Ajuste de la balanza

a Subgrid-Scale model (SGS). LES is computationally more expensive than RANS but provides better information, it can be also used to cal-ibrate RANS models. The use of parallel-architecture supercomputer makes possible the application of large-eddy simulation to real-scale coastal flow, where the complex flow physics requires a highly de-tailed analysis.

The first application of LES to the analysis of mixing in the oceans, was carried out by Skyllingstad and Denbo, 1995: in this work the authors studied the dynamics of plume under convective conditions in a simple Cartesian domain. Following, in D. Wang et al., 1996 authors used LES to analyse mixing in the open ocean. A review of applications of LES methodology for marine flow is presented in Scotti,2010.

In this thesis, water mixing and transport in the Panzano Bay is carried out by using a novel, state-of-art LES model (LES-COAST) (IEFLUIDS University of Trieste,Roman, Stipcich, et al.,2010), a high definition numerical model suitable for harbour and coastal areas.

The model has been recently applied to study water mixing and re-newal in Muggia Bay, the industrial harbour of Trieste in Italy (Petro-nio et al.,2013) and in Barcelona Harbour in Spain (Galea et al.,2014).

LES-COAST model solves the filtered form of three dimensional, non hydrostatic Navier-Stokes equations under the Boussinesq ap-proximation and the transport equation for salinity and temperature.

It makes use of large-eddy simulation approach to parametrize tur-bulence, the variables are filtered by application of a low-pass filter function represented by the size of the cells. The subgrid-scale fluxes (SGS), which come out from the filtering operation, are parametrized by a two-eddy viscosity anisotropic Smagorinsky model, to better adapt to coastal flow in which the horizontal length scale is much larger than the vertical one (Roman, Stipcich, et al., 2010). Complex geometry that characterizes coastal flows is treated by a combination of curvilinear grid and Immersed Boundary Method (IBM) (Roman, Napoli, et al.,2009).

In this section we describe LES-COAST model equations (Sec. 3.1 and3.2), we discuss the boundary conditions applied to simulate wa-ter dynamics in harbours and coastal areas (Sec. 3.3, 3.4, 3.5), finally we illustrate the numerical method implemented (Sec.3.6).

3.1 the governing equations

The fluid flow motion is governed by Navier-Stokes equations. In many environmental processes density anomalies are small compared with velocity gradient. Therefore Navier-Stokes equations can be written under the Boussinesq approximation, according to which den-sity variations can be neglected in continuity and in momentum equa-tions, except in the gravity term; so the fluid can be considered incom-pressible, meaning that volume variations are considered negligible in the fluid body.

3.1 the governing equations 34

LES-RESOLVED SGS-MODEL

∆

Figure 17: LES key idea: turbulent structures with length scale bigger than the grid size are directly resolved, the smaller ones are parametrized by SGS-model.

Using LES methodology, the variables of Navier-Stokes equations are filtered through an application of a low-pass filter. In three di-mensions the application of the filter to the variable u reads as:

u(x) =

Z

G(x, x⁰)u(x⁰)dx⁰; (76) where x is the Cartesian coordinate vector, G is the filter function,∆ is the filter width. We make use of the top-hat filter function:

G(x) =

 1/∆ if|x| <_∆/2;

0 otherwise; (77)

where the filter width ∆ is the grid cell size; turbulence structures, with length scales smaller than∆, have to be modelled. The filtering operation decomposes the variable u into the sum of two components:

u = u+u_SGS, u is the filtered (or resolved) component and it repre-sents large-eddy motion, while u_SGS is the residual one (or subgrid-scale), which is treated by means of a SGS model described in section 3.2. The concept of large-eddy simulation is shown in figure17, large scale eddies continuously break up into smaller eddies until they are too small and they dissipate into heat. The eddies with length scale greater than grid size ∆ are directly resolved, while the smaller ones are modelled.

In the Cartesian coordinate frame of reference the filtered form of Navier-Stokes equations under the Boussinesq approximation, to-gether with scalar transport equations, read as:

∂u_j

where ‘−‘ represents the filtering operation, u_i represents the i^th -component of the Cartesian velocity vector(u, v, w), x_irepresents the

3.1 the governing equations 35

Figure 18: Example of coordinate transformation from the physical to the computational space.

i^th-component of the Cartesian coordinates(x, y, z)(in the present the-sis x and z denote horizontal direction, and y vertical one), t is time, ρ₀ is the reference density, p is the filtered pressure, ν is the kine-matic viscosity, e_ijk is the Levi-Civita tensor, Ωi is the i^th-component of the Earth rotation vector, ∆ρ is the density anomaly, gi is the i^th -component of the gravity vector, and τ_ijis the SGS stress which comes from the non linearity of the advection term. T is temperature (K), S is salinity (PSU), k^T and k^S are respectively temperature and salinity diffusivity, λ^T_j are the SGS temperature fluxes, while λ^S_j are the salin-ity ones. Temperature and salinsalin-ity are related to denssalin-ity through the state equation:

∆ρ

ρ₀ = ^ρ−ρ₀

ρ₀ = −β^T(T−T0) +β^S(S−S0); (82) where ρ₀is the reference density at the temperature T₀and salinity S₀; β^T and β^S are respectively the coefficient of temperature expansion and saline contraction.

Equations (78)-(81) are written in Cartesian frame of reference. Since coastal area are characterized by complex geometry we use curvilin-ear coordinates to create a domain whose boundaries follow coastline and bathymetry. An example of coordinate transformation from the physical space to the computational one, is shown in figure18. Hence, in the curvilinear coordinate framework, equations (78)-(81) read as:

∂U_m of the Jacobian of the coordinate transformation, U_m is the volume