The microchannel flow of a micropolar fluid

Micro-channel flows have been computed to investigate the influence of Navier-Stokes formulation for the slip-flow boundary condition, and a micro-polar fluid model, respectively. The results of the slip boundary condition show that the current methodology is valid for slip-flow regime (i.e., for values of Knudsen number less than approximately 0.1). Drag reduction phenomena apparent in some micro-channels can be explained by slip-flow theory. These results are in agreement with some computations and experiments. An ad hoc micro-polar fluid model is developed to investigate the influence of micro effects, such as micro-gyration, in micro-scale flows. The foundation of the ad hoc micro-polar fluid is based on Eringen\u27s micro simple fluid, and is simplified for incompressible, two-dimensional, iso-thermal, and micro-isotropic case. Our model contains two material constants, μ and κ, one scale parameter, m × Kn, and one boundary condition parameter n. The number of parameters is significantly reduced from general micro-polar fluid model and makes the theory practical. The scale parameter m × Kn introduces the Knudsen number into the micro-polar fluid dynamics by statistical explanation. Therefore, the effect of rarefaction can be accounted into the model by modeling this parameter. The parameter μ is classical bulk viscosity. The vortex Viscosity κ is related to micro-gyration, and needs modeling at current time. It affects the flow field in two aspects, by modifying the apparent viscosity and by introducing the effect of microgyration. In the simplest case of fully-developed channel flow, the overall effect is equivalent to lessen the Reynolds number by (I + k/ 2). The current micro-polar fluid model explains the drag increase phenomenon in some micro-channel flows from both experimental and computational data. This result is exactly opposite to that predicted by slip-flow theory. The existence of micro-effect needs to be taken into account for the micro-scale flow. A projection method is used as a numerical technique for both models to solve the difficulty of implicit pressure equation, with the help of staggered grids. An explicit Euler scheme is used for solving the steady flow

Free surface flow and application to the filling and solidification of liquid metals into vessels of arbitrary shape

The research work presented herein addresses the problem of the mathematical modelling of the mould filling processes as encountered in the foundry industry. The quality of castings, especially aerospace components, is primarily pre-determined at the stage of mould filling within the entire casting process. The entrapment of oxide films, air voids and other impurities into the cast, caused by waves and the breaking of the molten metal surface during filling must be avoided. Otherwise, substandard casting products will result which cost the foundry industry millions of pounds in lost revenue. A three-dimensional control-volume, free surface flow technique known as the Scalar Equation Algorithm (SEA) has been developed as an attachment to the PHOENICS and Harwell-FLOW3D CFD codes for this study. The SEA technique uses a conserved scalar variable to represent the liquid, with an adaptation of the van Leer TVD scheme to define the instantaneous position of the interface. It is similar to the approach used by the well known Volume Of Fluid (VOF) method. However, the SEA technique deals with both air and liquid explicitly, whereas the VOF method does not. A technique has also been developed to allow the liquid temperature to be determined from a conserved `mixture' enthalpy. The liquid temperature is subsequently used in a solidification algorithm to simulate the effect of phase change. The filling model without heat transfer and solidification has been validated against experimental data in both water experiments and actual mould filling experiments. The capability of the SEA method in capturing convoluted waves and air voids has been successfully demonstrated in an example of filling part of a mould running system. It has also been compared against the predictions from the SOLution Algorithm-Volume Of Fluid (SOLA-VOF) and Marker And Cell (MAC) methods. Examples of the developed filling model coupled with heat transfer and solidification are also given. To increase computational speed, the filling model has been implemented into a parallelised version of the Harwell-FLOWSD CFD code. A speed-up of up to 80% has been achieved by using a network of heterogeneous processing nodes. Each node consists of an Intel i860 vector processor and an Inmos T800 transputer

Stabilized finite element formulations for the three-field viscoelastic fluid flow problem

The Finite Element Method (FEM) is a powerful numerical tool, that permits the resolution of problems defined by partial differential equations, very often employed to deal with the numerical simulation of multiphysics problems. In this work, we use it to approximate numerically the viscoelastic fluid flow problem, which involves the resolution of the standard Navier-Stokes equations for velocity and pressure, and another tensorial reactive-convective constitutive equation for the elastic part of the stress, that describes the viscoelastic nature of the fluid. The three-field (velocity-pressure-stress) mixed formulation of the incompressible Navier-Stokes problem, either in the elastic and in the non-elastic case, can lead to two different types of numerical instabilities. The first is associated with the incompressibility and loss of stability of the stress field, and the second with the dominant convection. The first type of instabilities can be overcome by choosing an interpolation for the unknowns that satisfies the two inf-sup conditions that restrict the mixed problem, whereas the dominant convection requires a stabilized formulation in any case. In this work, different stabilized schemes of the Sub-Grid-Scale (SGS) type are proposed to solve the three-field problem, first for quasi Newtonian fluids and then for solving the viscoelastic case. The proposed methods allow one to use equal interpolation for the problem unknowns and to stabilize dominant convective terms both in the momentum and in the constitutive equation. Starting from a residual based formulation used in the quasi-Newtonian case, a non-residual based formulation is proposed in the viscoelastic case which is shown to have superior behavior when there are numerical or geometrical singularities. The stabilized finite element formulations presented in the work yield a global stable solution, however, if the solution presents very high gradients, local oscillations may still remain. In order to alleviate these local instabilities, a general discontinuity-capturing technique for the elastic stress is also proposed. The monolithic resolution of the three-field viscoelastic problem could be extremely expensive computationally, particularly, in the threedimensional case with ten degrees of freedom per node. A fractional step approach motivated in the classical pressure segregation algorithms used in the two-field Navier-Stokes problem is presented in the work.The algorithms designed allow one the resolution of the system of equations that define the problem in a fully decoupled manner, reducing in this way the CPU time and memory requirements with respect to the monolithic case. The numerical simulation of moving interfaces involved in two-fluid flow problems is an important topic in many industrial processes and physical situations. If we solve the problem using a fixed mesh approach, when the interface between both fluids cuts an element, the discontinuity in the material properties leads to discontinuities in the gradients of the unknowns which cannot be captured using a standard finite element interpolation. The method presented in this work features a local enrichment for the pressure unknowns which allows one to capture pressure gradient discontinuities in fluids presenting different density values. The stability and convergence of the non-residual formulation used for viscoelastic fluids is analyzed in the last part of the work, for a linearized stationary case of the Oseen type and for the semi-discrete time dependent non-linear case. In both cases, it is shown that the formulation is stable and optimally convergent under suitable regularity assumptions.El Método de los Elementos Finitos (MEF) es una herramienta numérica de gran alcance, que permite la resolución de problemas definidos por ecuaciones diferenciales parciales, comúnmente utilizado para llevar a cabo simulaciones numéricas de problemas de multifísica. En este trabajo, se utiliza para aproximar numéricamente el problema del flujo de fluidos viscoelásticos, el cual requiere la resolución de las ecuaciones básicas de Navier-Stokes y otra ecuación adicional constitutiva tensorial de tipo reactiva-convectiva, que describe la naturaleza viscoelástica del fluido. La formulación mixta de tres campos (velocidad-presión-tensión) del problema de Navier-Stokes, tanto en el caso elástico como en el no-elástico, puede conducir a dos tipos de inestabilidades numéricas. El primer grupo, se asocia con la incompresibilidad del fluido y la pérdida de estabilidad del campo de tensiones, y el segundo con la convección dominante. El primer tipo de inestabilidades, se puede solucionar eligiendo un tipo de interpolación entre las incógnitas que satisfaga las dos condiciones inf-sup que restringen el problema mixto, mientras que la convección dominante requiere del uso de formulaciones estabilizadas en cualquier caso. En el trabajo, se proponen diferentes esquemas estabilizados del tipo SGS (Sub-Grid-Scales) para resolver el problema de tres campos, primero para fluidos del tipo cuasi-newtonianos y luego para resolver el caso viscoelástico. Los métodos estabilizados propuestos permiten el uso de igual interpolación entre las incógnitas del problema y estabilizan la convección dominante, tanto en la ecuación de momento como en la ecuación constitutiva. Comenzando desde una formulación de tipo residual usada en el caso cuasi-newtoniano, se propone una formulación no-residual para el caso viscoelástico que muestra un comportamiento superior en presencia de singularidades numéricas y geométricas. En general, una formulación estabilizada produce una solución estable global, sin embargo, si la solución presenta gradientes elevados, oscilaciones locales se pueden mantener. Con el objetivo de aliviar este tipo de inestabilidades locales, se propone adicionalmente una técnica general de captura de discontinuidades para la tensión elástica. La resolución monolítica del problema de tres campos viscoelástico puede llegar a ser extremadamente costosa computacionalmente, sobre todo, en el caso tridimensional donde se tienen diez grados de libertad por nodo. Un enfoque de paso fraccionado motivado en los algorítmos clásicos de segregación de la presión usados en el caso del problema de dos campos de Navier-Stokes, se presenta en el trabajo, el cual permite la resolución del sistema de ecuaciones que definen el problema de una manera completamente desacoplada, lo que reduce los tiempos de cálculo y los requerimientos de memoria, respecto al caso monolítico. La simulación numérica de interfaces móviles que envuelve los problemas de dos fluidos, es un tópico importante en un gran número de procesos industriales y situaciones físicas. Si se resuelve el problema utilizando un enfoque de mallas fijas, cuando la interfaz que separa los dos fluidos corta un elemento, la discontinuidad en las propiedades materiales da lugar a discontinuidades en los gradientes de las incógnitas que no pueden ser capturados utilizando una formulación estándar de interpolación. Un enriquecimiento local para la presión se presenta en el trabajo, el cual permite la captura de gradientes discontinuos en la presión, asociados a fluidos de diferentes densidades. La estabilidad y la convergencia de la formulación no-residual utilizada para fluidos viscoelásticos es analizada en la última parte del trabajo, para un caso linealizado estacionario del tipo Oseen y para un problema transitorio no-lineal semi-discreto. En ambos casos, se logra mostrar que la formulación es estable y de convergencia óptima bajo supuestos de regularidad adecuados.Postprint (published version

Local Fourier analysis for saddle-point problems

The numerical solution of saddle-point problems has attracted considerable interest in recent years, due to their indefiniteness and often poor spectral properties that make efficient solution difficult. While much research already exists, developing efficient algorithms remains challenging. Researchers have applied finite-difference, finite element, and finite-volume approaches successfully to discretize saddle-point problems, and block preconditioners and monolithic multigrid methods have been proposed for the resulting systems. However, there is still much to understand. Magnetohydrodynamics (MHD) models the flow of a charged fluid, or plasma, in the presence of electromagnetic fields. Often, the discretization and linearization of MHD leads to a saddle-point system. We present vector-potential formulations of MHD and a theoretical analysis of the existence and uniqueness of solutions of both the continuum two-dimensional resistive MHD model and its discretization. Local Fourier analysis (LFA) is a commonly used tool for the analysis of multigrid and other multilevel algorithms. We first adapt LFA to analyse the properties of multigrid methods for both finite-difference and finite-element discretizations of the Stokes equations, leading to saddle-point systems. Monolithic multigrid methods, based on distributive, Braess-Sarazin, and Uzawa relaxation are discussed. From this LFA, optimal parameters are proposed for these multigrid solvers. Numerical experiments are presented to validate our theoretical results. A modified two-level LFA is proposed for high-order finite-element methods for the Lapalce problem, curing the failure of classical LFA smoothing analysis in this setting and providing a reliable way to estimate actual multigrid performance. Finally, we extend LFA to analyze the balancing domain decomposition by constraints (BDDC) algorithm, using a new choice of basis for the space of Fourier harmonics that greatly simplifies the application of LFA. Improved performance is obtained for some two- and three-level variants

Modeling Particle-Laden Compressible Flows with an Application to Plume-Surface Interactions

During planetary descent, rocket plumes fluidize and eject surface granular matter. Consequently, ejected matter has been shown to obscure the landing site and even collide with the lander, causing serious damage. Given the high risk and cost of space exploration, the challenges associated with plume-surface interactions (PSI) are capable of jeopardizing future missions. The erosion, fluidization, and ejecta of granular matter during PSI occurs under transonic/supersonic, high Reynolds number conditions. These flow conditions pose significant challenges in both experimental and numerical analyses. To date, accurate and predictive physics-based models of PSI at relevant landing conditions do not exist. The objective of this project is to develop high-fidelity simulation capabilities to model compressible gas-particle flows at conditions relevant to PSI. To start, a rigorous derivation of the volume-filtered (locally averaged) compressible Navier--Stokes equations is presented for the first time. This derivation reveals many unclosed terms, for which models are either non-existent or not valid under the regimes of interest. To this end, key terms including pseudo-turbulent kinetic energy and pseudo-turbulent Reynolds stresses, are isolated and modeled via a transport equation in a new high-order finite difference Eulerian-Lagrangian framework. A new immersed boundary method is introduced to generate highly resolved, multi-particle simulations for model closure development. Using the proposed immersed boundary method and the Eulerian--Lagrangian framework, high-fidelity PSI simulations are performed. Single-phase jet impingement on flat surfaces is first shown for validation of the flow conditions. The work is then extended to PSI over a granular bed. For this case, it is shown that that ejected particles can exceed sonic speeds at high particle Reynolds numbers while the majority of the granular bed experiences subsonic particle Mach numbers. In addition, granular temperature is found to be most prevalent in region of high shear during crater formation. The uniqueness of this work lies in the combination of first principles physics and numerics to generate a modeling framework to improve predictions of plume-surface interactions for future missions involving entry, descent, and landing on planetary and satellite bodies.PHDMechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169969/1/grshall_1.pd

Teaching and Learning of Fluid Mechanics

This book contains research on the pedagogical aspects of fluid mechanics and includes case studies, lesson plans, articles on historical aspects of fluid mechanics, and novel and interesting experiments and theoretical calculations that convey complex ideas in creative ways. The current volume showcases the teaching practices of fluid dynamicists from different disciplines, ranging from mathematics, physics, mechanical engineering, and environmental engineering to chemical engineering. The suitability of these articles ranges from early undergraduate to graduate level courses and can be read by faculty and students alike. We hope this collection will encourage cross-disciplinary pedagogical practices and give students a glimpse of the wide range of applications of fluid dynamics

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described