Jasak, Hrvoje

Malenica, Šime

Vukčević, Vuko

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Assessment of Higher–Order Forces on a VerticalCylinder with Decomposition Model Based onSWENSE MethodVuko Vukcˇevic´1, Hrvoje Jasak1 and Sˇime Malenica21University of Zagreb, Zagreb/Croatia, {vuko.vukcevic, hrvoje.jasak@fsb.hr}2Bureau Veritas, Paris/France, sime.malenica@bureauveritas.com1 SummaryOff–shore structure design often relies on accurate assessment of higher–order forces because ofringing phenomena. In this work, calculation of higher–order forces on a vertical surface piercingcylinder is carried out by a decomposition CFD model based on Spectral Wave Explicit Navier–Stokes (SWENSE) approach [1]. First, mesh and time refinement studies are carried out for acase with intermediately steep (ka = 0.12) incoming wave train. Second, a set of waves withdifferent frequencies are simulated, keeping the wave steepness constant, ka = 0.06. Higher–order forces up to fourth order are compared with fully non–linear potential flow results [2]. Adiscussion on strong second order behaviour of vorticity effects in the vicinity of the cylinder ispresented.2 IntroductionCFD models based on Reynolds–Averaged Navier–Stokes (RANS) equations are progressivelygaining attention in the marine industry [4] due to increase in computational resources andtheir ability to tackle non–linear, coupled equation sets. However, verification and validationof RANS models is still under–way [3], both for steady and transient phenomena. Transientsimulations demand more CPU time compared to steady state simulations, making them moreexpensive to verify in terms of grid refinement and other sensitivity studies.In this paper, we shall simulate wave diffraction problem in regular waves and assess thehigher–order forces exerted on a vertical surface–piercing cylinder with a recently developeddecomposition model based on SWENSE [1] method. The model is implemented in foam–extend, a fork of OpenFOAM computational continuum mechanics software. Validation of themodel shall be achieved by comparing the results with fully non–linear potential flow, time–domain simulation by Ferrant et al. [2]. In order to verify the presented method, time–step andmesh refinement studies will be carried out for a representative test case.3 ApproachThe mathematical model of the incompressible, viscous, two–phase flow is modelled with mo-mentum (Navier–Stokes) and phase continuity equations. Interface capturing is achieved withimplicitly redistanced Level Set method [5] reformulated in a form suitable for Finite Volume(FV) discretisation [6]. The method does not need additional explicit redistancing. In orderto efficiently model incoming waves, solution is decomposed into incident and diffracted fieldsfollowing Ferrant et al. [1]. The method presented here is more general in a sense that diffractedfield may contain additional phenomena not taken into account by incident wave field, such as:viscosity, vorticity, two–phase effects, radiation effects, etc. Hence, perturbation around inci-dent wave field is calculated via fully non–linear, viscous, two–phase equation set in the presentCFD model. As a consequence, incident wave field is explicit in the CFD calculation, facilitatingwave transport [7]. Governing equations are briefly presented, the reader is referred to Vukcˇevic´and Jasak [6] for detailed derivation:• Volumetric continuity equation:∇•uD = −∇•uI , (1)where uD is the diffracted velocity field, and uI is the incident wave field. We stress thatindex D denotes perturbation around incident (I) field, modelling vorticity, viscosity,diffraction and possibly radiation effects.• Momentum equation:∂(ρuD)∂t+∇•(ρuuD)−∇•(µeff∇uD) = −∂(ρuI)∂t−∇•(ρuuI) +∇•(µeff∇uI)−∇pd − g•x∇ρ+∇u•∇µeff + σκ∇α ,(2)where ρ is the mixture density, µeff effective viscosity of the mixture, pd dynamic pressure,g gravitational acceleration, x position vector, σ surface tension coefficient, κ curvatureof the free surface and α is the volume fraction.• Level Set equation derived from Phase Field method [5]:∂ψD∂t+∇•(cψD)− ψD∇•c− b∇• (∇ψD) =− ∂ψI∂t−∇•(cψI) + ψI∇•c + b∇• (∇ψI) + b√2tanh(ψ√2),(3)where ψ denotes Level Set field, c convective flux [6], b stabilisation parameter and denotes smearing distance.Wave reflection is prevented using implicit relaxation zones [6], where the diffracted field isforced to vanish near the boundaries, leaving full potential flow solution.Numerical discretisation of governing partial differential equations, (1), (2) and (3) is achievedwith second–order accurate, polyhedral FV method in foam–extend. Terms related to incidentwave field are discretised explicitly as they are known at each time–step, while diffracted fieldsare solved in implicit manner. Temporal discretisation is achieved via blend of Euler and Crank–Nicholson schemes. Convective terms are discretised either with second–order accurate linearupwind scheme (for momentum) or with Van Leer’s TVD scheme (other convective terms). Dif-fusion terms are discretised using central differencing. Pressure–velocity–free surface couplingis achieved with a PIMPLE aglorithm, a combination of SIMPLE and PISO algorithms.4 Cylinder diffractionDiffraction of a 3–D vertical, surface piercing cylinder due to regular wave train is simulated. Adeep water wave with intermediate steepness ka = 0.12 (Table 1 presents other geometrical andwave properties) is chosen as a benchmark test case. Block–structured mesh extends two wavelenghts in the radial direction, while the symmetry plane is used to lower the CPU time. Thecells are heavily graded towards the free surface and the cylinder. There are 120 cells per wavelength in the area near the cylinder and 20 cells per wave height, while the maximum aspectratio is 166.7. The mesh consists of 552 000 cells.Table 1: Geometry and wave parameters for the benchmark case.Cylinder radius r, m 1Wave number–depth parameter kd 8Wave number–radius parameter kr 0.2Wave amplitude–depth parameter a/d 0.015Wave steepness parameter ka 0.124.1 Time resolution studyTime resolution study has been carried out by chaning the Courant–Friedrichs–Lewy (CFL)number from 5 (≈ 140 time–steps per period) to 0.125 (≈ 1600 time–steps per period). 10periods were simulated and the frequency domain analysis has been made via moving windowFFT. The statistically averaged inline force harmonics are calculated as the average of thelast 9 periods. The first period was excluded due to simulation start up. In–line forces arepresented in dimensionless form following Ferrant et al. [2]. ρ is set to 1000 kg/m3 and g is9.81 m/s2. Results of the time resolution study are presented in Figure 1. Figure 1a presentsthe normalized force harmonic coefficients for different CFL numbers, while Figure 1b presentsrelative errors compared to Ferrant et al. [2]. Relative errors for smallest CFL number arepresented in Table 2. First, third and fourth order forces have a relative error less than 3%.Steady second order drift force has the largest error, 16%. Second order force has the error ofapproximately 12%.0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5CFL number00.10.20.30.40.50.6|F i’||F2s’| - CFD|F2s’| - Ferrant et al.|F1’| /10- CFD|F1’|/10 - Ferrant et al.|F2’| - CFD|F2’| - Ferrant et al.|F3’| - CFD|F3’| - Ferrant et al.|F4’| - CFD|F4’| - Ferrant et al.(a) Normalized force harmonics.0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5CFL number-35-30-25-20-15-10-50510E r, %Er(|F2s’|)Er(|F1’|)Er(|F2’|)Er(|F3’|)Er(|F4’|)(b) Relative errors.Figure 1: Vertical cylinder diffraction: time refinement study.Table 2: Relative errors compared to Ferrant et al. (1999) [2] for CFL = 0.125 case.Er(|F ′1|),% Er(|F ′2s|),% Er(|F ′2|),% Er(|F ′3|),% Er(|F ′4|),%-2.75 -16.14 -12.39 -2.25 -1.004.2 Mesh refinement studyMesh refinement study has been carried out using four meshes. Finer meshes have 25%, 50%and 75% more cells in each direction compared to the benchmark case. Same settings are usedas in the previous subsection, while the CFL number was fixed at 0.25. Table 3 presents meshrefinement details. Figure 2a presents convergence of normalized force harmonics with respectto mesh refinement, while Figure 2b presents relative errors of higher order forces. It can beseen that the relative errors for higher order forces, namely third and fourth order forces exhibitoscillatory behaviour. Table 4 presents relative errors for the finest mesh.Table 3: Mesh refinement study parameters.Mesh refinement index 0 1 2 3Cell count increase, % 0 25 50 75Total cell count 552 000 1 005 700 1 854 000 2 964 5000 1 2 3 4Mesh refinement index00.10.20.30.40.50.60.7|F i’||F2s’| - CFD|F2s’| - Ferrant et al.|F1’|/10 - CFD|F1’|/10 - Ferrant et al.|F2’| - CFD|F2’| - Ferrant et al.|F3’| - CFD|F3’| - Ferrant et al.|F4’| - CFD|F4’| - Ferrant et al.(a) Normalized force harmonics.0 1 2 3Mesh refinement index-20-15-10-50510E r, %Er(|F2s’|)Er(|F1’|)Er(|F2’|)Er(|F3’|)Er(|F4’|)(b) Relative errors.Figure 2: Vertical cylinder diffraction: mesh refinement study.Table 4: Relative errors compared to Ferrant et al. (1999) [2] for the finest mesh.Er(|F ′1|),% Er(|F ′2s|),% Er(|F ′2|),% Er(|F ′3|),% Er(|F ′4|),%-1.82 -4.33 -9.80 -5.35 -1.224.3 Wave number studySimulations were performed for a wave with a steepness of ka = 0.06, while Table 5 presentsother geometrical and wave parameters. A different mesh is used for each case, such that thenumber of cells per wave height and length remains the same as in the base case. Figure 3presents normalized forces for different wave number–radius parameter kr. Black, red and bluelines represent real, imaginary and absolute value of the harmonic, respectively. Observing realand imaginary parts, slight phase shift difference is revealed. The phase shift is larger for higherwave numbers, while the absolute values of the harmonics are in good agreement for all orders.Table 5: Geometry and wave parameters for the wave number study.Cylinder radius r, m 1Wave number–depth parameter kd 8Wave amplitude–depth parameter a/d 0.0075Wave steepness parameter ka 0.06Wave number–radius parameter kr 0.05, 0.01, 0.15, 0.20, 0.25 and 0.300 0.05 0.1 0.15 0.2 0.25 0.3 0.35kr-6-4-20246F 1’Re(F1’) - CFDIm(F1’) - CFD|F1’| - CFDRe(F1’) - Ferrant et al.Im(F1’) - Ferrant et al.|F1’| - Ferrant et al.(a) Normalized first order harmonics.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35kr-0.6-0.4-0.200.20.40.6F 2’Re(F2’) - CFDIm(F2’) - CFD|F2’| - CFDRe(F2’) - Ferrant et al.Im(F2’) - Ferrant et al|F2’| - Ferrant et al.(b) Normalized second order harmonics.0 0.05 0.1 0.15 0.2 0.25 0.3kr-0.6-0.4-0.200.20.40.6F 3’Re(F3’) - CFDIm(F3’) - CFD|F3’| - CFDRe(F3’) - Ferrant et al.Im(F3’) - Ferrant et al.|F3’| - Ferrant et al.(c) Normalized third order harmonics.0 0.05 0.1 0.15 0.2 0.25 0.3kr-0.4-0.3-0.2-0.100.10.20.30.4F 4’Re(F4’) - CFDIm(F4’) - CFD|F4’| - CFDRe(F4’) - Ferrant et al.Im(F4’) - Ferrant et al.|F4’| - Ferrant et al.(d) Normalized fourth order harmonics.Figure 3: Vertical cylinder diffraction: wave number study.4.4 Vorticity effectsVorticity effects are assessed by measuring enstrophy at three probes positioned closely in thefront, at the side and in the back of the cylinder. Figure 4a presents enstrophy variation in timefor three probes. The highest amount of vorticity is present at the side of the cylinder, whichmight be related to orthogonality of incident and diffracted waves in that region. Figure 4bpresents same signals in frequency domain. Apart from the mean value, enstrophy at eachprobe exhibits significant second order behaviour. It was also observed that the magnitude ofthe enstrophy increases for higher frequencies: kr = 0.05 case measured enstrophy one orderof magnitude smaller than the kr = 0.3 case. As the potential flow model does not take intoaccount vorticity effects, authors believe that these effects are the reason for larger discrepancyfor second order forces, especially for cases with higher frequency.0 5 10 15 20 25 30 35t, s0100200300400500ε n, s-2Front probeSide probeBack probe(a) Time domain.0 1 2 3 4 5 6 7 8 9ω, rad/s050100150200F(ε n), s-2Front probeSide probeBack probe(b) Frequency domain.Figure 4: Enstrophy in the vicinity of the cylinder, kr = 0.3, ω = 1.718 rad/s case.4.5 ConclusionDecomposition model based on SWENSE method, implicitly redistanced Level Set and implicitrelaxation zones is briefly presented in this paper. Validation and verification of the modelis carried out by assessing higher order forces on a circular, surface piercing cylinder. Timeresolution and mesh refinement studies have been carried out for an intermediately steep waveka = 0.12, showing good convergence. A set of simulations with different wave frequencies hasbeen carried out. All results show satisfactory agreement compared to fully non–linear potentialflow solution. A qualitative study of vorticity effects near the cylinder is also presented, revealingstrong second order behaviour. Authors believe that vorticity effects could explain larger relativeerrors for second order forces, as the enstrophy is related to dissipation of the kinetic energy.References[1] P. Ferrant, L. Gentaz, and D. Le Touze´. A new RANSE/Potential Approach for WaterWave Diffraction. In Proc. Numerical Towing Tank Symposium, NuTTS, September 2002.[2] P. Ferrant, Sˇ. Malenica, and B. Molin. Nonlinear wave loads and runup on a vertical cylinder.In Advances in Fluid Mechanics, Nonlinear Water Wave Interaction. WIT Press, 1999.[3] L. Larsson, F. Stern, and M. Vissoneau. Numerical Ship Hydrodynamics: An assessment ofthe Gothenburg 2010 workshop. Springer, 2013.[4] F. Stern, J. Yang, Z. Wang, H. Sadat-Hosseini, M. Mousaviraad, Bhushan S., and T. Xing.Computational Ship Hydrodynamics: Nowadays and Way Forward. In Proceedings of the29th Symposium on Naval Hydrodynamics, pages 1–73, August 2012.[5] Y. Sun and C. Beckermann. Sharp interface tracking using the phase–field equation. J.Comput. Phys., 220:626–653, 2007.[6] V Vukcˇevic´ and H. Jasak. Decomposition Model for Naval Hydrodynamic Applications,Part I: Computational Method. Submitted to Ocean Engineering, ??:???–???, 2015.[7] V. Vukcˇevic´, H. Jasak, and Sˇ. Malenica. Solution and Domain Decomposition Model forNaval Hydrodynamics: RANS and Potential Flow Coupling. In VI International Conferenceon Computational Methods in Marine Engineering (MARINE 2015), pages 903–918, June2015.

Assessment of Higher–Order Forces on a Vertical Cylinder with Decomposition Model Based on SWENSE Method

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