The continued research and development of high-order methods in Computational Fluid Dynamics (CFD) is primarily motivated by their potential to significantly reduce the computational cost and memory usage required to obtain a solution to a desired level of accuracy. In this work, a high-order Central Essentially Non-Oscillatory (CENO) finite-volume scheme is developed for the Euler and Navier-Stokes equations in three dimensions. The proposed CENO scheme is based on a hybrid solution reconstruction procedure using a fixed central stencil. A solution smoothness indicator facilitates the hybrid switching between a high-order k-exact reconstruction technique, and a monotonicity preserving limited piecewise linear reconstruction algorithm. The resulting scheme is applied to the compressible forms of the Euler and Navier-Stokes equations in three dimensions. The latter of which includes the application of this high-order work to the Large Eddy Simulation (LES) of turbulent non-reacting flows.

Large-eddy simulation (LES) is emerging as a promising computational tool for reacting flows. High-order schemes for LES are desirable to achieve improved solution accuracy with reduced computational cost. In this study, a parallel, block-based, three-dimensional high-order central essentially non-oscillatory (CENO) finite-volume scheme for LES of premixed turbulent combustion is developed for Cartesian mesh. This LES formulation makes use of the flame surface density (FSD) for subfilter-scale reaction rate modelling. An algebraic model is used to approximate the FSD. A detailed explanation of the governing equations for LES and the mathematical framework for CENO schemes are presented. The CENO reconstruction is validated and is also applied to three-dimensional Euler equations prior to its application to the equations governing LES of reacting flows.

A block-based adaptive mesh refinement (AMR) finite-volume scheme is developed for solution of hyperbolic conservation laws on two-dimensional hybrid multi-block meshes. A Godunov-type upwind finite-volume spatial-discretization scheme, with piecewise limited linear reconstruction and Riemann-solver based flux functions, is applied to the quadrilateral cells of a hybrid multi-block mesh and these computational cells are embedded in either body-fitted structured or general unstructured grid partitions of the hybrid grid. A hierarchical quadtree data structure is used to allow local refinement of the individual subdomains based on heuristic physics-based refinement criteria. An efficient and scalable parallel implementation of the proposed algorithm is achieved via domain decomposition. The performance of the proposed scheme is demonstrated through application to solution of the compressible Euler equations for a number of flow configurations and regimes in two space dimensions. The efficiency of the AMR procedure and accuracy, robustness, and scalability of the hybrid mesh scheme are assessed.

Large-eddy simulation (LES) is emerging as a promising computational tool for reacting flows. High-order schemes for LES are desirable to achieve improved solution accuracy with reduced computational cost. In this study, a parallel, block-based, three-dimensional high-order central essentially non-oscillatory (CENO) finite-volume scheme for LES of premixed turbulent combustion is developed for Cartesian mesh. This LES formulation makes use of the flame surface density (FSD) for subfilter-scale reaction rate modelling. An algebraic model is used to approximate the FSD. A detailed explanation of the governing equations for LES and the mathematical framework for CENO schemes are presented. The CENO reconstruction is validated and is also applied to three-dimensional Euler equations prior to its application to the equations governing LES of reacting flows.

A block-based adaptive mesh refinement (AMR) finite-volume scheme is developed for solution of hyperbolic conservation laws on two-dimensional hybrid multi-block meshes. A Godunov-type upwind finite-volume spatial-discretization scheme, with piecewise limited linear reconstruction and Riemann-solver based flux functions, is applied to the quadrilateral cells of a hybrid multi-block mesh and these computational cells are embedded in either body-fitted structured or general unstructured grid partitions of the hybrid grid. A hierarchical quadtree data structure is used to allow local refinement of the individual subdomains based on heuristic physics-based refinement criteria. An efficient and scalable parallel implementation of the proposed algorithm is achieved via domain decomposition. The performance of the proposed scheme is demonstrated through application to solution of the compressible Euler equations for a number of flow configurations and regimes in two space dimensions. The efficiency of the AMR procedure and accuracy, robustness, and scalability of the hybrid mesh scheme are assessed.

Two immersed boundary methods (IBM) for the simulation of conjugate heat transfer problems with complex geometries are introduced: a finite element (IFEM) and a finite volume (IFVM) immersed boundary methods are discussed. In the IFEM a projection approach is presented for the coupled system of time-dependent incompressible Navier-Stokes equations (NSEs) and energy equation in conjunction with the immersed boundary method for solving fluid flow and heat transfer problems in the presence of rigid objects not represented by the underlying mesh. The IBM allows solving the flow for geometries with complex objects without the need of generating a body-fitted mesh. Dirichlet boundary constraints are satisfied applying a boundary force at the immersed body surface. Using projection and interpolation operators from the fluid volume mesh to the solid surface mesh (i.e., the “immersed” boundary) and vice versa, it is possible to impose the extra constraint to the NSEs as a Lagrange multiplier in a fashion very similar to the effect pressure has on the momentum equations to satisfy the divergence-free constraint. The IFEM approach presented shows third order accuracy in space and second order accuracy in time when the simulation results for the Taylor-Green decaying vortex are compared to the analytical solution. For the IFVM a ghost-cell approach with sharp interface scheme is used to enforce the boundary condition at the fluid/solid interface. The interpolation procedure at the immersed boundary preserves the overall second order accuracy of the base solver. The developed ghost-cell method is applied on a staggered configuration with the Semi-Implicit Method for Pressure-Linked Equations Revised algorithm. Second order accuracy in space and first order accuracy in time are obtained when the Taylor-Green decaying vortex test case is compared to the IFVM analytical solution. Computations were performed using the IFEM and IFVM approaches for the two-dimensional flow over a backward-facing step, two-dimensional flow past a stationary circular cylinder, three-dimensional flow past a sphere and two and three-dimensional natural convection in an enclosure with/without immersed body. The numerical results obtained with the discussed IFEM and IFVM were compared against other IBMs available in literature and simulations performed with the commercial computational fluid dynamics code STAR-CCM+/V7.04.006. The benchmark test cases showed that the numerical results obtained with the implemented immersed boundary methods are in good agreement with the predictions from STAR-CCM+ and the numerical data from the other IBMs. The immersed boundary method based of finite element approach is numerically more accurate than the IBM based on finite volume discretization. In contrast, the latter is computationally more efficient than the former.

The continued research and development of high-order methods in Computational Fluid Dynamics (CFD) is primarily motivated by their potential to significantly reduce the computational cost and memory usage required to obtain a solution to a desired level of accuracy. In this work, a high-order Central Essentially Non-Oscillatory (CENO) finite-volume scheme is developed for the Euler and Navier-Stokes equations in three dimensions. The proposed CENO scheme is based on a hybrid solution reconstruction procedure using a fixed central stencil. A solution smoothness indicator facilitates the hybrid switching between a high-order k-exact reconstruction technique, and a monotonicity preserving limited piecewise linear reconstruction algorithm. The resulting scheme is applied to the compressible forms of the Euler and Navier-Stokes equations in three dimensions. The latter of which includes the application of this high-order work to the Large Eddy Simulation (LES) of turbulent non-reacting flows.

Thin liquid films driven by surface tension gradients are studied in diverse applications, including the spreading of a droplet and fluid flow in the lung. The nonlinear partial differential equations that govern thin films are difficult to solve analytically, and must be approached through numerical simulations. We describe the development of a numerical solver designed to solve a variety of thin film problems in two dimensions. Validation of the solver includes grid refinement studies and comparison to previous results for thin film problems. In addition, we apply the solver to a model of surfactant spreading and make comparisons with theoretical and experimental results.

Thermal radiation is of significant importance in a broad range of engineering applications including high-temperature and large-scale systems. Although the governing equations of thermal radiation have been known for many years, the complexities inherent in the phenomenon, such as the multidimensionality and integro-differential nature of these equations, have made it difficult to obtain an accurate, efficient, and robust computational method. Developing the finite volume radiation method in the 1990s was a significant progress but not a panacea for computational radiation. The major drawback of this method, which is common among all methods that solve for directional intensities, is its slow convergence rate in many situations which increases the solution cost dramatically. These situations include large optical thicknesses, strongly reflecting boundaries, and any other factor that causes strong directional coupling like complex geometries. Several acceleration schemes have been developed in the heat transfer and neutron transport communities to expedite the convergence and reduce the solution cost, but none of them led to a general and reliable method. Among these available schemes, the two most promising ones, the multiplicative scheme and coupled ordinates method, suffer from failing on fine grids and being very complicated for complex scattering phase functions, respectively. In this research, a new computational method, called the QL method, has been introduced. The main idea of this method is using the phase weight concept to relate the directional and average intensities and re-arranging the Radiative Transfer Equation to find a new expression for the radiant heat flux. This results in an elliptic-type equation for the average intensity at each control volume which conserves the radiant energy in all directions in the control volume. This formulation gives the QL method a great advantage to solve for the average intensity while including the directional effects. Since the directional effects are included and the radiant energy is conserved in each control volume, this method is expected to be accurate and have a good convergence rate in all conditions. The phase weight distribution required by the QL method can be provided by a method like the finite volume method or discrete ordinates method. The QL method is applied to several 1D and 2D test cases including isotropic and anisotropic scattering, black and partially reflecting boundaries, and emitting absorbing problems; and its accuracy, convergence rate, and solution cost are studied. The method has been found to be very stable and efficient, regardless of grid size and optical thickness. This method establishes very accurate predictions on the tested coarse grids and its results approach the exact solution with grid refinement.

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