Quantification of Epistemic Uncertainties in the k-e Model ... -...

Quantification of Epistemic Uncertaintiesin the k − ε Model Coefficients

By:

Saeed Salehi

[email protected]

Gothenburg Region OpenFOAM User Group Meeting,

November 20th, 2019

Chalmers University of Technology

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mailto:[email protected]

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Introduction Uncertainty Quantification Efficient UQ Method Numerical Example

Introduction

Saeed Salehi Quantification of epistemic uncertainties in the k − ε model coefficients OFGBG19 0/ 19

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Uncertainty

“Uncertainty is the only certainty there is, and knowing how to livewith insecurity is the only security.”

— John Allen Paulos

Uncertainties are present in most engineering and practical applications.

Sources of Uncertainty:X physical propertiesX initial conditionsX boundary conditionsX geometryX model parametersX etc.

Despite these uncertainties can predictions be trusted?


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Uncertainty in Turbomachines

Turbomachines can be intensely sensitive to theuncertainties.

Aleatory uncertainties:X Operating conditions

X Geometry

Operational: Volumetric flow rate, turbulence properties,rotational speed.

Geometrical:X Manufacturing tolerances.

X Turbomachines are designed to run for years: in-serviceerosion and corrosion.


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Uncertainty Quantification Using Polynomial ChaosExpansion


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Uncertainty Quantification

How do input uncertainties affect objective functions are affected?Non-deterministic CFD is a growing field. (e.g. NUMECA)First step: identification of the sources of uncertainties and their PDFs.

Uncertainty Quantification

Quantitative characterization and reduction of uncertainties in applications.

Determinstic system

ξ1

...

ξd

Computational Modely = U(ξ1, ξ2, · · · , ξd)

y

Stochastic system

ξ1

fξ1

...

ξd

fξd

Computational Modely = U(ξ1, ξ2, · · · , ξd)

y

fy

Statistics:

〈y〉 =

∫ +∞

−∞zfy(z) dz

σ2

=

∫ +∞

−∞(z − E[y])

2fy(z) dz

µ3 =

∫ +∞

−∞(z − E[y])

3fy(z) dz

µ4 =

∫ +∞

−∞(z − E[y])

4fy(z) dz

UQ Methods

Introduce mathematical approachesto solve above integrals.


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Polynomial Chaos Expansion

Uncertainty Quantification approaches

X Sampling methods (non-intrusive)

X Quadrature methods (non-intrusive)

X Spectral methods (intrusive): Polynomial Chaos Expansion

The stochastic field U(x; ξ) is decomposed

PC Expansion

U(x; ξ) =∑

06|α|6puα(x)ψα(ξ)

The number of unknown coefficients (ui’s) is: P + 1 =

(p+ nsp

)Curse of dimensionality

Functions ψi(ξ)’s are the orthogonal polynomials with respect to input PDFs.

Regression approach is employed to calculate the PCE coefficients.

Sobol’ sampling scheme

Variance based sensitivity analysis using Sobol’ indices.


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Sparse Reconstruction of Polynomial Chaos ExpansionUsing Compressed Sensing


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Introduction

Industrial problems: large number of randomvariables.

Curse of dimensionality: Computational costof PCE grows exponentially with the numbervariables.

Remedy: efficient methods

Efficient methods

X Adaptive methods

X Reduced basis methods

X Multifidelity methods

X Sparse methods

Sparse reconstruction approaches:

X Hyperbolic sparse

X Compressed sensing

U(x; ξ) =∑

06|α|6puα(x)ψα(ξ)

Index of PC basis

PCE

coefficients

100 101 102 10310−10

10−8

10−6

10−4

10−2

100

Figure: PCE coefficient of drag coefficient ofthe RAE2822 airfoil with stochasticgeometry and operating condition


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Compressed Sensing

Definition

Recover a sparse signal from a set of incomplete observations

Optimization problem (`0-minimization):

u = argminu ‖u‖0 subject to Ψu = Y

`0-minimization is NP-hard! Hence, `1-minimization

u = argminu ‖u‖1 subject to Ψu = Y

Noisy signal:

u = argminu ‖u‖1 subject to ‖Ψu− Y‖2 6 ε

p = 1

u

u

A

p = 1.5

u

u

A

p = 2

u

u

A

p = 3

u

u

A

p = ∞

u

u

A

Original

Reduced to 10% of the waveletcoefficients


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Limitation of Classic PCE

PCE basis should be orthogonal with respect to thePDF of the uncertain input parameters.

The Wiener-Askey polynomial: exponential convergencefor a limited number of PDFs.

What about arbitrary PDFs?

Gram-Schmidt orthogonalization method.

Exponential convergence for arbitrary distributions.

The GSPCE is revisited and for the first time used withthe regression method.

Distribution Polynomials SupportGaussian Hermite (−∞,∞)Uniform Legendre [−1, 1]Gamma Laguerre [0,∞)Beta Jacobi [−1, 1]

U(x; ξ) =∑

06|α|6p

uα(x)ψα(ξ)

1.7 1.75 1.8 1.85 1.9 1.95 2 2.05 2.10

2

4

6

8

10

Cε2

PDF

Original

PDF of Cε2 in k − ε model.

−20 −10 0 10 20 30 400

0.01

0.02

0.03

0.04

0.05

Wind speed (m/s)

PDF of wind speed from measuredmeteorological data (www.umass.edu).


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Gram-Schmidt Orthogonalization Method

One-dimensional monic orthogonal polynomials:

ψj(ξ) = ej(ξ)−j−1∑k=0

cjkψj(ξ), j = 1, 2, · · · , p,

with

ψ0 = 1 cjk =〈ej(ξ), ψk(ξ)〉〈ψk(ξ), ψk(ξ)〉

,

where the polynomials ej(ξ) are polynomials ofexact degree j.

The polynomials are normalized as:

ψj(ξ) =ψj(ξ)

〈ψj(ξ), ψj(ξ)〉

Legendre (Uniform)

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

x

P3(x)

Legendre Polynomials

Gram-Schmidt

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

x

P4(x)


Gram-Schmidt

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

x

P5(x)


Gram-Schmidt

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

x

P6(x)


Gram-Schmidt

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

x

P7(x)


Gram-Schmidt

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

x

P8(x)


Gram-Schmidt


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Gram-Schmidt Orthogonalization Method

One-dimensional monic orthogonal polynomials:

ψj(ξ) = ej(ξ)−j−1∑k=0

cjkψj(ξ), j = 1, 2, · · · , p,

with

ψ0 = 1 cjk =〈ej(ξ), ψk(ξ)〉〈ψk(ξ), ψk(ξ)〉

,

where the polynomials ej(ξ) are polynomials ofexact degree j.

The polynomials are normalized as:

ψj(ξ) =ψj(ξ)

〈ψj(ξ), ψj(ξ)〉

Hermite (Gaussian)

−5 0 5

−20

−10

0

10

20

x

He3(x)

Hermite Polynomials

Gram-Schmidt

−5 0 5−50

0

50

x

He4(x)

Hermite Polynomials

Gram-Schmidt

−5 0 5−100

−50

0

50

100

x

He5(x)

Hermite Polynomials

Gram-Schmidt

−5 0 5−200

−100

0

100

200

x

He6(x)

Hermite Polynomials

Gram-Schmidt

−5 0 5−1000

−500

0

500

1000

x

He7(x)

Hermite Polynomials

Gram-Schmidt

−5 0 5−5000

0

5000

x

He8(x)

Hermite Polynomials

Gram-Schmidt


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Numerical Example: Quantification of epistemicuncertainties in the k − ε model coefficients in

OpenFOAM channel flow


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Epistemic Uncertainties in the k − ε Model Coefficients

Sources of uncertainties in RANS models:

Model formulation

Coefficients

The Launder-Shrama k − ε model:

X Boussinesq (eddy-viscosity) hypothesis:

uiuj = −νt(∂Ui

∂xj+∂Uj

∂xi

)+

2

3kδij

X Turbulent viscosity:

νt = Cµfµk2

ε

X To obtain νt, transport equations are solved:

∂

∂xj(Ujk) =

∂

∂xj

[(ν +

νt

σk

)∂k

∂xj

]+ Pk − ε− 2ν

(∂√k

∂xj

)2

∂

∂xj(Uj ε) =

∂

∂xj

[(ν +

νt

σε

)∂ε

∂xj

]+ Cε1

ε

σεPk − Cε2f2

ε2

k+ E (1)


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Launder-Sharma k − ε model coefficients

The empirical coefficients of low-Re Launder-Sharma k − ε model

Cµ σk σε Cε1 Cε20.09 1.0 1.3 1.44 1.92

The coefficients are related to some basic physical quantities (Durbin and Reif, 2011):

X The decay exponent in decaying homogeneous, isotropic turbulence, n

X The production to dissipation in homogeneous shear flow, P/εX The Von-Karman constant, κ

X The dimensionless turbulent kinetic energy in the logarithmic layer, klog/u2τ

Cµ =

(klog

u2τ

)−2

, Cε1 =Cε2 − 1

P/ε+ 1,

Cε2 =1− nn

, σε =κ2√

Cµ(Cε2 − Cε1).

The reported data for these quantities in the literature are collected and presented asPDFs (Margheri et al., 2014)


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PDFs of Basic Physical Quantities

−1.3 −1.2 −1.1 −10

0.2

0.4

0.6

0.8

1

1.2

1.4

n

1.4 1.5 1.6 1.7 1.8 1.90

0.2

0.4

0.6

0.8

1

1.2

1.4

P/ǫ

0.38 0.39 0.4 0.410

0.2

0.4

0.6

0.8

1

1.2

1.4

κ

2.8 3 3.2 3.4 3.6 3.8 40

0.2

0.4

0.6

0.8

1

1.2

1.4

klog/u2τ

Cµ =

(klog

u2τ

)−2

, Cε1 =Cε2 − 1

P/ε+ 1, Cε2 =

1− nn

, σε =κ2√

Cµ(Cε2 − Cε1).


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PDFs of k − ε Coefficients

0.04 0.06 0.08 0.1 0.12 0.14 0.160

5

10

15

20

Cµ

PDF

Original

1.3 1.4 1.5 1.6 1.7 1.80

1

2

3

4

5

6

7

8

Cε1

PDF

Original

1.7 1.75 1.8 1.85 1.9 1.95 2 2.05 2.10

2

4

6

8

10

Cε2

PDF

Original

0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

σǫ

PDF

Original

The PDFs of Launder-Sharma Coefficients are computed using a Monte-Carlo simulation

Reported standard values lie within its PDF range


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Test Case: Channel Flow

Fully-developed turbulent channel flow

Friction Reynolds number

Reτ =uτH

ν= 950

OpenFOAM Solver: boundaryFoam

Steady-state solver for incompressible, 1Dturbulent flow

grad p = divτ

Number of stochastic parameters ns = 4

UQ analyses preformed with p = 7

Reference solution: 8th order full PC

Periodic

Periodic

H

Wall

Wall

Flow

Turbulence model: LaunderSharmaKE

Discretization:

X gradSchemes: linear

X divSchemes: linear

X divSchemes: linear

Linear solvers:

X U: PCG, DIC

X k: smoothSolver, symGaussSeidel

X epsilon: smoothSolver, symGaussSeidel

10−1

100

101

102

103

0

5

10

15

20

25

y+U

+

Launder Sharma k − ε

DNS

10−1

100

101

102

103

0

0.5

1

1.5

2

2.5

3

y+

u′+


DNS

10−1

100

101

102

103

0

0.5

1

1.5

y+

v′+


DNS

10−1

100

101

102

103

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

y+

u′v′+


DNS


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Convergence of Friction Velocity (uτ) Statistics

Mean

101

102

103

10−4

10−3

10−2

10−1

100

101

102

Number of samples

Rel.Error

(%)

Full PC

Sparse PC (OMP)

Monte Carlo

Variance

101

102

103

10−4

10−2

100

102

104

106

Number of samples

Rel.Error

(%)

Full PC

Sparse PC (OMP)

Monte Carlo

PDF

0.02 0.03 0.04 0.05 0.06 0.070

20

40

60

80

100

uτ/Ub

Full PC, N=990

OMP, N=10

OMP, N=20

OMP, N=50

OMP, N=100

100

101

102

103

10−10

10−8

10−6

10−4

10−2

100

Index of PCE basis

PCE

coefficients

Full PC

Sparse PC, OMP

Sparse method converged muchfaster.

OMP with N = 100 sample methodhas successfully explored theimportant basis in the PCE.


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2D PDFs of Velocity Field

(a) OMP, N = 10 (b) OMP, N = 50

(c) OMP, N = 100 (d) Full PC, N = 990

Normalized 2D PDFs(PDF/max(PDF)).

Increasing number ofsamples improvesreconstructed 2D PDFs.

Using N = 100 samples theconstructed 2D PDF is verysimilar to the full PC.

The sparse method is 9times faster!


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2D PDFs of Turbulence Field

(a) OMP, N = 10 (b) OMP, N = 50

(c) OMP, N = 100 (d) Full PC, N = 990






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(a) OMP, N = 10 (b) OMP, N = 50

(c) OMP, N = 100 (d) Full PC, N = 990






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Comparing with DNS Data


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Quantification of Epistemic Uncertainties in the k-e Model ... -...

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