UNIVERSIDADE FEDERAL DO PARÁ INSTITUTO DE TECNOLOGIA PROGRAMA DE PÓS-GRADUAÇÃO EM ENGENHARIA MECÂNICA ÁREA DE CONCENTRAÇÃO-TÉRMICAS E FLUIDOS Cindy Stella Fernandes

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ABSTRACT

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LISTA DE ILUSTRAÇÕES

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01#" *( -'."#"+,- -$ . #: $ " 3-'.- 6 !-3 " ' = ."#"

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LISTA DE SÍMBOLOS

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SUMARIO

1 INTRODUÇÃO ... 17

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(8* Z [ 88888888888888888888888888888888888888888888888888888888888888888888888888888888(* (8( % \ 8888888888888888888888888888888888888888888888888888888888888888888888(( 2.2.1 Metodologia de Simulação Numérica Direta ( ! ) 22 2.2.2 Metodologia de Simulação por Grandes Escalas ( – ) 23 2.2.3 RANS – (Reynolds!Averaged Navier!Stokes) ... 24

(8; Z [ N Z ] % 88888888888888888888888888888888888888888888888888888888888888(X 2.3.1 Aproximação de Boussinesq ... 28

(8E % \ % % (D 2.4.1 Modelo Algébrico do Comprimento de Mistura ... 29

2.4.2 Modelos de Uma Equação ... 30

2.4.3 Modelo de Duas Equações (Modelo k! ε) ... 33

3 O MODELO DE TURBULÊNCIA − −ε ... 35

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4.4.1 Modificações realizadas neste trabalho ... 53

5 RESULTADOS E DISCUSSÕES ... 55

X8* % Z 8888888888888888XX 5.1.1 Introdução ... 55

5.1.2 Descrição da configuração... 56

5.1.3 Condições de contorno ... 58

5.1.4 Análise de Convergência de malha ... 59

5.1.5 Comparação dos resultados ... 61

X8( % % 888888888888888888888888888888888888888<< 5.2.1 Introdução ... 66

5.2.2 Descrição da configuração... 67

5.2.3 Condições de contorno ... 68

5.2.4 Comparação dos resultados ... 69

6 CONSIDERAÇÕES FINAIS ... 76

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1 INTRODUÇÃO

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$ "$ 6"# 76 $ $3- 2 3 "$5 -6"$ 6"# 76 $ $3- 2 3 "$ "."# 3 '8 $ & 32"'"

-P.#-/! '" : 32"' -Q8

2.3.1 Aproximação de Boussinesq

'& - - '" $ .-.1!"#' ".! 3" - ."#" $ # $-!6 # - .#-/! '" : 32"' - "

'- !"0 ' " 1#/1!= 3 " 1 ! L" " ".#-G '"+,- .#-.-$ " .-# %-1$$ $H ' *VWW5 H1

$ "/ ! 3 " # !"+,- # - $-# -! $ - 3"'.- '& - 6 !-3 " A Y 5 *DWXB8

.#-.-$ " -# 0 "! 3- $ #-1 " 2 .> $ H1 "$ $S $ -! $

$,-.#-.-#3 - " $ T :-#'"+,- - $3-"' - '& -5 "0 - '" #" $ ' !2" T$ $S $

6 $3-$"$5 " #"6&$ - 3- 3 - 6 $3-$ " 1#/1! "8

%-1$$ $H .#-.J$ 3 "!' H1 "$ $S $ 6 $3-$"$ '-! 31!"# $ .- # "' $ # -/ "$ 3-'

" "!-0 " 3-' - '- !- -M $8 %"$ " - -$ 3- 3 -$ 3 " $ :-#'1!" -$ .-# %-1$$ $H

32 0" $ "- $ 01 '- !- : 32"' - ."#" - $-# -! $9

ν δ _∂ _{∂ }  _  ′ ′  − = __ + _−  ∂ ∂     A*WB

- & " #0 " 3 & 3" 1#/1! " $3# " 3-'-9

(

)

′ ′ ′ ′ ′

= = + + A*VB

δ & - ! " N#- 3M #5 - "+,- : " 3-'-9

δ =  =

 ≠



A*DB

H1 $ #6 ."#" 3- 3 ! "# " : +,- 3-' " $-'" - #"+- A$-'" -$ 6"!-# $ " "0- "! -$

$-# $B - $-# $,- '- !" - " #"6&$ " 2 .> $ %-1$$ $H8 "L - $ =

' $ 9

ν ∂ δ

′ ′ ′ ′

− = − = − ⇒ =

(29)

(D

$ "3" - H1 : # " 6 $3-$ " '-! 31!"#5 " 6 $3-$ " 1#/1! " ν ,- & 1'"

.#-.# " :I$ 3" - :!1 - $ ' - $3-"' -5 & 1'" ' " !-3"! - I6 ! " 1#/1!= 3 "5

H1 6"# " .- - " .- - $3-"' - " $3-"' -5 . - $."+- - '.-8

#" ."# -$ '- !-$ 6 $3-$ " 1#/1! " 1 ! L" 1'" " "!-0 " 3-' " 6 $3-$ "

'-! 31!"#5 K7 H1 $ 3- $ #" " $,- 1#/1! " " 7!-0" " $,- 6 $3-$"5 ."#" : # "

6 $3-$ " 1#/1! " ν 8 "3-# - 3-' /#1 2-$" A());B5 " 6 $3-$ " 1#/1! " &

3- $ #" " .#-.-#3 - "! T '"$$" $. 3I: 3", T :!1 1"+,- 6 !3 " " 3'.# ' $3"!" 3"#"3 #I$ 3- " 1#/1!= 3 "8

- !-$ $ 01 - $ " 2 .> $ 3"!31!"' " 6 $3-$ " 1#/1! " ' 67# -$ I6 $

3-'.! G " 5 - $ $ '.! $ # !"+S $ "!0&/# 3"$ " & ' $'- H1"+S $ : # 3 " $8

$3# - '" $ "!2" "' ' !6" # # A())<B5 -$ '- !-$ 1#/1!= 3 " .- ' $ #

3!"$$ : 3" -$ 3-'- '- !-$ "!0&/# 3-$ '- !-$ : # 3 " $5 '- !-$ 1'" H1"+,-

'- !-$ 1"$ H1"+S $8

(8E % \ % %

2.4.1 Modelo Algébrico do Comprimento de Mistura

- $ #" - 1' $3-"' - $ '.! $5 1'" "$ .# ' #"$ " 6"$ $ '- !"# "$ $S $

1#/1! "$ :- .#-.-$ " .-# #" ! A*D(XB8 ! $ 6-!6 1 1'" 2 .> $ 3-'.# ' -

' $ 1#" .#-.J$5 /"$ " - !"5 1' '- !- - " 6 $3-$ " 1#/1! " & 3"!31!" " " #"6&$

G.# $$S $ "!0&/# 3"$8 '- !- '.# ' - ' $ 1#" & 2 3 - '- '- !- "!0&/#

3--1 '- !- T L #- H1"+,- A 5 ())<B8

"#" - '- !- "!0&/# 3- - 3-'.# ' - ' $ 1#" .# 3 $" $ $-' H1" " $

-3"'.- 6 !-3 " '& " - $3-"' -5 - H1 "3"/" # $1! " - ' 1'" 3 $$ " ' -#

# 31#$-$ 3-'.1 "3 - " $ " -$ -1 #-$ '- !-$5 3-'- -$ '- !-$ 1'" -1 1"$

(30)

-;)

.# 3I. - - H1 !I/# - !-3"!5 -1 $ K"5 H1 " #0 " 1#/1! " & $$ ." " .#- 1L " ' ' $'"

.#-.-#+,-8 $ '- !- & .# 3 $- :"L # "K1$ $ ."#" 6 "# H1 " 6 $3-$ " 1#/1! " $ K" 1!"

Aν = B H1" 0#" 6 !3 " '& "'/&' :# 1! ' # 0 S $ $3"'

-H1 $ ,- .! "' $ 6-!6 "$5 - -H1 ! 6" '1 "$ 6 L $ " # $1! " -$ ## " $8 '- !-

3-'.# ' - ' $ 1#" ".# $ " "'/&' -1 #"$ ! ' "+S $ 3-'- " :"! " 0 #"! " -$

" -$ 3"!31!" -$ A1'" "$ .# 3 ." $B5 K7 H1 & :I3 ! $. 3 : 3"# - 3-'.# ' - ' $ 1#" '

$3-"' -$ 3-'.! G-$ A 5 ())WB8

1$- - '- !- 3-'.# ' - ' $ 1#" & - "!' " H1" - ' $ 1"+S $ H1

6-!6 '5 .-# G '.!-5 # 0 S $ $ ."#"+,- - $3-"' -5 K7 H1 .-# 3"1$" -$ . H1 -$

0#" $ 6 !-3 " '& " - '- !- ,- 3- $ 01 .# 6 # -$ I6 $ ! 6" -$ 1#/1!= 3 "

6 # : 3" -$ G. # ' "!' ' # 0 S $ $ "0 "+,- - $3-"' - A 5

())<B8

#" ! A*DEXB .#-.J$ 1' '- !- ."#" ' !2-# .# L # "$ .#-.# " $ -$ $3-"' -$

1#/1! -$ ."#" $3# 6 # '" '" 3"' "$ $S $ 1#/1! "$ '" #" '" $ # "! $ "8

- $ '- !- " 6 $3-$ " 1#/1! " . " #0 " 3 & 3" "$ :!1 1"+S $

1#/1! "$k8 $$ ' $1#0 1 - 3- 3 - '- !- " 1'" H1"+,-8

2.4.2 Modelos de Uma Equação

- '- !- 1'" H1"+,- /1$3" $ $3# 6 # - 37!31!- 1'" "$ 0#" L"$ "

1#/1!= 3 "5 1$" "$ - 37!31!- " 6 $3-$ " 1#/1! "5 3-'- .-# G '.!-5 " #0 " 3 & 3"

1#/1! "5 $3# " . !" H8A*VB8 N-!'-0-#-6 #" ! A % 5 ());B .#-.1$ #"' "

$ 01 # !"+,- ."#" $ -/ # " 6 $3-$ " 1#/1! "9

ν_t = C k_# L A(*B

$ - C_# 1'" 3- $ " '.I# 3"5 " #0 " 3 & 3" 1#/1! " 1'" $3"!"

(31)

;*

- $ G.# $$"# " 6 $3-$ " 1#/1! " 3-'- - .#- 1 - 1'" 6 !-3 " 3"#"3 #I$ 3"

1'" $3"!" 3-'.# ' - 9

ν = A((B

N-!'-0-#-6 #" ! $10 # #"' " 1+,- 1'" H1"+,- #" $.-# ."#" 5 ' H1 $

-/ =' " H1"+,- : # 3 "! ."#" " #0 " 3 & 3" 1#/1! " " #"6&$ '" .1!"+S $ "$

H1"+S $ "6 # -M $8 -0-5 " H1"+,- #0 " 1#/1! " & 1L " 3-'- A

5 ())<B9

∂ ₊ ₌ _{+ +}

∂ Ck D Pk k ε

t A(;B

- $ : H1 9

∂ = ∂ k C x A(EB ν ρ  _ _′ _  ∂  _′ _ ∂  = − _ _ + __− _  ∂ _{ } _ ∂  ′   A(XB ∂ ′ ′ = − ∂ i

k i j

j

P u u

x A(<B ν _∂ _′ _{∂ }_′  _  = − _ _ ∂ ∂    ε A(WB $ #'-$ ∂

∂t Ck " H8A(;B 3"' " "G" 6"# "+,- !-3"! - #" $.-# .-# 3- 6

3+,-, ,- 3 $$ "' $ # '- !" -$8

k

D & - #'- H1 # .# $ " - #" $.-# .-# ' - :1$,-8 ' $1" $3#

+,-'" '7 3"5 H8A(XB5 - O! '- #'- # 3-!32 $ :"L "!1$,- "- #" $.-# :1$ 6- '-! 31!"#

5 $ - '.-# " $-' ' # 0 S $ /" G" $ " " 1#/1!= 3 "8 $ #'-$

# $ " $ H1 "."# 3 ' " :-#'1!"+,- D_k $ ,- ! 0" -$ "- #" $.-# :1$ 6- 1#/1! -

$,-".#-G '" -$ .-# ' - " & " - 3- 3 - 6 $3-$ " 1#/1! "9

(32)

;(

" H8A(VB5 γ & " :1$ 6 " #' " " " " "!-0 " -! $8 γ & " # !"+,- # "

:1$ 6 " H1"!H1 # .#-.# " ! "# 3-' " :1$ 6 " " H1" " '6 '

-A 5 ())<B5 -1 $ K"5

ν γ

σ

≃ t

k k

A(DB

- σ & - O' #- #" ! 1#/1! -5 H1 -#'"!' "$$1' - 6"!-# * ."#"

-#" $.-# 8

F7 - #'- P_k5 $3# - . !" H8A(<B # .# $ " - #'- .#- 1+,- #0 " 3 & 3"

1#/1! "8 $ #' G.# $$" " "G" #" $: #= 3 " #0 " $3"' '& ."#"

-' 3" $-'- 1#/1!= 3 "8 -$ '- !-$ /"$ " -$ " 2 .> $ " 6 $3-$ " 1#/1! "5 - $-#

' P_k & ".#-G '" - :"L - $ 1$- " H8A*WB8

O! '- #'- " H8A(;B $3# - 3-'- $ - " "G" $$ ."+,- #0 " 1#/1! " -1

"G" $$ ."+,- 6 $3-$" , & " 6"# 76 ! H1 .# 3 $" $ # '- !" "5 $ '" " .-#9

ε

ε≃k

L A;)B

. $"# H1 ' 3-'."#"+,- 3-' - '- "!0&/# 3- 3-'.# ' - ' $ 1#"5 - '-

1'" H1"+,- ' !2-# '" #" $ 0 : 3" 6" " H1"! " - 37!31!- "$ 0#" L"$

1#/1! "$5 $ '- !- "'/&' ".# $ " : 3 = 3 "$5 - $ $ "3" 1'" ' $. 3 "! H1 & "

3 $$ " $ .# $3# 6 # 1'" $3"!" 3-'.# ' - 3"#"3 #I$ 3- ."#" 3"#"3 # L"+,- "

1#/1!= 3 "8

:" - $ # G 0 - H1 $ $. 3 : H1 '. # 3"' - 3-'.# ' - $3"!" ."#"

-37!31!- " $$ ."+,- " 6 $3-$ " 1#/1! "5 -# " :I3 ! " ".! 3"+,- - '- !- 1'"

H1"+,-8 $$- ! 6-1 T #' "+,- - 3-'.# ' - $3"!" " #"6&$ " H1"+,- #" $.-#

1'" H1" " 1#/1! " 5 " ."# # - $ 1 6"!-#5 : # " 6 $3-$ " 1#/1! "ν_t " "G"

(33)

;;

2.4.3 Modelo de Duas Equações (Modelo k! ε)

" !"/-#"+,- '- !- 1"$ H1"+S $ '" &' $ " 1 ! L"+,- " H1"+,- " #0 "

3 & 3" 5 $$- .-# 3"1$" - .-13- '. # $'- 1$" - " $1" -/ +,-8 7# -$ .-$ '- !-$

:-#"' .#-.-$ -$5 # $ $ 1' -$ '" $ 3- 2 3 -$ 1 ! L" -$ & - '- !- 1"$ H1"+S $

ε

− #"/"!2" "$ .-# "1 # ."! 05 A*DWEB apud !6" # # A())<B5 H1 & /"$ " - " H1"+,- " #0 " 3 & 3" 1#/1! " " "G" $$ ."+,- " #0 " 3 & 3" 1#/1! "

ε

8

'- !- −ε #"/"!2" 3-' 1"$ 6"# 76 $ " 3 - " $ H1 $,- #- 1L "$ ."#" '- !"# "$

$S $ -! $8 $ '- !- # !"3 - " "$ $S $ -! $ 3-' -$ 0#" $

6 !-3 " '& " " 6 $3-$ " 1#/1! "5 3- 3 - - H1"! - '- !- $ /"$ " A2 .> $

%-1$$ $HB ".# $ "' 1"$ H1"+S $ #" $.-# H1 $,- '- !" "$5 1'" . "

-1 #"5 $,- $-!13 - " "$ 1'" ."#" " H1"+,- " #0 " 3 & 3" -1 #" ."#" - #'-

$$ ."+,- #0 " 3 & 3" 1#/1! "

ε

8

"#" $ -/ # 1'" H1"+,- G" " ."#" - #" $.-#

ε

5 '" .1!" $ "$ H1"+S $ "6 #

-M $8 $ #'-$ " H1"+,- #" $.-#

ε

.- ' $ # # 1 -$ '" #" " # .# $ "# '

' 3" $'-$ :I$ 3-$ :1$,-5 .#- 1+,- $ #1 +,-

ε

A 5 ())<B9

ε ε ε

ε ε

∂ ₊ ∂ ₌ ₊ ₊

∂ ∂ A;*B

" H8A;*B _ε & - ' 3" $'- :1$,-

ε

5 _ε & - ' 3" $'- .#- 1+,-

ε

_ε &

-' 3" $-'- $ #1 +,-

ε

8 "#" " '- !"0 ' -$ #'-$ " H1"+,-

ε

5 1$" $ 5 " " 7! $

' $ - "! " 1 +,- :I$ 3"8 :1$,-

ε

ε & ".#-G '" " 1$" - - 0#"

ε

9

   ∂ _ _ ∂  ≅ __ _{+ }_ _  ∂ ___ _∂ __ t ε

j ε j

ν ε

ν

x σ x A;(B

-# -1 #- !" -5 " .#- 1+,- 6 $ # /"!" 3 " " . !" .#- 1+,-

ε

_ε ."#" 6 "#

"1' - ! ' " - 8 $$ '5

ε

≈ A;;B

(34)

;E

#'- $ #1 +,-

ε

_ε 5 " H1"+,-

ε

5 6 # .#-. $,- " $ # : - H1"

-→ 5 $ ,- .- 6 # " $ -# "# 0" 6-8 $$ '9

ε

ε ε

≈ A;EB

! L" - - "$ "$ ".#-G '"+S $ K7 3 " "$5 - '- !- −ε ."#" "! -$ O' #-$ -! $

"$$1' " :-#'" 9

ν ν ε

  _∂ _∂  _∂  ∂ ₊ ∂ ₌ ∂ _ ∂ _₊ _ ₊ _{ } _₋

     

∂ ∂ ∂ __ ∂ __ __∂ ∂ _{ }_{ }∂ __ A;XB

ε ε

ε

ν

ε ε ε ε ε

σ

 

∂ ₋ ∂ ₌ ∂ _ ∂ _₊ ₋

 

∂ ∂ ∂ __ ∂ __ A;<B

- " ν & " 6 $3-$ " 1#/1! "5

ν

ε

= A;WB

5

ε ε $,- 3- $ " $ '.I# 3"$8

'- !- −ε "'/&' ' $1"$ : 3 = 3 "$8 '- !- & :"!2- ' .# 6 # $3-"' -$

":"$ " -$ " 3- +,- H1 !I/# - # .#- 1+,- $$ ."+,-5 ' :"! " 0 #"! " 5 K7 H1

' '1 "$ 3- $ " $ " $ # ' "K1$ " "$8 :-#'" 0 #"!5 -$ ##-$ - '- !- −ε$1#0 ' .-#

- $ '- 6-$5 1' .-# 3"1$" - 1$- 1'" # !"+,- # $S $ 1#/1! "$ "G"$

:-#'"+,- - $3-"' - '& -5 $ ' !2" " H1 & 1 ! L" " ."#" - $3-"' - !"' "#

-1 #- . !" .-13" :1 "' "+,- :I$ 3" " H1"+,- #" $.-#

ε

5 " H1"! 21'" "$

3-## +S $ $10 # "$ " & - '-' - :-# 3 1'" 0 #"! " $" $:" ># "8 $$"$ $&# "$

: 3 = 3 "$ '- $ #"' H1 - '- !- 6 $ # 1 ! L" - 3-' 3"1 !" " .# 6

(35)

;X

3 O MODELO DE TURBULÊNCIA

− −ε

;8*

1#/1!= 3 " ' " 1# L" #- "3 - "! $ 7 "$$-3 " " T $ "/ ! " - :!1 -8 " ' $'"

:-#'" H1 -$ $3-"' -$ !"' "# $5 -$ $3-"' -$ 1#/1! -$ "'/&' ".# $ "' "

:-#'"+,- 6># 3 $5 H1 $,- 3- $ #" -$ 3-'- " G.# $$,- '"3#-$3>. 3" $ "/ ! " $

$-# ' - :!1G- H1 $ 3 "' ' I6 ! '-! 31!"#8 - $ 3- $ #"# " 4' 3" 6># 3 $

3-'- 1'" "$ :-#'"$ ' !2-# 3-'.# $,- $ : J' $ # !"3 " $ " $3"'

-1#/1! -8 -# " -5 3- 2 3 # -$ "#01' -$5 2 .> $ $ :1 "' $ 1$" $ " :#'1!"+,

-3- 3 - 6># 3 $,- '.-# " $ ."#" : 3"#5 3"#"3 # L"# $3# 6 # "$ $ #1 1#"$

6-# 3" $ ' 6 $ 0"+S $ ># 3"$5 G. # ' " $ 1'&# 3"$ " 1#/1!= 3 " $ #1 1#"! -1

$3-"' -$ #- "3 - " $ ' #" $ +,- A 5 ())VB8

"3-# - 3-' "$ 3- $ #"+S $ M $ 1'! A*DW(Bapud #" A())VB & " 1#"!

H1 $ " "! $ $3-"' -$ 1#/1! -$ ' #" $ +,- 3-' /"$ " > 3" $ #1 1#"!5 ' 1'

3- G - # !"+,- T : +,- 6-# 3 " " 6 #$-$ 3- 3 -$ 6># 3 5 -1 1#/ !2,-5

c d -1 $ #1 1#" 3- # 3-'- & # : # 3 " - ' "!01'"$ ! #" 1#"$8

-/ " > 3" $ #1 1#"!5 " " 7! $ -$ $3-"' -$ & $ 6-!6 " 3-' "K1 " -+S $

1 6"$ 3- 3 -$ 0 -'& # 3-$ # !"3 - " -$ 3-' " 6-# 3 " A -# ' H1"! " 6"B H1

3- $ 01 ' 3-'.! ' "# "$ : +S $ -/K 6"$ A -# ' H1" " 6"B 6># 3 8

# H` ' "$ : +S $ 6># 3 $,- :-#'1!" "$ ' #'-$ - 6 -# 6-# 3 "

ω

5 "$

# 0 S $ - $3-"' - H1 $ 3"#"3 # L"' .-# # !"+S $ -/ "$ - $-# 0#"

6 !-3 " 5 "$ L- "$ 'I '" .# $$,- "$ #"K ># "$ ! 2"$ 3-## "$ ."# I31!"$

:!1 "$ 0 #" $ - $3-"' - A 5 ())VB8

-'" - "!01 $ 3- 3 -$ " 2 #- 4' 3"5 "'/ A*DDWB : 3" " #"K ># " 1'

! ' - :!1 - $1"$ 6 !-3 " $ ' 1' #' " . 5 #=$ "0 $9 '6 '

(36)

;<

32"'" - 6-# 3 " 8 '-6 ' - #" $!"+,- #" " - $!-3"' - - ! ' - 3-'- 1'

- -5 " :-#' $ # : # T :-#' - ! ' - 6 - " $,- " 0 3 "! " #-

"+,-'-$ #" " 6-!1+,- - ! ' - 3-'- 1' - - "- # -# 1' G- '-' 4 -5 3-' $1"$

6 !-3 " $ " 01!"# $ A R b 5 *DDVB8 1 #-$ "!2 $ $-/# 3- 3 -$

0 -'& # 3-$ '.-# " $ # !"3 - " -$ T 4' 3" " 6-# 3 " .- ' $ # 6 $ - ' !2-# '

#1 $ !! A*DXEB8

01 - 3 - -102 A())EB5 " 6-# 3 " 5 "!&' $ # 1' ."#4' #- '" '" 3"'

: - 3-'- $ - - 3"'.- 6 -# "! : - . !- #- "3 - "! - '-6 ' - 3-'- '-$ #" "

H8A;VB A-1 $ K"5 1'" 0#" L" :I$ 3" H1 H1" : 3" " #- "+,- 1' :!1 - H1" " '!&31!"

-:!1 - 0 #" ' -# - $ B5 & 3- $ #" " 1' 0# $$ 3 "! 1' $3-"' - 1#/1! -8

$$ '5 " 6-# 3 " '" '7 3" & 3- $ #1I " " ."# # 0#" $ 6 !-3 " " .#-.# "

- $3-"' - :I$ 3-5 $ $ 0#" $ $,- 3"!31!" -$ " ."# # 3"'.-$ 6 !-3 "

' -$5 ! 6" - " 1'" ' !2-# #' "+,- # " 6-# 3 " A 5())EB8

. $"# - :1 - # $$ 3- '.-#4 - -$ 6># 3 $5 27 .-13" 3- 3-# 4 3 " "

: +,- P6># 3 Q -1 P $ #1 1#" 3- # Q8 1 -$ 3- 3 -$ $,- #"/"!2" -$ ."#" " :

$$"$ $ #1 1#"$ ' #$"$ - $3-"' -5 - H1 0 #" : # +"$ $ 0 : 3" 6"$ "$ :-#'1!"+S $

# $1! " -$ -/ -$ A 5 ())VB8 -/ - .- - 6 $ " 1$$" A*DV;B5 $ #1 1#"$

3- # $ $,- '-6 ' -$ - $3-"' - 1#/1! - -#0" L" -$ ' 0#" $3"!"5 -1 $ K"5 1'

3-'.- -#0" L" - " 6-# 3 " H1 & 1'" :"$ 3-## !"3 - " " A $ - &5 3- # B $-/# "

G $,- " $ #1 1#"8 1$$" A*DV;B "'/&' ": #'" H1 - "'" 2- $ "$ $ #1 1#"$ 3- # $

$ 3-'."#" T ' $,- #" $6 #$"! - :!1G- 3 $"!2"' - A : 3" " . !" .# $ +"

6-# 3 " "! " ># "B5 ,- $ $-/# .- - $."3 "!' - 3" " 1'" $ 1 -'I - ! '

. 5 H1 "$ $ #1 1#"$ 3- # $ $,- "$ # $.- $76 $ . !- $ 0 : 3" 6- #" $.-#

'"$$"5 3"!-# '-6 ' - - $3-"' -5 $ ' 3 $$"# "' $ # "! "' #0& 3- ' $ 5

-1 $ K"5 1'" $ #1 1#" 3- # $ 3"#"3 # L" .-# "! -$ I6 $ 6-# 3 " 3- # 5 3- # $

$S $ -! $5 3- # .#- 1+,-5 1' 3- # #" $.-# 3"!-# '"$$"5 '"$

3 $$"# "' 1' ! 6" - I6 ! #0 " 3 & 3"8

7# "$ H1 $ S $ :1 "' " $ T # $. - -$ 6># 3 $ $,- # $.- "$ . !" -# " "6" +" "

N-!'-0-#-6 A*DE*B5 H1 & G.# $$" ' #=$ 2 .> $ $5 3-'- .- $ # 6 $ - ' '" -# $ "!2 $

' -. A());B8 $ "3" - " .# ' #" 2 .> $ 5 H1 L # $. - " $- #-. " -$ '-6 ' -$

(37)

$,-;W

": " -$ . !"$ 3- +S $ 3- -# - - $3-"' -8 N-!'-0-#-6 "#01' -1 H1 -$ $6 -$

# +,- "$ 0#" $ $3"!"$ $,- . # -$ - .#-3 $$- # 1+,- $3"!" 3"> 3"5 . !- H1"! "

#0 " & #" $: # " ."#" 6># 3 $ $13 $$ 6"' ' -# $ ' -# $5 3!"#" -5 .-# $$-5 H1

."#" $1: 3 ' "! - O' #- -! $5 -$ '-6 ' -$ ' . H1 " $3"!" 1#/1! "

$ " $ 3"' $- #>. 3-$ A 5 ());B8

F - 0 1$$" A*DDXB H1 $ - "' 67# "$ : +S $ 6># 3 $5 32"'" - " +,- ."#" "

6 #"3 " : +S $ 1 6"$8 '" "$ -+S $ 1 6"$ 6># 3 5 10 A*DDXB5

3- $ #" H1 1' 6># 3 3- $ $ ' 1' '-6 ' - 0 #" 1' 0#1.- ."# I31!"$ :!1 "$

' -# - 1' .- - ' 3-'1'5 $ " 3- $ #"+,- 3-'.! ' " - $ - 3!"$$ : 3"+S $

-/K 6"$5 :-#'" " 3"#"3 # L"# " 0 -' # " $ #1 1#"$ -/$ #6" "$ " " 1# L"5 '

G. # ' -$ " ."# # 6 $1"! L"+S $ -/ "$ 3-' $ '1!"+S $ 1'&# 3"$ !"/-#" -# " $8 1 #"

: +,- 1 6" : 3" 6># 3 $ 3-'- # 0 S $ ! 6" " 6-# 3 "

ω

5 '/-#" ,- G $ "

1' .# 3I. - " ."# # H1"! 6"!-# $ .-$$" ": #'"# H1" - " 6-# 3 " & ! 6" "8 6-# 3 "

! 6" " ' $ '.# 3" .# $ +" 6># 3 $5 K7 H1 $ .- ' # 6># 3 $ 3-' -1 $ '

6-# 3 " '

ω 5 3-'- "3- 3 - $3-"' - 3 $"!2" $ '.! $5 $ - H1 " ": #'"+,- "

G $ = 3 " 1' 6># 3 $ ' 6-# 3 " A6># 3 .- 3 "!B5 '/-#" 3- 3 1"!' '.-# " 5

' 1' 3"$- # "!5 ,- "3- 3 8

'

ω = _∂ −∂ _



∂ ∂

  A;VB

Figura 1 ! (a) um movimento de circulação sem rotação, vórtice sem vorticidade e (b) um movimento de circulação com rotação rígida, vórtice com vorticidade. Fonte:Andrade (2008)

01#" * !1$ #" - '-6 ' - 3 #31!"# 1"$ ."# I31!"$ :!1 "$ ' $ - " 2#7#

(38)

"-;V

!- 0- " #"K ># "8 - .# ' #- $ 2- " ."# I31!" '" =' $1" -# "+,-5 - H1 H1 # L # H1

- '-6 ' - #" $!"+,- & 1' '-6 ' - 3 #31!"+,- $ ' #- "+,-5 -1 $ K"5 - 6># 3 G $

$ ' 6-# 3 " " ."# I31!"8 F7 - 3"$- - $ 01 - $ 2-5 " -# "+,- " ."# I31!" ,- &

.# $ #6" "5 - H1 '-$ #" H1 - '-6 ' - #" $!"+,- & "3-'." 2" - 1'" #- "+,- #I0 "5

-1 $ K"5 - 6># 3 G $ 3-' 6-# 3 " " ."# I31!" : # L #- A 5 ())VB8

-# $$" 3 $$ " $ $ "/ ! 3 # 1'" ' " #- "3 - "! " 1' '-6 ' -5

#1 $ !! A*DXEB #- 1L 1 " H1" " " ' $ - "! 32"'" - O' #- 6-# 3 "

3 '7 3- 3-'- 1'" ' " #- "3 - "! " - '-6 ' -5

= A;DB

- - O' #- 6-# 3 " 3 '7 3- & : - 3-'- " # !"+,- # " "G" $ " 4 "

#- "+,- " "G" $ " 4 " $ #"' - -1 :-#'"+,- 8 O'

#-6-# 3 " 3 '7 3- "6"! " " H1"! " -1 - 0#"1 #- "3 - "! 1' '-6 ' -5 - $

3- $ #" -$ 6"!-# $ 6"# " - # L #-5 H1" - = 5 - '-6 ' - & -

3-'-##- "3 - "! : -5 H1" - = 5 : - 3-'- '-6 ' - #I0 -8 !&' "#01'

-3 '7 -3- #1 $ !! A*DXEB "'/&' #- 1L 1 - O' #- 6-# 3 " 4' 3-

3-'-:-#'" "6"! "# '& - -$ $-!1+,- H1"+S $ 2 #- 4' 3"$ /"$ " -$ " G $ = 3 " 1'

.- 3 "! 6 !-3 " 5 3-'- .- $ # 6 $ - ' '" -# $ "!2 $ ' #" A())VB8

- H1 " 0 -$ 3# &# -$ : 3"+,- 6># 3 $5 ' $ 67# -$ .-$5 3" " H1"! :

-1'" :1 +,- & 5 - " :1 +,- & & 1'" ' " - "'" 2- " 6-# 3 " ' # !"+,- T "G"

:-#'"+,- A 5 ())VB8 3# &# - : 3"+,- 6># 3 $3-!2 - ."#" $

#"/"!2- :- - 3# &# - .#-.-$ - .-# 1 5 _#" - A*DVVBapud #" A())VB5 32"'" -

3# &# - 8 : $ - O3! - - 6># 3 3-'- 1'" # 0 ,- 3- G" 1' $3-"' -

31K-$ 01 - 6"# " $3"!"# - $-# 0#" 6 !-3 " & .-$ 6- (

( )

= 9

(39)

;D

- & " $-'" -$ ! ' -$ " "0- "! .# 3 ."! " '" # L - $-# 0#"

6 !-3 " 8 0#" 6 !-3 " = = ∂ ∂ 5 01"# " :-#'"+S $ '.-# " $

$-/# $$"$ "G"$ $ " 4 "$ :-#'"+,-5 3 $"!2"' - #- "+,-8

' ) ' ' _∂ _∂ _{∂ }  _  _ _∂ _∂ _∂ _  _   _∂ _∂ _∂   _  = ∇ =_ __  ∂ ∂ _{∂ }  _ _∂ _∂ _∂ _    _  _  ∂ ∂ ∂   AE*B