Control of a sedimenting elliptical particle by electromagnetic forces
CControl of a sedimenting elliptical particle by electromagneticforces
Jianhua Qin , Guodan Dong , and Hui Zhang The State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences,Beijing 100190, China. University of Chinese Academy of Sciences, Beijing 100049, China. Key Laboratory of Transit Physics, Nanjing University of Science and Technology, Nanjing, Jiangsu 210094, China * Corresponding author at: The State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics,Chinese Academy of Sciences, Beijing 100190, China.University of Chinese Academy of Sciences, Beijing 100049, China.Email: [email protected] ** Corresponding author at: Key Laboratory of Transit Physics, Nanjing University of Science and Technology,Nanjing, Jiangsu 210094, ChinaEmail: [email protected]
In this paper, the effectiveness of electromagnetic forces on controlling the motion of a sedimenting elliptical particle isinvestigated using the immersed interface-lattice Boltzmann method (II-LBM), in which a signed distance function isadopted to apply the jump conditions for the II-LBM and to add external electromagnetic forces. First, mechanisms ofelectromagnetic control on suppressing vorticity generation based on the vorticity equation and vortex shedding basedon the streamwise momentum equation are discussed. Then, systematical investigations are performed to quantify andqualify the effects of the electromagnetic control by changing the electromagnetic strength, the initial orientation angleof the elliptical particle, and the density ratio of the particle to the fluid. To demonstrate the control effect of differentcases, comparisons of vorticity fields, particle trajectories, orientation angles, and energy transfers of the particles arepresented. Results show that the rotational motion of the particle can be well controlled by appropriate magnitudesof electromagnetic forces. In a relatively high solid to fluid density ratio case where vortex shedding appears, thesedimentation speed can increase nearly 40% and the motion of the particle turns into a steady descent motion oncean appropriate magnitude of the electromagnetic force is applied. When the magnitude of the electromagnetic force isexcessive, the particle will deviate from the center of the side walls. In addition, the controlling approach is shown tobe robust for various initial orientation angles and solid to fluid density ratios.
Keywords: rotational control, electromagnetic control, energy transfer, immersed interface-lattice Boltzmannmethod.
For flow around a blunt body, when the Reynolds number (the ratio of inertial forces to viscous forces) grows, theflow will be transformed from a steady state to an unsteady state and vortex shedding appears. Vortex shedding isbeneficial to harvest fluidic energies using piezoelectric materials. However, in most cases, the shedding vortices willbring bad effects like additional drag force for a travelling ship, aerodynamic noise for an airplane, undesired rotationof a missile, and structural damage for a bridge etc. Hence, it is imperative to study the suppression of vortex sheddingto lower the damage. In real life, the motions of airplanes, submarines and parachutes all involve multiple degreesof freedom, including the translational motion and the rotational motion, in which the rotational motion exists whenvortex shedding appears. Specifically, freely falling or rising of bodies are very common phenomena occurred insituations like falling leaves, floating bubbles and ball games etc. When a body sediments or rises in the fluid, many1 a r X i v : . [ phy s i c s . f l u - dyn ] F e b ossible motions can happen if the motion of the body is not controlled, including descending along a straight line,fluttering, and tumbling [1–3]. Therefore, it is meaningful to investigate the rotational control of a sedimenting body.Traditionally, the control of the rotational motion of a body is relied on imposing additional torques [4,5]. In this work,a novel idea of controlling the rotational motion of a rigid body in the fluid by the electromagnetic force is proposedand validated through numerical simulations.The electromagnetic control can be originated from the guess of Gialitis and Lielausis [6]. They put forward theidea that the electromagnetic force can change the structure of the boundary layer. The electromagnetic force, i.e. theLorentz force, is generated from staggered electrodes and magnets and is parallel to the flow direction [7–10]. Themagnitude of the electromagnetic force decreases exponentially as the penetration depth of the electromagnetic forceincreases. To optimize the control efficiency, Weier and Gerbeth [10] compared the control results of periodic elec-tromagnetic forces, static electromagnetic forces, and the traditional blowing control approach. Chen and Aubry [11]proposed to produce electromagnetic forces related to the azimuthal angle outside the cylinder, and validated that theapproach is able to suppress vortex shedding and reduce the drag force. Dousset and Poth´erat [12] investigated flowstructures for various Reynolds numbers and Hartmann numbers (the ratio of electromagnetic forces to viscous forces)under the effect of axial electromagnetic forces. Singhah and Sinhamahapatra [13] and Chatterjee et al. [14] investi-gated the influence of the Hartmann number for the control effect of flow around a blunt body. Zhang et al. [15–17]numerically studied the control of wake structures of a static circular cylinder or a circular cylinder undergoing trans-lational motions by different ways of imposing electromagnetic forces. Fisher et al. [18] presented the control of thehydraulic jump in a liquid channel flow with electromagnetic forces by experiments. The above mentioned electro-magnetic flow control approaches have only been applied to control ”constrained” boundaries such as static bodies orvortex-induced vibration of a spring mounted cylinder undergoing translational motions. To the best of our knowl-edge, research about the effectiveness of electromagnetic forces on controlling a freely moving body in the fluid thatinvolves the rotational motion is presently lacking.Due to the development of numerical methods and the improvement of experimental conditions, meticulous in-vestigations about bodies freely rising or falling in fluids become popular. On the numerical simulation domain, toavoid time-consurming mesh reconstruction and mesh distortion, the non-body fitted mesh methods are superior insimulating large amplitude motions like falling or rising of a body in the fluid. The immersed boundary method isa representative of these non-body fitted mesh methods [19–21]. Compared to the classical immersed boundary (IB)method [19], the direct forcing IB method [22] has an advantage in simulating rigid body dynamics because larger timesteps can be used. As mentioned before, the electromagnetic force is related to the penetration depth into the fluid.This requires the calculation of the distance between a fluid point and its orthogonal projection on the body surface.However, the standard direct forcing IB method does not involve the solution to distance between a fluid point andthe interface. The immersed interface-lattice Boltzmann method (II-LBM) proposed by Qin et al. [23] uses a signeddistance function to track the interface so that the distance between a fluid point and its orthogonal projection on theinterface can be calculated. The II-LBM is better than the immersed boundary-lattice Boltzmann method in volumeconservation and higher order of accuracy. For these reasons, this study uses the II-LBM to study the control of therotational motion of a sedimenting body via the azimuthal electromagnetic force [11].Particles of nonspherical shapes are commonly encountered in the nature [24], the industry [25] and scientificstudies [26–28]. Many interesting phenomena can appear when the fluid meets nonspherical particles. Feng et al. [29]reported that when settling between two parallel walls, an elliptical particle will translate itself to the center of wallsbecause of wall effects and orient its major axis to be orthogonal to gravity direction with various initial configura-tions. Huang et al. [30] showed that a settling ellipse will turn its major axis to the vertical direction for sufficientlysmall Reynolds numbers. Xia et al. [31] presented that fascinating modes including oscillating, tumbling, horizontal,vertical, and inclined sedimention could appear for an elliptical particle settling in narrow channels. By studyding thesedimentation of an ellipsoidal particle in circular and squares tubes, Huang et al. [32] showed that the geometry ofthe tube and the moment of inertia of the particle are significant to the sedimentation mode. Zhao et al. [33] demon-strated that preferential orientations of the nonspherical particles affect the rotation of aspherical particles. As a firststep towards understanding the controlling effectiveness, this paper studies the electromagnetic control of an ellipticalcylinder sedimenting in a relatively wide channel to avoid much more complicated phenomena that may appear fornarrow channels [31]. Particularly, our study is based on the sedimentation of an elliptical particle studied by Xia etal. [31], in which the aspect ratio of the channel width to the major axis is equal to four.2 Numerical method
In this study, we use the immersed interface-lattice Boltzmann method (II-LBM) [23] to simulate the sedimentationof an elliptical particle in the incompressible Newtonian fluid, and the azimuthal electromagnetic force to controlthe sedimenting process. This section introduces the numerical implementations of the II-LBM and the azimuthalelectromagnetic force.
The lattice Boltzmann equations for the incompressible fluid flow using the multi-relaxation-time (MRT) operatorscan be written as [23] ∂ f i ∂ t + e i · ∇ f i = − M − S d (cid:16) m i ( x , t ) − m ( eq ) i ( x , t ) (cid:17) + c ω i e i · g ( x , t ) + c ω i e i · g m ( x , t ) , (1)in which f i are the distribution functions, e i are directional velocity vectors. The first term on the right hand side (RHS)of Eq. (1) deals with the local collision process, in which M is the transformation matrix for the multi-relaxation-time(MRT) model [34], S d is the diagonal matrix. The space of distribution functions ( f = { f i } ) and the space of themoments of the distribution functions ( m = { m i } ) are related by m = M f . Similarly, we have m ( eq ) = M f ( eq ) , inwhich f ( eq ) and m ( eq ) are spaces of the equilibrium distribution functions and moments of the equilibrium distributionfunctions, respectively. The second term on the RHS of Eq. (1) includes the Eulerian force density g representing theadditional body forces imposed on the fluid by the immersed boundary [35]. The last term on the RHS of Eq. (1)accounts for the effect of the electromagnetic force on the fluid, in which g m is the electromagnetic force density.The interactions between the fluid and the immersed boundary are formulated by g ( x , t ) = (cid:90) Γ G ( l , t ) δ ( x − X ) d l , (2) G ( l , t ) = G ( X , U , l , t ) , (3) ∂ X ( l , t ) ∂ t = U ( l , t ) = (cid:90) Ω u ( x , t ) δ ( x − X ( l , t )) d x , (4)in which Γ and Ω are the immersed boundary and the fluid domain, respectively, l is the curvilinear coordinate thatparameterizes the interface, G ( l , t ) is the Lagrangian force density, X ( l , t ) and U ( l , t ) are the position and velocity ofthe boundary, and G is the functional that determines the interfacial force from the deformations and/or velocities ofthe immersed boundary. δ ( x − X ( l , t )) is the Dirac delta function used to spread the Lagrangian forces to the Eulerianforce and integrate the Eulerian velocity to obtain the Lagrangian velocity.The difference between the II-LBM and the IB-LBM is that the former formulation considers the jump condi-tions of the distribution functions so that higher order of accuracy and substantial better volume conservation can beachieved [23]. The jump conditions can be written as (cid:74) f i (cid:75) = ω i c G ( l , t ) · n | d X d l | , (5)in which ω i is the weighting function in each direction, n is the unit normal vector to the interface, c s is the soundspeed in lattice units, and | d X d l | = The lattice Boltzmann equation (LBE) is discretized via finite differences in both time and space, resulting in thefollowing discretized LBE f i ( x , t + ∆ t ) − f i ( x , t ) ∆ t + f i ( x , t ) − f i ( x − ∆ x i , t ) − (cid:74) f i ( x , t ) (cid:75) ∆ t = − M − S d (cid:16) m i ( x − ∆ x i , t ) − m ( eq ) i ( x − ∆ x i , t ) (cid:17) + c ω i e i · g || ( x , t ) + c ω i e i · g m ( x , t ) , (6)3n which g || ( x , t ) is the tangential part of the Eulerian force, and ∆ x i = e i ∆ t is the grid spacing in the e i direction.Here, we use the jump conditions to deal with the normal part of the Eulerian force [23],For the MRT D2Q9 model (i.e. i = , , , ..., ω i = , i = , , ≤ i ≤ , , otherwise , (7)and the directional velocity vectors e i are defined as ( , ) , ( ± , ) , ( , ± ) , ( ± , ± ) . Then the macroscopic density,pressure and velocity can be obtained via ρ f = ∑ i f i , p = ρ f and u = ρ f ∑ i f i e i . The transformation matrix anddiagonal matrix for the D2Q9 model are M = − − − − − − − − − − − − − − − − − −
10 0 − − −
10 1 − − − − , (8) and S d = diag ( s , s , ..., s ) , in which s = s = s , s = . ∆ t , s = . ∆ t , s = s = . ∆ t , and s = s = λ . When solvingthe equations, we use ∆ x = ∆ t = k th Lagrangian boundary point canbe obtained via a direct-focing IB approach as G k ( t ) = ρ f U D k ( t ) − U ∗ k ( t ) ∆ t , (9)where U D k ( t ) is the desired Lagrangian velocity, and U ∗ k ( t ) is the intermediate Lagrangian velocity obtained by U ∗ k ( t ) = ∑ x u ∗ ( x , t ) δ h ( x − X k ( t )) ∆ x , (10)where u ∗ ( x , t ) is the fluid velocity obtained by solving the fluid flow equations without considering the immersedboundary, and X k is the Lagrangian coordinate. Here, δ h ( x − X ) is a regularized delta function, which we define by δ h ( x − X ) = ( ∆ x ) Φ (cid:18) x − X ∆ x (cid:19) Φ (cid:18) y − Y ∆ x (cid:19) Φ (cid:18) z − Z ∆ x (cid:19) , (11)in which Φ ( r ) is a one-dimensional kernel function. We use Peskin’s four-point kernel function [19], Φ ( r ) = (cid:16) − | r | + (cid:112) + | r | − r (cid:17) , | r | ≤ , (cid:16) − | r | − (cid:112) − + | r | − r (cid:17) , ≤ | r | ≤ , , | r | > . (12)The Eulerian force is obtained from the distribution of the Lagrangian force via the same δ h g ( x , t ) = ∑ k G k ( t ) δ h ( x − X k ( t )) ∆ X , (13)in which ∆ X is the distance between two adjacent Lagrangian points. In this investigation, ∆ X = ∆ x is chosen. Then g || ( x , t ) can be calculated via g || ( x , t ) = ∑ k G || k ( t ) δ h ( x − X k ( t )) ∆ X , (14)in which G || k = G k − ( G k · n ) n is the tangential part of the Lagrangian force, and n is the outward normal directionwith respect to the interface. Then the only unknown in the discretized II-LBM equation is (cid:74) f i (cid:75) . To calculate the jump4onditions, n and G ( l , t ) in Eq. (5) have to be determined. The calculations of the two variables will be introducedalong with the identification of the interface later in Sec. 3.1.3.In this study, the translational and rotational motions of an elliptical particle are simulated. The discretized equa-tions for the motion of the elliptical particle are m a ( t + ∆ t ) = − ∑ X k G k ∆ X + ( ρ s − ρ f ) V s g e + ρ f V s U ( t ) − U ( t − ∆ t ) ∆ t , (15) I w ( t + ∆ t ) = − ∑ X k ( X k − X c ) × G k ∆ X + ρ f ρ s I W ( t ) − W ( t − ∆ t ) d t , (16)in which I , a , w , and W are moment of inertia, linear acceleration, angular acceleration and angular velocity of therigid body, respectively. The first terms on the RHS of Eqs. (15) and (16) represent the force and torque exerted on therigid body by the ambient fluid, respectively. The last terms in Eqs. (15) and (16) account for the motion of the fluidinside the rigid body [36]. Then the velocity of the mass center U c and angular velocity of the body W are updatedvia U c ( t + ∆ t ) = U c ( t ) + ∆ t a ( t + ∆ t ) , (17) W ( t + ∆ t ) = W ( t ) + ∆ t w ( t + ∆ t ) . (18)Because the numerical simulations performed in this paper are in two spatial dimensions, we only need to track theorientation angle in the z direction and the position of the center of mass of the rigid body. A forward Euler method isused to calculate the position of the center of mass and the orientation angle of the body X c ( t + ∆ t ) = X c ( t ) + U c ( t ) ∆ t + ( ∆ t ) a ( t + ∆ t ) , (19) θ ( t + ∆ t ) = θ ( t ) + W z ( t ) ∆ t + ( ∆ t ) w z ( t + ∆ t ) , (20)in which θ , W z , and w z are the orientation angle, z component of the angular velocity, and the angular acceleration ofthe rigid body, respectively. The signed distance function is used to represent the fluid-structure interface [23, 37, 38] and calculate n and G ( l , t ) in Eq. (5). Here, a brief introduction of the numerical implementation is given, the detailed implementation canbe referred to Li and Ito [37] and Qin et al. [23]. For an elliptical particle centered at ( x , y ) and immersed in abackground fluid, a signed distance function ϕ ( x ) can be defined as ϕ ( x ) = (cid:114) [( x − x ) cos θ + ( y − y ) sin θ ] a + [( x − x ) sin θ − ( y − y ) cos θ ] b − , (21)in which a and b are semi-major and semi-minor axes, respectively. Here, θ is the orientation angle of the ellipticalparticle. With the definition of ϕ ( x ) , the boundary of the particle, the exterior and interior fluid regions are representedby ϕ ( x ) = ϕ ( x ) > ϕ ( x ) <
0, respectively. Using the signed distance function, the outward unit normal vectorwith respect to the interface can be obtained via n ( x ) = ∇ ϕ | ∇ ϕ | , (22)where ∇ ϕ ( x ) = ( ϕ x ( x ) , ϕ y ( x )) is approximated using second order centered differences.If φ ( x ) φ ( x − ∆ x i ) < (cid:74) f i (cid:75) needs to be evaluated and implemented in Eq. (6). Otherwise, (cid:74) f i (cid:75) should be zero.The jump conditions for the distribution functions are accounted for through correction terms that are evaluated onthe Eulerian grid points. Consequently, the boundary point X ∗ corresponds to a particular Eulerian point x must bedetermined in order to obtain G ( l , t ) in Eq. (5). By using the same signed distance function, the orthogonal projectionof a point x close to the interface is X ∗ = x + r w ∇ ϕ ( x ) , (23)in which r w is the distance between x and X ∗ and is obtained by solving a quadratic equation obtained from thetruncated Taylor expansion [23]. Then G ( l , t ) on X ∗ can be calculated by using the direct forcing IB method [23, 39,40]. 5 .2 The azimuthal electromagnetic force In our present study, the azimuthal electromagnetic force proposed by Chen and Aubry [11] is applied on a certainrange of fluids exterior to the rigid body. As shown in Fig. 1, the electromagnetic force is generated from staggeredelectrodes and magnets and is parallel to the flow direction [10]. For rigid body simulations, a big difference of theII-LBM compared to conventional IB-LBMs is that the exterior and interior fluid regions of the interface are describedby a signed distance function for the II-LBM. Therefore, the electromagnetic force is taken into account by using thesame singed distance function used in the II-LBM.Figure 1. Schematic of electromagnetic forces generated from staggered electrodes and magnets.Fig. 2 is the schematic of the electromagnetic force applied on the exterior fluid of an elliptical particle. The semi-major and semi-minor axes of the particle are defined as a and b , respectively. The magnitude of the electromagneticforce on a fluid position M near the surface of the elliptical particle is g m = e − α m r w / a β m , (24)in which r w is the distance between M and the corresponding projection point on the surface and is calculated via thesigned distance function ϕ ( x ) [23]. α m is a constant showing the electromagnetic penetration into the flow which canbe controlled by electrode spacing [11]. In our present study, α m is chosen as 5. β m = N m ν ρ f ( a ) , in which N m is thedimensionless strength of the electromagnetic force.The setup of the problem studied within this paper is shown in Fig. 2. The electromagnetic force g m is parallelto the tangential direction of the interface. By using the signed distance function, the normal direction n = ( n x , n y ) of a fluid point M to the interface in two spatial dimensions can be obtained (Eq. (22)). Then the direction of theelectromagnetic force is perpendicular to n and can be written as τ = ( − n y , n x ) . (25)Finally, the density of the electromagnetic force can be calculated via g m = τ · g m . (26)It should be mentioned that the electromagnetic force is applied in a fluid region of0 ≤ ϕ ( x , t ) ≤ , (27)in which ϕ ( x , t ) is the level set function used in the immersed-interface method. Moreover, as shown in Fig. 2, theelectromagnetic force only exists in a portion of the above fluid region with Φ = π .It should be mentioned that the key idea of the electromagnetic force control is to suppress vortex generation andthe adverse pressure gradient in order to reduce the periodic vortex shedding in the wake of the elliptical particle.Therefore, the direction of electromagnetic force should be in the opposite direction of the adverse pressure gradient.This requires ( − n y , n x ) · ( − sin θ , cos θ ) ≥ , (28)in which θ is instantaneous orientation angle of the elliptical particle. Here, ( − sin θ , cos θ ) is the direction of theminor axis of the elliptical particle and the intersection angle between it and the gravity should not be less than π .6 a) (b) Figure 2. Schematic of electromagnetic control of a sedimenting elliptical particle. (a) Whole domain with the particleinitially centered at ( , ) ; (b) zoom in near the region of the particle during sedimentation. Electromagnetic control of the sedimentation of an elliptical particle is studied in this paper. Without the externalelectromagnetic force, the fluid-structure interaction (FSI) problem was proposed by Xia et al. [31]. As shown inFig. 2(a), the computational domain is chosen to be [ − a , a ] × [ − a , a ] and the elliptical particle initially centeredat ( , ) with an initial orientation angle of π . All boundaries of the outer domain are solid walls with Dirichletboundary conditions. As a result of the gravity force, the elliptical particle sediments downwards. The viscous forceand pressure on the particle surface drive it to move in the horizontal direction. Moreover, rotation of the particleis generated because of fluid torque on the particle. The density ratio between the elliptical particle and the fluid isrepresented by ρ ∗ and is chosen as 1.5. The kinematic viscosity is ˜ ν = .
01 m / s, and the gravity acceleration is˜ g e = ( , − . / s ) , in which the ‘tilde’ symbol represents the physical units. The semi-major and semi-minor axesof the elliptical particle are set to ˜ a = . × − m and ˜ b = ˜ a , respectively. The II-LBM used in this study hasbeen previously validated to be able to study this FSI problem without external electromagnetic forces [23].To check the mesh size on the study of electromagnetic control of the particle sedimentation, simulation results for N m = a = ∆ x , 40 ∆ x and 50 ∆ x are compared. Fig. 3 shows the trajectories ofthe elliptical particle and the relationship between the orientation angle and the displacement of the elliptical particlefor different meshes. The two curves shown in Fig. 3 are both periodic and are very close for three different meshes.However, the result obtained by 2 a = ∆ x shows larger differences between the latter two meshes. Considering theaccuracy of result and the computational cost, we choose 2 a = ∆ x to study the electromagnetic control of the particlesedimentation. Without the electromagnetic force, the simple sedimentation problem will result in an oscillating motion of the parti-cle [31]. The problem setup is the same as in Sec. 3.3 with ρ ∗ = . N m is set to zero here. As shown laterin Figs. 9(a) and 9(b), both the trajectory and the time response of the orientation angle of the particle are periodic. Thedensity ratio is within a relatively high range (( ρ ∗ − ) ∼ O ( )) where the convection of the fluid plays an importantrole in the terminal velocity ( U t ) of the particle [31]. To define the feature of the flow, the Reynolds number and thecharacteristic frequency of the flow should be specified. The Reynolds number is defined via Re = U t a ν . (29)7 a) (b) Figure 3. (a) Trajectories of the elliptical particle mass center, and (b) the orientation angle of the particle against thedisplacement of the particle mass center in the y direction for three meshes with N m = ρ ∗ = . Re ∼ ( ρ ∗ − ) . . In our present study, the resulting Re calculated by the II-LBM for ρ ∗ = . u x for a fixed position relative to the particle mass centerin the y direction in the wake, in which u x is the fluid velocity in the x direction. Here, the position is chosen as ( , X c , y ( t ) + a ) (point C in Fig. 5). u x for point C against the dimensionless time ¯ t is presented in Fig. 4(a), in which¯ t is the dimensionless time and is defined as t = t ( a ) / ν . (30)The dimensionless characteristic frequency can be defined as ¯ f = f ( a ) ν , where f is the frequency of vortex shedding.As shown in Fig. 4(b), the peak of the power spectral density (PSD) corresponds to ¯ f = u x of the characteristic point C against the dimensionless time, and (b) power spectral density of thefrequency for u x of point C.Four snapshots of vorticity fields and force distributions on the surface of the elliptical particle are shown in Fig. 5,in which the dimensionless vorticity is defined as ω = ων / ( a ) , (31)where ω is the vorticity, and the dimensionless Lagrangian force is written as G = G ν ρ f / ( a ) . One can see thatvortex shedding causes uneven vorticities attached to the surface of the particle and uneven forces on the left and rightsides of the surface. The shedding vortices are generated from flow separation when the boundary layer has travelledin an adverse pressure gradient for a long distance to result in a relative zero velocity on the surface. Because of theforces and resulting torques, the particle will rotate during the sedimentation process.8igure 5. Snapshots of vorticity fields (upper) and Lagrangian forces on the particle boundary (lower) for a sedimentingellpitical particle. From left to right, the four typical snapshots correspond to ¯ t , ¯ t , ¯ t , and ¯ t in Fig. 9(b), respectively. Electromagnetic control of a sedimenting elliptical particle is studied in this section. The rotation of the particle duringsedimentation is caused by vortex shedding. Let us explain why the electromagnetic force introduced in Sec. 3.2can control vortex shedding so that the rotational motion of the particle can be controlled. The mechanism behindthe electromagnetic control of vortex shedding is that the vorticity generation and the adverse pressure gradient onthe boundary layer of the particle can both be suppressed by this kind of electromagnetic forces. Fig. 6 shows theschematic of the boundary layer for uniform flow around an elliptical cylinder.Figure 6. Schematic of the boundary layer controlled by the electromagnetic force. The blue and red points representthe zero pressure gradient point and the stagnation point, respectively.The vorticity equation can be written asD ω D t = ( ω · ∇ ) u − ω ( ∇ · u ) + ρ ∇ ρ f × ∇ p + ∇ × ∇ · τ ρ f + ∇ × g m ρ f , (32)in which the first term on the right hand side represents vortex stretching and should be zero in two spatial dimensions.9or an incompressible fluid, the second and third terms on the right hand side of Eq. (32) are also zero. The viscousforce on the boundary layer which is generated by the motion of the particle obeys the Newton’s viscous law and isinversely proportional to d u d r w . Therefore, the direction of the viscous force is in contrast to the electromagnetic force.For this reason, the vorticity generation on the boundary layer in front of the separation point (the red point in Fig. 6)can be suppressed by the electromagnetic force. It should be noted that when the electromagnetic force is strongerthan the viscous force, the magnetic induced vorticity becomes the dominant vorticity in the fluid. It can be concludedthat appropriate electromagnetic forces can suppress vorticity generation to attenuate the energy so that the rotationalmotion of the particle can be suppressed.Because vortex shedding is caused by the adverse pressure gradient, a suppression of the adverse pressure gradientleads to a suppression of vortex shedding. Within the boundary layer, the streamwise momentum equation can beapproximately stated as u S ∂ u S ∂ s = − ρ f d p d s + ν ∂ u S ∂ n + g m , (33)in which u S is the streamwise velocity, and s and n c denote streamwise and normal coordinates. Without the effect ofthe electromagnetic force, a strong enough adverse pressure gradient (i.e. d p d s >
0) will cause u S to decrease along s andgoes to zero, resulting in the occurrence of flow separation. Due to the fact that the directions of the electromagneticforce and the adverse pressure gradient are opposite to each other, the electromagnetic force will suppress the adversepressure gradient. Fig. 7 shows the vorticity field of a sedimenting elliptical particle under different magnitudes of electromagnetic forcesat ¯ t =
1. As the magnitude of the electromagnetic force increases, the instantaneous displacement of the particle masscenter in the y direction ( y a ) are -32.04, -35.6, -37.44, -39.12, -42.28 and -43.60, respectively. This indicates that theexternal electromagnetic force accelerates the sedimentation. In the range of 0 ≤ N m ≤ . N m = N m = N m = ≤ N m ≤ N m = N m . A detailed comparison of the trajectories of the sedimentation for different N m isshown in Fig. 9(a). Without the electromagnetic force (i.e. N m = N m = x direction becomes smaller. As theelectromagnetic force increases, the oscillation amplitude in the x direction becomes less and less until N m = N m = − ≤ y / a ≤
0, the elliptical particle oscillates with a small amplitude in the x direction.Subsequently, the displacement in the x direction is almost zero.Interestingly, in the case of N m = N m = t = p = p − p ν / D . (34)Here p is the initial pressure of the fluid. When the rotation of the particle caused by the initial orientation angle endsfor N m = ≤ N m ≤ θ against time is periodic. As the electromagnetic forceincreases, the oscillation amplitude of the orientation angle decreases. The specific case of N m = N m >
0, the oscillation magnitudeof θ decreases with the increase of the sedimenting distance. For N m ≥ T ) is defined asKE T = KE x + KE y + KE R , (35)in which KE x and KE y are kinetic energies in the x and y directions, respectively. The kinetic energies are calculatedvia KE x = . ρ ∗ V s U , x a ν , (36)KE y = . ρ ∗ V s U , y a ν , (37)in which U c , x and U c , y are velocities of the particle mass center in the x and y directions, respectively. KE R is therotational energy defined by KE R = . I zz W z a ν . (38)When sedimenting, the gravitational potential energy of the elliptical particle becomes smaller as the mass centerof the particle is lower. Therefore, the y displacement of particle center can be seen as equivalent to the potential energyof the particle. The gravitational potential energy is converted into the kinetic energy of the particle and the kineticenergy of the fluid. Fig. 11(a) shows the relationship between the y displacement of particle center and the energies11igure 8. Sedimentation processes for the elliptical particles under the the effect of different magnitudes of electro-magnetic forces for ρ ∗ = . − ≤ y / a ≤
0, the total energy of the particle is larger without the electromagnetic force than N m = . N m = . N m = . N m = x direction and rotation energy become negligible, and the total energy of the particle is almostthe same as the kinetic energy in the y direction. The increase of the total energy and kinetic energy in the y directionwith the electromagnetic force indicates that the electromagnetic force is useful to reduce the energy loss duringsedimentation. Therefore, the sedimentation distance increases when N m increases (see Fig. 7). When building asedimenting device, it is also possible to accelerate the sedimentation of the device by adding the electromagnetic force.From the evolution of the sedimentation velocity shown in Fig. 11(b), the periodic oscillation has been suppressed withthe electromagnetic force. Moreover, the Reynolds number defined by Eq. (29) is increased from 32.7 for N m = N m = Anderson et al. [2] reported that the initial orientation angle will affect the dynamics of a sedimenting particle. There-fore, it is meaningful to study the effect of the initial orientation angle to the electromagnetic force control of theparticle. Here, the density ratio is set to 1.5 and the strength of the electromagnetic force is chosen as N m = [ , π ] with an interval of π . Fig. 12 shows the vorticity field at ¯ t = θ =
0, the particle almost moves in a straight line when y / a > −
30. Subsequently, the particle center deviates thecenter of the domain in the x direction with a small deviation amplitude. For θ >
0, including the nearly vertical case,the particle trajectories all show small oscillations around the x direction ( | x / a | < . a) (b) Figure 9. (a) Trajectories of the particle mass center during sedimentation, and (b) Orientation angle against thedimensionless time for different magnitudes of electromagnetic forces.Figure 10. Vorticity and pressure fields for N m = N m = t = . The Reynolds number is determined by the density ratio when the viscosity of the fluid and the geometry of theparticle remain unchanged. Three different density ratios of ρ ∗ = 1.1, 1.3 and 2 are chosen to check the robustness ofthe controlling approach. Fig. 14 presents the comparisons between the sedimentation processes for the three densityratios with and without the electromagnetic control. For ρ ∗ = 1.3 and 2.0 in the pure hydrodynamic case, periodicoscillations of the particle trajectories are similar to that of ρ ∗ =1.5. The electromagnetic force is shown to be effectivein controlling the rotational motions of the particles in these two density ratios. When the density ratio decreases to1.1, vortex shedding will not appear even without the electromagnetic force and the motion of the elliptical particletends to a steady descent [31]. Fig. 15 shows the instaneous vorticity fields of ρ ∗ = . ρ ∗ = . N m = x center of the domain.Therefore, the strength of the electromagnetic force needed for best control of the sedimentation process is one orderof magnitude smaller for ρ ∗ = . ρ ∗ = .
5. 13 a) (b)
Figure 11. (a) Comparison of particle energies for N m = N m = . ρ ∗ = .
5, and (b) Velocity of theparticle in the y direction against the dimensionless time for different magnitudes of electromagnetic forces.Figure 12. Vorticity fields at ¯ t = ρ ∗ = . N m = ρ ∗ = .
1. When 0 ≤ N m ≤ N m = x direction appears because the electromagnetic force is able to balance the pressure differences on the leftand right side of the particle even when the orientation angle is small. For this reason, to make the elliptical particlesediments stably, the electromagnetic force should also not be too excessive. It should be mentioned that the particledeviates towards the left wall instead of the right wall as shown when ρ ∗ = .
5. The difference is caused by the lowdensity ratio so that the boundary layer forms slowly even for N m = y / a = −
6, the total energy is almost equal to the kinetic energy in the y direction for all thecases. Because of the initial orientation angle, the elliptical particle will rotate to generate additional vorticities. Dueto the suppression of vorticity generation by the electromagnetic force, the total energy for the N m = N m = N m = N m = ρ ∗ = .
5, in which the total energy for the N m = . N m = ρ ∗ = . y directioncauses the particle to sediment slower for the N m = N m = a) (b) Figure 13. (a) Trajectories of particle center for different initial orientaton angles, and (b) Orientation angle againstthe dimensionless time for different initial orientation angles with ρ ∗ = . N m = ρ ∗ = . This paper has investigated the electromagnetic control of a sedimenting elliptical particle by using the immersedinterface-lattice Boltzmann method. Two main mechanisms that contribute to the electromagnetic control of the parti-cle motion are presented, i.e. suppressing vorticity generation on the boundary layer and reducing shedding sheddingvortices. The former mechanism is demonstrated by the vorticity equation, meanwhile the later one is explained bythe streamwise momentum equation. We consider various strengths of electromagnetic forces to control a sedimentingparticle of density ratio ( ρ ∗ ) equalling to 1.5. Without the electromagnetic force, vortex shedding behind the particleappears and the trajectory of the particle tends to be periodic. By adding electromagnetic forces, vortex shedding canbe suppressed and the elliptical particles sediment more stably. Also, the results show that particle motions are affectedby electromagnetic forces, including less rotations and smaller oscillation amplitudes in the x direction. As the strengthof the electromagnetic force increases, obvious magnetic induced vorticities appear. When the electromagnetic forceis excessive, the particle center will deviate from the center of the computational domain in the x direction. Becauseless potential energy is converted into the rotational energy and the kinetic energy in the x direction, the terminal ve-locity of the elliptical particle increased about 40% with the control of the electromagnetic force. Using N m = ρ ∗ = .
5, we change the initial orientation angles of the elliptical particle, in which N m is the dimensionless electro-magnetic force strength. We show that the electromagnetic force can effectively control the sedimentation for various15igure 15. Vorticity fields at ¯ t = ρ ∗ = . (a) (b) Figure 16. (a) Particle trajectories for different magnitudes of electromagnetic forces, and (b) orientation angle againstthe dimensionless time for different magnitudes of electromagnetic forces when ρ ∗ = . ρ ∗ changes to 1.1, the electromagnetic force can decrease the oscillation amplitude of the particle in the x direction. Like in the ρ ∗ = . ρ ∗ = . ρ ∗ = . The data that support the findings of this study are available from the corresponding author upon reasonable request.
This work is partially supported by NSFC Basic Science Center Program for “Multiscale Problems in NonlinearMechanics” (NO. 11988102). The authors would like to thank Dr. Xiaolei Yang for many discussions. The authorsalso thank the reviewers for the comments to improve this paper.16igure 17. Comparison of particle energies for N m =
0, 3e3 and 6e3 when ρ ∗ = . References [1] Patricia Ern, Fr´ed´eric Risso, David Fabre, and Jacques Magnaudet. Wake-induced oscillatory paths of bodiesfreely rising or falling in fluids.
Annual Review of Fluid Mechanics , 44:97–121, 2012.[2] A Andersen, U Pesavento, and Z Jane Wang. Unsteady aerodynamics of fluttering and tumbling plates.
Journalof Fluid Mechanics , 541:65–90, 2005.[3] Mattia Gazzola, Philippe Chatelain, Wim M Van Rees, and Petros Koumoutsakos. Simulations of singleand multiple swimmers with non-divergence free deforming geometries.
Journal of Computational Physics ,230(19):7093–7114, 2011.[4] Nalin A Chaturvedi, Amit K Sanyal, and N Harris McClamroch. Rigid-body attitude control.
IEEE controlsystems magazine , 31(3):30–51, 2011.[5] Firdaus E Udwadia and Aaron D Schutte. A unified approach to rigid body rotational dynamics and control.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences , 468(2138):395–414,2012.[6] A Gailitis and O Lielausis. On the possibility to reduce the hydrodynamic drag of a plate in an electrolyte.
Appl.Magnetohydrodynamics, Rep. Inst. Phys. Riga , 13:143–146, 1961.[7] A Tsinober. MHD flow drag reduction.
Viscous Drag Reduction in Boundary Layers , 123:327–49, 1990.[8] Catherine H Crawford and George Em Karniadakis. Reynolds stress analysis of emhd-controlled wall turbulence.Part I. Streamwise forcing.
Physics of Fluids , 9(3):788–806, 1997.[9] T Weier, G Gerbeth, G Mutschke, E Platacis, and O Lielausis. Experiments on cylinder wake stabilization in anelectrolyte solution by means of electromagnetic forces localized on the cylinder surface.
Experimental Thermaland Fluid Science , 16(1-2):84–91, 1998.[10] T Weier and G Gerbeth. Control of separated flows by time periodic lorentz forces.
European Journal ofMechanics-B/Fluids , 23(6):835–849, 2004.[11] Zhihua Chen and Nadine Aubry. Active control of cylinder wake.
Communications in Nonlinear Science andNumerical Simulation , 10(2):205–216, 2005.[12] Vincent Dousset and Alban Poth´erat. Numerical simulations of a cylinder wake under a strong axial magneticfield.
Physics of Fluids , 20(1):017104, 2008.[13] Sintu Singha and KP Sinhamahapatra. Control of vortex shedding from a circular cylinder using imposed trans-verse magnetic field.
International Journal of Numerical Methods for Heat & Fluid Flow , 21(1):32–45, 2011.[14] Dipankar Chatterjee, Kanchan Chatterjee, and Bittagopal Mondal. Control of flow separation around bluff ob-stacles by transverse magnetic field.
Journal of Fluids Engineering , 134(9), 2012.[15] Hui Zhang, Bao-chun Fan, Zhi-hua Chen, and Yan-ling Li. Effect of the lorentz force on cylinder drag reductionand its optimal location.
Fluid Dynamics Research , 43(1):015506, 2010.[16] Hui Zhang, Bao-chun Fan, Zhi-hua Chen, and Hong-zhi Li. Numerical study of the suppression mechanism ofvortex-induced vibration by symmetric lorentz forces.
Journal of Fluids and Structures , 48:62–80, 2014.1717] Hui Zhang, Mengke Liu, Yang Han, Jian Li, Mingyue Gui, and Zhihua Chen. Numerical investigations oftwo-degree-of-freedom vortex-induced vibration in shear flow.
Fluid Dynamics Research , 49(3):035506, 2017.[18] A E Fisher, E Kolemen, and M G Hvasta. Experimental demonstration of hydraulic jump control in liquid metalchannel flow using Lorentz force.
Physics of Fluids , 30(6):067104, 2018.[19] Charles S Peskin. The immersed boundary method.
Acta Numerica , 11:479–517, 2002.[20] Fotis Sotiropoulos and Xiaolei Yang. Immersed boundary methods for simulating fluid–structure interaction.
Progress in Aerospace Sciences , 65:1–21, 2014.[21] Boyce E Griffith and Neelesh A Patankar. Immersed methods for fluid–structure interaction.
Annual Review ofFluid Mechanics , 52:421–448, 2020.[22] Markus Uhlmann. An immersed boundary method with direct forcing for the simulation of particulate flows.
Journal of Computational Physics , 209(2):448–476, 2005.[23] Jianhua Qin, Ebrahim M. Kolahdouz, and Boyce E. Griffith. An immersed interface-lattice boltzmann methodfor fluid-structure interaction.
Journal of Computational Physics , 428:109807, 2021.[24] Anand Bala Subramaniam, Manouk Abkarian, L Mahadevan, and Howard A Stone. Non-spherical bubbles.
Nature , 438(7070):930–930, 2005.[25] Christiana E Udoh, Jo˜ao T Cabral, and Valeria Garbin. Nanocomposite capsules with directional, pulsed nanopar-ticle release.
Science Advances , 3(12):eaao3353, 2017.[26] Teng Li, Shuixiang Li, Jian Zhao, Peng Lu, and Lingyi Meng. Sphericities of non-spherical objects.
Particuology ,10(1):97–104, 2012.[27] Khuram Walayat, Nazia Talat, Saqia Jabeen, Kamran Usman, and Moubin Liu. Sedimentation of general shapedparticles using a multigrid fictitious boundary method.
Physics of Fluids , 32(6):063301, 2020.[28] Haibo Huang and Xi-Yun Lu. An ellipsoidal particle in tube poiseuille flow.
Journal of Fluid Mechanics ,822:664, 2017.[29] James Feng, Howard H Hu, and Daniel D Joseph. Direct simulation of initial value problems for the motion ofsolid bodies in a Newtonian fluid Part 1. Sedimentation.
Journal of Fluid Mechanics , 261:95–134, 1994.[30] Peter Huang, Howard Hu, and Daniel Joseph. Direct simulation of the sedimentation of elliptic particles inoldroyd-b fluids.
Journal of Fluid Mechanics , 362:297–326, 1998.[31] Zhenhua Xia, Kevin W Connington, Saikiran Rapaka, Pengtao Yue, James J Feng, and Shiyi Chen. Flow patternsin the sedimentation of an elliptical particle.
Journal of Fluid Mechanics , 625:249–272, 2009.[32] Haibo Huang, Xin Yang, and Xi-yun Lu. Sedimentation of an ellipsoidal particle in narrow tubes Sedimentationof an ellipsoidal particle in narrow tubes.
Physics of Fluids , 26(5):053302, 2014.[33] Lihao Zhao, Niranjan Reddy Challabotla, Helge I Andersson, and Evan A Variano. Rotation of nonsphericalparticles in turbulent channel flow.
Physical Review Letters , 115(24):244501, 2015.[34] Pierre Lallemand and Li-Shi Luo. Theory of the lattice boltzmann method: Dispersion, dissipation, isotropy,galilean invariance, and stability.
Physical Review E , 61(6):6546, 2000.[35] Xiaoyi He and Li-Shi Luo. Lattice boltzmann model for the incompressible navier–stokes equation.
Journal ofstatistical Physics , 88(3-4):927–944, 1997.[36] Zhi-Gang Feng and Efstathios E Michaelides. Robust treatment of no-slip boundary condition and velocityupdating for the lattice-Boltzmann simulation of particulate flows.
Computers & Fluids , 38(2):370–381, 2009.[37] Zhilin Li and Kazufumi Ito.
The immersed interface method: numerical solutions of PDEs involving interfacesand irregular domains . SIAM, 2006.[38] Lixing Zhu and Arif Masud. Variationally derived interface stabilization for discrete multiphase flows and re-lation with the ghost-penalty method.
Computer Methods in Applied Mechanics and Engineering , 373:113404,2021.[39] Jianhua Qin, Yiannis Andreopoulos, Xiaohai Jiang, Guodan Dong, and Zhihua Chen. Efficient coupling ofdirect forcing immersed boundary-lattice boltzmann method and finite element method to simulate fluid-structureinteractions.
International Journal for Numerical Methods in Fluids , 92(6):545–572, 2020.1840] Jianhua Qin, Xiaohai Jiang, Guodan Dong, Zeqing Guo, Zhihua Chen, and Yiannis Andreopoulos. Numericalinvestigation on vortex dipole interacting with concave walls of different curvatures.
Fluid Dynamics Research ,50(4):045508, 2018.[41] Rawad Himo and Charbel Habchi. Coherent flow structures and heat transfer in a duct with electromagneticforcing Coherent flow structures and heat transfer in a duct with electromagnetic forcing.