PHYSICAL REVIEW E 87, 023013 (2013)
Drag reduction of wake flow by shear-driven rotation Jianjun Tao* and Yan Bao SKLTCS and CAPT, Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, Beijing 100871, China (Received 16 November 2012; published 19 February 2013) This paper proposes a control strategy in which immersed bodies are driven to rotate not by external forces but by the flow itself. Due to the asymmetric distribution of the shear stress exerted on the surfaces, two side-by-side arranged cylinders rotate about their axes to produce moving fluid-solid boundaries. It is shown that the drag and the lift coefficients are significantly reduced in comparison with the stationary cylinder system. Different from the slip-boundary technique, the drag-reduction mechanism is explained in terms of a pressure-recovery effect introduced by the rotating surfaces. DOI: 10.1103/PhysRevE.87.023013
PACS number(s): 47.32.cb, 47.85.lb, 47.32.ck
I. INTRODUCTION
Since laminar flow and turbulence have different transport properties associated with energy, mass, and momentum, turbulence relaminarization or postponement of the laminarturbulent transition is required depending on different application purposes and can be achieved through different flow control strategies [1], e.g., blunt or streamlined bodies and blowing or suction boundaries. The mechanism of many control methods is to manipulate the viscous shear layers near the solid-fluid boundaries, which cause friction drag force, flow separation, hydrodynamic instabilities, transition to turbulence, etc. Accordingly, slip boundaries represent one of the first strategies capable of drag reduction. In reality, however, the no-slip boundary condition has been a proper approximation for almost all macroscopic flows. In order to release the no-slip condition, a moving (sliding) belt was suggested to replace parts of the rigid wall in turbulent [2] and laminar flows [3]. Recently, the combination of surface roughness and hydrophobicity has been used to produce a slip regime at the liquid-solid interface by utilizing the shear-free liquid-vapor interface [4], but the application of these superhydrophobic surfaces still has its limitations, such as inapplicability for the gas-solid interface, inherent fragility of the surface, etc. A flow past two cylinders is an elementary model to study flow patterns, flow control strategy, and aerodynamic characteristics around multiple bluff bodies in engineering practice. Stationary circular cylinders have been extensively investigated [5–11], and different flow patterns have been classified in the parameter space defined with the gap between two cylinder surfaces and the Reynolds number. Recently, the flow passing two cylinders rotating with constant angular velocities in side-by-side configuration has been investigated numerically and experimentally, and the effects of cylinder spacing, Reynolds number, and the rotating velocities of cylinders on the wake flow are discussed [12–16]. Both the inward and the outward rotations, where the cylinders’ surfaces near the gap move upstream and downstream, respectively, are found to suppress the vortex shedding phenomenon. All these investigations greatly improve our understanding of the wake flows. However, this rotation-control technique requires an
*
[email protected]
1539-3755/2013/87(2)/023013(4)
external force to drive the cylinders to rotate, and an excessive energy input will make it less practical and inefficient. In addition, it was shown that the uniform speed rotation of cylinders increased the mean lift coefficient of every cylinder [13], enhancing the transverse force experienced by these cylinders. In this paper, we propose a control scheme using shear-driven rotating cylinders, which reduce effectively both the drag coefficient and the lift coefficient without any energy input.
II. PHYSICAL MODEL
Let us consider a flow past two cylinders arranged side by side in the transverse direction as shown in Fig. 1. Different from superhydrophobic boundaries or moving belts, the cylinder surface has no displacement relative to its body but will rotate around its geometric center when the viscous shear stress exerted on the surface provides a net torque. The Reynolds number is defined as Re = U∞ D/ν, where D is the diameter of the two cylinders, U∞ is the upstream velocity, and ν is the kinematic viscosity. The distance between these cylinder surfaces is denoted by g = 0.05D. U∞ and D are used as characteristic velocity and length scales to nondimensionalize parameters in this flow system. ρ∞ and ρc are the densities of the fluid and the cylinders, respectively. When the wake flow becomes unstable, the dimensionless angular velocities of the upper and the lower cylinders 1 (t) and 2 (t) share the same oscillating amplitude and frequency but have a fixed phase difference of π . In addition, the oscillating amplitude, the oscillating frequency, and the mean values of 1 and 2 increase with Re. It is found that a larger density ratio M = ρc /ρ∞ will lead to a smaller oscillation amplitude of the angular velocities but have nearly no influence on their mean values; hence, the mean drag and the mean lift coefficients are found to be insensitive to the density ratio. In the following discussions, we will focus on the results for M = 5.0. The two-dimensional Navier-Stokes equations are solved by employing a characteristic-based split finite element method, and the no-slip boundary condition is used on the cylinder surfaces. The solid body rotation of the cylinders is governed by the angular momentum conservation law. For simplicity, the rotation damping is not considered. The computational domain extends from -35D at the upstream to 60D at the downstream in the streamwise direction and from
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©2013 American Physical Society
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PHYSICAL REVIEW E 87, 023013 (2013)
The drag-reduction effect caused by the shear-driven rotation is demonstrated in Fig. 3, in which the pressure drag coefficient, the shear drag coefficient, the lift coefficient, and the Strouhal number are defined as Cdp =
FIG. 1. (Color online) Schematic diagram of a flow past two sideby-side arranged cylinders. Every cylinder driven by the viscous shear stress exerted on its surface rotates about its axis.
-35D to 35D in the cross-flow direction. For details of the numerical schemes we refer to our previous paper [17]. III. RESULTS
The flow patterns behind these shear-driven rotating cylinders at different Reynolds numbers are shown in Fig. 2, in which the corresponding flows for stationary cylinders are illustrated as well for reference. It is shown that at Re = 21 the wake behind the stationary cylinders becomes unsteady, while for the shear-driven rotation case such unsteadiness is ´ an ´ completely suppressed. With the increase of Re, the Karm vortex street is formed for the stationary cylinder case, characterized with two rows of alternately staggered vortices, while for the shear-driven rotation system only one row of vortices is observed when Re is as high as 100, indicating less momentum loss of the mean flow. At Re = 170, the vortex distribution behind the stationary cylinders becomes irregular. In contrast, a regular vortex pattern is still maintained in the wake behind the rotating cylinders, implying that the transition from a two-dimensional wake to a three-dimensional one would be postponed by this self-driven rotation system.
FIG. 2. (Color) Isocontours of vorticity for the stationary (left) and the shear-driven rotating (right) cylinders at different Reynolds numbers. The viscous shear layers or the vorticity layers are indicated by arrows.
2Fdp fD 2Fdτ 2Fl , Cdτ = , Cl = , St = , 2 D 2 D 2 D ρU∞ ρU∞ ρU∞ U∞
where Fdp and Fdτ are the pressure force and the viscous shear force exerted on both cylinders in the streamwise direction, respectively. Fl is the net transverse force experienced by the cylinders, and f is the vortex shedding frequency obtained from the spectral analysis of the lift coefficient time series. In comparison with the stationary cylinder case, it is shown in Fig. 3(a) that Cdp for shear-driven cylinders is significantly reduced for all Re considered in this study, while Cdτ is slightly larger at small Re and almost coincides with the value of the stationary cylinder case as Re > 80. Consequently, the total drag coefficient Cd = Cdp + Cdτ decreases dramatically by applying such shear-driven boundaries; e.g., the time-averaged value of Cd is reduced by 30.5 % at Re = 100. Several interesting features should be noted. First, the unsteadiness of the wake flow causes an oscillation of the pressure drag coefficient, while the shear drag coefficient keeps almost invariant during a vortex shedding period both for the stationary and the rotating cylinder cases. In fact, the viscous shear drag of each cylinder varies periodically, but since the phase difference between 1 and 2 is π the sum of the shear drags exerted on both cylinders only oscillates slightly. Second, the fluctuation range of Cdp labeled by error bars in Fig. 3(a) for the shear-driven cylinders is apparently reduced in comparison to the fixed cylinder case. Third, it is shown in the inset of Fig. 3(a) that for the stationary cylinders there are multisolutions at 19 < Re < 21, indicating the existence of a subcritical bifurcation, which is found also in the single stationary cylinder wakes [18]. The multisolutions were obtained by gradually increasing or decreasing Re as shown by the arrows in the inset. However, it is difficult to find the multisolutions for the shear-driven rotation system. The lift or transverse force for unsteady wakes is also significantly reduced by the shear-driven rotation as shown in Fig. 3(b). For example, at Re = 100 the amplitude of the lift coefficient |Cl |max is only 52 % of the value for the stationary cylinders. Therefore, the vortex-induced vibration can be substantially suppressed. For the stationary cylinders, the Strouhal number becomes almost independent of Re at 70 < Re < 180 and close to a constant value 0.1, which is consistent with previous numerical observations [10]. In contrast, St continues to increase with Re for the shear-driven rotation system. This difference may be explained as follows. When the cylinders are stationary and the regular K´arm´an vortex street is formed, the wake flow is mainly dominated by the inertia force other than the viscous force at large Reynolds numbers and the vortex shedding frequency f is approximately proportional to the inertia time scale U∞ /D. As a consequence, St becomes insensitive to U∞ or Re in the studied Re regime. For the shear-driven rotation case, the vortex shedding is synchronized with the oscillation of the rotating velocities of the cylinders, which are caused by the viscous shear stress at the surfaces, and the oscillating frequency increases with
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FIG. 3. (Color) (a) Pressure drag coefficient Cdp (open square) and shear drag coefficient Cdτ (open triangle), (b) amplitude of lift coefficient |Cl |max , and (c) Strouhal number St as functions of Re. The blue and red symbols are results for the stationary and the shear-driven cylinder systems, respectively. In (a), the error bars indicate the variation ranges of drag coefficients and Cd = Cdp + Cdτ .
in Fig. 4(d), causing a pressure jump along the cylinder surface near θ = 270◦ [Fig. 4(b)] and hence recovering the pressure on the leeward side to some degree. This phenomenon is referred to as the pressure-recovery effect hereafter. The strong and reversed pressure gradient at the gap leads to a jet injecting from the gap to the upstream flow and causes a thrust force on the cylinder surface [Fig. 4(a), Re = 14]. At larger Reynolds 2
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Re. As a result, St continues to increase with Re as shown in Fig. 3(c). The underlying physics of the drag reduction by sheardriven rotation can be explained qualitatively as follows. The energy possessed by wake vortices, which are generated from the shear layers or vortex layers (shown by arrows in Fig. 2) attached to the cylinder surface, contributes to the drag force exerted on the cylinders. When the shear-driven rotation is permitted, the enstrophy of the vortex layers is transferred to not only the shed vortices but also the rotating cylinders, and hence the drag contributed by the downstream vortices is significantly reduced. It is shown in Fig. 2 that the vortex street of the rotating cylinder system has a narrower cross-flow length scale and weaker vortices than that of the stationary one. Accordingly, the vortex induced vibration can be suppressed as well. The detailed drag-reduction mechanism can be explained according to the results shown in Fig. 4. Since the timeaveraged wake flow is symmetric about the middle plane in the studied Re regime, only the results above the middle plane are illustrated. Because of the small gap ratio (g/D = 0.05), the wake flow behaves similar to the flow around a single-bluff body, with only a small amount of fluid going through the gap. Consequently, the mean shear stress τθ in the tangential direction of the surface reaches its maximum value at a windward position nearly θ = 125◦ as shown in Fig. 4(a), and the integral effect of such viscous layers around the surfaces provides torques to drive the upper and the lower cylinders to rotate around their axes in the clockwise and the anticlockwise directions, respectively. It is shown in Fig. 4(c) that there is a stagnation point on the windward surface (θ = 215◦ ) of the stationary cylinder, where the mean pressure coefficient 2 reaches its maximum. However, when Cp = (p − p∞ )/ 12 ρU∞ cylinders rotate the fluid near the surface moves along with the surface, and hence the windward stagnation point disappears and mean pressure coefficient Cp on the windward surface (90 θ 270◦ ) decreases in comparison with the stationary cylinder case. Also, the viscous shear layer near the rotating cylinder’s surface brings fluid to the rear neighborhood of the narrow gap, forming a stagnation point at a downstream position from the gap indicated by a white circle at the middle plane
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FIG. 4. (Color) (a) The mean shear stress τθ and (b) the mean 2 at the upper cylinder’s pressure coefficient Cp = (p − p∞ )/ 21 ρU∞ surface as functions of the azimuthal angle θ for the stationary (blue lines) and the shear-driven rotating (red lines) cylinder systems, where the dashed and the thick solid lines correspond to Re = 14 and 100, respectively. The mean pressure contours at Re = 100 for the stationary and the rotating cylinder systems are shown in (c) and (d), respectively, where several streamlines are shown to illustrate the flow patterns and the red points indicate the vortex cores of the mean flow fields.
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number, e.g., Re = 100, the flow becomes unsteady, and the reverse gradient of the mean pressure coefficient Cp at θ = 270◦ becomes smaller [Fig. 4(b)] and the jet velocity at the gap tends to be uniform, leading to a smaller shear stress in comparison with the case of small Reynolds number, e.g., Re = 14. When the cylinders are stationary, there is no pressure-recovery effect and the jet at the gap injects to the downstream direction as shown in Fig. 4(c) and hence exerts a local maximum drag force on both cylinders’ surfaces. Except the surfaces near the gap, the shear stress distributions on the rotating and the stationary cylinders are similar as shown in Fig. 4(a) and the integrated shear drag coefficients Cdτ are close to each other as well [Fig. 3(a)]. Therefore, the drag reduction in the rotating cylinder system is mainly contributed by the pressure-recovery effect caused by the shear-driven rotation. As discussed above, it is the strong asymmetric distribution of the mean viscous shear force that drives the cylinders to rotate, and hence the pressure-recovery effect is applicable to Stokes flows as well. It is checked that when Re is as low as 0.03 the total drag coefficient Cd is reduced by 3 % due to the shear-driven rotation. When the cylinders are moved further apart, the flow around each cylinder becomes more symmetric in the spanwise direction, and the driving force of rotation becomes weaker. Consequently, the pressure-recovery effect is reduced. It is found that when g/D is increased from 0.05 to 0.1 and 0.2 at Re = 100 the rotation-induced total drag reduction is decreased from 30.5 to 24.4 and 8.5 %, respectively. Recently, two side-by-side arranged cylinders rotating in the clockwise and anticlockwise directions with constant angular velocity were used to study the vortex interaction and the suppression of vortex shedding [12–16]. These previous researches focused on wakes with large gap ratios (g/D > 0.1), and the wake flow with a small gap ratio has not been studied with this uniform rotation method so far because its flow pattern is similar to that of a single bluff body [19]. However, the small gap ratio as discussed above is crucial for the self-driven rotation. We calculate the wake flow
of two cylinders rotating with constant angular velocities, which are chosen to be the same as the mean values of the shear-driven cylinders at Re = 100, the density ratio M = 5, and g/D = 0.05. It is found that both the total drag coefficient and the lift coefficient of each cylinder for the uniform rotation case are slightly larger than those of the shear-driven cylinders. In addition, it should be recalled that the uniform rotation technique requires an extra external power to keep the cylinders rotating with constant velocities in the unsteady wake flow.
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IV. CONCLUSIONS
In this paper, the proposed flow control strategy uses rotating cylinders driven by the flow itself rather than by external energy. In comparison with the stationary cylinder system, the drag coefficient and the vortex-induced vibration of the shear-driven rotating cylinders can be significantly reduced. In addition, the mean flow field has less vortices, and the main vortex is pushed further downstream to form a more streamlined circulation regime [Fig. 4(d)] because of the pressure-recovery effect. It is noted that the main contribution of these moving fluid-solid boundaries is to reduce the pressure drag, which is different from that of the superhydrophobic surfaces, where the friction drag is decreased. Though in our discussion the Reynolds number is not very high and the wake flow is confined to be two-dimensional, the pressure-recovery effect caused by rotation is applicable for three-dimensional and turbulent flows; hence, a postponement of the laminarturbulent transition and a substantial drag reduction at high Reynolds numbers can be expected when this passive control strategy is applied. ACKNOWLEDGMENTS
This work has been supported by the National Natural Science Foundation of China (Grants No. 11225209 and No. 10921202) and the Ministry of Science and Technology of China (Grant No. 2009CB724100).
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