Analyzing the hydrodynamic behavior of B-Series co-rotating tandem propellers

An Undergraduate Thesis on Analyzing the hydrodynamic behavior of B-Series co-rotating tandem propellers Course No: NAME 400 By Mohammad Ali (Student ID: 1912045) Md Shumit Islam Manik (Student ID: 1912048) Supervised By Dr. Md. Shahjada Tarafder Professor, Department of NAME, BUET Submitted to the Department of Naval Architecture and Marine Engineering, In partial fulfillment of the requirements for the degree of Bachelor of Science In Naval Architecture and Marine Engineering March 2025 Department of Naval Architecture and Marine Engineering, Bangladesh University of Engineering and Technology, Dhaka-1000, Bangladesh DECLARATION This is to certify that the work presented in the thesis is carried out by the authors under the supervision of Dr. Md. Shahjada Tarafder, Department of Naval Architecture and Marine Engineering, Dhaka. No portion of the work contained in this thesis has been submitted in support of an application for another degree or qualification of this or any other university or institution of learning. Authors Name: Mohammad Ali Name: Md Shumit Islam Manik Student ID: 1912045 Student ID: 1912048 Date: Date: Dr. Md. Shahjada Tarafder Department of Naval Architecture and Marine Engineering Bangladesh University of Engineering and Technology, Dhaka ACKNOWLEDGEMENT Dr. Md. Shahjada Tarafder, Professor of the Department of Naval Architecture and Marine Engineering at Bangladesh University of Engineering and Technology (BUET), Dhaka-1000, Bangladesh, is owed a great debt of gratitude for his helpful suggestions and cooperation throughout the course of this research project. Without his counsel and assistance, this initiative would not have been a success. The completion of this research was made possible by his constant supervision, constructive criticism, valuable advice, scholarly guidance, and momentary encouragement, as well as his wholehearted support by providing all papers, books, and internet materials related to this research and other facilities. We are quite grateful to the seniors who assisted us with the procedures. The department of Naval Architecture and Marine Engineering at Bangladesh University of Engineering and Technology (BUET), Dhaka-1000, Bangladesh, receives our best wishes. We are most grateful to Allah Almighty for the successful completion of the project. ABSTRACT This study investigates the hydrodynamic behavior of B-Series co-rotating tandem propellers, focusing on the influence of axial spacing and angular difference and hydrodynamic performance comparison with conventional propeller. A RANS-based CFD solver is used for numerical simulation. The results show that integrating the co-rotating tandem propeller increases thrust and torque coefficients across all loading ranges, although open water efficiency decreases slightly under moderate to high loadings and also an effective solution where problems like propeller cavitation is anticipated. Axial spacing shows minimal effect on open water efficiency, although a small increase on thrust and torque coefficients is observed at an axial spacing ratio of 0.2 between advance coefficient in range 0.2 to 0.4. Angular difference has minimal impact, except in bollard condition where thrust and torque are maximized at zero degrees and also provides higher open water efficiency than conventional propellers for specific advance coefficients. Keywords: Co-rotating Tandem Propellers Axial Spacing, Angular Difference, Diameter Ratio Bollard Condition Open Water Characteristics Tandem Propeller Performance TABLE OF CONTENTS ACKNOWLEDGEMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Objective Of Present Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2. THEORETICAL FRAMEWORK . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1 Open Water Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3. DESIGN AND ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.1 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2 Geometry Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.3.1 Domain Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.3.2 Boolean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.3.3 Meshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.3.4 Solver Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.3.4.1 Boundary Condition For Computational Domain . . . . . . . . . . . . . . . . 22 3.3.5 Residuals Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4. RESULTS AND DISCUSSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.1 Conventional Propeller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.2 Tandem Propeller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.3 Parametric Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.3.1 Influence of Axial Spacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.3.2 Influence of Angular Difference . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 6. REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 7. APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Appendix A: Conventional Propeller Open Water Characteristics . . . . . . . . . . . . . . . 47 Appendix B: Tandem Propeller Open Water Characteristics . . . . . . . . . . . . . . . . . . 48 Appendix C: Meshing Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Appendix D: Solver Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 LIST OF FIGURES Figure 3.1: Tandem Propeller 17 Figure 3.2: Conventional Propeller 17 Figure 3.3: Computational Domain 18 Figure 3.4: Meshing 20 Figure 3.5: Inflation Layer 20 Figure 3.6: Boundary Conditions 23 Figure 3.7: Residuals Vs Iterations at J=0.2 26 Figure 3.8 : Thrust Vs Iterations at J=0.2 27 Figure 3.9 : Moment Vs Iterations at J=0.2 28 Figure 4.1 : Open Water Characteristics of the B4-55 Propeller with P/D = 1.0. Validation study of the present CFD results with Oosterveld and van Oossanen, 1975. 30 Figure 4.2 : Open Water Characteristics of the Tandem Propeller. Validation study of the present CFD results with experiments presented in Sun Qin and Gu Yunde (1991). 31 Figure 4.2 : Open Water Characteristics of the Tandem Propeller. Validation study of the present CFD results with experiments presented in Sun Qin and Gu Yunde (1991). 31 Figure 4.3: Conventional Propeller Pressure Contour at Suction Side at J = 0.2 32 Figure 4.4: Tandem Propeller Pressure Contour at Suction Side at J = 0.2 32 Figure 4.5: Conventional Propeller Pressure Contour at Pressure Side at J = 0.2 32 Figure 4.6: Tandem Propeller Pressure Contour at Pressure Side at J = 0.2 32 Figure 4.7: Conventional Propeller Velocity Streamline at J = 0.2 33 Figure 4.8: Tandem Propeller Velocity Streamline at J = 0.2 33 Figure 4.9: Influence of Axial Spacing on the total thrust coefficient of the tandem propeller 36 Figure 4.10 : Influence of Axial Spacing on the total torque coefficient of the tandem propeller 37 Figure 4.11: Influence of axial spacing on open water efficiency of the tandem propeller 38 Figure 4.12: Influence of angular difference on the total thrust coefficient of the tandem propeller 41 Figure 4.13 : Influence of angular difference on the total torque coefficient of the tandem propeller 42 Figure 4.14: Influence of angular difference on open water efficiency of the tandem propeller 43 LIST OF TABLES Table 3.1 : Mesh Sensitivity Analysis 21 21 Table 4.1 : comparison between the experimental values, CFD simulation results for conventional propeller 29 29 Table 4.2 : comparison between the experimental values, CFD simulation results for Tandem propeller 31 31 Table 4.3 : Data for Analyzing the Effect of Axial Spacing on Thrust Coefficients, Torque Coefficients, and Open Water Efficiency for Axial Spacing Ratios of 0.2, 0.3, 0.4, and 0.5 35 Table 4.4 : Data for Analyzing the Effect of Angular Difference on Thrust Coefficients, Torque Coefficients, and Open Water Efficiency for Angular Difference at 0 degree, 30 degree and 50 degree 40 1. INTRODUCTION The maritime industry is increasingly competitive, demanding high speed, low power consumption, and improved propeller performance. To meet these demands, ongoing research and development are focused on optimizing marine propellers. Accurate performance testing is required under non-uniform flow conditions, which better represent real-world environments. Cavitation, which reduces propeller efficiency, is a critical factor and must be accurately estimated for effective propeller design. Computational Fluid Dynamics (CFD) simulations have become an essential tool in this process, allowing for better design predictions. Propeller design has traditionally relied on empirical charts and theoretical models, such as the Wageningen B-Series, which has been optimized using polynomial regression. Modern computational techniques now allow for more efficient and precise propeller designs, including B-Series co-rotating tandem propellers. This study investigates the hydrodynamic behavior of tandem propellers, focusing on the impact of axial spacing, angular difference, and diameter ratio on performance and compares them with conventional propellers. Literature Review The screw propeller has been the primary propulsion device for marine vessels throughout maritime history. The concept of multiple co-axial marine propellers dates back to the early 19th century, with the contra-rotating propeller patented by John Ericsson (1836). In contrast, tandem propellers, typically comprising two co-rotating propellers mounted on the same shaft, have a more recent application in marine propulsion. One of the earliest documented uses of tandem propellers was in 1897 by Sir Charles Parsons on the steam turbine-powered vessel Turbinia, where three co-axially mounted propellers helped reduce cavitation effects and increase thrust power. Tandem propellers are particularly suitable for vessels requiring high thrust with restricted propeller diameters, such as inland and coastal vessels with shallow drafts. These propellers distribute thrust loading between two propellers, maintaining a simpler and more cost-effective mechanism compared to contra-rotating propellers. Despite their potential advantages, tandem propellers have not received as much attention from propulsion hydrodynamicists as other types of screw propulsion. Earlier studies discussed the application of tandem propellers for high-powered merchant vessels. In 1966, Titoff and Biskup showed that tandem propellers could reduce the variable hydrodynamic loads transmitted to the propeller shaft, particularly on tankers. David and Hecker (1973) compared the open water characteristics of 3-bladed tandem propellers at different relative angular positions of the blades, finding that tandem arrangements could provide higher open water efficiency than conventional propellers for specific advance coefficients. Various design methodologies for tandem propellers have been developed over the years. Denny (1978) proposed a procedure for designing tandem propellers working in uniform inflow, while Sun et al. (1978) developed a theoretical method based on lifting line theory to calculate their performance. An open water test series for B-Series tandem propellers showed improved efficiency compared to conventional propellers, especially in cases of diameter restrictions and high power coefficients. Sun and Gu (1981) explored the optimal phase angle between the forward and aft propellers, demonstrating the benefits of tandem configurations in reducing stern vibrations. Several studies have continued to investigate different aspects of tandem propellers, including optimal phase angle, axial and angular spacing, diameter ratios, and pitch distribution. Sun and Gu (1991) developed simplified theoretical methods to evaluate the performance of tandem propellers and presented design charts for B4-40 and B4-55 propellers. Performance tests on various vessel types, including tugs and cargo ships, showed improvements in vessel speed and bollard pull with tandem propellers. With advancements in computational methods, CFD has become a vital tool for propeller design. Koronowicz et al. (2010) developed an algorithm for designing tandem propellers by accounting for the hydrodynamic loading and induced velocities of both propellers. More recent studies, such as those by Djahidi and Omar (2017), have explored the impact of geometrical parameters using RANS approaches, revealing that the optimal axial displacement is 0.6 times the propeller diameter, and that angular displacement has minimal effect. Cavitation studies by Djahidi and Omar (2019) highlighted that cavitation is more prominent on the forward propeller due to higher loading. Objective Of Present Research Analyzing the above researches and established full-scale results, it is evident that tandem propellers offer a promising solution for vessels with limitations on propeller diameters, high loading conditions, and concerns such as hull vibration and propeller cavitation. This study investigates key aspects of tandem propellers, aiming to enhance the understanding of the performance of both propellers in tandem under various design and loading configurations. The research aims to provide insights that will aid in the optimization of tandem propeller systems, particularly in terms of tailoring thrust distribution and improving overall efficiency. Objectives of this thesis are: To investigate the hydrodynamic behavior of co-rotating tandems propeller of which the distance between the propellers is in the range of 0.2D to 0.5D. To investigate effects of angular difference on hydrodynamic performance of tandem propellers. To Compare the hydrodynamic performance of tandem propellers with conventional propeller. 2. THEORETICAL FRAMEWORK 2.1 Open Water Characteristics The performance of a propeller is generally assessed in terms of its thrust output, torque, and efficiency. To evaluate these hydrodynamic characteristics, the open water performance of the propeller is studied. The open water characteristics are commonly expressed through several dimensionless quantities: the advance coefficient (J), the thrust coefficient (KT), the torque coefficient (KQ), and the open water efficiency (η0).These quantities are often represented graphically in the KT-KQ diagram, where KT, KQ, and η0 are plotted as functions of J. The following formulae are employed to calculate these dimensionless quantities for the tandem propeller. Thrust Coefficient,K_T=(T_f+T_a)/(ρn^2 D^4 ) 〖Torque Coefficient,K〗_Q=(Q_f+Q_a)/(ρn^2 D^5 ) Advance Coefficient,J=V_A/nD 〖Velcocity Of Advance V〗_A=V_S⋅(1-w) 〖Open Water Efficiency,η〗_0=K_T/K_Q ⋅J/2π 2.2 Governing Equation In the past decade, significant progress has been made in applying computational fluid dynamics (CFD) to the analysis and design of marine propellers. Various methods for modeling flow physics have been developed, with the Reynolds-Averaged Navier-Stokes (RANS) method gaining the most popularity due to its relatively shorter computational times compared to other approaches. CFD techniques are primarily based on the fundamental governing equations of fluid dynamics, which mathematically express the conservation laws of physics. The following physical principles are incorporated into a CFD analysis. Equation of Continuity: ∂ρ/∂t+∂(ρu_x )/∂x+∂(ρu_y )/∂y+∂(ρu_z )/∂z=0 Conversion of momentum along X axis: ∂(ρu_x )/∂t+∇⋅(ρu_x u)=-∂p/∂x+∇⋅(μ∇u_x )+ρg_x  "(2.2.1)" Conversion of momentum along Y axis: ∂(ρu_y )/∂t+∇⋅(ρu_y u)=-∂p/∂x+∇⋅(μ∇u_y )+ρg_y  "(2.2.2)" Conversion of momentum along Z axis∶ ∂(ρu_z )/∂t+∇⋅(ρu_z u)=-∂p/∂x+∇⋅(μ∇u_z )+ρg_z  "(2.2.3)" Equations of Turbulence Kinetic Energy(k): D/Dt (ρk)=∇⋅(ρD_k ∇k)+P-ρϵ Where k="Turbulent kinetic energy"  ["m" ^2 "s" ^(-2) ] D_k="Effective diffusivity for"  k P="Turbulent kinetic energy production rate"  ["m" ^2 "s" ^(-3) ] ϵ="Turbulent kinetic energy dissipation rate"  "m" ^2 "s" ^(-3) Equations of Energy Dissipation(ϵ): D/Dt (ρϵ)=∇⋅(ρD_ϵ ∇ϵ)+(C_1 ϵ)/k (P+C_3 2/3 k∇⋅u)-C_2 ρ ϵ^2/k where D_ϵ="Effective diffusivity for"  ϵ C_1="Model coefficient"   C_2="Model coefficient"   The turbulent viscosity equation〖,ν〗_t where C_μ = Model coefficient for the turbulent viscosity [-] ν_t= Turbulent viscosity m^2 s^(-1) 〖 ν〗_t= C_μ k2/ ϵ 3. DESIGN AND ANALYSIS 3.1 Design The design of a propeller is essential for ensuring a ship operates smoothly. It involves an iterative approach, and today, computer software is commonly employed for propeller design. The geometry of the Wageningen B-series propeller continues to be widely used as a foundational reference in propeller design. Therefore, the Wageningen B-series geometry has been adopted in this case. 3.2 Geometry Development The first step in the geometry development of the tandem propeller involves designing the individual propellers using the CAESES Wageningen B-Series Propeller Generator. This tool allows for precise customization of key parameters, such as the number of blades, diameter, and pitch, based on the well-established B-series formula. After creating the individual propellers, they are aligned along the same shaft using CAD software, specifically Rhinoceros. The two propellers are positioned one behind the other, ensuring proper alignment and spacing for optimal performance in the tandem configuration. Figure 3.1 : Tandem Propeller Figure 3.2: Conventional Propeller Where L = Axial Spacing and θ = Angular Displacement 3.3 Analysis The numerical analysis of the propeller is performed using Ansys Fluent. Prior to conducting the analysis, the propeller geometry is imported into the geometry module of the Ansys Fluent software for preparation and preprocessing. 3.3.1 Domain Generation The computational domain is defined by two distinct regions: an inner cylindrical rotating region that encompasses the two propellers and an outer cylindrical static region, which represents the surrounding fluid domain Figure 3.3 : Computational Domain The solution domain represents a cylinder with its inlet located at 4D upstream, and its outlet located at 6D downstream of propeller plane. In the radial direction, the domain was considered up to a distance of 4D from the axis of the hub. 3.3.2 Boolean The two domains are separated by constructing an interface using the Boolean subtract operation. This operation is utilized to add and subtract different solids, enabling the proper definition of the computational domains. For Both Conventional and Tandem propeller, two Boolean subtract operations were performed as follows: First Boolean Subtract: Tool body: Propeller Target body: Rotary domain Preserve: No Second Boolean Subtract: Tool body: Rotary domain Target body: Static domain Preserve: Yes 3.3.3 Meshing Before simulating the fluid flow around the propeller, proper meshing of the domains is crucial. The discretization of the rotary and static domains is done using the Finite Volume Method (FVM), with tetrahedral elements applied for better accuracy. The mesh type is unstructured, allowing for greater flexibility in meshing complex geometries. Both global and local meshing techniques have been utilized to ensure a high-quality mesh throughout the domain. Refined mesh zones are introduced around critical regions, such as the blade, hub, and near the blade tip of the propeller where smaller element sizes are applied due to the complex geometry, ensuring better results. These regions are crucial for capturing high flow gradients and accurately simulating the propeller's performance. Additionally, five inflation layers are applied, growing from the blade and hub surfaces. The transition ratio between the layers is set to 0.272, with a growth ratio of 1.1, ensuring a smooth transition and improved simulation accuracy. The Details of Mesh In Ansys Meshing is given in Appendix C. Figure 3.4 : Meshing Figure 3.5 : Inflation Layer Mesh Convergence A grid convergence test is conducted by varying the element size of the propeller from 20 mm to 60 mm, and observing the change in the results of KT, KQ and open water efficiency η0. Reducing the element size results in a reduction of the error percentage for open water efficiency. The least percentage of error is observed when the element size is 20 mm, but the simulation time is significantly longer compared to other element sizes. For better results the element size of 40 mm was chosen for the simulation considering efficient simulation time. This provides a reasonable error percentage while ensuring a more efficient simulation time. Element Size (mm) Total number of elements % Error KT % Error 10KQ % Error η 20 1630614 7.025 4.578 3.607 40 1197870 9.034 6.784 4.889 60 946214 12.482 9.464 8.338 Table 3.1 : Mesh Sensitivity Analysis 3.3.4 Solver Setting The numerical simulations are performed using the RANS solver for solving the fluid dynamics around the propeller. The flow is assumed to be steady and incompressible, which simplifies the analysis also viscous effect is considered here. The algorithm used for pressure-velocity coupling is SIMPLE, which is widely used for steady-state simulations. The propeller typically operates in water, which is considered a single-phase fluid, necessitating no need of using of a multiphase function. In the fluid zone, frame motion is applied to the rotary region, where an angular velocity of 12 revolutions per second (rps) is implemented. This setup is known as a moving reference frame. The turbulence model applied is the Realizable K-epsilon model, which is appropriate for a wide range of turbulent flows, ensuring accurate predictions for the flow around the propeller. To model the flow near solid boundaries, the Standard Wall Function is applied, which helps in resolving the velocity and turbulence profiles close to the wall. The solution method used in this analysis employs the SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) scheme for pressure-velocity coupling. The flux type is selected as Rhie-Chow with distance-based discretization. For spatial discretization, the gradient is calculated using the Least Squares Cell-Based method. The pressure is discretized with second-order accuracy, while the momentum is discretized using the Second-Order Upwind scheme. The Turbulent Kinetic Energy and Turbulent Dissipation Rate are both discretized using the First-Order Upwind scheme. The Details of Solver Setting In Ansys Fluent are given in Appendix D 3.3.4.1 Boundary Conditions for Computational Domain: In the computational fluid dynamics (CFD) analysis of tandem propellers, it is crucial to define the appropriate boundary conditions for the computational domain. The computational domain is divided into distinct regions, including a rotating region and a static region. The rotating region represents the area around the propellers, where the fluid experiences rotational motion due to the propeller’s action. In this region, a velocity inlet is applied to simulate the incoming flow, which is directed towards the propellers. The pressure outlet boundary condition is applied at the far end of the computational domain, allowing the flow to exit with a specified pressure value. A symmetry condition is applied along the sides of the domain. This symmetry assumption helps reduce computational complexity while maintaining the accuracy of the results. The boundary conditions specified here are crucial for ensuring accurate simulation results in tandem propeller studies. Figure 3.6: Boundary Conditions 1.Inlet Velocity V_inlet: \( V_inlet= f(J)\) where \( J \) is the Advance Coefficient.The inlet velocity varies with the advance coefficient. In the Cartesian coordinate system,the inlet velocity is expressed as: V_inlet= ( V_x,V_y,V_z )= (f_x (J),0,0 ) where f_x (J) represents the velocity component in the x -direction,and both \( V_y \) and \( V_z \) are zero (i.e.,no flow in the \( y \)- and \( z \)-directions). 2.Outlet Pressure P_outlet: P_{outlet} = 0 Pa (Gauge Pressure) The pressure at the outlet is set to zero.Since pressure is a scalar,it remains the same in all directions: P_{outlet} = 0 (for all directions: \( x \),\( y \),\( z \) ) 3.Rotating Region Rps ω_({rot}\)): ω_{rot} = 12 Rps The propeller rotates at 12 revolutions per second.The rotational velocity components in the Cartesian system are derived from the angular velocity: ω_{rot} = ( ω_x,ω_y,ω_z )= ( 0,0,12 ) Rps where the rotation is assumed to be along the z axis in the rotating frame of reference. 4.Propeller Wall (No Slip Condition): 〖 V〗_{wall} = 0 The no slip condition at the propeller wall implies that the velocity at the surface is zero. In the Cartesian coordinate system,this is expressed as: ( V_x,V_y,V_z )_{wall} = ( 0,0,0 ) 5.Symmetry Boundary: For the symmetry plane,the flow^' s normal component is zero, and the tangential velocity is mirrored across the plane.Mathematically, the symmetry boundary condition implies: n.V= 0 where n is the normal vector to the symmetry plane,and V is the velocity vector. In Cartesian components: ( V_x,V_y,V_z )_{symmetry} = ( -V_x,V_y,V_z ) ( for symmetry in \( x \)-axis,and similarly for \( y \)- and \( z \)-axis if applicable) 3.3.5 Residuals Study The process of ensuring the convergence of a simulation by checking the residuals and thrust force and torque convergence. It explains that all calculations are performed for the centroid of each individual cell, and the residuals are computed to check for force or energy imbalances. When these residuals approach zero, the results are considered accurate, although achieving this is challenging. In this case, the residuals are observed to stop changing after 1000 iterations, indicating that the simulation has converged. Figure 3.7 : Residuals Vs Iterations at J=0.2 Figure 3.8 : Thrust Vs Iterations at J=0.2 Figure 3.9 : Moment Vs Iterations at J=0.2 4. RESULTS AND DISCUSSION 4.1 Conventional Propeller The open water characteristics of the Wageningen B-Series propeller are obtained from the work of Oosterveld and van Oossanen (1975). The table below presents a comparison between the experimental values, CFD simulation results, and the corresponding error between these two sets of data. J kt(exp) kt(cfd) 10kq(exp) 10kq(cfd) n(exp) n(cfd) error(kt) error(10kq) error(n) 0.0001 0.4242 0.4323 0.6129 0.6231 0.0001 0.0001 -1.9021 -1.6683 -0.2299 0.2000 0.3716 0.3784 0.5477 0.5457 0.2159 0.2207 -1.8450 0.3682 -2.2214 0.4000 0.3038 0.3048 0.4655 0.4857 0.4155 0.3995 -0.3282 -4.3322 3.8378 0.6000 0.2241 0.2190 0.3657 0.3851 0.5852 0.5431 2.2606 -5.3111 7.1898 0.8000 0.1356 0.1305 0.2477 0.2542 0.6968 0.6538 3.7067 -2.6241 6.1690 1.0000 0.0413 0.0424 0.1111 0.1162 0.5914 0.5801 -2.5778 -4.5915 1.9253 Table 4.1 : comparison between the experimental values, CFD simulation results for conventional propeller The Experimental Data of Table 4.1 for Conventional Propeller is extracted from Appendix A Open Water Characteristics Diagram is plotted based on the data in the table -- Figure 4.1 : Open Water Characteristics of the B4-55 Propeller with P/D = 1.0. Validation study of the present CFD results with Oosterveld and van Oossanen, 1975. 4.2 Tandem Propeller The table below presents a comparison between the experimental values, CFD simulation results, and the corresponding error between these two sets of data for Tandem Propeller. The Experimental Data for Table 4.2 for Tandem Propeller is extracted from Appendix B J kt(exp) kt(cfd) 10kq(exp) 10kq(cfd) n(exp) n(cfd) error(kt) error(10kq) error(n) 0.0001 0.538 0.5124 0.73 0.7185 0.0001 0.0001 4.7553 1.5780 3.2283 0.2 0.44 0.4400 0.638 0.6350 0.2195 0.2206 0.0001 0.4719 -0.4740 0.4 0.358 0.3491 0.54 0.5390 0.4221 0.4123 2.4884 0.1863 2.3064 0.6 0.25 0.2299 0.41 0.3912 0.5823 0.5613 8.0202 4.5783 3.6070 0.8 0.15 0.1345 0.31 0.2901 0.6161 0.5903 10.3282 6.4194 4.1770 1 0.02 0.0162 0.158 0.1380 0.2015 0.1867 19.0418 12.6582 7.3087 Table 4.2 : comparison between the experimental values, CFD simulation results for Tandem propeller Figure 4.2 : Open Water Characteristics of the Tandem Propeller. Validation study of the present CFD results with experiments presented in Sun Qin and Gu Yunde (1991). Comparing the open water performance characteristics of the conventional and tandem propellers reveals that In the bollard condition, the thrust coefficient KT of the tandem propeller increases by 18.52% compared to the conventional propeller, which indicates that the tandem configuration produces more thrust under these conditions. Similarly, the torque coefficient KQ of the tandem propeller is increased by 15.24%, suggesting that the tandem arrangement generates more torque. However, the maximum efficiency of the tandem propeller decreases by 5.19% compared to the conventional propeller, indicating a trade-off in efficiency for the increased thrust and torque. Furthermore, the pressure distribution on the blade surface differs between the two configurations. The tandem propeller has less high-pressure area and low-pressure area on the blade surface compared to the conventional propeller, which results in reduced cavitation effects. This makes the tandem propeller a more effective option in terms of cavitation control, especially in conditions where cavitation is a concern. Overall, the tandem propeller exhibits higher thrust and torque but with a slight decrease in efficiency and a reduced cavitation effect, making it suitable for applications where high thrust is required without significantly compromising efficiency. 4.3 Parametric Investigations The hydrodynamics and open water performance of the tandem propeller are influenced by several geometrical parameters. This section focuses on the effects of two key parameters—axial spacing and angular difference—on the thrust, torque, and open water efficiency of the tandem configuration. These parameters, alongside the basic design of the propeller blades, contribute to a more comprehensive understanding of the secondary design elements that define a tandem propeller setup. For the parametric study, the standard pitch configuration used for the tandem propellers is (P/D)f=0.8 and (P/D)a=1.0 . 4.3.1 Influence of Axial Spacing In a tandem propeller configuration, the aft propeller operates within the contracted slipstream of the forward propeller, experiencing both axial and rotational induced velocities. These velocities directly affect the thrust, torque, and overall performance of the tandem propeller system. The axial spacing, defined by the ratio L/D, governs the inflow velocity field to the aft propeller. To study its effect, the axial spacing ratio is varied between 0.2, 0.3, 0.4, and 0.5, and the corresponding open water characteristics are presented in Figs. 4.9 – 4.11. L/D (0.2) j T kt(cfd) Q 10kq(cfd) n(cfd) 0.0001 0.4242 0.0013 0.6129 0.0826 0.0000 0.2000 142.0000 0.4210 4.7124 0.6350 0.2110 0.4000 110.2000 0.3267 3.9500 0.5323 0.3907 0.6000 75.2400 0.2230 3.2903 0.4434 0.4804 0.8000 42.0000 0.1245 2.5000 0.3369 0.4706 1.0000 0.4400 0.0013 0.7700 0.1038 0.0200 L/D (0.3) 0.0001 0.4242 0.0013 0.6129 0.0826 0.0000 0.2000 139.0000 0.4121 4.5700 0.6158 0.2130 0.4000 109.8000 0.3255 3.8500 0.5188 0.3994 0.6000 75.3450 0.2234 3.3200 0.4474 0.4768 0.8000 42.0700 0.1247 2.4850 0.3349 0.4742 1.0000 0.4400 0.0013 0.5400 0.0728 0.0285 L/D (0.4) 0.0001 0.4242 0.0013 0.6129 0.0826 0.0000 0.2000 140.0000 0.4150 4.5800 0.6171 0.2141 0.4000 110.1000 0.3264 3.8900 0.5242 0.3964 0.6000 75.0000 0.2223 3.3600 0.4528 0.4689 0.8000 42.3300 0.1255 2.5400 0.3423 0.4668 1.0000 0.4400 0.0013 0.5400 0.0728 0.0285 L/D (0.5) 0.0001 0.4242 0.0013 0.6129 0.0826 0.0000 0.2000 139.0000 0.4121 4.5100 0.6077 0.2158 0.4000 108.8000 0.3225 3.7000 0.4986 0.4118 0.6000 75.0400 0.2225 3.3250 0.4480 0.4741 0.8000 41.5050 0.1230 2.4650 0.3322 0.4716 1.0000 0.4400 0.0013 0.5400 0.0728 0.0285 Table 4.3 : Data for Analyzing the Effect of Axial Spacing on Thrust Coefficients, Torque Coefficients, and Open Water Efficiency for Axial Spacing Ratios of 0.2, 0.3, 0.4, and 0.5 Figure 4.9: Influence of Axial Spacing on the total thrust coefficient of the tandem propeller Figure 4.10 : Influence of Axial Spacing on the total torque coefficient of the tandem propeller Figure 4.11: Influence of axial spacing on open water efficiency of the tandem propeller The influence of axial spacing (L/D) on the performance of the tandem propeller shows minimal changes in the thrust and torque coefficients as the distance between the two propellers increases. However, a slight increase in both thrust and torque output is observed when the axial spacing ratio is set to 0.2, particularly in the advance coefficient range of J=0.2 to J=0.4 , where the effect becomes more pronounced. Despite these variations, the open water efficiency remains largely unaffected by the axial spacing of the propellers, indicating that the spacing does not significantly influence the efficiency of the system. 4.3.2 Influence of Angular Difference According to circulation theory, helicoidal vortex sheets are shed from the trailing edges of each propeller blade due to the rotating motion. In a tandem propeller setup, the vortex sheets shed by the forward propeller interact with the aft propeller, which can negatively impact its hydrodynamic performance. To mitigate this effect, a suitable angular spacing is proposed by Sun and Gu (1991), where the vortex sheet from the forward propeller aligns midway along the blades of the aft propeller. This optimal angular spacing is determined as a function of the axial spacing ratio (L/D), the pitch ratio (P/D) of the forward propeller, and the number of blades (Z), as given by the following expression: θ=(L\/D)/(P\/D)×360±180/Z The impact of angular difference on the thrust coefficient, torque coefficient, and open water characteristics of the tandem propeller is shown in Figs. 20–22. Three angular values are considered: 0° (with the forward and aft propeller blades overlapping), 30°, and 49.5° (the optimal angle based on the above formula). Angle (0) j T kt(cfd) Q 10kq(cfd) n(cfd) 0.0001 176.5000 0.5232 5.4170 0.7299 0.0001 0.2000 141.0000 0.4180 4.7000 0.6333 0.2101 0.4000 109.4400 0.3244 3.9796 0.5362 0.3852 0.6000 76.1400 0.2257 3.4150 0.4602 0.4886 0.8000 42.5000 0.1260 2.5400 0.3423 0.4687 1.0000 2.5850 0.0077 1.4871 0.2004 0.0609 Angle (30) 0.0001 170.4000 0.5051 5.2403 0.7061 0.0001 0.2000 143.0000 0.4239 4.6600 0.6279 0.2149 0.4000 110.5000 0.3276 3.9200 0.5282 0.3948 0.6000 78.5500 0.2329 3.3300 0.4487 0.4956 0.8000 42.6000 0.1263 2.5100 0.3382 0.4754 1.0000 4.4200 0.0131 1.4200 0.1913 0.1090 Angle (50) 0.0001 166.0000 0.4921 5.1203 0.6900 0.0001 0.2000 140.0000 0.4150 4.5400 0.6118 0.2159 0.4000 110.0000 0.3261 3.9000 0.5255 0.3950 0.6000 77.6000 0.2300 3.3100 0.4460 0.4925 0.8000 41.8000 0.1239 2.4900 0.3355 0.4702 1.0000 0.4400 0.0013 1.4000 0.1886 0.0110 Table 4.4 : Data for Analyzing the Effect of Angular Difference on Thrust Coefficients, Torque Coefficients, and Open Water Efficiency for Angular Difference at 0 degree, 30 degree and 50 degree Figure 4.12: Influence of angular difference on the total thrust coefficient of the tandem propeller Figure 4.13 : Influence of angular difference on the total torque coefficient of the tandem propeller Figure 4.14: Influence of angular difference on open water efficiency of the tandem propeller The influence of angular difference (θ) on the performance of the tandem propeller has minimal impact on the thrust and torque coefficients, with only a slight effect observed under bollard conditions (J=0). In this condition, the thrust and torque coefficients are highest when the angular difference is 0 degrees, and they decrease as the angular difference increases. Furthermore, the angular difference has little effect on open water efficiency, with only a minor influence seen at J=1, where the efficiency shows a slight variation. 5. CONCLUSIONS This study comprehensively evaluates the open water performance of conventional and tandem propellers, highlighting key differences in thrust, torque, and efficiency under various conditions. The tandem propeller demonstrates a significant increase in both thrust and torque coefficients compared to the conventional propeller, with an 18.52% increase in KT and a 15.24% increase in KQ in the bollard condition. However, this increase in thrust and torque comes at the cost of a 5.19% decrease in maximum efficiency, indicating a trade-off between performance and efficiency. The tandem configuration also offers improved cavitation control, with less high-pressure and low-pressure area on the blade surface, making it an effective solution in conditions prone to cavitation. The study further investigates the influence of axial spacing (L/D) and angular difference (θ) on the performance of the tandem propeller. The axial spacing has minimal effect on thrust and torque coefficients, although a slight increase is observed at an axial spacing ratio of 0.2, particularly in the advance coefficient range of J=0.2 to J = 0.4. The open water efficiency remains largely unaffected by axial spacing, indicating that the spacing does not significantly influence system efficiency. In contrast, the angular difference shows negligible impact on thrust and torque coefficients, with the most notable effect observed under bollard conditions, where the thrust and torque coefficients are maximized at 0° angular difference. Overall, the tandem propeller configuration is found to be advantageous for applications requiring high thrust and torque, with minimal compromise on efficiency. The study suggests that the tandem configuration, with its improved cavitation control and optimized thrust generation, is well-suited for vessels with high thrust demands, particularly in conditions where cavitation and high loading are concerns. Further optimization of the axial spacing and angular difference can refine performance, but the basic configuration already demonstrates significant potential for enhancing propeller performance in marine applications. 6. REFERENCES Open water performance of B-Series marine propellers in tandem configurations Sun, Qin, Gu, Yunde, Zheng, Shu-Zhen, 1978. Theoretical Calculation of Tandem Djahidi, Boucetta, Omar, Imine, 2017. Impact of some geometrical aspects on the tandem Co-rotating propeller hydrodynamic characteristics. Brodograndja Shipbuilding 68, 107–122. Uncertainty Analysis in CFD Verification and Validation Methodology and Procedures. ITTC – Recommended Procedures and Guidelines 7.5-03-01-01, 2017. Djahidi, Boucetta, Omar, Imine, 2019. Numerical simulation of the cavitating flow Carlton, J., 2018. Marine Propellers and Propulsion, fourth ed. Butterworth-Heinemann. Glover, E.J., 1981. Çontrarotating propellers and tandem propellers. In: Proceedings of Symposium Advance in Propeller Research and Design, Gdansk. Carlton, J., 2018. Marine Propellers and Propulsion, fourth ed. Butterworth-Heinemann. David, C.T., Hecker, R., 1973. Open-Water Performance of Tandem Propellers. DTNSRDC Report No. SPD-530-01 Denny, S.B., 1978. A Procedure for the Design of Tandem Propellers. DTNSRDC Report No. 530-H-04. Hadler, J.B., Pien, P.C., 1985. Results of Research on Tandem and Tip-attached Tandem Propellers. MARAD Report. Iriono, et al., 1994. Tandem propeller assembly for a marine propulsion unit. United States Patent Number 5, 310–371. Liu, Y., Gong, Q., 2020. Numerical investigation on the flow characteristics and hydrodynamic performance of tandem propeller. Appl. Ocean Res. 101, 102292. Morgan, W.B., 1960. The design of contra-rotating propellers using Lerbs’ Theory. Trans. SNAME. Sun, Qin, Gu, Yunde, 1981. ‘On the Tandem Propeller’. In: The 16th ITTC Proceedings, Session on Propulsion, Propellers, pp. 58–59. Sun, Qin, Gu, Yunde, Zheng, Shu-Zhen, 1978. Theoretical Calculation of Tandem Propellers and Their Open Water Test Series. Shanghai Ship and Shipping Research Zhang, J.L., Xu, H.M., 1981. Open water experimental investigations with tandem propellers in nozzle. Shanghai Ship and Shipping Research Institute. Report 81-1-5.