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OCR for page 329
Numerical and Analytical Investigations of a Stationary Flow
past a Self-PropeDed Body
N. P. Moshkin
Institute of Theoretical and Applied Mechanics
Novosibirsk, USSR
V. V. P~lchnachov and V. L. Senn~tskii
Lavrentyev Institute of Hydrodynamics
Novosibirsk, USSR
Abstract
A stationary viscous incompressible
liquid flow past a body is considered.
The liquid velocity distribution on the
surface of the body is prescribed so
that the total discharge through it
equals zero. The velocity vector tends
to a nonzero constant vector at infini-
ty. The following equations are to be
fulfilled:
= 0, (*)
~ = 0, (* *)
Here S and T are, respectively, the
momentum and the moment of momentum,
transfered by the body to the liquid
(for two-dimensional flows, S and ~
are the above-mentioned values, related
to the unit length). These conditions
form the boundary-value problem for the
Navier-Stokes equations, which we call
the problem of stationary flow past a
self-propelled body. Though the problem
of the flow past of a self-propelled bo-
dy has a natural origin~self propulsion
executes inhabitants,ships and airplan-
es~and though it has a practical impor-
tance,the number of works concerning it
is very limited.This work contains the
results of investigations of different
models of a self-propulsion of a body
in a liquid obtained by abalytical,nume-
rical and experimental methods.
§ 1.
1.1 Let £ be the bounded closed surfa-
ce in Rue and Q be the domain exter-
nal with respect to ~ . The problem
of defining vector-function The
difference between flow velocity at the
point Y hand constant velocity at in-
finity, en = `1, o, o' ~ and scalar
function pox) (pressure) which satis-
fy the stationary system of Navier-Sto-
kes equations in the domain ~
329
TV - Re av/Ox<-~vp _ Re v ·vv,`1 1
and the following boundary conditions:
v = amp (1.2)
v ~ O (1.3)
when x ~ a,
Here a is the vector-function satis-
~ a · n dS = 0, (1.4)
whilst still arbitrary in other respects'
~ is the nun' vector of the external
normal to the boundary of the domain Q.
Equations (1.1) have been written down
in dimensionless variables, so that
Re = V-1 ~ v is the Reynolds num-
ber, v is the kinematic viscosity
coefficient, l is the characteristic
linear scale (e.g., l = diem. Q ).
The value of V" (velocity of flow past)
is a natural velocity scale, and the
pressure scale is assumed equal to
p Via V / l , where p is the liquid
density.
Further the surface of ~ will be as-
sumed to belong to the Holder class
C23=, 0~ <1, and the components of
vector a - to belong to the class
C 20^ ~ ~ ). Problem-~1.1~-~1.4)
with the fixed function a was consi-
dered in a great number of works. The
most significant results were obtained
by R.Fin~n [8] and K.I.Babenko 623. If
a = - e' , we come to a classic prob-
lem of a flow past a body with an unmo-
vable impermeable boundary ~ . It is
well-known that in this case the resis-
tance force of the liquid exerted on
the body differs from zero. Thus, to
realize a stationary regime of flow
past a body, it is necessary to cr~nfi-
ne it to the liquid flow by external
forces. The classic problem of flow
OCR for page 330
pasr a body therefore should be called
the problem of flow past a towered body.
Here our particular emphasis will be
p l ac e d on the p rob l em of momentum! e ss
flow past a body, or problem ( 1. 1 ~
(1.4) with the additional conditioning.
In the terms v, p the latter is exp-
ressed by the equal ity
if_ i[-P4.n ~ Revs ~ elf n]dE=0~1.5)
Here F is the resistance force, PV
is the stress tensor corresponding_,to
the velocity and pressure fields ~ v
and p , respectively ), having the el-
ements
(PV)ij =-P6ij + aV~/aXj + ~V`~/axi
(i, j = 1, 2, 3~.
It is clear that in the model under con-
sideration condition ~ * ~ may be provid-
ed only by the mobility and (or) the per-
meability of the body boundary ~ . In
the latter case it is natural to be ex-
pec ted t hat the total discharge of a 1 i -
quid through the body surface is equal
to zero, that is expressed by ~ 1 . 4 ~ . The
condition of momentumless flow past a
body ~ ** ~ is not imposed here upon the
solution .
I t i s we l l-known that wi th f ixed a
problem ~ 1 . 1 ~ - ~ 1.4 ~ has at least one
solution for any Reynolds number, Redo
~2, 8, 14] . With low Re , its solution
is unique [23. From here it follows
that the problem of momentumless f low
past a body, (1.1~-~1.5) is solvable,
gene ral ly speak ing, only i f the_'addi t i o-
nal conditions on the function a ~ x
are fulfilled. In other words, this
problem is to be considered as a prob-
lem of a joint definition of functions
Hi, p and a from the relations ( 1. 1~-
-~1.5~. In this case condition (1.5),
equiva] ent to three scalar relations,
admits a wide Arbitrariness -in choosing
the function a , which may be consi-
dered as a determining function. It
should be noted that by virtue of the
law of momentum, condition ( 1. 5! is
equivalent to the equal i ty to zero of
the liquid pulse flow through any cont-
rol surface covering £ , in particu-
lar, an infinitely distanced co-,1;rol
surface t183.
1. 2. The existence of the solution to
the p roll em of morrientuml e ss f l ow pas t
a body has not been considered so far.
The ex i steno e p rob l em may be so lved ra-
ther effectively in the simplest case
when the Reynolds number is equal to
zero 619] . Let us denote system ( 1.1 )
when Re = 0 ( Stokes system) by ( 1.1 ) O
and the relation ( 1. 5 ) when Re = 0 by
(1.5)o. System (l.l)o, conditions (1.2)
- ( 1. 4) and equality ( 1. 5 ~ O form the
problem which will be denoted by ( 1.1 ) O
-~1 . 5) o and called the problem of momen-
tumless flow past a body in the Stokes
approximation .
In contrast to ~ 1. 1 ~ - ~ 1. 5 ), problem
(l.l~o-~1.5)o is linear, and the Stokes
operator generated by system ( 1.1 ) O is
a self-conjugated one t143. The above-
mentioned circumstances make it possib-
le to find an effective solution of
problem ( 1.1 ) o-( 1.5 ~ o in terms of eigen-
functions of some spectral problem. It
may be formulated as follows.
It is required to' find number ~ and
vector-function up ~ x ~ ~ O. determi-
ned on the surface 2,which satisfies con-
dition (1.4), from the relation
Here ~ denotes a linear operator,
which gluts in correspondence with func-
tion cp the valye of the trace of the
stress vector PV on the surface I,
calculated from the solution v' p_,of
problem ~ 1. 1 ~ 0 - ~ 1. 5 ~ O. where a = (p
Operator A , initially determined on
the functions tp ~ ~~ ~ °~( ~ ~ , admits
a self-conj~agated expansion up to the
operator, acting to H-~ (E ~ from
~ ^/2 ~ ~ ~ A tonality of the trace
of the functions u ~ H ~ ~ Q ~ on the
surface £ are denoted by H 4~2( ~ ~
and the closure in the Dirichlet integ-
ral norm of a set; of solenoidal vector-
-functions, smooth in Q , and equal
to zer o at sufficiently high values of
| x ~ is denoted by ~4 ~ Q ). The spa-
ce ~-~/2 ~ ~ ~ is conjugate with res-
pect to p - 2( ~ ). Using the theory, de-
veloped in t15], it is possible to find
the following properties of operator A:
~ i ~ it has an inverse A ~ , which is
c ant i nuous and se l f - c on j ugat e d; ~ i i ~ the
spectrum of operator ~ is discrete and
finite-fold; ~ iii ~ all the eigenvalues
\~<~2<~3 · · · are positive; ~ iv) Woo
when k ~ Go; the eigenfunctions (Pk
and (p3 corresponding to the eigenva-
lues As and \~ ;` Ak , are orthogo-
nal both in metrix I: ~ ~ £ ~ and that
gene rated by a sc al ar produc t ~ AM ~ ~2
The totality of eigenfunction~ { epic }
forms a full system both in ~2.( ~ ~ and
Hi/= (E ).
Now it is not difficult to formulate
the solution algorithm for the problem
of moment:umless flow past a body~in~the
Stokes approximation. Let as = (a, ~~)
denote the Fourier coefficients of ex-
pansion of the function a (x) in basis
{ Ilk } , orthonormalized to t~ ~ ~ ~
Then the condition (1.5)o may be written
as follows:
330
OCR for page 331
oo
~ ( Aka g Kid 2) - O (1 6)
where Ins . is; the i—th ~omnonent of
vector ' Ok . Condition (1.4) means
that
oo
~ (ak i ~~ · n d Z) = 0. (1.7)
k=1 ~
Let us choose an arbitrary element
() satisfying conditions (1.6),
(1.7) and then solve problem (l.l)o -
- (1.5)o. In so doing the velocity vec-
tor ~ is defined one-valued, and the
pressure p is defiend accurate to
the additive constant. Thus, in the so-
lution of (1.1)o~~1~5)o being She prob-
lem of determining functions v, p and
~ , there is a functional arbitra-
riness. Therefore, determining the fun-
ction ~ from the condition of mini-
mum power functional is considered to
be natural:
J = ~ ~ · ~ · ud Z.
The value of J is equal to the work
expended per unit line in sustaining a
stationary self-motion regime.
In the general case the problem of de-
termining minimum J under certain n~-
tural restrictions upon the function a
is solved by the method of Lagrange in-
definite coefficients. Let us consider
the examples of such restrictions:,the
function ~ + Alphas the support ~
which does not coincide with the whole
surface of ~ (physically the case
when the region ~ c £ is sufficient-
ly small is of interest); function
em has a zero normal component (the
boundary of a self-mov~ng body is im-
permeable); function al + em has a ze-
ro tangential component (the body sur-
face is unmovable). If the surface £
is a sphere, the minimization problem
of functional J for the second and
third type of the above-mentioned re-
strictions is solved explocitly with
the use of the results obtained in t10~.
In both cases the minimum of J is
achieved on the functions a , cor-
responding to the regime of potential
flow past a body. It is unknown if this
property of extremals of functional J
holds for an arbitrary surface ~ .
The matter of solvability of problem
(1.1~-~1.5) when Re > 0 is rather com-
plicated. However, it is hoped that its
solution may be positive for low Reynolds
numbers. The solution asymptotics for
a classical problem of flow past a body
~ ~ - -elf when Be ~ O to] is the
basis for such an optimism.
331
1.3. Now let, us consider an asymptoti-
cal behaviour of the solution to prob-
lem (1.~-~1.5) when r = |x | ~ ~ . For
now, let us ignore the additional con-
dition (1.~) and recall that the exis-
tence theorem for (1.~-(1.4) "as a
whole" is solid in the class of vector-
-functions v having Dirichlet fini-
te integral t14],
v : pvdx < (1.8)
Q
In 62] it was established that any so-
lution of the above-mentioned problem
satisfying the inequality (1.8) admits
the estimate
Iv~xyl< Cr-1/2- £ when r ~ ~ (1.9)
with some positive constants C and £
As was shown in t8], any solution of
problem (1.1~-~1.4) Ratifying inequa-
lity (1.9) has an asymptotical beha-
viour
v (x) = ~ ~ (x) + ~ (x). (1.10)
Here ~ is constant vector determined
by the formula (1.5), and ~ (~) is
the remaining term, for which the fol-
lowing estimate has been obtained ~ r
is high):
1: 1 < c1r~3/2+ ~ ~1 ~ S) .~1.11)
Here ~ = r - x , ~~ is arbitrarily
small, and Cal = const. > 0. symbolE(x)
denotes the fundamental tensor of Oseen
system, corresponding to (1.1~. The ex-
pressions for the elements of tensor E
may be found in t83.
It follows from (1.10),(1.11) that the-
re is a paraboloidal region of the wake
in the direction of e' , inside of
which v- Or. Beyond any circular
cone having the axis directed along ~ ,
= 0(r~9 ).The field asymptotics terms
vex) of the order of r - ~ /2 obtained
in 63] on the assumption of the colli-
nearity of vectors ~ and en . (This
assumption is fulfilled, for example.
in the case of axisymmetrical flow past
a surface of revolution ~ , thereby
the condition (**) being also fulfilled).
In 03, 6] the velocity vortex behaviour
at great distances from the body is in-
vestigated, and the vortex is shown to
decrease exponentially outside the wake
Formula (1.10) means that far from the
towed body, the velocity field distur-
bance (accurate to the smalls of higher
order) will be as that of Oseen flow
"flowing past" a concentrated force ~ .
OCR for page 332
The velocity field asymptotics in the
problem of momentumless flow past a bo-
dy proved to be determined by a much
higher number of functionals
characterizing both the body shape
and the way of realization of self-mo-
tion regime. The basic result may be
formulated as follows :203.
Let A, p be the solution of problem
(1.1~-~1.4) from the class (1.~! satis-
fying the additional condition (1.5~.
Then when I. - ~ the asymptotical repre-
sentation of vex) in the form
~~) - R ~ DE(x) + (1.12)
is valid. Here R = (Rz: ), Q = (Qi~)are
the constant tensors ~ i,j = 1,2,3~.
The elements of tensor R are express-
e~d explicitly in the terms of function
aid) . Symbols DE and VE denote
the third-rank tensors having the ele-
ments
(DE)ijk ~ 2 (a x + a ~ ~ '
and
(V E) _ `~ ,
i,j,k = 1,2,3. Summarizing in consoli-
dation R : DE , Q : ~ E is made with
respect to indices i and a . Func-
tion q(~) admits the estimate
1~ (x)1 ~ Car 2+£ (s + 1~-1/2 `1 13'
when r ~ = , where ~ and C2 are the
positive constants, ~ being arbitra-
rily small, s= r -x .
Formula (1.12) is derived on the basis
of integral reperentation of problem
(1.1~-~1.3) obtained in [9]. To estima-
te the volume integral
~(x) = /~) · v~y) ·= E(X - y) by
Q
when r ~ = , the results of t3] have
been used.
According to (1.12),(1.13), in the re-
gime of momentumless flow past a body
we have ~ - O(r~3/23in a paraboidal re-
gion of the trail and ~ = 0 (r~s/~)
beyond any cone having the axis e ~ .
Thus, a quicker decrease in velocity
disturbance at great distances from a
self-moving body, as compared to towed
one, is evident. Representation (1.12)
also means that (at least over the
wake region) the main terms of veloci-
ty field asymptotics in the problem of
momentumless flow past a body are cha-
racterized by 18 parameters, they are
the elements of tensors R and ~ .
In an axisymmetrical case the number of
these parameters decreases to eight.
Identification of the elements of tensor
Q being some functionals of the so-
lution to problem (1.1~-~1.5) is of par-
ticular interest. This problem has not
been solved so far.
One of paradoxical results, related to
the classsical problem of flow past a
body for the Navier-Stokes equations,
is as follows. Let in (1.2) a ~ - em (This
means immobility and impermeability of
the broody boundary). Then for any solut-
ion v, p of problem (1.1~-~1.3) satis-
fying the condition (1.8) we have
i 1~1'dx
Q
Such a statement related to the energy
of disturbed motion in the problem of
viscous flow past a body first appeared
in t73.
It is obvious that a self-moving cannot
contribute such a great disturbance in-
to a flow. An appropriate exact formu-
lation is as follows. Let V, p be the
solution of problem (1.1~-~1.4) satis-
fying the additional conditions (1.5),
(1.8~. Then
~ 1 V12 Ox ~ ~ (1.14)
Q
The property of the solution of the
problem of momentumless flow past a bo-
dy expressed by inequality (1.14-) dis-
tinguishes it among all the possible
solutions of problem (1.1~-~1.4) if
function a entering it is varied. We
hope that this property may be used to
investigate the existence of the solu-
tion of problem (1.1~-~1.5), if the lat-
ter is considered as some optimization
problem.
The two notes are to be made in connec-
tion with the velocity field asympto-
tics at great distances from a self-mo-
ving body. The first of them deals with
the velocity vortex behaviour with in-
creasing the distance from a body. When
r ~ ~ , the vorticity in the solution
of problem (1.1~-~1.5) decreases quicker
than in a calssical problem of flow past
a body both inside and outside of the
wake r63. In the exceptional cases the
wake may be absent. In [16] such a si-
tuation is illustrated by a plane poten-
tial viscous flow past a self-moving
"body". Its boundary consists of two
symmetric coupled components, on each
of them a normal velocity component be-
ing equal to zero, and a tangential one
being constant.
332
OCR for page 333
The second note is connected with the
velocity asymptotics in a turbulent flow
regime. In this case, based on the con-
sideration of ~203 and additional assum-
ptions related to the Reynolds stress
behaviour when r ~ ~ (for a self-moving
body these assumptions are discussed in
[53), it is possible to obtain represen-
tation (1.12) for the averaged velocity
field. However, here the elements of
tensors R and Q are the functionals
of the solution of unclosed system of
Reynolds equations. These equations are
treated as the Navier-Stokes equations
with the density of external mass for-
ces ~ =-2.Re . div ~ where ~ is the
Reynolds stress tensor having the ele-
ments USA =',i'v~' (i, j = 1,2,3~.
Here condition (*) changes its form in
comparison with (1.15) and is as follows:
t-Pi' · ~ + Be' (v + edgy · Ii +
§ 2.
~ = (l, o, o);Re - aV-/~ is the
Reynolds number ~ V is the kinematic
coefficient of the liquid viscosity);
~ is a function of Re; f is an
add function~of the angle ~ between
the vectors ~ wand X ~ + Y ~ ~ ~ =
(0, 1, o); ~ -,£.f,,' sin ~ 6; fit
are constants; t~ ~ O; f2 ~ 0~; ~ -
_ ~ ~ ~ = (0, 0, 1~; ~ is the
unit external normal to the cylinder
boundary).
The dependences f on ~ and ~ on
Re are prescribed so that ~ = 0.
In [213 the problem (2.1) was solved
approximately for low Re . The fol-
lowing asymptotical formula for the li-
quid velocity at great distances from
the body was obtained:
rat ~ 2 ~1f2 it a v1/2
' ~— ' L, ~ r ~ V4/2 Z;~/
- 1 ~
+ Re II · rid did = 0. _ x, jeep (-
2 v ~ 4v~
2.1. Liquid flows at great distances
from self-propelled and non-self-propel-
led bodies can be significantly diffe-
rent.
Let us consider the problems of a sta-
tionary flow of a viscous incompressib-
le liquid past self-propelled bodies
[21,223.
a). The body is a circular cylinder with
a moving body. The cylinder axis coinci-
des with the axis Z of the system of
the rectangular coordinates X, Y. Z.
The liquid flow is plane and symmetri-
cal in relation to the axis X in the
plane X, Y. The body boundary moves so
that the correlation (*) is fulfilled.
The correlation (**) is fulfilled by
the reason of the symmetry of the flow
in relation to the axis X.
The Navier-Stokes and the continuity
equations and the conditions on the cy-
linder boundary and at infinity have the
following form:
1
(7 · V)~ 2 ~ P + R 67' V. ~ = 0,
e (2.1)
~ = Off for r = 1, vet ~ for r ~ oo,
where V = V/V ~ V is the liquid ve-
locity; V is the X-component of ]be
liquid velocity Vex at infinity ~ r =
_ ~ V ,o, O) ;V04 so; = P/~V~)
~ P is the pressure in the liquid: _p
is the liquid density); r = FAX + Y7a
a is the radius of the cylinder);
333
(2.2)
for Pa ~ + ~ and constant Y /(aX),Re.
In accordance with (2.2) for the plane
flow past a self-propelled body, the
disturbance of the livid velocity va-
nishes by the law X (for the pla-
ne flow past a body which receives from
the liquid per unit length per unit ti-
me a non-zero momentum the disturbance
of the liquid velocity vanishes by the
law A/ rs~ my. The formula (2.2)
has such a form as the formula for the
velocity of plane liquid flow far from
a self-propelled body founded in ~5~.
b). The body is a ball with a liquid-
-permeable boundary. The ball centre
coincides with the origin of the coor-
dinates X, Y. Z. The liquid flow is sym-
metrical in relation to the axis X and
is non-swirling around it. On the body
boundary the liquid velocity component
normal to it is distributed so that
the liquid mass flux through this boun-
dary equals zero and the correlation
(*) is fulfilled. The correlation (**)
is fulfilled by the reason of the non-
-swirling of the flow around the axis
X.
The Navier-Stokes and the continuity
equations and the conditions on the
ball boundary and At infinity have the
following form
( - V V);~= -ha+ ale if' of' ~ ~ = 0'
(2.3)
v= Afn for r=1, vat i for rat Oo,
where r= ~ x2+y2+ zc /a ~ a is
the radius of-the ball); ~ is a
function of Re ; f is a function-~of
the angle ~ between the vectors ~ and
OCR for page 334
Z~ + YJ + Zig (f =~ ~ '~ (cos ~ );
fig ~ are~constants; I1 ; f2 /0; Pm
are the polynomials of Legendre).
The dependences f on ~ and ~ on
Re are prescribed so that ~ = 0.
In C22] the problem (2.3) was solved
approximately for low RO . The follo-
wing asymptotical formula for the li-
quid velocity at great distances from
the body was obtained: 2
~ ~ 3 foal y ~ z2
V ~ V [1 + 2 (1 - V" 4 Vie )x
x em (- V 00 - 4 j ] ~ (2.4)
fo/ (a ~ + ~ and constant(~2+Z2~/
In accordance with (2 .4 ~ for the axi-
symmetrical flow past a self-propelled
body the disturbance of the liquid ve-
locity vanishes by the law X~2 (for
the axisymmetrical flow past a body
which receives from the liquid per unit
time a non-zero momentum the disturban-
ce of the liquid velocity vanishes by
the law ~-1 t5J'.
2.2. A propeller work can significant-
ly affect a liquid flow near a body
at distances which do not exceed its
several transverse dimensions. This
may be used to organize required liquid
flow around a body (for example, a flow
with closed streamlines).
Let us consider the problem of a sta-
tionary flow of viscous incompressible
liquid past a pair of rotating cylin-
ders (a pair of identical parallel cir-
cular cylinders rotating around their
axes with opposite angular velocities)
6232. The axes of the cylinders are
parallel to the axis Z and intersect
the plane X, Y at points X = 0, Y = h
and X = 0, Y = -h. The liquid flow is
plane and symmetrical in relation to
the axis X in the plane X, Y.
The Navier-Stokes and the continuity
equations and the conditions on the
boundaries of the cylinders and at in-
finity have the following form:
( - v.~79v = PUP+ Re ~ or, v V :: 0,
v_ _k~ r+ for I r+ I =1 ~ A_ k x r ( 2 . ~ )
for Or ~ =1, v ~ £ i for r ~ ~ ,
where ~ = V/(Q a) ( Q is the mo-
dule of the angular velocities of the
cylinders; a is the radius of the
334
cylinders); p = ~ ~ pa 2a2~ ; r
r - ~~ + ~ +21/~2 £~) i,
£ = ~ (~) ; Be - a Q is the Rey-
nolds number.
In :23] the problem (2.5) was solved
approximately for small ~ . It was
ascertained that in the considered ap-
proximation the pair of rotating cylin-
ders is a self-propelled body.
,9
_: 1~]
it:
~=
Fig. ~
- x
:~ ~
In Fig. l the pattern of streamlines
of the flow around the cylindersis dis-
played(the sections of the cylinders
by the plane £ X , By are represented
by the points £X = o, £ y = l/2 and
SX = 0, By = -l/2; the cylinders are
surrounded by the liquid layer which is
streamed continuously by the liquid mo-
ving from infinity) This flow around
the cylinders is realized by the motion
of their boundaries, i.e. by the propel-
ler work.
The problem of a stationary flow of a
viscous incompressible liquid past a
pair of rotating cylinders was conside-
red in 623] in connection with the prob-
lem of decreasing the energy required
for a body to propel in a liquid. In
connection with the latter problem the
motion of a pair of rotating cylinders
in a liquid was also investi.g~.ted ex-
pe-rimentally 524, 253. The measurements
showed that the self-propelling of the
cylinders was energetically non-prof.i.-
`;able and a reduction of the energy
which was necessary for their propelling
OCR for page 335
could be achieved when the cylinders
simultaneously rotated and were af-
fected by an external force (the pro-
peller worked but not in a self-propel-
ling regime).
2,0
The power N which is necessary for a l8
pair of cylinders to proper is the sum
of two quantities:
~ _ FQ ~ W.
1,6
1,4
where ~ is the external force acting
onto the cylinders; Q is the veloci-
ty of the pair of cylinders; W is the
power which is necessary for the cylin- lo
ders to rotate.
],6
],4
i,2
i,0
0~6r
0,41
C
em _
~ ~ 6
_~
6
~'D,
ED ~
~ O
. 1 1
V ~>
D D ID D ID ~ ~ ID
~ ~e
.
0 0,8 I,6 2,4 3,2 4,0 4,B
i 0 3 (D 5 ,,9
me 40
0,2
Fig. 2
In Figs. 2, 3 the data obtained by mea-
surements on the dependences C =
F/~2La ~2) and w = W/ (2 ~ a p Q3)
on u = Q ~ Q are displayed ~ L is
the length of the cylinders; symbols
1-5 correspond to £ = 0.45;0.4~0.35;
0.3;0.25) :253. In accordance with the-
se data, the most energy profitable
propelling is realized at u % 1.3.The
power which is necessary for this pro-
pelling of the cylinders is approxima-
tely, 80% of the power required for
the propelling of the non-rotating
cylinders.
I,2
0,6
0,4
0,2
o
§ 3
335
W
I—
~ ° 2 e
3 ~ 4 0
5
0 0,8
1,6 2,4 3,2 4,0 4,8
Fig. 3
3.1. Let us come over to the problem
of numerical modelling of momentumless
flow past a body. It is based on the
Navier-Stokes equations with the use
of numerical methods t1,4,11,132. The
two cases are considered when relation
(*) is fulfilled. In one of them (down-
stream), there is a surface S behind
the ball (a part of sphere with a grea-
ter radius). The liquid flows over S
and gets thereby an additional pulse.
In the other case the ball surface is
permeable. On one its part, between two
cones with the divergence semi-angles
Q' , Q2 and a mutual axis 0 = ~ ,
the liquid is sucked in , and on the
other its part, "cut" by a cone Q3<65
the same quantity of the liquid is re-
turbed to the flow.
The other way to fulfil the condition
(*) are also possible, for example, by
prescribing a self-consistent distribu-
tion of volume forces, localized in a
small region behind the body. A numeri-
cal solution of the problem of momentum-
less flow past an extended ellipsoid of
OCR for page 336
revolution was obtained in this state-
ment in t123.
The solutions of the problem of flow
past a self-moving body becomes more
complicated due to nonlocality of con-
dition Gil. In the present paper a sta-
tiona.ry solution was found when solving
the nonstationary Navier-Stokes equa-
tions. In the spherical coordinate sys-
tem they have the form:
(u-cos6)aU + (~+sin6)au v(~+sinO)
r2E2 ~ [flu ~ ~ r 2 Bin ~ at X
x (sin ~ · V) ] , a~¢U-cos~' a
TV u(~+sin6) 2,~2ct~3~ ~ aP~
_ ~ _r __ _ ~
~~6 r ~ r r as (3.1)
+ReL69~ rid (3 - ae ], at +(U-c°S6)aa~+
Using ~'u-cosOw.(v~sin6)W Ct'Z:6 1 x
X(6w~ Use E3 )' r12 arfr U)''rsinSaG
resin E3 · v)=0, Re avV; r= Va a,
(A r2 ~3~(r ar) + r' sin (3 aa x
x-~ninO4~),
HereU, v and W are the projections
of the velocity vector to r , ~ and
~ directions, respectively, a is the
sphere radius, V. is the modulus of
inflowing flow velocity, ~ is the
characteristic value of swirling by the
propeller model. Equations (3.1) have
been written down in dimensionless v~-
riables. The values of a , Vat , pV-,
a ~ have been chosen as the scales
of length, velocity, pressure and
swirling. The boundary conditions are
prescribed as follows:
The sphere surface r = 1;
u = uO (e ~ + cos
v = - sin 6, W = WO(O ~
The axis of symmetry ~ = 0, ~ = a:
an/ as = o, v = o, w = o; (3.3)
The conditions on the sphere having a
large enough radius r = r :
~ = 3, W = 0, p = 0; (3.4)
The conditions on the surface
S ~ (r, S): r = rS, 6~< 0 < A}.
p+- p~_ [p], W~~ ~'V~- W (r 0 ~ (3'5)
Here an upper index + or - states to
the side of the surface S (e.g.
(rs' 0 ) ==llm+Ov (r, any.
The considerations dealing with the va-
lidity of such a statement can be
336
found in Ply.
S.~. To solve numerically the model
problem under consideration Ill;, the
methods such as [~AC, SUMAC and other
similar ones were generalized ~l,4,ll].
The main differences are due to a spe-
cial way of prescribing conditions
(3.4) when r=r" and the presence of
surface S having the pressure jump. The
unknown values were calculated in the
nodes of displaced network ~ u , v
in the middles of the cell sides; P ,
~ at the centers of them). The radi-
al velocity component u on the sphe-
re with a large radius was determined
from a difference analogy of the con-
tinuity equation. Surface S passed
through the centers of the cells. The
velocity components v and W pres-
cribed on S were found with the use of
(3.5), i.e. the equation at these points
was not used. When the problem with
surface S is considered, two pressures
P+ and P , are to be determined at
points of this surface, one of ~hem~be-
ing excluded with the help of P - P =
Opt. In the Poisson difference equation
for pressure, which is derived in a
usual way, in its righthand part there
appears an additional term const UP]
differing from zero only at the points
near surface S. The pressure equation
was solved by the method of upper re-
laxation. Fulfillment of (3.5) was ac-
hieved by correction of the pressure
jump En], prescribed on S. or the volu-
me of liquid flowing over the body sur-
face. The pressure jump or discharge
variations were made within the general
iteration process. The flow region was
mapped onto the rectangle by transfor-
mation r = exp Z. ~ _ 6~. The dimensio-
nal network was introduced in plane
~ Z. 0' ~
3.3. We made some calculations in or-
der to compare the basic hydrodynami-
cal characteristics for towed or self-
-moving sphere. The numerical experi-
ments were made under the following
boundary conditions. In the problem
with surface S we have
no ~ ~ ~ - O. WOL 6) - O. (3.6)
Ws~rs,(3~~
(1 e~33 R2Si=2 63~/(R . sinO)
(3^ ~ 6) ~ TC, (3.7)
O. for other ~
(here Rs is the dimensionless radius
of surface S). If self-motion is model-
led by the liquid flow over the sphere
surface, conditions (3.5) on S are not
used. Functions use ~ ~ and WOL e ~ were
OCR for page 337
Pig. 4a
.
. .
.
- ~ )~
ea -`
~ ~=~
!
n ~
Fig. id
prescribed in the following way:
-E f God, Q
USA- E1f1~.
~30 for other 0,
E'= ~ (Jo Si;E~0 d6)
1
2 ~ (r God) sin E3 d6)-'
~3
where f1~43) =~63-61)~632-6 )'£2~) ~ 3),
W (~3)~(1-e 63)/Si~1 ~ ~ 63< ~ < i,
° ~ O. for other ~ . ( 9
(3.8)
(3.7), (3.9) mean that the flow is swir-
ling only after it has passed through
the unit which models a propeller.
Presented in Fig. 4 are the isolines of
the stream function, swirling and pres-
sures for Re = 50. Fig. 4a illustrates
the towed sphere, and Figs.4b,c illust-
. .
at' <'=~ ~
Fig. 4b
337
\
\~ /
Fig. 4c
rate the self-moving sphere. The propel-
ler is modelled by the surface having
the pressure jump Rae = 1.4; 6~=162°.
Fig.4b illustrates the flow rotation
when ~ = 0; C P ~ = 3.00. Fig. 4c cor-
responds to ~ = 15, C P ~ = 4.38. Figs.
4d,e represents a self-moving sphere
with a permeable surface. The case when
~ = 0, 6' = 108°; 62= 132°; 63=
= 156° is shown in Fig.4d. Fig.4e corres-
ponds to the case when ~ = 20; 61 =
= 1125° ; 62 = 135°; 63 = 157
The calculations were made with the use
of different networks. The most detailed
network consisted of 40 nodes in a ra-
dial direction and 60 nodes in an angle.
In all the calculations r = 12.1 ~ Z =
= 2.5~. The number of iterations requi-
red for a stationary distribution of pa-
rameters to be achieved was dependent on
the choice of initial values Of r P ~
and U . When the rest state was assu-
med to be an initial state, not more
than 1500 iterations were to be found.
Contuniation in the number Re reduced
the required number of iterations by a
factor of 1.S or 2.
3.4. The problem of stationary motion
of an arbitrary body with a constant ve-
locity is tightly connected with the es-
timate of the consumption of energy ne-
cessary for providing the motion regime
under consideration. In the case of to-
wed body the required power ~ is simp-
ly expressed through the resistance ~ ;
OCR for page 338
OCR for page 340
Representative terms from entire chapter:
viscous incompressible
N = ~ · V . In the case of momentum-
less flow ~ - O. One can judge about
the value of required power by the los-
ses of mechanical energy, associated
with dissipation,
A- 2 ~ V r D: Ddx
Q
where D is the strain velocity tensor;
5~ is the flow region which is unbound
in the case of external problem of flow
past a body. Fig. 5 gives the values of
dimensionless dissipation ~ ~ ~ -
=~2~pV3a3~) for different cases of
stationary motion of a sphere having
the radius a with a constant velocity
V" in the liquid having the density p
for different Reynolds numbersRe_~/ Ah.
Fig. He
The solid line denotes the data for the
towed sphere ([ = Cd ~ 4) . A dashed
line denotes the data for a linear prob-
lem (Stokes approximation). The calcula-
ted values of dissipation for different
rotations for the problem with a per-
meable surface of the sphere, O' =112.5°.
62 = 135°; 03 = 157.5° are denoted
by rectangles. The problem with the sur-
face S ( rs = 1.4; 6~ = 162°) is deno-
ted by circles. Non-shaded circles cor-
respond to the problem with zero rota-
tion. It is easily seen that the flow
rotation amplification is accompanied
by the increase in dissipation of mecha-
nical energy, and, consequently, the
increase in required power.
For the problem with surface S the re-
sistivity coefficients C p and Cop may
be calculated in the usual way. Fig.6
presents the values of C p , C ~ for
different Re . It is interesting to no-
te that if Cop > Cp for a towed sphere,
Cop may be less than Cp for a self-
-moving one ire the Re range under
338
t.~T
I.4.
~ n.
. . _
0.6
0.2
7.~
'.4~
i.0
0.4f
up
C', Cp
consideration.
1.
A.
o
o
. ~~ ° °
\\\~
\
10 30 50 Re
Fig. 5
rl Polo
C' ~ O +
Cp c .
~ 0
\
\\ ~° 0
P ' _ _~
_ I I ~
0 30 50
Fig. 6
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339
