Subscribe

Online

Member: $140.00
Non-member: $2,425.00

Adding the item to your cart

Your cart has been updated.

Go to Cart Continue Browsing the AIAA ARC

Journal Information

EISSN: 1533-385X

The journal that started it all back in 1963. Look for pioneering theoretical developments and experimental results across a far-reaching range of disciplines.

Print issues of AIAA Journal were published 1963 – 2016 (ISSN: 0001-1452). The journal transitioned to online-only publication in January 2017.

Empirical Model of Wall Pressure Spectra in Adverse Pressure Gradients

Nan Hu*

DLR, German Aerospace Center, D-38108 Braunschweig, Germany

*Research Scientist, Department of Technical Acoustics, Institute of Aerodynamics and Flow Technology, Lilienthalplatz 7; . Member AIAA.

Copyright © 2018 by German Aerospace Center. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. All requests for copying and permission to reprint should be submitted to CCC at www.copyright.com; employ the ISSN 0001-1452 (print) or 1533-385X (online) to initiate your request. See also AIAA Rights and Permissions www.aiaa.org/randp.

Publication Date (online): June 28, 2018
Sections:

ABSTRACT
Next section

An empirical model of wall pressure spectra for adverse pressure gradient boundary layers is proposed. The effect of adverse pressure gradients on wall pressure spectra and the relationship between the boundary-layer parameters and the wall pressure are studied and demonstrated. A dataset for five experiments at four different test facilities, covering Reynolds number between 2.6103 and 1.9104 based on the boundary-layer momentum thickness, is selected to develop the new model. Goody’s model is served as the basis for the development. Predictions of the new model and comparisons with other published wall pressure spectral models for adverse pressure gradient boundary layers are made for the selected dataset. The new model shows good prediction accuracy for the selected dataset and a significant improvement compared to the other published models.

Nomenclature
Cf

skin friction coefficient, τw/Q

Cp

pressure coefficient, (pp)/(0.5ρU2)

H

boundary-layer shape factor, δ*/θ

k

wave number, m1

p, p

pressure and pressure at wind tunnel nozzle exit, Pa

Q

dynamic pressure, 0.5ρU02, Pa

RT

time-scale ratio, (δ/Ue)/(ν/uτ2)

Reδ, ReΔ

Reynolds number related to other boundary-layer parameters, (δ,Δ)Ue/ν

Reθ

Reynolds number based on boundary-layer momentum thickness, U0θ/ν

Reτ

Reynolds number based on wall shear stress, uτδ/ν

Ue, U0

boundary-layer edge velocity (0.99U0) and local freestream velocity

U

freestream velocity at wind tunnel nozzle exit

u

velocity, m/s

uτ

friction velocity, τw/ρ, m/s

u+

dimensionless velocity, u/uτ

x, y, z

spatial coordinates, m

y+

dimensionless wall-normal coordinate, yuτ/ν

βδ, βΔ

Clauser’s equilibrium parameter related to other boundary layer parameters, (δ,Δ)/Qdp/dx

βδ*, βθ

boundary-layer thickness and displacement thickness based Clauser’s equilibrium parameter, (δ*,θ)/τwdp/dx

Δ

boundary-layer defect thickness, δ*2/Cf

Δδ/δ*

boundary-layer related parameter, δ/δ*

δ

boundary-layer thickness, m

δ*

boundary-layer displacement thickness, m

θ

boundary-layer momentum thickness, m

κ

von Kármán constant

ν

kinematic viscosity, m2/s

Πδ*

Cole’s wake parameter, 0.8(βδ*+0.5)3/4

Πθ

Cole’s wake parameter related to boundary-layer momentum thickness, 0.8(βθ+0.5)3/4

ρ

density, kg/m3

τw

wall shear stress, Pa

ϕ(ω)

power spectral density of wall pressure fluctuations (single-sided), Pa2s

ω

angular frequency, rad/s

I. Introduction
Previous sectionNext section

Wall pressure fluctuations beneath a turbulent boundary layer are a fundamental topic in flow-induced noise. One major concern is the noise transmission through the surface structure due to the fluctuating pressure excitation on the surface. The pressure fluctuations caused by the fluctuating velocities exert on the surface, and consequently the induced surface vibration radiates noise, which makes the wall pressure fluctuations an important noise source for the cabin. Modeling of the wall pressure spectra is of fundamental and practical interest. The most used model for zero pressure gradient (ZPG) boundary layers is Goody’s [1] model, which uses self-similarity of wall pressure spectra and incorporates Reynolds number effects in the high-frequency range with a time-scale ratio; refer to Eq. (4). The spectra induced by non-ZPG boundary layers become more complicated. Experimental studies [210] for pressure gradient effects on wall pressure fluctuations showed that the wall pressure spectra lose their self-similarity. A group of sensors was installed in the streamwise direction to measure the spectral development due to the impact of pressure gradients [7,9,10]. For adverse pressure gradient (APG) boundary layers, the spectral slope at medium frequencies becomes successively steeper as the measurement position moves downstream. This is because the boundary layer experiences the APG for a longer distance at downstream positions. Furthermore, the stronger the pressure gradient, the steeper the midfrequency slope is.

Rozenberg et al. [11] (RRM) analyzed the spectral variation between ZPGs and APGs from selected experimental and numerical results and showed that a large inaccuracy in the spectral prediction can be caused if the Goody [1] model is applied for an APG case. Based on the observation that the APG increases the spectral peak level and the spectral drop in the midfrequency range compared to the ZPG case, RRM modified the Goody model involving some boundary-layer parameters, e.g., parameters based on boundary-layer thickness and Clauser’s [12] equilibrium parameter, to capture the spectral changing trend due to the presence of the pressure gradient, expressed as

ϕ(ω)Ueτw2δ*=[2.82Δδ/δ*2(6.13Δδ/δ*0.75+F1)A1][4.2(Πθ/Δδ/δ*)+1](ωδ*/Ue)2[4.76(ωδ*/Ue)0.75+F1]A1+[8.8RT0.57(ωδ*/Ue)]A2(1)
where
F1=4.76(1.4/Δδ/δ*)0.75[0.375A11],A1=3.7+1.5βθ,A2=min(3,19/RT)+7,βθ=θ/τwdp/dx,Πθ=0.8(βθ+0.5)3/4,Δδ/δ*=δ/δ*

Clauser’s [12] equilibrium parameter βθ is used to manage the slope variation at medium frequencies. The larger the value of βθ is, the steeper the slope is. The spectra shift to a higher frequency and a larger amplitude as Δδ/δ* increases. Both βθ and Δδ/δ* are in charge of the spectral amplitude.

Kamruzzaman et al. [13] (KBLWK) proposed a spectral model for prediction of the airfoil trailing edge noise. The trailing edge wall pressure spectra measured in the vicinity of the trailing edge from different investigations [6,1418] were used to develop the model. The formulation of the model reads

ϕ(ω)Ueτw2δ*=0.45[1.75(Πδ*2βδ*2)m+15](ωδ*/Ue)2[(ωδ*/Ue)1.637+0.27]2.47+[(1.15RT)2/7(ωδ*/Ue)]7(2)
where βδ*=δ*/τwdp/dx, Πδ*=0.8(βδ*+0.5)3/4, and m=0.5(H/1.31)0.3. Except for the midfrequency extension determined by RT, the formulated spectrum has a constant shape for different pressure gradient configurations, i.e., a constant slope of about ω2 at medium frequencies and a constant spectral peak location at the nondimensional frequency ωδ*/Ue0.44. The spectral amplitude is adjusted by a combination of Clauser’s equilibrium parameter [19] βδ*, Cole’s wake parameter [20,21] Πδ*, and boundary-layer shape factor H.

Catlett et al. [22] (CAFS) measured the wall pressure fluctuations on tapered trailing edge sections of a flat plate with three different wedge angles and proposed an empirical spectral model based on the measured data, which reads

ϕ(ω)Ueτw2δ=a(ωδ/Ue)b[(ωδ/Ue)c+d]e+[fRTg(ωδ/Ue)]h(3)
ln(aaG)=4.98(βΔReΔ0.35)0.13110.7,b=2,ccG=20.9(βδReδ0.05)2.76+0.162,ddG=0.328(βΔReΔ0.35)0.3100.103,eeG=1.93(βδReδ0.05)0.628+0.172,ffG=2.57(βδReδ0.05)0.224+1.09,ggG=38.1(βδH0.5)2.11+0.0276,hhG=0.797(βΔReΔ0.35)0.07240.310
where βδ,Δ=(δ,Δ)/Qdp/dx, Reδ,Δ=(δ,Δ)Ue/ν, and Δ=δ*2/Cf. Parameters ah are derived by fitting to the measured spectra. The parameters aGhG are the constants at the correspondent positions in the Goody [1] model, the values of which can be found in Eq. (4). The spectral amplitude, peak location, and slope at medium frequencies are affected by Clauser’s [12] equilibrium parameter and Reynolds numbers defined with different length scales.

Among others, Clauser’s [12] equilibrium parameter βδ,δ*,θ,Δ defined with different boundary-layer thicknesses is used as one important input quantity in all models. Equilibrium flows hold a constant βδ*, e.g., βδ*=0 for the specific case of a ZPG boundary layer [19]. For equilibrium boundary layers, Clauser demonstrated a clear dependence of the velocity profile shapes on this single parameter. Building upon Clauser’s work, Mellor and Gibson [19] predicted the measured equilibrium defect profiles from Clauser. Herring and Norbury [23] performed supplemental experiments on favorable pressure gradient (FPG) equilibrium boundary layers, in which βδ* possesses negative values. A good representation of their measured FPG velocity profiles was achieved using the theory of Mellor and Gibson. Accordingly, for equilibrium flows, the selection of βδ* is considered a well-suited parameter to reproduce the effect of a nonzero pressure gradient on the velocity profile and corresponding wall pressure spectra.

In the present paper, it is hypothesized that the boundary-layer shape factor H=δ*/θ represents a better-suited nondimensional quantity to cope with the prediction of wall pressures under arbitrary nonequilibrium flow conditions, in which also the history of the boundary-layer development is considered important. In the following, the effects of APGs on the boundary-layer mean velocity profile and on the wall pressure spectra are discussed based on the experimental results from Hu and Herr [10]. Parametric analysis and modification of the Goody [1] model are made to predict wall pressure spectra under APGs (including ZPGs). Four other experimental results of wall pressure spectra under APGs are selected to develop and assess the new model. Furthermore, comparisons of other published models are also given.

II. Parametric Analysis and Modification Approach
Previous sectionNext section

Hu and Herr [10] measured the wall pressure fluctuations with pinhole-mounted Kulite sensors on a plate model in the open-jet Aeroacoustic Wind Tunnel Braunschweig (AWB). The APG was generated by an adjustable NACA 0012 airfoil installed 120 mm above the plate surface, relative to the chord at zero deg angle of attack (AOA). The airfoil has a 400 mm chord length, and its leading edge was located at 850 mm behind the leading edge of the plate. Three geometrical AOAs at 6, 10, and 14 deg of the airfoil for realization of the APG were used. The spectral development in the streamwise direction was measured by a group of Kulite sensors placed in the range 1128  mmx1210  mm (x=0 for the leading edge tip of the plate). The mean flow velocity was measured by hot-wire anemometers at positions x=1128  mm and x=1210  mm, where the most upstream and downstream Kulite sensors were located.

Figure 1a shows the measured distributions of the pressure coefficient Cp on the plate model between 930  mmx1220  mm. Figure 1b shows the mean velocity profiles for ZPG and APG boundary layers at x=1210  mm for the freestream velocity of 30.2  m/s. The APG boundary layer shows a smaller velocity increase in the inner layer and a larger velocity increase in the outer layer when compared to the ZPG case. For a very strong APG boundary layer, e.g., the APG 14 deg case, an inflection point occurred at the transition region between the inner and outer layers, i.e., at about 0.10.2δ. The measured effects of APGs on the mean velocity profiles are consistent with the experimental results provided by White [21].

figure

Fig. 1 a) Cp distributions for APG and ZPG and b) mean velocity profiles at x=1210  mm.

Table 1 summarizes the relevant boundary-layer parameters for the two selected velocity measurement positions. Note that the local pressure gradients dp/dx decrease when the AOAs increase at x=1210  mm. An increase in dp/dx with an increasing AOA is observed at the more upstream locations; refer to Fig. 1a. From Fig. 1b, one could conclude that both the shape factor H and Clauser’s equilibrium parameter [19] βδ* can represent the ZPG and APG effects on the mean velocity profile shape. The larger the value of H and βδ*, the more strongly the mean flow profile is affected by the APGs. However, compared to H, βδ* is directly impacted by the local pressure gradient, and a stronger local dp/dx does not necessarily indicate the upstream boundary-layer development history of a much stronger initial pressure gradient.

Figure 2a shows the mean velocity profiles for the APG 10 deg and APG 14 deg cases at both measurement positions. The measured profile shapes indicate that the velocity profiles at x=1210  mm are still significantly affected by the strong initial APG conditions. The shape factor H is larger at the downstream position, which represents well the boundary-layer profile development from upstream to downstream. On the contrary, because of the larger value of the local dp/dx at x=1128  mm, the value of βδ* is larger at the upstream position. A plot of the corresponding defect profiles is shown in Fig. 2b. An equilibrium boundary layer presents a larger velocity gradient in the outer layer for a greater value of βδ* [12,19,23]. The measured APG boundary layers in the current study show reversed trends, indicating that Clauser’s equilibrium parameter [19] βδ* is not suited to define the shape of the velocity profiles for arbitrary nonequilibrium boundary layers, especially for cases with large pressure gradient changes.

figure

Fig. 2 Mean velocity profiles for ZPG and APG at 10 and 14 deg.

Figure 3 shows the wall pressure power spectral densities (PSDs) for APG and ZPG boundary layers at the freestream velocity of 30.2  m/s. Spectral levels in this paper are referenced to a 20μPa reference pressure. At low frequencies (<100  Hz), the measured spectra are contaminated by the open-jet free shear layer. The used 0.5 mm diameter pinhole arrangement provides a high spatial resolution, which ensures a small attenuation (<1.5  dB, according to Corcos’s [24] correction) until 8 kHz for the ZPG case. A Corcos correction is not applied in the present spectra.

figure

Fig. 3 Wall pressure spectra: a) ZPG and APG at 14 deg and b) spectra at x=1210  mm.

The dependence of the spectral shape on the initial pressure conditions is illustrated in Fig. 3. Compared to the ZPG spectra, the APG spectra feature an increase in peak level and a steeper slope at medium frequencies. For a given initial APG configuration, when moving downstream, the spectra shift toward lower frequencies and the slope at medium frequencies becomes successively steeper. The same changing trend also appears for a stronger initial APG configuration; see Fig. 3b. This indicates that the more strongly the boundary layer is affected by the APG, the steeper the spectral slope at medium frequencies is.

Figure 4a shows the scaled spectra for ZPG and APG boundary layers using the same scaling variables as the ones used in Goody’s [1] model. Unlike the good collapse of the ZPG spectra, normalized spectral peak levels diverge by up to more than 15 dB. Again, the evolution of the midfrequency slope from a ZPG boundary layer to an APG boundary layer is well illustrated. Note that the roll-off slope of ω5 at high frequencies appears to be unaffected for the APG spectra. A very good collapse for the ZPG and APG spectral maxima is found by scaling with uτ/Q2θ and θ/U0; see Fig. 4b. All the measured APG spectra at x=1128 and 1210 mm including the spectra for freestream velocities of 20.3 and 39.1  m/s are scaled using these variables, and a good collapse of the spectral maxima is shown in Fig. 4c. It seems to be more reasonable to scale the APG spectra using the outer pressure scale Q instead of τw due to the increasing importance of the boundary-layer outer layer for the spectral peak.

figure

Fig. 4 Scaled spectra: a and b) spectra at x=1210  mm and c) all measured spectra for APG.

From the previous analysis, we found a good collapse of the spectral maxima by using the outer variables to scale the spectra for ZPG and APG boundary layers. A clear trend for the slope at medium frequencies and a nearly unchanged high-frequency roll-off slope were observed. From these observations, we are encouraged to develop an empirical spectral model for APG wall pressure fluctuations based on a modification of the ZPG wall pressure spectrum.

To some extent, the APG spectra have trends similar to the ZPG spectra. The spectra increase first at low frequencies, then drop at medium frequencies and roll off at high frequencies. Goody’s [1] model can represent well these trends for ZPG spectra in the three different ranges, especially at medium and high frequencies. From this point, it is appropriate to take Goody’s model as the starting point. Goody’s model is expressed as

ϕ(ω)Ueτw2δ=a(ωδ/Ue)b[(ωδ/Ue)c+d]e+[fRTg(ωδ/Ue)]h(4)
where a=3, b=2, c=0.75, d=0.5, e=3.7, f=1.1, g=0.57, and h=7. Goody used Ue/τw2δ and δ/Ue as scaling variables for the ZPG spectra. However, based on the previous discussion, it is more appropriate to use uτ/Q2θ and θ/U0 as the scaling variables for the APG spectra. Thus, Goody’s model is rewritten as
ϕ(ω)uτQ2θ=a(ωθ/U0)b[(ωθ/U0)c+d]e+[f(ωθ/U0)]g(5)

The variables ag in this equation control the shape of the dimensionless spectra. Here, fRTg in Eq. (4) is expressed in one single variable. This is because the scaling variable is different from the one used in Goody’s [1] model, which may change the used parameter for the ratio of time scales. The amplitude of the spectra is adjusted by the value of a. The slopes in different frequency ranges are driven by the combination of b, c, e, and g. For example, the variable b fixes the slope at low frequencies. The function ceb is in charge of the slope at medium frequencies, and gb is in charge of the slope at high frequencies. The variable f determines the extension of the midfrequency range. The peak location is affected by the value of d, the slope at low frequencies, and the trend of the transition range between the increase and decrease at low and medium frequencies.

The first step of the modification is to represent the ZPG spectra that are supposed to have slopes of ω0.7 at medium frequencies and ω5 at high frequencies. These slopes can be realized by means of the combination of the variables, which follows bce=0.7 and bg=5. Goody [1] adopted b=2 from the Chase–Howe [25] model, which implies an ω2 increase at low frequencies. Panton and Linebarger [26] and Blake [27] calculated the spectra of the wall pressure fluctuations by solving a Poisson equation. The result is derived by integration of the contributions throughout all decks of the boundary layer. Because of the term k12/k2exp(2ky) in the solution, in which k1 is the wave number in the streamwise direction, k2=k12+k32, and y is the wall-normal distance to the wall, an ω2 increase at low frequencies is obtained. However, in this calculation, the fluctuation dynamics is assumed as frozen turbulence. Hu et al. [28] calculated the wall pressure in consideration of the decay of convective turbulence, and the result shows that the ω2 increase does not hold if nonfrozen turbulence is considered. The effect of convective decaying turbulence is only noticeable at higher frequencies for the velocity spectra. The wall pressure spectra are also affected at lower frequencies due to the extra term k12/k2exp(2ky). As a result, the slope of the low-frequency increase turns out to be smaller. This effect was also reported by Chase [29]. Furthermore, in the Poisson equation, the mean-shear turbulence term is normally considered as the dominant source term; thus, the turbulence–turbulence term is mostly discarded in the calculation. Kraichnan [30] and Meecham and Tavis [31] calculated the importance of the mean-shear term and claimed the dominance of the mean-shear term for the wall mean square pressure. However, numerical results [28,32,33] show that the mean-shear term and the turbulence–turbulence term have the same order of magnitude for the contribution to wall pressure fluctuations. The spectrum contributed from the turbulence–turbulence term possesses an almost plateaulike spectrum at low frequencies. Thus, the turbulence–turbulence term gains dominance at low frequencies. Consequently, the slope in the low-frequency range can be also affected by considering the effect of the turbulence–turbulence term. In the literature, the low-frequency spectral slope is reported between ω0.20.8 for the ZPG spectra, except for the work of Farabee and Casarella [34], who found an ω2 increase at the very lowest frequency range <10  Hz (ωδ/U0<0.08).

The measured APG spectra from Hu and Herr [10] show a slope of ω0.61.0 at low frequencies. A steeper slope was found in experiments from Catlett et al. [7] and Suryadi and Herr [9]. The results showed a larger low-frequency increase at a greater APG, and the increase can reach about ω1.4 as the APG boundary layer approaches separation. From those observations, it can be drawn that the low-frequency slope for an APG spectrum is dependent on the velocity profile of the boundary layer and probably also affected by the Reynolds number. However, because of the low-frequency contamination of most measured data, it is hard to figure out the dependence between the low-frequency slope and the possible important parameters. Thus, in this paper, a constant b=1.0, which is considered an averaged value for the APG spectra, is applied.

Since the value of b is fixed, we can determine g=b+5=6.0 and ce=b+0.7=1.7. To prescribe the spectral slope change at medium frequencies due to the pressure gradients, an additional variable h is added to govern the spectral decrease. The midfrequency spectral slope is determined with cehb; i.e., the slope is steeper as h increases. It is noted that a steeper decrease at medium frequencies follows a more rapid transition between the increase at low frequencies and the decrease at medium frequencies; refer to Fig. 4. This feature requires an increasing value at the position c in Eq. (5) as the decrease at medium frequencies steepens, because the spectrum in the transition range is primarily managed by (ωθ/U0)c. Finally, to determine the values of c and e and the proper way to introduce h, the values and the combination of those variables should be able to characterize the change of the spectral slope and the transition range at medium frequencies for weak and strong APGs.

Figure 5 shows the perfect fit between the measured spectra from Hu and Herr [10] under the weakest and strongest pressure gradient conditions and the formulated spectra using the expression

ϕ(ω)uτQ2θ=a(ωθ/U0)1.0[(ωθ/U0)1.5h1.6+d]1.13/h0.6+[f(ωθ/U0)]6.0(6)
where the variables are determined as follows: c=1.5 and e=1.13. It is found that the change of the transition between low and medium frequencies is too fast to be governed by a linear dependence of h, i.e., (ωθ/U0)1.5h. A formulation of (ωθ/U0)1.5h1.6 can identify the change of the transition range well. Furthermore, to keep a linear dependence between the change of the spectral slope at medium frequencies and the variable h, i.e., cehb1.7h1, a h0.6 is introduced as 1.13/h0.6. Consequently, four variables, namely, a, d, h, and f, remain to be determined, and this is done by fitting all measured APG spectra according to Eq. (6).

figure

Fig. 5 Comparison between the measured data and the formulated spectra based on Eq. (6).

Now, we need to find the decisive boundary-layer parameters and the dependencies between them and the undetermined variables. The considerations are as follows:

1)

The midfrequency slope controller h is directly impacted by the boundary-layer velocity profile. From the former discussion, the shape factor H is a proper choice as it directly correlates to the mean velocity profile.

2)

The variable d impacts the spectral peak location. It is considered that it could be dependent on both the mean velocity profile and the Reynolds number. During the tests, it was found that the variable d shows a high dependence of the combination of ReθH.

3)

The amplitude manager a also depends on the Reynolds number and the mean velocity profile, which could relate to the variable d.

4)

The variable f determines the extension of the midfrequency decrease range and should depend on Reτ. Goody [1] used ReT=ReτCf/2 to prescribe the extension, and good agreement with the experimental results is shown. However, for the applied time-scale variables θ/U0, Reτ shows better agreement to the results. Figure 6 shows the best-fit lines for the variables against the selected parameters. Thus, the variables as a function of the boundary-layer parameters can be determined as follows:

a=(81.004d+2.154)107(7)
d=105.8105ReθH0.35(8)
h=1.169ln(H)+0.642(9)
f=7.645Reτ0.411(10)

figure

Fig. 6 Determined values of the variables and best-fit lines according to Eqs. (710).

The curves of the resulting functions fit well for the determined values of the variables, especially for h and f. This indicates that the selected boundary-layer parameters can feature the change of the wall pressure spectra under APGs well. Note that one could relate the variable a directly to ReθH instead of relating to the variable d like Eq. (7). Also, different formulas for the variable a, which provide similar prediction accuracy within the Reynolds number range of the selected dataset, can be found. However, between the formulas, large differences can take place at large Reynolds numbers that are larger than the range of the selected dataset. The determination of the applied formula is based on assessment of prediction for measurements [35,36] at large Reynolds numbers.

In addition, Eq. (9), implies that the wall pressure spectrum for a FPG boundary-layer flow should have a flatter spectral decrease at medium frequencies, which is consistent with the experimental results from the literature [8,9]. Certainly, the dependence between the shape factor H and the spectral midfrequency slope for the FPG flow may differ from the case for the APG.

III. Comparison with Published Models
Previous sectionNext section

In this section, the formulation of Eq. (6) is compared with other published APG pressure spectral models. The major differences between the present formulation and the other models are as follows:

1)

The present formulation uses a more representative normalization with uτ/Q2θ for the APG spectra instead of Ue/τw2δ,δ* used in the other models.

2)

The shape factor H is used to operate on the midfrequency decrease. Contrarily, the other models, except the KBLWK model, which has a constant slope at medium frequencies, use Clauser’s [12] equilibrium parameter as the driving parameter, which, based on the former discussion, is not necessarily appropriate for applying in a nonequilibrium APG boundary layer.

3)

The variable b that manages the low-frequency increase is changed to be b=1. In the other models, b=2, which is adopted from Goody’s [1] model, is used.

Besides the data from Hu and Herr [10], four other experiments at four different facilities for measuring wall pressure fluctuations beneath APG boundary layers are selected to investigate the spectral models. Additionally, one case for ZPG boundary layers from Hu and Herr is also included. APG boundary layers of the selected experiments were realized on three different conceptions: on a flat plate with airfoils on top of it (Schloemer [2] and Hu and Herr), on tapered trailing edges of a flat plate (Catlett et al. [7,22]), and on airfoils (Herrig et al. [6] and Suryadi and Herr [9]). A brief summary of the experimental setups of the four other selected test cases will be provided here. For detailed description of the experiments, the reader is referred to the respective papers.

Schloemer [2] conducted measurements in the low-turbulence subsonic wind tunnel at Stevens Institute of Technology. A flat plate was installed in the closed test section. Wall pressure spectra were measured by flush-mounted Atlantic Research type LD 107-M transducers with approximately 1.5 mm diameter. An APG was achieved by a half NACA 0015 airfoil attached on the top channel wall. Wall pressure spectra and flow properties measured by hot-wire anemometers were only provided for one single position.

Catlett et al. [7,22] carried out measurements in the open-jet section of the Anechoic Flow Facility at the Naval Surface Warfare Center, Carderock Division. Wall fluctuating pressures were measured with flush-mounted sensors on tapered trailing edge sections of a flat plate with three different wedge angles (7, 12, and 17 deg related to the plate plane). Flow parameters were measured by hot-wire anemometers at several different streamwise positions. However, the flow measurements were limited to the wake region only, and the mean flow velocity of the inner layer was estimated by a best fit to the theoretical boundary-layer profiles.

Suryadi and Herr [9] measured the wall pressure fluctuations with pinhole Kulite sensors on a DU96 airfoil at chord positions from x/c=0.770.96 in the AWB. Boundary-layer parameters at the measurement positions were evaluated by XFOIL calculations. The values of pressure gradients were derived from the measured data. Data from three streamwise positions on the suction side of the airfoil at two AOAs of 0.8 and 3.2 deg were collected for the comparison.

Herrig et al. [6] measured the wall pressure spectra with flush-mounted 1.6 mm diameter Kulite sensors with the so-called B screen (eight 0.2 mm diameter holes around a 1.2 mm diameter circle) at the chord position x/c0.99 on a NACA 0012 airfoil in a closed test section in the Laminar Wind Tunnel of the University of Stuttgart. Flow properties were provided by XFOIL calculations. Data from AOAs of 0 and 4 deg on the suction side of the NACA 0012 airfoil are collected in this section.

Mean flow properties of turbulent boundary layers for the other selected experiments are summarized in Table 2. In the literature, the local freestream velocity U0 is provided for the flat plate boundary layers (the cases of Schloemer [2] and Hu and Herr [10]), and the boundary-layer edge velocity Ue is provided for the boundary layers measured on tapered trailing edges or on airfoils (the cases of Herrig et al. [6], Catlett et al. [7,22], and Suryadi and Herr [9]). For convenience, the boundary-layer edge velocity is converted into the local freestream velocity using the relationship Ue=0.99U0. The positions listed in the test case of Catlett et al. are the distance upstream of the trailing edge. Boundary-layer parameters from the test case of Catlett et al. were acquired by digitizing the plots of measured mean flow properties. However, pressure gradient values from this test case are not available, and these are estimated by making a best fit to the provided prediction of the CAFS model.

Figures 711 show comparisons of the predicted spectra for the APG test cases between the models, and Fig. 12 shows comparisons for the ZPG case. Spectra from the RRM model present no clearly different slopes between medium and high frequencies, except for the cases for ZPGs and very weak APGs, e.g., the test case of Hu and Herr [10], AOA=6  deg at x=1128  mm, where the spectra roll off at high frequencies with a much faster slope than the measured ones. The reason for that is the function A2=min(3,19/RT)+7 in Eq. (1), which could result in a faster roll-off at high frequencies for a small RT. A poor prediction of the spectral slope at medium frequencies for the test case of Catlett et al. [7,22] is shown in Fig. 8. For the test cases of Hu and Herr [10], Catlett et al. [7,22], and Suryadi and Herr [9], more than 5 dB discrepancy in the peak level is found, and the maximum discrepancy is about 12 dB found in the test case of Suryadi and Herr. Good agreement with the test cases of Schloemer [2] and Herrig et al. [6] is obtained.

figure

Fig. 7 Predictions of different models for the test case of Hu and Herr [10]: a) AOA=6  deg at x=1128  mm, b) AOA=6  deg at x=1210  mm, c) AOA=10  deg at x=1128  mm, d) AOA=10  deg at x=1210  mm, e) AOA=14  deg at x=1128  mm, and f) AOA=14  deg at x=1210  mm.

figure

Fig. 8 Predictions of different models for the test case of Catlett et al. [7,22]: a) open angle of 7 deg at x=50  mm, b) open angle of 7 deg at x=204  mm, c) open angle of 7 deg at x=406  mm, d) open angle of 12 deg at x=154  mm, e) open angle of 12 deg at x=210  mm, and f) open angle of 17 deg at x=106  mm.

figure

Fig. 9 Predictions of different models for the test case of Suryadi and Herr [9]: a) AOA=0.8  deg at x/c=0.77, b) AOA=0.8  deg at x/c=0.88, c) AOA=0.8  deg at x/c=0.96, d) AOA=3.2  deg at x/c=0.77, e) AOA=3.2  deg at x/c=0.88, and f) AOA=3.2  deg at x/c=0.96.

figure

Fig. 10 Predictions of different models for the test case of Schloemer [2].

figure

Fig. 11 Predictions of different models for the test case of Herrig et al. [6]: a) AOA=0  deg and b) AOA=4  deg.

figure

Fig. 12 Predictions of different models for the test case of Hu and Herr [10] for ZPG: a) U0=30.2  m/s and b) U0=39.2  m/s.

The KBLWK model formulates a constant spectral slope at low, medium, and high frequencies; only the extension of the midfrequency range and the spectral amplitude are governed by boundary-layer parameters. Therefore, a slope variation at medium frequencies due to APG effects shown in the test cases of Catlett et al. [7,22], Suryadi and Herr [9], and Hu and Herr [10] cannot be predicted, and the predicted slope at medium frequencies is too steep for ZPG cases. A good prediction of the peak amplitude is obtained except for the test case of Catlett et al., which shows a discrepancy of 10 dB. The spectral peak location is well predicted for most test cases. Good agreement with measured spectra at positions in the vicinity of trailing edge is shown in Figs. 9c, 9f, and 11.

The CAFS model underpredicts the spectral amplitude for all test cases except for the case of Catlett et al. [7,22]. The discrepancy can be larger than 15 dB. The trend of variation of the spectral slope at medium frequencies is not well predicted; e.g., a contradictory trend is shown in Figs. 7a, 7f, 9a, and 9f, in which the slope at medium frequencies should be steeper due to a stronger APG.

Equation (6) predicts well the spectral slope at medium frequencies and the roll-off frequency at high frequencies for the test cases of Catlett et al. [7,22], Suryadi and Herr [9], and Hu and Herr [10], except for one case with the wedge angle of 17 deg of Catlett et al., which may be caused by a boundary-layer separation occurring upstream of the measurement position. For the cases of Schloemer [2] and Herrig et al. [6], a much flatter spectral slope is predicted at medium and high frequencies. For those measurements, flush-mounted sensors with diameters of 1.5 and 1.6 mm were used to measure the wall pressure fluctuations, which cause an attenuation in spectral amplitude at medium and high frequencies due to the large sensor size. Although the measured spectra were corrected using the Corcos correction [24], uncertainties at higher frequencies could still be caused, which can explain the difference at higher frequencies between the prediction and those measurements. The Corcos correction assumed a uniform sensitivity for the sensors, whereas an actual sensor has a deflective sensitivity; e.g., for a condenser microphone, the sensitivity has the maximum at the center and decreases near the edge. Blake [27] showed that the measured acceptance of a condenser microphone at a higher wave number domain could be more than 5 dB smaller than the calculated theoretical acceptance with a uniform sensitivity. The difference is noticeable from ωr/Uc>1. This discrepancy at the acceptance will cause a smaller amplitude at higher frequencies even after using the Corcos correction. A new result from Hu and Erbig [37] for the wall pressure spectra measured by flush-mounted 2.54 mm diameter Kulite sensors with the so-called B screen for ZPG boundary layers shows an undercorrection using the Corcos correction from about ωr/Uc>0.5, and until ωr/Uc=1 the undercorrection can reach up to 3 dB. Another issue that should be also considered when using the Corcos correction is that the wall pressure convective velocity is much smaller for an APG boundary layer than for a ZPG boundary layer. It seems to make more sense to use the phase velocity determined from the closest distance (the order of the sensor size). The convective velocity Uc(ω) for an APG boundary layer at the closest distance could be less than 0.3U0 at high frequencies [10], which is much slower than the usually used 0.60.8U0. The too-large convective velocity used in the Corcos correction will lead to an undercorrection for the wall spectra. In addition, the APG increases the streamwise turbulence decay compared to the ZPG. A larger turbulence decay can further increase the attenuation due to the finite sensor size.

Furthermore, a good prediction of the peak amplitude is also obtained, except for the test case of Suryadi and Herr [9], which is mainly due to the imprecisely predicted spectral peak location. A slope of ω at low frequencies used in this formulation shows a better agreement with the measured spectra than the other models that possess an ω2 slope.

IV. Further Considerations
Previous sectionNext section

The results from the previous section show that Eq. (6) provides the most precise prediction among other models. However, for some cases, the prediction of the spectral peak position shows relatively large errors. This can also cause a large discrepancy for the prediction of the peak level, especially for the spectrum with a steeper midfrequency decrease. Therefore, an improvement of the prediction of the spectral peak position is essential. In this section, a new model, which aims to improve Eq. (6) in terms of the prediction of the peak location and level, will be proposed.

However, to figure out the decisive parameters for the peak position and the respective relationship between them is difficult. For example, the peak position also depends on the low-frequency spectral slope that varies in different APG conditions, whereas in the current formulation, a constant slope is applied. Note that the KBLWK model formulates a constant peak location in the nondimensional frequency domain and the prediction of the peak location shows good agreement with measured spectra. This indicates that the dependence of the peak position on the boundary-layer parameters may be weak and the peak position could be located in a narrow nondimensional frequency range.

Figure 13 shows the scaled spectra from the test cases used for the comparison. Configurations not included in this plot are as follows:

1)

Spectra measured in the vicinity of the trailing edge, i.e., measurements from Suryadi and Herr [9] at x/c=0.96 and Herrig et al. [6], are not included. For these configurations, spectra may likely be impacted by the trailing edge scattering effect. Furthermore, the boundary-layer parameter provided by XFOIL calculations may be imprecise in the vicinity of the trailing edge, especially for a larger AOA [38].

2)

The measurement position located not far downstream from a boundary-layer separation, i.e., the measurement from Catlett et al. [7,22] with the open angle of 17 deg, is not included.

figure

Fig. 13 Scaled spectra for the test cases of Hu and Herr, Schloemer (-), Catlett et al. (- -), and Suryadi & Herr (dotted dashe line).

A noteworthy finding from Fig. 13 is that the scaled spectra can be divided into three groups: group I for the cases of Schloemer [2] and Hu and Herr [10], with APG boundary layers developed at a flat plate with airfoils mounted above, 18.8  mmδ35.0  mm; group II for the case of Catlett et al. [7,22], with APG boundary layers developed at a tapered trailing edge section of a flat plate, 66.5  mmδ91.2  mm; group III for the case of Suryadi and Herr [9], with APG boundary layers developed on the airfoil suction side, 6.1  mmδ13.2  mm. The spectral peaks of each group collapse well by themselves, which may indicate a good scaling of the spectral peak is given when boundary layers experience a similar development history or the boundary-layer thicknesses have the same order. Although a good peak collapse is shown in each test case by itself, the differences in peak amplitude are still about 10 dB between different test configurations.

Nevertheless, the spectral peak location is located in a small range of ωθ/U0 of 0.2–0.35, i.e., a constant value for the variable d in the denominator in Eq. (6) could be used, with which the peak can be located in the range of ωθ/U0 of 0.2–0.35, and a value of 0.07 is found. The spectral amplitude in Eq. (6) is nearly independent of the choice of the value of d at higher frequencies. Therefore, the amplitude function can keep the form as it is. Thus, the model is rewritten as

ϕ(ω)uτQ2θ=(81.004d+2.154)107(ωθ/U0)[(ωθ/U0)1.5  h1.6+0.07]1.13/h0.6+[7.645Reτ0.411(ωθ/U0)]6(11)
where log10(d)=5.8105ReθH0.35 and h=1.169ln(H)+0.642.

Figures 1418 show predictions from the proposed model for the APG test cases, and Fig. 19 shows predictions for the ZPG case. The proposed model shows good agreement with the measured spectra. The spectral slope over the whole frequency range is well predicted. Exceptions are the slope at medium and high frequencies for the cases of Schloemer [2] and Herrig et al. [6]. As discussed previously, this discrepancy is probably caused by the large sensor size used in those measurements. Besides the sensor effect, the trailing edge scattering effect and a possible inaccuracy of the boundary-layer parameters in the vicinity of trailing edge provided by XFOIL calculation may likely produce the prediction uncertainty and increase the discrepancy compared to the measured spectra. This issue affects the results for the case of Herrig et al. and the case of Suryadi and Herr [9] at x/c=0.96, and the discrepancy is larger at a larger AOA. A poor prediction is made for the case of Catlett et al. [7,22] with the wedge angle of 17 deg, which is probably due to a separated boundary layer located near upstream of the measurement position. In addition, a good prediction is also obtained for the ZPG boundary layers.

figure

Fig. 14 Predictions of the new model for the test case of Hu and Herr [10]: a) AOA=6  deg at x=1128  mm, b) AOA=6  deg at x=1210  mm, c) AOA=10  deg at x=1128  mm, d) AOA=10  deg at x=1210  mm, e) AOA=14  deg at x=1128  mm, and f) AOA=14  deg at x=1210  mm.

figure

Fig. 15 Predictions of the new model for the test case of Catlett et al. [7,22]: a) open angle of 7 deg at x=50  mm, b) open angle of 7 deg at x=204  mm, c) open angle of 7 deg at x=406  mm, d) open angle of 12 deg at x=154  mm, e) open angle of 12 deg at x=210  mm, and f) open angle of 17 deg at x=106  mm.

figure

Fig. 16 Predictions of the new model for the test case of Suryadi and Herr [9]: a) AOA=0.8  deg at x/c=0.77, b) AOA=0.8  deg at x/c=0.88, c) AOA=0.8  deg at x/c=0.96, d) AOA=3.2  deg at x/c=0.77, e) AOA=3.2  deg at x/c=0.88, and f) AOA=3.2  deg at x/c=0.96.

figure

Fig. 17 Predictions of the new model for the test case of Schloemer [2].

figure

Fig. 18 Predictions of the new model for the test case of Herrig et al. [6]: a) AOA=0  deg and b) AOA=4  deg.

figure

Fig. 19 Predictions of the new model for the test case of Hu and Herr [10] for ZPG: a) U0=30.2  m/s and b) U0=39.2  m/s.

V. Conclusions
Previous sectionNext section

A new spectral model of wall pressure fluctuations including APG effects was proposed based on a dataset of five experiments at four different test facilities. The ZPG wall pressure spectral model from Goody [1] is taken as the basic form to develop the new model. This is because of the good collapse of the spectral peak using the dynamic pressure as the scaling parameter, a clear trend of the spectral slope change at medium frequencies due to the pressure gradient, and a nearly unchanged spectral slope at high frequencies compared to the ZPG cases. The proposed model in this work is compared to other published APG spectral models. There are three major differences between the proposed model and the other models. First, instead of Ue/τw2δ(δ*), uτ/Q2θ was used based on a good collapse of the spectral peaks for the measured spectra at ZPGs and APGs when scaling with this parameter, whereas an over 15 dB difference between the spectra was found when scaled using Ue/τw2δ. As discussed in this paper, the dynamic pressure Q could be more appropriate to scale the APG spectra than the usually used τw for ZPGs. Second, the boundary-layer shape factor was used to evaluate the spectral slope at medium frequencies instead of Clauser’s [12] equilibrium parameter. Overall, the wall pressure fluctuations are mostly affected by the boundary-layer mean velocity profile and the Reynolds stresses. The Reynolds stresses are again tightly related to the mean velocity profile. Therefore, the boundary-layer mean velocity profile could be an essential criterion to determine the shape of the wall pressure spectra. The connection between the mean velocity profile and the wall pressure spectral shape was demonstrated. On the one hand, the spectrum was almost only effected by the local boundary layer; on the other hand, the local boundary-layer parameters were predominantly determined by the upstream history of the flow. The measured data illustrated that the shape factor can represent well the boundary-layer profile development trend for different configurations and streamwise positions, whereas Clauser’s parameter may predict wrong development trends for different streamwise positions where the pressure gradients change rapidly. An excellent match of the spectral slope at medium frequencies between the predictions and the measured data was shown using the shape factor as the control parameter. Third, instead of an ω2 increase at low frequencies, a slope of ω, which was derived as an averaged value from the measured results from the literature for the APG cases, was used in the model. Arguments for replacement of the classic ω2 are as follows:

1)

The slope of ω2 is obtained by assuming a frozen flow and only counting the mean-shear source term. However, when dealing with a nonfrozen flow, the slope becomes flatter because the energy from higher frequencies spreads into lower frequencies.

2)

Wall pressure fluctuation spectra from the turbulence–turbulence term show a plateau at lower frequencies and take over the importance of the mean-shear term in the spectra for a ZPG boundary layer. However, exact knowledge of the importance of the turbulence–turbulence term for a non-ZPG boundary layer is still lacking.

The present model was validated by the selected dataset in a range of 2.7103<Reθ<1.5104, 1.4<H<2.15, 9.0<U0<62.7  m/s, and 6.1<δ<91.2  mm. Good prediction accurary was obtained, except for some specific configurations. These are as follows:

1)

Spectra were measured in the vicinity of a trailing edge. The reason for the unsatisfied prediction accuracy could be due to the trailing edge scattering effect and possible imprecise estimations of input boundary-layer parameters in the trailing edge area.

2)

Spectra were measured at positions not far downstream of a boundary-layer separation.

Good agreement for ZPG boundary layers from the proposed model was also obtained.

A. NaguibAssociate Editor
Acknowledgments
Previous sectionNext section

This work was conducted in the framework of the DLR, German Aerospace Center, project Comfort and Efficiency Enhancing Technologies (CENT).

References
Previous section
[1] Goody, M. C., "Empirical Spectral Model of Surface Pressure Fluctuations", _AIAA Journal_, Vol. 42, No. 9, 2004, pp. 1788-1794https://doi.org/10.2514/1.9433 [Abstract] [Google Scholar]
[2] Schloemer, H. H., "Effects of Pressure Gradients on Turbulent-Boundary-Layer Wall-Pressure Fluctuations", _Journal of the Acoustical Society of America_, Vol. 42, No. 1, 1967, pp. 93-113https://doi.org/10.1121/1.1910581 [Google Scholar]
[3] Burton, T. E., "Wall Pressure Fluctuations at Smooth and Rough Surfaces under Turbulent Boundary Layers with Favorable and Adverse Pressure Gradients", __, 1973, [Google Scholar]
[4] Blake, W. K., "A Statistical Description of Pressure and Velocity Fields at Trailing Edges of a Flat Strut", __, 1975, [Google Scholar]
[5] Simpson, R., Ghodbane, M., McGrath, B., "Surface Pressure Fluctuations in a Separating Turbulent Boundary Layer", _Journal of Fluid Mechanics_, Vol. 177, April 1987, pp. 167-186https://doi.org/10.1017/S0022112087000909 [Google Scholar]
[6] Herrig, A., "Validation and Application of a Hot-Wire Based Method for Trailing Edge Noise Measurements on Airfoils", __, 2012, [Google Scholar]
[7] Catlett, M. R., Forest, J. B., Anderson, J. M., Stewart, D. O., "Empirical Spectral Model of Surface Pressure Fluctuations Beneath Adverse Pressure Gradients", __, 2014, [Abstract] [Google Scholar]
[8] Salze, E., Bailly, C., Marsden, O., Jondeau, E., Juvé, D., "An Experimental Characterisation of Wall Pressure Wavevector-Frequency Spectra in the Presence of Pressure Gradients", __, 2014, [Abstract] [Google Scholar]
[9] Suryadi, A., Herr, M., "Wall Pressure Spectra on a DU96-W-180 Profile from Low to Pre-Stall Angles of Attack", __, 2015, [Abstract] [Google Scholar]
[10] Hu, N., Herr, M., "Characteristics of Wall Pressure Fluctuations for a Flat Plate Turbulent Boundary Layer with Pressure Gradients", __, 2016, [Abstract] [Google Scholar]
[11] Rozenberg, Y., Robert, G., Moreau, S., "Wall-Pressure Spectral Model Including the Adverse Pressure Gradient Effects", _AIAA Journal_, Vol. 50, No. 10, 2012, pp. 2168-2179https://doi.org/10.2514/1.J051500 [Abstract] [Google Scholar]
[12] Clauser, F. H., "Turbulent Boundary Layers in Adverse Pressure Gradients", _Journal of the Aeronautical Sciences_, Vol. 21, No. 2, 1954, pp. 91-108https://doi.org/10.2514/8.2938 [Abstract] [Google Scholar]
[13] Kamruzzaman, M., Bekiropoulos, D., Lutz, T., Würz, W., "A Semi-Empirical Surface Pressure Spectrum Model for Airfoil Trailing-Edge Noise Prediction", _International Journal of Aeroacoustics_, Vol. 14, No. 5–6, 2015, pp. 882-883https://doi.org/10.1260/1475-472X.14.5-6.833 [Google Scholar]
[14] Wolf, A., Kamruzzaman, M., Würz, W., Lutz, T., Krämer, E., "Wall Pressure Fluctuation (WPF) and Trailing-Edge Noise Measurements on a NACA64-418 Airfoil", __, 2009, [Google Scholar]
[15] Rozenberg, Y., "Modélisation Analytique du bruit Aérodynamique à Large Bande des Machines Tournantes: Utilisation de Calculs Moyennés de Mécanique des Fluides", __, 2007, [Google Scholar]
[16] Bertagnolio, F., "Boundary Layer Measurements of the NACA0015 and Implications for Noise Modeling", __, 2011, [Google Scholar]
[17] Garcia-Sagrado, A., Hynes, T., "Stochastic Estimation of Flow near the Trailing Edge of a NACA0012 Airfoil", _Experiments in Fluids_, Vol. 51, No. 4, 2011, pp. 1057-1071https://doi.org/10.1007/s00348-011-1071-9 [Google Scholar]
[18] Brooks, T. F., Hodgson, T. H., "Trailing Edge Noise Prediction from Measured Surface Pressure", _Journal of Sound and Vibration_, Vol. 78, No. 1, 1981, pp. 69-117https://doi.org/10.1016/S0022-460X(81)80158-7 [Google Scholar]
[19] Mellor, G. L., Gibson, D. M., "Equilibrium Turbulent Boundary Layers", _Journal of Fluid Mechanics_, Vol. 24, No. 2, 1966, pp. 225-253https://doi.org/10.1017/S0022112066000612 [Google Scholar]
[20] Cole, D., "The Law of the Wake in the Turbulent Boundary Layer", _Journal of Fluid Mechanics_, Vol. 1, No. 2, 1956, pp. 191-226https://doi.org/10.1017/S0022112056000135 [Google Scholar]
[21] White, F. M., _Viscous Fluid Flow_, 1991, pp. 411-421 [Google Scholar]
[22] Catlett, M. R., Anderson, J. M., Forest, J. B., Stewart, D. O., "Empirical Modeling of Pressure Spectra in Adverse Pressure Gradient Turbulent Boundary Layers", _AIAA Journal_, Vol. 54, No. 2, 2016, pp. 569-587https://doi.org/10.2514/1.J054375 [Abstract] [Google Scholar]
[23] Herring, H. J., Norbury, J. F., "Some Experiments on Equilibrium Turbulent Boundary Layers in Favorable Pressure Gradients", _Journal of Fluid Mechanics_, Vol. 27, No. 3, 1967, pp. 541-549https://doi.org/10.1017/S0022112067000527 [Google Scholar]
[24] Corcos, G. M., "Resolution of Pressure in Turbulence", _Journal of the Acoustical Society of America_, Vol. 35, No. 2, 1963, pp. 192-199https://doi.org/10.1121/1.1918431 [Google Scholar]
[25] Howe, M., _Acoustics of Fluid-Structure Interactions_, 1998, pp. 208 [Google Scholar]
[26] Panton, R. L., Linebarger, J. H., "Wall Pressure Spectra for Equilibrium Boundary Layers", _Journal of Fluid Mechanics_, Vol. 65, No. 2, 1974, pp. 261-287https://doi.org/10.1017/S0022112074001388 [Google Scholar]
[27] Blake, W. K., _Mechanics of Flow-Induced Sound and Vibration_, 1986, pp. 507-529 [Google Scholar]
[28] Hu, N., Reiche, N., Ewert, R., "Simulation of Turbulent Boundary Layer Wall Pressure Fluctuations via Poisson Equation and Synthetic Turbulence", _Journal of Fluid Mechanics_, Vol. 826, Sept. 2017, pp. 421-454https://doi.org/10.1017/jfm.2017.448 [Google Scholar]
[29] Chase, D. M., "Modeling the Wave-Vector Frequency Spectrum of Turbulent Boundary Layer Wall Pressure", _Journal of Sound and Vibration_, Vol. 70, No. 1, 1980, pp. 29-67https://doi.org/10.1016/0022-460X(80)90553-2 [Google Scholar]
[30] Kraichnan, R. H., "Pressure Fluctuations in Turbulent Flow over a Flat Plate", _Journal of the Acoustical Society of America_, Vol. 28, No. 3, 1956, pp. 378-390https://doi.org/10.1121/1.1908336 [Google Scholar]
[31] Meecham, W. C., Tavis, M. T., "Theoretical Pressure Correlation Functions in Turbulent Boundary Layer", _Physics of Fluids_, Vol. 23, No. 6, 1980, pp. 1119-1131https://doi.org/10.1063/1.863114 [Google Scholar]
[32] Kim, J., "On the Structure of Pressure Fluctuations in Simulated Turbulent Channel Flow", _Journal of Fluid Mechanics_, Vol. 205, Aug. 1989, pp. 421-451https://doi.org/10.1017/S0022112089002090 [Google Scholar]
[33] Chang, P., Piomelli, U., Blake, W. K., "Relationship Between Wall Pressure and Velocity-Field Sources", _Physics of Fluids_, Vol. 11, No. 11, 1999, pp. 3434-3448https://doi.org/10.1063/1.870202 [Google Scholar]
[34] Farabee, T. M., Casarella, M. J., "Spectral Features of Wall Pressure Fluctuations Beneath Turbulent Boundary Layers", _Physics of Fluids_, Vol. A3, No. 10, 1991, pp. 2410-2420https://doi.org/10.1063/1.858179 [Google Scholar]
[35] Ehrenfried, K., Koop, L., "Experimental Study of Pressure Fluctuations Beneath a Compressible Turbulent Boundary Layer", __, 2008, [Google Scholar]
[36] Klabes, A., Appel, C., Herr, M., Bouhaj, M., "Fuselage Excitation During Cruise Flight Conditions: Measurement and Prediction of Pressure Point Spectra", __, 2015, [Google Scholar]
[37] Hu, N., Erbig, L., "Effect of Flush-Mounted Sensors and Upstream Flow Development on Measured Wall Pressure Spectra", __, 2018, [Abstract] [Google Scholar]
[38] Suryadi, A., Martens, S., Herr, M., "Trailing-Edge Noise Reduction Technologies for Applications in Wind Energy", __, 2017, [Abstract] [Google Scholar]