  volumes of c e l l s  of cons t an t  geometric in- 
crements i n  c y l i n d r i c a l  coord ina tes  o f t e n  magnify and complicate  e r r o r s  
r e s u l t i n g  from us ing  d i f f e r e n t  numerical techniques.  Cont ro l  volumes do 
not  vary  w i t h  p o s i t i o n  i n  Car tes ian  coord ina te s ,  s o  t h a t  a program u t i l i z i n g  
such coord ina tes  i s  much more s u i t e d  t o  experimentat ion.  
To date, t h e  program has been developed t o  a po in t  where s u r f a c e  
waves are formed and p r e s s u r e  and v e l o c i t y  va lues  are computed, b u t  problems 
of s t a b i l i t y  of such s o l u t i o n s  s t i l l  have t o  be d e a l t  with.  It i s  a n t i c i -  
pated,  however, t h a t  such s t a b i l i t y  problems are common t o  a l l  t h e  phases 
of t h e  present  numerical  i n v e s t i g a t i o n ,  and t h a t  once such problems are 
solved f o r  t h e  gene ra t ion  of waves, t h e  same technique can be  appl ied  t o  the  
o the r  phases such as ob l ique  impact. 
Experimental Work 
A l a r g e  l u c i t e  tank  h a s  been cons t ruc t ed  and i n i t i a l  experiments per- 
formed t o  f i n d  s u i t a b l e  p l a s t i c  p a r t i c l e s .  
graphs have been taken  of t h e  motion of t h e s e  p a r t i c l e s  when a s p h e r i c a l  pro- 
j e c t i l e  is impacted v e r t i c a l l y  on the  water su r face .  
frame by frame wi th  a view t o  obta in ing  a t i m e  h i s t o r y  of t h e i r  v e l o c i t y  and 
a c c e l e r a t i o n .  
a c i r c u l a r  cy l inde r .  
High speed co lo r  movie photo- 
These are being examined 
A t t e n t i o n  has  now been turned t o  t h e  hydrodynamic impact of 
Nomenclature 
= r a d i u s  of computing system (no. of r a d i i )  
= he igh t  of computing system (no. of r a d i i )  
= l eng th  of p r o j e c t i l e  (no. of r a d i i )  
ot RP 
P % 
RP 
R,Z  = c y l i n d r i c a l  coord ina tes  
t = t i m e  
= d e n s i t y  of l i q u i d  
= dynamic v i s c o s i t y  o f  l i q u i d  
= i n i t i a l  impact v e l o c i t y  (axisymmetric) 
P 
P 
vO 
v = v e l o c i t y  of p r o j e c t i l e  
P 
P dV = d e c e l e r a t i o n  of p r o j e c t i l e  
d t  
A = impact area of p r o j e c t i l e  
P = pres su re  
U = r a d i a l  component of f l u i d  v e l o c i t y  
V 
W = v e l o c i t y  v e c t o r  
v*c = 0 = divergence of v e l o c i t y  
= a x i a l  component of f l u i d  v e l o c i t y  
+ 
L.2 +aq = Laplac ian  ope ra to r  
= material d e r i v a t i v e  - 
Dt 5-t 
References 
1. Harlow, F r a n c i s  H. and Welch, J. Eddie, "Numerical Ca lcu la t ion  of 
Time-Dependent Viscous Incompressible  Flow of F l u i d s  w i t h  F ree  Surface  ," 
The Physics  of F l u i d s ,  Vol. 8, No. 1 2 ,  pp. 2182-2189, December 1965. 
2. Welch, J. Eddie,  Harlow, F ranc i s  H. ,  et a l . ,  The MAC Method, A Computing 
Technique f o r  Solving Viscous,  Incompressible ,  Trans ien t  Fluid-Flow 
Problems Involving F ree  Surfaces ,  LA-3425, UC-32, Mathematics and 
Computers, TID-4500, March 1966. 
3. Harlow, F r a n c i s  H. and Welch, J. Eddie,  "Numerical Study of Large- 
Amplitude F r e e  Surface  Motions," The Phys ics  of F lu ids ,  Vol. 9 ,  No. 5, 
pp. 842-851, May 1966. 
I 
Figure  1. Diagram of Model and Computational 
Grid Assuming Axial Symmetry 
F l u i d  and pro j ec  t i l e  
p r o p e r t i e s  and i n i t i a l  cond i t ions  de f ined  
I 
u ,  v, known a t  1 t i m e  k- At) , t 
I 
AJV? , p known a t  .g t i m e  (t-At+,) 
L dt 
Find v at (ttat) fro!momentum equat ion  
F ind  v a t  t+a by averaging t 2 ) ;  
* 
I 
from c o n t i n u i t y  equat ion  v s q  = c) 
I 
f 
Solve for p res su re  i m p l i c i t l y  from 
& 1 
Find d e c e l e r a t i o n  of p r o j e c t i l e  from 
4 
I 
from momentum equa t ion  
I l o t  compatible 
t 
Move P a r t i c l e s  
4 
A .) L 
Increment t i m e  by S o l u t i o n  Complete 
1 
1 
STOP 
F igure  2. Br ie f  Ou t l ine  of Computational Procedure 
. 
" J 
1 
L - 
0 
l i  
0 
0 
m 
0 
0 
0 
c-4 
0 
0 
0 
d 
0 
0 
0 
0 
k 
w P, 
a, 
k 
7 
M 
d 
Frr 


Apollo water impact Status report, 1 Apr. - 1 Oct. 1969

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