Bab 3

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BAB 3. PEMBAHASAN

3.1 Kontruksi Model Matematika Persamaan Momentum ∀



𝜕𝜌𝑣𝜙𝑛 𝜕𝑦

𝜕𝜌𝑢𝜙𝑤 𝜕𝑥





𝜕𝜌𝑢𝜙𝑒 𝜕𝑥

𝜕𝜌𝑣𝜙𝑠 𝜕𝑦

Bentuk Umum : 𝜕𝜌𝜙0 + (𝑝𝑢𝑟𝑒 𝑟𝑎𝑡𝑒) = 𝛴𝐹 𝜕𝑡 ↔

𝜕𝜌𝜙0 + (𝑖𝑛 − 𝑜𝑢𝑡) = 𝛴𝐹 𝜕𝑡



𝜕𝜌𝜙0 𝜕𝜌𝑢𝜙𝑤 𝜕𝜌𝑢𝜙𝑒 𝜕𝜌𝑣𝜙𝑠 𝜕𝜌𝑣𝜙𝑛 𝜕𝜙𝑂 + (∀ −∀ ) + (∀ −∀ ) = −𝑔𝜌∀𝑈𝑢 − 𝜕𝑡 𝜕𝑥 𝜕𝑥 𝜕𝑦 𝜕𝑦 𝜕𝑥 𝑔𝜌∀𝑈𝑣

𝜕𝜙𝑂 + 𝑔(𝜌∀𝑈)2 (𝑆𝑥 + 𝑆𝑦 ) 𝜕𝑦

𝑡+∆𝑡 𝑦+∆𝑦 𝑥+∆𝑥







𝑡

𝑦

𝑥

𝜕𝜌𝜙0 𝜕𝜌𝑢𝜙𝑤 𝜕𝜌𝑢𝜙𝑒 𝜕𝜌𝑣𝜙𝑠 𝜕𝜌𝑣𝜙𝑛 + (∀ −∀ ) + (∀ −∀ ) 𝑑𝑥𝑑𝑦𝑑𝑡 𝜕𝑡 𝜕𝑥 𝜕𝑥 𝜕𝑦 𝜕𝑦

= 𝜌𝜙𝑂 ∆𝑥∆𝑦 + (∀𝜌𝑢𝜙𝑤 ∆𝑦∆𝑡 − ∀𝜌𝑢𝜙𝑒 ∆𝑦∆𝑡) + (∀𝜌𝑣𝜙𝑠 ∆𝑥∆𝑡 − ∀𝜌𝑣𝜙𝑛 ∆𝑥∆𝑡)

𝑡+∆𝑡 𝑦+∆𝑦 𝑥+∆𝑥





∫ −𝑔𝜌∀𝑈𝑢

𝑡

𝑦

𝑥

𝜕𝜙𝑂 𝜕𝜙𝑂 − 𝑔𝜌∀𝑈𝑣 + 𝑔(𝜌∀𝑈)2 (𝑆𝑥 + 𝑆𝑦 )𝑑𝑥𝑑𝑦𝑑𝑡 𝜕𝑥 𝜕𝑦

= −𝑔𝜌∀𝑈𝑢𝜙𝑂 ∆𝑦∆𝑡 − 𝑔𝜌∀𝑈𝑣𝜙𝑂 ∆𝑥∆𝑡 + 𝑔(𝜌∀𝑈)2 (𝑆𝑥 + 𝑆𝑦 )∆𝑥∆𝑦∆𝑡 Dari hasil diatas maka : 𝜌𝜙𝑂 ∆𝑥∆𝑦 + (∀𝜌𝑢𝜙𝑤 ∆𝑦∆𝑡 − ∀𝜌𝑢𝜙𝑒 ∆𝑦∆𝑡) + (∀𝜌𝑣𝜙𝑠 ∆𝑥∆𝑡 − ∀𝜌𝑣𝜙𝑛 ∆𝑥∆𝑡) = −𝑔𝜌∀𝑈𝑢𝜙𝑂 ∆𝑦∆𝑡 − 𝑔𝜌∀𝑈𝑣𝜙𝑂 ∆𝑥∆𝑡 + 𝑔(𝜌∀𝑈)2 (𝑆𝑥 + 𝑆𝑦 )∆𝑥∆𝑦∆𝑡 ↔ 𝜌𝜙𝑂 ∆𝑥∆𝑦 + (∀𝜌𝑢𝜙𝑤 ∆𝑦∆𝑡 − ∀𝜌𝑢𝜙𝑒 ∆𝑦∆𝑡) + (∀𝜌𝑣𝜙𝑠 ∆𝑥∆𝑡 − ∀𝜌𝑣𝜙𝑛 ∆𝑥∆𝑡) + 𝑔𝜌∀𝑈𝑢𝜙𝑂 ∆𝑦∆𝑡 + 𝑔𝜌∀𝑈𝑣𝜙𝑂 ∆𝑥∆𝑡 − 𝑔(𝜌∀𝑈)2 (𝑆𝑥 + 𝑆𝑦 )∆𝑥∆𝑦∆𝑡 = 0..........persamaan 1 Persamaan Massa

𝜕𝜌𝐴𝜙𝑛 𝜕𝑦

𝜕𝜌𝐴𝜙𝑒 𝜕𝑥

𝜕𝜌𝐴𝜙𝑤 𝜕𝑥

𝜕𝜌𝐴𝜙𝑠 𝜕𝑦

Bentuk Umum : 𝜕𝜌∀𝜙0 + (𝑖𝑛 − 𝑜𝑢𝑡) = 0 𝜕𝑡 ↔

𝜕𝜌∀𝜙0 𝜕𝜌∀𝜙𝑤 𝜕𝜌∀𝜙𝑒 𝜕𝜌∀𝜙𝑠 𝜕𝜌∀𝜙𝑛 + ( − )+ ( − )= 0 𝜕𝑡 𝜕𝑥 𝜕𝑥 𝜕𝑦 𝜕𝑦

𝑡+∆𝑡 𝑦+∆𝑦 𝑥+∆𝑥







𝑡

𝑦

𝑥

𝜕𝜌∀𝜙0 𝜕𝜌∀𝜙𝑤 𝜕𝜌∀𝜙𝑒 𝜕𝜌∀𝜙𝑠 𝜕𝜌∀𝜙𝑛 + ( − )+ ( − ) 𝑑𝑥𝑑𝑦𝑑𝑡 𝜕𝑡 𝜕𝑥 𝜕𝑥 𝜕𝑦 𝜕𝑦

= 𝜌∀𝜙𝑂 ∆𝑥∆𝑦 + (𝜌𝐴𝑢𝜙𝑤 ∆𝑦∆𝑡 − 𝜌𝐴𝑢𝜙𝑒 ∆𝑦∆𝑡) + (𝜌𝐴𝑣𝜙𝑠 ∆𝑥∆𝑡 − 𝜌𝐴𝑣𝜙𝑛 ∆𝑥∆𝑡) = 0

= 𝜌∀𝜙𝑂 ∆𝑥∆𝑦 = (𝜌𝐴𝑢𝜙𝑒 ∆𝑦∆𝑡 − 𝜌𝐴𝑢𝜙𝑤 ∆𝑦∆𝑡) + (𝜌𝐴𝑣𝜙𝑛 ∆𝑥∆𝑡 − 𝜌𝐴𝑣𝜙𝑠 ∆𝑥∆𝑡) = 𝜌𝜙𝑂 ∆𝑥∆𝑦 =

(𝜌𝐴𝑢𝜙𝑒 ∆𝑦∆𝑡 − 𝜌𝐴𝑢𝜙𝑤 ∆𝑦∆𝑡) + (𝜌𝐴𝑣𝜙𝑛 ∆𝑥∆𝑡 − 𝜌𝐴𝑣𝜙𝑠 ∆𝑥∆𝑡) … … 𝑝𝑒𝑟𝑠. 2 ∀

Substitusi persamaan 2 ke persamaan 1 (𝜌𝐴𝑢𝜙𝑒 ∆𝑦∆𝑡 − 𝜌𝐴𝑢𝜙𝑤 ∆𝑦∆𝑡) + (𝜌𝐴𝑣𝜙𝑛 ∆𝑥∆𝑡 − 𝜌𝐴𝑣𝜙𝑠 ∆𝑥∆𝑡) + ((∀𝜌𝑢𝜙𝑤 ∆𝑦∆𝑡 − ∀𝜌𝑢𝜙𝑒 ∆𝑦∆𝑡) + ∀ (∀𝜌𝑣𝜙𝑠 ∆𝑥∆𝑡 − ∀𝜌𝑣𝜙𝑛 ∆𝑥∆𝑡) + 𝑔𝜌∀𝑈𝑢𝜙𝑂 ∆𝑦∆𝑡 + 𝑔𝜌∀𝑈𝑣𝜙𝑂 ∆𝑥∆𝑡 − 𝑔(𝜌∀𝑈)2 (𝑆𝑥 + 𝑆𝑦 )∆𝑥∆𝑦∆= 0

Jadi persamaannya : 𝜙𝑤 (∀𝜌𝑢∆𝑦∆𝑡 −

𝜙𝑛 (

𝜌𝐴∆𝑦∆𝑡 𝜌𝐴∆𝑦∆𝑡 𝜌𝐴∆𝑥∆𝑡 − ∀𝜌𝑢∆𝑦∆𝑡) + 𝜙𝑠 (∀𝜌𝑣∆𝑥∆𝑡 − ) + 𝜙𝑒 ( )+ ∀ ∀ ∀

𝜌𝐴∆𝑥∆𝑡 − ∀𝜌𝑣∆𝑥∆𝑡) + 𝑔𝜌∀𝑈𝑢𝜙𝑂 ∆𝑦∆𝑡 + 𝑔𝜌∀𝑈𝑣𝜙𝑂 ∆𝑥∆𝑡 − 𝑔(𝜌∀𝑈)2 (𝑆𝑥 + 𝑆𝑦 )∆𝑥∆𝑦∆= 0 ∀

Diskritisasi Quick 1 3 3 𝜙𝑒 (𝑖, 𝑗) = − ∅(𝑖 − 1, 𝑗) + ∅(𝑖, 𝑗) + ∅(𝑖 + 1, 𝑗) 8 4 8 1 3 3 𝜙𝑛 (𝑖, 𝑗) = − ∅(𝑖, 𝑗 − 1) + ∅(𝑖, 𝑗) + ∅(𝑖, 𝑗 + 1) 8 4 8 1 3 3 𝜙𝑤 (𝑖, 𝑗) = − ∅(𝑖 − 2, 𝑗) + ∅(𝑖 − 1, 𝑗) + ∅(𝑖, 𝑗) 8 4 8 1 3 3 𝜙𝑠 (𝑖, 𝑗) = − ∅(𝑖, 𝑗 − 2) + ∅(𝑖, 𝑗 − 1) + ∅(𝑖, 𝑗) 8 4 8 Substitusi ke Persamaan Umum 𝜙𝑤 (∀𝜌𝑢∆𝑦∆𝑡 −

𝜙𝑛 (

𝜌𝐴∆𝑦∆𝑡 𝜌𝐴∆𝑦∆𝑡 𝜌𝐴∆𝑥∆𝑡 − ∀𝜌𝑢∆𝑦∆𝑡) + 𝜙𝑠 (∀𝜌𝑣∆𝑥∆𝑡 − ) + 𝜙𝑒 ( )+ ∀ ∀ ∀

𝜌𝐴∆𝑥∆𝑡 − ∀𝜌𝑣∆𝑥∆𝑡) + 𝑔𝜌∀𝑈𝑢𝜙𝑂 ∆𝑦∆𝑡 + 𝑔𝜌∀𝑈𝑣𝜙𝑂 ∆𝑥∆𝑡 − 𝑔(𝜌∀𝑈)2 (𝑆𝑥 + 𝑆𝑦 )∆𝑥∆𝑦∆= 0 ∀

1 3 3 𝜌𝐴∆𝑦∆𝑡 ↔ (− ∅(𝑖 − 2, 𝑗) + ∅(𝑖 − 1, 𝑗) + ∅(𝑖, 𝑗)) (∀𝜌𝑢∆𝑦∆𝑡 − ) 8 4 8 ∀ 1 3 3 𝜌𝐴∆𝑦∆𝑡 + (− ∅(𝑖 − 1, 𝑗) + ∅(𝑖, 𝑗) + ∅(𝑖 + 1, 𝑗)) ( − ∀𝜌𝑢 ∆𝑦∆𝑡) 8 4 8 ∀ 1 3 3 𝜌𝐴∆𝑥∆𝑡 + (− ∅(𝑖, 𝑗 − 2) + ∅(𝑖, 𝑗 − 1) + ∅(𝑖, 𝑗)) (∀𝜌𝑣∆𝑥∆𝑡 − ) 8 4 8 ∀ 1 3 3 𝜌𝐴∆𝑥∆𝑡 + (− ∅(𝑖, 𝑗 − 1) + ∅(𝑖, 𝑗) + ∅(𝑖, 𝑗 + 1)) ( − ∀𝜌𝑣∆𝑥∆𝑡) 8 4 8 ∀ + 𝑔𝜌∀𝑈𝑢𝜙𝑂 ∆𝑦∆𝑡 + 𝑔𝜌∀𝑈𝑣𝜙𝑂 ∆𝑥∆𝑡 − 𝑔(𝜌∀𝑈)2 (𝑆𝑥 + 𝑆𝑦 )∆𝑥∆𝑦∆= 0 1 𝜌𝐴∆𝑦∆𝑡 3 𝜌𝐴∆𝑦∆𝑡 ↔ (− ∅(𝑖 − 2, 𝑗) (∀𝜌𝑢∆𝑦∆𝑡 − )) + ( ∅(𝑖 − 1, 𝑗) (∀𝜌𝑢 ∆𝑦∆𝑡 − )) 8 ∀ 4 ∀ 3 𝜌𝐴∆𝑦∆𝑡 1 𝜌𝐴∆𝑦∆𝑡 + ( ∅(𝑖, 𝑗) (∀𝜌𝑢∆𝑦∆𝑡 − )) + (− ∅(𝑖 − 1, 𝑗) ( − ∀𝜌𝑢∆𝑦∆𝑡)) 8 ∀ 8 ∀ 3 𝜌𝐴∆𝑦∆𝑡 3 𝜌𝐴∆𝑦∆𝑡 + ( ∅(𝑖, 𝑗) ( − ∀𝜌𝑢∆𝑦∆𝑡)) + ( ∅(𝑖 + 1, 𝑗) ( − ∀𝜌𝑢 ∆𝑦∆𝑡)) 4 ∀ 8 ∀ 1 𝜌𝐴∆𝑥∆𝑡 + (− ∅(𝑖, 𝑗 − 2) (∀𝜌𝑣∆𝑥∆𝑡 − )) 8 ∀ 3 𝜌𝐴∆𝑥∆𝑡 3 𝜌𝐴∆𝑥∆𝑡 + ( ∅(𝑖, 𝑗 − 1) (∀𝜌𝑣∆𝑥∆𝑡 − )) + ( ∅(𝑖, 𝑗) (∀𝜌𝑣 ∆𝑥∆𝑡 − )) 4 ∀ 8 ∀ 1 𝜌𝐴∆𝑥∆𝑡 3 𝜌𝐴∆𝑥∆𝑡 + (− ∅(𝑖, 𝑗 − 1) ( − ∀𝜌𝑣 ∆𝑥∆𝑡)) + ( ∅(𝑖, 𝑗) ( − ∀𝜌𝑣∆𝑥∆𝑡)) 8 ∀ 4 ∀ 3 𝜌𝐴∆𝑥∆𝑡 + ( ∅(𝑖, 𝑗 + 1) ( − ∀𝜌𝑣∆𝑥∆𝑡)) 8 ∀ = −𝑔𝜌∀𝑈𝑢𝜙𝑂 ∆𝑦∆𝑡 − 𝑔𝜌∀𝑈𝑣𝜙𝑂 ∆𝑥∆𝑡 + 𝑔(𝜌∀𝑈)2 (𝑆𝑥 + 𝑆𝑦 )∆𝑥∆𝑦∆= 0 Sehingga kita dapatkan : 1 𝜌𝐴 ∅(𝑖 − 2, 𝑗) (− (∀𝜌𝑢 − ) ∆𝑦∆𝑡) … … . 𝐴 8 ∀ 1 𝜌𝐴 3 𝜌𝐴 ∅(𝑖 − 1, 𝑗) (− ( − ∀𝜌) ∆𝑦∆𝑡) + ( (∀𝜌𝑢 − ) ∆𝑦∆𝑡) … … . 𝐵 8 ∀ 4 ∀ 3 𝜌𝐴 3 𝜌𝐴 3 𝜌𝐴 ∅(𝑖, 𝑗) ( (∀𝜌𝑢 − ) ∆𝑦∆𝑡) + ( ( − ∀𝜌𝑢) ∆𝑦∆𝑡) + ( (∀𝜌𝑣 − ) ∆𝑥∆𝑡) 8 ∀ 4 ∀ 8 ∀ 3 𝜌𝐴 + ( ( − ∀𝜌𝑣) ∆𝑥∆𝑡) … … . 𝐶 4 ∀

3 𝜌𝐴 ∅(𝑖 + 1, 𝑗) ( ( − ∀𝜌𝑢) ∆𝑦∆𝑡) … … . 𝐷 8 ∀ 1 𝜌𝐴 ∅(𝑖, 𝑗 − 2) (− (∀𝜌𝑣 − ) ∆𝑥∆𝑡) … … . 𝐸 8 ∀ 1 𝜌𝐴 ∅(𝑖, 𝑗 − 1) (− ( − ∀𝜌𝑣) ∆𝑥∆𝑡) … … . 𝐹 8 ∀ 3 𝜌𝐴 ∅(𝑖, 𝑗 + 1) ( ( − ∀𝜌𝑣) ∆𝑥∆𝑡) … … . 𝐺 8 ∀ −𝑔𝜌∀𝑈𝑢𝜙𝑂 ∆𝑦∆𝑡 − 𝑔𝜌∀𝑈𝑣𝜙𝑂 ∆𝑥∆𝑡 + 𝑔(𝜌∀𝑈)2 (𝑆𝑥 + 𝑆𝑦 )∆𝑥∆𝑦∆ … … . 𝐻 Dengan skema dikritisasi sebagai berikut: (i,j+4) 

(i+1,j+4) 

(i+2,j+4) 

(i+3,j+4) 

(i+4,j+4) 

(i,j+3) 

(i+1,j+3) 

(i+2,j+3) 

(i+3,j+3) 

(i+4,j+3) 

(i,j+2) 

(i+1,j+2) 

(i+2,j+2) 

(i+3,j+2) 

(i+4,j+2) 

(i,j+1) 

(i+1,j+1) 

(i+2,j+1) 

(i+3,j+1) 

(i+4,j+1) 

(i,j) 

(i+1,j) 

(i+2,j) 

(i+3,j) 

(i+4,j) 

maka diperoleh matriks diskritisasi sebagai berikut:

3.2 Matlab

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