Indian Journal of Engineering & Materials Sciences Vol. 8, August 2001, pp. 189-197

Experimental and numerical investigation of an axisymmetric freejet

K B S N Viswanath & V Ganesan

Department of Mechanical Engineering,

Indian Institute of Technology Madras, Chennai 600 036, India

Received 9 May 2000; accepted 17 May 2001

The experimental and numerical investigations of an axisymmetric jet issuing from a nozzle are reported here. Experiments have been carried out using hot-wire anemometer and theoretical predictions made using the finite volume technique. The k-E models, with modified constants to take care of free shear flows, have been used for the physical modelling. The present investigations are aimed at comparing the predicted results with reported experimental measurements. The present numerical method along with k-E model has been found to predict the flow quite well. Therefore, the k-E model can be used with confidence for the evaluation of flow characteristics of similar turbulent flows.

Jet flow studies have been an interesting pursuit to research community for the past several decades owing to their ubiquitous occurrence in natural and engineering hydrodynamic systems. For more than a century, the theory of turbulent jets and its practical applications have attracted the attention of specialists working not only in the field of hydro-mechanics but also in such branches of science as oceanology, meteorology, and physics of the atmosphere. The axisymmetric jet represents a benchmark for research into the physics of turbulent fluid flow.

The vast quantum of knowledge presently available and the impetuous research currently being carried out, stand testimony to the importance associated with jet flows. The major reason for this sustained endeavour is the need for a proper understanding of free turbulent shear flows - the class of flows to which jets belong.

Different types of axisymmetric jets have been investigated in the past. For example, jet emanating from a nozzle, jet emanating from an orifice, jet issuing into a confined surrounding, and jet without any enclosure, have attracted the attention of researchers worldwide. The experimental investigations have been carried out using different techniques like Stationary Hot Wire Anemometry (SHW A), Flying Hot Wire Anemometry (FHW A), and Laser Doppler Anemometry (LDA), etc.

Weisgraber and Liepmann 1 studied the developing turbulent region of a round jet, using an improved implementation of Digital Particle Image Velocimetry (DPIV). The two-dimensional flow field in planes

normal and parallel to the axial velocity was measured at locations between 15 and 30 diameters downstream, for two Reynolds numbers of 5500 and 16,000. Their study consisted of instantaneous snapshots of the velocity correlations up to third order. Their examination of the turbulent velocity and vorticity fields revealed the dependence of the size of the fine scale structure on the Reynolds number. They found that the Reynolds number affected the rate of development of jet and shape of the mean velocity profiles. Profiles for Reynolds number of 16,000 were more characteristic of the profiles in the self-similar region, while the profiles of the Reynolds number of 5500 still retained a top hat shape near the centreline. They observed that the average velocity profiles at both Reynolds numbers exhibited the most similarity and the second and third order turbulent correlation ' s were still developing and increasing in magnitude in the downstream direction. They also observed lhat in the transitional region, some of the turbulent kinetic energy is still localized in the shear layer as was evident by the strong off-axis peaks in the profiles of the turbulent energy components. They concluded that DPIV provides the capability to study flows with geometries that are more complex and regions where the flow is not self-similar.

Boguslawki and Popiee took measurements of radial and axial distributions of mean velocity, turbulent intensities and kinetic energy as well as radial distributions of the turbulent shear stress in the initial region of a turbulent air jet issuing from a long round pipe into still air. The jet was transformed

190 INDIAN J ENG. MATER. SCI., AUGUST 2001

relatively smoothly into a jet flow. They found that in the core subregion, the mean centreline velocity decreases slightly and at distances beyond 8d the velocity profile becomes self-similar. In their investigations, they deduced the rate of velocity spread rate and entrainment coefficient. Within a distance of 4 to 8d from the nozzle, they observed rapid increase in the turbulence intensity. The highest turbulence level occurred at an axial distance of about 6d and radius of 0.7d to 0.8d. They attributed this to the extension of the mixing sub region over the whole cross-section of the jet. The radial profiles of the turbulent kinetic energy and shear stress reached a self-preserving form beyond a distance of about 75d and 60d respectively.

Pope3 investigated the turbulent round jet and plane jet and observed that with the values of constants appropriate to boundary layer flows, the velocity field in a two-dimensional plane jet was calculated quite accurately, but large errors occurred for axisymmetric jets. Specifically, the spreading rate of round jet was overestimated by about 40%. Experimental data indicated that the round jet spreads about 15% less rapidly than the plane jet, while its predicted spreading rate was 15% greater. An explanation was given to the anomaly, as in 2-d flows no vortex stretching can take place since the mean vorticity vector is normal to the plane of the flow ; for an axisymmetric jet, however, as the jet spreads rings and vortices are stretched. Accordingly, this causes the effective viscosity and hence the spreading rate is found to be lower in the round jet than in plane jets. This was an experimental observation. This made him suggest a modification to one of the constants of k-E model, i.e., CE3=0.79, which reproduced the measured spreadi ng rate and velocity profile predictions also showed an excellent agreement.

Rodi4 has discussed the merits and demerits of the turbulence models applied to free jets. According to him, the mixing length model has been widely used with considerable success for calculations of simple shear layers. However, he argues, that such a simple model like mixing length is not suitable whenever turbulence transport and history effects are important, and it is of little use for flows more complex than shear layers because of the great difficulties in specifying the mixing length distribution in such flows. One-equation models employing a transport equation fo r the kinetic energy of turbulence account for transport and history effects and hence they are

superior to the mixing length models for such flow conditions where the length-scale distribution can be prescribed realistically. They are, however, not very suitable for complex flows. The k-E model has been tested most widely and has been shown to predict many different flows with accuracy sufficient for practical purposes 15

• However, the use of an isotropic eddy viscosity in the k-E model does not describe certain important flow phenomena. Stress equation models simulate the turbulence processes more realistically and are capable of describing many of the features that defy simulation with an isotropic eddy viscosity model. However, they are rather complex and computationally expensive so that they are not very suitable for practical applications.

From the review of literature, it is seen that measurement of flow and turbulence quantities in a free jet are scanty. Further, it is also observed that the theoretical investigations are also very few. Hence, it is proposed to make measurements of both mean and turbulence quantities in this study. Further it is also proposed to numerically predict the flow field and compare the predicted results with experiments.

The present investigations are aimed at---(1) To study the flow field of the free jet by measuring the mean velocities in axial and radial direction using a hot-wire anemometer with a normal probe; (2) To investigate the turbulence parameters like longitudinal turbulence intensities, transverse intensities and Reynolds shear stress components, for three Reynolds numbers; (3) To make numerical predictions of the flow field using the finite volume technique and k-E turbulence model , and (4) To compare the predicted results with measurements and also with results reported in the literature.

Experimental Procedure The complete experimental set-up consists of a

wind tunnel, traversing mechanism, and constant temperature hot-wire anemometer. The details of these equipments and instruments are:

Wind Tunnel-It consists of a blower; a settling chamber with filters, a bell mouthed nozzle extending to a pipe and a convergent nozzle giving low turbulence flow. The outer diameter of the nozzle is 57mm. The velocity variation at the exit of nozzle is controlled by means of a throttle control at the exit of the blower.

Hot-wire anemometer-The anemometer basically works on the principle of convective heat transfer to

VISWANATH & GANES AN : INVESTIGATIONS OF AN AXISYMMETRIC FREE JET 191

the flow medium from an electrically heated thin wire. The magnitude of the heat transfer is dependent on a number of parameters such as properties of the wire, geometry of the wire, temperature, pressure and velocity of the medium. The present measurements were made using a constant temperature hot wire anemometer.

Measurement techniques The experimental procedure regarding the

calibration and linearisation of the probe is: Calibration of the wind tunnel - It was calibrated

first with the help of a pitot static tube and a micro manometer.

Calibration of the hot-wire probe-Calibration of the probe inter-links the input energy to the wire and the heat carried away by the flow medium.

Linearisation of hot-wire signals - The output from a constant temperature hot wire anemometer is a non-linear function of the velocity of the fluid passing over the probe. The measurements in highly turbulent flows are, therefo re, subject to errors. To reduce these errors, it is necessary to linearise the CT A signals. Therefore, a lineariser was used for this purpose.

Single wire measurements - The hot-wire probe used is DISA 55PJJ miniature single wire probe of 1.25 mm length made of platinum coated tungsten wi re of 2.5 microns diameter. The probe is connected to a DISA56C 17 anemometer operated in constant temperature mode. The mean velocity variations in the axial and radial directions have been measured using this probe.

Cross-wire measurements - The cross-wires are essentially designed for the measurement of the trans,erse components of velocity fluctuations . They can also be adopted for the determination of mean veloci ty components together with the turbulence intensi ties and shear stresses.

Numerical Prediction Method With the advent of high-speed digital computer and

improvement in the understanding of numerical techniques in recent years, there has been a major impact in the evaluation of complex flows and achieving their solution.

Computational fluid dynamics is a bridge between the experimental and theoretical studies. It endeavours to replace the governing partial differential equations of fluid flow with manageable algebraic equations and to obtain the solution as numbers and advancing these

numbers in space and time to obtain final numerical description of the flow field of interest. Numerical solution procedure requires the repetitive manipulation of the thousands or even millions of numbers. Therefore, advances in numerical procedure and its application to problem to obtain more and more details and sophistication are intimately related to advances in computer hardware, particularly with regard to storage and execution of speed.

The complete form of the equation governing the conservation of mass and momentum, in the time averaged form using Cartesian tensor notation can be written as5 Eqs (1) and (2) (Continuity/Momentum

Equations):

a(ur) + a(vr) = 0 ax ar

.. . (I)

. .. (2) where u and v are the velocities in the longitudinal

and radial directions, respectively, and v, is eddy

viscosity. The flow field details are obtained by solving the

continuity and momentum equations in a given domain of interest. The velocity component is governed by momentum equations. However, the real difficulty in the calculation of the velocity field lies in the unknown pressure field . The pressure gradient forms a part of the source term for a momentum equation. The difficulty associated with the determination of pressure has led to several methods in solving pressure from the governing equation. One such procedure, which widely adopted, is called SIMPLE, which stands for Semi-Implicit Method for Pressure Linked Equations. The solution procedure of this algori thm is that a pressure field is initially guessed and the corresponding velocity field is computed using momentum equation. On substituting these velocity fields in the continuity equation, a mass balance error arises, which is used to get pressure correction by solving the continuity equation written in terms of pressure correction using the corrected pressure, the above steps are repeated till converged solution is obtained.

The package used for the numerical prediction is FLUENT. This computer code has the capability for the numerical simulation of fluid flow, heat transfer.

192 INDIAN J ENG. MATER. SCI., AUGUST 2001

software, which can solve the conservation equations for mass, momentum, energy and species using a control volume based finite difference method. It has capabilities for incoq)orating various turbulence models.

Boundary conditions

The present problem has three types of boundaries. They are inlet, outlet, axis, and wall boundary. The appropriate boundary conditions have been prescribed, e.g.,'Outlet' - The boundary condition chosen for all the outlets was pressure outlet, which enables us to specify the static pressure at the outlet boundary. It also requires us to specify the turbulence parameters. There are three outlets in the present problem, so the static pressures and turbulent parameters have to be prescribed at all these three outlets.

Grid generation One of the most important steps required for

accurately solving a computational fluid dynamic problem using any of the discretisation procedures involves the proper location of nodal points in the flow region under considerations. The grid is generated with the help of a subroutine called GAMBIT. The mesh is conceptually made up of triangular bock grids (in mapped space) with each block corresponding to an important flow zone where independent mesh size control is required. Since the free jet is axisymmetric about the central axis, the cross section is discretised into an unstructured grid by mapping different parts of the geometry into a computational domain with triangular elements. A typical grid distribution of 25000 cells is used. A grid independence test between 20000 and 30000 cells was carried out and 25000 cells were found adequate.

Turbulence Model

In the present study, the two-equation, (k-E) turbulence model6

, which is based on the generalised Boussinesq eddy viscosity concepe is employed. This model uses two partial differential equations to estimate the turbulence kinetic energy (k) and turbulence kinetic energy dissipation rate (E) and )11 is the turbulent viscosity that may be related to the turbulence kinetic energy (k) and its dissipation

rate (E) by dimensional analysis. Thus )11 can be described as:

Jlt = cJjpeiE ... (3)

where pis density. The values of various constants used in k-E model are C1 = 1.44, C2 =1.92 and CJj = 0.09. Moreover, the turbulent Prandtl numbers for k and E are given by ak =0 •775 and af =0• 849, respectively.

In this study, predictions have been carried out for the axisymmetric free jet issuing from the wind tunnel by incorporating the details such as input conditions and geometry, etc from the experimental measurements. Profiles of mean velocity in axial and radial directions and turbulent kinetic energy variations are predicted. The k-E model is chosen in the present case in view of its success in turbulent flow predictions4

.

Results and Discussion The numerical and experimental investigations

have been carried out for an axisymmetric free jet issuing into free space. Three nozzle exit velocities (NEV) at the exit have been considered, viz. of 10 rnls, 20 rnls, and 30 rnls. The corresponding Reynolds numbers are 3.8 x 104

, 7.6 x . 104, 11.4 x 104

,

respectively. The results presented here are for the Reynolds number of 7.6 x 104

. In all the figures , symbols denote experimental results whereas the solid line denotes prediction .

The central core of the jet where the exit velocity is preserved up to some axial distance is called as potential core of the jet. The formation of the

Od

E' S O.OQ

>-

0.00 0.50 1.00

Fig.! -Axial velocity profiles at various xld showing potential core

VISWANATH & GANESAN: INVESTIGATIONS OF AN AXISYMMETRIC FREE JET 193

potential core in the jet is as shown in Fig. 1. One of the characteristic features of a free jet is its expansion and growth due to entrainment from ambient surroundings. The expansion of the jet from the exit of nozzle is shown in Fig. 2. It may be seen that predicted values agree quite well with experimental data.

The decay of mean velocity along the jet centreline at different stations downstream, from x/d = 0 to x/d =16, is shown in Fig. 3. As can be seen from Fig. 3, the velocity remains constant along the centreline up to around 5d and then starts decreasing with increasing x/d. The velocity, which is close to the exit velocity and remains constant in y direction, gives the map of potential core and the core is found to extend up to 5d in this study. In these plots, the centreline velocity has been normalised by the velocity at the jet

0.0 1.0 o.o U/Um 1.0 0.0 1.0

Fig. 2 - Radial distributions of axial velocity at various xld for 20 m/s

0

2 E

::>

1.20 --,-------------~

experimental

0.40 -

.... ~\ --- numerical

~

'~ <>..____--------

0.80 -- NEV=20m/s

o.oo -+----,..----~r--~-----"

0.00 10.00 xld

Fig. 3 -Centrel ine decay

20.00

exit. The axial distance has been normalised by the nozzle diameter. It is customary to find the decay constant in free jet in order to understand how fast the exit velocity is decaying. In order to get decay

U X constant, usually a plot of _!!!_ versus - is drawn. As

uo d can be seen from Fig. 4, the variation is found to be linear according to the relation: .

u/11 = i ~-~)-1 uo '1 d d

.. . (4) The distance between the origin and x-intercept of

the straight line is called the virtual origin. Virtual origin indicates the distance up to which centreline velocity is constant. Initially the central core of the jet

E :J 0 :J

4.00 ~--------

3.00

2.00

4.00

linear fit( numerical)

linear fit( experimental)

experimental

. NEV=20m/s

8.00 12.00 xld

Fig. 4- Virtual origin

16.00

+ xld•15

0 xld•12

<> lld•8

gaussian

-4.00 -2·00 0.00 2.00 I 4.00

r/r.5 Fig. 5 -Self-similarity plot for NEV = 20 m/s

194 INDIAN J ENG. MATER. SCI., AUGUST 2001

the jet exit value. It is found to be around 5d, which agrees well with that of Hussein et a/.9

In Eq. (4), A is called the decay constant of the centreline velocity and x01 is kinematic virtual origin. The decay constant depends on the exit conditions and is calculated from the Eq. (4). The value obtained in the present investigation is 5.78. It is grati fying to note that the value agrees well with the value of 5.8, found by Rodi4

, 5.95 by Abdel Rahman et a/. 8, 5.9 by

Hussein et a/.9 and 5.7 reported by Wygnanski and Fiedler10

.

The mean velocity profiles usually become self­similar at axial distances beyond 8d where dis nozzle exit diameter. Fig. 5 shows the similarity of the velocity profiles in all sections of the principal area of a round jet. Thus, at corresponding points on any two cross-sections of the main part of jet, the non­dimensional velocities are the same. The data are normalised by the centreline mean velocity U111 and profile half-width r0.5. The velocity profile half-width is the radial distance from the jet axis to a point where the velocity is half its centreline value. The theoretical profiles closely follow the experimental profiles. Mean velocity profiles illustrate that the jet flow is self-similar.

The amount of mixing that the free jet experiences with ambient surroundings, influences the spread rate of the jet. The variation of jet half-width with x/d is shown in Fig. 6. It can be seen from the figure that the half-width is increasing with downstream distance indicating the expansion of jet due to entrainment of ambient mass into itself. The jet half-width (r0 5)

2.00 1-~ linear fit( experimental)

I 1J experimental

l--- linear fit(numerical)~;

~ 1.00 _ iNEV=20m/s /

/

~/ I ,/-

0.00 ¥~~~-----,-- -----.---,----0.00 5.00 10.00 15.00

xld

Fig. 6- Half-width plot along axial direction

variation in the streamwise direction can be given by8:

ro.s _ {X Xoz ) -- ---d d d

... (5) where b is spread rate of the jet half-width and x02

is the geometrical jet virtual origin. Spread rate again depends on the exit conditions and the conditions of ambient surroundings and the value was found to be around 0.11, which shows good agreement with 0.097 of Abdel Rahman et a/.8

, 0.102 by Hussein et al.9

One of the basic characteristics of the jet is entrainment due to which there is a jet spread. When the fluid at the periphery comes in contact with the ambient fluid, the shear due to their difference in velocities causes the peripheral layer (shear layer) to roll up into a vortical structure, which during the process of roll-up, scoops ambient mass into itself. This phenomenon causes mass addition into the jet with axial distance and is called entrainment 11

•

Direct numerical integration of the mass flux data yields the total mass flux of air at that section which when compared to the initial mass flux of the jet is a measure of the entrainment. The mass flow rate rh is known to increase with distance x from the nozzle. The variation of kinematic mass flux wi th x/d is shown in Fig. 7. The coefficient of entrainment E is given by8

:

E=~ (6) d(xl d) .. ·

0 E e

6.00 -.'-----~~~~~~~---

4.00 • linear fit( numerical)

linear fit( experimental) / . 0 •• / ·

~"/* /

2.00 ,/ NEV=20m/s

0.00 5.00 10.00 15.00 xld

Fig. 7- Variation of mass flux

VISWANATH & GANESAN : INVESTIGATIONS OF AN AXISYMM ETRIC FREE JET 195

The ratio of mass flux at a section to the mass flux

m at the nozzle exit is found to be - =0.37xld. Thi s

constant of proportionality is independent of Reynolds number12 beyond 2.5x 104 and entrainment rate is slightly hi gher at lower Reynolds numbers. The value of the entrainment coeffi cient E found for free jet of the present study is in good agreement with reported values; 0.32 according to Ricou and Spalding 12

, as well as Hussein et al.9 An important test of any experimental data is to check whether it sati sfi es the equations believed to govern the flow9.0ne of the most important tests in fact for an ax isy mmetric jet in an infinite environment is whether the ve locity moment profiles sati sfy the momentum integral equation.

In a free isothermal round jet, kinematic momentum flu x across the jet, M, is conserved and is equal to its value at the je t exit M0 and is given bl:

:0 = 1:2 ('~' J'( ~'J . .. (7)

The trend in variation of momentum flu x is shown in Fig. 8. The total momentum flux is conserved within deviations of less than 9% of the initial value. As the pressure fie ld is assumed to be constant throughout the fl ow fi eld momentum flux should be constant. But, there exists a small negati ve pressure fi eld directly beneath the nozzle resulting in enhanced entrainment and a fall in mo mentum. The assumption of constant momentum flux is in apparent agreement with some experiments 13 whereas other measurements 14 found a slow decrease in momentum

2 . 00~ I e experimental

I

-- numerical

NEV=20m/s I

0.00 -t~---.--,-· --r-~~ 4.00 8.00 12.00 16.00

xld Fig. 8- Vari ation of momentum fl ux

-4 .00 -2.00 0.00 r/r.S

~ experimental

~ Boguwalski [1979]

present prediciton

4.00

Fig. 9a - Radial distribution of mean velocity at x/d fo r NEV = 20 m/s

4 .00 -2.00 0.00 r/r.5

Boguwalski [1979]

George [1994]

present predcition

experimental

4.00

4

Fig. 9b - Radial d istribution of mean velocity at xld = 8 fo r NEV = 20 rn!s

E :::> 3

-4.00 0.00 r/r.5

~

EB ~

experimental

present predcition

George [1994)

Boguwalski [1979]

4.00

Fig. 9c - Radi al distribution of mean velocity radi al direction at xld = 12 for NEV = 20 rn!s

196 INDIAN J ENG. MATER. SCI., AUGUST 2001

30.00 .• - -. -·------- -

numerical J

~ experimental

~ 2000 (~'\ f E

::::>

~ ""' I I o o~ ~¢

000 -r- r· --- -r-~ 4 r- -~00 ~00 400

r/r.5 Fig. lOa - Kinetic energy di stribution at x/d = 4 forNEY= 20 m/s

40.oo -r-- -··--. ----·----

a t ~~() ~ ~.00 ~r~ ., ,~ ~ \

I ~

numerical

experimental

0.00 -~-- -r- ---~~~--,---~00 ~00 400

r/r.5

Fig. lOb- Kinetic energy di stribution at x/d = 8 for NEV = 20 m/s

0 0 0 .­•

20.00 -

~ ~ i moo I

I 000 -t-

0.00 2.00 r/r.S

·-·r----

4.00

Fig. I Oc- Kinetic energy distribution at x/d = 12 forN EY= 20 m/s

distance from nozzle. Hussein et a/.9 attributed the wide variation of M!M0 found in the literature between 0.6 and 1.6 as evidence of the strong

influences of the jet experimental set-up and conditions on the jet momentum data.

In order to compare the present measurements and also to validate the predicted results velocity profiles at xld stations 4, 8 and 12 have been plotted along with that of Boguwalski and Popiel2 and Hussei n et a/. 9 experimental data. It is quite gratifying to note that the profiles are matching quite well as can be seen from Figs 9a-9c.

The variation of turbulent kinetic energy in radial direction is shown for the nozzle exi t velocity of 20m/s. The data are normalised with efflux velocity U111

2 and radial half-width. Figs. I Oa-lOc show the radial distribution of turbulent kinetic energy at different xld stations. The highest value of turbulent kinetic energy seems to occur at an axial distance of 8d and radius of (0.7 to 1.0) r0.5. The d ip near the axis occurs because of weak turbulence near the centre, which is expected, and the dip seems to recover as xld increases. The measurement also confirms the phenomenon. Further, in the developed region of the round jet, the profile of turbulent kinetic energy does not show a dip because the production of turbul ent kinetic energy is relatively large at the centreline. It is

also gratifying to note that the k-E model is able to handle such flows quite satisfactori ly .

Conclusions The k-E model is found to be a satisfactory physical

model to g ive good predictions for various flow characteristi cs of an axisymmetric free jet issuing into atmosphere. Most of the features of a free shear flow can be predicted with minor adjustments to constants in the model.

The reported calibration method and the measurement techniques used in calculating the various turbulence characteristics from the hot-wire data have worked well for the experimental investigations. The computational results clearly demonstrate the ability of finite volume method using SIMPLE procedure along with the k-E turbulence model in predicting the flow fi eld. The mean velocity profiles show self-similarity starting from ax ial di stance of 8d, which is in good agreement with other experimental results . Momentum flux variation with xld is found to be more or less constant.

Thus, the numerical method employed in the present investigations, together with k-E model can be used with confidence for the evaluation and the analysis of flow characteristics of similar turbulent fl ows.

YISWANATH & GANESAN : INVESTIGATIONS OF AN AXISYMMETRIC FREE JET 197

Nomenclature d = exit di ameter of nozzle k = turbulence kinetic energy

111 = mass flux variation in ax ial direc tion

1110

mass flux vari at ion at the ex it

M momentum flux in ax ial directio n M 0 momentum flu x at the ex it NEY = nozzle exi t ve loc ity U = mean velocity in radial direc tion Um = mean centreline velocity Un = ex it ve locity of nozzle £ = turbulence kinetic energy dissipation rate p = density Jl = effective viscosity Jl, turbulent viscosity }11 = laminar viscosity

References I Weisgraber T H & Liepmann D, Expts Fluid, 24 ( 1998) 210-

224.

2 Boguslawski L & Popiel C Z 0 , J Fluid Mech, 90 (1979) 531-539.

3 Pope S B, AIAA J, 16 ( 1978) 279-281.

4 Rodi W, DISA lnf, 17 (1975) 9-18.

5 Tulapurkara G E, NAL (India), Report No. SP 8715.

6 Launder B E & Spalding D B, Comput Method Appl Mech Eng, 3 (1974) 269-289.

7 Hinze J 0 , Turbulence (McGraw- Hill , New York), 1975. 8 Abdel-Rahman A, Chakroun W & Al-Fahed S F, Mech Res

Commun, 24 ( 1997) 277-288 . 9 Hussein J H, Steven P C & George W K, J Fluid Mech. 258

(1994) 3 1-75. 10 Wygnanski I & Fiedler H, J Fluid Mech, 38 ( 1969) 577-612. II Srinivasan K, Flow and noise characteristics of non-circular

j ets, Ph .D Thesis, Indian Inst itute of Technology. Kanpur. 1998.

12 Ricou F P & Spalding DB, J Fluid Mech, II ( 196 1) 21-32. 13 Bradbury L J S, J Fluid Mech, 23 ( 1965) 3 1-63. 14 Kotsovinos N E, J Fluid Mech, 87 (1978) 55-63. 15 Graham L J W & Bremhorst K, ASM£ J Fluid Eng , 115

(1993) 70-74.