Second-order theory for coupling 2D numerical and physical wave tanks: Derivation, evaluation and experimental validation

Abstract

a b s t r a c t a r t i c l e i n f oA full second-order theory for coupling numerical and physical wave tanks is presented. The ad hoc unified wave generation approach developed by Zhang et al. [Zhang, H., Schäffer, H.A., Jakobsen, K.P., 2007. Deterministic combination of numerical and physical coastal wave models. Coast. Eng. 54, 171-186] is extended to include the second-order dispersive correction. The new formulation is presented in a unified form that includes both progressive and evanescent modes and covers wavemaker configurations of the piston-and flap-type. The second order paddle stroke correction allows for improved nonlinear wave generation in the physical wave tank based on target numerical solutions. The performance and efficiency of the new model is first evaluated theoretically based on second order Stokes waves. Due to the complexity of the problem, the proposed method has been truncated at 2D and the treatment of regular waves, and the re-reflection control on the wave paddle is also not included. In order to validate the solution methodology further, a series of nonlinear, periodic waves based on stream function theory are generated in a physical wave tank using a piston-type wavemaker. These experiments show that the new second-order coupling theory provides an improvement in the quality of nonlinear wave generation when compared to existing techniques.

Velocity fields in breaking-limited waves on finite depth

“…Moreover, the results by Zhang and Schäffer are larger than what we have been able to generate in the present experiments for the similar wave periods. Yang et al [6] employed second-order potential theory to calculate the far field waves with successful comparison to experiments, obtaining a similar maximum wave amplitude as Zhang and Schäffer [5]. A rapid, fully nonlinear-dispersive potential theory formulation in three dimensions, including wave generation, may be used to calculate nonbreaking waves on finite or variable water depth [7,8].…”

“…With a wave generation where the wave breaking can be postponed, e.g. adding a second harmonic motion of the wave maker, the maximum far field wave amplitude can be somewhat enhanced, such as in Zhang and Schäffer [5] and Yang et al [6]. They were able to numerically generate periodic waves of height H/h = 0.6, similar to (about) the maximum wave height that Massel [21] indicated using estimates from potential theory, giving a far field amplitude that is 10-15 per cent higher than in field observations, e.g.…”

Catalogue of extreme wave events in Ireland: revised and updated for 14 680 BP to 2017

“…Note that field measurements of maximum wave heights (H ) in constant depth generally do not exceed 0.55h (where h is water depth) and tend to break before reaching this height (Nelson, 1994;Babanin et al, 2001). Experimental results by Grue et al (2014) closely agree with this for strongly breaking waves (H < 0.56h) but exceed this value for moderately breaking waves (H < 0.63h), while numerical results (Zhang and Schäffer, 2007;Yang et al, 2013) again exceed this value but not by much (H < 0.6h). Therefore, we expect H = 0.55h to be a good indicator of the maximum rogue wave heights in the ocean, while H < 0.63h would be the upper limit in idealised situations.…”

Numerical simulation of the second-order Stokes theory using finite difference method

“…The lateral boundary conditions and the initial conditions are taken as follows: Finally, the pressure field associated with a progressive wave is determined from the unsteady Bernoulli equation developed for an ideal fluid and the velocity potential appropriate to this case [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]:…”

“…Under the assumption of large spacing between the two cylinders, waves scattered by one cylinder may be replaced in the vicinity of the other cylinder by equivalent plane waves together with non-planner correction terms. Yang et al [12] formulated a full second-order theory for coupling numerical and physical wave tanks [13]. The new formulation is presented in a unified form that includes both progressive and evanescent modes and covers wavemaker configurations of the piston-and flap-type [14].…”