A model of the change in shape of a raindrop is presented. Raindrops measured by two orthogonal cameras were classified by shape and orientation to determine the nature of the oscillation. A physical model based on potential energy was then developed to study the amplitude variation of oscillating drops. The model results show that oscillations occur about the equilibrium axis ratio, but the time average axis ratio if significantly more spherical for large amplitudes because of asymmetry in the surface potential energy. A generalization of the model to oscillations produced by turbulence yields average axis ratios that are consistent with the camera measurements. The model results for average axis ratios were applied to rainfall studies with a dual polarized radar

Beard, K. V.

Welsh 

NASA Technical Reports Server

https://ntrs.nasa.gov/search.jsp?R=19820015576 2020-03-21T16:57:25+00:00Z
The l a t t e r  i s  a p p a r e n t l y  enhanced  by ae rodynamics  s i n c e  t h e  e x t k r n a l  p r e s s u r e  s h o u l d  
p r o v i d e  a  p o s i t i v e  f eedback  t o  t h e  h o r i z o n t a l  d e f o r m a t i o n .  Suppor t  f o r  a n  aerodynamic  
e f f e c t  i s  found i n  t h e  o b s e r v a t i o n  o f  s t a b l e  h o r i z o n t a l  " ~ r o l a t e s "  f o r  v e r y  l a r g e  d r o p s ' .  
The d r o p  was assumed t o  have  o n l y  s u r f a c e ,  ~ r a v i t a t i o n a l  and  k i n e t i c  e n e r g i e s .  The 
minimum i n  t h e  s u r f a c e  p l u s  g r a v i t a t i o n a l  e n e r g y  was c a l c u l a t e d  n u m e r i c a l l y .  Agreement 
w i t h  wind t u n n e l  d a t a  and  t h e  nume:*ical r e s u l t  o f  P ruppache r  and P i t t e r  f o r  t h c  e q u i l i b r i u m  
a x i s  r a t i o  was w i t h i n  1% f o r  d i a m e t e r s  l e s s  t h a n  5 mm ( F i g u r e  1 )  demons1;rat ing t h a t  t h e  
aerodynamic  and  hydrodynamic p r e s s u r e s  a r e  much l e s s  i m p o r t a n t  t h a n  t h e  h y d r o s t a t i c  p r e s -  
s u r e  i n  d e t e r m i n i n g  t h e  a x i s  r a t i o .  
The s d r f ~ c e  p l u s  g r a v i t a t i o n a l  ene rgy  
f u n c t i o ~  was used  as a  p o t e n t i a l  e n e r g y  "we l l "  
f o r  anhsnmonic o s c i l l a t i o n s  o f  an  e l l i p s o i d a l  
d r o p .  The e q u a t i o n  01' ax i symmet r i c  mo t ion  f o r  
t h e  fundamen tz l  !vile was o b t a i n e d  f rom t h e  
e n e r g y  r e l a t  i o n  
by  s o l v i n g  fo l .  
ra t  l o  
t h e  t e m p o r a l  change  i n  a x i s  
where 
Va = ( 5 ~ ~ / m ) l / ~  (1 + ~ A R ~ ) ' ~ / ~  ( 3 )  
and  
w i t h  a  c m s t s n t  volume c o n s t r a i n t  
An a n a l y t i c a l  s o l u t i o n  f o r  s m a l l  a m p l i t u d e  
y i e l d e d  t h e  R a y l e i g h  o s c : i l l a t i o n  t i m e  ( T R )  
u s i n g  t h e  f a c t o r  o f  1 / 5  i n  t h e  k i n e t i c  e n e r g y  
r e l a t i o n .  L i n e a r  d i s s i p a > i o n  was added  t o  t h e  
c a l c u l a t i o n  by a s s u n i n g  KE = B KE where  B was 
d e t e r m i n e d  f rom t h e  r e ~ u l t  o f  ~ a m b "  f o r  a 
comple t e  c y c l e .  
Numerica1. i n t e g r a t i o n s  were  made f o r  D = 
F i g u r e  1. Drop a x i a l  r a t i o  as a f u n c t i o n  0.12 cm t o  compare w i t h  t h e  Nav ie r -S tokes  
o f  e q u i v a l e n t  s p h e r i c a l  volume d i a m e t e r .  s i m u l a t i o n  o f  ~ o o t e ' ~  f o r  t h e  a x i s y m m e t r i c  
C i r c l e s  a r e  f o r  r a i n d r o p s  ( Jqnes! ,  and  o s c l l l a t l o n  w i t h o u t  r a t p i t y  w i t h  p = 1 g cm-3 
t h ~  c u r v e  f o r  e q u i l i b r i u m  shaped  w a t e r  and o = 73  dynes  cm- ~ L Z F  presen! model f o r  
d r o p s  ( P r u p p a h e r  and P i t t e r ) .  C m -  t h e  l a r g e  a m p l i t u d e  o s c i l l a t i o n  o f  Foo te  
f i d e n c e  i n t e r v a l s  f o r  t h e  mean have  (whe re  ARo = 1 /1 .7 ;  g i v e s  t h e  same o s c i l l a t i o n  
been auded  t o  t h e  o r i ~ i n a l  r a i n d r o p  t i m e s  ( i . e . ,  1 . 0 4  TR f o r  u = 0.06 cm2 sec-1  
d a t a  a t  t h e  95% s i g n l f l c a n c e  l e v e l  and  1 . 0 6  TR f o r  u = 0 1 ,  and  shows s i m i l a r  
ba sed  upon t h e  e s t i m a t e d  s t a n d a r d  de- t e m p o r a l  v a r i a t i o n s  i n  t h e  s u r f a c e  and  k i n e t i c  
v i a t i o n  and  sample s i z e .  T r i a n g l e s  e n e r g i e s ,  and  i n  a x i s  r a t i o  w i t h  57% o f  tk,e 
a r e  p r e s e n t  model f o r  e q u i l i b r i u m  ime r p e n t  a s  a  p r o l a t e  s p h e r o i d .  The d x i s  
s h a p e .  r a t i o  a v e r a b e d  o v e r  onz  c y c l e  f o r  t h e  shove  
two c a s e s ,  and  a l s o  f o r  D = 0 . 3  and  0 .6  c m  
w i t h  c r a v i t y  ( e  = 9E(O cm sec-2 ,  ow = 0.998 g cm-2 and  o a 7 3  dynes  cm-1) f o r  b o t h  u = 0 . 0 1  
cn2 sec-1  and  u = 0 ,  y i e l d e d  v a l u e s  o f  o n l y  s l i g h t l y  i e s s  t h a n  t h e  mean o f  t h e  undamped 
t u r n i n g  p o i n t s .  
A s y m ~ e t r i c  c s c i l l a t i o n s  o f  t h e  fundamen ta l  mode ( m  = 2 d e g e n e r a c y )  i s  c u r r e n t l ,  under  
~ n v e s t i g a t i o n .  A s e t  o f  e q u a t i o n s  s i m i l a r  t o  t h o s e  g i v e n  above  i s  used  t o  s o l v e  t h e  
t e m p o r a l  b e h a v i s r  o f  t h e  a x i s  r a t i o .  
A n a l y t i c a l  and n u m e r j c a i  s o l u t i o n s  a r e  more d i f f i c u l t  t o  a b t a i n  t h a n  t h e  ax i symmet r i c  
c a s e  s l f i c e  o s c i l l a t i o n s  o c c u r  i n  t h e  shape  o f  a  g e n e r a l  e l l i p s o i d .  P r e l i m i n a r y  a n a l y s i s  
l r d i c a t e s  t h a t  t h e  t i m e  a v e r a g e  a x i s  r a t i o  f o r  % h e  asymrnet,ric o s c i l l a t i o n  o f  a  r a i n d r o p  
can  b e  app rox ima ted  by t h e  mean o f  t h e  t u r n i n g  p o i n t s  d e f i n e d  by t h e  p o t e n t i a l  ene rgy  w e l l .  
ORIGINAL, r.A?G !S 
OF POOR ~i.i~,:-! a''?' 
Three  c a n d i d a t e s  f o r  t h e  c a u s e  of  r a i n d r o p  o s c i l l a t i o n s  were examined:  d r o p  i n t e r -  
a c t i o n s ,  t u r b u l e n c e  i n  t h e  a i r  and pumping by wake s h e d d i n g .  A d r o p  i n t e r a c t i o n  model was 
used t o  o b t a i n  t h e  o s c i l l a t i o n  e n e r g y  from a  b a l a n c e  between t h e  a v e r a g e  c o l l i s i o n a l  ene rgy  
and v i s c o u s  d i . s i p a t i o n .  The time a v e r a g e  a x i s  r a t i o  was found t o  b e  c o n s i d e r a b l y  l a r g e r  
t h a n  e q u i l i b r i u m  f o r  d r o p s  w i t h  D ? 3 mm i n  heavy r a i n f a l l .  I n  c o n t r a s t ,  p r e l i m i n a r y  
r e s u l t s  f o r  wake f o r c l n g  i n d i c a t e  t h a t  a p p r e c i a b l e  o s c i l l a t i o n s  o c c u r e  o n l y  f o r  D 5 2 mm. 
F s r  t u r b u l e n t  f o r c i n g  t h e  o s c i l l a t i o n  e n e r g y  was r e l a t e d  t o  t u r b u l e n c e  i n  t h e  I n e r t i a l  
sub range  u s i n g  t h e  d i s t a n c e  t r a v e l e d  by a  r a i n d r o p  d u r l n g  a n  o s c i l l a t i o n .  T h i s  gave  a  
unique  s c a l i n g  f o r  t h e  o s c i l l a t i o n  ene rgy  w i t h  r a i n a r o p  s i z e  and d i s s i p a t i o n  r a t e .  Mean 
a x i s  r ~ a t i o s  f o r  t h e  a v e r a g e  o s c i l l a t i o n  ene rgy  were c a l c u l a t e d  from a  b a l a n c e  between i n p u t  
o f  t u r b u l e n t  ene rgy  and d i s s i p a t i o n  Sy viscoq1r;y.  The r e s u l t s  p r o v i d e d  a  r e a s o n a b l e  f i t  t o  
t h e  mean a x i s  r a t i o s  o f  t h e  camer t  d a t a  and . s u i t l b l e  e n v e l o p e  f o r  t h e  obse rved  a m p l i t u d e s .  
It was concluded t h a t  a t m o s p h e r i c  t u r b u l e n c e  and a l s o  d r o p  i n t e r a c t i o n s  can  p roduce  
s i g n i f i c a n t  i n c r e a s e s  i n  a v e r a g e  a x i s  r a t i o  o f  r a i ~ d r o p s  r e s u l t i n g  i n  a n  a p p r e c i a b l e  change  
i n  r a d a  b a c k s c a t t e l  c r 3 s s  s e c t i o n s .  
I thank  A r t  Jameson and Dave Johnsol  f o r  t h e i r  h e l p f u l  s u g g e s t i o n s  d u r i n g  t ' l e  c o u r s e  o f  
t h i s  i n v e s t i g a t i o n .  T h i s  work was p a r ' i a l l y  s u p p o r t e d  by t h e  N a t i o n a l  S c i e n ~ .  w n d a t i o n  
under  g r a n t  NSF ATM79-18365. 
R e f e r e n c e s  
1. Pruppache r ,  H .  R . ,  and  K .  V.  Beard ,  "A Wind Tunne l  I n v e s t i g a t i o n  o f  t h e  I n t e r n a l  
C i r c u i a t i o n  and Shape o f  Water  Drops F a l l i n g  a t  Te rmina l  V e l o c i t y  i n  A i r , "  Q u a r t .  J .  Roy. 
Meteor.  S o c . ,  Vol. 96,  pp. 247-256. 1970.  
2. Magono, C . ,  "On t h e  S h i p e  of  Water  Drops F a l l i n g  i n  S t a g n a n t  A i r , "  J .  Meteor . ,  Vol.  
11. D D .  77-79. 1954.   - -  
3. p ruppache r ,  H .  R . :  and  R .  L. P i t t e r ,  "A Semi-Empirical  D e t e r m i n a t i o n  o f  t h e  Shape  o f  
Cloud anu R a i r d r o p s , "  J .  Atmos. S c i . ,  Vol. 28, pp.  86-94. 1971.  
4. Cunn, R . ,  "Yechanica l  Resonance i n  F r e e l y  F a l l i n g  Rain  Drops,"  J .  Geophys. R e s . ,  
Vol. 54,  pp. 383-385. 1949.  
5 .  B lanche rd ,  D. C . ,  "The Behav io r  o f  Water  Drops a t  T t r m i n a l  V e l o c i t y  i n  A i r , "  T r a n s .  
Rmer. Geophys. Union, Vol. 31,  pp .  83b-842. 1950.  
6 .  J o n e s ,  D .  M. A . ,  "The Shape  o f  Ra ind rops , "  J. Meteor . ,  Vol .  1 6 ,  pp.  504-510. 1959.  
7.  S e l i g a ,  T. A . ,  and V.  N .  B r i n g i ,  " P o t e n t i a l  Use of  Radar  D i f f e r e n t i a l  R e f l e c t i v i t y  
Measurements of Or thognna l  P o l a r i z a t i o n s  f o r  Measur ing  P r e c i p i t a t i o n , "  J .  Appl. mete of^, 
Vol. 1 5 ,  pp.  69-76. 1 9 7 s .  
8. Jameson, A .  R , ,  K .  V. B r w d ,  all; J .  Bresch ,  "Compl i ca t ions  i n  L'educing Ra in  Para-  
me te r s  from P o l a r i z a t i o n  Meajurements,"  Proc .  o f  t h e  2 0 t h  Conf.  on Rada r  Me teo r . ,  ( I n  p r e s s ) .  
1981. 
4 .  Mue l l e r ,  X .  A , ,  anti A .  L .  S ims,  I n v e s t 1  a t i o n  o f  t h e  u a n t i t a t i v e  D e t e r m i n a t i o n  o f  
P o i n t  and Area l  Yrec i  i t a t i o n  by Radar d u r e m e n t s .  Tzch.  Aepor t  ECOR-00032 - F 
TNTIS AD64)-21~& , 88 PP. 
1 0 .  Frown, R . ,  The Shape and S t a b i l i t y  o f  Three-Dimensional  I n t e r f a c e s .  Ph.D. D i s s c r t a -  
t i o n ,  Un! v e r s i t y  o f  Minnesota .  1979.  
11; Nelsan ,  A.  R . ,  and N. R .  Cokha?.e, " O s c i l l a t i o n  F r e q u e n c i e s  o f  F r e e l y  Suspended Water 
Drops,"  J. Ceophys. Res . ,  Vol.  77 ,  pp .  2724-2727. 1972.  
1 .  Lamb, H . ,  Hydrodynamics. Dover P u b l i c a t i o n s ,  New York, 738 pp .  1932.  
1 3 .  Poo te ,  G .  B . ,  "A Numerical  Method f o r  S tuZy ing  L iqu id  Drop Behav io r :  S imp le  
O s c i l l a t i o n , "  J .  Comput. Phys . ,  101. 11, pp.  507-530. 1973.  


Raindrop oscillations

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server