Cfd Manual Fluent

8/12/2019 Cfd Manual Fluent

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COMPUTATIONAL FLUID DYNAMICS

FLUENT

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1 INTRODUCTION.......................................3

2COMPUTATIONAL FLUID DYNAMICS(CFD)..............................................................7

3HEAT TRANSFER MODELLING............39

4TRANSIENT FLOW MODELLING.........46

5CFD EERCISE.........................................51P!"#$%& S'%*+,"-..........................................................................................................72

R%/$,76

P$", 0%$", 0%,"!...........................................................................................................76

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CHAPTER 1

1 INTRODUCTION

1.1 FLUID DYNAMICS:

F$/ -+& ,% %-% "* *$/ &","-. F$/ *$" "&&"-$ ,/% - "-%

"* ,!%% +

E'%!&%-,+$ *$/ -+&.

T%"!%,+$ *$/ -+&.

N/&%!+$$ "&'/,+,"-+$ *$/ -+& (CFD).

1. EXPERIMENTAL FLUID DYNAMICS (EFD)

W- ,/--%$ (/#"- ,!+-"- /'%!"- $"8,/!#/$%-% '!%/!%

:)

T"-; ,+-


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1.2.1. FLUID FLOW:

F$/ *$" %+$ , ,% *$" "* *$/("&'",,"- "* $>/ +- ;+). A *$/

"-, "* + $+!;% -/%! "* -=/+$ &"$%/$%.T%% "/$ - '!-'$% #% &"%$% +

-,%!+,-; "$ '+!,$%.

F;/!% 1.1 F$/ F$"

1.2.2. GRANULAR FLOW:

I, %+$ , *$" "* "$ '+!,$%. T% -,%!+,"- #%,%%- +?+%-, +$, ;!+- +-

+?+%-, *$/ '+!,$% >/,% **%!%-,.

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F;/!%1.2 G!+-/$+! F$"

1.3 FLOW CLASSIFICATIONS:

1.3.1. LAMINAR FLOW:

I, ,% "-% - ,% *$/ '+!,$% &"=% - $+%! ("! $+&-+) , "-% $+%! "**$/ $-; &"",$ "=%! +- +?+%-, $+%!. F$/ '+!,$% &"=% - %$$ %*-% '+, +-

,% !%,+- ,% +&% !%$+,=% '","- +, /%=% !" %,"- "* ,% *$" '++;%. T

,'% "* *$" "/! ;%-%!+$$ - &"", ''% %- ,% =%$", "* *$" $" +- +$" -

$>/ "* +=-; ; =",.

1.3.1.2. TURBULENT FLOW:

I, ,% "-% - ,% *$/ '+!,$% - +- %-,!%$ +'++! "! %!!+, &+--%!.

F$/ '+!,$% &"=% - +- /-'!%,+#$% '+, ,+, !%/$, - + !+' +- "-,-/"/ &-; "*

,% *$/ $%+-; ," &"&%-,/& ,!+-*%! + *$" "/!. I- / + *$" %% "! ="!,% +!%

'!%%-,. T%% %% +- ,%! !+-"& &"=%&%-, ;=% !% ," *$/,/+,"- - ,% =%$", +-

'!%/!% +, +- '"-, - ,% *$" *%$. T ,'% "* "/! - !=%! +-+$ ,!%+& +,%!

/''$ ''%.

1.3.1.3. STEADY FLOW:

I- ,%+ *$" =+!"/ +!+,%!, "* *$"-; *$/ / + =%$", '!%/!%

%-, ,%&'%!+,/!% %,.. +, + '"-, " -", +-;% , ,&%. I- ",%! "! + ,%+ *$"

&+ #% %*-% + ,+, - ,% =+!"/ +!+,%!, +!% -%'%-%-, "* ,&%. E;. p =

p(x,y,z) T


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T% *$/ "#% NEWTON@S LAW OF 0ISCOSITY( %+! ,!% !%,$

'!"'"!,"-+$ ," %+! ,!+-) +$$% N%,"-+- *$/.

1.3.1.!. NON NEWTONIANS FLUID:

T% *$/ "% -", "#% NEWTON@S LAW OF 0ISCOSITY( %+! ,!%

!%,$ '!"'"!,"-+$ ," %+! ,!+-) +$$% N"- N%,"-+- *$/.

1.3.1.". INCOMPRESSIBLE FLOW:

A *$" + ," #% -"&'!%#$% *$" * ,% %-, "-,+-, - + *$" *%$. F"!

+$$ '!+,+$ '/!'"% $>/ +- #% !%;+!% + -"&'!%#$% #%+/% ,% '!%/!% +-

,%&'%!+,/!% +-;% +=% $,,$% %**%, "- ,%! ="$/&%.

1.3.1.#. COMPRESSIBLE FLOW

A *$" + ," #% "&'!%#$% * ,% %-, +-;% *!"& '"-, ," '"-, /%

'!%/!% +- ,%&'%!+,/!%.

M+,%&+,+$$ -", %>/+$ ," "-,+-,.

1.3.1.$. SINGLE PHASE FLOW:

T% *$" - ,%!% -" +-;% "* '+%. I- , ,%!% "-$ "-% '+% "*

*$/ *$"(%,%! $>/ "! ;+).

1.3.1.1%. MULTI PHASE FLOW:

T% *$" - ,%!% +-;% "* '+% +$$% &/$,'+% *$". I- , ,%!%

+ &,/!% "* '+% *$"($>/8;+ $>/8"$ ;+8"$).

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CHAPTER 2

2 COMPUTATIONAL FLUID DYNAMICS (CFD)

1.4 COMPUTATIONAL FLUID DYNAMICS:

C"&'/,+,"-+$ *$/ -+& (CFD) ,% %-% "* '!%,-; *$/ *$" %+,

,!+-*%! &+ ,!+-*%! %&+$ !%+,"- +- !%$+,% '%-"&%-+ # "$=-; ,%

&+,%&+,+$ %>/+,"- ;"=%!- ,%% '!"%% /-; + -/&%!+$ '!"%.

T% !%/$, "* CFD +-+$% !%$%=+-, %-;-%%!-; +,+ /% -

C"-%',/+$ ,/% "* -% %;-.

D%,+$% '!"/, %=%$"'&%-,.

T!"/#$%"",-;.

R%%;-.

CFD +-+$ "&'$%&%-, ,%,-; +- %'%!&%-,+,"-.

R%/% ,% ,",+$ %**"!, !%>/!% - ,% $+#"!+,"!.

1.5 CFD How it wor!:

A-+$ #%;- , + &+,%&+,+$ &"%$ "* + '+$ '!"#$%&. C"-%!=+,"- "*

&+,,%! &"&%-,/& +- %-%!; &/, #% +,*% ,!"/;"/, ,% !%;"- "* -,%!%,. F$/

'!"'%!,% +!% &"%$% %&'!+$$. S&'$*-; +/&',"- +!% &+% - "!%! ," &+/+,"- +$;%#!+.

T% "$$%,"- "* %$$ +$$% ,% ;!.

T% %, "* +$;%#!+ %>/+,"- +!% "$=% -/&%!+$$ ("- + "&'/,%!) *"!

,% *$" *%$ =+!+#$% +, %+ -"% "! %$$.

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S,%& "* %>/+,"- +!% "$=% &/$,+-%"/$ ," '!"=% "$/,"-.

T% "$/,"- '",8'!"%% ," %,!+, >/+-,,% "* -,%!%, (%.;. $*, !+;

,"!>/% %+, ,!+-*%! %'+!+,"- '!%/!% $" %,.).

1." APPLICATIONS OF CFD:

A''$+,"- "* CFD +!% -/&%!"/B

F$" +- %+, ,!+-*%! - -/,!+$ '!"%% (#"$%! %+, %+-;%!

"/,"- %>/'&%-, '/&' #$"%! ''-; %,.).

A%!"-+& "* ;!"/- =%$% +!!+*, &$%.

F$& "+,-; ,%!&"*"!&-; - &+,%!+$ '!"%-; +''$+,"-.

F$" +- %+, ,!+-*%! - '!"'/$"- +- '"%! ;%-%!+,"- ,%&.

0%-,$+,"- %+,-; +- ""$-; *$" - #/$-;.

C%&+$ =+'"! %'","- (C0D) *"! -,%;!+,% !/, &+-/*+,/!-;.

H%+, ,!+-*%! *"! %$%,!"- '+


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CFD '!"=% ,% +#$, ," ,%"!%,+$$ &/$+,% +- '+$

"-,"-.

& A++*- * +6*' +' 08+*+8.

CFD +$$" ;!%+, "-,!"$ "=%! ,% '+$ '!"% +- '!"=% ,%

+#$, ," "$+,% '%* '%-"&%-+ *"! ,/.E+&'$% + %+, ,!+-*%!

'!"% +- #% %+$% , ++#+, "-,+-, %+, *$/ "! "-,+-,

,%&'%!+,/!% #"/-+!%.

& C7'9'8+,' +87*+8.

E'%!&%-, "-$ '%!&, +,+ ," #% %,!+,% +, + $&,% -/%! "*

$"+,"- - ,% ,%& (%.;. '!%/!% +- ,%&'%!+,/!% '!"#% %+, *$/

;+/;% LD0 %,.).

CFD +$$" ,% +-+$, ," %+&-% + $+!;% -/%! "* $"+,"- - ,%

!%;"- "* -,%!%, +- %$ + "&'!%%-=% %, "* *$" '+!+&%,%! *"!

%+&-+,"-.

1.& LIMITATIONS OF CFD:

& P9-+0 ':

CFD "$/,"- !%$ /'"- '+$ &"%$ "* !%+$ "!$ '!"%% (%.;.

,/!#/$%-% "&'!%#$, %&,! &/$,'+% *$" %,.).

T% CFD "$/,"- +- "-$ #% + +/!+,% + ,% '+$ &"%$ "-

,% +!% #+%.

& N6'7+0 '777:

S"$=-; %>/+,"- "- + "&'/,%! -=+!+#$ -,!"/% -/&%!+$ %!!"!.

R"/-8"** %!!"! /% ," *-,% "! % +=+$+#$% "- ,%

"&'/,%!.R"/-8"** %!!"! $$ +$+ %, (,"/; ,% +- #% &+$$

- &", +%).

T!/-+,"- %!!"! /% ," +''!"&+,"- - ,% -/&%!+$ &"%$.

T!/-+,"- %!!"! $$ ;" ," %!" + ,% ;! !%*-%. M% !%*-%&%-,

"-% + ," %+$ , ,!/-+,"- %!!"!.

& B687- 08+*+8:

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A , '+$ &"%$ ,% +/!+ "* ,% CFD "$/,"- "-$ +

;"" + ,% -,+$#"/-+! "-,"- '!"=% ," ,% -/&%!+$

&"%$.

E+&'$% *$" - + /, , /%- %'+-"-. I* *$" /''$% ,"

"&+- # + ''% "/ "/$ /% + */$$8%=%$"'% '!"*$% *"! =%$",

!+,%! ,+- +/&% /-*"!& "-,"-.

1.' DISC(ETI)ATION

T% +!, "* /#=-; + ,!/,/!% -," + "-=%-%-, -/%! "* &+$$%!

"&'"-%-,


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T% ;! -",-; #/, ,% "&+- ,+, %!%,% -," + *-,% %, "* "-,!"$ ="$/&%

"! %$$. T% %;!%% "* ;! !%"$/,"- +$" -%%+! ," *- "$/,"- *"! + '!"#$%& - %+

!%;"- "* ,% "&+-.

T%!% +!% ,!%% **%!%-, ,'% "* ;! - ;%-%!+$

F"!&+$ ;!.(,!+-;$% ,%,!+%!"- '!+& '!& %+%!"-

>/+!$+,%!+$ )

H#! ;!.

N"- *"!&+$ ;!.

2.!.2. TYPES OF CELL SHAPES OR GRIDS:

F;/!% 2.2 T'% "* G!

2.!.2.1. TETRAHEDRON MESH: T% &'$%, ,%, &% % ,%!& +- "!%! 4 "! $-%+! &% /% *"/!

'"-, , %*-% %+ ,%,!+%!"-. F"! "&'$% ;%"&%,!% >/+% &%% " -"

-/&%!+$ +=+-,+;% +- % +- +=% &%-; %**"!, # /-; + ,!,%, &%.

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F;/!% 2.3 T%,!+ M%

2.!.2.2. >UADRILATERAL MESH:

F"! &'$% ;%"&%,!% >/+% &%% +- '!"=% ;8>/+$,"$/,"- ,

*%%! %$$ ,+- + "&'+!+#$% ,!,%, &%.

F;/!% 2.4 /+ M%

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2.!.2.3. HYBRID MESH:

T ,% "-+,"- "* **%!%-, ,'% "* *"!&+$ ;! *"! + ="$/&% "* #". I- , +

'%* !%;"- &%% , **%!%-, ;! "! %$$ ,'%. T% %**%- +- +/!+ +!%

%-+-% !%$+,=% ," ,%,!+%!"- +- %+%!+$ &%. I, "-, "* #", &% !"/,%! +-&% $%-, - #", ,% ,'% "* -"% '%!*"!&-; !"/,-; +- *"!+!-;

*/-,"-+$,. T


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D"&+- ;!"/' "* -"% *+% +- %$$ "-%.

F;/!% 2.6 2D J 3D C%$$ T%!&-"$";

1.11 MESH +UALITY:

F"! ,% +&% %$$ "/-, %+%!+$ &%% $$ ;=% &"!% +/!+,% "$/,"- %'%+$$

* ,% ;! $-% +!% +$;-% , ,% *$". T% &% %-, "/$ #% ; %-"/; ,"

+',/!% +$$ !%$%=+-, *$" *%+,/!%. T% &% +?+%-, ," ,% +$$ "/$ #% *-% %-"/; ,"!%"$=% ,% #"/-+! $+%! *$". I- #"/-+! $+%! >/+ % +- '!&%;% %$$ +!%

'!%*%!!% "=%! ,! ,%, "! '!+&.

T%!% +!% ,!%% ,'% "* &% >/+$, SKEWNESS SMOOTHNESS (CHANGE IN SIE) ASPECT RATIO

2.#.1. MESH >UALITY: S?EWNESS

T" &%," *"! %,%!&--; UILATERAL @OLUME:

S


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"',&+$ %$$ %

T +''$+#$% "-$ ," ,!+-;$% +- ,%,!+%!"-. T ,% %*+/$, &%," *"!

,!+-;$% +- ,%,!+%!"-.

2.B' 8 *9' ',+*+8 7 87+' '6+*'7 8':

S/+$,

+$+ /% *"! '!& +- '!+&.

2.#.2. MESH >UALITY : SMOOTHNESS C+-;% - % - +- %$%&%-, "/$ #% ;!+/+$. I* ,% &%-; /!*+%

&"", ;"" +/!+ +- #% &+%. R"/; &%-; /!*+% %$%&%-, $%+ ,"

-+/!+.

F;/!% 2.7 S&"",-% - M%-;

2.#.3. MESH >UALITY : ASPECT RATIO

T %*-% + ,% !+," "* $+!;%, &%-"- "* ,% %$%&%-, ," ,% &+$$%,

&%-"-. I- &+- +% + ,% +'%, !+," -!%+% ,% -+/!+ "* ,% "$/,"-

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-!%+%. T% "-$/"- "* &+- !%%+!% ,+, ,% +'%, !+," "/$ #% $"% ,"

/-, + '"#$%. E>/+$ ," 1 (%+$) *"! +- %>/$+,%!+$ ,!+-;$% "! + >/+!%.

F;/!% 2. A'%, !+," - M%-;

1.12 ST(I$IN% FO( +UALITY:

A '""! >/+$, ;! $$ +/% -+/!+,% "$/,"- +-"! $" "-=%!;%-%. T" ;%, +

#%,,%! >/+$, &-&% $"+$ =+!+,"- - %$$ %. F"! %+&'$% +?+%-, %$$ "/$ -",

+=% Q+' 7*+ ;!%+,%! ,+- 2. I* / '""! >/+$, %, %$%,% ,% &%-;

;%"&%-,! " -%%+! %"&'","- +- "! '!% &% ,% %;% +- *+% +- !%&% ,%

%$%&%-,.

MINIMIUIANGLE S?EW:

H% +- >/+ %$$


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1.13 %(ID DESI%N:

2.1%.1. GRID DESIGN GUILDLINES: RESOLUTION

T% *%+,/!% "* '%!,-%-,(!%$%=+-,) *$" "/$ #% +%>/+,%$ !%"$=%. W%- ,%

*$" &/$, &%-"-+$ Q0' '0* 7*+ (!+," "* $+!;%, &%-"- ," ,% &+$$%,

&%-"-) "/$ #% -%+! "-%. A$" /+% +- #% ,!%,% %!% ,% *$" */$$

%=%$"'% +- %%-,+$$ "-%8&%-"-+$.

F;/!% 2.9 D%;- *"! G!

2.1%.2. GRID DESIGN SOLUTIONS:SMOOTHNESS

C+-;% - % - +- %$%&%-, "/$ #% ;!+/+$. I* ,% &%-; /!*+% &"",

;"" +/!+ +- #% &+%. R"/; &%-; /!*+% %$%&%-, $%+ ," -+/!+.

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F;/!% 2.1 S&"",-% *"! G!

I- +- &%-; %$%&%-, ,% &+&/& +-;% - ;! '+-; &/, #% $% ,+- 2.

2.1%.3. GRID DESIGN GUILDLINES: TOTAL CELL COUNT

I* ,% -/%! "* ;! "! %$$ +!% &"!% ,%- ,% +/!+ "* ,+, &%-; %$%&%-,

$$ #% ;%!. A, ,% +&% ,&% , $$ -!%+% &%&"! /+;% +- CPU ,&%. T%!% +!% ,"

+ /% ,"


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1E6 +!% $+!;%. S/ '!"#$%& +- #% %**%-,$ !/- /-;

&/$,'$% CPU #/, &% ;%-%!+,"- +- '",8'!"%-; &+

#%"&% $".

1E7 +!% /;% +- "/$ #% +="% * '"#$%. H"%=%! ,%

+!% "&&"- - +%!"'+% +- +/,"&",=% +''$+,"-.

1E +- &"!% +!% %'+!,&%-, "* %*%-% ,$% +''$+,"-.

2.1%.4. ADOPTION EXAMPLE: FINAL GRID AND SOLUTION

F;/!% 2.11 F-+$ G! *"! +- E$%&%-,

1.14 MAIN SOU(CES OF E((O(S IN MESHIN%:

M% ,"" "+!%.

H;


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I-,%!'"$+,"- %!!"! +, -"-8"-*"!&+$ -,%!*+%.

I-+''!"'!+,% #"/-+! $+%! &%.

1.15 %O$E(NIN% E+UATION:

T% ;"=%!--; %>/+,"- -$/% ,% *"$$"-; "-%!=+,"- $+ "* '

C"-%!=+,"- "* &+.

N%,"- %"- $+ ,% +-;% "* &"&%-,/& %>/+$ ,%

/& "* *"!% "- + *$/ '+!,$%.

F!, $+ "* ,%!&"-+& ("-%!=+,"- "* %-%!;) !+,% "*

+-;% "* %-%!; %>/+$ ,% /& "* !+,% "* %+, +,"- ," +-

"!< "-% "- *$/ '+!,$%.

T% *$/ ,!%+,% + + "-,-//&. F"! $%-;, +$% "* + 1 & +- $+!;%!. T% &"$%/$+! ,!/,/!% +- &","- &+ #% ;-"!%.

2.12.1. LAGRANGIAN DESCRIPTION:

T +''!"+ ",%!% +$$% + Q7*+0' 709. A *$/ *$" *%$ +- #%

,"/;, "* + #%-; "&'!% "* + $+!;% -/%! "* *-,% % *$/ '+!,$% +=%

&+ &"&%-,/& -,%!-+$ %-%!; +- ",%! '!"'%!,%. M+,%&+,+$ $+ +- ,%- #%

!,,%- *"! %+ *$/ '+!,$%. F$/ - &+,%!+$ ="$/&% (,%&) $$ &"=% ,"!, +-

+-;% % +- +'% #/, +$+ "-, "* ,% +&% *$/ '+!,$%.

'+,$-%

F;/!% 2.12 L+;!+-;+-

M"%$

2.12.2. EULERIAN

DESCRIPTION:

T ",%!% +$$% + Q+' 709. I- , &"%$ -=/+$ +!% -", $+#%$%

+- -", ,-;/% *!"& "-% +-",%!. I- ,% E/$%!+- %!',"- "* *$/ &","- %

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"-%! " *$" '!"'%!,% +-;% +, + *$/ %$%&%-, ,+, *% - '+% +- ,&%

(,) !+,%! ,+- *"$$"-; -=/+$ *$/ '+!,$%.

F;/!% 2.13 E/$%!+- M"%$

1.1" CONTINUITY E+UATION:

T% "-,-/, %>/+,"- ;"=%!-% *!"& ,% Q7+80+' 08'7,*+8 . I,

,+,% ,+, Q*9' 6+ /+8 *9769 *9' +' * 07 '0*+8 7'+8

08*8* + *9'7' + 8 6+ + ' 7 7',' 7 *9' +' .

MASS BALANCE:

R+,% "* -!%+% "* &+ - *$/ %$%&%-, %>/+$ ,% -%, !+,% "*

*$" "* &+ -," %$%&%-,. R+,% "* -!%+% "* &+ %'!%% +

1.1# MOMENTUM E+UATION:

T -",-; #/, QN'/*8 S'08 L/ M*+8 (F&+). T% -%, *"!% *"!% +,-; -

+- !%,"- %>/+$ ," !+,% "* +-;% "* &"&%-,/& - ,+, !%,"-.

L-%+! &'/$%+-;% - $-%+! &"&%-,/&.

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I- 8!%,"- -,+$ $-%+! &'/$%$-%+! &'/$%*-+$ $-%+! &"&%-,/&. R+,% "*

-!%+% "* 8 8 +- 8&"&%-,/&

T% *"$$"-; *"!% +,-; "- *$/ '+!,$% +!%

S/!*+% *"!% / + '!%/!% +- ="/ *"!%.

V" *"!% +, "- + ="$/&% / + ;!+=,

%-,!*/;+$ C"!"$ +- %$%,!"&+;-%, *"!%.

1.1& NA$IE(,STO-ES E+UATION:

T% %>/+,"- "* &","- ,"/, "-%!-; *"!% /% ," ,/!#/$%-% +-

"-%!-; ,% ;!+=, '!%/!% +- =", *"!% +!%


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,% *$/ %-, u ,% =%$", e ,% ,",+$ %-%!; '%! /-, ="$/&% p ,% '!%/!%

T ,% ,%&'%!+,/!% ,% "%**%-, "* =", +- ,% ,%!&+$ "-/,=,. T%

,",+$ %-%!; e -$/% -,%!-+$ %-%!; '%! /-, ="$/&% e (%!% e ,% -,%!-+$ %-%!;

'%! /-, &+) +- /!% "&'"-%-, "* ,% &+,%&+,+$ &"%$.

V"/-+!% +- -,%!-+$ /!*+% +!% !%'!%%-,% # *+% "-%. V"/-+! +,+ +!% +;-%

," *+% "-%.

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F;/!% 2.14 P'% *$" , "!*% '$+,%.

2.1!.1. NEUMANN AND DIRICHLET BOUNDARY CONDUTIONS:

W%- /-; + D!$%, #"/-+! "-,"- "-% '!%!#% ,% =+$/% "* + =+!+#$% +,

,% #"/-+! %.;. u(x) = "o#$ta#t. W%- /-; + N%/&+-- #"/-+! "-,"- "-%

'!%!#% ,% ;!+%-, -"!&+$ ," ,% #"/-+! "* + =+!+#$% +, ,% #"/-+! %.;. #u(x) =

"o#$ta#t. W%- /-; + &% #"/-+! "-,"- + */-,"- "* ,% *"!& au(x)%&u(x) =

"o#$ta#t +''$%. N",% ,+, +, + ;=%- #"/-+! **%!%-, ,'% "* #"/-+! "-,"-

+- #% /% *"! **%!%-, =+!+#$%.

2.1!.2. FLOW INLETS AND OUTLETS:

A % !+-;% "* #"/-+! "-,"- ,'% '%!&, ,% *$" ," %-,%! +- %, ,%

"$/,"- "&+-. I- ;%-%!+$ ,% %-,! '"!,"- -+,% '!%/!% -$%, +- %, '"!,"-

-+,% '!%/!% "/,$%,. F"! -"&'!%#$% *$" %-,! '"!,"- $$ #% &%-,"- + =%$",

-$%, +- %, + "/,*$". I- +% "* "&'!%#$% *$" %-",% + &+ *$" -$%, +-

'!%/!% *+!8*%$. I-$%, =%-, "/,$%, =%-, -,+


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S"/$ -", "#%!=% $+!;% ;!+%-, - !%,"- -"!&+$ ,"

#"/-+! -%+! -$%, +- "/,$%,. T -+,% +- -"!!%,

'!"#$%& '%*+,"-.

M-&% ;! /!% ,+, ;+/;% '!%/!% -'/,. T% "'%!+,-;

'!%/!% -'/, %, %'+!+,%$. T #"/-+! "-,"- $$ #% /%*/$ %- N%,%! ,%

*$" !+,% -"! ,% =%$", +!%


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2.1!.5. OUTFLOW BOUNDARY:

O/,*$" #"/-+! "-,"- +!% /% ," &"%$ *$" %, %!% ,% %,+$ "* ,%

*$" =%$", +- '!%/!% +!% -", /!% -*"!&+,"- *!"& -,%!"!.

2.1!.!. POROUS MEDIA CONDITIONS:

P"!"/ "-% &"%$ + '%+$ ,'% "* *$/ "-%. I- , '!%/!% $" - *$"

%,%!&-% =+ /%! -'/, "* !%,+-% "%**%-, ," $/&'% '+!+&%,%! &"%$. T /%

," &"%$ *$" ,!"/; '"!"/ &%+ +- ",%! ,!#/,%@ !%,+-% %.; '+


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1.21 SOLUTION METHODS:

2.1#.1. FINITE ELEMENT METHOD:

T + -/&%!+$ &%," *"! "$=-; '!"#$%& "* %-;-%%!-; +- &+,%&+,+$

+-+$. I- , &%," + #" "! + ,!/,/!% - ,% +-+$ ," #% +!!% "/,

/#=% -," &+$$%! %$%&%-, "* *-,% &%-"- +$$% *-,% %$%&%-,. T%- ,% #"

"-%!% + +- +%$+;% "* ,%% %$%&%-, "--%,% +, + *-,% -/%! "* ?"-, +$$%

Q8' "! N"+$ '"-,. V+% "- +''$+,"- ,% %$%&%-, '!"#$%& +!% $+*% +

*"$$"

S,!/,/!+$ '!"#$%&.

N"-8,!/,/!+$ '!"#$%&.

AD@ANTAGES:

H;%, +/!+ "- "+!% ;!.

E%$$%-, *"! **/"- "&-+,% '!"#$%& ="/ +- *!%% /!*+%

'!"#$%&.

DISAD@ANTAGES:

S$" *"! $+!;% '!"#$%&.

N", %$$ /,% *"! ,/!#/$%-, *$".

2.1#.2. FINITE DIFFERENCE METHOD:

T &%," /%*/$ *"! "$=-; %+, ,!+-*%! *$/ &%+- +- ,!/,/!+$

&%+- '!"#$%&. I, +''$+#$% ," +- '%-"&%-"- *"! **%!%-,+$ %>/+,"-

+$"-; , ,% #"/-+! "-,"- +=+$+#$%. I, "!/!%.T% ;"=%!--; %>/+,"- (- **%!%-,+$ *"!&) +!% !%,% ("-=%!,% ," +$;%#!+ *"!&).

F!, +- %"- %!=+,=% +!% +''!"&+,% # ,!/-+,% T+$"! %!% %'+-"-. T%

!%/$,-; %, "* $-%+! +$;%#!+ %>/+,"- "$=% %,%! ,%!+,=%$ "! &/$,+-%"/$. T

&%," **/$, ," /% %- !%;"- +=% /!=% "! !!%;/$+! #"/-+!% +- , **/$,

," !,% ;%-%!+$ "&'/,%! '!";!+&.

Page 27 of 81


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2.1#.3. FINITE @OLUME METHOD:

T% #+ %+ "* + *-,%8="$/&% &%," ," +,* ,% -,%;!+$ *"!& "* ,%

"-%!=+,"- $+ ," "&% %;!%% "* +''!"&+,"- *"! %+ "* &+- "-,;/"/ "-,!"$

="$/&% "=%! ,% "&+- "* -,%!%,. T &%," + +,,!+,=% #%+/% $%

=+!+#$% &+ -", #% "-,-/"/$ **%!%-,+#$% +!" "/+,"- (&"&%-,/& %-%!; %,.) +!% "#%% - +$$ %$$ ," +'%*% ,"$%!+-% ,% "$/,"- -" $"-;%! +-;% , +,"-+$ ,%!+,"-. M+

&"&%-,/& %-%!; +- +$+! #+$+-% +!% "#,+-%. R%/+$ &%+/!% +$+-% ("! %!!"!)

- "-%!=+,"- %>/+,"-. T% *"$$"-; ,-; "/$ #% "-%!% $% "-=%!;%-%

A$+ %-/!% '!"'%! "-=%!;%-% #%*"!% /-; + "$/,"-. /-"-=%!;%

"$/,"- +- #% &$%+-;.

S"$/,"- +!% "-=%!;% %- ,% *$" *%$ +- +$+! *%$ +!% -" $"-;%!

+-;-;.

2.1#.5. MONITOR RESIDUALS:

I* ,% !%/+$ +=% &%, ,% '%*% "-=%!;%-% !,%!"- #/, +!% ,$$ %!%+-;

,% "$/,"- &+ -", %, #% "-=%!;%. I* ,% !%/+$ -%=%! &%%, ,% "-=%!;%-% !,%!"-

#/, +!% -" $"-;%! %!%+-; +- ",%! "$/,"- &"-,"! " -", +-;% %,%! ,% "$/,"-

Page 28 of 81


29/81

"-=%!;%. F-+$ !%/+$ +!% "*,%- ;%! , ;%! "!%! !%,+,"- %&% ,+-

, *!, "!%! !%,+,"-. T+, "% -", &%+- ,+, ,% *!, "!%! "$/,"- #%,,%!.

R%/+$ +- #% &"-,"!% ;!+'+$$ +$".

F;/!% 2.16 M"-,"! R%/+$

1.22 P(ESSU(E $ELOCITY COUPLIN%:

P!%/!% +''%+! - +$$ ,!%% &"&%-,/& %>/+,"-. T% =%$", *%$ +$" + ,"

+,* ,% "-,-/, %>/+,"-. S" %=%- ,"/; ,%!% -" %'$, %>/+,"- *"! '!%/!% %

" +=% *"/! %>/+,"- *"! *"/! =+!+#$% +- ,% %, "* %>/+,"- $"%. S" '!%/!%8

=%$", "/'$-; +$;"!,& +!% /% ," %!=% %>/+,"- *"! ,% '!%/!% *!"& ,%

&"&%-,/& %>/+,"- +- ,% "-,-/, %>/+,"-. T% &", "&&"-$ /% +$;"!,& ,%

SIMPLE (S%&8I&'$, M%," *"!P!%/!%8L-/+,"-). A- +$;%#!+ %>/+,"- *"!

Page 29 of 81


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,% '!%/!% "!!%,"- p' %!=% - + *"!& &$+! ," ,% %>/+,"- %!=% *"! ,%

"-=%,"-8**/"- %>/+,"-

+''X +-#'X #X.

%!% ' '!%/!% "!!%,"-. # "-,-/, +$+-%.

E+ ,%!+,"- ,% '!%/!% *%$ /'+,% # +''$-; ,% '!%/!% "!!%,"-. T%"/!% ,%!& &' ,% "-,-/, +$+-%. T% ",%! "%**%-, %'%- "- ,% &% +- ,%*$" *%$.

2.1$.1. PRINCIPLE BEHIND SIMPLE:

T% '!-'$% #%- SIMPLE >/,% &'$%. I, #+% "- ,% '!%&% ,+, *$/

*$" *!"& !%;"- , ; '!%/!% ," $" '!%/!%. I- , *!, S,+!, , +- -,+$

'!%/!% *%$ ,%- L""< +, + %$$. I* "-,-/, -", +,*% #%+/% ,%!% &"!% &+

*$"-; -," ,+, %$$ ,+- "/, "* ,% %$$ ,% '!%/!% - ,+, %$$ "&'+!% ," ,%

-%;#"!-; %$$ &/, #% ,"" $". T/ ,% '!%/!% - ,+, %$$ &/, #% -!%+% !%$+,=% ,"

,% -%;#"!-; %$$. T% !%=%!% ,!/% *"! %$$ %!% &"!% &+ *$" "/, ,+- -. R%'%+,

, '!"% ,%!+,=%$ *"! +$$ %$$. T% ,!< - *--; + ;"" %>/+,"- *"! ,% '!%/!%

"!!%,"- + + */-,"- "* &+ +$+-%.

2.1$.2. IMPRO@EMENTS ON SIMPLE:

SIMPLE ,% %*+/$, +$;"!,& - &", "&&%!+$ *-,% ="$/&% "%. I&'!"=%

=%!"- +!%

SIMPLER (SIMPLE R%=%).

SIMPLEC (SIMPLE C"-,%-,).

PISO (P!%/!% I&'$, , S'$,,-; "* O'%!+,"!).

A$$ ,%% +$;"!,& +- '%% /' "-=%!;%-% #%+/% ,% +$$" *"! ,% /% "* $+!;%! /-%!

!%$++,"- *+,"! ,+- SIMPLE. A$$ "* ,%% $$ %=%-,/+$$ "-=%!;% ," ,% +&% "$/,"-.

Page 30 of 81


31/81

T% **%!%-% +!% - '%% +- ,+#$,. W +$;"!,& *+,%, %'%- "- ,% *$" +-

,%!% -" -;$% +$;"!,& ,+, +$+ *+,%! ,+- ,% ",%! "-%.

2.2*. SOLUTION P(OCEDU(E:

2.2%.1. SEGREGATED SOLUTION PROCEDURE:

F;/!% 2.17 S%;!%;+,% S"$/,"- P!"%/!%

2.2%.2. COUPLED SOLUTION PROCEDURE:

W%- ,% "/'$% "$=%! /% *"! ,%+ ,+,% +$/$+,"- , %%-,+$$ %&'$" +

&"*% ,&% %'%-%-, "$/,"- +$;"!,& /-; + ,&% ,%'

Z, CFL(UL).

Page 31 of 81


32/81

W%!% * 8 ,&% ,%'. CFL C"/!+-,8F!%!8L%= -/%!. 6 L"+$ =%$",.

L % "* ,% %$$.

2.21. TU(ULANCE MODELLIN%:

C+!+,%!+,"- "* T/!#/$%-, F$"

F!"& N+=%!8S,"/+,"- ," R%-"$8A=%!+;% N+=%!8S,"


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U-,%+ +'%!" &","- - +$$ ,!%% =%$", "&'"-%-, *$/,/+,% &-;

&+,,%! &"&%-,/& +- %-%!;. D%"&'"% =%$", -," &%+- +- *$/,/+,-; '+!,

U+(*) U+ J 6+(*).

2.21.1. TRANSITION IN BOUNDARY LAYER FLOW O@ER FLAT PLATE

F;/!% 2.1 T!+-,"- - V"/-+! $+%!

Page 33 of 81


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2.21.2. TRANSITION IN KET FLOW:

F;/!% 2.19 T!+-,"- - [%, F$"

2.21.3. TURBULENT FLOW CORRELATIONS:

EXTERNAL FLOWS:

R%\ 5]15 +$"-; + /!*+%.

R%D \ 2 +!"/- +- "#,+$%.

INTERNAL FLOWS:

R%D \ 122

W%!% R%L UL ^

L D D %,.

W%!% R% R%-"$ -/%!. _D%-, "* *$/. U _0%$", "* *$/. L _L%-;, "* ,/#%. _0", "* *$/. D _D+&%,%! "* ,/#%.

Page 34 of 81


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2.21.4. TURBULANT BOUNDARY LAYER:

I- ,/!#/$%-, *$" ,% #"/-+! $+%! %*-% + ,% ,- !%;"- "- ,% /!*+% "* +

#" - ="/ %**%, +!% &'"!,+-,. T% #"/-+! $+%! +$$" ,% *$/ ," ,!+-,"-

*!"& ,% *!%% ,!%+& =%$", U ," + =%$", "* %!" +, ,% +$$. T% =%$", "&'"-%-,-"!&+$ ," ,% /!*+% &/ &+$$%! ,+- ,% =%$", '+!+$$%$ ," ,% /!*+%. T% ;!+%-,

"* ,% *$" +!" ,% $+%! +!% &/ ;!%+,%! ,+- ,% ;!+%-, - ,% *$" !%,"-. T%

#"/-+! $+%! ,


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F;/!% 2.2 T/!#/$%-, V"/-+! L+%!

Page 36 of 81


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2.21.5. RANS M' D'07+*+8 8 B'9,+7 S67-

M"%$ D%!',"-

S'+$+!,

A$$&+!+

A -;$% ,!+-'"!, %>/+,"- &"%$ "$=-; !%,$ *"! + &"*% ,/!#/$%-, =",.

D%;-% '%*+$$ *"! +%!"'+% +''$+,"- -="$=-; +$$8#"/-% *$" "- +

*-% -%+!8+$$ &%. FLUENT &'$%&%-,+,"- +$$" ,% /% "* "+!%! &%%.

O',"- ," -$/% ,!+- !+,% - < '!"/,"- ,%!& &'!"=% '!%,"- "* ="!,+$

*$".S,+-+! /+,"- &"%$ "$=-; *"! < +- `. T ,% %*+/$,


38/81

S'+$+!,

A$$&+!+

E"-"&+$ *"! $+!;% &%%. P%!*"!& '""!$ *"! 3D *$" *!%% %+!

*$" *$" , ,!"-; %'+!+,"-. S/,+#$% *"! &$$ "&'$% (>/+82D)

%,%!-+$-,%!-+$ *$" +- #"/-+! $+%! *$" /-%! '!%/!% ;!+%-,

(%.;. +!*"$ -; +!'$+-% */%$+;% &$% ' /$$).

S,+-+!


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3 HEAT TRANSFER MODELLING

E/0r Etio/ I/tro67tio/

E/0r tr/!8ort 034tio/:

E/0r E 80r 4/it 9!! i! 60i/06 !:

Pr0!!4r0 wor /6 i/0ti7 0/0r r0 ;w! 77o4/t06 or wit0

iro/90/t w07tio/ /6 (6itio/

=o4/6r 7o/6itio/!.

W;; 9t0ri; /6 t


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Co/t0 H0t Tr/!0r E@98;0

Cir74it =or6 0@t0r/;; 7oo;06B *.1 W9G-

< 1.5 W92G-

T 2'& -

Air i/;0t

$ *.5 9!

T 2'& -

E;07tro/i7 Co98o/0/t

o/0


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M0!!. T


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Sritio/.

Wit t0r9 i! i/7;4606?

8r0!!4r0 r6i0/t /6 =o6 or70 t0r9

i/ t


44/81

Ntr; Co/>07tio/ t07tio/ ;ow! t0r? o4!!i/0!3 9o60; 94!t =0 4!06.

For 4/!t06 !o;>0r? o4!!i/0!3 9o60; or i60; ! ;w 7/ =0 4!06.

U!0r I/84t! or Nt4r; Co/>07tio/

D0i/0 tittio/; 770;0rtio/ 6o/0i/ O80rti/ Co/6itio/! 8/0;.

D0i/0 60/!it 9o60; !0>0r; o8tio/!r0 >i;=;0B.

o4!!i/0! 9o60;

E/=;0 r>it. S0t O80rti/ T0980rt4r0? T*.

S0;07t o4!!i/0!3 ! t


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(6itio/

(6itio/ 007t! !


46/81

4 TRANSIENT FLOW MODELLING

Tr/!i0/t CFD A/;!i! Si94;t0 tr/!i0/t ;ow i0;6 o>0r !807ii06 ti90 80rio6

I So;4tio/ 9 88ro7ri=;0! !to8 70ri=;0! ;47t4t0 wit< r080ti/ 8tt0r/

IYo4r o; 9 ;!o =0 !i98; to /;0 t0r 8r0!7ri=06 ti90

i/t0r>;.

Fr00 !4r70 ;ow!

Mo>i/ !0!

Et7.

E@tr7t 34/titi0! o i/t0r0!t

I Nt4r; r040/7i0! 0.. Stro40r06 /6or (MS >;40!

I Ti90,r0;t06 8r90t0r! 0.. ti90 r04ir06 to 7oo; it w>0!? !0!B

For706 tr/!i0/t!

I Ti90,6080/60/t =o4/6r 7o/6itio/!? !o4r70 t0r9! 6ri>0 ti/,H0;9


47/81

Tr/!i0/t F;ow Mo60;i/ Wor;ow

E/=;0 t0r.

S0t 48 8


48/81

S0;07ti/ t


49/81

Tr/!i0/t F;ow Mo60;i/ A/i9tio/!

Yo4 94!t !0t 48 / /i9tio/! EFO(E 80ror9i/ it0rtio/!.

I A/i9tio/ r90! r0 writt0/!tor06 o/,t


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P0ror9i/ It0rtio/!

T/7090/t!7


51/81

5 CFD EXERCISE

1.23 F6+ F/ 8 H'* T78'7 +8 M++8 T''

Strt i/ Wor=0/7asic initiali+ation imposes the same

&alues in all cells" #ou can impro&e on this in &arious 'ays for

e-ample% $y patchin! different &alues into different +ones"

Se&eral features% includin! patchin! and post.processin!% are

not a&aila$le until after initiali+ation"

This computes a &alue

for each &aria$le% $ased

on a&era!e conditions in

the select +one" This

&alue is used in e&ery

cell 'hen you press

;Initiali+e"

Page 58 of 81


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C;7;ti/

Wi0w r0!i64;! /6 9o/itor!

I 7


60/81


61/81

1.24 L+87 P+' F/ P7' S'0++0*+8

P7' S'0++0*+8

C"-%! *$/ *$"-; ,!"/; + !/$+! ''% "* "-,+-, !+/ + $$/,!+,%+#"=%. T% ''% +&%,%! .2 & +- $%-;,* &.

T% -$%, =%$", +z 1 &. C"-%! ,% =%$", ," #% "-,+-, "=%! ,%-$%, !"8%,"-.

T% *$/ %+/, -," ,% +%-, +,&"'%!% +, + '!%/!% "* 1

+,&. T+ Enery-!"" > Edit.... F"! -"&'!%#$% *$" ,% %-%!; %>/+,"- %"/'$% *!"& ,% "-,-/, +- &"&%-,/& %>/+,"-. W% -%% ," "$=% ,% %-%!;%>/+,"- "-$ * % +!% -,%!%,% - %,%!&--; ,% ,%&'%!+,/!% ,!#/,"-. W% $$ -",%+$ , ,%&'%!+,/!% - , %+&'$%. S" $%+=% ,%Enery E#uation%, ," "** +- $ 'luid > Create(Edit...

I- ,% Create(Edit Materials&%-/ %, ,%Density," 1 inlet > Edit...

N",% ,+, ,% +oundary Condition ,ype "/$ +=% #%%- +/,"&+,+$$ %, ,"velocity-inlet. N" ,% =%$", +, ,% #et$$ #% '%*%. I- ,% Velocity nlet&%-/ %,,% Velocity Speci"ication Met)od," Components +- %, ,%Axial-Velocity $m(s&," 1 &

T%- $< !," $"% ,% Velocity nlet&%-/.O6*'* B687- C8+*+8

F!, %$%, outet- ,%+oundary Conditions&%-/

A +- #% %%- - ,% &+;% +#"=% ,% ,ype"/$ +=% #%%- +/,"&+,+$$ %, ,"pressure-outlet. I* ,% ,ype -", %, ,"pressure-outlet ,%- %, , ,"pressure-outlet. N"-" */!,%! +-;% +!% -%%% *"! ,% outet#"/-+! "-,"-.

C'8*'7+8' B687- C8+*+8

S%$%, "e#te#e- ,%+oundary Conditions&%-/

A +- #% %%- - ,% &+;% +#"=% ,% ,ype+ #%%- +/,"&+,+$$ %, ," /all -", "!!%,. C+-;% ,% ,ype," axis

W%- ,% +$"; #"% +''%+! $< 0es," +-;% ,% #"/-+! ,'%. T%- $< !," +%', e%-,%!$-%e + ,% "-% -+&%.

Page 62 of 81
https://confluence.cornell.edu/display/SIMULATION/FLUENT+-+Laminar+Pipe+Flow+-+Problem+Specificationhttps://confluence.cornell.edu/display/SIMULATION/FLUENT+-+Laminar+Pipe+Flow+-+Problem+Specification


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P+' W B687- C8+*+8

F!, %$%,ppea- ,%+oundary Conditions&%-/

S"$/,"-

S%"- O!%! S%&%

A %"-8"!%! !%,+,"- %&% $$ #% /% ," +''!"&+,% ,% "$/,"-. I-"!%! ," &'$%&%-, ,% %"- "!%! %&% $< "- outo# eto$,%- $< "-oe#tu+- %$%, e"o# :e ;p#.

S%, I-,+$ G/%

H%!% ,% *$" *%$ $$ #% -,+$% ," ,% =+$/% +, ,% -$%,. I- "!%! ," +!! "/, ,%-,+$+,"- $< "-Solution nitiali1ation,%- $< "- Compute "rom+- %$%, inlet.

T%- $< ,%nitiali1e#/,,"- . T "&'$%,% ,% -,+$+,"-.

S'* C8,'7'80' C7+*'7+

FLUENT !%'"!, + !%/+$ *"! %+ ;"=%!--; %>/+,"- #%-; "$=%. T% !%/+$ + &%+/!% "* " %$$ ,% /!!%-, "$/,"- +,*% ,% !%,% *"!& "* %+ ;"=%!--;%>/+,"-. W%X$$ ,%!+,% ,% "$/,"- /-,$ ,% !%/+$ *"! %+ %>/+,"- *+$$ #%$" 1%86. I-"!%! ," '%* ,% !%/+$ !,%!+ $Clic%& Monitors > esiduals > Edit...

N%, +-;% ,% !%/+$ /-%! Converence Criterion*"! continuityx-velocity+-y-velocity +$$ ," 1%86

L+,$ $< !," $"% ,%esidual Monitors&%-/.

E%/,% C+$/$+,"-

P!"! ," !/---; ,% +$/$+,"- ,% &+&/& -/%! "* ,%!+,"- &/, #% %,. T"'%* ,% &+&/& -/%! "* ,%!+,"- $< "-un Calculation,%- %, ,%*um3er o"terations," 1.

Page 63 of 81


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1.25 F* P*' B687- L-'7

C"-%! + *$/ *$"-; +!" + *$+, '$+,% + $$/,!+,% +#"=%. O#,+- ,% =%$",

+- '!%/!% ,!#/,"- %- ,% R%-"$ -/%! #+% "- ,% '$+,% $%-;, 1.

T R%-"$ -/%! "#,+-% # /-; ,% *"$$"-; %,,-;.

o T% '$+,% $%-;, 1 &.

o T% -"&-; *$/ *$"-; - ,% 8!%,"- , + =%$", "* 1 &.

o T% %-, "* ,% *$/ 1


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T% %;, "* ,% "&+- $$ #% %, ," ,%- ,&% ,% #"/-+! $+%! ,


66/81

4.2 S'* C8,'7'80' C7+*'7+

FLUENT !%'"!, + !%/+$ *"! %+ ;"=%!--; %>/+,"- #%-; "$=%. T% !%/+$ + &%+/!% "* " %$$ ,% /!!%-, "$/,"- +,*% ,% !%,% *"!& "* %+ ;"=%!--;%>/+,"-. W%X$$ ,%!+,% ,% "$/,"- /-,$ ,% !%/+$ *"! %+ %>/+,"- *+$$ #%$" 1%86. I-

"!%! ," '%* ,% !%/+$ !,%!+

$Clic%& Monitors > esiduals > Edit....

N%, +-;% ,% !%/+$ /-%! Converence Criterion*"! continuityx-velocity+-y-velocity +$$ ," 1%86 + +- #% %%- #%$".

L+,$ $< !," $"% ,%esidual Monitors&%-/.

4.3 S'* I8+*+ G6'

H%!% ,% *$" *%$ $$ #% -,+$% ," ,% =+$/% +, ,% -$%,. T+, ,% -,+$ =+$/% "*+$$ ,% %$$ $$ #% %, ," 1 & +- P+ *"! =%$", +- ;+/;% '!%/!% !%'%,=%$. I-"!%! ," +!! "/, ,% -,+$+,"- $< "-Solution nitiali1ation,%- $< "- Compute"rom+- %$%, inlet+ "- #%$".

5. R'6*

5.1O/,$%, 0%$", P!"*$%

I- , %,"- % $$ *!, '$", ,% =+!+,"- "* ,% "&'"-%-, "* ,% =%$", +$"-; ,%"/,$%,. T%- % $$ '$", ,% V$+/ "$/,"- ," %% " ,% -/&%!+$ "$/,"- "&'+!%. I-

"!%! ," ,+!, ,% '!"% $Clic%& esults > 5lots > 70 5lot... > Set 6p.. + "- #%$".

Page 66 of 81


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W%X$$ !%+,% + ,!+;, =%!,+$ $-% *!"& ()(.5) ," (11)(.5.5). S%$%, Line,ool /-%! !ptions. E-,%! x80.5 y80x40.5 y40.5. E-,%! line1 /-%! *e/Sur"ace *ame. Y"/!Line(a%e Sur"ace&%-/ "/$ $""< %+,$ $


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1.2! S*'- F/ P* C-+8'7

P7' S'0++0*+8

C"-%! ,% ,%+ ,+,% +% "* + *$/ *$"-; '+, + $-%! + $$/,!+,% +#"=%.O#,+- ,% =%$", +- '!%/!% ,!#/,"- %- ,% R%-"$ -/%! "%- ,"#% 2.

I- "!%! ," &'$* ,% "&'/,+,"- ,% +&%,%! "* ,% ''% %, ," 1 & ,% "&'"-%-, "* ,% =%$", %, ," 1 & +- ,% %-, "* ,% *$/ %, ," 1/%,"-. W% &/,%,%!&-% +, "/! "$/,"- "&+- +- +, ,% #"/-+! "-,"- +!%.

S6*+8 D+8

F"! +- %,%!-+$ *$" '!"#$%& $


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