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❈❋❉ ④ ❙♦❧✉t✐♦♥ ❆❧❣♦r✐t❤♠s ◆✲❙ ❊q✉❛t✐♦♥s P■❙❖ ❙■▼P▲❊ ❈❋❉ ④ ❙♦❧✉t✐♦♥ ❆❧❣♦r✐t❤♠s ■♥✐t✐❛❧✴❇♦✉♥❞❛r② ❈♦♥❞✐t✐♦♥s ❙❖❊✸✷✶✸✴✹✿ ❈❋❉ ▲❡❝t✉r❡ ✸ ❋✉❧❧ s♦❧✉t✐♦♥ ④ r❡❝❛♣

❈❋❉ ④ ❙♦❧✉t✐♦♥ ◆✲❙ ❊q✉❛t✐♦♥s ❆❧❣♦r✐t❤♠s r ✿ ✉ ❂ ✵ ✭✶✮ ◆✲❙ ❊q✉❛t✐♦♥s ❅ ✉ ① ❅ t ✰ r ✿ ✭ ✉ ✉ ① ✮ ❂ � ✶ ❅ ♣ ❅ ① ✰ ✗ r ✷ ✉ ① P■❙❖ ✭✷✮ ✚ ❙■▼P▲❊ ❅ ✉ ② ❅ t ✰ r ✿ ✭ ✉ ✉ ② ✮ ❂ � ✶ ❅ ♣ ■♥✐t✐❛❧✴❇♦✉♥❞❛r② ❅ ② ✰ ✗ r ✷ ✉ ② ✭✸✮ ❈♦♥❞✐t✐♦♥s ✚ ❋✉❧❧ s♦❧✉t✐♦♥ ④ ❅ t ✰ r ✿ ✭ ✉ ✉ ③ ✮ ❂ � ✶ ❅ ✉ ③ ❅ ♣ r❡❝❛♣ ❅ ③ ✰ ✗ r ✷ ✉ ③ ✭✹✮ ✚ ❚r❛♥s♣♦rt ❡q✉❛t✐♦♥ ❢♦r♠✱ ✇✐t❤ s♦✉r❝❡ � ✶ ❅ ♣ ✚ ❅ ① ❚✇♦ ♠❛✐♥ ♣r♦❜❧❡♠s ✶ ❚❤❡ ❡q✉❛t✐♦♥ ✐s ♥♦♥✲❧✐♥❡❛r ✷ ❚❤❡ s♦✉r❝❡ t❡r♠ ✐♥✈♦❧✈❡s ♣

❈❋❉ ④ ❙♦❧✉t✐♦♥ ❆❧❣♦r✐t❤♠s ◆❙❊ ✑ ✸ ❡q✉❛t✐♦♥s ❢♦r ✹ ✈❛r✐❛❜❧❡s✱ ◆✲❙ ❊q✉❛t✐♦♥s ✰✶ ❝♦♥str❛✐♥t ❡q✉❛t✐♦♥✳ P■❙❖ ❙■▼P▲❊ ✎ ◆❡❡❞ ♣ t♦ ☞♥❞ ✉ ✱ ♥❡❡❞ ✉ t♦ ☞♥❞ ♣ ■♥✐t✐❛❧✴❇♦✉♥❞❛r② ✎ ❊q✉❛t✐♦♥s s♦❧✈❡❞ s❡q✉❡♥t✐❛❧❧② ❈♦♥❞✐t✐♦♥s ❋✉❧❧ s♦❧✉t✐♦♥ ④ ✎ ■t❡r❛t❡ ✉♥t✐❧ ❛❧❧ ☞❡❧❞s ❝♦rr❡❝t r❡❝❛♣ ❙❡✈❡r❛❧ ❞✐☛❡r❡♥t ❛♣♣r♦❛❝❤❡s ✉s❡❞ t♦ s♦❧✈❡ t❤❡s❡ ❡q✉❛t✐♦♥s✳ ✷ ♠❛✐♥ ♦♥❡s ✿ ✎ P■❙❖ ④ Pr❡ss✉r❡ ■♠♣❧✐❝✐t ❙♣❧✐tt✐♥❣ ♦❢ ❖♣❡r❛t♦rs ④ ❢♦r t✐♠❡ ❞❡♣❡♥❞❡♥t ✌♦✇s ✎ ❙■▼P▲❊ ④ ❙❡♠✐✲■♠♣❧✐❝✐t ▼❡t❤♦❞ ❢♦r Pr❡ss✉r❡ ▲✐♥❦❡❞ ❊q✉❛t✐♦♥s ④ ✉s❡❞ ❢♦r st❡❛❞② st❛t❡ ♣r♦❜❧❡♠s

❈❋❉ ④ ❙♦❧✉t✐♦♥ P■❙❖ ❆❧❣♦r✐t❤♠s ◆✲❙ ❊q✉❛t✐♦♥s ✎ ❊q♥ ✭✷✮ ✐❢ ✇❡ ❦♥♦✇ ♣ ✱ ✌✉①✱ ❝❛♥ ☞♥❞ ✉ ① ✳ P■❙❖ ❙■▼P▲❊ ✎ ❊q♥s ✭✷④✹✮✱ ✭✶✮ ❝♦♠❜✐♥❡❞ ✦ ❡q✉❛t✐♦♥ ❢♦r ♣ ❣✐✈❡♥ t❤❡ ■♥✐t✐❛❧✴❇♦✉♥❞❛r② ✌✉① ✿ ✔ ✶ ❈♦♥❞✐t✐♦♥s ✕ ❋✉❧❧ s♦❧✉t✐♦♥ ④ r ✿ ❆ r ♣ ❂ ✭✌✉① t❡r♠✮ r❡❝❛♣ ✎ ❋❧✉① ❤❛s t♦ s❛t✐s❢② ❝♦♥t✐♥✉✐t② ✭✶✮ P■❙❖ ❛❧❣♦r✐t❤♠ ❛s ❢♦❧❧♦✇s ✿ ✶ ●✉❡ss ♣ ✱ ✌✉① ✭✈❛❧✉❡s ❢r♦♠ ♣r❡✈✐♦✉s t✐♠❡st❡♣✮ ✷ ❯s❡ ❡q✉❛t✐♦♥ ✭✷✮ t♦ ☞♥❞ ✉ ① ✱ ✉ ② ❡t❝✳ ✸ ❙♦❧✈❡ ♣r❡ss✉r❡ ❡q✉❛t✐♦♥ ❢♦r ♣ ✹ ❈♦rr❡❝t ✌✉① t♦ s❛t✐s❢② ❝♦♥t✐♥✉✐t②

❈❋❉ ④ Start ❙♦❧✉t✐♦♥ ❆❧❣♦r✐t❤♠s ◆✲❙ ❊q✉❛t✐♦♥s Guess p, flux P■❙❖ (values from previous timestep) ❙■▼P▲❊ ■♥✐t✐❛❧✴❇♦✉♥❞❛r② Use equation (3) to find ❈♦♥❞✐t✐♦♥s ux, uy, uz ❋✉❧❧ s♦❧✉t✐♦♥ ④ r❡❝❛♣ Solve pressure equation to get p Correct flux to satisfy continuity Y Next timestep? N

❈❋❉ ④ ❙♦❧✉t✐♦♥ ❙■▼P▲❊ ❆❧❣♦r✐t❤♠s ❈♦✉❧❞ ✐t❡r❛t❡ P■❙❖ ✉♥t✐❧ st❡❛❞②✲st❛t❡✳ ▼♦r❡ ✉s✉❛❧ t♦ ✉s❡ ◆✲❙ ❊q✉❛t✐♦♥s ❙■▼P▲❊✳ P■❙❖ ❙■▼P▲❊ ■♥✐t✐❛❧✴❇♦✉♥❞❛r② ❈♦♥❞✐t✐♦♥s r ✿ ✉ ❂ ✵ ❋✉❧❧ s♦❧✉t✐♦♥ ④ r ✿ ✭ ✉ ✉ ① ✮ ❂ � ✶ ❅ ♣ r❡❝❛♣ ❅ ① ✰ ✗ r ✷ ✉ ① ✚ r ✿ ✭ ✉ ✉ ② ✮ ❂ � ✶ ❅ ♣ ❅ ② ✰ ✗ r ✷ ✉ ② ✭✺✮ ✚ r ✿ ✭ ✉ ✉ ③ ✮ ❂ � ✶ ❅ ♣ ❅ ③ ✰ ✗ r ✷ ✉ ③ ✚ ✶ ●✉❡ss ❛♥ ✐♥✐t✐❛❧ ♣r❡ss✉r❡ ☞❡❧❞ ♣ ✄ ✷ ❯s❡ ❡q✉❛t✐♦♥ ✭✺✮ t♦ ❝r❡❛t❡ ❛ ✈❡❧♦❝✐t② ☞❡❧❞ ✉ ✄ ❢r♦♠ t❤✐s ♣r❡ss✉r❡ ☞❡❧❞

❈❋❉ ④ ✸ ❋✐♥❞ ❛ ❝♦rr❡❝t✐♦♥ ♣ ✵ t♦ t❤❡ ♣r❡ss✉r❡ ☞❡❧❞ ❙♦❧✉t✐♦♥ ❆❧❣♦r✐t❤♠s ♣ ✄✄ ❂ ♣ ✄ ✰ ♣ ✵ ◆✲❙ ❊q✉❛t✐♦♥s P■❙❖ ✹ ❈♦rr❡❝t t❤❡ ✈❡❧♦❝✐t② ✭✌✉①✮ t♦ ♦❜❡② ❝♦♥t✐♥✉✐t②✳ ❙■▼P▲❊ ■♥✐t✐❛❧✴❇♦✉♥❞❛r② ■♥ t❤❡♦r②✱ ♣ ❛♥❞ ✉ s❤♦✉❧❞ ♥♦✇ ❜❡ t❤❡ ❞❡s✐r❡❞ s♦❧✉t✐♦♥✳ ■♥ ❈♦♥❞✐t✐♦♥s ♣r❛❝t✐❝❡✱ ✐t ✐s ♥❡❝❡ss❛r② t♦ r❡♣❡❛t t❤✐s ♣r♦❝❡❞✉r❡ ❛s ❛♥ ✐t❡r❛t✐✈❡ ❋✉❧❧ s♦❧✉t✐♦♥ ④ r❡❝❛♣ ♣r♦❝❡ss✳ ■❢ ♥❡✇ s♦❧✉t✐♦♥ ♣ ✄✄ ✱ ✉ ✄✄ ❛❞♦♣t❡❞ ❛t ❡❛❝❤ st❡♣ ✦ ❛❧❣♦r✐t❤♠ ❜❡❝♦♠❡s ✉♥st❛❜❧❡✳ ❚❤✉s✱ ✉s❡ ✉♥❞❡rr❡❧❛①❛t✐♦♥ ✿ ✺ ❯♥❞❡rr❡❧❛① s♦❧✉t✐♦♥ ♣ ♥ ✰✶ ❂ ☛ ♣ ✄✄ ✰ ✭✶ � ☛ ✮ ♣ ♥ ☛ ✐s ✉♥❞❡rr❡❧❛①❛t✐♦♥ ♣❛r❛♠❡t❡r✳ ❆t ❡❛❝❤ st❡♣ ♦❢ t❤❡ ✐t❡r❛t✐♦♥✱ t❤❡ ❡rr♦r ✐♥ t❤❡ s♦❧✉t✐♦♥ ✭t❤❡ r❡s✐❞✉❛❧✮ s❤♦✉❧❞ ❞❡❝r❡❛s❡ ④ r❡s✐❞✉❛❧s ❢♦r ❛❧❧ ✈❛r✐❛❜❧❡s s❤♦✉❧❞ ❜❡ ♠♦♥✐t♦r❡❞ t♦ ❡♥s✉r❡ t❤✐s✳

❈❋❉ ④ Start ❙♦❧✉t✐♦♥ ❆❧❣♦r✐t❤♠s Guess p* ◆✲❙ ❊q✉❛t✐♦♥s P■❙❖ ❙■▼P▲❊ * Find u from momentum eqn. ■♥✐t✐❛❧✴❇♦✉♥❞❛r② ❈♦♥❞✐t✐♦♥s ❋✉❧❧ s♦❧✉t✐♦♥ ④ Find pressure correction p’ r❡❝❛♣ Find flux correction Correct fields * p = p + p’ ** Update fields (underrelaxation) n Tolerance achieved? y

❈❋❉ ④ ❙♦❧✉t✐♦♥ ■♥✐t✐❛❧✴❇♦✉♥❞❛r② ❈♦♥❞✐t✐♦♥s ❆❧❣♦r✐t❤♠s ◆✲❙ ❊q✉❛t✐♦♥s P■❙❖ ■♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ④ st❛t❡ ♦❢ ♣ ❛♥❞ ✉ ❛t ❛❧❧ ♣♦✐♥ts ✐♥ t❤❡ ✌♦✇✳ ❙■▼P▲❊ ■♥✐t✐❛❧✴❇♦✉♥❞❛r② ❈♦♥❞✐t✐♦♥s ✎ ♠♦st ✌✉✐❞ ✌♦✇ ♣r♦❜❧❡♠s ♣❛r❛❜♦❧✐❝ ♦r ❤②♣❡r❜♦❧✐❝ ✦ ❋✉❧❧ s♦❧✉t✐♦♥ ④ s♦❧✉t✐♦♥ ❞❡♣❡♥❞s ✐♥ ♣❛rt ♦♥ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s r❡❝❛♣ ✎ s♣❡❝✐❢② ❦♥♦✇♥ ✈❛❧✉❡s ❢♦r ♣ ❛♥❞ ✉ ✱ ♦r ♠❛❦❡ s❡♥s✐❜❧❡ ❣✉❡ss ✎ ♠❛② ♥❡❡❞ t♦ t✐♠❡st❡♣ ✉♥t✐❧ t❤❡ ❡☛❡❝t ♦❢ t❤❡ ■✳❈✳ ❛❜s❡♥t ✎ ❙t❡❛❞② ✈✐s❝♦✉s ✌♦✇s ④ ❡❧❧✐♣t✐❝ ✎ ❤♦✇❡✈❡r ♠❡t❤♦❞s ♣❛r❛❜♦❧✐❝ ④ ♥❡❡❞ st❛rt✐♥❣ ❝♦♥❞✐t✐♦♥s ❇♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ④ ❧❛r❣❡❧② s♣❡❝✐❢② ✇❤❛t t❤❡ s♦❧✉t✐♦♥ ✇✐❧❧ ❜❡

❈❋❉ ④ ❙♦❧✉t✐♦♥ ❆❧❣♦r✐t❤♠s ◆✲❙ ❊q✉❛t✐♦♥s ✸ ♠❛✐♥ t②♣❡s ♦❢ ❜✳❝✳ ④ ✇❛❧❧s✱ ✐♥❧❡ts✱ ♦✉t❧❡ts P■❙❖ ❙■▼P▲❊ ✎ ❲❛❧❧s ④ ✉ ❂ ✵ ♣❛r❛❧❧❡❧ t♦ t❤❡ ✇❛❧❧ ✭✉♥❧❡ss t❤❡ ✇❛❧❧ ✐s ■♥✐t✐❛❧✴❇♦✉♥❞❛r② ❈♦♥❞✐t✐♦♥s ♠♦✈✐♥❣ ✐♥ ✇❤✐❝❤ ❝❛s❡ t❤❡ ✌✉✐❞ ✇✐❧❧ s❤❛r❡ t❤❡ ♠♦t✐♦♥✮ ❋✉❧❧ s♦❧✉t✐♦♥ ④ r❡❝❛♣ ✎ ❋❧✉✐❞ ✐♥❧❡t ④ s♣❡❝✐❢② ✉ ✳ ✎ ❋❧✉✐❞ ♦✉t❧❡t ④ s♣❡❝✐❢② ♣ ◆✳❇✳ ■❢ ❜✳❝✳ ✐♥❝♦rr❡❝t❧② s♣❡❝✐☞❡❞✱ ♣r♦❜❧❡♠ ❜❡❝♦♠❡s ✐❧❧ ♣♦s❡❞ ④ ♥♦ s♦❧✉t✐♦♥ ♣♦ss✐❜❧❡✳