Robust Monolithic - Multigrid FEM Solver for Three Fields Formulation Rising from non-newtonian Flow Problems
|
|
- April Shepherd
- 5 years ago
- Views:
Transcription
1 Robust Monolithic - Multigrid FEM Solver for Three Fields Formulation Rising from non-newtonian Flow Problems M. Aaqib Afaq Institute for Applied Mathematics and Numerics (LSIII) TU Dortmund 13 July 2017
2 Contents 1 Motivation 2 Governing Equations 3 Variational Formulation 4 Finite Element Approximation 5 Numerical Results 6 Summary
3 Contents 1 Motivation 2 Governing Equations 3 Variational Formulation 4 Finite Element Approximation 5 Numerical Results 6 Summary
4 Three Fields Formulation Formulation involving stress Bingham/Viscoplastic flows Viscoelastic flows Multiphase flows Saddle point problem Requires two inf-sup conditions Velocity-Pressure Stress-Velocity
5 Bingham Model with additional symmetric viscoplastic stress tensor D(u) ɛ W D(u) = 0 (2ηD(u) + τ s W) + p = 0 u = 0 u = g D in Ω in Ω in Ω on Γ D W = D(u) D(u) ɛ W = new stress tensor τ s = yield stress D = strain rate tensor
6 Viscoelastic Model with additional elastic stress tensor Oldroyd-B (γ = 0) Giesukus (γ = 1) u t + (u )u = p + η s u + η p eψ u = 0 ψ t + (u )ψ (Ωψ ψω) 2B = 1 (e ψ I) γe ψ (e ψ I) 2 ψ = log conformation tensor D = strain rate tensor η s = solvent viscosity η p = polymer viscosity
7 Multiphase Model with additional multiphase stress tensor τ = τ s + τ m ( ) u ρ(ψ) t + u u τ + p = 0 u = 0 ( ) ψ ψ τ m = σ, τ s = 2µ(ψ)D(u) ψ ϕ t + u ϕ γ nd (( ϕ ϕ ϕ 1) ϕ ϕ ) = 0 ψ τ + (γ ncψ(1 ψ) ϕ) (γ nd ( ψ ϕ) ϕ) = 0 τ m = multiphase stress τ s = viscous stress
8 Jump Discontinuity Rising bubble in non-newtonian fluids In viscoelastic liquid Volume exceeds a critical value Rise velocity jump discontinuity shape changes: convex to tear drop
9 Jump Discontinuity Prediction: Boundary condition changes from rigid to free Jump occurs due to negative wake Jump occurs due to shear dependence of the viscosity
10 Industrial Applications Bubble Phenomena Pipeline transport application Bio reactors Combustion engines Underwater explosions Medicine: Angioplasty Food industry Fermentation process Bread and yogurt Nature: Bubble fence
11 Industrial Applications Bubble Phenomena Chemical separator Dispersion of gas bubble Efficient mass transfer Polymer and sludge processes
12 Contents 1 Motivation 2 Governing Equations 3 Variational Formulation 4 Finite Element Approximation 5 Numerical Results 6 Summary
13 Goals to Achieve Efficient three fields solver Solver: Monolithic multigrid approach Constraints free for the choice of finite element spaces
14 Navier Stokes Equation Primitive Formulation (u )u 2ηD(u) + p = 0 in Ω u = 0 in Ω (1) u = g D on Γ D Three Field Formulation σ D(u) = 0 in Ω ( ) (u )u 2η(1 α)d(u) + 2ηασ + p = 0 in Ω u = 0 u = g D in Ω on Γ D (2)
15 Three Fields Formulation Stokes System σ D(u) = 0 in Ω ( ) 2η(1 α)d(u) + 2ηασ + p = 0 in Ω u = 0 u = g D in Ω on Γ D (3) σ = extra stress tensor α = parameter η = viscosity
16 Contents 1 Motivation 2 Governing Equations 3 Variational Formulation 4 Finite Element Approximation 5 Numerical Results 6 Summary
17 Variational Formulation V = H 1 0(Ω) := ( H 1 0 (Ω)) 2 velocity and dual space V Q = L 2 0 (Ω) pressure and dual space Q T = ( L 2 (Ω) ) 4 stress and dual space T sym A u defined on V V A u u, v := 2η(1 α) A σ defined on T T A σ σ, τ = 2ηα The associated bilinear forms Ω Ω D(u) : D(v) dx (4) σ : τ dx (5) a u (u, v) = A u u, v, a σ (σ, τ ) = A σ σ, τ (6)
18 Variational Formulation B and C defined on V Q respectively, V T Bv, q := v q dx (7) Ω Cv, τ :=2ηα τ : D(v) dx (8) Associated bilinear forms b(, ) and c(, ) defined on V Q R and V T R respectively Ω b(v, q) := Bv, q (9) c(v, τ ) := Cv, τ (10)
19 Two-Folds Saddle Point Problem A u C T B T u RHS σ C A σ 0 σ = RHS u (11) B 0 0 p RHS p The system (3) has the following weak formulation a σ (σ, τ ) + c(u, τ ) = RHS σ, τ τ T a u (u, v) + c(v, σ) + b(v, p) = RHS u, v v V b(u, q) = RHS p, q q Q (12)
20 Variational Formulation Introduce the null spaces Ker B Ker B = {v V; b(v, q) = 0 q Q} Let X := Ker B T A(u, σ), (v, τ ) = A u u, v + Cv, σ + A σ σ, τ Cu, τ (13) a(u, V) = a u (u, v) + c(v, σ) + a σ (σ, τ ) c(u, τ ) (14)
21 Solvability of Problem Find U X such that: a(u, V) = f, V V X (15) Theorem Let X be a Hilbert space and f X, topological dual space of X, and let a(.,.) be a bilinear form on X satisfying the following hypothesis: 1). a(, ) is continuous: there exists a constant C > 0 such that: a(u, V) C U V U, V X (16)
22 Solvability of Problem Theorem (cont...) 2). a(, ) is X-elliptic: there exists a constant β > 0 such that: a(v, V) β V 2 V X (17) then problem has a unique solution U X. Remark Besides from theory of saddle point problem it is easy to show that there exists a unique p Q such that (σ, u, p) is the unique solution of problem (12)
23 Solvability of the Problem Illustration for α 1 a(v, V) = a u (v, v) + a σ (τ, τ ) = 2η(1 α) D(v) 2 dx + 2ηα τ 2 dx Ω Ω C2η(1 α) v 2 V + 2ηα τ 2 T C (v, τ ) 2 X Remark When α = 1 we can show existence and uniqueness using saddle point theory
24 Contents 1 Motivation 2 Governing Equations 3 Variational Formulation 4 Finite Element Approximation 5 Numerical Results 6 Summary
25 Finite Element Approximation Domain Ω R d Quadrilaterals elements K, Ω = int ( K T h K) Finite element spaces approaximations T h = { τ h M, σ h K Q 2 (K) } V h = { v h V, v h K Q 2 (K) } { } Q h = q h Q, q h K P1 disc (K) (18)
26 Finite Element Approximation Approximation of Stokes problem for finite element spaces T h T, V h V, Q h Q a σ (σ h, τ h ) + c(u h, τ h ) = RHS σ, τ h τ h T h a u (u h, v h ) + c(v h, σ h ) + b(v h, p h ) = RHS u, v h v h V h b(u h, q h ) = RHS p, q h q h Q h (19) Two LBB conditions required Firstly, between velocity and stress Secondly, between velocity and pressure
27 Finite Element Approximation Approximation of Stokes problem for finite element spaces T h T, V h V, Q h Q a σ (σ h, τ h ) + c(u h, τ h ) = RHS σ, τ h τ h T h a u (u h, v h ) + c(v h, σ h ) + b(v h, p h ) = RHS u, v h v h V h b(u h, q h ) = RHS p, q h q h Q h Two LBB conditions required Firstly, between velocity and stress Secondly, between velocity and pressure
28 Stable Element Choice 1 Fortin and Fortin element (u, p) spaces LBB compatible σ and D(u) same FEM discontinuous space
29 Stable Element Choice 2 Marchal-Crochet element Subcell discretization to enrich the local d.o.f for σ n σ > n u condition satisfied Computational cost is increased due to more d.o.f
30 EO-FEM Stabilization Remedy: Jump term addition to ensure element pair is stable Penalizing the jump of the solution gradient over E J u (u h, v h ) = γ u 2ηαh [ u h ] : [ v h ] dω (20) e E e h A u + J u C B T u RHS u C T A σ 0 σ = RHS σ (21) B 0 0 p RHS p
31 Discrete Problem Theorem Find U h X h such that: a(u h, V h ) + j(u h, V h ) = f h, V h V h X h (22) V h 2 = V h 2 + j(v h, V h ) (23) Let X h be a Hilbert space and f h X h, topological dual space of X, and let a(.,.) be a bilinear form on X h satisfying the following hypothesis: 1). a(, ) is continuous: there exists a constant C h > 0 such that: a(u h, V h ) C h U h V h U h, V h X h (24)
32 Solvability of Problem Theorem (cont...) 2). There exists a constant β h > 0 such that : a(u h, V h ) sup β h U h U h X h (25) V h X h V h then problem has a unique solution U h X h. Remark Besides from theory of saddle point problem it is easy to show that there exists a unique p h Q h such that (σ h, u h, p h ) is the unique solution of problem (19) No constraints on the choice of discrete finite space for stress
33 Contents 1 Motivation 2 Governing Equations 3 Variational Formulation 4 Finite Element Approximation 5 Numerical Results 6 Summary
34 Benchmark Configuration Inlet: Dirichlet parabolic profile u(0, y) = 4 Uy(0.41 y) (0.41) 2 No-slip at upper and lower walls Γ 1 and Γ 3 Outlet: Neumann boundary Γ 2 Kinematic viscosity η = 10 3 Characteristic length of cylinder l c = 0.1
35 Numerical Results
36 Monolithic Multigrid Solver Three fields Stokes vs Stokes solver in primitive variables Level Lift Drag NL/LL Lift 1 Drag NL/LL / / / / / / / / /2 Three field solver performance efficient as primitive Stokes solver Check robustness and consistency! 1 Damanik. H FEM Simulation of Non-isothermal Viscoelastic fluids, PhD Thesis
37 EO-FEM : Consistency Consistency for case α = 0 Level α Lift Drag NL/LL Lift Drag NL/LL No Stab. Stab / / / / / / / /3 Edge Oriented FEM is consistent Side effect neither on solution nor on the solver
38 EO-FEM :Robustness Two extreme cases α=0 viscous contribution α=1 no viscous contribution Level α Lift Drag NL/LL Lift Drag NL/LL No Stab. Stab / / / / / / / / / / / /3 Robustness w.r.t. problem! Consistent and grid independent solver
39 Contents 1 Motivation 2 Governing Equations 3 Variational Formulation 4 Finite Element Approximation 5 Numerical Results 6 Summary
40 Summary Robust monolithic multigrid solver for three fields Stokes formulation using EO-FEM: Advantages Taking away the 2nd inf-sup condition (no constraints on the choice of the σ space) Allowing large class of discretiztation for the stress Taking the efficiency of the Stokes solver in primitive variables to Stokes solver in three fields formulation
FEM techniques for nonlinear fluids
FEM techniques for nonlinear fluids From non-isothermal, pressure and shear dependent viscosity models to viscoelastic flow A. Ouazzi, H. Damanik, S. Turek Institute of Applied Mathematics, LS III, TU
More informationA Multigrid LCR-FEM solver for viscoelastic fluids with application to problems with free surface
A Multigrid LCR-FEM solver for viscoelastic fluids with application to problems with free surface Damanik, H., Mierka, O., Ouazzi, A., Turek, S. (Lausanne, August 2013) Page 1 Motivation Polymer melts:
More informationEfficient 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,
More informationNewton-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
More informationFINITE ELEMENT APPROXIMATION OF STOKES-LIKE SYSTEMS WITH IMPLICIT CONSTITUTIVE RELATION
Proceedings of ALGORITMY pp. 9 3 FINITE ELEMENT APPROXIMATION OF STOKES-LIKE SYSTEMS WITH IMPLICIT CONSTITUTIVE RELATION JAN STEBEL Abstract. The paper deals with the numerical simulations of steady flows
More informationMonolithic FEM multigrid techniques for the simulation of viscoelastic flow
Monolithic FEM multigrid techniques for the simulation of viscoelastic flow A. Ouazzi, H. Damanik, S. Turek, J. Hron Institute of Applied Mathematics, LS III, TU Dortmund http://www.featflow.de European
More informationChapter 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
More informationDifferential relations for fluid flow
Differential relations for fluid flow In this approach, we apply basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of a flow
More informationExcerpt from the Proceedings of the COMSOL Users Conference 2006 Boston
Using Comsol Multiphysics to Model Viscoelastic Fluid Flow Bruce A. Finlayson, Professor Emeritus Department of Chemical Engineering University of Washington, Seattle, WA 98195-1750 finlayson@cheme.washington.edu
More informationA note on accurate and efficient higher order Galerkin time stepping schemes for the nonstationary Stokes equations
A note on accurate and efficient higher order Galerkin time stepping schemes for the nonstationary Stokes equations S. Hussain, F. Schieweck, S. Turek Abstract In this note, we extend our recent work for
More informationA Two-Grid Stabilization Method for Solving the Steady-State Navier-Stokes Equations
A Two-Grid Stabilization Method for Solving the Steady-State Navier-Stokes Equations Songul Kaya and Béatrice Rivière Abstract We formulate a subgrid eddy viscosity method for solving the steady-state
More informationDiscrete 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
More informationNumerical Simulation of Powder Flow
Numerical Simulation of Powder Flow Stefan Turek, Abderrahim Ouazzi Institut für Angewandte Mathematik, Univ. Dortmund http://www.mathematik.uni-dortmund.de/ls3 http://www.featflow.de Models for granular
More informationNumerical 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
More informationMixed Finite Element Methods. Douglas N. Arnold, University of Minnesota The 41st Woudschoten Conference 5 October 2016
Mixed Finite Element Methods Douglas N. Arnold, University of Minnesota The 41st Woudschoten Conference 5 October 2016 Linear elasticity Given the load f : Ω R n, find the displacement u : Ω R n and the
More informationREE Internal Fluid Flow Sheet 2 - Solution Fundamentals of Fluid Mechanics
REE 307 - Internal Fluid Flow Sheet 2 - Solution Fundamentals of Fluid Mechanics 1. Is the following flows physically possible, that is, satisfy the continuity equation? Substitute the expressions for
More informationOn a variational inequality of Bingham and Navier-Stokes type in three dimension
PDEs for multiphase ADvanced MATerials Palazzone, Cortona (Arezzo), Italy, September 17-21, 2012 On a variational inequality of Bingham and Navier-Stokes type in three dimension Takeshi FUKAO Kyoto University
More informationFundamentals of Fluid Dynamics: Elementary Viscous Flow
Fundamentals of Fluid Dynamics: Elementary Viscous Flow Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research
More informationWELL POSEDNESS OF PROBLEMS I
Finite Element Method 85 WELL POSEDNESS OF PROBLEMS I Consider the following generic problem Lu = f, where L : X Y, u X, f Y and X, Y are two Banach spaces We say that the above problem is well-posed (according
More informationVector 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
More informationA multiscale framework for lubrication analysis of bearings with textured surface
A multiscale framework for lubrication analysis of bearings with textured surface *Leiming Gao 1), Gregory de Boer 2) and Rob Hewson 3) 1), 3) Aeronautics Department, Imperial College London, London, SW7
More informationAn Extended Finite Element Method for a Two-Phase Stokes problem
XFEM project An Extended Finite Element Method for a Two-Phase Stokes problem P. Lederer, C. Pfeiler, C. Wintersteiger Advisor: Dr. C. Lehrenfeld August 5, 2015 Contents 1 Problem description 2 1.1 Physics.........................................
More informationNumerische Simulation zur Herstellung monodisperser Tropfen in pneumatischen Ziehdüsen
Numerische Simulation zur Herstellung monodisperser Tropfen in pneumatischen Ziehdüsen DFG SPP 1423 Prozess-Spray Prof. Dr. Stefan Turek, Dr. Institut für Angewandte Mathematik, LS III Technische Universität
More informationFEM-Level Set Techniques for Multiphase Flow --- Some recent results
FEM-Level Set Techniques for Multiphase Flow --- Some recent results ENUMATH09, Uppsala Stefan Turek, Otto Mierka, Dmitri Kuzmin, Shuren Hysing Institut für Angewandte Mathematik, TU Dortmund http://www.mathematik.tu-dortmund.de/ls3
More informationA Space-Time Multigrid Solver Methodology for the Optimal Control of Time-Dependent Fluid Flow
A Space-Time Multigrid Solver Methodology for the Optimal Control of Time-Dependent Fluid Flow Michael Köster, Michael Hinze, Stefan Turek Michael Köster Institute for Applied Mathematics TU Dortmund Trier,
More informationFluid Mechanics II Viscosity and shear stresses
Fluid Mechanics II Viscosity and shear stresses Shear stresses in a Newtonian fluid A fluid at rest can not resist shearing forces. Under the action of such forces it deforms continuously, however small
More informationMasters in Mechanical Engineering. Problems of incompressible viscous flow. 2µ dx y(y h)+ U h y 0 < y < h,
Masters in Mechanical Engineering Problems of incompressible viscous flow 1. Consider the laminar Couette flow between two infinite flat plates (lower plate (y = 0) with no velocity and top plate (y =
More informationA numerical approximation with IP/SUPG algorithm for P-T-T viscoelastic flows
Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 6 (2016, 152 161 Research Article A numerical approximation with IP/SUPG algorithm for P-T-T viscoelastic flows Lei Hou a, Yunqing Feng a,, Lin
More informationMonolithic Newton-multigrid solution techniques for incompressible nonlinear flow models
Monolithic Newton-multigrid solution techniques for incompressible nonlinear flow models H. Damanik a,, J. Hron b, A. Ouazzi a, S. Turek a a Institut für Angewante Mathematik, TU Dortmund, Germany b Institute
More information1. Introduction. The Stokes problem seeks unknown functions u and p satisfying
A DISCRETE DIVERGENCE FREE WEAK GALERKIN FINITE ELEMENT METHOD FOR THE STOKES EQUATIONS LIN MU, JUNPING WANG, AND XIU YE Abstract. A discrete divergence free weak Galerkin finite element method is developed
More informationMODELLING MULTIPHASE FLOWS OF DISCRETE PARTICLES IN VISCOELASTIC FLUIDS
MODELLING MULTIPHASE FLOWS OF DISCRETE PARTICLES IN VISCOELASTIC FLUIDS R. RIBEIRO 1, C. FERNANDES 1, S.A. FAROUGHI 2, G.H. McKINLEY 2 AND J.M. NÓBREGA 1 1 Institute for Polymers and Composites/i3N, University
More informationNumerical Analysis and Computation of Fluid/Fluid and Fluid/Structure Interaction Problems Jason S. Howell
J.S.Howell Research Statement 1 Numerical Analysis and Computation of Fluid/Fluid and Fluid/Structure Interaction Problems Jason S. Howell Fig. 1. Plot of the magnitude of the velocity (upper left) and
More informationFinite Element LES and VMS Methods on Tetrahedral Meshes
Finite Element LES and VMS Methods on Tetrahedral Meshes Volker John 1 Adela Kindl 2 and Carina Suciu 3 FR 6.1 Mathematik, Universität des Saarlandes, Postfach 15 11 50, 66041 Saarbrücken, Germany Abstract
More informationPressure corrections for viscoelastic potential flow analysis of capillary instability
ve-july29-4.tex 1 Pressure corrections for viscoelastic potential flow analysis of capillary instability J. Wang, D. D. Joseph and T. Funada Department of Aerospace Engineering and Mechanics, University
More informationAnalysis of the Oseen-Viscoelastic Fluid Flow Problem
Analysis of the Oseen-Viscoelastic Fluid Flow Problem Vincent J. Ervin Hyesuk K. Lee Louis N. Ntasin Department of Mathematical Sciences Clemson University Clemson, South Carolina 9634-0975 December 30,
More informationWe may have a general idea that a solid is hard and a fluid is soft. This is not satisfactory from
Chapter 1. Introduction 1.1 Some Characteristics of Fluids We may have a general idea that a solid is hard and a fluid is soft. This is not satisfactory from scientific or engineering point of view. In
More information- Marine Hydrodynamics. Lecture 4. Knowns Equations # Unknowns # (conservation of mass) (conservation of momentum)
2.20 - Marine Hydrodynamics, Spring 2005 Lecture 4 2.20 - Marine Hydrodynamics Lecture 4 Introduction Governing Equations so far: Knowns Equations # Unknowns # density ρ( x, t) Continuity 1 velocities
More informationFlow-Induced Vibration Analysis of Supported Pipes with a Crack
Flow-Induced Vibration Analysis of Supported Pipes with a Crack Jin-Hyuk Lee 1 *, Samer Masoud Al-Said,3 1 Department of Mechanical Engineering, American University of Sharjah, Sharjah, UAE, Department
More informationA Two-grid Method for Coupled Free Flow with Porous Media Flow
A Two-grid Method for Coupled Free Flow with Porous Media Flow Prince Chidyagwai a and Béatrice Rivière a, a Department of Computational and Applied Mathematics, Rice University, 600 Main Street, Houston,
More informationIntroduction to Marine Hydrodynamics
1896 1920 1987 2006 Introduction to Marine Hydrodynamics (NA235) Department of Naval Architecture and Ocean Engineering School of Naval Architecture, Ocean & Civil Engineering First Assignment The first
More informationOpen boundary conditions in numerical simulations of unsteady incompressible flow
Open boundary conditions in numerical simulations of unsteady incompressible flow M. P. Kirkpatrick S. W. Armfield Abstract In numerical simulations of unsteady incompressible flow, mass conservation can
More informationSome remarks on grad-div stabilization of incompressible flow simulations
Some remarks on grad-div stabilization of incompressible flow simulations Gert Lube Institute for Numerical and Applied Mathematics Georg-August-University Göttingen M. Stynes Workshop Numerical Analysis
More informationProposal for numerical benchmarking of fluid-structure interaction between an elastic object and laminar incompressible flow
Proposal for numerical benchmarking of fluid-structure interaction between an elastic object and laminar incompressible flow Stefan Turek and Jaroslav Hron Institute for Applied Mathematics and Numerics,
More informationChristel Hohenegger A simple model for ketchup-like liquid, its numerical challenges and limitations April 7, 2011
Notes by: Andy Thaler Christel Hohenegger A simple model for ketchup-like liquid, its numerical challenges and limitations April 7, 2011 Many complex fluids are shear-thinning. Such a fluid has a shear
More informationFEM for Stokes Equations
FEM for Stokes Equations Kanglin Chen 15. Dezember 2009 Outline 1 Saddle point problem 2 Mixed FEM for Stokes equations 3 Numerical Results 2 / 21 Stokes equations Given (f, g). Find (u, p) s.t. u + p
More informationINTRODUCTION OBJECTIVES
INTRODUCTION The transport of particles in laminar and turbulent flows has numerous applications in engineering, biological and environmental systems. The deposition of aerosol particles in channels and
More informationFast 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,
More informationSTEADY 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
More informationSimulation 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
More information1 The Stokes System. ρ + (ρv) = ρ g(x), and the conservation of momentum has the form. ρ v (λ 1 + µ 1 ) ( v) µ 1 v + p = ρ f(x) in Ω.
1 The Stokes System The motion of a (possibly compressible) homogeneous fluid is described by its density ρ(x, t), pressure p(x, t) and velocity v(x, t). Assume that the fluid is barotropic, i.e., the
More informationNumerical Simulation of a Bubble Rising in an Environment Consisting of Xanthan Gum
Numerical Simulation of a Bubble Rising in an Environment Consisting of Xanthan Gum Víctor, A, Aguirre 1, Byron, A, Castillo 1 and Christian Narvaez-Muñoz 1 a) 1 Departamento de Ciencias de la Energía
More informationMultigrid Methods for Saddle Point Problems
Multigrid Methods for Saddle Point Problems Susanne C. Brenner Department of Mathematics and Center for Computation & Technology Louisiana State University Advances in Mathematics of Finite Elements (In
More informationON THE SOLVABILITY OF A NONLINEAR PSEUDOPARABOLIC PROBLEM
Indian J. Pure Appl. Math., 44(3): 343-354, June 2013 c Indian National Science Academy ON THE SOLVABILITY OF A NONLINEAR PSEUDOPARABOLIC PROBLEM S. Mesloub and T. Hayat Mathematics Department, College
More informationFLUID MECHANICS. Atmosphere, Ocean. Aerodynamics. Energy conversion. Transport of heat/other. Numerous industrial processes
SG2214 Anders Dahlkild Luca Brandt FLUID MECHANICS : SG2214 Course requirements (7.5 cr.) INL 1 (3 cr.) 3 sets of home work problems (for 10 p. on written exam) 1 laboration TEN1 (4.5 cr.) 1 written exam
More informationTurbulent Boundary Layers & Turbulence Models. Lecture 09
Turbulent Boundary Layers & Turbulence Models Lecture 09 The turbulent boundary layer In turbulent flow, the boundary layer is defined as the thin region on the surface of a body in which viscous effects
More informationRHEOLOGY Principles, Measurements, and Applications. Christopher W. Macosko
RHEOLOGY Principles, Measurements, and Applications I -56081-5'79~5 1994 VCH Publishers. Inc. New York Part I. CONSTITUTIVE RELATIONS 1 1 l Elastic Solid 5 1.1 Introduction 5 1.2 The Stress Tensor 8 1.2.1
More informationDirect Simulation of the Motion of Solid Particles in Couette and Poiseuille Flows of Viscoelastic Fluids
Direct Simulation of the Motion of Solid Particles in Couette and Poiseuille Flows of Viscoelastic Fluids by P. Y. Huang 1, J. Feng 2, H. H. Hu 3 and D. D. Joseph 1 1 Department of Aerospace Engineering
More informationUNIVERSITY of LIMERICK
UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH Faculty of Science and Engineering END OF SEMESTER ASSESSMENT PAPER MODULE CODE: MA4607 SEMESTER: Autumn 2012-13 MODULE TITLE: Introduction to Fluids DURATION OF
More informationPDEs, part 1: Introduction and elliptic PDEs
PDEs, part 1: Introduction and elliptic PDEs Anna-Karin Tornberg Mathematical Models, Analysis and Simulation Fall semester, 2013 Partial di erential equations The solution depends on several variables,
More informationApproximation of fluid-structure interaction problems with Lagrange multiplier
Approximation of fluid-structure interaction problems with Lagrange multiplier Daniele Boffi Dipartimento di Matematica F. Casorati, Università di Pavia http://www-dimat.unipv.it/boffi May 30, 2016 Outline
More informationOn 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.
More informationChapter 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
More informationV (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t)
IV. DIFFERENTIAL RELATIONS FOR A FLUID PARTICLE This chapter presents the development and application of the basic differential equations of fluid motion. Simplifications in the general equations and common
More informationSTABILIZED FINITE ELEMENT METHODS OF GLS TYPE FOR MAXWELL-B AND OLDROYD-B VISCOELASTIC FLUIDS
European Congress on Computational Methods in Applied Sciences and Engineering ECCOMAS 2004 P. Neittaanmäki, T. Rossi, S. Korotov, E. Oñate, J. Périaux, and D. Knörzer (eds.) Jyväskylä, 24 28 July 2004
More informationA formulation for fast computations of rigid particulate flows
Center for Turbulence Research Annual Research Briefs 2001 185 A formulation for fast computations of rigid particulate flows By N. A. Patankar 1. Introduction A formulation is presented for the direct
More informationPlanar Geometry Ferrofluid Flows in Spatially Uniform Sinusoidally Time-varying Magnetic Fields
Presented at the 11 COMSOL Conference in Boston Planar Geometry Ferrofluid Flows in Spatially Uniform Sinusoidally Time-varying Magnetic Fields Shahriar Khushrushahi, Alexander Weddemann, Young Sun Kim
More informationAMS subject classifications. Primary, 65N15, 65N30, 76D07; Secondary, 35B45, 35J50
A SIMPLE FINITE ELEMENT METHOD FOR THE STOKES EQUATIONS LIN MU AND XIU YE Abstract. The goal of this paper is to introduce a simple finite element method to solve the Stokes equations. This method is in
More informationFLUID MECHANICS. Chapter 9 Flow over Immersed Bodies
FLUID MECHANICS Chapter 9 Flow over Immersed Bodies CHAP 9. FLOW OVER IMMERSED BODIES CONTENTS 9.1 General External Flow Characteristics 9.3 Drag 9.4 Lift 9.1 General External Flow Characteristics 9.1.1
More informationOptimal control in fluid mechanics by finite elements with symmetric stabilization
Computational Sciences Center Optimal control in fluid mechanics by finite elements with symmetric stabilization Malte Braack Mathematisches Seminar Christian-Albrechts-Universität zu Kiel VMS Worshop
More informationStochastic multiscale modeling of subsurface and surface flows. Part III: Multiscale mortar finite elements for coupled Stokes-Darcy flows
Stochastic multiscale modeling of subsurface and surface flows. Part III: Multiscale mortar finite elements for coupled Stokes-Darcy flows Ivan otov Department of Mathematics, University of Pittsburgh
More informationA unifying model for fluid flow and elastic solid deformation: a novel approach for fluid-structure interaction and wave propagation
A unifying model for fluid flow and elastic solid deformation: a novel approach for fluid-structure interaction and wave propagation S. Bordère a and J.-P. Caltagirone b a. CNRS, Univ. Bordeaux, ICMCB,
More informationUnified A Posteriori Error Control for all Nonstandard Finite Elements 1
Unified A Posteriori Error Control for all Nonstandard Finite Elements 1 Martin Eigel C. Carstensen, C. Löbhard, R.H.W. Hoppe Humboldt-Universität zu Berlin 19.05.2010 1 we know of Guidelines for Applicants
More informationNUMERICAL SIMULATION OF INTERACTION BETWEEN INCOMPRESSIBLE FLOW AND AN ELASTIC WALL
Proceedings of ALGORITMY 212 pp. 29 218 NUMERICAL SIMULATION OF INTERACTION BETWEEN INCOMPRESSIBLE FLOW AND AN ELASTIC WALL MARTIN HADRAVA, MILOSLAV FEISTAUER, AND PETR SVÁČEK Abstract. The present paper
More informationChapter 9: Differential Analysis of Fluid Flow
of Fluid Flow Objectives 1. Understand how the differential equations of mass and momentum conservation are derived. 2. Calculate the stream function and pressure field, and plot streamlines for a known
More informationComputational 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
More informationContinuum Mechanics and Theory of Materials
Peter Haupt Continuum Mechanics and Theory of Materials Translated from German by Joan A. Kurth Second Edition With 91 Figures, Springer Contents Introduction 1 1 Kinematics 7 1. 1 Material Bodies / 7
More informationUNIFORM PRECONDITIONERS FOR A PARAMETER DEPENDENT SADDLE POINT PROBLEM WITH APPLICATION TO GENERALIZED STOKES INTERFACE EQUATIONS
UNIFORM PRECONDITIONERS FOR A PARAMETER DEPENDENT SADDLE POINT PROBLEM WITH APPLICATION TO GENERALIZED STOKES INTERFACE EQUATIONS MAXIM A. OLSHANSKII, JÖRG PETERS, AND ARNOLD REUSKEN Abstract. We consider
More informationPerformance evaluation of different model mixers by numerical simulation
Journal of Food Engineering 71 (2005) 295 303 www.elsevier.com/locate/jfoodeng Performance evaluation of different model mixers by numerical simulation Chenxu Yu, Sundaram Gunasekaran * Food and Bioprocess
More informationDivergence-conforming multigrid methods for incompressible flow problems
Divergence-conforming multigrid methods for incompressible flow problems Guido Kanschat IWR, Universität Heidelberg Prague-Heidelberg-Workshop April 28th, 2015 G. Kanschat (IWR, Uni HD) Hdiv-DG Práha,
More informationComputer Fluid Dynamics E181107
Computer Fluid Dynamics E181107 2181106 Transport equations, Navier Stokes equations Remark: foils with black background could be skipped, they are aimed to the more advanced courses Rudolf Žitný, Ústav
More informationFOUR-WAY COUPLED SIMULATIONS OF TURBULENT
FOUR-WAY COUPLED SIMULATIONS OF TURBULENT FLOWS WITH NON-SPHERICAL PARTICLES Berend van Wachem Thermofluids Division, Department of Mechanical Engineering Imperial College London Exhibition Road, London,
More informationModelling of dispersed, multicomponent, multiphase flows in resource industries Section 4: Non-Newtonian fluids and rheometry (PART 1)
Modelling of dispersed, multicomponent, multiphase flows in resource industries Section 4: Non-Newtonian fluids and rheometry (PART 1) Globex Julmester 2017 Lecture #3 05 July 2017 Agenda Lecture #3 Section
More informationTable of Contents. II. PDE classification II.1. Motivation and Examples. II.2. Classification. II.3. Well-posedness according to Hadamard
Table of Contents II. PDE classification II.. Motivation and Examples II.2. Classification II.3. Well-posedness according to Hadamard Chapter II (ContentChapterII) Crashtest: Reality Simulation http:www.ara.comprojectssvocrownvic.htm
More informationPhan-Thien-Tanner Modeling of a Viscoelastic Fluid in the Stick-Slip Scenario. Andrew Griffith
Phan-Thien-Tanner Modeling of a Viscoelastic Fluid in the Stick-Slip Scenario Andrew Griffith Supervisor: Bruce A. Finlayson Department of Chemical Engineering University of Washington March 1, 7 Introduction
More informationNUMERICAL SIMULATION OF THE FLOW OF A POWER LAW FLUID IN AN ELBOW BEND. A Thesis KARTHIK KANAKAMEDALA
NUMERICAL SIMULATION OF THE FLOW OF A POWER LAW FLUID IN AN ELBOW BEND A Thesis by KARTHIK KANAKAMEDALA Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the
More informationWeierstraß-Institut. für Angewandte Analysis und Stochastik. Leibniz-Institut im Forschungsverbund Berlin e. V. Preprint ISSN
Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut im Forschungsverbund Berlin e. V. Preprint ISSN 2198-5855 On the divergence constraint in mixed finite element methods for incompressible
More informationDiscontinuous Galerkin methods for nonlinear elasticity
Discontinuous Galerkin methods for nonlinear elasticity Preprint submitted to lsevier Science 8 January 2008 The goal of this paper is to introduce Discontinuous Galerkin (DG) methods for nonlinear elasticity
More informationPurely irrotational theories of the effects of viscosity and viscoelasticity on capillary instability of a liquid cylinder
ve-march17.tex 1 Purely irrotational theories of the effects of viscosity and viscoelasticity on capillary instability of a liquid cylinder J. Wang, D. D. Joseph and T. Funada Department of Aerospace Engineering
More informationLecture Note III: Least-Squares Method
Lecture Note III: Least-Squares Method Zhiqiang Cai October 4, 004 In this chapter, we shall present least-squares methods for second-order scalar partial differential equations, elastic equations of solids,
More informationIntroduction and Fundamental Concepts (Lectures 1-7)
Introduction and Fundamental Concepts (Lectures -7) Q. Choose the crect answer (i) A fluid is a substance that (a) has the same shear stress at a point regardless of its motion (b) is practicall incompressible
More informationTHE 3D VISCOELASTIC SIMULATION OF MULTI-LAYER FLOW INSIDE FILM AND SHEET EXTRUSION DIES
THE 3D VISCOELASTIC SIMULATION OF MULTI-LAYER FLOW INSIDE FILM AND SHEET EXTRUSION DIES Kazuya Yokomizo 1, Makoto Iwamura 2 and Hideki Tomiyama 1 1 The Japan Steel Works, LTD., Hiroshima Research Laboratory,
More information3D CFD ANALYSIS OF HEAT TRANSFER IN A SCRAPED SURFACE HEAT EXCHANGER FOR BINGHAM FLUIDS
3D CFD ANALYSIS OF HEAT TRANSFER IN A SCRAPED SURFACE HEAT EXCHANGER FOR BINGHAM FLUIDS Ali S.* and Baccar M. *Author for correspondence Department of Mechanical Engineering, National Engineering School
More information1. Introduction, fluid properties (1.1, 2.8, 4.1, and handouts)
1. Introduction, fluid properties (1.1, 2.8, 4.1, and handouts) Introduction, general information Course overview Fluids as a continuum Density Compressibility Viscosity Exercises: A1 Fluid mechanics Fluid
More informationChapter 5. The Differential Forms of the Fundamental Laws
Chapter 5 The Differential Forms of the Fundamental Laws 1 5.1 Introduction Two primary methods in deriving the differential forms of fundamental laws: Gauss s Theorem: Allows area integrals of the equations
More informationLaminar Forced Convection Heat Transfer from Two Heated Square Cylinders in a Bingham Plastic Fluid
Laminar Forced Convection Heat Transfer from Two Heated Square Cylinders in a Bingham Plastic Fluid E. Tejaswini 1*, B. Sreenivasulu 2, B. Srinivas 3 1,2,3 Gayatri Vidya Parishad College of Engineering
More informationAdaptive 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
More informationNumerical Methods for the Navier-Stokes equations
Arnold Reusken Numerical Methods for the Navier-Stokes equations January 6, 212 Chair for Numerical Mathematics RWTH Aachen Contents 1 The Navier-Stokes equations.............................................
More informationA SHORT INTRODUCTION TO TWO-PHASE FLOWS Two-phase flows balance equations
A SHORT INTRODUCTION TO TWO-PHASE FLOWS Two-phase flows balance equations Hervé Lemonnier DM2S/STMF/LIEFT, CEA/Grenoble, 38054 Grenoble Cedex 9 Ph. +33(0)4 38 78 45 40 herve.lemonnier@cea.fr, herve.lemonnier.sci.free.fr/tpf/tpf.htm
More informationContents. Preface XIII. 1 General Introduction 1 References 6
VII Contents Preface XIII 1 General Introduction 1 References 6 2 Interparticle Interactions and Their Combination 7 2.1 Hard-Sphere Interaction 7 2.2 Soft or Electrostatic Interaction 7 2.3 Steric Interaction
More informationNumerical Study of Blood Flow through Symmetry and Non- Symmetric Stenosis Artery under Various Flow Rates
IOSR Journal of Dental and Medical Sciences (IOSR-JDMS) e-issn: 2279-0853, p-issn: 2279-0861.Volume 16, Issue 6 Ver. I (June. 2017), PP 106-115 www.iosrjournals.org Numerical Study of Blood Flow through
More information