Numerical Analysis of Electromagnetic Control of the Boundary Layer Flow on a Ship Hull ()
1. Introduction
The ability to manipulate a flow field to improve efficiency or performance is of immense technological importance. The intent of flow control may be to delay/advance transition, to suppress/enhance turbulence, or to prevent/promote separation. The resulting benefits include drag reduction, lift enhancement, mixing augmentation, heat transfer enhancement, and flow-induced noise. Flow control involves passive or active devices that have a beneficial change on the flow field. During the last decade, emphasis has been on the development of active control methods in which energy, or auxiliary power, is introduced into the flow. One of the active control methods is MHD control. If a fluid is electrically conducting, eelectromagnetic flow control permits to act directly within the boundary layer by applying directly local Lorentz forces.
Mainly three different force configurations have been investigated in order to control turbulent boundary layers: wall parallel streamwise, wall parallel spanwise and nominally wall normal forces. Wall parallel forces in streamwise direction have been applied, e.g., in the experiments of Henoch and Stace [1] and Weier et al. [2] as well as in the numerical analysis of Crawford and Karniadakis [3] . This force configuration increases instead of reducing wall shear stress, because the acceleration of the near wall fluid leads to a higher slope of the mean velocity profile in streamwise direction. However, the momentum gain due to the Lorentz force surpasses the friction drag rise.
Nosenchuck and Brown [4] used nominally wall normal, time dependent forces. Nosenchuck and co-workers reported several successful experiments with a multitude of electromagnetic actuators generating turbulent skin friction reductions.
A circular cylinder equipped with electrodes as well as permanent magnets generating a wall parallel force in streamwise direction was used in the experiments and numerical calculations of Weier et al. [5] . Similar configurations have later been investigated by Kim and Lee [6] , Posdziech und Grundmann [7] , and Chen and Aubry [8] . While skin friction drag is increased by this force configuration, form drag is strongly reduced for an initially separated flow at Reynolds numbers Re = O(100). For stronger forcing, the increase in skin friction drag dominates the form drag decrease. Shatrov and Yakovlev [9] studied numerically the flow around a sphere with a mainly wall parallel Lorentz force at Re up to 1000. For increasing interaction parameter, i.e. the ratio of electromagnetic to inertial forces, the size of the separation region is first reduced. Later, separation is suppressed completely resulting in a strong decrease of form drag. Shatrov and Yakovlev [10] extended the investigated Reynolds number range up to 105 treating the problem of a steady and axially averaged flow. For large Reynolds numbers, the total drag of the sphere was reduced by 4 times and despite the moderate electrical efficiency of 
, the total energy consumption was reduced as well.
In present study, electromagnetic control of turbulent boundary layer on a ship hull, without free surface effects, is numerically investigated. For this purpose, a combination of electric and magnetic fields is applied to a region of Turbulent boundary layer on stern so that produce wall parallel Lorentz forces in streamwise direction as body forces in stern flow and their effects on flow parameters such as boundary layer thickness, flow separation, wake distribution on propeller plane, pressure distribution on ship hull and resistance is investigated. This study is done on the geometry of a tanker model hull. Very detailed experimental data are available for this ship [11] . For investigation of Lorentz forces effects, calculations are conducted for two cases without applying electromagnetic field and with applying electromagnetic field locally to stern flow. For this simulation, RANS equations with SST k-ω turbulent model coupled with electric potential equation are used. This governing equation is numerically solved in computational grid generated around ship hull by using Ansys Fluent 14.5 finite volume codes. Accuracy of SST k-ω turbulent model of Fluent in predicting Turbulent flow around a ship is also tested by comparing with available experimental results that it shows a good agreement with experimental data.
2. Governing Equations
2.1. RANS Equations
In this study, it assumed that fluid is incompressible. The governing equations are the mass and momentum conservations. Using the Reynolds averaging approach, the Navier-Stokes equation can be stated as:
(1)
(2)
where 
represents Reynolds stresses and
is Lorentz body force included in momentom equation for MHD flow
2.2. Turbulent Model
The Reynolds-averaged approach to turbulence modeling requires that the Reynolds stresses in Equation (2) are appropriately modeled. A common method employs the Boussinesq hypothesis to relate the Reynolds stresses to the mean velocity gradients:
(3)
In this article, the two-equation Shear-Stress Transport (SST) k-ω turbulence model is used for modeling turbulent viscosity.
The turbulence kinetic energy, k, and the specific dissipation rate, ω, are obtained from the following transport equations:
(4)
(5)
where 
represents the generation of turbulence kinetic energy due to mean velocity gradients:
(6)
represents the generation of ω :
(7)
and 
are the turbulent Prandtl numbers for k and ω, respectively:
(8)
(9)
is cross-diffusion term.
The turbulent viscosity, 
, is computed as follows:
(10)
where S is the strain rate magnitude, the coefficient 
damps the turbulent viscosity causing a low-Reynoldsnumber correction .
and 
are blending functions.
2.3. Electric Potential Equation
In general, the electric field 
can be expressed as:
(11)
where 
and 
are the scalar potential and the vector potential, respectively. For a static field and assuming 
in which 
is induced magnetic field and 
is imposed magnetic field, Ohm's law can be written as:
(12)
For sufficiently conducting media, the principle of conservation of electric charge gives:
(13)
The electric potential equation is thus given by:
(14)
The current density can then be calculated from Equation (12).
With the knowledge of the induced electric current, the MHD coupling is achieved by introducing additional source terms to the fluid momentum equation. This additional source term is the Lorentz force given by:
(15)
which has units of 
in the SI system.
For MHD flow around a ship hull, the magnetic field induced by the motion of sea water can be neglected compared with the imposed magnetic field. On the other hand, when the electric field is applied, a magnetic field is generated around the electric current. The magnitude of this generated magnetic field is evaluated by integrating Maxwell equation, 
, in the region where the electric field is applied:
(16)
where I is total electric current and S is area of applied electric field.
is of 
for sea water, and assuming that I is of 103 Amperes at most, this induced magnetic field can again be neglected, in comparison with the imposed magnetic field. So potential electric equation in which the induced magnetic field is neglected, can be a proper method for MHD flow around a ship hull.
Finally, the governing equations to solve become seven equations including four RANS equations, two transport equations for k and ω and one potential equation that must be solved simultaneously.
3. Computational Grid and Boundary Conditions
In present study, a tanker model hull is used for calculate on of flow around ship. This model was presented at CFD workshop in Tokyo [11] . Very detailed experimental data of this ship is available for public use. Main dimensions of this model are shown in Table 1.
Flow around the hull has been considered symmetrical to the center plane of the ship so the calculation is only conducted in on side of the symmetrical plane. Figure 1 shows the computational domain and boundary conditions applied for turbulent flow calculations. The distance of domain boundaries from the ship hull has been considered large enough in order to apply correct boundary conditions. These boundaries include the surface bellow:
1) A spherical surface with a radius of 
, that it's center is on the intersection of aft perpendicular and waterplane 2) A cylindrical surface with a radius of 
, that extends for 
from aft perpendicular to downstream flow.
For studying of electromagnetic effects on flow, an external magnetic field is applied to a region of flow limited to following boundaries (by assuming that the origin is on intersection of aft perpendicular and waterplane):
• In longitudinal direction from x = 0.3 m plane to x = 1.3 m plane
• In transverse direction from hull surface to y = 0.5 m plane
• In vertical direction from z = −0.05 m plane to z = 0.25 m plane
Figure 2 represents MHD region and conditions applied to this region. These conditions are:
1) An external magnetic field of 
applied in z-axis negative direction.
2) “Conducting Wall” boundary condition with electric potential of 
applied on ship hull in MHD region.
Since we have a large change in velocity in the wall normal direction in boundary layer, we need to use
Table 1. Principle dimensions of a tanker model.
Figure 1. Solution domain and boundary conditions.