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The ADV is a remote-sensing, three-dimensional velocity sensor, originally developed and tested for use in physical model facilities (Kraus et al., 1994;

Lohrmann et al., 1994), and its operation is based on the Doppler shift effect.

High levels of noise and spikes have been reported in measurements with ADV velocimeter (Nikora and Goring, 1998; McLelland and Nicholas, 2000). In turbulent flows, the ADV velocity fluctuations characterise the combined effects of the Doppler noise, signal aliasing, velocity fluctuations, installation vibrations and other disturbances (Chanson et al., 2008). Both the spectra and the probability distributions indicate that as a first approximation, the Doppler noise is Gaussian white noise (Nikora and Goring, 1998). Chanson et al. (2002) discussed the noise of an ADV system. The main source of Doppler noise is the random target distribution in the sampling volume, which induces an instantaneous Lagrangian deviation of the position of the target from the mean position determined by the spatially averaged velocity. This creates instantaneous random Doppler-phase noise that is added to the spatially averaged Doppler phase (Doroudian et al., 2010). McLelland and Nicholas (2000) explained the physical processes causing spikes by aliasing of the Doppler signal. For steady flow situations a few techniques to eliminate the spikes have been developed (Nikora and Goring, 1998; Goring and Nikora, 2002; Wahl, 2003). Wahl (2000) also developed the WinADV program for post-processing and analysis of ADV data. Due to the presence of intermittent spikes contaminating time series measured by acoustic Doppler velocimetry, statistical properties, and power spectral density of such data sets can have unrealistic values (Parsheh et al., 2010). These spikes appear when the flow velocity exceeds the preset velocity range of the equipment, the turbulence intensities are high, or there are contaminations from previous pulses reflected from the flow boundaries.

Voulgaris and Trowbridge (1998) evaluated accuracy of the acoustic Doppler velocimeter (ADV) by measurements of open channel flow using an ADV and a laser Doppler velocimeter. The results were qualitative and showed a good agreement between the mean values obtained by the two sensors. They examined the degree of accuracy in measuring mean velocity, variance, and covariance of the flow field together with the effect of the proximity to the boundary in flow measurements.

The types and magnitudes of noise involved in flow measurements using an ADV sensor were also examined. They concluded that ADV is suitable for accurate measurements of mean flow even at positions close to the boundary. However, Dombroski and Crimaldi (2007) stated that the accuracy of ADVs is limited when making measurements close to the bed or in flows where large spatial gradients are present. To validate the use of ADVs for the measurement of turbulent flows, Khorsandi et al. (2012) conducted experiments in an axisymmetric turbulent jet and in approximately homogenous isotropic turbulence with zero mean flow. The jet experiments showed that the horizontal RMS velocities measured by the ADV were overestimated compared to both flying hot-film anemometry measurements and the accepted values in the literature. However, the vertical component of the RMS velocity agreed well with those of other studies. To correct the data, post-processing filters and a Doppler noise-reduction method were applied to the jet data. Despite decreasing the RMS velocities, they remained erroneously higher than the accepted values. Their results showed no clear relationship between the Doppler noise and the mean flow.

Goring and Nikora (2002) assumed that good data can be found within a cluster and that points located outside the cluster are spikes. They suggested a method based on iterative phase-space thresholding as the most suitable solution for spike detection. They reported that the method worked extremely well confirmed by successful application of the method to numerous ADV data. Doroudian et al. (2010) combined a spike-removal procedure on the beam velocities with a noise-reduction method on the flow velocities to improve turbulence measurements with ADVs. It was shown that spikes were best removed from ADV beam velocity data before calculating flow velocities, thereby correcting all three flow velocity components at the source. Yin et al. (2001) developed a method to analyse a time series of velocity signals in order to obtain a time series of a moving-averaged velocity. They described the turbulent velocity fluctuations with a Gaussian probability distribution,

and the method was developed to calculate the local mean velocity of unsteady tidal flow from the experimental data using ADV with noisy signals. de Nijs et al. (2009) used a 10 minute moving average filter to determine the main flow and turbulence statistics.

Snyder and Castro (1999) assessed the use of ADVs in a stratified tank with variable density of saltwater over the transmitting path of the probe. The tests showed that the indicated distance to the boundary was directly proportional to the specific gravity of the saltwater. A correction scheme was developed for highly concentrated saltwater, and its suitability was verified in homogeneous saltwater solutions.

2.3

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If a model is only to be applied for a strongly stratified water body, so that it maintains a two-layer system of flow throughout the simulation, a two-layer system, which represents the simplest case of a stratified flow (Harleman, 1961), may be considered. The flow in a two-layer system is divided into two homogenous layers with only a density difference at the interface (Karelse et al., 1974). Multi-layer systems have been discussed in Vreugdenhil (1994). However, these models cannot simulate problems involving three-dimensionality and vertical eddies or flows with continuous-density gradients. In two-layer numerical model simulations of saline intrusion and sediment transport in the Rotterdam Waterways, HR Wallingford (1979) concluded that tidal processes in the waterways can be simulated satisfactorily for engineering purposes by schematising the flow into two layers. The main process, which determines the longitudinal and vertical distribution of salt within the layered system, is the vertical turbulent exchange of salt across the interface. Castro et al. (2007) used a two-layer, finite volume model for simulations of stratified flows through channels with irregular geometries. The flow was assumed to be composed of two shallow layers of immiscible fluids with constant densities, and was presumed to be one dimensional. In a similar study, Castro et al. (2004) investigated maximal and tidally induced two-layer exchange flows through the Strait of Gibraltar. Chen and Peng (2006) applied a two-dimensional explicit finite volume method for solving the two-layer shallow water equations for confluence simulation.

A successful modelling of strong advection is one of the most challenging problems. Although traditional first-order finite difference methods are monotonic and stable, they are also strongly dissipative, and suffer from severe inaccuracies due to truncation error. On the other hand, traditional high-order difference methods are less dissipative but are susceptible to numerical instabilities, which cause non-physical oscillations in advection-dominated regions and zones of large gradient of the variables. Incorporating artificial diffusion into the numerical scheme to dampen spurious oscillations in regions of large gradients, results in smearing out the solution elsewhere. The ULTIMATE QUICKEST scheme (Leonard, 1991) gives results which are probably entirely adequate for most practical situations (Fig. 2.1). In a study of comparing a series of numerical schemes for one-dimensional advection-diffusion problems, Wang and Hutter (2001) concluded that the modified TVD Lax-Friedrichs method is the most competent method for advectively-dominated problems with a steep spatial gradient of the variables. However, they are at most first-order accurate at local extrema and highly dependent on the slope limiters used in the model in some cases. Bruneau et al. (1997) also proposed TVD schemes from a family of second- and third-order Lax-Wendroff-type schemes.

Figure (2.1) - ULTIMATE QUICKEST results (adapted from Leonard, 1991) Finite difference method has been deployed in a number of numerical models using structured grids (POM, Blumberg and Mellor, 1987; TRIM, Casulli and Cheng, 1992; Lin and Falconer, 1996; ECOMSED, HydroQual, Inc., 2004; ROMS, Shchepetkin and McWilliams, 2005; NCOM, Barron et al., 2006; Anthonio and Hall, 2006). Unstructured grids have also been employed in a number of studies deploying finite element method (ADCIRC, Luettich et al., 1991; QUODDY, Lynch and Werner, 1991; SEOM, Iskandarani et al., 2003), or using hybrid approaches involving finite volume method (UnTRIM, Casulli and Walters, 2000; FVCOM, Chen et al., 2003; ELCIRC, Zhang et al., 2004).

2.3.1 2DH Models

Two-dimensional horizontal (2DH) numerical models are very popular due to their acceptable accuracy and high efficiency. There is a wide variety of numerical models developed on the basis of depth-averaged Navier-Stokes equations, for simulating the tidal circulation and flushing in harbours (Falconer, 1980b), for steady shallow water flows (Zhou, 1995), and for modelling water quality processes and fate and transport of oil spills (Sarhadi Zadeh and Hejazi, 2010, 2012). Estuaries are transitional areas which trap significant quantities of particulate and dissolved matter through a wide variety of physical and biogeochemical processes. Harbour planform effects and investigation of effective harbour geometry parameters on cohesive sediment transport and sedimentation has also been studied by means of 2DH models (Mojabi and Hejazi, 2011; Mojabi et al., 2011, 2013). Karimi et al. (2012) presented an integrated 2DH numerical model for interactive simulations of oil spills, sedimentation and transport of oil in sediment laden marine waters. In some cases depth integrated models are preferred as they are computationally economical, easier to program and provide sufficiently accurate results. In depth-averaged models, however, the vertical distribution of currents is not known and the bed friction is expressed in terms of the mean velocity rather than the velocity near the bottom.

In addition to employing high-order numerical methods to increase the accuracy of solution of shallow-water equations, nested grid refinement may be implemented in order to improve the solution through resolution of computational grids. Peng et al. (2010) developed a nested-block, finite volume based Cartesian grid, method for simulating the unsteady viscous incompressible flows with complex immersed boundaries using a two-step fractional step procedure. Hadaeghi and Hejazi (2013) developed a 2DH finite volume nested model to solve vorticity transport in viscous fluid flows. The solution was based on a fractional step method and an iterative successive-over-relaxation (SOR) scheme was employed to solve the two-way nested grid normalised equations. The discretisation scheme provided second-order accuracy in space and time.

Attempting to obtain higher efficiency and affording accuracy, three-dimensional Reynolds averaged Navier-Stokes equations may be depth-averaged but retain the non-hydrostatic pressure terms (Stelling and Zijlema, 2003; Walters, 2005;

Bai and Cheung, 2013). Marshall et al. (1997) discussed the use of ocean models based on hydrostatic, quasi-hydrostatic, and non-hydrostatic equation sets.

Nece and Falconer (1989a) stated that in complex studies where vertical density gradients are important and single-fluid models are not appropriate, then the three-dimensional equations should be solved directly. Lin and Falconer (1997) presented a three-dimensional layer-integrated model with flooding and drying, and studied the tidal motion in lower reaches of the Humber estuary.

2.3.2 2DV Models

Perrels and Karelse (1982) introduced a finite difference two-dimensional, laterally averaged model with hydrostatic pressure assumption, for salt intrusion simulations in estuaries. Daubert et al. (1982) and Daubert and Cahouet (1984) developed 2DV models using a three-step fractional method. Haque and Berlamont (1998) developed a 2DV finite element model to predict the flow, density, and turbulence fields of a stratified tidal medium. Zhou and Stansby (1999) developed a 2DV numerical model based on arbitrary Lagrangian-Eulerian (ALE) description and in the σ-coordinate system, using a semi-implicit time-stepping method for solving unsteady Navier-Stokes equations. Yuan and Wu (2004a) developed an implicit finite difference model in the σ-coordinate system for non-hydrostatic, two-dimensional vertical plane free surface flows using the solution method presented in Namin et al. (2001). Zijlema and Stelling (2005) introduced a semi-implicit 2DV numerical model which solves the incompressible Euler equations with the aid of a projection method and splitting the pressure into hydrostatic and non-hydrostatic components by the finite volume technique. More 2DV models were developed by Stelling and Busnelli (2001), Stansby and Zhou (1998), and Memarzadeh and Hejazi (2012). Hejazi et al. (2013) introduced a 2DV numerical model to simulate wave-mud interaction. The fully non-linear Navier-Stokes equations with complete set of kinematic and dynamic boundary conditions at free surface and interface with the two-equation k-ε turbulence model with buoyancy terms were solved. A finite volume method based on an ALE description was utilised for the simulation of wave motion in a combined system of water and viscous mud layer. The propagation of irregular waves in a two-layer viscous fluid system and interaction of water waves with a muddy bed bounded below by a horizontal rigid plane was also investigated by the model and spectral method (Hejazi et al., 2014b). The model was further modified to study the interaction between wave, current and mud bed (Hejazi et al., 2014a).

2.3.3 3D Models

Undoubtedly the vast variety of the three-dimensional models developed to date and extensive features of their groups and each cannot be distinctly categorised.

However, to formulate the present review, and to have more useful conclusions, this section follows the main parts of the structure of the numerical model developed herein which have been identified to be the solution method, the pressure treatment approach, and the coordinate system.