2.1. Simulation datasets

The DNS and LES datasets are gathered or re-generated from cases in four prior studies: Pargal et al. (Reference Pargal, Wu, Yuan and Moreau2022), Wu et al. (Reference Wu, Moreau and Sandberg2019), Na & Moin (Reference Na and Moin1998) and Wu & Piomelli (Reference Wu and Piomelli2018). Details of the flows in these datasets are listed in table 1. The first three are DNS while Wu & Piomelli (Reference Wu and Piomelli2018) is a LES study. The data of Pargal et al. (Reference Pargal, Wu, Yuan and Moreau2022) and Wu et al. (Reference Wu, Moreau and Sandberg2019) are collected directly from simulations of a turbulent boundary layer on a flat plate and that on a controlled-diffusion (CD) aerofoil with matched non-equilibrium APG distributions along the streamwise direction. Comparison between these two flows reveal the effects of the convex wall curvature and the trailing edge on WPS, which were partially discussed in Pargal et al. (Reference Pargal, Wu, Yuan and Moreau2022) and will be further discussed for the WPS herein. Na & Moin (Reference Na and Moin1998) and Wu & Piomelli (Reference Wu and Piomelli2018) conducted simulations of flat-plate boundary layers with suction and blowing free-stream velocities, leading to boundary-layer separation and then reattachment; these two cases are rerun to collect boundary-layer parameters, streamwise mean velocity and wall-pressure statistics at the same streamwise locations, as these data were not fully available from the original publications. For the case of Wu & Piomelli (Reference Wu and Piomelli2018), this work provides new data as the wall pressure was not discussed previously.

Table 1.

List of simulation datasets. For the cases of Na & Moin (Reference Na and Moin1998) and Wu & Piomelli (Reference Wu and Piomelli2018), the boundary-layer separation leads to $\beta$ values between $-\infty$ and $\infty$. The Reynolds-number values are slightly different from those in Na & Moin (Reference Na and Moin1998) and Wu & Piomelli (Reference Wu and Piomelli2018) due to difference in the definitions of the boundary-layer edge.

A brief summary of the four simulations is as follows. The case of Wu et al. (Reference Wu, Moreau and Sandberg2019) provides DNS data on a boundary layer developing on the pressure side of a CD aerofoil, at a free-stream Mach number of 0.25. The compressible Navier–Stokes equations are solved for the flow around an aerofoil with the multi-block structured code HiPSTAR (High Performance Solver for Turbulence and Aeroacoustics Research) (Sandberg Reference Sandberg2015). An initial two-dimensional Reynolds-averaged Navier–Stokes (RANS) simulation was run to provide boundary and initial conditions to the DNS simulation. Details of the problem formulation are provided by Wu et al. (Reference Wu, Moreau and Sandberg2019). The simulation was validated against experimental data (Jaiswal Reference Jaiswal2020; Jaiswal et al. Reference Jaiswal, Moreau, Avallone, Ragni and Pröbsting2020) for wall-pressure spectral data and flow statistics at different streamwise locations. The case of Pargal et al. (Reference Pargal, Wu, Yuan and Moreau2022) is an incompressible DNS of a flat-plate turbulent boundary layer to emulate the boundary-layer development on the downstream portion of the CD aerofoil flow studied by Wu et al. (Reference Wu, Moreau and Sandberg2019). A finite difference solver on a staggered grid was used. To match the pressure-gradient parameter ($K$) of the aerofoil boundary layer, a streamwise pressure gradient was imposed by prescribing a streamwise-varying $U_\infty (x)$ at the top boundary of the domain. A fully turbulent boundary-layer flow at the inlet of the domain was obtained using the recycling/rescaling method. A convective outflow boundary condition was used at the outlet and periodic boundary conditions were used in the spanwise direction. Similar discretization methods and boundary conditions were used in Wu & Piomelli (Reference Wu and Piomelli2018) and Na & Moin (Reference Na and Moin1998) with slight variations in details. Simulations of the flows in these two studies were rerun, based on the methodologies of Pargal et al. (Reference Pargal, Wu, Yuan and Moreau2022). The meshes in the new simulations were similar to those in the original studies. The same domain lengths and similar boundary conditions were used. The rerun simulation of Na & Moin (Reference Na and Moin1998) has been validated against results reported in the original work on flow statistics and wall-pressure spectra at different streamwise locations. For the rerun LES simulation of Wu & Piomelli (Reference Wu and Piomelli2018), the governing equations were solved for the filtered velocities at scales larger than the low-pass filter. A different dynamic eddy-viscosity model based on the Lagrangian-averaging procedure (Meneveau, Lund & Cabot Reference Meneveau, Lund and Cabot1996) was used for the present simulation. Boundary-layer developments in the rerun simulations will be compared with those reported in the original studies in § 3.

2.2. Experimental datasets

The DNS and LES simulations are limited to comparatively low Reynolds numbers $(Re_{\theta }=300$ to 7000). Experimental datasets are gathered from the studies of Hu (Reference Hu2018), Fritsch et al. (Reference Fritsch, Vishwanathan, Todd Lowe and Devenport2022b) and Goody & Simpson (Reference Goody and Simpson2000), which provide ZPG or pressure-gradient flow data with $Re_\theta$ of up to 23 400. Only existing datasets with both mean velocity profile data and WPS data measured at the same streamwise locations are included, as these quantities are required to calibrate and test the proposed model to be introduced in § 5.2.

A brief description of the experimental set-up of each case is given below. Hu (Reference Hu2018) carried out experiments in an open-jet anechoic test section of the Acoustic Wind Tunnel Braunschweig (AWB). Adverse and favourable pressure gradients in flat-plate boundary layers were achieved by placing a rotatable NACA 0012 aerofoil above the flat plate. Wall-pressure statistics were measured with sub-miniature pressure transducers and boundary-layer velocity profiles were obtained using hot-wires. The value of $Re_{\theta }$ was up to 19,000, with $\beta =-0.9$ to $16$. The study is among the few experimental studies that measured wall-pressure statistics across very different flows due to the very wide ranges of pressure gradient and Reynolds number. Similarly, Fritsch et al. (Reference Fritsch, Vishwanathan, Todd Lowe and Devenport2022b) carried out experiments in a subsonic wind tunnel with a NACA 0012 aerofoil installed in the centre of the test section. The boundary layer was tripped at the upstream section, to ensure a fully turbulent boundary layer in the test section. Wall-pressure statistics were measured for non-equilibrium pressure gradients ranging from a $\beta$ of $-0.5$ to 0.5, with $Re_{\theta }$ reaching 18 000. Goody & Simpson (Reference Goody and Simpson2000) carried out measurements in the boundary-layer tunnel of the Aerospace and Ocean Engineering department of Virginia Tech. The wall-pressure statistics measurement was limited to ZPG flows but data reached Reynolds numbers as high as $Re_{\theta }=23\,400$.

5.1. Performance of existing wall-pressure spectral models

Most existing wall-pressure spectral models are developed for regions with zero and adverse pressure gradients. Figure 9 compares a number of existing WPS models introduced in § 1 against the present datasets of ZPG and APG (attached regions only) flows (marked by blue solid lines) for six different $Re_\theta$-$\beta$ combinations. Among them, figures 9(e) and 9(f) show two examples near boundary-layer separation. The models and their legend are listed in table 4, including a proposed model (shown by red solid lines) to be formulated in § 5.2 to address the issues of the existing models observed in this section. All model predictions are re-normalized with the optimal $\phi _{pp}$ normalization ($\rho |\overline {u'v'}|_{max}$, $U_e$ and $\delta$) identified in § 4. The comparison of the measurement results among these six flows demonstrates dependencies of the WPS on the APG and the Reynolds number, as discussed in § 4. Under the present normalization, the main variations are in the width and slope of the overlap range.

Figure 9.

Comparison between predictions of WPS models and numerical or experimental measurements (——, blue solid line), for different types of flow: (a) ZPG, high-$Re_\theta$ flow (Fritsch et al. Reference Fritsch, Vishwanathan, Todd Lowe and Devenport2022b, $2^\circ$), (b) weak-APG, high-$Re_\theta$ flow (Fritsch et al. Reference Fritsch, Vishwanathan, Todd Lowe and Devenport2022b, $12^\circ$), (c) strong-APG, low-$Re_\theta$ flow (Wu et al. Reference Wu, Moreau and Sandberg2019), (d) very-strong-APG flow at intermediate $Re_\theta$ (Wu & Piomelli Reference Wu and Piomelli2018), (e) flow near separation point at a low $Re_\theta$ (Na & Moin Reference Na and Moin1998) and (f) flow near separation point at an intermediate $Re_\theta$ (Wu & Piomelli Reference Wu and Piomelli2018). Predictions of all models are re-normalized by $\rho |\overline {u'v'}|_{max}$, $U_e$ and $\delta$ for comparison purposes. Legend of model results is given in table 4. The proposed model (——, red solid line) will be introduced in § 5.2.

Table 4.

List of WPS models examined with the present datasets.

First, the overall performance of the WPS models in predicting the wall-pressure fluctuation intensity is analysed based on the predicted $p_{rms}$ value. Table 5 lists the prediction error of each of the tested model in the six flows examined in figure 9. Good $p_{rms}$ predictions (with errors up to 26 %) were made by all models in the ZPG flow (row a). Under stronger APG (with $\beta$ up to 23, rows b,c), over- or under-prediction by a large fraction (40 % to 60 %) is seen in four out of the five existing models. However, in flows with very strong APG, including the region near the separation point (rows d–f), very large errors are seen in all five existing models, as they were not formulated or calibrated for non-equilibrium strong-APG flows. The proposed model calibrated from some of these flows, on the other hand, is shown to reproduce very well the $p_{rms}$ values (with up to 8 % error) in all six flows. A good prediction of wall-pressure fluctuation intensity does not necessarily mean equally good prediction in the WPS shape, which is analysed next. The large model errors in rows (d) to (f) are also explained below.

Table 5.

Wall-pressure r.m.s. prediction errors from the existing models tested and the proposed models, examined for the six flows in figure 9. The error is calculated as $(\,p_{rms}-p_{rms,a})/p_{rms,a}$, where $p_{rms,a}$ is the actual value from measurement. Rows (a)–(f) correspond to the flows in figure 9(a–f), respectively. In (e,f), the Lee (Reference Lee2018) and Rozenberg et al. (Reference Rozenberg, Robert and Moreau2012) models yield undefined values from double-precision floating-point number operations.

In a ZPG flow (figure 9a), almost all existing models (except the Kamruzzaman et al. (Reference Kamruzzaman, Bekiropoulos, Lutz, Würz and Krämer2015) model) give overall good predictions in the spectrum. For the weak-APG flow (figure 9b), the tested models are shown to give reasonably good predictions overall. Goody's model over-predicts mid-to-high-frequency WPS content due to lack of sensitivities of the overlap-range width and slope to the pressure gradient. Kamruzzaman et al. (Reference Kamruzzaman, Bekiropoulos, Lutz, Würz and Krämer2015) model does not produce an overlap range and, as a result, yields significantly under-predicted high-frequency contents. Moreover, except for the Hu (Reference Hu2018) model, all existing models give noticeable differences in the overlap-range slope from the experimental measurement.

For the flow under relatively weak APG ($\beta \approx 8$), figure 9(c) shows that the Rozenberg et al. (Reference Rozenberg, Robert and Moreau2012) and Lee (Reference Lee2018) models give very good overall predictions of the spectrum, whereas the Hu (Reference Hu2018) and Kamruzzaman et al. (Reference Kamruzzaman, Bekiropoulos, Lutz, Würz and Krämer2015) models under-predict the WPS at low frequencies. The Goody (Reference Goody2004) model under-predicts the WPS in the whole frequency range, which is expected, as this model was developed and calibrated for ZPG flows only.

In strong-APG flows (figure 9d–f), especially near the boundary-layer separation point, the existing models give large errors. This is because of the model parameters used in these models: $R_t$ and $\beta$ tend towards zero and infinity, respectively. Another source of error is the inappropriate pressure scaling (i.e. $\tau _w$) used in the models. For instance, Rozenberg et al. (Reference Rozenberg, Robert and Moreau2012) model

(5.1)\begin{equation} \frac{\phi_{pp} (\omega) U_e}{\tau_w^2 \delta^*}=\frac{0.78(1.8\varPi \beta+6)(\omega \delta^*/U_e)^2}{[(\omega \delta^*/U_e)^{0.75}+C_1']^{3.7}+[C_3'(\omega \delta^*/U_e)]^7}, \end{equation}

where $C_3'=3.76R_T^{-0.57}$, reaches a singularity as $\tau _w$ and $R_t$ become zeros and $\beta$ becomes infinity. Employing $\tau _w$ as the pressure scale for $\phi _{pp}$ also renders the dimensionless spectrum excessively sensitive to strong APGs, even if the flow stays attached. This issue is common in existing models. In addition, most of these models were fitted to limited types of flows, such as low-Reynolds-number aerofoil boundary layers in Kamruzzaman et al. (Reference Kamruzzaman, Bekiropoulos, Lutz, Würz and Krämer2015), flat-plate boundary layers in Hu (Reference Hu2018) and ZPG flows in Goody (Reference Goody2004). Moreover, sometimes the boundary-layer flow properties used for model calibration were estimated from lower-fidelity methods such as XFOIL or RANS calculations, as opposed to DNS/LES or experimental measurements. In § 5.2, these limitations are addressed to develop a well-behaved WPS model for both ZPG and APG flows, which can be attached or separated.

5.2. A new generalized WPS model for non-equilibrium boundary layers

In this section, a new wall-pressure spectral model is developed, with the Goody (Reference Goody2004) model as a base. First, various components of the modification are introduced and justified. Then, prediction of the new model is compared with the datasets, first for attached APG flows and next for separated flows, reattached flows and FPG flows. Finally, a sensitivity analysis of the new model prediction to the model parameters is carried out, to verify that the intended dependencies are indeed captured.

The Goody (Reference Goody2004) model is a good starting point as it has been shown to be successful in WPS prediction for ZPG boundary layers; it is also relatively simple compared with some other existing models. In summary, the Goody (Reference Goody2004) model is modified by removing the Reynolds-number parameter ($R_t$) and using model inputs that derive directly from the local mean velocity distribution, $U^+(x,y)$. Only inputs that are quantifiable from engineering predictive approaches (such as RANS models) are considered, so the model is of practical use in engineering applications. The dependence of these inputs on the $U(y)$ distribution at a given $x$ location allows the model to sense the local state of the turbulent boundary layer.

The Goody (Reference Goody2004) model reads

(5.2)\begin{equation} \tilde{\phi}_{pp}\left(\tilde{\omega}\right) =\frac{a\tilde{\omega}^b}{(h\,\tilde{\omega}^c+d)^{e}+(\,f \mathcal{F}\,\tilde{\omega})^g}, \end{equation}

where $\tilde {\phi }_{pp}=\phi _{pp}/\phi _{pp}^*$ and $\tilde {\omega }=\omega /\omega ^*$ are dimensionless WPS and frequency, respectively, based on a spectrum scale of $\phi ^*_{pp}$ and a frequency scale of $\omega ^*$, as listed in table 6. The coefficients $a$ to $h$ and the function $\mathcal {F}$ are defined in table 6. The model was developed based on the observed dependencies of the low-frequency spectrum on the outer scales, as well as that of the high-frequency spectrum on the inner scales, in a ZPG or weak-pressure-gradient flow. The model was known to predict well the overlap range of the spectrum for ZPG boundary layers (as also shown in figure 9a), which depends on the Reynolds number only. The dependence is captured by including $R_t$ in the model. However, the overlap-range spectrum also depends on the pressure gradient in strong-pressure-gradient boundary layers. In figure 10(a), the prediction of the original Goody's model is examined at several $x$ locations of the Wu & Piomelli (Reference Wu and Piomelli2018) case in the attached-flow region with weak to strong APGs. In this region, $\beta$ ranges from 0 to 200 and $Re_{\theta }$ is from 2000 to 6000. With the increase in APG, Goody's model shows increasing under-prediction in the whole frequency range.

Table 6.

Comparison between Goody (Reference Goody2004) model and the proposed model. The parameters $\phi _{pp}^*$ and $\omega ^*$ are the chosen scalings for WPS and frequency, respectively. Special treatments for a flow region characterized by a separated boundary layer or a limited logarithmic layer are listed.

Figure 10.

Comparison between model predictions of (5.2) with progressive changes described in text (——, red solid line) and Wu & Piomelli (Reference Wu and Piomelli2018) (——, thick solid line) data in attached flows with weak to strong APGs, to show improvement brought by each model change. (a) The Goody (Reference Goody2004) model (predictions are plotted using the optimal normalizations), (b) pressure scale changed from $\tau _w$ to $\rho |\overline {u'v'}|_{max}$, (c) additionally replacing dependency on $R_t$ with that on $y_w^+$, (d) further addition of Coles’ parameter. Thicker lines indicate increase in $x$ (corresponding to increasing $\beta$).

Table 6 lists the changes from the original Goody (Reference Goody2004) model to the proposed model. The modified expressions of the model parameters were obtained by fitting a subset of the datasets: the APG and ZPG data of Pargal et al. (Reference Pargal, Wu, Yuan and Moreau2022), Wu & Piomelli (Reference Wu and Piomelli2018), Hu et al. (Reference Hu, Buchholz, Herr, Spehr and Haxter2013), Fritsch et al. (Reference Fritsch, Vishwanathan, Todd Lowe and Devenport2022b) and Goody (Reference Goody2004), as well as separated-flow data of Na & Moin (Reference Na and Moin1998) and Wu & Piomelli (Reference Wu and Piomelli2018). Datasets that were not used for model development were used in testing the model in figure 9(c,e,f) for attached APG flows, figure 13(b) for a reattached flow and figure 14(b) for a FPG flow.

In the following, the main changes are introduced and progressively applied to demonstrate the improvement of each of them. First, the pressure scale (i.e. $\tau _w$) on the left-hand side of (5.2) is replaced with $\rho |\overline {u'v'}|_{max}$, with the additional change of replacing the constant $d=0.5$ with 0.7 for a better low-frequency collapse with the current data. The effect of these modifications is shown in figure 10(b), where the low-frequency range is shown accurately predicted for all $x$ locations. The use of $\rho |\overline {u'v'}|_{max}$ ensures that the spectral values are finite near the separation point.

Next, to capture the variation of overlap and high-frequency range in APG flows, a new parameter is needed to replace $R_t$ to model the width change of the overlap range. Past studies showed that turbulent fluctuations in the logarithmic layer form the main contributor to the overlap range of the WPS. For example, Farabee & Casarella (Reference Farabee and Casarella1991) showed that, for ZPG boundary layers, with an increase in Reynolds number accompanied by a thickening of the logarithmic layer, the WPS overlap range becomes wider and its integral contribution to $p_{rms}^2/\tau _w^2$ varies as $6.5+1.85\ln (Re_{\tau }/333)$ at sufficiently high Reynolds numbers. Additional evidence in APG flows is provided by Jaiswal et al. (Reference Jaiswal, Moreau, Avallone, Ragni and Pröbsting2020), who showed that the logarithmic layer yields the highest contribution to the overlap range of $\phi _{pp}$ based on analyses of the velocity sources of the wall-pressure Poisson equation. Here, a new model input is introduced: $y_w^+(x)$, defined as the local elevation of the upper edge of the logarithmic layer, to sensitize the model spectrum to the change in logarithmic-layer thickness due to Reynolds number and/or pressure gradients. The subscript $w$ in $y_w$ represents the width of the logarithmic layer. Specifically, the term $1.1R_t^{-0.57}$ is replaced with $(y_w^+)^{-0.37}$. The new expression is obtained by assuming a power function of $y_w^+$ and performing a nonlinear least square fit (using both ZPG and APG datasets) to calculate the exponent.

The parameter $y_w^+(x)$ is dynamically determined based on the boundary-layer mean velocity, $U(x,y)=\bar {u}$, as the $y^+$ location where $U^+(x,y) -[\kappa (x)^{-1} \log y^+ + B(x)]$ departs from $0$ at the upper limit of the logarithmic layer (as shown in figure 11c). Here, $\kappa$ is the von Kármán constant and $B$ is the log-law intercept, both of which are allowed to vary along $x$ in a non-equilibrium boundary layer. Note that, due to the use of local viscous scaled units in $y_w^+$ and $U^+$ etc., the present modification needs special treatment at the separation point and inside the separated-flow region, which will be discussed later. To determine $\kappa (x)$, the diagnostic function $I(x,y)=y^+ \partial U^+/\partial y^+$ (figure 11b) is calculated from the mean velocity profile (figure 11a) for data of Wu et al. (Reference Wu, Moreau and Sandberg2019). The local minima of $I(x,y)$ at a given $x$ is taken as $1/\kappa (x)$. Following the determination of $\kappa (x)$, $B(x)$ is calculated using $U^+(x,y) -[\kappa (x)^{-1} \log y^+ + B(x)]=0$. Figure 11(d) shows that the correlation between $\kappa (x)$ and $B(x)$ obtained for all cases in the present attached-flow datasets is consistent with that observed by Nagib & Chauhan (Reference Nagib and Chauhan2008) from a large collection of flows with or without pressure gradients. As shown in figure 10(c) compared with figure 10(b), the modelling of overlap-range width based on $y_w^+$ is successful for the present APG datasets: the high-frequency range is now better predicted with the corrected width. As the APG becomes stronger, $y_w^+$ decreases as shown in figure 11(c), reducing the overlap-range width as shown in figure 10(c). The main WPS prediction error is now predominantly an inaccurate slope of the overlap range, as shown in figure 10(c).

Figure 11.

(a–c) Calculation of $\kappa (x)$, $B(x)$ and model parameters shown using Wu et al. (Reference Wu, Moreau and Sandberg2019) data at locations $x/\theta _o=50$, 105, 120 and 130. Increasing line thickness indicates increase in $x$. (a) Mean velocity profiles in inner units. (b) Diagnostic function, $I=y^+\partial U^+/\partial y^+$. (c) Velocity profiles with logarithmic relation subtracted. Calculations of $\kappa$, $\varPi$ and $y_w^+$ are indicated for $x/\theta _o=105$ in (b,c). (d) Correlation between calculated $\kappa B$ and $B$, compared with the fitted relation from Nagib & Chauhan (Reference Nagib and Chauhan2008) (– – –, dashed line): $\circ$ (red) Wu & Piomelli (Reference Wu and Piomelli2018) (attached flow before separation), $\circ$ Wu & Piomelli (Reference Wu and Piomelli2018) (attached flow downstream of reattachment), $\Diamond$ (red) Hu (Reference Hu2018), $\times$ (red) Fritsch et al. (Reference Fritsch2022a), $+$ (red) Goody (Reference Goody2004), $\square$ (red) Pargal et al. (Reference Pargal, Wu, Yuan and Moreau2022), $\triangleright$ (red) Wu et al. (Reference Wu, Moreau and Sandberg2019) and $\Delta$ (red) Na & Moin (Reference Na and Moin1998).

The change in WPS overlap-range slope (in addition to the change of the width of this range as is characterized by $y_w^+$) is assumed to be caused by the variation in the strength of the outer-layer turbulent motions. This assumption arises from the understanding that an APG leads to more energized large turbulence motions in the outer layer and a thinner logarithmic layer. In addition, an APG is known to cause an increase in low-frequency WPS contents and a decrease in mid-to-high-frequency ones (e.g. Rozenberg et al. Reference Rozenberg, Robert and Moreau2012; Catlett et al. Reference Catlett, Anderson, Forest and Stewart2016; Lee Reference Lee2018), which is reflected in a steeper overlap-range slope. To account for the variation in the strength of the wake region, an additional model input is used: Coles’ parameter, $\varPi (x)$. An augmentation of $\varPi$ signals stronger turbulent intensity and mixing in the outer layer. Here, $\varPi (x)$ is evaluated based on $U^+(x,y)$, by measuring the peak value of $U^+ -[\kappa ^{-1} (\log (y^+) + B)]$ (as shown in figure 11c) and dividing it by $2/\kappa (x)$. In the generic model form in (5.2), the coefficient $c$ is known to impose the slope of the overlap range of $\phi _{pp}$ (Rozenberg et al. Reference Rozenberg, Robert and Moreau2012; Lee Reference Lee2018; Thomson & Rocha Reference Thomson and Rocha2022). The present datasets suggest that $c$ is a function of both $y_w^+$ and $\varPi$: for flows with larger $y_w^+$ values, the same pressure gradient difference leads to a larger slope variation compared with cases with smaller $y_w^+$ values. Therefore, the constant $c$ is replaced by a function of both $\varPi$ and $y_w^+$. A posynomial functional form is assumed; the constant coefficients and exponents are fitted based on the present datasets. Additionally, for boundary layers with very strong APG or near the separation point, the logarithmic layer diminishes and the WPS overlap range is very narrow or absent. At these flow locations the modelled WPS is weakly sensitive to the $c$ value. The present datasets yield a maximum value of $c\approx 0.98$ in strong-APG flows. For simplicity, $c$ is capped at 1.0. This treatment, however, may be improved in the future when more high-Reynolds-number strong-APG data become available. The final form of the generalized WPS model is

(5.3)$$\begin{gather} \frac{\phi_{pp} (\omega) U_e}{(\rho |\overline{u'v'}|_{\max})^2 \delta} =\frac{3(\omega \delta/U_e)^2}{[(\omega \delta/U_e)^c+0.7]^{3.7}+[(y_w^+)^{-0.37}(\omega \delta/U_e)]^7}, \quad \text{where} \end{gather}$$

(5.4)$$\begin{gather}c = \min\left[1.0,0.8+3.34\times 10^{-4} \varPi^{1.86} (y_w^+)^{0.76}\right]. \end{gather}$$

For the ZPG flow shown in figure 9(a), the Goody (Reference Goody2004) model and the proposed model give similar predictions and both compare well with the experimental measurements. The small difference between the predictions of the two models is because of the use of different model parameters and that the fitting of the proposed model is conducted for both ZPG and APG flows. For the strong-APG flows in the present datasets, figure 10(d) and figure 9 show that the generalized model predicts the WPS very well. Note, however, that the present strong-APG data used to calibrate this model are from limited-Reynolds-number flows with a rather narrow WPS overlap range. Additional data from high-Reynolds-number strong-APG flows are not available, but are needed to validate the use of the model of its present form in flows with higher Reynolds numbers.

A few scenarios require special treatments, as listed in table 6. For cases with very low Reynolds numbers and extreme APGs, which practically remove the logarithmic region from the boundary layer (i.e. if $y_w^+<15$), $y_w^+$ and $c$ are set to constant values: $y_w^+=15$ and $c=0.85$ as calibrated from the present datasets, to reflect the insensitivity of $\phi _{pp}$ to either Reynolds number or pressure gradient, as shown in figure 7(a). Moreover, in case of boundary-layer separation, modification of the model is needed for $x$ locations at the separation point and inside the separation bubble. Inspired by the similar WPS profiles across the normalized frequency range in the separated region as shown in figure 8(b), $y_w^+$ and $c$ are set to constants: $y_w^+=2$, $c=0.75$, and $d$ is set to 0.5 (table 6), calibrated based on the data of Na & Moin (Reference Na and Moin1998) and Wu & Piomelli (Reference Wu and Piomelli2018). The separation modification is activated for $x$ regions where $C_f(x)$ is calculated as zero or negative, corresponding to the region of mean-flow separation. With the aforementioned separated-flow treatment, the proposed model is evaluated at a number of streamwise locations inside the separation regions of the flows of Na & Moin (Reference Na and Moin1998) and Wu & Piomelli (Reference Wu and Piomelli2018) in figure 12. The WPS prediction does not vary with $x$ in this region, since the model parameters are set to constants. Overall good comparison with the simulation data is achieved. Although the separation treatment introduces a discontinuity in $c$ value at the separation point (as $c$ approaches 1 towards the point while $c=0.75$ at that point), it is shown not to affect the prediction significantly, as the overlap range is short in the strong-APG region in the vicinity of the detachment.

Figure 12.

Comparison between model prediction (——, red solid line) and simulation data (—— Na & Moin (Reference Na and Moin1998) and —— (light grey solid line) Wu & Piomelli Reference Wu and Piomelli2018) in the separated-flow regions. Increasing line thickness indicates an increase in $x$ values as used in figure 8(a).

Although the model is primarily developed for attached or separated APG flows, it is examined in other regions of a non-equilibrium boundary layer to explore its extendibility to more universal applications. First, the model is evaluated in the region downstream from the flow reattachment point until a near-equilibrium ZPG state in figure 13 against Wu & Piomelli (Reference Wu and Piomelli2018) data, at four $x$ locations between $x/\theta _o=250$ and $x/\theta _o=350$. Figure 13(a) shows that, near the reattachment point (shown by the thinnest lines), the local mean velocity departs significantly from a canonical boundary-layer profile, without a clear logarithmic layer. With increasing $x$, the logarithmic layer gradually recovers towards the equilibrium ZPG state and the overlap range of the WPS thickens gradually (figure 13b). The model is shown to capture such a trend of WPS variation. The overall spectral levels are well predicted due to the approximate scaling of $p_{rms}$ on $\rho |\overline {u'v'}|_{max}$, while the spectral shape is captured by $y_w^+(x)$ and $\varPi (x)$ representing the local thickening of the logarithmic layer and the weakening of wake, respectively, during recovery.

Figure 13.

Comparison between model prediction (——, red) and Wu & Piomelli (Reference Wu and Piomelli2018) data (——) in the attached-flow region downstream of reattachment point (at $x/\theta _o$ from 250 to 350, the most downstream location corresponding to near-equilibrium flow). (a) Mean velocity profiles with the logarithmic relation subtracted; – – – (blue dashed line) value corresponding to logarithmic layer. (b) Wall-pressure PSD comparison. Increasing line thickness indicates an increase in $x$.

In addition, the model is tested in FPG flows, as shown in figure 14, against the experimental data of Hu (Reference Hu2018) and Fritsch et al. (Reference Fritsch, Vishwanathan, Todd Lowe and Devenport2022b). Figure 14(a) shows that, under FPGs (shown in blue) as compared with the ZPG profiles (shown in grey), the main change is a reduction of $\varPi$, which leads to a milder slope of the WPS overlap range. Figure 14(b) shows that this change in WPS is globally captured well by the proposed model.

Figure 14.

Comparison between model prediction (——, red solid line) and experimental data in FPG flows: $\circ$ (cyan) Hu (Reference Hu2018, $\beta =-0.5$) and - - - (cyan dotted line) Fritsch et al. (Reference Fritsch, Vishwanathan, Todd Lowe and Devenport2022b, $\beta =-0.5$). (a) Mean velocity profile with the logarithmic relation subtracted; the ZPG profile in each of the two studies (shown in grey) is compared. – – – Value corresponding to logarithmic layer. (b) Wall-pressure PSD comparison.

Figure 15 shows results of sensitivity analyses carried out for the parameters $y_w^+$ and $\varPi$ of the proposed model. The WPS predictions obtained with $\pm 30$ % change of each of the two parameters (marked by the highlighted region) are compared with the datasets, for three types of flows with different ranges of Reynolds number and $\beta$. Figure 15(a,c,e) shows that the variation of $y_w^+$ has an effect on the overlap-range and high-frequency contents, by controlling the width of the overlap range. The effect appears to be particularly strong in a weak-APG flow. Figure 15(b,d,f) shows that $\varPi (x)$ modifies the slope of the overlap range, with the model particularly sensitive to its value in high-Reynolds-number flows where the overlap range is pronounced. These results show that the introduced parameters affect the WPS prediction in their intended ways. Furthermore, slight variations in quantifying $y_w^+$ and $\varPi$ do not significantly worsen WPS prediction.

Figure 15.

The WPS prediction of the proposed model (——, red solid line) compared with measurements (——, blue solid line) in the following cases: (a,b) high-Reynolds-number and weak-APG flow (Fritsch et al. Reference Fritsch, Vishwanathan, Todd Lowe and Devenport2022b, $\beta =0.58$), (c,d) low-Reynolds-number and strong-APG flow (Wu et al. Reference Wu, Moreau and Sandberg2019, $\beta =8.3$) and (e,f) high-Reynolds-number and strong-APG flow (Hu Reference Hu2018, $\beta =6$). Yellow regions mark prediction variations with $\pm 30$ % change in input parameters $y_w^+$ (a,c,e) or $\varPi$ (b,d,f). In (d), variation of $\varPi$ does not change WPS prediction as $c$ is set to a constant at this location due to $y_w^+<15$ (see table 6).