Drag reduction of wall bounded incompressible turbulent flow based on active dimples/pimples＊

Ming-wei Ge (葛铭纬), Le Fang (方乐), Yong-qian Liu (刘永前)

1.State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, China, E-mail: gmwncepu@163.com

2.Sino-French Engineering School, Beihang University, Beijing 100191, China

Drag reduction of wall bounded incompressible turbulent flow based on active dimples/pimples＊

Ming-wei Ge (葛铭纬)1, Le Fang (方乐)2, Yong-qian Liu (刘永前)1

1.State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, China, E-mail: gmwncepu@163.com

2.Sino-French Engineering School, Beihang University, Beijing 100191, China

The control of turbulence by dimples/pimples has drawn more and more attention. The objective of this paper is to investigate the effectiveness of the active dimples/pimples for the drag reduction in the incompressible turbulent flow. Firstly, the drag reduction by the opposition control based on active dimples/pimples at the lower wall is studied via the direct numerical simulation of the turbulent channel flow. It is found that large active dimples/pimples can not suppress the streamwise vortices significantly and thus almost no drag reduction is achieved. Small active dimples and pimples with the diameter of one fourth of the streak width can both reduce the friction drag, but pimples will induce a larger pressure drag than dimples. Then the suboptimal control scheme is examined based on small active dimples using the spanwise wall shear information only. It is shown that the friction drag decreases by about 4.5% but the total drag is only reduced by about 2.7% abated by the pressure drag. Compared with the actuation of the all-point blowing/suction or the all-point wall movement, the effectiveness of the turbulent drag reduction based on active shallow dimples is much smaller.

Turbulent drag reduction, active dimple, opposition control, suboptimal control

Introduction

Generally, the turbulent friction drag contributes a large proportion of the total drag for the vehicles moving in the fluid. For the conventional ships or the aircraft under the cruising state, the turbulent frictional drag can attribute 50% of the total drag, while for the torpedoes, the submarines and other underwater vehicles, the ratio can be as high as 70%. The issue of high friction of the turbulence is also encountered in other engineering applications, such as the high-speed train and the pipeline. Studies show that for a transport aircraft, if the total drag is reduced by 10 percent, $3 billion’s fuel might be saved in a year for the U.S. airline industry. For an underwater vehicle, under a certain power and energy condition, if the resistance is reduced by 10%, its cruising speed and range can increase approximately by 3.75%[1]. Therefore, the study of the turbulent drag reduction is very important, especially in the modern society with problems of growing energy shortage and environmental pollution.

In view of the convenience of the application, many passive control methods were proposed, such as the bio-inspired riblets, the compliant wall, and the superhydrophobic surfaces[2,3]. But most of the passive control techniques have quite limited effects. Hence, the active turbulent drag reduction becomes a new research focus aiming to improve the situation. The discovery of the coherent structures near the wall makes a theoretical breakthrough for the turbulent drag reduction, which may serve as a theoretical guidance for the turbulence control design and application. Based on the close relationship between the near-wall coherent structures and the high skin-friction in the wall turbulence[4], many active drag-reduction control schemes were developed by instantaneously interfering with the evolution of the near-wall coherent structures. Based on the physical model of streamwise vortices, Choi et al.[5]proposed the opposition control by app-lying the reverse blowing/suction on the wall to inhibit the vortices on both sides of the ejection and sweep, reducing the strength of the vortices with a considerable drag reduction. With regard to the exciting effect on the drag reduction, various control schemes were proposed[6-8]. However, this kind of control is unpractical in real applications since the information of the flow field is required, which is hard to be measured. With the opposition control in mind, Lee et al.[9]developed a so called suboptimal control scheme by optimizing a cost function in a single time step based on the wall pressure and the spanwise wall friction. Many other control schemes were also developed, as reviewed by Kim[4]and Kasagi et al.[10].

In recent years, the development of the microelectromechanical systems (MEMS) technology provides us with the possibility for the practical application of the active turbulence control[10]. Using the technology of MEMS, the “smart skin” for drag reduction shows very good prospects[11,12]. As an excellent candidate for the shape of MEMS, the flow over dimples/ pimples has drawn more and more attentions in the flow controls in recent years[13,14]. But most of studies only focus on the heat exchange enhancement and the flow structures induced by static dimples. Through experimental measurements of turbulent flows over surfaces with a regular arrangement of static shallow dimples, Alekseev el al.[15]pointed out that the shallow dimples can lead to a decrease of the skin-friction up to 20% apart from the heat-transfer enhancement. However, a reverse conclusion was made by Lienhart et al.[16]through the investigation via both simulation and experiment for the same arrangement as that employed by Alekseev et al.. With complementary evidence, they pointed out that the shallow dimples on a flat surface does not lead to drag reduction but rather to a slight total drag increase. After that, the flow structures over a static dimple were studied by Ge et al.[13]in detail via direct numerical simulations and a slight total drag increase was observed by Lienhart et al.[16]. The active dimples on the wall were first used for drag reduction in the incompressible flow by Yang et al.[17]based on both the opposition control and the suboptimal control scheme, achieving drag reductions of about 12% and 11.4%, respectively. However, the result seems a little encouraging since the deformation of the dimple is approximately represented by a disturbing velocity. To clarify the effectiveness of the active pimples/dimples for the turbulence drag reduction, the direct numerical simulation is carried out to study the turbulent flow over the active dimples/pimples in the present paper. Different arrangements of the dimples/pimples are studied by the opposition control to find the suitable diameter of the dimples/pimples to achieve better drag reduction. Then, the suboptimal control scheme based on small active dimples is investigated in detail.

1. Numerical method

As many previous studies in the field of turbulent drag reduction, the turbulent channel flow is adopted as a research objective in the present study. The Navier-Stokes equation and the continuity equation for the incompressible Newtonian fluid are taken as the governing equations:

For a channel with active dimples/pimples on the wall, the wall geometry is time-dependent and complex. Let the upper and lower walls are located atandrepresent the amount of deformation at the corresponding walls, respectively. The computational coordinate systemis defined as:

By the above coordinate transform, the temporal and spatial derivatives can be represented by

By the coordinate transformation, the Navierstokes equation can be written as

In the solution of the governing equations, the step is advanced in time by the 3-order time-splitting method. In this method, one whole time step is split into three sub-steps, i.e., the nonlinear step, the pressure step and the viscous step. In the nonlinear step, an intermediate velocityis introduced, and it is determined by

In the above equation, we setIn the pressure step, an intermediate velocityis introduced. The pressure correction is considered and the continuity constraint is imposed on the intermediate velocity, which satisfies the following relations:

By taking the divergence of Eq.(12), we can obtain the Poisson equation for the total pressure. In the computational space, it becomes

with the no-slip velocity conditions at the wall, whereIn the computational space, the equation for the viscous correction becomes

Fig.1 Dimples/pimples on the lower wall of the channel

Fig.2 Profile of the dimples/pimples

2. Opposition control by active dimples/pimples

2.1Physical problem and opposition control schemes

First, many cosine-shaped dimples/pimples are imposed on the lower wall of the channel, as shown in Fig.1. The profile of the dimples or pimples is shown in Fig.2, which can be described by the following function:

The velocity of the profile of each dimple/dimple is:

Fig.3 Time traces of

Two simple control schemes are used to calculate the velocity of the wall in the centre of the dimples/ pimples. The first one is the opposition control scheme with damping, which can be described mathematically by[11]

A monitor plane is put above the lower wall at a distancewhereis the velocity at the centre of the dimples/pimples at the time stepis the vertical velocity on the plane at the time stepIn this study, we selectThe last term of the equation is the damping term,is the damping coefficient andis the displacement of the center of the dimples/pimples. Ifthe damping term disappears and the definition of the velocity at the center of the dimples/pimples is reduced to that used by Kang and Choi[12], if, the damping term slows down the dimples or pimples when they move awayfrom the flat position of the lower wall or will accelerate them when they move towards the flat position. Figure 3(a) shows the variations ofduring a time period, where we can observe that the correlation between the two variables becomes weaker with the increase of the damping coefficient. Figure 3(b) shows the variation ofwith time for different damping coefficients. With the increase of the damping coefficient, the displacement of the dimples/pimples becomes smaller.

Table 1 Parameters and results for different control cases

Fig.4 Time traces of drag coefficient (denotes the total drag and the friction drag,denotes the pressure drag)

The second control scheme is the opposition control with relaxation

For this scheme, the velocity of the centre of the dimples/pimples is computed using the velocity in the monitor plane and the velocity of the wall at the former step,is the relaxation coefficient to avoid the abrupt movement of the dimples/pimples. Under such a control scheme, we restrict the maximum amplitude of the wall deformation. In other words, when at a certain place along the wall, the maximum amplitude is approached or reached, its movement speed is reduced or the place becomes inactive, and this happens, despite the fact that a certain speed is required to obtain the drag-reduction. Thus, this restriction may deteriorate the control performance. However, with large amplitudes, the boundary condition given in the control scheme may not be physically valid due to the non-negligible surface gradient. Without this restriction, the wall-deformation magnitude is rapidly increased up to 10 wall units and the simulation will finally break down. Four cases are simulated to study the opposition control based on the movement of dimples/pimples on the lower wall. The parameters of these cases are listed in Table 1.

2.2Results of the opposition control schemes

The time traces of the total drag, the friction drag and the pressure drag at the lower wall under the opposition control with damping are shown in Fig.4. In Case 1,, which is about half of the distance between two low speed streaks, the averaged total drag remains almost unchanged after the control. In Case 2,, which is about one fourth of the distance between two low speed streaks, the averaged friction drag reduction is about 3.1%, while the total drag is reduced by about 2%. From the comparison between Case 1 and Case 2, it can be concluded that the opposition control is very sensitive to the size of the dimples/pimples, and thus, the size of the dimples/pimples should be much smaller than the distance of low speed streaks in order to have a significant drag reduction.The pressure drag seems very small in these two cases, with the averaged values of 4.5×10−5and 5.84×10−5, respectively, corresponding to 1.1% and 1.4% of the total drag.

Fig.5 Instantaneous deformations of the lower wall

Fig.6 Contours ofplane

The deformation of the wall in Case 1 and Case 2 are shown in Fig.5. Because of the damping effect, the movements of the dimples/pimples are restrained and the displacements of most dimples/pimples are not very important, hence, the pressure drag caused by the dimples/pimples is relatively small. However, this restriction reduces the range of the effective movement of the dimples/pimple to control the coherent structure. Figure 6 shows the comparison ofplane between Case 1, Case 2 and a flat channel without any control. In Case 1, the dimples/pimples cannot counteract the sweeping or throwing of the fluid. In Case 2 the small dimples/pimples have obviously restrained the streamwise vortices.

Fig.7 Time traces of drag coefficient

The effect of the drag reduction in cases of using active small dimples/pimples as well as with only dimples is studied in Case 3 and Case 4. Figure 7 shows the time traces of the total drag, the friction drag and the pressure drag under the opposition control with relaxation. In Case 3, the averaged friction drag reduction is 5.1%, but the total drag seems to remain unchanged. This control successfully reduces the friction drag of the wall, however causes an increase of the pressure drag. In Case 4, the friction drag reduction reaches 4.8%, the total drag reduction reaches 3.4 % and the pressure drag caused by the wall deformation is only 1.4% of the total drag. The active dimples can reduce the friction drag effectively because it causes a smaller pressure drag than the pimples. Figure 8 shows the comparison ofplane between Case 3, Case 4 and a flat channel without any control. The near wall streamwise vortices are restrained by the small-sized active dimples/pimples. Figure 9 shows thedeformation of the instantaneous wall in Case 3 and Case 4. In Case 3, a great number of pimples have a large displacement to cause a much larger pressure drag. Thus, under the opposition control with relaxation, both in Case 3 and Case 4, a friction drag reduction is achieved. However, the pimples cause an important increase of the pressure drag which reduces the gain of the total drag. In contrast, the small active dimples can cause a relatively important friction drag reduction with a small pressure drag increase. We consequently recommend that the future set of actuators should be based on the dimple geometry instead of the pimple one.

Fig.8 Contours ofplane

Fig.9 Instantaneous deformations of the lower wall

For the simple opposition control schemes, the velocity of the centre of the dimples/pimples is determined using the wall-normal component of the velocity at, resulting in a 3.4% drag reduction in the numerical simulations. This approach, however, is not applicable since the required velocity information atis not measurable by a real sensor. For any practical implementation, the control scheme must be solely based on quantities measurable at the wall.

3. Suboptimal control based on small active dimples

For the simple feedback control schemes, the velocity of the wall at the centre of the dimples/pimples should be calculated using the wall-normal component of the velocity atresulting in 3.4% reduction in the numerical simulation. This approach, however, is impractical since the required velocity information atis not normally available. For any practical implementation, a control scheme should be based solely on quantities measurable on the wall. By applying a suboptimal control theory to a turbulent channel flow, a simple feedback control law was developed by Lee et. al.[9], in which only the wall shear stresses are involved.

It should be noted that the suboptimal control algorithm is expressed in terms of the Fourier coefficients (in the wave-number space). We havewheredenote the streamwise and spanwise wave numbers in thedirections, respectively.is a positive scale factor that directly determines the speed of the movement of the channel wall.denotes the spanwise shear stress on the wall in the time step. Variables with the capare the Fourier coefficients of the initial variables. The control algorithm can also be written in the physical space with an inverse transformation ofnumerically[9].

3.1Coarse grid

The number of the dimples on the lower wall is 32×32, the diameterwhich are the same as Case 4 in the opposition control. The computational domain isin accordance withgrids. Here two different values ofare selected to predict the effect of the drag reduction by the suboptimal control scheme.

Fig.10 Time traces of drag coefficients

Figures 10(a) and 10(b) show the time traces of the total drag, the friction drag and the pressure drag under the suboptimal control with(Case 5 in Table 1) and(Case 6 in Table 1), respectively. For, the averaged friction drag reduction is about 1.1%, but the total drag is only reduced by 0.4%. In this case, the pressure drag induced by the deformation of the wall is very small, only 0.7% of the total drag. For, the friction drag reduction is about 3.2% and the total drag reduction reaches about 2.2%. Similar to the case of0.002, the pressure drag owing to the active dimples is also very small, only 1.0% of the total drag. For the cases using the suboptimal control algorithm, the active dimples can reduce the friction drag effectively with very small pressure drag, and hence a net drag reduction is achieved.

Figure 11 shows the comparison ofin theplane between the controlled cases and a flat channel without any control. The near wall streamwise vortices are significantly restrained by the suboptimal control based on the active dimples imposed on the lower wall, the drag reduction increases with the increase ofFigure 12 shows the deformation of the instantaneous wall in the cases using the suboptimal control, which shows a similar behavior as Case 4 in the opposition control.

Fig.11 Contours ofplane

3.2Fine grid

In the opposition control, only one sensor is needed and it is placed above an actuator at, but for the suboptimal control, the actuation is a weightedaverage of the wall information of the entire grid points and hence many sensors are needed for each actuator. This means that the control scheme depends on the settings of the grids. If more grids are used on the wall, more sensors should be used for a single actuation. To reach a solid conclusion of the suboptimal control scheme based on the active dimples, fine grids are used forgrid points are used for a computational domain of. Figure 13 shows the time traces of the total drag, the friction drag and the pressure drag with the fine grid. In this case, the friction drag decreases by 4.5% and the total drag reduces by 2.7% with the pressure drag increased by 1.8%. It should be pointed out that both the coarse and fine grids are fine enough to capture the small flow structures of the turbulence and to obtain reliable results via the direct numerical simulation. Similar or even coarser grids are ever used by Ge et al.[6], Deng et al.[7]and Fang et al.[19]to obtain the turbulence database of high quality. The difference between the results from the coarse grid and the fine grid can be attributed to the grid dependence of the control scheme. This test on the fine grid confirms the effectiveness of the suboptimal control based on active dimples.

Fig.12 Wall deformations in the cases using suboptimal control

Fig.13 Time traces of drag coefficients for suboptimal control using the fine grid

Compared with the smart skin studied by Endo et al.[11]based on an opposition control and by Kang et al.[12]based on the suboptimal control , in which a drag reduction of about 10%-13% is achieved, the friction drag reduction rate in the present study is only about 4.5% with a total drag reduction of about 2.7% , which is substantially lower. Unlike Endo and Kang’s application of active control on the entire wall points, the gaps between the dimples reduce the total wall area of the active control significantly. To study the area effect, the opposition control is applied only to a portion of the wall area based on the blowing/suction by Choi et. al.[4], and it was found that the drag reduction is proportional to the surface area being controlled. In this study, the total area of the dimples is 11.57, only about 58.6% of the total wall area. It means that if the active dimples are arranged fully on the lower wall in the present study, the friction drag reduction will be 7.6% with a total drag reduction of about 4.6%. The constraint of freedom can be another explanation for the lower drag reduction. Compared with the arbitrary deformation with freedoms of the entire wall points, less “actuators” are used in the dimpled surface with freedoms only on the central points of the dimples. Obviously, the constraint of freedom will reduce the effect of control, for example, when the sweep motion occurs on the edging region of the dimples, the active dimples would not be effective. Besides the previous explanation, the pressure drag is also an important factor for the low total drag reduction due to the movement of the active dimples while the streamwise elongated grooves are formed in the smart skin by Endo et al.[11]and Kang et al.[12]to avoid this disaster.

Some feedback control systems were designed to explore its effectiveness for the turbulent drag reduction through physical experiments, but few of them achieved net drag reduction. Examples include the linear feedback control systems using two rows of three wall-mounted hot-film sensors and a single row of three synthetic jet actuators developed by Rathnasingham and Breuer[20], the feedback control system with a single hot-wire sensor and a single blowing/ suction actuator created by Rebbeck and Choi[21]. In 2008, about 6% drag reduction was achieved in a physical experiment for the first time by Yoshino et al.[22]with arrayed micro hot-film wall shear stress sensors and wall-deformation magnetic actuators in a wind tunnel. A genetic-algorithm-based optimal scheme is adopted in the laboratory experiment. The differences between the experiment by Yoshino et al.[22]and the present numerical study might be due to many factors, including the shape of the actuators, the arrangement of the actuators, the amplitude of actuations, and the control algorithm. Anyway, both results from the numerical simulations and the physical experiment involve a rather low drag reduction rate. Hence, there isstill a lot of work to do on the optimization of the design of MEMS. As an excellent candidate for the shape of MEMS, the flow control using active dimples is worth a further investigation in the future.

4. Summary and conclusions

The drag reduction is studied via direct numerical simulations of the turbulent channel flow with active dimples/pimples. Various shapes of the dimples/pimples are defined using a cosine function.

Firstly, the drag reduction by the opposition control based on active dimples/pimples at the lower wall is studied. To investigate the effect of the diameter of the dimple, two cases with the dimple diameter of 0.24 and 0.12 are studied under the opposition control scheme with a damping. It is shown that, large active dimples/pimples can not suppress the streamwise vortices significantly and thus almost no drag reduction is achieved while the smaller ones result in an averaged friction drag reduction of about 3.1% corresponding to the total drag reduction of about 2%. With the damping, the pressure drag in both cases is very small because of the small deformation of the active dimples/ pimples. Another opposition control scheme without the damping is applied to Case 3 and Case 4 with the central point of the dimples/pimplesrespectively. It is found that, without the damping, the deformation of the dimples/pimple is larger and the pressure drag can not be ignored. Active dimples can reduce the friction drag effectively because they cause a smaller pressure drag while with a relatively larger friction drag reduction. On the contrary, the pimples cause an important increase of the pressure drag which reduces the gain on the total drag.

Based on the above results, the small dimples are arranged on the lower wall of the channel and the suboptimal control scheme is used for the active dimples based on the spanwise wall shear information only. Two kinds of grids are used for the simulation with consistent results. It is shown that the friction drag decreases by 4.5% but the total drag is only reduced by 2.7% due to the pressure drag increases caused by the dimple movement.

Acknowledgements

This work was supported by the State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University (Grant No. LAPS17007), “the Fundamental Research Funds for the Central Universities”. The authors would like to express their gratitude to Prof. Xu Chun-xiao and Prof. Cui Gui-xiang in Tsinghua University for their help in this work.

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(Received March 7, 2015, Revised October 21, 2015)

* Project supported by the National Natural Science Foundation of China (Grant Nos. 11402088, 51376062).

Biography: Ming-wei Ge (1984-), Male, Ph. D.,

Associate Professor