In wall-bounded shear flows the transition to turbulence through localized disturbances goes through a pattern starting with a development of shear layers. The localized normal velocity fluctuations induce normal vorticity through the lift-up effect. These shear layers become unstable to secondary disturbances, and if the amplitudes of the disturbances are large enough, a turbulent spot develops. Investigations of the spot in boundary layers has shown that the turbulent part of the spot is very similar to a fully developed boundary layer. Wygnanski et al. (1976) showed that the mean profile at the center-symmetry plane has a logarithmic region and Johansson et al. (1987) showed that both the higher-order statistics and flow structures in the spot were the same as in the corresponding fully developed flow. In what respects the turbulence inside the Poiseuille spot is similar to fully developed turbulent channel flow is studied. The numerically simulated spot is used, where the characteristics inside the spot are compared to those of the wave packet in the wingtip area. A recent experimental investigation of the velocity field associated with the Poiseuille spot by Klingmann et al. is used for comparison

Alfredsson, P. Henrik

Henningson, D. S.

Kim, John

English

NASA Technical Reports Server

Center for ilLrbulence Rerearrh 
Proceedingr of the Summer Program 1988 
179 
Turbulence characteristics inside a 
turbulent spot in plane Poiseuille flow 
B y  D.S. Henningson', J. Kim2, & P. H. Alfredsson3 
1. Introduction 
In wall-bounded shear flows the transition to turbulence through localized distur- 
bances goes through a pattern starting with a development of shear layers. The lo- 
calized normal velocity fluctuations induce normal vorticity through the lift-up effect 
(Gustavsson, 1978; Henningson, 1988; Landahl, 1975). These shear layers become 
unstable to secondary disturbances (Breuer, 1988), and if the amplitudes of the dis- 
turbances are large enough, a turbulent spot develops (Chambers & Thomas, 1983). 
The spot develops in approximately a self-similar manner with an average spreading 
angle of about 10" for most wall-bounded shear flows (Riley & Gad-el-Hak,l985). 
In plane Poiseuille flow spanwise wingtips of a spot consist of large-amplitude wave 
packets that breakdown as they propagate into the spot. The breakdown of the 
wave packets has been observed both in experiments (Henningson & Alfredsson, 
1987) and in a numerical simulation (Henningson et al.,1987). The experiments 
have shown that the wave packets consist of the least stable Tollmien-Schlichting 
(T-S) mode. Analysis of the simulation data using kinematic wave theory has also 
shown that the wave breakdown is closely related to the spot spreading (Henning- 
son,1988). 
Investigations of the spot in boundary layers has shown that the turbulent part of 
the spot is very similar to fully-developed boundary layer. Wygnanski et al. (1976) 
showed that the mean profile at the center-symmetry plane has a logarithmic region 
and Johansson et al. (1987) showed that both the higher-order statistics and flow 
structures in the spot were the same as in the corresponding fully-developed flow. 
The aim of the present investigation is to study in what respects the turbulence 
inside the Poiseuille spot is similar to fully developed turbulent channel flow. The 
numerically simulated spot (Henningson et al. 1987) will be used in the present 
investigation, where the characteristics inside the spot will be compared to those 
of the wave packet in the wingtip area. A recent experimental investigation of the 
velocity field associated with the Poiseuille spot by Klingmann et al. (1988) will be 
used for comparison. 
2. Eduction of Mean Velocity 
1 Massachusetts Institute of Technology 
2 NASA Ames Research Center 
3 Royal Institute of Technology 
https://ntrs.nasa.gov/search.jsp?R=19890015183 2020-03-20T03:02:14+00:00Z
180 
ORIGINAL PAGE IS 
OF POOR QUALITY 
D.S. Eenningson, J .  Kim & P .  E .  Alfreddaon 
FIGURE 1. Surfaces of streamwise mean velocity at the centerline, a) conical 
averaging, b) and c) Gaussian filters with different filter widths at t=258. Note 
that only half the spot is shown. 
Turbulent spot in plane Poiaeuille flow 181 
The numerical simulation of a turbulent spot in plane channel was performed 
at Reynolds number of 1500 based on the centerline velocity (Uc) and the channel 
half-width (h). The reader is referred to Henningson et al. (1987) for details re- 
garding the simulation. Because of its enormous cost of computation, no attempt 
was made to run several cases to obtain ensemble-averaged statistics. In order to 
compare the simulation results from this single realization to the ensemble-averaged 
results obtained from experiments, a mean velocity has to be defined. Analysis of 
the fluctuating components also requires a well defined mean. Since the spot is ap- 
proximately self similar, a conical averaging procedure was tried first. One hundred 
and twenty velocity fields between t = 222 and t = 258 were interpolated into a 
conical coordinate system and added together. The horizontal conical coordinates 
were defined as f = z / t  and C = z / t .  The mean velocity obtained from this pro- 
cedure was not smooth enough to be regarded as a mean as shown in Fig. la, 
where the surface representing the streamwise velocity at the centerline (y = 0) is 
plotted. The conical average of the normal velocity revealed more small-scale struc- 
tures that survived the averaging process. Spatial averaging was also tried using 
Gaussian filters. Figures l b  and IC show the results obtained by using two different 
filter widths. The Gaussian-filtered velocity field shown in Fig. IC was chosen to 
represent the mean because the velocity fluctuations from the other two averages 
were judged to be too large in the turbulent area of the spot. The problem with 
a fairly wide Gaussian-filter function (in physical space) is that sharp gradients in 
the mean flow are smoothed out as well when the small scales are removed. This 
was observed at the transition region where sharp gradients exist. 
The ensemble-averaged results from Klingmann et al. (1988) and those obtained 
from the procedure mentioned above are shown in Figs. 2 and 3, respectively. 
Agreement between the two results are remarkable: the sharp transition region with 
the dip in the velocity, the arrow head front part of the disturbance, and the speed- 
up of the laminar fluid around the spot. One may conclude from this comparison 
that the spatial average can reasonably well represent the true ensemble average. 
The spanwise mean velocity at the centerline is shown in Fig. 3c. Most of the fluid 
inside the spot can be seen to move outward while the laminar fluid in the front 
and back of the spot is directed towards the spanwise symmetry line, z = 0. 
The fluctuating components can be defined as the full simulated velocities with 
the spatially averaged mean subtracted. Fig. 4 shows contours of streamwise and 
normal components. In Fig. 4a, we notice an artificial oblique structure at the 
spanwise wingtip of the spot, resulting from too much smoothing in the sharp 
gradient between the laminar and turbulent part. The streamwise fluctuating com- 
ponent closer to the wall is shown in Fig. 4b. The long streaky structure typical of 
near-wall turbulent flows can be seen inside the spot. The wingtip wave packet is 
discernible at this position, but it can be more clearly seen in the normal velocity 
at the centerline as shown in Fig. 4c. 
The rms fluctuations are also computed from the fluctuating components. Con- 
tours of the rms fluctuation of streamwise velocity at the centerline are shown in 
182 
0.20 
0.15 
I- 
N 
\ 0 .10  
0.05 
D.S. Henningson, J.  Kim d P. E. Aljkdsson 
I I I I I 
0 . 4  0 .5  0 . 6  0 .7  0 . 8  0 .9  1 . 0  
0.20 
0.15 
I- 
N 
\ 0.10 
0 . 0 5  
0 
6 
I I I I I 
2 
4 0 .5  0 . 6  0 . 7  0 . 8  0 .9  1 . 0  
0 .20  I I I I I 
0 .15 
0 . 4  0 .5  0 . 6  0 . 7  0 . 8  0 .9  1 . 0  
X / T  
at the centerline (in conical coordinates), a) mean velocity measured at z = 100, b) 
mean velocity measured at z = 200, c) rms velocity measured at z = 200. Contour 
spacing is 0.02 in a) and b) and 0.01 in c). 
Turbulent spot in plane Poiseuille flow 183 
12 contours, min: 0.800: mox: 1.020: incr: 0.020 Y 
0 
0 
T 
0 
a 
s 
x 
8 
0 
0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0 
X/T 
21 contours, min: 0.700: mox: 1.100: incr: 0.020 Y 
IO 
FIGURE 3. Contours of mean Gaussian filtered velocity at centerline, a) streamwise 
velocity for t = 138, b) streamwise velocity for t = 258, c) spanwise velocity for t 
= 258. 
2 contours. min:-0.040; max: 0.040; incr: 0.080 n 
0 
2 contours, min:-0.010; mox: 0.010; incr: 0.020 Y 
0 
FIGURE 4. Contours of disturbance velocities for t = 258, a) streamwise velocity at 
y = 0, b) streamwise velocity at y = -0.83, c) normal velocity at y = 0. Dashed lines 
in this and following figures indicate negative contour levels unless stated otherwise. 
Turbulent spot in plane Poiseudle f i w  185 
0 
B 
0- 
In 
0 
s 
9- 
0 
0 
9 
I 
8 contours, min: 0.010; rnax: 0.080; incr: 0.010 Lo 
30 
P 
0 
Lo 
0 
I 
s 
9 
0 
0 
9 
X/T 
FIGURE 5.  
normal velocities. 
Contours of rms velocities for t = 258 at y = 0, a) streamwise and b) 
Fig. 5a. It increases substantially high inside the turbulent part. The correspond- 
ing rms fluctuation obtained from experiments by Klingmann et al. is shown in Fig. 
2c. They are in good agreement with each other, including the s m d  peak in the 
wingtip region. This indicates that the peak is not due to the excess disturbance 
velocity induced by the Gaussian smoothing of the sharp transition gradient. The 
rms fluctuation of normal velocity at the the centerline is shown in Fig. 5b. It is 
also substantially larger in the turbulent area, but it rises to twice that value in 
the spanwise wave region. This is a consequence of the high amplitude T-S waves 
present at the wingtip. 
0 
3. Comparison with fully-developed turbulent channel flow 
In order to compare turbulence characteristics inside the spot with those in fully 
developed channel flow turbulence, one would like to scale the flow variables with 
186 D.S. Henningson, J.  K i m  & P. H .  Aljkdsson 
V +  
FIGURE 6. a) Contours of u, at lower channel wall, b) mean velocity in the lower 
channel half averaged over the turbulent area compared with the law of the wall 
and the log-law. Dashed curves are u+ = y+ and u+ = 2.5 lny+ + 5.5. 
the inner-wall scaling, i.e., with the friction velocity u: = d- and the 
kinematic viscosity v. Contours of u, = u:/Uc~ are shown in Fig. 6a. A mean 
< 0.70 
and -0.08 < C < 0.0 to yield about 0.05. (The above designated area will in the 
following be referred to as the turbulent area). The y-dependence of the mean 
is u+ = y+ and the line u+ = 2.51n(y+) + 5.5, where the superscript + denotes 
a quantity scaled by the inner-wall scaling. The log-region is very small since the 
Reynolds number is low (Re, = u,h/v M 70). Considering the low Reynolds 
number the shape is in good agreement with other simulation and experimental 
results (Kim et al.,1987). The y-dependence of the rms values of three velocity 
components averaged over the turbulent area are shown in Fig. 7a. They are in 
good agreement with those corresponding to the fully-developed turbulent channel 
flow, except that the peak in the streamwise velocity close to the wall appears to 
I value of uLs inside the spot was obtained by averaging it in the area 0.58 < 
streamwise velocity in the turbulent area is shown in Fig. 6b. The dotted curve I 
I 
90 
Turbulent spot in plane Poiseuille flow 187 
be a bit low. To contrast the findings in the turbulent area, a wave area is defined 
as 0.70 < f < 0.74 and -0.12 < < < -0.08. The y-dependence of the rms values 
in the wave area is shown in Fig. 7b. All of the fluctuating components are larger 
in the wave area, and the normal velocity has a different shape, which is somewhat 
similar to the mode shape of the least stable T-S wave. The Reynolds shear stress 
both in the turbulent and the wave area is shown in Fig. 7c. In the turbulent area 
it shows the behavior expected from a fully developed channel flow; in the wave 
area it is substantially higher, indicating higher turbulence production. 
Another characteristic of wall-bounded turbulent shear flows is the sharp shear- 
layer structures found in the near-wall region. In order to examine the structure of 
the shear layers, the VISA (Variable-Time Space-Average) technique was used to 
detect them. This technique has been used by Kim (1985) and Johansson, Alfreds- 
son & Kim (1987) to detect the shear layers in fully-developed turbulent channel 
flow. Islands of high variance of streamwise velocity were identified for a given flow 
field, and an ensemble average was obtained by aligning the maximum local vari- 
ance in the islands for every individual event detected. The results obtained from 
this procedure is given in Figs. 8 and 9. They appear to be in good agreement with 
those obtained from the full turbulent simulation. Figures 8a and 8b show the typi- 
cal inclined shear-layer structure associated with the lift-up of low speed fluid in the 
downstream and the downward sweep in the upstream. The spanwise structure of 
the shear layer is shown in Fig. 8c. The spanwise distance between the outer loops 
is slightly above 100 viscous units, indicating that the average spacing between the 
low- and high-speed streaks are captured correctly (Kim et a.l.,1987). The ensemble 
average of the shear-layer structure in the wave area shows a similar behavior in 
the normal plane (Fig. 9a), but the horizontal structure clearly reflects the wave- 
like characteristics (Fig. 9b). The oblique character of the waves can be seen, and 
more importantly a structure of peaks and valleys in the cross wave direction. This 
is typical of the secondary breakdown seen in vibrating-ribbon experiments, (Kle- 
banoff et al., 1962; Nishioka et al., 1976). This supports the idea that the secondary 
instability is responsible for the breakdown of the large-amplitude waves and the 
subsequent rapid transition to turbulence. 
4. Summary and discussion 
Using a horizontal Gaussian-filter function a mean and fluctuating components 
were computed from instantaneous velocity fields obtained from a single realization. 
The streamwise component of the mean was in good agreement with the ensemble- 
averaged velocity field obtained from the experiments by Klingmann et al. (1988). 
The contours of streamwise rms fluctuation also showed a good agreement between 
the experimental and simulated results. 
In the turbulent part of the spot the velocity field was very similar to that of 
fully-developed turbulent channel flow. The mean velocity and the rms fluctuations 
as well as the Reynolds shear-stress profiles all showed good agreements. The shear- 
layer structure obtained from the VISA technique also revealed similar behavior, 
indicating that the turbulence inside the spot behaves as if it were a fully-developed 
turbulent channel flow. 
188 D.S. Henningson, J.  Kim 8 P. H. Aljizdsson 
'1 
1.5 
1 
> a 0  
-1 
-1.5 
L. /- *. \ 
-1.0 -.75 -.50 -25 0 .25 .50 .75 1.0 
Y 
FIGURE 7. a) Rms velocities averged over the turbulent part, b) rms velocities 
averaged over the wave area (rms velocities are scaled with mean u: from the 
turbulent area: - 9 % m a ;  ---- 3 vtma; . . . . . . . . , wtma. C) - , Reynolds shear 
stress in turbulent area; ---- , wave area; -.-.---- , uv = -y. 
Turbulent spot in plane Pobeuille flow 189 
12 conts, min:-0.150; max: 0.150; incr: 0.025 
9 - 
_ _ _ _ _ - - - - -  - - - _ _  _ _ _ - -  
*2 
2 
.O 
X 
- 
>2- 
9 
0 
\ --_ 
c 
I 
I I  
-4 -- c -  
c - 2 ,  > -- 
- - - - /  
N 
I 
SI(- 
-:* - - _  - - - - - - _ _ _  I----  - _ _ _ _ _ - - -  
9 I 
-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 
X 
FIGURE 8. Contours of conditionally averaged velocities: a) streamwise velocity in 
the zy-plane along the center of the shear layer; b) normal velocity in the zy-plane; 
c) streamwise velocity in the zz-plane at y = 0.22. Note that y is measured from 
the bottom of the channel, and the tick marks denote 70 wall units. 
The results from the wave area were somewhat different. The fluctuating compo- 
nents were larger, and the profile of rms fluctuations indicated the typical shape of 
the least stable T-S wave profile. The Reynolds stress was much higher in the wave 
area than inside the turbulent part, indicating high turbulence production in the 
190 D.S. Henningson, J .  Kim d P. H. Alfredsson 
13 conk, min:-0.045; rnax: 0.02a incr: 0.005 
I 
N 
-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 
X 
FIGURE 9. Contours of conditionally averaged velocities: a) streamwise velocity 
in the zy-plane; b) normal velocity in the zy-plane; c) streamwise velocity in the 
zz-plane at y = 0.22. 
wave area. This is a property in common to T-S wave transition, and has been ob- 
served both in experiments (Klebanoff et al.,1962) and simulation (Gilbert,1988). 
The appearance of shear layers in the wave area is also typical of the T-S wave 
transition. The small-scale structure in the cross wave direction captured by the 
conditional averaging procedure is typical of secondary instability associated with 
T-S wave breakdown. Thus the transition from laminar to turbulent flow seems 
to follow a similar pattern in the wingtip area of the spot as in the 2D vibrating 
ribbon experiments. 
Acknowledgment 
We would like to thank Arne Johansson for many fruitful discussions. 
Turbulent spot in plane Poiseuille flow 191 
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Turbulence characteristics inside a turbulent spot in plane Poiseuille flow

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