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 (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
