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

ON THE ROLE OF THE KELVIN-HELMHOLTZ WAVE OF TURBULENTFLOWS OVER PERMEABLE POROUS WALL

Yusuke Kuwata
Department of Mechanical Engineering,Osaka Prefecture University
Sakai, Osaka 599-8531, Japan.
kuwata@me.osakafu-u.ac.jp

ABSTRACT
Direct numerical simulations (DNSs) of porous-walled
channel flows with computational domain size constraints
were performed to clarify the role of large-scale turbulent motions associated with the Kelvin–Helmholtz (K–H) instability
on turbulent flows over porous walls. The constraints of thedomain size was imposed to control the large-scale motions,enabling us to discuss the isolated roles of the near-wall turbulent motions and large-scale motion. In addition, we considered thick and thin porous walls to isolate the effects of roughness and permeability. It is found that near-wall turbulent motions, including streaks and longitudinal vortices, are predominantly altered by the surface roughness rather than the wall
permeability, whereas the wall permeability is essential for thedevelopment of the large-scale motions. The large-scale motion, which is organized into spanwise rollers, contributes toan increase in turbulent friction drag, and induces the largescale streamwise perturbation near the thick porous wall. Discussions in the logarithmic mean velocity profile based on theJackson model confirms that the large-scale motion increases
the von Kármán constant, which is considered to reflect an increase in the characteristic vortex size due to the large-scale
motions.

Figure 1.
flow.

Flow configuration of a porous-walled channel

associated with the K–H instability.

Flow conditions
The configuration of a porous-walled channel flow is presented in Figure 1, where H and h are the channel and porous
wall heights, respectively. We considered a solid wall forthe top and bottom walls at y = H and y = −h, respectively,
whereas a porous wall was considered in lower part of thechannel −h < y < 0. The flow was periodic in the streamwise (x) and spanwise (z) directions, and the flow was driven
by a mean streamwise pressure gradient. The highly isotropic
permeable porous media of the porosity φ = 0.95 was modeled as arrays of Kelvin cells (Thomson, 1887). The size ofthe single unit of the Kelvin cell was set to L = H/3, and theresulting permeability was K/H 2 = 6.4 × 10−4 . In this study,
to discuss isolated effects of surface roughness and wall permeability, we considered two porous walls: one was the casewith sufficiently thick porous wall of h = H, and the other was
the case with thin porous wall of h = H/3.
As for the computational domain size, the computational
domain of Lx (x) × (H + h)(y) × Lz (z) with Lx = 6H and Lz =3H was considered for the baseline case as in Kuwata & Suga
(2016b, 2017). In addition to the baseline case, we performed
simulations with the streamwise-reduced domain of Lx = H
and the spanwise-reduced domain of Lz = H, both of which
was designed such that the domain was large enough to sustain near-wall turbulent vortices but was too small to prop-

Introduction
A turbulent flow around a porous wall is of special interest in engineering because of considerable heat and masstransfer performance due to larger specific surface area. As
the surface roughness and wall permeability of porous media have a large impact on turbulent flows over porous media,
particular attention is paid to control turbulent boundary layer
by modification of porous walls characteristics. It is well established that porous media usually increases turbulence andmomentum exchanges across the porous/fluid interface (Sugaet al., 2010), whereas the drag reduction is sometimes achievedby streamwise-preferential porous walls (Gómez-de Segura &Garcı́a-Mayoral, 2019). The distinctive features of the porous
wall turbulence is the presence of the large-scale motions associated with the K–H instability (Kuwata & Suga, 2016b,
2017). However, the turbulent flows over porous walls consists of near-wall streaks and longitudinal vortices with a superimposition of large-scale motions, and the coexistence offine-scale and large-scale motions makes it difficult to identify
the role of the large-scale motion. The goal of this study is toclarify how roughness and permeability alter turbulent flows
over porous walls and identify the role of large-scale motion

1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
erly reproduce the streamwise perturbations modulated by the
spanwise rollers. As the streamwise length of Lx = H isshorter than the wavelength of the streamwise perturbations,the streamwise-reduced domain does not accommodate the
spanwise rollers. In contrast, the spanwise rollers are enhancedby the spanwise-reduced domain because the coherence of the
spanwise rollers is reinforced by the spanwise periodical conditions with insufficient spanwise separation. We considered
six simulation cases with different computational domain sizes
and porous wall heights, as shown in Figure 2. In the following, for the sake of brevity, cases were named according to
AxBz-C, where A = Lx /H is the streamwise domain length,
B = Lz /H is the spanwise domain length, and C is either “thin”(h = H/3) or “thick” (h = H), as shown in Fig.2.
The numerical method for the present DNS study is thelattice Boltzmann method (LBM). Owing to the high efficiency of massive parallel computing and easiness of handling complex geometry, the LBM is highly suited to DNS of
turbulent flows over resolved porous walls (Kuwata & Suga,
2016b, 2017; Kuwata et al., 2020). There are several possible
choices for the LBM models, the D3Q27 multiple-relaxationtime LBM (Suga et al., 2015) was used, which has been extensively validated against canonical flows and successfully applied to turbulent flows over porous and rough walls (Kuwata& Suga, 2016b, 2017; Kuwata et al., 2020; Kuwata & Nagura,2020). As the LBM is discretized with uniform spacing grid,exceedingly fine grid that satisfies the near-wall resolution requirement should be used throughout the computational domain. This requires a large investment of computer resources
for the DNSs. To circumvent the the computational demands,
we applied the local grid refinement proposed by Kuwata &Suga (2016a). The grid resolution near the porous wall was
determined such that the grid spacing in wall units are less than
2.0 wall units as in the previous lattice Boltzmann DNS studies(Kuwata & Suga, 2016b, 2017; Kuwata et al., 2020; Kuwata& Nagura, 2020), and the resulting total number of grid points
for the baseline case of 6x3z-thick is approximately 800 millions. In this resolution, the single unit of the Kelvin cell was
resolved by 101 grid points along each direction. The grid independence test confirmed that further refinement of the grid
resolution by a factor of 1.5 yielded a change in the skin friction coefficient by 0.4%.

Figure 2.

Computational domain for all simulation cases.

porous wall seems to be different from that over the thin porous
wall. For case 6x3z-thick, the turbulent structure is characterized by the relatively larger high- and low-speed regions whichhave a connection in the spanwise direction rather than in thestreamwise direction. The first look of the snapshots of thevelocity fluctuations confirms that the spanwise rollers associated with the K-H instability develops for the baseline case of
6x3z-thick and spanwise-reduced domain case of 6x1z-thick,and the coherence of the spanwise rollers is more significant
for the case 6x1z-thick.
The presence of the large-scale motion can be quantitatively detectable from the one-dimensional streamwise premultiplied energy spectra of streamwise velocity fluctuations
at y p+ ≃ 10 as shown in Figure 4. It is evident from Fig.4 (a,b)
that the energy spectra for cases 6x3z-thick and 6x1z-thick exhibit a pronounced peak around λx /δ p = 3 − 5. Here, δ p denotes the boundary layer thickness over the porous wall, which
is defined as the distance from y = 0 to the position at thezero-total shear stress. Since the wavelength of λx /δ p = 3 − 5roughly corresponds to the characteristic wavelength of the K–
H instability (Kuwata & Suga, 2017; Suga et al., 2018), we
can conclude that the relatively large-scale turbulence structures as observed for cases 6x1z-thick and 6x3z-thick in Fig.3
is attributed to the K–H instability. Another noticeable observation from the figure is that, for the streamwise-reduced
domain cases (1x3z) in Fig.4(c), the energy spectra for thethin and thick porous wall cases perfectly collapse onto eachother. For the streamwise-reduced domain cases (1x3z), thestreamwise length is long enough to sustain near-wall turbulence including the streaks and longitudinal vortices but too
short to accommodate the spanwise rollers. Given that only
the near-wall turbulence develops for cases 1x3z, we can assert that near-wall turbulence is unaffected by the thickness ofthe porous wall. In other words, the near-wall turbulence ispredominantly altered by the surface roughness rather than thewall permeability.
To better understand the effect of the large-scale motion
on mean flow properties, Figure 5 depicts streamwise mean
velocity profiles normalized by the friction velocity. For com-

Results and discussions
Figure 3 shows snapshots of the streamwise velocity flucp
tuations above the porous wall at y p+ ≃ 10 where y p+ = yuτ /νpis the inner-scaled distance from the porous wall with uτ being
the friction velocity at the porous wall. For the streamwisereduced cases (1x3z), near wall turbulence is sustained for the
thick and thin porous wall cases, and we observe no distinct
difference in turbulent vortex structure between the thin andthick wall cases. For the spanwise-reduced cases (6x1z), a
clear difference is detectable between the thin and thick wallcases; turbulence structure for the thick porous wall case ischaracterized by streamwise alternating low- and high-speed
regions, whereas no such streamwise perturbation is observed
from the thin porous wall case. This streamwise perturbation
is similar to what has been reported in the DNS of porous
wall turbulence by Kuwata & Suga (2016b, 2017), who concluded that the streamwise perturbation was the footprint of the
spanwise rollers associated with the K–H instability. For the
full-domain cases (6x3z), the streamwise alternating low- andhigh-speed regions for 6x3z-thick are rather obscure than thosefor 6x1z-thick, whereas turbulence structure over the thick2

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

6x3z-thin
6x3z-thick

6x1z-thin
6x1z-thick

1x3z-thin
1x3z-thickFigure 4. Pre-multiplied one-dimensional streamwise spectra just above the porous wall at y p+ ≃ 10: (a) for cases 6x3z,(b) for cases 6x1z, and (c) for cases 1x3z.

suggests that the friction drag at the porous wall is increased
not only by the surface roughness but also by the spanwise
rollers associated with the K-H instability. .
The effects of the domain size and porous wall thickness on turbulence intensity are presented in Figures 6 and
7 where the streamwise component and wall-normal component normalized by the friction velocity at the porous walls areshown. For comparison, the DNS data for a smooth wall at
Reτ = 590 from Vreman & KuertenVreman & Kuerten (2014)
are included.
For the thin porous walls in Figs.6(a) and 7(a), the maximum peak just above the porous wall is absent, while thestrength of the wall-normal component is maintained. This
trend is similar to what has been found for the rough wall turbulent boundary layer Ligrani & Moffat (1986); Flack et al.
(2005); Kuwata & Kawaguchi (2019). Regarding the effectsof the domain size, the profiles near the thin porous walls areinsensitive to the domain size. For the thick porous walls in
Figs.6(b) and 7(b), the modification of the Reynolds stress appears qualitatively similar to thin porous wall cases, while thedomain size has a substantial influence on the turbulence intensity unlike the thin porous wall cases. Inside the porous
wall (y < 0), the turbulence intensity are the largest for the
spanwise-reduced domain case (6x1z), followed by the full-Figure 3. Snapshots of streamwise velocity fluctuations justabove the porous wall at y p+ ≃ 10.

parison, DNS data for smooth wall turbulence at Reτ = 590 byVreman & Kuerten (2014) is also included. As the porous wall
increases the friction drag, the mean velocity profile exhibits
significant asymmetry, and the profile is lower than that of thesmooth wall result. For the streamwise-reduced domain cases(1x3z) in Fig.5(c), no visible difference in the mean velocity
profile is detectable between the thin and thick porous wallcases. In contrast, for the full-domain cases (6x3z) in Fig.5(a)and the spanwise-reduced cases (6x1z) in Fig.5(b), the profile
over the thick porous wall is considerably lower than that over
the thin porous wall. The possible explanation for this trend
is responsible for the spanwise rollers, which develop for the
thick wall cases but not for the thin wall cases, as observedin Fig.3. For the full domain cases of 6x3z, the skin friction
coefficient at the thin porous wall is 5.7 times larger than thesmooth wall result from Vreman & Kuerten (2014), and it isfurther increased by 43% for the thick porous wall case. This3

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

6x3z-thin
6x3z-thicksmooth6x1z-thin
6x1z-thicksmooth1x3z-thin
1x3z-thicksmoothFigure 6. Streamwise turbulence intensity normalized by thefriction velocity at the porous wall: (a) thin porous wall casesand (b) thick porous wall cases.

Figure 5. Comparison of the inner-scaled mean velocity profile: (a) for cases 6x3z, (b) for cases 6x1z, and (c) for cases
1x3z.

domain case (6x3z), and the smallest for the streamwisereduced domain case (1x3z). Over the thick porous wall, theincreased wall-normal component and decreased streamwise
component are more prominent for the 6x1z-thick where thestrong spanwise rollers are artificially induced by the periodic conditions with an insufficient separation. Hence, it issuggested that the spanwise rollers increases the wall-normal
Reynolds stress while decreasing the streamwise component.
Finally, we discuss the effect of the spanwise rollers on
the logarithmic mean velocity profile. The logarithmic velocity profile for turbulent flows over a porous wall is customary
written as follows:

U p+ =

)
(
1
y+d
,
ln
κ
hr

(1)

where U p+ is the mean velocity normalized by the friction velocity at the porous wall, and the parameters κ , d, and hr are
the von Kármán constant, zero-plane displacement, and equivalent roughness height, respectively. There are several possible
routes for determining those parameters. One of the widelyused approach in the field of porous wall turbulence is fitting
the mean velocity profile to the logarithmic law of Eq.1, where

Figure 7. Wall-normal turbulence intensity normalized by
the friction velocity at the porous wall: (a) thin porous wallcases and (b) thick porous wall cases

412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
the zero-plane displacement d is first determined based on the
diagnostic function γ :γ = (y + d) p+

dU p+
.
dy p+

smoothporous(2)

The value of d is determined such that the diagnostic function
γ takes plateau in the logarithmic region. The von Kármánconstant κ is given as the inverse of the plateau value of γ (i.e.,κ = 1/γ ), and the equivalent roughness height is obtained byfitting Eq.(1) to the mean velocity profile. Another well-known
approach to obtain the value of d is using the Jackson model
Jackson (1981). In this approach, d is assumed to be the levelat which the height-averaged drag force acts:smoothporous∫ 0

dJ = ∫−h0

y fx dy

−h

.

(3)

fx dy

where fx is the x-z plane-averaged drag force. Since this approach does not assume the logarithmic velocity profile ofEq.(1), it is particularly useful for relatively low Reynolds
number flows where the logarithmic region is not guaranteed.
Figure 8 presents the diagnostic function with the zero-plane
displacement obtained by the Jackson model Jackson (1981).
The DNS result for the smooth wall case at Reτ = 590 from
Vreman & Kuerten (2014) is shown for comparison. Note thatthe results for the reduced domain cases of 6x1z and 1z3z are
not shown here because the logarithmic mean velocity profiles cannot be properly represented by the reduced domain
cases. We see that the diagnostic function γ for smooth wallcase does not take a plateau in the logarithmic region because
the Reynolds number under consideration is not so sufficiently
high that the mean velocity profiles follows the logarithmic
law. This implies that fitting to the log-law profile of Eq.(1)yield unphysical log-law parameters of κ , d, and hr . The figure shows that for the porous wall cases, the diagnostic function for the thick wall case in Fig.8(a) is shifted downward
with respect to the smooth wall result, while that for the thinwall case in Fig.8(b) is close to the smooth wall result. Given
that the plateau value of the diagnostic function corresponds toinverse of the von Kármán constant, the observation suggeststhat the von Kármán constant for the thin porous wall case isalmost the same as that for the smooth wall case, while that for
the thick porous wall case takes a somewhat larger value.
The logarithmic velocity profiles with the log-law parameters by the fitting approach and Jackson model are comparedin Figure 9. The zero-plane displacement by the fitting app+p+
proach d f it is larger than that by the Jackson model dmJ , andthe corresponding von Kármán constant is considerably lower
than the standard value of κ = 0.41. We observe that the mean
velocity profiles with the scaling of (y + d f it ) p+ perfectly follows the log-law profile with a considerably steeper slope incomparison with the smooth wall case. It should be however
remarked that the low-law parameters obtained by the fitting
approach do not have any physical implications because thoseparameters are simply the results of overfitting of the lowReynolds number results to the log-law profile. Concerning
the profiles with dmJ , although the agreement is not perfect,
the profile over the thin porous wall can be reasonably approximated by the log-law profile with κ = 0.41, whereas a somewhat larger value of κ = 0.49 offers a good approximation for
the velocity profile over the thick porous wall.

Figure 8. Comparison of the diagnostic function γ with thezero-plane displacement by the Jackson model: (a) thin porous
wall cases and (b) thick porous wall cases. The DNS result forthe smooth wall case at Reτ = 590 from Vreman & Kuerten(2014) is shown for comparison.

The larger value of κ for the thick porous wall case isphysically interpreted as the results of the spanwise rollers. It
has been well established that the vortex size is directly related
to Prandtl’s classical mixing length, which scales with the distance from the wall with the proportional constant of κ . In
the context of the turbulence over the thick porous wall, turbulent motions consist of the near-wall turbulent motion witha superimposition of the spanwise rollers; thus, the characteristic vortex size for the thick porous wall appears to be larger
than that for the thin porous wall, as observed in Fig.3. The
increase in the vortex size by the spanwise rollers means anincrease in the mixing length, implying an increase in the proportional constant of the mixing length κ from the standard
value of 0.41. Slight deviation for the log-law profile is dueto the low-Reynolds number effect as can be seen in the profile of γ in Fig.8. It is conceivable that an increase in the vonKármán constant for the thick porous wall case is the resultof an increase in the characteristic vortex size due to the presence of the large-scale motion. Indeed, although the results are
not shown here, the mixing-length for the thick porous wall islarger than that for the thin porous wall case, and the increased
mixing-length for the thick porous wall case can be reasonable represented by the classical mixing-length model with theincreased von Kármán constant.

Conclusions
The influence of spanwise rollers associated with K–H instability on a turbulent flow over a porous wall was systematically studied using DNSs of porous-walled channel flows
with constraints in the size of the computational domain. We
performed the simulations with the truncated domains, whichwere large enough to sustain the near-wall turbulence includ-5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
thin porous wall is the same as that for a smooth wall, whilethat for the thick porous wall case is increased by 20% for the
standard value. The increased von Kármán constant for the
thick porous wall case is considered to reflect an increase in
the characteristic vortex size owing to the spanwise rollers.
The relatively large-scale spanwise rollers, which dominate
turbulent motions for the thick porous wall cases, increase thecharacteristic vortex size, i.e., the mixing length, leading to anincrease in the von Kármán constant.

REFERENCES
Flack, Karen A, Schultz, Michael P & Shapiro, Thomas A
2005 Experimental support for townsend s reynolds number similarity hypothesis on rough walls. Phys. Fluids
17 (3), 035102.
Jackson, PS 1981 On the displacement height in the logarithmic velocity profile. J. Fluid Mech. 111, 15–25.
Kuwata, Y. & Kawaguchi, Y. 2019 Direct numerical simulation of turbulence over systematically varied irregular roughsurfaces. J. Fluid Mech. 862, 781–815.
Kuwata, Y. & Nagura, R. 2020 Direct numerical simulationon the effects of surface slope and skewness on rough-wall
turbulence. Phys. Fluids 32 (10), 105113.
Kuwata, Y. & Suga, K. 2016a Imbalance-correction gridrefinement method for lattice Boltzmann flow simulations.
J. Comput. Phys. 311, 348–362.
Kuwata, Y & Suga, K 2016b 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., Tsuda, K. & Suga, K. 2020 Direct numericalsimulation of turbulent conjugate heat transfer in a porouswalled duct flow. J. Fluid Mech. 904.
Ligrani, P. M. & Moffat, R. J. 1986 Structure of transitionallyrough and fully rough turbulent boundary layers. J. Fluid
Mech. 162, 69–98.
Gómez-de Segura, G. & Garcı́a-Mayoral, R. 2019 Turbulent drag reduction by anisotropic permeable substrates–
analysis and direct numerical simulations. J. Fluid Mech.
875, 124–172.
Suga, K., Kuwata, Y., Takashima, K. & Chikasue, R. 2015 A
D3Q27 multiple-relaxation-time lattice Boltzmann method
for turbulent flows. Comput. Math. Appl. 69, 518–529.
Suga, K., Matsumura, Y., Ashitaka, Y., Tominaga, S. &Kaneda, M. 2010 Effects of wall permeability on turbulence. Int. J. Heat Fluid Flow 31, 974–984.
Suga, K., Okazaki, Y., HO, U. & Kuwata, Y. 2018 Anisotropic
wall permeability effects on turbulent channel flows. J.
Fluid Mech. 855, 983–1016.
Thomson, William 1887 Lxiii. on the division of space withminimum partitional area. The London, Edinburgh, andDublin Philosophical Magazine and Journal of Science
24 (151), 503–514.
Vreman, A.W. & Kuerten, J.G.M. 2014 Statistics of spatial
derivatives of velocity and pressure in turbulent channel
flow. Phys. Fluids 26 (8), 085103.

Figure 9. Inner-scaled mean velocity profile with the logarithmic scaling: (a) thin porous wall cases and (b) thick porous
wall cases. The scaling with the fitting approach (y + d f it ) p+
and Jackson model (y + dmJ ) p+ are shown. For comparison,
the DNS result for the smooth wall case at Reτ = 590 from
Vreman & Kuerten (2014) is included.

ing the streaks and longitudinal vortices but too small to properly capture spanwise rollers by the K-H instability. For eachdomain case, we considered thick and thin porous walls forthe bottom wall. The thin porous wall effectively attenuates
the effect of wall permeability, maintaining the effect of thesurface roughness. The constrained DNSs aims to clarify how
surface roughness and wall permeability alter turbulent flows
over porous walls and identify the role of the spanwise rollers
associated with the K–H instability.
A turbulent flow over a porous wall consists of nearwall turbulent motions with a superimposition of the spanwise
rollers owing to the K–H instability. The near-wall turbulent
motions are found to be insensitive to the thickness of a porous
wall, and a modification of the the near-wall turbulent motions
by a porous wall is principally due to the role of the surface
roughness of a porous wall. The spanwise rollers with streamwise alternating low- and high-speed regions develop over the
thick porous wall and contribute to an enhancement of the turbulence inside a porous wall, leading to an increase in the skinfriction coefficient at a porous wall. Hence, wall permeability
plays an important role in sustaining the spanwise rollers but
not in modification of the near-wall turbulent motions.
To discuss the mean flow away from a porous wall, we
discuss the logarithmic scaling of the mean velocity profiles
with two commonly used definitions of zero-plane displacement. The fitting approach, in which the zero-plane displacement is obtained by fitting the mean velocity profile to the logarithmic law, yields a considerably smaller von Kármán constant, irrespective of the presence of spanwise rollers. For thescaling with the Jackson mode, the von Kármán constant for a6