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Computational Fluids Mechanics

Part 1： Numerical methodsPart 2： Turbulence modelsPart 3： Practice of numerical simulation

N.Oshima A5-32 [email protected] A5-33 [email protected] of Mechanical and Space EngineeringComputational Fluid Mechanics Laboratoryhttp://www.eng.hokudai.ac.jp/labo/fluid/index_e.html

Computational Fluid MechanicsPart1： Numerical Methods

Objective：

Numerical methods for fluids mechanics and their accuracy

1. Introduction2. Numerical methods for fluid mechanics (1) ~ Basic equations3. Numerical methods for fluid mechanics (2) ~Discretizing schemes4. Numerical methods for fluid mechanics (3) ~Coupling algorism 5. Numerical methods for fluid mechanics (4) ~Additional problems6. Reliability of numerical simulationSummary

Prof. Oshima A5-32 [email protected]

Computational Fluid Mechanics

Part1： Numerical Methods (2a)1. Introduction

2. Numerical methods for fluid mechanics (1) ~ governing equations– Governing equations of fluid flow– Typical solutions of fluid flow– Additional models for complex flow phenomena

3. Numerical methods for fluid mechanics (2) ~Discretizing schemes4. Numerical methods for fluid mechanics (3) ~Coupling algorism5. Numerical methods for fluid mechanics (4) ~Additional problems6. Reliability of numerical simulation

Governing Eqs. of Fluid mechanics

IDtD

qTpIDtDTcv 2

RTp

Continuity Eq.(mass conservation)

Momentum eq.

Energy Eq.

Eq. of state

iiii

i fuxI

xp

DtDu

2

3

jj xu

tDtD

;

j

iij

j

i

xuI

xu

3

;

kk xx

2

2;

k

k

xuI

;

Governing Eqs. of Fluid Flow

Non-dimensional Parameters

– Reynolds no. ～Flow instability（length）×（velocity）・Turbulence

（viscosity）・Viscous flow– Mach no. ～Ｈｉｇｈ speed flow（velocity）／（Sound speed）・Shock wave

・Acoustics– Knussen no. ～ Continuum（free pass） /（length）・Rarefied gas

・Micro flowlength （log）

Velocity

spaceship

MicroMachine

Airplane

Automobile（log）

TurbulenceSound wave

MD

Non-dimensional Parameters - in the governing eq.-

*

**

*

**

*

*

0 jj

j xu

xu

tUtL j

*

0

*

0

2

02*

*2

0

**

00

0

*

***

*

**

0

qUc

LQTcU

ULxT

ULcIp

Tcp

xTu

tT

UtL

vvvv

j

j

j

*** Tp

**22*

*2

*

*

0*

*

20

0*

***

*

**

0 31

ij fLUF

x

uxI

ULxp

Up

xu

utu

UtL

j

i

iij

ii

RTp

00

0

**

0*

0*

0***

0*

,

,,,,,

qQqFff

TTTpppUuuLxxttt iijj

Non-dimensional Parameters - in the governing eq.-

StUtL

0

Re1

0

UL

22

2

20

0 10

MUa

Up

(Strouhal no.)

(Reynolds No.)

(M:Mach No)

100

00

00

0

TcRT

Tcp

vv

PrRePe0

ULcv

（Pe:Peclet No.、Pr:Prandtl No.)

2

0

2

1 MTcU

v

Parameter of fluid flow

(g: specific heat ratio)

Lagrangean vs. Eularian formulations

Lagrangean

Eularian

utDt

D

u

u

tDtD

t

=0 (Mass conservation)

V(t) :Volume per unit mass

on unit mass [*/kg]

on unit volume [*/m3]

u: convectionvelocity

uconvectionflux

dV

dV

(x,t):Mass per unit volume

Conservation lowMass flux vector

u

t

yv

xu

t

dSdVt

nu xvvyuuyx

t nsew

Δx

Δy

t

w

w

uvu

0,1,nu

e

e

uvu

0,1,nu

n

n

vvu

1,0,nu

s

s

vvu

1,0,nu

w e

n

s

Conservation lowFlux vector vs. Stress tensor

ttt

tfPTuuu

⇒ xxx fu

tu

pτu

vut u ， xxt fff

vvvuuvuu

vut

uuuu ，

pp

yx 00

ppP ijpp I

yp

xpp tP

SττT

2

2

2

yv

xv

yu

xv

yu

xu

yx

Conservation lowMomentum flux vector

xpresdiffconv ftu

... JJJ

uvuu

uconv

uJ . ，

xvyuxu

diff 2

.J ，

0.

ppresJ

dVfdSdVtu

xpresdiffconv nJJJ ...

Δx

Δy

xftu ,

wpres.wpres.

wdiff.wdiff.

wconv.wconv.

pJxu

J

uuJ

nJ

nJ

nJ

2

epres.epres.

ediff.ediff.

econv.econv.

pJxu

J

uuJ

nJ

nJ

nJ

2w e

n

s

Exercise 2aCalculate each component of momentum flux through the vertical surface (n & s in fig.) of considering volume.Derive the conservation law for y-direction component of momentum (v)


Part1： Numerical Methods (2b)1. Introduction



Typical solutions of fluid flow

k

k

xu

jj xTk

x

RTp

ii

Fxp

jj xρu

t

j

jj

i

xu

utu

jjvv xTuc

tTc

ijijk

k

j

Sxu

x

23

Qxupk

k

convection interaction(volume source)

diffusion(surface flux)

Solution of convection

xu

t

)(),( 0 utxtx

x

t0⇒ ｔ

•Model of momentum convection（Burger’s Eq.）

xuu

tu

x

fast (ｕ=large）

•Model of scalar convection（linear eq.）

slow (ｕ=small）

・shape deformed･high frequency produced

•keep linear solution•adaptive to steep variation

Solution of diffusion and external force

2

2

xx

u

Diffusion of scalar （Linear）

St

Source term （Linear）⇒ St exp 0

⇒ const

x

u

2

2

xt

⇒

tinxn exp

time dependent heat transfer problem

steady convective – diffusion problem

Pecret No.： Pe=uL/

Solution of Interaction step

const

p

xu

t

xp

tu

Compressible （sound wave）

（S＝const. ）

xu

t

xa

tu

2

p

ddpa

s

2

2

22

2

2

xua

tu

atxgatxfu

Sound wave

Mach No. ：Ma = U / aa： sound speedU： flow velocity


const

0

i

i

xuu

ii

i Fxp

tu

1 Fu

t

p1

Pressure poison eq.)(

i

iji xuuF

Incompressible flow

if Ma→0a（sound speed）→∞


Compressible （1D nozzle）

TdTd

pdp

0AdA

udud

0dpudu

0 d

pdp

AdA

MMd

12

2

AdA

Mudu

112

AdA

MM

pdp

12

2

AdA

MM

TdT

112

2

A

dAM

Mada

1212

2

AdA

MM

MdM

112

2

2

0

0.5

1

1.5

2

-2 -1 0 1 2

M

A

T/T0

p/p0

r/r0

ru/rc0

u/c*

Solution of 1D compressible flow

Solution of Interaction stepCompressible （1D shock wave）

Mass ： 2211 uu ①

Momentum：2

2222

111 upup ②

Energy ：2

22

221

1

1

21

121

1upup

③

2

1

1

2

1

2

1

2

11

111

uu

pp

pp

＞１ Rankine－Hugoniot eq.

1111

11

111

2

1

1

2

1

2

1

2

1

2

2

1

1

2

1

2

pppp

pp

pp

pp

pp

TT

＞１

2222 ,,, MTp1111 ,,, MTp

Shock wave

Profiles of shock wave

M∞ medium M∞ large

Exercise 2bSolve the following equations of and draw the profiles for different parameter values.

2

2

xx

u

are constant parameter ,u

LxatxatLx

1000

x


Part1： Numerical Methods (2a)1. Introduction



Complexity of systemStrong coupling

k

k

xu

QxTk

x jj

RTp

ixp

jj xρu

t

j

jj

i

xu

utu

jjvv xTuc

tTc

iijijj

FSIx

23

k

k

xup

convection interaction diffusion external-force

3) parabolic（incompressible）

M

2) harmonic（sound wave）1) hyperbolic（compressible）

mass

momentum

energy

Ma<1

Ma>1 Combined solutionIs needed

Complexity of systemWeak coupling

k

k

xu

QxTk

x jj

RTp

ixp

jj xρu

t

j

jj

i

xu

utu

jjvv xTuc

tTc

iijijj

FSIx

23

k

k

xup

M

・Turbulencemodel

・NonNewtonianfluid

・Buoyancy force・Multi-phase

model・Chemical reaction

mass

momentum

energy

interactionconvction diffusion production

Additional eqs.

Alternative solutionIs available

Mass:

Momentum:

Chemical spices

Energy:

Eq.of gas state:

Arrhenius’ lawoxfu

ox

ox

fu

fufu M

YMY

RTEB

exp

＋

＋

Chemical reaction

Model of Combustion Flow - Basic eqs. of ｒｅａｃｔｉｖｅ flow-

Detail reactions of H2･O2 flameSpices N2 O2 H2 H2O OH

Lewis No. 1 1.11 0.3 1.12 0.73Spices O H H2O2 HO2 -

Lewis No. 0.7 0.18 1.12 1.1 -

OHO21H 222

･･･ 21 reactions

Model of Combustion Flow- non-dimensional parameters of mass & heat transfer-

• Nusselt no. h： heat transfer coefficient

l：Thermal conductivity

• Prandtl no. ν：kinetic viscosity

• Schmidt no. D：mass diffusion conductivity

• Lewis no. α： temp. diffusion conductivity

Pr

LhNu

DSc

DLe

Model of Combustion Flow- non-dimensional parameters of reactive flow-

• (primal) Damköhler no. tr： time scale of flow

tc： time scale of reaction

• Karlovits no. Tickness ofheating zone:

Combustion speed：S

• turbulent Damköhler no.

ｌ：turbulent length scaleu’：velocity fractuation、δ: flame thickness

dydU

UK

c

rD

SScp

S

uDa

c

t

'

Model of Combustion- mechanics of reactive flow-

time

Enthalpy： H＝E＋PV ⇒ dH＝(TdS－PdV)＋d(PV)＝ TdS＋VdP

Free energy： G＝H－TS ⇒ dG＝(TdS＋VdP)－d(TS) ＝－SdT＋VdP

Enthalpy components： pp

cTH

0

STG

p

TemperatureDensity

24

4

OCH

CH

YYY

Mixture fraction

Chemical potential＋thermal energy

dTcdnhdH p

0

0

TSH

p

Mass transport

Vt

VYtY

dYYtY

DtDY vV

Y

VVVVv ; 　　　d

0 ,0 , dY v＝ 0

vtDt

D

convection diffusion

YDYtY

DtDY V

YDY d v Fick’s law

Sc

D

Sc

Le Pr

TT

YD

YD

DYX

YYp

ppXY

DYX

X

TT

ff

vv

WY

WYX

TT

DD

pXXYY

ppXY

XXYYYDY

T

T

2,

1,21

21

221

1121

2111211

ffv

In case of two spices

1

d

ddddd

Y

YYYYYYYYYYY

V

VVVVVvv

Distribution of mole fraction

Fick’s law

Mass transport

Energy transport

T

T pp dTchhdTYcdYhhYdh0

,, ,

reacRpp QQTcTDtDp

DtDTC

j

Enthalpy eq.

dtdpp

tp

DtDpp

v0Low Ma No. apprrox.

T q

Mass fux （Fick’s law）

Heat flux （Fourier’s law）

YD j

,pp CYC Specific heat

Temperature eq.

RQYLeDhDtDp

DtDh

1

Thermal conductivitypC

DDCLe p

.reacQh ：Reactive heat

Tdsdpdh Theory.

Reactiveheat

Solution of combustion flow

k

k

xu

QxTk

x jj

RTp

ixp

jj xρu

t

j

jj

i

xu

utu

jjvv xTuc

tTc

iijijj

FSIx

23

k

k

xup

convection interaction diffusion external-force

M mass

momentum

energy

Additional eqs.

jj xYD

xjj xYu

tY

chemical

spices

Heat sourceWeak relation by low Ma approx.

Exercise 2cConsider a system of equations for complex flow phenomena and analyze its coupling process to fluid dynamics;

Ex. Multi-phase flowBuoyancy flowMagneto-hydrodynamicsFlow in porous mediaPlasma flow etc.

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