Various realizations of variational multiscale (VMS) methods for simulating turbulent incompressible flows have been proposed in the past fifteen years. All of these realizations obey the basic principles of VMS methods: they are based on the variation al formulation of the incompressible Navier–Stokes equations and the scale separation is defined by projections. However, apart from these common basic features, the various VMS methods look quite different. In this review, the derivation of the different VMS methods is presented in some detail and their relation among each other and also to other discretizations is discussed. Another emphasis consists in giving an overview about known results from the numerical analysis of the VMS methods. A few results are presented in detail to highlight the used math ematical tools. Furthermore, the literature presenting numerical studies with the VMS methods is surveyed and the obtained results are summarized.

In this paper, we resolve several long-standing issues dealing with optimal pointwise in time error bounds for proper orthogonal decomposition (POD) reduced order modeling of the heat equation. In ...

Proper orthogonal decomposition (POD) stabilized methods for the Navier--Stokes equations are considered and analyzed. We consider two cases: the case in which the snapshots are based on a non inf-...

In this paper, we propose a new stabilized projection-based proper orthogonal decomposition reduced order model (POD-ROM) for the numerical simulation of incompressible flows. The new method draws inspiration from successful numerical stabilization techniques used in the context of finite element (FE) methods, such as local projection stabilization (LPS). In particular, the new LPS-ROM is a velocity-pressure ROM that uses pressure modes as well to compute the reduced order pressure needed for instance in the computation of relevant quantities, such as drag and lift forces on bodies in the flow. The new LPS-ROM circumvents the standard discrete inf-sup condition for the POD velocity-pressure spaces, whose fulfillment can be rather expensive in realistic applications in computational fluid dynamics (CFD). Also, the velocity modes do not have to be either strongly or weakly divergence-free, which allows the use of snapshots generated, for instance, with penalty or projection-based stabilized methods. The numerical analysis of the fully Navier--Stokes discretization for the new LPS-ROM is presented by mainly deriving the corresponding error estimates. Numerical studies are performed to discuss the accuracy and performance of the new LPS-ROM on a two-dimensional laminar unsteady flow past a circular obstacle.

Currently, there is a lack of accurate simulation tools for the thermal performance modeling of courtyards due to their intricate thermodynamics. Machine Learning (ML) models have previously been used to predict and evaluate the structural performance of buildings as a means of solving complex mathematical problems. Nevertheless, the microclimatic conditions of the building surroundings have not been as thoroughly addressed by these methodologies. To this end, in this paper, the adaptation of ML techniques as a more comprehensive methodology to fill this research gap, covering not only the prediction of the courtyard microclimate but also the interpretation of experimental data and pattern recognition, is proposed. Accordingly, based on the climate zoning and aspect ratios of 32 monitored case studies located in the South of Spain, the Support Vector Regression (SVR) method was applied to predict the measured temperature inside the courtyard. The results provided by this strategy showed good accuracy when compared to monitored data. In particular, for two representative case studies, if the daytime slot with the highest urban overheating is considered, the relative error is almost below 0.05%. Additionally, values for statistical parameters are in good agreement with other studies in the literature, which use more computationally expensive CFD models and show more accuracy than existing commercial tools.