Mathematical models are widely recognized as a valuable tool for cardiovascular diagnosis and the study of circulatory diseases, especially to obtain data that require otherwise invasive measurements. To correctly simulate body hemodynamics, the viscoelastic properties of vessels walls are a key aspect to be taken into account as they play an essential role in cardiovascular behavior. The present work aims to apply the augmented fluid-structure interaction system of blood flow to real case studies to assess the validity of the model as a valuable resource to improve cardiovascular diagnostics and the treatment of pathologies. First, the ability of the model to correctly simulate pulse waveforms in single arterial segments is verified using literature benchmark test cases. Such cases are designed taking into account a simple elastic behavior of the wall in the upper thoracic aorta and in the common carotid artery. Furthermore, in-vivo pressure waveforms, extracted from tonometric measurements performed on four human common carotid arteries and two common femoral arteries, are compared to numerical solutions. It is highlighted that the viscoelastic damping effect of arterial walls is required to avoid an overestimation of pressure peaks. An effective procedure to estimate the viscoelastic parameters of the model is herein proposed, which returns hysteresis curves of the common carotid arteries dissipating energy fractions in line with values calculated from literature hysteresis loops in the same vessel.variations are determined by the physical and mechanical properties of blood and vessels walls, which are the essence of a complex fluid-structure interaction (FSI) mechanism, as well as by the anatomy of the entire cardiac network [33,48]. Viscoelastic properties of vessels play an essential role in the cardiovascular behavior [39,33,20]. In fact, viscoelasticity is one of the features that must be realistically included in the mathematical model when accurate numerical results are sought [20,1,26]. Vessel walls manifest viscoelastic properties that are summed up in three main attributes: creep, stress relaxation and hysteresis [6,39,22]. Among the existing linear viscoelastic models, the Standard Linear Solid (SLS) model provides a better representation of the arterial wall mechanics than the generally adopted Kelvin-Voigt model [1,26,46,32,23], being the latter unable to describe an exponential decay of stress over time [49,10,45]. On the other hand, when modeling the vessel mechanics by means of an elastic behaviour, the information related to hysteresis (i.e. the energy dissipated by viscoelastic effects) vanishes and pressure peaks are overestimated [1,6,20].The augmented FSI (a-FSI) system for blood flow modeling, presented in [7,8], is herein extended to real case studies in single arteries, to assess the capability of the model to serve as a valuable tool for practical medical applications, cardiovascular diagnosis and the study of circulatory pathologies. The extension of the model underlines the importanc...

In this work, an arbitrary order HLL-type numerical scheme is constructed using the flux-ADER methodology. The proposed scheme is based on an augmented Derivative Riemann solver that was used for the first time in [A. Navas-Montilla, J. Murillo, Energy balanced numerical schemes with very high order. The Augmented Roe Flux ADER scheme. Application to the shallow water equations, J. Comput. Phys. 290 (2015) 188-218]. Such solver, hereafter referred to as Flux-Source (FS) solver, was conceived as a high order extension of the augmented Roe solver and led to the generation of a novel numerical scheme called AR-ADER scheme. Here, we provide a general definition of the FS solver independently of the Riemann solver used in it. Moreover, a simplified version of the solver, referred to as Linearized-Flux-Source (LFS) solver, is presented. This novel version of the FS solver allows to compute the solution without requiring reconstruction of derivatives of the fluxes, nevertheless some drawbacks are evidenced. In contrast to other previously defined Derivative Riemann solvers, the proposed FS and LFS solvers take into account the presence of the source term in the resolution of the Derivative Riemann Problem (DRP), which is of particular interest when dealing with geometric source terms. When applied to the shallow water equations, the proposed HLLS-ADER and AR-ADER schemes can be constructed to fulfill the exactly well-balanced property, showing that an arbitrary quadrature of the integral of the source inside the cell does not ensure energy balanced solutions. As a result of this work, energy balanced flux-ADER schemes that provide the exact solution for steady cases and that converge to the exact solution with arbitrary order for transient cases

Turbulent shallow flows are characterized by the presence of horizontal large-scale vortices, caused by local variations of the velocity field. Apart from these 2D large vortices, small scale 3D turbulence, mainly produced by the interaction of the flowing water with the solid boundaries, is also present. The energy spectrum of turbulent shallow flows shows the presence of a 2D energy cascade at low wave numbers and a 3D energy cascade at high wave numbers, with a well-defined separation region between them. Horizontal flow movements (e.g. 2D large-scale vortical structures) at low wave numbers mostly determine the hydrodynamic behavior of these flows. Moreover, the generation of standing waves often occurs closely associated to the interaction of 2D horizontal flows with lateral boundaries, this is the case of seiches. To adequately reproduce these phenomena, a mathematical and numerical model able to resolve 2D turbulence is required. We herein show that depth-averaged (DA) unsteady Reynolds averaged Navier Stokes (URANS) models based on the Shallow Water Equations (SWE) are a suitable choice for the resolution of turbulent shallow flows with sufficient accuracy in an affordable computational time. The 3D small-scale vortices are modelled by means of diffusion terms, whereas the 2D large-scales are resolved. A high order numerical scheme is required for the resolution of 2D large eddies. In this work, we design a DA-URANS model based on a high order augmented WENO-ADER scheme. The mathematical model and numerical scheme are validated against observation of complex experiments in an open channel with lateral cavities that involve the presence of resonant phenomena (seiching). The numerical results evidence that the model accurately reproduces both longitudinal and transversal resonant waves and provides an accurate description of the flow field. The high order WENO-ADER scheme combined with a SWE model allows to obtain a powerful, reliable and efficient URANS simulation tool.

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Powerful numerical methods have to consider the presence of source terms of different nature, that intensely compete among them and may lead to strong spatiotemporal variations in the flow. When applied to shallow flows, numerical preservation of quiescent equilibrium, also known as the wellbalanced property, is still nowadays the keystone for the formulation of novel numerical schemes. But this condition turns completely insufficient when applied to problems of practical interest. Energy balanced methods (E-schemes) can overcome all type of situations in shallow flows, not only under arbitrary geometries, but also with independence of the rheological shear stress model selected. They must be able to handle correctly transient problems including modeling of starting and stopping flow conditions in debris flow and other flows with a non-Newtonian rheological behavior. The numerical solver presented here satisfies these properties and is based on an approximate solution defined in a previous work. Given the relevant capabilities of this weak solution, it is fully theoretically derived here for a general set of equations. This useful step allows providing for the first time an E-scheme, where the set of source terms is fully exercised under any flow condition involving high slopes and arbitrary shear stress. With the proposed solver, a Roe type first order scheme in time and space, positivity conditions are explored under a general framework and numerical simulations can be accurately performed recovering an appropriate selection of the time step, allowed by a detailed analysis of the approximate solver. The use of case-dependent threshold values is unnecessary and exact mass conservation is preserved.

In this work, numerical solvers based on extensions of the Roe and HLL schemes are adapted to deal with test cases involving extreme collapsing conditions in elastic vessels. To achieve this goal, the system is transformed to provide a conservation-law form, allowing to define Rankine-Hugoniot conditions. The approximate solvers allow to describe the inner states of the solution. Therefore, source term fixes can be used to prevent unphysical values of vessel area and, at the same time, the eigenvalues of the system control stability. Numerical solvers of different order are tested using a wide variety of Riemann problems, including extreme vessel collapse and blockage. In all cases, the robustness of the approximate solvers presented here is checked using first and third order methods in time and space, using the WENO reconstruction scheme in combination with the TVDRK3 method.

In this work, an arbitrary order augmented WENO-ADER scheme for the resolution of the 2D Shallow Water Equations (SWE) with geometric source term is presented and its application to other shallow water models involving non-geometric sources is explored. This scheme is based in the 1D Augmented Roe Linearized-ADER (ARL-ADER) scheme, presented by the authors in a previous work and motivated by a suitable compromise between accuracy and computational cost. It can be regarded as an arbitrary order version of the Augmented Roe solver, which accounts for the contribution of continuous and discontinuous geometric source terms at cell interfaces in the resolution of the Derivative Riemann Problem (DRP). The main novelty of this work is the extension of the ARL-ADER scheme to 2 dimensions, which involves the design of a particular procedure for the integration of the source term with arbitrary order that ensures an exact balance between flux fluctuations and sources. This procedure makes the scheme preserve equilibrium solutions with machine precision and capture the transient waves accurately. The scheme is applied to the SWE with bed variation and is extended to handle non-geometric source terms such as the Coriolis source term. When considering the SWE with bed variation and Coriolis, the most relevant equilibrium states are the still water at rest and the geostrophic equilibrium. The traditional well-balanced property is extended to satisfy the geostrophic equilibrium. This is achieved by means of a geometric reinterpretation of the Coriolis source term. By doing this, the formulation of the source terms is unified leading to a single geometric source regarded as an apparent topography. The numerical scheme is tested for a broad variety of situations, including some cases where the first order scheme ruins the solution.