3
e
colloque INCA, 17
-
18 novembre 2011
1
M. Rochoux
, B. Cuenot, S. Ricci, A. Trouvé, B. Delmotte,
S. Massart, R. Paoli, R.
Paugam
.
Data assimilation applied to combustion
Thèse de doctorat (2010
-
2013)
2
Data assimilation
Physical system
↘ MEASUREMENTS
Observable quantity
Uncertainty quantification
↘ NUMERICAL MODEL
Control variables
Uncertainty quantification
Integrate observations into a running model in such way
as to
minimize the erro
r, using known error statistics on
both simulated and
observed
data.
↘ PRINCIPLE
3
Data assimilation
Physical system
↘ MEASUREMENTS
Observable quantity
Uncertainty quantification
↘ NUMERICAL MODEL
Control variables
Uncertainty quantification
Compare experiment and simulation.
Quantify and reduce uncertainties.
Optimize an observation network.
↘ BENEFITS
4
Application to combustion
Sources of
uncertainties in CFD
•
Simplification of the physics
↘
Ex.: Turbulent flame speed, burner extinction.
•
Physical and numerical parameters.
•
Boundary conditions
↘
Ex.: Heat transfer to the walls, spray injection.
•
Initial condition
↘
Ex.: Burner ignition.
Potential of data assimilation
•
Improve simulations via
improved
initial/
boundary
condition.
•
Improve physical
models
via
calibrated
parameters.
•
Optimize
place and time of probe
measurements
.
•
Bénédicte Cuenot
•
Sophie Ricci
•
Denis Veynante
•
Nasser Darabiha
•
Arnaud Trouvé
5
Data assimilation for wildfire spread
Towards a more accurate prediction of flame propagation.
Research framework
Today’s outline
↘
Data
-
driven model parameter estimation for
flame propagation
.
①
Description of the physical system.
②
Data assimilation algorithm.
③
Validation on a synthetical case.
④
Application to a real case of fire propagation.
6
1. Description of the physical system
Model of flame propagation
•
Objective
↘
Build a simplified model of premixed flame that gives the time
-
evolution of the flame front.
•
Scalar progress variable c
↘
Interface between burnt and fresh fuel (c = 0.5).
↘
Front propagating at the local flame speed
ϒ
.
ϒ
c =
1
c =
0
c = 0.5
FRESH
AREA
FRONT
BURNT
AREA
Level
-
Set equation
Local flame speed
↘
Parameterization in terms of a
reduced number of parameters.
↘
Linked to fuel mixture and
flow conditions.
2
-
D computational domain
BURNT AREA
FRESH AREA
FRONT
Proportionality
coefficient (m/s)
7
1. Description of the physical system
Model of flame propagation
•
Objective
↘
Build a simplified model of premixed flame that gives the time
-
evolution of the flame front.
•
Scalar progress variable c
↘
Interface between burnt and fresh fuel (c = 0.5).
↘
Front propagating at the local flame speed
ϒ
.
ϒ
c =
1
c =
0
c = 0.5
FRESH
AREA
FRONT
BURNT
AREA
Level
-
Set equation
Local flame speed
Random
field
of fuel
mass fraction
2
-
D computational domain
BURNT AREA
FRESH AREA
FRONT
8
1. Description of the physical system
Simulation vs. Experiments
•
Example of fire spread simulation
↘
Constant coefficient:
P = 0.1 m/s
.
↘
Size of computational domain:
300m x 300m
.
↘
Initial condition:
semi
-
circular front
.
Input data
↘ Random fuel mass fraction
Simulation outputs
↘ Time
-
evolving location of
the flame front (from t=0 to
t=800s)
Observation
at
t
=800s
o
How to compare quantitatively simulation
and experiments?
o
How to make simulations more reliable?
Data assimilation for parameter calibration
9
2. Data assimilation algorithm
Variables of interest
STEP. 1: Description of the physical system
•
Observation vector
y
o
↘
2
-
D coordinates of the points defining the
observed
fronts.
↘
Several
observed
fronts over the assimilation time window [0,T].
10
2. Data assimilation algorithm
Variables of interest
STEP. 1: Description of the physical system
•
Observation error covariance matrix
R
↘
Observation error following a Gaussian distribution N(0,
R
).
↘
Uncorrelated errors in space and time:
diagonal
matrix R.
11
2. Data assimilation algorithm
Variables of interest
STEP. 1: Description of the physical system
•
Control vector
x
↘
Contains the control parameters that are to be optimized.
↘
Estimate of the
true
value
x
t
, starting from an a priori value
x
b
.
Background
12
2. Data assimilation algorithm
Variables of interest
STEP. 1: Description of the physical system
•
Background error covariance matrix
B
↘
Background error following a Gaussian distribution N(0,
B
).
↘
Diagonal elements > Error variance on each control parameter.
13
2. Data assimilation algorithm
Variables of interest
STEP. 2:
Definition of the observation operator
H
•
Non
-
linear
operator, resulting from a 2
-
step operation:
↘
Model integration over the assimilation time window.
↘
Selection operator (from field c to front isocontour c=0.5).
2. Data assimilation algorithm
Variables of interest
STEP. 3:
Distance between simulated and observed fronts
•
Formulation of the
innovation vector
d
ob
↘
At each observation time, projection of the simulated front
onto the
observed
front.
2. Data assimilation algorithm
Variables of interest
STEP. 4:
Gain matrix
K
•
Weight of the data assimilation correction.
•
Requires the Jacobian of the observation operator
H
↘
Not available analytically.
↘
Jacobian matrix called the tangent linear
H
.
here,
2. Data assimilation algorithm
Variables of interest
STEP. 5: Analysis
x
a
•
Feedback information for the model (inverse
problem
).
•
Optimal estimation of the
true
control vector
x
t
, such that the
variance
of its distance to
x
t
gets a minimum
.
16
17
2. Data assimilation algorithm
Best Linear Unbiased Estimator (BLUE)
•
Kalman filter approach
↘
Analysis
x
a
as the correction of the background vector
x
b
↘
Assumption of a
linear
observation operator
H
.
Analysis error covariance matrix
Adaptation of the BLUE algorithm:
iterative process
Reduction of the variance
of the distance between
fronts.
18
2. Data assimilation algorithm
Best Linear Unbiased Estimator (BLUE)
•
Kalman filter approach
↘
Analysis
x
a
as the correction of the background vector
x
b
↘
Assumption of a
linear
observation operator
H
.
Analysis error covariance matrix
In the observation space
Reduction of the variance
Validation criteria
19
3. Validation
Synthetical case of flame propagation
•
Observation System Simulation Experiment (OSSE)
↘
Known
true
control vector, b
ackground
and observation errors.
↘
Generation of
synthetical observations
using the numerical model
o
Integrate numerical model using the
true
control vector.
o
Add noise to the fronts positions.
o
Quantify the quality of the
background correction.
o
Validate the data assimilation
algorithm.
20
3. Validation
Synthetical case of flame propagation
•
Validation experiment for one parameter calibration
↘
True
control vector:
x
t
= P
t
= 0.1
↘
Fixed
observation
standard deviation: σ
o
= 0.0073m.
o
Observed front discretized with 200 points.
o
16 observation times each 50s.
↘
Assimilation time window [0,800s].
↘
Background
x
b
varying
from 0.02 to 0.18
SCENARIO: High confidence in the
observation.
VARIABLE background standard
deviation σ
b
True value
From 20%
to 80%
From
-
20%
to
-
80%
Analysis
x
a
equal to
x
t
with less than 0.1%
(very low number of external loops
-
2 or 3).
RESULTS
Error standard deviation on the
analysis
x
a
systematically reduced
Ignition times
21
4. Application to a real case
Natural
fire
propagation
•
Real observations of flame positions
↘
Domain of propagation: 4m x 4m
↘
Homogeneous
grass
vegetation
o
Height: 8cm
o
Fuel loading: 0.4 kg/m
2
o
Moisture content: 21.7%
Wind (
1.3 m/s)
Data from Ronan
Paugam
,
Dept
. Of
Geography
,
King’s
College
of London
Rate of spread
From 0 to
0.02m/s
22
4. Application to a real case
Natural
fire
propagation
•
Model for the fire rate of spread
↘
New parameterization of the flame speed
o
vegetation characteristics (moisture content M
f
, surface
-
area
-
to
-
volume ratio Σ).
o
wind velocity along the normal direction to the front
u
.
Rothermel’s model
23
4. Application to a real case
Natural
fire
propagation
•
Calibration of 2 vegetal parameters
↘
Parameters that are critical for the rate of spread and that are
embedded with important uncertainties.
o
Moisture content.
o
Surface
-
area
-
to
-
volume ratio.
↘
Assimilation time window [51s,78s]
o
1 observation time at t=78s.
o
observation error: linked to spatial resolution 0.047m.
o
background error: 30% uncertainty.
Background
parameters
Background
variance
Mf
Σ
0.22
4921
4.4 x 10
-
3
2.1 x 10
6
Analysis
parameters
Analysis
variance
Mf
Σ
0.11
13193
1.18 x 10
-
5
1.6 x 10
5
Reduction of the variance on the
control parameters.
RESULTS
Control parameters
still
in their
domain of validity.
o
Σ far from typical value.
o
over
-
correction to compensate for the
uncertainties in the rate of spread
Γ.
24
4. Application to a real case
Natural
fire
propagation
•
Assimilation & Forecast
↘
Assimilation time window [51s,78s]
Analysis
parameters
Standard
deviation OMA
Mf
Σ
0.11
13193
0.0978m
Background
parameters
Standard
deviation
OMB
Mf
Σ
0.22
4921
0.331m
25
4. Application to a real case
Natural
fire
propagation
•
Assimilation & Forecast
↘
Forecast time window [78s,106s]
Analysis
parameters
Standard
deviation OMA
Mf
Σ
0.11
13193
0.415m
Background
parameters
Standard
deviation
OMB
Mf
Σ
0.22
4921
0.680m
↘
Assimilation time window [51s,78s]
ANALYSIS
t = 78s
FORECAST
t = 106s
Perspectives
↘ Utilisation de l’assimilation de données comme un
outil d’aide à la modélisation de la propagation des feux.
↘ Extension possible à d’autres problématiques en
combustion.
Résultats
↘
Gain
significatif
d’informations
sur
le
système
en
prenant
en
compte
les
observations
.
↘
Réduction
et
estimation
de
l’incertitude
sur
le
résultat
de
l’assimilation
de
données
.
↘
Prédiction
de
l’évolution
du
système
comme
en
météorologie
.
Data assimilation for flame propagation
•
Algorithm able to
retrieve
more
accurate
value of the
control parameters
↘
Turbulent flame speed, burner extinction.
•
Physical and numerical parameters.
•
Boundary conditions
↘
Ex.: Heat transfer to the walls, spray injection.
•
Initial condition
↘
Ex.: Burner ignition.
Conclusion
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