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A 2D+T VOF Fully Coupled Formulation for Calculation of Breaking Free Surface Flow Yann Andrillon, Bertrand A:lessandrini (Ecole Centrale de Nantes - FRANCE) ~ Abstract In this paper, we present some results of dimensional stea- dy how. To obtain this results a 2D+T method have been used with a two dimensional CFD code. The feature of our approach, is the used of a 2D Navier-Stokes solver with an interface capturing method to simulate complex free surface. This code solves Navier-Stokes equation us- ing a finite volume method adapted to structured or un- structured mesh and used a fully coupled method to dis- cretize the equation. The capturing method employed to follow the interface is the "Volume Of Fluid" method. This approach allow to simulate complex flow as break- . . 1ng or merging wave. 2 Introduction The main objective of our work is to be able to predict the flow past a complex hull with breaking and splashing waves. today, different code exist to predict steady wave field. But less code could take into account the break- ing bow splash and they are particularly computationally expensive. Recently M. Landrini E1] have presented in- teresting results from a 2D+T decomposition techniques t2, 31. This approach present the advantage of being able to give accurate wave field using a simple two dimen- sional unsteady flow solver. However, it has much rec- ommended for fine ships like combatants, and especially for higher froude number where massive breaking of di- vergent waves is observed. Then initially we restrict sim- ulation to last kind of hull. The solver used must solve complex unsteady free sur- face flow. It solve the Navier-Stokes equation for incom- pressible fluid and used a finite volume decomposition to discretise the equations. Then structured as unstructured mesh is accept to grid the simulation domain. The novelty in our code is the used at the same time a Volume Of Fluid capturing method and a verry accurateFully Coupled for- mulation to solve the Navier Stokes equation. In order to simulate an interface flow two different methods exist the front capturing method and the front tracking method. The last one have been investigated by several authors, and it have been showed their limitation to predict merg- ing wave. Harlow and Welch have introduced the first cap- turing approach: the Marker and Cell (MAC). Later The Volume Of Fluid method from Hirt and Nichols proved the versatility and the accuracy of the capturing approach. It will be described more precisely below and details can be found in 0. Ubbink t4] and E. Didier ted. The Fully Coupled Method [6, 73 have been recently investigated. It has been showed that the method is particularly robust and accurate. Numerically this technique present the ad- vantage to accelerate the velocity-pressure coupling and to converge faster than other weak coupling algorithms as SIMPLE or PISO. To check the solver capabilities, different two dimen- sional flow case have been tested. The first one is the liquid sloshing in a tank with various amplitude. The sim- ulation has been compared to theoretical and experimental data. Second one are the flow made by a collapsing col- umn water and a Rayleigh Taylor instability. The results illustrate the robustness of the solver and the ease to pre- dict merging and rupturing interfaces. Third, the results from a numerical two dimensional wave tank is presenting to showed the capabilities of the code to propagate a wave field. At least, a dimensional steady flow past a Wialey ship at different froude number with breaking wave has been simulated. Results prove the ability of the method to predict breaking bow wave. . . ~ , 3 Volume Of Fluid methoc! The "Volume Of Fluid" method is an interface capturing method. The computation is performed on a fixed grid, which extends beyond the surface. The shape of the inter- face is determined by cell which are partially filled. The main advantage of this approach is the ability to simu- late complex free surface geometries like wave breaking. Also, a new variable is introduced to the system, the vol- ume fraction, which indicates, whether a cell is filled by
fluid 1 (c = 0), filled by fluid 2 (c = 1) or filled by a mixture, (0 < c < 1~. The physical properties of fluid is deducted from the volume fraction by the equations (1 ) and (2~. p = cop + (1 - cope (1) ,~ = car + (1 - c),u2 (2) This variable is ruled by a advection equation (3) which are obtained from the second Navier-Stokes equation ( 1 1~. The integral form of this advection equation is used with a finite volume method for the spatial discretization. In order to approximate the time integrals a Euler explicit scheme is applied (4~. And a cell centered arrangement is employed. rt+~trt ~ ~ ~ ,03~ + ~ (c~) = 0 (3) dt -S+~;Cft~L=0 (4) The critical issue in this type of method is the choice of the differencing schemes for the convective term (4~. Low order schemes like central differencing scheme are not suitable because a bounded solution is not ensured. Other differencing schemes like first-order upwind scheme are too diffusive, smear the interface and introduce artifi- cial mixing of the two fluids over a wide region. Then high order mixing scheme have been designed for this kind of application by Peric (HRIC scheme) [8, 9, 10] or by Ubbink (CICSAM scheme) [43. The last one is partic- ularly accurate to keep a sharp interface and do not create non-physical deformation. This interpolation is a High Resolution differencing scheme based on the Normalized Variable Diagram t11, 12~. This approach is particularly effective in the case of unstructured mesh. use / Figure 1: normalization variable cd = cdcu ~5' C'dCu The CICSAM scheme use a mix of the compressible scheme (Hyper-C) and Ultimate-Quickest scheme t11~. This scheme use the Courant Number and weighting fac- tor by, based on the angle between the interface and the direction of motion. The aim of these factor is to avoid stability problems and to limit overshoot or undershoot. Another factor, kit introduce the control of the dominance of the different scheme. More concretely, the UQ operates where Hyper-C fails to preserve the gradient in the inter- face and the Hyper-C operates where UQ fails to maintain the sharpness of the interface. CfCbc = { min (1, NC ~ if 0 < cd ~ 1 (6) cot else - ~ mitt ~ ~ ~ Cfcbc) CfUq = ~ if 0 < cat < 1 (7) cot else Nc = Am, max(O, V ~ (8) of = mitt (ok C05(~2§f + 1 1\) `9 At Least, the CICSAM have been evaluated on few convection test with accurate results. The free surface contour is sharply defined on two or three cells. How- ever the boundedness criterion is checked only for weak Courant Number (Nc < 0.1~. An additional subroutine has been implemented to the last advection routine to treat the unbounded volume fraction value. 4 Navier-Stokes Equations The motion of the fluid is driven by the incompressible Navier-Stokes equations. The dimensionless conserva- tion mass and the conservation of momentum equations (10,1 1) are presented below under the conservative form. /v 0t /. dS+/ dV is Rep ~ Ad) dS = +/ Fr2 ~dV (10) /v Rep (~) (( At') dV / Aids = 0 s with · ~ is the velocity field vector · p is the pressure · g is the acceleration of gravity · p is the mass density (1 1)
. ~ . . . · p IS dynamic VlSCOSlty · Re is the Reynolds number · En is the Froude number In order to solve the governing equations, a fully im- plicit finite volume method is choose. It could be apply on unstructured mesh made by triangle or quadrilateral. A cell centered arrangement is used to stored all depen- dent variables. The integrals are calculated with a second order approximation, and the flux approximation is eval- uated with a deferred correction to obtain a second order accuracy. The different term are evaluated at the new time level, while the old values appear only in the second order correction and in the approximation of the time derivative. Uip + U'p + 9aP . Up + ~ 9i7t .Pn = 0 (12) .0 iterations Figure 2: Convergence of the system during no-linear iteration * an Fi chosen in order to use unstructured mesh. Nevertheless, UiP + ~ a Uin = a (13) if the convection equation allow to transport an interface zone wide of two or three elements, the discretisation of the Navier-Stokes equation require in unstructured mesh a wider zone. About 10 or more element is necessary to define the free surface without creating abnormal veloc- ity. Then in order, to make accurate simulation main of our grid are structured one. The pressure equation is derived from the discrete con- servation momentum equation using a Fully-coupled me- thod [7, 131. The new variable u*, v* are introduced in the discrete momentum conservation equation (121. The Rhie and Chow t14l velocity flux reconstruction applied to the conservation of momentum Navier-Stokes equation produce the discrete pressure equation presented below. Pp + ~ c Pn¢.4~ <7 = (14) The different discrete equation give the following lin- ear system. ~ G ] (U) (Fu) (15) O D DO p Fp In opposition to weak coupling, the fully coupled me- thod do not require correction step, a single numerical system is solved to obtain the solution (15~. An iterative algorithm BiCGSTAB-w is used to solve the system pre- conditionned by incomplete LU decomposition. One ad- vantage of this technique is the numerical accelerates of the coupling between velocity and pressure. So, the con- vergence is fast, and it allow a reduction of 10 order in a few number of non linear iteration as the figure 4 shows. 4.! unstructured aspect The different method used for the discretization of the Navier-Stokes equation and advection equations have been 5 2D+T method This method has been introduced by M. Munk into aero- dynamics for the prediction of loads on inclined slender bodies of revolution and low aspect ratio wings. This method has been extended in hydrodynamics domain, in order to predict wave field due to a hull. More recently, M. Tulin and M. Landrini [1, 2] have presented result of bow- wave radiation. The analysis is limited to slender ships, with a sharp stem. The idea exploited is that longitudinal gradients of relevant flow quantities are small compared with vertical and transverse gradients. Then a three di- mensional steady flow past a ship hull could be generated by a sequence of unsteady two dimensional problems in vertical sections. The two dimensional flow are generated by a deformable body, which correspond to a cross section of the passing through ship. Then, the section is deducted from the hull by a cut at the x=u.t coordinate, where u it the boat speed and t the 2D solver time. It has been show that this technique give accurate results for flow about a ship at high Fn (> 1), and for low aspect ratio planning boats or slender displacement hull((B/L)2 << 1~.
S.:l wavemaker The deformable body described before used to generate the 2D+T flow could be ranking as a special wavemaker. To allow for our code to simulate this special wavemaker a simple moving mesh technic has been implemented. We define the displacement of the frontier node 16, and the nodes inside the domain are moved using a translation cal- culate as a linear function of the x-coordinate, like on the figure 3. J (`Ybody ~ = Figure 3: example of moving boundary t=To and at t=To+n.dt The Navier-Stokes equation and the advection equa- tion of volume fraction equation are modified as described in Eland 18. (iv cdv) + is Of (ifUgrid)) ~dS = 0 `17' f5t (/v REV) + is ~ (( - ugrid~.~) dS n iv P + is Rep It\) dS (18) iv Fr2 9 dV iv Rep (~ ( ~) dV However, to limit the deformation of the cell near the bottom of the hull, the ship used in simulation would be as thin as possible. 6 2D Computation Procedure The developed numerical method is applied to simulate several kinds of unsteady free surface flows, from simple to complex interface topologies. In order, to demonstrate the versatility of the code, different cases as have been simulated. · Oscillating flow in a tank has been simulated t151. This problem is used to compare our code to an- other CFD code using front tracking method, and to experimental data, showing good results. · Collapse of a dam [16] is a typical problem used for illustrating the versatility of the current solver and the capability to predict merging and breaking interfaces A two dimensional Rayleigh Taylor instabilities. (16) · 2D wave tank has been simulated numerically with success. This 2D unsteady simulation is a particular 2D+T case. 6.l 2D sloshing tank In this section, we present the flow in a sloshing tank. Tad- jbakhsh & Keller t17] have developed a theory on slosh- ing liquid in a tank under the influence of gravity. Ini- tially the fluid has an interface defined by a half cosine period with amplitude 0.005 m. The simulation domain is a structured mesh 1 m long and 0.5 m height. The fluids is respectively air and water without viscosity. The theo- retical period of sloshing of the first mode is given by the equation (19), where h is the depth of the tank and k is the number of wave. Ps = 27T>/gk tanh~kh) = 0.3739s (19) Simulation ...... Theory 0.2' _ . _ - 0 . 7 5 ~ _ 0.5 1 t(s) 1.5 2 Figure 4: height of free surface on the left side Theoretical result and numerical simulation are com- pared on the the fig. 4. It show plots of position of the interface at the left boundary against time for the six peri- ods. It could be pointed out a good agreement. Another sloshing flow have been simulated. it have been compared to experiment made by Corrignan [151. The tank 6.1 is moved with an horizontal displacement.
The movement is ruled by the equ. (20~. The calculations were performed on a structured and a Restructured mesh of about 3500 control volume and the time step is about dt = 0.001. 0.4 m Figure 5: tank dimension ~ A.(~sin(~27rf~t)sin(21rf2t)) x(t)= ~ siO<t<3.43s (20) Osinon A = 7.5.10-3m fit = 1.598Hz (21) f1 = 1.307Hz The exact interface shape are available at several time step. The figures 6.1, give a comparison between the ex- perimental and the numerical free surface. It can be no- ticed that the transfers between the potential energy and kinetic energy is correct and that the numerical diffusion seems to be neglected. Numerical sloshing tank present a good results. 6.2 wave breaking The collapse of a dam [16, 18] is a typical test case used to demonstrate the ability of the code to compute transient fluid flow with breaking free surface. The dimensions of the tank and the water column correspond with those used in the experiment carried out by Koshizuka et al.[l81. The tank's dimensions are specified below (fig. 7~. The mea- surements of the exact free surface is not available. how- ever, some secondary data as the speed of the wave front on the vertical and the horizontal wall are available. Dif- ferent photographs is available too, and showed the time evolution of the breaking water wave. The simulation of this breaking waves has been made on a unstructured mesh using approximatively 5000 ele- ments. The figure 8 showed the comparison between numer- ical and experimental results. The predicted height is in good accordance with the experiment, but the non dimen- sional positions of the leading edge are less accurate. As 0.05 O 4.05 ~ X 0.05 O _ 0,1 _.__ O 4.05 _ >. = - e__ ~3 ~A Get, _ o~2 0.3 ·.05 _ 0.1 . ,,, I .,, . I, ., . 1 . . . , I 0 0.1 02 0.3 o.4 Figure 6: free surface at t=0.605s ;0.9625s;1.652s; 2.004s other researchers show, the reasons of this difference seem to come from the difficulty to capture the interface. the fluid moves on the bottom in a thin layer. Another simulation of a collapsing dam has been made with an obstacle in the tank. The difference with the last simulation is the addition of an obstacle in the bottom of the tank creating a water set of water which bounce up from the upper left corner of the obstacle in the direction of the opposite wall. Then the free surface has a very complex interface as it is presented in the fig. 9.
2a Figure 7: montage colonne d'eau 1.2r 11 o.8t Cal 0 . 6 0.4 _ 0.2 _ n _ _ : 4 _ 3.5 3 0.5 _ simulation · exp. result I ~ 1 . , . . . . . . . . . . . . . . . 1 2 3 4 t~75 ,~ simulation · exp. result 1 2 t\/;77; , 3 Figure 8: The position of the leading edge versus time 6.3 Rayleigh Taylor instability This simulation showed the versatility of our code to dif- ferent application domain. The figures 12 give the initial configuration of the instability and the evolution of the free surface at different time. The mesh used for the sim- ulation is a Cartesian grid width 256x64 elements. The time step is 0.001 s, and the running time is about 30 mn Figure 9: collapse of a dam with an obstacle on a HP station with an 500 Mhz Alpha processor for 6000 iterations. The numerical domain is box 1 x4 m, filled by two fluid. the density of fluid in the bottom are 0.1694 kg/m3 and for the other it is 1.225 kg/m3. they have the same viscosity 0.00313 kg/m.s . The numerical results have been compared to other one obtained by Puckett and al.~191) and it shows a good concordance. 6.4 two dimensional wave tank The two dimensionnal wave tank allows to test the abil- ity of the method to propagate wave without amplitude damping or phase lag that are the two recurrent problems for CFD hydrodynamics simulations. The first step try to reproduce a monochromatic wave field, without break- ing wave and check the accuracy with theoretical solu- tion. The wave is generated with a piston wave makers as it have been discussed previously in the wavemaker sec- tion. The geometry of the numerical tank is 100 m long, 15 m depth and 10 m height. We use a damping zone of 200 m long downstream the outlet boundary to avoid the wave reflection. The motion of the piston wave makers is defined by: x((t) = A.(~cos(<w * t)1) with A=O.5 m and ~ = 1.57 rad.s-i Numerical results is compared to the linearity water wave theory. Then the dispersion relationship reduces in the following manner eq.(22). Where, k is the wave num- ber, h the water depth and ~ the wave length. The per- manent solution for the movement of the wavemaker de- scribe above is easily calculable. The wave amplitude is calculated with the eq. (23) L_ 2~ 1.56T2 (22) Aw = 2A (23)
T = 4s ~ = 25m (24) On the figures 13, the evolution of the free surface dur- ing the installation of the established mode. On the fig. 14 the numerical solution is compared to the solution of the linear water potential theory. The wave length don't be the same one as that predicted by the theory. An largest length seems to be more appropriate . Moreover, the wave amplitude is undershoot against the theory too. Along the wave tank, the amplitude is attenuated. However, it is im- portant to notice that the simulation used a particularly inaccurate mesh as it can be viewed on the fig. 14. Only 25 elements to define a wave length. Then, the wave is correctly propagated for this simulation condition. To test more qualitatively the cfd code to this kind of application, it seems to be necessary to refine the grid. 7 3D Compulation Procedure The well-known Wigley hull is chosen for the first hull for, since it is a mathematical model made of parabolic curves as eq. 25. X2=B2 (1-(E) ) (1 (~) ) where x is the offset of the hull and L,B,d are the length, breadth and draft of the ship.the length-to-breadth ratio and length-to-draft ratio is set as below. L, = 0.1 AT = 0.1 Figure 10: Wigley Hull the flow past the ship has been calculated for different Froude number. The computation domain is 2 m long and 1.2 m wide. The number of element is approximately of 18000. The time step adopted is function of Froude num- ber and betwen 0.001 and 0.00025. The different CPU time is about 1 6h on a HP DEC 500 MHz station. Figure IS shows the free surface elevation along the hull for different Froude Number. For the Froude Number 0.289 the result is compared to the experiment [20, 211. The difference could be explain by the inaccuracy of the mesh use for the simulation, and also due to the 2D+T approximation. Moreover, the oscillation of the elevation curve arise from the difficulties to locate the free surface across the interface. The following figures is the wave field made the wigley hull. More precisely on the figure for 0.7 and 1.0 Froude Number, we not wave breaking on the bow. In fact, even more results with refined mesh is neces- sary to approve the quality of the simulation, the presented result give an idea on the power of the globally approach and the robustness of our two dimensional flow solver. ~ Conclusion and Perspective We have presented here a new numerical method using both advantages: The accuracy of the Fully Coupled algo- rithm first tested on Finite Differences convective formu- lation for tracking interface method and the robustness of Volume of Fluid capturing method to simulate very com- plex interfaces. The two dimensional test cases show that we achieved our first objective.: the two dimensional solver is robust and evade over a wide range of hydrodynamics appli- cations. The three dimensional test case present a method to (25) obtain the flow past a fine hull, with accurate result. Near future work will give more accurate results using unstructured grids. Then examine the problem discussed in the section "unstructured aspect" .To allow the code to simulate 2D+T flow on more complex hull as serie 60 or DTMB model... Then thereafter it is planned to make evolve the cfd code to a three dimensional flow solver.
- - - amp ~ -A -an aim - - Figure 1 1: collapse of a dam at different instant: t = 0.2, 0.2, 0.4, 0.6, 1.0
Figure 12: Rayleigh Taylor instability at instant: O.Os;0.4s;0.8s;1.2s;1.6s;2.0s;2.4s;2.8s
3 ~ 2 - . .. . ... .. . .. I t: 9.0000 . .. . . . . .. .. .. ... . so - ~ 10 20 30 40 5XO o -2 -3 3 2 1 o -1 -2 3 3 2 1 o -1 -2 -3 1 ) 10 ~0 ~ . , . ... .............. . t~5.0000 t:33.0000 - 10 20 30 40 5XO 60 io 80 90 100 Figure 13: numerical wave tank at different instant.
2 'I ~~ ~11411~ O ~ I _ -1 _ _ _ _ ~ ~ - m _ _ _,9 _ _ L _ ~ _ ~ _ _ ~,~ ~ .~ ~ .~ ~ .~ ~.~ I .~ _ _ _ . . 111111~i ~l~Llr~lTR~ . ~~ ~~ ~ PUTTY i L7 f@; ~ 20 30 40 Figure 14: comparison between numerical result and theoretical. 0.04 0.03 0.02 0.01 o -O.Ot -0.02 -0.03 11111111111111111111111111 11111111111111111 11~:111111~111~111111111111 1111111111111 11 I 1~1~1 LL - tTI I t~ ~ 1~1 T I IllF 111111111~111111111111~111 · t:45.0000 ~ I.lllll~l.lT.11.7117iT.71,T71TIl',l,l,l,l,!lllll ~ 60 70 80 90 100 W~ N~ ~ I, N`~ ~~` Fn=0.289 exp. results Fn=0.289 . Figure 15: free surface elevation along the hull Figure 16: Wave field around the Wigley hull: Fn=0.289
Figure 17: Wave field around the Wigley hull: Fn=0.7 Figure 18: Wave field around the Wigley hull: Fn=l.O
Acknowledgement The authors express their thanks to the Delegation Generate pour l'Armement (DGA) which is supporting this work. References t1] M. P. Tulin M. Landrini, A. Colagrossi, "Numerical studies of breaking compared to experimental obser- vations," 4th Numerical Towing Tank Symposium, Hamburg, Germany, 2001. [2] M. Wu M. Tulin, "Divergent bow waves," 2Ist Symposium on Naval Hydrodynamics, 1997. [31 M. Wu M. Tulin, "Bow waves on fine ships - nonlin- ear numerical studies," 9th Int. Workshop on Water Waves and Floating Structures,Kyushu, Japan, 1994. [4] O Ubbink, Numerical prediction of two fluid systems with sharp interface. PhD thesis, University of London, 1997. [5] E Didier, "Simulation d'ecoulements a surface libre sur des maillages Restructures," These de Doctorat, 2001. t6] X. Vasseur, Etude Numerique de techniques d'acceleration de convergence lors de la resolution des equations de Navier-Stokes en Formulation Decouplee ou Fortement Couplee. PhD thesis, Uni- versite de Nantes, 1991. [7] M. Ferry, Resolution Des Equations de Navier-Stockes Incompressibles En Formulation t19 Vitesse-Pression Fortement Couplee. PhD thesis Universite de Nantes, 1991. [8] M Peric and J H Ferziger, Computational Methods for Fluid Dynamics. Springer,second edition, 1997 [9] R Azcueta, S Muzaferija, and M Peric, "Computa- tion of water and a~r flow around ships," Euromech 374, Poitiers, 1998, pp. 121-132. [21] [10] M Peric, "Basics of viscous flow CFD," CFD for ship and offshore design, 31st Wegemt School, 1999. [11] B P Leonard, "Bounded higher-order upwind mul- tidimensional finite-volume convection-diffusion agorithms," dans W.J. Minkowycz, E.M. Sparrow (eds), Advances in Numerical Heat Transfer, Chap 1, Taylor and Francis, New York, 1997, pp. 1-57. [12] H Jasak, H C Weller, R I Issa, and A D Gosman, "High resolution NVD differencing scheme for ar- bitrarily unstructured meshes," Web published, Vol. 0, 1996. t13l B Alessandrini and G Delhommeau, "A multi- grid velocity-pressure-free surface elevation fully coupled solver for turbulent incompressible flow around a hull calculations," Proc 9th International Conference on Numerical Methods in Laminar and Turbulent Flows, Atlanta, 1995, pp. 1173-1184. rl4] C M Rhie and W L Chow, "A numerical study of turbulent flow past an isolated airfoil with trailing edge separation," AIAA Journal, Vol. 21, 1983, pp. 179-195. P Corrignan, "Analyse physique des phenomenes as- socies au ballotement de liquide dans les reservoirs (sloshing)," These de Doctorat, 1994. [16] W.J. Moyce J.C. Martin, "An experimental study of the collapse of liquid columns on a rigid hori- zontal plane.," Philos. Trans. Roy. Soc. London, Vol. A244, 1952, pp. 312-324. t17] J.B. Keller I. Tadjbakhsh, "Standing surface waves of finite amplitude," J. Fluid Mech., Vol. 442-451, 1960, p. 8. [18] S Koshizuka, H Tamako, and Y Oka, "A particle method for incompressible viscous flow with fluid fragmentation," Computational Fluid Dynamics JOURNAL, Vol. 4~1), 1995, pp. 29~6. J.B. Bell E.G. Puckett, A.S. Almgren, "A high or- der projection method for tracking fluid interfaces in variable density incompressibl flows," J. Comp. Phys., Vol. 100, 1997, pp. 269-282. F. Larrarte, Etude experimentale et theorique des profies de vagues le long d'une carene. PhD thesis, Universite de Nantes, 1994. C.Y. Chen F. Noblesse, "Comparaison beetween theorical predictions of wave-resistance and exper- imental data for the wigley hull," J. of Ship Research, Vol. 27, 1991, pp. 215-226.
DISCUSSION Hoyte C. Raven MARIN, Netherlands In the discussion on Figure 15, you attributed the imperfect agreement of the hull wave profile at Fn=0.289 to an insufficient mesh density. However, at such Froude numbers transverse waves still play a role; and transverse waves cannot be modeled by 2D+t. Transverse wave length here is about 0.5L, and this may well explain some of the disagreement.
