Calculations of the Hydrodynamic Characteristics of a Ducted Propeller Operating in Oblique Flow

Resumen Cálculos de las características hidrodinámicas de una hélice con ductos que funcionan en flujo oblicuo Date Received: November 29th 2016 Fecha de recepción: Noviembre 29 de 2016 Date Accepted: December 19th 2016 Fecha de aceptación: Diciembre 19 de 2016 Calculations of the Hydrodynamic Characteristics of a Ducted Propeller Operating in Oblique Flow 1 Department of Maritime Engineering, Amirkabir University of Technology, Tehran. Email: iran. gasemi@aut.ac.ir 2 Department of Marine Engineering, Imam Khomeini University, Nowshahr, Iran. Email: sohrabmajd@gmail.com 3 Department of Marine Engineering, Imam Khomeini University, Nowshahr, Iran. Email: hamid.foruzan1348@gmail.com Ship Science & Technology Vol. 10 n.° 20 (31-40) January 2017 Cartagena (Colombia) 32 Diff erent types of the marine vessels are widely equipped with ducted propellers (Duct_Ps). Th rust is generated by propellers and ducts which are lifting bodies. Th e duct surrounds the propeller with a small slope angle. A Duct_P used on the back of the ship and also vectors that fl ow into the duct and the propeller are shown in Fig. 1. To give more effi ciency, it is recommended using Duct_P in ships such as tugs and trawlers (a type of fi shing vessel) that work in heavy conditions. Ducts produce thrust augmentation especially in heavy conditions. Generally speaking, it is said that in extreme conditions, the duct may generate 50% of the total thrust. Th erefore, it is necessary to focus on the Duct_P to understand that the duct and propeller generate all forces and moments. (Ghassemi et al.) have carried out the numerical method on the many types of propellers (contrarotating propeller, podded draive, surface piercing propeller and ducted propeller). Many researchers have conducted various experimental and numerical approaches on Duct_Ps. (Haimov et al. 2010) carried out the prediction calculation method using the RANSE code to calculate the open water behavior of the Duct_P. (Kerwin et al. 1987) have used a combination of BEM to model the duct and a vortex lattice method (VLM) for the propeller. Th e most popular choice in Duct_P numerical studies has been the RANSE method coupled with an isotropic turbulence model like the SST k-ω model, i.e. employed by (Menter 1994). (Pangusión 2005) has developed the design of a Duct_P and model tests of a fi shing research vessel. (Falco 1983) conducted a study about the analysis of the performance of Duct_P in axisymmetric fl ows. A RANS-based analysis tool for Duct_P systems in open water conditions was presented. (Zondervan et al. 2006) carried out a study on the fl ow analysis, design and model testing of Duct_P. Using a vortex lattice and fi nite volume methods, (Gu & Kinnas 2003) modeled the fl ow around a Duct_P. (Broglia et al. 2013) analyzed diff erent propeller models and their eff ect on maneuvering prediction. (Bhattacharyya et al. 2015) conducted a study on the hydrodynamic characteristics of open and Duct_Ps by transitional fl ow modeling. (Bosschers et al. 2015) were the scholars who conducted the study in open water conditions and they applied a combination of the RANS and BEM to study the Duct_Ps. (Dubbioso et al. 2013, 2014) analyzed the performance of a four blade propeller INSEAN E779A model in oblique fl ow using CFD code in two load conditions and various oblique angles. (Yao 2015) researched the hydrodynamic performance of a marine propeller in oblique fl ow. (A. Vega and D.L Martinez 2015) used computational fl uid dynamics to conduct a bollard pull test for a double propeller tugboat. Th ey simulated a bollard pull test for a specifi c Introduction Fig. 1. Illustrative of the Duct_P a) DP behind a ship b) Flow fi eld into the Duct_P Ghassemi, Majdfar, Forouzan Ship Science & Technology Vol. 10 n.° 20 (31-40) January 2017 Cartagena (Colombia) 33 tugboat and evaluated the results with the real test results to which it was subjected after construction. In this study, the performance of a rotating Duct_P set at two oblique angles (10 and 20 degrees) with respect to the inflow is researched by means of an approach based on the RANSE numerical solution. This propeller type may help to improve the maneuvering of the ship. The selected propeller is the four bladed Kaplan type with 19A duct and the experimental data was obtained from (Carlton 2013). In this paper, the propeller’s rotational velocity of the propeller is imposed bymodeled using aa moving reference frame (MRF) applied to the inner region of the domain because of the high speed and accuracy of computing and simulating. The MRF is a relatively simple, strong, and efficient, CFD modeling method to simulate rotating machines. Given that the fluid flow becomes stable and transient after times a period of time and using MRF is often meaningful significant in this kind of flows, this frame was applied. In the case event of unsteady flows and when MRF is being used, the CFD codes like such as ANSYS CFX and Fluent may solve the unsteady flow. In this situation, it is necessary to add unsteady terms to all the governing applicable transport equations. The finite volume method (FVM) is used to discretize the governing applicable incompressible Navier-Stokes equations, the finite volume method (FVM) is used while in the hybrid SST-Kω turbulence model is chosen for the case of the numerical treatment of turbulence. The SST-Kω model, introduced by Menter (1994), is more accurate and reliable for a wider class types of flows since it combines the robust and precise formulation of the k-ω model in the near-wall region with the free stream independence of the k-ε model in the far field. Continuity Equation Continuity of Flow, as one of the key principles used in the analysis of uniform flow, is the result of the fact that mass is always conserved in fluid systems without taking into consideration the pipeline complexity or direction of flow. Euler proposed the following basic law of fluid: And if fluid density is constant it will be as follows: The equation is regardless of time for stable currents. So ∂ρ / ∂ρ = 0 and continuity equation is as follows: Momentum Conservation Equation According to Newton's second law, the change rate of momentum of a particle and the resulting fluid forces acting on the particle are equal. So momentum conservation equation, in general, is as follows: where fs and fb are surface forces and volume forces, respectively. Here the volume forces are equal to gravity, as well as the surface forces acting on fluid caused by tension viscosity (shear stress) and the fluid pressure. So the 4th equation is expressed as follows: So that τij includes the vertical and shear stresses. If i = j, the stress will be normal stress and in other cases the stress will be shear stress. In cases where fluid flow is transient, the applicable equations must be discretized in both space and time by implicit and explicit time integration Methodology Temporal Discretization (1)

Diff erent types of the marine vessels are widely equipped with ducted propellers (Duct_Ps).Th rust is generated by propellers and ducts which are lifting bodies.Th e duct surrounds the propeller with a small slope angle.A Duct_P used on the back of the ship and also vectors that fl ow into the duct and the propeller are shown in Fig. 1.To give more effi ciency, it is recommended using Duct_P in ships such as tugs and trawlers (a type of fi shing vessel) that work in heavy conditions.Ducts produce thrust augmentation especially in heavy conditions.Generally speaking, it is said that in extreme conditions, the duct may generate 50% of the total thrust.Th erefore, it is necessary to focus on the Duct_P to understand that the duct and propeller generate all forces and moments.(Ghassemi et al.) have carried out the numerical method on the many types of propellers (contrarotating propeller, podded draive, surface piercing propeller and ducted propeller).Many researchers have conducted various experimental and numerical approaches on Duct_Ps.(Haimov et al. 2010) carried out the prediction calculation method using the RANSE code to calculate the open water behavior of the Duct_P.(Kerwin et al. 1987) have used a combination of BEM to model the duct and a vortex lattice method (VLM) for the propeller.Th e most popular choice in Duct_P numerical studies has been the RANSE method coupled with an isotropic turbulence model like the SST k-ω model, i.e. employed by (Menter 1994).tugboat and evaluated the results with the real test results to which it was subjected after construction.
In this study, the performance of a rotating Duct_P set at two oblique angles (10 and 20 degrees) with respect to the inflow is researched by means of an approach based on the RANSE numerical solution.This propeller type may help to improve the maneuvering of the ship.The selected propeller is the four bladed Kaplan type with 19A duct and the experimental data was obtained from (Carlton 2013).
In this paper, the propeller's rotational velocity of the propeller is imposed bymodeled using aa moving reference frame (MRF) applied to the inner region of the domain because of the high speed and accuracy of computing and simulating.The MRF is a relatively simple, strong, and efficient, CFD modeling method to simulate rotating machines.Given that the fluid flow becomes stable and transient after times a period of time and using MRF is often meaningful significant in this kind of flows, this frame was applied.In the case event of unsteady flows and when MRF is being used, the CFD codes like such as ANSYS CFX and Fluent may solve the unsteady flow.In this situation, it is necessary to add unsteady terms to all the governing applicable transport equations.The finite volume method (FVM) is used to discretize the governing applicable incompressible Navier-Stokes equations, the finite volume method (FVM) is used while in the hybrid SST-Kω turbulence model is chosen for the case of the numerical treatment of turbulence.
The SST-Kω model, introduced by Menter (1994), is more accurate and reliable for a wider class types of flows since it combines the robust and precise formulation of the k-ω model in the near-wall region with the free stream independence of the k-ε model in the far field.

Continuity Equation
Continuity of Flow, as one of the key principles used in the analysis of uniform flow, is the result of the fact that mass is always conserved in fluid systems without taking into consideration the pipeline complexity or direction of flow.Euler proposed the following basic law of fluid: And if fluid density is constant it will be as follows: The equation is regardless of time for stable currents.So ∂ρ / ∂ρ = 0 and continuity equation is as follows:

Momentum Conservation Equation
According to Newton's second law, the change rate of momentum of a particle and the resulting fluid forces acting on the particle are equal.So momentum conservation equation, in general, is as follows: where f s and f b are surface forces and volume forces, respectively.
Here the volume forces are equal to gravity, as well as the surface forces acting on fluid caused by tension viscosity (shear stress) and the fluid pressure.So the 4 th equation is expressed as follows: So that τ ij includes the vertical and shear stresses.
If i = j, the stress will be normal stress and in other cases the stress will be shear stress.
In cases where fluid flow is transient, the applicable equations must be discretized in both space and time by implicit and explicit time integration

Methodology
Temporal Discretization (1) (2) (3) (4) (5) Since has a higher effi ciency than a propeller with a normal blade outline when operating in a duct, it is the most common propeller for Duct_ Ps.Having a large chord at the tip is one of the physical characteristics of a Kaplan type propeller.
Th e Kaplan propeller with a P/D ratio of 0.8 is used in the entire analysis of this paper.Geometric modeling of a Kaplan propeller was done with Propcad and Solidworks software.Kaplan geometric data and Duct is shown in Table 1.
Due to the favorable hydrodynamic characteristics, 19A is the most common accelerator type of duct and when it is used with the Ka propeller series it will be more useful in the designing of Duct_Ps.Th e ducts' length is equal to half of the propellers' diameter (L P =0.5DP) and the distance between the propellers' tip and the inner surface of the duct is equal to one percent of the propellers' diameter (gap=1%D P ).Fig. 2 shows the three-dimensional model of the Duct_P created in Solidworks.
ICEM software is applied to generate unstructured mesh in Duct_P and domains.Th e generated mesh size grows outwards at a 1.2 ratio.Boundary conditions include inlet, outlet, rotating domain, open water, propeller and duct.Fig. 3 shows the meshes near the propeller and the duct.Due to the complex geometry of the propeller and its gap with the duct, this surface is fi rst divided into seven separate surfaces and a structured quadrilateral surface grid is generated for each one.Th e average size of the mesh elements on the propellers' surface at its fi nest generated mesh is 0.005Dp.Th en, this surface mesh is extruded in a wall-normal direction with a transition ratio of 1.1 (for fi nest generated mesh), an initial layer thickness of 2.5 x 10 -5 m (corresponding to Y + = 1) and a layer numbers of n = 50 that results in a streamlined structured hexahedral mesh volume in near-fi eld fl ow.In the fi rst step, 4 million meshes were used for meshing the model; then, using smaller meshes, 5, 5.5, 6, 6.2 and 6.5 million-cell models were tested and the results were compared at an advanced ratio of 0.4.Comparison of the results shows that the minimum number of cells for this model is 6 million.Fig. 4 shows the mesh convergence of the thrust (6)  Also, a second order backward Euler for transient scheme in continuity and momentum equations and upwind method to solve advection turbulence kinetic energy k and specifi c dissipation rate ω are applied.Th e SST turbulence model is selected in most of the articles, because of its popularity due to its higher accuracy.Total duration of time is set to 10 seconds and the time steps are selected based on the propellers' rotating speed for each run.
Hydrodynamic performance of Duct_P is defi ned as follows: Advance ratio: Th rust coeffi cient (duct and propeller and total): Torque coeffi cient: Effi ciency: where V A is advanced velocity, n is revolution per second (RPS), D is propellers' diameter in meters, T is the total thrust, T P and T d are the thrust of the propeller and duct, ρ is water density and Q is total torque.Comparison of hydrodynamic performance of the Duct_P in open fl ow is shown in Fig. 5. Th e relative error is about 8% at heavy loaded condition (J=0.1)but at the design condition (J=0.45) the error is less than 3%.
Th e propeller has been tested for two diff erent advance ratios (J=0.3 and 0.5), covering a relatively A tri-linear interpolation of the pressure and velocity is applied to solve the RANS equations.A high-resolution method is applied in order to solve velocity-pressure coupling and advection scheme in continuity and momentum equations.0.17 0.17 0.17 Oblique angle 20 J=0.5 Oblique angle 20 J=0.5 Oblique angle 20 J=0.5 0.1 0.1 0.1 0 0 0 0.2 0.2 0.2 0.3 0.3 0.3 0.4 0.4 0.4 0.5 0.5 0.5 0.6 0.6 0.6 0.8 0.8 0.8 0.7 0.7 0.7 0.9 0.9 0.9 1 1 1 In this article, the Kaplan propeller with duct 19A is analyzed at two advance ratios of 0.3 and 0.5 by the numerical method in open water and oblique fl ow conditions (10 and 20 degrees).Th e following results can be drawn: • Hydrodynamic performance of Duct_P is compared with the experimental data and shows good correlation at high advance coeffi cient but some discrepancy at low advance coeffi cient.
• Pressure distributions are obtained on the propeller blade and duct.Low pressure and high pressure are at the rear side of the blade and duct.• Hydrodynamic performances of the Duct_P are compared at oblique angles (10 and 20 degrees).When oblique angle increases torque does not change, while effi ciency signifi cantly changes especially at high advance coeffi cients.

Conclusions
Numerical computations presented here have been performed on the parallel machines of the High Performance Computing Research Center (HPCRC) of Amirkabir University of Technology; their support is greatly recognized.

Fig. 3 .
Fig. 3. Mesh cells on the propeller and duct

Fig. 5 .FigFig. 7 .
Fig. 5. Comparison of the numerical and experimental hydrodynamics performance of Duct_P Fig. 11.Pressure distribution on duct profile at J=0.3 Where values at the start and end of the time step are assigned as the superscripts n+½ and n-½, respectively, φ is a scalar quantity, ρ is viscosity, n+1/2 value at the next time level, n-1/2 value at the previous time level, V is control volume.
Calculations of the Hydrodynamic Characteristics of a Ducted Propeller Operating in Oblique Flow Ship Science & Technology -Vol. 10 -n.° 20 -(31-40) January 2017 -Cartagena (Colombia)methods.In order to control volumes that do not deform in time, the general conservative approximation of the transient term for the nth time step is:

Table 1 .
Kaplan geometric parameters and duct