modeling and uncertainty quantification for airfoil icing · 2016-08-10 · modeling and...

Motivation/BackgroundData-Based UQ

Computational UQ

Modeling and Uncertainty Quanti�cation for Airfoil Icing

Anthony DeGennaroClarence W. Rowley III

Luigi MartinelliPrinceton University

JUP Quarterly MeetingAthens, OHAugust 2016

Anthony DeGennaroClarence W. Rowley IIILuigi MartinelliPrinceton UniversityModeling and Uncertainty Quanti�cation for Airfoil Icing


Computational UQ

Outline

1 Motivation/Background

2 Data-Based UQ

3 Computational UQ



Computational UQ

Topic


2 Data-Based UQ

3 Computational UQ



Computational UQ

Introduction

Wing icing deteriorates aerodynamics

Leading edge �ow separationbubble

Lower lift, higher drag

Unpredictable stall

Figure: Leading edge horn separation



Computational UQ

Introduction

Signi�cant ice shape variation, sensitivity to physical parameters1

Complex physics (aero-thermodynamics, macro/micro scale physics)

Uncertainty in physical parameters

Nondeterministic variation (�same conditions, di�erent shapes�)

−0.1 0.0 0.1 0.2−0.06

0.00

0.06

Figure: Wind tunnel experimental ice shapes

1Addy, H.E. Ice Accretions and Icing E�ects for Modern Airfoils. NASA TR 2000-210031.



Computational UQ

Introduction

Di�erent types of ice accretion2

�Rime� ice: cold temperature, low liquid water content

�Horn� ice: warm temperature, high liquid water content

Uncertainty in physical conditions can create uncertainty in performance

(a) Rime Ice (b) Horn Ice

2Beaugendre et. al. Development of a Second Generation In-Flight Simulation Code. J.Fluids Engineering, 2006.



Computational UQ

Introduction

Observations

Many experimental ice accretion results exist, but no one has modeled thisdata or inferred anything from it

Numerical methods/codes exist, but no one has systematically quanti�edstatistical e�ects of parametric uncertainty in physics

Questions

Can we infer/identify prominent ice shape features from database ofshapes?

We can link physical information about data to ice shape features?

Can we create a statistical modeling of icing using only data?

Can we investigate parametric uncertainty in numerical icing codes?



Computational UQ

Introduction

Research Goals

Data-driven, equation-free model of ice accretionModel ice accretion using only data, no physicsStudy random ice shapes corresponding to a range of icing conditionsCould be used to benchmark/correct numerical calculationsCould be used to explore non-deterministic ice shape variations

Computational model of ice accretionBuild numerical code to calculate ice accretion given physical inputsCompute ice shape for a range of physical conditionsPerform parametric UQ and study e�ects on aerodynamic performance



Computational UQ

Introduction

Research Approach

Data-driven, equation-free modeling of ice accretionCollect large number of ice shapes into a databaseModel database shape features using Proper Orthogonal Decomposition(POD)Link physical information to shape featuresBuild a purely data-driven, statistical ice accretion model (no equations)Perform parametric UQ, study performance variations, etc.

Computational model of ice accretionBuild droplet advection/impact simulator (C++)Build icing thermodynamics simulator (C++)Interface code with aerodynamic solver (FLO103)Perform parametric UQ, study performance variations, etc.



Computational UQ

Topic


2 Data-Based UQ

3 Computational UQ



Computational UQ

Dataset

−0.1 0.0 0.1 0.2−0.06

0.00

0.06


Dataset consists of 145 experimental ice shapes

Obtained in icing wind tunnel at NASA Glenn1

Representative of a wide variety of icing conditions (temperature, LWC,accretion time, etc.)



Computational UQ

Data-Driven Model

−0.1 0.0 0.1 0.2−0.06

0.00

0.06


Goal: Make a purely data-driven model of icing (no equations)Approach:

Build low-dimensional model of shape using POD

Correlate POD coe�cients to temperature, accretion time, LWC

Generate random ice shapes corresponding to given conditions



Computational UQ

Proper Orthogonal Decomposition (POD)

Goal/Utility

Statistical analysis tool

Extracts linear, orthogonal basis that optimally explains dataset

Method

Singular value decomposition of data matrix gives POD modes andeigenvalues

X =

x1 · · · xS

X = ΦΣV∗

x ≈M∑i

ciφi



Computational UQ

POD Eigenvalues

100 101 102

i

100

101

102

103

104

λi

(a) Magnitude.

100 101 102

M

0.0

0.2

0.4

0.6

0.8

1.0

M ∑ i=1λi/∑ λ

i

(b) Cumulative sum.

Figure: POD eigenvalues.

2/3 of cumulative sum contained in �rst 10 modes

2/3 of statistical variation contained in �rst 10 modes

Truncate model at 10 modes



Computational UQ

POD Modes

Figure: Mean and POD modes.

Represent ice shapes as composite sum of these pictures

Modes 1 and 2 simply add ice mass

Modes 3 and 4 switch between upper/lower surface horns and rime

Higher order modes contain more extreme shape perturbations



Computational UQ

POD Reconstructions

Figure: POD reconstructions.

Agreement is great for shapes close to mean, less good for extreme shapes



Computational UQ

Link Physical Conditions to Modes

−100−50 0 50 100 150 200 250Mode 1

−100

−50

0

50

100

150

Mod

e 2

0

20

Time (m

in)

−150−100−50 0 50 100 150 200Mode 3

−150−100−50

050100150

Mod

e 4

0

20

Time (m

in)

−100−50 0 50 100 150 200 250Mode 1

−100

−50

0

50

100

150

Mod

e 2

-20

-5

Tempe

rature (C

)

−150−100−50 0 50 100 150 200Mode 3

−150−100−50

050

100150

Mod

e 4

-20

-5

Tempe

rature (C

)

−100−50 0 50 100 150 200 250Mode 1

−100

−50

0

50

100

150

Mod

e 2

0.3

0.6

LWC (g/m

3)

−150−100−50 0 50 100 150 200Mode 3

−150−100−50

050100150

Mod

e 4

0.3

0.6

LWC (g/m

3)

Figure: POD coe�cients, colored with parameters.

Statistically relate time, temperature and LWC to POD modesInput conditions, output POD coe�cients that respect the data



Computational UQ

Data-Driven Icing Model

Input

� Time� Temperature

� LWC

Database

� Ice shapes� Conditions

Statistics

� Filtered coe�s� Random samples

Shape

X =∑

Miciφi

Figure: Flowchart of data-driven model.

Input physical condition ranges

Filter database for shapes that match conditions

Create POD coe�cient distributions for downselected dataGenerate random samples from these distributions

Reconstruct ice shape using data-inferred POD coe�cients



Computational UQ

Random Shapes

−0.100 0.000 0.075X/c

−0.06

0.00

0.06

Y/c

(c) Time > 10 min; temperature > -10 C; LWC > 0.45 g/m3

−0.100 0.000 0.075X/c

−0.06

0.00

0.06

Y/c

(d) Time > 10 min; temperature <-10 C; LWC < 0.45 g/m3

Figure: Random data-driven ice shapes.

These shapes were generated at random, no physics



Computational UQ

Uncertainty Quanti�cation

−0.1 0.0 0.1 0.2−0.06

0.00

0.06


Goal: Quantify performance variation with POD modesApproach:

Generate random samples in POD space with Latin Hypercube Sampling(LHS)

Test corresponding shapes with �ow solver

Quantify lift/drag statistics



Computational UQ

Latin Hypercube Samples

−0.15 0.00 0.15X/c

−0.075

0.000

0.075

Y/c

0.34

0.4

CL

0.25 0.30 0.35 0.40CL

0100200300400500600

Coun

t

0.00 0.05 0.10CD

02004006008001000

Coun

tFigure: Latin Hypercube samples.

1,921 LHS samples from 10-D modal space

LHS statistics re�ect database statistics



Computational UQ

Latin Hypercube Samples

−0.15 0.00 0.15X/c

−0.075

0.000

0.075

Y/c

0.34

0.4

CL

−0.15 0.00 0.15X/c

−0.075

0.000

0.075

Y/c

0.0125

0.02

σ(CL )

Figure: Spatial average and variance.

Spatial average: lower surface rime accretions are relatively benign

Spatial variance: performance sensitive to upper surface horns



Computational UQ

Topic


2 Data-Based UQ

3 Computational UQ



Computational UQ

Motivation

Investigate uncertainty in the physical process of icing

What is the statistical e�ect of uncertainty in physical parameters?Free-stream temperatureLiquid water content (LWC)Accretion timeDroplet diameter distribution (MVD)Surface roughness

Want to quantify how perturbations of the physics a�ects aerodynamics



Computational UQ

Airfoil Icing Code Flowchart

−0.02 0.00 0.02 0.04 0.06 0.08 0.10−0.04

−0.03

−0.02

−0.01

0.00

0.01

0.02

0.03

0.04

Clean Airfoil

Mesh/Flow Solver

−0.1 0.0 0.1 0.2 0.3−0.20

−0.15

−0.10

−0.05

0.00

0.05

0.10

0.15

Droplet Simulation

∂F∂t + ∇ · F = S

Thermodynamics

−0.02 0.00 0.02 0.04 0.06 0.08 0.10−0.04

−0.03

−0.02

−0.01

0.00

0.01

0.02

0.03

0.04

Iced Geometry

−0.02 0.00 0.02 0.04 0.06 0.08 0.10−0.04

−0.03

−0.02

−0.01

0.00

0.01

0.02

0.03

0.04

Final Geometry



Computational UQ

Droplet Advection

Advection Equations:

dx

dt= v

mdv

dt=

1

2ρgCDπr

2||vg − v||(vg − v) + mg

−0.05 0.00 0.05 0.10 0.15 0.20x/c

−0.05

0.00

0.05

0.10

y/c

Figure: R = 10µm

−0.05 0.00 0.05 0.10 0.15 0.20x/c

−0.05

0.00

0.05

0.10

y/c

Figure: R = 118µm



Computational UQ

Thermodynamics

Conservation Equations:

ρw

{∂hf∂t

+∇ · (ufhf )

}= mimp − mevap − mice

ρw

{∂(hf cWT )

∂t+∇ · (ufhf cWT )

}=

[cWTd +

u2

d

2

]mimp

− Levapmevap

+ (Lfus + ciceT )mice

+ cH(TRec − T )

MassEnters through impinging dropletsExits via evaporation/sublimation and freezing

EnergyEnters through impinging droplets, freezing of iceExits via evaporation/sublimation, radiation, convection

Solution procedure: explicit marching, �nite volume discretization withupwinded derivatives



Computational UQ

Thermodynamics Solution Procedure

∂U

∂t+∂F

∂s= S∫

Bk

(∂U

∂t+∂F

∂s

)ds =

∫Bk

S ds∫Bk

∂U

∂tds +

(Fk+1/2 − Fk−1/2

)=

∫Bk

S ds

∂Uk

∂t=

1

∆sk

∫Bk

S ds −∆Fk︸︷︷︸δu

Ut+∆tk = U

tk −∆tδu

KK-1 K+1

Fk-1/2 Fk+1/2Sk

Figure: Finite volume method.

Finite volume discretization of equations

Roe scheme �ux calculation (upwinding)



Computational UQ

Validations

Figure: Experiment (red), LEWICE (green), and CATFISh (blue).



Computational UQ

UQ Study: Temp + LWC

250 260 270T∞ (K)

0.30

0.65

1.00

LWC (g/m

3)

(a) Quadrature points (col-orscale identical to (b)).

(b) PCE surrogate.

4.0 4.5 5.0 5.5 6.0 6.5CLα

0.00.20.40.60.81.01.21.41.61.8

PDF(CLα)

(c) PCE surrogate statistics.

Figure: Quadrature points, PCE surrogate, and statistics for the 2 parameter (T∞ andLWC) study on lift slope.



Computational UQ

UQ Study: Temp + LWC + Time

T∞ (K)250

270

LWC (g/m

3)

0.3

1.0

T (min)

1.0

5.0


(b) PCE surrogate (parame-ter units identical to (a)).

−0.05 0.00 0.05 0.10 0.15X/c

−0.06

−0.04

−0.02

0.00

0.02

0.04

0.06

Y/c

(c) Ice shapes at quadra-ture points (colorscale iden-tical to (c)).

4.0 4.5 5.0 5.5 6.0 6.5CLα

0.00.20.40.60.81.01.21.41.6

PDF(CLα)

(d) PCE surrogate statistics. (e) Statistics of CLα as afunction of ∆T.

Figure: Quadrature points, PCE surrogate, and statistics.



Computational UQ

UQ Study: Temp + LWC + Roughness

T∞ (K)

250

270

LWC (g/m

3)

0.3

1.0

ks (m

m)

0.1

1


(b) PCE surrogate (parame-ter units identical to (a)).

−0.05 0.00 0.05 0.10 0.15X/c

−0.06

−0.04

−0.02

0.00

0.02

0.04

0.06

Y/c

(c) Ice shapes at quadra-ture points (colorscale iden-tical to (c)).

4.0 4.5 5.0 5.5 6.0 6.5CLα

0.0

0.2

0.4

0.6

0.8

1.0

PDF(CLα)

(d) PCE surrogate statistics. (e) Statistics of CLα as afunction of ∆T.

Figure: Quadrature points, PCE surrogate, and statistics.



Computational UQ

Conclusions

Problems:

Wing icing deteriorates icing aerodynamics, danger to safe �ight

Ice shapes are diverse and complex

Not clear what the exact aerodynamic e�ects of di�erent shapes are

Would like systematic way of exploring airfoil icing through data

Solutions:

Data-driven approachBuild empirical model of ice accretion from dataPerform parametric UQ to quantify range of performance

Computational approachBuild computational ice accretion code from scratchPerform parametric UQ to quantify e�ects of physics


modeling and uncertainty quantification for airfoil icing · 2016-08-10 · modeling and...

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