The design of direct-drive marine engines, ship propellers and micro-space propulsions have reached an advanced stage of development, and only incremental advances in performance may be expected in the future without introduction of new technology or new propulsion concepts. So the EMHD (electromagnetic hydrodynamics) propulsion has achieved more attention in recent years.

If navigation immersed in conductive fluid (seawater, plasma, etc.), electromagnetic force or Lorentz force would be generated by applied electric field and magnetic field around the surface, which could control and propel the surround fluid. Flow control with Lorentz force was first proposed by Rice in 1961, during the following years researchers have developed and applied this methods on ship propulsion [1M.C. Gourdine, "Recent advances in magnetohydro-dynamic propulsion", ARS J., vol. 31, no. 12, pp. 1670-1677, 1961.
[http://dx.doi.org/10.2514/8.5890] , 2S. Molokov, and R. Moreau, Magnetohydrodynamics: Historical Evolution and Trends., Berlin Heidelberg Press: New York, 2007.
[http://dx.doi.org/10.1007/978-1-4020-4833-3] ]. Then a large amount of works have been devoted to study the structure and field distribution of electromagnetic actuators, in weakly conductive fluid. For electrolyte solution like seawater, Lorentz forces provide several attractive approaches in flow control, such as boundary layer control [3K.S. Breuer, J. Park, and C. Henoch, "Actuation and control of a turbulent channel flow using Lorentz forces", Phys. Fluids, vol. 16, pp. 897-907, 2004.
[http://dx.doi.org/10.1063/1.1647142] , 4V. Shatrov, and G. Gerbeth, "Magnetohydrodynamic drag reduction and its efficiency", Phys. Fluids, vol. 19, p. 035109, 2007.
[http://dx.doi.org/10.1063/1.2714077] ], drag reduction [5J. Pang, and K.S. Choi, "Turbulent drag reduction by Lorentz force oscillation", Phys. Fluids, vol. 16, pp. L35-L38, 2004.
[http://dx.doi.org/10.1063/1.1689711] , 6J. Pang, K. Choi, and A. Aessopos, "Control of near-wall turbulence for drag reduction by Spanwise oscillating Lorentz force", In: 2^nd AI-AA Flow Control Conference, Oregon: Portland, 2004, p. 28.], lift enhancement [7Y.H. Chen, B.C. Fan, and Z.H. Chen, "Flow pattern and lift evolution of hydrofoil with control of electro-magnetic forces. Science in China Series G: Physics", Mech. Astron., vol. 52, pp. 1364-1374, 2009.
[http://dx.doi.org/10.1007/s11433-009-0172-4] ] as well as noise suppression [8Z. Chen, and N. Aubry, "Closed-loop control of vortex induced vibration", Commun. Nonlinear Sci. Numer. Simul., vol. 10, pp. 287-297, 2005.
[http://dx.doi.org/10.1016/S1007-5704(03)00127-8] ]. Furthermore, Lorentz force has been successfully used in satellite electric propulsion (attitude control and station keeping, etc.) [9L. Garrigues, and P. Coche, "Electric propulsion: comparisons between different concepts", Plasma Phys. Contr. Fusion, vol. 53, p. 124011, 2011.
[http://dx.doi.org/10.1088/0741-3335/53/12/124011] -11S. Tsujii, M. Bando, and H. Yamakawa, "Spacecraft formation flying dynamics and control using the geomagnetic Lorentz force", Control, and Dynamics., vol. 36, pp. 136-148, 2013.
[http://dx.doi.org/10.2514/1.57060] ], in this way plasma or charged particle can be accelerated and injected to adjust satellite attitude. So if the navigation is covered with body-fitted electromagnetic actuators, Lorentz force would be generated and propelled the surround fluid, this is the basic principle of EMHD propulsion by surface. EMHD propulsion is a kind of static electric magnetic propulsion system, which had attracted the attention of several inventors over the past several years [12Z.K. Liu, B.M. Zhou, H.X. Liu, and Z.G. Liu, "The analysis of effects and theories for electromagnetic hydrodynamics propulsion by surface", Wuli Xuebao, vol. 60, pp. 84701-084701, 2011., 13Z.K. Liu, J.L. Gu, and B.M. Zhou, "Investigation of electromagnetic hydrodynamics propulsion and vector control by surfaces based on a rotational navigation body", Wuli Xuebao, vol. 63, pp. 74704-074704, 2014.]. The main reason for this attraction has been the seemingly elegant operating principle using no moving parts. Proposed applications include the silent propulsion of naval submarines, the use in high-speed cargo submarines, and the propulsion of future high-speed surface ships without the danger of cavitation.

During the EMHD propulsion, the presence of an external magnetic field B in an electrically conducting flow gives birth to an induced magnetic field and induced electrical currents. When the magnetic Reynolds number Re is much smaller than 1, as in small liquid metal experiments, the induced magnetic field is negligible, but not the induced currents and the total magnetic field is the imposed one [14P.A. Davidson, An Introduction to Magnetohydrodynamics., Cambridge University Press: UK, 2001.
[http://dx.doi.org/10.1017/CBO9780511626333] ]. The Lorentz force generated by the interaction between the magnetic field and the induced currents tends to damp velocity variations along the streamlines [15J.L. Sommeria, and R. Moreau, "Why, how, and when, MHD turbulence becomes two-dimensional", J. Fluid Mech., vol. 118, pp. 507-518, 1982.
[http://dx.doi.org/10.1017/S0022112082001177] ]. EMHD propulsion by surface would also be applied widely, such as high-speed aircraft flight in ionosphere or outer space. For some special circumstances, in order to achieve better propulsion effects the environment parameters such as the electrical conductivity and permeability could be improved by release special medium (such as plasma). The recently dramatic developments in high-temperature superconducting materials and the successful operation of a large and sophisticated magnet system at Laboratory now make it desirable to reconsider EMHD propulsion. Under these conditions, the disturbance of induced item to propulsion effects must be considered. So in this paper numerical investigations have been conducted on the basic of this special environment.

To gain some insight into the effect of magnetic induction intensity to propulsion character, numerical simulations have been conducted at Re= 1 × 10⁴ with three different interaction parameters. 1. N=40, 1⁄κ=8, 2. N=160, 1⁄κ=2, 3. N=240, 1⁄κ=4/3. The selecting of these parameters is to reflect the influence of three different magnetic intensities to propulsive characteristics. During this simulation, the electric field intensity is kept constant.

Fig. (3) shows the distributions of velocity amplitude isosurface range from 5.5 to 6.5, which are slightly larger than U_∞=5. For unforced case, t=1.8 as Fig. (3a) shows that velocity isosurface just appears around the leading edge. However, when the force applied the isosurface near the leading edge as well as the navigation body surface is extended. The largest velocity isosurface appears in Fig. (3c) at t=3.8, and then the Fig. (3b and d). It could be concluded that the change of propulsion force with magnetic field intensity presents nonlinear characters. In order to reveal this influence, further analysis will be listed in the following paper.

In order to display the velocity changes better with wall normal distance, a spherical surface is defined as Fig. (4) shows with center (L/2, 0, 0) and radius 2R. The distributions of fluid field over this surface would be mainly analyzed in the following paper.

Fig. (5a) is the u velocity distribution on the spherical surface at t=3.8 (force stability action time). Fig. (5b) is the curve of u velocity along wall normal distance from the center location, where Fig. (5a) shows that on the spherical surface relatively low speed boundary layer still appears near the surface, due to the fluid viscous. With wall distance increasing red high speed region appears, but flow velocity on the further region gradually attenuates to U_∞. The high velocity range on the left side of spherical surface is narrower than the rightbecause the downstream fluid continuously obtained the kinetic energy of Lorentz force, which leads to an increase of near wall fluid velocity. It could also be seen from Fig. (5b), that a low speed. Meanwhile, u velocity is increasing away from the surface with the action of Lorentz force, and the maximum velocity 5.3 appears within the high speed range corresponding to the high speed red area shown in Fig. (5a).

Fig. (3) The distributions of velocity isosurface (5.5~6.5).

Fig. (4) Schematic of the spherical surface.

In order to offer more detail interpretations of velocity distribution in Fig. (5), Fig. (6) is drawn to display the distribution curves of Lorentz force and its each component along wall normal distance, at t = 3.8 (Force in Fig. (5a) denotes the total Lorentz force, Force1 is the conducting item intensity, Force2 denotes the induced item intensity and Force=Force1−Force2). The force action range. And the distribution of Force1 is related only to the intensity of electric field and magnetic field, which does not change with the fluid flow velocity. The fluid around the surface is of lower u velocity, so force intensity is relatively small, about 0.5. However, with wall distance increasing, Force2 reaches to the maximum 3.2 at d₁/R ≈ 0.25. The u velocity intends to U_∞ in farther distance, but Force2 decreases to zero due to the attenuation of total force along wall normal distance. Fig. (6c) is the distribution curve of Lorentz force. Owing to the negative effect of induced items on Lorentz force, it could befound that the total force decreases. By comparing Fig. (6a and c), we could find that there is no remarkable difference between the distribution curves of Force1 and Force except for their gradients.

Fig. (5) The flow velocity distribution on spherical surface, and changes curve along the normal distance.

Fig. (6) Distribution curves of the Lorentz force and its each component along the wall normal distance.

Fig. (7a) is the u velocity distribution on the spherical surface at t = 3.8. Fig. (7b) is the curve of u velocity along wall normal distance from the center location, whereUnder the action of Lorentz force, high speed fluid is appeared and surrounded the navigation. Meanwhile, the fluid velocity is increased under this force control, so as illustrated in Fig. (7b) the red high velocity range is extended slightly larger than the former case (N = 40, 1⁄κ = 8, t = 3.8).

Fig. (7) The flow velocity distribution on spherical surface, and changes curve along the normal distance.

As Fig. (7b) shows the near wall (0.1 < d₁/R < 1.6) fluid is of larger velocity than U_∞. That the u velocity along the wall normal reaches to the maximum 6.35. Moreover, the low velocity (u < 5) range is constantly contracted to 0 < d₁/R < 0.1. This illustrates that when keeping the electric field intensity constant, the increase of magnetic induction intensity within a certain range can promote the Lorentz force.

Fig. (8) is the distribution curves of Lorentz force and its each component, t=3.8. As the Fig. (8a) shows Force1 intends to decay exponentially with wall distance. The maximum of Force1 appears at about 160, near the surface, but then decay to zero gradually with wall distance increases. The Fig. (8b) displays that when Force2 intends to 12, but with the wall distance increasing the peak 74 of Force2 occurred at approximately way from the surface, and then decay to zero gradually.

Fig. (8) Distribution curves of the Lorentz force and its each component along the wall normal distance.

Near wall, the flow velocity is rather low even N is bigger; but the product of N and u velocity is also very small. Compare Fig. (8c) with (8a) it could note that Force in range of 0 < d₁/R < 1 has more sharply attenuation gradient, so Force1 ≈ 55 at d₁/R = 0.5, but Force ≈ 27, which is mainly due to the inhibition of magnetic induced intensity.

Fig. (9a) is the u velocity distribution on the spherical surface at t = 3.8 (Lorentz force stability action time). Fig. (9b) is the curve of u velocity along wall normal distance from the center location of navigation surface,. It could be found that the total force intensity decreases with further increasing of η. Accordingly, u velocity of near wall fluid is decreased and the red high velocity range (0.12 < d₁/R < 1.2, u > 5) is contracted. For this case, the maximum of u velocity is 5.95 larger than the first case but smaller than the second one, which causes the boundary layer thickness (0 < d₁/R < 0.12, u < 5) narrower than the former.

Fig. (9) The flow velocity distribution on spherical surface, and change curve along the normal distance.

Fig. (10) is the distribution curves of Lorentz force and its each component, t=3.8. As this figure shows, Force1 decreases exponentially with wall normal distance. So near the surface, Force1 appears the maximum about 160, but then decreases to zero gradually with the increase of wall distance. As shown in Fig. (10b) the induction of electromagnetic force Force2 near the surface is almost zero, which is caused by the boundary layer. However, the flow velocity increases with the increase of wall normal distance and reaches to the maximum 160 at 0.15. Comparing with Force1, the proportion of induced item Force2 to total force increases gradually. The Fig. (10c) is the distribution curve of Force. The intensity of induced item is larger near the surface, which leads to a sharply decrease of Force in. Then with the increasing of wall distance the gradient of Force decrease intends to slowly. In fact, the above investigations suggest that the increasing of η in center range could promote the total Lorentz force. However, when the percent of induced item exceeds a certain value, the total force will decrease with the further increase of induced item.

Fig. (10) Distribution curves of the Lorentz force and its each component along the wall normal distance.

The force in each direction would be mainly analyzed based on flow fluid structure, in the following paper. We set f_t = f_prx + f_shx the total force, where f_prx and f_shx are the pressure force and shear force in x-direction, respectively; while f_lorx and f_lorx1 denote the total Lorentz force and applied item, respectively. The force histories are recorded from flow stable time t=0.5, and the force action time range is from t=2 to 4.

Fig. (11) Time histories of each forces (N = 40, 1/κ = 1, case1).

Fig. (11a and b) are time histories of f_t, f_lorx and f_lorx, f_lorx1, respectively. As Fig. (11a) shows f_t intends to 0.075 without force control but then increases sharply to 0.11 since t = 2. During this time, it could be found that f_lorx increases from 0 to 0.27 (at t=2.05) and then decreases slightly to 0.266. This is due to that while the Lorentz force is applied, the velocity u increases from zero, which brings about a continue increase influence of induced item on total Lorentz force. Finally, this influence intends to stability. As Fig. (11b) shows, a peak value of f_lorx appears at the beginning of force applied. But the applied item f_lorx1 keeps constant at about 0.31, which is larger than the total Lorentz force f_t.

Fig. (12a and b) are time histories of f_t, f_lorx and f_lorx, f_lorx1, respectively. For this case, with the increase of magnetic field, the influence of induced item is expanded.

Fig. (12) Time histories of each forces (N = 160, 1/κ = 2, case2).

The Fig. (12a) displays that f_lorx jumps to peak value 0.64 while the force applied but then decreases slowly to 0.56 at about t=2.3. This change is similar to the former case, but the stable value of f_lorx is 0.56 (N = 160, 1/κ = 2) larger than 0.266 (N = 40, 1/κ = 8). The increase of magnetic field leads to a significant rise in fluid velocity and drags force thus f_t increases and becomes stabilized at about 0.15. Compare Fig. (12b) with (11b), it could be found that the applied item f_lorx1 ≈ 1.24 (N = 160, 1/κ = 2) keeps constant over t = 2-4, whichis about four times larger than f_lorx1t ≈ 0.31(N = 40, 1/κ = 8).

Fig. (13) Time histories of each forces (N = 240, 1/κ = 4/3, case3).

It can be seen in the Fig. (13) that, with further increase of magnetic field, more significant inhibition effect for navigation propulsion is observed. So comparing with the former two cases f_lorx reaches to the maximum 0.56 and then falls sharply to a plateau 0.44 less than 0.64 (case 2). Accordingly, f_t approximately stabilizes at 0.14 also less than the former cases and the difference between f_lorx ≈ 0.44 and f_lorx1 ≈ 1.92 is enlarged.

Then we introduce C_pro as thrust coefficient of navigation

The C_pro for the three cases at Re = 1 × 10⁴ are listed in Table 1. It could be found that the C_pro ≈ 0.0328 of case 2, is 60.97% and 26.83% larger than that of case1 and case 3, respectively.

Table 1
The C_pro for the three cases.

For larger Reynolds number or magnetic Reynolds number environment, the influence of induced item on navigation surface propulsion effects could not be ignored anymore. In this paper, based on navigation model, the influence of three different magnetic intensities (keep the electric field intensity constant) on flow field characteristic and propulsion effects have been analyzed. These conclusions reveal that while keeps the electric field constant, the thrust coefficient of navigation rises with the increasing of magnetic field at first but then drops more quickly. The change of thrust coefficient with magnetic field intensity also presents a nonlinear character, which depends not only on magnetic field but also on the Re and Re_mag.