HIGH ACCURACY MULTISCALE MULTIGRID COMPUTATION FOR PARTIAL DIFFERENTIAL EQUATIONS

Scientific computing and computer simulation play an increasingly important role in scientific investigation and engineering designs, supplementing traditional experiments, such as in automotive crash studies, global climate change, ocean modeling, medical imaging, and nuclear weapons. The numerical simulation is much cheaper than experimentation for these application areas and it can be used as the third way of science discovery beyond the experimental and theoretical analysis. However, the increasing demand of high resolution solutions of the Partial Differential Equations (PDEs) with less computational time has increased the importance for researchers and engineers to come up with efficient and scalable computational techniques that can solve very large-scale problems. In this dissertation, we build an efficient and highly accurate computational framework to solve PDEs using high order discretization schemes and multiscale multigrid method. Since there is no existing explicit sixth order compact finite difference schemes on a single scale grids, we used Gupta and Zhang’s fourth order compact (FOC) schemes on different scale grids combined with Richardson extrapolation schemes to compute the sixth order solutions on coarse grid. Then we developed an operator based interpolation scheme to approximate the sixth order solutions for every find grid point. We tested our method for 1D/2D/3D Poisson and convection-diffusion equations. We developed a multiscale multigrid method to efficiently solve the linear systems arising from FOC discretizations. It is similar to the full multigrid method, but it does not start from the coarsest level. The major advantage of the multiscale multigrid method is that it has an optimal computational cost similar to that of a full multigrid method and can bring us the converged fourth order solutions on two grids with different scales. In order to keep grid independent convergence for the multiscale multigrid method, line relaxation and plane relaxation are used for 2D and 3D convection diffusion equations with high Reynolds number, respectively. In addition, the residual scaling technique is also applied for high Reynolds number problems. To further optimize the multiscale computation procedure, we developed two new methods. The first method is developed to solve the FOC solutions on two grids using standardW-cycle structure. The novelty of this strategy is that we use the coarse level grid that will be generated in the standard geometric multigrid to solve the discretized equations and achieve higher order accuracy solution. It is more efficient and costs less CPU and memory compared with the V-cycle based multiscale multigrid method. The second method is called the multiple coarse grid computation. It is first proposed in superconvergent multigrid method to speed up the convergence. The basic idea of multigrid superconvergent method is to use multiple coarse grids to generate better correction for the fine grid solution than that from the single coarse grid. However, as far as we know, it has never been used to increase the order of solution accuracy for the fine grid. In this dissertation, we use the idea of multiple coarse grid computation to approximate the fourth order solutions on every coarse grid and fine grid. Then we apply the Richardson extrapolation for every fine grid point to get the sixth order solutions. For parallel implementation, we studied the parallelization and vectorization potential of the Gauss-Seidel relaxation by partitioning the grid space with four colors for solving 3D convection-diffusion equations. We used OpenMP to parallelize the loops in relaxation and residual computation. The numerical results show that the parallelized and the sequential implementation have the same convergence rate and the accuracy of the computed solutions

Spectral methods for CFD

One of the objectives of these notes is to provide a basic introduction to spectral methods with a particular emphasis on applications to computational fluid dynamics. Another objective is to summarize some of the most important developments in spectral methods in the last two years. The fundamentals of spectral methods for simple problems will be covered in depth, and the essential elements of several fluid dynamical applications will be sketched

Efficient Multigrid Preconditioners for Atmospheric Flow Simulations at High Aspect Ratio

Many problems in fluid modelling require the efficient solution of highly anisotropic elliptic partial differential equations (PDEs) in "flat" domains. For example, in numerical weather- and climate-prediction an elliptic PDE for the pressure correction has to be solved at every time step in a thin spherical shell representing the global atmosphere. This elliptic solve can be one of the computationally most demanding components in semi-implicit semi-Lagrangian time stepping methods which are very popular as they allow for larger model time steps and better overall performance. With increasing model resolution, algorithmically efficient and scalable algorithms are essential to run the code under tight operational time constraints. We discuss the theory and practical application of bespoke geometric multigrid preconditioners for equations of this type. The algorithms deal with the strong anisotropy in the vertical direction by using the tensor-product approach originally analysed by B\"{o}rm and Hiptmair [Numer. Algorithms, 26/3 (2001), pp. 219-234]. We extend the analysis to three dimensions under slightly weakened assumptions, and numerically demonstrate its efficiency for the solution of the elliptic PDE for the global pressure correction in atmospheric forecast models. For this we compare the performance of different multigrid preconditioners on a tensor-product grid with a semi-structured and quasi-uniform horizontal mesh and a one dimensional vertical grid. The code is implemented in the Distributed and Unified Numerics Environment (DUNE), which provides an easy-to-use and scalable environment for algorithms operating on tensor-product grids. Parallel scalability of our solvers on up to 20,480 cores is demonstrated on the HECToR supercomputer.Comment: 22 pages, 6 Figures, 2 Table

Turbulent Cells in Stars: I. Fluctuations in Kinetic Energy and Luminosity

Three-dimensional (3D) hydrodynamic simulations of shell oxygen burning (Meakin and Arnett, 2007b) exhibit bursty, recurrent fluctuations in turbulent kinetic energy. These are shown to be due to a general instability of the convective cell, requiring only a localized source of heating or cooling. Such fluctuations are shown to be suppressed in simulations of stellar evolution which use mixing-length theory (MLT). Quantitatively similar behavior occurs in the model of a convective roll (cell) of Lorenz (1963), which is known to have a strange attractor that gives rise to chaotic fluctuations in time of velocity and, as we show, luminosity. Study of simulations suggests that the behavior of a Lorenz convective roll may resemble that of a cell in convective flow. We examine some implications of this simplest approximation, and suggest paths for improvement. Using the Lorenz model as representative of a convective cell, a multiple-cell model of a convective layer gives total luminosity fluctuations which are suggestive of irregular variables (red giants and supergiants (Schwarzschild 1975)), and of the long secondary period feature in semi-regular AGB variables (Stothers 2010, Wood, Olivier and Kawaler 2004). This "tau-mechanism" is a new source for stellar variability, which is inherently non-linear (unseen in linear stability analysis), and one closely related to intermittency in turbulence. It was already implicit in the 3D global simulations of Woodward, Porter and Jacobs (2003). This fluctuating behavior is seen in extended 2D simulations of CNeOSi burning shells (Arnett and Meakin 2011b), and may cause instability which leads to eruptions in progenitors of core collapse supernovae PRIOR to collapse.Comment: 30 pages, 13 figure

Coolant side heat transfer with rotation. Task 3 report: Application of computational fluid dynamics

An experimental and analytical program was conducted to investigate heat transfer and pressure losses in rotating multipass passages with configurations and dimensions typical of modern turbine blades. The objective of this program is the development and verification of improved analysis methods that will form the basis for a design system that will produce turbine components with improved durability. As part of this overall program, a technique is developed for computational fluid dynamics. The specific objectives were to: select a baseline CFD computer code, assess the limitations of the baseline code, modify the baseline code for rotational effects, verify the modified code against benchmark experiments in the literature, and to identify shortcomings in the code as revealed by the verification. The Pratt and Whitney 3D-TEACH CFD code was selected as the vehicle for this program. The code was modified to account for rotating internal flows, and these modifications were evaluated for flow characteristics of those expected in the application. Results can make a useful contribution to blade internal cooling

Numerical resolution of turbulent flows on complex geometries.

This thesis aims at developing a numerical methodology suitable for the direct numerical simulation (DNS) and large-eddy simulation (LES) of turbulent flows in order to be used in complex flows, currently encountered in industrial application. At the same time, the study of such turbulent flows can be an opportunity for gaining insight into the complex physics associated with them. To accomplish these goals, the mathematical formulation, conservative spatial discretization on unstructured grids and time- integration scheme for solving the Navier-Stokes equations are presented. The spatial discretization proposed preserves the symmetry properties of the continuous differential operator and ensure both, stability and conservation of the global kinetic energy balance on any grid. Furthermore, the time-integration technique proposed is an efficient self-adaptive strategy, based on a one-parameter second-order-explicit scheme, which has been successfully tested on both Cartesian staggered and unstructured collocated codes, leading to CPU cost reductions of up to 2.9 and 4.3, respectively. After presenting the general methodology for computing flows in complex geometries with unstructured grids, different LES models and regularization models suitable for these kind of meshes are presented and assessed by means of the analysis of different flows. First, regularization models are tested by means of the simulation of different cases with different level of complexity of the mesh. From a structured grid to a very complex mesh, with zones composed of prism and tetrahedral control volumes. It has been shown, that regularization models are very dependent on the quality of the filtering process. Although good results can be obtained with structured or smooth unstructuredmeshes, their performance is affected under fully irregular unstructured grids. A possible remedy to circumvent this issue is also presented. The main idea is to formulate the C4 model within a LES template. Although preliminary results are promising, further testing is still required. After regularization model assessment, LES models are also tested in a natural convection flow. It is shown that, although first order statistics are well solved for most of the models tested (with the exception of the Smagorinsky model), QR- and dynamic-Smagorinsky models present a better prediction of the second-order statistics. However, if CPU time is considered, then QR model is the best alternative. The second part of the thesis is devoted to the study of turbulent flows past bluff bodies. The cases studied are: the flow past a sphere, the flow past a circular cylinder and the flow past a NACA 0012 airfoil. All these cases shares some characteristics encountered in turbulent flows with massive separations, i.e., flow separation, transition to turbulence in the separated shear-layers and turbulent wakes with periodic shedding of vortices. However there are intrinsic characteristics of the turbulence in each of them, which make them interesting for the studying of the turbulence. Furthermore, the results presented for the flow past a sphere at Re = 3700 and 10000, together with the flow past a NACA 0012 at Re=50000 and AoA = 8 are the first DNS results presented in the literature for both flows. Conclusions drawn from the good results obtained point out that the use of the conservative formulation presented in this thesis, is one of the keys for the success of the SGS models used. This formulation, together with the use of unstructured grids might be a step towards the use of LES models for solving industrial flows on complex geometries at high Reynolds numbers.La present tesi proposa una metodologia apte per a realitzar simulacions directes de la turbulència (DNS) i simulacions de les grans escales (LES) de fluxos turbulents en geometries complexes. Tanmateix també s'estudia detalladament els mecanismes bàsics de funcionament dels fluxos turbulents en diferents situacions d'interès industrial i acadèmic. Per acomplir aquest objectiu s'ha desenvolupat una innovadora formulació matemàtica que permet conservar discretament les propietats continues de les equacions governants en malles no estructurades. La formulació proposada preserva la simetries originals dels operadors diferencials, assegurant així l'estabilitat i la conservació de l'energia cinètica turbulent en qualsevol mallat. Posteriorment s'ha proposat una metodologia d'integració temporal basada en una formulació explicita de segon ordre. Aquesta nova tècnica ha demostrat ser entre 2.9 i 4.3 més rapida que les tècniques anteriorment utilitzades per la comunitat. Un cop presentada la formulació per a simular fluxos turbulents en geometries complexes, s'han validat diferents models LES adaptats a malles no estructurades. Els models s'han testejat usant diferents solucions de referencia de la literatura i simulacions d'alt nivell generades en el context de la present tesi. Finalment s'ha conclòs que la conjunció de la formulació bàsica proposada amb alguns del models LES sorgits en els darrers anys es molt efectiva per a simular fluxos turbulents en situacions complexes, essent el Variational Multiscale WALE i el model QR els més adequats per a simular situacions de interes industrial. La segona part de la tesi es dedicada a l'estudi aerodinàmic del flux turbulent al voltant de diferents perfils. El perfils seleccionats son: el flux al voltant d'una esfera, flux al voltant d'un cilindre i flux al voltant d'un perfil NACA 0012. Els tres casos comparteixen fenomenologies com ara separació massiva de capes límits, esteles turbulentes i desprendiment periòdic de remolins. Tot i així cadascun d'ells es comporta diferent a nivell turbulent així que es d'interès estudiar-los i entendre quins son les causes de les diferencies físiques que es troben. Cal recordar que la física estudiada es la que es pot trobar posteriorment en ales d'avió, perfils de turbines de vent, aerodinàmica de cotxes, etc. Finalment recalcar que els resultats DNS del flux al voltant de l'esfera a Re=3700 i Re=10000 conjuntament amb els DNS del flux al voltant del perfil NACA a Re=50000 i AoA =8 son els primers presentats en la literatura internacional en el seu àmbit. Finalment es pot concloure que la formulació conservativa presentada en la tesis juntament amb els diferents models LES d'última generació testejats en la tesis, han demostrat ser una eina eficaç tan per a resoldre fluxos turbulents d'interès acadèmic com per simular situacions d'interès industrial.Postprint (published version

Numerical Analysis of Transient Teflon Ablation with a Domain Decomposition Finite Volume Implicit Method on Unstructured Grids

This work investigates numerically the process of Teflon ablation using a finite-volume discretization, implicit time integration and a domain decomposition method in three-dimensions. The interest in Teflon stems from its use in Pulsed Plasma Thrusters and in thermal protection systems for reentry vehicles. The ablation of Teflon is a complex process that involves phase transition, a receding external boundary where the heat flux is applied, an interface between a crystalline and amorphous (gel) phase and a depolymerization reaction which happens on and beneath the ablating surface. The mathematical model used in this work is based on a two-phase model that accounts for the amorphous and crystalline phases as well as the depolymerization of Teflon in the form of an Arrhenius reaction equation. The model accounts also for temperature-dependent material properties, for unsteady heat inputs and boundary conditions in 3D. The model is implemented in 3D domains of arbitrary geometry with a finite volume discretization on unstructured grids. The numerical solution of the transient reaction-diffusion equation coupled with the Arrhenius-based ablation model advances in time using implicit Crank-Nicolson scheme. For each time step the implicit time advancing is decomposed into multiple sub-problems by a domain decomposition method. Each of the sub-problems is solved in parallel by Newton-Krylov non-linear solver. After each implicit time-advancing step, the rate of ablation and the fraction of depolymerized material are updated explicitly with the Arrhenius-based ablation model. After the computation, the surface of ablation front and the melting surface are recovered from the scalar field of fraction of depolymerized material and the fraction of melted material by post-processing. The code is verified against analytical solutions for the heat diffusion problem and the Stefan problem. The code is validated against experimental data of Teflon ablation. The verification and validation demonstrates the ability of the numerical method in simulating three dimensional ablation of Teflon

HPTAM, a two-dimensional Heat Pipe Transient Analysis Model, including the startup from a frozen state

A two-dimensional Heat Pipe Transient Analysis Model, 'HPTAM,' was developed to simulate the transient operation of fully-thawed heat pipes and the startup of heat pipes from a frozen state. The model incorporates: (a) sublimation and resolidification of working fluid; (b) melting and freezing of the working fluid in the porous wick; (c) evaporation of thawed working fluid and condensation as a thin liquid film on a frozen substrate; (d) free-molecule, transition, and continuum vapor flow regimes, using the Dusty Gas Model; (e) liquid flow and heat transfer in the porous wick; and (f) thermal and hydrodynamic couplings of phases at their respective interfaces. HPTAM predicts the radius of curvature of the liquid meniscus at the liquid-vapor interface and the radial location of the working fluid level (liquid or solid) in the wick. It also includes the transverse momentum jump condition (capillary relationship of Pascal) at the liquid-vapor interface and geometrically relates the radius of curvature of the liquid meniscus to the volume fraction of vapor in the wick. The present model predicts the capillary limit and partial liquid recess (dryout) in the evaporator wick, and incorporates a liquid pooling submodel, which simulates accumulation of the excess liquid in the vapor core at the condenser end