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Numerical simulation of two dimensional breaking waves past a submerged hydrofoil R. Muscari, A. Di Mascio (INSEAN, Italian Ship Model Basin, :~taly) ABSTRACT A numerical model for the simulation of two- dimensional spilling breaking waves is described. It is derived from a previous model which, in turn, takes the underlying ideas from the Cointe and Tulin's theory of steady breakers. With respect to the former model, the present one is local, i.e. the inception, the extension and the geometry of the breaker are determined through the local shape of the water surface. The model has been imple- mented in a RANSE code, developed for the simu- lation of ship flows, through a modification in the boundary conditions. This yields an effective and simple way to reproduce the breaker influence on the underlying flow. The resulting code has been used for the simulation of the flow past a submerged hydrofoil. The numerical results are compared with those of the previous model and with the experi- mental data obtained by Duncan. INTRODUCTION The aim of this work is the development and valida- tion of a numerical model for the simulation of flows with breaking waves. In particular, we are inter- ested in flows with steady or quasi-steady spilling breakers. Examples of this kind of flows are the waves ap- proaching to a shore, the wave train generated by a two-dimensional object in uniform motion near the free surface or the wave pattern generated by a ship moving through calm water. Despite the importance of the effects of the breaking on these flows has been generally acknowledged (see Ban- ner and Peregrine (1993~; Melville (1996~; Duncan (2001) for comprehensive reviews), very little has been done, from a numerical point of view, in or- der to include these effects in naval hydrodynamics codes. The first attempt, to our knowledge, to practically include the breaking in a numerical code was done by Cointe and Tulin (1994), who elaborated a the- ory describing the shape and effects of the breaker, and then implemented it in a potential code for two-dimensional flows. One of the advantages of Cointe and Tulin's theory is that it yields some boundary conditions to model the breaker which are simple but effective and readily applicable to free surface RANSE codes. This feature has been exploited by Rhee and Stern (2002) and Muscari and Di Mascio (2002), who have implemented two different models in their already existing RANSE codes, developed for the study of flows past ship hulls. In particular, Rhee and Stern (2002) adopt the same flat-top geometry for the breaker and the same relationship between its height and that of the breaking wave as suggested by Cointe and Tulin. Furthermore, they specify the velocity distribution along the wave-breaker interface by assuming that, at the toe of the breaker, the horizontal component of velocity u undergoes a discontinuity, whose in- tensity depends on the wave steepness. Beyond the toe, u is computed by assuming constant the total head. In Muscari and Di Mascio (2002) some modifica- tions in the original theory were introduced, with the goal of developing a stable and accurate numer- ical algorithm when coupling the model of break- ing to their steady state free surface RANSE code. First of all, it was assumed a different relation be- tween the height of the breaking region and that of the following waves. Furthermore, the sharp trian- gular shape of the breaker adopted by Cointe and
Tulin and Rhee and Stern was smoothed out in or- der to mitigate the abrupt transition in the free surface dynamic boundary condition and hence en- hance convergence rate to steady state. The models cited so far have been implemented only for two-dimensional flows and share the disadvan- tage that are non-local. This means that, for ex- ample, in order to compute the hydrostatic pres- sure exerted by the breaker on a free surface point we need to know the height of the following waves (Cointe and Tulin, 1994; Rhee and Stern, 2002) or the location of the trough and the crest of the break- ing wave (Muscari and Di Mascio, 2002~. It is not evident how these geometrical properties of a two- dimensional wave train should be interpreted in a three-dimensional context and, however, the result- ing algorithm would be inefficient and prone to er- rors. With the ultimate objective of simulating spilling breaking in general naval flows, we develop in this work a "localized" version of the model described in Muscari and Di Mascio (2002~. The selection of the points where breaking occurs, the local height of the breaker and, hence, the hydrostatic pressure are all determined by the free surface elevation and its first derivative. In fact, we think that this is a necessary step before the model could be general- ized to three-dimensional flows. We do not consider viscous effects which deserve further attention and will be the subject of future work. The model is described in the next session and, then, its results are compared to the ones of the "non-local" model and to the experiments by Duncan (1983~. A LOCAL MODEL FOR SPILLING BREA- KING Following the Cointe and Tulin's idea, we want to model the effects of the breaker through a suitable pressure, applied on the patches of free surface in- volved in the process, and simulating the weight of the breaker itself. To this purpose, the main prob- lems to be solved are the detection of the afore- mentioned patches and the calculation of the local height of the breaker and, hence, of the pressure to be applied. Cointe and Tulin assume that the top of the breaker coincides with the crest of the wave (see fig. 1) and compute the vertical distance between the top and the toe as: he = 2 - Zl°P where Fr is the Froude's number. Then, assuming a flat-top shape for the breaker, they prescribe the hydrostatic pressure to be applied at the free sur- face points between Toe and atop: hi = Stopzfx) ~ pax) = Fr2 Pb being the density of the air-water mixture in the breaker, which is taken equal to 0.6. X~T ~~ I (top X Figure 1: Geometry of the breaker in the Cointe and Tulin's theory. In Muscari and Di Mascio (2002) some modifica- tions to the Cointe and Tulin's theory are intro- duced in order to gain robustness and accuracy for their RANSE code. Their expression for the verti- cal distance between the toe and the top, obtained by Duncan's (1981) experimental data, is: hm<~ = 0.64 ab (1) where ab is the vertical distance between the trough and the crest of the breaking wave. Furthermore, the shape of Cointe and Tulin's breaker is smoothed by an exponential function, in order to enhance con- vergence to the steady state: hm~(x) = hC~(x) {1exp ~5 ( toe )] } StopXtoe Apart from pros and cons of the two formulations, both of them need the detection of the crest of the breaking wave and, from this, of the toe of the breaker. Unfortunately, in a three-dimensional case these geometrical entities are not easily de- tected and, moreover, they do not uniquely locate the breaking zone. For the development of a local model we need to define a detector of breaking A(x) in every point of the free surface. This can be set equal to: A(x) = N|Z2 + (Z~42 which would yield the wave amplitude in the case of single sine wave 2
z(x) =A cos ~ x (2) Evaluating the wave number K as for a plane pro- gressive wave ~ = F 2 (Newman, 1977), we get: A(x) = 4~/z + (ax Fr ~ (3) Figure 2 shows that, for the experimental setup de- scribed in the following, eq. (2) is a very good ap- proximation of the actual wave probe. 0.,: 0.05 _ U ~ -0.05 ~ 1 I. - ~C''''"'"""" \ \ I at\ _ .. ............ \ . ~ , \ ~ . \\ \~. .......... Figure 2: Actual wave profile (solid line) and eq. (2) (dashed line) for depth = 18.5 cm. By eq. (3), we can establish a local criterion for detecting the inception of breaking, that is: 2 A(x) ~ 0.69 Fr2 (4) This equation is the natural extension of ab ~ 0.69Fr2, proposed in Duncan (1983) and used in Muscari and Di Mascio (2002), but it is not enough, by itself, to locate correctly the breaking zone. In particular, on the forward face of the leading wave it detects a zone much smaller than the experimen- tal investigations would suggest and, on the other side, it includes points of the backward face which are not directly involved in the breaking process. However, on the forward face of the wave we have u ax ~ 0 (5) where u is the horizontal component of the velocity. We found that an extremely effective solution is to activate the model in all free surface points where eq. (5) is verified and where eq. (4) holds at least in one point of the considered wave region. Finally, to evaluate the pressure distribution due to the breaker, we prescribe the following expression: hl,oc (X) = /C z2 (X) (ZZtoe) (6) which is obviously equal to zero at the toe and at the crest of the wave. Ztoe is found through Ztoe = stop - h* and the height of the breaker is determined by eq. (1). In order to close the problem, we assume stop = 2 and resort to the experimental data by Duncan (1981) to calculate ab: ab = 0.586 Fr2 The last free parameter, I;, is determined by en- forcing that the area of the breaker for the present model and that for the Cointe and Tulin's one be the same: [Xtop / thct(X)hi do = 0 Jxtoe Assembling all elements of the model, we come to the final expression for the local breaker height hoc(X) = 14.35 z2 (z + 0.117 Fr2) (7) which is illustrated in fig. 3. Figure 3: Geometry of the breaker for the present model. From eq. (7) we get the pressure to be applied on all free surface points between xtOe and Stop to sim- ulate the presence of the breaker: p(x)= Fry This final expression conjugates very good results with the desired locality of the model. In particu- lar, to calculate the height of the breaker only the wave elevation and its first derivative are necessary. The proposed model will be examined in the fol- lowing in order to verify its capabilities in a two- dimensional case. The extension to a more gen- eral three-dimensional case will be done in a future work. 3
TEST CASE In order to validate the proposed model we chose to simulate the flow described in Duncan (1981, 1983) and illustrated in fig. 4. This flow has been also used for the validation of the non-local model in Muscari and Di Mascio (2002), so that we can bene- fit, together with extensive experimental data, from other numerical results. Re = 1.423 x 10 Fr = 0.5672 Figure 4: Experimental setup. A submerged hydrofoil, a NACA 0012 profile whose chord is 20.3 cm, is towed in a tank with speed 0.8 m/s and 5° angle of attack. The leading wave of the train can break or not depending on the depth of submergence. This latter is varied through the water level in the tank, whereas the profile is kept fixed with respect to the bottom. The numerical solution was computed on a multi- block fine grid of about 60.000 cells, and on two coarser grids, each obtained by removing every other point from the previous finer one. The nu- merical uncertainty U was evaluated as suggested in ITTC Quality Manual (2001~; Roache (1997), by U = NIGH + UI2T where UH is the contribution from the grid size | hone _ hmediUm | (~ = 2 being the theoretical convergence order and r = 2 the grid refinement ratio for all the compu- tations reported), whereas UIT is the contribution from incomplete iterative convergence UIT= 2h Ah being the oscillation amplitude of the solution. Unfortunately, experimental uncertainty was not available for this test case, and therefore a com- plete validation as required in ITTC Quality Man- ual (2001) was not possible. Nevertheless, use- ful information on the reliability of the model can be gained when comparing the numerical solutions with the towing tank data. RESULTS AND DISCUSSION First, the depth = 19.3 cm is considered. As re- ported in Duncan (1983), for this submergence the steady breaking is not spontaneous but must be triggered by a disturbance (a surface current cre- ated by dragging a cloth on the water surface in front of the hydrofoil). ~o2 s 10.3 1o" t ~ DEPTH = 19.3 CM ~~_~ 1 _ _ ., . . , . j ~ ~~7,k~ - fX t- I ~~'t;l.`,. 1 ~ it. ~ ~ ~?~- - .................. , ,, - if; _ 0.38 _M _ 0.33 _ 028 _ 0.23 0.18 I ~ 1 ~ .. .08 iter Figure 5: Residuals and resistance histories. _ n 1~ In fig. 5, L2-norm of residuals and non-dimensional resistance histories are shown. The solution was computed by means of a Full Multigrid - Full Ap- proximation Scheme (FMG-FAS) with four grid lev- els, and the solutions on the two finer grid were used to estimate the uncertainty. It can be seen from the figure that a stationary regime is reached on each grid level, and therefore the contribution to uncer- tainty from incomplete convergence was negligible. The comparison with the non-local model, fig. 6, shows that, despite of the different breaker geome- tries, the computed wave profiles are substantially equal, with only a very slight phase difference. The top of the breaker obtained with the local model is set between the toe and the crest of the wave, not at the crest itself as it is assumed in the Cointe and Tulin's theory. This is an obvious consequence of the locality of the model, but we do not think that it represents a real drawback. The wave profile is also in very good agreement with experiments, fig. 7. The slight difference at the troughs, which are overpredicted, can be well within 4
the experimental uncertainties. As for the numer- ical uncertainties, their very small value is due to the fact that convergence to steady state has been reached (UIT ~ O) and that the wave profile has almost reached grid independence on the medium grid (UH ~ O) O.1 ... . . . ... 0.05 coos -0.1 _ DEPTH = 19.3: CM _ ..... . . . o _ . ~ _. .. . ., .. . . .. . .. . . . .. .. J : 1 _~. ,,,,, . , ,. ,, . .,, . . 1 1 2 3 4 ·f~s Figure 6: Wave profiles with breaker's geome- tries. Solid line: local model; dotted line: non- local model. our. O.Oc o O.O: -0.1 1 · . ! . .. . . . . . DEPTH = 19.3 CM ~ ~ . ~ ~ ~ ~ ~~k ~~ ~ offs _~: W. ., ~ . · . ................................................................................................................................................................................. ·... . . . The last case is depth = 15.9 cm. For this submer- gence the breaking is very intense and it is not clear, from the experimental wave profile, if a steady or quasi-steady state can be actually reached, so the comparison with the numerical data is rather ques- tionable. The numerical solution too does not at- tain convergence to steady state and the residuals oscillates around a small but constant value. Even so, it is interesting to compare the different data sets, fig. 9 and fig. 10. With respect to the non- local model, the local one produces a higher pres- sure with a consequent reduction of the amplitude of the wave train. Furthermore, a second breaker appears on the first following wave. In fact, in the proposed model there is no need to know a pr~ori which wave breaks. On the contrary, the inception of breaking is essentially determined by the local steepness of the water surface, eq. (3~-~4), and is propagated to the neighboring points with a sim- ple, robust but very effective criterion. As could be expected, comparison with experiments can be only qualitative. In this sense, the new solution performs better then the non-local one with a lesser wave amplitude and a larger extension of the breaking area. n ~ _ r DEPTH = 18.5 CM . . . ~ : ~~! .O . T -lo '- - ~ ''''1"':" ' ''' ' ' '' ' ''' '' '' . ~ . / . . e_f - . .. I ~- Figure 7: Computed wave profile with numerical uncertainties vs. experiments. The test case with depth = 18.5 cm is the the first experimental setup for which breaking spon- taneously occurs, and it is mild enough to have a regular steady following wave train. The gen- eral considerations done for the previous case still hold. Here, we report only the comparison with the experiments, fig. 8, which is again very good. Although the leading wave is not captured, the fol- lowing waves show a remarkably good agreement with the experiments. Even the troughs, that for the depth = 19.3 cm case were overpredicted, are well reproduced. Figure 8: Computed wave profile with numerical uncertainties vs. experiments. CONCLUSIONS A local model for simulating two-dimensional spilling breaking has been proposed. It is de- rived from the non-local model described in Mus- cari and Di Mascio (2002) and represents a nec- essary premise towards the simulation of three- dimensional flows with general breaking patterns. The inception and the shape of the breaker depend on local characteristics of the free surface, the el- evation and the gradient, and, as a consequence, 5
the extension to three-dimensional flows does not present any major theoretical difficulties. The model has been applied to the study of a wave train created by a hydrofoil towed under the free surface. For the cases with milder breaking the re- sults are similar to those obtained by the non-local model and in very good agreement with the experi- ments. For the most severe case, the new algorithm performs better then the old one. Even if it can not cope with an apparently unsteady flow, nonetheless the inception of a secondary breaker indicates that the extension of the phenomenon can not be re- stricted to the forward face of the leading wave. 01r nor O ; -0.05 ~.1 ~ , ~ . -., DEPTH = 15.9 CM .W _ . I. \ .. ~ ' \ _ . .... it. .. .,\ . 1 ~ . ·. : : . . .. . . . . . I: :,. : ~ / ·/ Figure 9: Wave profiles with breaker's geome- tries. Solid line: local model; dotted line: non- local model. 0.05 At. .... . . ~. ........ .. . '''''I 'my ~. I' /~ :. . . . .. . . . ... Figure 10: Computed wave profile vs. experi- ments. The natural evolution of this work will be the ap- plication to a three-dimensional flow and, possibly, the inclusion of the phenomenology due to the vis- cosity. ACKNOWLEDGMENTS This work was sponsored by the Italian "Ministero delle Infrastrutture e dei liasporti" through the IN- SEAN Research Program 2000-2002, and by the Office of Naval Research contract N00014-00-1-0344 under the administration of Dr. Patrick Purtell. REFERENCES Banner, M. L. and Peregrine, D. H. (1993). Wave breaking in deep water. Arson. Rev. Fluid Mech., 25:373-397. Cointe, R. and Tulin, M. P. (1994~. A theory of steady breakers. J. Fluid Mech., 276: 1-20. Duncan, J. H. (1981~. An experimental investiga- tion of breaking waves produced by a towed hy- drofoil. Proc. R. Soc. Lond. A, 377:331-348. Duncan, J. H. (1983~. The breaking and non- breaking wave resistance of a two-dimensional hy- drofoil. J. Fluid Mech., 126:507-520. Duncan, J. H. (2001~. Spilling breakers. Aran. Rev. Fluid Mech., 33:519-547. ITTC Quality Manual (2001~. Resistance Commit- tee of 23th ITTC. Melville, W. K. (1996~. The role of surface-wave breaking in air-sea interaction. Ann. Rev. Fluid Mech., 28:279-321. Muscari, R. and Di Mascio, A. (2002~. A model for the simulation of spilling breaking waves. Sub- mitted to J. Ship Research. Newman, J. N. (1977~. Marine Hydrodynamics. The MIT Press, Cambridge, MA. Rhee, S. H. and Stern, F. (2002~. Rans model for spilling breaking waves. J. Fluid Erlg., 124:1-9. Roache, P. J. (1997~. Quantification of uncertainty in computational fluid dynamics. Ar7~rz. Rev. Fluid Mech., 29:123. 6
DISCUSSION L.J. Doctors The University of New South Wales, Australia I would like to thank the authors for a very interesting paper on the question of wave breaking, which should have application to breaking wave system behind a transom in stern at low speeds. My question relates to the equation at the top of column 2 of page 58 of your paper, in which the height of the wave is given as a constant times the Froude number squared. Could you kindly chart the dimensions of this equation? AUTHORS' REPLY Thank you very much for the comment. Regarding the question, all the equations in the paper are written in non-dimensional form. In particular, the hydrofoil chord length is the reference length for the wave height ab
