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

THE STRUCTURE OF TURBULENT CHANNEL FLOW OVER
RATCHET-TYPE ROUGHNESSAngela BusseJames Watt School of Engineering
University of Glasgow
Glasgow, United Kingdom, G12 8QQangela.busse@glasgow.ac.uk

Oleksandr ZhdanovJames Watt School of Engineering
University of Glasgow
Glasgow, United Kingdom, G12 8QQ
oleksandr.zhdanov@glasgow.ac.uk

ABSTRACTof roughness features than with their leeward sides. In mostcases, this insensitivity to the sign of the slope is not relevant,since for the majority of rough surfaces the probability density function of ∂ h(x, y)/∂ x is not skewed, and therefore nosignificant orientation dependency is expected.

Ratchet surfaces are a form of regular two-dimensional
spanwise bar roughness where the bars have a scalene triangular cross-section. The structure of turbulent channel flow overratchet surfaces is investigated using direct numerical simulations at Reτ = 395 for a set of five ratchet surfaces withratchet lengths from 2k to 16k, where k is the ratchet height, inwindward and leeward orientation giving a total of ten cases.
The focus of this paper is the turbulent kinetic energy andthe Reynolds stress anisotropy measured at the mid-cavity andthe ratchet crest location. Both show strong dependence on
the ratchet length and orientation. Visualisations of the instantaneous wall-normal vorticity field show that the near-wall
turbulence activity is strongest at the mid-cavity location for
ratchets in leeward orientation, whereas the highest turbulence
activity is observed upstream of the roughness crests for ratchets in windward orientation.

However, there are rough surfaces where a significant imbalance in the slope distribution can be observed. For example,
geomorphical forms such as transverse sand dunes typicallyhave a low windward slope and a high leeward slope which isa result of their formation processes (Gao et al., 2015). Further examples from a geophysical context, where skewed slope
distributions can emerge, include sand ripples and some typesof ocean waves. Other roughness generation processes may
also produce orientation dependent roughness. For example,
some icing processes experienced by aircraft (e.g., run-backicing) can result in roughness forms that have higher windward
slopes than leeward slopes (Whalen et al., 2007).

INTRODUCTION
The effect of a rough surface on a wall-bounded turbulent
flow strongly depends not only on the roughness height but
also on the roughness topography. One of the key topographical parameters in this context is the effective slope of a roughsurface (Chung et al., 2021), which is defined as (Napoli et al.,2008)
1ES =AZ  ∂ h(x, y)  dxdy ∂x 

Ratchet-type roughness is an idealised two-dimensional
form of roughness that is comprised of transverse bars withscalene triangular cross-section. By varying the relative slopeof the leeward versus the windward side of the triangle, roughsurfaces can be constructed that have identical effective slope
but different windward and leeward slopes. Using direct numerical simulations of channel flow we showed that the roughness function ∆U + of ratchets in high windward slope orientation is significantly higher than for otherwise identical ratchetsurfaces in low windward slope orientation (Busse & Zhdanov,2021). Previously, Jiang et al. (2018) demonstrated for turbulent convection over ratchet roughness that large-scale convective structures are strongly influenced by the orientationof ratchet surfaces. The present investigation therefore aims
to determine whether structures in turbulent channel flow arealso strongly influenced by ratchet orientation. To this end, thestructure of turbulent channel flow over ratchet-type roughness
is investigated from both an Eulerian and a Lagrangian viewpoint.

(1)

where h(x, y) is the height of the surface as a function of thestreamwise (x) and the spanwise (y) coordinate and A is theplanform area of the heightmap.
One of the weaknesses of the effective slope parameteris that, by definition, it is insensitive to the orientation of therough surface: upslope sections, where ∂ h(x, y)/∂ x > 0, makethe same absolute contribution to ES as downslope sections,where ∂ h(x, y)/∂ x < 0, although wall-bounded turbulent flows
tend to interact much more strongly with the windward sides

1

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

2k

2k

4k

4k

8k

8k

12k

12k

16k

16k

a/ℓ

1/8

7/8

1/8

7/8

1/8

7/8

1/8

7/8

1/8

7/8

orientation

ww

lw

ww

lw

ww

lw

ww

lw

ww

lw

ES

1

1

0.5

0.5

0.25

0.25

0.167

0.167

0.125

0.125

Ssk∂ h/∂ x+2.3−2.3+2.3−2.3+2.3

−2.3

+2.3

−2.3

+2.3

−2.3

Table 1. Key geometrical and topographical parameters of the ratchet surface: ℓ - base length of the triangular cross-section; a x-coordinate of apex; orientation of high slope section (ww - windward; lw - leeward); ES - effective slope; Ssk∂ h/∂ x - skewness of
streamwise slope distribution.

Figure 1. (a) Schematic cross-section of a triangular bar. kis the roughness height, ℓ is the length of the bar. By moving
the apex location, a, a high windward slope or a high leeward
slope can be achieved. The mean flow direction is from left toright. (b) Measurement locations for statistics: apex location(solid line) and mid-cavity location (dashed line); statistics arephase averaged over repeating units.

Figure 2. Schematic illustration of the computational domain(not to scale). The ratchet roughness is applied to both channelwalls. On the upper wall, the pattern is shifted by ℓ/2 in thestreamwise direction. A ratchet case in leeward orientation isshown.

METHODOLOGY

pressible turbulent channel flow was conducted at Reτ = 395
using the code iIMB (Busse et al., 2015) which uses an iterative version of the embedded boundary method of Yang &Balaras (2006) to resolve the roughness. The ratchet surfaces
were applied to both walls of the channel and the flow wasdriven by a constant mean streamwise pressure gradient. Aschematic illustration of the domain is shown in Figure 2. The
wall reference location was set to the roughness mean plane,i.e., the total wall-normal extent of the simulation domain was
2.1δ . For cases with ℓ = 2n k the domain size was set to 6.4δin the streamwise and 3.2δ in the spanwise direction (see Table 2). For the case ℓ = 12k the domain size was increased
to 7.2δ in the streamwise direction to fit an integer numberof transverse triangular bars into the domain. Uniform grid
spacing was applied in the streamwise and spanwise direction,using ∆y+ = 4.94 and ∆x+ ≤ 4.94. For short ratchets, thestreamwise grid spacing was reduced to keep a minimum of
128 grid points per ratchet. In the wall-normal direction, uniform grid spacing was applied across the height of the ratchet∆z+min = 0.667 and the grid was gradually stretched above attaining ∆z+
max = 4.22 at the channel centre.

Ten different ratchet surfaces were constructed using barsof different triangular cross-sections (see schematic shown inFigure 1(a)). In all cases, the height of the triangles was set tok = 0.1δ , where δ is the mean channel half-height. The basesize of the triangle, ℓ, was varied from 0.2δ to 1.6δ . The entire surface was covered by the triangular bars, i.e., the plansolidity, λ p , is unity for all cases. For roughness composed oftriangular transverse bars at λ p = 1 the effective slope is givenby (2k)/ℓ, i.e., for the present cases ES was varied from 1 to0.125 as ℓ was increased. An imbalance between the slopes of
the two sides of the triangle interacting with the flow was introduced by positioning the x-coordinate of apex of the triangle,
a, at 1/8ℓ to create triangles with high windward and low leeward slope (windward oriented ratchet). To create equivalentleeward oriented ratchets, a was positioned at 7/8ℓ, i.e., effectively mirroring the triangle with respect to the vertical axis.
Table 1 gives an overview of key topographical parameters ofthe surfaces. Due to the simple triangular form of the present
surfaces, all of them have the same height distribution, and theroot mean square roughness height and skewness√of the height
distribution are identical for all cases (Sq = k/(2 3), Ssk = 0).
The skewness of the streamwise slope distribution Ssk∂ h/∂ x is
positive for ratchets in windward orientation
p and negative in
leeward orientation (Ssk∂ h/∂ x = (ℓ − 2a)/ a(ℓ − a) for triangular transverse bars at λ p = 1).
For each case, a direct numerical simulation of incom-RESULTS
Local statistics for the turbulent kinetic energy measured
at the apex location and at the mid-cavity location are shown212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Table 2. Simulation parameters for the direct numerical simulations. Lx - domain size in streamwise direction; Ly - domain size inspanwise direction; Lz - domain size in wall-normal direction; Nx × Ny × Nz - grid size; ∆x+ - grid spacing in streamwise direction; ∆y++- grid spacing in spanwise direction; ∆z+min - minimum wall-normal grid spacing; ∆zmax - maximum wall-normal grid spacing.

Lx /δ

Ly /δ

Lz /δNx × Ny × Nz

∆x+

∆y+

∆z+
min

∆z+
max

ℓ = 2k

6.4

3.2

2.1

4096 × 256 × 432

0.67

4.94

0.67

4.22

ℓ = 4k

6.4

3.2

2.1

2048 × 256 × 432

1.23

4.94

0.67

4.22

ℓ = 8k

6.4

3.2

2.1

1024 × 256 × 432

2.47

4.94

0.67

4.22

ℓ = 12k

7.2

3.2

2.1

1152 × 256 × 432

2.47

4.94

0.67

4.22

ℓ = 16k

6.4

3.2

2.1

1024 × 256 × 432

2.47

4.94

0.67

4.22

smooth

8.0

4.0

2.0

640 × 320 × 360

4.94

4.94

0.50

3.98

ences between the trajectories for surfaces in windward and inleeward orientations extend significantly further into the flow.
This is also evident from the profiles of the anisotropy function F = 1 − 27η 2 + 54ξ 3 (F = 1 for three-dimensional and
F = 0 for two-dimensional turbulence), where lower levels of
isotropy can be observed for long ratchets in windward orientation in the outer part of the profile up to the channel centre
whereas the leeward oriented cases show a good collapse on
the smooth-wall case for z/δ ⪆ 0.2 for all ratchet lengths.
Figure 5 shows a visualisation of the wall-normal component of the instantaneous vorticity field closely above theratchets at a distance of approximately 12 wall units above theroughness crests. Ratchet length and orientation have a strong
effect on how the structure of the vorticity field is affected bythe roughness. With decreasing ratchet length the organisation of the ωz contours is increasingly reduced. The reduction in the organisation of structures in the instantaneous wallnormal vorticity field is in line with observations for transverse bar roughness (Ashrafian & Andersson, 2006; Orlandi& Leonardi, 2009).
For ratchets in leeward orientation, the strongest disruption of structures can be observed at the mid-cavity locations,whereas for ratchets in windward orientation, the strongest interaction between the vorticity field and the ratchets can beobserved just upstream of the crest locations. The orientationdependent differences are most obvious for the longest ratchets with ℓ = 16k and become less pronounced for the shorterratchets. This is in line with the observations for the turbulent kinetic energy profiles, which show elevated levels at thecrest location for ratchets in windward orientation and elevatedlevels at the mid-cavity location for ratchets in leeward orientation. This dependency of turbulent kinetic energy levels on
the measurement location is also most strongly developed forthe longest ratchets.

in Figure 3. Closely above the ratchets, the streamwise measurement location has the strongest effect for flow over longratchets (ℓ ≥ 8k) in leeward orientation – the maximum observed for the turbulent kinetic energy profile is significantly
higher when measured at the mid-cavity location compared tothe equivalent measurements taken at the apex location. The
opposite behaviour can be observed for ratchets in windward
orientation. Here, for long ratchets significantly higher levels of turbulent kinetic energy can be observed above the apex
location compared to the mid-cavity location.
The turbulent kinetic energy profiles for the shorter ratchets show a good collapse on the smooth-wall turbulent kinetic
energy profiles for z/δ ⪆ 0.2. For long ratchets (ℓ ≥ 8k), thedifferences in the turbulent kinetic energy profiles extend further into the flow. This deviation from the smooth-wall behaviour tends to be more pronounced for ratchets in windward
orientation and for the mid-cavity location.
The Reynolds stress anisotropy measured at the apex andthe mid-cavity locations was analysed using the invariants ξand η which are derived from the anisotropy tensor bi j (Pope,2000)
6η 2 = bi j b ji ,36ξ = bi j b jk bki .

(2)
(3)

The anisotropic invariant maps of η plotted versus ξ are shownin Figure 4 based on the Reynolds stress profiles measured atthe apex location (see Figure 4 (a,b)) and at the mid-cavity
location (see Figure 4 (c,d)). When the profiles are sampled
at the apex location, in all cases the trajectories first approach
the left boundary of the Lumley triangle, i.e., the Reynolds
stress ellipsoid attains an approximately axisymmetric, oblate
spheroid shape (‘disk’-like turbulence). A similar behaviour
was reported for roughness comprised of transverse squarebars (Ashrafian & Andersson, 2006).
Disk-like turbulence is also observed within the cavitiesof the surface, however, above the reference crest location(indicated by the square symbols on the maps) the trajectories corresponding to the mid-cavity location remain close to
the right side of the triangle and follow the reference smoothwall case (‘rod’-like turbulence). For short ratchets, there areno significant differences between trajectories for the ratchetroughness in leeward and in windward orientation at locations
above z/δ ≈ 0.15.
For long ratchets, the windward oriented cases experienceat the mid-cavity location a stronger tendency towards twodimensional turbulence above the roughness and the differ-CONCLUSIONSThe structure of turbulence over ratchet-type roughness
has been analysed using local phase- and time-averaged
profiles of the turbulent kinetic energy and the invariant mapsof the anisotropy tensor. Both ratchet length and ratchetorientation influence the local profiles of the turbulent kinetic
energy and the anisotropic invariant maps. Flow visualisationsof the instantaneous wall-normal vorticity field show thatthe streamwise locations of elevated turbulence activity arestrongly influenced by the ratchet orientation. The influenceof ratchet length and orientation on further statistics, such asejections and sweeps, the wall-normal transport velocity of3

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

Figure 3. Turbulent kinetic energy profiles sampled at the apex location (a,b) and the mid-cavity (c,d) location for surfaces in leeward
orientation (a,c) and in windward orientation (b,d). Smooth wall data is shown for comparison. The vertical dotted line indicates theapex height.

the turbulent kinetic energy, and tracer particle statistics will
be considered in the next stage of this study.

ical simulation of turbulent flow over a rough surface basedon a surface scan. Computers & Fluids 116, 129–147.
Busse, A. & Zhdanov, O. 2021 Turbulent channel flow overratchet-type roughness. In Proceedings of the 13th International ERCOFTAC symposium on Engineering, Turbulence,
Modelling and Measurements (ETMM 13), Rhodes, Greece,
15-17 Sep 2021.
Chung, D., Hutchins, N., Schultz, M. P. & Flack, K. A. 2021
Predicting the drag of rough surfaces. Annual Review ofFluid Mechanics 53, 439–471.
Gao, X., Narteau, C. & Rozier, O. 2015 Development andsteady states of transverse dunes: A numerical analysis ofdune pattern coarsening and giant dunes. Journal of Geophysical Research: Earth Surface 120 (10), 2200–2219.
Jiang, H., Zhu, X., Mathai, V., Verzicco, R., Lohse, D. & Sun,C. 2018 Controlling heat transport and flow structures inthermal turbulence using ratchet surfaces. Physical Review
Letters 120 (4), 044501.
Napoli, E., Armenio, V. & De Marchis, M. 2008 The effect of
the slope of irregularly distributed roughness elements on
turbulent wall-bounded flows. Journal of Fluid Mechanics

ACKNOWLEDGEMENTS
This work used the Cirrus UK National Tier-2 HPC
Service at EPCC (http://www.cirrus.ac.uk) funded by theUniversity of Edinburgh and the Engineering and PhysicalSciences Research Council (EP/P020267/1). A.B. gratefully
acknowledges support via a Leverhulme Trust Research
Fellowship for this work.

REFERENCES
Ashrafian, A. & Andersson, H. I. 2006 The structure of turbulence in a rod-roughened channel. International Journal ofHeat and Fluid Flow 27, 65–79.
Busse, A., Lützner, M. & Sandham, N. D. 2015 Direct numer-4

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

Figure 4. Anisotropic invariant maps (AIM) based on Reynolds stresses measured at the apex location (a,b) and at the mid-cavity
location (c,d) for ratchets in leeward (a,c) and windward (b,d) orientation. For the trajectories based on Reynolds stresses measured at
the mid-cavity location, the wall-normal location corresponding to the ratchet height is indicated using squares. The inset panels show
the corresponding profiles of the anisotropy function F. Lines styles as in Figure 1.

613, 385–394.
Orlandi, P. & Leonardi, S. 2009 DNS of turbulent channel
flows with two- and three-dimensional roughness. Journalof Turbulence 7, 53.
Pope, S. B. 2000 Turbulent Flows. Cambridge University
Press.
Whalen, E. A., Broeren, A. P. & Bragg, M. B. 2007 Char-acteristics of runback ice accretions and their aerodynamiceffects. Tech. Rep. DOT/FAA/AR-07/16. U.S. Departmentof Transportation, Federal Aviation Administration.
Yang, J. & Balaras, E. 2006 An embedded-boundary formulation for large-eddy simulation of turbulent flows interactingwith moving boundaries. Journal of Computational Physics
215 (1), 12–40.

512th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 5. Structure of instantaneous ωz at approximately 12 wall units above the roughness crest for ratchets with length ℓ = 2k, 4k,8k, and 16k (from top to bottom) in leeward (left column) and windward (right column) orientation. The black dotted lines indicate thelocations of the roughness crests.

6