Computational Fluids Mechanics

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

N.Oshima A5-32 oshima@eng.hokudai.ac.jpM.Tsubokura A5-33 mtsubo@eng.hokudai.ac.jpDivision 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 oshima@eng.hokudai.ac.jp

Computational Fluid Mechanics

Part1： Numerical Methods (1a)

1. Introduction– What do we want to know ? – How to do numerical simulation ? – Basic mathematics for Fluid Mechanics

2. 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 simulation

What we want to know ？

• 3-D and unsteady features of fluid flow– turbulence– vortex interactions

• Realistic design of fluid machine– detail design (ex. parts design, flow conditions, etc.) – unsteady conditions (ex. cascade, sloshing, etc.)

• Dynamic interaction in complex flows– reactive flows / multiphase flows– flow induced instability and acoustics problems

Instability of free shear layer

Transition eddies

LES result (Re=3900, case 2)Instantaneous enstrophy distribution

Transition eddies

Experiment (Corke Re=8000[13])Streaklines visualization(smoke wire)

blade

Vortex structures in wall share layer

Figure 4: Instantaneous eddy structures at the stagnation region of plane(left) and round (right) impinging jets when six waves are imposed at the inlet

(left: lz/pD=1/6; right: lq/pD=1/6, iso-surface: Dp’=4.5)

Design of Turbomachine(1)- 3D Measuring of Torque Converter -

Leading edge

Y

Z

X

laser sheet

laser sheet

shel

l sid

e

core

sid

e

shel

l sid

e

core

sid

e

Meridional flow direction

(main flow direction)

Input shaft rotating speed 400 [r/min]Speed ratio 0.05 / 0.3 / 0.8

Measurement Result Speed Ration 0.05

shel

l sid

e

core

sid

e

* These flow field planes are viewfrom upstream

* 3 [ mm ] downstream from leadingedge of stator blade

Meridional Flow Velocity Distribution

shel

l sid

e

core

sid

e[m/s]

Stream Lines

shel

l sid

e

core

sid

e

Secondary Flow Velocity Vectors

* Starting points of stream linesare located at 3 [mm] upstreamfrom leading edge of stator blade.

[m/s]

Design of Vehicle (1)

stream line

Single Vehicle

LES Grid

1,000,000 points

Platoon running

– Shape: tetrahedron– Element numbers: 117,060,909– Node numbers: 20,957,323

– LOLA B03/51– Formula NIPPON (2003~05)

Unsteady flow design of Formula car

Design of Vehicle (2)

Sources of Flow Noiseby Pantographs

Shinkan-sen (Bullet Train）300km/h → 360km/h(2002) (2012)

Prediction of Far-field Sound Spectra & Sound Source Distribution

Identified Instantaneous Sound Sources

■ Div・(u x ω)

Far-Field Sound Pressure Spectra

MOVIE

combustor

turbine

compressor

Combustion flow in Engineering

Solution for Engineering problems•Lean premixed combustion•Lower emission / high efficiency

• Technical Elements – spray– turbulence– heat transfer– material design– detail reaction– Ignition / extinction– radiation– noise / vibration

Ex. Design of Gasturbine

Turbulence model・Swirl / Recirculation・Heat / Mass flux

Turbulent mixing Flame dynamics Chemical reactionFlame propagation・Premix / Diffusion flame・Emission / Blow out

Reaction model・Rarefied combustion・Alternative fuels

Structural design

Strategy for turbulent combustion design

1103Flamelet approach

／LESElement reaction

／DNS

Detail of chemical element reaction

PDF approach／RANS

NOOHCO：：

Structural analysis・Stress / vibration・Thermal stripingSound analysis・Noise / instability

ex. combustion flow modeling

Exercise 1aConsider an application of computational fluid mechanics on to your interest design or research problem, and tell its targeting objective you want to know.

Computational Fluid MechanicsPart1： Numerical Methods (1b)

1. Introduction– What do we want to know ? – How to do numerical simulation ? – Basic mathematics for Fluid Mechanics

2. Numerical methods for fluid mechanics (1) ~ Basic equations3. 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

How to do numerical simulation?

• Basic elements of numerical simulation

Governing eq. Numerical schemeex. incompressible assumption ex. upwind scheme

turbulence model ･･･ MAC algorithm

Boundary condition Numerical gridex. inlet / outlet ex. grid resolution

wall condition unstructured mesh

• Matrix solver ／ parallel computing • Post processing ／ visualization

physics mathematics

global

local

Get a numerical solution of differential equation

Governing eq.

Generalized convection-diffusion eq.

St

ju

etc.e,temperaturcomponent,velocityvariable: ＝

ctorvelociy ve:termconvection: uu ，

rflux vectodiffusion :termdiffusion: jj ，

.,termon)(dissipatiproduction:

etcsourceheatforcegravityS

，

Ex. Thermal equation T(x,t)

pcQ

xTk

xxTu

tT

pcQkutT

tT ,,,,facorts;Constant,,;variable

fluxconvection:, uuu TxuTTT

fluxdiffusionTk,

jjxTk

x

capacity:cdensity:sourceheat :, p，， QcQS p

Numerical solution of deferential eq.Deferential eq. （+ B.C.） → Algebraic eq.Ex.

Equation system （incl. B.C.） → sequence of values

00,02 fxxdxdf 2xxf

12120201

00

ffffff

f

,9,4,1,0,3,2,1,0

ffff

Approx. of solutionApprox. of equation

bx A

Numerical solution of 2D/3D problems

Numerical Grid Approximation

Conservation laws forMass, momentum & energy

Values on millions gird points should be solved！

Solution

Algorism for solutionEx. 1D incompressible flow

011

1

dxuu ki

ki

Algorism 1: Solve u by eq.1 then solve p by eq.2 (successive)

Eq.1

Eq.2 dxpp

dxuuu

dxuuu

dtuu k

iki

ki

ki

ki

ki

kik

i

ki

ki

11

1

2111

1 12

dxFF

dtD

dxppp k

iki

ki

ki

ki

ki

1

2

11

111 21

Eq.3

Algorism 2: Solve p by eq.3 then solve u by eq.2 (successive)

Algorism 3: Solve u by eq.1 and solve p by eq.3 (parallel)

Time marching k → k+1

Numerical solution of complex problems

Distribution ofLevelset Scalar G

Distribution ofMixture fraction

Distribution ofTemperature &Density

Ex. Combustion flow by flamelet model

Velocity fieldFeedback

Numerical gridB.C.

Descretized eq.Differential eq.

Approximation & Error

Solution Num. solution

Setting conditions

Real physics

(Correct value)

approx. approx.

Prediction

Experiment Theory Numerical simulation

Error ＝ Analyzable discrepancy from correct answerdue to approximation and procedures

Measured value

(Correct value)(Correct value)proc. proc. proc.

Reliability of numerical solution

Case1 2nd Central Scheme

dWiggle

Over-estimation of Reynolds stresses

・ Cubic obstacle in channel- Park ＆ Taniguchi (1995)- LES-WS (1995, Tegernsee)

Mean velo.<U> experiment

Problem: How to remove numerical errors from a correct solution ?

Case2 QUICK with numerical diffusion

Fluid vs. Solid

Solid structural analysis

・Analyze shape deformation・Arrangement of elements is fixed

and observable

・Equilibrium physics ・Internal mechanism is structural,anisotropic and inhomogeneous

・Analyze shape deformation rate・Arrangement of elements is

not fixed or unobservable

・Unequilibrium physics ･ Internal mechanism is statistical,

isotropic and homogeneous

Fluid flow analysis

Exercise 1bSolve the three algorisms for 1D incompressible flow under the following condition

i.c.

b.c.

01

dxuuDki

kik

i at t=tk

0, 010

dxppUu at x=x0

0,01

NNN p

dxuu

at x=xN

Computational Fluid MechanicsPart1： Numerical Methods (1c)

1. Introduction– What do we want to know ? – How to do numerical simulation ? – Basic mathematics for Fluid Mechanics

2. Numerical methods for fluid mechanics (1) ~ Basic equations3. 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

Equations of flow phenomena

tt

ttt ptDt

D fΣuuuu

Momentum

Mass uu

tDt

D

EnergyDtDvp

DtDsTqe

te

DtDe

u

diffusion heat source

pressurework

1,v2

vpDtDp

DtDp u

100010001

,232, ISIuΣu vvut

Basic mathematics for fluid mechanics

Operations of vector and tensor variables

2

1

aa

a ，

2

1

bb

b ， 21 aat a ， 21 bbt b ，

2221

1211

aaaa

A ，

2212

2111

aaaatA

2211

2

1

21 bababb

aat

baba aaaaa t2

222121

212111

2

1

2221

1211

babababa

bb

aaaa

Ab

222121212111

2221

1211

21 ababababaaaa

bbt

AbAb

2212

2111

21

2

1

babababa

bbaatab ttt

abababab

aabb

abba

2212

2111

21

2

1

tt bbaa

cc abcbac

21

2

1

21

Derivative operators for vector variables

yx

，

vu

u ⇒ yv

xu

u ，

yfxf

f

yvyuxvxu

vuyxtu

ya

xa

ya

xa

aaaa

yx22122111

2221

1211A

uuuuu

y

vx

ut， fff uuu

uuu Convection of scalar

jjj

jjj

uxx

uux

ttt uuuuuu Convection of vector u

jj

ij

ijij

j

ux

uxuuuu

x

Derivative operators for vector variables(continued)

yx

，

vu

u ⇒ yv

xu

u ，

yfxf

f

yvyuxvxu

vuyxtu

fyf

xf

yfxf

yxf 2

2

2

2

2

tt

yv

xv

yu

xu

yvyuxvxu

yxuu 2

2

2

2

2

2

2

2

2

Scalar Lapｌacian operator

Vector Lapｌacian operator

Derivative operators for vector variables(continued)

ΩSu t

zw

zv

yw

zu

xw

yw

zv

yv

yu

xv

xw

zu

xv

yu

xu

xu

xu

i

j

j

i

2

2

2

21

21S

0

0

0

21

21

zv

yw

zu

xw

yw

zv

yu

xv

xw

zu

xv

yu

xu

xu

i

j

j

iΩ

00

0

12

13

23

，

3

2

1

uω

Strain tensor (symmetric component)

Voricity tensor (anti-symmetric component)

Other Eqs. of flow phenomena

Kinetic energy 2222

21

21

21 wvuuuK kk u

u

fuuuuuuuu

tpt

fuΣuuu

tuppKtK

ijj

iiji

j

t ΣxuΣu

x

,Σu

Vorticity

fSuuu

tpt

2

uω

（Momentum eq.）t

Exercise 1cComplete the component formulation of vorticity equation

?

zw

yv

xw

tiiii

Confirm the following relation between Stress tensor formulations

tt uu

SIuΣ

2

3

232