This paper presents topology optimization of capillary, the typical two-phase flow with immiscible fluids, where the level set method and diffuse-interface model are combined to implement the proposed method. The two-phase flow is described by the diffuse-interface model with essential no slip condition imposed on the wall, where the singularity at the contact line is regularized by the molecular diffusion at the interface between two immiscible fluids. The level set method is utilized to express the fluid and solid phases in the flows and the wall energy at the implicit fluid-solid interface. Based on the variational procedure for the total free energy of two-phase flow, the Cahn-Hilliard equations for the diffuse-interface model are modified for the two-phase flow with implicit boundary expressed by the level set method. Then the topology optimization problem for the two-phase flow is constructed for the cost functional with general formulation. The sensitivity analysis is implemented by using the continuous adjoint method. The level set function is evolved by solving the Hamilton-Jacobian equation, and numerical test is carried out for capillary to demonstrate the robustness of the proposed topology optimization method. It is straightforward to extend this proposed method into the other two-phase flows with two immiscible fluids.

An investigation of natural convective flow and heat transfer inside a three dimensional rectangular cavity containing an array of discrete heat sources is carried out. The array consists of a row and columnwise regular arrangement of identical square shaped isoflux discrete heaters and is flush mounted on a vertical wall of the cavity. A symmetrical isothermal sink condition is maintained by cooling the cavity uniformly from either the opposite wall or the side walls or the top and bottom walls. The other walls of the cavity are maintained adiabatic. A finite volume method based on the SIMPLE algorithm and the power law scheme is used to solve the conservation equations. The parametric study covers the influence of pertinent parameters such as the Rayleigh number, the Prandtl number, side aspect ratio of the cavity and cavity heater ratio. A detailed fluid flow and heat transfer characteristics for the three cases are reported in terms of isothermal and velocity vector plots and Nusselt numbers. In general it is found that the overall heat transfer rate within the cavity for Ra=107 is maximum when the side aspect ratio of the cavity lies between 1.5 and 2. A more complex and peculiar flow pattern is observed in the presence of top and bottom cold walls which in turn introduces hot spots on the adiabatic walls. Their location and size are highly sensitive to the side aspect ratio of the cavity and hence offers more effective ways for passive heat removal.

A high-order finite difference scheme has been developed to approximate the spatial derivative terms present in the unsteady Poisson-Nernst-Planck (PNP) equations and incompressible Navier-Stokes (NS) equations. Near the wall the sharp solution profiles are resolved by using the combined compact difference (CCD) scheme developed in five-point stencil. This CCD scheme has a sixth-order accuracy for the second-order derivative terms while a seventh-order accuracy for the first-order derivative terms. PNP-NS equations have been also transformed to the curvilinear coordinate system to study the effects of channel shapes on the development of electroos-motic flow. In this study, the developed scheme has been analyzed rigorously through the modified equation analysis. In addition, the developed method has been computationally verified through four problems which are amenable to their own exact solutions. The electroosmotic flow details in planar and wavy channels have been explored with the emphasis on the formation of Coulomb force. Significance of different forces resulting from the pressure gradient, diffusion and Coulomb origins on the convective electroosmotic flow motion is also investigated in detail.

The significance of flow optimization utilizing the lattice Boltzmann (LB) method becomes obvious regarding its advantages as a novel flow field solution method compared to the other conventional computational fluid dynamics techniques. These unique characteristics of the LB method form the main idea of its application to optimization problems. In this research, for the first time, both continuous and discrete adjoint equations were extracted based on the LB method using a general procedure with low implementation cost. The proposed approach could be performed similarly for any optimization problem with the corresponding cost function and design variables vector. Moreover, this approach was not limited to flow fields and could be employed for steady as well as unsteady flows. Initially, the continuous and discrete adjoint LB equations and the cost function gradient vector were derived mathematically in detail using the continuous and discrete LB equations in space and time, respectively. Meanwhile, new adjoint concepts in lattice space were introduced. Finally, the analytical evaluation of the adjoint distribution functions and the cost function gradients was carried out.

The unsteady flow separation of airfoil with a local flexible structure (LFS) is studied numerically in Lagrangian frames in detail, in order to investigate the nature of its high aerodynamic performance. For such aeroelastic system, the characteristic-based split (CBS) scheme combined with arbitrary Lagrangian-Eulerian (ALE) framework is developed firstly for the numerical analysis of unsteady flow, and Galerkin method is used to approach the flexible structure. The local flexible skin of airfoil, which can lead to self-induced oscillations, is considered as unsteady perturbation to the flow. Then, the ensuing high aerodynamic performances and complex unsteady flow separation at low Reynolds number are studied by Lagrangian coherent structures (LCSs). The results show that the LFS has a significant influence on the unsteady flow separation, which is the key point for the lift enhancement. Specifically, the oscillations of the LFS can induce the generations of moving separation and vortex, which can enhance the kinetic energy transport from main flow to the boundary layer. The results could give a deep understand of the dynamics in unsteady flow separation and flow control for the flow over airfoil.

A localized flexible airfoil at low Reynolds numbers is modeled and the aerodynamic performance is analyzed numerically. With characteristic based split scheme, a fluid solver for two dimensional incompressible Navier-Stokes equations is developed under the ALE framework, coupled with the theory of shallow arch, which is approximated by Galerkin method. Further, the interactions between the unsteady flow and the shallow arch are studied in detail. In particular, the effect of the self-excited vibration of the structure on aerodynamic performance of the airfoil is investigated deeply at various angles of attack. The results show that the lift-to-drag ratio has been increased greatly compared with the rigid airfoil. Finally, the relationship between the self-excited vibration and the evolution of the flow is analyzed using FFT tools.