Computational Fluid Dynamics

Transcription

1 Computational Fluid Dynamics Dr.Eng. Reima Iwatsu Phone: NACO Building Room Time Summer Term Lecture: Tuesday 7:30-9:00 (every two weeks) LG4/310 Exercise: Tuesday 7:30-9:00 (every four weeks)lg4/310 Evaluation: 10% Attendance 90% Exercise and Report 90% Speeking time Tuesday 9:00-10:30 Lehrstuhl Aerodynamik und Strömungslehre (LAS) Fakultät 3, Maschinenbau, Elektrotechnik und Wirtschaftsingenieurwesen Brandenburgische Technische Universität Cottbus Karl Liebknecht-Straße 102,D Cottbus

2 Terminplanung für die Vorlesung Computational Fluid Dynamics (Di., 7:30 9:00 Uhr, LG 4 Raum 310) Date Contents of the lecture Introduction The mathematical nature of the flow equations Finite Difference Method (FDM) Finite Element Method (FEM) Finite Volume Method (FVM), Fourier/Spectral method Exercise Time integration, Stability analysis Iterative methods for algebraic systems Convection-diffusion equation Exercise Incompressible Navier Stokes(NS) equations Some remarks on incompressible fows Heat and fluid flow Turbulence model Grid generation Exercise Example CFD results Lecture from Dr.Ristau

3 Contents of the lecture Mathematical Property of the PDEs 1 Introduction 2 The Mathematical Nature of the Flow Equations Various Discretization Method 3 Finite Difference Method (FDM) 4 Finite Element Method (FEM) 5 Finite Volume Method (FVM) 6 Fourier/Spectral Method Numerical Method for Time Marching and System of Equations 7 Time Integration 8 Stability Analysis 9 Iterative Methods for Algebraic Systems 10 Convection-Diffusion Equation Incompressible Flows 11 Incompressible Navier Stokes(NS) Equations 12 Some Remarks on Incompressible Fows 13 Heat and Fluid Flow 14 Turbulence Model Grid Generation / CFD Examples 15 Grid Generation 16 Example CFD Results 17 Lecture on Applicational Computation (Dr. Ristau)

4 Contents of the lecture for today 1 Introduction 1.1 Introductnion Motivation Computational Fluid Dynamics: What is it? The Role of CFD in Modern Fluid Dynamics The Objective of This Course 1.2 The Basic Equations of Fluid Dynamics Fluid and Flow Mathematical Model Conservation Law The Continuity Equation The Momentum Equation: Navier-Stokes Equations The Energy Equation Thermodynamic Considerations Submodel 2 The Mathematical Nature of the Flow Equatnions 2.1 Linear Partical Differential Equations(PDEs) Classification of the Second Order Linear PDEs General Behaviour of the Different Classes of PDEs 2.2 The Dynamic Levels of Approximation Inviscid Flow Model: Euler Equations Parabolized Navier-Stokes Equations, Boundary Layer Approximation Potential Flow Model, Incompressible Fluid Flow Model

5 1.1 Introduction Motivation: Why should you be motivated to learn CFD? The flowfield over a supersonic blunt-nosed body Artist's conception of next generation supersonic aircraft

6 Vehicle aerodynamics, combustion and DNS of turbulence

7 Some More Examples

8 Computational Fluid Dynamics Computational Fluid Dynamics (CFD) is a discipline that solves a set of equations governing the fluid flow over any geometrical configuration. The equations can represent steady or unsteady, compressible or incompressible, and inviscid or viscous flows, including nonideal and reacting fluid behavior. The particular form chosen depends on the intended application. The state of the art is characterized by the complexity of the geometry, the flow physics, and the computer time required to obtain a solution. Fluid Mechanics and Acoustics Division NASA Langley Research Center in Hampton, VA.

9 Computational Fluid Dynamics: What is it? The physical aspects of any fluid flow are governed by the following fundamental principles: (a) mass is conserved (b) F = ma (Newton s second law) (c) energy is conserved. These fundamental principles are expressed in terms of mathematical equations (partial differential equations). CFD is the art of replacing the governing equations of fluid flow with numbers, and advancing these numbers in space and/or time to obtain a final numerical description of the complete flow field of interest. The high-speed digital computer has allowed the practical growth of CFD.

10 The Role of CFD in Modern Fluid Dynamics Pure experiment Pure theory Computational Fluid dynamics Fig. The three dimensions of fluid dynamics A new third approach. Equal partner, but never replace either.

11 The Objective of This Course To whom: The completely uninitiated student To provide what: (a) an understanding of the governing equations (b) some insight into the philosophy of CFD (c) a familiarity with various popular solution techniques; (d) a working vocabulary in the discipline At the conclusion of this course: will be well prepared to understand the literature in this field, to follow more advanced lecture series, and to begin the application of CFD to your special areas of concern. I hope...

12 1.2 The Basic Equations of Fluid Dynamics Fluid and Flow Gas, air, fuel, oxigen, CO 2 Fluid: Liquid water, oil, liquid metal that flows (Solid) (geophysical flows)

13 Eulerian description of the flow Eulerian and Lagrangian framework the velocity of the fluid u at the position x at time t u (x, t ) u (x, t ) Lagrangian labels the state and motion of the point particles a = (a 1,a 2,a 3 ) Lagrangian labels u (x, t ) = u ( X (a, t ), t ) = U (a, t ) b a U (a, 0 ) U (a, t )

14 Finite Control Volume Mathematical Model A closed volume in a finite region of the flow: a controle volume V; a controle surface S, closed surface which bounds the volume Control surface S S Control volume V V fixd in space moving with the fluid Infinitesimal Fluid Element An infinitesimally small fluid element with a volume dv volume dv

15 The Material Derivative The velocity of a point particle = the rate of change of its position X substantial, material, or convective derivative and is denoted by D/Dt The Lagrangian set (a, t ) and the Eulerian set (x, t ) The relationship between the partial derivatives of a function f The chain rule (material derivative) (temporal & spatial derivatives with respect to the Eulerian variables)

16 Control Volume Fixed in Space The Continuity Equation Physical principle: Mass is conserved. (Integral form) Gauss Divergence Theorem

17 The Continuity Equation (continued) Control Volume Moving with the Fluid The relationship between the divergence of Vand dv

18 The Continuity Equation (summary) The meaning of divergence V

19 The Momentum equation Physical principle: F = m a (Newton s second law) Navier Stokes Equations

20 The Momentum equation

21 The Energy Equation Physical principle: Conservation of energy E: total energy (= e + V 2 /2 ), e: internal energy

22 Thermodynamic Considerations Unknown flow-field variables: /rho, p, u, v, w, E( or e) Closure conditions for state variables ( e: specific internal energy, s: entropy, p: pressure, /rho: density, T: temperature) Equation of State Ideal Gas (a perfect gas) pv = nr T ( p = rho R T ) (1) V: volume, n:number of kilomoles, R=8.134 KJ: Universal Gas Constant, T: temperature [K]. pv = RT, R = R / w( p = rho RT ) (1#) v : specific volume (=V/m, V: Volume, m: mass. v=1/ rho ) w: relative molecular mass, m = n w, R: Specific Gas Constant. a thermodynamic relation e = e(t,p) (2) a perfect gas e = c v T (2#) c v : spesific heat at constant volume

23 2 The Mathematical Nature of the Flow Equatnions 2.1 Linear Partial Differential Equations (PDEs) Second Order Linear PDEs Navier Stokes Equations Second Order PDEs

24 2.1.1 Classification of the Second Order Linear PDEs F(x,y) af xx + bf xy + cf yy +df x + ef y + g = 0 b 2 4ac > 0 b 2 4ac = 0 b 2 4ac < 0 hyperbolic parabolic elliptic

25 2.1.2 General Behaviour of the Different Classes of PDEs Hyperbolic Equations Characteristic curves Region II Domain of influence y Left-running characteristic Initial data upon which p depends a P Region I Influenced by point p -Region of influence- b c Right-running characteristic Region influenced by point c x

26 Hyperbolic Equations (x,y,z) y Characteristic surface Initial data in the yz plane upon which p depends Volume which influences point p P Volume influenced by point p x z

27 Parabolic Equations Only one characteristic direction, Marching-type solutions Initial data line y c a Boundary conditions known d P Region influenced by P Boundary conditions known b x y y x=0 x=t

28 No limited regions of influence; Elliptic Equations information is propagated everywhere in all directions (at once). y b c P Boundary conditions a b x A specification of the dependent variables along the boundary. Dirichlet condition A specification of derivatives of the dependent variables along the boundary. Neumann condition A mix of both Dirichlet and Neumann conditions.

29 2.2 The Dynamic Levels of Approximation Inviscid Flow Model: Euler Equations Steady inviscid supersonic flows Wave equation

30 2.2.2 Parabolized Navier-Stokes Equations, Boundary Layer Approximation Unsteady thermal conduction Boundary-layer flows Parabolized viscous flows

31 2.2.3 Potential Flow Model, Incompressible Fluid Flow Model Physical picture consistent with the behavior of elliptic equations Potential Flow: Steady, subsonic, inviscid flow Incompressible Fluid Flow: the Mach number M = V/c 0 Flow over an airfoil Flow over a cylinder cylinder

32 Discretization techniques route map PDFs System of allgebraic equations Finite Difference Method (FDM) Finite Element Method (FEM) Finite Volume Method (FVM) Fourier / Spectral Method Basic derivations Discretization errors Time integration Initial value problem Types of solutions: Explicit and implicit Stability analysis Iterative methods Boundary value problem I.V.&B.V. problem

33 2 Finite Difference Method 2.1 Basic Concept > to discretize the geometric domain > to define a grid > a set of indices (i,j) in 2D, (i,j,k) in 3D > grid node values

34 The definition of a derivative

35 2.2 Approximation of the first derivative Taylor series expansion Expansion at x i+1 Expansion at x i-1 Using eq. at both x i+1 and x i-1

36 Truncation error

37 The Forward (FDS), backward (BDS) and central difference (CDS) Approximations truncating the series Truncation errors >for small spacing the leading term is the dominant one >The order of approximation m, m-th order accuracy Second order approximation

38 2.2.2 Polynomial fitting To fit the function an interpolation curve and differentiate it; Piecewise linear interpolation: FDS, BDS A parabola: A cubic polynomial and a polynomial of degree four: Third order BDS, third order FDS and fourth order CDS

39 2.3 Approximation of the second derivative Approximation from x i+1, x i

40 second derivative

41 2.4 Approximation of mixed derivatives non-orthogonal coordinate system combining the 1D approximations The order of differentiation can be changed

42 2.5 Implementation of boundary conditions

43 2.6 Discretization errors Truncation error (the imbalance due to the truncation of Taylor series) The exact solution of L h Discretization error Relationship between the truncation error and the discretization error Richardson extrapolation for sufficiently small h the exact solution: the exponent p (order of the scheme): Approximation for the discretization error on grid h:

44 3 Finite Element Method

45 3.1 Interpolation function Approximation by linear combinations of basis functions (shape, interpolation or trial functions) Mehods based on defining the interpolation function on the whole domain: trigonometric functions: collocation and spectral methods loccaly defined polynomials: standard finite element methods

46 One dimensional linear function

47 3.2 Method of weighted residuals

48 4.1 Introduction 4 Finite Volume Method

49 4.2 Approximation of surface integrals

50 4.3 Approximation of volume integrals

51 4.4 Interpolation practices

52 Linear interpolation

53 Quadratic Upwind Interpolation (QUICK)

54 Higher-order schemes

55 5.1 Basic concept A discrete Fourier series 5 Spectral Methods Fourier series for the derivative: Method of evaluating the derivative -- Given f(x), use (36) to compute ^f ; -- Compute the Fourier coefficient of df/dx ; ik q ^f (k q ) ; -- Evaluate the series (37) to obtain df/dx at the grid points. ++ higher derivatives; d 2 f/dx 2 ; -k q2^f (k q ). ++ The error in df/dx decreases exponentially with N when N is large. ++ The cost of computing ^f scales with N 2 (expensive!). The method is made practical by a fast method of computing Fourier transform (FFT); N log 2 N.

56 5.2 A Fourier Galerkin method for the wave equation

57 Example

58 Error

59 7 Time integration Unsteady flows Initial value problem (Initial boundary value problem) Steady flows Boundary value problem t Solution at time t=t t=0 time t=0 Initial condition Steady solution Boundary condition Initial value problem Boundary value problem

60 7.1 Methods for Ordinary Differential Equation (ODE) Two-level methods

61 Two-level methods

62 Predictor-Corrector method

63 Adams-Bushforth methods Mutipoint methods

64 The second order Runge-Kutta method Runge-Kutta methods The fourth order Runge-Kutta method

65 Other methods An implicit three-level second order scheme

66 One dimensional convection equation 8 Stability analysis Finite difference equation; forward in time, centered in space

67 Q: Does a solution of FDE converges to the solution of PDE? PDE approximate FDE Solution of PDE 0? Solution of FDE Ans.: Even if we solve the FDE that approximate the PDE appropriately, the solution may not always be the correct approximation the exact solution of PDE.

68 Consistency 8.1 Consistency, stability and convergence PDE 0 FDE Stability Convergence

69 Lax s equivalence theorem PDE consistent FDE Solution of PDE 0 convergent Solution of FDE stable L : linear operator

70 Truncation error

71 8.2 Von Neumann s method Fourier representation of the error on the grid points

72 FTCS method for 1D convection equation Fourier series of the error Amplification factor G G > 1: Unconditionally unstable

73 Forward in time, forward in space(upwind scheme) CFL (Courant Freedrichs, Levey) condition

74 BTCS method (Backward in time, centered in space) G < 1: Unconditionally stable

75 Stability limit of 1D diffusion equation

76 8.3 Hirt s method

77 Matrix form 8.4 The matrix method Spectral radius of the matrix C (maximum eigenvalue of C)

78 9 Iterative methods for algebraic systems

79 Linear equations : matrix form

80 9.0 Direct methods Gauß elimination A = A 21 /A 11 Forward elimination upper triangular matrix Back substitution + The number of operations (for large n) ~ n 3 / 3 (n 2 / 2 in back substitution) + pivoting (not sparse large systems)

81 9.0.2 LU decomposition Solution of Ax = Q (0) Factorization into lower (L) and upper (U) triangular matrices A = LU (1) Into two stages: U x = y (2) L y = Q (3)

82 9.0.3 Tridiagonal matrix Thomas algorithm / Tridiagonal Matrix Algorithm (TDMA) + the number of operations ~ n (cf n 3,Gauß elimination)

83 Iterative methods : Basic concept Matrix representation of the algebraic equation A u = Q (1) After n iterations approximate solution u n, residual r n : r n = Q -A u n (2) The convergence error: e n = u u n (3) Relation between the error and the residual: A e n = r n (4) The purpose: to drive r n to zero. e n 0

84 Iterative scheme Iterative procedure A u = Q M u n+1 = N u n + B (5) Obvious property at convergence : u n+1 = u n A = M N, B = Q (6) More generally, PA = M N, B = PQ (7) P : pre-conditioning matrix An alternative to (5): -M u n M (u n+1 u n ) = B (M N) u n (8) or M d n = r n d n = u n+1 u n : correction

85 9.1 Jacobi, Gauß-Seidel, SOR method Poisson equation u = f (u i+1,j 2 u i,j + u i-1,j )+(u i,j+1 2 u i,j + u i,j-1 )=f i,j h Jacobi method

86 9.1.3 SOR method Gauß-Seidel method

87 9.1.4 SLOR method Red-Black SOR method

88 9.1.5 Zebra line SOR method

89 9.1.6 Incomplete LU decomposition : Stone s method Idea : an approximate LU factorization as the iteration matrix M M = LU = A + N Strongly implicit procedure (Stone) N (non-zero elements on diagonals corresponding to all non-zero diag. of LU ) N u ~ 0 u* NW ~ a ( u W + u N u P ), u* SE ~ a ( u S + u E u P ) a < 1

90 Elliptic problem parabolic problem ADI method trapezoidal rule in time and CDS in space at time step n+1 The last term ~ O((dt) 3 ) alternating direction implicit (ADI) method splitting or approximate factorization methods

91 9.2 Conjugate Gradient (CG) method Non-linear solvers Newton-like methods global methods Minimization problem descent method Steepest descents Conjugate gradient method A: positive definite with p 1 and p 2 conjugate condition number of A preconditioning C -1 AC -1 Cp=C -1 Q

92 9.3 Multi Grid method

93 Spectral view of errors

94 Fourier modes

95 Restriction

96 Interpolation

97 Coarse grid scheme

98 V-cycle, W-cycle & FMV scheme

99 9.4 Non-linear equations and their solution Newton s method linearization new estimate y y = f(x) x 0 o x 1 x

100 Newton s method System of non-linear equations linearization Matrix of the system: the Jacobian The system of equations

101 Picard iteration approach Other techniques Newton s method

102 9.5 Examples

103 Transonic flow over an airfoil

104 11 Incompressible Navier Stokes (NS) equations Incompressible fluid Incompressible fluid flow Compressible fluid flow Ma < 0.3 Ma > 0.3 density variation d ~ Ma 2 Mach number Ma = (v/c) v : velocity, c : sound speed ex. Ma < 0.3 c air (101.3hPa, 300K) ~ 340 m/s, v < 100 m/s ( 360 km/h ), c water ~ 1000 m/s, v < 300 m/s.

105 Dynamic similitude Reynolds number Re = ex. U = 10 m/s, L = 1 m, v = 0.15 St, Re ~ U = 0.1 m/s, L = 100 m, Re ~ ρ ul ul = η ν u : velocity scale, l : length scale, v : kinematic viscosity

106 11.2 The pressure Poisson equation method Governing equations Navier-Stokes equations Explicit Euler method Pressure Poisson equation

107 11.3 The projection method A system of two component equations The pressure P a projection function The projection step (BTCS) Poisson equation with the Neumann boundary condition :

108 11.4 Implicit Pressure-Correction method The momentum equations (implicit method) Discrete Poisson equation for the pressure : Outer iteration (iterations within one time step): Pressure-correction Modification of the pressure field The (tentative) velocity at node P The relation between the velocity and pressure corrections : For convenience, The discretized continuity equation Pressure-correction equation : The (final) corrected velocities and pressure : Common practice; neglect unknowns ~u SIMPLE algorithm more gentle way

109 Implicit Pressure-Correction method Approximate u by a weighted mean of the neighbour values Neglect ~u in the first correction step. The second correction to the velocity u : Approximate ~u by : The second pressure correction equation: Approximate relation between u and p : The coefficient in the pressure-correction equation A A +... And the last term disappears. SIMPLEC algorithm Essentially an iterative method for pressurecorrection equation with the last term treated explicitly; PISO algorithm Pressure-correction with the last term neglected. p correct the velocity field to obtain u m i. The new pressure field is calculated from pressure equation using ~ u m i instead of ~ um* i SIMPLER algorithm (Patankar 1980)

110 Implicit Pressure-Correction method The SIMPLE algorithm does not converge rapidly due to the neglect of ~u in the pressurecorrection equation.it has been found by trial and error that convergence can be improved if : SIMPLEC, SIMPLER and PISO do not need under-relaxation of the pressure-correction. An optimum relation between the under-relaxation factors for v and p : The velocities are corrected by i.e., ~u is neglected. By assuming that the final pressure correction is a p p : By making use of correction equation, expression for a p : If we use the approximation used in SIMPLEC, the equation reduces to : In the absence of any contribution from source terms, if a steady solution is sought, a p = 1 -a v which has been found nearly optimum and yields almost the same convergence rate as SIMPLEC method.

111 11.5 Other methods Streamfunction-vorticity methods Stream function Kinematic equation Vorticity transport equation -NS equations have been replaced by a set of two PDEs. -A problem : the boundary conditions, especially in complex geometries. -The values of the streamfunction at boundaries. -Vorticity at the boundary is not known in advance. -Vorticity is singular at sharp corners.

112 Artificial compressibility methods Artificial continuity equation beta : an artificial compressibility parameter The pseudo-sound speed: should be much much faster than the vorticity spreads criterion on the lowest value of beta. Typical values are in the range between 0.1 and 10. Obviously, should be small.

113 12 Some remarks on incompressible flows