Trans RINA, Vol 161, Part A4, Intl J Maritime Eng, Oct-Dec 2019
There are also some other studies in the literature that k-ε model has been applied to simulate the turbulent flow, such as, Cakici et al. (2017), Delen et al. (2017), Terziev et al., (2018). In order to represent the surface gravity waves on the interface between the heavy fluid and light fluid, the Volume of Fluid (VOF) method has been chosen in this study. The VOF method is a suitable method both for tracking the free surface between immiscible fluids and for capturing a high-resolution interface (CD-adapco, 2017). A second order convection scheme has been preferred to capture the surface interfaces between the two phases quite well. The phases are chosen as fresh water and air. The temperature of fluids is set to 15ºC (ρwater=999.1 kg/m3, νwater=1.14E-06 m2/s). The calm water condition has been defined by the flat VOF wave. In order to avoid wave reflections on free surface from the surrounding boundaries of computational domain, the VOF wave damping model has been activated far enough away from hull. The coupling between fluid and body has been established with the Dynamic Fluid-Body Interaction (DFBI). The DFBI provides the motion of the form, taking into account the balance of forces and moments at the boundary. In CFD simulations, the motion of hull has been set to free to trim and sinkage. The time step (Δt) in numerical analyses has been determined as ∆ = 0.005 ⁄ recommended by ITTC (2011b) for each model. Detailed information on the solution strategy are given in the user guide of CD-adapco (2017). The wall y+ values for the underwater surfaces of models are around 100 for each hull. A sample image of wall y+ contour has been shown for model 1 in Figure 5.
21/32) must be examined. If the R value is between 0 and 1, the solution of numerical model converges
monotonically (Stern et al., 2002). This means that numerical uncertainty can be estimated quantitatively by the GCI method. The apparent order of method (p) is calculated by,
= 1 ln(21) |ln|32 21⁄ |+ ()|
() = ln(21 −
= {−1 3 +1
32 − )
32 21⁄ < 0 32 21⁄ > 0 } 32 = √2 3,⁄3 (12) (13) (14)
where, r21 and r32 are refinement factor calculated by 21 = √1 2,⁄
. It is recommended that
refinement factor greater than 1.3 (Celik et al., 2008). denotes the mesh number of ith grid. The extrapolated value is calculated by,
21 = 21
1 − 2 21
− 1
The approximate relative error can be expressed as follows:
21 = |1 − 2 1
|
Figure 5. Wall y+ distributions on the model 1 surface at Fr=0.26.
6. V&V STUDY
A verification study has been conducted on the numerical model in terms of grid size to estimate numerical uncertainty. GCI method which is frequently used in CFD applications, has been used to determine the appropriate mesh structure, e.g., De Marco et al. (2017), Dogrul et al. (2017), Usta and Korkut (2018). In this study, the total resistance coefficient has been selected as key parameter. Since the estimation of uncertainty of grid structure for each model would require a lot of solution time, the uncertainty study has been conducted only for Model 1. For the other models, grid size of the hull surface and free surface region have been refined systematically.
GCI method has been briefly described as follows. 32 = 3 −2 & 21 = 2 − 1, represent the solution of key parameter of ith grid. The convergence condition ( =
Then, the relative error can be obtained by,
21 = | 21 − 1
21
|
The fine-grid convergence index is estimated by,
21 = 1.25 21
21 − 1
(18)
Detailed information of GCI method can be found in (Celik et al., 2008). Three significantly different sets of grids, called fine, medium and course, corresponding to cell numbers N1, N2 and N3, have been generated. The analyses have been simulated to examine the change of key parameter (CT). The converged condition (R) value has been found as 0.262. The other coefficients of GCI method have been given in Table 2. As a result, the numerical uncertainty of fine grid has been estimated as 1.06% and fine grid type has been then selected for rest of analyses.
A-472 ©2019: The Royal Institution of Naval Architects (16) (15)
(17)
Page 1 |
Page 2 |
Page 3 |
Page 4 |
Page 5 |
Page 6 |
Page 7 |
Page 8 |
Page 9 |
Page 10 |
Page 11 |
Page 12 |
Page 13 |
Page 14 |
Page 15 |
Page 16 |
Page 17 |
Page 18 |
Page 19 |
Page 20 |
Page 21 |
Page 22 |
Page 23 |
Page 24 |
Page 25 |
Page 26 |
Page 27 |
Page 28 |
Page 29 |
Page 30 |
Page 31 |
Page 32 |
Page 33 |
Page 34 |
Page 35 |
Page 36 |
Page 37 |
Page 38 |
Page 39 |
Page 40 |
Page 41 |
Page 42 |
Page 43 |
Page 44 |
Page 45 |
Page 46 |
Page 47 |
Page 48 |
Page 49 |
Page 50 |
Page 51 |
Page 52 |
Page 53 |
Page 54 |
Page 55 |
Page 56 |
Page 57 |
Page 58 |
Page 59 |
Page 60 |
Page 61 |
Page 62 |
Page 63 |
Page 64 |
Page 65 |
Page 66 |
Page 67 |
Page 68 |
Page 69 |
Page 70 |
Page 71 |
Page 72 |
Page 73 |
Page 74 |
Page 75 |
Page 76 |
Page 77 |
Page 78 |
Page 79 |
Page 80 |
Page 81 |
Page 82 |
Page 83 |
Page 84 |
Page 85 |
Page 86 |
Page 87 |
Page 88 |
Page 89 |
Page 90 |
Page 91 |
Page 92 |
Page 93 |
Page 94 |
Page 95 |
Page 96 |
Page 97 |
Page 98 |
Page 99 |
Page 100 |
Page 101 |
Page 102 |
Page 103 |
Page 104 |
Page 105 |
Page 106 |
Page 107 |
Page 108 |
Page 109 |
Page 110 |
Page 111 |
Page 112 |
Page 113 |
Page 114 |
Page 115 |
Page 116 |
Page 117 |
Page 118 |
Page 119 |
Page 120 |
Page 121 |
Page 122 |
Page 123 |
Page 124 |
Page 125 |
Page 126 |
Page 127 |
Page 128 |
Page 129 |
Page 130 |
Page 131 |
Page 132 |
Page 133 |
Page 134 |
Page 135 |
Page 136 |
Page 137 |
Page 138 |
Page 139 |
Page 140 |
Page 141 |
Page 142 |
Page 143 |
Page 144 |
Page 145 |
Page 146 |
Page 147 |
Page 148 |
Page 149 |
Page 150 |
Page 151 |
Page 152 |
Page 153 |
Page 154 |
Page 155 |
Page 156 |
Page 157 |
Page 158 |
Page 159 |
Page 160 |
Page 161 |
Page 162 |
Page 163 |
Page 164 |
Page 165 |
Page 166
