Numerical Computations And Simulations For Hemodynamics

Arterial diseases,such as atherosclerosis and thrombus,have been a leading threat to human health in the modern society.The main goal of hemodynamics is to investigate blood flows in arteries,which can improve our comprehension to occurrence and development of arterial diseases.Previous investigations present local alterations of hemodynamical features and complex flow structures have a tight link with arterial diseases which often occurs near to locations with special geometric shape,such as branches.Blood is a suspension consisting of various cells and plasma which results in a non-Newtonian fluid.In addition,the arterial wall is elastic.Blood flows in arteries appear pulsatile mainly because of pulsatility of the heart beat.Pulsatile flows and elastic walls lead to a complicated instantaneous fluid-structure interaction system which goes beyond the capacity of both theoretical and experimental methods.With development of computational fluid dynamics,the relevant numerical simulations in association with CAD technique or 3D digital image reconstruction technique become a prevailing fashion to analyze blood flows.In essence,numerical simulations are to numerically solve equations governing flows.Incompressible Newtonian fluid flows are mathematically modeled by using incompressible Navier-Stokes equations.Replacing the diffusion terms with the viscous stress terms,incompressible Navier-Stokes equations are turned into equations governing the incompressible non-Newtonian flows.In terms of numerical solutions,the viscous stress term causes no intrinsic obstacles for discretization.So,the solution to incompressible Navier-Stokes equations is the primary job.The SIMPLE method and the projection method are two popular algorithm for solving incompressible flows.Owing to pulsatility of the blood flow,the SIMPLE method originally designed for steady flows has lower efficiency than the projection method does.Apart from this,complex geometries of diseased arteries demand numerical method on unstructured meshes.So,the finite volume method and the finite element method are major selections.Concerning the finite volume discretization of the momentum equations,many excellent and mature non-oscillatory schemes can be employed to reconstruct convective velocity of the interface.However, the Galerkin finite element may cause numerical oscillations which can be suppressed by using SUPG method or GLS method.These methods result in complex implementation on unstructured meshes.On the other hand,published FVM projection methods all use the FVM discretization of the pressure Poisson equation on unstructured meshes, which is not the merit of the finite method but of the finite element method.So,it is an appealing job to propose a numerical method combined with two advantages and push it into the solution of hemodynamics.Additionally,non-oscillatory high order schemes are necessary for suppressing unphysical wiggles of numerical solutions in discretizing the convection velovity.The NVF high resolution scheme,one of many non-oscillatory schemes,is of our interest mostly because of its concise construction and simple implementation.Moreover,with employing the 3-point stencil to evaluate the convective velocity of the interface,the NVF scheme can be readily implemented on unstructured meshes.A newλformulation and a new definition of the normalized variable on unstructured meshes bring about more efficiency of the NVF scheme.Fundamental problems in hemodynamics is briefly introduced in Chapter 1.In this chapter,numerical methods for hemodynamics,including incompressible Newtonian and non-Newtonian fluid flows,are also reviewed.Chapter 2 presents modeling equations governing incompressible flows which involves the continuity equation, the momentum equations and energy equation with their relevant initial and boundary conditions.Additionally,constitutive equations often used to depict non-Newtonian behavior of blood flows are summarized.Numerical schemes in the normalized formulation (NVF) for the convection-diffusion equation is presented in Chapter 3.And the convective boundedness criterion(CBC) to design non-oscillatory NVF schemes is described in detail.A newλformulation for NVF schemes is proposed and implemented readily on unstructured meshes by defining a new normalized variable.In Chapter 4,various projection methods are thoroughly reviewed and the rotational incremental pressure projection method is revised to solve the incompressible non-Newtonian flows.In addition,full discussion of constructions of the finite volume control volume and their relevant fashions of the variable arrangement.On the basis of published researches,a new vertex-centered finite volume/finite element projection method is proposed for viscous incompressible flows on unstructured meshes.To validate the present hybrid method,typical numerical test problems are utilized.Good arrangements with benchmark solutions are obtained.In Chapter 5,the proposed FVM/FEM method is applied to numerical solution to blood flows in channels with(multiple) constrictions,branches.The significant factors having close correlation with arterial diseases,such the wall shear stress and the pressure drop,are discussed by using numerical results.Summarizations on the present dissertation and prospects on numerical solution and simulation of hemodynamics are performed in the last chapter.Some unsolved issues are also listed.