Coupling of a swirl-type resonant sloshing and a mean rotational flow
Referring to experimental results by Prandtl (1949) and Hutton (1964) as well as more recent model tests by Royon-Lebeaud, Hopfinger & Cartellier (2007) and Reclari (2013), a Moiseev-type asymptotic almost periodic (steady-state) solution of a resonant sloshing problem is derived to show...
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Інститут математики НАН України
2017
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| author | Timokha, A. N. Тимоха, Н. Н. Тімоха, О. М. |
| author_facet | Timokha, A. N. Тимоха, Н. Н. Тімоха, О. М. |
| author_institution_txt_mv | [
{
"author": "A. N. Timokha",
"institution": "Institute of Mathematics"
}
] |
| author_sort | Timokha, A. N. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2018-02-13T11:53:50Z |
| description | Referring to experimental results by Prandtl (1949) and Hutton (1964) as well as more recent model tests by Royon-Lebeaud, Hopfinger & Cartellier (2007) and Reclari (2013), a Moiseev-type asymptotic almost periodic (steady-state) solution of a resonant sloshing problem is derived to show that a time-averaged rotational liquid flow, if occurs, becomes nonlinearly coupled with the dominant swirl-type wave component. The coupling appears as a necessary solvability condition and consists of nonlinear (differential) equations with respect to four amplitude parameters of the two lowest natural sloshing modes and the time-averaged velocity field, which is governed by a partial differential equation of the first order. Finding its unique solution requires to know the (Stokes) steady-streaming. |
| first_indexed | 2026-08-04T01:06:04Z |
| format | Article |
| fulltext |
Збiрник праць Iнституту математики НАН України 2017, т. 14, №2, 205–219
УДК 532.595
Coupling of a swirl-type resonant
sloshing and a mean rotational flow∗
A.N. Timokha 1,2
1 Institute of Mathematics of NAS of Ukraine, Kyiv;
2 Centre of Excellence AMOS, Norwegian University of Science and
Technology, Trondheim, Norway; atimokha@gmail.com
Referring to experimental results by Prandtl (1949) and Hutton (1964) as
well as more recent model tests by Royon-Lebeaud, Hopfinger & Cartel-
lier (2007) and Reclari (2013), a Moiseev-type asymptotic almost periodic
(steady-state) solution of a resonant sloshing problem is derived to show
that a time-averaged rotational liquid flow, if occurs, becomes nonlinearly
coupled with the dominant swirl-type wave component. The coupling ap-
pears as a necessary solvability condition and consists of nonlinear (dif-
ferential) equations with respect to four amplitude parameters of the two
lowest natural sloshing modes and the time-averaged velocity field, which
is governed by a partial differential equation of the first order. Finding its
unique solution requires to know the (Stokes) steady-streaming.
Iз посиланням на експериментальнi результати Прандтля (1949) та
Хаттона (1964), а також на бiльш новi модельнi випробування Ройон–
Лебод, Хопфiнгера & Картельєра (2007) та Рекларi (2013), виводится
асимптотичний майже перiодичний (усталений) розв’язок типу Моiсє-
єва резонансної задачi про хлюпання рiдини з метою показати, що усе-
реднена кругова течiя (якщо така виникає) є нелiнiйним чином пов’я-
заною з домiнантною компонентою кругової хвилi. Цей зв’язок виникає
як необхiдна умова розв’язностi задачi та приймає форму нелiнiйних
диференцiальних рiвнянь вiдносно чотирьох амплiтудних параметрiв
для двох перших натуральних форм коливань рiдини та усередненого
поля швидкостей, яке описується диференцiальним рiвнянням у ча-
стинних похiдних першого порядку. Аби мати единий розв’язок остан-
нього рiвняння, треба знати усталену вторинну течiю, обумовлену по-
верхневим шаром.
∗The work was partly supported by the Grant № 0117U004077. The author also
acknowledges the financial support of the Centre of Autonomous Marine Operations
and Systems (AMOS) whose main sponsor is the Norwegian Research Council (Project
number 223254–AMOS).
c© Timokha A. N., 2017
206 Timokha A. N.
Introduction
Mentioning an anecdote about P.-S. Laplace, who, as the anecdote tells,
used a glass of wine to show that resonantly forced swirl-type sloshing
leads, contradictory to some hydrodynamic theorems, to a non-negligible
rotational fluid-particles transport, Prandtl (1949) [10] conducted a series
of dedicated model tests on resonant sloshing in an upright circular cylin-
drical tank, which performs an orbital (circular) horizontal motion with
the forcing frequency close to the lowest natural sloshing frequency. The
experimental observations confirmed the aforementioned phenomenon by
Laplace. In 1964, Hutton [5] repeated Prandtl’s experiments by using a
longitudinally excited tank and the fact that the swirl-type stable steady-
state sloshing is possible in a small vicinity of the primary resonant zone
for that excitation type. The Hutton’ paper reports visual observations
and measurements of angular particle-transport velocity versus the ver-
tical/horizontal probe position and the surface wave amplitude. Because
this velocity was measured far from the tank walls/bottom and its nondi-
mensional values had a clearly quadratic character with respect to the
nondimensional wave amplitude, Hutton logically related the mean angu-
lar particle transport to the angular Stokes drift, which is also a second-
order hydrodynamic phenomenon in the Lagrangian specification [4].
However, his idea was not supported by the experiments. The most
attractive discrepancy between the measured and theoretical particle-
transport velocity was that the Stokes drift exponentially decays to the
bottom but the experimental values weakly depend on the vertical probe
position. Royon-Lebeaud, Hopfinger & Cartellier [14] followed Hutton [5]
by applying a harmonic horizontal excitation to the tank. They ob-
served and documented the Lagrange–Prandtl–Hutton phenomenon writ-
ing down that “nonlinear (swirling) waves can transfer angular momen-
tum to the whole liquid column that starts to rotate”. Because, in con-
trast to the orbital tank excitation [10], the longitudinal forcing causes
the swirl-type sloshing only in the primary resonance zone, the works [5]
and [14] were not able to examine whether the angular mean transport
is actually due to the resonant sloshing.
A growing interest to the liquid sloshing dynamics in laboratory tanks
(incl. bioreactors) initialised a series of dedicated model tests whose
particular focus was a mean (steady) rotational flow accompanying the
swirl-type wave motions. Because the related applications deal with or-
bital tank excitations of various frequencies and amplitudes (radii of the
Coupling of a swirl-type resonant sloshing and a mean rotational flow 207
orbits), namely, with the Prandtl’ case, the corresponding novel experi-
mental reports in [1,2,11,12,15] documented what happens in the primary
resonance zone, away from any resonances (linear sloshing theory is appli-
cable), and where the so-called secondary resonances matter [3]. All the
authors discussed the steady-streaming flow due to the oscillatory Stokes
boundary layers on the wetted tank walls/bottom and below the free sur-
face but, since theoretical description of the steady-streaming looks an
open problem in three-dimensional context [1,13], the actually-done the-
oretical estimates were based on the angular Stokes drift prediction (the
Lagrangian contribution to the mean flow). As matter of fact, they fol-
lowed Hutton [5] and, therefore, a satisfactory agreement was found away
from the resonance cases, but, as expected, the measured and theoretical
results differ in the studied resonant zones.
Staying within the framework of the Eulerian specification of fluid
flows, the present paper constructs a mathematical theory, which clari-
fies coupling the time-averaged (mean rotational) flow and the resonant
swirling waves in an upright circular cylindrical tank when the forcing fre-
quency is close to the lowest natural sloshing frequency (the primary reso-
nance). Elliptical orbital horizontal tank motions are assumed, for which
rotary and longitudinal excitations are two limit cases. The constructed
theory assumes defined the Stokes steady-streaming at the wetted tank
surface and the mean free surface and uses the Moiseev-type asymptotic
technique [9]. The latter technique is normally employed for analysing
the steady-state sloshing with irrotational flows when the forcing fre-
quency is close to the lowest natural sloshing frequency and there are no
secondary resonances [3]. The present study admits rotational flows of
an incompressible inviscid liquid. According to the Moiseev asymptotic
technique, the lowest-order asymptotic wave component is of the order
O(ǫ1/3), where O(ǫ) characterises the forcing amplitude. This means that
the lowest-order steady flow component W equals to O(ǫ2/3). Focusing,
as in [3], on four amplitude parameters associated with the O(ǫ1/3) ampli-
fications of the two lowest natural sloshing modes by the primary cosine-
and sine- harmonic components and W , derives four differential equa-
tions with respect to the four amplitude parameters (differentiation by
the slow time) and a nonlinear partial differential equation of the first
order with respect to W . The latter equation is defined in the hydro-
static liquid domain and appear as a necessary solvability condition of
the original free-surface problem. The coupling occurs, if and only if, a
swirl-type wave is realised. The differential equation for W requires the
208 Timokha A. N.
corresponding inhomogeneous boundary condition, which should, most
probably, follow from the Stokes boundary layer steady-streaming. Be-
cause this boundary condition is the only inhomogeneous quantity in
the solvability condition with respect to W , the constructed theory de-
scribes, in fact, a physical mechanism how and why the Stokes boundary
layer and associated steady-streaming generate the mean rotational flow
in bulk.
1 Statement
(b)
1
(t)
η
2
(t)
0
0
0
0
x
y
(t)Σ
Q (t)
S(t)
z
Σ
z
x
y
Q
V
B
(a)
η
Figure 1. Panel (a): Upright circular cylindrical tank moves horizontally
along an elliptic orbit so that the nondimensional tank translatory velocity is
vO(t) = (vO1(t), vO2(t), 0) = (η̇1(t), η̇2(t), 0), where η1(t) = η1a cos t, η2(t) =
η2a sin t. Coordinate system is rigidly fixed with the tank so that the mean free
surface Σ0 belongs to Oxy and the origin is in the centre of Σ0. Panel (b): The
hydrostatic liquid shape: Q0 is the mean liquid domain, Σ0 is the mean free
surface, V0 is the hydrostatically wetted walls, and B0 is the bottom.
An upright circular cylindrical rigid tank is partially filled with an
inviscid incompressible liquid; rotational flows are allowed. The tank
moves horizontally and translatory along an elliptic orbit to excite a res-
onant liquid sloshing (the orbital frequency is close to the lowest natural
sloshing frequency). The liquid sloshing dynamics is considered in the
non-inertial tank-fixed coordinate system Oxyz (cylindrical coordinates
(r, θ, z)) so that Oxy-plane coincides with the mean free surface Σ0 and
Oz passes through the centre of Σ0 (V0 is the mean wetted tank walls, Q0
is the mean liquid domain and B0 is the tank bottom). Figure 1 intro-
Coupling of a swirl-type resonant sloshing and a mean rotational flow 209
duces the main geometric notations including the translatory tank veloc-
ity vO(t) = (η̇1(t), η̇2(t), 0) = (−η1a sin t, η2a cos t, 0), the time-depending
liquid domain Q(t), and the free surface Σ(t). The nondimensional forc-
ing amplitudes η1a and η2a are small, i.e.
η1a ∼ η2a = O(ǫ) ≪ 1. (1)
The nondimensional sloshing problem is formulated by adopting the
characteristic linear size, time and mass
l∗ = r0/k, t∗ = 1/σ and m∗ = ρl3∗, (2)
respectively, where r0 is the tank radius, σ is the forcing frequency, ρ is
the liquid density, and k is the lowest positive root of the transcedental
equation J ′
1(k) = 0 (J1 is the Bessel function of the first kind).
Remark 1.1. Using the denominator k in (2) (typical characteristic di-
mension equals to r0 [3]) is needed to get analytically simpler expressions
for the two lowest degenerating natural sloshing modes, which take now
the form
J1(r)Z(z) cos θ and J1(r)Z(z) cos θ; Z(z) = cosh(z + h)/ sinhh, (3)
where h is the nondimensional mean liquid depth; the natural sloshing
frequencies are computed by
σ2
Mi =
g
l∗
kMi tanh
(
kMi
h
l∗
)
, J ′
M (kMik) = 0; M ≥ 0, i ≥ 1 (4)
(according to the chosen normalisation, k11 = 1; g is the gravity acceler-
ation).
Within the framework of the Eulerian specification, the liquid flow is
described by the velocity field v(r, θ, z, t), the equation z = ζ(r, θ, t) de-
termines the free surface and the pressure field is defined by p(r, θ, z, t).
These three unknowns should be found from the corresponding free-
surface problem, which consists of the kinematic and dynamic parts. The
kinematic part includes the continuity equation and the normal-velocity
boundary conditions
div v = 0 in Q(t), (5a)
v · n = vO · n on S(t), (5b)
210 Timokha A. N.
v · n = vO · n+ ζ̇/
√
1 + (∇ζ)2 on Σ(t), (5c)
(henceforth, the dot denotes the partial time derivative), where n is the
outer normal. In addition, one should require the volume conservation
condition ∫
Q(t)
dQ = const. (6)
The dynamic part includes the Euler equation and the dynamic (con-
stant–pressure) condition. The nondimensional Euler equation in the
(tank-fixed) non-inertial coordinate system takes the form [6]:
v̇ − (v − vO)× rotv +∇
[
1
2 |v|
2 − v · vO + p+ ḡz
]
= 0 in Q(t), (7)
where ḡ = g/(l∗σ
2) is the nondimensional gravity acceleration. The
constant-pressure condition reads as
p = 0 on Σ(t). (8)
Remark 1.2. Because the vorticity ω = rotv is not zero, one can
introduce the vorticity equation written (after applying the rotor operation
to the Euler equation (7)) in the form
ω̇ = rot [(v − vO)× ω] in Q(t). (9)
After solving (9), v may be restored by using the Biot-Savart law.
2 Almost steady-state asymptotic solution
The Moiseev-type asymptotic solution [3, 9] implies that, if the
forcing amplitude has the order O(ǫ) (1), the dominant wave amplitude
response is associated with the lowest (primary excited) natural slosh-
ing modes (3) of the order O(ǫ1/3). In addition, the Moiseev detuning
condition
Λ =
σ2
11
σ2
− 1 = O(ǫ2/3) (10)
is required, which expresses the closeness of the forcing frequency to
the lowest natural sloshing frequency σ11. Employing the multi-timing
procedure also introduces the “quick” 2π periodic oscillations by t and
the slow time variable [7, 8]
τ = 1
2ǫ
2/3t. (11)
Coupling of a swirl-type resonant sloshing and a mean rotational flow 211
Asymptotic almost-periodic solution of the free-surface problem (5)–
(8) is posed in terms of O(ǫ1/3) where each summand is a function of
spatial variables, τ , and it is 2π-periodic by t. The velocity field
v = v1/3(r, θ, z, t; τ) + v2/3(r, θ, z, t; τ) + v3/3(r, θ, z, t; τ) + ...
= ∇φ1/3 +
[
∇φ2/3 +w2/3
]
+
[
vO +∇φ3/3 +w3/3
]
+ ... (12)
has the lowest-order asymptotic quantity identical to that for irrota-
tional flows but the lowest-order rotational flow component emerges in the
second-order approximation. The free-surface elevations and the pressure
are posed as
ζ(r, θ, t; τ) = ζ1/3 + ζ2/3 + ζ3/3 + ..., (13a)
p(r, θ, z, t; τ) = −ḡz + p1/3 + p2/3 + p3/3 + ... , (13b)
where p0/3 = −ḡz is the hydrostatic pressure.
The lowest-order asymptotic component can be taken from [3].
By introducing the four slowly-varying amplitude parameters a(τ) ∼
ā(τ) ∼ b̄(τ) ∼ b(τ) = O(ǫ1/3) at the primary harmonics by t, the compo-
nent becomes expressed by the two lowest natural sloshing modes and,
accounting for specific normalisation (2), it takes the form
φ1/3(r, θ, z, t; τ) = Z(z)J1(r)
[
cos θ (−a(τ) sin t+ ā(τ) cos t)
+ sin θ (−b̄(τ) sin t+ b(τ) cos t)
]
, (14a)
ζ1/3(r, θ, t; τ) = J1(r)
[
cos θ (a(τ) cos t+ ā(τ) sin t)
+ sin θ (b̄(τ) cos t+ b(τ) sin t)
]
, (14b)
p1/3(r, θ, z, t; τ) = −φ̇1/3 = Z(z)J1(r)
[
cos θ (a(τ) cos t+ ā(τ) sin t)
+ sin θ (b̄(τ) cos t+ b(τ) sin t)
]
; (14c)
v1/3 = ∇φ1/3 = [−a(τ) sin t+ ā(τ) cos t] va(r, θ, z)
+
[
−b̄(τ) sin t+ b(τ) cos t
]
vb(r, θ, z), (14d)
212 Timokha A. N.
where
va(r, θ, z) =
(
J ′
1(r) cos θZ(z),−
J1(r)
r
sin θZ(z), J1(r) cos θZ
′(z)
)
,
vb(r, θ, z) =
(
J ′(r) sin θZ(z),
J1(r)
r
cos θZ(z), J1(r) sin θZ
′(z)
) (15)
are considered in the cylindrical coordinate frame whose three unit vec-
tors are r̂, θ̂, and ẑ, respectively.
The lowest-order solution (14) satisfies (5)–(7) within to higher-order
terms. Inserting (14) into the dynamic boundary condition (8) leads to
(Z(0)− ḡ)
[
cos θ(a(τ) cos t+ ā(τ) sin t)+ sin θ(b̄(τ) cos t+ b(τ) sin t)
]
= 0,
which is, globally, of the order O(ǫ/3/3) due to the Moiseev asymptotic
detuning (10), i.e.
Z(0)− ḡ = Z(0)
[
1− g
l∗Z(0)
1
σ2
]
= Z(0)
[
1− σ2
11
σ2
]
= O(ǫ2/3). (16)
Classification of the resonant surface wave regimes may be
done by using the lowest-order approximation of the free surface motions
(14b). As explained in [3], one classifies either standing or swirling wave
type associated with
ab− āb̄ ≡ 0 and ab− āb̄ 6≡ 0, (17)
respectively. Swirling (rotary) wave implies a progressive angular wave
motions. Both wave types are extensively discussed in [3] and Ch. 9 of [4].
The mean (time-averaged) flow. According to (12), (13), the
rotational flow velocity field component w(r, θ, z, t; τ) and the vortex
ω(r, θ, z, t; τ) are of the order O(ǫ2/3) so that
w = w2/3 +w3/3 + ..., ω = rotv = rotw = ω2/3 + ω3/3 + ... .
Furthermore, irrotational wave motions in bulk (associated with ∇φk/3)
cannot generate a mean flow and, therefore, one can relate the mean
(time-averaged) velocity field to w2/3, i.e.
W (r, θ, z; τ) = w2/3 = 〈v〉t , Ω(r, θ, z; τ) = ω2/3 = 〈ω〉t = rotw2/3. (18)
Coupling of a swirl-type resonant sloshing and a mean rotational flow 213
Because w can be restored from ω by using the Biot–Savart law, our
forthcoming focus will be on the vortex ω. Using the vorticity equation
(9) in the (3/3)-approximation,
ω̇3/3 = rot
[
v1/3 ×Ω
]
, (19)
where v1/3 is given by (14d) and
〈
ω3/3
〉
t
=
〈
ω̇3/3
〉
t
= 0 (according to
definition (18)), gives the solution
ω3/3(r, θ, z, t; τ) = (a(τ) cos t+ ā(τ) sin t) rot [va(r, θ, z)×Ω(r, θ, z; τ)]
+
(
b̄(τ) cos t+ b(τ) sin t
)
rot [vb(r, θ, z)×Ω(r, θ, z; τ)] , (20)
which is a function of the mean vortex Ω(r, θ, z; τ).
To find Ω(r, θ, z; τ), the time-averaged (4/3)-approximation
〈
1
2ǫ
2/3∂τΩ+ ω̇4/3 = rot
[
v1/3 × ω3/3 + (∇φ2/3 +W )×Ω
]〉
t
(21)
is needed, which leads, accounting for (18), the necessary solvability con-
dition
1
2ǫ
2/3∂τΩ = rot [W ×Ω]
+ 1
2 (ab− āb̄) rot [vb × rot (va ×Ω)− va × rot (vb ×Ω)] in Q0, (22)
where divΩ = div rotW ≡ 0 and vb,va are defined by (15).
Remark 2.1. Getting a unique W from Ω by using the Biot-Savart law
requires an appropriate boundary condition for W . If exist, the restored
W converts (22) to a nonlinear integral-and-differential equation with
respect to Ω. Alternatively, substituting Ω = rotW into (22) and addiing
the aforementioned boundary conditions transform (22) to a boundary
value problem.
Remark 2.2. The mean vortex Ω is coupled with sloshing, if and only
if, ab− āb̄ 6≡ 0, namely, only for swirl-type wave regimes (17).
Remark 2.3. Within the framework of the adopted hydrodynamic model,
one can suggest the zero boundary condition
W · n = 0 on S0 +Σ0, k ≥ 2. (23)
214 Timokha A. N.
However, this condition would lead, generally speaking, to the trivial solu-
tion W = Ω = 0. To have a non-trivial one, an inhomogeneous boundary
condition is needed instead of (23). Appropriate inhomogeneous condi-
tion follows from the Stokes steady-streaming at the wetted tank surface
and beneath the free surface. Because the steady-streaming is of the sec-
ond order O(ǫ2/3) with respect to the lowest-order velocity field (14d), its
usage for the second-order variable W is consistent with the Moiseev-type
asymptotic technique and one can postulate
W = 1
2 (ab − āb̄)W 0 on walls/bottom/free surface walls. (24)
Dedicated studies on the analytical form (24) are needed.
Remark 2.4. When introducing
Ω(r, θ, z; τ) = Ωr r̂ +Ωθ θ̂ +Ωz ẑ (25)
the rot-term at the 1
2 (ab − āb̄) multiplier of (22) reads as
−
[
Z2(z)f(r)− J2
1 (r)
r2 sinh2 h
]
∂θΩ
− θ̂
(
2Ωr
[
3g(r)Z2(z) +
g1(r)
sinh2 h
]
− rf(r) [Z2(z)]′zΩz
)
(26)
where
g(r) =
(rJ ′
1(r) − J1(r))
2
r4
, f(r) = g(r) +
J2
1 (r)
r2
,
g1(r) =
J1(r)(rJ
′
1(r)− J1(r))
r2
. (27)
The graphs of f(r) > 0, g(r) > 0 and g1(r) by (27) are demonstrated in
figure 2. One can prove that
f(r) =
1
4
exp
(
−6
∫ r
0
g(r)
rf(r)
dr
)
. (28)
The secular (solvability) equations for the four dominant ampli-
tude parameters a(τ), ā(τ), b̄(τ), b(τ) were derived in [3] for the potential
flow model. The procedure can be generalised to the studied case.
Coupling of a swirl-type resonant sloshing and a mean rotational flow 215
k
2
1
3
−0.15
−0.05
0
0.1
0.15
0.2
0.25
1 2 3 4 5 0
0.05
−0.1
Figure 2. The graphs of f(r) (marked by 1), g(r) (by 2) and g1(r) (by 3).
Using the kinematic boundary condition (5c) gives
∂zφ3/3 = ζ̇3/3 +
1
2ǫ
2/3∂τ ζ1/3
+ ∂rζ1/3Wr +
1
r∂θζ1/3Wθ − ∂zWzζ1/3 + n.p. on Σ0, (29)
where the framed quantity corresponds to the nonlinear potential flow
components.
The Euler equation (7) leads to
(
ẇ3/3 −∇φ1/3 ×Ω−∇Ψ
)
+∇
[
p3/3 + φ̇3/3 +
1
2ǫ
2/3∂τφ1/3
+v̇O · x+∇φ1/3 · ∇φ2/3 +∇φ1/3 ·W +Ψ
]
= 0 in Q0, (30)
where x = (r, θ, z) and an auxiliary function Ψ is introduced to provide
the both summands equal to zero. Applying the divergence to ẇ3/3 −
∇φ1/3×Ω−∇Ψ = 0 and adding the zero boundary conditionsw3/3·n = 0
on Σ0 +B0 + V0 give
Ψ = [a sin t− ā cos t]Ψa(r, θ, z; τ) + [b̄ sin t− b cos t]Ψb(r, θ, z; τ), (31)
where
∇2Ψa,b = div[va,b×Ω] in Q0, ∇Ψa,b ·n = [va,b×Ω] ·n on V0+Σ0+B0.
(32)
The second summand of (30) derives the (3/3) pressure approximation
p3/3 = −φ̇3/3 − 1
2ǫ
2/3∂τφ1/3
216 Timokha A. N.
− v̇O · x−∇φ1/3 · ∇φ2/3 −∇φ1/3 ·W −Ψ − C(t), (33)
where C(t) is an arbitrary time-depending function.
The dynamic boundary condition (8) in the (3/3)-approximation trans-
forms to
p3/3 + ∂zp0/3ζ3/3 + ∂zp2/3ζ1/3 + ∂zp1/3ζ2/3 +
1
2∂
2
zp1/3ζ
2
1/3
+
[
p1/3 + ∂zp0/3ζ1/3
]
= C(t) on Σ0, (34)
where the square brackets expression is of the O(ǫ3/3) order due to the
Moiseev detuning (as remarked ahead of (16)) but the rotational flow
components are only due to p3/3 by (33).
Finding φ1/3 from (29), substituting the result into (33) and us-
ing all together in (34) to get projections of (34) on J1(r) cos θ cos t,
J1(r) cos θ sin t, J1(r) sin θ cos t, and J1(r) sin θ sin t on Σ0 derive the fol-
lowing four secular equations (the solvability condition)
ǫ2/3ā′ + a
(
Λ +m1(a
2 + ā2 + b̄2) +m3b
2
)
+ ā
(
(m1 −m3)bb̄+ Va,cos[W ]
)
+ bVb,cos[W ] = ǫx, (35a)
− ǫ2/3a′ + ā
(
Λ +m− 1(a2 + ā2 + b2) +m3b̄
2
)
+ a
(
(m1 −m3)bb̄− Va,cos[W ]
)
− b̄Vb,cos[W ] = 0, (35b)
− ǫ2/3b̄′ + b
(
Λ +m1(b
2 + b̄2 + ā2) +m3a
2
)
+ b̄ ((m1 −m3)aā− Vb,sin[W ])− aVa,sin[W ] = ǫy, (35c)
ǫ2/3b′ + b̄
(
Λ +m1(b
2 + b̄2 + a2) +m3ā
2
)
+ b ((m1 −m3)aā+ Vb,sin[W ]) + āVa,sin[W ] = 0, (35d)
where m1 and m3 are the h-dependent coefficients, which are, within to
the adopted normalisation, the same as in [3],
ǫx = η1aP, ǫy = η2aP ; P =
1
Z(0)||J1||2
∫ k
0
r2J1(r)dr,
||J1||2 =
∫ k
0
rJ2
1 (r)dr,
(36)
Coupling of a swirl-type resonant sloshing and a mean rotational flow 217
and Va,cos[W ], Vb,cos[W ], Va,sin[W ] and Vb,sin[W ] and the linear opera-
tors acting on W as follows
V(∗1),(∗2)[W ] =
1
π||J1||2
∫ k
0
∫ π
−π
r [ (∗2)θ ] J1(r)V(∗1)[W ] dθdr, (37)
in which
Va[W ] =
[
2J ′
1(r) cos θWr −
2
r
J1(r) sin θWθ
+
J1(r) cos θ
Z(0)
(Wz − ∂zWz)−
Ψa[W ]
Z(0)
]
z=0
, (38a)
Vb[W ] =
[
2J ′
1(r) sin θWr +
2
r
J1(r) cos θWθ
+
J1(r) sin θ
Z(0)
(Wz − ∂zWz)−
Ψb[W ]
Z(0)
]
z=0
. (38b)
In summary, the (Stokes) steady-streaming causes a mean rotational
flow in the liquid bulk for the swirl-type sloshing. The swirl-type sloshing
and the mean rotational flow are coupled and described by the nonlinear
differential equations (35) with respect to the dominant wave amplitude
parameters a(τ), ā(τ), b̄(τ), b(τ) ∼ O(ǫ1/3) and (22) governing W .
3 Conclusion
By assuming known a steady-streaming at the wetted tank walls/bottom
and beneath the free surface and using an inviscid incompressible liq-
uid with rotational flows, we derive the governing equations coupling the
wave amplitude parameters and the corresponding mean (rotational) liq-
uid flow in bulk. The Moiseev-type asymptotic technique is employed
for an upright circular cylindrical tank orbitally forced with the forcing
frequency close to the lowest natural sloshing frequency. The coupling
occurs, if and only if, a swirl-type sloshing is realised.
218 Timokha A. N.
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on flow characteristics // AIChE Journal.–– 2014.–– 60.–– P. 3951–3968.
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1. Голуб А.П., Лисенко Л.О.
2. Кіфоренко Б.М., Ткаченко Я.В., Васильєв І.Ю.
Постановка задачі
Оптимальні керування
Результати розрахунків
3. Коломійчук О.П., Новицький В.В.
Вступ
Ідентифікація системи з кососиметричною невиродженою матрицею коефіцієнтів методом квазілінеаризації
4. Кононов Ю.Н., Джуха Ю.А.
Введение
Постановка задачи
Метод решения
Собственные частоты совместных колебаний упругой мембраны и жидкости
Устойчивость осесимметричных колебаний упругой мембраны при перегрузке
5. Константінов О.В., Новицький В.В.
Математична модель механічної системи ``резервуар – рідина з вільною поверхнею''
Побудова програмного керування та керування зі зворотним зв'язком
Результати чисельного моделювання
6. Лимарченко О.С., Нефьодов О.О.
Вступ
Математична модель системи
Результати чисельного моделювання
Висновки
7. Лимарченко В.О., Лимарченко О.С., Сапон М.М.
Вступ
Математична модель системи
Аналіз числових прикладів
Висновки
8. Луковський І.О.
Крайова задача теорії просторового руху резервуара, цілком заповненого ідеальною нестисливою рідиною
Варіаційний принцип у задачі про просторовий рух пружного резервуара, цілком заповненого рідиною
Визначення сил взаємодії між пружними стінками резервуара і рідиною
9. Мазко О.Г., Кусій С.М.
Вступ
Допоміжні твердження
Динамічний регулятор по вимірюваному виходу
Зважений рівень гасіння обмежених збурень.
Динамічний регулятор зі збуреннями
Дискретні системи з керованими і спостережуваними виходами
Приклад. Двомасова механічна система
Висновок
10. Працьовитий М.В., Свинчук О.В.
Вступ
Основний об'єкт
Розподіл значень функції f(x) при заданому розподілі випадкового аргументу
11. Сатур О.Р.
Існування граничних координат
Нерухомі точки динамічної системи конфлікту з притягальною взаємодією
12. Солодун А.В.
Постановка задачи
Нелинейные модальные системы
Модальные представления и
Кинематические и динамические уравнения
Нелинейная форма модальных уравнений
Бесконечномерная система нелинейных асимптотических модальных уравнений третьего порядка
Асимптотика Моисеева-Нариманова
Общие бесконечномерные нелинейные асимптотические модальные уравнения
13. Сосницький C.П.
Рівняння руху для обмеженої задачі трьох тіл
Про деякі важливі рівності в еліптичній обмеженій задачі трьох тіл
Про рух малої частки по координаті
14. Троценко Ю.В.
Постановка задачи
Вариационная формулировка задачи
Построение решений
Некоторые результаты расчетов
15. Тугай Г.В.
Попередні відомості
Побудова матриці Якобі, асоційованої з сингулярно збуреним оператором
Висновки
16. Raynovskyy I.A., Timokha A.N.
Introduction
Statement of the problem
Linear damping coefficients
Steady-state resonant solution
Response curves in the (/11,A,B) space
Conclusions
17. Timokha A.N.
Statement
Almost steady-state asymptotic solution
Conclusion
|
| id | oai:trim.imath.kiev.ua:article-344 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:06:04Z |
| publishDate | 2017 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/83/4e9d3d9351c9299aa23ce634476d8683.pdf |
| spelling | oai:trim.imath.kiev.ua:article-3442018-02-13T11:53:50Z Coupling of a swirl-type resonant sloshing and a mean rotational flow Сопряжение вихревого резонансного плескания и усредненного вращательного потока Поєднання вихрових резонансних коливань та усередненого обертового потоку Timokha, A. N. Тимоха, Н. Н. Тімоха, О. М. Referring to experimental results by Prandtl (1949) and Hutton (1964) as well as more recent model tests by Royon-Lebeaud, Hopfinger & Cartellier (2007) and Reclari (2013), a Moiseev-type asymptotic almost periodic (steady-state) solution of a resonant sloshing problem is derived to show that a time-averaged rotational liquid flow, if occurs, becomes nonlinearly coupled with the dominant swirl-type wave component. The coupling appears as a necessary solvability condition and consists of nonlinear (differential) equations with respect to four amplitude parameters of the two lowest natural sloshing modes and the time-averaged velocity field, which is governed by a partial differential equation of the first order. Finding its unique solution requires to know the (Stokes) steady-streaming. Ссылаясь на экспериментальные результаты Прандтля (1949) и Хаттона (1964), а также на более новые Модельные испытания Район-Лебод, Хопфингера & Картельера (2007) и Реклари (2013), выводится асимптотические почти периодические (установившиеся) решение типа Моисеева резонансной задачи о плесканиях жидкости с целью показать, что усредненное круговое течение (если возникает) является нелинейным образом связано с доминантной компонентой круговой волны. Эта связь возникает как необходимое условие разрешимости задачи и принимает форму нелинейных дифференциальных уравнений относительно четырех амплитудних параметров для двух первых натуральных форм колебаний жидкости и усредненного поля скоростей, которое описывается дифференциальными уравнениями в частных производных первого порядка. Чтобы иметь единственное решение последнего уравнения, надо знать устойчивое вторичное течение, обусловленное поверхностным слоем. Iз посиланням на експериментальнi результати Прандтля (1949) та Хаттона (1964), а також на бiльш новi модельнi випробування Ройон–Лебод, Хопфiнгера & Картельєра (2007) та Рекларi (2013), виводится асимптотичний майже перiодичний (усталений) розв’язок типу Моiсєєва резонансної задачi про хлюпання рiдини з метою показати, що усереднена кругова течiя (якщо така виникає) є нелiнiйним чином пов’язаною з домiнантною компонентою кругової хвилi. Цей зв’язок виникає як необхiдна умова розв’язностi задачi та приймає форму нелiнiйних диференцiальних рiвнянь вiдносно чотирьох амплiтудних параметрiв для двох перших натуральних форм коливань рiдини та усередненого поля швидкостей, яке описується диференцiальним рiвнянням у частинних похiдних першого порядку. Аби мати единий розв’язок останнього рiвняння, треба знати усталену вторинну течiю, обумовлену поверхневим шаром. Інститут математики НАН України 2017-10-31 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/344 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 14 No. 2 (2017): Mathematic problems of mechanics and computational mathematics; 205-219 Сборник Трудов Института математики НАН Украины; Том 14 № 2 (2017): Математические проблемы механики и вычислительной математики; 205-219 Збірник Праць Інституту математики НАН України; Том 14 № 2 (2017): Математичні проблеми механіки та обчислювальної математики; 205-219 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/344/355 Авторське право (c) 2017 Праці Інституту математики НАН України |
| spellingShingle | Timokha, A. N. Тимоха, Н. Н. Тімоха, О. М. Coupling of a swirl-type resonant sloshing and a mean rotational flow |
| title | Coupling of a swirl-type resonant sloshing and a mean rotational flow |
| title_alt | Сопряжение вихревого резонансного плескания и усредненного вращательного потока Поєднання вихрових резонансних коливань та усередненого обертового потоку |
| title_full | Coupling of a swirl-type resonant sloshing and a mean rotational flow |
| title_fullStr | Coupling of a swirl-type resonant sloshing and a mean rotational flow |
| title_full_unstemmed | Coupling of a swirl-type resonant sloshing and a mean rotational flow |
| title_short | Coupling of a swirl-type resonant sloshing and a mean rotational flow |
| title_sort | coupling of a swirl-type resonant sloshing and a mean rotational flow |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/344 |
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