1 Introduction

The wake instability of bluff bodies cannot only induce unexpected vibrations in the bluff bodies, which can potentially damage structures^[1-2], but also cause flow transition thus increasing field noise. It is found that wake instability is developed along with vortex shedding, and is related to the streamwise momentum transfer between the recirculation bubble and the main flow. Just like suppression of instability, reducing the transfer can decrease the form drag^[3]. Also, the instability will cause symmetry breaking of the two-dimensional time-periodic cylinder wake, leading to the three-dimensional transition^[4]. Hence, the control of the wake instability is necessary in the engineering application. The wake instability and control, especially for the cylinder wake, as important research areas have been studied extensively^[5-10]. Related control techniques can be subdivided into passive and active ones, both of which gain much favor. The passive control typically modifies the geometry of the body to achieve the goal. The well-known examples include adding an end plate^[11], a splitter plate^[12], or a secondary small cylinder^[13]. On the contrary, the active control preserves the bluff body and applies energy addition by an actuator^[14] like electromagnetic actuators^[15], dielectric barrier discharge plasma actuators^[16], and synthetic jets^[17]. These traditional control tactics pay more attention on the achievements, but cannot provide physical insights into control mechanism.

Recently, a growing number of researchers have been interested to apply global stability analysis to flow control^[18-22]. Actually, to date global stability analysis has dealt successfully with a wide range of applications arising in aerospace engineering, physiological flows, food processing, and nuclear-reactor safety. Theofilis^[23-24] systematically reviewed the advances in global stability analysis and its applications. The specific methods were reviewed by Sipp et al.^[21] and Camarri^[22], mainly containing two strategies, i.e., the control acting on the base flow and the control acting on the perturbations. Furthermore, the control inputs have two frames: two-dimensional and three-dimensional^[25]. The two-dimensional frame means that the control inputs cannot vary along the spanwise direction. For example, Marquet et al.^[26] took a global stability analysis of the base flow modified by a two-dimensional control, which is placing a small secondary cylinder in the separation shear layer. Their results indicated that this method is beneficial to stabilizing the near wake global instabilities. Hwang et al.^[25] performed a linear stability analysis of the spanwise wavy base flow. The base flow is modified by a threedimensional control named spanwise blowing and suction. The control of perturbations was studied in Refs.[18]-[20].

Though the control tactics inspired by the base flow modification are insightful and rigorous, the Reynolds number in most studies was relatively low. As we know, there are two instability modes, the modes A and B in the cylinder wake transition regime^[27]. The critical Reynolds numbers of modes A and B are Re₂=188.5 and Re′₂=259, respectively, according to the Floquet stability analysis^[28]. Therefore, the three-dimensional instability exists at a higher Reynolds number. Of course, we are sure that the three-dimensional control inputs can suppress the growth of the three-dimensional instability according to Ref.[25]. However, the effects of the two-dimensional control on the three-dimensional instability still remain to be explored. Therefore, it is very necessary for us to investigate the instability of the cylinder wake under a two-dimensional control at a higher Reynolds number. The investigation can also be deemed as the application of global stability analysis in the transition control of the cylinder wake. The electromagnetic actuator will be a good choice in this paper to produce the two-dimensional control inputs due to its good performances in the flow control of conductive fluid (sea water, blood et al.)^{[15, 29-32]}. Generally, bright prospects of electromagnetic flow control in poor conductors are reviewed in Ref.[33].

In the present paper, an open-loop active control on the base flow at the Reynolds number 300 by the electromagnetic actuator will be designed to stabilize the cylinder wake instability, reduce the potential damages (oscillating drag and lift) to the cylinder, and to realize transition delay. The global stability analysis will be done on the modified base flow, to investigate the instability. The relationship between the instability mode and the control factor, along with the physical mechanism of the results will be explored and discussed in detail.

2 Methodology 2.1 Governing equations

The electromagnetic actuator model is shown in Fig. 1. The streamwise Lorentz force is generated from the electromagnetic field excited by the arranged alternately electrodes and poles. Therefore, the steady non-dimensional Lorentz force can be written as follows:

(1)

where j₀ is the current density on electrode surface, and B₀ is the magnetic induction on the pole surface. D is the cylinder diameter and the characteristic length. ρ is the conductive fluid density. U_∞ is the free-stream velocity and the characteristic velocity. j and B are the vectors of current density and magnetic induction nondimensionalized by j₀ and B₀, respectively. N is the control factor or interaction number, representing the strength of the Lorentz force relative to the inertial force of the fluid. The energy addition under active control usually acts on Navier-Stokes equations as a body force. The base flow is governed by the non-dimensional incompressible Navier-Stokes equation (NSE),

(2)

(3)

where u is the velocity field, p is the static pressure, and Re is the Reynolds number. Though pulsation of the streamwise Lorentz force exists along the spanwise direction, we can adjust the width of electrode and pole to drop it. The control inputs can be approximated into the two-dimensional frame. The current density j equals κ(E + u × B), in which κ is the conductivity, and E is the electronic field vector. |u × B|/|E| has the order of 10^-3, therefore, the Lorentz force |f_L| can be approximately simplified as Nexp ()^{[29, 32]}. r is the nondimensional distance normal to the cylinder surface. a is the non-dimensional width of electrode and pole, and a/π is defined as the penetration length^{[15, 29, 32]}. Generally, the penetration length is close to the boundary layer thickness. Therefore, we choose a=0.2 in this paper. The space distributions of the Lorentz force in a spanwise section are presented in Fig. 1. The instability of the modified base flow is studied by solving the non-dimensional linearized Navier-Stokes equations (LNSE),

(4)

(5)

where U=(u, v) is the base flow solutions, u′ is the three-dimensional perturbation velocity, and p′ is the perturbation pressure.

2.2 Global stability analysis

The perturbation velocity u′ can be expressed as^[28]

(6)

and similarly for p′. β is the perturbation wavenumber. Furthermore, the perturbations can be written as follows^[28]:

(7)

In global stability analysis, whether the base flow is steady or time-periodic, L_E in Eq.(4) can be considered as the linear time-periodic operator. Therefore, Eq.(4) satisfies the Floquet type, and its modal solution can be decomposed as^[28]

(8)

where the complex number λ=σ + iω, and σ is the growth rate of perturbations. Of course, if the base flow is steady, the period can be any time interval. After evolving over several periods τ, defining a state transition operator, i.e.,

(9)

(10)

will lead to an eigenvalue problem (EVP),

(11)

in which µ=exp (λτ) is the eigenvalue of M(τ).

If the base flow is periodic with the period T, the global stability analysis is called the Floquet stability analysis, and the eigenvalue µ_f=exp (λT) is called the Floquet multiplier. The magnitude of µ_f dictates the stability of the flow, i.e., |µ_f| > 1 means σ > 0, perturbations will grow, and the flow is unstable, |µ_f| < 1 means σ < 0 and the flow is called Floquet stable. The critical condition occurs at |µ_f|=1, that is neutral stability and Hopf bifurcation. Generally speaking, the essence of the global stability analysis is solving the eigenvalue of the state transition operator in this paper.

2.3 Spectral-element method (SEM)

Both the NSE and LNSE are solved with the highly accurate SEM^[34] in a domain as shown in Fig. 2. The spatial domain is split into 1 144 quadrilateral finite elements, in which both the geometry and solution variables employ 8th-order tensor-product Lagrange polynomial expansions. The polynomial order satisfies the convergence requirements (see Table 1). Gauss-Lobatto Legendre polynomials are adopted as a basis for the numerical solution. The “timestepper’s approach”^[35], containing the Krylov subspace method and the Arnoldi decomposition method, is used to resolve the full complex leading eigenvalue system.

For the base flow solution, a Dirichlet condition is set at Γ_∞(u=1, v=0), and a Neumann condition is set at Γ₀ (∇_nu=0, ∇_nv=0). A no-slip Dirichlet condition is set at Γ_b (u=0, v=0). A high-order Neumann condition^[36] for the normal component along the boundary calculated from the NSE is set for the pressure at all boundaries except Γ₀ where the pressure is set to be zero. For the LNSE solution, the boundary conditions are similar with the conditions of the base flow solution, however, the perturbation velocity u′ is set to be zero at Γ_∞ and Γ_b.

2.4 Convergence and verification tests

Results of convergence test are shown in Table 1, where C_d is the mean drag, and δ represents the relative difference compared with the result of K=10. We can observe that the difference order is less than 10^-4 when K≥6. Therefore, K=8 is a satisfactory polynomial order to guarantee calculation accuracy. Furthermore, the Strouhal numbers of the cylinder wake at different Reynolds numbers are calculated to perform a verification test of the algorithm, and the results are shown in Fig. 3. Values of St agree with the available numerical and experimental results^{[28, 37-38]}. Figure 3 indicates the effectiveness of the numerical algorithm.

3 Results and discussion 3.1 Mode transformation

The base flow is still periodic under the open-loop active control, and the Strouhal number St increases with the growth of the interaction number (see Fig. 4). Results of the EVP are shown in Fig. 5. Profiles at N≤0.8 containing two crests demonstrate that the wake flow is unstable with the mode B. The unstable mode will be transformed from the mode B to the mode A when N increases to 1.0, while the wake flow is Floquet stable at N=1.3.

Figure 6 shows the growth rates of instability with the mode A and the mode B, varying with the interaction number. The perturbation wavenumber of the mode A and the mode B is around 1.70 and 7.70, respectively. Intersections between the two lines and the x-axis, N_A ≈ 1.20 and N_B ≈ 0.98 are the critical interaction number. It means that the further secondary instability still exists when N < N_B, while the further secondary instability is suppressed, however, the secondary instability still exists when N_A < N < N_B. The secondary instability will be restrained when N > N_A. Figure 7 describes the phase diagram of the lift coefficient with the drag coefficient at different interaction numbers. The shrink of the phase diagram indicates that the oscillating amplitude of drag and lift can be reduced with the increase of the interaction number. Moreover, the left shift of the phase diagram suggests the drag reduction can also be achieved. Figure 7 illustrates that the potential damage to the cylinder caused by the wake decreases with the increase of the interaction number. In other words, the streamwise Lorentz force cannot only increase the critical Reynolds number to achieve transition delay, but also reduce the potential damages to the cylinder.

Figure 8 displays the Floquet multiplier modes at N=0.0 and 1.0, presenting the magnitude of perturbation velocities. The perturbation wavelength 2π/β of the mode A and the mode B is 3.93 and 0.82, respectively, according to Fig. 5. Contour plots are at levels 0-0.03. As revealed by the representation pattern, perturbations in the mode B mainly occur in the braid region connecting vortex cores and the area near the cylinder (see Fig. 8(a)), while perturbations in the mode A largely concentrate in the vortex core (see Fig. 8(b)).

3.2 Elliptic instability and hyperbolic instability

Since the physical mechanism of the mode A and the mode B is elliptic instability and hyperbolic instability, respectively^[39-40], the mode transformation is greatly relevant with variations of elliptic and hyperbolic instability regions. To explore the main cause for mode transformation, we define a parameter ζ to describe eccentricity^[41],

(12)

where

(13)

are the strain rate tensor and the rotation rate tensor in the two-dimensional flow field, respectively. is the strain rate magnitude, and is the vorticity magnitude, “tr” denotes the matrix trace, and the superscript “T” refers to transposition. The inviscid growth rate σ_i of the most unstable perturbations is given by^[40]

(14)

Therefore, the fluid deformation plays a major role in the hyperbolic instability, and the fluid rotation makes important contributions to elliptic instability. Actually, the inviscid growth rate of hyperbolic instability is the square root of Okubo-Weiss function^[35], because ||S||² can be written as

(15)

where are the strain rates denoting the compression along the normal direction and the shear along the tangent direction, respectively.

Figure 9 shows evolutions of the hyperbolic instability regions within half a shedding period. These regions are full of possible three-dimensional instability predicted by the inviscid growth rate. The non-dimensional period T_A for N=0.0 is 4.725, and T_B for N=1.0 is 4.348. The phases of the vortices plotted with the solid line (the vorticity is 1) and the dash line (the vorticity is -1) are similar with each other at t_A and t_B. The hyperbolic instability regions in the near wake mainly concentrate in two parts agreeing with Fig. 8(a), i.e., one is the braid region connecting two vortex cores, and the other is the area between the cylinder surface and the primary vortex pair. It is worth mentioning that the “vortex” here represents the relative rotation of particles in the Euler frame. The near wake inviscid growth rate varies more obviously than that near the front half cylinder. The hyperbolic instability regions will increase when the vortex pair is stretched. Comparing with the lateral two figures, we can find that hyperbolic instability regions in the near wake decrease under the active control.

Similarly, Fig. 10 shows evolutions of the elliptic instability regions predicted by the inviscid growth rate. Variations of elliptic instability regions mainly occur in the vortex core, and this conforms to Fig. 8(b). Evolutions of the instability regions in Figs. 9 and 10 illustrate that the elliptic and hyperbolic instabilities are the main cause for the wake transition. The inviscid growth rate increases in the growing vortex core, while decreases in the shedding vortex core. Anyway, like changes of hyperbolic instability regions, the elliptic instability regions in the vortex core also decrease in the modified base flow.

Figures 9 and 10 indicate that both elliptic and hyperbolic instability regions predicted by the inviscid growth rate in the near wake are reduced by the streamwise Lorentz force. Actually, the total growth rate σ_t=σ_i-σ_v contains the inviscid growth rate σ_i, and the viscous decay rate σ_v, which is the effect of viscosity on the growth rate. σ_v has been evaluated in Refs.[30]-[31] and given by

(16)

where γ is the perturbation wavelength equalling 2π/β, and θ is the angle between the wave vector of the most unstable perturbation and the axis of the rotation of the base flow. ν is the kinematic viscosity which is equal to Re^-1 in this paper. Compared with the inviscid growth rate, the viscous decay rate is small scale, because the spanwise perturbation wavelength of the most unstable perturbation is around ≈ 3.696 for the mode A and ≈ 0.816 for the mode B in Fig. 6. Therefore, the main cause for mode transformation is the decrease of instability regions predicted by the inviscid growth rate. However, the viscous damping effects on the total growth rate will increase with the decrease of the inviscid growth rate. Theoretically, σ_v equals σ_i at the critical interaction number, which is N_A in the elliptic flow and N_B in the hyperbolic flow.

3.3 Essential physical mechanism

Although Subsection 3.2 successfully explains the mode transformation, the reason for the decrease of instability regions still bothers us. Therefore, we decide to hunt for the underlying causes. As analyzed above, elliptic and hyperbolic instabilities rely on the fluid deformation and rotation. Physically, the deformation and rotation result from the relative motion of particles. Hence, particle tracing is beneficial for us to find the essential physical mechanism of the mode transformation, though it is very expensive in the direct numerical simulation. Without loss of generality, we put some particles (the black cycles in Fig. 11), into the near wake at t_A and t_B. Path lines record the particle motion trails over a short non-dimensional time interval 0.5. The velocity magnitude and motion direction of particles are transparently seen from the path lines. Admittedly, the length of the path line can be treated as the dynamic pressure or the force acting on the fluid, and its shape can report the fluid motion form. Alternatively, elliptic and hyperbolic flows will be marked at a glance. Remarkable regions, except the region Ⅱ, correspond to the hyperbolic instability regions at the moment t_A and t_B in Fig. 9. The region Ⅱ is marked to be compared with the region Ⅱ. Seeing that the inviscid growth rate largely decreases at the next moment ( and ), the relatively low growth rate regions below the upper vortex core in Fig. 9 are not marked in Fig. 11.

Path lines, especially of the upmost particles illustrate that the flow separation is suppressed to a certain extent under the active control. Both the interaction space between the vortex and the upper flow, and the distance from the vortex center to the cylinder surface increase, in contrast of two vortex center locations O and O′. Consequently, the effects of the cylinder surface on the fluid deformation are lowered, compared with the regions Ⅱ′, Ⅲ′ and Ⅱ, Ⅲ. Incidentally, “vortex” in this section is the pattern of the fluid rotation in the Lagrange frame. Besides, comparing the region Ⅰ with Ⅰ′, we can find lengths of the path lines are shorter under the active control. This means that the fluid deformation and the rotation are hindered so that elliptic and hyperbolic regions decrease in the modified base flow. Ultimately, the suppression of flow separation is the essential physical mechanism of mode transformation.

4 Conclusions

In summary, we numerically investigate the instability of the cylinder wake under the openloop active control by the streamwise Lorentz force, and explore the effects of two-dimensional control inputs on the three-dimensional instability. Results of Floquet stability analysis suggest that the instability mode (or the transition mode) at Re=300 can be transformed from the mode B to the mode A with the interaction number N=1.0, and the wake flow is Floquet stable when N is increased to 1.3. Moreover, Re=300 will be the critical Reynolds number of the mode A with N_A=1.20 and the mode B with N_B=0.98, respectively. Besides, oscillating amplitudes of drag and lift are reduced with the increase in the interaction number. Therefore, transition delay can be achieved, along with the decreasing potential damage to the cylinder from the wake, under the open-loop active control by the streamwise lorentz force. As the mainspring of mode transformation, the decrease of elliptic and hyperbolic instability regions is obtained, because of suppression of flow separation hindering fluid deformation and rotation. Overall, the open-loop active control with the streamwise Lorentz force is beneficial to the wake stabilization and the transition delay.