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

MODELLING SECONDARY STRUCTURES DEVELOPING OVERLONGITUDINAL RIDGES WITH DIFFERENT GEOMETRIES

Gerardo Zampino, Davide Lasagna, Bharath Ganapathisubramani
Aeronautics and Astronautics
Faculty of Engineering and Physical Sciences
University of Southampton,
Southampton, SO17 1BJ, UK
g.zampino@soton.ac.uk

INTRODUCTION

the ridge geometries considered is reported in figure 1(a,b,c),
where S is the spacing between the ridges and W is the ridge
width. The entire methodology is based on decomposing theflow field into a homogeneous flow over the flat channel andthe flow perturbation induced by the ridges. Defining ε as
the peak-to-peak distance of the ridge, we expand any timeaveraged flow quantity q using a Taylor series in ε as

Secondary flows are generated when a lateral variationof the topography, such as streamwise aligned ridges, is imposed to a turbulent wall-bounded flow. In this case, the flow
field is characterized by vortical structures developing along
the streamwise direction known as Prandt’s vortices of the second kind (Prandtl, 1952). These surfaces have gained increasing importance in industrial applications because secondary
flows alter and modify the performances of fluid dynamics surfaces (Jiménez, 2004; Mejia-Alvarez & Christensen, 2013). It
has been shown that the flow organization and the strength ofsuch structures can depend on the shape of the ridges (Medjnoun et al., 2020) but how the geometry affects the generation mechanism of secondary structures and their strength is
not fully clear. Some authors (Wu & Christensen, 2007; Castro et al., 2021) observed that, in some cases, secondary flows
over different geometries appear to be similar and this suggests
a possible scaling of the vortical structures with a geometricalproperty of the surface. In particular, Castro et al. (2021) concluded that the strength of the secondary flows developing overrectangular ridges depends on the ratio between the ridge spacing and width and the flow organization is independent of thespacing when scaled with the channel height. Other scaling
geometrical parameters have been proposed in literature, suchas the ratio between the wetted area above and below the ridge
mean height by Medjnoun et al. (2020). In this paper, we address these questions by developing a tool based on linearized
Reynolds-Averaged Navier-Stokes (RANS) equations with the
Spalart-Allmaras (SA) turbulence model to close the system.
To contrast and compare the most common studied ridge geometries, we apply the present tool to elliptical, triangular andrectangular ridges.

q(x2 , x3 ) = q(0) (x2 ) + εq(1) (x2 , x3 ) + O(ε 2 ),(1)

where the zero-order term q(0) is the base flow solution in theflat channel and the first-order term q(1) is the flow response
per unit of ridge height. Following Russo & Luchini (2016),
in the limit of small ridges, we assume that the intensity of thesecondary currents is proportional to ε. Thus, in the definition (1) higher order terms in ε are neglected. For streamwisealigned ridges, we consider a streamwise invariant flow, thus
∂ · /∂ x1 = 0. Substituting the Taylor expansion (1) in the governing equations written using the streamwise velocity u1 andstreamfunction ψ formulation, and considering the terms at order one in ε, we obtain(1)∂ ψ (1)
1
−
Γ=
∂ x3
Reτ

∂2
∂2
+
∂ x22 ∂ x32

!

1
0=
Reτ

∂2
∂2
+ 2
2
∂ x2 ∂ x3

!2

(1)

u1 +

(1)

∂ τ12 ∂ τ13
+
,
∂ x2
∂ x3

(2)

ψ (1)

(3)
!
∂2
∂2
∂ 2  (1) (1) (1)
τ −τ22 +−τ ,+∂ x2 ∂ x3 33∂ x22 ∂ x32 23

where Γ is the zero-order streamwise velocity wall-normal gra(1)dient, Reτ = uτ h/ν is the friction Reynolds number and τi j
is the Reynolds stress tensor perturbation. Similarly, to close
the system of equations (2) and (3), the SA transport equation(1)
for the perturbation of the eddy viscosity νt induced by theridges is linearized (here omitted for brevity).
(1)(1)
The tensor τi j must be expressed as a function of νt
and the other mean quantities. As already discussed in literature of noncircular ducts (Perkins, 1970; Bottaro et al., 2006),
when the linear Boussinesq’s hypothesis is used, no secondary
flows can be predicted because the streamwise momentum
equation (2) and the streamfunction equation (3) are decoupled. Hence, a nonlinear Reynolds stress model is necessary

LINEARIZED RANS EQUATIONSA pressure-driven channel with streamwise aligned ridges
is studied. A symmetric channel configuration is here considered. In the following, the streamwise, wall-normal and spanwise directions, normalized with the channel half height h, areidentified by the cartesian coordinates (x1 , x2 , x3 ) respectively.
The coordinate system is centred on the channel mid-plane.
The flow is governed by the Reynolds-Averaged continuity and the momentum equations for the velocity components (u1 , u2 , u3 ) scaled with the friction velocity uτ . The oneequation SA turbulence model (Spalart & Allmaras, 1994) isused as closure model for the eddy viscosity νt . A sketch of1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
(downwelling) motion above the ridges (inside the troughs).
The secondary vortices occypy only a quarter of the channelheight and they are similar in size and strength for all threegeometries considered. Although some differences are predicted in the very near-wall region where the ridge geometry
affects the local flow organization, these differences are weakand negligible. For increasing ridge width, the vortices growin size and strength until they occupy the entire channel halfheight (central column). Some differences are here evident in
the strength of secondary flows and stronger downwash velocity is predicted at the gap centre for rectangular ridges. For
W = 1, very large turbulent structures are observed for all geometries (right panels). In particular, for rectangular ridges,
the secondary flows are locked at the ridge edge due to thestrong discontinuity introduced by the ridge geometry. The
same aspect is also observed for elliptical ridges where thesecondary flows develop in proximity of the ridge edge. By
contrast, for triangular ridges, the secondary currents occur at
the flank of the ridge. For this case, tertiary flows at the centreof the trough are predicted only for triangular ridges, and produce a local change of the flow direction. For specific spacings
and widths, tertiary flows can also be observed at the centre of
the trough for the other geometries when the gap between theridges is large enough to allow the turbulent structures to fully
develop. In fact, for large spacing, the vortical structures reach
their maximum size and a further increase of the spacing allows tertiary flows to emerge.
For similar reasons, tertiary flows can be predicted over
the ridges when the ridge width is large enough. This only occurs for rectangular and elliptical ridges, because of the sharp
discontinuity in the height of the ridge topography. By contrast, no tertiary flows over the ridge are observed for the triangular shape because the deflection of the spanwise velocity
component induced by the flank of the ridges is weaker.
Studies not reported here show that the Reynolds number
affects only the strength of the secondary flows but the flow organization is unaffected. More specifically, for high Reynolds
number, the solution of the present model becomes Reynolds
invariant because of the turbulence model used. In fact, theSpalart-Allmaras model (Spalart & Allmaras, 1994) is built
in order to obtain a collapse of the eddy viscosity profile in
the logarithmic layer for high Reynolds numbers. As a consequence, the eddy viscosity profile and the Reynolds stresses
are asymptotically Reynolds number independent when scaledwith the friction velocity. For this reason, the results forReτ = 1000 were here reported and they are representativeof the secondary flows generated by the ridges also at highReynolds numbers.

for the correct prediction of anisotropic stresses that are thesource of the secondary flows. In order to have a compact andsimple model to be manipulated, the Quadratic Constitutive
Relation (QCR) nonlinear stress model (Spalart, 2000) is here
integrated in the SA turbulence model. The Reynolds stresses
becomehi
τi j = τiLj −Cr1 Oik τ Ljk + O jk τikL ,(4)where Oik is the normalized rotation tensor and τiLj = νt Si j is
the linear Reynolds stress tensor from the Boussinesq’s approximation with Si j the mean velocity gradient tensor. The
constant Cr1 = 0.3 is calibrated to match the anisotropy inthe outer region over wall-bounded turbulent flows following
Spalart (2000).
The effect of the ridges is introduced using linearizedboundary conditions (Luchini, 2013; Busse & Sandham,2012). Expanding the velocity near the surface in a Taylor
series in x2 and enforcing the no-slip condition at the physicalsurface, the streamwise velocity component at the lower domain boundary is given by the inhomogeneous boundary condition
(0)

(1)

ui

x2 =−1+ f (x3 )

∂ ui
∂ x2

= 0,

(5)

x2 =−1

i.e. the perturbation velocity at the boundary of the numerical
domain is proportional to the wall-normal gradient of the velocity in the plane channel to preserve the no-slip condition on
the modulated topography. Similarly, inhomogeneous boundary conditions are derived for the perturbation eddy viscosity(1)
νt which vanishes at the physical surface. Finally, any periodic wall modulation with an unitary ridge height is modelledby the zero-mean modulation∞f (x3 ) = ∑ f n cos(nk3 x3 ),

(6)

n=1

where f n is the amplitude of the n-th wavenumber mode andk3 = 2π/S is the fundamental wavenumber. The flow field overany complex geometry can be reconstructed following the superposition principle. Since the first-order equations are linear,a complex solution is given by the constructive (or destructive)
interference of solutions at independent wavenumbers. The
entire methodology is extensively described in Zampino et al.
(2022).

Scaling of the dispersive stresses
In many experimental and numerical works, dispersive
stresses

RESULTS
Linear prediction of flow organization

(1)

(1)

σi j (x2 , x3 ) = ui (x2 , x3 )u j (x2 , x3 )Secondary flows are predicted over elliptical, rectangular and triangular ridges at Reτ = 1000 for duty cycle DC =W /S = 0.5 and W = 0.25 (left panels), 0.5 (central panels)
and 1.0 (right panels) in figure 1. The contour lines of theperturbation streamfunction ψ (1) and the colour map of the(1)wall-normal velocity perturbation u2 are also reported to better display the flow organization. Because symmetric channel
configurations are studied, subjected to a periodic wall modulation, only the bottom half-channel and a single ridge period is here shown. For small spacing (left column), vorticalstructures predicted for all three geometries show an upwellingare commonly used to study the generation of secondary flows.
In order to characterize their global strength using a single
scalar quantity, we introduce the integral
Rli j =Z SZ l0−l

σi j (x2 , x3 )dx2 dx3 ,(7)where we take l = 0.9 to discard the contribution in the nearwall region. The map of R0.9
12 is plotted in the panels (d,e,f)

2

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

Figure 1. Contour lines of the perturbation streamfunction ψ (1) at Reτ = 1000 and for elliptical ridges (top panels), rectangular ridges(central panels) and triangular ridges (bottom panels). Dashed lines are used for negative values. A sketch of the cases studied isreported in panels (a,b,c). The flow organization is shown for the bottom half channels in the region delimited by the red lines. The(1)colour map of the wall-normal velocity perturbation u2 is also reported to better display the flow organization above the ridges. The
duty cycle DC = 0.5 for all cases while the width varies from 0.25 (left column) 0.5 (central column) and 1 (right column). To help thereader, a simplified representation of the ridges is reported at the bottom of each plot (bold black lines).

0.9 (d,e,f) and scaled dispersive stress R̂0.9 (g,h,i) are reported as a function of the ridge spacing
Figure 2. Integral dispersive stress R12
12
S and ridge width W for Reτ = 1000 for elliptical ridges (top panels), rectangular ridges (central panels) and triangular ridges (bottom
panels). The dashed lines identify the peak amplification configuration at (S,W ) = (1.25, 0.67) A sketch of the cases studied is reported
0.9 is reported in panel (l) where the black contours are plotted for ∆R̂0.9 % = 2, 5, 10 and 15%. The
in panels (a,b,c). The quantity ∆R̂12
12
straight lines for W = 0.1S (blue) and W = 0.2S (red) are also shown.

3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
15% (top panels) and (S,W ) = (2, 1) where ∆R̂0.9
12 % ≈ 150%
(bottom panels).
For the first configuration, we predict two counterrotating vortices that occupy the entire half-height of the channel. The fact that the flow organization is similar for all geometries for regimes where the ridges are narrow and isolated
explains why the strength of secondary flows can be scaledwith the mean ridge height. If the ridge is wide or tightly
packed, for (S,W ) = (2, 1) (bottom panels), the flow topology
depends on the ridge shape and scaling the velocity components with the ridge mean height does not lead to a collapse of
the response. Generally, an upwash (downwash) is predicted at
the centre of the ridge (trough) for the three ridge geometries
but this is not sufficient to produce a scaling of the secondary
flows. The position and the orientation of the vortices are different.

of figure 2 for Reτ = 1000 and as a function of S and W .
For all cases studied, we predict i) a region of high amplification at (S,W ) ≈ (1.25, 0.67), denoted by the intersection of
the dashed lines, where the strength of the secondary flows is
maximum, and ii) high amplification along a line at constantW ≈ 0.67. However, the peak values across the three cases aredifferent (stronger for rectangular ridges), confirming that thestrength of secondary currents depends on the ridge geometry.
For high spacing S or width W , the quantity R0.9
12 for rectangular ridges also displays a second amplification peak not visible
for the other geometries and corresponding to the maximum
strength of the tertiary flows developing at the ridge centre (ortrough centre).
Despite the difference in the strength of the response, theflow organization observed for the narrow ridge case W = 0.2
shown in figure 1 is similar for all geometries, suggesting thatthe cross-stream velocities and dispersive stresses might bescaled using an appropriate geometrical parameter. In this particular regime, one can assume that the surface perturbation
generating the secondary currents is localised in a narrow region at the wall and the effect of the ridge is proportional toits cross-sectional area but not its geometry. For this reason,
we introduce the mean ridge height H as the ratio between
the cross-sectional area and the spacing S and use this quantity to scale the perturbation velocity field. In addition, since
the dispersive stresses are given by the product of the veloc0.9 is scaled with the square of
ity perturbations, the quantity R12

Velocity profiles
For a better characterization of the scaling of the secondary flows, the velocity components profiles divided by H
for elliptical, triangular and rectangular ridges are reported infigure 4. Three cases are here considered for a constant spacing S = 1.25 and varying width in order to display the scaling
behaviour for narrow and isolated ridges for W = 0.2 (left column), and the scaling breakdown for W = 0.67 (central column), corresponding to the maximum amplification configuration, and for W = 1.0 (right column), corresponding to wideridges. The wall-normal velocity profiles for the three geometries at the ridge centre and at the centre of the trough are reported in the top and central panels, respectively. The spanwise velocity component at the ridge edge is also provided in(1)
the bottom panels. The wall-normal velocity u2 /H at theridge centre (top panels) collapses in the far-wall region only
for W = 0.2 whereas for increasing ridge width the scaled profiles differ and no collapsing is observed. Close to the wall
(panel a), the velocity profiles show a peak value that dependon the ridge geometry. This is due to the proximity to theridge that locally modify the flow field. For increasing width(panel b and c), the velocity profiles differ. The ridge geometry affects the peak value (higher for triangular ridges). For
increasing W , the strength of the vortices decreases, too.
Similarly, the wall-normal velocity at the centre of thetrough is plotted in panels (d,e,f) of figure 4. These velocity
profiles collapse for the entire channel height only for the isolated ridge configuration (W = 0.2). To explain the collapse of
the profiles at the centre of the trough for narrow and isolated
ridges, one can observe that the secondary flows at the centre of the trough are not affected by the ridge geometry, since
the distance from the nearest ridge is large compared with W .
The peak value slightly changes for W = 0.67. For W = 1.0
where the gap is reduced, the velocity profiles are strongly dependent on the ridge geometry and its effect is not negligible
when moving towards the centre of the channel. However, acollapse of the curves is still observed in the near wall region
where the influence of the ridges is weak.
For all three geometries and for W = 0.2, the spanwise
velocity profiles in the bottom panels collapse along the entire
wall-normal direction. A negative peak is predicted at the wall
over the ridge edge where the Reynolds stresses are stronger.
Except for the case W = 0.2, the peak value depends on theridge geometry. Moving towards the channel centre, the velocity decreases in magnitude. For increasing width, the scaling
breaks down and some differences can be observed across thethree geometries.

2

0.9 = R0.9 /H obtained
the mean ridge height H. Maps of R̂12
12
for the three geometries are plotted in panels (g,h,i) of figure
2 as a function of the spacing S and the ridge width W . These
maps show strong similarities for the region characterized byhigh ratio S/W and small W . We define these two areas of theparameter space as the “isolated ridge” regime, where ridges
are small in width compared to the distance from one another,and the “narrow ridge” regime, where the ridge is narrow compared to the boundary layer thickness. To better visualise thisbehaviour, we introduce the quantity

0.9
∆R̂12
%=

0.9 − min R̂0.9
max R̂12
12· 100,
0.9
mean R̂12(8)

where the function “max” is the maximum value obtained
across the three geometries for a fixed configuration (S,W ).
Similarly, we define the functions “min” and “mean” as theminimum and the mean value, respectively. The quantity∆R̂0.912 % can be interpreted as the difference in secondary flows
strength across the three geometries for the same width andspacing. This quantity is plotted in figure 2(l) where selected
contours are reported for the difference of 2, 5, 10 and 15%,
to ease the visualisation. We can observe that the relative difference in scaled strength is small if the ridges are either narrow or isolated. If this condition is not met, for configurations
where the ridges are wide or tightly packed, the difference inthe scaled dispersive stresses across the three geometries increases.

Flow organization and scaling behaviour
In order to compare the flow response for the ridge configurations showing a scaling behaviour of the secondary flows
and the configurations where it does not hold, the contours of
the streamfunction ψ (1) and the map of the scaled wall-normal(1)velocity u2 /H are reported in figure 3. Two configurations
0.9 % ≈
are here considered: (S,W ) = (1.25, 0.2) where ∆R̂12

4

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

(1)

Figure 3. Colour maps of the wall-normal velocity u2 scaled with H for elliptical, triangular and rectangular ridges. The Reynolds
number is Reτ = 1000 for (S,W ) = (1.25, 0.2) (top panels) and (S,W ) = (2, 1) (bottom panels). The contour To help the reader, a
simplified representation of the ridges is reported at the bottom of each plot (bold black lines). The black lines are the contours ofconstant streamfunction ψ (1) (solid lines for positive values and dashed lines for negative values).

CONCLUSIONS

the effect of the ridge geometry is negligible or confined to thevery near-wall region and the scaled velocity profiles collapse.
Finally, the scaled velocity components profiles for different
configurations are also compared. The velocity profiles collapse only for the case corresponding to narrow and isolated
ridges. In particular, the collapse of the profiles at the centreof the trough can be easily explained as the consequence ofthe low influence of the nearest ridge. For increasing width,
the scaling behaviour breaks down because the flow topology
strongly depends on the ridge geometry.

The present approach provides an approximation of secondary flows produced by different ridge geometries using alinearized RANS based model. The model allows to quickly
predict the flow organization over a large variety of different ridge configurations, at a reduced computational cost.
Here, secondary flows over elliptical, triangular and rectangular ridges are investigated. We assume that any flow quantity
can be decomposed as a zero-order and a first-order term corresponding to the flow perturbation generated by the ridges. In
the limit of small ridges, linearized equations are obtained forthe perturbation of the streamwise velocity u1 and the streamfunction ψ and they are solved by imposing inhomogeneous
boundary conditions to model the effect of the ridges. Since
the equations are linear, the superposition principle is applied
to reconstuct the flow field over complex geometries. Resultsfor Reτ = 1000 are reported as a representative case.
The flow organization is displayed for the three geometries. Strong similarities in size of the secondary flows are observed for narrow ridges with small width compared with thespacing. For increasing width, the vortices grow in strengthand size until they entirely occupy the half channel height.
The flow organization differs and for wide ridges where thesecondary structures depend on the ridge geometry.
These similarities in the flow topology for narrow ridges
suggest that secondary structures can be scaled using an appropriate parameter that depends on the ridge geometry. Since
the ridges are localized at the wall, the effect of the wall perturbation is assumed to be proportional to the cross-sectional
area of the ridges not their geometry. For this reason, the meanridge height H is here proposed as the scaling parameter.
The integral of the dispersive stresses are here used tocharacterize the strength of secondary flows with a single
0.9 are are scaled with H 2scalar quantity. The maps of R12
and they collapse for the ridge configurations with small width
W , defining the ”narrow ridges” regime, and high ratio S/W ,defining the ”isolated ridges” regime. For these configurations

REFERENCES
Bottaro, A., Soueid, H. & Galletti, B. 2006 Formation of secondary vortices in turbulent square-duct flow. AIAA Journal
44, 803–811.
Busse, A. & Sandham, N. D. 2012 Influence of an anisotropic
slip-length boundary condition on turbulent channel flow.
Phys. Fluid 24, 055111.
Castro, I. P, Kim, J. W., Stroh, A. & Lim, H. C. 2021 Channel
flow with large longitudinal ribs. J. Fluid. Mech. 915, A92.
Jiménez, J. 2004 Turbulent flows over rough walls. Annu. Rev.
Fluid Mech. 36, 173–196.
Luchini, P. 2013 Linearized no-slip boundary conditions atrough surface. J. Fluid Mech. 737, 349–367.
Medjnoun, T., Vanderwel, C. & Ganapathisubramani, B. 2020
Effects of heterogeneous surface geometry on secondary
flows in turbulent boundary layers. J. Fluid Mech. 886, A31.
Mejia-Alvarez, R. & Christensen, K. 2013 Wall-parallel stereo
particle-image velocimetry measurements in the roughness
sublayer of turbulent flow overlying highly irregular roughness. Phy. Fluids 25, 115109.
Perkins, H. J. 1970 The formation of streamwise vorticity inturbulent flow. J. Fluid Mech. 44, 721–740.
Prandtl, L. 1952 Essentials of fluid dynamics. Hafner.
Russo, S. & Luchini, P. 2016 The linear response of turbulent

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

(a)

(b)

(c)

0

0

0

-0.5

-0.5

-0.5

-1

0

10

20

30

-1

40

(d)

0

5

10

-1

15

(e)

0

0

0

-0.5

-0.5

-0.5

-5

0

(g)

-1
-10

-5

-1
-10

0

(h)
0

0

-0.5

-0.5

-0.5

-10

0

10

20

-5

0

(i)

0

-1
-20

10

(f)

0

-1
-10

5

-1
-20

-10

0

10

20

-1
-20

-10

0

10

20

Figure 4. Scaled profiles of the wall-normal velocity component at the centre of the ridge (top panels) and at the centre of the gap(central panels). The spanwise velocity component is extrapolated at the ridge edge (bottom panels). The spacing is S = 1.25 and theridge width W = 0.2 (left column), 0.67 (central column) and 1.0 (right column). The profiles are obtained for the elliptical “Ellip”,
triangular “Tri” and rectangular “Rect” ridges for Reτ = 1000.

flow to a volume force: comparison between eddy-viscosity
model and dns. J. Fluid Mech. 790, 104–127.
Spalart, P. R. 2000 Strategies for turbulence modelling andsimulations. Int. J. Heat Fluid Flow 21, 252–263.
Spalart, P. R. & Allmaras, S. R. 1994 A one-equation turbulence model for aerodynamic flows. Rech. Aerosp. 1, 5–21.
Wu, Y. & Christensen, K. T. 2007 Outer-layer similarity in thepresence of a practical rough-wall topography. Phys. Fluid
19, 085108.
Zampino, G., Lasagna, D. & Ganapathisubramani, B. 2022
Linearised reynolds-averaged predictions of secondary currents in turbulent channels with topographic heterogeneity.
arXiv:2111.08459.

6