Large Eddy Simulations in haemodinamics: models and applications

In the human cardiovascular system the blood flow is usually laminar. Some cardiovascular districts in pathological conditions are characterized by com- plex geometries (bends, bifurcations, areas with an abrupt increase in diam- eter) and high blood velocity. This two characteristics, combined with the flow pulsatility due to the heart activity, can generate high frequency veloc- ity fluctuations within the blood flow, leading to a transitional or turbulent flow field. This transitional/turbulent state has important implications for the pathophysiology of vascular diseases and the design of blood-bearing devices (N.J. Quinlan and P.N. Dooley. Models of flow-induced loading on blood cells in laminar and turbulent flow, with application to cardiovas- cular device flow. Annals of Biomedical Engineering, 35(8):1347-1356, 2007; K. Anupindi Y.T. Delorme and S.H. Frankel. Large Eddy Simulation of FDA’s idealized medical device. Cardiovasc. Eng. Technol., 4(4):1-26, 2013). Three main turbulence modeling approaches exist in literature: 1. the Reynolds-Averaged Navier-Stokes (RANS) modeling, which aims at calculating the time-averaged flow; 2. the Large Eddy Simulation (LES) technique, which aims at computing only the larger spatial scales of the flow and at modeling the smaller ones; 3. the Direct Numerical Simulation (DNS), characterized by the com- plete simulation of all the scales characterizing a fluid-dynamic prob- lem. The context of the present work is the numerical approximation of the partial differential equations underlying blood flow phenomena, by means of computational methods, in particular of Finite Elements. The main inter- est is the study with such methods of the blood fluid-dynamics in patient- specific stenotic carotids, characterized by a geometrical complexity that can trigger transition to turbulence in the blood flow. These vessels have been studied in literature only using the RANS turbulence models (A. De Champlain Y. Douville M. King F. Ghalichi, X. Deng and R. Guidoin. Low Reynolds number turbulence modeling of blood flow in arterial stenoses. Biorheology, 35(4,5):281-294, 1998; S. A. Berger V. L. Rayz and D. Sa- loner. Transitional flows in arterial fluid dynamics. Comput. Methods Appl. Mech. Engrg., 196:3043-3048, 2007) and the DNS (S.E. Lee, S.W. Lee, P.F. Fischer, H.S. Bassiouny, and F. Loth. Direct numerical simula- tion of transitional flow in a stenosed carotid bifurcation. Jour. Biomech., 41(11):2551-2561, 2008). As for LES models, only ideal geometries of stenotic carotids have been considered so far. In this thesis, we address for the first time the problem of the numerical modeling of blood transitional effects in patient-specific stenotic carotids by means of LES techniques. Since we are considering medium-large arteries, the blood is modelled as an incompressible homogeneous Newtonian fluid. We use clinical data (Echo- Color Doppler measurements) on blood velocity to impose patient-specific boundary conditions, and medical images (MRI) to reconstruct the three- dimensional geometries we use in our simulations. We implemented the LES framework in the parallel C++ Finite Element open-source code LIFEV developed at MOX - Politecnico di Milano, IN- RIA - Paris, CMCS - EPF of Lausanne and Emory University - Atlanta. All the numerical results have been obtained using this code. The main goals of this work are: • to build a new suitable DNS to be used in haemodynamics in pres- ence of transition to turbulence phenomena. This is done by means of the stability analysis of a viscous, linear, incompressible and two- dimensional shear layer; • to validate the implemented LES framework using the well known decaying isotropic turbulence test [19]; • to simulate transition to turbulence in a patient-specific stenotic carotid bifurcation and to compare numerical results coming from different LES models (Smagorinsky model [122], Sigma model [84], their mixed variants [143, 149] and their dynamic versions [40, 67]) with respect to the DNS ones. The simulation of an ascending aorta in presence of a bicuspid aortic valve using LES is also discussed; • to apply the explicit differential filtering [36,37] on unstructured meshes using Finite Elements for the first time in literature and to compare nu- merical results obtained with differential filtered-based LES models and with the implicit filtered-based ones.