Turbulent flow over partially superhydrophobic underwater structures: The case of flow over sphere and step

Ocean Engineering 195 (2020) 106688

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Turbulent flow over partially superhydrophobic underwater structures: The case of flow over sphere and step Behrad Zeinali, Jafar Ghazanfarian * Mechanical Engineering Department, Faculty of Engineering, University of Zanjan, Zanjan, Iran

A R T I C L E I N F O

A B S T R A C T

Keywords: Superhydrophobicity Janus surface Backward-facing step Sphere Turbulence

Results of the large eddy simulations (LES) over two partially superhydrophobic 3D geometries, including the backward-facing step and the sphere at the Reynolds number of 400, 1000, 2000 have been presented using the open-source OpenFOAM Cþþ libraries. The effect of imposing the locally partial-slip on the surface of the mentioned cases has been investigated. The results confirmed that the superhydrophobicity decreases the drag coefficient and increases the Strouhal number of the sphere by up to 46% and 25%, respectively. In the meantime, the superhydrophobic backward-facing step experiences an increment of almost 4% and 3% in both the reattachment length placement and the recirculation length, respectively. In addition, the results have been analyzed to improve the model’s efficacy by locating the region on the geometry’s surface with a dominant role on the resulting vicissitudes of the drag coefficient, the Strouhal number, and the reattachment length. To obtain such an optimum model, the shear stress distribution diagram on the surface of the geometries have been studied. It is found that the areas with utmost changes in the shear stress were half of the sphere that faces the flow and the side of the step that the flow reattaches to it.

1. Introduction

reduction of the drag force and an increase of the Strouhal number as a result of the superhydrophobicity imposition. Zeinali et al. concluded that the Janus concept is capable of reducing the drag coefficient and increasing the Strouhal number by approximately 20%. After a thorough investigation of numerous cases, they pointed out that only a fraction of the surface of the cylinder contributes to the changes mentioned above, and imposing such boundary condition on only 50% of the surface, yields similar results in comparison to those for the whole surface being superhydrophobic. A series of experiments proved that at the Reynolds numbers above 24, the vortex rings appear in the flow wake (Taneda, 1956). Up to the Reynolds number of 210, the wake remains axisymmetric and steady and above that, this axisymmetric structure will be lost due to the appearance of fluctuating lift force that will increase substantially as the Reynolds number increases (Margarvey and Bishop, 1961; Wu and Faeth, 1993). While the planar symmetry still exists, the vortices began shedding periodically at Re ¼ 300 (Sakamoto and Haniu, 1990). At higher Reynolds numbers, the vortex loops start to shed in a nonplanar symmetrical pattern (Tomboulides et al., 1993). Sakamoto and Haniu (1990), Chomaz et al. (1993) and Kim and Durbin (1988) found out that the Reynolds number of 800 is another critical point in flow past

Reattachment of flows occurs in many engineering cases. This phe­ nomenon causes the boundary layer to separate from the surface to which it was attached before and upon reattachment a recirculation region will emerge. Among many geometries with such behavior as diffusers and combustor, the most popular one is the backward-facing step (Le et al., 1997). Study of flow past submerged basilar geometries have proven to be of great significance since such disquisitions can contribute to better understanding the behavior of flow in more so­ phisticated and realistic applications, whether they are engaged with the moving boundaries (Ghazanfarian et al., 2016) or the heat transfer (Ghazanfarian and Taghilou, 2018). Such a practice can also elevate our knowledge of paramount exceptional cases, such as the work of Du et al. (2017) who investigated different methods of maintaining air bubble on a surface to preserve its’ superhydrophobic characteristics. The superhydrophobicity is long bonded to drag reduction in several situations and geometries. The flow past a superhydrophobic cylinder (equivalent to a cylinder with sanded Teflon coating) was numerically investigated using the LES method (Zeinali et al., 2018) at Reynolds numbers ranging from 830 to 4160. The reported data expressed a

* Corresponding author. E-mail address: [email protected] (J. Ghazanfarian). https://doi.org/10.1016/j.oceaneng.2019.106688 Received 15 June 2019; Received in revised form 13 September 2019; Accepted 3 November 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.

B. Zeinali and J. Ghazanfarian

Ocean Engineering 195 (2020) 106688

Fig. 1. (a) Schematic view and geometrical details of the computational domain for flow over sphere, (b) a close snapshot of the boundary layer mesh near body, (c) a three-dimensional perspective view of the mesh. Table 1 Details of mesh topology and results of grid convergence study at the Reynolds number of 400 for flow over sphere.

Coarse mesh Fine mesh Finest mesh

Grid count

Smallest mesh-size

Mesh expansion factor

St

CD

2; 217; 013

0.0100

1.15

0.1541

0.7016

2; 931; 467

0.0050

1.20

0.1658

0.6398

3; 887; 506

0.0005

1.12

0.1661

0.63003

Fig. 2. (a) Schematic view and dimensions of the backward-facing step, (b) a close snapshot of the boundary layer mesh near walls.

Table 2 Results of the time-step independence study of the mesh with 2 931 467 grids at Re ¼ 400 for flow over sphere. Dimensionless time-step

Courant number

St

CD

50.0 16.7 10.0

2.119 0.999 0.483

0.2590 0.1658 0.1656

0.7328 0.6398 0.6398

Table 3 Details of the mesh topology and results of grid convergence study for flow over backward-facing step at the Reynolds number of 300. All of the length values are non-dimensionalized using the step size. Grid count

Smallest meshsize

Largest meshsize

Reattachment length

Coarse mesh Medium mesh Fine mesh

134; 210

0.0204

0.286

7.867

242; 210

0.1633

0.204

7.334

318; 980

0.0102

0.0816

6.861

Finest mesh

353; 019

0.00204

0.0204

6.866

Fig. 3. Variation of the normalized lift coefficient of the flow over cylinder as a function of the Reynolds number for various SHF factors. SHF ¼ 0:425 (Zeinali et al., 2018) is seen to have the best agreement with the experimental data of Ref. (Daniello et al., 2013). Table 5 Comparison of Strouhal number and drag coefficient for the no-slip sphere with the existing experimental data. Margarvey and Bishop (Margarvey and Bishop, 1961) Tomboulides et al. (Tomboulides et al., 1993) Kim and Durbin (Kim and Durbin, 1988) Achenbach (Achenbach, 1974) Present study

Table 4 Results of the time-step-size independence study of the mesh with 318 980 grids used in simulation of flow over the step at Re ¼ 300. Dimensionless time-step

Courant number

Reattachment length

0.192 0.118 0.096

1.867 0.998 0.623

6.859 6.861 6.861

St

CD

0.143 – 0.168 0.168 0.166

– 0.62 0.6398

the sphere. At this stage, two dominant modes of unsteadiness exist in the wake behind the sphere that influence the measurement of the Strouhal number, and that is why two different Strouhal numbers (low-mode and high-mode) appear. 2

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Ocean Engineering 195 (2020) 106688

Fig. 4. Flow schematics, geometrical details and partitioning (a1 to a5) of the backward-facing step. X1 , X2 and X3 correspond to the non-dimensional reat­ tachment length and the starting and the ending points of the recirculation region, respectively. All these lengths are converted into the dimensionless form using the height of the step.

Table 6 Comparison of computed values of the reattachment length and the recirculation region for the backward-facing step with the existing experimental data.

Armaly et al. (Armaly et al., 1983) Current study

Reynolds Number

X1 =s

300 600 1200 300 600 1200

6.87 11.48 17.67 6.58 10.73 16.33

X2 =s

X3 =s

8.92 14.40

15.98 23.26

9.08 13.49

14.65 25.02

Fig. 6. Dominant frequencies in the power density spectrum of the lift coeffi­ cient obtained from the FFT spectral analysis of flow past the sphere at Re ¼ 2000 for both the no-slip condition and the superhydrophobic spheres using SHF ¼ 0.425.

They have also reported that rotation will also decrease the Strouhal number. Another critical parameter in external flows is the drag coefficient. Pan et al. (2018) numerically investigated the influence of imposed stream-wise magnetic field of different degrees on the drag coefficient, the wake structure, and the transition of the flow at the Reynolds numbers less than 300. Their results suggest that up to a specific Lorentz force, the recirculation length decreases while the separation angle increases. For larger Lorentz forces, an opposite trend has been observed. They also found that there is a linear dependence between Cd and the root of the Lorentz force. Ghazanfarian et al. (2015) conducted a numerical study on a rotating cylinder and found out that such rotation causes the drag reduction that is in line with the findings of Nobari and Gha­ zanfarian (2009) who simulated flow past a rotating and oscillating cylinder and stated that although the drag coefficient decreases by increasing the dimensionless angular velocity (up to 2.25), the amount of this drag suppression decreases by increasing the amplitude of oscillations. The effect of hydrophobicity on water entry of solid objects has been experimentally investigated by Korkmaz and Gazel (2017). They used simple geometries (including cylinder and sphere) and determined that the strain on the surface upon entering the water is reduced in hydro­ phobic cases, and the pressure is distributed more evenly, which resulted to a 3% and 40% increment in the penetration and splash velocity of the sphere model for the drop height of 50 cm, respectively. Choi et al. (2006) experimentally investigated the impact of dimples on the surface of the sphere on drag coefficient reduction at high Rey­ nolds numbers. Since these dimples cause the local flow separation, they contribute to the shear-layer instability and the production of significant turbulence intensity. Choi et al. reported that the amount of drag reduction would be almost 50%. Zhou et al. (2015) experimentally investigated the effect of groove and dimples on the flow past a circular cylinder and reported a drag reduction of 18% to 29% in dimpled cyl­ inder and 10% to 30% reduction in ridged cylinders. Another way to suppress the drag force is by placing splitters upstream of the flow. Amiraslanpour et al. (Amiraslanpoura et al., 2017) investigated the ef­ fect of various splitter placement layout on the drag suppression of an oscillating cylinder. They pointed out that the maximum drag reduction occurs for the lock-on condition at the dimensionless oscillation

Fig. 5. Variation of the mean dimensionless wall distance, yþ , as a function of the Reynolds number for the no-slip and the superhydrophobic spheres for coarse, fine, and finest meshes.

An important parameter in studying flow past bluff bodies is the Strouhal number. Numerous studies have investigated the effect of different variables on the Strouhal number. An example would be the study of Bouzari and Ghazanfarian (2017), who studied the impact of radial fins on unsteady convective heat transfer and the Strouhal num­ ber. They reported that as the number of fins increases from 0 to 6, the Strouhal number decreases to the point that the Strouhal number of the cylinder with six fins is 28% less than that of a cylinder with only one fin. There are also studies (Ghazanfarian and Nobari, 2009) that have investigated the effect of rotation of cylinder on the Strouhal number. 3

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Ocean Engineering 195 (2020) 106688

Fig. 7. Variation of the mean dimensionless shear stress versus the dimensionless radius for the case of smooth, partially and totally superhydrophobic sphere at the Reynolds numbers of (a) 400, (b) 1000, (c) 2000. The superhydrophobic Janus angle is 90∘ . As shown in the figure, x indicates the horizontal distance of the border of superhydrophobic section and the center of the sphere.

Fig. 8. Variation of (a) the drag coefficient and (b) the Strouhal number as a function of Reynolds number for the no-slip surface, the case with SHF ¼ 0.425, and Partially SHF sphere with Janus angle 90∘ . Experimental data from References (Sakamoto and Haniu, 1990; Kim and Durbin, 1988; Achenbach, 1974; Roos and Willmarth, 1971).

4

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local drug diffusion control, reduction of infections and bacterial adhesion on newly implanted body organ, etc. (Efremov et al., 2013). Ou et al. (2004) investigated the effect of ultra-hydrophobicity on the pressure drop of the flow in microchannels. They conducted a series of experiments to find out the impact of micropost size and their spacing on the pressure drop of the flow and reported that as the spacing of the microposts increases from 15 μm to 150 μm, the amount of pressure drop decreases by about 30%. Ou et al. (2004) stated that the efficiency of these surfaces increases by increasing the spacing of the posts. George et al. (2016) created a superhydrophobic surface by repli­ cating laser written nanoscale hierarchical structures on PMMA. They concluded that the wettability and the contact angle of a water droplet on such fabricated surface is only a function of the laser fluence. Daniello et al. (2009) studied the turbulent flow over two types of micro-patterned superhydrophobic surfaces by the PIV technique and the direct pressure measurements. They found that the slip velocities and the drag reduction increase with increasing the Reynolds number and the slip velocities, and the drag reduction tends to grow more rapidly with increasing the feature spacing. They also noticed through experiments that the viscous sublayer is a limiting factor in choosing the micro-ridges’ spacing, deducting that if the micro-ridges’ spacing is higher than the viscous sublayer, the reduction of the drag coefficient stops. In the present paper, the effects of superhydrophobicity on the drag coefficient and the Strouhal number of the three-dimensional sphere and its impact on the reattachment length and the recirculation region of flow over a two-dimensional backward-facing step have been investi­ gated. Furthermore, the sections of these geometries that contribute the most to such alterations are identified. These findings will enable us to extend such trends to predict results on other internal or external geometries. The first step is to numerically model the slip coefficient, which is done by validating the numerical results of flow past the threedimensional benchmark case by experiments presented by Daniello et al. (2013). The details of computing such engineering model are presented in Ref. (Zeinali et al., 2018). The result of (Zeinali et al., 2018) suggests that for the SHF value of 0.425, the corresponding experimental results (Daniello et al., 2013) can be achieved. Based on the numerically found slip coefficient, other essential pa­ rameters such as the drag coefficient and the Strouhal number of the sphere and the reattachment length and recirculation region of the backward-facing step are computed. For validation of the numerical results of flow past a three-dimensional sphere, the data presented by Tomboulides et al. (1993), Margarvey and Bishop (1961), Kim and Durbin (1988) and Achenbach (1974) and for flow over the backward-facing step, the experimental results of Armaly et al. (1983) have been used. The procedure for producing superhydrophobic material is a chal­ lenging exercise (Zeinali et al., 2018). Due to high manufacturing costs and the complexities of manufacturing the superhydrophobic surfaces, it is crucial to identify the most effective segments of the mentioned ge­ ometries. For this purpose, the skin friction coefficient is calculated along the surface of both sphere and backward-facing step in the framework of the Janus concept. The sections along the sphere and the backward-facing step surface with the highest contribution to the total drag force reduction and reattachment length relocation have been identified. The main focus of the present study is to examine the effect of superhydrophobicity on the flow past sphere and over a backwardfacing step. The superhydrophobicity boundary condition that equal­ izes the sanded Teflon surface has been numerically modeled and used throughout this work. By analyzing the shear stress diagram and the Janus concept, the fraction of these geometries’ surface, that play the key role on the observed changes, are spotted. Such study results in a more effective and optimal usage of the superhydrophobicity, which will, in turn, reduce the expenditure of using this new technology.

Fig. 9. Distribution of the mean dimensionless slip-velocity on the surface of the partially superhydrophobic sphere with Janus angle of 90∘ at different Reynolds numbers at the dimensionless time of 0.06.

frequency of 1.1. Flow past steps is a famous case in fluid dynamics. One of the most comprehensive experiments concerning steps has been conducted by Armaly et al. (1983), who examined how the reattachment length is affected by the Reynolds number in a span of 70–8000. They also con­ ducted numerical investigations. Ikohagi and Shin (1991), Gartling (1990), and Kim and Moin (1985) all used different computer codes to simulate flow over a two-dimensional backward-facing step up to the Reynolds number of 1200. Williams and Baker (1997) also simulated flow up to the Reynolds number of 1200 but in a three-dimensional computational domain and their work is in excellent accordance with the experiments of Armaly et al. (1983). Meri and Wengle (2002) focused on differences between the spatial discretization order (the fourth-order Hermitian and the second-order central schemes) of the numerical simulations, including both the DNS and the LES, on a case involving the backward facing steps at the Reynolds number of 3300. They found that using a fourth-order discretization technique in their LES simulations would yield more accurate results but is not as precise as those of the DNS. Terekhov and Terekhov (2017) studied the effect of suction and in­ jection on the reattachment length at the Reynolds number of 5000 and 50000. They reported that the normalized injection velocity of 0.02, would move the reattachment length further downstream by almost 10% while the normalized suction velocity of 0.02 decreases this value by 30%. The drag suppression has been sought for various applications. This phenomenon can be achieved in multiple ways, for instance, the rotation of the body (namely cylinder) (Nobari and Ghazanfarian, 2010), impo­ sition of electromagnetic fields on the surface (Jiang et al., 2019), making the surface superhydrophobic or other methods that are wholly reviewed by Rashidi et al. (2016). The superhydrophobicity is an exciting characteristic of surfaces, first discovered by high water repel­ lency of lotus leaves (Barthlott and Neinhui, 1997). Among many applications, this phenomenon can reduce the drag coefficient (Ou et al., 2004), demonstrate the self-cleaning properties (George et al., 2016) and has an anti-corrosion effect (Das et al., 2018), that can increase the lifespan of the products in the maritime industry. Medical science is another field where the superhydrophobicity has growing application (Faldea et al., 2016). Superhydrophobicity is of diverse interest for a variety of medical applications that include the 5

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Ocean Engineering 195 (2020) 106688

Fig. 10. The pressure coefficient distribution (left column) and the pressure contours (right column) at Re ¼ 2000, for smooth sphere, (a) the beginning of a cycle at t � ¼ 10:95 � 104 , (b) the shedding phase ¼ π=2, t � ¼ 11:4 � 104 , (c) the shedding phase ¼ π and t � ¼ 11:85 � 104 (d) shedding phase ¼ 3π=2.

The present paper is organized as follows. Sec 2 presents the details of geometry and boundary conditions and the numerical structure of the solution in the OpenFOAM codes. Sec 3 describes the implementation of the engineering model for the superhydrophobic surfaces. Sec 4 presents the computational results, and finally, Sec 5 concludes the paper.

subgrid-scale (SGS) stress tensor, which represents the motion of fluid at scales smaller than the filter width. To obtain the closure, the SGS model of one equation eddy-viscosity model is used in the present paper (Legendre et al., 2009). The eddy viscosity is calculated from the value of the SGS turbulent kinetic en­ ergy, kSGS , derived from the following equation, � � ∂kSGS ∂kSGS ∂ 2 ∂kSGS ; (2) ¼P εþ þ ui ∂xi ReT ∂xi ∂t ∂xi

2. Mathematical formulation and numerical models The governing equations for the continuity and momentum conser­ vation law in their non-dimensional form are as follows.

∂ui ¼0 ∂xi � ∂p ∂ui ∂ 2 ∂Sij uu þ ¼ þ ∂t ∂xj i j ∂xi Re ∂xj

where, 3

∂τrij ∂xj

P ¼ 2νT Sij Sij ;

(1)

1

l ¼ ðΔxΔyΔzÞ3 ;

where , p, u, ∂∂x, ν, Sij and t denote filtered variable, filtered pressure field, filtered velocity, derivation with respect to x, the kinematic viscosity, the rate of strain tensor, and time, respectively. τrij is the residual

ReT ¼

6

UL

νT

ε ¼ Cε

Sij ¼ ;

k2SGS l

� � 1 ∂ui ∂uj ; þ 2 ∂xj ∂xi

νT ¼ Ck k0:5 SGS l:

(3)

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Ocean Engineering 195 (2020) 106688

Fig. 11. The pressure coefficient distribution (left column) and the pressure contours (right column) at Re ¼ 2000, for partially superhydrophobic sphere with the Janus angle of 90∘ (a) the beginning of a cycle at t � ¼ 10:95 � 104 , (b) the shedding phase ¼ π=2, t � ¼ 11:4 � 104 , (c) the shedding phase ¼ t � ¼ π and 11:85 � 104 , (d) the shedding phase ¼ 3π=2.

The current study has numerically modeled the boundary condition that replicates the sanded Teflon superhydrophobic surface, which the related data have been experimentally produced by Daniello et al. (2013) and the extracted model has been used throughout the present work. The overall view of the computational domain and the geometrical details of the flow over the sphere are depicted in Fig. 1. To be sure about the reliability of this model, the mesh-independence studies have been conducted using different grid sizes at the Reynolds number of 400. The model has also been verified by the time-step-size independence study at the same Reynolds number. The results of the grid study and the timestep independence tests have been presented in Tables 1 and 2, respectively. It should be noted that the Courant number is kept under unity and the inlet boundary condition for the studied cases involve a developed velocity profile with turbulence inlet boundary condition, which would simulate turbulence at the entrance with the scale of 0:05% of the mean flow. Although it should be mentioned that the flow over step has been

where U and L are the characteristic velocity and length, respectively. The details of one-equation LES model can be found in Ref. (Legendre et al., 2009). In order to model the superhydrophobic condition over walls, a third-type mixed (Robin) boundary condition has been introduced and implemented into the OpenFOAM codes. Equation (1) has been solved with respect to the superhydrophobic boundary condition, � �� ∂u u�slip þ ð1 βÞ ¼0 (4) ∂y� wall where uslip is the fluid’s relative velocity at wall. β, the super­ hydrophobicity factor (SHF), is an adjustable coefficient which has been computed based on the curve-fitting process to the experimental data presented by Daniello et al. (2013) (Zeinali et al., 2018). The dimen­ sionless parameters are the Reynolds number (Re ¼ ρUDμ 1 ) and the SHF (β) coefficient. It is found that this coefficient does not depend on the flow regime, the Reynolds number, or the shear stress of the wall. 7

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Ocean Engineering 195 (2020) 106688

Fig. 12. The right column in vorticity and the left column denotes to Q-Crite­ rion at Q ¼ 100. (a, c, e, g) are for smooth sphere and (b, d, f, h) are for partially superhydrophobic sphere with Janus angle of 90∘ . The non-dimensional time of the simulations are as follows. (a, b) t ¼ 10:85 � 104 that corresponds to first phase of shedding, (c, d) t ¼ 11:3 � 104 corresponds to π=2 phase of shedding, (e, f) 11:75 � 104 corre­ sponds to π phase of shedding and (g, h) t ¼ 12:2 � 104 ) which corresponds to 3π=2 phase of shedding.

investigated in pseudo-3D case. All simulations in OpenFOAM are con­ ducted using 3D meshes with a specific thickness in the third dimension. Details of the geometry of planar flow inside a channel containing a backward-facing step have been given in Fig. 2. The mesh- and the timestep-independence studies have been accomplished at the Reynolds number of 300, and the results are available in Table 3 and Table 4, respectively. In these tables, the lengths have been non-dimensionalized using the height of the step. The velocity-slip boundary condition has been integrated into the open-source OpenFOAM Cþþ libraries. The PisoFoam solver has been chosen as the basic solver (OpenFOAM User’s Guide, 2018). The PISO algorithm (Pressure Implicit with Splitting of Operators) is found to be suitable for solving the pressure-velocity coupling. The number of iter­ ations for the pressure equation and the momentum corrector in each step is set to 2. The tolerance of convergence is 1 � 10 7 . The PisoFoam algorithm is relatively fast, but it comes at the expense of it being very sensitive to the time-step size. So, for the stability issues, the Courant number is kept less than 1 throughout the simulations.

3. Numerical validation and verification 3.1. Case 1: flow over cylinder Value of the superhydrophobicity factor (SHF) as a function of the Reynolds number has been calculated based on the data presented by (Daniello et al., 2013), and the details are presented in (Zeinali et al., 2018), which suggest that the most compatible results with the experi­ mental data can be obtained using the SHF value equal to 0.425. The same superhydrophobicity constant and boundary condition will be applied to the surfaces of the sphere and the backward-facing step. Fig. 3 shows the details of the computation of the SHF factor using the data corresponding to the case of a three-dimensional cylinder with super­ hydrophobic surfaces. 3.2. Case 2: flow over sphere Table 5 presents the Strouhal number and the drag coefficient calculated from the present LES simulation of flow over the sphere in comparison with the existing experimental results (Margarvey and Bishop, 1961; Tomboulides et al., 1993; Kim and Durbin, 1988; Achenbach, 1974) using the computational domain with details described in Table 1 at the Reynolds number of Re ¼ 400. The 8

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Ocean Engineering 195 (2020) 106688

comparison shows an agreement between the current predictions and the experimental results with the mean relative error less than 5% and 3% for the drag coefficient and the Strouhal number, respectively.

Table 7 Reattachment length and recirculation region for the no-slip backward-facing step, the fully superhydrophobic case, and the case with a1 to a5 made super­ hydrophobic. In this figure every element (a1 to a5) are given the super­ hydrophobic wall condition individually. Armaly et al. (Armaly et al., 1983)

Current study

Current study Current study

Current study

Current study

Noslip Noslip Noslip Noslip Noslip Noslip All SH All SH All SH A1 SH A2 SH A3 SH A4 SH A5 SH A1 SH A2 SH A3 SH A4 SH A5 SH A1 SH A2 SH A3 SH A4 SH A5 SH

Re

X1 =s

300

6.87

X2 =s

X3 =s

600

11.48

8.92

15.98

1200

17.67

14.40

23.259

300

6.58

600

10.73

9.08

14.65

1200

16.33

13.487

25.023

300 600 1200 300 300 300 300 300 600 600 600 600 600 1200 1200 1200 1200 1200

6.85 10.93 16.63 6.58 6.58 6.83 6.58 6.58 10.73 10.73 10.93 10.73 10.73 16.33 16.33 16.59 16.33 16.33

9.18 13.96

14.93 25.65

9.08 9.08 9.15 9.08 9.08 13.487 13.487 13.92 13.487 13.487

3.3. Case 3: flow over step Fig. 4 shows the geometrical details and symbols defined to measure the reattachment length and the recirculation region for flow over a backward-facing step, which has been previously experimentally stud­ ied by Armaly et al. (1983). These lengths have been non-dimensionalized by the height of the step, which is shown in Fig. 4. Table 6 lists the results of the LES simulation for the mentioned geom­ etry, which have been compared with the existing experimental results of Armaly et al. (1983) at the Reynolds numbers of Re ¼ 300, 600, 1200. The comparison proves an agreement between the current predictions and the experimental results with an average relative error of approxi­ mately 6%. Table 8 Comparison of reattachment length and recirculation region for the no-slip, the fully superhydrophobic, and the partially superhydrophobic backward-facing step. Only one-fourth of the a3 element (40.82–51) has a superhydrophobic wall condition.

14.65 14.65 14.92 14.65 14.65 25.023 25.02 25.61 25.02 25.02

Smooth Superhydrophobic Partial SH

Reynolds Number

x1 =s

300 600 1200 300 600 1200 300 600 1200

6.58 10.73 16.33 6.85 10.93 16.63 6.85 10.92 16.61

x2 =s

x3 =s

9.08 13.49

14.65 25.02

9.18 13.96

14.93 25.65

9.18 13.95

14.93 25.65

Fig. 13. Variation of the dimensionless shear stress at non-dimensional time of 15.3 on a1 (0–40.82), a3 (40.82–81.64) of the backward-facing step for both no-slip and superhydrophobic cases at Reynolds numbers of (a) 300 [a1, a3], (b) 600 [a1, a3] and (c) 1200 [a1, a3], and on the vertical side of the step (d) a2 at various Reynolds numbers. The horizontal axis is non-dimensionalized by the height of the step. 9

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Ocean Engineering 195 (2020) 106688

Fig. 14. Variation of the dimensionless slip-velocity versus the dimensionless position on (a) a1 and (b) a3 at the Reynolds number of 300, 600, and 1200, when all surfaces are superhydrophobic. The effect of superhydrophobicity on the slip velocity increases by the Reynolds number.

Fig. 15. Streamlines of flow over the backward-facing step at Re ¼ 300 (a) the no-slip, (b) the superhydrophobic cases, Re ¼ 600 (c) the no-slip, (d) the super­ hydrophobic cases, Re ¼ 1200 (e) the no-slip, (f) the superhydrophobic cases.

4. Results and discussions

superhydrophobic sphere with the SHF factor of β ¼ 0:425 will be pre­ sented in this section. First, the adequacy of the grid resolution near the wall should be assessed. As a fact, for the boundary layer created along a flat plate, the first grid point should be inside the viscous sub-layer.

Case I. Sphere As the flow past the sphere with the no-slip condition has been validated using the experimental data, the results of flow over a 10

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Ocean Engineering 195 (2020) 106688

Fig. 16. Comparison of vorticity contours (upper) and the Q-criterion at Q ¼ 24 (lower) for smooth (a,c,e) and super­ hydrophobic (b,d,f) backward-facing step, at the Reynolds number of 300 (a,b), 600 (c,d) and 1200 (e,f) and dimensionless time of 30.7. Comparison of the smooth and super­ hydrophobic cases unveils that the super­ hydrophobicity imposition moves the reattachment length and the recirculation region further downstream, and also ex­ pands the width of the recirculation regions.

1975). Fig. 6 shows the spectral analysis of the lift coefficient of the sphere at the Reynolds number of 2000. Fig. 6 displays the variation of the power density versus the dimensionless frequency or the Strouhal number of flow past the sphere with the no-slip and the super­ hydrophobic conditions. The spectrum is obtained by taking the fast Fourier transformation of the time-history of the lift coefficient, and it is an indicator of the dominant vortex shedding frequencies in the wake of the sphere. Comparison of the dominant frequencies of the no-slip and the superhydrophobic spheres concludes that the normalized vortex shed­ ding frequency of the superhydrophobic surface experiences a growth of about 22% and 10% for the first and the second peaks, respectively. So, the superhydrophobicity not only increases the frequency of the periodic flow over the sphere but also weakens the value of the peaks in the spectrum. Since the drag coefficient is closely related to the variation of the shear stress along the wall and the angle of separation point at which the

Based on the wall scale, this condition is satisfied as long as yþ ¼ 5, where yþ is defined as yþ ¼ uτ y=ν where uτ , y and ν are the friction velocity, the wall normal distance, and the fluid kinematic viscosity, respectively. For flow around the sphere with a curved surface, a conservative approach is to have the first grid point at yþ < 1 to ensure that the nearwall grid resolution is fine enough to capture the wall shear stress correctly. The plots of the mean value of yþ as a function of the Reynolds number over the surfaces of the no-slip and the superhydrophobic spheres with coarse, fine, and finest meshes are reported in Fig. 5. As it is obvious from the figure, the value of yþ has been conservatively kept less than 0.3 for all cases. As mentioned earlier, the experimental data (Sakamoto and Haniu, 1990; Kim and Durbin, 1988; Achenbach, 1974) have shown two different modes in the Strouhal number spectrum of the sphere for the Reynolds numbers higher than 800. This high-mode exists up to the Reynolds number of 104 (Moller, 1938), while at higher Reynolds numbers, only the low-mode Strouhal number can be seen (Taneda, 11

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Ocean Engineering 195 (2020) 106688

shear stress becomes zero, the variation of shear stress along the surface of the sphere is studied in this section. Fig. 7 illustrates the distribution of the mean dimensionless shear stress versus the dimensionless distance between the border of the superhydrophobic section and the center of sphere at various Reynolds numbers in the range 400–2000. Three cases of the smooth sphere, the superhydrophobic sphere, and the partially superhydrophobic sphere with the Janus angle of 90� measured from the upstream stagnation point at the Reynolds number of 400, 1000, 2000 are considered. The Janus degree can easily be computed using the data presented in Fig. 7. It is found that in all cases, the superhydrophobicity reduces the magnitude of the local shear stress on the surface. Results show that the maximum normalized shear stress has been reduced from 0.38, 1.55, 3.6 to 0.34, 1.4, 3.2 by increasing the Reynolds number. Comparison of totally and partially superhydrophobic spheres implies that imposing the superhydrophobicity behind the sphere has a minor effect on the distribution of the shear stress along the entire body. So, it can be ex­ pected that using the superhydrophobic materials behind the sphere where the shear stress is near zero is not beneficial. Another point in Fig. 7 is the influence of superhydrophobicity on the location of the separation point. Comparison of the data obtained from the smooth and the superhydrophobic cases depicts that the super­ hydrophobicity shifts the point of separation towards the upstream in all cases. The magnitude of the shift varies between 18� at the Reynolds number of 400 to almost 2� at the Reynolds number equal to 2000. So, it can be concluded that increasing the Reynolds number weakens the effect of the superhydrophobicity on the position of the separation point. In order to better understand the mechanism of the shear stress reduction, it shall be understood that due to the lower surface free en­ ergy of the coating or the trapped air between the ridges on the super­ hydrophobic surface, the lower friction and thus the slip velocity appears. Increase of the slip velocity on the surface is then responsible for the reduction of the slope of the velocity profile or the velocity gradient and the shear stress on the wall. Here, the partially superhydrophobic case with the Janus angle of 90∘ is considered. As Fig. 7 suggested, the major changes in the mean dimensionless shear stress occur between angles 0–90� . So, we use a new term called the partial superhydrophobicity, which means that only half of the sphere’s surface (0 to 90∘ , measured from the leading edge) is superhydrophobic. This kind of semi-optimization method was first presented by Zeinali et al. (2018), and as they suggested, this can be tested for other complex and multi-dimensional geometries. Fig. 8a presents the effects of the Reynolds number on flow past the superhydrophobic, the partially superhydrophobic and the smooth sphere. It is evident from this figure that the superhydrophobicity de­ creases the drag coefficient by 25% at the Reynolds number of 400 to 46% at Re ¼ 2000. The amount of the drag coefficient reduction and the difference between the partially and the totally superhydrophobic cases increases with the Reynolds number. The Strouhal number variation with respect to the Reynolds number is presented in Fig. 8b. At the Reynolds numbers above 800, the Strouhal number diagram is divided into two branches, the high-mode and the low mode (Sakamoto and Haniu, 1990; Chomaz et al., 1993; Kim and Durbin, 1988). While in this figure, the Strouhal number of Reynolds 1000 yields only one of them, both of these modes (high-mode and low-mode) are captures at Re ¼ 2000. The Strouhal number for partially and totally superhydrophobic cases are almost the same at Reynolds 400, 1000 and the low-mode Strouhal number at Reynolds number 2000, but this overlap does not exist in high-mode Strouhal number at Re ¼ 2000. The Strouhal number at Re ¼ 400 remains almost the same as the smooth case, while at the Reynolds number of 1000 and 2000, the Strouhal number increases by about 25%. The experimental results are also included in Fig. 8. For the drag coefficient, the experiments of Roos and Willmarth (1971), and regarding the Strouhal number, the results of Sakamoto and Haniu

(1990) and Kim and Durbin (1988) have been used. In addition to further validation of results, it is obvious from this figure that imposing the superhydrophobicity to half and all of the sphere’s surface, approximately yields the same result, which again emphasizes the noteworthiness of the Janus concept. The mean dimensionless velocity-slip distribution over the surface of the sphere is presented in Fig. 9. Abscissa of the curve corresponds to the smooth, non-superhydrophobic, slipless sphere. The slip-velocity shown in the figure has been normalized using the free-stream velocity. It is found that for all three cases the maximum slip-velocity takes place around x=R � 0:4, and its value changes from 0.12 to 0.25 at the Rey­ nolds number of 400–2000, respectively. As expected after the separa­ tion point regardless of the Reynolds number, the slip-velocity is nearly zero, which confirms the necessity of using the partial super­ hydrophobicity on the surface. In order to demonstrate the origin of drag suppression based on the Janus concept, the pressure contours and distribution of the pressure coefficient over the surface of smooth and the partially super­ hydrophobic sphere with the Janus angle of 90∘ at the Reynolds number of 2000 are shown in Figs. 10 and 11, respectively. The fluctuations in pressure coefficient are due to the fact that the diagram depicts 0 to 180∘ , but it represents both sides of the sphere, and these fluctuations are because of the difference between the two sides. If inspected closely, the difference between the leading edge and the trailing edge pressure coefficients (left column) of the smooth sphere (Fig. 10) is more than that of the partially superhydrophobic sphere (Fig. 11), which in turn can result in drag coefficient reduction. These figures show the variation of the pressure coefficient and the pressure contour during one cycle of vortex shedding. The right column of Fig. 12 compares the vorticity contours of in the wake of the sphere at the Reynolds number of 2000 during a vortex shedding cycle, while the left column displays the instantaneous isosurface of the second invariant of the velocity-gradient tensor at the same Reynolds number. The second invariant of the velocity-gradient tensor (Q-criterion) ∂ b ð∂ b u i =∂xj is defined as Q ¼ u i =∂xj Þð∂ b u j =∂xi Þ. The Q-criterion contours (shown in Fig. 12) correspond to Q ¼ 100. Cases a, c, e, g are for smooth sphere and sub-figures b, d, f, h are for a partially superhydrophobic sphere with the Janus angle of 90∘ . The normalized times are (a, b) t� ¼ 10:85 � 104 that corresponds to the beginning of the cycle, (c, d) t � ¼ 11:3 � 104 corresponds to π =2 phase of shedding, (e, f) t � ¼ 11:75 � 104 corresponds to π phase of shedding, and (g, h) t� ¼ 12:2 � 104 ) corresponds to 3π=2 phase of shedding. Comparison of these figures shows that by imposing the super­ hydrophobicity condition on walls of the sphere, the width and the length of the wake behind the sphere become narrower and more pro­ longed, respectively. Also, in the regions of positive Q, the vorticity is significant, so the iso-surface of Q is an identifier of the coherent vortices in the flow structure (Lesieur et al., 2005). Fig. 12 shows that the flow structure is three-dimensional, and has strong longitudinal vortices spread between the big Kelvin-Helmholtz billows. It also demonstrates that the superhydrophobicity wall condition does not alter the presence of the 3D coherent structures. Case II. Backward-facing step In order to investigate the impact of superhydrophobicity on the reattachment length and the recirculation region, walls of the step have been divided into five sections, which were named a1 to a5, as shown in Fig. 4. At first, the superhydrophobicity has been imposed to all sections, and then in order to identify the most influential section, each of 5 sections was given the same boundary condition individually, and the results have been reported in Tab 7. The shear stress on a4 and a5 is zero, and therefore, they are not crucial components for superhydrophobicity imposition. Although the shear stress is not zero on a1 and a2, they also have an insignificant impact on the reported changes. It is obvious from the data presented in Table 7 that imposing superhydrophobicity on a1, a2, a4, 12

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Ocean Engineering 195 (2020) 106688

a5 results in data that is equal to that of the no-slip condition. It is the a3 element which is the most influential section and thus imposing the superhydrophobicity on only a3 will present results equal to that of the whole geometry (a1 to a5) having complete superhydrophobic wall condition. The mean dimensionless shear stress on the walls of the backwardfacing step is studied thoroughly and Fig. 13 shows the distribution of this variable at the Reynolds number of 300, 600, 1200. It is evident from this figure that the changes in mean dimensionless shear stress are maximum on the a3 element. The shear stress is zero on a4 and a5, and therefore, they have no impact on the reattachment length and the recirculation area. Although the changes in the shear stress are non-zero on a1 and a2, their impact on such changes is still negligible, and thus the section which plays the most influential role on these changes is the a3 element. The oscillations near the corner of the step seen in Fig. 13 are because of the instantaneous creation, shedding, and interference of the vortex rings with the up/downstream flows. Also, the change of patch at xs ¼ 40 (before and after the step) may create sudden changes in flow pattern, and hence in the shear stress. From Fig. 13a–c, it is clear that the most significant amount of change of shear stress on a3 element takes place between 40.82 and 51. Now the a3 element shall be split into two sections (40.82–51 super­ hydrophobic and 51–81.64 smooth). The results are presented in Table 8. The results suggest that the impact of only one-fourth of the a3 element is equal to 95% of the impact of a complete superhydrophobic backward-facing step in which all elements are superhydrophobic. Influence of imposing the slip-velocity on a1 and a3 surfaces has also been investigated. This slip-velocity is normalized by the inlet velocity. It can be seen in Fig. 14 that by increasing the Reynolds number, the dimensionless slip-velocity also increases. The slip velocity at a1 in­ crease from 0.02 at Re ¼ 400 to 0.09 at Re ¼ 2000, while the average values in a3 are 0.01, 0.02 and 0.05 at Reynolds number 300, 600 and 1200, respectively. This means that the effect of superhydrophobicity on the slip velocity increases by the Reynolds number. It is clear that the minimum point in Fig. 14b, where the velocity reaches zero belongs to the location of reattachment length. Streamlines of flow in inside a backward-facing channel are shown in Fig. 15 at Re ¼ 300, 600, 1200 for both smooth and superhydrophobic cases. It is obvious that by increasing the fluid’s inlet velocity and consequently the Reynolds number, more stretched and powerful recirculation region will be formed. Imposing the superhydrophobicity on the walls of the backward-facing step also increases the length of the recirculation region, which will diminish by increasing the Reynolds number. Fig. 16 compares the vorticity and the Q-criterion contours of the smooth and the superhydrophobic step. It is obvious from these figures that the superhydrophobic wall condition will enlarge the width of the recirculation region by almost 10%. Analysis of these figures also reveals that the superhydrophobicity also moves the reattachment length further downstream by about 8%.

dominant segments in the mentioned changes. The numerical results, including the surface shear stress and the velocity-slip over the wall, have shown that imposing the superhydrophobic wall condition on almost half of the sphere surface and three-fourths of the backwardfacing step will have little effect on drag reduction and augmentation of the reattachment length. That is while the remaining 50% and 25% sections reduce the drag coefficient and increase the reattachment length almost identical to the whole surfaces being superhydrophobic. References Achenbach, E., 1974. Vortex shedding from spheres. J. Fluid Mech. 62, 209–221. Amiraslanpoura, M., Ghazanfarian, J., Razavi, S.E., 2017. Drag suppression for 2D oscillating cylinder with various arrangement of splitters at Re¼100: a highamplitude study with OpenFOAM. J. Wind Eng. Ind. Aerodyn. 164, 128–137. Armaly, B.F., Durst, F., Pereira, J.C., Schounung, B., 1983. Experimental and theoretical investigation of backward facing step flow. 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5. Conclusions The effects of superhydrophobicity on flow characteristics over the three-dimensional sphere and the two-dimensional backward-facing step is investigated numerically using the large eddy simulation method. After validating the computational simulation results, the correlation for the boundary condition is implemented using a new third-type Robin boundary condition. It is found that the superhydrophobicity in the case of the sphere will result in a reduction of the drag coefficient and an augmentation of the Strouhal number, and in the backward-facing step case moves the reattachment length further downstream and increases the length strength of the recirculation zone. Furthermore, the concept of Janus surface is also implemented in both cases to identify the 13

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Nobari, M.R.H., Ghazanfarian, J., 2010. Convective heat transfer from a rotating cylinder with inline oscillation. Int. J. Therm. Sci. 49, 2026–2036. OpenFOAM User’s Guide, 2018. https://www.openfoam.com/documentation/user-gu ide. Ou, J., Perot, J.B., Rothstein, J.P., 2004. Laminar drag reduction in microchannels using ultrahydrophobic surfaces. Phys. Fluids 16, 4635–4643. Pan, J.H., Zhang, N.M., Ni, M.J., 2018. The wake structure and transition process of a flow past a sphere affected by a streamwise magnetic field. J. Fluid Mech. 842, 248–272. Rashidi, S., Hayatdavoodi, H., Esfahani, J.A., 2016. Vortex shedding suppression and wake control: a review. Ocean Eng. 126, 57–80. Roos, W.F., Willmarth, W.W., 1971. Some experimental results on sphere and disk drag. AIAA J. 9, 285–291. Sakamoto, H., Haniu, H., 1990. A study of vortex shedding from spheres in uniform flow. J. Fluids Eng. 112, 386–392. Taneda, S., 1956. Experimental investigation of the wake behind a sphere at low Reynolds numbers. J. Phys. Soc. Jpn. 1104–1108.

Taneda, S., 1975. Visual observations of the flow past a sphere at Reynolds numbers between 10 4 and 10 6. J. Fluid Mech. 85, 187–192. Terekhov, V.V., Terekhov, V.I., 2017. Effect of surface permeability on the structure of a separated turbulent flow and heat transfer behind a backward-facing step. J. Appl. Mech. Tech. Phys. 58, 254–263. Tomboulides, A.G., Orszag, S., Karniadakis, G., 1993. Direct and Large-eddy simulations of axisymmetric wakes. Reno, NV, January 11-14. In: Proceedings of the Aerospace Sciences Meetings. AIAA Paper No. 93-0546. Williams, P.T., Baker, A.J., 1997. Numerical simulations of laminar flow over a 3D backward-facing step. Int. J. Numer. Methods Fluids 24, 1159–1183. Wu, J.S., Faeth, G.M., 1993. Sphere wakes in still surroundings at intermediate Reynolds numbers. AIAA J. 31, 1448–1455. Zeinali, B., Ghazanfarian, J., Lessani, B., 2018. Janus surface concept for threedimensional turbulent flows. Comput. Fluid 170, 213–221. Zhou, B., Wang, X., Guo, W., Min Gho, W., Keat Tan, S., 2015. Experimental study on flow past a circular cylinder with rough surface. Ocean Eng. 109, 7–13.

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