A Lattice Boltzmann and Immersed Boundary Scheme for Model Blood Flow in Constricted Pipes:Part 2-Pulsatile Flow

One viable approach to the study of haemodynamics is to numerically model this flow behavior in normal and stenosed arteries.The blood is either treated as Newtonian or non-Newtonian fluid and the flow is assumed to be pulsating,while the arteries can be modeled by constricted tubes with rigid or elastic wall.Such a task involves formulation and development of a numerical method that could at least handle pulsating flow of Newtonian and non-Newtonian fluid through tubes with and without constrictions where the boundary is assumed to be inelastic or elastic.As a first attempt,the present paper explores and develops a time-accurate finite difference lattice Boltzmann method(FDLBM)equipped with an immersed boundary(IB)scheme to simulate pulsating flow in constricted tube with rigid walls at different Reynolds numbers.The unsteady flow simulations using a time-accurate FDLBM/IB numerical scheme is validated against theoretical solutions and other known numerical data.In the process,the performance of the time-accurate FDLBM/IB for a model blood flow problem and the ease with which the no-slip boundary condition can be correctly implemented is successfully demonstrated.

A Lattice Boltzmann and Immersed Boundary Scheme for Model Blood Flow in Constricted Pipes:Part 1-Steady Flow

Hemodynamics is a complex problem with several distinct characteristics;fluid is non-Newtonian,flow is pulsatile in nature,flow is three-dimensional due to cholesterol/plague built up,and blood vessel wall is elastic.In order to simulate this type of flows accurately,any proposed numerical scheme has to be able to replicate these characteristics correctly,efficiently,as well as individually and collectively.Since the equations of the finite difference lattice Boltzmann method(FDLBM)are hyperbolic,and can be solved using Cartesian grids locally,explicitly and efficiently on parallel computers,a program of study to develop a viable FDLBM numerical scheme that can mimic these characteristics individually in any model blood flow problem was initiated.The present objective is to first develop a steady FDLBM with an immersed boundary(IB)method to model blood flow in stenoic artery over a range of Reynolds numbers.The resulting equations in the FDLBM/IB numerical scheme can still be solved using Cartesian grids;thus,changing complex artery geometry can be treated without resorting to grid generation.The FDLBM/IB numerical scheme is validated against known data and is then used to study Newtonian and non-Newtonian fluid flow through constricted tubes.The investigation aims to gain insight into the constricted flow behavior and the non-Newtonian fluid effect on this behavior.

A Discrete Flux Scheme for Aerodynamic and Hydrodynamic Flows

The objective of this paper is to seek an alternative to the numerical simulation of the Navier-Stokes equations by a method similar to solving the BGK-type modeled lattice Boltzmann equation.The proposed method is valid for both gas and liquid flows.A discrete flux scheme(DFS)is used to derive the governing equations for two distribution functions;one for mass and another for thermal energy.These equations are derived by considering an infinitesimally small control volume with a velocity lattice representation for the distribution functions.The zero-order moment equation of the mass distribution function is used to recover the continuity equation,while the first-order moment equation recovers the linear momentum equation.The recovered equations are correct to the first order of the Knudsen number(Kn);thus,satisfying the continuum assumption.Similarly,the zero-order moment equation of the thermal energy distribution function is used to recover the thermal energy equation.For aerodynamic flows,it is shown that the finite difference solution of the DFS is equivalent to solving the lattice Boltzmann equation(LBE)with a BGK-type model and a specified equation of state.Thus formulated,the DFS can be used to simulate a variety of aerodynamic and hydrodynamic flows.Examples of classical aeroacoustics,compressible flow with shocks,incompressible isothermal and non-isothermal Couette flows,stratified flow in a cavity,and double diffusive flow inside a rectangle are used to demonstrate the validity and extent of the DFS.Very good to excellent agreement with known analytical and/or numerical solutions is obtained;thus lending evidence to the DFS approach as an alternative to solving the Navier-Stokes equations for fluid flow simulations.