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Ring bursting behavior en route to
turbulence
in quasi two-dimensional Taylor-Couette flows
Кольцевой взрывной эффект
как промежуточная стадия при формировании турбулентности
в квази-двумерном потоке Куэтта-Тейлора
Sebastian Altmeyer,
Younghae Do and Ying-Cheng Lai
(2015)
We
investigate the quasi two-dimensional Taylor-Couette system in the regime where
the radius ratio is close to unity - a transitional regime between three and
two dimensions.
В данной работе мы
рассматриваем квази-двумерную систему Куэтта – Тейлора, находящуюся в
состоянии, когда отношение радиусов близко к единице, то есть в промежуточном
качестве между трехмерным и двумерным.
By
systematically increasing the Reynolds number we observe a number of standard transitions,
such as one from the classical Taylor vortex flow (TVF) to wavy vortex flow
(WVF), as well as the transition to fully developed turbulence.
Систематически увеличивая число
Рейнольдса, мы отслеживаем стандартные переходы, такие, как переход от
классического тейлоровского вихревого потока (TVF) к волновому режиму вихревого потока (WVF), а также переход к полностью развитой
турбулентности.
Prior
to the onset of turbulence we observe intermittent burst patterns of localized
turbulent patches, confirming the experimentally observed pattern of very short
wavelength bursts (VSWBs).
Непосредственно перед началом
стадии турбулентности мы наблюдаем узоры с пульсирующими взрывами
локализованной турбулентности, подтверждающие экспериментально полученную
картинку для взрывов на ультракоротких волнах (VSWB).
A
striking finding is that, for Reynolds number larger than the onset of VSWBs, a new type of intermittently bursting behaviors emerge:
burst patterns of azimuthally closed rings of various orders.
Замечательный результат
заключается в том, что при числе Рейнольдса, большем, чем соответствующее для VSWB, возникает новый тип пульсирующего
взрывного поведения: структуры, образованные взрывами, имеют вид азимутально
замкнутых колец различных порядков.
We
call them ring-burst patterns, which surround the cylinder completely but
remain localized and separated by non-turbulent mostly wavy structures in the
axial direction.
Назовем их кольцевыми взрывными
структурами, плотно окружающими цилиндр, но остающимися локализованными и, в
осевом направлении, разделенными нетурбулентными (преимущественно - волновыми)
структурами.
We
use a number of quantitative measures, including the cross-flow energy, to
characterize the ring-burst patterns and to distinguish them from the
background flow.
Мы используем несколько
количественных мер, в частности –
энергию поперечного течения, для оценки кольцевых взрывных структур и
выделения их из основного потока.
The
ring-burst patterns are interesting because it does not occur in either three-
or two-dimensional Taylor-Couette flow: it occurs only in the transition, quasi
two-dimensional regime of the system, a regime that is less studied but
certainly deserves further attention so as to obtain deeper insights into turbulence.
Кольцевые взрывные структуры
интересны тем, что они не встречаются в трех- и двумерных потоках
Куэтта-Тейлора, зато их можно наблюдать в переходном, квази-двумерном
состоянии системы, состоянии малоизученном, но, безусловно, требующем внимания
для более глубокого осознания сути феномена турбулентности.
Введение
Characteristics of
turbulence in three- and two-dimensional flows are typically quite distinct.
For example, in three-dimension flows, the energy spectrum of fully-developed
turbulence obeys the well-known Kolmogorov 1941 scaling law [1] of k-5/3,
while in two dimensions the scaling [2, 3] is k-3. Although two-dimensional flow systems offer great advantages from the
standpoint of computation and mathematical analysis as compared with
three-dimensional flows, the former are not merely a kind of toy model of
turbulence. In fact, two-dimensional turbulence is relevant to the dynamics of
oceanic currents, origin of the ozone hole through mixing of chemical species
in the polar stratosphere, the existence of polar vortex, strong eddy motions
such as tropical cyclones, and other large-scale motions of planetary
atmospheres [4, 5].
Характеристики турбулентности трех- и двумерных
потоков достаточно просты. Например, для трехмерных потоков спектр энергии
развитой турбулентности подчиняется хорошо известному степенному (k-5/3) закону
Колмогорова от
Turbulence is
arguably one of the most difficult problems in science and engineering, and the
vast literature on turbulence was mostly focused on three or two dimensions [6].
To gain new insights into turbulence, it is of interest to study the
transitional regime between three and two dimensions. In such a regime,
properties of both three- and two-dimensional flows are relevant, and one
naturally wonders whether there are any unexpected features associated with,
for instance, transition to turbulence [7, 8]. The purpose of this paper is to
report a new phenomenon in a paradigmatic quasi-two-dimensional system: the
Taylor-Couette flow [9] with gap between the inner and outer cylinders so
narrow that the system is neither completely three-dimensional nor exactly
two-dimensional. In fact, this regime has not been investigated systematically
previously. The new type of intermittent dynamics occurs en route to turbulence
as the Reynolds number is increased.
Турбулентность является, пожалуй, одной из
труднейших проблем в науке; обширная литература по этой теме в основном
сфокусирована на случаях двух или трех измерений [6]. Необходим новый подход,
интересно исследовать переходные состояния между двумерными и трехмерными
моделями. В этих состояниях проявляются свойства, характерные и для двумерного,
и для трехмерного случая, и, что особенно удивляет, так это некоторые
неожиданные явления, связанные, например, с переходом к турбулентности [7, 8]. Цель данной работы – рассказать о новом феномене квази-двумерной
системы, где поток Куэтта-Тейлора [9] существует в зазоре между внешним и
внутренним цилиндрами, достаточно узком, чтобы система не являлась ни вполне
трехмерной, ни вполне двумерной. В самом деле, такой режим до сих пор не
был достаточно тщательно исследован. Этот новый тип пульсирующей динамики
встречается при зарождении турбулентности в результате увеличения числа
Рейнольдса.
The Taylor-Couette
system, a flow between two concentric rotating cylinders, has been a paradigm
in the study of complex dynamical behaviors of fluid flows, especially
turbulence [10-18]. The flow system can exhibit a large variety number of
ordered and disordered behaviors in different parameter regimes. For cylinders
of reasonable length, the effective dimensionality of the system is determined
by a single parameter - the ratio between the radii of the two cylinders. If
the ratio is markedly less than unity, the flow is three dimensional. As the
ratio approaches unity, the flow becomes two dimensional.
Most previous
studies focused on the setting where the radius ratio is below, say about 0.9,
the so-called wide-gap regime [13], or when the ratio approaches unity so that
the geometry is locally planar, resulting in an effectively two-dimensional
Couette flow [19]. Here, we are interested in the narrow-gap case, where the
radius ratio is close to unity but still deviates from it so that the flow is
quasi two-dimensional. To be concrete, we fix the radius ratio to be 0:99 and,
without loss of generality, restrict our study to the case where the outer
cylinder is stationary. In fact, regardless of whether the outer cylinder is
rotational or stationary, transition to turbulence can occur with increasing
Reynolds number. In particular, for systems of counter-rotating cylinders, an
early work [10] showed that transition to turbulence can be sudden as the
Reynolds number is increased through a critical point. For fixed outer
cylinder, the transition from laminar flow to turbulence can occur through a
sequence of instabilities of distinct nature [14, 20].
For Taylor-Couette
system of counter-rotating cylinders, spatially isolated flow patterns, the
so-called localized patches, can emerge through the whole fluid domain and
decay [13, 21]. Depending on the parameters, the localized patches can be
laminar or more complex patterns such as inter-penetrating spirals. In the
wide-gap regime (three-dimensional flow), numerical simulations [22] revealed
the existence of so-called Goertler vortices [23], small scale azimuthal
vortices that can cause streaky structures and form herringbone-like patterns
near the wall. Localized turbulent behaviors can arise when the Goertler
vortices concentrate and grow at the outflow boundaries of the Taylor vortex
cell [22].
For narrow gap
(quasi two-dimensional) flows, there was experimental evidence of the
phenomenon of very short wavelength bursts (VSWBs) [14]. One contribution of
our work is clear computational demonstration of VSWBs. Remarkably,
we uncover a class of solutions in quasi two-dimensional Taylor-Couette flow en
route to turbulence. These are localized, irregular, intermittently bursting,
azimuthally closed patterns that manifest themselves as various rings located
along the axial direction. For convenience, we refer to the states as
“ring-bursts”. The number of distinct rings can vary depending on the parameter
setting but their extents in the axial direction are similar. The ring bursts
can occur on some background flow that is not necessarily regular. For example,
in typical settings the background can be wavy vortex flows (WVFs) with
relatively high azimuthal wave numbers. Because of the coexistence of complex
flow patterns, to single out ring bursts is challenging, a task that we
accomplish by developing an effective mode separation method based on the
cross-flow energy. We also find that ring bursts are precursors to turbulence,
signifying a new route to turbulence uniquely for quasi two-dimensional
Taylor-Couette flows. To our knowledge, there was no prior report of ring
bursts patterns or likes. This is mainly due to the fact that the quasi
two-dimensional regime is a less explored territory in the giant landscape of
turbulence research. It would be interesting to identify precursors to
turbulence in quasi two-dimensional flow systems in general.
In Sec. II, we
outline our numerical method and describe a number of regular states in quasi
two-dimensional Taylor-Couette flows. In Sec. III, we present our main results:
numerical confirmation of experimentally observed VSWBs and more importantly,
identification and quantitative confirmation of intermittent ring bursts as
precursors to turbulence. In Sec. IV, we present conclusions and discussions.