The title of this presentation is the title of our research grant. While transition study is the objective of the work, the results to date are principally on the properties of turbulent boundary layers at low Reynolds numbers are discussed. Testing was done in a small return wind tunnel. Mean boundary layer development is given. The skin friction behavior of the turbulent points are considered. No standard laminar flow was observed. Furthermore, the turbulent mean flow data seem reasonable for the elevated disturbance levels of the tests in the sense that there is no discernible wake component to any of the profiles and that the variation of skin friction with R sub theta is consistent with zero wake strength. The no-grid data are in all likelihood transitional

Paik, D. K.

Reshotko, E.

English

NASA Technical Reports Server

TRANSITION I N  A DISTURBED ENVIRONMENT 
E. Reshotko and D.K. Paik 
Case Western Reserve U n i v e r s i t y  
Cleveland, Ohio 44106 
The t i t l e  o f  t h i s  p resenta t ion  i s  t h e  t i t l e  o f  our  research gran t .  While 
t r a n s i t i o n  study i s  t h e  o b j e c t i v e  o f  t h e  work, t h e  r e s u l t s  t o  da te  a re  p r i n c i -  
p a l l y  on t h e  p rope r t i es  o f  t u r b u l e n t  boundary l aye rs  a t  low Reynolds numbers. 
EXPERIMENTAL EQUIPMENT 
The work was done i n  a  smal l  open-return wind tunne l  o f  poor f l o w  q u a l i t y .  
Furthermore i t s  f l o w  q u a l i t y  could e a s i l y  be made worse. The tunne l  ( f i g .  1 )  
i s  19.7 cm (7-3/4 i n )  h igh.  The t e s t  sec t i on  i s  about 66 cm (26 i n )  long. The 
f l o w  v e l o c i t y  i s  o f  t h e  order  o f  6  m/sec (20 f t / s e c )  so t h a t  u n i t  Reynolds num- 
bers a re  about 366 000/m (120 0 0 0 / f t ) .  The t e s t  p l a t e  spans t h e  tunne l  and 
f i l l s  t h e  l eng th  o f  t h e  t e s t  sec t ion .  The p l a t e  lead ing  edge i s  e l l i p t i c a l ,  
and a  smal l  f l a p  i s  at tached t o  t h e  downstream end o f  t h e  p l a t e  t o  s e t  t h e  
attachment l i n e  on t h e  working s ide  o f  t h e  lead ing  edge o f  t h e  p l a t e .  Measure- 
ments were made over  t h e  f i r s t  18 cm (0.6 f t )  downstream o f  t h e  lead ing  edge by 
us ing  a  t r a n s l a t i n g  hot-wi re probe i n  t h e  spanwise centerplane. No spanwise 
v a r i a t i o n s  a re  measured. Length Reynolds numbers f o r  a l l  o f  t h e  data recorded 
were under 75  000. Test sec t i on  tu rbu lence l e v e l s  were a l t e r e d  by p l a c i n g  
g r i d s  a t  t h e  l o c a t i o n  ahead o f  t h e  c o n t r a c t i o n  i n d i c a t e d  i n  f i g u r e  1. The 
g r i d s  used and t h e  corresponding tes t - sec t i on  tu rbu lence l e v e l s  a re  shown i n  
f i g u r e  2. 
MEAN BOUNDARY-LAYER RESULTS 
The mean boundary-layer development i s  shown i n  f i g u r e  3. I n  t h e  absence 
o f  a  g r i d  t h e  growth o f  Ree ( f i g .  3 (a ) )  seems t o  be laminar,  bu t  t h e  
values are  a t  about tw i ce  t h e  B las ius  l e v e l  f o r  t h e  same d is tance f rom t h e  
l ead ing  edge. With g r i d  1  t h e  boundary l a y e r  s t a r t s  growing a t  a  more r a p i d  
r a t e  (shaded p o i n t s )  beyond 9  cm f rom t h e  lead ing  edge. For g r i d  2 t h e  more 
r a p i d  growth begins a t  about 6  cm f rom t h e  lead ing  edge. The corresponding 
shape f a c t o r s  a re  shown i n  f i g u r e  3(b) .  I n  t h e  absence o f  a  g r i d  t h e  shape 
f a c t o r s  decrease cont inuous ly  f rom about 2.6 t o  1.9 over  t h e  range o f  measure- 
ment. The no-gr id  data are be l ieved t o  be t r a n s i t i o n a l  f o r  reasons t o  be e lab-  
o ra ted  l a t e r .  With g r i d s  1  and 2  t h e  shaded p o i n t s  corresponding t o  t h e  more 
r a p i d  growth i n  momentum th ickness i n  f i g u r e  3 (a)  have shape f a c t o r s  below 1.7 
and are  be l ieved t o  be t u r b u l e n t .  For these p o i n t s  a  shear v e l o c i t y  u, 
cou ld  be obta ined by f i t t i n g  t h e  p r o f i l e s  t o  a  law o f  t h e  w a l l .  
An example o f  t h e  de termina t ion  o f  u, i s  shown i n  f i g u r e  4 f o r  t h e  
g r i d  2 p r o f i l e  a t  x  = 9 cm. The law-of- the-wal l  expression used i s  t h e  a lge-  
b r a i c  form developed by Musker ( r e f .  1 )  t h a t  inc ludes  t h e  laminar  sublayer  and 
t h e  b u f f e r  l a y e r  and i n  t h e  l o g - l i n e a r  reg ion  d i sp lays  Coles'  constants 
( r e f .  2 ) .  By comparing t h e  experimental  p o i n t s  made dimensionless w i t h  d i f -  
f e r e n t  values o f  u, w i t h  t h e  Musker curve, a  va lue o f  u, o f  0.34 m/sec 
(1.12 f t / s e c )  was chosen based p r i m a r i l y  on t h e  p o i n t s  i n  t h e  near l o g - l i n e a r  
https://ntrs.nasa.gov/search.jsp?R=19850023123 2020-03-20T17:22:37+00:00Z
reg ion.  A more impressive f i t  i s  shown i n  f i g u r e  5. To be noted i n  both 
f i g u r e s  4  and 5  i s  t h e  absence o f  a  wake component t o  t h e  v e l o c i t y  p r o f i l e .  
This  i s  t r u e  o f  a l l  t h e  p o i n t s  obtained w i t h  g r i d s  1  and 2  t h a t  a re  i d e n t i f i e d  
as t u r b u l e n t .  
For t h e  no-gr id  case ( f i g .  6) a  shear v e l o c i t y  de termina t ion  :auld n o t  be 
made i n  t h i s  way as t h e  p r o f i l e s  d i d  n o t  d i s p l a y  law-of- the-wal l  s i m i l a r i t y .  
Hence what i s  p l o t t e d  i s  u/ue versus y/6. The boundary-layer th ickness  
6 i s  n o t  a  measured th ickness.  Rather i t  i s  computed from t h e  displacement 
th ickness  and shape f a c t o r  by us ing t h e  r e l a t i o n  6/6* = (H + 1 ) / (H  - 1). Wi th 
t h i s  no rma l i za t i on  t h e  l o c a t i o n  where y/b = 1  corresponds t o  a  0.95 veloc-  
i t y  r a t i o .  The p r o f i l e s  a re  co l lapsed f o r  y  > 6, w h i l e  f o r  y  < 6 t h e  
v e l o c i t y  p r o f i l e s  f i l l  o u t  w i t h  d is tance downstream. This  supports t h e  i d e n t -  
i f i c a t i o n  o f  t h e  no-gr id  data as t r a n s i t i o n a l .  
Next, we consider  t h e  s k i n  f r i c t i o n  behavior  o f  t h e  t u r b u l e n t  p o i n t s  
obta ined w i t h  g r i d s  1 and 2. Shown i n  f i g u r e  7 i s  a  semilog p l o t  o f  ue/u, 
versus Ree. The present  data a re  t h e  shaded p o i n t s  i n  t h e  lower l e f t  o f  t h e  
f i g u r e ,  which correspond t o  t h e  shaded p o i n t s  i n  f i g u r e  3. The o the r  p o i n t s  
come from p r i o r  i n v e s t i g a t i o n s  o f  f l a t - p l a t e  t u r b u l e n t  boundary l aye rs  a t  low 
tu rbu lence l e v e l s  and i nc lude  t h e  Wieghardt data as reduced by Coles ( r e f .  3) ,  
t h e  data o f  P u r t e l l ,  K lebanof f ,  and Buckley ( r e f .  4). and t h e  data o f  M u r l i s ,  
Tsai,  and Bradshaw ( r e f .  5) .  A l l  these data a t  low tu rbu lence l e v e l  seem con- 
s i s t e n t  w i t h  each o the r  and w i t h  Coles'  ( r e f .  2) model based on a  d im in i sh ing  
wake s t reng th  as Ree reduces toward a  value j u s t  below 500. The present  
data d i s p l a y  bo th  t h e  l e v e l  and slope f o r  ue/u, versus Ree o f  a  
boundary l a y e r  t h a t  has no wake s t rength .  This  i s  seen by comparison w i t h  t h e  
ca l cu la ted  curve from t h e  Musker p r o f i l e  f o r  n = 0. 
I n  1962, Coles ( r e f .  2)  d i d  a l l u d e  t o  t h e  reduc t ion  i n  wake s t reng th  and 
e levated s k i n  f r i c t i o n  t o  be expected w i t h  an increase i n  tu rbu lence l e v e l .  A 
more p l a u s i b l e  phys ica l  hypothesis has been advanced by Huffman and Bradshaw 
( r e f .  6) through t h e i r  comparison o f  t u r b u l e n t  boundary l aye rs  developing under 
a  quiescent  i r r o t a t i o n a l  f r e e  stream w i t h  t h e  t u r b u l e n t  f l o w  i n  pipes, where 
t h e  t u r b u l e n t  core i s  n e i t h e r  quiescent nor  i r r o t a t i o n a l .  I n  t h e  l a t t e r  f l o w  
t h e r e  i s  no observable wake; i n  t h e  former t he re  i s  d e f i n i t e l y  a  wake component 
t o  t h e  v e l o c i t y  p r o f i l e s .  For t h e  t u r b u l e n t  boundary l a y e r  a t  low Reynolds 
numbers i n  quiescent  environments, Huffman and Bradshaw a t t r i b u t e  t h e  eros ion  
o f  t h e  wake component t o  t h e  increased importance o f  t h e  viscous super layer  - 
t h e  i n t e r f a c e  between t h e  boundary l a y e r  and t h e  i r r o t a t i o n a l  ex te rna l  f l o w  - 
i n  eroding t h e  wake component. For ex te rna l  f lows a t  e levated free-stream 
tu rbu lence - by analogy t o  t h e  s i t u a t i o n  i n  pipes - t h e  wake component i s  
eroded more severe ly  i f  n o t  e n t i r e l y  as i n  t h e  present  r e s u l t s .  An at tempt  
w i l l  be made i n  our  f u t u r e  work t o  develop t h i s  argument f u r t h e r  i n  con junc t ion  
w i t h  o the r  data sets  f o r  t u r b u l e n t  boundary l aye rs  developing i n  d i s tu rbed  
streams. 
DISTURBANCE FLOW RESULTS 
Figures 8 and 9 show t h e  l o n g i t u d i n a l  tu rbu lence i n t e n s i t y  d i s t r i b u t i o n  
a t  var ious values o f  Ree f o r  t h e  g r i d  1  and g r i d  2  data, respec t i ve l y .  
The peak value o f  ul/u, i s  about 2.35 t o  2.4 a t  y+ = 13. The approximate 
s i m i l a r i t y  shown i n  t h e  f i g u r e s  i s  i n  good agreement w i t h  t h e  r e s u l t s  o f  
P u r t e l l  e t  a l .  ( r e f .  4) .  Beyond t h e  peak, t h e  data p o i n t s  f a l l  u n t i l  t hey  
eventual  l y  l e v e l  a t  u;/u, corresponding t o  t h e  f ree-stream d is tu rbance 
l e v e l .  Taken together  w i t h  t h e  ue/u, values f o r  these p r o f i l e s ,  they  g i v e  a  
measure o f  t h e  free-stream d is tu rbance l e v e l ,  which t u r n s  o u t  t o  be about 
5.5 percent  f o r  g r i d  1  and 6.7 percent  f o r  g r i d  2. The P u r t e l l  e t  a l .  ( r e f .  4) 
da ta  shown i n  f i g u r e  9  i n d i c a t e  s i g n i f i c a n t l y  lower free-stream dis turbance 
l e v e l s  and a l s o  some spreading w i t h  Reynolds number beyond t h e  peak. One can 
perhaps surmise t h e  Reynolds number dependence o f  v iscous super layer  e f f e c t s  
i n  t h e  P u r t e l l  data t h a t  a re  absent i n  t h e  present  r e s u l t s  f o r  h igh  free-stream 
d is tu rbance l e v e l s .  
For t h e  no-gr id  case ( f i g .  l o ) ,  we do n o t  have u, as a  reference quan- 
t i t y .  P l o t t e d  i n  f i g u r e  10 t h e r e f o r e  i s  u ' / ue  versus y/&*. The p r o f i l e s  
f i l l  o u t  w i t h  d is tance downstream, b u t  t h e  peak values occur c o n s i s t e n t l y  a t  
y/&* o f  about 1.2, very c lose  t o  t h e  l o c a t i o n  o f  y/6* = 1.33 f o r  which t h e  
low-frequency u '  peak was observed i n  laminar  f lows a t  low free-stream t u r -  
bulence l e v e l s  by Klebanoff  ( r e f .  7) and others.  It would be o f  i n t e r e s t  t o  
see i f  t h e  Klebanof f  argument when app l i ed  t o  law-of- the-wal l  t u r b u l e n t  p ro-  
f i l e s  exp la ins  t h e  observed peak a t  y+ = 13. 
It i s  o f  i n t e r e s t  a t  t h i s  p o i n t  t o  look a t  d is turbance spectra.  F igure  11 
shows spec t ra  taken i n  t h e  f r e e  stream 6  cm downstream o f  t h e  lead ing  edge o f  
t h e  p l a t e  f o r  t h e  th ree  free-stream d is tu rbance l e v e l s .  Note t h e  increase i n  
i n t e n s i t y  a t  t h e  low f requencies w i t h  increase i n  tu rbu lence l e v e l  due t o  t h e  
g r i d s .  I n s i d e  t h e  boundary layer ,  t h e  spectra f o r  g r i d s  1  and 2  ( f i g s .  12 and 
13, respec t i ve l y )  a re  e s s e n t i a l l y  unchanged w i t h  downstream d is tance a t  these 
h igh  d is turbance l e v e l s .  For t h e  no-gr id  case, however, t he re  a re  progress ive 
changes i n  t h e  shape and i n t e n s i t y  o f  t h e  spectra ( f i g .  14). Although t h e  
i n t e n s i t i e s  t he re  a re  smal l ,  t h e  l a r g e s t  growth ra tes  occur i n  t h e  f r e  uency 
range 50 t o  100 HZ ( f i g .  15), corresponding t o  BV/U$ between 120x10- 8 
and 250x10-6. It i s  premature t o  asc r i be  any l i n e a r  i n s t a b i l i t y  connotat ions 
t o  t h i s  r e s u l t .  
SUMMARY 
The fo rego ing  i s  an i n t e r i m  r e p o r t  o f  our  i n v e s t i g a t i o n .  It i s  apparent 
t h a t  t he re  i s  much y e t  t o  be done. One t h i n g  t h a t  i s  f a i r l y  c l e a r  i s  t h a t  no 
standard laminar  f l o w  was observed. Furthermore t h e  t u r b u l e n t  mean f l o w  data 
seem reasonable f o r  t h e ' e l e v a t e d  d is turbance l e v e l s  o f  our  t e s t s  i n  t h e  sense 
t h a t  t he re  i s  no d i s c e r n i b l e  wake component t o  any o f  t h e  p r o f i l e s  and t h a t  
t h e  v a r i a t i o n  o f  s k i n  f r i c t i o n  w i t h  Ree i s  cons i s ten t  w i t h  zero wake 
s t rength .  The no-gr id  data a re  i n  a l l  l i k e l i h o o d  t r a n s i t i o n a l .  This  case 
requ i res  a d d i t i o n a l  concentrated study i n  o rder  t o  o b t a i n  more d e f i n i t e  i n f o r -  
mation regard ing t h e  t r a n s i t i o n  process i n  a  d i s tu rbed  environment. 
REFERENCES 
1. Musker, A.J.: E x p l i c i t  Expression f o r  t h e  Smooth Wall V e l o c i t y  D i s t r i b u -  
t i o n  i n  a  Turbulent  Boundary Layer. A I A A  J., vo l .  17, no. 6, June 1979, 
pp. 655-657. 
2. Coles, D.E.: The Turbulent  Boundary Layer i n  a  Compressible F lu id ,  Rand 
Corp., Rep. R-403-PR, 1962. 
3. Coles, D.E.:  The Young Person's Guide t o  t h e  Data. I n  Proceedings o f  1968 
AFOSR-IFP-Stanford Conf. (Vo l .  11), S tan ford  Univ., 1969, pp. 1-45. 
4 .  P u r t e l l ,  L.P.; K lebanof f ,  P.S.; and Buckley, F.T.: Turbu len t  Boundary 
Layer a t  Low Reynolds Number. Phys. F lu ids ,  v o l .  24, no. 5, May 1981, pp. 
802 -81 1 . 
5. M u r l i s ,  J.; Tsai ,  H.M.; and Bradshaw, P.: The S t r u c t u r e  o f  Turbulent  
Boundary Layers a t  Low Reynolds Numbers. J. F l u i d  Mech., v o l .  122, 1982, 
pp. 13-56. 
6. Huffman, G.D.; and Bradshaw, P., A Note on von Karman's Constant i n  Low 
Reynolds Number Turbulent  Flows. J.  F l u i d  Mech., v o l .  53, 1972, pp. 54-60. 
7. K lebanof f ,  P.S.: P r i v a t e  communication. 
r T r a v e r s i n g  hot-wire probe 
Grid location 
Figure 1.  - Schematic diagrams o f  wind tunnel and tes t  p late .  (Dimensions are  i n  inches.) 
Grid 2 
Figure 2. - Shapes and dimensions of  grids. 
~;ms"Je# 
percent 
0 No qrid -2 
  rid 1 5-6 
Grid 2 6-7 
- 
Distance from leading edge, x, cm 
(a) Boundary-layer growth. 
(b) Shape factor variation. 
Figure 3. - Mean boundary-layer development. 
Muske- -"-'- 
15 (ref I 
Figure 4. - Mean-velocity profile at 9 an - grid 2. u, = 1.12 ft/sec. 
Figure 5. - Mean-velocity profile at 15 an - grid 2. u, = 1.09 ft/sec. 
Figure 
Figure 6 .  - Development o f  boundary layer - no grid.  
28 
26 
24 
+- 
'I 22 
2'1 =Ir 
0 Coles (ref. 3) 
20 Purtell, Klebanoff, and Buckley (ref. 
18  
16 
203 2000 5000 10000 
Re0 
7. - Skin f r i c t i o n  o f  turbulent boundary layers a t  low Reynolds 
4) 
numbers. 
Figure 8. - Distribution of  longitudinal fluctuating velocity - grid 1. 
0 375 
- 
• a 280 
. 465 (ref. 4) 
- 
- n 
0 
0. 
a 
Figure 9. - Distribution of longitudinal fluctuating velocity - grid 2. 
Y / &  
Figure 10. - Disturbance profiles for no-grid case. 
(a) Nogrid.  (b) Grid 1. (c) Grid 2. 
Figure 11. - Free-stream turbulence spectra, 0-250 Hz. 
200 .- 
> 
E O ?  (a) I I I I 1 
ai m  
g 400 
> 
300 
200 '- 
(bl I I I I (d) I I I 1 I 
0 50 100 150 200 250 0 H) 100 150 200 250 
Frequency, Hz 
Frequency, Hz 
(a) x = 18 an. (c) x = 12 an. 
(b) x = 15 an. (d) x = 9 an. 
(e)  x = 6.m. 
Figure 12. - Spectra of  u' with increasing distance from leading edge - grid 1. 
6 
- 
(b) I I I I I (d) I I I I I 
0 50 100 150 200 250 0 50 100 150 200 250 
Frequency, Hz 
0 50 100 150 200 250 
Frequency, Hz 
(a) x = 18 an. (c) x = 12 an. 
(b) x = 15 an. (d) x = 9 an. 
(el  x = 6 an. 
Figure 13. - Spectra of  u' with increasing distance from leading edge - grid 2. 
Frequency. Hz 
(a) x = 18 an. (c) x = 12 an. 
(b) x = 15 an. (d) x = 9 an. 
(e) x = 6 an. 
Figure 14. - Disturbance spectral data inside boundary layer - no grid. 
Frequency, rHz 10
A 19.5 
a 29 
0 39 
v 49 
0 68 
0 88, 98 
Figure 15. - Relative amp1 i-tudes o f  selected frequencies - no grid. 


Transition in a disturbed environment

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