Specific features of application of perturbation techniques in problems of nonlinear oscillations of a liquid with free surface in cavities of noncylindrical shape
We consider the problem of nonlinear oscillations of an ideal incompressible liquid in a tank of a body-of-revolution shape. It is shown that the ordinary way of application of perturbation techniques results in the violation of solvability conditions of the problem. To avoid this contradiction we i...
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Institute of Mathematics, NAS of Ukraine
2007
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509353894739968 |
|---|---|
| author | Limarchenko, O. S. Лимарченко, О. С. |
| author_facet | Limarchenko, O. S. Лимарченко, О. С. |
| author_sort | Limarchenko, O. S. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:50:22Z |
| description | We consider the problem of nonlinear oscillations of an ideal incompressible liquid in a tank of a body-of-revolution shape. It is shown that the ordinary way of application of perturbation techniques results in the violation of solvability conditions of the problem. To avoid this contradiction we introduce some additional conditions and revise previously used approaches. We construct a discrete nonlinear model of the investigated problem on the basis of the Hamilton-Ostrogradskii variational formulation of the mechanical problem, preliminarily satisfying the kinematic boundary conditions and solvability conditions of the problem. Numerical examples testify to the efficiency of the constructed model. |
| first_indexed | 2026-03-24T02:39:46Z |
| format | Article |
| fulltext |
UDC 532.595
O. S. Limarchenko (Int. Math. Center Nat. Acad. Sci. Ukraine, Kyiv)
PECULIARITIES OF APPLICATION
OF PERTURBATION TECHNIQUES
IN PROBLEMS OF NONLINEAR OSCILLATIONS
OF LIQUID WITH A FREE SURFACE
IN CAVITIES OF NON-CYLINDRICAL SHAPE
OSOBLYVOSTI ZASTOSUVANNQ METODIV ZBUREN\
U ZADAÇAX PRO NELINIJNI KOLYVANNQ RIDYNY
Z VIL\NOG POVERXNEG V POROÛNYNAX
NECYLINDRYÇNO} FORMY
We consider the problem about nonlinear oscillations of ideal incompressible liquid in a tank of revolution. It
is shown that the ordinary way of application of perturbation techniques results in the violation of solvability
conditions of the problem. To avoid this contradiction we state some additional conditions and revise previously
used approaches. The construction of a discrete nonlinear model of the investigated problem is done on the basis
of the Hamilton – Ostrogradsky variational formulation of the mechanical problem with preliminary satisfying
of kinematical boundary conditions and solvability conditions of the problem. Numerical examples are evidence
of effectiveness of the constructed model.
Rozhlqda[t\sq zadaça pro nelinijni kolyvannq ideal\no] nestyslyvo] ridyny v rezervuari v formi tila
obertannq. Pokazano, wo zvyçajnyj ßlqx zastosuvannq metodiv zburen\ pryvodyt\ do porußennq umov
rozv’qznosti zadaçi. Dlq unyknennq ci[] supereçnosti vvodqt\sq dodatkovi umovy i perehlqdagt\sq
pidxody, qki vykorystovuvalysq raniße. Pobudova dyskretno] nelinijno] modeli vykonu[t\sq na osnovi
formulgvannq mexaniçno] zadaçi u vyhlqdi variacijnoho pryncypu Hamil\tona – Ostrohrads\koho z
poperednim vykonannqm kinematyçnyx hranyçnyx umov i umov rozv’qznosti zadaçi. Çyslovi pryklady
pidtverdΩugt\ efektyvnist\ pobudovano] modeli.
1. Introduction. The problem about oscillations of ideal incompressible liquid with a
free surface in a cavity of arbitrary geometrical shape is stated in the form of the following
boundary problem [1 – 5, 8, 12, 18, 23, 29]:
∆ϕ = 0 in τ, (1)
∂ϕ
∂n
= 0 on Σ, (2)
∂ϕ
∂n
= −
∂η
∂t
‖�∇η‖
on S, (3)
∂ϕ
∂t
+
1
2
(
�∇ϕ
)2
+ U = 0 on S, (4)
here motion is described in the Cartesian reference frame Oxyz connected with the tank,
ϕ is the velocity potential of liquid, τ is the domain occupied by liquid,
∂
∂n
is the external
normal derivative to a surface, Σ is the boundary of contact of liquid with tank walls
in perturbed motion (for convenience we introduce also Σ0, which corresponds to the
boundary of contact of liquid with tank walls in unperturbed motion and ∆Σ variation of
the contact boundary caused by liquid perturbation, Σ = Σ0 + ∆Σ), S is a free surface
of liquid in its perturbed motion (S0 is a free surface of liquid in unperturbed motion),
η(x, y, z, t) is the equation of a free surface of liquid, ‖�∇η‖ is the norm of gradient of the
function η, U is the function of the potential energy of liquid, t is time. For simplification
of the following analysis of some general properties of the boundary problem (1) – (4)
initially we limit our consideration by the case of an immovable tank.
c© O. S. LIMARCHENKO, 2007
44 ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 1
PECULIARITIES OF APPLICATION OF PERTURBATION TECHNIQUES IN PROBLEMS ... 45
Practical investigations showed that the unique possible way of analytical investiga-
tion of the boundary value problem (1) – (4) is connected with application of perturbation
technique [5, 18, 26]. In this case we suppose that elevations of a free surface are small
values, and further we can use both power expansions of nonlinear conditions (3), (4)
by elevations of a free surface and projection of these conditions onto unperturbed free
surface S0. At the same time it is necessary to pay additional attention to the boundary
condition (2), which, being linear by the form of relations, is set on variable boundary Σ
that depends on liquid motion. Thus, in spite of linearity of the relation (2) this condition
in essence is nonlinear similar to the conditions (3), (4).
In the case when the tank cavity represents a cylindrical domain one succeeded to
resolve the equation of the free surface relative to the variable z, which corresponds to
vertical direction. Then the equation of a free surface takes the form of
η(x, y, z, t) = z − ξ(x, y, t),
the corresponding linear problems admit analytical solution on the basis of the method of
variables separation, and the problem (1) – (4) becomes essentially simpler. In the case of
cylindrical domains occupied by liquid it is possible to construct effective algorithms for
solving nonlinear problems of dynamics tanks with liquid with a free surface, including
the case of translational and rotational motion of the carrying body [1, 5, 7 – 20, 23, 25,
27, 29, 30]. The most substantial results in this direction were obtained on the basis of
variational algorithms of statement and solving of nonlinear problems of dynamics of
tanks with liquid. Thus, for tanks of cylindrical shapes various problems of dynamics of
steady and transient modes of motion of reservoirs with liquid were investigated.
In spite of the fact that the question about spreading results obtained for cylindri-
cal tanks on the cases of tanks of complex geometry seemed for many researchers as a
question of numerical realization, until now cases of tanks of non-cylindrical shape are
investigated rather superficially. Attempt of creation of unified highly universal approach
for solving problems of nonlinear dynamics of liquid in cavities with inclined walls was
made in publications of I. Lukovsky [21, 29], where the problem about motion of liquid
in tank was formulated for non-Cartesian parametrization of the liquid domain. How-
ever, until now this method have not found practical application and investigations in this
direction are not continued.
Further attempts of solving problems of dynamics of liquid in tanks of non-cylindrical
shapes showed that there is a number of unclarified problems, which fundamentally make
difficult solving of the problem. Most clearly this become apparent on application of
methods of formal point-wise discretization [6, 7, 9, 31], when during one period of os-
cillations violation of laws of mass and energy conservation was about 20 %. Taking into
account that laws of conservation of mass and energy are not only physically evident for
this class of problems, but they practically coincide with the mathematical condition of
solvability of the problem (1) – (4), these methods as well as some analytical methods
collapse even for small time intervals. Moreover, most frequently methods of point-wise
approximation are applied for nonlinear 2D or axis-symmetrical problems, because they
are based on essential usage of limited computer resources, which is insufficient for in-
vestigation of most complicated nonlinear processes.
For further progress in solving applied problems of this class it is necessary to analyze
deeper mechanical and mathematical essence of the problem. This analysis will be done
from the point of view of further application of the variational method of the problem
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 1
46 O. S. LIMARCHENKO
Fig. 1. General scheme of denotations.
solving, which is based on formulation of the mechanical problem on the basis of the
Hamilton – Ostrogradsky variational principle. It is known that this approach was suffi-
ciently successfully applied for solving nonlinear problems of dynamics of reservoirs of
cylindrical shape with liquid with a free surface [12, 14 – 20, 25]. The mechanical analy-
sis of this approach shows that a part of the problem conditions (kinematical constraints)
should be satisfied before solving the variational problem of the stage of construction of
decompositions of desired variables, and dynamic boundary conditions and motion equa-
tions for the carrying body are obtained from the variational relation. It is significant to
note that in the subsequent procedure of problem solving no other increase of accuracy
takes place in satisfying kinematical requirements of the problem. For cylindrical do-
mains the solvability condition of the Neumann problem for the Laplace equation (for
the problem about oscillations of liquid with a free surface (1) – (4)) is quite trivial. This
condition is equivalent to requirement of conservation of a liquid volume in its perturbed
motion for every natural mode of oscillations. The analysis conducted in the present ar-
ticle shows that simple transfer of this form of the solvability condition to the case of
oscillations of liquid in non-cylindrical cavities is insufficient.
Hence, construction of a nonlinear discrete resolving model for the problem about
oscillations of ideal liquid with a free surface in a tank of non-cylindrical shape will be
done according to the following scheme:
1. The analysis of the solvability condition.
2. Construction of decompositions of desired variables, which hold linear kinematic
boundary conditions.
3. Construction of decompositions of desired variables, which hold nonlinear kine-
matic boundary conditions.
4. Construction of a resolving system of motion equations relative to amplitude pa-
rameters of liquid motion and parameters of translational motion of the carrying body.
2. Problem statement. We consider a problem about oscillations of liquid with a
free surface in a reservoir, which cavity is of revolution shape. We consider the case when
reservoir is movable and can perform finite translational movements. Basic denotations,
which were described in Introduction, are shown in Fig. 1.
For specification of liquid motion we introduce non-Cartesian parametrization of the
domain, occupied by liquid, according to the following scheme:
α =
r
f(z)
, β =
z
H
, (5)
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PECULIARITIES OF APPLICATION OF PERTURBATION TECHNIQUES IN PROBLEMS ... 47
where r = f(z) is the equation of the generatrix of a body of revolution, H is filling
depth of liquid. Here we suppose that the origin of the reference frame is in the center
of the undisturbed free surface of liquid, the axis Oz is directed upward, (r, θ, z) is the
system of cylindrical coordinates, which according to the relations (5) is substituted for
the new non-Cartesian system of coordinates (α, θ, β) (α ∈ [0, 1]; θ ∈ [0, 2π] and for
unperturbed state β ∈ [−1, 0]). For the accepted system of parametrization the domain of
liquid takes cylindrical shape and the equation of a free surface can be resolved relative
to the coordinate β and it takes the form of
β =
1
H
ξ(α, θ, t). (6)
The problem about motion of the bounded volume of liquid takes now the form of
∆ϕ = 0, (7)
∂ϕ
∂n
=
1√
1 + f ′2
(
∂ϕ
∂r
− f ′ ∂ϕ
∂z
)
= 0 for r = f(z), (8)
∂ξ
∂t
+
1
f2
∂ξ
∂α
∂ϕ
∂α
+
1
α2f2
∂ξ
∂θ
∂ϕ
∂θ
− αf ′
f
∂ξ
∂α
∂ϕ
∂z
− ∂ϕ
∂z
= 0 for β =
1
H
ξ(α, θ, t).
(9)
We note that the underlined term appeared in the kinematical boundary condition (9) ow-
ing to non-cylindrical shape of the domain occupied by liquid and reflects non-Cartesian
property of the accepted parametrization.
The equations (7) – (9) don’t present the complete formulation of the problem about
motion of liquid with a free surface, but they present the system of kinematical restric-
tions of the problem. Similar to the approach used for cylindrical tanks we shall ob-
tain dynamical boundary conditions and motion equations for the carrying body from the
Hamilton – Ostrogradsky variational principle.
Translational motion of the reservoir relative to a conventionally immovable reference
frame is described by the vector of displacements �ε.
In contrast to publications [21, 29] further for realization of transformations in an
invariant form we apply vector calculus instead of tensor calculus.
3. Analysis of the solvability condition of the problem. Taking into account that the
problem (1) – (4) as well as the corresponding linear boundary problem [2, 8, 12, 22, 24,
28] has no exact analytical solution for arbitrary cavities of revolution, we must come from
the fact that boundary conditions of the problem (1) – (4) will be realized approximately.
According to the general theory of solvability of the Neumann boundary problems for the
Laplace equation the solvability condition for the problem (1) – (4) can be given as∫
Σ0
∂ϕ
∂n
dΣ +
∫
∆Σ
∂ϕ
∂n
dΣ +
∫
S
∂ϕ
∂n
dS = 0. (10)
Let us analyze term by turn the expression (10). The first addend represents require-
ment of satisfying (in weak sense) the non-flowing condition on an unperturbed boundary
of contact of liquid with tank walls Σ0. Therefore, holding of this boundary condition of
non-flowing on Σ0 should be performed with improved accuracy. In our appearance on
realization of different procedures of solving this class of problems insufficient attention
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 1
48 O. S. LIMARCHENKO
was paid to this question. Sometimes natural modes of oscillations with errors of satisfy-
ing of boundary non-flowing condition about 20% and more were applied for numerical
realization. Correspondingly, such violation of the non-flowing conditions, and, therefore,
the solvability condition, results in instability of realization of numerical procedures.
The second addend corresponds to the requirement of realization in weak sense of the
non-flowing condition on the boundary ∆Σ, i.e., on wave crests of liquid over level of the
undisturbed free surface of liquid. This physically evident kinematic boundary condition
is not consequence of statement of the linear problem about oscillations of liquid in a
tank, which is usually applied for construction of decompositions of desired variables.
Normally this condition is not taken into consideration at all on analysis of nonlinear
oscillations of liquid in tanks of non-cylindrical shape, although we suppose that this
condition is the dominant one in the analysis of the physical sense of the considered
problem. In accordance with the maximum principle for harmonic functions the solution
tends to violate realization of non-flowing condition, and this corresponds to overflow
of liquid through the tank wall (namely this causes “loss” of liquid in methods of point-
wise discretization). In spite of the property that this condition is expressed by linear
mathematical relation, according to its nature it is nonlinear, because it corresponds to
realization of the kinematic condition on a nonlinear perturbed surface, and it is evident
that this condition does not enter the linear statement of the problem.
In order to analyze the third addend in the solvability condition of the problem we
perform its transformation. It is known that one of the form of the kinematic boundary
condition on a free surface has the form
∂ϕ
∂n
=
1√
1 +
(
�∇ξ
)2
∂ξ
∂t
on S.
After substituting this condition into the third addend of the condition (7) we obtain
∫
S
∂ϕ
∂n
ds = −
∫
S0
∂ξ
∂t
dS = − ∂
∂t
∫
S0
ξ dS,
i.e., the integral over a perturbed free surface of liquid S can be transformed to the un-
perturbed free surface S0. The latter form of the third addend corresponds to requirement
of the liquid volume conservation in its perturbed motion. Realization of this require-
ment will be considered below, where we shall show that realization of the requirement of
liquid volume conservation for every separately taken natural mode of liquid oscillation,
which corresponds to linearized requirement, is not sufficient for realization on a whole
of the requirement of volume conservation in its perturbed motion.
Thus, the analysis of solvability conditions of the nonlinear boundary problem shows,
that for correct solving of the nonlinear problem it is necessary to
a) realize with high precision the non-flowing requirement of liquid on the moisten
in the unperturbed state boundary of contact liquid – tank;
b) satisfy requirement of non-flowing of liquid on tank walls over the level of an
unperturbed free surface where crests of nonlinear waves reach;
c) satisfy requirements of liquid volume conservation in its perturbed nonlinear mo-
tion;
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 1
PECULIARITIES OF APPLICATION OF PERTURBATION TECHNIQUES IN PROBLEMS ... 49
d) realize all these requirements on the stage of construction of decompositions of
desired variables, which satisfy all kinematic boundary conditions of the problem, before
solving the variational problem.
4. Construction of decompositions of desired variables, which hold linear kine-
matic boundary conditions. As it follows from the analysis of solvability conditions
of the problem for successful realization of an algorithm of construction of the nonlinear
finite-dimensional model of dynamics of liquid with a free surface in cavity of revolution
it is necessary to construct a system of coordinate functions satisfying with high accuracy
requirements of non-flowing on the moisten boundary Σ, which consists of the unper-
turbed moisten boundary Σ0 and its certain prolongation ∆Σ, until which wave crests
can rise.
Traditionally the problem about determination of this system of coordinate functions
was identified with the classical problem about determination of natural frequencies and
modes of oscillations of ideal liquid with a free surface in cavities of different geometri-
cal shape. As it is follows from the mentioned above analysis the system of coordinate
functions for solving the nonlinear problem does not coincide with natural modes of os-
cillations, since it must supplementary satisfy nonflowing conditions on ∆Σ. At the same
time the problem about determination of natural frequencies and modes of oscillations of
liquid with a free surface has independent theoretical and applied significance. First of
all, namely on the basis of this problem calculation of natural frequencies of oscillations
of a free surface of liquid and hydrodynamic coefficients of the motion equations (asso-
ciated or virtual masses) is realized. Here virtual masses are expressed in an open form
as quadratures of natural modes of oscillations. Natural frequencies and virtual masses
of liquid are the basic parameters for construction of linear dynamic systems for control
of bodies with liquid. At present different methods for determination of natural frequen-
cies and modes of oscillations of liquid in reservoirs are developed [1, 4, 5, 12, 22, 24,
28], which give suitable results for practice. At that practically all approaches use the
traditional boundary eigenvalue problem
∆ϕ = 0 in τ0,
∂ϕ
∂n
= 0 on Σ0,
∂ϕ
∂z
= λϕ on S0, (11)
or its variational analog
δI = 0, where I =
∫
τ0
(�∇ϕ)2dτ − λ
∫
S0
ϕ2dS. (12)
As it was shown in [22], the solution of the problem (11) or its variation analog (12) con-
tains an analytical singularity on the contour L0. Availability of this singularity mainly
clarifies sufficiently rapid manifestation of numerical instability on increase on the num-
ber of coordinate functions, which approximate a solution of the problem (11). Prelim-
inary analytical isolation of the singularity on the contour L0 and further search of the
problem solution in the form of sum of singular and regular components essentially com-
plicates the algorithm of solving the problem and reduces its practical applicability. Pres-
ence of this mathematical singularity is predetermined by existence of angular contour on
L0 and alteration of the form of the boundary condition in a vicinity of L0. Partial cause
of manifestation of this singularity is caused by linearization of the problem, as a result
of which certain mechanical contradiction appears, i.e., the problem is solved for the im-
mutable domain τ0, however, for points of the boundary S0 of the domain τ0 non-zero
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 1
50 O. S. LIMARCHENKO
Fig. 2. General scheme of clarification of supplementary conditions
and the method of an auxiliary domain.
normal velocities are admitted. This contradiction is absent in the nonlinear statement of
the problem where singular properties of the solution are manifested weaker.
In connection with development of analytical and numerical-analytical methods for
solving nonlinear problems of dynamics of bodies with liquid, where application of nat-
ural modes of oscillations as coordinate functions is assumed, solutions of the problem
similar to (11) must meet requirements of not only integral character (as in the case of
the linear theory), but of differential character too. Sense of supplementary requirements
consists in the property that the nonflowing boundary condition on ∆Σ, i.e., on prolon-
gation of the surface Σ0 outside the domain τ (Fig. 2), must hold. Practically this must
be provided in the mentioned methods of solving nonlinear problems by existence and
vanishing derivatives of the following type
∂k+1ϕ
∂n∂lk
, where �l is the unit tangent vector to
the surface Σ0 on the angular contour (k = 1, 2, . . . depends upon order of maximally
taken nonlinearities on simulation of motiuon of liquid with a free surface).
It is evident that in the general case solutions of the problem (11) does not hold this
requirements, since the initial statement of the boundary problem admits only conditions
with differential operators of the first order. In this connection it is expedient to certain
extent to refuse from the traditional problem of determination of natural frequencies and
modes and construct approximately the system of coordinate functions ψi, which is close
to solutions of the problem (11) ψi with correspondingly close parameters λi and λi,
but which in addition holds with high accuracy nonflowing condition on Σ0 and on the
surface ∆Σ.
For realization of this goal we suggest two techniques, i.e., successive refinement of
the solution of the problem (11) and the method of an auxiliary domain for reduction of
influence of the singular points on behavior of the solution.
As it is known, success of application of direct methods of solving variational prob-
lems depends essentially on properties of coordinate functions. However, in most cases it
is not possible to construct effective coordinate functions, which in advance hold exactly
a part of conditions of the problem (11). So, mainly researchers use for solving problems
for different simply connected domains the harmonic polynomials w
(m)
k as coordinate
functions, which do not hold specificity of geometrical shape of a cavity at all.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 1
PECULIARITIES OF APPLICATION OF PERTURBATION TECHNIQUES IN PROBLEMS ... 51
For partial account of geometrical and physical characteristics of natural functions
(desired solution) we suggest the following algorithm of successive refinement of coordi-
nate functions (it is necessary to note that in non-successive form the method with some
similar elements was suggested in [6]). After solving the variational problem (12) and
the algebraic eigenvalue problem we propose to select the obtained eigenfunctions (natu-
ral modes of oscillations of a free surface of liquid) as the new coordinate functions and
repeat the procedure of solving the problem (12). In the case of necessity this step can
be repeated in addition. However, for next step of successive refinement of coordinate
functions it is not necessary to calculate quadratures again, since the above described
techniques is equivalent to the following calculation scheme.
Let
(A− λB)x = 0 (13)
be the algebraic eigenvalue problem, which is obtained by numerical realization of the
variational problem (12) and D is the matrix composed of eigen-vectors, i.e., solutions
of the algebraic problem (13). Then, one step of the described successive process is
equivalent to transition from the problem (13) to the problem
[
DT (A− λB)D
]
x = 0. (14)
Moreover, with considering non-degeneracy of the matrix D (otherwise solutions will
be dependent) eigenvalues of the problems (13) and (14) coincide. Since solutions of the
problem (11) possess property of orthogonality, then the matrix B∗ = DTBD tends to a
diagonally scattered structure (non-zero elements are located only on the main diagonal,
part of rows are completely zero). This property of the matrix B∗ created additional pre-
conditions for more precise numerical solving the problem (13). We note that according
to the method of the book [6] the matrix of eigen-vectors of the matrix B or A was used
as the matrix D (in our case we use eigen-vectors of the problem (13) as a whole).
Actually effectiveness of the described technique is based on imperfections of the nu-
merical solution of (13). Moreover, advantages of this approach manifest vividly on tend-
ing to a boundary of divergence of calculation procedure of the problem (12) on increase
of the number of coordinate functions. Thus, in the case of solving the problem about os-
cillations of liquid with a free surface in the spherical cavity with H = 0,5R for N = 12
coordinate functions with the given calculation accuracy ε = 10−6 and for single itera-
tion we obtain results, which are practically coincide with solutions of the problem with
ε = 10−13 without iterations. However, for ε = 10−19 successive refinement of solving
the problem (13) for N = 12 is not practically manifested. This technique becomes ap-
parent most effectively for 23 ≤ N ≤ 28, when for filling levels of liquid close to H = R
calculation convergence of solutions of the problem (13) appears. In this case results of
calculations testify the described iteration procedure possesses properties of contracting
mapping for errors of solving the problem (13), i.e., for certain parameters after reaching
calculation instability on the initial step as early as on the first step of the iteration process
recovery to a domain of stable calculation takes place. Results of numerical experiments
in the domain of divergence of calculations (23 ≤ N ≤ 28 for sphere) application of the
iterative scheme makes it possible to improve the solution of the problem (13) obtained
according to the classical scheme 1,5 – 2 times, and in the distance from a free surface it is
improved 3 – 10 times (in some cases 50 times). Let us note that the property of solution
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 1
52 O. S. LIMARCHENKO
continuability along the surface ∆Σ, as a rule, become worst after application of iteration
procedure.
For the purpose of construction of coordinate functions for solving the nonlinear prob-
lem of dynamics of liquid with a free surface we apply the following technique for reduc-
tion of influence of singularity on angular contour on character of behavior of the solu-
tion. Numerical experiments show that solving the problem by the variational method
with application of regular coordinate functions presence of singularity in statement of
the boundary problem (11) becomes apparent in poor convergence, which further results
in instability of numerical procedure.
Hence, we propose to solve the problem about searching coordinate functions close to
natural modes of oscillations but having the mentioned above supplementary properties
in the following way. We solve the problem (12) for the domain τ + ∆τ (Fig. 2), where
∆τ is selected on the basis of requirements for validity range of the nonlinear theory for
desirable boundaries of satisfying of nonflowing condition of wave crests on ∆Σ. Here
singularities in the constructed solution become apparent in a vicinity of the point A′.
However, in a vicinity of the point A, which is an internal point for the domain τ + ∆τ,
the solution possesses regular properties. Next important stage consists in a method of
projection of the obtained values of ψ′
i and λ′
i onto the surface S0, i.e., with reference to
the domain τ. If we use regular solutions ψ′
i for the domain τ ′ as coordinate functions of
the problem (12) for the domain τ, then singularities in the problem statement (12) result
in weak convergence for the domain ∆Σ including the point A. Therefore, we propose
to consider values ψ′
i on the surface S0 as ψi. Moreover, as numerical experiments show
according to character of behavior ψi are close to ψ′
i and ψi, and eigenvalues, determined
for ψi by the Rayleigh method, differ from λi very slightly.
Let us show application of methods of successive refinement and auxiliary domains
on examples of numerical realization. We consider a spherical reservoir filled by liquid
with the level H = 0,5R. Table 1 shows results of determination of first natural modes
of oscillations (k = 1) for peripheral numbers m = 1 and m = 2. The solution of the
problem was constructed on the basis of decompositions by N = 28 harmonic polyno-
mials. Algebraic eigenvalue problem was solved for ε = 10−19. Here λ are eigenvalues,
δc is ratio error of realization of non-flowing condition at angular point, δb is ratio error
of realization of non-flowing condition at the point ∆z = 0, 2R higher the level of a free
surface of liquid. Ratio error was calculated according to the following formula
δ =
∂ϕ
∂n
∣∣∣
Σ
/
max
∂ϕ
∂n
∣∣∣
S0
.
The variant 1 corresponds to classical realization of the variational method, the variant
2 corresponds to the method of successive refinement and the variant 3 corresponds to the
method of an auxiliary domain applied in the aggregate with the method of successive
refinement. As it follows from Table 1 application of the method of successive refinement
does not influence the frequency, and application of the method of an auxiliary domain
for ∆z = ∆H = 0,2R raises the frequency no more than on 10−4. Moreover, in the case
∆H = R; λ = 1,210119, i.e., distinction by frequency does not exceed 0,2%. This shows
potential of approximate determination of frequencies in tanks of revolution according
to the following scheme: we solve the problem for maximally required filling level with
improved determination of natural modes; then, by known eigenfunctions for the maximal
level we determine the frequency for arbitrary intermediate leven on the basis of the the
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PECULIARITIES OF APPLICATION OF PERTURBATION TECHNIQUES IN PROBLEMS ... 53
Table 1
Parameter
Variant
1 2 3
m = 1
δc 1,6 · 10−4 7,4 · 10−5 6,1 · 10−5
δb 0,172 4,64 0,956 · 10−3
λ 1,20772 1,20772 1,20782
m = 2
δc 4,7 · 10−5 7,2 · 10−5 3,2 · 10−5
δb 0,212 0,942 1,4 · 10−3
λ 2,30753 2,30753 2,30781
Rayleigh method, i.e., by calculation of two quadratures from similar coordinate function
for the maximal level taken on the cross-section, which corresponds to the required free
surface. Here it is necessary to note that the coordinate function satisfies the non-flowing
condition on Σ0 with high accuracy.
On the basis of analysis of errors of realization of non-flowing condition presented
in Table 1 it is possible to note that in cases of the classical method and the method of
successive refinement behavior of the solution above the free surface is unsatisfactory
from the point of view of potentials of application of this solution as coordinate functions
for solving the nonlinear problem. This is perfectly explicable, since statement of the
boundary problem (11) obtained on the basis of the linear theory and its hypotheses does
not impose restrictions on character of behavior of the solution above the free surface
of liquid. The solution obtained by the method of auxiliary domain with the acceptable
accuracy “follows” the contour above the free surface and values of deviation δ on Σ0
also decrease.
Graph results reflecting behavior of the solution on ∆Σ are showed in Fig. 3. Here
curves are enumerated in the following way: 1 corresponds to the classical method; 2 is
associated with the method of an auxiliary domain. In the case of the spherical tank with
liquid filling H = 0,5R we take into consideration N = 22 functions w(m)
k . For defining
the auxiliary domain it was accepted ∆H = 0,25R for the mode m = 1, k = 1 and
∆H = 0,15R for the mode m = 2, k = 1. The law of alteration of ratio error δ of non-
flowing condition on Σ0 for the mode m = 1, k = 1 is schematically shown in Fig. 3.
Character of alteration of δ along the surface Σ0 is such, that δ reaches the maximum
in a vicinity of L0, i.e., errors on the contour of the generatrix are much smaller than at
the corner point. So, errors obtained on the basis of the classical method and the method
of an auxiliary domain are approximately of the same order. However, behavior of the
solutions on ∆Σ differs fundamentally. For the classical method errors δ on the upper
boundary ∆Σ are much greater than errors obtained by the method of auxiliary domain
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54 O. S. LIMARCHENKO
Fig. 3. Character of behavior of the solution above the free surface of liquid.
(in 800 times for the mode m = 1; k = 1 and in 1800 times for the mode m = 2; k = 1),
in spite of closeness of both solutions by frequency parameter and values of δ on Σ0.
We note that as a result of application of the auxiliary procedure of projection of
function ψ′
i onto S0 the property of orthogonality of functions ψ̄i, which correspond to
same m and different k, is violated.
Closeness of the obtained coordinate functions and the corresponding frequency pa-
rameters to natural modes and natural frequencies of the problem (11) is fundamentally
significant for application of methods of solving nonlinear problems owing to widely
used division of natural modes of liquid oscillations into classes [17, 19, 20]. This divi-
sion is realized on the basis of mechanical analysis of potential contribution of coordinate
functions into formation of wave processes with taking into account values of the corre-
sponding frequencies. Later on this division is used for introduction of hypothesis about
ways of simulation of nonlinear terms for amplitudes, which correspond to different natu-
ral modes. This division into classes was analyzed qualitatively and quantitatively in [17,
20], where effectiveness of this techniques was shown.
In contrast to the classical approach, when the problem statement does not contain
information about character of variation of the surface ∆Σ above the level of the undis-
turbed free surface, the method of an auxiliary domain reflects dependence of the solution
of the problem about determination of frequencies and coordinate functions on character-
istics of ∆Σ.
The suggested methods of successive refinement of coordinate functions and the
method of an auxiliary domain according to their essence are engineering approaches
for construction of coordinate functions, which do not possess mathematical stringency.
At the same time simplicity of numerical realization of the desribed approaches for ar-
bitrary cavities of revolution, their mechanical clearness and obtaining good final results
of realization of the non-flowing condition more precise than by classical methods of
solving the problem (11) make it possible to recommend the proposed methods for con-
struction of coordinate functions of nonlinear problem, search of reductive dependence of
the frequency on filling depth.
For realization of the described techniques of improved determination of coordinate
functions close to natural modes of oscillation of liquid in cavity of revolution we suggest
the following algorithm:
1. The initial eigenvalue problem is solved by the method of an auxiliary domain.
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PECULIARITIES OF APPLICATION OF PERTURBATION TECHNIQUES IN PROBLEMS ... 55
2. Refinement of the problem solution is realized according to the successive ap-
proach.
3. In the case, when we include into the system of coordinate function several func-
tions with the same circular number, then we produced their orthogonalization (functions
with different circular numbers are automatically orthogonal).
4. For every function we determine the frequency parameter by the Rayleigh method
(these parameters are mainly used for verification and estimation of frequency character-
istics).
For solving the nonlinear problem of dynamics of combined motion of a reservoir and
liquid, which partially fills cavity of revolution, we applied the following discretization
parameters n1 = 10, n2 = 6, n3 = 3. Moreover, the coordinate functions are ordered in
the following way:
ψ1 = ψ∗
1,1 sin θ, ψ2 = ψ∗
1,1 cos θ, ψ3 = ψ∗
0,1,
ψ4 = ψ∗
2,1 sin 2θ, ψ5 = ψ∗
2,1 cos 2θ, ψ6 = ψ∗
0,2, ψ7 = ψ∗
3,1 sin 3θ, (15)
ψ8 = ψ∗
3,1 sin 3θ, ψ9 = ψ∗
1,2 sin θ, ψ10 = ψ∗
1,2 cos θ,
where ψ∗
m,k is the solution of the problem about refined determination of the coordinate
function closed to the natural mode of oscillations for the circular number m, to which
the k-the eigenvalue corresponds (if eigenvalues are put in ascending order).
Tables 2 – 4 shows results of numerical determination of coordinate functions (15) for
cavities of spherical (two filling levels) and conic shapes. Here we use the following de-
notations ∆z is the level, which determines the value of auxiliary increase of the domain
τ. We see that all coordinate functions with sufficiently high accuracy “follow” the profile
of ∆Σ. Moreover, it is necessary to note that on solving of the nonlinear problem ampli-
tudes of oscitation of natural modes have order 0,1 – 0,3, which reduces in 3 – 10 times
errors of realization of the non-flowing condition on the surface Σ + ∆Σ.
The suggested technique of determination of coordinate functions is based on solving
the linear problem, but it is supplemented by a number of requirements, which lie outside
the scope of the linear statement of the problem and reflect a part of kinematic require-
ments and solvability conditions of the nonlinear problem. Finally this makes it possible
to construct the system of coordinate functions, which in improved way holds the non-
flowing condition on the perturbed moisten boundary of the domain occupied by liquid.
Further this system of functions was successfully used for solving the nonlinear problem.
5. Construction of decompositions of desired variables, which hold nonlinear
kinematic boundary conditions. According to results of [19, 20] we select decompo-
sitions of the desired variables of excitation of a free surface of liquid ξ and the velocity
potential ϕ in the following form:
ξ = ξ̄(t) +
∑
i
aiψ̄i(α)Ti(θ), Φ = �̇ε · �r + ϕ0,
ϕ =
∑
biψi(α, β)Ti(θ), (16)
where
ψ̄i(α) =
∂ψi
∂z
∣∣∣
β=0
. (17)
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56 O. S. LIMARCHENKO
Table 2
Sphere H = 0,5R, ∆z = 0,25
m; k 1; 1 1; 2 0; 1 0; 2 2; 1 3; 1
λ 1,2079 4,5120 3,1057 5,2670 2,3080 3,3703
δi 2,4 · 10−4 7,5 · 10−5 1,7 · 10−4 4,3 · 10−4 2,8 · 10−4 3,1 · 10−4
δc 2,1 · 10−4 7,1 · 10−5 1,6 · 10−4 4,1 · 10−4 3,2 · 10−4 3,9 · 10−4
δb 3,1 · 10−3 9,1 · 10−4 2,6 · 10−3 6,7 · 10−3 4,1 · 10−3 5,2 · 10−3
Table 3
Sphere H = R, ∆z = 0,2
m; k 1; 1 1; 2 2; 1 0; 1 0; 2 3; 1
λ 1,5622 5,2315 2,8246 3,7345 6,8658 4,0020
δi 2,9 · 10−3 7,2 · 10−3 3,1 · 10−3 5,9 · 10−3 1,4 · 10−2 4,0 · 10−3
δc 2,3 · 10−3 4,1 · 10−3 1,2 · 10−3 1,7 · 10−3 2,1 · 10−3 3,4 · 10−4
δb 8,9 · 10−2 1,1 · 10−1 4,8 · 10−2 7,3 · 10−2 1,7 · 10−1 5,8 · 10−2
Table 4
Cone H = R, ∆z = 0,25
m; k 1; 1 1; 2 0; 1 0; 2 2; 1 3; 1
λ 1,0000 2,0272 1,2972 2,9844 1,7675 2,5054
δi 4 · 10−6 4,2 · 10−4 8,0 · 10−6 5,2 · 10−4 6 · 10−6 9 · 10−6
δc 10−6 5,2 · 10−5 2,0 · 10−6 3,9 · 10−5 1 · 10−6 3 · 10−6
δb 1 · 10−6 5,2 · 10−4 1,9 · 10−5 6,2 · 10−4 7 · 10−6 2 · 10−5
Due to specificity of a cavity shape we separate the circular coordinate in the decom-
positions (16). Here Ti(θ) are trigonometric functions, which selection is predetermined
by distribution of functions (15). The functions ψi(α, β) are constructed by the described
above technique, therefore, they are harmonic and satisfy with high accuracy the non-
flowing conditions on the moisten border Σ including a part of this domain, where waves
can reach above the level of the unperturbed free surface. The functions ψ̄i(α) possess
the property of completeness on S0 [22, 28]. We note also that in contrast to the case of
cylindrical domains decompositions (16) contain the term ξ̄(t), which is determined from
the requirement of conservation of the liquid volume in its perturbed motion.
Following the approach of publications [15 – 20, 27] we select the amplitude pa-
rameters ai as independent variables. Therefore, we shall determine interdependence
bi = bi(aj , ȧk) from the kinematic boundary condition on a free surface (9), and the de-
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PECULIARITIES OF APPLICATION OF PERTURBATION TECHNIQUES IN PROBLEMS ... 57
pendence ξ = ξ(ai, t) from the requirement of conservation of the liquid volume (in the
case of absence of liquid outflow this dependence transforms into the form ξ = ξ(ai)).
To this end we represent ξ and bi as
ξ = ξ
(1)
+ ξ
(2)
+ ξ
(3)
; bi = b
(1)
i + b
(2)
i + b
(3)
i + b
(4)
i , (18)
where upper indexes correspond to the order of smallness of values, if we accept the order
of ai as a small value.
On computation of variation of the liquid volume in its perturbed motion we make use
of the above introduced non-Cartesian parameterization α, θ, β
∆V =
∫
τ
dτ −
∫
τ0
dτ =
2π∫
0
1∫
0
ξ/H∫
−1
[f(Hβ)]2dβαH dαdθ−
−
2π∫
0
1∫
0
0∫
−1
[f(Hβ)]2dβαH dαdθ =
=
2π∫
0
1∫
0
ξ/H∫
−1
f2(Hβ) dβ
αHdαdθ.
For calculation of the last integral we make use of the formula for integration of expres-
sions over variable volume, which is based on application of the Taylor expansion of the
integral with variable limits of integration [20]. After realization of this type of integra-
tion in the analytical form and substitution of decompositions for ξ from requirements of
volume conservation we obtain the following expressions for ξ
(i)
:
ξ
(1)
= 0, ξ
(2)
= − f ′(0)
πf(0)
∑
i,j
aiajβ
v
ij ,
ξ
(3)
= −f ′2(0) + f(0)f ′′(0)
3πf2(0)
∑
i,j,k
aiajakγ
v
ijk.
(19)
As it is seen from the relations (19), the requirement of volume conservation in amplitude
parameters represents certain holonomic constraint, which makes it possible to eliminate
parameters ξ. Moreover, the linear term in representation of ξ is absent. Character of
variation of coefficients βv
ij (a positively defined matrix for the non-orthogonalized system
and a diagonal one with positive elements for the orthogonal system) shows that ξ
(2)
is a
function of constant sign and its sign is opposite to sign of f ′(0), i.e., ξ2 is determined by
amplitudes a2
i and inclination angle of tank walls in a vicinity of a free surface. The value
ξ
(3)
depends on curvature of the surface Σ0 in a vicinity of L0.
Realization of the procedure of elimination of the kinematic boundary condition on
a free surface is similar to the procedure for a cylindrical tank [15 – 20]. Distinction of
this procedure consists in the property that in derivations it is necessary to keep terms
ξ(ai) presented in (19), take into account additional terms of (9), which appear owing
to non-cylindrical shape of the tank cavity, and produce decompositions into series with
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58 O. S. LIMARCHENKO
respect to ξ not only unknown functions, but their products with the Jacobian of transition
to non-Cartesian parametrization, which is also a function of β.
By omitting intermediate derivation we write down dependencies bi = bi(aj , ȧk),
which are similar to the case of cylindrical cavity represent non-holonomic constraints.
However, by application of methods of nonlinear mechanics and the Galerkin method
they admit elimination from consideration the set of dependent parameters bi :
b(1)p = ȧp, b(2)p =
∑
i,j
ȧiajγ
c
ijk,
b(3)p =
∑
i,j,k
ȧiajakδ
c
ijkp, b(4)p =
∑
i,j,k,l
ȧiajakalh
c
ijklp.
(20)
The relations (19) and (20) include different coefficients, which are determined by quadra-
tures from functions ψi and Ti over the unknown free surface of liquid S0. Their explicit
form will be given below.
After determination of dependencies (19) and (20) parameters ai can be supposed as
the complete independent system of variables, which characterizes motion of the limited
volume of liquid. Now we can pass to immediate realization of the Hamilton – Ostro-
gradsky variational principle for unconstrained mechanical system, which motion is set
by the generalized coordinates ai and εk.
6. Construction of resolving system of motion equations relative to amplitude
parameters of liquid motion and parameters of translational motion of the carrying
body. We shall derive the motion equations of the system on the basis of the Hamilton –
Ostrogradsky variational principle applied to dynamics of bounded liquid volume and
a rigid body with cavity of revolution. For transition from continuum structure of the
initial model of the system rigid body – liquid to its discrete model we make use of
the Kantorovich method. Here spatial and surface integration in separate terms of the
Lagrange function is realized in variables α, θ, β. We note that calculation of volumetric
integrals in variables α, θ, β can be reduced to successive integration with analytical
derivation of integrals over liquid depth [20].
Finally this volumetric integrals are reduced to surface integrals over the undisturbed
free surface S0 from expressions in the form of expansions relative to powers of ξ. In con-
trast to the case of the cylindrical cavity occupied by liquid not only terms, which contain
the velocity potential ϕ are differentiated by β, but also terms, which contain products of
the velocity potential and the Jacobian of transition to non-Cartesian parametrisation of
the liquid domain τ0.
On the whole the general procedure of transition from the continuum structure of the
initial mechanical system body – liquid to its discrete model (it is based on application
of the Kantorovich method to the variational formulation of the problem in the form of
the Hamilton – Ostrogradsky variational principle) differs insignificantly from the case of
the cylindrical domain occupied by liquid [15 – 20]. Therefore, we do not describe this
procedure in details. Main distinction consists not in the techniques of such a transition,
but in the following two properties: first, decomposition of the velocity potential relative
to coordinate function is realized with respect to the system holds the non-flowing condi-
tion approximately, second, a group of geometrical nonlinearities defined by the relation
(19) is supplemented, which predetermine additional dependence of all natural modes of
oscillations.
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PECULIARITIES OF APPLICATION OF PERTURBATION TECHNIQUES IN PROBLEMS ... 59
As the result of application of the proposed technique we obtain the following La-
grange equation of the second kind, i.e., the motion equations of the system body – liq-
uid in amplitude parameters ai and parameters of translational motion of the carrying
body εi :
∑
i
äi
(
V 1
ir +
∑
j
ajV
2
irj +
∑
j,k
ajakV
3
irjk
)
+
+�̈ε ·
(
�U1
r +
∑
i
ai
�U2
ri +
∑
j,k
aiaj
�U3
rij +
∑
i,j,k
aiajak
�U4
rijk
)
=
=
∑
i,j
ȧiȧjV
2∗
ijr +
∑
i,j,k
ȧiȧjakV
3∗
ijrk+
+�̇ε ·
( ∑
i
ȧi
�U2∗
ir +
∑
i,j
ȧiaj
�U3∗
ijr +
∑
i,j,k
ȧiajak
�U4∗
ijkr
)
−
−g
( ∑
i
aiW
2
ir +
3
2
∑
i,j
aiajW
3
ijr + 2
∑
i,j,k
aiajakW
4
ijkr
)
, (21)
r = 1, 2, . . . , N,
ρ
Mr + Ml
∑
i
äi
(
�U1
i +
∑
j
aj
�U2
ij +
∑
j,k
ajak
�U3
ijk
)
+ �̈ε =
=
�F
Mr + Ml
+ �g − ρ
Mp + M
∑
i,j
ȧiȧj
(
�U2
ij + 2
∑
k
ak
�U3
ijk
)
. (22)
The mentioned system of equation represents the nonlinear model of dynamics of a
body and liquid, which partially fills its cavity, in the case of translational motion of the
carrying body. For construction of the system of equations (21), (22) it is necessary to
compute the following quadratures from the coordinate functions ψi and ψi :
Np =
2π∫
0
1∫
0
ψ
2
iT
2
i αdα dθ,
βv
ij = Niδij ,
γv
ijk =
2π∫
0
1∫
0
ψiψjψkTiTjTkαdα dθ,
δv
ijkl =
2π∫
0
1∫
0
ψiψjψkψlTiTjTkTlαdα dθ,
γb0
ijp =
2π∫
0
1∫
0
[(
αA0
1i
∂ψj
∂α
+ α2A0
3i
∂ψj
∂α
− αA1
4iψj
)
TiTj+
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60 O. S. LIMARCHENKO
+
1
α
A0
2iψjT
′
iT
′
j
]
ψpTp dαdθ,
γb1
ijp =
2π∫
0
1∫
0
[(
αA1
1i
∂ψj
∂α
+ α2A1
3i
∂ψj
∂α
− αA2
4iψj
)
TiTj+
+
1
α
A1
2iψjT
′
iT
′
j
]
ψpTpdα dθ,
δb1
ijkp =
2π∫
0
1∫
0
[(
A1
1iα
∂ψj
∂α
+ α2A1
3i
∂ψj
∂α
− αA2
4iψj
)
TiTj +
+
1
α
A1
2iψjT
′
iT
′
j
]
ψjkψpTkTpdα dθ,
hb2
ijklp =
2π∫
0
1∫
0
[(
αA2
1i
∂ψj
∂α
+ α2A2
3i
∂ψj
∂α
− αA3
4iψj
)
TiTj+
+
1
α
A2
2iψjT
′
iT
′
j
]
ψkψlψpTkTlTp dαdθ,
βb1
ip =
2π∫
0
1∫
0
A1
4iψpTiTpαdαdβ,
βf2
ij =
2π∫
0
1∫
0
ψiψjTiTjαdαdθf2(0),
βf3
i =
2π∫
0
1∫
0
[
α
(
∂ψj
∂α
)2
T 2
i +
1
α
ψ2
i T
′2
i + α f2ψ
2
iT
2
i
] ∣∣∣∣∣
β=0
dαdθ, (23)
γf3
ijk =
2π∫
0
1∫
0
[
α
∂ψi
∂α
∂ψj
∂α
TiTj +
1
α
ψiψjT
′
iT
′
j + α f2ψ
2
iψjTiTj
]∣∣∣∣∣
β=0
ψkTkdαdθ,
δf4
ijkl =
2π∫
0
1∫
0
[
α
2H
(
∂2ψi
∂α∂β
∂ψj
∂β
+
∂ψi
∂α
∂2ψj
∂α∂β
)
TiTj +
+
1
2αH
(
∂ψi
∂β
ψj + ψi
∂ψj
∂β
)
T ′
iT
′
j+
+
α
2H
(
2Hff ′ ∂ψi
∂z
∂ψj
∂z
+ f2 ∂
2ψi
∂z∂β
+ f2 ∂ψi
∂z
∂2ψj
∂z∂β
)
TiTj
]∣∣∣∣∣
β=0
ψkψlTkTldαdθ,
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PECULIARITIES OF APPLICATION OF PERTURBATION TECHNIQUES IN PROBLEMS ... 61
αx1
i =
2π∫
0
1∫
0
α2f
∣∣
β=0
ψiTi cos θdαdθ,
αx2
i =
2π∫
0
1∫
0
(
αf
∂ψi
∂α
Ti cos θ − fψiT
′
i sin θ
) ∣∣∣
β=0
dαdθ,
βx2
ij =
2π∫
0
1∫
0
(
αf
∂ψi
∂α
Ti cos θ − fψiT
′
i sin θ
)
ψjTj dαdθ,
γx3
ijk =
1
2H
2π∫
0
1∫
0
[ (
αHf ′ ∂ψi
∂α
+ αf
∂2ψi
∂α∂β
)
Ti cos θ−
−
(
Hf ′ψi + f
∂ψi
∂β
)
T ′
i sin θ
]∣∣∣∣∣
β=0
ψjψkTjTkdαdθ,
δx4
ijkl =
1
6H2
2π∫
0
1∫
0
[ (
H2f ′′α
∂ψi
∂α
+ 2Hf ′α
∂2ψi
∂α∂β
+ αf
∂3ψi
∂α∂β2
)
Ti cos θ−
−
(
H2f ′′ψi + 2Hf ′ ∂ψi
∂β
+ f
∂2ψi
∂β2
)
T ′
i sin θ
]∣∣∣∣∣
β=0
ψjψkψpTjTkTpdαdθ,
βz2
ij = f2(0)Niδij ,
γz3
ijk =
1
2H
2π∫
0
1∫
0
[
2ff ′Hψi + f2 ∂
2ψi
∂z∂β
]∣∣∣∣
β=0
ψjψkTiTjTkαdαdθ,
δz4
ijkl =
1
6H2
2π∫
0
1∫
0
[
4ff ′Hf2 ∂
2ψi
∂z∂β
+ 2H2
(
f ′2 + f ′′f
)
f2 ∂ψi
∂z
+
+f2 ∂3ψi
∂z∂β2
]∣∣∣∣
β=0
ψjψkψkTiTjTkTlαdαdθ.
We note that in the mentioned quadratures components of integration by α and θ
are always separated, y-components of vectors are calculated similar to x-components,
but it is necessary to substitute for − cos θ instead of sin θ and sin θ instead of cos θ (in
relations (23) expressions for y-components are omitted). On calculation of integrals we
use the following denotations, i.e., δij is the Kronecker symbol, f (i) = f (i)(0), where i
is the order of differentiation. Numerical algorithm of determination of quadrature was
based on the Gauss quadrature formula with 96 points of division. We introduced also the
following functions of the argument α in the relations (23)
A0
1i =
1
f2
ψi, A1
1i =
1
Hf2
(
∂2ψi
∂α∂β
− 2Hf ′
f
∂ψi
∂α
) ∣∣∣∣
β=0
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62 O. S. LIMARCHENKO
A2
1i =
1
2H2f2
[
∂3ψi
∂α∂β2
− 4Hf ′
f
∂2ψi
∂α∂β
+ H2
(
6
f ′2
f2
− 2
f ′′
f
)
∂ψi
∂α
] ∣∣∣∣
β=0
,
A0
2i =
1
f2
ψi
∣∣∣∣
β=0
, A1
2i =
1
Hf2
(
∂ψi
∂β
− 2Hf ′
f
ψi
) ∣∣∣∣
β=0
,
A2
2i =
1
2H2f2
[
∂2ψi
∂β2
− 4H
f ′
f
∂ψi
∂β
+ H2
(
6
f ′2
f2
− 2
f ′′
f
)
ψi
] ∣∣∣∣
β=0
,
A0
3i = −f ′
f
ψi, A1
3i =
1
Hf
[(
−Hf ′′ + H
f ′2
f
)
∂ψi
∂z
− f ′
f
∂2ψi
∂z∂β
] ∣∣∣∣
β=0
, (24)
A2
3i =
1
2H2
[ (
−H2 + 3H2 f
′′f ′
f2
− 2H2 f
′3
f3
)
∂ψi
∂z
+
+H
(
− f ′
f
+
f ′2
f2
)
∂2ψi
∂z∂β
− f ′
f
∂3ψi
∂z∂β2
]∣∣∣∣
β=0
,
A1
4i =
1
H
∂2ψi
∂z∂β
∣∣∣∣
β=0
,
A2
4i =
1
2H2
∂3ψi
∂z∂β2
∣∣∣∣
β=0
, A3
4i =
1
6H3
∂4ψi
∂z∂β3
∣∣∣∣
β=0
.
On the basis of the mentioned quadratures (23), which are calculated with application of
denotations (24), the coefficients of the motion equations (21) and (22) are determined
according to the following algorithm:
γc
ijp =
1
Np
γb0
ijp,
δc
ijkp =
1
Np
( ∑
i
γc
ijmγb0
mkp + δb1
ijkp +
g2
πg1
βb1
ipNjδjk
)
,
hc
ijklp =
1
Np
[∑
m
(
δc
ijkmγb0
mlp + δb1
mlkpγ
c
ijm+
+
g2
πg1
γc
ijmβb1
mpNkδkl
)
+ βb1
ip γ
v
jkl
g3
πg1
γb1
ijpNpδkl
]
,
V 1
ij = βf2
ij , V 2
ijk = γf3
ijk + 2
∑
m
γc
ikmβf2
mj ,
V 3
ijkl = δf4
ijkl −
g2
πg1
Nkδklβ
f3
i δij+
+
∑
m
(
2γf3
mjkγ
c
ilm + 2βf2
mjδ
c
iklm + γc
jkmγc
ilmβf2
mm
)
,
�U1
i = α1
i , U2
ij = �β2
ij +
∑
m
γc
ijmα1
m,
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PECULIARITIES OF APPLICATION OF PERTURBATION TECHNIQUES IN PROBLEMS ... 63
�U3
ijk = �γ3
ijk +
∑
m
(
δc
ijkm�α1
m + γc
ijm
�β2
mk
)
− g2
πg1
�α2
iNjδjk,
�U4
ijkl =
∑
m
(
hc
ijklm�α1
m−
− g2
πg1
γc
ijm�α2
mNkδkl + δc
ijkm
�β2
ml + γc
ilm�γc
mjk
)
− (25)
− g3
πg1
γv
jkl�α
2
i + �δ4
ijkl,
W 2
ij = e1Niδij , W 3
ijk = e2Niγ
v
ijk,
W 4
ijkl = e3δ
v
ijkl + NiNkδijδkl
(
e1g
2
2
πg2
1
− 3g2e2
πg1
)
,
V 2∗
ijr =
1
2
V 2
ijr − V 2
irj , V 3∗
ijkr = V 3
ijkr − 2V 3
irjk,
�U2∗
ir = �U2
ir − �U2
ri,
�U3∗
ijr = 2
(
�U3
ijr − �U3
rij
)
,
�U4∗
ijkr = 3
(
�U4
ijkr − �U4
rijk
)
, e1 = f2, e2 =
4
3
f ′f,
e3 =
1
2
(f ′′f + f ′2), g1 = πe1, g2 = πff ′,
g3 =
π
3
(f ′2 + ff ′′).
Basic stages of numerical realization of the suggested approach mainly coincide with
the case of cylindrical tank. However, distinction consists in the property that approxi-
mate determination of coordinate functions close to natural modes of oscillations becomes
significant component of the procedure. For determination of parameters of an auxiliary
domain ∆τ we conventionally select ∆z = 0,25, i.e., excess of the filling level of liquid
z related to the radius of the undisturbed free surface. This parameter defines the do-
main ∆τ and the surface ∆Σ on application of the method of an auxiliary domain (for
comparison we consider also variants ∆z = 0, ∆z = 0,2, ∆z = 0,3).
We consider the problem about determination of coefficients of the motion equations
for nonlinear oscillation of liquid in circle cylindrical tank as a testing one. Good con-
cordance with results of publications [10, 20, 23] was obtained, i.e., coefficients of the
motion equations coincide accurate to four significant digits.
7. Numerical examples. For verification of the suggested approach we investigate
problems about transient processes in the system tank — liquid in the case when cavity
partially filled by liquid is of revolution shape. We realize the following three variants:
the conic tank with the half-angle α =
π
4
and H = R; spherical tank for H = 0,5R
and H = R. Calculations were realized on the basis of the suggested scheme with
the following parameters of the nonlinear model n1 = 10, n2 = 6, n3 = 3 and with
application of the above determined coordinate functions for ∆z = 0,25. It is neces-
sary to note that for comparison we consider also a variant, when coordinate functions
were determined according to the classical approach, which corresponds to ∆z = 0. In
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64 O. S. LIMARCHENKO
Fig. 4. Free oscillations of liquid in the immovable conic tank.
Fig. 5. Free oscillations of liquid in the immovable spherical tank.
these cases for sufficiently short time interval numerical instability takes place (for am-
plitudes of waves about 0,2R and 0,3R we have stable numerical solution only until 5 s
and 2 s, correspondingly). Obviously, the reason of manifestation of this instability con-
sists in violation of requirement of conservation of liquid volume, what was analyzed
above.
For determination of peculiarities of development of wave generation on a free surface
of liquid three groups of problems were considered:
1) free oscillations of liquid in the immovable conic (Fig. 4) and spherical H = 0,5R
(Fig. 5) tanks, caused by initial perturbations of a free surface of liquid relative to the
natural mode a1(0) = 0,25R;
2) free oscillations of liquid in movable conic (Fig. 6), spherical H = 0,5R (Fig. 7)
and spherical H = R (Fig. 8) tanks, which can perform translational motion in the hor-
izontal plan, caused by the initial perturbation of a free surface of liquid relative to the
natural mode a1(0) = 0,25R;
3) forced oscillations of liquid in movable conic (Fig. 9) and spherical H = 01) 5R
(Fig. 10) tanks, which can perform translational motion in the horizontal plane, caused by
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PECULIARITIES OF APPLICATION OF PERTURBATION TECHNIQUES IN PROBLEMS ... 65
Fig. 6. Free oscillations of liquid in the movable conic tank.
Fig. 7. Free oscillations of liquid in the movable spherical tank H = 0,5R.
Fig. 8. Free oscillations of liquid in the movable spherical tank H = R.
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66 O. S. LIMARCHENKO
Fig. 9. Forced oscillations of liquid in movable conic tank.
Fig. 10. Forced oscillations of liquid in movable conic tank.
sudden application of rectangular force impulse with the amplitude F = 1,2(Mr + Ml)
and duration τ = 0,5 s to the quiescent system.
In all cases we accept Mr = 0, 2Ml; R = 1 m; σ and ρ were selected for water.
Figures also include time, which corresponds to the observed states of a free surface of
liquid.
From the analysis of behavior of a free surface of liquid in the case of free oscillations
of liquid in immovable conic and spherical reservoirs it is seen that the property of excess
of the wave crest over the depth of trough is manifested not so notably as in the case of
a cylindrical tank, and in some time instants it is violated at all. This is caused first of
all by the property that ξ2 < 0 as well as by the property that relatively large inclination
of tank walls in outside direction promotes conditions for liquid displacement sideways
over the undisturbed free surface larger than below it. It is seen from Figures 4 and 5
that for certain time instants perturbation of axis-symmetric natural modes is manifested
essentially, which is only consequence of presence of internal nonlinear constraints in the
system.
In the case of free oscillations of liquid in movable tank (Figures 6 and 7) we observe
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PECULIARITIES OF APPLICATION OF PERTURBATION TECHNIQUES IN PROBLEMS ... 67
Fig. 11. Development of spatial wave generation in spherical tank.
approximately the same qualitative picture of development of wave generation. Here con-
tribution of natural modes ψ9 and ψ10, which correspond to m = 1 and k = 2, is mani-
fested essentially. This is caused by the property that in movable tank perturbation of high
anti-symmetric modes occur even within the framework of the linear model. Especially
contribution of modes ψ9 and ψ10 is noticeable for the sphere H = R (Fig. 8).
On impulse force excitation of tank motion (Figures 9 and 10) nonlinear interdepen-
dencies are manifested in greater extent, i.e., contribution of axis-symmetrical and high
anti-symmetric modes is more substantial. The effect of excess of height of wave crest
over depth of trough becomes apparent more clearly than in preceding examples.
For the analysis of influence of liquid filling on motion of a carrying body we investi-
gate alteration in time of the reservoir velocity and the main vector of liquid pressure on
tank walls. As it follows from numerical results in general influence of liquid filling on
motion of a carrying body is essential. We note that at certain time instants after force
disturbance the carrying body performs motion in the direction opposite to the direction
of the applied force. This motion is caused by internal liquid sloshing and is manifested
most clearly for the sphere at t ≈ 0,8 s, i.e., on the second half-period of oscillations
of a liquid free surface. Moreover, by character oscillations differ from harmonic ones.
Variation of the horizontal component of the main vector of forces of liquid pressure on
tank walls is close to harmonic law. however, for conic cavity in a vicinity of time 2,5 s
we observe significant distortions of the harmonic law.
Results of investigation of development of spatial wave generation in movable spher-
ical reservoir (H = 0,5R) are shown in Fig. 11. We assume that initially a free surface
of liquid is excited relative to the second natural mode with the amplitude a2(0) = 0,1R
and the force impulse F = 1,1(Mr + Ml) with duration τ = 0,5 s is applied to the
tank. Figure 11 shows the picture of waves observed for time instants 0,5 s and 2,25 s
in the form of cross-sections in mutually orthogonal planes. It is easy to see that in this
case wave crests exceed depth of wave trough, contribution of axis-symmetric and high
anti-symmetric natural modes of oscillations is noticeable (they are excited only owing to
manifestation of internal nonlinear constraints in the system).
Results of the comparative analysis of behavior of reservoirs with spherical H =
= 0,5R, conical H = R and cylindrical cavities under perturbation of their motion from
the quiescent state by the rectangular force impulse F = 1,2(Mr + Ml) with duration
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68 O. S. LIMARCHENKO
Table 5
Parameter
Sphere Cone Cylinder
H = 0,5R H = R H = R
ξmax 0,25 0,331 0,230
t(ξmax) 0,55 0,60 0,56
a1 max 0,188 0,249 0,205
Vmax 0,760 0,846 0,704
t(Vmax) 0,35 0,50 0,50
Rmax 1,716 1,753 1,210
t(Rmax) 0,50 0,50 0,54
τ = 0,5 s and Mr = 0,2Ml are presented in Table 5. Here ξmax is the maximal per-
turbation of a liquid free surface on the considered time interval (until 3 s); t(ξmax) is
time when ξ = ξmax; a1max is the maximal value of amplitude of the first anti-symmetric
mode; Vmax is the maximal velocity of motion of the reservoir; t(Vmax) is time when
V = Vmax; Rmax is the maximal value of the horizontal component of the main vector of
pressure forces on tank walls; t(Rmax) is time when R = Rmax. The realized comparison
makes it possible to state that for the same relative loading and the same ratio of masses
of a reservoir and liquid the case of cylindrical reservoir results in minimal manifestation
of effects connected with internal sloshing of liquid, which becomes apparent for all con-
sidered parameters. So, the cylindrical shape of the tank superimposes greater restriction
on liquid mobility when spherical (H = 0,5R) and conical (H = R) shapes. Compari-
son of spherical (H = 0,5R) and conic (H = R) tanks testifies to the property that for
the considered filling levels the spherical shape superimposes greater restriction on liquid
mobility when the conical one. It is interestingly to note that, apparently, these restric-
tions are determined first of all by inclination of tank walls of a reservoir in a vicinity of
the undisturbed free surface.
8. Conclusions. We state the problem about oscillations of ideal liquid with a free
surface in a tank of non-cylindrical shape. Owing to specificity of the the problem we
introduce non-Cartesian coordinate system for description of the system motion. Due to
the property that it is impossible to find the exact solution of the problem about determi-
nation of natural modes of liquid oscillations in arbitrary vessel, we focus our attention on
approximate construction of coordinate functions of the problem, which are close to natu-
ral modes of liquid oscillation but in addition hold solvability conditions of the Neumann
boundary problem for the Laplace equation describing the problem about determination
of natural modes of oscillations. We propose to apply the method of successive refinement
of the solution coupled with the method of an auxiliary domain. The suggested approach
makes it possible to avoid some singular properties (of non-physical sense) of the ini-
tial problem, remove some mechanical contradictions in the mathematical statement of
the linear problem. This approach was generated on the basis of the analysis of the me-
chanical nature of the statement of the problem about free oscillations of liquid with a free
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PECULIARITIES OF APPLICATION OF PERTURBATION TECHNIQUES IN PROBLEMS ... 69
boundary and on properties of the solution of this problem obtained by different analytical
and numerical approaches.
The constructed coordinate functions satisfy boundary conditions with the required
accuracy, they “follow” tank walls above the level of the undisturbed free surface of liquid.
Application of this system of boundary function results in providing stability of numerical
procedures.
We construct decompositions of unknown variables and realize the procedure of an-
alytical elimination of the kinematical boundary condition on the free surface of liquid.
So, in this way we construct the complete system of amplitude parameters suitable for
description of dynamics of the system and which possesses property of minimality (the
number of dynamic parameters is equal to the number of mechanical degrees of freedom).
For construction of the system of motion equations we make use of the Hamilton – Os-
trogradsky variational principle. Realization of the suggested approach is conducted for
arbitrary number of coordinate functions entrained into construction of discrete model of
the system liquid — tank.
The suggested procedure was numerically realized for modes of free and forced os-
cillations. The obtained results about wave generation of a free surface, alteration of
the tank velocity and dynamic interaction of liquid with tank walls are evidence of good
reflection of general regularities of the system behavior. Comparison of properties of dy-
namic mobility of liquid in conic, spherical and cylindrical reservoirs made it possible to
draw conclusion about factors, which determine restriction of liquid mobility in tanks.
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Received 20.09.2006
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|
| id | umjimathkievua-article-3291 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:39:46Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/bc/1d3da85e643fa7681325f8d12f9b23bc.pdf |
| spelling | umjimathkievua-article-32912020-03-18T19:50:22Z Specific features of application of perturbation techniques in problems of nonlinear oscillations of a liquid with free surface in cavities of noncylindrical shape Особливості застосування методів збурень у задачах про нелінійні коливання рідини з вільною поверхнею в порожнинах нециліндричної форми Limarchenko, O. S. Лимарченко, О. С. We consider the problem of nonlinear oscillations of an ideal incompressible liquid in a tank of a body-of-revolution shape. It is shown that the ordinary way of application of perturbation techniques results in the violation of solvability conditions of the problem. To avoid this contradiction we introduce some additional conditions and revise previously used approaches. We construct a discrete nonlinear model of the investigated problem on the basis of the Hamilton-Ostrogradskii variational formulation of the mechanical problem, preliminarily satisfying the kinematic boundary conditions and solvability conditions of the problem. Numerical examples testify to the efficiency of the constructed model. Розглядається задача про нелінійні коливання ідеальної нестисливої рідини в резервуарі в формі тіла обертання. Показано, що звичайний шлях застосування методів збурень приводить до порушення умов розв'язності задачі. Для уникнення цієї суперечності вводяться додаткові умови і переглядаються підходи, які використовувалися раніше. Побудова дискретної нелінійної моделі виконується на основі формулювання механічної задачі у вигляді варіаційного принципу Гамільтона - Остроградського з попереднім виконанням кінематичних граничних умов і умов розв'язності задачі. Числові приклади підтверджують ефективність побудованої моделі. Institute of Mathematics, NAS of Ukraine 2007-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3291 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 1 (2007); 44–70 Український математичний журнал; Том 59 № 1 (2007); 44–70 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3291/3327 https://umj.imath.kiev.ua/index.php/umj/article/view/3291/3328 Copyright (c) 2007 Limarchenko O. S. |
| spellingShingle | Limarchenko, O. S. Лимарченко, О. С. Specific features of application of perturbation techniques in problems of nonlinear oscillations of a liquid with free surface in cavities of noncylindrical shape |
| title | Specific features of application of perturbation techniques in problems of nonlinear oscillations of a liquid with free surface in cavities of noncylindrical shape |
| title_alt | Особливості застосування методів збурень у задачах про нелінійні коливання рідини з вільною поверхнею в порожнинах нециліндричної форми |
| title_full | Specific features of application of perturbation techniques in problems of nonlinear oscillations of a liquid with free surface in cavities of noncylindrical shape |
| title_fullStr | Specific features of application of perturbation techniques in problems of nonlinear oscillations of a liquid with free surface in cavities of noncylindrical shape |
| title_full_unstemmed | Specific features of application of perturbation techniques in problems of nonlinear oscillations of a liquid with free surface in cavities of noncylindrical shape |
| title_short | Specific features of application of perturbation techniques in problems of nonlinear oscillations of a liquid with free surface in cavities of noncylindrical shape |
| title_sort | specific features of application of perturbation techniques in problems of nonlinear oscillations of a liquid with free surface in cavities of noncylindrical shape |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3291 |
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