The concept of Flettner rotor, a rotating cylinder immersed in a fluid current, with a top-mounted disk, has been analyzed by means of unsteady Reynolds averaged Navier-Stokes simulations, with the aim of creating a suitable tool for the preliminary design of the Flettner rotor as a ship’s auxiliary propulsion system. The simulation has been executed to evaluate the performance sensitivity of the Flettner rotor with respect to systematic variations of several parameters, that is, the spin ratio, the rotor aspect ratio, the effect of the end plates, and their dimensions. The Flettner rotor device has been characterized in terms of lift and drag coefficients, and these data were compared with experimental trends available in literature. A verification study has been conducted in order to evaluate the accuracy of the simulation results and the main sources of numerical uncertainty. All the simulation results were used to achieve a surrogate model of lift and drag coefficients. This model is an effective mathematical tool for the preliminary design of Flettner rotor. Finally, an example of assessment of the Flettner rotor performance as an auxiliary propulsion device on a real tanker ship is reported.

In the era in which most of the world’s attention is focused on improving energy savings, the use of spinning cylinders as an auxiliary naval propulsion system has become a reality. Flettner rotors are rotating cylinders that, when immersed in a fluid stream, are able to produce fluid dynamic lift using the Magnus effect. This idea is due to the German engineer Anton Flettner who studied in the 1920s the effectiveness of spinning cylinders as a ship’s propulsion system. This kind of propulsion systems was enrolled for the first time in 1925-1926 on the Buckau ship, shown in Figure

Buckau, first Flettner’s ship (a), and E-Ship 1 by Enercon wind company (b).

The capability of infinite length rotating cylinders to produce aerodynamic forces was studied for the first time at the Langley NACA Laboratory by Reid [

The aerodynamic coefficients of an FR depend on various parameters (geometrical and functional). In the following subsections the most important parameters are briefly summarized.

The amount of aerodynamic force generated by a rotating cylinder, that is, an FR, is mainly dependent on the

Sketch of the FR with end plate and main relevant parameters.

Swanson [

The main shape factor of an FR is the aspect ratio

The idea of applying an end plate on FR to optimize its aerodynamic efficiency was first suggested by Prandtl [

The FR with end plate, also called Thom disk, is able to produce almost double the lift at high velocity ratios; for example,

A discussion of how the end plate size is related to

Concerning the use of FR for marine applications, not many research papers are available in the literature. An overview of the applications of the Magnus effect devices in the marine field is given in Morisseau [

Traut et al. [

The present research extends previous investigations presented in De Marco et al. [

To choose the ranges of the key parameters some constraints have been taken into account. For instance one has to consider the currently available technology as well as the practicality in marine applications. Another important factor is the vortex shedding risk (first and second mode) which depends on the mutual interaction between

The known installations of FR and their reference data are summarized in Table

Geometric, performance, and structural related parameters collected from all-known rotor ships.

Ship (year) | Buckau (1924) | Barbara (1926) | E-Ship 1 (2010) | Estraden (2014) |
---|---|---|---|---|

Type | Retrofit | Newbuild | Newbuild | Retrofit |

Height (m) | 15.6 | 17.0 | 27.0 | 19.0 |

Diameter (m) | 2.8 | 4.0 | 4.0 | 3.0 |

Aspect ratio | 5.6 | 4.3 | 6.8 | 6.3 |

End plate | Yes | Yes | Yes | Yes |

Material | Zinc coated steel | Aluminum | NA | Composite |

Max rpm | 135.0 | 150.0 | NA | 250.0 |

Estraden ship: latest FR installation.

Systematic variations of FR configurations have been investigated by means of URANS simulations in incompressible flow. All the analyses have been performed using the commercially available computational fluid dynamics software CD adapco STAR-CCM+ v. 9.06. The simulations were conducted using the same approach, such as the overset/chimera grid technique, described in De Marco et al. [

Values of analyzed variables.

Variables | Values | |||
---|---|---|---|---|

SR | 1.0 | 1.5 | 2.0 | 3.0 |

AR | 2.0 | 4.0 | 6.0 | 8.0 |

| 1.0 | 2.0 | 3.0 | / |

Sketch of all geometry of FRs tested.

The rotating motion of the FR was simulated using the overset/chimera mesh methodology with distance-weighted interpolation method. This method, which is especially suitable for rotational movements, uses an interpolation factor inversely proportional to the distance from acceptor cell to donor cell, as indicated in CD adapco User’s Guide [

Hybrid mesh approach, coupling unstructured and structured mesh, has been used for all the simulations. The computational domain contains two regions: the background, nonrotating, region and the overlapped, rotating, region (Figure

Section of the computational grid (a). Close-up views of the hybrid mesh (b).

The chosen URANS-solving algorithm uses a first-order forward Euler scheme for the temporal discretization, an implicit element-based finite volume method, and a segregated flow approach with second-order upwind discretization of the convective terms. A fully turbulent approach with

Summary of the numerical simulation setup.

Pressure link | Pressure | Convection term | Temporal discretization | Time step (s) | Iteration per time step | Turbulence model | Overset interpolation scheme |
---|---|---|---|---|---|---|---|

Simple | Standard | 2nd order | 1st order | Function of angular speed (Ω) | 11 | | Distance weighted |

It has to be noted that the simulation time step is a function of the angular speed

A box-shaped domain has been created around the cylinder geometry, as seen in Figure

Boundary conditions and domain dimensions in function of the main dimensions of FR (

In order to assess the numerical setup and to evaluate the simulation numerical uncertainty

The benchmark experimental data are derived from Badalamenti and Prince [

According to the Oberkampf and Blottner [

The numerical uncertainty evaluation was performed using two different methods: the grid convergence index (GCI) method and the correction factor (CF) method. The general form of the uncertainty evaluation, based on the generalized Richardson extrapolation (RE) method, can be written as follows:

The GCI method proposed by Roache [

The other method used is the CF described in Stern et al. [

The verification study has been carried out for the critical points of

The iterative uncertainties are estimated by the fluctuations of the time-history of the results in the last few periods, as indicated in Stern et al. [

The three grids tested.

Grids | Cells | |
---|---|---|

Grid A | Coarse | |

Grid B | Medium | |

Grid C | Fine | |

Grid and iterative uncertainty for

SR | Grids | Grid ratio | | | | % | % | % | % | % | |
---|---|---|---|---|---|---|---|---|---|---|---|

GCI | CF | ||||||||||

| 2.0 | A-B-C | √2 | 0.80 | | 1.20 | 8.91 | 4.26 | 0.26 | 8.91 | 19.30 |

2.5 | A-B-C | √2 | 0.96 | | 1.04 | 4.40 | 3.23 | 0.55 | 4.43 | 6.24 | |

| |||||||||||

| 2.0 | A-B-C | √2 | 0.97 | | 1.01 | 20.06 | 15.60 | 1.37 | 20.11 | 36.23 |

2.5 | A-B-C | √2 | 0.96 | | 1.04 | 19.18 | 14.02 | 2.15 | 19.30 | 31.36 |

Because

The values of simulation uncertainty reported in Table

Iterative convergence is achieved for all simulations and

Moreover, the verification procedure cannot be completed with the validation phase due to the lack of experimental uncertainty data. Therefore, only the results related to simulation numerical uncertainty are reported in Table

The numerical results of the verification study have been compared with the experimental data, as shown in Figure

Computational time required for the grids tested.

Comparison between CFD results and experimental data for lift (a) and drag (b) coefficient using mesh Case B and two different turbulence models.

For the reasons mentioned above, the grid of Case B has been assumed as the reference mesh. This grid guarantees an acceptable solution without an excessive computational effort (such as in Case C).

The comparison between experimental data and numerical results for

As shown in Table

The results of the simulations, performed into the variable ranges indicated in Table

Response curves for

The response curves in Figure

About the variation of

The ratio of these two formulas allows evaluating the aerodynamic efficiency of such devices. In both equations, the coefficients

The values of

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0 | 0 | 0 | | | | | | | |

0 | 0 | 1 | | | | | | | |

0 | 0 | 2 | | | | | | | |

0 | 1 | 0 | | | | | | | |

0 | 1 | 1 | | | | | | | |

0 | 1 | 2 | | | | | | | |

0 | 2 | 0 | | | | | | | |

0 | 2 | 1 | | | | | | | |

0 | 2 | 2 | | | | | | | |

0 | 3 | 0 | | | | | | | |

0 | 3 | 1 | | | | | | | |

0 | 3 | 2 | | | | | | | |

| |||||||||

| | | | | | | | | |

| |||||||||

2 | 0 | 0 | | | | | | | |

2 | 0 | 1 | | | | | | | |

2 | 0 | 2 | | | | | | | |

2 | 1 | 0 | | | | | | | |

2 | 1 | 1 | | | | | | | |

2 | 1 | 2 | | | | | | | |

2 | 2 | 0 | | | | | | | |

2 | 2 | 1 | | | | | | | |

2 | 2 | 2 | | | | | | | |

2 | 3 | 0 | | | | | | | |

2 | 3 | 1 | | | | | | | |

2 | 3 | 2 | | | | | | | |

A test of the reliability of the surrogate model has been performed using the experimental data available in Pearson [

Comparison of experimental and predicted values of

To evaluate the potentiality of the FRs as marine propulsion devices, it is important to consider that for FR on a ship the resulting wind speed is the vector sum of the environmental wind and the ship speed. If the resulting wind relative to the ship coordinate system is coming from the bow quarter direction as illustrated in Figure

Total force and thrust delivered by FR at different apparent wind directions: from bow quarter (a), from stern quarter (b).

In Figures

Figure

In (a)

As strongly suggested by Figure

Therefore the comparison of resistance of a ship with the thrust data, as shown in Figure

The aim of these considerations is a rough evaluation of the potentiality of the FR as a marine propulsion device. For a more comprehensive analysis of the FR effectiveness, more aspects have to be taken into account. These include, for instance, the asymmetric hydrodynamic flow condition produced by the transversal component of

In this paper, a systematic approach to identify the most influencing parameters on FR performance and the applicability of such device for marine application has been presented. Results from the numerical simulations have been used to achieve a surrogate model useful for preliminary FR designs.

The uncertainty analysis shows that the highest numerical uncertainty and error are for

Regarding the capability of FR to act as a propulsion device, it is confirmed that, in terms of magnitude,

Finally, the potentiality of the FR as a marine propulsion device, here evaluated on a tanker ship, highlights that, in a wide range of wind angles, a couple of FR can give a thrust whose magnitude is up to 30% of the ship resistance in the range of operational speed. Also, the assessment of the aerodynamic efficiency is less significant than the evaluation of each of the aerodynamic force components, as the drag gives a positive contribution to the thrust in a wide range of apparent wind angles.

Apparent wind angle (deg)

Reference area (

Aspect ratio

Drag coefficient

Lift coefficient

Thrust coefficient

Courant-Friedrichs-Lewy number

Correction factor

Correction factor method

Drag (N)

Cylinder diameter (m)

End plate diameter (m)

Iteration convergence error

Grid convergence error

Time step convergence error

Error evaluated by RE method

Dead weight tonnage (t)

Solution change

Comparison error

Frequency (Hz)

Factor of safety

Flettner rotor

Grid convergence index

Cylinder length (m)

Lift (N)

Large Eddy Simulation

Dynamic viscosity (Pa s)

Angular velocity (rad/s)

Air density (kg/m^{3})

Estimated order of accuracy

Reynolds number

Richardson extrapolation method

Refinement ratio

Spin ratio

Strouhal number

Effective thrust (N)

Total force (N)

Free stream or flux velocity (m/s)

Grid uncertainty

Iteration uncertainty

Simulation uncertainty

Time step uncertainty

Unsteady Reynolds averaged Navier-Stokes

Nondimensional wall distance.

The authors declare that they have no competing interests.

The authors gratefully acknowledge the availability of the Calculation Centre SCoPE of the University of Naples “Federico II” and thanks are due to SCoPE academic staff for the given support.