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

PERMEABILITY EFFECTS ON TURBULENT CHANNEL FLOWS OVER
POROUS RIB-ROUGHNESS

Y. Okazaki
Department of Mechanical Engineering
Osaka Prefecture University, Japanokazaki@htlab.me.osakafu-u.ac.jp

Y. TakaseDepartment of Mechanical Engineering
Osaka Prefecture University, Japan
takase@htlab.me.osakafu-u.ac.jp

Y. Kuwata
Department of Mechanical Engineering
Osaka Metropolitan University, Japankuwata@me.osakafu-u.ac.jp

K. Suga
Department of Mechanical Engineering
Osaka Metropolitan University, Japansuga@me.osakafu-u.ac.jp

ABSTRACT

Huang et al., 2013; Rubol et al., 2018; Ming et al., 2021). The
difference between the definitions of porous and rough walls
hence sometimes becomes ambiguous. The important parameters for porous media are however porosity (void fraction) andpermeability (Whitaker, 1986, 1996). The permeability is amacroscopic measure of porous media and it can be considered
to include the effects of all the aforementioned topographicalroughness parameters on drag through porous structures.
For flows around porous media, permeability plays a determinate role for drag. Zippe & Graf (1983); Kuwata & Suga
(2016a) confirmed that overall friction loss was greater thanthat over equivalent impermeable rough surfaces. By the measurements of the momentum flux across the porous interface,
Pokrajac & Manes (2009) recognised that due to the permeability, the momentum exchange process was enhanced near
the porous interface. By systematic direct numerical simulations (DNSs) and particle-image-velocimetry (PIV) measurements (Breugem et al., 2006; Kuwata & Suga, 2016a; Sugaet al., 2011, 2017), turbulence over permeable walls was investigated in detail and it was confirmed that due to the weakened wall blocking effects by the wall permeability, the wallnormal fluctuating velocity was not completely damped leading to the high drag over the permeable walls. By analysing the
DNS data, Kuwata & Suga (2016b) confirmed that while themain source of the turbulent energy was the mean shear overthe porous layer, the contribution from the dispersion became
dominant inside the porous layer.
To the best of the authors’ knowledge, however, except
for the study by the present authors’ group (Okazaki et al.,2020), there has been no systematic study that reports turbulence over rib-roughened porous walls in the literature. To discuss the porous roughness effect, Okazaki et al. (2020) carriedout PIV measurements of turbulent flows over rib-roughened
porous walls. Their case was for regularly aligned squareribs at w/k = 9 where w and k are the rib-spacing and therib-height, respectively. Following Perry et al. (1969), it is
widely accepted that “impermeable” wall roughness is categorized into two types: k- and d-types and w/k = 9 is in the
k-type roughness regime. The experiments confirmed that as
the permeability increased, the recirculation region formed behind the rib became smaller and moved down into the porous
wall leading to weakened turbulence and decreased hydraulic

To understand permeable roughness effects on turbulence,
PIV measurements of turbulent channel flows over porousroughness are carried out. We consider fully developed turbulent channel flows over porous bottom walls with transverse square porous-ribs whose heights are 10% of the channel height. The considered ratios of the rib spacing w to therib height k are w/k =1, 3, 7, 9 and 19. The porous ribs
are also made of the same material as that for the bottom
wall. Three kinds of metallic foam materials are applied as
the porous media. The permeability of the most permeable
medium is approximately five times as large as that of the least
permeable one while the porosities of all three media are 0.95.
Solid impermeable rib-roughness cases are also measured forcomparison. The measured flows are in the range of the bulk
Reynolds numbers of 5000–20000. At w/k = 1, the levels of
turbulence quantities and the drag coefficient become larger
as the permeability increases while such a trend becomes unclear at w/k = 3 and a reversed trend is seen at w/k ≥ 7. Ata higher permeability, however, it is found that the sensitivity
to the rib spacing on the profiles of the turbulence quantities
and the drag coefficient is weakened. Even at a lower permeability, turbulence tends to be less sensitive to the rib spacing
at w/k > 3. The equivalent sand grain roughness height hence
follows these trends. It is found that there is an almost linear correlation between the zero-plane displacement and thehydraulic roughness scale of the logarithmic law of the mean
velocity over the roughness. It is also shown that the effective displacement, which is introduced for isolating the permeability effects, and the von Kármán constant have reasonable
correlations with the permeability Reynolds numbers.

INTRODUCTION
Fundamental knowledge on rough wall turbulence accumulated by many studies has been summarized several times
in review articles (e.g., Raupach et al., 1991; Jiménez, 2004;
Flack & Schultz, 2010; Piomelli, 2018; Chung et al., 2021).
From those studies, to characterise rough wall turbulence werecognize that there are many parameters as described in thelatest review by Chung et al. (2021). In case of very highroughness, the surfaces such as vegetation and urban canopies
are sometimes treated as porous media (e.g., Lin et al., 2012;1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
ሺܽሻroughness heights. The turbulent drag hence became small
as the permeability increased. It was however still unknown
whether such a trend was maintained in other w/k cases. Also,
there was no knowledge on how the permeability affected therib-spacing effects on turbulence.
Therefore, to further explore the above-mentioned trends
and to understand turbulence over permeable roughness, thisstudy has carried out PIV measurements of turbulent flows in
the porous rib-roughened channels of w/k =1–19 at a rangeof the Reynolds numbers: Reb =5000–20000. We apply threekinds of metallic foam porous media of the porosity of 0.95,
but their permeabilities and mean pore diameters are significantly different. These porous media are considered to be
isotropic media.

y

flow

x

test section

0.06m(2H )porous wall
3.0m

1.0m

flow meter

valve

pump
ሺܾሻ
Laser sheet

Flow

ሺܿሻ

Double pulse
Nd-YAG laser

water tank
Acrylic Plate

CCD camera

‫ܪ‬

Flow

‫ݕ‬

H

݇

m
0 .3

y
MeasuringdomainPorous medium

z

ߣ
‫ݓ‬

݇

‫ݔ‬

x
Flow

Porous MediaEXPERIMENTAL METHOD
Fig.1 shows our flow facility. The PIV experimental setup is the same as that used in our previous studies (e.g., Sugaet al., 2010). It consists of a conditioning section and driving
and test sections whose cross-sectional size is 0.06 m (height)
× 0.3 m (width) as shown in Fig.1(a). Rib-roughened porous
slabs fill the bottom half of these sections so that the clear
channel height is H=0.03 m. Turbulent flows are fully developed through the driving section (100H long) and enter the test
section. As seen in Fig.1(b), the measurements are conductedat the symmetry plane of the channel where two dimensionality is confirmed (Suga et al., 2013). This study applies threekinds of metallic foam porous media of porosity φ = 0.95 aslisted in Table 1. The porosity is defined as the ratio of the
intrinsic (fluid phase) and the superficial (total phase) volumesof porous media. The permeabilities are obtained using theDarcy-Forchheimer equation:−µ Ud ρ CF 2
∆P
=
+
U ,
∆x
K
K d

Figure 1. Experimental setup: (a) schematic view of the flow
facility, (b) overall view of the test section, (c) side view of thetest section.

RESULTS and DISCUSSION
Streamlines
To illustrate the general flow patterns, Figs.2 and 3 show
streamlines by plotting contour lines of the stream functions
calculated with the measured mean velocity distributions at
Reb ≃15000. In the low permeability case (case #30m), although the confined recirculation is still observed between theribs at w/k = 1, the centre of the recirculation bubble shifts
downstream. When w/k becomes large, the recirculation tends
to relax and seems to submerge into the bottom porous wall. In
the high permeability case (case #13m), the trend observed advances and almost no recirculation bubbles can be seen. Thisis significant at w/k = 1 while the flow patterns at w/k = 7
look similar to those of case #30m. Obviously, as the permeability increases, the flows through the ribs increase leading
to the relaxation of the recirculating flows. The bottom wall
permeability also weakens the recirculation by the by-passing
flows below the ribs and it also allows penetration of the flows
between the ribs.

(1)

F
where ∆P∆x , µ , ρ ,C and Ud are the pressure gradient in streamwise direction, fluid viscosity, density, the Forchheimer coefficient and the mean velocity, respectively. The mean pore diameter D p was obtained by averaging the equivalent circle areas
converted from the measured cross section areas of pores. The
most permeable medium #13m has approximately five times
larger permeability than the least permeable one #30m. Accordingly, we call #13m, #20m and #30m “high”, “medium”
and “low” permeable media, respectively. On their surface,square-shaped porous ribs whose height k = 3 mm are set withthe same spacings of w/k = 1, 3, 7, 9, 19 as shown in Fig.1(c).
Note that the porous ribs are made of the same material as thatfor the bottom wall and hence the mean pore diameter of themost permeable medium is 77% of the rib height. To compare
the data, the solid impermeable cases are also measured. The
measured Reynolds number range is Reb = 5000–20000. The
Reynolds number is defined as Reb = Ub H/ν where ν is thefluid kinematic viscosity and Ub is the bulk velocity that is calculated by integrating the measured mean velocities from therib-bottom position to the top wall.

Turbulence distribution
To generally observe the turbulence trends, Figs. 4 and 5
show spatial distributions of in-plane turbulent kinetic energy
q2xy /2 = (u′2 + v′2 )/2 normalized by the bulk mean velocity
Ub at Reb ≃15000. Here, u′2 and v′2 are the streamwise andwall-normal Reynolds normal stresses, respectively.
It is seen that the turbulence level of the low permeability
case (case #30m) at w/k = 1 shown in Fig. 4 (a) is significantly
lower than those of the other cases. As w/k becomes larger,
generation of turbulence is enhanced above the shear layers as
seen in Fig. 4 (b) and (c). This trend is consistent with thatshown for the Reynolds shear stress of impermeable cases by
Cui et al. (2003). In the high permeability case (case #13m),
as shown in Fig. 5, the general turbulence level at w/k = 1 iscomparable to that at the larger w/k. This is because turbulence generation in the region near the rib top position (y = 0)
increases significantly depending on the increase of the permeability as Fig. 5 (a) indicates. The increasing trend of turbulence level, however, is not monotonic and at w/k = 7 thegeneral turbulence level looks slightly reduced as seen in Fig.

Table 1. Characteristics of the porous media; φ : porosity, K:
permeability, D p : mean pore diameter.

Porous Med.

φ

K(mm2 )D p (mm)

#30m
#20m
#13m

0.95

0.007
0.013
0.033

0.8
1.7
2.3
2

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

y/k

(a)

(b)




2
q xy



(a)



2U b2

(b)













x/k



í







x/k







y/k

(c) 
í

y/k























x/k









í

(c)Figure 2. Streamlines in the low permeability case (case #30)
at Reb ≃ 15000: (a) w/k = 1, (b) w/k = 3, (c) w/k = 7.
(b)





x/k





í












x/k





















y/k

(a)



y/k

í








í



x/k





í





x/k







í

y/k(c) 









x/k









x/k







Figure 4. In-plane turbulent kinetic energy q2xy /2 distributions in the low permeability case (case #30m) at Reb ≃ 15000:(a) w/k = 1, (b) w/k = 3, (c) w/k = 7.

í





2
q xy

Figure 3. Streamlines in the high permeability case (case#13) at Reb ≃ 15000: (a) w/k = 1, (b) w/k = 3, (c) w/k = 7.

(a)

2U b2

(b)











5 (c). More detailed turbulence trends are discussed focusing
on each turbulence quantity in the following section.







Drag coefficient

Fig.6 shows the total drag coefficient CD = 2τw /(ρ Ub2 )at Reb ≃ 15000, where τw is the total drag on the rib bottom.

íy/k



Here, τw is obtained by extrapolating total stress (sum of the
Reynolds stress, viscous stress and dispersion stress) profile tothe bottom porous wall at y = −k. The presently observed general trend of case solid agrees with the data of Leonardi et al.
(2003) while the present values at w/k = 3, 9 are somewhat
larger. In the present study, the peak appears around w/k ≃ 9,
although the difference between the values of w/k = 7 and 9
is not significant.
The total drag coefficients CD for the permeable cases
show characteristic behaviours which correspond to the trendsof turbulence quantities. Since the mean pore diameter of thehigh permeability case (case #13) is 77% of the rib height andits porosity is 0.95, its permeability is considered to be close
to the maximum for the rib-roughened wall geometry. Thisimplies that the range of the observed permeability (or porescale) dependent trends for turbulence quantities covers thoseof most permeable rib-roughness cases (from the impermeable
to highest permeable cases). At w/k = 1, CD of the low permeable case (case #30m) is significantly smaller than thoseof the medium and high permeability cases (cases #20m and#13m). As w/k increases, CD of case #30m increases significantly up to w/k = 3 like in case solid. After catching up withthose of the higher permeable cases, CD of case #30m gradually increases toward w/k = 9, then turns to decrease slowly
to w/k = 19. As the permeability increases, CD increases at
w/k = 1 while it seems to decrease at w/k ≥ 9 . This trend

(c)



x/k





í












x/k

















y/k







í



x/k







Figure 5. In-plane turbulent kinetic energy q2xy /2 distributions in the high permeability case (case #13m) at Reb ≃15000: (a) w/k = 1, (b) w/k = 3, (c) w/k = 7.

is confirmed for all measured Reynolds numbers. The above
observation suggests that as the permeability or the pore-scale
increases, the d-type characteristics become weak and the transition to the k-type roughness takes place. Since the larger permeability and pore-scale reduce pressure drag of the rib roughness, turbulence tends to decrease gradually at the larger w/k
as seen at w/k > 9.

3

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

0.06

200
w/k #30m #20m #13m
1
3
7
9
19

0.05

CD

0.04
150

㻌

0.03

0.01
0

5

10
w/k

15

#20

#13

0.4
0.35

#06
0.3

solid
#30m
#20m
#13m
Leonardi et al. (2003)

0.02

flat

h/d =0.45

h

+

100

20

㻌

solid w/k = 7

Figure 6.

Drag coefficient at Reb ≃ 15000.

solid w/k = 950solid w/k = 19

 20

ܷା

15

solid
#30
#20
#13

0

10

0

100

200

300

d

400

500

+

5

/
0
10

100



ା

1

Figure 8. Correlation between roughness scale h and zeroplane displacement d. The data of the flat cases are from Sugaet al. (2010).

1000



for obtaining κ , d and h. Fig.7(a) shows velocity profiles inthe semi-logarithmic chart and the fitted lines. Note that asreported in the previous studies (e.g., Breugem et al., 2006;
Suga et al., 2010), the von Kármán constant becomes lower
than 0.41. This suggests that the roughness function which
normally denotes the downward shift of the semi-logarithmic
velocity profile cannot be simply applied to the porous walled
turbulence.
For the rough wall turbulence, another logarithmic formula:

݇௦ା

1000

㻌

100
solid
#30
#20
#13

10
0 2 4 6 8 10 12 14 16 18 20
/㻌

U + = κ −1 ln

Roughness parameters
In the following, we discuss the mean velocity profiles ofthe logarithmic formula:
y+d
,
h

(3)

may be applied with the equivalent sand grain roughness
height ks and the wall-normal distance ŷ from the origin whichshould be determined empirically (Jiménez, 2004) and we apply ŷ = y + d. Consequently, coupling equations (2) and (3)we can convert h to the equivalent sand grain roughness ks as
ks = h exp(8.5κ ). Fig. 7(b) shows the w/k dependency of ks+ at
Reb ≃ 15000. The overall trends are hence similar to those observed in the CD profiles shown in Fig.6 while the decreasing
trend by the increased permeability at w/k ≥ 9 is not observed.
Among the parameters of the lag-law formula, we focus
on the relation between d and h. Since both parameters are
related to the surface geometry, there may be a relationship(Okazaki et al., 2021). Fig.8 shows h vs. d for all measured
Reynolds numbers including the plots for the flat porous wallcases of Suga et al. (2010). Note that porosities in Suga et al.
(2010) were φ ≃0.8 and slightly smaller than the present one.
While the plots are somewhat scattered, a generally reasonable correlation around h/d = 0.4 is seen for all the porous

Figure 7. (a) Semi-logarithmic velocity profiles of w/k = 1
and (b) Equivalent sand grain roughness, at Reb ≃ 15000.

U + = κ −1 lnŷ+ 8.5,
ks(2)

with d and h which are the zero-plane displacement and theroughness scale, respectively. In the literature, discussions offlows over porous media and canopies have applied this equation. The method described in Breugem et al. (2006); Sugaet al. (2010) is applied to fitting the profiles to the equation 2

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

350

0.0

250

flat

#20

#13

case solid
Leonardi (2003)

0.2

dS /k

300

w/k #30m #20m #13m
1
3
7
9
19

0.4
0.6

#06

0.8

 ା

(k-km)/k

1.0

200

0

2

4

6

w/k

150
100

Figure 10. Zero-plane displacement for impermeable roughness. The solid line is from Leonardi et al. (2003) while thedashed line denotes the position of the mean roughness height.

50
0

8 10 12 14 16 18 20

0

5

10

15

20

0.45

Re௄

0.40
0.35

Figure 9. Distribution of the effective displacement dˆ against
the permeability Reynolds number ReK . The thin solid blackline denotes dˆ+ = 14ReK .

0.30

 0.25
0.20
0.15
0.10

cases. For the impermeable case (case solid), although a linear correlation may exist depending on w/k, the ratio of h/d
is significantly smaller than 0.4. This implies that although hand d have an almost linear correlation, the ratio may change
depending on the permeability and the roughness spacing.
Since d depends on both wall permeability and roughness
geometry, following (Okazaki et al., 2021), we introduce theeffective displacement dˆ to separate the effect of the roughness geometry as dˆ = d − dS , where dS denotes the zero-plane
displacement of the corresponding impermeable case. To describe permeability effects on flow characteristics,
the perme√ability Reynolds number, ReK = u∗ K/ν , has been usuallyapplied (e.g., Breugem et al., 2006). Fig.9 hence shows therelation between dˆ+ and ReK . The flat porous wall cases arealso included with dS = 0. It is seen that all plots in Fig.9 distribute around the thin black line which denotes dˆ+ =14 ReK .
This indicates that generally, dˆ+ has reasonable linear relationship with ReK . It is noticeable that the wall permeability
effect can be separated by simply subtracting the impermeablezero-plane displacement from the zero-plane displacement ofporous roughness.
As for dS , Fig.10 shows the presently obtained dS and theresults of Leonardi et al. (2003) who calculated dS with thecentroid of the moment of forces acting on the elements using the method of Jackson (1981). Note that dS /k = 0 and
1 correspond to the rib-top and rib-bottom positions, respectively. The dashed curved line shows dS /k = (k −km )/k, where
km = k/(k + w) is the mean roughness height. At the d-typeroughness at w/k < 3, the present plots of dS /k suggest thatthe line of Leonardi et al. (2003) is reasonable, while for the
k-type roughness at w/k ≥ 7, the present plots far deviate fromthe solid line but agree well with the dashed line. We thus suggest using the Jackson model for the d-type roughness but for

0.05
0.00

0

2

4

6

8

10

12

14

16

18

20

Re௄
Figure 11. Distribution of κ against the permeability
Reynolds number ReK . Symbols are as in Fig. 9.
the k-type roughness we recommend applying the simple relation: dS = k − km .
Considering the presently accumulated data, we try tofind an empirical correlation formula for porous rib-roughness.
Fig.11 shows the relation of κ to ReK . It is seen that regardless of the rib spacing, κ distributes more or less around thethin solid lines which indicate the formula:κ = κ0 − A exp[−(ReK /B)C ](4)

with κ0 = 0.41 and A = 0.165, B = 2.5 and C = 2. It is therefore noticeable that approximated values of all the parametersof log-law formulae can be estimated using ReK and dS . Thisis very helpful for engineering. Indeed, this strategy can be applied to wall functions for turbulence models in computational
fluid dynamics codes which are routinely utilized for designing industrial devices.

CONCLUSIONSTo describe the effects of permeable roughness on turbulence, we have carried out PIV measurements of fully de-512th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
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veloped turbulent channel flows over three kinds of isotropic
porous media with square-rib roughness at the bulk Reynolds
numbers of 5000–20000. Since the flow rate through the ribs
increases as the permeability increases, the recirculation bubbles between the ribs shift downstream and to the bottom wall.
The change of the flow patterns at w/k < 3 enhances turbulence around the rib-top region. For the high permeability
case, even at w/k = 1, it is observed that the enhanced rib
permeability leads to the vanishment of the recirculation bubbles and the significant increase of turbulence over the ribs. At
w/k = 1, because the permeability is considered to enhance thetransition from the d-type to k-type roughness, the magnitudeof turbulence becomes larger depending on the permeability.
At w/k > 3, which is characterized as the k-type roughness forthe impermeable case, the increase of turbulence by the permeability becomes unclear since the transition to the k-type
is already completed. At w/k ≥ 7, since the permeability reduces pressure drag of the rib roughness, the turbulence level
reduces slightly as the permeability increases. These trends arereflected in the profiles of total drag and the equivalent sandgrain roughness height. Since the permeability also enhances
the transition to the fully rough regime, the Reynolds number effects on turbulence quantities tend to vanish at a higher
Reynolds number. It is found that there are almost linear correlations between the zero-plane displacement d and the hydraulic roughness scale h of the logarithmic law of the mean
velocity over the roughness. The ratio h/d however varies
depending on the permeability and the rib spacing. For thepresently measured permeable cases at w/k ≤ 9, most plots
distribute around h/d = 0.4. It is confirmed that the wall permeability effect on the zero-plane displacement can be separated by simply subtracting the impermeable zero-plane displacement from the zero-plane displacement of porous roughness. The effective displacement, which is obtained by thisprocedure, has an approximately linear correlation to the permeability Reynolds number. In the permeable cases, the vonKármán constant becomes smaller than 0.41 and varies significantly depending on the permeability and the rib spacing. Its
plots suggest there is a reasonable correlation with the permeability Reynolds number.

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