FEniCS Course. Lecture 6: Incompressible Navier Stokes. Contributors Anders Logg André Massing
|
|
- Lucas Johnston
- 5 years ago
- Views:
Transcription
1 FEniCS Course Lecture 6: Incompressible Navier Stokes Contributors Anders Logg André Massing 1 / 11
2 The incompressible Navier Stokes equations u + u u ν u + p = f in Ω (0, T ] u = 0 in Ω (0, T ] u = g D on Γ D (0, T ] ν u n pn = g N on Γ N (0, T ] u(, 0) = u 0 in Ω u is the fluid velocity and p is the pressure divided by the density ρ ν = µ/ρ is the kinematic viscosity, µ dynamic viscosity f is a given body force per unit mass g D is a given boundary flow g N is a given boundary function for the natural boundary condition u 0 is a given initial velocity 2 / 11
3 Variational problem Multiply the momentum equation by a test function v and integrate by parts: ( u + u u) v dx + ν u : u dx p v dx Ω Ω Ω = f v dx + g N v ds Ω Γ N Short-hand notation: ( u + u u, v) + ν( u, v) (p, v) = (f, v) + (g N, v) ΓN Multiply the continuity equation by a test function q and sum up: find (u, p) V such that ( u + u u, v) + ν( u, v) (p, v) (q, u) = (f, v) + (g N, v) ΓN for all (v, q) ˆV 3 / 11
4 Discrete variational problem Time-discretization leads to a saddle-point problem on each time step: [ ] [ ] [ ] M + ta + tn(u) tb U b tb = 0 P 0 Efficient solution of the saddle-point problem relies on the efficiency of special-purpose preconditioners (Uzawa iteration, Schur complement preconditioners,... ) We will use another approach (simpler and often more efficient) 4 / 11
5 The classical Chorin-Teman projection method Step 1: Compute tentative velocity u solving u u n ν u + (u )u = f n+1 t u = g D u n = 0 in Ω on Ω D on Ω N Step 2: Compute a corrected velocity u n+1 and a updated pressure p n+1 solving u n+1 u + p n+1 = 0 t in Ω u n+1 = 0 in Ω u n+1 n = 0 on Ω 5 / 11
6 Computing the tentative velocity In principle, the term (u )u can be approximated in several ways Explicit: u = u = u n diffusion-reaction equation Semi-implicit u = u n and u = u n+1 convection-diffusion-reaction equation Fully-implicit u = u = u n retaining the basic non-linearity in the Navier-Stokes equations The natural outflow condition ν n u pn = 0 is artificially enforced by requiring n u = 0 on Ω N in step 1 p n+1 = 0 on Ω N in step 2 6 / 11
7 Solving the projection step Applying to un+1 u + p n+1 = 0 and using requirement t u n+1 = 0 yields p n+1 = 1 t u in Ω We already required p = 0 on Ω N Multiplying un+1 u + p n+1 = 0 with n and restricting to t Ω D gives Compute u n+1 by p n+1 n = 0 on Ω D u n+1 = u t p n+1 including boundary conditions for u at t = t n+1 7 / 11
8 Chorin-Teman projection method Summary 1 Compute tentative velocity u by ( u u n, v) + ((u )u, v) + ν( u, v) (f, v) = 0 t including boundary conditions for the velocity. 2 Compute new pressure p n+1 by ( p n+1, q) + 1 t ( u, q) = 0 including boundary conditions for the pressure. 3 Compute corrected velocity by (u n+1 u, v) + t( p n+1, v) = 0 including boundary conditions for the velocity. 8 / 11
9 Useful FEniCS tools (I) Note vs. : dot ( grad (u), u) dot (u, nabla_grad (u)) Solving linear systems: solve (A, x, b) solve (A, x, b, " gmres ", " ilu ") solve (A, x, b, "cg", " amg ") Defining a and L based on residual formulation: F1 = ( (1/k)* inner (u - u0,v) + inner ( grad ( u0)*u0,v) + nu* inner ( grad (u), grad (v)) - inner (f,v) )* dx a1 = lhs ( F1) L1 = rhs ( F1) 9 / 11
10 The FEniCS challenge! Implement a famous benchmark simulating a laminar flow around a cylinder. The geometry is described by Set the kinematic viscosity ν = m 2 /s and ρ = 1.0 kg/m 3. A do-nothing boundary condition is assumed at the outlet. Defining U m = 1.5 m/s, the time-dependent inflow condition is given by U = 4U m y(h y) sin(πt/8)/h 2, V = 0. Schäfer/Turek, Benchmark Computations of Laminar Flow Around a Cylinder (1996) 10 / 11
11 The FEniCS challenge! The inflow boundary lies at x = 0.2 and the outflow boundary at x = 2.0. Compute the flow on the time interval [0, 8] with time-step dt = Test your implementation first for a larger time-step dt = 0.01 and the same channel problem but with the cylinder removed. If everything goes fine you should get something like Happy coding! Schäfer/Turek, Benchmark Computations of Laminar Flow Around a Cylinder (1996) 11 / 11
Numerical methods for the Navier- Stokes equations
Numerical methods for the Navier- Stokes equations Hans Petter Langtangen 1,2 1 Center for Biomedical Computing, Simula Research Laboratory 2 Department of Informatics, University of Oslo Dec 6, 2012 Note:
FEniCS Course. Lecture 0: Introduction to FEM. Contributors Anders Logg, Kent-Andre Mardal
FEniCS Course Lecture 0: Introduction to FEM Contributors Anders Logg, Kent-Andre Mardal 1 / 46 What is FEM? The finite element method is a framework and a recipe for discretization of mathematical problems
Efficient Augmented Lagrangian-type Preconditioning for the Oseen Problem using Grad-Div Stabilization
Efficient Augmented Lagrangian-type Preconditioning for the Oseen Problem using Grad-Div Stabilization Timo Heister, Texas A&M University 2013-02-28 SIAM CSE 2 Setting Stationary, incompressible flow problems
Math background. Physics. Simulation. Related phenomena. Frontiers in graphics. Rigid fluids
Fluid dynamics Math background Physics Simulation Related phenomena Frontiers in graphics Rigid fluids Fields Domain Ω R2 Scalar field f :Ω R Vector field f : Ω R2 Types of derivatives Derivatives measure
Two-Dimensional Unsteady Flow in a Lid Driven Cavity with Constant Density and Viscosity ME 412 Project 5
Two-Dimensional Unsteady Flow in a Lid Driven Cavity with Constant Density and Viscosity ME 412 Project 5 Jingwei Zhu May 14, 2014 Instructor: Surya Pratap Vanka 1 Project Description The objective of
Preconditioners for the incompressible Navier Stokes equations
Preconditioners for the incompressible Navier Stokes equations C. Vuik M. ur Rehman A. Segal Delft Institute of Applied Mathematics, TU Delft, The Netherlands SIAM Conference on Computational Science and
Conservation of Mass. Computational Fluid Dynamics. The Equations Governing Fluid Motion
http://www.nd.edu/~gtryggva/cfd-course/ http://www.nd.edu/~gtryggva/cfd-course/ Computational Fluid Dynamics Lecture 4 January 30, 2017 The Equations Governing Fluid Motion Grétar Tryggvason Outline Derivation
Fast Solvers for Unsteady Incompressible Flow
ICFD25 Workshop p. 1/39 Fast Solvers for Unsteady Incompressible Flow David Silvester University of Manchester http://www.maths.manchester.ac.uk/~djs/ ICFD25 Workshop p. 2/39 Outline Semi-Discrete Navier-Stokes
Parabolic Flow in Parallel Plate Channel ME 412 Project 4
Parabolic Flow in Parallel Plate Channel ME 412 Project 4 Jingwei Zhu April 12, 2014 Instructor: Surya Pratap Vanka 1 Project Description The objective of this project is to develop and apply a computer
OpenFOAM selected solver
OpenFOAM selected solver Roberto Pieri - SCS Italy 16-18 June 2014 Introduction to Navier-Stokes equations and RANS Turbulence modelling Numeric discretization Navier-Stokes equations Convective term {}}{
fluid mechanics as a prominent discipline of application for numerical
1. fluid mechanics as a prominent discipline of application for numerical simulations: experimental fluid mechanics: wind tunnel studies, laser Doppler anemometry, hot wire techniques,... theoretical fluid
Chapter 2. General concepts. 2.1 The Navier-Stokes equations
Chapter 2 General concepts 2.1 The Navier-Stokes equations The Navier-Stokes equations model the fluid mechanics. This set of differential equations describes the motion of a fluid. In the present work
PDE Solvers for Fluid Flow
PDE Solvers for Fluid Flow issues and algorithms for the Streaming Supercomputer Eran Guendelman February 5, 2002 Topics Equations for incompressible fluid flow 3 model PDEs: Hyperbolic, Elliptic, Parabolic
Vector and scalar penalty-projection methods
Numerical Flow Models for Controlled Fusion - April 2007 Vector and scalar penalty-projection methods for incompressible and variable density flows Philippe Angot Université de Provence, LATP - Marseille
20. A Dual-Primal FETI Method for solving Stokes/Navier-Stokes Equations
Fourteenth International Conference on Domain Decomposition Methods Editors: Ismael Herrera, David E. Keyes, Olof B. Widlund, Robert Yates c 23 DDM.org 2. A Dual-Primal FEI Method for solving Stokes/Navier-Stokes
A Robust Preconditioned Iterative Method for the Navier-Stokes Equations with High Reynolds Numbers
Applied and Computational Mathematics 2017; 6(4): 202-207 http://www.sciencepublishinggroup.com/j/acm doi: 10.11648/j.acm.20170604.18 ISSN: 2328-5605 (Print); ISSN: 2328-5613 (Online) A Robust Preconditioned
HIGHER-ORDER LINEARLY IMPLICIT ONE-STEP METHODS FOR THREE-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATIONS
STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume LIII, Number 1, March 2008 HIGHER-ORDER LINEARLY IMPLICIT ONE-STEP METHODS FOR THREE-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATIONS IOAN TELEAGA AND JENS
An advanced ILU preconditioner for the incompressible Navier-Stokes equations
An advanced ILU preconditioner for the incompressible Navier-Stokes equations M. ur Rehman C. Vuik A. Segal Delft Institute of Applied Mathematics, TU delft The Netherlands Computational Methods with Applications,
ON USING ARTIFICIAL COMPRESSIBILITY METHOD FOR SOLVING TURBULENT FLOWS
Conference Applications of Mathematics 212 in honor of the 6th birthday of Michal Křížek. Institute of Mathematics AS CR, Prague 212 ON USING ARTIFICIAL COMPRESSIBILITY METHOD FOR SOLVING TURBULENT FLOWS
Laminar Boundary Layers. Answers to problem sheet 1: Navier-Stokes equations
Laminar Boundary Layers Answers to problem sheet 1: Navier-Stokes equations The Navier Stokes equations for d, incompressible flow are + v ρ t + u + v v ρ t + u v + v v = 1 = p + µ u + u = p ρg + µ v +
A Review of Solution Techniques for Unsteady Incompressible Flow
Zeist 2009 p. 1/57 A Review of Solution Techniques for Unsteady Incompressible Flow David Silvester School of Mathematics University of Manchester Zeist 2009 p. 2/57 PDEs Review : 1966 1999 Update : 2000
PALADINS: Scalable Time-Adaptive Algebraic Splitting and Preconditioners for the Navier-Stokes Equations
2013 SIAM Conference On Computational Science and Engineering Boston, 27 th February 2013 PALADINS: Scalable Time-Adaptive Algebraic Splitting and Preconditioners for the Navier-Stokes Equations U. Villa,
Discrete Projection Methods for Incompressible Fluid Flow Problems and Application to a Fluid-Structure Interaction
Discrete Projection Methods for Incompressible Fluid Flow Problems and Application to a Fluid-Structure Interaction Problem Jörg-M. Sautter Mathematisches Institut, Universität Düsseldorf, Germany, sautter@am.uni-duesseldorf.de
A numerical study of heat transfer and fluid flow over an in-line tube bank
Fluid Structure Interaction VI 295 A numerical study of heat transfer and fluid flow over an in-line tube bank Z. S. Abdel-Rehim Mechanical Engineering Department, National Research Center, Egypt Abstract
Newton-Multigrid Least-Squares FEM for S-V-P Formulation of the Navier-Stokes Equations
Newton-Multigrid Least-Squares FEM for S-V-P Formulation of the Navier-Stokes Equations A. Ouazzi, M. Nickaeen, S. Turek, and M. Waseem Institut für Angewandte Mathematik, LSIII, TU Dortmund, Vogelpothsweg
Numerical Simulations of the Fluid-Structure Coupling in Physiological Vessels Mini-Workshop I
Numerical Simulations of the Fluid-Structure Coupling in Physiological Vessels INRIA Rocquencourt, projet REO miguel.fernandez@inria.fr CRM Spring School, Montréal May 16, 2005 Motivations It is generally
Adaptive C1 Macroelements for Fourth Order and Divergence-Free Problems
Adaptive C1 Macroelements for Fourth Order and Divergence-Free Problems Roy Stogner Computational Fluid Dynamics Lab Institute for Computational Engineering and Sciences University of Texas at Austin March
Turbulence Modeling I!
Outline! Turbulence Modeling I! Grétar Tryggvason! Spring 2010! Why turbulence modeling! Reynolds Averaged Numerical Simulations! Zero and One equation models! Two equations models! Model predictions!
Homework 4 in 5C1212; Part A: Incompressible Navier- Stokes, Finite Volume Methods
Homework 4 in 5C11; Part A: Incompressible Navier- Stokes, Finite Volume Methods Consider the incompressible Navier Stokes in two dimensions u x + v y = 0 u t + (u ) x + (uv) y + p x = 1 Re u + f (1) v
Fast Iterative Solution of Saddle Point Problems
Michele Benzi Department of Mathematics and Computer Science Emory University Atlanta, GA Acknowledgments NSF (Computational Mathematics) Maxim Olshanskii (Mech-Math, Moscow State U.) Zhen Wang (PhD student,
A Review of Preconditioning Techniques for Steady Incompressible Flow
Zeist 2009 p. 1/43 A Review of Preconditioning Techniques for Steady Incompressible Flow David Silvester School of Mathematics University of Manchester Zeist 2009 p. 2/43 PDEs Review : 1984 2005 Update
IMPLEMENTATION OF A PARALLEL AMG SOLVER
IMPLEMENTATION OF A PARALLEL AMG SOLVER Tony Saad May 2005 http://tsaad.utsi.edu - tsaad@utsi.edu PLAN INTRODUCTION 2 min. MULTIGRID METHODS.. 3 min. PARALLEL IMPLEMENTATION PARTITIONING. 1 min. RENUMBERING...
SIMPLE Algorithm for Two-Dimensional Channel Flow. Fluid Flow and Heat Transfer
SIMPLE Algorithm for Two-Dimensional Channel Flow Fluid Flow and Heat Transfer by Professor Jung-Yang San Mechanical Engineering Department National Chung Hsing University Two-dimensional, transient, incompressible
Fast solvers for steady incompressible flow
ICFD 25 p.1/21 Fast solvers for steady incompressible flow Andy Wathen Oxford University wathen@comlab.ox.ac.uk http://web.comlab.ox.ac.uk/~wathen/ Joint work with: Howard Elman (University of Maryland,
Project 4: Navier-Stokes Solution to Driven Cavity and Channel Flow Conditions
Project 4: Navier-Stokes Solution to Driven Cavity and Channel Flow Conditions R. S. Sellers MAE 5440, Computational Fluid Dynamics Utah State University, Department of Mechanical and Aerospace Engineering
On a class of numerical schemes. for compressible flows
On a class of numerical schemes for compressible flows R. Herbin, with T. Gallouët, J.-C. Latché L. Gastaldo, D. Grapsas, W. Kheriji, T.T. N Guyen, N. Therme, C. Zaza. Aix-Marseille Université I.R.S.N.
Cranfield University, Cranfield, Bedfordshire, MK43 0AL, United Kingdom. Cranfield University, Cranfield, Bedfordshire, MK43 0AL, United Kingdom
MultiScience - XXX. microcad International Multidisciplinary Scientific Conference University of Miskolc, Hungary, 21-22 April 2016, ISBN 978-963-358-113-1 NUMERICAL INVESTIGATION OF AN INCOMPRESSIBLE
AE/ME 339. K. M. Isaac. 9/22/2005 Topic 6 FluidFlowEquations_Introduction. Computational Fluid Dynamics (AE/ME 339) MAEEM Dept.
AE/ME 339 Computational Fluid Dynamics (CFD) 1...in the phrase computational fluid dynamics the word computational is simply an adjective to fluid dynamics.... -John D. Anderson 2 1 Equations of Fluid
Heat and Mass Transfer Prof. S.P. Sukhatme Department of Mechanical Engineering Indian Institute of Technology, Bombay
Heat and Mass Transfer Prof. S.P. Sukhatme Department of Mechanical Engineering Indian Institute of Technology, Bombay Lecture No. 18 Forced Convection-1 Welcome. We now begin our study of forced convection
Explicit algebraic Reynolds stress models for boundary layer flows
1. Explicit algebraic models Two explicit algebraic models are here compared in order to assess their predictive capabilities in the simulation of boundary layer flow cases. The studied models are both
Bridging the Gap: Harnessing advanced Computational Libraries for coupled multi-physics problems
Bridging the Gap: Harnessing advanced Computational Libraries for coupled multi-physics problems Marc Spiegelman 1,2 and Cian R Wilson 1 Plus enormous contributions from PETSc, FEniCS and AMCG groups 1
Hybrid Discontinuous Galerkin methods for incompressible flow problems
Hybrid Discontinuous Galerkin methods incompressible flow problems Christoph Lehrenfeld, Joachim Schöberl MathCCES Computational Mathematics in Engineering Workshop Linz, May 31 - June 1, 2010 Contents
Efficient FEM-multigrid solver for granular material
Efficient FEM-multigrid solver for granular material S. Mandal, A. Ouazzi, S. Turek Chair for Applied Mathematics and Numerics (LSIII), TU Dortmund STW user committee meeting Enschede, 25th September,
Soft Bodies. Good approximation for hard ones. approximation breaks when objects break, or deform. Generalization: soft (deformable) bodies
Soft-Body Physics Soft Bodies Realistic objects are not purely rigid. Good approximation for hard ones. approximation breaks when objects break, or deform. Generalization: soft (deformable) bodies Deformed
7.4 The Saddle Point Stokes Problem
346 CHAPTER 7. APPLIED FOURIER ANALYSIS 7.4 The Saddle Point Stokes Problem So far the matrix C has been diagonal no trouble to invert. This section jumps to a fluid flow problem that is still linear (simpler
STEADY AND UNSTEADY 2D NUMERICAL SOLUTION OF GENERALIZED NEWTONIAN FLUIDS FLOW. Radka Keslerová, Karel Kozel
Conference Applications of Mathematics 1 in honor of the th birthday of Michal Křížek. Institute of Mathematics AS CR, Prague 1 STEADY AND UNSTEADY D NUMERICAL SOLUTION OF GENERALIZED NEWTONIAN FLUIDS
Module 2 : Convection. Lecture 12 : Derivation of conservation of energy
Module 2 : Convection Lecture 12 : Derivation of conservation of energy Objectives In this class: Start the derivation of conservation of energy. Utilize earlier derived mass and momentum equations for
Simulation of T-junction using LBM and VOF ENERGY 224 Final Project Yifan Wang,
Simulation of T-junction using LBM and VOF ENERGY 224 Final Project Yifan Wang, yfwang09@stanford.edu 1. Problem setting In this project, we present a benchmark simulation for segmented flows, which contain
Computation of Incompressible Flows: SIMPLE and related Algorithms
Computation of Incompressible Flows: SIMPLE and related Algorithms Milovan Perić CoMeT Continuum Mechanics Technologies GmbH milovan@continuummechanicstechnologies.de SIMPLE-Algorithm I - - - Consider
A monolithic FEM solver for fluid structure
A monolithic FEM solver for fluid structure interaction p. 1/1 A monolithic FEM solver for fluid structure interaction Stefan Turek, Jaroslav Hron jaroslav.hron@mathematik.uni-dortmund.de Department of
JMBC Computational Fluid Dynamics I Exercises by A.E.P. Veldman
JMBC Computational Fluid Dynamics I Exercises by A.E.P. Veldman The exercises will be carried out on PC s in the practicum rooms. Several (Matlab and Fortran) files are required. How these can be obtained
Lecture 30 Review of Fluid Flow and Heat Transfer
Objectives In this lecture you will learn the following We shall summarise the principles used in fluid mechanics and heat transfer. It is assumed that the student has already been exposed to courses in
Basic hydrodynamics. David Gurarie. 1 Newtonian fluids: Euler and Navier-Stokes equations
Basic hydrodynamics David Gurarie 1 Newtonian fluids: Euler and Navier-Stokes equations The basic hydrodynamic equations in the Eulerian form consist of conservation of mass, momentum and energy. We denote
10. Buoyancy-driven flow
10. Buoyancy-driven flow For such flows to occur, need: Gravity field Variation of density (note: not the same as variable density!) Simplest case: Viscous flow, incompressible fluid, density-variation
Efficient Solvers for the Navier Stokes Equations in Rotation Form
Efficient Solvers for the Navier Stokes Equations in Rotation Form Computer Research Institute Seminar Purdue University March 4, 2005 Michele Benzi Emory University Atlanta, GA Thanks to: NSF (MPS/Computational
Game Physics. Game and Media Technology Master Program - Utrecht University. Dr. Nicolas Pronost
Game and Media Technology Master Program - Utrecht University Dr. Nicolas Pronost Soft body physics Soft bodies In reality, objects are not purely rigid for some it is a good approximation but if you hit
Assessment of a vorticity based solver for the Navier-Stokes equations
Assessment of a vorticity based solver for the Navier-Stokes equations Michele Benzi Maxim A. Olshanskii Leo G. Rebholz Zhen Wang Abstract We investigate numerically a recently proposed vorticity based
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 1 / 43 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS Treatment of Boundary Conditions These slides are partially based on the recommended textbook: Culbert
Finite Element Decompositions for Stable Time Integration of Flow Equations
MAX PLANCK INSTITUT August 13, 2015 Finite Element Decompositions for Stable Time Integration of Flow Equations Jan Heiland, Robert Altmann (TU Berlin) ICIAM 2015 Beijing DYNAMIK KOMPLEXER TECHNISCHER
Algebraic Flux Correction III
Algebraic Flux Correction III Incompressible Flow Problems Stefan Turek and Dmitri Kuzmin Abstract This chapter illustrates the use of algebraic flux correction in the context of finite element methods
( ) II. Why statistics? SIO 221B, Rudnick adapted from Davis!1
SIO 221B, Rudnick adapted from Davis 1 II. Why statistics? This course addresses the analysis of oceanographic observations and, as they become more complex, ocean models. Much of the material involves
Space-time XFEM for two-phase mass transport
Space-time XFEM for two-phase mass transport Space-time XFEM for two-phase mass transport Christoph Lehrenfeld joint work with Arnold Reusken EFEF, Prague, June 5-6th 2015 Christoph Lehrenfeld EFEF, Prague,
Numerical Methods for Problems with Moving Fronts Orthogonal Collocation on Finite Elements
Electronic Text Provided with the Book Numerical Methods for Problems with Moving Fronts by Bruce A. Finlayson Ravenna Park Publishing, Inc., 635 22nd Ave. N. E., Seattle, WA 985-699 26-524-3375; ravenna@halcyon.com;www.halcyon.com/ravenna
Numerical Methods for Incompressible Viscous Flow
Numerical Methods for Incompressible Viscous Flow Hans Petter Langtangen Kent-Andre Mardal Dept. of Scientific Computing, Simula Research Laboratory and Dept. of Informatics, University of Oslo Ragnar
2 Equations of Motion
2 Equations of Motion system. In this section, we will derive the six full equations of motion in a non-rotating, Cartesian coordinate 2.1 Six equations of motion (non-rotating, Cartesian coordinates)
Vector Penalty-Projection Methods for the Solution of Unsteady Incompressible Flows
Author manuscript, published in "5th International Symposium on Finite Volumes for Complex Applications, Aussois : France 2008)" Vector Penalty-Projection Methods for the Solution of Unsteady Incompressible
Flow simulation on moving boundary-fitted grids and application to fluid-structure interaction problems
Flow simulation on moving boundary-fitted grids and application to fluid-structure interaction problems Martin Engel and Michael Griebel Institute of Numerical Simulation, University of Bonn, Wegelerstr.
FEniCS Course. Lecture 1: Introduction to FEniCS. Contributors Anders Logg André Massing
FEniCS Course Lecture 1: Introduction to FEniCS Contributors Anders Logg André Massing 1 / 30 What is FEniCS? 2 / 30 FEniCS is an automated programming environment for differential equations C++/Python
The solution of the discretized incompressible Navier-Stokes equations with iterative methods
The solution of the discretized incompressible Navier-Stokes equations with iterative methods Report 93-54 C. Vuik Technische Universiteit Delft Delft University of Technology Faculteit der Technische
FREE BOUNDARY PROBLEMS IN FLUID MECHANICS
FREE BOUNDARY PROBLEMS IN FLUID MECHANICS ANA MARIA SOANE AND ROUBEN ROSTAMIAN We consider a class of free boundary problems governed by the incompressible Navier-Stokes equations. Our objective is to
Implicit Solution of Viscous Aerodynamic Flows using the Discontinuous Galerkin Method
Implicit Solution of Viscous Aerodynamic Flows using the Discontinuous Galerkin Method Per-Olof Persson and Jaime Peraire Massachusetts Institute of Technology 7th World Congress on Computational Mechanics
FEniCS Course. Lecture 1: Introduction to FEniCS. Contributors Anders Logg André Massing
FEniCS Course Lecture 1: Introduction to FEniCS Contributors Anders Logg André Massing 1 / 15 What is FEniCS? 2 / 15 FEniCS is an automated programming environment for differential equations C++/Python
Introduction to Heat and Mass Transfer. Week 9
Introduction to Heat and Mass Transfer Week 9 補充! Multidimensional Effects Transient problems with heat transfer in two or three dimensions can be considered using the solutions obtained for one dimensional
Numerical Modelling in Fortran: day 10. Paul Tackley, 2016
Numerical Modelling in Fortran: day 10 Paul Tackley, 2016 Today s Goals 1. Useful libraries and other software 2. Implicit time stepping 3. Projects: Agree on topic (by final lecture) (No lecture next
A Fast, Parallel Potential Flow Solver
Advisor: Jaime Peraire December 16, 2012 Outline 1 Introduction to Potential FLow 2 The Boundary Element Method 3 The Fast Multipole Method 4 Discretization 5 Implementation 6 Results 7 Conclusions Why
1 Introduction. J.-L. GUERMOND and L. QUARTAPELLE 1 On incremental projection methods
J.-L. GUERMOND and L. QUARTAPELLE 1 On incremental projection methods 1 Introduction Achieving high order time-accuracy in the approximation of the incompressible Navier Stokes equations by means of fractional-step
Transport processes. 7. Semester Chemical Engineering Civil Engineering
Transport processes 7. Semester Chemical Engineering Civil Engineering 1. Elementary Fluid Dynamics 2. Fluid Kinematics 3. Finite Control Volume Analysis 4. Differential Analysis of Fluid Flow 5. Viscous
Numerical Simulation of Newtonian and Non-Newtonian Flows in Bypass
Numerical Simulation of Newtonian and Non-Newtonian Flows in Bypass Vladimír Prokop, Karel Kozel Czech Technical University Faculty of Mechanical Engineering Department of Technical Mathematics Vladimír
15 The coffee cup J Nonlinear Dynamics II: Continuum Systems Lecture 15 Spring 2015
18.354J Nonlinear Dynamics II: Continuum Systems Lecture 15 Spring 2015 15 The coffee cup Let s try and apply our knowledge of fluid dynamics to a real observation, to test whether the theory actually
Une méthode de pénalisation par face pour l approximation des équations de Navier-Stokes à nombre de Reynolds élevé
Une méthode de pénalisation par face pour l approximation des équations de Navier-Stokes à nombre de Reynolds élevé CMCS/IACS Ecole Polytechnique Federale de Lausanne Erik.Burman@epfl.ch Méthodes Numériques
Computational Fluid Dynamics 2
Seite 1 Introduction Computational Fluid Dynamics 11.07.2016 Computational Fluid Dynamics 2 Turbulence effects and Particle transport Martin Pietsch Computational Biomechanics Summer Term 2016 Seite 2
DELFT UNIVERSITY OF TECHNOLOGY
DELFT UNIVERSITY OF TECHNOLOGY REPORT 13-10 Comparison of some preconditioners for the incompressible Navier-Stokes equations X. He and C. Vuik ISSN 1389-6520 Reports of the Delft Institute of Applied
Towards a Numerical Benchmark for 3D Low Mach Number Mixed Flows in a Rectangular Channel Heated from Below
Copyright 2008 Tech Science Press FDMP, vol.4, no.4, pp.263-269, 2008 Towards a Numerical Benchmark for 3D Low Mach Number Mixed Flows in a Rectangular Channel Heated from Below G. Accary 1, S. Meradji
Fluid-Particles Interaction Models Asymptotics, Theory and Numerics I
Fluid-Particles Interaction Models Asymptotics, Theory and Numerics I J. A. Carrillo collaborators: T. Goudon (Lille), P. Lafitte (Lille) and F. Vecil (UAB) (CPDE 2005), (JCP, 2008), (JSC, 2008) ICREA
Math 575-Lecture Viscous Newtonian fluid and the Navier-Stokes equations
Math 575-Lecture 13 In 1845, tokes extended Newton s original idea to find a constitutive law which relates the Cauchy stress tensor to the velocity gradient, and then derived a system of equations. The
A TAXONOMY AND COMPARISON OF PARALLEL BLOCK MULTI-LEVEL PRECONDITIONERS FOR THE INCOMPRESSIBLE NAVIER STOKES EQUATIONS
University of Maryland Department of Computer Science R-4867 University of Maryland Institute for Advanced Computer Studies R-2007-20 A AXONOMY AND COMPARISON OF PARALLEL BLOCK MULI-LEVEL PRECONDIIONERS
Block-Structured Adaptive Mesh Refinement
Block-Structured Adaptive Mesh Refinement Lecture 2 Incompressible Navier-Stokes Equations Fractional Step Scheme 1-D AMR for classical PDE s hyperbolic elliptic parabolic Accuracy considerations Bell
Implementing a k-ε Turbulence Model in the FEniCS Finite Element Programming Environment
Implementing a k-ε urbulence Model in the FEniCS Finite Element Programming Environment Kristian Valen-Sendstad 1,3 1 Simula School of Research and Innovation e mail: kvs@simula.no Mikael Mortensen 2,
Chapter 9: Differential Analysis
9-1 Introduction 9-2 Conservation of Mass 9-3 The Stream Function 9-4 Conservation of Linear Momentum 9-5 Navier Stokes Equation 9-6 Differential Analysis Problems Recall 9-1 Introduction (1) Chap 5: Control
Entropy 2011, 13, ; doi: /e OPEN ACCESS. Entropy Generation at Natural Convection in an Inclined Rectangular Cavity
Entropy 011, 13, 100-1033; doi:10.3390/e1305100 OPEN ACCESS entropy ISSN 1099-4300 www.mdpi.com/journal/entropy Article Entropy Generation at Natural Convection in an Inclined Rectangular Cavity Mounir
Colloquium FLUID DYNAMICS 2012 Institute of Thermomechanics AS CR, v.v.i., Prague, October 24-26, 2012 p.
Colloquium FLUID DYNAMICS 212 Institute of Thermomechanics AS CR, v.v.i., Prague, October 24-26, 212 p. ON A COMPARISON OF NUMERICAL SIMULATIONS OF ATMOSPHERIC FLOW OVER COMPLEX TERRAIN T. Bodnár, L. Beneš
6.2 Governing Equations for Natural Convection
6. Governing Equations for Natural Convection 6..1 Generalized Governing Equations The governing equations for natural convection are special cases of the generalized governing equations that were discussed
MATHEMATICAL MODELING AND NUMERICAL SOLUTION OF 3D ATMOSPHERIC BOUNDARY LAYER
, Vol, Pt, Special Issue Proceedings of International Conference RDAMM 585 MATHEMATICAL MODELING AND NUMERICAL SOLUTION OF D ATMOSPHERIC BOUNDARY LAYER L. Beneš, K. Kozel Department of Technical Mathematics,
Vector penalty-projection methods for outflow boundary conditions with optimal second-order accuracy
Vector penalty-projection methods for outflow boundary conditions with optimal second-order accuracy Philippe Angot, Rima Cheaytou To cite this version: Philippe Angot, Rima Cheaytou. Vector penalty-projection
2. FLUID-FLOW EQUATIONS SPRING 2019
2. FLUID-FLOW EQUATIONS SPRING 2019 2.1 Introduction 2.2 Conservative differential equations 2.3 Non-conservative differential equations 2.4 Non-dimensionalisation Summary Examples 2.1 Introduction Fluid
Design and Modeling of Fluid Power Systems ME 597/ABE Lecture 7
Systems ME 597/ABE 591 - Lecture 7 Dr. Monika Ivantysynova MAHA Professor Fluid Power Systems MAHA Fluid Power Research Center Purdue University Content of 6th lecture The lubricating gap as a basic design
Partitioned Methods for Multifield Problems
C Partitioned Methods for Multifield Problems Joachim Rang, 6.7.2016 6.7.2016 Joachim Rang Partitioned Methods for Multifield Problems Seite 1 C One-dimensional piston problem fixed wall Fluid flexible
Coupling Stokes and Darcy equations: modeling and numerical methods
Coupling Stokes and Darcy equations: modeling and numerical methods Marco Discacciati NM2PorousMedia 24 Dubrovnik, September 29, 24 Acknowledgment: European Union Seventh Framework Programme (FP7/27-23),
Pressure corrected SPH for fluid animation
Pressure corrected SPH for fluid animation Kai Bao, Hui Zhang, Lili Zheng and Enhua Wu Analyzed by Po-Ram Kim 2 March 2010 Abstract We present pressure scheme for the SPH for fluid animation In conventional
OUTLINE ffl CFD: elliptic pde's! Ax = b ffl Basic iterative methods ffl Krylov subspace methods ffl Preconditioning techniques: Iterative methods ILU
Preconditioning Techniques for Solving Large Sparse Linear Systems Arnold Reusken Institut für Geometrie und Praktische Mathematik RWTH-Aachen OUTLINE ffl CFD: elliptic pde's! Ax = b ffl Basic iterative