Chapter Introduction Chapter 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 1      Chapter Introduction 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 2      Chapter Introduction 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 3      Chapter Introduction 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 4      Chapter Introduction 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 5      Chapter Introduction 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 6      Chapter Introduction 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. 7      Chapter Introduction 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 8      Chapter Introduction 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 9      Chapter Introduction 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 10      Chapter Introduction 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 13      Chapter Introduction 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 14      Chapter Introduction introduced in order to approximate or enforce the no-slip condition on the immersed boundaries. So, how to accurately evaluate the force term becomes a critical issue for a successful implementation of the pressure-velocity formulation-based IBM. The calculation of body force depends on the characteristics of the immersed boundary: elastic and rigid boundaries require different treatments. The present thesis mainly focuses on objects with rigid boundary. The elastic boundaries (Zhu & Peskin 2002; Kim & Peskin 2007; Francois & Shyy 2003; Udaykumar et al.1997; Kim & Peskin 2006) can be handled perfectly by using the constitutive law as what Peskin’s original study did (Peskin 2002). However, utilizing the constitutive law to rigid boundaries would pose problems (Mittal & Iaccarino, 2005). As a result, tremendous efforts have been spent to tackle the issue of how to properly calculate the force density on the solid boundaries. Several remarkable strategies are introduced in the following. Lai & Peskin (2000) proposed a penalty force scheme to calculate the singular Lagrangian force density. They assumed that the boundary points of the immersed object were being attached to their equilibrium positions by springs with high stiffness. When the boundary deviates from the equilibrium location, a restoring force will be generated according to the Hooke’s law so that the 15      Chapter Introduction boundary points will stay close to the target boundary position. Goldstein et al. (1993) generalized the above idea and provided a two-mode feedback forcing scheme for the control of velocity condition at the immersed boundary. This forcing term reflects the velocity difference between the desired boundary value and the interpolated one, and behaves in a feedback loop such that the boundary velocity remains close to the desired value. The approach has been employed for the simulation of low Reynolds number turbulent flows over a riblet-covered surface (Goldstein et al. 1995) and self-propelled fish-like swimming (Wang et al. 2014). Later, Mohd-Yusof (1997) suggested a forcing evaluation approach in which the body force was directly derived from the transformed momentum equation. The method is frequently termed the direct forcing method. Fadlun et al. (2000) successfully applied the approach to large-eddy simulation of turbulent flow. Although this force evaluation scheme was initially suggested in a sharp interface method, Uhlmann (2005) has successfully generalized this idea into Peskin’s immersed boundary method, by incorporating the regularized delta function into the force calculation and spreading process. Shu et al. (2007) discovered that most of the aforementioned methods cannot accurately satisfy the no-slip boundary condition and penetration of some 16      Chapter Introduction streamlines happens on the solid boundary. The penetration may degrade the accuracy of the method and also the boundary force calculation. To remove flow penetration through the solid boundary, a velocity correction approach was proposed by Shu et al. (2007), where the velocity correction is made in the vicinity of the boundary points such that the no-slip boundary condition is directly enforced. Following the idea of Shu et al. (2007), Wu & Shu (2009) recently proposed an implicit velocity correction-based immersed boundary solver, in which the velocity correction is determined implicitly in a way that the velocity at the boundary interpolated from the corrected velocity field through the discrete delta function accurately satisfies the no-slip boundary condition. It should be noted that although the immersed boundary solvers of Shu et al. (2007) and Wu & Shu (2009) can guarantee the accurate satisfaction of no-slip condition on the immersed boundary, the flow field in both works is delivered by the Lattice Boltzmann method (LBM), whose computational efficiency is always constrained by its intrinsic limitation of requiring special uniform lattice, thus making it challenging to offer high reolution near a solid body and/or include far-field boundary. In this regard, it would make good sense to explore how to implement the velocity correction-based IBM into the Navier-Stokes solver and how well it would perform. 17      Chapter Introduction 1.3.2 Stream function-vorticity formulation-based immersed boundary method Literature shows that almost all the established immersed boundary methods are inherited from Peskin's original approach. Their common feature is to introduce the momentum forcing terms (i.e., body force term) into the governing equation to represent the effect of the immersed boundary. In this regard, the IBM seems to be more effective under the framework of velocity-pressure (primitive-variable) formulation. On the other hand, we have to indicate that for incompressible flows, the primitive-variable formulation suffers from some difficulties in the solution process (Shu & Wee, 2002). As can be seen, the velocity components appear in both the momentum and continuity equations, but the pressure only appears in the momentum equation. When the momentum equations are used to compute the velocity components, there is no guarantee that the obtained velocity components would satisfy the continuity equation. In addition, there is no transport equation for computing the pressure. This brings difficulty for coupling between the velocity field and the pressure field (Chaviaropoulos & Ciannakoglou, 1996). To remove this difficulty, special techniques are required, and a pressure Poisson equation is usually introduced. The solution of elliptic Poisson equation often takes a lot of computational time. Furthermore, it requires the use of staggered grid, which could bring complexity in programming. In contrast, the stream function-vorticity formulation of Navier-Stokes equations has been well 18      Chapter Introduction recognized to be more efficient for 2D incompressible flows. In this form, the momentum equations are combined and transformed into a transport equation for vorticity ω , where the pressure gradient term disappears. This clearly eliminates the critical issue of velocity-pressure coupling. Besides, the continuity equation is automatically satisfied due to the introduction of stream function ϕ , so the normal grid can be effectively utilized. Furthermore, two variables ( ω , ϕ ) rather than three variables ( u, v, p ) are solved. This would reduce the computational effort. To make a good use of the respective merits inherited from the stream function-vorticity formulation and the IBM, it is desirable to develop a stream function-vorticity formulation-based IBM solver. However, the basic idea of original IBM is that the forcing terms are added to the momentum equations to show the effect of immersed boundary. Obviously, this idea cannot be directly incorporated into the stream function-vorticity formulation as the momentum equations are differentiated and combined into a single equation for the vorticity. Due to this difficulty, there is little work in the literature in this development. To the best of our knowledge, Wang et al. (2009) made an effort to study this problem. In their method, the vorticity-velocity formulation is adopted and the spirit of Peskin's original IBM is directly applied. Both velocity and vorticity are divided into two parts. One is the velocity and vorticity without the influence of immersed boundary, and the other is the 19      Chapter Introduction velocity and vorticity correction. In order to correct the velocity, the multi-direct forcing scheme suggested by Wang et al. (2008) and Luo et al. (2007) is utilized, which, as a matter of fact, is an iterative process to approximately satisfy the no-slip condition at the immersed boundary. In the solution process, the no-slip condition is not enforced. For vorticity correction, source terms, which are derived from the momentum equations with forcing terms by differentiating with respect to y (for x-momentum equation) and x (for y-momentum equation), are added into the vorticity transport equation. These terms are very complicated, and involve the first, second and third order spatial derivatives. Numerical dicretization of these spatial derivatives brings complexity into the computation. In view of simplifying the computational complexity, an efficient stream function-vorticity formulation-based IBM solver for simulating 2D incompressible viscous flows is desired. 1.3.3 Applications of immersed boundary method The popularity of immersed boundary method has seen increased for flow problems with complex or/and moving boundaries. Although being created to study the fluid dynamics of cardiovascular physiology such as flow in heart valves (McQueen & Peskin 2000), flow in arterioles (Arthurs et al. 1998), deformation of red blood cells in a shear flow (Eggleton & Popel 1998), etc., the method has been applied to various biological flows including wave propagation in the cochlea of the inner ear (Beyer Jr. 1992), thrust generation 20      Chapter Introduction of swimming fish (Wu & Wang 2009) and lift production of flying insects (Miller & Peskin, 2005). In addition, a variety of applications concerning fluid-structure interaction (Sotiropoulos & Yang 2014), turbulent flow (Shun et al. 2014), multiphase flow (Li et al. 2012) and multi-component flow (Du et al. 2014) have also been reported. Examples may refer to parachutes inflation (Kim & Peskin, 2009), particle sedimentation (Feng & Michaelides, 2004), topological changes of wet foam (Kim & Seol 2013), flapping motion of turbine blades (Kalitzin & Iaccarino, 2003), even flow around a walking person (Choi et al. 2007). Apart from the applications to fluid flows, there are some other studies with a relevance to heat transfer problems. Due to considerable importance of heat transfer in a wide range of engineering applications, such as convection in buildings, solar energy systems, electronic cooling equipment, crystal growth processes and nuclear reactor, a better understanding of the involved flow and thermal physics in these phenomena has a fundamental significance to their improvement. However, most of these thermal problems are suffering from complex or moving configuration, which stimulates scholars to generalize the traditional immersed boundary method and extend it to the thermal flow field. Since the basic idea of IBM is that the effect of the immersed boundary on its surrounding fluid is realized through introducing forcing terms in the momentum equations, it has a difficulty to be directly applied to heat transfer 21      Chapter Introduction problems since they also involve the energy equation. In fact, an inspection of literatures shows that there is relatively little work on applying IBM to heat transfer problems as compared to its application for fluid dynamic problems. Nevertheless, there are some efforts to extend IBM for application to thermal flow problems with Dirichlet-type boundary condition. Among notable achievements, Zhang et al. (2008) extended their early work for fluid flows (Zhang & Zheng 2007) to the heat-transfer problem. Similar to the introduction of a forcing term into the momentum equations, they incorporated a heating term in the energy equation to represent the virtual heat source. It is evaluated from the difference between the given temperature and the computed one at the Lagrangian point, which is then mapped back to the Eulerian grid points using the same idea as in the traditional feedback-forcing IBM (Saiki & Biringen 1996). Wang et al. (2009) investigated problems of natural convection between concentric cylinders as well as forced convection around a stationary circular cylinder using a multi-direct heat source scheme, which is similar to the multi-direct forcing method (Wang et al. 2008) proposed previously for isothermal flows. Young et al. (2009) combined the direct-forcing approach with moving-grid process under arbitrary Lagrangian-Eulerian (ALE) framework to simulate moving boundary problems with heat transfer effect. Feng et al. (2009) combined direct numerical simulation (DNS) with IBM to simulate the natural convection in particulate flows. Kim et al. (2008) studied the natural convection induced by 22      Chapter Introduction a temperature difference between a cold outer square enclosure and a hot inner circular cylinder at different vertical locations using IBM, in which the heat source/sink is introduced. Later, Lee et al. (2010) extended the work of Kim et al. (2008) by considering various locations of inner cylinder along horizontal and diagonal directions. In all the above works, the common feature is that a heat source term is introduced in the energy equation. However, like the conventional IBM, the heat source term is treated explicitly and pre-calculated. In most of the works, there is no mechanism to enforce the boundary condition for temperature. As a consequence, the boundary condition for temperature is not accurately satisfied, which would affect the accuracy of numerical results. Besides, almost all the efforts were made for heat transfer problems with Dirichlet boundary condition for temperature, i.e., the temperature on the immersed boundary is specified (without special illustration, the Dirichlet and Neumann boundary conditions mentioned below specially refer to those on immersed boundaries). The only notable one in applying IBM to thermal flows with Neumann boundary condition was given by Zhang et al. (2008). In their work, a layer of assistant points which are placed one-grid spacing away from the immersed boundary along its outward normal direction is firstly defined. Then the temperature at assistant points is calculated through interpolation from the temperature at Eulerian points. After that, the normal derivative of temperature in the Neumann condition is approximated by the first order 23      Chapter Introduction one-sided finite difference scheme, from which the wall temperature can be computed. With the calculated wall temperature, the algorithms used in the work of Zhang & Zheng (2007) for fluid flows are applied to correct the surrounding temperature field at Eulerian points. From the solution process of Zhang et al. (2008), the Neumann condition is discretized to give the wall temperature. Then the algorithms of IBM with Dirichlet condition are applied. So in their method, the immersed boundary solver is employed to deal with the Dirichlet condition. Technically, their method is not a real immersed boundary solver for treating the Neumann condition. To the best of our knowledge, there is no available work on the application of traditional IBM (Peskin’s original IBM) for thermal flow problems with Neumann (given heat flux) boundary condition for temperature in reported literatures. Compared to the Dirichlet temperature condition case, heat transfer problem with Neumann-type temperature condition presents an even a bigger challenge. 1.4 Objective of this thesis The above literature review shows that remarkable works have been done on the development of the immersed boundary method. However, there are still rooms for the improvement of methodology and these stimulate the studies in the present thesis. The objective of this thesis is to develop several novel immersed boundary methods for fluid and thermal dynamics problems with complex or moving boundaries, aiming to further refine the existing IBM. The 24      Chapter Introduction principal goals of the current study are as follows: ¾ The conventional IBM cannot guarantee an exact satisfaction of no-slip condition on the immersed boundary, so the present thesis will firstly extend the velocity correction-based IBM originally developed within the framework of Lattice Boltzmann (LB) solver to the Navier-Stokes (NS) solver-based version where the primitive variable formulation is solved. ¾ Although the stream function-vorticity formulation is more efficient for two-dimensional incompressible flow, the published immersed boundary models are so complicated that high computational effort is needed. The present thesis will develop a computationally efficient stream function-vorticity formulation-based immersed boundary method; ¾ The reported immersed boundary methods for thermal flows are not capable of exactly satisfying the Dirichlet thermal condition on the immersed boundary. In this thesis, a boundary condition-enforced IBM for thermal flows with Dirichlet boundary condition will be presented. Furthermore, two simple and convenient ways for the calculation of Nusselt number in the framework of IBM are suggested; ¾ Very few work was reported by extending IBM to solve problems with Neumann boundary condition in the framework of Peskin’s original IBM. In the present thesis, an efficient heat flux correction-based immersed boundary method will be proposed for heat transfer problems with Neumann condition; 25      Chapter Introduction Detailed discussions about the above topics are presented in the following chapters. 1.5 Organization of this thesis This thesis is organized as follows: Chapter extends the boundary condition-enforced LB-IBM to the NS solver-based version, with the emphasis placed on how to implement the boundary condition-enforced IBM within the framework of Navier-Stokes equation in the primitive variable form such that the no-slip condition on the immersed boundary is exactly satisfied. The solution procedure and critical techniques are carefully described and numerical experiments are performed. Chapter presents a detailed description of the efficient stream function-vorticity formulation-based IBM. Attentions are mainly paid to the satisfying of no-slip condition on the immersed boundary and evaluation of vorticity source term. Two numerical approximations to the vorticity source term are suggested and their performances are tested by several numerical validation cases. Chapter extends the IBM to solve heat transfer problems and suggests a boundary condition-enforced method for thermal flows with Dirichlet 26      Chapter Introduction boundary conditions based on a temperature correction procedure. Additionally, two convenient ways for evaluation of Nusselt number are provided which are tested to be accurate and efficient. Chapter presents a challenging work. A novel IBM which is directly inherited from Peskin’s original IBM is developed for problems with Neumann boundary conditions. Particularly, heat transfer problems with heat flux condition are taken as examples to illustrate and then to validate the methodology. Chapter demonstrates the capability of the proposed methods by applying them to two-dimensional fluid and thermal flow problems. Three test cases including insect hovering flight near or far away from the ground, elliptical or cold circular particle sedimentation through a narrow channel, and forced convective heat transfer from a transversely oscillating circular cylinder in the wake of a stationary one are considered. Chapter shows the performance of the proposed methods to three-dimensional thermal flows. Specifically, forced convective heat transfer from an isolated stationary sphere, two tandem stationary spheres and a single rotating sphere as well as natural convective heat transfer inside concentric or vertically eccentric annuluses are simulated and analyzed. 27      Chapter Introduction Chapter shows the capability of the present methods to three-dimensional moving boundary problems. Two biomimmtric flow dynamic problems concerning flow behaviors around a heaving and pitching flapping wing and hydrodynamic performance of swimming fish are studied. Chapter summarizes the major contributions of this thesis and gives some suggestions for future research. 28    [...]... These “so-called immersed boundary methods are not real immersed boundary methods and therefore fall out of the scope of the present 13       Chapter 1 Introduction 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,... of the immersed boundary The previously proposed methods reveal that the boundary/ body force is 14       Chapter 1 Introduction introduced in order to approximate or enforce the no-slip condition on the immersed boundaries So, how to accurately evaluate the force term becomes a critical issue for a successful implementation of the pressure-velocity formulation-based IBM The calculation of body force... regularized delta function into the force calculation and spreading process Shu et al (2007) discovered that most of the aforementioned methods cannot accurately satisfy the no-slip boundary condition and penetration of some 16       Chapter 1 Introduction streamlines happens on the solid boundary The penetration may degrade the accuracy of the method and also the boundary force calculation To remove flow... done on the development of the immersed boundary method However, there are still rooms for the improvement of methodology and these stimulate the studies in the present thesis The objective of this thesis is to develop several novel immersed boundary methods for fluid and thermal dynamics problems with complex or moving boundaries, aiming to further refine the existing IBM The 24      Chapter 1 Introduction... function-vorticity formulation is more efficient for two-dimensional incompressible flow, the published immersed boundary models are so complicated that high computational effort is needed The present thesis will develop a computationally efficient stream function-vorticity formulation-based immersed boundary method; The reported immersed boundary methods for thermal flows are not capable of exactly satisfying... Dirichlet thermal condition on the immersed boundary In this thesis, a boundary condition-enforced IBM for thermal flows with Dirichlet boundary condition will be presented Furthermore, two simple and convenient ways for the calculation of Nusselt number in the framework of IBM are suggested; Very few work was reported by extending IBM to solve problems with Neumann boundary condition in the framework of. .. When the boundary deviates from the equilibrium location, a restoring force will be generated according to the Hooke’s law so that the 15       Chapter 1 Introduction boundary points will stay close to the target boundary position Goldstein et al (19 93) generalized the above idea and provided a two-mode feedback forcing scheme for the control of velocity condition at the immersed boundary This forcing... best of our knowledge, Wang et al (2009) made an effort to study this problem In their method, the vorticity-velocity formulation is adopted and the spirit of Peskin's original IBM is directly applied Both velocity and vorticity are divided into two parts One is the velocity and vorticity without the influence of immersed boundary, and the other is the 19       Chapter 1 Introduction velocity and vorticity... for simulating 2D incompressible viscous flows is desired 1. 3.3 Applications of immersed boundary method The popularity of immersed boundary method has seen increased for flow problems with complex or /and moving boundaries Although being created to study the fluid dynamics of cardiovascular physiology such as flow in heart valves (McQueen & Peskin 2000), flow in arterioles (Arthurs et al 19 98), deformation... satisfying of no-slip condition on the immersed boundary and evaluation of vorticity source term Two numerical approximations to the vorticity source term are suggested and their performances are tested by several numerical validation cases Chapter 4 extends the IBM to solve heat transfer problems and suggests a boundary condition-enforced method for thermal flows with Dirichlet 26      Chapter 1 Introduction . immersed boundary methods and therefore fall out of the scope of the present  Chapter 1 Introduction 14   thesis. 1. 3 Brief review of Immersed boundary methods The immersed boundary method. 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, . IBM solver for simulating 2D incompressible viscous flows is desired. 1. 3.3 Applications of immersed boundary method The popularity of immersed boundary method has seen increased for flow problems