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

RESTRICTED NONLINEAR SCALES OF TURBULENT SECONDARYFLOWS OVER SPANWISE HETEROGENEOUS ROUGHNESSBenjamin A. Minnick
Dept. of Mechanical Eng.
Johns Hopkins University
3400 N. Charles St.
Baltimore, MD, 21218, USA
bminnic2@jhu.edu

Xiaowei Zhu
Atmospheric & Oceanography Science
Princeton University
300 Forrestal Rd.
Princeton, NJ, 08540, USA
xz18@princeton.edu

ABSTRACT

Dennice F. Gayme
Dept. of Mechanical Eng.
Johns Hopkins University
3400 N. Charles St.
Baltimore, MD, 21218, USA
dennice@jhu.edu

high and low roughness regions (Willingham et al., 2014; Anderson, 2015; Bai et al., 2018; Chung et al., 2018). This bodyof work classified the non-zero time-averaged streamwise vorticity as the second kind of Prandtl’s secondary flow (Prandtl,
1952). This secondary flow is generated from spanwise variations in the Reynolds stresses and as a result induces transverse mixing that modifies momentum transfer in the boundary layer. These structures also lead to a secondary outer-layer
peak in the streamwise energy spectra (Wangsawijaya, 2020).

The natural scale separation in the restricted nonlinear(RNL) modelling approach is exploited to build upon recent
studies, e.g., Wangsawijaya (2020), that have used scale separation to provide insight into mechanisms underlying secondary motions in turbulent flow over spanwise heterogeneousroughness. In the RNL decomposition the large-scale comprises the streamwise averaged mean and the small-scales aredefined through a dynamical restriction that leads to computational tractability, while providing good agreement with salient
flow features. In agreement with the experimental work, ourresults indicate that energy of the large-scales is amplified overthe low roughness region due to the secondary flow. The
small-scales are shown to play a dominant role in the Reynolds
stresses responsible for generation of the secondary flow. Conditional averaging of the RNL mean field reveals stronger
momentum pathways over low roughness regions experiencing downwash in instances that differ from the time-averaged
trends. Further analysis of the large scale indicates that meandering of low speed streaks in the RNL flow is in response to
secondary flow momentum mixing.

Unsteady behavior of the momentum pathways and secondary flow was historically masked by time-averaging ofdata. However, analysis using correlations and conditional averaging of an experimental study of flow over herringbone riblets, uncovered meandering of the low-speed streaks (Kevin,2017). Subsequent work has attributed this meandering to a
streak-vortex instability described in Jeong et al. (1997), andlater observed in the outer-layer at larger scales (Kevin et al.,2019). There is also evidence that these large-scale streaks
modulate small-scales near the wall (Awasthi & Anderson,2018). This modulation is stronger in the LMP, as is the meandering.
Spanwise heterogeneous rough surfaces which effectively“lock” coherent streaks and rolls in a spanwise location andmay provide insight regarding smooth-wall boundary layers,
where such structures exist but are permitted to move freely.
A reduced-order model shown useful in the study of such coherent structures is the restricted nonlinear (RNL) model, seee.g. Farrell et al. (2017); Gayme & Minnick (2019). The
RNL representation decomposes the flow into a streamwise
averaged mean component that is permitted to interact nonlinearly and perturbations whose nonlinear interactions are limited to those contributing to the mean. RNL turbulence is selfsustaining despite its reduced streamwise varying perturbation
field (Thomas et al., 2014). The model also reproduces accurate turbulent statistics at moderate Reynolds numbers when
the perturbation field is restricted to wavelengths that coincidewith the outer-layer surrogate dissipation spectra (Minnick &Gayme, 2019).

INTRODUCTION
Turbulent flow over spanwise heterogeneous roughness
has long been of interest due to its relevance in a wide range
of applications. Although the problem has been well studied,a full understanding of how these surfaces affect the structuralproperties of the turbulent boundary layer has yet to be realized. This understanding is complicated by the difficulty inextending common techniques to evaluate skin-friction dragpredictions over homogeneous rough surfaces, such as the
Moody diagram or Hama roughness function (Nikuradse &Nikuradse, 1933; Moody, 1944; Hama et al., 1954) to heterogeneous rough surfaces. Even the assumption of outer-layer
similarity that turbulent statistics for flow over rough walls
are comparable to smooth-wall predictions (Townsend, 1980;
Schultz & Flack, 2005; Flack et al., 2007) does not apply.
Recent studies aimed at expanding the understanding ofthese flows have focused on how the topography affects flow
structure. In an early investigation along these lines, Barros & Christensen (2014) identify time-averaged alternating
low- and high-momentum pathways (LMPs and HMPs respectively) flanked by streamwise vortices over a turbine blade
eroded by foreign materials. This finding inspired others to
study flow over topographies with sharp interfaces between

In this work, we employ the RNL large eddy simulation (RNL-LES) model introduced in Bretheim et al. (2018) tostudy turbulent flow over spanwise heterogeneous strip roughness. First we demonstrate that this reduced-order model predicts time-averaged secondary flow and momentum pathway
trends consistent with observations in the literature. We thenuse the RNL decomposition to assess the contributions of thelarge-scale mean and small-scale perturbation field to the mix-1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
The streamwise (i = 1) and spanwise (i = 3) components ofthe total wall-stress are prescribed as,

ing and momentum transfer mechanisms in these flows. Our
RNL analysis approach enables us to also examine the dynamics of the structures, which is carried out using a conditional
averaging approach similar to that in Kevin (2017). The results
uncover similar meandering behavior, which suggests momentum mixing is primarily due to the secondary flow.

(w)

τi,2 (x, z,t) = −u2∗ũT,i (x, y1 , z),|u|where y1 is the first grid point from the wall,

THE RNL-LES MODEL
We take (x, y, z) to denote the streamwise, wall-normal,
and spanwise spatial coordinates with respective domain extents (Lx , Ly , Lz ). t denotes the temporal coordinate. The total
filtered velocity field, ũT (x, y, z,t), is decomposed into a mean
component, Ũ(y, z,t) = ⟨ũT ⟩x , where the angle brackets withthe subscript x indicate streamwise averaging, and a streamwise perturbation component, ũ(x, y, z,t). The governing equations consist of divergence free conditions, ∇ · Ũ = ∇ · ũ = 0,
and momentum equations,u∗ (z) =|u|, and |u|(z) =log(y1 /y0 (z))

qŨ12 (y1 , z) + Ũ32 (y1 , z).

This equilibrium wall-model differs from the traditional wallmodel used in LES in that |u| is the magnitude of the streamwise mean velocity, i.e. |U| rather than the total velocity. The
top boundary (y = δ ) is taken to be a stress-free, slip surface.
Periodic boundary conditions are applied in the streamwise
and spanwise directions.

∂t Ũ + Ũ · ∇Ũ + ∇P̃ + ∇ · ⟨τ⟩x = −⟨ũ · ∇ũ⟩x ,∂t ũ + Ũ · ∇ũ + ∇ p̃ + ∇ · (τ − ⟨τ⟩x ) = −ũ · ∇Ũ,NUMERICAL APPROACH
which are derived from the filtered Navier-Stokes equations with nonlinear interactions between perturbations areomitted (except those contributing to the mean), i.e.,⟨ũ · ∇ũ⟩x − ũ · ∇ũ = 0. This dynamical restriction achieves theorder-reduction wherein the modelled streamwise wavelengths
are isolated and only interact with the streamwise constantmean.
The sub-grid scale stress tensor is modelled using thestandard Smagorinsky model (Smagorinsky, 1963) with aneddy viscosity formulation for the deviatoric component of
this tensor

Simulations are performed in a half channel configuration
with domain extents [Lx , Ly , Lz ]/δ = [2π, 1, 2π]. The crossplane grid resolution used for all simulations is [Ny , Nz ] =[144, 144]. The non-zero streamwise wavenumber retained in
the dynamics is kx δ = 8, and corresponds to a point in theouter-layer peak region of the vorticity spectra of an LES withequivalent grid resolution for a homogeneous rough wall configuration as detailed in Bretheim et al. (2018). Here the dimensional streamwise wavenumber is defined, kx = 2πn/Lx
for some integer n.
We take advantage of the reduced number of streamwise
wavenumbers in the RNL-LES dynamics by solving the governing equations in (kx , y, z,t) space. This involves computing
the components of the nonlinear terms involving streamwiseinteractions as a convolution instead of as a product in physicalspace, which reduces the need to apply transforms and saves
computational time and resources, see Bretheim et al. (2018)
for details.
The governing equations are solved in the pseudo-spectral
code LESGO (https://lesgo.me.jhu.edu). Spectral derivatives are applied in the streamwise and spanwise directions,while second-order central finite differencing is used forwall-normal derivatives. The second-order explicit AdamsBashforth method is used for time-marching. The 3/2-rule isused for dealiasing. The pressure Poisson equation is solved
exactly by direct inversion in the streamwise and spanwise
Fourier space, then discretizing the resulting ordinary differential equation using finite differencing in the wall-normal direction and applying the tri-diagonal matrix algorithm.
We consider a topography comprised of three repeating
units, each of width S = (2π/3)δ and consisting of a high andlow roughness strip. The topography is shown in the contour
plot in Fig. 1, where black and white rectangles respectively
indicate the high and low roughness regions. Within each unit
the high roughness strip width is ℓ = 0.3125 S. This topography is imposed through an equilibrium wall-model wherey0,h = 0.001δ for the high roughness and y0,l = 0.01y0,h forthe low roughness. This particular topography was selected
because secondary flows have been reported to be dominant
for the spacing S ∼ δ (Wangsawijaya, 2020) and the particularroughness heights, y0,h and y0,l , are similar to those used inother LES studies, e.g. Anderson (2015).

1
τ − tr(τ)I = −2νe S̃T .
3Since the RNL methodology clearly defines streamwise scalesin terms of large-scales represented by the streamwise averaged mean and small-scales as the streamwise varying perturbations, subgrid-scales in the cross-plane are modelled using
the mean component. The eddy viscosity is therefore taken to
be using the product of the streamwise averaged filtered strain
rate tensor,
νe (y, z,t) = (Cs ∆)2

q2⟨S̃T ⟩x : ⟨S̃T ⟩x .

Mason wall-damping is used for the Smagorinsky coefficient,Cs ∆ =

n i−1/nh 1 n 
1
+
,
C0 ∆
κ(y + y0 (z))

with n = 2, von Kármán constant, κ = 0.4, and a spanwise
varying roughness height y0 to prescribe the spanwise heterogeneous roughness. The filter size is taken to be, ∆ =(∆y∆z)1/2 , instead of the traditional ∆ = (∆x∆y∆z)1/3 , andtherefore we use a slightly higher constant of C0 = 0.23 rather
than the common value of C0 = 0.16 (Bretheim et al., 2018).
The spanwise heterogeneous roughness on the bottom
boundary (y = 0) is modelled using the equilibrium wallmodel, following previous LES studies of spanwise heterogeneous roughness (Willingham et al., 2014; Anderson, 2015).

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

Figure 1: Time-averaged (a) velocity field and (b) swirl strength. The contour in (a) denotes the streamwise velocity field,

arrows indicate the cross-plane velocity field and do not represent grid resolution.

Figure 2: Instantaneous streamwise velocity field. Thick black lines in panel (a) show the borders of the high roughness

regions centered at black dashed lines. Shaded regions in panel (a) are shown up close in panels (b) and (c). The xz-planein (b) is taken at height y = δ /2.

Figure 3: Instantaneous streamwise velocity field, ũT decomposed into the streamwise averaged mean, Ũ, and streamwise
varying perturbation, ũ, components. The mean component is further decomposed into a time-average, Ũ, and fluctuation,
Ũ ′ , component. All quantities are non-dimensionalized by uτ .

RESULTSover the high roughness region, which is a consequence of theboundary condition imposed. This trend changes far from thewall, where HMPs and LMPs are observed over the high andlow roughness regions respectively. The formation of thesepathways is highly dependent on the secondary flow, visualized by arrows in Fig. 1a. A pair of time-averaged streamwise
rolls with alternating directions is observed for each high/low

Fig. 1 demonstrates that the RNL-LES model is capable of reproducing time-averaged features of turbulent flow
over spanwise heterogeneous roughness. The time-averaged
streamwise velocity reported in Fig. 1a shows similar behavior to experimental and numerical studies, see e.g., Anderson(2015). Near the wall, lower streamwise velocity is observed

312th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 4: Total Reynolds stress decomposed into RNL-LES streamwise averaged mean (large-scale) and streamwise varying perturbation (small-scale) components. Results shown over one repeating roughness unit.

roughness region, centered at y ≈ δ /2. The location and orientation of this secondary flow is comparable to results reported
in Anderson (2015), as is the intensity, visualized using theswirl strength in Fig. 1b. The ability of the RNL-LES model toaccurately predict the time-averaged secondary flow and corresponding momentum pathways is a result of the streamwise
constant nonlinearity retained in the model dynamics.

cross plane snapshots indicate that, as expected, the streamwise average mean velocity acts as a large-scale in that it effectively filters small scale activity. The perturbation field is
supported by a single streamwise wavenumber but full scalevariation is observed in the cross plane. The intensity of theperturbation component is particularly high just over the highroughness strip. The RNL mean component is further decomposed into a time-average, Ũ, and fluctuation, Ũ ′ , components
in the bottom row of Fig. 3. Note that since time-averaged
quantities are independent of streamwise location, ũT = Ũand ũ = 0. This decomposition reveals that while the timefluctuating component of the streamwise average mean contributes to the total fluctuations, it comprises larger cross-planescales than the perturbation component.

Fig. 2 shows a three dimensional perspective of an instantaneous snapshot of the streamwise velocity field at the
mid-plane of the half-channel. Here the boundaries of thehigh roughness regions are indicated by solid black lines. Aty ≈ δ /2 where the secondary flow is dominant, high and low
speed streaks represented by the RNL mean are shown to alignwith high and low roughness regions respectively. Althoughthese high and low speed large-scale structures are organizedpreferentially over the respective roughness regions, they meander instantaneously - a phenomenon introduced in Kevin(2017). Among these large-scale streaks, a single small-scalestreamwise wavelength, represented by the perturbation component, is shown to be prevalent. This perturbation component, which is loosely coupled to the mean, is thought to reenergize the large-scale streaks. In the cross-plane, the sum ofthese components is shown to produce multi-scale behavior.

Having illustrated the characteristics of the RNL meanand perturbation components for a given instance, we nextreport the contributions of each towards various Reynolds
stresses in Fig. 4. Since both the mean and perturbation components of the RNL model are time-fluctuating, they both contribute to the total Reynolds stress. Furthermore with only
two streamwise wavenumbers, kx δ = 0, 8, simulated in the
RNL-LES model these results are equivalent to modelled onedimensional spectra varying in the spanwise and wall-normal
directions. We therefore treat Reynolds stress involving thestreamwise averaged mean as spectra from cumulative largescales and Reynolds stresses involving streamwise perturba-

In Fig. 3 we show the RNL mean and perturbation components and how they contribute to the total streamwise velocity at a particular streamwise location and time instance. The

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

Figure 5: Conditionally averaged velocity fields based on direction of locally averaged wall-normal velocity performed
over the entire high roughness region (indicated by black rectangle) and low roughness regions (centered by solid
black line and bounded by dotted lines). Direction of flow in these regions indicated by +/- respectively denoting upwash/downwash. All five other possible combinations [(+,+,+),(+,+,-),(-,+,+),(-,+,-),(-,-,-)] not shown in panels (a-c) are
included in (d). Color bar for streamwise velocity contour same as Fig. 2.

neighboring low roughness regions. The spanwise boundaries
in which these local averages are performed, are indicated bysolid black lines in Fig. 5, with the center location indicatedby the dashed black line. The local averages are performed
over the entirety of the corresponding roughness region andthe wall-normal direction. It should be noted that this analysis can be sensitive to the chosen width of the local averages,
and the one chosen here is larger than the one used in Kevin(2017), however the overall trends illustrating the secondary
flow direction and streak meandering will remain relatively unaffected.
Fig. 5 shows the results from this conditional averaging
across a total of 70000 samples organized into four bins respectively shown in panels (a)– (d): bin 1 (+,-,+), bin 2 (,-,+), bin 3 (+,-,-), and bin 4, which includes the other fivecombinations. Here the +/- sign is used to denote the upward/downward direction of the locally averaged wall-normal
velocity of the streamwise averaged mean component in eachregion. The order of the combination denotes the location ofeach local average, from left to right: over the low roughness
strip, the high roughness strip, then the other low roughness
strip. Included in this figure is the percentage of samples used
to develop each conditional average. Additionally, we denotethe spanwise position in these figures as zc , taking zc = 0 to
denote the center of the high roughness region and zc = ±0.5S
as the center of the neighboring low roughness regions.
The case (+,-,+) which is most comparable to the timeaverage velocity field is shown to be predominant, more-so
than the case considered in Kevin (2017), however this is likely
due to the choice of conditional averaging width. Regardless,
it is clear that a flow behavior different than the time-averaged
sense occurs relatively often. Similar to the results in Kevin(2017), cases where downwash exists simultaneously over thehigh roughness region and a neighboring low roughness region, i.e., those denoted (-,-,+) and (+,-,-), occur often.
The downwash region of the (-,-,+) and (+,-,-) cases continue to show a time-averaged vortex, however reduced in size
and magnitude. Interestingly, the streamwise momentum inthis region is notably higher than the overall time-average fieldtions as spectra from small-scales.
The large-scale component of the streamwise Reynolds
stress, Ũ ′Ũ ′ , shows higher energy in the outer-layer over the
low roughness region near the center of the time-averaged secondary flow. This is consistent with observations in the experimental work of Wangsawijaya (2020), which suggests thesecondary flow supplies additional energy to the very largescale motions. The small-scale component, ũ′ ũ′ , contributes
little to this outer-layer energy, and only peaks near the wallparticularly over the high roughness region. A similar trend
is observed for the dominant Reynolds stress that acts in thestreamwise momentum equation, ũ′T ṽ′T . Both the RNL meanand perturbation components contribute to this Reynolds stressand, as expected, the small-scale perturbations exchange momentum more in the near wall region and the large-scale mean
exchanges momentum further from the wall. Similar to thestreamwise Reynolds stress, a larger exchange of momentum
is observed further from the wall over the low roughness strip,
particularly in the mean component, indicating the large-scalesecondary motions enhance mixing in this region.
The ṽ′T w̃′T Reynolds stress is of significance as its gradients are responsible for the production of the secondary flow
Anderson (2015). The RNL decomposition of this Reynolds
stress reveals the small-scale component, ṽ′ w̃′ , is dominant,particularly near the wall at the roughness transition. The
sharp sign change in this Reynolds stress at the roughness transition is not observed in the large-scale component, Ṽ ′W̃ ′ , suggesting only small-scales play a role in the generation of secondary flow.
Isolating the large-scales of this flow into the streamwise
average mean representation of the RNL decomposition enables structural features of the momentum pathways and secondary flow to be interrogated directly. We next exploit thisstructure in performing a conditional averaging analysis similar to that of Kevin (2017) to illustrate streak meanderingand other features masked by time-averaging. Instantaneous
flow samples are sorted according to the direction of the locally averaged wall-normal velocity of the large-scale streamwise mean centered over a high roughness region and the two512th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
indicating a conditionally averaged high momentum pathway
between the high and low roughness region coinciding withthe conditionally averaged downwash. This behavior, well described within the RNL framework, is a result of the streamwise component of the mean equation, U responding to thesecondary flow, (V,W ), as a passive scalar only subject tosmall-scale forcing. In other words, streak meandering andsecondary flow have a one-way coupling where the secondary
flow drives streak meandering. This coupling is imposed in the
RNL mean but given the similarity in behavior observed in experiments this type of coupling may persist in the full system.

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CONCLUSIONAn RNL-LES model is shown to reproduce the momentum pathways formed from secondary flow induced by spanwise heterogeneous strip roughness. The natural scale decomposition of this framework is then exploited to isolate the roleof large and small streamwise scales in these secondary motions. An analysis of scale dependent streamwise Reynolds
stress indicates that large-scale low speed streaks over the lowroughness accumulate energy from the secondary flow. The
small-scale component of the ṽ′T w̃′T Reynolds stress is observed to be dominant near the wall at the roughness transition, suggesting small-scales are primarily responsible for thegeneration of the secondary flow.
Conditional averaging based on the direction of the wallnormal velocity of the RNL mean component over the low
and high roughness regions is performed to quantify streak
meandering. Upwash over low roughness regions and downwash over the high roughness region, as observed in the timeaveraged flow field, is predominant however not the only instance observed. The other cases, that occur approximately aquarter of the time, show less intense secondary flow coinciding with an intensified streamwise momentum streak.
The RNL dynamics appears to provide a minimal representation of the streak-vortex instability model describedin Jeong et al. (1997) and reasoned to be the mechanism
for outer-layer streak meandering in spanwise heterogeneousroughness (Kevin et al., 2019). The construction of the RNL
model and its ability to reproduce this meandering suggests a
one-way coupling wherein secondary flow leads to meandering. Further analysis of this interaction is a direction of ongoing work.
Another possibility for future study further investigationof the role of small-scales and their response to streak meandering. The RNL-LES model could prove useful for such a
study as it readily represents smaller scales as a streamwise
perturbation component. It may also prove insightful to investigate this meandering behavior for varying high roughness
width to low roughness width ratio to further investigate thefindings in Wangsawijaya (2020), where spanwise wavelengthof the roughness was varied.

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Awasthi, A. & Anderson, W. 2018 Numerical study of turbulent channel flow perturbed by spanwise topographic heterogeneity: Amplitude and frequency modulation within
low- and high-momentum pathways. Phys. Rev. Fluids 3,044602.

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