Data assimilation applied to combustion

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22 Φεβ 2014 (πριν από 3 χρόνια και 3 μήνες)

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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
.



Description of the physical system.


Data assimilation algorithm.


Validation on a synthetical case.


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