Calculation of turbulent boundary layer wall pressure spectra

Calculation of turbulent boundary layer wall pressure spectra Dean E. Capone Applied Research Laboratory, Pennsylvania State University, P.O. Box 30, State College, Pennsylvania 16804

Gerald C. kauchle

Graduate Program in Acoustics and Applied Research Laboratory, Pennsylvania State University, P.O. Box 30, State College, Pennsylvania 16804

(Received 19 December 1993; revised 11 February 1995; accepted 30 April 1995)

The suitability of various wave-vector-frequency spectral models for predicting the point wall pressure spectrum due to developed turbulent boundary layer flow over planar, rigid surfaces is investigated. The early Corcos model [G. Corcos, J. Fluid Mech. 18, 353-378 (1964)] along with the incompressible and compressible models developed by Chase [D. Chase, J. Sound Vib. 112, 124-147 (1987)] are given particular attention. The effect of finite-sized measurement transducers is included in the integration of the theoretical wave-vector-frequency spectrum over the in-plane wave numbers in order to arrive at an attenuated point pressure frequency spectrum that can be compared directly to existing experimental data obtained in air, water, and glycerine. The selected experiments cover a wide range of fluid properties and Reynolds numbers. It is found that the Corcos model does not predict the measured data as well as the Chase model. An optimum set of empirical constants needed to exercise the Chase models are presented, but found to depend on the experiment considered. The need to consider viscous effects in future modeling is borne out when dealing with viscous fluids like glycerine. ̧ 1995 Acoustical Society of America.

PACS numbers: 43.30.Lz, 43.50.Ed

INTRODUCTION

The development of a comprehensive model of the wave-vector-frequency spectrum for turbulent boundary layer (TBL) wall pressure spectra has been the focus of considerable work. Modeling the subconvective or low wave- number region of the turbulent boundary layer has proven to be a very complex and unresolved task. This is partially due to a lack of experimental data for this region and partially due to the fact that most of the energy in the boundary layer is concentrated in the convected region. Corcos • developed an empirical model of the turbulent boundary layer wall pressure spectrum, P(k,to), which depends upon the wave vector k and the frequency to. The flow under consideration in the Corcos paper is a homogeneous TBL flow at low Mach number over a smooth, stationary, rigid plane with zero pressure gradient. In 1980, Chase 2 developed a more rigorous model of the wave-vector-frequency spectrum for the incompressible, inviscid domain which includes convective and higher subconvective wave numbers. In 1987, Chase 3 reformulated the model in an attempt to characterize P(k, to) from the convective domain down into the acoustic domain.

When considering the acoustic domain, the effects of com- pressibility of the fluid are taken into consideration.

Models of the wave-vector-frequency spectrum rely on experimental data to help establish values for dimensionless coefficients developed in the theory. Until recent advances in the areas of sensor technology and signal processing, the accurate measurement of boundary layer wall pressure fluctuations had proved difficult. Low-frequency components of the TBL were often masked by the noise of the measurement facility. Often times, due to the large size of the pressure

transducers employed, the high-frequency components were subject to attenuation due to area averaging over the face of the transducer. Because the established theoretical models

rely upon experimental data for implementation, the accuracy of these models, particularly in the high- and low- frequency limits, is affected by the accuracy of the experiment. Because of the advances in sensor technology and signal processing, better experimental estimates of the TBL wall pressure point spectrum are available. Examination of these recent experimental findings suggests that changes in the empirical constants used in the theoretical models may be in order.

Most recent analytical and computational work examin- ing turbulent boundary layer pressure fluctuations has utilized the Corcos wave-vector-frequency spectrum. For example, a paper by Ko and Schloemer 4 examined the transmission of a Corcos turbulent boundary layer pressure spectrum into a viscoelastic layer. Other papers by Ko and Nuttall 5 and Thompson and Montgomery 6 evaluated the response of an array of flush-mounted hydrophones to the Cor- cos turbulent boundary layer pressure spectrum.

Mellen 7 examined the effects of the rhombic Fourier

window used by Corcos in his formulation of the TBL space/ frequency correlation function. The Fourier transform of the space/frequency correlation function yields the Corcos wave- vector-frequency spectrum. Plots of the Corcos wave- vector-frequency spectrum for various azimuthal angles revealed irregular behavior of the spectrum for wave numbers below the convective peak. Calculations of the wave-vector- frequency spectrum resulting from an elliptic windowing on the space/frequency correlation function were also performed. Upon comparison of the results for the elliptic and

2226 J. Acoust. Sec. Am. 98 (4), October 1995 0001-4966/95/98(4)/2226/9/$6.00 ¸ 1995 Acoustical Society of America 2226

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rhombic windows, Mellen concluded that the irregular behavior of the Corcos model in the wave-vector-frequency domain was a result of the rhombic window. In'a following letter, Mellen 8 noted that Chase was the first to propose an elliptic model for the space/frequency spectrum. Mellen ex- tends his comparison to include the effect of rhombic and elliptic windows on measurement of turbulence using wave- vector filter arrays. Comparison of array response as a function of nondimensional frequency for the elliptic and Corcos models shows that the Corcos model predictg higher levels at all frequencies than the elliptic model. Mellen concludes that although the Corcos model has been more widely used, due to its simplicity, the elliptic model is preferable.

It is the purpose of this paper to evaluate the suitability of applying various prediction methods for TBL point wall pressure spectra. At the time this work was started, the Corcos, • Chase, 2'3 Witting, 9 and Ffowcs Williams •ø models were the most recent available. The Witting model is based on a dipole description of the burst-sweep flow structures known to exist in turbulent boundary layers. The predictions made with this model are in very close agreement with the predictions performed using the Corcos model. 9 The Ffowcs Williams model •ø is similar to those of Chase but requires additional constants to exercise. The Corcos and Chase mod-

els were thus selected for this study because much recently published work 4-6'11 refers to the Corcos model, the Witting model will yield results similar to those of the Corcos model, and the Chase models predict considerably different behavior in the low wave-number region using fewer empirical constants than the Ffowcs Williams model. Comparisons will be made between experimental point pressure spectra and calculated point pressure spectra utilizing the Corcos and Chase models. Based upon these initial results, additional effort will focus on prediction of the TBL wall pressure spectra using the Chase models only. A comprehensive set of new empirical constants will be provided for these predictions based on the most recent experimental findings.

I. MODELING THE TBL WALL PRESSURE SPECTRUM

A. Autospectrum of the turbulent boundary layer wall pressure fluctuation

Based on the work of Uberoi and Kovasznay, •2 the measured one-sided autospectrum of the TBL wall pressure fluctuations is given by

Grr(ro)=2 P(k• ,k3,ro)lH(k• ,k3)l 2 dk• dk3,

(1)

where P(k•,k3,ro) is the wave-vector-frequency spectral density of the TBL wall pressure fluctuations and IH(k• ,k3)l 2 is the in-plane wave-number response function of the measurement sensor. Equation (1) assumes the pressure fluctuations are acting directly on the face of a single transducer and that the TBL is homogeneous and infinite in extent. The wave-number response function accounts for area averaging of the pressure spectrum that results from a finite-sized transducer. The in-plane wave-number response for a circular transducer is given by •

(2)

where R is the active radius of the sensing element and J1 is a Bessel function of the first kind and order one. The result-

ant magnitude of the wave number in the plane of the wall is

(3) where k• is the stream•ise wave number and k 3 is the cross- stream wave number. Also, k=(k• ,k3) is the in-plane wave vector.

B. Wave-vector-frequency spectrum models due to Chase

Chase 2 begins his effort to model the wave-vector- frequency spectrum with two basic premises: P(k, ro) should tend to zero as K approaches zero, with functional form P(k, ro)ocK 2. For K>>ro/c, the acoustic wave number, the most general form for the wave-vector-frequency spectrum is

p(k, ro)=p2u 6 -3f( u*k rø•v / ,to ,k& • , (4)

where it is assumed that dependence on Reynolds number is weak enough to be ignored. The parameters in Eq. (4) include the density p, the TBL friction velocity u,, the kine- matic viscosity v, radian frequency to, and the boundary layer thickness 8.

One important point to note is that the final models de- rived by Chase 2'3 do not include the viscous domain, which is characterized by the viscous variable roy/u2, in Eq. (4). Chase points out that as long as roy/u2 •< 2 the effect of the viscous domain on the convective ridge will be small; and as long as roy/u2, •< 1/2, the effect on the low wave-number region will also be small.

The final form that Chase 2 proposes for the wave- vector-frequency spectrum in the incompressive domain is

P(k, ro) 2 3 2 -5 = C2uk•K2u ), (5) p u,( +CrK2K• 5 where

Ki2:(oo-Uck•)2/hi2u 2 +K2+(b 8) 2, q• i i-M,r. (6)

In Eq. (5), M indicates the turbulence-mean shear compo- nent, T indicates the turbulence-turbulence contribution, and

Uc the convection velocity, typically 0.6 U• to 0.8 U•, where U• is the free-stream velocity. One can see by exam- ining Eq. (5) that the resulting model is quadratic in wave number. The model contains six dimensionless coefficients,

C•u, Cr, b2u, br, h2u, and hr, which must be determined by fitting the model to experimental data. Chase integrates Eq. (]) with IH(k• ,k3)l 2- ].0 to establish the point power spectrum. Then using the measured "point" spectra of Bull •3 he establishes

a+=0.766, rr=0.389, b2u=0.756,

br=0.378 ' /•2u=/•r= 0.176 (7) and calculates the value of the six dimensionless coefficients

using

2227 J. Acoust. Soc. Am., Vol. 98, No. 4, October 1995 D.E. Capone and G. C. Lauchle: Turbulent pressure spectra 2227


a +----(2 rr/3)(Crhr+

rr---- Crhr/( Crhr+ CMh/u), (8)

r/u = 1-r r, I•r------hru,/u, I•/u----h/uu,/u.

Chase 2 assumed that P(k,to) would tend to zero qua- dratically as K went to zero. In a refined version of the theory, 3 the restrictions on the form of P(k, to) in the low wave-number region are not as severe where fluid compress- ibility effects are more important, and consideration is given to functional forms which vary from K ø to K 2. The result is a wall pressure spectrum that is nearly wave vector white for 8-•<k•<(bS)-• where b is a scale factor. In the region (bS)-•<k•<to/Uc, the 1987 model is similar to the 1980 model in that P(k,to) varies as k•.

The incompressible flow model of the wall pressure spectrum from Chase 3 is

p2u3 P(k, to)=[K2++(bS)_2]s/2 CT K2 K2+ + (b S) -2 K 2 + ( b (5) - 2

(9)

where C/u and Cr are constants analogous to those of the 1980 model. In contrast, the compressible flow model for the wave-vector-frequency spectrum is given by

/O2U3 { P(k, to)- jr2+ q- (b•)_215/2

+1-c2-c 3 Cr K2

C 2 q-C 3

K2+ + (bS) -2 ]

k , (10)

where c2 and c 3 are additional empirical constants. The com- pressibility enters into Eq. (10) in the (g/Igcl) terms, where K c is

K2- to2/c2 K> to/c Igcl 2- ' (11) to2/c2- K 2, K< to/c'

and

g2+-(to - Uckl)2/(hu,)2+ g 2. (12) Chase 2 uses the data of Martin and Leehey TM to determine the values of the empirical constants; in summary,

h=3.0, Cr=.0047, C/u-0.155,

b=0.75, c2-c3=0.17. (13)

C. Wave-vector-frequency spectrum model due to Corcos

The model of the wave-vector-frequency spectrum proposed by Corcos • is

P(k, to)- 7r2[ ( k l _ kc) 2 q_ ( Ol l kc) 2][ k32 q_ ( o12kc) 2] ,

where P(to) is the point pressure spectrum:

(14)

P( to) - aop2u4,/to. (15) In our calculations of Sec. III, the numerical values used for

the dimensionless coefficients a 0, a•, and a2 were those used by Ko and Nuttall. 5 These values yield a low wave- number region which is some 40 dB below the convective ridge.

II. NUMERICAL INTEGRATION

Predictions of the TBL wall pressure spectra as measured by finite-sized transducers in three common fluids (air, water, and glycerine) are obtained by numerically integrating Eq. (1). Using Eq. (1) to compute the point wall pressure spectra removes the uncertainties introduced by different experiments using different transducer sizes for the measurement of the TBL wall pressure spectra. The integration was carried out in the k• and k 3 directions utilizing Simpson's 1/3 rule for two dimensions: •5

f Xi+l•yVi•lf(x,y I= )dx dy xi-1 i-

h

•- (f i + •,j + • + 4f i,j + • + f i- •,j + • ) h h

+ 4 •- (fi + •,j + 4fi,j + fi- 1,j) q- •- (fi + •,j- • + 4fi,j- •

q- fi-l,j-1) ß (16)

At each point in wave-number space where the integration is performed, a rectangle consisting of nine locations is de- fined. This rectangle has dimensions of twice the step size in the k• direction by twice the step size in the k 3 direction, where h and k in Eq. (16) are the step sizes in the k• and k 3 directions, respectively. The function f(x,y) is the sensor response-weighted wave-number-frequency spectrum. For Chase's 1980 model, Eqs. (5) and (6) are used in Eq. (1) for P (k, to). For the incompressible model in Chase's 1987 paper, Eq. (9) is integrated, for Chase's compressible model, Eq. (10) is integrated, and for the Corcos model, Eq. (14) is integrated.

Based upon an examination of the wave-number dependence of the turbulent wall pressure spectrum, and from pre-

16 vious experience in evaluating Eq. (1), values for the upper and lower limits of integration were established. For all cases, in both the k• and the k 3 directions, the double integral is evaluated at 80 locations in the range -10kc to 10kc where kc = to/Uc. The Chase models were assumed continu- ous across the discontinuity at k•=0. This assumption has very little effect on the integration results because the spectral level at k• =0, for the Chase models, is extremely small in comparison to the levels near the convective wave number which is expected to dominate the point pressure spectrum prediction for small transducers. For each case, the spectral level for 30 frequency points between the lowest and highest frequencies of interest is calculated. Due to the finite limits necessary to carry out the numerical integration the results



represent a lower bound on the magnitude of the wall pressure spectra. The frequency range of interest depends upon the data available for each fluid. Depending on the author, various frequency-dependent values of the convection velocity have been established. Ko and Schloemer 4 and Thompson and Montgomery 6 use a convective velocity of 0.6 Uo•, while Chase 3 uses a convection velocity of 0.65 Uo•. A value of 0.65 Uo• has been used in this study. Because this study was concerned with measurements made with single point pressure sensors, the effect of choice for U c was found to be minimal. It is recognized, however, that when an array of sensors is used to spatially filter the wall pressure fluctuations, the frequency-dependent nature of Uc becomes important.

One point to note is that the experiment in water was performed in a rectangular channel facility, and the glycerine experiment was performed in a pipe flow facility. For both of these cases, the TBL was not a free boundary layer. The boundary layer thicknesses for the bounded cases were there- fore considered to be the half-height of the corresponding facility.

III. RESULTS--ORIGINAL MODELS

A. Comparison to experimental results--Chase models

A comparison of the spectral levels calculated using the constants proposed by Chase with the experimental results are presented in this section. Results are presented in abso- lute values of frequency and wall pressure level to allow direct comparison with the published experimental results. Because the integration results obtained with Eqs. (9) and (10) were essentially identical, only the compressible flow model results from Eqs. (10) and (13) will be displayed, along with the results from Eqs. (5) and (7).

An experiment in a turbulent pipe flow of glycerine was performed by Lauchle and Daniels 17 utilizing both small pressure transducers and advanced signal processing techniques. The flow velocity at the centerline of the pipe ranges from 6.8 to 7.5 m/s, and the bulk fluid temperature was varied between 35 and 46 øC as to permit a pipe diameter Rey- nolds number (Reo) range from 10000 to 25 000. Fluid properties at 35 øC and a flow velocity of 7.15 m/s (Reo=10 890) were used for the calculations, boundary layer thickness was assumed to be the radius of the tunnel, and the sensor diameter is given as 0.5 mm. The flow con- ditions and pressure transducer size result in 0.7•<d +•<1.5, where d + is the transducer diameter d, normalized on viscous wall units u,d/v. The small pressure transducers com- bined with the large viscous scales of glycerine flow allowed better resolution of the high-frequency components of the wall pressure fluctuations than had previously been achieved. Also, by postprocessing the acquired data, Lauchle and Daniels 17 were able to remove extraneous low-frequency facility noise from the transducer signals.

Figure 1 shows a comparison between the calculated and experimental spectra for the glycerine experiment. The spectral levels predicted using the original Chase constants are considerably higher than those measured. The predicted peak

Exp

1980 Model

......... 1987 Model• 1 lO lOO lOOO

Frequency (Hz)

FIG. 1. Calculated and measured point wall pressure spectra for glycerine experiment (see Ref. 17) using the Chase (see Refs. 2 and 3) models.

level occurs at around 10 Hz, while the measured peak level occurs at about 20 Hz. The measured levels decrease rapidly after the peak value while the predicted values decrease at a much slower rate. The marked difference in spectral level above 100 Hz may be a result of the fact that Chase's models do not take into account the viscous domain of the TBL.

Based on the work of Farabee and Casarella, 18 the source of the high-frequency wall pressure fluctuations is the overlap region within the boundary layer and the small-scale velocity fluctuations in this region are affected by the viscosity of the fluid. By neglecting the effect of viscosity, the theories may be overpredicting the small-scale, high-frequency, velocity fluctuations. Another issue to be considered for these dis-

crepancies is the use of an external flow prediction method for an internal flow.

Home and Handler 19 performed an experiment in a water-filled rectangular channel flow facility. The same type of pressure transducers as those used by Lauchle and Daniels 17 were utilized in this experiment although 20•<d + •<40 because of the lower viscosity of water. The postprocessing techniques utilized were similar to those used by Lauchle and Daniels and the investigators were able to remove the low-frequency facility noise contamination from the measured TBL wall pressure spectra.

A comparison between the measured and predicted spectral levels for the water channel experiment of Home and Handler 19 at a channel width-based Reynolds number of 25 000 is shown in Fig. 2. The agreement between the measured and predicted values is much better for this experiment in comparison to Fig. 1. In the higher frequency range, above about 100 Hz, the models predict the rate of decrease of the spectral levels much more accurately for the water experiment than for the glycerine experiment. In the low-frequency region, below about 5 Hz, the predictions decrease more rapidly than the measurements.

An air experiment utilizing 1-mm-diameter and larger pressure transducers was performed by Schewe 2ø in a wind tunnel. Schewe developed electrostatic transducers of the Sell type to achieve 19•<d + •<333. Results used for comparison purposes in this paper correspond to a Reynolds number of Re0 = 1400 based on a boundary layer momentum thickness of 0=3.3 mm.



140

130

120 110

"/ ..... 1980 ,/ ,/

/ ......... 1987 Model

0.1 1 10 100 1000 10000

Frequency (Hz)

FIG. 2. Calculated and measured point wall pressure spectra for water experiment (see Ref. 19) using the Chase (see Refs. 2 and 3) models.

Figure 3 shows the measured and calculated spectral levels for the wind tunnel experiment of Schewe. 2ø The predictions using the 1987 Chase models show relatively good agreement with the actual measured levels. Once again, the high-frequency region (f•>900 Hz) is where the agreement between the measured and predicted levels is the poorest.

In general for all three of Chase's models using his originally proposed set of constants, the agreement between measured and predicted levels for the air and water data is good. The predictions for the glycerine experiment show much poorer agreement with the experimental data. The predictions with the 1987 models, both incompressible and compressible, show better agreement with experimental results than the predictions using the 1980 model. In all cases the models tend to overpredict the measured high-frequency levels.

B. Comparison to experimental results---Corcos model

Results of numerical integrations using the Corcos model, and the values for the dimensionless constants proposed by Ko and Nuttall, 5 were also compared to the experimental results from the air, water, and glycerine studies. These constants are a0 = 1.064, a•=0.09, and a2=0.63. Fig-

170

• 160

• 150 ,. 140

• 130 • 120

• 110

100 Corcos • 1 10 100 1000 10000

Frequency (Hz)

FIG. 4. Calculated and measured point wall pressure spectra for glycerine experiment (see Ref. 17) using the Corcos (see Ref. 1) model.

ure 4 shows a comparison of calculated and experimental spectra for the glycerine experiment. •7 As with Chase's models, the Corcos model overpredicts the measured pressure levels. Below the peak measured value at 10 Hz the predicted levels continue to increase as frequency decreases while the measured levels decrease.

Figure 5 is a comparison of calculated and experimental results for the water experiment. •9 There is good agreement between the model and experimental results in the frequency range above 200 Hz. As frequency decreases below 200 Hz, the Corcos model predicts a steady increase in pressure levels while the experimental results show a slight decrease.

Figure 6 is a comparison of experimental and calculated results for the air experiment. 2ø From 100 to 1000 Hz the predicted levels are below the measured levels, although in a gross sense; the slope of the predicted curve is similar to the measured data in this region. Above 1000 Hz, and below 100 Hz, the model does not agree with the spectrum.

C. Discussion

Both the Corcos and Chase models appear to be best suited for prediction of measured TBL point wall pressure spectra in the midfrequency range. Neither model accounts for the effects of viscosity on the high-frequency portion of

lOO

80

• 70

50

-, %

......... 1987 Model

1 10 lO0 lOOO 10000

Frequency (Hz)

FIG. 3. Calculated and measured point wall pressure spectra for air experiment (see Ref. 20) using the Chase (see Refs. 2 and 3) models.

150

140

130 120

110 • Exp

...... Corcos

0.1 1 10 100

Frequency (Hz)

FIG. 5. Calculated and measured point wall pressure spectra for water experiment (see Ref. 19) using the Corcos (see Ref. 1) model.



lOO lOO

8O

'70

..... Corcos •

I lO lOO lOOO 10000

Frequency (Hz)

FIG. 6. Calculated and measured point wall pressure spectra for air experiment (see Ref. 20) using the Corcos (see Ref. 1) model.

the pressure spectrum. Figures 1 and 4 show clearly the effect of viscosity on the measured high-frequency portion of the wall pressure spectra. The most significant difference between the Corcos and Chase models is evident in the low-

frequency portion of the predicted TBL wall pressure spectrum. The Corcos model predicts pressure levels which continually increase as frequency decreases [a consequence of Eq. (15)], while the. Chase model predicts pressure levels that decrease with decreasing frequency below a flow- dependent maxima. The predictions using the Chase models appear to be more accurate in the low-frequency region.

The numerical values assigned to the empirical constants affect the predicted results and have thus been investigated. This has been performed for only the Corcos model and the Chase 1987 compressible model. It was determined for the Corcos model that a 0, a•, and a 2 affect only the level of the entire spectrum, not the shape or slope. Attention has there- fore been focused on the Chase 1987 compressible model.

IV. RESULTS--MODIFIED CHASE MODEL

A. Empirical constants

In order to obtain a better agreement between the experimental and calculated results, the six parameters of Chase's compressible model [Eq. (10)] were systematically varied. The first step was to vary each parameter individually to quantify the effect each parameter has on the calculated wall pressure spectrum. As an example, Fig. 7 shows that the overall spectral level decreases by about 5 dB as a result of lowering the value of Ca4 by a factor of 3 while keeping the other parameters constant. Reducing C r by the same factor as Ca4 has a much lesser impact on the predicted levels. This is to be expected since the contribution to the TBL wall pressure by the turbulence-mean shear interaction term is significantly larger than the contribution from the turbulence-turbulence interaction term.

Figure 8 compares the calculation results for the original set of constants to the values predicted by the model when b is reduced by a factor of 3. The effect of b, the scale coefficient, on the low-frequency predicted levels is considerable. This scale coefficient affects the low wave-number region of the pressure spectrum which in turn affects the integrated

80

70

60

• Given ..... CM/3

1 10 lOO lOOO lOOOO

Frequency (Hz)

FIG. 7. Calculated spectral levels for original constants proposed by Chase (see Ref. 3) vs Ca4/3 for air experiment (see Ref. 20).

low-frequency spectral levels and spectral shape. The effect of the velocity dispersion coefficient h on the calculated wall pressure spectra was found to be similar to that of Ca4; the level of the entire spectrum is reduced, but the slope of the predicted curve remains the same. Varying the parameters c• and c2 had an extremely small effect on the calculated wall pressure spectrum. None of the parameters have any impact on the slope of the high-frequency end of the spectrum. This fact will limit the effectiveness of fitting the model to some of the experimental data.

B. Empirical best fit to experimental data

Using the information gained by varying the constants individually, an empirical best fit to the experimental data was obtained by varying combinations of the empirical parameters. The combinations used are obviously not unique because of the number of unknowns, but in general the scale coefficient b was used to fit the slope of the low-frequency end of the spectrum and the overall level was adjusted using combinations of h and C•t. Due to the relatively minor impact of CT on the calculated pressure spectrum, its value was left unchanged from that noted in Sec. I B.

Figure 9 is a comparison of the experimental spectrum for the air experiment 2ø with the best fit obtained by varying

100

• 8o • 70

5o

Given

1 lO lOO lOOO lOOOO

Frequency (Hz)

FIG. 8. Calculated spectral levels for original constants proposed by Chase (see Ref. 3) vs b/3 for air experiment (see Ref. 20).



100

'- 80

":' 70

50

0.01 0.1 1 10

• Exp x x ..... Best Fit

1 10 100 1000 10000

Frequency (Hz)

FIG. 9. Experimental data for air experiment (see Ref. 20) vs best fit of 1987 compressible flow model (see Ref. 3).

the empirical parameters of Eq. (9). The nondimensional frequency (tov/u 2 ,) is included on this and subsequent figures to help identify the frequencies where viscous effects become important. Although b was altered by a significant amount in order to adjust the low-frequency slope of the spectrum, the remainder of the values were changed very little. Above 1000 Hz, the model still overpredicts the wall pressure levels because none of the theoretical parameters can affect solely the high-frequency end of predicted pressure spectrum.

Figure 10 is a comparison of the Home and Handler 19 experimental data for the water experiment with the best fit obtained by a new set of empirical parameters. There is rea- sonable agreement between the calculated and measured values for the water experiment in both the low- and high- frequency regions.

Figure 11 is a comparison of the wall pressure spectrum measured in the glycerine experiment of Lauchle and Daniels 17 with the best fit of the compressible flow theory. Above 100 Hz, there are still large discrepancies between measured and predicted spectral levels. Glycerine, having the highest viscosity of the three fluids under consideration, ex-

140

130

120 110

90

0.001 0.01 0.1 1 10 , , , ,,,,,1 , , , ,,, ..............................

Exp

..... Fi Best

0.1 1 10 100 1000 10000

Frequency (Hz)

FIG. 10. Experimental data for water experiment (see Ref. 19) vs best fit of 1987 compressible flow model (see Ref. 3).

0.01

160 .....

150

130

120

110

90

0.1 1 10

• Exp

..... Best Fit

1 10 100 1000 10000

Frequency (Hz)

FIG. 11. Experimental data for glycerine experiment (see Ref. 17) vs best fit of 1987 compressible flow model (see Ref. 3).

hibits the poorest high-frequency agreement between measured and predicted spectral levels.

Table I shows the original and final empirical constants used in Eq. (10), and used in the predictions of Figs. 1-3 and 9-11. Figure 12 is provided to show that the predicted low wave-number levels are altered slightly due to the change in parameters. These calculations were performed using the Chase 1987 compressible model and parameters from the air experiment. This particular example is for the air experiment. The change in low wave-number behavior due to changing the constants h, b, and C M is clearly evident. We note however, that the low wave-number levels predicted for the glycerine experiment using the modified constants were some 15 dB below those predicted using the original constants. The in-water changes are comparable to those of Fig. 12.

v. CONCLUSIONS

This paper has attempted to evaluate the suitability of applying various wave-vector-frequency spectrum models 1-3 to the prediction of measured TBL point wall pressure spectra. Particular attention has been given to the effects of transducer spatial averaging; such effects are ac- counted for in the predictions as to remove the ambiguity associated with varying values of d + from experiment to experiment. As a result of this and other considerations, new sets of empirical constants that are needed to exercise the Chase theory have been determined. In the course of this investigation, a number of conclusions have been reached.

(1) The Chase models 2'3 for the TBL wall pressure wave-vector-frequency spectrum, when incorporated into

TABLE I. Original and final parameter values used in numerical calculations.

C• C r b h c2 c3

Original 0.155 0.0047 0.75 3.0 0.1667 0.1667 Glycerine 0.10 0.0047 0.40 1.0 0.1667 0.1667 Water 0.0777 0.0047 1.40 3.0 0.1667 0.1667

Air 0.2330 0.0047 0.50 3.0 0.1667 0.1667



10

0

q- -•o CM

• -20 o -30

.•. -40

-• , •-..

-7o •,,,.. ........ --"" -•sLnt ,,,,.,.... -80

0.01 0.1 k• / (• / U•) 1 10

FIG. 12. Wave-number-frequency spectrum from 1987 compressible flow model (see Ref. 3) for air experiment (see Ref. 20) at a frequency of 100 Hz.

Eq. (1), predict point wall pressure spectrum levels that are much closer to measured levels than those predicted using Eq. (1) with the Corcos • model.

(2) The 1987 compressible and incompressible Chase models 3 predict point wall pressure spectral levels that are closer to measured levels than does the 1980 model. 2

(3) Chase 2'3 states that the effect of viscosity on the convective ridge of the wave-vector-frequency spectrum will be small provided that tov/u 2 •< 2 and that the effect of viscos- , ,

ity on the low wave-number domain will be small if tov/u 2 •< 1/2 For the glycerine experiment, this corresponds , ß

to viscous effects becoming important in the low wave- number domain for frequencies above 90 Hz and for the convective domain at frequencies above 360 Hz. The final results for the glycerine experiment, Fig. 11, show the calculated values deviating from the measured values at frequencies above about 90 Hz. For the water experiment, where viscosity is expected to play less of a role, the above criteria correspond to the much higher frequencies of f=627 and 2508 Hz, for the low and convective wave-number regions, respectively. The predictions and experimental results show some divergence at these frequencies. For the experiment performed in air, a medium in which the viscosity is between those of glycerine and water, viscous effects become important at frequencies off=393 and 1571 Hz. As seen in Fig. 9 these important frequencies coincide with the frequencies where deviation from prediction and experiment occur. In all cases, the models tend to overpredict the measured spectral levels in the high-frequency region. The cause of this appears to be that the models do not take into account the viscous

domain of the TBL.

We conclude that in comparison to the Corcos model, the Chase model provides the best method for predicting the TBL wall pressure spectra at convective and subconvective wave numbers for the low- and midfrequency ranges. The

2 noted above predict accurately the fre- limits on wv/u, quency range where the models break down.

Future research may include similar studies using the

Witting 9 and Ffowcs Williams •ø models. Both are relatively unexplored in comparison to the Corcos and Chase theories, so work along the lines presented here will perhaps enable these models, too, to be exercised with limitations revealed.

2G. M. Corcos, "The structure of the turbulent pressure field in boundary layer flows," J. Fluid Mech. 18, 353-378 (1964). D. M. Chase, "Modeling the wavevector-frequency spectrum of turbulent boundary layer wall pressure," J. Sound Vib. 70, 29-67 (1980).

3D. M. Chase, "The character of the turbulent wall pressure spectrum at subconvective wavenumbers and a suggested comprehensive model," J. Sound Vib. 112, 125-147 (1987). S. H. Ko and H. H. Schloemer, "Calculations of turbulent boundary layer pressure fluctuations transmitted into a viscoelastic layer," J. Acoust. Soc. Am. 85, 1469-1477 (1989). S. H. Ko and A. H. Nuttall, "Analytical evaluation of flush-mounted hy- drophone array response to the Corcos turbulent wall-pressure spectrum," J. Acoust. Soc. Am. 90, 579-588 (1991).

6W. Thompson and R. E. Montgomery, "Approximate evaluation of the spectral density integral for a large planar of rectangular sensors excited by turbulent flow," J. Acoust. Soc. Am. 93, 3201-3207 (1993). R. H. Mellen, "On modeling convective turbulence," J. Acoust. Soc. Am. 88, 2891-2893 (1990).

8R. H. Mellen, "Wave-vector filter analysis of turbulent flbw," J. Acoust. Soc. Am. 95, 1671-1673 (1994).

9j. M. Witting, "A spectral model of pressure fluctuations at a rigid wall bounding an incompressible fluid, based on turbulence structures in the boundary layer," Noise Control Eng. J. 26, 28-43 (1986). j. E. Ffowcs Williams, "Boundary layer pressures and the Corcos model: A development to incorporate low-wavenumber constraints," J. Fluid Mech. 125, 9-25 (1982). S. H. Ko, "Performance of various shapes of hydrophones in the reduction of turbulent flow noise," J. Acoust. Soc. Am. 93, 1293-1299 (1993).

•2M. S. Uberoi, and L. S. G. Kovasznay, "On mapping and measurement of random fields," Q. Appl. Math. 10, 375-393 (1953). M. K. Bull, "Wall-pressure fluctuations associated with subsonic turbulent boundary layer flow," J. Fluid Mech. 28, 719-754 (1967).

14N. C. Martin and P. Leehey, "Low wavenumber wall pressure measurements using a rectangular membrane as a spatial filter," J. Sound Vib. 52, 95-120 (1977). j. W. McCormick and M. G. Salvadori, Numerical Methods in FORTRAN (Prentice-Hall, Englewood Cliffs, NJ, 1965). j. F. McEachem and G. C. Lauchle, "A study of flow induced noise on a bluff body," Structure and Flow Sound Interactions, edited by T. M. Fara-



bee and M.P. Paidoussis (ASME, New York, 1992), NCA Vol. 13, pp. 23-38.

•7 G. C. Lauchle and M. A. Daniels, "Wall-pressure fluctuations in turbulent pipe flow," Phys. Fluids A 30, 3019-3024 (1987).

•ST M. Farabee and M. J. Casarella, "Spectral features of wall-pressure fluctuations beneath turbulent boundary layers," Phys. Fluids A 3, 2410- 2420 (1991).

•9 M.P. Home and R. A. Handler, "Note on the cancellation of contaminat- ing noise in the measurement of turbulent wall pressure fluctuations," Exp. Fluids 12, 136-139 (1991).

2øG. Schewe, "On the structure and resolution of wall-pressure fluctuations associated with turbulent boundary-layer flow," J. Fluid Mech. 134, 311- 328 (1983).



Calculation of turbulent boundary layer wall pressure spectra

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