Experimental and Numerical Analysis of Surface Magneto-Hydrodynamic Propulsion Induced by NdFeB Magnets

The so-called surface Magneto-hydro-dynamic (MHD) propulsion relies on the Lorentz force induced in weak electrolyte solutions (such as seawater or plasma) by NdFeB Magnets. The Lorentz force plays an important role in such dynamics as it directly affects the structures of flow boundary layers. Previous studies have mainly focused on the development of such boundary layers and related fluid-dynamic aspects. The main focus of the present study is the determination of electromagnetic field distributions around the propulsion units. In particular dedicated experiments and numerical simulations (based on the finite volume method) are conducted considering a NACA0012 airfoil immersed in seawater. The results show that, along the propulsion unit, the field strength undergoes a rapid attenuation in the direction perpendicular to the wall.

force, parallel to the magnetic or electrical poles. In previous studies, the flow field structures around the wall Lorentz force were analyzed through numerical simulation and experiments [4]. Globally, researchers have proposed using stripes of electrodes and magnets formed an electromagnetic actuator to prevent boundary layer growth, reduce skin friction, suppress vortices, and postpone flow separation [5,6].
With the continuous development of new technological materials and superconducting technology researchers can obtain a stronger magnetic field more easily, which can establish a sufficiently large electromagnetic field; thus, the electromagnetic fluid surface propulsion can be realized [7,8]. Electromagnetic fluid surface propulsion implies that the Lorentz force can be applied as the driving force generated over the outer surface of the submarine. This method has a certain guiding significance for optimizing the overall outer structure of the submarine and improving the submarine's propulsion efficiency [9][10][11].
In addition to using sonars, in future battlefields, the accurate location of submarines can be detected using electromagnetic fields [12]. Seawater is an electrically conducting fluid, and electromagnetic propulsion devices can be used for practical applications. In this work, we discuss the distribution and penetration characteristics of electric fields, magnetic fields, and the Lorentz force around the electromagnetic propulsion unit in seawater. Moreover, owing to the high relative dielectric constant of seawater, electromagnetic fields require a large magnetic field energy propagating through seawater. Therefore, to assess the actual magnetic field strength, further studies should be conducted to improve the electromagnetic safety for underwater navigation and reduce the probability of being electromagnetic field detected by the enemy's detection equipment. It is necessary to evaluate the electromagnetic safety before the MHD propulsion units are practically applied to underwater navigation systems.
Previous studies on electromagnetic actuators have focused on establishing certain macroscopic mathematical models [13]. Although these efforts can reflect the effects of the Lorentz force and flow fields to a certain extent, few studies have quantitatively analyzed the field strength around the electromagnetic actuator and the electromagnetic environment characteristics in a particular medium, such as seawater. Therefore, in this paper, Section 3 presents the numerical method used for simulation. Section 4 presents the numerical results of investigation of the field strength distribution characteristics above the electromagnetic propulsion unit in seawater. Section 5 describes the simplified electromagnetic field mathematical model. Section 6 presents the numerical and experimental results of flow field evolution comparisons under the Lorentz force control at Reynolds number (Re) = 2000.

Structure of the Electromagnetic Propulsion Unit
The electromagnetic propulsion unit used in the numerical simulation was designed with the general geometric characteristics, as shown in Fig. 1. Stripes of eight electrodes and magnetic poles (4 pairs) are alternately arranged in parallel on the surface of an insulator substrate. During the numerical simulation of the electromagnetic field, the coordinate origin was selected at the center of the left edge and maintained 1 mm above the upper surface of the unit. The x-and z-axis are parallel to the horizontal and electromagnetic pole directions of the model, respectively, and the y-axis points directly to the upward direction.
Four white normal lines in the center and one horizontal spanwise line are marked, as shown in Fig. 1. The field strength along some of these lines will be discussed. The four white normal lines in the center, 160 mm in length, start from the center points that the starting points remain 1 mm above the center of the N, -, S, and + poles, repectively. The horizontal spanwise line 1 is maintained 1 mm away from the surface and has a length of 70 mm. The black solid curves and dotted curves represent the electric and magnetic field lines, respectively. The two large arrows along the electric and magnetic pole strips denote the direction of the induced Lorentz force.

Numerical Methods
The magnetic induction H (A/m) can be transformed from the magnetic induction value B using where l 0 is the relative magnetic permeability in a vacuum, i.e., l 0 ¼ 4p Â 10 À7 T Á m=A. The relative magnetic permeability l r in seawater tends to be 1.
Based on Maxwell's equations, the mutual conversion relationship between the electric field E and magnetic field B can be expressed as follows: From Ohm's law, we obtain where J, U, q e , l, r, e and M P represent the current density, fluid velocity vector, dielectric constant, charge density, conductivity, permeability, and atomic magnetic kinetic energy, respectively.
Eqs. (2) and (3) are the common electromagnetic field equations. However, in this work, the voltage can be adjusted with time. For time-varying electromagnetic fields, Eqs. (2) and (3) can be rewritten as vector wave Eqs. (5) and (6). The distributions of the electric field, magnetic field, and electromagnetic force around the actuator are analyzed using the finite element method (Vector finite element method-edge element method) [14].
Here, e r denotes the relative dielectric constant. The variation formulations of Eqs. (5) and (6) can be written as: FðHÞ ¼ 1 2 where V m denotes the volume of the different dielectric layers. To realize the spatial dispersion of FðEÞ and FðHÞ, the different dielectric layers are divided into several small tetrahedral units (K 1 is the number of different dielectric layers and m ¼ 1; 2; 3; …K 1 ). V e m ðe ¼ 1; 2; 3; …M 1 Þ represents the volume of each small tetrahedral unit, where M 1 is the total number of small computing units into which each medium layer is divided. The electric field in each unit can be approximated by: where N e i;m Á E e i;m and n represent the vector basis function, the basis function expansion coefficient, and number of edges in the body of the component, respectively.
Here, l e r and e e r are the relative conductivity and relative dielectric constant, respectively. After an entire marking, which will give a number to each of the edges and the components. Then the Eq. (10) can be written in matrixes form: To solve Eq. (11), we applied Rayleigh-Ritz method, where F(E) is considered as a partial derivative with respect to each of the unknown divided edge fields, and then, the equations are set to zero. Thus, the eigenvalue equations can be obtained, and eigenvalue k 2 0 and corresponding eigenvectors can also be solved by solving the matrix equations. Thereafter, the electric field and other parameters can be obtained. Using the same method, the corresponding magnetic field can also be calculated.

Boundary Conditions
Practically, the electromagnetic parameters are not constants for different latitudes of the global ocean. Most of these values change with the local seawater salinity and temperature. Seawater is a nonferromagnetic substance, and its magnetic permeability is approximately equal to the permeability of vacuum. The conductivity of seawater ranges from 3 to 5 S/m. At 17°C, the standard seawater conductivity ranges between 4.54 S/m and 4.82 S/m, and the relative dielectric constant is approximately 81.
There are two types of ocean background electromagnetic fields: natural and induced. The natural electromagnetic field primarily refers to the geomagnetic field, which varies based on the earth's latitude. The earth's magnetic field in the north or south poles can reach an extreme value of approximately 7 À 8 Â 10 4 nT; however, the magnetic field near the equatorial region is approximately 3 À 4 Â 10 4 nT. The daily variation of the geomagnetic field can only reach dozens of nT, regardless of the changes in the season and temperature. In addition, an induced electromagnetic field must be generated when the seawater (as a conductor) moves in the earth's magnetic field. It is estimated that it has a period of approximately 16-17 s. The induced magnetic fields are produced by waves of 1 m height and day and night tidal waves that are negligibly small. The average geomagnetic field is approximately 5:5 Â 10 4 nT. Based on the seawater parameters, this study concentrates on the distribution characteristics of the magnetic field, electric field, and electromagnetic force around the propulsion unit made with NdFeB magnets, including the electromagnetic environment characteristics of the airfoil propeller.
In this numerical simulation, the overall electromagnetic propeller unit length, width, and height were 160 mm, 120 mm, and 9 mm, respectively. The electrode and magnetic pole stripes were of the same size; the length, width, and thickness are 120 mm, 2 mm, and 0.1 mm, respectively. Glass_PTFEreinf was used as the substrate material, neodymium iron boron magnetic was used as the magnetic pole, and copper was utilized as the electrode. NdFeB magnet remanence was 1 T, the coercive force in this work was −900000 A/m, and the positive and negative electrode voltages were selected as +10 V and −10 V, respectively.   Fig. 1. The a y is the width of the electrode and magnetic pole stripe. It can be observed that the voltages over the positive and negative electrodes are approximately 8.8 V and −8.8 V, respectively; however, at the magnetic pole center, the voltages tend to zero. Fig. 4 shows the voltage and electric field change with the wall-normal direction above the positive electrode center (the white center normal line in Fig. 1). Fig. 4 shows that the voltage decays rapidly with the increase in the wall-normal distance from the positive center surface. The voltage is approximately  9 V at a 1 mm distance (y=a y % 0) away from the wall. When the distance y=a y increases to 1, the voltage attenuates to approximately 2 V. As the distance y=a y continues to increase to approximately 4, the voltage tends to 0. The electric field distribution along the normal line is shown in Fig. 4. Comparing Fig. 4, it can be concluded that the electric field and voltage exhibit a similar exponential decay tendency with the wall distance. The electric field peak value is also exhibited near the unit surface, which is approximately 1300 V/m. When y=a y % 1, the electric field strength attenuates to 260 V/m. Similarly, the electric field strength is mostly attenuated to zero, while y=a y > 4. Fig. 5 shows the permeability regions of the magnetic induction field around the propulsion unit. It can be observed that near the magnetic pole edges, the magnetic induction intensity is significantly larger than those at the central zones. Moreover, the magnetic induction lines emerge from the N pole and end to the S pole outside the magnet (as shown in Fig. 1). The inward arrow on the surface of the S pole indicates the magnetic induction line entering the S pole, and the other end of the line emerges from the N pole.      Fig. 8 that, at the near-wall surface, the magnetic field strength has a maximum value and is continuously attenuated as the normal distance increases. It decays to 30 kA/m when y=a y tends to 1 and further decreases to zero at y=a y % 2. Simultaneously, the magnetic induction is approximately 0.36 T when it is nearest to the wall; it sharply decays to 0.04 T until y=a y equals 1; then, it gradually decreases and tends to zero at y=a y > 2 Therefore, the magnetic induction strength will decay near or below the ocean background magnetic field value, while y=a y > 6:6. Fig. 9 presents two distribution curves of the Lorentz force along the wall-normal direction above the N and S magnet poles. From the figure, the maximum Lorentz force is exhibited at the wall surface, which is approximately 1.6 and 1.8 kN/m 3 at the N and S poles' surfaces, respectively. It decays slowly in the normal direction of the N or S pole. Furthermore, the electromagnetic force primarily acts at 0 < y=a y < 1:1.

Field Strength Distribution Characteristics
It can be observed from the above figures that, in seawater, the propeller unit that can produce the Lorentz force exhibits a periodic variation along the span direction and an exponential decay in the wallnormal direction. For the electromagnetic environment characteristics of electromagnetic propulsion units in seawater, the N and S poles are adjacent and alternately arranged, thereby the magnetic field lines originated from the N poles will ended to the nearest S poles forming a short loop of the magnetic field. Therefore, the magnetic lines from N to S are not too long on the surface of the propulsion device without an evident magnetic leakage phenomenon. While y=a y % 6:6 away from the wall surface, the field strength will attenuate to the average geomagnetic field strength.

Simplified Mathematical Model of the Electromagnetic Field
This study primarily focuses on the electromagnetic field vector wave equations, which consider the fluctuation of the electromagnetic fields and their mutual intercoupling relationships. However, in the numerical simulation of the flow field, we should further simplify the above equations to better analyze the interaction mechanism between the electromagnetic field and weak electrolyte solution. In addition, as seawater is a weak electrolyte solution, the fluctuation information of electric and magnetic fields over time is excluded here. A simplified mathematical model is required in the numerical calculation of the flow field evolution under the action of the Lorentz force. Since the research object is a weakly conductive fluid (seawater) and is electrically neutral, during the numerical investigation using Eqs. (2) and (3), q e can be set as zero. Furthermore, the electromagnetic field is considered to be in a steady state and does not change with time. Therefore, Eqs. (2) and (3) can be rewritten as follows: The electric and magnetic fields are irrotational; thus, they can be denoted as the electric potential function U and magnetic potential function Φ, respectively. The detailed deduction process can be found in Crawford et al. [15]. Thereafter, Eqs. (12) and (13) can be further simplified to two Laplace equations as follows: To obtain U and Φ, the finite difference numerical method is utilized for a discrete solution based on the Laplace equations [16]. Subsequently, the distributions of the electric and magnetic fields can be obtained. Figs. 10-12 demonstrate the distributions of the electric field, magnetic induction field, and Lorentz force around the actuator, respectively. From the figure, the electric field, magnetic field, and Lorentz force change periodically along the spanwise direction, and the field strength appears to be largest near the wall surface, which decreases continuously with the increase in the wall-normal distance.   Because the distribution of the Lorentz force along the normal direction plays an important role in flow control and in the 2D model or simple 3D model, the crosswise evolution characteristics of the Lorentz force and the influence of the induction term for the entire Lorentz force can be neglected. In the electromagnetic fluid control, the distribution of this force is usually simplified as an exponential function that only changes with the wall-normal distance: where N denotes the dimensionless Lorentz force action parameter, which can be expressed as where e Àðp=aÞd represents the attenuation function of the Lorentz force in the normal direction. Moreover, d and e* are the distance to the surface and the vector in the wall-normal direction, respectively. The Lorentz force action parameter N denotes the ratio of the Lorentz force to the inertial force, and l is the character or dimensionless reference length. In addition, a is a constant representing the effective penetration depth [17]; in simulation, it is usually set to 1/10 of the character length. Further, U ∞ , ρ, J 0 and B 0 represent the inlet flow velocity, fluid density, surface maximum current density, and magnetic induction intensity, respectively.

Numerical and Experimental Comparisons
To understand the application of electromagnetic fluid surface propulsion or control, the structure and evolution characteristics of the flow field around the hydrofoil influenced by the streamwise Lorentz force are analyzed according to experiments and numerical simulation. The results of the flow around the rudder plane are presented in Liu et al. [18]. The hydrofoil structure and its experimental model are presented in Figs. 13a and 13b. The experiment was conducted in a rotating water tank. Potassium permanganate microtubules were used to mark the fluid flow paths. The conductive fluid density was 1003 kg/m 3 , electrical conductivity was 10 S/m, electrode voltage was 8 V, current density was 1210 A/m 2 , and Re was 2000. The incoming flow velocity and the chord length of the hydrofoil are 1 m/s and 0.1 m, respectively.
Figs. 13a and 13b illustrate the hydrofoil structure and experimental model covered by the actuators. The design of the hydrofoil is based on the NACA 0012 airfoil/hydrofoil. Its chord length is 100 mm, the spanwise length is 170 mm, and the distance from the leading-edge point to the trailing-edge point is 80.8 mm. The dimensions of the slots on the hydrofoil surface to install the magnetic poles are 75 × 10 × 2 mm. Fig. 13b presents a physical diagram of the hydrofoil covering with the electromagnetic poles. The yellow parts indicate the copper electrodes. The two adjacent copper electrodes were employed with positive and negative voltages. The Lorentz force intensity can be regulated by changing the voltage. Fig. 14 shows the photographed flow field evolution around the hydrofoil with an attack angle. The Lorentz force is applied both on the leeside and windward side for 10 s. It can be observed from Fig. 14a that, when the force is removed, the flow field separation on the leeward side of the hydrofoil is relatively significant and evident. The flow vortex is generated and detached on the leeward side, and the trailing edge vortex on the windward side is coiled up counterclockwise. Under the action of the Lorentz force, the flow separation at the front edge of the leeward side began to be inhibited, and the separation point was pushed downstream. When the force is constantly applied, the separation point is gradually removed from the hydrofoil (Figs. 14d and 14e), the flow separation is completely inhibited, and vortices cease to be formed and detached from the leeside.

Conclusions
This study investigated the field-intensity distribution characteristics surrounding an electromagnetic propulsion unit in seawater (a weak electrically conducting fluid environment). The results indicated the following: 1. The alternatively arranged electric or magnetic poles can produce a Lorentz force with periodic fluctuations in the spanwise direction, and its maximum strength is exhibited right above the edges of the electrode and magnetic poles; however, the average force strength rapidly decays along the wall-normal direction. 2. A simplified mathematical model of the electromagnetic field was developed, and the 3D numerical simulation results were obtained and presented. 3. Based on the simplified Lorentz force distribution function in the source term of the Navier-Stokes equations, numerical simulation was performed for the flow field evolution around a hydrofoil (NACA0012). Meanwhile, a water tunnel experiment was conducted to verify the numerical results and the effectiveness of the Lorentz force action. 4. Therefore, the Lorentz force propulsion method can suppress the magnetic leakage phenomenon and can offer appropriate electromagnetic safety. In this study, some of the variables are non-dimensional and only discuss the general relative distribution or changing trends of flow fields. However, the grid independence test should also be discussed. In further studies, the influence of the current intensity on the electromagnetic environment and quantification of MHD propulsion efficiency should receive significant attention. Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.