A COUPLED RANS-VOF AND FINITE ELEMENT MODEL FOR WAVE INTERACTION WITH HIGHLY FLEXIBLE VEGETATION

Transcription

1 A COUPLED RANS-VOF AND FINITE ELEMENT MODEL FOR WAVE INTERACTION WITH HIGHLY FLEXIBLE VEGETATION Haifei Chen 1, Qingping Zou * an Zhilong Liu 1 This paper presents a couple wae-egetation interaction moel suitable for ery flexible egetation with large eflections. The wae hyroynamics is moele by a Naier-Stokes flow soler along with a Volume of Flui surface capturing metho. The goerning equation for flexible egetation motion is sole by a Finite Element Metho using a semi-implicit time ifferencing scheme. The coupling between wae hyroynamics an egetation motion is achiee using an immerse bounary metho. The moel is aliate with experimental measurements for a single-stem egetation an a large-scale egetation patch in a wae flume. Keywors: wae interaction with egetation; flexible egetation; large eflection; IHFOAM; OpenFOAM; RANS- VOF; finite element metho; immerse bounary metho INTRODUCTION Vegetation plays an important role in protecting natural shoreline against storm surge an waes. Majority of preious stuies of wae-egetation interaction focus on fixe egetation. Li & Yan (2007) an Marsooli & Wu (2014) consiere rigi egetation in their Reynols Aerage Naier-Stokes (RANS) moels an neglecte the swaying motion of the egetation. Maza et al. (2013) use a twoimensional RANS moel couple with a submerge egetation moel, which soles only the isplacement at the top of each stem an assumes a linear ariation of eflection along the stem. Zhu an Chen (2015) couple a non-hyrostatic phase resoling wae moel, NHWAVE, with a Finite Element Metho (FEM) base egetation moel. The complete force balance equation for the egetation motion was sole. Their egetation moel is suitable for small eflections only. The objectie of this stuy is to eelop a new couple wae an egetation moel suitable for flexible egetation with large sway of motion. A Reynols Aerage Naier-Stokes soler with Volume of Flui surface capturing scheme (RANS-VOF) is extene to inclue the effect of flexible egetation on flow. The egetation is assume to behae like a slener ro, whose goerning equations are sole by Finite Element Metho (Garrett 1982; Chen et al. 2011). There is no constraint with regar to the magnitue of the egetation motion in this couple moel as with most preious wae-egetation moels. NUMERICAL MODEL FEM Moel for Flexible Vegetation Figure 1. Coorinate system of the ro moel for flexible egetation. r(s,t) is a position ector of the ro s centerline, s is the arc length, an q(s,t) is istribute force per unit length. 1 Ciil an Enironmental Engineering, Uniersity of Maine, 5711 Boarman Hall, Orono, ME 04473, USA. * qingping.zou@maine.eu 1

2 2 COASTAL ENGINEERING 2016 The flexible egetation is treate as a slener ro. The goerning equation for the egetation motion consiers the buoyancy, stiffness, an graity as restoring forces, rag an inertia as riing forces, an is gien by EIr r q m r (1) where r s,t is the position ector of the ro, the prime an ot enote the ifferentiation with respect to arc length s an time t, respectiely, EI is bening stiffness, m mass per unit length, 2 s, t T EI with T being the local effectie tension an r ro curature. The external e e istribute loa on the egetation, q w F s F, is the sum of self-weight, w, hyrostatic force, s an hyroynamic force, F, acting on a unit length of single-stem egetation. The hyroynamic force on a single-stem egetation is gien by the Morrison equation, 1 n n n n n n F U r U r U 1 2 C b b t C b C U r (2) D M A 2 4 where C, C, an C D M A are the rag, inertia, an ae mass coefficients, is the ensity of the egetation, b, t are the with an thickness of the rectangular cross-section of the egetation blae, U, U are flui elocity an acceleration, r, r are the ro elocity an acceleration, the superscript n inicates the normal component of the ariable perpenicular to the ro. FEM is use to iscretize the goerning equations. For static problems such as egetation reconfiguration uner current action, the Newton-Raphson metho is use to sole the equations. For ynamic problems where transient wae motion is inole, a semi-implicit time ifferencing scheme is use. Reaers shoul refer to Garret (1982) an Ran (2000) for more etails. RANS-VOF Wae Moel The Reynols Aerage Naier-Stokes Soler couple with a Volume of Flui surface capturing scheme has been use extensiely to sole free surface flow problems with large an complex eformations of free surface (Lin an Liu 1998; Lubin et al. 2006; Christensen 2006; L et al. 2009; Wang et al. 2009; Zou an Peng 2011). A Naier-Stokes soler built on the OpenFOAM platform is aopte to simulate the wae hyroynamics. The open-source package, IHFOAM, can actiely generate an absorb waes at the inlet an outlet bounaries (Higuera et al. 2013). The Volume of Flui (VOF) metho is use to capture the air-water interface. The goerning equations for the RANS-VOF soler are gien by U 0 (3) U UU U eff p * g X U eff F (4) t U U 1 r 0 (5) t * where is the flui ensity, p the pseuo-ynamic pressure, g the graitational acceleration, X the position ector, is the effectie ynamic iscosity, which takes into account of molecular eff t ynamic iscosity an the turbulent ey iscosity. Note that the egetation-inuce t hyroynamic forces, F, are ae as a source term in the momentum equation (4). Eq. (5) is the aection equation for the olume of flui function,, with an extra term to achiee a sharp interface (Weller 2002). Coupling Wae an Vegetation Moels Using an Immerse Bounary Metho The egetation motion, Eq. (1), is sole by Finite Element Metho consiering rag an inertia as riing forces, while the RANS-VOF wae moel consiers the rag an inertia as a source term in the momentum equation (4). The two moels are hence couple through the egetation-inuce hyroynamic forces. The common coupling approach use in the literature is base on the porous meia flow theory, which consiers the egetate region as a porous meia continuum an is thus concerne with only the olume-aerage flow fiel. Different from preious stuies using the porous meia approach, we use an immerse bounary metho suitable for a ery thin object to account for the two way coupling of wae an egetation (Perskin 2002). The force exerte by the egetation on the flow, -F, is incorporate as a source term F,

3 COASTAL ENGINEERING in the momentum equation (4), an is applie only whereer stans the egetation stem. The immerse bounary metho is a popular approach in simulating flui-structure interactions (Lin 2007; Zhang et al. 2009). Fig. 2 illustrates the flow chart of the proceure of soling the couple wae-egetation interaction moel. At each time step of the computation, the egetation motion is sole first with the known wae elocities. The egetation eflection an isplacement elocity are fe back to the wae moel to calculate new flow elocities. Both the flow elocity an egetation motion will then be fe to the egetation moel to obtain the new egetation eflection an elocity at the next time step. The turbulence transport equations may be sole at the en of each time step. Vegetation motion - FEM Next time step VOF aection PISO loop Momentum preictor Pressure correction Immerse bounary metho Turbulence Figure 2. Flow chart of soling the couple RANS-VOF flow moel an FEM egetation moel. RESULTS AND DISCUSSION In this section, both the iniiual moel components an the couple wae-egetation moel is aliate against the aailable experimental measurements. The IHFOAM wae moel has been aliate extensiely by Higuera et al. (2013). The FEM moel for flexible egetation will be first aliate using measurements for single-stem egetation. The couple moel is then aliate by a large-scale flume experiment of wae interaction with a egetation patch. Wae Interaction with a Single-stem Flexible Vegetation Abelrhman (2007) photographe Z. marina blaes expose to three ifferent current spees U c = 0.06, 0.12 an 0.14 m/s. Fig. 3 shows a comparison of the preicte an obsere blae postures for this aquatic egetation. The blae with an thickness are b =0.8 cm an t =0.35 mm, blae length 40 cm, an ensity =700 kg/m 3. The rag coefficient use is ientical to that for flat plates, C D =1.95. The preicte two blae postures are obtaine using the higher (E = 2.4 GPa) an lower (E = 0.4 GPa) boun alues of the elastic moulus for the blae. It is apparent that a higher elastic moulus correspons to a more upright configuration. Gien the uncertainty in the blae geometry an mechanical properties in the real egetation, it can be state that reasonable agreement is achiee between the preicte postures an the obserations.

4 4 COASTAL ENGINEERING 2016 Figure 3. Comparison of moel preictions for blae posture (left column) with the obserations (right column) by Abelrhman (2007) for the seagrass Zostera marina expose to a current of spee U c (a) 0.06 m/s, (b) 0.12 m/s, an (c) 0.14 m/s. Maza et al. (2013) conucte a set of experiments to measure the extent of plant motion uner wae action. The egetation mimics were compose of four PVC strips, one pair 45 cm long an another pair 27.5 cm. The PVC strips were 1 mm thick an 1 cm wie, with a Young's moulus E=0.9 GPa an a ensity =700 kg/m 3. Since the two-pair, four-stem egetation mimics respon freely to the wae action, it s ifficult to etermine the extent to which the four stems bin together to resist the wae action. To simplify the moel simulation, an equialent single-stem egetation is use an moel tests are thus use to etermine the geometric an mechanical properties of the equialent egetation. Figure 4. Instantaneous configuration of single-stem egetation at eery one tenth of the wae perio uner a perioic wae T=4.0 s, H=0.2 m, =0.8 m. Re squares correspon to the measurements of the two extreme isplacements uner the wae motion by Maza et al. (2013). Fig. 4 shows the preicte instantaneous configurations for the flexible egetation oer one wae perio. The rag coefficient use is ientical to that for flat plates, C =1.95. The bening stiffness D

5 COASTAL ENGINEERING use is equialent to that when approximately 2 PVC strips were bining together to resist the wae motion. The obsere two maximum isplacements of egetation in the same an opposite irection as the wae irection are also shown as re squares in Fig. 4. The present moel preictions are in goo agreement with the measurements. Due to the asymmetry of wae kinematics, the egetation flexibility an nonlinear wae-egetation coupling, the egetation motion is asymmetric relatie to the ertical axis, i.e. the egetation has a larger sway uner wae crest (~0.25 m) than uner trough (~0.17 m). Wae Interaction with a Patch of Flexible Vegetation The experimental measurements by Stratigaki et al. (2011) are use to aliate the couple waeegetation moel. Fig. 5 shows the present moel setup simulating a regular wae, T= 3.5 s an H= 0.5 m, propagating through a egetation meaow. The water epth is 2.4 m at the waemaker bounary an transitions, through a 12 m long, 1:17 sany slope, to 1.7 m at the onshore bounary. A 10.7 m long artificial egetation meaow was place oer the flat portion of the sany bottom. The beginning of this egetation patch was locate at m from the wae pale. The remaining sany sloping beach beyon the egetation patch is ignore in the simulation. The waes propagating through the egetation fiel are absorbe by using proper outlet bounary conitions. The egetation mimics use in this experiment were the same as escribe aboe (Maza et al. 2013). The egetation ensity for this special case is N=180 stems/m z (m) m : x (m) Figure 5. Moel setup for couple wae-egetation interaction (Stratigaki et al. 2011). Fig. 6 shows the spatial eolution of wae height oer the egetation patch. The x-coorinate of the egetation patch has been shifte to compare with experimental measurements easily. It is eient that the present moel reprouces the tren of wae height ecay as well as the magnitue of wae height oscillation oer the egetation fiel. Howeer, the oscillation pattern is shifte probably ue to the unknown wae reflection patterns in the physical flume not present in the numerical flume. Figure 6. Preicte an measure (Stratigaki et al. 2011) spatial ariation of wae height along the egetation patch. Fig. 7 presents the ertical profiles of horizontal an ertical wae elocity components at a location 2.0 m into the egetation patch. It s seen that the ertical istribution of wae elocities are

6 6 COASTAL ENGINEERING 2016 obiously altere by the presence of the egetation, see Fig. 8 for illustration. Both the numerical preictions an the measurements show that the horizontal elocity increases aroun the upper en of the egetation fiel. This skimming flow results from the strong iscontinuity in the rag force between the area occupie by the meaow an the free flow aboe it. The strong iscontinuity woul cause a non-zero wae shear stress, leaing to a mean current flowing in the irection of the wae propagation. Figure 7. Moel-ata comparisons of ertical istribution of maximum, mean, an minimum (a) horizontal an (b) ertical wae elocities at a location 2.0 m into the egetation patch. Soli lines: moel preiction; squares: measurements by Stratigaki et al. (2011). Figure 8. Snapshot of instantaneous configuration of the egetation patch an color contour of wae horizontal elocity at t=53 s. CONCLUSIONS A couple RANS-VOF an finite element moel is eelope to simulate the wae-egetation interaction. The wae hyroynamics is simulate by a RANS-VOF moel. A Finite Element Metho is aopte for soling the egetation motion, which is applicable to flexible egetation with either small or large eflections. The flexible egetation moel was aliate with experiments for single-stem egetation. The moels of wae hyroynamics an egetation motion is couple through a iffuse immerse bounary metho. The couple moel was aliate with measurements for a large-scale flume experiment, where a regular wae propagating though a egetation meaow was inestigate. Both the obsere wae height ecay an oscillation magnitue an wae kinematics were reasonably preicte by the couple moel. REFERENCES Abelrhman, M.A Moeling coupling between eelgrass Zostera marina an water flow. Mar. Ecol. Prog. Ser. 338, Chen, H., S. Xu, an H. Guo Nonlinear analysis of flexible an steel catenary risers with internal flow an seabe interaction effects. Journal of Marine Science an Application, 10(2),

7 COASTAL ENGINEERING Christensen, E.D Large ey simulation of spilling an plunging breakers. Coastal Engineering, 53 (5-6), Garrett, D.L Dynamic analysis of slener ros, Journal of Energy Resources Tech., 104(4), Higuera, P., J.L Lara, an I.J. Losaa Realistic wae generation an actie wae absorption for Naier-Stokes moels: application to OpenFOAM. Coast. Eng. 71, Li, C.W., an K. Yan Numerical inestigation of wae-current-egetation interaction. J. Hyraul. Eng., (2007)133:7(794), Lin, P.Z., an P.L.-F. Liu A numerical stuy of breaking waes in the surf zone. J. Flui Mech., 359, Lin, P.Z A fixe-gri moel for simulation of a moing boy in free surface flows. Computers & fluis, 36(3), Lubin, P., S. Vincent, S. Abaie, an J.P. Caltagirone Three-imensional large ey simulation of air entrainment uner plunging breaking waes, Coastal Engineering, 53, L, X., Q.P. Zou, D.E. Reee, an Y. Zhao A noel couple leel set an olume of flui metho for sharp interface capturing on 3D tetraheral gris, Journal of Computational Physics, 229(7), Marsooli, R., an W. Wu Numerical inestigation of wae attenuation by egetation using a 3D RANS moel. Aances in Water Resources, 74, Maza, M., J.L. Lara, an I.J. Losaa A couple moel of submerge egetation uner oscillatory flow using Naier-Stokes equations. Coast. Eng., 80, Peskin, C.S The immerse bounary metho. Acta numerica, 11, Ran, Z Couple ynamic analysis of floating structures in wae an current. Ph.D thesis, Texas A&M Uniersity, College Station, TX. Stratigaki, V., E. Manca, P. Prinos, I.J. Losaa, J.L. Lara, M. Sclao, C.L. Amos, I. Caceres, an A. Sanchez-Arcilla Large-scale experiments on wae propagation oer Posionia oceanica. Journal of Hyraulic Research, 49, Wang, Z.Y., Q.P. Zou, an D.E. Reee Simulation of spilling breaking waes using a two phase flow CFD moel, Computers & Fluis, 38 (10), Weller, H.G Deriation, moelling an solution of the conitionally aerage two-phase flow equations. Technical Report TR/HGW/02, OpenCFD Lt. Zhang, Y.L., Q.-P. Zou, D. Greaes, D. Reee, A. Hunt-Raby, D. Graham, P. James, an X. L A leel set immerse bounary metho for water entry an exit. Communications in Computational Physics, 8(2), Zhu, L., an Q. Chen Numerical Moeling of Surface Waes oer Submerge Flexible Vegetation. Journal of Engineering Mechanics, A Zou, Q.-P., an Z. Peng Eolution of wae shape oer a low-creste structure, Coastal Engineering, 58(6),