# A generalized wall-pressure spectral model for non-equilibrium boundary
layers

Saurabh Pargal1,2  Junlin Yuan1<EMAIL_ADDRESS>Stephane Moreau2  1Michigan
State University, Michigan, USA 2Université de Sherbrooke, Quebec, Canada

###### Abstract

This study uses high-fidelity simulations (DNS or LES) and experimental
datasets to analyse the effect of non-equilibrium streamwise mean pressure
gradients (adverse or favourable), including attached and separated flows, on
the statistics of boundary layer wall-pressure fluctuations. The datasets
collected span a wide range of Reynolds numbers ($Re_{\theta}$ from 300 to
23,400) and pressure gradients (Clauser parameter from $-0.5$ to 200). The
datasets are used to identify an optimal set of variables to scale the wall
pressure spectrum: edge velocity, boundary layer thickness, and the peak
magnitude of Reynolds shear stress. Using the present datasets, existing semi-
empirical models of wall-pressure spectrum are shown unable to capture effects
of strong, non-equilibrium adverse pressure gradients, due to inappropriate
scaling of wall pressure using wall shear stress, calibration with limited
types of flows, and dependency on model parameters based on friction velocity,
which reduces to zero at the detachment point. To address these short-comings,
a generalized wall-pressure spectral model is developed with parameters that
characterize the extent of the logarithmic layer and the strength of the wake.
Derived from the local mean velocity profile, these two parameters inherently
carry effect of the Reynolds number, as well as those of the non-equilibrium
pressure gradient and its history. Comparison with existing models shows that
the proposed model behaves well and is more accurate in strong-pressure-
gradient flows and in separated-flow regions.

###### keywords:

##  1 Introduction

The fluctuation in time of the wall pressure beneath a turbulent boundary
layer is one of the major sources of flow-induced noise and vibrations.
Accurate modeling of the statistics of wall-pressure fluctuations is important
for noise prediction in a wide range of applications such as wind turbines
(Avallone et al., 2018; Deshmukh et al., 2019; Venkatraman et al., 2023),
cooling fans (Sanjosé & Moreau, 2018; Swanepoel et al., 2023; Luo et al.,
2020), propellers (Lallier-Daniels et al., 2021; Casalino et al., 2021),
unmanned/manned air vehicles or drones (Lauzon et al., 2023; Pargal et al.,
2023; Celik et al., 2021), and cabin noise (Samarasinghe et al., 2016; Borelli
et al., 2021), etc., as well as for prediction of flow-induced structure
fatigue (Franco et al., 2020). In these applications, the boundary layer flows
are often turbulent and non-equilibrium, due to surface curvature and
significant pressure gradients that vary in the streamwise direction, which
may induce boundary layer separation and can be found in a large range of
Reynolds number. Therefore, the generation of noise in non-equilibrium
turbulent boundary layers is physically complex and challenging to model.

The modeling of wall-pressure loading as a noise source predominantly depends
on the power spectral density (PSD) of wall-pressure fluctuations, as well as
its spanwise correlation length and the convection velocity of turbulent
structures (Amiet, 1976; Roger & Moreau, 2005; Moreau & Roger, 2009; Lee et
al., 2021). The focus here is on modeling wall-pressure spectrum (WPS). It is
established that the WPS of a boundary layer with zero or minimal pressure
gradient consists of three ranges (Goody, 2004; Farabee & Casarella, 1991;
Chang III et al., 1999): (i) a range with $\omega^{2}$ variation at low
frequencies (where $\omega$ is the frequency), (ii) a range with $\omega^{-5}$
behavior at high frequencies, and (iii) an overlap range with a $\omega^{-1}$
decay rate between the above two ranges. Based on data primarily in
equilibrium flows, the width of the overlap range was found to increase with
Reynolds number (Farabee & Casarella, 1991; Goody, 2004).

Contributions from different layers of wall turbulence to the WPS has been
studied and are summarized below. The experimental studies of Farabee &
Casarella (1991) suggested different dominant sources for different wavenumber
ranges of the WPS: the high-wavenumber range is mainly attributed to turbulent
activities in the logarithmic region, while the low-wavenumber range is
attributed to large-scale turbulent motions in the outer layer. Van
Blitterswyk & Rocha (2017) quantified the correlations between the
fluctuations of wall pressure and those of velocity in different layers of
boundary layer and observed that high-frequency and overlap ranges of the WPS
are associated with flows in the buffer and logarithmic regions, respectively.
As opposed to earlier studies performed on channel flow or canonical flat-
plate boundary layer data, Jaiswal et al. (2020) analyzed data collected near
the trailing edge of a cambered airfoil with a strong mean adverse pressure
gradient (APG) in a highly non-equilibrium turbulent boundary layer, to
compare contributions from various velocity sources at different wall-normal
locations to the wall-pressure fluctuations based on the pressure Poisson’s
equation. They found that the mean shear (MS) term in the inner and
logarithmic regions is the dominant contributor, especially in the mid-to-high
frequency range.

Past studies were mostly on zero-pressure-gradient (ZPG) turbulent boundary
layers. They showed that the wall-pressure fluctuations are intensified under
a higher Reynolds number, mainly due to the increase in overlap-range spectral
contribution. For a ZPG flat-plate boundary layer, Farabee & Casarella (1991)
integrated the pressure spectrum over various frequency ranges and showed that
the low-to-mid frequency range and the high-frequency range were not sensitive
to a change in Reynolds number, whereas the significance of the overlap range
increases with Reynolds number, leading to an augmentation of $p_{rms}$. They
proposed that $p_{rms}^{2}/\tau_{w}^{2}=6.5+1.86\ln(Re_{\tau}/333)$, which was
tested with ZPG boundary layer and channel flow data. Panton & Linebarger
(1974) also demonstrated that the overlap range is correlated with the
Reynolds number.

The current understanding of the WPS in non-zero pressure gradient flows is
summarized as follows. Review of the earlier work before the mid 1990s are
provided by Willmarth (1975) and Bull (1996). Schloemer (1967) showed that
under an APG low-frequency contents of the WPS become more prominent as large
eddies are energized, while the high-frequency contents become less important.
Under a favourable pressure gradient (FPG), however, the opposite applies,
with stronger high-frequency contents. The WPS slope in the overlap range also
varies with the pressure gradient. Cohen & Gloerfelt (2018) investigated the
effects of mild pressure gradient using large-eddy simulations (LES) and
showed scale-based dependencies of the WPS on FPG similar to those observed
before. Na & Moin (1998) conducted direct numerical simulation (DNS) of a
boundary layer with prescribed freestream suction and blowing to induce flow
separation and reattachment. They showed that none of the outer, inner or
mixed scaling collapsed the wall pressure spectra in all regions of the flow.
Normalization with the local maximum magnitude of the Reynolds shear stress,
however, was shown to collapse the low-frequency range of WPS for APG flows
including those with separation (Abe, 2017; Ji & Wang, 2012; Caiazzo et al.,
2023).

Modeling of turbulent WPS is broadly classified in two categories: (i) semi-
empirical modelling and (ii) analytical modeling based on solution of the
Poisson’s equation of pressure (Kraichnan, 1956; Panton & Linebarger, 1974;
Jaiswal et al., 2020; Grasso et al., 2022; Palani et al., 2023; Hales & Ayton,
2023). The focus of this paper is on the first approach, which requires a
smaller amount of inputs from the flow field in comparison to the analytical
modelling approach. Existing semi-empirical WPS closures mostly model the
magnitude and shape of the WPS normalized by some boundary-layer parameters
that are either internal, external or mixed, such as the boundary layer
thickness ($\delta$), the edge velocity ($U_{e}$) and the wall shear stress
($\tau_{w}=\rho u_{\tau}^{2}$, where $u_{\tau}$ is the friction velocity),
etc. For some of these studies see Kraichnan (1956), Corcos (1964), Willmarth
(1975), Amiet (1976),Bull & Thomas (1976),Chase (1980), Goody (2004),
Rozenberg et al. (2012), Kamruzzaman et al. (2015), Lee (2018), Hu (2018), and
Pargal et al. (2022). One of the most widely used semi-empirical models for
ZPG boundary layers was proposed by Goody (2004), which captures accurately
the Reynolds number effect on the spectrum for these flows. But for boundary
layers under pressure gradients, Goody’s model may yield large errors, as
shown by a number of studies (Rozenberg et al., 2012; Kamruzzaman et al.,
2015; Catlett et al., 2016; Hu et al., 2013; Lee, 2018; Rossi & Sagaut, 2023).
For instance, in APG flows the more prominent low-frequency contents and
reduced mid-to-high-frequency ones of the WPS were not captured by this model.
Rozenberg et al. (2012) integrated additional boundary layer flow parameters
to sensitize the model to pressure gradient effects, especially those of APG.
The additional parameters include Clauser’s parameter ($\beta$) (Clauser,
1954) and Cole’s wake parameter ($\Pi$) (Coles, 1956). The former includes the
local effect of mean pressure gradients, while the latter represents the
cumulative effect of the history of mean pressure gradient up to the
considered location in the boundary layer. Several later models developed
modifications of the model that capture effects of other complexities such as
wall curvature and FPG. Kamruzzaman et al. (2015) developed a model by fitting
it on a large amount of experimental WPS data collected in various non-
equilibrium boundary layer flows on airfoils. Hu (2018) used the shape factor
($H$) and Reynolds numbers ($Re_{\theta}$ or $Re_{\tau}$) instead of $\beta$
to incorporate the effect of non-equilibrium pressure gradients, as $\beta$ —
a descriptor of local pressure gradient — does not carry the history effect of
a spatially varying pressure gradient. Lee (2018) improved Rozenberg’s model
based on experimental data gathered from a wide range of flows with different
Reynolds numbers and pressure gradients. Thomson & Rocha (2022) proposed a new
model for flows with FPG. Recently, machine learning approaches such as gene
expression programming and artificial neural networks were used to model WPS
as a function of boundary layer parameters (Fritsch et al., 2022a; Dominique
et al., 2022; Shubham et al., 2023; Ghiglino et al., 2023).

Despite the success of the models mentioned above in the specific flows for
which they were developed, these models are not universally applicable to both
ZPG flows and those with non-equilibrium pressure gradients and/or surface
curvature, due to the following reasons. (i) Models developed by curve-fitting
to data of a limited type of flows do not naturally apply to other flows, such
as Goody’s model, which works for ZPG flows only. (ii) Normalizations of wall
pressure statistics used for ZPG flows (e.g. $\tau_{w}$) may not be
appropriate for strong-APG flows (e.g. a boundary layer close to separation
where $\tau_{w}$ approaches zero). (iii) Local boundary layer parameters do
not account sufficiently for the history effect of the pressure gradient. In
addition, some existing models were developed based on experimental wall-
pressure measurements that are supplemented with low-fidelity flow-field data,
such as those estimated from XFOIL (Drela, 1989).

The objective of this study is therefore to develop a general WPS model that
is tunable for both ZPG and non-equilibrium, strong-pressure-gradient
turbulent boundary layers, as well as special cases such as flow separation
and reattachment. To this end, model parameters that derive from the local
mean velocity profile are incorporated to sensitize the model to the
streamwise pressure gradient and its history. An appropriate pressure
normalization for flows with and without pressure gradients is used. The model
is calibrate based on a large and inclusive database, containing both
experimental measurements and DNS/LES data (existing or new) of flows over a
wide range of Reynolds number, with or without separation.

The organization of the paper is as follows. Section 2 describes the database,
Section 3 presents the boundary layer development of the cases in the
datasets, Section 4 discusses the wall-pressure fluctuations and WPS in the
datasets, Section 5 discusses the performances of existing WPS models and
introduces a new generalized WPS model, and conclusions are presented in
Section 6.

##  2 Datasets collection

The first step to develop a generalized WPS model is to collect and analyze
high-fidelity datasets in a wide range of flows. The goal is to collect
datasets for both equilibrium and non-equilibrium boundary layers, including
ZPG, FPG, and APG flows, with or without wall curvature (as in boundary layers
developed on airfoils) and boundary layer separation and reattachment, across
a wide range of Reynolds number based on momentum thickness ($Re_{\theta}=300$
to 23,400).

###  2.1 Simulation datasets

DNS and LES datasets are gathered or re-generated from cases in four prior
studies: Pargal et al. (2022), Wu et al. (2019), Na & Moin (1998) and Wu &
Piomelli (2018). The first two are DNS cases. Data are collected directly from
simulations of turbulent boundary layer on a flat-plate and that on a
controlled-diffusion (CD) airfoil with matched non-equilibrium APG
distributions along the streamwise direction, as discussed in Pargal et al.
(2022). The other two studies are DNS and LES studies, respectively, of flat-
plate boundary layers with suction and blowing freestream velocities which
lead to boundary layer separation and then reattachment; these two cases are
rerun to collect velocity and wall-pressure statistics. For the case of Wu &
Piomelli (2018), this work provides new data as the wall pressure was not
discussed previously.

Here, the streamwise, wall-normal and spanwise directions are denoted as $x$,
$y$ and $z$. $u$, $v$ and $w$ are the velocity components in those directions,
$t$ is time, $P$ is the static pressure, $\rho$ is the density and $\nu$ is
the kinematic viscosity. An instantaneous flow variable $\phi(x,y,z,t)$ is
decomposed as $\phi=\overline{\phi}(x,y)+\phi^{\prime}(x,y,z,t)$, where
$\overline{(\cdot)}$ denotes averaging in $z$ and time.

The case of Wu et al. (2019) provides DNS data on a boundary layer developing
on the pressure side of a CD airfoil. The compressible Navier-Stokes equations
are solved with the multi-block structured code HiPSTAR (High Performance
Solver for Turbulence and Aeroacoustics Research) (Sandberg, 2015). The
spatial discretization involves both a five-point fourth-order central
standard-difference scheme with Carpenter boundary stencils in the streamwise
and crosswise directions (Carpenter et al., 1999), and a spectral method using
the FFTW3 library in the spanwise direction. The time discretization is
achieved by an ultra-low-storage five-step fourth-order Runge-Kutta scheme
(Kennedy et al., 1999). Characteristic-based boundary conditions are used to
prevent spurious reflections at the computational domain boundaries (Sandberg
& Sandham, 2006; Jones et al., 2008). Details of the problem formulation are
provided by Wu et al. (2019).

The case of Pargal et al. (2022) is an incompressible DNS of a flat-plate
turbulent boundary layer to emulate the boundary layer development on the
downstream portion of the CD airfoil flow studied by Wu et al. (2019). A
finite difference solver on a staggered grid was used. It employs second-
order, central differences for all spatial derivatives and second-order Adams-
Bashforth semi-implicit time advancement. To match the pressure gradient
parameter ($K$) of the airfoil boundary layer, a streamwise pressure gradient
is imposed by prescribing a streamwise-varying $U_{\infty}(x)$ at the top
boundary of the domain. The wall-normal freestream velocity $V_{\infty}(x)$ is
then obtained based on the conservation of mass. A fully turbulent boundary
layer flow at the inlet of the domain is obtained using the
recycling/rescaling method (Lund et al., 1998). A convective outflow boundary
condition (Orlanski, 1976) is used at the outlet and periodic boundary
conditions are used in the spanwise direction. Similar discretization methods
and boundary conditions were used in Wu & Piomelli (2018) and Na & Moin (1998)
with slight variations in details. Simulations of the flows in these two
studies were rerun, based on the methodologies of Pargal et al. (2022). For
the LES case of Wu & Piomelli (2018), the governing equations were solved for
the filtered quantities. The sub-grid stress tensor was modeled using a
dynamic eddy-viscosity model (Germano et al., 1991; Lilly, 1992), in which the
coefficient was adjusted using the Lagrangian-Averaging procedure (Meneveau et
al., 1996). Boundary layer developments in the rerun simulations will be
compared to those reported in the original studies in Section 3.

###  2.2 Experimental datasets

DNS and LES simulations are limited to comparatively low Reynolds numbers.
Experimental datasets are gathered from the studies of Hu & Herr (2016),
Fritsch et al. (2022b), and Goody (2004), which provide ZPG or pressure
gradient flow data with $Re_{\theta}$ of up to 23,400. A brief description of
the experimental setup of each case is given below. Hu (2018) carried out
experiments in an open-jet anechoic test section of Acoustic Wind Tunnel
Braunschweig (AWB). Adverse and favorable pressure gradients in flat-plate
boundary layers were achieved by placing a rotatable NACA 0012 airfoil above
the flat plate. Wall-pressure statistics were measured with sub-miniature
pressure transducers and boundary layer velocity profiles were obtained using
hot wires. $Re_{\theta}$ 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. (2022b) carried out
experiments in a subsonic wind tunnel with a NACA 0012 airfoil installed in
the center 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 $\beta$ of $-0.5$ to 0.5, with $Re_{\theta}$ reaching 18,000.
Goody & Simpson (2000) 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 $Re_{\theta}=23,400$.

##  3 Boundary layer development

In this section, the streamwise developments of pertinent flow and boundary
layer variables are presented for cases in the database. The goal is to
provide insights on the appropriate choice of scaling variables for WPS
modeling in non-equilibrium flows.

Figure 1: Streamwise development of pressure coefficient (a) and friction
coefficient (b) in the simulated cases:  Pargal et al. (2022),  Wu et al.
(2019),  Wu & Piomelli (2018), and  Na & Moin (1998), compared to original
data of Wu & Piomelli (2018) and Na & Moin (1998) ($\circ$). (c) Contour of
mean streamwise velocity normalized by $U_{o}$ in Wu & Piomelli (2018) case,
with streamlines shown at stream-function values of $\psi_{o}=0.5$, 1, 1.5 and
2.

Figure 1 shows the variation of the mean wall-pressure coefficient,
$C_{p}=(p|_{y=0}-p_{e})/(0.5\,\rho\,U_{e}^{2})$, and skin friction
coefficient, $C_{f}=(u_{\tau}/U_{e})^{2}/2$, where $p_{e}$ is edge static
pressure at the location of $x=0$, which corresponds to the reference (ZPG)
location defined in each of the studies. In Figure 1(b), the $C_{f}$ is
normalized by its value at the reference location to better compare all cases.
Only simulated cases are presented as the boundary parameters for a continuous
range of $x$ are available. Figures 1(a) shows that the variations of $C_{p}$
in the flat-plate (Pargal et al., 2022) and airfoil (Wu et al., 2019) cases
are matched, as required by the flat-plate case setup; comparison between
these two flows reveal the effects of convex wall curvature on WPS. In Figures
1(b), the variations of $C_{f}$ are shown to match overall for these two
cases, with some differences near the airfoil trailing edge due to the convex
wall curvature and trailing-edge effects. These comparisons are discussed in
Pargal et al. (2022). In the cases of Na & Moin (1998) and Wu & Piomelli
(2018) where the flows undergo freestream suction and blowing, the $C_{p}$
variation indicates three phases of a separated boundary layer flow (marked in
Figures 1(b)): (1) attached APG flow, (2) separated region and (3) reattached
flow under FPG. This is also reflected in $C_{f}$ variations: $C_{f}$ first
decreases toward zero in the APG region, reaching negative values in the
separated flow region, and increases near the flow reattachment in the FPG
region. The contour of mean streamwise velocity of the case of Wu & Piomelli
(2018) in Figure 1(c) confirms these flow stages. The coefficients are
compared between the results of the present rerun simulations and those of the
original studies (Na & Moin, 1998; Wu & Piomelli, 2018); good match is
obtained in both cases.

Figure 2: Development of boundary layer parameters in the simulated cases. For
legend see Figure 1.

In Figure 2, the variations of boundary layer thickness ($\delta$), momentum
thickness ($\theta$) and displacement thickness ($\delta^{*}$) along the
streamwise direction are shown. As expected with an increase in APG, the
boundary layer becomes thicker. For the cases with suction and blowing, the
thicknesses reach their maxima near the end of the APG zone and then decrease
with FPG. The development of the shape factor, $H$, in Figure 2(d) shows a
similar response, which reflects that the displacement thickness is more
sensitive to the pressure gradients compared to the momentum thickness.

Reynolds numbers based on different velocity scales are compared. The one
based on the inner velocity, $Re_{\tau}=u_{\tau}\delta/\nu$, shows a similar
trend as that of $C_{f}$ (Figure 2(e)), while the one based on edge velocity,
$Re_{\theta}=U_{e}\theta/\nu$, shows a variation similar to that of $\theta$
(Figure 2(f)). The DNS cases are conducted in low Reynolds numbers
($Re_{\theta}\approx 300$ to 1200), while higher values are reached for the
LES case ($Re_{\theta}\approx 2000$ to 7000).

Figure 3: Development of $R_{t}$ and $\beta$ in the simulated cases. For
legend see Figure 1. For Wu & Piomelli (2018) and Na & Moin (1998) cases, only
the attached-flow region upstream of the separation point is shown;
(vertical): location of separation point.

Figure 3 shows the streamwise developments of two boundary layer parameters
that are used in most existing WPS models to sensitize the modeled spectrum to
the Reynolds number and the pressure gradient: $R_{t}$ $\equiv
Re_{\tau}(u_{\tau}/U_{e})$ (Figure 3 (a)) and $\beta$ (Figure 3 (b)),
respectively. As the separation point is approached, $R_{t}$ tends to 0 and
$\beta$ to infinity. This indicates issues in many existing WPS models when
used for strong-APG flows near incipient separation (Caiazzo et al., 2023),
which are examined in detail in Section 5.1.

Cases | $Re_{\theta}$ | $H$ | $C_{f}$ | $\beta$
---|---|---|---|---
Hu (2018), ZPG | 4889 | 1.41 | 0.0025 | 0
Hu (2018), APG ($-6^{\circ}$) | 6979 | 1.61 | 0.0017 | 3.8
Hu (2018), APG ($-10^{\circ}$) | 8670 | 1.75 | 0.0012 | 6
Hu (2018), APG ($-14^{\circ}$) | 11046 | 2.12 | 0.0006 | 12.5
Hu (2018), FPG ($14^{\circ}$) | 1940 | 1.26 | 0.0068 | -0.5
Fritsch et al. (2022b), ZPG ($2^{\circ}$) | 16000 | 1.29 | 0.0026 | -0.02
Fritsch et al. (2022b), APG ($12^{\circ}$) | 18606 | 1.31 | 0.0024 | 0.58
Fritsch et al. (2022b), FPG ($-10^{\circ}$) | 14000 | 1.26 | 0.0028 | -0.47
Goody & Simpson (2000), ZPG | 7300 | 1.29 | 0.0026 | 0
Goody & Simpson (2000), ZPG | 23400 | 1.29 | 0.0022 | 0
Table 1: List of experimental datasets and values of boundary layer parameters
at measurement locations of available data. For Hu (2018) and Fritsch et al.
(2022b) datasets, the angle of attack of the airfoil imposed to generate mean
pressure gradient is indicated.

For most of the experimental datasets, streamwise variations of the boundary
layer parameters are available at discrete locations only. Representative
values of $Re_{\theta}$, $H$, $C_{f}$ and $\beta$ are tabulated in Table 1.
Specifically, the datasets of Hu (2018) experiments contain five cases: one
ZPG flow, three APG flows with $\beta$ varying from 4 to 12, and one FPG flow,
at $Re_{\theta}=5,000$ to 11,000. The data show that boundary layer
thicknesses (as indicated here by $Re_{\theta}$; for other thicknesses see the
original studies) and the shape factor increase with APG and decrease in FPG,
whereas $C_{f}$ decreases with APG and increases in FPG. The cases from
Fritsch et al. (2022b) are non-equilibrium APG and FPG flows but with
comparatively milder APG compared to Hu (2018). As a result, the variations in
boundary layer parameters are more limited. Also included are measurements by
Goody & Simpson (2000), which were carried out for ZPG flows only but reached
higher Reynolds numbers.

##  4 Wall-pressure statistics

Figure 4: (a) Streamwise variation of the wall-pressure r.m.s, normalized by
its value at the reference location. (b) Streamwise variation of the local
peak magnitude of Reynolds shear stress, normalized by its value at the
reference location. For legend see Figure 1.

In this section, the development of wall-pressure statistics along $x$ is
discussed using the datasets. Different normalizations are used to analyze the
wall-pressure scaling. Effects of pressure gradient, boundary layer separation
and reattachment on the wall-pressure spectrum are examined.

For the simulation datasets, streamwise variations of the root-mean-square
(r.m.s.) of wall-pressure fluctuations, normalized by their specific values at
$x=0$, are compared in Figure 4(a). Convex wall curvature and trailing edge
effect intensify the wall pressure fluctuations, as shown by the comparison
between the flat-plate case of Pargal et al. (2022) and the airfoil case of Wu
et al. (2019) for $x/\theta_{o}>200$. In the separated-flow region, a drop in
wall-pressure fluctuations is seen, which was also observed by Abe (2017). The
dip is attributed to the departure of turbulent eddies from the wall, with
mainly large recirculating eddies interacting with the near-wall region. As
the separated shear layer reattaches, the re-emergence of intense turbulent
motions near the wall leads to an augmentation of wall-pressure fluctuations,
shown by the $p_{rms}(x)$ maximum shortly after the reattachment point (at
$x/\theta_{o}\approx 280$ for Wu & Piomelli (2018) and $x/\theta_{o}\approx
500$ for Na & Moin (1998)). Interestingly, Figure 4(b) shows that the $x$
variation of the local maximum magnitude of the Reynolds shear stress profile,
$|\overline{u^{\prime}v^{\prime}}|_{\max}(x)$, is very similar to that of
$p_{rms}$: the decrease near the separation point and the peak near the flow
reattachment occur at almost the same $x$ locations. Downstream from the
reattachment point, as the flow recovers towards the equilibrium ZPG flow,
both $p_{rms}$ and $|\overline{u^{\prime}v^{\prime}}|_{\max}$ reduce towards
the ZPG values at $x=0$.

Figure 5:  Wall-pressure r.m.s. variation normalized by (a) local wall shear
stress ($\tau_{w}$), (b) local dynamic pressure ($q_{e}$), and (c) local peak
magnitude of Reynolds shear stress. In (a),  (vertical): locations of
separation points. In (b) and (c), separated-flow regions are marked. For
legend see Figure 1.

In Figure 5 different quantities are used to normalize wall-pressure r.m.s. as
it varies along $x$. The r.m.s. normalized by $\tau_{w}$ (Figure 5(a))
increases with APG and tends towards infinity as the separating point is
approached. The use of $\tau_{w}$ to scale $p_{rms}$ in WPS models is,
therefore, inappropriate for strong-APG boundary layers. The r.m.s. normalized
by $q_{e}=0.5\,\rho\,U_{e}^{2}$ (Figure 5(b)) displays a significant increase
in the APG zone before the flow separation. This is because wall-pressure
fluctuations are augmented in the APG region, while the edge velocity
decreases. In comparison,
$p_{rms}/(\rho|\overline{u^{\prime}v^{\prime}}|_{\max})$ stays almost constant
as long as the boundary layer is attached, regardless of the pressure gradient
(Figure 5(c)). In the separated flow regions, however, a dip of
$p_{rms}/(\rho|\overline{u^{\prime}v^{\prime}}|_{\max})$ is observed, caused
by a faster damping of $p_{rms}$ inside the recirculation bubble than that of
the Reynolds shear stress in the detached shear layer (as shown in Figure 4).
These observations indicate that wall-pressure r.m.s. scales better with
$\rho|\overline{u^{\prime}v^{\prime}}|_{\max}$ than with $q_{e}$ or
$\tau_{w}$, in attached flows under strong pressure gradients; similar
observations were made by Na & Moin (1998), Abe (2017) and Caiazzo et al.
(2023). However, the appropriate wall-pressure r.m.s. scaling for the
separated flow region remains yet to be found; but this is out of the scope of
the present work.

Cases | Legend
---|---
Pargal et al. (2022), $x/\theta_{o}=0$ , ZPG ($\beta=0$) |
Pargal et al. (2022), $x/\theta_{o}=100$, Low APG ($\beta=0.3$) |
Pargal et al. (2022), $x/\theta_{o}=290$, APG ($\beta=5$) |
Pargal et al. (2022), $x/\theta_{o}=340$, High APG ($\beta=8$) |
Wu et al. (2019), $x/\theta_{o}=0$ , ZPG ($\beta=0.01$) |
Wu et al. (2019), $x/\theta_{o}=100$, Low APG ($\beta=0.28$) |
Wu et al. (2019), $x/\theta_{o}=290$, APG ($\beta=4.8$) |
Wu et al. (2019), $x/\theta_{o}=340$, High APG ($\beta=8.3$) |
Wu & Piomelli (2018), $x/\theta_{o}=50$, Low APG ($\beta=0.9$) |
Wu & Piomelli (2018), $x/\theta_{o}=105$, Very high APG ($\beta=22.8$) |
Wu & Piomelli (2018), $x/\theta_{o}=120$, Very high APG ($\beta=46.28$) |
Wu & Piomelli (2018), $x/\theta_{o}=130$, Before flow separation ($\beta=171$) |
Na & Moin (1998), $x/\theta_{o}=50$, ZPG ($\beta=0.018$) |
Na & Moin (1998), $x/\theta_{o}=210$, High APG ($\beta=8$) |
Na & Moin (1998), $x/\theta_{o}=230$, Before flow separation ($\beta=162$) |
Hu (2018), ZPG ($\beta=0.1$) |
Hu (2018),High APG ($-10^{\circ}$) ($\beta=6$) |
Hu (2018), Very high APG ($-14^{\circ}$) ($\beta=12.5$) |
Hu (2018), FPG ($14^{\circ}$) ($\beta=-0.5$) |
Fritsch et al. (2022b), ZPG ($2^{\circ}$) ($\beta=0$) |
Fritsch et al. (2022b), APG ($12^{\circ}$) ($\beta=0.5$) |
Fritsch et al. (2022b), FPG ($-10^{\circ}$) ($\beta=-0.5$) |
Goody & Simpson (2000), ZPG (7300) ($\beta=0$) |
Goody & Simpson (2000), ZPG (23400 ) ($\beta=0$) |
Table 2:  Datasets in attached-flow regions (under ZPG or APG) that are
considered in analyses of wall-pressure spectrum.  Figure 6: Power spectral
density (PSD) of wall-pressure fluctuations for attached-flow datasets with
ZPG and APGs listed in Table 2, normalized by three different scalings
involving $\rho|\overline{u^{\prime}v^{\prime}}|_{\max}$ (as the pressure
scaling) and outer scales: (a) $\rho|\overline{u^{\prime}v^{\prime}}|_{\max}$,
$\delta$ and $U_{e}$, (b) $\rho|\overline{u^{\prime}v^{\prime}}|_{\max}$,
$\delta^{*}$ and $U_{e}$ and (c)
$\rho|\overline{u^{\prime}v^{\prime}}|_{\max}$, $\theta$ and $U_{e}$.
Normalizations with (d) inner scales ($\tau_{w}$, $\delta_{\nu}$ and
$u_{\tau}$), (e) mixed scales ($\tau_{w}$, $\delta$ and $U_{e}$), and (f)
outer scales ($q_{e}$, $\delta$ and $U_{e}$). The PSD is evaluated in dB.
Insets show ZPG profiles, demonstrating high-frequency collapse in (d) and
low-frequency collapse in (e), respectively. Legend is listed in Table 2.

The PSD of the wall-pressure fluctuations (denoted by $\phi_{pp}$) is computed
for all simulated and experimental cases and compared in Figure 6. Only
attached-flow regions, with ZPG or non-equilibrium APG, are considered here.
The $x$ location, $\beta$ value, and legend for each PSD curve in Figure 6 are
listed in Table 2. Different normalizations are compared. Both $Re_{\theta}$
($Re_{\theta}=300$ to $23,400$) and $\beta$ ($\beta=0$ to $200$) vary greatly
among these data. The high $\beta$ values occur near the separation points.

Figure 6(a) compares the results using
$-\rho|\overline{u^{\prime}v^{\prime}}|_{\max}$ as the pressure scale,
$\delta$ the length scale, and $U_{e}$ the velocity scale (or equivalently
$p_{rms}$, $\delta^{*}$ and the Zagarola-Smits velocity, as shown by Caiazzo
et al. (2023)). Note that in the experimental datasets the Reynolds shear
stress data were missing. For these cases, the wall shear stress ($\tau_{w}$)
at a mild-APG ($\beta<1$) location immediately upstream of the APG region,
instead of the local Reynolds shear stress, is used to form the pressure scale
for the strong-APG region. This approximation is based on the observation that
$|\overline{u^{\prime}v^{\prime}}|_{\max}(x)$ does not vary significantly in
the attached-flow region upstream of the separation point (Figure 4(b)). The
value of
$|\overline{u^{\prime}v^{\prime}}|_{\max}(x)\approx|\overline{u^{\prime}v^{\prime}}|_{\max}(0)$
can then be approximated as $\tau_{w}(0)$ due to the existence of a constant-
stress layer in a boundary layer under zero or mild pressure gradients. The
treatment mentioned above is employed for the experimental cases only. Under
such normalization, Figure 6(a) shows that approximate low-frequency collapse
is obtained. This is expected as the low-frequency contents are the main
contributor to $p_{rms}$, which in turn scales with
$\rho|\overline{u^{\prime}v^{\prime}}|_{\max}$. Swapping the length scale for
$\delta^{*}$ or $\theta$, however, gives more scatter in the low-frequency
range (Figures 6(b) and (c)), as also shown by Caiazzo et al. (2023). Even
though previous works (Kamruzzaman et al., 2015; Abe, 2017; Caiazzo et al.,
2023) have shown $\rho|\overline{u^{\prime}v^{\prime}}|_{\max}$ to be the best
pressure scaling for wall-pressure spectra, most of them were limited to low
Reynolds number cases with mild pressure gradients. Here, the chosen set of
scaling is shown to collapse low-frequency portion of the PSD for a large
range of Reynolds number with strong non-equilibrium APG also.

In comparison, normalization based on inner velocity and length scale only
(i.e. using $\tau_{w}$, $\delta_{\nu}\equiv\nu/u_{\tau}$ and $u_{\tau}$ as
shown in Figure 6(d)) gives a high-frequency collapse for the ZPG spectra, but
a large scatter for the APG ones. When mixed variables are used (i.e. using
$\tau_{w}$, $\delta$ and $U_{e}$ as shown in Figure 6(e)), which is commonly
applied in existing WPS models, the low-frequency range collapses for the ZPG
spectra only, but not for cases with strong APG, as $p_{rms}$ does not scale
with $\tau_{w}$. On the other hand, normalization based on outer variables
only (i.e. using $q_{e}$, $\delta$ and $U_{e}$ as shown in Figure 6(f)) gives
a better collapse than that based purely on the inner variables; however, it
still fails to collapse the low-frequency range. Based on these observations,
the best $\phi_{pp}$ scaling among these options is thus
$(\rho|\overline{u^{\prime}v^{\prime}}|_{\max})^{2}\delta/U_{e}$.

Figure 7:  Wall-pressure PSDs in APG boundary layers with different Reynolds
number ranges: (a) low-$Re_{\theta}$ range ($Re_{\theta}=300$ to $1000$), (b)
mid-$Re_{\theta}$ range ($Re_{\theta}=2000$ to $8000$), and (c)
high-$Re_{\theta}$ range ($Re_{\theta}=8000$ to $23,400$). (d) PSDs in FPG
flows (blue) compared to ZPG ones (gray). See Table 2 for legend.

The effects of Reynolds number are analyzed next. Figures 7(a-c) categorize
the wall pressure PSDs in APG and ZPG flows into three Reynolds number groups:
low-$Re_{\theta}$ ($Re_{\theta}\approx 300$ to 1000), mid-$Re_{\theta}$
($Re_{\theta}\approx 200$ to 8,000) and high-$Re_{\theta}$
($Re_{\theta}\approx 8,000$ to 23,400) groups. Note that in the
high-$Re_{\theta}$ group, only ZPG or mild-APG flows are available in the
present datasets. Figure 7(a) shows that all low-$Re_{\theta}$ spectra
collapse very well in the majority of frequency range. This is because the
overlap range is limited and the low-frequency range is well collapsed by
using $\rho|\overline{u^{\prime}v^{\prime}}|_{\max}$ as the pressure scaling.
At higher Reynolds numbers, the overlap range appears and grows with
$Re_{\theta}$ (Figures 7(b,c)). The width of the overlap range is shown to
decrease with APG and the slope of this range increases with APG.

The PSDs in FPG flows are shown in Figure 7(d) using the two FPG datasets (in
blue) of Hu (2018) and Fritsch et al. (2022b), as compared to the
corresponding ZPG spectra (in gray) from these two studies. Under
$\beta\approx-0.5$ (relatively mild FPG), both spectra show a milder slope in
the overlap range than the ZPG PSDs. This is consistent with the increase in
slope for APG flows discussed above. In addition, the overlap ranges are
slightly widened under FPG with the low-frequency limit moving towards lower
frequencies, especially for the lower-Reynolds-number case (Hu, 2018). This is
associated with a weaker mean-flow wake region under FPG.

Figure 8: (a) Wall-pressure PSDs in separated-flow regions ( ) compared to
those at respective reference ZPG locations ( ); gray lines show Wu & Piomelli
(2018) data at $x/\theta_{o}=150$, 175, 200, 220 and 240; black lines show Na
& Moin (1998) data at $x/\theta_{o}=270$, 300 and 400. (b) Wall-pressure PSDs
in reattached-flow region of Wu & Piomelli (2018), at $x/\theta_{o}=275$, 300,
350 and 450. Increase in line thickness indicates downstream direction.

To analyze the WPS associated with separated and reattached flows, Figures
8(a) and 8(b) compare the spectra extracted, respectively, from the separated-
flow regions and the regions downstream of the boundary layer reattachment in
the cases of Na & Moin (1998) and Wu & Piomelli (2018). Figures 8(a) shows
that in the separated-flow regions both overlap-range and high-frequency wall-
pressure fluctuations are reduced compared to those in respective reference
ZPG locations (dashed lines), due to the departure of intense turbulent
motions from the wall following the detachment of the shear layer. The scaling
does not collapse the low-frequency range as it does for attached flows. This
is expected as the wall pressure r.m.s does not scale with
$|\overline{u^{\prime}v^{\prime}}|_{\max}$ in this region (Figures 5(c)).
However, it is interesting that the shape of the spectrum does not vary
significantly in the separated-flow region: the spectra in Figures 8(a) all
display a narrow low-frequency peak with greatly reduced high-frequency
contribution. Downstream from the reattachment point, Figure 8(b) shows that
the spectrum recovers gradually from the low-frequency-dominant state inside
the separated-flow region towards the equilibrium state, with augmented mid-
to high-frequency contents.

##  5 Wall-pressure spectra modelling

###  5.1 Performance of existing wall-pressure spectra models

Model | Legend
---|---
Goody (2004) |
Lee (2018) |
Rozenberg et al. (2012) |
Hu (2018) |
Kamruzzaman et al. (2015) |
Proposed model |
Table 3: List of WPS models examined with the present datasets.  Figure 9:
Comparison between predictions of WPS models and numerical or experimental
measurements ( ), for different types of flow: (a) weak-APG and high-Reynolds-
number flow (Fritsch et al., 2022b, $12^{\circ}$), (b) strong-APG and low-
Reynolds-number flow (Wu et al., 2019), (c) very-strong-APG flow at
intermediate Reynolds number (Wu & Piomelli, 2018), (d) flow near separation
point at a low Reynolds number (Na & Moin, 1998), and (e) flow near separation
point at an intermediate Reynolds number (Wu & Piomelli, 2018). All models
predictions are re-normalized in chosen scaling for comparison purposes.
Legend of model results is given in Table 3.

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 Section 1 against the present datasets of ZPG and APG
(attached regions only) flows (marked by blue solid lines) for a few
$Re_{\theta}$-$\beta$ combinations. Among them, Figures 9(d,e) show two
examples near boundary-layer separation. The models and their legend are
listed in Table 3. The optimal $\phi_{pp}$ normalization as identified in
Section 4 is used. Results of the new generalized model (shown by red solid
lines) to be introduced in Section 5.2 is also compared. The comparison
demonstrates dependencies of the WPS on the APG and the Reynolds number as
discussed in Section 4. Under the present normalization, the main variations
are in the width and slope of the overlap range.

In the (almost) ZPG region (Figure 9 (a)), all tested models are shown to give
reasonably good overall predictions . Kamruzzaman et al. (2015) model does not
produce an overlap range and, as a result, yields significantly under-
predicted high-frequency contents. Moreover, except for Hu (2018) model, all
existing models over-predict the spectrum in the overlap range, partially
because of a mismatch in the slope.

For a flow under relatively weak APG ($\beta\approx 8$), Figure 9(b) shows
that Rozenberg et al. (2012) and Lee (2018) models give very good overall
predictions, whereas Hu (2018) and Kamruzzaman et al. (2015) models under-
predict the WPS at low frequencies. Goody (2004) 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 9(c-e)), especially near the boundary layer
separation point, the existing models give large errors. This is because 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. (2012) model:

$\frac{\phi_{pp}(\omega)U_{e}}{\tau_{w}^{2}\delta^{*}}=\frac{0.78(1.8\Pi\beta+6)(\omega\delta^{*}/U_{e})^{2}}{[(\omega\delta^{*}/U_{e})^{0.75}+C_{1}^{\prime}]^{3.7}+[C_{3}^{\prime}(\omega\delta^{*}/U_{e})]^{7}},$
(1)

where $C_{3}^{\prime}=3.76R_{T}^{-0.57}$, reaches a singularity as $\tau_{w}$
and $R_{t}$ become zeros and $\beta$ becomes infinity. 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 airfoil boundary layers in
Kamruzzaman et al. (2015), flat-plate boundary layers in Hu (2018), and ZPG
flows in Goody (2004). Moreover, sometimes the boundary layer flow properties
used for model calibration were estimated from lower-fidelity methods such as
XFOIL or RANS calculations. Employing $\tau_{w}$ as the pressure scale for
$\phi_{pp}$ also renders the dimensionless spectrum complicated to model, even
for attached flows at strong APGs, as the dimensionless $\phi_{pp}$ displays
large variations. In Section 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

Goody (2004) model is modified by replacing the Reynolds number and pressure
gradient parameters with new model inputs that derive directly from the local
mean velocity distribution, $U(x,y)$, and are not defined based on $u_{\tau}$.
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.

Goody (2004) model reads

$\widetilde{\phi}_{pp}\left(\widetilde{\omega}\right)=\frac{a\,\widetilde{\omega}^{b}}{(h\,\widetilde{\omega}^{c}+d)^{e}+(f\mathcal{F}\,\widetilde{\omega})^{g}},$
(2)

where $\widetilde{\phi}_{pp}=\phi_{pp}/\phi_{pp}^{*}$ and
$\widetilde{\omega}=\omega/\omega^{*}$ are dimensionless WPS and frequency,
respectively, based on a spectrum scaling of $\phi^{*}_{pp}$ and a frequency
scaling of $\omega^{*}$ as listed in Table 4. The coefficients $a$ to $h$ and
the function $\mathcal{F}$ are defined in Table 4. 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
9(a)), 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 (2018) 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.

Parameters in Equation0(2) | Goody (2004) | Present model
---|---|---
$\phi_{pp}^{*}$ | $U_{e}/(\tau_{w}^{2}\delta)$ | $U_{e}/[(\rho|\overline{u^{\prime}v^{\prime}}|_{\max})^{2}\delta]$
$\omega^{*}$ | $\delta/U_{e}$ | $\delta/U_{e}$
$a$ | 3 | 3
$b$ | 2 | 2
$c$ | 0.75 | $\min[1,0.8+(3.34e^{-4})\Pi^{1.86}(y_{w}^{+})^{0.76}]$
$d$ | 0.5 | 0.7
$e$ | 3.7 | 3.7
$f$ | 1.1 | 1
$g$ | 7 | 7
$h$ | 1 | 1
$\mathcal{F}$ | $R_{t}^{-0.57}$ | $(y_{w}^{+})^{-0.37}$
Separated flow ($\tau_{w}\leq 0$) | - |  $y_{w}^{+}=2$, $c=0.75$, $d=0.5$
Limited-log-layer flow ($y_{w}^{+}<15$) | - |  $y_{w}^{+}=15$, $c=0.85$
Table 4: Comparison between Goody (2004) model and the proposed model.
$\phi_{pp}^{*}$ and $\omega^{*}$ are the chosen scalings for WPS and
frequency, respectively. Special treatments for flow region characterized by a
separated boundary layer or a limited logarithmic layer are listed.  Figure
10: Comparison between model predictions of Equation (2) with progressive
changes described in text ( ) and Wu & Piomelli (2018) ( ) data in the
attached-flow region with weak to strong APGs, to show improvement brought by
each model change. (a) Goody (2004) model, (b) pressure scaling changed from
$\tau_{w}$ to $\rho|\overline{u^{\prime}v^{\prime}}|_{\max}$, (c) additionally
replacing dependence on $R_{t}$ with that on $y_{w}^{+}$, (d) further addition
of Coles’ parameter. Thicker lines indicate increase in $x$ (corresponding to
increasing $\beta$).

In the following, the main model changes (as listed in Table 4) are introduced
and progressively applied to demonstrate the improvement of each change.
First, the pressure scale (i.e. $\tau_{w}$) on the left-hand-side of Equation
(2) is replaced with $\rho|\overline{u^{\prime}v^{\prime}}|_{\max}$, with the
additional change of replacing the constant $d=0.5$ with 0.7 for a better low-
frequency collapse with current data. The effect of these modifications is
shown in Figure 10(b), where the low-frequency range is shown accurately
predicted for all the $x$ locations. The use of
$\rho|\overline{u^{\prime}v^{\prime}}|_{\max}$ ensures that the spectral
values are finite near the separation point.

Figure 11:  Variation of wall-pressure r.m.s. normalized by $\tau_{w}$ in ZPG
flows:  $\Diamond$ Wu et al. (2019),  $\Diamond$ Pargal et al. (2022),
$\Diamond$ Wu & Piomelli (2018), $\triangle$ Blake (1970),  $\triangle$
Simpson et al. (1987), $\circ$ Farabee & Casarella (1991) and $\square$ Bull &
Thomas (1976).  $p_{rms}^{2}/\tau_{w}^{2}=6.5$ for $Re_{\tau}<333$ and
$p_{rms}^{2}/\tau_{w}^{2}=6.5+1.86\ln(Re_{\tau}/333)$ for $Re_{\tau}\geq 333$
(Farabee & Casarella, 1991).

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. Turbulent fluctuations in the logarithmic layer are
known as the main contributor to the overlap range of the WPS. For example,
Farabee & Casarella (1991) showed that for ZPG boundary layers, with an
increase in Reynolds number accompanied by a thickening of the logarithmic
layer, the overlap range becomes wider and $p_{rms}/\tau_{w}$ increases. This
dependency is also shown by the current ZPG datasets in Figure 11. Additional
evidence is provided by Jaiswal et al. (2020), 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 wall-pressure Poisson’s 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. Specifically, the term $1.1R_{t}^{-0.57}$ is
replaced with $(y_{w}^{+})^{-0.37}$, with the constant fitted from present
data.

Figure 12: (a-c) Calculation of $\kappa(x)$, $B(x)$ and model parameters shown
using Wu et al. (2019) data at locations $x/\theta_{o}=50$, 105, 120 and 130.
(a) Mean velocity profiles in inner units. Increasing line thickness indicates
increase in $x$. (b) Diagnostic function, $I=y^{+}\partial U^{+}/\partial
y^{+}$. (c) Velocity profiles with logarithmic relation subtracted.
Calculations of $\kappa$, $\Pi$ and $y_{w}^{+}$ are indicated for
$x/\theta_{o}=105$ in (b,c). (d) Correlation between calculated $\kappa B$ and
$B$, compared to the fitted relation from Nagib & Chauhan (2008) ( ):  $\circ$
Wu & Piomelli (2018) (attached flow before separation), $\circ$ Wu & Piomelli
(2018) (attached flow downstream of reattachment),  $\Diamond$ Hu (2018),
$\times$ Fritsch et al. (2022a),  $+$ Goody (2004) ,  $\square$ Pargal et al.
(2022),  $\rhd$ Wu et al. (2019) and  $\bigtriangleup$ Na & Moin (1998).

The parameter $y_{w}^{+}(x)$ is dynamically determined based on the boundary
layer mean velocity $U(x,y)=\overline{u}$, 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 12(c)). Here, $\kappa$ is
the von Kármán constant and $B$ is the log-law intercept, both of which
allowed to vary along $x$ in a non-equilibrium boundary layer. Note that due
to the use of local plus 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 12(b)) is calculated from the mean velocity profile (Figure
12(a)) for data of Wu et al. (2019). 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 12(d) shows that the correlation between $\kappa(x)$ and $B(x)$
obtained for all cases in the present datasets in the attached flow regions is
consistent with that observed by Nagib & Chauhan (2008) and Nickels (2004)
from a large collection of flows with or without pressure gradients. As shown
in Figure 10(c) compared to Figure10(b), the modeling 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. The main WPS
prediction error is now predominantly an inaccurate slope of the overlap
range, as shown in Figure 10(c).

Recognizing that an APG leads to more energized large turbulence motions in
the outer layer and a thinner logarithmic layer, 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. To account for the variation in
the strength of the wake region, an additional model input is used: Coles’
parameter, $\Pi(x)$. An augmentation of $\Pi$ signals stronger turbulent
intensity and mixing in the outer layer. Here, $\Pi(x)$ is evaluated based on
$U^{+}(x,y)$, by measuring the peak value of
$U^{+}-[\kappa^{-1}(\log(y^{+})+B)]$ (as shown in Figure 12(c)) and dividing
it by $2/\kappa(x)$. In the generic model form in Equation (2), the
coefficient $c$ is known to impose the slope of the overlap range of
$\phi_{pp}$ (Thomson & Rocha, 2022; Lee, 2018; Rozenberg et al., 2012).
Therefore, the constant $c$ is replaced by a function of both $\Pi$ and
$y_{w}^{+}$, fitted based on the present datasets. To keep the model stable,
especially in cases with extreme pressure gradient or in the region near flow
separation, $c$ is limited to a maximum value of 1. The final form of the
generalized WPS model is

$\frac{\phi_{pp}(\omega)U_{e}}{(\rho|\overline{u^{\prime}v^{\prime}}|_{\max})^{2}\delta}=\frac{3(\omega\delta/U_{e})^{2}}{[(\omega\delta/U_{e})^{c}+0.7]^{3.7}+[(y_{w}^{+})^{-0.37}(\omega\delta/U_{e})]^{7}},\quad\text{where}$
(3) $c=\min\left[1,\;0.8+3.34e^{-4}\Pi^{1.86}(y_{w}^{+})^{0.76}\right].$ (4)

Figure 10(d) shows that the generalized model predicts the WPS very well in
both ZPG and strong-APG flows in the present datasets. 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.

Figure 13: Comparison between model prediction ( ) and simulation data ( Na &
Moin (1998) and  Wu & Piomelli (2018)) in the separated-flow regions.
Increasing line thickness indicates an increase in $x$ as marked in Figure
8(a).

A few scenarios require special treatments as listed in Table 4. For cases
with very low Reynolds numbers with extreme adverse pressure gradient which
practically removes 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 almost
identical WPS profile 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 4), calibrated based on
the data of Na & Moin (1998) and Wu & Piomelli (2018). 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 above-mentioned separated-flow treatment, the proposed model is evaluated
at a number of streamwise locations inside the separation regions of the flows
of Na & Moin (1998) and Wu & Piomelli (2018) in Figure 13. The WPS prediction
does not vary with $x$ in this region, since the model parameters are set to
constants. 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 14: Comparison between model prediction ( ) and Wu & Piomelli (2018)
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;  value corresponding to logarithmic layer. (b) Wall-
pressure PSD comparison. Increasing line thickness indicates an increase in
$x$.

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 till a
near-equilibrium ZPG state in Figure 14 against Wu & Piomelli (2018) data, at
four $x$ locations between $x/\theta_{o}=250$ and $x/\theta_{o}=350$. Figure
14(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 14 (b)). The model is
shown to capture such trend of WPS variation. The overall spectral levels are
well predicted due to the approximate scaling of $p_{rms}$ on
$\rho|\overline{u^{\prime}v^{\prime}}|_{\max}$, while the spectral shape is
captured by $y_{w}^{+}(x)$ and $\Pi(x)$ representing the local thickening of
the logarithmic layer and the weakening of wake, respectively, during
recovery.

Figure 15:  Comparison between model prediction ( ) and experimental data in
FPG flows (Hu, 2018; Fritsch et al., 2022b). For legend see Table 2. (a) Mean
velocity profile with the logarithmic relation subtracted; FPG and ZPG states
are shown in blue and gray, respectively.  Value corresponding to logarithmic
layer. (b) Wall-pressure PSD comparison.

In addition, the model is tested in FPG flows as shown in Figure 15, against
the experimental data of Hu (2018) and Fritsch et al. (2022b). Figure 15(a)
shows that, under FPGs (shown in blue) as compared to the ZPG profiles (shown
in gray), the main change is a reduction of $\Pi$, which leads to a milder
slop of the WPS overlap range. Figure 15(b) shows that this change in WPS is
globally captured well by the proposed model.

Figure 16: WPS prediction of the proposed model ( ) compared to measurements (
) in the following cases: (a,b) high-Reynolds-number and low-APG flow (Fritsch
et al., 2022b, $\beta=0.58$), (c,d) low-Reynolds-number and strong-APG flow
(Wu et al., 2019, $\beta=8.3$), and (e,f) high-Reynolds-number and strong-APG
flow (Hu, 2018, $\beta=6$). Yellow regions mark prediction variations with
$\pm 30$% change in input parameters $y_{w}^{+}$ (a,c,e) or $\Pi$ (b,d,f). In
(d), variation of $\Pi$ does not change WPS prediction as $c$ is set to a
constant at this location due to $y_{w}^{+}<15$ (see Table 4).

Figure 16 shows results of sensitivity analyses carried out for the parameters
$y_{w}^{+}$ and $\Pi$ 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 $\beta$. Figures 16(a,c,e) show 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. Figures 16(b,d,f) show
that $\Pi(x)$ 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 $\Pi$ do not significantly worsen
WPS prediction.

##  6 Conclusions and discussions

In this study, datasets collected from numerical (DNS and LES) and
experimental studies are used to characterize the variation of wall-pressure
statistics in various types of boundary layer flows, attached or separated and
then reattached, with zero, adverse or favorable pressure gradients at
different ranges of Reynolds number. The numerical data in the datasets were
validated in various flow quantities. They are not prone to errors originating
from installation effects as in many experimental studies and are free from
modeling errors as in RANS or boundary layer closures used in the development
of some existing WPS models. Strongly non-equilibrium streamwise pressure
gradient variations are included in the datasets. By comparing different sets
of variables used to normalize the wall-pressure spectrum ($\phi_{pp}$), an
optimal set of scaling is identified: $U_{e}$, $\delta$ and
$\rho|\overline{u^{\prime}v^{\prime}}|_{\max}$, and is used for wall-pressure
spectrum model development.

The performances of various existing wall-pressure spectrum models are
evaluated in the flows contained in the datasets. These models are shown to
fail to predict the wall-pressure spectra in non-equilibrium strong-APG flows.
The failures are caused by the use of inappropriate pressure scaling
($\tau_{w}$), being fitted to limited types of flows, and the dependencies on
$u_{\tau}$-based model parameters, as $u_{\tau}$ reduces to zero at the
detachment point.

Next, more robust model parameters are proposed and used to modify Goody’s
model. These parameters are (i) the logarithmic-layer extent, $y_{w}^{+}$, and
(ii) Coles’ parameter, $\Pi$. These parameters carry information on the local
state of the boundary layer flow as carried in the mean velocity profile. The
parameters are quantifiable from Reynolds-averaged Navier-Stokes (RANS)
calculation of the mean velocity. This is a more direct approach to model the
change in contributions of wall-layer and outer-layer turbulent flows to the
wall-pressure spectrum, compared to existing approaches based on local
pressure gradient (e.g. Clauser’s parameter, $\beta$) and/or local Reynolds
number (e.g. $R_{t}$). In addition, $y_{w}^{+}$ and $\Pi$ capture the history
effect of non-equilibrium pressure gradients as recorded in the velocity
prediction (from a RANS simulation, for example), which is not directly
represented by the local pressure gradient.

Comparison with available numerical and experimental measurements shows that
the proposed model gives good predictions for ZPG, APG (attached-flow region)
and FPG flows. For strong APG flows with boundary-layer separation and
reattachment, the wall-pressure spectra are shown to display similar shapes
and magnitudes across the separation bubble. There, the model is shown to give
overall good predictions, if the overlap-range width and slope are set to
constants fitted based on present data. A qualitatively good prediction is
also obtained downstream of flow reattachment where the boundary layer departs
significantly from its equilibrium state. Hence, the new model is considered
as a generalized wall-pressure spectral model for a wide range of equilibrium
and non-equilibrium boundary layers, as opposed to existing models designed
for limited types of flows.

## Acknowledgments

SP is grateful for the funding provided by the Consortium for the Development
of Ultra-High Efficiency Quiet Fans at Université de Sherbrooke. SP and JY
also gratefully acknowledge the additional financial support by Office of
Naval Research (Award No. N00014-17-1-2102). Computational support was
provided by Michigan State University’s Institute for Cyber-Enabled Research
and the Digital Research Alliance of Canada.

## Declaration of interests

The authors report no conflict of interest.

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