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126  Progress in Computational Fluid Dynamics, Vol. 16, No. 2, 2016  

Copyright © 2016 Inderscience Enterprises Ltd. 

Investigation of a wake formation for flow over acylinder using Lagrangian coherent structures

 Ali Bahadir Olcay

Department of Mechanical Engineering,Yeditepe University,

Istanbul, 34755, Turkey

Email: bahadir.olcay@yeditepe.edu.tr

Abstract: In this study, flow over a cylinder was investigated to understand the physics of fluid

motion in the wake region. Eulerian velocity field attained from computational fluid dynamicsmodel while Lagrangian information obtained from finite time Lyapunov exponent fields

illustrated how a vortex formation takes place once it is detached from the cylinder.

Keywords: wake evolution behind the cylinder; Lagrangian coherent structures; LCSs.

Reference to this paper should be made as follows: Olcay, A.B. (2016) ‘Investigation of a wake

formation for flow over a cylinder using Lagrangian coherent structures’,  Progress in

Computational Fluid Dynamics, Vol. 16, No. 2, pp.126–130.

Biographical notes:  Ali Bahadir Olcay received his BS, MS and PhD degrees in Mechanical

Engineering from Middle East Technical University, Southern Illinois University Edwardsvilleand Southern Methodist University, respectively. He is currently an Assistant Professor in the

Department of Mechanical Engineering at Yeditepe University.

1 Introduction

Flow field over a blunt body has attracted many researchers’

attention due to the wake region taking place behind the

 body. The wake region becomes responsible for lower pressure at the downstream and this pressure variation

 between upstream and downstream eventually results in

large drag.

In earlier work of Roshko (1952), wake development

 behind circular cylinder has been studied. They used a low

speed wind tunnel to investigate the flow behind the

cylinder for Reynolds number from 40 to 10,000. They

reported that in the ranges of 40 to 150, 150 to 300, and 300

to 10,000, the wake patterns are stable, transitory

and irregular, respectively. Saiki and Biringen (1996)

investigated stationary as well as moving cylinders in

uniform flow at low Reynolds number using a virtual

 boundary technique. They compared separation angle, dragcoefficient and Strouhal number with previous experimental

and numerical results and concluded that virtual boundary

technique provides promising results for steady and

unsteady flow problems. Recently, Benim et al. (2008)

investigated turbulent flow over a circular cylinder by

employing RANS, URANS, LES and DES. Their findings

showed that while two dimensional RANS were under

 predicting drag coefficient obtained from experiments for a

wide range of Reynolds number, two dimensional URANS

was over predicting drag coefficients.

The physics of the fluid motion in the wake of the flow

over cylinder is the primary focus of the present study.Wake formation has been investigated using the Lagrangian

coherent structure (LCS) method to understand how the

wake forms behind the cylinder. Specifically, emphasis will

 be on how the separated shear layer forms the wake and

once the vortex is formed, how it grows and becomes part

of the vortex street at the downstream.

2 Numerical simulation

2.1 Numerical model

Flow over cylinder is simulated using the flow domain

shown in Figure 1(a). In the figure, the  x- y plane represents

two-dimensional domain with dimensions of 50 D in length

and 20 D  in height and  D  is the cylinder diameter. To

simulate the flow field over the cylinder, a uniform

horizontal inlet velocity, U , was provided over the left side

of the domain while ambient pressure conditions werespecified over the domain’s right side boundary. Symmetry

was defined over the upper and the lower part of the domain

and the no-slip condition was applied to the wall of cylinder

to consider viscosity as shown in Figure 1(a).

The domain was discretised for both  x and  y directions

with increased mesh density around the cylinder wall and

 behind the cylinder at the downstream as shown in

Figure 1(b) so that wall gradients can be properly resolved

during the flow evolution process. 45,760 quadrilateral cells

were employed for the solution domain. The two

dimensional, unsteady, incompressible Navier-Stokes

equations with zero swirl given in equation (1) were used to

simulate the flow evolution.

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   Investigation of a wake formation for flow over a cylinder using Lagrangian coherent structures 127 

2 2

2 2

2 2

2 2

0

1

1

 y x

 x x x x x x y

 y y y y y x y

uu

 x y

u u u p u uu u   ν

t x y   ρ  x x y

u u u u u pu u   ν

t x y   ρ  y x y

∂∂+ =

∂ ∂

⎡ ⎤∂ ∂ ∂ − ∂ ∂ ∂+ + = + +⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦

∂ ∂ ∂ ∂ ∂⎡ ⎤− ∂+ + = + +⎢ ⎥

∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦

  (1)

The non-dimensional number used to characterise the flow

field is the Reynolds number ( Re) defined as Re  ρUD

 μ=  

where U  is the inlet velocity, D is the cylinder diameter, ρ is

the fluid density and µ is the dynamics viscosity of the fluid.

In this study, Reynolds number is taken to be 10,000 so that

 boundary layer remains laminar since it is less than critical

Reynolds number of 300,000 and Karman vortex street is

expected to appear at the downstream (i.e., behind the

cylinder). Strouhal number (Sr ) defined as , fD

Sr U 

=  where

 f  is vortices shedding frequency, is another non-dimensional

number which describes the flow field oscillations due to

the vortex shedding frequency.

Table 1  Convergence analysis and model validation results

for domain size of 50 D by 20 D with 45,760 elements

 Flow features Present results Literature results

(Williamson, 1996)

θ  at Re = 104  80.50 800

Sr  at Re = 104  0.19 0.2

Flow features Present results Literature results*C d  at Re = 10

4  1.24 1.17

C d  at Re = 1.56 × 104  1.26 1.18

C d  at Re = 2.92 × 104  1.27 1.19

Source:  *Data extracted from Figure 8 ofBenim et al. (2008)

Domain and grid convergence analysis have been performed

to determine domain and mesh sizes. The separation angle

(θ  ), drag coefficient (C d ) and Sr   were used to monitor

domain and space convergence. Domain convergence was

tested for three different domain sizes with  L  = 25 D  and

W  = 10 D, L = 50 D and W  = 20 D, L = 100 D and W  = 40 D,where  L  and W   are the domain length and width,

respectively. The domain size of 50 D  by 20 D  resulted in

0.66% and 3% differences from documented θ   and Sr  

number as in Table 1. Grid convergence was tested for three

different meshes with elements of 24,880, 45,760 and

92,000. θ , Sr   and C d   for 45760 elements showed only

0.59%, 1.18% and 1.62% differences from that of 92,000

elements, respectively. Model also has been tested at

different Re and results are given in Table 1.

Figure 1  (a) The solution domain (b) The solution domainshowing 45,760 quadrilateral cells placed with

increased mesh density around the cylinder wall and behind the cylinder at the downstream (see onlineversion for colours)

40D 

50D 

DInlet

Wall

Outlet 20D

Symmetry

Symmetry

 x 

y  

2.2 LCSs technique

A fluid particle’s trajectory at position x0 at time t 0 is given

as the solution of the initial value problem,

( ) ( )( )0 0 0 0; , ; , ,t t t t t  =x x V x x   (2)

( )0 0 0 0; ,t t    =x x x  

where x(t ; t 0, x0) is the position of the fluid particle at time t  

which was at x0 at time t 0 and V is the velocity of the fluid particle at time t , which was at x0  at time t 0. The velocity

field on the right hand side of equation (2) can be derived

from the CFD solution of the flow problem. The solution to

the initial value problem given by equation (2) can be

treated as a flow map 00

0( )t T 

t + xφ   that describes the position

information of the fluid particle at time t  = t 0 + T  which was

initially (i.e., at t  = t 0) at x0. The flow map can be expressed

as

( )   ( )00

0 0 0 0; , .t T 

t t T t + = +x x xφ    (3)

The finite time Lyapunov exponent (FTLE) is then defined

as

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128  A.B. Olcay 

0max

1( ) ln

| |T t σ λ

T ≡x   (4)

where λmax is the maximum eigenvalue of

( ) ( )0 00 0

*( ) ( )t T t T  

t t + +∇ ∇x xφ φ    (5)

where ( )*  implies the adjoint (transpose) operation. It can

 be shown (Shadden et al., 2005) that the separation of

 particles advected by the flow is proportional to 0( )| |

.T t 

σ  T e

x 

Figure 2  Stable and unstable manifolds showing transport barriers in the flow over a cylinder at t * = 0.50

Unstablemanifolds

Stablemanifolds

 Note: Both stable and unstable manifolds have the sameintegration length of |T | = 50.

When distance between two moving fluid particles in a flow

field are considered, FTLE can be seen as a finite time

average measure of the maximum expansion rate of particle

 pairs advected by the flow. Besides, the ridges in the FTLE

field characterise LCS. Shadden et al. (2005) have shown

that the flux across a LCS varies with1

| |T  and therefore,

for large |T | LCSs can be treated as transport barriers in the

flow, and thus material lines. Typically, once the velocity

field data [e.g., right side of equation (2)] is integrated

forward and backward in time, one can calculate LCSs of

the solution domain. When LCSs are obtained, the stable

manifolds named as repelling LCSs (i.e., T   > 0) and

unstable manifolds named as attracting LCSs (i.e., T  < 0) at

the solution domain can be identified. This technique can be

used to identify wake development behind a circular

cylinder because the formulation applies well forunsteady flows. LCSs calculations were performed with the

help of ManGen software (http://mmae.iit.edu/shadden/

LCS-tutorial/mangen.html). ManGen gives the FTLE field

through equation (4) for the flow domain formed by a grid

of massless particles using advected velocity field.Computation of σ   was performed with a uniform grid of

0.005 D  resolution and at the location of –1 ≤  x/ D ≤ 8 and

 –1.50 ≤  y/ D ≤ 1.50 to produce sharp ridges for the attracting

and repelling LCS in the current investigation. The data as

obtained from LCS identified the boundaries of the growing

wake as shown in Figure 2. Specifically, repelling LCS

calculated by taking |T | = 50 identified upstream whileattracting LCS obtained by taking |T | = 50 revealed the

downstream. Stable manifolds shown in the upstream

divides the flow field in two regions. Fluid initially between

the stable manifolds is enforced to remain in the wake

region later in time while fluid initially outside the stable

manifolds continues to flow as the free stream without being

affected by the viscous effects near the wall region.

Unstable manifolds of the downstream, on the other hand,

identify vortex boundaries of the vortices in the Karman

vortex street as well as joining the forming vortices.

3 Results and discussion

Once the upstream fluid meets with a blunt body, fluid

moves over the surface of the body until it loses all of its

momentum. Then, flow separates from the wall and forms a

wake region. In this study, flow over a cylinder has been

studied to investigate wake formation which causes a large

 pressure drop across the body.

The ridges seen in plots of Figure 3 are referred to LCSsand act as flow barriers. Specifically, fluid residing on the

upper (or left) side of these LCSs is not allowed to cross

these lines. Similarly, fluid residing on the lower (or right)

side of these LCSs cannot cross these ridges to pass to the

upper (or left) side. Evolution of the first several vortices is

fundamentally different from the rest of the vortices.

Specifically, Figure 3(a) shows the first two vortices

rotating in opposite directions at t * = 0.10. In here, t * is the

non-dimensional time defined as the ratio of the

instantaneous time to the total simulation time. It is also

seen that the unstable manifold at  y/ D = 0 and 0.5 ≤  x/ D ≤ 

8.0 separates the upper flow ( y/ D > 0) from the lower flow

( y/ D < 0). These rotating vortices grow in size by drawing

more fluid in until the horizontal unstable manifold, y/ D = 0

and 0.5 ≤  x/ D ≤ 8.0, shows a wavy behaviour as shown in

Figure 3(b). Right after the first two vortices detach from

the cylinder, third and forth vortices first form, then grow

and finally detach from the cylinder and move in the

downstream direction. The evolution of the sixth and the

seventh vortices are given in Figure 3(c) while Figure 3(d)

demonstrates the evolution of the eighth and ninth vortices.

The vortex shown in Figure 3(d), which was born as a

consequence of boundary layer separation, starts to draw the

surrounding fluid towards its centre and continues to draw

during its evolution. Starting with this vortex (i.e., ninthvortex), evolution of vortices is predominantly governed by

fluid entrainment in an organised way. To describe a typical

evolution process of this nature, the evolution of the ninth

vortex right after the boundary layer separation is depicted

in Figures 3(d) and 3(e). More specifically, as is observed in

Figure 3(e), the ninth vortex grows gradually by entraining

more and more fluid towards its central region. Physically,

fluid downstream the cylinder moves back towards the low

 pressure region, i.e., towards the cylinder, while pushing the

already formed preceding vortex, in this case, the eighth

vortex. This process can be seen clearly in Figure 3(e). At

later times, Figure 3(e) the growing vortex is pushed

upward and further deformed by the ridge of LCS. This may

 be attributed to the fact that time came for the tenth vortex

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   Investigation of a wake formation for flow over a cylinder using Lagrangian coherent structures 129 

to form and while developing will also be drawn into it. It

may then be imagined that the ridge somehow pushes the

ninth vortex up for the sake of opening room for the newly

emerging tenth vortex. In the meantime, it should be noted

that the centres of the eighth and ninth vortices get closer to

each other due to the lower pressure just behind the cylinder

( x/ D ≈  1 and  y/ D ≈  0). Finally, the ninth vortex shown in

Figure 3(e) detaches from the cylinder and starts to travel in

the downstream direction by Biot-Savart induction.

Figure 4(a) illustrates the pressure contour plot at

t * = 0.42. It can be seen that while pressure gradient is small

in the vicinity of the vortex centre, pressure variation

 becomes significant in the upstream and downstream sides

of the ninth vortex [enlarged plot is given in Figure 4(b)].

Arrows are superimposed on the pressure contour plots to

show the route of the fluid before it is entrained by the

vortex. The growing vortex actually pulls fluid from

surroundings and gets larger in volume. This in return

reduces the pressure in the region from where the fluid is

entrained. Once the vortex is formed, Biot-Savart induction

 provided by shear layer associated with the flow separation

drives the vortex in the downstream direction. Lastly, the

velocity field together with the LCS structure is plotted in

Figure 5. Velocity vectors marking fluid  particles demonstrate

how surrounding fluid is drawn into the vortex centre

through the entrainment path.

Figure 3  Time evolution1 of vortex formation behind the cylinder at (a) t * = 0.10, (b) 0.23, (c) 0.35, (d) 0.40, (e) 0.43

First vortex

Second vortex

(a) (b)

Fifth vortex

Sixth vortex

Seventh vortexInitial formation of ninth

vortex

Fifth vortex

Sixth vortex

Seventh vortex

Eighth vortex

(c) (d)

Ridge causing deformation of

ninth vortex

Formation oftenth vortex

(e)

 Note: Short animation of this simulation can be accessed at http://youtu.be/LCY5txrQ4Ek.

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130  A.B. Olcay 

Figure 4  (a) Pressure contour plot of flow over cylinder att * = 0.42 (b) Enlarged view of the pressure contour plot

for ninth vortex

LCSs

LCSs

(a)

(b)

 Note: Grey thick lines show the LCSs.

Figure 5  Velocity vector field of flow over cylinder at t * = 0.42

 Notes: Grey thick lines show the LCSs. Every fifthvelocity vector is shown in the figure.

References

Benim, A.C., Pasqualotto, E. and Suh, S.H. (2008) ‘Modelling

turbulent flow past a circular cylinder by RANS, URANS,LES and DES’,  Progress in Computational Fluid Dynamics,

Vol. 8, No. 5, pp.299–307.

Roshko, A. (1952) On the Development of Turbulent Wakes fromVortex Streets, PhD thesis, Department of Mechanical

Engineering, California Institute of Technology, California,USA.

Saiki, E.M. and Biringen, S. (1996) ‘Numerical simulation of a

cylinder in uniform flow: application of a virtual boundarymethod’, Journal of Computational Physics, Vol. 123, No. 2, pp.450–465.

Shadden, S.C., Lekien, F. and Marsden, J.E. (2005) ‘Definitionand properties of Lagrangian coherent structures from

finite-time Lyapunav exponents in two-dimensional aperiodicflows’, Physica D, Vol. 212, Nos. 3–4, pp.271–304.

Williamson, C.H.K. (1996) ‘Vortex dynamics in the cylinder

wake’,  Annu. Rev. Fluid Mech., Vol. 28, pp.477–539, DOI:10.1146/annurev.fl.28.010196.002401.

Nomenclature

CFD Computational fluid dynamics

 D Diameter of the cylinder [m]

FTLE Finite time Lyapunov exponents

LCS Lagrangian coherent structure

 Re Reynolds number

t Instantaneous time [s]

t total   Total time for flow simulation [s]

t *

  Non-dimensional time defined as t * = t /ttotal  U Uniform inlet velocity

|T | Integration time length [s]

( )00 0t T 

t + xφ    Flow map describes the position information of

the fluid particle at time t  = t 0 + T  

0( )T t σ  x   The finite time Lyapunov exponent defined as

0 max

1( ) ln

| |T t σ λ

T ≡x  

 ρ  Density of fluid

 µ Dynamics viscosity of fluid