Introduction to Finite Volume projection methods
On Interfaces with non-zero mass ux
Rupert Klein
Mathematik & Informatik, Freie Universität Berlin
Summerschool SPP 1506
Darmstadt, July 09, 2010
On interfaces with non-zero mass ux
Rankine-Hugoniot conditions
Lax' entropy condition
Flame speed laws: sharp vs. diffuse interfaces
Multiscale ame structure modelling
Discontinuities
Rankine-Hugoniot conditions
t
Conservation law
ut + f (u)x = 0
u = ul
u = ur
Admits shocks with jump conditions
D [ u]] = [ f ]
x
On interfaces with non-zero mass ux
Rankine-Hugoniot conditions
Lax' entropy condition
Flame speed laws: sharp vs. diffuse interfaces
Multiscale ame structure modelling
Discontinuities
Lax' entropy condition
19
!"#########$%&'(#)*+,&-!,.&,
0
01'
3"########*+45!67!,.&,
0
02'
01'
02'
02'
0
02'
01'
/
0
01'
/
Unknowns:
2n for
+ 1shocks and
Unknowns:
2n + 1 (b).
Figure 3.
Characteristic diagrams
detonations (a) and deflagrations
Jump conditions:
n
Jump conditions:
n
Incoming characteristics: n + 1
Incoming characteristics:
n
and temperature
(or density) are known. (For
a non-reactive front, such as
somethings required
ame speed law required
a shock wave, these reduce to the condition that the species mass fractions
do not change across the front.)
Differentiating the fluxes ρYi u in the species transport equations (2) and
Discontinuities
Lax' entropy condition
19
!"#########$%&'(#)*+,&-!,.&,
0
01'
3"########*+45!67!,.&,
0
02'
01'
02'
02'
0
02'
01'
/
0
01'
/
Unknowns:
2n for
+ 1shocks and
Unknowns:
2n + 1 (b).
Figure 3.
Characteristic diagrams
detonations (a) and deflagrations
Jump conditions:
n
Jump conditions:
n
Incoming characteristics: n + 1
Incoming characteristics:
n
and temperature
(or density) are known. (For
a non-reactive front, such as
somethings required
ame speed law required
a shock wave, these reduce to the condition that the species mass fractions
do not change across the front.)
Differentiating the fluxes ρYi u in the species transport equations (2) and
On interfaces with non-zero mass ux
Rankine-Hugoniot conditions
Lax' entropy condition
Flame speed laws: sharp vs. diffuse interfaces
Multiscale ame structure modelling
The choice of G0 is arbitrary but fixed for a single combustion event. The
flame surface(s) G = G0 naturally decompose the flow domain into unburnt
gas (G < G0 ) and burnt gas regions (G > G0 ). Differentiating (59) with
respect to time and using (58) one finds
Discontinuities
∂G dxf
∂G
+
· ∇G =
+ D · ∇G = 0 ,
∂t
dt
∂t
Typical ame
speed law
the G-equation.
(61)
Flame Front
G(x,t) = G0
Temperature
Fuel
ns
burnt
G > G0
unburnt
G < G0
Figure 4.
Schematic representation of premixed flame front propagation
The key physical ingredients of the level set approach are the burning
Aturb
velocity law determining s as a function of thermo-chemical
and flow conS
=
S
(p
,
T
)
`
u
u
ditions and some local features of the flame
geometry. It is important to
Aresolved
notice that s is defined as the relative velocity between points on the front
and the unburnt gas immediately in front of it. The relative velocity sb
between the burnt
gas and of
theunburnt
front differs
from
s because of the thermal
Function
gas
conditions
gas expansion within the flame front and the associated jump of the normal
velocity. Because of mass conservation the mass flux density normal to the
⇓
front does not change across the discontinuity and the burnt gas relative
speed is easily computed as
Smearing of the interface is dangerous
ρ
ρs = (ρs)b = ρu s
⇒
sb =
u
ρb
s.
(62)
Although both the flow velocity and the relative speed between flow field
and front change across the flame, their sum, namely the vector D appear-
The choice of G0 is arbitrary but fixed for a single combustion event. The
flame surface(s) G = G0 naturally decompose the flow domain into unburnt
gas (G < G0 ) and burnt gas regions (G > G0 ). Differentiating (59) with
respect to time and using (58) one finds
Discontinuities
∂G dxf
∂G
+
· ∇G =
+ D · ∇G = 0 ,
∂t
dt
∂t
Typical ame
speed law
the G-equation.
(61)
Flame Front
G(x,t) = G0
Temperature
Fuel
ns
burnt
G > G0
unburnt
G < G0
Figure 4.
Schematic representation of premixed flame front propagation
The key physical ingredients of the level set approach are the burning
Aturb
velocity law determining s as a function of thermo-chemical
and flow conS
=
S
(p
,
T
)
`
u
u
ditions and some local features of the flame
geometry. It is important to
Aresolved
notice that s is defined as the relative velocity between points on the front
and the unburnt gas immediately in front of it. The relative velocity sb
between the burnt
gas and of
theunburnt
front differs
from
s because of the thermal
Function
gas
conditions
gas expansion within the flame front and the associated jump of the normal
velocity. Because of mass conservation the mass flux density normal to the
⇓
front does not change across the discontinuity and the burnt gas relative
speed is easily computed as
Smearing of the interface is dangerous
ρ
ρs = (ρs)b = ρu s
⇒
sb =
u
ρb
s.
(62)
Although both the flow velocity and the relative speed between flow field
and front change across the flame, their sum, namely the vector D appear-
On interfaces with non-zero mass ux
Rankine-Hugoniot conditions
Lax' entropy condition
Flame speed laws: sharp vs. diffuse interfaces
Multiscale ame structure modelling∗
Smiljanowski et al., Comb. Theor. & Mod., 1, (1997); Schmidt & Klein, Comb. Theor. & Mod., 7, (2003)
Towards a generalized Level-Set/In-Cell
Reconstruction Approach for Accelerating
Turbulent Premixed Flames
Matthias Münch1, Heiko Schmidt1, Rupert Klein1,2
1
2
Mathematik & Informatik, Freie Universität Berlin
Data & Computation, Potsdam Institut für Klimafolgenforschung
email: [email protected] [email protected]
The authors gratefully acknowledge fruitful, stimulating discussions with as well as direct software support by Dr. Alan Kerstein and Dr. Scott Wunsch
(SANDIA Nat. Lab. Livermore), and Dr. Fan Zhang and Dr. Paul Thibault (Combustion Dynamics, Ltd., Canada). This project has been partly supported
by the European Commission (grant FI4S-CT96-0025) and the Deutsche Forschungsgemeinschaft (grant KL 611/7)
Overview
•
Motivation
•
Governing Flow Equations
•
Level-Set Approach
•
Flame-Flow Coupling by In-Cell-Reconstruction
•
Unsteady Flame Structure Modelling
•
Some Results
•
Conclusions and Outlook
Motivation
•
Capturing/Tracking schemes are able to model accelerating premixed Flames
with the following features [10]
– flame is thin compared to the large geometrical scales
– Flamelet ansatz; flame physics condensed into a burning rate law
– standard Rankine-Hugoniot jump conditions apply
•
They have the following limitations
– inherent assumption of a quasi stationary flame structure
– depend on availability of burning rate law
•
Goal
– Modular numerical approach for general unsteady flame structures
Governing Equations
ρ̄t
+ ∇ · (ρ̄ṽ)
= 0
(ρ̄ṽ)t + ∇ · (ρ̄ṽ ◦ ṽ) + ∇p̄ + 23 ρ̄k̃
(ρ̄Ẽ)t + ∇ · (ṽ(ρ̄Ẽ + p̄ + 23 ρ̄k̃))
1
= ( Re
+ µt)∇ · σ
1
= ( Re
+ µt)∇ · (σ · ṽ)
+
γ
γ−1 ∇
1
· (( RePr
+ σµt )∇ ρ̄p̄ )
h
1
+ σµt )∇k̃)
+ ∇ · (( Re
k
(ρ̄Ỹ )t + ∇ · (ρ̄Ỹ ṽ)
ρ̄Ẽ
=
p
γ−1
1
= ∇ · (( ReSc
+ σµt )∇Ỹ ) + ρ̄!
ωY
Y
+ 12 ρ̄ṽ · ṽ + ρ̄Ỹ Q + ρ̄k̃
Characteristic numbers
Re =
ρref uref lref
µ
Pr =
µcp
λ
Sc =
µ
ρref D
Level-Set Method
Illustration for the flamelet regime
• Next to the flame front
Gt + (&vu + st&n) · ∇G = 0
(1)
G<0
unburnt
sl
st
G>0
burnt
• At larger distances
|∇G| = 1
(2)
• Standard methods can be used in
both steps, [9], [11].
dA
dA
Damköhler’s ansatz:
st =
δA
slam
δA
Flame-flow coupling by in-cell-reconstruction
Reconstruction of burnt and unburnt states
• Integral constraint for conserved quantities
α U u + (1 − α) U b = U
• Rankine-Hugoniot conditions
Dfront [[U ]] − [[F (U )]] = 0
1−α
(3)
(4)
α
Ub
Fb
Area-weighted flux composition
• Flux densities for states “u” & “b”
Fν = F
God.
(U lν , U rν )
(ν ∈ {u, b})
discrete
Flame Front
Uu
(5)
β
Fu
Net fluxes for “mixed cells”
• Net flux
F = β F u + (1 − β) F b
(6)
Unsteady Flame Structures
Integral Conservation Laws
" x2
∂U
dx + (F (x2, t) − F (x1, t)) = 0 (7)
∂t
x1
In a moving co-ordinate system
" ξ2
∂U
Dbox [[U ]] − [[F (U )]] =
ξ1 ∂t
δ << L
Temperature
τ
Fuel
#
#
# dξ
#
(8)
ξ
x
∂U /∂t ≡ (Dbox − Dfront) ∂U /∂ξ
Dfront [[U ]] − [[F (U )]] = 0
ξ
dx = D
dt
box
• Stationary Structure
(9)
(Standard Rankine-Hugoniot Conditions)
• Unsteady Structure
t
Eq. (8) is a generalized jump condition!
Flame with unsteady internal structure
Results so far:
Conservation Laws
LevelSet Approach
Generalized in-cell-reconstruction
Ansatz of
Coupling
• Jump conditions from (8) instead of (4)
Smiljanovski,
via
Moser and Klein
InCell
Reconstruction
• Calculate
ρs, Q and source terms
" ξ2
#
∂U #
∂t ξ dξ by solving quasi-1D un-
strategy
ξ1
steady flow equations in moving frame
of reference (module).
burning velocity s
Boundary Conditions
specific heat release Q
• Choice of module depends on the turbulent combustion regime
• The structure calculation accounts for
internal physical effects, which are not
active in the outer flow but essential for
the front motion and its effect on the
surrounding fluid.
R.H.Integrals
generalized modular
coupling procedure
Interface to
Flame Structure Module
Compressible Flow
Incompressible Flow
Equations
Equations
?
GLEMAnsatz
(Kerstein, Menon)
Description of quasi
onedimensional
laminar Flames
Our choice of
pdfAnsatz
ODTAnsatz
LEMAnsatz
(Pope)
(Kerstein)
(Kerstein)
stochastic
Turbulence Models
Scales and Regimes in turbulent combustion
Time Scales
Da =
tturb
tchem
Length Scales
lF2
Ka = 2
η
! !
Borghi-diagram: Regimes in turbulent combustion
Test Case: Laminar Flame
1.04
Y
timestep 10
timestep 30
timestep 100
timestep 220
p
(pressure)
Flame structure module
• 1D compressible Navier-Stokes solver
1.02
1
• explicit “shock capturing” technique
• 2nd order inner/outer interpolation
0
2
4
x
1
• one-step Arrhenius kinetics
• actvation energy (Ea/RTu = 20, Tb/Tu = 4)
Molecular transport
• Reynolds No’s. Reoutside = 104, Remodule = 102
• Prandtl No. P r = 0.72, Lewis No. Le = 1
8
10
timestep
0
timestep
50
timestep 100
timestep 1000
0.8
Reaction model
6
0.6
Y
(fuel mass fraction )
0.4
0.2
0
0
0.02
0.04
ξ
0.06
0.08
0.1
0.8
1
0.016
0.012
s
0.008
(burning velocity)
0.004
Ignition of laminar flame:
Large scale pressure field (top); Fuel mass fraction(middle); Effective burning velocity (bottom).
0
0
0.2
0.4
t
0.6
Turbulent Combustion I
Flame structure module
• Hybrid algorithm:
– 1D low-Mach RANS solver (PISO)
1.2
Turbulent combustion model
• Pope’s pdf-Ansatz, [7], [8]
• 8-reaction H2–O2 scheme (Maas/Warnatz)
Open questions:
• Deterministic / non-deterministic model coupling
p
1.15
(pressure)
1.1
1.05
1
0
– pdf-Monte-Carlo for enthalpy and species, [12]
• k − . turbulence model
Mass Fraction Y
50 * dt
120 * dt
190 * dt
1
2
3
4
5
6
0.03
7
8
9 10
x[m]
1000*dt
4000*dt
7000*dt
0.025
0.02
0.015
0.01
0.005
0
0
Y
(fuel mass fraction)
5
10
15
20
25 30
x[cm]
0.7
0.6
low turbulence intensity
high turbulence intensity
D
(effective flame speed)
0.5
0.4
0.3
Ignition of a flame in a turbulent flow field:
Large scale pressure (top); Hydrogen mass fraction (middle); Effective Flame velocity (bottom).
0.2
0.1
0
0
1
2
3
4
5
6
7
t
Turbulent Combustion II
1
0.8
Flame structure module
• 1D zero Mach number solver (projection
method)
• turbulent transport with Linear Eddy Model
(LEM), [3]
• Coupling to outside solution via prescribed inflow data and integral mean values
0.6
Y
0.4
0.2
0
4.6
4.8
5
5.2
5.4
x
1
0.8
0.6
Y
0.4
0.2
Reaction model
• one-step Arrhenius kinetics
• aktvation energy (Ea/RTu = 20, Tb/Tu = 4)
0
4.7
4.8
4.9
5
5.1
5.2
x
1
0.8
0.6
Y
0.4
Flame Structures for varying turbulent Reynolds Numbers Ret:
Fuel mass fraction for fixed chemical time and turbulent length scale;
Ret = 3 (top), 10 (middle), 100 (bottom)
0.2
0
3
4
5
x
6
7
First 3D Test Case
Simplified Flame structure module
• GLEM-Method [5]
• Flame is described by a Level-Set Function
• turbulent transport by Linear Eddy Model is applied to the G-Field (gridless implementation)
• prescribed turbulence parameters
• Each Flamelet propagates with a prescribed
laminar burning speed
• Model only valid in Flamelet Regime
Flame kernel evolution in prescribed turbulence field: Snapshots of Level Set G = 0.
Work in Progress and Outlook
Flame structure modules
• Full coupling to outer solution using its (k − .)-turbulence model
data to determine turbulent length and time scales
• LEM / GLEM as a flame structure module in 3D channel flows with
obstacles
• ODT-based combustion module for DDT applications (OneDimensional-Turbulence, Kerstein et al.)
• Flame structure module based on turbulent combustion models for
the ,,thin reaction zone“-regime, [6]
Flame kernel evolution over obstacles: Turbulent length and time scales for
structure module are provided by k − " turbulence model; Plots of isoline G = 0
at different times and vector plot of flow field
Conclusions
The proposed generalized numerical In-Cell-Reconstruction strategy ...
•
extends our level set technique to complex internal front structures
•
handles non-stationary flame structures in a modular fashion
•
flexibly models a wide range of different combustion regimes
•
provides a basis for the modelling of Deflagration-to-Detonation-Transition (DDT)
in large scale systems
References
[1] Bourlioux, A., Private Communication, University of Montreal, Canada
[2] Fedkiw, R., Aslam, T., Merriman, B. and Osher, S., A Non-oscillatory Eulerian Approach to interface in Multimaterial Flows (the Ghost Fluid
Method), Journal of Computational Physics, 152, 457-492, 1999
[3] Kerstein, A. R., Linear-eddy model of turbulent transport and mixing, Combustion Sci. and Tech., 60, 391-421, 1988
[4] Maas, U., Warnatz, J.: Ignition processes in hydrogen-oxygen mixtures, Combustion and Flame, 74, 53-69, 1988
[5] Menon, S. und Kerstein, A. R., Stochastic Simulation of the structure and propagation rate of turbulent premixed flames, Twenty-Fourth Symposium
(International) on Combustion, 443-450, The Combustion Institute, Pittsburgh, 1992
[6] Peters, N.: Turbulent Combustion. Cambridge University Press, 2000
[7] Pope, S. B.: A Monte Carlo method for the PDF equations of turbulent reactive flow. Combustion Science and Technology, 25, 159-174, 1981
[8] Pope, S. B.: PDF methods for turbulent reactive flows. Prog. Energy Combustion Science, 11, 119-192, 1985
[9] Sethian, J. A., Level Set Methods and Fast Marching Methods, Cambridge University Press, 1999
[10] Smiljanovski, V., Moser, V. and Klein, R., A Capturing-Tracking Hybrid Scheme for Deflagration Discontinuities, Journal of Combustion Theory and
Modelling, 2(1), 183-215, 1997
[11] Sussman, M., Smereka, P. and Osher, S., A Level Set Approach for Computing Solutions to Incompressible Two-Phase Flow, Journal of Computational
Physics, 114, 146-159, 1994
[12] Zhang, F. und Thibault, P., A PDF-Module for Turbulent Flame Structure Computation, Final Report on subcontracted work for EC-Project FI4SCT96-0025, Combustion Dynamics Ltd., Halifax, Canada, 1998