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

EFFECTS OF FRONTAL AND PLAN SOLIDITY ON ROUGH-WALL
TURBULENT CHANNEL FLOW OVER PYRAMID ROUGHNESSOleksandr ZhdanovJames Watt School of Engineering
University of Glasgow
Glasgow, United Kingdom, G12 8QQ
oleksandr.zhdanov@glasgow.ac.uk

Angela BusseJames Watt School of Engineering
University of Glasgow
Glasgow, United Kingdom, G12 8QQangela.busse@glasgow.ac.uk

ABSTRACTe.g., Cheng et al., 2007; Leonardi & Castro, 2010). Whilethese investigations provide valuable insights, their main limitation is the fact that by definition of a cube λ f = λ p ; thus,
it is impossible to explore the relative effects of these two parameters. Placidi & Ganapathisubramani (2015, 2018) used a
different approach and separated the effects of frontal and plansolidity by combining LEGO R bricks in different configurations keeping one of the parameters (either λ f or λ p ) constant
while varying the other. In contrast to studies with cubicalroughness in standard configurations, they found a peak in theroughness effect only for variations in λ f at constant λ p , whileat constant λ f drag monotonically decreased with an increasein λ p , showing that these two parameters can influence roughwall behaviour in differing ways.
Another feature of surfaces comprised of cubical roughness elements is the fact that in the extreme case of plan solidity approaching unity (λ p → 1), the rough surface merges intoa smooth wall elevated by the roughness height. However, using other 3D roughness elements, e.g., pyramids, it is possible
to create fully covered surfaces that remain rough. Schultz &Flack (2009) experimentally studied rough surfaces comprisedof close-packed pyramids (λ p = 1) while systematically varying their frontal solidity. Their results show that at low λ f
the roughness function does not scale with roughness height
and term surface ‘waviness’ was proposed to describe the behaviour of these surfaces. However, the effects of plan solidity have not been considered and it is not clear whether theseobservations will hold for λ p < 1. In this context it is alsointeresting to note that Nugroho et al. (2021) showed that for
irregular surfaces with low effective slope (ES = 2λ f ) non ktype behaviour is observed when the in-plane roughness wavelength is comparable to the boundary layer thickness.
In the present study the effects of plan and frontal solidity are studied numerically for surfaces covered with pyramidshaped roughness elements in different configurations. The
aim of this investigation is to explore individual effects of λ fand λ p on the fluid dynamic roughness effect. The focus of thecurrent study is the low effective slope or ‘sparse’ regime, i.e.,surfaces that would fall under the ‘waviness’ regime following
the classification of Schultz & Flack (2009).

Direct numerical simulations of turbulent flow over roughsurfaces comprised of pyramids are conducted in a closed
channel configuration at Reτ = 395. The effects of frontal
and plan solidity on the fluid dynamic roughness effect in the‘waviness’ regime are investigated. The frontal and plan solidity are systematically varied from 0.05 to 0.1 and from 0.25
to 1, respectively. The results show that both parameters havesignificant effect on the roughness function and mean flow andturbulence statistics. Frontal solidity influences the roughness
function at any value of plan solidity, while plan solidity has astrong impact only at low frontal solidity. In addition, surfaces
with peak-to-peak spanwise spacing comparable to the outerscale of the flow are found to generate secondary currents.

INTRODUCTION
Prediction of the drag of rough surfaces based on their
topography is a key challenge in fluid mechanics due thewide range of roughness topographies found in engineeringand geophysical applications (Jiménez, 2004; Flack & Schultz,
2010). Despite a significant body of work in this field, existing
empirical formulae cover only a limited range of roughness topographies and further systematic studies are required (Chunget al., 2021).
The effect of a rough surface on the mean flow is commonly expressed in terms of the Hama roughness function∆U + (Hama, 1954), which represents the downward shift ofthe mean inner-scaled velocity profile in the log region compared to the smooth-wall case. Since the pioneering work of
Schlichting (1937), rough surfaces composed of regular roughness elements are typically characterised by two density parameters, namely the frontal and the plan solidity. The frontal
solidity (λ f ) is the ratio of the total projected frontal area ofthe roughness elements to the total plan area. In turn, the plansolidity (λ p ) is the fraction of the total plan area covered byroughness elements.
Most previous experimental and numerical investigations
on the effects of λ f and λ p have been conducted for surfaces
with regular arrangements of cubical roughness elements (see,112th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
Table 1. Key topographical parameters of the investigated surfaces: λ f - frontal solidity, λ p - plan solidity, Sz - maximum peak tovalley height, z0 - smooth-wall displacement, which is equal to the roughness mean height hh(x, y)i in the present study, W - spanwise
peak-to-peak distance.

Case

1

2

3

4

5

6

7

8

9

λf

0.05

0.075

0.1

0.05

0.075

0.1

0.05

0.075

0.1

λp

1

1

1

0.444

0.444

0.444

0.25

0.25

0.25

Sz /δ

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

−z0 /δ

0.033

0.033

0.033

0.01480.01480.01480.00833

0.00833

0.00833

W /δ

1

0.667

0.5

0.667

0.444

0.333

0.5

0.333

0.25

Line styleMarkerMETHODOLOGY

wise and spanwise directions is set to 8δ × 4δ . For case 5 aslightly lower domain size (7.11δ × 3.55δ ) is used to maintainan integer number of pyramids. The grid spacing is uniform inthe streamwise and spanwise directions with ∆x+ = ∆y+ < 5
and is adjusted to maintain at least 36 grid points per pyramidin each direction. The wall-normal grid spacing is constantacross the roughness height ∆z+
min = 2/3 and is gradually increased above, reaching its maximum value ∆z+max < 5 at thechannel centre.

Nine different surfaces were constructed using pyramidshaped roughness elements. The pyramids are similar to thoseused by Schultz & Flack (2009), i.e., they have a square basewith an edge facing the flow and a staggered arrangement is
employed. In all cases, the roughness elements have a fixedheight of 0.1δ , where δ is the mean channel half-height.
The number and slope of the pyramids are systematicallyvaried to change either the frontal solidity at constant plansolidity or vice versa. The topographical parameters for thesurfaces are summarised in Table 1. The plan solidity, λ p , isvaried from 1 (closely packed) to 0.25 by moving the pyramids further apart (see Figure 1). For fixed plan solidity, thefrontal solidity, λ f , is varied by changing the height-to-baseratio of the pyramids, i.e., by modifying the inclination angle of the pyramids’ edges. All values of the frontal solidity
fall into the low frontal solidity regime (λ f ≤ 0.1), where a
reduction in the roughness effect is expected with decreasing
λ f (Jiménez, 2004). Based on their effective slope values, allnine cases should fall under the ‘waviness’ regime. The peakto-peak spacing of the pyramids, W , decreases with decreasing
λ p and increasing λ f . The roughness mean height, hh(x, y)i,
which in the present study is equal to the smooth-wall offset,
z0 , applied in each simulation, is only influenced by the plansolidity λ p but independent of λ f due to the fixed maximum
peak-to-valley height, Sz , which corresponds for the present
surfaces to the pyramid height.
For each configuration a direct numerical simulation ofa fully developed turbulent channel flow at friction Reynolds
number Reτ = uτ δ /ν = 395 is conducted to obtain the fluid
dynamic properties of the surface using the DNS code iIMB
(Busse et al., 2015). Flow statistics are acquired in each casefor over 100 flow through times after a statistically stationary flow has been attained. To compute statistical quantities
the double-averaging approach of Raupach & Shaw (1982) isused. The rough surfaces are resolved using an iterative version of the embedded boundary method of Yang & Balaras(2006). The pyramid roughness is applied to both the lower
and the upper wall of the channel. To minimise any local
blockage effects, the roughness on the upper wall is shiftedby half of the minimum repeating unit length in the streamwise direction. An offset z0 , is applied so that the z = 0 location corresponds to the roughness mean plane, setting themean channel height to 2δ in all conducted simulations.
For all cases except case 5, the domain size in stream-

RESULTS AND DISCUSSION
Mean velocity profiles
Mean streamwise velocity profiles for all consideredcases are presented in Figure 2a together with smooth-wallreference data at the same friction Reynolds number. With theincrease of λ f at constant plan solidity the downward shift inthe mean streamwise velocity profile increases, indicating anenhancement of the roughness effect. With a decrease of λ p atfixed frontal solidity, the observed effect is similar. However,
the mean velocity profiles for the two lowest values of λ p at
the two highest values of λ f are very close to each other (cases
5 & 8 and 6 & 9), suggesting that the effect of plan solidity on
the mean velocity profile saturates for low values of λ p .
Streamwise velocity profiles in the defect form are plottedin Figure 2b. A good collapse between smooth- and roughwall data is observed at wall-normal locations z/δ > 0.5.
Closer to the roughness crest, the profile shapes differ between
most cases, which can be attributed to the presence of large
secondary currents in cases 1, 2, and 4 or a shift of the virtual
origin of the studied surfaces. This is in line with observations
by Chan et al. (2018) for egg-carton surfaces with decreasing
wavelength.

Roughness function
The roughness function for all considered surfaces is plotted against the effective slope, which is directly related tofrontal solidity (ES = 2λ f ), in Figure 3a and against plan solidity in Figure 3b. In addition, data by Napoli et al. (2008),
Schultz & Flack (2009), and Chan et al. (2018) is also included.
The ∆U + values were determined based on the differencein centreline velocity of the rough and smooth wall profiles.
An alternative method for measuring ∆U + was also tested: theroughness function was evaluated at the height of 50 wall units

212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

Increasing λp

Increasing λf
Case 1Case 2

Case 3

Case 4

Case 5

Case 6

Case 7

Case 8

Case 9Figure 1. Illustration of the studied surfaces composed of pyramid-shaped roughness elements. Surfaces with varying frontal solidity
at constant plan solidity are presented in horizontal rows. Surfaces with varying plan solidity at constant frontal solidity are presented
in vertical columns.

(a)

(b)

Figure 2. (a) Mean streamwise velocity profiles; (b) Streamwise velocity defect profiles for all studied surfaces. Line styles andcolours are specified in Table 1. Reference smooth-wall data is also shown with a solid black line. The location of the maximum
roughness height for each surface is indicated by a thin vertical line with a corresponding line style.

above the roughness crest, as in Chan et al. (2015) for eggcarton roughness in the transitionally rough regime, to checkthe influence of the applied approach to determine ∆U + . The
difference in ∆U + determined by both methods is below 10%
for all cases except for case 1, where the difference is 17%.
The higher difference for case 1 can be attributed to the presence of strong secondary flows. However, this does not affectthe observed trends in ∆U + with variation of frontal and plansolidity.

values of roughness function at any plan solidity. Increase in
λ f at constant plan solidity results in higher drag (Figure 3a).
A more complex picture emerges for the dependency of ∆U +on plan solidity: while for high plan solidities range an increase in the roughness function with a decrease in λ p is observed (see Figure 3b), the influence of plan solidity starts tosaturate for surfaces with λ p ≤ 0.444. The effect of plan solidity is more pronounced at low values of frontal solidity. Cases
with λ p = 1 and / or λ f = 0.05 generate significant secondary
currents. This can be attributed to the fact that these cases havea high spanwise peak-to-peak distance between pyramids.

Both λ f and λ p have a clear effect on the roughness function but, as can be seen from Figure 3, these parameters influence ∆U + in opposite ways. Frontal solidity influences theComparing to the published data for other rough surfaces,3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022(a)strate a good collapse in the outer layer for all cases (Figure4a). Compared to the smooth-wall case, a reduction of hu0 u0 i+
is observed in the near wall region. With the increase in frontal
solidity at constant λ p the peak value of the streamwise turbulent fluctuations drops. The same effect is found when plansolidity is reduced while λ f is kept constant. This is consistent with the observed increase of the roughness function andthe results of Leonardi & Castro (2010) for cube roughness,where with increase of roughness solidity a reduction of thepeak value of the streamwise Reynolds stress was reported.
The peaks are located above the roughness crest, except forcase 1, where the peak falls within roughness canopy.
The wall-normal Reynolds stresses are much less affectedby the variation in frontal and plan solidity (Figure 4c). Theirprofiles also demonstrate a reasonable collapse in the outerlayer for all cases. For rough surfaces comprised of staggered
cubes, Leonardi & Castro (2010) found that the peak value ofthe wall-normal Reynolds stress decreases with the increase inroughness density. However, this behaviour is not consistently
observed for the surfaces considered in the present study. With
the increase in λ f at constant plan solidity the peak value ofwall-normal turbulent fluctuations slightly decreases. However, surfaces where high levels of dispersive stresses and consequently strong secondary currents are observed (cases 1, 2,
and 4), do not follow this trend and their peak values are reduced compared to the surfaces with the same λ p but without
strong secondary currents. As a result, the trend is reversed forcases 1, 2, and 3, where surfaces with higher frontal solidity
have higher peak values of hw0 w0 i+ .
Reynolds shear stress profiles are plotted in Figure 4e. In
all cases Reynolds shear stress levels are reduced compared tothe smooth-wall case and show a good collapse in the outerlayer. The most notable outlier is case 1, where a reductionof the Reynolds shear stress is observed up to wall-normal location z/δ ≈ 0.63. Overall, the wall-normal location wherethe −hu0 w0 i+ profile of the rough surface collapses onto thesmooth-wall data depends on the pyramids’ peak-to-peak distance in spanwise direction. Surfaces with different solidities
but equal W /δ (see Table 1) collapse at approximately thesame z/δ onto the smooth-wall case, e.g., cases 2 and 4. With
the decrease in W /δ the location where the rough-wall profile collapses onto the smooth-wall data moves closer to theroughness crest. For surfaces with constant plan solidity, thepeak value of the Reynolds shear stress rises with increase in
λ f , while for surfaces with constant frontal solidity peak values are reduced when λ p is increased. However, case 1 does
not follow these trends and its peak value of hu0 w0 i+ is of similar magnitude as for the surfaces with the same plan solidity
but higher λ f .

(b)Figure 3. Roughness function as a function of (a) effective
slope (ES = 2λ f ) and (b) plan solidity. The data from literature for different rough surfaces is included for comparison.
Present data markers are specified in Table 1.

the same increasing trend of ∆U + with effective slope / frontal
solidity is observed in all cases. A good match between present
results and the irregular 2D roughness by Napoli et al. (2008)
is observed at ES ≥ 0.15. However, a significant scatter in thevalues of roughness function between present data and the simulations of Napoli et al. (2008) is observed at low values of ES,
which can be attributed to the effect of plan solidity. In addition, the value of the roughness function for case 3 (ES = 0.2,
λ p = 1) is close to those reported by Schultz & Flack (2009)
for the same type of roughness as considered in this study withsimilar ES, but at much higher Reτ .

For all studied surfaces the peak value of −hu0 w0 i+ is observed above the maximum roughness height. The distancefrom the roughness crest to the peak location increases with areduction in λ f at constant plan solidity, and with an increasein λ p at constant frontal solidity. Case 1 is an outlier fromthese trends again, and its peak location is closer to the roughness crest compared to the cases with the same λ f or λ p . In
case 1 two peaks are observed, but due to similar values of−hu0 w0 i+ at these peaks, the Reynolds shear stress profile appears almost constant above the maximum roughness height.
Profiles of dispersive stresses for all surfaces are shownin Figures 4b, d, and f. For cases 1, 2, 3, 4, and 7, where the
spanwise peak-to-peak distance between pyramids is comparable to the channel half-height, elevated levels of all dispersive
stresses can be observed well above roughness crest, indicating
the presence of secondary currents. This is consistent with re-

Turbulence statistics
Double-averaged profiles of streamwise, wall-normal,
and shear Reynolds and dispersive stresses are presented in
Figure 4. To facilitate observations of the peak behaviour
with variation of frontal and plan solidity, in the insets toparts of Figure 4 the profiles for each surface are shiftedby their smooth-wall displacement, z0 , so that the locationof the roughness crest coincides for all cases. The doubleaveraged streamwise normal Reynolds stress profiles demon-

412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
Chan, L, MacDonald, M, Chung, D, Hutchins, N & Ooi, A
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Chan, L, MacDonald, M, Chung, D, Hutchins, N & Ooi, A
2018 Secondary motion in turbulent pipe flow with threedimensional roughness. Journal of Fluid Mechanics 854, 5–
33.
Cheng, H., Hayden, P., Robins, A. G. & Castro, I. P. 2007 Flowover cube arrays of different packing densities. Journalof Wind Engineering and Industrial Aerodynamics 95 (8),715–740.
Chung, D., Hutchins, N., Schultz, M. P. & Flack, K. A. 2021
Predicting the drag of rough surfaces. Annual Review of
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Flack, K. A. & Schultz, M. P. 2010 Review of hydraulic roughness scales in the fully rough regime. Journal of Fluids Engineering 132 (4), 041203.
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Leonardi, S. & Castro, I. P. 2010 Channel flow over large cube
roughness: a direct numerical simulation study. Journal of
Fluid Mechanics 651, 519–539.
Napoli, E, Armenio, Vincenzo & De Marchis, M 2008 The
effect of the slope of irregularly distributed roughness elements on turbulent wall-bounded flows. Journal of Fluid
Mechanics 613, 385–394.
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A1.
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and plan solidities on aerodynamic parameters and theroughness sublayer in turbulent boundary layers. Journal of
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Meteorology 167 (1), 99–121.
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215 (1), 12–40.
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currents and very-large-scale motions in open-channel flow
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A17.

sults for other rough surfaces, where secondary currents werereported when the spanwise spacing of the roughness elements(e.g., Zampiron et al., 2020) or the roughness wavelength (e.g.,
Chan et al., 2018) were similar to the outer scale of the flow.
For case 1, which has the highest value of W /δ , significant
ewi
e + and −hee + are observed up to z/δ ≈ 0.7,
values of hw
uwi+
while heuuei is elevated up to z/δ ≈ 0.5. The extent of all dispersive stresses in the wall-normal direction is dependent on
W /δ and decreases as W is reduced.
The peak value of streamwise dispersive stress is locatedbelow the roughness crest in all cases. In general, the peakvalues follow the same behaviour as the streamwise Reynolds
stress, i.e., with an increase in frontal solidity at constant λ p
the peak value drops. Once again, case 1 does not follow thistrend and its peak value is lower than for case 2, which has
higher λ f .
ewi
e + fall within the
While for most cases peaks of hw
roughness canopy, the peak value for case 1 is located at theroughness crest (Figure 4d). No consistent behaviour is found
for the wall-normal dispersive stress peak values with variationof λ f or λ p . While for surfaces with λ p = 0.25 the maximumewi
e + drops with an increase of λ f , for cases 1, 2,
value of hw
ewi
e + increases withand 3, where λ p = 1, the peak value of hw
frontal solidity.
e + proFor all cases with secondary currents, the −heuwifiles exhibit high levels above the roughness crest, with a second peak emerging in some cases. Case 1 again shows distinctive behaviour as the only case where a minimum and negae + can be observed within the roughness
tive values for −heuwi
canopy.

CONCLUSIONS
Frontal and plan solidity of pyramid roughness has a significant effect on mean flow and turbulence statistics of roughwall channel flow. The present study focuses on the ‘waviness’ regime, i.e., low effective slope cases. The roughness
effect of the surface increases with frontal solidity at constantplan solidity, and decreases with increasing plan solidity atconstant frontal solidity. Frontal solidity has a strong influence on ∆U + at any considered value of plan solidity, while
a significant effect of λ p is found at low λ f , and the effectof plan solidity saturates once low plan solidity values are attained. Another significant parameter is the spanwise peakto-peak distance between pyramids, which affects the extent
of the dispersive stresses and formation of secondary currents.
Dispersive stresses extend furthest into the flow for a spanwise
peak-to-peak distance of the order of the channel half-height
and their extent decreases with increase in frontal solidity due
to reduction in the distance between adjacent peaks.
In the next stage of this project, an in-depth investigationof the secondary currents will be conducted and the structureof the turbulent flow will be investigated. In addition, simulations at different Reτ are planned to test for the Reynolds
number dependency of the results.

ACKNOWLEDGEMENTS
We gratefully acknowledge support by the United Kingdom’s Engineering and Physical Sciences Research Council
under grant number EP/V002066/1.

REFERENCES
Busse, A., Lützner, M. & Sandham, N. D. 2015 Direct numerical simulation of turbulent flow over a rough surface basedon a surface scan. Computers & Fluids 116, 129–147.

5

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

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4. Double-averaged profiles of (a) Streamwise Reynolds normal stress; (b) Streamwise dispersive normal stress; (c) Wallnormal Reynolds normal stress; (d) Wall-normal dispersive normal stress; (e) Reynolds shear stress; (f) Dispersive shear stress for all
studied surfaces. Line styles and colours are specified in Table 1. Reference smooth-wall data is also shown with a solid black line.
The location of the maximum roughness height for each surface is indicated by a thin vertical line with a corresponding line style. An
intrinsic averaging is employed in all cases.

6