Chapter Conclusions and Recommendations     Chapter Conclusions and Recommendations 9.1 Conclusions In this thesis, several novel IBM solvers have been developed for fluid and thermal flows involving complex geometries or moving objects/boundaries: one for two dimensional incompressible viscous flows and two for thermal flows respectively with Dirichlet and Neumann temperature conditions. Additionally, a new version of boundary condition-enforced IBM in the framework of NS solver is presented. Their abilities for dealing with fluid and thermal problems have been well demonstrated by applications to various fluid and thermal flows with geometric and dynamic complexities. Note that the four methods proposed employ Dirac delta function in modeling the immersed boundaries. In other words, the immersed boundaries are assumed to be have finite thickness. Therefore they are truly immersed boundary methods. The critical issue of the IBM lies in the accurate evaluation of body forces. The conventional IBM generally calculates body forces explicitly, so the flow penetration to the surface of the immersed objects happens. On the other hand, 308      Chapter Conclusions and Recommendations the velocity correction-based IBM, which was originally developed within the framework of LBM solver, has proven to be effective in exactly enforcing the no-slip condition on the immersed boundary. In this regard, the present thesis firstly extends the method to a Navier-Stokes (NS) solver-based version where the popular primitive variable formulation is solved. Combined with the fractional step technique, the body force in the modified momentum equation is implicitly determined so that the no-slip condition on the immersed boundary is accurately satisfied. The good performance of the new IBM is extensively verified, first through the classical problems of flow over a single stationary circular cylinder and flow interference between two side-by-side circular cylinders, and then by the moving boundary cases such as vortex-structure interaction around a transversely oscillating cylinder and vortex-induced-vibration of an elastically mounted circular cylinder. Most IBM solvers are established under the framework of pressure-velocity formulation. For incompressible flows, the pressure-velocity formulation suffers from some difficulties and is usually rather expensive to solve. In contrast, the stream function-vorticity formulation has been recognized to be more efficient for two dimensional incompressible viscous flows. Inspired by the merits of the stream function-vorticity formulation and IBM techniques, a stream function-vorticity formulation-based IBM solver is proposed, in which the immersed object is modeled as localized vorticity sources and plays the 309      Chapter Conclusions and Recommendations role of vorticity corrections in the stream function equation. The main idea of the method is to accurately satisfy the governing equation and boundary condition, which is realized through velocity correction and vorticity correction procedures. The velocity corrections, in the present model, are evaluated in the principle of guaranteeing the no-slip condition on the immersed boundary, i.e., the velocity at the boundary (Lagrangian) point interpolated from the physical velocity at the Eulerian points equals to the given boundary velocity. The vorticity corrections, on the other hand, are directly evaluated from the first-order derivatives of velocity corrections by two proposed approximation methods. The whole fluid solver and the two vorticity correction methods were first validated by their applications to a stationary boundary problem of flow over a circular cylinder. The obtained flow characteristics agree very well with the published data in the literature both qualitatively and quantitatively. Additionally, no streamline penetration appears on the cylinder surface, implying that the boundary condition is accurately satisfied. Then the proposed solver was applied to examples of flow over a left moving circular cylinder and flow over an inline oscillating circular cylinder, to further examine its capability in handling moving boundary problems. Furthermore, a fluid-structure interaction problem, sedimentation of a circular particle between two parallel walls, was examined. All the obtained numerical results for the considered moving boundary cases were in very good agreement with the previous reported ones in the literature, indicating the 310      Chapter Conclusions and Recommendations promising potential of the stream function-vorticity formulation-based IBM for solving two dimensional incompressible viscous flows involving complex geometries and moving boundaries. To extend the IBM for heat transfer problems which are suffering from complex or moving/deforming boundaries, two thermal IBM solvers were developed. Firstly, a boundary condition-enforced IBM solver was suggested for heat transfer problems with Dirichlet-type boundary condition, where the heated immersed boundary is modeled as a set of heat sources added to the energy equation as the source term. All the previous IBM solvers treated the heat source term explicitly and no mechanism was implemented to enforce the boundary condition for temperature. Thus the prescribed temperature on the immersed boundary was not accurately satisfied and the accuracy of numerical results was negatively affected. In the present boundary condition-enforced IBM, the heat source/sink term is considered as unknown and determined implicitly such that the energy equation and the corresponding thermal boundary condition can be accurately satisfied. Furthermore, the critical issue of how to effectively calculate the average Nusselt number in IBM was properly addressed, as an additional contribution. Numerical experiment on accuracy analysis showed a second-order accuracy of the proposed IBM solver. The present method and proposed techniques to compute the Nusselt number were then validated by both forced and natural convections where the obtained 311      Chapter Conclusions and Recommendations numerical results compared considerably well with available data in the literature, implying that the suggested thermal solver and Nusselt number evaluation techniques provide a wonderful tool for thermal problems with Dirichlet condition. Secondly, a heat flux correction-based IBM was proposed for thermal flows with prescribed heat flux condition. Note that this is the first time IBM is applied to solve thermal flow problems with Neumann-type boundary conditions. In the proposed solver, the heated immersed boundary was regarded as localized heat sources whose evaluations and contributions were considered through a heat flux correction procedure. To compensate the differences between the calculated heat fluxes and the prescribed ones, boundary heat sources arising from the heat flux differences were introduced which were then distributed to the surrounding fluid as volumetric heat sources to correct the temperature field. The whole process was intelligible and easy to implement. The present solver, through a numerical analysis on a model problem of heat conduction, was recognized to be of second order in accuracy. Then its capability and efficiency were validated by well-established examples like forced convection over a stationary heated circular cylinder and natural convection in a horizontal concentric and eccentric annulus between two circular cylinders. The numerical results obtained matched very well with the reported ones, showing the high potential of the proposed IBM solver for problems with Neumann conditions. 312      Chapter Conclusions and Recommendations Subsequently, the developed methods were applied to various two- and three-dimensional fluid and thermal flows to further check their capabilities for handling geometric complexity and moving boundaries. In two dimensions, insect hovering flight where the insect wing undergoes a prescribed harmonic translational and rotational motion near or away from the ground was firstly simulated, which was followed by particulate flows where the motion of the involved particle is not given prior but coupled and mutually determined with the surrounding fluid. Obtained results showed that both types of moving boundary problems are correctly predicted and good agreements with available experimental and numerical results were observed. With the confidence achieved from the above validations, the methods were utilized to study the forced convective heat transfer from a transversely oscillating circular cylinder in the wake of a stationary one, where it was found that the flow behaviors and heat transfer characteristics were greatly affected by the mutual influences of spacing and downstream cylinder excitation. After successfully verifying their exciting performance in two-dimensional applications, the proposed temperature correction-based and heat flux correction-based IBM solvers are examined for their capabilities in dealing with three-dimensional thermal flows involving complex or moving boundaries. Examples related to forced convective heat and mass transfer from stationary or streamwise rotating spheres in uniform cross flow were simulated. 313      Chapter Conclusions and Recommendations The good agreement between the obtained results and the published ones in the literature clearly shows the easy implementation and accurate natures of the proposed IBM solvers for three-dimensional heat transfer problems. Finally, the simulation of complex moving boundary flows was conducted, where they were either with complex geometry or in complex movement, or both. One is the solid finite-span foil heaving and pitching in the air, and the other is the flexible-body fish swimming in the water. Both problems were accurately predicted as compared to the available numerical results, showing the great potential of the immersed boundary method for three-dimensional practical flow problems with complex solid or flexible moving boundaries. 9.2 Recommendations for the Future Work The temperature correction-based IBM is proposed to improve the accuracy of the existing IBM solvers for heat transfer problems subject to Dirichlet-type boundary conditions and the developed heat flux correction-based IBM is considered as an initial attempt to extend the IBM for solving problems with Neumann-type boundary conditions. While validation tests indicate their precious features such as accurate, efficient, simple and easy implementation, they can be extensively adopted to solve a large number of thermal flows with complex or moving geometries. However, both thermal solvers are limited to the cases where the temperature condition on the immersed boundary, either in Dirichlet or Neumann type, is prescribed in advance. From the viewpoint of 314      Chapter Conclusions and Recommendations methodology development, it may be desirable to generalize the solvers to more general cases where the temperature conditions change with time and are to be determined during the solution process. From the viewpoint of practical and engineering applications, the present study conducted simulations for three-dimensional forced and natural convective flows with complex geometries (Chapter 7) and three-dimensional moving boundary isothermal problems with prescribed motion (Chapter 8). Fluid and thermal problems with high dynamic complexity, such as mix convections, three-dimensional moving boundary thermal flows or fully fluid-structure interactions, are not included in the present thesis. 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