Date of Award

12-2018

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematical Sciences

Committee Member

Leo Rebholz, Committee Chair

Committee Member

Timo Heister

Committee Member

Qingshan Chen

Committee Member

Fei Xue

Abstract

The Navier-Stokes equations model the evolution of water, oil, and air flow (air under 220 m.p.h.), and therefore the ability to solve them is important in a wide array of engineering design problems. However, analytic solution of these equations is generally not possible, except for a few trivial cases, and therefore numerical methods must be employed to obtain solutions. In the present dissertation we address several important issues in the area of computational fluid dynamics. The first issue is that in typical discretizations of the Navier-Stokes equations such as the mixed finite element method, the conservation of mass is enforced only weakly, and this leads to discrete solutions which may not conserve energy, momentum, angular momentum, helicity, or vorticity, even though the physics of the Navier-Stokes equations dictate that they do. It is widely believed in the computational fluid dynamics community that the more physics is built into the discretization, the more accurate and stable the discrete solutions are, especially over longer time intervals. In chapter 3 we study conservation properties of Galerkin methods for the incompressible Navier-Stokes equations, without the divergence constraint strongly enforced. We show that none of the commonly used formulations (convective, conservative, rotational, and skew-symmetric) conserve each of energy, momentum, and angular momentum (for a general finite element choice). We aim to construct discrete formulations that conserve as many physical laws as possible without utilizing a strong enforcement of the divergence constraint, and doing so leads us to a new formulation that conserves each of energy, momentum, angular momentum, enstrophy in 2D, helicity and vorticity (for reference, the usual convective formulation does not conserve most of these quantities). In chapter 3 we also perform a number of numerical experiments, which verify the theory and test the new formulation. To study the performance of our novel formulation of the Navier-Stokes equations, we need reliable reference solutions/statistics. However, there is not a significant amount of reliable reference solutions for the Navier-Stokes equations in the literature. Accurate reference solutions/statistics are difficult to obtain due to a number of reasons. First, one has to use several millions of degrees of freedom even for a two-dimensional simulation (for 3D one needs at least tens of millions of degrees of freedom). Second, it usually takes a long time before the flow becomes fully periodic and/or stationary. Third, in order to obtain reliable solutions, the time step must be very small. This results in a very large number of time steps. All of this results in weeks of computational time, even with the highly parallel code and efficient linear solvers (and in months for a single-threaded code). Finally, one has to run a simulation for multiple meshes and time steps in order to show the convergence of solutions. In the second chapter we perform a careful, very fine discretization simulations for a channel flow past a flat plate. We derive new, more precise reference values for the averaged drag coefficient, recirculation length, and the Strouhal number from the computational results. We verify these statistics by numerical computations with the three time stepping schemes (BDF2, BDF3 and Crank-Nicolson). We carry out the same numerical simulations independently using deal.II and Freefem++ software. In addition both deal.II/Q2Q1 and Freefem/P2P1 element types were used to verify the results. We also verify results by numerical simulations with multiple meshes, and different time step sizes. Finally, in chapter 4 we compute reference values for the three-dimensional channel flow past a circular cylinder obstacle, with both time-dependent inflow and with constant inflow using up to 70.5 million degrees of freedom. In contrast to the linearization approach used in chapter 2, in chapter 4 we numerically study fully nonlinear schemes, which we linearize using Newton's method. In chapter 4 we also compare the performance of our novel EMAC scheme with the four most commonly used formulations of the Navier-Stokes equations (rotational, skew-symmetric, convective and conservative) for the three-dimensional channel flow past circular cylinder both with the time-dependent inflow and with constant inflow.

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