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.

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.

Molecular dynamics (MD) and computational fluid dynamics (CFD) allowresearchers to study fluid dynamics from two very different standpoints. From a microscopic standpoint, molecular dynamics uses Newton's second law of motion to simulate the interatomic behavior of individual atoms, using statistical mechanics as a tool for analysis. In contrast, CFD describes the motion of a fluid from a macroscopic level using the transport of mass, momentum, and energy of a system as a model. This thesis investigates both MD and CFD as a viable means of studying viscous flow on a nanometer scale. Specifically, we investigate a pressure-driven Poiseuille flow. The results of the MD simulations are processed using software we created to measure velocity, density, and pressure. The CFD simulations are run on numerical software that implements the MacCormack method for the Navier-Stokes equations. Additionally, the CFD simulations incorporate a local definition of viscosity, which is usually uncharacteristic of this simulation method. Based on the results of the simulations, we point out similarities and differences in the obtained steady-state solutions. / Master of Science

The solution to partial differential equations generally requires approximations that result in numerical error in the final solution. Of the different types of numerical error in a solution, discretization error is the largest and most difficult error to estimate. In addition, the accuracy of the discretization error estimates relies on the solution (or multiple solutions used in the estimate) being in the asymptotic range. The asymptotic range is used to describe the convergence of a solution, where an asymptotic solution approaches the exact solution at a rate proportional to the change in mesh spacing to an exponent equal to the formal order of accuracy. A non-asymptotic solution can result in unpredictable convergence rates introducing uncertainty in discretization error estimates. To account for the additional uncertainty, various discretization uncertainty estimators have been developed. The goal of this work is to evaluation discretization error and discretization uncertainty estimators based on Richardson extrapolation for computational fluid dynamics problems. In order to evaluate the estimators, the exact solution should be known. A select set of solutions to the 2D Euler equations with known exact solutions are used to evaluate the estimators. Since exact solutions are only available for trivial cases, two applications are also used to evaluate the estimators which are solutions to the Navier-Stokes equations: a laminar flat plate and a turbulent flat plate using the k-Ï SST turbulence model. Since the exact solutions to the Navier-Stokes equations for these cases are unknown, numerical benchmarks are created which are solutions on significantly finer meshes than the solutions used to estimate the discretization error and uncertainty. Metrics are developed to evaluate the accuracy of the error and uncertainty estimates and to study the behavior of each estimator when the solutions are in, near, and far from the asymptotic range. Based on the results, general recommendations are made for the implementation of the error and uncertainty estimators. In addition, a new uncertainty estimator is proposed with the goal of combining the favorable attributes of the discretization error and uncertainty estimators evaluated. The new estimator is evaluated using numerical solutions which were not used for development and shows improved accuracy over the evaluated estimators. / Master of Science

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.

In this thesis we develop tools for carrying out computer assisted proof of the stability of traveling wave solutions of the spatially one-dimensional compressible Navier-Stokes equations with an ideal gas equation of state. In particular, we obtain rigorous, tight error bounds on a high-accuracy numerical approximation of the traveling wave profile for parameters corresponding to air, and we obtain rigorous representations in a neighborhood of positive and negative infinity of the solution to the first order ODE associated with linearizing the PDE equations about the traveling wave solution. We also develop supporting tools for rigorous verification of wave stability.

The interaction of oblique shocks and oblique compression waves with a laminar boundary layer on an adiabatic flat plate is analyzed by solving the Navier-Stokes equations in conservation-law form numerically. The numerical scheme is based on the Beam and Warming’s implicit method with approximate factorization. We examine the flow of Bethe-Zel’dovich-Thompson (BZT) fluids at pressures and temperatures on the order of those of the thermodynamic critical point. A BZT fluid is a single-phase gas having specific heat so large that the fundamental derivative of gas dynamics, Γ, is negative over a finite range of pressures and temperatures. The equation of state is the well-known Martin-Hou equation. The main result is the demonstration that the natural dynamics of BZT fluids can suppress boundary layer separation. Physically, this suppression can be attributed to the decrease in adverse pressure gradients associated with the disintegration of compression discontinuities in BZT fluids. / Ph. D.

The two-dimensional, confined, laminar flow over a downstream facing step was studied using a finite element, Navier-Stokes equation solver. The weak form of the stationary, incompressible Navier-Stokes equations in primitive variable form was obtained using the conventional Galerkin technique for mixed problems. Biquadratic Lagrange interpolating polynomials were used to construct the basis functions that generated the finite-dimensional subspace containing the approximate solutions to the velocity field, while the pressure field was represented by a discontinuous, piecewise-linear approximation. This particular combination of solution subspaces was previously shown in a mathematically rigorous fashion to yield stable, consistent solutions to the Navier-Stokes equations. The results of the computations were benchmarked against the experimental data of Denham and Patrick, and also compared to earlier calculations by Ecer and Thomas, both of whom utilized alternative, unconventional formulations. These comparisons indicate that with the proper choice of basis functions, a conventional Galerkin scheme can yield results that are in as good and in many cases better agreement with the available experimental data than those of unconventional schemes that rely upon an infusion of artificial dissipation to enhance their numerical stability. The computational algorithm was also used to ascertain the cause of the noticeable lack of development and skewness that characterized the experimental data of Denham and Patrick both at and upstream of the step. The results of this study indicated that as suspected by Denham and Patrick, the skewness as well as the lack of development of the velocity profiles near the step were caused by the geometry of the test apparatus upstream of the step rather than by the presence of the step itself. The numerical experiments conducted here have been carefully documented so as to facilitate future comparisons intended to assess the relative efficiency of the present method of computation. / Doctor of Philosophy

Supersonic slot injection problems were studied by a finite volume method. The numerical technique used is the upwind method of Roe’s flux difference splitting (FDS) with vertical line Gauss-Seidel relaxation applied to the thin layer Navier-Stokes equations. To test the accuracy of the numerical methods without the complications and uncertainties of turbulence modeling, two sample cases were chosen with laminar flows. The sample problems were the compressible laminar boundary layer flow over a flat plate and the laminar boundary layer - shock interaction problem. For these problems, both the results from Roe’s FDS and van Leer’s flux vector splitting (FVS) are compared with exact solutions and experimental data. For the sample problems, comparisons showed that Roe’s FDS method is more accurate than van Leer’s FVS method. Because of the very complicated wave patterns and strong viscous-inviscid interaction produced by supersonic slot injection, an adaptive grid based on the equidistribution law was combined with the solution algorithm. The results from Roe’s FDS method with the adaptive grid showed good results for the supersonic slot injection over a flat plate. For the slot injection over a 10-degree wedge surface case, there is a significant difference between the numerical and experimental wall pressure distribution. Some potential reasons for the discrepancy including 3D effects and/or transition in the reattachment region in the experiments and possibly a need for a much finer grid in the calculations are discussed. / Ph. D.