A COMPARATIVE ANALYSIS OF TURBULENCE MODELS UTILISED FOR PREDICTION OF TURBULENT AIRFLOW THROUGH A SUDDEN EXPANSION

A COMPARATIVE ANALYSIS OF TURBULENCE MODELS UTILISED FOR PREDICTION OF TURBULENT AIRFLOW THROUGH A SUDDEN

EXPANSION

Engr. Karinate Valentine Okiy

School of Process, Environmental and Materials Engineering, Faculty of Engineering, University of Leeds, United Kingdom

Email: [email protected]

Keywords: Turbulence, recirculation length, sudden expansion, Turbulence models.

Abstract. The turbulent airflow in a circular duct with sudden expansion was investigated utilizing

three turbulence models. The turbulence models chosen are: the k-epsilon model, the shear stress

transport model and the Reynolds-stress model. The performance of the models was investigated

with respect to the flow parameter-recirculation length. The turbulent kinetic energy and velocity

predictions were compared between the turbulence models and with experimental data, then

interpreted on the basis of the recirculation length. From the results, the shear stress transport model

predictions of recirculation length had the closest agreement with the experimental result compared

to the other model. Likewise, the convergence rate for the shear stress transport model was

reasonable compared to that of the Reynolds model which has the slowest convergence rate. In light

of these findings, the shear stress transport model was discovered to be the most appropriate for the

investigation of turbulent air flow in a circular duct with sudden expansion.

1.0 Introduction

Turbulent fluid flow usually occurs fully at Reynolds numbers above the critical value of

5000. At this stage fluid behaviour is chaotic, random and unsteady accompanied by the occurrence

of unsteady vortices (eddies) with varying ranges of length scales that that interact with each other

and transfer kinetic energy in the process (Figure 1). This process is characterized by radical

changes in flow characteristics such as large rapid variations in pressure and velocity, low

momentum diffusion and high momentum convection [1, 2].

Figure 1: Flow visualization of a turbulent flow jet [3].

Turbulent flows undergoing sudden expansion in circular ducts is a frequent occurrence in many

industrial applications such as dust collectors, expansion joints, orifices, burners in process heaters

and steam boilers. In this type of flow, the flow field consists of the separated boundary layer, a

recirculation zone (mixing region) and a reattachment point [4] as depicted in Figure 2 below.

International Journal of Engineering Research in Africa Vol. 16 (2015) pp 64-78 Submitted: 2015-04-22© (2015) Trans Tech Publications, Switzerland Revised: 2015-06-03doi:10.4028/www.scientific.net/JERA.16.64 Accepted: 2015-06-03

All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TransTech Publications, www.ttp.net. (ID: 41.220.69.166-12/06/15,16:41:29)

http://dx.doi.org/10.4028/www.scientific.net/JERA.16.64

http://www.ttp.net

Figure 2: Flowing fluid undergoing expansion in a circular duct [5].

The sudden expansion passage is a typical geometry where the sudden change in the flow

cross section causes generation of turbulent energy leading to flow instability which culminates in

turbulence [4]. This sudden expansion of turbulent flow has an important effect on the turbulent

flow structure and produces large velocity gradients which increase the entropy generation rate.

This has an effect on the momentum, heat and mass transfer processes that take place in the fluid

[6,7]. Previous studies carried out to determine the physics underlying the flow has revealed that

with reduction in the inlet duct cross-section, the reattachment point (length of the mixing region

consisting of eddies) increases downstream of the expansion and turbulence intensity downstream

of the expansion also increases [8]. Studies carried out on the heat transfer effects in the region of

sudden expansion has shown significant enhancement of convective heat transfer signified by large

increases in Nusselt number up to nine times greater in magnitude than that for fully developed

flows in pipes with similar diameter downstream [8].

In this work, compressible turbulent flow of air undergoing sudden expansion in a circular

duct was simulated using FLUENT 13.0 and the effectiveness of the following turbulence models

(Shear Stress transport model, K-epsilon turbulence model and Reynolds Stress model) in

predicting the turbulent airflow field analysed.

2.0 Simulation Methodology

Computational Details. The simulated problem is the turbulent flow of air in a sudden expansion.

The physical properties of air are given in Table 1 below.

Table 1: Air Properties and Flow characteristics

Density [kg/m3] 1.225

Kinematic Viscosity [m2/s] 1.46073 × 10

-5

Mass Flowrate [kg/s] 0.01628

Bulk Velocity at inlet [m/s] 1.75

Reynolds Number 11980

Temperature [K] 288

The sudden expansion geometry consists of a cylindrical tube of length 2m and inlet

diameter 0.1m. The expansion ratio is 1:2 and The Reynolds number at the pipe inlet is 11,980. The

schematic of the cylindrical duct with sudden expansion is shown in Figure 3.

International Journal of Engineering Research in Africa Vol. 16 65

Figure 3: Representation of turbulent air flow through cylindrical duct with sudden expansion.

The boundary conditions specified were the inlet conditions given (Velocity = 1.75 m/s), the

turbulent intensity at the inlet (5%), hydraulic diameter (0.1m).The outlet boundary conditions used

were turbulent intensity (5%), hydraulic diameter (0.1m) and at the wall no slip condition was

assumed.

Mathematical Models. The following assumptions were made;

The air is assumed to behave as an ideal gas.

Isothermal steady turbulent compressible flow

The gravity effect is negligible.

No-slip condition is assumed at the pipe wall.

Governing Equations. The governing equations of continuity and momentum for this case are

given below;

Continuity Equation

. (1)

Momentum Equations

X-Directed Momentum

- +

ρgx . (2)

Y-Directed Momentum

−

+ ρgy . (3)

Z-Directed Momentum

− +

ρgz . (4)

Where τijtot

= τijl + τij

t . (5)

τijl = laminar values

τijt = additional turbulent stresses (Reynolds-stresses)

66 International Journal of Engineering Research in Africa Vol. 16

τijt = ρ . where i = j (Normal Stresses) (6)

where i ≠ j (Shear stresses)

U = Average mean flow velocity obtain from the Reynolds time averaging equation (7);

(7)

Where p is the static pressure, ρ the fluid density and ν the molecular viscosity;

ui = U + ui’ (8)

ui = instantaneous velocity

ui’ = turbulent velocity flunctuation

U = average velocity

The Reynolds-stresses given in equation (6) are unknown and have to be specified for the

aforestated turbulent transport equations in order to obtain unique solutions by applying turbulence

closure models. In all three turbulence models were utilized for the determination of the unknown

Reynolds-stresses, these are k-epsilon turbulence model, Shear Stress transport model, Reynolds-

stress model.

The Standard k-epsilon model

For turbulent kinetic energy ;

+ Pk + Pb − ρϵ − YM + Sk . (9)

For dissipation ;

+ (Pk + ) − + . (10)

Where turbulent viscosity is given by;

= . (11)

Production of ;

. (12)

Model constants are given as;

C1ϵ = 1.44, C2ϵ = 1.92, Cµϵ= 0.09, σk = 1.0, σϵ = 1.3

Shear Stress Transport model

. (13)

. (14)


. (15)

. (16)

. (17)

The turbulent eddy viscosity is obtained from equation (18);

. (18)

= density,

(turbulent kinematic viscosity)

= molecular dynamic viscosity

= magnitude of vorticity.

The Reynolds-stress model (RSM)

The transport equations for the transport of Reynolds stresses is given by;

+

(19)

The redistributive fluctuating pressure term that extracts energy from the mean flow and

redistributes it between the three fluctuating velocity components is given as;

+ k +

− − . (20)

The wall reflection term that is associated which the pressure reflections from any solid

surface that modifies the fluctuating pressure field away from the surface and in the main body of

the flow is given as;

. (21)

Finally, the turbulence energy dissipation rate equation is given as;

. (22)

The turbulence model constants are obtained through a data fitting exercise.

Numerical Procedure. Fluent 13.0 solver was utilised for this study. The structured grid was built

using Gambit 2.4.6. The mesh consisted of 107,011 structured cells. The transport equations were

solved using the second order upwind discretisation scheme for the convection term together with

the k-epsilon, shear stress transport and Reynolds-stress model for turbulence respectively in order

to determine the best model prediction of the flow field under investigation. The boundary

conditions aforestated in the previous section were applied.


Mesh Independence Test. A series of refinements were made to the original mesh employed using

the adaptation feature in Fluent, each time crosschecking the mass fluxes and residuals values for

close similarity after each simulation run .Finally an optimized mesh size of 220,520 cells was

obtained after the third successive refinement using the adaptation feature in Fluent 13.0. The

comparison of the recirculation length predicted employing this mesh size with the preceding mesh

size of 170,400 cells showed close agreement of 1.15m .The value for the net mass flux obtained

was 1.5528 × 10-8

. The corresponding plot of residuals history is shown in Figure 4 for

convergence criteria of 1.0 × 10-4

.

Figure 4: Plot of residuals history for final adapted mesh.

Figure 5: Plot of comparative axial velocity profiles obtained for mesh of size 170,400 and 220,

520 cells (● – Mesh 1, + – Mesh 2).


3.0 Results and Discussion The main objectives of this study is the prediction of the turbulent airflow field in the

circular duct with a sudden expansion using the k-epsilon turbulence model and the evaluation of

the performance of the three turbulence models chosen for predicting of the flow field. The

computational results of recirculation length for each model are then compared with the

experimental data for similar investigation carried out using diesel fluid [9] in order to determine

the best turbulence model. The parameter utilised for the comparison of the turbulence models is the

recirculation length, which is a major parameter for the analysis of separated flows [7].

Figure 6: Plot of axial velocity [m/s] predictions for the K-epsilon model.

Figure 7: Contour plot of axial flow velocity [m/s] prediction for k-epsilon model.


Figure 8: Contour plot of turbulent kinetic energy [m

2/s

2] prediction for k-epsilon model.

Figure 9: Turbulent kinetic energy [m

2/s

2] prediction for the k-epsilon model.

Figure 8 shows the contour plot of the turbulent kinetic energy variation for the k-epsilon

model (expansion ratio 2.0). This depicts the magnitude of the turbulence occurring at the sudden

expansion of the cylindrical duct. From the contour plots, It is evident that the generation of


turbulent kinetic energy begins at the region of sudden expansion of the cylindrical duct and the

degree of turbulence is strongest around point Z= 0.85m (Figure 6). Subsequently it decreases in

magnitude till equilibrium flow conditions is achieved at position Z= 1.15m (Figure 9). This point

marks the end of the recirculation (intense mixing) zone, corresponding to a recirculation length of

1.15m for the flow field predicted by k-epsilon model. As shown in the contour plot (Figure 8), the

maximum turbulent kinetic energy occurs in the recirculation region. However, close to the walls of

the cylinder, the turbulent kinetic energy becomes negligible due to the resistance offered to the

flow by the presence of the laminar sub-layer [4]. Comparison of the predicted recirculation length

(Z=1.15m) and experimental data (Z= 1.35m) for similar study done using diesel [9] showed that

the k-epsilon model under-predicted the experimental recirculation length (14.8% error). This

means that the k-e model predicts that equilibrium conditions are achieved faster than in reality. The

symmetry in flow conditions is evident from Figure 9 above, velocity profile being the similar on

both sides of the curve plot. This confirms the similarity in flow behaviour throughout the cross

section of the circular duct.

From the investigation of the flow field using the shear stress transport turbulence model,

the following figures and plots were obtained:

Figure 10: Plot of axial velocity predictions [m/s] for shear stress transport model.


Figure 11: Contour plot of axial velocity predictions [m/s] for shear stress transport model.


2/s

2] for shear stress transport model.


Figure 13: Plot of turbulent kinetic energy [m

2/s

2] for shear stress transport model.

Figure 12 shows the contour plot of the turbulent kinetic energy variation for the shear

stress transport model).Again, It is evident that the generation of turbulent kinetic energy begins at

the region of sudden expansion of the cylindrical duct and the degree of turbulence is strongest

around point Z= 0.75m (Figures 11 & 13). Then, subsequently it decreases in magnitude till

equilibrium flow conditions is achieved at approximately position Z= 1.36m (Figure 10). This point

marks the end of the recirculation (intense mixing) zone, corresponding to a recirculation length of

1.36m for the flow field predicted by shear stress transport model. As shown in the contour plot,

the maximum turbulent kinetic energy occurs in the recirculation region. Close to the walls of the

cylinder, the turbulent kinetic energy becomes negligible due to the resistance offered to the flow by

the presence of the laminar sub-layer [4]. Comparison of the predicted recirculation length

(Z=1.36m) and experimental data (Z= 1.35m) for similar study done using diesel [9], showed that

the shear stress transport model over-predicted the recirculation length (-0.74 % error). This means

the shear stress transport model predicts that equilibrium conditions are achieved a bit slower than

in reality.

From the investigation of the flow field using Reynolds-stress model, the following figures

and plots were obtained:


Figure 14: Plot of axial velocity [m/s] predictions for the Reynolds-stress model.

Figure 15: Contour plot of axial flow velocity [m/s] prediction for Reynolds-stress model.



2/s

2] for Reynolds-stress model.

Figure 17: Plot of turbulent kinetic energy [m

2/s

2] for Reynolds-stress model.

Figure 16 shows the contour plot of the turbulent kinetic energy variation for the Reynolds-

stress model. Again, it is evident that the generation of turbulent kinetic energy begins at the region

of sudden expansion of the cylindrical duct and the degree of turbulence is strongest around point

Z= 0.90m (Figure 17). Subsequently, it decreases in magnitude till equilibrium flow conditions is

achieved at approximately position Z= 1.25m (Figure 14). This point marks the end of the

recirculation (intense mixing) zone, corresponding to a recirculation length of 1.25 m for the flow


field predicted by Reynolds-stress model. As shown in the contour plot, the maximum turbulent

kinetic energy occurs in the recirculation region. Close to the walls of the cylinder, the turbulent

kinetic energy becomes negligible due to the resistance offered to the flow by the presence of the

laminar sub-layer [4]. Comparison of the predicted recirculation length (Z=1.25m) and

experimental data (Z= 1.35m) for similar study done using diesel [9], showed that the Reynolds-

stress model under predicted the recirculation length (7.4 % error). This means that the Reynolds-

stress model predicts that equilibrium conditions are achieved faster than in reality.

Table 2: Predicted recirculation lengths and relative errors

Experiment [7] k-epsilon model Shear stress

transport model

Reynolds-stress

model

Recirculation

length [m]

1.35 1.15 1.36 1.25

Relative error [%] - 14.8 -0.74 7.4

The predicted recirculation lengths for the three turbulence models and the experimental

measured recirculation length are shown in Table 2. The results obtained using the shear stress

transport model is in closest agreement with the experimental results. But, the k-epsilon model has

the largest discrepancy from the experimental data, it grossly under-predicts the recirculation length

by a relative error margin of 14.8% .This implies that the shear stress transport model is the ideal

model for prediction of the flow field under investigation. However, it is important to add that exact

experimental data for turbulent air flow from sudden expansion in circular ducts could not be found

in literature for thorough validation of the computational results. This is a likely source of error in

the comparison of the performance of the turbulence models in the prediction of the turbulent flow

field.

4.0 Conclusion

The use of the each of three turbulence models for the prediction of the flow field under

investigation has led to a number of conclusions.

In general, the turbulence models utilised failed to predict accurately the flow field after the sudden

expansion going by the discrepancies in their predicted values of the parameter used for flow

analysis (recirculation length) and the experimental results used for validation. Two of the models

(k-epsilon and Reynolds stress model) wrongly predicted that the flow will achieve equilibrium

conditions in less time than in real conditions.

In all, the shear stress transport model gave the closest agreement with experimental results

for recirculation length (-0.74% relative error), while the k-epsilon model gave the largest deviation

(14.8% relative error) from the experimental result. Hence, it follows that the shear stress transport

model is the most appropriate for prediction of the flow field.

However, it is pertinent to add that the convergence rate is slower for the shear stress transport

model than the k-epsilon model which is less accurate.

The Reynolds-stress turbulence model is intermediate between the two models in terms of

accuracy (7.4% relative error) but has the slowest convergence rate of all the models. This entails

having the highest computational cost of the three models.

On a final note, the model recommended for the investigation of the problem of turbulent air

flow in a circular duct with sudden expansion (prediction of velocity and turbulent kinetic energy

profiles) is the shear stress transport model. This model gives the most accurate result and has

moderate computational cost compared with the Reynolds-stress model and k-epsilon model.


References

[1] H.K. Versteeg and W. Malalasekera.1995. An Introduction to Computational Fluid Dynamics.

Essex. Longman Group Limited.

[2] K. Avila, D. Moxey, A. de Lozer, M. Avila, D. Barkey and B. Hof. 2011. The Onset of

Turbulence in Pipe Flow. Science. 333(6039). pp.192-196.

[3] Wikipedia.2011. Turbulence [online]. [Accessed 5 March 2011]. Available from:

http://wikipedia.org/wiki/turbulence.

[4] R. Vikram, M. Snehamoy and S. Dipankar.2010. Analysis of the Turbulent Fluid flow in an

axi-symmetric sudden expansion. International Journal of Engineering Science and

Technology. 2(6). pp.1569-1574.

[5] S. Kangovi and R.H.Page. 1977.The Turbulent flow through a sudden enlargement at

subsonic speeds. 6th Australasian Hydraulics and Fluid Mechanics Conference. Adelaide.

pp.213-216.

[6] H. Tennekes and J.L. Lumley. 1972. A First Course in Turbulence. Cambridge: MIT Press.

[7] V.I. Terekhov and M.A.Pakhomov.2011. Particle Dispersion and Turbulence modification in

a dilute mist non-isothermal Turbulent flow downstream of a sudden pipe expansion. Journal

of Physics.318(9), pp.1-10.

[8] N.A.Khalid. 2011. Flow and Heat Transfer characteristics downstream of a porous sudden

expansion : A numerical study. Thermal Science.15(2).pp.389-396.

[9] E.D. Koronaki, H.H. Liakos, M.A. Founti and N.C. Markatos.2001. Numerical Study of

Turbulent diesel flow in a pipe with sudden expansion. Applied Mathematical Modelling.25.

pp.319-333.


Documents

A COMPARATIVE ANALYSIS OF TURBULENCE MODELS UTILISED FOR PREDICTION OF TURBULENT AIRFLOW THROUGH A SUDDEN EXPANSION