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Turbulence modelling

As it occurs in the RANS approach, the volume-averaged Reynolds stresses cannot be simulated and need to be modelled instead. Therefore, turbulence modelling in VARANS requires volume-averaged turbulence equations that will yield a volume-averaged turbulent eddy viscosity. However, the volume-averaging process, similarly to what happened in the conservation of momentum equations, causes the appearance of a number of terms which cannot be simulated. Thus, an additional model is also needed to close the turbulence equations.

So far, the only volume-averaged turbulence model that has a published closure is кe. The closure model was developed by Nakayama and Kuwahara (1999) for heat transfer studies and was later applied in Hsu et al. (2002) for wave interaction with a composite breakwater. Moreover, del Jesus et al. (2012) volume-averaged the кw SST turbulence model, although no closure model was developed. The reader is referred to Higuera (2015) for the expressions and the complete derivation of the volume-averaged кe and к — ш SST turbulence models.

Discussion

The VARANS equations just presented are a mathematical model that has been volume averaged. Consequently, it is important to understand the scope of applicability and the limitations that they pose.

First of all, from a physical perspective, porosity needs to be bounded between 0 and 1, both included. However, upon inspection of the equations it is clear that porosity cannot be zero, which means that this set of equations cannot represent an impermeable obstacle. For example, the convective term in Equation 15, will be infinity if ф = 0. Moreover, practical experience indicates that the VARANS equations will also diverge even for small values of porosity, as the drag forces will grow unbounded even before ф approaches 0.

It must be remarked that the equations and the form of the closure terms presented in this chapter are not the only ones available in the literature, as other authors (Hsu et al., 2002; Hur et ah, 2008; del Jesus et ah, 2012; Nikora et ah, 2013; Jensen et ah, 2014) have followed different approaches and averaging techniques, obtaining accurate results in all cases. There is also a wide variety of friction factors to choose from in the literature. The reader is referred to Losada et ah (2016) for a catalogue of the most important ones used. A feasible explanation for the general good results in literature is the fact that the friction factors include variables that serve as tuning factors, and the authors are selecting the values that make the model results fit better with the experimental data. Unfortunately, currently there is no other way to estimate the parameters, as their physical significance is not completely understood and their calibration is flow-dependent. Moreover, as shown in Jensen et ah (2014) and Higuera (2015), there is often a problem selecting the friction factors, as there is no unique solution. Since both friction factors are linked together, an excess of the linear friction force can be balanced with a defect of the nonlinear friction force and vice versa.

Further research also needs to be pursued to shed light in the flow kinematics and dynamics through interfaces. It has been observed (Dimakopoulos, Allsop and Pullen, 2019, personal communication) that there are significant uncertainties in quantifying transitions of porosity both at the interface between the structure and the clear flow region (e.g., armour layers roughness) and between internal interfaces (e.g., transition between underlayer, core, filters, land fillings). For realistic applications, it is possible to calibrate the Darcy- Forchheimer coefficients against known empirical relations for run-up (e.g., Eurotop manual, van der Meer et ah (2018)) or internal pressure transmission, e.g., Troch et ah (2002). However, this calibration will reflect the uncertainties of the empirical relations and, in addition, it may not always be feasible and practical to apply, as flow kinematics at the interfaces are generally mesh dependent and there may be other elements involved (e.g., geotextiles) that produce additional challenges to model. Thus, for such studies, additional testing of sensitivity is recommended to explore different, possible outputs. A better mathematical (and ultimately numerical) description of the flow dynamics at porous interfaces will certainly help quantify and address these uncertainties.

The real need for turbulent dissipation inside porous media (via a turbulence model) or even the turbulence enhancement produced by the closure terms in к — e (which к — ш SST is lacking), is also an interesting open topic. Jensen et ah (2014) argues that “when the actual turbulence levels are of minor interest the effect of the turbulence can be included via the Darcy-Forchheimer equation”. In fact, this is because slightly larger friction factors may account for the expected turbulent dissipation from a mathematical point of view due to their tuning factor behaviour, but from a physical point of view, it is clear that the turbulent terms should be included.

The final and most compromising factor is modelling prototype scale coastal structures with the VARANS equations. So far, detailed information about the flow inside structures in laboratory tests is widely available but the data inside real structures is very limited. Hence, modelling a prototype scale requires an extrapolation of the friction factors at laboratory scale. The fundamental difference between both scales, assuming Fioude number similarity to make sure that accelerations (i.e., gravity forces) are scaled correctly, is that given a length scale factor Л, Reynolds number scales with a factor A3/2. Therefore, laminar or mildly turbulent cases at laboratory scale can often be fully turbulent at prototype scale, changing the physics driving the problem completely.

 
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