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A direct method of nding the 
ow potential for inverse boundary-value problems, including for cavity 
ows is proposed.
The method is based on the theory of functions of complex variable and make it possible to nd the functions of complex
velocity and the derivative of the complex potential from the values of their modulus an argument at the boundary of the

ow region. The method has been evaluated when solving the problem on the unsteady cavity 
ow around a wedge (a
plate as a special case).
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ˆ­â¥£à¨àãï ¢ëà ¦¥­¨¥ (2), ¬®¦­® ¯®«ãç¨âì
¢ëà ¦¥­¨¥ ª®¬¯«¥ªá­®£® ¯®â¥­æ¨ « 
W(u;t) = N(t)
Z
uexp
2
4
1

1
Z
0
@
@
ln(u
2
+
2
)d
3
5
du:
(7)
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ˆ­â¥£à « Š®è¨-‹ £à ­¦ , § ¯¨á ­­ë©¤«ïâ®ç-
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á⨠á ãç¥â®¬ £à ¢¨â æ¨¨ g ¨ ¯®¢¥àå­®áâ­®£® ­ -
â殮­¨ï  ¨¬¥¥â ¢¨¤
@
@t
+
v
2
2
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p
c

+gy BnZr

R
=
v
2
0
2
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p
c

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£¤¥ R { à ¤¨ãá ªà¨¢¨§­ë ᢮¡®¤­®© ¯®¢¥àå­®áâ¨;
y { ª®®à¤¨­ â  ¢ ­ ¯à ¢«¥­¨¨, ¯à®â¨¢®¯®«®¦­®¬
£à ¢¨â æ¨¨ g; p
c
{ ¤ ¢«¥­¨¥ ¢ ª ¢¥à­¥;  { ¯«®â-
­®áâì.
®â¥­æ¨ « ¢ â®çª¥ O ¯à¨­¨¬ ¥âáï à ¢­ë¬ ­ã-
«î, ¯®í⮬ã
@
@t
j
u=0
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Ž¯à¥¤¥«ïï ¨§ ¢ëà ¦¥­¨ï (7)
(;t) = Re[W(u;t)]
u=i
¨ ¤¨ää¥à¥­æ¨àãï ¯®«ã祭­®¥ ¢ëà ¦¥­¨¥ ¯® ¢à¥-
¬¥­¨, ¬®¦­® ¯®«ãç¨âì ¯à®¨§¢®¤­ãî ¯®â¥­æ¨ « 
¯® ¢à¥¬¥­¨, ¢å®¤ïéãî ¢ ¤¨­ ¬¨ç¥áª®¥ £à ­¨ç­®¥
ãá«®¢¨¥:
@
@t
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N
(;t)BnZr
N(t)


Z
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0
ln(
02
BnZr
2
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(9)
exp
2
4
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Z
0
@
@
0
ln(
02
BnZr
2
)d
0
3
5
d;
£¤¥ â®çª  ­ ¤ ¯¥à¥¬¥­­®© ®¡®§­ ç ¥â ¯à®¨§¢®¤-
­ãî ¯® ¢à¥¬¥­¨.
„¨­ ¬¨ç¥áª®¥ £à ­¨ç­®¥ ãá«®¢¨¥ (8) á ãç¥â®¬
¢ëà ¦¥­¨ï (9) ¯®§¢®«ï¥â®¯à¥¤¥«¨âì äã­ªæ¨î ¬®-
¤ã«ï ᪮à®á⨠­  ᢮¡®¤­®© £à ­¨æ¥, ¢å®¤ïéãî ¢
¢ëà ¦¥­¨¥ ª®¬¯«¥ªá­®© ᪮à®á⨠(1):
v
2
= v
2
0
BnZr2
@
@t
BnZr2
pBnZrp
c

: (10)
96 ž. €. ‘¥¬¥­®¢
ISSN 1561-9087 à¨ª« ¤­  £÷¤à®¬¥å ­÷ª . 2000. ’®¬ 2 (74), N 3. ‘. 94{99
„ ¢«¥­¨¥ ­  ª®­âãॠª ¢¥à­ë p = p
á
, ­  ª®­âãà¥
§ ¬ëª ­¨ï äã­ªæ¨ï p = p[s(;t)] ®¯à¥¤¥«ï¥âáï ¨§
à áç¥â  ­¥áâ æ¨®­ à­®£® âãà¡ã«¥­â­®£® â¥ç¥­¨ï
¢ á«¥¤¥ §  ª ¢¥à­®©. „¨ää¥à¥­æ¨àãï ¢ëà ¦¥­¨¥
(10) ¯® ª®®à¤¨­ â¥  á ãç¥â®¬
ds
d
=




dz
du




u=i
;
¬®¦­® ¯®«ãç¨âì ¨­â¥£à®-¤¨ää¥à¥­æ¨ «ì­®¥ ãà ¢-
­¥­¨¥ ¤«ï ®¯à¥¤¥«¥­¨ï ä㭪樨 v():
@v
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v
exp
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4
1
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0
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@
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lnj
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BnZr
2
jd
0
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N

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:
Šˆ…Œ€’ˆ—…‘ŠŽ…ƒ€ˆ—Ž…
“‘‹Ž‚ˆ…
 áᬮâਬ í«¥¬¥­â dz ᢮¡®¤­®© £à ­¨æë.
“£®« ­ ª«®­  ᢮¡®¤­®© £à ­¨æë ª ®á¨ X ¢ â¥-
ªã騩 ¬®¬¥­â ¢à¥¬¥­¨ ¥áâì arg(dz). ˆ§¬¥­¥­¨¥
㣫  ­ ª«®­  £à ­¨æë ¢ ­¥ª®â®à®© â®çª¥ ¢ë§¢ -
­® ¨§¬¥­¥­¨¥¬ ­®à¬ «ì­®© ª®¬¯®­¥­âë ᪮à®áâ¨
¢¤®«ì £à ­¨æë, ¯®í⮬㠬®¦­® § ¯¨á âì á«¥¤ãî-
饥 ãà ¢­¥­¨¥:
@[arg(dz)]
@t
=
dv
n
ds
; (11)
£¤¥ @s = jdzj.
„«ï ­¥áâ æ¨®­ à­ëå ª ¢¨â æ¨®­­ëå â¥ç¥­¨©
᪮à®áâì ¤¢¨¦¥­¨ï £à ­¨æë áãé¥á⢥­­® ¬¥­ìè¥
᪮à®á⨠¦¨¤ª®á⨠­  £à ­¨æ¥, ¯®í⮬㠬®¦­® ¯®-
«®¦¨âì v
s
= v,   v
n
=BnZrv. “ç¨â뢠ï, çâ®
@s
@y
=




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u=i
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1
v




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¯à¥¤áâ ¢¨âì ¢ ¢¨¤¥
@[arg(dz)]
@t
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dW
du




u=i
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BnZr1

dv
n
d
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(12)
Ž¯à¥¤¥«ïï ¨§ ¢ëà ¦¥­¨ï (3) arg(dz) ­  ᢮¡®¤-
­®© £à ­¨æ¥ (u = i) ¨ ¯®¤áâ ¢«ïï ¢ ¢ëà ¦¥­¨¥
(12), ¯®«ã稬 ¨­â¥£à®-¤¨ää¥à¥­æ¨ «ì­®¥ ãà ¢­¥-
­¨¥ ¤«ï ä㭪樨 (;t):
@
@
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@ lnv
@
BnZr
1
v
2




dW
du




u=i
 (13)

0
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‚ëà ¦¥­¨ï (11), (14), ¯à¥¤áâ ¢«ïîâ ᮡ®© á¨-
á⥬㠤¢ãå ¨­â¥£à®-¤¨ää¥à¥­æ¨ «ì­ëå ãà ¢­¥-
­¨© ¤«ï ®¯à¥¤¥«¥­¨ï ä㭪権 _v(;t) ¨ ¯à¨ § -
¤ ­­ëå ­ ç «ì­ëå ãá«®¢¨ïå. „¨ää¥à¥­æ¨àãï ¯®
¢à¥¬¥­¨ ãá«®¢¨ï (4), (6), ¬®¦­® ®¯à¥¤¥«¨âì ¯à®-
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_
N ¨ _
c
. ’ ª¨¬ ®¡à §®¬,
§ ¤ ç  ᢮¤¨âáï ª á¨á⥬¥ ®¡ëç­ëå ¤¨ää¥à¥­-
樠«ì­ëå ãà ¢­¥­¨© ®â­®á¨â¥«ì­® ä㭪権 N(t)
¨ 
c
(t) ¯à¨ ¨­â¥£à¨à®¢ ­¨¨ ª®â®à®© ­  ª ¦¤®¬
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á¨â¥«ì­® _v(;t). ®«ã祭­ ï á¨á⥬  ¨­â¥£à®-
¤¨ää¥à¥­æ¨ «ì­ëå ãà ¢­¥­¨© ¬®¦¥â ¡ëâì à¥è¥-
­  ¯à¨ § ¤ ­­®© ä㭪樨 p(s;t) ¢¤®«ì ª®­âãà  § -
¬ëª ­¨ï ª ¢¥à­ë. â  äã­ªæ¨ï ¬®¦¥â ¡ëâì ­ ©-
¤¥­  ¨§ à¥è¥­¨ï § ¤ ç¨ ­¥áâ æ¨®­ à­®£® â¥ç¥­¨ï
¢ á«¥¤¥ §  ª ¢¥à­®© ¢ ¯à¨¡«¨¦¥­¨¨ ⥮ਨ ¯®£à -
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HECT€–ˆŽ€Ž…’…—…ˆ…‚ ‘‹…„…
’¥ç¥­¨ï ᮡࠧ®¢ ­¨¥¬¯ à®¢ë媠¢¥à­¢á« ¡®-
¢ï§ª¨å ¦¨¤ª®áâïå ¯® ᢮¥© ¯à¨à®¤¥ ïîâáï ­¥-
áâ æ¨®­ à­ë¬¨ [6, 7]. ¥áâ æ¨®­ à­®áâì â¥ç¥­¨ï
®¡ãá«®¢«¥­  ¯ã«ìá æ¨ï¬¨ ¦¨¤ª®á⨠¢ ®¡« á⨠§ -
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ïîâáï ॠ«ì­ë¥ ᢮©á⢠ ¦¨¤ª®áâ¨: ¢ï§ª®áâì,
¯®¢¥àå­®áâ­®¥ ­ â殮­¨¥, £ §®­ áë饭¨¥. ‚ ¤ ­-
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¤  ¢ ¢¨¤¥ § á⮩­®© ®¡« áâ¨, ¢ ª®â®à®© ᪮à®áâì
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X ¬®¦­® ¯à¥­¥¡à¥çì:

0
@v
x
@t
= BnZr
@p
@x
+
@
@y
; (14)
£¤¥ 
0
{®á।­¥­­ ï ¯®¯¥à¥ª á«¥¤  ¯«®â­®áâ줢ãå-
ä §­®© á।ë.
Š á â¥«ì­®¥ ­ ¯à殮­¨¥  ¢ ᮮ⢥âá⢨¨ á
¯¥à¢®© ¬®¤¥«ìî âãà¡ã«¥­â­®á⨠à ­¤â«ï ®¯à¥-
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£¤¥  { ª®íää¨æ¨¥­â âãà¡ã«¥­â­®áâ¨; v

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‘।­¥¥ §­ ç¥­¨¥ @=@y ¯®¯¥à¥ª á«¥¤  ®¯à¥¤¥-
«ï¥âáï ¢ëà ¦¥­¨¥¬
@
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;
£¤¥ 

{ ⮫騭  ¢ï§ª®£® á«®ï.
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¯à¨â®ª  ¦¨¤ª®á⨠¢ ª ¢¥à­ã «¨¡® §  áç¥â ®â⮪ 
¦¨¤ª®á⨠¨§ ª ¢¥à­ë. ®í⮬㠮á।­¥­­ ï ¯®¯¥-
४ á«¥¤  ᪮à®áâì v
x
á¢ï§ ­  á ¨§¬¥­¥­¨¥¬ ¤«¨­ë
ª ¢¥à­ë ¢ëà ¦¥­¨¥¬
v
x
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dl
c
dt
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l
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; (16)
£¤¥
l
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0
dz
du




u=i
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:
‘ ãç¥â®¬ ¢ëà ¦¥­¨© (15), (16) ãà ¢­¥­¨¥ ª®«¨ç¥-
á⢠ ¤¢¨¦¥­¨ï (14) ¬®¦­® § ¯¨á âì ¢ ¢¨¤¥

0

l
c
= BnZr
@p
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ˆ­â¥£à¨àãï ¢ëà ¦¥­¨¥ (17) ¢¤®«ì ®á¨ X á ãç¥-
⮬ £à ­¨ç­®£® ãá«®¢¨ï, ¯®«ãç ¥¬ ãà ¢­¥­¨¥ ¤«ï
à áç¥â  ¯à®¤®«ì­®£® £à ¤¨¥­â  ¤ ¢«¥­¨ï ¢ áâàã©-
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p(s;t) = 
0

s
Z
l
c
(v

(s
0
;t)BnZr
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l
c
)
2


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0
;t)
ds
0
BnZr
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(sBnZrl
c
)

l
c
: (18)
„«¨­  § á⮩­®© ®¡« á⨠®¯à¥¤¥«ï« áì á ãç¥â®¬
í¬¯¨à¨ç¥áª®£® ª®íää¨æ¨¥­â  ­ à áâ ­¨ï è¨à¨­ë
§®­ë ᬥ襭¨ï ¨§ ãá«®¢¨ï 
1
(l
z
) = 0, £¤¥ 
1
­ å®-
¤¨âáï ¨­â¥£à¨à®¢ ­¨¥¬ ãà ¢­¥­¨ï
d
1
dx
=
d

dx
BnZr
b
c
:
Ž…„…‹…ˆ…—ˆ‘‹€ Š€‚ˆ’€–ˆˆ„‹Ÿ
…‘’€–ˆŽ€ŽƒŽ’…—…ˆŸ
‚ ¤ ­­®¬ à¥è¥­¨¨ ¤ ¢«¥­¨¥ ­  ¡¥áª®­¥ç­®áâ¨
¨¬¥¥â ®á®¡¥­­®áâì ¯¥à¢®£® ¯®à浪 . â® á¢ï§ ­®
á ⥬, çâ® ¨§¬¥­¥­¨¥ ¯«®é ¤¨ ®¡« áâ¨, ®£à ­¨-
祭­®© ª®­âã஬ ª ¢¥à­ë ¨ ª®­âã஬ § ¬ëª ­¨ï
¡¥áª®­¥ç­®. ˆ­â¥£à « Š®è¨-‹ £à ­¦ , á¢ï§ë¢ -
î騩 ¯ à ¬¥âàë ­ ¡¥£ î饣® ¯®â®ª  ¨ ¯ à ¬¥-
âàë â¥ç¥­¨ï ¢ â®çª¥ O, ¨¬¥¥â ¢¨¤
@
@t




x=BnZr1
+
p
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2
1
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+
p
c

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v
2
0
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;
£¤¥ p
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{ ¤ ¢«¥­¨¥ ¨ ᪮à®áâì ¢ ­ ¡¥£ î饬
¯®â®ª¥; ¯®â¥­æ¨ « ¢ â®çª¥ O ¯à¨­ïâ (0;t) = 0,
á«¥¤®¢ â¥«ì­®,
@
@t




u=0
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‚á«¥¤á⢨¥ ®á®¡¥­­®á⨠¢ ¡¥áª®­¥ç­®áâ¨, ¢ ¤ ­-
­®© § ¤ ç¥ ç¨á«® ª ¢¨â æ¨¨ ®¯à¥¤¥«¥­® á ãç¥â®¬
¯à®¨§¢®¤­®© ¯®â¥­æ¨ «  ¯® ¢à¥¬¥­¨:
(t) =
p
1
+
@
@t
j
x=BnZr1
BnZrp
c
1
2
v
2
1
; (19)
⮣¤  ᪮à®áâì ¢ â®çª¥ O ¡ã¤¥â
v
0
(t) = v
1
p
1+ (t):
«®áª¨¥ â¥ç¥­¨ï ïîâáï ¨¤¥ «¨§ æ¨¥© ॠ«ì-
­®£® âà¥å¬¥à­®£® â¥ç¥­¨ï ¨¬®£ãâॠ«¨§®¢ë¢ âì-
áï ¢ ®£à ­¨ç¥­­®© ç á⨠¯à®áâà ­á⢠, £¤¥ íä-
䥪⠬¨ âà¥å¬¥à­®áâ¨, ᦨ¬ ¥¬®á⨠¦¨¤ª®á⨠¨
¯®¤ â«¨¢®áâìî £à ­¨æ ¯®â®ª  ¬®¦­® ¯à¥­¥¡à¥çì.
„«ïâ ª¨åâ¥ç¥­¨© ¯à¨§ ¤ ­­®¬¤ ¢«¥­¨¨ ­ ­¥ª®-
â®à®¬ à ááâ®ï­¨¨ ®â ⥫  p
1
ç¨á«® ª ¢¨â æ¨¨ ¡ã-
¤¥â ­ å®¤¨âáï ¢ ¯à®æ¥áᥠà¥è¥­¨ï § ¤ ç¨ ¨§ ãà ¢-
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p
1
(t)BnZrp
v
0;5v
2
1
+
2
v
2
1
@
@t
:
—¥¬ ¤ «ìè¥ ®â ¯« á⨭ª¨ á«¥¢  § ¤ ­® ¤ ¢«¥­¨¥
p
1
(t), ⥬ ¡®«ì訩 ¢ª« ¤ ¡ã¤¥â ¢­®á¨âì ¯à®¨§¢®¤-
­ ï ¯®â¥­æ¨ «  ¨ ⥬ ¬¥­ìè¥ ¡ã¤¥â ¨§¬¥­¥­¨¥ ç¨-
á«  ª ¢¨â æ¨¨ ,   á«¥¤®¢ â¥«ì­®, ᪮à®á⨠v
0
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â®çª¥ O. ”¨§¨ç¥áª¨ íâ® ®¡êïá­ï¥âáï ¨­¥à樮­-
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  à¨á. 2. ¯à¥¤áâ ¢«¥­ë § ¢¨á¨¬®á⨠ª®íää¨-
樥­â  ᮯà®â¨¢«¥­¨ï ¯« á⨭ª¨ ( = =2), ª®-
íää¨æ¨¥­â  ¤ ¢«¥­¨ï ¢ â®çª¥ H(C
H
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H
BnZr
p
c
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¢à¥¬¥­¨ t = T

v
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樨.
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t = 2 ª®íää¨æ¨¥­â ᮯà®â¨¢«¥­¨ï ¯à¨­¨¬ ¥â áâ -
樮­ à­®¥ §­ ç¥­¨¥. „«¨­  ª ¢¥à­ë ®áâ ¥âáï ¯®-
áâ®ï­­®© ¤«ï t < l
c
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0
¨ ­ ç¨­ ¥â 㬥­ìè âìáï,
ª®£¤  ¢®§¬ã饭¨¥ ᪮à®á⨠­  ᢮¡®¤­®© £à ­¨æ¥
98 ž. €. ‘¥¬¥­®¢
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u-
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