# Wall-bounded flow over a realistically rough superhydrophobic surface

###### Abstract

Direct numerical simulations (DNS) are performed for two wall-bounded flow configurations: laminar Couette flow at and turbulent channel flow at , where is the shear stress at the wall. The top wall is smooth and the bottom wall is a realistically rough superhydrophobic surface (SHS), generated from a three-dimensional surface profile measurement. The air-water interface, which is assumed to be flat, is simulated using the volume-of-fluid (VOF) approach. The laminar Couette flow is studied with varying interface heights to understand the effect on slip and drag reduction (). The presence of the surface roughness is felt up to of the channel height in the wall-normal direction. A nonlinear dependence of on is observed with three distinct regions. is most sensitive to in the region , where is the root-mean-square (rms) of roughness height. The DNS results are used to obtain a nonlinear curve fit for gas fraction as a function of , where determines the amount of slip area exposed to the flow. A power law linear regression fit is used to obtain as a function of . For the turbulent channel flow, statistics of the flow field are compared to that of a smooth wall to understand the effects of roughness and . Two interface heights, and are simulated to study their effect on the behaviour of the flow, where and are the rms and maximum peak of the roughness heights respectively. Results show that the presence of trapped air in the cavities significantly alters near wall flow physics. The fully wetted roughness increases the peak value in turbulent intensities, whereas the presence of the interface suppresses them, as evident from vorticity and turbulent kinetic energy. The peak values shift towards the rough wall in the presence of the interface. Overall, there exists a competing effect between the interface and the asperities, where the interface suppresses turbulence whereas the asperities enhance them.

## 1 Introduction

Superhydrophobicity is a property attributed to surface roughness (ridges, grooves, posts or random textures) and surface chemistry which maintains large contact angles for sessile drops, thus producing low wettability, termed as the Cassie-Baxter state (Cassie & Baxter 1944). The interface meniscus creates an air mattress that acts like a lubricant for the outer flow (Rothstein 2010). When the interface fails, the liquid fills the surface cavities and the superhydrophobic effect is lost. This is referred to as the Wenzel state (Wenzel 1936).

Nature provides numerous examples of superhydrophobic surfaces (SHS), which can be exploited for practical applications. For example, the lotus leaf is believed to take advantage of superhydrophobicity for a self cleaning mechanism (Barthlott & Neinhuis 1997). Frictional drag reduction is central to the performance of marine vessels, and anti-biofouling, anti-icing and microfluidic devices (Furstner et al. 2005; Genzer & Efimenko 2006; Fang et al. 2008; Jung et al. 2011). Any impact on skin friction drag reduction substantially improves the overall performance and yields savings in fuel cost (Choi & Kim 2006). Various other mechanisms can be used for drag reduction such as adding polymers, riblets, compliant walls, active suction and blowing, and injecting gas bubbles (Lumley 1969; Bechert et al. 1997; Hahn et al. 2002; Park & Choi 1999; Ceccio 2010). In the present work we focus on drag reduction using SHSs in laminar Couette flow and turbulent channel flow.

With recent developments in three-dimensional printing and microfabrication processes, it is possible to create surfaces exhibiting superhydrophobic characteristics when coupled with chemical treatments. Laminar flow over SHSs have been studied both numerically and experimentally. SHS have been shown to achieve drag reduction (Ou et al. 2004; Ou & Rothstein 2005; Joseph et al. 2006; Choi & Kim 2006; Maynes et al. 2007; Woolford et al. 2009; Emami et al. 2011). Analytical models relate the slip lengths to various surface parameters such as groove width, pitch and height (Lauga & Stone 2003; Ybert et al. 2007) or the slip velocities to geometry (Seo & Mani 2016). In general the SHSs are considered to be simple grooved geometries, and numerically the interface is typically assumed to be flat and represented using zero-shear boundary conditions. Others have included the effect of viscosity on the interface (Vinogradova 1995; Belyaev & Vinogradova 2010; Nizkaya et al. 2014). Several papers have investigated the effect of the curvature due to the meniscus and modified the analytical solutions to take curvature into account (Cottin-Bizonne et al. 2003; Sbragaglia & Prosperetti 2007; Wang et al. 2014; Li et al. 2017).

Turbulent flows over textured surfaces have been studied extensively in the past. Experimentally it becomes difficult to make measurements near the wall and to maintain a stable interface, but drag reduction and slip lengths have been investigated. Some past work report that SHSs had no effect on turbulent statistics (Zhao et al. 2007; Peguero & Breuer 2009), while others reported otherwise (Gogte et al. 2005; Henoch et al. 2006; Daniello et al. 2009; Jung & B. 2009; Aljallis et al. 2013; Bidkar et al. 2014; Park et al. 2014; Srinivasan et al. 2015). Rosenberg et al. (2016) showed that the turbulent skin friction is reduced over air- and liquid-impregnated surfaces (SLIPS) for Taylor-Couette flows. Numerically the interface is assumed to be flat and modeled using zero-shear boundary conditions (Min & Kim 2004; Fukagata et al. 2006; Martell et al. 2009, 2010; Frohnapfel et al. 2010; Busse & Sandham 2012; Park et al. 2013; Jelly et al. 2014; Turk et al. 2014). Investigation of the interface stability was studied using post processed pressure fluctuations to displace the interface using stagnation pressure (Seo et al. 2015). The profiles of mean velocity, turbulence intensities and Reynolds shear stresses where characterized for the inner part of turbulent boundary layers over several SHSs with varying textures and a range of (Ling et al. 2016). Recently, Fu et al. (2017) used the level-set method to study SHSs with varying viscosity ratios over spanwise and streamwise grooves. Seo et al. (2018) studied the propagation of capillary waves in the air-water interface that lead to different modes of failure due to capillary pressure modes.

All past work on flow over SHSs have considered simple geometries such as longitudinal and transverse grooves or posts. Realistically rough surfaces are devoid of any such regular pattern. To the best of our knowledge, none of the past numerical work has simulated a multiphase flow over realistically rough surfaces as done in this paper. The main goal of the present work is to perform DNS of (i) laminar Couette flow and (ii) turbulent channel flow, where the bottom wall is realistically rough surface. We aim to demonstrate the DNS capability of DNS for such SHSs and explore the effect of interface height on slip, drag reduction, near-wall flow field and turbulence statistics. The rest of the paper is organized as follows: §2 describes simulation details including the numerical method, parameters, and problem formulation. Results are described in §3 which include flow visualizations, drag reduction in laminar Couette flow and mean flow statistics of the turbulent flow. Finally, the work is summarized in §4.

## 2 Simulation details

### 2.1 Numerical method

Direct numerical simulations (DNS) are performed using a mass conserving volume-of-fluid (VOF) methodology on structured grids to study the effect of an air-water interface over a realistically rough surface. The governing equations are solved using the finite volume algorithm developed by Mahesh et al. (2004) for solving the incompressible Navier-Stokes equations on unstructured grids. The governing equations for the momentum and continuity equations are given by the Navier-Stokes equations:

(2.0) |

(2.0) |

where and are the -th component of the velocity and position vectors respectively, denotes pressure, is density and the viscosity of the fluid. The fluids are assumed to be immiscible. Additionally in eq. 2.1, is the Kroenecker delta, is the body force which is only active in the liquid phase and is the surface tension force. The algorithm is robust and emphasizes discrete kinetic energy conservation in the inviscid limit which enables it to simulate high-Reynolds number flows without adding numerical dissipation. The solution is advanced in time using a second order Adams-Bashforth method. A predictor-corrector methodology is used where the velocities are first predicted using the momentum equation, and then corrected using the pressure gradient obtained from the Poisson equation yielded by the continuity equation. The Poisson equation is solved using a multigrid pre-conditioned conjugate gradient method (CGM) using the Trilinos libraries (Sandia National Labs). The implicit time advancement uses the Crank-Nicholson discretization with a linearization of the convection terms.

The volume fraction is represented by a colour function () to keep track of two different fluids. The function () varies between the constant value of one in a filled cell to zero in an empty cell, with an intermediate value between zero and one to define an interface cell. The volume of each fluid cell is tracked in a two-step process for each time iteration, they are termed as the reconstruction and advection steps. The reconstruction step is based on a set of analytic relations proposed by Scardovelli & Zaleski (2000); given the volume fraction in each cell, the reconstruction of the interface shape uses a local normal vector and a piecewise linear interface calculation (PLIC) to approximate the interface shape. This step geometrically conserves the volume in each cell and therefore conserves mass and prevents over- and under-shoots in the value of the colour function. This ensures boundedness. The colour function is then advected by the fluid velocity field, while geometrically conserving the area using a directionally split flux scheme to exchange the reference phase volume across the boundary of neighbouring cells. The governing equations for the colour function material derivative are given by

(2.0) |

and the density and viscosity are evaluated as

(2.0) |

(2.0) |

The surface tension force is modeled as a continuum surface force as proposed by Brackbill et al. (1992):

(2.0) |

where is the surface tension constant, and is the curvature calculated using the height function which has been shown to significantly reduce numerical errors that are associated with surface tension. These errors are known as spurious currents (Cummins et al. 2005). The gradient of the colour function, , is representative of the surface normals. The gradient term in the surface tension force is discretized in the same manner as the pressure gradient term in the projection step:

(2.0) |

where denotes the outward normal of the face with respect to the control volume on which the summation is performed. This method of discretization ensures proper pressure jump recovery across the interface. The surface is represented by obstacle cells where they are masked out. Fluid cells are flagged as and obstacle cells as . The wetted masked cells (cells that share a face between a fluid and obstacle cell) enforce a zero face-normal velocity . The cell-centered velocities satisfy a no-slip boundary condition, with the exception of corner cells that take a weighted average of the neighbouring cell-centered values. The algorithm has been validated for a variety of flows and most recently in the context of superhydrophobicity (Li et al. 2016, 2017). In this study, we enforce a zero face-normal velocity at the interface . The condition models a high surface tension regime with a stable flat interface. This was done to focus on the effect of varying interface heights for a finite viscosity lubricant. The assumption made is valid for flow regimes where the interfacial surface tension dominates the interface dynamics. Surface tension effects will likely become important at larger Reynolds number flows. The statistics of the turbulent channel flow were averaged over a period of 300 flow through times after the discharge had reached a steady state value.

### 2.2 Surface generation

The roughness used in the present work is obtained from a real surface manufactured at UT Dallas (courtesy Prof. Wonjae Choi), with a 3D surface profile measurement using a 20X objective lens obtained from MIT (courtesy Prof. Gareth McKinley). The sample is Aluminum 6061 sandblasted using 150 grit, etched for 25 seconds, boehmetized for 30 minutes and hydrophobized using Ultra Ever Dry top coat in isopropanol. Figure 1 provides an illustration of the scanned surface data coloured with height.

We begin with a pre-processing step by reading the scanned surface data. The number of pixels in the scan width and height are stored as the number of nodes in the streamwise and spanwise directions respectively. The values of the roughness height and spatial location are then interpolated to cell centers given our domain of choice. Proper scaling of the roughness height is ensured given the parameters we are trying to simulate. Any obstacle cell which shares an edge with a fluid cell is tagged as a boundary cell. Boundary cells can either be an edge cell (if the boundary cell borders exactly one fluid cell) or a corner cell (if the boundary cell shares a corner with two or more fluid cells). The momentum equations are solved inside the fluid domain while the pressure is solved everywhere. No-slip boundary conditions are applied at the edge cells and a weighted average of the neighbouring cells is applied at the corners. This does not affect the pressure equation since we use collocated grids where the face-normal velocities are set to zero at the boundaries independent of the cell center value. This ensures a proper pressure jump recovery at the obstacle walls where the values inside the obstacle domain do not affect the pressure values in the fluid domain.

The surface statistics are computed and compared to the experimental values. The power spectral density (PSD) of the surface height obtained from the scan is shown in figure 2(a), where the visible cross pattern is due to the aliasing effects at the non-periodic boundaries of the unfiltered surface. The probability distribution function (PDF) of the surface height is compared to a Gaussian distribution in figure 2(b). Note that the real surface has a slight negative skewness. The detailed statistics of the original surface are reported in table 1. The roughness height was scaled accordingly to each of the canonical problems’ domain requirement.

Parameter | Description | Formula | Value |
---|---|---|---|

Average Roughness Height | 1.59 m | ||

RMS Roughness Height | 2.03 m | ||

Maximum Valley Depth | -10.0 m | ||

Maximum Peak Height | 8.31 m | ||

Maximum Peak to Valley Height | 18.38 m | ||

Mean Peak to Valley Height | 12.75 m | ||

Skewness | -0.32 | ||

Kurtosis (Flatness) | 3.47 | ||

RMS Slope of Roughness | 0.547 | ||

Wenzel Roughness | 1.129 | ||

Directional Derivative |

### 2.3 Problem description

Simulations are performed for two canonical problems: (i) laminar Couette flow and (ii) turbulent channel flow, where the surface described in §2.2 is used as the bottom surface. The case name, domain definitions and grid resolution used are described in table 2.

For the laminar Couette flow, the height of the top wall was chosen such that the rms roughness height is around 2% of . Note that the original surface is scaled to achieve the roughness height ratios described above. The reference system is chosen such that the origin coincides with the rms roughness height. The schematic diagram shown in figure 3(a) illustrates the flow domain. No-slip boundary conditions are prescribed on the bottom surface and a constant velocity in the streamwise -direction is prescribed at the top wall. The streamwise and spanwise -directions are periodic and a non-uniform grid is used. The Reynolds number , where and are taken to be unity and the liquid phase being the reference material property. A smooth planar Couette flow (Case L-S) is used as a baseline where the reference shear stress , where is the reference viscosity in the water phase. First, a fully wetted case (L-RFW) is simulated to baseline the effect of roughness on drag when compared to the smooth wall. In the experiments performed by (Ling et al. 2016), the tunnel pressure is increased to control the interface location which compresses the air-layer into the SHS and in turn exposes more asperities, thereby reducing the extent of drag reduction. An idealized way to model this effect numerically is to progressively increase the interface height from the maximum valley depth to the maximum peak height of the surface . Case L-RI1 denotes and Case L-RI2 denotes , where and . The change in shear stress due to the roughness and is used to compute the drag reduction:

(2.0) |

For the turbulent channel flow, no-slip boundary conditions are applied on both the top smooth wall and bottom rough wall with periodicity in the streamwise and spanwise directions. A constant body force in the liquid phase is applied such that the reference . This was validated (not shown here) with the results of the flat smooth channel (Kim et al. 1987). The original surface is scaled such that The viscosity ratio of is used in all the simulations reported in this paper, where is representative of an air-water interface which is assumed to be flat. Three cases were considered: (i) fully wetted rough channel for Case T-RFW, (ii) two-phase rough channel with for Case T-RI1, and (iii) for Case T-RI2. A schematic diagram describing the turbulent channel domain is given in figure 3(b).

The maximum interface height is unphysical in a realistic scenario, but it serves the purpose of providing the largest amount of slip that is theoretically achievable. It also helps describe the trend between limiting cases. Non-uniform grids are used in all the simulations with grid clustering near the surface region. Table 2 provides the case names, grid information and domain extents of the laminar and turbulent problems considered in this paper. Out of the cases considered in the laminar Couette flow, three conditions were given specific case names to facilitate their description in the mean velocity profile plots.

Case | h | |||||||
---|---|---|---|---|---|---|---|---|

Laminar Couette Flow | 1-18 | - | ||||||

L-S | - | |||||||

L-RFW | - | |||||||

L-RI1 | ||||||||

L-RI2 | ||||||||

Turbulent Channel Flow | T-S | - | ||||||

T-RFW | - | |||||||

T-RI1 | ||||||||

T-RI2 |

## 3 Results

### 3.1 Laminar Couette Flow

#### 3.1.1 Steady state flow field

Initially, the fully wetted Case L-RFW is considered. Figure 4 shows the flow field after it is fully developed. The wall-normal velocity (figure 4a) and the vorticity magnitude along with surface pressure (figure 4b) are shown. A wall-normal velocity component into the flow is induced due to the surface asperities. Additional vorticity is generated due to the surface roughness, and large variations of pressure on the surface are evident due to the presence of peaks and valleys.

The penetration effect of the surface roughness is illustrated in figure 5, where the instantaneous streamwise velocity is shown for the fully wetted rough case at four wall parallel planes varying from to . Notice that even at the location of , the velocity field is still inhomogeneous suggesting that the surface roughness effects can penetrate up to that distance.

#### 3.1.2 Mean flow field properties

Simulations are performed for each of the interface heights varying from to . Drag reduction () is shown for all the interface heights in figure 6. The increase in is not linear when is varied from to . Note that the fully wetted case has negative indicating that the absence of the interface has increased drag due the exposed asperities.

The presence of the interface produces nearly the same drag reduction when is in the region which we will refer to as region . This suggests that the value of is insensitive to in region . From the increase in drag reduction is rapid, we will refer to this as region . This is because the increase in the interface height in region exposes less asperities to the outer flow. The slip is enhanced due to a much larger area of air-water interface. In region beyond , hits a plateau and becomes insensitive to the interface height since it covers most of the asperities. is therefore sensitive to in the vicinity of mean roughness . It is therefore evident from figure 6 that we can classify the interface cases into three distinct regions:

(3.0) |

Out of the three regions presented above, one can extract a representative case from each region. The baseline smooth wall is denoted as Case L-S, the fully wetted case in region is denoted as Case L-RFW, the interface at in region is denoted by Case L-RI1 and the interface at in region is denoted by Case L-RI2. The velocity profiles are extracted and compared in figure 7. Case L-RFW exhibits an increase in velocity gradient when compared to Case L-S indicating an increase in drag. Once the interface is introduced, the effect is reversed and the velocity gradient decreases for Case L-RI1. The effect is more pronounced in Case L-RI2 since it corresponds to the interface being at the highest peak where most of the asperities are covered and the largest slip effect is achieved.

The increase in interface height reduces the amount of rough surface area exposed to the flow. As a result, the flow is subjected to an increase in slip area. The asperities exposed to the outer flow can be represented by a solid fraction , which is found by calculating the area of the rough surface above the interface normalized by the projected area of the bottom wall. Alternatively, can be represented as a function of slip area instead of interface height by using the gas fraction defined as . This dependence is shown in figure 8 where plotted as a function of gas fraction .

Given the above, it is evident that there exists a relationship between the interface height and gas fraction . This is useful since is not known a priori and is prescribed as an initial condition. A simple nonlinear fit relates to and is described by the following equation:

(3.0) |

Figure 9 shows a comparison between the actual data and eq. 3.1.2 for as a function of . Note the negative skewness shifts the origin to which normally would correspond to .

The trend in as a function of is similar to the previously published analytical solutions of slip lengths as functions of for longitudinal and transverse grooves (Ybert et al. 2007). This suggests considering a power law relation between the effective slip length and . Based on the definition of in eq. 2.3, it can be shown that is related to by the following equation:

(3.0) |

therefore,

(3.0) |

The computed are plotted in figure 10 against . A linear regression is performed on the calculated data when and the scaling coefficients of the power law are rounded to the nearest number and given by the following equation:

(3.0) |

This formula provides a simple expression for slip length over the rough surface given the amount of gas fraction present when . The fully wetted roughness remains an outlier.

### 3.2 Turbulent channel flow

#### 3.2.1 Instantaneous flow field

The limiting cases used in the laminar Couette flow are used to study turbulence. The instantaneous flow field obtained from the turbulent channel flow simulations for all the cases are visualized in figure 11 and 12, where the instantaneous streamwise velocity and surface pressure on the bottom wall are shown. Figure 11(a) represents Case T-S where the bottom wall is smooth and will serve as a baseline. Figure 11(b) represents Case T-RFW, a fully wetted rough channel. Figure 12(a) corresponds to Case T-RI1, the turbulent channel flow with an interface at and figure 12(b) corresponds to Case T-RI2, the turbulent channel flow with an interface at .

Comparing figures 11(b) and 11(a), the higher streamwise momentum region slightly shifts away from the wall due to the presence of surface roughness. The surface pressure fluctuations appear to be have smaller structures in the presence of roughness when compared to a smooth wall. The overall magnitude of instantaneous pressure fluctuations does not change, however the scale of the fluctuations appears to be broken down into smaller scales with roughness. In figure 11(a), the pressure regions impinging on the flat wall have a distinct size, whereas in the fully wetted case 11(b) those regions are broken to smaller scales. As observed in figures 12(a) and 12(b), the presence of an interface has a significant effect on both the velocity field and surface pressure. In figure 12(a), we see a shift in the high momentum velocity field towards the rough wall due to the slip effect at the interface. In figure 12(b), this effect is more apparent where the slow momentum region is suppressed near the rough wall region due to the interface. It is observed from figures 12(a) and 12(b) the region of slow velocity field near the wall is thinner with increasing interface height (cases T-RI1 and T-RI2) when compared to the flat and fully wetted cases (T-S and T-RFW) shown in figures 11(a) and 11(b) respectively.

Figures 13 and 14 show the instantaneous vorticity magnitude plots in the plane at the streamwise location . The presence of random roughness does not seem to have a significant effect on the vorticity magnitude when compared to a flat channel as seen in figure 13(a) and 13(b) with the exception of a shift in vorticity away from the wall. Once an interface is introduced at the height , figure 14(a) clearly shows a suppression in the vorticity magnitude for Case T-RI1. This is further enhanced once the interface reaches the maximum peak height as observed in figure 14(b) for Case T-RI2. This is due to the increase in the slip which directly relates to the vorticity in the spanwise direction . As slip increases, the shear rate decreases which in turn suppresses the spanwise vorticity.

Figures 15 and 16 show the instantaneous magnitude in the plane extracted at the spanwise location . Since we have a constant body force, momentum needs to be conserved. The top wall shear stress needs to balance the change in the bottom wall. When the roughness is fully wetted, the high momentum shifts away from the rough wall towards the center. This effect can be seen in figure 15(b) when compared to figure 15(a). The top wall high vorticity magnitude region is thinner than a flat wall due to the upward shift. When an interface is present, the opposite happens. Slip causes the higher momentum fluid to shift down towards the rough wall. This effect is evident from the suppressed vortex structures near the interface as observed in figure 16(a) and figure 16(b) being more pronounced due to the fact that the bottom wall has the largest slip area. For cases T-RI1 and T-RI2, the top wall exhibits shorter less coherent wall structures that tend to look more chaotic when compared to cases T-S and T-RFW and where the long structures are broken up into smaller less coherent ones.

#### 3.2.2 Mean flow field statistics

Mean velocity profiles, pressure profiles, and Reynolds stresses are computed for four cases (T-S; T-RFW; T-RI1 and T-RI2). Figure 17(a) shows the mean velocity profile normalized by the friction velocity of the baseline case (T-S). The fully wetted simulation shows a slightly lower centreline mean velocity with a slight shift away from the rough wall, but not much difference overall. This is mostly evident in the centreline velocity and less evident in the near wall region. The small difference is due to the fact that the mean roughness height is hydrodynamically smooth where . Typically a hydrodynamically smooth channel should not show a difference, this can be explained by using the surface property , which is reported by Busse et al. (2017) to be a more suitable equivalent to the sand-grain roughness . The value is larger than 5 hence can be considered in the early transition from hydrodynamically smooth to rough regime. When an air-water interface is present, we see clear evidence of slip due to the interface. Cases (T-RI1 and T-RI2) clearly show a slip effect in . There is not much difference observed in the near wall region for Case T-RI1, but a clear increase in maximum centreline velocity. Case T-RI2 shows the largest slip due to the maximum slip area exposed to the liquid where . Notice as the interface height increases, a reverse flow is observed near the bottom wall. The profile is partially skewed towards the region of the wall roughness due to the slip effect. This indicates that the high momentum fluid is moving closer to the bottom wall. This was also observed qualitatively in figure 12. The skewness in the mean velocity profile for Case T-RI2 is more pronounced. The maximum velocity is shifts from the center towards the rough wall region.

Figure 17(b) shows the mean pressure profile. Case T-RFW does not show any difference in the bottom half of the domain when compared to Case T-S. The value starts to increase near the center of the domain and the top half region. The roughness tends to affect the mean pressure away from the wall. Cases (T-RI1 and T-RI2) show an increase in mean pressure as we move away from the wall. The change in mean pressure decreases as it nears the center of the domain but then follows the same trend as the bottom wall when it approaches the top wall. Case T-RI1 shows a larger increase in mean pressure in the lower half and a smaller decrease near the top wall region. Case T-RI2 displays the largest variation when compared to the rest. The mean pressure gradient increases with the presence of the interface since the pressure gradient is related to the vertical velocity fluctuations through the lateral mean-momentum equation given by:

(3.0) |

where is directly affected by the interface and asperities as will be demonstrated later.

The Reynolds stresses that result from the turbulent flow over a random rough SHSs are shown in figure 18. The turbulent intensities in the streamwise direction are shown in figure 18(a), Case T-RFW does not show much difference at the peak when compared to Case T-S. The bottom rough wall is almost equivalent in value to the smooth wall with the exception of the top flat wall that shows a slightly smaller peak than that of the flat channel. Away from the wall, the streamwise turbulent intensity is slightly higher. Once the interface is introduced for the Case T-RI1, we see a clear drop in the peak stress near the rough wall region, and an increase in peak stress near the top smooth wall. There is a slight skew of the centreline stress values towards the rough wall region. For the Case T-RI2 where the interface is at the maximum peak of the roughness, the streamwise turbulent intensity overshoots the baseline peak significantly. The top smooth flat wall region has a lower peak when compared to the rough wall region ( due to momentum balance given a constant body force), the peak is still higher than that of the baseline case. The skewness of the centreline value is also much more pronounced in this case, with the values shifting towards the rough wall region where there is highest slip.

Figure 18(b) shows the wall-normal turbulent intensity. Case T-RFW has a higher peak near the bottom rough wall and lower peak near the top flat wall when compared with the baseline case (T-S). The centreline value is equivalent in both cases. When a fluid interface is introduced in Case T-RI1, the peak drops in the near rough wall region and increases near the top flat wall region. The centreline value of the wall-normal turbulent intensity is skewed towards the bottom rough wall region. As the interface reaches the height of the maximum roughness asperity for Case T-RI2, the wall-normal turbulent intensity peak in the rough wall region is reduced significantly. Since we are enforcing a zero normal velocity at the interface, we notice an increase in the wall-normal stress within the air region, it then goes to zero at the interface, peaks then starts decreasing again as we move away from the wall. The centreline is significantly skewed towards the rough wall region. The turbulent intensities go to a larger peak at the top flat wall region to balance out the constant body force. Case T-RI2 does not show the same smooth transition behavior in turbulent intensity from near wall regions to the channel centreline when compared to all the other cases.

The spanwise turbulent intensity exhibit a similar behaviour as the wall-normal turbulent intensity with the exception of Case T-RI2 as seen in figure 18(c). Note that Case T-RFW has a higher peak stress in the near rough wall region, and lower peak at the flat top wall region balancing out the constant pressure gradient. This is reversed in Case T-RI1. The peak is suppressed near the rough wall, and enhanced in the top flat wall region. However in Case T-RI2, the peak at the rough wall region is not further suppressed, but in fact higher than all the other cases. The peak in the top wall region behaves as expected and is significantly higher than the rest.

Case T-RFW has a lower peak in Reynolds stresses near the rough wall when compared to Case T-S as shown in figure 18(d). This is due to the increase in the vertical velocity fluctuations which enhances the mixing mechanism near the rough surface. As we get closer to the top wall, the behaviour follows the same trend as the baseline case. The peak value of the Reynolds stresses remains lower when compared to the baseline case. Once an interface is introduced in Case T-RI1, we get the opposite effect. This is evident in both cases (T-RI1 and T-RI2) where an interface is placed at or . The Reynolds stress value near the wall for cases (T-RI1 and T-RI2) has a higher peak when compared to both cases (T-S and T-RFW). When the interface is at , the peak of Reynolds stress is highest when compared to all the other cases. It is therefore evident that there are competing effects between the interface suppressing vertical velocity fluctuations and the asperities doing the opposite by enhancing them.

#### 3.2.3 TKE, production and dissipation

The governing physics of superhydrophobicity in a turbulent channel is related to high momentum fluid shifting towards the rough wall when an interface is present or away when the channel is fully wetted. Therefore a few select terms are computed which pertain directly to the velocity field fluctuations. We investigate the turbulent kinetic energy (), the production () and viscous diffusion () terms, defined as:

(3.0) |

(3.0) |

(3.0) |

Note that denotes the average over time and the streamwise and spanwise directions. Figure 19(a) shows the plot as a function of the wall-normal distance. exhibits a similar trend to the streamwise turbulent intensities where Case T-RFW does not show much difference at the peak when compared to Case T-S for both the bottom or top wall. Roughness does not have a significant effect on . Case T-RI1 shows a clear drop in the peak near the rough wall region, and an increase in near the top smooth wall. There is a slight skew of the centreline stress values towards the rough wall region. For the Case T-RI2 where the interface is at the maximum peak of the roughness, the overshoots the baseline peak significantly. The top flat wall region has a lower peak than that of the rough wall region. The skewness of the centreline value is also much more pronounced in this case, where the values tend towards the rough wall region, the area with largest slip.

Figure 19(b) shows the production term . Case T-RFW shows an increase in peak production near the rough wall due to the peak reduction in Reynolds stresses near the bottom wall. The presence of an interface in cases (T-RI1 and T-RI2) suppress production near the rough wall due to the slip effect.

The roughness for Case T-RFW reduces the peak of viscous diffusion at the bottom wall. Case T-RI1 exhibits a similar behavior, except that it shift away from the rough wall due to the presence of an interface. This is further pronounced when the interface is at the maximum height for Case T-RI2. The is further pushed away from the wall since the interface is at a higher location, it is interesting to note however that the peak in overshoots the peaks of the other cases as we move away from the wall which explains why the vortex structures observed in figures 14 and 16 are dissipated near the interface.

The spanwise vorticity is shown in figure 20. Described in 20(a) and 20(b), the behavior of near the rough wall shows a consistent decrease in peak value over all cases. The presence of roughness suppresses the peak value for Case T-RFW and is shifted away from the wall due to the presence of roughness when compared to Case T-S. The interface further suppresses the near wall spanwise vorticity as evident from the peaks for cases (T-RI1 and T-RI2) but is shifted towards the rough wall where the interface is present due to slip. To see the effect mentioned earlier more clearly, is averaged over the entire domain and is plotted in 20(b). For all the cases considered, the corresponding vorticity plot is normalized by its own maximum magnitude. Figure 20(b) clearly shows the trends. Case T-RFW shifts away from the rough wall to the right of Case T-S, while cases (T-RI1 and T-RI2) are to the left of Case T-S indicating a shift towards the rough wall where the interface is present due to slip.

## 4 Summary

DNS of laminar Couette flow at and turbulent channel flow at are performed, where the bottom wall is a realistically rough SHS. The surface scan reproduced computationally, and the surface statistics are verified with experiments. Simulations are also performed for a smooth wall to serve as a baseline. Simulations of the fully wetted case and an air-water interface at various heights are compared to the smooth channel. The effect of roughness and interface heights are discussed in detail.

Simulations of laminar Couette flow show a penetration effect up to in the wall-normal direction due to the roughness. Various interface heights were considered and a nonlinear dependence of drag reduction on is observed. The dependence can be categorized into three distinct regions. The drag is very sensitive to the interface location in region described in section 3.1.2 where is in the range of . The negative skewness of the roughness profile indicates that the surface contains more valleys than peaks and asperities. More than half of the surface roughness is filled with gas When is within the vicinity of since a large number of valleys become wetted. The solid fraction decreases and the gas fraction increases with increasing where less asperities are exposed to the outer flow. Therefore can be plotted as a function of . To demonstrate the relation between and , we first calculate by measuring the amount non-wetted area above the interface. Based on the definition , the gas fraction is obtained and plotted as a function of . A nonlinear fit is given by and shows good agreement with the data. This is useful since is not known a priori and is prescribed as an initial condition. Effective slip can be directly related to using the definition . A power law using a linear regression fit is found and corrected for the fully wetted case, the relation is given by which shows a good agreement with the data and provides a useful model for the slip length given .

Based on the observations of the three distinct regions made in the laminar Couette flow, two interface heights are chosen from those regions. Simulations of a fully wetted rough case and two interface heights and are performed and the results are discussed. The mean velocity profile shows a small overall decrease in the fully wetted case due to roughness, whereas the presence of an interface shows an increase (up to ) in . The fully wetted case shows an increase in shear at the bottom rough wall and decreases at the top wall since a constant body force is prescribed. The streamwise , wall-normal and spanwise turbulent intensities, show an increase in peak value for the fully wetted roughness and a suppression in peak values with the presence of an interface. This is consistent with the observations of the total kinetic energy () which exhibits a similar behaviour. The Reynolds shear stress increases in peak intensity for the fully wetted rough wall case which is indicative of enhanced mixing. When the interface is present, the shear stress is further suppressed with decreasing solid fraction . This is also observed in the production term where the fully wetted rough case increases production and therefore enhances mixing. The presence of an interface suppresses and in turn the near wall vorticity. The peak in mean velocity, turbulent intensities and Reynolds shear stress shift towards the slip wall. This indicates a shift in high momentum fluid towards the rough wall with increasing slip. Overall there exists a competing effect between the interface suppressing vertical fluctuations and the exposed asperities of the rough surface enhancing them.

## Acknowledgments

This work was supported by the United States Office of Naval Research (ONR) MURI (Multidisciplinary University Research Initiatives) program under Grant N00014-12-1-0874 managed by Dr. Ki-Han Kim. Computing resources were provided by the Minnesota Supercomputing Institute (MSI). We are grateful to Prof. W. Choi at University of Texas Dallas and Prof. G. H. McKinley at MIT for providing us with the scanned surface data used in the present work. The authors would like to thank Dr. P. Kumar and Ms. Y. Li for their helpful discussions and suggestions.

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