wing-tip vortices are reviewed and quantlfied i n terms of the Rossby number. c r i t e r i on. ..... number, swirl and axial velocity is similar. In addition, the ...
i
NASA Contractor Report 173232%
ICASE REPORT NO. 87-3
ICASE ON A CRITERION FOR VORTEX BREAKDOWN
CN A C B I l E l i l C b F O R V O B T E X F i n a l heport (bASA) 24 p CSCL 20D
N87- 1 8 0 2 8
(EASA-Cli-778232) E8liAKDOMN
63/31, R. E . S p a l l T . B . Gatski
t
C . E. Grosch
Contract No. NASl-18107 January 1987
INSTITUTE FOR COMPUTER APPLICATIONS I N S C I E N C E AND ENGINEERING NASA Langley R e s e a r c h Centcr, Hampton, Virginia 23665 Operated by the Universltics Space Research A s s o c i a t i o n
NASA
National Aeronautics and Swce Administration
Lsr@eyR~rchCmtw Hampton,Virgmla 23665
Unclas 43279
OH A
CRITERIOS t O R YGZTEX EREAKDOUN
R . E. S p a 1 1 Department of Mechanical Engineering and Mechanics Old Dominion University, Norfolk, VA 23508
T. B. Gatski Viscous Flow Branch NASA Langley Research Center, Hampton, VA
23665-5225
C. E. Grosch Departments of Oceanography and Computer Science Old Dominion Universlty, Norfolk, VA 23508 and ICASE, NASA Langley Research Center, Hampton, VA 23665-5225
Abstract A c r i t e r i o n f o r the onset o f vortex breakdown i s proposed.
Based upon
previous experimental, computational, a n d t h e o r e t i c a l s t u d i e s , an appropriately defined local Rossby number i s used t o d e l f n e a t e the region where breakdown occurs.
In addition, new numerical r e s u l t s a r e presented which f u r t h e r validate
this criterion.
A number of previous t h e o r e t i c a l s t u d i e s concentrating on
i n v i s c i d standi ng-wave analyses for t r a i 1 in g wi n g - t i p vorti ces are rev?ewed a n d r e i n t e r p r e t e d i n terms of the Rossby number c r i t e r i o n .
Consistent w i t h previous
s t u d i e s , the physical basis for the onset of breakdown i s i d e n t i f i e d as the a b i l i t y o f the f l o w t o sustain such waves.
Previous computational r e s u l t s are
reviewed and re-evaluated in terms of t h e proposed breakdown c r i t e r i o n .
As a
r e s u l t , the cause o f breakdown occurring near the inflow computational boundary, common' t o several numerical studies, 1 s i d e n t i f i e d .
F i n a l l y , previous
experimental studies of vortex breakdown f o r both leading edge and t r a i l i n g wing-tip vortices are reviewed and quantlfied i n terms of the Rossby number c r i t e r i on.
PACS
-
40, 47, 47.30 Y, 47.20 F t .
R e s e a r c h f o r t h e t h i r d a u t h o r w a s s u p p o r t e d u n d e r NASA C o n t r a c t N o . NASl-18107 w h i l e h e w a s i n r e s i d e n c e a t t h e I n s t i t u t e f o r Computer A p p l i c a t i o n s i n S c i e n c e a n d E n g i n e e r i n g ( I C A S E ) , NASA L a n g l e y R e s e a r c h C e n t e r , Hampton, VA 2 3 6 6 5 . i
1.
Introduction
Vortices can be generated i n many ways.
Of s p e c i f i c i n t e r e s t a r e v o r t i c e s
generated by a f i n i t e p l a t e or sharp-edged body a t a non-zero angle of a t t a c k . These vortices are often highly s t a b l e s t r u c t u r e s characterized by a s t r o n g axial flow.
Other examples o f v o r t i c e s w i t h a s t r o n g axial velocity component
include tornadoes and waterspouts, intake v o r t i c e s , a n d swirling flow in pipes and tubes. Leading-edge v o r t i c e s shed from a d e l t a wing induce a velocity f i e l d t h a t r e s u l t s i n increased l i f t and s t a b i l i t y o f the wing.
However, under c e r t a i n
conditions related t o the angle o f attack of the w i n g , these v o r t i c e s can undergo a sudden and d r a s t i c change in s t r u c t u r e known as vortex breakdown. This breakdown can adversely a1 t e r t h e aerodynamic c h a r a c t e r i s t i c s of t h e wing.
A similar vortex bursting phenomena has been observed f o r t r a i l i n g w i n g -
t i p v o r t i c e s , which i s d e s i r a b l e because these v o r t i c e s represent a hazard t o smaller a i r c r a f t i n areas of dense a i r t r a f f i c .
The fundamental d i f f e r e n c e
between these two c l a s s e s of vortices 1 i e s i n t h e i r c i rcumferenti a1 velocity distributions.
Far downstream, as was shown by Batchelor,' the circumferential
velocity p r o f i l e o f the w i n g - t i p vortex behaves l i k e the two-dimensional Burgers' vortex; whereas Hal 1 has shown t h a t the circumferential velocity d i s t r i b u t i o n o f the leading-edge vortex can be approximated using the concept o f
a viscous subcore very near the axis surrounded by an inviscid r o t a t i o n a l conical flow region.
Thus, the r a d i a l gradients of the circumferential velocity
near the axis o f the leading-edge vortices are much larger than those of the wi n g - t i p vortices. The a b i l i t y t o control these vortical s t r u c t u r e s i s an important and active area of research.
For example, i t i s desirable t o delay the process over a
d e l t a w i n g and accelerate i t f o r t r a i l i n g - t i p vortices.
Unfortunately, a
2
comprehensive scheme t o d e s c r i b e t h e breJkdown process z i d the parameters e f f e c t i n g i t i s p r e s e n t l y l a c k i n g , a1 though s e v e r a l t h e o r i e s have been proposed. Vortex breakdown was f i r s t observed e x p e r i m e n t a l l y by Peckham and A t k i n ~ o n . ~They observed t h a t v o r t i c e s shed from a d e l t a wing a t h i g h angles of a t t a c k appeared t o " b e l l out'' and d i s s i p a t e s e v e r a l c o r e diameters downstream f r o m t h e t r a i l i n g edge o f t h e wing.
S i nce then, v o r t e x breakdown has been
observed i n s w i r l i n g f l o w s i n s t r a i g h t pipes, n o z z l e s and d i f f u s e r s , combustion chambers, and tornadoes.
Seven types o f breakdown have been id e n t i f ied
e ~ p e r i m e n t a l l y ,ranging ~ from a m i l d " s p i r a l " t y p e t o a s t r o n g "bubble" t y p e breakdown.
Observations i n t h e e a r l y 1960's s p u r r e d c o n s i d e r a b l e e f f o r t t o
develop a t h , e o r e t i c a l e x p l a n a t i o n f o r t h e v o r t e x breakdown phenomena.
Three
d i f f e r e n t classes of phenomena have been suggested as t h e cause o r e x p l a n a t i o n o f breakdown.
These are:
(1) the concept o f a c r i t i c a l ~ t a t e , ~( 2- )~ analogy
t o boundary-layer ~ e p a r a t i o n , ' ~(~3 ) hydrodynamic i n s t a b i l i t y . 10-12 The c r i t i c a l s t a t e t h e o r y i s based upon t h e p o s s i b i l i t y t h a t a columnar v o r t e x can support axisymmetric s t a n d i n g waves.
The s u p e r c r i t i c a l s t a t e has
l o w - s w i r l v e l o c i t i e s and t h e f l o w i s unable t o support these waves. f l o w s have h i g h - s w i r l v e l o c i t i e s and a r e a b l e t o s u p p o r t waves.
Subcritical
Vortex
breakdown can be thought o f as t h e a b i l f t y o f t h e f l o w t o s u s t a i n s t a x d i n g waves. I n Hal 1 s 2 theory, t h e breakdown phenomena i s taken t o correspond t o a f a i l u r e o f the q u a s i - c y l i n d r i c a l
approximation.
The i d e a b e i n g t h a t when
s t r e a m i s e gradients i n t h e w r t e x become l a r g e t h e quasi -cy? i n d r i c a ? a p p r o x i m a t i o n must f a i l , thus s i g n a l i n g breakdown.
T h i s i s c o n s i d e r e d t o be
analogous t o the f a i l u r e o f t h e boundary-layer e q u a t i o n s which s i g n a l s an impending separation. S t a b i l i t y theory o n l y a l l o w s one t o f n v e s t i g a t e t h e a m p l i f i c a t i o n o r decay
3
o f in f i n i tesrnal l y sinal 1 d i sturbances imposed on t h e base v o r t e x f l o w . i s then assumed t o be analogous t o l a m i n a r - t u r b u l e n t t r a n s i t i o n .
Breakdown
O f course, as
p o i n t e d o u t by Leibovich,13 breakdown can occur w i t h 1 it t l e s i g n o f i n s t a b i l i t y and a v o r t e x f l o w may become u n s t a b l e and n o t undergo breakdown. The purpose of t h i s paper i s t o show t h a t t h e e x i s t e n c e o f a c r i t i c a l c o n d i t i o n f o r v o r t e x breakdown can be d e s c r i b e d i n terms o f a fundamental l o c a l f l o w parameter.
A common i d e a among several p r e v i o u s t h e o r e t i c a l s t u d i e s w i l l
be i d e n t i f i e d and an analogy t o t h e ideas o f T a y l o r 1 4 c o n c e r n i n g t h e s t a b i l i t y o f Couette f l o w w i l l be noted.
The i d e n t i f i c a t i o n o f t h e Rossby number as t h e
key parameter w i l l be discussed and i t s c l o s e r e l a t i o n s h i p w i t h t h e i d e a s s e t f o r t h i n previous s t u d i e s by o t h e r s i s examined.
F i n a l l y , t h e Rossby number
c r i t e r i o n w i l l be a p p l i e d t o p r e v i o u s c o m p u t a t i o n a l and e x p e r i m e n t a l r e s u l t s and t h e numerical r e s u l t s o f the p r e s e n t i n v e s t i g a t o r s .
2.
Previous Theoretical Resul ts
Throughout the remainder o f t h i s paper we use a c y l i n d r i c a l p o l a r
( r , e, z ) ,
c o o r d i n a t e system,
and corresponding v e l o c i t y components, U i n the
r a d i a l ( r ) direction, V i n the circumferential axial ( z ) direction.
( e ) d i r e c t i o n , and W i n t h e
I n d i s c u s s i n g previous work, we adopt t h e ( r ,
e, z)
convention. Squire8 appears t o be t h e f i r s t t o have performed a t h e o r e t i c a l a n a l y s i s o f v o r t e x breakdown.
He suggested that i f s t a n d i n g waves were a b l e t o e x i s t on a
v o r t e x c o r e then m a l 1 d i s t u r b a n c e s , p r e s e n t downstream, c o u l d propagate upstream and cause breakdown.
This i s analogous t o t h e e a r l i e r work o f T a y l o r 14
on t h e s t a b i l i t y of c i r c u l a r Couctte flow.
There, a l i n e a r s t a b i l i t y a n a l y s i s
was performed t o a s c e r t a i n the a b i l i t y of t h e base f l o w t o s u p p o r t axisymmetric standing-wave disturbances.
I n a l l o f t h e cases s t u d i e d , S q u i r e assumed t h a t
t h e v o r t e x flow was i n v i s c i d and axisymmetric.
He t h e n sought t o determine
4
c o n d i t i o n s under which an i n v i s c i d , a x i s y m n c t r i c , steady p e r t u r b a t i o n t o t h e T h i s c o n d i t i o n , which was necessary f o r t h e e x i s t e n c e of a
flow c o u l d e x i s t .
s t a n d i n g wave, was taken t o mark t h e t r a n s i t i o n between s u b c r i t i c a l and s u p e r c r i t i c a l states.
Two o f t h e cases s t u d i e d by S q u i r e a r e r e l e v a n t t o t h e
p r e s e n t study. I n t h e f i r s t case W was taken t o be a c o n s t a n t .
V was taken t o be t h a t o f
a s o l i d body r o t a t i o n i n s i d e a c o r e o f u n i t r a d i u s and t h a t o f a p o t e n t i a l v o r t e x outside.
That i s O S r\c 1
V = V o r V = Vo/r w i t h Vo a constant.
(1)
r>,l
He found t h a t f o r s t a n d i n g waves t o e x i s t a s w i r l
parameter, "k" the r a t i o o f t h e maximum s w i r l speed t o t h e a x i a l speed, had t o satisfy a criterion k = Vmax/W
(21
>, 1.20
When k = 1.20 the wave i s i n f i n i t e l y l o n g b u t has a f i n i t e wavelength f o r
k > 1.20. I n t h e second case W was a l s o taken t o be a c o n s t a n t , b u t
w i t h Vo a nondimensional parameter.
Again, S q u i r e found t h a t t h e r e was a
c o n d i t i o n on the s w i r l parameter "k" f o r t h e e x i s t e n c e of a s t a n d i n g wave. c o n d i t i o n was
k = Vmax/w >/ 1.00,
(4)
VmaX = 0.638 V,
(5)
where we note t h a t
Benjamin5 examined t h i s phenomena from a d i f f e r e n t p o i n t o f view.
He
The
5
considered v o r t e x breakdown t o be a f i n i t e t r a n s i t i o n between two d y n a m i c a l l y There i s s u b c r i t f c a l flow,
c o n j u g a t e s t a t e s of flow.
w h i c h i s d e f i n e d as t h e
s t a t e t h a t i s a b l e t o support standing waves, and a c o n j u g a t e s u p e r c r i t i c a l f l o w which i s unable t o s u p p o r t standing waves.
I n t h i s c o n t e x t t h e work o f S q u i r e
g i v e s a c o n d i t i o n m a r k i n g t h e i n t e r f a c e between these two s t a t e s .
As i n t h e
work o f Squire, a u n i v e r s a l c h a r a c t e r i s t i c parameter was d e f i n e d which delineates the c r i t i c a l regions o f the flow.
T h i s parameter, denoted by
N, i s
t h e r a t i o o f t h e a b s o l u t e phase v e l o c i t i e s o f l o n g w a v e l e n g t h waves which propogate i n t h e a x i a l d i r e c t i o n , i.e.,
N =
Here C+ and
c,
+
C
,
c-
C- a r e t h e phase v e l o c i t i e s o f t h e waves w h i c h propogate w i t h and
against the flow, respectively. and f o r
-F For N > 1 t h e f l o w c o n d i t i o n s a r e s u p e r c r i t i c a l
N < 1, s u b c r i t i c a l .
Benjamin a p p l i e d t h i s theory t o a s p e c i f i c v o r t e x f l o w , d e f i n e d by W a c o n s t a n t and
If
R
+
-, t h i s
V = V o r
O c r d l
V = Vo/r
l\
