This open access book brings the reader a pedestrian pace to grasp the main bulk of fluid flow problems and contains exercises and solutions

AbstractThe Prandtl’s equations for laminar boundary layer are obtained via dimensional analysis. The case of the flat plate is treated as a suitable example for the development of the boundary layer on a simple geometry. Various thicknesses are introduced. The integration of Prandtl’s equation across the boundary layer produces the von Kármán integral equation which allows the elaboration of the approximate von Kármán-Pohlhausen method where the velocity profile is given as a polynomial. The use of a third degree polynomial for the flat plate demonstrates the feasibility of the approach.

AbstractThis chapter treats the theory of irrotational flows of perfect fluids by the use of complex variables. The theory is based on a complex velocity and the related concepts like circulation, flow rate, complex potential. Several simple examples are given. More elaborated is the flow around a circular cylinder with and without circulation . Using conformal mapping and especially the Joukowski transformation, it is possible to consider an aerodynamics application, namely the flow around an airfoil. Blasius theorem allows for the computation of the forces and moment generated by the flow around an immersed body. It is applied to the case of the cylinder and Joukowski profile.

We consider the Rosenbrock methods, namely a family of methods for Differential Algebraic Equations, for the solution of the unsteady three-dimensional Navier-Stokes equations. These multistage schemes are attractive for non-linear problems because they achieve high order in time, ensuring stability properties and linearizing the system to be solved at each timestep. Moreover, as they provide inexpensive ways to estimate the local truncation error, adaptive timestep strategies can be easily devised. In this work we test the Rosenbrock methods for the solution of three-dimensional unsteady incompressible flows. We derive the correct essential boundary conditions to impose at each stage in order to retain the convergence order of the schemes. Then, we consider two benchmark tests: a flow problem with imposed oscillatory pressure gradient whose analytical solution is known and the classical flow past a cylinder. In the latter case, we especially focus on the accuracy in the approximation of the drag and lift coefficients. In both benchmarks we test the performance of a time adaptivity scheme. (C) 2018 Elsevier Ltd. All rights reserved.

The book presents a snapshot of the state-of-art in the field of turbulence modeling and covers the latest developments concerning direct numerical simulations, large eddy simulations, compressible turbulence, coherent structures, two-phase flow simulation and other related topics. It provides readers with a comprehensive review of both theory and applications, describing in detail the authors own experimental results. The book is based on the proceedings of the third Turbulence and Interactions Conference (TI 2012), which was held on June 11-14 in La Saline-les-Bains, La Runion, France and includes both keynote lectures and outstanding contributed papers presented at the conference. This multifaceted collection, which reflects the conferences emphasis on the interplay of theory, experiments and computing in the process of understanding and predicting the physics of complex flows and solving related engineering problems, offers a practice-oriented guide for students, researchers and professionals in the field of computational fluid dynamics, turbulence modeling and related areas.

Cet ouvrage présente les concepts de base pour la modélisation des milieux continus classiques, fluides et solides (tenseurs cartésiens, cinématique, dynamique, thermodynamique, lois de comportement, lois des fluides classiques newtoniens, mécanique des solides et mécanique des fluides newtoniens). Des exemples d’applications simples permettent d’obtenir des solutions analytiques qui illustrent la puissance de la modélisation ainsi proposée. Chaque chapitre propose une série d’exercices et des annexes donnent les compléments nécessaires pour suivre l’ouvrage et résoudre les problèmes. Clair et pédagogique, cet ouvrage s’adresse plus particulièrement aux étudiants de deuxième et troisième cycle en mécanique, mathématique, physique, ainsi qu’aux ingénieurs, physiciens et scientifiques qui souhaitent acquérir les principes de base de la mécanique des milieux continus.

Leonardo wrote, Mechanics is the paradise of the mathematical sciences, because by means of it one comes to the fruits of mathematics; replace Mechanics by Fluid mechanics and here we are.- From the Preface to the Second EditionAlthough the exponential growth of computer power has advanced the importance of simulations and visualization tools for elaborating new models, designs and technologies, the discipline of fluid mechanics is still large, and turbulence in flows remains a challenging problem in classical physics. Like its predecessor, the revised and expanded Second Edition of this book addresses the basic principles of fluid mechanics and solves fluid flow problems where viscous effects are the dominant physical phenomena.Much progress has occurred in the half a century that has passed since the edition of 1964. As predicted, aspects of hydrodynamics once considered offbeat have risen to importance. For example, the authors have worked on problems where variations in viscosity and surface tension cannot be ignored. The advent of nanotechnology has broadened interest in the hydrodynamics of thin films, and hydromagnetic effects and radiative heat transfer are routinely encountered in materials processing. This monograph develops the basic equations, in the three most important coordinate systems, in a way that makes it easy to incorporate these phenomena into the theory.The book originally described by Prof. Langlois as \"a monograph on theoretical hydrodynamics, written in the language of applied mathematics\" offers much new coverage including the second principle of thermodynamics, the Boussinesq approximation, time dependent flows, Marangoni convection, Kovasznay flow, plane periodic solutions, Hele-Shaw cells, Stokeslets, rotlets, finite element methods, Wannier flow, corner eddies, and analysis of the Stokes operator.

In the framework of the spectral element method, a comparison is carried out on turbulent firstand second-order statistics generated by large eddy simulation (LES), under-resolved (UDNS) and fully resolved direct numerical simulation (DNS). The LES is based on classical models like the dynamic Smagorinsky approach or the approximate deconvolution method. Two test problems are solved: the lid-driven cubical cavity and the differentially heated cavity. With the DNS data as benchmark solutions, it is shown that the numerical results produced by the UDNS calculation are of the same accuracy, even in some cases of better quality, as the LES computations. The conclusion advocates the use of UDNS and calls for improvement of the available algorithms.

Summary This study presents the vortex structure and numerical instability increase occurring when the level of elasticity is enhanced in inertial flows in planar contraction configuration for finitely extensible nonlinear elastic model by Peterlin (FENE‐P) fluid . The re‐entrant corner effect on corner vortices is also considered. The calculations are performed using extended matrix logarithm formulation described in a previous paper: A. Jafari et al . A new extended matrix logarithm formulation for the simulation of viscoelastic fluids by spectral elements. Computer & Fluids 2010; 39 (9):1425–1438. In that reference, the proposed algorithm has been tested for simple geometry such as Poiseuille flow. In this study, we are interested in the capability of this algorithm for more complex geometry. This formulation helps to reach higher values of the Weissenberg number when compared with the classical one. Copyright © 2015 John Wiley & Sons, Ltd.

In the framework of the spectral element method, a comparison is carried out on turbulent first-and second-order statistics generated by large eddy simulation (LES), under-resolved (UDNS) and fully resolved direct numerical simulation (DNS). The LES is based on classical models like the dynamic Smagorinsky approach or the approximate deconvolution method. Two test problems are solved: the lid-driven cubical cavity and the differentially heated cavity. With the DNS data as benchmark solutions, it is shown that the numerical results produced by the UDNS calculation are of the same accuracy, even in some cases of better quality, as the LES computations. The conclusion advocates the use of UDNS and calls for improvement of the available algorithms.

Mathematical Modeling for Complex Fluids and Flows provides researchers and engineering practitioners encountering fluid flows with state-of-the-art knowledge in continuum concepts and associated fluid dynamics. In doing so it supplies the means to design mathematical models of these flows that adequately express the engineering physics involved. It exploits the implicit link between the turbulent flow of classical Newtonian fluids and the laminar and turbulent flow of non-Newtonian fluids such as those required in food processing and polymeric flows. The book develops a descriptive mathematical model articulated through continuum mechanics concepts for these non-Newtonian, viscoelastic fluids and turbulent flows. Each complex fluid and flow is examined in this continuum context as well as in combination with the turbulent flow of viscoelastic fluids. Some details are also explored via kinetic theory, especially viscoelastic fluids and their treatment with the Boltzmann equation. Both solution and modeling strategies for turbulent flows are laid out using continuum concepts, including a description of constructing polynomial representations and accounting for non-inertial and curvature effects. Ranging from fundamental concepts to practical methodology, and including discussion of emerging technologies, this book is ideal for those requiring a single-source assessment of current practice in this intricate yet vital field.

In nuclear safety, some severe accident scenarios lead to the presence of fission products in aerosol form in the closed containment atmosphere. It is important to understand the particle depletion process to estimate the risk of a release of radioactivity to the environment should a containment break occur. As a model for the containment, we use the three-dimensional differentially heated cavity problem. The differentially heated cavity is a cubical box with a hot wall and a cold wall on vertical opposite sides. On the other walls of the cube we have adiabatic boundary conditions. For the velocity field the no-slip boundary condition is applied. The flow of the air in the cavity is described by the Boussinesq equations. The method used to simulate the turbulent flow is the large eddy simulation (LES) where the dynamics of the large eddies is resolved by the computational grid and the small eddies are modelled by the introduction of subgrid scale quantities using a filter function. Particle trajectories are computed using the Lagrangian particle tracking method, including the relevant forces (drag, gravity, thermophoresis). Four different sets with each set containing one million particles and diameters of 10 mu m, 15 mu m, 25 mu m and 35 mu m are simulated. Simulation results for the flow field and particle sizes from 15 mu m to 35 mu m are compared to previous results from direct numerical simulation (DNS). The integration time of the LES is three times longer and the smallest particles have been simulated only in the LES. Particle statistics in the LES and the DNS were similar and the settling rates were practically identical. It was found that for this type of flow no model was necessary for the influence of the unresolved flow scales on the particle motions. This can be explained by the dominant nature of gravity settling compared to turbophoresis which is negligible for the particle sizes of the present study. (C) 2013 Elsevier B.V. All rights reserved.

In this paper, a time dependent one-dimensional linear advection–diffusion equation with Dirichlet homogeneous boundary conditions and an initial sine function is solved analytically by separation of variables and numerically by the finite element method. It is observed that when the advection becomes dominant, the analytical solution becomes ill-behaved and harder to evaluate. Therefore another approach is designed where the solution is decomposed in a simple wave solution and a viscous perturbation. It is shown that an exponential layer builds up close to the downstream boundary. Discussion and comparison of both solutions are carried out extensively offering the numericist a new test model for the numerical integration of the Navier–Stokes equation.