FENG Xue-mei

School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

Marine Design and Research Institute of China, Shanghai 200011, China, E-mail: koalafengfeng@163.com

LU Chuan-jing

School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

WU Qiong, CAI Rong-quan

Marine Design and Research Institute of China, Shanghai 200011, China

(Received October 31, 2011, Revised March 6, 2012)

NUMERICAL RESEARCHES ON INTERACTION BETWEEN PROPELLERS IN UNIFORM FLOW*

FENG Xue-mei

School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

Marine Design and Research Institute of China, Shanghai 200011, China, E-mail: koalafengfeng@163.com

LU Chuan-jing

School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

WU Qiong, CAI Rong-quan

Marine Design and Research Institute of China, Shanghai 200011, China

(Received October 31, 2011, Revised March 6, 2012)

Numerical simulations of the flow around two bidirectional staggered propellers are conducted in uniform flow. The computed open water performance of the fore-propeller is compared with the corresponding experimental results, and the influence of the fore-propeller on the aft one is carefully investigated. It is found that the inflow around the aft propeller close to the side of the fore is especially affected by the fore one, leading to abnormal circumferential distribution of force on the blade in the shade region. For either forces or velocity distributions, the abnormal changes behave contrarily for cases with the rotating speed larger or smaller than the idle. Moreover, the more the rotating speed of the fore differs from the idle, the larger the abnormal values become.

numerical simulation, interaction, propellers, open water performance

Introduction

The Computational Fluid Dynamics (CFD) for ship hydrodynamics has been developed rapidly over the last decade with the advancement of computing power and numerical algorithm. The numerical simulations of flow around marine propellers with true geometries have also become popular, especially in the aspect of predicting open water performance in both steady and unsteady ways[1-4]. Recently more and more detailed flow phenomena around propellers including different quadrant flows and cavitations[5-8]have been studied. In the 1990s, simulations of coupled ship and true propeller flow came forth and gradually replaced the body force method, Interactions among hull/propellers/free surface also began to be considered[9-12]. The CFD research group of the authors has also conducted a series of related compu-tations, and experienced the similar course from numerical investigations of open water performance of single propeller and interactions of hull and propeller, to simulations of flow around self-propelled ship with free surface[13-15]. Simulating the flow around a hull propelled by multi-propellers is a further step and four propellers is the future object to be computed. Obviously, the numerical simulation of the flow around multi-propellers in uniform flow is the primary work.

In this paper numerical simulations are carried out at model scale on bidirectional staggered propellers in uniform inflow using the CFD software Fluent. The simulated open water characteristics of the forepropeller are compared with the corresponding experimental results, showing the effectiveness of the model and simulation.

Influence of the fore-propeller on the aft exists for all cases except idle state of the fore. The inflow of the aft close to the side of the fore is especially affected, leading to abnormal circumferential distributionof force on the blade in the shadow region. For either forces or velocity distributions, the abnormal changes of the aft behave contrarily for cases with the rotating speed larger or smaller than the idle, and moreover, the more the rotating speed of the fore deviates from the idle, the larger the abnormal value appears.

1. Governing equations

Ship hydrodynamic problems are generally solved with the numerical code in the framework of the Reynolds-Averaged Navier-Stokes (RANS) equations. The continuity equation is

and the momentum equations are

where p is the static pressure,τ is the stress tensor, and ρg and F are the gravitational body force and external body forces (e.g., that arise from interaction with the dispersed phase), respectively,F also contains other model-dependent source terms such as porous-media and user-defined sources.

In the Reynolds averaging, the dependent variables in the instantaneous (exact) Navier-Stokes equations are decomposed into the mean (ensemble-averaged or time-averaged) and fluctuating components. So the Eqs.(1) and (2) can be written in Cartesian tensor form as

where δijis the Kronecker delta andare the unknown Reynolds stresses

The equations are made closed with the turbulence model, and here the SST k-ω model is employed:

In these equations, Gkrepresents the generationof turbulence kinetic energy due to mean velocity gradients, Gωthe generation of ω, Γkand Γωthe effective diffusivities of k andω, respectively, Ykand Yωthe dissipations of kandω due to turbulence,Dωthe cross-diffusionterm,and Skand Sωthe user-defined source terms.

2. Numerical methods

The RANS formulations are used and equations aresolved in a moving reference frame for steadyflow mode, and a sliding interface method used for unsteady-flow mode[16]. The pressure-velocity coupling and the overall solution procedure are based on the SIMPLE algorithm. The second-order scheme is used for pressure convection terms and second-order upwind difference scheme for diffusion terms.

Fig.1 Propeller model

All the propellers used in this paper have the samegeometries except the difference of left-handed or right-handed, as shown in Fig.1. The propeller has 5 blades with diameter D of 0.159 m, blade area ratio of 0.731, and boss/diameter ratio of 0.167.

3. Computational schemes

3.1 One blade

For steadysimulation, due to periodicity in thecircu mferential direction, only 1/5 sector of the full domain needs to be computed for the study of open water performance. The computational domain is created as one passage surrounding a blade: inlet at 1.5D upstream, outlet at 4D downstream, solid surfaces on the blade and hub aligned with uniform inflow, cylinder outer boundary at 1.5D from the hub axis;and two rotationally periodic boundaries with 72obetween. Hybrid meshesare generated:prismatic cells attached to the blade and hub surface and tetrahedral cells in the rest domain. Figure 2 shows the computational domain and grids used.

F ig.2 Computational domain and grids for one blade case

Boundary conditions are set as follows. On the inlet and outer boundary, velocity components of uniform stream with the given inflow speed are imposed. On the outlet boundary, the static pressure is set to be a constant value. On the blade and hub surfaces, the no-slip conditions are imposed. And on the periodic boundaries, rotational periodicity is ensured.

F ig.3 Computational domain and grids for one propeller case

Fig.4 Sketch of two bidirectional staggered propellers

A series of advance coefficientJ,0.1≤J≤1.1, are investigated, and compared with the measured data. Various values of J are obtained by keeping a constant flow speed of Va=-2.503m/s but varying the rotational speeds

Fig.5 Grids of two propellers case

Fig.6 Comparisons of open water performance

Fig.7Open water characteristics of the fore-propeller

3.2 One propeller

In the case ofunsteady simulation, the whole domain should be computed with the sliding mesh technique. The computational domain is defined with a cylinder of 8.3D diameter surrounding the propeller and hub. The inlet and outlet boundaries are located at 2.5D upstream and 5.3D downstream the center of the propeller respectively. The domain is split into global stationary part and moving part which is specified by a smaller cylinder enclosing the blades and hub entirely. Tetrahedral cells are generated for the global stationary block. The grid in the rotationalblock is built by using the rotational transformation of one passage used in the steady flow case four times, to turn at an angle of 72oper time, as illustrated in Fig.3. Boundary conditions are set similar with those of the steady simulation except periodic boundaries can not be used and the rotational speeds should be denoted to the moving part. The time step length is set aos 0.0003589 s corresponding to the propeller rotating 2.

Fig.8 Thrusts on five blades of the aft-propeller in cases of different rotational speeds

Fig.9 Sketch of phase angle

3.3 Two propellers

Two propellers in the open water are studied with the longitudinal distance of2.2Dand the transverse distance of 1.1D between each other, for which the upstream right-handed one iscalled the fore-propeller and another left-handed the aft-propeller, as shown in Fig.4. The computational domain is also defined with a cylinder with the diameter of 8.3D. The inlet is placed at 2.5D upstream the center of the forepropeller, and the outlet 5.3D downstream the center of theaft-propeller.

Table 1 Abnormal values of blade forces

Because of the presence of two propellers, the who le domain is split into a global stationary part and two moving parts surrounding each propellers respectively, completely the same as that in the case of one propeller. The stationary block is filled with tetrahedral cells also, and the moving parts are meshed in the manner of rotational transform adopted in one propeller case, as displayed in Fig.5. Boundary conditions are the same as those in the single propeller case in open water.

While theaft-propeller rotates at a constant rate of 15.48 rps, which is expressed as 1.0 w here, five rotational speeds are computed for the fore, 0 w (i.e., the fore-propeller holds still), 0.4 w, 0.7 w, 0.8 w, 0.9 w and 1.0 w, which are also indexed in the corresponding computation. In addition, two left-handed propellers are also studied at the same rotating speed of 1.0 w, which is marked with -1.0 w. The time step of each computation is soet as a value just making the fore-propeller rotating 2 per time step. For the case of 0 w, the time step is set equal to 0.0003589 s.

4. Comparison of the open water performance

The computed and measured open water dataare compared in Fig.6, in which three groups of computed KT, KQand η are included: data from one blade stead y simulation, one propeller unsteady simulation and data of the fore-propeller in the case of two-propellers. As can be seen from the figures, the three groups of computed plots have similar shapes with the corresponding experimental results, and the unsteady results for one propeller case agree better than the steady results especially at smaller J values, such as at J=0.9 with the differences of smaller than 2% for KTand KQand 4% for the open water efficiency. Butas Jgrows larger, the error for KTbecomes larger even to 10% at the point J=1.08, which leads to the larger error for the open water efficiency. The reason of the big differences may partly be due to the very small absolute value of KT, smaller than 0.1 as J＞1.0, and the doubted experimental results at large values of J may be also considered.

As to the unsteady results of the fore-propeller in the twopropellers case, the open water performance plots lie almost perfectly on those of one propeller case,providing a basis for the analysis in the following text.

5. Results and analysis

5.1 Forces

All the computed open water data of the forepropeller are plotted in Fig.7. As can be seen from the figure, the curves of KTand KQtend monotonously downward in the range of J[1.017,2.542], except the open water efficiency. In fact, the open water efficiency can not be illustrate d by one continuous curve but two discrete ones becauseof the interruption induced by the infinity of the value when the corresponding KQequals to zero, at about J=1.287, which is very close to J=1.27 for the case of 0.8 w. As is known, the torque of a propeller is zero when it runs idle, that is, the fore-propeller is almostat idle motion when rotating at 0.8 w. Here the 0.8 w case is approximately regarded as idle status to be convenient for the following analysis, and the rotational speed is expressed asωf. Because of the negative value of KTat 0.8 w, the curves of the open water characteristics extend from the first quadrant to the forth, for either force or torque.

Figure 8 gives the forces on blades of the aft-propeller in a blade-period at six different rotational speeds for the fore-propeller. The torques are not given for the reason of almost the same behavior as the forces.

Fig.10 Velocity vectors

It is noticeable that the forces for five cases behave very differently with those of single propeller case besides the 0.8 w case, of which forces on the fiveblades are almost the same with each other neglectingtheallowablenumericalerror,thatis,theaftpropeller is hardly influenced by the fore. However, in other cases, forces on five blades present abnormal in turn in a blade-period, moreover, the abnormality occurs when blade just approaches to the siode of forepropeller, that is, the position of about 0 shown in Fig.9. When the rotational speed of the fore-propeller is smaller than ωf, such as in the cases of 0 w, 0.7 w, the abnormal force on the blade approaching to 0ois larger than the mean force of the other four blades. On the contrary, in the cases of 0.9 w, 1.0 w and -1.0 w, for which the rotational speed is larger than ωf, the abnormal force is smaller than the mean force of the other four blades. Furthermore, the more the rotational speed of the fore deviates from the idle, the larger the abnormal value of the force, especially in thecase of 0 w, the relative abnormal value reaches 25.71%, as listed in Table 1. In addition, slight difference between abnormal values of the case 1.0 w and -1.0 w is observed.

Fi g.11 Relative axial velocity Vx/Va

5.2 Velocity

As was mentioned above, the abnormal forces on blades directly reflect the effect of the fore-propeller on the aft, which may be partly explained through the observation of the wake of the fore and inflow of the aft.

Fig .12 Relative radial velocity Vr/Va

Fig .13 Relative tangential velocity Vt/Va

To make it easy to analyze, a plane perpendicular tothe axis of the aft-propeller is chosen at the position of 0.11Dupstream the aft, i.e.,2.1D downstream the fore, and white homocentric circles are also plotted with radii of 0.7R, 0.8R, 0.9R , 1.0R and 1.2R respectively. The velocity vectors in the plane are illustrated from a view downstream in Fig.10 for the four cases of 0 w, 0.7 w, 0.8 w and1.0w, with color meaning the value of relative axial velocity Vx/Va, where Vxis the axial velocity in relative cylindrical coordinates with axis pointing to upstream. It is clear that wake of the fore in the plane beha-ves counterclockwise flow as the rotational speed is smaller than ωf, just like the cases of 0 w and 0.7 w, and clockwise flow for case of larger rotational speed than ωf, like 1.0 w, while hardlyrevolving at idle motion like 0.8 w. Furthermore, it is also found that the inflow of the aft has been affected by the wake, especially the flow field at the side near the fore with obvious centrifugal or centripetal flows relative to the aft-propeller axis in some circumferential and radial range. In other words, the influence of the fore wake has extended to the aft propeller disc and changed the distributions of the velocity components.

The distributions of the relative axial, radial and tangential velocities marked with Vx/Va, Vr/Vaand Vt/Vain the plane may be more easily explained in details. Figure 9 describes the angular coordinate along circles θ around the aft-propeller from the downstream view, and the zero angle is defined as the direction exactly pointing to the fore side. Based on the description, the distributions of Vx/Va, Vr/Vaand Vt/Vaalong the circles of 0.7R, 0.8R, 0.9R and 1.0R are shown in Figs.11-13. It must be noted that the positive value of Vx/Va, Vr/Vaand Vt/Vameans flow along with the incoming flow, centripetal and clockwise flow respectively, attributing to the negative value of Va.

It canbe seen that the three velocity components all keepregular distributions in the idle 0.8 w case same as the single propellercase, but not as the other cases. For the cases of 0 w, 0.7 w and 1 w, for relative axial, radial or tangential velocities, the uniformity of the distribution alongcircles is broken by abnormal chan ges near zero angle, consistent with the analysis of the velocity vectors in Fig.10. And furthermore, the abnormal reduces gradually from 1.0R to 0.7R and almost concentrate in the range of -36o-36o, which is also indicated in Fig.9. For Vx/Vaand Vt/ Va, the further the rotational speed of the fore-propeller from the idle, the larger the abnormal values of the velocity components, especially for the case of 0 w. And the opposite changes for the cases of larger rotational speeds than the idle present for those of smaller ones. Comparisons of Vr/Vashow some differences bet ween 0 w case and others. The values of Vr/Vaat 0.8R, 0.9R and 1.0R develop from positive to negative with angle changing from about -36ofor the case of 0 w, which means centripetal to centrifugal flows, and not the same as the centripetal flows in the region at large values of r/R in other cases, again consistent with the results in Fig.10.

By connecting with the forces on blades, it is found that the blade with abnormal force just incurs in the region close to the abnormal inflow, and the velocity distributions of the inflow have inherent relations with the forces, which is consistent with the traditional theory of propeller.

6. Conclusions

Computations have been conducted to simulate the flow around one blade, one propeller and two bidirectional staggered propellers in uniform flow respectively. The open water characteristics obtained in all the three cases have been compared and found in agreement with those of experimental results. The idle rotational speed is also sought as the torque equaling to zero, which is found to be a critical state of the interactions between the two propellers.

The investigation of the flow around two propellers reveals that the influence on the aft-propeller of the fore is almost nigligible at the idle rotational speed of the fore. In other cases, the forces on the aft blades behave abnormal in turn in a blade-period, and the abnormal blade is observed just located at the nearest position to the fore-propeller. The velocity distributions of the inflow around this blade are also different from the others with abnormal velocity components. Furthermore, for either forces or velocity distributions, the abnormal changes behave contrarily for cases with rotational speed larger or smaller than the idle, and moreover, the more the rotational speed of the fore deviates from the idle, the larger the abnormal values.

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10.1016/S1001-6058(11)60291-6

* Biography: FENG Xue-mei (1976-), Female, Ph. D., Engineer

DOI: 10.1016/S1001-6058(11)60292-8