Numerical Study of Discharge Coefficients for Side-Damaged Compartments
Abstract
:1. Introduction
2. Computational Principles and Numerical Modeling
2.1. Principle of Discharge Coefficient Calculation
- (1)
- Averaging:
- (2)
- Integral method:
2.2. Control Equations
2.3. Turbulence Modeling
2.4. Free Level Treatment
3. Simplified Modeling and Design of Working Conditions
3.1. Basic Model
- The breakthrough surface is vertical or nearly vertical to the horizontal plane, in line with the typical characteristics of the ship-side shell plate arrangement.
- The breakthrough is completely located below the water line, enabling the seawater to pass through the entire breakthrough into the cabin.
- The size of the damaged opening should not be too small or too large because the unsuitable damaged opening size makes the study lose its practical significance.
- The damaged opening chamber must be open and spacious to avoid the impact of the jet stream on the bulkheads or other structures on the calculation.
3.2. Simplified Model
3.3. Design of Working Conditions
4. Numerical Simulation Points
Boundary Condition Settings and Post-Processing
5. Mesh Division and Irrelevance Test
6. Simulation
7. Analysis of Results and Summarization of Patterns
7.1. Influence of the Feature Size and Central Depth of Damaged Openings on Inlet Discharge Coefficients
7.2. Influence of Shape and Area of Damaged Openings on Inlet Discharge Coefficients
7.3. Fitting Empirical Equations for Discharge Coefficients
8. Conclusions
- The calculations determined the patterns of influence that the central depth, characteristic dimensions, and shapes of the damaged openings have on the discharge coefficient of water ingress through a ship’s side damaged by anti-ship weapons. Regarding the central depth of the damaged opening, within the water depth range of 1 m to 3 m, the influence of water depth on the discharge coefficient is minimal, with a maximum difference of only 0.003 (about 0.5%). But from the perspective of the damaged opening’s characteristic dimensions and shapes, the variations in discharge coefficients are more pronounced. Overall, the triangular-damaged opening has a higher coefficient than the circular-damaged opening, which in turn is higher than that of the square-damaged opening; the coefficient decreases as the size of the damaged opening increases.
- Based on the results of the numerical calculations, two sets of semi-empirical formulas for the discharge coefficient of ship-side damage are provided using the Π theorem and polynomial fitting method, respectively; the fitting effects are good. These formulas can serve as a reference for determining discharge coefficients in the time-domain simulation of ship-damaged opening scenarios using the quasi-static method and for the expedited evaluation of buoyancy stability.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Main Guns | Torpedoes | Anti-Ship Missiles | |||
---|---|---|---|---|---|
Ship Classification | Caliber (mm) | Type | Caliber (mm) | Type | Caliber (mm) |
Arleigh Burke | 127 | MK48 | 533 | AGM-109 | 520 |
Ticonderoga | 127 | Sailfish | 533 | Hsiung Feng III | 460 |
Sejong Daewang | 127 | Varunastra | 533 | AGM-84 | 340 |
Chilung | 127 | MK46 | 324 | Type 12 | 350 |
Murasame | 76 | MU-90 | 324 | P-J10 | 670 |
Type | Size (m) | Number in Different Depth of Damaged Opening Center | ||||
---|---|---|---|---|---|---|
1.0 m | 1.5 m | 2.0 m | 2.5 m | 3.0 m | ||
Circular | 0.1 | No. 1 | No. 11 | No. 21 | No. 31 | No. 41 |
Circular | 0.2 | No. 2 | No. 12 | No. 22 | No. 32 | No. 42 |
Circular | 0.3 | No. 3 | No. 13 | No. 23 | No. 33 | No. 43 |
Circular | …… | …… | …… | …… | …… | …… |
Circular | 1.0 | No. 10 | No. 20 | No. 30 | No. 40 | No. 50 |
Square | 0.1 | No. 51 | No. 61 | No. 71 | No. 81 | No. 91 |
Square | 0.2 | No. 52 | No. 62 | No. 72 | No. 82 | No. 92 |
Square | 0.3 | No. 53 | No. 63 | No. 73 | No. 83 | No. 93 |
Square | …… | …… | …… | …… | …… | …… |
Square | 1.0 | No. 60 | No. 70 | No. 80 | No. 90 | No. 100 |
Triangular | 0.1 | No. 101 | No. 111 | No. 121 | No. 131 | No. 141 |
Triangular | 0.2 | No. 102 | No. 112 | No. 122 | No. 132 | No. 142 |
Triangular | 0.3 | No. 103 | No. 113 | No. 123 | No. 133 | No. 143 |
Triangular | …… | …… | …… | …… | …… | …… |
Triangular | 1.0 | No. 110 | No. 120 | No. 130 | No. 140 | No. 150 |
Feature Size (m) | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1 | |
---|---|---|---|---|---|---|---|---|---|---|---|
A = 5% | Absolute value of target surface size/mm | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 |
mesh count | 750k | 720k | 700k | 700k | 690k | 690k | 690k | 690k | 690k | 690k | |
A = 1% | Absolute value of target surface size/mm | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 1 |
mesh count | 920k | 890k | 820k | 850k | 890k | 790k | 800k | 810k | 830k | 840k | |
A = 0.5% | Absolute value of target surface size/mm | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 | 4.5 | 5 |
mesh count | 1150k | 1100k | 970k | 1050k | 1140k | 930k | 970k | 1000k | 1040k | 1090k | |
A = 0.1% | Absolute value of target surface size/mm | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1 |
mesh count | 3490k | 2900k | 3850k | 2510k | 2940k | 3310k | 2120k | 2310k | 2510k | 2730k | |
A = 0.05% | Absolute value of target surface size/mm | 0.05 | 0.1 | 0.15 | 0.2 | 0.25 | 0.3 | 0.35 | 0.4 | 0.45 | 05 |
mesh count | 8150k | 6010k | 8830k | 4990k | 6200k | 7140k | 3900k | 4280k | 4800k | 5280k |
Condition | Total Number of Divided Meshes | Percentage of Better Quality Meshes |
---|---|---|
1 | 3,699,176 | 99.537% |
75 | 3,313,313 | 99.909% |
150 | 2,728,580 | 99.768% |
No. | Characteristic Size of the Damaged Opening (m) | t = 0.02 s | t = 0.01 s | t = 0.005 s | |||
---|---|---|---|---|---|---|---|
Mass Flow (kg/s) | Cd | Mass Flow (kg/s) | Cd | Mass Flow (kg/s) | Cd | ||
1 | 0.1 | 12.0625 | 0.631 | 11.7499 | 0.614 | 11.7493 | 0.614 |
2 | 0.2 | 47.4681 | 0.621 | 47.4725 | 0.621 | 47.4775 | 0.621 |
3 | 0.3 | 106.887 | 0.621 | 107.049 | 0.622 | 107.038 | 0.622 |
4 | 0.4 | 190.75 | 0.623 | 190.705 | 0.623 | 190.726 | 0.623 |
5 | 0.5 | 298.143 | 0.624 | 298.068 | 0.623 | 298.043 | 0.623 |
6 | 0.6 | 429.153 | 0.623 | 428.857 | 0.623 | 428.832 | 0.623 |
7 | 0.7 | 581.992 | 0.621 | 581.018 | 0.620 | 581.016 | 0.620 |
8 | 0.8 | 757.61 | 0.619 | 757.265 | 0.619 | 757.531 | 0.619 |
9 | 0.9 | 956.957 | 0.618 | 956.298 | 0.617 | 956.265 | 0.617 |
10 | 1.0 | 1178.24 | 0.616 | 1176.44 | 0.615 | 1176.29 | 0.615 |
Feature Size (m) | Central Depth = 1 m | Central Depth = 1.5 m | Central Depth = 2 m | Central Depth = 2.5 m | Central Depth = 3 m | |||||
---|---|---|---|---|---|---|---|---|---|---|
Mass Flow (kg/s) | Cd | Mass Flow (kg/s) | Cd | Mass Flow (kg/s) | Cd | Mass Flow (kg/s) | Cd | Mass Flow (kg/s) | Cd | |
0.1 | 21.0423 | 0.607 | 25.7237 | 0.606 | 29.6843 | 0.605 | 33.1695 | 0.605 | 36.3102 | 0.604 |
0.2 | 84.9736 | 0.612 | 104.002 | 0.612 | 120.019 | 0.612 | 134.113 | 0.611 | 146.833 | 0.611 |
0.3 | 191.866 | 0.615 | 234.961 | 0.615 | 271.194 | 0.614 | 303.076 | 0.614 | 331.845 | 0.614 |
0.4 | 340.639 | 0.614 | 417.515 | 0.614 | 482.060 | 0.614 | 538.815 | 0.614 | 590.004 | 0.614 |
0.5 | 531.413 | 0.613 | 652.102 | 0.614 | 753.108 | 0.614 | 841.911 | 0.614 | 921.874 | 0.614 |
0.6 | 762.600 | 0.611 | 937.116 | 0.613 | 1082.85 | 0.613 | 1210.43 | 0.613 | 1325.47 | 0.613 |
0.7 | 1029.91 | 0.606 | 1267.53 | 0.609 | 1465.09 | 0.61 | 1638.22 | 0.61 | 1793.45 | 0.609 |
0.8 | 1337.63 | 0.603 | 1649.34 | 0.607 | 1907.15 | 0.608 | 2132.20 | 0.607 | 2334.33 | 0.607 |
0.9 | 1681.33 | 0.598 | 2077.22 | 0.604 | 2402.71 | 0.605 | 2686.44 | 0.605 | 2940.79 | 0.604 |
1 | 2042.68 | 0.589 | 2549.51 | 0.6 | 2949.95 | 0.601 | 3298.05 | 0.601 | 3609.84 | 0.601 |
Feature Size (m) | Central Depth = 1 m | Central Depth = 1.5 m | Central Depth = 2 m | Central Depth = 2.5 m | Central Depth = 3 m | |||||
---|---|---|---|---|---|---|---|---|---|---|
Mass Flow (kg/s) | Cd | Mass Flow (kg/s) | Cd | Mass Flow (kg/s) | Cd | Mass Flow (kg/s) | Cd | Mass Flow (kg/s) | Cd | |
0.1 | 27.0993 | 0.614 | 33.1543 | 0.613 | 38.2832 | 0.613 | 42.7587 | 0.612 | 46.8080 | 0.612 |
0.2 | 109.280 | 0.619 | 133.767 | 0.618 | 154.410 | 0.618 | 172.566 | 0.618 | 188.909 | 0.617 |
0.3 | 246.704 | 0.621 | 302.272 | 0.621 | 349.012 | 0.621 | 390.068 | 0.621 | 427.234 | 0.621 |
0.4 | 437.600 | 0.619 | 536.720 | 0.620 | 619.837 | 0.620 | 693.003 | 0.620 | 758.784 | 0.620 |
0.5 | 682.197 | 0.618 | 837.909 | 0.620 | 968.037 | 0.620 | 1082.39 | 0.620 | 1185.02 | 0.620 |
0.6 | 977.003 | 0.615 | 1202.27 | 0.617 | 1389.44 | 0.618 | 1553.49 | 0.618 | 1700.87 | 0.618 |
0.7 | 1316.88 | 0.609 | 1623.77 | 0.613 | 1877.54 | 0.614 | 2100.19 | 0.614 | 2299.20 | 0.613 |
0.8 | 1709.89 | 0.605 | 2113.09 | 0.610 | 2444.56 | 0.612 | 2733.25 | 0.616 | 2991.76 | 0.611 |
0.9 | 2142.21 | 0.599 | 2655.29 | 0.606 | 3073.29 | 0.607 | 3435.82 | 0.607 | 3760.66 | 0.607 |
1 | 2616.39 | 0.592 | 3253.41 | 0.601 | 3766.78 | 0.603 | 4211.06 | 0.603 | 4608.45 | 0.602 |
Feature Size (m) | Central Depth = 1 m | Central Depth = 1.5 m | Central Depth = 2 m | Central Depth = 2.5 m | Central Depth = 3 m | |||||
---|---|---|---|---|---|---|---|---|---|---|
Mass Flow (kg/s) | Cd | Mass Flow (kg/s) | Cd | Mass Flow (kg/s) | Cd | Mass Flow (kg/s) | Cd | Mass Flow (kg/s) | Cd | |
0.1 | 11.7499 | 0.614 | 1176.44 | 0.614 | 16.5797 | 0.613 | 18.5307 | 0.613 | 20.2856 | 0.613 |
0.2 | 47.4725 | 0.621 | 14.3713 | 0.620 | 67.0424 | 0.620 | 74.9126 | 0.619 | 82.0587 | 0.619 |
0.3 | 107.049 | 0.622 | 58.0955 | 0.622 | 151.219 | 0.621 | 168.908 | 0.621 | 184.936 | 0.620 |
0.4 | 190.705 | 0.623 | 131.047 | 0.623 | 269.551 | 0.623 | 301.280 | 0.623 | 329.866 | 0.622 |
0.5 | 298.068 | 0.624 | 233.539 | 0.624 | 421.496 | 0.623 | 471.291 | 0.624 | 515.780 | 0.623 |
0.6 | 428.857 | 0.623 | 365.076 | 0.624 | 606.969 | 0.623 | 678.444 | 0.623 | 742.886 | 0.623 |
0.7 | 581.018 | 0.620 | 525.731 | 0.621 | 822.979 | 0.621 | 920.329 | 0.621 | 1007.48 | 0.621 |
0.8 | 757.265 | 0.619 | 712.717 | 0.620 | 1074.34 | 0.621 | 1201.32 | 0.621 | 1315.01 | 0.620 |
0.9 | 956.298 | 0.617 | 929.729 | 0.620 | 1358.08 | 0.620 | 1518.17 | 0.620 | 1661.83 | 0.619 |
1 | 1176.44 | 0.615 | 1175.69 | 0.618 | 1672.80 | 0.619 | 1870.24 | 0.619 | 2046.65 | 0.618 |
Feature Size (m) | Central Depth = 1 m | Central Depth = 1.5 m | Central Depth = 2 m | Central Depth = 2.5 m | Central Depth = 3 m | |||||
---|---|---|---|---|---|---|---|---|---|---|
Cd | Δ | Cd | Δ | Cd | Δ | Cd | Δ | Cd | Δ | |
0.1 | 0.608 | 0.001 | 0.608 | 0.002 | 0.608 | 0.003 | 0.608 | 0.003 | 0.610 | 0.006 |
0.2 | 0.611 | −0.001 | 0.612 | 0.000 | 0.612 | 0.000 | 0.611 | 0.000 | 0.609 | −0.002 |
0.3 | 0.612 | −0.003 | 0.613 | −0.002 | 0.614 | 0.000 | 0.615 | 0.001 | 0.615 | 0.001 |
0.4 | 0.611 | −0.003 | 0.613 | −0.001 | 0.614 | 0.000 | 0.616 | 0.002 | 0.617 | 0.003 |
0.5 | 0.610 | −0.003 | 0.611 | −0.003 | 0.613 | −0.001 | 0.614 | 0.000 | 0.616 | 0.002 |
0.6 | 0.609 | −0.002 | 0.609 | −0.004 | 0.611 | −0.002 | 0.612 | −0.001 | 0.614 | 0.001 |
0.7 | 0.607 | 0.001 | 0.607 | −0.002 | 0.608 | −0.002 | 0.610 | 0.000 | 0.611 | 0.002 |
0.8 | 0.605 | 0.002 | 0.604 | −0.003 | 0.605 | −0.003 | 0.606 | −0.001 | 0.608 | 0.001 |
0.9 | 0.603 | 0.005 | 0.602 | −0.002 | 0.602 | −0.003 | 0.603 | −0.002 | 0.605 | 0.001 |
1 | 0.600 | 0.011 | 0.599 | −0.001 | 0.599 | −0.002 | 0.600 | −0.001 | 0.603 | 0.002 |
Feature Size (m) | Central Depth = 1 m | Central Depth = 1.5 m | Central Depth = 2 m | Central Depth = 2.5 m | Central Depth = 3 m | |||||
---|---|---|---|---|---|---|---|---|---|---|
Cd | Δ | Cd | Δ | Cd | Δ | Cd | Δ | Cd | Δ | |
0.1 | 0.614 | 0.000 | 0.613 | 0.000 | 0.613 | 0.000 | 0.612 | 0.000 | 0.611 | −0.001 |
0.2 | 0.616 | −0.003 | 0.617 | −0.001 | 0.618 | 0.000 | 0.618 | 0.000 | 0.616 | −0.001 |
0.3 | 0.617 | −0.004 | 0.618 | −0.003 | 0.619 | −0.002 | 0.621 | 0.000 | 0.621 | 0.000 |
0.4 | 0.616 | −0.003 | 0.617 | −0.003 | 0.619 | −0.001 | 0.620 | 0.000 | 0.621 | 0.001 |
0.5 | 0.614 | −0.004 | 0.615 | −0.005 | 0.617 | −0.003 | 0.618 | −0.002 | 0.620 | 0.000 |
0.6 | 0.612 | −0.003 | 0.613 | −0.004 | 0.614 | −0.004 | 0.615 | −0.003 | 0.617 | −0.001 |
0.7 | 0.610 | 0.001 | 0.610 | −0.003 | 0.610 | −0.004 | 0.612 | −0.002 | 0.614 | 0.001 |
0.8 | 0.608 | 0.003 | 0.607 | −0.003 | 0.607 | −0.005 | 0.608 | −0.008 | 0.610 | −0.001 |
0.9 | 0.605 | 0.006 | 0.603 | −0.003 | 0.604 | −0.003 | 0.605 | −0.002 | 0.607 | 0.000 |
1 | 0.602 | 0.010 | 0.600 | −0.001 | 0.600 | −0.003 | 0.602 | −0.001 | 0.605 | 0.003 |
Feature Size (m) | Central Depth = 1 m | Central Depth = 1.5 m | Central Depth = 2 m | Central Depth = 2.5 m | Central Depth = 3 m | |||||
---|---|---|---|---|---|---|---|---|---|---|
Cd | Δ | Cd | Δ | Cd | Δ | Cd | Δ | Cd | Δ | |
0.1 | 0.618 | 0.004 | 0.620 | 0.006 | 0.625 | 0.012 | 0.635 | 0.022 | 0.653 | 0.040 |
0.2 | 0.621 | 0.000 | 0.621 | 0.001 | 0.620 | 0.000 | 0.619 | 0.000 | 0.616 | −0.003 |
0.3 | 0.622 | 0.000 | 0.623 | 0.001 | 0.624 | 0.003 | 0.623 | 0.002 | 0.622 | 0.002 |
0.4 | 0.623 | 0.000 | 0.624 | 0.001 | 0.625 | 0.002 | 0.626 | 0.003 | 0.626 | 0.004 |
0.5 | 0.622 | −0.002 | 0.624 | 0.000 | 0.625 | 0.002 | 0.627 | 0.003 | 0.627 | 0.004 |
0.6 | 0.622 | −0.001 | 0.623 | −0.001 | 0.625 | 0.002 | 0.626 | 0.003 | 0.628 | 0.005 |
0.7 | 0.621 | 0.001 | 0.622 | 0.001 | 0.624 | 0.003 | 0.625 | 0.004 | 0.627 | 0.006 |
0.8 | 0.620 | 0.001 | 0.621 | 0.001 | 0.622 | 0.001 | 0.623 | 0.002 | 0.625 | 0.005 |
0.9 | 0.618 | 0.001 | 0.619 | −0.001 | 0.620 | 0.000 | 0.621 | 0.001 | 0.623 | 0.004 |
1 | 0.617 | 0.002 | 0.617 | −0.001 | 0.618 | −0.001 | 0.619 | 0.000 | 0.621 | 0.003 |
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Tian, S.; Peng, F.; Wang, Z.; Li, J. Numerical Study of Discharge Coefficients for Side-Damaged Compartments. J. Mar. Sci. Eng. 2024, 12, 1502. https://doi.org/10.3390/jmse12091502
Tian S, Peng F, Wang Z, Li J. Numerical Study of Discharge Coefficients for Side-Damaged Compartments. Journal of Marine Science and Engineering. 2024; 12(9):1502. https://doi.org/10.3390/jmse12091502
Chicago/Turabian StyleTian, Siwen, Fei Peng, Zhanzhi Wang, and Jingda Li. 2024. "Numerical Study of Discharge Coefficients for Side-Damaged Compartments" Journal of Marine Science and Engineering 12, no. 9: 1502. https://doi.org/10.3390/jmse12091502