Reynolds-number scaling of wall-pressure–velocity correlations in wall-bounded turbulence

Abstract Wall-pressure fluctuations are a practically robust input for real-time control systems aimed at modifying wall-bounded turbulence. The scaling behaviour of the wall-pressure–velocity coupling requires investigation to properly design a controller with such input data so that it can actuate upon the desired turbulent structures. A comprehensive database from direct numerical simulations (DNS) of turbulent channel flow is used for this purpose, spanning a Reynolds-number range $Re_\tau \approx 550\unicode{x2013}5200$. Spectral analysis reveals that the streamwise velocity is most strongly coupled to the linear term of the wall pressure, at a Reynolds-number invariant distance-from-the-wall scaling of $\lambda _x/y \approx 14$ (and $\lambda _x/y \approx 8$ for the wall-normal velocity). When extending the analysis to both homogeneous directions in $x$ and $y$, the peak coherence is centred at $\lambda _x/\lambda _z \approx 2$ and $\lambda _x/\lambda _z \approx 1$ for $p_w$ and $u$, and $p_w$ and $v$, respectively. A stronger coherence is retrieved when the quadratic term of the wall pressure is concerned, but there is only little evidence for a wall-attached-eddy type of scaling. An experimental dataset comprising simultaneous measurements of wall pressure and velocity complements the DNS-based findings at one value of $Re_\tau \approx 2$k, with ample evidence that the DNS-inferred correlations can be replicated with experimental pressure data subject to significant levels of (acoustic) facility noise. It is furthermore shown that velocity-state estimations can be achieved with good accuracy by including both the linear and quadratic terms of the wall pressure. An accuracy of up to 72 % in the binary state of the streamwise velocity fluctuations in the logarithmic region is achieved; this corresponds to a correlation coefficient of $\approx$0.6. This thus demonstrates that wall-pressure sensing for velocity-state estimation – e.g. for use in real-time control of wall-bounded turbulence – has merit in terms of its realization at a range of Reynolds numbers.


Introduction
Inspiration for this work was born out of practical considerations associated with the implementation of real-time flow control for the reduction of skin-friction drag in wallbounded turbulence.Efforts in turbulence control comprise both passive and active methods † Email addresses for correspondence: w.j.baars@tudelft.nl,leemk@uh.eduarXiv:2307.06449v2[physics.flu-dyn]10 Jan 2024 2 Baars, Dacome, and Lee to target near-wall structures that scale in viscous units,  * ≡ /  , where   ≡ √︁   / is the friction velocity and  and  are the fluid's density kinematic viscosity, respectively.Leaving passive techniques aside, most studies on active skin-friction control tailor (statistical) forcing techniques to the inner scales (not requiring any sensing, e.g., Choi et al. 1998;Kasagi et al. 2009;Choi et al. 2011;Bai et al. 2014); we refer to this as 'predetermined forcing-control.'Only a few studies do incorporate sensing in numerical (Choi et al. 1994;Lee et al. 1998) and experimental (Rathnasingham & Breuer 2003;Qiao et al. 2018) 'real-time control' efforts of the near-wall structures.This approach requires that sensors and actuators are sized to said near-wall structures, which are roughly 500 * in length and 100 * in width.On practical engineering systems such as on an aircraft in cruise,   ≈ 80k at a typical location for control on the fuselage (here   ≡   /, with  being the boundary layer thickness).As such, control of the near-wall scales requires sensors/actuators with a spatial scale of ∼ 0.1 mm (e.g., the size of sensors/actuators) and with a temporal scale in excess of 30 kHz (e.g., the frequency of actuation).This required spatial-temporal resolution of sensors and actuators is out-of-reach for current technologies.In addition, even if near-wall scales can be sensed and partially disrupted, the flow is expected to recover to the uncontrolled state within a streamwise distance that scales in viscous units (recall the auto-regeneration mechanisms of near-wall turbulence (Hamilton et al. 1995), and see the study by Qiao et al. (2018) in which the controlled skin-friction drag recovers in less than 100 viscous units).Hence, the number of control-stations for achieving streamwise-persistent control is also impractical.To overcome these issues, alternative control strategies emerged in tandem with studies of higher-Reynolds number wall-turbulence.A direct numerical simulation (DNS) study by Schoppa & Hussain (1998) utilized spanwise jets for predetermined, large-scale forcing-control and reported drag reductions of up to 50%.However, the low Reynolds number of   = 180 yielded a negligible large-scale energy content in terms of the bulk turbulence kinetic energy (TKE), and recent investigations debate the effectiveness at higher   (Canton et al. 2016;Deng et al. 2016;Yao et al. 2018).With higher-Reynolds number studies available to date, experimental efforts to control flow structures within the logarithmic region in real-time were undertaken (Abbassi et al. 2017).Even though the mean friction was favorably affected, the net energy savings of such control systems are difficult to assess and generalize.
Independent of the implementation or large-scale control, the theoretical work of Renard & Deck (2016) outlines the potential of this pathway in an elegant manner: their kinetic-energy budget analysis reveals how the skin-friction drag relates to different physical phenomena across the boundary layer.Due to the decay of the relative contributions of the buffer and wake regions to the TKE production with increasing   (Smits et al. 2011a), the generation of the turbulence-induced excess friction is dominated by the dynamics in the logarithmic region.Hence, the work of Renard & Deck (2016) suggests that drag reduction strategies targeting the TKE production mechanisms in that layer are worth investigating.
A controller's ability to selectively target certain turbulent structures depends on the degree of coupling between off-the-wall flow quantities within the linear, buffer, and/or logarithmic regions of the flow, and observable wall-quantities (sensors should be flush within the wall to avoid parasitic drag).When working towards realistic setups, a real-time control system with discrete sensors and actuators is the most logical (components should still be wall-embedded to minimize parasitic drag, but the entire wall itself is not used for conducting the control action).In addition, when bringing in the requirement to (selectively) manipulate structures (e.g., Abbassi et al. 2017), a controller has to operate with an unavoidable wall-normal separation, Δ, between the sensing location and the 'target point' (figure 1).Likewise, a separation in the streamwise direction, Δ, must be adopted to account for control latency.These spatial separations result in a loss-of-correlation between the controller's input and the grazing turbulent velocities.Therefore, a sound understanding of (the scaling of) wall- the grazing turbulent velocities.Therefore, a sound understanding of (the scaling of) wallpressure-velocity correlations is a pre-requisite for designing a successful control system based on wall-pressure input data.
The notation in this paper is as follows.Coordinates , and denote the streamwise, wall-normal, and spanwise directions of the flow, and , , and represent the Reynolds decomposed fluctuations of the three velocity components and the static pressure, respectively.The wall-pressure is denoted as .Overlined capital symbols, e.g., , are used for the absolute mean.When quantities are presented in outer-scaling, length scale and velocity scale ∞ are used, while a viscous scaling is signified with superscript '+' and comprises a scaling with length scale * = / and velocity scale .

Wall-pressure-velocity correlations
When attempting large-scale control of wall-bounded turbulence based on wall-pressure input data, a first step is to estimate a dynamic state of logarithmic region-turbulence from said fluctuating wall-pressure, .The velocity fluctuations themselves form the true state.Many studies are concerned with scaling laws and modeling attempts of pressure fluctuations (e.g., Willmarth 1975;Farabee & Casarella 1991;Klewicki et al. 2008;Hwang et al. 2009), but only very few works are concerned with an assessment of the instantaneous or statistical coupling between the velocity fluctuations and the wall-pressure.Thomas & Bull (1983) revealed characteristic wall-pressure signatures associated with the near-wall burst-sweep events.
Throughout the last decade, high-resolution mappings of the spatio-temporal pressurevelocity correlation have been reported (Ghaemi & Scarano 2013;Naka et al. 2015), but for a specific condition.More recently, input-output linear time-invariant (LTI) system analyses have been conducted in the experimental works of Van Blitterswyk & Rocha (2017) and Gibeau & Ghaemi (2021).Nevertheless, no studies exist to date addressing the scaling of input-output relations, as a function of and .It is challenging to simultaneously acquire velocity and (noise-free) pressure data, particularly in wall-parallel planes as shown in figure 1, while such data are needed for examining wall-pressure-velocity correlations as a function of the streamwise and spanwise wavenumbers.Novel experiments can be designed to yield 2D coherence spectra (see Deshpande et al. (2020) for velocity-velocity correlations), but these are not trivial to perform.Given the current knowledge gap on wall-pressurevelocity coupling, our work covers an LTI-system analysis using high-fidelity data available from DNS campaigns.
Figure 1: Wall-based quantities are to be used for a linear time invariant (LTI) system analysis, to estimate the state of the off-the-wall turbulent velocities.A sparse implementation considers a limited number of sensors/actuators, and includes typical offsets in the wall-normal (Δ) and streamwise (Δ) directions between the sensing location and the controller's 'target point'.
pressure-velocity correlations is a pre-requisite for designing a successful control system based on wall-pressure input data.The notation in this paper is as follows.Coordinates ,  and  denote the streamwise, wall-normal, and spanwise directions of the flow, and , ,  and  represent the Reynolds decomposed fluctuations of the three velocity components and the static pressure, respectively.The wall-pressure is denoted as   .Overlined capital symbols, e.g., , are used for the absolute mean.When quantities are presented in outer-scaling, length scale  and velocity scale  ∞ are used, while a viscous scaling is signified with superscript '+' and comprises a scaling with length scale  * = /  and velocity scale   .

Wall-pressure-velocity correlations
When attempting large-scale control of wall-bounded turbulence based on wall-pressure input data, a first step is to estimate a dynamic state of logarithmic region-turbulence from said fluctuating wall-pressure,   .The velocity fluctuations themselves form the true state.Many studies are concerned with scaling laws and modeling attempts of pressure fluctuations (e.g., Willmarth 1975;Farabee & Casarella 1991;Klewicki et al. 2008;Hwang et al. 2009), but only very few works are concerned with an assessment of the instantaneous or statistical coupling between the velocity fluctuations and the wall-pressure.Thomas & Bull (1983) revealed characteristic wall-pressure signatures associated with the near-wall burst-sweep events.Throughout the last decade, high-resolution mappings of the spatiotemporal pressure-velocity correlation have been reported (Ghaemi & Scarano 2013;Naka et al. 2015), but for a specific   condition.More recently, input-output linear timeinvariant (LTI) system analyses have been conducted in the experimental works of Van Blitterswyk & Rocha (2017) and Gibeau & Ghaemi (2021).Nevertheless, no studies exist to date addressing the scaling of input-output relations, as a function of  and   .It is challenging to simultaneously acquire velocity and (noise-free) pressure data, particularly in wall-parallel planes as shown in figure 1, while such data are needed for examining wallpressure-velocity correlations as a function of the streamwise and spanwise wavenumbers.Novel experiments can be designed to yield 2D coherence spectra (see Deshpande et al. (2020) for velocity-velocity correlations), but these are not trivial to perform.Given the current knowledge gap on wall-pressure-velocity coupling, our work covers an LTI-system analysis using high-fidelity data available from DNS campaigns.
A model transfer kernel between wall-pressure and velocity is particularly useful for wallbased estimations of turbulent velocities.Upon confining the analysis to stochastic estimationbased techniques, these have proven to be useful for estimating large-scale features in turbulent flows using sparse input data.First-order techniques, known as Linear Stochastic Estimation (LSE), were initially introduced in the turbulence community to inspect coherent turbulent structures in shear flows (Adrian 1979;Adrian & Moin 1988).LSE can be implemented following a single-or multi-offset approach (Ewing & Citriniti 1999), in which the latter is identical to a spectral approach (see Tinney et al. 2006).The transfer kernel of the spectral LSE approach accounts for a gain and offset (or phase) per temporal and/or spatial scale, depending on the implementation.In view of figure 1, an LSE of an off-the-wall velocity field (,   , ) at an estimation position   starts with a 2D spatial Fourier transform of the unconditional input field   (, ), (1.1) A spectral-domain estimate is then formulated as and the physical-domain estimate is found through the inverse Fourier transform, (1.3) The stochastic and complex-valued linear kernel in (1.2) is computed a priori and is equal to the wall-pressure-velocity cross-spectrum,  To deduce how much energy of the LTI-system's output can be estimated, the linear coherence spectrum (LCS) between the input and output data is insightful, and is defined as Using (1.4) the coherence can be rewritten as In a stochastic sense, the coherence magnitude is thus interpreted as the energy in the estimated output signal (|  | 2      ), relative to the true output energy (  ).An assessment of the coherence between the wall-pressure and the turbulent velocities will address whether a substantial amount of energy in the turbulent fluctuations can be estimated.
So far, only the linear wall-pressure term was considered and this confines the analysis to scale-by-scale interactions.LSE must be extended to a Quadratic Stochastic Estimation (QSE) when nonlinearities manifest themselves in the input-output relation.Naguib et al. (2001) methodized a time-domain QSE and showed that the quadratic terms are critical for satisfactory estimates of the conditional streamwise velocity based on wall-pressure events in turbulent boundary layer (TBL) flows.Another example of improved estimates with QSE over LSE includes the estimate of velocities in a cavity shear layer, based on wall-pressure (Murray & Ukeiley 2003, 2004;Lasagna et al. 2013).By including the second-order, quadratic term in the stochastic estimate, a QSE procedure for the off-the-wall velocity field is formulated as When the skewness of the wall-pressure is zero, it can be shown that the linear kernel in (1.6) is identical to (1.4), thus  ′  =   .For details, we refer to Naguib et al. (2001) and reported observations of negligible wall-pressure skewness (Gravante et al. 1998;Tsuji et al. 2007;Klewicki et al. 2008).Regarding the quadratic term, this one consists of the Fourier transform of the quadratic wall-pressure,   2 (  ,   ) = F  2  (, ) .The pressure-squared term  2  is computed as the square of the de-meaned wall-pressure field.The quadratic kernel   (  ,   ,   ) is computed based on the same wall-pressure-squared term, following (1.7), and under the condition that the wall-pressure skewness is negligible. .
(1.7) Naguib et al. (2001) attributed the significant improvement of their conditional estimates of streamwise velocity fields based on wall-pressure events-with the inclusion of the quadratic pressure term-to the non-negligible turbulent-turbulent source (the 'slow' nonlinear pressure source associated with large-scale motions).This was analysed by considering the wallpressure dependence on the turbulent flow field, following the solution of Poisson's equation for incompressible flow.When estimates were based on wall-shear stress, quadratic terms were less crucial (Adrian et al. 1987;Guezennec 1989).Naguib et al. (2001) hypothesized that the portion in the estimate from the quadratic term represents a flow structure obeying outer scaling.However, they remained inconclusive due to their relatively small Reynolds number range.

Present contribution and outline
In summary, the extent of coupling between wall pressure and velocity at various friction Reynolds numbers in wall-bounded turbulence remains undetermined.Detailing the stochastic coupling, with linear and quadratic wall-pressure terms, is critical for gaining practical insight into whether a control system that relies on wall-pressure input has merit for realtime estimating (and controlling) large-scale structures in the logarithmic region.Albeit a wide variety of estimation techniques can be applied, such as physics-informed models (e.g., based on linearized Navier-Stokes equations (NSE) Madhusudanan et al. 2019), resolventbased methods assimilating nonlinearity of the NSE (Arun et al. 2023) or neural networks (Guastoni et al. 2021), the simplicity of the LTI-system analysis allows for interpretability.
This work is structured as follows.First, DNS data are presented in § 2.1 and analysed through an input-output LTI-system approach in § 3 to infer Reynolds-number scaling relations for the wall-pressure-velocity correlations.Experimental wall-pressure and velocity data are also described ( § 2.2) and assessed to identify whether the correlations can be replicated with experimental pressure data that are subject to significant levels of (acoustic) facility noise.Subsequently, the accuracy of velocity-state estimations is covered in § 4, and this aspect of wall-pressure sensing for estimation is of high practical relevance when addressing whether control based on wall-pressure is a realistic route forward for real-time control.

Numerical and experimental data
2.1.Direct numerical simulation of turbulent channel flow Four incompressible turbulent channel flow datasets are used and span one decade in Reynolds number,   ≈ 550 -5 200.Details on the numerical scheme, resolution, and turbulence statistics are documented by Lee & Moser (2015, 2019), and parameters of the DNS data and case names are summarized in table 1.In relevance to the current work involving pressure, the static pressure fields were obtained by solving the Poisson equation, based on the full velocity field at each instant snapshot.Neumann boundary conditions were applied at the walls, following Finally, the average pressure in the homogeneous directions was set according to, Here, ⟨•⟩ denotes the average in the homogeneous direction, and integration constant  is set to zero for convenience.Later on, not only wall-pressure fields are used, but also fields of the wall-pressure squared  2  .Such nonlinear products during simulations and post-processing steps were generated using a 3 /2 de-aliasing rule with zero-padding (Orszag 1971).Details of the pressure field computations are described in Panton et al. (2017).
For reference, 1D spectrograms of ,  and  are presented in figures 2(-), with isocontours of the pre-multiplied spectrum, e.g.,  +   +  .Four sets of coloured iso-contours at the two values indicated in the caption correspond to the four Reynolds number cases, and the small lines at the ordinates correspond to each   ≡  + .Only for the highest respectively, for all cases (an increased colour intensity corresponds to an increase in ; the channel half-widths + = are indicated along the ordinate).The grey scale contour shows a finer discretization of iso-contours for the R5200 case only.
Reynolds number case, R5200, a finer discretization of grey-filled contours is shown.Data are presented in a similar manner later on in § 3.
As is well-known, the most energetic content resides at + ≈ 15 and + ≈ 10 3 for the streamwise velocity, and + ≈ 250 for the wall-normal velocity and wall-pressure.The pressure spectrum remains constant for + 5, reflecting the wall-pressure spectrum.For all fluctuating quantities in figure 2, their variance, at a given + , grows due to additional energy at large + .For instance, see Mathis et al. (2009) for the scaling of -spectrograms and Panton et al. (2017) for the scaling of the mean-square pressure fluctuations.

Experimental wall-pressure-velocity data
Experiments were carried out in the W-tunnel facility within the Faculty of Aerospace Engineering at the Delft University of Technology.This open-return tunnel has a contraction ratio of 4.7:1, with a square cross-sectional area of 0.60 × 0.60 m 2 at the inlet of the test section, and can produce a maximum flow velocity of roughly 16.5 m/s.For generating a TBL flow, a setup was developed with a relatively long streamwise development length, consisting of a metal frame with poly-carbonate walls for optical access (figure 3 ).The bottom wall has a width of 0.60 m and a total length of 3.75 m.The TBL was initiated just downstream of the leading edge at = 0 with a trip of P40-grit sandpaper over a length of 0.115 m and over the full perimeter.The floor was suspended 0.08 m above the bottom wall of the upstream tunnel contraction, allowing for a clean start of the TBL flow without leading-edge separation.A flexible, 4 mm thick poly-carbonate ceiling was configured for a zero-pressure gradient (ZPG) development of the TBL.This ZPG condition was assessed using the acceleration parameter = ( / 2 )(d /d ), where ( ) is the boundary layer edge-velocity.A Pitot-static tube in the potential flow at = 3.07 m provided measures of the total ( 0 ) and static ( ) pressures, and thus ∞ = at (the static temperature and barometric pressure were also recorded for inferring the air density and viscosity).Using two streamwise rows of (in total) 100 static pressure taps in the floor, providing ( ), the variation in ( ) was computed as ( ) = 2 ( 0 − ( )) / , given that the total pressure remains constant along a streamline.For the nominal free-stream velocity of the current study ( ∞ ≈ 15 m/s), the acceleration parameter remained within an acceptable range for a ZPG condition along the entire test section ( < 1.6 respectively, for all   cases (an increased colour intensity corresponds to an increase in   ; the channel half-widths  + =   are indicated along the ordinate).The grey scale contour shows a finer discretization of iso-contours for the R5200 case only.
Reynolds number case, R5200, a finer discretization of grey-filled contours is shown.Data are presented in a similar manner later on in § 3.
As is well-known, the most energetic content resides at  + ≈ 15 and  +  ≈ 10 3 for the streamwise velocity, and  +  ≈ 250 for the wall-normal velocity and wall-pressure.The pressure spectrum remains constant for  + ≲ 5, reflecting the wall-pressure spectrum.For all fluctuating quantities in figure 2, their variance, at a given  + , grows due to additional energy at large  +  .For instance, see Mathis et al. (2009) for the scaling of -spectrograms and Panton et al. (2017) for the scaling of the mean-square pressure fluctuations.

Experimental wall-pressure-velocity data
Experiments were carried out in the W-tunnel facility within the Faculty of Aerospace Engineering at the Delft University of Technology.This open-return tunnel has a contraction ratio of 4.7:1, with a square cross-sectional area of 0.60 × 0.60 m 2 at the inlet of the test section, and can produce a maximum flow velocity of roughly 16.5 m/s.For generating a TBL flow, a setup was developed with a relatively long streamwise development length, consisting of a metal frame with poly-carbonate walls for optical access (figure 3𝑎).The bottom wall has a width of 0.60 m and a total length of 3.75 m.The TBL was initiated just downstream of the leading edge at  = 0 with a trip of P40-grit sandpaper over a length of 0.115 m and over the full perimeter.The floor was suspended 0.08 m above the bottom wall of the upstream tunnel contraction, allowing for a clean start of the TBL flow without leading-edge separation.A flexible, 4 mm thick poly-carbonate ceiling was configured for a zero-pressure gradient (ZPG) development of the TBL.This ZPG condition was assessed using the acceleration parameter  = (/ 2  )(d  /d), where   () is the boundary layer edge-velocity.A Pitot-static tube in the potential flow at   = 3.07 m provided measures of the total ( 0 ) and static () pressures, and thus  ∞ =   at   (the static temperature and barometric pressure were also recorded for inferring the air density and viscosity).Using two streamwise rows of (in total) 100 static pressure taps in the floor, providing (), the variation in   () was computed as   () = √︁ 2 (  0 − ()) /, given that the total pressure remains constant along a streamline.For the nominal free-stream velocity of the current study ( ∞ ≈ 15 m/s), the acceleration parameter remained within an acceptable range for a ZPG condition along the entire test section ( < 1.6 • 10 −7 following Schultz & Flack 2007).Finally, measurements are performed near the aft of the setup around   = 3.07 m.Here, the free-stream turbulence intensity at the nominal free-stream velocity, based on the hot-wire measurement described later, is2 / ∞ ≈ 0.3%.
Hot-wire anemometry measurements were performed with a Dantec Dynamics 55P15 miniature-wire boundary layer probe.This single-wire probe only yields the streamwise velocity (given that ≫ in boundary layer flows) and comprised a plated Tungsten wire with a diameter of = 5 µm and a sensing length of = 1.25 mm (resulting in / = 250).The viscous-scaled wire-length of + = 42.4 yields an acceptable spatial resolution (Hutchins et al. 2009), given that this study concentrates primarily on the logarithmic region for wall-pressure-velocity correlations (the measurement is fully-resolved in that region, as shown later on).Hot-wire traversing was done with a Zaber X-LRQ300HL-DE51 traverse, with an integrated encoder and controller yielding a positional accuracy of 13 µm (a resolution better than 0.4 * ).A Taylor-Hobson micro alignment telescope was used to position the hotwire at the most near-wall position before performing a wall-normal traverse spanning 40 logarithmically spaced positions in the range 10.2 + 1.2 .
The hot-wire was operated in constant temperature mode using a TSI IFA-300 bridge at an overheat ratio of 1.8.For each hot-wire position, signals were sampled (simultaneously with the wall-pressure sensors) at a rate of Δ + ≡ 2 / / = 0.36, where = 51.2kHz is the sampling frequency.The anemometer system low-pass filtered the voltage signal with a spectral cut-off at 20 kHz.Sampling was done with a 24-bit A/D conversion through an NI 9234 module embedded in an NI cDAQ.Relatively long signals were acquired with an uninterrupted acquisition time of = 150 seconds at each wall-normal position, resulting in more than 32 860 boundary layer turnover times; this is more than sufficient for converged spectral statistics at the lowest frequencies of interest.An in-situ calibration of the hot-wire was performed in the potential flow-region using the reference velocity provided by the Pitotstatic tube.A correction method for hot-wire voltage drift due to variations in barometric pressure and static temperature was also implemented (Hultmark & Smits 2010).
Profiles of and 2 are plotted in figure 4( ) and the boundary layer parameters are listed in table 1.These parameters were obtained by fitting the mean velocity profile to a composite profile with log-law constants of = 0.384 and = 4.17 (Chauhan et al. 2009).On the basis of these values, = 2 280 and the Reynolds number based on the momentum thickness is free-stream turbulence intensity at the nominal free-stream velocity, based on the hot-wire measurement described later, is √︁  2 / ∞ ≈ 0.3%.Hot-wire anemometry measurements were performed with a Dantec Dynamics 55P15 miniature-wire boundary layer probe.This single-wire probe only yields the streamwise velocity (given that  ≫  in boundary layer flows) and comprised a plated Tungsten wire with a diameter of   = 5 µm and a sensing length of  = 1.25 mm (resulting in /  = 250).The viscous-scaled wire-length of  + = 42.4 yields an acceptable spatial resolution (Hutchins et al. 2009), given that this study concentrates primarily on the logarithmic region for wallpressure-velocity correlations (the measurement is fully-resolved in that region, as shown later on).Hot-wire traversing was done with a Zaber X-LRQ300HL-DE51 traverse, with an integrated encoder and controller yielding a positional accuracy of 13 µm (a resolution better than 0.4 * ).A Taylor-Hobson micro alignment telescope was used to position the hot-wire at the most near-wall position before performing a wall-normal traverse spanning 40 logarithmically spaced positions in the range 10.2 ≲  + ≲ 1.2  .
The hot-wire was operated in constant temperature mode using a TSI IFA-300 bridge at an overheat ratio of 1.8.For each hot-wire position, signals were sampled (simultaneously with the wall-pressure sensors) at a rate of Δ + ≡  2  //   = 0.36, where   = 51.2kHz is the sampling frequency.The anemometer system low-pass filtered the voltage signal with a spectral cut-off at 20 kHz.Sampling was done with a 24-bit A/D conversion through an NI 9234 module embedded in an NI cDAQ.Relatively long signals were acquired with an uninterrupted acquisition time of   = 150 seconds at each wall-normal position, resulting in more than 32 860 boundary layer turnover times; this is more than sufficient for converged spectral statistics at the lowest frequencies of interest.An in-situ calibration of the hot-wire was performed in the potential flow-region using the reference velocity provided by the Pitotstatic tube.A correction method for hot-wire voltage drift due to variations in barometric pressure and static temperature was also implemented (Hultmark & Smits 2010).
Profiles of  and  2 are plotted in figure 4() and the boundary layer parameters are listed in table 1.These parameters were obtained by fitting the mean velocity profile to a composite profile with log-law constants of  = 0.384 and  = 4.17 (Chauhan et al. 2009).On the basis of these values,   = 2 280 and the Reynolds number based on the momentum thickness is -law with constants = 0.384 and = 4.17, and the -TKE profile is corrected for spatial resolution effects.( ) Premultiplied energy spectrogram + + (filled iso-contours 0.2:0.2:1.8); the scale-axis is converted to a wavelengthdependence using ≡ ( )/ .
From the streamwise turbulence intensity profile, it is clear that the experimental data are attenuated due to the hot-wire's spatial resolution.When correcting for this limited resolution via the method of Smits et al. (2011b), it matches well with the DNS profile in the buffer region and above, but a slight overestimate of the expected peak-value at + ≈ 15 is observed, following 2 max = 0.63 ln ( ) + 3.80 (Lee & Moser 2015;Smits et al. 2021).We ascribe this mismatch to wall-proximity effects (Hutchins et al. 2009).
Prior to computing frequency spectra ( ), the experimental time series ( , ) were down-sampled with a factor of 5 to match the corrected wall-pressure data (see Appendix A).Larger scales are more energetic in the logarithmic region, although the Reynolds number is not yet high enough for a discernible outer-spectral peak to appear (Baars & Marusic 2020).
Wall-pressure measurements were simultaneously made with the hot-wire ones, and thus sampled with the parameters listed in table 1.Eight GRAS 46BE 1 /4 in.CCP free-field microphones were employed.Seven of them formed a spanwise array for wall-pressure measurements, while one was used in the potential flow region to measure facility noise.The microphone sets have a nominal sensitivity of 3.6 mV/Pa and a frequency response range with an accuracy of ±2 dB for 4 Hz to 80 kHz, while for the range 10 Hz to 40 kHz the accuracy is -law with constants  = 0.384 and  = 4.17, and the -TKE profile is corrected for spatial resolution effects.() Premultiplied energy spectrogram  +   +  (filled iso-contours 0.2:0.2:1.8); the scale-axis is converted to a wavelengthdependence using   ≡  ()/  .matches a Coles-Fernholtz relation,   = 2 [ 1 /0.38 ln (  ) + 3.7] −2 , to within 4.5%.The mean velocity profile compares well to the R2000 case of the DNS, up to the wake-region.From the streamwise turbulence intensity profile, it is clear that the experimental data are attenuated due to the hot-wire's spatial resolution.When correcting for this limited resolution via the method of Smits et al. (2011b), it matches well with the DNS profile in the buffer region and above, but a slight overestimate of the expected peak-value at  + ≈ 15 is observed, following  2 max = 0.63 ln (  ) + 3.80 (Lee & Moser 2015;Smits et al. 2021).We ascribe this mismatch to wall-proximity effects (Hutchins et al. 2009).
Prior to computing frequency spectra   (  ), the experimental time series (, ) were down-sampled with a factor of 5 to match the corrected wall-pressure data (see Appendix A).One-sided spectra   (;  ) = 2⟨ (;  ) * (;  )⟩ were computed by way of ensembleaveraging a total of 180 FFT partitions of  = 2 14 samples (50 % overlap and a Hanning window applied to them).This yields a spectral resolution of d  = 0.625 Hz or d  + = 3.4 • 10 −5 .For interpretive purposes, frequency spectra are converted to wavenumber spectra using a single convection velocity   (taken as the local mean velocity  () unless stated otherwise).With wavenumber   = 2  /  the wavenumber spectrum becomes   (  ) =   (  )d  /d, where group d  /d =   /(2) converts the energy density from a 'per unit frequency' to a 'per unit wavenumber' (the spectral resolution becomes d ≈ 0.88 m −1 or d + ≈ 2.6 • 10 −5 at the lowest position of  + = 10).Throughout this paper, the scale axis is either presented in terms of   or wavelength   = 2/  .Figure 4() presents the streamwise energy spectrogram     for validation of the experimental setup.The innerspectral peak is clearly visible and is identified with the × marker at  +  = 10 3 and  + = 15.Larger scales are more energetic in the logarithmic region, although the Reynolds number is not yet high enough for a discernible outer-spectral peak to appear (Baars & Marusic 2020).
Wall-pressure measurements were simultaneously made with the hot-wire ones, and thus sampled with the parameters listed in table 1.Eight GRAS 46BE 1 /4 in.CCP free-field microphones were employed.Seven of them formed a spanwise array for wall-pressure measurements, while one was used in the potential flow region to measure facility noise.The microphone sets have a nominal sensitivity of 3.6 mV/Pa and a frequency response range with an accuracy of ±2 dB for 4 Hz to 80 kHz, while for the range 10 Hz to 40 kHz the accuracy is ±1 dB.The dynamic range is 35 dB to 160 dB (with a reference pressure of  ref = 20 µPa).For our current wall-pressure-velocity correlation study, the primary frequencies of interest lay between roughly 5 Hz and 800 Hz, and the measured pressure intensity is on the order of 105 dB, thus making these microphones suitable for these types of measurements.
The spanwise array of seven equally-spaced pinhole-mounted microphones had an interspacing of 20 mm, or 0.30, with the total width spanning Δ = 1.78.The hot-wire profile was measured above the center pinhole.Each microphone was screwed inside a cavity (after removal of the microphone grid-cap) so that the sensing diaphragm formed the bottom of the cavity.On the back side, underneath the wind tunnel floor, a box surrounding the microphones prevented any pressure fluctuations at their venting holes.On the TBL side, a pinhole with a diameter of  + = 13.6 ( = 0.40 mm) ensured a sufficient spatial resolution of the measurement (Gravante et al. 1998).The pinhole depth was  = 0.80 mm and the cavity diameter matched the microphone-body outer diameter ( = 6.35 mm).The cavity length was designed as  = 2.0 mm, so that the Helmholtz resonance frequency of the cavity was above the frequency range of interest (  +  =   / 2  = 0.15 or   = 2 750 Hz).Raw signals of the pinhole-microphone measurements required post-processing to yield valid time series of the wall-pressure fluctuations.The post-processing steps are described in Appendix A.

Scaling of the wall-pressure-velocity coupling
This section utilizes the DNS data to assess the coupling between the fluctuations of  and , and the wall-pressure field   .First, we proceed with a 1D spectral analysis in the streamwise direction, which is reminiscent of the data available from typical experiments.

1D analysis in the streamwise direction
Cross-spectra of wall-pressure and velocity yields an indication of the coupling of absolute energy.We only examine the gain of the complex-valued 1D cross-spectrum,     (  , ); the phase is beyond the scope of our current work and is only relevant for spatial/temporal lags.The gain of the cross-spectrogram is presented with iso-contours of | +    | in figure 5().One particular contour value is chosen and the iso-contours correspond to the four Reynolds number cases of the DNS (table 1).It is evident that the region of cross-spectral energy grows along the black solid line with an increase in   .Since the black line indicates a constant ratio of   /, this trend is representative of distance-from-the-wall scaling.
A normalized coherence is now considered to explore how the coupling scales, independent of the scaling of  energy).The coherence is presented in figure 6() using one iso-contour of  2    for all four   of the DNS.For the highest Reynolds number case, R5200, grey-filled contours show that a spectral band of strong coherence appears at a self-similar scaling of   / ≈ 14; this scaling also appears to be Reynolds-number invariant.To further accentuate these observations, individual coherence spectra are superimposed in figure 6().Here, each bundle of lines with the same colour corresponds to one Reynolds-number dataset.Coherence spectra are plotted corresponding to a relatively coarse grid of  positions that are logarithmically spaced between lower-and upper bounds of an extended logarithmic region, chosen as  + = 80 and  + = 0.16  , respectively (resulting in 1 curve for the R0550 data, and 15 curves for the R5200 data).A wall-scaling of   / is adopted on the abscissa.It is evident that the coherence spectra collapse; signifying a Reynolds-number invariant location of the coherence peak and a constant coherence magnitude.This coherence-peak location at / = 14 closely matches the Reynolds number invariant aspect ratio of wall-attached coherent structures of , see Baars et al. (2017).The relatively low magnitude indicates a weak linear coherence, but this is expected given that (extreme) events in the wall-pressure are primarily caused by sweeps, ejections, and thus motions associated with strong vertical velocity fluctuations (and less so with fluctuations).This can, for instance, be observed in the work by Ghaemi & Scarano (2013) where high amplitude near-wall pressure events are considered by visualizing the associated conditional pressure and velocity structures.In this regard, it is evident from figure 7( ) that the 'self-similar scaling' of coherence appears at its smallest scale near the wall with a peak at + ≈ 15 and + ≈ 14 × 15 ≈ 210 (visualized with a round marker).This streamwise scale agrees with the typical (conditional) near-wall turbulent structure associated with these high-amplitude pressure peaks observed by Ghaemi & Scarano (2013).
By further inspection, it is intriguing to note the appearance of a region of zero-coherence at relatively large wavelengths and at positions below the ridge, even though this region resides closer to the wall.It can thus be concluded that large-scale fluctuations in close proximity to the wall do not comprise a phase-consistent linear relationship with the wall-pressure, whereas the fluctuations further up in the wall-bounded turbulent flow do.Presumably, this is due to the stronger dispersive (and random) nature of the convection of large-scale structures near the wall (Liu & Gayme 2020).The wall-pressure footprint for each scale is thus only linearly coupled to fluctuations that are statistically self-similar and that govern the footprint from each respective height.This is consistent with the fact that the pressure scalar is influenced by the velocity field in the entire domain; this is opposite to, for instance, wall-attached velocity fluctuations that are coherent throughout the full wall-normal extent (Baars et al. 2017).
Finally, two dominant regions of non-zero coherence appear: (1) very near the wall at wavelengths + 200 and at + 5, and (2) at large outer-scaled wavelengths (e.g., for ≈ 5 200 for wavelengths + 3 • 10 4 ).Although these regions have a non-zero coherence, it is important to note that has no significant energy in region 1 (figure 2 ) and neither has in region 2 (figure 2 ).Likewise, the cross-spectral energy is low in these regions (recall figure 5 ).Although insignificant in terms of absolute energy, very large global modes of velocity fluctuations spanning the entire wall-normal extent (and simulation constant coherence magnitude.This coherence-peak location at   / = 14 closely matches the Reynolds number invariant aspect ratio of wall-attached coherent structures of , see Baars et al. (2017).The relatively low magnitude indicates a weak linear coherence, but this is expected given that (extreme) events in the wall-pressure are primarily caused by sweeps, ejections, and thus motions associated with strong vertical velocity fluctuations (and less so with  fluctuations).This can, for instance, be observed in the work by Ghaemi & Scarano (2013) where high amplitude near-wall pressure events are considered by visualizing the associated conditional pressure and velocity structures.In this regard, it is evident from figure 7() that the 'self-similar scaling' of coherence appears at its smallest scale near the wall with a peak at  + ≈ 15 and  +  ≈ 14 × 15 ≈ 210 (visualized with a round marker).This streamwise scale agrees with the typical (conditional) near-wall turbulent structure associated with these high-amplitude pressure peaks observed by Ghaemi & Scarano (2013).
By further inspection, it is intriguing to note the appearance of a region of zero-coherence at relatively large wavelengths and at positions below the ridge, even though this region resides closer to the wall.It can thus be concluded that large-scale  fluctuations in close proximity to the wall do not comprise a phase-consistent linear relationship with the wall-pressure, whereas the  fluctuations further up in the wall-bounded turbulent flow do.Presumably, this is due to the stronger dispersive (and random) nature of the convection of large-scale structures near the wall (Liu & Gayme 2020).The wall-pressure footprint for each scale is thus only linearly coupled to  fluctuations that are statistically self-similar and that govern the footprint from each respective height.This is consistent with the fact that the pressure scalar is influenced by the velocity field in the entire domain; this is opposite to, for instance, wall-attached velocity fluctuations that are coherent throughout the full wall-normal extent (Baars et al. 2017).
Finally, two dominant regions of non-zero coherence appear: (1) very near the wall at wavelengths  +  ≲ 200 and at  + ≲ 5, and (2) at large outer-scaled wavelengths (e.g., for   ≈ 5 200 for wavelengths  +  ≳ 3 • 10 4 ).Although these regions have a non-zero coherence, it is important to note that  has no significant energy in region 1 (figure 2𝑎) and neither has   in region 2 (figure 2𝑐).Likewise, the cross-spectral energy is low in these regions (recall figure 5𝑎).Although insignificant in terms of absolute energy, very large global modes of velocity fluctuations spanning the entire wall-normal extent (and simulation    shown in figures 5() and 6(), respectively.As for   and , coherence spectra for   with  are superimposed in figure 6(), as function of   / and for a range of  positions for all Reynolds-number datasets, to bring attention to the self-similar and Reynolds numberinvariant coherence-peak near   / = 8.As for the  fluctuations, a significant bulk of the coherence resides at small wavelengths very near the wall when   and  are concerned, but this coherence is insignificant given the absence of cross-spectral energy (figure 5𝑏).The main finding that the wall-pressure-coherent energy in  resides at smaller wavelengths than their streamwise counterpart is interpreted as follows.Fluctuations in  induce wall-pressure fluctuations through a 'flow stagnation' when directed towards the wall.When wall-pressure coherent velocity fluctuations are thought of as wall-attached motions (e.g., hairpins or packets of them), their centre regions contain  < 0, while the -fluctuations induced by the vortical motions of vortex heads inducing wall-pressure reside at shorter   (statistically thus nearly half the size).This is evident from spectra: while the peak in the  cospectra resides around   / ≈ 15 (dominated by the higher energy in , residing at relatively large streamwise scales), the dominant energy in the spectra of  alone appears at much shorter scales throughout the logarithmic region, along   / ≈ 2 (Baidya et al. 2017).In addition, Jiménez & Hoyas (2008) showed that pressure spectra scale relatively well with the local Reynolds shear stress  2 (representative of the intensity of eddying motions).Given the relatively large streamwise scales in  and the fact that the wall-pressure spectrum embodies a footprint of the global pressure fluctuations, the peak-coherence of   with  does indeed reside at slightly larger scales   / ≈ 8 (than the peak location of the  spectra at   / ≈ 2, see Baidya et al. 2017).
Higher order terms of the wall-pressure-velocity coupling are of relevance to the analysis in § 4, when   forms the input for estimates of the velocity fluctuations.For this reason, coherence spectra of the wall-pressure squared, with both the  and  velocities, are presented as spectrograms in figures 7() and 7(), respectively.It is important to note here that  2  is the square of the de-meaned wall-pressure (this is not the same as the de-meaned wallpressure-squared).Note that this form of a nonlinear correlation does address interactions of different scales (e.g. an interscale interaction of pressure with velocity), although not explicitly in terms of triadic (or higher-order) scale interactions.Bipectral analysis is required to address those interactions (Baars & Tinney 2014;Cui & Jacobi 2021), which is beyond the scope of this manuscript.Focusing on the coherence of  2  with , Naguib et al. (2001) had hypothesized that this quadratic pressure-interaction represents a flow structure obeying outer-scaling.Trends of  2   2  show a self-similar scaling in the logarithmic region ( + ≳ 100) and coherence only appears for   ≳ 14; note that this corresponds to the region where  2    starts to decrease from its peak-ridge at   / = 14.A distance-from-the-wall scaling is also seen in the trends of  2   2  , which appears to become relevant at scales slightly larger than   / = 8 (the lowest locations at which    coherence appears is not fixed in outer-scaling).
Physically, the coherence at relatively large scales involving  2  would be reminiscent of nonlinearities associated with an intensity-modulation phenomenon (Tsuji et al. 2007).This modulation of the near-wall quantities in wall-bounded turbulence is induced by largescale velocity fluctuations that are most energetic in the logarithmic region (and are most pronounced at high values of   ) and that leave a direct imprint on the wall.This imprint changes the local, large-scale friction velocity and, thus the viscous scale.Consecutively, this yields modulated near-wall pressure and small-scale velocity fluctuations given that these near-wall quantities are universal in viscous scaling (Zhang & Chernyshenko 206;Chernyshenko 2021).Thus, when the intensity of (wall-)pressure fluctuations are modulated by the large-scale  (or ) fluctuations, the intensity-modulation 'envelope' (which by itself has no energy contribution in   ) becomes energetic in the square of the demeaned wall-pressure signal,  2  .At the same time, the usage of  2  results in linear coherence at wavelengths larger than where the coherence peaks with the linear pressure term (figure 7,).Further inspection of figure 7 reveals that the coherence is strongest for the velocity fluctuations taken in the logarithmic region; this reflects the general consensus in the community that the modulation of wall-quantities is driven by energetic large-scale motions in the logarithmic region of the flow.As a final note, we confirmed that the coherence with any higher-order pressure terms ( 3  with either  or ) was zero.

2D analysis in the streamwise-spanwise plane
In the 1D spectral analysis, all spanwise information was lumped together.To inspect the full spectral picture in both homogeneous directions, we move towards a 2D analysis.At first the 2D spectrograms of the streamwise velocity,   (  ,   , ), and wall-pressure,      (  ,   ), are shown for  + ≈ 80 at   ≈ 5 200 in figures 8() and 8(), respectively.A solid black line in figure 8 than spanwise scales.This is reminiscent of (very) large-scale motions in the streamwise direction.The wall-pressure reflects a more isotropic behaviour in the two homogeneous directions, owing to the fact that wall-pressure has a stronger coupling with the motions, which are themselves more isotropic than motions.Nevertheless, the scope of the current paper is to inspect the normalized coherence, since this reveals the degree of phaseconsistency and thus the energy-fraction of the velocity fluctuations that are stochastically coupled to the wall-pressure.Coherence in 2D is a generalization of (3.1), following: Coherence for the data used to present the 2D (auto-)spectrograms in figures 8( , ) is shown in figure 8( ).Notably, the coherence only appears for + 400 (when taking a threshold of 2 = 0.05), agreeing to what was observed from the 1D analysis in figure 6.
A coherence-peak in the 2D view still resides close to / ≈ 14 (for figure 8( ) equivalent to + = 14 + = 1120) and is reasonably symmetric around / = 2.The latter implies that wall-pressure-coherent motions are roughly twice as long as they are wide.Another region of peak-coherence resides at very large , but the energy there is insignificant and hence this high coherence is physically irrelevant.For instance, at ( + , + ) = (10 3 , 10 4 ) in figure 8( ), 2 ≈ 0.25 even though the absolute energy of is invisible in figure 8( ).For the motions (auto-spectrogram shown in figure 8 ) the 2D coherence in figure 8( ) shows symmetry around / = 1.A further statistical interpretation is reserved for when the 2D coherence is presented in a different manner, described next.
So far, 2D auto-spectrograms were pre-multiplied with and visualized with isocontours of in a log-Cartesian space.This visualization ensures that the 'areaunder-the-contour' is proportional to the energy residing in logarithmic ranges of and .Lee & Moser (2019) pointed out that this method of visualization also comes with two primary shortcomings.In summary, the distortion of the 2D wavevector orientation complicates observing the alignment of modes with the and directions: all lines of constant / (or / ) have a slope of 1 and are only marginally offset on the log-log plot, for different constants of / (see figure 8 ).In addition, vector wavelengths with a similar magnitude = 2 + 2 1 /2 follow a non-trivial trend line that creates difficulties in assessing scale isotropy.That is, when turbulence quantities contain an equal energy content at vector wavelengths (or wavenumbers) that have the same magnitude in the wall-parallel plane, this energy is said to be distributed isotropically in scale.Because of the two shortcomings, than spanwise scales.This is reminiscent of (very) large-scale motions in the streamwise direction.The wall-pressure reflects a more isotropic behaviour in the two homogeneous directions, owing to the fact that wall-pressure has a stronger coupling with the  motions, which are themselves more isotropic than  motions.Nevertheless, the scope of the current paper is to inspect the normalized coherence, since this reveals the degree of phaseconsistency and thus the energy-fraction of the velocity fluctuations that are stochastically coupled to the wall-pressure.Coherence in 2D is a generalization of (3.1), following: Coherence for the data used to present the 2D (auto-)spectrograms in figures 8(,) is shown in figure 8().Notably, the coherence only appears for  +  ≳ 400 (when taking a threshold of  2    = 0.05), agreeing to what was observed from the 1D analysis in figure 6.A coherence-peak in the 2D view still resides close to   / ≈ 14 (for figure 8() equivalent to  +  = 14 + = 1120) and is reasonably symmetric around   /  = 2.The latter implies that wall-pressure-coherent  motions are roughly twice as long as they are wide.Another region of peak-coherence resides at very large   , but the energy there is insignificant and hence this high coherence is physically irrelevant.For instance, at ( +  ,  +  ) = (10 3 , 10 4 ) in figure 8(),  2    ≈ 0.25 even though the absolute energy of  is invisible in figure 8().For the  motions (auto-spectrogram shown in figure 8𝑐) the 2D coherence in figure 8() shows symmetry around   /  = 1.A further statistical interpretation is reserved for when the 2D coherence is presented in a different manner, described next.
So far, 2D auto-spectrograms were pre-multiplied with     and visualized with isocontours of      in a log-Cartesian space.This visualization ensures that the 'area-underthe-contour' is proportional to the energy residing in logarithmic ranges of   and   .Lee & Moser (2019) pointed out that this method of visualization also comes with two primary shortcomings.In summary, the distortion of the 2D wavevector orientation complicates observing the alignment of modes with the  and  directions: all lines of constant   /  (or   /  ) have a slope of 1 and are only marginally offset on the log-log plot, for different constants of   /  (see figure 8𝑏).In addition, vector wavelengths with a similar magnitude follow a non-trivial trend line that creates difficulties in assessing scale isotropy.That is, when turbulence quantities contain an equal energy content at vector wavelengths (or wavenumbers) that have the same magnitude in the wall-parallel plane, this energy is said to be distributed isotropically in scale.Because of the two shortcomings, Lee Hence, to preserve the total energy in plotting, quantity 2 / is visualized with linear axes # and # .Hence, lines of constant / possess slopes of / , and a curve of constant becomes an arc. Figure 9 re-plots all 2D spectrograms of figure 8 using the log-polar format, with + ref = 1/50 000.Five arcs of constant = 2 / indicating the trend of scale-isotropy are visualized with the dash-dotted lines on all plots and correspond to + = 10 1 , 10 2 , 10 3 , 10 4 and 10 5 from the largest to the smallest arc.Lines of constant / correspond to the = 2 trend (solid line) and = trend (dashed line).
For a detailed interpretation of the 2D energy spectrograms of and in log-polar format (figures 9 , ), we refer to Lee & Moser (2019).From the wall-pressure spectrogram (figure 9 ) it becomes evident that a significant part of the energy is distributed close to & Moser (2019) introduced a 'log-polar' format for the 2D spectrum.When considering wavenumbers,  (  ,   ) can be expressed in polar coordinates, yielding  ( cos ,  sin ) with   =  cos () and   =  sin ().When defining  = log 10 (/ ref ), where  ref is a reference wavenumber smaller than the smallest non-zero wavenumber considered, a set of logarithmic polar coordinates would be  and , with associated Cartesian coordinates  #  =  cos  =    / and  #  =  sin  =    /, respectively.Following Parseval's theorem, the spectral-integral equals Hence, to preserve the total energy in plotting, quantity  2 / is visualized with linear axes  #  and  #  .Hence, lines of constant   /  possess slopes of   /  , and a curve of constant  becomes an arc. Figure 9 re-plots all 2D spectrograms of figure 8 using the log-polar format, with  + ref = 1/50 000.Five arcs of constant  = 2/ indicating the trend of scale-isotropy are visualized with the dash-dotted lines on all plots and correspond to  + = 10 1 , 10 2 , 10 3 , 10 4 and 10 5 from the largest to the smallest arc.Lines of constant   /  correspond to the   = 2  trend (solid line) and   =   trend (dashed line).
For a detailed interpretation of the 2D energy spectrograms of  and  in log-polar format (figures 9,), we refer to Lee & Moser (2019).From the wall-pressure spectrogram (figure 9𝑏) it becomes evident that a significant part of the energy is distributed close to Baars, Dacome, and Lee 16Baars, Dacome, and Lee 10 5 + = 10 4 10 3 10 2 10 1 Figure 9: ( , , ) Similar to figures 8( , , ) but now presented in log-polar format as described in the text; grey scale contours show 7 iso-contours ranging up to the maximum value.( , ) Similar to figures 8( , ) but now presented in log-polar format.For indicating trends of scale-isotropy, five dash-dotted arcs of constant = 2 / are plotted in all sub-figures and correspond to + = 10 1 , 10 2 , 10 3 , 10 4 and 10 5 (from the most outer-arc going inward).
isotropically in scale, with a ridge at + ≈ 200.The peak energy resides close to = 2 , meaning that the pressure modes are, statistically, slightly elongated in the streamwise direction.Larger-scale energy (e.g., around the arc of + = 10 3 ) tends to be concentrated towards the # axis and thus resides in Fourier modes with / lower than 1 (or / lower than 1).When bringing in the coherence, for the wall-pressure with in figure 9( ), it peaks with a trend of scale-isotropy around the + = 10 3 arc for structures that are not strongly elongated in the streamwise direction (residing at < 2 ); contrary, the energy in is relatively low at these large scales (figure 9 ).Because the coherence is relatively strong below the = line (in fact, constant at vector wavenumbers with the same magnitude), while it weakens beyond 2 , the following can be concluded.Fluctuations in that are elongated in the spanwise direction (residing at ) are energetically weak (see figure 9 ) but are coupled to the wall-pressure imprint more strongly than motions that are strongly elongated in (residing at 2 ).The former would represent coherent spanwise vorticity, while the latter is representative of coherent streamwise vorticity.Concerning the motions, 2 and the wall-pressure energy peak for streamwise elongated scales, along = 2 .The scale at which the coherence peaks is larger since only the larger scales are coherent.Note that the single wall-normal position considered here is near + = 80 and does not provide a full-view: lower positions cause the peak-coherence to reside at smaller scales isotropically in scale, with a ridge at  + ≈ 200.The peak energy resides close to   = 2  , meaning that the pressure modes are, statistically, slightly elongated in the streamwise direction.Larger-scale energy (e.g., around the arc of  + = 10 3 ) tends to be concentrated towards the  #  axis and thus resides in Fourier modes with   /  lower than 1 (or   /  lower than 1).When bringing in the coherence, for the wall-pressure with  in figure 9(), it peaks with a trend of scale-isotropy around the  + = 10 3 arc for    structures that are not strongly elongated in the streamwise direction (residing at   < 2  ); contrary, the energy in  is relatively low at these large scales (figure 9𝑐).Because the coherence is relatively strong below the   =   line (in fact, constant at vector wavenumbers with the same magnitude), while it weakens beyond   ≳ 2  , the following can be concluded.Fluctuations in  that are elongated in the spanwise direction  (residing at   ≲   ) are energetically weak (see figure 9𝑐) but are coupled to the wall-pressure imprint more strongly than  motions that are strongly elongated in  (residing at   ≳ 2  ).The former would represent coherent spanwise vorticity, while the latter is representative of coherent streamwise vorticity.Concerning the  motions,  2    and the wall-pressure energy peak for streamwise elongated scales, along   = 2  .The scale at which the coherence peaks is larger since only the larger scales are coherent.Note that the single wall-normal position considered here is near  + = 80 and does not provide a full-view: lower positions cause the peak-coherence to reside at smaller scales Figure 10: ( , , ) 2D linear coherence of and , for all cases and for three different inner-scaled wall-normal positions.Note that the wavenumber representation on the axes includes a wall-scaling with .The solid, coloured iso-contours correspond to the same two contour values as were considered in figure 9( ), 2 = [0.05;0.25].( , , ) Similar to sub-figure ( ) but now for and , with the solid, coloured iso-contours corresponding to the same two contour values as were considered in figure 9( ).For indicating trends of scale-isotropy, dash-dotted arcs correspond to constant values of / = 1, 3, 8, 14, 85, and 140 (from the most outer-arc going inward).
(as was already observed in the 1D analysis, see figure 6 ).Hence it will be beneficial to (as was already observed in the 1D analysis, see figure 6𝑎).Hence it will be beneficial to consider the scaling of coherence with  and   , before drawing further conclusions.
To present the 2D coherence for   and  as a function of  and   , we reside to a representation based on distance-from-the-wall scaling.First, all cases of   are shown for a height of  + ≈ 80 in figure 10(); the darkest red contour re-shows the contour of figure 9() for   ≈ 5 200.Here the axes of the log-polar format are adapted to account for the wallscaling: instead of  + ref = 1/50 000, the reference wavenumber is now made -dependent following  yref = 1/(1 000).This reference wavenumber is used in the new definition of the log-polar axes,  *  =    / and  *  =    /, with  = log 10 / yref .Using the same format, two other wall-normal positions of  + ≈ 20 (figure10) and  + ≈ 320 (figure 10𝑒) are considered.Alongside, the 2D coherence for   and  is shown in the exact same format.All plots include six arcs of constant   (or constant / with the values stated in the caption), indicating trends of scale-isotropy.Lines of constant   /  still correspond to the ones shown in figure 8, and indicate the   = 2  trend (solid lines in figures 10- ) and   =   trend (dashed lines in figures 10,,  ).
Iso-contours of  2    in figures 10(,,) collapse well for all   , in particular for the  2 = 0.05 iso-contour.These plots reveal a wall-scaling of the 2D coherence because the iso-contours appear at the same position for all three  positions, increasing consecutively by a factor of four ( + ≈ 20 → 80 → 320).Note that iso-contours are less well bundled at  + ≈ 320; this is ascribed to only the very large (less-converged) scales being coherent.Figures 10(,,  ) present iso-contours of coherence for   and , and reveal a Reynolds number-invariant wall-scaling at the small-wavelength end.In contrast to , the  iso-contours of coherence adhere to trends of scale-isotropy for    structures that are not strongly elongated in  (e.g., for scales residing at   ≲ 2  ).A maximum of the coherence resides near / = 8 and is largely invariant with increasing   /  , except that the coherence shows a significant drop in amplitude for strongly-stretched structures in  (this is more visible in figure 9  ).Finally, the wall-pressure is correlated stronger with  than with  ( 2    >  2    ).As for the 1D analysis in § 3.1, the 2D coherence analysis is here extended to the coherence between the  and  motions and the wall-pressure squared.Contours of  2

𝑤
does not obey a perfect wall-scaling, but for  + ≈ 80 and  + ≈ 320, the contours nearly collapse for all   and appear at similar locations in the plots.In general, the coherence with the wall-pressure-squared term resides at larger scales than the linear wall-pressure term, as was also the case in the 1D analysis.Again, iso-contours at the highest  position correspond to much larger scales with less smooth spectral statistics, and for the two lowest Reynolds numbers, this position is beyond the logarithmic region.In the buffer region (at  + ≈ 20 in figure 11𝑎), the coherence remains nearly Reynolds number-invariant and does no longer abide by a wall-scaling.This was also apparent from figure 7().For the coherence of the wall-pressure-squared term with the  motions, observations are similar.

Exploring velocity-state estimation using sparse experimental data
Now that Reynolds-number scalings of the wall-pressure-velocity were identified, experimental wall-pressure and velocity data are assessed for exploring whether velocity-state estimation based on wall-pressure input data is feasible (in practice).At first, the experimental findings are compared to the DNS-inferred correlations in § 4.1, after which the quadratic stochastic approach for velocity-state estimation is outlined in § 4.2.Finally, in § 4.3, the accuracy in the velocity-estate estimates is discussed.

Coherence of single-point input-output data
With the available experimental time series of the streamwise velocity, (, ), and the corresponding post-processed wall-pressure   (), the linear coherence  2    is computed in frequency space.Note that we only consider the wall-pressure-velocity correlation Figure 11: Similar to figure 10, but now for ( , , ) and 2 (wall-pressure squared) and ( , , ) and 2 .
with the component since the single hot-wire probe measurements performed only provide streamwise velocity time series.Seemingly, the experimental spectra agree well in terms of the onset of coherence at a wallscaling of around / = 3 and its peak residing near / = 14.For lower positions with the  component since the single hot-wire probe measurements performed only provide streamwise velocity time series.Figure 12() shows the coherence spectrogram after conversion from a frequency-to-wavelength dependence, using the local mean velocity ( § 2.2).Overlaid is one iso-contour computed from the DNS R2000 data, highlighting that the ridge of strong coherence was well-captured by the experiment.Inspection of the slightly noisier experimental spectrogram is facilitated by showing individual coherence spectra in figure 12(), for 11 positions within the logarithmic region taken as 80 ≲  + ≲ 0.15  .Seemingly, the experimental spectra agree well in terms of the onset of coherence at a wallscaling of around   / = 3 and its peak residing near   / = 14.For lower  positions and quantify the estimation accuracy.Stochastic estimation procedures are performed in frequency space (not wavelength space) since the experimental data comprise time series.When concentrating on LSE, the time-domain estimate of the logarithmic region fluctuations at can be formed through the convolution: where the temporal kernel is the inverse Fourier transform of the frequency-domain kernel, The expression (4.2) is equivalent to (1.4), but for a 1D frequency-dependence, and the kernel's gain can be related to the coherence, according to For QSE, the time-domain estimate can be written the first higher-order term of the input pressure (Naguib et al. 2001): As noted in § 1.1, when the probability density function (PDF) of the input is symmetric, the linear kernel in the QSE is equal to the one in the LSE (for our experimental wall-pressure data the skewness is negligible, see figure 21 in Appendix A).The quadratic kernel also follows from the inverse Fourier transform of the frequency-domain kernel, ℎ = F −1 , with Figure 13: () Linear coherence spectrogram of  and  2  , for the experimental data (grey scale filled isocontours 0.04:0.04:0.24)and DNS R2000 data (a single red contour at  2   2  = 0.08).() Linear coherence spectra for 11 positions of the experimental data within the logarithmic region, in the range 80 ≲  + ≲ 0.15  .Results in frequency space are converted to a wavelength-dependence using   ≡  ()/  .and quantify the estimation accuracy.Stochastic estimation procedures are performed in frequency space (not wavelength space) since the experimental data comprise time series.When concentrating on LSE, the time-domain estimate of the logarithmic region  fluctuations at   can be formed through the convolution: where the temporal kernel is the inverse Fourier transform of the frequency-domain kernel, The expression (4.2) is equivalent to (1.4), but for a 1D frequency-dependence, and the kernel's gain can be related to the coherence, according to For QSE, the time-domain estimate can be written the first higher-order term of the input pressure   (Naguib et al. 2001): As noted in § 1.1, when the probability density function (PDF) of the input is symmetric, the linear kernel in the QSE is equal to the one in the LSE (for our experimental wall-pressure data the skewness is negligible, see figure 21 in Appendix A).The quadratic kernel also follows from the inverse Fourier transform of the frequency-domain kernel, ℎ  = F −1   , with .
(4.5)For completeness, the linear and quadratic kernels can also be written just in terms of the two-point correlations: . (4.6) Estimates are now performed, for which transfer kernels were first generated from 50 % of the available data (thus from time series of /2 long, still spanning more than 16 000 boundary layer turnover times).The remaining time series data were used for the estimation.
Results for an unconditional estimate of the fluctuations at + ≈ 80 are shown in figure 14( ).Time series are shown for a total duration of Δ + = 3000, based on the LSE and QSE.Estimates are compared to the true (measured) time series: the raw measured time series is shown with the grey line, while a large-scaled filtered version, , is shown with the black line.This latter signal only retains the wall-attached scales, which are defined as the streamwise velocity fluctuations that are correlated with the wall-shear stress (or friction velocity) fluctuations.Practically, is a large-scale pass-filtered signal of , and the filter was derived from velocity-velocity correlations and was confirmed to be Reynolds-number invariant (Baars et al. 2017).This spectral filter has a definitive cut-off at / = 14 and thus also comprises a wall-scaling ∼ , meaning that only progressively larger scales are retained for larger positions.Further details of this filter can be found in the literature (figure 9 and pp.16-17 of Baars & Marusic 2020).Note that will serve as a reference case for comparing the estimations to since the filtered fluctuations following a wall-scaling are more representative of the fluctuations that can physically be estimated using stochastic estimation.When inspecting figure 14( ), it is evident that the QSE procedure better estimates (and thus also ), but this is further quantified in the next section.

Accuracy of velocity-state estimation
We here assess the accuracy of the velocity-state estimates using a binary-state approach as well as a conventional correlation coefficient.When flow control systems would operate with on/off (binary) actuators only (Abbassi et al. 2017), actuators would, in its simplest form, operate based on the estimated signal's sign (e.g., with a zero-valued threshold).In this For completeness, the linear and quadratic kernels can also be written just in terms of the two-point correlations: Estimates are now performed, for which transfer kernels were first generated from 50 % of the available data (thus from time series of   /2 long, still spanning more than 16 000 boundary layer turnover times).The remaining time series data were used for the estimation.Results for an unconditional estimate of the  fluctuations at  +  ≈ 80 are shown in figure 14().Time series are shown for a total duration of Δ + = 3000, based on the LSE and QSE.Estimates are compared to the true (measured) time series: the raw measured time series is shown with the grey line, while a large-scaled filtered version,   , is shown with the black line.This latter signal   only retains the wall-attached scales, which are defined as the streamwise velocity fluctuations that are correlated with the wall-shear stress (or friction velocity) fluctuations.Practically,   is a large-scale pass-filtered signal of , and the filter was derived from velocity-velocity correlations and was confirmed to be Reynolds-number invariant (Baars et al. 2017).This spectral filter has a definitive cut-off at   / = 14 and thus also comprises a wall-scaling   ∼ , meaning that only progressively larger scales are retained for larger  positions.Further details of this filter can be found in the literature (figure 9 and pp.16-17 of Baars & Marusic 2020).Note that   will serve as a reference case for comparing the estimations to since the filtered fluctuations following a wall-scaling are more representative of the fluctuations that can physically be estimated using stochastic estimation.When inspecting figure 14(), it is evident that the QSE procedure better estimates   (and thus also ), but this is further quantified in the next section.

Accuracy of velocity-state estimation
We here assess the accuracy of the velocity-state estimates using a binary-state approach as well as a conventional correlation coefficient.When flow control systems would operate with on/off (binary) actuators only (Abbassi et al. 2017), actuators would, in its simplest form, operate based on the estimated signal's sign (e.g., with a zero-valued threshold).In this context, the goodness of the estimates is quantified with a binary accuracy (BACC).When binarizing   (here considered as the true signal) and the estimated signal (  LSE or  QSE ) at every time instant, only four events are possible: a true positive (TP) occurs when both signals are positive, whereas both signals being negative will yield a true negative (TN).Additionally, false positive (FP) and false negative (FN) linear estimates occur for   () < 0 and  LSE () ⩾ 0, or vice versa, respectively.The BACC defined as BACC =  TP +  TN  , (4.7) represents the cumulative time that the estimate is true positive and negative ( TP +  TN ), relative to the total duration of the signal.Note that a BACC of unity does not mean that the estimate is perfect (that would be  LSE () =   ()), but only that sgn [  LSE ()] = sgn [  ()] ∀. Figure 14() presents the time-percentage of each of the four binary events for the full estimate of the case shown in figure 14().The total BACC for the LSE procedure is 66.7 %, while the QSE improves this to 71.7 %.This improvement comes from an increase of TN instances at the expense of less FP instances (such occasions appear around  + •10 −3 = 0.6 and  + • 10 −3 = 1.8 in figure 14𝑎).
To quantify the goodness of the estimate further, the BACC is plotted for a range of   , starting at the lower end of the logarithmic region,  +  ≈ 80, up to the start of the intermittent region at  +  = 0.4  .Figure 15() includes three profiles: the red ones are based on wall-pressure input, using both LSE and QSE, while the orange is based on an LSE with a representative wall-friction velocity input.This latter case considers data acquired in an identical experiment as the current one, except that the wall-pressure quantity was replaced with a single hot-film sensor on the wall, yielding a voltage signal as a surrogate for the fluctuations in friction velocity (see Abbassi et al. 2017;Dacome et al. 2023).It is wellknown that hot-film yields relatively clean signals (not subject to facility noise as is the wall-pressure) and that the linear correlation between the wall-signal and the off-the-wall velocity fluctuations is relatively strong (e.g., Hutchins et al. 2011;Baars et al. 2017).Note that it was confirmed that for this case with a friction-velocity input, a QSE does not result in an improved estimate per the findings of Guezennec (1989).Finally, before interpreting the results, it is important to put the BACC magnitude into perspective.An estimate with a BACC of 50 % would reflect a random process and would thus be, on average, impractical when attempting real-time control based on such an estimate.In the real-time control work of Abbassi et al. (2017) it was shown that real-time control with a BACC level of around 70 % was sufficient for targeting specific structures (e.g., positive or negative excursions in streamwise velocity).They used a friction-velocity input for their off-the-wall velocity state estimation, and their case is represented by the 'LSE,   input' curve in figure 15().Hence, this case serves as a (successful) reference case.
For an LSE procedure with wall-pressure input, the BACC in the logarithmic region remains below 67 %.However, a QSE procedure results in a very similar BACC (mostly in excess of 72 %) as the reference case with the LSE based on a hot-film sensor-input.Hence, the current analysis shows that velocity-state estimation of the turbulent flow in the logarithmic region is viable with wall-based pressure sensing (provided that the quadratic pressure term is included), even when significant levels of facility noise are present.
Similar trends as for the BACC curves are found when concentrating on a conventional cross-correlation coefficient between the estimated signals and the true (large-scale) filtered signal   .Figure 15() presents profiles of the correlation coefficient of the measured wallattached  fluctuations and the linear estimate (  with  LSE ), and the quadratic estimate (  with  QSE ).A correlation coefficient of   with a wall-friction velocity signal is also shown for reference.As for the observations made based on the BACC, the inclusion of the

Concluding remarks
This work has been motivated by the potential feasibility of using wall-pressure input data for estimating off-the-wall turbulent velocities.Such an estimation capability is of high interest to realizing real-time controllers for wall-bounded turbulence-solely relying on wall pressureinput data-as long as they have a sufficient velocity-state estimation capability and are robust with variations in Reynolds number.The latter requires a firm understanding of the Reynolds-number scaling of wall-pressure-velocity correlations.In this regard, we examined the scaling behaviour of the coherence between the turbulent velocity fluctuations and the quadratic term of the wall-pressure improves the correlation of the estimated signal with the true signal,   : the normalized correlation coefficient reaches 0.60 (at  +  = 80) with the QSE, while the LSE only results in a correlation coefficient of 0.48.Thus, not only the BACC show a good promise for wall-based pressure sensing to perform off-the-wall velocity estimates, but further details of the velocity fluctuations themselves-as captured by a correlation coefficient-are also well-captured.In this regard, a final comparison between the binary fluctuations obtained through QSE with the wall-pressure, and LSE with the hotfilm sensor-input is made in figure 16.Near the geometric centre of the logarithmic region, at  +  ≈ 190, the BACC and correlation coefficient are nearly identical (see figures 15 and 15).Binary fluctuations of the estimates, as well as the true large-scale signal   are shown in figure 16().Note that these two sets of time series are not synchronized, since they come from different experiments.Differences in the estimations are summarized in terms of a PDF of all uninterrupted time spans for which the binary signals equal unity (denoted as time span Δ 1 ).This PDF is presented in premultiplied form (figure 16𝑏), so that longer time spans are weighted accordingly, e.g., the area-under-the-curve is representative of how much a range of Δ 1 contributes to the total time series-duration.PDFs corresponding to the true signals (the two grey binary fluctuations on the left) are equal as expected.Even though the BACC is equal for the two estimates, the binary fluctuations obtained with QSE (  input) results in more 'short' events, while the LSE (  input) results in more 'long' events (cross-over at Δ + 1 ≈ 100).So although the current findings show the feasibility of velocity-state estimation based on wall-pressure sensing, further research should address this accuracy in more detail by considering temporal-accuracy characteristics, its application in real-time control, and robustness to actuator noise and other external sources of noise.

Concluding remarks
This work has been motivated by the potential feasibility of using wall-pressure input data for estimating off-the-wall turbulent velocities.Such an estimation capability is of high interest to realizing real-time controllers for wall-bounded turbulence-solely relying on wall pressureinput data-as long as they have a sufficient velocity-state estimation capability and are robust with variations in Reynolds number.The latter requires a firm understanding of the Reynolds-number scaling of wall-pressure-velocity correlations.In this regard, we examined the scaling behaviour of the coherence between the turbulent velocity fluctuations and the (i) Not only the 1D coherence analysis ( § 3.1), but also a 2D coherence analysis ( § 3.2) revealed that the coherence between and , and and , adhere to a Reynolds number-invariant wall-scaling with .
(ii) A 1D streamwise data analysis with and velocity fluctuations at a given position revealed that they are most strongly coupled to the linear term of the wall-pressure fluctuations, , at a scale of / ≈ 14 and / ≈ 8, respectively.
(iii) With the 2D extension of the analysis as function of and , it was shown that the peak-coherence for and remains near / ≈ 14 and is reasonably symmetric around / = 2.The 2D coherence for and peaks around / = 1.0.
(iv) When statistically combining the results from the 1D and 2D analyses, it can be concluded that the ridge of coherence for and scales following : : ∝ 14 : 7 : 1, while for and the scaling is : : ∝ 8 : 8 : 1.0.
(v) Based on the 2D analysis, turbulent motions in that are coherent with show a much stronger isotropic behaviour than the motions, and also possess a higher coherence magnitude.While the motions elongated in following / ≈ 2 are most coherent, the motions that are most coherent reside near a 2D wave-vector of / = 8 and is largely invariant with increasing / up to only very stronglystretched structures in beyond 3 .
(vi) The coherence with the wall-pressure-squared term 2 resides at larger scales than the linear wall-pressure term, as was also the case in the 1D analysis.The coherence involving 2 is reminiscent of a (large-scale) intensity-modulation of the pressure fluctuations.This is because the coherence dominates at a spatial scale that is a multitude of the characteristic (linearly correlated) fluctuations and because the coherence with the quadratic term, 2 , is strongest for the velocity fluctuations taken in the logarithmic region.The latter reflects the general consensus that the modulation of wall-quantities is driven by energetic large-scale motions in the logarithmic region of the flow.
With the DNS data spanning a decade in friction Reynolds number and the appearance of a clear Reynolds number invariance when adopting a wall-scaling ∼ , the current work suggests strong evidence that an extrapolation of the scaling laws to higher conditions can be made.Moreover, an experimental dataset comprising simultaneous measurements of wall-pressure field, based on DNS data of turbulent channel flow with a Reynolds-number range   ≈ 550 -5 200.Several findings are summarized as follows: (i) Not only the 1D coherence analysis ( § 3.1), but also a 2D coherence analysis ( § 3.2) revealed that the coherence between   and , and   and , adhere to a Reynolds number-invariant wall-scaling with .(ii) A 1D streamwise data analysis with  and  velocity fluctuations at a given  position revealed that they are most strongly coupled to the linear term of the wall-pressure fluctuations,   , at a scale of   / ≈ 14 and   / ≈ 8, respectively.(iii) With the 2D extension of the analysis as function of   and   , it was shown that the peak-coherence for   and  remains near   / ≈ 14 and is reasonably symmetric around   /  = 2.The 2D coherence for   and  peaks around   /  = 1.0.(iv) When statistically combining the results from the 1D and 2D analyses, it can be concluded that the ridge of coherence for   and  scales following   :   :  ∝ 14 : 7 : 1, while for   and  the scaling is   :   :  ∝ 8 : 8 : 1.0.(v) Based on the 2D analysis, turbulent motions in  that are coherent with   show a much stronger isotropic behaviour than the  motions, and also possess a higher coherence magnitude.While the  motions elongated in  following   /  ≈ 2 are most coherent, the  motions that are most coherent reside near a 2D wave-vector of / = 8 and is largely invariant with increasing   /  up to only very stronglystretched structures in  beyond   ≳ 3  .(vi) The coherence with the wall-pressure-squared term  2  resides at larger scales than the linear wall-pressure term, as was also the case in the 1D analysis.The coherence involving  2  is reminiscent of a (large-scale) intensity-modulation of the pressure fluctuations.This is because the coherence dominates at a spatial scale that is a multitude of the characteristic (linearly correlated) fluctuations and because the coherence with the quadratic term,  2  , is strongest for the velocity fluctuations taken in the logarithmic region.The latter reflects the general consensus that the modulation of wall-quantities is driven by energetic large-scale motions in the logarithmic region of the flow.With the DNS data spanning a decade in friction Reynolds number and the appearance of a clear Reynolds number invariance when adopting a wall-scaling   ∼ , the current work suggests strong evidence that an extrapolation of the scaling laws to higher   conditions can be made.Moreover, an experimental dataset comprising simultaneous measurements of wall-pressure and velocity provided ample evidence, at one value of   ≈ 2k, that the DNS-inferred correlations can be replicated with experimental pressure data subject to significant levels of (acoustic) facility noise.It was furthermore shown that in order to reach similar levels of estimation accuracy in the wall-pressure based estimates, compared to estimates based on an input resembling friction velocity fluctuations, it is critical to include the quadratic pressure term.This is consistent with earlier observations of Naguib et al. (2001) who explored a time-domain QSE for the estimates of the conditional streamwise velocity.An accuracy of up to 72 % in the binary state of the streamwise velocity fluctuations in the logarithmic region is achieved; this corresponds to a correlation coefficient of ∼ 0.6.Since measuring the fluctuating wall-pressure is relatively robust in practice and is a viable quantity to measure on an aircraft fuselage, the current study is a step towards the implementation of a reliable flow state estimation framework for wall-bounded turbulence based on wall-pressure.
the Helmholtz correction has a minimal impact on the pressure fluctuations in this range.Nevertheless, the correction is still required for the validity of the wall-pressure spectrum at smaller scales.In this regard, the correction following (A 7) can amplify high-frequency noise due to the fast decay of the gain beyond   .An application of a low-pass filter prevents this and was implemented following the literature (Tsuji et al. 2007;Gibeau & Ghaemi 2021) with a cutoff frequency of  / 2  = 0.25 (  = 3 kHz).A spectrum of the Helmholtz-corrected pressure fluctuations  3 is shown in figure 20() and shows the removal of the amplified energy near   .
Step 3: Remove facility noise.A final step involves the removal of any remaining facility noise with the aid of the free-stream acoustic measurement,   ().This measurement was achieved by mounting a microphone in the potential flow region.A GRAS RA0022 1 /4 in. the nosecone was installed to remove as much as possible the pressure fluctuations from the turbulence in the stagnation point (see photograph in figure 17𝑐).The noise removal procedure was implemented according to the description in Gibeau & Ghaemi (2021) and is only briefly summarized here.A subtraction of an estimate of the facility noise, denoted as   , from the wall-pressure signal at the inlet of the pinhole-inlet,  3 , is implemented following  4 () =  3 () −   () .
(A 8) The estimate of the facility noise does not equal the measured facility noise,   , given that the noise is measured at a different position (in the potential flow region).Moreover, even though a nosecone was installed, the measurement is still intrusive and can be subject to self-induced pressure fluctuations.To generate the estimate, the Wiener noise cancelling filter coefficients can be derived from the measurement data and be implemented through the convolution of a digital FIR filter:   (  ) =  ⊛   (  ).To emphasize the discrete-time dependence, subscript 'd' is used.Filter coefficients (  ) are defined for  time instants and they are obtained through the Wiener-Hopf equations, R = , in which  is a symmetric Toeplitz-matrix of size  ×  with the auto-correlations of   , and  is the two-point crosscorrelation vector of  3 and   and has size  × 1.The unique solution  = R −1  yields the filter coefficients.The filter step is implemented with an order of  = 30 • 10 3 on the signals that were down-sampled by a factor of 5, thus leaving an effective sampling frequency of   = 51.2/5= 10.24 kHz.A spectrum of the pressure fluctuations after the removal of the facility noise through (A 8) is shown in figure 20() and highlights the removal of noise peaks that were still present in the  3 signal.Finally, it was also attempted to directly remove all tunnel noise from only the center pinhole-mounted microphone (without first performing step 1), however, that resulted in a noisier final spectrum.The spectrum of  4 is compared to the 1D spatial spectrum of DNS in figure 20().Here frequency was transformed to wavelength using   =   /  and a convection velocity of  +  = 10.In terms of integrated energy, the inner-scaled wall-pressure intensity from the experiments,  ′+ 4 ≈ 3.04, compares well with (A 1) (Farabee & Casarella 1991;Klewicki et al. 2008) predicting a value of 3.31.The spectral density itself compares reasonably well at the mid-scale range.At the large wavelength-end of the spectrum the attenuation is expected due to the over-excessive removal of energy in step 1.At the small wavelength-end, the energy is amplified instead, presumably due to shortcomings in removing the Helmholtz resonance.That is, the transfer function in step 2 was generated using an acoustic calibration.However, a resonator behaves differently when excited by a grazing TBL flow, in comparison to an acoustic wave-excitation, since the 'end correction' is changed and thus the resonance frequency and effective damping (Panton & Miller 1975).Nevertheless, the mismatch of the spectral shape does not affect any conclusions of the current work, given that the resonance is

(Figure 1 :
Figure1: Wall-based quantities are to be used for a linear time invariant (LTI) system analysis, to estimate the state of the off-the-wall turbulent velocities.A sparse implementation considers a limited number of sensors/actuators, and includes typical offsets in the wall-normal (Δ ) and streamwise (Δ ) directions between the sensing location and the controller's 'target point'.
Figure 3: ( ) Setup for the ZPG-TBL studies in the W-tunnel.( , ) Experimental arrangement with a single hot-wire probe and a pinhole-mounted microphone array measuring the wall-pressure at 7 spanwise positions.In addition, one microphone is placed in the free-stream to measure the facility (acoustic) noise.

Figure 3 :
Figure 3: () Setup for the ZPG-TBL studies in the W-tunnel.(,) Experimental arrangement with a single hot-wire probe and a pinhole-mounted microphone array measuring the wall-pressure at 7 spanwise positions.In addition, one microphone is placed in the free-stream to measure the facility (acoustic) noise.
Figure4: ( ) Experimental boundary layer profiles of the streamwise mean velocity and turbulence intensity, compared to the DNS R2000 case.The mean velocity profile is compared to the log.-law with constants = 0.384 and = 4.17, and the -TKE profile is corrected for spatial resolution effects.( ) Premultiplied energy spectrogram + + (filled iso-contours 0.2:0.2:1.8); the scale-axis is converted to a wavelengthdependence using ≡ ( )/ .
One-sided spectra ( ; ) = 2 ( ; ) * ( ; ) were computed by way of ensembleaveraging a total of 180 FFT partitions of = 2 14 samples (50 % overlap and a Hanning window applied to them).This yields a spectral resolution of d = 0.625 Hz or d + = 3.4 • 10 −5 .For interpretive purposes, frequency spectra are converted to wavenumber spectra using a single convection velocity (taken as the local mean velocity ( ) unless stated otherwise).With wavenumber = 2 / the wavenumber spectrum becomes ( ) = ( )d /d , where group d /d = /(2 ) converts the energy density from a 'per unit frequency' to a 'per unit wavenumber' (the spectral resolution becomes d ≈ 0.88 m −1 or d + ≈ 2.6 • 10 −5 at the lowest position of + = 10).Throughout this paper, the scale axis is either presented in terms of or wavelength = 2 / .Figure 4( ) presents the streamwise energy spectrogram for validation of the experimental setup.The innerspectral peak is clearly visible and is identified with the × marker at + = 10 3 and + = 15.

Figure 4 :
Figure4: () Experimental boundary layer profiles of the streamwise mean velocity and turbulence intensity, compared to the DNS R2000 case.The mean velocity profile is compared to the log.-law with constants  = 0.384 and  = 4.17, and the -TKE profile is corrected for spatial resolution effects.() Premultiplied energy spectrogram  +   +  (filled iso-contours 0.2:0.2:1.8); the scale-axis is converted to a wavelengthdependence using   ≡  ()/  .
Figure 5: ( ) 1D gain of the cross-spectrogram, computed using and .Solid, coloured iso-contours correspond to one contour value indicated in the figure for all cases (an increased colour intensity corresponds to an increase in ); the greyscale contour shows a finer discretization of iso-contours for the R5200 case only.A black trend line indicates a wall-scaling of / = 14.( ) Similar to sub-figure ( ), but now for and .A black-dashed trend line indicates a wall-scaling of / = 8.

Figure 5 :
Figure 5: () 1D gain of the cross-spectrogram, computed using  and   .Solid, coloured iso-contours correspond to one contour value indicated in the figure for all   cases (an increased colour intensity corresponds to an increase in   ); the greyscale contour shows a finer discretization of iso-contours for the R5200 case only.A black trend line indicates a wall-scaling of   / = 14.() Similar to sub-figure (), but now for  and   .A black-dashed trend line indicates a wall-scaling of   / = 8.
Figure 6: ( ) Linear coherence of and .Solid, coloured iso-contours correspond to one contour value for all cases (an increased colour intensity corresponds to an increase in ); the greyscale contour shows a finer discretization of contours for ≈ 5 200 only.( ) Similar to sub-figure ( ), but now for the linear coherence of and .( ) Linear coherence spectra 2 (one bundle of lines per condition) within the logarithmic region, in the range 80 + 0.15 .Results in frequency space are converted to a wavelength-dependence using ≡ ( )/ .( ) Similar to sub-figure ( ), but now with the linear coherence spectra 2 .
Figure 8: 2D spectrograms of ( ) , ( ) and ( ) at one and + , as indicated in the sub-figures; grey-filled contours show 7 iso-contours ranging up to the maximum value.( ) 2D linear coherence of and , for the same and + as ( , ); the solid, coloured iso-contours correspond to two contour values of 2 = [0.05;0.25].( ) Similar to sub-figure ( ), but now for the 2D linear coherence of and ; the solid, coloured iso-contours correspond to two contour values of 2 = [0.10;0.30].Lee & Moser (2019) introduced a 'log-polar' format for the 2D spectrum.When considering wavenumbers, ( , ) can be expressed in polar coordinates, yielding ( cos , sin ) with = cos ( ) and = sin ( ).When defining = log 10 ( / ref ), where ref is a reference wavenumber smaller than the smallest non-zero wavenumber considered, a set of logarithmic polar coordinates would be and , with associated Cartesian coordinates # = cos = / and # = sin = / , respectively.Following Parseval's theorem, the spectral-integral equals

Figure 8 :
Figure 8: 2D spectrograms of () , ()   and ()  at one   and  + , as indicated in the sub-figures; grey-filled contours show 7 iso-contours ranging up to the maximum value.() 2D linear coherence of  and   , for the same   and  + as (,); the solid, coloured iso-contours correspond to two contour values of  2    = [0.05;0.25].() Similar to sub-figure (), but now for the 2D linear coherence of  and   ; the solid, coloured iso-contours correspond to two contour values of  2    = [0.10;0.30].

Figure 9 :
Figure9: (,,) Similar to figures 8(,,) but now presented in log-polar format as described in the text; grey scale contours show 7 iso-contours ranging up to the maximum value.(,) Similar to figures 8(,) but now presented in log-polar format.For indicating trends of scale-isotropy, five dash-dotted arcs of constant  = 2/ are plotted in all sub-figures and correspond to  + = 10 1 , 10 2 , 10 3 , 10 4 and 10 5 (from the most outer-arc going inward).
Figure10: (,,) 2D linear coherence of  and   , for all   cases and for three different inner-scaled wall-normal positions.Note that the wavenumber representation on the axes includes a wall-scaling with .The solid, coloured iso-contours correspond to the same two contour values as were considered in figure9(),  2    = [0.05;0.25].(,,  ) Similar to sub-figure () but now for  and   , with the solid, coloured iso-contours corresponding to the same two contour values as were considered in figure9().For indicating trends of scale-isotropy, dash-dotted arcs correspond to constant values of / = 1, 3, 8, 14, 85, and 140 (from the most outer-arc going inward).
Figure13: ( ) Linear coherence spectrogram of and 2 , for the experimental data (grey scale filled isocontours 0.04:0.04:0.24)and DNS R2000 data (a single red contour at 2 2 = 0.08).( ) Linear coherence spectra for 11 positions of the experimental data within the logarithmic region, in the range 80 Figure 14: ( ) Sample of the stochastic estimate of the fluctuations at + ≈ 80, based on wall-pressure, in comparison to a true (measured) time series.The measured hot-wire time series ( , ) is shown with a grey line, and is filtered to only retain the wall-attached fluctuations, ( , ), shown with the black line.Estimates using LSE, LSE ( , ), and QSE, QSE ( , ), are shown with orange and red lines, respectively.Correlation coefficients of the measured wall-attached fluctuations and the linear estimate ( with LSE ), and the quadratic estimate ( with QSE ), equal 0.48 and 0.60, respectively (see figure 15( ) for + ≈ 80).( ) Time-percentage of binary events in the LSE (orange dash-dotted) and QSE (red solid), relative to .

Figure 14 :
Figure 14: () Sample of the stochastic estimate of the  fluctuations at  +  ≈ 80, based on wall-pressure, in comparison to a true (measured) time series.The measured hot-wire time series (  , ) is shown with a grey line, and is filtered to only retain the wall-attached  fluctuations,   (  , ), shown with the black line.Estimates using LSE,  LSE (  , ), and QSE,  QSE (  , ), are shown with orange and red lines, respectively.Correlation coefficients of the measured wall-attached  fluctuations and the linear estimate (  with  LSE ), and the quadratic estimate (  with  QSE ), equal 0.48 and 0.60, respectively (see figure 15() for  +  ≈ 80).() Time-percentage of binary events in the LSE (orange dash-dotted) and QSE (red solid), relative to   . figure16

Figure 15 :
Figure 15: () BACC for a range of estimation positions,   .() Similar to sub-figure (), but now for the correlation coefficient between the LSE-and QSE-based estimates and the true large-scales, taken as   .
Figure 16: ( ) Binary fluctuations of the QSE in solid dark red (top row) and LSE in dashed orange (bottom row) near the geometric centre of the logarithmic region, compared to the binary fluctuations of the true large-scales, , in grey.( ) Premultiplied probability density functions of all uninterrupted time spans for which the binary signals equal unity (Δ 1 ).wall-pressure field, based on DNS data of turbulent channel flow with a Reynolds-number range ≈ 550 -5 200.Several findings are summarized as follows:

Figure 16 :
Figure 16: () Binary fluctuations of the QSE in solid dark red (top row) and LSE in dashed orange (bottom row) near the geometric centre of the logarithmic region, compared to the binary fluctuations of the true large-scales,   , in grey.() Premultiplied probability density functions of all uninterrupted time spans for which the binary signals equal unity (Δ 1 ).

Table 1 :
Parameters of data sets used: channel DNS data and experimental turbulent boundary layer data.Note that  for DNS of the channel flows and the boundary layer experiment denote the channel halfwidth and the boundary layer thickness, respectively.†Total simulation time without transition.