Abstract
The flow structure in the aerated region of the roller generated by breaking waves remains a great challenge to study, with large quantities of entrained air and turbulence interactions making it very difficult to investigate in details. A number of analogies were proposed between breaking waves in deep or shallow water, tidal bores and hydraulic jumps. Many numerical models used to simulate waves in the surf zone do not implicitly simulate the breaking process of the waves, but are required to parameterise the wave-breaking effects, thus relying on experimental data. Analogies are also assumed to quantify the roller dynamics and the energy dissipation. The scope of this paper is to review the different analogies proposed in the literature and to discuss current practices. A thorough survey is offered and a discussion is developed an aimed at improving the use of possible breaking proxies. The most recent data are revisited and scrutinised for the use of most advanced numerical models to educe the surf zone hydrodynamics. In particular, the roller dynamics and geometrical characteristics are discussed. An open discussion is proposed to explore the actual practices and propose perspectives based on the most appropriate analogy, namely the tidal bore.
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The authors thank their students, former students and co-workers for their work and input. The financial support of the Australian Research Council (Grant DP120100481) is acknowledged.
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Appendices
Appendix 1
See Table 4.
Appendix 2: Basic momentum and energy considerations for bores and jumps
A bore is an abrupt rise in water depth (Fig. 1) and its front may be analysed as a hydraulic jump in translation [41, 118, 174]. In a system of reference in translation with the bore front, the integral form of the continuity and momentum equations gives a relationship between the cross-section area upstream and downstream of the roller, A1 and A2 respectively, and the upstream Froude number Fr1 [114]
where Fr1 is the bore Froude number: Fr = (U + V1)/(g × A1/B1)1/2, V is the cross-sectional averaged velocity positive downstream as shown in Fig. 1, U is the bore celerity positive upstream, the channel cross-sectional area A measured perpendicular to the flow direction, ρ is the water density, g is the gravity acceleration, Ffric is the friction force, W is the weight force, θ the angle between the bed slope and horizontal, and the subscripts 1 and 2 refer to the flow conditions immediately before and after the bore roller respectively (Fig. 1). In Eq. (9), B and B′ are characteristic widths functions of the bathymetry:
with d the flow depth (Fig. 1b). For a smooth rectangular horizontal channel, Eq. (9) yields to the Bélanger equation:
where d1 and d2 are respectively the upstream and downstream flow depth, and Fr1 simplifies into: Fr1 = (U + V1)/(g × d1)1/2 (Fig. 1). First presented in 1841 [12, 36], Eq. (12) is inappropriate in an irregular channel [41]. The roller height hr is in first approximation: hr = d2 − d1 and Eq. (12) may rewritten as:
Note that the above development (Eq. 9, 12 and 13) assumes implicitly that the bore celerity is uniform and the roller shape two-dimensional. Field and laboratory observations suggested that these assumptions are simplistic [114].
The application of the energy principle across the bore roller gives an expression of the energy dissipation [4, 21]:
assuming hydrostatic pressure upstream and downstream. For a smooth horizontal rectangular channel, the rate of energy dissipation becomes [5]:
where E1 = d1 + (V1 + U)2/(2 × g). The rate of energy dissipation ranges from 0 for Fr1 = 1 to more than 70 % for Fr1 > 9 [32, 88]. The power dissipated in the bore is:
where L is the transverse length of the bore roller.In the above equations, the solution for a stationary hydraulic jump is obtained for U = 0.
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Lubin, P., Chanson, H. Are breaking waves, bores, surges and jumps the same flow?. Environ Fluid Mech 17, 47–77 (2017). https://doi.org/10.1007/s10652-016-9475-y
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DOI: https://doi.org/10.1007/s10652-016-9475-y