The development of computational fluid dynamics has given us a very powerful tool for investigation of fluid dynamics. However, in order to maintain the progress, it is necessary to improve the numerical algorithms. Nowadays, the high-order methods based on the discontinuous projection seem to have the largest potential for the future. In the work, we used open-source framework Nektar++, which provides the high-order discretization method. We tested the abilities of the framework for computing the compressible sonic and transonic flow. We successfully obtained simulations of the viscous and inviscid flow. We computed the lift and the drag coefficients and showed that for a higher polynomial order we can obtain the same accuracy with less degrees of freedom and lower computational time. Also, we tested the shock capturing method for the computation of the inviscid transonic flow and confirmed the potential of the high order methods. 1

京都大学 / 0048 / 新制・課程博士 / 博士(農学) / 甲第21151号 / 農博第2277号 / 新制||農||1059(附属図書館) / 学位論文||H30||N5125(農学部図書室) / 京都大学大学院農学研究科地域環境科学専攻 / (主査)教授 村上 章, 教授 川島 茂人, 教授 藤原 正幸 / 学位規則第4条第1項該当 / Doctor of Agricultural Science / Kyoto University / DFAM

The implementation of a hybrid spectral/finite-element discretization on the unsteady, incompressible, Navier-Stokes equations with a semi-implicit time-stepping method, an explicit treatment of the advective terms, and an implicit treatment of the pressure and viscous terms leads to an algorithm capable of calculating 3D flows over complex 2D geometries. This also results in multiple Fourier mode linear systems which must be solved at every timestep, which naturally leads to two parallelization approaches: Fourier space partitioning, where each processor individually and simultaneously solves a linear system, and physical space partitioning, where all processors collectively solve each linear system, sequentially advancing through Fourier modes. These two parallelization approaches are compared based upon computational cost using multiple solvers: direct sparse LU, smoothed aggregation AMG, and single-level ILUT preconditioned GMRES; and on two supercomputers of different memory architecture(distributed and shared memory). This study revealed Fourier space partitioning outperforms physical space partitioning in all problems analyzed, and scales more efficiently as well. These differences were more dramatic on the distributed memory platform than the shared memory platform. Another study compares the previously mentioned solvers along with one additional solver, pointwise AMG, in Fourier space partitioning without parallelization to better understand computational scaling for problems with large meshes. It was found that the direct sparse LU solver performed well in terms of computational time, scaled linearly, but had very high memory usage which scaled in a super-linear manner. The single-level ILUT preconditioned GMRES solver required the least amount of memory, which also scaled linearly, but only had acceptable performance in terms of computational time for coarse meshes. Both AMG methods scaled linearly in both memory usage and time, and were comparable to the direct sparse LU solver in terms of computational time. The results of these studies are particularly useful for implementation of this algorithm on challenging and complex flows, especially direct numerical and large-eddy simulations. Reducing computational cost allows the analysis and understanding of more flows of practical interest.

We consider three different phenomena governing the fluid flow in the presence of charged particles: electroconvection in fluids, electroconvection in porous media, and electrodiffusion. Electroconvecton in fluids is mathematically represented by a nonlinear drift-diffusion partial differential equation describing the time evolution of a surface charge density in a two-dimensional incompressible fluid. The velocity of the fluid evolves according to Navier-Stokes equations forced nonlinearly by the electrical forces due to the presence of the charge density. The resulting model is reminiscent of the quasi-geostrophic equation, where the main difference resides in the dependence of the drift velocity on the charge density. When the fluid flows through a porous medium, the velocity and the electrical forces are related according to Darcy’s law, which yields a challenging doubly nonlinear and doubly nonlocal model describing electroconvection in porous media. A different type of particle-fluid interaction, called electrodiffusion, is also considered. This latter phenomenon is described by nonlinearly advected and nonlinearly forced continuity equations tracking the time evolution of the concentrations of many ionic species having different valences and diffusivities and interacting with an incompressible fluid. This work is based on [1, 2, 3] and addresses the global well-posedness, long-time dynamics, and other features associated with the aforementioned three models. REFERENCES:[1] E. Abdo, M. Ignatova, Long time dynamics of a model of electroconvection, Trans. Amer. Math. Soc. 374 (2021), 5849–5875. [2] E. Abdo, M. Ignatova, Long Time Finite Dimensionality in Charged Fluids, Nonlinearity 34 (9) (2021), 6173–6209. [3] E. Abdo, M. Ignatova, On Electroconvection in Porous Media, to appear in Indiana University Mathematics Journal (2023). / Mathematics

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This thesis describes a simple efficient and robust numerical technique for solving two-dimensional incompressible laminar steady flows at moderate-to-high Reynolds numbers. The method uses an incremental multigrid method and an extrapolation procedure based on minimum residual concepts to accelerate the convergence rate of a robust block-line-Gauss-Seidel solver for the vorticity-stream function equations. Results are presented for the driven cavity flow problem using uniform and nonuniform grids and for the flow past a backward facing step in a channel. / M.S.

M.S.

This dissertation discusses two aspects of computational fluid dynamics: high order spatial accuracy and convergence-rate acceleration through system preconditioning. Concerning high-order accuracy, the computational qualities of various spatial methods for the finite-volume solution of the Euler equations are presented. The two-dimensional essentially non-oscillatory (ENO), k-exact, and dimensionally split ENO reconstruction operators are discussed and compared in terms of reconstruction and solution accuracy and computational cost. Standard variable extrapolation methods are included for completeness. Inherent steady-state convergence difficulties are demonstrated for adaptive-stencil algorithms. Methods for reconstruction error analysis are presented and an exact solution to the heat equation is used as an example. Numerical experiments presented include the Ringleb flow for numerical accuracy and a shock-reflection problem. A vortex-shock interaction demonstrates the ability of the EN 0 scheme to excel in capturing unsteady high-frequency flow physics. Concerning convergence-rate acceleration, characteristic-wave preconditioning is extended to include generalized finite-rate chemistry with non-equilibrium thermodynamics Additionally, the proper preconditioning for the one-dimensional Navier-Stokes equations is presented. Eigenvalue stiffness is resolved and convergencerate acceleration is demonstrated over the entire Mach-number range from the incompressible to the hypersonic. Specific benefits are realized at low and transonic flow speeds. The extended preconditioning matrix accounts for thermal and chemical non-equilibrium and its implementation is explained for both explicit and implicit time marching. The effects of high-order spatial accuracy and various flux splittings are investigated. Numerical analysis reveals the possible theoretical improvements from using preconditioning at all Mach numbers. Numerical results confirm the expectations from the analysis. The preconditioning matrix is applied with dual time stepping to obtain arbitrarily high-order accurate temporal solutions within an implicit formulation. Representative test cases include flows with previously troublesome embedded high-condition-number regions. / Ph. D.

The central goal of this work is to define and analyze least squares finite element methods for the Stokes and Navier-Stokes equations that are practical and optimal in a systematic and rigorous way. To accomplish this task we begin by developing the least squares theory for the linear Stokes equations. We introduce least squares methods based on the minimization of functionals that involve residuals of the equations of an equivalent first order formulation for the Stokes problem. We show that for the Stokes equations there are two general types of boundary conditions. For the first type, practical least squares methods can be defined and analyzed in a fairly standard way, based on application of the Agmon, Douglis and Nirenberg a priori estimates. For the second type of boundary conditions this task is more difficult and involves mesh dependent (weighted) least squares functionals. Among the main results are the optimal error estimates for the weighted least squares method in two and three space dimensions. Then, we formulate two least squares methods for the nonlinear Navier-Stokes equations written as a first order system. We consider the first method as a conforming discretization of an abstract nonlinear problem and the second weighted one, which is more practical, as a nonconforming discretization of the same abstract problem. As a result, the analysis of the first method fits into the framework of the approximation theory of Brezzi, Rappaz and Raviart and the analysis of the second does not. Thus, we develop an abstract approximation theory that is suitable for nonconforming discretizations of the abstract problem. The central result is based on the application of our abstract theory to the weighted least squares method. We prove that this method results in optimally accurate approximations for the Navier-Stokes equations. We believe that these error analyses of Chapter are the first treatment of a least squares formulation for a nonlinear problem in the current literature. We then discuss various implementation issues, including theoretical and numerical estimates of condition numbers and the presentation of numerical examples. In particular, we study the numerical convergence rates of various implementations of least squares methods and demonstrate that the weights are necessary for the optimal rates to hold. Finally, we compare numerical results for the driven cavity flow problem with some benchmark results reported in the literature. / Ph. D.

The velocity tracking problem is motivated by the desire to match a desired target flow with a flow which can be controlled through time dependent distributed forces or time dependent boundary conditions. The flow model is the Navier-Stokes equations for a viscous incompressible fluid and different kinds of controls are studied. Optimal distributed and boundary controls minimizing a quadratic functional and an optimal bounded distributed control are investigated. The distributed optimal and the bounded control are compared with a linear feedback control. Here, a unified mathematical formulation, covering several specific classes of meaningful control problems in bounded domains, is presented with a complete and detailed analysis of all these time dependent optimal control velocity tracking problems. We concentrate not only on questions such as existence and necessary first order conditions but also on discretization and computational aspects. The first order necessary conditions are derived in the continuous, in the semidiscrete time approximation and in the fully finite element discrete case. This derivation is needed to obtain an accurate meaningful numerical algorithm with a satisfactory convergence rate. The gradient algorithm is used and several numerical computations are performed to compare and understand the limits imposed by the theory. Some computational aspects are discussed without which problems of any realistic size would remain intractable. / Ph. D.