Huang, L

Li, M

Thomas, G

Tukovic, Z

UCL Discovery

The 13th OpenFOAM Workshop (OFW13), June 24-29, 2018, Shanghai, ChinaSIMULATION OF THE HYDROELASTIC RESPONSE OF A FLOATING ICE SHEETLUOFENG HUANG1, MINGHAO LI2, ZˇELJKO TUKOVIC´3, GILES THOMAS41University College London, UK, ucemlhu@ucl.ac.uk2Chalmers University of Technology, Sweden, limi@student.chalmers.se3University of Zagreb, Croatia, Zeljko.Tukovic@fsb.hr4University College London, UK, giles.thomas@ucl.ac.ukKeywords: Fluid-Structure Interaction, Hydroelasticity, Sea Ice, Ocean Surface Wave, OpenFOAMIntroductionIn the polar region, ice sheets can be several kilometres long and subjected to the effects of ocean waves. As its thicknessto length ratio is very small, an ice sheet may experience significant localised vibrations under continuous wave excitation.In such situations, the vibratory response of the large ice sheet is dominated by an elastic deformation rather than rigidbody motions, which is also known as the hydroelastic response of sea ice. A review on this phenomenon has been givenby Squire [1], indicating that understanding it fully will be a key challenge of polar engineering. The hydroelastic wave-ice interaction can induce wave reflection and transmission [2], which makes the wave field nearby very different fromthat in the open ocean and can lead to a considerable influence on any adjacent structures and/or passing vessels.Previous studies on the hydroelasticity of sea ice have mainly been conducted analytically. Fox and Squire [2] consideredthe reflection and transmission of waves from open water to an ice sheet. They adopted the method of eigenfunctionexpansion for the velocity potentials of the water, and used the conjugate gradient method to impose continuity at theinterface and edge condition. The reflection and transmission coefficients of the incident waves were obtained and foundto be dependent on the wave period, ice thickness and water depth. A similar analytical study was also conducted byChung and Fox [3] using the Wiener-Hopf method.The analytical works have provided great insight into this problem. However, in order to formulate the case inmathematics, they were built upon specific ideal assumptions, where the fluid viscosity and non-linear behaviours wereneglected. This limitation makes an analytic analysis insufficiently realistic and accurate, which motivates an improvementof developing a numerical model that has the ability to obtain a higher-order solution and capture the phenomena that havenot been included in the analytical models.Fluid-Structure Interaction (FSI) simulation through Computational Fluid Dynamics (CFD) is an advanced numericalmethod to model solid deformations induced by fluid, but has not to date been applied to ice hydroelasiticity. To developa reliable CFD tool to fill this gap, Tukovic et al. [4] designed an FSI package using the opensource CFD platformOpenFOAM, and Huang [5] extended this package into multiphases so that it can model the hydroelastic problem wherethe floating structure is in contact with both air and water.As shown in Figure 1, this work will use the the developed code [5] to simulate the hydroelastic response of a floating icesheet, and assess its accuracy by comparing the computational results with the experimental data of Sree et al. [6]. Theinfluence of environmental variables will also be investigated, including the incident wave period, ice thickness and waterdepth. Furthermore, the wave field affected by the ice sheet will be compared with its open water counterpart.Figure 1: Schematic of the case: a thin ice sheet is floating on the water surface and subjected to incoming waves, with itselastic deformation induced. Three probes (A, B and C) are set at 0.1 m, 0.3 m, and 0.5 m inside the ice edge respectively tomeasure the local vertical vibration, and two other probes (D and E) are set at 0.1 m and 0.3 m outside the ice edge respectivelyto measure the free surface elevation.213The 13th OpenFOAM Workshop (OFW13), June 24-29, 2018, Shanghai, ChinaNumerical ApproachA partitioned FSI scheme is applied to simulate the hydroelastic case, where the fluid and solid sides are solved separatelyand coupled via the fluid-solid interface. The fluid side is solved through the Reynolds-averaged Navier-Stokes (RANS)equations together with the Volume of Fluid (VOF) method. On the solid side, the ice deformation is solved accordingto the Saint-Venant Kirchoff constitutive model with the plane strain assumption. An extended introduction of this FSIscheme has been given by Tukovic et al. [7]. The solution procedure of this work is illustrated in Figure 2, with the detailsbeing explained in the steps outlined below:• To start a new time step loop, the structural displacement is first updated according to the results of the previous timestep. Then, to keep the fluid mesh in accordance with the solid mesh, the Aitken coupling scheme is employed, whichintroduces an Aitken Relaxation Factor (ARF ), as defined in Equation (1).ARF− = ARF ×∑(Rest=n−1 ·∆Res)∑(∆Res ·∆Res) (1)Thus the ARF is updated according to the residual (Res), which is the difference between the structural interfacedisplacement (SID) and the fluid interface displacement (FID), namely Res = SID − FID. Afterwards, the fluidmesh is adjusted with the updated ARF value, as in Equation (2).FluidMesh+ = ARF ×Res (2)• The FID is extracted from the adjusted fluid mesh. Then its differential produces the velocity of the fluid interfaceand the mesh motion of the rest fluid region is obtained according to this interface velocity.• Based on the moved mesh, the fluid solver calculates the velocity and pressure field. Meanwhile, the pressure andviscous force on the fluid interface can be obtained.• The fluid load on the fluid interface is transferred to the solid interface, considered as the fluid load on the structure.• According to the load on the interface, the structure solver calculates the displacement of the structure.• The SID can be extracted from the structural displacement and then compared with the FID to obtain a new residual,Res. The solver switches to the next time step when either the residual criteria is satisfied or the pre-defined maximumFSI iteration time has been reached, otherwise it continues looping in the current time step.Figure 2: Flow chart of the FSI solution procedure.Case setupThe general setup of the simulation case is shown in Figure 1. The case is set as a two-dimensional rectangularcomputational domain. It is filled with fresh water of density ρw = 1000 kg m −3 to a depth of h = 0.3 m. On thetop of the domain, a static pressure boundary condition is applied to model the atmosphere. The bottom boundary isdefined as a no-slip wall to account for the presence of the seabed. Using the waves2Foam tool [8], regular waves ofamplitude a = 0.0085 m and period T = 0.7 s are constantly generated at the inlet boundary without any current speed,and a wave absorption zone is set up in front of the outlet boundary to avoid waves being reflected here.A thin ice sheet is initialised as one side free floating on the water surface and the other side fixed, with its length L of1 m and thickness t of 0.01 m. The ice rheology is set as density ρ = 910 kg m −3, Young’s modulus E = 870 MPa and214The 13th OpenFOAM Workshop (OFW13), June 24-29, 2018, Shanghai, ChinaPoisson ratio ν = 0.3. These properties may be varied to study their influence. Three probes (A, B and C) are set at 0.1 m,0.3 m, and 0.5 m inside the ice edge respectively to measure the local vertical vibration, and two other probes (D and E)are set at 0.1 m and 0.3 m outside the ice edge respectively to measure the free surface elevation.Results and discussionTo validate the proposed model, the vertical vibrations at the point A, B, C were computed and compared with theexperimental data of Sree et al. [6]. In spectral analysis, both the computational results (CFD) and the experimentalresults (Exp.) are plotted in Figure 3, in which the proposed model reveals very good agreement with the experimentalmeasurements at all the three points. The main oscillating frequency of the ice sheet (where the peaking amplitude occurs)equals to the frequency of the incident wave, which means the vertical oscillation of the ice has approached a steady state.Besides, the peaking amplitude gets smaller from the ice edge to the middle section of the ice (from point A to C).To investigate the influence of environmental variables on the hydroelastic response of the ice sheet, the ratio of thelocal vertical oscillation amplitude (A) to the incident wave amplitude (a) was calculated and plotted out, as a function ofdifferent incident wave period, ice thickness and water depth, in Figure 4, 5 and 6 respectively. Generally, the deformationof the ice sheet is found to increase as the wave period is increased, and decrease with increasing ice thickness. However,the variation is undistinguished when the water depth is changed, with a slight increase in the shallow water situation andnearly constant in deep water.Figure 4 shows the ice deformation reducing dramatically when the incident wave period is less than 0.5 s, which isbecause the waves become very short then, corresponding to a wavelength of less than 0.4 m. As the wave period increases,the vibration amplitude of the ice can approach the incident wave amplitude. Figure 5 shows the ice deformation becomesvery weak when the ice thickness increases to a level of the incident wave height, and when the thickness keeps decreasing,the oscillating amplitude can be larger than the ice thickness.The ice sheet is found to have a significant influence on the wave field, where there is a superposition of the ice-reflectedwaves and the incident waves. The free surface elevation at points D and E is plotted in Figure 7, alongside a comparisonwith the open water situation of no ice. It shows that the existence of the ice sheet weakened the waves at point D butinduces stronger waves at point E, which suggests the ice sheet can influence the wave field to a large degree and theinfluence is dependent on the location.ConclusionsThis work provided an available CFD code to simulate the wave-induced FSI problems. After validation againstexperiments, the code reveals very good accuracy on predicting the hydroelastic response of a floating ice sheet. Itcan also be applied to other deformable floating bodies, not just the sea ice case.AcknowledgmentsThe code was developed during the OSCFD project organised by Ha˚kan Nilsson, and it is publicly downloadable at http://dx.doi.org/10.17196/OS_CFD#YEAR_2017, with a report [5] including its detailed development process. Inthis manner, we hope to promote the development of opensource CFD tools.References[1] V. A. Squire, “Past, present and impendent hydroelastic challenges in the polar and subpolar seas,” PhilosophicalTransactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 369, no.1947, pp. 2813–2831, 2011.[2] C. Fox and V. A. Squire, “Reflection and transmission characteristics at the edge of shore fast sea ice,” Journal ofGeophysical Research: Oceans, vol. 95, no. C7, pp. 11 629–11 639, 1990.[3] H. Chung and C. Fox, “Calculation of wave-ice interaction using the wiener-hopf technique,” New Zealand J. Math,vol. 31, no. 1, pp. 1–18, 2002.[4] Z. Tukovic, P. Cardiff, A. Karac, H. Jasak, and A. Ivankovic, “Openfoam library for fluid structure interaction,” in9th OpenFOAM Workshop, vol. 2014, 2014.[5] L. Huang, “An opensource solver for wave-induced fsi problems,” In Proceedings of CFD withOpenSource Software,Tech. Rep., 2018.[6] D. K. Sree, A. W.-K. Law, and H. H. Shen, “An experimental study on the interactions between surface waves andfloating viscoelastic covers,” Wave Motion, vol. 70, pp. 195–208, 2017.[7] Z. Tukovic, A. Karac, P. Cardiff, H. Jasak, and A. Ivankovic, “Openfoam finite volume solver for fluid-solidinteraction,” Tractions of FAMENA, 2018.[8] N. G. Jacobsen, D. R. Fuhrman, and J. Fredsøe, “A Wave Generation Toolbox for the Open-Source CFD Library:OpenFoam R©,” International Journal for Numerical Methods in Fluids, vol. 70, no. 9, pp. 1073–1088, 2012.215The 13th OpenFOAM Workshop (OFW13), June 24-29, 2018, Shanghai, ChinaFigure 3: Spectral analysis (discrete Fourier transform,| Y (σ) |) of the vertical vibration at different locations alongthe ice sheet: a comparison between computational andexperimental results.Figure 4: Local vibrations of the ice sheet, as a function of theincident wave period (T ).Figure 5: Local vibrations of the ice sheet, as a function of theice sheet thickness (t).Figure 6: Local vibrations of the ice sheet, as a function of thewater depth (h).Figure 7: Free surface elevation at different locations outsidethe ice sheet: a comparison between the computational resultsand corresponding open water counterpart.216

Simulation of the hydroelastic response of a floating ice sheet

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Simulation of the hydroelastic response of a floating ice sheet

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