The Bateman–Luke variational formalism in a sloshing with rotational flows
Based on a presentation of the velocity field in terms of Clebsch potentials, the Bateman–Luke variational formalism is generalized for the sloshing of an ideal incompressible liquid with rotational flows.
Збережено в:
| Дата: | 2016 |
|---|---|
| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Видавничий дім "Академперіодика" НАН України
2016
|
| Назва видання: | Доповіді НАН України |
| Теми: | |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/99860 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | The Bateman–Luke variational formalism in a sloshing with rotational flows / A.N. Timokha // Доповiдi Нацiональної академiї наук України. — 2016. — № 4. — С. 30-34. — Бібліогр.: 10 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-99860 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-998602016-05-08T03:02:11Z The Bateman–Luke variational formalism in a sloshing with rotational flows Timokha, A.N. Математика Based on a presentation of the velocity field in terms of Clebsch potentials, the Bateman–Luke variational formalism is generalized for the sloshing of an ideal incompressible liquid with rotational flows. Базуючись на представленнi поля швидкостей через потенцiали Клєбша, узагальнюється варiацiйний формалiзм Бейтмена–Люка для задачi про хлюпання iдеальної нестисливої рiдини з вихоровими течiями. Базируясь на представлении поля скоростей через потенциалы Клебша, обобщается вариационный формализм Бейтмена–Люка для задачи о плескании идеальной несжимаемой жидкости с вихревыми течениями. 2016 Article The Bateman–Luke variational formalism in a sloshing with rotational flows / A.N. Timokha // Доповiдi Нацiональної академiї наук України. — 2016. — № 4. — С. 30-34. — Бібліогр.: 10 назв. — англ. 1025-6415 http://dspace.nbuv.gov.ua/handle/123456789/99860 532.595 en Доповіді НАН України Видавничий дім "Академперіодика" НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Математика Математика |
| spellingShingle |
Математика Математика Timokha, A.N. The Bateman–Luke variational formalism in a sloshing with rotational flows Доповіді НАН України |
| description |
Based on a presentation of the velocity field in terms of Clebsch potentials, the Bateman–Luke variational formalism is generalized for the sloshing of an ideal incompressible liquid with rotational flows. |
| format |
Article |
| author |
Timokha, A.N. |
| author_facet |
Timokha, A.N. |
| author_sort |
Timokha, A.N. |
| title |
The Bateman–Luke variational formalism in a sloshing with rotational flows |
| title_short |
The Bateman–Luke variational formalism in a sloshing with rotational flows |
| title_full |
The Bateman–Luke variational formalism in a sloshing with rotational flows |
| title_fullStr |
The Bateman–Luke variational formalism in a sloshing with rotational flows |
| title_full_unstemmed |
The Bateman–Luke variational formalism in a sloshing with rotational flows |
| title_sort |
bateman–luke variational formalism in a sloshing with rotational flows |
| publisher |
Видавничий дім "Академперіодика" НАН України |
| publishDate |
2016 |
| topic_facet |
Математика |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/99860 |
| citation_txt |
The Bateman–Luke variational formalism in a sloshing with rotational flows / A.N. Timokha // Доповiдi Нацiональної академiї наук України. — 2016. — № 4. — С. 30-34. — Бібліогр.: 10 назв. — англ. |
| series |
Доповіді НАН України |
| work_keys_str_mv |
AT timokhaan thebatemanlukevariationalformalisminasloshingwithrotationalflows AT timokhaan batemanlukevariationalformalisminasloshingwithrotationalflows |
| first_indexed |
2025-07-07T10:00:56Z |
| last_indexed |
2025-07-07T10:00:56Z |
| _version_ |
1836982091878760448 |
| fulltext |
UDC 532.595 http://dx.doi.org/10.15407/dopovidi2016.04.030
Corresponding Member of the NAS of Ukraine A.N. Timokha
Institute of Mathematics of the NAS of Ukraine, Kiev
E-mail: atimokha@gmail.com
The Bateman–Luke variational formalism in a sloshing
with rotational flows
Based on a presentation of the velocity field in terms of Clebsch potentials, the Bateman–
Luke variational formalism is generalized for the sloshing of an ideal incompressible liquid with
rotational flows.
Keywords: Bateman–Luke variational principle, sloshing, Clebsch potentials.
Using the Bateman–Luke variational formulation [1, 2] is a commonly accepted approach to
analytic studies of nonlinear liquid sloshing [3–5]. The formulation suggests an ideal incompressi-
ble liquid with an irrotational velocity field that is both consistent with Kelvin’s circulation
theorem and supported by experiments for a clean (without internal structures) rigid tank.
According to Kelvin’ and Laplace’ results on the invariance of circulation, the vorticity cannot
be generated in the interior of a viscous incompressible fluid subject to a conservative extraneous
force, but is necessarily diffused inward from the boundaries; for clean tanks, the shedding vortices
remain localized near the smooth wetted tank surface.
Referring to his friend, Carl Runge, Ludwig Prandtl [6] writes that Pierre-Simon marquis
de Laplace demonstrated, in personal communications with colleagues, a “glass–wine” paradox
showing that a steady-state swirl sloshing generates, almost immediately, a stable sloshing-sup-
ported vortex in the liquid domain by a conversion of the wave angular momentum to the vortex
angular momentum. The conversion phenomenon makes inconsistent the existing mathematical
theories of circulation, since the vortex is neither appears at nor diffuses from the tank walls.
Prandtl [6] conducted a dedicated model test confirming the paradox. Analogous experimental
observations were reported in [7, 8], where the authors attempted to explain the paradox, without
a serious success, by the angular Stokes drift [3, Sect. 9.6.3], which together with the wave
breaking could be a trigger of the conversion, but not its driver. Studying the phenomenon
requires a revision of the existing analytic methods to include the rotational solenoidal flows
into account. Using Clebsch potentials [9, 10] and ideas by Bateman [1, p. 164–166], the present
paper generalizes the Bateman–Luke variational formalism to a sloshing with rotational flows.
A mobile rigid tank is considered partly filled with an inviscid incompressible liquid (the
mass density ρ = const). Figure 1 shows the liquid domain Q(t) bounded by the free surface
Σ(t) and the wetted tank surface S(t), an absolute (inertial) coordinate system O′x′1x
′
2x
′
3, and a
non-inertial (tank-fixed) coordinate system Ox1x2x3. The Ox1x2x3-system moves (relatively to
O′x′1x
′
2x
′
3) with the absolute translatory velocity vO(t) and the instant angular velocity ω(t), so
that any fixed point in the Ox1x2x3 has the absolute velocity
vb = vO + ω × r, (1)
where r = (x1, x2, x3) is the tank-fixed radius-vector. The gravity potential can be written as
U(x1, x2, x3, t) = −g · r′, r′ = r′O + r,
© A.N. Timokha, 2016
30 ISSN 1025-6415 Dopov. Nac. akad. nauk Ukr., 2016, №4
Fig. 1
where r′ is the radius-vector of a point of the body–liquid system with respect to O′, r′O is the
radius-vector of O with respect to O′, and g is the gravity acceleration vector. The free surface
Σ(t) is implicitly defined in the tank-fixed coordinate system by the equation Z(x1, x2, x3, t) = 0
so that the outer normal n to Σ(t) is −∇Z/|∇Z|. The function Z is the unknown that satisfies
the volume (mass) conservation condition∫
Q(t)
dQ = Vl = const (2)
treated as a geometric constraint.
The liquid motions are described by three Clebsch potentials φ(x1, x2, x3, t), m(x1, x2, x3, t),
and ϕ(x1, x2, x3, t), so that the absolute velocity field v = (v1, v2, v3, t) reads
v = ∇φ+m∇ϕ. (3)
Even though relation (3) does not give a unique representation of the velocity field (substitution
m := Cm, ϕ := ϕ/C, where C is a non-zero constant, confirms that), the Clebsch potentials
are henceforth assumed being three independent functions. The case of irrotational flows implies
either m = 0 or ϕ = const.
As remarked in [3, p. 47], the spatial derivatives in the introduced inertial (∂′i) and non-
inertial (∂i) coordinate systems remain the same, but the time-derivatives (∂′t and ∂t, respectively)
change, i. e.,
∂′i = ∂i; ∂′t = ∂t − vb · ∇; d′t = ∂′t + v · ∇ = ∂t + (v − vb) · ∇. (4)
Based on relations (4) and [1, p. 164], the Lagrangian
L(φ,m, ϕ, Z) =
∫
Q(t)
P dQ = −ρ
∫
Q(t)
[
∂′tφ+m∂′tϕ+
1
2
|v|2 + U
]
dQ =
= −ρ
∫
Q(t)
[
∂tφ+m∂tϕ− vb · v +
1
2
|v|2 + U
]
dQ (5)
ISSN 1025-6415 Доп. НАН України, 2016, №4 31
and the action
W (φ,m, ϕ, Z) =
t2∫
t1
[
L− p0
∫
Q(t)
dQ
]
dt =
t2∫
t1
∫
Q(t)
(P − p0) dQdt (6)
are introduced for any fixed instant times t1 < t2. Functional (6) acts on the independent
Clebsch potentials and Z. The Lagrange multiplier p0 is a consequence of the volume conservation
constraint (2).
Henceforth, the assumption is that the Clebsch potentials are smooth functions in Q(t),
which admit, for any instant time t, an analytic continuation through the smooth (provided by
the admissible Z) free surface Σ(t). Using the calculus of variables, Reynolds’ transport and
divergence theorem makes it possible to prove the following three lemmas.
Lemma 1. Under the assumption on the smoothness of the Clebsch potentials and the free
surface Σ(t), the zero first variation
δφW = 0 subject to δφ|t=t1,t2 = 0 (7)
is equivalent to the kinematic relations of the sloshing problem consisting of the continuity equati-
on
∇ · (v − vb) ≡ ∇ · v = 0 in Q(t), (8)
as well as the kinematic boundary conditions
(v − vb) · n = 0 on S(t), and (v − vb) · n = − ∂tZ
|∇Z|
on Σ(t), (9)
expressing that the normal velocity is defined by the rigid wall motions and the fluid particles
remain on the free surface Σ(t).
Lemma 2. Under the assumption on the smoothness of the Clebsch potentials and the free
surface Σ(t), the zero first variation
δmW = 0 (10)
is equivalent to the equation
d′ϕ ≡ ∂′tϕ+ v · ∇ϕ ≡ ∂tϕ+ (v − vb) · ∇ϕ = 0 in Q(t), (11)
which indicates that the Clebsch potential ϕ remains constant during the motions of a liquid
particle (a vortex line moves with the liquid and always contains the same particles).
Lemma 3. Under the assumption on the smoothness of the Clebsch potentials and the free
surface Σ(t), the zero first variation
δϕW = 0 subject to δϕ|t1,t2 = 0, (12)
and the kinematic problem (8), (9) is equivalent to
d′m ≡ ∂′tm+ v · ∇m ≡ ∂tm+ (v − vb) · ∇m = 0 in Q(t), (13)
which has the same meaning that (11), but for the Clebsch potential m.
32 ISSN 1025-6415 Dopov. Nac. akad. nauk Ukr., 2016, №4
Remark 1. In contrast to the Bateman–Luke formulation for potential liquid flows, the
function P adopted in the definition of Lagrangian (5) is, generally speaking, not the pressure
and cannot be treated as the pressure for arbitrary Clebsch potentials. One can show that the
pressure p = P + f(t) (f(t) is an arbitrary function), when (11) and (13) are satisfied. In other
words, when assuming that (11) and (13) are fulfilled, the Euler equation
d′v = −1
ρ
(∇P +∇U) in Q(t) (14)
holds true. This fact follows from the expression for the left-hand side of (14)
d′(∇φ+m∇ϕ) = [∇(∂′tφ) +m∇(∂′tϕ) + ∂′tm∇ϕ] + v · ∇(∇φ+m∇ϕ)︸ ︷︷ ︸
v·∇∇φ+mv·∇∇ϕ+∇ϕ(∇m·v)
=
= ∇(∂′tφ) +m∇(∂′tϕ) + v · ∇∇φ+mv · ∇∇ϕ+∇ϕ(∇m · v) +∇ϕ[d′m]
and the right-hand side (after annihilating the U -term)
∇
(
∂′tφ+m∂′tϕ+
1
2
|v|2
)
= [∇(∂′tφ) +m∇(∂′tϕ) + ∂′tϕ∇m] + v · ∇∇φ+mv · ∇∇ϕ+
+∇m(∇ϕ · v)= ∇(∂′tφ)+m∇(∂′tϕ)+v · ∇∇φ+mv · ∇∇ϕ+∇ϕ(∇m · v)+∇m[d′ϕ],
in which the framed terms are identical, but the residual terms vanish, as (11) and (13) hold true.
Using Lemmas 1–3 and Remark 1 makes it possible to prove the following theorem.
Theorem 1. Under the assumption on the smoothness of the Clebsch potentials and the free
surface Σ(t), the zero first variation of action (6)
δW = δφW + δmW + δϕW + δZW = 0 (15)
subject to
δφ|t1,t2 = δϕ|t1,t2 = 0 (16)
is equivalent to the sloshing problem, which includes the kinematic relations (8) and (9), two
equations (11) and (13) expressing the fact that the Clebsch potentials ϕ and m are constant
along the vortex lines, as well as the dynamic boundary condition
p− p0 = −ρ
(
∂tφ+m∂tϕ− vb · v +
1
2
|v|2 + U
)
− p0 = 0 on Σ(t) (17)
establishing that the pressure equals to the ullage pressure p0 on the free surface. The volume
conservation condition (2) should be added to the sloshing problem.
In summary, utilizing the Clebsch potentials and the Bateman–Luke principle (the Lagrangi-
an is a “pressure integral”) makes it possible to derive the full system of governing equati-
ons (8), (11), (13) and the boundary conditions (9) and (17) for the sloshing of an ideal incom-
pressible liquid with rotational flows. Specifically, the principle (integrand in Lagrangian (5) is
the pressure) holds true if and only if the vorticity equations (11) and (13) are a priori satisfied.
The generalized Bateman–Luke formulation can be a background for the nonlinear multimodal
method.
ISSN 1025-6415 Доп. НАН України, 2016, №4 33
The author 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).
References
1. Bateman H. Partial differential equations of mathematical physics, New York: Dover, 1944.
2. Luke J.G. J. Fluid Mech., 1967, 27: 395–397.
3. Faltinsen O.M., Timokha A.N. Sloshing, New York: Cambridge Univ. Press, 2009.
4. Lukovsky I. A. Nonlinear dynamics: Mathematical models for rigid bodies with a liquid, Berlin: de Gruyter,
2015.
5. Takahara H., Kimura K. J. Sound Vibr., 2012, 331, No 13: 3199–3212.
6. Prandtl L. ZAMM, 1949, 29, No 1/2: 8–9.
7. Hutton R.E. J. Appl. Mech., Trans. ASME, 1964, 31, No 1: 145–153.
8. Royon-Lebeaud A., Hopfinger E., Cartellier A. J. Fluid Mech., 2007, 577: 467–494.
9. Clebsch A. J. Reine Angew. Math., 1857, 54: 293–313.
10. Clebsch A. J. Reine Angew. Math., 1869, 56: 1–10.
Received 25.11.2015
Член-кореспондент НАН України О.М. Тимоха
Iнститут математики НАН України, Київ
E-mail: atimokha@gmail.com
Формалiзм Бейтмена–Люка в задачах хлюпання рiдини
з вихоровими течiями
Базуючись на представленнi поля швидкостей через потенцiали Клєбша, узагальнюється
варiацiйний формалiзм Бейтмена–Люка для задачi про хлюпання iдеальної нестисливої рi-
дини з вихоровими течiями.
Ключовi слова: варiацiйний принцип Бейтмена–Люка, хлюпання рiдини, потенцiали
Клєбша.
Член-корреспондент НАН Украины А.Н. Тимоха
Институт математики НАН Украины, Киев
E-mail: atimokha@gmail.com
Формализм Бейтмена–Люка в задачах плескания жидкости
с вихревыми течениями
Базируясь на представлении поля скоростей через потенциалы Клебша, обобщается ва-
риационный формализм Бейтмена–Люка для задачи о плескании идеальной несжимаемой
жидкости с вихревыми течениями.
Ключевые слова: вариационный принцип Бейтмена–Люка, плескание жидкости, потенци-
алы Клебша.
34 ISSN 1025-6415 Dopov. Nac. akad. nauk Ukr., 2016, №4
|