ACTA MECHANICA SINICA, Vol. 4. No. 4, November, 1988

Science Press, Beijing, China

Allerton Press, INC., New York, U. S. A.

ISSN 0567--7718

B U R S T I N G F R E Q U E N C Y IN T U R B U L E N T B O U N D A R Y L A Y E R S *

Shu Wei Tang Ning

( Tianjin University)

ABSTRACT: Employing laser Doppler anemometry and VITA techniques, the bursting frequency in

turbulent boundary layers has been measured over the Reynolds-number range 320 to 1470. The result

indicates that the mean and non-dimensional bursting frequency scaled with the variables appropriate for the

wall region was constant and independent of Reynolds number. When the same data are plotted using the

outer variables of boundary layer tO normalize the bursting frequency, the non-dimensional frequency

increases as the Reynolds number increases. This is in agreement with the results of Blaekwelder et al. (1983)

who used hot wire anemometry and VITA technique.

KEY WORDS: turbulent boundary layer, bursting frequency, conditional sample.

I. INTRODUCTION Bursting frequency in turbulent boundary layer has been measured by a number of workers.

The earlier studies showed that the bursting period 7' B scaled with inner variables (friction velocity

v*, kinematic viscosity v) increases as Reynolds number (characteristic length is momentum

thinckness 0 of boundary layer) increases. The mean bursting period should be constant when

normalized with outer variables (velocity of free flow U~, boundary-layer-thickness 6). Summing

many experimental results Kim et al. (1971) Ill got the formula below:

D 0"73 TBUo~/6 = 5 (600 < Reo<9000) 7"By . 2 / v = 0.65 1re0

To the contrary, the experiments of Blackwelder et al. (1983) lzl showed that ~Bv* 2 / v is

indenpedent of Reynolds number, and that 7" B U~ / 6 decreases as Reynolds number increases. It

was pointed ou t in the paper of Blackwelder et al. (1983) that the sensor length in the spanwise

direction is important in detecting bursts, especially if the sensor length exceeds approximately 20

v/v*. Willmarth et al. (1983) TM used a hot wire probe of 0.3 viscous length scale and got consistent

conclusion with that of Blackwelder et al..

To compare with Blackwelder et al. and Willmarth et al. whose experiments were both carried

out with hot wire anemometry in wind tunnel, in this paper LDA was employed in an open water

channel , which was the same as the equipments used by several investigators earlier. While flow

visualization and hot wire were usually the techniques preferred in the past, non-contact ing velocity

measuring technique was used in this experiment, so new results were expected.

Received 11 June 1987.

* The project is supported by the National Natural Science Foundation of China

292 ACTA MECHANICA SINICA 1988

II. EXPERIMENTAL APPARATUS AND MEASUREMENT TECHNIQUES The experiments were performed in a free surface warter channel of 240 ram in width, 400 mm

in depth and 6000 mm in length. For convenience of free movement of LDV's traversing system, the

channel is installed without any supports under test section, as shown in Fig. 1.

Fig. 1 Experimental equipment

The flow speed in the channel is of the range from 3 to 30 em / s, and the free-stream turbulence

level is 2 - - 3 % . The water level was controlled by a multiple-hole end valve and was being detected

during theexperiment by an automatie water-level-tracking-meter to ensure a constant water depth.

The flat plate was rigidly suspended in the water.channel and measurement carried out at a place 4.5

meters from the leading edge. According to the proposal of Blackwelder et al. (198.3) [21, measuring

points for detecting bursting frequency were chosen at y+ = 15, where y+ was the vertical distance

from the plate scaled with the viscous length yv*/v . In the course of velocity measurement, the analog output of LDA was sampled and processed

by "Data Sampling System of Turbulence signals ''Is[, which had 4k memory for storing data. The

system's sampling points could be varied over a range (1--16).256 and sampling interval from 157

gs to 0.15 sec. After sampling the velocity versus time curve was figured out on CRT and the data

chosen would be stored on disc for calculation later.

Detection of bursting frequency (reciprocal of mean bursting period 7"B), as Bogard et al.

(1986) tr pointed out, is one of the sources of controvercy. In this paper only relationship between

bursting frequency and Reynolds number was concerned, so that the absolute value of frequency

was not considered very important. To compare with the result of Blackwelder et al., VITA

(variable-interval-time-average) technique was used and its detection function is:

~1 if Vat > k * u 2 a n d d u ( t ) / d t > 0 D(t) r m s

otherwise

where u is root-mean-square of u {t} which is fluctuating streamwise velocity and Vat is the VITA r m s . ~ . - -

measure of turbulent energy. Vat is defined as

Va/"r(x,, t, z) = u 2 (x,, t, ~) -- [ u(x,, t, z)] 2

where denote " ^ " means variable interval time average. The variable-interval-time-average of a

fluctuating quantity Q (xl, t, z) is defined by

-~ J , - , /2 Q(x,, s) ds

Obviously, as z --+ ~ , it results in the conventional time average quantity

lira Vat(x,, t, , ) = U 2

Vol. 4. No. 4 Shu & Tang: Bursting Frequency in Turbulent Boundary Layers s

With detect ion function shown above burs t ing frequency was obtained by computer . Since

sometimes several chaotic motions existing in one burs t could be mis taken as several burs ts by the

original VITA, in a certain t ime interval only one case o f the chaotic motions was responded with

others being ignored, as first supposed by Wi l lmar th et al. (1984) tal. There are two parameters in

VITA which can be arbi t rar i ly chosen and for compar ison with Blackwelder et al., k = 1.21 and z +

= 10 (3 + = z v? 2 / v) were taken in this paper, bu t z + = 50 might be more reasonable according to

the calculation here.

HI . EXPERIMENT A L RESULTS

3.1 Time mean veloeity profiles in boundary layers The mean velocity profile in Fig. 2 was measured at x = 4500 m m by LDA. Because of bursts

both the variance of instantaneous streamwise velocity and the t ime scale of the flow were large, long

Statistic average t ime was needed. The least average t ime for mean velocity measurement at one

point was so chosen that two t imes ' sampling for one k ind of free flow could give near ly the same

values and it was usually more than twenty minutes in the present experiment . A special p rogram for the purpose was developed and a profile was schematical ly given in Fig. 2.

1.0

+

4- *t~ ,4.

+

4- +

.4. +

0.+++ .t-++, +J ++++7 , , t

0.2 0.4 0.6 0.8 i .0 U/U|

Fig. 2 Time mean velocity profde in turbulerit boundary layer (R~ - 470)

0.8

0.6

0.4

0.2

0 i

Table 1

R~ U~ [cm / s) v* (cm/s) r = 1/](s) f+ f6/U|

320 6.64 .379 9.46 .010 .119

400 7.57 .420 7.22 .110 .139

470 8.50 .460 6.20 .011 .146

570 9.99 .520 5.23 .010 .151,

830 13.50 .670 2.92 .011 .211

990 16.55 .820 2.07 .011 .253

1170 19.90 .990 1.49 .010 .311

1270 27.00 1.280 .88 .010 .420

1470 28.50 1.383 .73 .011 .495

294 ACTA MECHANICA SINICA 1988

Momentum thickness was calculated according to its definition and velocity profiles measured.

The friction velocity was determined by fitting the data of this experiment to Clanse's form of log

law. The experimental results of this paper are shown in Table 1.

Table 2

+ = rv* 2 / y 8.9 17.9 26.8 50 62.5 71.4 80.4

7" B (s) 2.63 1.45 1.40 1.40 1.40 1.27 1.30

f+ .004 .005 .011 .011 .011 .012 .013

(U~ = 19.9 cm/s,R~0 = 1170)

3.2 Streamwise velocity during bursts and digital filter As shown in Fig. 3(a), the streamwise velocity during burst consists Of patches of higb-

frequency fluctuations and low-frequency fluctuations of large scale. Because bursts concerned here

were low-frequency large scale turbulent signals, low-pass digital filter was employed in the first

stage of the processing of raw data, and the filtered configuration was shown in Fig. 3(b) in which the

original low-frequency information was preserved. Then by VITA technique with suitable threshold

k and time interval z, average bursting frequency f and its non-dimensional values f + were

calculated.

j' ::.~./, :::'.'.:.... ."" " ":.~': k':'" ":' ; ",'!':.'.....

(a) "-::II"~(':":Y

..." "...'...."...,

(b)

......

% ...... j "..........

Fig. 3 Streamwise velocity during bursts (at y+ = 15)

(a) the original configuration

(b) the filtered configuration

3.3 Relationship between bursting frequency and Reynolds number I f the bursting frequency was scaled with inner parameters of boundary layer v* and y, the

nomalized frequency f+ =fT/v *z was independent of Reynolds number,, exhibiting constant

values of 0.011, as shown in Fig. 4.

Vol. 4. No. 4 Shu & Tang: Bursting Frequency in Turbulent Boundary Layers 295

0.01 O

0 50 I a i I I 300 0 700 900 1100 1300 1500

Re0

Fig. 4 Dependence of the mean burst ing frequency scaled with the inner variables

of boundary layer upon Reynolds number

When outer flow variables of boundary layer Uoo and 6 were used to scale bursting frequency,

the non-dimensional value increases as Reynolds number increases, as shown in Fig. 5. Results

above are in agreement with those of Blackwelder et al. (1983) [2] and Willmarth et al. (1984) TM, but

contrary to that of Kim et al. (1971) ElI.

8

o.~

0 I L I t * I 300 500 700 900 1100 1300 1500

Nee

Fig. 5 Dependence of the bursting frequency scaled with outer flow parameters upon Reynolds number

IV. DISCUSSION

4.1 With LDA and VITA techniques, dependence of mean bursting frequency upon

Reynolds number in flat plate boundary layer was investigated over~a Reynolds number range of Re0

= 300 to 1470. The bursting frequency scales with the wall parameters and is independent of

Reynolds number; non-dimensional frequency normalized by outer flow variables increases as

Reynolds number increases. This result agrees well with that of Blackwelder et al. (1983) lzl, whose

experiment was carried out in wind tunnel by hot wire anemometer over a Reynolds number range

different from that of this paper. So conclusions here are new and complementory to that of

Blackwelder et al.

4.2 It was suggested and proved by Blackwelder et al. that hot wire probe of too large length

scale was the reason that caused the different result of Kim et al. (1971). In this experiment the

longer axis of LDA measuring volume was much less than 20v / v*, and the same result as those of

Blackwelder et al. and Willmarth et al. was obtained.

4.3 Although VITA is not the most reliable method detecting bursting frequency, in this

paper only the variance of bursting frequency over a certain Reynolds number range was considered,

the absolute values of frequencies were not important and did not affect the final result. For

comparison, the same threshold k and time interval r + as used by Blackwelder et al. were used. It

296 ACTA MECHANICA SINICA 1988

was found by authors that k = 1.21 and k = 1.44 brought neligible difference and that when

z + >I 5ffthe values of burs t ing frequency calculated became steady. Table 2shows the effect of r +.

+ = 10 a n d f + = 0.004 were the case of Blackwelder et al. In this paper �9 § = 50 was chosen a n d f +

= .012 was required. The present results are the same quanti tat ively, bu t the authors th ink that ~ +

= 50 is more reasonable for the consistency of data.

REFERENCES I l l Kim, H. T., Kline, S. J., Reynolds, W. C., J~ Fluid Mech:, 50(1971), 133.

[2] Bhckwelder, R. F., Haritonidi, J. H., J. Fluid Mech., 152 (1983), 87.

[3] Willmarth, W. ~/\., Sharma, L. K., J. Fluid Mech., 142(1984), i21.

[4] Shu Wei, Guo Xiaoming, J. Mech., 19, 1(1987), 15 (in Chinese).

[5] Sben Yah, Thesis for Master Degrees of Tianjin University, (1986) (in Chinese).

[6] Bogard, D. G., Tiederman, W. G., J. Fluid Mech., 162 (1986), 389.

[7] Luchik, T. S., Tiederman, W. G., J. Fluid Mech., 174(1987), 529.