The EMAC scheme for Navier-Stokes simulations, and application to flow past bluff bodies

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\u27s 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

Weak Form of Stokes-Dirac Structures and Geometric Discretization of Port-Hamiltonian Systems

We present the mixed Galerkin discretization of distributed parameter port-Hamiltonian systems. On the prototypical example of hyperbolic systems of two conservation laws in arbitrary spatial dimension, we derive the main contributions: (i) A weak formulation of the underlying geometric (Stokes-Dirac) structure with a segmented boundary according to the causality of the boundary ports. (ii) The geometric approximation of the Stokes-Dirac structure by a finite-dimensional Dirac structure is realized using a mixed Galerkin approach and power-preserving linear maps, which define minimal discrete power variables. (iii) With a consistent approximation of the Hamiltonian, we obtain finite-dimensional port-Hamiltonian state space models. By the degrees of freedom in the power-preserving maps, the resulting family of structure-preserving schemes allows for trade-offs between centered approximations and upwinding. We illustrate the method on the example of Whitney finite elements on a 2D simplicial triangulation and compare the eigenvalue approximation in 1D with a related approach.Comment: Copyright 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0

Numerical analysis of a structure-preserving space-discretization for an anisotropic and heterogeneous boundary controlled N-dimensional wave equation as port-Hamiltonian system

The anisotropic and heterogeneous N-dimensional wave equation, controlled and observed at the boundary, is considered as a port-Hamiltonian system. The recent structure-preserving Partitioned Finite Element Method is applied, leading directly to a finite-dimensional port-Hamiltonian system, and its numerical analysis is done in a general framework, under usual assumptions for finite element. Compatibility conditions are then exhibited to reach the best trade off between the convergence rate and the number of degrees of freedom for both the state error and the Hamiltonian error. Numerical simulations in 2D are performed to illustrate the optimality of the main theorems among several choices of classical finite element families.Comment: 36 pages, 1 figure, submitte

Solution of Differential Equations with Applications to Engineering Problems

Over the last hundred years, many techniques have been developed for the solution of ordinary differential equations and partial differential equations. While quite a major portion of the techniques is only useful for academic purposes, there are some which are important in the solution of real problems arising from science and engineering. In this chapter, only very limited techniques for solving ordinary differential and partial differential equations are discussed, as it is impossible to cover all the available techniques even in a book form. The readers are then suggested to pursue further studies on this issue if necessary. After that, the readers are introduced to two major numerical methods commonly used by the engineers for the solution of real engineering problems

High-Resolution Mathematical and Numerical Analysis of Involution-Constrained PDEs

Partial differential equations constrained by involutions provide the highest fidelity mathematical models for a large number of complex physical systems of fundamental interest in critical scientific and technological disciplines. The applications described by these models include electromagnetics, continuum dynamics of solid media, and general relativity. This workshop brought together pure and applied mathematicians to discuss current research that cuts across these various disciplines’ boundaries. The presented material illuminated fundamental issues as well as evolving theoretical and algorithmic approaches for PDEs with involutions. The scope of the material covered was broad, and the discussions conducted during the workshop were lively and far-reaching

Hybridizable compatible finite element discretizations for numerical weather prediction: implementation and analysis

There is a current explosion of interest in new numerical methods for atmospheric modeling. A driving force behind this is the need to be able to simulate, with high efficiency, large-scale geophysical flows on increasingly more parallel computer systems. Many current operational models, including that of the UK Met Office, depend on orthogonal meshes, such as the latitude-longitude grid. This facilitates the development of finite difference discretizations with favorable numerical properties. However, such methods suffer from the ``pole problem," which prohibits the model to make efficient use of a large number of computing processors due to excessive concentration of grid-points at the poles. Recently developed finite element discretizations, known as ``compatible" finite elements, avoid this issue while maintaining the key numerical properties essential for accurate geophysical simulations. Moreover, these properties can be obtained on arbitrary, non-orthogonal meshes. However, the efficient solution of the resulting discrete systems depend on transforming the mixed velocity-pressure (or velocity-pressure-buoyancy) system into an elliptic problem for the pressure. This is not so straightforward within the compatible finite element framework due to inter-element coupling. This thesis supports the proposition that systems arising from compatible finite element discretizations can be solved efficiently using a technique known as ``hybridization." Hybridization removes inter-element coupling while maintaining the desired numerical properties. This permits the construction of sparse, elliptic problems, for which fast solver algorithms are known, using localized algebra. We first introduce the technique for compatible finite element discretizations of simplified atmospheric models. We then develop a general software abstraction for the rapid implementation and composition of hybridization methods, with an emphasis on preconditioning. Finally, we extend the technique for a new compatible method for the full, compressible atmospheric equations used in operational models.Open Acces

Numerical Studies of Regularized Navier-Stokes Equations and an Application of a Run-to-Run Control model for Membrane Filtration at a Large Urban Water Treatment Facility

This dissertation consists of two parts. The first part consists of research on accurate and efficient turbulent fluid flow modeling via a family of regularizations of the Navier-Stokes equation which are known as Time Relaxation models. In the second part, we look into the modeling application for the filtration/backwash process at the River Mountains Water Treatment Facility in Henderson, NV. In the first two chapters, we introduce the Time Relaxation models and their associated differential filter equations. In addition, we develop the regularization method which employs the Nth van Cittert deconvolution operator, which gives rise to the family of models. We also justify theoretically and computationally the use of an effective averaging length scale δ in the time relaxation model when using the van Cittert operator for higher orders of deconvolution N, by presenting experimental results from our use of this model in the Shear Layer Roll Up benchmarking problem. In addition, we will perform a sensitivity analysis with respect to the time relaxation coefficient χ which appears as a scaling factor for the regularization term in the model, and show how sensitivities with respect to χ are improved when utilizing the effective averaging length scale δ. In the third chapter, we develop the time relaxation model with the newly proposed energy-momentum-angular momentum conservation (EMAC) discretization of the non-linear term. We will present energy, momentum and angular momentum balances for the continuous formulation of the TRM with EMAC as well as the full discretized scheme using TRM with EMAC, and we will show that the fully discrete balances for TRM with EMAC reduce to the fully discrete analogues of the conservation of energy, momentum and angular momentum for the continuous Navier-Stokes equations under the assumption of no viscosity, no regularization, and no body force. In addition, we will present the stability and error estimate of the TRM with EMAC, and we will compare these results with the stability and error estimate for the TRM with the well known skew symmetric formulation for the non-linear term. We show that the error estimate for the EMAC scheme under high Reynolds number is much improved over the skew-symmetric scheme. In particular, we will show that the error for the EMAC scheme is O(eν−1), while under the same conditions, the skew-symmetric scheme error is O(eν−3), which is a significant improvement for high Reynolds number, i.e. low values of kinematic viscosity ν. We will then present numerical experiments on the Taylor Green vortex problem to verify the convergence rates for our error estimates, and experiments on the 3D Ethier-Steinman problem and the 2D Lattice Vortex problem to show that the numerical errors produced by EMAC are much smaller than the skew-symmetric scheme. In the fourth chapter, We begin the work of part two by introducing the general run-to-run control model, which is used in a wide array of applications in addition to water treatment. Then we will introduce the specific run-to-run control model which is formulated specifically for a general filtration/backwash system, and we will modify it to fit the parameters, specifications, and measured data that is available from the River Mountains facility. In particular, we will discuss the implementation of a least squares problem to fit parameters for a filtration cycle ODE model. We will also discuss the backwashing component of the proposed model, and the difficulties in implementing such a model with real time plant data. From there, we formulate a cost of power objective function for the whole filtration/backwashing cycle in terms of the setpoints of filtration/backwashing operation, namely the length of both filtration and backwashing cycles, and the setpoints for the fluid flux during each of these cycles. We show the process of minimizing this objective function with respect to the setpoints of plant operation, including our various implementations of this model using standard function optimization with constraints, genetic algorithms, and MCMC methods. In the final chapter, we will draw some conclusions and summarize the findings of all the work contained in the dissertation