12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

WALL-PRESSURE–VELOCITY TRANSFER KERNEL IN HIGH REYNOLDSNUMBER TURBULENT CHANNEL FLOWS

Woutijn J. Baars
Faculty of Aerospace Engineering
Delft University of Technology
2629 HS Delft, The Netherlands
w.j.baars@tudelft.nl

Myoungkyu Lee
Department of Aerospace Engineering and Mechanics
The University of Alabama
Tuscaloosa, AL 35487, U.S.A.
mklee@ua.edu

ABSTRACT

velocity fluctuations (consider those as the control ‘targets’) and a set of input sensors measuring the dynamic
state of the boundary layer (Rathnasingham & Breuer,
2003; Brunton & Noack, 2015). For the case of wallbounded turbulence, these sensors should be integrated
within the wall itself to avoid form drag and, in this regard, a wall-pressure measurement is practically-robust
and thus ideal in practise. Analyzing the statistical coupling between the velocity fluctuations in the boundary
layer and the wall-pressure field is a pre-requisite to implementing a control system with a high ‘observability’(Abbassi et al., 2017; Samie et al., 2020). Our work
scrutinizes the statistical coupling, in the context of alinear time invariant system analysis. The notation inthis paper is as follows. Coordinates x, y and z denotethe streamwise, wall-normal and spanwise directions of
the flow, and lower-case u, v, w and p represent the
Reynolds decomposed fluctuations of the three velocity
components and the static pressure, respectively. The
wall-pressure is denoted as pw . The friction Reynolds
number Reτ ≡ δuτ /ν or δ + is the ratio of the channel
half-width δ to the viscous length scale ν/uτ .
Many works are concerned with scaling laws andmodeling attempts of pressure fluctuations in wallbounded turbulence (e.g., Willmarth, 1975; Farabee &Casarella, 1991; Klewicki et al., 2008; Hwang et al.,2009), but a direct assessment of the instantaneous or
statistical coupling between the fluctuating velocity field
and the wall-pressure is required for wall-based velocity field estimations. Though, simultaneous measurements of turbulent velocities and pw have already revealed characteristic wall-pressure signatures associatedwith burst-sweep cycle events (Thomas & Bull, 1983).
Throughout the last decade, high-resolution mappingsof the spatio-temporal pressure-velocity correlation havealso been reported (Ghaemi & Scarano, 2013; Nakaet al., 2015), and have been extended to the spectral
domain with the aid of coherence and phase spectra(most notably in the experimental works of Van Blitterswyk & Rocha, 2017; Gibeau & Ghaemi, 2021). Ex-

Since wall-pressure fluctuations would form apractically-robust input to a real-time active controllerof wall-bounded turbulence, it is of high practical interest to study the scaling behavior of the wall-pressure–velocity coupling. This work investigates the couplingof the wall-pressure fluctuations with the streamwise
and wall-normal velocity fluctuations. Both the gain(or coherence) and phase spectra of the wall-pressure–velocity transfer kernel are assessed using a comprehensive database, available from direct numerical simulations of turbulent channel flow. With data spanning adecade in friction Reynolds number Reτ ∼ 550 − 5200, a
1D analysis (in terms of the streamwise wavelength, λx )reveals that the streamwise velocity and wall-pressure
are most strongly coupled at a self-similar wall-scalingof λx /y ≈ 14. For the wall-normal velocity component,
the strongest coupling appears at approximately half
this ratio (λx /y ≈ 8.5). An analysis of the kernel’s
phase demonstrates that both the coherent fluctuationsof streamwise and wall-normal velocity obey a forwardleaning inclination angle of α ≈ 30◦ . When extending
the analysis to 2D (as a function of λx and λz ), the peakcoherence for pw and u still resides close to λx /y ≈ 14
and is reasonably symmetric around λx /λz = 2.3. The
2D coherence for pw and v peaks around λx /λz = 1.0.
Both the 2D coherence for pw and u, and pw and v,
adhere to a wall-scaling with y. Scaling behaviours
identified in this work will aid the efficacy of real-time
controllers, by for instance the implementation of dataderived FIR filters to only control velocity structuresthat are captured through wall-pressure measurements.

INTRODUCTION AND CONTEXT
Inspiration for this work was born out of practical considerations associated with the implementation ofreal-time flow control of wall-bounded turbulence. Realtime opposition control is only effective when there isa significant degree of correlation between the grazing

1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
(a)

(b)

(c)

102

102

102

101

y+

103

y+

103

y+

103

101

101

kx+ φ+
uu
10

0

102

103

104

105

kx+ φ+
ppkx+ φ+
vv
10

0

102

103

λ+
x

104

105

10

0

102

λ+
x

103

104

105

λ+
x

Figure 1: 1D spectrograms of (a) u, (b) v and (c) p. Two clusters of solid, coloured iso-contours on (a-c)+ ++ +correspond to two contour values of k + φ+
uu = [0.2; 1.2], k φvv = [0.05; 0.3] and k φpp = [0.45; 2.25], respectively,
for all Reτ cases (an increased colour intensity corresponds to an increase in Reτ ; the channel half-widths
δ + = Reτ are indicated along the ordinate). The gray-scale contour shows a finer discretization of iso-contours
for the highest Reynolds number case only (Reτ ≈ 5200).

perimental studies are subject to the inherent challenges
associated with wall-pressure measurements, and performing those over a range of Reτ in the absence of
spatial/temporal resolution effects. In addition, it ischallenging to simultaneously acquire velocity data (in astreamwise–spanwise plane) and the wall-pressure field.
These data are required to investigate the wall-pressure–velocity coupling as a function of the energetic scales
decomposed in their streamwise and spanwise Fouriermodes. Novel experiments can be designed to yield
2D coherence spectra (e.g., for velocity–velocity correlations, Deshpande et al., 2020), but these are nottrivial to conduct at a range of Reτ without resolution effects entering the problem. We here analyse thewall-pressure–velocity kernel using the vast amount ofhigh-fidelity data that is currently available from direct
numerical simulation (DNS) campaigns.
Bode plots of the scale-dependent gain (or coherence) and phase for pw and u, and pw and v, are particularly useful for wall-based control. That is, a routine action for control would, for instance, involve an
estimate of the streamwise velocity component u, at awall-normal location ye , based on pw . When we for
now confine ourselves to the x dimension only, a linear stochastic estimate ue(x, ye ) can be generated, either with a time (Naguib et al., 2001; Lasagna et al.,2013) or spectral (Tinney et al., 2006; Baars et al.,2016) approach. An estimate in the spectral domaine (λx , ye ) = HL (λx , ye )Pw (λx ), where HL
is denoted as U
e (λx , ye ) and Pw (λx )is a complex-valued kernel and U
are the Fourier transforms of ue(x, ye ) and pw (x), respectively. Note that λx = 2π/kx denotes the streamwise wavelength. The kernel is found from two-pointcalibration data and equals the complex-valued crossspectrum, divided by the wall-pressure spectrum:
HL (λx ) =φupw (λx )= |HL (λx ) | · expjψ(λx ) ,φpw pw (λx )Note that the argument ye is omitted in eqs. (1) and (2)for brevity. Even though the u velocity component wasused above, the same can be done for v and w. And, as
shown later, the analysis can be extended to 2D (e.g.,
Encinar & Jiménez, 2019). Now this work investigatesthe scaling behavior of the kernel HL , for pw and the uand v velocities, with the aid of DNS data of turbulent
channel flow.

DIRECT NUMERICAL SIMULATION DATA
Four channel flow datasets are used and span one
decade in Reynolds number, Reτ ∼ 550 − 5200 (see table 1 for exact values). Details on the numerical scheme,
resolution and turbulence statistics can be found in Lee& Moser (2015, 2019). In relevance to the current work
involving pressure, the static pressure fluctuations wereobtained by solving a Poisson equation (Panton et al.,2017). For reference, 1D spectrograms of u, v and p areshown in figs. 1a-c, respectively. Gray scale iso-contours
correspond to the Reτ ≈ 5200 dataset only, while thetrends of how the spectrograms change with Reτ areshown with two sets of coloured iso-contours, drawn for
all Reτ (see figure caption). The pressure spectrum remains constant for y + . 5, reflecting the wall-pressure
spectrum. For all fluctuating quantities in fig. 1 it isevident that the total energy at a given y + grows dueto the additional energetic scales at large λ+x . For instance, see Mathis et al. (2009) for the scaling of uspectrograms and Panton et al. (2017) for the scaling of
the mean-square pressure fluctuations.

Table 1: Channel DNS data (Lee & Moser, 2015).

(1)

where the kernel’s phase ψ(λx ) equals the phase of thecross-spectrum. The gain is often expressed in terms ofa normalized, linear coherence with bounds of 0 and 1,
φp p (λx )2.
(λx ) = |HL (λx ) |2 · w w
γupw
φuu (λx )Case name

R0550

R1000

R2000

R5200

Reτ

544

1000

1995

5186

WALL-PRESSURE–VELOCITY COUPLING
Given the energy growth of the spectrograms infig. 1, with an increase in Reτ , it is of high interest to assess the Reτ -trend in both the upw and vpw coherence.

(2)2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
−7
kx+ |φ+upw | = 5 · 10

(a)

(b)

102

102

λx /y = 14

10

101
100

102

101

102

103

104

105

inclination α
3

y+

10

y+

3

y+

3

10

(c)

2
= 0.08γup
w

100

α = 10◦ , 30◦ , 50◦

101

102

103

104

105

100

102

103

λ+
x

λ+
x

104

105

λ+
x

Figure 2: (a) 1D gain of the cross-spectrogram, computed using u and wall-pressure pw . Solid, coloured isocontours correspond to one contour value indicated in the figure, for all Reτ cases (an increased colour intensity
corresponds to an increase in Reτ ); the gray scale contour shows a finer discretization of iso-contours forReτ ≈ 5200 only. (b,c) Similar to sub-figure (a), but now for (b) the linear coherence of u and pw and (c) the
forward-leaning geometric phase angle α, as shown in fig. 4.

(b)
10

102

102

y+

3

λx /y = 8.5

10

101

inclination α
3

102

y+

3

10

(c)

2
= 0.08
γvp
w

y+

(a)

101

α = 10◦ , 30◦ , 50◦ , 70◦

101−8kx+ |φ+vpw | = 5 · 10

100

102

103

104

105

100

102

103

104

105

100

102

λ+
x

λ+
x

103

104

105

λ+
x

Figure 3: Similar to fig. 2 but for v and pw , instead of u and pw .

(Liu & Gayme, 2020). There is a region of non-zero coherence obeying an outer-scaling behavior at very large
λx . This indicates that only the very large global modesin the velocity fluctuations, spanning the entire logarithmic region (del Álamo & Jiménez, 2003), leave a
coherent wall-pressure footprint.
When focusing on the kernel’s phase ψ(λx ) weconvert the phase angle into a geometric inclination
angle for ease of interpretation, following α(λx ) ≡tan−1 (y/∆x), where ∆x = λx ψ(λx )/(2π) is the geometric phase shift. This procedure yields a geometric angle
for each wavelength λ+x , and for each wall-normal location y + , at which the turbulent velocity is assessed inrelation to pw . Fig. 2c shows iso-contours of constantα(λx ), for all Reτ cases. Fluctuations of u that are
most wall-pressure-coherent (residing along λx /y = 14)lean forward with α ≈ 30◦ . Moreover, structures withhigher and lower aspect ratios are less and more inclined,
respectively. Finally, a perfect collapse of the differentReτ curves indicates Reynolds number invariant behavior; this is of high practical importance when extrapolating wall-pressure–velocity filters to higher Reτ flow
control experiments.
Similar to the correlation analysis of pw and u, theanalysis can be performed with considering the v fluctuations instead. Findings are presented in fig. 3 andhighlight that the wall-pressure-coherent energy in v resides at a smaller aspect ratio, around λx /y ≈ 8.5 (notethat the ridge of maximum coherence does not necessarily correspond to the ridge of maximum energy, as isFirst we assess the coherence in the 1D, streamwise direction, since this is reminiscent of the confined view
available from typical experiments.

1D analysis in the streamwise direction
Following eq. (1), the magnitude of the complexvalued cross-spectrum yields a first indication of thewall-pressure–velocity coupling. From the iso-contoursof |φ+
upw | in fig. 2a it is evident that the degree of coupling grows self-similarly with an increase in Reτ : theridge in cross-spectral energy grows along the black solidline, which is representative of a wall scaling (constant
λx /y = 14). This scaling is even more distinct when con2
in fig. 2b. Clearly, pw andsidering the coherence γupw
u are most strongly coupled at the self-similar scalingof λx /y ≈ 14. This particular wall-scaling matches the
Reynolds number invariant aspect ratio of wall-attachedcoherent structures of u, see Baars et al. (2017). The
ridge of coherence is relatively narrow-band and only
grows to larger viscous-scaled heights with increasing
Reynolds number, not in width. It is surprising tonote the appearance of a region of zero-coherence at
relatively large wavelengths, and at locations below theridge (which are in fact closer to the wall-pressure signature). This suggests that large-scale u fluctuationsin close proximity to the wall do not comprise a phaseconsistent relationship with the wall-pressure. Possibly
due to the stronger dispersive (and random) nature in
the convection of large-scale structures near the wall3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022zhairpinwall-pressure coherent velocity fluctuations
u-fluctuations

u-fluctuations

v-fluctuations

packets

v-fluctuations
vorticity

y
e
tur
uc
str

x

α (scale & height dependent)zxwall-pressure fluctuations

Figure 4: Illustration of hairpin packets and the statistical correlation between the wall-pressure, pw , and the u-

and v-velocity. Since the correlation and forward-leaning geometric phase angle α are dependent on scale andheight, the coherent fluctuations are visualized as harmonic waves of varying streamwise scale.

≈ 80 & Reτ ≈ 5200

(c)

kx+ kz+ φ+
pw pw

104

104

103

103

103

102

102

102

102

103

104

105

2
γup
w

λ+
z

@y

+

λ+
z

104

(b)

kx+ kz+ φ+
uu

λ+
z

(a)

102

103

104

105

λ+
x

λ+
x

λx /λz = 2.3
102

103

104

105

λ+
x

Figure 5: 2D spectrograms of (a) u and (b) p at one Reτ and y + , as indicated in the figure. The gray scale contour

shows 7 iso-contours ranging up to the maximum value. (c) 2D linear coherence of u and pw , for the same Reτ
2
and y + as (a,b); the solid, coloured iso-contours correspond to two contour values of γup= [0.05; 0.25].
w
velocity u(x, ye , z) and wall-pressure pw (x, z) (where ye
is an evaluation location for the velocity):

shown by the λx /y ≈ 8.5 on fig. 1b). The v fluctuations
also consistently lean forward with the same angle as
was found for u, namely α ≈ 30◦ .
Observations made on the basis of figs. 2 and 3
are summarized in fig. 4. When coherent motions
are thought of as hairpins (and packets of them), thespanwise centre is associated with a negative streamwise fluctuation while the v-fluctuations induced by
the vortical motions of the packet reside at a shorter
streamwise wavelength. This is reflective of the ridges
of peak-coherence for u and v at λx /y ≈ 14 and 8.5,
respectively. Mechanistically, v fluctuations induce
wall-pressure through a pressure stagnation when directed towards the wall, while u fluctuations are anticorrelated to pressure fluctuations because of the balance/exchange of pressure and momentum.

2γup(λx , λz ) =w|φupw (λx , λz ) |2.
φuu (λx , λz ) φpw pw (λx , λz )(3)Note again that the ye argument of the wall-normal location is omitted in eq. (3) for brevity. When computing the 2D coherence for the same data as for which the
2D (auto-)spectrograms were plotted in figs. 5a,b, thecoherence-contour of fig. 5c is obtained. Notably, the coherence only becomes significant (when taking a thresh2
old of γup= 0.05) for λ+x > 400, agreeing to what wasw
previously observed in fig. 2. More interesting is that thepeak-coherence in 2D still resides close to λx /y ≈ 14 (for+fig. 5c equivalent to λ+x = 14y = 1120) and is reasonably symmetric around λx /λz = 2.3. The latter implies
that the wall-pressure-coherent u velocity structures areroughly twice as long as they are wide.
An important note of caution for interpreting thecoherence is that a high value is not always relevant interms of absolute energies (since the coherence is energy+
34normalized). For instance, at (λ+x , λz ) = (10 , 10 ) in2
fig. 5c, the coherence is just over γup
=
0.25,
even
w
though the absolute energy in the u fluctuations isextremely low (and not visible in the contour plot offig. 5a).
In order to now assess the trend of the 2D coherence
for pw and u with height and with Reτ , we move towards
the representation in fig. 6 and start with fig. 6b, which

2D analysis in the streamwise–spanwise plane
A 1D spectral analysis along x lumps all spanwise
information together. In order to inspect which spanwise scales of velocity fluctuations are coherent with thethe spanwise scales of wall-pressure, we move towards a
2D analysis in the streamwise–spanwise plane. At firstthe 2D spectrograms of streamwise velocity fluctuations,φuu (λx , λz ), and pressure fluctuations, φpw pw (λx , λz ),are shown for y + = 80 at Reτ ≈ 5200 in figs. 5a and 5b.
When integrated out its λz dependence, the 1D spectra of figs. 1a,c are obtained at y + = 80 (for u) and at
the wall (for pw ). A 2D coherence is a generalization ofeq. (2) and results in the following for the streamwise4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
(a)
103

2
= [0.05; 0.25] @ y + ≈ 20
γup
w

(b)
103

2
= [0.05; 0.25] @ y + ≈ 80
γup
w

(c)
103

2
= [0.05; 0.25] @ y + ≈ 320
γupw

λx /λz = 2.3

λz /y

101

100
100

102

λz /y

102

λz /y

102

101

101

102

103

100
100

101

101

102

λx /y

103

100
100

101

λx /y

102

103

λx /y

Figure 6: 2D linear coherence of u and pw , for all Reτ cases and for three wall-normal positions, indicated ineach of the three sub-figures (a-c). Note that the wavelengths λx and λz are represented with a wall-scaling(normalized by y). The solid, coloured iso-contours correspond to the same two contour values as were considered2= [0.05; 0.25].
in fig. 5c, γup
w
(a)
103

2
= [0.10; 0.40] @ y + ≈ 20
γvp
w

(b)
103

2
= [0.10; 0.40] @ y + ≈ 80
γvp
w

(c)
103

2
= [0.10; 0.40] @ y + ≈ 320
γvpw

λx /λz = 1.0

λz /y

101

100
100

102

λz /y

102

λz /y

102

101

101

102

103

100
100

101

101

102

λx /y

103

100
100

101

λx /y

102

103

λx /y

Figure 7: Similar to fig. 6 but for v and pw , instead of u and pw . The solid, coloured iso-contours correspond to2
two contour values of γvp= [0.10; 0.40].
w

is valid for a height of y + ≈ 80. For all cases of Reτ ,two sets of iso-contours of the coherence are shown; thesame colour scale as in fig. 2 is adopted. For the mostdark-red case (Reτ ≈ 5200), the two iso-contours are
similar to the ones shown in fig. 5c. Similar plots canbe generated for lower y + locations (fig. 6a for y + ≈ 20)
and higher locations (fig. 6c for y + ≈ 320). Since theaxes in all sub-figures are scaled with the respective ylocation, a match of the iso-contours corresponds to aself-similar scaling in y. This is the case when comparing
figs. 6a-c. Moreover, these graphs are direct evidence ofa Reynolds number invariant behaviour of the coherence
because all Reτ iso-contours collapse.

scales as λx : λz : y ∝ 8.5 : 8.5 : 1.0 (note that a coherence analysis for turbulent pipe flow at high Reynolds
numbers resulted in an aspect ratio of 7:1:1 for the velocity fluctuations with their skin-friction footprint, see
Baidya et al., 2019). Finally, the coherence magnitudeof v is roughly a factor two higher (note the different
iso-contour between figs. 6 and 7.

CONCLUDING REMARKSThis work examined the scaling behaviour of the coherence between the turbulent velocity fluctuations andthe wall-pressure field. For streamwise data of u andv (in relation to the wall-pressure pw ), the strongest
coupling appears at λx /y ≈ 14 and λx /y ≈ 8.5, respectively. The 2D extension of the analysis (as a functionof λx and λz ) revealed that the peak-coherence for pw
and u still resides close to λx /y ≈ 14 and is reasonably
symmetric around λx /λz = 2.3. The 2D coherence forpw and v peaks around λx /λz = 1.0. Both the 2D coherence for pw and u, and pw and v, adhere to a wallscaling with y. Statistically this can be summarized asthat the ridge of coherence for pw and u scales following λx : λz : y ∝ 14 : 6 : 1, while for pw and v the scalingis λx : λz : y ∝ 8.5 : 8.5 : 1.0. An analysis of the kernel’s phase demonstrated that both the coherent fluctuations of the streamwise and wall-normal velocity obey
a forward-leaning inclination angle of α ≈ 30◦ . With
the data spanning a decade in friction Reynolds num-

Note that when the 2D coherence analysis is repeated for pw and v (instead of u), the resultant assessment for the scaling behavior is shown in fig. 7. Since the2collapse also, it can be concluded
iso-contours of γvp
w
that the 2D coherence for pw and u, and pw and v, adhere to a Reynolds number-invariant wall-scaling withy. Because it was already derived in the 1D analysis
that the wall-pressure-coherent v fluctuations are half
the size of the corresponding u fluctuations, the same isobserved in the 2D coherence. While the 2D coherenceof pw and u peaks around λx /λz = 2.3, the coherenceof pw and v peaks around λx /λz = 1.0, illustrating thatthe shorter streamwise extent of wall-coherent v fluctuations are roughly of the same width. That is, we
can conclude that the ridge of coherence for pw and u
scales as λx : λz : y ∝ 14 : 6 : 1, while for pw and v this5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
319, 199–217.
Klewicki, J. C., Priyadarshana, P. J. A. & Metzger,
M. M. 2008 Statistical structure of the fluctuating wall pressure and its in-plane gradients at highreynolds number. J. Fluid Mech. 609, 195–220.
Lasagna, D., Orazi, M. & Iuso, G. 2013 Multi-time delay, multi-point linear stochastic estimation of a cavity shear layer velocity from wall-pressure measurements. Phys. Fluids 25, 017101.
Lee, M. & Moser, R. D. 2015 Direct numerical simulation of turbulent channel flow up to Reτ = 5200. J.
Fluid Mech. 774, 395–415.
Lee, M. & Moser, R. D. 2019 Spectral analysis of thebudget equation in turbulent channel flows at highReynolds number. J. Fluid Mech. 860, 886–938.
Liu, C. & Gayme, D. F. 2020 An input–output based
analysis of convective velocity in turbulent channels.
J. Fluid Mech. 888, A32.
Mathis, R., Hutchins, N. & Marusic, I. 2009 Large-scale
amplitude modulation of the small-scale structures inturbulent boundary layers. J. Fluid Mech. 628, 311–
337.
Naguib, A. M., Wark, C. E. & Juckenhöofel, O. 2001
Stochastic estimation and flow sources associatedwith surface pressure events in a turbulent boundary
layer. Phys. Fluids 13 (9), 2611–2626.
Naka, Y., Stanislas, M., Foucaut, J.-M., Coudert, S.,
Laval, J.-P. & Obi, S. 2015 Space–time pressure–velocity correlations in a turbulent boundary layer.
J. Fluid Mech. 771, 624–675.
Panton, R. L., Lee, M. & Moser, R. D. 2017 Correlationof pressure fluctuations in turbulent wall layers. Phys.
Rev. Fluids 2, 094604.
Rathnasingham, R. & Breuer, K. S. 2003 Active control of turbulent boundary layers. J. Fluid Mech. 495,
209–233.
Samie, M., Baars, W. J., Rouhi, A., Schlatter, P., Örlu,
R., Marusic, I. & Hutchins, N. 2020 Near wall coherence in wall-bounded flows and implications for flow
control. Int. J. Heat Fluid Flow 86, 108683.
Thomas, A. S. W. & Bull, M. K. 1983 On the role ofwall-pressure fluctuations in deterministic motions in
the turbulent boundary layer. J. Fluid Mech. 128,
283–322.
Tinney, C. E., Coiffet, F., Delville, J., Glauser, M. N.,
Jordan, P. & Hall, A. M. 2006 On spectral linear
stochastic estimation. Exp. Fluids 41 (5), 763–775.
Van Blitterswyk, J. & Rocha, J. 2017 An experimental study of the wall-pressure fluctuations beneath
low Reynolds number turbulent boundary layers. J.
Acoust. Soc. Am. 141 (2), 1257–1268.
Willmarth, W. W. 1975 Pressure fluctuations beneathturbulent boundary layers. Annu. Rev. Fluid Mech.
7, 13–36.

ber Reτ ∼ 550 − 5200 and a very good Reynolds number
invariant trend, the current work allows for an extrapolation of the identified scaling laws to higher Reτ conditions. The scaling behaviours will therefore aid theefficacy efficacy of real-time controllers, by for instancethe implementation of data-derived FIR filters to only
control velocity structures that are captured throughwall-pressure measurements.

REFERENCES
Abbassi, M. R., Baars, W. J., Hutchins, N. & Marusic, I.
2017 Skin-friction drag reduction in a high-Reynoldsnumber turbulent boundary layer via real-time control of large-scale structures. Int. J. Heat Fluid Flow67, 30–41.
Baars, W. J., Hutchins, N. & Marusic, I. 2016 Spectral
stochastic estimation of high-Reynolds-number wallbounded turbulence for a refined inner-outer interaction model. Phys. Rev. Fluids 1, 054406.
Baars, W. J., Hutchins, N. & Marusic, I. 2017 Selfsimilarity of wall-attached turbulence in boundary
layers. J. Fluid Mech. 823, R2.
Baidya, R., Baars, W. J., Zimmerman, S., Samie, M.,
Hearst, R. J., Dogan, E., Mascotelli, L., Zheng, X.,
Bellani, G., Talamelli, A., Ganapathisubramani, B.,
Hutchins, N., Marusic, I., Klewicki, J. & Monty, J. P.
2019 Simultaneous skin friction and velocity measurements in high Reynolds number pipe and boundary
layer flows. J. Fluid Mech. 871, 377–400.
Brunton, S. L. & Noack, B. R. 2015 Closed-loop turbulence control: Progress and challenges. Appl. Mech.
Rev. 67 (5), 050801.
del Álamo, J. C. & Jiménez, J. 2003 Spectra of the verylarge anisotropic scales in turbulent channels. Phys.
Fluids 15 (6), L41–L44.
Deshpande, R., Chandran, D., Monty, J. P. & Marusic, I. 2020 Two-dimensional cross-spectrum of thestreamwise velocity in turbulent boundary layers. J.
Fluid Mech. 890, R2.
Encinar, M. P. & Jiménez, J. 2019 Logarithmic-layer
turbulence: A view from the wall. Phys. Rev. Fluids
4, 114603.
Farabee, T. M. & Casarella, M. J. 1991 Spectral features of wall pressure fluctuations beneath turbulent
boundary layers. Phys. Fluids A 3 (10), 2410–2420.
Ghaemi, S. & Scarano, F. 2013 Turbulent structure ofhigh-amplitude pressure peaks within the turbulent
boundary layer. J. Fluid Mech. 735, 381–426.
Gibeau, B. & Ghaemi, S. 2021 Low- and mid-frequency
wall-pressure sources in a turbulent boundary layer.
J. Fluid Mech. 918, A18.
Hwang, Y. F., Bonness, W. K. & Hambric, S. A. 2009
Comparison of semi-empirical models for turbulent
boundary layer wall pressure spectra. J. Sound Vibr.

6