Smith Mi Nathan Dally

7/31/2019 Smith Mi Nathan Dally

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The "Round Jet Inflow-Condition Anomaly" for the k-ε

Turbulence Model

Elizabeth Smith1, Jianchun Mi, Graham Nathan and Bassam Dally

School of Mechanical Engineering

The University of Adelaide, SA, 5005 AUSTRALIA

Abstract

The capability of the k-ε (epsilon) Turbulence model to predict the influence on

the downstream flow caused by variations to the nozzle source flow in turbulent round

jets is assessed. The numerical model is compared with previously published

experimental data for three jets issuing from a smooth contraction, a sharp-edged orifice

plate and a long pipe, respectively. It is found that the predicted trends in the rates of

spread and decay caused by changes to nozzle type, and hence the jet source flow, are

the opposite from those obtained by experiments. Likewise key aspects of the near-field

flow are predicted incorrectly, and the trends in the RMS field are also in error. These

errors are consistently traced to the simplifying assumptions by which two-equation

models assume homogeneous isotropic turbulence. In contrast, coherent large-scale

structures are known to exist in these three flows, at least in the near field and propagate

downstream. The results also show that the turbulence coefficients used in the k-ε

turbulence model for a round jet are not sufficient to account for this discrepancy.



1. Introduction

Townsend [1] argued that turbulent flows should achieve true self-similarity

when they become asymptotically independent of initial conditions. This has led to the

hypothesis that `turbulence forgets its origins .̀ However, the analytical results of

George [2], and subsequently George and Davidson [3], dispelled this hypothesis by

showing that the entire flow is influenced by the initial (or upstream) conditions,

resulting in a variety of initial-condition-dependent self-similar states in the far field.

George’s analytical work is supported by experiments [4-7]. Mi et al. [4 & 5] compared

downstream scaling mixing characteristics for round jets issuing from a smooth

contraction (SC) nozzle, a sharp edged orifice plate (OP) and a long pipe (LP). Xu and

Antonia [6] compared the downstream velocity decay between round jets issuing from a

LP and SC nozzle. Mi et al. [7] compared velocity decay between round jets issuing

from a LP, SC and OP. These investigators concluded that differences seen in the

downstream decay are directly related to the underlying turbulence structure of the jet,

which propagate downstream from the nozzle exit. Those initial conditions known to

affect the downstream characteristics include the Reynolds number, Re, and the initial

turbulence field, as characterised by the nozzle exit radial profiles of mean velocity and

turbulence intensity, and density ratio between jet and co-flow fluid. These above

experimental studies [4-7] showed that the flow emerging from the OP exhibited the

highest decay rate and the widest spreading angle, followed by the SC nozzle and then

the LP.



Stokes (RANS) models to reproduce known trends appears to have been performed.

The present paper seeks to fill this gap by assessing the flow fields predicted to emerge

from jets issuing from LP, SC and OP nozzles using the k-ε model.

2. Different Round Nozzle Flows

It is well established that the exit velocity profiles from the three types of round

nozzle are distinctly different, as shown schematically in Fig. 1. The radial velocity

profile (U(r)) from the SC nozzle is approximately uniform (i.e. "top hat") while that

from the LP is initially fully developed, and so is well described by the power law

velocity distribution. The radial velocity profile from the OP is described as "saddle

backed" with the highest velocity at the edge of the jet. The initial turbulence intensity

profiles (U’(r)/U cl) from each nozzle are also different. For the SC nozzle the mean

turbulence intensity is low (about 0.5%) except in the shear layer at the edge (r < 0.45d

where it increases to ~8%). In contrast, the relative turbulence intensity from the LP is

higher throughout the exit plane, especially in the shear layer and typically varies

between 3% to 9.5% [5]. The exit turbulence intensity of the OP is between these two

extremes.

The above differences in exit flow is influenced by differences in the flow

upstream from the exit planes. Firstly, the parallel walls of the LP nozzle results in

parallel mean flow, with a high intensity generated by the walls. The SC nozzle reduces

the intensity of the turbulence fluctuations and increases the uniformity of the mean



so that the initial flow has a slight radial inflow at the edge, resulting in the well-known

"vena contracta".

The emerging flow from a round jet nozzle can be divided into four regions [9]:

The core region where the centerline velocity is equal to the outlet velocity, a transition

region where velocity begins to decay, then a profile similarity region where the flow

will become self-similar, and independent of axial distance (Fig. 2). The fourth region

is the termination region where velocity rapidly decays. The core region, or the

“potential core” only exists for the SC nozzle, since this is the only nozzle which creates

an initial region of uniform velocity, and hence potential flow. However, due to its

wide usage to describe the near-field region of SC jets, we adopt it here, in parenthesis,

to refer to the near-field flow region for all three jets, upstream from the onset of

velocity decay.

3. k-εεεε Turbulence Model

Today, even with the successful development of DNS and LES (large eddy

simulation) for turbulent flows, the most popular models for turbulent round jet flows,

especially those at industrial scale and/or involving combustion, are the two-equation

Reynolds Averaged Navier Stokes (RANS) turbulence models. Of these, the k-ε two-

equation model accounts for 95% or more of the industrial usage at present [10]. This

form of model is easy to solve, converges relatively quickly, is numerically robust and



All two-equation models are based on the Boussinesq approximation, Eq. (1),

and the turbulent kinetic energy equation, Eq. (2). The Boussinesq approximation is

used to approximate the Reynolds stress tensors introduced by the Reynolds averaging

of the conservation equations (where isotropic turbulence is assumed). The turbulent

kinetic energy equation describes the physical processes of the turbulence throughout

the flow.

The second equation of the k-ε model, the specific dissipation rate equation, Eq.

(3), contains the dissipation rate, ε , which describes the rate of energy transfer from the

large energy containing scales, characterised by integral scales, to the smaller

dissipating scales, characterised by the Kolmogorov scales. Turbulent flows contain a

spectrum of length scales, the intensity and distribution of which, depends upon the

initial and boundary conditions.

ijijT ij k S δ υ τ 3

22 −=

∂

∂

+

∂

∂+−−

∂

∂=

∂

∂+

∂

∂

jk

T

j j

i

ij

j

j x

k

x D

x

U

x

k U

t

k

σ

υ υ ε τ

E x

k

xk f C

x

U

k C

xU

t j

T

j j

iij

j

j +

∂

∂

+

∂

∂+−

∂

∂=

∂

∂+

∂

∂

ε

ε ε σ

υ υ ε

ε τ

ε ε ε 221

The closure coefficients and auxiliary relations for the standard k-ε model are

defined by Launder et al. [11], where the empirical turbulence coefficients within the

dissipation rate term are defined as C ε 1=1.44 and C ε 2=1.92. However, C ε 1 and C ε 2 are

(1)

(2)

(3)



The standard k-ε model with the standard coefficients predicts quite accurately

the velocity field of a two-dimensional plane jet, but results in large errors for

axisymmetric round jets, where the spreading rate is overestimated by 40% [12]. This

"round-jet plane-jet anomaly" results from the numerous simplifying assumptions in all

RANS models, and is further evidence of the non-universality of turbulence. It is also

this work which prompted the title of the present investigation.

To tailor the k-ε model for solving round jet flows the turbulence coefficients

C ε 1 and C ε 2 can be modified. Modifications to the turbulence coefficients have been

suggested by McGuirk and Rodi [13], Morse [14], Launder et al. [11], and Pope [12].

All of these modifications involve the turbulence coefficients becoming functions of the

velocity decay rate and jet width. For self-similar round jets it was found that

modifications made by Morse [14], and Pope [12] lead to C ε 1 having a fixed value of

1.6.

To examine the impact of the modifications on the accuracy of the k-ε model

when used for self-similar round jets, Dally et al. [15] compared the use of the Morse

[14] and Pope [12] modifications with the standard k-ε coefficients (C ε 1 =1.44 and C ε 2

=1.92) and a fixed value for C ε 1 =1.6 with C ε 2 =1.92. It was found that the

modifications by Morse [14] and Pope [12] did improve the accuracy of the k-ε model

relative to the standard k-ε coefficients. However the fixed value of C ε 1 =1.6 with C ε 2

=1.92 matched the experimental results the closest. The k-ε model with C ε 1 =1.6 with



More recently, George and Davidson [3], regarded RANS models to be missing

the necessary physics to account for the differences in initial conditions, and the

spreading rate is entirely determined by the model coefficients C ε 1 and C ε 2. However,

up to now there has been no direct comparison of the performance of a k-ε model for

the above three types of round jet. Hence it is unknown whether the modified k-ε model

is suitable for all round jet flows, or if further model modification is required to predict

the flow from each class of round jet nozzle.

4. Numerical Method and Code Validation

The present numerical investigation was performed in a low velocity co-flow,

with a low ratio of co-flow velocity (U a) to bulk jet exit velocity (U b), U a /U b=2%, rather

than in ambient air, to provide more definitive boundary conditions. This co-flow

satisfies the velocity criterion of Maczynski [16] and Nickels and Perry [17] in which

the effect of a slight co-flow on the jet mixing is deemed to be negligible. As such, it

allows the calculations to be compared with the relevant experiments, since all direct

comparisons of the effect of varying a jet’s inflow conditions have been performed with

no co-flow.

The Reynolds number based on nozzle diameter, d (d = 9.45mm), and bulk

velocity at jet exit (U b) for all three nozzle types was Re = 28,200. Water was used as

the working fluid within the control volume and also as the fluid in the jet and co-flow,

thus the density ratio between jet (j) and co-flow (a) streams is j / a = 1 This allows



The computational domain extended 50 diameters upstream from the jet exit to

ensure fully developed pipe flow, for the pipe jet nozzle, and a developed co-flow. It

also extended 105 diameters downstream from the jet exit, to ensure capture of data in

the self-similar region, and 37 diameters in the radial direction to ensure that wall

effects are negligible. A schematic diagram of the computational domain is shown in

Fig. 3. Grid cells were placed closer together near to the jet walls and further apart with

increasing distance from the jet exit. The commercially available CFD program Fluent

6.1.22 was used for all calculations. Fluent uses a finite volume formulation over a

structured mesh.

The total number of nodes in the long pipe geometry is 303,012 and 299,113 in

the smooth contraction/orifice plate geometry, with 25 cells at the jet exit plane. Grid

independence was ensured for both geometries, such that the number of cells at the jet

exit plane and at the near field region did not impact the length of the potential core and

the mean rates of spread and decay of the emerging jet flow. A dense mesh was applied

to both geometries to ensure the entire flow area was captured with significant detail.

To further improve the numerical accuracy of the model, a second order upwind

differencing scheme was applied.

A steady state k-ε model was applied with 2-D axisymmetric assumption; the k-

ε model is modified for improved prediction of round jet flows by using the coefficients

C ε 1 = 1.6 and C ε 2 = 1.92 recommended by Dally et al. [15]. Convergence was

considered to be complete when the ratio of mass residuals to mass entering the jet was



mean and RMS mixture fraction. Temperature and density were under-relaxed to

prevent heat release so that temperature remained constant at 293K.

To obtain a fully developed pipe flow at the jet exit (Fig. 3) the flow was

initiated 50 diameters upstream from the jet exit. The resulting velocity profile at the

exit of the long pipe follows the power law velocity distribution described by Eq. (4),

where n=6.62 (corresponding to Re=28,200). The calculated turbulence intensity

profile of the LP is similar in trend, but slightly different in detail to the measured

profile of Mi et al. [5] (Fig. 4).

n

cld

r

U

U 1

21

−= (4)

For prediction of the downstream flow emerging from the OP and SC nozzles,

the in-flow boundary condition at the jet exit was fixed. The appropriate mean velocity

and turbulence intensity profile for each jet was specified directly to match the

experimental data of Mi et al. [5] at the upstream inlet of the pipe jet and at the exit of

the orifice and smooth contraction jet (Fig. 3). A top hat profile is used for the SC

nozzle. For the OP the same profile used by Boersma et al. [8] in their DNS

calculations is used, which is represented in Eq. (5), where = 0.2 is a constant and U in

is selected so that the bulk velocity of each jet is identical.



The turbulence intensity produced by the orifice plate and smooth contraction

jets are sufficiently low, and their mean profile sufficiently simple to allow it to be

defined directly at the nozzle exit plane. This approach also reduces the complexity of

the initial geometric shape to be modeled. The resulting profiles for jet exit velocity and

turbulence intensity are shown in Fig. 4.

Other boundary conditions were kept unchanged and the same as those applied

to the LP. This approach ensures that the co-flow around the nozzle is consistent for all

jets, since it does not introduce differences in the external shape of the supply pipe. As

such it allows reasonable comparison with experimental data obtained in ambient air or

water where there is no external boundary layer, since there is no co-flow.

5. Results and Discussion

A comparison of the predicted downstream velocity decay from the three round

nozzles is shown in Fig. 5. Axial distance is normalised with the effective diameter, d ε .

Where the effective diameter, d ε = 0.99d TN (long pipe), d ε = 0.97d TN (smooth

contraction) and d ε = d TN (orifice plate) [4]. Here the Thring-Newby [19] diameter, d TN ,

has been used to account for any differences in the initial fluid density of the jet and co-

flow, where d TN = d ( j / a)1/2. This correction for density ratio is necessary because

much of the experimental data for the scalar field has been performed at a non-unity

density ratio (notably using a slightly heated jet). The effective diameter, d ε is used in



Fig 5 reveals that the centre-line velocity of the LP begins to decay immediately

from the nozzle exit, i.e. it has no "potential core", unlike the SC nozzle. This trend

agrees with previous experimental data by Mi and Nathan [7]. However the “potential

cores” of the SC and OP extend to x/d ε ≈ 10 so that their lengths are about twice those

of the measured values. This dramatic over-prediction of the length of the potential

core for the SC and OP nozzles is consistent with the k-ε model not accounting for the

role of large-scale coherent motions, whose growth and pairing at the edge of the

potential core is well-known to dominate the near-field entrainment of ambient fluid by

the jet, and hence also the near-field velocity decay. That the length of the potential

cores of the OP and SC jets are predicted to be about the same is likewise consistent

with the model not accounting for the significant differences in large-scale, coherent

motions found in the two jets [5]. Nevertheless, the k-ε model does reproduce some

trends. The centre-line velocity of the OP jet varies with x/d through the “potential

core”, as it does in reality. Downstream from the slight initial increase in velocity with

axial distance is a region of fairly constant centre-line velocity, which decays slightly

upstream from that of the SC nozzle. This earlier decay is also consistent with

experimental trends [6 & 7]. Taken together, this near-field assessment shows that

accounting for the differences in exit velocity profiles of the three jets by the k-ε model

results in a limited capacity to reproduce real differences in the flow. However, simply

reproducing the mean and RMS inflow profiles is insufficient to reliably model these

near field flows where the underlying turbulent structures are both quite different and



concentration values drop quickly for the LP with a short uniform “potential core”, the

length of which matches reasonably well with experimental measurements. Again, the

predicted “concentration potential core” for the SC and OP is over-estimated by the

model.

The rates of decay in the far field are seen more easily in the inverse plotting of

velocity, as shown in Fig. 7, where axial profiles of velocity (U(x)) are normalised (U n)

by three different velocity scales U n = U m, U b and Uco. this method of normalising

(using three velocity scales) demonstrates how decay rates can vary depending on the

normalising method [7]. Here U m is the maximum of U c(x), U b is the bulk mean velocity

across the jet exit and U co (or U cl(0)) is the centerline velocity at the jet exit. For

comparison, previous published data is also included. For each method of normalising

the LP is predicted to have the greatest rate of decay.

When normalised with bulk mean exit velocity (Fig 7a) the predicted decay rates

for the three jets are quite similar, although the decay rates of the SC and OP are slightly

lower than that of the LP. Predicted values are compared with the experimental results

of Mi and Nathan [7] and Xu and Anonia [6]. These experimental results show that the

decay rate is highest for the OP and lowest for the LP, a trend not replicated by the k- ε

model.

Although the normalization by the bulk-mean velocity is arguably the most

relevant, many previous investigators have normalized by the centre-line exit velocity.

When normalised with centerline velocity at the jet exit (Fig 7b) the LP is predicted to



higher rate of decay than the OP and SC and that the decay rate of the SC is greater than

the OP. This difference in trend by this method of normalization results from the fact

that the centre-line velocity is higher than the mean for the pipe jet, but lower for the

orifice jet. Although by this normalization the predicted trend is consistent with

experiments, the difference in decay rates between each jet is significantly greater for

the predicted data.

When normalising with the maximum centerline velocity (Fig 7c) the predicted

LP is again shown to have the greatest rate of decay and the OP the lowest. Predicted

values are compared with the experimental results of Mi and Nathan [7] and Quinn [20].

The predicted trends are again shown to be opposite to those found experimentally.

In Fig. 8, the normalised mean mixture fraction along the jet centerline is

compared with the experimental results of Parham [18] and Mi et al. [7]. Consistent

with the velocity data, the numerical results for the long pipe are in reasonable

agreement with the experimental results of Parham [18]. Also consistent with the

velocity data, the trends in decay rates between the three nozzles is opposite to the

measured ones.

The rate of decay along the axis of the velocity and scalar fields in the far field

( x/d > 20) may be determined from Equations 6 and 7 respectively, where K 1,u is the

velocity decay constant (based on the bulk velocity) and K 1,c is the mixture fraction

decay constant

( )

−− x xU U uab ,1,01 (6)



The present numerical study predicted the velocity decay constants to be K 1,u = 7.25

(for LP), 7.41 (for SC) and 7.38 (for OP) and the concentration decay constants to be

K 1,c = 6.52 (for LP), 6.81 (for SC) and 6.57 (for OP) . The predicted values are

compared with experimental data in Table 1. There is little difference between the

predicted decay rates of each jet. It should be noted that care has been taken in the

present study to ensure a consistent normalising procedure in the data of Table 1 due to

sensitivity of results to the method of normalising, as shown in Fig. 7.

The numerical model has predicted the velocity and concentration decay

constants of the pipe jet reasonably well - within 8% for K 1,u relative to the average of

experimental measurements and 3% for K 1,c relative to Parham [18] (using comparable

initial conditions to current numerical study) However the decay constants of the

smooth contraction and orifice plate jets have been overestimated for both the scalar and

velocity fields by the k-ε model. The velocity decay constants have been overestimated

by 21% for the SC and 17% for the OP, compared with the average of decay from

various experimental measurements. The scalar decay constants have been

overestimated by 29% for SC and 32% for the OP. compared with the average of

experimental measurements.

Fig. 9 & 10 present the rates of spread of the velocity and scalar fields,

respectively, for three jets, as characterized by the half-width, along with experimental

data. The LP jet is predicted to have a higher spreading rate than the SC and OP, which

are almost identical, although that predicted for the SC is marginally lower. Not only are



The jet , velocity half-width, r 1/2,u and the jet concentration half-width, r 1/2,c

provide a measure of jet spreading, defined in Eq. (8) and Eq. (9) respectively, where

K 2,u is the velocity spreading rate constant and K 2,c is the scalar spreading rate constant.

−=

ε ε d

x xK

d

r u

u

u ,2,0

,2

,2 / 1 (8)

−=

ε ε d

x xK

d

r c

c

c ,2,0

,2

,2 / 1 (9)

Values for the velocity spreading constant, K 2,u predicted by the current

numerical study are K 2,u = 0.083 (LP), 0.082 (SC) and 0.082 (OP). Values for the scalar

spreading constant are K 2,c =0.074 (LP), 0.073 (SC) and 0.074 (OP) . The corrected

values are compared with experimental data in Table 1. The numerical model has

underestimated the concentration and velocity decay constants of all three jets, and the

difference between predicted spreading rates is small. The velocity spreading rate

constants have been underestimated by 5% for the LP, 15% for the SC and 24% for the

OP, compared with average of the experimental results. The scalar decay constants

have been overestimated by 31% for the LP, 35% for SC and 58% for the OP relative to

the average of the experimental results. Although the modified k-ε model has

underestimated the velocity and scalar spreading rate constants for the long pipe, it still

provides a solution closer to the measured values than does the standard k-ε model.



dramatically until x/d ε ~ 10, from where it exhibits a x-1

decay. For the SC and OP

nozzles, the predicted TKE is much lower within the “potential core and remains close

to zero throughout. Like the LP, the predicted TKE of the SC and OP jets increases

dramatically from the end of the potential core and then decays at a rate of x-1

. However,

their peak values are 25% lower than those of the LP and occur slightly further

downstream. These trends are consistent with the trends in initial turbulence intensity at

the exit plane. The LP jet has the highest initial turbulence intensity and is predicted to

have the highest peak at x/d ~ 10.

The axial evolution of the normalised RMS, C rms-cl /C cl, for simple jets is known

from experiments to exhibit a rapid initial increase in the initial region and then, for x/d

> 50, to asymptote to a constant value. The asymptotic value obtained from

experimental measurements are, 0.21<C rms-cl /C cl < 0.23 (LP), 0.23<C rms-cl /C cl < 0.24

(SC) and C rms-cl /C cl ~ 0.220 (OP) (Table 1).

A comparison of the evolution of C rms-cl /C cl predicted for the three round jet

nozzles, is shown in Fig. 12. The numerical model predicts the constant asymptotic

values to be C rms-cl /C cl, = 0.226 (LP), 0.225 (SC) and 0.224 (OP). The asymptotic

values are reasonably well matched for the LP, but underestimated by ~4% for the SC

and overestimated by ~2% for the OP. This predicted trend is consistent with the

downstream propagation of the initial turbulence intensity, since the LP has the highest

initial turbulence intensity and the SC the lowest. Nevertheless, this effect is weak, since

the predicted asymptotic values differ by less than 1%. As already noted, this trend is



The numerical model also predicts all three jets to have the same general shape

in axial evolution of C rms-cl /C cl downstream from the end of the “potential core”, with

only the “origin” of each curve being translated axially. Further, this predicted shape

matches the actual shape of the pipe flow with a monotonic increase in C rms-cl /C cl. In

contrast, the experimental data shows that the SC has a strong near-field hump just

downstream from the end of the “potential core”. This hump in C rms-cl /C cl for the SC

jet is believed to be due to the instability caused by the breakdown of the large-scale

coherent structures formed in the shear-layer around the potential core, which are not

present in the pipe jet. Again these errors, although less significant than in the near field,

are consistent with the failure of RANS models to account for the role of different

underlying large-scale motions in the three jets [4 & 6]

6. Further Discussion

The difference in spread and decay rates between each jet, although small, is

evident and the k-ε model has been shown to predict the opposite trends to those found

experimentally. To explain this, it is noted that the calculated trend in spreading rate

matches the trend in the total amount of initial turbulence intensity. That is, the LP has

the highest initial turbulence intensity and is also predicted to have the highest rate of

spread and decay, while the SC has the lowest initial turbulence, and is predicted to

have the lowest rate of spread and decay. An inverse relationship between the predicted



bU

k

I 3

2

= (10)

Next we note that the present calculations show that the k-ε model predicts that

differences in the source flow are propagated downstream. That is, it does predict that

turbulence does not entirely “forget” its origins, even though it fails to reproduce the

“memory” correctly. However, unlike the real flow, in the model the differences in

source flow are not propagated by the underlying turbulence structure. This is especially

evident in the predictions of the “potential core” flow, where the roles of the large-scale

motions are most dominant, but also in the failure of the k-ε model to predict the near-

field hump in normalized scalar fluctuations for the SC nozzle (Fig. 12). However, it is

also evident in the incorrect trends in all the asymptotic values. These discrepancies are

consistent with the knowledge that RANS models do not directly model turbulence

structure, and so cannot directly account for those differences in this structure which

have been deduced to be responsible for the observed differences in the mixing

characteristics of the three jets [5 & 6]. This failure is consistent with the conclusion of

George and Davidson [3] who argue that RANS models cannot correctly account for

dependencies in initial conditions, in contrast to LES, which does simulate large-scale

motions.

Finally, the present results provide further evidence that the coefficients of the k-

ε model are not absolute, even as turbulence is non-universal. That is, no one set of



differences in source flow, while also retaining the low computational expense of

RANS modeling. However, this approach cannot fully overcome the limitations of the

model. For example, the underlying turbulence structure will also depend on other

upstream boundary conditions, such as the effect of a co-flow or external boundary

layer. Hence any calibration can only be presumed to be reliable for a narrow range of

flows. Furthermore, even with appropriate calibration, the k-ε model is unable to

account for all effects, such as the near field differences in normalized RMS shown in

Fig. 12.

6. Conclusion

The present investigation has shown that it is not possible to optimize the

coefficients of the k-ε model in a way that is general for round jets, i.e. to optimise them

for all types of round nozzle. For example, the Modified coefficients of Dally et al. [15]

work well for the present pipe jet flow, the type of jet for which they were optimized,

but differ in the values of K 1,u , K 1,c , K 2,u and K 2,c for the SC nozzles by 21%, 29%,

15% and 35%, respectively, and for the OP by 17%, 32%, 24% and 58%, respectively.

The study also shows that, while differences in the nozzle exit flow, as

characterized by the mean and turbulence intensity profiles, are predicted to propagate

downstream by the k-ε model to affect the entire flow (consistent with turbulence not

“forgetting” its origin), they are not propagated correctly. This is attributed to the fact

that steady-state equations do not model turbulence structure and so cannot directly



However, even the trends in the mean far-field flow also have significant errors.

Notably, measurements have consistently shown that the mean velocity decays most

rapidly for the orifice jet and slowest for the pipe jet [6 & 7]. This trend is also

supported by direct numerical simulation [8]. However when applying the Modified k-ε

model to calculate the flow from the three round nozzle types, the long pipe is predicted

to decay most rapidly. In addition the model is unable to reproduce the differences in

the evolution of normalized RMS of the scalar field – either in the near-field shape or

the trends in asymptotic values. The model predicts that all jets have the same shape in

axial evolution of normalized RMS, with differences only appearing in the location of

their virtual origins. In contrast, measurements show that the SC nozzle produces a

near-field hump in normalized RMS for the SC jet, but not for the pipe, and that the SC

nozzle produces a 4% higher asymptotic value than the LP.

The failure of the present steady-state model to reproduce the measured trends in

a jet flow caused by differences in the source flow is consistent with earlier deductions

[4 & 5] that the unsteady nature, or underlying structure, in each of these three types of

jet is different, while the coefficients of the model are calibrated for only one type of jet,

and hence only one class of structure. Probably the most simplistic approach is to

recalibrate the coefficients for each class of jet flow. This will improve accuracy,

provided that the boundary conditions are well matched, whilst retaining the chief

advantage of two-equation models, namely their low computational expense. However

such an approach must always be taken with caution, since it is well known that flow



7. Acknowledgements

The authors wish to thank the Australian Research Council (ARC), the Sugar

Research Industry (SRI) and Fuel and Combustion Technology (FCT) for their support

of this work.



9. References

[1] Townsend AA. The Structure of Turbulent Shear Flow, 2nd

Edition, Cambridge

University Press, 1996.

[2] George WK. The Self-Preservation of Turbulent Flows and Its Relation to Initial

Conditions and Coherent Structures. Recent Advances in Turbulence, Ed. Arndt,

REA, George WK, 1989.

[3] George GK. Davidson L. Role of Initial Conditions in Establishing Asymptotic

Flow Behavior. AIAA Journal 2004; 47: 438-446.

[4] Mi J, Nobes DS, Nathan GJ. Influence of jet exit conditions on the Passive Scalar

Field of an Axisymmetric Free Jet. J. Fluid Mech 2001a; 432: 91-125.

[5] Mi J, Nathan GJ, Nobes DS. Mixing Characteristics of Axisymmetric Free Jets

From a Contoured Nozzle, an Orifice Plate and a Pipe. Journal of Fluids

Engineering, ASME 2001b; 123: 878-883.

[6] Xu G, Antonia RA. The Effect of Different Initial Conditions on a Turbulent

Round Jet. Experiments in Fluids 2002; 33: 677-683.

[7] Mi, J, Nathan GJ. Momentum Mixing Characteristics of Turbulent Axisymmetric

Jets with Different Initial Velocity Profiles 2006. J. Fluid Mech. in preparation.

[8] Boersma BJ, Brethouwer G, Nieuwstadt FTM. A numerical Investigation on the

Effect of the Inflow Conditions on the Self Similar Region of a Round Jet. Phys

Fluids 1998; 10: 899-909.



[10] George WK, Wang H, Wollblad C, Johansson TG. Homogeneous Turbulence and

its Relation to Realizable Flows, 14th

AFMC, Adelaide, Australia, 2001.

[11] Launder BE, Morse AP, Rodi W, Spalding DB. The Prediction of Free Shear

Flows – A comparison of Six Turbulence Models. NASA SP-311, 1972.

[12] Pope SB. An Explanation of the Round Jet/Plane Jet Anomaly. AIAA Journal

1978;16 (3): 279-281.

[13] McGuirk JJ, Rodi W. The Calculation of Three-Dimensional Turbulent Free Jets.

1st

Symp. On Turbulent Shear flows, Ed. Durst F, Launder BE, Schmidt FW,

Whitelaw JH, 1979, 71-83.

[14] Morse AP. Axisymmetric Turbulent Shear Flows with and without Swirl. Ph.D.

Thesis, London University, 1977.

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Body Jets and Flames. Combust. Theory Modelling 1998; 2: 193-219.

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597-608.

[17] Nickels TB, Perry AE. An experimental and theoretical study of the turbulent co-

flowing jet. J. Fluid Mech 1996; 309:157-182.

[18] Parham JJ. Control and Optimisation of Mixing and Combustion from a Precessing

Jet Nozzle. Ph.D. Thesis. School of Mechanical Engineering, The University of

Adelaide, Australia, 2000.



[20] Quinn, W R. Streamwise Evolution of a Square Jet Cross Section. AIAA Journal

1992; 30: 2852-2857

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Turbulent Non-premixed Propane Jet. Sandia National Laboratories Report

SAND87-8610, 1987.

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particle image velocimetry and planar laser induced fluorescence, Exp. Therm.

Fluid Sci. 2000; 22: 213-229.

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axisymmetric turbulent jets. Expt. Fluids 1991; 11: 125-134.

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of turbulent free jets. J. Fluid Mech. 1993; 254: 417-435.

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School of Mechanical Engineering, The University of Adelaide, Australia. 1997.



Nomenclature

C ε 1 Dissipation rate equation production coefficientC ε 2 Dissipation rate equation dissipation coefficient

C Scalar concentration

d Pipe exit diameter

d e Effective diameter

d TN Thring-Newby diameter

D

2

2

1

2

∂

∂=

i x

k D µ

E

22

2

∂∂

∂=

ii

T x x

U E µν

I Turbulence intensity

k Kinetic energy of turbulent fluctuation per unit mass

r Radial distance

Re Reynolds number

K 1 Decay constant

K 2 Spreading Rate constantr 1/2, Half width, value of radius at which axial velocity/Concentration (respectively) is half the

centerline value

U Axial component of velocity

U a Bulk mean velocity of co-flow

U b Bulk mean velocity

U’ Velocity RMS

U cl Centerline velocity

U co Centerline velocity at jet exit

U in Input velocity for OP velocity profile

U m Maximum centerline velocityU n Velocity normalizing method

S ij Mean strain rate tensor,

∂

∂+

∂

∂=

i

j

j

iij

x

U

x

U S

2

1

x Axial distance

x0,1 Virtual origin based on the inverse concentration

x0,2 Virtual origin based on the jet concentration half-width

δ ij Kronecker delta

ε Dissipation per unit mass

µ Dynamic molecular viscosityυ Kinematic molecular viscosity

Density

υ T Kinematic eddy viscosity

σ k Turbulent Prandtl number for kinetic energy (k-ε)

σ ε Turbulent Prandtl number for dissipation rate (k-ε)

τ Specific Reynolds stress tensor ( ′′ )



Figure Captions

Fig. 1. Schematic diagram showing flow upstream and downstream from the (a) long

pipe, (b) smooth contraction, and (c) orifice plate nozzles.

Fig. 2. Schematic diagram showing downstream regions of a round jet flow

Fig. 3. Schematic diagram of the computational domain.

Fig. 4: Radial profiles of (a) mean velocity (U(r)/U cl); and (b) turbulence intensity

(U`(r)/U cl) at x/d=0.05 for the three jets. That issuing from the long pipe (LP) was

calculated, while those from the smooth contraction (SC) and orifice plate (OP) were

specified to match the measured profiles of Mi et al [5].

Fig. 5: Predicted axial decay of mean velocity for jets issuing from the smooth

contraction nozzle (SC), orifice (OP) and long pipe (LP) compared with experimental

data from the literature [6 & 20].

Fig. 6: Predicted axial decay of mean mixture fraction for jets issuing from the smooth





literature [6, 7 & 20]. Velocity normalised with (a) bulk velocity (U b), (b) jet exit

velocity at the centerline (U co) and (c) maximum centerline velocity (U m).


contraction (SC), orifice (OP) and long pipe (LP) compared with experimental data

from the literature [5 & 18].

Fig. 9: Predicted radial half width decay of velocity for jets issuing from the smooth



Fig. 10: Predicted radial half width decay of mean mixture fraction for jets issuing

from the smooth contraction (SC), orifice (OP) and long pipe (LP) compared with

experimental data from the literature [5 & 18].

Fig. 11: Predicted axial evolution of turbulent kinetic energy on the centre-line for jets

issuing from the smooth contraction (SC), orifice (OP) and long pipe (LP) compared

with experimental data from the literature [21].

Fig. 12: Predicted axial decay of RMS mixture fraction for jets issuing from the smooth




Table Caption

Table 1. Comparison between numerical and measured mixing statistics of the scalar

concentration field and the velocity flow field for jets issuing from the smooth

contraction (SC), orifice (OP) and long pipe (LP). Note that all values of K have been

corrected for density ratio and initial momentum as per Eq. (6)-(9).



Fig. 1. Schematic diagram showing flow upstream and downstream from the (a) long

pipe, (b) smooth contraction, and (c) orifice plate nozzles.

Mean RMS

U`(r)/U cl U ( r)

b. Smooth Contraction Nozzle

a. Long Pipe Nozzle

c. Orifice Plate Nozzle



Fig. 2. Schematic diagram showing downstream regions of a round jet flow.

Core Transition Profile Similarity x

r

U cl ( x)

r1/2_u

U cl /2Nozzle



Fig. 3. Schematic diagram of the computational domain.

x

r

105 d

37 d

50 d

JET EXIT

Upstream flow

development

Downstream flow

U a

U

U a

x/d =0



Fig. 4: Radial profiles of (a) mean velocity (U(r)/U cl); and (b) turbulence intensity

(U`(r)/U cl) at x/d=0.05 for the three jets. That issuing from the long pipe (LP) was

calculated, while those from the smooth contraction (SC) and orifice plate (OP) were

specified to match the measured profiles of Mi et al [5].

(b)

0

2

4

6

8

10

-0.5 -0.25 0 0.25 0.5

r/d

U ' ( r ) / U c l ( % )

(a)

0

0.2

0.4

0.6

0.8

1

1.2

-0.5 -0.25 0 0.25 0.5

r/d

U ( r ) / U c l

LP

SC

OP

LP _Mi et al. [5]

SC _Mi et al. [5]

OP _Mi et al. [5]



Fig. 5: Predicted axial decay of mean velocity for jets issuing from the smooth



0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40 50 60

x/d e

( U c l - U a

) / ( U c o - U a

)

OP

SCLPLP_Xu & Antonia [6]SC_Xu & Antonia [6]SC_Quinn [20]OP_Quinn [20]






0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40 50 60

x/d e

C c l

LP

SC

OP

LP _Mi et al. [5]

SC _Mi et al. [5]

OP _Mi et al. [5]

LP_ Parham [18]



35

Fig. 7: Predicted axial decay of velocity for jets issuing from the smooth contraction (SC), orifice (OP) and long pipe (LP) compared with

experimental data from the literature [6, 7 & 20]. Velocity normalised with (a) bulk velocity (U b), (b) jet exit velocity at the centerline

(U co) and (c) maximum centerline velocity (U m).

(a) (b) (c)

U n =U b

0

1

2

3

4

5

6

7

8

9

10

0 20 40 60

x/d e

( U n - U a

) / ( U c l - U a

)

LP

SC

OP

LP _Mi et al. [7]

SC _Mi et al. [7]

OP _Mi et al. [7]

LP_Xu & Antonia [6]

SC_Xu & Antonia [6]

U n =U co

0

1

2

3

4

5

6

7

8

9

10

0 20 40 60

x/d e

( U n - U a

) / ( U c l - U a

)

LP

SC

OP

LP_Xu & Antonia [6]

SC_Xu & Antonia [6]

SC_Quinn [20]

OP_Quinn [20]

U n =U m

0

1

2

3

4

5

6

7

8

9

10

0 20 40 60

x/d e

( U n - U a

) / ( U c l - U a

)

LP

SC

OP

LP _Mi et al. [7]

SC _Mi et al. [7]

OP _Mi et al. [7]

SC_Quinn [20]

OP_Quinn [20]






0

2

4

6

8

10

12

0 10 20 30 40 50 60

x/d e

1

/ C c l

LP

SC

OP

LP _Mi et al. [5]

SC _Mi et al. [5]

OP _Mi et al. [5]

LP_ Parham [18]



Fig. 9: Predicted radial half width decay of velocity for jets issuing from the smooth



0

1

2

3

4

5

6

7

0 10 20 30 40 50 60

x/d

r 1 / 2_

u / d

LP

SCOPLP _Mi et al. [7]SC _Mi et al. [7]OP _Mi et al. [7]LP_Xu & Antonia [6]SC_Xu & Antonia [6]



Fig. 10: Predicted radial half width decay of mean mixture fraction for jets issuing

from the smooth contraction (SC), orifice (OP) and long pipe (LP) compared with

experimental data from the literature [5 & 18].

0

1

2

3

4

5

6

7

0 10 20 30 40 50 6

x/d

r 1 / 2_

c / d

LPSC

OP

LP _Mi et al. [5]

SC _Mi et al. [5]

OP _Mi et al. [5]

LP_ Parham [18]



Fig. 11: Predicted axial evolution of turbulent kinetic energy on the centre-line for jets

issuing from the smooth contraction (SC), orifice (OP) and long pipe (LP) compared

with experimental data from the literature [21].

0

2

4

6

8

10

12

14

16

18

20

0 10 20 30 40 50 60

x/d e

k / U b 2 x 1 0 0 0

LP

SC

OP

Schefer et al. [21]



Fig. 12: Predicted axial decay of RMS mixture fraction for jets issuing from the smooth



0

0.05

0.1

0.15

0.2

0.25

0 10 20 30 40 50 60

x/d e

C r m

s_

c l / C c l

LP

SCOP

LP_Mi et al. [5]

SC_Mi et al. [5]

OP_Mi et al. [5]

LP_ Parham [18]

Smith Mi Nathan Dally

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