A New Simulation Method for the Installation of Subsea Structures from the Splash Zone to the Ultra Deep

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1 A New Simulation Method for the Installation of Subsea Structures from the Splash Zone to the Ultra Deep Bas Buchner and Tim Bunnik MARIN (Maritime Research Institute Netherlands) Haagsteeg 2 / P.O. Box AA Wageningen, The Netherlands b.buchner@marin.nl, t.bunnik@marin.nl Dick Honig and Geert Meskers Heerema Marine Contractors Vondellaan 47, 2332 AA Leiden, The Netherlands dhonig@hmc-heerema.com, gmeskers@hmc-heerema.com Abstract Existing simulation methods are not able to determine in detail the wave loads on a complex subsea structure when it is passing through the splash zone. To determine these loads, model tests are necessary. Otherwise only simplified formulations or empirical relations for added mass and damping can be used. The improved Volume Of Fluid (ivof) method presented in this paper is a potential candidate for the better numerical prediction of the behaviour of a subsea structure in the splash zone. The simulated flow around and through the structure looks very realistic and shows a strong resemblance with observations from model tests. The quantitative comparison of the vertical load on the subsea structure shows that the total load levels are well predicted. This good initial comparison shows the potential of the improved Volume Of Fluid (ivof) method for the simulation of the behaviour of subsea structures in the splash zone. However, significant further development and validation is needed before a fully coupled simulation of a subsea structure and its lifting vessel in waves can be carried out. This possibility is also affected by the long simulation times required at the moment. As an intermediate step the method might be used to determine the wave loads and added mass in an uncoupled simulation. Introduction For the development of deep and ultra deep fields, the safe and economical installation of subsea equipment is of vital importance. The practically continuous swells West of Africa result in significant motions of the installation vessels, in other areas the possible wind seas can induce significant wave loads on the subsea structure when it is lowered through the splash zone. These subsea structures have a large variety of shapes and their shape is typically very complex, see Figure 1. Consequently, the prediction of the motions and loads during the installation is not an easy task. The state-of-the-art approach for the evaluation of such operations is based on time domain simulations of the combined installation vessel and subsea structure. However, with the existing simulation methods it is impossible to determine in detail the wave loads either during the passage through the splash zone, or the added mass and damping when the subsea structure is submerged or close to the seabed. In order to determine these loads, model tests are necessary. Otherwise only simplified formulations or empirical relations for added mass and damping can be used.

2 The present paper presents a new methodology, which was initially developed for the simulation of sloshing in tanks and green water loading on the deck of ships (Reference [1]). This improved Volume Of Fluid (ivof) method is able to simulate the non-linear wave loads on structures in the wave zone, including the flow in and out of the structure. The present paper is part of a research project that has the objective to come to a fully coupled lift simulation: the time domain simulation of the moving lifting vessel with its lifting gear on one side and the subsea structure in the splash zone on the other side. To achieve this objective, first the loads of the waves on the subsea structure have to be studied in detail. The present paper focuses on this aspect. The results of model tests will be used for validation. In the next phase of the project the resulting motions of the subsea structure and the coupling with the behaviour of the lifting vessel will be studied. The paper first summarises the improved Volume Of Fluid (ivof) method included in the ComFLOW program. Then results of simulations with a typical subsea structure in the splash zone are presented. A comparison is made with results of dedicated model tests with the same structure. The comparison gives good insight in the special capabilities of the method. The improved Volume Of Fluid (ivof) Method ComFLOW is an improved Volume Of Fluid (ivof) Navier-Stokes solver. The program has been developed initially by the University of Groningen/RuG (Prof.dr. Arthur Veldman) to study the sloshing of liquid fuel in satellites. This microgravity environment requires a very accurate and robust description of the free surface. Coupled dynamics between the sloshing fluid and the satellite were investigated as well (References [2] and [3]). In close co-operation with MARIN, this methodology was later extended to the calculation of green water loading on a fixed bow deck (Reference [4]), see Figure 2. Also anti-roll tanks, including the coupling with ship motions (Reference [5]), were investigated. Furthermore, the entry of a wedge in a fluid was studied as part of the RuG-MARIN co-operation (Reference [6]), as well as the wave impact loads on fixed structures (see Figure 3). The Volume Of Fluid (VOF) algorithm as developed by Hirt and Nichols (see Reference [7]) is used as a basis for the fluid advection. The method solves the incompressible Navier-Stokes equations with a free-surface condition on the free boundary. In the VOF method a VOF function F (with values between 0 and 1) is used, indicating which part of the cell is filled with fluid. The VOF method reconstructs the free surface in each computational cell. This makes it suitable for the prediction of all phases of the local free surface problem. First the mathematical and numerical model will be summarised. This will be limited to the main aspects, because the detailed numerical aspects are outside the scope of the present paper. Excellent overviews of the numerical details of the method can be found in References [1] through [8]. To distinguish between the original VOF method of Hirt and Nichols (1981) and the present method with its extensive number of modifications, the name improved-vof (ivof) method will be used in the rest of this paper. Mathematical model The incompressible Navier-Stokes equations describe the motions of a fluid in general terms. They are based on conservation of mass (Expression 1) and momentum (Expressions 2 through 4). u v w + + = 0 x y z (1) u u u u 1 p u u u + u + v + w = +ν + + F t x y z ρ x x y z v v v v 1 p v v v + u + v + w = +ν + + F t x y z ρ y x y z w w w w 1 p z z z + u + v + w = +ν + + F z t x y z ρ z x y z x y (2) (3) (4) F =(F x, F y, F z ) is an external body force, such as gravity.

3 With: p = pressure t = time u = velocity in x-direction v = velocity in y-direction w = velocity in z-direction x = x-position y = y-position z = z-position ν = kinematic viscosity ρ = fluid density The Navier-Stokes equations can also be written in a shorter notation as: u = 0 u + p = R t (5) (6) R now contains all convective, diffusive and body forces. Numerical model: geometry and free surface description For the discretisation of a computational domain in numerical simulations a large number of different methods is available. Basically, they can be divided into: Structured and unstructured grids Boundary fitted and non-boundary fitted grids In the improved-vof method a structured (Cartesian) non-boundary fitted grid (not necessarily equidistant) is chosen. This has the following advantages related to the use of the method for the prediction of wave loading: Easy generation of the grid around complex structures A lot of research on surface tracking on orthogonal grids has been carried out Moving objects in the fluid can be dealt with in a similar way as fixed boundaries, without re-gridding The main disadvantage of this discretisation method is the fact that the boundary and free surface are generally not aligned with the gridlines. This requires special attention in the solution method, as will be shown below. An indicator function is used in the form of volume and edge apertures to track the amount of flow in a cell and through a cell face: Volume aperture: the geometry aperture F b indicates which fraction of a cell is allowed to contain fluid ( 0 Fb 1). For bodies moving through the fluid, the geometry aperture may vary in time. The time-dependent fluid aperture F s indicates which fraction of a cell is actually occupied by fluid and satisfies the relation 0 Fs Fb. Edge aperture: the edge apertures A x, A y, and A z define the fraction of a cell surface through which fluid may flow in the x, y and z direction respectively. Obviously, these apertures are between zero and one. Figure 4 shows a two-dimensional example with F b = 0.8 and F s = 0.3. After the apertures have been assigned to the grid cells and the cell edges, every cell is given a label to distinguish between boundary, air and fluid. Two classes of labelling exist: Geometry cell labels and fluid cell labels. The geometry labelling at each time step divides the cells into three classes: F(low)-cells : All cells with F b 0 B(oundary)-cells : All cells adjacent to a F-cell (e)x(ternal)-cells : All remaining cells

4 The free-surface cell labelling is a subdivision of the F-cells. The subdivision consists of 3 subclasses: E(mpty) cells : All cells with F s = 0 S(urface) cells : All cells adjacent to an E-cell F'(luid)-cells : All remaining F-cells Figure 5 shows an example of geometry cell labelling and free-surface cell labelling for a wedge entering a fluid. The discretisation of the Navier-Stokes equations is done on a staggered grid, which means that the pressure will be set in the cell centres and the velocity components in the middle of the cell faces between two cells. This is shown in 2D in Figure 6. The Navier-Stokes equations are discretised in time according to the explicit first order Forward Euler method as follows:.u n+ 1 = 0 u n+ 1 u n n 1 n + p + = R t (7) (8) t is the time step and n+1 and n denote the new and old time level. The conservation of mass in Expression (7) and the pressure in Expression (8) are treated on the new time level n+1 to assure that the new u is divergence-free (no loss of fluid). The spatial discretisation will now be explained using the computational cell shown in Figure 7. Expression (7) is applied in the centres of the cells and a central discretisation is used. In the cell with centre w the discretised equation becomes: n+ 1 n+ 1 n+ 1 n+ 1 C W NW SW u u v v + = 0 h h x y (9) The momentum Expression (8) is applied in the centres of the cell faces, thus the discretisation in point C becomes: n+ 1 n n+ 1 n+ 1 C C e w u u p p + = R t h x n C (10) In the detailed work of Gerrits (Reference [3]) other aspects of the numerical method are described in detail, such as: Discretisation of R c n Discretisation near the free-surface In- and outflow discretisation Pressure Poisson equation Free surface reconstruction and displacement Use of the Courant-Friedrichs-Levy (CFL) number Calculation of forces Summarising, the following functionalities are presently available in ComFLOW (see References [1] through [8]): - Calculation of the fluid motion by solving the incompressible Navier-Stokes equations. - One type of fluid flow is considered, with a void where no fluid is present. - Possibility to model an arbitrary number of fixed objects in the fluid. The objects are defined piecewise linearly. - Options to use no-slip or free-slip boundary conditions at the solid boundaries. At the free surface continuity of tangential and normal stresses (including capillary effects) is prescribed. Inflow and outflow boundary conditions for fluid velocities and/or pressures can be defined.

5 - The fluid simulations are carried out on a Cartesian grid with user-defined stretching. The Cartesian grid is fixed in the domain. When the domain is moving a virtual body force is added to the forcing term in the Navier-Stokes equations. The fluid motions are thus solved in a domain-fixed co-ordinate system. - To distinguish between the different characters of grid cells, the cells are labelled. The Navier-Stokes equations are discretised and solved in cells that contain fluid. The free-surface displacement is described by the Volume Of Fluid method with a local height function. - The generation of waves, which has been accomplished by specifying fluid velocities at the inflow boundary of the fluid domain. The fluid velocities are obtained from potential flow. Linear waves (Airy waves) and fifth-order waves (Stokes waves) have been implemented. The linear waves have been stretched towards the actual position of the free surface using Wheeler stretching. - Several absorbing boundary conditions at the outflow boundaries. - The possibility to use the velocity field from a (de-coupled) linear diffraction theory. Linear diffraction theory is used to compute body motions and fluid velocities which are then used in ComFLOW to prescribe the body motions and fluid velocities on the inflow and outflow boundaries. - An off-line coupling with a structural-analysis code has been established to compute the response of a structure to high peak loads. Simulation and model test results The ivof method has now been applied to the problem of a subsea structure in the splash zone. Specific model tests were carried out to validate the simulations. Figure 8 shows the model of the subsea structure in the model basin. It was build at scale 1:40 and its main dimensions are given in Figure 9 and Table 1. Two series of tests were carried out: A. Tests with the subsea structure captive in a force and moment frame to determine the wave exciting loads in 6 degrees of freedom. B. Tests with the subsea structure free-hanging in slings. For the present paper only the results for Series A with the captive model are used. Series B will be used in a later phase to check the fully coupled motion of the subsea structure in waves. Figure 10 shows the captive force measurement set-up in the basin and Figure 11 presents the two extreme positions tested: the subsea structure just above the free surface and just fully submerged. In addition to these positions, also an intermediate draft was tested. Figure 12 shows several examples of the complex flow in and around the subsea structure in waves observed during the tests. This flow is characterised by aspects such as: Added mass and damping effects of the water under, above and in the subsea structure Buoyancy effects in the waves Impact loads against horizontal plates in the subsea structure Flow through openings in the top and bottom of the subsea structure Complex flow through the equipment of the subsea structure Water captured temporarely in certain parts of the subsea structure In the free-hanging situation during the lift, this can result in unwanted oscillations, slack lines and larger vertical loads in the slings than expected. The numerical model of the subsea structure above the approaching wave is shown in Figure 13. This model was built with basic shapes such as beams, boxes and cylinders. The dominant components in the subsea structure were all modelled. The computational domain contained a total number of 170 * 100 * 100 (= ) cells. Figure 14 shows a number of snapshots of the flow around the subsea structure in different phases during a simulation in waves. The flow around and through the structure looks very realistic and shows a strong resemblance with the observation from the model tests. A quantitative comparison of the vertical load on the subsea structure is given in Figure 15. The simulated load is given in blue and measured load is given in red. In the simulations only one wave oscillation was calculated. This was done because the calculation is extremely time consuming: 65 hours on a 2.4 MHz PC for a simulation of 1 wave period! The comparison between the simulation and model test is reasonably good. The total load levels are reasonably predicted. However, the simulation shows load peaks that are not present in the model tests and lachs some of the oscillations that are observed in the tests. These differences need further investigation.

6 Conclusions The improved Volume Of Fluid (ivof) method presented in the present paper is a potential candidate for the better numerical prediction of the behaviour of a subsea structure in the splash zone. Based on the initial comparison between dedicated model tests and simulations the following conclusions seem justified: - The simulated flow around and through the structure looks very realistic and shows a strong resemblance with observations from model tests. - The quantitative comparison of the vertical load on the subsea structure shows that the total load levels are well predicted. - However, the simulation shows load peaks that are not present in the model tests and misses some of the oscillations that are observed in the tests. These differences need further investigation. This good initial comparison shows the potential of the improved Volumes Of Fluid (ivof) method for the simulation of the behaviour of subsea structures in the splash zone. However, significant further development and validation is needed before a fully coupled simulation of the subsea structure and its lifting vessel in waves can be carried out. This possibility is also affected by the long simulation times required at the moment. As an intermediate step the method might be used to determine the wave loads and added mass in an uncoupled simulation. References 1. Buchner, B.; Green Water on Ship-type Offshore Structures, PhD thesis Delft University of Technology, Gerrits, J., Loots, G.E., Fekken, G. and Veldman, A.E.P.; Liquid Sloshing on Earth and in Space, In: Moving Boundaries V (Sarler, B., Brebbia, C.A. and Power, H. Eds.) WIT Press, Southampton, pp , Gerrits, J.; Dynamics of Liquid-Filled Spacecraft, Numerical Simulation of Coupled Solid-Liquid Dynamics, PhD thesis, RuG, Fekken, G., Veldman, A.E.P. and Buchner, B.; Simulation of Green Water Flow Using the Navier-Stokes Equations, Seventh International Conference on Numerical Ship Hydromechanics, Nantes, Daalen, E.F.G. van, Kleefsman, K.M.T., Gerrits, J., Luth, H.R. and Veldman, A.E.P.; Anti Roll Tank Simulations with a Volume Of Fluid (VOF) based Navier-Stokes Solver, 23rd Symposium on Naval Hydrodynamics, Val de Reuil, September Buchner, B., Bunnik, T.H.J., Fekken, G. and Veldman, A.E.P.; A Numerical Study on Wave Run Up on an FPSO Bow, OMAE2001, Rio de Janeiro, Hirt, C.W. and Nichols, B.D.; Volume Of Fluid (VOF) Method for the Dynamics of Free Boundaries, Journal of Computational Physics, 39, pp , Kleefsman, K.M.T., Fekken, G., Veldman, A.E.P., Bunnik, T., Buchner, B. and Iwanowski, B.; Prediction of green water and wave impact loading using a Navier-Stokes Based Simulation Tool, OMAE conference, Oslo, 2002.

7 Table 1: Main dimensions and weight distribution of the subsea structure (1:40 scale) M 266 t KG 4.96 m K xx 2.68 m K yy 4.96 m K zz 4.80 m

8 Figure 1: Examples of different complex subsea structures in the splash zone

9 Figure 2: Example of earlier application of the method: green water on the deck of an FPSO Figure 3: Example of earlier application of the method: wave in deck loading on a fixed platform

10 y F -F = 0.5 b s A = x 0.2 F = 0.3 s A = 0.5 y x Figure 4: Two-dimensional example of a grid cell using apertures Figure 5: Geometry cell labelling (left) and free-surface cell labelling (right) for a wedge entering a fluid p v p u p u p v p Figure 6: Location of the pressure and velocity components in the staggered grid

11 N NW NE h y W w C e E SW SE S h x Figure 7: Spatial discretisation cell, using compass indication for cell phases Figure 8: Model of a subsea structure in the model basin Figure 9: Main dimensions of the subsea structure model (dimensions in m full-scale, scale model 1:40)

12 Figure 10: Subsea structure in captive force measurement set-up Figure 11: Subsea structure in captive force measurement set-up at the two extreme positions tested: just above the free surface and just fully submerged

13 Figure 12: several examples of complex flow in and around the subsea structure in waves

14 Figure 13: Numerical model of subsea structure

15 Figure 14: The subsea structure in different phases during the simulation in waves

16 Fz [kn] time [s] Figure 15: Simulated (blue) and measured vertical force on the subsea structure

Simulation of Offshore Wave Impacts with a Volume of Fluid Method. Tim Bunnik Tim Bunnik MARIN

Simulation of Offshore Wave Impacts with a Volume of Fluid Method Tim Bunnik Tim Bunnik MARIN Outline Part I: Numerical method -Overview Part II: Applications - Dambreak - Wave run-up - Sloshing loads