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

SLIP-TYPE BOUNDARY CONDITIONS FOR TURBULENT FLOWS OVER
HETEROGENEOUS ROUGHNESSJonathan Neuhauser, Kay Schäfer, Davide Gatti, Bettina Frohnapfel
Institute of Fluid Mechanics
Karlsruhe Institute of Technology (KIT)
Kaiserstraße 12, 76131 Karlsruhe, Germany
jonathan.neuhauser / kay.schaefer / davide.gatti / bettina.frohnapfel@kit.edu

Table 1: Grid properties. The box size Lx × Ly × Lz is8δ ×2δ ×4δ , except for cases with b/δ = 4 (Lz = 16δ ).

ABSTRACT
Turbulent flow over a surface with laterally varying
roughness is studied by means of direct numerical simulations(DNS) in a channel. It is known that this configuration gives
rise to secondary flows which affect the mean momentum andheat transport. We prescribe spanwise varying wall properties
by means of anisotropic slip, which allows to study both ridgeand strip-type roughness without the need for a geometricalrepresentation of the surface structure.

INTRODUCTION∂w∂ y y=0

∆y+
min

∆z+

180

256 × 193 × 128

5.625

1.001

5.625

540

768 × 451 × 384

5.625

1.003

5.625

Slip lengths are a well-researched model for textured surfaces, mostly in form of protrusion heights (Luchini et al.,1991). It has been established that a transversal slip increases
near-wall turbulence and thus skin-friction drag; while a longitudinal slip shifts the mean velocity profile (increasing the flowrate under CPG) (Min & Kim, 2004). A combination of spanwise and streamwise slip may serve as a virtual wall (Gómezde-Segura & Garcı́a-Mayoral, 2020). In order to relate the dimensionless spanwise slip length to well-known properties ofhomogeneously rough surfaces, a series of DNS simulations
at Reτ = 180 and Reτ = 540 were carried out, the results ofwhich are displayed in Fig. 2. The relative velocity deficit

We solve the Navier-Stokes equations for the incompressible flow of a Newtonian fluid subject to spanwise heterogeneous slip-length boundary conditions (Navier, 1823)

wy=0 = ls,w (z)∆z+

RESULTS
Spanwise slip

METHODOLOGY∂u,∂ y y=0

Nx × Ny × Nz

by means of direct numerical simulation (DNS). Streamwise,
wall-normal and spanwise directions are denoted through x,
y and z, respectively; the corresponding instantaneous velocity components are u, v and w. The problem considered is adoubly-periodic channel as sketched in Fig. 1 subject to constant pressure gradient (CPG) forcing. The boundary conditions employed at top and bottom wall are symmetric aroundthe channel centerline with a spanwise strip width s.
In order to characterize the correspondence of ls,w to traditional roughness measures, homogeneous reference cases are
considered. The DNS in this work are performed using thesolver XCompact3D (Bartholomew et al., 2020) at Reτ = 180
and Reτ = 540 (homogeneous reference only). In the streamwise and spanwise directions, the mesh is homogeneous; inthe wall normal direction stretching is used to refine the meshtowards the wall such that the first node is placed at y+ = 1.
The mesh properties are given in Table 1. All turbulent properties denoted with ⟨·⟩ correspond to averages that exploit all
symmetries of the set-up, including phase averaging. Additionally, · denotes quantities that are averaged in such a waythat they are
independent of the spanwise coordinate z, i.e.
1 R 2s+ indicates nondiu(y) = 2s
0 ⟨u⟩(y, z) dz. The superscript
mensionalization in wall units.

Secondary flows of Prandtl’s second kind, i.e. nonzero
mean values of the transverse velocity components are known
to occur in wall-bounded flows with spanwise inhomogeneousroughness distribution (strip-type roughness, Hinze (1967))
and streamwise aligned ridges (Vanderwel & Ganapathisubramani, 2015). Their strength amounts to a few percent of themean flow velocity only. Despite this relatively weak intensity, they can strongly alter the mean flow field, and, in consequence, the drag and heat transfer properties of a turbulent
boundary layer (see e.g. Stroh et al. (2020a)).
While the numerical resolution of surface roughness canbe achieved, it often induces stricter mesh requirements thanfor smooth wall simulations. Therefore, roughness models to
be used within flow-scale resolving simulations can enable theinvestigation of roughness effects at higher Reynolds numbers
and in complex flow scenarios. In the present study we employ
a slip length boundary condition to prescribe spanwise inhomogeneous roughness. Such a boundary condition has beenused as a model for textured surfaces such as drag-reducing riblets (Luchini et al., 1991), superhydrophobic surfaces (Min &Kim, 2004) and general rough surfaces (Luchini, 2013). Thisboundary condition is chosen because a nonzero slip length forthe spanwise velocity component enhances near-wall turbulence (Min & Kim, 2004), while at the same time leaving laminar flow unaffected. The present contribution aims to investigate the suitability of a boundary condition based on spanwise
slip for modelling spanwise heterogeneous roughness. In addition, we investigate to which extent a streamwise slip length
may be used to account for spanwise elevation differences.

uy=0 = ls,u (z)

Reτ

(1)

1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
∆U + = ⟨u⟩+ δ − ⟨u⟩+ δ ,NSBC monotonously increases withflow occupies only a small part of the channel (at the interface
between the patches) and has thus little influence on C f , so (2)holds again. Further elaboration on the interpretation of theskin friction coefficient in a spanwise inhomogeneous setting
is given in Neuhauser et al. (2022).
The results of the cases with spanwise slip are compared with strip-type roughness IBM data presented in Schäfer
et al. (2022). Their DNS simulation featured IBM-resolved
roughness strips, while the elevation of the smooth strip waschanged to investigate the influence of the roughness protrusion. A direct comparison between the results is shown in Fig.
5. The mean velocity profile, Reynolds stress and secondary
flow agree well, in particular with the IBM case of sligtly recessed roughness. (The IBM-resolved roughness has a mean
(meltdown) height of k̄/δ = 0.043, with tips protruding tokmax /δ = 0.1; the smooth surface is located at y = 0.07525δ ).
Shown alongside the IBM and SLBC results are the results ofa parametric forcing (PFA) model from Schäfer et al. (2022).
In general, the turbulence properties of the SLBC case show
good qualitative agreement with the ones found over slightly
recessed roughness strips (for PFA: h = 2.5k̄, displayed in Fig.
5(c), (f), (i)). The gray area represents the PFA modelledroughness area and the white area the elevated smooth wall.
Note that the friction Reynolds number is higher for PFA andIBM data (Reτ = 500) than for the present cases. While themore complex PFA model is able to capture the effects at thecorners of the roughness, the bulk of the flow is well captured
with the SLBC model as well.
The question immediately arises whether it’s possible tomodel different roughness heights or, by extension, strip-typeroughness (at least for small elevations) in a slip length framework as well. In this context, we investigate the combinationof spanwise and streamwise slip.

the slip length and collapses well for both Reynolds numbers.1
The strip-type configuration is investigated in a parameter study for different patch sizes s/δ = 0.25..16 at Reτ = 180+ = 9). The
with a spanwise slip length of ls,w = 0.05δ (ls,w
results reveal – for all strip size ratios – a pair of secondary
vortices that yields upwelling in the no-slip (smooth) region
and downwelling in the spanwise-slip (rough) region, independent of the strip width. The vortices are of approximately
square size, i.e. confined to a region near the wall in case ofnarrow strips (see Fig. 3(a)), and space-filling only for s ≈ δ(Fig. 3(c)). The momentum distribution is strongly dependent
on the strip width and shows different behavior in the limitingcases of small and large strips. For narrow strips (s = 0.25δ ),the influence of the inhomogeneity is limited to a region near
the wall; the outer layer is nearly homogeneous.
For s ≫ δ , the flow prefers the path of least resistance(i.e. the no-slip region) and consequently a high momentum
pathway is located over this region (see Fig. 3(b)). For s ≈ δ ,the secondary flow has the strongest impact on the momentum distribution in the channel. The secondary motion induces
an upwelling above the no-slip region, thus generating a low
momentum pathway in this region. The location of high- andlow-momentum pathways is therefore opposite for s ≈ δ ands ≫ δ . These results are consistent with the boundary layer
experiments by Wangsawijaya et al. (2020) and also partly inagreement with DNS results based on the prescription of wall
shear stress (Chung et al., 2018).
In contrast to those existing DNS studies the present configuration allows a direct evaluation of the skin friction coefficient which is significantly increased through the presence ofthe secondary flow for s ⪆ δ . The global skin friction coefficient is not only governed by the relationship between C f andls,w , but also by additional contributions from the secondary
flow. In order to quantify the latter, we assume that the skinfriction coefficient in flows with roughness strips but without
any secondary motion can be computed based on the limiting case of two non-interacting channel parts (one smooth and
one rough) that are both driven by the same pressure gradient (same uτ ). The identical pressure gradient in both channelpart ensures that no spanwise pressure gradient is generated
which in turn could drive a spanwise mean flow. The resulting flow rate for these two channel parts will naturally differ and the mean flow rate between the channels is given by
Ub = 12 (Ub,rough + Ub,smooth ). The definition of the skin friction coefficient C f = 2uτ 2 /Ub2 then yields an expected global
skin friction coefficient of

Cf =

1
2

1

1

p
+p
C f ,roughC f ,smooth

Spanwise and streamwise slip
Based on a virtual origin framework (Gómez-de-Segura& Garcı́a-Mayoral, 2020), a combination of slip lengths forthe three velocity components may be used to impose a wall
(rough or smooth) at a given elevation, as long as the elevation is small compared to the channel height. The spanwise
slip (also wall-normal slip, but this is not considered here) sets
the origin of turbulence, while the streamwise slip sets the origin of the mean flow. These virtual origins are the (typicallynegative) wall-normal lenghts by which the mean profile or
the Reynolds stresses would have to be shifted in order to collapsed with those of a canonical channel. If the origin of turbulence lT+ and the origin of the mean flow lU+ both are equaland positive, this corresponds to a recessed wall. (Increasing
lT+ further represents a rough recessed wall.) This can be used
to model ridge-type roughness, assuming that the height of theridges is below h+ ⪅ 10, because otherwise the near-wall turbulence cycle would be clipped by the simulation box. As
discussed in Gómez-de-Segura & Garcı́a-Mayoral (2020), the+ and l + are not identical, the underlyspanwise slip length ls,wT
ing saturation effect is also discussed in Fukagata et al. (2006)+ = 9 and l + = 13.5, suchand displayed in Fig. 2. For ls,us,w
a recessed wall is achieved. Indeed, such a combined boundary condition is able to reverse the direction of the secondary
flow, see Fig. 3(e). This matches IBM data from von Deynet al. (2021) for ridges of relatively small height; in the configuration of Fig. 3(e), the ridge corresponds to the no-slip partof the wall. It also bears resemblance to slightly protruding
roughness strips as seen in Stroh et al. (2020b) and Schäfer
et al. (2022), suggesting that the flow topology in such a caseis dominated by the protrusion and not by the roughness. In

!!−2
,

(2)

The obtained value in comparison with the computed C f fordifferent s/δ is is displayed in Fig. 4 in which C f of thesmooth and rough homogeneous reference cases is also included. which is the (p = − 21 ) power mean of the local skinfriction coefficients. For the limiting case of small patches,
eqn. (2) correctly predicts the global skin friction coefficient.
For patch sizes s/δ ≈ 1, the pronounced secondary flow is expected to increase the global skin friction coefficient, which isindeed the case. For very large patches s/δ ≫ 1, the secondary

1 Due to the relatively low Reynolds number, the difference of meancenterline velocity is used for the computation of ∆U + instead of evaluating the logarithmic law.

2

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

this case, no pronounced high- or low momentum pathways
can be recognized.
It is apparent that the case of a recessed wall (3(e)) can
be constructed as the superposition of the cases with only one+ = 13.5 (Fig. 3(c) displays l + = 9)component of slip, i.e. ls,ws,w+and ls,u = 9 (displayed in Fig. 3(d)). While it is difficult tointerpret the boundary condition for Fig. 3(d) as a physicalwall; in particular, there are no deflections of spanwise fluctuations, since the boundary condition for w is identical over both
strips. The results are still relevant as part of a decompositionof ridge-type roughness. The part of the channel with elevatedstreamwise slip (ESS) naturally has higher streamwise velocity⟨u⟩ at y = 0, but the velocity at the centerline is higher over the
no-slip region. The orientation of the streamwise secondary
vortices is consistent with this observation, i.e. downwelling(upwelling) in the z region of the high (low) momentum pathway, respectively. The ESS strips increase the overall flow rateof the channel, reducing the skin friction coefficient. The flow
response to the combined boundary condition (Fig. 3(e)) also
appears to be well-captured by a superposition of the cases
Fig. 3(c) and Fig. 3(d), which feature slip only in one direction, respectively. The streamwise velocity and skin friction
coefficient behaves as the average of the two cases, while thesecondary flow resembles the sum of the respective secondary
flow patterns.
By switching the relative position of the regions of elevated u and w slip, the secondary flow can be strongly enhanced; both effects – due to roughness and ESS – induce
identically oriented streamwise rolls. This is shown in Fig.
3(f). Such an increase in secondary flow magnitude should
correspond to an increase in skin friction, and indeed, the skinfriction coefficient is approx. 25% higher in case of Fig. 3(f)than in Fig. 3(e), even after correcting for the lower w slip
length.
Generally, the skin friction coefficient in absence of secondary flow can be computed with Eq. (2). The decrease ofskin friction in areas of streamwise slip must be taken into account, which avoids establishing an effective channel height.
The decomposition of the skin friction coefficient in this casewill be be further analyzed at the conference.
Note that Fig. 3(f) is not equal to a rough ridge; this would
require lT+ = lU+ in the region of streamwise slip.

We gratefully acknowledge the financial support of DFG
through project 237267381-TRR150 and support by the stateof Baden-Württemberg through bwHPC.

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 acartesian mesh 12, 100550.
Chung, D., Monty, J. P. & Hutchins, N. 2018 Similarity andstructure of wall turbulence with lateral wall shear stressvariations. Journal of Fluid Mechanics 847, 591–613.
von Deyn, L. H., Gatti, D., Frohnapfel, B. & Stroh, A. 2021
Parametric study on ridges inducing secondary motions inturbulent channel flow. Proc. Appl. Math. Mech. 20 (1),e202000139.
Fukagata, Koji, Kasagi, Nobuhide & Koumoutsakos, Petros
2006 A theoretical prediction of friction drag reduction in
turbulent flow by superhydrophobic surfaces. Physics of
Fluids 18 (5), 051703.
Gómez-de-Segura, G. & Garcı́a-Mayoral, R. 2020 Imposing
virtual origins on the velocity components in direct numerical simulations. International Journal of Heat and Fluid
Flow 86, 108675.
Hinze, J. O. 1967 Secondary currents in wall turbulence. Phys.
Fluids 10 (9), S122.
Luchini, P. 2013 Linearized no-slip boundary conditions at arough surface. Journal of Fluid Mechanics 737, 349–367.
Luchini, P., Manzo, F. & Pozzi, A. 1991 Resistance of agrooved surface to parallel flow and cross-flow. Journal of
Fluid Mechanics 228, 87.
Min, T. & Kim, J. 2004 Effects of hydrophobic surface onskin-friction drag. Physics of Fluids 16 (7), L55–L58.
Navier, C. L. M. H. 1823 Mémoire sur les lois du mouvement
des fluides. Mémoires de l’Académie Royale des Sciences
de l’Institut de France 6 (1823), 389–440.
Neuhauser, J., Schäfer, K., Gatti, D. & Frohnapfel, B. 2022
Simulation of turbulent flow over roughness strips. submitted to Journal of Fluid Mechanics .
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. 2020a
Secondary flow and heat transfer in turbulent flow overstreamwise ridges. International Journal of Heat and Fluid
Flow 81, 108518.
Stroh, A., Schäfer, K., Frohnapfel, B. & Forooghi, P. 2020b
Rearrangement of secondary flow over spanwise heterogeneous roughness. J. Fluid Mech. 885.
Vanderwel, C. & Ganapathisubramani, B. 2015 Effects of
spanwise spacing on large-scale secondary flows in roughwall turbulent boundary layers. Journal of Fluid Mechanics
774.
Wangsawijaya, D. D., Baidya, R., Chung, D., Marusic, I. &Hutchins, N. 2020 The effect of spanwise wavelength of
surface heterogeneity on turbulent secondary flows. J. Fluid
Mech. 894.

CONCLUSION AND OUTLOOKThe current study demonstrates the possibility to model
spanwise inhomogeneous roughness based on spanwise andstreamwise slip. The boundary condition correctly predicts
the orientation and size of the secondary flow and the location of low and high momentum pathways for strip-type roughness (spanwise slip only) and roughness consisting of shallow
ridges (combination of spanwise and spanwise slip). One important property of the proposed model for strip-type roughness is the fact that it does not alter laminar flow behavior.
It thus enables to study the impact of heterogeneous surface
roughness onto turbulence without artefacts that are generated
when the laminar flow behavior is modified through the boundary conditions as it is e.g. the case with the prescription of laterally varying shear stress. Additionally, the estimation of theskin friction coefficient C f is discussed. Further comparisons
between data for elevated/recessed roughness strips, (shallow)
ridge type roughness and the corresponding slip length models will be shown at the conference. Slip lengths thus provide a very simplified, but unified framework to study different
canonical roughness configuration in DNS.

3

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

Lz
∆U +

Lx

y
Ly

2
1

z

Reτ = 180Reτ = 540

x
s

s

s

0

no-slip

s

0

2

4

ls,w > 0⟨u⟩+ |δ ,NSBC for Reτ = 180 and Reτ = 540 and (solidline) Logarithmic approximation 1.38 ln(lsw + 1).

able slip length. The configuration shown here corresponds to strip-type roughness (Fig. 3(c)).

z/b

z/b
−2

0

2

4

0.8

0.6

0.6

0.4

0.4

0.2

0.2

ls+

y/δ

(a) s/δ = 0.25

−0.5

0

0.5

1

0.6

0.6

0.4

0.4

0.2

0.2

−0.5

0

0.5

1

(e) s/δ = 4

y/δ

−0.6

−0.4

−0.2

0

0.5

1

0

0.5

1

(b) s/δ = 4−0.5(d) s/δ = 4

0.8

9
0
−1
1

−0.8

9
0
−1
1

0.8

ls+

y/δ

(c) s/δ = 1

ls+

1

0.8

9
0
−1
1

10

Figure 2: Relative velocity deficit ∆U + = ⟨u⟩+ |δ −Figure 1: A sketch of the channel with spanwise vari-

−4
1

8

6
+
ls,w

9
0
−1
1

−0.5

(f) s/δ = 4

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

13.5
0

9
0
0 2 4 6 8 10 12 14 16 18 20 22 24
Streamwise velocity ⟨u⟩/uτFigure 3: Momentum distribution and secondary flow for different cases of piecewise constant slip length. Contours: time

averaged streamwise velocity in the yz plane. The secondary flow [⟨v⟩, ⟨w⟩]/uτ̄ in the plane is indicated with vectors, withequally scaled length among the plots. Below the plots the respective boundary configuration is given; solid line: ls,w ,dashed line: ls,u . (a), (b), (c): Different spanwise wavelenghts of ls,w ; the first two show the limiting behavior, the third anintermediate case. (d): u-slip only. (e) and (f) are combined cases; (e) represents recessed ridges based on a virtual origin
approach (lT ≈ lU ), (f) increases the strength of the secondary flow by constructive interference of the flow topology forstreamwise and spanwise slip only.
412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
·10−2
heterogeneous roughness
homogeneous roughness
smoothprediction of (2)

Cf

1.2

1

0.8
100

101
s/δ

Figure 4: Skin friction coefficient C f as a function of the strip width s/δ with step-wise change of the boundary condition++ = 9 and theand ls,w,max= 9 in the rough patch. C f for the standard smooth wall, the homogeneous rough case with ls,w
prediction based on Eqn. (2) are included for reference.

SLBC
z/s
−1 −0.5 0 0.5 1
1⟨u⟩= 16.6+max(a)⟨u⟩+ ; (⟨v⟩, ⟨w⟩)y/δIBM (h = 1.7k̄)z/s−1 −0.5 0 0.5 1

0.8

⟨u⟩+
max = 14.6|(⟨v⟩, ⟨w⟩)|+
max = 0.193(b)|(⟨v⟩, ⟨w⟩)|+
max = 0.193

PFA (h = 2.5k̄)z/s−1 −0.5 0 0.5 1⟨u⟩+max = 14.1|(⟨v⟩, ⟨w⟩)|+
max = 0.46555

(c)

0.6
0.4
0.2
0
⟨u⟩+

0
1

4

8

12 16

(d)

⟨u′ w′ ⟩+max = 0.292(g)⟨v′ w′ ⟩+max = 0.0543

0

4

8

12

16

(e)

⟨u′ w′ ⟩+max = 0.254(h)⟨v′ w′ ⟩+max = 0.310

0

4

8

12

16
·10−1

(f)

⟨u′ w′ ⟩+max = 0.641(i)⟨v′ w′ ⟩+max = 0.412⟨u′ w′ ⟩+
y/δ

0.8
0.6
0.4
0.2
0

y/δ

⟨v′ w′ ⟩+

·10−1

7
5
3
1
−1
−3
−5
−7
1.4
1
0.6
0.2
−0.2
−0.6
−1
−1.4

+
+ = 0 with reference data in form of IBM-resolved roughness from Schäfer
Figure 5: Comparison of SLBC (ls,w,max= 9, ls,u

et al. (2022). All cases at s/δ = 0.5; the simulations are arranged in columns. s/δ = 0.5. First row: Mean streamwise
velocity (⟨u⟩+ resp. (⟨u⟩ − ū(y = 0))+ for SSBC), secondary flow (vectors, same scaling in viscous units), second row:
⟨u′ w′ ⟩+ , third row: ⟨v′ w′ ⟩+ .

5