1 Introduction

The nature and significance of organized large-scale structures,e.g.,streaky structures and vortical structures,in turbulent boundary layers have been extensively studied for over six decades^{[1, 2, 3, 4, 5]}. It has been recognized that most high skin-friction drags are attributed to the existence of the near-wall vortical structures,e.g.,quasi-streamwise vortices (QSVs),hairpin vortices,and soliton-like coherent structures (SCSs)^{[6, 7]},and the associated ejection/sweep events.

Kim and Bewley^[8] and Kim^[9] proposed that the near-wall streamwise vortices were the most important turbulent structure in drag manipulation. Various methods have been proposed, focusing on the suppression of the near-wall coherent structures,e.g.,wall oscillation^{[10, 11, 12]}, deformed wall^{[13, 14, 15]},blowing and suction wall^{[16, 17]},and turbulent flows around cylinders^{[18, 19]}.

In this paper,the spatially oscillating spanwise Lorentz force is used to control the near- wall turbulence. A statistically steady flow field with well-organized streamwise vortices is maintained via the Lorentz force. The drag decreases considerably due to the suppression of the near-wall coherent structures. A formulation expressing the relation between the Reynolds shear stress and the skin-friction drag is derived for the further discussion of the drag reduction.

2 Numerical calculations 2.1 Governing equation

The discussed problem is the control of the Lorentz force for a fully developed turbulent flow in a weakly conductive fluid in a channel mounted with electromagnetic actuators on both the upper and lower walls. The flow is governed by the incompressible Navier-Stokes (N-S) equations with the externally imposed Lorentz force term written as follows:

where all variables are nondimensionalized with respect to the channel half width h and the center line velocity U_c. u is the velocity vector. p is the pressure. Re is the Reynolds number. f is the Lorentz force per unit mass,and it is given by where in which A is the non-dimensional amplitude of excitation (or interaction parameter),δ is the effective penetration of the Lorentz force,and Lx is the streamwise length of the computational domain.k_xis the streamwise wave number,and it is defined by where λ_x is the wavelength along the streamwise direction.

The numerical method adopted here is based on the standard Fourier-Chebyshev spectral method. The usual no-slip and no-penetration conditions are applied at the wall,and the periodic conditions are imposed along the homogeneous directions^{[20, 21]}. The time advance- ment is carried out by a semi-implicit back-differentiation formula method with the third-order accuracy.

Figure 1 shows a schematic diagram of the computational domain. The computational domain size is (4φ/3)× 2 × (2φ/3) (approximately 754×360×377 wall units),corresponding to the streamwise,normal,and spanwise directions,respectively. The grid spacing is uniform in the homogeneous directions. More details are referred to Refs. ^[18] and ^[19].

The flow control by the Lorentz force begins at t = 200 when the initial disturbance in- duced for generating the fully developed turbulence has been essentially eliminated. Then,the turbulence statistics is performed from t = 500 to t = 1 000.

2.2 Skin frictional drag in turbulent channel flow

The skin frictional drag of a wall-bounded turbulent flow is attributed to the existence of the near-wall vortical structure and the associated ejection/sweep events. For the channel flow with a constant flux,a relation between the skin friction coefficient and the Reynolds stress distribution is derived as follows.

If the Reynolds decomposition is applied in Eq.(1) in the x-direction,based on the base- fluctuation decomposition with the base flow U(y) = 1 − (1 − y)²,we have

where

Based on the assumption of the periodic flow structures in the homogeneous directions and the no-slip boundary conditions,which are the basic features of the turbulent wall-bounded flow,we average Eq.(5) by the integration in both the x-direction and the z-direction,i.e.,the homogeneous directions,and obtain

where the overbar “-” denotes an average over a plane with a constant y.

Integrating Eq.(6) over y gives

where C_f is the skin friction coefficient defined by in which U_c is the center line velocity.

Substituting Eq.(7) into Eq.(6) yields

Substituting the triple integration,i.e.,into Eq.(8),we have For the stationary turbulence, Therefore, where all variables are normalized by the channel half width and the center line velocity. u′v′ is the Reynolds shear stress produced directly by the ejection (v′ > 0) and sweep (v′ < 0) activities. This equation shows that the skin-friction drag in a turbulent channel flow consists of the laminar drag and the y-weighted integration of u′v′,which is regarded as a streamwise momentum transform in the wall-normal direction. The sweep activities are more significant than the ejection activities in the vicinity of the wall due to the effects of the bursting events, which can create the regions of the high skin-friction and cause the y-weighted integration of u′v′ to be always positive.

2.3 Flow induced by spatially oscillating spanwise Lorentz force

The spatially oscillating Lorentz force f_z defined by Eqs. (3) and (4) is independent of time. The amplitude of the Lorentz force oscillates sinusoidally in the x-direction,and decays exponen- tially along the y-direction (see Fig. 2(a)). When this force is introduced into a static flow field, a generalized Stokes-layer is then created. The distributions of the induced spanwise velocity are shown in Fig. 2(b),where the red areas (deep-colored areas at the front tips) refer to the positive velocity,and the blue areas (deep-colored areas at the back tips) refer to the negative velocity. The distributions of the induced spanwise velocity,when a laminar flow is introduced,are shown in Fig. 2(c). Due to the main flow,the red and blue areas (the deep-colored areas) are overlaid at nλ_x/2 (n ∈ N) to form the inclined shear layers (see Fig. 2(d)). From Fig. 2,we can see that there is no vortex in the induced Stokes-layer.

3 Results and discussion

Within the viscous sublayer (y⁺ < 5),assume that the mean velocity profile is linear. Then, the wall shear stress is

Therefore,the skin-friction drag can be obtained from the measured mean streamwise velocity for a position in the viscous sublayer.

The percentage of the drag reduction rate is simply given by

where (τ_w)i denotes the average wall shear stress with the Lorentz force control,and (τ_wn)ni denotes the average wall shear stress without control.

The calculated drag reduction,as a result of the Lorentz force control,is presented in Fig. 3, where Fig. 3(a) shows the drag reduction variation with the wave numberk_xwhen δ = 0.02 and A = 1.0,and Fig. 3(b) shows the drag reduction variation with the amplitude A when δ = 0.02 andk_x= 3. The results indicate that the largest drag reduction is about 62%,occurring at kx = 3 (see Fig. 3(a)) and about 62% at A = 1.0 (see Fig. 3(b)). Therefore,the skin-friction drag can be reduced significantly by an appropriate combination of the amplitude and the wave number.

Due to the spatially oscillating spanwise Lorentz force when A = 1.0,k_x= 3,and δ = 0.02, the near-wall turbulent flow is modified by an additional induced flow. The evolutions of the streaks and the vortex structures in the near-wall region in the controlling process are illustrated in Fig. 4,where the blue areas (deep-colored areas in the right-side subfigures of Figs. 4(a),4(b), and 4(c),the middle part of the left-side subfigure of Fig. 4(a),and the left-side subfigure of Fig. 4(c)) represent the low-speed streaks,and the red areas (deep-colored areas in the upper and lower parts of the left-side subfigure of Figs. 4(a) and the left-side subfigure of Fig. 4(b)) represent the high-speed streaks. The left column shows the streaky structures. The right column shows the vortex structures when 0 < y⁺ < 100. At t = 200,before actuating,the meandering and irregular streaky structures and the characteristic structures of the longitudinal vortices (e.g.,quasi-streamwise vortices and horseshoes vortices) are exhibited. Subsequently, these intrinsic coherent structures become more and more well-organized. Finally,the flow is maintained at a statistically steady state via the oscillating Lorentz force (see Fig. 4 at t = 340), where the vortex shape is very similar to each other,and all vortices are well organized.

The typical streamwise vortex structures are shown in Fig. 5,where the red area (the lower deep-colored area) and the blue area (the upper deep-colored area) represent,respectively,the positive and negative values of the vorticity,which are denoted by S_P and S_N,respectively. It is shown that the positive vortex is tilted with the negative tilting angle of about −32®,while the negative vortex is titled with the positive tilting angle of about 32®. Each vortex has an averaged streamwise length of 0.5λ_x (about 200 wall units). The intervals of 0.5λ_x and 0.3λ_x in the streamwise and spanwise directions,respectively,overlay with each other with opposite signs at nλ_x/2 (n ∈ N).

The profiles of the Reynolds shear stress are shown in Fig. 6. It is shown that the Reynolds shear stress u′v′ is weakened considerably by the control,especially in the near-wall area.

4 Conclusions

A formulation of the skin-friction drag in the turbulent channel flow related to the Reynolds shear stress is derived. It shows that the turbulent drag consists of the laminar drag and the y-weighted integration of u′v′,which are directly related to the turbulent coherent structures.

The spatially oscillating spanwise Lorentz force with several proper control parameters can control the near-wall turbulence well. The controlled flow is finally stabilized at a statisti- cally steady state via the Lorentz force so that the streamwise vortices can be well organized, whereas few hairpin vortices are found. Consequently,the Reynolds shear stress is weakened considerably in all controlled flow fields,resulting in the drag reduction.