The Study on the Flow Generated by an Array of Four Flettner Rotors: Theory and Experiment

We present an immersed array of four rotors whose promoted flow can be mathematically modeled with a creeping flow solution from the incompressible Navier-Stokes equations. We show that this solution is indeed representative of the two-dimensional experiment and validate such class of solution with experimental data obtained through the Particle Image Velocimetry technique and time-lapsed particles visualizations.

success in reducing fuel consumption and carbon dioxide (CO 2 ) emissions. Recent examples such as Enercon's E-ship 1 have proven seaworthy and economically viable along major shipping routes [4].
Recently, some preliminary assessments of numerical simulations have been conducted by comparison with experimental investigation of Flettner rotors in order to evaluate the functional relationship and the interaction between the control factors [5], the preliminary design of the Flettner rotor as a ships auxiliary propulsion system [6], its evaluation with another wind power technology, namely, the towing kite [7], and characterized in terms of lift and drag coefficient [8].
In contrast to the previous studies, in this article, a simple model derived from the Navier-Stokes equations is obtained and compared with a simple experimental model that represents the Flettner rotors from the Enercon's E-Ship 1, with four large rotor sails [9] when the ship is at rest and no flow is incident to it. The objective is the study of the behaviour of the flow patterns due to the rotating cylinders.

Statement of the Problem
The theoretical approach begins with the mass and momentum conservation for real fluids [10] 0 (2) where the velocity vector is denoted by u , p is the pressure, ρ is the density, ν is the kinematic viscosity and t represents time. Considering that the surface is flat, and that the motions is laminar, thus the motion of the flow occurs on the x-y plane and the perpendicular velocity is negligible, thus we assume that the flow is two-dimensional. As a first approach, we consider a single rotor. Locating the origin in the geometrical center of the rotor and using polar coordinates, the flow can be considered as symmetric around the origin, that is, independent of the θ-direction. Thus the velocity vector is , which automatically satisfies the continuity Equation (1) and the Navier-Stokes Equation (2) for the r and θ components are Equation (4) is a homogeneous second order differential equation, thus we must impose two boundary conditions. The first one is that the velocity at infinity is zero, that is The second one is a non-slip condition, as the cylinder rotates with a uniform angular velocity ω , the tangential velocity at the cylinder's radius i R is: The stream lines can be obtained with the following relation Once the velocity field is known, the pressure distribution can be calculated from Equation (3) ( ) where C is a constant that can be obtained by evaluating ( ) p ∞ . The solution (5) and stream line function (7) in Cartesian coordinates are As stated previously, Equations (5) and (7) correspond to single rotor located at the origin. However, as the solution is linear, we can sum the solutions to consider the flow of four rotating cylinders, that is we can perform a superposition of rotors. In Cartesian coordinates, the total stream lines solutions Ψ for the four contemplated sets of rotation are as follows: x y x a y x y a x a y x y a x y x a y x y a x a y x y a x y x a y x y a x a y x y a x y x a y x y a x a y x y a where the function ψ represents the flow generated by a single rotor with negative (clockwise) or positive (counter clockwise) rotation depending on its sign, and a is the shifting length from the origin. Next sections show the experimental procedure and the comparison between this theoretical model and the experimental measurements.

Experimental Procedure
The

Main Results
In this section, the experimental results are compared with the simple theoretical model obtained from the Navier-Stokes equations. Figure 2 and Figure 3 show   demonstrates that the flow regime is laminar and can be compared with the theoretical model. The theoretical predictions are one order of magnitude higher than experimental measurements. This can be attributed to the friction of the container's bottom with the fluid. Since the model is two-dimensional this friction is not taken into account, thus the velocity in the theoretical model is higher than the experimental obtained through PIV.

Conclusions and Suggestions
A solution for the incompressible Navier-Stokes equations was derived in the creeping regime for the case of a single rotation rotor. Considering its linearity, it was superposed to consider an array of four rotors in an unbounded domain. An investigation of the validity of this solution was carried out experimentally. It turned out that this solution represents qualitatively the experimental problem. A verification was established through PIV and time-lapse particles visualization.