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This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics 1
A generalized wall-pressure spectral model
for non-equilibrium boundary layers
Saurabh Pargal1,2, Junlin Yuan1†and Stephane Moreau2
1Michigan State University, Michigan, USA
2Universit´e de Sherbrooke, Quebec, Canada
(Received xx; revised xx; accepted xx)
This study uses high-fidelity simulations (DNS or LES) and experimental datasets to1
analyse the effect of non-equilibrium streamwise mean pressure gradients (adverse or2
favourable), including attached and separated flows, on the statistics of boundary layer3
wall-pressure fluctuations. The datasets collected span a wide range of Reynolds numbers4
(Reθfrom 300 to 23,400) and pressure gradients (Clauser parameter from −0.5 to 200).5
The datasets are used to identify an optimal set of variables to scale the wall-pressure6
spectrum: edge velocity, boundary layer thickness, and the peak magnitude of Reynolds7
shear stress. Using the present datasets, existing semi-empirical models of wall-pressure8
spectrum are shown unable to capture effects of strong, non-equilibrium adverse pressure9
gradients, due to inappropriate scaling of wall pressure using wall shear stress, calibration10
with limited types of flows, and dependency on model parameters based on friction11
velocity, which reduces to zero at the detachment point. To address these short-comings,12
a generalized wall-pressure spectral model is developed with parameters that characterize13
the extent of the logarithmic layer and the strength of the wake. Derived from the local14
mean streamwise velocity profile, these two parameters inherently carry effect of the15
Reynolds number, as well as those of the non-equilibrium pressure gradient and its16
history. Comparison with existing models shows that the proposed model behaves well17
and is more accurate in strong-pressure-gradient flows and in separated-flow regions.18
Key words:19
†Email address for correspondence: junlin@msu.edu
2S. Pargal et al.
1. Introduction20
The fluctuation in space and time of the wall pressure beneath a turbulent boundary21
layer is one of the major sources of flow-induced noise and vibrations. Accurate modeling22
of the statistics of wall-pressure fluctuations is important for noise prediction in a wide23
range of applications such as wind turbines (Avallone et al. 2018; Deshmukh et al.24
2019; Venkatraman et al. 2023), cooling fans (Sanjos´e & Moreau 2018; Swanepoel et al.25
2023; Luo et al. 2020), propellers (Lallier-Daniels et al. 2021; Casalino et al. 2021),26
unmanned/manned air vehicles or drones (Lauzon et al. 2023; Pargal et al. 2023; Celik27
et al. 2021), and cabin noise (Samarasinghe et al. 2016; Borelli et al. 2021), etc., as28
well as for prediction of flow-induced structure fatigue (Franco et al. 2020). In these29
applications, the boundary layer flows are often turbulent and non-equilibrium, due30
to surface curvature and significant pressure gradients that vary in the streamwise31
direction, which may induce boundary layer separation and can be found in a large32
range of Reynolds number. Here, a non-equilibrium boundary layer is defined as that33
with streamwise (i.e. x) variation of the Clauser parameter, β(x) = (δ∗/τw)(dpe/dx),34
where δ∗(x) is the displacement thickness, τw(x) is the wall shear stress, and pe(x) is35
the static pressure at the edge of the boundary layer. Therefore, the generation of noise36
in non-equilibrium turbulent boundary layers is physically complex and challenging to37
model.38
The modeling of wall-pressure loading as a noise source predominantly depends on39
the power spectral density (PSD) of wall-pressure fluctuations, as well as its spanwise40
correlation length and the convection velocity of turbulent structures (Amiet 1976; Roger41
& Moreau 2005; Moreau & Roger 2009; Lee et al. 2021). The focus here is on modeling42
wall-pressure spectrum (WPS). It is established that the WPS of a boundary layer with43
zero or minimal pressure gradient consists of three ranges (Goody 2004; Farabee &44
Casarella 1991; Chang III et al. 1999): (i) a range with ω2variation at low frequencies45
(where ωis the frequency), (ii) a range with ω−5behavior at high frequencies, and (iii) an46
Wall-pressure spectral model for non-equilibrium boundary layers 3
overlap range with an approximate ω−1decay between the above two ranges. Based on47
data primarily in equilibrium flows, the width of the overlap range was found to increase48
with Reynolds number (Farabee & Casarella 1991; Goody 2004).49
Contributions from different layers of wall turbulence to the WPS has been studied50
and are summarized below. The experimental studies of Farabee & Casarella (1991)51
suggested different dominant sources for different wavenumber ranges of the WPS: the52
high-wavenumber range is mainly attributed to turbulent activities in the logarithmic53
region, while the low-wavenumber range is attributed to large-scale turbulent motions54
in the outer layer. Van Blitterswyk & Rocha (2017) quantified the correlations between55
the fluctuations of wall pressure and those of streamwise velocity in different layers of56
boundary layer and observed that high-frequency and overlap ranges of the WPS are57
associated with flows in the buffer and logarithmic regions, respectively. As opposed to58
earlier studies performed on channel flow or canonical flat-plate boundary layer data,59
Jaiswal et al. (2020) analyzed data collected near the trailing edge of a cambered60
aerofoil with a strong mean adverse pressure gradient (APG) in a highly non-equilibrium61
turbulent boundary layer, to compare contributions from various velocity sources (i.e. the62
mean-shear and turbulence-turbulence terms) at different wall-normal locations to the63
wall-pressure fluctuations based on the pressure Poisson’s equation. They found that the64
mean shear (MS) term in the inner and logarithmic regions is the dominant contributor,65
especially in the mid-to-high frequency range.66
Past studies were mostly on zero-pressure-gradient (ZPG) turbulent boundary layers.67
They showed that the wall-pressure fluctuations (evaluated by the root-mean-square68
values, prms) are amplified under a higher Reynolds number, mainly due to the increase69
in overlap-range spectral contribution. For a ZPG flat-plate boundary layer, Farabee &70
Casarella (1991) integrated the pressure spectrum over various frequency ranges and71
showed that the low-to-mid frequency range and the high-frequency range were not72
sensitive to a change in Reynolds number, whereas the significance of the overlap range73
increases with Reynolds number, leading to an augmentation of prms. They proposed74
that p2
rms/τ2
w= 6.5 + 1.86 ln(Reτ/333), which was tested with ZPG boundary layer and75
channel flow data. Panton & Linebarger (1974) also demonstrated that the overlap range76
4S. Pargal et al.
is correlated with the Reynolds number.77
The current understanding of the WPS in non-zero pressure gradient flows is sum-78
marized as follows. Review of the earlier work before the mid 1990s are provided by79
Willmarth (1975) and Bull (1996). Schloemer (1967) showed that under an APG low-80
frequency contents of the WPS become more prominent as large eddies are energized,81
while the high-frequency contents become less important. Under a favourable pressure82
gradient (FPG), however, the opposite applies, with stronger high-frequency contents.83
The WPS slope in the overlap range also varies with the pressure gradient. Cohen84
& Gloerfelt (2018) investigated the effects of mild pressure gradient using large-eddy85
simulations (LES) and showed scale-based dependencies of the WPS on FPG similar86
to those observed before. Na & Moin (1998) conducted direct numerical simulation87
(DNS) of a boundary layer with prescribed freestream suction and blowing to induce88
flow separation and reattachment. They showed that none of the outer, inner or mixed89
scaling collapsed the wall-pressure spectra in all regions of the flow. Normalisation with90
the local maximum magnitude of the Reynolds shear stress, however, was shown to91
collapse the low-frequency range of WPS for APG flows including those with separation92
(Abe 2017; Ji & Wang 2012; Caiazzo et al. 2023).93
Modeling of turbulent WPS is broadly classified in two categories: (i) semi-empirical94
modelling and (ii) analytical modeling based on solution of the Poisson’s equation of95
pressure (Kraichnan 1956; Panton & Linebarger 1974; Jaiswal et al. 2020; Grasso et al.96
2022; Palani et al. 2023; Hales & Ayton 2023). The focus of this paper is on the first97
approach, which requires a smaller amount of inputs from the flow field in comparison to98
the analytical modelling approach. Existing semi-empirical WPS closures mostly model99
the magnitude and shape of the WPS normalized by some boundary-layer parameters100
that are either internal, external or mixed, such as the boundary layer thickness (δ),101
the edge velocity (Ue) and the wall shear stress (τw=ρu2
τ, where uτis the friction102
velocity and ρis the density), etc. For some of these studies see Kraichnan (1956), Corcos103
(1964), Willmarth (1975), Amiet (1976), Bull & Thomas (1976), Chase (1980), Goody104
(2004), Rozenberg et al. (2012), Kamruzzaman et al. (2015), Lee (2018), Hu (2018), and105
Pargal et al. (2022). Goody (2004) proposed a model for ZPG boundary layers, which106
Wall-pressure spectral model for non-equilibrium boundary layers 5
accurately models the Reynolds number effect on the wall-pressure spectrum for these107
flows. To capture the pressure gradient effect, several other models have been proposed108
(Rozenberg et al. 2012; Kamruzzaman et al. 2015; Catlett et al. 2016; Hu et al. 2013;109
Lee 2018; Rossi & Sagaut 2023). Rozenberg et al. (2012) integrated additional boundary110
layer flow parameters to sensitize the model to pressure gradient effects, especially those111
of APG. The additional parameters include Clauser’s parameter (β) (Clauser 1954) and112
Cole’s wake parameter (Π) (Coles 1956). The former includes the local effect of mean113
pressure gradients, while the latter represents the cumulative effect of the history of mean114
pressure gradient up to the considered location in the boundary layer. Several later models115
developed modifications of the model that capture effects of other complexities such as116
wall curvature and FPG. Kamruzzaman et al. (2015) developed a model by fitting it on117
a large amount of experimental WPS data collected in various non-equilibrium boundary118
layer flows on aerofoils. Hu (2018) used the shape factor (H) and Reynolds numbers (Reθ
119
or Reτ) instead of βto incorporate the effect of non-equilibrium pressure gradients, as β120
— a descriptor of local pressure gradient — does not carry the history effect of a spatially121
varying pressure gradient. Lee (2018) improved Rozenberg’s model based on experimental122
data gathered from a wide range of flows with different Reynolds numbers and pressure123
gradients. Thomson & Rocha (2022) proposed a new model for flows with FPG. Recently,124
machine learning approaches such as gene expression programming and artificial neural125
networks were used to model WPS as a function of boundary layer parameters (Fritsch126
et al. 2022a; Dominique et al. 2022; Shubham et al. 2023; Ghiglino et al. 2023).127
Despite the success of the models mentioned above in the specific flows for which128
they were developed, these models are not universally applicable to both ZPG flows129
and those with non-equilibrium pressure gradients and/or surface curvature, due to130
the following reasons. (i) Models developed by curve-fitting to data of a limited type131
of flows do not naturally apply to other flows, such as Goody’s model, which works132
for ZPG flows only. (ii) Normalisations of wall-pressure statistics used for ZPG flows133
(e.g. τw) may not be appropriate for strong-APG flows (e.g. a boundary layer close to134
separation where τwapproaches zero). (iii) The choices of local boundary layer parameters135
do not account sufficiently for the history effect of the pressure gradient. In addition,136
6S. Pargal et al.
some existing models were developed based on experimental wall-pressure measurements137
that are supplemented with low-fidelity flow-field data, such as those estimated from138
XFOIL (Drela 1989).139
The objective of this study is therefore to develop a general WPS model that is140
tunable for both ZPG and non-equilibrium, strong-pressure-gradient turbulent boundary141
layers, as well as special cases such as flow separation and reattachment. To this end,142
model parameters that capture the local characteristics of the mean streamwise velocity143
profile (which evolves under a history of the pressure gradient variation) are derived and144
incorporated to sensitize the model to the streamwise pressure gradient and its history.145
An appropriate pressure normalisation for flows with and without pressure gradients is146
used. The model is calibrated based on a large and inclusive database, containing both147
experimental measurements and DNS/LES data (existing or new) of flows over a wide148
range of Reynolds number, with or without separation.149
The organization of the paper is as follows. Section 2 describes the database, Section 3150
presents the boundary layer development of the cases in the datasets, Section 4 discusses151
the wall-pressure fluctuations and WPS in the datasets, Section 5 discusses the perfor-152
mances of existing WPS models and then introduces a new generalized WPS model, and153
conclusions are presented in Section 6.154
2. Datasets collection155
The first step to develop a generalized WPS model is to collect and analyze high-fidelity156
datasets in a wide range of flows. The goal is to collect datasets for both equilibrium157
and non-equilibrium boundary layers, including ZPG, FPG, and APG flows, with or158
without wall curvature (as in boundary layers developed on aerofoils) and boundary159
layer separation and reattachment, across a wide range of Reynolds number based on160
momentum thickness (Reθ= 300 to 23,400).161
2.1. Simulation datasets162
DNS and LES datasets are gathered or re-generated from cases in four prior studies:163
Pargal et al. (2022), Wu et al. (2019), Na & Moin (1998) and Wu & Piomelli (2018).164
Wall-pressure spectral model for non-equilibrium boundary layers 7
Cases ReθβK(106)
Wu et al. (2019), DNS 300-1200 0 to 12 -4 to 0
Pargal et al. (2022), DNS 300-1200 0 to 10 -4 to 0
Na & Moin (1998), DNS 300-1300 - -1.4 to 1.0
Wu & Piomelli (2018), LES 2100-7000 - -25 to 25
Table 1. List of simulation datasets. For the cases of Na & Moin (1998) and Wu & Piomelli
(2018), the boundary layer separation leads to βvalues between −∞ and ∞. The Reynolds
number values are slightly different from those in Na & Moin (1998) and Wu & Piomelli (2018)
due to difference in the definitions of the boundary layer edge.
Details of the flows in these datasets are listed in Table 1. The first three are DNS165
while Wu & Piomelli (2018) is a LES study. The data of Pargal et al. (2022) and Wu166
et al. (2019) are collected directly from simulations of turbulent boundary layer on a167
flat-plate and that on a controlled-diffusion (CD) aerofoil with matched non-equilibrium168
APG distributions along the streamwise direction. Comparison between these two flows169
reveal the effects of the convex wall curvature and the trailing edge on WPS, which were170
partially discussed in Pargal et al. (2022) and will be further discussed for the WPS171
herein. Na & Moin (1998) and Wu & Piomelli (2018) conducted simulations of flat-plate172
boundary layers with suction and blowing freestream velocities, leading to boundary layer173
separation and then reattachment; these two cases are rerun to collect boundary layer174
parameters, streamwise mean velocity and wall-pressure statistics at the same streamwise175
locations, as these data were not fully available from the original publications. For the176
case of Wu & Piomelli (2018), this work provides new data as the wall pressure was not177
discussed previously.178
A brief summary of the four simulations is as follows. The case of Wu et al. (2019)179
provides DNS data on a boundary layer developing on the pressure side of a CD aerofoil,180
at a freestream Mach number of 0.25. The compressible Navier-Stokes equations are181
solved for the flow around an aerofoil with the multi-block structured code HiPSTAR182
(High Performance Solver for Turbulence and Aeroacoustics Research) (Sandberg 2015).183
An initial 2D RANS simulation was run to provide boundary and initial conditions to the184
DNS simulation. Details of the problem formulation are provided by Wu et al. (2019). The185
simulation was validated against experimental data (Jaiswal et al. 2020; Jaiswal 2020)186
8S. Pargal et al.
for wall-pressure spectral data and flow statistics at different streamwise locations. The187
case of Pargal et al. (2022) is an incompressible DNS of a flat-plate turbulent boundary188
layer to emulate the boundary layer development on the downstream portion of the CD189
aerofoil flow studied by Wu et al. (2019). A finite difference solver on a staggered grid190
was used. To match the pressure gradient parameter (K) of the aerofoil boundary layer,191
a streamwise pressure gradient was imposed by prescribing a streamwise-varying U∞(x)192
at the top boundary of the domain. A fully turbulent boundary layer flow at the inlet193
of the domain was obtained using the recycling/rescaling method. A convective outflow194
boundary condition was used at the outlet and periodic boundary conditions are used195
in the spanwise direction. Similar discretization methods and boundary conditions were196
used in Wu & Piomelli (2018) and Na & Moin (1998) with slight variations in details.197
Simulations of the flows in these two studies were rerun, based on the methodologies198
of Pargal et al. (2022). The meshes in the new simulations were similar to those in the199
original studies. The same domain lengths and similar boundary conditions were used.200
The rerun simulation of Na & Moin (1998) has been validated against results reported201
in the original work on flow statistics and wall-pressure spectra at different streamwise202
locations. For the rerun LES simulation of Wu & Piomelli (2018), the governing equations203
were solved for the filtered velocities at scales larger than the low-pass filter. A different204
dynamic eddy-viscosity model based on the Lagrangian-averaging procedure (Meneveau205
et al. 1996) was used for the present simulation. Boundary layer developments in the206
rerun simulations will be compared to those reported in the original studies in Section 3.207
2.2. Experimental datasets208
DNS and LES simulations are limited to comparatively low Reynolds numbers209
(Reθ=300 to 7000). Experimental datasets are gathered from the studies of Hu (2018),210
Fritsch et al. (2022b), and Goody & Simpson (2000), which provide ZPG or pressure-211
gradient flow data with Reθof up to 23,400. Only existing datasets with both mean212
velocity profile data and WPS d