Infraestructura y Construcción:

As pointed in [219], in shallow water, complex events can be observed related to turbulent processes. One of these processes corresponds to the breaking of waves near the coast. As it will be seen in the numerical tests proposed in this work, the models presented here cannot describe this process without an additional term which allows the model to dissipate the required amount of energy on such situations. When breaking processes occur, mostly close to shallow areas, two different approaches are usually employed when dispersive Boussinesq-type models are considered.

Close to the coast where breaking starts, the SWE propagates breaking bores at the correct speed, since kH is small, and dissipation of the breaking wave is also well reproduced. Due to that, the simplest way to deal with breaking waves, when considering dispersive systems, consists in neglecting the dispersive part of the equation. This means to force the non-hydrostatic pressure to be zero where breaking occurs. Due to that, this technique has the advantage that only a breaking criterion is needed to stop and start it. However, the main disadvantage is that the grid-convergence is not ensured when the mesh is refined, and global and costly breaking criteria should be taken into account (see [163]).

The other strategy, that will be adopted in this dissertation, consists in the dissipation of breaking bores with a diffusive term. Again, breaking criteria to switch on/off the dissipation is needed. Usually, an eddy viscosity approach (see [219]) solves the matter, where an empirical parameter is defined, based on a quasi-heuristic strategy to determine when the breaking occurs. The main difficulty that presents this mechanism is that usually the diffusive term must be discretized implicitly due to the high order derivatives from the diffusion. Otherwise, it will lead to a severe restriction on the CFL number. As a consequence, an extra linear system has to be solved, losing efficiency. In any case, this challenge is overcome by C. Escalante et al. in [112] for the one layer non-hydrostatic system derived in [259], and a natural extension of this procedure for the two layer case was given by C. Escalante et al. in [118]. In this section the two breaking criteria will be

described for the one layer non-hydrostatic pressure systems (YAM-2D), (NHyp-2D). An extension of this technique is proposed for the case of two layer system (NH-2L).

A breaking mechanism for the one layer non-hydrostatic pressure

systems

One space dimension

For the sake of clarity, the breaking mechanism is described for the system (NH-1L) in one space dimension. Later, the extension of the procedure for two dimensional domains for the system (YAM-2D) and (NHyp-2D) will be given. Let us consider a simple and well-known eddy viscosity approach similar to the one introduced in [219], by adding a diffusive term in the horizontal momentum equation of system (NH-1L):

                                 ∂th + ∂x(hu) = 0, ∂t(hu) + ∂x  hu2₊1 2gh 2_{+ hp}  = (gh + 2p) ∂xH − τb,x+ ∂x(νh∂xu) , ∂t(hw) + ∂x(uhw) = 2p, ∂xu + w − wb h/2 = 0, wb = −∂tH − u∂xH, ν being the eddy viscosity

ν = Bh|∂x(hu)|, B = 1 − ∂x(hu) U1 , (1.5.1) where U1 = B1 p gh, U2 = B2 p gh,

denote the flow speeds at the onset and termination of the wave-breaking process and B1, B2 are calibration coefficients that should be fixed through laboratory experiments

(see [219]). Wave energy dissipation associated with breaking begins when |∂x(hu)| ≥ U1

and continues as long as |∂x(hu)| ≥ U2. The proposed definition of the viscosity ν requires

a positive value of B. To satisfies that, for negative values of B, the viscosity ν is set to zero.

It is a known fact that using an explicit scheme for a parabolic equation requires a time step restriction of type ∆t = O(∆x2_{). The breaking mechanism has this nature and}

this would mean a too restrictive time step. This is the reason for choosing an implicit discretization of this term. This can be solved by considering an implicit discretization

of the eddy viscosity term, evaluating the term ∂x(νh∂xu) at the right-hand side of the

momentum discrete equation in (NH-1L). The implicit discretization involves solving an extra tridiagonal linear system, leading to a loss of efficiency.

In [112] a new efficient treatment of the eddy viscosity term was present by C. Escalante et al. for depth averaged non-hydrostatic models. To do that, the horizontal momentum equation is rewritten as ∂t(hu) + ∂x  hu2+ 1 2gh 2_{+ hp − νh∂} xu  = (gh + 2p) ∂xH − τb,x (1.5.2) and define p =p + ν∂_e xu. (1.5.3)

Thus, replacing p by _ep + νux, the system can be rewritten as

                                 ∂th + ∂x(hu) = 0, ∂t(hu) + ∂x  hu2₊ 1 2gh 2_{+ h} e p  = (gh + 2p) ∂_e xH − τb,x+ 2ν∂xu∂xH, ∂t(hw) + ∂x(uhw) = 2p + 2ν∂e xu, ∂xu + w − wb h/2 = 0, wb = −∂tH − u∂xH. (1.5.4)

Note that the terms 2ν∂xux∂xH, in the horizontal momentum equation, and 2ν∂xu, in the

vertical velocity equation, are essentially first order derivatives of u, and can be discretized explicitly without the aforementioned severe restriction on the CFL condition. That gives us an efficient discretization of the eddy viscosity terms.

Moreover, since the incompressibility equation in (1.5.4) holds, then one has that

2ν∂xu = 4ν

wb − w

h .

Since it is of interest in this dissertation to obtain relatively simple systems, a simplification on the breaking mechanism described in (1.5.4) is proposed by assuming a mild slope bottom for the breaking terms, that is

∂xH ≈ 0, ∂tH ≈ 0, and thus, wb ≈ 0.

system (NH-1L), only the source term S(U ) has to be modified as follows S(U ) = −      0 τb Rbr      (1.5.5) Rbr being Rbr = −4B|∂x(hu)|w, B = 1 − ∂x(hu) U1 . (1.5.6)

In Chapter 4 numerical experiments and comparisons with laboratory data will show that breaking waves can be accurately described with this simple and efficient breaking mechanism.

Due to the form of the breaking terms, they can be discretized as the friction with the bottom terms appearing at the horizontal momentum equation, τb. In this thesis this

terms will be discretized in a semi-implicit manner.

Note that following the same procedure, a similar breaking mechanism can be introduced for the system (NHyp) modifying the source term

S(U ) = −          0 τb − 2p + Rbr 2c2_{(w + ∂} tH)          , (1.5.7)

and similarly for the system (YAM) replacing the vertical velocity equation by ∂tw = 2

p

h + Rbr, being Rbr the term defined in (1.5.6)

Remark 1.5.1. Reinterpretation of the eddy viscosity approach: • Let us consider the vertical component of the stress-tensor

τzz = 2ν(x, z, t)∂e zW (x, z, t),

where ν(x, z, t) is a positive function and W is the vertical velocity. Now, following_e the same process carried out in [259] to depth-average the vertical momentum equation from Euler equations. To do so, let us integrate the vertical component of the stress-tensor along z ∈ [−H, η]:

Z η −H ∂zτzz dz = 2 Z η −H ∂zeν∂zW +eν∂zzW dz.

Due to the assumption of a linear vertical profile for the vertical velocity W, then ∂zzW = 0 and ∂zW does not depend on z and thus

Z η

−H

∂zτzz dz = 2ς∂zW,

where ς = R_−Hη ∂zeν is the eddy viscosity. Using again the linearity of the vertical profile for W, and the no-penetration boundary condition:

∂zW = w − wb h/2 , w = 1 h Z η −H W dz.

Using the incompressible condition: w − wb

h/2 = −∂xu. Thus, Z η

−H

∂zτzz dz = −4ςux.

Finally, it remains to choose a closure for ς in the system with the described depth- averaged vertical component of the stress-tensor.

• Note that seting

ς = −1 2ν,

where ν is the eddy viscosity described in (1.5.4), then coincides with the same term 2ν∂xu introduced in the vertical momentum equation in (1.5.4).

Two space dimension

A new, simple and efficient breaking mechanism can be considered for two dimensional domains following the procedure presented in Remark 1.5.1. The same process as in [21] is used, to depth-average the vertical component of the stress-tensor. Due to the linearity on the vertical profile of the vertical velocity within the fluid layer, it is defined:

Rbr = Z η −H ∂zτzz dz = 2 Z η −H ∂zeν∂zW +eν∂zzW dz = 2ς∂zW,

where ς = −R_−Hη ∂zν is the eddy viscosity. Using the incompressibility condition frome (YAM-2D):

Rbr = 4ς

w − wb

h , wb = −u∂xH − v∂xH − ∂tH.

In this dissertation, as in [112], [219] and inspired in the definition of ς in Remark 1.5.1, ς is choosen to be

ς = −1

where B is a coefficient related to the breaking criteria. Following a natural and simpler extension of the criteria proposed by [219],

B = 1 −∂x(hu) + ∂y(hv) U1

,

wave energy dissipation associated with breaking begins when |∂x(hu) + ∂y(hv)| ≥ U1 and

continues as long as |∂x(∂x(hu) + ∂y(hv)| ≥ U2, where

U1 = B1

p

gh, U2 = B2

p gh,

denote the flow speeds at the onset and termination of the wave-breaking process and B1, B2 are calibration coefficients that should be calibrated through laboratory

experiments. In this work, as in [112], [219], B1 = 0.5 and B2 = 0.15 for all the test

cases studied.

Similarly to the case of one dimensional domains, a simplification is made by assuming a mild slope bottom for the breaking terms, that is

∇H ≈ 0, ∂tH ≈ 0, and thus, wb ≈ 0.

Therefore the breaking mechanism considered consists in adding at the right hand side of the vertical equation of the systems (YAM-2D) or (NHyp-2D), the term:

Rbr = −4B|∂x(hu) + ∂y(hv)|w, B = 1 −

∂x(hu) + ∂y(hv)

U1

. (1.5.8)

A breaking mechanism for the two layer non-hydrostatic pressure

system (NH-2L)

Although the generalization of the breaking mechanism presented above to the case of the two layer system straightforward, in [118] C. Escalante et al. designed an ad-hoc breaking mechanism for the case of the two layer non-hydrostatic pressure system (NH-2L). To do so, following the same ideas as for the case of the one layer system, let us consider the vertical component of the stress-tensor

τzz = 2eν∂zw,

where _eν(x, z, t) is a positive function. The same process as in Subsection 1.2.1 is used to depth-average the vertical component of the stress-tensor. By taking into account the incompressibility condition and that the vertical velocity has a linear profile within each layer:

Z

Lα

where ςα =

R

Lα∂zeν.

Now the system is closed defining ςα(x, t). In the subsequent, ςα are computed

assuming the equations in one space domains, to better clarify the procedure. To do so, it is observed that the linear combination of the non-hydrostatic pressures, pb and pI

appearing at the right hand side of the vertical equations, can be expressed in terms of uα

and its derivatives ∂xxuα, ∂xtuα, ∂xuα and ∂tuα. The proposed election of ςα in this work

is based on the idea of cancelling those aforementioned ∂xuα terms with −ςαhα∂xuα. The

procedure is detailed in Appendix A. The following definition is proposed:        ς1 = w1− 3u1∂xH + 2∂tH, ς2 = w2+ 3u2∂xzI+ 2∂tH. (1.5.10)

Note that the deduced breaking terms are essentially first order derivatives of u1, u2,

and can be discretized explicitly, as in the spirit of the case for the one-layer systems. Finally, a breaking criteria to switch on/off the dissipation is needed. Following a natural and simpler extension of the criteria proposed by [219] and used in the case of the one layer system, wave energy dissipation associated with breaking begins when |∂x(l1hu1+ l2hu2)| ≥ U1 and continues as long as |∂x(l1hu1+ l2hu2)| ≥ U2, where

U1 = B1

p

gh, U2 = B2

p gh,

denote the flow speeds at the onset and termination of the wave-breaking process and B1, B2 are calibration coefficients that should be calibrated through laboratory

experiments. In this work, as in [219] B1 = 0.5 and B2 = 0.15 for al the test cases

studied.

Note that the breaking criteria to switch on/off the dissipation is a simplified version of the one proposed in [219], that includes some improvements such as: taking into account a residence time for the activation/deactivation of the criteria to accounting a continuous dissipation; or computing in the breaking criteria |∂x(hu)| as 1₂(|∂x(hu)| + ∂x(hu)) which

would automatically become zero on the back of the crest of the wave.

The breaking mechanism proposed in this work can be considered with this improved breaking criteria given in [219], as well as the one proposed in [163], which are more sophisticated and expensive.

Nevertheless, although a fast and straightforward breaking criterion has been chosen, the numerical tests in Chapter 4 will show that this technique performs adequately. Moreover, the simple breaking mechanism considered in this work: corrects the classical overshoot that dispersive models present for the run-up of waves (see Fig. 4.10); ensures the grid convergence even if breaking mechanism is switching on/off during the experiment (see Fig. 4.13). Although the simple breaking criterion does not detect hydraulic jumps, is observed that the proposed system without the breaking dissipation can handle well

with hydraulic jumps. In any case, a more sophisticated breaking detector that reveals hydraulic jumps can be considered as well.