Superhydrophobic surfaces can significantly reduce hydrodynamic skin drag by accommodating large slip velocity near the surface due to entrapment of air bubbles within their microscale roughness elements. While there are many Stokes flow solutions for flows near superhydrophobic surfaces that describe the relation between effective slip length and surface geometry, such relations are not fully known in the turbulent flow limit. In this work, we present a phenomenological model for the kinematics of flow near a superhydrophobic surface with periodic postpatterns at high Reynolds numbers. The model predicts an inverse square root scaling with solid fraction, and a cube root scaling of the slip length with pattern size, which is different from the reported scaling in the Stokes flow limit. A mixed model is then proposed that recovers both Stokes flow solution and the presented scaling, respectively, in the small and large texture size limits. This model is validated using direct numerical simulations of turbulent flows over superhydrophobic posts over a wide range of texture sizes from L^{+} ≈ 6 to 310 and solid fractions from ϕ_{s} = 1/9 to 1/64. Our report also embarks on the extension of friction laws of turbulent wallbounded flows to superhydrophobic surfaces. To this end, we present a review of a simplified model for the mean velocity profile, which we call the shiftedturbulent boundary layer model, and address two previous shortcomings regarding the closure and accuracy of this model. Furthermore, we address the process of homogenization of the texture effect to an effective slip length by investigating correlations between slip velocity and shear over patternaveraged data for streamwise and spanwise directions. For L^{+} of up to O(10), shear stress and slip velocity are perfectly correlated and well described by a homogenized slip length consistent with Stokes flow solutions. In contrast, in the limit of large L^{+}, the patternaveraged shear stress and slip velocity become uncorrelated and thus the homogenized boundary condition is unable to capture the bulk behavior of the patterned surface.
I. INTRODUCTION
Superhydrophobic surfaces (SHSs) are textured hydrophobic surfaces capable of entrapping air pockets in their grooves when immersed in water.^{1} In such a state, called the CassieBaxter state, direct contact between water and the solid phase is suppressed.^{2} In regions where the solid contact is replaced by a gas interface, the water can sustain a slip boundary condition. As a result, the effective macroscopic (or homogenized) boundary condition recognized by the fluid is a slip boundary condition that can be characterized by a slip length. The first theoretical investigations aiming to determine the slip length considered Stokes flow equations.^{3–9} These analyses often considered simple SHS texture geometries such as periodic ridges^{3–6} or posts^{7,8} with sharp edges that would hold up a flat airwater interface as shown in Figure 1(a). The effective macroscopic boundary condition is shown to follow the form
where u_{s} is the averaged velocity on the boundary, called the slip velocity, and ∂u/∂n is the wallnormal derivative of the mean velocity profile and represents the mean shear when multiplied by the fluid viscosity. b is the effective slip length, which depends solely on the texture geometry in the Stokes flow limit. More specifically, it was demonstrated^{3–8} that the slip length scales linearly with the texture length scale
where L is the SHS pattern wavelength, and C_{b} is a dimensionless number, in most cases of order one, that depends only on the texture shape, and not its size. For example, Ybert et al.^{7} showed that for a SHS texture involving isotropic square posts, shown in Figure 1, the coefficient C_{b} is
where ϕ_{s} is the area fraction of the interface with solid contact.
In the lowReynolds number regimes these models, consistent with experimental investigations,^{10–13} demonstrated that SHS can lead to significant drag reduction when b is comparable to the length scale of the mean velocity gradients (e.g., channel height or boundary layer thickness). For example, reducing the drag by only 30% in a laminar channel flow would require a SHS texture size of about 15% of channel height with ϕ_{s} ≈ 10%. With such design, the percentage of drag reduction remains independent of the flow speed as long as the flow remains laminar. In this case, Eq. (3) predicts that the resulting slip length would be on the order of 7% of the channel height.
Given the success of the developed theoretical foundations for SHS drag reduction in laminar flows, researchers started exploring this phenomenon in the context of turbulent flows.^{14–21} The main contrast between laminar and turbulent flows is the separation of scale between the length scale of the mean velocity gradients and that of macroscopic geometry. Specifically, the near wall region of turbulent flows is controlled by the shear length scale, δ_{ν}, which is much smaller than the macroscopic geometry inversely proportional to the Reynolds number. For better clarification, this contrast is schematically demonstrated in Figure 2. In turbulent boundary layers, the majority of the mean velocity gradient occurs over a thin layer ∼O(10δ_{ν}) next to the wall, while the bulk of the flow has a relatively flat profile in comparison to a counterpart laminar profile. Therefore, while turbulent flows induce significantly larger absolute drag compared to laminar flows, when it comes to percentage drag reduction, turbulent flows can utilize much smaller SHS textures. For instance, for the aforementioned example of a channel flow, the same 30% drag reduction would require SHS textures size of about 1% channel height, when the flow is turbulent with Reynolds number of Re_{τ} ≃ 400.
While a key bottleneck in the development of SHS for drag reduction of turbulent flows is the robustness of the airwater interface, many questions still remain at the kinematic level. For example, even in the limit of flat airwater interface, which corresponds to an infinitely robust SHS, the following questions are still subject of debate:

How does the slip length, b, scale with SHS texture size and flow rate in the turbulent flow regime?

Can a homogenized slip length model, over a flat surface, represent the same effect of a patterned surface when it comes to turbulent flows?

How should the friction law of turbulent boundary layers be modified so that it could be extended to flows over superhydrophobic surfaces?
Our study presents new models and data aiming to address these questions. To this end, we present Direct Numerical Simulations (DNSs) of turbulent flows over superhydrophobic surfaces over a wide range of texture sizes. Specifically, we have considered flows over superhydrophobic posts, with pattern wavelength in plus units from L^{+} = 6 to 310. This is the first compilation of data for turbulent flows over threedimensional SHS textures (i.e., posts, as opposed to twodimensional textures such as streamwise extruded ridges) with such a wide range of texture scales and gas fractions. The present range recovers the Stokes flow sliplength model in the small texture limit, while a different power law, predicted by our phenomenological model, emerges in the large texture limit.
Additionally, we address the question of how DNS data over patterned surfaces can be used to measure the slip length in both streamwise and lateral directions. To this end, we present correlations between slip velocity and shear over patternaveraged data in both streamwise and lateral directions and assess whether simpler simulations in which SHS texture effects are lumped into a homogenized slip length would be valid. We show that in the limit of small L^{+} the correlations in both directions can be well represented by a single slip length for both mean and fluctuation quantities consistent with linear Stokes flow theories. However, in the limit of large L^{+} not only shear and slip fluctuations become uncorrelated (in a patternaveraged sense), but also homogenized models, over any reasonable slip length, fail to predict the correct mean velocity profile.
The paper is organized as follows. The Introduction section continues by providing an overview of past developments in investigation of SHS effects on the mean velocity profile of turbulent boundary layers. In Section II, we describe details of our numerical simulations followed by a derivation of our phenomenological model. Section III presents our DNS results and collapse of data based on our predicted scaling followed by our investigation of homogenization approximation. Finally, summary and conclusions are provided in Section IV.
A. Shiftedturbulent boundary layer (TBL) model
Before presenting our analysis, we present an overview of a simple model for the mean velocity profile of turbulent flows over superhydrophobic surfaces. The starting point is the wellknown loglaw that determines the relation between the freestream velocity and the shear velocity in a turbulent boundary layer (without SHS),^{22}
In Eq. (4), U_{∞} is the freestream velocity (or centerline velocity in channels), u_{τ} is the shear velocity equal to square root of mean wall shear stress normalized by fluid density ($ \tau w / \rho $), and Re_{τ} is the shear Reynolds number based on u_{τ} and boundary layer thickness (or channel halfwidth). κ is the von Karman constant and the values of κ = 0.41 and B = 5.0 have been widely used for channel flows. Π is the Coles’ wake parameter which describes the excess velocity of the outer layer,^{23} and is 0.47 for zeropressuregradient flat plate boundary layers, and zero for channels.
We now consider a scenario in which the applied pressure gradient (and thus u_{τ}) is held fixed and the walls are replaced by SHS with b ≠ 0. The most simple model that one could imagine is to assume that the velocity profile remains the same, but its magnitude is shifted by a constant slip velocity, u_{s},
We note that by definition, $ u s =b d U d y  wall $, and thus $ u s + = b + $. The first establishment of the abovestated model starts from DNS of turbulent flows over hydrophobic walls with homogenized slip.^{14} Min and Kim^{14} observe the upward shift of the mean velocity profile for turbulent flow by $ u s + $ when the slip length is active only in the streamwise direction (see Figure 3).
In reality, the mean velocity profile is not simply shifted upward but also modified in response to lateral slip, $ b l + $. Min and Kim^{14} showed that the buffer layer is shortened due to SHS induced mixing via lateral slip. As a result, the loglayer is shifted downward while its slope is preserved as similarly observed in roughwall bounded flows.^{24–27} This downward shift, denoted by ΔU^{+} (Fig. 3), acts in the opposite direction of the slip effect and results in reduction of a fraction of gained momentum due to SHS slip. Taking this effect into account, Fukagata et al.^{28} presented a model for skin friction of turbulent flows over SHS as
Previous data, as well as our present data, indicate that this model, Eq. (6), presents a suitable framework to quantify impact of SHS on the mean velocity profile of turbulent flows, and we hereafter call it shiftedTBL model. For a wide range of slip lengths up to order hundreds, Fukagata et al.^{28} present a fitting model for ΔU^{+} as a function of lateral slip. Later, Busse and Sandham^{29} suggest a modified prediction of ΔU^{+}, as
However, this relation is obtained using simulations of homogenized surface models in which b and b_{l} are directly specified. Therefore, while the applicability of Eq. (7) is subject to validity of the homogenization approximation, this relation does not yet fully close Eq. (6), since one would still need a relation describing slip lengths, b^{+} and $ b l + $ in terms of the pattern size and flow conditions.
B. Dependence of slip length on texture size
Recent experimental studies of turbulent flows over SHS report successful drag reduction mostly on the order of 20% or less,^{16,17,30,31} and show that larger feature size causes larger drag reduction.^{16,19} Direct numerical simulations^{15,18,20,21,32,33} with patterned slip/noslip boundary conditions enable more detailed and systematic investigation into interaction of the texture with overlying turbulence. The first DNS, in which SHS texture was resolved, was performed by Martell et al.^{15} They showed significant slip lengths b^{+} ≈ 15–20 for two canonical geometries of isotropic posts and ridges with L^{+} ≈ 33–220. According to their results, larger slip length can be obtained by increasing the texture size, but the slip length is less than that predicted by the Stokes flow theory. A consecutive work by Martell et al.^{18} showed that the slip length, when normalized in terms of plus units, is independent of Re_{τ}, and only depends on L^{+}, and SHS texture topology. This further confirms the hypothesis that the SHS pattern only modifies the inner region of the turbulent boundary layer (as long as b ≪δ), and thus Re_{τ}, which expresses the outer length in terms of inner units, should not have an influence on the statistics associated with SHS flow modification. Park et al.^{32} investigated streamwise ridges and obtained a nonlinear slip length relation over a wide range of surface size, L^{+} ≈ 30–1300. They also reported that slip length sharply increases with increasing L^{+} at small texture size limit, and eventually saturates at large L^{+} limit, L^{+} > 200. Türk et al.^{20} examined streamwise ridges with texture size in the range L^{+} ≈ 9–550 and reported that in the small texture limit, the slip length can be reasonably predicted by the Stokes flow model of Lauga and Stone.^{4}
While these studies explored the effect of texture size on the slip length, the scaling of slip length with texture size is not fully known yet, particularly for texture topologies relevant to practical applications. While many of the aforementioned investigations considered streamwise ridges,^{16,17,19,20,33} their applicability to practical scenarios is fairly limited since fabrication of perfectly streamwisealigned ridges at scaled up geometries is very expensive. A more economic and scalable method of fabrication such as spray coating^{30,31,34} inevitably involves threedimensional textures, which can be more suitably modeled as isotropic posts.
II. METHODOLOGY
A. Model problem and computational method
We performed direct numerical simulations of turbulent channel flow over superhydrophobic surfaces. In our simulations, the threedimensional NavierStokes equation is discretized and solved using a secondorder finitedifference scheme on a staggered mesh. We use uniform mesh spacing in the streamwise (x) and spanwise (z) directions, and a stretched mesh in the wallnormal (y) direction. The secondorder AdamsBashforth and CrankNicholson schemes are used for time advancement with standard fractional step method for pressure treatment.^{35} Computational domain has dimensions 2π × π × 2 in the streamwise, spanwise, and wall normal directions, respectively. The mean pressure gradient is set at a constant value so that the friction Reynolds number, Re_{τ} is fixed a priori. Our simulation parameters are listed in Table I.
Case .  Symbol .  Re_{τ} .  L^{+} .  ϕ_{s} .  $ D x + \xd7 D z + $ .  N_{x} × N_{z} × N_{y} .  

P06  ○  197.5  6.5  1/9  1240.9 × 620.5  2304 × 1152 × 128  
P13  ○  197.5  12.9  1/9  1240.9 × 620.5  1152 × 576 × 128  
P26  ○  197.5  25.9  1/9  1240.9 × 620.5  1152 × 576 × 128  
P38  ○  197.5  38.8  1/9  1240.9 × 620.5  576 × 288 × 128  
P77  ○  197.5  77.6  1/9  1240.9 × 620.5  384 × 192 × 128  
P155  ○  197.5  155.1  1/9  1240.9 × 620.5  192 × 192 × 128  
P155SF64  □  197.5  155.1  1/64  1240.9 × 620.5  192 × 192 × 128  
P155RE  ●  395  155.1  1/9  2481.9 × 1240.9  384 × 384 × 192  
P155SF16  ▾  395  155.1  1/16  2481.9 × 1240.9  512 × 512 × 192  
P310  ●  395  310.2  1/9  2481.9 × 1240.9  384 × 384 × 192  
P310SF36  ▴  395  310.2  1/36  2481.9 × 1240.9  384 × 384 × 192  
Case  Symbol  Re_{τ}  b^{+}  $ b l + $  Slip lengths  $ D x + \xd7 D z + $  N_{x} × N_{z} × N_{y} 
HP06  ♢  197.5  3.1  3.1  $ b + = b l + = b P 06 + $  1240.9 × 620.5  192 × 192 × 128 
HP13  ♢  197.5  5.3  5.3  $ b + = b l + = b P 13 + $  1240.9 × 620.5  192 × 192 × 128 
HP38  ♢  197.5  10.3  10.3  $ b + = b l + = b P 38 + $  1240.9 × 620.5  192 × 192 × 128 
HP38S  ⊳  197.5  10.3  20.2  $ b + = b P 38 + , b l + = b S P 38 + $  1240.9 × 620.5  192 × 192 × 128 
S  197.5  Smooth  1240.9 × 620.5  192 × 192 × 128 
Case .  Symbol .  Re_{τ} .  L^{+} .  ϕ_{s} .  $ D x + \xd7 D z + $ .  N_{x} × N_{z} × N_{y} .  

P06  ○  197.5  6.5  1/9  1240.9 × 620.5  2304 × 1152 × 128  
P13  ○  197.5  12.9  1/9  1240.9 × 620.5  1152 × 576 × 128  
P26  ○  197.5  25.9  1/9  1240.9 × 620.5  1152 × 576 × 128  
P38  ○  197.5  38.8  1/9  1240.9 × 620.5  576 × 288 × 128  
P77  ○  197.5  77.6  1/9  1240.9 × 620.5  384 × 192 × 128  
P155  ○  197.5  155.1  1/9  1240.9 × 620.5  192 × 192 × 128  
P155SF64  □  197.5  155.1  1/64  1240.9 × 620.5  192 × 192 × 128  
P155RE  ●  395  155.1  1/9  2481.9 × 1240.9  384 × 384 × 192  
P155SF16  ▾  395  155.1  1/16  2481.9 × 1240.9  512 × 512 × 192  
P310  ●  395  310.2  1/9  2481.9 × 1240.9  384 × 384 × 192  
P310SF36  ▴  395  310.2  1/36  2481.9 × 1240.9  384 × 384 × 192  
Case  Symbol  Re_{τ}  b^{+}  $ b l + $  Slip lengths  $ D x + \xd7 D z + $  N_{x} × N_{z} × N_{y} 
HP06  ♢  197.5  3.1  3.1  $ b + = b l + = b P 06 + $  1240.9 × 620.5  192 × 192 × 128 
HP13  ♢  197.5  5.3  5.3  $ b + = b l + = b P 13 + $  1240.9 × 620.5  192 × 192 × 128 
HP38  ♢  197.5  10.3  10.3  $ b + = b l + = b P 38 + $  1240.9 × 620.5  192 × 192 × 128 
HP38S  ⊳  197.5  10.3  20.2  $ b + = b P 38 + , b l + = b S P 38 + $  1240.9 × 620.5  192 × 192 × 128 
S  197.5  Smooth  1240.9 × 620.5  192 × 192 × 128 
In our simulations, both top and bottom walls are treated as superhydrophobic surfaces. Our main set of DNSs resolves texture with patterned slip and noslip condition. We assume shearfree boundary condition on gasliquid interface, which is an appropriate condition for an airwater boundary,^{36} and noslip on solidliquid contact. In addition to these simulations, we have performed a set of reducedorder simulations in which the SHS surfaces are represented via homogenized boundary conditions with uniform slip lengths. In our patternresolved DNS, the SHS is modeled as an isotropic post with texture wavelength ranging from L^{+} = 6 to 310. The nominal solid fraction for such simulations is fixed at ϕ_{s} = 1/9, and we additionally consider different solid fractions from ϕ_{s} = 1/16 to ϕ_{s} = 1/64. In these simulations, the grid resolution is fine enough to resolve both features of the overlying turbulence as well as the texture geometry. The simulation code has been verified against simulation of Kim et al.^{37} and Martell et al.^{18} and we have also established grid convergence through mesh refinement studies.^{21} In the case of the coarsest texture, the grid resolution is Δx^{+} = 6.4 in the streamwise direction and Δz^{+} = 3.2 in the spanwise direction as listed in Table I. For wall normal direction, the minimum grid resolution is Δy^{+} = 0.12 near the wall and maximum is Δy^{+} = 15 at the centerline.
B. A phenomenological model
Before presenting our analysis, we briefly make a remark on the effects of channel Reynolds number on the slip length. Considering a fixed L^{+} = 155, we examined variation of the Reynolds number from Re_{τ} ≈ 200 to Re_{τ} ≈ 400. This change in the Reynolds number has resulted in only 1% difference in the measured slip length from b^{+} = 20.2 to b^{+} = 20.4. Smaller texture sizes are expected to be even less sensitive to channel Reynolds number, since their influence on the flow structure remains more confined to the near wall region.^{21} We have also examined the impact of Re_{τ} on the higher order statistics,^{21} which further confirms that the effect of texture on flow is dictated by the feature size L^{+} and not Re_{τ} when all quantities are measured in plus units.^{38} Next, we present a phenomenological model that results in a scaling for b^{+} versus L^{+} in the limit of large L^{+}.
We consider a SHS texture consisting of isolated posts with very small solid fraction, ϕ_{s} ≪ 1. The core foundation of our model is to assume that the flow on the posts can be represented by laminar boundary layers, as schematically depicted in Figure 4. The width of the post is defined as w. The appropriate scaling of the freestream velocity encountering these boundary layers is the slip velocity, u_{s}, as opposed to channel mean velocity, given that the thickness of these boundary layers is much less than the channel height. Considering the boundary layer scaling, $\delta \u223c \nu w / u s $, the shear on the solid surface is then scaled as
where μ is the viscosity of overlaying fluid. The mean shear on the interface can be obtained by averaging τ_{s} over both air and solid interfaces. This will result in the introduction of geometric factor ϕ_{s} = (w/L)^{2},
Rearranging terms while noting $ \tau w /\rho = u \tau 2 $ and $ u s + = b + $, results in
As a result, slip length scales with the cube root of L^{+} while it follows inverse square root scaling with the solid fraction. We expect this scaling to hold in the large L^{+} limit, as long as L/δ is small enough not to influence the outer region of the turbulent boundary layer. In a dimensional form, the slip length scales with
which implies that for a fixed geometry increasing Reynolds number (i.e., due to faster flow rate) would lead to smaller dimensional slip length, but larger b^{+}.
The scaling predicted by this phenomenological model is expected to hold for large L^{+}. In the small texture limit, however, we expect dominance of viscous effects with no inertial boundary layers on posts in Stokes flow regime, the length scale associated with the flow over the posts is the width of texture, w, when ϕ_{s} ≪ 1. Therefore, a proper scaling for shear on the post would be
This scaling expressed in dimensionless form will lead to
which is consistent with the Stokes flow solution reported by Ybert et al.^{7} Interestingly, both scaling laws expressed in Eqs. (10) and (13) have the same inverse square root dependence on the solid fraction, ϕ_{s}, but the dependence on the dimensionless texture size, L^{+}, is different. A plot of b^{+} versus L^{+} plot is expected to recover Ybert et al.’s relation in the small L^{+} limit and merge to the cube root law in the large L^{+} limit.
III. DNS RESULTS
A. Model validation and development of a mixed scaling
From DNS results, we show slip length relation with the texture size from L^{+} ≈ 6 to 310 in Figure 5. When b^{+} is scaled against $ \varphi s $, as shown in Figure 5(b), a remarkable collapse is achieved. This result confirms our predicted scaling with ϕ_{s}. Furthermore, the data show asymptotic matches with the predicted linear scaling for the small L^{+} limit, as well as the cube root scaling for the large L^{+} limit. These are the most important highlights of our investigation. An additional observation is that for the fixed ϕ_{s}, the upper limit of the validity of the Stokes solution is at relatively high L^{+} of order O(10), leading to b^{+} of up to about 5.
Next, we seek a universal relation that would match the two scalings in the small and large L^{+} limits, as well as small ϕ_{s} limit. One of the simplest ways of constructing such a formula is to consider a superposition in the form
with C_{b} analytically available for a wide range of texture geometries. Equation (14) matches automatically the expected behavior in the small texture limit, it also provides scaling (10) in the large texture limit. The only fitting parameter, α, should be tuned by DNS data. The final form of the equation is
Using this relation, the slip length, b^{+}, can be computed as a function of texture size L^{+}, and solid fraction ϕ_{s}. This relation is plotted in Figure 5(b) and is shown to predict the DNS data with excellent agreement. Equation (15) is the first universal representation of slip length for turbulent flows over SHS, in such a wide range of texture sizes and solid fractions. We note that the transition region between the linear and third root dependence on L^{+} can be fitted with a square root scaling. Such a fit has been recently proposed by Rastegari and Akhavan.^{39} However, the square root law is not backed by any mechanistic physical description, and we believe it has been the artifact of transition between the presented linear law and third root law.
B. Validity limits of homogenized slip length and estimation of ΔU^{+}
In this section, we address the remaining issue regarding the closure of the shiftedTBL model, by shedding light into the applicability of models that correlate ΔU^{+} with the lateral slip length, such as that in Eq. (7). To this end, by recognition of the fact that these models use homogenized slip lengths, we address the applicability of the homogenization assumption itself, and investigate its validity bounds. Concurrently, we determine the proper lateral slip length, $ b l + ( L + , \varphi s ) $ that would close these models within their validity limits.
As shown in Figure 3, ΔU^{+} represents a downward shift of the loglayer due to shortening of the buffer region. As previously studied by Min and Kim^{14} and suggested by Eq. (7), the main contributor to the modification of the buffer layer is the slip length in the lateral direction, b_{l} since it can strongly affect momentum mixing via modification of streamwise vortices.^{14} However, the dependence of b_{l} on L^{+} has not been investigated for turbulent flows. Quantification of b_{l} involves extra challenge compared to determining b since both mean shear and slip velocity are zero in the lateral direction, and thus the problem leads to a 0/0 ambiguity.
As the starting point, we investigate whether the homogenized slip simulations with properly chosen slip lengths can effectively capture the mean kinematic properties, most importantly ΔU^{+}, of patterned surfaces. For the streamwise direction, the proper homogenized slip length, b, must be the same as that measured from corresponding pattern resolved DNS in order to ensure correct capture of mean slip velocity. Since b_{l} cannot be measured from the mean data of DNS, as a starting guess, we consider a scenario in which the homogenized b_{l} is set to be equal to b, given isotropy of the patterns considered here.
Figure 6 presents a comparison of the mean velocity profiles obtained from simulations of homogenized slip length models and that of the corresponding patternresolved DNS. In order to allow a visual quantification of ΔU^{+}, the slip velocity is subtracted from all data and the plots are accompanied by the mean velocity profile of the conventional smoothwall channel flow (noslip). In this form, the difference between smoothwall model and SHS model at the outer region represents the corresponding ΔU^{+}. Figures 6(a) and 6(b), representing cases with L^{+} = 6 and 13, respectively, indicate a remarkable matching between the velocity profiles obtained from the proposed homogenized sliplength model and that of the corresponding patternresolved DNS. This matching confirms that for L^{+} up to O(10) the isotropic SHS can be well represented by a single slip length acting in both streamwise and lateral directions.
The above conclusions change for cases with L^{+} larger than O(10). For example, for the case of L^{+} = 38, Figure 6(c) shows that the DNS with patterned boundary condition predicts a ΔU^{+} significantly larger than that predicted by the homogenized slip length model. For this case, any physically acceptable $ b l + $ leads to an underprediction of the ΔU^{+} from DNS with patterned slip. For instance, we demonstrate an additional simulation for this case, HP38S, in which b_{l} is retuned based on Eq. (2), which is the upper limit of the two presented scalings. Despite an increase by a factor of two in b_{l}, the resulting velocity profile has barely changed. Tuning to even higher b_{l} is hardly justifiable physically, and based on our discussion below, we conclude that for large patterns above L^{+} ∼ O(10) the homogenization approximation becomes invalid.
Next, we investigate details of data from DNS of patternresolved simulations in order to infer a statistical representation of slip length. Since both mean shear and mean slip in the lateral direction are zero, the lateral slip cannot be inferred from the mean data. To remedy this limitation, we study fluctuations of shear and slip velocity after averaging data over unit periods of patterns. Each instant in time, the shear and slip velocity data are averaged within each pattern unit $0\u2264 x \u0303 <1$, $0\u2264 z \u0303 <1$, where $ x \u0303 =modulo ( x , L x ) / L x $ and $ z \u0303 =modulo ( z , L z ) / L z $. These instantaneous singlepatternaveraged quantities are denoted by “∼” symbol, and their joint distributions are presented in Figure 7. In order for the homogenized slip length model to be valid, the joint distribution should lay on a straight line passing through the origin.
Figures 7(a) and 7(b) show that for small texture with L^{+} = 6, the wall shear and slip velocity are reasonably correlated in both streamwise and lateral directions and are consistent with the constraints of the homogenized slip model. For the streamwise statistics, the least square linear fit almost passes through the origin. While in this case the offset is negligible compared to scatter width of the data, we here take this change to introduce the distinction between the dynamic slip length and static slip length. The standard definition of the slip length, which is the ratio of the mean slip velocity to mean shear strain rate at the wall, b = 〈u〉/〈∂u/∂y〉_{y=0}, is indeed a static definition. However, instantaneous fluctuations of slip and shear relative to their mean can be related by a different ratio, which we call the dynamic slip length. Static and dynamic slip length can be geometrically represented based on the data shown in Figure 7(a). The static slip length is the ratio of the coordinates of the centroid of the distribution, and the dynamic slip length is the slope of the fitted line.
An improved homogenization model should use the equation of the line fit in the figure to relate wall shear to wall slip and thus incorporate both static and dynamic effects. However, for small L^{+} limit, dynamic and static slip lengths are very close and neglecting their distinction is justifiable. For the case of L^{+} = 6, the measured static slip length is 3.1 and the measured dynamic slip length is 2.9, which are both in reasonable agreement with analytical slip length predicted from Stokes flow solution (Eq. (2)), $ b S + =3.0$.
For the lateral statistics, the static slip length is not defined since the centroid is on the origin. For L^{+} = 6 the measured dynamic slip length is 2.6, which is still close to the analytical prediction by Stokes flow models (Eq. (2)). These results further confirm that for L^{+} up to order O(10) the homogenization approximation is valid, and that the effective slip lengths in both streamwise and lateral directions can be predicted from either Stokes flow theory or our universal model presented in Eq. (15).
For larger L^{+}, the fluctuation in slip velocity and shear becomes highly uncorrelated for both streamwise and lateral directions as shown in Figures 7(c) and 7(d). Therefore, regardless of the values of fitted dynamic and static slip lengths, the homogenization model faces a limitation since it cannot support uncorrelated slip and shear data. Furthermore, we confirmed that a homogenized model based on linear fits to these data indeed fails to predict the correct ΔU^{+} in the large L^{+} limit.
Based on these observations, we conclude that the ΔU^{+}(b^{+}) relation in Eq. (7) must be limited to small texture sizes with L^{+} < O(10), which translates to a limit based on slip length of b^{+} ≲ 5.
IV. SUMMARY
We presented a comprehensive investigation of the kinematic properties of turbulent flows over superhydrophobic surfaces with microposts. In this report, we first provided a summary of the previous models for turbulent flows over SHS, and introduced an emerging model which we called the shiftedTBL model. We then identified two existing shortcomings in terms of closure of this model, and motivated the need for predictive relations that can describe the streamwise and lateral slip lengths in terms of known input parameters. We then presented, for the first time, a phenomenological model predicting the scaling of the slip length in terms of texture size and solid fraction in the limit of highly turbulent flows with large L^{+}. The proposed scaling predicted a cube root dependence of b^{+} with L^{+} and an inverse square root dependence on ϕ_{s}. Combining this scaling with the Stokes flow scaling for the small L^{+} limit led to a universal relation that predicted b^{+} over a wide range of L^{+} and ϕ_{s} (Eq. (15)) with a remarkable accuracy. Incorporating this relation in the shiftedTBL model (Eq. (6)) results in an algebraic equation that can be solved for the friction velocity, u_{τ}, and thus provides a method of predicting the friction coefficient for turbulent flows over SHS.
Our study also addresses the validity of the homogenization process in which the patterned geometry is represented by an effective slip length. To shed insight on this topic, we presented for the first time the joint distributions of instantaneous slip velocity and wall shear strain from patternaverage data of patternresolved DNS over a wide range of L^{+}. Our results indicate that for patterns smaller than L^{+} ∼ O(10) the slip and shear are well correlated and the flow can be represented by a single slip length predicted by the Stokes flow theory (or our universal model) acting in both streamwise and lateral directions. For L^{+} larger than O(10), however, the data showed lack of correlations between fluctuations of slip velocity and shear, suggesting that homogenized sliplength models are invalid for this range as confirmed by example simulations. For this regime, ΔU^{+} cannot be predicted via previous fits to homogenized models (i.e., Eq. (7)) and a predictive scaling for ΔU^{+} is yet to be proposed.
We also introduced the concept of static and dynamic slip lengths and formalized a procedure for their quantification. However, in regimes that homogenization process is useful we found that dynamic and static slip lengths are close to each other.
The presented study considered only a subset of SHS geometries limited to isotropic square posts and different from commonly investigated streamwise ridges. Our choice was motivated by practical applications in which the most economical method of SHS fabrication involves spraying the material that inherently would lead to statistically isotropic structures. At this point, it is unclear how the presented results can be fitted to anisotropic structures and such an extension requires future investigation.
Acknowledgments
The authors would like to thank the Office of Naval Research for support of this research under Grant No. 3002451214. The authors thank Kwanjeong Education Foundation for the partial funding support for Jongmin Seo.
REFERENCES
While in our simulations we maintained desired L^{+} by adjusting the texture size, we should note in practice Reynolds number is change by changing the flow rate, while keeping the fluid and geometry fixed, which inevitably leads to change of L^{+}.