Development of a high-order parallel solver for direct and large eddy simulations of turbulent flows

Turbulence is inherent in fluid dynamics, in that laminar flows are rather the exception than the rule, hence the longstanding interest in the subject, both within the academic community and the industrial R&D laboratories. Since 1883, much progress has been made, and statistics applied to turbulence have provided understanding of the scaling laws which are peculiar to several model flows, whereas experiments have given insight on the structure of real-world flows, but, soon enough, numerical approaches to the matter have become the most promising ones, since they lay the ground for the solution of high Reynolds number unsteady Navier-Stokes equations by means of computer systems. Nevertheless, despite the exponential rise in computational capability over the last few decades, the more computer technology advances, the higher the Reynolds number sought for test-cases of industrial interest: there is a natural tendency to perform simulations as large as possible, a habit that leaves no room for wasting resources. Indeed, as the scale separation grows with Re, the reduction of wall clock times for a high-fidelity solution of desired accuracy becomes increasingly important. To achieve this task, a CFD solver should rely on the use of appropriate physical models, consistent numerical methods to discretize the equations, accurate non-dissipative numerical schemes, efficient algorithms to solve the numerics, and fast routines implementing those algorithms. Two archetypal approaches to CFD are direct and large-eddy simulation (DNS and LES respectively), which profoundly differ in several aspects but are both “eddy-resolving” methods, meant to resolve the structures of the flow-field with the highest possible accuracy and putting in as little spurious dissipation as possible. These two requirements of accurate resolution of scales, and energy conservation, should be addressed by any numerical method, since they are essential to many real-world fluid flows of industrial interest. As a consequence, high order numerical schemes, and compact schemes among them, have received much consideration, since they address both goals, at the cost of a lower ease of application of the boundary condition, and a higher computational cost. The latter problem is tackled with parallel computing, which also allows to take advantage of the currently available computer power at the best possible extent. The research activity conducted by the present author has concerned the development, from scratch, of a three-dimensional, unsteady, incompressible Navier-Stokes parallel solver, which uses an advanced algorithm for the process-wise solution of the linear systems arising from the application of high order compact finite difference schemes, and hinges upon a three-dimensional decomposition of the cartesian computational space. The code is written in modern Fortran 2003 — plus a few features which are unique to the 2008 standard — and is parallelized through the use of MPI 3.1 standard’s advanced routines, as implemented by the OpenMPI library project. The coding was carried out with the objective of creating an original CFD high-order parallel solver which is maintainable and extendable, of course within a well-defined range of possibilities. With this main priority being outlined, particular attention was paid to several key concepts: modularity and readability of the source code and, in turn, its reusability; ease of implementation of virtually any new explicit or implicit finite difference scheme; modern programming style and avoidance of deprecated old legacy Fortran constructs and features, so that the world wide web is a reliable and active means to the quick solution of coding problems arising from the implementation of new modules in the code; last but not least, thorough comments, especially in critical sections of the code, explaining motives and possible expected weak links. Design, production, and documentation of a program from scratch is almost never complete. This is certainly true for the present effort. The method and the code are verified against the full three-dimensional Lid-Driven Cavity and Taylor-Green Vortex flows. The latter test is used also for the assessment of scalability and parallel efficiency

Improvement in Computational Fluid Dynamics Through Boundary Verification and Preconditioning

This thesis provides improvements to computational fluid dynamics accuracy and ef- ficiency through two main methods: a new boundary condition verification procedure and preconditioning techniques. First, a new verification approach that addresses boundary conditions was developed. In order to apply the verification approach to a large range of arbitrary boundary condi- tions, it was necessary to develop unifying mathematical formulation. A framework was developed that allows for the application of Dirichlet, Neumann, and extrapolation bound- ary condition, or in some cases the equations of motion directly. Verification of boundary condition techniques was performed using exact solutions from canonical fluid dynamic test cases. Second, to reduce computation time and improve accuracy, preconditioning algorithms were applied via artificial dissipation schemes. A new convective upwind and split pressure (CUSP) scheme was devised and was shown to be more effective than traditional precon- ditioning schemes in certain scenarios. The new scheme was compared with traditional schemes for unsteady flows for which both convective and acoustic effects dominated. Both boundary conditions and preconditioning algorithms were implemented in the context of a strand grid solver. While not the focus of this thesis, strand grids provide automatic viscous quality meshing and are suitable for moving mesh overset problems

Towards a robust Terra

In this work mantle convection simulation with Terra is investigated from a numerical point of view, theoretical analysis as well as practical tests are performed. The stability criteria for the numerical formulation of the physical model will be made clear. For the incompressible case and the Terra specific treatment of the anelastic approximation, two inf-sup stable grid modifications are presented, which are both compatible with hanging nodes. For the Q1hQ12h element pair a simple numeric test is introduced to prove the stability for any given grid. For the Q1h Pdisc 12h element pair and 1-regular refinements with hangig nodes an existing general proof can be adopted. The influence of the slip boundary condition is found to be destabilizing. For the incompressible case a cure can be adopted from the literature. The necessary conditions for the expansion of the stability results to the anelastic approximation will be pointed out. A numerical framework is developed in order to measure the effect of different numerical approaches to improve the handling of strongly varying viscosity. The framework is applied to investigate how block smoothers with different block sizes, combination of different block smoothers, different prolongation schemes and semi coarsening influence the multigrid performance. A regression-test framework for Terra will be briefly introduced

Implicit schemes and parallel computing in unstructured grid CFD

The development of implicit schemes for obtaining steady state solutions to the Euler and Navier-Stokes equations on unstructured grids is outlined. Applications are presented that compare the convergence characteristics of various implicit methods. Next, the development of explicit and implicit schemes to compute unsteady flows on unstructured grids is discussed. Next, the issues involved in parallelizing finite volume schemes on unstructured meshes in an MIMD (multiple instruction/multiple data stream) fashion are outlined. Techniques for partitioning unstructured grids among processors and for extracting parallelism in explicit and implicit solvers are discussed. Finally, some dynamic load balancing ideas, which are useful in adaptive transient computations, are presented

Development of A Direct-Forcing Immersed-Boundary Method on Unstructured Meshes for Multi-Body Interactions in Air-Water Two-Phase Flows

A direct-forcing immersed boundary method (IBM) is developed in the framework of a finite-volume incompressible solver for high-Reynolds-number flows. The method solves governing equations on a background mesh whose grid lines do not conform to the concerned surface geometry, whereby the difficulty of generating high-quality body-fitted meshes is significantly reduced. The boundary conditions on the surface of the geometry are enforced through interpolation. A unique aspect of the proposed IBM is that the method is compatible with unstructured meshes, and as such can be combined with body-fitted meshes, so that some geometries can be represented by body-fitted meshes, and other geometries are represented by the IBM. The method provides an accurate solution for the cases of moving objects in both single-phase and air-water two-phase flows. The method can also be applied to both steady and unsteady, laminar and turbulent flows. In the current work, the method is implemented for solving the Reynolds-Averaged Navier-Stokes equations, and for turbulent flows, the Spalart-Allmaras turbulence model is used. A noticeable challenge of using IBMs is the difficulty in resolving boundary layers at high Reynolds numbers. In this thesis a universal wall function is implemented, which provides a smooth velocity profile from the outer edge of the logarithmic region down to the wall. The wall function improves accuracy when the mesh is not sufficiently fine to resolve the viscous sublayer. As a result, the stringent requirement of near-wall cell spacing for high-Reynolds-number flows is significantly alleviated. The Volume-of-Fluid (VoF) method is used for air-water two-phase flows. A field extension method is used to enforce the boundary condition of the volume fraction on the immersed surface. Detailed verification and validation studies are performed to demonstrate that the current method is second-order accurate. A careful comparison is presented between the results of the IBM, the experimental data, and other numerical results. The comparison fully demonstrates the accuracy and feasibility of the method by examining the flow field and the force on the immersed surface. The validation case of a ship advancing with a rotating rudder is also performed. The results demonstrate the accuracy, flexibility and efficiency when the IBM is used combined with unstructured body-fitted meshes.PHDNaval Architecture & Marine EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/162867/1/hxye_1.pd

A weakly compressible hybridizable discontinuous Galerkin formulation for fluid-structure interaction problems

A scheme for the solution of fluid-structure interaction (FSI) problems with weakly compressible flows is proposed in this work. A novel hybridizable discontinuous Galerkin (HDG) method is derived for the discretization of the fluid equations, while the standard continuous Galerkin (CG) approach is adopted for the structural problem. The chosen HDG solver combines robustness of discontinuous Galerkin (DG) approaches in advection-dominated flows with higher order accuracy and efficient implementations. Two coupling strategies are examined in this contribution, namely a partitioned Dirichlet-Neumann scheme in the context of hybrid HDG-CG discretizations and a monolithic approach based on Nitsche's method, exploiting the definition of the numerical flux and the trace of the solution to impose the coupling conditions. Numerical experiments show optimal convergence of the HDG and CG primal and mixed variables and superconvergence of the postprocessed fluid velocity. The robustness and the efficiency of the proposed weakly compressible formulation, in comparison to a fully incompressible one, are also highlighted on a selection of two and three dimensional FSI benchmark problems.Comment: 49 pages, 20 figures, 2 table

Variational Multiscale Modeling and Memory Effects in Turbulent Flow Simulations

Effective computational models of multiscale problems have to account for the impact of unresolved physics on the resolved scales. This dissertation advances our fundamental understanding of multiscale models and develops a mathematically rigorous closure modeling framework by combining the Mori-Zwanzig (MZ) formalism of Statistical Mechanics with the variational multiscale (VMS) method. This approach leverages scale-separation projectors as well as phase-space projectors to provide a systematic modeling approach that is applicable to complex non-linear partial differential equations. %The MZ-VMS framework is investigated in the context of turbulent flows. Spectral as well as continuous and discontinuous finite element methods are considered. The MZ-VMS framework leads to a closure term that is non-local in time and appears as a convolution or memory integral. The resulting non-Markovian system is used as a starting point for model development. Several new insights are uncovered: It is shown that unresolved scales lead to memory effects that are driven by an orthogonal projection of the coarse-scale residual and, in the case of finite elements, inter-element jumps. Connections between MZ-based methods, artificial viscosity, and VMS models are explored. The MZ-VMS framework is investigated in the context of turbulent flows. Large eddy simulations of Burgers' equation, turbulent flows, and magnetohydrodynamic turbulence using spectral and discontinuous Galerkin methods are explored. In the spectral method case, we show that MZ-VMS models lead to substantial improvements in the prediction of coarse-grained quantities of interest. Applications to discontinuous Galerkin methods show that modern flux schemes can inherently capture memory effects, and that it is possible to guarantee non-linear stability and conservation via the MZ-VMS approach. We conclude by demonstrating how ideas from MZ-VMS can be adapted for shock-capturing and filtering methods.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/145847/1/parish_1.pd