NASA/CR-2005-213938 

Modeling of Turbulent Boundary Layer Surface 

Pressure Fluctuation Auto and Cross Spectra – 

Verification and Adjustments Based on 

TU-144LL Data 

Robert Rackl and Adam Weston 

The Boeing Company, Seattle, Washington 

 

December 2005 
The NASA STI Program Office . . . in Profile

Since its founding, NASA has been dedicated to the 

advancement of aeronautics and space science. The 

NASA Scientific and Technical Information (STI) 

Program Office plays a key part in helping NASA 

maintain this important role. 

The NASA STI Program Office is operated by 

Langley Research Center, the lead center for NASA’s 

scientific and technical information. The NASA STI 

Program Office provides access to the NASA STI 

Database, the largest collection of aeronautical and 

space science STI in the world. The Program Office is 

also NASA’s institutional mechanism for 

disseminating the results of its research and 

development activities. These results are published by 

NASA in the NASA STI Report Series, which 

includes the following report types: 

• TECHNICAL PUBLICATION. Reports of 

completed research or a major significant phase 

of research that present the results of NASA 

programs and include extensive data or 

theoretical analysis. Includes compilations of 

significant scientific and technical data and 

information deemed to be of continuing 

reference value. NASA counterpart of peerreviewed formal professional papers, but having 

less stringent limitations on manuscript length 

and extent of graphic presentations. 

• TECHNICAL MEMORANDUM. Scientific 

and technical findings that are preliminary or of 

specialized interest, e.g., quick release reports, 

working papers, and bibliographies that contain 

minimal annotation. Does not contain extensive 

analysis. 

• CONTRACTOR REPORT. Scientific and 

technical findings by NASA-sponsored 

contractors and grantees. 

• CONFERENCE PUBLICATION. Collected 

papers from scientific and technical 

conferences, symposia, seminars, or other 

meetings sponsored or co-sponsored by NASA. 

• SPECIAL PUBLICATION. Scientific, 

technical, or historical information from NASA 

programs, projects, and missions, often 

concerned with subjects having substantial 

public interest. 

• TECHNICAL TRANSLATION. Englishlanguage translations of foreign scientific and 

technical material pertinent to NASA’s mission. 

Specialized services that complement the STI 

Program Office’s diverse offerings include creating 

custom thesauri, building customized databases, 

organizing and publishing research results ... even 

providing videos. 

For more information about the NASA STI Program 

Office, see the following: 

• Access the NASA STI Program Home Page at 

http://www.sti.nasa.gov

• E-mail your question via the Internet to 

help@sti.nasa.gov 

• Fax your question to the NASA STI Help Desk 

at (301) 621-0134 

• Phone the NASA STI Help Desk at 

(301) 621-0390 

• Write to: 

 NASA STI Help Desk 

 NASA Center for AeroSpace Information 

 7121 Standard Drive 

 Hanover, MD 21076-1320
NASA/CR-2005-213938 

Modeling of Turbulent Boundary Layer Surface 

Pressure Fluctuation Auto and Cross Spectra – 

Verification and Adjustments Based on 

TU-144LL Data 

Robert Rackl and Adam Weston 

The Boeing Company, Seattle, Washington 

National Aeronautics and 

Space Administration

Langley Research Center Prepared for Langley Research Center 

Hampton, Virginia 23681-2199 under Contract NAS1-20220

 

December 2005 
Available from: 

NASA Center for AeroSpace Information (CASI) National Technical Information Service (NTIS) 

7121 Standard Drive 5285 Port Royal Road 

Hanover, MD 21076-1320 Springfield, VA 22161-2171 

(301) 621-0390 (703) 605-6000 
Preface 

The random fluctuations generated by the turbulent boundary layer are a dominant source 

of interior cabin noise at subsonic as well as supersonic speeds; added treatment to manage this noise usually is a significant weight consideration. Fluctuating surface pressure 

measurements were conducted on flight tests of the Tupolev 144LL Flying Laboratory in 

late 1997 and early 1998 in Russia. These flight tests used pressure sensors mounted in 

cabin window blanks in different locations down the length of the fuselage. The recorded 

data were returned to the U.S. for analysis by NASA and Boeing. 

The literature on turbulent boundary layer pressure fluctuations provides several empirical 

models which were compared to the measured TU-144 data. The model developed by 

Efimtsov showed the best agreement. Adjustments were made to improve its agreement 

further, consisting of the addition of a broad band peak in the mid frequencies, and a minor 

modification to the high frequency roll-off. The adjusted Efimtsov predicted and measured 

results are compared for both subsonic and supersonic flight conditions. Measurements in 

the forward and middle portions of the fuselage have better agreement with the model than 

those from the aft portion. For High Speed Civil Transport supersonic cruise, interior levels 

predicted by use of this model are expected to increase by 1 to 3 dB due to the adjustments to the Efimtsov model. 

The space-time cross-correlations and cross-spectra of the fluctuating surface pressure 

were also investigated. This analysis is an important ingredient in structural acoustic models of aircraft interior noise. Once again the measured data were compared to the predicted levels from the Efimtsov model. No adjustments were made to the cross-correlation 

portion of the prediction model. 

The auto-spectral model presented in this report can be used as improved input data to 

structural acoustic prediction models. However, further investigation of the crossfunctional model is recommended to fully utilize the information available from the TU-144 

measured data. 

iii 
Table of Contents 

1. Introduction.................................................................................................................................1

2. Models ........................................................................................................................................1

2.1 Robertson [Ref. 1] ...............................................................................................................1

2.2 ESDU [Ref. 2] ......................................................................................................................1

2.3 Laganelli [Ref. 3]..................................................................................................................2

2.4 Goodwin [Ref. 4]..................................................................................................................2

2.5 Efimtsov [Ref. 5 and 13] ......................................................................................................3

3. Boundary Layer Data..................................................................................................................3

3.1 Data Analysis.......................................................................................................................6

3.1.1 Data Reduction.............................................................................................................6

3.1.2 Data Normalization.......................................................................................................6

3.1.3 Kulite Flushness ...........................................................................................................6

3.1.4 Outlying Fluctuating Pressure Levels...........................................................................6

4. Data Correction...........................................................................................................................8

5. Comparisons of Measured Data with Models ...........................................................................10

5.1 Auto-Spectra......................................................................................................................10

5.2 Cross-Functions ................................................................................................................16

5.2.1 Cross-Correlations .....................................................................................................16

5.2.1.1 Cross-Correlations at Front Fuselage .................................................................16

5.2.1.2 Cross-Correlations at Center Fuselage...............................................................19

5.2.1.3 Cross-Correlations at Rear Fuselage..................................................................22

5.2.2 Cross-Spectra ............................................................................................................25

5.2.2.1 Cross-Spectra at Forward Window Blank ...........................................................25

5.2.2.2 Cross-spectra at Center Window Blank Cluster..................................................34

5.2.2.3 Cross-Spectra at Aft Window Blank ....................................................................52

6. Efimtsov Auto Spectrum Model Adjustments............................................................................61

6.1 Broad Peak........................................................................................................................61

6.2 High Frequency Roll-Off ....................................................................................................62

6.3 Efimtsov Adjustment Functions .........................................................................................62

6.4 Comparisons of Adjusted Efimtsov Prediction to Measured TU-144LL Data ....................63

6.5 Comparison to Other Flight Test Data...............................................................................70

7. Consequences of Model Adjustments to HSCT Interior Noise .................................................72

8. Recommendations for Future Work..........................................................................................72

iv 
9. References ...............................................................................................................................73

10. Appendix A - ESDU Equations..............................................................................................74

11. Appendix B - Fourier Transform of a Turbulence Cross-Correlation Model..........................75

v 
List of Figures 

Figure 1 TU-144 Flight Envelope and Experiment 2.1 Test Points....................................................4

Figure 2 Location and Identification of Window Blanks (WB) Instrumented with Kulite Pressure 

Transducers (N1.1 through N7.5, S1, S2) ..................................................................................5

Figure 3 Geometry of Face of Kulite Transducer XCS-190 ..............................................................9

Figure 4 Example of a Corcos Correction.......................................................................................10

Figure 5 Comparison of the Five Investigated Models, Robertson, ESDU, Laganelli, Goodwin, and 

Efimtsov, and Measured Data on Tu144 (M 0.74, 16,400 ft, Window Blank 4). .......................11

Figure 6 Comparison of the Five Investigated Models, Robertson, ESDU, Laganelli, Goodwin, and 

Efimtsov and Measured Data on Tu144 (M 1.95, 55,400 ft, Window Blank 4). ........................12

Figure 7 Normalized Comparison of the Five Investigated Models, Robertson, ESDU, Laganelli, 

Goodwin, and Efimtsov and Measured Data on Tu144 (M 1.95, 55,400 ft, Window Blank 4)..13

Figure 8 Closer Comparison of Efimtsov and Goodwin Models and In-Flight SPL Data (M 0.74, 

16,400 ft, Window Blanks 1, 4, and 7)......................................................................................14

Figure 9 Closer Comparison of Efimtsov and Goodwin Models and In-Flight SPL Data (M 1.95, 

55,400 ft, Window Blanks 1, 4, and 7)......................................................................................15

Figure 10 Cross-correlations - Longitudinal, Window Blank 1, Wide Band, Mach 0.74..................17

Figure 11 Cross-correlations - Longitudinal, Window Blank 1, Wide Band, Mach 1.95..................17

Figure 12 Cross-correlations - Lateral, Window Blank 1, Wide Band, Mach 0.74 ..........................18

Figure 13 Cross-correlations - Lateral, Window Blank 1, Wide Band, Mach 1.95 ..........................18

Figure 14 Cross-correlations - Longitudinal, Window Blanks 3/4/5, Wide Band, Mach 0.74 ..........19

Figure 15 Cross-correlations - Longitudinal, Window Blanks 3/4/5, Wide Band, Mach 0.88 ..........20

Figure 16 Cross-correlations - Longitudinal, Window Blanks 3/4/5, Wide Band, Mach 1.2 ............20

Figure 17 Cross-correlations - Longitudinal, Window Blanks 3/4/5, Wide Band, Mach 1.6 ............21

Figure 18 Cross-correlations - Longitudinal, Window Blanks 3/4/5, Wide Band, Mach 1.95 ..........21

Figure 19 Convection Velocity as a Function of Separation Distance and Mach Number, Window 

Blanks 3/4/5..............................................................................................................................22

Figure 20 Cross-correlations - Longitudinal, Window Blank 7, Wide Band, Mach 0.74..................23

Figure 21 Cross-correlations - Longitudinal, Window Blank 7, Wide Band, Mach 1.95..................23

Figure 22 Cross-correlations - Lateral, Window Blank 7, Wide Band, Mach 0.74 ..........................24

Figure 23 Cross-correlations - Lateral, Window Blank 7, Wide Band, Mach 1.95 ..........................24

Figure 24 Cross-Spectra - Longitudinal, Window Blank 1, Measured and Predicted at Mach 0.74, 

32 to 100 Hz .............................................................................................................................26

Figure 25 Cross-Spectra - Longitudinal, Window Blank 1, Measured and Predicted at Mach 0.74, 

125 to 500 Hz ...........................................................................................................................27

Figure 26 Cross-Spectra - Longitudinal, Window Blank 1, Measured and Predicted at Mach 0.74, 

600 to 2500 Hz .........................................................................................................................28

vi 
Figure 27 Cross-Spectra - Longitudinal, Window Blank 1, Measured and Predicted at Mach 1.95, 

32 to 100 Hz .............................................................................................................................29

Figure 28 Cross-Spectra - Longitudinal, Window Blank 1, Measured and Predicted at Mach 1.95, 

125 to 500 Hz ...........................................................................................................................30

Figure 29 Cross-Spectra - Longitudinal, Window Blank 1, Measured and Predicted at Mach 1.95, 

600 to 2500 Hz .........................................................................................................................31

Figure 30 Cross-Spectra - Lateral, Window Blank 1, Measured and Predicted at Mach 0.74........32

Figure 31 Cross-Spectra - Lateral, Window Blank 1, Measured and Predicted at Mach 1.95........33

Figure 32 Cross-Spectra - Longitudinal, Central Window Blanks, Measured and Predicted at Mach 

0.74, 32 to 100 Hz ....................................................................................................................35

Figure 33 Cross-Spectra - Longitudinal, Central Window Blanks, Measured and Predicted at Mach 

0.74, 125 to 500 Hz ..................................................................................................................36

Figure 34 Cross-Spectra - Longitudinal, Central Window Blanks, Measured and Predicted at Mach 

0.74, 600 to 2500 Hz ................................................................................................................37

Figure 35 Cross-Spectra - Longitudinal, Central Window Blanks, Measured and Predicted at Mach 

0.88, 32 to 100 Hz ....................................................................................................................38

Figure 36 Cross-Spectra - Longitudinal, Central Window Blanks, Measured and Predicted at Mach 

0.88, 125 to 500 Hz ..................................................................................................................39

Figure 37 Cross-Spectra - Longitudinal, Central Window Blanks, Measured and Predicted at Mach 

0.88, 600 to 2500 Hz ................................................................................................................40

Figure 38 Cross-Spectra - Longitudinal, Central Window Blanks, Measured and Predicted at Mach 

1.2, 32 to 100 Hz ......................................................................................................................41

Figure 39 Cross-Spectra - Longitudinal, Central Window Blanks, Measured and Predicted at Mach 

1.2, 125 to 500 Hz ....................................................................................................................42

Figure 40 Cross-Spectra - Longitudinal, Central Window Blanks, Measured and Predicted at Mach 

1.2, 600 to 2500 Hz ..................................................................................................................43

Figure 41 Cross-Spectra - Longitudinal, Central Window Blanks, Measured and Predicted at Mach 

1.6, 32 to 100 Hz ......................................................................................................................44

Figure 42 Cross-Spectra - Longitudinal, Central Window Blanks, Measured and Predicted at Mach 

1.6, 125 to 500 Hz ....................................................................................................................45

Figure 43 Cross-Spectra - Longitudinal, Central Window Blanks, Measured and Predicted at Mach 

1.6, 600 to 2500 Hz ..................................................................................................................46

Figure 44 Cross-Spectra - Longitudinal, Central Window Blanks, Measured and Predicted at Mach 

1.95, 32 to 100 Hz ....................................................................................................................47

Figure 45 Cross-Spectra - Longitudinal, Central Window Blanks, Measured and Predicted at Mach 

1.95, 125 to 500 Hz ..................................................................................................................48

Figure 46 Cross-Spectra - Longitudinal, Central Window Blanks, Measured and Predicted at Mach 

1.95, 600 to 2500 Hz ................................................................................................................49

Figure 47 Cross-Spectra - Lateral, Central Window Blanks, Measured and Predicted at Mach 0.7450

Figure 48 Cross-Spectra - Lateral, Central Window Blanks, Measured and Predicted at Mach 1.9551

vii 
Figure 49 Cross-Spectra - Longitudinal, Window Blank 7, Measured and Predicted at Mach 0.74, 

32 to 100 Hz .............................................................................................................................53

Figure 50 Cross-Spectra - Longitudinal, Window Blank 7, Measured and Predicted at Mach 0.74, 

125 to 500 Hz ...........................................................................................................................54

Figure 51 Cross-Spectra - Longitudinal, Window Blank 7, Measured and Predicted at Mach 0.74, 

600 to 2500 Hz .........................................................................................................................55

Figure 52 Cross-Spectra - Longitudinal, Window Blank 7, Measured and Predicted at Mach 1.95, 

32 to 100 Hz .............................................................................................................................56

Figure 53 Cross-Spectra - Longitudinal, Window Blank 7, Measured and Predicted at Mach 1.95, 

125 to 500 Hz ...........................................................................................................................57

Figure 54 Cross-Spectra - Longitudinal, Window Blank 7, Measured and Predicted at Mach 1.95, 

600 to 2500 Hz .........................................................................................................................58

Figure 55 Cross-Spectra - Lateral, Window Blank 7, Measured and Predicted at Mach 0.74........59

Figure 56 Cross-Spectra - Lateral, Window Blank 7, Measured and Predicted at Mach 1.95........60

Figure 57 Excerpt from Ref. 16: ‘Corrected power spectral densities ...’........................................61

Figure 58 Sample Adjustments for Mach 2, * δ =1 ft , U0=1800 fps................................................63

Figure 59 Adjusted Efimtsov Model and In-Flight SPL Data (M 1.95, 55,400 ft, Window Blanks 1, 4, 

and 7)........................................................................................................................................64

Figure 60 Adjusted Efimtsov Model and Tu144 In-Flight SPL Data (M 0.74, 16,400 ft, Window 

Blanks 1, 4, and 7)....................................................................................................................65

Figure 61 Adjusted Efimtsov Model and Tu144 In-Flight SPL Data (M 0.88, 29,500 ft, Window 

Blanks 1, 4, and 7)....................................................................................................................66

Figure 62 Adjusted Efimtsov Model and Tu144 In-Flight SPL Data (M 1.2, 41,000 ft, Window 

Blanks 1, 4, and 7)....................................................................................................................67

Figure 63 Adjusted Efimtsov Model and Tu144 In-Flight SPL Data (M 1.6, 45,000 ft, Window 

Blanks 1, 4, and 7)....................................................................................................................68

Figure 64 Adjusted Efimtsov Model and Tu144 In-Flight SPL Data (M 1.95, 55,400 ft, Window 

Blanks 1, 4, and 7)....................................................................................................................69

Figure 65 Adjusted Efimtsov Predicted SPL and Measured Flight Test Data from Concorde (Mach 

2)...............................................................................................................................................71

List of Tables 

Table 1 Outlying Fluctuating Pressure Measurements. ....................................................................7

viii 
1. Introduction 

The motivation for this work stems from the need for an aerodynamic loads model for predicting 

passenger aircraft interior noise, at subsonic as well as supersonic cruising speeds. The turbulent 

boundary layer on the fuselage causes structural vibrations which radiate random noise into cockpit and passenger cabin. Simple interior noise prediction models use auto spectra of the pressure 

fluctuations. More complex models require space-time cross-correlations (or cross-spectra) of 

pressure fluctuations. 

This work was performed at the Boeing Company under NASA Contract 20220, High Speed Research Program, Phase II, Task 27, Structural Acoustics. The objective was the validation of fluctuating pressure turbulent boundary layer models or model, both for single signals (auto spectra) 

and signal pairs (cross spectra). 

The data set underlying this study came from measurements during flight tests of the Tupolev 144 

Supersonic Flying Laboratory in late 1997 and early 1998 [Ref. 10]. 

2. Models 

Several models available in the literature are discussed and then used to compare the present 

data with their predictions. A model by Smol’yakov-Tkachenko [Ref. 17] is not used because it 

was built with marine applications in mind and is not well suited to high speed flight [Ref. 11]. As 

discussed in the Sections below, several authors have built different models for power spectra 

(auto spectra) at a single point of the surface. In those references, models for space-time crosscorrelation or frequency-wave number cross spectra are either not discussed, or are cast in only 

one form (exponential decay with spatial separation) which is discussed in Section 2.5 (Efimtsov). 

The equations presented here are meant to convey the flavor and complexity of the models. If the 

reader wishes to actually implement any of these, please refer to the complete references. 

2.1 Robertson [Ref. 1]

This prediction of the turbulent boundary layer fluctuating pressure power spectrum is derived from 

an empirical evaluation of sub-sonic and super-sonic wind tunnel experimental results. The normalized equation for the frequency spectrum is a function of an estimate of the mean squared pressure and the Strouhal number. 

( )

2

2

2 * 2 * 0.9

0

0

1

P

U q

q

U

ω

δ ω δ ω

ω

∞ ∞

∞

∞

Φ = ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ + ⎜ ⎟ ⎝ ⎠ ⎝ ⎠

Where: 

0 * 0.5U

ω

δ

∞ = and ( )

( )

2 2

2 2 2

0.006

1 0.14

P

q∞ M

=

+

2.2 ESDU [Ref. 2]

The predicted power spectrum is based on an analysis of flight test results. Fluctuating pressure 

levels, flow field, and boundary layer parameters from flight testing is curve fitted to determine the 

1 
predicted octave and third octave band levels. The prediction is a function of: Mach number, 

Strouhal number, and Reynolds number. The curve fitting equations appear in Section 10 (Appendix A). 

2.3 Laganelli [Ref. 3] 

Predicted power spectra are derived from experimental and fluid dynamic principles. Experimental 

data includes: AFWAL Mach 3 facility, Blake, Laganelli, Raman, Coe, and Bull (these do not appear to include flight data). The equation is an expansion of Robertson's equation, but adds friction coefficient considerations: 

( ) 5 0.5733

2 * 2 *

2.867

10 2.293

1

C

C

U F

q

F

U

ω

δ δ

ω

− −

∞

∞

Φ

= ×

⎛ ⎞

+ ⎜ ⎟ ⎝ ⎠

where Fc is a transformation from compressible to incompressible flow states: 

*

1 1 1 1 2 2 0.22 2 2 2 2

i

c

f w

c

f e aw

C h h F r M

C h h

⎛ ⎞ γ γ − − = = = + ⎜ ⎟ + + ⎝ ⎠ r M

2.4 Goodwin [Ref. 4]

This fluctuating pressure predicted spectrum was derived from previous prediction equations and 

flight test data on three super-sonic test aircraft: XB-70, A3J, and Concorde. A very small portion 

of the XB-70 data and even less of the A3J data was usable for this type of prediction due to a 

number of experimental errors. Additionally, the Concorde data, which used Kulites in window 

blanks very similar to those used for the Tu144 testing, is somewhat suspect due to the experimenter not paying close attention to the sensitivity of the Kulites to the flushness of the sensor's 

surface relative to the surrounding surface (discussed further in Section 3.1.3). The Kulite measurements varied as much as 9 dB between sensors in the same window blank. This test also experienced trouble duplicating results for repeated test conditions. Goodwin adjusted the Robertson 

and Laganelli predictions to account for the measured in-flight data using the following equations: 

( )

( ) 2 3 0.45

2

, 7 2 1.2 0.1

7

1.2 1

Prms w w T Pw n

q f

U

ρ

π ρ

θω

∞ ∞

∞

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ − ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ + ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ Φ =

⎛ ⎞

+ ⎜ ⎟ ⎝ ⎠

10 4.0 2.5log 631,000

Rex n

⎛ ⎞

= + ⎜ ⎟ ⎝ ⎠

( ) 2 2 0 rms pp

A U P

π θ B

∞ = Φ

( )

3 0,45

*

7 1 0.08 , 1.2

, 7 w w

n A B Constant

T P

ρ θ

ρ δ

∞

∞

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ − = + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

=

2 
2.5 Efimtsov [Ref. 5 and 13]

Professor B. M. Efimtsov of TsAGI (Central Aerohydrodynamic Institute in Moscow, Russia) updated the existing model of turbulent boundary layer fluctuating pressure levels. This new model 

utilized data from the low and high speed TsAGI wind tunnels, Tu144, and what appears to be a 

TU-22 (twin engined, supersonic military aircraft) in-flight measurements. Details of the flight conditions and sensor locations are given; they appear to be similar to those measured on the Tu144 

by NASA-Boeing. However, no actual flight data is presented for direct comparison to the NASABoeing tests. 

The analytical derivation of the TsAGI predicted power spectrum references Laganelli's calculations; the principal independent variables are Strouhal number and Reynolds number. Efimtsov's 

equation is: 

( )

( )

3 2

10 1 3 3 2 3

2 ,

1 8

U

Sh Sh Re Re

τ

τ τ

β ω πα ρ δ

α αβ

Φ =

⎛ ⎞ + + ⎜ ⎟ ⎝ ⎠

1

3 3

0 1 , 0.01 Re

Re τ

τ

β α ⎡ ⎤ ⎛ ⎞ = + ⎢ ⎜ ⎟ ⎥ = ⎝ ⎠ ⎣ ⎦, Sh

Uτ

ωδ = (Strouhal Number) 

The model for cross-spectral density assumes that dependencies on frequency and spatial separation can be mathematically separated as follows: 

( )

1 2

1

1 2 , , ( ) 1 2

i U

p

e

ξ ξ

ωξ

ξ ξ ω ω

⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦

− − − Λ Λ

Φ = Φ

where subscript 1 refers to the longitudinal (flow-wise) direction, and subscript 2 to the lateral 

(cross flow along the surface) direction. 1 ξ and 2 ξ are separation distances between points on the 

surface, and and are the correlation length scales which strongly depend on frequency (see 

Ref. 13 for details). 

Λ1 Λ2

3. Boundary Layer Data

The reader is referred to Ref. 10 for details on how TU-144 data were acquired. Figure 1 shows the 

flight conditions for which data was acquired. Due to lack of time for this study, only a representative subset of the data was utilized. 

3 
Figure 2 shows the layout and nomenclature of the transducers in the window blanks and in the 

fuselage skin.

300 400 500 600 700 800 900 1000

5

0

10

Altitude, km

Indicated Air S

15

20

peed, km/hr

4 

M=0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

2.0

Vmaxe

TU-144 FLIGHT ENVELOPE

1993 Concorde Test

10

10, 17

11

11

11

11 11

9, 16

9 9 9

Flights 9, 10, 11, 16, 17

17 11

17

17

Conditions on which

this report concentrates

 Figure 1 TU-144 Flight Envelope and Experiment 2.1 Test Points 
5 

WB7 WB6 WB5 WB4 WB3 WB2 

Figure 2 Location and Identification of Window Blanks (WB) Instrumented with Kulite Pre

Transducers (N1.1 through N7.5, S1, S2) 

N3.1 N2.1

S1

N4.7 N4.2 N4.6 N4.3 N4.1 N4.5 N4.8 N4.4 N4.9 

S2

N6.1 N5.1

 N7.2 N7.3 N7.1 N7.5  N7.4 
3.1 Data Analysis 

3.1.1 Data Reduction 

Pressure level spectra were calculated from one minute samples recorded for each transducer and 

for each flight condition. These spectra were calculated both on a narrow band and a third octave 

band basis. The narrow bands have a uniform constant bandwidth of 2.44 Hz; spectral levels reported here were corrected to a 1Hz bandwidth. The third octave band levels were also corrected 

to a 1 Hz bandwidth for direct comparison between the narrow band and the third OB and to compare levels to some of the published results from previous investigations. 

All signal analysis was carried out using the MATLAB software package on an HP-UX work station. 

Analysis scripts were provided by Dr. Stephen A. Rizzi of NASA Langley Research Center. 

3.1.2 Data Normalization 

Two forms of data normalization were investigated to help compare between channels and between flight conditions, and for comparing data to previous tests. The first normalization is what is 

seen in most of the literature on turbulent boundary layer spectral shapes [Ref. 6 and 7] 

( ) *

0

2 *

0 0

. U

vs

q U

ω ωδ

δ

Φ

where U0 is the free stream velocity, q0 is the dynamic pressure, and * δ is the displacement 

boundary layer thickness. Efimtsov [Ref. 5] suggests a slightly different normalization for a better 

collapse of the TsAGI measured data: 

( )

2 3 .

w

vs

U U τ τ

ω ωδ

ρ δ

Φ

where Uτ is the friction velocity defined by the quotient of wall shear stress over fluid density at the 

wall ρ w , and δ is the (velocity deficit) boundary layer thickness. We usually use the Efimtsov normalization in our work. 

3.1.3 Kulite Flushness 

An exhaustive wind tunnel test was performed by Professor Efimtsov at TsAGI [Ref. 8] to determine the effect sensor flushness would have on the measured results at both subsonic and supersonic speeds. For measured flushness ranging from recessed to protruding, deviation from expected third octave band levels were reported. Variations of less than 2 dB can be expected for 

sensors deviating less than 20x10-6m from flush, the flushness deviation being measured in the 

stream-wise direction. The Tu144 flight test Kulite flushness measurements are discussed in Ref. 

10. Only four of the Kulites measured this much of a deviation from flushness. However, when 

these four Kulites were compared to Kulites in the same window blanks (Section 3.1.4), only one of 

them (N4.1) varied significantly from the average measured SPL within that blank; data from Kulite 

N4.1 was not used in the auto spectrum analysis in this study. 

3.1.4 Outlying Fluctuating Pressure Levels 

Within window blank comparisons were conducted by plotting the third octave band levels for all of 

the sensors within each window blank for 22 selected steady-state flight conditions. Channels that 

6 
showed significant deviation from the other channels within the same window blank were noted as 

deviant and were not used for further auto-spectral analysis (Table 1). Causes for the deviation in 

measured SPL may included: line noise, damaged sensors, sensor flushness, localized flow disturbances. 

Table 1 Outlying Fluctuating Pressure Measurements. 

Flight & 

Run # 

Altitude 

(kft) 

Mach # Window 

Blank 1 

Only 

Window 

Blank 4 

Only 

Window 

Blank 7 

Only 

Central 

Transducer 

on all 7 

Window 

Blanks 

Bad Channel #'s

f9r17 16.4 0.648 OK 4.1, 4.9 OK OK 5 26

f9r21 16.4 0.877 OK 4.1, 4.9 OK 7.1 5 16 26

f9r23 13.1 0.563 OK 4.1, 4.9 OK OK 5 26

f10r29 45 1.6 OK 4.1, 4.4,

4.5 

7.5 4.1 5 8 9 30

f10r31 46.6 1.6 OK 4.1, 4.5 7.5 4.1 5 9 30

f10r33 55.4 1.95 OK 4.1 7.5 4.1 5 30

f10r34 55.4 1.95 4.1, 4.3 4.1 5 7 

f10r35 56.8 1.95 OK 4.1, 4.3 7.5 4.1 5 7 30

f10r36 56.8 1.95 4.1, 4.3 4.1 5 7

f10r37 56.8 1.95 4.1, 4.3 4.1 5 7

f11r40 52.5 1.8 OK 4.1, 4.5 OK 4.1 5 9

f11r42 54.5 1.7 OK 4.1, 4.3,

4.5 

OK 4.1 5 7 9

f11r44 50.9 1.7 OK 4.1, 4.5 OK 4.1 5 9

f11r48 41.3 1.4 OK 4.1, 4.5 OK 4.1 5 9

f11r50 41 1.2 1.2 4.1 OK 4.1 5 22

f11r53 16.4 0.75 OK 4.1 OK 4.1 5

f17r66 53.1 2 OK 4.3 OK 4.1 5 7

f17r68 55.4 2.05 OK 4.3 OK 4.1 5 7

f17r70 53.8 1.75 OK 4.1, 4.3,

4.5 

OK 4.1 5 7 9

f17r71 49.2 1.6 N/A 4.3 N/A OK 7

Single sensors located on window blanks by themselves (blanks 2,3,5, and 6) and the two sensors 

on the skin (S1 and S2) were more difficult to compare for outlying SPL. Normalized levels plotted 

for sensors down the length of the plane could be used to some extent. However, the uncertainties 

in the data were too great for meaningful interpolations so that data from single sensors were not 

included in the auto-spectral analyses in this report. 

7 
4. Data Correction 

A correction was added to all auto-spectra of boundary layer surface pressure fluctuations. It accounts for the lack for spatial turbulence scale resolution of the transducer at high frequencies, according to Corcos [Ref. 14]. Ideally, one would follow the theory presented in Ref. 14 and develop 

the correction formula based on the transducer’s actual geometry as shown in Figure 3. This was 

attempted but abandoned due to mathematical complexity and because an assumption made in 

the theory was not satisfied which says that the sensing areas on the transducer face contribute to 

the total instantaneous measured pressure without phase shift. In the present case, the 12 small 

circles in Figure 3 are just through holes to a small volume behind the face containing a single 

strain gage to which each hole contributes with unknown phase shifts. Instead, a circular transducer sensing surface was assumed in order to use the calculated results in Ref. 14. 

Then, the crucial question is what transducer radius to use in the correction procedure. Efimtsov 

[Ref. 8] stated that the transducer he investigated in the wind tunnel behaved with regard to a Corcos correction as if its diameter was about 8 mm, i.e. double its outer diameter. However, when 

we used a radius of 4mm for the Corcos correction, the high frequency behavior of the corrected 

spectra became unreasonable: instead of a more or less constant roll-off slope on a log-log plot, 

the spectrum started to curve upwards again at higher frequencies. Instead, Philippe Spalart [Ref. 

15] suggested that an equivalent radius be determined on the basis of calculating root-meansquare distances from points on the sensing device to the center. With a ring, the root-meansquare equals the radius of the ring. With a disk, it is the disk radius divided by square root of two. 

Therefore, we multiplied the radius of the circle on which the sensing holes are located (the “ring”) 

by 2 resulting in 0.104 2 1.87

2

inches = mm. This resulted in reasonable corrections. An example is shown in Figure 4. 

8 
0.006”

 Faceplate 

Thickness is 

0.006” 

0.104”

0.139”

 Figure 3 Geometry of Face of Kulite Transducer XCS-190 

9 
101 102 103 104

85

90

95

100

105

SPL, dB (ref 2e−5Pa)

Frequency, Hz

Corcos Correction Comparison, M=1.95, Alt=54000 ft

No Corcos Correction

Corcos Correction Applied

 Figure 4 Example of a Corcos Correction 

5. Comparisons of Measured Data with Models 

5.1 Auto-Spectra 

Figure 5 and Figure 6 (absolute levels) and Figure 7 (normalized) show comparisons between all 

the models and the measured data, for window blank 4 (in the middle of the fuselage). The 

Efimtsov and Goodwin models are the most accurate for predicting the measurements on the 

Tu144. Figure 8 and Figure 9 contain comparisons of TU-144LL data with Efimtsov and Goodwin 

predictions for window blanks 1, 4, and 7. These data are typical of what we saw: at subsonic 

speeds, the Efimtsov model predicts well, and at supersonic speeds both the Efimtsov and Goodwin models provide good estimates. All models but Efimtsov’s predict a steeper high frequency rolloff than the TU-144LL data (which has been corrected for transducer size). Efimtsov's model was 

chosen for further adjustment, as it had the best overall match to the measured data. 

10 
101 102 103 104 80

85

90

95

100

105

110

Frequency, Hz

Φ(f), dB, (ref: 2x10−5Pa)

Predicted and Measured SPL Data, Mach 0.74, Alt=16400 ft

Measured Window Blank 4

Predicted Efimtsov

Predicted Goodwin

Predicted ESDU

Predicted Laganelli

Predicted Robertson

 Figure 5 Comparison of the Five Investigated Models, Robertson, ESDU, Laganelli, Goodwin, and Efimtsov, and Measured Data on Tu144 (M 0.74, 16,400 ft, Window Blank 4). 

11 
101 102 103 104 80

85

90

95

100

105

110

Frequency, Hz

Φ(f), dB, (ref: 2x10−5Pa)

Predicted and Measured SPL Data, Mach 1.95, Alt=55400 ft

Measured Window Blank 4

Predicted Efimtsov

Predicted Goodwin

Predicted ESDU

Predicted Laganelli

Predicted Robertson

 Figure 6 Comparison of the Five Investigated Models, Robertson, ESDU, Laganelli, Goodwin, and Efimtsov and Measured Data on Tu144 (M 1.95, 55,400 ft, Window Blank 4). 

12 
100 101 102 103 −20

−15

−10

−5

ω δ /Uτ

10 log [

Φ(ω) / (ρ2

w

 U3

τ δ)]

Normalized Predicted and Measured Data, Mach 1.95,Alt=55400 ft

Measured Window Blank 4

Predicted Efimtsov

Predicted Goodwin

Predicted ESDU

Predicted Laganelli

Predicted Robertson

 Figure 7 Normalized Comparison of the Five Investigated Models, Robertson, ESDU, Laganelli, Goodwin, and Efimtsov and Measured Data on Tu144 (M 1.95, 55,400 ft, Window

Blank 4). 

13 
101 102 103 104 80

85

90

95

100

105

110

Frequency, Hz

Φ(f), dB, (ref: 2x10−5Pa)

Predicted and Measured SPL Data, Mach 0.74, Alt=16400 ft

Measured Window Blank 7

Goodwin Predicted WB 7

Efimtsov Predicted WB 7

Measured Window Blank 4

Goodwin Predicted WB 4

Efimtsov Predicted WB 4

Measured Window Blank 1

Goodwin Predicted WB 1

Efimtsov Predicted WB 1

 Figure 8 Closer Comparison of Efimtsov and Goodwin Models and In-Flight SPL Data (M 

0.74, 16,400 ft, Window Blanks 1, 4, and 7) 

14 
101 102 103 104 80

85

90

95

100

105

110

Frequency, Hz

Φ(f), dB, (ref: 2x10−5Pa)

Predicted and Measured SPL Data, Mach 1.95, Alt=55400 ft

Measured Window Blank 7

Goodwin Predicted WB 7

Efimtsov Predicted WB 7

Measured Window Blank 4

Goodwin Predicted WB 4

Efimtsov Predicted WB 4

Measured Window Blank 1

Goodwin Predicted WB 1

Efimtsov Predicted WB 1

 Figure 9 Closer Comparison of Efimtsov and Goodwin Models and In-Flight SPL Data 

(M 1.95, 55,400 ft, Window Blanks 1, 4, and 7) 

15 
5.2 Cross-Functions 

All measured cross-functions presented here were computed using the entire 1 minute time history 

records available in the TU-144LL data. The cross-correlations use the entire frequency band 

available: from about 2 Hz to 11.2 KHz. The cross-spectra were calculated on a narrow band basis (2.44 Hz) and then averaged over one third octave bands, corrected to a 1 Hz bandwidth. 

A correction for transducer size was not applied (analogous to the Corcos [Ref. 14] correction for 

auto spectra). This would have required applying digital filters to amplify the high frequency portion 

of the two input time histories being cross-analyzed. 

5.2.1 Cross-Correlations 

Cross-correlations are defined by 

( ) [ ] [ ]

1 ( , ) ( ) 2 , ( ) ( )

lim

T

T T

rms rms

p t p t d

T

p p

τ τ

φ τ →∞ −

+ +

= +

∫ x x

x

x x

ξ,

, ξ ξ

where p is the fluctuating surface pressure, x is a location on the fuselage, 2 ξ ξ ) 1, ξ = ( is a spatial 

separation vector, t is the time, and τ is a time delay. In the following text the location on the fuselage is omitted and implied by context. 

All the cross-correlations exhibit expected behavior: 

• For longitudinal separations (stream-wise) the peak occurs at a time delay corresponding to the 

convection velocity, and the magnitude at the peak decreases with increasing separation. The 

cross-correlations are well behaved and show discernible correlations even at the largest separation distance of 1741 mm (5.7 feet). The convection velocity can be calculated by dividing 

the separation distance into the time delay of the correlation peak. The result is shown in 

Figure 19. 

• For lateral separations, the peaks occur close to zero time delay. 

• For equal separation distances, the magnitude for longitudinal separations is much greater 

than for lateral separations. 

The angle of local surface stream lines was not known; therefore, no corrections have been applied for the flow not being aligned with the transducers. For some lateral separations, the peak 

occurs slightly before zero time delay which may be explained by the flow direction not being perpendicular to the line between the transducers. 

5.2.1.1 Cross-Correlations at Front Fuselage 

Figure 10 and Figure 11 show cross-correlations with longitudinal (stream-wise) separation distance as a parameter, for transducers on window blank 1. 

Figure 12 and Figure 13 show cross-correlations, with lateral (cross stream) separation distance as 

a parameter, for transducers on window blank 1. 

16 
0 0.5 1 1.5 2 2.5 3

x 10−3

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

φ(ξ

1, τ)/φ(0,0)

Time Delay, τ, seconds

W1

Longitudinal Separation

 ξ

1

, mm 

 40

 60

100

 Figure 10 

Crosscorrelations - 

Longitudinal, 

Window

Blank 1, Wide 

Band, Mach 

0.74 

0 0.5 1 1.5 2 2.5 3

x 10−3

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

φ(ξ

1, τ)/φ(0,0)

Time Delay, τ, seconds

W1

Longitudinal Separation

 ξ

1

, mm 

 40

 60

100

 Figure 11 

Crosscorrelations - 

Longitudinal, 

Window

Blank 1, Wide 

Band, Mach 

1.95 

17 
−1 −0.5 0 0.5 1 1.5 2

x 10−3

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

φ(ξ

2, τ), Pa2

Time Delay, τ, seconds

Lateral Separation

 ξ

2

, mm 

W1

 60

 90

150

 Figure 12 

Crosscorrelations - 

Lateral, Window

Blank 1, Wide 

Band, Mach 

0.74 

−1 −0.5 0 0.5 1 1.5 2

x 10−3

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

φ(ξ

1, τ)/φ(0,0)

Time Delay, τ, seconds

Lateral Separation

 ξ

2

, mm 

W1

 60

 90

150

 Figure 13 

Crosscorrelations - 

Lateral, Window

Blank 1, Wide 

Band, Mach 

1.95 

18 
5.2.1.2 Cross-Correlations at Center Fuselage 

Figure 14 through Figure 18 show cross-correlations which are functions of time delay, with separation distance as a parameter, for transducers on window blanks 3 through 5, and between them. 

The increase in peak value of the correlation going from 60mm to 120mm spacing, as shown in 

Figure 14 and in Figure 16, can be explained by the inaccurate auto spectrum measured on transducer N4.1 (discussed in Section 3.1.3). 

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 −0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

φ(ξ

1, τ)/φ(0,0)

Time Delay, τ, seconds

Longitudinal Separation

 ξ

1

, mm 

 60

 120

 435

 496

 811

 871

 931

1307

1741

 Figure 14 

Crosscorrelations - 

Longitudinal, 

Window Blanks 

3/4/5, Wide 

Band, Mach 

0.74 

19 
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 −0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

φ(ξ

1, τ)/φ(0,0)

Time Delay, τ, seconds

Longitudinal Separation

 ξ

1

, mm 

 60

 120

 435

 496

 811

 871

 931

1307

1741

 Figure 15 

Crosscorrelations - 

Longitudinal, 

Window Blanks 

3/4/5, Wide 

Band, Mach 

0.88 

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 −0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

φ(ξ

1, τ)/φ(0,0)

Time Delay, τ, seconds

Longitudinal Separation

 ξ

1

, mm 

 60

 120

 435

 496

 811

 871

 931

1307

1741

 Figure 16 

Crosscorrelations - 

Longitudinal, 

Window Blanks 

3/4/5, Wide 

Band, Mach 1.2 

20 
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 −0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

φ(ξ

1, τ)/φ(0,0)

Time Delay, τ, seconds

Longitudinal Separation

 ξ

1

, mm 

 60

 120

 435

 496

 811

 871

 931

1307

1741

 Figure 17 

Crosscorrelations - 

Longitudinal, 

Window Blanks 

3/4/5, Wide 

Band, Mach 1.6 

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 −0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

φ(ξ

1, τ)/φ(0,0)

Time Delay, τ, seconds

Longitudinal Separation

 ξ

1

, mm 

 60

 120

 435

 496

 811

 871

 931

1307

1741

 Figure 18 

Crosscorrelations - 

Longitudinal, 

Window Blanks 

3/4/5, Wide 

Band, Mach 

1.95 

21 
0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0 500 1000 1500 2000

Longitudinal Separation Distance, mm

Convection over Free Stream Velocity

0.74

0.88

1.2

1.6

1.95

 Figure 19 Convection Velocity as a Function of Separation Distance and Mach Number, 

Window Blanks 3/4/5 

5.2.1.3 Cross-Correlations at Rear Fuselage 

Figure 20 and Figure 21 show cross-correlations, with longitudinal separation distance as a parameter, for transducers on window blank 7. 

Figure 22 and Figure 23 show cross-correlations, with lateral separation distance as a parameter, 

for transducers on window blank 7. 

22 
0 0.5 1 1.5 2 2.5 3

x 10−3

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

φ(ξ

1, τ)/φ(0,0)

Time Delay, τ, seconds

W7

Longitudinal Separation

 ξ

1

, mm 

 40

 60

100

 Figure 20 

Crosscorrelations - 

Longitudinal, 

Window

Blank 7, Wide 

Band, Mach 

0.74 

0 0.5 1 1.5 2 2.5 3

x 10−3

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

φ(ξ

1, τ)/φ(0,0)

Time Delay, τ, seconds

W7

Longitudinal Separation

 ξ

1

, mm 

 40

 60

100

 Figure 21 

Crosscorrelations - 

Longitudinal, 

Window

Blank 7, Wide 

Band, Mach 

1.95 

23 
−1 −0.5 0 0.5 1 1.5 2

x 10−3

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

φ(ξ

1, τ)/φ(0,0)

Time Delay, τ, seconds

Lateral Separation

 ξ

2

, mm 

W7

 60

 90

150

 Figure 22 

Crosscorrelations - 

Lateral, Window

Blank 7, Wide 

Band, Mach 

0.74 

−1 −0.5 0 0.5 1 1.5 2

x 10−3

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

φ(ξ

1, τ)/φ(0,0)

Time Delay, τ, seconds

Lateral Separation

 ξ

2

, mm 

W7

 60

 90

150

 Figure 23 

Crosscorrelations - 

Lateral, Window

Blank 7, Wide 

Band, Mach 

1.95 

24 
25 

( )

Figure 30 and Figure 31 show cross-spectral comparisons (magnitude only) for laterally separated

transducers. Again, the measured magnitudes and decay rates are lower than the predicted ones. 

At Mach 1.95, predicted cross-spectral values coincide for all frequencies; the measured values

tend to cluster closer together than at Mach 0.74.

Figure 24 through Figure 29 contain measured and predicted cross-spectral values as a function of

longitudinal separations with frequency as the parameter, one set for a mach number of 0.74, another set for mach 1.95. Measured magnitudes are generally lower than predicted ones. At the 

subsonic Mach number, decay rates with distance are comparable. At the supersonic speed, the

measured cross spectra show little decay. Phases compare well in all cases.

5.2.2.1 Cross-Spectra at Forward Window Blank 

All cross-spectra have dimensions Pascal-squared per Hertz. Ref. 13 was used to compute the 

predicted values.

As shown in Figure 2, window blanks 1 and 7 only offer three longitudinal and two lateral separation distances. The cluster of transducers in and between window blanks 3 to 5 offer a much

greater range of longitudinal separations, but only two in the lateral direction.

where f is frequency. Again, the location vector x is omitted below and is implied by context (window blank).

The cross-spectral density is defined as the Fourier transform of the cross-correlation. In Section

5.2.1, the cross-correlation is defined with normalization. Here, we define the cross-spectrum without normalization:

5.2.2 Cross-Spectra 

1 2 , ( , ) ( ) 2 lim

T

if

T T

f p t p t dt e d

T

π τ τ τ

∞

−

−∞ →∞ −

⎡ ⎤ Φ = ⎢ ⎥ + +

⎣ ⎦ ∫ ∫ x x , ξ x ξ,
40 60 100 200 600 2000

0

0.5

1

1.5

2

Measured φ(ξ1,0,f=const) M=0.74 Alt=16400 ft

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

W1

40 60 100 200 600 2000

−3.14−1.57

 0

 1.57 3.14

ξ1, mm

Phase(φ), Radians

40 60 100 200 600 

0

0.5

1

1.5

2

Predicted φ(ξ1,0,f=const) M=0.74 Alt=16400 f

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

40 60 100 200 600 

−3.14−1.57

 0

 1.57 3.14

Phase(φ), Radians

ξ1, mm

 Figure 24 Cross-Spectra - Longitudinal, Window Blank 1, Measured and Predicted at Mach 0.74

26 
40 60 100 200 600 2000

0

0.5

1

1.5

2

Measured φ(ξ1,0,f=const) M=0.74 Alt=16400 ft

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

W1

40 60 100 200 600 2000

−3.14−1.57

 0

 1.57 3.14

ξ1, mm

Phase(φ), Radians

40 60 100 200 600 

0

0.5

1

1.5

2

Predicted φ(ξ1,0,f=const) M=0.74 Alt=16400 

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

40 60 100 200 600 

−3.14−1.57

 0

 1.57 3.14

Phase(φ), Radians

ξ1, mm

 Figure 25 Cross-Spectra - Longitudinal, Window Blank 1, Measured and Predicted at Mach 0.74, 

27 
40 60 100 200 600 2000

0

0.5

1

1.5

2

Measured φ(ξ1,0,f=const) M=0.74 Alt=16400 ft

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

W1

40 60 100 200 600 2000

−3.14−1.57

 0

 1.57 3.14

ξ1, mm

Phase(φ), Radians

40 60 100 200 600 

0

0.5

1

1.5

2

Predicted φ(ξ1,0,f=const) M=0.74 Alt=16400 f

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

40 60 100 200 600 

−3.14−1.57

 0

 1.57 3.14

Phase(φ), Radians

ξ1, mm

 Figure 26 Cross-Spectra - Longitudinal, Window Blank 1, Measured and Predicted at Mach 0.74, 

28 
40 60 100 200 600 2000

0

0.5

1

1.5

2

Measured φ(ξ1,0,f=const) M=1.95 Alt=55400 ft

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

W1

40 60 100 200 600 2000

−3.14−1.57

 0

 1.57 3.14

ξ1, mm

Phase(φ), Radians

40 60 100 200 600 

0

0.5

1

1.5

2

Predicted φ(ξ1,0,f=const) M=1.95 Alt=55400 ft

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

40 60 100 200 600 

−3.14−1.57

 0

 1.57 3.14

Phase(φ), Radians

ξ1, mm

 Figure 27 Cross-Spectra - Longitudinal, Window Blank 1, Measured and Predicted at Mach 1.95

29 
40 60 100 200 600 2000

0

0.5

1

1.5

2

Measured φ(ξ1,0,f=const) M=1.95 Alt=55400 ft

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

W1

40 60 100 200 600 2000

−3.14−1.57

 0

 1.57 3.14

ξ1, mm

Phase(φ), Radians

40 60 100 200 600 

0

0.5

1

1.5

2

Predicted φ(ξ1,0,f=const) M=1.95 Alt=55400 f

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

40 60 100 200 600 

−3.14−1.57

 0

 1.57 3.14

Phase(φ), Radians

ξ1, mm

Figure 28 Cross-Spectra - Longitudinal, Window Blank 1, Measured and Predicted at Mach 1.95, 

30 
40 60 100 200 600 2000

0

0.5

1

1.5

2

Measured φ(ξ1,0,f=const) M=1.95 Alt=55400 ft

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

W1

40 60 100 200 600 2000

−3.14−1.57

 0

 1.57 3.14

ξ1, mm

Phase(φ), Radians

40 60 100 200 600 

0

0.5

1

1.5

2

Predicted φ(ξ1,0,f=const) M=1.95 Alt=55400 ft

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

40 60 100 200 600 

−3.14−1.57

 0

 1.57 3.14

Phase(φ), Radians

ξ1, mm

Figure 29 Cross-Spectra - Longitudinal, Window Blank 1, Measured and Predicted at Mach 1.95, 6

31 
60 90 150

0

0.5

1

1.5

Measured φ(0,ξ2,f=const) M=0.74 Alt=16400

Mag(φ(0,ξ

2

,f)),Pa2/Hz

ξ2, mm

W1

60 90 150

0

0.5

1

1.5

Predicted φ(0,ξ2,f=const) M=0.74 Alt=16400

Mag(φ(0,ξ

2

,f)),Pa2/Hz

ξ2, mm

60 90 150

0

0.5

1

1.5

Predicted φ(0,ξ2,f=const) M=0.74 Alt=16400

Mag(φ(0,ξ

2

,f)),Pa2/Hz

ξ2, mm

60 90 150

0

0.5

1

1.5

Measured φ(0,ξ2,f=const) M=0.74 Alt=16400

Mag(φ(0,ξ

2

,f)),Pa2/Hz

ξ2, mm

 Figure 30 Cross-Spectra - Lateral, Window Blank 1, Measured and Predicted at Mach 

32 
60 90 150

0

0.5

1

1.5

Measured φ(0,ξ2,f=const) M=1.95 Alt=55400

Mag(φ(0,ξ

2

,f)),Pa2/Hz

ξ2, mm

60 90 150

0

0.5

1

1.5

Predicted φ(0,ξ2,f=const) M=1.95 Alt=55400

Mag(φ(0,ξ

2

,f)),Pa2/Hz

ξ2, mm

60 90 150

0

0.5

1

1.5

Predicted φ(0,ξ2,f=const) M=1.95 Alt=55400

Mag(φ(0,ξ

2

,f)),Pa2/Hz

ξ2, mm

60 90 150

0

0.5

1

1.5

Measured φ(0,ξ2,f=const) M=1.95 Alt=55400

Mag(φ(0,ξ

2

,f)),Pa2/Hz

ξ2, mm

 Figure 31 Cross-Spectra - Lateral, Window Blank 1, Measured and Predicted at Mach 

33 
34 

5.2.2.2 Cross-spectra at Center Window Blank Cluster 

Figure 32 through Figure 46 show cross-spectral values plotted as a function of longitudinal 

(stream-wise) separation distance obtained at window blanks 3 through 5, and at transducers S1 

and S2 between windows. The measured narrow band cross spectral real and imaginary parts 

were averaged over third octave bands. Measured and predicted (using Efimtsov’s model) magnitude and phase are shown for three different frequency regions on three separate graphs for each 

of five Mach numbers. We chose not to normalize the data (by dividing by the rms value for the 

selected frequency band) in order to provide magnitude comparisons between frequencies and between Mach numbers. Here are some observations: 

• Comparisons of measured with predicted data is more favorable at Mach numbers away from 

Mach 1 (0.74 and 1.95) than for Mach 0.88 and 1.2. 

• Predicted magnitude is usually higher than measured at smaller separations. Regarding decay 

rates, predicted magnitude decreases with increasing separation faster than measured. 

• The Mach 1.6 case (starting at Figure 41) shows abnormally high cross-spectral values. The 

measured low frequency phase behaves much more like the predicted than at other Mach 

numbers. We have no explanation for this occurrence. 

• Measured cross spectral values often peak around 500 Hz for all separation distances. This is 

in agreement with predictions. 

• The measured phase behaves much more erratically than the predicted one in the low frequencies. In the mid frequencies, measured and predicted phases often compare well. As one 

would expect, phase behavior at the higher frequencies is hard to discern. 

Figure 47 and Figure 48 show cross-spectral values (magnitude only) plotted as a function of lateral (cross-stream) separation distance obtained at window blank 4. Only two such separations 

were available: 60 and 120 mm. Some observations: 

• At Mach 0.74, the magnitude is much smaller than predicted, whereas it is comparable at Mach 

2. 

• The slopes with separation distance are generally comparable to the prediction for Mach 0.74, 

but not for Mach 1.95. 

• At Mach 1.95, the prediction almost independent of frequency; the measured data shows a 

definite frequency dependence. 

• Measured and predicted data both exhibit a trend towards lower values with increasing Mach 

number. 

No attempt was made to improve upon the cross functional model. 
60 200 600 2000

0

0.5

1

1.5

2

Measured φ(ξ1,0,f=const) M=0.74 Alt=16400

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

60 200 600 2000

−3.14−1.57

 0

 1.57 3.14

ξ1, mm

Phase(φ), Radians

60 200 600 

0

0.5

1

1.5

2

Predicted φ(ξ1,0,f=const) M=0.74 Alt=16400

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

60 200 600 

−3.14−1.57

 0

 1.57 3.14

Phase(φ), Radians

ξ1, mm

Figure 32 Cross-Spectra - Longitudinal, Central Window Blanks, Measured and Predicted at Mach 0

35 
60 200 600 2000

0

0.5

1

1.5

2

Measured φ(ξ1,0,f=const) M=0.74 Alt=16400

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

60 200 600 2000

−3.14−1.57

 0

 1.57 3.14

ξ1, mm

Phase(φ), Radians

60 200 600 

0

0.5

1

1.5

2

Predicted φ(ξ1,0,f=const) M=0.74 Alt=16400

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

60 200 600 

−3.14−1.57

 0

 1.57 3.14

Phase(φ), Radians

ξ1, mm

Figure 33 Cross-Spectra - Longitudinal, Central Window Blanks, Measured and Predicted at Mach 0.

36 
60 200 600 2000

0

0.5

1

1.5

2

Measured φ(ξ1,0,f=const) M=0.74 Alt=16400

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

60 200 600 2000

−3.14−1.57

 0

 1.57 3.14

ξ1, mm

Phase(φ), Radians

60 200 600 

0

0.5

1

1.5

2

Predicted φ(ξ1,0,f=const) M=0.74 Alt=16400

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

60 200 600 

−3.14−1.57

 0

 1.57 3.14

Phase(φ), Radians

ξ1, mm

Figure 34 Cross-Spectra - Longitudinal, Central Window Blanks, Measured and Predicted at Mach 0.7

37 
60 200 600 2000

0

0.5

1

1.5

2

Measured φ(ξ1,0,f=const) M=0.877 Alt=29500

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

60 200 600 2000

−3.14−1.57

 0

 1.57 3.14

ξ1, mm

Phase(φ), Radians

60 200 600 

0

0.5

1

1.5

2

Predicted φ(ξ1,0,f=const) M=0.877 Alt=29500

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

60 200 600 

−3.14−1.57

 0

 1.57 3.14

Phase(φ), Radians

ξ1, mm

Figure 35 Cross-Spectra - Longitudinal, Central Window Blanks, Measured and Predicted at Mach 0

38 
60 200 600 2000

0

0.5

1

1.5

2

Measured φ(ξ1,0,f=const) M=0.877 Alt=29500

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

60 200 600 2000

−3.14−1.57

 0

 1.57 3.14

ξ1, mm

Phase(φ), Radians

60 200 600 

0

0.5

1

1.5

2

Predicted φ(ξ1,0,f=const) M=0.877 Alt=29500

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

60 200 600 

−3.14−1.57

 0

 1.57 3.14

Phase(φ), Radians

ξ1, mm

Figure 36 Cross-Spectra - Longitudinal, Central Window Blanks, Measured and Predicted at Mach 0.

39 
60 200 600 2000

0

0.5

1

1.5

2

Measured φ(ξ1,0,f=const) M=0.877 Alt=29500

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

60 200 600 2000

−3.14−1.57

 0

 1.57 3.14

ξ1, mm

Phase(φ), Radians

60 200 600 

0

0.5

1

1.5

2

Predicted φ(ξ1,0,f=const) M=0.877 Alt=29500

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

60 200 600 

−3.14−1.57

 0

 1.57 3.14

Phase(φ), Radians

ξ1, mm

Figure 37 Cross-Spectra - Longitudinal, Central Window Blanks, Measured and Predicted at Mach 0.8

40 
60 200 600 2000

0

0.5

1

1.5

2

Measured φ(ξ1,0,f=const) M=1.2 Alt=41000

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

60 200 600 2000

−3.14−1.57

 0

 1.57 3.14

ξ1, mm

Phase(φ), Radians

60 200 600 

0

0.5

1

1.5

2

Predicted φ(ξ1,0,f=const) M=1.2 Alt=41000

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

60 200 600 

−3.14−1.57

 0

 1.57 3.14

Phase(φ), Radians

ξ1, mm

Figure 38 Cross-Spectra - Longitudinal, Central Window Blanks, Measured and Predicted at Mach 1

41 
60 200 600 2000

0

0.5

1

1.5

2

Measured φ(ξ1,0,f=const) M=1.2 Alt=41000

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

60 200 600 2000

−3.14−1.57

 0

 1.57 3.14

ξ1, mm

Phase(φ), Radians

60 200 600 

0

0.5

1

1.5

2

Predicted φ(ξ1,0,f=const) M=1.2 Alt=41000

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

60 200 600 

−3.14−1.57

 0

 1.57 3.14

Phase(φ), Radians

ξ1, mm

Figure 39 Cross-Spectra - Longitudinal, Central Window Blanks, Measured and Predicted at Mach 1

42 
60 200 600 2000

0

0.5

1

1.5

2

Measured φ(ξ1,0,f=const) M=1.2 Alt=41000

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

60 200 600 2000

−3.14−1.57

 0

 1.57 3.14

ξ1, mm

Phase(φ), Radians

60 200 600 

0

0.5

1

1.5

2

Predicted φ(ξ1,0,f=const) M=1.2 Alt=41000

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

60 200 600 

−3.14−1.57

 0

 1.57 3.14

Phase(φ), Radians

ξ1, mm

Figure 40 Cross-Spectra - Longitudinal, Central Window Blanks, Measured and Predicted at Mach 1.

43 
60 200 600 2000

0

0.5

1

1.5

2

Measured φ(ξ1,0,f=const) M=1.6 Alt=45000

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

60 200 600 2000

−3.14−1.57

 0

 1.57 3.14

ξ1, mm

Phase(φ), Radians

60 200 600 

0

0.5

1

1.5

2

Predicted φ(ξ1,0,f=const) M=1.6 Alt=45000

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

60 200 600 

−3.14−1.57

 0

 1.57 3.14

Phase(φ), Radians

ξ1, mm

Figure 41 Cross-Spectra - Longitudinal, Central Window Blanks, Measured and Predicted at Mach 1

44 
60 200 600 2000

0

0.5

1

1.5

2

Measured φ(ξ1,0,f=const) M=1.6 Alt=45000

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

60 200 600 2000

−3.14−1.57

 0

 1.57 3.14

ξ1, mm

Phase(φ), Radians

60 200 600 

0

0.5

1

1.5

2

Predicted φ(ξ1,0,f=const) M=1.6 Alt=45000

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

60 200 600 

−3.14−1.57

 0

 1.57 3.14

Phase(φ), Radians

ξ1, mm

Figure 42 Cross-Spectra - Longitudinal, Central Window Blanks, Measured and Predicted at Mach 1

45 
60 200 600 2000

0

0.5

1

1.5

2

Measured φ(ξ1,0,f=const) M=1.6 Alt=45000

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

60 200 600 2000

−3.14−1.57

 0

 1.57 3.14

ξ1, mm

Phase(φ), Radians

60 200 600 

0

0.5

1

1.5

2

Predicted φ(ξ1,0,f=const) M=1.6 Alt=45000

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

60 200 600 

−3.14−1.57

 0

 1.57 3.14

Phase(φ), Radians

ξ1, mm

Figure 43 Cross-Spectra - Longitudinal, Central Window Blanks, Measured and Predicted at Mach 1.

46 
60 200 600 2000

0

0.5

1

1.5

2

Measured φ(ξ1,0,f=const) M=1.95 Alt=55400

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

60 200 600 2000

−3.14−1.57

 0

 1.57 3.14

ξ1, mm

Phase(φ), Radians

60 200 600 

0

0.5

1

1.5

2

Predicted φ(ξ1,0,f=const) M=1.95 Alt=55400

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

60 200 600 

−3.14−1.57

 0

 1.57 3.14

Phase(φ), Radians

ξ1, mm

Figure 44 Cross-Spectra - Longitudinal, Central Window Blanks, Measured and Predicted at Mach 1

47 
60 200 600 2000

0

0.5

1

1.5

2

Measured φ(ξ1,0,f=const) M=1.95 Alt=55400

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

60 200 600 2000

−3.14−1.57

 0

 1.57 3.14

ξ1, mm

Phase(φ), Radians

60 200 600 

0

0.5

1

1.5

2

Predicted φ(ξ1,0,f=const) M=1.95 Alt=55400

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

60 200 600 

−3.14−1.57

 0

 1.57 3.14

Phase(φ), Radians

ξ1, mm

Figure 45 Cross-Spectra - Longitudinal, Central Window Blanks, Measured and Predicted at Mach 1.

48 
60 200 600 2000

0

0.5

1

1.5

2

Measured φ(ξ1,0,f=const) M=1.95 Alt=55400

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

60 200 600 2000

−3.14−1.57

 0

 1.57 3.14

ξ1, mm

Phase(φ), Radians

60 200 600 

0

0.5

1

1.5

2

Predicted φ(ξ1,0,f=const) M=1.95 Alt=55400

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

60 200 600 

−3.14−1.57

 0

 1.57 3.14

Phase(φ), Radians

ξ1, mm

Figure 46 Cross-Spectra - Longitudinal, Central Window Blanks, Measured and Predicted at Mach 1.9

49 
60 120

0

0.20.40.60.8

1

Measured φ(0,ξ2,f=const) M=0.74 Alt=16400

Mag(φ(0,ξ

2

,f)),Pa2/Hz

ξ2, mm

60 120

0

0.20.40.60.8

1

Measured φ(0,ξ2,f=const) M=0.74 Alt=16400

Mag(φ(0,ξ

2

,f)),Pa2/Hz

ξ2, mm

60 120

0

0.20.40.60.8

1

Predicted φ(0,ξ2,f=const) M=0.74 Alt=16400

Mag(φ(0,ξ

2

,f)),Pa2/Hz

ξ2, mm

60 120

0

0.20.40.60.8

1

Predicted φ(0,ξ2,f=const) M=0.74 Alt=16400

Mag(φ(0,ξ

2

,f)),Pa2/Hz

ξ2, mm

 Figure 47 Cross-Spectra - Lateral, Central Window Blanks, Measured and Predicted at Ma

50 
 Figure 48 Cross-Spectra - Lateral, Central Window Blanks, Measured and Predicted at Ma

51 
52 

5.2.2.3 Cross-Spectra at Aft Window Blank 

Figure 49 through Figure 54 contain cross-spectral values for longitudinal separations. At Mach 

0.74, measured and predicted values (magnitude and phase) are comparable; measured decay 

rates are shallower. At Mach 1.95, the measured low frequency magnitudes for the smallest separation (40mm) are exceptionally high. At higher frequencies the range of values is much wider 

than the predicted ones. Phases compare very well except at the low frequencies. 

Figure 55 and Figure 56 show cross-spectral values (magnitude only) for lateral separations. At 

Mach 0.74, the low frequency magnitudes and decay rates are comparable; high frequency measured magnitudes are significantly lower. As noted before, at Mach 1.95, the prediction is almost 

independent of frequency in contrast to the measurements. 
40 60 100 200 600 2000

0

0.5

1

1.5

2

Measured φ(ξ1,0,f=const) M=0.74 Alt=16400 ft

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

W7

40 60 100 200 600 2000

−3.14−1.57

 0

 1.57 3.14

ξ1, mm

Phase(φ), Radians

40 60 100 200 600 

0

0.5

1

1.5

2

Predicted φ(ξ1,0,f=const) M=0.74 Alt=16400 f

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

40 60 100 200 600 

−3.14−1.57

 0

 1.57 3.14

Phase(φ), Radians

ξ1, mm

 Figure 49 Cross-Spectra - Longitudinal, Window Blank 7, Measured and Predicted at Mach 0.74

53 
40 60 120 200 600 2000

0

0.5

1

1.5

2

Measured φ(ξ1,0,f=const) M=0.74 Alt=16400 ft

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

W7

40 60 120 200 600 2000

−3.14−1.57

 0

 1.57 3.14

ξ1, mm

Phase(φ), Radians

40 60 120 200 600 

0

0.5

1

1.5

2

Predicted φ(ξ1,0,f=const) M=0.74 Alt=16400 f

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

40 60 120 200 600 

−3.14−1.57

 0

 1.57 3.14

Phase(φ), Radians

ξ1, mm

 Figure 50 Cross-Spectra - Longitudinal, Window Blank 7, Measured and Predicted at Mach 0.74, 

54 
40 60 120 200 600 2000

0

0.5

1

1.5

2

Measured φ(ξ1,0,f=const) M=0.74 Alt=16400 ft

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

W7

40 60 120 200 600 2000

−3.14−1.57

 0

 1.57 3.14

ξ1, mm

Phase(φ), Radians

40 60 120 200 600 

0

0.5

1

1.5

2

Predicted φ(ξ1,0,f=const) M=0.74 Alt=16400 ft

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

40 60 120 200 600 

−3.14−1.57

 0

 1.57 3.14

Phase(φ), Radians

ξ1, mm

 Figure 51 Cross-Spectra - Longitudinal, Window Blank 7, Measured and Predicted at Mach 0.74, 

55 
40 60 100 200 600 2000

0

0.5

1

1.5

2

Measured φ(ξ1,0,f=const) M=1.95 Alt=55400 ft

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

W7

40 60 100 200 600 2000

−3.14−1.57

 0

 1.57 3.14

ξ1, mm

Phase(φ), Radians

40 60 100 200 600 

0

0.5

1

1.5

2

Predicted φ(ξ1,0,f=const) M=1.95 Alt=55400 f

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

40 60 100 200 600 

−3.14−1.57

 0

 1.57 3.14

Phase(φ), Radians

ξ1, mm

 Figure 52 Cross-Spectra - Longitudinal, Window Blank 7, Measured and Predicted at Mach 1.95

56 
40 60 100 200 600 2000

0

0.5

1

1.5

2

Measured φ(ξ1,0,f=const) M=1.95 Alt=55400 ft

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

W7

40 60 100 200 600 2000

−3.14−1.57

 0

 1.57 3.14

ξ1, mm

Phase(φ), Radians

40 60 100 200 600 

0

0.5

1

1.5

2

Predicted φ(ξ1,0,f=const) M=1.95 Alt=55400 f

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

40 60 100 200 600 

−3.14−1.57

 0

 1.57 3.14

Phase(φ), Radians

ξ1, mm

 Figure 53 Cross-Spectra - Longitudinal, Window Blank 7, Measured and Predicted at Mach 1.95, 

57 
40 60 100 200 600 2000

0

0.5

1

1.5

2

Measured φ(ξ1,0,f=const) M=1.95 Alt=55400 ft

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

W7

40 60 100 200 600 2000

−3.14−1.57

 0

 1.57 3.14

ξ1, mm

Phase(φ), Radians

40 60 100 200 600 

0

0.5

1

1.5

2

Predicted φ(ξ1,0,f=const) M=1.95 Alt=55400 ft

Mag(φ(ξ

1

,0,f)),Pa

2

/Hz

ξ1, mm

40 60 100 200 600 

−3.14−1.57

 0

 1.57 3.14

Phase(φ), Radians

ξ1, mm

 Figure 54 Cross-Spectra - Longitudinal, Window Blank 7, Measured and Predicted at Mach 1.95, 

58 
60 90 150

0

0.5

1

1.5

Measured φ(0,ξ2,f=const) M=0.74 Alt=16400

Mag(φ(0,ξ

2

,f)),Pa2/Hz

ξ2, mm

60 90 150

0

0.5

1

1.5

Predicted φ(0,ξ2,f=const) M=0.74 Alt=16400

Mag(φ(0,ξ

2

,f)),Pa2/Hz

ξ2, mm

60 90 150

0

0.5

1

1.5

Predicted φ(0,ξ2,f=const) M=0.74 Alt=16400

Mag(φ(0,ξ

2

,f)),Pa2/Hz

ξ2, mm

60 90 150

0

0.5

1

1.5

Measured φ(0,ξ2,f=const) M=0.74 Alt=16400

Mag(φ(0,ξ

2

,f)),Pa2/Hz

ξ2, mm

 Figure 55 Cross-Spectra - Lateral, Window Blank 7, Measured and Predicted at Mach 

59 
60 90 150

0

0.5

1

1.5

Measured φ(0,ξ2,f=const) M=1.95 Alt=55400

Mag(φ(0,ξ

2

,f)),Pa2/Hz

ξ2, mm

60 90 150

0

0.5

1

1.5

Predicted φ(0,ξ2,f=const) M=1.95 Alt=55400

Mag(φ(0,ξ

2

,f)),Pa2/Hz

ξ2, mm

60 90 150

0

0.5

1

1.5

Predicted φ(0,ξ2,f=const) M=1.95 Alt=55400

Mag(φ(0,ξ

2

,f)),Pa2/Hz

ξ2, mm

60 90 150

0

0.5

1

1.5

Measured φ(0,ξ2,f=const) M=1.95 Alt=55400

Mag(φ(0,ξ

2

,f)),Pa2/Hz

ξ2, mm

 Figure 56 Cross-Spectra - Lateral, Window Blank 7, Measured and Predicted at Mach 

60 
6. Efimtsov Auto Spectrum Model Adjustments

Comparisons of the measured flight data and the predictions from the Efimtsov model showed two 

characteristics which the Efimtsov model did not predict. The first was a broad band spectral peak 

around a Strouhal number of 0.6 where the Strouhal number is calculated as 

*

0

2 f

U

π δ . The measured data also showed a slightly steeper roll-off at high frequencies (above 1 kHz) than the predicted roll-off. 

6.1 Broad Peak 

The broad band peak’s magnitude is of the order of 2 dB. In some of the data it is obvious and 

prominent, in some of the data it is non-existent. Here are arguments for the presence of the 

broad peak: 

• Intuitively, one would expect that certain frequency regions would contribute more strongly to 

turbulence energies according to the length scales imposed on the flow by the boundary layer 

thickness; such behavior has been shown to exist for free shear layers in jet plumes. The 

measured data’s behavior supports this reasoning by exhibiting its spectral peak at a constant 

Strouhal number. 

• Schewe [Ref. 16] shows a figure (here reproduced as Figure 57) with spectra that exhibit the 

broad spectral peak. 

 Figure 57 Excerpt from Ref. 16: ‘Corrected power spectral densities ...’

• The scatter in the low frequency auto spectral data can be attributed - at least in part - to transducers not being completely flush mounted. Efimtsov’s [Ref. 8] wind tunnel data indicates that 

the probability is higher of measuring a higher level due to flushness deviations than one below 

the ‘true’ value. One might therefore believe the lower portion of the scattered points more, 

leading to the broad peak asserting its presence. 

61 
• The low frequency data for window blanks in the back of the aircraft may be contaminated by 

engine noise, air as well as structure borne, leading to higher measured levels than would be 

measured in a ‘pure’ boundary layer. 

Here are arguments against the presence of the broad peak: 

• The broad peak’s magnitude is small; it is within the scatter of the data. 

• The majority of past models exhibit no broad peaks (the exception is the ESDU model). 

To the authors, the arguments ‘for’ the peak were more compelling. Therefore, a procedure was 

developed to add an appropriate adjustment to Efimtsov’s model. The peak occurs in a frequency 

region which is weighted strongly in human response to noise metrics for interior noise; it is therefore important to model loads there as accurately as possible. 

6.2 High Frequency Roll-Off 

The Efimtsov model predicts a much shallower high frequency negative slope than all other models, as well as the TU-144LL data. We decided to adjust the Efimtsov model to follow the TU144LL after correcting for transducer size (see Section 4, ‘Data Correction’). 

6.3 Efimtsov Adjustment Functions 

Two analytical functions were used to adjust the predicted curves to more accurately reproduce the 

measured SPL spectra. 

The broadband peak was modeled by a log-normal distribution equation: 

( ( ) ( ) ( ) ) 2

1 1 C S exp − − log t log St

where St is the Strouhal number, and St1 is a reference value. 

To account for the slope of the high frequency roll-off, another function is added to the prediction 

equation. The Efimtsov model over-predicts for flight conditions below Mach 1.65 and under -

predicts for conditions above 1.65 which leads to the following adjustment function: 

( 1 M − .65)log10 ( f )

Since only the high frequency slope needs adjustment, a tanh function centered at 1000 Hz is included. The final adjustment equation is: 

( ) ( )

( ) ( ) ( ) ( )

2 *

10 10

0

2 2.5exp ln ln 0.6 1 tanh log 1 1.65 log 4 1000

adjusted predicted f f

f f M f ( ) U

πδ

Φ − Φ =

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ − − ⎜ ⎟ ⎜ ⎟ + ⎜ ⎟ ⎜ ⎟ + − ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

Figure 58 shows an example of this function. 

62 
0

0.5

1

1.5

2

2.5

3

10 100 1000 10000

Frequency, Hz

Adjustment, dB

Broad Peak

Slope

Together

 Figure 58 Sample Adjustments for Mach 2, * δ =1ft , U0=1800 fps. 

Figure 59 shows the original Efimtsov model and the adjusted model for Mach 1.95 55,400ft flight 

condition as well as measured flight test data. Other conditions are shown in Figure 60 - Figure 

64. 

6.4 Comparisons of Adjusted Efimtsov Prediction to Measured TU-144LL Data 

Some observations: 

• The adjusted model follows the measured data better on forward and center window blanks 

than on the aft. 

• The high frequency roll-off of the data is well modeled, except that the model does not follow a 

data trend where further aft sensors exhibit a greater high frequency slope, crossing over the 

spectra of further forward sensors. 

• At low frequencies, no adjustments were made to the Efimtsov model because there was no 

trend to be found in the widely scattered data. 

63 
101 102 103 104 80

85

90

95

100

105

Frequency, Hz

Φ(f), dB, (ref: 2x10−5Pa)

Predicted and Measured SPL Data, Mach 1.95, Alt=55,400 ft

Measured WB 7

Efimtsov Predicted WB 7

Efimtsov Adjusted WB 7

Measured Middle

Efimtsov Predicted WB 4

Efimtsov Adjusted WB 4

Measured Forward

Efimtsov Predicted Forward

Efimtsov Adjusted Forward

 Figure 59 Adjusted Efimtsov Model and In-Flight SPL Data (M 1.95, 55,400 ft, Window

Blanks 1, 4, and 7) 

64 
101 102 103 104 80

85

90

95

100

105

Frequency, Hz

Φ(f), dB, (ref: 2x10−5Pa)

Predicted and Measured SPL Data, Mach 0.74, Alt=16400 ft

Measured Window Blank 7

Predicted Window Blank 7

Measured Window Blank 4

Predicted Window Blank 4

Measured Window Blank 1

Predicted Window Blank 1

 Figure 60 Adjusted Efimtsov Model and Tu144 In-Flight SPL Data (M 0.74, 16,400 ft, Window Blanks 1, 4, and 7) 

65 
101 102 103 104 80

85

90

95

100

105

Frequency, Hz

Φ(f), dB, (ref: 2x10−5Pa)

Predicted and Measured SPL Data, Mach 0.877, Alt=29500 ft

Measured Window Blank 7

Predicted Window Blank 7

Measured Window Blank 4

Predicted Window Blank 4

Measured Window Blank 1

Predicted Window Blank 1

 Figure 61 Adjusted Efimtsov Model and Tu144 In-Flight SPL Data (M 0.88, 29,500 ft, Window Blanks 1, 4, and 7) 

66 
101 102 103 104 80

85

90

95

100

105

Frequency, Hz

Φ(f), dB, (ref: 2x10−5Pa)

Predicted and Measured SPL Data, Mach 1.2, Alt=41000 ft

Measured Window Blank 7

Predicted Window Blank 7

Measured Window Blank 4

Predicted Window Blank 4

Measured Window Blank 1

Predicted Window Blank 1

 Figure 62 Adjusted Efimtsov Model and Tu144 In-Flight SPL Data (M 1.2, 41,000 ft, Window

Blanks 1, 4, and 7) 

67 
101 102 103 104 80

85

90

95

100

105

Frequency, Hz

Φ(f), dB, (ref: 2x10−5Pa)

Predicted and Measured SPL Data, Mach 1.6, Alt=45000 ft

Measured Window Blank 7

Predicted Window Blank 7

Measured Window Blank 4

Predicted Window Blank 4

Measured Window Blank 1

Predicted Window Blank 1

 Figure 63 Adjusted Efimtsov Model and Tu144 In-Flight SPL Data (M 1.6, 45,000 ft, Window

Blanks 1, 4, and 7) 

68 
101 102 103 104 80

85

90

95

100

105

Frequency, Hz

Φ(f), dB, (ref: 2x10−5Pa)

Predicted and Measured SPL Data, Mach 1.95, Alt=55400 ft

Measured Window Blank 7

Predicted Window Blank 7

Measured Window Blank 4

Predicted Window Blank 4

Measured Window Blank 1

Predicted Window Blank 1

 Figure 64 Adjusted Efimtsov Model and Tu144 In-Flight SPL Data (M 1.95, 55,400 ft, Window Blanks 1, 4, and 7) 

69 
6.5 Comparison to Other Flight Test Data 

A comparison of other measured data could be used in order to determine if the adjustments are 

specific to the NASA-Boeing Tu144LL test or if the adjustments could be applied to all flight vehicles. The existing supersonic flight data is limited to the Concorde [Ref. 9], and XB-70 [Ref. 8]. 

Several sources of wind tunnel data exist but will likely be of limited use for direct comparison purposes. 

Figure 65 shows the adjusted Efimtsov predicted SPL and those measured on the Concorde [Ref. 

9]. A broad spectral peak seems to be present in the Concorde data as well. The gross outlier in 

window blank 2 probably needs to be attributed to a flushness problem. If the Corcos correction 

were to be applied to the Concorde data the measured and predicted high frequency roll-offs would 

match a lot better. 

70 
101 102 103 104 80

85

90

95

100

105

Frequency, Hz

Φ(f), dB, (ref: 2x10−5Pa)

Concorde Predicted and Measured SPL Data, Mach 1.98, Alt=52300 ft

No Corcos Correction Added to Measured Data

Measured Window Blank 4

Predicted Window Blank 4

Measured Window Blank 3

Predicted Window Blank 3

Measured Window Blank 2

Predicted Window Blank 2

Measured Window Blank 1

Predicted Window Blank 1

Figure 65 Adjusted Efimtsov Predicted SPL and Measured Flight Test Data from Concorde 

(Mach 2) 

71 
7. Consequences of Model Adjustments to HSCT Interior Noise 

What follows is a qualitative assessment of what changes one might expect in HSCT interior noise 

predictions that are based on the Efimtsov model: 

• The broad spectral peak is added in a frequency range roughly from 200 to 3000 Hz. In the 

higher end of this range the human ear is particularly sensitive. In the lower end of that range, 

fuselage structures often exhibit coincidence effects (particularly honeycomb) or panel resonances (particularly skin/stringer). 

• The small change of the high frequency roll-off slope decreases the high frequency portion at 

Mach numbers below 1.65, and increases it above that Mach number. 

Without the benefit of a detailed quantitative analysis, one might engage in educated speculation 

that high Mach number cruise interior levels will increase somewhere between 1 to 3 dB. 

For most of the length of the fuselage (middle and aft) the model adjustment adds noise energy 

mostly at frequencies below 1000 Hz, regardless of Mach number. This is the area where active 

noise control shows the most promise. 

8. Recommendations for Future Work 

The following items should be considered for future work: 

• Auto-Spectra: 

• Use Boeing data (777, MD-90) to further evaluate subsonic performance of the adjusted 

model. 

• Explanation of higher than currently predicted levels in the aft portion of the fuselage, 

found in Concorde as well as TU-144 measurements. Possibly further adjust the 

model. 

• Use A. Frendi’s numerical prediction method to investigate the high Reynolds number 

and the thick boundary layer situation on the aft fuselage. 

• Include in the model effects of pressure gradients on the boundary layer, particularly the 

negative gradient on the tail cone section. 

• Cross-Spectra: 

• Correct the measured cross-functions for the influence of local flow direction (measured 

on the TU-144 in 1999 [Ref. 18]) 

• Adjust the cross-functional model using measured cross-functions using more of the 

TU-144 data than presented here 

• Use ideas from Smol’yakov-Tkachenko and further develop a model for high Reynolds and 

Mach numbers without the assumption of multiplicative separability of the functions of the time 

and space separation variables. 

72 
73 

9. References 

Ref. 1 "Prediction of In-Flight Fluctuating Pressure Environments Including Protuberance Induced Flow" Wyle Laboratories Research Staff Report, WR71-10, (1971). 

Ref. 2 "Estimation of the Surface Pressure Fluctuations in the TBL of a Flight Vehicle," Engineering Sciences Data Unit, Item Number 75021, (1975). 

Ref. 3 Laganelli, A. L., Wolfe, H., "Prediction of Fluctuating Pressure in Attached and Separated TBL Flow," AIAA 12th Aeroacoustics Conference, AIAA Paper 89-1064, (1989). 

Ref. 4 Goodwin, P. W., "An In-Flight Supersonic TBL Surface Pressure Fluctuation Model", 

Contract No. NAS1-20013, Boeing Document No. D6-81571 (1994). 

Ref. 5 Efimtsov, B. M., "The Prediction of the Pressure Fluctuation Field Characteristics of the 

TBL", Contract No. BTRC-101F, Document No. BTRC101F-5/95, (1995) 

Ref. 6 Blake, W. K., Mechanics of Flow-Induced Sound and Vibration, Academic Press, Inc., 

New York, 1986. 

Ref. 7 Bies, D. A., "A Review of Flight and Wind Tunnel Measurements of Boundary Layer 

Pressure Fluctuations and Induced Structural Response", NASA CR-626, (1965). 

Ref. 8 Efimtsov, B. M., Munin, A. G., "Determination of Optimal Fluctuation Pressure Transducer Mount Configuration and Frequency Response for Tu144LL" proprietary report by 

Central Aerohydrodynamic Institute (TsAGI) in Moscow, Russia, to The Boeing Company, August 1996. 

Ref. 9 Lohman, K. T., "Cabin Noise and Turbulent Boundary Layer Pressure Fluctuations on a 

British Airways Concorde Aircraft – Test Report", Boeing Document No. T6-6970, (1994) 

Ref. 10 Rizzi, S.A., Rackl, R.G., Andrianov, E.V., “Flight Test Measurements from the Tu-144LL 

Structure/Cabin Noise Experiment, NASA TM-2000-209858, January 2000. 

Ref. 11 Efimtsov, B.M., Personal Communication, October 1998. 

Ref. 12 Duff, G.F.D., Naylor, D., “Differential Equations of Applied Mathematics”, Wiley 1966, 

Fourier transform tables, p.411 ff. 

Ref. 13 Efimtsov, B.M., “Characteristics of the field of turbulent wall pressure fluctuations at large 

Reynolds numbers”, Sov. Phys. Acoust. 28(4), July-August 1982, pp 289-292. 

Ref. 14 Corcos, G.M., “Resolution of Pressure in Turbulence”, J. Acoust. Soc. Amer. 35(2), February 1963. 

Ref. 15 Spalart, P. (Boeing Associate Technical Fellow), Personal Communications, September/October 1998. 

Ref. 16 Schewe, G., “On the structure and resolution of wall-pressure fluctuations associated 

with turbulent boundary-layer flow,” J. Fluid Mech. (1983), vol. 134, pp. 311-328. 

Ref. 17 Smol’yakov, A.V., Tkachenko, V.M., “Model of a field of pseudosonic turbulent wall pressures and experimental data”, Sov. Phys. Acoust. 37(6), Nov.-Dec. 1991. 

Ref. 18 Rizzi, S.A., Rackl, R.G., Andrianov, E.V., “Flight Test Measurements from the Tu-144LL 

Structure/Cabin Noise Follow-On Experiment, NASA TM-2000-209859, February 2000. 
( )( ( )( ( )( ( ))))

( )( ) ( )( ) ( )( ) ( )

( ) ( ) ( )

10 10 10 10

10 10 10 10

10 10 10

178.441947 log 39.29984 log 3.0984996 log 25.32020 25.18335 log

5.2735 log 0.19002 log 2.29509 log 4.83039 5.93327 log

0.16237 log 0.00510 log 0.01416 log 0.30915 0.378

a

b

c

L M M M ML M M M ML M M M

= + + − + ⋅

= − − + − + ⋅

= − ( ) − ( ) − ( ) + ( )

( ) ( ) ( )

( )

( ( )( ( )( ( )( 

( )( 

( )( 

( ) 

10

10 2

1 10 10

10 *

10

10 10 10 10 10 10

94 log

1481.34 20 log 1

2

log log

10 log

8.24 log

12.84 log 6.95 log 24.75 log 29.30 log 14.26 log 3.17 0.27 log

p

a b c p

u

s

M

L

c

L L Re L Re L L

U L

L St

St St St St St St

ρ

δ

∞

∞

⋅

⎛ ⎞

= × ⎜ ⎟ ⎜ ⎟ ⎝ ⎠

= + × + × −

⎛ ⎞ = × ⎜ ⎟ ⎝ ⎠

= − − ×

+ + + + + + ⋅

Φ = ( ) fi s L + L1 1 − Lu +10×log 0 ( ) bandwidth( fi)

Equations for calculating fluctuating pressure levels derived by ESDU [Ref. 3]: 

74 

10. Appendix A - ESDU Equations 
11. Appendix B - Fourier Transform of a Turbulence Cross-Correlation 

Model 

This work was performed with the expectation of creating an improved cross-spectral pressure 

fluctuation model, but could not be completed due to lack of time. The partial effort is documented 

here for the time when it can be continued. 

When computing cross functions using the TU-144 data, cross-correlations turn out to be nicely 

smooth curves to which a model can be fitted with little uncertainty. Cross-spectra are harder to fit 

since they contain more noise and consist of real and imaginary parts. The approach to modeling 

cross-spectra is therefore to model the cross-correlations and to Fourier transform that model to 

obtain the cross-spectrum. 

One way of modeling a cross-correlation function as a function of time delay τ is shown in Eq. 1. 

This form was chosen such as to be able to fit measured correlations with functions that have relatively simple Fourier transforms. Ref. 12 provides Fourier transform formulas. 

R r e r r r ( ) f

r s

s

s

( ) τ cos ( )

τ τ

τ

π τ τ

τ τ

τ

τ τ

τ = + +

⎛ −

⎝

⎜ ⎞

⎠

⎟ + −

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+

+

⎛ −

⎝

⎜ ⎞

⎠

⎟

−

⎛ −

⎝

⎜ ⎞

⎠

⎟

0

2

1 2

2

2

3 0

4

3

2

1 2

1

    Eq. 1 

Meaning of Symbols: 



τ = time delay of peak of cross-correlation 

τ si = time delay scale factors 

ri = magnitude constants 

f 0 = frequency of spectral peak 

The objective is to find the Fourier transform of this function, defined by 

F R{ } R e d if ( ) τ τ( ) π τ = −

−∞

∞

∫ 2 τ

I

Eq. 2 

Eq. 1 can be re-written: 

R r ( ) I I I I . . τ = +012 + + 3 1 + 3 2 + 4

where: I r e s

1 1

2

1 = −

⎛ −

⎝

⎜ ⎞

⎠

⎟ τ τ

τ



, I r e s

s

2 2

2

2

2

1 = ⎛ −

⎝

⎜ ⎞

⎠

⎟ −

⎛ −

⎝

⎜ ⎞

⎠

⎟ τ τ

τ τ τ

τ

 

I

r

e e s if

3 1

3

2

2

2

1 0

.

( ) = −

⎛ −

⎝

⎜ ⎞

⎠

⎟ −

τ τ

τ π τ τ





, I

r

e e s if

3 2

3

2

2

2

1 0

.

( ) = −

⎛ −

⎝

⎜ ⎞

⎠

⎟ − −

τ τ

τ π τ τ





I

r

s

4

4

3

2

1

=

+

⎛ −

⎝

⎜ ⎞

⎠

⎟ τ τ

τ



Eq. 3 

75 
Each element on the right hand side of Eq. 3 can be Fourier transformed separately: 

F r{ 0 0 } = = r δ ( 0 f ) Eq. 4 

F I { } r e d s

if

1 1

2 1

2

= = −

⎛ −

⎝

⎜ ⎞

⎠

⎟ −

−∞

∞

∫

τ τ

τ π τ

τ



r e d s

x if x s

1 1

2 2 1 τ − − π τ +τ

−∞

∞

∫ ( ) x 

where: x s

=

⎛ −

⎝

⎜ ⎞

⎠

⎟ τ τ

τ



1

, τ = + τ τ s x1



, d dx s τ = τ 1

F I { }1 = r e dx s

x if x if s

1 1

2 2 2 1 τ − − π τ − π τ

−∞

∞

∫ 

= − + + − +

−∞

∞

∫ r e s

x if x if if if s s s

1 1

2 2 2 1 1 2 1 2

τ ( ( π τ π τ ) (π τ ) π τ dx )



= − − − +

−∞

∞

∫ r e e dx s

f if x if s s

1 1

2 2 1

2 2 1 2

τ π τ π τ π τ  ( ) = − − r e s

f if s

1 1

2 2 1

2 2

τ π π τ π τ



= F I { 1} Eq. 5 

F I { } r e d s

if s

2 2

2

2

2 1

2

= ⎛ −

⎝

⎜ ⎞

⎠

⎟ = −

⎛ −

⎝

⎜ ⎞

⎠

⎟ −

−∞

∞

∫ τ τ

τ

τ

τ τ

τ π τ  

r e s

s s

if s

2

1

2

2

1

2

2 1

2

τ

τ

τ τ

τ

τ

τ τ

τ ⎛ π τ

⎝

⎜ ⎞

⎠

⎟ ⎛ −

⎝

⎜ ⎞

⎠

⎟ −

⎛ −

⎝

⎜ ⎞

⎠

⎟ −

−∞

∞

∫ d

 

= ⎛

⎝

⎜ ⎞

⎠

⎟ − −

−∞

∞

∫ r x e dx s

s

s

x if

2

1

2

2

1

2 2 τ 2

τ

τ π τ = ⎛

⎝

⎜ ⎞

⎠

⎟ − − −

−∞

∞

∫ r x e s

s

s

x if x if s

2

1

2

2

1

2 2 2 2 1 τ

τ

τ π τ π τ

dx

= ⎛

⎝

⎜ ⎞

⎠

⎟ − − − +

−∞

∞

∫ r

r

r e x e dx s

s

s

f if x if 2 s s

1

1

2

2

1 1

2 2 2 1

2 2 1 τ 2

τ

τ π τ π τ π τ  ( ) { } = ⎛

⎝

⎜ ⎞

⎠

⎟ r

r

i

d F I

df

s

s

2

1

1

2

2

2

2

1

2

τ

τ

τ π π [ ] ( ) πτ τ

τ

τ π τ π τ  

e f i

df

d

r

r

r s

f if

s

s

s s − − ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = − − − 2

1

2

1 1

2

2

1

1

2 2 2 1 2

2

[( ) ( )] 2

1

2 2

1

2 2

1 1

2

2

1

1

2 2 2 1 2

(2 ) s s

f if

s

s

s r e f i

r

r s τ π π πτ τ πτ

τ

τ π τ π τ − − + − ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = − − −  

= F{I2} Eq. 6 

2

0

1

1,2

2 ( ) 2

3

3 2

s

if if r I e d

τ τ π τ τ π τ τ τ

⎛ ⎞ − ∞ − ± ⎜ ⎟ − −

⎝ ⎠

−∞ = ∫

  2 0 1 1 3 1 2 2 2

2

s s r s x if x if x if e d π τ π τ π τ x

τ ∞ − ± − −

−∞ = ∫ 

( ) 2 1 0 2 ( ) 2 3 1

2

s x i x f f if s r e dx

τ ∞ − + π τ + π τ

−∞ = ∫  ∓ ( ) 2 2 2 2 2 2 2 1 0 1 0 1 0 2 ( ) ( ) ( ) 2 3 1

2

s s s x i x f f f f f f if s r e d π τ π τ π τ π τ

x

τ ∞ − + − + +

−∞ = ∫  ∓ ∓ ∓

( )2 2 2 2 1 0 1 0 3 1 ( ) 2 ( )

2

s s f f if x i f f s r

e e dx

τ π τ π τ π τ ∞ − − − +

−∞ = ∫

 ∓ ∓ 2 2 2 1 0 ( ) 2

3 1 2

s f f if

s r e

π π τ π τ τ − − =  ∓

1,2 3 = I Eq. 7 

{ }

2

4 4 2

3

1

if

s

e F I r d

π τ

τ

τ τ

τ

− ∞

−∞ = ⎛ ⎞ − + ⎜ ⎟ ⎝ ⎠

∫ 

Substitute: x = − τ τ

 , τ = +x τ , d d τ = x

76 
{ }

2

2

4 4 2

3

1

ifx

if

s

e F I r e dx

x

π

π τ

τ

− ∞ −

−∞ = ⎛ ⎞

+ ⎜ ⎟ ⎝ ⎠

∫

 2

2 2

4 3 2 2

3

ifx

if

s

s

e

r e dx

x

π

π τ τ

τ

− ∞ −

−∞ = + ∫  3 2 2

4 3 s if fs r e e π τ πτ πτ − − = 

{ } 3 2 ( )

4 4 3 s f i F I s r e π τ τ πτ − + = 

Eq. 8 

77 
REPORT DOCUMENTATION PAGE Form Approved 

OMB No. 0704-0188 

The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing

data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or 

any other aspect of this collection of information, including suggestions for reducing this burden, to Department of Defense, Washington Headquarters Services, Directorate

for Information Operations and Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302. Respondents should be aware that

notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently

valid OMB control number. 

PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS.

1. REPORT DATE (DD-MM-YYYY) 2. REPORT TYPE 3. DATES COVERED (From - To) 

4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER 

5b. GRANT NUMBER 

5c. PROGRAM ELEMENT NUMBER 

6. AUTHOR(S) 5d. PROJECT NUMBER 

5e. TASK NUMBER 

5f. WORK UNIT NUMBER 

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION

REPORT NUMBER 

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORING/MONITOR'S ACRONYM(S)

11. SPONSORING/MONITORING

REPORT NUMBER 

12. DISTRIBUTION/AVAILABILITY STATEMENT

13. SUPPLEMENTARY NOTES 

14. ABSTRACT

15. SUBJECT TERMS 

16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 

ABSTRACT

18. NUMBER 

OF

PAGES

19a. NAME OF RESPONSIBLE PERSON

a. REPORT b. ABSTRACT c. THIS PAGE

19b. TELEPHONE NUMBER (Include area code) 

Standard Form 298 (Rev. 8-98)

Prescribed by ANSI Std. Z39-18 

01-12-2005 Contractor Report Sep 1997 - Nov 1998

Modeling of Turbulent Boundary Layer Surface Pressure Fluctuation Auto 

and Cross Spectra – Verification and Adjustments Based on TU-144LL Data

NAS1-20220

537-06-37-20

Rackl, Robert

Weston, Adam

The Boeing Company

Seattle, WA

National Aeronautics and Space Administration

Washington, DC 20546-0001

NASA

NASA/CR-2005-213938

Unclassified-Unlimited

Subject Category 71

Availability: NASA CASI (301) 621-0390

Langley Technical Monitor: Stephen A. Rizzi. An electronic version can be found at http://ntrs.nasa.gov

The literature on turbulent boundary layer pressure fluctuations provides several empirical models which were compared to the 

measured TU-144 data. The Efimtsov model showed the best agreement. Adjustments were made to improve its agreement 

further, consisting of the addition of a broad band peak in the mid frequencies, and a minor modification to the high frequency rolloff. The adjusted Efimtsov predicted and measured results are compared for both subsonic and supersonic flight conditions.

Measurements in the forward and middle portions of the fuselage have better agreement with the model than those from the aft 

portion. For High Speed Civil Transport supersonic cruise, interior levels predicted by use of this model are expected to increase 

by 1-3 dB due to the adjustments to the Efimtsov model. The space-time cross-correlations and cross-spectra of the fluctuating

surface pressure were also investigated. This analysis is an important ingredient in structural acoustic models of aircraft interior 

noise. Once again the measured data were compared to the predicted levels from the Efimtsov model.

High speed research; Tupolev 144; Tu-144LL; Supersonic turbulent boundary layer pressure fluctuations

U U U UU 88

STI Help Desk (email: help@sti.nasa.gov)

(301) 621-0390