WAVE-LENGTH-DEPENDENT REARRANGEMENT OF SECONDARY VORTICES OVER SUPERHYDROPHOBIC SURFACES WITH STREAMWISE GROOVES

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1 June 3 - Jul 3, Melbourne, Australia 9 9A-1 WAVE-LENGTH-DEPENDENT REARRANGEMENT OF SECONDARY VORTICES OVER SUPERHYDROPHOBIC SURFACES WITH STREAMWISE GROOVES A. Stroh 1, Y. Hasegawa, J. Kriegseis 1 and B. Frohnapfel 1 1 Institute of Fluid Mechanics, Karlsruhe Institute of Technolog, Karlsruhe, German alexander.stroh / jochen.kriegseis / Institute of Industrial Science, The Universit of Toko, Toko, Japan sk@iis.u-toko.ac.jp ABSTRACT An investigation of the flow dnamics over a superhdrophobic surface (SHS) carring streamwise micro grooves is performed in a full developed turbulent channel flow under a constant pressure gradient (CPG). The SHS is modeled as a flat boundar with alternating and slip conditions. A series of direct numerical simulations (DNS) is carried out sstematicall varing the spanwise periodicit of the streamwise grooves. It is observed that the alternating and boundar conditions cause a spanwise inhomogeneit of the Renolds shear stress near the SHS, and consequentl generate Prandtl s second kind of secondar flow characteried b coherent streamwise vortices. The arrangement and rotational direction of these motions are shown to be strongl affected b the dimension of the introduced SHS. Therefore, the detailed turbulent statistics are obtained and the effect of the secondar flow on the resultant flow rate increase is discussed. For the clarification of the secondar flow nature, the vortical structures are investigated b means of streamline-based topolog analsis and complementar application of the vorticit transport equation. INTRODUCTION The reduction of skin friction drag in turbulent flows is an important issue linked to the efficient use of available energ resources. Superhdrophobic surfaces are shown to be an interesting approach to this issue in water flows. These surfaces are able to enforce local -like conditions at the wall due to the coating with a certain roughness pattern, which entraps air in the surface cavities. Since the shear stress at a liquid-gas interface is much lower than the stress at a full wetted liquid-solid interface, the overall skin friction drag is significantl reduced. The effects of SHS in the laminar flow regime have been analticall analed b Philip (197), while investigations in the turbulent regime have been conducted onl in the last decade b means of experiments (Daniello et al., 9) and numerical simulations (Min & Kim, 4; Martell et al., 9). The recent work b Türk et al. (14) states a possible flow rate increase up to % under constant pressure gradient conditions in a full developed turbulent channel flow with SHS in comparison to plain channel flow and proposes a model for SHS simulations based on an edd viscosit approach. This work also shows the appearance of secondar motions over SHS due to the spanwise inhomogeneit of the boundar conditions at the wall. The results of Türk et al. (14) show an opposite rotational direction of the induced vortical motions when the spanwise extent of the surface grooves, L, as depicted in Figure 1 is increased from 1 to 8. This change of the rotational direction is found to affect the resultant drag reduction effect. In their stud, the authors hpothesie that this change occurs due to the absence of direct interaction between the small-scale secondar motions at the / interface for larger L. Instead, largescale tertiar motions are introduced b a pair of secondar vortices. In order to further investigate this hpothesis we focus on the analsis of the SHS-induced secondar motions in dependence of the / area sie for 1 < L < 8, where the switch of the secondar motion rotation occurs. It should be noted that the present stud assumes the existence of a stable and flat liquid-gas interface. This model assumption is probabl limited to small values of L (Seo et al., 13; Türk et al., 14). Nevertheless, similar phenomenon for secondar motion is also observed in the work of Goldstein & Tuan (1998) where a cross-plane secondar motion appears in turbulent flow over streamwise oriented riblets if a certain riblet spacing is exceeded. PROCEDURE A series of DNS has been carried out in a full turbulent channel flow driven b a CPG. For an incompressible Newtonian fluid, the flow satisfies the continuit and Navier-Stokes equations: u i x i =, (1) 1

2 d L x flow.πδ Figure 1. Schematic of the numerical domain with / boundar conditions at the walls. πδ δ Du i Dt = 1 p ρ xi + ν u i x, () j x j where p is the static pressure and ν is the kinematic viscosit. The asterisk denotes a dimensional quantit, while otherwise quantities are non-dimensionalied b the friction velocit, u τ = τw/ρ, such that u = u /u τ, x i = xi u τ/ν and t = t u τ /ν (viscous units). Note that u τ remains unchanged across the investigated parameter range, since the pressure gradient is kept constant during the simulation. The same applies for the friction Renolds number given b Re τ = u τδ /ν = 18 where δ is the half channel height. The Navier-Stokes equations are numericall integrated b a finite difference method on a staggered grid with a fractional step method for pressure decoupling. For temporal advancement, the convection and viscous terms are discretied using the nd order Adams-Bashforth and Crank-Nicholson methods, respectivel. The schematic of the numerical domain is depicted in Figure 1. Periodic boundar conditions are applied in streamwise (x) and spanwise () directions, while the wall-normal () extension of the domain is bounded b alternating / boundar conditions at the upper ( = δ) and lower wall ( = ). An impermeabilit condition is applied for the wall-normal velocit component on the entire area of both channel walls. The wave-length, L, represents the sie of the alternating structure with a solid fraction Φ = d/l =.. The secondar motion is observed in the cross sectional plane perpendicular to the main flow direction. For the variation of L the number of grid nodes in spanwise direction is adjusted, while the spanwise resolution, and, is kept constant. Consequentl, the spanwise extension of computational domain varies with L and the wall area is alwas covered b four wave-lengths (Table 1). Since the change of the secondar flow rotation occurs in the range 1 < L < 8, following wave-lengths are selected for simulations: L = 8, 16, 144, 148, 3, 8, 16, 171, 18,, and. Additionall, the data-sets from Türk et al. (14) for L = 3.,7.8,141.4 and L = 8.8 with =.9 are used in the present investigation. Table 1. Numerical properties of the performed DNS. N x N N x x min The statistical data-set is obtained for an integration time of approximatel edd turn-overs. Considering the periodic nature of the streamwise grooves, we introduce the phase averaging operator as ϕ (θ,) = 1 N ( ( ) ) θ N ϕ x,,l n=1 t x π + n,t dxdt, (3) where ϕ is an arbitrar variable as a function of space and time, whilst θ is a phase with respect to the periodic structure and N is a number of periods in the computational domain. Hence, an flow quantit can be decomposed into the phase averaged and the random components: ϕ(x,,,t) = ϕ(θ,) + ϕ (x,,,t). (4) In this paper, the secondar flow is regarded as the phase averaged flow field. Additionall, using an assumption of smmetrical velocit distribution with respect to the middle-line of the or the region as well as to the channel half-height, we perform corresponding spatial averaging in order to obtain smoother statistical data. Flow field analsis based on topolog considerations as comprehensivel discussed e.g. b Foss (4) is applied to the resultant vector fields in order to understand the arrangement of the appearing secondar motions. One such example of secondar flow is illustrated in Figure (a) for L = 16 and Φ =.. Line integral convolution (LIC, Cabral & Leedom (1993)) is used for the identification of the singular points in the vector field such as nodes (N), saddles (S) and half-saddles (S ). Finall, a topological map is derived from the position of the singular points and LIC (Figure (b)). Considering velocit invariant methods, an analsis of the secondar flow dnamics based on the vorticit production terms is performed. Utiliing the definition for the streamwise vorticit ω x = w v, () combined with the Navier-Stokes equations () for the wallnormal (i = ) and spanwise (i = 3) velocit component we obtain the following transport equation for the phase averaged streamwise vorticit: v ω x + w ω x = v w w w + v v + v w }{{ } streamwise vorticit production, P ωx + 1 ( ω x Re τ + ) ω x. (6) The terms on the left-hand-side represent the convection of ω x due to the secondar flow. The first four terms on the right-hand-side are production terms of ω x, while the last term represents the viscous diffusion of ω x. Note that the production terms have non-ero values onl when the and surfaces are resolved. Therefore, the effective slip models which are homogeneous in the spanwise direction (Min & Kim, 4; Fukagata et al., 6) are not able to predict the formation of the secondar flow.

3 node u 18 saddle half-saddle (a) vector plot with streamwise mean velocit Figure (b) topological map with LIC in background Secondar motions over SHS in --plane for L = 16 and Φ =.. TOPOLOGICAL REARRANGEMENT with opposite rotational direction of the dominant structures the secondar motion strength increases again and reaches 1 % of Ub for < L < 88. The influence of the secondar motion on the increase of bulk mean velocit is shown in Figure 4(b). The plot compares the present DNS results with a model prediction without secondar flow and with consideration of secondar motion, where v and w extracted from the DNS data is used for the evaluation of convective terms. The model utilies a uniform -dependent turbulent viscosit profile evaluated from the empirical relation b Renolds & Hussain (197) with adjustment of van Driest constant (Tu rk et al., 14). The model without secondar motion over-predicts the increase of Ub b 4 %, while the overshoot increases for larger L. The model with secondar motion improves the prediction of Ub especiall for larger L and cases where secondar motion intensit is significant (e.g. L = 8, Fig. 4(a)). When the secondar motion is taken into account the model over-prediction does not exceed % and remains < % for L < 16 and L >. The strongest deviation from the DNS data is observed for 16 < L < where the rearrangement of the secondar structures occurs. In order to clarif the influence of the solid fraction on the secondar motion formation a parametric stud with variation of Φ at constant L = 16 is performed. Figure illustrates the change of the secondar motion topolog when the solid fraction is slightl reduced or increased. It is evident that a slight reduction of the solid fraction at L = 16 (Φ =.43 with d = 7) corresponding to an increase of the spacing leads to a simplification of the motion topolog with two smaller-scale edge vortices and the larger-scale vortex-pair above. This configuration strongl resembles the topolog present for Φ =. and wave-length L 171 (Figure 3). An increase of the solid fraction at L = 16 (Φ =.7 with d = 9) also meaning a reduction of the spacing shows the opposite effect: the large-scale edge vortex-pair dominates the topolog while the vortex-pair above the region is reduced in sie. This topolog is similar to the cases with Φ =. at L < 16 (e.g. Figure 3, L = 149). The observation suggests that the secondar flow topolog is rather determined b the dimension of the area while the influence of the spacing is minor. Figure 3 illustrates the change of the flow topolog for increasing L b introducing color coding for the same vortical structures at different wave-lengths. The SHS with wave-lengths up to L = 1 show similar flow topolog with two counter-rotating large-scale secondar vortices causing an upward motion above the area and a downward motion above the area. Starting from L = 144 the flow topolog significantl changes, for L = 149 a formation of two additional vortices above the middle of the noslip area and two arising vortices above the / freeslip edge can be clearl identified. Further increase of the wave-length to L = 16 demonstrates an even more complex topolog map with an additional vortex-pair above the region. An expansion of the previousl observed near-wall vortical structures above the region occurs, while the edge vortices grow and move towards the wall. The large-scale motions are pushed farther to the outer flow region. For L = 171 we see that the newl appeared vortex-pair is absorbed b the growing vortices above the region, while the vortex-pair which was dominant for smaller L is shrinking. Starting from L = 18 the sie of the edge vortices is successivel reduced and the vortices above the area dominate the cross section of the flow field. For L = onl two vortex-pairs can be observed - the small edge vortices which completel disappear for L > and the major vortex-pair occuping the entire domain half-height. Comparing the topolog map of the flow over SHS at L = 141 and L = one can clearl see that the rotational direction of the dominating vortex-pair is reversed. For larger L, flow is pushed towards the wall above the region, while an upward motion occurs above the area. Hence, the switch of the rotation is a consequence of a complete reorganiation of secondar motions caused b the increase of the wave-length. It has to be emphasied that the strength of the observed motions varies depending on L. Figure 4(a) shows the maximum magnitude of the secondar motion normalied b the corresponding bulk mean velocit, Ub. For the cases up to L = 8 the magnitude of the secondar motion increases to 1% of the bulk mean velocit followed b a decrease of the magnitude between 16 and 18 where the rearrangement of vortical structures occurs. For the larger L 3

4 L=7 L=141 L=149 S'4 S'4 S1 S S1 N1 N N1 S3 N N1 N N3 N4 S' S' S' S' L=18 L= S' S' N N S3 N3 N4 N3 N4 N N6 N N6 S'7 S' S' S'7 S' S' S'8 Figure 3. Topolog map of the considered flow: spatial evolution of the singularities and vortical structures over SHS for increasing L with Φ =.. Color code marks the identical vortical structures. DNS model w/o model w/ secondar flow v + w [%] 1 3 L (a) maximum magnitude normalied b the U b at Φ =.. Ub/Ub, [%] 3 L (b) flow rate increase due to SHS at Φ =.. Figure 4. Properties of the secondar motions. Figure. Topolog map of the considered flow: spatial evolution of the singularities and vortical structures over SHS for changing Φ at L = 16. Color code marks the identical vortical structures. 4

5 VORTICITY ANALYSIS Figure 6 shows the distribution of ω x in the vicinit of the wall for the wave-lengths L = 8,16 and. Distribution of ω x agrees well with the location of the vortical structures presented previousl in the topological map (Figure 3). Figure 7 presents the distribution of the total vorticit production, P ωx, from Equation (6) with the spanwise coordinate normalied b L (Fig. 7(a)) and the viscous units (Fig. 7(b)), whereas the wall-normal coordinate is normalied with viscous unit in both figures. In the latter plot we demonstrate onl the left half of the periodic SHS (/L = 1/ in Fig. 7(a)) with horiontall aligned / edge for all L in order to enable easier comparison. The highest magnitude of the vorticit production can be observed around the / edges, confirming the fact that the secondar motions are inherentl triggered b the alteration of the wall boundar condition, which translates into inhomogeneit of the Renolds stress. Interestingl, the distribution of S ω,x remains qualitativel and quantitativel ver similar for all investigated L in spite of the obvious differences in distributions of vorticit and topolog of the secondar flow. The wall-normal extent of the vorticit production spots is slightl dependent on L, reaching 6 8. Considering spanwise extent, a triangle-shaped distribution can be observed above the freeslip region, which seems to scale with L as suggested from Figure 7(a). The part of the distribution above the region rapidl vanishes within 7 from the / edge due to the presence of condition at the wall. Presumabl, the wall-normal extent of the vorticit production distribution and its spanwise extent over the region decas within approximatel 6 viscous units from the / edge, while the spanwise extent of the distribution above area scales with the sie of the region or L. Figure 8 reveals a complex composition of the governing terms in the vorticit production from Equation (6) for L = 16. The main contribution to the total vorticit production arises from the first two terms of S ω,x, v w and w w. These terms originate from the momentum equation for the spanwise velocit component. The contribu- tion from the fourth term, v w, is rather minor, while the third term, v v, shows significantl smaller values than the governing terms and is therefore neglected. Latter terms originate from the momentum equation for the wall-normal velocit component. The fact that the vorticit production mainl emerges from the equation for the spanwise velocit component is reasonable due to the definition of the imposed boundar conditions: the introduction of SHS implies onl changes of the boundar condition for the streamwise and spanwise component, while the impermeabilit in the wall-normal direction still holds at and regions. L = 8 L = 16 L = /L ω x Figure 6. Vorticit distribution for for L = 8, 16, and Φ =.. v w w w /L v w Figure 8. Governing terms of the vorticit production for L = 16 and Φ =.. SUMMARY AND OUTLOOK A detailed investigation of the statistical data of turbulent channel flow DNS with streamwise oriented SHS for 3 < L < 88 reveals a series of significant changes in the structure of the occurring secondar motions leading to topological transition in this wave-length region. It is shown that secondar flow can account for up to % of U b, for larger L and its consideration in modelling approach significantl improves prediction of U b. The singu- lar point analsis shows that the onl counter-rotating vortex pair appearing for L < 1 is displaced b a complex entit of other structures for 144 < L < and then completel replaced b a different vortex pair with opposite rotational direction for L. The analsis of the streamwise vorticit transport equation shows that the secondar flow is driven b the vorticit production distribution concentrated at the edge between and region, which originates from the switch of boundar conditions at the wall translated into the inhomogeneit of Renolds

6 P ωx L = 8 L = 16 L = /L (a) wave-length normalied spanwise extent L = 8 L = 16 L = (b) spanwise extent in viscous units P ωx Figure 7. Vorticit production, P ωx, for L = 8,16, and Φ =.. shear stress. This causes a formation of a vortex-pair located above the / edge with L-independent rotation direction inducing local motion towards the solid region. The vortex-pair interacts with the surrounding flow triggering a formation of additional tertiar structures in the outer parts of the flow. The organiation of the structures depends on the wave-length as well as on the solid fraction of the introduced superhdrophobic surface. The spanwise and wall-normal vorticit production distribution extents remain ver similar in viscous units around the / noslip edges for different L, while the spanwise extent over the region scales with L. Comparison of the particular production terms unveils the fact that the vorticit production is dominated b two terms originating from the spanwise component of momentum equation. This is reasonable considering that the spanwise velocit component is significantl affected b the switch of the boundar conditions from to. The present knowledge provides useful information for modelling the secondar flow, and thus the resultant flow rate increase over SHS. ACKNOWLEDGEMENTS The authors greatl acknowledge the support b Karlsruhe House of Young Scientists (KHYS) through scholarship for research abroad and DFG project FR83/-1. This work was performed on the computational resource ForHLR Phase I funded b the Ministr of Science, Research and the Arts Baden-Württemberg and DFG ( Deutsche Forschungsgemeinschaft ) within the framework program bwhpc. REFERENCES Cabral, B. & Leedom, L. C Imaging vector fields using line integral convolution. In Proceedings of the th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH 93, pp New York, NY, USA: ACM. Daniello, R., Waterhouse, N. & Rothstein, J. 9 Drag reduction in turbulent flows over superhdrophobic surfaces. Phs. Fluids 1 (8), 83. Foss, J. F. 4 Surface selections and topological constraint evaluations for flow field analses. Exp. Fluids 37 (6), Fukagata, K., Kasagi, N. & Koumoutsakos, P. 6 A theoretical prediction of friction drag reduction in turbulent flow b superhdrophobic surfaces. Phs. Fluids 18 (), Goldstein, D. B. & Tuan, T.-C Secondar flow induced b riblets. J. Fluid Mech. 363, 1 1. Martell, M. B., Perot, J. B. & Rothstein, J. P. 9 Direct numerical simulations of turbulent flows over superhdrophobic surfaces. J. Fluid Mech., Min, T. & Kim, J. 4 Effects of hdrophobic surface on skin-friction drag. Phs. Fluids 16 (7), L. Philip, J. 197 Integral properties of flows satisfing mixed and no-shear conditions. Zeitschrift für angewandte Mathematik und Phsik ZAMP 3 (6), Renolds, W. C. & Hussain, A. K. M. F. 197 The mechanics of an organied wave in turbulent shear flow. part 3. theoretical models and comparisons with experiments. J. Fluid Mech. 4 (), Seo, J., García-Maoral, R. & Mani, A. 13 Pressure fluctuations in turbulent flows over superhdrophobic surfaces. In Annual Research Briefs, pp Center for Turbulence Research, Stanford Universit. Türk, S., Daschiel, G., Stroh, A., Hasegawa, Y. & Frohnapfel, B. 14 Turbulent flow over superhdrophobic surfaces with streamwise grooves. J. Fluid Mech. 747,