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1. REPORT DATE (DD-MM-YYYY) November 2014

2. REPORT TYPEBriefing Charts

3. DATES COVERED (From - To) November 2014- November 2014

4. TITLE AND SUBTITLE

5a. CONTRACT NUMBER N/A

Optimal Numerical Schemes for Time Accurate Compressible Large Eddy Simulations: Comparison of Artificial Dissipation and Filtering Schemes

5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

6. AUTHOR(S)

5d. PROJECT NUMBER

Edoh, A., Karagozian, A., Merkle, C. and Sankaran, V. 5e. TASK NUMBER

5f. WORK UNIT NUMBER Q12J

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION REPORT NO.

Air Force Research Laboratory (AFMC) AFRL/RQR 5 Pollux Drive. Edwards AFB CA 93524-7048

9. SPONSORING / MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S)Air Force Research Laboratory (AFMC) AFRL/RQR 5 Pollux Drive. 11. SPONSOR/MONITOR’S REPORT

Edwards AFB CA 93524-7048 NUMBER(S)

AFRL-RQ-ED-VG-2014-321

12. DISTRIBUTION / AVAILABILITY STATEMENT Distribution A: Approved for Public Release; Distribution Unlimited.

13. SUPPLEMENTARY NOTES Briefing Charts presented at 67th American Physical Society (APS) Meeting, San Francisco, CA, 23-25 Nov, 2014. PA#14565

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Venke Sankaran

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661-275-5634 Standard Form

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November 23-25, 2014 Distribution A – Approved for public release; distribution is unlimited

Optimal Numerical Schemes for Time Accurate Compressible Large Eddy Simulations:

Comparison of Artificial Dissipation and Filtering Schemes

67th American Physical Society (APS) Meeting Division of Fluid Dynamics

San Francisco, CA

Ayaboe Edoh, Ann Karagozian (UCLA) Charles Merkle (Purdue), Venke Sankaran (AFRL)

ENERGY & PROPULSION RESEARCH

LABORATORY

Research Supported by AFOSR (Drs. Fahroo and Li, PMs)

Distribution A – Approved for public release; distribution is unlimited

Challenges: Reactive LES

CHARLES (Stanford)

LESLIE3D (Georgia Tech)

OpenFOAM (OpenCFD)

Fluent (Ansys)

Ref: 2013 - Cocks, Sankaran, Soteriou, “Is LES of reacting flow predictive? Part 1: Impact of Numerics”

Need to determine BEST discretization schemes for Reacting LES

Algorithm comparisons: - identical subgrid modeling - differences reside in numerics

2


Objectives GOAL:

damp high frequency errors while preserving low wave content (ie: low-pass response)

1) compare damping character of Artificial Dissipation and Filtering 2) formulate filter as an equivalent Artificial Dissipation scheme - consequence of filter damping for stiff problems 3) insight on achieving “ideal” low-pass response for general problems •  von Neumann Analysis

•  Crank-Nicolson w/ 6th order central differencing

3

von Neumann Analysis

Eigenvalues of the amplification matrix specify growth factor and phase errors.

Phase Error Growth Factor

Qn+1 = G θ( )Qn

∂Q∂t

+ A ∂Q∂x

= 0

gi 2φ

φexact=− tan−1 Im gi( ) / Re gi( ){ }

CFLi ×θ

Qi = Q̂ k( )eikxk∑

Qi+i = Q̂ k( )eik x+Δx( )

k∑ = Q̂ k( )eikΔxeikx

k∑

1D Euler System (quasi-linear form):

kΔx = θ with θ ∈ −π ,π[ ]

4 Distribution A – Approved for public release; distribution is unlimited

Artificial Dissipation ∂Q∂t

+ A ∂Q∂x

= −1( )m−1 Δx( )2m−1 ε2m λu+c∂2mQ∂x2mm

∑

ε2 = 1/ 2ε4 = 1/12ε6 = 1/ 60

Qi+1 −Qi−1

2Δx− 12

Qi+1 − 2Qi +Qi−1

Δx⎛⎝⎜

⎞⎠⎟ =

Qi −Qi−1

Δx−Qi+2 + 8Qi+1 − 8Qi−1 +Qi−2

12Δx+ 112

Qi+2 − 4Qi+1 + 6Qi − 4Qi−1 +Qi−2

Δx⎛⎝⎜

⎞⎠⎟ =

4Qi+1 + 6Qi −12Qi−1 + 2Qi−2

12Δx

m = 1: m = 2: etc…

governing equations augmented with dissipation terms

upwind biased stencils

Distribution A – Approved for public release; distribution is unlimited 5

Filtering (Explicit)

represent total amplification of filter scheme as: with

Qi = 1+ S2m Δx( )2m ∂2m

∂x2mm∑⎡

⎣⎢

⎤

⎦⎥Qi

*

G θ( ) = R θ( )G* θ( )

R θ( ) 2≤1

R θ = ±π( ) = 0

NOTE: filter is purely dissipative and does not alter original scheme’s phase behavior

smoothing of solution as post-process of integration step


Filtering (Implicit)

represent total amplification of filter scheme as: with

G θ( ) = R θ( )G* θ( )

R θ( ) 2≤1

R θ = ±π( ) = 0

NOTE: filter is purely dissipative and does not alter original scheme’s phase behavior

smoothing of solution as post-process of integration step

1+ S '2m Δx( )2m ∂2m

∂x2mm∑⎡

⎣⎢

⎤

⎦⎥Qi = 1+ S2m Δx( )2m ∂2m

∂x2mm∑⎡

⎣⎢

⎤

⎦⎥Qi

*


Artificial Dissipation: Growth Factor

2nd Order AD

4th Order AD

6th Order AD

no AD

2nd Order AD

4th Order AD 6th Order AD

no AD


Explicit Filter: Growth Factor

Qi = 1+ Sm Δx( )2m ∂2m

∂x2m

⎡

⎣⎢

⎤

⎦⎥Qi

* with Sm =−1( )m−1

22mShapiro Filter (1975)

500th Order

10th Order

6th Order

2nd Order

4th Order

500th Order

10th Order

6th Order

4th Order

2nd Order


2nd Order Implicit Filter: Growth Factor

1+ 1−δ( )S Δx( )2 ∂2

∂x2

⎡

⎣⎢

⎤

⎦⎥Qi = 1+ S Δx( )2 ∂2

∂x2

⎡

⎣⎢

⎤

⎦⎥Qi

* with δ ∈ 0,1]Long Filter (1971)

δ = 1e−1

δ = 1e− 2

δ = 1e− 3

δ = 1e− 4 δ = 1e− 5


2nd vs. 4th Order Implicit Filters: Growth Factor Error

1+ S Δx( )2 ∂2

∂x2

⎡

⎣⎢

⎤

⎦⎥Qi = 1+ S Δx( )2 ∂2

∂x2 +−δ

4 − 8δ⎛⎝⎜

⎞⎠⎟ S Δx( )4 ∂4

∂x4

⎡

⎣⎢

⎤

⎦⎥Qi

* with δ ∈ 0,14th order Lele Filter (1992)

* 500th Order expFil

decreasing delta


Filtering as an Artificial Dissipation Scheme

Qin+1 = 1+ 1

4Δx( )2 ∂2

∂x2

⎡

⎣⎢

⎤

⎦⎥Qi

*

= 1+ 14

Δx( )2 ∂2

∂x2

⎡

⎣⎢

⎤

⎦⎥ Qi

n − Δt 1−θ( ) ∂En

∂x+ Δtθ ∂E*

∂x⎡

⎣⎢

⎤

⎦⎥ with θ ∈ 0,1[ ]

Qin+1 −Qi

n

Δt+ 1−θ( ) ∂E

n

∂x+ θ( ) ∂E

*

∂x= 14

Δx( )2Δt

∂2Qn

∂x2− 1−θ( ) 1

4Δx( )2 ∂

3En

∂x3− θ( ) 1

4Δx( )2 ∂

3E*

∂x3

dispersive terms: •  restore phase of original scheme

dissipation term scales as •  increased damping w/ decreasing time-step

ε2 ~ 1 /CFLu+c


Effect of CFL

6th Order expFil

10th Order expFil

AD (CFL = 1e0)

AD (CFL = 1e-1)

* AD shown is 6th order

AD (CFL = 1e-2)

10th Order expFil

6th Order expFil

AD (CFL = 1e0)

AD (CFL = 1e-1)

AD (CFL = 1e-2)


Cumulative Low CFL Damping

CFLu+c = 10−4

Nsteps = 104

6th Order AD

10th Order expFil

6th Order expFil

2nd Order impFil δ = 1e− 7


Summary •  Filtering is a form of artificial dissipation

–  damping behavior more predictable and tunable •  Explicit filters require very high order for low-pass response

–  overly dissipative for small time-steps •  Implicit filters can be efficiently designed for low-pass response

–  superior to artificial dissipation or explicit filters

Acknowledgments guidance from advisors and collaborators: •  Dr. Venke Sankaran (Edwards AFRL) •  Professor Ann Karagozian (UCLA) •  Professor Charles Merkle (Purdue) * support from Dr. Fariba Fahroo and Dr. Chiping Li (AFOSR)



THANK YOU


Back-Up Slides

Staggered Grid Von Neumann Analysis

Eigenvalues of the amplification matrix specify growth factor and phase errors.

Continuity/Energy Momentum

Staggered Grid Scheme/ Quasi-Linear Form

Phase Error Growth Factor

1D Euler Eqns


Runge Kutta 4: von Neumann Collocated Staggered

No damping of highest modes

No convection of highest modes

No damping of PARTICLE WAVE’s highest modes Highest ACOUSTIC

modes damped

No convection of PARTICLE WAVE’s highest modes

Slow convection of highest ACOUSTIC modes


Kinetic Energy Preservation (KEP)

∂∂t

12

ui2⎛

⎝⎜⎞⎠⎟+ ∂∂x j

12

ui2uj

⎛⎝⎜

⎞⎠⎟= −

∂ui P∂xi

+ ui

∂τ ij

∂x j

⎛

⎝⎜

⎞

⎠⎟

ui

∂ui

∂t+∂uiu j

∂x j

= − ∂P∂xi

+∂τ ij

∂x j

⎛

⎝⎜

⎞

⎠⎟

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

Incompressible Flow: 0j

j

ux

∂=

∂

Compressible Flow: 2

02i

j i i i j ijj j i j

u u u u u u Pt x t x x xρ ρ ρ ρ τ

⎧ ⎫ ⎧ ⎫− ∂ ∂ ∂ ∂ ∂ ∂⎪ ⎪ ⎪ ⎪+ + + + − =⎨ ⎬ ⎨ ⎬∂ ∂ ∂ ∂ ∂ ∂⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭

∂∂t

12ρui

2⎛⎝⎜

⎞⎠⎟+ ∂∂x j

ρuj

ui2

2⎛

⎝⎜⎞

⎠⎟= −ui

∂P∂xi

+ ui

∂τ ij

∂x j

⎛

⎝⎜

⎞

⎠⎟

-  K = ½ ui2 bounded and constant at inviscid limit

-  KEP schemes satisfy secondary equation discretely -  Richtmeyer & Morton (1967) -  Arakawa (1966)

-  Discrete analogue seeks: -  Accurate transport of KE accurate physical transfer of energy: E = KE + Uint

-  “in computations of turbulent flow fields, dissipative errors show up at the level of kinetic energy” (Mahesh 2004)

-  Robust at inviscid limit (Re ∞)


KEP: Applied to 1D Euler

21

-‐‑  compare Crank-‐‑Nicolson (CN) with Fully KEP scheme (F-‐‑KEP)

-‐‑ discrete secondary equation satisfied to machine zero if KEP

(CN) (F-‐‑KEP)

with

(ρφk )in+1 − (ρφk )i

n

Δt+ 1

Vi

φ( )f∑

f

m(ρ u j ) f

n+1/2 ⋅Si +1Vi

∂pυk , j

∂x j

⎛

⎝⎜

⎞

⎠⎟

f∑

f

n+1/2

⋅Si = 0

11 ( )2

m n nφ φ φ+= +1

1

( ) ( )( ) ( )

n nm

n n

ρφ ρφφρ ρ

+

+

+=

+

2 1 2 21/2 1/2

, , , ,( ) ( ) 1 1( u ) ( )

2 2

mn nn m nk i k i kj f f i k i k j f i

f fi if

S p S RESIDUALt V V

ρφ ρφ φρ φ υ+

+ +⎛ ⎞− + ⋅ + ⋅ =⎜ ⎟Δ ⎝ ⎠∑ ∑

2 12 2 2 2

m m mk k k

i nbrf

φ φ φ⎛ ⎞ ⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

(Collocated Grid)

Subbareddy/Candler(2009) Merkle (2013)

1

k

ueY

φ

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦


Evaluating KEP: u2

22

Mach = 0.859 disturb = 0.01%

disturb = 1%

-  F-KEP always secondary conservative -  CN is KEP for low compressibility effects


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