(JNAOÇ

© SNAK, 2015

Int. J. Nav. Archit. Ocean Eng. (2015) 7:540~558 http://dx.doi.org/10.1515/ijnaoe-2015-0039 pISSN: 2092-6782, eISSN: 2092-6790

Virtual maneuvering test in CFD media in presence of free surface

ABSTRACT: Maneuvering oblique towing test is simulated in a Computational Fluid Dynamic (CFD) environment to obtain the linear and nonlinear velocity dependent damping coefficients for a DTMB 5512 model ship. The simulations are carried out in freely accessible OpenFOAM library with three different solvers, raslnterFoam, LTSInterFoam and interDyMFoam, and two turbulence models, k-s and SST k-m in presence of free surface. Turning and zig-zag maneuvers are simulated for the DTMB 5512 model ship using the calculated damping coefficients with CFD. The comparison of simulated results with the available experimental shows a very good agreement among them.

KEY WORDS: Computational fluid dynamic (CFD); OpenFOAM; Linear hydrodynamic coefficients; Nonlinear hydro-dynamic coefficients; Maneuver.

INTRODUCTION

Maneuverability is an important quality of marine vehicles. It should be controlled during various design stages and at the end of building the vessels. It has influences on efficiency and safety of marine transportation system. Maneuvering of a marine vehicle is judged based on its course keeping, course changing and speed changing abilities. The regulation bodies and international marine organizations such as IMO recommend criteria to investigate ship and other marine vehicles maneuvering quality (IMO, 2002a; 2002b).

Maneuverability of a ship or another marine vehicle may be predicted by model tests, mathematical models or both. Mathematical models for prediction of marine vehicle maneuverability may be divided into two main categories called as hydrodynamic models, and response models. The hydrodynamic models are of two types and recognized as the Abkowitz (Abkowitz, 1969) and MMG (Yoshimura, 2005) models. The Abkowitz model is based on the Taylor series expansion of hydrodynamic forces and moments about suitable initial conditions. The MMG model, also called as modular model, decomposes hydrodynamic forces and moments into three components namely: the bare hull; rudder; and propeller. The response model investigates the relationship for the motion responses of the vehicle to the rudder action and used to investigate the course control problems (Nomoto, 1960)

The hydrodynamic models, especially the Abkowitz formulation, are more suitable for computer simulation. It contains several derivatives that are known as the hydrodynamic coefficients. These hydrodynamic coefficients should be determined in advance to proceed into the predicting the maneuvering characteristics of a marine vehicle. These hydrodynamic coefficients are named as added mass and damping coefficients. All of them are function of the geometry of the vessel but the added mass

Corresponding author: S. HosseinMousavizadegan, e-mail: hmousavi@aut.ac.ir

This is an Open-Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Ahmad Hajivand and S. Hossein Mousavizadegan

Maritime Engineering Department, Amirkabir University of Technology, Tehran, Iran

Received 23 August 2014; Revised 7 February 2015; Accepted 24 March 2015

coefficients depend on the acceleration of the vessel while the damping coefficients are velocity dependent. The added mass coefficients can be computed through the solution of the non-viscous fluid flow around the vessel. The damping coefficients are due to the wave formation in the free surface of the water and the effect of the viscosity. The total damping coefficients may be obtained through the solution of viscous fluid flow around the vessel.

There are several methods to obtain hydrodynamic coefficients such as theoretical approach, semi empirical formulas, captive model tests, and CFD. Theoretical approach is limited to slender bodies and do not consider the interaction between the hull and the appendages. It can provide the added mass coefficients and the part of damping coefficients due to the wave formation on the free surface of water. Semi empirical formulas are obtained using linear regression analysis of captive model test data. They can only provide the linear coefficients for some specific geometrical shape and are inaccurate when the particulars of vessel are outside of the database. The captive model tests provide the hydrodynamic coefficients through the running the tests: Oblique Towing Test (OTT), Rotating Arm Test (RAT) or Circular Motion Test (CMT) and Planar Motion Mechanism (PMM) test. OTT and PMM test are done in a towing tank and RAT is run in a maneuvering basin. Oblique towing-tank tests provide the damping coefficients depend on the translational velocities while rotating arm tests give the angular velocity dependent coefficients. Planar motion mechanism tests can provide all the damping and added mass coefficients (Lewis, 1988). These model tests are expensive, time consuming and their results include the scaling effects due to inconsistency of Reynolds number between the ship and the model.

CFD can also be applied to obtain the hydrodynamic coefficients of a marine vehicle such as a ship. CFD methods are used DNS, LES and RANS approaches to solve the fluid flow equation for a viscous flow such as the flow around a maneuvering ship. DNS and LES need very high computational cost. Therefore RANS model is employed.

Application of RANS to solve the maritime problems goes back to Wilson et al. (1998) and Gentaz et al. (1999) where the results are largely unsatisfying. By the increasing growth of computing capacities and recent progress in RANS models, stunning advances in this field are achieved. Nowadays, CFD is crucial tool for various aspect of a marine vehicle hydrodynamics such as ship resistance and propeller performance not only for research but also as a design tool. One of the most recently and important application of CFD in marine industry is computation of hydrodynamic coefficients of marine vehicles by simulating the captive model tests. Sarkar et al. (1997) develop a new computationally efficient technique to simulate the 2-D flow over axisymmetric AUVs by Using the CFD software PHOENICS. Nazir and Wang (2010) and Zhang and Cai (2010) apply the commercial CFD software Fluent to obtain hydrodynamic coefficients of 3-D fins and an AUV, respectively. Tyagi and Sen (2006) compute transverse hydrodynamic coefficients of an AUV using a CFD commercial software. The hydrodynamic forces and moments on an AUV due to the deflection of control surfaces are investigated using ANSYS Fluent commercial CFD software by Dantas and De Barros (2013). Ray et al. (2009) applies CFD software Fluent to compute linear and nonlinear hydrodynamic coefficients of the SUBOFF submarine in an unrestricted fluid flow. There are very few works where CFD is used to predict the maneuvering of surface ships. Simonsen et al. (2012) simulate the fixed OTT for the KCS model by employing the commercial CFD software STAR-CCM+ to calculate the hydrodynamic coefficients.

The OpenFOAM software is applied to simulate the OTT for a DTMB 5512 model ship, shown in Fig. 1 with the particulars given in Table 1, in presence of free surface. OpenFOAM is an open source library that numerically solves a wide range of problems in fluid dynamics from laminar to turbulent flows. It contains an extensive set of standard solvers to solve various ranges of CFD problems. Jasak (2009) describes the objected oriented libraries of OpenFOAM package.

The fluid flow around a ship body is usually turbulent in presence of the free surface. The suitable OpenFOAM solvers for such cases are: raslnterFoam, interDyMFoam, and LTSInterFoam. The raslnterFoam solver is for the unsteady, incompressible, immiscible fluid flows. It applies Volume of Fluid (VOF) for tracking free surface and library of Reynolds-Averaged Simulation (RAS) turbulence models to consider effects of turbulence. In addition to this unsteady solver, computations are carried out using quick and reliable quasi-steady VOF solver known as LTSInterFoam (user guide of OpenFOAM). Finally, interDyM Foam solver is applied to investigate the effects of dynamic trim and sinkage on damping coefficients. InterDyMFoam applied 6DOF solver to perform translations and rotations in space and solve vessel motion equations.

Fig. 1 DTMB 5512 bare hull model (Yoon, 2009).

Table 1 Geometrical data for DTMB 5512 model (Yoon, 2009).

LPP [m] 3.048

B [m] 0.410

T [m] 0.136

V [m3] 0.086

S [m2] 1.371

Cb [-] 0.506

The turbulence models k-s and SST k-ra are used and the simulations are done for restrained and free conditions, to investigate dynamic trim and draft effects on hydrodynamic coefficients. The computations are done up to large drift angles to provide the possibilities of finding the nonlinear coefficients. Finally free running maneuver tests are simulated for three solvers based on the hydrodynamic coefficients obtained from CFD. The results are compared with the available data based on experimental results (Yoon, 2009). It is found that the results of simulations comply with the existing results especially for raslnterFoam solver.

FLUID FLOW MODELLING

The unsteady viscous flow around a marine vehicle is governed by the Navier-Stokes equations. Navier-Stokes equations can be applied to both laminar and turbulent flow but a very fine meshing is necessary to capture all the turbulence effects in a turbulent flow regime. The Reynolds-averaged Navier Stokes (RANS) equations can also be applied to model the turbulent flow. The RANS equations may be given according to Rusche, 2002 as follows for an incompressible flow.

V-U = 0 (1)

P + V- (pUU) - V- (Vef VU) = -Vp - g -RVp+VU - VMeff + K7c (2)

where U is the velocity vector, p the density, Jueff the effective viscosity which can be defined as ßeif=ß+pvlurb(m is the dynamic viscosity and vIUrb is the turbulent kinetic viscosity), p* the pressure, g the gravity acceleration vector, R the position vector, c the surface tension coefficient, k the free surface curvature. In addition, cis the volume fraction that is defined as (Vajr / Vtotal) and is obtained by solution of the advection equation (Rusche, 2002).

— + V-cU + V-c(1 -c)Ua = 0 (3)

where Ua is velocity field suitable to compress the interface, |ua| = min[ac |u| ,max(|u|)] in which ac is a constant which specifies the enhancement of interface compression. For further reference regarding the governing equations see (Ubbink, 1997).

Transport equation is solved for volume fraction to track free surface. At free surface the fluid density, p, and viscosity, are calculated as follows (Hirt et al., 1981)

P = PairC + Pwater (1 - c) (4)

ß = ßa<rC + ßwa,er (1 - c) (5)

There is no a general accepted turbulence model for all kinds of fluid flow problems. The k-e and SST k-m model are used to model the turbulence effects. The two equation turbulence model k-e is the most frequently used turbulence model where the effect of Reynolds stresses is considered as an additional eddy viscosity which is a property of the flow. Eddy viscosity expressed as:

= PCM — (6)

where k is the turbulence kinetic energy per unit mass, s is the rate of the dissipation of the turbulence kinetic energy per unit mass and is a dimensionless constant of a normal value of 0.09. The turbulent kinetic energy and the dissipation rate are calculated from the solution of transport equations (Ferziger and Peric, 2002). The SST k-ra turbulence model has a precise formulation and uses the standard k-ra model in the inner part of the boundary layer, with the standard k-e in the free stream. The notation ra is the specific dissipation rate. The SST k-ra is accurate and reliable for a wider class of flows especially for boundary layer regions and adverse pressure gradient flows (Menter et al., 2003)

Mesh generation

Finite Volume Method (FVM) is the common approach that is applied to solve RANS equations in computational domain. OpenFOAM implements a cell-centered FVM. Domain dimensions are selected sufficient large to avoid back flow at high drift angles. Distance of the inlet and outlet boundary from ship center is considered 2.5 LPP and 4 LPP, respectively. The side boundaries are located at 3.5 LPPand the top and bottom boundary is located at 1 LPP and 1.5 LPP from the free surface, respectively (Fig. 2).

There are different meshing strategies to discretize the computational domain (Seo et al., 2010). One of the common method in OpenFOAM is to apply SnappyHexMesh method. In this method first a hexahedral background grid is created and then the mesh around boundaries are refined. The overall view of the mesh around the hull bow is displayed in Fig. 3. To solve the boundary layer close to the ship hull the flow nearby to the boundary is modeled by empirical wall function to save a large number of grid points. Park et al. (2013) investigate implementation of the wall function for the prediction of ship resistance. The wall function is applicable if the non-dimensional wall distance, y+, be in the range 30<y+<100. The y+ values on the hull surface is around 30 that show boundary layer prediction is well. Distribution of y+ for medium mesh on hull is shown in Fig. 4.

Fig. 3 Mesh around the hull. Fig. 4 y+ distribution around hull.

Boundary conditions

Appropriate boundary conditions on the free surface, fluid domain boundaries and ship's hull must be applied to create a well-posed system of equations. The boundaries of domain split into patches as shown in Fig. 2. The boundary conditions are chosen such that to avoid back flow and lateral wall effects. The velocity and pressure conditions for each patch are presented in Table 2. The fixed value condition, Dirichlet condition, is applied for velocity condition at inlet and hull boundaries. For sides boundaries the symmetry plane condition is a Neumann condition which means pressure, tangential velocities and turbulence quantities have a zero gradient normal to the surface but for the normal velocity component, a Dirichlet condition, is applied. For the zero gradient boundary condition, the near wall cell value is set for boundary value. The fixed flux pressure condition is set for inlet and ship hull boundaries. This condition modifies the pressure gradient in order that the boundary flux matches the velocity boundary condition. These conditions are same for three different solvers except for interDyMFoam the moving wall velocity condition is applied for ship hull boundary.

Table 2 OpenFOAM built-in boundary conditions.

boundary Velocity Pressure

Sides symmetryPlane symmetryPlane

Inlet fixedValue fixedFluxPressure

Outlet zeroGradient zeroGradient

Ship fixedValue fixedFluxPressure

GRID CONVERGENCE

Mesh sensitivity examination is the most straight-forward and the most consistent technique for determining the order of discretization error in numerical simulation. In other words, numerical results can be considered as precise and valid if its solution be independent of the grid. A mesh sensitivity study involves implementation solution on the CFD model, with sequentially refined grids of reduced mesh size, until the solutions become independent of the mesh size. Three different meshes with constant grid refinement factor in all three spatial directions, r = h2 / h = h3 / h2 = 1.8 , are employed. The notation h is a measure of the mesh discretization. Based on experiments, it is desirable that r > 1.3 , this reduces the errors arising from extrapolation. These cases are labeled al al, a2 and <r3 from finest to coarsest mesh. Corresponding solution for these cases are designated S1, S2 and S3, respectively.

The oblique towing test is simulated with OpenFOAM with three solvers, raslnterFoam, LTSInterFoam and interDyM Foam, using these grids. The corresponding forces and moment are obtained for a drift angle p = -6 at Fn = 0.28. The number of meshes and calculated non-dimensional forces and moment coefficients are shown in Table 3-5. The forces and moment are made dimensionless with water density p , inflow speed V , lateral underwater area TLPP and length between perpendiculars LPP :

X =-V--(7)

0.5pV2TLff v '

Y 0.5pV2TLPP (8)

N =-z—:r- (9)

0.5pV2TL2PP K)

Convergence ratio defined as follows.

s21 = s2 - s1 is the difference between solution of fine and medium grid; s32 = s3 - s2 is the difference between solution of medium and coarse grid.

The possible convergence situations are: R>1 : Grid divergence R<0 : Oscillatory convergence 0<R<1 : Monotonic grid convergence

Table 3 Dimensionless forces and moment for different grid (raslnterFoam).

Number of grids X ' Y ' N '

1,578,257 -0.0192 0.0368 0.0187

3,211,764 -0.0177 0.0333 0.0173

5,685,054 -0.0172 0.0325 0.0166

EFD* -0.0169 0.0314 0.0154

Experimental Fluid Dynamic

Table 4 Dimensionless forces and moment for different grid (LTSInterFoam).

Number of cells X ' Y ' N '

1,578,257 -0.0223 0.0394 0.0227

3,211,764 -0.0191 0.0367 0.0206

5,685,054 -0.0179 0.0354 0.0194

EFD -0.0169 0.0314 0.0154

Table 5 Dimensionless forces and moment for different grid (InterDyMFoam).

Number of cells X ' Y ' N '

1,578,257 -0.0215 0.0381 0.0211

3,211,764 -0.0186 0.0358 0.0190

5,685,054 -0.0177 0.0344 0.0181

EFD -0.0169 0.0314 0.0154

If grid convergence occurs, Richardson extrapolation also called h2 extrapolation is used to estimate convergence rate. Order of discretization estimated as follows:

ln (g32/ g21 )

After that, Grid Convergence Index (GCI) is defined

oct.. = fs^l , (11)

' S rp -1

where VS is a safety factor with a value of VS = 1.25 as recommends by Roache (1997) for convergence study with minimum three grids or more. The notation GCI indicates that computed value how far away from exact value. On the other hand, GCI is a measure of solution changes with more grid refinement. Small value of GCI means that the solution is in exact value range. Computed convergence ratio, order of discretization and GCI are illustrated in Table 6. Theoretical value for convergence is p=2. The difference is due to grid orthogonally, problem nonlinearities, turbulence modeling. The predicted water elevation along the plane at y=0.3 m and y=-0.3 m (Fig. 5) for the coarse, medium and fine grid is compared in Figs. 6 and 7, respectively. It is seen that difference between the water elevation of medium and fine grids is lower than difference between the water elevation of coarse and medium grids, especially at midship.

Table 6 Estimated convergence ratio, order of discretization and GCI for different solvers.

rasInterFoam LTSInterFoam interDyMFoam

X' Y' N' X' Y' N' X' Y' N'

R 0.3333 0.2286 0.5000 0.3750 0.4815 0.5714 0.3103 0.6087 0.4286

P 1.8691 2.5110 1.1792 1.6687 1.2435 0.9521 1.9906 0.8446 1.4415

GCIfine 0.0182 0.0091 0.0527 0.0503 0.0426 0.1031 0.0286 0.0791 0.0466

Fig. 6 Comparison of water elevation along the cut at y=0.3 for different grids for p = 6 at Fn=0.28.

Fig. 7 Comparison of water elevation along the cut at y=-0.3 for different grids for p = 6 at Fn=0.28

Difference between the simulation results of fine grids and medium grids are shown in Table 7. It is seen that the average change is approximately 2.4-6.7% but the computational time is significantly increased from medium to fine grids. Therefore, the medium grid is applied throughout this study to obtain solutions with minimum computational effort.

Table 7 Difference between fine and medium results.

Solver E % X ' E % Y ' E % N '

rasInterFoam 2.9 2.4 4.2

LTSInterFoam 6.7 3.5 5.8

InterDyMFoam 5.0 4.0 4.7

COMPUTATIONAL FLUID DYNAMIC SIMULATIONS

The fluid flow around DTMB 5512 model ship is simulated with and without drift angle with respect to the fluid flow direction. For the case without drift angle, the resistance, dynamic trim and sinkage can be obtained. This is called as resistance simulation. For the case with drift angle which is called as OTT, the lateral velocity dependent damping coefficients can be obtained. All computation are done with PIMPLE (merged PISO-SIMPLE model) algorithm for pressure-velocity coupling. The second-order upwind scheme is applied for advection term in momentum equation.

Resistance simulation

The resistance tests are simulated to investigate the effect of dynamic trim and sinkage on ship resistance and validate interDyMFoam results with available EFD data. In this solver, the relative motion is expressed by the grid deformation. The deformation of grids on hull is obtained from the equation of motion solution. On the domain boundaries grid are considered fixed. The solution algorithm of interDyMFoam is given in Fig. 8. The resistance, dynamic trim and sinkage of DTMB 5512 model ship is computed for Froude number Fn = 0.05 - 0.45 with an increment of 0.05 at zero drift angle. The resistance

coefficient is defined as C =-T—-, where R is the total resistance that is equal to the drag force, S is the wetted sur-

T 0.5 pSV ^

face of the model ship and V is the inflow velocity.

Fig. 8 InterDyMFoam solution algorithm (Schmode et al., 2009).

The resistance coefficient is obtained by finding the solution with all three solvers and compared with the experimental data (EFD) given in Olivieri et al. (2001) in Fig. 9. The solution with raslnterFoam provides good prediction with an error up to 10%

for different Froude numbers. The interDyMFoam solver gives a good prediction of CT for Fn < 0.2 with an average error of 4% with respect to EFD but for Fn > 0.2 the error increases up to 14%. The solver LTSInterFoam provides the solution with an error around 12%.

0 EFD ■ ■■ill.

- CFD(raslnlerFoam) ---CFDCLTSMerFoami ......... CFD(interDyMFoam) / / ■ / /

• / / / y

//i / 9 . - ' ^ $ o '

0.05 0.1 0,15 0.2 0,25 0,3 0.35 0.4 0.45 Froude number

Fig. 9 Compare computed and experimental resistance coefficients Vs. Froude number.

Fig. 10 Comparison of computed and experimental trim angle variation for various Fn.

The dynamic trim and sinkage results are obtained from interDyMFoam solver and compared with EFD in Figs. 10 and 11. The CFD solution for moderate Froude numbers (Fn < 0.4) have good agreement with EFD. It indicates that the dynamic simulations using interDyMFoam solver gives reasonably accurate predictions especially for Fn < 0.4.

■ EFD

- CFD(iiileiDyMFoam)

/ ■ ! "

, „ .

0.05 0,1 0 15 0.2 0,25 0.3 0.35 0,4 0,4.^

Froude number

Fig. 11 Comparison of computed and experimental sinkage variation for various Fn.

Pure drift simulation

The OTT is simulated in OpenFOAM to evaluate the linear and nonlinear velocity dependent damping coefficients. OTT is done with a constant inflow speed of 7at various drift angles p. A right handed coordinate system fixed to the body is defined so that x — and y — axis are longitudinal and transverse axes as depicted in Fig. 12. The z — axis is the vertical axis and positive downward. The components of the flow velocity along the x — and y — axis are u = —V cos p and v = — V sin p . The body is acted by a hydrodynamic force with components x and y along the longitudinal and transverse axes respectively. The body is also acted by a moment N about the vertical axis z . If the initial condition is defined when the drift angle p is zero and considering the port and starboard symmetry, the components of hydrodynamic force and moment may be given as follows using Taylor series expansion.

x = x + xv2

Y = Yv + Yv3

N = Nv + Nv

where Xvv, Yv, Yvvv, Nv and Nm are transverse velocity dependent damping coefficients. The coefficients Yv and Nv are the linear coefficients and the rest are nonlinear ones. Simulation of OTT at various drift angle p provides the forces x and Y and moment N . By using a curve fitting to the data of forces and moment as a function of p, the hydrodynamic derivatives or coefficients Xv, Yv ,Yvvv, Nv and Nvvv are obtained.

The Simulation of OTT on CFD environment with OpenFOAM is done at drift angle p = 0, 2, 6, 9, 10, 11, 12, 16, 20 degrees with two Froude numbers Fn = 0.138, 0.28. Furthermore to investigate the port-starboard symmetry on hydrodynamic forces, simulation is also done at drift angle p = -6 degrees. These correspond to the model test program has been done by Yoon (2009) at Iowa Institute of Hydraulic Research (IIHR) to provide the validation tool.

To choose the appropriate turbulence model, simulations for OTT at Fn = 0.28 are done with rasInterFoam solver using k-e and SST k-ffl turbulence models. The predicted wave patterns around the body are depicted in Fig. 13 with zero drift angle for using k-e and SST k- ® turbulence models. The non-dimensional transverse force y ' and yaw moment N' compared with EFD in Figs. 14 and 15. The forces x and y and moment N are made non-dimensional using (7), (8) and (9), respectively. The turbulence model SST k-® gives a more accurate solution. Accordingly, all simulations are done with SST k-® turbulence model for the solvers rasInterFoam, LTSInterFoam and interDyMFoam at different drift angle and Froude numbers. The interDyMFoam solver are applied to investigate the effects of dynamic trim and sinkage on hydrodynamic forces, moment and derivatives.

Fig. 13 Comparison of predicted wave pattern for the k -e (bottom) and SST k-w (top).

O EFD V

- CFCHk-c)

--- CFEHSST fc-w) 0 s''

Dritt angled eg)

Fig. 14 Computed and experimental dimensionless transverse force for static maneuver.

g 0.02 =

1 0 ■s

O EFD 0

- CFD(k-f)

---CFD(SST b-wj

.-'''o

-20 -IJ -10

-S 0 5

['mil ;iiu.Yi.ky:

Fig. 15 Computed and experimental dimension-less yaw moment for static maneuver.

Predicted wave patterns are shown in Figs. 16 to 19 for p = 6, 9, 16, 20 degrees and Fn = 0.28. The contours in these figures are the iso-elevation lines. The water elevation around the body is changing in a nonlinear pattern with variation of the drift angle. The numerical solutions for non-dimensional longitudinal and transverse forces are shown in Figs. 20 and 21as a function of drift angle p for Fn = 0.138 with solvers rasInterFoam, LTSInterFoam and interDyMFoam. The experimental results for fixed condition, without dynamic trim and sinkage, are also depicted in these figures for comparison.

Fig. 16 Comparison of predicted wave pattern for p = 6 at Fn=0.28.

Fig. 18 Comparison of predicted wave pattern for p = 16 at Fn=0.28.

Fig. 17 Comparison of predicted wave pattern for p = 9 at Fn=0.28.

Fig. 19 Comparison of predicted wave pattern for p = 20 at Fn=0.28.

Drift anide{ilefc)

Fig. 20 Computed and experimental longitudinal force for static maneuver at Fn=0.138.

- CFD(wintcrFovii) /

---CFDfmteiDyMFiUm» / 0

CFD(LTSInterFoam> y

s / Ï'

-20 -I? -10 -3 0 5 10 IS 20

DriH an£lc(dcg)

Fig. 21 Computed and experimental transverse force for static maneuver at Fn=0.138.

The solutions for non-dimensional longitudinal force should be symmetrically about p = 0 for identical drift angle to port or to starboard due to the symmetrical shape of the body. The experimental solutions do not show such a trend at Fn = 0.138. The non-dimensional transverse force should have identical value with different sign for identical drift angle to port and starboard due to the symmetrical shape of the body. The experimental data show also such a trend approximately. The LTSInter-Foam solver does not give accurate results for non-dimensional transverse force in compare with EFD especially at large p but the interDyMFoam solver provides relatively accurate solutions.

Fig. 22 Computed and experimental longitudinal force for static maneuver at Fn=0.28.

Fig. 23 Computed and experimental transverse force for static maneuver at Fn=0.28.

o.os 0.06 o.o-i | 0.02 0

I -0.02

-0.04 -0.06 -oos

0 EFD /

- CFD(rasIiiI«Foani) /

---CFD (ml ctDyViF o are ) / <5

CFn<T-TSIntcrFoam;i y j,

f 1 1 1 I L 1

-5 0 S

Drift angle{de£)

Fig. 24 Computed and experimental yaw moment for static maneuver at Fn=0.138.

Fig. 25 Computed and experimental yaw moment for static maneuver for Fn=0.28.

The numerical and experimental results for non-dimensional longitudinal and transverse forces are shown in Figs 22 and 23 for Fn = 0.28 as a function of p. The numerical solutions with the rasInterFoam solver are more accurate for both non-dimensional longitudinal and transverse forces in compare with EFD. The non-dimensional yaw moment is also depicted in Figs 24 and 25 as a function of p forFn = 0.138,0.28, respectively. The N- p graph should demonstrate a symmetrical shape with respect about p = о . The experimental results show approximately such a trend for both Fn as depicted in Figs. 24 and 25. The solver rasInterFoam gives more accurate results in compare with EFD.

The Solver interDyMFoam provides the solutions for Hydrodynamic forces and moment while the dynamic trim and sinkage exist. The solutions with the interDyMFoam are different than the EFD as shown in Figs. 21, 23, 24 and 25. The differences may exist due to the effect of dynamic trim and sinkage. By increasing the drift angle the difference between transverse force and yaw moment obtained by interDyMFoam and experimental results is increased.

The derivatives Yv and Nv can be obtained from the transverse force and yaw moment curves against p from chain rule as follows.

ddß dv

= - Y I

V ß|ß=0

The derivatives Yp and Np are the slope of the transverse force and yaw moment curves against drift angle at p = 0. The values of Yv and Nvare obtained using (15, 16) and are given in Tables 8 and 9 for Fn = 0.138, 0.28, respectively, with various solvers. The experimental values of these derivatives are also tabulated for comparison. Difference between the solvers results and EFD are shown in Table 10. It is seen that the rasInterFoam solver provides more accurate results.

Table 8 Linear hydrodynamic coefficients (Fn=0.138).

coefficients EFD CFD (rasInterFoam) CFD (LTSInterFoam) CFD (interDyMFoam)

Yv -0.2637 -0.2442 -0.3010 -0.2996

Nv -0.1396 -0.1321 -0.1508 -0.1484

Table 9 Linear hydrodynamic coefficients (Fn=0.280).

Coefficients EFD CFD (rasInterFoam) CFD (LTSInterFoam) CFD (interDyMFoam)

Yv -0.2961 -0.2694 -0.3405 -0.3346

Nv -0.1667 -0.1550 -0.1833 -0.1824

Table 10 Difference between EFD and CFD for linear HDC.

Fn=0.138 Fn=0.28

Coefficients error rasInterFoam LTSInterFoam interDyMFoam rasInterFoam LTSInterFoam interDyMFoam

E % Yv 7.39 14.14 13.61 9.02 14.99 13.00

E % Nv 5.37 8.02 6.30 7.02 9.96 9.42

The nonlinear derivatives, Xvv, Yvvv, and Nvvv are obtained from the longitudinal and transverse forces and yaw moment curves against p by using chain rule of differentiation.

dß dv

d 3Y ß

'd£ . dv ,

ßßß

= 53 N

ß = 0

ß (1 dv ) I v j --ßßßiß=c

^ßßßL (i9)

The nonlinear derivative Xpp is obtained by finding the second derivative of the longitudinal force curve against drift angle at p = 0. This can be obtained by using a curve fitting and finding the second derivatives of the fitted curve. The derivatives Yppp and Nppp are also obtained by calculating the third derivative of the transverse force and yaw moment curves against drift angle at p = 0. These are obtained by using curve fittings to the related data. The solutions for these derivatives are given in Tables 11 and 12 for Fn = 0.138, 0.28, respectively, with various solvers. The experimental results are also given in these tables for comparison. The differences among the numerical solutions and experimental solutions are more for nonlinear derivatives than the linear ones. Difference between the solvers results and EFD are shown in Table 13. However, the raslnterFoam solver provides more accurate results than the other two solvers. The interDyMFoam provides the less accurate results than the others. It may be due to the effects of dynamic trim and sinkage that exist in solution with interDyMFoam solver.

Table 11 Non-linear hydrodynamic coefficients (Fn=0.138).

Coefficients EFD CFD (raslnterFoam) CFD (LTSInterFoam) CFD (interDyMFoam)

Y J- vvv -1.6256 -1.3278 -2.0329 -2.0970

N vvv -0.3426 -0.4076 -0.4375 -0.4450

Xvv -0.0301 -0.0363 -0.0385 -0.0392

Table 12 Non-linear hydrodynamic coefficients (Fn=0.280).

coefficients EFD CFD (rasInterFoam) CFD (LTSInterFoam) CFD (interDyMFoam)

Y vvv -1.9456 -2.3150 -2.4397 -2.5487

N vvv -0.4355 -0.3574 -0.5504 -0.5681

Xvv -0.1528 -0.1812 -0.1949 -0.1983

Table 13 Difference between EFD and CFD for Non-linear HDC.

Fn=0.138 Fn=0.28

Coefficients error raslnterFoam LTSInterFoam interDyMFoam rasInterFoam LTSInterFoam interDyMFoam

E % Yvvv 18.32 20.04 22.48 14.49 20.25 23.66

E % Nvvv 15.95 21.69 23.01 13.75 20.88 23.34

E % Xvv 17.08 21.82 23.21 14.32 21.60 22.95

MANEUVERING SIMULATION BASED ON CFD DATA

The regulations bodies assign some standard maneuvers to evaluate the maneuvering qualities of a marine vehicle. The steady turning and zig-zag maneuvers are the maneuvers that are designed to provide the turning, yaw checking and course-keeping abilities of a marine vehicle. Steady turning maneuver is done at a desired speed by deflecting the rudder to a maximum angle (35 deg) to port or starboard from a zero yaw angle until a steady turning circle is obtained. Tactical diameter, advance, transfer and steady turning radius are the essential parameters that are obtained from this maneuver. Zig-zag maneuver is done by deflecting the rudder angle to a desired angle such as 20° to port or starboard and keep it until heading angle approach to 20° then the rudder angle shifted to other side. Overshoot angles and initial turning time to second execute are essential parameters that are obtained from the zig-zag maneuver.

The simulations of these two maneuvers are obtained through the solution of the maneuvering equation in horizontal plane. Using the body coordinate system defined in Fig. 12, the dynamic motion equation of the body are defined in horizontal plane as follows.

(m - Xu )ù = X

(m - Yv )v + (mxG - Yr )r = Y (21)

(mxG - Nv )v + (IZ - Nr )r = N (22)

where m is the mass of the body, IZ is the moment of inertia of the body about z - axis, ù and v are the velocity of the body along x and y directions, respectively. The notations ù and v are the acceleration of the body along x and y directions, respectively, and r and r are the angular velocity and acceleration around the z -axis of the body and xG is the longitudinal position of the center of gravity. The notations X and Y are the external forces on the body along x and y directions, respectively and N is the external moment on the body about z -axis.

The external forces and moments may be divided into hydrodynamic forces and moments due to the surrounding fluid, the environmental forces and moments due to the wind and waves and the other forces and moments due to the action of propulsion and steering systems. The hydrodynamic forces and moments are also divided into added mass forces and moments due to the fluid accelerations, damping forces and moments due to fluid velocity and restoring forces and moments due to the interaction of the buoyancy and gravity forces acting on the body. The steering forces and moments are the forces and moments acting on the body due to the action of the rudder (s) or other maneuvering devices. It is assumed that there are no wind and wave forces and the body is equipped with a rudder at the stern.

The motion equations that are used to simulate the turning and zig-zag maneuvers are (21) and (22). These two equations are called as the steering equations for ships. The steering equations may be given as following (Yoon, 2009).

1 3 1 2

(m - Yv )v + (mxG - Yr )r = Yvv +- Ymvv + - Yvrrvr

+Yvùv(ù - V) + (Yr -mV)r +1Y r3 +1Y rv2 + Y S +1Y S3

T , J rrr' T , J rvv' v T 1 Su T , 1 SSSU 6 2 6

1 3 1 2

(mxG - N )v + (Iz - Nr )r = Nvv + - N^v + - Nvrrvr

+Nvùv(ù - V) + (Nr - mV)r +1 n r3 +1N rv2 + N â +1N S3

T , Jvrrr' T , rvv'v T JVS" T , JVSSS" 6 2 6

where Yv and Nv are the derivative of the transverse force and yaw moment with respect to accelerationv. The notations Yr and Nr are the derivative of the transverse force and yaw moment with respect to acceleration r. The parameters Yv, Yvvv, Yvvr, Yu, Yr, Yrrr, Yvrr and Yrvv are the various order derivatives of the transverse force with respect to the variables written as indices. The notations Nv, Nvv, Nvr, Nu, Nr, Nrrr, Nvrr and Nrvv are the different order derivatives of yaw moment with respect to the variables written as indices. The parameter S is rudder angle and Ys, Ysss, NS and Nsss are the various order derivatives of transverse force and yaw moment with respect to the rudder angle. The hydrodynamic coefficients Yv, Yvvv, Nv and Nvvv are obtained from CFD simulation by OpenFOAM. The others are taken from available model test data given in Yoon (2009) as shown in Table 12. The mass of the model ship is 86 kg and the mass moment of inertia is 49.99 kgm2 (Yoon, 2009).

To calculate the ship path during each maneuver, initial values for v, r and S are set to zero. A time step such as h is considered and the new values for v, r at time tx = t0 + h are found from the solution of (23) and (24). The procedure is repeated using the values of v, r at time tn to obtain the new v, r for a time tn+1 = tn + h and so on. The difference of turning and zig-zag maneuver is about definition of the rudder deflection as a function of time. To simulate turning maneuver the rudder deflects with constant deflection rate 0.04 rad/s up to maximum rudder angle, 35 deg, and then rudder angle set to this angle. But for zig-zag maneuver first rudder deflects with constant rate 0.04 rad/s up to 20 deg and keep it until the ship heading achieved 20 deg. After that the rudder is deflected to other side.

The Fourth-order Runge-Kutta method and Euler algorithm are applied to simulate the turning and zig-zag maneuvers, respectively. The time step is set to be equal to h = 0.1 s in simulation of the maneuvers. After finding the values of v, r for each maneuver at various times t, the yaw angle and the position of the ship relative to a fixed coordinate system are calculated by numerical integration of the following equations during each maneuver.

W(f) = \[r(t )dt

x(t) = f (u(t)cos^(t)-v(t)sinK(t))dt

I (25)

_y(t) = Jo(w(t) sin^(t) + v(t) cos^(t)) dt

The resultant trajectory of turning and zig-zag maneuver are shown in Figs. 26 and 27, respectively. The parameters of turning maneuver are given in Table 13 for different solvers at Fn = 0.28. All results are compared with each other and with EFD. The raslnterFoam solver provides a good prediction of the maneuvers in compare with EFD.

Fig. 26 Simulation of turning circle of ship with 5=35 deg for Fn=0.28.

Fig. 27 Simulation of 20/20 deg zigzag of ship for Fn=0.28.

Table 14 Hydrodynamic derivatives of steering equation Yoon (2009).

-0.1111

-0.0136

Y vrr -1.3683

Ym -0.0242

Y r -0.0457

Y rrr -0.0570

Yvr -1.7067

Ys 0.0586

Y 1 333 -0.0097

Nv -0.0131

Nr -0.0096

N„rr -0.4011

Nvu -0.0397

Nr -0.0487

Nrrr -0.0342

N„r -0.5512

Ns -0.0293

N ly 333 0.0048

Table 15 Turning test characteristics for Fn=0.28.

raslnterFoam interDyMFoam LTSInterFoam EFD

Tactical Diameter 10.97 11.33 13.13 11.81

Advance 17.25 16.01 19.86 18.04

Steady turning radius 5.70 5.52 6.66 5.91

Transfer 3.36 2.43 3.76 3.40

CONCLUSION

Maneuverability is an important hydrodynamic quality of a marine vehicle. The maneuvering characteristics of a marine vehicle should be predicted during the various design stages and validated after construction of the vessel during the trial tests. There are various models to predict the maneuvering properties of a marine vehicle and among them the Abkowitz model is used more than the others. In this model, the external forces and moments are defined using hydrodynamic derivatives or coefficients based on Taylor series expansion. These hydrodynamic coefficients should be found in advance to predict the maneuvering properties of a marine vehicle. Computational Fluid Dynamics (CFD) is used to found some of these hydrody-namic coefficients of a model ship by virtual simulating OTT.

OpenFOAM is applied to simulate OTT and finding the lateral velocity dependent damping coefficients of a DTMB 5512 model ship. The solutions are obtained by three different solvers: raslnterFoam (unsteady solver), LTSInterFoam (steady solver) and interDyMFoam (dynamic solver). These solvers are based on RANS formulation and it is required to use an appropriate turbulence model. Two different well known models k-e and SST k-w are examined in simulations and the results indicate that

SST k-ю gives more accurate turbulence model because of its good performance to predict separated flow at high drift angles.

Comparison of the numerical results with EFD shows that the raslnterFoam solver gives more accurate solutions than two other solvers but needs much more computational time. Although the LTSInterFoam solver gives less accurate results than raslnterFoam solver but it reaches to steady-state solution quickly by manipulating the time step for each grid. The computational time for LTSInterFoam is usually 15-25% less than raslnterFoam. The interDyMFoam solver provides an accurate prediction of dynamic motion for a moderate Fn and is useful to calculate the effect of dynamic trim and sinkage on hydrodynamic coefficients.

The hydrodynamic forces and moments have nonlinear variations and therefore, the linear and nonlinear coefficients should be obtained to simulate a maneuver accurately. Virtual simulation by CFD can be done in a wide range of drift angle and consequently, the linear and nonlinear coefficients can be obtained more precisely. This can help at the preliminary design stage to obtain optimal maneuvering performance, since CFD is a precise and affordable tool.

It should be indicated that application of CFD to calculate hydrodynamic coefficients has been limited to underwater marine vehicles without the effect of the free surface. The presence of free surface makes the fluid flow a two phase flow and needs much more computational efforts. This research work is unique due to the applications of CFD to find the hydrodynamic coefficient to a model ship and of OpenFOAM software to simulate the fluid flow around the body. The source code of OpenFOAM is freely accessible which affords a robust and very flexible advance environment for a viscous ship maneuvering simulation.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the computing time granted by the High Performance Computing Research Center (HPCRC) at Amirkabir University of Technology.

REFERENCES

Abkowitz, M.A., 1969. Stability and motion control of ocean vehicles. Massachusetts and London: MIT, Cambridge. Beck, R. and Reed, A. Modern seakeeping computations for ships. Twenty Third Symposium on Naval Hydrodynamics,

Val de Reuil, France, 17-22 September 2000, pp.1-45. Dantas, J.L.D. and de Barros, E.A., 2013. Numerical analysis of control surface effects on AUV manoeuvrability. Journal

of Applied Ocean Research, 42, pp.168-181. Ferziger, H. J. and Peric, M., 2002. Computational methods for fluid dynamics. 3rd edition. Berlin: Springer. Gentaz, L., Guillerm, P.E., Alessandrini, B. and Delhommeau, G. Three-dimensional free-surface viscous flow around a ship in forced motion. Proceedings of Seventh International Conference Numerical Ship Hydrodynamic, Nantes, France, 19-22 July 1999, pp.1-12. Hirt, C.W. and Nichols, B.D., 1981. Volume of fluid (VOF) method for the dynamics of free boundaries. Journal of Computational Physics, 39, pp.201-225. IMO (International Maritime Organization), 2002a. Resolution standards for ship maneuverability, MSC.137 (76). London: IMO.

IMO (International Maritime Organization), 2002b. Explanatory notes to the standards for ship maneuverability, MSC/ Circ 1053. London: IMO.

Jasak, H., 2009. OpenFOAM: Open source CFD in research and industry. International Journal of Naval Architecture and

Ocean Engineering, 1(2), pp.89-94. Lewis, E.V., 1988. Principles of naval architecture. Jersey City, NJ: The Society of Naval Architects and Marine Engineers. Menter, F.R., Kuntz, M. and Langtry, R. 2003. Ten Years of industrial experience with the SST turbulence model. Turbulence, Heat and Mass Transfer 4. edited by K. Hanjalic, Y. Nagano, and M. Tummers. New York: Begell House, Inc. Nazir, Z., Su, Y. and Wang, Z., 2010. A CFD based investigation of the unsteady hydrodynamic coefficients of 3-D fins in

viscous flow. Journal of Marine Science and Application, 9(3), pp.250-255. Nomoto, K., 1960. Analysis of Kempf's standard maneuver test and proposed steering quality indices. Proceedings of 1st Symposium on Ship Maneuverability, Department Of The Navy, Maryland, United State of America, 24 -25 May 1960, pp.275- 304.

Olivieri, A., Pistani, F., Avanzini, A., Stern, F. and Penna, R., 2001. Towing tank experiments of resistance, sinkage and trim, boundary layer, wake, and free surface flow around a naval combatant insean 2340 model, IIHR Technical Report No. 42. Iowa: Iowa institute of hydrolic research, The University of Iowa.

Park, S., Park, S.W., Rhee, S.H., Lee, S.B., Choi, J. and Kang, S.H., 2013. Investigation on the wall function implementation for the prediction of ship resistance. International Journal of Naval architecture and Ocean Engineering, 5(1), pp.33-46.

Ray, A., Singh, S.N. and Seshadri, V., 2009. Evaluation of linear and nonlinear hydrodynamic coefficients of underwater vehicles using CFD. Proceedings of the ASME 28th International Conference on Ocean, Offshore and Arctic Engineering, Honolulu, Hawaii, 31May - 5 June 2009, pp.257-265.

Roache, P.J., 1997. Quantification of uncertainty in computational fluid dynamics. Annual Review of Fluid Mechanics, 29, pp.123-160.

Rusche, H., 2002. Computational fluid synamics ofsispersed two-phase flows at high phase fractions. Ph.D thesis. Department of Mechanical Engineering, Imperial College of Science, Technology & Medicine, London.

Sarkar, T., Sayer, P.G. and Fraser, S.M., 1997. A study of autonomous underwater vehicle hull forms using computational fluid dynamics. International Journal for Numerical Methods in Fluids, 25(11), pp.1301-1313.

Schmode, D., Bertram, V. and Tenzer, M., 2009. Simulating ship motions and loads using OpenFOAM. 12th Numerical Towing Tank Symposium, Cortona, Italy, 4-6 October 2009, pp. 148-152.

Seo, J.H., Seol, D.M., Lee, J.H. and Rhee, S.H., 2010. Flexible CFD meshing strategy for prediction of ship resistance and propulsion performance. International Journal of Naval Architecture and Ocean Engineering, 2(3), pp.139-145.

Simonsen, C.D., Otzen, J.F., Klimt, C., Larsen, N.L. and Stern, F., 2012. Maneuvering predictions in the early design phase using CFD generated PMM data. 29th Symposium on Naval Hydrodynamics, Gothenburg, Sweden, 26-31 August 2012.

Tyagi, A. and Sen, D., 2006. Calculation of transverse hydrodynamic coefficients using computational fluid dynamic approach. Journal of Ocean Engineering, 33, pp.798-809.

Ubbink, O., 1997. Numerical prediction of two fluid systems with sharp interfaces. Ph.D thesis. Department of Mechanical Engineering, Imperial College of Science.

Wilson, R., Paterson, E. and Stern, F. Unsteady RANS CFD method for naval combatants in waves. Proceedings of 22nd Symposium Naval Hydrodynamic, Washington, D C., 9-14 August 1998, pp.532-549.

Yoon, H., 2009. Phase-averaged stereo-PIV flow field and Force/moment/motion measurements for surface combatant in PMM maneuvers. Ph.D thesis. The University of Iowa.

Yoshimura, Y., 2005. Mathematical model for maneuvering ship motion (MMG Model). Workshop on Mathematical Models for Operations involving Ship-Ship Interaction, Tokyo, August 2005, pp.1-6.

Zhang, H., Xu, Y. and Cai, H., 2010. Using CFD software to calculate hydrodynamic coefficients. Journal of Marine Science and Application, 9, pp.149-155.