February 2011 

NASA/TM-2011-217060 

Measurement of the Correlation and Coherence 

Lengths in Boundary Layer Flight Data 

Daniel L. Palumbo 

Langley Research Center, Hampton, Virginia 
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February 2011 

NASA/TM-2011-217060 

Measurement of the Correlation and Coherence 

Lengths in Boundary Layer Flight Data 

Daniel L. Palumbo 

Langley Research Center, Hampton, Virginia 
Available from: 

NASA Center for AeroSpace Information 

7115 Standard Drive 

Hanover, MD 21076-1320

443-757-5802 
1

I. Abstract

Wall pressure data acquired during flight tests at several flight conditions are analyzed

and the correlation and coherence lengths of the data reported. It is shown how the

frequency bandwidth of the analysis biases the correlation length and how the convection of

the flow acts to reduce the coherence length. Coherence lengths measured in the streamwise

direction appear much longer than would be expected based on classical results for flow over

a flat plat. 
2

II. Introduction

Correlation length is a measure of the extent of spatially coherent power in the wall pressure and is thus an

important parameter in studies of structural response to boundary layer excitation1,2,3

. It has been established, based

on data acquired over several years4, that the coherent power in the wall pressure decays exponentially with distance.

This behavior is reflected in most models of the wall pressure cross spectrum5,6,7

. The models also assume that the

wall pressure statistics are homogeneous and stationary5,8

, that is, the wall pressure can be considered a single process

and statistical quantities such as the power spectrum and cross spectrum are constant over the space/time dimensions

of the analyses. Assuming stationarity, the cross spectrum can be derived by scaling the power spectrum according to

a coherence length*

 and adjusting the phase according to the convection wavenumber. This is the basis of the Corcos

model5. As explained in later sections, the coherence length is similar, but not equivalent, to the correlation length.

In the Corcos model, the coherence length is inversely proportional to frequency leading to the unrealistic (in light of

available data) prediction that the coherence length in the wall pressure would tend towards infinity as the frequency

approached zero. This shortcoming in the Corcos model was addressed by Efimtsov6

 who used boundary layer

thickness, δ, friction velocity, uτ, and Strouhal number, Sh, to fit the coherence length to available data. Several other

models have since been proposed (see Graham9 and Hwang10 for overviews). The purpose of this study is not to

select the best model, but to establish a better understanding of the underlying behavior that these models attempt to

capture. The Corcos/Efimtsov model will be used to relate the current data and analyses to established norms. The

data that forms the basis for the study were acquired in a flight test that was a cooperative venture between the

Gulfstream Aerospace Corporation, Boeing Commercial Airplanes and the National Aeronautics and Space

Administration (NASA). Previous flight data had issues relating to the installation of sensors flush to the surface of

the fuselage despite best efforts to achieve a smooth surface11,12. The subject data set was acquired using a pinhole

array machined in a single piece of stainless steel sheet metal yielding a surface with minimal irregularities. As will

be shown, the data are remarkably consistent over the array. The high quality of the data raises confidence in results

that show coherence lengths in the streamwise direction several times longer than expected based on previous studies.

Values often quoted in the literature for the Efimtsov model’s three parameters for streamwise coherence length (a1,

a2, a3) are 0.1, 75.4 and 1.546 and predict a coherence length of 0.1 m at 1 kHz. Analysis of the data reveal a

coherence length of 0.45 m at 1 kHz requiring parameters of 0.06, 15 and 3. Another recent study also reports a

similar trend towards increased measured coherence lengths requiring modifications to model parameters13. The

observed increased coherence lengths may be due to improved instrumentation12 as well as the conditions under

which the data were acquired as will be discussed in the text. It will be shown that the value of a derived power decay

parameter depends on the analysis, i.e., whether it is cross correlation or cross spectrum, and that these analyses can

return different results depending on the bandwidth and bin width employed. 

The components of the flight test instrumentation and flight test conditions will be described in the next section.

The quality of the data will then be evaluated followed by a discussion on the difference between the streamwise and

cross stream results. The theoretical basis for computing the correlation and coherence lengths is established

followed by a demonstration of the biased behavior of the cross correlation and cross spectrum when operating on

turbulent boundary layer wall pressure data.

III. The Flight Test

The flight test was conducted as a cooperative effort between the Gulfstream Aerospace Corporation, Boeing

Commercial Airplanes and NASA. The Gulfstream aircraft (a G550) was chosen primarily because its large windows

provide a good platform for a long, streamwise, array of sensors that could be built into a window blank.

* The term ‘coherence length’ is introduced here to remove ambiguities on how the quantity is calculated and used.
3

A. The test bed and instrumentation

The sensor array was located in the first window of the G550, 6.7 m from the nose of the aircraft on the starboard

side, Fig. 1(a). The array was inclined at an angle of 5.5° to the fuselage axis to account for the angle of attack and

flow over the body during cruise, Fig. 1(b). The window immediately aft of the test window was marked with a line

in the same relative position as the streamwise array and tufts fitted so that the streamline could be observed and the

aircraft made to hold the required angle of attack during data acquisition. There were a total of 43 sensors in the

streamwise array spanning 47.7 cm and 15 sensors in the cross stream array spanning 8.1 cm. The sensors in the high

density portions of the arrays were spaced 3 mm apart. Additional sensors spaced at 6 mm, 12 mm and 36 mm were

added to increase the length of the arrays beyond the point of expected significant correlation. The pressure sensors

were 1.6 mm in diameter with piezoresistive elements operated in differential mode, vented to the cabin. These type

of sensors have a wide temperature range (-55o C to 120o C) and a wide dynamic range with an upper frequency limit

of 150 kHz. The sensors were AC coupled. The pinholes were 0.5 mm in diameter and 0.254 mm in depth. The

cavity above the sensors was 1.8 mm in depth and 1.45 mm in diameter. The usable bandwidth of the array is limited

by the Helmholtz resonance of the cavity and the pinhole dimensions. The Helmholtz resonance frequency for the

cavity is approximately 15 kHz. At a nominal convection velocity of 150 m/s the correction factor for a sensor

diameter of 0.5 mm is less than 5% at 5 kHz8. The data were acquired at 204.8 kHz to maximize time resolution and

-300

-200

-100

0

100

200

300

-400 -300 -200 -100 0 100 200 300 400

x, mm

y, mm

Flow

Fig. 1 Flight test vehicle, (a) and position of array in window blank, (b).

(a)

(b)
4

for 60 seconds to provide adequate time for averaging. The frequency band of primary interest in this study is limited

to 3 kHz to avoid the need for correction while still capturing the important behavior. Unless otherwise noted, 250

averages were performed in computing the spectra.

B. Flight test conditions

Data were acquired over a wide range of test conditions in which Mach number, angle of attack and altitude were

varied. For the purposes of studying coherence lengths, only those datasets taken under cruise conditions are useful.

Those 3 flight conditions are summarized in Table 1 where c is the speed of sound, U∞ is the free stream velocity, x is

the distance from the nose to the window blank, ν is the kinematic viscosity, Re is the Reynolds number based on x,

Cf is the coefficient of skin friction, uτ is the friction velocity and δ is the boundary layer thickness. The aircraft was

held at constant Mach number and angle of attack during data acquisition. The following relationships were used to

derive the quantities shown in Table 1 given the altitude and Mach number taken from the aircraft’s air data computer

and the distance from the nose to the window blank. The speed of sound in air, c, and the kinematic viscosity, ν, are

taken from the Aerospaceweb’s Atmospheric Properties Calculator* which is based on U.S. Standard Atmosphere

1976.

(1)

(2)

(3)

(4)

(5)

IV. The Data

Before evaluating the data, it is instructive to review characterizations of wall pressure spectra that have formed

over the years. These are summarized by Hwang10 based on observations by Blake14, Farabee15 and Bull4 among

others. The expected shape of the wall pressure power spectrum is based on a determination of which scaling

variables work best in a particular frequency range. Of note is the division of the spectrum into 4 sections, Fig. 2.

* http://www.aerospaceweb.org/design/scripts/atmosphere

Table 1: Flight test conditions and flow parameters

Flight

Number

Mach

Number

Altitude

(ft) c (m/s) U∞ (m/s) x (m) ν (m2/s) Re Cf uτ (m/s) δ (m)

B101 0.86 48000 295 252 6.7 6.9E-5 2.5E7 2.0E-3 7.9 8.5E-2

B106 0.7 32000 301 210 6.7 3.4E-5 4.1E7 1.8E-3 6.3 7.7E-2

B109 0.57 24000 311 174 6.7 2.7E-5 4.3E7 1.8E-3 5.2 7.6E-2

U∞ = M c ×

Re xU∞ = ν⁄

δ 0.382xRe–0.2 =

Cf 0.0594Re–0.2 =

uτ 0.5U

2

∞

Cf     0.5=
5

The low frequency (up to ωδ/ut

=5) and mid-frequency (ωδ/ut<100) sections are dominated by structure in the outer

layer. The high frequencies (ων/ut

2>0.3) are dominated by structure in the inner layer. The center section exhibits

behavior characteristic of the inner and outer layers. It is instructive to note that the wall pressure spectrum in the low

and mid frequency ranges are composed of pressure fluctuations impressed on the wall by a physical process that

occurs largely in the outer layer, away from the wall while the higher spectral frequencies reflect physical behavior

occurring close to wall. This duality contributes to the non-homogeneous nature of the wall pressure. Using the flow

parameters in Table 1 and frequency in Hertz, (Hz), the power spectrum level is expected to increase with a slope

proportional to f

 2 in the low frequency region, up to 65 Hz. The spectrum is then expected to continue to increase in

the mid frequencies until it reaches a peak at about 650 Hz. The slope than gradually decreases until it enters the

center frequency range at 1.3 kHz where the slope becomes proportional to approximately f

 -1. The inner scale region

is reached at 55 kHz above which the slope becomes proportional to f

 -5. A well behaved data set should exhibit these

characteristic behaviors. The power spectrum for the sensor at the intersection of the streamwise and cross stream

arrays for the 0.7M case is shown in Fig. 2 across the full bandwidth. The frequencies mentioned above are marked

and the expected slopes indicated as red dashed lines. The spectrum exhibits the expected behavior except for the

low frequency region (below 65 Hz) where the spectrum does not roll off with a slope proportional to f 2. The

spectrum below the peak (at about 1 kHz) appears to decay with a slope closely proportional to f 0.3. This is in

reasonable agreement with the f 0.2 slope observed by Leclercq in this region16. An anticipated defect occurs above

10 kHz where the effect of the Helmholtz cavity resonance perturbs the spectrum. The true behavior of the spectrum

in this region cannot be known, but the shape of the spectrum outside the influence of the cavity resonance appears to

be as expected. 

A. Data Consistency

A subset of sensors in both the streamwise and cross stream arrays were sampled and the power spectra and cross

spectra compared at each frequency. The cross spectra were taken for pairs of sensors spaced at 3.6 cm for the

streamwise array and 0.9 cm for the cross stream array. The distributions of the difference in the power spectra

between the sensors in the streamwise array over a band from 100 Hz to 3 kHz are shown in Fig. 3 for the 0.7 M case.

Fig. 2 Power spectrum of a typical sensor at 0.7 M; ---, expected slope; ---estimated slope. 
6

This result is representative of the other 2 flight conditions. For all cases sampled in the streamwise array, 90% of the

differences between the power spectra were less than +/- 1 dB and 95% of the differences between the cross spectra

were less than +/- 1 dB.

The same analysis (except for the cross spectrum being taken at 0.9 cm) for the cross stream array produced less

consistent results, Fig. 4. The 0.7 M and 0.56 M cases exhibited similar behavior with 45% of the power spectrum

differences below 1 dB and 35% of the cross spectrum differences below 1 dB. The 0.86 M case had much tighter

distributions with 70% of the power spectrum differences below 1 dB, and 65% of the cross spectrum differences

below 1 dB.

The variation in power over the cross stream array is better understood by viewing the individual power spectra.

These are shown in Fig. 5 for the 0.7 M case (a) and the 0.86 M case (b). The direction of increasing y dimension is

indicated. As can be seen, the levels of both spectra decrease with increasing y. For the 0.7M case (and the 0.56 M

case) the overall reduction in level is accompanied by a gradual elimination of the peak at 1 kHz. The power at 1 kHz

decreases by nearly 5 dB over the 8.1 cm length of the array at 0.7 M. The spectra at 0.86 M retains its shape, but the

Fig. 3 Differences in power spectrum (a) and cross spectrum at 3.6 cm (b) for sensors in streamwise array at 

0.7 M.

(a) (b)

(a) (b)

Fig. 4 Differences in power spectrum (a) and cross spectrum at 0.9 cm (b) for sensors in cross stream array 

at 0.7 M.
7

peak is shifted slightly to lower frequencies, decreasing just over 2 dB. The tighter grouping and retention of its basic

shape explains why the power differences for the 0.86 M case are less than the 0.7 M and 0.56 M cases. The

flattening of the low speed pressure curves with increasing y may indicate a reduction in outer scale influence on the

wall pressure that appears to persist at the 0.86M case. The change in pressure over the span of the cross stream array

is possibly due to the curvature of the aircraft fuselage. In a study by Heenan17, data were acquired around the

circumference of a cylinder placed at an angle of attack to the flow. This study showed that the pressure spectra on

the cylinder exhibited a behavior very similar to that observed here as the sensor locations transitioned from the

windward to the leeward side of the cylinder. The difference in behavior between the low speed (0.7 M and 0.56 M)

cases and the high speed, 0.855 M, case may be due to a slight difference in angle of attack or the difference in

Reynolds number (from Table 1, the Reynolds number for the 0.855 M case is 35% below the lower speed cases).

In a previous flight test18 it was noted that flush mounted sensors, despite best efforts to achieve flushness, could

affect sensor phase angle. The phase behavior of the data taken using the pin hole sensors is quite good from very

low frequencies up to 10 kHz where the effect of the Helmholtz resonance begins to perturb the phase, Fig. 6.

(a) (b)

Fig. 5 Power spectra over cross stream array for 0.7 M (a) and 0.86 M (b).

Fig. 6 Cross spectrum phase angle between 2 sensors 

separated by 2 cm.
8

B. Confidence Intervals

The primary measurement is that of the instantaneous pressure. The mean of this value is artificially forced to

zero as the data were acquired AC coupled. Better measures, and ones more related to the subject matter of this

paper, are the mean square power and the mean cross power. One sample of the mean power is taken from 4096

pressure measurements. This length ensemble was chosen because it was the smallest length at which the effects of

convection are reduced (as will be explained in the following sections). Using this ensemble length, it is possible to

take 3,000 samples of the mean power. The large number of samples guarantees a normal distribution according to

the Central Limit Theorem. This is shown to be the case for the mean square power in Fig. 7(a). The mean cross

power was similarly distributed. The confidence interval for a normally distributed random variable with unknown

standard deviation is19

,

where ci is the confidence interval, α is the complement of the desired probability that the measurement is in the

interval, t is the Student’s t distribution, n is the number of samples and s is the standard deviation of the sample set.

Using n=3000, s = 0.618 and α=0.01, the mean square power is 127 dB with a 99% confidence interval of 0.029 dB.

The mean cross power is found to be 125 dB with a 99% confidence interval of 0.026 dB. A similar analysis was

done in the frequency domain with the results shown in Fig. 7(b). The frequency domain confidence intervals are an

order of magnitude greater than the time domain values, but are still only a fraction of a dB.

V. Correlation and Coherence Length Analyses

The term ‘correlation length’ is often used to describe coherent power decay rates derived by both the cross

correlation and the cross spectrum. The resulting rates are quite different, however, and using the same term for both

quantities introduces ambiguity. In this study, the term correlation length will be used to describe power decay rates

computed using the cross correlation and ‘coherence length’ for rates computed using the cross spectrum. The spacetime cross correlation in one spatial dimension is written as

ci( ) α, , n s t

1 α

2 – ---, n 1–

s

n

------     = ±

Fig. 7 Distribution of samples of mean square power, (a), confidence intervals in frequency domain, (b).

(a) (b)
9

(6)

Here p(x,t) is the pressure at some point in space and time and the brackets, , denotes the average over N

samples in time. The power decay rate is determined from cross correlation analysis by fitting an exponential

distribution to the peaks in the cross correlation taken from a reference sensor to other sensors in the array, see Eq. 7

and Fig. 8(a).

(7)

The correlation length, L, is determined by the value that produces the best fit over all the correlation peaks. Note

that the cross correlation is normalized to the RMS power in each sensor so that the peak in the auto correlation is 1.

 The space-time correlation is related to the cross spectrum by the Fourier transform.

(8)

For a discrete Fourier transform, the integral is taken over N samples in time and produces one ensemble of cross

spectrum. If the data were acquired at a sample rate fs, the bandwidth of the spectrum would be fs /2 with (N+1)/2

frequency bins of width fs /N. The cross spectrum magnitudes are normalized by the square root of the respective

auto spectra so that the resulting quantity is equivalent to the square root of the coherence.

(9)

The coherence length for a particular frequency is found by fitting an exponential to γ at that frequency.

(10)

 An example of an exponential fit to γ in the streamwise direction at 1.5 kHz is shown in Fig. 8(b). Two things are

immediately obvious when comparing the coherence length in Fig. 8(b) to the correlation length in Fig. 8(a). First,

the coherence length has a much better fit to the exponential than the correlation length. This is true at all

frequencies. Second, the coherence length at this frequency is much longer than the correlation length. Both these

observations can be understood by considering the plot in Fig. 9 of the coherence length over the band from 100 Hz to

30 kHz.

χ x0 x1 t0 ( ) , ,, τ p x0 t0 ( ) , p x1 t0 =   ( ) ⋅ ( ) , τ +

  ...

max χ x0 xi t0 ( ) ( ) ,,, τ e

x0 x – i

L –-----------------

≅

φ x0 x1 t0 ( )χ , ,, f x0 x1 t0 ( ) , ,, τ e

–i2πfτ

dτ

N

 =

γ x0 x1 t0 ( ) , ,, f φ x0 x1 t0 ( ) , ,, f

φ x0 x0 t0 ( ) , ,, f 0.5 φ x1 x1 t0 ( ) , ,, f 0.5 = ------------------------------------------------------------------------------------

γ x0 xi t0 fj ( ) ,,, e

x0 x – i

Lj

–-----------------

≅
10

The coherence length is strongly frequency dependent reaching its maximum at frequencies corresponding to the

peak in the power spectrum, i.e., ~ 1 kHz, and decreasing rapidly at higher frequencies. At high frequencies,

structures in the inner layer which decay rapidly dominate the wall pressure behavior. At low frequencies, structures

in the outer layer dominate with much longer decay lengths. The cross correlation is a wide band analysis in that its

result is a combination of these behaviors over all frequencies. At short distances the short decay lengths of the

structures at the higher frequencies causes a rapid fall off in correlated power and at greater distances, the longer

decay lengths of the low frequency structures sustain the correlated power. The lack of consistency in the correlation

decay processes across the frequency band results in a behavior that cannot be adequately modeled by a single

exponential curve. This is the reason for the less than optimal fit in Fig. 8(a) at greater distances. 

The disparity in behavior can be reduced by narrowing the analysis frequency band to a region of interest, for

example, around the 1 kHz peak in the power spectrum. The result of deriving the correlation length from data that

was band limited to the frequency range of 100 Hz to 3 kHz is shown in Fig. 10(a). At this analysis bandwidth, the

correlation length (0.23m) is far greater then that obtained using the data’s full bandwidth, shown in Fig. 8(a), and

illustrates the effect of removing most of the high frequency, inner layer, behavior. The correlation length will

approach the coherence length as the frequency band is narrowed. The coherence length at this bandwidth is shown

Fig. 8 Correlation length determined by exponential fit to peaks in cross correlation, (a) and coherence 

length determined by exponential fit to cross spectrum magnitude at 1500 Hz, (b)

(a) (b)

Fig. 9 Coherence length 100 Hz to 30 kHz.
11

in Fig. 8(b) and is, of course, unchanged. At this point, one might conclude that the cross spectrum is the more

reliable statistic. That is not entirely the case as will be demonstrated in the following sections.

A. Effects of convection and ensemble length.

The coherence length result shown in Fig. 10(b) was computed with an ensemble length of 2048 samples resulting

in a bin width of 100 Hz. Unfortunately, the coherence length computation depends on ensemble length, converging

with longer lengths, Fig. 11. Care must be taken to use ensemble lengths long enough to accurately estimate the

coherence length. The quality of fit of the exponential to the coherence decay is not altered as the ensemble length is

varied and remains very good, e.g. see Fig. 8(b). 

Fig. 10 Correlation length, (a), and coherence length, (b), computed over a frequency band from 100 Hz to 3 

kHz.

Fig. 11 Coherence length computed at several 

ensemble lengths.
12

VI. Data Analysis

A cross spectrum with a 4096 ensemble length and 50 Hz bin was used to analyze the flight data. The coherence

length will be compared to historic values by comparison to Corcos and Efimtsov model parameters most often

quoted in the literature, for example by Graham1

.

The Corcos model5

 for the cross spectrum assumes an exponential decay of correlated power with distance and

can be written as:

(11)

Here r1,α1 and r2,α2 are the distance between two points and the decay constant for the streamwise and cross

stream directions, respectively. The coherence length, Lcoh, is then

(12)

The predicted coherence length is inversely proportional to frequency leading to over estimation of the value at

low frequency when compared to actual data. Efimtsov6 offered a correction for the coherence length as

, (13)

where δ is the boundary layer thickness, Sh is the Strouhal number, uτ is the friction velocity and parameters a1..3

are defined specifically for the streamwise and cross stream directions. The values of these parameters most often

used in the literature are given in Table 2 along with values derived from the flight test data. The Efimtsov model is

constrained by the Corcos model (through a1) at mid to high frequencies. Parameter a2 controls where the Efimtsov

model breaks away from the Corcos curve. Parameter a3 controls the low frequency roll off. The frequency and

height of the peak in the coherence length curve can then be set by adjusting a1, a2 and a3. 

Fig. 12(a) illustrates the fit of the Corcos and Efimtsov models to the 0.7 M streamwise coherence length data.

The fit is a compromise in that some error must be tolerated at the high frequencies to raise the curve to match the

peak at lower frequencies. The changes in the parameters reflect an observed increase in coherence length. The

original Efimtsov parameters predicted a maximum streamwise coherence length of 0.1 m at 1.5 kHz. Analysis of the

Table 2: Coherence length parameters.

Streamwise Cross Stream

a1=α1 a2 a3 a1=α2 a2 a3

Graham1 0.1 72.8 1.54 0.77 548 13.5

0.57 M 0.06 5.0 1.0 0.77 260 4.0

0.7 M 0.06 5.0 1.0 0.77 260 4.0

0.86 M 0.07 4.0 1.0 0.85 60 4.0

φ r1 r ( ) , , 2 ω e

–α1ω r1 Uc ⁄

e

–α2ω r2 Uc ⁄

e

iωr1 Uc ⁄ =

Lcoh C,

Uc

αω

-------=

Lcoh E, δ

a1Sh

Uc uτ ⁄ ---------------     a2

2

Sh2 a2 a3 ( ) ⁄ 2 +

+ -------------------------------------

1 2⁄

=
13

flight data has the coherence length peaking at close to 0.5 m at 750 Hz. Note that all flight conditions returned

similar values for a1, a2 and a3 indicating a consistent result. The cross stream cases were less well behaved, Fig.

12(b). The Corcos parameter, a1, was unchanged for the 0.57 M and 0.7 M cases and required a slight increase for the

0.86 M case. The a2 parameter had to be reduced by 50% for the lower speed cases and nearly 90% for the 0.86 M

case. The reduction in a2 for the 0.86 M cross stream case is of the same order as that required in a2 for the

streamwise case. The reason for this, as described earlier in section “IV.A. Data Consistency”, is that the 0.86M

cross stream spectrum maintained a pressure response indicative of outer layer coherent structure that diminishes at

the slower speeds. The behavior of the wall pressure under these flight conditions appears to be much different than

that observed under the controlled laboratory conditions in which the historic models and parameters where derived.

VII. Conclusions

For the purpose of determining the parameters for models which estimate the wall pressure of the turbulent

boundary layer over the surface of an aircraft fuselage, the current data are inadequate. Although it has been argued

that the data are of high quality, the variation in pressure levels and spectra shape along the cross stream array leads

one to conclude that a single set of parameters extracted from data taken at one location on the fuselage are

insufficient. Thus, more tests are necessary to validate the observed behavior including the very long coherence

lengths and especially the hypothesis that fuselage curvature modulates the influence of outer layer structure on the

wall pressure. Given the available data, the Efimtsov model is able to adequately describe the coherence length

frequency dependency although the result is a compromise between best fit for the lower versus the higher frequency

ranges. To match observed coherence lengths that were almost five times longer than predicted, it was necessary to

make considerable changes to the accepted Efimtsov parameters.

The different turbulent mechanisms that are at work in the inner and outer layers produce a frequency dependence

that may bias wide band analyses such as the cross correlation. Error in the cross spectrum due to a shift in the data

caused by convection may be reduced by using long ensemble lengths and narrow bin widths.

Fig. 12 Coherence length dependence on frequency for 0.7 M streamwise case (a), and, comparison of 0.7 M 

and 0.86 M cross stream cases, (b)

(a) (b)
14

VIII. References

1

Graham, W.R., “Boundary layer induced noise in aircraft, Part I: The flat panel model”, Journal of Sound and Vibration, Vol.

192(1), pp. 101-120, 1996.

2

Maury, C., Gardonio, P., and Elliott, S. J., “A wavenumber approach to modelling the response of a randomly excited panel,

Part II: Application to aircraft panels excited by a turbulent boundary layer”, Journal of Sound and Vibration, Vol. 252(1), pp.

115-139, 2002.

3

Hambric, S.A., Hwang, Y.F., and Bonness, W.K., “Vibrations of plates with clamped and free edges excited by low-speed

turbulent boundary layer flow”, Journal of Fluids and Structures, Vol. 19, pp. 93-110, 2004.

4

Bull, M.K., “Wall-pressure fluctuations beneath turbulent boundary layers: some reflections on forty years of research”,

Journal of Sound and Vibration, Vol. 190(3), pp. 299-315, 1996.

5

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19Canavos, G.c., Applied Probability and Statistical Methods, Little, Brown and Co., Boston, 1984.
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Technical Memorandum

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Measurement of the Correlation and Coherence Lengths in Boundary Layer 

Flight Data

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Palumbo, Daniel L.

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NASA Langley Research Center

Hampton, VA 23681-2199

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14. ABSTRACT

Wall pressure data acquired during flight tests at several flight conditions are analyzed and the correlation and coherence 

lengths of the data reported. It is shown how the frequency bandwidth of the analysis biases the correlation length and how 

the convection of the flow acts to reduce the coherence length. Coherence lengths measured in the streamwise direction appear 

much longer than expected. 

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Correlation length; Coherence length; Boundary layer

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