Numerical Investigation of Unsteady Turbulent Flow of

Incompressible Fluid around a Cylinder

M. Essahraoui

, A. Ederouich

, R. El Bouayadi

and A. Saad

(4) SEALAB, Systems Engineering Advanced Laboratory, Kenitra, Morocco

National School of Applied Sciences, Ibn Tofail University, Kenitra, Morocco

Keywords: Turbulent, Unsteady Flow, Cylinder, Drag, Lift, FVM, 𝑘𝜀.

Abstract: In this work we have investigated a numerical study of unsteady two-dimensional turbulent flow of

incompressible fluid around and behind a cylinder with various diameter



𝐷



50𝑚𝑚, 𝐷





100𝑚𝑚



𝑎𝑡 𝑅



1010



𝑎𝑛𝑑 𝑅



3010



, respectively. A numerical simulation will be used by CFD

code to visualize the phenomenon of turbulent wakes generated behind a cylinder and the separation region

around the surface cylinder. The standard 𝑘𝜀 model is the numerical method used to manage the turbulent

flow. This study led us to focus on the velocity contours, pressure contours, the static pressure distribution

along the x-direction, the profiles of skin friction coefficients, and the variation of drag and lift coefficients

as a function of time. The results obtained for the two cases considered gave us satisfactory results. The

development of an oscillation region behind the cylinder, the variation of static pressure and skin friction

coefficient along the cylinder, the mean velocity fields, pressure fields, all will be captured by the present

simulation.

1 INTRODUCTION

The study of the characteristics flow of fluid around

an object is essential for automotive, aeronautical and

oil industries, but also for civil engineering. In

particular, it is a prerequisite for the understanding of

erosion phenomena in earthen dams, whether internal

or external. The flow around or behind a cylinder is a

classical fluid dynamics problem, and serves as a

framework validating of new numerical methods.

This academic problem has received a renewed

interest during the last decade, both experimentally

and numerically, due to the emergence of new

methods for solving the equations of dynamics.

The studies have been done by the recent

experiments demonstrated that the flow field is

symmetrical upon the circular cylinder at values of

𝑅



5, and as we increase the value of Reynolds

number, the flow becomes unstable and the rate of

mixing layers of fluid become more intense which

causing the vortex shedding. (Dou, 2015)

Most of the experimental studies investigated the

steady and unsteady behaviours of the alternating

vortices in the wake. The work of (Tritton, 1971),

(Lourenco & Shih, 1993), and (Braza, Chassaing, &

Minh, 1990) should be mentioned. Besides these

theoretical and numerical investigations, some

experimental visualizations have been described by

(Honji & Taneda , 1969), (Coutanceau & Defaye,

1991). (Rahman, Karim, & Alim, 2007)

The study of Hydrodynamic and aerodynamic

phenomena of a wake modification behind an

obstacle remains a fundamental problem in fluid

dynamics fields. Thus, the vortex structures generate

behind obstacles is of great interest in engineering

practice. Indeed, the comprehension of the vortex

structures produced behind these obstacles and their

different regimes are of primary use in

constructing structures exposed to fluid flows. For a

better visualization of the hydrodynamic and

aerodynamic phenomena, a cylindrical obstacle is

required because of their geometrical simplicity

allowing numerical and experimental facilities.

(Essel, Sharkey, & Tachine, 2014)

The objective of this numerical study, is to

investigate the phenomena of unsteady two-

dimensional turbulent flow of incompressible fluid

around and behind a cylinder of diameter 𝐷





50𝑚𝑚 𝑎𝑛𝑑 𝐷



100𝑚𝑚 at 𝑅



10



𝑎𝑛𝑑 𝑅



3010



respectively. Reynolds

number based on the diameter of the cylinder, in our

study we have focused to change the diameter of the

Essahraoui, M., Ederouich, A., El Bouayadi, R. and Saad, A.

Numerical Investigation of Unsteady Turbulent Flow of Incompressible Fluid around a Cylinder.

DOI: 10.5220/0010733200003101

In Proceedings of the 2nd International Conference on Big Data, Modelling and Machine Learning (BML 2021), pages 311-315

ISBN: 978-989-758-559-3

311

cylinder and visualize how this change influence on

the wake region behind the cylinder, and in the

variation of drag, lift, static pressure, velocity profile

and skin friction coefficient. Our work gives us a

satisfactory result compared to the recent studies.

2 PHYSICAL MODEL

2.1 Mathematical Formulations

𝜕𝑢



𝜕𝑥



0 (1)

𝜕𝑢



𝜕𝑡

𝑢



𝜕𝑢



𝜕𝑥





𝜌

𝜕𝑝

𝜕𝑥



𝜈

𝜕



𝑢



𝜕𝑥







𝜕𝑢





𝑢







𝜕𝑥



(2)

𝜕𝑢





𝑢







𝜈





𝜕𝑢



𝜕𝑥





𝜕𝑢



𝜕𝑥





𝑘𝛿



(3)

𝜕𝜌𝑘

𝜕𝑡



𝜕𝜌𝑘𝑢



𝜕𝑥





𝜕

𝜕𝑥



𝜇 

𝜇



𝜎





𝜕𝑘

𝜕𝑥



𝐺



𝜌𝜀 (4)

𝜕𝜌𝜀

𝜕𝑡



𝜕𝜌𝜀𝑢



𝜕𝑥





𝜕

𝜕𝑥





𝜇

𝜇



𝜎





𝜕𝜀

𝜕𝑥





𝐶



𝜀

𝑘



𝐺





𝐶



𝜌

𝜀



𝑘

(5)

Where 𝑖,𝑗  1,2



𝑥



:ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛

𝑥



:𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛

𝑢



,𝑢



:𝑚𝑒𝑎𝑛 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡𝑠

𝑢





𝑢







:𝑅𝑒𝑦𝑛𝑜𝑙𝑑𝑠 𝑠𝑡𝑟𝑒𝑠𝑠 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡𝑠

𝑢

,



:𝐹𝑙𝑢𝑐𝑡𝑢𝑎𝑡𝑖𝑛𝑔 𝑝𝑎𝑟𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦

𝑝:𝐷𝑦𝑛𝑎𝑚𝑖𝑐 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒

𝜌:𝑇ℎ𝑒 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑙𝑢𝑖𝑑

𝑘:𝑇ℎ𝑒 𝑡𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑡 𝑘𝑖𝑛𝑒𝑡𝑖𝑐 𝑒𝑛𝑒𝑟𝑔𝑦

𝜀:𝑇ℎ𝑒 𝑑𝑖𝑠𝑠𝑖𝑝𝑎𝑡𝑖𝑜𝑛 𝑟𝑎𝑡𝑒

𝜈



:𝑇𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑡 𝑣𝑖𝑠𝑐𝑜𝑐𝑖𝑡𝑦

𝐺



: 𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑐𝑒 𝑘𝑖𝑛𝑒𝑡𝑖𝑐 𝑒𝑛𝑒𝑟𝑔𝑦

𝐶



,𝐶



: 𝐸𝑚𝑝𝑖𝑟𝑖𝑐𝑎𝑙 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡𝑠 𝑜𝑓 𝑘 𝜀

𝜎



,𝜎



:𝑇𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑡 𝑝𝑟𝑎𝑛𝑑𝑡𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑓𝑜𝑟 𝑘 𝑎𝑛𝑑 𝜀

For our study, we have chosen the standard 𝑘𝜀

model, so the values of empirical coefficients are:



1.44,C



1.92,σ



1.0,σ



1.3,C



0.09

The governing equations have been discretized by

the finite volume method (FVM). The SIMPLE

scheme algorithm has been used by the coupling of

pressure, and velocity. The Reynolds numbers values

are depended on the variation of the cylinder

diameter. (Kaur & Kumar, 2018)

2.2 Geometric and Meshing

Figure 1: The computational Domain.

The computational domain is a surface of

dimension 800x2300. The diameter of each cylinder

is (𝐷



50𝑚𝑚 𝑎𝑛𝑑 𝐷



100𝑚𝑚 .The upstream

length is 800mm and the downstream length is

1500mm from the centre of the cylinder.

Figure 2: The grid around the cylinder.

A fine refined mesh around the cylinder is required to

have better accuracy of the results by the CFD solver,

and the method used for our mesh is the Rectangular

domain with smooth quadrilateral grids. (Salehi,

Mazaheri, & Kazeminezhad, 2018)

2.3 Characteristic Parameter

Drag and lift coefficients are the fundamental

characteristics parameters for better understanding

the motion of fluid around a cylinder. (Shim, Sharma,

& Richards, 2009) The Strouhal number is describing

the oscillating flow mechanism in order to study

vortex shedding. The pressure coefficient is a

dimensionless term, which describes the relative

BML 2021 - INTERNATIONAL CONFERENCE ON BIG DATA, MODELLING AND MACHINE LEARNING (BML’21)

312

pressure through the flow field. Moreover, the skin

friction coefficient dimensionless parameter

describes the aerodynamic resistance due to the

contact of moving fluid on the surface of the cylinder.

All these parameters are defined as follows:

𝐶





2𝐹



𝜌𝑉





𝐴

(6)

𝐶





2𝐹



𝜌𝑉





𝐴

(7)

𝐶





𝜏



𝜌𝑉





(8)

𝐶





𝑃𝑃



𝜌𝑉





(9)

𝑆





𝑓

𝐿

𝑉



(10)

𝑨∶𝑃𝑟𝑜𝑗𝑒𝑐𝑡𝑒𝑑 𝐴𝑟𝑒𝑎

𝒇∶𝑇ℎ𝑒 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑣𝑜𝑟𝑡𝑒𝑥 𝑠ℎ𝑒𝑒𝑑𝑖𝑛𝑔

𝑳: 𝑇ℎ𝑒 𝑐ℎ𝑎𝑟𝑎𝑐𝑡𝑒𝑟𝑖𝑠𝑡𝑖𝑐 𝑙𝑒𝑛𝑔𝑡ℎ 𝑖.𝑒 𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟

𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟

𝝉

𝒘

∶𝑆𝑘𝑖𝑛 𝑠ℎ𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑠𝑠 𝑜𝑛 𝑎 𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟 𝑠𝑢𝑟𝑓𝑎𝑐𝑒

𝑷: 𝑇ℎ𝑒 𝑠𝑡𝑎𝑡𝑖𝑐 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑎𝑡 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡 𝑎𝑡

𝑤ℎ𝑖𝑐ℎ 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒

𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 ℎ𝑎𝑠 𝑏𝑒𝑒𝑛 𝑒𝑣𝑎𝑙𝑢𝑎𝑡𝑒𝑑.

𝑷



:𝑇ℎ𝑒 𝑠𝑡𝑎𝑡𝑖𝑐 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑟𝑒𝑒 𝑠𝑡𝑟𝑒𝑎𝑚.

𝑭

𝒅

,𝑭

𝒍

∶𝐷𝑟𝑎𝑔 𝑎𝑛𝑑 𝐿𝑖𝑓𝑡 𝑓𝑜𝑟𝑐𝑒 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦

3 RESULTS

Since we are studying the effect of fluid flow on two

various cylinder diameter with an unsteady turbulent

incompressible flow, the velocity and pressure

contours do not manifest a sizeable difference with

the literature, and it can be notice that there are big

vortices that develop downstream of the cylinder

whose they are ejected in alternation sometimes

towards the upper and the lower wall. See figures 3,

4, 5 and 6. The velocity and pressure profiles around

the cylinder



𝑫

𝟏

𝟓𝟎𝒎𝒎 𝒂𝒏𝒅 𝑫

𝟐

𝟏𝟎𝟎𝒎𝒎



different x positions have been visualized by the

present simulation. See figures 7 and 8. The drag and

lift coefficients around the cylinder vary with time in

the form of quasi sinusoidal curve. See figures 9 and

10. The skin friction coefficient increases relatively

linearly from the stagnation point 𝐱𝟎.𝟎𝟓



𝐃

𝟐



𝟏𝟎𝟎𝐦𝐦



𝐚𝐧𝐝 𝐱𝟎.𝟎𝟐𝟓



𝐃

𝟏

𝟓𝟎𝐦𝐦



, until

it reaches a maximum level at approximately

𝟎.𝟎𝟑𝟓𝐱𝟎.𝟎𝟏𝟓 at different Reynolds

number values. This location is upstream of the

cylinder, approximately 0,35≤𝑪𝒇≤0,045, as we will

observe in the figures below, and then it gradually

decreases before stabilizing to an asymptotic value.

See figure 11.

3.1 Velocity Contours

Figure 3: Velocity contours at 𝑅



1010



Figure 4: Velocity contours at 𝑅



3010



3.2 Pressure Contours

Figure 5: Pressure contours at 𝑅



1010



Figure 6: Pressure contours at 𝑅



3010



Numerical Investigation of Unsteady Turbulent Flow of Incompressible Fluid around a Cylinder

313

3.3 Velocity Profiles

Figure 7: Velocity profiles x=±1.5 at different Reynolds

number.

3.4 Pressure Profiles

Figure 8: Pressure Profiles X=±0.06 at Different Reynolds

NUMBER.

3.5 Drag & Lift Coefficient at

𝑹

𝒆

𝟏𝟎𝟏𝟎

𝟑

Figure 9: Variation of the coefficient of drag and lift in the

case of D=50MM.

3.6 Drag and Lift Coefficient at

𝑹

𝒆

𝟑𝟎𝟏𝟎

𝟑

Figure 10: Variation of the coefficient of drag and lift in the

case of D=100mm.

3.7 Skin Friction Coefficient

Figure 11: Variation of Skin friction coefficient at different

𝑅



4 CONCLUSIONS

The study of unsteady 2D turbulent flow around

obstacles was the objective of the present work.

Numerical simulation has been adopted by resolve the

equations of an unsteady flow of an incompressible

fluid of a turbulent regime. We compared our results

with those obtained in the literature for a flow around

a cylinder. We have visualized the phenomenon of

vortex shedding, which has been generated by

unfavorable pressure gradient at the separation points

on the cylinder surface. Thus, the pressure values are

lower above and below the cylinder and an oscillation

region behind the cylinder was observed for both

Reynolds number. In the middle of the vortex area,

the velocity and pressure are low compared to the

extremity. The variation of the lift and drag

coefficients are simultaneous with the vortex

shedding detachment. This detachment is responsible

for the creation of large vortices behind the cylinder.

We can notice from the graphs of static pressure and

skin friction coefficient that the values of pressure

and skin friction coefficient are not identical around

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314

the cylinder, and the flow is separated on the cylinder

surface.

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