RELAMINARIZATION OF TURBULENT CHANNEL FLOW USING TRAVELING WAVE-LIKE WALL DEFORMATION

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1 RELAMINARIZATION OF TURBULENT CHANNEL FLOW USING TRAVELING WAVE-LIKE WALL DEFORMATION Rio Nakanishi, Koji Fukagata, Hiroa Mamori Department of Mechanical Engineering Keio Universit Hioshi --, Kohoku-ku, Yokohama -85, Japan ABSTRACT Effect of traveling wave-like wall deformation (i.e. peristalsis) in a full developed turbulent channel flow is investigated b means of direct numerical simulation. We not onl demonstrate that the friction drag is reduced b wavelike wall deformation traveling toward the downstream direction, but also show that the turbulence is completel killed (vi. the flow is re-laminaried) under some sets of parameters. It is also found that at higher amplitude of actuation the re-laminaried flow is unstable and exhibits a periodic ccle between high and low drag. Introduction Drag reduction in wall-bounded turbulent flow is of great importance for mitigating environmental impact through the efficient utiliation of energ. The large friction drag in turbulent flows on a solid wall, as compared to that in laminar flows, is attributed to the vortical structure in the region near the wall (Robinson, 99). Passive devices such as riblets (Walsh, 98) and structured roughness (Sirovich and Karlsson, 997) are known to have friction drag reduction effect on the order of % b interacting with these vortical structure. In order to gain larger effects, recent efforts have been made to activel suppress these structures to reduce the friction drag in wall-turbulence, e.g. b predetermined spanwise traveling wave (Du and Karniadakis, ) and feedback control using actuation from the wall (Kim, ; Kasagi et al., 9). Mathematicall, the friction drag coefficient C f of a full-developed turbulent flow in a two-dimensional channel is expressed b using a weighted integration of the Renolds shear stress ( u v ), i.e. (Fukagata et al., ) C f = + ( )( u Re v )d, () b where Re b = U b h/ν (with U b, h, and ν being the bulk-mean velocit, the channel half-width, and the kinematic viscosit, respectivel) is the bulk Renolds number and denotes the non-dimensionalied distance from the wall. This identit equation not onl suggests that the friction drag can be reduced b reducing the Renolds shear stress, especiall in the region near the wall, but also implies that the drag can reach a sub-laminar level if the second term can be made largel negative. Based on this implication, Min et al.(6) proposed a novel predetermined control method using traveling wave-like blowing and suction from the walls. The showed b means of numerical simulation that the blowing and suction traveling toward the upstream direction (hereafter referred to as the upstream traveling wave) significantl reduce the near-wall Renolds shear stress. The also demonstrated that under some sets of parameters the drag can be reduced to a sublaminar level b making the near-wall Renolds shear stress largel negative. Min et al.(6) s control is attractive in the sense that it does not require massive sensors and is much simpler than the advanced feedback control methods (Kim, ; Kasagi et al., 9). Therefore, several follow-up studies have been made to investigate the stabilit (Lee et al., 8; Moarref & Jovanović, ; Lieu et al., ) and scaling (Mamori et al., ) of this control. However, in terms of fabrication of actual devices, the traveling wave-like blowing and suction is still complicated. One possibilit toward the practical application is to replace it b a traveling wave-like deforming surface, which ma be driven b a smaller number of actuators. The primar mechanism of drag reduction b traveling wave-like blowing and suction is considered to be pumping from the wall. Hœpffner & Fukagata (9) found in their numerical simulations in the absence of base flow that the traveling wave-like deforming wall (more generall known b the term peristalsis ) has the pumping effect in the same direction as the wave travels, while the traveling wave-like blowing and suction has the pumping effect in the opposite direction. This difference suggests that the drag reduction using the deformable wall is expected not b the upstream traveling wave but b the downstream traveling wave (note that ver recent studies (Moarref & Jovanović, ; Lieu et al., ) show that high-speed downstream traveling wave-like blowing and suction also re-laminalies the flow: in that case, the primar mechanism responsible for the drag reduction ma not be the pumping from the wall but stabiliation). Drag reduction effect b such downstream traveling

2 wave-like wall deformation was also experimentall found almost ears ago. Taneda and Tomonari (97) conducted an experiment on a spatiall developing boundar laer on a plate waving like the swimming motion of fish. The observed laminariation effect b a downstream traveling wave at a wavespeed much faster than the uniform flow. The also found that the waving plate under some conditions makes the boundar laer cclicall laminar and turbulent. Shen et al.() investigated a flow around a similar wav wall b means of direct numerical simulation (DNS). The showed that the total drag as well as the net power can be reduced b downstream traveling wave. Both studies deal with wav walls with relativel large amplitude and discuss drag reduction as compared to the case of stationar wave. From the viewpoint of drag reduction control, however, it is important to clarif how the drag is reduced as compared to that of a turbulent flow on a flat surface. In the present stud, we investigate b means of DNS the drag reduction effect of the traveling wave-like deforming wall in a turbulent channel flow. We not onl confirm that the drag is reduced b the downstream traveling wave, but also demonstrate that the flow is re-laminaried under some sets of parameters. Indeed, as has been proved mathematicall (Bewle, 9; Fukagata et al., 9), re-laminariation is the ultimate goal of drag reduction control in terms of energ saving. Direct Numerical simulation The DNS code is based on the channel flow code of Fukagata et al.(6) adapted to boundar fitted coordinates. In order to express the wall deformation, the Cartesian coordinates u i are transformed into ξ i following Kang & Choi (). The traveling wave-like deformation of wall is given as the boundar condition. At the lower wall (ξ = ), for instance, it reads, Traveling wave deforming wall Figure. Schematic of a channel with traveling-wave like wall deformation. dp/dx 8 6 x Case (solid line) Laminar flow (dotted line) 6 8 t Case no control Case Figure. Time trace of mean pressure gradient dp/dx: Case (a =., c =, k = ), ordinar drag reduction; Case (a =., c =, k = ), stable re-laminariation; Case (a =., c =, k = ), unstable re-laminariation. Re b = U b h/ν = 56, corresponding to the friction Renolds number of Re τ 8 in the uncontrolled (i.e. solid wall) case. u = u =, u = η d t = acos(k(x ct)), () where a, k, and c denote the velocit amplitude, the wavenumber, and the phase-speed of wall deformation, respectivel. The upper wall deforms in-phase with the lower wall (i.e. varicose mode). The schematic of traveling wave-like wall deformation is shown in Fig.. The displacement of lower and upper wall, η d and η u, is obtained b integrating Eq. () in time, i.e. η d = a kc sin(k(x ct)), η u = a sin(k(x ct)). () kc The sie of computational box is π.5 in the streamwise (ξ ), wall-normal (ξ ), and spanwise (ξ ) directions and the corresponding number of computational cells is The time step is set to be t =. in most cases, while t =.5 for k or a.5 cases in order to sufficientl resolve the temporal change of boundar condition and the near-wall flow induced thereb. Hereafter, all the variables are non-dimensionalied b using the channel half-width h and twice the bulk-mean velocit U b. All DNS is performed under a constant flow rate. The bulk Renolds number is set at Results and Discussion Drag reduction effects The DNS runs with different sets of parameters a, k, and c revealed that as a rule the drag is decreased in the cases of c > (i.e. downstream traveling waves) and increased for c < (i.e. upstream traveling waves). Figure shows the time trace of mean pressure gradient dp/dx in tpical cases of drag reduction. Case is an example of ordinar drag reduction, in which the drag is reduced but the flow remains turbulent. Case exemplifies the case of re-laminariation: the turbulent fluctuations completel vanish and dp/dx converges close to the value of laminar channel flow. At the stead state, the pressure gradient is slightl lower than the laminar value, which indicates that a small amount of pumping is replaced b the peristaltic pumping. Although the peristaltic pumping is about twice expensive (i.e. power-consuming) than the external pressure gradient (Hœpffner & Fukagata, 9), the present result shows that the contribution of peristaltic pumping is considerabl small and thus the net energ saving rate is close to the drag reduction rate (about 7%). Case is a case of unstead re-laminariation: dp/dx decreases toward the laminar value but rapidl increases again to the uncontrolled value. Figure summaries the control effect under different

3 wave number k c = c = wall deforming velocit amplitude a Figure. Drag change b the traveling wave-like deforming wall:, drag increase;, ordinar drag reduction;, stable re-laminariation;, unstable re-laminariation. The value noted together with the smbols (except for the unstead relaminariation cases) is the drag reduction rate R D defined b Eq. () 7 Displacement amplitude η + = k a+ + c Deformation period T + = k π + c + Figure 5. Response of the flow to the traveling wave-like deforming wall: smbols are the same as those in Fig.. wave number k c = c = wall deforming velocit amplitude a 7.6 Figure. Net power saving effect b the traveling wave-like deforming wall: smbols are the same as those in Fig.. The value noted together with the smbols (except for the drag increasing cases and the unstead re-laminariation cases) is the net power saving rate S defined b Eq. (5) sets of parameters. The drag reduction rate R D noted together with the smbols is defined as R D = P P P [%], () where P = dp/dx denotes the mean pressure gradient and P is that in the uncontrolled flow. The map indicates that for both phase-speeds (c = and c = ) the flow is re-laminaried b actuation of relativel high wave-number (k) and amplitude (a), but becomes unstable again if the amplitude is further increased. Figure shows the net power saving effect b wall deformation. The net power saving rate S is defined as S = W p (W p +W a ) W p [%], (5) where W p and W a are the pumping power and power required for wall deformation, respectivel; W p denotes the pumping power of the uncontrolled flow. It can be noticed that the net power is reduced in the cases of c = (left of figure ). The net power saving effect is noticeable in the re-laminariation cases: the maximum net power saving rate is 6 %. In the cases of c = (right of figure ), however, the net power is not saved even in the re-laminariation cases. This means that faster traveling wave need larger power for actuation than the pumping power reduced. Figure 5 summaries the control effect under different sets of parameters. The horiontal axis is the deformation period and the vertical axis is the displacement amplitude, both in wall units. Roughl speaking, the drag decreases with T + < and increases with T + >. The drag is nearl unchanged at < T + <. Also noticed is that relaminariation is achieved when the displacement amplitude 5 < η + < 5 and the flow is unstable when η + > 5. This is different from Taneda and Tomonari (97) s result, where the displacement achieving the drag reduction is on the order of boundar laer thickness. Instantaneous flow field Figure 6 illustrates four instantaneous flow fields during the process of re-laminariation in the case of (a,k,c) = (.,, ). For each time instant, the mean velocit profiles and the contour of Renolds shear stress u v in an x plane are shown in the left figure; the cross-sectional velocit vectors and the streamwise velocit contour are shown in the right figure. In the uncontrolled flow (Fig. 6(a)), we observe a flattened mean velocit profile (left) and streak and vortical structures (right), which are tpical to turbulent channel flows. As the time goes (Figs. 6(b)-(c)), an organied Renolds shear stress distribution is formed near the wall due to the actuation, the velocit profile becomes less flattened thereb (left), and the streak and vortical structures are weakened (right). Finall, after a sufficientl long time (Fig. 6(d)), the velocit profile becomes similar to the Poiseuille profile (left) and the streak and vortical structures completel vanish (right). In the re-laminaried state, the Renolds shear stress distribution has a cellular structure (Fig. 6(d), left) akin to that observed in the laminar flow with traveling wave-like blow-

4 (a) φ/(π/) (b) φ/(π/) (c) φ/(π/) (d).5.5 φ/(π/) Figure 6. Instantaneous flow fields during the process of re-laminariation (the case of a =., c =, and k = ): (a) uncontrolled; (b) t = 5.7; (c) t = 56.; (d) t = 68.. Figures in the left column shows the streamwise velocit and Renolds shear stress in the x plane: black line, streamwise velocit averaged at each phase (φ =,π/,π,π/): red line, uncontrolled turbulent profile; blue line, laminar Poiseuille profile; color contour, Renolds shear stress u v. Figures in the right column shows the velocit field in the plane at φ =, at which the deformation velocit is maximum: vector, wall-normal and spanwise velocit; color contour, streamwise velocit..6.. ing/suction (Mamori et al., ). The peak of u v is located close to the wall due to the wall-normal velocit induced b the wall deformation, whereas u v is found to be nearl ero in the region far from the wall. Although the mean velocit profile is close to the Poiseuille profile, a small amount of reverse flow is also observed near the wall in the phases of φ/(π/). and.5 φ/(π/). This is due to the wall deformation, which carries the fluid particles back and forth so that the draw l-shaped drifting tracks (Hœpffner & Fukagata, 9). In the unstead re-laminariation cases, the process of re-laminariarion is almost the same as that in the stead relaminariation cases. Figure 7 shows the evolution of flow in which the re-laminaried flow (7(a)) is destabilied (7(b)- (d)) and stabilied again (7(e)-(f)). When the pressure gradient begins to increase, the low-speed fluid is ejected from the near-wall region and vortical structure appears (Fig. 7(b), right.) Similar phenomena are observed at different places (Fig. 7(c), right), accompaning the development of Renolds shear stress. Accordingl, the mean streamwise velocit pro-

5 (a) φ/(π/) (b) φ/(π/) (c) φ/(π/) (d) φ/(π/) (e) φ/(π/) (f) φ/(π/) Figure 7. Instantaneous flow fields during the process that re-laminaried flow return to turbulence rapidl (the case of a =., c =, and k = ): (a) t = 56.; (b) t = 58.; (c) t = 5.; (d) t = 55.; (e) t = 565.5; (f) t = 6.6. The contents are same as Fig. 6 5

6 file is distorted (left). At the time when the mean pressure gradient is maximied (Fig. 7(d)), low-speed region, vortices, and Renolds shear stress spread in whole channel. The Renolds shear stress distribution and the mean velocit profile looks simiar to those of the uncontrolled flow (Fig. 6(a)) After that time, the low-speed region and the vortices weakened (Fig. 7(e)-(f), right) and the flow is re-laminaried again. As compared to the stead re-laminaried case, Fig. 6(d), the reverse flow near the wall in Fig. 7(a) is stronger due to the higher amplitude of wall deformation. This stronger reverse flow makes an inflection point in the mean streamwise velocit profile to cause the instabilit as observed in Fig.. Conclusions We demonstrated b means of DNS that the downstream traveling wave-like wall deformation significantl reduces the drag. The primar drag reduction mechanism can be explained b the pumping effect in the same direction as the wave (Hœpffner & Fukagata, 9). In the present stud, we observe that under several sets of parameters this control can also re-laminarie the low Renolds number turbulent channel flow. Although similar observation has been made in previous studies, e.g., Taneda and Tomonari (97), a clear distinction with those is that the re-laminaliation is achieved with much smaller amplitude of wall deformation, i.e., on the order of viscous sublaer. An interesting phenomenon additionall observed is that with stronger actuation the re-laminaried flow is unstable and exhibits a periodic ccle between high and low drag. The cause and dnamics of this peculiar phenomenon, however, is open for future work. Acknowledgments The authors are grateful to Dr. Shinnosuke Obi (Keio Universit), Dr. Jerome Hœpffner (UPMC), Drs. Nobuhide Kasagi and Yosuke Hasegawa (The Universit of Toko), and Dr. Kaoru Iwamoto (Toko Universit of Agriculture and Technolog) for fruitful discussions. This work was supported through Grant-in-Aid for Scientific Research (A) (No. 66) b Japan Societ for the Promotion of Science (JSPS) and Keio Universit Global COE program of Center for Education and Research of Smbiotic, Safe and Secure Sstem Design. REFERENCES Bewle, T. R. 9, A fundamental limit on the balance of power in a transpiration-controlled channel flow, Journal of Fluid Mechanics, Vol. 6, pp. -6. Du, Y. and Karniadakis, G. E., Suppressing wall turbulence b means of a transverse traveling wave, Science, Vol. 88, pp. -. Fukagata, K., Iwamoto, K. and Kasagi, N., Contribution of Renolds stress distribution to the skin friction in wall-bounded flows, Phsics of Fluids, Vol., L7-L76. Fukagata, K., Kasagi, N. and Koumoutsakos, P. 6, A theoretical prediction of friction drag reduction in turbulent flow b superhdrophobic surfaces, Phsics of Fluids, Vol. 8, 57. Fukagata, K., Sugiama, K. and Kasagi, N. 9, On the lower bound of net driving power in controlled duct flows, Phsica D, Vol. 8, pp Hœpffner, J. and Fukagata, K. 9, Pumping or drag reduction? Journal Fluid Mechanics, Vol. 65, pp Kang, S. and Choi, H., Active wall motions for skin-friction drag reduction, Phsics of Fluids, Vol., pp. -. Kasagi, N., Suuki, Y. and Fukagata, K. 9, Microelectromechanical sstems-based feedback control of turbulence for skin friction reduction, Annual Review of Fluid Mechanics, Vol., pp. -5. Kim, J., Control of turbulent boundar laers, Phsics of Fluids, Vol. 5, pp Lee, C., Min, T. and Kim, J. 8, Stabilit of a channel flow subject to wall blowing and suction in the form of a traveling wave, Phsics of Fluids, Vol., 5. Lieu, B., Moarref R. & Jovanović, M. R., Controlling the onset of turbulence b streamwise traveling waves. Part : Direct numerical simulations, Journal of Fluid Mechanics, Vol 66, -9. Mamori, H., Fukagata, K. and Hœpffner, J., Phase relationship in laminar channel flow controlled b travelingwave-like blowing or suction, Phsic Review E, Vol. 8, 6. Min, T., Kang, S. M., Speer, J. L. and Kim, J. 6, Sustained sub-laminar drag in a full developed channel flow, Journal of Fluid Mechanics, Vol. 558, pp Moarref, R. and Jovanović, M. R., Controlling the onset of turbulence b streamwise traveling waves. Part : Receptivit analsis, Journal of Fluid Mechanics, Vol. 66, pp Robinson, S. K. 99, Coherent motions in the turbulent boundar laer, Annual Review of Fluid Mechanics, Vol., pp Shen, L., Zhang, X., Yue, D. K. P. and Triantafllou, M. S., Turbulent flow over a flexible wall undergoing a streamwise traveling wave motion, Journal of Fluid Mechanics, Vol. 8, pp Sirovich, L. and Karlsson, S. 997, Turbulent drag reduction b passive mechanisms, Nature, Vol. 88, pp Taneda, S. and Tomonari, Y. 97, an Experiment on the Flow around a Waving Plate, Journal of the Phsical Societ of Japan, Vol. 6, pp Walsh, M. J. 98, Riblets as a viscous drag reduction technique, AIAA Journal, Vo., pp