Trans RINA, Vol 152, Part A4, Intl J Maritime Eng, Oct-Dec 2010 3. NUMERICAL SIMULATIONS 3.1 CALCULATION METHOD


All tested damage cases were calculated with a time- domain flooding simulation tool in the NAPA software. The details of the applied algorithm are described by Ruponen, [6] and [7]. In principle the simulation is based on Bernoulli's equation (conservation of momentum), which is solved simultaneously with the equation of continuity by using a pressure-correction algorithm with implicit time integration. The pressure losses in the openings are taken into account by using semi-empirical discharge coefficients. The following gives a brief introduction to the applied method.


At each time step the conservation of mass must be satisfied in continuity is:


each Ω ∂ Ω = − ρv S⋅ d ∫ ∂ρ t d ∫ S


where ρ is density, t is time, v is the velocity vector and S is the surface that bounds the control volume Ω. The normal vector of the surface points outwards from the control volume.


The velocities in the openings are calculated by applying Bernoulli’s equation for a streamline from point A that is in the middle of a flooded room to point B in the opening:


∫ () () = 0 2


B A


dp ρ


+


2 1


u − uA + g hB − hA + 2 B


2 1


k uL B 2 (2)


where p is air pressure, u is flow velocity and h is the water level height from the common reference level. All losses in the opening are represented by the non- dimensional pressure-loss coefficient kL. It is assumed that uA = 0. Consequently, the water flow through an opening with area dS is:


dQ C g h h=− +dA B


⎡⎤ 2 () AB


⎢⎥ ⎣⎦


pp


where the discharge coefficient is: 1


Cd = 1+ kL


ρ dS −


(3)


The results of a flooding simulation depend on the applied parameters for the openings and the flooded compartments. These are discussed in the following.


3.2 (a) Discharge coefficients


The flow rate through an opening is directly proportional to the applied discharge


coefficient, equation (3).


Typically in literature, for example [10], a constant discharge coefficient Cd = 0.6 is used for all openings. But


obviously


significantly since the discharge coefficient depends on the shape and size of the opening.


In this study the results of the full-scale experiments within the FP7 European Union funded research project FLOODSTAND,


[11], were used. For manholes, the


rough estimation corresponds rather well with the measurements for free discharge into air. However, when the discharge was into water, somewhat larger values (up to 0.70) were obtained in the full-scale tests for a manhole, [11].


(4)


The calculation within a time step is iterative, based on the linearized Bernoulli’s equation, [6] and [7]. The algorithm corrects the hydrostatic and air pressures in the flooded rooms until both Bernoulli’s equation for each opening and the conservation of mass for each room is satisfied with sufficient accuracy. This is controlled by the applied convergence criterion. The floating position


A-200


Special attention was paid to the two (almost identical) valves since they included a short pipe. After the flooding tests, one valve was removed and extensively tested in the flume of the Water Engineering Group of the Aalto University School of Science and Technology. Most notably, different values were eventually used for these valves


due to the completely different flow


conditions. The “damage hole” valve is submerged very rapidly and discharges to water with a large pressure head most of the flooding process. The valve that is located between the side tank and the equipment room discharges to air with a small pressure head for a very long time. This is taken into account by applying a smaller discharge coefficient. On the other hand, when


the flow characteristics can vary flooded room. The equation of (1)


of the ship is calculated on the basis of the distribution of floodwater in the compartments. The progressive flooding is considered as added weight. In this study, the dynamic roll motion was calculated with the assumption of linear damping. The other degrees-of-freedom (trim and draft) were considered to be quasi-stationary.


The airflows and air pressures are solved by using the approximation of perfect gas and Bernoulli’s equation for compressible fluid. Furthermore, the flooding process is considered to be isothermal. All water levels are assumed to be


flat and horizontal. The pressure-correction


algorithm solves a combination of hydrostatic pressures (water level heights) and air pressures that satisfy both the conservation of mass and momentum (Bernoulli’s equation for each opening). This method has proven to be very efficient and numerically stable, even with very complex flooding cases, [8]. Previously this simulation method has been successfully validated against model test experiments, [9].


3.2 INPUT PARAMETERS


©2010: The Royal Institution of Naval Architects


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