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

ON THE ORIGIN OF SECONDARY FLOWS IN TURBULENT BOUNDARYLAYERS OVER ROUGH WALLS

Ioannis K. Kaminaris, Elias Balaras,
Department of Mechanical and Aerospace Engineering
The George Washington University
Washington, DC, USA
kaminaris@gwu.edu; balaras@gwu.edu

Ralph J. Volino and Michael P. Schultz
United States Naval Academy
Annapolis, MD, USA
volino@usna.edu; mschultz@usna.edu

ABSTRACT

They found no correlation, however, between the locations of
the momentum pathways and the local topography. Instead
they noted that a correlation may exist with the leading part of
the roughness.
In the present study Direct Numerical Simulations (DNS)
of a turbulent boundary layer with zero pressure gradient overa rough wall are presented. Roughness is generated by staggered and random arrangements of truncated cones. The
roughness geometries and parametric space are identical to theones utilized in the experiments reported by Womack et al.
(2022). The latter is also used to validate the DNS. The study
aims to address some of the remaining questions about the nature and the origin of secondary flows over rough walls. Emphasis is given on their correlation to the topographical statistics. The impact of the secondary motions on the flow statistics
is also discussed.

Boundary layers with irregular roughness generate andsustain mean-flow heterogeneities in a cross-stream plane,
which manifest as high and low momentum pathways in themean streamwise velocity. Herein, to investigate their origin
and evolution we report direct numerical simulations of a turbulent boundary layer with a zero pressure gradient evolving
over truncated cones in staggered and random configurations.
A correlation of the momentum pathways with the topographical statistics is detected and mainly attributed to the leadingedge of the roughness and the strong presence of highly clustered areas throughout the roughness domain. The high andlow momentum pathways are present on both staggered andrandom topographies, but are much weaker and confined near
the roughness crest in the former case. In the random cases
they depart from the wall and approach the outer edge of theboundary layer.

Methodologies and parametric space
Introduction

Following Womack et al. (2022) the rough surface wascreated utilizing truncated cones in random and staggered configurations, which resemble barnacles found on fouled ship
hulls. For the staggered arrangement a coverage level of 39%(S39) is considered, while for the random we will report threecoverage levels: 10% (R10), 39% (R39) and 57% (R57). The
computational domain was also designed to match the experiment and is shown in Figure 1: the boundary layer develops
over a flat plate of 200D × 28D, in the streamwise (x) andspanwise (z) directions respectively (D is the base diameterof the truncated cone) and the freestream boundary is positioned 20D from the wall. The inflow boundary is positioned
upstream of the rough area to allow the boundary layer to develop over the smooth portion first. A turbulent boundary layer
with zero pressure gradient matching the experiment is specified as an inflow boundary condition which is extracted froman earlier computation over a smooth wall. The Reynolds
number at the inflow plane is set to Reθ = U∞ θo /ν = 1600
(θo = 0.16D is the momentum thickness at the inflow plane,
U∞ the freestream velocity and ν the kinematic viscosity of thefluid). The domain is discretized using 6000 × 1200 × 350 grid
points in the streamwise (x), wall-normal (y) and spanwise (z)In recent years several studies have focused on the presence of secondary flows over rough-wall boundary layers,which are mean flow features observed in the cross-stream
plane perpendicular to the dominant flow direction and havelong been studied in non-circular ducts. However, their origin
and correlation to the roughness topography remain open questions. Initially it was generally accepted that topographies withregular roughness arrangements generate stronger secondary
flows when compared to topographies with irregular arrangements. Mejia-Alvarez & Christensen (2013) showed evidenceof secondary flows over irregular topographies, which weredirectly associated to regions with mean streamwise momentum deficit or low momentum pathways (LMP), and regionswith mean streamwise momentum surplus or high momentum
pathways (HMP). In a followup study, Barros & Christensen(2014) correlated the spanwise locations of the LMP with recessed topography, and those of the HMP with elevated topography. Recently Womack et al. (2022) conducted an experimental study over regular and irregular topographies andobserved secondary motions similar to those of Mejia-Alvarez& Christensen (2013) for the case of random arrangements.

1

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

Figure 1: Computational setup corresponding to the experiments reported byWomack et al. (2022). (a) Precursor simula-tion to generate inflow boundary conditions; (b) Computational domain used for all production runs; (c) Truncated cones
used to generate all topographies.
directions respectively, resulting in ∆x+ = 11, ∆y+ = 1.9 and∆z+ = 9.3 (based on the average friction velocity uτ for thesmooth part of the plate. Each barnacle is discretized by approximately 36 × 42 × 67 points in the streamwise, spanwise
and wall-normal directions respectively.
The roughness elements were immersed in the Cartesian grid and the non-slip condition was enforced using an
immersed boundary formulation. In particular, an in-house,finite-difference, Navier–Stokes solver is utilized for all cases
presented in this work. All spatial derivatives are discretizedusing second-order central differences on a staggered gird.
Equations are advanced in time with a semi-implicit fractional
step method. A direct-forcing, immersed-boundary method
is used to impose the no-slip condition on the roughness elements. The details on the overall formulation can be found in
Yang & Balaras (2006).

(a)Results
All computations were initialized using smooth-wall data
and were advanced in time until the effects of the initial conditions were washed out. Time averaged statistics were accumulated over four flow-through times for all cases. We foundthis sample size to be a good compromise between convergence and cost. To validate the DNS we compare the mean
velocity profiles and velocity fluctuations to the experimental
measurements. Comparisons are done at the streamwise location (x/D = 103 from the leading edge of the roughness) where
the experimental measurements by Womack et al. (2022) are
reported. Figure 2 shows the mean velocity profiles for thetwo random arrangements: R10 and R57. Outer coordinates
are used to eliminate uncertainties coming from the computation of the wall stress. The agreements with the experiments isvery good. Figure 3 shows the corresponding velocity fluctuations u′ u′ , v′ v′ in the streamwise and wall-normal directions
respectively. Also in this case the agreement to the experiments is good and the observed discrepancies are within the(b)
Figure 2: Mean streamwise velocity profiles at x/D =103 in outer coordinates
DNS; experiment by
Womack et al. (2022). (a) random topography, R10; (b)random topography, R57.

uncertainty of the measurements and the statistical sampling
error in the simulations.
Having established confidence in the DNS results, next
we conducted detailed analysis of the secondary flow patterns

212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
in the random arrangements and none in the case of the staggered arrangements.
Close inspection of the the evolution of the HMP/LMP in
the boundary layer revealed that at their origin (i.e. the leadingedge of the roughness) they are located very close to the roughness elements, while as one moves downstream they grow,lift-off the wall and occupy a substantial part of the boundary layer. A three-dimensional view of the secondary flows
and related momentum pathways is shown in Figure 5, where
the mean cross-plane vectors normalised by their magnitudeand colored by the mean streamwise vorticity, alongside with
momentum pathways at a region close the experimental measurement location (x/D = 103) are plotted. It can be seen thatstrong, streamwise coherent counter-rotating secondary flows
are present on both sides of the HMP base accounting for thelateral momentum transfer: high momentum fluid is laterallytransferred from the HMP regions to the core of the LMP. The
counter-rotation of the secondary flows is demonstrated by thesign difference in the streamwise vorticity, consistent with the
spanwise alternation of the HMP and LMP. Similar behaviour
was observed in all random arrangements with different percentage coverage. For the staggered case (not shown here forsimplicity) the counter-rotating secondary flow motions followed the topography and were confined below the roughness
crest.
The origin of the momentum pathways and hence of thesecondary flows is still an open question. Barros & Christensen (2014) suggested that the LMP and HMP correlate withthe local recessed and elevated roughness, respectively. Womack et al. (2022), however, found no correlation between the
momentum pathways and the local topography and presented
evidence that these structures may originate from leading edgeof the roughness. To investigate this conjecture utilizing the
DNS database we first need to quantify the correlation between the topography and HMP/LMP. To capture a representative sample of the surface we define sample volumes thatcover the rough surface and have a streamwise and spanwise
length of ∼ 3d min (where d min is mean minimum distance toneighbor roughness elements), and varying height (i.e. 2 to
20 times the roughness height, k) depending of their streamwise location in order to overlap with the HMP/LMP in theboundary layer. Within each volume we compute averaged
local topographical statistics, as well as an average of ũ/Ue ,which depending on the sign indicates the presence of a low
or high momentum pathway. Their correlation is reported at aparticular streamwise location where consecutive volumes arearranged along the spanwise direction. Figure 6, for example
demonstrates the relation between the local topography andpresence of HMP/LMP. The latter is captured by the frontal
area coverage, λ f , which is computed as follows:

(a)

(b)
Figure 3: Normal Reynolds stress profiles at x/D = 103

in outer coordinates.
u′ u′ ,
v′ v′ , u′ u′ , ◦ v′ v′ ;Lines are from the DNS and symbols from the experiment by Womack et al. (2022). (a) random topography,
R10; (b) random topography, R57.

that may be present in this class of flows. In the experimental work by Mejia-Alvarez & Christensen (2013) and Womack
et al. (2022), for example, secondary flows were observed in
the cross-stream plane at the measurement station, and theirimpact on the momentum transport was clearly visible on the”bending” of the mean streamwise velocity isolines on theplane. This essentially generates ”channeling” of the mean
flow in alternating high and low momentum areas, typicallycalled HMP and LMP respectively. These are mean flows features and to visualize them one can plot the fluctuations of thetime averaged streamwise velocity in the spanwise direction:ũ = u− < u >z ,(1)

where . is the time averaging operator and < . >z is the
spanwise averaging operator. Areas of ũ > 0 correspond to
HMP and of ũ < 0 to LMP. In Figure 4 a top view of isosurfaces of ũ/Ue ∼ ±5% are shown for two cases of staggered(S10) and random (R39) arrangements. The momentum pathways detected over the staggered arrangement (Figure 4a) arevery weak and confined to the roughness crest throughout theboundary layer evolution. This may also be the reason thatthey have not been detected in the experiments by Womack
et al. (2022), as the first measurement point is typically above
the roughness crest. For the random cases, on the other hand,
HMP and LMP are very coherent and occupy over the full
extent of the plate in the streamwise direction (Figure 4b).
Clearly most of these structures originate directly from theleading edge of the roughness and remain fairly coherent in
the streamwise direction. These observations are in agreementwith the findings of Womack et al. (2022), where they also observed strong secondary flows throughout the boundary layer(n⃗u · ⃗S,Sf =0,n⃗u · n⃗s < 0, where ns = ⃗S/ ⃗S .
n⃗u · n⃗s ≥ 0(2)Note ns is the surface unit vector, nu is the unit vector in the
direction of the mean flow (i.e. x-direction) and ⃗S the surface
area vector. The frontal area coverage, λ f , is then computed
by:

λf =

Sf
100%, where So = n⃗o · ⃗S,So(3)

and no is the unit vector in the wall-normal direction. Figure 6a,c shows the relation between λ f and the momentum3

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

(a)

(b)
Figure 4: Top view of HMP/LMP visualized by isosurfaces of ũ:ũ/Ue = 5% HMP;ũ/Ue = −5% LMP. (a)staggered arrangement S39; (b) random arrangement R39.

Figure 5: Three dimensional view of the mean cross-plane velocity vectors in the vicinity of HMP and LMP (also shown

as isosurfaces) for the case of the random arrangement, R39. Velocity vectors are normalised by magnitude and coloredby the streamwise vorticity. Green circles mark the counter-rotating vortices.
pathways (captured by ±eu) at the leading edge of the roughness for both random cases considered: R10 and R57. It isclear that high positive values of ue, representative of HMP, areonly present in areas where λ f = 0. Large negative values ofue, representative of LMP on the other hand, are only present inareas of large λ f . The high levels of correlation between the
topography and momentum pathways are the leading edge are
not presnet further downstream. Figure 6b,d shows the same
quantities further downstream at x/D = 103. No such correlation is observed and LMP can be found in non-clustered or
lower frontal area coverage areas. The same applies to HMP
that are also found in high frontal area coverage areas. This
result supports the conjecture that HMP/LMP, at least for thisparticular family of roughness topographies, are generated at
the leading edge of the roughness and remain coherent for several boundary layer thicknesses downstream. In addition, thedownstream topography just underneath does not significantly

affect their evolution.
To further investigate the role of the leading edge topography on the structure of the momentum pathways over thewhole roughness patch we conducted several numerical experiments around the R39 case and: i) changed the distributionof λ f along the spanwise direction at the leading edge of theroughness by rearranging the truncated cones over a small strip
while keeping the coverage the same (the topography further
downstream remained unchanged); ii) removed all roughness
elements downstream of the leading edge (see Figure 7c). In
the former case (not shown here) the locations of HMP/LMP
shifted to coincide with the new distribution of λ f and established a similar correlation with the one demonstrated in
Figure 6a,c. A visualization of HMP/LMP for both cases isshown in Figure 7. Despite the elimination of the downstream
roughness, the momentum pathways are still formed and their
original spanwise locations are preserved. It is noted that al-

4

0.0
z/D

7.0

8.00
3.75
0.50
4.75
9.00
14.0

(a)

-7.0

0.0
z/D

f%
f%

51
38
25
13
0
-14.0
50
37
25
12
0
-14.0

7.0

12.0
4.5
3.0
10.5
18.0
14.0

-7.0

0.0
z/D

7.0

6.00
3.25
0.50
2.25
5.00
14.0

(b)

u/Ue %

f%-7.0

u/Ue %

14
10
7
3
0
-14.0

7.0

9.00
2.75
3.50
9.75
16.00
14.0

u/Ue %

20
15
10
5
0
-14.0

u/Ue %f%

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

-7.0

0.0
z/D

(c)

(d)

Figure 6: Correlation of the momentum pathways with the topography underneath.

Frontal area coverage, λ f ;HMP; LMP. (a) random topography (R10) at the leading of the roughness; (b) random topography (R57) at the leadingof the roughness; (c) random topography (R10) at x/D = 110; (d) random topography (R58) at x/D = 110.

(a)

(b)

(c)

(d)

Figure 7: Frontal view of (a) the original random, R39 topography configuration, (c) the LE of the roughness only. Isosur-

faces of the time-averaged spanwise fluctuations of the streamwise velocity for the values of ũ/Ue = 5% HMP
andũ/Ue = −5% LMP
in the case of the random, R39 topography; for (b) the original arrangement. (d) the LE of theroughness only.

though the strength of the momentum pathways seems to beaffected by the absence of the downstream topography, theirstreamwise coherence is still significant and extends throughthe whole roughness domain (i.e. 60δo , where δo is the incoming boundary layer thickness). Furthermore, the stream-

wise coherence of the momentum pathways observed in theoriginal arrangement can be found in the current modified caseexactly the same in lower strength values (e.g. |ũ/Ue | ≈ 2%).
The latter observations strongly contribute to the idea that theobserved momentum pathways are entirely defined by the very5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
teracting with moving boundaries. J. Comput. Phys 215 (1),
12–40.

upwind part of the roughness and that the downstream topography may slightly affect their strength.
The impact of these secondary flows on the structure of
the boundary layer over a rough wall remains an open question. Womack et al. (2022), for example, observed heterogeneity among the mean streamwise velocity profiles along
the spanwise direction highlighting the effect of the secondary
flows in the mean flow statistics. They noted that the biggest
deviation occurs in the random arrangements, while quick convergence within one roughness height, k, above the roughness
crest was detected in the case of staggered arrangements. The
same behaviour is observed in the DNS reported here. Mean
velocity profiles at the same streamwise location and Reynolds
number, are shown in Figure 8 for the cases of the staggered(S39) and random (R10 and R57) arrangements. The velocity
profiles for the staggered arrangement (Figure 8a) show verysmall deviation with this scaling, while significant difference
almost across the whole boundary layer height is observed forthe random arrangements (Figure 8b-c). This is an indicationthat the outer-layer similarity breaks down in these cases, andthe strong secondary motions are the primary contributor.

U/Ue

1.0
0.9
0.8
0.7
0.6
0.5
0.4

0.4

0.7 1.01.3

0.4

0.7 1.01.3

0.4

0.7 1.01.3(y-d)/( -d)
(a)

U/Ue

1.0
0.9
0.8
0.7
0.6
0.5
0.4

0.1

Summary and Conclusions
We present DNS of a zero pressure gradient boundary
layer evolving over different configurations of truncated cones.
Excellent agreement is achieved for the flow statistics compared to the experimental work by Womack et al. (2022). Presence of significant and coherent secondary flows due to roughness irregularity is demonstrated for the cases with random topographies. The lateral momentum transfer mechanism between the momentum pathways is investigated and attributed
to counter-rotating vortices lying in the base of the HMP. Correlation of the locations of the HMP and LMP with the lowerand higher values of the surface statistics respectively in theleading edge of the roughness is demonstrated. This correlation is very weak further downstream.

(y-d)/( -d)
(b)

U/Ue

1.0
0.9
0.8
0.7
0.6
0.5
0.4

0.1

REFERENCES
Barros, J. M. & Christensen, K. T. 2014 Observations of turbulent secondary flows in a rough-wall boundary layer. J.
Fluid Mech. 748, R1.
Mejia-Alvarez, R & Christensen, K. T. 2013 Wall-parallel
stereo particle-image velocimetry measurements in theroughness sublayer of turbulent flow overlying highly irregular roughness. Physics of Fluids 25, 129901.
Womack, K. M., Volino, R. J., Meneveau, C. & Schultz, M. P.
2022 Turbulent boundary layer flow over regularly and irregularly arranged truncated cone surfaces. J. Fluid Mech.
933, A38.
Yang, Jianming & Balaras, Elias 2006 An embedded-boundary
formulation for large-eddy simulation of turbulent flows in-

0.1(y-d)/( -d)(c)Figure 8: Mean streamwise velocity profiles in outer co-ordinates at x/D = 110 for various spanwise coordinates. (a) staggered topography (S39); (b) random topography (R10); (c) random topography (R57).
represents the spanwise-averaged velocity profile.

6