Recent conjectures concerning the correlation between regions of high local helicity and low dissipation are examined from a rigorous theoretical standpoint based on the Navier-Stokes equations. It is proven that only the solenoidal part of the Lamb vector omega x u (which is directly tied to the nonlocal convection and stretching of vortex lines) contributes to the energy cascade in turbulence. Consequently, it is shown that regions of low dissipation can be associated with either low or high helicity, a result which disproves earlier speculations concerning this direct connection between helicity and the energy cascade. Some brief examples are given along with a discussion of the consistency of these results with the most recent computations of helicity fluctuations in incompressible turbulent flows

Speziale, Charles G.

English

NASA Technical Reports Server

r, 3 
NASA Contractor R e p &  178403 
ICASE REPORT NO. 87-69 
ICASE 
ON H E L I C I T Y  FLUCTUATIONS AND 
THE ENERGY CASCADE I N  TURBULENCE 
(#AS A-CR- 1 78 4 r) 3) ON HEL I C  ITY FLUCTUA T .IO NS ~ a 8 -  1 2 ~ 2 8  
AND THE E N E R G Y  CASCADE I N  TURBULENCE F i n a l  
Report {BASA) 1 3  p Avail:  NTIS RC 
A03/HF A01 CSCL 20D Uncla s 
G3/34 0108401 
Charles G. Speziale 
# 
Contract No. NAS1-18107 
November 1987 
INSTITUTE FOR COMPUTER APPLICATIONS IN SCIENCE AHD ENGINEERING 
NASA Langley Research Center, HMpton, Virginia 23665 
Operated by the Universities Space Research Association 
https://ntrs.nasa.gov/search.jsp?R=19880002646 2020-03-20T08:41:37+00:00Z
ON EIELICITY FLUCTUATIONS AND TKE ENERGY 
CASCADE IN TURBULENCE 
Charles  G. Spez ia le  
ICASE, NASA Langley Research Center 
Hampton, VA 23665 
ABSTRACT 
Recent con jec tu res  concerning the  c o r r e l a t i o n  between reg ions  of h igh  
l o c a l  h e l i c i t y  and low d i s s i p a t i o n  a r e  examined from a r igorous  t h e o r e t i c a l  
s tandpoin t  based on the  Navier-Stokes equat ions.  It i s  proven t h a t  only t h e  
s o l e n o i d a l  p a r t  of t h e  Lamb vec to r  u) x u (which i s  d i r e c t l y  t i e d  t o  t h e  
nonloca l  convect ion and s t r e t c h i n g  of vo r t ex  l i n e s )  c o n t r i b u t e s  t o  t h e  energy 
cascade i n  turbulence.  Consequently, i t  is  shown t h a t  regions of low d i s s ipa -  
t i o n  can be a s soc ia t ed  with e i t h e r  low o r  high he l ic i ty- -a  r e s u l t  which d i s -  
proves e a r l i e r  specu la t ions  concerning t h i s  d i r e c t  connect ion between h e l i c i t y  
and the  energy cascade. Some b r i e f  examples a r e  g iven  along with a d i s c u s s i o n  
of  t h e  cons is tency  of t h e s e  r e s u l t s  wi th  t h e  most recent  computations of h e l i -  
c i t y  f l u c t u a t i o n s  i n  incompressible  tu rbu len t  flows. 
Th i s  r e sea rch  was supported by t h e  Nat ional  Aeronaut ics  and Space Administra- 
t i o n  under NASA Contract  No. NAS1-18107 while the  au thor  w a s  i n  res idence  a t  
t h e  I n s t i t u t e  f o r  Computer Appl ica t ions  i n  Science and Engineering (ICASE), 
NASA Langley Research Center ,  Hampton, VA 23665. 
i 
During the past decade, helical structures have been a subject of in- 
creasing interest among researchers in turbulence. Various speculations 
concerning the relationship between helicity fluctuations and turbulence 
activity have been put forth beginning with the papers of Levich and his co- 
worke rs . In particular, claims have been made concerning the direct corre- 
lation between regions of high helicity and low dissipation. More recent 
theoretical investigations and numerical simulations of incompressible turbu- 
lent flows have cast grave suspicions on the existence of such a correlation 
but speculations along these lines continue to persist in the literature. The 
purpose of this paper is to show conclusively, based on a rigorous analysis of 
the Navier-Stokes equations, that there is no direct correlation between re- 
gions of high helicity and low dissipation in turbulence. Specific examples 
will be given along with a discussion of the consistency of the results to be 
derived herein with those obtained previously from direct numerical simula- 
tions of the Navier-Stokes equations. Since the results of any given direct 
numerical simulation of the Navier-Stokes equations, at high Reynolds numbers, 
are subject to some doubt, it is essential that a corroborating theoretical 
argument be provided if this issue is to be resolved. 
We will consider the incompressible turbulent flow of a homogeneous, vis- 
cous fluid governed by the Navier-Stokes equation 
au 2 - + u*vu = -vp + vv u at 
and continuity equation 
v*u = 0. 
-2- 
Here, u is the velocity vector, p is the modified pressure (which can in- 
clude a gravitational body force potential), and v is the kinematic vis- 
cosity of the fluid. The acceleration on the left-hand side of Eq. (1) can be 
written in the alternative form 9 
- au  + u.vu 
at at au + 0 x u + 1- 1 vlul 2 ( 3 )  
where 
is the vorticity vector and o x u is usually referred to as the Lamb 
vector. This gives rise to the alternative form of the Navier-Stokes equation 
- au + 0 x u = -V(p + I u1 2 ) + UP 2 u. ( 5 )  at 
Since the cascade of energy is a nonlinear velocity effect, it is clear that 
it must emanate from the Lamb vector o x U. Furthermore, since the vector 
identity 
(U.d2 + In x u12 = 1u12lol2 ( 6  1 
holds, it is clear that regions of high helicity density 
are associated with small Lamb vectors 10  x u1 when each are normalized 
with respect to 1u1 1 0 1 .  This has motivated several researchers to con- 
jecture that regions of high helicity have an associated low dissipation due 
-3- 
to a reduction in the energy cascade. 1-5 Such conjectures have led to the 
hypothesis that helical structures are coherent and endure for relatively long 
times . 
It will now be shown that the speculations made in the literature con- 
cerning helicity and dissipation are not supported by the Navier-Stokes equa- 
tions. In order to accomplish this task, we will decompose the Lamb vector 
into irrotational and solenoidal parts (i.e., a Helmholtz decomposition) as 
follows 
Y x u = Vu + V x 8 (7 1 
where a is scalar field and B is a vector field which can be made 
solenoidal (V*@ = 0) with no loss of generality. After using this 
Helmholtz decomposition, Eq. ( 6 )  can be written in the form 
The scalar potential a is absorbed by the pressure and has no effect on the 
evolution of the velocity field u which determines the energy cascade. It 
is thus clear that the energy cascade arises only from the solenoidal part of 
the Lamb vector 0 x B .  Consistent with Eq. ( 6 1 ,  small magnitudes of the 
solenoidal part of w x u need not be associated with large values of U*W 
(i.e., IV x 81 can be small, with large !Val and correspondingly small 
U ' W )  
For flow in an infinite domain, 
3 d XI 
-4- 
where the  n o t a t i o n  o’ : w(x’,t) and u’ E u(x’,t) i s  u t i l i z e d .  Eq. ( 9 )  
fol lows from a d i r e c t  a p p l i c a t i o n  of the vec to r  i d e n t i t y  
(where we have made use of t he  f a c t  t h a t  B is s o l e n o i d a l )  along with t h e  
cons t ruc t ion  of the Green‘s func t ion  s o l u t i o n  of t he  r e s u l t i n g  Poisson equa- 
t ion. 10 Since,  
0 x (m x u) = u*Vw - @*Vu (11) 
i t  fol lows t h a t  
P h y s i c a l l y ,  t h e  vec to r  f i e l d  u*Vo - @*Vu accounts f o r  t he  convection and 
s t r e t c h i n g  of vo r t ex  l i n e s  i n  the  f l u i d .  Hence, i t  is  obvious t h a t  t he  energy 
cascade resul ts  from the  nonlocal  convection and s t r e t c h i n g  of vo r t ex  l i n e s .  
It should be noted t h a t  the v o r t i c i t y  s t r e t c h i n g  term o*Vu g i v e s  rise t o  
the  usua l  energy cascade from l a r g e  t o  small s c a l e s ,  whereas, the convect ive 
v o r t i c i t y  term u*Vo can g ive  rise t o  a r eve r se  energy cascade from small 
s c a l e s  t o  l a r g e  s c a l e s  ( t h i s  has been r igo rous ly  proven f o r  two-dimensional 
turbulence l 1  ) 
It w i l l  now be demonstrated by p resen t ing  counterexamples t h a t  t h e r e  i s  
no d i r e c t  c o r r e l a t i o n  between the turbulence d i s s i p a t i o n  (which i s  d i r e c t l y  
t i e d  t o  t h e  energy cascade) and l o c a l  f l u c t u a t i o n s  i n  the  h e l i c i t y .  Consider 
t he  case of two-dimensional turbulence with the v e l o c i t y  f i e l d  
t -5- 
and i t s  a s soc ia t ed  v o r t i c i t y  f i e l d  
10 = w(x,y, t )k .  
For such a two-dimensional flow, the  h e l i c i t y  d e n s i t y  vanishes ,  i .e.,  
However, i t  is well  known t h a t  two-dimensional turbulence e x h i b i t s  low d i s s i -  
p a t i o n  a t  high Reynolds numbers due t o  s p e c t r a l  blocking (i.e.,  energy actual- 
l y  undergoes a r eve r se  cascade from the  small  scales t o  t h e  l a r g e  scales which 
d r a m a t i c a l l y  reduces t h e  d i s s i p a t i o n ;  see F j  o r to f t ' ' ) .  I n  a Beltrami flow, 
and, hence, t he  normalized h e l i c i t y  d e n s i t y  hN assumes i ts  maximum value of 
one, i.e., 
( i n  gene ra l ,  hN can assume the range of values  0 - < - < 1). However, 
s i n c e  t h e  energy cascade arises from the  Lamb v e c t o r  10 x u (which van i shes  
f o r  a Beltrami f low),  i t  fol lows t h a t  Beltrami flows have low d i s s i p a t i o n  a t  
h i g h  Reynolds numbers. Hence, Beltrami flows and two-dimensional flows each 
have low d i s s i p a t i o n ,  a t  high Reynolds numbers, but  have h e l i c i t y  d e n s i t i e s  
- 6- 
that are at the opposite extremes (the former has a small normalized heli- 
city hN = 0 while the latter has a large normalized helicity hN = 1). It 
is thus clear that regions of low dissipation can be associated with either 
high or low helicity density which disproves the earlier claims made concern- 
ing the direct correlation between helicity and dissipation. In fact, it can 
be shown that for a given turbulence dissipation there can be arbitrarily dif- 
ferent normalized helfcity fluctuations. For example, consider two turbulent 
flows with velocities u and u that are related as follows 
* 
* 
u = u + U,(t), 
where Uo(t) can be any random function of time alone. If we take Uo(t) 
to satisfy the constraint 
Iuoct>I >> Iu(x,t)l 
f o r  all x, t, it follows that 
and, hence, 
where X : Uo/lUoI is - an arbitrary unit vector that is independent of the 
vorticity vector (6. Hence, the normalized helicity hi associated with 
the velocity field u can be varied independently of the corresponding 
* 
- 7- 
h e l i c i t y  a s soc ia t ed  wi th  t h e  v e l o c i t y  f i e l d  U. However, both v e l o c i t y  
f i e l d s  u and u have t h e  same d i s s i p a t i o n ,  i.e., * 
* *  
auk auk - a% a u k  v --- 
ax  II ax, ax, ax, 
where an overbar  r ep resen t s  an average and t h e  E i n s t e i n  summation convention 
a p p l i e s  t o  repeated ind ices .  It is  thus c l e a r  t h a t ,  i n  gene ra l ,  t h e r e  is no 
d i r e c t  c o r r e l a t i o n  between l o c a l  f l u c t u a t i o n s  i n  the  h e l i c i t y  and t h e  d i s s i p a -  
t i o n  ra te  of turbulence.  
F i n a l l y ,  comparisons w i l l  be made wi th  some recent  computational s t u d i e s  
of h e l i c i t y  based on d i r e c t  numerical s imula t ions  of t he  Navier-Stokes equa- 
t i o n s  f o r  homogeneous t u r b u l e n t  f lows as w e l l  as f o r  t u rbu len t  channel flow. 
The recent  computations of Rogers and Moin7 f o r  t u rbu len t  channel f low ind i -  
cate  t h a t  peaks i n  t h e  p r o b a b i l i t y  d e n s i t y  func t ion  f o r  t h e  normalized h e l i -  
c i t y  d e n s i t y  are - not  a s soc ia t ed  with regions of low d i s s i p a t i o n .  To be more 
s p e c i f i c ,  t h e  pdf f o r  t he  normalized h e l i c i t y  d e n s i t y  condi t ioned on low d is -  
s i p a t i o n  w e r e  e i ther  f a i r l y  f l a t  o r  s l i g h t l y  peaked a t  hN = 0. This is i n  
s t a r k  c o n s t r a s t  t o  t h e  r e s u l t s  of Pelz ,  e t  a l .4  who claimed t h a t  i n  t h e  in- 
t e r i o r  of t he  channel ,  t h i s  pdf f o r  hN is s t r o n g l y  peaked a t  hN = 1 ( in -  
d i c a t i v e  of a Beltrami flow where u x w = 0 ) .  This  d i s p a r i t y  i n  resul ts  is  
most l i k e l y  t o  have a r i s e n  due t o  inadequate  r e s o l u t i o n  i n  the  computations of 
P e l z ,  e t  a l .4  ( t h e  computations of Pe lz ,  e t  a l e 4  were conducted on a 323 g r i d  
S i m i l a r l y ,  
Rogers and Moin7 found t h a t  t h e  pdf f o r  t h e  normalized h e l i c i t y  d e n s i t y  (con- 
d i t i o n e d  on low d i s s i p a t i o n )  obtained from d i r e c t  numerical s imula t ions  of 
homogeneous tu rbu len t  shea r  flow were f a i r l y  f la t - -another  example of t h e  
whereas those of Rogers and Moin7 were conducted on a 128 3 g r i d ) .  
-8- 
l ack  of a d i r e c t  c o r r e l a t i o n  between high h e l i c i t y  and low d i s s i p a t i o n .  The 
recent  d i r e c t  s imula t ions  of e r r 8  ( i . e . ,  a 128 3 forced  pseudo-spectral  simu- 
l a t i o n  of turbulence i n  a pe r iod ic  box) a l s o  ind ica t ed  t h a t  t he re  is no evi-  
dence t h a t  regions wi th  enhanced normalized h e l i c i t y  are d i r e c t l y  involved i n  
r e t a r d i n g  the energy cascade process.  It i s  thus c l e a r  t h a t  the  t h e o r e t i c a l  
r e s u l t s  presented h e r e i n  a r e  c o n s i s t e n t  with the  most recent  and h ighly  re- 
solved numerical s t u d i e s  of h e l i c i t y  i n  turbulence.  
I n  conclusion,  i t  has  been r igo rous ly  demonstrated t h a t  t h e r e  i s  no di-  
r e c t  c o r r e l a t i o n  between regions of high h e l i c i t y  and low d i s s i p a t i o n .  I n  
f a c t ,  regions of low d i s s i p a t i o n  can be a s soc ia t ed  with e i t h e r  low or high 
h e l i c i t y .  Furthermore, i t  was proven t h a t  t he  t h e o r e t i c a l  argument which 
i n i t i a t e d  t h i s  con jec tu re  was d e f e c t i v e  i n  i t s  neg lec t  of t he  f a c t  t h a t  low 
magnitudes of t he  so l eno ida l  p a r t  of t h e  Lamb vec to r  (which i s  the  only con- 
t r i b u t o r  t o  the  energy cascade process)  a r e  not  n e c e s s a r i l y  a s soc ia t ed  wi th  
h igh  va lues  of h e l i c i t y .  It is t h e r e f o r e  h igh ly  doubt fu l  t h a t  h e l i c i t y  
f l u c t u a t i o n s  can p lay  an important r o l e  i n  t h e  c o r r e l a t i o n  of tu rbulence  
a c t i v i t y  t h a t  is d i r e c t l y  t i e d  t o  the  energy cascade process .  
-
-9- 
REFERENCES 
E. Levich and A. Tsinober ,  Phys. L e t t . ,  z, 293 (1983). 
* A. Tsinober and E. Levich, Phys. L e t t . ,  s, 321 (1983). 
E. Levich, Ann. New York Acad. Sci.,  404, 73 (1983)- 
R. B. Pe lz ,  V. Yakhot, S. A. Orszag, L. Shtilman, and E. Levich, Phys. Rev. 
Le t t . ,  54, 2505 (1985). -- 
R. B. Pe lz ,  L. Shtilman, and A. Tsinober,  Phys. F lu ids ,  - 29, 3506 (1986). 
C. G. Spezia le ,  Quart. Appl. Math., _. 45, 123 (1987). 
' M. Rogers and P. b i n ,  Phys. F lu ids ,  - 30, 2662 (1987). 
R. M. Kerr, Phys. Rev. L e t t . ,  ( i n  press ) .  
G. K. Batchelor ,  An In t roduc t ion  t o  F lu id  Dynamics, (Cambridge Univ. P res s ,  
1967). 
lo P. M. Morse and H. Feshbach, Methods of Theore t i ca l  Physics ,  ( M c G r a w - H i l l ,  
1953). 
l 1  R. F j o r t o f t ,  Te l lu s ,  -' 5 225 (1953). 
Report Documentation Page 
1. Report No. 
NASA CR-178403 
ICASE Report No. 87-69 
2. Government Accession No. 
7. Authork) 
Charles  G. Speziale  
17. Key Words (Suggested by Author(s)) 
9. Performing Organization Name and Address 
I n s t i t u t e  f o r  Computer Applicat ions i n  Science 
Mail Stop 132C, NASA Langley Research Center 
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11 A0 2 
5. Supplementary Notes 
Langley Technical Monitor: 
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87-69 
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NAS1-18107 
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Submitted t o  Physical  Review 
Letters 
6. Abstract 
Recent con jec tu res  concerning t h e  c o r r e l a t i o n  between regions of high loca 
h e l i c i t y  and low d i s s i p a t i o n  a r e  examined from a r igorous t h e o r e t i c a l  s tandpoin 
based on the  Navier-Stokes equat ions.  It i s  proven t h a t  only the s o l e n o i d a l  par ,  
of t h e  Lamb v e c t o r  o x u (which is  d i r e c t l y  t i e d  t o  the  nonlocal  convectiol 
and s t r e t c h i n g  of vo r t ex  l i n e s )  c o n t r i b u t e s  t o  the  energy cascade i n  turbulence 
Consequently, i t  i s  shown t h a t  regions of low d i s s i p a t i o n  can be a s s o c i a t e d  w i t 1  
e i t h e r  low o r  high he l i c i ty - - a  r e s u l t  which disproves e a r l i e r  s p e c u l a t i o n s  con- 
ce rn ing  t h i s  d i r e c t  connection between h e l i c i t y  and the  energy cascade. Somc 
b r i e f  examples a r e  given along with a d i s c u s s i o n  of t he  consis tency of thesc 
r e s u l t s  with the most r ecen t  computations of h e l i c i t y  f l u c t u a t i o n s  i n  incompres- 
s i b l e  t u r b u l e n t  flows. 
turbulence , h e l i c i  t y  , coherent  
s t r u c t u r e s ,  energy cascade 
34 - Fluid Mechanics and Heat 
T rans fe r  
I Unclas s i f i ed  - unlimited 
NASA-Langley, 1987 ASA FORM 1626 OCT 86 


On helicity fluctuations and the energy cascade in turbulence

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