BEM for compressible fluid dynamics
|
|
- Eunice Garrett
- 5 years ago
- Views:
Transcription
1 BEM for compressible fluid dynamics L. gkerget & N. Samec Faculty of Mechanical Engineering, Institute of Power, Process and Environmental Engineering, University of Maribor, Slovenia Abstract The fully developed boundary element method (BEM) numerical model of compressible fluid dynamics is presented. In particular, the singular boundary domain integral approach, which has been established for the viscous incompressible flow problem, is modified and extended to capture the compressible fluid state. As test cases a natural convection of compressible fluid in a closed cavity and an L-shape cavity are studied. 1 Introduction This paper deals with a BEM numerical scheme developed for simulation of motion of compressible viscous fluid. The method is based on the approximate solution of the set of Navier-Stokes equations in the velocity-vorticity formulation. Particular attention is given to proper transformation of the governing differential equations into corresponding integral representations, which satisfy the continuity equation exactly, e.g. velocity and mass density field functions. The pressure field is computed by using the Poisson equation for pressure for known velocity, vorticity and mass density functions. 2 Conservation laws The analytical description of the continuous homogenous medium motion is based on conservation of mass, momentum and energy with associated rheological models and equation of state. The present development will be focused on laminar flow of compressible viscous isotropic fluid in solution domain $2 bounded by boundary T. The field functions of interest are velocity
2 124 Boutrdary Elcmatr~~ XXV vector field vi(rj,t), pressure field p(rj,t), mass density field p(rj,t) and temperature field T(rjt) such that the mass, momentum and energy equations are satisfied, written in the Cartesian tensor notation xi, where p and c,, are fluid mass density and isobaric specific heat per unit mass, t is time, gi is gravitational acceleration vector, qi denotes the components of the total stress tensor, g; stands for heat flux vector. Using the Stokes mass time derivative in general form --- ~t at the following alternative form of the conservation laws can be stated where c is the isobaric specific heat per unit volume, c -- cp P. For an incompressible fluid, the rate of change of mass density following the motion is zero, that is and the mass conservation equation takes simple form V4=0, expressing the solenoidality constraint for the velocity vector. The set of field eqns (4)-(6) has to be closed and solved in conjunction with appropriate rheological models of the fluid and boundary and initial conditions
3 of the flow problem. Boundary conditions in general depend on the dependent variables applied, i.e. primitive or velocity-vorticity variables formulation. For a compressible fluid, the Cauchy total stress au can be decomposed into a pressure contribution plus an extra deviatoric stress tensor field function 0.. = -p6.. +r.. 1 'I II ' (9) where fiij is the Kronecker delta function. 3 Rheological models In general, real fluid in motion sustains shear stresses. The most general relationship between the extra stress tensor qj and the strain rate tensor E, is given by the Reiner-Rivlin model r0 = a6,i + P&ii + y2ik&kj, (10) where the coefficients a, p and y are functions of three scalar invariants of strain rate tensor Eij. For a simple viscous shear compressible fluid in motion one can consider the relations a = -2qD / 3 and P = 2q, such that the following constitutive model can be stated n where quantity D = div6 = Cii, and r] is dynamic viscosity. For a most heat transfer problems of practical importance Fouier model of heat diffusion is accurate enough where k is heat conductivity. 4 Summary of governing equations Combining constitutive models for stress tensor and heat diffusion flux, eqns (1 1) and (12) in conservation eqns (5) and (6) the following system of nonlinear equations is developed
4 126 Bnurzdary Elcmatzt~ XXV Using an extended form of the operator divz, i.e. the momentum eqn (14) can be written in the form appropriate for development of the velocity-vorticity formulation DU 4 p- = -grad p -I- pi - rot(r]g)+ -grad(q~)+ 2grad6 gradv Dt 3 (17) +2 grad 17 X c grad 7. 5 Velocity-vorticity formulation The divergence and the curl of a vector field function are fundamental differential operators in vector analysis. Applied to the velocity vector field they give local rate of expansion D and local vorficity vectoor representing a solenoidal vector by definition, the fluid motion computation procedure is partitioned into its kinetics and kinematics [l]. The vorticity transport in fluid domain is governed by non-linear parabolic difhsive-convective equation obtained as a curl ot the momentum eqn (17), i.e. written in general vector form D ac =-+(~V)CZ=V,AW+(U~~V)~~-WD+-VXP, (20) ~t at PO or in Cartesian notation form The pseudo body force vector Fincludes the effects of variable material properties F=~"(~-Z)-~~VXG-GXV~+~VU.V~-~DV~, (22) the following tensor notation form is also valid
5 where a = DV l Dt represents acceleration vector. For the two dimensional plane notation the vorticity vector i7i has just one component perpendicular to the plane of the flow, and it can be treated as a scalar field function. The stretching-twisting term is identically zero, reducing the vector vorticity eqn (20) to a scalar one for the vorticity w where pseudo body force term is - am a17 avi aq F;. = p(gi -a,)-r7e..-+e..m d-, 317 (25) rj lj axj axj ax, axi or by putting together last three terms represented also as In eqns above, the material properties are considered as a sum of constant and variable part, i.e. 17 =% +ff and p=po+p. (27) Applying the curl operator to the vorticity deffinition PX~~~=TX(V~V)=V(V.~)-A~~, (28) and by using the continuity eqn (13), the following elliptic Poisson eqn is obtained AU+VX~~~-VD=O, (29) or in tensor notation form a *V, am, m -0 +eij, a.,' axj axi The eqn (30) represents the kinematics of a compressible fluid motion expressing the compatibility and restriction conditions between velocity and solenoidal vorticity vector field functions at a given point in space and time. TO accelerate the convergence of the coupled velocity-vorticity iterative scheme the false transient approach is applied. Thus, in the solution scheme the eqn (30) is rewritten as parabolic diffusion eqn for velocity vector with a as a relaxation parameter. It is obvious that the governing velocity eqn (30) is exactly satisfied only at the steady state (t+-) when the false time derivative or false accumulation term vanishes.
6 6 Pressure equation Let us rewrite momentum cqn (17) for pressure gradient gradp vp=fp =P(X-ri)-~x(rlui)+~~(rl~)+2~~~~0+2~~xoi-2~~ (32) 3 whcrc in vector function fr, inertia, gravitational, diffusion and non-linear malcrial effects are incorporated. To derive pressure equation dependent on known field function values the divergence of eqn (32) should bc considered A~-V.&, =O. (33) Considering the normal component of the eqn (32) thc Neuman pressure boundary conditions ar specified. 7 Integral representations The unique advantage of BEM originates from the application of Green fundamental solutions as particular weighting functions. Sincc they only consider the linear transport phenomena, an appropriate selection of a linear differential operator is of main importance in establishing stable and accurate singular integral representations of thc original differential conservation equations. 7.1 Kinematics Consider an integral representation of false transient velocity eqn (31), which can be recognized as a non-homogenous parabolic PDE of the form the following corresponding boundary-domain integral eqn can be obtained by applying weighted residual statement, e.g. written in a time incremental form with a timc step At = tf - tf-, where U* is the parabolic diffusion fundamental solution
7 Equating pseudo body force term b with rotational and compressible part of fluid motion and assuming constant variation of all field functions within individual time increment, one can derive the following integral statement c(<@({,r,)+ J(vu* = ~(vu* xii)xi#+ JGXVU*~Q r I- R R t~ where U* =a ju*dt DVU*~~ + J"F-lu;-,d~, t~-l The kinematics of planar fluid motion is given by R (37) Eqn (37) is equivalent to continuity equation also recognized as compatibility and restriction conditions between velocity and mass density field functions, and vorticity definition expressing the kinematics of general compressible fluid motion in the integral form. Velocity boundary conditions are incorporated in the boundary integrals, while the first two domain integrals express the influence of the local vorticity and expansion rate on the velocity field. The last domain integral is taking into account the influence of initial velocity conditions of false transient phenomena. In eqn (37) normal and tangential derivative of fundamental solution au * /an and au * /at are employed. To compute boundary values of field fuctions normal or tangential form of vector eqn (37) [2] is demanded. The boundary vorticity values are expressed in integral form within the domain integral. One has to use tangential component of eqn (37) to determine the boundary vorticity values ~(<~(~)xc(<.t,)+n(~)xj(vu*.ri)im= r 7.2 Kinetics Considering the vorticity kinetics in an integral representation one has to consider the parabolic diffusive-convective character of the vorticity eqn. Since
8 130 Boutrdary Elcmatr~~ XXV only the linear parabolic diffusion differential operator is employed in this work, the vorticity eqn can be formulated as a non-homogenous parabolic diffusion eqn as follows with the following integral representation wher U* is again the parabolic diffusion Green function. The domain integral in eqn (41) of the non-homogenous non-linear contribution b, represented as includes the convection and effects of variable material properties. Assuming again constant variation of all field functions within the individual time increment, the final integral statement reads as 1 1 au* -dq 770 r. '70 a axj --~pwvjnj(l*~+-jpm Applying similar procedure to heat transport equation, one derives the following integral representation
9 7.3 Pressure equation Pressure eqn (33) is recognized as an elliptic Poisson equation, thus the following can be stated with the corresponding singular integral representation where U* is the Laplace fundamental solution. By equating body forces with the expression the following final integral statement can be obtained For known Neuman boundary conditions, given by eqn (34), scalar pressure field function can be computed by solving eqn (33) or (48) for known velocity and vorticity fields and material properties for a given instant of time. The solution to eqn (34) is written in an explicit manner of boundary and domain integrals. 8 Computational scheme If one is to solve singular boundary-domain integral representations to get values of field functions one has to transform the derived integral eqn into its discrete algebraic forms. The key to this is partitioning of computational external boundary into boundary elements and interior domain into domain cells [3]. Use of Green fundamental solution results in boundary discretization of linear transport phenomena part, while internal cells consider non-linear transport phenomena part. In the present work, quadratic interpolation functions are used for boundary elements and internal cells. 9 Numerical solution 9.1 Natural convection in closed cavity As a first numerical example natural convection in a closed cavity (shown in Fig. 1 with corresponding boundary conditions) is examined for ~a=10~. The numerical simulation results (velocity components v, and V, at x=0.5 and y=0.5) for the full compressible form of Navier-Stokes eqns are compared with those based on the Bussinesq approximation of Navier-Stokes eqn in Fig. 2 and Fig. 3
10 132 Boutrdary Elcmatr~~ XXV Figure l: Closed cavity with corresponding boundary conditions coordinates of the velocity components comparisons. and Figure 2: Velocity component v, of compressible (com) and non-compressible (ncom) fluid flow at x=0.5 in closed cavity. The comparison of overall Nusselt number values obtained by De Vahl Davis et al. [4], [S] and BEM for Bussinesq approximation (B) and full compressible flow (C) case is shown in Table 1 for I?a=1o4,
11 X Figure 3: Velocity component V, of compressible (corn) and non-compressible (ncom) fluid flow at y=0.5 in closed cavity. Tablc I : Comparison of Nusselt number values. Dav~s (B) BEM (B) BEM (C) fi I Figure4: L-shaped cavity with corresponding boundary conditions and coordinates of the velocity components comparisons.
12 9.2 Natural convection in an L-shaped cavity Natural convection in an L- shaped cavity (shown in Fig. 4) has been applied as the second example case to analyze the differences between velocity field of compressible and non-compressible fluid flow. Velocity components at x=0.75 and y=0.75 have been compared in Figs 5 and Vx Figure 5: Velocity component v, of compressible (com) and non-compressible (ncom) fluid flow at x=0.75 in L-shaped cavity. Figure 6: Velocity components vy of compressible (corn) and non-compressible (ncom) fluid flow at y=0.75 in L-shaped cavity.
13 10 Conclusions BEM numerical model of the compressible fluid dynamics has been developed successfully. In particular, singular boundary domain integral approach, which has been established for viscous incompressible flow problem, is modified and extended to capture compressible fluid. As test cases, natural convection of compressible fluid in closed cavity and L-shaped cavity are studied at ~a=10~. In addition, developed model represents good basis for BEM model currently under development for multi-component compressible reacting flows. References [l] Skerget, L., HriberSek, M., Kuhn, G. Computational fluid dynamic by boundary-domain integral method. Int. J. Numer. Meth. Engnng., 46, [2] $kerget, L. AlujeviE, A., Brebbia, CA., Kuhn, G. Matural and forced convection simulation using the velocity-vorticity approach. Topics in Boundary Element Research, [3] Wrobel, L.C. The boundary element method. Vol l., Applications in thermo-fluids and acoustics, Wiely, [4] Davis, G.D.V. Natural convection in a square cavity: A bench mark numerical solution. Int. Jou. for Num. Meth. in Fluids, 3, 1983, [5] Davis, G.D.V. Natural convection in a square cavity: A comparison exercise. Int. Jou. for Num. Meth. in Fluids, 3, 1983,
14
BEM for turbulentfluidflow. L. Skerget & M. Hribersek Faculty of Mechanical Engineering, University ofmaribor, Slovenia.
BEM for turbulentfluidflow L. Skerget & M. Hribersek Faculty of Mechanical Engineering, University ofmaribor, Slovenia. Abstract Boundary-Domain Integral Method is applied for the solution of incompressible
More informationFluid flow in a channel partially filled with porous material
Fluid flow in a channel partially filled with porous material M. Hriberkk', R. Jec12& L. Skerget' 'Faculty of Mechanical Engineering, University of Maribor, Smetanova 17, SI-2000 Maribor, Slovenia 2Facultyof
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 informationNavier-Stokes Equation: Principle of Conservation of Momentum
Navier-tokes Equation: Principle of Conservation of Momentum R. hankar ubramanian Department of Chemical and Biomolecular Engineering Clarkson University Newton formulated the principle of conservation
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 informationGetting started: CFD notation
PDE of p-th order Getting started: CFD notation f ( u,x, t, u x 1,..., u x n, u, 2 u x 1 x 2,..., p u p ) = 0 scalar unknowns u = u(x, t), x R n, t R, n = 1,2,3 vector unknowns v = v(x, t), v R m, m =
More informationCHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION
CHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION 7.1 THE NAVIER-STOKES EQUATIONS Under the assumption of a Newtonian stress-rate-of-strain constitutive equation and a linear, thermally conductive medium,
More informationPDE 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
More informationEntropy generation and transport
Chapter 7 Entropy generation and transport 7.1 Convective form of the Gibbs equation In this chapter we will address two questions. 1) How is Gibbs equation related to the energy conservation equation?
More informationReview of fluid dynamics
Chapter 2 Review of fluid dynamics 2.1 Preliminaries ome basic concepts: A fluid is a substance that deforms continuously under stress. A Material olume is a tagged region that moves with the fluid. Hence
More informationINDEX 363. Cartesian coordinates 19,20,42, 67, 83 Cartesian tensors 84, 87, 226
INDEX 363 A Absolute differentiation 120 Absolute scalar field 43 Absolute tensor 45,46,47,48 Acceleration 121, 190, 192 Action integral 198 Addition of systems 6, 51 Addition of tensors 6, 51 Adherence
More informationSoft 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
More informationDetailed Outline, M E 521: Foundations of Fluid Mechanics I
Detailed Outline, M E 521: Foundations of Fluid Mechanics I I. Introduction and Review A. Notation 1. Vectors 2. Second-order tensors 3. Volume vs. velocity 4. Del operator B. Chapter 1: Review of Basic
More informationOpenFOAM 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 {}}{
More informationContents. Part I Vector Analysis
Contents Part I Vector Analysis 1 Vectors... 3 1.1 BoundandFreeVectors... 4 1.2 Vector Operations....................................... 4 1.2.1 Multiplication by a Scalar.......................... 5 1.2.2
More informationChapter 2: Fluid Dynamics Review
7 Chapter 2: Fluid Dynamics Review This chapter serves as a short review of basic fluid mechanics. We derive the relevant transport equations (or conservation equations), state Newton s viscosity law leading
More informationFluid Dynamics: Theory, Computation, and Numerical Simulation Second Edition
Fluid Dynamics: Theory, Computation, and Numerical Simulation Second Edition C. Pozrikidis m Springer Contents Preface v 1 Introduction to Kinematics 1 1.1 Fluids and solids 1 1.2 Fluid parcels and flow
More information2.29 Numerical Fluid Mechanics Fall 2011 Lecture 7
Numerical Fluid Mechanics Fall 2011 Lecture 7 REVIEW of Lecture 6 Material covered in class: Differential forms of conservation laws Material Derivative (substantial/total derivative) Conservation of Mass
More informationCHAPTER 8 ENTROPY GENERATION AND TRANSPORT
CHAPTER 8 ENTROPY GENERATION AND TRANSPORT 8.1 CONVECTIVE FORM OF THE GIBBS EQUATION In this chapter we will address two questions. 1) How is Gibbs equation related to the energy conservation equation?
More informationChapter 1. Continuum mechanics review. 1.1 Definitions and nomenclature
Chapter 1 Continuum mechanics review We will assume some familiarity with continuum mechanics as discussed in the context of an introductory geodynamics course; a good reference for such problems is Turcotte
More informationContents. I Introduction 1. Preface. xiii
Contents Preface xiii I Introduction 1 1 Continuous matter 3 1.1 Molecules................................ 4 1.2 The continuum approximation.................... 6 1.3 Newtonian mechanics.........................
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 informationConservation 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
More informationTHREE-DIMENSIONAL DOUBLE-DIFFUSIVE NATURAL CONVECTION WITH OPPOSING BUOYANCY EFFECTS IN POROUS ENCLOSURE BY BOUNDARY ELEMENT METHOD
J. Kramer et al., Int. J. Comp. Meth. and Exp. Meas., Vol. 1, No. (013) 103 115 THREE-DIMENSIONAL DOUBLE-DIFFUSIVE NATURAL CONVECTION WITH OPPOSING BUOYANCY EFFECTS IN POROUS ENCLOSURE BY BOUNDARY ELEMENT
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 informationNumerical 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:
More information3. FORMS OF GOVERNING EQUATIONS IN CFD
3. FORMS OF GOVERNING EQUATIONS IN CFD 3.1. Governing and model equations in CFD Fluid flows are governed by the Navier-Stokes equations (N-S), which simpler, inviscid, form is the Euler equations. For
More informationNatural Convection in Parabolic Enclosure Heated from Below
www.ccsenet.org/mas Modern Applied Science Vol. 5, No. 3; June 011 Natural Convection in Parabolic Enclosure Heated from Below Dr. Ahmed W. Mustafa (Corresponding auther) University of Tikrit, College
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 informationGame 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
More informationCH.9. CONSTITUTIVE EQUATIONS IN FLUIDS. Multimedia Course on Continuum Mechanics
CH.9. CONSTITUTIVE EQUATIONS IN FLUIDS Multimedia Course on Continuum Mechanics Overview Introduction Fluid Mechanics What is a Fluid? Pressure and Pascal s Law Constitutive Equations in Fluids Fluid Models
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 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 informationFluid Dynamics Exercises and questions for the course
Fluid Dynamics Exercises and questions for the course January 15, 2014 A two dimensional flow field characterised by the following velocity components in polar coordinates is called a free vortex: u r
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 informationEKC314: TRANSPORT PHENOMENA Core Course for B.Eng.(Chemical Engineering) Semester II (2008/2009)
EKC314: TRANSPORT PHENOMENA Core Course for B.Eng.(Chemical Engineering) Semester II (2008/2009) Dr. Mohamad Hekarl Uzir-chhekarl@eng.usm.my School of Chemical Engineering Engineering Campus, Universiti
More informationCourse Syllabus: Continuum Mechanics - ME 212A
Course Syllabus: Continuum Mechanics - ME 212A Division Course Number Course Title Academic Semester Physical Science and Engineering Division ME 212A Continuum Mechanics Fall Academic Year 2017/2018 Semester
More informationAPPH 4200 Physics of Fluids
APPH 4200 Physics of Fluids Review (Ch. 3) & Fluid Equations of Motion (Ch. 4) September 21, 2010 1.! Chapter 3 (more notes) 2.! Vorticity and Circulation 3.! Navier-Stokes Equation 1 Summary: Cauchy-Stokes
More informationBasic 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
More informationModule 1: Introduction to Finite Difference Method and Fundamentals of CFD Lecture 1: Finite Difference Method
file:///d:/chitra/nptel_phase2/mechanical/cfd/lecture1/1_1.htm 1 of 1 6/19/2012 4:29 PM The Lecture deals with: Classification of Partial Differential Equations Boundary and Initial Conditions Finite Differences
More information2.29 Numerical Fluid Mechanics Fall 2011 Lecture 5
.9 Numerical Fluid Mechanics Fall 011 Lecture 5 REVIEW Lecture 4 Roots of nonlinear equations: Open Methods Fixed-point Iteration (General method or Picard Iteration), with examples Iteration rule: x g(
More information3 2 6 Solve the initial value problem u ( t) 3. a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1
Math Problem a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1 3 6 Solve the initial value problem u ( t) = Au( t) with u (0) =. 3 1 u 1 =, u 1 3 = b- True or false and why 1. if A is
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 informationMath 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
More informationReview of Vector Analysis in Cartesian Coordinates
Review of Vector Analysis in Cartesian Coordinates 1 Scalar: A quantity that has magnitude, but no direction. Examples are mass, temperature, pressure, time, distance, and real numbers. Scalars are usually
More information( ) Notes. Fluid mechanics. Inviscid Euler model. Lagrangian viewpoint. " = " x,t,#, #
Notes Assignment 4 due today (when I check email tomorrow morning) Don t be afraid to make assumptions, approximate quantities, In particular, method for computing time step bound (look at max eigenvalue
More informationRelativistic Hydrodynamics L3&4/SS14/ZAH
Conservation form: Remember: [ q] 0 conservative div Flux t f non-conservative 1. Euler equations: are the hydrodynamical equations describing the time-evolution of ideal fluids/plasmas, i.e., frictionless
More informationAA210A Fundamentals of Compressible Flow. Chapter 1 - Introduction to fluid flow
AA210A Fundamentals of Compressible Flow Chapter 1 - Introduction to fluid flow 1 1.2 Conservation of mass Mass flux in the x-direction [ ρu ] = M L 3 L T = M L 2 T Momentum per unit volume Mass per unit
More information1 Exercise: Linear, incompressible Stokes flow with FE
Figure 1: Pressure and velocity solution for a sinking, fluid slab impinging on viscosity contrast problem. 1 Exercise: Linear, incompressible Stokes flow with FE Reading Hughes (2000), sec. 4.2-4.4 Dabrowski
More informationPrimitive variable dual reciprocity boundary element method solution of incompressible Navier Stokes equations
Engineering Analysis with Boundary Elements 23 (1999) 443 455 Primitive variable dual reciprocity boundary element method solution of incompressible Navier Stokes equations Božidar Šarler a, *,ünther Kuhn
More informationComputational Fluid Dynamics Prof. Sreenivas Jayanti Department of Computer Science and Engineering Indian Institute of Technology, Madras
Computational Fluid Dynamics Prof. Sreenivas Jayanti Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture 20 Equations governing fluid flow with chemical reactions
More informationLECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES)
LECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES) RAYTCHO LAZAROV 1 Notations and Basic Functional Spaces Scalar function in R d, d 1 will be denoted by u,
More information7 The Navier-Stokes Equations
18.354/12.27 Spring 214 7 The Navier-Stokes Equations In the previous section, we have seen how one can deduce the general structure of hydrodynamic equations from purely macroscopic considerations and
More informationPartial Differential Equations. Examples of PDEs
Partial Differential Equations Almost all the elementary and numerous advanced parts of theoretical physics are formulated in terms of differential equations (DE). Newton s Laws Maxwell equations Schrodinger
More informationARTICLE IN PRESS. Engineering Analysis with Boundary Elements
Engineering Analysis with Boundary Elements 33 (29) 561 571 Contents lists available at ScienceDirect Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound BEM
More informationCHAPTER 2 THERMAL EFFECTS IN STOKES SECOND PROBLEM FOR UNSTEADY MICROPOLAR FLUID FLOW THROUGH A POROUS
CHAPTER THERMAL EFFECTS IN STOKES SECOND PROBLEM FOR UNSTEADY MICROPOLAR FLUID FLOW THROUGH A POROUS MEDIUM. Introduction The theory of micropolar fluids introduced by Eringen [34,35], deals with a class
More informationComputational Astrophysics
Computational Astrophysics Lecture 1: Introduction to numerical methods Lecture 2:The SPH formulation Lecture 3: Construction of SPH smoothing functions Lecture 4: SPH for general dynamic flow Lecture
More informationIntroduction to Fluid Mechanics
Introduction to Fluid Mechanics Tien-Tsan Shieh April 16, 2009 What is a Fluid? The key distinction between a fluid and a solid lies in the mode of resistance to change of shape. The fluid, unlike the
More informationLecture 8: Tissue Mechanics
Computational Biology Group (CoBi), D-BSSE, ETHZ Lecture 8: Tissue Mechanics Prof Dagmar Iber, PhD DPhil MSc Computational Biology 2015/16 7. Mai 2016 2 / 57 Contents 1 Introduction to Elastic Materials
More informationFluid Mechanics Prof. S. K. Som Department of Mechanical Engineering Indian Institute of Technology, Kharagpur
Fluid Mechanics Prof. S. K. Som Department of Mechanical Engineering Indian Institute of Technology, Kharagpur Lecture - 15 Conservation Equations in Fluid Flow Part III Good afternoon. I welcome you all
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 information2. Conservation Equations for Turbulent Flows
2. Conservation Equations for Turbulent Flows Coverage of this section: Review of Tensor Notation Review of Navier-Stokes Equations for Incompressible and Compressible Flows Reynolds & Favre Averaging
More information2 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)
More informationAE/ME 339. K. M. Isaac Professor of Aerospace Engineering. 12/21/01 topic7_ns_equations 1
AE/ME 339 Professor of Aerospace Engineering 12/21/01 topic7_ns_equations 1 Continuity equation Governing equation summary Non-conservation form D Dt. V 0.(2.29) Conservation form ( V ) 0...(2.33) t 12/21/01
More informationChapter 0. Preliminaries. 0.1 Things you should already know
Chapter 0 Preliminaries These notes cover the course MATH45061 (Continuum Mechanics) and are intended to supplement the lectures. The course does not follow any particular text, so you do not need to buy
More informationComputational Fluid Dynamics-1(CFDI)
بسمه تعالی درس دینامیک سیالات محاسباتی 1 دوره کارشناسی ارشد دانشکده مهندسی مکانیک دانشگاه صنعتی خواجه نصیر الدین طوسی Computational Fluid Dynamics-1(CFDI) Course outlines: Part I A brief introduction to
More informationChapter 6: Incompressible Inviscid Flow
Chapter 6: Incompressible Inviscid Flow 6-1 Introduction 6-2 Nondimensionalization of the NSE 6-3 Creeping Flow 6-4 Inviscid Regions of Flow 6-5 Irrotational Flow Approximation 6-6 Elementary Planar Irrotational
More informationTwo-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
More informationKINEMATICS OF CONTINUA
KINEMATICS OF CONTINUA Introduction Deformation of a continuum Configurations of a continuum Deformation mapping Descriptions of motion Material time derivative Velocity and acceleration Transformation
More informationViscous Fluids. Amanda Meier. December 14th, 2011
Viscous Fluids Amanda Meier December 14th, 2011 Abstract Fluids are represented by continuous media described by mass density, velocity and pressure. An Eulerian description of uids focuses on the transport
More informationProject 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
More informationLecture 3: 1. Lecture 3.
Lecture 3: 1 Lecture 3. Lecture 3: 2 Plan for today Summary of the key points of the last lecture. Review of vector and tensor products : the dot product (or inner product ) and the cross product (or vector
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 informationINTRODUCTION TO PDEs
INTRODUCTION TO PDEs In this course we are interested in the numerical approximation of PDEs using finite difference methods (FDM). We will use some simple prototype boundary value problems (BVP) and initial
More informationFINITE ELEMENT ANALYSIS OF MIXED CONVECTION HEAT TRANSFER ENHANCEMENT OF A HEATED SQUARE HOLLOW CYLINDER IN A LID-DRIVEN RECTANGULAR ENCLOSURE
Proceedings of the International Conference on Mechanical Engineering 2011 (ICME2011) 18-20 December 2011, Dhaka, Bangladesh ICME11-TH-014 FINITE ELEMENT ANALYSIS OF MIXED CONVECTION HEAT TRANSFER ENHANCEMENT
More informationCartesian Tensors. e 2. e 1. General vector (formal definition to follow) denoted by components
Cartesian Tensors Reference: Jeffreys Cartesian Tensors 1 Coordinates and Vectors z x 3 e 3 y x 2 e 2 e 1 x x 1 Coordinates x i, i 123,, Unit vectors: e i, i 123,, General vector (formal definition to
More informationCartesian Tensors. e 2. e 1. General vector (formal definition to follow) denoted by components
Cartesian Tensors Reference: Jeffreys Cartesian Tensors 1 Coordinates and Vectors z x 3 e 3 y x 2 e 2 e 1 x x 1 Coordinates x i, i 123,, Unit vectors: e i, i 123,, General vector (formal definition to
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 informationElements of Rock Mechanics
Elements of Rock Mechanics Stress and strain Creep Constitutive equation Hooke's law Empirical relations Effects of porosity and fluids Anelasticity and viscoelasticity Reading: Shearer, 3 Stress Consider
More information2 GOVERNING EQUATIONS
2 GOVERNING EQUATIONS 9 2 GOVERNING EQUATIONS For completeness we will take a brief moment to review the governing equations for a turbulent uid. We will present them both in physical space coordinates
More informationSeveral forms of the equations of motion
Chapter 6 Several forms of the equations of motion 6.1 The Navier-Stokes equations Under the assumption of a Newtonian stress-rate-of-strain constitutive equation and a linear, thermally conductive medium,
More informationexamples of equations: what and why intrinsic view, physical origin, probability, geometry
Lecture 1 Introduction examples of equations: what and why intrinsic view, physical origin, probability, geometry Intrinsic/abstract F ( x, Du, D u, D 3 u, = 0 Recall algebraic equations such as linear
More information1. Introduction, tensors, kinematics
1. Introduction, tensors, kinematics Content: Introduction to fluids, Cartesian tensors, vector algebra using tensor notation, operators in tensor form, Eulerian and Lagrangian description of scalar and
More informationFluid Mechanics Abdusselam Altunkaynak
Fluid Mechanics Abdusselam Altunkaynak 1. Unit systems 1.1 Introduction Natural events are independent on units. The unit to be used in a certain variable is related to the advantage that we get from it.
More informationThe 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
More information2.20 Fall 2018 Math Review
2.20 Fall 2018 Math Review September 10, 2018 These notes are to help you through the math used in this class. This is just a refresher, so if you never learned one of these topics you should look more
More information3.5 Vorticity Equation
.0 - Marine Hydrodynamics, Spring 005 Lecture 9.0 - Marine Hydrodynamics Lecture 9 Lecture 9 is structured as follows: In paragraph 3.5 we return to the full Navier-Stokes equations (unsteady, viscous
More informationBenchmark solutions for the natural convective heat transfer problem in a square cavity
Benchmark solutions for the natural convective heat transfer problem in a square cavity J. Vierendeels', B.Merci' &L E. Dick' 'Department of Flow, Heat and Combustion Mechanics, Ghent University, Belgium.
More informationQuick Recapitulation of Fluid Mechanics
Quick Recapitulation of Fluid Mechanics Amey Joshi 07-Feb-018 1 Equations of ideal fluids onsider a volume element of a fluid of density ρ. If there are no sources or sinks in, the mass in it will change
More informationAA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 1 / 29 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS Hierarchy of Mathematical Models 1 / 29 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 2 / 29
More informationPressure-velocity correction method Finite Volume solution of Navier-Stokes equations Exercise: Finish solving the Navier Stokes equations
Today's Lecture 2D grid colocated arrangement staggered arrangement Exercise: Make a Fortran program which solves a system of linear equations using an iterative method SIMPLE algorithm Pressure-velocity
More informationfluid 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
More informationContinuum Mechanics Fundamentals
Continuum Mechanics Fundamentals James R. Rice, notes for ES 220, 12 November 2009; corrections 9 December 2009 Conserved Quantities Let a conseved quantity have amount F per unit volume. Examples are
More informationIntroduction. Finite and Spectral Element Methods Using MATLAB. Second Edition. C. Pozrikidis. University of Massachusetts Amherst, USA
Introduction to Finite and Spectral Element Methods Using MATLAB Second Edition C. Pozrikidis University of Massachusetts Amherst, USA (g) CRC Press Taylor & Francis Group Boca Raton London New York CRC
More informationAE/ME 339. Computational Fluid Dynamics (CFD) K. M. Isaac. Momentum equation. Computational Fluid Dynamics (AE/ME 339) MAEEM Dept.
AE/ME 339 Computational Fluid Dynamics (CFD) 9//005 Topic7_NS_ F0 1 Momentum equation 9//005 Topic7_NS_ F0 1 Consider the moving fluid element model shown in Figure.b Basis is Newton s nd Law which says
More informationMaxwell's Equations and Conservation Laws
Maxwell's Equations and Conservation Laws 1 Reading: Jackson 6.1 through 6.4, 6.7 Ampère's Law, since identically. Although for magnetostatics, generally Maxwell suggested: Use Gauss's Law to rewrite continuity
More informationASTR 320: Solutions to Problem Set 2
ASTR 320: Solutions to Problem Set 2 Problem 1: Streamlines A streamline is defined as a curve that is instantaneously tangent to the velocity vector of a flow. Streamlines show the direction a massless
More informationThe Kinematic Equations
The Kinematic Equations David Roylance Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge, MA 0139 September 19, 000 Introduction The kinematic or strain-displacement
More informationin this web service Cambridge University Press
CONTINUUM MECHANICS This is a modern textbook for courses in continuum mechanics. It provides both the theoretical framework and the numerical methods required to model the behavior of continuous materials.
More informationEffects of Viscous Dissipation on Unsteady Free Convection in a Fluid past a Vertical Plate Immersed in a Porous Medium
Transport in Porous Media (2006) 64: 1 14 Springer 2006 DOI 10.1007/s11242-005-1126-6 Effects of Viscous Dissipation on Unsteady Free Convection in a Fluid past a Vertical Plate Immersed in a Porous Medium
More information