Sigma transformation and ALE formulation for three-dimensional free surface flows

HAL Id: hal-00275463

https://hal.archives-ouvertes.fr/hal-00275463

Preprint submitted on 25 Apr 2008

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Sigma transformation and ALE formulation for three-dimensional free surface flows

Astrid Decoene, Jean-Frédéric Gerbeau

To cite this version:

Astrid Decoene, Jean-Frédéric Gerbeau. Sigma transformation and ALE formulation for three-

dimensional free surface flows. 2008. �hal-00275463�

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z = 0

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‘>1$'BC!#/w^m-301A1!#&9/:-31AL>-3v/:-31A$'&U!#^6>9l^%-34!#21$w^m-3<-3B!#1A1$w+t&<$:$,0&M+}!3/:$[R K4!3AAL>-32]

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01<$:A

z

!3l(*$'6>68!30&-301`$'&&<-3&<=R“!#G9n/:-3B{"!#&<-31"!=EF$D^m$:$'pBC!3A${^m$'K(T$:$'<-3m-3?[&U!#G9WI}+.-36>6-(5>?

9L<$'BDD!#1A1$

z

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-=EF$'&MO"-(&<-/:$:<<$:Q!#1Ay^m-3<-3B 67!=9F$'&@AL9W"!#B{/:R

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BD-[*/:-3B{BD-36>9C01<$:A

σ

(5>6>62&<$J+}$'&5<-C!351$l/'67!3</=!#6 >?[BC!v&U!#1M+}-3&BZR

,.-, =89; 478 ; 4 2 6 2 4 6 6!4 ! 2 #"%$& 8'(

81$/'67!3</=!#6

σ

σ

σ (x, y, z, t) = z − b (x, y, t)

h (x, y, t) .

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b _h (x, y, t)

h _h (x, y, t)

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ω

-#+H<01/<’!4BC!#>?1R

Ω Ω(t)

1

0

η (x,y,t)

b(x,y,t)

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<-v1$l(T!#<$'&xA$'y+.01/'-3

J ˆ _t = ∂z

∂ˆ z = h (x, y, t) .

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z

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<>?[BC!†IK/:-W-3&<AL>"!#<$;&<$:<0&<$l?[&U!3AL$'WX$'&&<-3&=R

, - = 4 8 2 476 = :<2 = 2 = 2 <4>=6 = 2 2 2

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z

(5&><$:

g ∇ ₂ ( Z η

z

∆ρ ρ 0

dδ),

(51$'&<$

∆ρ

ρ ₀

∇ ₂

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!3,m-3>G<$:Aˆ-30T^G9 z$'6>6-3&’æBòñ7ÿ

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g ∇ ₂ ( Z _η

z

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3

> /=!3<$:v(51$'&<$ˆ&<-3?p&U!#"/=!#-31C!#&<$‰&<$:<$'WZh„ƒ #kKR z-3&<$:-=EF$'&=]w1$'9!#%$=!#&D$'EF$'(51$' 1$

!#&U$x+t&<-3B1$P>G<$'&m-367!#-3’$'&&<-3&<@-3‰1$P^01-9G!#1/'9ˆ<$'&B•h„ƒ ]ƒLiJk2>1$'&<$'GQ<-v1$ 1><$$'6$'BD$'W

BD$'1-AsR;n-3&<A$'&P<-y/'67!#&r+t9ZPBC!#<$'&=]26$'x01lA$'EF$'6-3n1$v/=!3<$4(51$'&<$v1${A$'1>9 "$'6A“E[!#&$:

VW0"!3AL&U!#/=!#6>6>9y>y1$lEF$'&/=!#6)AL>&<$:/'-3z-36>9F]L+}-3&5>1U!#1/:$

∆ρ (x, y, z, t) = a z ² + b

Ω _t × I,

(51$'&<$

a

b

∇ ₂ ( Z η

z

∆ρ

ρ ₀ dδ) = 0 .

\x<0BD$1-=( "!#Q1$;A$'1>9yE3!#&7!#-31P!#&<$>W<$'&%-367!#<$:A’^G9 1><$4$'6$'BD$'GX+t01/'-31@(5/<n!#&<$

$:/:$'(5U$P6>>1$=!#&5>‰1$xEF$'&/=!#62AL>&<$:/'-32Rw\881$PAL</'&<$'<$P6$'EF$'6N]L01>?v1$x&U!#m$'‹:>0B &06$@+}-3&81$

EF$'&/=!#6*>G<$'?[&U!#-3gd(5/<o$JY!3/'+.-3&l6>>1$=!#&l+.01/'-31¬g]H!#1ApA$'1-3>?d^W9

N _l

6$'EF$'6@>y1$lBD$:2](T$l1$'Z"!=EF$l"!#

∇ ₂ ( Z η

z

∆ρ _h

ρ ₀ dz) = 1 ρ ₀ ∇ ₂

N

−1

X

k=1

∆z _k+1/2 ∆ρ (z _k ) + ∆ρ (z _k+1 ) 2

!

= 1 ρ ₀ ∇ ₂

N

−1

X

k=1

∆z _k+1/2 a (z _k ² + z _k+1 ² ) + 2 b 2

!

= 1 ρ ₀

N

−1

X

k=1

a (z _k ² + z ² _k+1 ) + 2 b

2 ∇ ₂ ∆z _k+1/2

! .

(51$'&<$

z _k

k

∆z _k+1/2 = z _k+1 − z _k

k = 1, . . . , N _l

‹=-31$:H(51$'&<$81$T?[&Av$'6$'BD$'G<*!#&<$*1-3LIK-3&1-3?F-3"!#6N]G1$T1-3&>‹=-3GU!#61?[&U!3AL$'Ww-#+"1$8A$'

∆z _k+1/2

^m$'K(T$:$'y1$x67!:9F$'&<8T1-3*‹=$'&<-R,Ky!3AAL>-32]L^%$:/=!#01U$X-#+)1$X>G<$'&m-367!#-3d$'&&<-3&5-3

∆ρ

&-301|G0BD$'&/=!#6A$'1<>K9/'0&&<$'G<R 8ˆ!3/'<C6>>‘F$“!#Š!#&"/'7!#6l<-30&</:$d<$'&BZ]Q/'&<$=!#>?<0&-301

EF$'6-/'>$:=]s!#1Azr+w1$v$'&&<-3&xX>B{m-3&U!#G=]m6>>‘F$4>z<"!#6>6-=(!#&<$=!3x(5>“<<$:$'’^m-3<-3B 6-3%$:]">P/=!#

AL<0&^Z1$&U!#"/=!#-31x!#1AZ$'EF$'Z1$P+.&<$:$l0&M+N!3/:$[R

_2$'ˆ01C1-( >6>6>01&U!#<$z|G0BD$'&/=!#6l1$'1-3BD$'1-3 &<-30?[€<-3BD$’&<$:06><|-3^U!#>1$:A -3Š1$

<>B{6$v<$:l/=!3<$v-#+8!‰U!#1AL>?ˆ^"!3>n01>?ˆ1$v)$'6$'BC!3/JI

3

!#?[067!#&x^"!3>Z(5>“!v6$'?[’-#+

500

100

−25

−50

> >?[0&<$ 1RD"!3l^m$:$'-3^U!#>1$:Ap01>?d1$D/'67!3</=!#6,<>?[BC!ˆ&U!#1M+}-3&BC!#-3!#1An>6>>?’0p1$

(*-#IKAL>BD$'1-3"!#6H?[&Ad!#

11

y = 50

81$;^"!3>Z16>6$:Ad(5>’!4O10A’/:-3GU!#>>?y!#’!3/'>EF$l&U!3/:$'&

T

ρ

1$6>>1$=!#&@U!#<$$:VW0"!#-3

ρ = ρ ₀ + 0.749979 c,

(51$'&<$

c

T

^W9zU$'>?’!‰"!#&/'067!#&>>7!#6,AL&>^0-3o-#+T1$v&U!3/:$'&=R p$vBC!#‘F${1$v&<$'67!#>EF$CA$'1>K9

∆ρ/ρ ₀

X

0 200

400 600

Y 0

50 100

Z

-8 -6 -4 -2 0

X Y

Z

x

0 100 200 300 400 y 500

0 50

100

z

-40 -20 0

X Y

Z

>?[0&<$ w81$l>>7!#6 j3Œ BD$:2R

z(m)

-200 -150 -100 -50 0

x(m)

0 100 200 300 400 500

f>?[0&<$ w$'&/=!#6 /'&<-[<IK<$:/'-3d-#+H1$P>>7!#6fj3ΠBD$:d!#

y = 50

>’1$v $'6$'BC!3/JI

3

1$vO10Anl‹=$'&<-RCK!3AAL>-32] 1${ALrq%01<>EW>9“-#+81$v&U!3/:$'&;1$'?[6$:/'<$:A!#1An1-Z$JYL<$'&-3&;<-30&</:$

<$'&B !#6>$:AsR`81$D!#"!#6>9/=!#6w<-36>0-3p-#+TP<$:PPG01/'6$=!#&;!#1A“>B{6$x1$4+.&<$:$4<0&M+}!3/:$

!#1Ay1$lA$'1>9 "$'6AZ&<$'BC!#>d01AL0&^m$:As]"!#1Ay1$xO10AyEF$'6-/'>K9ˆ5‹=$'&<-R

p$P$'B{"!3>‹=$P-3ˆ1$@+N!3/'T"!#T8<$:<*T&<$'6$'E[!#G8+}-3&8!#6>/=!#-315>ˆW9AL&U!#06>/:=R,K|+}!3/'=]>

/=!#Z^m$<$:$'’!3x!{>B{6>"$:Ad/:-3@1?[0&U!#-3n-#+w!v67!#‘F$;-3&x!#’-/:$=!#’!#@$:VW0>6>>^&>0BZRT\X1Ay>XQ$:<<$'LI

7!#6s"!#Q1$PG0BD$'&/=!#6f</<1$'BD$&<$:<$'&EF$:QQ$:VW0>6>>^&>0BZ]>d-3&<A$'&5<-D^%$l!#^6$l<-{>B`067!#<$lBD-3&<$

/:-3B{6$JY|!#1AC&<$=!#6></@/=!3<$:=RH8*<$:*/=!3U$Q,<>B{>67!#&,<-1$X(*$'6>6rIN‘1-=(5ˆA$=!#6>>‹=$:A‰<$=!†INBD-30GT<$:

/=!3<$Dh>i[] L]ƒ 3k)01<$:Ay^W9‰BC!#W9d!#01-3&<Q(51$'zA$=!#6>>?C(5>Z1$;1-3&>‹=-3GU!#6f&<$:<0&<$l?[&U!3AL$'WX$'&&<-3&=R

,X-(*$'EF$'&=]*1$ˆ<$=!†INBD-30G{<$:v>WEF-36>EF$:D!dEF$'&9<$:$'<-3%-3?[&U!#W9FR p$ˆ1-( (5>-30&v<$:v"!#

>B{%-3&U!#WX$'&&<-3&<X/=!#d!#6<-C!#m$=!#&5>Z1$l&<$:<$'1/:$-#+HB{>6A$'&Q^m-3<-3B ?[&U!3AL$'G<R

f>&<=][(T$Q<$'*!PA$'1>9 "$'6A{E3!#&9>?l6>>1$=!#&6>9v>{1$5EF$'&/=!#6AL>&<$:/'-3|-36>9FRb81$8>>7!#6A$'1>9

AL&>^0-32]s!3@(T$'6>6w!3@1$4AL&>^0-3n!†+t<$'&

100

A$'1>K9zl>W<$'&%-367!#<$:An6>>1$=!#&6>9F])1$v1-3&>‹=-3GU!#6*&<$:<0&<$v?[&U!3AL$'G`l/:-3B{0<$:Ap$JY1!3/'6>9FR{K1A$:$:As]

!3Q$JYL%$:/'<$:As]1$l<9<$'B &<$'BC!#>1@01/<"!#?F$:A“!#1Aˆ1-D0&-301@EF$'6-W/'>$:@!#&<$;A$'<$:/'<$:AsR

81$P>>7!#6m&U!3/:$'&QAL&>^0-3ZT1$'yBD-AL"$:Ay<-;"!#51$PA$'1>93"$'6A‰E[!#&$:5VG0"!3AL&U!#/=!#6>6>9

f>?[0&<$d… ˆ>7!#656>>1$=!#&CA$'1>9AL&>^0-3Š!#1AAL&>^0-3~-3^U!#>1$:A~!#

t = 100

y = 50

>W<$'&%-367!#-32]!#1ADT6$=!3AT<-`!#ˆ$'&&<-3&*>‰1$X/:-3B{0U!#-3y-#+21$X1-3&>‹=-3GU!#6s&<$:<<0&<$Q?[&U!3AL$'G=R

81$D>>7!#6wA$'1>K9nAL&>^0-32]f!3P(*$'6>6*!3l1${AL<&>^0-3o!†+t<$'&

100

f>?[0&<$@ŽRf81$ , ~$'&&<-3&b"!3H1-x>B{"!3/'w-3v1$8A$'1>K9vAL&>^0-32][^0b>H&<-EF-3‘F$:b1$8/'&<$=!#-3

f>?[0&<$vŽ ,>7!#6)VG0"!3AL&U!#/`A$'1>K9yAL&>^0-3n!#1AZAL&>^0-3z-3^U!#>1$:Az!#

t = 100

y = 50

-#+b0&-301@EF$'6-W/'>$:=R p$l/=!#d$'E3!#6>0"!#<$1$$'&&<-3&

e _h

1$;0&-301QEF$'6-/'>K9

e _h = ||U _h || .

f>?[0&<$ 41-(Q51$l&<- 16$l-#+H5$'&&<-3&

e _h

x = 100

1$4$'&&<-3&P@1$`BD-[P>?["/=!#G=RP81$`BC!†Y>BC!#6fE[!#6>01$:x-#+w1$4$'&&<-3&l!#&<$`-3^U!#>1$:An!#x1$;^m-3<-3B

!#1A’!#Q1$x+t&<$:$l<0&M+}!3/:$-#+H1$l^"!3>2]L(51$'&<$>Q&<$=!3/<1$:

2.04 10 ⁻³

81$|0&-3014EF$'6-/'>$:4>1/'&<$=!3<$|<>?["/=!#G6>9(51$' !Z<&U!#"$:AoA$'1>9 "$'6Ao`<$'=R’Ko1$

1$JYL,<$:<=]

ρ

>1<U!#1/:${>p&<$:<$'1/:$D-#+Q!ˆ1-3&>‹=-3WU!#6,<$'B{m$'&U!#0&<$C-3&U!#6>>>9“<&U!#"/=!#-32R|81${>>7!#6*A$'1>K9

<&U!#"/=!#-3{)&<$:<$'W<$:A46$J+tf> f>?[0&<$8X!#1A`1$TA$'1>K94AL&>^0-3{-3^U!#>1$:Av!#

t = 100

, $'&&<-3&8/'&<$=!#<$:867!#&?F$'&50&-3018EF$'6-W/'>$:8(5/Uˆ%$'&0&^ˆ1$PA$'1>K9‰&U!#"/=!#-32R f>?[0&<$vi

&<$:U$'G<w1$8&<- 16$@-#+%1$5$'&&<-3&

e _h

x = 100

0, 12

−400 −35 −30 −25 −20 −15 −10 −5 0 0.2

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2x 10⁻³

depth [m]

error eh [m/s]

linear density quadratic density

>?[0&<$ b&<- 16$-#+H1$$'&&<-3&

e _h

x = 100

t = 100

f>?[0&<$* 2>7!#6WA$'1>9&U!#"/=!#-3{!#1A;A$'1>9AL&>^0-3v!#)

= 100

!#

y = 50

K+2(*$X?F-+.0&1$'&*(5>‰1$@<>B;067!#-32]L(*$X1-3/:$x"!#*1$x9L<$'B <$'1A*<-v!l1$'( <$=!3AL9CU!#<$[]

>’(5/U“1$;&U!3/:$'&PAL&>^0-3“^%$:/:-3BD$:PBD--32]%<-C"!#x1$ , $'&&<-3&lA$:/'&<$=!3<$:=R f>?[0&<$|i[i

1-=(Q@1$P>BD$$'EF-36>0-3’-#+H1$$'&&<-3&

E _h (t)

L ²

E _h (t) = Z

Ω(t)

||U || ²

! 1/2

.

81$`$'&&<-3&l-[</'>6>67!#<$:]m!3l!|/:-31<$:VW01$'1/:$4-#+*1$`$'EF-36>0-3p-#+w1$vA$'1>9d&U!#"/=!#-32])!#1Az<$:$'BD

<-|/:-3WEF$'&?F$4<-‰!C1-3’‹=$'&<-|E[!#6>01$[R f>?[0&<$ˆi=ƒC1-(Qx1$`A$'1>9dAL&>^0-3n-3^U!#>1$:An!†+t<$'&

1000

<$:/:-31A=Rb\xw(T$X/=!#ˆ<$:$[]W1$Q1-3&>‹=-3GU!#62<&U!#"/=!#-3‰,1-=(!#6>BD-[TBD-W-32Rb81$X?[&U!3AL$'G5-#+21$

A$'1>K9’E3!#&7!#-3“"!3x1$'&<$J+}-3&<$4A$:/'&<$=!3U$:As])!#1A’U-C"!3l!#6<-‰1$4$'&&<-3&x>“1$4/:-3B{0U!#-3-#+,1$

1-3&>‹=-3GU!#6T&<$:<0&<$v?[&U!3AL$'W=R|\x;!y/:-31<$:VW01$'1/:$[] 1$D0&-301lEF$'6-/'>$:`!#&<$D<BC!#6>6$'&=]f!3<1-=(5

> 1?[0&<$‰i=jLRQ81$`BC!†Y>BC!#6fE3!#6>01$`-#+w1$6-/=!#6 $'&&<-3&

e _h

6. 10 ⁻²

A$'1>K9yAL&>^0-3zA-W$:81-3@^%$:/:-3BD$l6>>1$=!#&=]11$$'&&<-3&@A-$:51-3@/=!#1/:$'6NR

8{<$:D/=!3U$:D/'6$=!#&6>9&<$'EF$=!#651$d!#%$=!#&U!#1/:$d-#+l!# æWêWêŸé7êz(51$'~1$yA$'1>K9 "$'6A

−400 −35 −30 −25 −20 −15 −10 −5 0 0.02

0.04 0.06 0.08 0.1 0.12

depth [m]

error eh [m/s]

density stratification

f>?[0&<$yi b&<- 16${-#+,1${$'&&<-3&

e _h

x = 100

t = 100

(5>z!#Z>>7!#6)A$'1>9ˆ<&U!#"/=!#-32R

0 100 200 300 400 500 600 700 800 900 1000

0 0.5 1 1.5 2 2.5 3 3.5 4

t [s]

ERROR Eh

norm of the spurious horizontal velocity

f>?[0&<$Xi[i28>BD$*$'EF-36>0-34-#+1$,$'&&<-3&

E _h

>1/'&<$=!3U$:P(5>z1$4E[!#6>01$`-#+*1$`>>7!#6HEF$'&/=!#6bA$'1>9d?[&U!3AL$'G=R`K1A$:$:As]%1$4BC!†YL>BC!#6H<0&-301

EF$'6-/'>$:!#&<$4-3^U!#>1$:A’(51$'“1$4A$'1>K9 "$'6AdPAL</:-3W>G01-301=]2G01x(51$'“><XEF$'&/=!#6H?[&U!3AL$'W

Q>@1><$[R

Z1$l-31$'&Q"!#1As]"1$'9ˆ!#m$=!#&@$'EF$'Z(51$'d1$lG9LAL&<-[U!#//:-31<<$'1/'9ˆ/:-31AL>-3’

+.0616>6$:As]"!#1Ay1$'9y/=!#y^m$l>?["/=!#Gx-=EF$'&X^%-3<-3BD5(5>dB{>6Ay?[&U!3AL$'G<=R

SW>1/:$“1$z<&U!#"/=!#-31d!#&<$z1$=!#&6>9~1-3&>‹=-3GU!#6;>ŠBD-[<ˆ<>0"!#-31=]x!

z

A$'1<>K9o!#&<$‰$:<<$'W7!#6N]H`(T-306ALv^%$ˆ!yE[!#6>Ao-3-32Rz\X1-31$'&49L<$'B !:EF-3AL>?“1$C&<$:U0&<$

?[&U!3AL$'W{&<-3^6$'B `1$‰<-#IK/=!#6>6$:A<-3W9/'/ˆ/:--3&<AL>"!#<$‰<9<$'BZ]H(5/U01<$:;1$ˆA$'1>9o!341$

f>?[0&<$Di=ƒ wŒP$'1>9ˆAL<&>^0-3’!#Q

= 1000

y = 50

−400 −35 −30 −25 −20 −15 −10 −5 0

0.01 0.02 0.03 0.04 0.05 0.06

depth [m]

horizontal velocity [m/s]

density stratification

f>?[0&<$|i=j b&<- 16$4-#+w1$;$'&&<-3&

e _h

x = 100

t = 1000

(5>’!#y>>7!#6 A$'1<>K9y&U!#"/=!#-32R

EF$'&/=!#6@/:--3&<AL>"!#<$[RD*0{C9<<$'B v1-3‰!3A$:VG0"!#<$ˆ<-n01AL9^"!#&<-3&<-3/ˆO"-(QC-3&{(51$'1$'EF$'&

<-3%-3?[&U!#W9y67!:9L@!4BC!--3&Q&<-36$[R

81$'&<$J+}-3&<$[]2>B{m-3&U!#G;$Jqs-3&<P"!:EF${^m$:$'“>GEF$:<$:Ap>n1$4A$'EF$'6-3BD$'W`-#+TG0BD$'&/=!#6,BD$'1-A

>B{&<-=E>?C1$4!#&<-:YL>BC!#-3z-#+H1-3&>‹=-3GU!#6f?[&U!3AL$'G<=RT\‡>B{6$;!#1AZ<>?["/=!#Gx>B{&<-=EF$'BD$'Gx

-3^U!#>1$:AC^W9C0^&U!3/'>?41$Q1-3&>‹=-3WU!#6%BD$=!#‰E[!#6>01$5+.&<-3B 1$P!3/'0"!#6%A$'1>9 "$'6AC^m$J+}-3&<$@/:-3B{0MI

>?y1$ , +.-3&</:$yh„ƒ[j3kKR4\+t0&1$'&l>B{&<-EF$'BD$'G4&<$:06><P+.&<-3B 1$v01<${-#+*1$C<-#IK/=!#6>6$:AnA$'1>K9WI

G!3/:-3^7!#Z+}-3&B;067!#-3z-#+ 1$l&<$:U0&<$P?[&U!3AL$'Gvc.<$:${h„ƒ#k !#1Anh„j #kteJRb5Q-3^U!#>1$:Ay^W9y!#6>9>?D1$

_2$'>^>‹;&06$P<-D1$l^"!#&<-W/'6>>/"!#&@-#+f1$^01-=9G!#1/'9ˆ<$'&B cFe

g ∇ ₂ Z η

z

∆ρ ρ ₀ dδ

= g Z _η

z

∇ ₂ ∆ρ

ρ ₀

dδ + g ∆ρ

ρ ₀ ∇ ₂ η .

81$yA$'1>K9oO101/'0"!#-3<$'&B DALrqs$'&<$'G7!#<$:A1&<C!#1A>G<$'?[&U!#<$:A~!†+.<$'&=]w(5"!#‰!#6>6-=(QC!’BD-3&<$

!3/:/'0&U!#<$l!#&<-:YL>BC!#-32R 1-3&8>1U!#1/:$[]r+)1$xA$'1>K93"$'6ACE[!#&$:T6>>1$=!#&6>9F]L><TEF$'&/=!#6%>W<$'?[&U!#-3

&<$:06><>!y<$:/:-31AIK-3&<A$'&4<$'&BZR|8W01=]b!y<$:/:-31AIK-3&<A$'&{!3/:/'0&U!#<$CALrqs$'&<$'1/'>?dBD$'1-Ap"!3<-Z^%$

01<$:Ay<-D!3/U$'EF$`!#d$JY1!3/'@/:-3B{0U!#-3’-#+b1$l?[&U!3AL$'W@01>?C1$l$JY&<$:<<-3Z6$J+t@>cMi=ƒFeJR Z1$

>W<$'?[&U!#-3 BD$'1-WA{/=!#^m$C01<$:AsRz84<$:/<VW01$‰"!3`^m$:$'/:-3GE>1/'>?“+.-3&41$B1><$ˆ$'6$'BD$'W

<9<$'B )$'6$'BC!3/JI

3

∆ρ

1-3LIN6>>1$=!#&`-3&AL</:-3G>W01-301`A$'1>9 "$'6A]s$'&&<-3&<>6>6,!#m$=!#&=R 1$'&l<$:/UVG01$:`/:-31P>n>BvI

6$'BD$'W>?>?[1$'&MIK-3&<A$'&DBD$'1-A{<-“$:>BC!#<$y1$‰&<$:<<0&<$ˆ?[&U!3AL$'GD<$'&B c.$[R?1RŠh>iJkX!#1A€h„ƒ[ƒ†kteJ]

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Ω ˆ

Ω _t

A ˆ _t : ˆ Ω −→ Ω _t , x(ˆ x, t) = ˆ A _t (ˆ x)

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x = ˆ x, y = ˆ y

z = Z (x, y, z, t) ˆ ,

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Z (x, y, z, t) ˆ

∂Z/∂ˆ z ≥ 0

Z (x, y, 0, t) = η(x, y, t), ∀ (x, y, t) ∈ ω × I, Z (x, y, 1, t) = b(x, y, t).

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c = (0, 0, c) ^T

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