Lattice Boltzmann Simulation Of The Flow Control By Electromagnetic Forces

When the viscous flow passes a bluff body, the flow separates on the downstream side, developing two counter-rotating vortices which become more and more elongated as time increases. Finally, the shedding of the vortices presents behind the bluff body. The flow of bluff body wakes changes a lot. The appearance of vortices shedding is accompanied by power consumption, which may cause destruction. Meanwhile, the boundary layer separation will lead to a large fluctuation of force, causing drag increment. So, from the engineering point of view, it is of great significance to effectively control the flow around a bluff body.In the past ten years, people studied application of an electromagnetic force to control the flow around a bluff body. Its principle is to use the electromagnetic force to change the flow of the boundary layer, control the wake structure, prevent the boundary layer from separation, suppress the shedding of vortices, and reduce the drag. Unlike ordinary magneto-hydro-dynamics (MHD), in which in liquid metal Magnetohydrodynamicsσ~ 10 6 S /m, the Lorentz force density is strongly coupled with the flow, since the flow induces currents via the u×B term, and the induced currents produce Lorentz forces strong enough to change the flow. For seawater and other electrolytes,σis rather small (σ~ 10 S /m). Therefore, the induced currents are very low for moderate applied magnetic fields and the induced Lorentz forces due to these currents are negligible and too weak for a noticeable influence on the flow. An external electric field has to be applied in order to generate forces large enough to influence the flow. This special application of MHD is usually called electro-magneto-hydro-dynamics (EMHD). For EMHD control problems, this paper presents the basic equations of electromagnetic field and hydrodynamics. The electrodes and permanent magnets are flush mounted on the the bluff body in such a way that the Lorentz force (electromagnetic body force) is generated in a streamwise direction. If the magnets and the electric currents are fixed, it satisfies Maxwell equationswhere E denotes the electric field, u is macroscopic value of the fluid velocity, andσis the electrical conductivity, respectively.σ( u×B ) denotes the electric current induced by magnetic field. In the case of seawater and weak electrolytes,σis small. Therefore, the induced currentσ( u×B ) is much too low to produce any noticeable effect on the flow and is negligible. The first equation is written as j =σE , Hereby, we assume that only the externally applied electric field E and the magnetic field B contribute to the Lorentz force. ,The Lorentz force F appears as an electromagnetic body force term of the Navier–Stokes equation for incompressible flow Based on chord length l of the hydrofoil and freestream velocity u∞, the governing equations in dimensionless form read for the 2-D case the electromagnetic force density acts as a momentum source for the flow. In the equations, the Reynolds is defined as Re = l ? u∞/υ, where p is the macroscopic pressure,υis the kinematic viscosity of the fluid, F is the total force acting on the fluid particle, and pointing in the streamwise direction which decays exponentially in wall-normal direction.For the governing equations mentioned before, it is often used finite volume method, finite difference, finite element method and other numerical method to solve them. But it is difficult to deal with complex boundaries, achieve the effect of non-continuous, and implement parallelity in the simulation of electro-magneto-hydro-dynamics. Therefore, we consider them in the mesoscopic level, and apply the lattice Boltzmann method to solve the problem of electro-magneto-hydro-dynamics, and study the flow of the boundary layer. Unlike macromethods, the main advantages of LBM are easy to implement boundary treatment and achieve non-continuous effect, easy parallelity, and suitable for simulating small-scale micro-electromagnetic fluid. Due to above advantages, this paper systematicly researched on the electromagnetic flow control using the Lattice Boltzmann method. It is important that the numerical EMHD models are able to simulate accurately the evolution of relatively small-scale features within large complex domains.Lattice Boltzmann method (LBM) establishes evolutionary equation of the particle distribution function based on kinetic theory. The macroscopic quantities (such as mass densityρand momentum densityρu) are capable of being obtained by evaluating the hydrodynamic moments of the distribution function. The main idea of the LBM is to simplify the process of collision and the streaming express characteristic of fluid dynamics using equation of microscopic particle motion instead of using the equation of macroscopic flow. It has been indicated that the LBM is a reliable and easy way to implement complex boundaries for simulating fluid flows.Therefore, We solve the governing equations using a lattice-Boltzmann algorithm: where, eαdenotes the particle velocity in theαt hdirection, denotes the particle distribution function along the eαdirection, fα( eq)( x i, t) denotes corresponding equilibrium distribution function, x i describes the spatial position, t denotes the time,δt represents time step,τis the dimensionless relaxation time, and gα( xi , t) is related to the total body force acting on the fluid particle, and described as: The equilibrium distribution function fα( eq)( x i, t)in the lattice Boltzmann equation is written as whereδxis lattice length, xThe fluid densityρand the velocity u can be evaluated by the following two equations:For the curved boundary of the bluff body surface, we use bounce-back boundary conditions and immersed boundary method to deal with curved boundary. Meanwhile, we utilize momentum exchange method and implicit velocity correction-based immersed boundary to determine the force acted on the bluff body surface.Firstly, the flow around a circular cylinder in the electromagnetic field is simulated numerically by the Lattice Boltzmann method. The effect of Lorentz force (electromagnetic force) acting on the flow of weak electric medium solution is investigated. The treatment of curved boundary and the evaluation of the boundary forces are analyzed. The streamlines, vorticity contours, vorticity, drag and lift coefficients are presented. Numerical results verify that Lorentz force can modify the boundary layer around a circular cylinder and prevent the boundary layer from separation. Lorentz force is able to suppress the shedding of vortices, and reduce the drag. If the direction of Lorentz force is along the mean flow, and Lorentz force has a positive effect for drag reduction and damping. If the direction of Lorentz force is opposite to the mean flow, then both the drag and oscillation amplitude of lift force increase.Secondly, the flow around an elliptic cylinder in the electromagnetic field is simulated numerically by the Lattice Boltzmann method. The effect of Lorentz force (electromagnetic force) acting on the flow of weak electric medium solution is investigated. The treatment of curved boundary and the evaluation of the boundary forces are analyzed. The flow past an elliptic cylinder with different shapes is computed by fixing the vertical semiaxis length and varying the horizontal semiaxis length. Numerical results verify that Lorentz force can modify the boundary layer around an elliptic cylinder and prevent the boundary layer from separation. At the same time, the effect of different Reynolds number, different the ratio of the horizontal axis and vertical axis, different attack angle, and different magnitude of electromagnetic force on boundary layer control are taken into account. For the same value of the Reynolds number, at the zero angle of attack, when the electromagnetic force is turned off, the drag coefficient decreases with the increasing of the ratio of the horizontal axis and vertical axis. When electromagnetic force is applied, the drag coefficient decreases. But as the ratio of the horizontal axis and vertical axis increases, the degree of the drag coefficient reduction falls. For the same value of the ratio of horizontal and vertical axis, without the electromagnetic force, if the angle of attack becomes larger, then the shock of vortex becomes larger too. When the electromagnetic force is applied, if the angle of attack gets larger, then the drag reduction gets worse too. For the same value of the angle of attack, without electromagnetic force, if value of the Reynolds number grows larger, then the boundary layer separation grows easier as well. When the electromagnetic force is applied, if the Reynolds number is smaller, then the suppression of boundary layer separation is easier.Thirdly, the control of electromagnetic of the flow around hydrofoil at a larger Reynolds number Re = 2.4×105has been investigated by applying Lattice Boltzmann Method to Large Eddy Simulation. The treatment of curved boundary and the evaluation of the boundary forces are analyzed. Numerical results show that without electromagnetic force, the flow is unsteady, and the boundary separation occurs. Furthermore, as the angle of attack increases, the drag of the hydrofoil surface increases. The stall angle of attack obtained by Large Eddy Simulation-lattice Boltzmann method is12 . When the electromagnetic force is applied, the electromagnetic force can effectively control the flow around hydrofoil, suppress the the boundary separation, suppress the shedding of vortices, and reduce drag. As the electromagnetic force increases, the leading edge separation is gradually suppressed, the lift can gradually increase, and the drag can gradually reduce. Stronger electromagnetic force leads to larger gain in lift but also to larger reduce drag.Finally, using immersed boundary-lattice Boltzmann the flow around hydrofoil in electromagnetic field is simulated. Immersed boundary method and the lattice Boltzmalm method are effective combined. Applying the advantages of two methods to calculate the fluid problem, the algorithm is simpler, the boundary treatment is easier. At a small Reynolds number Re = 500, a certain attack angle of hydrofoil, the effects of different electromagnetic force magnitude on distribution of velocity, the boundary layer separation, the drag and the lift are investigated. Numerical results show that the electromagnetic force applied on the hydrofoils surface in a streamwise direction can modify the boundary layer. The higher values of electromagnetic force are of greater help to suppress the leading edge vortices, the tailing edge vortices and the boundary layer from separation. By applying a strong enough Lorentz force, vortex shedding can completely disappear, boundary layer separation can be completely suppressed, the lift is considerably increased, and the drag is considerably decreased. Since our results are consistent with the theory of the fluid flow past the hydrofoil, we use them to produce asymptotic relations for variation of the studied quantities as a function of the interaction parameter (N). The value of the drag coefficient is found varying according to , and the lift coefficient is according to e c1 N + c2.