Computers and Fluids 250 (2023) 105711
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Computers and Fluids
journal homepage: www.elsevier.com/locate/compfluid
Vortex collision against static and spinning round cylinders: A lattice
Boltzmann study
Alessandro De Rosis
Department of Mechanical, Aerospace and Civil Engineering, The University of Manchester, Oxford Road, Manchester, M13 9PL, UK
ARTICLE
INFO
Keywords:
Lattice Boltzmann method
Vortex dynamics
ABSTRACT
In this paper, the flow physics generated by the collision of a vortex dipole that moves against a spinning round
cylinder is investigated numerically. Fluid dynamics is predicted by a combined central-moments-based lattice
Boltzmann-immersed boundary method. First, the model is validated against well established consolidated
benchmark problems, showing very high accuracy properties. Then, results from a comprehensive numerical
campaign are presented. A wide set of values of the Reynolds number (Re) is investigated, ranging from
10 to 1000. The cylinder is forced to spin around its centre with different angular velocities, which are
obtained by varying the spinning number (Sp) between 0 (corresponding to the static case) and 0.75. The
generation of secondary vortices as a consequence of the impact is elucidated and linked to the time evolution
of the kinetic energy, enstrophy and hydrodynamic forces. Interestingly, we find that the flow physics changes
drastically when Re ≥ 250, independently from the value of Sp. Through a closer look at the vorticity field, we
find that the impact creates two primary–secondary structures and a second impingement takes place when
Re ≥ 250. Interestingly, the normalised drag force (𝐶𝑑 ) is found to constantly fluctuates around a mean value.
Oscillations are due to the vorticity created by the rotation of the cylinder and are more emphasised as Sp
grows. Specifically, 𝐶𝑑 can achieve marked negative values as a consequence of the velocity field created by
the cylinder during its rotation.
1. Introduction
The impact of a vortex against a solid surface is a well known
generic event in two-dimensional incompressible flows and it is routinely experienced in many areas of practical interest. For instance,
a knowledge of the time evolution of trailing vortices generated by
an aircraft interacting with the ground is crucial in the landing/takeoff phases of a flight. Moreover, vortex–wall collisions are of direct
relevance to jet-impingement-based heat transfer applications. In addition, other real-world examples range from jet engine start-ups to
jellyfish propulsion. The interaction between a vortex and a solid
boundary is definitely a fundamental topic of fluid dynamics whose
understanding can help scientists and engineers to get insights into
a rich class of problems. Motivated by this opportunity, researchers
have been devoting an impressive amount of efforts to study this
phenomenon during the last fifty years, idealising the vortex as a dipole
consisting of two monopoles, i.e. two vortices rotating with opposite
directions. Experimental measurements and numerical simulations are
reliable approaches to investigate the vortex–wall collision.
On the one hand, experimental measurements have provided very
deep insights into this problem. In 1971, Harvey and Perry [1] performed an experimental investigation of the vortex rebound against
ground and highlighted the mechanism responsible of creating a secondary vortex. Later, Barker and Crow [2] observed vortex rebounding
against a shear–stress free surface in laboratory experiments. Van Heijst
and Flor [3] gave further insights into the dipole–wall collision. Interestingly, their experimental tests showed that the two monopoles get
separated and asymmetric dipoles appear as the result of oppositelysigned vorticity production in the boundary layer. The newly formed
dipoles roll back, collide and divide into two parts: a smaller dipole
that moves away from the wall, and a larger one that impacts the wall
again. Chu et al. [4] shed light on the forces acting upon the wall
as a result of the collision. Specifically, they proved that a small net
surface force during the impact can arise from the cancellation of a
force pushing on the surface and a suction force due to the induced
boundary layers. Wells and Afanasyev [5] carried out a laboratory
investigation of quasi-two-dimensional turbulence decaying in a rectangular container. They found that variations of net angular momentum
Alessandro De Rosis, Linkedin: alessandro-de-rosis-175464120.
E-mail address: alessandro.derosis@manchester.ac.uk.
URL: https://www.research.manchester.ac.uk/portal/alessandro.derosis.html.
https://doi.org/10.1016/j.compfluid.2022.105711
Received 29 June 2022; Received in revised form 5 October 2022; Accepted 28 October 2022
Available online 9 November 2022
0045-7930/© 2022 Elsevier Ltd. All rights reserved.
Computers and Fluids 250 (2023) 105711
A. De Rosis
Heijst [26], Clercx et al. [27], Sansón and Sheinbaum [28], and Clercx
and Van Heijst [29] for further numerical results and discussions.
In contrast to the adoption of aforementioned numerical techniques, Latt and Chopard [30] successfully employed the lattice Boltzmann method (LBM) [31] to investigate the collision of a vortex dipole
against a straight no-slip wall. In short, the LBM idealises the fluid by
collections (also known as distributions or populations) of fictitious
particles colliding and streaming along the links of a fixed Cartesian
lattice [32]. The computational simplicity of the algorithm [33], its
natural suitability for multiphysics [34] and the possibility to perform
easier parallel implementation due to the locality of the scheme [35]
have promoted its wide application. Latt and Chopard [30] validated
their approach against findings in [15], highlighting the accuracy of
their approach. Interestingly, moment-based boundary conditions were
tested against the dipole–wall collision for no-slip boundaries [36] and
slip boundaries [37], showing very high numerical properties. More
recently, De Rosis and Coreixas [38] proposed a collision operator in
the space of central moments (Geier et al. [39], Asinari [40], Premnath
and Banerjee [41], De Rosis [42], Saito et al. [43], De Rosis and
Luo [44], Saito et al. [45], De Rosis et al. [46], De Rosis and Tafuni
[47], De Rosis and Enan [48]) and tested it against the dipole–wall
collision. Their results showed a very good agreement with findings
in [15] and [36]. Central-moments-based collision operators were also
adopted to perform accurate simulations of thermal flows (Hajabdollahi
and Premnath [49] and Hajabdollahi et al. [50]). Notably, the collision
between a vortex dipole and a flat wall in an electrically conductive
fluid has been investigated very recently in [51], demonstrating that
the presence of a magnetic field can significantly modify the flow field.
The literature review discussed above confirms that significant efforts have been devoted to study the collision of a dipole against walls
of different shapes (e.g., straight, round or V-shaped). Motivated by the
symmetry-breaking mechanisms due to moving walls [20] and by the
emission of vortices in circular flows around a cylinder [52], we pose
the following question: during the collision of a vortex against a round
cylinder, how much will the flow physics be affected by the rotation
of the solid body? We aim to reply to this question through the results
of a numerical campaign where the impact of a dipole against a round
cylinder is considered under different flow conditions. To the best of
our knowledge, this particular configuration has not been investigated
so far, apart from few results available in a bachelor thesis [53]. In
our simulations, we vary the Reynolds number in the range 10–1000.
Moreover, the round cylinder is assumed to spin counter-clockwise
around its centre. A large range of angular velocities is investigated
by varying the spinning number between 0 (corresponding to a static
case) to 0.75. Insights into the flow physics are given by having a close
look at the vorticity field at salient time instants. Our considerations
are linked to the time evolution of the kinetic energy, kinetic enstrophy and hydrodynamic drag force acting upon the immersed body.
Numerical simulations are carried out by employing a two-dimensional
central-moments-based lattice Boltzmann method [54]. This particular
collision operator has been preferred to the standard Bhatnagar–Gross–
Krook (BGK) [55] and multiple-relaxation-time (MRT) [56] ones due to
the superior properties in terms of stability (Geier et al. [57], Ning et al.
[58], De Rosis [59], Yahia et al. [60]). The presence of the (possibly
moving) curved surface is accounted for by the Immersed Boundary
method (IBM) (Peskin [61], Feng and Michaelides [62], Mittal and
Iaccarino [63], Wu and Shu [64], Suzuki and Inamuro [65]). The
choice of the IBM over the interpolated bounce-back scheme (Mei et al.
[66], Bouzidi et al. [67], Mei et al. [68]) is motivated by the more
general implementation and superior stability properties of the former
(De Rosis et al. [69], Favier et al. [70], Peng et al. [71]). Before
presenting the results of the dipole–cylinder collisions, the model is
validated against consolidated reference data involving the collision of
a vortex against a flat wall. Moreover, the accuracy of the combined
LBM-IBM approach is assessed against a benchmark problem involving
the harmonic oscillations of a cylinder in a viscous fluid.
occur due to the interactions of coherent vortex dipoles with the walls.
A more recent experimental study of vortex-ring collisions upon round
cylinders with different cylinder-to-vortex-ring diameter ratios revealed
that the collision behaviour is very different from the one associated
with flat surfaces [6]. The authors also showed that the flow physics
is very sensitive to the aforementioned ratio. New et al. [7] presented
an investigation of collisions against V-shaped walls and highlighted
the significant flow differences with respect to the impact against flat
inclined surfaces. The impingement of a vortex pair against a wavy wall
in [8] elucidated the fundamental mechanism behind the breakup and
decay of trailing vortices formed in the wake of an aircraft.
On the other hand, the development of progressively more flexible
and reliable numerical methods have helped to overcome the limitations arising in laboratory experiments (e.g., a usually large amount
of time and resources, the sensitivity of the adopted tools and the
accuracy of the measurement techniques), allowing us to simulate an
impressively large set of scenarios. In 1990, Orlandi [9] proposed the
first numerical effort to predict the flow physics induced by the vortex
dipole–wall collision. His analyses confirmed the experimental observations by Harvey and Perry [1] about vortex rebounds. Later, Coutsias
and Lynov [10] described fundamental processes connected to vorticity
detachment from boundary layers caused by proximity of vortical
structures with the adoption of spectral methods. Verzicco et al. [11]
presented an experimental and numerical study of a vortex impacting
a round cylinder, addressing the evolution of the vorticity field and
the influence of the cylinder diameter in relation to the dipole. Clercx
et al. [12] revealed that the no-slip boundaries play a crucial role
in the flow evolution with the rise of a spontaneous spin-up of the
flow due to normal and shear stresses exerted on the fluid by the
walls. Two-dimensional turbulence in square containers was further
investigated by Clercx et al. [13]. They showed that flow physics differs
largely from the case where periodic boundary conditions are adopted.
In fact, boundary layers act as sources of small-scale vorticity that is
continuously injected into the flow domain. Notably, vortex statistics
related to this problem were presented in [14]. Clercx and Bruneau
[15] used finite differences and a pseudospectral code and determined
the minimum necessary number of grid points or Chebyshev collocation
points to capture the flow dynamics accurately. They provided a very
reliable set of results concerning the normal and oblique dipole–wall
collision that nowadays is widely adopted to validate computational
algorithms [16]. Keetels et al. [17] used Fourier spectral and wavelet
solvers in combination with volume penalisation method. They proved
that the dipole–wall collision has a non-zero energy dissipation in the
vanishing viscosity limit [18]. By adopting compact differences and
Fourier methods, Sutherland et al. [19] found no indication of the
persistence of energy dissipating structures in the limit of vanishing
viscosity. However, the results in [18] were recovered when the slip
length was varied inversely with Reynolds number. They concluded
that the presence of energy dissipating structures in the inviscid limit
is due to both wall slip length and viscosity going to zero, not being
treated independently in the use of the volume penalisation method.
While the impact against a straight static wall is characterised by symmetric flow patterns, the interaction with a tangentially moving surface
breaks this symmetry. Distinctive flow regimes for different wall speeds
were shown by Guzmán et al. [20]. Through the study of the interaction
of a vortex and a semi-infinite plate, Peterson and Porfiri [21] proved
that OpenFOAM is a valid tool to tackle the dipole–wall collision. The
same solver for the fluid domain was adopted by Zivkov et al. [22], who
evaluated the fluid forces exerted by a vortex dipole on a deformable
plate with an estimation of the experienced loadings and displacements.
The importance of numerical simulations is confirmed by a very recent
study by New et al. [23]. In fact, while corroborating the experimental
observations in [7], the authors revealed additional flow details on the
vortex dynamics and vortex-core trajectories. The interested reader can
refer to Clercx and Van Heijst [24], Clercx et al. [25], Clercx and van
2
Computers and Fluids 250 (2023) 105711
A. De Rosis
The rest of the paper is organised as follows. In Section 2, the
problem is stated and the adopted numerical approach is recalled.
A validation of our numerical scheme is reported in Section 3. In
Section 4, results from our numerical campaign are discussed. Some
conclusions are drawn in Section 5. The interested reader can find the
code to reproduce our results in the Supplementary Material.
where 𝑈 = 1 is the root-mean-square of the velocity field in Eqs. (3).
The cylinder spins around its centre with a constant uniform counterclockwise angular velocity 𝜃. As a consequence, it is possible to define
another governing parameter, the spinning number Sp, that is
2. Problem statement and numerical approach
2.2. Lattice Boltzmann method
In this section, we first state the problem and its governing equations. Then, the adopted numerical scheme is recalled.
Let us consider the D2Q9 discretisation [31], where the lattice
directions 𝒄 𝑖 = [𝑐𝑖𝑥 , 𝑐𝑖𝑦 ] are defined as
2.1. Problem definition
𝑐𝑖𝑥 = [0, 1, 0, −1, 0, 1, −1, −1, 1],
Sp =
𝜃𝐷
.
2𝑈
(7)
𝑐𝑖𝑦 = [0, 0, 1, 0, −1, 1, 1, −1, −1],
Let us consider a two-dimensional Cartesian coordinate system
(𝑥, 𝑦). Let us adopt the following nomenclature: 𝑡 is the time, 𝜌 is the
mass density, 𝑝 is the fluid pressure, 𝜈 is the fluid kinematic viscosity,
𝒖 = [𝑢𝑥 , 𝑢𝑦 ] and 𝑭 = [𝐹𝑥 , 𝐹𝑦 ] are the flow velocity and external body
force vectors, respectively. The incompressible flow of a viscous fluid
is governed by the Navier–Stokes equations:
𝜕𝑡 𝒖 + (𝒖 ⋅ 𝛁) 𝒖 = −
𝛁𝑝
𝑭
+ 𝜈𝛥𝒖 + ,
𝜌
𝜌
with 𝑖 = 0, … , 8. The lattice Boltzmann equation (LBE) with the forcing
term can be generally expressed as in Premnath [72], Premnath and
Abraham [73], McCracken and Abraham [74], Premnath and Abraham
[75], that is
eq
𝑓𝑖 (𝒙 + 𝒄 𝑖 , 𝑡 + 1) = 𝑓𝑖 (𝒙, 𝑡) + 𝜦[𝑓𝑖 (𝒙, 𝑡) − 𝑓𝑖 (𝒙, 𝑡)]
+ (𝐈 − 𝜦∕2)𝐹𝑖 (𝒙, 𝑡),
(1)
1
2 ∫0
40 𝑟0
1
𝛺(𝑡 = 0) =
2 ∫0
40 𝑟0
30 𝑟0
∫0
30 𝑟0
∫0
|𝒖|2 (𝒙, 𝑡 = 0) d𝑥 d𝑦 = 2,
The LBE can be divided into two steps, i.e., collision:
eq
𝑓𝑖⋆ (𝒙, 𝑡) = 𝑓𝑖 (𝒙, 𝑡) + 𝜦[𝑓𝑖 (𝒙, 𝑡) − 𝑓𝑖 (𝒙, 𝑡)]
+ (𝐈 − 𝜦∕2)𝐹𝑖 (𝒙, 𝑡),
𝑓𝑖 (𝒙 + 𝒄 𝑖 , 𝑡 + 1) = 𝑓𝑖⋆ (𝒙, 𝑡).
𝑈𝐷
,
𝜈
(12)
By shifting the lattice directions by the local fluid velocity [39], it is
possible to obtain the shifted lattice directions 𝒄̄ 𝑖 = [𝑐̄𝑖𝑥 , 𝑐̄𝑖𝑦 ], which are
defined as
(4)
𝑐̄𝑖𝑥 = 𝑐𝑖𝑥 − 𝑢𝑥 ,
|𝜔|2 (𝒙, 𝑡 = 0) d𝑥 d𝑦 = 800,
𝑐̄𝑖𝑦 = 𝑐𝑖𝑦 − 𝑢𝑦 .
(13)
The collision stage in the space of central moments (CMs) is performed
by adopting a rotation matrix, allowing us to transform populations into
central moments (and vice versa). We adopt the matrix in [79]:
⎡ 1, … , 1 ⎤
⎥
⎢ 𝑐̄
𝑖𝑥
⎥
⎢
⎢ 𝑐̄𝑖𝑦 ⎥
⎢𝑐̄2 + 𝑐̄2 ⎥
𝑖𝑦 ⎥
⎢ 𝑖𝑥
⎢𝑐̄2 − 𝑐̄2 ⎥
𝑖𝑥
𝑖𝑦
𝐓=⎢
⎥.
⎢ 𝑐̄𝑖𝑥 𝑐̄𝑖𝑦 ⎥
⎥
⎢ 2
⎢ 𝑐̄𝑖𝑥 𝑐̄𝑖𝑦 ⎥
⎢ 𝑐̄ 𝑐̄2 ⎥
⎢ 𝑖𝑥 𝑖𝑦 ⎥
⎢ 𝑐̄2 𝑐̄2 ⎥
⎣ 𝑖𝑥 𝑖𝑦 ⎦
(5)
being the vorticity. These initial values of 𝐸 and 𝛺 are obtained by
setting the strength of the monopoles as 𝑤𝑒 = 299.56. A round cylinder
of diameter 𝐷 = 4𝑟0 is placed in the fluid domain with its centre
located at (𝑥𝑐 , 𝑦𝑐 ) = (22 𝑟0 , 15 𝑟0 ). The problem mimics the one in [15].
Specifically, in both the cases the distance between the centre of the
dipole and the left edge of the domain is equal to 10 𝑟0 . Moreover,
the distance between the centre of the dipole and the leftmost point
of the cylinder is equal to 10 𝑟0 , that in turn is equal to the distance
between the centre of the dipole and the right (impacted) edge when
a straight wall is considered [15]. Therefore, the dipole is expected
to hit the cylinder at the same time as it would do if a straight wall
was considered. The height of the domain is three times larger than
the one in [15]. This choice is motivated by the need to avoid the
possible influence of the upper and lower boundaries of the domain.
The characteristic Reynolds number is
Re =
(11)
and streaming:
respectively, with
𝜔 = 𝛁 × 𝒖 = 𝜕𝑥 𝑢𝑦 − 𝜕𝑦 𝑢𝑥
(9)
𝐈 being the unit tensor. Equilibrium populations are evaluated by
expanding onto the full basis of Hermite polynomials up to the fourth
order [76]. The benefits of this choice over the classical second-order
truncated expression are discussed with details in [77] and [44]. The
term 𝐹𝑖 accounts for external body forces and it is written according
to Huang et al. [78]. Macroscopic variables are computed as
∑
∑
𝑭
(10)
𝜌=
𝑓𝑖 ,
𝜌𝒖 =
𝑓𝑖 𝒄 𝑖 + .
2
𝑖
𝑖
(2)
[
]
2
2
where 𝜕 denotes a partial derivative, 𝛁 = 𝜕𝑥 , 𝜕𝑦 and 𝛥 = 𝜕𝑥 + 𝜕𝑦 are
the gradient and Laplacian operators, respectively. Let us consider a
rectangular domain (𝑥, 𝑦) ∈ (0 ∶ 40𝑟0 , 0 ∶ 30𝑟0 ), enclosed by no-slip
walls at each side. The velocity is initialised as
[ (
)
)2 ]
1 (
𝑢𝑥 (𝒙, 𝑡 = 0) = − 𝑤𝑒 𝑦 − 𝑦1 exp − 𝑟1 ∕𝑟0
2
[ (
)
)2 ]
1 (
+ 𝑤𝑒 𝑦 − 𝑦2 exp − 𝑟2 ∕𝑟0
,
2
[ (
)
)2 ]
1 (
𝑢𝑦 (𝒙, 𝑡 = 0) = 𝑤𝑒 𝑥 − 𝑥1 exp − 𝑟1 ∕𝑟0
2
[ (
)
)2 ]
1 (
− 𝑤𝑒 𝑥 − 𝑥2 exp − 𝑟2 ∕𝑟0
,
(3)
2
forcing the dipole to move horizontally rightward. 𝒙 is the spatial
position. The two monopoles are located at (𝑥1 , 𝑦1 ) = (10 𝑟0 , 16 𝑟0 ) and
(𝑥
=
(10 𝑟0 , 14 𝑟0 ). The √
radii are 𝑟0
=
0.1, 𝑟1
=
√2 , 𝑦2 )
(
)2 (
)2
(
)2 (
)2
𝑥 − 𝑥1 + 𝑦 − 𝑦1 and 𝑟2 =
𝑥 − 𝑥2 + 𝑦 − 𝑦2 . The initial
values of the kinetic energy and enstrophy are
𝛁 ⋅ 𝒖 = 0,
𝐸(𝑡 = 0) =
(8)
(14)
The relaxation matrix in the populations space then reads as 𝜦 =
𝐓−1 𝐊𝐓, where
𝐊 = diag[1, 1, 1, 1, 1∕𝜏, 1∕𝜏, 1, 1, 1]
(15)
is the relaxation matrix in the CMs space, and 𝜏 (
is the relaxation
time.
)
√
1 2
It is linked to the fluid dynamic viscosity by 𝜈 = 𝜏 −
𝑐𝑠 , 𝑐𝑠 = 1∕ 3
2
being the lattice sound speed [31].
(6)
3
Computers and Fluids 250 (2023) 105711
A. De Rosis
Let us collect pre-collision, equilibrium and post-collision CMs as
[
]⊤
𝑘𝑖 = 𝑘0 , … , 𝑘 𝑖 , … , 𝑘 8 ,
[
eq
eq
eq
eq ]⊤
𝑘𝑖 = 𝑘0 , … , 𝑘 𝑖 , … , 𝑘 8
,
]
[ ⋆
⋆
⋆
⋆ ⊤
𝑘𝑖 = 𝑘0 , … , 𝑘 𝑖 , … , 𝑘 8 ,
(16)
matrix 𝐓 can be written as 𝐓 = 𝐌𝐍 [81], where 𝐌 is
⎡ 1, … , 1 ⎤
⎥
⎢ 𝑐
𝑖𝑥
⎥
⎢
⎢ 𝑐𝑖𝑦 ⎥
⎢𝑐 2 + 𝑐 2 ⎥
𝑖𝑦 ⎥
⎢ 𝑖𝑥
2⎥
⎢ 2
𝐌 = ⎢𝑐𝑖𝑥 − 𝑐𝑖𝑦 ⎥ ,
⎢ 𝑐𝑖𝑥 𝑐𝑖𝑦 ⎥
⎥
⎢ 2
⎢ 𝑐𝑖𝑥 𝑐𝑖𝑦 ⎥
⎢ 𝑐 𝑐2 ⎥
⎢ 𝑖𝑥 𝑖𝑦 ⎥
⎢ 𝑐2 𝑐2 ⎥
⎣ 𝑖𝑥 𝑖𝑦 ⎦
respectively, ⊤ denoting the transpose operator. The first two quantities
are evaluated as
eq
𝑘𝑖 = 𝐓𝑓𝑖 ,
eq
𝑘𝑖 = 𝐓𝑓𝑖 .
(17)
In the space of central moments, the collision process is written as
(
)
𝐊
𝑒𝑞
𝑘⋆
(18)
𝑖 = (𝐈 − 𝐊) 𝑘𝑖 + 𝐊 𝑘𝑖 + 𝐈 − 2 𝐓𝐹𝑖 .
Post-collision central moments read as follows
and the lower triangular matrix 𝐍 is obtained as 𝐍 = 𝐌−1 𝐓. 𝐍 allows
us to transform post-collision central moments into post-collision raw
]
[ ⋆
−1 ⋆
⋆
⋆ ⊤
by 𝑟⋆
moments 𝑟⋆
𝑖 = 𝐍 𝑘𝑖 . The resultant
𝑖 = 𝑟0 , … , 𝑟 𝑖 , … , 𝑟 8
expressions are
𝑘⋆
= 𝜌,
0
𝐹
⋆
𝑘1 = 𝑥 ,
2𝜌
𝐹𝑦
⋆
𝑘2 =
,
2𝜌
⋆
𝑘3 = 2𝜌𝑐𝑠2 ,
)
(
1
𝑘4 ,
𝑘⋆
= 1−
4
𝜏
)
(
1
𝑘⋆
𝑘5 ,
= 1−
5
𝜏
2
𝐹𝑦 𝑐𝑠
𝑘⋆
=
,
6
2𝜌
2
𝐹𝑥 𝑐𝑠
,
𝑘⋆
7 =
2𝜌
𝑘⋆
= 𝜌𝑐𝑠4 .
8
𝑟⋆
= 𝜌,
0
𝑟⋆
= 𝑘⋆
+ 𝜌𝑢𝑥 ,
1
1
𝑟⋆
= 𝑘⋆
+ 𝜌𝑢𝑦 ,
2
2
𝑟⋆
= 𝑘⋆
+ 2𝑢𝑦 𝑘⋆
+ 2𝑢𝑥 𝑘⋆
+ 𝜌(𝑢2𝑥 + 𝑢2𝑦 ),
3
3
2
1
𝑟⋆
= 𝑘⋆
− 2𝑢𝑦 𝑘⋆
+ 2𝑢𝑥 𝑘⋆
+ 𝜌(𝑢2𝑥 − 𝑢2𝑦 ),
4
4
2
1
𝑟⋆
= 𝑘⋆
+ 𝑢𝑥 𝑘⋆
+ 𝑢𝑦 𝑘⋆
+ 𝜌𝑢𝑥 𝑢𝑦 ,
5
5
2
1
𝑢𝑦 ⋆
⋆
⋆
⋆
𝑟6 = 𝑘6 + 2𝑢𝑥 𝑘5 + (𝑘3 + 𝑘⋆
) + 𝑢2𝑥 𝑘⋆
+ 2𝑢𝑥 𝑢𝑦 𝑘⋆
+ 𝜌𝑢2𝑥 𝑢𝑦 ,
4
2
1
2
𝑢𝑥 ⋆
⋆
⋆
2 ⋆
⋆
⋆
2
𝑟⋆
7 = 𝑘7 + 2𝑢𝑦 𝑘5 + 2 (𝑘3 − 𝑘4 ) + 2𝑢𝑥 𝑢𝑦 𝑘2 + 𝑢𝑦 𝑘1 + 𝜌𝑢𝑥 𝑢𝑦 ,
𝑘4 ⋆ 2
⋆
⋆
2
𝑟⋆
= 𝑘⋆
+ 2(𝑢𝑥 𝑘⋆
7 + 𝑢𝑦 𝑘6 ) + 4𝑢𝑥 𝑢𝑦 𝑘5 − 2 (𝑢𝑥 − 𝑢𝑦 )
8
8
⋆
𝑘
+ 3 (𝑢2𝑥 + 𝑢2𝑦 ) + 2𝑢𝑥 𝑢𝑦 (𝑢𝑥 𝑘⋆
+ 𝑢𝑦 𝑘⋆
) + 𝜌𝑢2𝑥 𝑢2𝑦 .
2
1
2
(19)
It should be noted that the force vector accounts for terms resulting
from the enforcement of the no-slip condition at the fluid–solid interface by the IBM. The interested reader can refer to Suzuki and Inamuro
[65], De Rosis et al. [69] and De Rosis and Lévêque [80] for further
details about the implementation.
From an algorithmic viewpoint, one could be tempted to compute
pre-collision CMs 𝑘4,5 as
∑
2
2
𝑘4 =
𝑓𝑖 (𝑐̄𝑖𝑥
− 𝑐̄𝑖𝑦
),
∑
𝑓𝑖 𝑐̄𝑖𝑥 𝑐̄𝑖𝑦 .
𝑓0⋆ = 𝜌 − 𝑟⋆
+ 𝑟⋆
,
3
8
⋆
𝑓1⋆ = (𝑟⋆
+ 𝑟⋆
)∕4 + (𝑟⋆
− 𝑟⋆
7 − 𝑟8 )∕2,
3
4
1
𝑓2⋆ = (𝑟⋆
− 𝑟⋆
)∕4 + (𝑟⋆
− 𝑟⋆
− 𝑟⋆
)∕2,
3
4
2
6
8
⋆
𝑓3⋆ = (𝑟⋆
+ 𝑟⋆
)∕4 + (−𝑟⋆
+ 𝑟⋆
7 − 𝑟8 )∕2,
3
4
1
𝑓4⋆ = (𝑟⋆
− 𝑟⋆
)∕4 + (−𝑟⋆
+ 𝑟⋆
− 𝑟⋆
)∕2,
3
4
2
6
8
(20)
⋆
𝑓5⋆ = (𝑟⋆
+ 𝑟⋆
+ 𝑟⋆
7 + 𝑟8 )∕4,
5
6
𝑖
⋆
𝑓6⋆ = (−𝑟⋆
+ 𝑟⋆
− 𝑟⋆
7 + 𝑟8 )∕4,
5
6
However, it should be recognised that (i) the summation over the index
𝑖 is a time-consuming operations and (ii) the pre-collision central moments are related to the pre-collision raw moments, whose computation
might be faster. Therefore, we suggest to compute 𝑘4,5 as follows. First,
we define the pre-collision raw moments as
∑
2
2
𝑟4 =
𝑓𝑖 (𝑐𝑖𝑥
− 𝑐𝑖𝑦
),
⋆
𝑓7⋆ = (𝑟⋆
− 𝑟⋆
− 𝑟⋆
7 + 𝑟8 )∕4,
5
6
⋆
𝑓8⋆ = (−𝑟⋆
− 𝑟⋆
+ 𝑟⋆
7 + 𝑟8 )∕4.
5
6
∑
𝑓𝑖 𝑐𝑖𝑥 𝑐𝑖𝑦 ,
(21)
𝑖
Algorithm of computation
Summing up, the following actions are required within the typical
time step:
and then we do not implement the evaluation of these quantities
through the summation over 𝑖, but simply as follows
𝑟4 = 𝑓0 + 𝑓1 − 𝑓2 − 𝑓3 ,
𝑟5 = 𝑓4 + 𝑓5 − 𝑓6 − 𝑓7 .
• get the force-free macroscopic variables in Eqs. (10) by neglecting
the force vector 𝑭 ;
• perform the IBM to obtain 𝑭 (see [65], [69], [80]) and correct
the fluid velocity by adding the corresponding term;
• get pre-collision central moments by Eqs. (23);
• apply the collision step by Eqs. (19);
• obtain the post-collision raw moments by Eqs. (25);
• reconstruct post-collision populations by Eqs. (26);
• stream by Eq. (12) and advance in time.
(22)
Pre-collision CMs are then obtained as
(
)
𝑘4 = 𝑓0 + 𝑓1 − 𝑓2 − 𝑓3 − 𝜌 𝑢2𝑥 − 𝑢2𝑦 ,
𝑘5 = 𝑓4 + 𝑓5 − 𝑓6 − 𝑓7 − 𝜌𝑢𝑥 𝑢𝑦 .
(26)
Notably, the central-moments-based collision operator can be seen
as a general multiple-relaxation-time LBM, where the (classical) rawmoments-based MRT LBM is recovered by simply setting 𝐍 = 𝐈.
𝑖
𝑟5 =
(25)
Eventually, post-collision populations can be reconstructed as 𝑓𝑖⋆ =
𝐌−1 𝑟⋆
𝑖 , i.e.
𝑖
𝑘5 =
(24)
(23)
Collision takes place by Eqs. (19). Before transforming post-collision
CMs into populations, it is important to notice that the transformation
4
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Fig. 1. Collision of a dipole against a straight wall: contour lines of the vorticity field at Re = 625.
Table 1
Collision of a dipole against a straight wall: kinetic energy at representative time
instants for different values of the Reynolds number. LBM denotes data from [36].
FD and SM stand for finite difference and spectral method, respectively, by Clercx and
Bruneau [15]. Relative discrepancies between our results (Present) and reference ones
are reported too.
Let us mention that spatial derivatives in Eq. (5) are evaluated as
1 ∑
𝜕𝑥 𝑢 𝑦 =
𝑤𝑖 𝑢𝑦 (𝒙 + 𝒄 𝑖 )𝑐𝑖𝑥 ,
𝑐𝑠2 𝑖
1 ∑
𝜕𝑦 𝑢 𝑥 =
𝑤𝑖 𝑢𝑥 (𝒙 + 𝒄 𝑖 )𝑐𝑖𝑦 ,
(27)
𝑐𝑠2 𝑖
where the weighting factors are 𝑤0 = 4∕9, 𝑤1,2,3,4 = 1∕9 and 𝑤5,6,7,8 =
1∕36.
Re
Time
Present
LBM
FD
SM
𝜀LBM [%]
𝜀FD [%]
𝜀SM [%]
625
0.25
0.5
0.75
1.501
1.008
0.762
1.501
1.013
0.767
1.502
1.013
0.767
1.502
1.013
0.767
0.00
0.49
0.65
0.07
0.49
0.65
0.07
0.49
0.65
1250
0.25
0.5
0.75
1.720
1.308
1.052
1.719
1.312
1.061
1.721
1.313
1.061
1.720
1.313
1.061
0.06
0.30
0.85
0.06
0.38
0.85
0.00
0.38
0.85
3. Validation
In this section, we assess the accuracy of the proposed approach.
First, we compare our findings against reference data by Clercx and
Bruneau [15] and Mohammed et al. [36], where the collision of a
dipole against a straight wall is considered. Then, the effectiveness
of the combined LBM-IBM to estimate forces acting upon a moving
immersed solid body is evaluated against the well-known benchmark
problem proposed by Dütsch et al. [82], where a round cylinder undergoes harmonic horizontal oscillations. In all the runs, the density is
initialised to 𝜌0 = 1 everywhere.
findings and (i) the LBM study by Mohammed et al. [36] and (ii)
finite-differences and spectral analyses by Clercx and Bruneau [15]. The
contour lines of the vorticity magnitude at representative time instants
are sketched in Figs. 1 and 2 for Re = 625 and 1250, respectively.
These pictures are in strong agreement with those depicted in [15]
and Mohammed et al. [36].
3.2. Oscillating cylinder in a quiescent fluid
3.1. Collision of a dipole against a straight wall
In this test, a rigid round cylinder of diameter 𝐷𝑐 undergoes harmonic horizontal oscillations of velocity 𝑣(𝑡) = −𝑉 cos (2𝜋𝑡∕𝑇 ). 𝑉
and 𝑇 denotes the maximum velocity and the period of oscillation,
respectively. The domain consists of 55𝐷𝑐 and 35𝐷𝑐 lattice points in
the horizontal and vertical directions, respectively. At 𝑡 = 0 the cylinder
is placed in the centre of the domain and the fluid is at rest. Outflow
boundary conditions are prescribed at each side of the domain. This
flow is governed by two parameters: the Reynolds number, Re =
𝑉 𝐷𝑐 ∕𝜈 = 100, and the Keulegan–Carpenter number, KC = 𝑉 𝑇 ∕𝐷𝑐 = 5.
The diameter of the cylinder is represented by 100 points. In order to
reduce detrimental compressibility effects proportional to the second
power of the Mach number, the maximum velocity is set equal to
𝑉 = 0.01. According to Morison et al. [83], the horizontal force 𝐹𝑥
acting on the cylinder can be computed as
Let us consider a square domain (𝑥, 𝑦) ∈ (−1 ∶ 1, −1 ∶ 1) enclosed by
no-slip walls at each side. The velocity is initialised as in Eqs. (3). Here,
the positions of the two monopoles are (𝑥1 , 𝑦1 ) = (0, 0.1) and (𝑥2 , 𝑦2 ) =
(0, −0.1). The characteristic Reynolds number is Re = 𝑈 𝐿∕𝜈, where 𝐿 =
1 is the half width of the domain. Two values of the Reynolds number
are considered: Re = 625 and 1250. Following the grid independence
analysis in [36], each side of the domain is discretised by 512 points
when Re = 625 and 768 points when Re = 1250. The half-way bounce
back rule is adopted to enforce the no-slip boundary condition. In
Table 1, the values of the energy at representative time instants are
reported. We compare our results to reference data in [15] and findings
obtained by the LBM analyses in [36]. Notably, present results show a
very slight mismatch (< 1%) with respect to those in the LBM study
by Mohammed et al. [36], which, in turn, are closer to the reference
data in [15]. We link this behaviour to the fact that a more accurate
boundary condition was adopted in [36]. Specifically, we introduce
𝜀LBM , 𝜀FD and 𝜀SM , which are the relative discrepancy between our
𝜕𝑣(𝑡)
1
1
𝐹𝑥 (𝑡) = − 𝑐𝑑 𝜌0 𝐷𝑐 𝑣(𝑡)|𝑣(𝑡)| − 𝑐𝑚 𝜋𝜌0 𝐷𝑐2
,
(28)
2
4
𝜕𝑡
where 𝑐𝑑 and 𝑐𝑚 denote the drag and added-mass coefficients, respectively [83]. These are obtained by a non-linear least-square fit of the
5
Computers and Fluids 250 (2023) 105711
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Fig. 2. Collision of a dipole against a straight wall: contour lines of the vorticity field at Re = 1250.
Fig. 3. Primary and secondary vortex structures emerging during the collision of a dipole against a round cylinder. Axes of the cylinder are sketched too.
Table 2
Oscillating cylinder in a quiescent fluid: 𝑐𝑑 and 𝑐𝑚 estimated by our approach (Present)
against reference values from the literature (Dütsch et al. [82], Uzunoğlu et al. [84],
Yuan et al. [85]).
𝑐𝑑
𝑐𝑚
Present
Dütsch et al.
[82]
Uzunoğlu
et al. [84]
Yuan et al.
[85]
2.10
1.45
2.09
1.45
2.10
1.45
2.10
1.45
number as Sp = 0.01, 0.05, 0.1, 0.25, 0.5, 0.75. For each value of
Sp, runs are repeated by varying the Reynolds number as Re = 10,
50, 100, 250, 500, 750, 1000. In all the 49 simulations, the density
is initialised as 𝜌0 = 1 everywhere and the boundary conditions at the
sides of the domain are enforced by the half-way bounce-back rule. The
computational grid consists of 1200 and 900 points in the horizontal
and vertical directions, respectively. This corresponds to adopt 30
points to represent 𝑟0 , that is consistent with the grid independence
analysis in [36]. The reference velocity in lattice units is set to 0.002,
that corresponds to a time step equal to 6.66 × 10−6 . The initial dipole is
represented by a positive monopole and a negative one. We will refer to
it as primary dipole. Upon impact, two new vortical structures will form
due to the interaction of the initial dipole with the boundary layers.
We will refer to these as secondary dipoles, where the top one and the
bottom one will be located above and below the horizontal axis of the
cylinder, respectively. Primary and secondary structures are sketched in
Fig. 3, together with the axes of the cylinder. Before going any further,
force signal. Table 2 compares findings obtained by the present method
to reference values from the literature (Dütsch et al. [82], Uzunoğlu
et al. [84], Yuan et al. [85]). One can immediately appreciate that an
excellent agreement is found.
4. Numerical tests
In this section, we discuss the results of our numerical campaign.
First, the rotation of the cylinder is neglected and this corresponds
to Sp = 0. Then, its motion is accounted for by varying the spinning
6
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Fig. 4. Static cylinder: vorticity field. (a)–(c): Re = 50. (d)–(f): Re = 100. (g)–(i): Re = 250. (a), (d), (g): 𝑡 = 0.3. (b), (e), (h): 𝑡 = 0.6. (c), (f), (i): 𝑡 = 0.9.
let us define the spatial maximum of the vorticity as
𝜔max (𝑡) = max𝒙 [|𝜔(𝒙, 𝑡)|] .
4.1. Static cylinder
(29)
Let us consider the setup presented in Section 2 by neglecting the
rotation of the cylinder (i.e., Sp = 0). In Fig. 4, the vorticity field
at representative time instants is reported in scenarios where Re =
50, 100, 250. Snapshots corresponding to Re = 10 are not reported
because the flow physics is strongly dominated by the high diffusion
and the vorticity vanishes too soon. At Re = 50, vortices rapidly
tend to dissipate after the impact. A humble presence of vorticity
in the boundary layers is found. Upon the impact, this vorticity is
Moreover, let us introduce the normalised the drag force 𝐶𝑑 experienced by the cylinder, that is
𝐶𝑑 =
𝐹𝑑
𝜌0 𝑈 2 𝐷
,
(30)
𝐹𝑑 being the drag force computed by the IBM [69].
7
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Fig. 5. Static cylinder: vorticity field. (a)–(c): Re = 500. (d)–(f): Re = 750. (g)–(i): Re = 1000. (a), (d), (g): 𝑡 = 0.3. (b), (e), (h): 𝑡 = 0.6. (c), (f), (i): 𝑡 = 0.9.
pushed away from the cylinder at Re = 100. Two secondary vortices
are generated. These surround the initial monopoles and form two
primary–secondary structures that counter-rotate with respect to these.
At Re = 250, the flow physics becomes different. After the impact, the
two new couples of primary–secondary dipoles roll back together and
produce a second impact. Tertiary vortical structures are created and
tend to surround primary vortices, even if they dissipate rapidly.
High-Re fluid dynamics is considerably more interesting and Fig. 5
depicts the vorticity field for the remaining cases. At Re = 500, we
see again that secondary vortices detach from the boundary layers and
follow the primary ones. However, other smaller vortices are ejected
from the boundary layers and remain closer to the cylinder. The two
systems of primary–secondary dipoles rotate and impinge the cylinder.
The second impact is more evident than the one experienced at Re =
250. In fact, primary vortices interact again with the boundary layers
and create tertiary structures of larger magnitude. The primary–tertiary
dipoles have trajectories very similar to those of the primary–secondary
systems. During the second impingement, there is a strong interaction
of the primary and secondary vortices. A part of the latter is pushed towards the cylinder, while the remaining fraction changes the direction
8
Computers and Fluids 250 (2023) 105711
A. De Rosis
Fig. 6. Static cylinder: time histories of (a) kinetic energy, (b) kinetic enstrophy, (c) spatial maximum of the vorticity, and (d) normalised drag force at Re = 10 (grey), 50 (black),
100 (red), 250 (green), 500 (blue), 750 (magenta), and 1000 (cyan).
Fig. 7. Static cylinder: horizontal component of the velocity field at salient time instants corresponding to the up-down-up trend of the drag coefficient when Re = 1000.
of its motion and points towards the left edge of the domain. The flow
physics characteristics described in this case are even more emphasised
at Re = 750 and 1000. All the vortices tend to rotate faster and to
last longer into the domain. By creating more violent impacts, primary
vortices are responsible of the generation of secondary and tertiary
structures of considerably larger magnitude and size. After the second
impact, secondary structures move faster leftward. Notably, the tertiary
structures born during the second impact become more prominent and
travel a larger distance from the cylinder. The flow physics described
above corroborates the experimental observations by New and Zang
[6], where the authors showed that the primary vortex cores flatten and
spread radially upon impact. Thereafter, the secondary vortices turn
and move towards the collision axis.
Fig. 6 reports the time history of the energy, enstrophy, spatial maximum of the vorticity and the normalised drag force. As expected, the
Fig. 8. Spinning cylinder: normalised peak of the enstrophy for different values of Re
and Sp.
9
Computers and Fluids 250 (2023) 105711
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Fig. 9. Spinning cylinder at Sp = 0.25: time histories of (a) kinetic energy, (b) kinetic enstrophy and (c) spatial maximum of the vorticity at Re = 10 (grey), 50 (black), 100 (red),
250 (green), 500 (blue), 750 (magenta), and 1000 (cyan).
Table 3
Static cylinder: normalised maxima of the enstrophy and time instants when these are
achieved.
Re
10
50
100
250
500
750
100
max (𝛺)
𝛺(𝑡 = 0)
1
1
1
1.058
1.847
2.2923
2.581
𝑡 [max (𝛺)]
0
0
0
0.3658
0.3372
0.328
0.323
whose time history is reported in Fig. 6(d). The diagram is relatively
flat at Re = 10 due to the rapid dissipation of the primary dipole. At
Re = 50, there is a humble increment of 𝐶𝑑 caused by the impact. The
normalised drag force exhibits a more marked variation at Re = 100
that corresponds to a stronger impact and a higher enhancement of
the maximum of the vorticity. For larger values of Re, the first impact
generates a different behaviour, where a first local maximum is rapidly
followed by a local minimum and, again, by a local maximum. We link
this up-down-up pattern to the dichotomous presence of a rightward
force induced by the dipole and another one exerted by the boundary
layers that acts in the opposite direction. To further shed light on it,
Fig. 7 reports the horizontal component of the fluid velocity, 𝑢𝑥 , when
Re = 1000 at the time instants corresponding to the ups and downs of
𝐶𝑑 . At 𝑡 = 0.28, the cylinder is surrounded by a positive (rightward)
horizontal velocity, that is responsible of the local maximum of 𝐶𝑑 . We
can observe portions of negative (leftward) horizontal velocity along
the cylinder at 𝑡 = 0.32, that promote a decrement of 𝐶𝑑 and its local
minimum. Negative velocities move rapidly far from the cylinder, that
remains largely surrounded by positive values of 𝑢𝑥 , generating the
second local maximum of 𝐶𝑑 . Present findings corroborate the observation in [4], where a joint numerical and experimental investigation
measured a small net force on a surface upon the impingement of a
vortex. A similar trend is experienced later in time during the second
impact. However, the lower strength of the impact is responsible of a
less evident up-down-up trend.
energy dissipates faster as the Reynolds number reduces (see Fig. 6(a)).
It is of interest to observe the presence of jumps into its monotonic
decreasing trend if Re ≥ 100. This decrement of 𝐸 can be clearly linked
to the impact, when smaller-scale vortical structures are created. This is
confirmed by observing the non-monotonic time history of the kinetic
enstrophy in Fig. 6(b), where the time when a peak in 𝛺 is found
corresponds to a knee of 𝐸. Similarly to the energy, enstrophy decreases
faster in time if Re is lower. The impact against the cylinder generates a
very strong increment of the enstrophy at higher values of the Reynolds
number. Indeed, 𝛺 multiplies its values by almost three times when
Re = 1000. It should also be noted that the maxima of the enstrophy
and the knees of the energy are localised earlier in time as Re grows.
Table 3 reports the values of the maxima of the enstrophy (normalised
by its initial value) and the time when these manifest. Interestingly, the
time evolutions of 𝛺 exhibit a second peak later in time, that should be
linked to the second impingement. The magnitude of the second impact
is considerably smaller than the one of the first one, resulting in a less
sharp spike of 𝛺 and a more gentle knee of 𝐸. This corroborates the
observations in Figs. 4 and 5, where (i) the primary dipoles appear to
hit the cylinder earlier when the kinematic viscosity reduces and (ii)
stronger vorticity arises in the boundary layers for increasing values
of the Reynolds number. The peaks in 𝛺 reflect an even more sudden
variation of the vorticity, that is confirmed by Fig. 6(c). In fact, 𝜔max
gets amplified by a factor larger than four during the first impact when
Re = 1000.
We conclude the discussion about the impact against the static cylinder by drawing some considerations about the normalised drag force,
4.2. Spinning cylinder
Let us now consider a spinning cylinder. In Fig. 8, the peak of the
enstrophy normalised by its initial value (see Eqs. (4)) is plotted as
a function of the Reynolds and the spinning numbers. Independently
from Re, the behaviour is substantially overlapped to the static case
until Sp ≤ 0.1, suggesting that the role of the rotation of the cylinder
on the encompassing fluid dynamics is negligible if its velocity is
smaller than the 10% of the one of the initial dipole. Interestingly,
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Computers and Fluids 250 (2023) 105711
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Fig. 10. Spinning cylinder at Sp = 0.25: vorticity field. (a)–(c): Re = 100. (d)–(f): Re = 500. (g)–(i): Re = 1000. (a), (d), (g): 𝑡 = 0.3. (b), (e), (h): 𝑡 = 0.6. (c), (f), (i): 𝑡 = 0.9.
all the curves show a marked change when Re = 250, indicating a
drastic variation of the flow physics. To elucidate the presence of any
underlying mechanism, the time histories of the energy, enstrophy and
spatial maximum of the vorticity are reported in Fig. 9 at Sp = 0.25
for different values of the Reynolds number. Similarly to the static
case, these graphs suggest that the dipole vanishes before hitting the
cylinder when Re = 10, 50. At Re = 100, a very humble increment of
𝛺 is experienced at 𝑡 ∼ 0.46, corresponding to the time instant when
the dipole impacts the cylinder. As Re ≥ 250, we observe a spike in
the enstrophy and a knee in the energy. We link this behaviour to
the fact that progressively stronger initial dipoles hit the surface. It
is of interest to compare the time histories of the enstrophy to those
experience in the static case. While in the latter scenario only a second
local maximum is found, two peaks are found at Sp = 0.25. This can
be more remarkably appreciated by looking at the spatial maximum
of the vorticity, that undergoes considerably larger oscillations. This
suggest that the rotation of the cylinder modifies the second impact.
A closer look at the vorticity field in Fig. 10 allows us to corroborate
these statements by getting further insights into the flow physics. The
maps confirm that the dipole dissipates rapidly at Re = 100. Upon
11
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Fig. 11. Spinning cylinder at Sp = 0.75: vorticity field. (a)–(c): Re = 50. (d)–(f): Re = 100. (g)–(i): Re = 250. (a), (d), (g): 𝑡 = 0.3. (b), (e), (h): 𝑡 = 0.6. (c), (f), (i): 𝑡 = 0.9.
impact, vortices are rotated counter-clockwise. Interestingly, the second
of progressively finer flow structures as the fluid kinematic viscosity
reduces.
Figs. 11 and 12 show the vorticity field at representative time
instants by considering Sp = 0.75. Vortices in the boundary layers
surrounding the cylinder are more evident. Upon the first impact, the
primary dipole is clearly rotated counter-clockwise, even for low values
of the Reynolds number. At Re = 100, only the positive monopole
bounces back. The interaction between the negative monopole and the
boundary layer makes the former sliding along the latter (and along
the cylinder’s surface). As a consequence, it overcomes the vertical axis
impact is a double impact when Re ≥ 250. The rotation of the cylinder
pushes farther the primary–secondary vortex system on the top, but it
also receives spin. The primary–secondary vortex system on the bottom
clashes with the cylinder first, and then the top one hits it. This results
in two subsequent spikes of 𝜔max . Intriguingly, we observe the presence
of a positive vortex appears to be surrounded by three negative satellite
structures at 𝑡 = 0.9 when Re = 1000, which are ejected by the
boundary layers during the second impact. This indicates the presence
12
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Fig. 12. Spinning cylinder at Sp = 0.75: vorticity field. (a)–(c): Re = 500. (d)–(f): Re = 750. (g)–(i): Re = 1000. (a), (d), (g): 𝑡 = 0.3. (b), (e), (h): 𝑡 = 0.6. (c), (f), (i): 𝑡 = 0.9.
of the cylinder. At Re = 250, only the positive dipole bounces back
again. Intriguingly, the behaviour of the negative dipole is radically
different from the case when Re = 100. In fact, it does not slide along
the cylinder, but collides against it. Vorticity from the boundary layer
detaches and a primary–secondary structure is formed. This impinges
the cylinder (giving rise to a second impact) and now the negative part
slides along the cylinder. At Re = 500, the primary dipole performs
the first collision. Upon this, vorticity from the boundary layers detaches and forms two primary–secondary structures by coupling with
the primary dipole. The bottom primary–secondary vortex is closer to
the cylinder. It impacts the surface, while the top primary–secondary
vortex is not able to do it. In fact, its negative part is blocked by
the positive part of the bottom counterpart. The positive contribution
is pushed away by the boundary layer. Interestingly, the negative
part of the initial dipole forms a primary–tertiary structure with the
vortex detaching during the second impact. After the first impact, the
two primary–secondary structures lasts longer in the domain at Re =
750 and 1000. The interaction is considerably more compelling and
promotes smaller-scale vortices departing from the boundary layers
during the subsequent impacts.
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Fig. 13. Spinning cylinder at Sp = 0.75: time histories of (a) kinetic energy, (b) kinetic enstrophy and (c) spatial maximum of the vorticity at Re = 10 (grey), 50 (black), 100 (red),
250 (green), 500 (blue), 750 (magenta), and 1000 (cyan).
5. Conclusions
The time histories in Fig. 13 provide us with further details about
the flow physics. Specifically, the kinetic energy and enstrophy plateau
to values which are considerably larger than those experienced before.
Clearly, this is a consequence of the higher velocity of rotation of
the cylinder. Notably, the time evolution of 𝛺 changes drastically for
Re ≥ 250. In fact, the enstrophy shows high average values spiking in
correspondence of the impacts. By making a comparison against the
same quantities plotted in Fig. 9 at Sp = 0.25, we can observe that the
cylinder injects considerable amounts of energy and enstrophy in the
system. As a consequence, 𝐸 and 𝛺 increase in time, especially if Re
grows. The peaks of the enstrophy and those of the spatial maximum
of the vorticity are less sharp, due to the presence of existing strong
vorticity produced by the rotation of the cylinder.
Eventually, we observe the time histories of the normalised drag
force at Sp = 0.25 and 0.75 reported in Fig. 14. In the former case, the
average trend of all the curves is very similar to the one experienced in
the static case. However, it should be noted that the curves appears to
fluctuate around a mean value. The rotation of the cylinder is clearly
responsible of this behaviour, that is expected to be more emphasised
as Sp grows. In fact, these oscillations are clearly more marked Sp =
0.75. In addition, it is possible to observe that the time evolution
of 𝐶𝑑 at higher values of Sp are characterised by the presence of
more frequent local maxima and minima. This is particularly more
evident for increasing values of Re. Indeed, the interaction between
the impacting dipoles, the boundary layers and the larger rotation
of the cylinder promotes the presence of progressively finer, more
complex and compelling vortical structures, which are responsible of
the experienced strong oscillations. For the sake of curiosity, Fig. 15
reports the time history of the normalised lift force, 𝐶𝑙 , that is defined
as
𝐹𝑙
(31)
𝐶𝑙 =
𝜌0 𝑈 2 𝐷
In this paper, we examined the flow physics induced by the collision
of a vortex dipole against a round cylinder numerically. Our analyses
were carried out by using a lattice Boltzmann method equipped with
a central-moments-based collision operator. The presence of the cylinder was accounted for through the immersed boundary method. The
adopted approach was validated against two test cases. The former
test involved the collision of a dipole against a straight wall. Our
model generated results that are in very close agreement with consolidated benchmark data obtained by high-resolution pseudo-spectral
simulations, as well as with findings achieved by using the LBM with
sophisticated boundary conditions. In the latter case, we assessed the
accuracy of the coupled LBM-IBM by computing the drag force acting
upon an oscillating cylinder. Again, the very good accuracy of the
scheme was demonstrated.
After the validation phase, the impact against a cylinder was investigated by considering different values of the Reynolds and spinning
numbers. Findings in terms of kinetic energy, kinetic enstrophy, spatial
maximum of the vorticity and normalised drag force experienced by
the cylinder were discussed in details. Our statements were supported
by having a closer look at the flow physics through the vorticity and
velocity maps. We found that the flow field is progressively more rotated counter-clockwise (that is the direction of rotation of the cylinder)
if Re and Sp grow. The dependence on Re should be linked to the
fact that the vortices last longer into the domain if the fluid kinematic
viscosity reduces. Regarding the dependence on the spinning number,
this is due to the fact that the velocity of the rotation of the cylinder
becomes comparable to the one of the initial dipole. After the first
impact, vorticity ejected from the boundary layers interacts with initial
dipole forming two primary–secondary structures. In the static case,
these roll back and are responsible of a second impact, where the two
structures hit the cylinder simultaneously. As Sp grows, the second
impingement is a double impact because the primary–secondary vortex
systems on the top and the bottom collide the walls at different times.
In fact, the bottom one is kept close to the cylinder as a consequence
of the rotation. Energy, enstrophy and vorticity do not vanish in time,
for the aforementioned values of Sp. 𝐹𝑙 is the lift force that is obtained
by IBM (analogously to the drag force 𝐹𝑑 ). Similarly to what experienced for the drag force, it appears evident that the lift force undergoes
large oscillations and fluctuations around a mean value, which are due
to the mutual interplay between the rotation of the cylinder and the
vortex impact/detachment.
14
Computers and Fluids 250 (2023) 105711
A. De Rosis
Fig. 14. Time histories of the normalised drag force at Re = 10 (grey), 50 (black), 100 (red), 250 (green), 500 (blue), 750 (magenta), and 1000 (cyan) for two representative
values of the spinning numbers.
Fig. 15. Time histories of the normalised lift force at Re = 10 (grey), 50 (black), 100 (red), 250 (green), 500 (blue), 750 (magenta), and 1000 (cyan) for two representative values
of the spinning numbers.
Appendix A. Supplementary data
but plateau to values generated by the work done by the rotating
cylinder. The drag force constantly fluctuates around a mean value.
Its oscillations are due to the vorticity created by the rotation of the
cylinder and are more evident for higher values of Sp, resulting from
the vorticity generated in the boundary layers. Moreover, we found that
𝐶𝑑 can achieve marked negative values as a consequence of the high
leftward velocity created by the cylinder during its motion.
In a future work, we aim to investigate flow regimes characterised
by higher values of the spinning number, i.e. Sp ≥ 1. In fact, when
the velocity of the rotation of the cylinder is larger than the one of
the dipole, the role of the cylinder is more prominent and the fluid
dynamics is also affected by the presence of vortices that ejected from
the boundary layers due to the rotation, rather than be caused by the
impacting dipole.
Supplementary material related to this article can be found online
at https://doi.org/10.1016/j.compfluid.2022.105711.
References
[1] Harvey J, Perry FJ. Flowfield produced by trailing vortices in the vicinity of the
ground. AIAA J 1971;9(8):1659–60. http://dx.doi.org/10.2514/3.6415.
[2] Barker SJ, Crow SC. The motion of two-dimensional vortex pairs in a
ground effect. J Fluid Mech 1977;82(4):659–71. http://dx.doi.org/10.1017/
S0022112077000913.
[3] Van Heijst G, Flor J. Laboratory experiments on dipole structures in a stratified
fluid. In: Elsev Ocean Serie. Vol. 50, Elsevier; 1989, p. 591–608. http://dx.doi.
org/10.1016/S0422-9894(08)70209-5.
[4] Chu C-C, Wang C-T, Chang C-C. A vortex ring impinging on a solid plane
surface-vortex structure and surface force. Phys Fluids 1995;7(6):1391–401.
http://dx.doi.org/10.1063/1.868527.
[5] Wells J, Afanasyev YD. Decaying quasi-two-dimensional turbulence in a rectangular container: laboratory experiments. Geophys Astro Fluid 2004;98(1):1–20.
http://dx.doi.org/10.1080/030919209410001648390.
[6] New T, Zang B. Head-on collisions of vortex rings upon round cylinders. J Fluid
Mech 2017;833:648–76. http://dx.doi.org/10.1017/jfm.2017.599.
[7] New T, Long J, Zang B, Shi S. Collision of vortex rings upon V-walls. J Fluid
Mech 2020;899. http://dx.doi.org/10.1017/jfm.2020.425.
[8] Morris SE, Williamson C. Impingement of a counter-rotating vortex pair on a
wavy wall. J Fluid Mech 2020;895. http://dx.doi.org/10.1017/jfm.2020.263.
[9] Orlandi P. Vortex dipole rebound from a wall. Phys Fluids A-Fluid
1990;2(8):1429–36. http://dx.doi.org/10.1063/1.857591.
[10] Coutsias E, Lynov J-P. Fundamental interactions of vortical structures with
boundary layers in two-dimensional flows. Physica D 1991;51(1–3):482–97.
http://dx.doi.org/10.1016/0167-2789(91)90254-7.
[11] Verzicco R, Flór J-B, Van Heijst G, Orlandi P. Numerical and experimental study
of the interaction between a vortex dipole and a circular cylinder. Exp Fluids
1995;18(3):153–63. http://dx.doi.org/10.1007/BF00230259.
CRediT authorship contribution statement
Alessandro De Rosis: Conceptualization, Methodology, Software,
Writing.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to
influence the work reported in this paper.
Data availability
A script to generate all the data in this paper is attached as supplementary material.
15
Computers and Fluids 250 (2023) 105711
A. De Rosis
[39] Geier M, Greiner A, Korvink J. Cascaded digital lattice Boltzmann automata for
high Reynolds number flow. Phys Rev E 2006;73(6):066705. http://dx.doi.org/
10.1103/PhysRevE.73.066705.
[40] Asinari P. Generalized local equilibrium in the cascaded lattice Boltzmann
method. Phys Rev E 2008;78(1):016701. http://dx.doi.org/10.1103/PhysRevE.
78.016701.
[41] Premnath K, Banerjee S. Incorporating forcing terms in cascaded lattice Boltzmann approach by method of central moments. Phys Rev E 2009;80(3):036702.
http://dx.doi.org/10.1103/PhysRevE.80.036702.
[42] De Rosis A. Nonorthogonal central-moments-based lattice Boltzmann scheme in
three dimensions. Phys Rev E 2017;95(1):013310. http://dx.doi.org/10.1103/
PhysRevE.95.013310.
[43] Saito S, De Rosis A, Festuccia A, Kaneko A, Abe Y, Koyama K. Color-gradient
lattice Boltzmann model with nonorthogonal central moments: Hydrodynamic
melt-jet breakup simulations. Phys Rev E 2018;98(1):013305. http://dx.doi.org/
10.1103/PhysRevE.98.013305.
[44] De Rosis A, Luo KH. Role of higher-order Hermite polynomials in the centralmoments-based lattice Boltzmann framework. Phys Rev E 2019;99(1):013301.
http://dx.doi.org/10.1103/PhysRevE.99.013301.
[45] Saito S, De Rosis A, Fei L, Luo KH, Ebihara K-i, Kaneko A, Abe Y. Lattice
Boltzmann modeling and simulation of forced-convection boiling on a cylinder.
Phys Fluids 2021;33(2):023307. http://dx.doi.org/10.1063/5.0032743.
[46] De Rosis A, Huang R, Coreixas C. Universal formulation of central-momentsbased lattice Boltzmann method with external forcing for the simulation of
multiphysics phenomena. Phys Fluids 2019;31(11):117102. http://dx.doi.org/10.
1063/1.5124719.
[47] De Rosis A, Tafuni A. A phase-field lattice Boltzmann method for the solution of
water-entry and water-exit problems. Comput-Aided Civ Inf 2022;37(7):832–47.
http://dx.doi.org/10.1111/mice.12651.
[48] De Rosis A, Enan E. A three-dimensional phase-field lattice Boltzmann method
for incompressible two-components flows. Phys Fluids 2021;33(4):043315. http:
//dx.doi.org/10.1063/5.0046875.
[49] Hajabdollahi F, Premnath KN. Central moments-based cascaded lattice Boltzmann
method for thermal convective flows in three-dimensions. Int J Heat Mass Tran
2018;120:838–50. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.12.085.
[50] Hajabdollahi F, Premnath KN, Welch SW. Cascaded lattice Boltzmann method
based on central moments for axisymmetric thermal flows including swirling
effects. Int J Heat Mass Tran 2019;128:999–1016. http://dx.doi.org/10.1016/j.
ijheatmasstransfer.2018.09.059.
[51] De Rosis A, Skillen A. Vortex dynamics in an electrically conductive fluid
during a dipole–wall collision in presence of a magnetic field. Phys Fluids
2022;34(8):081704. http://dx.doi.org/10.1063/5.0107338.
[52] Kizner Z, Makarov V, Kamp L, Van Heijst G. Instabilities of the flow around
a cylinder and emission of vortex dipoles. J Fluid Mech 2013;730:419–41.
http://dx.doi.org/10.1017/jfm.2013.345.
[53] Grotegoed R. Vortex dipole collision with a rotating cylinder. Eindhoven University of Technology; 2019, URL: https://research.tue.nl/en/studentTheses/vortexdipole-collision-with-a-rotating-cylinder.
[54] Tafuni A, De Giorgi MG, De Rosis A. Smoothed particle hydrodynamics vs lattice
Boltzmann for the solution of steady and unsteady fluid flows. Comput Part Mech
2022;9:1049–71. http://dx.doi.org/10.1007/s40571-021-00447-5.
[55] Bhatnagar PL, Gross EP, Krook M. A model for collision processes in gases. I.
Small amplitude processes in charged and neutral one-component systems. Phys
Rev 1954;94(3):511. http://dx.doi.org/10.1103/PhysRev.94.511.
[56] d’Humières D. Multiple-relaxation-time lattice Boltzmann models in three dimensions. Philos Trans R Soc A 2002;360(1792):437–51. http://dx.doi.org/10.1098/
rsta.2001.0955.
[57] Geier M, Greiner A, Korvink J. A factorized central moment lattice Boltzmann
method. Eur Phys J-Spec Top. 2009;171(1):55–61, doi:epjst/e2009-01011-1.
[58] Ning Y, Premnath K, Patil D. Numerical study of the properties of the central moment lattice Boltzmann method. Int J Numer Methods Fluids 2016;82(2):59–90.
http://dx.doi.org/10.1002/fld.4208.
[59] De Rosis A. A central moments-based lattice Boltzmann scheme for shallow water
equations. Comput Methods Appl Mech Eng 2017;319:379–92. http://dx.doi.org/
10.1016/j.cma.2017.03.001.
[60] Yahia E, Schupbach W, Premnath KN. Three-dimensional central moment lattice
Boltzmann method on a cuboid lattice for anisotropic and inhomogeneous flows.
Fluids 2021;6(9). http://dx.doi.org/10.3390/fluids6090326.
[61] Peskin C. The immersed boundary method. Acta Numer 2002;11(2):479–517.
http://dx.doi.org/10.1017/S0962492902000077.
[62] Feng Z-G, Michaelides EE. The immersed boundary-lattice Boltzmann method for
solving fluid-particles interaction problems. J Comput Phys 2004;195(2):602–28.
http://dx.doi.org/10.1016/j.jcp.2003.10.013.
[63] Mittal R, Iaccarino G. Immersed boundary methods. Annu Rev Fluid Mech
2005;37:239–61. http://dx.doi.org/10.1146/annurev.fluid.37.061903.175743.
[64] Wu J, Shu C. Implicit velocity correction-based immersed boundary-lattice
Boltzmann method and its applications. J Comput Phys 2009;228(6):1963–79.
http://dx.doi.org/10.1016/j.jcp.2008.11.019.
[65] Suzuki K, Inamuro T. Effect of internal mass in the simulation of a moving
body by the immersed boundary method. Comput Fluids 2011;49(1):173–87.
http://dx.doi.org/10.1016/j.compfluid.2011.05.011.
[12] Clercx H, Maassen S, Van Heijst G. Spontaneous spin-up during the decay
of 2D turbulence in a square container with rigid boundaries. Phys Rev Lett
1998;80(23):5129. http://dx.doi.org/10.1103/PhysRevLett.80.5129.
[13] Clercx H, Maassen S, Van Heijst G. Decaying two-dimensional turbulence
in square containers with no-slip or stress-free boundaries. Phys Fluids
1999;11(3):611–26. http://dx.doi.org/10.1063/1.869933.
[14] Clercx H, Nielsen A. Vortex statistics for turbulence in a container with
rigid boundaries. Phys Rev Lett 2000;85(4):752. http://dx.doi.org/10.1103/
PhysRevLett.85.752.
[15] Clercx HJ, Bruneau C-H. The normal and oblique collision of a dipole with a
no-slip boundary. Comput Fluids 2006;35(3):245–79. http://dx.doi.org/10.1016/
j.compfluid.2004.11.009.
[16] Kramer W, Clercx H, Van Heijst G. Vorticity dynamics of a dipole colliding
with a no-slip wall. Phys Fluids 2007;19(12):126603. http://dx.doi.org/10.1063/
1.2814345.
[17] Keetels GH, D’Ortona U, Kramer W, Clercx HJ, Schneider K, van Heijst GJ.
Fourier spectral and wavelet solvers for the incompressible Navier–Stokes equations with volume-penalization: Convergence of a dipole–wall collision. J Comput
Phys 2007;227(2):919–45. http://dx.doi.org/10.1016/j.jcp.2007.07.036.
[18] Nguyen van yen R, Farge M, Schneider K. Energy dissipating structures produced by walls in two-dimensional flows at vanishing viscosity. Phys Rev Lett
2011;106:184502. http://dx.doi.org/10.1103/PhysRevLett.106.184502.
[19] Sutherland D, Macaskill C, Dritschel D. The effect of slip length on vortex
rebound from a rigid boundary. Phys Fluids 2013;25(9):093104. http://dx.doi.
org/10.1063/1.4821774.
[20] Guzmán J, Kamp L, Van Heijst G. Vortex dipole collision with a sliding wall.
Fluid Dyn Res 2013;45(4):045501. http://dx.doi.org/10.1088/0169-5983/45/4/
045501.
[21] Peterson SD, Porfiri M. Impact of a vortex dipole with a semi-infinite rigid plate.
Phys Fluids 2013;25(9):093103. http://dx.doi.org/10.1063/1.4820902.
[22] Zivkov E, Peterson SD, Yarusevych S. Combined experimental and numerical
investigation of a vortex dipole interaction with a deformable plate. J Fluid
Struct 2017;70:201–13. http://dx.doi.org/10.1016/j.jfluidstructs.2017.01.012.
[23] New T, Gotama G, Vevek U. A large-eddy simulation study on vortex-ring
collisions upon round cylinders. Phys Fluids 2021;33(9):094101. http://dx.doi.
org/10.1063/5.0057475.
[24] Clercx H, Van Heijst G. Energy spectra for decaying 2D turbulence in a bounded
domain. Phys Rev Lett 2000;85(2):306. http://dx.doi.org/10.1103/PhysRevLett.
85.306.
[25] Clercx H, Nielsen A, Torres D, Coutsias E. Two-dimensional turbulence in square
and circular domains with no-slip walls. Eur J Mech B-Fluid 2001;20(4):557–76.
http://dx.doi.org/10.1016/S0997-7546(01)01130-X.
[26] Clercx H, van Heijst G. Dissipation of kinetic energy in two-dimensional bounded
flows. Phys Rev E 2002;65(6):066305. http://dx.doi.org/10.1103/PhysRevE.65.
066305.
[27] Clercx H, Van Heijst G, Molenaar D, Wells M. No-slip walls as vorticity sources
in two-dimensional bounded turbulence. Dyn Atmos Oceans 2005;40(1–2):3–21.
http://dx.doi.org/10.1016/j.dynatmoce.2004.10.002.
[28] Sansón LZ, Sheinbaum J. Elementary properties of the enstrophy and strain
fields in confined two-dimensional flows. Eur J Mech B-Fluid 2008;27(1):54–61.
http://dx.doi.org/10.1016/j.euromechflu.2007.04.002.
[29] Clercx H, Van Heijst G. Dissipation of coherent structures in confined twodimensional turbulence. Phys Fluids 2017;29(11):111103. http://dx.doi.org/10.
1063/1.4993488.
[30] Latt J, Chopard B. A benchmark case for lattice Boltzmann: turbulent dipole-wall
collision. Internat J Modern Phys C 2007;18(04):619–26. http://dx.doi.org/10.
1142/S0129183107010863.
[31] Succi S. The Lattice Boltzmann equation for fluid dynamics and beyond.
Numerical mathematics and scientific computation, Oxford: Clarendon Press;
2001.
[32] Krüger T, Kusumaatmaja H, Kuzmin A, Shardt O, Silva G, Viggen EM. The Lattice
Boltzmann method: Principles and practice. Springer; 2017, http://dx.doi.org/10.
1007/978-3-319-44649-3.
[33] Chen S, Doolen GD. Lattice Boltzmann method for fluid flows. Annu Rev Fluid
Mech 1998;30(1):329–64. http://dx.doi.org/10.1146/annurev.fluid.30.1.329.
[34] Guo Z, Shu C. Lattice Boltzmann method and its applications in engineering.
World Scientific; 2013, http://dx.doi.org/10.1142/8806.
[35] Aidun CK, Clausen JR. Lattice-Boltzmann method for complex flows. Annu Rev
Fluid Mech 2010;42:439–72. http://dx.doi.org/10.1146/annurev-fluid-121108145519.
[36] Mohammed S, Graham D, Reis T. Assessing moment-based boundary conditions
for the lattice Boltzmann equation: A study of dipole-wall collisions. Comput
Fluids 2018;176:79–96. http://dx.doi.org/10.1016/j.compfluid.2018.08.025.
[37] Mohammed S, Graham D, Reis T. Modeling the effects of slip on dipole–wall
collision problems using a lattice Boltzmann equation method. Phys Fluids
2020;32(2):025104. http://dx.doi.org/10.1063/1.5131865.
[38] De Rosis A, Coreixas C. Multiphysics flow simulations using D3Q19 lattice
Boltzmann methods based on central moments. Phys Fluids 2020;32(11):117101.
http://dx.doi.org/10.1063/5.0026316.
16
Computers and Fluids 250 (2023) 105711
A. De Rosis
[76] Malaspinas O. Increasing stability and accuracy of the lattice Boltzmann scheme:
recursivity and regularization. 2015, arXiv preprint arXiv:1505.06900 URL:
https://arxiv.org/abs/1505.06900.
[77] Coreixas C, Chopard B, Latt J. Comprehensive comparison of collision models
in the lattice Boltzmann framework: Theoretical investigations. Phys Rev E
2019;100(3):033305. http://dx.doi.org/10.1103/PhysRevE.100.033305.
[78] Huang R, Wu H, Adams NA. Eliminating cubic terms in the pseudopotential
lattice Boltzmann model for multiphase flow. Phys Rev E 2018;97(5):053308.
http://dx.doi.org/10.1103/PhysRevE.97.053308.
[79] De Rosis A. Non-orthogonal central moments relaxing to a discrete equilibrium:
A D2Q9 lattice Boltzmann model. Europhys Lett 2017;116(4):44003. http://dx.
doi.org/10.1209/0295-5075/116/44003.
[80] De Rosis A, Lévêque E. Central-moment lattice Boltzmann schemes with fixed
and moving immersed boundaries. Comput Math Appl 2016. http://dx.doi.org/
10.1016/j.camwa.2016.07.025.
[81] Fei L, Luo KH. Consistent forcing scheme in the cascaded lattice Boltzmann
method. Phys Rev E 2017;96(5):053307. http://dx.doi.org/10.1103/PhysRevE.
96.053307.
[82] Dütsch H, Durst F, Becker S, Lienhart H. Low-Reynolds-number flow around an
oscillating circular cylinder at low Keulegan-Carpenter numbers. J Fluid Mech
1998;360:249–71. http://dx.doi.org/10.1017/S002211209800860X.
[83] Morison J, Johnson J, Schaaf S. The force exerted by surface waves on piles. J
Pet Technol 1950;2(05):149–54. http://dx.doi.org/10.2118/950149-G.
[84] Uzunoğlu B, Tan M, Price W. Low-Reynolds-number flow around an oscillating
circular cylinder using a cell viscousboundary element method. Internat J Numer
Methods Engrg 2001;50(10):2317–38. http://dx.doi.org/10.1002/nme.122.
[85] Yuan R, Zhong C, Zhang H. An immersed-boundary method based on the
gas kinetic BGK scheme for incompressible viscous flow. J Comput Phys
2015;296:184–208. http://dx.doi.org/10.1016/j.jcp.2015.04.052.
[66] Mei R, Luo L, Shyy W. An accurate curved boundary treatment in the lattice
Boltzmann method. J Comput Phys 1999;155(2):307–30. http://dx.doi.org/10.
1006/jcph.1999.6334.
[67] Bouzidi M, Firdaouss M, Lallemand P. Momentum transfer of a Boltzmann-lattice
fluid with boundaries. Phys Fluids 2001;13(11):3452–9. http://dx.doi.org/10.
1063/1.1399290.
[68] Mei R, Yu D, Shyy W, Luo L. Force evaluation in the lattice Boltzmann method
involving curved geometry. Phys Rev E 2002;65(4):041203. http://dx.doi.org/
10.1103/PhysRevE.65.041203.
[69] De Rosis A, Ubertini S, Ubertini F. A comparison between the interpolated
bounce-back scheme and the immersed boundary method to treat solid boundary
conditions for laminar flows in the Lattice Boltzmann framework. J Sci Comp
2014;61(3):477–89. http://dx.doi.org/10.1007/s10915-014-9834-0.
[70] Favier J, Revell A, Pinelli A. A lattice Boltzmann-immersed boundary method to
simulate the fluid interaction with moving and slender flexible objects. J Comput
Phys 2014;261:145–61. http://dx.doi.org/10.1016/j.jcp.2013.12.052.
[71] Peng C, Ayala OM, Wang L-P. A comparative study of immersed boundary
method and interpolated bounce-back scheme for no-slip boundary treatment in the lattice Boltzmann method: Part I, laminar flows. Comput Fluids
2019;192:104233. http://dx.doi.org/10.1016/j.compfluid.2019.06.032.
[72] Premnath KN. Lattice Boltzmann models for simulations of drop-drop collisions (Ph.D. Thesis), Purdue University, West Lafayette, IN; 2004, URL: https:
//docs.lib.purdue.edu/dissertations/AAI3178542/.
[73] Premnath KN, Abraham J. Simulations of binary drop collisions with a multiplerelaxation-time lattice-Boltzmann model. Phys Fluids 2005;17(12):122105. http:
//dx.doi.org/10.1063/1.2148987.
[74] McCracken ME, Abraham J. Multiple-relaxation-time lattice-Boltzmann model
for multiphase flow. Phys Rev E 2005;71:036701. http://dx.doi.org/10.1103/
PhysRevE.71.036701.
[75] Premnath KN, Abraham J. Three-dimensional multi-relaxation time (MRT)
lattice-Boltzmann models for multiphase flow. J Comput Phys 2007;224(2):539–
59. http://dx.doi.org/10.1016/j.jcp.2006.10.023.
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