Empirical Model of Wall Pressure Spectra in Adverse Pressure Gradients
Sections: 
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 boundarylayer parameters and the wall pressure are studied and demonstrated. A dataset for five experiments at four different test facilities, covering Reynolds number between $2.6\cdot {10}^{3}$ and $1.9\cdot {10}^{4}$ based on the boundarylayer 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  
${C}_{f}$  skin friction coefficient, ${\tau}_{w}/Q$ 
${C}_{p}$  pressure coefficient, $(p{p}_{\infty})/(0.5\rho {U}_{\infty}^{2})$ 
$H$  boundarylayer shape factor, ${\delta}^{*}/\theta $ 
$k$  wave number, ${\mathrm{m}}^{1}$ 
$p$, ${p}_{\infty}$  pressure and pressure at wind tunnel nozzle exit, Pa 
$Q$  dynamic pressure, $0.5\rho {U}_{0}^{2}$, Pa 
${R}_{T}$  timescale ratio, $(\delta /{U}_{e})/(\nu /{u}_{\tau}^{2})$ 
$R{e}_{\delta}$, $R{e}_{\mathrm{\Delta}}$  Reynolds number related to other boundarylayer parameters, $(\delta ,\mathrm{\Delta}){U}_{e}/\nu $ 
$R{e}_{\theta}$  Reynolds number based on boundarylayer momentum thickness, ${U}_{0}\theta /\nu $ 
$R{e}_{\tau}$  Reynolds number based on wall shear stress, ${u}_{\tau}\delta /\nu $ 
${U}_{e}$, ${U}_{0}$  boundarylayer edge velocity ($0.99{U}_{0}$) and local freestream velocity 
${U}_{\infty}$  freestream velocity at wind tunnel nozzle exit 
$u$  velocity, $\text{m}/\text{s}$ 
${u}_{\tau}$  friction velocity, $\sqrt{{\tau}_{w}/\rho}$, $\text{m}/\text{s}$ 
${u}^{+}$  dimensionless velocity, $u/{u}_{\tau}$ 
$x$, $y$, $z$  spatial coordinates, m 
${y}^{+}$  dimensionless wallnormal coordinate, $y{u}_{\tau}/\nu $ 
${\beta}_{\delta}$, ${\beta}_{\mathrm{\Delta}}$  Clauser’s equilibrium parameter related to other boundary layer parameters, $(\delta ,\mathrm{\Delta})/Q\cdot dp/dx$ 
${\beta}_{{\delta}^{*}}$, ${\beta}_{\theta}$  boundarylayer thickness and displacement thickness based Clauser’s equilibrium parameter, $({\delta}^{*},\theta )/{\tau}_{w}\cdot dp/dx$ 
$\mathrm{\Delta}$  boundarylayer defect thickness, ${\delta}^{*}\sqrt{2/{C}_{f}}$ 
${\mathrm{\Delta}}_{\delta /{\delta}^{*}}$  boundarylayer related parameter, $\delta /{\delta}^{*}$ 
$\delta $  boundarylayer thickness, m 
${\delta}^{*}$  boundarylayer displacement thickness, m 
$\theta $  boundarylayer momentum thickness, m 
$\kappa $  von Kármán constant 
$\nu $  kinematic viscosity, ${\text{m}}^{2}/\text{s}$ 
${\mathrm{\Pi}}_{{\delta}^{*}}$  Cole’s wake parameter, $0.8\cdot {({\beta}_{{\delta}^{*}}+0.5)}^{3/4}$ 
${\mathrm{\Pi}}_{\theta}$  Cole’s wake parameter related to boundarylayer momentum thickness, $0.8\cdot {({\beta}_{\theta}+0.5)}^{3/4}$ 
$\rho $  density, $\mathrm{kg}/{\text{m}}^{3}$ 
${\tau}_{w}$  wall shear stress, Pa 
$\mathrm{\varphi}(\omega )$  power spectral density of wall pressure fluctuations (singlesided), ${\text{Pa}}^{2}\cdot \text{s}$ 
$\omega $  angular frequency, $\mathrm{rad}/\text{s}$ 
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 boundarylayer parameters, e.g., parameters based on boundarylayer thickness and Clauser’s [12] equilibrium parameter, to capture the spectral changing trend due to the presence of the pressure gradient, expressed as
$$\frac{\mathrm{\varphi}(\omega ){U}_{e}}{{\tau}_{w}^{2}{\delta}^{*}}=\frac{[2.82{{\mathrm{\Delta}}_{\delta /{\delta}_{*}}}^{2}\cdot {(6.13{{\mathrm{\Delta}}_{\delta /{\delta}_{*}}}^{0.75}+{F}_{1})}^{{A}_{1}}][4.2\cdot ({\mathrm{\Pi}}_{\theta}/{\mathrm{\Delta}}_{\delta /{\delta}_{*}})+1]{(\omega {\delta}^{*}/{U}_{e})}^{2}}{{[4.76\cdot {(\omega {\delta}^{*}/{U}_{e})}^{0.75}+{F}_{1}]}^{{A}_{1}}+{[8.8{R}_{T}^{0.57}\cdot (\omega {\delta}^{*}/{U}_{e})]}^{{A}_{2}}}$$  (1) 
$${F}_{1}=4.76\cdot {(1.4/{\mathrm{\Delta}}_{\delta /{\delta}_{*}})}^{0.75}\cdot [0.375\cdot {A}_{1}1],\phantom{\rule{0ex}{0ex}}{A}_{1}=3.7+1.5{\beta}_{\theta},\phantom{\rule{0ex}{0ex}}{A}_{2}=\mathrm{min}(3,19/\sqrt{{R}_{T}})+7,\phantom{\rule{0ex}{0ex}}{\beta}_{\theta}=\theta /{\tau}_{w}\cdot dp/dx,\phantom{\rule{0ex}{0ex}}{\mathrm{\Pi}}_{\theta}=0.8\cdot {({\beta}_{\theta}+0.5)}^{3/4},\phantom{\rule{0ex}{0ex}}{\mathrm{\Delta}}_{\delta /{\delta}^{*}}=\delta /{\delta}^{*}$$ 
Clauser’s [12] equilibrium parameter ${\beta}_{\theta}$ is used to manage the slope variation at medium frequencies. The larger the value of ${\beta}_{\theta}$ is, the steeper the slope is. The spectra shift to a higher frequency and a larger amplitude as ${\mathrm{\Delta}}_{\delta /{\delta}_{*}}$ increases. Both ${\beta}_{\theta}$ and ${\mathrm{\Delta}}_{\delta /{\delta}_{*}}$ 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 [6,14
$$\frac{\mathrm{\varphi}(\omega ){U}_{e}}{{\tau}_{w}^{2}{\delta}^{*}}=\frac{0.45[1.75\cdot {({{\mathrm{\Pi}}_{{\delta}^{*}}}^{2}\cdot {{\beta}_{{\delta}^{*}}}^{2})}^{m}+15]{(\omega {\delta}^{*}/{U}_{e})}^{2}}{{[{(\omega {\delta}^{*}/{U}_{e})}^{1.637}+0.27]}^{2.47}+{[{(1.15{R}_{T})}^{2/7}\cdot (\omega {\delta}^{*}/{U}_{e})]}^{7}}$$  (2) 
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
$$\frac{\mathrm{\varphi}(\omega ){U}_{e}}{{\tau}_{w}^{2}\delta}=\frac{a\cdot {(\omega \delta /{U}_{e})}^{b}}{{[{(\omega \delta /{U}_{e})}^{c}+d]}^{e}+{[f{R}_{T}^{g}\cdot (\omega \delta /{U}_{e})]}^{h}}$$  (3) 
$$\mathcal{ln}(a{a}_{G})=4.98\cdot {({\beta}_{\mathrm{\Delta}}R{e}_{\mathrm{\Delta}}^{0.35})}^{0.131}10.7,\phantom{\rule{0ex}{0ex}}b=2,\phantom{\rule{0ex}{0ex}}c{c}_{G}=20.9\cdot {({\beta}_{\delta}R{e}_{\delta}^{0.05})}^{2.76}+0.162,\phantom{\rule{0ex}{0ex}}d{d}_{G}=0.328\cdot {({\beta}_{\mathrm{\Delta}}R{e}_{\mathrm{\Delta}}^{0.35})}^{0.310}0.103,\phantom{\rule{0ex}{0ex}}e{e}_{G}=1.93\cdot {({\beta}_{\delta}R{e}_{\delta}^{0.05})}^{0.628}+0.172,\phantom{\rule{0ex}{0ex}}f{f}_{G}=2.57\cdot {({\beta}_{\delta}R{e}_{\delta}^{0.05})}^{0.224}+1.09,\phantom{\rule{0ex}{0ex}}g{g}_{G}=38.1\cdot {({\beta}_{\delta}{H}^{0.5})}^{2.11}+0.0276,\phantom{\rule{0ex}{0ex}}h{h}_{G}=0.797\cdot {({\beta}_{\mathrm{\Delta}}R{e}_{\mathrm{\Delta}}^{0.35})}^{0.0724}0.310$$ 
Among others, Clauser’s [12] equilibrium parameter ${\beta}_{\delta ,{\delta}^{*},\theta ,\mathrm{\Delta}}$ defined with different boundarylayer thicknesses is used as one important input quantity in all models. Equilibrium flows hold a constant ${\beta}_{{\delta}^{*}}$, e.g., ${\beta}_{{\delta}^{*}}=0$ 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 ${\beta}_{{\delta}^{*}}$ 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 ${\beta}_{{\delta}^{*}}$ is considered a wellsuited 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 boundarylayer shape factor $H={\delta}^{*}/\theta $ represents a bettersuited nondimensional quantity to cope with the prediction of wall pressures under arbitrary nonequilibrium flow conditions, in which also the history of the boundarylayer development is considered important. In the following, the effects of APGs on the boundarylayer 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.
Hu and Herr [10] measured the wall pressure fluctuations with pinholemounted Kulite sensors on a plate model in the openjet 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 $1128\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}\le x\le 1210\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}$ ($x=0$ for the leading edge tip of the plate). The mean flow velocity was measured by hotwire anemometers at positions $x=1128\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}$ and $x=1210\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}$, where the most upstream and downstream Kulite sensors were located.
Figure 1a shows the measured distributions of the pressure coefficient ${C}_{p}$ on the plate model between $930\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}\le x\le 1220\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}$. Figure 1b shows the mean velocity profiles for ZPG and APG boundary layers at $x=1210\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}$ for the freestream velocity of $30.2\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{m}/\mathrm{s}$. 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 $0.1\u20130.2\delta $. The measured effects of APGs on the mean velocity profiles are consistent with the experimental results provided by White [21].
Table 1 summarizes the relevant boundarylayer parameters for the two selected velocity measurement positions. Note that the local pressure gradients $dp/dx$ decrease when the AOAs increase at $x=1210\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}$. 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] ${\beta}_{{\delta}^{*}}$ can represent the ZPG and APG effects on the mean velocity profile shape. The larger the value of $H$ and ${\beta}_{{\delta}^{*}}$, the more strongly the mean flow profile is affected by the APGs. However, compared to $H$, ${\beta}_{{\delta}^{*}}$ is directly impacted by the local pressure gradient, and a stronger local $dp/dx$ does not necessarily indicate the upstream boundarylayer 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 $x=1210\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}$ are still significantly affected by the strong initial APG conditions. The shape factor $H$ is larger at the downstream position, which represents well the boundarylayer profile development from upstream to downstream. On the contrary, because of the larger value of the local $dp/dx$ at $x=1128\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}$, the value of ${\beta}_{{\delta}^{*}}$ 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 ${\beta}_{{\delta}^{*}}$ [12,19,23]. The measured APG boundary layers in the current study show reversed trends, indicating that Clauser’s equilibrium parameter [19] ${\beta}_{{\delta}^{*}}$ 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 3 shows the wall pressure power spectral densities (PSDs) for APG and ZPG boundary layers at the freestream velocity of $30.2\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{m}/\mathrm{s}$. Spectral levels in this paper are referenced to a $20\mu {P}_{a}$ reference pressure. At low frequencies ($<100\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{Hz}$), the measured spectra are contaminated by the openjet free shear layer. The used 0.5 mm diameter pinhole arrangement provides a high spatial resolution, which ensures a small attenuation ($<1.5\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{dB}$, according to Corcos’s [24] correction) until 8 kHz for the ZPG case. A Corcos correction is not applied in the present spectra.
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 rolloff slope of ${\omega}^{5}$ 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 ${u}_{\tau}/{Q}^{2}\theta $ and $\theta /{U}_{0}$; see Fig. 4b. All the measured APG spectra at $x=1128$ and 1210 mm including the spectra for freestream velocities of 20.3 and $39.1\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{m}/\mathrm{s}$ 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 ${\tau}_{w}$ due to the increasing importance of the boundarylayer outer layer for the spectral peak.
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 highfrequency rolloff 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
$$\frac{\mathrm{\varphi}(\omega ){U}_{e}}{{\tau}_{w}^{2}\delta}=\frac{a\cdot {(\omega \delta /{U}_{e})}^{b}}{{[{(\omega \delta /{U}_{e})}^{c}+d]}^{e}+{[f{R}_{T}^{g}\cdot (\omega \delta /{U}_{e})]}^{h}}$$  (4) 
$$\frac{\mathrm{\varphi}(\omega ){u}_{\tau}}{{Q}^{2}\theta}=\frac{a\cdot {(\omega \theta /{U}_{0})}^{b}}{{[{(\omega \theta /{U}_{0})}^{c}+d]}^{e}+{[f\cdot (\omega \theta /{U}_{0})]}^{g}}$$  (5) 
The variables $a\u2013g$ in this equation control the shape of the dimensionless spectra. Here, $f{R}_{T}^{g}$ 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 $c\cdot eb$ 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 ${\omega}^{0.7}$ at medium frequencies and ${\omega}^{5}$ at high frequencies. These slopes can be realized by means of the combination of the variables, which follows $bc\cdot e=0.7$ and $bg=5$. Goody [1] adopted $b=2$ from the Chase–Howe [25] model, which implies an ${\omega}^{2}$ 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 ${k}_{1}^{2}/{k}^{2}\mathrm{exp}(2ky)$ in the solution, in which ${k}_{1}$ is the wave number in the streamwise direction, ${k}^{2}={k}_{1}^{2}+{k}_{3}^{2}$, and $y$ is the wallnormal distance to the wall, an ${\omega}^{2}$ 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 ${\omega}^{2}$ 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 ${k}_{1}^{2}/{k}^{2}\mathrm{exp}(2ky)$. As a result, the slope of the lowfrequency increase turns out to be smaller. This effect was also reported by Chase [29]. Furthermore, in the Poisson equation, the meanshear 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 meanshear term and claimed the dominance of the meanshear term for the wall mean square pressure. However, numerical results [28,32,33] show that the meanshear 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 lowfrequency range can be also affected by considering the effect of the turbulence–turbulence term. In the literature, the lowfrequency spectral slope is reported between ${\omega}^{0.2\u20130.8}$ for the ZPG spectra, except for the work of Farabee and Casarella [34], who found an ${\omega}^{2}$ increase at the very lowest frequency range $<10\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{Hz}$ ($\omega \delta /{U}_{0}<0.08$).
The measured APG spectra from Hu and Herr [10] show a slope of ${\omega}^{0.6\u20131.0}$ 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 lowfrequency increase at a greater APG, and the increase can reach about ${\omega}^{1.4}$ as the APG boundary layer approaches separation. From those observations, it can be drawn that the lowfrequency 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 lowfrequency contamination of most measured data, it is hard to figure out the dependence between the lowfrequency 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 $c\cdot e=b+0.7=1.7$. 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 $c\cdot e\cdot hb$; 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 ${(\omega \theta /{U}_{0})}^{c}$. 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
$$\frac{\mathrm{\varphi}(\omega ){u}_{\tau}}{{Q}^{2}\theta}=\frac{a\cdot {(\omega \theta /{U}_{0})}^{1.0}}{{[{(\omega \theta /{U}_{0})}^{1.5\cdot {h}^{1.6}}+d]}^{1.13/{h}^{0.6}}+{[f\cdot (\omega \theta /{U}_{0})]}^{6.0}}$$  (6) 
Now, we need to find the decisive boundarylayer 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 boundarylayer 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 $R{e}_{\theta}H$.  
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 $R{e}_{\tau}$. Goody [1] used $R{e}_{T}=R{e}_{\tau}\sqrt{{C}_{f}/2}$ to prescribe the extension, and good agreement with the experimental results is shown. However, for the applied timescale variables $\theta /{U}_{0}$, $R{e}_{\tau}$ shows better agreement to the results. Figure 6 shows the bestfit lines for the variables against the selected parameters. Thus, the variables as a function of the boundarylayer parameters can be determined as follows:

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 boundarylayer parameters can feature the change of the wall pressure spectra under APGs well. Note that one could relate the variable $a$ directly to $R{e}_{\theta}H$ 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 boundarylayer 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.
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 ${u}_{\tau}/{Q}^{2}\theta $ for the APG spectra instead of ${U}_{e}/{\tau}_{w}^{2}\delta ,{\delta}^{*}$ 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 lowfrequency 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 lowturbulence subsonic wind tunnel at Stevens Institute of Technology. A flat plate was installed in the closed test section. Wall pressure spectra were measured by flushmounted Atlantic Research type LD 107M 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 hotwire anemometers were only provided for one single position.
Catlett et al. [7,22] carried out measurements in the openjet section of the Anechoic Flow Facility at the Naval Surface Warfare Center, Carderock Division. Wall fluctuating pressures were measured with flushmounted 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 hotwire 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 boundarylayer profiles.
Suryadi and Herr [9] measured the wall pressure fluctuations with pinhole Kulite sensors on a DU96 airfoil at chord positions from $x/c=0.77\u20130.96$ in the AWB. Boundarylayer 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 flushmounted 1.6 mm diameter Kulite sensors with the socalled B screen (eight 0.2 mm diameter holes around a 1.2 mm diameter circle) at the chord position $x/c\approx 0.99$ 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 ${U}_{0}$ is provided for the flat plate boundary layers (the cases of Schloemer [2] and Hu and Herr [10]), and the boundarylayer edge velocity ${U}_{e}$ 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 boundarylayer edge velocity is converted into the local freestream velocity using the relationship ${U}_{e}=0.99{U}_{0}$. The positions listed in the test case of Catlett et al. are the distance upstream of the trailing edge. Boundarylayer 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 7–11 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], $\mathrm{AOA}=6\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}$ at $x=1128\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}$, where the spectra roll off at high frequencies with a much faster slope than the measured ones. The reason for that is the function ${A}_{2}=\mathrm{min}(3,19/\sqrt{{R}_{T}})+7$ in Eq. (1), which could result in a faster rolloff at high frequencies for a small ${R}_{T}$. 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.
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 boundarylayer 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 rolloff 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 boundarylayer 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, flushmounted 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 $\omega r/{U}_{c}>1$. 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 flushmounted 2.54 mm diameter Kulite sensors with the socalled B screen for ZPG boundary layers shows an undercorrection using the Corcos correction from about $\omega r/{U}_{c}>0.5$, and until $\omega r/{U}_{c}=1$ 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 ${U}_{c}(\omega )$ for an APG boundary layer at the closest distance could be less than $0.3{U}_{0}$ at high frequencies [10], which is much slower than the usually used $0.6\u20130.8{U}_{0}$. The toolarge 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 $\omega $ at low frequencies used in this formulation shows a better agreement with the measured spectra than the other models that possess an ${\omega}^{2}$ slope.
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 lowfrequency 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 boundarylayer 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 boundarylayer 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 boundarylayer separation, i.e., the measurement from Catlett et al. [7,22] with the open angle of 17 deg, is not included. 
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, $18.8\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}\le \delta \le 35.0\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}$; 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, $66.5\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}\le \delta \le 91.2\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}$; group III for the case of Suryadi and Herr [9], with APG boundary layers developed on the airfoil suction side, $6.1\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}\le \delta \le 13.2\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}$. 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 boundarylayer 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 $\omega \theta /{U}_{0}$ 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 $\omega \theta /{U}_{0}$ 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
$$\frac{\mathrm{\varphi}(\omega ){u}_{\tau}}{{Q}^{2}\theta}=\frac{(81.004d+2.154)\cdot {10}^{7}\cdot (\omega \theta /{U}_{0})}{{[{(\omega \theta /{U}_{0})}^{1.5\text{\hspace{0.17em}\hspace{0.17em}}{h}^{1.6}}+0.07]}^{1.13/{h}^{0.6}}+{[7.645R{e}_{\tau}^{0.411}\cdot (\omega \theta /{U}_{0})]}^{6}}$$  (11) 
Figures 14–18 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 boundarylayer 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.
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 ${U}_{e}/{\tau}_{w}^{2}\delta ({\delta}^{*})$, ${u}_{\tau}/{Q}^{2}\theta $ 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 ${U}_{e}/{\tau}_{w}^{2}\delta $. As discussed in this paper, the dynamic pressure $Q$ could be more appropriate to scale the APG spectra than the usually used ${\tau}_{w}$ for ZPGs. Second, the boundarylayer 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 boundarylayer mean velocity profile and the Reynolds stresses. The Reynolds stresses are again tightly related to the mean velocity profile. Therefore, the boundarylayer 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 boundarylayer parameters were predominantly determined by the upstream history of the flow. The measured data illustrated that the shape factor can represent well the boundarylayer 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 ${\omega}^{2}$ increase at low frequencies, a slope of $\omega $, 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 ${\omega}^{2}$ are as follows:
1)  The slope of ${\omega}^{2}$ is obtained by assuming a frozen flow and only counting the meanshear 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 meanshear term in the spectra for a ZPG boundary layer. However, exact knowledge of the importance of the turbulence–turbulence term for a nonZPG boundary layer is still lacking. 
The present model was validated by the selected dataset in a range of $2.7\cdot {10}^{3}<R{e}_{\theta}<1.5\cdot {10}^{4}$, $1.4<H<2.15$, $9.0<{U}_{0}<62.7\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{m}/\mathrm{s}$, and $6.1<\delta <91.2\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}$. 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 boundarylayer parameters in the trailing edge area.  
2)  Spectra were measured at positions not far downstream of a boundarylayer separation. Good agreement for ZPG boundary layers from the proposed model was also obtained. 
This work was conducted in the framework of the DLR, German Aerospace Center, project Comfort and Efficiency Enhancing Technologies (CENT).