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

DYNAMICS OF SECONDARY MOTIONS IN TURBULENT CHANNELS
WITH STREAMWISE RIDGES

K. Schäfer, B. Frohnapfel, D. Gatti
Institute of Fluid Mechanics,
Karlsruhe Institute of Technology,
Karlsruhe, Germany
kay.schaefer / bettina.frohnapfel / davide.gatti@kit.edu

ABSTRACT

While many of the previous studies have investigated secondary motions in a time-averaged sense, a few recent studies
focused on the time-dependent properties of secondary motions, which might be masked by the averaging procedure(Vanderwel et al., 2019; Zampiron et al., 2020; Wangsawijaya
et al., 2020). To these characteristics belong, that high andlow momentum regions found in turbulent secondary motions
appear to meander in a manner similar to large-scale motion(LSM) and very-large-scale motions (VLSM) found in highReynolds-number flows above smooth walls (Zampiron et al.,2020). Additionally, secondary motions and LSM/VLSM arecharacterized by large-scale roll circulation in the instantaneous flow field, however the flow structures of the secondary
motions are spanwise locked at the surface transition and timepersistent (Wangsawijaya & Hutchins, 2022). The reason of
the spatial meandering of the secondary motions remains unclear as well as the dynamical characteristics of their instantaneous flow structures.
In the present study we aim to contribute to a better
understanding of these characteristics by performing direct
numerical simulations of turbulent open-channel flow withstreamwise-aligned ridges. The wide aspect ratio of the
adopted open-channel domain allows the occurrence of meandering of the VLSM. The spanwise spacing of the ridges isvaried, such that the different flow regimes found in the experiments by Zampiron et al. (2020) are reproduced and extended to larger spacings. The temporally and spatially highly
resolved data provide an excellent basis to better investigate
the spatial and dynamical properties of the instantaneous flow
structures associated with secondary motions. In addition, tostudy the influence of these flow structures on the heat transfer,an additional passive scalar is considered in the simulations.

The present study investigates secondary motions in turbulent open-channel flow with streamwise-aligned ridges bydirect numerical simulations. The large domain size chosen
for this study allows the presence of very-large-scale motions(VLSM) for smooth wall conditions. The spanwise spacingof the ridges S is systematically varied to investigate its influence on the interaction of secondary motions and VLSM, aswell as their coexistence. In the recent experiments by Zampiron et al. (2020) no contribution of VLSM is found for denseridge spacings S ≤ δ , where δ is the half-channel height, while
for wide ridge spacings VLSM and secondary motion can coexist. However, the spectra reveals a contribution of VLSM
for S = 2δ , which corresponds to the spanwise wavelength ofnaturally occuring VLSM and was not observed in the experiments, suggesting that coexistence of secondary motions and
VLSM might be possible for this ridge spacing.

INTRODUCTION
Many technical and environmental flows occur over spatially heterogeneous surfaces, while smooth surfaces tend tobe the exceptions. In particular, heterogeneous surfaces thatvary in the direction normal to the mean flow can cause theemergence of so-called turbulent secondary motion. These are
streamwise-aligned counter-rotating vortices and can significantly alter the momentum and heat transport of the turbulent flow (Stroh et al., 2020). Secondary motions were foundover various heterogeneous surface types, such as streamwisealigned ridges (Vanderwel et al., 2019; Zampiron et al., 2020)or alternating rough and smooth wall strips (Wangsawijaya
et al., 2020; Schäfer et al., 2022). It has been found, for instance, that the characteristic spanwise wavelength or distanceof the surface inhomogeneity influences the spatial extent of
the secondary motion in the time-spatial mean (Wangsawijaya
et al., 2020; Zampiron et al., 2020). For wavelengths of manytimes the outer length scale of the flow (the thickness of a turbulent boundary layer, for instance), the flow splits into regions
affected by the secondary motions (e.g., at the smooth-roughtransition or close to the ridges) and homogeneous flow regions not influenced by the wall inhomogeneity. If the wavelength is of the order of the boundary layer thickness, the secondary motion is the strongest (Wangsawijaya et al., 2020).
A further reduction of the spacing leads to a reduction of thespatial extent of the secondary motions which scales with the
spanwise distance S.

METHODOLOGY
The direct numerical simulation of the fully developed
turbulent open-channel flow is driven at a constant pressure gradient (CPG) corresponding to the effective friction
Reynolds numbers Reτ,eff. = 540, based upon the friction velocity uτ , the domain height δeff. and the kinematic viscosity ν.
The effective domain height δeff. takes into account the presence of the Gaussian ridges, which reduces the cross-sectional
area of the channel. In addition, a temperature transport
equation is solved, representing a passive scalar with Prandtl
number Pr = 1. The simulation domain is Lx × Ly × Lz =36δ × δ × 12δ and the number of grid points for discretiza-

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

12δ

u
δ

y
x
z

S

36δ

hg

Figure 1.

Schematic of open-channel flow domain with streamwise aligned Gaussian ridges.

tion is Nx × Ny × Nz = 3072 × 257 × 1152. In the wall-parallel
directions periodic boundary conditions are applied, while noslip boundary conditions are imposed at the bottom wall andsymmetry boundary conditions on the top wall. Streamwisealigned Gaussian ridges are placed on the wall of the channel,
as depicted in Figure 1, with a fixed ridge height hg = 0.1δ .
Three different spanwise spacings S of the Gaussian ridges are
explored with S/δ = 1, 2, 4. At the bottom wall, a constanttemperature is applied and at the top of the domain a zeroflux condition. In order to avoid a constant heating of the flow
volume and to reach a stationary state of the temperate field asource term is added to the temperature transport equation.
The incompressible Navier-Stokes equations are solved
with the open source code Xcompact3d (Bartholomew et al.,2020). Sixth-order compact finite difference schemes are used
for the spatial discretization and time integration is performed
with a third-order Runge-Kutta method. The representationof the Gaussian ridges is achieved by an immersed boundary
method (IBM) based on a high order polynomial reconstruction method to guarantee a smooth solution between fluid andsolid region (Gautier et al., 2014).
The velocity and temperature field is split by means of a
triple decomposition into:ui (x, y, z,t) = hui i(y) + ũi (y, z) + u00i (x, y, z,t)(1)

= hui i(y) + u0i (x, y, z,t)(2)

spanwise extent, while for S/δ = 0.5 and 1 the spanwise andvertical extent of the secondary motion cell is reduced, whichscales with S. The smooth wall case still displays some upwardbulging of the mean streamwise velocity at specific spanwise
positions, e.g. z/δ ≈ 2.5, which is the statistical footprint of
the slow dynamics of VLSM, which appear due to the limited
time integration for two-dimensional statistics. For the specific case S/δ = 4 an additional upward bulging of the mean
streamwise velocity can be observed in the center between adjacent ridges. A similar spatial inhomogeneity is also found
for the mean temperature field for all ridge cases, resulting inhigher temperature values which extend into the bulk region
above the ridges (not shown here).

Table 1.

where hui i is the temporal, streamwise and spanwise averaged
velocity, ũi is the coherent component, which is the spatial
fluctuation in z about hui i, and u00i is the random fluctuation.
The sum of coherent component and random fluctuation results in the turbulent fluctuation u0i . Streamwise and temporal
averages are indicated by · and spanwise averaging by angularbrackets h·i.

Global flow properties of simulation cases.

Reτ,eff.

S/δ

Reb

u+
b

C f (·10−3 )

Nu

540

∞

9845

18.22

6.03

15.05

540

12

9786

18.13

6.08

15.11

540

4

9612

17.80

6.32

15.31

540

2

9411

17.45

6.57

15.51

540

1

9233

17.11

6.83

15.85

540

0.5

9098

16.84

7.05

16.84

The instantaneous flow structures, such as low- and highmomentum regions, present for the smooth and ridge cases areshown in figure 3 for the instantaneous streamwise velocity
fluctuations u0 in the horizontal plane at the wall-normal position y = 0.5δ . For the smooth wall case S/δ = ∞ the spatial
meandering of the VLSM can be observed by long streamwise coherent low- and high-velocity streaks, e.g. at z/δ ≈ 10
a high-momentum region is found with a streamwise wavelength λx ≈ 20δ . For the largest ridge spacing S = 12δ a strong
and streamwise coherent low-speed streak is found at the spanwise position of the ridge, while at some spanwise distanceof the ridge the spatial meandering of the high-speed streaksof VLSM are visible again. The strength of the low-speed
streak at the ridge is amplified compared to the ones found forsmooth wall conditions, which is also the case for S = 4δ . For
S = 4δ less intense low-speed streaks with some streamwise
coherence can be observed between adjacent ridges. At denser
ridge spacing S = 2δ and S = δ the low-speed streaks are also
located in correspondence with the ridges and are pronounced,
however their streamwise extent is interrupted. The same iseven more evident for denser ridge spacings. Finally, for thedensest ridge spacing S = 0.5δ the streamwise coherence of

RESULTS
The introduction of streamwise aligned ridges in a turbulent channel leads to the formation of secondary motions.
Since the simulations are run at constant pressure gradient, theintroduction of Gaussian ridges leads to a decrease in bulk velocity u+b (superscript + represents viscous units) and consequently to a reduced bulk Reynolds number Reb = ub δeff /ν
and an increased friction coefficient C f = 2u2τ /u2b . Secondary
motions introduce a spanwise inhomogeneity of the mean flow,
as can be seen for the present Gaussian ridges in figure 2. The
upward motion of the secondary motion is centered at the ridge
position for all rough cases. For the largest spanwise ridge
spacings S/δ = 2, 4, 12 the secondary motion have their largest

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

Figure 2. Mean streamwise velocity in y-z cross section: (a) smooth wall; (b) S = 12δ ; (c) S = 4δ ; (d) S = 2δ ; (e) S = δ ; ( f )S = 0.5δ .

Figure 3. Instantaneous streamwise velocity fluctuations in horizontal plane at y = 0.5δ for smooth wall case S/δ = ∞ and different
spanwise ridge spacings S. The black horizontal lines at the outer right figure frame indicate the spanwise position of the ridges.

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

Figure 4. Pre-multiplied two-dimensional streamwise energy spectra κx κz Φu00 u00 as a function of the streamwise wavelength λx andspanwise wavelength λz at y = 0.5δ for smooth wall case S/δ = ∞ and ridge cases.

tribution of the mean streamwise energy at the same spanwise wavelength. For the denser ridge spacings S = δ and
S = 0.5δ this energy contribution for large λx cannot be foundanymore, which is consistent with the experimental observations of Zampiron et al. (2020). This indicates that a spanwise
ridge spacing which is in the range of spanwise wavelengthof the natural occurring VLSM allows spatial meandering oflarge-scale motion, while this is inhibited for S = 4δ and notpresent for denser ridge spacings. For S = 2δ an additionalenergy peak for smaller streamwise wavelengths λx ≈ 6δ andλz ≈ 1.8δ can be found, which also appears for denser ridge
spacings in a similar λz range.
To illustrate at which spanwise position the large-scale
motions occur, figure 5 shows the one-dimensional energy
spectrum κx Φu00 u00 across the spanwise coordinate z at two different wall-normal positions y = 0.25δ and y = 0.5δ . The
smooth wall case S = ∞ depicts large values for λx ≥ 10δ at all
spanwise position with some strong peaks due to slow dynammics of VLSM. Case S = 12δ in contrast, shows a clear separation of the energy for large λx , where no energy contribution isfound at the spanwise position of the ridge, while high energy
similar to the smooth wall case is found in the wide smoothwall region unaffected by the secondary motions. However,high energy contributions are found for λx ≤ 10δ at the spanwise position of the ridge. The energy deficit of large λx at
the ridges is also observed for S = 4δ at y = 0.25δ and partially at y = 0.5δ , where for instance large-scale contributions
are found on both lateral sides of the ridge at z = 2δ . Interestingly, high energy contributions are also found in the middle
between two ridges at y = 0.5δ for all λx , which depict similar values as the contribution at the ridges. For case S = 2δstronger energy peaks between 3δ ≤ λx ≤ 10δ occur at theridges. Unlike the coarse ridge spacings, energy contributions
from high λx now appear constantly at the ridges, while in thevalleys no contributions is present. The main contribution of
streamwise energy of case S = δ is in the range of intermediate
λx , while contributions of large λx are less frequent.
The one-dimensional energy spectrum κz Φu00 u00 across thewall-normal coordinate y is shown figure 6 (a). The generalshape of the energy spectrum is similar among the smooth walland ridge cases. The smooth wall case features an additional
high energy contribution at λz ≈ 1.8δ , which is not visible forthe low-speed streaks is broken and the spanwise ridge position cannot be inferred anymore. Comparison of the streamwise velocity fluctuations with the temperature fluctuations T 0
reveal a strong correlation between both fields and the position and spatial extent of the low-speed streaks coincides withintense values of T 0 (not shown here).
The contribution of the meandering VLSM and low- andhigh-velocity streaks in the instantaneous snapshots can bequantified by the two-dimensional power spectral density ofthe streamwise turbulent fluctuations Φu00 u00 in the streamwise
and spanwise direction, which is defined as

hu00 u00 i(y) =Z ∞Z ∞
00

Φu00 u00 (κx , y, κz ) dκx dκz ,(3)where the streamwise and spanwise wavenumber are definedas κx ≡ 2π/λx and κz ≡ 2π/λz and λx and λz are the streamwise and spanwise wavelength. Figure 4 presents the energy
spectra at the wall-normal position y = 0.5δ for the smoothwall case and ridge cases. Note that the energy concentrationof the streamwise mean (contribution of κx = 0) has been artificially placed at the streamwise wavelength λx = 72 to present
the contribution of stationary and non-meandering structures.
The inclined gray line indicates that an increase of streamwise
wavelength λx of the flow structures goes in hand with an increase of the spanwise wavelength λz up to λz ≈ 2δ .
The VLSM for the smooth wall case show a clear concentration of energy in the range of λz ≈ 2δ (indicated by
the dashed gray line) with two strong and distinct peaks at
λx = 18δ and λx = 36δ . The coarsest ridge case S = 12δ
also displays high energy concentration for λx = 18δ , where
one peak appears at slightly larger λz compared to the smoothwall case. These strong peaks for large λx cannot be found for
S = 4δ , however high values are now concentrated in the stationary component of the mean streamwise energy for wavelength λz = 2 and λz = 4, which corresponds to the spanwise
wavelengths of the low-speed streaks observed in figure 3 of
S = 4δ . This illustrates that VLSM cannot strongly meander for S = 4δ as it is the case for the smooth wall case and
S = 12δ . Interestingly, the meandering behaviour of VLSM
at λx ≈ 18 reappears for S = 2δ together with some con-4

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

Figure 5. Pre-multiplied streamwise energy spectra κx Φu00 u00 as a function of the streamwise wavelength λx and spanwise position z
at two wall-normal positions: top panel y/δ = 0.25 and bottom panel y/δ = 0.5. The spanwise position of the ridges is indicated by
the vertical lines at the outer top figure frame.

S = 12δ , S = 4δ and S ≤ δ . For S = 2δ a high energy contributions reappears in a similar range of λz , which extends tohigher wall-normal positions. As shown in the streamwise energy spectra in figure 4 and 5 at y = 0.5δ this contribution atλz ≈ 2 is related to flow structures with large streamwise wavelengths λx . To further illustrate the contribution of these verylarge scale structures across the wall-normal height the streamwise energy spectrum κz Φu00 u00 is integrated only for structureswith λx ≥ 10δ and presented in figure 6 (b). As can be seen
for the smooth wall case the additional peak in figure 6 (a) isdue to the contribution of the large-scale streamwise structures(λx ≥ 10δ ) with δ ≤ λz ≤ 2.2δ , which have high energy contributions up to y = 0.7δ . The range of the energy contribution
is similar for the coarse ridge spacing case S = 12δ , which features lower maximum values and with decreasing ridge spacing S the contribution area shrinks. Finally for the small ridge
spacings S ≤ δ no contributions of very-large scales is found.
The energy contribution of the time- and streamwiseaveraged secondary motions in figure 2, which do not meander in the streamwise direction, is not reflected in the energy contribution of u00 u00 . Similar to the energy spectrum of
the streamwise random fluctuation Φu00 u00 , the one-dimensional
power spectral density of the streamwise coherent component
Φũũ is defined ashũũi(y) =lapping with the contribution of large streamwise wavelengthsof the fluctuating random velocity contribution. In addition,
sub-harmonics with smaller wall-normal extent appear. For
S = δ the contributions at λz ≈ 2.3 vanish and the primary energy contributions of the stationary secondary motions is concentrated at λz ≈ δ . In addition, the wall-normal extent of thispeak is reduced compared to S = 2δ , which is consistent withthe reduced spatial bulging of the secondary motions for smallvalues of S in figure 2.

CONCLUSION & OUTLOOKThe present study investigates secondary motions in turbulent open-channel flow with streamwise aligned ridges. The
spanwise spacing of the ridges S is varied to study its influence on the interaction of secondary motions and very-largescale motions and their coexistence. The recent experimental
study of Zampiron et al. (2020) suggest that VLSM cannot
coexist with ridge-induced secondary motions for ridge spacings S / 2δ , but can coexist for wider ridge spacing as shown
for S ≈ 4δ . In agreement with these experimental findings,
the present DNS reveal the appearance of VLSM for wideridge spacing S = 12δ in the broad smooth wall region between the ridge-induced secondary motion. They meander
similarly to the VLSMs in the smooth channel. However,for S = 4δ we observe for the present case that the spectral signature of VLSM is weakened compared to the smoothwall and ridge case S = 12δ , and for this case the VLSM are
not able to strongly meander. This observation might be related to the smaller Reynolds number used for the simulations
compared to the experiments, since VLSM intensify with increasing Reynolds number (Hutchins & Marusic, 2007), andthereby might surpass the influence of secondary motions.
While VLSM are weaker for S = 4δ , the velocity spectrum
shows stronger contribution of turbulent structures in the center between the ridges for a similar range as for the secondary
motion with δ ≤ λx ≤ 10δ . This suggests that for S = 4δturbulent structures are also preferentially located between theridges, which contributes to an additional upward bulging of
the mean velocity in this region, which is however weaker than
the secondary motion (compare with figure 2).
Also in line with the experimental results no contribution

Z ∞0Φũũ (y, κz ) dκz ,(4)which represents the energy contribution of the stationary
secondary motions shown in figure 6 (c) (Wangsawijaya &Hutchins, 2022). The energy spectrum Φũũ does not show any
contribution for the smooth wall case, since no coherent motion is present in the time- and streamwise-averaged velocity
field. The coherent contribution for case S = 12δ is weak, because large parts of the spanwise domain are unaffected by thesecondary motions appearing at the single ridge. Interestingly,
for case S = 4δ an energy contribution of the coherent velocity
appears at λz ≈ 2.3δ instead of λz ≈ 4δ corresponding to the
spanwise spacing of the ridges. This is related to the appearance of the additional upward bulging between the ridges seenin figure 2. This energy contributions for λz ≈ 2.3δ is intensified for case S = 2δ showing a peak region at y ≈ 0.45δ , over-5

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

Figure 6. Pre-multiplied streamwise energy spectra κz Φu00 u00 as a function of the wall-normal position y and spanwise wavelength λz
for all streamwise wavelengths λx in (a) and only for λx ≥ 10δ in (b). The pre-multiplied energy spectra of the coherent streamwise
velocity κz Φũũ is shown in (c).

of VLSM is found for the two densest ridge spacings S = δ and
S = 0.5δ . However, for S = 2δ the spectra reveal contributionsof VLSM at large streamwise wavelengths λx and λz ≈ 2δ ,which appear closer to the ridges in the spanwise direction
than seen for larger S. The contribution of these structures(λz ≈ 2δ ) occurs over a higher wall-normal range comparedto the contributions for S > 2δ and is not connected to thewall. For this case the spanwise spacing S = 2δ corresponds
to the spanwise wavelength of VLSM naturally occurring inthe smooth wall case. This suggests that for this specific ridge
spacing, the appearance of the low-speed streaks at the ridge
location related to secondary motion allow the simultaneous
formation of VLSM. Of particular interest for future analysis
will be cases where secondary motions and very-large-scale
motions can coexist to investigate their similarities and differences.

Gautier, R., Laizet, S. & Lamballais, E. 2014 A DNS studyof jet control with microjets using an immersed boundary
method. International Journal of Computational Fluid Dynamics 28 (6-10), 393–410.
Hutchins, N. & Marusic, Ivan 2007 Evidence of very longmeandering features in the logarithmic region of turbulent
boundary layers. Journal of Fluid Mechanics 579, 1–28.
Schäfer, K., Stroh, A., Forooghi, P. & Frohnapfel, B. 2022
Modelling spanwise heterogeneous roughness through aparametric forcing approach. Journal of Fluid Mechanics
930.
Stroh, A., Schäfer, K., Forooghi, P. & Frohnapfel, B. 2020 Secondary flow and heat transfer in turbulent flow over streamwise ridges. International Journal of Heat and Fluid Flow81.
Vanderwel, C., Stroh, A., Kriegseis, J., Frohnapfel, B. & Ganapathisubramani, B. 2019 The instantaneous structure of secondary flows in turbulent boundary layers. Journal of Fluid
Mechanics 862, 845–870.
Wangsawijaya, D. D., Baidya, R., Chung, D., Marusic, I. &Hutchins, N. 2020 The effect of spanwise wavelength of
surface heterogeneity on turbulent secondary flows. Journalof Fluid Mechanics 894.
Wangsawijaya, D. D. & Hutchins, N. 2022 Investigation of
unsteady secondary flows and large-scale turbulence in heterogeneous turbulent boundary layers. Journal of Fluid Mechanics 934, publisher: Cambridge University Press.
Zampiron, A., Cameron, S. & Nikora, V. 2020 Secondary
currents and very-large-scale motions in open-channel flow
over streamwise ridges. Journal of Fluid Mechanics 887,
A17.

ACKNOWLEDGEMENT
This work was performed on the supercomputer ForHLR
II and the storage facility LSDF funded by the Ministry of
Science, Research and the Arts Baden-Württemberg and by
the Federal Ministry of Education and Research. The funding
through DFG project number 423710075 is greatly acknowledged.

REFERENCES
Bartholomew, P., Deskos, G., Frantz, R. A. S., Schuch, F. N.,
Lamballais, E. & Laizet, S. 2020 Xcompact3D: An opensource framework for solving turbulence problems on a
Cartesian mesh. SoftwareX 12, 100550.

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