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

PRESSURE SOURCE CONTRIBUTION AND NEAR WALL FLOW STRUCTURES
RELATED TO HIGH AMPLITUDE WALL PRESSURE PEAKS
Yoshitsugu Naka
Department of Mechanical Engineering
Meiji University
1-1-1 Higashimita Tama-ku Kawasaki, Japanemail: naka@meiji.ac.jp

Kosuke Osawa
School of Aeronautics
Universidad Politecnica de Madrid
Plaza del Cardenal Cisneros 3, Madrid, Spainemail: kosawa@torroja.dmt.upm.es

ABSTRACT
Coherent flow structures contributing to high amplitude wallpressure peaks are identified using a direct numerical simulation(DNS) database of a turbulent channel flow. A new theoreticalformulation for the local pressure source contribution is derived
from the Poisson equation for pressure using the delta function.
A high amplitude wall pressure peak is defined as a local
maximum in magnitude larger than three times of the overallroot mean square value. The high amplitude wall pressure peaks
are mostly contributed from the near wall region up to the buffer
layer for both positive and negative pressure peaks. The
conditionally averaged flow fields around the high amplitude
wall pressure peaks are evaluated. The spanwise asymmetry of
the instantaneous flow field is carefully treated. The
conditionally averaged field associated with the positive wallpressure peaks indicates the inclined streamwise vortices lining
in the streamwise direction. The negative pressure peak is foundlocated right under the inclined streamwise vortex. Clustering
analyses highlight that the positive high amplitude wall pressure
peaks are related to multiple vortices in different positions. The
wall pressure peaks are active in the large-scale high-speed
region indicating a link between the high amplitude wallpressure peaks and the large-scale structures.

The present study aims to clarify the relation between thewall pressure fluctuations and the turbulent flow structures in the
turbulent channel flow. Our interest lies in the local source
contribution to the wall pressure fluctuation, which is evaluatedfrom the spatial distribution of the contribution to the wallpressure at a certain point based on the Poisson equation ofpressure. Appropriate statistical procedures will also be applied
to reveal the characteristics of the flow field associated with highamplitude wall pressure peaks.
CHANNEL FLOW DNS DATABASE AND
FORMULATION OF LOCAL WALL PRESSURE
CONTRIBUTION
The turbulent channel flow database for 𝑅𝑅𝑅𝑅𝜏𝜏 = 178 and 360
were established. To obtain the data of fully developed turbulent
channel flows, direct numerical simulations (DNS) were
performed using Incompact3D (Laizet and Lamballais, 2009).
The computational conditions are summarized in Table 1. The
computational parameters, i.e., the domain size and the grid
resolution, were determined as in existing studies. The present
results were validated against the existing DNS database (Del
Alamo and Jiménez, 2003), and it was confirmed that the
average velocity, the RMS value of velocity fluctuations, and thepower spectrum are in good agreement.
The pressure field is obtained by separately solving the
Poisson equation for pressure. Since the turbulent channel flow
is periodic in the streamwise and spanwise directions, it isreasonable to use the Fourier transformed form of the Poisson
equation as in Kim (1989),

INTRODUCTION
Wall pressure fluctuations are like footprints of near wall
turbulent flow structures. Johansson et al. (1987) showed that thepositive high amplitude wall pressure peak is associated with astrong shear layer near the wall, while the negative highamplitude wall pressure peak is related to sweeps. Ghaemi and
Scarano (2013) clarified the three-dimensional flow structure
linked to high amplitude wall pressure peaks by conditional
averaging. They showed that the positive pressure peak isrelevant to the shear layer induced by the upstream sweep andthe downstream ejection. It is also shown that the sweep motion
originates from the outer layer and the ejection is caused by thehairpin vortex structure. Ghaemi and Scarano (2013) alsomentioned that the negative pressure peak can be associated withthe streamwise and spanwise vortices. Mehrez et al. (2019)
visualized the turbulence structure associated with the wallpressure fluctuation using the second invariant of the velocity
gradient tensor, and showed that the peak of the fluctuatingpressure is linked to the turbulent structure in both inner andouter layers. Although the vortex structure related to thefluctuating wall pressure obtained by their conditional averaging
exhibits a hairpin-like shape, the location and size of thestructure are different from the typical hairpin vortex structure
model proposed by Adrian et al. (2000).

𝑝𝑝̂𝑤𝑤 (𝒌𝒌) = � 2𝑄𝑄� (𝑦𝑦 ′ , 𝒌𝒌)𝐺𝐺�𝑤𝑤 (𝑦𝑦 ′ , 𝒌𝒌) 𝑑𝑑𝑦𝑦 ′(1)𝛿𝛿𝑝𝑝𝒙𝒙 (𝒙𝒙𝒐𝒐 ) = 𝑄𝑄(𝒙𝒙𝒐𝒐 )𝐺𝐺� (𝒙𝒙, 𝒙𝒙𝒐𝒐 )(2)

where 𝑝𝑝̂𝑤𝑤 , 𝐺𝐺�𝑤𝑤 and 𝑄𝑄� are the wall pressure in the wave number
space with the wave number 𝒌𝒌, the Green’s function, and thesecond invariant of the velocity gradient tensor, respectively. 𝑦𝑦 ′stands for the wall normal coordinate. A pressure source
contribution at a location 𝒙𝒙𝒐𝒐 to the wall pressure at a point 𝒙𝒙 isrepresented by the delta function, and substituted it into Eq. (1)yields the local source contribution to the wall pressure as
follows

where 𝛿𝛿𝑝𝑝𝒙𝒙 (𝒙𝒙𝒐𝒐 ) expresses the contribution to the wall pressure at
𝒙𝒙 from the source at 𝑥𝑥𝑜𝑜 , and hereafter 𝛿𝛿𝑝𝑝𝒙𝒙 (𝒙𝒙𝒐𝒐 ) is called the local
pressure contribution. 𝑄𝑄(𝒙𝒙𝒐𝒐 ) indicates the second invariant of1

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

𝑅𝑅𝑒𝑒𝜏𝜏

Table 1. Computational parameters.

180
400

𝐿𝐿𝑥𝑥

𝐿𝐿𝑦𝑦

4𝜋𝜋

2

4𝜋𝜋

2

𝐿𝐿𝑧𝑧

𝑁𝑁𝑥𝑥

𝑁𝑁𝑦𝑦

𝑁𝑁𝑧𝑧

4𝜋𝜋/3

256

129

128

288

287

180

4𝜋𝜋/3

(a)

(a)

(b)

(b)

Figure 2. Contribution of rapid (line) and slow terms 𝑠𝑠𝑖𝑖𝑖𝑖 (a)for 𝑝𝑝max and (b) for 𝑝𝑝min. 𝑅𝑅𝑒𝑒𝜏𝜏 = 180.
the buffer layer. The contribution to 𝑝𝑝max shows a maximum at
𝑦𝑦 + = 5 and minimum at 𝑦𝑦 + = 25 . Here, superscript “+”indicate the wall unit. On the other hand, the contribution to 𝑝𝑝min
indicates a minimum at 𝑦𝑦 + = 5 and a maximum around 𝑦𝑦 + =15.
The pressure source can be decomposed to the rapid term
𝜕𝜕𝑦𝑦 𝑈𝑈𝜕𝜕𝑥𝑥 𝑣𝑣 and the slow terms 𝑠𝑠𝑖𝑖𝑖𝑖 = 𝜕𝜕𝑗𝑗 𝑢𝑢𝑖𝑖 𝜕𝜕𝑖𝑖 𝑢𝑢𝑗𝑗 . Figures 2(a) and2(b) show the contributions of these components on 𝑝𝑝max and𝑝𝑝min, respectively. The rapid term works positive to 𝑝𝑝max , whichmeans that 𝜕𝜕𝑥𝑥 𝑣𝑣 around 𝑝𝑝max is positive since 𝜕𝜕𝑦𝑦 𝑈𝑈 is positive.
For 𝑝𝑝min, the rapid term is negative, that means 𝜕𝜕𝑥𝑥 𝑣𝑣 negative. In
fact, Naka et al. (2015) reported that the positive wall pressure
is correlated with the sweep (𝑢𝑢 > 0 and 𝑣𝑣 < 0) at upstream; thenegative wall pressure with the downstream sweep. Among the

Figure 1. Wall normal profiles of averaged local
contribution to the wall pressure fluctuation (a) for 𝑝𝑝maxand (b) for 𝑝𝑝min.
the velocity gradient tensor at 𝒙𝒙𝒐𝒐 in the physical space, and𝐺𝐺� (𝒙𝒙, 𝒙𝒙𝒐𝒐 ) is the representation of the Green’s function in physicalspace. One can evaluate the local pressure contribution using
Eq. (2). It is noted that 𝐺𝐺� at 𝒙𝒙 ≈ 𝒙𝒙𝒐𝒐 is negative, thus positive 𝑄𝑄
usually gives negative contribution to 𝑝𝑝̂𝑤𝑤 and vice versa.

2slow terms, the diagonal components, �𝜕𝜕𝑗𝑗 𝑢𝑢𝑗𝑗 � , contribute
positive to the pressure by definition. Thus, they work positive
for 𝑝𝑝max and negative for 𝑝𝑝min . On the other hand, the offdiagonal components generally contribute in opposite. 𝑠𝑠23 is thelargest in magnitude while it turns to be positive in 𝑦𝑦 + < 3. A
noteworthy thing is that gradients in 𝑦𝑦 and 𝑧𝑧 have higher
magnitude than that in 𝑥𝑥. For example, the magnitudes of 𝑠𝑠22
and 𝑠𝑠33 are larger than that of 𝑠𝑠11 , and 𝑠𝑠23 is larger than 𝑠𝑠12 and𝑠𝑠31 . This suggest that the pressure peaks are associated with thestructures elongated in the streamwise direction. The other thing
to note is that all the terms exhibit peak within the buffer layer,
up to 𝑦𝑦 + ≈ 20.

AVERAGE SOURCE CONTRIBUTION AND
TURBULENCE STRUCTURES ASSOCIATED WITH
HIGH AMPLITUDE WALL PRESSURE PEAKS
The wall pressure maxima and minima, 𝑝𝑝max and 𝑝𝑝min ,having magnitudes greater than three times of the RMS value ofthe overall wall pressure fluctuation, 𝑝𝑝rms , are considered for thepresent analysis. Figure 1(a) and 1(b) show the wall normal
distribution of the local pressure contribution to 𝑝𝑝max and 𝑝𝑝min
averaged in the 𝑥𝑥-𝑧𝑧 plane, respectively. For the present Reynolds
number range, the local pressure contributions are significant in2

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

(a)

(b)

(b)

Figure 3. Conditionally averaged distribution around thehigh amplitude wall pressure peaks (a) for 𝑝𝑝max and (b) for𝑝𝑝min. Yellow: 𝛿𝛿𝛿𝛿 = rms(𝛿𝛿𝛿𝛿); cyan: 𝛿𝛿𝛿𝛿 = −rms(𝛿𝛿𝛿𝛿); red:𝑢𝑢 = 𝑢𝑢𝜏𝜏 . Bottom wall: 𝑝𝑝𝑤𝑤 (darker to brighter), and contour
at 𝑝𝑝𝑤𝑤 = 0 (white line). 𝑅𝑅𝑅𝑅𝜏𝜏 = 360.

Figure 4. Probability density functions of the relative
position from an intense pressure peak to the closest peak.
Area within white line contains 50% of data; (a) 𝑝𝑝min
around 𝑝𝑝max , (b) 𝑝𝑝max around 𝑝𝑝max . 𝑅𝑅𝑅𝑅𝜏𝜏 = 360.

Figures 3(a) and 3(b) show the conditionally averaged
distributions around 𝑝𝑝max and 𝑝𝑝min , respectively. To accountfor the asymmetry of the flow field in the spanwise direction, theorientation of the individual flow fields was aligned based on the
distribution of 𝑄𝑄 so that the spatial averaged value of 𝑄𝑄 for
subdomain in 𝑧𝑧′ < 0 has larger than that for the subdomain in𝑧𝑧′ > 0, where 𝑧𝑧′ is the spanwise coordinate after the alignment.
From the region where 𝛿𝛿𝛿𝛿 < 0 (approximately corresponding to
the region where 𝑄𝑄 > 0), the flow pattern associated with 𝑝𝑝max
is reminiscent of longitudinal vortex structures tilted toward theflow direction. These longitudinal vortices are located slightly
away from the pressure peak and surround the positive local
pressure contribution. In contrast, for 𝑝𝑝min, the negative pressure
contribution exists directly above the negative peak location.
The characteristic shape of the negative pressure contribution
seems to be decomposed to an elongated structure in thestreamwise direction and a rather rounded shape region near theorigin. For the elongated structure in the streamwise direction, it
is similar to the longitudinal vortex structure observed in the
distribution for 𝑝𝑝max . Ghaemi and Scarano (2013) pointed outthat a spanwise vortex structure is observed just above thenegative pressure peak, and that this corresponds to the head of
the hairpin vortex. In the present study, although the significant
negative contribution region of a round shape is found just above
the peak, no clear hairpin-like structure is observed. Thissuggests that different coherent structures can be extracted bydifferent averaging procedures.
It is found that 𝑝𝑝max and 𝑝𝑝min are respectively located at thefront and back of the sweep motion, which is indicated by the
red transparent iso-surface in Fig. 3. Furthermore, the mean wallpressure changes its sign in the streamwise direction with thespacing of Δ𝑥𝑥 + ≈ ±75 . This suggests that the positive andnegative peaks form a pair.
PAIRING CHARACTERISTICS OF POSITIVE AND
NEGATIVE WALL PRESSURE PEAKS
The conditionally averaged flow distributions presented inFig. 3 suggest that 𝑝𝑝max and 𝑝𝑝min are likely to be associatedwith streamwise vortices aligned in the streamwise direction.
The pairing characteristics of 𝑝𝑝min and 𝑝𝑝max are investigated.
For each pressure peak, the closest peak with the opposite sign3

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

Figure 5. Behaviour of gap statistics against the number ofclusters, 𝑛𝑛grp with different filter widths.
is extracted. Figure 4 shows probability density function (PDF)of the relative positions of such pairs. Two combinations of thepressure peaks are considered: 𝑝𝑝min around 𝑝𝑝max shown in Fig.
4(a), and 𝑝𝑝max around 𝑝𝑝max in Fig. 4(b). Figure 4(a) indicatesthat 𝑝𝑝max tend to have 𝑝𝑝min at 50+ in upstream. Although thisdistance is somewhat shorter than the distance measured fromthe conditional average, 75+ , shown in Fig. 3(a), they coincide
well when considering the difference of the instantaneous peaks
and the peaks in the averaged wall pressure. These results
confirm that the pressure peaks form a pair in the streamwise
direction in the instantaneous distribution.
The PDF of 𝑝𝑝max around 𝑝𝑝max , shown in Fig. 4(b), has apeak in the spanwise direction, although these peaks are lessclear compared to those of 𝑝𝑝min - 𝑝𝑝max pairs. The most probabledistance is 25+ -30+ , and therefore the pressure peaks can beassociated with streamwise elongated structure in the buffer
layer.
CLUSTERING ANALYSES OF SOURCE
DISTRIBUTION ASSOCIATED WITH POSITIVE WALL
PRESSURE PEAKSIn the previous section, the representative flow pattern is
extracted based on the conditional sampling taking into accountof asymmetry of the flow in the spanwise direction. Whendefining the representative flow structure associated with a flow
event, we are in a dilemma that averaging smears out
instantaneous features, and a subjectively chosen instantaneous
snapshot cannot represent the whole phenomena. To overcome
such a situation and to further investigate the representative
structures of the source distributions contributing to the pressure
peaks, the clustering analyses are introduced. The clustering
analysis is widely used to separate data into groups (Hastie et al.,2009) based on the pairwise distance. Here, the source
distributions around 𝑝𝑝max are served for the K-means clustering
analysis. When 𝑝𝑝max can be associated with some representative
distribution patterns, it is expected that they are grouped todifferent clusters.
When performing the K-means clustering, one should
specify the number of clusters, 𝑘𝑘, as an input parameter while itis not known a priori. The behaviour of the sum of the pairwise
distance of each cluster gives a hint to choose a good value of 𝑘𝑘,
and there is a common practice so-called “the elbow method”

Figure 6. Representative structures around 𝑝𝑝max identified
by the K-means clustering: the ensemble average of eachcluster. The red and blue isosurfaces indicate the intense
positive and negative local pressure contributions.

412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 7. Streamwise velocity averaged over 𝑦𝑦+ ≥ 100 (colormap) with 𝑝𝑝max (red circle) and 𝑝𝑝min (blue circle). Size of circle is
proportional to the magnitude of pressure peaks. 𝑅𝑅𝑒𝑒𝜏𝜏 = 400.
finding a point ending the rapid decrease of the sum of distance.
The gap statistic (Tibshirani et al., 2001) is a quantitative
measure for reasonable determination of the number of clusters.
The gap value is defined as the logarithmic ratio of the sum of
the pairwise distance of reference data over that of the served
data. The reference data are white noise having the uniform
distribution along with the data axes or along with the principal
component axes of the data. When the sum of the distance of theserved data is significantly smaller than the reference, it gives
the large gap value. The behaviour of the gap value for different
number of clusters facilitates a reasonable determination of 𝑘𝑘.
In the present study, instead of the white noise reference,
randomly sampled data in the flow field are used. Reference data
are taken in the same way as the data around 𝑝𝑝max but they aretaken at randomly sampled points. This can highlight themeaningful cluster of the data around 𝑝𝑝max against thebackground wall pressure fluctuations. The positive gap value
indicates the sum of the distance for 𝑝𝑝max is smaller than that of
the randomly sampled data, which means the data of 𝑝𝑝max are
meaningfully clustered.
The data are filtered in space before being served for theclustering so that the data can contain a certain size of thestructures with a smaller number of dimensions. Here, thenumber of dimensions is fixed to 8, i.e., 2 × 2 × 2 in space, andthe different filter widths are tested. In 𝑥𝑥 and 𝑧𝑧 directions, theboxes are defined in the negative and positive regions, and in 𝑦𝑦
direction, the box in the near wall region, 0 ≤ 𝑦𝑦 + ≤ 15, and thebox further from the wall, 𝑦𝑦 + > 15 , where the overall
contribution changes its sign as presented in Fig. 1, are used. The
number of samples is 13901. Figure 5 presents the behaviour ofthe gap values with different filter widths in 𝑥𝑥 and 𝑧𝑧 directions.
The gap values are higher for smaller filter widths. The threegroups are observed for small (𝑙𝑙+ ≤ 12), medium (25 ≤ 𝑙𝑙+ ≤100 ) and large ( 150 ≤ 𝑙𝑙 + ≤ 200 ) filter widths, where 𝑙𝑙 +denotes the filter width in the wall unit length. This suggests thatthe features detected with the filter width of 25+ - 100+ play a
specific role in clustering.
Figure 6 shows the ensemble average of the source
distribution among the samples belonging to the same cluster.
The clustering is performed with the filter width 𝑙𝑙 + = 100. All
clusters have positive source near the origin, which directly
contributes to 𝑝𝑝max . For the different clusters, the negative
contribution exists around the origin at different quadrants. The
level of the iso-surface is set to visualize the intense local source

contribution corresponding to 150% of the RMS value of source
within a box of dimension 200+ × 100+ × 200+ . Figure 6
indicates that 𝑝𝑝max is primarily associated with a strong inclined
streamwise vortex, and weaker vortices co-exist around theprimary one. The strong vortex (or vortices) induces the intense
shear layer structure at 𝑧𝑧 = 0 plane, which is often associatedwith positive wall pressure in the literature (Johansson et al.,
1989; Ghaemi and Scarano, 2013; Mehrez et al., 2019).
AMPLIFICATION OF WALL PRESSURE PEAKS WITHLARGER SCALE STRUCTURES
Although our results show that the pressure peak is decided
by the near-wall events beneath the buffer layer, Naka et al.
(2015) and Gibeau and Ghaemi (2021) observed the associationof the large-scale velocity structure. Figure 7 shows that thestreamwise velocity averaged in the wall normal direction in theslab of 𝑦𝑦 + ≥ 100, hereafter denoted as 𝑢𝑢slab , which is a proxyof the large-scale motion (LSM) in the outer layer. The intensepressure peaks overlayed in the figure with circles show
preferential amplification in the high momentum side, e.g., 𝑥𝑥 + ≈500, 𝑧𝑧 + ≈ 800. Figure 8(a) shows the PDF of the 𝑢𝑢slab above
the pressure peaks. The pressure peaks more intense than 3𝑝𝑝rms
tend to be located under the high momentum region, and thepreferential amplification is evident. Figure 8(b) shows the same
PDF but for the velocity in the near-wall slab of 0 ≤ 𝑦𝑦 + ≤ 30.
It gives the consistent result as the far-wall slab, suggesting thatthe associated LSM is attached to the wall. As in Figs. 7 and 8,
𝑝𝑝max and 𝑝𝑝min are observed more in the large-scale high-speed
region. It is noticeable that the intense peaks are located in thecenter part of the high-speed region as well as the interface
between the high-speed and the low-speed regions. It alsosuggests that these wall pressure peaks are not directly linked tothe hairpin packets which tend to exist around the low-speed
region.
Though the positive and negative high amplitude pressure
peaks are mostly contributed from the near wall region, i.e., up
to the buffer layer, it is known that the characteristics of the wallpressure fluctuation cannot be scaled only with the near wallquantity (Tsuji et al., 2007). The contribution of large-scale
structure attached to the wall is evident. Such structures extend
further from the wall and they consist of the large-scale and verylarge-scale motions. The large-scale motion itself can create
large scale wall pressure but not small-scale high amplitude

512th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
highlight the representative turbulent flow structures having asingle vortex or multiple vortices around positive wall pressure
peaks. The pressure peaks tend to exist more under the largescale high-speed region. It suggests that the high amplitude wallpressure peaks are indirectly enhanced in the large-scale highspeed region.

(a)

ACKNOWLEDGEMENT
This work was supported by JSPS KAKENHI Grant
Number 19K04176.
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(b)Figure 8. PDFs of the streamwise velocity conditioned on
the magnitude of the pressure peaks. The velocity data
above the pressure peak locations are sampled; (a) 𝑦𝑦+ ≥ 100
and (b) 0 ≤ 𝑦𝑦+ ≤ 30. 𝑅𝑅𝑒𝑒𝜏𝜏 = 400.
peaks. Even though the high amplitude pressure peaks are notdirectly contributed by the structures far from the wall, they canbe correlated since the large-scale structure is wall attached andit extends further from the wall. This can be confirmed by theconditionally averaged flow pattern. The conditionally averaged
velocity distribution shows meaningful sweep motion in theregion higher than the channel half height.
CONCLUSION
The near wall turbulent flow structures related to the highamplitude wall pressure peaks are clarified using DNS data of aturbulent channel flow. The local pressure contribution isderived from the source term of the Poisson equation forpressure using the delta function. It is shown that most of thecontribution to the high amplitude wall pressure peak is found inand below the buffer layer. The conditionally averaged flow
fields around the high amplitude wall pressure peaks areevaluated. It is found that the extracted turbulent flow structurearound the positive wall pressure peak is a longitudinal vortex
structure surrounding the pressure peak position, and that of thenegative wall pressure peak is a longitudinal vortex structure
directly above the peak position. The positive and negative
pressure peaks prone to conform pairs. The clustering analyses

6