1 Introduction

A boundary layer can be produced on surfaces of moving objects in a viscous fluid, which leads to deceleration, vibration, and instability. Especially, the friction drag will increase significantly in turbulent flow. Therefore, many control approaches have been studied to modify the structures of the boundary layer, and then decrease the friction drag of moving objects, which has a widely applied prospect and great practical value in the aviation and navigation to improve propulsive efficiency, energy loss, and kinetic stability^[1-4].

For flow control, the active methods can be distinguished from passive methods by energy input. Electromagnetic control has been considered one of the most practical active methods due to the flexible design and easy installation^[5]. As early as the 1860s, 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 turbulent channel flows were investigated numerically by Lim et al.^[6] with electromagnetic force along the streamwise, spanwise, and wall-normal directions, respectively. The results indicated that the streamwise vortex can be suppressed by both the spanwise and wall-normal electromagnetic forces, and the former has more remarkable effects in suppressing streamwise vortices. Jiménez and Pinelli^[7] further found that the regenerating cycle mechanism of turbulence depends on the two structures, i.e., quasi-streamwise vortices and streaks. Next, Satake and Kasagi^[8], Du and Karniadakis^[9], and Du et al.^[10] 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 were instead of a wide `ribbon' of low-speed velocity when parameters of electromagnetic force matched well, which result in a drag reduction rate up to 30%. Lee and Kim^[11] investigated the vorticity dynamics in the viscous sublayer with the application of electromagnetic force. The results indicated that the turbulent structure can be improved near the wall, and the skin drag can also be decreased by suppressing the streamwise vortices of the viscous sublayer. Moreover, the streamwise travelling and standing waves of velocity imposed at the walls of a plane turbulent channel flow were investigated^[12-13] with direct numerical simulation (DNS) methods, and they obtained drag reductions about 48% and 52% for travelling waves and standing waves, respectively. Moubarak and Antar^[14] and Habchi and Antar^[15] investigated two scales on the energy cascade in a two-dimensional (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 and Lee^[16] calculated three-dimensional (3D) turbulence 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 of 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 was 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 Altintaş and Davidson^[17] by introducing 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 to wall-turbulence were investigated by our research group^[18-19]. The results indicated that the negative streamwise vortex induced by electromagnetic force existed in the flow field with control, which can suppress the inherent positive streamwise vortex and merge the inherent negative streamwise vortex in turbulent flow field. With the effects of the periodic action, the number of streamwise vortices and streaks in the flow field was decreased, which further led to the decrease of skin drag.

The above-mentioned studies indicate that the significant effects of turbulence control can be achieved by using the electromagnetic force with spanwise oscillating or traveling wave. However, the characteristic structures of the steady flow field dependent on the distribution of the electromagnetic force cannot be obtained due to the time-varying electromagnetic force. With the application of space-dependent electromagnetic force, some characteristic structures have been found in our research group^[20]. Therefore, more in-depth studies are necessary, especially for different Reynolds numbers.

With the Fourier-Chebyshev spectral method applied, the turbulent channel flow controlled by the space-dependent electromagnetic force is investigated by the DNS in this paper. The spanwise electromagnetic force with sinusoidal distribution along the streamwise direction is selected to control flow field, and the characteristic structures and their effects in flow field with the electromagnetic force are analyzed. Moreover, a formulation is derived to express the relation between the drag and the Reynolds shear stress for the further discussion of the potential relations among characteristic structures of flow field, the distribution of Reynolds shear stress, and the effect of drag reduction at different Reynolds numbers.

2 Numerical calculations 2.1 Governing equations

Based on the fully developed turbulent flow of a weakly conductive fluid, the flow can be controlled by electromagnetic actuators which are installed on the lower wall of the channel. With the electromagnetic force imposed as a source term, the dimensionless Navier-Stokes equations of incompressible 3D channel flow can be written as follows:

(1)

(2)

Here, all variables are nondimensionalized with respect to the channel half width h and the center line U_c. u is the velocity vector, p is the pressure, is the Reynolds number with the kinematic viscosity coefficient ν, and f is the electromagnetic force given by

(3)

where

(4)

Here, f_z is the spanwise electromagnetic force with sinusoidal distributions along the streamwise direction, whose variation depends on space rather than time. A is the nondimensional amplitude, Δ is the effective penetration of electromagnetic force, and L_x and k_x are the channel length and wave number along the streamwise direction, respectively, where L_x/k_x=λ_x is the streamwise wave length.

The standard Fourier-Chebyshev spectral method is applied to spatial directions for Eqs. (1) and (2). With the periodic character of turbulent channel flow in the streamwise and spanwise directions, a dealiased Fourier method is applied in these two directions, while the Chebyshev-tau method is used in the normal direction of the wall. Moreover, the usual no-slip and no-penetration conditions are used on the wall. The time advancement is performed by a semi-implicit back-differentiation formula method with third-order accuracy. To eliminate residual divergence, the pressure term and the linear term are solved with a Chebyshev-tau influence matrix method. For the non-linear term, a spectral truncation method referred to as the 3/2-rule is employed to remove aliasing errors.

The computational domain is shown in Fig. 1, where the sizes of the streamwise, normal, and spanwise directions correspond to 4π/3×2×2π/3 (approximately 754×360×377 wall units), respectively. Uniform grid spacing is employed in the streamwise and spanwise directions, while the non-uniform grid space is employed in the normal direction. More details about the numerical methods can be found in Refs.[18]-[19].

The turbulent channel flow is controlled with the application of electromagnetic force at t=300 when the initial disturbance induced for generating a fully developed turbulence has been essentially eliminated, which is steady from t=500 to t=1 000. The cases are selected with different Reynolds numbers from 4 000 to 7 000 (the density of fluid ρ=1.0×10³ kg/m³, and kinematic viscosity ν=1.0×10^-6 m²/s).

2.2 Friction drag along the streamwise direction in turbulent channel flow

For channel flow, we have

(5)

(6)

(7)

(8)

where U(y) represented by the parabolic equation is the streamwise velocity distribution of the base flow

(9)

Here, y is the distance from the wall. u, v, and w are velocities along the x-, y-, and z-directions, respectively. The superscript "＇" indicates the fluctuation. Π(t) is the real-time adjustment of pressure for keeping flow stable.

From Eq. (1), the momentum equation in the x-direction is

(10)

Substituting Eqs. (5)-(8) into Eq. (10), one obtains

(11)

Based on the assumption of the periodic flow structures in the homogeneous directions and the no-slip boundary conditions, which are the basic characters of the turbulent wall-bounded flow, we average Eq. (11) by integrating the equation in the x- and z-directions, and have

(12)

where the overbar "—" indicates the average value of corresponding plane for a constant y.

The mean pressure gradient between the entrance and the exit of the channel is balanced by the shear-stress gradient. Therefore, integrate Eq. (12) from 0 to 1 in y -direction with the friction drag coefficient defined as C_f=τ_w/(ρU_c²/2), then

(13)

Substituting Eq. (13) into Eq. (12), we have

(14)

With the triple integral of Eq. (14), one obtains

(15)

For the steady turbulence,

(16)

Finally, the friction drag coefficient is obtained as

(17)

The equation indicates that the friction drag coefficient is composed of two parts for the steady turbulent channel flow. The first term is the laminar drag coefficient, which depends on the Reynolds numbers. The second term is the y-weighted integration of , which is the Reynolds shear stress due to the ejection and sweep in the turbulence.

Define

(18)

(19)

By the derivative of Eq. (19), the distribution of C_f(y) with y is

(20)

The contribution of the Reynolds shear stress on the different locations in the wall-normal direction can be described in Eq. (20).

2.3 Program verification

For the sake of testing the code and algorithm, the turbulent channel flow without control for Re=4 000 is selected to compare with the results from Kim et al.^[21]. The results indicate that the mean velocity profile in the paper agrees with those in Ref.[21] and the classic wall law. More details about the validation of the code can be found in Refs.[18]-[19].

3 Results and discussion 3.1 Induced flow fields

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 k_x=3, A=1.0, and Δ =0.02, as a case, is shown in Fig. 3. The electromagnetic force can induce spanwise motion in the conductive fluid, and produce shear layers called Stokes layer near the wall. When the conductive fluid is affected by electromagnetic force, as a case of the laminar flow field for Re=4 000, 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 x-axis, the distribution of spanwise velocity is inclined significantly to the downstream, which is asymmetric but periodic with the electromagnetic force. The incline of the spanwise velocity distribution produces spanwise shear layers.

3.2 Control effects at different Reynolds numbers

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]

(21)

where 〈τ_w〉 and 〈τ_wn〉 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 Figs. 5(a), 5(b), 5(c), and 5(d) correspond to the cases of Re=4 000, 5 000, 6 000, and 7 000, respectively. From the figures, the variations of D_r with the parameters of electromagnetic force have the 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 drag decrease and increase 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 the each case of the four Reynolds numbers. The parameters of electromagnetic force corresponding to the optimal drag reduction rate in Fig. 5 are shown in Table 1. From the table, the optimal drag reduction rate decreases with the increase in Reynolds numbers, and the corresponding wave number of electromagnetic force increases, while the variations of the amplitude are around 1.0.

To find the reasons that the drag reduction rate decreases with the increase in Reynolds number, it is necessary to study the in-depth mechanisms from the structures of flow fields before and after control, respectively. The streaks and streamwise vortex structures are the basic elements of near wall turbulence coherent structures. Therefore, the distribution of streaks and streamwise vortex structures of the flow field without control is investigated at first and shown in Fig. 6, where Figs. 6(a), 6(b), 6(c), and 6(d) are spanwise velocities, streaks, streamwise vortex structures in the flow field in the cases of Re=4 000, 5 000, 6 000, and 7 000, respectively. The first column describes the distributions of spanwise velocity at y⁺ = 5, where red areas denote positive values and blue areas denote negative values, respectively. It can be seen that the spanwise velocity has an irregular distribution, of which the amplitude and regions increase with the increase in Re, which indicates that the random characteristic of the flow field is enhanced. The second column describes the distribution of streaks at y⁺=5, where red and blue areas represent high-speed and low-speed streaks, respectively. As shown in the figure, high-speed and low-speed streaks are alternate with different lengths and curvatures. Moreover, the fluctuations of streamwise velocity are enhanced further with the increase in Re, which lead to the increased number of high-speed and low-speed streaks. The third column describes the distribution of streamwise vortex structures at 0 < y⁺ < 108, where red areas denote positive values of streamwise vorticity, and blue areas denote negative values of streamwise vorticity. From the figure, there are a large number of irregular quasi-streamwise vortices near the wall, and more vortices appear on a denser distribution and a smaller scale with the increase in Re.

The cases in Fig. 6 can be controlled by the optimal Lorentz force in Table 1, and the corresponding results after control are shown in Fig. 7. Similar with Fig. 6, the first, second, and third columns represent distributions of spanwise velocity at y⁺ = 5, streaks at y⁺ = 5, and vortex structures at 0 < y⁺ < 108, together with red and blue areas representing positive and negative values, respectively. In the first column figures, the distribution of spanwise velocity shows a regular and alternate positive-negative character, of which the period is equal to that of the electromagnetic force applied for Re=4 000. With the increase in the Reynolds number, the regularity of the distribution is weakened, the amplitude is decreased, and the period of spanwise velocity is decreased corresponding to that of the optimal electromagnetic force. In the second column figures, the streaks in the flow field controlled by electromagnetic force at Re=4 000 are distributed alternately along spanwise with a regular and periodical curving characteristic. The periods of the streaks are the same as that of the electromagnetic force, and the fluctuating value of velocity is small. Moreover, the distributions of streaks are similar with the increase in Reynolds number, while the fluctuating values of velocity are increased, and the regularity of the streaks is weakened gradually. In the third column figures, the number of irregular quasi-streamwise vortices of the flow field controlled by electromagnetic force at Re=4 000 is decreased significantly. The remaining quasi-streamwise vortices, which are induced by electromagnetic force, are distributed regularly, and the period of the alternate positive-negative is equal to that of the electromagnetic force. Therefore, the period of the quasi-streamwise vortices is decreased corresponding to that of electromagnetic force with the increase in Reynolds number. However, irregular quasi-streamwise vortices reappear farther away from the wall, of which the number increases with the increase in Reynolds number, which indicates that the control effect of the optimal electromagnetic force is weakened with the increase in Reynolds number.

The relationship between the electromagnetic force and the regular quasi-streamwise vortices induced near the wall is investigated further on a study of the flow at Re=4 000. As shown in Fig. 8, the top part shows the distribution of electromagnetic force along the streamwise direction, and the middle and bottom parts offer views of the regular quasi-streamwise vortex structures near the wall from the xz- and xy-planes, where red and blue areas represent positive and negative values of vorticity, respectively. Including the above-mentioned characters (the period of the regular quasi-streamwise vortices is equal to that of the electromagnetic force), it can be found that the negative streamwise vortices are induced by positive spanwise electromagnetic force and inclined toward positive spanwise, while the positive streamwise vortices are induced by negative spanwise electromagnetic force and inclined toward negative spanwise direction.

To investigate the development of quasi-streamwise vortices in detail, a pair of positive and negative vortex structures in Fig. 8 is selected as shown in Fig. 9. In the figure, the left side is 3D figure, where red areas represent positive vorticity marked as S_P, and blue areas represent negative vorticity marked as S_N. For the right side, D₀-D₀, D₁-D₁, D₂-D₂, and D₃-D₃ are the corresponding sectional views along the x-direction sequentially. Moreover, the curves on the right of the sectional views are isolines of streamwise vorticities, where the solid lines represent positive vorticities with clockwise rotation and the dotted lines represent negative vorticities with counterclockwise rotation. The development of the quasi-streamwise vortices in the figure can be summarized as follows: the positive streamwise vortex is defined as parent vortex, and its tail, the lower end part turned inward, is induced by the upstream vortex. The middle is the main part of the vortex structure, which is lifted gradually and inclined with the effect of spanwise electromagnetic force. Then, the vorticity grows and reaches the maximum at D₀. When the streamwise vortex is developed to D₁, a new negative streamwise vortex called offspring vortex is generated at the lower left under the effect of entrainment of the parent vortex. Next, the parent vortex decays while the offspring vortex grows, and the parent vortex is developed to the top point at D₂, which is the head of parent vortex at the part of D₁-D₂. Subsequently, the parent vortex is vanished by turbulent burst while the offspring vortex is fully developed at D₃. The generation mechanism of streamwise vortex in a turbulent flow has been discussed in detail by Schoppa and Hussain^[22-23]. Combined with Fig. 8, it is obvious that the location f_z = 0 corresponds to the tail and head of the vortex with the small absolute value of vorticity, while the location (|f_z| reaches the maximun value) corresponds to the middle part of the vortex with the maximum absolute value of vorticity.

For the cases with different Reynolds numbers, there are similar quasi-streamwise vortex structures with the application of electromagnetic force. However, the scale of quasi-streamwise vortex is distinct due to the different parameter values of the optimal electromagnetic force as shown in Fig. 10, where y represents the height from lower wall, x represents the size of vortices along streamwise and quasi-streamwise vortex structures induced by electromagnetic force corresponding to the cases of Re=4 000, 5 000, 6 000, and 7 000 from left to right, respectively. From the figure, the scale of the induced quasi-streamwise vortex structures shrinks with the increase in Reynolds number due to the increase of wave number of the optimal electromagnetic force. It is well known that the scales of irregular quasi-streamwise vortices decrease with the increase in Reynolds number. Therefore, the scale of the induced vortex can agree with that of irregular quasi-streamwise vortex in the turbulence, and the maximum rate of drag reduction can be obtained for the cases of different Reynolds numbers.

The variations of flow field lead to the variations of drag force. According to Eq. (20), the turbulent drag coefficient generated by the random fluctuations depends on the y-weighted integration of , and the near the wall has greater weight in the component of turbulent drag. The distributions of C_f^y(y) and before and after control by the optimal electromagnetic force for different Reynolds numbers are shown in Fig. 11, where Figs. 11(a), 11(b), 11(c), and 11(d) correspond to the computational results for Re=4 000, 5 000, 6 000, and 7 000, respectively. From the case Re=4 000 in Fig. 11(a), the absolute value of before control increases rapidly at first and then decreases slowly with the increase in y, and C_f^y(y) has the similar variations. Moreover, the maximum of C_f^y(y) is closer to the wall. After control, the absolute value of is decreased significantly, especially near the wall, which indicates that the suppression effect originates mainly from the quasi-streamwise vortex induced by the electromagnetic force. Therefore, the coefficient distribution of turbulent drag is decreased with the decrease in the absolute value of . Furthermore, the suppression effect of electromagnetic force to the Reynolds shear stress is weakened with the increase in Reynolds number as shown in Figs. 11(b), 11(c), and 11(d). The reason is that the height of the quasi-streamwise vortex induced by the optimal electromagnetic force is decreased as shown in Fig. 10, which means the decrease of the effective depth with the increase in Reynolds number. Therefore, the value of the Reynolds shear stress increases and the distribution of that close to the wall, which leads to the weakness effect of drag reduction in Fig. 11.

4 Conclusions

The control of turbulent channel flow by space-dependent electromagnetic force and the mechanism of drag reduction are investigated with the 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, and the similar structures, i.e., the regular quasi-streamwise vortex structures, which appear in the flow field, have the same period as that of the electromagnetic force. The random velocity fluctuations are suppressed by these structures, which leads to the absolute value of mean Reynolds shear stress decreasing and the distribution of that moving away from the wall. Therefore, the effect of drag reduction can be obtained. Moreover, the wave number of optimal electromagnetic force increases with the increase in Reynolds number, resulting in the decrease of scale of the regular quasi-streamwise vortex structures, which further leads to the decrease of control depth. Finally, the rate of drag reduction decreases due to the decay of control effect for mean Reynolds shear stress with the increase in Reynolds number.