computational fluid dynamics cfd · 2015. 4. 1. · computational fluid dynamics cfd 1. something...

Computational Fluid Dynamics

CFD

1

Something to discuss:

Why do we need CFD?

Are there alternative ways?

Two views on CFD

”We computer advocates don’t

think wind tunnels will become

useless. They will always be

great places to store computer

printout in” D.R Chapman,

1991

”CFD is like a dog walking on its hind

legs; it’s doing it badly, but it’s

amazing that it can do it at all” Anon

ExamplesDiesel engine simulation

- Scania D12 mod. optical engine

- 2500bar injection pressure

- 6bar IMPEg (20% load)

- N-heptane fuel

- 45°, 90° and 135° inter-jet angle

- 2.18 Swirl ratio

- 15.1:1 Compression ratio

Model assumptions:

- Primary atomization not modeled

- Particle size distribution

- Constant wall temperature

- Top-hat rail-pressure profile

- Valve motions not included

ExamplesDiesel engine simulation

Sample Results

Instantaneous axial velocity field

Traces from blades

Wake downstream

the tower

H

Staggered configuration

DY=0 D

DY=0.36 D

DY=0.71 D

DY=1.43 D

Staggered configuration

Mean velocity Rms of fluctuation

Relation to Hemodynamics

• Blood flow rate (Re)

• Wall shear-stresses- Magnitude

- Spatial- and temporal-fluctuations

– Basically local in character

– Intermediate arteries i.e. Re-dependent

– Near bifurcations

Velocity Field

WSS-Magnitude

Closed building

•4 wind directions

•60 m/s

•k-e RNG

•770000 cells

•0.5 m node distance close to

surface

•Domain size 1000X1000X500

m

Closed building

Path lines

0 deg. 90 deg.45 deg.

Open building

•2 blade configurations

•4 m/s

•k-e RNG

•800000 cells

•0.5 m node distance close to

surface

•Domain size 1000X1000X500 m

Open building

Case 1 Case 2

Axial Compressor

Aeroelastic simulation of compressorblades using ANSYS 12

Axial Compressor

Mean Mach number

Axial Compressor

Aeroelastic simulation of compressorblades using ANSYS 12

Axial Compressor

Deformation of the blade tip

0i

i

x

u

i

j

ij

j

i

jij

jiiF

xx

u

xx

p

x

uu

t

u

1

Models for

turbulence,

combustion etc.

Geometry

Mathematical

description Results

For example speed, pressure, temperature Numerical

methods

Governing equations

and boundary

conditions

Discretisation,

choice of grid

System of

algebraic

equations

Equation

system solver

Approximate

solution

Mathematical

description of

physical ”reality”

FV, FD, FE?

All these steps introduces errors!

How can we guarantee that the approximate

solution is close to the exact one and close to

”reality”

Governing equations

The governing equations will depend on what assumptions can

be made regarding the flow. For example is it incompressible or

compressible?

The flow situation will determine the character of the system of

equations. This will in turn influence the choice of numerical

method.

Discretisation and grid

Questions:

•How complex is the geometry?

•What accuracy is required? Grid quality?

•What about stability?

•Grid refinement?

Examples

Prismatic airfoil

ExamplesPrismatic airfoil

Inviscid flow

Grid refinement

Coarse grid Refinement 1 Refinement 2

Coarse: Cdp=0.0500

R1: Cdp=0.0518

R2: Cdp=0.0530

ExamplesPrismatic airfoil

Inviscid vs viscous

Inviscid

Cdp=0.0500

Viscous

Cdp=0.0500

Cd,tot=0.0539

ExamplesPrismatic airfoil

Wall refinement

coarserefined

Coarse: Cdp=0.0500, Cd,tot=0.0539

Refined:Cdp=0.0531, Cd,tot=0.0569

ExamplesPrismatic airfoil

Wall refinement

Converged: Cdp=0.0500, Cd,tot=0.0539

Non-converged:Cdp=0.0495, Cd,tot=0.0533

Residual 10-8 Residual 10-3

System of equation solver

Questions:

•Speed of the computation?

•What accuracy is required?

•What about stability?

Approximate solution

Questions:

•Is the solution physically reasonable? Conservation?

•How to determine the accuracy of the solution?

Simulation Measurement

Governing

equations

Governing equationsDifferentialrelationer

System of equations:

VfVppVVTkqDt

De

pfDt

VD

0

V

t

Mass

Momentum

Energy

Governing equations

System of equations:

ii

i

i

i

ijj

iii

i fux

pu

x

u

x

Tk

xt

q

x

Eu

t

E

j

ij

j

i

j

jii

xx

pf

x

uu

t

u

0

i

i

x

u

t

Mass

Momentum

Energy

Governing equations

ii

i

i

i

ijj

iii

i fux

pu

x

u

x

Tk

xt

q

x

Eu

t

E

j

ij

j

i

j

ij

i

xx

pf

x

uu

t

u

0

i

i

i

ix

u

xu

t

Mass

Momentum

Energy

Non-conserved forms

Governing equations

Conserved form

ii

i

i

i

ijj

iii

i fux

pu

x

u

x

Tk

xt

q

x

Eu

t

E

j

ij

j

i

j

jii

xx

pf

x

uu

t

u

0

i

i

x

u

t

Mass

Momentum

Energy

Governing equationsConserved form

i

j

iji Jx

F

t

U

Mass

Momentum

Energy

Jx

F

t

U

i

i

Jx

F

t

U

j

j

E

uU i

jkk

j

jj

ijijji

i

ux

TkpuEu

puu

u

F

t

qfu

fJ

kk

i

0

Classification of

PDEs

Classification of PDEs

A comment on characteristic curves

Characteristic curves are curves along which signals

are propagated.

Example, an initial value problem.

)()0,(

0

0 xuxu

x

uc

t

u

By using the chain rule one

can find:

cdt

du

dt

dxby defined lines along 0

or,

ctxu on constant

The ”constancy of u” is the

signal carried along the

characteristic curves

Solution: )(, 0 ctxutxu

x

t

Charateristic curves

Classification of PDEs

22222

11111

fy

vd

x

vc

y

ub

x

ua

fy

vd

x

vc

y

ub

x

ua

P

dx

dy

Differentials:

dyy

vdx

x

vdv

dyy

udx

x

udu

Classification of PDEs

dv

du

f

f

y

vx

v

y

ux

u

dydx

dydx

dcba

dcba

A

2

1

2222

1111

00

00

dydx

dydx

dcbf

dcbf

B

00

00

2222

1111

A

B

x

u

Characteristic curves correspond to 0A

21221122112212

1221 dxdbdbdxdycbcbdadadycacaA

Classification of PDEs

21221122112212

1221 dxdbdbdxdycbcbdadadycacaA

0122112211221

2

1221

dbdb

dx

dycbcbdada

dx

dycaca

0

2

c

dx

dyb

dx

dya

a

acbb

dx

dy

2

42

Three situations:

04

04

04

2

2

2

acb

acb

acb Hyperbolic, two real characteristics

Parabolic, one real characteristic

Elliptic, no real characteristic

Classification of PDEs

Example: 2D inviscid steady flow of a compressible gas under small perturbations.

0''

0''

1 2

v

v

y

u

y

v

x

uMa

'

'

vv

uUu

0 ;1 ;1 ;0

1 ;0 ;1

2222

1112

1

dcba

dcbMaa

Characteristic equation:

01221121221

2

1221

cba

dbdbdx

dycbdada

dx

dycaca

Classification of PDEs

Characteristic equation:

01221121221

2

1221

cba

dbdbdx

dycbdada

dx

dycaca

1

0

1 2

c

b

Maa 011

22

dx

dyMa

12

14

1

12

2

2

Ma

Ma

Madx

dyCompare:

a

acbb

dx

dy

2

42

hyperbolic 1

elliptic 1

2

2

Ma

Ma

Classification of PDEs

Hyperbolic

PDomain of

dependence Region of influence

Charateristic lines

y

xExamples: Inviscid supersonic flow, unsteady inviscid flow

Classification of PDEs

Parabolic

PDomain of

dependence Region of influence

y

x

Known boundary conditions

Known boundary conditions

Examples: Steady boundary layer flow, unsteady heat conduction

Classification of PDEs

Elliptic

P

y

x

Every point influences all

other points

Examples: Steady subsonic inviscid flow, incompressible inviscid flow