1. Introduction
In the contest of hydraulic engineering, the simulation of hydrodynamic fields, turbulence, and concentration fields of suspended solid particles under wave breaking permits the analysis of the effects produced by the structures on sea bottom and shoreline modifications. The definition of turbulent closure relations under wave breaking is significant in order to adequately represent sediment particle suspension and settling phenomena.
One of the most used approaches is based on depth-averaged motion equations [
1,
2,
3,
4], which are obtained by assuming a simplified distribution of the hydrodynamic quantities along the vertical direction (depth-averaged models).
In order to quantify the dissipation mechanism in the surf zone, two strategies are considered in the literature. The first strategy is based on an eddy viscosity approach; in particular, Kazolea and Ricchiuto [
5] used a turbulent kinetic energy equation to calculate a breaking viscosity coefficient.
In the context of the second strategy, the extended Boussinesq equations [
1,
2,
3] are used to simulate the propagation of waves in deep water, where dispersive effects are dominant; in the surf zone, the nonlinear shallow water equations are used to simulate shoaling and wave breaking, because the dispersive terms of the Boussinesq equations are switched off. In the surf zone, the nonlinear shallow water equations, integrated by a shock-capturing numerical scheme, are used to simulate the wave breaking, where the nonlinear effects are dominant [
4]: the dissipation of the total energy obtained in a shallow water shock is used to implicitly model the energy dissipation due to the breaking. It is evident that it is necessary to introduce a criterion to establish where and when to switch from one system of equations to the other.
In the context defined by the second strategy, the turbulence models are used just in order to determine a distribution of the eddy viscosity coefficient that does not intervene in the simulation of the wave resolving depth-averaged velocity fields but intervene in the simulation of only the concentration fields of suspended solid particles to follow the sea bottom evolution. In this case, the turbulent closure relations that derive from the calculation of the eddy viscosity vertical distribution are carried out by following the line proposed by Fredsøe et al. and Deigaard et al. [
6,
7] and used by Rakha [
8]. In particular, under breaking waves, the instantaneous eddy viscosity vertical distribution is calculated by taking into account the turbulence contribution due to the wave boundary layer and the current and wave breaking [
3].
These models are not able to adequately represent the three-dimensional hydrodynamic quantities.
The first three-dimensional models for free surface flows numerically solve the Navier–Stokes equations by adopting the volume of fluid technique (VOF) to track the location of the free surface [
9,
10]: in the VOF technique, the calculus cells are arbitrarily crossed by the vertical fluxes. Therefore, the pressure and kinematic boundary conditions are not correctly assigned on the free surface.
Another category of models overcomes the above-mentioned difficulty by mapping the physical domain that varies in time along the vertical direction in order to follow the free surface elevation (
-coordinate transformation) [
11] in a computational domain, which has always a rectangular prismatic shape. In this way it is possible to correctly assign the pressure and kinematic boundary condition on the upper face of the top computational cell [
12,
13].
These two above-mentioned categories of models empirically introduce, in the surf zone, an appropriate energy dissipation, so that it is possible to represent the decrease of the wave height in the aforementioned zone. Consequently, the location of the initial wave breaking point is settled “a priori”. The definition of the point is where the empirical energy dissipation must be introduced, and therefore the identification of the initial wave breaking point is settled “a priori”.
Shock-capturing methods make it possible to model the energy dissipation due to the breaking by the total energy dissipation obtained in a shallow water shock. The -coordinate shock-capturing methods locate wave breaking without requiring any “a priori” criterion.
In this paper we adopt the new approach proposed by Gallerano et al. [
14] in which the motion equations are expressed in terms of variables that are Cartesian based: only the vertical coordinate is expressed as a function of a time-dependent curvilinear coordinate that follows the free surface movements.
In this model the motion equations are numerically solved, on a time dependent curvilinear coordinate system, where the vertical coordinate can vary over time, by using a finite-volume shock-capturing scheme, which uses an approximate HLL (Harten, Lax and van Leer)-type Riemann solver [
15].
The shock-capturing scheme [
14] does not need to use any “a priori” criterion to identify the location of the initial wave breaking point, and it is able to correctly assign the boundary conditions on the free surface. In this model, by using a computational domain that has a limited number of points along the vertical direction, the numerical wave height fits the experimental measurements before the breaking point and the wave breaking is tracked correctly.
Ma et al. [
12] demonstrated that in general the shock-capturing model, defined in the context of the
-coordinate, underestimates the energy dissipation at the breaking point and consequently overestimates the wave height in the surf zone.
Therefore, it is necessary to introduce turbulence models that are able to adequately represent the specific energy dissipation at the breaking point.
The definition of turbulent closure relations under wave breaking implies the necessity to consider the production of turbulent kinetic energy associated with the breaking in order to represent sediment particle suspension and settling phenomena.
In order to develop a wave resolving model for nearshore suspended sediment transport, Ma et al. [
16] implemented a
model in a three-dimensional
-coordinate model to simulate the velocity and particles concentration fields.
In general, standard models are located in the context of Reynolds-Averaged Navier–Stokes equations (RANS), in which all the unsteady velocity fluctuations are expelled from the simulation of the Reynolds time-averaged velocity fields.
In this approach the effects of all unsteady periodic vortex structures and unsteady stochastic velocity fluctuations are represented in terms of the total transfer of energy dissipation from the averaged motion to all the scales of turbulent motion, through the Reynolds tensor.
In the standard models, the Reynolds tensor is related to all the energy of the unsteady periodic vortex structures and unsteady stochastic velocity fluctuations that are expelled from the representation of the Reynolds averaged motion. Consequently, these models are not able to directly simulate unsteady periodic vortex structures because they overestimate the Reynolds tensor and produce an excessive dissipation of energy and non-physical wave height reduction.
The velocity field in the surf zone is very complex and is characterized by unsteady periodic vortex structures and unsteady stochastic velocity fluctuations.
Outside the surf zone, the production of turbulent kinetic energy is limited to the oscillating wave boundary layer; inside the surf zone there is the production of turbulent kinetic energy in the oscillating wave boundary layer and the production of turbulent kinetic energy near the wave crest, and between the oscillating wave boundary layer and the wave crest, there is a phenomenon of mainly turbulent kinetic energy dissipation.
In general, the stochastic turbulent fluctuations are superimposed on the periodic unsteady motion of the vortex structures. Bosch and Rodi [
17] proposed the simulation of the velocity fields characterized by the above-mentioned unsteady vortex structures by decomposing the instantaneous flow quantities in a time mean component, in a periodic component, and in a turbulent fluctuating component. Following this approach, the sum of the time mean and the periodic part gives rise to the ensemble-average component of the flow quantities. The latter are calculated by the numerical integration of the ensemble-average continuity and momentum equations; the complete spectrum of the stochastic motion is simulated by a statistical turbulence model. The models coherent with the above-mentioned approach are named in the literature as Unsteady Reynolds-Averaged Navier–Stokes (URANS) models.
In this paper, we propose, for the simulation of the wave motion and the turbulence under wave breaking, to expel the turbulent kinetic energy associated with unsteady stochastic velocity fluctuations from the ensemble-average motion. This model can be considered in the context of the Unsteady Reynolds-Averaged Navier–Stokes (URANS) models. The modifications to the model with respect to the standard one consist of the following:
The closure relation of the production of turbulent kinetic energy takes into account the nonlinear terms, following the procedure of [
18,
19];
The closure relation of the destruction of the dissipation rate is expressed by means of a dynamic procedure, used by Yakhot et al. and Derakty et al. [
20,
21];
The production of the turbulent kinetic energy dissipation is calculated by a dynamic procedure. In the surf zone the production of turbulent kinetic energy dissipation is limited by the dynamic coefficient, related to the local derivative of the free surface. Consequently, differently from the standard model, it is possible to limit the non-physical decrease of the wave height.
The paper is structured as follows: in
Section 2 and
Section 3 are presented the motion equations and modified turbulent
model, in which vector and tensor quantities are expressed in Cartesian components on a general boundary conforming curvilinear time dependent coordinate system that includes a time-varying vertical coordinate. In
Section 4 the results are shown and discussed. Conclusions are drawn in
Section 5.
2. Motion Equations
We consider a transformation , from a Cartesian coordinate system , to a curvilinear coordinate system and the inverse transformation, .
In this paper we adopt a particular transformation from Cartesian to curvilinear coordinates in order to follow the free surface movements [
14].
in which
is the total water depth;
is the undisturbed water depth and
is the free surface elevation;
is the horizontal boundary conforming curvilinear coordinates and
is the time varying vertical coordinate that spans from
(at the bottom) to
(at the free surface). The irregular varying domain in the physical space is mapped into a regular fixed domain in the transformed space, as shown in
Figure 1.
Let
and
be the
covariant and contravariant base vectors, respectively, and let
and
be the metric tensor and its inverse, respectively, whose components are
The Jacobian of the transformation is given by .
The equations are expressed in terms of
e
, the spatial averaged values over volume elements defined in the form
where
is the contour line of the control area
in the transformed space.
The motion equations, expressed in integral form in a time-dependent curvilinear coordinate system, are as follows:
where
is the fluid velocity;
is the velocity of the moving coordinate lines;
is the outer product;
is the scalar product; and
,
and
are, respectively, the constant of gravity, the fluid density, and the turbulent eddy viscosity.
is the dynamic pressure;
is the stress tensor and
is the strain rate tensor; the
(
are cyclic) is the contour line of the area
on which
is constant and which is located at the larger and smaller value of
, respectively.
where
is the contour line of area
on which
is constant and which is located at the larger and smaller value of
, respectively.
Equations (5) and (6) are numerically solved, on a time dependent curvilinear coordinate system, where the vertical coordinate varies in time, by a finite volume shock capturing scheme, which uses an approximate HLL-type Riemann solver [
15].