J Mar Sci Teclinol (2012) 17:422-445 D O I 10.1007/S00773-012-0178-X
O R I G I N A L A R T I C L E
URANS simulations of static and dynamic maneuvering
for surface combatant: part 1. Verification and validation
for forces, moment, and hydrodynamic derivatives
Nobuaki Sakamoto • Pablo M. Carrica • Frederick Stern
Received: 18 August 2010/Accepted: 3 M a y 2012/Published online: 16 June 2012 © J A S N A O E 2012
Abstract Part I of this two-part paper presents the
verifi-cation and validation results of forces and moment coeffi-cients, hydrodynamic derivatives, and reconstrucdons of forces and moment coefficients from resultant hydrody-namic derivatives for a surface combatant Model 5415 bare hull under static and dynamic planar motion mechanism simulations. Unsteady Reynolds averaged Navier-Stokes (URANS) computations are carxied out by a general purpose URANS/detached eddy simulation research code CFDShip-lowa Ver. 4. The objective of this research is to investigate the capability of the code in regards to the computational fluid dynamics based maneuvering prediction method. In the cuiTent study, the ship is subjected to static drift, steady turn, pure sway, pure yaw, and combined yaw and drift motions at Froude number 0.28. The results are analyzed in view of: (1) the veriflcation for iterative, grid, and time-step convergence along with assessment of overall numerical uncertainty; and (2) validations for forces and moment coefficients, hydro-dynamic derivatives, and reconstruction of forces and moment coefficients from resultant hydrodynamic deriva-tives together with the available experimental data. Part 2 provides the validation for flow features with the experi-mental data as well as investigations for flow physics, e.g., flow separation, thi-ee dimensional vortical structure, and reconstructed local flows.
N , Sakamoto • P. M . Carrica • F. Stern ( H )
UHR-Hydroscience and Engineering, C. M a x w e l l Stanley Hydraulics Laboratory, The University o f Iowa, Iowa, l A 52242-1585, U S A
e-mail: fredei-ick-stern@uiowa.edu Present Address:
N . Sakamoto
National Maritime Research Institute, 6-38-1 Shinkawa, Mitaka, T o k y o 181-0004, Japan
Keywords URANS • P M M • Verification and validation
1 Introduction
In recognition of the importance of ship maneuverability as a major factor for navigational safety the International Maritime Organization (IMO) has developed Standards for Ship Maneuverability [1]. Meeting these standards has placed greater emphasis on maneuvering prediction meth-ods, which historically have been more empirical than those developed for resistance, propulsion, and seakeeping [2]. Among several methods for maneuvering prediction, static and dynamic planar' motion mechanism (PMM) tests are one of the most commonly used approaches. They provide hydrodynamic derivatives by focusing on the creation of a mathematical model. The P M M tests can be feasible in a conventional towing tank equipped with a P M M motion generator or a basin with rotating arm capability. However, the tests contain several disadvantages; (1) expensive test facilities and complexity in the experimental settings; (2) considerable scale effect arising from the impossibility in practice to achieve Froude number (Fn) and Reynolds number {Rn) similarities simultatieously; and (3) hmitations in obtaining physical understanding of flow flelds around a ship in maneuvering motions.
Computational fluid dynamics (CFD) based maneuvering prediction methods significantly contribute to resolve these disadvantages. Since the viscous effects are very important for accurate maneuvering prediction, unsteady Reynolds averaged Navier-Stokes (URANS) simulation and detached eddy simulation (DES) have been considered to be the most promising approach rather than inviscid approaches. The URANS/DES simulations replace the static and dynamic P M M experiments to obtain hydrodynamic derivatives and
J Mar Sci Teclinol (2012) 17:422-445 423 provide detail local flow physics around the hull under
maneuvering motions.
The objective of this research is to investigate the capabihty of general purpose URANS/DES research code CFDShipTowa Ver. 4 [3-5] simulating surface combatant Model 5415 under static and dynamic PMM tests. Part 1 of this two-part paper presents the veriflcation and validation (V&V) results of forces and moment coefficients, valida-tion for hydrodynamic derivatives, and reconstrucvalida-tions of forces and moment coefficients f r o m resultant hydrody-namic derivatives. Overall results in the present study are extensive [6], and, thus, the most important outcomes are presented herein and at SIMMAN 2008 [7] by IIHR-Hydroscience and Engineering [8] for the CFD-based method. Part 2 provides the detailed validation for flow fea-tures with the experimental data [9] as well as investigations for flow physics, e.g., flow separation, three dimensional vortical structure and reconstructed local flows [10].
1.1 CFD-based maneuvering prediction at S I M M A N 2008
For K V L C C I & 2 tankers, Brogha et al. [11] perform dynamic P M M simulations with steering rudder and body-force propeller. They figure out that the stern region is more effective in producing lateral hydrodynamic force. Toxopeus and Lee [12] and Cura Hochbaum et al. [13] show that the hydrodynamic derivatives determined from URANS simulations are able to predict ship trajectories with enough accuracy when they are compared with the free sailing data. Canica and Stem [14] demonstrate the capability o f the URANS/DES method with moving rudder and discretized rotating propeller to simulate f u l l time domain maneuvers.
For a KCS container ship, Simonsen and Stern [15] perform a V & V study f o r the forces and moment coeffi-cients for pure yaw motion. Due to the relative small grid refinement ratio and having two degrees of freedom (heave and pitch), they have difficulties in applying the verifica-tion method [16] to the time series of forces and moment coefficients.
For a Model 5415 naval surface combatant, Sakamoto et al. [8] and Guilmineau et al. [17] perform static and dynamic P M M simulations for the bare hull. Sakamoto et al. [8] identify the vortical structures around the hull, and Guilmineau al. [17] show better resolution of local flow around vortex cores. Miller [18] uses both bare and frflly appended huUs for static/dynamic P M M simulations, and shows that the eiTors of forces and moment coefficients in fully appended hull is greater than the bare hull results. Carrica et al. [19] perform f u l l time domain URANS/DES maneuvering simulations in calm water and in waves with moving rudder and body-force propeller, showing detail
vortical structures around appendages during the maneuvers.
1.2 Conclusion from past research
Reviews for the SIMMAN 2008 [7, 20] lists several pre-liminary conclusions and issues for the CFD-based maneuvering prediction method: (1) number of study for surface combatant is much less than commercial type ships; (2) grid, turbulence model, and inclusion of free surface may play important roles to predict forces, moment, and local flow quantities; (3) blockage effect may not be negligible for ships at larger amplitude P M M tests; (4) hydrodynamic derivatives are usually not computed from resultant forces and moment coefficients, although a few cases show that URANS methods can accurately pre-dict linear hydrodynamic derivatives; (5) attention is not paid to evaluation of non-linear and cross-coupling deriv-atives in most of the cases; (6) local flow physics are analyzed in limited cases, and no systematic validations together with the experimental fluid dynamics (EFD) data are made, and; (7) f u l l time domain URANS/DES maneuvering simulation is possible but still challenging, thus, the practical approach is to combine viscous CFD simulations of static/dynamic P M M tests with systems-based maneuvering simulation.
2 Test overviews
2.1 Geometry
The geometry used in the current study is the David Taylor Model Basin (DTMB) Model 5512 (the length between perpendicular Lpp = 3.048 m), which is the preliminary design for a surface combatant ca. 1980 and a geosym of the larger Model 5415 (Lpp = 5.73 m). Model 5415 have been chosen as one of the benchmark hulls by the Inter-national Towing Tank Conference (ITTC) Resistance Committee [21, 22] and Maneuvering Comnuttee [2]. It has been used in several ship hydrodynamics workshops [23, 24] and is also adopted in S I M M A N 2008. The model used in the cuiTent study does not have appendages but with fltted bilge keels at port and starboard.
2.2 Static and dynamic P M M tests
Figure 1 describes the coordinate system utilized for cur-rent static and dynamic P M M simulations. In the figures,
and )'E denote the ZT-plane in the earth-fixed system, and Xs and ys denote the XT-plane in the ship-fixed system. The rest of the nomenclature is defined i n a later part of this section. The coordinate system is different from what is
424 J M a r Sci Teclinol (2012) 1 7 : 4 2 2 ^ 4 5
Fig. 1 Coordinate system o f the static and dynamic P M M simula-tions (notice that the positive direction i n x is f r o m FP to A P )
usually leveraged in the maneuvering field in that the positive direction of x is from forward perpendicular (FP) to aft perpendicular (AP). This is due to the fact that the positive direction of AE is set to be identical to the direction of free-stream incoming velocity to the ship. Positive direction of y is pointing from port to starboard, and thus the direction of rotation is counter-clockwise in accordance with the right-handed coordinate system.
During the static dnft test, the model is towed in a conventional towing tank at a constant velocity UQ with the initial dnft angle ^ relative to the ship's axis.
During the steady turn test, a yaw angular velocity r is imposed on the model by fixing it to the end of a radial arm and rotating the aim with its length R about a vertical axis fixed in the tank [25]. The yaw angular velocity r is given by
During the pure sway test, the ship axis is always parallel to the tank centerhne and the model is given sway position sway velocity and sway acceleration v as a function of titne 3' V V sin (co/) -1'max COs(cor) Vmax Sm{(Dt) (2)
where m, is the angular frequency of sway motion, Vmax is maximum sway velocity, and v^ax is maximum sway acceleration. The coiresponding drift angle /^corr. of flow relative to the ship is defined as
tan (3)
During the pure yaw test, the ship is towed down the tank with the ship axis always tangent to its path. The model is given not only sway position, velocity, and acceleration by Eq. (2), but also yaw angle \p, yaw angular velocity r, and yaw angular acceleration /• as a function of time
lAn,ax C0S(CÖ/)
sin(co;)
cos(cot)
lAmax (4)
where co, is the angular frequency of yaw motion which is equal to the angular frequency of the sway motion, i/^^a^ is maximum yaw amphtude, ij/^^^ is maximum yaw angular velocity, and t/z^a^ is maximum yaw angular acceleration. During the combined yaw and drift test, the ship is given r and r as a function of time by Eq. (4) with constant drift angle P thus the ship axis is not always tangent to its towing path.
The computational results of surge force X, sway force
Y, and yaw moment N are subjected to the validation with
the available experimental data [7, 9].
3 Computational method
3.1 Modeling
The CFD solver utilizes an absolute/relative inertial coor-dinate system and a non-inertial ship-fixed coorcoor-dinate system to describe prescribed/predicted ship motions [5]. The flow field is solved in the absolute/relative inertial coordinate system while the ship motions are solved in the non-inertial ship-fixed coordinate system. The code solves an incompressible URANS equation with a single-phase level-set method as a free surface modeling, and isotropic blended k - elk - co (BKW) model or BKW-based alge-braic Reynolds stress (ARS) model with DES option as a turbulence modeling [3, 26].
A l l governing equations are made non-dimensional by f/o. ^pp> fluid density p, gravitational acceleration g, and the dynamic viscosity /.t which yield the definitions in Fn =
UQ/y^gLpp and in Rn = pUoLpp/fi. This provides the
fol-lowing non-dimensionalization in i', v, /•, X, Y, and A'^ as
(5)
• /
-V V
J Mar Sci Technol (2012) 17:422-445 425 X y' 0.5pVlT^Lpp Y O.SpUlT^Lpp N (6) (7) O.SpUlT^Llp
where is a draft of a model ship i n full-load condition.
3.2 Numerical methods and high-performance computing
A second-order Euler backward difference is used for a temporal discretization of all variables. The finite-differ-ence method is utilized for a spatial discretization, e.g., a second-order upwind scheme (ED2) or second-order total variation diminishing with "superbee" (TVD2S) scheme [27] in momentum convection, a first-order upwind scheme in turbulence convection, and ahybrid first and second-order upwind scheme in the level-set convection. The viscous terms in momentum and turbulence equations are computed using a second-order central difference scheme. The pressure implicit split operator (PISO) algorithm is used to couple the momentum and continuity equations. The code is made parallel using message passing interface (MPI) with a domain decomposition technique.
Overset grid technique is adopted to simulate dynamic ship motions and local grid refinements [28]. Figure la, b show the typical overset grid ari'angement in the current study. The external software SUGGAR is used to obtain the grid connectivity between overlapping grids. It runs as a separate process from the flow solver, and is called every time when the ship motions are prescribed in time to pro-vide the interpolation information between the overset grids to the flow solver [4]. Another preprocessing software USURP [29] provides weights to the active points on the overlapped region of no-slip surfaces, e.g., between the hull grid and the bilge-keel grid in the present study. This avoids counting the same area in space more than once, and, thus, the flow solver is able to calculate the correct area, forces, and moments.
4 Simulation design
4.1 Test cases
(e.g., static drift and steady turn) and (x/Lpp, }'/Lpp, z/Lpp) — (0.50515, 0, -0.004) for dynamic PiVlM simulations (e.g., pure sway, pure yaw and combined yaw. and drift), respectively. In both static and dynamic P M M simulations, the center of rotation (CoR) is set to (x/Lpp, y/Lpp, z/Lpp) = (0.5, 0, —0.00208) where the yaw moment around the z-axis is computed at the location. At flrst, all the simula-tions except steady turn are performed with side walls since it is considered to be important to accurately reproduce the experimental condition. However, the time histories of forces and moment coefficients for the simulations with side walls show very large and slowly damped oscillation due to the spurious waves which are partially reflected by the upstream, downstream, and side wall boundaries which yields slow convergence [6]. Therefore, the simulations without walls are caiTied out.
4.2 Grid, domain size and time step
Figure 2a-d present the overview of the computational grids, their domain size and boundary conditions. Table 2 summarizes the size of the fine grids. The commercial software GRIDGEN® with hyperbolic extrusion for the curvilinear grids is used to generate all the grids. A t the solid surfaces the first grid point is set at)'+ < 1 as required by the k — elk — co turbulence model. The grid 1 is iru-tially designed to include the side walls of the IIHR towing tank. The grid 1' is prepared due to the necessity of increasing boundary layer and free surface resolution fol-lowed by the result of the straight ahead simulation with grid I [6], still maintaining the same domain size as grid I . To exclude the side walls, the grid lNv//grid \'^^ is designed whose base structure is the same as grid 1/grid 1' but the distance from the centerline to side wall boundaries is 20 times larger. The medium and coarse grids for veii-fication study are coarsened from fine grids using non-integer refinement ratio \/2. Prescribed sway/yaw motions are applied to all the blocks except the outer boundary which remains stationary during the dynamic P M M simulations.
For the static P M M simulations, the non-dimensional time step is set to Ar = O.OI. For dynamic P M M simula-tions, At2 = 0.00979 is used which allows 384 time steps per one sway/yaw period. For the verification study, sys-tematically refined time steps with refinement ratio 2 are used, resulting in A / j = 0.00489 and Afs = 0.01957.
Table 1 summarizes the test cases presented in this article. In all the cases Rn is 4.67 x 10* and Fn is 0.28. The hull configuration are either fixed at even-keel (FXQ) or fixed at sunk and trimmed ( F X ^ ^ ) . The CoG is set to {xlLpp, J'/Lpp,
zJLpp) = (0.5, 0, -0.004) for static P M M simulations
4.3 Boundary conditions
The boundary conditions utilized in the current study are inlet, outlet, no-slip, and far-field conditions for which their mathematical descriptions can be found in Cairica et al. [3]
Table 1 Test matrix f o r the static and dynamic P M M simulations
Grid Condition Presc. 6 D 0 F
Pai'am. Value Conv. schm. turb. model EFD data Static d r i f t 1.1 1', 2', 3' 1 2 1' 9' 3' •'•^ -^NW' ^IW ^NW 1 -3 3x111;
Steady turn (R is non-dimensional by L„„.) 2.1 31,-Pure sway 3.1 3' Pure yaw 4.1 1 , 2 , 3 4.2 T 9'iNW' ^NW' -^NW V V
Combined yaw and d r i f t 5.1 10° 0°, 2 ° , 6°, 9°, 10°, 11° 12°, 16°, 2 0 ° 1.67, 3.33, 6.67 2°, 4 ° , 10° 0.3 0.15, 0.3, 0.6 9°, 10°, 11° F D 2 - B K W F D 2 - B K W T V D 2 S - A R S
Surge, sway, yaw T V D 2 S - A R S
Sway Sway, yaw T V D 2 S - A R S F D 2 - B K W T V D 2 S - A R S D r i f t , sway, yaw T V D 2 S - A R S X',
r,
N' X',r,
N', y„ 7„„ X',r,
N', Xrr, Y„ Y„ X',r,
N',x,„
Yi, y„ X', Y, N' X',r,
N', Yr, Yr, X', Y, N'. X,„ Y„r, Yr N„„ Ai'„,„ Purpose V & V w. walls V & V w.o. wallsForces and moment coefficients, hydrodynamic derivatives Forces and moment coefficients,
hydrodynamic derivatives
Forces and moment coefficients, hydrodynamic derivatives V & V w. walls
V & V w.o. walls
Forces and moment coefficients, hydrodynamic derivatives
Forces and moment coefficients, hydrodynamic derivatives
J Mar Sci Technol (2012) 17:422-445 427
Fig. 2 Grid and boundary conditions: a Grid capable f o r dynamic motions, b Grid f o r the ship fixed at sunk and trimmed, c Boundary conditions
with walls, d Boundary conditions without walls
and Paterson et al. [30]. In the absolute inertial coordinate system for steady turn simulation, the ship's surge and sway coordinate values are prescribed with a constant time interval as well as the velocity components on the no-slip surface. Since the ship moves with the prescribed velocity, the velocity components at the inlet boundary are all zero. In the relative inertial coordinate system for static drift and all the dynamic P M M simulations, the surge motion is not imposed to the ship thus the velocity components at inlet are {U,V,W) = (1,0,0). For straight ahead and static drift simulations, no motions are prescribed thus the velocity components at no-slip suiface are {U,V,W) = (0,0,0). For the dynamic P M M simulations, the ship has prescribed lateral velocity by sway motion and linear components of
axial and lateral velocity due to yaw motion. They are brought into the U and y-components of no-slip condition.
4.4 Analysis method
4.4.1 Fourier analysis
For the dynamic P M M tests, the sway and yaw motions are prescribed by sine and cosine functions, thus, the response of forces and moment coefficients are assumed to be reconstructed as a Fourier series (FS) with the non-dimensional angular frequency of sway/yaw motion
co(=2nLpp/TpMMUo), TpuM is a dimensional period of
prescribed sway/yaw motion period, as
428
J M a r Sci Teclinol (2012) 17:422-445
Table 2 Description Of fine grids
Block name B l o c k type Fine 1 Fine Ij^w Fine 1' Fine I'^^y
Block name B l o c k type
Blocks Grid points Blocks Grid points Blocks G r i d points Blocks Grid points
Boundary layer Body-fitted 4 0.51 M 4 0.51 M 12 1.52 M 16 4.30 M
Bilge keel Body-fitted 4 0.51 M 4 0.51 M 4 0.51 M 8 1.44 M
1st refinement Orthogonal 8 1.0 M 8 1.03 M 16 1.95 M 24 5.52 M
2nd refinement Body-fitted 12 1.49 M 12 1.49 M _ _ _
2nd refinement Orthogonal
-
-
_ _ _Background Orthogonal 4 0.51 M 8 1.01 M 8 1.49 M 36 8.74 M
Total 32 4.02 M 36 4.55 M 40 5.47 M 84 20 M
Domain size - 8 . 6 < xlL pp < 8.6 - 8 . 6 < A'/Lpp < 8.6 Same as fine 1 Same as fine I N W - 0 . 5 < ylL pp < 0.5 - 1 0 . 0 < yLpp < 10.0
- 1 . 0 < dl pp < 0.25 - 1 . 0 < z/Lpp < 0.25 oo oo
F{t) = flo -I- ^ a„ cos{ncot)+ ^ b„ sm{ncot) (8)
"=i 11=1
where F(t) represents the time series of forces and moment coefficients, a„ and b„ is «th-order Fourier sine and cosine coefficients, respecfively. Equation (5) can be re-written as
oo
F{t) = flo + Xl^o cos{nojr + (p,J
(9) with An = sjal + bl cp„ = tan~\-a„/b„) where A„, is the /ïth-order Fourier cosine harmonics and (p„ is the phase angle. Equation (9) is used to evaluate the iterative error for the harmonics of the forces and moment coefficients from the dynamic PMM simulations by marching harmonic analysis [9].
4.4.2 Hydrodynamic derivatives
The hydrodynamic derivatives from static and dynamic P M M tests are calculated based on the Abkowitz-type mathematical model [31] with tlu'ee degrees of freedom (3DOF), e.g., surge, sway, and yaw. The model describes the forces and moment coefficients at ship-fixed coordinate system with bare hull condition as
hydrodynamic derivatives can be calculated as foUows. Notice that the LHS and independent variables in the RHS are ah non-dimensional as defined in [8, 9], and thus the resultant hydrodynamic derivatives are non-dimensional as well.
For the static drift results, the forces and moment coefficients are the function of v thus the right hand side (RHS) of Eq. (10) are simplified as
' A + Sv'2 Y
=
Cv' + Dv^_Ev' + Fv\
with A = Z „ 5 = X„, ,C=Y,,D = ,E=N,,F = N„„. (11) The hydrodynamic derivatives shown in Eq. (11) are obtained as polynomial coefficients by the least-square curve fitting method.
For the pure sway results, the resultant forces and moment coefficients are the function of both v' and v' thus, the RHS of Eq. (11) are simplified as
A + 5v'2
Y'
=
Gv +Cv' +Dv'^ with G=Y,;, H = N^._N' j _ Hv + Ev + Fv'^
(12)
X , + X „ „ v ' 2 + Z , , ; • ' 2 + X „ , l / r '
T/./•' + YiV + Y,v' + y„„,v'3 -f 7,„.,v','2 + Y,.r + T,,.,;'^ +
Ni-r' + Ni,v + N„v + N„,,v^ + N,„rv'r^- + N,.r + N.-n-r'^ -f Yn.rv^
N,,,r'v'^
(10)
The hydrodynamic derivatives in Eq. (10) are of the interest in the current study. Taking the results of static drift and pure sway cases as an example, the
Using v' and v' representation as Eqs. (2) and (5) to Eq. (12) results in the FS representation of the forces and moment coefficients as
J M a r Sci Teclinol (2012) 17:422-445 429 Y'
=
Gvmax sin{cot) N' _ - {cv: - (Ev' cos(2w?) ? ö v l x ) cos(cüO f F v l J cos(cor) '3 cos(3co?) 4 - ' m a x C O s ( 3 r o O (13) 4 maxFourier sine and cosine coefficients are associated witfi Eq. (13) as X2,cos Y,,sin • M . s i n Tl.cos A^l,cos Ts.cos A'^3,cos 1 0 G" H 'C E 2 max 2 max I F ' A ' max >/3 (14) (15) (16) (17)
To calculate hydrodynamic derivatives from the results of pure sway tests, there are two approaches, e.g., (1) linear and nondinear curve fitting methods (LCF and NLCF, respectively), and (2) single run method (SR). For the CF methods, Fourier sine and cosine coefficients of forces and moment coefficients are obtained from multiple pure sway tests, and the polynomial functions with respect to v'^^^ or ''max ^1'^ '•'^^'1 t° calculate the hydrodynamic derivatives with least-square curve fitting. For the SR method, solving Eqs. (14-17) algebraically, hydrodynamic derivatives respect to v' and v' are calculated from a single result of a pure sway test.
For the rest of hydrodynamic derivatives from the other static and dynamic P M M tests, they are calculated f o l -lowing the similar manner as explained above [6].
4.4.3 Reconstruction of forces and moment coefficients
The current "reconstruction" approach evaluates how well the mathematical model with hydrodynamic derivatives can reproduce originally computed forces and moment coefficients instead of performing trajec-tory simulations by resultant hydrodynamic derivatives [32, 33]. The "reconstruction"procedure is given as follows taking static drift and the pure sway cases as examples.
For the static drift, Eq. (9) can reproduce the forces and moment coefficients with the hydrodynamic derivatives. Reconstructed computational results are termed SR. Sim-ilarly for the pure sway, the time history of the forces and moment coefficients over 1 sway period can be repro-duced by Eq. (10) with the hydrodynamic derivatives.
Reconstructed computational results are instantaneous, and they are termed 8^. The comparison error between the experimental data and SR or S/R is the only available one, and for the cuiTent cases, the most acceptable mea-sure to evaluate the quality of the hydrodynamic deriva-tives from the CFD simulations.
4.4.4 Definition of comparison error
The comparison eiTor E between the computational results
S and the experimental data D is defined as
E{%D) = X 100. (18)
The error definition by Eq. (18) is used when the computational results of forces and moment coefficients from static drift tests and hydrodynamic derivatives from all the cases are compared with the experimental data. For the forces and moment coefficients from dynamic P M M tests, it is difficuh to apply Eq. (18) since the profiles of Y and N' cross 0 at certain planar motion phases. To avoid this, the average comparison error EX,Y,N over 1 planar motion period is defined as
E w ( % | ö | ) = ^ I A X 100 (19)
where A' is the total number of experimental data points,
Di is the instantaneous experimental data, and 5,- is
the instantaneous computational results. To match the instantaneous time between the simulation and the experiment, the computational results are subjected to cubic spline interpolation before the eiTor is calculated. The average reconstruction eiTor for the experimental data FRJPJ, is defined following a similar manner, where 5;, is replaced by DR, the instantaneous and reconstructed experimental data. The average comparison error between the reconstructed computational results and the original experimental data FRCFDIS also defined using Eqs. (18) or (19) where S or St is replaced by SR or SjR. The total average comparison error FAVB. for all forces and moment coefficients is defined as
(20)
430 J Mar Sci Teclinol (2012) 17:422^45
5 Uncertainty analysis for forces and moment coefficients
5.1 Verification and validation procedure
Uncertainty analysis is performed using the V & V method following the procedure by Stern et al. [16] with improved factor of safety [34]. Verification procedures identify the most important numerical error sources such as iterative eiTor Sl, grid size error SQ and time-step eiror ÖT and provide eiTor and estimates of simulation numerical uncertainty
Usyi-The forces and moment coefficients are subjected to the uncertainty analysis in the cuirent study. The iterative uncertainty Uj is estimated for all the cases. The grid uncertainty UQ is estimated for the static drift cases at
j] = 10° with and without walls. Both the UQ and the time
step uncertainty Ur is estimated for the pure yaw cases at ' m a x = 0.3 with and without walls. For the static drift and the pure yaw cases, the USN and the validation uncertainty
Uy is also estimated utilizing the experimental uncertainty Uo.
5.2 Iterative convergence
5.2.1 Dependency for inner iteration in pure yaw
The pure yaw is selected to study the 5i depending on the number of iterations to couple the non-linear terms in turbulence and momentum equations (termed inner iter-ation hereafter). The error is evaluated by performing three simulations using medium grid (grid 2') with med-ium time step (Afa), and changing the number of inner iterations from 3 to 4 to 6. Using the mean of longitudinal force coefficient {XQ) and most dominant harmonics {X2,
Yl and A''i), the solution changes of harmonics ( A F S )
based on the solution with inner iteration 6 are computed. The A F S for all the harmonic amplitudes are at least one order of magnitude smaller than the C/j which w i l l be discussed in the next section, and thus the iterative error depending on the number of inner iteration is considered to be negligible. In all the cases, the number of inner iteration is 4.
5.2.2 Solution iterative convergence
5.2.2.1 Static drift and steady turn Two quantities are
extracted from the time history of the forces and moment coefficients to study the statistical convergence, e.g., (1) the running mean ( R M ) , and (2) the magnitude of root mean square of organized oscillation (RMSQ) defined as
RMSo = y (21) where A' is a total number of data points and Ri is the
instantaneous forces and moment coefficients. The average of maximum and minimum R M is considered Ui. Notice that the quantities of RMS,, and Ui referenced in this sec-tion are extracted from Sakamoto [6].
The RMSo of static drift simulations with walls are up to about 29 %M where M is a mean value of forces and moment coefficients in time. It indicates relatively large damped oscillation due to the spurious free surface waves which are partially reflected by the upstream, downstream, and side wall boundaries. When the wide external domains are used in order to leverage the numerical dissipation associated with large grid spacing at inlet/exit/side boundaries, the levels of R M S Q are at most four times smaller than those of the results with walls. This ensures the faster statistical convergence and smaller Ui (less than 0.7 %M). Statistical convergence for steady turn case is similar to the static drift cases with walls.
5.2.2.2 Pure sway, pure yaw and combined yaw and drift In the dynamic P M M cases, the R M S Q responds
mainly to the imposed sway/yaw motion. Once the simu-lation reaches periodic, the amplitude of forces and moment coefficients should be constant and independent from the number of iteration. In order to quantify the Ui of the harmonics, the R M of time histories of firstly- and secondary-dominant harmonic amplitudes are utilized. Overall, the iterative convergence in XQ, YI and is achieved evidenced by the Ui less than 4 % M for all the dynamic PMM cases wMle it is difficult to achieve iterative convergence in X2 in pure sway/yaw and Xi in combined yaw and drift. In pure sway/yaw, although the F3 is more than 20 %Yi, small C/j in Y^ automatically ensures the statistical convergence in T3 (The same discussion can be applied between A^i and A^3).
5.3 Grid and time step convergence
5.3.1 Static drift
Table 3 shows the results of verification in forces and moment coefficients. The X', Y and A^' are separated into friction and pressure components described with the sub-script f and p, respectively.
The Ul for X'^ has the relative highest value since the oscillation in the pressure field decay is much slower than velocity fluctuations [35], thus a sufficient number of iterations is required to reduce the fluctuations in the pressure component. The solution change between fine grid
J Mar Sci Teclinol (2012) 17:422-445
Table 3 Grid convergence f o r , ^^^^^^1 ,,3^ p Convergence (%Sj)
forces and moment coefficients . . for static d r i f t [1 = 10°w/w.o.
walls w. walls 4 1.87e-3 1.68 1.73 0.97 0.16 0.08 M C 145.42
K
1.36 4.78 7.79 0.61 2.82 1.41 M C 63.10 X' 0.23 2.92 4.16 0.70 2.03 1.02 M C 12.87K
0.046 9.93 0.09 115.46 --
M D-r
0.053 9.60 0.13 74.30 - M D-K
0.051 2.19 1.42 1.54-
- M D -0.043 2.15 1.43 1.50-
- M D -w.o. walls 0.08 2.27 0.37 6.14 --
OD-K
0.24 13.33 11.61 - 1 . 1 5 --
OD -X' 0.26 2.82 4.04 - 0 . 7 0 --
OC 2.1K
0.014 0.16 0.02 - 7 . 9 2-
-
OD-r
0.015 0.19 0.06 - 3 . 0 2 --
OD-K
0.020 0.98 0.84 - 1 . 1 6 - - O D -iV 0.020 0.97 0.91 - 1 . 0 6-
- OD-and medium grid £021 is at least one order of magnitude lai-ger than most of the Ui except X'^, indicating that the effect of iteration is almost negligible compared to the effect of grid refinement. In the P = 10° results, the con-vergence ratio Ra shows that only X'p X'^ and X' are monotonically converged (MC) while the rest of coeffi-cients are either oscillatory converged (OC), monotonically or oscillatory diverged (MD and OD, respectively). The grid utilized in the current verification study is not Ificely to be fine enough since Bhushan et al. [36] who utilize up to 250 M show MC/OC in X' and Y with the acceptable level of UG.
5.3.2 Pure yaw
Table 4 summarizes the resuhs of the grid and time step convergence study for XQ, X2, Yi and A^i.
5.3.2.1 Grid convergence A general trend shows that ZQ, X2 and Tl are relatively sensitive to the grid resolution
which is evidenced by the £ 0 2 j up to 10 %Si where the is the fine grid solution. In contrast, A^i is fairly insensitive to the grid resolution since the maximum £ 0 2 1 is up to 3.8 %Si. The Ui for XQ, T I and Nx are 2-20 times smaller than £ G 2 I while the Ui for X2 is nearly the same or
some-times larger than £ 0 2 1 - As a result, the effect of iteration is almost negligible compared to grid refinement in XQ, YI and A ' l , while it is not in X 2 . The RQ values show that i t is difficult to achieve M C in most of the harmonics.
5.3.2.2 Time step convergence A general trend shows
that X', Y and N' are all very sensitive to the size of time
step which is evidenced by the £721 up to 57 %Si. The £721 for the most dominant harmonics is up to 10 %Si and 57 %Si for the case with and without walls, respectively. As well as the effect of grid refinement to the iteration, the
Ul in XQ, YI and A^i is one order of magnitude smaller than
£x2i while it is not in X2. In consequence, the effect of iteration is almost negligible in XQ, YI and A^i compared to the size of time step. Opposite to the results in the grid convergence study, most of the dominant harmonics are MC but with large UT in X2 with/without wafis up to 230 %Si. In addition to the time step convergence study for the dominant harmonics, the U-p is computed as a function of time over 1 yaw motion period, and Table 8 summarizes the locally time-averaged f/x excluding unacceptably spiked Ur- The level of Uj varies in between 2.6 %Si to 7.7 %Si.
5.4 Validation
Table 5 summarizes the validation results of the static drift and the pure yaw cases. For the static drift, due to poor grid convergence in Y' and A ' ' , only X' is of the interest for the validation. The \E\ is slightly smaller than
Uy for the case with walls and thus X' is validated at
15.2 %D interval, while it is not for the case without walls. For the pure yaw, it is difficult to use the FS decomposed forces and moment coefficients for valida-tion due to the poor grid convergence, thus locally time-averaged /7D and USN are adopted to compute C/y [6]. Notice that the large U^, in Y' is due to the electronic noise from the AC servomotor at the P M M carriage [9]. The results show that X' is not validated whereas Y' and
432 J Mar Sci Technol (2012) 1 7 : 4 2 2 ^ 4 5
Table 4 Verification f o r
dominant harmonics of forces and moment coefficients f o r pure yaw /'max = 0.3 w/w.o. walls Grid/time step Installation FS (%) l£k2l/ Rk Si\ X 100 Pk G r i d convergence
4.1 Grid 1, 2, 3 FXo, w. A'o 1.7 5.57 - 0 . 6 1 - — OC
with ht2 walls X2 6.3 4.15 - 1 . 7 5 - - OD )'l 2.2 6.29 - 2 . 0 9 - - O D /V, 0.6 2.10 - 0 . 9 2 -
-
OC 4.2 Grid l^^v. FXo, w.o. 0.3 5.73 - 0 . 6 1-
- OC 2mv> 3 N W walls 25.5 9.55 0.05 8.38 4.19 M C with A?2 6.58 i i 0.5 6.58 - 2 . 3 6 --
OD Ni 0.4 1.69 - 0 . 7 9 --
OC rime-step convergence 4.1 Grid 1, 2, 3 FXo, w. Xo 1.1 2.50 0.32 1.66 0.83 M C w i t h At2 walls X2 33.3 8.51 0.25 1.97 0.99 M C I . l 9.89 0.57 0.82 0.41 M C A'l 1.8 3.85 0.45 1.16 0.57 M C 4.2 Grid I N W , FXo, w.o. A'o 0.2 2.59 0.37 1.43 0.72 M C 2 N W . 3 N W walls X2 25.7 56.59 - 0 . 1 2 _ _ OC w i t h At2 i i 0.9 10.49 0.57 0.82 0.41 M C A'l 2.9 3.74 0.46 1.13 0.57 M C Convergence Uk (%5,) 1,79 0.09 1.80 29.84 0.23 2.01 4.68 27.27 6.14 2.81 230.45 28.71 6.16A'' are validated at 37 %D and 11 %D interval, respectively.
6 Validation of forces and inoment coefficients, hydrodynamic derivatives and reconstruction
6.1 Static P M M tests
6.1.1 Static drift
Figure 3 shows the experimental and computational results of forces and moment coefficients, as well as reconstructed computational results. The figure also includes the friction-pressure ratio (R/Rp) for X', Y' and N'. Table 6 summarizes the hydrodynamic derivatives and Table 7 presents the averaged and maximum values of E, FRCFD and FREFO •
6.1.1.1 Validation of forces, moment and liydrodyuamic derivatives The computational results show overall
agreement to the experimental data with i i up to 10 %D at /? < 12°, and then they tend to become larger than the experimental results. At 0° < P < 20°, the Y and A/', which are dominated by the pressure force, agree better to the experimental data than X' which is dominated by the viscous force. For the R^/R^, as the ship encounters stronger cross flow at larger P, the X'^ becomes significant, and at
P = 20° it almost balances the X'f. In Y and A'' the pressure
component is almost two and three orders of magnitude larger, respectively, than the friction over the entire p.
Table 5 Validation o f forces and moment coefficients f o r static d r i f t
/? = 10° w/w.o. walls and f o r pure yaw r^jax = 0.3 w. walls along 1 yaw motion period
l£l (%0) Uy (%D) Static d r i f t W i t h walls X' 14.3 Y 0.7 N' 4.2 Without walls X' 4.4 y 10.0 iV 2.0 Pure yaw W i t h walls Ë (%Df r 17.08 r 34.03 N' 8.75 15.2 4.1 Uy (%Df 9.96 37.43 10.90 3.6 5.4 2.6 3.6 5.4 2.6 C/D {%Df 6.4 14.6 4.1 [ / S N {%D) 14.8 2.0 f/sN {%DT 7.64 34.46 10.11 %Ta\D,\iN.
The computational results of the linear derivatives agree very well to the experiment within £ of 5 %D as well as the length of the de-stabilizing arm Ny/Yy, but the non-linear derivatives show relatively large E . The non-linear derivatives are likely to be independent from the range of P while the non-linear derivatives are not.
J M a r Sci Technol (2012) 17:422-445 433
Fig. 3 Original/reconstructed forces and moment coefficients f o r static d r i f t at different drift angles {left) and pressure-friction ratio {right)
6.1.1.2 Reconstruction The computational and the
exper-imental results show that the derivatives obtained from the NLCF at 0° < /? < 20° give the smallest £ R , ^ in both |lAve. and £niax • It indicates that the extrapolation should be avoided. In the
I^^Ave. |> the £R(;PJ, is slightly larger than the FREFO but still in the same order of magnitude. This implies that the current CFD simulation for the static drift up to ^ = 20° may able to be a replacement of the experiment provided that
the [iiniax I of the computational resuhs, especially in X' and V, decreases to the similar level for the experiment.
6.1.2 Steady turn
Figure 4 shows the experimental and computational results of forces and moment coefficients, as well as reconstructed computational results. The figure also includes the Rf/R^
434 J Mar Sci Technol (2012) 17:422-445
Table 6 Hydrodynamic derivatives f r o m static d r i f t tests
Static d r i f t (#1.3) L C F N L C F N L C F 0° < P <2° 0° < p < 10° 0° < P < 20° C F D EFD E CFD E F D E CFD EFD E - - - - 0 . 1 3 0 - 0 . 0 9 5 - 3 6 . 8 - 0 . 1 4 8 - 0 . 1 0 2 - 4 5 . 1 41.0 - 0 . 2 8 0 - 0 , 2 6 4 - 6 . 1 - 0 . 2 8 1 - 0 . 2 7 1 - 3 . 7 - 0 . 3 1 2 - 0 . 2 9 7 - 5 . 1 4.9 Y -
-
- - 2 . 6 1 2 - 2 . 0 2 3 - 2 9 . 1 - 1 . 5 3 7 - 1 . 2 9 2 - 1 9 . 0 24.0 N,. - 0 . 1 4 5 - 0 . 1 3 8 - 3 . 6 - 0 . 1 4 4 - 0 . 1 4 9 3.4 - 0 . 1 5 1 - 0 . 1 6 1 6.2 4.4-
-
-
- 0 . 5 0 7 - 0 . 4 9 4 - 2 . 6 - 0 . 2 3 4 - 0 . 1 1 7 - 1 0 0 . 0 51.3 0.518 0.523 1.0 0.512 0.550 6,9 0.484 0.542 10.7 6.2 E, \E\ (%D)Table 7 Average and maximum comparison eiTor of forces and moment coefficients between original EFD/CFD and reconstructed E F D / E F D
#1.3 Original 0° < /3 < 20° E Average error (%D) \Ëx\ 5.67 \EY\ 6.63 \M 3.10 |lAve.| 5.13 M a x i m u m error (%£)) /?max,X - 1 4 . 7 2 £max,y - 9 - 8 3 fimax.A' 7.29 |£maxl 10.61
Hydrodynamic derivatives used f o r reconstnrction L C F 0° < « < 2° -RCFD 16.53 10.50 13.52 37.39 16.57 26.98 - R E F D 21.53 15.10 18.32 41.03 20.74 30.89 N L C F 0° < < 10° -RCFD 3.60 10.60 4.68 6.29 - 7 . 4 1 - 3 1 . 1 3 - 1 7 . 0 9 - 1 8 . 5 4 - R E F D 1.14 4.65 4.18 3.32 3.29 -13.61 18.87 11.92 N L C F 0° < /3 < 2 0 ° -RcHi 6.09 6.90 2.56 5.18 -14.83 -13.46 4.57 -10.95 ^ R E I 0.63 2.30 2.55 1.83 1.32 -7.47 -6.16 4.98 R reconstructed
for X', Y and N'. Table 8 summarizes the hydrodynamic derivatives and Table 9 presents the averaged and maxi-mum values of E,E-B^^^ and
FREPD-6.1.2.1 Validation of forces, moment and liydrodynamic derivatives Due to the limited experimental data, it is
difficult to discuss the trend of the computational results for the experiment. Yet the computational results of the Y and A'' give better agreement than X' within the E of 10 %D. For the
Rf/Rp, as the ship encounters more cross flow due to the larger
yaw rate theZp becomes significant, and at /• — 0.6 it almost balances to the X'^. In Y and A'' the pressure component is almost tln-ee orders of magniUide lai'ger than the friction over the enthe
The computational results of the hnear derivatives show fair agi-eement to the experiment wiflirn E of 14 %Z) as well as the length of the stabilizing ami (N,- - mxc/Yr — m), but
the non-linear derivatives show relatively large \E\. Although the experimental and the computational results utilize the same number of data points to calculate derivatives, the range of r is different between the two which makes the systematic comparison difficult. Together with the N^/Y^ from the static drift results, the {N,- — mxa/Y,- — m) — Ny/Y^ is approxi-mately —0.3 wliich indicates that the slnp is naturally course unstable [25].
6.1.2.2 Reconstruction The computational results show
that the derivatives obtained from the NLCF give almost
J M a r Sci Teclinol (2012) 17:422-445 435
Forces and moment coeff.
Rf/Rp from raw CFD
-0.015 -0.02 X -0.025 -0.03 0.04 0.03 >- 0.02 X=-0.0162-0.0327r' - G - - - E F D ( B E C ) - • C F D , raw C F D , Quadratic fit, -0.6<r<0 —1 ! : ; I : , : 5?--0,6 -0.4 -0.2 • O - - - E F D ( B E C ) C F D , rav/ C F D , linear fit,-0.15<r<0 C F D , Cubic fit,-0.6<r<0 Y=-0.0506r-0.0192r^ 0.01 h Ol 1-0.C08 >-0.004 0.03 0.02 0.01 •---G---- E F D ( B E C ) CFD.raw C F D , Linear fit, 0<r<0.15 C F D , Cubic fit, 0<r<0.6 E F D ( B E C ) CFD.raw C F D , Linear fit, 0<r<0.15 C F D , Cubic fit, 0<r<0.6 E F D ( B E C ) CFD.raw C F D , Linear fit, 0<r<0.15 C F D , Cubic fit, 0<r<0.6 E F D ( B E C ) CFD.raw C F D , Linear fit, 0<r<0.15 C F D , Cubic fit, 0<r<0.6 / X / / /
\ \ :
— N = - 0 . 0 4 5 1 r - 0 . 0 2 0 1 r ^ N=-0.0424r 1 1 , \ 0.012 0.008 0.004 •0.6 -0.4 -0.2F i g . 4 Original/reconstructed forces and moment coefficients f o r steady turn at d i f f e r e n t yaw rates ( J e f f ) and pressure-friction ratio {right)
identical FR^^^ in 7 in both ^Ave. ( 6 %U) and F^ax ( 9 %D) 6.2 Dynamic P M M tests compared to the results from the LCF. Since it is opposite
to the conclusion obtained from the static drift and the 62.1 Pure sway experiment, diagnostics for the experimental data and more
CFD simulations with the same range of to the experi- Figure 5 shows the experimental and computational results ment would be necessary. of the forces and moment coefficients at three different
436 J Mar Sci Teclinol (2012) 1 7 : 4 2 2 ^ 4 5
Table 8 Hydrodynamic derivatives f r o m steady turn tests
Hydrodynamic derivative EFD^ CFD (#2.1) E (%£)) \E\
L C F N L C F L C F N L C F L C F N L C F ( % D ) 0.15 < ;•' < 0.3 0.15 < / < 0.3 0 < )^ < 0.15 0 < )^ < 0.60 _ - 0 . 0 1 6 2 _ - 0 . 0 1 6 2 - 0.00 0.0 X,,. - - 0 . 0 5 9 1
-
- 0 . 0 3 2 7-
44.67 44.7 i',- - 0 . 0 5 1 - 0 . 0 4 5 2 - 0 . 0 4 9 2 - 0 . 0 5 0 6 3.53 - 1 1 . 9 5 7.7 Y,,,. - - 0 . 0 9 2 5-
- 0 . 0 1 9 2 - 79.24 79.2 Nr - 0 . 0 4 9 - 0 . 0 3 9 3 - 0 . 0 4 2 4 - 0 . 0 4 5 1 13.47 - 1 4 . 7 6 14.1 Krr-
- 0 . 0 9 5 5 - - 0 . 0 2 0 1 - 78.95 79.0 (N, - inxaViY, - m) 0.262 0.211 0.227 0.240 13.36 - 1 3 . 7 4 13.6 " Raw data by Bassin d'Essai des Carenes (BEC), http://www.sininian2008.dk/Table 9 Average and maximum comparison error o f forces and moment coefficients between original E F D / C F D and reconstructed E F D / E F D
#2.1 Original Hydrodynamic derivatives used f o r reconstruction
L C F N L C F
0.15 <i^< 0.3 0 < / < 0.15 (CFD), 0.15 < Z < 0.3 (EFD) 0 < / < 0.6 (CFD), 0.15 < / < 0.3 (EFD) E ^ R C F D ^ R E F D - Ë ' R O T ^ R E F D Average en'or (%D) \Ëx\ 6.83 - - 7.27 4.74 \Ey\ 6.19 6.06 3.17 5.19 0.91 \ËN\ 0.50 6.29 7.37 5.36 1.02 |£Ave.| 4.51 6.18 5.27 5.94 2.22 M a x i m u m error (%D) Emux.X 13.27 - - 9.61 -8.53 Emax.y 8.02 7.94 - 6 . 2 9 - 8 . 2 2 -1.68 Emax.A- - 0 . 9 7 11.64 - 1 2 . 1 1 - 8 . 5 8 -2.01 | I n , a x | 7.42 9.79 9.20 8.80 4.07 R reconstructed
jSniax in one sway motion period. Tlie figure also includes the reconstructed computational results of the forces and moment coefficients using the hydrodynamic derivatives obtained from the static drift and current pure sway sim-ulations. Table 10 summarizes the experimental and com-putational results of the hydrodynamic derivatives, and Table 11 presents the E, FAVC-! ^RCFD and
FREPD-6.2.1.1 Validation of forces, moment and hydrodynamic derivatives The overall trend shows that the
computa-tional results agree well to the experimental data within £Ave. of 10 %D. In both the experimental and the compu-tational results, the dominant harmonic is 2nd inX' and 1st in Y' and A'' which agree to the Abkowitz's approximation. The y shows phase lead by about 25° with respect to imposed sway motion, and a similar trend is observed for KVLCC1&2 under the same pure sway motion [ I I ] . The virtual mass force is proportional to the acceleration which
has a maximum at the maximum y position of the ship (//T = 0.25 and 0.75), while all other forces peak at the maximum velocity point at t/T — 0 and 1, i.e., at the center of the towing tank. This causes Y' to peak approximately at
t/T = 0.9. Different from Y*, N' is in phase with the sway
motion. Since the yaw moment is mostly caused by lateral forces acting through the CoR, a symmetric A'' should have been caused by a symmetric Y. The simulation shows that, after the peak in y , N' is increasing while y is decreasing. This is most likely due to a local decrease of the lateral force near the stern which causes an increase of the yaw moment.
The computational and the experimental results show that the CF and the SR methods give consistent value in y , but not in Since the A^,; is coupled added inerda and is nearly zero as long as the ship is geometrically symmetric between port and starboard, it is difficult for both the simulation and the experiment to calculate it accurately.
J Mar Sci Technol (2012) 1 7 : 4 2 2 ^ 4 5 437 -0.012 k-0.015 -0.018 0.01 -0.01 • EFD • CFD, raw CFD, p^,,=2', reconst. CFD, p^„=4°,reconst. • CFD,p^„-10' CFD,Cuivefl:0'<p^„<10" • CFD, Sialic drift: 0'<p<20' 0.2 0.4 0.6 VT 0.8
F i g . 5 Original/reconstructed forces and moment coefficients f o r pure sway at different /?nia.x
For linear derivatives, the CF and the SR methods give consistent values in both 7,, and A^„ while the trend is opposite in non-linear derivatives. The cuirent jSniax range is almost within the linear range in connection to the results from static drift (see Fig. 2) which makes it difficult to calculate non-linear derivatives. The length of the stabi-hzing arm is slightly longer (e.g., A/,/F,, ~ 0.6) than the result from the static drift, and it tends to be shorter as jSmax becomes larger.
6.2.1.2 Reconstruction The computational results show
that the E-p,^^^ is small only when the reconstruction is done using its own derivaUves which is the same conclusion as is obtained from the experiment. The Ep^.^ from the NLCF using pure sway is about 5 %D which is almost the same level as FREPD. It implies that the CFD simulation of pure sway test at /J^ax up to 10° can be a replacement of the experiment.
6.2.2 Pure yaw
Figure 6 shows the experimental and computational results of the forces and moment coefficients at thr'ee different i-'^^y. in one yaw motion period. The figure also includes the reconstructed computational results of forces and moment coefficients using the hydrodynamic derivatives obtained from the steady turn and the pure yaw simulations. Table 12 summarizes the experimental and computational results of the hydrodynamic derivatives, and Table 13 presents the Ë, Ë^ye-, Ep^^^ and ER^^^.
6.2.2.1 Validation of forces, moment and hydrodynamic derivatives The overall trend shows that the
computa-tional results show fair agreement to the experimental data within iÏAve. of 20 %D. The larger £Ave. compared to the pure sway is mostly due to large £ in T'. In Fig. 6, the experimental and the computational results show the same trends about the dominant frequency of forces and moment coefficients over one yaw motion period to the pure sway results. The peaks of N' is likely to appear prior to the peak of 7-' (i.e., before t/T = 0.25 and 0.75) since the added hydrodynamic moment of inertia increases as the yaw rate becomes larger.
The computational and the experimental results show that the CF and the SR methods give a different result in F,--but a consistent result inN,-. Similar to A',; in the pure sway,
Yi- is coupled added inertia and is a very small quantity and,
thus, it is difficult to calculate. For linear derivatives, the CF and the SR methods give different Y,. but gives con-sistent A',.. For non-linear derivatives, the CF and the SR methods give different derivatives. The length of the
IS)-I
4-ca
Table 10 Hydrodynamic derivatives f r o m pure sway tests
Pure sway (#3.1)
SR SR SR N L C F
B — 9°
Hmax — ^max ~ ^
iSmax = 10° 0° < /?„.,x < 10°
CFD EFD CFD EFD E CFD EFD E CFD EFD E
- 0 . 0 1 6 7 - 0 . 0 1 6 1 - 3 . 8 3 - 0 . 0 1 6 9 - 0 . 0 1 6 1 - 5 . 3 3 - 0 . 0 1 9 0 - 0 . 0 1 8 6 - 2 . 4 4 - 0 . 0 1 6 3 - 0 . 0 1 6 2 - 1 . 0 7 3.17
-
--
-
--
-
-
-
- 0 . 2 4 0 - 0 . 0 7 0 - 1 2 6 . 7 126.7 0.346 - 0 . 7 5 3 146.0 0.062 - 0 . 0 0 1 4650.0 - 0 . 0 0 8 0.053 116.2 - 0 . 0 0 6 0.051 112.4 1256.15 - 0 . 1 0 2 - 0 . 0 9 2 - 1 1 . 0 - 0 . 1 0 4 - 0 . 0 8 0 - 3 0 . 4 - 0 . 1 1 1 - 0 . 1 0 2 - 9 . 2 - 0 . 1 1 0 - 0 . 0 9 9 - 1 1 . 6 15.55 - 0 . 2 3 6 - 0 . 2 4 4 3.1 - 0 . 2 4 3 - 0 . 2 6 9 9.6 - 0 . 2 7 6 - 0 . 2 9 1 5.0 - 0 . 2 5 5 - 0 . 2 6 8 4.9 4.86 4>y n - 1 4 . 5 3 - 1 2 . 7 5 - 1 4 . 0 - 1 4 . 3 8 - 1 0 . 0 8 - 4 2 . 9 - 1 3 . 5 5 - 1 1 . 0 4 - 1 3 . 5 8-
-
-
23.45 i'vvv, 1-
--
-
-
-
-
-
-
- 3 . 8 8 7 - 2 . 4 4 3 - 5 9 . 2 59.20 Yvvv,3 - 1 6 . 6 9 4 - 1 2 . 3 6 0 - 3 5 . 1 - 7 . 7 3 5 - 3 . 4 8 5 - 1 2 1 . 9 - 2 . 9 4 5 - 1 . 4 4 2 - 1 0 4 . 3 - 2 . 9 6 7 - 1 . 4 5 1 - 1 0 4 . 4 91.43 - 0 . 0 1 2 - 0 . 0 1 2 - 0 . 6 - 0 . 0 1 1 - 0 . 0 0 7 - 5 8 . 3 - 0 . 0 1 1 - 0 . 0 0 8 - 3 8 . 9 0.011 - 0 . 0 0 8 - 3 9 . 4 34.30 - 0 . 1 5 0 - 0 . 1 6 3 0.7 - 0 . 1 4 7 - 0 . 1 6 4 0.7 - 0 . 1 4 6 - 0 . 1 5 9 - 1 3 . 1 - 0 . 1 5 4 - 0 . 1 6 2 5.0 4.80 - 7 . 5 0 - 6 . 8 9 - 8 . 9 - 7 . 2 6 - 4 . 1 2 - 7 5 . 9 - 7 . 0 7 - 4 . 6 8 - 5 1 . 2 3-
-
-
45.34-
-
-
-
-
-
-
-
-
- 0 . 3 6 9 - 0 . 0 8 3 - 3 4 2 . 7 342.70 ^' vvv,3 - 6 . 1 6 2 3.021 - 1 0 4 . 0 - 2 . 1 6 6 - 0 . 2 2 3 - 8 3 3 . 9 - 0 . 7 4 5 - 0 . 2 4 6 402.5 - 0 . 7 5 4 - 0 . 2 4 3 - 2 0 8 . 1 185.88 0.636 0.668 4.9 0.605 0.610 0.8 0.529 0.546 3.2 0.604 0.604 0.1 3.SJ M a r Sci Teclinol (2012) 17:422^45 439 Stabilizing arm is similar to the solution from the steady
turn, and it tends to be longer as 74ax becomes larger. Using the lengths of the stabilizing and de-stabilizing arm obtained from the pure sway and the pure yaw, respec-tively, (A',- — mxa/Y,- — m) — Ny/Yy is approximately —0.4
thus the ship is naturally course unstable which is the same conclusion obtained from the static P M M tests.
6.2.2.2 Reconstruction The computational results show
that the FR^pn is small only when the reconstruction is done using its own derivatives. The FRCFD from the NLCF using pure yaw is at least two times larger than FREPD, thus it is still inconclusive to state that the CFD simulation of pure yaw test at r'^^-^ up to 0.6 can be a replacement of the experiment. The Ep^^ from the NLCF with the steady turn results is two times smaller than the Ep^^^ from the NLCF with the pure yaw results.
6.2.3 Combined yaw and drift
Figure 7 shows the experimental and computational results of the forces and moment coefficients at three different fi with constant i-'^^ in one yaw motion period. The figure also
includes the reconstructed computational results of forces and moment coefficients using the hydrodynamic derivatives obtained from the static drift, pure sway, and pure yaw sim-ulations. Table 1 4 summarizes the experimental and compu-tational results of the hydrodynamic derivatives, and Table 1 5
presents the £ , ËAve., SRCFD andER^p^,.
6.2.3.1 Validation for forces, moment and hydrodynamic derivatives The overall trend shows that the
computa-tional results show fair agreement to the experimental data within iÏAve. of 1 6 %D regardless of the different installa-tion condiinstalla-tion between the experiment (FXQ) and the simulation (FX<j^). Relatively large FAVB. is mostly due to large E in Z'. According to the towing path of the com-bined yaw and drift test, at 0 < //T < 0.5 the direction of the yaw rotation is towards the leeward side which reduces the cross flow to the ship induced by the given ji. During this period the apparent /? become smaller than the given fi. At Q.5 <t/T < 1, the yaw rotation is towards the windward side which increases the cross flow towards the ship. During this motion period the instantaneous /? is up to 2 1 . 2 ° at t/T = 0 . 7 5 when the given /5 = 11°, thus, the E in X' becomes larger. In Fig. 7, both the experiment and the
Table 11 Average comparison eiTor over 1 sway motion period between original and reconstructed forces and moment coefficients f r o m pure
sway tests
E (%D) Original pure sway Hydrodynamic derivatives used f o r reconstruction
Static drift" (#1.3) Pure sway (#3.1)
N L C F SR SR SR N L C F 0° < /? < 2 0 ° /? — 2° )5max = 10° 0° < /?max < 10° ft max E ^ R E F D ^ R E F D ^ R C F D EREFD ^ R C E D ^ R E F D ^ R C F D ^ R E F D UD 2° Ex 6.93 5.77 3.7 4.16 4,7 4.13 4.7 15.34 12.4 3.50 4,4 Er 6.56 20.98 15.1 6.17 1,6 7.17 9.0 12.85 12.8 10.45 5.8 -EM 3.72 5.71 4.8 3,28 1,7 7.11 5.9 8.71 4.7 3.66 5.0 -E\vo. 4° Ex 5.74 10.82 7.9 4.54 2,7 6.14 6.5 12.30 10.0 5.87 5.1 -E\vo. 4° Ex 9,05 6.04 7.3 6,60 12,5 6.99 5.3 18.58 14.8 6.95 5.5 — Ey 13.99 18.61 11.9 14.90 8,3 13.38 1.7 17.01 12.3 16.99 10.9 -£,v 6.41 7.79 1.2 8,94 8.6 6.13 0.8 9.69 2.8 5.75 1.2
-Efi.ve, 9.82 10.81 6.8 10,15 9.8 8.83 2.6 15.09 10.0 9.90 5.9 -10° Ê A ' 7.86 11.33 3.1 35,76 55.5 10.06 9.5 7.86 6.2 14,37 13.4 6.8 ËY 7.01 6.97 3.2 70.02 49.8 23.16 14.0 7.09 2.8 7.41 5.9 5.2 EN 3.22 5.67 3.3 68.62 38.7 18.43 3.6 3.54 1.1 3.49 3.0 4.9 EAVC. 6.03 7.99 3.2 58,13 48.0 17.22 9.0 6.16 3.4 8,42 7.4 5.6 Yi and Ny are taken f r o m curve fit results o f pure sway440 J Mar Sci Teclinol (2012) 17:422-445 0 0.2 0.4 0.6 0.8 trr 0 0.2 0.4 0.6 0.8 1 tn-X 0 0.2 0.4 0.6 0.8 1
F i g . 6 Original/reconstructed forces and moment coefficients f o r pure yaw at different /^^^
computational results show that the Ist-harmonic is the most dominant in X', Y, and N' which is different trend from the pure yaw cases, but it mathematically agrees with the Abkowitz's approximation [6].
The computational results show that the CF and the S R methods give a different result in all the cross-coupling derivatives which is the similar trend for the experiment, with the exception in T,,„. and N„,:
6.2.3.2 Reconstruction The computational results show
that the Ep^^ from the NLCF is about 1.2 times larger than ERPPJ, from the SR, but it is nearly the same level of ER^^ from the NLCF. This implies that the CFD simulation of combined yaw and drift test at r ^ a x 0.3 with up to 11° may able to be a replacement o f the experiment, although more diagnostics are necessary for the pure yaw results which provide the hydrodynamic derivatives (7„ T,.,.„ N,-,
Nrrr) foT tlic reconsttuction.
7 Conclusions
Static and dynamic P M M simulations of a surface com-batant Model 5415 are performed using a viscous CFD solver with dynamic overset interface. The objective of this research is to investigate the capability of the current solver for CFD-based maneuvering prediction.
The V & V study is performed for forces and moment coefficients in the static drift at /J = 10° and the pure yaw with ;4ax ~ The use of wide background domain is strongly recommended in static drift simulation in order to dissipate spurious free surface waves so that faster statis-tical convergence can be achieved, although the systematic quantification of the blockage effect to the forces and moment [37] would be suggested as one for future work. In the pure yaw (and the other dynamic P M M cases), the statistical convergence of the forces and moment c o e f f i -cients in terms of their firsdy and secondary dominant harmonics is mostly ensured. For grid and time step con-vergence, the c u i T e n t static drift results show difficulties in obtaining grid convergence in most of the forces and moment coefficients. As reported by Bhushan et al. [36] who utilize the DES simulation with finer grid (up to 250 M grid points) to simulate Model 5512 in staric drift
(P = 10° and 20°), the resolution of vordcal structure and
its unsteadiness is the key for better grid and time-step convergence. Thus, such simulafions would be suggested as future work to achieve grid convergence with acceptable
USN in X' and F . In the pure yaw, the effect of grid is likely
to have a stronger influence to the forces and moment coefficients than the effect of the size of time steps.
J Mar Sci Teclinol (2012) 17:422-445 441
Table 12 Hydrodynamic derivatives f r o m pure yaw tests
Pure yaw (#4.3)
SR SR SR NLCF
'^nax = 0-15 C „ = 0.30 ' i n . = 0.60 0 < r" — max
< 0.60
CFD EFD CFD EFD E CFD EFD E CFD EFD E
-0.0162 -0.0166 2.21 -0.0173 -0.0179 3.07 -0.0183 -0.0210 12.73 -0.0159 -0.0157 -1.7 4.92 Xrrfl -0.074 0.092 180.4 -0.037 0.010 453.6 -0.029 -0.021 -39.2 -0.0293 -0.0289 - 1 4 168.65 -0.024 0.091 126.6 0.0003 0.039 99.1 -0.013 0.004 445.2 0.0129 -0.006 311.5 245.60 1 niot -0.006 -0.002 -189.6 -0.009 -0.001 -549.3 -0.013 -0.005 -188.2 -0.008 -0.006 -36.0 240.78 Yr -0.038 -0.050 24.8 -0.030 -0.047 35.0 -0.009 -0.023 60.9 -0.042 -0.053 20.0 35.18 h O 74.94 85.87 127 63.09 87.09 27.55 17.96 67.31 73.30 - - - 37.85 i',-,r,l _ - - -
-
-0.019 -0.021 9.8 9.80 Yrrr,3 -0.348 -0.076 -363.8 -0.211 -0.119 -76.8 -0.144 -0.130 -11.4 -0.145 -0.130 -12.2 116.05 -0.008 -0.007 -18.4 -0.008 -0.006 -21.3 -0.008 -0.005 -39.1 -0.008 -0.006 -36.0 28.70 A', -0.041 -0.043 5.1 -0.042 -0.045 6.6 -0.041 -0.047 11.6 -0.042 -0.046 7 4 7.68<i>H n
71.45 74.97 4.69 72.66 76.48 4.99 72.96 75.98 3.9 - - - 4.53 Nrrr.l _ - - - --
-
--
-0.032 -0.036 12.0 12.00 Nrrr.3 -0.161 -0.037 76.9 -0.043 -0.050 13.9 -0.0360 -0.0328 -9.6 -0.0361 -0.0332 -8.9 27.33 (Nr - IILXaV 0.233 0.229 -1.89 0.251 0.244 -2.85 0.280 0.294 4.48 0.234 0.241 3.15 3.09 (}', - m)" E = D ~ S (%D), <PY = tan~\-Yf/coYi), same for (p„. [9]
Table 13 Average comparison error over 1 yaw motion period between original and reconstructed forces and moment coefficients f r o m pure
yaw tests
E (%D) Original pure yaw Hydrodynamic derivatives used f o r reconstruction (%D) Steady turn (#2.1) Pure yaw (#4.3)
N L C F SR SR SR N L C F 0 < Z < 0.60 max 0.15 ;•' = max 0.30 r' = max 0.60 0 < — max < 0.60 'max E ^ R C F D ^ R C F D ^ R E F D ^ R C F D ^ R E F D ^ R C F D ^ R E F D ^ R C E D ^ R E F D Uo 0.15 Ex 8.24 8.43 10.18 4.8 11.37 12.1 50.58 34.8 7.76 7.7 Ëy 21.99 9.61 22.92 8.9 44.88 9.8 89.35 52.5 37.35 10.3
-EN 9.65 4.40 9.77 2.2 7.50 3.1 8.50 5.6 6.75 4.6
-EA\e. 13.29 7.48 14.29 5.3 21.25 8.3 49.48 31.0 17.29 7.5
-0.3 ËX 8.85 14.11 13.26 22.4 8 4 1 1.1 48.21 29.2 12.86 12.0 6.4 Ëy 29.21 9.26 21.87 3.6 29.61 2.3 76.47 43.5 39.82 9.0 14.6 EN 8.16 4.23 10.94 11.1 8.24 1.3 9.74 3.5 8.43 2.9 4.1 E^ve. 15.41 9.20 15.36 1 2 4 15.42 1.6 44.81 2 5 4 20.37 8.0 8 4 0.6 Ex 10.57 21.76 16.22 98.5 15.08 46.6 26.83 2.8 19.14 18.5 _ Ëy 37.20 18.37 117.26 25.7 51.82 39.0 36.92 4.0 40.52 17.7
-EN 10.68 8.23 13.07 49.6 1041 6.8 10.86 3.8 11.25 3,8
-Ep^wt. 19.48 16.21 48.85 57.9 25.77 30.8 24.87 3.5 23.64 13.3 -<Ö Springer
442 J Mar Sci Technol (2012) 1 7 : 4 2 2 ^ 4 5 -0.014 ^-0.021 -0.028 0.09 ^ 0.06 0.03 0.045 ^ 0.03 0.015 b P=io° 0 0.2 0.4 0.6 trr c p = i r -0.014 -0.021 -0.028 0.12 0.04 0.06 0.04 0.02 0.4 0.6 0.8 1 f/T
Fig. 7 Original/reconstructed forces and moment coefficients f o r
combined yaw and d r i f t at different [i
