[PPT] - Fu Future Direction in Turbulence Modeling: Oleg V. Vasilyev & PowerPoint Presentation

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Mechanical Engineering Department of

& Simulation Laboratory & Simulation Laboratory Multi-Scale Modeling Multi-Scale Modeling

F u t u r e D i r e c t i

n

s i n C F D R e s e a r c h , A u g u s t 8 , 2 1 2 FuFuture Direction in Turbulence Modeling:

Oleg V. Vasilyev & AliReza Nejadmalayeri

Department of Mechanical Engineering University of Colorado Boulder

1

Wednesday, August 8, 12

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Mechanical Engineering Department of

& Simulation Laboratory & Simulation Laboratory Multi-Scale Modeling Multi-Scale Modeling

F u t u r e D i r e c t i

n

s i n C F D R e s e a r c h , A u g u s t 8 , 2 1 2 FuFuture Direction in Turbulence Modeling: FuDynamic Two-way Coupling of

Numerical Methods and Physical Models

Oleg V. Vasilyev & AliReza Nejadmalayeri

Department of Mechanical Engineering University of Colorado Boulder

1

Wednesday, August 8, 12

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Mechanical Engineering Department of

& Simulation Laboratory & Simulation Laboratory Multi-Scale Modeling Multi-Scale Modeling

F u t u r e D i r e c t i

n

s i n C F D R e s e a r c h , A u g u s t 8 , 2 1 2 FuFuture Direction in Turbulence Modeling: FuDynamic Two-way Coupling of

Numerical Methods and Physical Models

Oleg V. Vasilyev & AliReza Nejadmalayeri

Department of Mechanical Engineering University of Colorado Boulder

1

m(q)-LES Adaptive LES with Model Refinement

Wednesday, August 8, 12

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Mechanical Engineering Department of

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Future Directions in CFD Research?

2

Parallel adaptive high order numerical methods New/improved turbulence models

Wednesday, August 8, 12

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Future Directions in CFD Research?

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Parallel adaptive high order numerical methods New/improved turbulence models Not Not

Wednesday, August 8, 12

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Future Directions in CFD Research?

2

Parallel adaptive high order numerical methods New/improved turbulence models Not Not

Reasons:

Spatial/temporal intermittency of turbulent flows is not used
Inhomegeneous fidelity
a-priori large/small scale separation
under-resolves energetic structures
over-resolves in between them

Wednesday, August 8, 12

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Mechanical Engineering Department of

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Future Directions in CFD Research?

2

Parallel adaptive high order numerical methods New/improved turbulence models Not Not

Wednesday, August 8, 12

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Mechanical Engineering Department of

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Future Directions in CFD Research?

2

Parallel adaptive high order numerical methods New/improved turbulence models

Wednesday, August 8, 12

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Mechanical Engineering Department of

& Simulation Laboratory & Simulation Laboratory Multi-Scale Modeling Multi-Scale Modeling

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Future Directions in CFD Research?

2

Parallel adaptive high order numerical methods New/improved turbulence models New direction/philosophy/paradigm:

Wednesday, August 8, 12

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Mechanical Engineering Department of

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Future Directions in CFD Research?

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Parallel adaptive high order numerical methods New/improved turbulence models Direct physics-based coupling of & that takes advantage of spatio-temporal intermittency

f turbulent flows

New direction/philosophy/paradigm:

Wednesday, August 8, 12

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What does direct coupling bring?

the active control of the fidelity/accuracy of the simulation
near optimal spatially adaptive computational mesh
the “desired” flow-physics is captured by considerably smaller

number of spatial modes

considerably smaller Reynolds scaling exponent,
robust general mathematical framework for spatial/temporal

model-refinement (m-refinement) that can be extended to LES with AMR approach

mathematical framework for epistemic uncertainty quantification

3

Reα, α < 9/4

Wednesday, August 8, 12

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Wavelet-based Turbulence Modeling Hierarchy

4

u

>✏

i (x) =

X

l2L0

c0

l φ0 l (x) + +1

X

j=0 2n1

X

µ=1

X

k 2 Kj |dj

k| ✏ kuk

dµ,j

k ψµ,j k (x)

Wavelet thresholding filter:

Wednesday, August 8, 12

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Mechanical Engineering Department of

& Simulation Laboratory & Simulation Laboratory Multi-Scale Modeling Multi-Scale Modeling

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Wavelet-based Turbulence Modeling Hierarchy

4

u

>✏

i (x) =

X

l2L0

c0

l φ0 l (x) + +1

X

j=0 2n1

X

µ=1

X

k 2 Kj |dj

k| ✏ kuk

dµ,j

k ψµ,j k (x)

Physical Space

X 0.2 0.4 0.6 0.8 1

1
0.5

0.5 1 1.5 ( X)

Local Support

20 40 60 80 100 1 2 3 4 5 x 10

5
(

K)

Local

Support

Wave Number Space

Wavelet thresholding filter:

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Wavelet-based Turbulence Modeling Hierarchy

4

u

>✏

i (x) =

X

l2L0

c0

l φ0 l (x) + +1

X

j=0 2n1

X

µ=1

X

k 2 Kj |dj

k| ✏ kuk

dµ,j

k ψµ,j k (x)

Wavelet thresholding filter:

Wednesday, August 8, 12

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Mechanical Engineering Department of

& Simulation Laboratory & Simulation Laboratory Multi-Scale Modeling Multi-Scale Modeling

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Wavelet-based Turbulence Modeling Hierarchy

4

u

>✏

i (x) =

X

l2L0

c0

l φ0 l (x) + +1

X

j=0 2n1

X

µ=1

X

k 2 Kj |dj

k| ✏ kuk

dµ,j

k ψµ,j k (x)

Wavelet thresholding filter: u(x, t) = u

>✏(x, t) + u ≤✏(x, t)

Wednesday, August 8, 12

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e l s , A u g u s t 8 , 2 1 2

Wavelet-based Turbulence Modeling Hierarchy

4

u

>✏

i (x) =

X

l2L0

c0

l φ0 l (x) + +1

X

j=0 2n1

X

µ=1

X

k 2 Kj |dj

k| ✏ kuk

dµ,j

k ψµ,j k (x)

Wavelet thresholding filter:

Choice of :

WDNS -
CVS* -
SCALES† - ✏ > ✏opt

✏ ≈ ✏opt ✏ ⌧ 1 ✏

*Coherent Vortex SImulation (CVS): Farge M, Schneider K, Kevlahan N. Phys. Fluids 11:2187–201, 1999.

†Stochastic Coherent Adaptive Large Eddy Simulations (SCALES): Goldstein, D.E. and

Vasilyev, O.V., Phys. Fluids 16: 2497-2513, 2004.

u(x, t) = u

>✏(x, t) + u ≤✏(x, t)

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e l s , A u g u s t 8 , 2 1 2

Wavelet-based Turbulence Modeling Hierarchy

4

u

>✏

i (x) =

X

l2L0

c0

l φ0 l (x) + +1

X

j=0 2n1

X

µ=1

X

k 2 Kj |dj

k| ✏ kuk

dµ,j

k ψµ,j k (x)

Wavelet thresholding filter:

Choice of :

WDNS -
CVS* -
SCALES† - ✏ > ✏opt

✏ ≈ ✏opt ✏ ⌧ 1 ✏

*Coherent Vortex SImulation (CVS): Farge M, Schneider K, Kevlahan N. Phys. Fluids 11:2187–201, 1999.

†Stochastic Coherent Adaptive Large Eddy Simulations (SCALES): Goldstein, D.E. and

Vasilyev, O.V., Phys. Fluids 16: 2497-2513, 2004.

Wednesday, August 8, 12

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Mechanical Engineering Department of

& Simulation Laboratory & Simulation Laboratory Multi-Scale Modeling Multi-Scale Modeling

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u

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e l s , A u g u s t 8 , 2 1 2

Wavelet-based Turbulence Modeling Hierarchy

4

u

>✏

i (x) =

X

l2L0

c0

l φ0 l (x) + +1

X

j=0 2n1

X

µ=1

X

k 2 Kj |dj

k| ✏ kuk

dµ,j

k ψµ,j k (x)

Wavelet thresholding filter:

Choice of :

WDNS -
CVS* -
SCALES† - ✏ > ✏opt

✏ ≈ ✏opt ✏ ⌧ 1 ✏

*Coherent Vortex SImulation (CVS): Farge M, Schneider K, Kevlahan N. Phys. Fluids 11:2187–201, 1999.

†Stochastic Coherent Adaptive Large Eddy Simulations (SCALES): Goldstein, D.E. and

Vasilyev, O.V., Phys. Fluids 16: 2497-2513, 2004.

Large Eddies Small Eddies Coherent Incoherent Increasing Wave # DNS WDNS CVS SCA LES LES DNS

Wednesday, August 8, 12

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e l s , A u g u s t 8 , 2 1 2

Wavelet-based Turbulence Modeling Hierarchy

4

u

>✏

i (x) =

X

l2L0

c0

l φ0 l (x) + +1

X

j=0 2n1

X

µ=1

X

k 2 Kj |dj

k| ✏ kuk

dµ,j

k ψµ,j k (x)

Wavelet thresholding filter:

Large Eddies Small Eddies Coherent Incoherent Increasing Wave # DNS WDNS CVS SCA LES LES DNS

Wednesday, August 8, 12

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Mechanical Engineering Department of

& Simulation Laboratory & Simulation Laboratory Multi-Scale Modeling Multi-Scale Modeling

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u

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e l s , A u g u s t 8 , 2 1 2

Wavelet-based Turbulence Modeling Hierarchy

4

u

>✏

i (x) =

X

l2L0

c0

l φ0 l (x) + +1

X

j=0 2n1

X

µ=1

X

k 2 Kj |dj

k| ✏ kuk

dµ,j

k ψµ,j k (x)

Wavelet thresholding filter:

Wednesday, August 8, 12

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Mechanical Engineering Department of

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u

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N u m e r i c a l M e t h

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e l s , A u g u s t 8 , 2 1 2

Wavelet-based Turbulence Modeling Hierarchy

4

u

>✏

i (x) =

X

l2L0

c0

l φ0 l (x) + +1

X

j=0 2n1

X

µ=1

X

k 2 Kj |dj

k| ✏ kuk

dµ,j

k ψµ,j k (x)

Wavelet thresholding filter: Simulate the evolution of the most energetic coherent vortices (track them), while modeling the effect of the subgrid scales. ∂u>✏

i

∂t + ∂u>✏

i u>✏ j

∂xj = −∂p>✏ ∂xi + 1 Re ∂2u>✏

i

∂xj∂xj + ∂τij ∂xj

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Adaptive Wavelet Collocation Method

5

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Adaptive Wavelet Collocation Method (AWCM)

Single-mode Rayleigh-Taylor Instability (incompressible limit)

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Shock Wave Propagation over the Cylinder

7

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Shock Wave Propagation through the Cylinder Array

8

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Acoustic Timescale Detonation Initiation

9

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Acoustic Timescale Detonation Initiation

1

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Acoustic Timescale Detonation Initiation

1 1

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Acoustic Timescale Detonation Initiation

1 2

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Acoustic Timescale Detonation Initiation

1 3

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Acoustic Timescale Detonation Initiation

1 4

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Hierarchical Variable Fidelity

Multiscale Turbulence Modeling

1 5

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Turbulence Resolution

1 6

Kinetic Energy Based: SGS dissipation Based:

F =

Π εres+Π

F =

ksgs kres+ksgs

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Turbulence Resolution

Fidelity of the simulation is a function of Turbulence

Resolution

Objective - control the level of fidelity

1 6

Kinetic Energy Based: SGS dissipation Based:

F =

Π εres+Π

F =

ksgs kres+ksgs

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Turbulence Resolution

1 6

Kinetic Energy Based: SGS dissipation Based:

F =

Π εres+Π

F =

ksgs kres+ksgs

Wednesday, August 8, 12

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Turbulence Resolution

1 6

Kinetic Energy Based: SGS dissipation Based:

F =

Π εres+Π

F =

ksgs kres+ksgs

Homogeneous Turbulence:
LES with fixed complexity
LES with fixed complexity

FKE FD ∼ Re0 = 1 ∼ Re9/4

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Spatial Variable Thresholding

Fully Adaptive Wavelet Thresholding Filter Scales Dependency

Buildup scales Losing some scales

= ⇒ ↑ ↓ = ⇒

✏ ✏ In General:

Wavelet-Threshold-Filtered Velocity Depends on: 1) Threshold Level 2) Velocity Scale Goal: wherever

↓

✏ Idea:

Threshold is determined on-the-fly By Tracking areas of Locally Significant: 1) SGS Dissipation or 2) Any other Physical quantity

εres Π↑

↑

Π > ΠGoal ✏ = ✏ (Π) u>

i

∼

&

kuik

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Lagrangian “Variable Thresholding” SCALES

Spatial Variable Thresholding

If changed in spatial space Then @ next time-step that flow-structures will move in space, it will face to either a smaller or greater Recommended Solution: Track within a Lagrangian frame by “Lagrangian Path-Line Diffusive Averaging” Approach (Similarly to Vasilyev et al., [JOT, 9(11), 2008] Lagrangian SGS SCALES] ) Similarly to Meneveau et al. [JFM, 319, 1996] : Linear Averaging Along Characteristics Diffusion Term can be ignored Because “Linear Averaging” itself will create required diffusion. Lagrangian Path-Line Diffusive Averaging Evolution equation for

✏ ✏ ✏

1 ∆t

⇥ new (x, t + ∆t) − old (x − u>∆t, t) ⇤ = −forcingterm ⇤t + u>

j ⇤xj = −forcingterm + ⇥⇤2 xjxj

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

τeddy Total FSGSD

1 9

F⇥ =

hΠi hεresi+hΠi

F =

Π εres+Π

τ −1

= |Sij

>|⇥

τ −1

= |Sij

>|

5τeddy

Hybrid CVS & SCALES (Hierarchical Multiscale Adaptive Variable Fidelity) – Time Varying Goal Benchmark

⇤t + u>

j ⇤xj = −forcingterm + ⇥⇤2 xjxj 1 ∆t

⇥ new (x, t + ∆t) − old (x − u>∆t, t) ⇤ = −forcingterm forcingterm = old (x − u>✏∆t, t) 1

⌧ ✏ (F − G)

G ∈ {0.2, 0.25, 0.3, 0.2, 0.3, 0.25} τeddy = u02

hεi =

2 3 K

2QK = 1 3Q

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Mechanical Engineering Department of

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

τeddy Total FSGSD

1st order Interpolation 3rdOrder Interpolation

2

F⇥ =

hΠi hεresi+hΠi

τ −1

= |Sij

>|

5τeddy G ∈ {0.2, 0.25, 0.3, 0.2, 0.3, 0.25}

Hybrid CVS & SCALES (Hierarchical Multiscale Adaptive Variable Fidelity) – Time Varying Goal Benchmark

Interpolation Approach 1st & 3rd Order

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Hybrid CVS / SCALES – Threshold Animation

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Hybrid CVS / SCALES – Threshold Animation

1st Interpolation

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Hybrid CVS / SCALES – Threshold Animation

1st Interpolation 3rd Interpolation

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Hybrid CVS / SCALES – Threshold Animation

1st Interpolation 3rd Interpolation

ν✏ = 0.05

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Hybrid CVS / SCALES – Threshold Animation

1st Interpolation 3rd Interpolation

ν✏ = 0.1 ν✏ = 0.05

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Hybrid CVS / SCALES – Threshold Animation

1st Interpolation 3rd Interpolation

ν✏ = 0.1 ν✏ = 0.05 ν✏ = 4

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Hybrid CVS / SCALES – Threshold Animation

1st Interpolation 3rd Interpolation

ν✏ = 0.1 ν✏ = 0.05 ν✏ = 5 ν✏ = 4

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Hybrid CVS / SCALES – Threshold Animation

1st Interpolation 3rd Interpolation

ν✏ = 0.1 ν✏ = 0.05 ν✏ = 5 ν✏ = 4

This was a Benchmark to Test Methodology + Time-Response + Accuracy

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Hybrid CVS / SCALES – Threshold Animation

1st Interpolation 3rd Interpolation

ν✏ = 0.1 ν✏ = 0.05 ν✏ = 5 ν✏ = 4

This was a Benchmark to Test Methodology + Time-Response + Accuracy Perspective: In Reality Goal can change in Space as well not just in Time

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

τeddy Total FSGSD

Evolution νε=0.05 Evolution νε=0.1 Evolution νε=4 Evolution νε=5

2 2

5τeddy G ∈ {0.2, 0.25, 0.3, 0.2, 0.3, 0.25}

Hybrid CVS & SCALES (Hierarchical Multiscale Adaptive Variable Fidelity) – Time Varying Goal Benchmark

Solving Evolution Equation Directly

F⇥ =

hΠi hεresi+hΠi

τ −1

= |Sij

>|

ν✏ ∈ {0.05, 0.1, 4, 5}

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Reynolds Scaling and its

Dependence on “Desired” Captured Flow Physics

2 3

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Time-Averaged Energy Spectra – CVS and SCALES

Linear Forcing Coefficient : Adaptive Grid corresponds to (at highest level of resolution) Taylor micro-scale Reynolds number :

Q = 6.¯ 6 Reλ ∼ = 70, 120, 190, 320 ν = 0.09, 0.035, 0.015, 0.006 2563, 5123, 10243, 20483 Jmax = 6, 7, 8, 9 = 0.2, 0.43 Reλ ∼ = 320 Reλ ∼ = 190 Reλ ∼ = 120 Reλ ∼ = 70 Ŋ/Ŋmax = 2.1 × 10−4 Ŋ/Ŋmax = 3.3 × 10−3 Ŋ/Ŋmax = 6.8 × 10−3 Ŋ/Ŋmax = 8.9 × 10−4

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Time-Averaged Energy Spectra – CVS and SCALES

Linear Forcing Coefficient : Adaptive Grid corresponds to (at highest level of resolution) Taylor micro-scale Reynolds number :

Q = 6.¯ 6 Reλ ∼ = 70, 120, 190, 320 ν = 0.09, 0.035, 0.015, 0.006 2563, 5123, 10243, 20483 Jmax = 6, 7, 8, 9 = 0.2, 0.43

10 10

1

10

2

10

−10

10

−8

10

−6

10

−4

10

−2

10 10

2

10

4

Wavenumber Time Averaged Energy Spectra

k−5/3

Cf=6.6667

2563 CVS Re=70 5123 CVS Re=120 10243 CVS Re=190 2563 SCALES Re=70 5123 SCALES Re=120 10243 SCALES Re=190 20483 SCALES Re=320

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SCALES CVS 512³

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512³ 2048³

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Taylor Microscale Reynolds number Number of Points

Reλ

9/2

Reλ

3.25

Reλ

2.75

SCALES CVS DNS

2 8

Computational Complexity –

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Taylor Microscale Reynolds number Number of Points

Reλ

9/2

Reλ

3.25

Reλ

2.75

SCALES CVS DNS

2 8

Computational Complexity –

4.5 − 3.25 = 1.25

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Taylor Microscale Reynolds number Number of Points

Reλ

9/2

Reλ

3.25

Reλ

2.75

SCALES CVS DNS

2 8

Computational Complexity –

4.5 − 2.75 = 1.75 4.5 − 3.25 = 1.25

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Taylor Microscale Reynolds number Number of Points

Reλ

9/2

Reλ

3.25

Reλ

2.75

SCALES CVS DNS

2 9

Computational Complexity – Fractal Dimension

Spatial DOF† :

†Paladin G,

Vulpiani A, 1987. Phys. Rev. A 35:1971–1973

Re3DF /(DF +1) DF ≤ 3 DFCVS . 13

11 = 1.18

DFSCALES . 11

13 = 0.846153

DFSCALES < 1

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Fraction SGS Dissipation – SCALES

Linear Forcing Coefficient : Adaptive Grid corresponds to (at highest level of resolution) Taylor micro-scale Reynolds number :

Q = 6.¯ 6 Reλ ∼ = 70, 120, 190, 320 ν = 0.09, 0.035, 0.015, 0.006 2563, 5123, 10243, 20483 Jmax = 6, 7, 8, 9 = 0.43

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

eddy Total FSGSD

f

0.32318 0.47587 0.59473 0.74504 0.32318 0.47587 0.59473

2563 Re=70 5123 Re=120 10243 Re=190 20483 Re=320

F⇥ =

hΠi hεresi+hΠi

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70 120 190 320 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8

Taylor Microscale Reynolds number Total FSGSD

SCALES

3 1

Complexity – How %Fraction-SGSD Scales as Reynolds?

Fully Adaptive Wavelet Thresholding Filter ✏ = ✏ (Π)

Drawback: as Solution: Spatial Variable Thresholding

Π

↑ ↑

Re F⇥ = ⇣

hΠi hεresi+hΠi

⌘

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Computational Complexity –

G = F⇥2563 with =0.43 = 0.32

70 120 190 320 10

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Taylor Microscale Reynolds number Number of Points

Reλ

9/2

Reλ

3.25

Reλ

2.75

SCALES SCALES Time Varying ε SCALES Spatially Adaptive ε CVS DNS

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Computational Complexity – Different

G

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 0.1 0.2 0.25 0.32 0.4 0.5 0.8

τeddy Total FSGSD G = 0.2 G = 0.25 G = 0.32 G = 0.4 G = 0.5

2563 Reλ=70
5123 Reλ=120
10243 Reλ=190

20483 Reλ=320

F⇥ =

hΠi hεresi+hΠi

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9

10

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Taylor Microscale Reynolds number Number of Points

Reλ

9/2

Reλ

3.25

Reλ

2.75

SCALES ǫ = 0.43 SCALES G = 0.2 SCALES G = 0.25 SCALES G = 0.32 SCALES G = 0.4 SCALES G = 0.5 CVS ǫ = 0.2 DNS 3 4

Computational Complexity – Different G

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Taylor Microscale Reynolds number Number of Points

Reλ

9/2

Reλ

3.25

Reλ

2.75

SCALES ǫ = 0.43 SCALES G = 0.2 SCALES G = 0.25 SCALES G = 0.32 SCALES G = 0.4 SCALES G = 0.5 CVS ǫ = 0.2 DNS 3 4

Computational Complexity – Different G Perspective: Very High Reynolds + 3D WDNS + True CVS

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Ultimate Goal of SCALES – Data Mining

70 120 190 320 0.5 1 1.5 2 2.5 x 10

7

Taylor Microscale Reynolds number Number of Points

Reλ

2.75

SCALES ǫ = 0.43 SCALES G = 0.2 SCALES G = 0.25 SCALES G = 0.32 SCALES G = 0.4 SCALES G = 0.5

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How to Incorporate
Dynamic Coupling into existing LES

3 6

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G SGS CFD R h

(x)

Dependency Diagram – LES

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G SGS AWCM R

num

K.E. K.E. G (K.E., )

Dependency Diagram – SCALES

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Hybrid WDNS/CVS/SCALES (Hierarchical Multiscale Adaptive Variable Fidelity) – m-SCALES

G SGS AWCM R m

G (K.E., (R,F)) (R,F) K.E.

num

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model refinement is not limited to SCALES – m-LES

G SGS CFD R h m

(F (R)) F (R)

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Conclusions

the active control of the fidelity/accuracy of the simulation
near optimal spatially adaptive computational mesh for the

user-defined fidelity

the “desired” flow-physics is captured by considerably smaller

number of spatial modes

considerably smaller Reynolds scaling exponent, that depends
n the captured flow physics (KE or SGS dissipation)
robust general mathematical framework for spatial/temporal

model-refinement (m-refinement) that can be extended to AMR approach

4 1

Demonstrated:

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Perspectives

4 2

The proposed philosophy/paradigm of dynamic coupling of AMR and turbulence modeling is the FUTURE!

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