List of tables
1.3 Introduction of Lattice Boltzmann method
In the past two decades, the Lattice Boltzmann method(LBM) has gained popularity for the simulation of fluid in both engineering and academic fields. LBM is based on microscopic models and mesoscopic kinetic equations. Besides, it can also be viewed as finite difference method for solving the Boltzmann transport equation [8]. In LBM, the fluid is taken as fictitious particles, which stream along certain directions and collide at lattice sites. The collision and streaming processes are local. Hence, it can be programmed naturally for parallel processing machines [49]. Another advantage of LBM is in dealing with complex boundaries. Through many years of development, LBM has been used in various fields, such as multi-phase flow, chemical reaction, magnetic fluid, biomedical dynamics, porous media as well as other aspects.
The basic concepts and models of LBM will be introduced in the following subsections.
1.3.1 Basic principal and model
In the Boltzmann equation, the fluids can be imagined as consisting a large number of particles.
It describes the time evolution of the particle distribution function as in Eq.1.1[8].
∂f(x,t)
∂t +u·∇f(x,t) =Ω (1.1)
where f(x,t)is the particle distribution function,uis the particle velocity, andΩis the collision operator. Lattice Boltzmann method, inheriting the idea of Lattice Gas Automata, confines the particles on the nodes of lattice and only selected directions of motion are possible.
The collision operator was originally a complex matrix. A particularly simple version of collision operator based on Bhatnagar-Gross-Krook(BGK) collision model was independently introduced by several authors [15,34,54]:
fi(x+ci∆t,t+∆t)−fi(x,t) =−1
τ(fi(x,t)−fieq(x,t)) (1.2) where fieq(x,t)is the equilibrium distribution function.ciis the particle velocity along the i-th direction andτ is the relaxation time toward local equilibrium. Eq.1.2is updated in the following two steps [52]:
Collision step:
fi′(x,t) = fi(x,t)−1
τ(fi(x,t)−fieq(x,t)) (1.3) Streaming step:
fi(x+ci∆t,t+∆t) = fi′(x,t) (1.4)
1.3 Introduction of Lattice Boltzmann method 7 where fiand fi′denote the pre- and post-collision states of the distribution function. fieqis defined as
RT is related to the speed of sound. The streaming process is illustrated in Fig.1.4.
Fig. 1.4 Illustration of the streaming process [8]
For the two dimensional model, the number of the directions ofci can be 5,7 or 9. The most common one is the D2Q9 model, meaning the two-dimensional nine-velocity model. The parameters of D2Q9 are listed as below.
ci=
3 and c= ∆x∆t is the lattice speed. The relaxation time is related to the kinematic viscosityν by
ν = 2τ−1 6
(∆x)2
∆t (1.8)
The macroscopic quantities are obtained by the distribution functions and particle velocities as follows:
Through Chapman-Enskog expansion, Eq.1.2can be derived to Navier-Stokes equation.
Three dimensional models (see for instance [48]) is not introduced here for the sake of simplic-ity.
1.3.2 Boundary conditions
Boundary conditions (BCs) are critical to the stability and the accuracy of any numerical solution. LBM has particular advantage in handling complex geometries. Bounce-back boundary condition is one of the most widely used BCs. It is quite simple and preserves a decent numerical accuracy [8].
Bounce-back model is typically used for the no-slip BCs. The basic idea is that during the streaming step, if fiencounters a boundary node, it would be reflected and go back to the node where it left. The value of firemains same, but its velocity is reversed. The illustration is shown in Fig.1.5.
Fig. 1.5 Illustration of on-grid bounce-back BC
1.4 Objectives 9 In this thesis, we use the bounce-back model for the fully resolved stent. Many other boundary conditions are seen in [76,17,28,75].
1.3.3 Palabos
In spite of the particular advantages of LBM, the complicated mathematic models of LBM still cause many difficulty its application. Some CFD-solver based on LBM are developed to simu-late various kinds of problems in engineering and academy. Palabos(http://www.palabos.org) provides a good choice for this purpose. It integrates the most widely used fluid models, boundary conditions, pre- and post-processing as well as other necessary ingredients for the CFD simulation. Programs written with Palabos can be automatically parallelized. Due to the many facilities that Palabos brings, the researchers can concentrate on actual physical problems instead of staying stuck in tedious software development. All the simulations in this thesis are carried out with Palabos.
1.4 Objectives
The overall aim of the present thesis is to develop, explore and exploit a hydrodynamic model which can represent a FDS in aneurysm with fully consideration of its geometrical parameters, like local porosity, strut diameter, deployment effect and so on in a coarse scale. The model needs to be studied and tested from the hydrodynamical prototypes to the ultimate application on patient-specific aneurysms. Specific goals and research questions are:
• Develop a FDS model prototype to replace the fully resolved stent in cerebral aneurysm numerical simulation
From the aspect of the geometrical shape, though porous media has a same porous feature as FDS, it still shows up obvious difference in the thickness. Is there any other porous objects which are more close to the FDS in geometry and whose hydrodynamic characteristics have been well studied?
The expected model should be capable of dealing with various deployed FDSs with specific geometrical parameters.
• Test the FDS model with various conditions
Given that FDS is deployed in various kinds of aneurysms and arteries which may cause the deformation of the FDS, and great different velocity distribution of the blood, the proposed model should be tested and validated with similar environments that FDSs are implanted in.
• Test the FDS model with specific-patient stented aneurysms
Patient-specific stented aneurysms are always more complicated than the idealized ones due to the geometries of aneurysms of the patient and the actual braided FDS. Therefore, how to implement the model on patient-specific aneurysms? Compared with the fully resolved flow diverters, how does the model performs for the patient-specific aneurysms?
How much computational time can the model save?
To answer the first question, we review the equations for the pressure drop of porous media, including Darcy’s law and Forchheimer’s equation. These equations are then compared with the drag forces of 2D stent-like object obtained by numerical experiments, but the result show that all the porous media equations highly overestimate the pressure drop of the stent. Through many investigations, we find that the hydrodynamic characteristics of the device known as screen is closer to that of the FDS. Therefore, the screen becomes the prototype for the research of the model for FDS in intracranial aneurysms. The result of this part was not included in the incorporated paper, so we can’t show it in the thesis.
Based on this, we develop an effective model which can properly represent, at continuum scale, a stent in an aneurysm. Compared with the porous medium method adopted by previous authors in the literature, we propose the novel idea of applying screen models to the field of stent modeling. Besides, we also take into consideration the deflection effect caused by the stent. This model is explained in detail in Chapter 2.
Then, the performance of the screen based flow diverter model (SFDM) is tested in various conditions such as different stent placements inside an artery, varying homogeneous porosi-ties, and pulsatile inlet velocities in Chapter 3. All the studies above are carried out in two dimensions.
Finally, Chapter 4 extends the framework of screen-based flow diverter model (SFDM) to 3D flows and validates it using actual medical flow diverters in patient specific aneurysms.
The numerical tests show that the 3D SFDM can reproduce the results of direct numerical simulation both qualitatively and quantitatively with high precision, and are capable of reducing the simulation time by an order of magnitude or more.