study of turbulent flow downstream from a

International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –

6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 1, Jan - Feb (2013) © IAEME

8

STUDY OF TURBULENT FLOW DOWNSTREAM FROM A

LINEAR SOURCE OF HEAT PLACED INSIDE THE

CYLINDER WAKE

D. Tcheukam-Toko*1

, B. S. Tagne-Kaptue2, A. Kuitche

2, R. Mouangue

1, P. Paranthoën

3

1Department of Energetic Engineering, IUT, University of Ngaoundere, P. O. Box 455

Ngaoundere, Cameroon 2

Departments of Energetic and Electrical Engineering, ENSAI. P. O. Box 455 Ngaoundere,

Cameroon. 3

CNRS UMR 6614 CORIA, University of Rouen, P. O. Box 12 – 76801 Saint-Etienne du

Rouvray, France.

* Corresponding author. Email: [email protected]

ABSTRACT

A turbulent flow downstream from a linear source of heat placed inside the cylinder

wake has been studied numerically in this paper. Special attention has been paid to the

cylinder wake effect on the source of heat diffusion in downstream flow. The turbulent model

has been applied a standard κ-ε two equations model and the two-dimensional Reynolds

Averaged Navier–Stokes (RANS) equations are discretized with the second order upwind

scheme. The SIMPLE algorithm, which is developed using control volumes, is adopted as the

numerical procedure. Calculations were performed for a wide variation of the Reynolds

numbers. The investigations reveal that with increasing Reynolds number, the instabilities

appear in the wake zone, showing an oscillatory flow, also called von Karman Vortex Street.

His geometry has an important influence on the thermal field and the diffusion process.

Comparison of numerical results with the experimental data available in the literature is

satisfactory.

Keywords: Passive scalar, linear source of heat, Cylinder wake, Turbulent flow, CFD.

INTERNATIONAL JOURNAL OF MECHANICAL

ENGINEERING AND TECHNOLOGY (IJMET)

ISSN 0976 – 6340 (Print)

ISSN 0976 – 6359 (Online)

Volume 4 Issue 1 January- February (2013), pp. 08-21

© IAEME: www.iaeme.com/ijmet.asp

Journal Impact Factor (2012): 3.8071 (Calculated by GISI)

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IJMET

© I A E M E



9

I. INTRODUCTION

The dispersion of passive contaminant generated of locale fashion in a turbulent flow,

is an important phenomenon funded in many problems of heat and mass transfer (Warhaft,

[1]). His industrials applications are the dryer, the heat pump, the boilers, the air

conditioning, the refreshes of electronics components, the reactor conception, etc... The terms

passive and locale means respectively that the contaminant emitted does not modified the

characteristics of main flow and the scale, at which the scalar is injected, is always very lower

than the integral scale of turbulence. In many practices situations, these diffusions

phenomenon’s appeared in some complex turbulent flows which are perturbed by the

obstacles and are characterized by the higher structures.

Many studies carried out in turbulence during these last decade, have showed the existence of

coherent structures inside the stress flows, even at the high Reynolds numbers. Veeravalli and

Warhaft, [2], carried out a study of thermal dispersion from a line source in a shearless

turbulence mixing layer. They did not associate the instabilities phenomenon caused by the

existence of wake. Le Masson, [3], has worked on the control of Bénard Von-Karman

instabilities downstream from a heated obstacle at low Reynolds number, but he does not

defined all the control parameters of instabilities. Brajon-Socolescu, [4], has carried out a

numerical study on the Bénard Von-Karman instabilities behind a heated cylinder. Lecordier

and al. [5], also, who have worked on the transition control downstream from a 2D obstacle

using a source of heat located inside his neared wake. These last two studies were limited

because of lack of critical Reynolds number. Weiss [6], has studied a passive scalar diffusion

inside the neared obstacle wake. He demonstrated that the thermal field is strongly influenced

by the geometry of Vortexes Street, but he worked only with one Reynolds number.

Paranthoën and al. [7], have carried out a dynamic field experimental study of Bénard von-

Karman Street downstream from a heated or not heated 2D obstacle. This study used only

one Reynolds number. Many others recent studies were carried out by Champigny and

Simoneau, [8], on the mixed convection around a wide vertical cylinder. They did not take in

account, the wake effects on the thermal field dispersion and the choice of turbulence model.

Aloui, [9], in the studies carried out on the flow control, does not take in account the choice

of parameters control and the source of heat. However, it is clear that few of these studies

have been dedicated on the influence of structures in the diffusion and transport phenomenon,

with the exception of Crow and al. [10], who have worked on the solids particles dispersion.

The number of studies carried out in this domain is not enough, however, it’s aroused many

interest, because of his responsibility on the existence of counter-gradient zones in these

flows. In this case, the flux of passive contaminant has the same direction and the same way

with the mean temperature gradient (FuIachier and al. [11]; Sreenivasan and al. [12],

Veeravalli and Warhaft, [2]). Corsin [13], has showed that it is not possible to model the heat

transport with the linear model of gradient transport using a turbulent diffusivity.

In order to explain the influence of structures on the thermal transfer phenomenon and

diffusion process, we have carried out a numerical study of turbulent flow downstream from

a linear source of heat placed inside the cylinder wake, by using several Reynolds numbers.

To lead well this study, we are going to analyze the temperature and velocity profiles

respectively inside the cylinder wake and downstream from the linear source of heat. Then,

we will analyze the means temperature gaps profiles, the transversal flux of heat profiles, in

function of transversal gradient of mean temperature. We will end this analyze by doing the

comparison between the numerical and the experimental results.



10

II. MATERIAL AND METHODS

II.1 Mathematical models used The continuity equation is given by the equation bellow: �� 0 (1)

The conservation equations of the average quantity of movement of Navier-Stockes known by the

name RANS are for compressible fluid and Newtonian given by the formula bellow.

��

��

�� !"#$%&!

� ' �(��)*

+�,��- /0� &

1,�--��-

� ��

23 4�0)��

� �0��)

' 56 7��

�0)��)

89��-��-��: +�,��-

� ��

;'�<′�=′>>>>>?��+�,��-

@A#A!" AB CD �0,E0F��

� G� (2)

'�<′�=′>>>>> are the components of the Reynolds stress. Its expression is bellow as given by the

Boussinesq J. (1897), hypothesis:

'�<′�=′>>>>> � 3� 4�0)��

� �0��)

8 ' 56 4H � �0)

��8 7�� (3)

The k- є turbulence models used by the software FLUENT [14] are:

• the k-є standard model

• the k-є RNG model

• the k-є realisable model

We are going to use the k-є realisable model to carry out calculations in the software

FLUENT. The turbulence k-є realisable model proposed by Shih and al., [15], was proposed to make

up for the insufficiency of the other k-є models such as the k- є standard model, the k-є RNG model,

etc..., by adopting a new formula for the turbulent viscosity while implicating a variable Cµ at the

origin (proposed by Reynolds) and a new equation for the disposed based on the dynamic equation of

the vortices fluctuations. The equations of its transporting equations are:

• The turbulent kinetic energy transport equation, which is given by the formula bellow.

• IIJ �KL� � I

IMN�KLOP � I

IMN2;Q � QJ

RL? IL

IMP9 � SL � ST ' KU ' VW (4)

• The transport equation of the dissipation rate of turbulent kinetic energy, which is given by

the formula bellow.

IIJ �KU� � I

IMP�KUOP � I

IMP2;Q � QJ

RU? IU

IMP9 � KXYZU ' KX[

U[L\√^U � XYU

ULX_U (5)

Where:

a � bcd e0.43, jj\k l, with, m � n o

p (6)

Gk represent the turbulent kinetic energy due to the average gradient of velocity. Gb represent the

generation of kinetic energy due to floating. YM represent the contribution of the fluctuating

dilatation. C2 and C1ε are the constants; σk and σε are the numbers of turbulent Prandtl relative to k

and ε.

The values of constants are represented on the table 1 below.

Table 1: The constants of model

C1ε 5 qo qp

1.44 1.9 1.0 1.2



11

The turbulent transport of heat is modelled by the usage of the analogy concepts of Reynolds to the

turbulent transfer. The energy equation is given as:

�� u� � �

��)v��u � w�x � �

��4H�++

�y��

� ��z�� ++8 � n{ (7)

E is the total energy, its expression is:

u � | ' (� � })~

5 (8)

Keff, the coefficient of effective thermal conductivity and K, the coefficient of laminar thermal

conductivity, expressed as:

H�++ � H � ��(,� (9)

�z�� ++ is the Tension Newtonian effective of vicious stress. Its expression is given by the formula

below:

�z�� ++ � 3�++ 4��)

� �0)��

8 ' 56 3�++

�0)��)

7�� (10)

II.2 Boundaries conditions We have based our study on the experimental study of Paranthoën and al. [16],

carried out inside the air by choosing as first value of Reynolds number (Re = 15). The

figure 1 bellow shows the configuration of that experiment. The Reynolds number is

obtained from the following relation, Re = UD/ν, where D represented the cylinder

diameter, and U, the air longitudinal velocity. The value of Re, at which the vortexes

street appears is 48, and it is considered as the critical Reynolds number (Rec). The

electric power by length unity (P/L), supply to linear source is about 10W/m, which is

corresponding to a temperature of 393K, higher than the temperature of the upstream

flow. For this threshold difference (Re - Rec), for this level of heating P/L, for these

positions inside the vortexes street (Xs+

= 7 ; Ys+

= 0), and for this ratio D/d = 100, the

linear source do not modified the instability as shown in Lecordier and al.[5], d is the

linear source of heat diameter.

Figure 1: Experimental configuration of Paranthoën and al. [16].

The calculation domain is a cobbled of length 300 mm, and of height 32 mm. On this domain,

the linear source of heat is located at 14 mm behind the cylinder, at the same axis. The

principal flow is emitted longitudinally across a rectangular section of width 64 mm and of

height 32 mm. The cylinder diameter is 2 mm, and the linear source of heat diameter is 0.02

mm, which is well satisfied by the ratio D/d = 100. In this study, the sign “+” in quote,



12

indicates a normalised quantity. The heights are normalised by D and the temperature gaps

are normalised by the reference temperature gap ∆Tref. The molecular effects being negligible

in front of the turbulence, the relationship ∆T/∆Tref can be assimilated to a concentration C,

which will vary between 1 at emission and 0 at the infinity. ∆T is the difference between the

initial temperature of the principal flow and the temperature of the linear source of heat at an

instant t. The velocities are normalized by the sound velocity at 300K, when air is assimilated

as a perfect gas. This domain of calculation represented on figure 2 bellow, is meshed with

the Gambit program. It is a regular grid type with its cells in the quadrilateral form, with

185,054 cells. The principal flow is introduced longitudinally through the left of the cylinder.

Air comes out at 300 mm from the input.

a)

b)

c)

Figure 2: Mesh of calculation domain:

a): Calculation domain, b): Zoom around the

cylinder wake, c): Zoom around the linear

source of heat

We have imposed the atmospheric pressure conditions at the output. The different values of

Reynolds number applied at the input are: Re = 63, 126, 252, 504, 700 and 900. The wall

cylinder temperature and the ambient temperature chosen are 300K.

III. RESULTS AND DISCUSSION

III.1 Dynamic field The figures 3a, 3b, 3c, 3d, 3e and 3f, represented the fields of dimensionless velocities

iso-value, respectively for the following Reynolds number 63, 126, 252, 504, 700 and 900.

For a low Reynolds number (Re = 63), we observe the formations of turbulent boundaries

layers around of cylinder. Inside the cylinder wake, the velocities remains weak and the flow

is propagated progressively to the linear source of heat direction, located at the position X+ =

7. This propagation has a spherical wave form which appear in upstream and downstream



13

from this linear source of heat. When the flow velocities increase (Re = 126), the vortexes

tables appears inside the cylinder wake and becomes slightly oscillatory in downstream from

the linear source of heat, where the smalls vortexes street are beginning to appear.

For the middle velocities (Re = 252, and Re = 504), the vortexes numbers are increasing, and

these small vortexes alternated are more than more periodicals. The vortexes tables are

increasing along the longitudinal axis, showing the formation of the Bénard von-Karman

vortex street.

a)

b)

c)

d)

e) f)

Figure 3: Dimensionless velocities iso-values.

a) Re = 63, b): Re = 126, c): Re = 252, d): Re = 504, e): Re = 700, f): Re = 900.

When the flow velocities are increased (Re = 700 and Re = 900), the coherent structures are

becoming more than more periodicals, because of the concave angle of the boundary layer

around the cylinder walls which decreases. For Re = 700, the periodical removing model of

vortex is changing. The wake symmetry is decreasing with a production of a secondary



14

periodical removing of vortex. For Re = 900, the secondary periodical removing does not

appear again inside the vortexes twins. For these two Reynolds numbers, there is a strong

apparition of instabilities generating also an oscillatory flow which evolved as small alternate

vortexes called Bénard Von-Karman Vortex Street. The thickness of these alternate vortexes

is decreasing with their longitudinal propagation. We also noted a net adherence between the

cylinder lateral wall and the fluid, because of low values of velocities (blue color zone, U+ =

0.0625).

The figures 4a and 4b represented the longitudinal variation of dimensionless velocities (U+),

near the cylinder, respectively at the positions Y+

= -1, and Y+ = +1, for the different

Reynolds number (Re = 63, 126, 252, 504, 700 and 900). We observed a strong augmentation

of the velocity which decreased suddenly in the neared cylinder wake. This strong gradient of

velocity approved the presence of turbulent boundaries layers around the cylinder. These

profiles show that the cylinder is an obstacle which generated the instabilities in the flow

when the velocities are increasing.

a) b)

Figure 4: Dimensionless longitudinal velocity profiles : a) Y+

= -1.5, b) Y+

= +1.5.

III.2 Thermal field The figures 5a, 5b, 5c, 5d, 5e and 5f, represented the flow thermal field, principally the area

of the linear source of heat, for the different Reynolds numbers. When the Reynolds number

is increasing, the heat propagation is decreasing. The heat reached the position (X+, Y

+) =

(+9, ± 0.25), for Re = 63, while it’s reached a position less than (X+, Y

+) = (+8, 0.02), for Re

= 900. This give a difference of (∆X+, ∆Y

+) = (+1, 0.23), on the thickness of the thermal

field. This strongly diminution shows the incapacity of thermal field to have more resistance

when the flow becomes more than more turbulent. This means that the Reynolds numbers

increased the passive scalar dispersion in the turbulent flow (Tcheukam-Toko and al., [19]).

The figures 6 bellow represented the longitudinal temperature profiles for different Reynolds

number at a certain positions around of cylinder. Theses profiles reveals the existence of

symmetry between the temperatures evolutions with the origin axis Y+ = 0. For the low

Reynolds number (Re < 504), the temperature of linear source of heat remains higher along a

large distance, then his value decreased from the position X+ = 16, where it’s not changing,

and evolve longitudinally to his minima value. For the higher Reynolds numbers, the

temperature of linear source of heat remains weak and stays minima as from the position X+

= 18, where it’s longitudinally evolve. This shows that the passive scalar total dispersion is

developing between the position X+ = 7 (linear source of heat position), and the position X

+ =

20 (from 40 mm of cylinder and from 26 mm of the linear source of heat). For the higher

Reynolds number, the longitudinal flow is predominating and the linear source of heat

remains weak.



15

a)

b)

c)

d)

e)

f)

Figure 5: Dimensionless Temperature Iso-value.

a) Re = 63, b): Re = 126, c): Re = 252, d): Re = 504, e): Re = 700, f): Re = 900.



16

a)

b)

c)

d)

Figure 6: Dimensionless longitudinal temperature profiles.

a): Y+= -1, b) Y

+= +1. c) Y

+= -1.5, d) Y

+= +1.5

III.3 Comparison of numerical and experimental results To valid our results, we compared the dimensionless mean temperature gaps profiles

(∆T+), and transversal velocity – temperature correlation profiles (<v’T’>+), with the experimental

results.

The figure 7 bellow, shows that the dimensionless mean temperature gaps profiles are in

accordance with the experimental result when Re = 63. These accordance are more important for

∆X+ = 1. The figure 8 shows a similar accordance for the positions ∆X+ = 2, and ∆X+ = 4.

The figure 9 shows that the comparison of transversal velocity – temperature correlation profiles

(<v’T’>+), with experimental data, is also satisfactory.

a)

b)

Figure 7: Comparison of dimensionless mean temperature gaps profiles numerical and

experimental for Re = 63. a): at ∆X+

= 1, b): at ∆X+

= 16.

International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976

6340(Print), ISSN 0976 – 6359(Online) Volume

Figure 8: Comparison of dimensionless mean temperature

Numerical (multicolor)

Figure 9: Comparison of transversal velocity

for Re = 63: numerical (multi color) and experimental (black on white).

Figure 10: Comparison of transversal flux of heat profiles in

temperature.

International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976

6359(Online) Volume 4, Issue 1, Jan - Feb (2013) © IAEME

17

Figure 8: Comparison of dimensionless mean temperature gaps profiles.

umerical (multicolor) and Experimental (black)

transversal velocity – temperature correlation profiles (<v’T’>

: numerical (multi color) and experimental (black on white).

Figure 10: Comparison of transversal flux of heat profiles in function of transversal gradient of mean

temperature. –v’+T’

+ = f((dT/dY)

+), at the position X

+ = 9


) © IAEME

profiles (<v’T’>+)

function of transversal gradient of mean



18

The Richardson number is very low for our all simulations Ri << 1. This means that, the

forced convection predominated. Moreover, the dynamic perturbation and the gravity effects

are very low, because of linear source Reynolds number and Ri which are respectively less

than 1 and 10-3

, as approved by Lecordier and al., [5], and Godard and al., [17]. The

calculated value of Peclet number confirmed that, the heat exchanges are only by convection

(Pe >> 1), as in the studies carried out by par Husson, [18].

Until present, the existence of counter-gradient zones was, observed in the heated flows,

showing the dissymmetry of the velocity and temperature profiles, characterized by a

minimum or a maximum (FuIachier and al. [11], Sreenivasan and al. [12], Veeravalli and

Warhaft, [2]). The counter-gradient observed when the linear source is placed on the

central line of the vortex street, shows that, the dissymmetry of the velocity and mean

temperature profiles is not his necessary condition of existence. This last is depending at

the same time, of the fluctuations form v/u, of the location of the source of heat, and of

the thickness of the linear source (Paranthoën and al., [16]). In these conditions, the heat

emitted by the source of much localized fashion, undergo a preferential convection in

these two corresponding directions. These shows the presence of a maxima, observed on

the mean temperature profiles which has a symmetry position with the central line. This

could not be the same case if the prevision density of dynamic field parameters were

Gaussian.

The counter-gradient is coming out from a simply situation where the small dimensions

of heated fluid zones (relatively at velocity field scale), are carrying preferentially in

some directions different to the principal flow. In this case, the heat flux created

downstream from a linear source and the mean temperature profiles are not still

compatible with the transit by gradient. This variation can be dissymmetric as in the

experimental works carried out by Veeravalli & Warhaft, [2], or can be symmetry as in

this present numerical work. The necessary condition is that, this variation must be

maximal for one or many values different of zero at the position of air injection.

VI. CONCLUSIONS

These results reveal that, the stability of wake zone is influenced by the behavior of

the physicals properties in function of temperature and of the geometry configuration

considered. In fact, these show that, the thermal field is strongly influenced by the vortex

street. The diffusion process seems to be in two phases connected to the filling time of

Vortexes Street. Moreover, in this case where the mean temperature profiles is generated by

the thermal transfer; we could rather name these counter-gradient zones by “counter-flux

mean temperature profiles”. The different comparisons makes between the numerical and

experimental profiles are satisfactory, but the difference observed, is located at the maximal

level. In perspectives, it would be interesting to associate the heated air jets to this present

study, in order to analyze their influence on the stability of thermal and dynamic fields.

ACKNOWLEDGEMENT

The authors acknowledge the CORIA UMR 6614 CNRS University of Rouen-France,

and The University of Ngaoundere, Cameroon.



19

NOMENCLATURE

Small letters x longitudinal coordinate (m)

y vertical coordinates (m)

Capital letters

D Cylinder diameter (m)

d source of hat diameter (m)

P Pressure (Pa)

T Temperature (K)

u,v velocities components (m/s)

∆Tréf Temperature difference between heat source and the ambient domain (K)

0x longitudinal axis

0y vertical axis

Greek symbols

ν Kinetic viscosity of air (m2.s

-1)

3 Dynamic viscosity of air (Pa.s)

� Dissipation ratio of the turbulent kinetic energy

K Turbulent kinetic energy (J.kg-1

)

Volume mass (m3.s

-1)

No Dimensional numbers Re Reynolds number

w�� Turbulent Prandtl number

Res Reynolds number of the linear source

Grs Grashof number of the heat linear source

Gr Grashof number

σk and σε Turbulent Prandtl number relative to k and �

Exponents, indices and specials characters

+ Dimensionless values (with D for the lengths) and (with ∆Tréf for the Temperatures)

Cp Thermal capacity at constant pressure

3� Turbulent viscosity

� Thermal conductivity

��++ Effective thermal conductivity

�z�� ++ Effective Newtonians tensor of viscous constraints

x relative to the longitudinal component

y relative to the vertical component

eff effective

n�� Stress ratio of mean Tensor

3� Turbulent kinematic viscosity

�� Turbulent dynamic viscosity

�,�+ Referenciel turbulent dynamic viscosity

σij Stress tensor



20

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21

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Fins” International Journal of Mechanical Engineering & Technology (IJMET), Volume1,

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[21] Cherian Paul and Parvathy Venugopal, “Modelling Of Interfacial Heat Transfer

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International Journal of Mechanical Engineering & Technology (IJMET), Volume1, Issue1,

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[22] Kavitha T , Rajendran A , Durairajan A and Shanmugam A, “Heat Transfer

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Journal of Mechanical Engineering & Technology (IJMET), Volume3, Issue 2, 2012,

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[23] Er. Pardeep Kumar, Manoj Sain and Shweta Tripathi, “Enhancement Of Heat Transfer

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Mechanical Engineering & Technology (IJMET), Volume3, Issue 2, 2012, pp. 85 - 91,

Published by IAEME

[26] Ashok Tukaram Pise and Umesh Vandeorao Awasarmol, “Investigation Of

Enhancement Of Natural Convection Heat Transfer From Engine Cylinder With Permeable

Fins” International Journal of Mechanical Engineering & Technology (IJMET), Volume1,

Issue1, 2010, pp. 238 - 247, Published by IAEME

study of turbulent flow downstream from a

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