Trans RINA, Vol 161, Part A4, Intl J Maritime Eng, Oct-Dec 2019
conducted with Grid Convergence Study (GCI) method for KCS hull at a sample scale ratio. The numerical results obtained for three scales have been validated with the experiments carried out at Istanbul Technical University and available data in the literature. The relationship between total resistance coefficient and Re number has been correlated by Telfer’s GEOSIM method. The results computed by Telfer’s GEOSIM method have been found to be less than 1% at full scale ship while the results obtained by ITTC extrapolation method have been found to be larger than those of Telfer’s GEOSIM method. Additionally, Telfer’s GEOSIM method has then been extended to the computation of nominal wake at full scale by using wake values at three different model scale ratios. A good correlation has also been found for nominal wake values.
2. 2.1 METHODOLOGY NUMERICAL MODEL
Three dimensional unsteady Reynolds Averaged Navier- Stokes (RANS) equation for incompressible flow along with continuity equation are given below.
= 0
+
= −1
+ (−
′ ̅ ′̅̅̅̅̅)
+ 2
2 (1) (2)
Here Ui is the mean velocity in the ith direction of the Cartesian coordinate; ρ is the density, P is the mean pressure, ui
In these equations, 1 = 1.44 and 2 = 1.92 are constant, = 1.0 and = 1.3 are turbulent Prandtl numbers for k and ε (Jones and Launder, 1972)
2.2 TELFER’S GEOSIMMETHOD
As it is known, resistance coefficient of geometrically similar hulls is based on Froude and Reynolds numbers analogy. The non-dimensional resistance coefficient of a hull can be expressed as:
ρSV2 = f(
√
, )
(6)
Here the first term in the parenthesis of functional expression is Froude (Fr) number and represents gravity wave resistance, the second termis Reynolds (Re) number and represents viscous resistance. The extrapolation of viscous phenomena from model scale to largest conceivable ship has been expressed as 1/Re since viscous forces have not significant contribution in higher Re numbers (Telfer, 1927). In Telfer’ s GEOSIM method, the total resistance of a ship is expressed as a function of Re number, using geometrically similar models. Since the Fr number is the same in the model scale and the full scale, the inertia (residual) resistance coefficient is assumed to be the same for both model and ship. The non-dimensional resistance values (CT) are then plotted against a wide range of Re numbers of the model family. Finally, the model resistance can be extrapolated to full scale resistance by CT-Re curve. The curve of ship resistance versus Re number can be expressed as;
′uj ̅ ′̅̅̅̅ is the Reynolds stress and ν the kinematic
viscosity. In the present study, two equation k-ε turbulence model is used for modelling the Reynolds stress() . In standard k-ε model, the Reynolds stress is related to strain rate linearly as follows:
= ′ ̅ ′̅̅̅̅̅ = 2
3 − (
+
) (3)
where represents the eddy kinematic viscosity ( = 2
⁄). is an empirical constant and its value is equal
to 0.09. The turbulence kinetic energy (k) and its rate of dissipation (ε) are obtained from the following transport equations.
Dk Dt = −ui
′uj ̅ ′̅̅̅̅ ∂Ui
∂xi
Dε Dt = −Cε1
ε k ui
−ε+ ∂ ∂xj
′uj ̅ ′̅̅̅̅∂Ui
∂xi
[(νt σk
− Cε2
+ ν) ∂k ∂xj
ε2 k + ∂
∂xj ]
[(νt σk
+ν) ∂ε ∂xj
] (4) (5)
= ( ) +
(7)
In this equation, power x is generally taken as 1/3, then a linear system of two equations with two unknowns (a and b) is solved. However, in this study, power x has also been assumed to be an unknown parameter to increase the accuracy of Telfer’s GEOSIM method, especially in high Re number. Three unknowns (a, b and x) can be solved by resistance values at three different model scales for a corresponding Fr number. Equation 7 generates a nonlinear system which can be solved by a least square method (Nocedal and Wright, 2006).
2.3
1978 ITTC PERFORMANCE PREDICTION METHOD
The results of ITTC 1978 method were compared with those of Telfer's GEOSIM method. The total resistance coefficient of a ship without bilge keel can be given as,
= (1 + ) +∆ + + + (8)
©2019: The Royal Institution of Naval Architects
A-469
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