Development of immersed boundary methods for isothermal and thermal flows 1

schemes; 3 implementation of boundary condition – the boundary conditions on the boundary grids/cells are reconstructed, which, together with the aforementioned discrete equations, form

Chapter 1

Introduction

1.1 Background of computational fluid dynamics

Computational fluid dynamics, frequently known as CFD, is a branch of fluid dynamics which uses numerical methods to predict problems including fluid flows, heat transfers and other related phenomena Presently, the great strides

in computers have driven CFD as an important alternative to expensive experiments and bewildering theoretical fluid dynamics Researchers and engineers are strongly encouraged to rely on CFD for the analysis of fluid dynamics-related problems and technologies

The principle of CFD is to pursue an approximate numerical solution for the governing equations of the flow field, i.e Navier-Stokes (N-S) equations The general procedure for this includes: (1) mesh or grid generation – the fluid region of interest is divided into a collection of finite cells or discrete points; (2) discretization of governing equations – the Navier-Stokes equations, which are generally partial differential equations, are discretized into the discrete equations on the interior grids/cells by employing some appropriate numerical

schemes; (3) implementation of boundary condition – the boundary conditions

on the boundary grids/cells are reconstructed, which, together with the aforementioned discrete equations, form a set of well-defined algebraic equation system; (4) solution of resultant equations – the set of algebraic equations are then solved numerically at each cell or point to get numerical solutions for the fluid domain We can see clearly that the numerical solution strongly depends on the grid generation process and the discretization method for the governing equations

Traditionally, body-fitted mesh is often used, in conjunction with the classical finite difference (FD) and finite volume (FV) method These traditional body-fitted methods perform well and enjoy certain popularity in many areas

of scientific research and engineering analysis

1.1.1 Limitations of traditional body-fitted method

Despite the good performance and popularity of the traditional body-fitted methods, their wider applications have been limited due to the geometrical complexities frequently encountered in flow problems Many scientific and engineering practices involve bodies with complex geometries, or objects under moving and/or deformation, which would present considerable computational difficulties for the body-fitted method For example, mesh generation of the computational domain, could be a very troublesome issue, if

considering its significant impact on convergence rate, solution accuracy and CPU time required To overcome the difficulties associated with the geometrical complexity, two techniques have been introduced: structured curvilinear mesh for FD and FV methods and unstructured mesh for FV and

FE (finite element) methods

Structured curvilinear mesh allows boundaries to be aligned with constant coordinate lines and is capable of providing a good representation of boundaries and surface boundary layers, simplifying the treatment of boundary conditions and reducing the numerical “false-diffusion” errors, etc The construction of structured curvilinear mesh always resorts to the coordinate transformation and mapping techniques which would transform a complex physical domain into a rectangular computational domain However, during the projection process, a highly accurate method is required to calculate the transformation Jacobian matrix Otherwise, additional geometrical errors will

be introduced and the accuracy of the domain is thus degraded Furthermore, the coordinate transformation is problem-dependent and tedious Even for seemingly simple geometries, generating a good-quality body-fitted structured mesh can always be an iterative process with a substantial amount of time, not

to mention more complicated ones

In comparison, the unstructured mesh for FV and FE methods makes use of

arbitrarily shaped polygons (such as triangles, quadrilaterals in two-dimension,

or tetrahedral, pyramids, prisms in three-dimension) and thus seems to offer greater flexibility to fit the complex shape of the physical domain Although meshing effort can be saved by using the unstructured mesh, there is some memory and CPU overhead for unstructured referencing since a list of connectivity pattern which specifies how a given set of vertices make up individual elements is required to be stored In addition, the grid quality and robustness can be aggravated with increasing complexity in the geometry It is also noted that the unstructured grid method originally emerged as a feasible alternative to the structured grid technique for discretizing complex geometries However, owing to the inapplicability of powerful line/block iteration and geometrical multi-grid techniques to unstructured grid, unstructured grid methods are in general slower on a per-grid-point basis than structured grid methods

On the other hand, moving boundary problems pose an even greater challenge

to grid generation, especially when they are combined with geometrical complexity With the movement of bodies or objects, the physical fluid domain changes continuously In view of the body-fitted concept, the grid/mesh should be moving correspondingly to conform to the configurations However, in most cases, the mesh deforms to such an excessive distortion that the computation would break down To avoid this, successive re-meshing of

the domain is required This inevitable grid/mesh regeneration is remarkably expensive and unsatisfactory Additionally, the solution variables need to be projected from the old mesh to the new one after re-meshing This interpolation process not only brings forth heavy computational burden, but also leads to undesirable degradations of solution accuracy, robustness and stability

1.1.2 The concept of non-body-conforming method

In the last two decades, a group of so-called non-body-conforming Cartesian grid methods have been proposed, in an attempt to overcome the weakness of the body-fitted grid methods As its name implies, the non-body-conforming methods are specially designed to eliminate the necessity of adapting the underlying computational mesh to the physical configuration of the fluid domain One of the key advantages of non-body-conforming Cartesian grid methods lies in time and human-labor savings on the mesh construction Since the Cartesian grid is generally utilized, the grid complexity is relieved from the geometric complexity For moving objects, there is no need for grid re-generation at each time step At the same time, the method retains most of the favorable properties of structured grids such as easy application of line/block iterative method and geometric multi-grid method In this way, the non-body-conforming Cartesian grid methods can tackle flows involving complex geometries or moving boundaries with relative ease That is why the

non-body conforming method has become popular in recent years

1.2 Non-body-conforming method

The introduction of non-body-conforming Cartesian grid method is credited to Peskin who proposed an immersed boundary method (IBM) in 1972 when studying the blood flow and cardiac mechanisms inside the human heart (Peskin, 1972) Since then, more and more scholars have been attracted, showing strong interests in improving the method and widening its application

As a result, various non-body-conforming Cartesian grid methods have been springing up in the last two decades Generally, the non-body-conforming method always takes a regular region, which may frequently be a rectangular one, as its computational domain The domain is sufficiently large to cover the entire problem region inside it The complex-geometric and/or moving bodies, under such circumstances, are regarded as interfaces or boundaries immersed

in the domain

As learnt from its name, the non-boundary-conforming Cartesian grid is not aligned with the geometry of the physical domain Therefore, imposing the boundary conditions is not as straightforward as the traditional body-fitted method As a result, a procedure which is capable of incorporating the boundary condition (or the effects of the boundary) into the overall algorithm and, at the same time, does not affect the accuracy or significantly increase the

computational cost, is definitely required It is this challenging procedure that distinguishes one method from the other Based on whether the immersed boundary is treated as an interface with a finite thickness or not, the existing non-body-conforming Cartesian grid methods can be broadly classified into two categories: sharp interface method and diffuse interface method In the sharp interface method, the boundary is viewed as a zero-thickness sharp interface The ghost-cell method, cut-cell method, immersed interface method fall into this category In the diffuse interface method, the effect of boundary is smeared out across the interface to a thickness of the order of the mesh width The immersed boundary method mentioned above is among this category

1.2.1 Sharp interface method

The sharp interface methods are capable of accurately capturing the solid interfaces and enforcing the boundary conditions on them, at the expense of complicated algorithms for accurate implementation of boundary conditions Some representatives like ghost cell method, cut cell method and immersed interface method are reviewed in the following

1.2.1.1 Ghost cell method

In the ghost cell method, the boundary conditions on the fluid-solid interface are imposed through the flow variables at the “ghost-cells”, whose cell centers are falling inside the solid region but having at least one neighboring fluid cell

Employing an appropriate local reconstruction scheme (interpolation or extrapolation), the flow variable values of the ghost cells are calculated in such a way that the prescribed boundary condition at the interface is satisfied Different reconstruction schemes such as linear, bilinear and quadratic ones (Majumdar, 2001; Tseng & Ferziger 2003; Pan & Shen 2009) require different reconstruction stencils, and their complexity determines the methodology complexity For example, a linear reconstruction model (Tseng & Ferziger 2003) can be employed, utilizing the projection point of the ghost cell on the immersed boundary and two fluid points nearest to the projection point as the stencil for extrapolation However, when any of the two fluid points in the stencil is too close to the interface, numerical instability will arise Furthermore, it is more likely to introduce spurious oscillations with more stencil points So the ghost cell method may be troubled by the robustness issue associated with supporting stencils for the reconstruction scheme

1.2.1.2 Cut-cell method

The cut-cell method is another typical Cartesian grid-based sharp interface method In the cut-cell method, a series of irregular truncated cut cells which exist immediately adjacent to the boundary play the role of implementing the boundary conditions In practice, the truncated cut cells may be arbitrarily small (especially for highly curved or complex boundary) and would lead to severe numerical instability To avoid an impractical time step size, a cell

merging technique in which the cut cell was absorbed by an appropriately selected neighboring cell is usually necessary (Ye et al 1999; Chung 2006) After the cut cells are reshaped, the governing equations are discretized in these merged cells based on their actual shape However, due to the various manners the boundary may intersect with the background regular mesh, numerous scenarios for shapes of the merged cut cells should be accounted for There is another difficulty which frequently disturbs the application of cut-cell method – the presence of degenerate cut cells (Je et al., 2008) In two-dimension, the degenerate cut-cells are those that (1) have more than two intersection points with the boundary curve or (2) have more than one intersection point with any cell face Further, as pointed out by Mittal & Iaccarino (2005), “successful implementation of the cut-cell method to three-dimensional geometries has not yet been accomplished.”

1.2.1.3 Immersed interface method

The immersed interface method (IIM) was originally proposed by LeVeque &

Li (1994) for elliptic equations with discontinuous coefficients, and was later extended to account for two-dimensional incompressible flows with interfaces

or immersed boundaries (LeVeque & Li, 1997; Li & Lai, 2001; Xu & Wang, 2006; Le et al 2006) In practice, the existence of interfaces or immersed boundaries may lead to jumps in pressure and in the derivatives of both pressure and velocity at the interface/boundary The basic principle of IIM for

fluid dynamic problems is that the jump conditions in the flow variables and/or their derivatives are explicitly incorporated into the difference equations to achieve second or even higher order of accuracy However, the determination of jump conditions across the immersed boundary is not an easy job at present Firstly, they normally have a very complicated form even for the simple membrane flow system Secondly, the derivation of the jump conditions always requires the immersed interface to be a closed structure, i.e.,

a closed curve in two-dimension or a closed surface in three-dimension (Xu & Wang, 2006) The IIM is also troubled with the drawback that special finite-difference stencils need to be particularly designed for the discretization

of Navier-Stokes equations near the immersed boundaries

In summary, the success of any sharp interface Cartesian grid methods depends strongly on how the boundary conditions are implemented and how the discretization schemes are modified at the immersed boundary, which is frequently accompanied by an iterative data reconstruction procedure and elaborate efforts for special mesh treatment

1.2.2 Diffuse interface method

It can be recognized that a common difficulty for various sharp interface methods is the requirement of irregular stencils near the immersed boundary for derivative approximation or data reconstruction scheme Compared to the

sharp interface Cartesian grid methods, the diffuse interface methods are relieved from these troubles and relatively easy to implement Representatives are fictitious domain method and immersed boundary method

1.2.2.1 Fictitious domain method

The fictitious domain method enforces the conditions on the immersed boundaries in a weak form by means of Lagrangian multipliers (Glowinski et al., 1994) Using the fictitious domain method, the physical solution in the fluid domain is required to extend into the solid domain (frequently referred to

as the fictitious or artificial fluid domain) in a continuous manner (Glowinski

et al 1994) For example, the fictitious fluid solution in Glowinskin et al (1995) and Yu et al (2006) was directly enforced to be the same as the solid solution For problems where the solid bodies have prescribed motions, the constraint together with corresponding Lagrangian multipliers only need to be set along the fluid-solid interface, as suggested by Glowinski et al (1997) However, if the solid body motion is not given in advance but caused by the hydrodynamic forces and torques, the constraint of rigid-body motion on the fluid-solid interface alone is not enough In this case, Glowinski (1999) exploited a distributed Lagrangian multiplier-based fictitious domain method (DLM/FD) in which the constraint of rigid-body motion was extended and imposed on the fictitious fluids as well Although successfully applied to problems like particulate flows (Patankar et al 2000; Yu et al 2006), the

Lagrangian multiplier in the fictitious domain method is normally calculated implicitly from the rigid-body motion constraint This implicit determination causes DLM/FD method to suffer from expensive computations

1.2.2.2 Immersed boundary method

A very popular and attractive diffuse interface method in last decades is the immersed boundary method (IBM), which was developed by Peskin when he studied blood flow in human heart (Peskin, 1972) In his work, the human heart was modeled as an elastic membrane immersed in a rectangular flow domain He used a fixed Eulerian Cartesian mesh to describe the blood flow and a set of elastic fibers (represented by a series of Lagrangian points which can move and deform freely through the underlying Eulerian mesh) for the heart motion The interaction between the heart and blood flow was realized through the introduction of Dirac delta function Once the heart moves or deforms, singular forces are generated along the heart wall These singular forces at the Lagrangian points are then spread to their surrounding Cartesian Eulerian grids as body forces via a discrete delta function The incompressible Navier-Stokes equations with the additional body forces are then solved on the entire domain including both the interior and exterior of the human heart After the velocity on the fixed Eulerian gird are calculated, the heart is updated to its new shape and location according to the no-slip condition between the blood flow and the heart wall In this way, the coupling between

the governing equation solver and the boundary condition implementation is eliminated, and dynamically updating the geometry changes becomes straightforward As such, the solution to the whole system (blood flow + heart motion) is easily yielded

From the above illustration, it can be observed that the immersed boundary method is conceptually independent of the spatial discretization and is simple

to implement in an existing Navier-Stokes solver By modeling the immersed boundaries as force sources, it can handle complex geometries easily without any special mesh treatment, even for flexible boundaries undergoing a complicated movement or shape variation In fact, the method has proven to

be a versatile and successful tool for problems with complex geometries and moving boundaries In this regard, the immersed boundary method attracts our attention and is studied in the present thesis It should be noted that although some methods (Deng et al 2006; Choi et al 2007; Zhang & Zheng 2007; Paravento et al 2008; Liao & Lin 2012; Noor et al 2009; Ghias et al 2007; Chen et al 2013; Mittal et al 2008; etc.) in the literature are also claimed to be the immersed boundary methods and introduce the momentum forces into the governing equations to represent the effect of the immersed objects, they treat the immersed boundary as a sharp one, which is quite different from Peskin’s original method These “so-called immersed boundary methods” are not real immersed boundary methods and therefore fall out of the scope of the present

thesis

1.3 Brief review of Immersed boundary methods

The immersed boundary method has received great attention since being published Following Peskin’s pioneer contribution, abundant variations of the method have come forth Among them, some are devoted to the improvement

or refinement of the fluid solver while others concentrated on widening the application fields of the method

In general, the diverse immersed boundary methods based on the Navier-Stokes solvers are established in two frameworks according to the underlying form of Navier-Stokes equation utilized: pressure-velocity formulation-based immersed boundary method, and stream function-vorticity formulation-based immersed boundary method Consequently, researches on algorithm improvement or refinement have been proceeding along the two directions

1.3.1 Pressure-velocity formulation-based immersed boundary method

The pressure-velocity formulation-based immersed boundary method follows Peskin’s original work, in which the body force term is explicitly incorporated into the momentum equation to represent the effect of the immersed boundary The previously proposed methods reveal that the boundary/body force is