CFD

Computational Fluid Dynamics

Dr.Eng. Reima Iwatsu

Phone: 0355 69 4875

e-mail: [email protected]

NACO Building Room 53-107

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 CottbusKarl Liebknecht-Straße 102,D-03046 Cottbus

Terminplanung für die Vorlesung „Computational Fluid Dynamics“ (Di., 7:30 – 9:00 Uhr, LG 4 Raum 310)

• Date Contents of the lecture• 3. 4. 2001 Introduction

The mathematical nature of the flow equations• 17. 4. 2001 Finite Difference Method (FDM)

Finite Element Method (FEM)Finite Volume Method (FVM), Fourier/Spectral method

• 24. 4. 2001 Exercise• 8. 5. 2001 Time integration, Stability analysis• 22. 5. 2001 Iterative methods for algebraic systems

Convection-diffusion equation• 29. 5. 2001 Exercise• 5. 6. 2001 Incompressible Navier Stokes(NS) equations

Some remarks on incompressible fows• 12. 6. 2001 Heat and fluid flow

Turbulence model Grid generation

• 19. 6. 2001 Exercise• 26. 6. 2001 Example CFD results• 3. 7. 2001 Lecture from Dr.Ristau•

Contents of the lecture

Mathematical Property of the PDEs1 Introduction2 The Mathematical Nature of the Flow EquationsVarious Discretization Method3 Finite Difference Method (FDM)4 Finite Element Method (FEM)5 Finite Volume Method (FVM)6 Fourier/Spectral MethodNumerical Method for Time Marching and System of Equations7 Time Integration8 Stability Analysis9 Iterative Methods for Algebraic Systems10 Convection-Diffusion EquationIncompressible Flows11 Incompressible Navier Stokes(NS) Equations12 Some Remarks on Incompressible Fows13 Heat and Fluid Flow14 Turbulence ModelGrid Generation / CFD Examples15 Grid Generation16 Example CFD Results17 Lecture on Applicational Computation (Dr. Ristau)

• 1 Introduction

Contents of the lecture for today

• 1.1 Introductnion– 1.1.1 Motivation

– 1.1.2 Computational Fluid Dynamics: What is it?

– 1.1.3 The Role of CFD in Modern Fluid Dynamics– 1.1.4 The Objective of This Course

• 1.2 The Basic Equations of Fluid Dynamics– 1.2.1 Fluid and Flow

– 1.2.2 Mathematical Model

– 1.2.3 Conservation Law– 1.2.4 The Continuity Equation

– 1.2.5 The Momentum Equation: Navier-Stokes Equations– 1.2.6 The Energy Equation

– 1.2.7 Thermodynamic Considerations– 1.2.8 Submodel

• 2 The Mathematical Nature of the Flow Equatnions

• 2.1 Linear Partical Differential Equations(PDEs)– 2.1.1 Classification of the Second Order Linear PDEs

– 2.1.2 General Behaviour of the Different Classes of PDEs

– 2.2.1 Inviscid Flow Model: Euler Equations– 2.2.2 Parabolized Navier-Stokes Equations, Boundary Layer

Approximation– 2.2.3 Potential Flow Model, Incompressible Fluid Flow Model

• 2.2 The Dynamic Levels of Approximation

1.1 IntroductionMotivation: Why should you be motivated to learn CFD?

The flowfield over a supersonic blunt-nosed body Artist's conception of next generation supersonic aircraft

Vehicle aerodynamics, combustion and DNS of turbulence

Some More Examples

http://www2.icfd.co.jp

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.

• http://fmad-www.larc.nasa.gov/aamb/

• Fluid Mechanics and Acoustics Division

• NASA Langley Research Center in Hampton, VA.

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.

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.

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.

II hope...hope...

1.2 The Basic Equations of Fluid DynamicsFluid and Flow

Gas, air, fuel, oxigen, CO2

Fluid: Liquid

(Solid)

water, oil, liquid metal

(geophysical flows)

that flows

Eulerian and Lagrangian frameworkEulerian description of the flow

the velocity of the fluid u at the position x at time t

u (x , t )

Lagrangian labels

the state and motion of the point particles

a = (a1,a

2,a

3) Lagrangian labels

u (x , t ) = u ( X (a, t ) , t ) = U (a, t )

u (x , t )

ab U (a, 0 )

U (a, t )

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

The Material DerivativeThe 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 (temporal & spatial derivativesderivative) with respect to the Eulerian variables)

The Continuity EquationControl Volume Fixed in Space Physical principle: Mass is conserved.

(Integral form)

Gauss Divergence Theorem

The Continuity Equation (continued)

Control Volume Moving with the Fluid

The relationship between the divergence of

Vand dV

The Continuity Equation (summary)

The meaning of divergence V

The Momentum equation

Physical principle: F = m a (Newton‘s second law)

Navier Stokes Equations

The Momentum equation

The Energy EquationPhysical principle: Conservation of energy

E: total energy (= e + V 2/2 ) , e: internal energy

Thermodynamic ConsiderationsUnknown 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 = cvT (2#)

cv

: spesific heat at constant volume

2 The Mathematical Nature of the FlowEquatnions

2.1 Linear Partial Differential Equations (PDEs)

Second Order

Linear PDEs

Navier Stokes

Equations

Second Order PDEs

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

b2

– 4ac > 0 hyperbolic

b2

– 4ac = 0 parabolic

b2

– 4ac < 0 elliptic

2.1.2 General Behaviour of the Different Classes ofPDEs

• Hyperbolic Equations

• Characteristic curves

Initial data upon which pdepends

Region II Domain of influence

y

a P

b

Left-running characteristic

Region I Influenced by point p

-Region of influence-

c

Region influenced by point c

Right-running characteristic

x

Hyperbolic Equations

(x,y,z)

Characteristic surface

y

Initial data in the yz plane upon which p

depends

Volume which influences point p

PVolume

influenced by point p

x

z

Parabolic Equations• Only one characteristic direction,• Marching-type solutions

y Boundary conditions known

c d

Initial data line Pa Region influenced by

P

Boundary conditions known b

x y

y

x=0 x=t

Elliptic Equations• No limited regions of influence;

• 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. Neumanncondition

• A mix of both Dirichlet and Neumann conditions.

2.2 The Dynamic Levels of Approximation2.2.1 Inviscid Flow Model: Euler Equations

Steady inviscid supersonic flows

Wave equation

2.2.2 Parabolized Navier-Stokes Equations, BoundaryLayer Approximation

Unsteady thermal conduction

Boundary-layer flows

Parabolized viscous flows

2.2.3 Potential Flow Model, Incompressible Fluid FlowModel

Physical picture consistent with the behavior of elliptic equations

Potential Flow: Steady, subsonic, inviscid flow

Flow over an airfoil

Incompressible Fluid Flow: the Mach number M = V/c 0

Flow over a cylinder cylinder

Discretization techniques route mapPDFs Æ System of allgebraic equations

Finite Difference

Method (FDM)

Finite Element

Method (FEM)

Finite Volume

Method (FVM)

Fourier /

Basic derivations

Spectral Method

Discretization errors

Time integration

Initial value problem

Types of solutions: Explicit

and implicit

Stability analysis

Iterative methods

Boundary value problemI.V.&B.V. problem

2 Finite Difference Method2.1 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

The definition of a derivative

2.2 Approximation of the first derivative2.2.1 Taylor series expansion

Expansion at xi+1

Expansion at xi-1

Using eq. at both xi+1

and xi-1

Truncation error

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

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

2.3 Approximation of the second derivative

Approximation from xi+1

, xi

second derivative

2.4 Approximation of mixed derivatives¾non-orthogonal coordinate system¾combining the 1D approximations¾The order of differentiation can be changed

2.5 Implementation of boundary conditions

2.6 Discretization errors

Truncation error (the imbalance due to the truncation of Taylor series) The

exact solution of Lh

Discretization error

Relationship between the truncation error and the discretization error

Richardson extrapolationfor sufficiently small h

the exact solution:

the exponent p (order of the scheme):

Approximation for the discretization error on grid h:

3 Finite Element Method

3.1 Interpolation functionApproximation 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

One dimensional linear function

3.2 Method of weighted residuals

4 Finite Volume Method• 4.1 Introduction

4.2 Approximation of surface integrals

4.3 Approximation of volume integrals

4.4 Interpolation practices

Linear interpolation

Quadratic Upwind Interpolation (QUICK)

Higher-order schemes

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 ; ikq

^f (kq) ;

-- Evaluate the series (37) to obtain df/dx at the grid points.

2++ higher derivatives; d2f/dx2 ; - k

q ^f (k

q).

++ The error in df/dx decreases exponentially with N when N is „large“.

++ The cost of computing ^f scales with N2(expensive!).

The method is made practical by a fast method of computing Fourier transform (FFT);

N log2

N.

5.2 A Fourier Galerkin method for the wave equation

Example

Error

7 Time integrationUnsteady flows Initial value problem (Initial boundary value problem) Steady flows

Boundary value problem

tSolution at time t=t

t=0

time t=0Initial condition

Steady solution

Boundary condition

Initial value problem Boundary value problem

7.1 Methods for Ordinary Differential Equation (ODE)

Two-level methods

Two-level methods

Predictor-Corrector method

Mutipoint methods

Adams-Bushforth methods

Runge-Kutta methodsThe second order Runge-Kutta method

The fourth order Runge-Kutta method

Other methods

An implicit three-level second order scheme

8 Stability analysisOne dimensional convection equation

Finite difference equation;forward in time, centered in space

Q: Does a solution of FDE converges to the solution ofP

DE?

approximate

PDE FDE

Æ 0

Solution of PDE 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.

8.1 Consistency, stability and convergenceConsistency

Æ 0

PDE FDE

Stability

Convergence

Lax‘s equivalence theorem

consistent

PDE FDE

Solution of PDEÆ 0

convergent

stable

Solution of FDE

L : linear operator

Truncation error

8.2 Von Neumann‘s methodFourier representation of the error on the grid points

FTCS method for 1D convection equation

Fourier series of the error

Amplification factor G

|G| > 1: Unconditionally unstable

Forward in time, forward in space(upwind scheme)

CFL (Courant Freedrichs, Levey) condition

BTCS method (Backward in time, centered in space)

|G| < 1: Unconditionally stable

Stability limit of 1D diffusion equation

8.3 Hirt‘s method

Matrix form

8.4 The matrix method

Spectral radius of the matrix C (maximum eigenvalue of C)

9 Iterative methods for algebraic systems

Linear equations : matrix form

9.0 Direct methods9.0.1 Gauß elimination

A = A

21

/A11

Forward elimination

upper triangular matrix

Back substitution

+ The number of operations (for large n) ~ n3

/ 3 (n2

/ 2 in back substitution)

+ pivoting (not sparse large systems)

9.0.2 LU decomposition

Solution ofAx = 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)

9.0.3 Tridiagonal matrix

Thomas algorithm / Tridiagonal Matrix Algorithm (TDMA)

+ the number of operations ~ n (cf n3

,Gauß elimination)

Iterative methods : Basic concept

Matrix representation of the algebraic equationA 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

Æ0

Iterative scheme

Iterative procedureA 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 matrixAn 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

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

h2

9.1.1 Jacobi method

9.1.3 SOR method

9.1.2 Gauß-Seidel method

9.1.4 SLOR method

9.1.5 Red-Black SOR method

9.1.5 Zebra line SOR method

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 ~ 0u*

NW

~ a ( uW

+ uN

– uP

), u*SE

~ a ( uS

+ uE

– uP

)a < 1

9.1.7 ADI method

Elliptic problem Æ parabolic problem

trapezoidal rule in time and CDS in space

at time step n+1

alternating direction implicit (ADI) method

The last term ~ O((dt)3)

splitting or approximate factorization

methods

9.2 Conjugate Gradient (CG) methodNon-linear solvers Newton-like methodsglobal methods descent method

Minimization problem

Steepest descents

Conjugate gradient method

with p1

and p2

conjugate

condition n

umber of AA: positive definite

preconditioning

C-1

AC-1

Cp=C-1

Q

9.3 Multi Grid method

Spectral view of errors

Fourier modes

Restriction

Interpolation

Coarse grid scheme

V-cycle, W-cycle & FMV scheme

9.4 Non-linear equations and their solution

9.4.1 Newton‘s method

linearization

new estimate

y y = f(x)

x0

ox

1

x

Newton‘s method

System of non-linear equations

linearization

Matrix of the system: the Jacobian

The system of equations

9.4.2 Other techniques

Picard iteration approach

Newton‘s method

9.5 Examples

Transonic flow over an airfoil

11 Incompressible Navier Stokes (NS) equations

11.1.0 Incompressible fluid

Incompressible fluid flow Compressible fluid flow

Ma < 0.3 Ma > 0.3

density variation d ~ Ma2

Mach number Ma = (v/c)

v : velocity, c : sound

speedex.

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.

11.1.1 Dynamic similitude

Reynolds number ρ ul u lRe = η

= ν

ex.

u : velocity scale, l : length scale, v : kinematic viscosity

U = 10 m/s, L = 1 m, v = 0.15

St, Re ~ 105. U = 0.1 m/s, L =

100 m, Re ~ 105.

11.2 The pressure Poisson equation method

Governing equations

Navier-Stokes equations

Explicit Euler method

Pressure Poisson equation

11.3 The projection method

A system of two component equations

The pressure P Æ a projection function

The projection step (BTCS)

Poisson equation with the Neumann boundary condition :

11.4 Implicit Pressure-Correction method

The momentum equations (implicit method)

Outer iteration (iterations within one time step):

Modification of the pressure field

The (tentative) velocity at node P

For convenience,

The discretized continuity equation

The (final) corrected velocities and pressure :

Discrete Poisson equation for the pressure :

Pressure-correction

The relation between the velocity and pressure corrections :

Pressure-correction equation :

Common practice; neglect unknowns ~u‘

SIMPLE algorithm more gentle way Æ

Implicit Pressure-Correction method

Approximate u‘ by a weighted mean of the neighbour values

Approximate ~u‘ by :

Neglect ~u‘ in the first correction step. The second correction to the velocity u‘‘ :

The second pressure correction equation:

Approximate relation between u‘ and p‘ : Essentially an iterative method for pressure- correction equation with the last term treated explicitly;

Implicit Pressure-Correction method

PISO algorithmThe coefficient in the pressure-correction equation A Æ A + ...And the last term disappears.

SIMPLEC algorithm

Pressure-correction with the last term neglected.i

p‘ Æ correct the velocity field to obtain um .

The new pressure field is calculated from

ipressure equation using ~ um

instead of ~ um*i

SIMPLER algorithm (Patankar 1980)

The SIMPLE algorithm does not converge rapidly due to the neglect of ~u‘ in the pressure- correction 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

app‘ : 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

vwhich has been found nearly optimum and yields almost the same convergence rate asSIMPLEC method.

11.5 Other methods

11.5.1 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.

11.5 Other methods

-The values of the streamfunction at boundaries.-Vorticity at the boundary is not known in advance.-Vorticity is singular at sharp corners.

11.5.2 Artificial compressibility methodsArtificial continuity equation

beta : an artificial compressibility parameterThe pseudo-sound speed:

should be much much faster than the vorticity spreadsÆcriterion on the lowest value of beta. Typical values are in the range between 0.1 and 10. Obviously,

should be small.

12 Some remarks on incompressible flows