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Direct pressure and pressure-marching methods

Direct methods are those in which pressure is solved directly (sometimes exactly). This is in contradistinction to the projection method where the value for the pressure is solved approximately by the iterative solution of an elliptic Poisson equation (PPE). Because the PPE is elliptic it necessitates the use of an implicit solution technique, thus introducing another layer of complexity to parallel codes when compared with simple explicit techniques. The explicit time-marching (ETM) of pressure falls into the direct methods class as defined here. This is because it is not an iterative method. The ETM approach, which includes both the penalty approach of Temarn (1984) and the artificial compressiblitv approach devised independly by Yanenko (1971) and Chorin (1967), is particularly attractive for free surface flows that involve (potentially) violent fluid structure interactions. The equations in ETM approaches, such as the penalty approach (Teniam, 1984), the artificial compressibility approach (Chorin, 1967; Yanenko, 1971), or combinations of these two approaches, are necessarily stiff ancl thus necessitate the use of a small time-step to ensure numerical stability. This means that such approaches are not competitive with implicit iterative-type approaches for flows without free surfaces, relatively slow moving free surface flows or flows that tend toward a steady state (Dukowicz, 1994). For highly unsteady free surface flows, however, rapid change (which can only be captured by a small time-step) is an inherent feature of the flow motion; thus, the main disadvantage of ETM approaches, namely the requirement of a small time-step, is no longer a disadvantage in this context. Moreover, large benefits can be gained from combining the penalty and artificial compressibility approaches together; this was first understood by Yanenko et al. (1984) who combine the two aforementioned methods by adding a linear combination of the pressure and pressure evolution terms to the velocity divergence. Later this technique was refined and improved by Ramshaw and Mesina (1991) who added the pressure and pressure evolution contributions to the pressure itself. Very little work has been done on combined penalty-artificial compressibility approaches applied to free surface flows. It is noted here that this avenue appears to offer a lot of promise as purely explicit codes are relatively simple to write and straightforward to parallelize.

Machine learning

Machine learning (ML) is a series of techniques that allow computers to perform tasks without being explicitly programmed to do so, by “learning” insights from input datasets. ML has a vast potential and is currently a hot topic in multiple scientific fields; CFD and fluid mechanics in general are not an exception. Although the variety of approaches that machine learning encompasses is very wide, the number of works of machine learning applied in coastal engineering is not very large yet. Two review papers, one focussed in general ML and another focussed in Deep Reinforcement Learning applied to fluid mechanics can be found in Brunton et al. (2019) and Gamier et al. (2019). Other examples of particular applications are provided as follows.

ML has proven to be a very useful tool to rationalise the use of computational resources in CFD. In this sense, Stefanakis et al. (2014) explored the maximum runup behind a conical island, a system that depends on multiple inputs, and ML identified the interdependence between them and allowed reducing the total number of simulations required to gain valuable insights. Hennigh (2017) used ML to reduce the computational time and memory required to run Lattice Boltzmann CFD simulations and, most recently,Bar-Sinai et al. (2019) presented an approach to obtain accurate representations of PDEs on coarse grids, which in the future could be applied to NS equations.

One of the most recent trends and promising approaches within ML is Deep Learning (DL), which is expected to play a critical enabling role in the future of modelling complex flows (Kutz, 2017). This technique mimics how brains work, representing the connections between neurons; therefore, the input data gets assimilated with so-called neural networks, which can have different architecture depending on their purpose. The shape of the neural network determines its suitability to perform certain functions, and the internal weights of such neurons produce the output.

Due to its complexity, and the fact that it is still not fully understood, turbulence analysis and modelling is the perfect target to be attacked with DL. Most of the relevant numerical modelling studies that have been performed, unfortunately outside the coastal engineering field, aim at improving turbulence modelling with experimental or numerical (i.e., DNS) turbulence data. Examples applied to RANS turbulence modelling include Ling et al. (2016); Moghaddam and Sadaghiyani (2018); Zhu et al. (2019), whereas LES subgrid closure is studied in Beck et al. (2018).

In view of the results of these studies, it can be concluded that sooner rather than later, ML will have a noticeable impact in CFD models for coastal flows.

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