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Quantitative study of the motion of a rigid body in an ideal fluid
MECHANICS RESEARCH COMMUNICATIONS Vol. 17(4), 249-254, 1990. Printed in the USA. 0093-6413/90
$3.00 + .00
QUANTITATIVE
STUDY
Copyright (c)
OF THE MOTION
1990 Pergamon Press plc
O F A RIGID B O D Y
IN A N I D E A L F L U I D
G. BOUCHER Laboratoire de M~canique Th(~orique, Universit(," de Franche-Comt~ 250];0 Besan(;on C~dex, France
(Received 22 January 1990; accepted for print 10 April 1990)
Introduction
The equations of the motion of a rigid body in an ideal" incompressible fluid in an i r r o t a t i o n a l motion, at rest at i n f i n i t y , can be put in a simple from which shows their close analogy with the equations of a non-immersed rigid body. The numerical solution of these equations allows to visualize various trajectories, especially when the forces other than aerodynamic ones are g r a v i t a t i o n a l forces. In the f i r s t paragraph, we write the equations of the three-dimensional motion of rigid body and in the second one we study more p a r t i c u l a r l y the case of a plan profile without a sharp edge with circulation.
1. Equations of the motion
We consider the motion of a rigid body S, with mass M, to which is attached a coordinate
system
R
with
origin
O.
S moves
in a n
ideal
incompressible
fluid
.+
in an irrotational motion~ at rest at i n f i n i t y . We denote by
Vo the
velocity of
..).
0 a n d by co t h e a n g u l a r v e l o c i t y o f S. -4*
.+
If we consider that T = (R, Mp) represents the resultant vector and the resultant moment
about
P of a v e c t o r
field, the kinematic
f i e l d is v = (co, ,.~), t h e
field is Y: = ( M V G , ~o) and the dynamic field is A = (M~G, ~o), with tions. 249
kinetic
usual
nota-
250
G. BOUCHER
1.1. It is easy to show that there is a m a t r i x H, such t h a t
Z
: H(v)
; H=
f'l
M_G_~__M_I_) LJo -MG )
w h e r e Jo is the m a t r i x of i n e r t i a of S, 13 the unit m a t r i x and IS the a n t i s y m m e t r i c m a t r i x of the l i n e a r mapping u
~,
G(G) :
u
A OGo On
A = d____~_Z, and if we p r o j e c t in R, we have A : H ( v ) dt the square b r a c k e t s of L i e of x and y, [1]
1.2. The a e r o d y n a m i c forces
: We show t h a t
the
other
hand,
we
+ [v,H(v)]~ w h e r e
the k i n e t i c
energy of
have
Ix, y]
the
is
f l u i d is
T : 1/2 f I gr~d , 12 d'[ : 1/2 t v L * v w h e r e ~ is the v e l o c i t y p o t e n t i a l of the Fluid fluid~ and L * a s y m m e t r i c a l m a t r i x depending only on the g e o m e t r y of S. The a e r o d y n a m i c forces, are F : - d g
f_o, ] :-4 -13
with L : L*.
. [2]
withg
dt
=
Z L*..v.
i j:i
,j/
i : 1, ..., 6 so t h a t g :
L(v)
tJ31 o ) By
analogy with
i f we l e t
Q :
kinematics,
we
can w r i t e
F = -(L(¢1) + [ v , L ( v ) ] )
in R. F i n a l l y ,
H + L, the e q u a t i o n s of the m o t i o n of the body S are w r i t t e n •
Q(~) + [v,Q(v)] : K
(I)
w h e r e K represents the forces o t h e r than a e r o d y n a m i c a l ones. We can see t h a t
if
we denote by P
the
transition
matrix
from
the
frame
r e f e r e n c e R 1 to IR, and by x the p o s i t i o n of 0 in R1, we have x = P v o and P = P £
¢o
of ,
ql
-r
where
f~=| r 0 - P / , ( p , q , r ) being the c o m p o n e n t s of ~0 in R. Thus it is not k-q P 02 necessary to specify the p o s i t i o n - p a r a m e t e r s of the r i g i d body, in o r d e r to i n t e g r a t e the system (1) and to o b t a i n x and P. f_
Lastly, if Q :[A kL.
~1, A , B , C , D
being 4 (3 x 3 ) m a t r i c e s , w i t h B and C s y m m e t r i c a l ~
LJJ
the e q u a t i o n s reduce to : A
+ B¢o + coA (Aco + Bvo) = R e s u l t a n t of K
c~
+ Dec + c0A(Cco + Dye) - vo A (Arc + Bvo) = R e s u l t a n t m o m e n t of K.
1.3. Integrals and positions of equilibrium for (1) : * If K derives f r o m a p o t e n t i a l U, t h e n we have the i n t e g r a l of the k i n e t i c energy~ T If
total
=U+h.
K = 0~ we have :
* I A ~ + B:,,I *(A~
+
: C1
B~.).(C~
+
D~o)
= C 2
* In some p a r t i c u l a r cases~ t h e r e is a f o u r t h i n t e g r a l [3].
RIGID BODY IN IDEAL FLUID
251
The positions of e q u i l i b r i u m satisfy ~ = 0. I t must exist t w o reals :~ and P which
satisfy the system
(C- pl 3) ~ + (D + :kl3) Vo = 0 (A-
XI 3) ~ + B~o = O.
1.4. Particular cases ." * M o t i o n of E u l e r - P o i n s o t : 0 is fixed and the resultant moment of K is zero. The equation giving the r o t a t i o n is = Cco + co ACco -- 0. I t
is
the
same
as
in
the
absence of a fluid. *
Motion
of
Lagrange-Poisson
." The rigid
body
is d y n a m i c a l l y revolving about
the axis z (OG = ~ z ) . 0 is f i x e d and the solid is subjected to the g r a v i t y forces.
The equation giving the r o t a t i o n is _).
C
-y
-~
+ m h Cm = mgR,.Pz
-)*
A z
C is s y m m e t r i c a l . There are two matrices U and D, D diagonal so t h a t C = U -1 DU, and D•'
+ ~' A D ~ ' = m g ~ , U P ~
(~'
;
= LI~).
This is the equation corresponding to the motion of any rigid body w i t h a fixed point~ subjected
to
gravity
forces.
The e f f e c t
of
the presence of
the f l u i d is
to s h i f t the position of G.
2. Bi-dimensional case of a profile without a sharp edge
A
reasoning
analogous to
that
of
section
1 allows the d e r i v a t i o n of
a matrix
equation s i m i l a r to (1) but in three dimensions. We denote by (~,,m) the components -4~ of
--~
--)-
vo in R and by co = ~0 z 1 the r o t a t i o n ; then v = ( ~ , m~ co) and
Q =
M
MXGI
+ the m a t r i x from aerodynamical fo~'ces.
/
-MY G
MX G
lo
,J ÷ AR
We o b t a i n Q(~,) + [ v ~ Q ( v ) ] + 0 £ J ( v )
= K with now [v,T] =
.+
where
R
is
roAR the resultant the circulation
v e c t o r o f T. p I" 3(v) is I" is not e q u a l to z e r o :
the
additional
aerodynamical
force
when
252
G BOUCHER
0 3 =
-1 - XC
center,
1
X C "]
0
YCI
-Yc
' XC
and Y C are the c o o r d i n a t e s of the p r o f i l e
O
p the density of f l u i d .
C o u c h e t [/4] showed the e x i s t e n c e of a c o o r d i n a t e system
R in w h i c h the m a t r i x
Q is d i a g o n a l ; in this way we have the w e l l - k n o w n equations A~.
=
Bc0m - p £(m + c0XC) + X
Bfn
= -AR0J + p £(R - c0Yc) + Y
C~
= (A-B)
2.1. N u m e r i c a l
w i t h B >-- A > 0
and C > 0
(2)
Rm + p F ( R X C + m Y C) + Z
study when X = Y = Z = O :
The system (2) a d m i t s two i n t e g r a l s : AR 2 + Bm 2 + C J
- K1
(3)
A2R 2 + B2m 2 + 2pF (ARY C - B m X C + Cw)= K 2 The t r a j e c t o r y
of
(2) is the
intersection
(4)
of e l l i p s o i d (3) and the p a r a b o l o i d (4).
i t is closed, and hence it is p e r i o d i c a l . On an e l l i p s o i d w i t h an energy K 1> 0, we
can
have
2,
3, 4,
5 or 6 positions of
e q u i l i b r i u m for the system (2), d e f i n e d by Re
P ['O~eYc -
me
;
p[" _Ac~ e
PFU)eX C ;
we
=
coe
BoJe - p£
coe must s a t i s f y the e q u a t i o n P (coe) = AR e + B m e + Cw e - K 1 = 0. It is an e q u a t i o n equivalent
to
a polynomial
because P ( - / / ~ ) > 0 , v
C
of
P(0) < 0
degree
6 which
always at
least
two
real roots,
and p ( / ' K _ ~ ] ) > 0.
It is easy to see on a graph the stable positions of equilibrium
and the unstable
positions, fig (1).
2.2. Numerical
study when K represents
the 9ravity
forces :
X = - M g sin e, Y -- - M g cose, Z = - M g ( X G c o s 8 - YG sine),
with ~
de dt
Then we have the i n t e g r a l A (R---~Mg cos6)2 +B(m +-~-MgsinS)2 + Cc02 = 2Mg [(X C - X G) sine + ( Y c - YG ) c ° s 8] - (Mg) 2 ( A c o s 2 0 + B sin2O)+ h
P£
RIGID BODY IN IDEAL FLUID
253
L e t F(0) + h be the second member ; F is C co and 2~-periodical. We deduce t h a t the motion is possible only when 0 satisfies F(0) + h > 09 fig (2).
Positions of e q u i l i b r i u m f o r (2). m =-
~.:m:~=6:0
i m p l i e s 0 : 0 e , 6o = 0 , ~, = -Mg - cos 0e r #F There are at least two positions of e q u i l i b r i u m ,
nrMgs i n e e and F ' ( e e) = 0.
because-F'
always has two roots. One of these positions is stabler the s t a b i l i t y
being assured i f F " ( e e) < 0. [4]
2.3. Study of the motion around a stable equilibrium position : We choose h~ so t h a t F(0 e) + h is
small
and
we
choose
the
following
initial
conditions : eo = 0e + E, 4o = Mg cOS0o + (F (eo) + h)1/2 cos U2o cos % pF
6oo -- (F(eo) + h) 1/2 sin ~2o r
mo = - Mg sin 0o + iF (0o) + h) 1/2 cos d2o sin % , pF ~, ~°o, ~2o are a r b i t r a r y parameters.
By i n t e g r a t i n g the system (2)~ to which we add the equation
dO :- 6o, we
obtain the
curve t ÷ 0(t) and the t r a j e c t o r y of point 0r fig (3). Moreoverrwe can observe the v a r i a t i o n s of J~,r mr
6o. in the space (~,r m, 6o)rwecut
the t r a j e c t o r i e s w i t h the ellipsoid A(~, - ~,e)2 + B (m-me)2 + C 6o2 -- F(0 e) + h.
The
i n t e r s e c t i o n s are situated on polhodes s i m i l a r to the previous ones. fig ( 4 ) .
References 1.
D. C h e v a l l i e r and 3.M. H e l m e r . Annales des Ponts et Chauss~es l e r t r i m e s t r e . France (1984).
2.
N.E.
Kotchin,
I.A.
IKibel,
N.V.
Roze.
Theorical
hydromechanics.
Intersience
publishers (1964). 3.
V.N. Rubanovskii. PMM U.S.S.R., Vol. 52, n ° 3 (1988).
4.
G.
Couchet.
3oukowski. 5.
Les p r o f i l s Librairie
en aerodynamique
scientifique
et technique
P. Capodanno. Les #quations d'Appel d'un solide dans un fluide.
instationnaire Albert
et la condition
Blanchard.
et d'Euler-Lagrange
M a t e m a t i c a 21 (44)r 9-32 (1979).
de
Paris (1976).
pour le mouvement
254
G. BOUCHER
i
FIG. i
FIG. 2
Trajectories on the ellipsoid
Curve F(8)+h 8e position of equilibrium
L~
5.,, i¸ ,:
~ ......
.::..,.
ii~ /i,!i~ ....... I
FIG. 3
FIG. 4
Trajectory of point 0
Intersection trajectories with the ellipsoid
$3.00 + .00
QUANTITATIVE
STUDY
Copyright (c)
OF THE MOTION
1990 Pergamon Press plc
O F A RIGID B O D Y
IN A N I D E A L F L U I D
G. BOUCHER Laboratoire de M~canique Th(~orique, Universit(," de Franche-Comt~ 250];0 Besan(;on C~dex, France
(Received 22 January 1990; accepted for print 10 April 1990)
Introduction
The equations of the motion of a rigid body in an ideal" incompressible fluid in an i r r o t a t i o n a l motion, at rest at i n f i n i t y , can be put in a simple from which shows their close analogy with the equations of a non-immersed rigid body. The numerical solution of these equations allows to visualize various trajectories, especially when the forces other than aerodynamic ones are g r a v i t a t i o n a l forces. In the f i r s t paragraph, we write the equations of the three-dimensional motion of rigid body and in the second one we study more p a r t i c u l a r l y the case of a plan profile without a sharp edge with circulation.
1. Equations of the motion
We consider the motion of a rigid body S, with mass M, to which is attached a coordinate
system
R
with
origin
O.
S moves
in a n
ideal
incompressible
fluid
.+
in an irrotational motion~ at rest at i n f i n i t y . We denote by
Vo the
velocity of
..).
0 a n d by co t h e a n g u l a r v e l o c i t y o f S. -4*
.+
If we consider that T = (R, Mp) represents the resultant vector and the resultant moment
about
P of a v e c t o r
field, the kinematic
f i e l d is v = (co, ,.~), t h e
field is Y: = ( M V G , ~o) and the dynamic field is A = (M~G, ~o), with tions. 249
kinetic
usual
nota-
250
G. BOUCHER
1.1. It is easy to show that there is a m a t r i x H, such t h a t
Z
: H(v)
; H=
f'l
M_G_~__M_I_) LJo -MG )
w h e r e Jo is the m a t r i x of i n e r t i a of S, 13 the unit m a t r i x and IS the a n t i s y m m e t r i c m a t r i x of the l i n e a r mapping u
~,
G(G) :
u
A OGo On
A = d____~_Z, and if we p r o j e c t in R, we have A : H ( v ) dt the square b r a c k e t s of L i e of x and y, [1]
1.2. The a e r o d y n a m i c forces
: We show t h a t
the
other
hand,
we
+ [v,H(v)]~ w h e r e
the k i n e t i c
energy of
have
Ix, y]
the
is
f l u i d is
T : 1/2 f I gr~d , 12 d'[ : 1/2 t v L * v w h e r e ~ is the v e l o c i t y p o t e n t i a l of the Fluid fluid~ and L * a s y m m e t r i c a l m a t r i x depending only on the g e o m e t r y of S. The a e r o d y n a m i c forces, are F : - d g
f_o, ] :-4 -13
with L : L*.
. [2]
withg
dt
=
Z L*..v.
i j:i
,j/
i : 1, ..., 6 so t h a t g :
L(v)
tJ31 o ) By
analogy with
i f we l e t
Q :
kinematics,
we
can w r i t e
F = -(L(¢1) + [ v , L ( v ) ] )
in R. F i n a l l y ,
H + L, the e q u a t i o n s of the m o t i o n of the body S are w r i t t e n •
Q(~) + [v,Q(v)] : K
(I)
w h e r e K represents the forces o t h e r than a e r o d y n a m i c a l ones. We can see t h a t
if
we denote by P
the
transition
matrix
from
the
frame
r e f e r e n c e R 1 to IR, and by x the p o s i t i o n of 0 in R1, we have x = P v o and P = P £
¢o
of ,
ql
-r
where
f~=| r 0 - P / , ( p , q , r ) being the c o m p o n e n t s of ~0 in R. Thus it is not k-q P 02 necessary to specify the p o s i t i o n - p a r a m e t e r s of the r i g i d body, in o r d e r to i n t e g r a t e the system (1) and to o b t a i n x and P. f_
Lastly, if Q :[A kL.
~1, A , B , C , D
being 4 (3 x 3 ) m a t r i c e s , w i t h B and C s y m m e t r i c a l ~
LJJ
the e q u a t i o n s reduce to : A
+ B¢o + coA (Aco + Bvo) = R e s u l t a n t of K
c~
+ Dec + c0A(Cco + Dye) - vo A (Arc + Bvo) = R e s u l t a n t m o m e n t of K.
1.3. Integrals and positions of equilibrium for (1) : * If K derives f r o m a p o t e n t i a l U, t h e n we have the i n t e g r a l of the k i n e t i c energy~ T If
total
=U+h.
K = 0~ we have :
* I A ~ + B:,,I *(A~
+
: C1
B~.).(C~
+
D~o)
= C 2
* In some p a r t i c u l a r cases~ t h e r e is a f o u r t h i n t e g r a l [3].
RIGID BODY IN IDEAL FLUID
251
The positions of e q u i l i b r i u m satisfy ~ = 0. I t must exist t w o reals :~ and P which
satisfy the system
(C- pl 3) ~ + (D + :kl3) Vo = 0 (A-
XI 3) ~ + B~o = O.
1.4. Particular cases ." * M o t i o n of E u l e r - P o i n s o t : 0 is fixed and the resultant moment of K is zero. The equation giving the r o t a t i o n is = Cco + co ACco -- 0. I t
is
the
same
as
in
the
absence of a fluid. *
Motion
of
Lagrange-Poisson
." The rigid
body
is d y n a m i c a l l y revolving about
the axis z (OG = ~ z ) . 0 is f i x e d and the solid is subjected to the g r a v i t y forces.
The equation giving the r o t a t i o n is _).
C
-y
-~
+ m h Cm = mgR,.Pz
-)*
A z
C is s y m m e t r i c a l . There are two matrices U and D, D diagonal so t h a t C = U -1 DU, and D•'
+ ~' A D ~ ' = m g ~ , U P ~
(~'
;
= LI~).
This is the equation corresponding to the motion of any rigid body w i t h a fixed point~ subjected
to
gravity
forces.
The e f f e c t
of
the presence of
the f l u i d is
to s h i f t the position of G.
2. Bi-dimensional case of a profile without a sharp edge
A
reasoning
analogous to
that
of
section
1 allows the d e r i v a t i o n of
a matrix
equation s i m i l a r to (1) but in three dimensions. We denote by (~,,m) the components -4~ of
--~
--)-
vo in R and by co = ~0 z 1 the r o t a t i o n ; then v = ( ~ , m~ co) and
Q =
M
MXGI
+ the m a t r i x from aerodynamical fo~'ces.
/
-MY G
MX G
lo
,J ÷ AR
We o b t a i n Q(~,) + [ v ~ Q ( v ) ] + 0 £ J ( v )
= K with now [v,T] =
.+
where
R
is
roAR the resultant the circulation
v e c t o r o f T. p I" 3(v) is I" is not e q u a l to z e r o :
the
additional
aerodynamical
force
when
252
G BOUCHER
0 3 =
-1 - XC
center,
1
X C "]
0
YCI
-Yc
' XC
and Y C are the c o o r d i n a t e s of the p r o f i l e
O
p the density of f l u i d .
C o u c h e t [/4] showed the e x i s t e n c e of a c o o r d i n a t e system
R in w h i c h the m a t r i x
Q is d i a g o n a l ; in this way we have the w e l l - k n o w n equations A~.
=
Bc0m - p £(m + c0XC) + X
Bfn
= -AR0J + p £(R - c0Yc) + Y
C~
= (A-B)
2.1. N u m e r i c a l
w i t h B >-- A > 0
and C > 0
(2)
Rm + p F ( R X C + m Y C) + Z
study when X = Y = Z = O :
The system (2) a d m i t s two i n t e g r a l s : AR 2 + Bm 2 + C J
- K1
(3)
A2R 2 + B2m 2 + 2pF (ARY C - B m X C + Cw)= K 2 The t r a j e c t o r y
of
(2) is the
intersection
(4)
of e l l i p s o i d (3) and the p a r a b o l o i d (4).
i t is closed, and hence it is p e r i o d i c a l . On an e l l i p s o i d w i t h an energy K 1> 0, we
can
have
2,
3, 4,
5 or 6 positions of
e q u i l i b r i u m for the system (2), d e f i n e d by Re
P ['O~eYc -
me
;
p[" _Ac~ e
PFU)eX C ;
we
=
coe
BoJe - p£
coe must s a t i s f y the e q u a t i o n P (coe) = AR e + B m e + Cw e - K 1 = 0. It is an e q u a t i o n equivalent
to
a polynomial
because P ( - / / ~ ) > 0 , v
C
of
P(0) < 0
degree
6 which
always at
least
two
real roots,
and p ( / ' K _ ~ ] ) > 0.
It is easy to see on a graph the stable positions of equilibrium
and the unstable
positions, fig (1).
2.2. Numerical
study when K represents
the 9ravity
forces :
X = - M g sin e, Y -- - M g cose, Z = - M g ( X G c o s 8 - YG sine),
with ~
de dt
Then we have the i n t e g r a l A (R---~Mg cos6)2 +B(m +-~-MgsinS)2 + Cc02 = 2Mg [(X C - X G) sine + ( Y c - YG ) c ° s 8] - (Mg) 2 ( A c o s 2 0 + B sin2O)+ h
P£
RIGID BODY IN IDEAL FLUID
253
L e t F(0) + h be the second member ; F is C co and 2~-periodical. We deduce t h a t the motion is possible only when 0 satisfies F(0) + h > 09 fig (2).
Positions of e q u i l i b r i u m f o r (2). m =-
~.:m:~=6:0
i m p l i e s 0 : 0 e , 6o = 0 , ~, = -Mg - cos 0e r #F There are at least two positions of e q u i l i b r i u m ,
nrMgs i n e e and F ' ( e e) = 0.
because-F'
always has two roots. One of these positions is stabler the s t a b i l i t y
being assured i f F " ( e e) < 0. [4]
2.3. Study of the motion around a stable equilibrium position : We choose h~ so t h a t F(0 e) + h is
small
and
we
choose
the
following
initial
conditions : eo = 0e + E, 4o = Mg cOS0o + (F (eo) + h)1/2 cos U2o cos % pF
6oo -- (F(eo) + h) 1/2 sin ~2o r
mo = - Mg sin 0o + iF (0o) + h) 1/2 cos d2o sin % , pF ~, ~°o, ~2o are a r b i t r a r y parameters.
By i n t e g r a t i n g the system (2)~ to which we add the equation
dO :- 6o, we
obtain the
curve t ÷ 0(t) and the t r a j e c t o r y of point 0r fig (3). Moreoverrwe can observe the v a r i a t i o n s of J~,r mr
6o. in the space (~,r m, 6o)rwecut
the t r a j e c t o r i e s w i t h the ellipsoid A(~, - ~,e)2 + B (m-me)2 + C 6o2 -- F(0 e) + h.
The
i n t e r s e c t i o n s are situated on polhodes s i m i l a r to the previous ones. fig ( 4 ) .
References 1.
D. C h e v a l l i e r and 3.M. H e l m e r . Annales des Ponts et Chauss~es l e r t r i m e s t r e . France (1984).
2.
N.E.
Kotchin,
I.A.
IKibel,
N.V.
Roze.
Theorical
hydromechanics.
Intersience
publishers (1964). 3.
V.N. Rubanovskii. PMM U.S.S.R., Vol. 52, n ° 3 (1988).
4.
G.
Couchet.
3oukowski. 5.
Les p r o f i l s Librairie
en aerodynamique
scientifique
et technique
P. Capodanno. Les #quations d'Appel d'un solide dans un fluide.
instationnaire Albert
et la condition
Blanchard.
et d'Euler-Lagrange
M a t e m a t i c a 21 (44)r 9-32 (1979).
de
Paris (1976).
pour le mouvement
254
G. BOUCHER
i
FIG. i
FIG. 2
Trajectories on the ellipsoid
Curve F(8)+h 8e position of equilibrium
L~
5.,, i¸ ,:
~ ......
.::..,.
ii~ /i,!i~ ....... I
FIG. 3
FIG. 4
Trajectory of point 0
Intersection trajectories with the ellipsoid