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SLIDE 1

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

❙✐♠✉❧❛t✐♦♥ s✉r ❧❡s ♥♦✉✈❡❧❧❡s ❛r❝❤✐t❡❝t✉r❡s ♠❛ss✐✈❡♠❡♥t ♣❛r❛❧❧è❧❡s ✿ ✉♥ ❡①❡♠♣❧❡ ❡♥ ♠é❝❛♥✐q✉❡ ❞❡s ✢✉✐❞❡s

❆✳ ❈❛❞✐♦✉✱ ▼✳ ❇✉✛❛t✱ ▲✳ ▲❡ P❡♥✈❡♥✱ ❏✳ ▼♦♥t❛❣♥✐❡r

▲❛❜♦r❛t♦✐r❡ ❞❡ ▼é❝❛♥✐q✉❡ ❞❡s ❋❧✉✐❞❡s ❡t ❞✬❆❝♦✉st✐q✉❡ ❈◆❘❙✱ ❯♥✐✈❡rs✐té ▲②♦♥ ■✱ ➱❝♦❧❡ ❈❡♥tr❛❧❡ ❞❡ ▲②♦♥✱ ■◆❙❆ ❞❡ ▲②♦♥

▲②♦♥❈❛❧❝✉❧

slide-2
SLIDE 2

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

P❛rt ■ Prés❡♥t❛t✐♦♥ ❞❡ ❧❛ ♣r♦❜❧é♠❛t✐q✉❡ ♣❤②s✐q✉❡

◆✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥ ✐♥ ✢✉✐❞ ♠❡❝❤❛♥✐❝s

slide-3
SLIDE 3

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

▲❡ ▲❛❜♦r❛t♦✐r❡ ❞❡ ▼é❝❛♥✐q✉❡ ❞❡s ❋❧✉✐❞❡s ❡t ❞✬❆❝♦✉st✐q✉❡

❉♦♠❛✐♥❡s ❞❡ r❡❝❤❡r❝❤❡ P❤②s✐q✉❡ ❡t ♠♦❞é❧✐s❛t✐♦♥ ❞❡ ❧❛ t✉r❜✉❧❡♥❝❡✱ ✐♥st❛❜✐❧✐tés ❤②❞r♦❞②♥❛♠✐q✉❡s✱ é❝♦✉❧❡♠❡♥ts ❞✐♣❤❛s✐q✉❡s✱ ♠é❝❛♥✐q✉❡ ❞❡s ✢✉✐❞❡s ❡♥✈✐r♦♥♥❡♠❡♥t❛❧❡✱ ❛ér♦❞②♥❛♠✐q✉❡ ✐♥t❡r♥❡✱ ♣❤é♥♦♠è♥❡s t❤❡r♠✐q✉❡s ❝♦✉♣❧és✱ ❛ér♦❛❝♦✉st✐q✉❡✱ ♣r♦♣❛❣❛t✐♦♥ ❛❝♦✉st✐q✉❡✱ ♠ét❤♦❞❡s ❞❡ rés♦❧✉t✐♦♥ ♥✉♠ér✐q✉❡ ❞❡s éq✉❛t✐♦♥s ❞❡ ◆❛✈✐❡r✲❙t♦❦❡s✱ ❝♦♥trô❧❡ ❛❝t✐❢ ♦✉ ♣❛ss✐❢ ❞❡s é❝♦✉❧❡♠❡♥ts✱ ♠✐❝r♦✢✉✐❞✐q✉❡✳

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SLIDE 4

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

▲❡ ▲❛❜♦r❛t♦✐r❡ ❞❡ ▼é❝❛♥✐q✉❡ ❞❡s ❋❧✉✐❞❡s ❡t ❞✬❆❝♦✉st✐q✉❡

❙❡❝t❡✉rs ❞✬❛♣♣❧✐❝❛t✐♦♥ ❡t ♣❛rt❡♥❛✐r❡s ✐♥❞✉str✐❡❧s ❆ér♦♥❛✉t✐q✉❡ ❡t ❙♣❛t✐❛❧ ✿ ❙❆❋❘❆◆✴❙◆❊❈▼❆✱ ❈◆❊❙✱ ❖◆❊❘❆✱ ❚✉r❜♦♠❡❝❛✱ ❊❆❉❙✱ ❉❛ss❛✉❧t ❆✉t♦♠♦❜✐❧❡ ❘❡♥❛✉❧t✱ ❱♦❧✈♦✱ ❱❛❧❡♦✱ ❉❡❧♣❤✐✱ ■❆❱✱ ❈◆❘❚✱ P❖✱ Pô❧❡s ▲❯❚❇✱ ▼♦✈❡♦✱ ■❉✹❝❛r ❊♥✈✐r♦♥♥❡♠❡♥t ✲ ❇r✉✐t ❉●❆✱ ❈◆❊❙✱ ❖◆❊❘❆✱ ❙◆❈❋✱ ❊✉r♦❝♦♣t❡r✱ ❊❆❉❙✱ P❙❆✱ ❈❊❆ ❉❆▼ ❊♥✈✐r♦♥♥❡♠❡♥t ✲ ❛t♠♦s♣❤ér✐q✉❡✱ ❤②❞r♦❧♦❣✐q✉❡ ❊❉❋✱ ■◆❊❘■❙✱ ❈❊❆✱ ❈❊▼❆●❘❊❋✱ ❚♦t❛❧✱ ▲♦♥❞r❡s✱ ❚✉r✐♥✱ ❆✐r P❛r✐❢✳✳✳

  • é♥✐❡ ❞❡s ♣r♦❝é❞és✱ ❊♥❡r❣✐❡ ❈❊❆✱ ❆✈❡♥t✐s✱ ●❡♦s❡r✈✐❝❡s✱ ❆♥❞r✐t③✳✳✳
slide-5
SLIDE 5

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

▲❛♠✐♥❛r✴t✉r❜✉❧❡♥t tr❛♥s✐t✐♦♥

❏❡t ✢♦✇✿ ▲❛♥❞✐s✲❙❤❛♣✐r♦✱ ♦♠❜r♦s❝♦♣② ▲❆▼■◆❆❘ ❚❘❆◆❙■❚■❖◆ ❚❯❘❇❯▲❊◆❚ ❚❯❘❇❯▲❊◆❚ ❂ ✸❉✱ ♠✉❧t✐♣❧❡ s♣❛❝❡✲t✐♠❡ s❝❛❧❡s✱ s❡❡♠✐♥❣❧② r❛♥❞♦♠ ❜❡❤❛✈✐♦✉r✱ ❢♦r ♣r❛❝t✐❝❛❧ ❛♣♣❧✐❝❛t✐♦♥s ✿ ❞❡t❛✐❧s ❛r❡ ♥♦t ♣r❡❞✐❝t❛❜❧❡✳

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SLIDE 6

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s

❈❧❛✉❞❡ ▲♦✉✐s ◆❆❱■❊❘ ✭✶✼✽✺✲✶✽✸✻✮

❋♦r ✐♥❝♦♠♣r❡ss✐❜❧❡✱ ❤♦♠♦❣❡♥❡♦✉s ✢✉✐❞✿ ✈❡❧♦❝✐t② ❱ ✱ ♣r❡ss✉r❡ P ∂t❱ + ❱ .∇❱ = −✶/ρ ∇♣ + ν∆❱ ∇.❱ = ✵ ❞❡♥s✐t② ρ✱ ✈✐s❝♦s✐t② ν ✰ ✐♥✐t✐❛❧ ❛♥❞ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s NS(❱ ) = ✵

  • ❡♦r❣❡ ●❛❜r✐❡❧ ❙❚❖❑❊❙

✭✶✽✶✾✲✶✾✵✸✮

❲❡❧❧✲❦♥♦✇♥ ❡q✉❛t✐♦♥s✱ ❜✉t ♦♣❡♥ ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦❜❧❡♠s ✳✳✳✳

slide-7
SLIDE 7

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

▲❛♠✐♥❛r✴t✉r❜✉❧❡♥t tr❛♥s✐t✐♦♥

▲❆▼■◆❆❘ ❆ ✧❧❛♠✐♥❛r✧ s♦❧✉t✐♦♥ ❡①✐sts ✿ NS(❱ ✵) = ✵✳ ❙t❛❜✐❧✐t② ♦❢ ❱ ✵ ✉♥❞❡r ♣❡rt✉r❜❛t✐♦♥s❄ ❱ = ❱ ✵ + ✈ ▲✐♥❡❛r st❛❜✐❧✐t② ♣r♦❜❧❡♠ ✿

❞N S ❞❱ |❱ ✵✈ = ✵ s❤♦✇s t❤❛t |✈| ր✿ ❱ ✵ ✉♥st❛❜❧❡✳

slide-8
SLIDE 8

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

▲❛♠✐♥❛r✴t✉r❜✉❧❡♥t tr❛♥s✐t✐♦♥

❙t❛❜✐❧✐t② ❛♥❛❧②s✐s✿ ❧✐♥❡❛r✱ ✇❡❛❦❧② ♥♦♥✲❧✐♥❡❛r ❊①♣❡r✐♠❡♥ts✱ ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts

slide-9
SLIDE 9

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

▲❛♠✐♥❛r✴t✉r❜✉❧❡♥t tr❛♥s✐t✐♦♥

❙t❛❜✐❧✐t② ❛♥❛❧②s✐s✿ ❧✐♥❡❛r✱ ✇❡❛❦❧② ♥♦♥✲❧✐♥❡❛r ❊①♣❡r✐♠❡♥ts✱ ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ■❞❡♥t✐❢② s❝❡♥❛r✐✐ ❢♦r tr❛♥s✐t✐♦♥ t♦ t✉r❜✉❧❡♥❝❡✱ ❊①♣❧❛✐♥ ✇❤② t✉r❜✉❧❡♥❝❡ ✐s s❡❧❢✲s✉st❛✐♥✐♥❣ ❛s ✐t ❝♦♥✈❡❝ts ❞♦✇♥str❡❛♠✱ ❉❡s✐❣♥ st❛t✐st✐❝❛❧ ♠♦❞❡❧s ❢♦r ✇❡❧❧✲❞❡✈❡❧♦♣♣❡❞ t✉r❜✉❧❡♥❝❡✳

→ ❊q✉❛t✐♦♥s ❢♦r t❤❡ ❛✈❡r❛❣❡❞ ✈❛r✐❛❜❧❡s✿ < ❱ (①, t) >✱ ✳✳✳

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SLIDE 10

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

Pr❛❝t✐❝❛❧ ❝♦♥s❡q✉❡♥❝❡s ♦❢ t✉r❜✉❧❡♥❝❡

Pr♦♠♦t❡ ❡♥❡r❣② tr❛♥s❢❡r t♦ s♠❛❧❧❡r s❝❛❧❡s ❛♥❞ ✐♥❝r❡❛s❡ ❞✐ss✐♣❛t✐♦♥✳

▲ ✉

= ⇒

▲ η ≈ ❘❡✸/✹ t

η

❘❡t = ✉▲

ν

■♥❝r❡❛s❡ s♣❛t✐❛❧ tr❛♥s❢❡r ❢♦r ♠❛ss✱ ♠♦♠❡♥t✉♠✱ ❡♥❡r❣②✳

♠♦♠❡♥t✉♠ tr❛♥s❢❡r ♦♥ s♦❧✐❞ ✇❛❧❧ ⇒ ❢♦r❝❡✱

❚ = µ❞<❯>

❞②

❚ = µ❞❯

❞②

❈♦♥s❡q✉❡♥❝❡s ♠❛② ❜❡ ❢❛✈♦r❛❜❧❡ ♦r ♥♦t✱ ❞❡♣❡♥❞✐♥❣ ♦♥ ❛♣♣❧✐❝❛t✐♦♥s✳

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SLIDE 11

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

❊①❛♠♣❧❡ ✿ ♣r❡ss✉r❡ ❧♦ss ✐♥ ❝✐r❝✉❧❛r ♣✐♣❡s

P✷ − P✶ = λ ▲

❉ ρ❯✷

✷ ✱ ❘❡ = ❯❞ ❉ ν

Pr❡ss✉r❡ ❧♦ss ❝♦❡✣❝✐❡♥t λ

▲ ❉ P✶ P✷ ① ❯❞

✭✶✾✾✽✲✷✵✵✹✮ ✭✷✵✵✷✮

λ = ✻✹

❘❡

❧❛♠✐♥❛r t✉r❜✉❧❡♥t

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SLIDE 12

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

❚❤❡ ♣r❡s❡♥t st✉❞②

  • 2h

U0 L W

boundary layer boundary layer

  • y

x z ❉❡✈❡❧♦♣♣✐♥❣ ✢♦✇ ✐♥ ❛ ♣❧❛♥❡ ❝❤❛♥♥❡❧ ✭✷ ♣❛r❛❧❧❡❧ s♦❧✐❞ ✇❛❧❧s✱ ∆P✮✳ ❋♦r ❧❛r❣❡ ❯✵✱ t✉r❜✉❧❡♥t tr❛♥s✐t✐♦♥ ✐♥s✐❞❡ t❤❡ ❜♦✉♥❞❛r② ❧❛②❡rs✳ ❊✈♦❧✈❡s t♦✇❛r❞s ✇❡❧❧✲❞❡✈❡❧♦♣♣❡❞ st❛t❡ ✭ ❞<.>

❞①

= ✵✮✳ ◆✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥s ✿

▲♦♥❣ ❣❡♦♠❡tr✐❡s ✭▲ ∼ ✶✵✵❤✮✱ ❣r✐❞ r❡✜♥❡♠❡♥t ♥❡❛r t❤❡ ✇❛❧❧s✳ ❊✛❡❝ts ♦❢ ❡♥tr❛♥❝❡ ♣❡rt✉r❜❛t✐♦♥s✳

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SLIDE 13

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

❚r❛♥s✐t✐♦♥ ✐♥ ❜♦✉♥❞❛r② ❧❛②❡rs

boundary layer

TURBULENT LAMINAR

instabilities transition ❘❡❝ = ✺✷✵ ❘❡①❝ = ✾✶✵✽✵ ❘❡′

❝ = ✹✻✶

❘❡′

①❝ = ✼✶✻✽✵

✶✵✺ < ❘❡①t < ✸.✶✵✻

① δ∗

❘❡① = ❯∞① ν ❘❡ = ❯∞δ∗ ν ▼✳ ▼❛ts✉❜❛r❛ ❡t P✳❍✳ ❆❧❢r❡❞ss♦♥

♠♦❞❡r❛t❡ ❧❡✈❡❧ ♦❢ ♣❡rt✉r❜❛t✐♦♥ ✧str❡❛❦s✧ ✭✸❉✮ → ✧❜②✲♣❛ss✧ tr❛♥s✐t✐♦♥ ✭❧♦✇❡r ❘❡①✮

❍✳ ❲❡r❧é

❧♦✇ ❧❡✈❡❧ ♦❢ ♣❡rt✉r❜❛t✐♦♥ ❚♦❧❧♠✐❡♥✲❙❝❤❧✐❝❤t✐♥❣ ✇❛✈❡s ✭✷❉✮ → tr❛♥s✐t✐♦♥

slide-14
SLIDE 14

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

◆✉♠❡r✐❝❛❧ ♠❡t❤♦❞

◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥t✿ ♥❡❡❞ t♦ r❡s♦❧✈❡ t❤❡ ✢♦✇ ❛t ❛❧❧ s❝❛❧❡s ✳ ❆s

η

✸ ∼ ❘❡✾/✹ ր✱ ✐♥❝r❡❛s✐♥❣❧② str✐♥❣❡♥t ❝♦♥❞✐t✐♦♥ ❢♦r t✉r❜✉❧❡♥❝❡✳ ❙♣❡❝tr❛❧ ♠❡t❤♦❞s ❛r❡ ❛ttr❛❝t✐✈❡✱ ❞✉❡ t♦ t❤❡✐r ❤✐❣❤ s♣❛t✐❛❧ ❛❝❝✉r❛❝②✳

❙♣❛t✐❛❧ ❞❡r✐✈❛t✐✈❡s ❛r❡ ❝❛❧❝✉❧❛t❡❞ ❡①❛❝t❧②✳ ❊①♣♦♥❡♥t✐❛❧ ❝♦♥✈❡r❣❡♥❝❡ ❢♦r s♠♦♦t❤ s♦❧✉t✐♦♥s ✭❢❛st❡r t❤❛♥ ❋❊✱ ❋❉ ✳✳✳✮✳

1 2 3 kh 1 (du

dx) h −du dx

spectral DF order 2 DF order 4

❙✐♥❝❡ t❤❡ ✼✵✬s✱ ❡①t❡♥s✐✈❡❧② ❛♣♣❧✐❡❞ t♦ s✐♠✉❧❛t✐♦♥ ♦❢ t✉r❜✉❧❡♥t ✢♦✇s ❜✉t✱ t❤❡✐r ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦♥ ♥❡✇ ❍P❈ ♠✉st ❜❡ ❝❛r❡❢✉❧❧② ❝♦♥s✐❞❡r❡❞✳

slide-15
SLIDE 15

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

P❛rt✐❡ ■■ ▼✐s❡ ❡♥ ♦❡✉✈r❡ s✉r ❧❡s ♣❧❛t❡❢♦r♠❡s ❍P❈

❆❞❛♣t❛t✐♦♥ ❞✉ ❝♦❞❡ ◆❛❞✐❛❙♣❡❝tr❛❧ ❛✉① ♥♦✉✈❡❧❧❡s ❛r❝❤✐t❡❝t✉r❡s

slide-16
SLIDE 16

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

■♥❝♦♠♣r❡ss✐❜❧❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s

  • ♦✈❡r♥✐♥❣ ❡q✉❛t✐♦♥s

∂❯ ∂t + ❯.∇❯ = −∇♣ + ✶ ❘❡ ∆❯ ∇.❯ = ✵ ❯(t = ✵) = ❯✵ ❯|∂Ω

  • ❛❧❡r❦✐♥ ❢♦r♠✉❧❛t✐♦♥ ✉s✐♥❣ ❛♥ ♦rt❤♦❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ✈❡❧♦❝✐t②

❯ = ❯❖❙(❯.❡②) + ❯❙◗((∇ × ❯).❡②) s♣❡❝tr❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ❯(t, ①, ②, ③) =

ˆ ❯✐(t)α✐(①, ②, ③)

slide-17
SLIDE 17

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

◆✉♠❡r✐❝❛❧ ♠❡t❤♦❞

❙♣❡❝tr❛❧ ❝♦❡✣❝✐❡♥ts ✇✐t❤ ◆① × ◆② × ◆③ ♠♦❞❡s ❯(①, ②, ③, t) =

◆①/✷

  • ♠=−◆①/✷

◆③/✷

  • ♣=−◆③/✷

 

◆② −✶

  • ♥=✵

α♠♣

❖❙,♥ ˆ

❯♠♣

❖❙,♥ + ◆② −✶

  • ♥=✵

α♠♣

❙◗,♥ ˆ

❯♠♣

❙◗,♥

  ❖♣t✐♠❛❧ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ s♦❧❡♥♦✐❞❛❧ ✈❡❧♦❝✐t② ✜❡❧❞ ❊❧✐♠✐♥❛t✐♦♥ ♦❢ t❤❡ ♣r❡ss✉r❡ ❙♣❡❝tr❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ❋♦✉r✐❡r✲❈❤❡❜②s❤❡✈ ❛♣♣r♦①✐♠❛t✐♦♥ ✇✐t❤ ❛ ●❛❧❡r❦✐♥ ❢♦r♠✉❧❛t✐♦♥ ❚✐♠❡ ✐♥t❡❣r❛t✐♦♥ ✇✐t❤ ❈r❛♥❦ ◆✐❝♦❧s♦♥ ✴ ❆❞❛♠s ❇❛s❤❢♦rt❤ s❝❤❡♠❡

slide-18
SLIDE 18

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

❘❡s♦❧✉t✐♦♥ ♦❢ ❝♦✉♣❧❡❞ s②st❡♠s ❢♦r ♥♦♥✲❧✐♥❡❛r ❛❞✈❡❝t✐✈❡ t❡r♠s

❆t ❡❛❝❤ t✐♠❡ st❡♣✱ ◆① × ◆③ ❧✐♥❡❛r s②st❡♠s ♦❢ ❞✐♠❡♥s✐♦♥ ◆② − ✸ ❛r❡ s♦❧✈❡❞ ❆♠♣

❖❙α♠♣ ❖❙

= ❜♠♣

❖❙

❆♠♣

❙◗α♠♣ ❙◗

= ❜♠♣

❙◗

❆♠♣

❖❙ ❛♥❞ ❆♠♣ ❙◗ ❛r❡ s♣❛rs❡ ♠❛tr✐❝❡s ✭r❡s♣✳ ✼❉ ❛♥❞ ✺❉✮

❜♠♣ = ❜♠♣(α♠♣

❙◗, α♠♣ ❖❙)

❝♦♥t❛✐♥s ♥♦♥✲❧✐♥❡❛r t❡r♠s ✭❝♦♥✈♦❧✉t✐♦♥ ♣r♦❞✉❝ts ❝♦✉♣❧✐♥❣ ❡✈❡r② α♠♣

♥ ✮

⇒ ❜ ✐s ❝❛❧❝✉❧❛t❡❞ ✐♥ ♣❤②s✐❝❛❧ s♣❛❝❡ ⇒ ♠✉st ♣❡r❢♦r♠ ❋❋❚s ✐♥ ❡❛❝❤ ❞✐r❡❝t✐♦♥ P❡r ✐t❡r❛t✐♦♥✱ ✐✳❡✳ ❛t ❡❛❝❤ t✐♠❡ st❡♣✱ ✷✼ ❋❋❚ ✭❞✐r❡❝t ♦r ✐♥✈❡rs❡✮ ❛r❡ ♣❡r❢♦r♠❡❞

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SLIDE 19

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

❈❤❛❧❧❡♥❣❡✿ ❢r♦♠ ✶✵✵ t♦ ✶✵✵✵✵ ❝♦r❡s

Pr❡s❡♥t ❝♦♥✜❣✉r❛t✐♦♥✿ ❝♦♠♣✉t❛t✐♦♥❛❧ ❞♦♠❛✐♥ s✐③❡ ✶✺✵ × ✷ × ✸.✷ ✶✼✷✽✵ × ✶✾✷ × ✸✽✹ ♠♦❞❡s ✭∼ ✶.✸ ❜✐❧❧✐♦♥ ♦❢ ♠♦❞❡s✮ tr❛✈❡❧ ✶ ❧❡♥❣t❤ ✇✐t❤ ✐t❂✸✵✵✵✵✵ ✐t❡r❛t✐♦♥s✳ ✭∼ ✶✻ ♠✐❧❧✐♦♥s ♦❢ ❋❋❚✮ ▼❡♠♦r② ❝♦♥str❛✐♥t ◆ = ◆① × ◆② × ◆③ ✱ ✇✐t❤ ◆ ✈❡r② ❧❛r❣❡ ✲ ❧❛r❣❡ ♠❡♠♦r② r❡q✉✐r❡♠❡♥t ✭∼ ✺✷✵●♦✮ ✲ ❇❧✉❡●❡♥❡✴P ✵.✺ ●♦ ♣❡r ❝♦r❡ ⇒ ∼ ✶✵✵✵ ❝♦r❡s ♥❡❡❞❡❞ ◆① >> ◆②, ◆③✱ ❡❧♦♥❣❛t❡❞ ✐♥ ♦♥❡ ❞✐r❡❝t✐♦♥ ✲ ✶❉ ❞♦♠❛✐♥ ❞❡❝♦♠♣♦s✐t✐♦♥ ⇒ ❧✐♠✐t❡❞ t♦ ∼ ✶✵✵ ❝♦r❡s ✲ ❝❛♥ ♦♥❧② s✐♠✉❧❛t❡ ❛ ✶✵ t✐♠❡s s❤♦rt❡r ❝❤❛♥♥❡❧ ❧❡♥❣t❤ ❲❛❧❧ ❝❧♦❝❦ t✐♠❡ ❝♦♥str❛✐♥t ❈P❯ t✐♠❡ ✶✵✵❤ ∼ ✹ ❞❛②s ♦♥ ∼ ✽✵✵✵ ❝♦r❡s ✲ ✇✐t❤ ✶✵✵ ❝♦r❡s ✭✐❢ ♣♦ss✐❜❧❡✮✱ ✽✵ t✐♠❡s s❧♦✇❡r✱ ✽✵✵✵❤ ∼ ✶ ②❡❛r

slide-20
SLIDE 20

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

❖✉t❧✐♥❡

■♠♣❧❡♠❡♥t❛t✐♦♥ ♦♥ ❍P❈ ♣❧❛t❢♦r♠s ▼P■ str❛t❡❣② t♦ s❝❛❧❡ ❢r♦♠ ❖(✶✵✵) t♦ ❖(✶✵ ✵✵✵) ❝♦r❡ ❍②❜r✐❞ str❛t❡❣② t♦ ♠✐❣r❛t❡ ♦♥ ♠❛♥②✲❝♦r❡ ♣❧❛t❢♦r♠ ❆❞❞✐t✐♦♥❛❧ ❝♦♥str❛✐♥t ❢♦r ♦♣t✐♠✐③❛t✐♦♥ ❉❛t❛ ♠❛♥✐♣✉❧❛t✐♦♥ ❞✉r✐♥❣ s✐♠✉❧❛t✐♦♥ ❉❛t❛ ♠❛♥✐♣✉❧❛t✐♦♥ ❢♦r ❛♥❛❧②s✐s ❛♥❞ ♣♦st✲tr❡❛t♠❡♥t

slide-21
SLIDE 21

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

❖✉t❧✐♥❡

■♠♣❧❡♠❡♥t❛t✐♦♥ ♦♥ ❍P❈ ♣❧❛t❢♦r♠s ▼P■ str❛t❡❣② t♦ s❝❛❧❡ ❢r♦♠ ❖(✶✵✵) t♦ ❖(✶✵ ✵✵✵) ❝♦r❡ ❍②❜r✐❞ str❛t❡❣② t♦ ♠✐❣r❛t❡ ♦♥ ♠❛♥②✲❝♦r❡ ♣❧❛t❢♦r♠ ❆❞❞✐t✐♦♥❛❧ ❝♦♥str❛✐♥t ❢♦r ♦♣t✐♠✐③❛t✐♦♥ ❉❛t❛ ♠❛♥✐♣✉❧❛t✐♦♥ ❞✉r✐♥❣ s✐♠✉❧❛t✐♦♥ ❉❛t❛ ♠❛♥✐♣✉❧❛t✐♦♥ ❢♦r ❛♥❛❧②s✐s ❛♥❞ ♣♦st✲tr❡❛t♠❡♥t

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SLIDE 22

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

✷❉ ❞♦♠❛✐♥ ❞❡❝♦♠♣♦s✐t✐♦♥

SPECTRAL SPACE SPECTRAL SPACE PHYSICAL SPACE udu/dx Non linear terms udu/dx Non linear terms u du/dx FFT inverse axe x FFT axe x FFT inverse axe z FFT axe z

❈❤❡❜②s❤❡✈ ❜❡t✇❡❡♥ ✇❛❧❧s ✭② ❞✐r❡❝t✐♦♥✱ ◆② + ✶ ♠♦❞❡s✮ ✷❉ ❋❋❚ ✐♥ ♣❡r✐♦❞✐❝❛❧ ❞✐r❡❝t✐♦♥s ✭① ❞✐r❡❝t✐♦♥ ❛♥❞ ③ ❞✐r❡❝t✐♦♥✮ ❚r❛♥s♣♦s❡ ❢r♦♠ ②−♣❡♥❝✐❧ t♦ ①−♣❡♥❝✐❧✱ ①−♣❡♥❝✐❧ t♦ ③−♣❡♥❝✐❧ ❛♥❞ ❜❛❝❦ ■♥❝r❡❛s❡ t❤❡ ♥✉♠❜❡r ♦❢ ▼P■ ♣r♦❝❡ss❡s ❛♥❞ r❡❞✉❝❡ ✇❛❧❧ ❝❧♦❝❦ t✐♠❡ ✶❉ ❞❡❝♦♠♣♦s✐t✐♦♥✿ ▼P■ ≤ ◆② ✶✼✷✽✵ × ✶✾✷ × ✸✽✹ → ♠❛①✳ ♦❢ ▼P■ ♣r♦❝❡ss❡s✿ ♥♣r♦❝❂✶✾✷ ✷❉ ❞❡❝♦♠♣♦s✐t✐♦♥✿ ▼P■ ≤ ◆② × ◆③ ✶✼✷✽✵ × ✶✾✷ × ✸✽✹ → ♠❛①✳ ♦❢ ▼P■ ♣r♦❝❡ss❡s✿ ♥♣r♦❝❂✼✸ ✼✷✽ P❡r❢♦r♠ ❞❛t❛ ❝♦♠♠✉♥✐❝❛t✐♦♥s ❛♥❞ r❡♠❛♣♣✐♥❣ ❈❤♦♦s❡ ❞❛t❛ r❡❛rr❛♥❣❡♠❡♥t t♦ ❧✐♠✐t t❤❡ ✐♥❝r❡❛s❡ ✐♥ ❝♦♠♠✉♥✐❝❛t✐♦♥s

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SLIDE 23

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

❖✉t❧✐♥❡

■♠♣❧❡♠❡♥t❛t✐♦♥ ♦♥ ❍P❈ ♣❧❛t❢♦r♠s ▼P■ str❛t❡❣② t♦ s❝❛❧❡ ❢r♦♠ ❖(✶✵✵) t♦ ❖(✶✵ ✵✵✵) ❝♦r❡ ❍②❜r✐❞ str❛t❡❣② t♦ ♠✐❣r❛t❡ ♦♥ ♠❛♥②✲❝♦r❡ ♣❧❛t❢♦r♠ ❆❞❞✐t✐♦♥❛❧ ❝♦♥str❛✐♥t ❢♦r ♦♣t✐♠✐③❛t✐♦♥ ❉❛t❛ ♠❛♥✐♣✉❧❛t✐♦♥ ❞✉r✐♥❣ s✐♠✉❧❛t✐♦♥ ❉❛t❛ ♠❛♥✐♣✉❧❛t✐♦♥ ❢♦r ❛♥❛❧②s✐s ❛♥❞ ♣♦st✲tr❡❛t♠❡♥t

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SLIDE 24

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

❈♦♥str❛✐♥ts r❡❧❛t❡❞ t♦ ♠♦❞❡r♥ ♠❛♥②✲❝♦r❡s ♣❧❛t❢♦r♠s

❚❡♥❞❛♥❝② t♦✇❛r❞s ♠❛♥②✲❝♦r❡s ♣❧❛t❢♦r♠s ▲✐♠✐t❡❞ ♥✉♠❜❡r ♦❢ ♥♦❞❡s ■♥❝r❡❛s❡ ♦❢ ❝♦r❡s ♣❡r ♥♦❞❡ ✭❇❧✉❡●❡♥❡✴P = ✹ ✲ ❙✉♣❡r▼❯❈ = ✶✻✮ ■♥❝r❡❛s❡ ▼P■ ♣r♦❝❡ss❡s ❛❧❧♦✇ ❧❛r❣❡r ♥✉♠❜❡r ♦❢ ♠♦❞❡s ✇✐t❤✐♥ t❤❡ s❛♠❡ ✇❛❧❧ ❝❧♦❝❦ t✐♠❡ ❧✐♠✐t t❤❡ ♠❡♠♦r② ❛✈❛✐❧❛❜❧❡ ♣❡r ♣r♦❝❡ss✉s

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SLIDE 25

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

❈♦♥str❛✐♥ts r❡❧❛t❡❞ t♦ ♠♦❞❡r♥ ♠❛♥②✲❝♦r❡s ♣❧❛t❢♦r♠s

❚❡♥❞❛♥❝② t♦✇❛r❞s ♠❛♥②✲❝♦r❡s ♣❧❛t❢♦r♠s ▲✐♠✐t❡❞ ♥✉♠❜❡r ♦❢ ♥♦❞❡s ■♥❝r❡❛s❡ ♦❢ ❝♦r❡s ♣❡r ♥♦❞❡ ✭❇❧✉❡●❡♥❡✴P = ✹ ✲ ❙✉♣❡r▼❯❈ = ✶✻✮ ■♥❝r❡❛s❡ ▼P■ ♣r♦❝❡ss❡s ❛❧❧♦✇ ❧❛r❣❡r ♥✉♠❜❡r ♦❢ ♠♦❞❡s ✇✐t❤✐♥ t❤❡ s❛♠❡ ✇❛❧❧ ❝❧♦❝❦ t✐♠❡ ❧✐♠✐t t❤❡ ♠❡♠♦r② ❛✈❛✐❧❛❜❧❡ ♣❡r ♣r♦❝❡ss✉s

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SLIDE 26

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

❈♦♥str❛✐♥ts r❡❧❛t❡❞ t♦ ♠♦❞❡r♥ ♠❛♥②✲❝♦r❡s ♣❧❛t❢♦r♠s

❚❡♥❞❛♥❝② t♦✇❛r❞s ♠❛♥②✲❝♦r❡s ♣❧❛t❢♦r♠s ▲✐♠✐t❡❞ ♥✉♠❜❡r ♦❢ ♥♦❞❡s ■♥❝r❡❛s❡ ♦❢ ❝♦r❡s ♣❡r ♥♦❞❡ ✭❇❧✉❡●❡♥❡✴P = ✹ ✲ ❙✉♣❡r▼❯❈ = ✶✻✮ ■♥❝r❡❛s❡ ▼P■ ♣r♦❝❡ss❡s ❛❧❧♦✇ ❧❛r❣❡r ♥✉♠❜❡r ♦❢ ♠♦❞❡s ✇✐t❤✐♥ t❤❡ s❛♠❡ ✇❛❧❧ ❝❧♦❝❦ t✐♠❡ ❧✐♠✐t t❤❡ ♠❡♠♦r② ❛✈❛✐❧❛❜❧❡ ♣❡r ♣r♦❝❡ss✉s

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SLIDE 27

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

❈♦♥str❛✐♥ts r❡❧❛t❡❞ t♦ ♠♦❞❡r♥ ♠❛♥②✲❝♦r❡s ♣❧❛t❢♦r♠s

❚❡♥❞❛♥❝② t♦✇❛r❞s ♠❛♥②✲❝♦r❡s ♣❧❛t❢♦r♠s ▲✐♠✐t❡❞ ♥✉♠❜❡r ♦❢ ♥♦❞❡s ■♥❝r❡❛s❡ ♦❢ ❝♦r❡s ♣❡r ♥♦❞❡ ✭❇❧✉❡●❡♥❡✴P = ✹ ✲ ❙✉♣❡r▼❯❈ = ✶✻✮ ■♥❝r❡❛s❡ ▼P■ ♣r♦❝❡ss❡s ❛❧❧♦✇ ❧❛r❣❡r ♥✉♠❜❡r ♦❢ ♠♦❞❡s ✇✐t❤✐♥ t❤❡ s❛♠❡ ✇❛❧❧ ❝❧♦❝❦ t✐♠❡ ❧✐♠✐t t❤❡ ♠❡♠♦r② ❛✈❛✐❧❛❜❧❡ ♣❡r ♣r♦❝❡ss✉s

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SLIDE 28

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

❈♦♥str❛✐♥ts r❡❧❛t❡❞ t♦ ♠♦❞❡r♥ ♠❛♥②✲❝♦r❡s ♣❧❛t❢♦r♠s

❚❡♥❞❛♥❝② t♦✇❛r❞s ♠❛♥②✲❝♦r❡s ♣❧❛t❢♦r♠s ▲✐♠✐t❡❞ ♥✉♠❜❡r ♦❢ ♥♦❞❡s ■♥❝r❡❛s❡ ♦❢ ❝♦r❡s ♣❡r ♥♦❞❡ ✭❇❧✉❡●❡♥❡✴P = ✹ ✲ ❙✉♣❡r▼❯❈ = ✶✻✮ ■♥❝r❡❛s❡ ▼P■ ♣r♦❝❡ss❡s ❛❧❧♦✇ ❧❛r❣❡r ♥✉♠❜❡r ♦❢ ♠♦❞❡s ✇✐t❤✐♥ t❤❡ s❛♠❡ ✇❛❧❧ ❝❧♦❝❦ t✐♠❡ ❧✐♠✐t t❤❡ ♠❡♠♦r② ❛✈❛✐❧❛❜❧❡ ♣❡r ♣r♦❝❡ss✉s

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SLIDE 29

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

❍②❜r✐❞ ❖♣❡♥▼P✴▼P■

❙✉✐t❛❜❧❡ ❢♦r r❡❝❡♥t ♠❛♥②✲❝♦r❡ ♣❧❛t❢♦r♠s ❘❡❞✉❝❡s t❤❡ ♥✉♠❜❡r ♦❢ ▼P■ ♣r♦❝❡ss❡s

❘❡❞✉❝❡s t❤❡ ♥✉♠❜❡r ♦❢ ❝♦♠♠✉♥✐❝❛t✐♦♥s ■♥❝r❡❛s❡s t❤❡ ❛✈❛✐❧❛❜❧❡ ♠❡♠♦r② s✐③❡ ♣❡r ♥♦❞❡

▼♦❞✐✜❝❛t✐♦♥ ❢♦r ♠❛♥② t❤r❡❛❞s ❚✐♠❡ ♦❢ t❤r❡❛❞ ❝r❡❛t✐♦♥ ❡①❝❡❡❞s ✐♥♥❡r ❧♦♦♣ t✐♠❡ ❡①❡❝✉t✐♦♥ ■♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ❡①♣❧✐❝✐t ❝r❡❛t✐♦♥ ♦❢ t❤r❡❛❞s ❘❡❝♦✈❡r ❢✉❧❧ ▼P■ ♣❡r❢♦r♠❛♥❝❡ ❛♥❞ ❛❧❧♦✇ ❢✉rt❤❡r ✐♠♣r♦✈♠❡♥t✳

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SLIDE 30

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

❖✉t❧✐♥❡

■♠♣❧❡♠❡♥t❛t✐♦♥ ♦♥ ❍P❈ ♣❧❛t❢♦r♠s ▼P■ str❛t❡❣② t♦ s❝❛❧❡ ❢r♦♠ ❖(✶✵✵) t♦ ❖(✶✵ ✵✵✵) ❝♦r❡ ❍②❜r✐❞ str❛t❡❣② t♦ ♠✐❣r❛t❡ ♦♥ ♠❛♥②✲❝♦r❡ ♣❧❛t❢♦r♠ ❆❞❞✐t✐♦♥❛❧ ❝♦♥str❛✐♥t ❢♦r ♦♣t✐♠✐③❛t✐♦♥ ❉❛t❛ ♠❛♥✐♣✉❧❛t✐♦♥ ❞✉r✐♥❣ s✐♠✉❧❛t✐♦♥ ❉❛t❛ ♠❛♥✐♣✉❧❛t✐♦♥ ❢♦r ❛♥❛❧②s✐s ❛♥❞ ♣♦st✲tr❡❛t♠❡♥t

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SLIDE 31

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

▼♦r❡ t❤❛♥ ❞♦♠❛✐♥ ❞❡❝♦♠♣♦s✐t✐♦♥ ✳✳✳

❚❛s❦s ♣❛r❛❧❧❡❧✐③❛t✐♦♥ ✿ ♠❛s❦ ❝♦♠♠✉♥✐❝❛t✐♦♥s ❜② ❡①❡❝✉t✐♦♥ t✐♠❡ r❡❞✉❝❡s ❜② ✷✵% t✐♠❡ ♣❡r ✐t❡r❛t✐♦♥ ❧❡ss ❧♦ss ✐♥ ❝♦♠♠✉♥✐❝❛t✐♦♥s ✲ ✇❛✐st ∼ ✶✵% P❧❛❝❡♠❡♥t ♦❢ ♣r♦❝❡ss✉s s♣❡❝✐✜❝ ♦♥ ❡❛❝❤ ♣❧❛t❢♦r♠✱ ♦♣t✐♠✐③❡ ✐♥t❡r❝♦♥♥❡❝t✐♦♥ ❝♦♠♠✉♥✐❝❛t✐♦♥s ❛✈♦✐❞ t❤r❡❛❞s t♦ ♠✐❣r❛t❡ ❢r♦♠ ♦♥❡ ❝♦r❡ t♦ ❛♥♦t❤❡r

❡①❛♠♣❧❡✿ ❚❖❘❯❙ ✈❡rs✉s ▼❊❙❍ ✐♥ ❇❧✉❡●❡♥❡✴P ♣❧❛t❢♦r♠ ✲ ✺✵% ❢❛st❡r

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SLIDE 32

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

❖✉t❧✐♥❡

■♠♣❧❡♠❡♥t❛t✐♦♥ ♦♥ ❍P❈ ♣❧❛t❢♦r♠s ▼P■ str❛t❡❣② t♦ s❝❛❧❡ ❢r♦♠ ❖(✶✵✵) t♦ ❖(✶✵ ✵✵✵) ❝♦r❡ ❍②❜r✐❞ str❛t❡❣② t♦ ♠✐❣r❛t❡ ♦♥ ♠❛♥②✲❝♦r❡ ♣❧❛t❢♦r♠ ❆❞❞✐t✐♦♥❛❧ ❝♦♥str❛✐♥t ❢♦r ♦♣t✐♠✐③❛t✐♦♥ ❉❛t❛ ♠❛♥✐♣✉❧❛t✐♦♥ ❞✉r✐♥❣ s✐♠✉❧❛t✐♦♥ ❉❛t❛ ♠❛♥✐♣✉❧❛t✐♦♥ ❢♦r ❛♥❛❧②s✐s ❛♥❞ ♣♦st✲tr❡❛t♠❡♥t

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SLIDE 33

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

Pr♦❜❧❡♠s r❡❧❛t❡❞ t♦ t❤❡ ✈❡r② ❧❛r❣❡ ❝❛❧❝✉❧❛t✐♦♥s ❉❛t❛ ♠❛♥✐♣✉❧❛t✐♦♥ ❞✉r✐♥❣ s✐♠✉❧❛t✐♦♥

❉❛t❛ ■♥♣✉t✴❖✉t♣✉t ❛♥❞ st♦r❛❣❡ ▲❛r❣❡ ❞❛t❛

✲ ❝❛s❡ ✶✼✷✽✵ × ✶✾✷ × ✸✽✹ ✿ ♦♥❡ ✈❡❧♦❝✐t② ✜❡❧❞ ∼ ✸✵ ●♦

⇒ ❯s❡ ♣❛r❛❧❧❡❧ ■❖ ✭❡❛❝❤ ♣r♦❝❡ss❡s ✇r✐t❡s ✐ts ♦✇♥ ❞❛t❛✮ ▲❛r❣❡ ❛♠♦✉♥t ♦❢ ✜❧❡✱ ❝♦✉❧❞ r❛♣✐❞❧② ❡①❝❡❡❞s ✐♥♦❞❡ ♦r q✉♦t❛ ❧✐♠✐t

✲ ❝❛s❡ ✶✼✷✽✵ × ✶✾✷ × ✸✽✹ ✿ st❛t✐st✐❝s ✽ ✜❡❧❞s ♦♥ ∼ ✽✵✵✵ ♣r♦❝❡ss❡s ✭∼ ✸✵✵ ●♦✮ ✲ ✇r✐t❡ ∼ ✶✺✵ t✐♠❡s ❞❛t❛ ❞✉r✐♥❣ tr❛✈❡❧ ❧❡♥❣t❤ ✭▲① = ✶✺✵✮ ✭❞✐s❦ q✉♦t❛ ∼ ✸ ❚♦✮

⇒ ✇r❛♣ ✐♥ t❛r ❛r❝❤✐✈❡ ✜❧❡ ♦r s❡♣❛r❛t❡❞ ❞✐r❡❝t♦r② ▼❛♥❛❣❡ t❤❡ ❧❛r❣❡ ❛♠♦✉♥t ♦❢ ❞❛t❛ ❣❡♥❡r❛t❡❞ ⇒ ❯s❡ ♦❢ ♣r❡❞❡✜♥❡❞ ♣❛r❛❧❧❡❧ ❢♦r♠❛t ✭❱❚❑✱ ❍❉❋✺✱ ◆❡t❈❋❉✱ ✳✳✳✮ ❜❡✇❛r❡ ♥♦t t♦ ❛❞❞ ✉s❡❧❡ss ❝♦♠♣❧❡①✐t② ❢♦r r❡❣✉❧❛r str✉❝t✉r❡❞ ❞❛t❛ ⇒ ❖♣t✐♠✐③❡ ❞❛t❛ tr❛♥s❢❡rt ❜❡t✇❡❡♥ ♣❧❛t❢♦r♠

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SLIDE 34

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

❍P❈ s✐♠✉❧❛t✐♦♥s r❡q✉✐r❡ ❡✈❡r② ❧❛②❡r ♦❢ ❍P❈ r❡ss♦✉r❝❡s

❚✐❡r✲✵✱ P❘❆❈❊

✶ ❏❯●❊◆❊ ❛♥❞ ❏❯◗❯❊❊◆✱ ❏ü❧✐❝❤✱ ●❡r♠❛♥② ✷ ❈❯❘■❊✱ ❇r✉②èr❡s✲❧❡✲❈❤ât❡❧✱ ❋r❛♥❝❡ ✸ ❙✉♣❡r▼❯❈✱ ●❛r❝❤✐♥❣✱ ●❡r♠❛♥②

❚✐❡r✲✶✱ ●❊◆❈■

✶ ■❉❘■❙✱ ❖rs❛② ✷ ❈■◆❊❙✱ ▼♦♥t♣❡❧❧✐❡r ✸ ❚●❈❈✱ ❇r✉②èr❡s✲❧❡✲❈❤ât❡❧

❚✐❡r✲✷✱ ❋é❞ér❛t✐♦♥ ▲②♦♥♥❛✐s❡ ❞❡ ▼♦❞é❧✐s❛t✐♦♥ ❡t ❙❝✐❡♥❝❡s ◆✉♠ér✐q✉❡s P✷❈❍P❉✱ ❧❛ ❉♦✉❛ ▼❛♥② t❤❛♥❦s t♦ ❈❤r✐st♦♣❤❡ Pér❛

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SLIDE 35

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

Pr♦❜❧❡♠s r❡❧❛t❡❞ t♦ t❤❡ ✈❡r② ❧❛r❣❡ ❝❛❧❝✉❧❛t✐♦♥s ❉❛t❛ ♠❛♥✐♣✉❧❛t✐♦♥ ❛❢t❡r s✐♠✉❧❛t✐♦♥

❉❛t❛ ♣r♦❝❡ss✐♥❣ P❛rt ♦❢ t❤❡ ❛♥❛❧②s✐s ✐s ♣❡r❢♦r♠❡❞ ❞✉r✐♥❣ s✐♠✉❧❛t✐♦♥ P❛rt ♦❢ ✐t ✐s ❡①♣❧♦r❡❞ ❛❢t❡r✇❛r❞s

❊♥t❛✐❧s s♣❛t✐❛❧ ❞❡r✐✈❛t✐♦♥✱ ❡✐❣❡♥✈❛❧✉❡s ❡✈❛❧✉❛t✐♦♥ ✳✳✳ Pr❡s❡r✈❡ t❤❡ s❛♠❡ ❛❝❝✉r❛❝② t❤❛♥ ✐♥ t❤❡ s✐♠✉❧❛t✐♦♥ ❙❤♦✉❧❞ ❜❡ ✐♥t❡r❛❝t✐✈❡ ❛♥❞ ✇❤❡♥ r❡❛❞② ♦♥ ❜❛t❝❤ ♠♦❞❡

⇒ ❙❤♦✉❧❞ ❜❡ ❞♦♥❡ ❧♦❝❛❧❧② ❜❡❝❛✉s❡ ♦❢ ❞❛t❛ st♦r❛❣❡ ⇒ ▼✉st ❜❡ ♣❛r❛❧❧❡❧✱ ❜✉t ♦♥ ❛ s♠❛❧❧❡r s❝❛❧❡

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SLIDE 36

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

Pr♦❜❧❡♠s r❡❧❛t❡❞ t♦ t❤❡ ✈❡r② ❧❛r❣❡ ❝❛❧❝✉❧❛t✐♦♥s ❉❛t❛ ♠❛♥✐♣✉❧❛t✐♦♥ ❛❢t❡r s✐♠✉❧❛t✐♦♥

❉❛t❛ ✸❉ ✈✐s✉❛❧✐③❛t✐♦♥ ❈❛♥♥♦t ❜❡ ♣❡r❢♦r♠❡❞ ❞✐r❡❝t❧② ♦♥ ❍P❈ ♣❧❛t❢♦r♠s Pr❡s❡r✈❡ ❛❝❝✉r❛❝② ♦❢ t❤❡ s✐♠✉❧❛t✐♦♥ ❙❤♦✉❧❞ ❜❡ ✐♥t❡r❛❝t✐✈❡ ❛♥❞ ✇❤❡♥ r❡❛❞② ♦♥ ❜❛t❝❤ ♠♦❞❡ ⇒ ▼✉st ❜❡ ❞♦♥❡ ✇✐t❤ r❡♠♦t❡ ❛❝❝❡ss ⇒ ▼✉st ❜❡ ♣❛r❛❧❧❡❧✱ ❜✉t ♦♥ ❛ s♠❛❧❧❡r s❝❛❧❡

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SLIDE 37

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

Pr♦❜❧❡♠s r❡❧❛t❡❞ t♦ t❤❡ ✈❡r② ❧❛r❣❡ ❝❛❧❝✉❧❛t✐♦♥s ❉❛t❛ ♠❛♥✐♣✉❧❛t✐♦♥ ❛❢t❡r s✐♠✉❧❛t✐♦♥

❖r❣❛♥✐③❛t✐♦♥ ❉❛t❛ tr❛♥s❢❡rr❡❞ ♦♥ ❛ ❧♦❝❛❧ ❞❛t❛ s❡r✈❡r ✲ ✇✐t❤ r❛♣✐❞ ♥❡t✇♦r❦ ❛❝❝❡ss Pr♦❝❡ss✐♥❣ ✐s ♣❡r❢♦r♠❡❞ ♦♥ ❛ ▼❡❝♦❝❡♥t❡r ♣❧❛t❢♦r♠ ❈❧✐❡♥t✴❙❡r✈❡r s②st❡♠ t♦ ❛❝❝❡ss ❢r♦♠ ❧❛♣t♦♣

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SLIDE 38

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

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SLIDE 39

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

❲❤❛t ✇❛s ❛❝❤✐❡✈❡❞ ❢♦r ❍P❈ s✐♠✉❧❛t✐♦♥s

❆ s✉✐t❛❜❧❡ ❞❡✈❡❧♦♣♠❡♥t ❛♥❞ s♦❢t✇❛r❡ ❡♥✈✐r♦♥♠❡♥t ❝♦❞❡ ❈✰✰ ❇▲❆❙✱ ●❙▲ ▼P■✴❖♣❡♥▼P ✲ ♦♣t✐♠✐③❡❞ ❧✐❜r❛r✐❡s ✭❡✳❣✳ ❋❋❚❲✱ ▼❑▲✮ ❝♠❛❦❡✱ ❣✐t

s✇✐❣ ✐♥t❡r❢❛❝❡ P②t❤♦♥ ❛♥❞ ❛ ❈✰✰ ❧✐❜r❛r② ❞❡r✐✈❡❞ ❢r♦♠ t❤❡ ❝♦❞❡ ♣②t❤♦♥✱ ♠♣✐✹♣②✱ ♥✉♠♣②✱ ♠❛t♣❧♦t❧✐❜✱ ♠❛②❛✈✐ ✳✳✳

❉❡✈❡❧♦♣♠❡♥t ♦❢ ❛ ♣❛r❛❧❧❡❧ str❛t❡❣② ❢♦r t❤❡ ❝♦❞❡ r❡✈✐s✐t ♣❛r❛❧❧❡❧ str❛t❡❣② ♦❢ t❤❡ ❝♦❞❡ r❡✈✐s✐t str❛t❡❣② ❢♦r t❤❡ ❛♥❛❧②s✐s

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SLIDE 40

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

❘❡s✉❧t✐♥❣ ♠❡t❤♦❞

❈❤❛r❛❝t❡r✐❝t✐❝s ❊✣❝✐❡♥t s♦❧✈❡r ❢♦r ❤②❜r✐❞ ♠✉❧t✐❝♦r❡ ♠❛ss✐✈❡❧② ♣❛r❛❧❧❡❧ ♣❧❛t❢♦r♠s

❖r✐❣✐♥❛❧ ❝♦❛rs❡ ❣r❛✐♥❡❞ ▼P■✴❖♣❡♥▼P str❛t❡❣② ❚❛s❦s ♦✈❡r❧❛♣♣✐♥❣

Pr❡✲ ❛♥❞ ♣♦st✲ ♣r♦❝❡ss✐♥❣ t♦♦❧s ❢♦r s♠❛❧❧❡r ▼P■ ♣❧❛t❢♦r♠s

♣❛r❛❧❧❡❧ ❱❚❑ ❢♦r♠❛t ✭♣❛r❛✈✐❡✇✮ P❛r❛❧❧❡❧ ❈❧✐❡♥t✴❙❡r✈❡r ♣r♦❣r❛♠s ✐♥ P②t❤♦♥ ❝❛❧❧✐♥❣ ❛ s♣❡❝tr❛❧ ❧✐❜r❛r② ✷❉✴✸❉ ♣❛r❛❧❧❡❧ ✈✐s✉❛❧✐③❛t✐♦♥ ✲ ✭♠❛t♣❧♦t❧✐❜✴♠❛②❛✈✐✮

Pr♦♣❡rt✐❡s ❋❛✐r❧② ♣♦rt❛❜❧❡ ❙♠❛❧❧ t✐♠❡ s♣❡♥t ✐♥ ❝♦♠♠✉♥✐❝❛t✐♦♥s ∼ ✶✵% ❘❛♣✐❞ ✇❛❧❧ ❝❧♦❝❦ t✐♠❡ ❢♦r ❛ ❣❧♦❜❛❧ s❝❤❡♠❡ ✭✶.✸s✴✐t ♦♥ ❇❧✉❡●❡♥❡✴P ✲ ✵.✷s✴✐t ♦♥ ❙✉♣❡r▼❯❈✮

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SLIDE 41

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

❉◆❙ ♦❢ t✉r❜✉❧❡♥t tr❛♥s✐t✐♦♥ ✐♥ ❝❤❛♥♥❡❧ ❡♥tr❛♥❝❡ ✢♦✇

❋✐rst tr❛♥s✐t✐♦♥ ❙❡❝♦♥❞ tr❛♥s✐t✐♦♥

x y

■♥st❛♥t❛♥❡♦✉s ✈❡❧♦❝✐t② ✐♥ (②, ③) ♣❧❛♥❡

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SLIDE 42

❈♦♥t❡①t P❛r❛❧❧❡❧✐③❛t✐♦♥

❉◆❙ ♦❢ t✉r❜✉❧❡♥t tr❛♥s✐t✐♦♥ ✐♥ ❝❤❛♥♥❡❧ ❡♥tr❛♥❝❡ ✢♦✇

❚♦ r❡❛❞ ♠♦r❡ ✿ ❏✳ ▼♦♥t❛❣♥✐❡r✱ ❆✳ ❈❛❞✐♦✉✱ ▼✳ ❇✉✛❛t✱ ▲✳ ▲❡ P❡♥✈❡♥✱ ❚♦✇❛r❞s ♣❡t❛s❝❛❧❡ s♣❡❝tr❛❧ s✐♠✉❧❛t✐♦♥s ❢♦r tr❛♥s✐t✐♦♥ ❛♥❛❧②s✐s ✐♥ ✇❛❧❧ ❜♦✉♥❞❡❞ ✢♦✇ ✭✷✵✶✷✮✱ ■♥t✳ ❏♦✉r♥❛❧ ❢♦r ◆✉♠❡r✐❝❛❧ ▼❡t❤♦❞s ✐♥ ❋❧✉✐❞s✱ ❞♦✐✿✶✵✳✶✵✵✷✴✢❞✳✸✼✺✽