Computational Fluids Mechanicscfml.eng.hokudai.ac.jp/wp-content/uploads/2017/09/...Computational...
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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)
Computational Fluid Mechanics
Part1: Numerical Methods (2b)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
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
Solution of Interaction step
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)→∞
Solution of Interaction step
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
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
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.