Simplified formulas of heave added mass coefficients at high frequency for various two-dimensional bodies in a finite water depth

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4 4 Int. J. Nav. Archit. Ocean Eng. (2015) 7:115~127 SIMPLIFIED FORMULA FOR ADDED MASS COEFFICIENTS Calculation of basis values for added mass coefficients To build a simplified formula, the added mass induced by vertical motion of a simple geometric body at high frequency was calculated using a developed two-dimensional frequency-domain NWT. The calculated added masses from the selected geometric conditions were used as basis values to formulate the correction factor in the simplified formula. To confirm the accuracy of the basis values, the added masses calculated by the NWT were compared with those by other methods. Fig. 2 shows the comparison of the added masses and damping coefficients from various methods in deep water conditions. It was found that the present numerical results agreed well with the numerical results of Ursell (1948) and the experimental data of Vugts (1968). (a) (b) (c) Fig. 2 Comparison of heave added mass and damping coefficients for various simple geometric bodies in deep water: (a) rectangle (B/T = 2), (b) semicircle (B/T = 2), (c) triangle (B/T = 1.155), BEM denotes the present results.

5 Int. J. Nav. Archit. Ocean Eng. (2015) 7:115~127 5 After verification of the results calculated using NWT, the added masses and damping coefficients of simple geometric bodies at various water depth ratios (h/t) were compared (Fig. 3). The comparison indicates that the trends in hydrodynamic coefficients for all geometric bodies are similar in the high frequency region. Compared to the rectangular body, the change rate of added mass for a semicircle was found to be small. In other words, the difference in added masses between the respective water depth ratios (h/t) was smaller than the added mass for a rectangular body. This is because the effective wetted surface of a semicircle is smaller than that of a rectangle in the direction of prescribed vertical motion. Thus, the effect of the sea bottom was relatively small and the heave added mass of a semicircle did not change significantly. This was also expected to be true for a triangular body. Since the effective wetted surface of a triangle is small, the interaction between the body and flat sea bottom decreases during vertical movement of the body. Thus, the difference in the heave added mass of a triangle corresponding to various relative body submergences was smaller than for other geometric shapes at high frequencies. In particular, the variation in added mass at deep water conditions (h/t = 30 in this study) was not significant in the high frequency region greater than 1.6, which indicates that the effect of the sea bottom was relatively small and little interactions occurred between the body and the rigid sea bottom. However, at the same frequency, the added mass increased at shallow water depths, indicating that the existence of the sea bottom may strengthen the wave-body interaction. (a) (b) (c) Fig. 3 Comparison of heave added mass and damping coefficient for various water depths: (a) rectangle (B/T = 2), (b) semicircle (B/T = 2), (c) triangle (B/T = 1.155).

7 Int. J. Nav. Archit. Ocean Eng. (2015) 7:115~127 7 If ξ ( = h/ T) > 7 α ( ψ, ξ) = R + R ψ + R ξ + R ψ + R ψξ + R ξ + R ψ + R ψ ξ + R ψξ (7) c b1 b2 b3 b4 b5 b6 b7 b8 b9 where ψ = B/ T (ratio of wetted body shape), ξ = h/ T (ratio of water depth or relative body submergence), and the coefficients (R i ) of the correction factor for a rectangular body are listed in Table 1. Table 1 Coefficients of correction factor for a rectangular body. ξ ( = h/ T) 7 ξ ( = h/ T) > 7 R a R b R a R b R a R b R a R b R a R b R a R b R a R b R a R b R a R b R a R a Correction factor for a semicircular body Since a semicircle has a constant ratio of body shape (B/T = 2), the correction factor is much simpler than for the rectangular body. Hence, the factor can be a function of water depth ratio only, which was developed in the previous study (Koo and Kim, 2013) h αc = T (8) Therefore, the added mass for a semicircle for specific water depths can be predicted as mas = mad (1.093ξ ). The normalized added mass of a semicircular body with the deep water condition at infinite frequency was known to be When the relative body submergence (h/t) is given, the heave added mass for various water depths can be easily predicted by Eq. (8). Correction factor for a triangular body The correction factor for a triangle is also expressed as a function of the relative water depth and body beam/draft ratios. Similar to the rectangular body, the correction factor for a triangle should be expressed for two different water depth regions (Eq. (9) and Eq. (10)). Fig. 5 also shows a three-dimensional view of the added mass change for various body shapes and water depths at high frequency ( ω B/2g =2).

8 8 Int. J. Nav. Archit. Ocean Eng. (2015) 7:115~127 h/ T 7 h/ T > 7 Fig. 5 Three-dimensional view of heave added mass coefficients of a triangle for various body beam/draft ratio (B/T) and water depth ratio (h/t) at high frequency ( ω B/2g =2), black dot represents basis values of added mass. If ξ ( = h/ T) 7 a ( ψ, ξ) = T + T ψ + T ξ + T ψξ + T ξ + T ψξ + T ξ + T ψξ + T ξ + T ψξ + T ξ (9) c a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 If ξ ( = h/ T) > 7 α ( ψ, ξ) = T + T ψ + T ξ + T ψ + T ψξ + T ξ + T ψ ξ + T ψξ + T ξ (10) c b1 b2 b3 b4 b5 b6 b7 b8 b9 where the coefficients (T i ) of the correction factor for a triangular body are listed in Table 2. Table 2 Coefficients of correction factor for a triangular body. ξ ( = h/ T) 7 ξ ( = h/ T) > 7 T a T b T a T b T a T b T a T b T a T b T a T b T a T b T a T b T a T b T a T a

10 10 Int. J. Nav. Archit. Ocean Eng. (2015) 7:115~127 Fig. 6 Comparison of added mass coefficients from simplified formulae (symbols) with numerical calculation (lines) for a rectangular body with ω ( B 2 g) = Fig. 7 Comparison of added mass coefficients from simplified formulae with numerical calculation (solid line) for a semi-circle body with ω ( B 2 g) = Fig. 8 shows results for a triangular body obtained using the same procedure of added mass prediction and verification. Likewise, for other geometric bodies, the added mass predicted by the simplified formula was accurate to within 10% of the numerical results by NWT. Fig. 8 Comparison of added mass coefficients from simplified formulae (symbols) with numerical calculation (lines) for a triangular body with ω ( B 2 g) = 1.99.

12 12 Int. J. Nav. Archit. Ocean Eng. (2015) 7:115~127 Fig. 11 shows the comparison of the added mass coefficients for a triangular body. As with other simple geometries, the predicted added masses agreed with the calculated values when the normalized wave frequency was greater than 1.0. Since the correction factor was developed by interpolating the basis added mass obtained at the high frequency, the accuracy of the predicted value increased as the frequency and the water depth increased. Fig. 11 Comparison of added mass coefficients in shallow water ( h/ T 7); Symbol: Predicted values, Lines: Numerical calculation. Table 4 Comparison of added mass coefficients of a semicircle for various water depth ratios and ω ( B 2 g) = h/t Numerical calculation ( a 33 / ρ A) Simplified formula ( a 33 / ρ A ) Difference (%) The added mass coefficients of a simple geometric body were accurately predicted for various ratios of wetted body shapes and water depths (relative body submergence) to within 10% of the numerical calculations. The predicted coefficients were also valid for various wave frequencies to within 10% until the normalized wave frequency was greater than 1.0. Therefore, the use of the simplified formula developed in this study can be useful to predict the added mass of a simple geometric floating body in various environmental conditions. CONCLUSIONS In the present study, we developed a simplified formula for added mass coefficients induced by the vertical movement of various two-dimensional body sections in a finite water depth. The developed formula reflects changes in added mass due to the

13 Int. J. Nav. Archit. Ocean Eng. (2015) 7:115~ effects of the sea bottom and wave frequencies. A two-dimensional frequency-domain NWT technique was used to calculate the basis values of the added masses for various body sections at the high (infinite) wave frequency. The calculated basis values were then used to formulate the correction factor of the simplified formula. For better accuracy, the correction factor was determined by dividing the system into two regions of water depth (relative body submergence to water depth). It was found that the accuracy of the predicted added mass increased as the frequency and water depth increased because the correction factor was formulated with the basis added mass values obtained at the high frequency. The comparative study showed that the added mass coefficient of a simple geometric body was accurately predicted for various ratios of body shapes and water depths to within 10% of the full-domain BEM-based numerical calculation. The predicted coefficients were also found to be valid for various wave frequencies to within 10% until the normalized frequency was greater than 1.0. Therefore, the simplified formula of added mass for various body sections developed in this study can be used for a quick analysis of a two-dimensional based floating structure section response (ship and ocean structure maneuverability and operation) in the shallow water regions such as harbor and canal. ACKNOWLEDGEMENT This work was supported by INHA UNIVERSITY Research Grant. (INHA ) and this work was also supported by a Special Education Program for Offshore Plant by the Ministry of Trade, Industry and Energy Affairs (MOTIE). REFERENCES Frank, W., Oscillation of Cylinders in or below the free surface of deep fluids. Hydromechanics laboratory research and development report Washington: Naval Ship Research and Development Centre. Kim, C.H., Clement, A.H. and Tanizawa, K., Recent research and development of numerical wave tanks-a review. International Journal of Offshore and Polar Engineering, 9(4), pp Koo, W.C. and Kim, J.D., Development of simplified formulae for added mass of a 2-D floating body with a semicircle section in a finite water depth. Journal of Ocean Engineering and Technology, 27(1), pp Sutulo, S., Rodrigues, J.M. and Soares, C.G., Hydrodynamic characteristics of ship sections in shallow water with complex bottom geometry. Ocean Engineering, 37, pp Shen, Y.M., Zheng, Y.H. and You, Y.G., On the radiation and diffraction of linear water waves by a rectangular structure over a sill. Part I. Infinite domain of finite water depth. Ocean Engineering, 32, pp Ursell, F., On the heaving motion of a circular cylinder on the surface of a fluid. Quarterly Journal of Mechanics and Applied Mathematics, 2(2), pp Vugts, J.H., The Hydrodynamic coefficients for swaying, heaving and rolling cylinders in a free surface. Netherlands Ship Res. Centre TNO. Report No Delft: Shipbuilding Lab., Delft University of Technology. Zhou, Z.X., Edmond, Y.M.Lo and Tan, S.K., Effect of shallow and narrow water on added mass of cylinders with various cross-sectional shapes. Ocean Engineering, 32, pp

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