Reduced Order Models (ROMs), also known as Reduced Basis Methods (RBMs), have received considerable attention in recent years for their ability to drastically reduce CFD cost, particularly when dealing with parametrised problems in a multi-query setting.

This Special Issue gathers recent advances in ROM/RBM techniques for complex flow problems relevant to applications in mechanical and aerospace engineering, as well as medical and applied sciences.

Manuscripts have been selected focusing on methodological developments, with an emphasis on mathematical modelling and applications in areas such as nonlinear inverse problems, optimal flow control, shape optimisation and uncertainty quantification. Advanced developments are proposed to cover broader applications in multiphysics contexts, such as fluid-structure interaction problems and such coupled phenomena involving inviscid, viscous and thermal flows in the incompressible and compressible flow regimes.

This Special Issue provides an ideal and timely context to highlight some state-of-the-art methodologies ready to be applied in industrial and medical problems, including aeronautical, mechanical, naval, offshore, wind, sport, biomedical engineering and cardiovascular surgery, combining elements of high-performance computing and advanced ROM/RBM, real time computing, data management and visualisation.

Kaveh and Habashi, by means of ROM, show how CFD costs can be drastically reduced. In addition, a more complete investigation of a continuous design space obtained by adding experimental fluid dynamics and flight fluid dynamics data leads to a better integration of physical testing and computational data.

Pascarella and co-authors show how accurate solutions of unsteady flows during the design process of an aircraft can be a highly demanding task. RBMs
are used to cut down the number of degrees of freedom, while preserving high accuracy in the reconstruction of the temporal nonlinear dynamics of the flow for a specific class of impulsively started lifting bodies.

Hess and co-authors propose a ROM to accurately approximate the steady-state Navier-Stokes equations, for a wide range of parametric geometries and kinematic viscosity values, in the onset of asymmetries (i.e. symmetry breaking bifurcation) in blood flow through a regurgitating mitral valve, depending on the Reynolds number and valve shape.

Glas and co-authors derive an a posteriori energybased error estimator for the second-order parametric wave equation, used to derive a POD-Greedy RBM approach.

Ferrero and co-authors show how ROM can reduce the cost of shape optimisation problems, dealing with compressible turbulent flows around a gas turbine blade, simulated by a Discontinuous-Galerkin scheme.

Mou and co-authors deal with a data-driven correction ROM in the numerical simulation of the quasi-geostrophic equations. They employ available data to model the correction term, which is generally adopted to represent the missing information in low-dimensional ROMs.

Brandes Costa Barbosa and Perotto resort to isogeometric hierarchically ROMs for modelling the blood flow in healthy and stenotic patient-specific artery segments, by discretising the Stokes equations, with a significant gain in terms of computational time. They additionally verify the reliability of the ROM when computing indices of clinical interest.

These seven contributions allow us to provide a 'snapshot' of the state-of-the-art in the field of ROMs cum RBMs, from the theoretical and implementation viewpoints, while identifying current challenges and highlighting the future landscape in ROMs.