# A Family of Switching Pursuit Strategies
for a Multi-Pursuer Single-Evader Game

Marco Casini, Andrea Garulli M. Casini and A. Garulli are with the
Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, Università
di Siena, via Roma 56, 53100 Siena, Italy. E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

A new family of pursuit strategies is introduced for a multi-pursuer single-
evader game. By exploiting the optimal solution of the game involving two
pursuers, conditions are derived under which the multi-pursuer game becomes
equivalent to the two-pursuer one. This opens the possibility of designing a
number of pursuit strategies in which the pursuers first try to enforce the
satisfaction of the aforementioned condition and then switch to a two-pursuer
game as soon as it is verified. The contribution is useful in two ways. First,
new winning pursuit strategies can be devised starting from simple plans, such
as pure pursuit. Moreover, the performance of existing pursuit strategies,
like those based on Voronoi partitions, can be significantly improved by
resorting to the corresponding switching version.

Keywords: Pursuit-evasion games; multi-agent systems; autonomous agents.

##  1 Introduction

Pursuit-evasion games provide a general framework for a number of problems in
the broad research area of multi-agent systems, with applications in mobile
robotics [1, 2], optimal robot control [3], space operations [4, 5], predator-
prey interactions in living species [6], and many other fields. In recent
years, there has been a growing interest towards games involving either
multiple pursuers [7, 8, 9], multiple evaders [10], or both [11], with a wide
variety of alternative problem formulations and solutions.

The standard way to tackle pursuit-evasion problems is to formulate them as
differential games [12, 13]. However, finding an optimal solution within this
setting is often challenging, especially when multiple agents are involved. A
notable exception is that of the game involving two pursuers and one evader,
moving in simple motion within a two-dimensional environment, for which a
minimum-time solution is available [14, 15, 16]. Conversely, for the three-
pursuer single-evader game, only suboptimal solutions have been proposed [17].
Moreover, it has been observed that optimal strategies must involve switching
among different linear sub-paths [18, 19]. For games involving more than three
pursuers, a minimum-time open-loop solution is proposed in [20], showing that
all agents have to travel along linear paths. However, a closed-loop solution
of the differential game has not been found yet.

Motivated by the above observations, a new family of pursuit strategies is
proposed in this paper, for a multi-pursuer single-evader game in which planar
agents move in simple motion with equal maximum speed. The main idea is to
derive and exploit conditions under which the multi-pursuer game becomes
equivalent to a game involving only two pursuers, and then devise strategies
for the pursuer team in order to enforce such conditions and switch to a two-
pursuer game as soon as they are verified. The resulting switching pursuit
strategies, albeit suboptimal, always reduce the capture time with respect to
the corresponding strategies without switching. Therefore, they are able to
improve the performance of existing pursuit techniques and allow one to design
new successful multi-pursuer strategies.

The contribution of the paper is threefold: i) all the agents configurations
in which capture of the evader can be achieved are characterized, thus
providing a complete solution of the game of kind; ii) conditions are derived
under which the minimum-time pursuit strategy boils down to that of a two-
pursuer single-evader game; iii) by exploiting the aforementioned conditions,
new families of switching pursuit strategies guaranteeing capture are devised.
In particular, very simple pursuit strategies (like, e.g., pure pursuit),
which in general may end up with the evader escape, are transformed into
winning pursuit strategies by switching to a suitable two-pursuer game.
Moreover, it is shown via numerical simulations that the switching mechanism
can remarkably reduce the capture time of techniques that are guaranteed to
achieve capture, like the one based on Voronoi partitions proposed in [7].

The paper is organized as follows. The considered multi-pursuer single-evader
game is formulated in Section 2, which provides also a review of the optimal
solution of the two-pursuer game. The solution of the game of kind and the
conditions for switching to the two-pursuer game are derived in Section 3. New
switching pursuit strategies are proposed in Section 4 and analyzed via
numerical simulations in Section 5. Finally, some concluding remarks are given
in Section 6.

###  1.1 Notation and definitions

Given a vector $V$, its transpose is denoted by $V^{\prime}$, while $\|V\|$ is
its Euclidean norm. For $V,W\in\mathbb{R}^{2}$, we denote by $\overline{VW}$
the segment with $V$ and $W$ as endpoints.
$\mathcal{C}(P,r)=\\{Q\in\mathbb{R}^{2}\colon\|Q-P\|\leq r\\}$ denotes a
circle centered in $P\in\mathbb{R}^{2}$ with radius $r$. For a closed set
$\mathcal{A}$, $\partial\mathcal{A}$ is the boundary of $\mathcal{A}$, while
$\mathbb{H}\left\\{\mathcal{A}\right\\}$ denotes the convex hull of
$\mathcal{A}$. The interior of $\mathcal{A}$ is represented by
$\mathtt{int}\left\\{\mathcal{A}\right\\}=\mathcal{A}\backslash\partial\mathcal{A}$.

##  2 Problem formulation

In this paper, we consider a _multi-pursuer one-evader game_ (hereafter
referred to as MP1EG) where $p$ pursuers aim at capturing one evader. The
positions of the pursuers and the evader at time $t$ are denoted by
$P_{k}(t)\in\mathbb{R}^{2},k=1,\ldots p$, and $E(t)\in\mathbb{R}^{2}$,
respectively. The players move in the plane in simple motion, that is

$\left\\{\begin{array}[]{l}\dot{E}(t)=v_{E}(t)\,,\\\
\dot{P}_{k}(t)=v_{P_{k}}(t)\,,~{}k=1,\ldots,p\end{array}\right.$

where $v_{E}(t),v_{P_{k}}(t)\in\mathbb{R}^{2}$ are the velocity vectors of the
players. It is assumed that all agents have the same maximum velocity, which
w.l.o.g. is set to $1$, that is $\|v_{E}(t)\|\leq 1$, $\|v_{P_{k}}(t)\|\leq
1,~{}k=1,\ldots,p$, $\forall t\geq 0$. At each time $t$, $v_{E}(t)$ and
$v_{P_{k}}(t)$ are computed as functions of the state of the game
$\xi(t)=[E^{\prime}(t)~{}P_{1}^{\prime}(t)~{}\dots~{}P_{p}^{\prime}(t)]^{\prime}$.
Notice that the agents do not have information on the velocities of the other
players. The _strategies_ of the agents are functions mapping the current
state into their chosen velocity vector, $v_{E}=\phi(\xi)$,
$v_{P_{k}}=\psi_{k}(\xi)$ (dependence on time is omitted when it is clear from
the context).

The aim of the pursuers is to capture the evader, while the evader tries to
avoid capture or to protract the game as long as possible. Capture occurs when
the distance between at least one pursuer and the evader is equal to the
radius of capture $r>0$. Hereafter, only initial conditions such that

$\|P_{k}(0)-E(0)\|>r,\quad k=1,\ldots,p$ (1)

will be considered. A _winning pursuit strategy_ is a strategy adopted by the
pursuers which guarantees that capture occurs in finite time whatever is the
strategy adopted by the evader. The capture time $T$ is such that

$\|P_{k}(T)-E(T)\|=r,\text{~{}for some~{}}k\in\\{1,\ldots,p\\}.$ (2)

A _minimum time pursuit strategy_ is a strategy
$\psi=[\psi_{1},\dots,\psi_{p}]$ which is a solution of the min-max problem

$\min_{\psi}\max_{\phi}~{}T.$ (3)

For brevity, we will refer to functions $\psi$ and $\phi$ solving problem (3)
as _optimal strategies_. To the best of our knowledge, (3) is still an open
problem for $p\geq 3$. Conversely, for $p=2$ the problem has been fully solved
[14, 15, 16]. In the following, the solution of the _two-pursuer one-evader
game_ (2P1EG) given in [16] is recalled. It will play a key role in the
definition of the class of winning strategies for the MP1EG proposed in this
work.

###  2.1 Minimum-time solution of the two-pursuer one-evader game

Consider two pursuers $P_{1},P_{2}$ trying to capture one evader. Let us
define the region

$\mathcal{D}_{12}=\mathtt{int}\left\\{\mathbb{H}\left\\{\mathcal{C}(P_{1},r)\cup\mathcal{C}(P_{2},r)\right\\}\right\\}.$
(4)

First, a solution to the game of kind is provided [16].

######  Proposition 1.

There exists a winning pursuit strategy for the 2P1EG if and only if
$E\in\mathcal{D}_{12}$.

Hence, set $\mathcal{D}_{12}$ is the _2-pursuer capture region_ associated to
$P_{1}$ and $P_{2}$; if the evader belongs to this region it will be captured
by the pursuers in finite time. An example is depicted in Fig. 1. To simplify
the treatment, w.l.o.g. we will refer to the reference frame reported in Fig.
1, where the origin is the midpoint of the pursuers, and they lie on the
$x$-axis, i.e., $P_{1}=[-d,0]^{\prime}$, $P_{2}=[d,0]^{\prime}$,
$E=[x,y]^{\prime}$. Let
$E\in\mathcal{D}_{12}\backslash(\mathcal{C}(P_{1},r)\cup\mathcal{C}(P_{2},r))$
and define

$T_{12}=\displaystyle\frac{\kappa
r+|y|\sqrt{\kappa^{2}-4x^{2}(r^{2}-y^{2})}}{2(r^{2}-y^{2})}$ (5)

where $\kappa=d^{2}-x^{2}+y^{2}-r^{2}$. Now, let us define the point $H_{12}$
satisfying

$\|E-H_{12}\|=\|P_{1}-H_{12}\|-r=\|P_{2}-H_{12}\|-r=T_{12}.$ (6)

Within the considered reference frame, one has $H_{12}=[0,\,h_{12}]^{\prime}$
where

$\displaystyle
h_{12}=\begin{cases}y+\text{sign}(y)\sqrt{T_{12}^{2}-x^{2}}&\text{~{}~{}~{}if
}y\neq 0\\\\[5.69054pt] \sqrt{T_{12}^{2}-x^{2}}&\text{~{}~{}~{}if
}y=0.\end{cases}$

The following result provides a solution to the minimum-time 2P1EG [16].

######  Proposition 2.

Let $p=2$. The strategies solving problem (3) require each player to go along
a straight line to the point $H_{12}$ defined by (6). Then, the evader will be
captured in $H_{12}$, at time $T_{12}$ given by (5).

Hereafter, the optimal strategies of the pursuers and the evader in the 2P1EG
will be denoted by $\psi_{k}^{12}$ and $\phi^{12}$, respectively.

Figure 1: The 2-pursuer capture region for the 2P1EG.

##  3 Multi-pursuer games: game of kind and switching conditions

In this section, the solution of the game of kind for the MP1EG is presented.
Then, a condition is derived under which the MP1EG boils down to a 2P1EG.

For a pair of pursuers $P_{i},P_{j}$, let $\mathcal{D}_{ij}$ be the
corresponding 2-pursuer capture region, given by (4). If
$E\in\mathcal{D}_{ij}$, let us denote by $\psi^{ij}$ the minimum-time pursuit
strategy for the 2P1EG played by $P_{i}$ and $P_{j}$, and by $\phi^{ij}$ the
corresponding optimal evader strategy. Moreover, we denote by $T_{ij}$ the
time needed by $P_{i}$ and $P_{j}$ to capture $E$ when the agents $P_{i}$,
$P_{j}$ and $E$ play the optimal 2P1EG strategies. The capture time $T_{ij}$
and capture point $H_{ij}$ are computed according to (5)-(6), for each pair
$i,j$.

Let us define the convex hull of the pursuer locations

$\mathcal{P}=\mathbb{H}\left\\{P_{1},P_{2},\ldots,P_{p}\right\\}$ (7)

and introduce the _multi-pursuer capture region_ , defined as

$\mathcal{M}=\mathtt{int}\left\\{\mathbb{H}\left\\{\bigcup_{i=1}^{p}\mathcal{C}(P_{i},r)\right\\}\right\\}.$
(8)

The following result provides the solution of the game of kind for the MP1EG.

######  Theorem 1.

If $E\in\mathcal{M}$ in (8), then there exists a winning pursuit strategy. On
the contrary, if $E\notin\mathcal{M}$, then the evader can avoid capture
indefinitely.

###### Proof.

Let us prove that capture occurs if $E\in\mathcal{M}$. It is known that if the
evader lies inside the pursuer convex hull $\mathcal{P}$, there exist pursuit
strategies which guarantee capture in finite time, as for instance the
Voronoi-based strategy proposed in [7]. So, it remains to prove that the
evader can be captured also when $E\in\mathcal{M}\backslash\mathcal{P}$. It is
easy to note that if $E\in\mathcal{M}\backslash\mathcal{P}$ there exist at
least two pursuers, say $P_{i},P_{j}$, such that $E\in\mathcal{D}_{ij}$. Then,
the evader will be captured if $P_{i},P_{j}$ play $\psi^{ij}$.
Let us now consider the case $E\notin\mathcal{M}$. Since $\mathcal{M}$ is
convex there exists a hyperplane which separates $E$ from $\mathcal{M}$. Thus,
the evader can escape by going straight orthogonally to such hyperplane, in
the opposite direction with respect to $\mathcal{M}$ (an example is shown in
Fig. 2). In fact, if $P_{i},P_{j}$ is the pair of pursuers which is closest to
the separating hyperplane, the evader will be able to escape from them
according to Theorem 1. The other pursuers will neither be able to capture the
evader, being further from it than $P_{i},P_{j}$. ∎

Figure 2: Example of multi-pursuer capture region $\mathcal{M}$ for $p=3$.
Since $E\notin\mathcal{M}$, the evader can avoid capture by moving along the
direction $v$.

The next theorem reports conditions under which the MP1EG reduces to the
2P1EG, in the sense that only two pursuers can provide capture in minimum
time, irrespectively of the strategy played by the other pursuers.

######  Theorem 2.

Let $E\in\mathcal{D}_{ij}$ for some $i,j\in\\{1,2,\ldots,p\\}$. If

$\|H_{ij}-P_{k}\|\geq T_{ij}+r,\quad k=1,\ldots,p$ (9)

then the optimal MP1EG strategies for $P_{i},P_{j},E$ are $\psi^{ij}$ and
$\phi^{ij}$, respectively, guaranteeing capture at time $T_{ij}$. The
strategies adopted by the other pursuers are irrelevant to the game duration.

###### Proof.

If $P_{i},P_{j}$ and $E$ play their optimal strategies for the 2P1EG, capture
will occur at the location $H_{ij}$ at time $T_{ij}$. So,
$\|H_{ij}-P_{i}\|=\|H_{ij}-P_{j}\|=T_{ij}+r$. Let $P_{k}$ denote a generic
pursuer not involved in the considered 2P1EG, i.e., $k\neq i,j$. We prove
that, if condition (9) holds, any strategy adopted by $P_{k}$ cannot reduce
the capture time below $T_{ij}$. This is equivalent to show that $P_{k}$
cannot capture the evader during its path to $H_{ij}$. W.l.o.g. assume $t=0$.
Let $v_{E}=\frac{H_{ij}-E(0)}{\|H_{ij}-E(0)\|}$ denote the velocity vector of
the evader. Since $E$ travels towards $H_{ij}$ following a straight path, one
has $E(\tau)=E(0)+\tau v_{E}$, $0\leq\tau\leq T_{ij}$. Since
$\|H_{ij}-E(0)\|=T_{ij}$, then
$\mathcal{C}(E(0),r)\subset\mathcal{C}(H_{ij},T_{ij}+r)$ and
$\mathcal{C}(E(\tau),r)\subset\mathcal{C}(H_{ij},T_{ij}+r-\tau),~{}0\leq\tau\leq
T_{ij}$ (see Fig. 3).

Consider the case $\|H_{ij}-P_{k}(0)\|>T_{ij}+r$, where strict inequality is
assumed in (9). It follows that $P_{k}(0)\notin\mathcal{C}(H_{ij},T_{ij}+r)$
and then for any possible trajectory of $P_{k}$ one has
$P_{k}(\tau)\notin\mathcal{C}(H_{ij},T_{ij}+r-\tau),~{}0\leq\tau\leq T_{ij}$.
Hence, $P_{k}(\tau)\notin\mathcal{C}(E(\tau),r)$, $0\leq\tau\leq T_{ij}$ and
then $P_{k}$ cannot capture $E$ before $T_{ij}$.

It remains to consider the case when condition (9) holds with equality, i.e.,
$\|H_{ij}-P_{k}\|=T_{ij}+r$. Let $Q=E(0)-rv_{E}$. To avoid capture at time
$0$, $P_{k}(0)\neq Q$. By going straight towards $H_{ij}$, the $k$-th pursuer
will always lie on the border of $\mathcal{C}(H_{ij},T_{ij}+r-\tau)$, thus
leading again to $P_{k}(\tau)\notin\mathcal{C}(E(\tau),r)$ for
$0\leq\tau<T_{ij}$. When $\tau=T_{ij}$ one will have
$\|P_{k}(T_{ij})-H_{ij}\|=r$, meaning that $P_{i}$, $P_{j}$ and $P_{k}$ will
capture the evader simultaneously at time $T_{ij}$.

Figure 3: Sketch of the proof of Theorem 2.

∎

######  Corollary 1.

If there exist two pairs of pursuers $(P_{i},P_{j})$ and $(P_{h},P_{l})$
satisfying the conditions of Theorem 2, then $T_{ij}=T_{hl}$.

###### Proof.

By contradiction, assume $T_{hl}<T_{ij}$. By Theorem 2 applied to
$P_{i},P_{j}$, if the evader plays $\phi^{ij}$ it cannot be captured before
reaching $H_{ij}$ at time $T_{ij}$. However, again from Theorem 2, if
$P_{h},P_{l}$ play $\psi^{hl}$ they will capture the evader at most at time
$T_{hl}$, leading to a contradiction. The case $T_{hl}>T_{ij}$ is specular. ∎

If $P_{i}$, $P_{j}$ and $E$ play their optimal 2P1EG strategies, one has
$\|H_{ij}-P_{i}\|=\|H_{ij}-P_{j}\|=T_{ij}+r$ and so (9) can be rewritten as

$\|H_{ij}-P_{k}\|\geq\|H_{ij}-P_{i}\|,\quad k=1,\ldots,p.$ (10)

Assume that at time $t$, condition (9) or (10) is satisfied by a pair of
pursuers $P_{i},P_{j}$. Theorem 2 and Corollary 1 state that the strategies of
these two pursuers must switch to $\psi^{ij}$ in order to be optimal from time
$t$ onwards, irrespectively of what the other pursuers will do. Similarly, the
evader must switch to strategy $\phi^{ij}$ in order to maximize its survival
time. This fact will be exploited in the next section to devise a new family
of winning pursuit strategies for the MP1EG.

##  4 Switching pursuit strategies

In this section, the concept of _switching pursuit strategy_ is introduced.
The aim of such strategies is to guarantee that condition (9) holds at some
finite time, and hence capture is assured by switching to the appropriate
2P1EG strategies.

######  Definition 1.

A pursuit strategy is a _potential switching strategy_ if there exists a pair
of pursuers $P_{i},P_{j}$ for which condition (9) holds at some finite time,
for any possible strategy of the evader. A potential switching strategy is
referred to as _switching strategy_ if $P_{i},P_{j}$ play the optimal 2P1EG
strategy as soon as condition (9) holds.

By the previous definition, a switching strategy leads to the evader capture
in finite time, and so it is a winning strategy. In fact, from the switching
time onwards the pursuers will play the optimal 2P1EG strategy, guaranteeing
the evader capture.

It is easy to see that for every winning strategy, capture is achieved
simultaneously by at least two pursuers. The following result holds.

######  Theorem 3.

Let the pursuers adopt a winning strategy and assume that capture occurs at
time $\bar{t}$. Then, at time $\bar{t}$ condition (9) is satisfied.

###### Proof.

Let $P_{i}$ and $P_{j}$ be two pursuers achieving capture at time $\bar{t}$.
According to the notation adopted in Section 2.1, at time $\bar{t}$ one
clearly has $H_{ij}=E$, $T_{ij}=0$ and $\|H_{ij}-P_{i}\|=\|H_{ij}-P_{j}\|=r$.
Any other pursuer $P_{k}$, $k\neq i,j$, will be at distance greater or equal
than $r$ to the evader (otherwise, $P_{k}$ has captured the evader before time
$\bar{t}$). Hence,

$\begin{array}[]{l}\|H_{ij}-P_{k}\|=\|E-P_{k}\|\geq r=\|H_{ij}-P_{i}\|=\\\
=\|H_{ij}-P_{j}\|,\quad k=1,\ldots,p,~{}k\neq i,j,\end{array}$

which corresponds to condition (9). ∎

Theorem 3 states that for a winning strategy condition (9) must be satisfied
_at least_ when capture occurs. However, this turns out to be a very special
case, as in most games such a condition is satisfied long before capture
occurs (it may be even satisfied when the game starts). Since Theorem 2
guarantees that when condition (9) holds it is always advantageous for the
pursuers to switch to the 2P1EG strategy, it can be concluded that every
winning strategy is a potentially switching strategy and its capture time
cannot be increased by switching.

In general, it is not easy to design pursuit strategies that guarantee
satisfaction of condition (9). Therefore, it is useful to derive sufficient
conditions under which (9) holds, that the pursuers can enforce whatever is
the strategy played by the evader. Hereafter, two such conditions are
presented.

Let us define the union of all the 2-pursuer capture regions as

$\widehat{\mathcal{D}}=\bigcup_{i,j\in\\{1,\ldots,p\\}}\mathcal{D}_{ij}.$

An example of set $\widehat{\mathcal{D}}$ is shown in Fig. 4-(a), while the
corresponding set
$\widehat{\mathcal{D}}\backslash\mathtt{int}\left\\{\mathcal{P}\right\\}$ is
depicted in Fig. 4-(b). The next theorem provides a sufficient condition for
switching to 2P1EG.

(a) (b)

Figure 4: (a) Set $\widehat{D}$ denoting the union of all the 2-pursuer
capture regions; (b) set
$\widehat{\mathcal{D}}\backslash\mathtt{int}\left\\{\mathcal{P}\right\\}$.

######  Theorem 4.

If
$E\in\widehat{\mathcal{D}}\backslash\mathtt{int}\left\\{\mathcal{P}\right\\}$,
then there exists a pair of pursuers $P_{i},P_{j}$ for which condition (9)
holds.

###### Proof.

See Appendix. ∎

######  Corollary 2.

If $E\in\partial\mathcal{P}$, then there exists a pair of pursuers
$P_{i},P_{j}$ for which condition (9) holds.

###### Proof.

Since
$\partial\mathcal{P}\in\widehat{\mathcal{D}}\backslash\mathtt{int}\left\\{\mathcal{P}\right\\}$,
the proof follows directly by Theorem 4. ∎

Theorem 4 suggests that if the pursuers play a strategy which reduces the size
(in some sense) of $\mathcal{P}$ over time, then satisfaction of (9) is
eventually assured. For example, the following result holds.

######  Theorem 5.

Let us consider a pursuit strategy such that

$\max_{i,j=1,\ldots,p}\|P_{i}(\bar{t})-P_{j}(\bar{t})\|\leq 2r$ (11)

at a certain time $\bar{t}$. Then, there exist a time $\hat{t}<\bar{t}$ and a
pair of pursuers $P_{i},P_{j}$ for which condition (9) holds at $\hat{t}$.
Hence, the considered pursuit strategy is a potential switching strategy.

###### Proof.

If
$E(0)\in\widehat{\mathcal{D}}(0)\backslash\mathtt{int}\left\\{\mathcal{P}(0)\right\\}$,
by Theorem 4 condition (9) holds. So, let us analyze the case
$E(0)\in\mathtt{int}\left\\{\mathcal{P}(0)\right\\}$. It is easy to note that
(11) implies
$\mathcal{P}(\bar{t})\subset\bigcup_{i=1}^{p}\mathcal{C}(P_{i}(\bar{t}),r)$.
This means that $E(\bar{t})\notin\mathcal{P}(\bar{t})$ otherwise capture has
already occurred. Since $E(0)\in\mathtt{int}\left\\{\mathcal{P}(0)\right\\}$,
by continuity of the agents’ trajectories $\exists\hat{t}<\bar{t}$ such that
$E(\hat{t})\in\partial\mathcal{P}(\hat{t})$ and then, by Corollary 2, (9)
holds at $\hat{t}$. ∎

Results in Theorems 4 and 5 can be exploited to design families of switching
strategies, starting from very simple pursuit strategies that would not be
successful without switching. An example is that of a Fixed-Point Pursuit
Strategy (FPPS), in which each pursuer $P_{k}$ goes straight towards a fixed
point $M\in\mathbb{R}^{2}$, with speed $\|v_{P_{k}}(t)\|\geq\varepsilon$, for
some $0<\varepsilon\leq 1$, and stops once it reaches $M$.

######  Theorem 6.

Any FPPS is a potential switching strategy.

###### Proof.

Let $d=\max_{k=1,\ldots,p}\|P_{k}(0)-M\|$. Then, at time
$\tau=\frac{d-r}{\varepsilon}$ one has $\|P_{k}(\tau)-M\|\leq r$,
$k=1,\ldots,p$, and then
$\|P_{i}(\tau)-P_{j}(\tau)\|\leq\|P_{i}(\tau)-M\|+\|M-P_{j}(\tau)\|\leq 2r$,
for any pair of pursuers $P_{i},P_{j}$. Thus, the result follows by Theorem 5.
∎

An example of FPPS is the _centroid-based_ strategy, where all pursuers move
with same velocity towards the center of the minimum-radius circle which
includes them.

Another simple pursuit strategy is the so-called _pure pursuit_. Such a
strategy is defined as the one in which all pursuers always point towards the
evader at maximum velocity, i.e.

$v_{P_{k}}(t)=\frac{E(t)-P_{k}(t)}{\|E(t)-P_{k}(t)\|},\quad k=1,\ldots,p.$
(12)

It is well known that pure pursuit does not guarantee capture of the evader.
However, it becomes a winning strategy if switching to 2P1EG is adopted.

######  Theorem 7.

The pure pursuit strategy is a potentially switching strategy.

###### Proof.

Let $\delta_{k}(t)=\|E(t)-P_{k}(t)\|$. Then, by using (12),

$\begin{array}[]{l}\displaystyle\frac{1}{2}\displaystyle\frac{d\delta^{2}_{k}(t)}{dt}=(E(t)-P_{k}(t))^{\prime}(\dot{E}(t)-\dot{P}_{k}(t))=(E(t)-P_{k}(t))^{\prime}\dot{E}(t)-\delta_{k}(t).\end{array}$

Being $\|\dot{E}(t)\|=\|v_{E}(t)\|\leq 1$, one has
$\frac{d\delta^{2}_{k}(t)}{dt}\leq 0$ and in particular
$\frac{d\delta^{2}_{k}(t)}{dt}=0$ if and only if $v_{E}(t)=v_{P_{k}}(t)$ given
by (12). Hence, $\lim_{t\rightarrow+\infty}\delta_{k}(t)=\bar{\delta}_{k}\geq
0$ and $\lim_{t\rightarrow+\infty}\dot{\delta}_{k}(t)=0$, which in turns leads
to
$\lim_{t\rightarrow+\infty}v_{P_{k}}(t)=\lim_{t\rightarrow+\infty}v_{E}(t)$,
$\forall k=1,\dots,p$. But this means that all pursuers and the evader are
asymptotically aligned, with the evader lying outside the segment containing
the pursuers, i.e., $\lim_{t\rightarrow+\infty}E(t)\notin\mathcal{P}$. By
continuity of the agents’ trajectories, $E(t)$ must have crossed
$\partial\mathcal{P}$ at some time instant and hence the conclusion follows by
Corollary 2. ∎

##  5 Numerical simulations

In this section, some simulations are reported to show the improvement (in
terms of capture time) when a pursuit switching strategy is adopted w.r.t. a
non-switching one. Three pursuer strategies are considered: pure pursuit
strategy (PPS), centroid-based strategy (CS), and the strategy proposed in
[7], referred to as Voronoi-based strategy (VS), in which the pursuers aim at
minimizing the area of the evader Voronoi cell. All these strategies are
simulated both in their original form (non-switching) and in switching form,
i.e. by applying the optimal 2P1EG strategy when condition of Theorem 2 holds.
Switching strategies are denoted by the prefix “S-”, e.g., S-PPS denotes the
switching pure pursuit strategy.

In the following, simulated games will be graphically illustrated. In the
figures, a square denotes the initial condition of a player and a dot its
final position. Blue and red lines represent the trajectories of the pursuers
and the evader, respectively. Capture occurs when at least one pursuer reaches
the circle of radius $r$ centered at the evader, drawn in dashed black. If the
evader manages to avoid capture, the simulation is stopped as soon as it exits
from the capture region $\mathcal{M}$.

######  Example 1.

Three pursuers are initially located at the vertices of an equilateral
triangle centered at the origin with sides of length 20, that is
$P_{1}(0)=[-10,-10/\sqrt{3}]^{\prime}$, $P_{2}(0)=[10,-10/\sqrt{3}]^{\prime}$,
$P_{3}(0)=[0,20/\sqrt{3}]^{\prime}$. The capture radius is $r=1$ and the
maximum speed of the agents is fixed to 1. The evader moves downwards with
speed 1, i.e., $v_{E}(t)=[0,-1]^{\prime}$, independently of the position of
the pursuers.

Let the pursuers play PPS, and let the evader initial position be
$E(0)=[0,~{}0]^{\prime}$. In Fig. 5 (left), the trajectories of the players
are depicted until time $t=5.59$, when the evader goes outside the capture
region $\mathcal{M}$. So, the evader can indefinitely escape by going
downwards.

A different situation occurs when all the players switch to the 2P1EG strategy
as soon as condition (9) is satisfied. In fact, thanks to Theorem 7, capture
in finite time is assured if $E(0)\in\mathcal{M}(0)$. This game is reported in
Fig. 5 (right). At time $t=4.47$, condition (9) holds and capture occurs at
time $t=20.64$.

Figure 5: Example 1. Pursuers play PPS (left) and S-PPS (right).

######  Example 2.

Under the same setting of Example 1, the pursuers play CS, i.e., they point
towards their circumcenter, that is towards the origin. Without switching, the
evader can escape, see Fig. 6 (left). In fact, at $t=4.52$ it holds
$E(t)\notin\mathcal{M}(t)$, and the evader can avoid capture by going
downwards. Assuming that the pursuers play S-CS, capture occurs at time
$t=25.57$, as shown in Fig. 6 (right).

Figure 6: Example 2. Pursuers play CS (left) and S-CS (right).

######  Example 3.

Consider the same setting as in Examples 1 and 2, and let the pursuers play
VS. In [7], it has been proved that capture occurs in finite time
independently on the strategy adopted by the evader. The result of such a game
is reported in Fig. 7 (left), in which capture occurs at time $t=16.98$. If
the pursuers play S-VS, when $E\simeq[0,-9]^{\prime}$ condition (9) occurs and
the agents start playing the corresponding optimal 2P1EG strategy. The evader
changes its direction and starts moving upwards. Capture occurs simultaneously
by the three pursuer at $t=14.35$. Hence, S-VS is able to achieve capture in a
time about $15\%$ less than VS.

Figure 7: Example 3, $E(0)=[0,0]^{\prime}$. Pursuers play VS (left) and S-VS
(right).

It is worthwhile to note that the improvement of S-VS w.r.t. VS can be much
larger, even for simple games involving 3 pursuers. For instance, under the
same setting, let us set the initial evader position to
$E(0)=[0,-5.5]^{\prime}$. The capture time required by VS turns out to be
$t=169.28$, while that needed by S-VS is $t=96.37$ (see Fig. 8). In this case,
switching to the 2P1EG strategy leads to a capture time reduction of about
$43\%$.

Figure 8: Example 3, $E(0)=[0,-5.5]^{\prime}$. Pursuers play VS (left) and
S-VS (right).

######  Example 4.

Let us now consider games involving more than three pursuers. The initial
position of the evader is $E(0)=[0,0]^{\prime}$, while the initial position of
the pursuers is randomly chosen on a circumference centered at the origin with
a radius $R_{p}$. The capture radius is fixed to $r=1$, and the maximum speed
of the players is set to $1$. We compare the capture time obtained when
pursuers play the Voronoi-based strategy (namely, $T_{{}_{VS}}$), with that
obtained when playing the switching version S-VS ($T_{{}_{S\text{-}VS}}$). In
particular, we focus on the ratio between these two quantities, i.e.,

$\rho=\frac{T_{{}_{VS}}}{T_{{}_{S\text{-}VS}}}.$

In all games, the evader strategy is to point towards the farthest vertex of
its Voronoi cell at each time $t$. A total of $N=100$ games are played for any
combination of the number of pursuers $p\in\\{5,8\\}$, and of the radius of
their initial condition $R_{p}\in\\{10,25,50,75,100\\}$. The average and
maximum values of $\rho$ are reported in Table 1. Notice that, in general, the
average capture time when pursuers use the switching strategy is much less
than that obtained by adopting the original Voronoi-based strategy, with a
maximum ratio $\rho$ which is greater than $1.6$. Thus, it is apparent that
the capture time can be significantly reduced if pursuers switch to 2P1EG once
condition (9) occurs.

Table 1: Example 4. Mean and maximum value of $\rho$ for different game configurations (over $100$ games) | $p=5$ | $p=8$
---|---|---
| $\mathrm{mean}(\rho)$ | $\max(\rho)$ | $\mathrm{mean}(\rho)$ | $\max(\rho)$
$R_{p}=10$ | 1.180 | 1.405 | 1.074 | 1.320
$R_{p}=25$ | 1.262 | 1.376 | 1.145 | 1.341
$R_{p}=50$ | 1.317 | 1.488 | 1.193 | 1.346
$R_{p}=75$ | 1.346 | 1.546 | 1.219 | 1.579
$R_{p}=100$ | 1.365 | 1.587 | 1.236 | 1.620

##  6 Conclusions

A new family of pursuit strategies for a multi-pursuer single-evader game has
been introduced. The main property that has been exploited is the fact that
such a game must eventually boil down to a two-pursuer single-evader game, for
which a minimum-time solution exists. Hence, conditions for the switching
between the two game settings have been derived, allowing one to define new
winning pursuit strategies and to improve the performance of existing ones.
Future developments concern the investigation of pursuit strategies achieving
the switching condition in such a way that the overall capture time is
minimized. The extension of the proposed approach to the case of multi-pursuer
multi-evader games will also be addressed.

## Appendix

In order to prove Theorem 4, the following lemmas are introduced.

######  Lemma 1.

Let $P_{1},P_{2}$ and $E$ be such that
$E\in\mathcal{D}_{12}\backslash(\mathcal{C}(P_{1},r)\cup\mathcal{C}(P_{2},r))$,
and set $P_{3}=P_{2}+\alpha(P_{2}-P_{1})$, $\alpha>0$. Then, $T_{13}>T_{12}$.

###### Proof.

W.l.o.g., let us refer to the reference frame introduced in Section 2.1, see
Fig. 9. Hence, from
$E=[x,y]^{\prime}\in\mathcal{D}_{12}\backslash(\mathcal{C}(P_{1},r)\cup\mathcal{C}(P_{2},r))$,
one has $-d<x<d$ and $(x\pm d)^{2}+y^{2}-r^{2}>0$. Moreover, since
$E\in\mathcal{D}_{12}\backslash(\mathcal{C}(P_{1},r)\cup\mathcal{C}(P_{2},r))$,
one has also
$E\in\mathcal{D}_{13}\backslash(\mathcal{C}(P_{1},r)\cup\mathcal{C}(P_{3},r))$.
Therefore, the capture time $T_{13}$ of pursuers $P_{1}$ and $P_{3}$ can be
computed by using (5), in which $d$ and $x$ are replaced by $d+\alpha d$ and
$x-\alpha d$, respectively. One gets

$T_{13}=\frac{\kappa_{\alpha}r+|y|\sqrt{\kappa_{\alpha}^{2}-4(x-\alpha
d)^{2}(r^{2}-y^{2})}}{2(r^{2}-y^{2})}$ (13)

in which $\kappa_{\alpha}=\kappa+2\alpha d(d+x)$. In order to prove
$T_{13}>T_{12}$, it is sufficient to show that $T_{13}$ in (13) is a strictly
increasing function of $\alpha$, for $\alpha>0$. First, one has

$\frac{d\kappa_{\alpha}}{d\alpha}=2d(d+x)>0.$

Then, it remains to show that $\kappa_{\alpha}^{2}-4(x-\alpha
d)^{2}(r^{2}-y^{2})$ is strictly increasing in $\alpha$. Through some
straightforward manipulations, one obtains

$\begin{array}[]{l}\\!\\!\\!\displaystyle\frac{d}{d\alpha}\left\\{\kappa_{\alpha}^{2}-4(x-\alpha
d)^{2}(r^{2}-y^{2})\right\\}=4\left\\{d(d-x)+2d^{2}\alpha\right\\}\left\\{(x+d)^{2}+y^{2}-r^{2}\right\\}\end{array}$

which is positive for all $\alpha>0$. ∎

######  Lemma 2.

Consider $P_{1}$, $P_{2}$ and $E$ such that
$E\in\mathcal{D}_{12}\backslash(\mathcal{C}(P_{1},r)\cup\mathcal{C}(P_{2},r))$.
Let $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ be the lines containing the
segments $\overline{P_{1}E}$ and $\overline{P_{2}E}$, respectively. Then, let
$Q_{1}$ and $Q_{2}$ be the intersections of $\mathcal{L}_{1}$ and
$\mathcal{L}_{2}$ with the circumference
$\partial\mathcal{C}(H_{12}^{+},T_{12}+r)$, opposite to $P_{1}$ and $P_{2}$,
respectively. Define the region $\mathcal{W}_{12}$ as the portion of the
circle $\mathcal{C}(H_{12}^{+},T_{12}+r)$ bounded by the segments
$\overline{Q_{1}E}$, $\overline{Q_{2}E}$ and the arc of circumference
$\wideparen{Q_{2}Q_{1}}$. Then, if a pursuer $P_{3}\in\mathcal{W}_{12}$, then
either $E\in\mathcal{D}_{13}$ or $E\in\mathcal{D}_{23}$.

###### Proof.

Let us consider the region $\mathcal{W}_{12}$ depicted in Fig. 10. Let
$S=\partial\mathcal{C}(E,r)\cap\partial\mathcal{C}(H_{12}^{+},T_{12}+r)$ be
the point of tangency between the two circumferences. Notice that the segment
$\overline{ES}$ divides $\mathcal{W}_{12}$ in two disjoint regions. Thus,
either $\overline{P_{1}P_{3}}$ or $\overline{P_{2}P_{3}}$ will intersect
$\overline{ES}$. Let us assume that
$\overline{P_{2}P_{3}}\cap\overline{ES}\neq\emptyset$. Then, the minimum
distance between $E$ and $\overline{P_{2}P_{3}}$ is less than $r$, and hence
$E\in\mathcal{D}_{23}$. A similar argument can be adopted if
$\overline{P_{1}P_{3}}\cap\overline{ES}\neq\emptyset$.

Figure 9: Sketch of the proof of Lemma 1. The gray region denotes
$\mathcal{D}_{12}\backslash(\mathcal{C}(P_{1},r)\cup\mathcal{C}(P_{2},r))$.
Figure 10: Sketch of the proof of Lemma 2.

∎

Proof of Theorem 4. Let $P_{i},P_{j}$ be the pair of pursuers such that
$T_{ij}\leq T_{kh},~{}k,h\in\\{1,\ldots,p\\}$. We show that for this pursuer
pair, condition (10), and hence (9), holds. Condition (10) states that
$\|P_{k}-H_{ij}\|\geq\|P_{i}-H_{ij}\|=T_{ij}+r$, $k=1,\ldots,p$. By
contradiction, assume that there exists a pursuer $P_{z}$ which satisfies
$\|P_{z}-H_{ij}\|<T_{ij}+r$, i.e.,
$P_{z}\in\mathtt{int}\left\\{\mathcal{C}(H_{ij},T_{ij}+r)\right\\}$. Let us
define $\mathcal{L}_{iE}$ and $\mathcal{L}_{jE}$ as the lines passing through
$P_{i},E$ and $P_{j},E$, respectively. Since $P_{z}\in\mathcal{P}$ and by
assumption $E\notin\mathtt{int}\left\\{\mathcal{P}\right\\}$, then $P_{z}$
must belong to the region $\mathcal{W}_{ij}$, shown in Fig. 11. By Lemma 2,
either $E\in\mathcal{D}_{iz}$ or $E\in\mathcal{D}_{jz}$. W.l.o.g., assume
$E\in\mathcal{D}_{jz}$. Let $\mathcal{L}_{jz}$ be the line crossing $P_{j}$
and $P_{z}$. It holds
$\mathcal{L}_{jz}\cap\mathcal{C}(H_{ij},T_{ij}+r)=\\{P_{j},P_{q}\\}$, where
$P_{q}$ denotes a fictitious pursuer. Then, by Lemma 1 one has
$T_{jz}<T_{jq}=T_{ij}$. Since by assumption $T_{ij}\leq T_{kh}$ for all
$k,h\in\\{1,\ldots,p\\}$, a contradiction occurs. $\blacksquare$

Figure 11: Sketch of the proof of Theorem 4. The gray area $\mathcal{W}_{ij}$
denotes the admissible region of $P_{z}$.

## References

  * [1] T. H. Chung, G. A. Hollinger, and V. Isler, “Search and pursuit-evasion in mobile robotics,” _Autonomous Robots_ , vol. 31, no. 4, pp. 299–316, 2011\.
  * [2] N. Noori, A. Beveridge, and V. Isler, “Pursuit-evasion: A toolkit to make applications more accessible,” _IEEE Robotics & Automation Magazine_, vol. 23, no. 4, pp. 138–149, 2016.
  * [3] V. Macias, I. Becerra, R. Murrieta-Cid, H. M. Becerra, and S. Hutchinson, “Image feedback based optimal control and the value of information in a differential game,” _Automatica_ , vol. 90, pp. 271–285, 2018.
  * [4] C. Venigalla and D. J. Scheeres, “Delta-v-based analysis of spacecraft pursuit–evasion games,” _Journal of Guidance, Control, and Dynamics_ , vol. 44, no. 11, pp. 1961–1971, 2021.
  * [5] M. Shi, D. Ye, Z. Sun, W. Zhong, and H. Deng, “Spacecraft orbital pursuit–evasion games with j2 perturbations and direction-constrained thrust,” _Acta Astronautica_ , vol. 202, pp. 139–150, 2023.
  * [6] R. Lin and M. Egerstedt, “Predator-prey interactions through heterogeneous coverage control using reaction-diffusion processes,” in _2023 62nd IEEE Conference on Decision and Control (CDC)_. IEEE, 2023, pp. 5431–5436.
  * [7] Z. Zhou, W. Zhang, J. Ding, H. Huang, D. M. Stipanović, and C. J. Tomlin, “Cooperative pursuit with Voronoi partitions,” _Automatica_ , vol. 72, pp. 64–72, 2016.
  * [8] J. Chen, W. Zha, Z. Peng, and D. Gu, “Multi-player pursuit-evasion games with one superior evader,” _Automatica_ , vol. 71, pp. 24–32, 2016.
  * [9] M. Kothari, J. G. Manathara, and I. Postlethwaite, “Cooperative multiple pursuers against a single evader,” _Journal of Intelligent & Robotic Systems_, vol. 86, pp. 551–567, 2017.
  * [10] W. L. Scott and N. E. Leonard, “Optimal evasive strategies for multiple interacting agents with motion constraints,” _Automatica_ , vol. 94, pp. 26–34, 2018.
  * [11] V. G. Lopez, F. L. Lewis, Y. Wan, E. N. Sanchez, and L. Fan, “Solutions for multiagent pursuit-evasion games on communication graphs: Finite-time capture and asymptotic behaviors,” _IEEE Transactions on Automatic Control_ , vol. 65, no. 5, pp. 1911–1923, 2019.
  * [12] R. Isaacs, _Differential games_. Wiley, 1965.
  * [13] T. Başar and G. J. Olsder, “Pursuit-evasion games,” in _Dynamic Noncooperative Game Theory_ , ser. Mathematics in Science and Engineering. Elsevier, 1982, vol. 160, pp. 344–398.
  * [14] E. Garcia, Z. E. Fuchs, D. Milutinovic, D. W. Casbeer, and M. Pachter, “A geometric approach for the cooperative two-pursuer one-evader differential game,” _IFAC-PapersOnLine_ , vol. 50, no. 1, pp. 15 209–15 214, 2017.
  * [15] M. Pachter, A. Von Moll, E. Garcia, D. W. Casbeer, and D. Milutinović, “Two-on-one pursuit,” _Journal of Guidance, Control, and Dynamics_ , vol. 42, no. 7, pp. 1638–1644, 2019.
  * [16] M. Casini and A. Garulli, “A two-pursuer one-evader game with equal speed and finite capture radius,” _Journal of Intelligent & Robotic Systems_, vol. 106, p. 77, 2022.
  * [17] B. Pshenichnyi, “Simple pursuit by several objects,” _Cybernetics and Systems Analysis_ , vol. 12, no. 3, pp. 484–485, 1976.
  * [18] G. Ibragimov, S. Yunus, and I. A. Alias, “Effective pursuit strategies in simple motion differential game of three pursuers and one evader,” _International Journal of Pure and Applied Mathematics_ , vol. 111, no. 3, pp. 409–417, 2016.
  * [19] M. Casini and A. Garulli, “On the advantage of centralized strategies in the three-pursuer single-evader game,” _Systems & Control Letters_, vol. 160, p. 105122, 2022.
  * [20] A. Von Moll, D. Casbeer, E. Garcia, D. Milutinović, and M. Pachter, “The multi-pursuer single-evader game,” _Journal of Intelligent & Robotic Systems_, vol. 96, pp. 193–207, 2019.


# Significant influence of fluid viscoelasticity on flow dynamics past an
oscillating cylinder

Faheem Hamid1  C. Sasmal1<EMAIL_ADDRESS>1Department of Chemical
Engineering, Indian Institute of Technology Ropar, Rupnagar, Punjab,
India-140001.

###### Abstract

This study presents a numerical investigation of how fluid viscoelasticity
influences the flow dynamics past a transversely forced oscillating cylinder
in the laminar vortex shedding regime at a fixed Reynolds number of $Re=100$.
In particular, we examine how fluid viscoelasticity influences the boundary
between the lock-in and no lock-in zones and the associated wake topology
compared to that seen in a simple Newtonian fluid. All in all, we find that
the fluid viscoelasticity facilitates the synchronization of the vortex street
with the cylinder motion at lower oscillation frequencies than that required
for a Newtonian fluid. Consequently, the boundary of the lock-in region for a
viscoelastic fluid differs from the Newtonian one and broadens in the non-
dimensional cylinder oscillation amplitude and frequency plane. Furthermore,
we propose that excess strain generated due to the stretching of polymer
molecules in a viscoelastic fluid results in a marked difference in the wake
structure from that seen in a Newtonian fluid. For a Newtonian fluid, only
’2S’ (two single vortices) and ’P+S’ (a pair and a single vortex) vortex
shedding modes are detected in the primary synchronization region. However, a
’2P’(two pairs of vortices) vortex mode is detected for a viscoelastic fluid
in this region. We also employ the data-driven dynamic mode decomposition
(DMD) reduced order modeling technique to extract and analyze underlying
coherent flow structures and their associated frequencies to understand the
differences in the flow dynamics of Newtonian and viscoelastic fluids past an
oscillating cylinder.

##  1 Introduction

The flow past a fixed cylinder is a widely studied problem in fluid mechanics,
both from the pragmatic and fundamental points of view. This benchmark problem
provides significant information on several aspects of flow past a bluff body,
such as boundary layer separation and vortex shedding, hydrodynamic drag and
lift forces, transition among several flow states, etc. The flow physics
becomes more complex and interesting when the cylinder oscillates either in
the stream-wise or span-wise direction of the free stream (Griffin & Ramberg,
1974). Numerous studies have been carried out in the past on the flow past a
forced oscillating cylinder due to its immense engineering applications in
structural, offshore, and thermal power engineering applications. In
particular, the flow past a sinusoidally oscillating cylinder has received
considerable interest because of the difference in the wake formation and
vortex shedding phenomena from that of a stationary cylinder (Anagnostopoulos,
2000b). This dissimilarity manifests in terms of the synchronization or the
lock-in condition and an increased vortex strength, which eventually impact
the lift and drag forces acting on the cylinder (Kumar et al., 2013; Bishop &
Hassan, 1964b). In the lock-in condition, the vortex shedding frequency
diverges from the natural vortex shedding frequency corresponding to a
stationary cylinder (known as the Strouhal frequency) and synchronizes with
the imposed oscillating frequency of the cylinder (Bishop & Hassan, 1964a).
Numerous experimental (Bishop & Hassan, 1964a; Koopmann, 1967) and numerical
(Blackburn & Henderson, 1999; Karniadakis & Triantafyllou, 1989;
Anagnostopoulos, 2000a) investigations have been performed to understand and
characterize the lock-in phenomenon for a wide range of Reynolds numbers
(Jones Jr, 1968), cylinder oscillation amplitude and frequency (Koopmann,
1967). All these stated works aimed to determine the boundary between the
lock-in and no lock-in conditions on the cylinder oscillation amplitude and
frequency space. Also, sufficient information is now available on various flow
aspects such as different wake structures (e.g., ’2S’-two single vortices,
’2P’-two pairs of vortices, and ’P+S’-a pair and a single vortices shed in one
oscillation cycle) (Williamson & Roshko, 1988; Leontini et al., 2006),
mechanisms of vorticity production (Blackburn & Henderson, 1999), and
hydrodynamic forces acting on the cylinder.

However, to date, almost all of these studies have dealt with simple Newtonian
fluids like water. On the other hand, most of the fluids encountered from our
daily lives to several industrial settings exhibit various non-Newtonian
behaviors, such as shear-thinning, shear-thickening, viscoplasticity,
viscoelasticity, etc (Chhabra & Richardson, 2011). Despite this, almost no
study is available on how the complex rheological behavior of a fluid could
influence the flow dynamics past a forced oscillating cylinder, particularly
the lock-in and no lock-in boundary and the associated vortex dynamics and
hydrodynamic forces. Only very recently, Alam et al. (Alam et al., 2021)
investigated the effect of the shear-thinning rheological behaviour and found
a significant difference in the vortex structures depending upon the values of
the cylinder oscillation amplitude and frequency. Viscoelasticity is also
another non-Newtonian behaviour that is often seen in many complex fluids
(Bird et al., 1987). Adding a minute amount of solid polymers, even in parts
per million (ppm) quantity, into a Newtonian solvent like water dramatically
changes the flow behaviour of the resulting polymer solution compared to that
seen in water alone. This is due to the presence of non-linear elastic
stresses in these fluids, originating from the stretching and relaxation of
polymer molecules in a deformed flow field. To the best of our knowledge, how
this fluid viscoelasticity would tend to influence the flow dynamics of a
periodically forced oscillating cylinder is not investigated yet. The present
study aims to fill that knowledge gap in the literature. In particular, this
study investigates how one can alter the boundary between the lock-in and no
lock-in regimes and the associated vortex dynamics in flows past a forced
oscillating cylinder by inducing elasticity in a viscous fluid. This could be
useful in many practical applications dealing with oscillating structures and
viscous fluids. We also employ one of the widely used reduced order modeling
technique, the dynamic mode decomposition (DMD) technique, in the present
study for visualizing and better understanding the difference between the
viscoelastic and Newtonian fluid flow dynamics, especially in the lock-in and
no lock-in regimes.

Figure 1: (a) Schematic of the present problem, (b) temporal variation of drag
coefficients ($C_{D}$) for three different mesh densities, and (c) comparison
of drag coefficient between the present results with that of Alam et al. (Alam
et al., 2021).

##  2 Problem statement and governing equations

We investigate a two-dimensional, laminar, and unbounded flow of viscoelastic
fluid past a periodically forced oscillating cylinder of diameter $d$, as
shown schematically in figure 1(a). The cylinder is placed at the center of a
fictitious circular domain of viscoelastic fluid of diameter $100d$, wherein
the fluid enters the domain with a uniform velocity of $U_{\infty}$. The
cylinder is considered to be infinitely long in the z-direction so that no
gradient exists in this direction,
i.e.,$\dfrac{\partial{()}}{\partial{z^{*}}}=0$ and also there is no flow in
this direction, i.e., $u_{z}^{*}=0$. Furthermore, the cylinder is forcefully
oscillated by applying a harmonic oscillation with an amplitude of $A^{*}$
(normalized) and a frequency of $f_{y}^{*}$ such that the normalized
displacement $(Y)$ of the cylinder with respect to time $(t^{*})$ is given by
$Y=A^{*}$Sin$(2\pi f_{y}^{*}t^{*})$. Here, the cylinder oscillation amplitude
and frequency are normalized with the cylinder diameter ($A^{*}=\frac{A}{d}$)
and the Strouhal frequency ($f^{*}=\frac{f_{y}^{*}}{f_{st}}$), respectively.
The incompressible flow of the viscoelastic fluid, whose rheological behaviour
is mimicked with the Oldroyd-B constitutive equation (Oldroyd, 1950), is
governed by the following dimensionless equations:
Continuity equation:

$\nabla\cdot\textbf{u}=0$ (1)

Cauchy momentum equation:

$\dfrac{\partial{\textbf{u}}}{\partial{t}}+\textbf{u}\cdot\nabla{\textbf{u}}=-\nabla{p}+\frac{\beta}{Re}\nabla\cdot\bm{\tau_{s}}+\frac{(1-\beta)}{Wi}\frac{1}{Re}\nabla\cdot\bm{\tau_{p}}$
(2)

Oldroyd-B constitutive equation:

$Wi\left(\frac{\partial\bm{\tau_{p}}}{\partial
t}+\bm{u}\cdot\nabla\bm{\tau_{p}}-\bm{\tau_{p}}\cdot\nabla\bm{u}-\left(\nabla\bm{u}\right)^{T}\cdot\bm{\tau_{p}}\right)+\bm{\tau_{p}}=\left(1-\beta\right)\left(\nabla\bm{u}+\nabla\bm{u}^{T}\right)$
(3)

The above equations have been non-dimensionalized using parameters $d$,
$\frac{d}{U}_{\infty}$, $U_{\infty}$, $\rho\,U_{\infty}^{2}$,
$\frac{\eta_{0}\,U_{\infty}}{d}$ for length ($\bm{x}^{*}$), time ($t^{*}$),
velocity ($\textbf{u}^{*}$), pressure ($p^{*}$), and stress tensor
($\bm{\tau}^{*}$), respectively. Here, $\rho$ is the fluid density, $\eta_{0}$
is the zero-shear rate viscosity of the polymer solution given by the sum of
the solvent viscosity, and the contribution of the polymer, i.e.,
$\eta_{0}=\eta_{s}+\eta_{p}$. The total deviatoric stress tensor in the fluid
is a linear combination of the viscous term $\bm{\tau_{s}^{*}}$ (the solvent
contribution) and the viscoelastic extra-stress term $\bm{\tau_{p}^{*}}$ (the
polymer contribution). The Newtonian solvent contribution to the stress tensor
is given by $\bm{\tau_{s}}^{*}=2\eta_{s}\textbf{D}^{*}$, where
$\textbf{D}^{*}$ is the rate of deformation tensor given by
$\frac{1}{2}\left[\left(\nabla^{*}{\textbf{u}^{*}}\right)+\left(\nabla^{*}{\textbf{u}^{*}}\right)^{T}\right]$
and $\eta_{s}$ is the viscosity of the Newtonian solvent. It can be seen that
the present flow is governed by three dimensionless numbers, namely, Reynolds
number defined by $Re=\frac{\rho\,U_{\infty}\,d}{\eta_{0}}$, Weissenberg
number defined by $Wi=\frac{\lambda\,U_{\infty}}{d}$, and polymer viscosity
ratio defined as $\beta=\frac{\eta_{s}}{\eta_{s}+\eta_{p}}$. Here, $\lambda$
is the characteristic relaxation time of polymer molecules. Apart from these,
normalized cylinder oscillation amplitude $(A^{*})$ and frequency $(f^{*})$
are two additional dimensionless numbers that will govern the present flow.
The present investigation is carried out at fixed values of $Re=100$, $Wi=2$,
$\beta=0.5$ and for a range of values of $0.4\leq\,A^{*}\,\leq 1.2$ and
$0.6\leq\,f^{*}\,\leq 1.2$. Additionally, simulations for a Newtonian fluid
have been carried out to directly compare the flow dynamics in viscoelastic
fluids under the same conditions.

##  3 Numerical method and validation

A detailed description of the numerical method employed in the present study
is available elsewhere (Hamid et al., 2022; Khan et al., 2020), and hence, it
is not repeated here for the sake of brevity. Broadly, we have utilized the
finite volume method (FVM) based open source code OpenFoam coupled with
RheoTool (Pimenta & Alves, 2016) to solve the governing mass, momentum, and
Oldroyd-B viscoelastic constitutive equations. Details of the discretization
techniques applied to various terms of the governing equations are provided in
our earlier study (Hamid et al., 2022). Additionally, an automatic mesh motion
scheme (dynamicFvMesh) has been used to treat the fluid-structure interaction,
wherein the mesh at the outer boundary is fixed. At the same time, it deforms
in the vicinity of the cylinder due to its motion. A grid independence study
was conducted by making three different grids consisting of regular hexahedral
elements, namely, Mesh 1(54000), Mesh 2 (102000), and Mesh 3 (206800), to
choose an optimal mesh density. Figure 1(b) shows the temporal variation of
the drag coefficient $C_{D}$ at extreme values of oscillation parameters used
in the present study. After carefully inspecting the results obtained with
three different mesh densities, Mesh 2 was chosen for the present study as
Mesh 2 (102000) and Mesh 3 (206800) values show a nearly perfect match.
Furthermore, a time step size of $\Delta t=0.0001\frac{d}{U_{\infty}}$ was
chosen to carry out all the simulations after performing a rigorous time
independence study. Some validation studies have also been carried out to
establish the accuracy and reliability of the present numerical setup. Figure
1(c) shows the comparison of the drag coefficient with the results of Alam et
al. (Alam et al., 2021). A very good correspondence can be seen between the
two results. Finally, the following boundary conditions were employed to
facilitate the present simulations. At the inlet, $u_{x}=1$ and $u_{y}=0$ are
used. Furthermore, the pressure gradient and the extra stresses due to
polymeric contribution are set to zero at this boundary. At the Outlet, the
pressure is set to zero value, and the Neumann boundary condition is used for
the rest of the variables. At the cylinder surface, the standard no-slip
boundary condition is applied along with a zero gradient for the pressure. The
polymeric stresses are extrapolated linearly onto this surface.

The DMD analysis has been carried out using the algorithm proposed by Schmid
(Schmid, 2011), which is also detailed in our earlier study (Hamid et al.,
2022). Briefly put, temporally equispaced (with 0.1s interval) snapshots ( a
total of $N=501$) of the vorticity field in the unsteady periodic regime
($t\geq 500$) are vectorized and assembled to form a matrix
$S_{1}^{N}=\\{s_{j}\\}_{j=1}^{N}$. DMD assumes a linear map $\mathbf{M}$ that
connects the consecutive vorticity fields as $s_{j+1}=\mathbf{M}s_{j}$.
Therefore, the system can be arranged as
$S_{2}^{N}=\mathbf{M}S_{1}^{N-1}=\mathbf{C}S_{1}^{N-1}+\mathbf{r}$. Here,
$\mathbf{r}$ is the residual, which is minimized to compute the eigenvalues
(Ritz Values denoted by $\lambda_{j}$) and eigenvectors (DMD modes denoted by
$\phi_{j}$) of $\mathbf{C}$ by singular value decomposition (SVD). The DMD
modes capture the spatial coherent features of the flow field, whereas the
Ritz values ($\lambda_{j}$) provide information about the growth rate
($\sigma_{j}=Re(log(\lambda_{j}/t))$) and frequency
($St_{j}=Im(log(\lambda_{j})/2\pi\,t)$) of these modes. The energy
contribution of each mode is quantified using the mode amplitude $b_{j}$,
which is obtained by $\mathbf{b}=\mathbf{\Phi}^{{\dagger}}s_{1}$. Here,
$\mathbf{b}$ is the matrix of amplitude vectors, $\Phi^{{\dagger}}$ denotes
the Moore-Penrose pseudoinverse of mode matrix, and $s_{1}$ is the initial
snapshot vector.

##  4 Results and discussion

We start the discussion by presenting the comparison of the lock-in and no
lock-in states seen between viscoelastic and Newtonian fluids at selected
combinations of $A^{*}$ and $f^{*}$. Subsequently, for the same combinations,
we illustrate in detail how the associated vortex dynamics differs in
viscoelastic and Newtonian fluids. It is followed by the dynamic mode
decomposition (DMD) analysis of two cases when i) both Newtonian and
Viscoelastic fluids are in the lock-in condition ($A^{*}=0.4$ and
$f^{*}=0.8$), and ii) viscoelastic fluid deviates from the Newtonian one (no
lock-in) and exhibits the lock-in condition ($A^{*}=0.4$ and $f^{*}=0.6$). In
the present study, we have chosen seven sets of $A^{*}$ and $f^{*}$ based on a
recent study by Alam et al. (Alam et al., 2021) for elucidating the effect of
the fluid viscoelasticity on the lock-in and no lock-in behaviours. First, the
non-dimensional frequency-amplitude map exhibiting the lock-in and no lock-in
behaviors of Newtonian and viscoelastic fluids is presented in figure 2(a).
From this plot, one can clearly observe that at a combination of $A^{*}=0.4$
and $f^{*}=0.6$, viscoelastic fluid exhibits the lock-in state, whereas
Newtonian fluid is in the no lock-in state. However, as the value of $f^{*}$
increases to 0.8 or 1 at the same value of $A^{*}=0.4$, both fluids display
the lock-in behavior. On further increasing $f^{*}$ to 1.2, both fluids move
to the no lock-in zone. This suggests a stark difference in the
synchronization behaviour of the wake structures for the value of $f^{*}<0.8$
between viscoelastic and Newtonian fluids. This information is obtained from
the power spectral density (PSD) plot of the temporal variation of the lift
coefficient presented in figure 2(b-e). As can be seen, for $f^{*}<1$, the
viscoelastic fluid in both cases (b corresponding to point 1 in the map, and c
corresponding to point 2 in the map) has a dominant peak at the cylinder
oscillation frequency and another peak at its integer multiple, thereby
fulfilling the criteria of the lock-in condition. Contrary to this, at a low
frequency of $f^{*}=0.6$, the power spectrum of the lift coefficient of
Newtonian fluids has several peaks at non-integer multiples of the cylinder
oscillation frequency, indicating a no lock-in behavior of the wake.
Furthermore, the lock-in condition of the wake for an Oldroyd-B fluid at
$f^{*}=0.6$ is easily discernible from the vortex patterns shown in figure
3(a). Here, the wake in a Newtonian fluid is in the non-synchronized state,
with a pair of vortices (2P) being shed in one oscillation cycle. On the other
hand, for an Oldroyd-B fluid, two single vortices (2S) are shed into the wake,
which is again synchronized with the cylinder oscillation. Moreover, these
vortices are found to be stretched in the stream-wise direction with a
connecting region of vorticity between the two subsequent vortices in the near
wake zone. On further moving to higher oscillation frequencies, i.e.,
$f^{*}=0.8$ and $f^{*}=1$, both fluids exhibit the lock-in behavior (see
figure 2(c) and (d) for the corresponding PSD plots) with 2S mode of vortex
shedding. However, the vortex patterns are significantly different from each
other. While for a Newtonian fluid, the wake at $f^{*}=0.8$ resembles the von-
Karman vortex street, the vortices in the far wake organize into rows for an
Oldroyd-B fluid. Moreover, this trend of a viscoelastic fluid is replicated by
the Newtonian fluid at $f^{*}=1$, although with a higher rate of vortex
dissipation. At this point ($f^{*}=1$), vortices in an Oldroyd-B fluid display
substantial stretching in the downstream direction before being shed with
higher lateral spacing. Much similar trend is observed at the frequency ratio
of $f^{*}=1.2$ where the PSD plots (figure 2(e)) show a number of small
amplitude peaks, which are again more pronounced in an Oldroyd-B fluid than in
the Newtonian one. In both fluids, the non-synchronized vortices advect behind
the cylinder with a 2P configuration; however, the Oldroyd-B vortices are
noticeably stronger than the Newtonian ones.

Figure 2:  (a) Map illustrating the lock-in and no lock-in states for
viscoelastic and Newtonian fluids at selected points in the non-dimensional
amplitude ($A^{*}$) and frequency ($f^{*}$) plane. Power spectrum density
plots of the lift coefficients at $A^{*}=0.4$ and frequency of: b)
$f^{*}=0.6$, c) $f^{*}=0.8$, d) $f^{*}=1$, e) $f^{*}=1.2$.

On the other hand, at a fixed frequency of $f^{*}=1$, as the oscillation
amplitude increases from $A^{*}=0.4$ to $A^{*}=1.2$, both Newtonian and
viscoelastic fluids stay in the lock-in state. However, a notable difference
is again seen in the wake dynamics, as illustrated in figure 4. First, at
$A^{*}=0.6$, for an Oldroyd-B fluid, the vortices are shed with ’2S’ mode but
in parallel rows from the near wake itself, whereas this partition occurs at
some distance for a Newtonian fluid. With the further increment in the
amplitude ratio to $A^{*}=1$, a notable difference appears in the wake
topology. While for a Newtonian fluid, the wake still remains in ’2S’ mode,
the wake shifts to ’2P’ mode for a viscoelastic fluid. This is a significant
deviation from the Newtonian behavior where only ’2S’ or ’P+S’ modes have been
observed in the primary synchronization region at multiple Reynolds numbers in
earlier studies (Leontini et al., 2006). A possible explanation for this,
already provided by Govardhan and Willianmson (Govardhan & Williamson, 2000),
is that the single vortices in the near wake split due to excess strain
resulting in the pairing of vortices. Since polymer molecules also undergo
considerable stretching in the cylinder vicinity in a viscoelastic fluid, the
excess straining effect enhances the pairing of the vortices at lower cylinder
oscillation amplitude and frequency values than the Newtonian fluid. For a
Newtonian fluid, a synchronized ’P+S’ wake state is observed at a higher
amplitude ratio of $A^{*}=1.2$, as reported by many earlier studies.

Figure 3: Vortex structures in Newtonian and viscoelastic fluids corresponding
to points 1, 2, 3, and 4 in the map as shown in figure 2(a). Figure 4: Vortex
structures in Newtonian and viscoelastic fluids corresponding to points 3, 5,
6, and 7 in the map as shown in figure 2(a).

Next, we utilize the dynamic mode decomposition (DMD) technique to analyze the
differences in the coherent flow structures of both fluids at $f^{*}=0.6$,
where they exhibit different synchronization behavior. We also perform the DMD
analysis at $f^{*}=0.8$ as well, where both fluids are in the lock-in state.
DMD extracts the underlying structures of dynamic relevance from the global
flow field data and their associated frequencies prevalent in the domain. This
will aid in visualizing the competition between the cylinder oscillation and
the natural vortex shedding frequencies in the wake. First, the Ritz values of
all the cases under consideration are plotted in figure 5(a). Irrespective of
the fluid type and oscillation parameters, most of these neutrally stable
values are clustered around the unit circle $|\lambda_{j}|=1$ with a few
strongly damped values lying inside it. It denotes the convergence towards a
linear representation of the nonlinear flow. Also, these values are
symmetrical with respect to the real axis due to real flow field data. In our
analysis, the sorting of dominant modes is done based on the amplitude
($b_{j}$), which is determined by projecting the modes back to the original
data sequence. The modes with larger projections are more significant and are
depicted by larger sizes in figure 5(a).

Figure 5: (a) The Ritz values ($\lambda_{j}$) both for Newtonian and
viscoelastic fluids at $A^{*}=0.4$ and different values of $f^{*}$, (b) the
normalized magnitudes corresponding to the most dominant modes (excluding the
mean mode) are plotted against the non-dimensional frequency of each mode.
Fundamental frequencies in the fluid flow are denoted by the dashed lines. The
rest of the frequencies are higher harmonics of these fundamental frequencies
in the flow.

The mode with zero imaginary Ritz value is shown by the largest circle for all
fluids and captures the mean vorticity field. For other modes, the normalized
magnitudes are plotted against the associated frequencies ($St_{j}$) in figure
5(b). From this plot, it is obvious that only a few modes capture the maximum
flow energy in all four scenarios, which is expected as the flow is periodic.
To begin, for the lock-in case (i.e., $f^{*}=0.8$), both Oldroyd-B and
Newtonian fluids have a pronounced peak at the cylinder oscillation frequency.
The rest of the frequencies are higher harmonics of this frequency and
contribute negligibly to the data sequence as reflected from their amplitudes.
Moreover, the natural vortex shedding frequency ($f_{st}$) of both fluids does
not appear in this spectrum. It further confirms that the vortex shedding
synchronizes with the cylinder oscillation in the lock-in condition.
Therefore, the cylinder oscillation dominates while the natural shedding
frequency perishes in the flow field.

Figure 6 shows the real parts of the DMD modes corresponding to the mean flow
and the cylinder oscillation frequency both for Newtonian and viscoelastic
fluids.

Figure 6: Visualization of the DMD modes of the vorticity ﬁelds both for
Newtonian and viscoelastic fluids at the lock-in state in the non-dimensional
amplitude-frequency plane ($A^{*}=0.4$ and $f^{*}=0.8$). (a, c) mean mode, (b,
d) mode 1.

In both fluids, the mean mode ($St_{j}=0$) comprises two shear layer-type
structures arising behind the cylinder and extending symmetrically into the
wake. However, these structures are more prominent for a viscoelastic fluid
and extend further into the wake. Mode 1 provides typical structures to
capture the vortex shedding phenomenon and its associated frequency. For both
fluids, the bubble-like structures convect in an anti-symmetric fashion away
from the cylinder with its oscillation frequency. However, there is a marked
difference in the far wake structures between the two fluids. Contrary to the
flow of viscoelastic fluids past a stationary cylinder (Hamid et al., 2022)
where the strength of these structures is higher than Newtonian fluids, the
opposite trend is observed here. This is because, in a Newtonian fluid, single
vortices (’S’) sustain to a greater distance in the wake than in a
viscoelastic fluid. In this fluid, the vortex street is partitioned into two
rows in the far wake (see figure 3), which is successfully captured by the DMD
structures. Figure 7 depicts the modes for Newtonian no lock-in and
viscoelastic lock-in cases at $A^{*}=0.4$ and $f^{*}=0.6$. As seen before, in
the lock-in condition, the DMD extracts only one dominant frequency in
viscoelastic flows, i.e., the cylinder oscillation frequency (see figure
5(b)), and the other low magnitude frequencies are its higher harmonics.
However, for the Newtonian no lock-in case, two fundamental frequencies are
visible due to the non-synchronization in the vortex formation. Hence, from
the flow field data, it is obvious that these two frequencies, i.e., the
cylinder oscillation frequency and the vortex shedding frequency, compete in
the flow domain. The associated flow structures at these frequencies are
visualized in figure 7, where the mean mode for both fluids has similar
structures as explained in the earlier case. Moving to mode 1 in a
viscoelastic fluid, again, the same vortex-shedding structures appear;
however, these resemble the DMD structures of the Newtonian lock-in case (see
figure 5(b)), albeit more concentrated due to the viscoelastic effect. This
observation of similarity in the DMD structures is also reinforced by the
vortex topology at two points in figure 3 (see bottom: $A^{*}=0.4$ and
$f^{*}=0.6$, and top: $A^{*}=0.4$ and $f^{*}=0.8$), and also suggests that
inducing viscoelasticity in a Newtonian fluid expedites the transition to the
lock-in region for a transversely oscillating cylinder. Now looking at the DMD
mode 1 for a Newtonian fluid, which corresponds to the cylinder oscillation
frequency, the concentrated structures in the cylinder vicinity signify the
dominance of forced oscillations in the near wake zone. These structures fade
as we move downstream of the cylinder. Competing with this frequency, the
anti-symmetric structures in mode 2 move with the natural shedding frequency
and are more stretched in the span-wise direction as compared to the lock-in
vortex shedding structures, see figure 7(c).

Figure 7: Visualization of the DMD modes of the vorticity ﬁeld both for
Newtonian (no lock-in) and viscoelastic (lock-in) fluids at the deviation
point in the non-dimensional amplitude-frequency plane ($A^{*}=0.4$ and
$f^{*}=0.6$). (a, d) mean mode, (b, e) mode 1, (c) mode 2.

Finally, we propose that this deviation of viscoelastic fluid behavior from
the Newtonian one is attributed to the stretching of the polymer molecules. We
quantify and visualize this polymer stretching using the non-dimensional
conformation tensor, figure 8. It can be seen that highly stretched strands of
polymer molecules are formed in the cylinder vicinity, which further extend
into the downstream wake. These strands suppress the vortex shedding as the
polymer molecules sustain the stresses for a longer duration, which results in
the elongation of the vortices (see figure 3 $A^{*}=0.4$ and $f^{*}=0.6$). As
a result, the inertial effects generated by the cylinder are not dissipated
but sustained in the fluid. Therefore, the cylinder oscillation dominates the
flow field in viscoelastic fluids, leading to an early transition into the
lock-in condition on a non-dimensional amplitude-frequency plane.

Figure 8: Stretching of polymer molecules in viscoelastic ﬂuid at $A^{*}=0.4$
and frequency ratios of $f^{*}=0.6$ and $f^{*}=0.8$.

##  5 Conclusions

In summary, we have numerically investigated the flows of Newtonian and
viscoelastic fluids over a circular cylinder oscillating transversely to the
streaming fluid at a fixed Reynolds number of $Re=100$. In particular, the
effect of the fluid viscoelasticity on the boundary between the lock-in and no
lock-in regions and the underlying vortex structures are analyzed in detail
for a range of cylinder oscillation amplitudes and frequencies. Our study has
revealed that in the case of a viscoelastic fluid, the lock-in region is found
to occur at a low non-dimensional cylinder oscillation frequency at which a no
lock-in condition is seen for a Newtonian fluid. Furthermore, a substantial
difference in the wake structures is also observed. We report a ’2P’ (two
pairs of vortices in one oscillation cycle) vortex shedding mode for a
viscoelastic fluid in the lock-in zone, which does not occur in a Newtonian
fluid. This deviation is ascribed to the extra elastic stresses arising from
the stretching of the polymer molecules present in viscoelastic fluids.
Additionally, the data-driven dynamic mode decomposition (DMD) analysis has
been employed to provide a better insight into the competition between the
cylinder oscillation frequency and natural vortex shedding frequency in the
wake structure. DMD has successfully extracted the coherent flow structures
associated with each of these frequencies for both fluids in the lock-in and
no lock-in zones.

## References

  * Alam et al. (2021) Alam, Md Irshad, Raj, Apurva, Khan, Piru Mohan, Kumar, Subrata & Roy, Somnath 2021 Numerical simulation of flow of a shear-thinning carreau fluid over a transversely oscillating cylinder. Journal of Fluid Mechanics 921.
  * Anagnostopoulos (2000a) Anagnostopoulos, Petros 2000a Numerical study of the flow past a cylinder excited transversely to the incident stream. part 1: Lock-in zone, hydrodynamic forces and wake geometry. Journal of Fluids and Structures 14 (6), 819–851.
  * Anagnostopoulos (2000b) Anagnostopoulos, P 2000b Numerical study of the flow past a cylinder excited transversely to the incident stream. part 2: timing of vortex shedding, aperiodic phenomena and wake parameters. Journal of Fluids and Structures 14 (6), 853–882.
  * Bird et al. (1987) Bird, Robert Byron, Armstrong, Robert Calvin & Hassager, Ole 1987 Dynamics of polymeric liquids. vol. 1: Fluid mechanics .
  * Bishop & Hassan (1964a) Bishop, Richard Evelyn Donohue & Hassan, AY 1964a The lift and drag forces on a circular cylinder in a flowing fluid. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 277 (1368), 32–50.
  * Bishop & Hassan (1964b) Bishop, Richard Evelyn Donohue & Hassan, AY 1964b The lift and drag forces on a circular cylinder oscillating in a flowing fluid. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 277 (1368), 51–75.
  * Blackburn & Henderson (1999) Blackburn, Hugh M & Henderson, Ron D 1999 A study of two-dimensional flow past an oscillating cylinder. Journal of Fluid Mechanics 385, 255–286.
  * Chhabra & Richardson (2011) Chhabra, Raj P & Richardson, John Francis 2011 Non-Newtonian flow and applied rheology: engineering applications. Butterworth-Heinemann.
  * Govardhan & Williamson (2000) Govardhan, R & Williamson, CHK1789246 2000 Modes of vortex formation and frequency response of a freely vibrating cylinder. Journal of Fluid Mechanics 420, 85–130.
  * Griffin & Ramberg (1974) Griffin, Owen M & Ramberg, Steven E 1974 The vortex-street wakes of vibrating cylinders. Journal of Fluid Mechanics 66 (3), 553–576.
  * Hamid et al. (2022) Hamid, Faheem, Sasmal, Chandi & Chhabra, RP 2022 Dynamic mode decomposition analysis and fluid-mechanical aspects of viscoelastic fluid flows past a cylinder in laminar vortex shedding regime. Physics of Fluids 34 (10), 103114.
  * Jones Jr (1968) Jones Jr, George W 1968 Unsteady lift forces generated by vortex shedding about a large, stationary, and oscillating cylinder at high reynolds numbers. In ASME Symposium on Unsteady Flow. NASA-TM-X-61214.
  * Karniadakis & Triantafyllou (1989) Karniadakis, George Em & Triantafyllou, George S 1989 Frequency selection and asymptotic states in laminar wakes. Journal of Fluid Mechanics 199, 441–469.
  * Khan et al. (2020) Khan, Mohd Bilal, Sasmal, C & Chhabra, RP 2020 Flow and heat transfer characteristics of a rotating cylinder in a fene-p type viscoelastic fluid. Journal of Non-Newtonian Fluid Mechanics 282, 104333.
  * Koopmann (1967) Koopmann, GH 1967 The vortex wakes of vibrating cylinders at low reynolds numbers. Journal of Fluid Mechanics 28 (3), 501–512.
  * Kumar et al. (2013) Kumar, S, Lopez, C, Probst, O, Francisco, G, Askari, D & Yang, Y 2013 Flow past a rotationally oscillating cylinder. Journal of Fluid Mechanics 735, 307–346.
  * Leontini et al. (2006) Leontini, Justin Scott, Stewart, Bronwyn Elaine, Thompson, Mark Christopher & Hourigan, Kerry 2006 Wake state and energy transitions of an oscillating cylinder at low reynolds number. Physics of fluids 18 (6), 067101.
  * Oldroyd (1950) Oldroyd, James G 1950 On the formulation of rheological equations of state. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 200 (1063), 523–541.
  * Pimenta & Alves (2016) Pimenta, F. & Alves, M.A. 2016 rheoTool. https://github.com/fppimenta/rheoTool.
  * Schmid (2011) Schmid, Peter J 2011 Application of the dynamic mode decomposition to experimental data. Experiments in Fluids 50 (4), 1123–1130.
  * Williamson & Roshko (1988) Williamson, Charles HK & Roshko, Anatol 1988 Vortex formation in the wake of an oscillating cylinder. Journal of Fluids and Structures 2 (4), 355–381.