The boundary layer is produced on surfaces of moving objects in the viscous fluid, which leads to deceleration, vibration, and instability. In particular, the friction drag will increase significantly in turbulent flow. Therefore, many control approaches have been used to modify the structures of the boundary layer, which can reduce the friction drag of moving objects. These approaches have a wide prospect of application and great practical value in aviation and navigation.^[1–5]

For flow control, the active methods can be distinguished from passive methods by energy input. The electromagnetic control has been considered as one of the most practical active methods due to the flexible design and easy installation.^[6] As early as the 1960s, Gailitis found that the boundary layer can be modified by electromagnetic actuators with alternating electrodes and magnets. Subsequently, the control effects of streamwise vortex in turbulence channel flow were investigated numerically by Lim and Choi^[7] with electromagnetic force along the streamwise, spanwise, and wall-normal, respectively. The results indicated that the streamwise vortex can be suppressed by both the spanwise and wall-normal electromagnetic force, and the former has more remarkable effects in suppressing streamwise vortices. Jiménez and Pinelli^[8] further found that the regenerating cycle mechanism of turbulence depends on the two structures; i.e., quasi-streamwise vortexes and streaks. Next, Satake and Du et al.^[9–11] studied the turbulent control by electromagnetic force with a spanwise wave traveling along the streamwise direction. They found that the initial near-wall streak structures would replace a wide “ribbon” of low-speed velocity when parameters of electromagnetic force match well, which results in a drag reduction rate up to 30%. Lee et al.^[12] investigated the vorticity dynamics in the viscous sublayer with the application of electromagnetic force. The results indicated that turbulent structures can be improved near the wall and the skin drag can also be decreased by suppressing the streamwise vortexes of the viscous sublayer. Yin and You^[13] studied the characteristics for a cylinder wake with a turbulent boundary layer by using electromagnetic force, and found the force can suppress the flow separation. Moreover, the streamwise traveling and standing waves of velocity imposed at the walls of a plane turbulent channel flow were investigated by Viotti and Quadrio^[14,15] with direct numerical simulation (DNS) methods, and the drag reductions were obtained at about 48% and 52% for traveling waves and standing waves, respectively. Moubarak and Habchi^[16,17] investigated two scales on the energy cascade in a two-dimension (2D) turbulent flow with electromagnetic force, and found that there exists a linear relationship between the large-scale motion in the atmosphere and the very small ones. Ostillamónico^[18] calculated three-dimension (3D) turbulent with electromagnetic force by fully solving the Navier–Stokes equations which were coupled with the Poisson–Nernst–Planck equations. The results showed that an ion concentration-dependent viscosity leads to the emergence of a quiescent layer with a higher ion concentration, and the presence of this layer could play a role in disrupting the turbulence generation cycle as one would expect that the turbulence is weaker in regions of higher viscosity, thus potentially decreasing friction. Recently, a DNS study for a fully developed turbulent channel flow was carried out by Altıntaş et al.,^[19] which introduced a spanwise oscillating electromagnetic force near the lower wall. They observed that the sweeps and ejections moved away from the wall in the fully turbulent region with the application of electromagnetic force. The control effects of streamwise traveling wave and bidirectional wavy electromagnetic force on the wall-turbulence were investigated by our research group.^[20,21] The results indicated that the negative streamwise vortex induced by electromagnetic force exists in the flow field with control, which can suppress the inherent positive streamwise vortex and merge the inherent negative streamwise vortex in the turbulent flow field. With the effects of the periodic action, the number of streamwise vortexes and streaks in the flow field was reduced, which further led to the decrease of skin drag.

These studies indicate that the significant effects of turbulence control can be achieved by using electromagnetic force with spanwise oscillating or traveling wave. However, the character structures of the steady flow fields, which depend on the distribution of the electromagnetic force, cannot be obtained due to the time-varying force. With the application of space-dependent electromagnetic force, the regular quasi-streamwise vortex structures in the flow field have been found in our research group. The random velocity fluctuation and mean Reynolds stress can be suppressed by these structures at Re=4000, which further leads to the drag reduction.^[22] Therefore, further investigations are necessary on the relations among fluctuating velocities, Reynolds stress, and drag reduction, especially for different Reynolds numbers.

In this paper, the Fourier–Chebyshev spectral method is used to investigate the turbulence channel flow controlled by the space-dependent electromagnetic force using DNS. The spanwise electromagnetic force with sinusoidal distribution along the streamwise direction is selected to control the flow field. Then, the variations of fluctuating velocities, the distributions of Reynolds stress, and the effects of drag reduction of flow field with different Reynolds numbers before and after control are discussed. Moreover, the potential relations among these three are further analyzed by deriving a formula that expresses the relation between the coefficient of friction drag and the fluctuating velocities.

The induced flow field is a stable laminar flow field with the effect of electromagnetic force. The spanwise electromagnetic force introduced in the paper is defined by Eq. (4), which is independent of time. The spatial distribution of electromagnetic force with parameters , A=1.0, and , as a case, is shown in Fig. 3. The electromagnetic force can induce spanwise motion in the conductive fluid, and produce layers called Stokes layer near the wall. When the conductive fluid is affected by the electromagnetic force, as a case of the laminar flow field for Re=4000, the spanwise velocity distribution of the induced laminar flow field is presented in Fig. 4, where the red and blue areas refer to the positive and negative values, respectively. As a result of the main flow along the positive direction of the x-axis, the distribution of the spanwise velocity is inclined significantly to the downstream, which is asymmetric but periodic with the electromagnetic force.

Fig. 3.

	Figure Option ViewDownloadNew Window
	Fig. 3. The spatial distribution of electromagnetic force.

Fig. 4.

	Figure Option ViewDownloadNew Window
	Fig. 4. The spanwise velocity distribution of the induced laminar flow field.

In the flow control, the rate of drag reduction is usually introduced to evaluate the control effect of electromagnetic force, which is defined as 18 where and are the average values of the wall shear stress with and without control, respectively. The variations of D_r with k_x and A are shown in Fig. 5, where Figure 5(a), 5(b), 5(c), and 5(d) correspond to the cases of Re=4000, 5000, 6000, and 7000, respectively. From the figures, the variations of D_r with the parameters of electromagnetic force have similar distributions for different Reynolds numbers. At low wave numbers, the values of drag reduction rate vary dramatically with the amplitude (i.e., the absolute values of the reduction and increase in drag are large), whereas the values of that vary slowly at high wave numbers. Moreover, the maximum drag reduction can be obtained with the optimal parameter combination in each case of the four Reynolds numbers. The parameters of electromagnetic force correspond to the maximum drag reduction rate in Fig. 5, as shown in Table 1. From the table, the maximum drag reduction rate decreases as the Reynolds numbers increase, and the corresponding wave number of electromagnetic force increases, while the variation of the amplitude is around 1.0.

Fig. 5.

	Figure Option ViewDownloadNew Window
	Fig. 5. The variations of drag reduction rate with amplitude and wave number for different Reynolds numbers: (a) Re=4000, (b) Re=5000, (c) Re=6000, and (d) Re=7000.

Table 1.

Table 1.

Table 1. The parameter combinations corresponding to the maximum drag reduction rate for different Reynolds numbers. . Reynolds number Maximum drag reduction rate Wave number Amplitude 4000 58.9% 3 1.0 5000 49.4% 4 1.0 6000 45.2% 5 1.0 7000 42.4% 6 1.0

Table 1. The parameter combinations corresponding to the maximum drag reduction rate for different Reynolds numbers. .

To find the reasons that the maximum drag reduction rate decreases with the increase of Reynolds number, it is necessary to study the in-depth mechanisms from the flow fields before and after control, respectively. The streaks are the basic elements of near wall turbulence coherent structures. Therefore, the distribution of streaks in the flow field at y⁺=5 is first investigated, as shown in Fig. 6, where Figure 6(a), 6(b), 6(c), and 6(d) represent the cases of Re=4000, 5000, 6000, and 7000, respectively. The left figures indicate the distribution of streaks before control, and the right figures indicate the distribution of streaks after control with the electromagnetic force in Table 1, where red and blue represent high-speed and low-speed streaks, respectively. As shown in the left figures, high-speed and low-speed streaks are alternate with different lengths and curvatures in the flow field for Re=4000. With the increase of Reynolds number, the strength of high-speed and low-speed streaks is enhanced, and the number of these streaks is increased. However, in the right figures, the strength of high-speed and low-speed streaks controlled by electromagnetic force at Re=4000 is decreased significantly in the flow field. The distributions of these streaks are alternate along the spanwise direction with the well-organized characters. In particular, the periods of these streaks along the streamwise direction are the same as that of the electromagnetic force. Moreover, the distributions of high-speed and low-speed streaks are similar to the increase of Reynolds number, while the strength of these streaks is enhanced and the regularity of the streaks is gradually weakened. The control effects of the optimal electromagnetic force decay with the increase of Reynolds number.

Fig. 6.

	Figure Option ViewDownloadNew Window
	Fig. 6. The distribution of streaks at y +=5 in the flow field before and after control with the optimal electromagnetic force: (a) Re=4000, (b) Re=5000, (c) Re=6000, and (d) Re=7000.

The variations of fluctuating velocities (i.e., , , and ) can be further investigated and their values can be obtained from the value on each grid point before and after control. Therefore, with the coordinate axes , , and , the discrete distributions of , , and for y⁺=5 are obtained from values of fluctuating velocities on these locations (y⁺=5), as shown in Fig. 7. Here, black and red discrete points represent the distributions of fluctuating velocities before and after control, respectively. Figure 7(a), 7(b), 7(c), and 7(d) correspond to the results of Re=4000, 5000, 6000, and 7000, respectively. For the case of Re=4000 before control, the distributions of , , and are dispersed. After control, the distributions of and converge to the origin, especially for , which means that their absolute values are decreased. However, the absolute value of is increased dramatically and the distribution region of is enlarged with the effect of the spanwise electromagnetic force. These variations eventually lead to the straight cylindrical distribution of fluctuating velocities after control as red discrete points in Fig. 7(a). From Eq. (12), drag is a function with respect to and rather than . Therefore, the increase of does not lead to the increase of drag, while the decrease of and leads to the decrease of drag. Moreover, the distribution regions of the three fluctuating velocities gradually increase with the increase of Reynolds number, as shown in Figs. 7(b), 7(c), and 7(d), which means that some values of the fluctuating velocities increase. After control, the central tendency of the fluctuating velocities is weakened, which means that the control effect of the optimal electromagnetic force is weakened.

Fig. 7.

	Figure Option ViewDownloadNew Window
	Fig. 7. The discrete distribution of fluctuating velocities at y + = 5 before and after control with the optimal electromagnetic force: (a) Re=4000, (b) Re=5000, (c) Re=6000, and (d) Re=7000.

The Reynolds stresses depend on the product of and rather than . Therefore, the variations of the product can be described by the probability distributions of with y⁺, where the left and right figures in Fig. 8 respectively represent the cases before and after control, and Figure 8(a), 8(b), 8(c), and 8(d) correspond to the cases of Re=4000, 5000, 6000, and 7000, respectively. P is the value of probability at a constant y⁺, which is defined as 19 Here, is a subinterval of [−0.002, 0.002] which is the range of the value of . is the number of the value of distributed in . N_x and N_z are the numbers of grid points in the x and z directions, respectively. Red and blue areas represent the large and small values, respectively. From the case before control in Fig. 8(a), the distribution of almost converges to , and the peak value of P is large due to the flow dominated by the effect of viscosity in the area . With the increase of y⁺, the influence of turbulent fluctuation dramatically strengthens and exceeds the effect of viscosity, which leads to the increase of the value in some regions. Therefore, the distribution of is more homogeneous and the peak value of P is decreased rapidly. However, with the optimal electromagnetic force, the distribution of is significantly close to in near-wall regions ( ), which indicates that the values of have been suppressed effectively. In the region , the central tendency of distribution is weakened, which means the control effect is weakened, because the electromagnetic force decays rapidly with the increase of y according to Eq. (4). As the Reynolds number increases, similar variations are obtained in Figs. 8(b), 8(c), and 8(d). However, the central tendency of distribution is weakened (particularly in the regions ), which means that the control effects of the optimal electromagnetic force decrease with the increase of Reynolds number.

Fig. 8.

	Figure Option ViewDownloadNew Window
	Fig. 8. The variation of probability distributions of u′v′ with y+ before and after controlling the optimal electromagnetic force: (a) Re=4000, (b) Re=5000, (c) Re=6000, and (d) Re=7000.

The Reynolds stresses can be obtained from the mean value of . According to Eq. (17), the turbulent drag coefficient depends on the y⁺-weighted integration of , which has a larger weight near the wall in the component of turbulent drag. The distributions of and before and after control by the optimal electromagnetic force for different Reynolds numbers are shown in Fig. 9, where Figure 9(a), 9(b), 9(c), and 9(d) correspond to the computational results for Re=4000, 5000, 6000, and 7000, respectively. From the case of Re=4000 in Fig. 9(a), the absolute value of before control increases rapidly at first then decreases slowly with the increase of y⁺, and has similar variations. After control, the absolute value of decreases significantly, especially near the wall. The reason is that the values of have been suppressed effectively with the effect of the optimal electromagnetic force, particularly in the region of , as shown in Fig. 8. Therefore, decreases with the decrease of the absolute value of based on Eq. (17). Moreover, the suppression effect of the optimal electromagnetic force for is weakened with the increase of Reynolds number, as shown in Figs. 9(b), 9(c), and 9(d), due to the weakening effect of the central tendency of distribution in Figs. 8(b), 8(c), and 8(d). Therefore, the absolute values of increase, which leads to the increase of and indicates the weakening effect of drag reduction with the increase of Re in Fig. 9.

Fig. 9.

	Figure Option ViewDownloadNew Window
	Fig. 9. The distribution of turbulent drag and Reynolds stress before and after control by the optimal electromagnetic force with y + for different Reynolds numbers: (a) Re=4000, (b) Re=5000, (c) Re=6000, and (d) Re=7000.

The relations on fluctuating velocities and the friction drag coefficient with the spanwise space-dependent electromagnetic force in turbulent channel flow and the mechanism of drag reduction are investigated with DNS for different Reynolds numbers. The results show that the maximum drag reductions are obtained with an optimal combination of parameters for each case of different Reynolds numbers. With the application of the optimal electromagnetic force, and (i.e., the fluctuating velocities along the streamwise and normal directions) are suppressed, while (i.e., the fluctuating velocity along the spanwise direction) is enhanced dramatically due to the spanwise electromagnetic force. However, the values of Reynolds stress depend on and rather than . Therefore, the distributions of and converge to the origin point, and the distributions of the product also converge to , which further leads to the decrease of Reynolds stress and drag reduction. Moreover, the control effect of the optimal electromagnetic force for the three fluctuating velocities gradually decays with the increase of Reynolds numbers, resulting in the weakening effects of central tendency for . Finally, the maximum drag reduction decreases due to the decay of control effect for Reynolds stress with the increase of Reynolds number.