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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.

Boats ships have crossed the oceans around the globe for centuries by the use of paddling, sails, engine propellers or even turbine propellers. An alternative to the latter propulsion systems is the rotor ship. Through this alternative technique, the ship is propelled, at least in part, by large vertical rotors, sometimes known as rotor sails. German engineer Anton Flettner was the first to build a ship which attempted to tap this force for propulsion, and the ships are sometimes known as Flettner ships [_{2}) emissions. Recent examples such as Enercon’s E-ship 1 have proven seaworthy and economically viable along major shipping routes [

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 [

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 [

The theoretical approach begins with the mass and momentum conservation for real fluids [

∇ ⋅ u = 0 (1)

∂ u ∂ t + ( u ⋅ ∇ ) u = − 1 ρ ∇ p + ν ∇ 2 u (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 q-direction. Thus the velocity vector is u = u θ ( r ) , which automatically satisfies the continuity Equation (1) and the Navier-Stokes Equation (2) for the r and θ components are

u θ 2 r = 1 ρ d p d r (3)

d 2 u θ d r 2 + d d r ( u θ r ) = 0 (4)

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 u θ ( ∞ ) = 0 . 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 R i is: u θ ( R i ) = ω R i , where ω = 2 π f and f is the rotating cylinder frequency. The solution to Equation (4) is

u θ ( r ) = ω R i 2 r (5)

The stream lines can be obtained with the following relation

d ψ d r = u θ (6)

where ψ ( r ) is the stream function. The boundary condition is ψ ( R i ) = 0 , so we get

ψ ( r ) = ω R i 2 ( ln ( r ) − ln ( R i ) ) (7)

Once the velocity field is known, the pressure distribution can be calculated from Equation (3)

p ( r ) = − ρ ω 2 R i 4 2 r 2 + C (8)

where C is a constant that can be obtained by evaluating p ( ∞ ) . The solution (5) and stream line function (7) in Cartesian coordinates are

u ( x , y ) = ω R i 2 x 2 + y 2 cos ( arctan ( y / x ) ) (9)

v ( x , y ) = ω R i 2 x 2 + y 2 sin ( arctan ( y / x ) ) (10)

ψ ( x , y ) = ω R i 2 ( ln ( x 2 + y 2 ) − ln ( R i ) ) (11)

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:

Ψ 1 ( x , y ) = ψ ( x + a , y ) + ψ ( x , y + a ) + ψ ( x − a , y ) + ψ ( x , y − a ) Ψ 2 ( x , y ) = ψ ( x + a , y ) + ψ ( x , y + a ) − ψ ( x − a , y ) + ψ ( x , y − a ) Ψ 3 ( x , y ) = − ψ ( x + a , y ) + ψ ( x , y + a ) − ψ ( x − a , y ) + ψ ( x , y − a ) Ψ 4 ( x , y ) = − ψ ( x + a , y ) − ψ ( x , y + a ) + ψ ( x − a , y ) + ψ ( x , y − a ) (12)

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.

The experiment set up consist of four rotating cylinders immersed in a fluid layer. The working fluid is water at room temperature (20˚C). The fluid is contained in a 30 × 30 × 10 cm glass box which is open in the upper side, see

sufficiently far away from the flow. The height of the water level in the glass container is 8 cm. Above the fluid, a rigid table is placed at the top of the box with the four electric DC motors (6 V with gearbox) embedded in it. Considering the geometric center of the rigid table as the origin of coordinates, the electric motors are located at a distance of 2.5 cm from the origin. A rigid cylinder L = 10 cm is coupled to the shaft of every motor so that the cylinder is partially submerged in the liquid. The radius of every cylinder is R i = 0.4 cm . The DC motors are connected to a power supply whose output voltage is 6 V. As a single cylinder rotates, it promotes a circular fluid flow around itself. The direction of rotation of the cylinders can be changed by inverting the polarity of each motor. As every cylinder rotates independently, four sets of rotation are studied. Every set of rotation produces a different flow pattern. The frequency of rotation of the rotors is f = 0.667 Hz . The surface of the fluid is seeded with hollow glass spheres, with an approximated diameter of 10 mm. Images of the surface are recorded with a photographic camera Nikon D90 with a AF micro-nikkor 60mm f/2.8D lens placed underneath the glass box. Streamlines of the flow cases are obtained by using an exposition time of 30 s. The Particle Image Velocimetry (PIV) technique was used in order to obtain velocity vector fields. The experimental data gathered was used to perform a comparative with the theoretical results.

In this section, the experimental results are compared with the simple theoretical model obtained from the Navier-Stokes equations.

the experimental stream lines and the vector fields, respectively, for the four sets of rotation. The stream lines allow to easily identify the flow patterns. The first case,

is R e = U R i / ν = 8 , where U is the maximum velocity in the flow field, which 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.

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.

This research was supported by CONACYT, Mexico, under project 258623. A.F. thanks the Cátedras program from CONACYT.

Garzón, F. and Figueroa, A. (2017) The Study on the Flow Generated by an Array of Four Flettner Rotors: Theory and Experiment. Applied Mathematics, 8, 1851-1858. https://doi.org/10.4236/am.2017.812132