International Conference on Computational Methods in Marine Engineering MARINE 2007 P. Bergan, J. Garc´ıa, E. O˜nate and T. Kvamsdal (Eds) c
CIMNE, Barcelona, 2007
FREE-SURFACE FLOW MODELLING WITH INTERFACE CAPTURING TECHNIQUES M. HOEKSTRA∗ , G. VAZ∗ , B. ABEIL∗∗ AND T. BUNNIK∗ ∗ Maritime
Research Institute Netherlands (MARIN) P.O. Box 28, 6700 AA Wageningen, The Netherlands e-mail:
[email protected], web page: http://www.marin.nl ∗∗ Now
MARIN, formerly Chalmers University, Sweden
Key words: Marine Engineering, Computational Techniques, Free-Surface Modelling. Summary. MARIN, HSVA and TUHH are jointly developing the RANS-code F RE SC O. The freesurface modelling part of the code is based on interface capturing. Results with well-known discretization schemes, like CICSAM and HRIC, are presented for artificial benchmark problems and two dam-break problems. They will be compared with experimental data as well as results obtained with an alternative code, based on interface tracking. A critical evaluation is finally given, leading to the conclusion that the interface capturing scheme in F RE SC O should not be governed by the Courant number. 1 INTRODUCTION Under the EU Project Virtue, MARIN, HSVA and TUHH are jointly developing a RANS-code for flow problems in ship and offshore hydrodynamics. The code is called F RE SC O1 . It uses finite-volume discretization with cell-centered co-located variables and can be applied to structured as well as unstructured grids. The free-surface modelling refers to an interface capturing technique, i.e. a volume-fraction transport equation is solved on a fixed grid, requiring special discretization schemes to keep the sharpness of the air-water interface tolerable. We have tried some well-known interface capturing schemes (CICSAM2 and HRIC3 ) and present representative results of application in two areas: artificial standard problems and dam-break problems. Comparisons with experimental data and with a reference result from another code (COMFLOW4 ) will be shown. We conclude with an evaluation of the results and the features of current schemes. 2 INTERFACE CAPTURING SCHEMES Free surface modelling by interface capturing is based on the solution of the transport equation: ∂α + ∇ · (αV) = 0, ∂t
(1)
in which α is the volume fraction, which specifies for each grid cell to what extent the cell is occupied with one of the fluids, and V is the velocity field. If α is associated with the water phase, the air fraction is (1 − α). For implicit, backward time-integration, Eq.(1) can be discretized as
c1 αcn
+
c2 αcn−1
+
c3 αcn−2
∆V
∆t
+
nf h X
i
αfn (Vf · Af ) = 0,
(2)
f =1
where ∆V is the cell volume, Af the face area vector, ∆t the time step and n the time iteration index. The subscripts c and f refer to cell-averaged and face-averaged data, respectively. After some rearrangement, 1
M. Hoekstra, G. Vaz, B. Abeil and T. Bunnik
a characteristic quantity appears from Eq.(2) as Cf = (Vf · Af ) ∆t/∆V, the Courant number. For a certain velocity field, two discretization aspects remain to be analyzed: the time discretization, and the spatial discretization, i.e. the construction of αf , based on the values of neighboring αc ’s. For the time-discretization we consider both the implicit 1st order backward Euler scheme (c1 = 1.0, c2 = −1.0, c3 = 0.0) and the 2nd order three-time-level scheme (c1 = 1.5, c2 = −2.0, c3 = 0.5). With regard to the construction of αf , it is well known that downwind discretization is needed in particular circumstances. Widely used schemes, like CICSAM2 and HRIC3 , and also BICS5 , contain therefore elements of downwinding. We have implemented the CICSAM and HRIC schemes in F RE SC O and have checked their performance. 3 RESULTS OF APPLICATION We have applied F RE SC O interface capturing to artificial standard problems and dam-break flows. In the latter category, comparisons with experimental data and with reference results from another code called COMFLOW4 (interface tracking method being used at MARIN in green water impact simulations) will be made. 3.1 Artificial Problems The validation of interface capturing schemes is commonly done by placing arbitrarily-shaped fluid interfaces in a known velocity field. From the several cases that the third author has investigated in detail6 , and which were studied earlier by Ubbink2 and Rudman7 , we have selected one example. The case consists of the transport of a square-shaped bubble embedded in another fluid of the same density in an oblique velocity field. The grids are rectangular Cartesian, the base grid having 100 × 100 cells and we kept the Courant number C fixed at 0.20 in most calculations. Figs.(1)-(3) show the shape of the bubble at the end of the simulation under various conditions. Fig.(1) shows the effect of the construction method for αf , Fig.(2) the effect of the time-step, represented by the Courant number, for CICSAM, while Fig.(3) illustrates grid density effects. CICSAM performs better than the other schemes, but has a strong dependence on the Courant number and grid refinement. Also, CICSAM results show a flow trail, which is not visible for Courant independent schemes as QUICK. 3.2 2D and 3D Dam-Break Problem We consider here both the original 2D dam-break tests by Martin&Moyce8 and the 3D dam-break with an obstacle tested at MARIN, and described in the work by Kleefsman4 . Wall-type boundary conditions are used: Dirichlet for velocity and Neumann for pressure and water volume fraction. Rectangular Cartesian grids with 55 × 33, 100 × 60 and 200 × 120 are used for the 2D problem, together with time steps of ∆t = 0.001, 5 × 10−4 and 2.5 × 10−4 s. The initial water height 2a is used as a reference length. For the 3D test case grids with 16058, 143696 and 1027712 elements and time steps of ∆t = 2 × 10−3 , 1 × 10−3 , 5 × 10−4 s were adopted. Fig.(4) shows that for the 2D problem a good correlation with experimental data and COMFLOW results is obtained, while the grid resolution has little effect. For the 3D problem the computed time-evolution of the pressure at two locations on the obstacle is compared with measurements and COMFLOW results. For the fine and medium grids, F RE SC O results agree satisfactorily with the experiments. 4 CONCLUSIONS We have seen in our exercises a quite acceptable performance of the CICSAM scheme. It would seem natural to adopt CICSAM as the preferred scheme in F RE SC O. However, all the interface capturing schemes mentioned have the curious property that the discretization is dependent on the local Courant number. It is well known that the Courant number plays an essential role in the discretization for explicit time integration in the transport equation of α, i.e. when the new time-level n appears only in the ∂/∂t term, not in the advection part of the equation. Stability analysis then shows that the time step must be 2
M. Hoekstra, G. Vaz, B. Abeil and T. Bunnik
QUICK
HRIC
IDEAL Ideal
CICSAM
Figure 1: Convection schemes behaviour.
C= 0.10
C= 0.20
C=Ideal 0.60
C= 0.40
Figure 2: Courant number C influence on CICSAM behaviour.
Grid 50x50
Grid 100x100
Grid 200x200
Figure 3: Grid influence on CICSAM behaviour. C = 0.20.
chosen so as to keep the Courant number below 1. For implicit time integration schemes such a Courant number limitation does not exist; the time step must be chosen to get a certain accuracy, but stability of the scheme is guaranteed also for time steps yielding high Courant number. While CICSAM, HRIC and BICS are all designed for implicit time integration, it is unexpected that the discretization depends on the Courant number. Indeed, the implication is that when solving a steady free-surface problem, the discretization and thus the result will depend on the choice of the time step, which is highly undesirable. If a steady flow is solved in a transient process, the time becomes a quasi-time, and the final solution obtained with any convenient choice of the quasi-time step should not depend on that choice. Further study of the building blocks of CICSAM revealed that this scheme is founded on mistakes and misunderstandings. In Ubbink’s work the transition to so-called normalised variables is erroneous, but brings the scheme - apparently not recognised by the developer - in the field of explicit time integration methods. Ultimate QUICKEST9 , a corner stone of CICSAM, has been designed for explicit time integration, while Ubbink advocates the implicit Crank-Nicolson scheme for CICSAM. We have concluded that CICSAM is a proper scheme in the context of explicit time integration methods, but is not the method of choice for implicit time integration. The Courant-number dependence of HRIC and BICS is less, for both schemes allow time steps giving a local Courant number above 1. But the dependence is still there, which is essentially wrong. This has led us to construct a Courant-number independent free-surface capturing scheme. A full description of this scheme and a demonstration of its performance is left for another paper.
3
M. Hoekstra, G. Vaz, B. Abeil and T. Bunnik
1.2
4 55x33 100x60 200x120 ComFLOW Experiments
1
3.6 3.2 2.8
r/a
h/(2a)
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0.4
55x33 100x60 200x120 ComFLOW Experiments
1.6 1.2
0.2 0
1
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0
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4
t(g/a) t(2g/a) Figure 4: 2D Dam-break validation results. Water height (left) and water-front position (right). 10 14
Sensor 1
Sensor 2
12
8
p [kPa]
10 8
p [kPa]
Coarse Medium Fine ComFLOW Experiments
6
Coarse Medium Fine ComFLOW Experiments
6 4
4 2 2 0
0
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t [s]
0
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t [s]
Figure 5: 3D Dam-break validation results. Pressure at two different positions in front of the obstacle.
REFERENCES [1] D. Schmode and D. Hafermann. Comparison of Wind Tunnel Measurements and Computations with RANS-Solver FreSCo, NuTTS Symposium, Varna, Bulgaria (2005). [2] O. Ubbink. Numerical Prediction of Two-Fluid Systems with Sharp Interfaces, PhD thesis, Imperial College of Science, Technology and Medicine, London (1997). [3] S. Muzaferija and M. Peri´c. Computation of Free-Furface Flows using Interface-Tracking and Interface-Capturing Methods, Comp. Mech. Publications, in Nonlinear Water Wave Interaction (1998). [4] T. Kleefsman. Water Impact Loading on Offshore Structures, PhD thesis, University of Groningen, The Netherlands (2005). [5] G. Deng, P. Queutey, M. Visonneau. Report on Numerical Free-Surface Modelling, Deliverable D1.1.5.1 of the Virtue Project (2006). [6] B. Abeil. Validation of a RANS Code in the Handling of Free-Surface Flows, MSc thesis, Chalmers University, Gothenburg, Sweden (2007). [7] M. Rudman. Volume-Tracking Methods for Interfacial Flow Calculations, International Journal for Numerical Methods in Fluids, Vol. 24 (1997). [8] J. Martin and W. Moyce. An Experimental Study of the Collapse of Liquid Columns on a Rigid Horizontal Plate, Philos. Trans. Roy. Soc. London, Vol. 244 (1952). [9] B. Leonard. The ULTIMATE Conservative Difference Scheme Applied to Unsteady OneDimensional Advection, Computer Methods in Appl. Mech. and Eng., Vol. 88 (1991). 4
