Data assimilation applied to combustion

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