The present study considers the two-dimensional case of an array of N rectilinear, like-sign vortices above an infinite flat boundary. The method of images can be employed with this configuration to reduce the problem to that of 2N vortices in free space, constrained by 2N symmetry relations. This system is Hamiltonian and therefore certain invariants of the motion are known. Further, from the Hamiltonian constant, the equations of motion are readily derived and may be integrated numerically to determine the vortex trajectories. This knowledge of the time-dependent vortex motion then allows the resulting noise radiation to be computed by standard aeroacoustic techniques

Adelman, J. L.

Hardin, J. C.

English

The present study considers the two-dimensional case of an array of N rectilinear, like-sign vortices above an infinite flat boundary. The method of images can be employed with this configuration to reduce the problem to that of 2N vortices in free space, constrained by 2N symmetry relations. This system is Hamiltonian and therefore certain invariants of the motion are known. Futher, from the Hamiltonian constant, the equations of motion are readily derived and may be integrated numerically to determine the vortex trajectories. This knowledge of the time-dependent vortex motion then allows the resulting noise radiation to be computed by standard aeroacoustic techniques

NASA Technical Reports Server

J. L. Adelman 
George Washington Universi ty  
J o i n t  I n s t i t u t e  f o r  Advancement of F l i g h t  Sciences 
il2mpton, Virgtnia  23665 
and 
J. C. Hardin 
NASA Laqgley Research Center 
Haapton, Virginia  23665 
ABSTRACT 
Knowledge of vortex dynamics is crucial. t o  t he  underFtanding of boundary 
l aye r  flow 2nd its noise  production, as w e l l  as s t n * . c t u r a l  f a t i g u e  caused by 
i n t e r i c t i o n s  between turbulent  flows and various surfaces .  The growing use  of , 
i 
high-performance a i r c r a f t ,  r o t o r c r a f t ,  and o the r  high-speed t r a n s p o r t a t i o n  
sys tems i n  recent years  has emphasized the need t o  understand such dynamics, 
s ime l a a i n a r  flow con t ro l ,  i n t e r i o r  nc i se  reduct ion,  and s t r u c t u r a l  f a t i g u e  
c h e r a c t e r i s t j c s  may be c r i t i c a l  t o  the  success of such vehicles .  
Turbulent baundary l aye r s  are compriced of v o r t i c i t y  whose c h a r a c t e r i s t i c s  
a r e  defined b;? the flow d i r e c t i o n  and surface geometry. As 3 s i m p l e  model of a 
boundary layer, the present  study considers the two-dimensional case of an a r r a y  
of N r e c t i l i n e a r ,  l ike-sign v o r t i c e s  above an i n f i n i t e  f l a t  boundary. The 
method of images can be employed with t h i s  configurat ion t o  reduce the  problem 
t o  t h a t  of 2N v o r t i c e s  i n  f r e e  space, constrained by 2N symmetry r e l a t i o n s .  
Th i s  sys tem is  Hamiltonian and the re fo re  c e r t a i n  i n v a r i a n t s  of the motion are 
known. Further ,  from the Samiltonian constant ,  the equat ions of motion are 
readi ly  derived and may be in t eg ra t ed  numerically t o  determine the vortex 
t r a j e c t o r i e s .  This knowledge of t he  time-dependent ve r t ex  motion then allows 
the r e s u l t i n g  noise  r a d i a t i o n  t o  be computed by s tandard ae roacous t i c  
techniques 
(YASP-T~I-83475) VORTBX STUDIBS RELATING TO 185-271 G9 
BOUXDABP LAYd.B TObdULEYICB AND IJCISJJ (#ASA)  
5 p RC A42/HP A G I  CSCL 2QD Uncla s 
63/34 23378 
https://ntrs.nasa.gov/search.jsp?R=19850018858 2020-03-20T19:04:37+00:00Z
2 
The model has been examined ex tens ive ly  f o r  =any d i f f e r e n t  i n i t i a l  vortex 
conf igura t ions ,  including both t r a j e c t o r y  and noise  ca l cu la t ions  . Several  
a n a l y t i c a l  and numerical c h a r a c t e r i s t i c s  of t he  i n t e r a c t i o n s  are observed. For 
example, t o r  W2, a c r i t e r i o n ,  
y Yy q x 2  +. i' 2 )  > y y + q x 2  + Y2) ,  
1 2  0 
where Y1 and Y2 are the  i n i t i a l  he ights  of tire two v o r t i c e s  above the  plane,  
X and Y are t h e i r  i n i t i a l  separa t ions  i n  the respzc t ive  coordinate  d i r e c t i o n s ,  
Po is twice the  average he ight  of the  vo r t i ce s  above the  boundary, y is t he  
r a t i o  of the  
motion of trre vor t i ce s  can be derived. Such motion is pe r iod ic  and the re fo re  
produces sound spec t r a  containing ocly harmonically r e l a t e d  d i s c r e t e  frequency 
components. Figiire la is an example of the  vo r t e r  t r a j e c t o r i e s  i n  t h i s  case 
while Figure l b  is t h e  r e su l t i ng  noise  r ad ia t ion  t o  a f ixed  observer. The 
apparent modulation of the  noise  si,.ial is due t o  the  d i r e c t i v i t y  of the  noise  
source as it moves with respect  t o  the  f ixed  observer.  When t h i s  c r i t e r i o n  is 
not s a t i s f i e d ,  the vortex mcrtion i p  non-osc i l l a t ing  i n  na ture  and the re fo re  
produces very l i t t l e  noise.  Figure 2a is an example of the  trajectcries i n  t h i s  
type of motion while Figure 2b dizplays the r e s u l t i n g  noise .  In  t h i s  case, the  
inf luence  of the  image of the  vortex c loses t  t o  t he  boundary causes i t  t o  
convect rap id ly  away from the  o ther  vor tez  r e s u l t i n g  i n  i reql igible  i n t e r a c t i o n .  
The d i f f e rences  between these  tvo cases are similar t o  those between laminar and 
turbulen t  boundary layers . 
c i r c u l a t i o n s ,  and P is the  cent ro id  of v o r t i c i t y ,  fo i  o s c i l l a t i n g  
The ana lys i s  is extended f o r  K,i. I n  t h i s  case,  the phenomena of 
non-integrabi l i ty  and non-de termids t ic  "chaotic" so lu t ions  occur. 
t h i s  type of morion are included. 
Examples of 
3 
The r e s u l t s  of the  s tudy i n d i c a t e  t h a t  the  sepLrat ion of v o r t i c e s  is an 
important f a c t o r  i n  the noise  production of boundary l a y e r  flow. Thus, t h e  
p o s s i b i l i t y  of con t ro l  of t h i s  separa t ion  has impl ica t ions  t o  the  development of 
turbulence wi th in  the  boundary l a y e r  and its noise  r ad ia t ion  and may o f f e r  a 
p o t e n t i a l  for drag reduction. 
small degree of con t ro l  over vortex spacing wi th in  a boundary layer .  
p o s s i b i l i t y  of doing t h i s  mst be the  subjec t  of f u t u r e  experimental  and 
a n a l y t i c a l  work . 
However, i t  may prove d i f f i c u l t  t o  obtair .  even a 
The 
X 
Figure la: Vortex Trajectories in Oscillating Case 
n 1 1 
Figure lb: Acoristic Time History in Oscillating Case 
Figure 2a: Vortex Trajectories in Nonoscillating Case 
al 
L 
3 
0 
4 
‘0 
al 
N 
4 
d 
i! 
0 
2 
Figure 2b: Acoustic Time Hsstory in Nonoscillating Case 


Vortex studies relating to boundary layer turbulence and noise

VORTEX STUDIES RELATING TO BOUNDARY LAYER TURBULENCE AND NOISE
J.L. Adelman
George Washington University
Hampton, Virginia 23665
J.C. Hardin
National Aeronautics and Space Administration
Langley Research Center
Hampton, Virginia 23665
Knowledge of vortex dynamics is crucial to the understanding of boundary
layer flow and its noise production, as well as structural fatigue caused by
interactions between turbulent flows and various surfaces. The growing use of
high-performance aircraft, rotorcraft, and other high-speed transportation
systems in recent years has emphasized the need to understand such dynamics,
since laminar flow control, interior noise reduction, and structural fatigue
characteristics may be critical to the success of such vehicles.
Turbulent boundary layers are comprised of vorticity whose characteristics
are defined by the flow direction and surface geometry. As a simple model of a
boundary layer, the present study considers the two-dimensional case of an array
of N rectilinear, like-sign vortices above an infinite flat boundary. The
method of images can be employed with this configuration to reduce the problem
to that of 2N vortices in free space, constrained by 2N symmetry relations.
This system is Hamiltonian and therefore certain invariants of the motion are
known. Further, from the Hamiltonian constant, the equations of motion are
readily derived and may be integrated numerically to determine the vortex
trajectories. This knowledge of the time-dependent vortex motion then allows
the resulting noise radiation to be computed by standard aeroacoustic
techniques.
25
https://ntrs.nasa.gov/search.jsp?R=19860002548 2020-03-20T16:09:22+00:00Z
The model has been examined extensively for many different initial vortex
configurations, including both trajectory and noise calculations. Several
analytical and numerical characteristics of the interactions are observed. For
example, for N=2, a criterion,
yITY21/T(X 2 + y 2) > _T+I/T(X2 + y2)O
where YI and Y2 are the initial heights of the two vortices above the plane,
X and Y are their initial separations in the respective coordinate directions,
Yo is twice the average height of the vortices above the boundary, T is the
ratio of the circulations, and _ is the centroid of vorticity, for oscillating
motion of the vortices can be derived. Such motion is periodic and therefore
produces sound spectra containing only harmonically related discrete frequency
components. Figure la is an example of the vortex trajectories in this case
while Figure ib is the resulting noise radiation to a fixed observer. The
apparent modulation of the noise signal is due to the directivity of the noise
source as it moves with respect to the fixed observer. When this criterion is
not satisfied, the vortex motion is non-oscillating in nature and therefore
produces very little noise. Figure 2a is an example of the trajectories in this
type of motion while Figure 2b displays the resulting noise. In this case, the
influence of the image of the vortex closest to the boundary causes it to
convect rapidly away from the other vortex resulting in negligible interaction.
The differences between these two cases are similar to those between laminar and
turbulent boundary layers.
The analysis is extended for N>2. In this case, the phenomena of
non-integrability and non-deterministic "chaotic" solutions occur. Examples of
this type of motion are included.
26
The results of the study indicate that the separation of vortices is an
important factor in the noise production of boundary layer flow. Thus, the
possibility of control of this separation has implications to the development of
turbulence within the boundary layer and its noise radiation and may offer a
potential for drag reduction. However, it may prove difficult to obtain even a
small degree of control over vortex spacing within a boundary layer. The
possibility of doing this must be the subject of future experimental and
analytical work.
m
Figure la: Vortex Trajectories in Oscillating Case
15--
-IS 0 I0 20 3Q 40 SO 1SO
t
Figure Ib: Aco.stic Time History in Oscillating Case
27
10--
Y
5
• 4)
0 _, 10 '15 2'0 25 30 35 40 ,15 50 _ 80 I_ 7'0 75
o I'"'1'"'1""1''"1""1''"1''"1''"1'"'1''"1''"1""1""1'"'1"*'1
X
Figure 2a: Vortex Trajectories in Nonoseillating Case
rJl
U
-rl
tn
0
¢.J
OJ
.r,..l
r'-I
m
O
Z
--I 0 10 20 30 _ JSO mr*i,,,,,,,,,i,,,,+,,,,i,,,,,0,,,I,,,,,''''t'''''''''l'''''''''l
t
Figure 2b: Acoustic Time History in Nonoscillating Case
28


Vortex Studies Relating to Boundary Layer Turbulence and Noise

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