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.

Active fluid control is typically used to mitigate the drag force and suppress the noise produced by underwater weapons. Active flow boundary layer control can be based on physical mechanisms such as vortex suppression, separation postponement, transition delay, and velocity adjustment, in the turbulence boundary layer. Electrically conducting fluids (e.g., seawater) can be controlled using the Lorentz force. This may be regarded as a novel and very promising active fluid control mode. The Lorentz force can be generated along different directions over a body; thereby, its outer surface can be used as the working surface of a propeller. The process by which the Lorentz force propels a body is called electromagnetic fluid surface propulsion. One of the advantages associated with magneto-hydro-dynamic MHD control or propellers is the possibility to exert a flexible control directly on the fluid boundary layer by only adjusting the applied voltage [

Gailitis [

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 [

In addition to using sonars, in future battlefields, the accurate location of submarines can be detected using electromagnetic fields [

Previous studies on electromagnetic actuators have focused on establishing certain macroscopic mathematical models [

The electromagnetic propulsion unit used in the numerical simulation was designed with the general geometric characteristics, as shown in

Four white normal lines in the center and one horizontal spanwise line are marked, as shown in

The magnetic induction

where

Based on Maxwell’s equations, the mutual conversion relationship between the electric field

From Ohm’s law, we obtain

where

Here,

where _{m} denotes the volume of the different dielectric layers. To realize the spatial dispersion of _{1} is the number of different dielectric layers and _{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

Here,

To solve

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 non-ferromagnetic 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

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.

^{3}. Similar to the magnetic induction distribution, the Lorentz force exhibits several significant fluctuations with the change in the N, +, S, and – poles. Compared with _{y} positions; however, the Lorentz force maximum values do not show evident correlation with the voltage distribution.

^{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

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 wall-normal 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

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

The electric and magnetic fields are irrotational; thus, they can be denoted as the electric potential function

To obtain

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

where _{∞}, _{0} and _{0} represent the inlet flow velocity, fluid density, surface maximum current density, and magnetic induction intensity, respectively.

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. [^{3}, electrical conductivity was 10 S/m, electrode voltage was 8 V, current density was 1210 A/m^{2}, and

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:

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.

A simplified mathematical model of the electromagnetic field was developed, and the 3D numerical simulation results were obtained and presented.

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.

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.