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

SLIP AND DRAG IN TURBULENT FLOWS OVER SUPERHYDROPHOBIC
SURFACES WITH SURFACTANT

Samuel D. Tomlinson
Department of Mathematics
University of Manchester
Oxford Rd, Manchester M13 9PL, UK
samuel.tomlinson@manchester.ac.uk

Ruby Gans
Department of Mechanical Engineering
University of California, Santa Barbara
Santa Barbara CA 93106, USA
rubygans@ucsb.edu

Frederic Gibou
Oliver E. Jensen
Department of Mechanical Engineering
Department of Mathematics
University of California, Santa Barbara
University of Manchester
Santa Barbara CA 93106, USA
Oxford Rd, Manchester M13 9PL, UK
fgibou@ucsb.edu
oliver.jensen@manchester.ac.uk

Julien R. Landel
Department of Mathematics
University of Manchester
Oxford Rd, Manchester M13 9PL, UK
julien.landel@manchester.ac.uk

Paolo Luzzatto-Fegiz
Department of Mechanical Engineering
University of California, Santa Barbara
Santa Barbara CA 93106, USA
fegiz@engineering.ucsb.edu

François Peaudecerf
Institute of Environmental Engineering
ETH Zürich
8063 Zürich, Switzerland
peaudecerf@ifu.baug.ethz.ch

Fernando Temprano-Coleto
Department of Mechanical Engineering
University of California, Santa Barbara
Santa Barbara CA 93106, USA
ftempranocoleto@ucsb.edu

ABSTRACTINTRODUCTION

Superhydrophobic surfaces (SHSs) can reduce friction
drag in turbulent flows. In the laminar regime, it has beenshown that trace amounts of surfactant can negate this drag reduction, at times rendering these surfaces no better than solid
walls (Peaudecerf et al. (2017)). However, surfactant effects
on the drag-reducing properties of SHSs have not yet beenstudied under turbulent flow conditions. Predicting the effectsof surfactant in numerical simulations remains expensive by
today’s standards. We present a model for turbulent flow, ineither a channel or boundary layer, over a periodic array of
longitudinal ridges (of period P and gas fraction φ ) inclusiveof surfactant. To acquire an expression for the drag reduction,
we adopt a technique based upon a shifted log law. The homogenised streamwise and spanwise slip lengths are derived
by introducing a local laminar model within the viscous sublayer, whereby the effect of surfactant is modelled by modifying the streamwise slip length. We compare the predictions ofour model with numerical and experimental results from theliterature and discuss the potential effect of surfactants on turbulent flows over SHSs. Our model agrees with available data
for small P+ (subscript + denotes wall units), showing a clear
departure at large P+ , where the drag reduction data asymptotically approach φ for P+ → ∞.

SHSs owe their performance to a combination of hydrophobic chemistry and surface roughness, which acts toentrap gas layers in their surface and reduce the drag whencompared to solid walls. Harnessing this feature in turbulent flows could benefit numerous marine and industrial applications. Maritime shipping alone contributes to over 2%of CO2 emissions, and over 13% of NOx and SOx emissions(Smith et al. (2015)), with up to 80% of the energy expended
to overcome friction drag (Fukuda et al. (2000); Xu et al.
(2020)). Early investigations into the laminar regime modelled the SHS as a mixture of no-slip and shear-free boundaries(where the liquid-gas interface is assumed flat), thereby predicting large reductions in drag (Rothstein (2010)). However,more recent experimental studies in laminar flow conditions
have shown that trace amounts of surfactant, unavoidable inpractice, can strongly impair the drag-reducing effect of SHSs
(Kim & Hidrovo (2012); Bolognesi et al. (2014); Peaudecerf
et al. (2017); Song et al. (2018)). Motivated by these experimental findings, laminar theories have been developed forthe fluid and surfactant in channels bounded by SHSs (Landel et al. (2020); Temprano-Coleto et al. (2021)), where it has
been shown that model predictions can be improved relative tosurfactant-free theories.

1

W
Marangoni
force

P

φ=W/P

Gas

ViscoussublayerSurfactant
gradient

Uw
λx

∂U
∂y w

SHS

Surfactant

Log layer

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

Figure 1. The mechanism by which the presence of surfactant can negatively impact the drag reduction for a flow over a SHS, withperiod P, gas ridge (plastron) width W and gas fraction φ = W /P. A build up of surfactant at the downstream stagnation point induces anadverse Marangoni force due to the reduction in surface tension (Peaudecerf et al. (2017)). The adverse Marangoni force acts to reduce
the spanwise-averaged streamwise slip length λx and slip velocity Uw at the interface. The smaller spanwise-averaged streamwise slip
length (or slip velocity) reduces the drag reduction when compared to a surfactant-free flow over a SHS.

Numerous studies have investigated the performance ofSHSs, both experimentally (Daniello et al. (2009); Park et al.
(2014); Xu et al. (2021)) and computationally (Park et al.
(2013); Türk et al. (2014); Rastegari & Akhavan (2015)), in
turbulent flows without added surfactant. A recent reviewof 14 experiments shows broad inconsistencies (Gose et al.
(2018)); the drag reduction ranges from −90% (i.e. drag increase) to +90%, with five studies finding little (< 20%) orno drag reduction. There exist a number of possible causes
for this disagreement in the literature due to the complicatedphysics associated with flows over SHSs, which are discussed
in detail in Park et al. (2021). As an example, the liquid-gas
interface at the SHS can deform due to pressure differencesin the bulk fluid and gas cavity, which has been shown to alter the drag reduction in laminar flows depending on the signand magnitude of the protrusion angle (Teo & Khoo (2009)).
Alternatively, turbulence may induce partial or complete wetting of the plastron containing the gas subphase, where theflow would no longer benefit from a shear-free surface (Rastegari & Akhavan (2019)). We neglect both these features of
SHSs here, as apart from in certain asymptotic limits (see, forexample, Crowdy (2017)), the laminar solutions found in theviscous sublayer would need to be found numerically.

longitudinal slip length is commonly defined as

λx =

Uw
,
γw

(1)

where Uw is the spanwise-averaged slip velocity at the SHS
boundary y = 0 and γ w is the spanwise-averaged shear rate
at y = 0 (see Figure 1 and 2). The average longitudinal slip
length λx represents the extrapolated distance, below the SHS,
where the time-averaged longitudinal velocity U vanishes (seeFigure 1).
Secondly, for flows under the constant flow rate condition(equivalent to U = U0 , where U and U0 are the cross-plane
average velocities for the flow over a SHS and a solid wall
respectively), we define the drag reductionDR =τ0 − τ
,τ0(2)

where τ is the spanwise-averaged wall shear stress of the SHS
flow, and τ0 is the spanwise-averaged wall shear stress of the
no-slip flow. Alternatively, the SHS and no-slip flows can be
driven by imposing the same constant pressure gradient, suchthat the average shear stresses at the boundaries in both flows
are equal (Türk et al. (2014)). To evaluate the global performance of a SHS geometry, the relationship between the dragreduction and the relevant independent non-dimensional parameters is sought in the form DR = f (Re, P/H, φ ), where Re
is the Reynolds number, P is the pitch, H is the channel height(which is replaced by the boundary layer thickness δb , if weare considering external flows such as Park et al. (2014); Xu
et al. (2021)) and φ = G/P is the gas fraction, with G the plastron width (see Figure 2).

This study aims to investigate the potential effects of surfactant in turbulent flow conditions, for both internal and external geometries, over longitudinally ridged SHSs (see Figure1). We consider longitudinal ridges only as they are generally considered optimal when compared to transverse ridges,
pillars or random surface roughness (Park et al. (2021)). We
assume the liquid-gas interfaces are flat and can be approximated as shear-free, which allows us to relate the geometry of
the SHS to the drag reduction using a technique based on ashifted log law (Fukagata et al. (2006)).

MODEL

SHS PERFORMANCE

As noted earlier, turbulent flows can be characterised by
two regions of variation: an inner viscous sublayer and anouter log-law layer (see Figure 1). We assume that the viscous sublayer thickness, of order 10δτ (where δτ = ν/uτ is

There are two main quantities of interest that characterize
the local and global performance of the SHS flow compared toa no-slip flow over solid walls. Firstly, the spanwise-averaged2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
the period of the SHS (see Figures 1 and 2), then
λx+ =

 

P+
πφ
ln sec,π2(5)where λx+ is modelled using steady unidirectional laminar flow
(Philip (1972); Türk et al. (2014)), since the flow in the viscous
sublayer is dominated by viscosity. Various approaches havebeen employed to prescribe g(λz+ ). The saturation as λz+ →∞ is captured using a formula inspired by riblet technologies(Luchini et al. (1991); Busse & Sandham (2012))g(λz+ ) =the viscous length and Uτ = (τ/ρ)1/2 is the friction velocity), is much larger than the SHS texture period P. Thus, theSHS texture affects the turbulent bulk flow via averaged quantities only, such as the average longitudinal and spanwise slip
lengths (λx and λz ). Our turbulent flow model over the SHS isbased on a semi-empirical modification to the log law, where
the mean velocity profile, U + (y+ ) in wall units (+), is shifted
up by an average slip velocity at the interface and down due
to turbulent momentum transfer (in accordance with Fukagata
et al. (2006))
1
ln(y+ ) + B + ∆U + ,κ(3)where κ ≈ 0.41 is the von Kármán parameter, B ≈ 5.3 is anempirical constant, and ∆U + is the combined shift relative tothe no-slip flow. For the no-slip flow we instead have the classical log law (Pope (2000))
U0+ (y+
0)=

1
ln(y+
0 ) + B,κ

(6)An alternative relationship for g(λz+ ) was proposed by Fukagata et al. (2006), whereby g depended exponentially on λz+ .
We choose to employ the relationship of Busse & Sandham(2012) because of its simplicity and of its connections withthe literature on drag reduction using riblets (Luchini et al.
(1991)).
We close the above model by assuming that both λx+
and λz+ are based on the laminar solution (5) due to Philip(1972), for longitudinal and transverse shear flows over parallel grooves, such that λz+ = λx+ /2 with λx+ in (5). A similar
closure hypothesis was made for flows over riblets (Luchini
et al. (1991); Ibrahim et al. (2021)). The normalisation of
the average longitudinal slip length in wall coordinates is welldefined through λx+ = λx /δτ . However, the normalisation of
the average spanwise slip length in wall coordinates, λz+ , ismore subtle (Türk et al. (2014); Seo & Mani (2016)). We assume that the spanwise velocity fluctuations at the origin of
the spanwise turbulent momentum transfer scale with the longitudinal velocity fluctuations. This assumption is commonly
made for wall turbulent boundary layers (Pope (2000)). This
assumption implies that the outer flow is homogenised in sucha way that the average and fluctuating bulk shear stresses in thelongitudinal and spanwise directions are of the same order ofmagnitude. Therefore, we can normalise both the longitudinaland spanwise average slip lengths using the viscous length δτ .
To summarise, we can determine the bulk Reynolds number of the SHS flow Re from the shifted log law (3), employing
the analytical solutions due to Philip (1972) that describe thestreamwise and spanwise slip length in terms of the SHSs geometry, φ , P and H (or δb ). Next, we can evaluate the bulk
Reynolds number of the no-slip flow Re0 from the standard
log law (4). The expressions for the SHS and no-slip Reynolds
numbers are substituted into the constant-flow-rate assumption, Re = Re0 , which when combined with (2) allows one toevaluate the drag reduction.
To incorporate surfactant effects in turbulent flows over
SHSs, we modify the slip lengths to account for Marangonistresses generated by surfactant adsorbed on the air-waterinterfaces. We leverage recent progress on the modellingof surfactant-induced Marangoni stresses on SHSs for twodimensional and three-dimensional periodic geometries (Landel et al. (2020); Temprano-Coleto et al. (2021)). By linearising for small concentrations of surfactant, we can relate theaverage streamwise slip length λx to the average Marangoni
shear rate along the interface γMa , by solving two-dimensionalboundary value problems using conformal mappings (Philip(1972)). The surfactant thereby affects the log law through achange in ∆U + . As γMa increases, the surface becomes immobile and as γMa → 0 we recover the nominal slip length for

Figure 2. Schematic of the channel flow configuration. An
incompressible liquid flows through a plane channel assumed
infinitely wide in the transverse z direction. Top and bottom
walls are made of long, parallel, periodic superhydrophobic
ridges, such that the liquid is in the (suspended) Cassie-Baxter
state. A shear-free condition is assumed at the plastrons and ano-slip condition is assumed at the ridges. The time-averaged
fully-developed flow velocity in the longitudinal x direction U
is assumed invariant with x, and periodic in the z direction withperiod P.

U + (y+ ) =−4λz+.
4 + λz+(4)

where the subscript 0 implies that these variables have instead
been normalised with wall units that are associated with theflow over a solid wall. For the boundary layer flows consideredherein, the log laws (3)–(4) could be extended to include awake function (Pope (2000)). However, if we assume that thewake function is the same over both a SHS and solid wall, thenthese terms will cancel in the drag reduction calculation.
The combined shift ∆U + has been shown to be well represented by the streamwise slip length (which is equal to theslip velocity in wall units) and by a function of the spanwise slip length, which we denote λx+ and g(λz+ ), such that∆U + = λx+ + g(λz+ ) (Fukagata et al. (2006); Busse & Sandham (2012)). When the viscous sublayer is thick compared to3

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

Experiments

Figure 3. Comparison of our model with (a) numerical and (b) experimental results in the literature, for different gas fractions, φ ,where the average Marangoni shear rate γMa = 0 and surfactant based effects are neglected. DR is the drag reduction, P+ is the pitch
in wall units and Re is the Reynolds number. Open symbols and dashed lines represent laminar simulations and theory in panel (a),whereas, filled symbols and solid lines represent turbulent simulations and theory in panels (a) and (b).

a clean (surfactant-free) interface. The dependence of γMa on
the geometry and properties of fluid and surfactant can be attained using the scaling theory detailed in Landel et al. (2020)and Temprano-Coleto et al. (2021). We study the effect of
surfactant by changing the average Marangoni shear rate γMa ,thus modifying the average streamwise slip length λx+ .

the thickness of the viscous sublayer is reduced and the SHSs
will have a greater effect on the streamwise velocity (that acts
to reduce the drag) whilst reducing spanwise momentum transfer (that acts to increase the drag). Returning to our model, we
propose taking the drag reduction to be the minimum valuemin(DR, φ ), which improves model predictions significantly
for large P+ .
We also performed numerical simulations for turbulent
channel flows over SHSs on a dynamically adapted forest of
Octree grids in a parallel environment based on Egan et al.
(2021) - see the black data point in Figure 3(a). An extensionof this algorithm, which couples the velocity and pressure tobulk and interfacial surfactant evolution, is currently under development to give further insight into the role of surfactants inturbulent flows over SHSs.
We compare the drag reduction predicted by our modelwith experimental data in Figure 3(b) for Re ∈ [103 , 105 ] andφ ∈ [0.31, 0.96]. The experimental works of Park et al. (2014)
and Xu et al. (2021) consider external flows, for which a turbulent boundary layer thickness must first be obtained in orderto evaluate DR. We assume that the turbulent boundary layer
thickness may be approximated by the classical result from turbulent boundary-layer theory (Schlichting & Gersten (2003))

RESULTS AND DISCUSSIONIn Figure 3(a) we observe a comparison between the dragreduction predicted by our model (excluding surfactant, suchthat the average Marangoni shear rate γMa = 0) and direct numerical simulations (exclusive of surfactant), as a function of
the SHS period in wall units P+ ∈ [0, 200] and gas fraction
φ = 0.5, 0.75, 0.88 and 0.94. We note a regime transition
from those laminar simulations which do not vary with P+(open symbols and dashed lines), to those with a thick viscous sublayer (filled symbols and solid lines), where the dragreduction increases rapidly with P+ . The transition from laminar to thick viscous sublayer turbulent flow is associated withthe formulation of a turbulent boundary layer, characterised by
the viscous length scale δτ . Due to the limited amount of numerical data for φ > 0.5, it is not possible to comment on thedependence of the transition on the gas fraction. In Figure 3(a),the drag reduction increases with increasing gas fraction. The
agreement is good for 0 ≲ P+ ≲ 50, however, for P+ ≳ 50, another regime transition seems to occur, with the drag reduction
tending to saturate, in contrast with the model which increases
to 100% drag reduction for P+ → ∞. This is not unexpected,
as our model assumes that the viscous sublayer thickness, oforder 10δτ , is much larger than the SHS texture period P.
As previously noted, a limitation of the above model isthat DR → 1 as the Reynolds number Re → ∞. Recent simulations, however, suggest that the drag reduction saturates forincreasing P+ (Park et al. (2013); Türk et al. (2014); Rastegari & Akhavan (2015)) - see Figure 3(a). It seems reasonable
to assume that in the limit as Re → ∞, there exists an upperbound given by DR → φ , as also discussed by Daniello et al.
(2009); Rothstein (2010). As the Reynolds number increases,

δb =

0.37x
Rex 1/5(7)

where Rex = Ux/ν is the Reynolds number (U is the bulk
velocity and ν is the kinematic viscosity). Equation (7) isevaluated at a fixed central location down the test section, x,
in order to calculate the drag reduction. This highlights animportant difference in external flows, where the ratio δb /Pvaries greatly comparing the brown curves of Park et al. (2014)
and the grey curves of Xu et al. (2021). The large change inδb /P is due to the test section being much longer in Xu et al.
(2021), which causes the drag reduction to be much smaller
even though φ = 0.9 does not change. Our model captures thevariation of drag reduction with respect to the gas fraction in4

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

(b)
0.6

0.8
0.7

0.5
0.6
0.4
0.5
0.4

0.3

0.3
0.2
0.2
0.1
0.1
0
0.2

0.4

0.6

0
0

0.8

2000

4000

6000

8000

Figure 4. Comparison of our model with the experimental results in turbulent boundary layers (a) Park et al. (2014) and (b) Xu
et al. (2021), for different average Marangoni shear rates, γMa . DR is the drag reduction, φ is the gas fraction and Reτ0 is the friction
Reynolds number (normalised with wall units associated with the flow over a solid wall). Filled symbols and solid lines represent
turbulent experimental data and theory respectively in panels (a) and (b).

DRModel to the drag reduction predicted by experimental data
DRData in Table 1. We see that for the data in Park et al.
(2014) and Xu et al. (2021), the predictions for small γMa givea smaller root mean squared error than those for a clean channel γMa = 0. However, any decrease in εRMS is not statistically significant and lies within the experimental uncertaintyof the results. The limited data and lack of experiments including surfactant makes the experimental results difficult tointerpret. More accurate experiments are therefore required toinfer whether surfactants will be important in turbulent applications, as several additional features could be involved andcausing the changes in drag; e.g. liquid-gas interface curvature, the gas subphase, loss of plastron, or ridge alignment(Park et al. (2021)). Our findings also call for turbulent direct numerical simulations inclusive of surfactants, which, aspreviously mentioned, we are currently pursuing.

the experiments of Park et al. (2014) (see the green, orange,
yellow, brown and pink curves in Figure 3(b)). There is significant spread in the original experimental data of Daniello et al.
(2009) which could be due to a number of features of SHSs,
e.g. liquid-gas interface curvature, the gas subphase, loss ofplastron, ridge alignment (Park et al. (2021)). In Figure 3(b),we show an ensemble average of the DR data extracted from
Daniello et al. (2009) over Re, in a similar manner to Park et al.
(2021), in order to ease the comparison between this data andthe other experiments.
We investigate the effect of average Marangoni shear rate
γMa on the model predictions, where we find that the introduction of surfactant impairs the drag reduction DR, but does
not generally improve the model predictions when compared
to a clean channel - see Figures 4 and Table 1. One would expect the effect of surfactant to be more prominent in fieldwork
than in a laboratory setting, where the water is relatively clean.
Surfactant effects are quantified via the root mean squared error εRMS comparing the drag reduction predicted by our model

Park et al. (2014)

CONCLUSIONS
Motivated by recent developments that demonstrate theimportance of surfactants in laminar flows over SHSs (Kim
& Hidrovo (2012); Bolognesi et al. (2014); Peaudecerf et al.
(2017); Song et al. (2018); Landel et al. (2020)), we have proposed a model for turbulent flow that includes surfactant basedon a shifted log law (Fukagata et al. (2006)). We consider
both channel and boundary layer flows over SHSs, in order
to compare with the wide range of numerical and experimental results in the literature. The model assumes that the viscous sublayer thickness is much larger than the SHS texture
period P and, therefore, that the SHS texture affects the turbulent bulk flow via averaged quantities only, such as the average
longitudinal and spanwise slip lengths. Our model employs anempirical relationship for the saturation of the transverse slip
length based on riblet theory (Luchini et al. (1991); Busse &Sandham (2012)). Using the analytical solutions due to Philip(1972), we can then use our model to relate the drag reduction directly to the geometry of the SHS and properties of thesurfactant.

Xu et al. (2021)γMa (s−1 )εRMSγMa (s−1 )

εRMS

0

0.005200.00101 × 10−1

0.0052

5 × 10−2

0.00097.5 × 10−1

0.0090

2.5 × 10−1

0.0019

1.5 × 100(a)0.02101 × 100

0.0215(b)

Table 1. The root mean square error of our model, εRMS ,comparing the drag reduction predicted by our model DRModelto the drag reduction predicted by experimental data DRData ,considering experimental results in turbulent boundary layers(a) Park et al. (2014) or (b) Xu et al. (2021), for different average Marangoni shear rates, γMa .

512th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
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We compare our model predictions with direct numericalsimulations, where there is an agreement in the drag reduction
for small P+ (in wall units +). The model captures the dependence of the drag reduction on the geometry of the SHS
and channel height (or boundary layer thickness). This holds
until we transition into a different regime where the drag reduction from the direct numerical simulations asymptoticallyapproaches the gas fraction for P+ → ∞. We also compare ourmodel predictions with experimental data, which allows us toinvestigate surfactant effects in turbulent flows over SHSs. In
addition, the model demonstrates that the presence of surfactant is detrimental to the drag reduction at all gas fractions,where greater reductions in drag are seen at smaller Reynolds
numbers. From the comparison between a surfactant-inclusivemodel and the experimental data found in the literature, we
do not have sufficient evidence to conclude that surfactants
have affected the drag reduction performance of the SHS studied. The differences between experimental findings are better explained by changes in the ratio between the boundary
layer thickness and the SHS texture period. For shorter gratings (which are necessary at high speeds) and higher surfactant concentrations (which are common in maritime environments), our model suggests that surfactant may become important again. More numerical and experimental work should
be conducted to further disentangle the effect of surfactant in
turbulent flows over SHSs.

ACKNOWLEDGEMENTS
We gratefully acknowledge support from CBET-EPSRC(NSF #2054894, EPSRC Ref. EP/T030739/1), and partial support from ARO MURI W911NF-17-1-0306.

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