numerical approach on hydrogen detonation: fundamentals and applications -part 1- · 2007-07-26 ·...
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Numerical Approach on Hydrogen Numerical Approach on Hydrogen Detonation: Fundamentals and Detonation: Fundamentals and
ApplicationsApplications--Part 1Part 1--
2007.08.022007.08.02Nobuyuki TSUBOINobuyuki TSUBOIISAS/JAXA, JapanISAS/JAXA, Japan
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1.Motivations
2. Numerical simulation for compressiblehigh speed flow
(1) Scalar equations(2) System equations(3) System equations with chemical reactions
3.Summary
OverviewOverview
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MotivationsMotivations1.Hydrogen/air mixture: detonable gas
2.Detonation: shock induced combustion-Pressure behind detonation increasesabout 10 times ambient pressure
3.Closed environment such as a tunnel causes serious accident.
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Numerical SimulationsNumerical Simulations1.Detonation as well as thermal and gas dynamic
phenomena of airplane and aerospace plane are elucidated numerically.
2. Main issues (1) Robust numerical scheme for high pressure and temperature(2) Stiff problem: chemical characteristic time is much faster than the fluid characteristic time(3) Chemical reaction model for high pressure combustion(4) Unsteadiness, turbulence, and three-dimensional problems
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Flow Chart of Numerical SimulationsFlow Chart of Numerical Simulations
Computational Grid
Program
Visualization
Physical Model?
Flowfield to Simulate?
Mathematical Model?
Discretization?
Submit Job
・・Compressible? Incompressible?Chemical reaction? Turbulence?
・・Euler equations?Navier-Stokes equations?
・・Finite difference?Finite Volume?Finite element?
・・Structure grid?Unstructure grid?Cartesian grid?
・・Supercomputer? Workstation?
・・Workstation?Personal computer?
・・Fortran? C?
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Conceptual Conceptual designdesign
w/o separationw/o separationless turbulenceless turbulence
Panel MethodPanel MethodPotential Potential EqEq..
Academic Academic researchresearch
Academic Academic researchresearch
Academic Academic research,research,Detailed Detailed designdesign
Conceptual Conceptual designdesign
LevelLevel
Solve all vorticesSolve all vorticesDNS(DirectDNS(Direct Numerical Numerical Simulation)Simulation)
Solve scale Solve scale vortices smaller vortices smaller than gridthan grid
LES(LargeLES(Large Eddy Simulation)Eddy Simulation)
Boundary layer, Boundary layer, separationseparation
RANS(ReynoldsRANS(Reynolds averaged averaged NavierNavier-- Stokes Stokes EqEq.).)
ShockShockEuler Euler EqEq..
TargetTargetMathematical ModelMathematical Model
Numerical ModelNumerical Model
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Structured grid ?Unstructured grid ?Cartesian grid ?
How to make grids ?・・Commercial software ?・・Make by oneself ?
Computational GridsComputational Grids
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Flowchart of program is defined after discretization and grid are decided
Start
Initialization
Boundary condition
Define time step
Calculate numerical flux
Calculate time integration
Convergence ?
Output data
End
No
Read grid, parameter, and set initial condition
Output simulation results
CFL numberConvective flux, viscous termUpwind method, central difference, Higher-order
Explicit or implicit time integration
Numerical analysis for fluid: finite different methodNumerical analysis for fluid: finite different method
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Scalar EquationScalar Equation
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Wave equation ・・ wave propagates with speed, c
u
x
c
u2
u1
Finite Difference for Scalar EquationFinite Difference for Scalar Equation
0u uct x
∂ ∂+ =
∂ ∂
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1111
( ) )(22
11111
1 nj
nj
nj
nj
nj uu
xtcuuu −++−
+ −⎟⎠⎞
⎜⎝⎛ΔΔ
−+=
)(2 11
1 nj
nj
nj
nj uu
xtcuu −+
+ −⎟⎠⎞
⎜⎝⎛ΔΔ
−=
FTCS (Forward in Time and Central Difference in Space)・・Explicit+central difference:unstable
⎟⎠⎞
⎜⎝⎛ΔΔ
=xtcν
・・CFL number:parameter to govern numerical stability
First-order upwind method (Upwind difference)
)( 11 n
jnj
nj
nj uu
xtcuu −
+ −⎟⎠⎞
⎜⎝⎛ΔΔ
−= ・・Find direction which wave propagatesand use upwind data only
Lax methodMore stable than FTCSLarge numerical dissipation
Lax-Wendroff method 2nd order in space and time
0u uct x
∂ ∂+ =
∂ ∂
( ) ( )22
11 1 1 12
2 2n n n n n n nj j j j j j j
c t c tu u u u u u ux x
++ − + −
Δ Δ⎛ ⎞ ⎛ ⎞= − − + − +⎜ ⎟ ⎜ ⎟Δ Δ⎝ ⎠ ⎝ ⎠
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Pseudo-finite volume method・Conservation form
・・Non-linear equation with discontinuous wave can be calculate and discreted
・・Compressible equations is based on this concept
0,u f f cut x
∂ ∂+ = =
∂ ∂
1−j
Flux
j 1+j
21−j 2
1+j
( )11/ 2 1/ 2
n n n nj j j j
tu u f fx
++ −
Δ⎛ ⎞= − −⎜ ⎟Δ⎝ ⎠% % n
jf 2/1~
+ :Numerical flux
njf 2/1
~+1/ 2
njf −%
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( )11/ 2 1/ 2
n n n nj j j j
tu u f fx
++ −
Δ⎛ ⎞= − −⎜ ⎟Δ⎝ ⎠% %
( ) ( )1/ 2 1 112 2
n n n n nj j j j j
cf f f u u+ + += + = +%
For example, FTCS is applied,
( )
( ) ( ){ }
( )
11 1
1 1
1/ 2 1/ 2
212
n n n nj j j j
n n n n nj j j j j
n n nj j j
c tu u u ux
tu cu cu cu cux
tu f fx
++ −
+ −
+ −
Δ⎛ ⎞= − −⎜ ⎟Δ⎝ ⎠Δ⎛ ⎞= − + − +⎜ ⎟Δ⎝ ⎠
Δ⎛ ⎞= − −⎜ ⎟Δ⎝ ⎠% %
Therefore
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There exist many numerical schemes, however, these schemes are designed how to estimate numerical flux n
jf 2/1~
+
FTCS˜ f j+1 / 2
n =12
( fj+1n + fj
n) =c2
(uj+1n + uj
n )
Lax
)(21)(
2~
112/1nj
nj
nj
nj
nj uu
txuucf −⎟⎠⎞
⎜⎝⎛ΔΔ
−+= +++
1st order upwind difference˜ f j+1 / 2
n =c2
(uj+1n + uj
n ) −c2
(uj+1n − uj
n ) = cujn = fj
n
)])(1(21[)1(
2)1(
2~
112/1 jjjjjnj uuucucucf −−+=−++= +++ ννν
Lax-Wendroff
⎟⎠⎞
⎜⎝⎛ΔΔ
=xtcν
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1st order upwind difference
˜ f j+1 / 2n =
c2
(uj+1n + uj
n ) −c2
(uj+1n − uj
n ) = cujn = fj
n
⎪⎩
⎪⎨⎧
≤≤
=+
+ 0,0,~
12/1 cf
cff n
j
njn
j
When sign of c is unclear, summarized as follows:
[ ]
[ ])()(21
)()(21
22~
11
11
12/1
jjjj
jjjj
jjnj
uucff
uuccucu
ucc
ucc
f
−−+=
−−+=
++
−=
++
++
++
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1st order upwind difference
˜ f j+1 / 2n = cuj
n
Lax-Wendroff
)])(1(21[
)1(2
)1(2
~
1
12/1
jjj
jjnj
uuuc
ucucf
−−+=
−++=
+
++
ν
νν
L-W scheme = 1st order upwind + modified flux-> 2nd order
TVD(TotalTVD(Total Variation Variation DimishingDimishing) Method) Method
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)])(1(21[~
12/1 jjjnj uuucf −−+= ++ ν
modified flux
L-W scheme generates numerical oscillation.Another scheme added non-linear flux is introduced.
~ [ ( )( )]/ /f c u B u ujn
j j j j+ + += + − −1 2 1 2 112
1 ν
flux limiter functionWhen B select well, monotone scheme, which is almost higher order and becomes 1st order near discontinuity, can be created・・Modified flux method proposed by Harten
1( ) (1 )2
c cσ ν≅ −
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u
xMonotone imply a property to become 1st order near discontinuity
monotone solution
non-monotone solution
What is monotone scheme?What is monotone scheme?
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TV(un ) = uj+1n − uj
n
j∑
TV(un+1) ≤ TV(un )
Total variation in space at a time defines as follows:
TV stability:Total variation decrease with time (diminishing)・・TVD condition
Scheme which is satisfied this condition is called TVD scheme.
:Total variation
TVD(TotalTVD(Total Variation Diminishing) MethodVariation Diminishing) Method
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2020
TVD scheme means to preserve monotone profile
For example, 1st order upwind difference is TVD scheme.
In order to preserve monotone,
ujn+1 = uj
n +ΔtΔx
[cj+1 / 2− (uj+1 − uj) + cj−1/ 2
+ (uj − uj−1)]
cj+1/ 2− > 0, cj−1/ 2
+ > 0,ΔtΔx
(cj+1/ 2− − cj−1/ 2
+ ) ≤ 1
monotoneTVD scheme
CFL condition
TVD(TotalTVD(Total Variation Diminishing) MethodVariation Diminishing) Method
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⎪⎩
⎪⎨⎧
≤=≤=
=++
+ 0,0,~
112/1 ccuf
ccuff n
jnj
nj
njn
j
Frompiecewise constant in celltopiecewise linear in cell・・cause oscillate solutions!
piecewise constant piecewise linearj-2 j-1 j j+1
uj-2
uj-1
uj
uj+1
j-1 j j+1 j+2
uj-1
uj
uj+1
uj+2
1/ 2Lju +
1/ 2Rju +
Higher order Numerical FluxHigher order Numerical Flux
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1st order 2nd order
1st order 2st order
Non-MUSCL+limiter, TVD
MUSCL+limiter,TVD
u u
)(~ uf )(~ uf
MUSCL:Monotone Upstream-centredSchemes for Conservation Laws
Higher order Numerical FluxHigher order Numerical Flux
j-1 j j+1 j+2
uj-1
uj
uj+1
uj+2
1/ 2Lju +
1/ 2Rju +
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System EquationSystem Equation
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Define flowchart of program after discretization and grid are decided
Start
Initialization
Boundary condition
Define time step
Calculate numerical flux
Calculate time integration
Convergence ?
Output data
End
No
Read grid, parameter, and set initial condition
Output simulation results
CFL numberConvective flux, viscous termUpwind method, central difference, Higher-order
Explicit or implicit time integration
Numerical Analysis for Fluid: Finite Different MethodNumerical Analysis for Fluid: Finite Different Method
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Solution of system equations
Solution of scalar equation
・Non-linear-Previous discussion assumed as linear equation ・Simultaneous equations-Difficult to find the upwind direction
Therefore,・Linearization is necessary・Decompose to some wave equations in order to define the upwind direction
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Mass conservation
Momentum conservation
Energy conservation
1-D Compressible Euler Equations
e p h Hρ+ = =Enthalpy per unit mass
p RTρ=
Equation of State(Ideal Gas) where
( ) 0u
t xρρ ∂∂
+ =∂ ∂
( ) ( )2
0u pu
t xρρ ∂ +∂
+ =∂ ∂
( )( )0
e p uet x
∂ +∂+ =
∂ ∂
DiscretizationDiscretization of System Equationsof System Equations
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where, (ideal gas)
1 11 1 v
pE RT C Tγ ρ γ
= = =− −
Internal energy per unit mass
Enthalpy per unit mass
2
2 2 2
1 11 2
1 1 11 2 1 2 2p
h e p pH u p
p u RT u C T u
ρρ ρ ρ γ
γ γγ ρ γ
⎛ ⎞+= = = + +⎜ ⎟−⎝ ⎠
= + = + = +− −
Enthalpy per unit volume
22
21
1)
2( uPuEe ρ
γρ +
−=+=
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0Q Et x
∂ ∂+ =
∂ ∂
Vector form of governing equations:
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
++=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=
upeup
uE
euQ
)(, 2ρ
ρρρ
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
++
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+
+=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
vpevp
vuv
F
upeuv
upu
E
evu
Q
)(
,
)(
, 2
2
ρρρ
ρρ
ρ
ρρρ
1-D
2-D
0Q E Ft x y
∂ ∂ ∂+ + =
∂ ∂ ∂
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Governing equation is linearlized
⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
−−−−
−−−
−=∂∂
=
uuhuhu
uuQEA
ρρρ
ργρ
22
2
)1()2
1(
1)3(2
3010
0Q Et x
∂ ∂+ =
∂ ∂0Q QA
t x∂ ∂
+ =∂ ∂
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Property of 1-D Euler equations
Derive eigen valuewwA rr λ=
λ
0
)1()2
1(
1)3(2
301
22
2 =
−−−−−
−−−−
−
−
λρρρ
ρλγρλ
uuhuhu
uu
0)()( 2 =⎟⎟⎠
⎞⎜⎜⎝
⎛−−−
ργλλ puuthen,
cucuu −+= ,,λEigen values are
ργ pc =Speed of sound:
For 2-D, eigen values include another u
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How to derive Jacobian matrix:
2
2
, /
( )
1( 1)2
mQ m E m p
e me p
mp e
ρρ
ρ
γρ
⎛ ⎞⎜ ⎟⎛ ⎞ ⎜ ⎟⎜ ⎟= = +⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝ ⎠ ⎜ ⎟+⎜ ⎟⎝ ⎠
⎛ ⎞= − −⎜ ⎟
⎝ ⎠
2 2 2( / ) ( / ) ( / )
(( ) / ) (( ) / ) (( ) / )
m m mm e
E m p m p m pAQ m e
e p m e p m e p mm e
ρρ ρ ρρ
ρ ρ ρρ
∂ ∂ ∂⎛ ⎞⎜ ⎟∂ ∂ ∂⎜ ⎟⎜ ⎟∂ ∂ + ∂ + ∂ +
= = ⎜ ⎟∂ ∂ ∂ ∂⎜ ⎟⎜ ⎟∂ + ∂ + ∂ +⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠
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衝撃波膨張波 接触面
Three eigen values corresponds to the propagation velocities in flow
Contact surface(Slip line or contact discontinuity):
pressure : balance
density, temperature,velocity normal to surface : different
Expansion wave Contact surface Shock wave
t
x
uu-c u+c
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Q 1st order Q 2nd order
E 1st order E 2nd order
Non-MUSCL+limiter
MUSCL+limiter
Another schemes not to belong these procedures:・Lax method・Lax-Wendroff method・2-step L-W method・MacCormack method
Flux Difference Splitting(FDS):has to solve Riemannproblem
Flux Difference Splitting(FDS):has to solve Riemannproblem
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The shock tube problem is called the Riemann Problem.This problem presents an exact solution of the fully system of one-dimensional Euler equations containing simultaneously a shock wave, a contact discontinuity, and an expansion fan.
Accurate and physical solution is obtained by solving one-dimensional Euler equations approximately.
Qjn+1 = Qj
n −ΔtΔx
[ ˜ E j+1/ 2 − ˜ E j−1/ 2 ]
Local shock tube problem
Time
x
Qshock
contact discontinuityexpansion
t=nΔt
t=(n+1)Δt
jj-1 j-1/2RL
Approximate Riemann SolverApproximate Riemann Solver
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Qjn+1 = Qj
n −ΔtΔx
[ ˜ E j+1/ 2 − ˜ E j−1/ 2 ]
1)Flux Difference Splitting (FDS)
2)Flux Vector Splitting (FVS)
has to need approximate Riemann solver
does not need approximate Riemann solver.
Relation between Approximate Riemann Relation between Approximate Riemann Solver and Numerical FluxSolver and Numerical Flux
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Qjn+1 = Qj
n −ΔtΔx
[ ˜ E j+1/ 2 − ˜ E j−1/ 2 ]
A R Rj j j j+ + + +−=1 2 1 2 1 2 1 2
1/ / / /Λwhere,
˜ E j+1/2 =12
[Ej+1 +Ej − A j+1 /2(Qj+1−Qj )]
Upwind process on eigen values
difference!
necessary to calculatethis average
Finite Difference Splitting MethodFinite Difference Splitting Method
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Define average states (j±1/2) as a non-linear function,
The following conditions have to be satisfied to calculate Roe average:
E(QR) − E(QL ) = A(QR,QL)(QR − QL )= Aave (QR − QL)
1.
has real eigen values with linearly independent eigenvectors
A(QR,QL)2.
A(Q,Q) = A(Q)3.
Shock wave automatically generates.Characteristic wave speed is correctly calculated.
necessary in smooth (differentiable) region
Conservation law is satisfied.
Roe AverageRoe Average
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ave L Rρ ρ ρ=
Roe Average
cave2 = (γ −1)Have −
12
uave2where,
The results are as follows:
Roe average means a weighted average.However, the integral average form exists(Chakaravathy & Osher).
Roe AverageRoe Average
L L R Rave
L R
u uu
ρ ρρ ρ
+=
+
L L R Rave
L R
H HH
ρ ρρ ρ
+=
+
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Previous discussion is based on 1st order in space.Then how to increase accuracy in space?
1.Approach by using MUSCL
2.Approach by using non-MUSCL
Monotone-Upstream centered Schemes for Conservation Laws
Higher Order FDSHigher Order FDS
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Qjn+1 = Qj
n −ΔtΔx
[ ˜ E j+1/ 2 − ˜ E j−1/ 2 ]
˜ E j+1/ 2 =12
[ER + EL − A j+1 / 2 (QR − QL)]
where, is calculated by left and right side quantities which are interpolated
˜ E j+1/ 2,L RQ Q
Higher Order FDS by MUSCLHigher Order FDS by MUSCL
piecewise linearj-1 j j+1 j+2
uj-1
uj
uj+1
uj+2RQ
LQ
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Q Q s s Q s Qj L j j j+ − += + − + +1 2 4 1 1/ ( ) ( )κ κΔ Δ
Q Q s s Q s Qj R j j j+ + + + − += − − + +1 2 1 1 14 1 1/ ( ) ( )κ κΔ Δ
For example, slope limiter(Van Albada limiter):
s=2Δ+Δ− +ε
(Δ+)2 +(Δ−)2 +ε
slope
Interpolated variables in MUSCL are:・Conservative variables : density, momentum, energy・Primitive variables : density, velocity, pressure・Characteristic variables : variables transported by waves
such as entropy
Higher Order FDS by MUSCLHigher Order FDS by MUSCL
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x
Q
j j-1 j+1 j+2
QL
QR
Q Q Q Qj L j j j+ − += + − + +1 214
1 1/ ( ) ( )κ κΔ Δ
Q Q Q Qj R j j j+ + + + − += − − + +1 2 1 1 114
1 1/ ( ) ( )κ κΔ Δ
Because a extreme causes numerical instability, spatial order decrease 1st order. (monotone)
x
Q
j j-1 j+1 j+2
MUSCL and role of limiterHigher Order FDS by MUSCLHigher Order FDS by MUSCL
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* Yee’s Upwind-TVD
˜ E j+1/ 2 =12
[(Ej+1 + Ej ) + Rj+1 / 2Φ j+1/ 2 ]
φ j+1/ 2l = σ (cj+1 / 2
l )(gjl + gj+1
l ) −ψ (cj+1/ 2l +γ j+1/ 2
l )α j+1/ 2l
limiter Modified flux
NonNon--MUSCL TVD MethodMUSCL TVD Method
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• Hirsch, C., Numerical Computation of Internal and External Flows Vol.2,John Wiley & Sons,1990
• Yee,H., Upwind and Symmetric Shock-Capturing Scheme, NASA TM 89464, 1987.
ReferencesReferences
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System Equation with Chemical System Equation with Chemical ReactionsReactions
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Define flowchart of program after discretization and grid are decided
Start
Initialization
Boundary condition
Define time step
Calculate numerical flux
Calculate time integration
Convergence ?
Output data
End
No
Read grid, parameter, and set initial condition
Output simulation results
CFL numberConvective flux, viscous termUpwind method, central difference, Higher-order
Explicit or implicit time integration
Chemical reaction
Numerical Analysis for Fluid: Finite Different MethodNumerical Analysis for Fluid: Finite Different Method
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Problems for simulation of combustion
• Computational code with combustion is few because:- reaction mechanism is complex- hard task to construct computational code (especially point implicit on reaction source term)
- computational time is large - stiffness problem for detailed reaction model: restrict time step
• Detailed reaction model is at most proposed for hydrogen system, 2H2+O2→2H2O(global (one-step) reaction)H2+O→HO+H(detailed reaction)
Low-order simulation may done when vaporization, condensation, and heterogeneous combustion are include.
About Simulation of CombustionAbout Simulation of Combustion
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-Specific heat ratio-Mach number-Reynolds number
-First Damköhler number aI
c
D ττ
= =Characteristic time of fluid
Characteristic time for chemical reaction
Combustion:10-3~10-2secReaction locally (combustion)
a cτ τ≈ Reaction near mixing zone (Catalysis near accelerator, photochemical smog)Uniform reaction in field(in a small scale experimental device)
Similarity law for combustion phenomena is not discovered!
ca ττ ⟨⟨
ca ττ ⟩⟩
Normalized Parameter for Combustion Normalized Parameter for Combustion PhenomenaPhenomena
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From non-reactive code (constant specific heat ratio)
to gas-phase reactive code:
1.Specific heat ratio depends on temperature
2.Mass equations for chemical species(H2,O2,..) are
added
3.Source term including reaction is added
4.viscous coefficients, heat conduction coefficients,
and diffusive coefficients depend on temperature
5.Reaction model has to select
Modified Points for Fluid Calculation CodeModified Points for Fluid Calculation Code(Compressible Equations)(Compressible Equations)
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1.Detailed reaction model vs. global reaction modelFor example, reaction of hydrogen:
2H2+O2→2H2Ois global reaction. Most Hydrogen reaction systems haveabout 8 species and 20 elemental reactions.
2.Detailed reaction modelElemental reactions should be included as much as
possible after sensitivity analysis for elemental reactions are done.
Chemical Reaction ModelChemical Reaction Model
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Graph showing temperature dependence of the specific reaction rate constant k
ln ln( )b a
u
Ek ATR T
= −
k is decided by experimental data
However, k depends on both temperature and temperature range although many reactions follows the Arrhenius law.
Detailed reaction model does not constructed for all cases. Therefore you should select valid reaction model to reproduce phenomena.
ln k
1/T
Slope a
u
ER
= −
Pressure dependence also exists
ln k
1/T
Low temperature data
High temperature data
expb a
u
Ek ATR T
⎛ ⎞= −⎜ ⎟
⎝ ⎠
Chemical Reaction ModelChemical Reaction Model
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Pressure-temperature explosion diagram of a stoichiometric H2/O2 mixture in a spherical vessel.(explosion peninsula)
Explosion Limit of H2O2 SystemExplosion Limit of H2O2 System
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First limit : The destruction of HO2 on the wall causes. This is dependent on the size of vessel.
HO2+H→H2+O2(wall)HO2+OH→H2O+O2(wall)
Second limit : The following reactions dominate. HO2 is relatively unreactive.
O2+H→O+OH (low pressure)O2+H+M→HO2+M (high pressure: terminating reaction)
Third limit : HO2 can collide and react with H2molecules to form H2O2 and H atm. H2O2 can dissociate to effectively generate OH.
HO2+H2→H2O2+H2OH
Explosion Limit of H2O2 SystemExplosion Limit of H2O2 System
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Reaction kA kn kaE comments
(1) 2O H H OH+ + 45 00 10. × 2.70 6290 (2) 2 2H O M HO M+ + + 182 80 10. × -0.90 0 i (3) 2 2 2 2H O O HO O+ + + 203 00 10. × -1.70 0 (4) 2 2 2 2H O H O HO H O+ + + 189 38 10. × -0.80 0 (5) 2 2 2 2H O N HO N x+ + + 192 60 10. × -1.20 0 (6) 2H O O OH+ + 138 30 10. × 0.00 14413 (7) 2 2 2H HO O H+ + 132 80 10. × 0.00 1068 (8) 2H HO OH OH+ + 141 34 10. × 0.00 635 (9) 2 2 2 2H H O HO H+ + 71 21 10. × 2.00 5200 (10) 2 2OH H H O H+ + 82 16 10. × 1.50 3430 (11) 2 2OH OH M H O M+ + + 137 40 10. × -0.40 0
infa bk ,
182 30 10. × -0.90 -1700 0k (12) 2 2 2OH HO O H O+ + 132 90 10. × 0.00 -500 (13) 2 2 2 2OH H O HO H O+ + 121 75 10. × 0.00 320 c
ak
145 80 10. × 0.00 9560 cbk
(14) 2 2 2 2 2HO HO O H O+ + 111 30 10. × 0.00 -1630 dck
144 20 10. × 0.00 12000 ddk
(15) 2O O M O M+ + + 171 20 10. × -1.00 0 e (16) O H M OH M+ + + 175 00 10. × -1.00 0 f (17) 2H OH M H O M+ + + 222 20 10. × -2.00 0 g (18) 2H H M H M+ + + 181 00 10. × -1.00 0 h
Petersen and Hanson model
Reaction (11):pressure dependence defined by Troe’s formula
[M] : Mole fraction rate of 3rd body
L. Petersen and K. Hanson “Reduced Kinetics Mechanism for Ram Accelerator Combustion”Journal of propulsion and power Vol.15, No.4, July-August 1999
(11) OH+OH+M=H2O2+M
Rat
e C
oeffi
cien
t(cm
3/m
ol-s
)[M](mol/cm3)
Pressure(atm)0.1 1 10 100 1000
w/o pressure dependence
w/ pressure dependence
Example of Detailed Reaction ModelExample of Detailed Reaction Model
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5555
v v vE F GQ E F G St x y z x y z
∂ ∂ ∂∂ ∂ ∂ ∂+ + + = + + +
∂ ∂ ∂ ∂ ∂ ∂ ∂
Compressible Navier-Stokes Equations
( ) ( ) ( )
2
2
2, , ,
i i i i
u v wu u p uv uwv uv v p vw
Q E F Gw uw vw w p
e e p u e p v e p wu v w
ρ ρ ρ ρρ ρ ρ ρρ ρ ρ ρρ ρ ρ ρ
ρ ρ ρ ρ
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥+⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥+
= = = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥+⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥+ + +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦
00 0 000
, , ,00
yxxx zx
yyxy zy
v v vyzxz zz
yx yy yz yxx xy xz x zx zy zz z
ii ii ii i
E F G Su v w qu v w q u v w q
YY YDD Dyx z
ττ τττ τττ τ
τ τ ττ τ τ τ τ τ
ρ ωρ ρ
⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢⎢ ⎥⎢ ⎥ ⎢ ⎥= = = = ⎢⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢⎢ ⎥⎢ ⎥ ⎢ ⎥+ + −+ + − + + −⎢ ⎥⎢ ⎥ ⎢ ⎥
∂∂ ∂⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣⎢ ⎥⎢ ⎥ ⎢ ⎥∂∂ ∂⎣ ⎦ ⎣ ⎦⎣ ⎦&
⎤⎥⎥⎥⎥⎥
⎢ ⎥⎢ ⎥⎢ ⎥⎦
Species conservation equations are included
Governing Equations for Combustion Governing Equations for Combustion SystemSystem
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5656
( )2 2 2
1 2
N
i ii
e h p u v wρρ=
= − + + +∑
∑∑==
==N
i ii
N
iii T
WRTRp
11ρρ
R : universal gas constant
Equation of State
Energy
CR
a a T a T a T a Tpi
ii i i i i= + + + +1 2 3
24
35
4
hR T
aa
Ta
Ta
Ta
TaT
i
ii
i i i i i= + + + + +12 3 2 4 3 5 4 6
2 3 4 5sR
a T a Ta
Ta
Ta
T ai
ii i
i i ii
0
1 23 2 4 3 5 4
72 3 4= + + + + +ln
Specific heat of constantpressure
Enthalpy
Enthopy
Coefficients in Cp, h, and s are calculated from JANAF Table by least square method. Coefficients for low-temperature (T<1000K) and high-temperature (T>1000K) exist.
Temperaturedependence!
p
p
C const
h C T
=
=
If is constant,γ21 1
1 2pe uρ
γ ρ= +
−
Governing Equations for Combustion Governing Equations for Combustion SystemSystem
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5757
′=∑ν ik ii
N
x1
′′=∑ν ik ii
N
x1
The most general opposing chemical reactions,
For example,:The number of elemental reactions is k.
2H+O O+OH↔then, 2,1 ,1 ,1 ,11, 1, 1, 1H O O OHν ν ν ν′ ′ ′′ ′′= = = =
The number of elemental reaction is k. -> Table is created.
Stoichiometric coefficients ofi-species, k-th reaction
( )∑=
′−′′=K
kkikikii RPW
1ννω&
i-species’ production rate [kg/m3/s]
Reaction rate [mol/m3/s](Two-body reaction)
( ) ( )RP k c k ck f k ii
N
b k ii
Nik ik
= −′
=
′′
=∏ ∏, ,χ
ν
χ
ν
1 1
i-th species’mole concentration
Forward rate constantBackward rate constant
Governing Equations for Combustion Governing Equations for Combustion SystemSystem
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Forward rate constant (Modified Arrhenius)
k A T EaRTf k k
n kk, exp= −⎛
⎝⎜⎞⎠⎟
Activation EnergyCollision frequencyBackward rate constant
kkKcb k
f k
k,
,= Equilibrium constant
of concentration( )
Kc KppRTk k
atmik ik
i
N
=⎛⎝⎜
⎞⎠⎟∑ ′′ − ′=
ν ν1
( ) ( )KpsR
hR Tk ik ik
i
iik ik
i
ii
N
i
N
= ′′ − ′⎧⎨⎩
⎫⎬⎭− ′′ − ′
⎧⎨⎩
⎫⎬⎭
⎡
⎣⎢
⎤
⎦⎥
==∑∑exp ν ν ν ν
0
11
(patm=1 atm)
Equilibrium constant of pressure : function of entropy s and enthalpy h
They are calculated using experimental data
Governing Equations for Combustion Governing Equations for Combustion SystemSystem
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ˆˆEAQ
∂∂
=ˆˆ
GQ
∂∂
ˆˆ
FQ∂∂, , =
( )( )
( )( ) ( )
1
1
1
1
1 1 1
0 0 0 0
1
x y z
x p u x M y x N z x L x e x x N
y p x y M y N z y L y e y y N
z p w x z M y N z L z e z z N
x M y N z L e N
x y
k k kk p k u p k u k p k u k p k p k p k pk p k k p k p k k p k p k p k pk p k w k p k w p k w p k p k p k p
p H k H p k H p k H p p p p
Y k Y k Y k
ρ ρ
υ ρ ρ
ρ ρ
ρ ρ ρ
θ θθ υ θ υ υθ θ
θ θ θ θ θ θ θ
θ
− + + + +− + + + +− + + + +
− + + + +
−
L
L
L
L
L
1 0 0
0 0
z
N x N y N z N
Y
Y k Y k Y k Y
θ
θ θ
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥−⎢ ⎥⎣ ⎦
L
M M M M M M O M
L
Jacobian matrix:
e pHρ+
= wkvkuk zyx ++=θ,
Increase with species
θ θ θ θ θ θ, , , , ,+ − ⋅ ⋅ ⋅⎡
⎣⎢⎢
⎤
⎦⎥⎥
ak akN個
123
( )2 2 2 2
1
N
j j ej
a p Y p p H u v wρ ρ=
= + + − − −∑
Frozen speed of sound
Eigenvalue
pC R
hC
RR Ti
j pjj
N
j jj
N i
j pjj
N
j jj
N iρ
ρ ρ
ρ
ρ=
−−
⎧
⎨⎪⎪
⎩⎪⎪
⎫
⎬⎪⎪
⎭⎪⎪= =
=
=∑ ∑
∑
∑1
11 1
1
1
Pressure differential by i-species’ density
Governing Equations for Combustion Governing Equations for Combustion SystemSystem
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Equations have a stiffness problem for reactive flow
-> Point implicit method is applied.
Though the point implicit affects on time accuracy for DNS simulation, an explicit integration is used with significantly small time step.
Calculation for source term:
Governing Equations for Combustion Governing Equations for Combustion System: Source TermSystem: Source Term
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1[ ( ) ]n n n n nj j j
EQ t E Q tSx Q
+∂ ∂Δ + Δ + Δ + = Δ
∂ ∂L
2
nn nj j
j
t S EI Q t tSQ x
⎡ ⎤Δ ∂ ∂⎛ ⎞− Δ = −Δ + Δ⎜ ⎟⎢ ⎥∂ ∂⎝ ⎠⎣ ⎦
Unknown
Known
Governing Equations for Combustion Governing Equations for Combustion System: Source TermSystem: Source Term
Crank-Nicholson type implicit method
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ˆˆSQ∂
=∂ ( ) ( ) ( )
( ) ( ) ( )
( ) ( )
1 1 1 1 1 1 1 1
1
1
0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0
j N
i i i i i i i i
j N
N N N
u v w e
u v w e
u v
ω ω ω ω ω ω ω ωρ ρ ρ ρ ρ ρ ρ
ω ω ω ω ω ω ω ωρ ρ ρ ρ ρ ρ ρ
ω ω ωρ ρ ρ
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂∂ ∂ ∂
L L
L L
L L
L L
L L
& & & & & & & &L L
M M M M M M M M
& & & & & & & &L L
M M M M M M M M
& & & &
( ) 1
N N N N N
j Nw eω ω ω ω ωρ ρ ρ ρ
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥∂ ∂ ∂ ∂⎢ ⎥
∂ ∂ ∂ ∂ ∂⎢ ⎥⎣ ⎦
& & & &L L
Jacobian matrix for source term is neccesaryfor point implicit.
1 1
000ˆ00
i
S J S J
ω
− −
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥
= = ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦&
Increase with species
Governing Equations for Combustion Governing Equations for Combustion System: Source TermSystem: Source Term
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( ) ( ) ( ), ,
1 1 1
lk lkN NK
f k b ki ii ik ik l l
k l l
dk dkT T W c cT dT dT
ν ν
χ χω ∂ω ν νρ ρ ∂ ρ
′ ′′
= = =
⎡ ⎤⎧ ⎫∂ ∂ ∂ ′′ ′= = ⋅ − −⎨ ⎬⎢ ⎥∂ ∂ ∂ ⎩ ⎭⎣ ⎦∑ ∏ ∏
& &
Example of each component in Jacobian matrix for source term
( )2 2 2
1 1 1
1 1 121
N N N
j j j pj j jj j j
T u v wR C Rρ ρ ρ ρ
= = =
⎡ ⎤⎢ ⎥∂ ⎧ ⎫⎢ ⎥= − + +⎨ ⎬∂ ⎢ ⎥⎩ ⎭−⎢ ⎥⎣ ⎦
∑ ∑ ∑
Differential of forward rate constant
( ) ( )dkdT Kc
dkdT
kT
hR T
b k
k
f k f kik ik
i
iik ik
i
N
i
N, , ,= − ′′ − ′
⎧⎨⎩
⎫⎬⎭− ′′ − ′
⎡
⎣⎢
⎤
⎦⎥
⎡
⎣⎢⎢
⎤
⎦⎥⎥==
∑∑111
ν ν ν νDifferential of backwardrate constant
dkdT
kT
nEaRT
f k f kk
k, ,= +⎛⎝⎜
⎞⎠
Governing Equations for Combustion Governing Equations for Combustion System: Source TermSystem: Source Term
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Inversion of Jacobian matrix for source term:
21 22
ˆˆ2
n
I O
t SM IQ
A A
⎛ ⎞⎜ ⎟⎜ ⎟⎛ ⎞Δ ∂ ⎜ ⎟= − =⎜ ⎟⎜ ⎟∂ ⎜ ⎟⎝ ⎠⎜ ⎟⎜ ⎟⎝ ⎠
M
L L L L
M
M
M
Inverse matrix is calculated by Gauss Jordan method.Matrix is divided into four.In the case of 9 species, 14x14 matrix becomes 9x9 matrix (3D).
1
1
1 121 22 22 21 22
I O I O
MA A A A A
−
−
− −
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟= =⎜ ⎟ ⎜ ⎟
− −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
M M
L L L L L L L L
M M
M M
M M
Inverse matrix
Governing Equations for Combustion Governing Equations for Combustion System: Source TermSystem: Source Term
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Definition of temperature : energy is a polynomial of temperature.Therefore it is impossible to calculate temperature directly.
( )2 2 2
1 1 2
N N
j j j jj j
e h R T u v wρρ ρ= =
= − + + +∑ ∑
hR T
aa
Ta
Ta
Ta
TaT
i
ii
i i i i i= + + + + +12 3 2 4 3 5 4 6
2 3 4 5
( ) ( )2 2 2
1 1 2
N N
j j j jj j
F T h R T u v w eρρ ρ= =
= − + + + −∑ ∑
Newton method is applied to obtain temperature iteratively.
Two or three iterations are enough.
Governing Equations for Combustion Governing Equations for Combustion System: Determination of TemperatureSystem: Determination of Temperature
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SummarySummary
Numerical simulation on detonation has to use Numerical simulation on detonation has to use compressible governing equations.compressible governing equations.But viscous effects are quite limited except DDT.But viscous effects are quite limited except DDT.Chemical reaction model including high pressure Chemical reaction model including high pressure effects is better but should be more research.effects is better but should be more research.
Numerical simulation on detonation using a Numerical simulation on detonation using a detailed reaction model requires high hardware detailed reaction model requires high hardware performance and high numerical techniques yet. performance and high numerical techniques yet.
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• K.K. Kuo, Principle of Combustions• M. Hishida, Ph.D Thesis in Nagoya University,
1993
ReferencesReferences
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•To Prof. Vladimir Molkov for the invitation
•To Morley Robert for assist of visiting Belfast
•To Prof A.Koichi Hayashi for his special advices
ThanksThanks