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

ACOUSTIC LINERS AND THEIR INDUCED DRAG

Haris Shahzad
Faculty of Aerospace Engineering
Delft University of Technology
Kluyverweg 2, 2629 HS Delft,
The Netherlands
h.shahzad@tudelft.nl

Stefan Hickel
Faculty of Aerospace Engineering
Delft University of Technology
Kluyverweg 2, 2629 HS Delft,
The Netherlands
s.hickel@tudelft.nl

Davide Modesti
Faculty of Aerospace Engineering
Delft University of Technology
Kluyverweg 2, 2629 HS Delft,
The Netherlands
d.modesti@tudelft.nl

ABSTRACT

impermeability. Permeable surfaces have been studied far
less than canonical roughness geometries, and it is not clear
whether several aspects of rough wall turbulent flow also apply
to permeable surfaces. Manes et al. (2009) and Breugem et al.
(2006) studied the differences between permeable surfaces andcanonical rough surfaces and showed how permeable surfaces
might have a much more profound effect on the flow. Manes
et al. (2009) further noted that the drag increase over a permeable surface is higher than over a rough surface with thesimilar geometry. Breugem et al. (2006) and Kuwata & Suga(2017) have also noted a breakdown of outer layer similarity
over permeable surfaces.
Furthermore, the interaction of a turbulent flow with apermeable surface can change dramatically depending upon
the geometry, and unlike canonical (i.e homogeneous) permeable surfaces, acoustic liners have not been studied extensively.
Experiments and numerical simulations of the flow over realistic liners are challenging. The diameter of the orifices
d is significant with respect to the boundary layer thickness(d/δ ≈ 0.1) and much larger than the viscous length scale(d + = d/δv ≈ 200), where δ is the boundary layer thickness andp δv = νw /uτ is the viscous length scale based onuτ = τw /ρw where νw , ρw and τw are the wall kinematic
viscosity, wall density and the drag per plane area, respectively. Simultaneously satisfying these constraints on the diameter implies high computational cost. Therefore, previous
numerical studies have avoided resolving the entire geometry.
For instance, Scalo et al. (2015) performed Large Eddy Simulation of turbulent channel flow with an impedance boundary
condition modelling the liner. They performed simulations ata bulk Reynolds number, Reb = 6900, and bulk Mach number,
Mb = 0.02–0.5, and observed a drag increase of up to 350%.
Tam et al. (2014) opted for 2D Reynolds Averaged Navier
Stokes (RANS) simulations of an array of acoustic liners andestimated a drag increase of about 4% in the presence of 140dB
acoustic waves. Zhang & Bodony (2016) aimed at replicating the geometry studied at the Grazing Flow Impedance Tube
(GFIT) facility at NASA (Jones et al., 2004) and carried out
Direct Numerical Simulation (DNS) of a turbulent boundary

In order to reduce the noise emitted by aircraft engines,
the nacelle is coated with acoustic liners. An undesirable effectof these surfaces is that they increase the aerodynamic drag. In
the present work, we characterize this type of surface roughness by performing Direct Numerical Simulations of fully resolved acoustic liner geometries. We find evidence of a fullyrough regime, whose onset is determined by the value of theviscous-scaled Forchheimer coefficient. Moreover, the intensity of the wall-normal velocity fluctuations at the wall also
scales with the viscous-scaled wall-normal permeability, leading to a relation between fluctuations and added drag.

INTRODUCTION
Aircraft engines are the primary source of noise during
take-off and landing. In order to meet the noise regulations,the nacelle of modern engines is coated with acoustic liners, which represent the state-of-the-art technology for engine
noise abatement. Acoustic liners are panels with a sandwich
structure, consisting of a honeycomb core bounded by a perforated facesheet and a backplate. They typically cover the inner
nacelle surface in front of the fan and in the by-pass flow. The
working principle of acoustic liners is based on the idea of theHelmholtz resonator, namely a cavity and a throat which dissipates the energy of incoming acoustic waves. An undesirable
side effect of these surfaces is that they increase the total aircraft drag, essentially behaving as distributed surface roughness. The acoustic absorption of acoustic liners has been studied extensively as far back as the work of Sivian (1935) andIngård & Labate (1950), while added drag has been accepted
as inevitable. However, moving towards cleaner aviation requires a more in-depth understanding of the drag increase overthese surfaces.
Acoustic liners can have a significant effect on the turbulent mean flow as the liner facesheet can not be consideredhydraulically smooth. Furthermore, unlike canonical roughness, acoustic liners present a permeable surface to the incoming turbulent flow resulting in a relaxation of the wall-1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022shows the top view of a single cavity and the distribution of theorifices. We compare the results of the liner simulations withsmooth-wall simulations at approximately matching friction
Reynolds number. Quantities that are non-dimensionalised by
δv and uτ are denoted by the ‘+’ superscript.

layer over an isolated cavity. Zhang & Bodony (2016) performed DNS at Reθ = 2300 and Mach number, M = 0.5, noting a drag increase of 4.2% without acoustic waves and 25%
with 140dB acoustic waves.
The added drag has been extensively documented using
experiments at the GFIT (Howerton & Jones, 2015, 2016,2017) resulting in a large database for acoustic liners geometries and the drag they induce. These studies have helped identify geometrical parameters that can be fine-tuned to reduce
acoustic liner drag. The experiments note that the additionaldrag can be as low as 10% or as high as 350%, depending
upon the geometry considered. They further note that small
modifications in the geometry can be made without altering
acoustic attenuation. Therefore, there is significant room to
aerodynamically optimise acoustic liners, which has largely
been neglected until now, without compromising on acoustic
performance.
Previous studies have attempted to characterise acoustic
liner drag. However, numerical studies applied significantly
simplifying models, such as the use of modelled boundary conditions and isolated cavities, and experiments are affected bysignificant uncertainties in the drag measurement and can notprovide detailed information about turbulent flow structures.
The discrepancies between previous studies is therefore verylarge and it remains unknown whether acoustic liners behave
as canonical roughness or porous surfaces. In this work, we
aim to accurately quantify the drag variation induced by acoustic liners in the absence of incoming acoustic waves. We tackle
the problem by performing DNS of turbulent channel flow overfully resolved acoustic liner geometries.

RESULTS
Mean Velocity Profile and Drag Increase
Figure 2 shows the mean velocity profiles for all flow
cases, where e· is the Favre averaging operator. Liner cases
show a downward shift of the viscous-scaled mean velocity
profile with respect to the smooth-wall, ∆U + in the logarithmic region, which is a clear symptom of drag increase. The
flow case with low porosity, σ = 0.0357 and d + = 40 (circles), shows a smooth-wall-like behaviour with minor changes
in the velocity profile. However, differences from the smoothwall velocity profile become evident as either σ or d + is increased. The effect of the liner is restricted to the near-wallregion, and velocity profiles are parallel to those obtained witha smooth wall in the logarithmic and in the outer layer, supporting Townsend’s outer layer similarity hypothesis.
A fundamental question is whether acoustic liners behave as canonical roughness, that is, whether the Hama roughness function can characterise the drag increase, ∆U + =κ −1 log(ℓ+ ) + B(ℓ+ ), where κ = 0.4 is the von Kármán constant, and ℓ is a suitable roughness length scale. In canonicalk-type roughness, ℓ is simply the roughness height. However,for acoustic liners, different choices are possible. We consider
first the orifice diameter. Figure 2(a) shows ∆U + evaluated aty+ ≈ 100 as a function of d + . The figure shows that d + is nota suitable length scale because ∆U + increases for a constantd + as the porosity increases. The drag increase is, therefore,
not only a function of the viscous-scaled orifice diameter, motivating the use of the permeability as a possible parameter
for acoustic liners. The permeability represents the ease withwhich fluid passes through a porous surface and incorporates
the effects of changes to the diameter and the porosity into asingle variable. We consider the square root of the viscous√ +scaled wall-normal Darcy permeability K and the inverseof the wall-normal Forchheimer coefficient, 1/α + , also referred to as non-linear permeability. The Darcy coefficient represents the pressure drop through a permeable surface within
the limit of Stokes flow, and the Forchheimer coefficient is relevant if the inertial effects are more significant. Figure 3 shows∆U + as a function of the square root of the wall-normal permeability and the Forchheimer coefficient. The former does notlead to a monotonic curve and is, therefore, also not a suitableparameter. Instead, we find that the inverse of the wall-normal
Forchheimer coefficient, 1/α, is more appropriate for characterising the additional drag. We see that by increasing 1/α + ,∆U + tends towards κ −1 log(1/α + ) − 3.5, suggesting the existence of a fully rough regime. Such a trend suggests thatthe ease with which the momentum transfer occurs between
the two regions of the flow, above and below the facesheet, isdominated by inertial effects.
Further evidence that the inverse of the Forchheimer coefficient is the defining length scale for acoustic liner behaviouris that essentially no change in ∆U + is observed if the spacingof the holes is modified. The Darcy and Forchheimer coefficients are geometrical parameters of a particular geometry anddepend primarily on the porosity σ and the thickness to diameter ratio t/d (Bae & Kim, 2016). Therefore, changing the
distribution of the holes does not change the permeability coefficients, as observed by Bae & Kim (2016). Case Lu4 has

METHODOLOGY
We perform DNS of turbulent channel flow with constantbulk velocity using the solver STREAmS (Bernardini et al.,2021). The channel is a rectangular box of size Lx × Ly × Lz =3δ × 2δ × 1.5δ , where δ is the channel half-width. We use
a uniform mesh spacing in the streamwise and spanwise directions. In the wall-normal direction, the mesh is clustered
towards the wall and coarsened towards the backplate and thechannel centre. The simulations are performed at bulk Mach
number, Mb = ub /cw = 0.3, where ub is the bulk flow velocity
and cw is the speed of sound at the wall. The upper and lower
channel walls are replaced by acoustic liners using a ghostpoint immersed boundary method (Vanna et al., 2020).
We choose the liner geometry to match as close as possible the realistic parameters of acoustic liners in operating
conditions. Our cavity geometry has a square cross-section
with a side length λ = 0.335δ , depth, k = 0.5δ , and the orifices have a diameter d = 0.08δ , where δ is the channel halfwidth. The computational domain comprises a total of 64 cavities: an array of 8 × 4 in the streamwise and spanwise direction, respectively, covering both upper and lower walls of
the channel. We vary the liner porosity (i.e. open area ratio), between σ = 0.03–0.32 by varying the number of orifices
per cavity between 1 and 9, while keeping fixed the friction
Reynolds number Reτ = δ /δv = 500. Additionally, we carryout simulations at fixed porosity σ = 0.32, and increase thefriction Reynolds number to Reτ = 1000 and 2000. This corresponds to a viscous-scaled orifice diameter ranging between
d + = 40 − 160. Details of the flow cases are shown in Table
1. The orifice configurations within a cavity, along with an instantaneous flow visualisation of the flow field for σ = 0.32
and Reτ = 2000, are shown in Figure 1, where vortical structures are visualised using the Q-Criterion. The figure also212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
√ +K
1/α + ∆U + ∆x+ ∆y+
∆y+
Reb
Reτ
d+
σ
max
min

∆z+

S1

9268

506.1

0

0

0

0

-

5.1

0.80

3.83

5.1

S2

21180

1048

0

0

0

0

-

5.2

0.80

4.45

5.2

S3

45240

2060

0

0

0

0

-

5.2

0.80

6.67

5.2

L1

9139

503.5

40.3

0.03571.04

0.0528

0.14

1.1

0.80

5.81

1.1

L2

8794

496.4

39.7

0.142

2.06

0.859

0.56

1.0

0.80

5.81

1.0

L3

8264

505.3

40.4

0.322

3.22

5.14

1.90

1.0

0.81

5.81

1.0

L4

19505

1038

83.0

0.142

4.30

1.718

0.96

2.1

0.83

6.30

2.1

Lu4

19505

1044

83.5

0.142

4.32

1.727

0.98

5.9

0.84

6.10

5.9

L5

17810

1026

82.1

0.322

6.53

10.4

2.78

2.1

0.82

6.29

2.1

L6

35470

2044

164.0

0.322

13.0

20.8

4.44

4.1

0.82

6.70

4.1

Table 1. DNS dataset comprising smooth, (Sn ) and liner (Ln ) cases. σ is the porosity (open area ratio), d + is the orifice diameter,
Ky is the Darcy permeability, α is the Forchheimer coefficient, and ∆U + is the Hama roughness function. Simulations are performedin computational a box with dimensions Lx × Ly × Lz = 3δ × 3δ × 1.5δ . ∆x+ and ∆z+ are the viscous-scaled mesh spacing in the+streamwise and spanwise direction. ∆y+min and ∆ymax are the minimum and the maximum mesh spacing in the wall-normal direction.
Liner cases (L1 )–(L6 ) have equispaced orifices in the streamwise and spanwise direction. Case Lu4 has the same porosity and orificesize of L4 , but the holes are not equispaced in the streamwise direction, see bottom right configuration of Figure 1.

the same porosity and diameter as case L4 but the orifices are
not equispaced in the streamwise direction. However, approximately the same ∆U + is observed. Acoustic liners, therefore,
act as a permeable substrate.

permeable walls (Kuwata & Suga, 2019). The liner induces
a similar effect on the turbulent flow. As shown in Figure
5 (a), which shows the velocity fluctuations with respect to
the Reynolds average, streaks become shorter over the liner.
Even in the liner’s presence, the streaky structures can still be
discerned, suggesting a modulation rather than a complete replacement of the near-wall cycle.

Reynolds Stresses
The effect of the liner can also be seen in the Reynolds′′ u′′ , shown in Figure 4 for liner cases L
Stresses, τi j = ρ ug
5i j
and L6 , compared to their respective smooth wall cases, where
the double prime symbol indicates fluctuations with respect to
the Favre average and · is the Reynolds averaging operator. Significant changes in velocity fluctuations are seen only
close to the wall, where non-zero turbulence intensities are observed. An increase in the maximum τ33 is observed in thepresence of the liner, whereas the maximum τ11 tends to decrease compared to the smooth wall. The maxima of the velocity fluctuations move slightly closer to the wall, irrespectiveof the component. Together with the non-zero intensity at thewall, enhance momentum transfer near the wall. The total dragincreases because of the higher Reynolds stresses and the adverse pressure gradient experienced by the flow at the downstream lip of each orifice. Much smaller differences can beseen in the outer layer. These differences are more pronounced
for flow case L6 with 1/α + ≈ 20.8 and not as noticeable forflow case L5 with 1/α + ≈ 10.4 suggesting a departure from
Townsend’s outer layer similarity hypothesis as the inverse of
the Forchheimer coefficient is increased.
The trend of the velocity fluctuations is in line with previous research on rough (Endrikat et al., 2021) and permeable
walls (Kuwata & Suga, 2016) on turbulent flow. For permeable
walls, the wall-normal velocity fluctuations at the wall increase
because of a relaxation of the wall blockage effect (Kuwata& Suga, 2019). The reduction in the peak of the streamwise
Reynolds Stress results from these high-wall normal velocity
fluctuations perturbing the classical near-wall turbulence cycle. High-speed and low-speed streaks, typical of near-wall
turbulence, are perturbed by the significant wall-normal velocity fluctuations at the wall and, thus, might break down overAcoustic Liners as Porous Surfaces
Although the trend of the Reynolds stresses is similar totrends observed for permeable surfaces, acoustic liners are different from most canonical porous surfaces. Unlike canonical porous surfaces, the cavity walls prevent a net flow underneath the surface of the facesheet as there is no streamwise
and spanwise permeability for acoustic liners. This can leadto significantly altered flow physics. For instance, Kuwata& Suga (2017) performed DNS over permeable surfaces withanisotropic permeability and noted spanwise invariant Kelvin–
Hemholtz-like rollers as the streamwise and spanwise permeability is relaxed. No such rollers are observed for the current
simulations, possibly because of the geometrical differences
between acoustic liners and canonical porous surfaces.
Furthermore, we show supporting evidence that theForchheimer coefficient is the relevant length scale for the flowin contrast to previous studies of turbulence over canonicalporous surfaces (Gómez-de Segura & Garcı́a-Mayoral, 2020;
Rosti et al., 2015) that highlight the importance of the Darcy
coefficient. Canonical porous surfaces may exhibit Darcian
velocities inside their pores. However, as acoustic liners arelarge with respect to the viscous length scale, inertial effects
are significant inside the orifices of an acoustic liner. Based on
our DNS results, we believe that the Forchheimer drag might
become the dominant drag on canonical porous surfaces for
sufficiently high Reynolds numbers.
Figure 5 (b) shows the wall-normal velocity in a wallnormal plane, where we observe that the effect of the liner on
the flow is concentrated near the wall and inside the orifices.
Inside the orifices, high wall-normal velocity fluctuations are312th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Howerton, B. M. & Jones, M. G. 2016 Acoustic Liner Drag:
Measurements on Novel Facesheet Perforate Geometries. In
AIAA paper 2016-2979.
Howerton, B. M. & Jones, M. G. 2017 A Conventional Liner
Acoustic/Drag Interaction Benchmark Database. In AIAA
paper 2017-4190.
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22 (2), 211–218.
Jones, M., Watson, W. R., Parrott, T. & Smith, C. 2004 Design
and Evaluation of Modifications to the NASA Langley Flow
Impedance Tube. In AIAA paper 2004-2837.
Kuwata, Y. & Suga, K. 2016 Lattice Boltzmann direct numerical simulation of interface turbulence over porous and roughwalls. Int. J. Heat Fluid Flow 61, 145–157.
Kuwata, Y. & Suga, K. 2017 Direct numerical simulation of
turbulence over anisotropic porous media. J. Fluid Mech.
831, 41–71.
Kuwata, Y. & Suga, K. 2019 Extensive investigation of theinfluence of wall permeability on turbulence. Int. J. Heat
Fluid Flow 80, 108465.
Manes, C., Pokrajac, D., McEwan, I. & Nikora, V. 2009 Turbulence structure of open channel flows over permeable
and impermeable beds: A comparative study. Phys. Fluids
21 (12), 125109.
Orlandi, P. & Leonardi, S. 2006 DNS of turbulent channel
flows with two- and three-dimensional roughness. J. Turbul. 7, N73.
Rosti, Marco E., Cortelezzi, Luca & Quadrio, Maurizio 2015
Direct numerical simulation of turbulent channel flow overporous walls. J. Fluid Mech. 784, 396–442.
Scalo, C., Bodart, J. & Lele, S. K. 2015 Compressible turbulent channel flow with impedance boundary conditions.
Phys. Fluids 27 (3), 035107.
Gómez-de Segura, Garazi & Garcı́a-Mayoral, Ricardo 2020
Imposing virtual origins on the velocity components in direct numerical simulations. Int. J. Heat Fluid Flow 86,108675.
Sivian, L. J. 1935 Acoustic Impedance of Small Orifices. J.
Acoust. Soc 7 (2), 94–101.
Tam, C. K. W, Pastouchenko, N. N., Jones, M. G. & Watson,
W. R. 2014 Experimental validation of numerical simulations for an acoustic liner in grazing flow: Self-noise andadded drag. J. Sound Vib. 333 (13), 2831–2854.
Vanna, F. D., Picano, F. & Benini, E. 2020 A sharp-interface
immersed boundary method for moving objects in compressible viscous flows. Comput. Fluids 201, 104415.
Wilkinson, S. 1983 Influence of wall permeability on turbulent boundary-layer properties. In AIAA paper 1983-294.
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layers. J. Fluid Mech. 792, 936–980.

visible, and they are notably higher at the downstream edge.
A jet-like flow is observed inside the cavities due to the highwall-normal velocity fluctuations, indicating important inertial
effects inside the orifices.
Figure 6 (a) shows the intrinsic average of the wallnormal velocity fluctuations for flow case L6 with 1/α + ≈20.8. High wall-normal velocity fluctuations are observed andare responsible for momentum exchange between the outerflow and flow inside the liner, enhancing mixing and increasing drag. This is in line with prior work, where high values of
the wall-normal velocity fluctuations have been often related
to drag increase, both over roughness (Orlandi & Leonardi,
2006) and permeable surfaces (Wilkinson, 1983). Wilkinson(1983) focuses on acoustic liner geometries and proposes thisblowing and suction effect as one of the possible mechanismsof the drag increase.
For this reason, we report the mean wall-normal velocity
fluctuations at the wall as a function of permeability coefficients. Very similar to the trend observed for ∆U + , the maximum average wall-normal
velocity does not show a monotonic√
trend with the K + . In comparison, an almost linear trend isobserved with 1/α + for the wall-normal velocity - suggestinga correlation of the ∆U + with the wall-normal velocity fluctuations similar to the findings of Orlandi & Leonardi (2006)
and Wilkinson (1983).

CONCLUSION AND FUTURE WORK
We performed unprecedented DNS of turbulent flow overfully resolved acoustic liner geometries to study their effect on
turbulence and added drag. The parametric study comprises
liner porosities in the range σ = 0.0357 − 0.322 and viscousscaled orifice diameters in the range d + = 40 − 160. Our results show that the inverse of the viscous-scaled Forchheimer
coefficient is the relevant length scale for the added drag ofthese surfaces, suggesting that acoustic liners can be regarded
as a permeable substrate, within which inertial effects are significant. Our DNS data further supports the existence of a fullyrough regime, although simulations at higher 1/α + would be
required to fully confirm this trend.

ACKNOWLEDGEMENTS
We acknowledge PRACE for awarding us access to Piz
Daint, at the Swiss National Supercomputing Centre (CSCS),
Switzerland.

REFERENCES
Bae, Y. & Kim, Y. I. 2016 Numerical modeling of anisotropic
drag for a perforated plate with cylindrical holes. Chem.
Eng. Sci. 149, 78–87.
Bernardini, Matteo, Modesti, Davide, Salvadore, Francesco &Pirozzoli, Sergio 2021 STREAmS: a high-fidelity accelerated solver for direct numerical simulation of compressible
turbulent flows. Comput. Phys. Commun. 263, 107906.
Breugem, W. P., Boersma, B. J. & Uittenbogaard, R. E. 2006
The influence of wall permeability on turbulent channel
flow. J. Fluid Mech. 562, 35–72.
Endrikat, S., Modesti, D., Garcı́a-Mayoral, R., Hutchins, N. &Chung, D. 2021 Influence of riblet shapes on the occurrenceof Kelvin–Helmholtz rollers. J. Fluid Mech. 913.
Howerton, B. M. & Jones, M. G. 2015 Acoustic Liner Drag: A
Parametric Study of Conventional Configurations. In AIAA
paper 2015-2230.

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

on
ti
rec
Di
w
o
Fl

Figure 1. Instantaneous flow field from DNS of turbulent channel flow at Reτ = 2000 and bulk Mach number Mb = 0.3. Vorticalstructures are visualised using the Q-Criterion. The orifice configurations for the three porosities, σ , studied are shown on the right.

(a)

(b)

30

5
4

ũ+

∆U +

20

10

3
2
1

0 0
10

101

102
y

0 1
10

103

102
+

+

d

Figure 2. Mean streamwise velocity as a function of the wall-normal distance (a) and ∆U + as a function of the viscous-scaled orifice
diameter (b). Symbols indicate different porosities: σ = 0.0357 (circles), σ = 0.143 (squares) and σ = 0.322 (triangles). The dashedline in (a) indicates smooth wall streamwise velocity profiles.

(a)

(b)

10

4

8

3

6

∆U +

5

2
1
0 0
10

4
2

√ +
K

0
10−2

101

10−1

100

101

102

ks+ ≈ 1/α+Figure 3. ∆U + as a function of the square root of the Darcy Coefficient (a) and the inverse of the Forchheimer coefficient (b). The
dotted line in (b) indicates ∆U + = κ −1 log(1/α + ) − 3.5. Filled circles in (b) represent Nikuradse’s data.

5

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

(b)

10

τij /τw

8
6
4
2
0 0
10

101

102

103

100

101

102

103

y+

y+

Figure 4. Reynolds stresses as a function of the viscous-scaled wall-normal distance for flow case L5 with 1/α + = 10.4 (a) and flowcase L6 with 1/α + = 20.8 (b). Lines indicate the smooth-wall cases and triangles indicate the liner cases. Solid lines indicate τ11 /τw ,dashed lines indicate τ22 /τw and dashed-dotted lines indicate τ33 /τw .

u′ /uτ

(a)

1.5

(b)

5

z/δ

4

v ′ /uτ

1.5

3
2
0.0

1
y/δ

−1

1.5

0.0

−2
z/δ

−3
−4
0.0
0.0

1.5
x/δ

−1.5
0.0

−5

3.0

1.5
x/δ

3.0

2.0
1.6
1.2
0.8
0.4
−0.4
−0.8
−1.2
−1.6
−2.0

Figure 5. Instantaneous streamwise velocity fluctuations in a wall-parallel plane at y+ = 12 (a) for flow case L3 (top) and flow case
L6 (bottom) and instantaneous wall-normal velocity for flow case L6 (b). In (a), the position of the orifices is shown at the bottom left
corner, for one cavity.

(a)

(b)

0.10
′′ v ′′ imax(σ 2 hρvg
y/δ<0 /τw )

1.5

0.08

′′ v ′′ i/τ
hρvg
w

1.0

0.06
0.04

0.5

0.02

0.0
−0.2

0.00

−0.1

0.0
y/δ

0.1

0.2

0

5

10
15
√ +
K , 1/α+20

25

Figure 6. Intrinsic average of the wall-normal velocity fluctuations as a function of the wall distance (a) for flow case L6 and maximum√ +of the wall-normal velocity fluctuations below the wall (b) as a function of 1/α + (solid line with filled symbols) and K (dotted linewith empty symbols). The dashed line in (a) indicates the smooth-wall case.

6