shape. In such cases, an adequate modelling of the turbulent … · 2014. 5. 14. · The finite element predictions obtained with different turbulence models are evaluated by detailed

Finite element analysis of turbulence

models for confined swirling flows using

LDV-measurements for verification

J. Goerres, A. Fingerle, H.-C. Magel, U. Schnell

Institut fur Verfahrenstechnik und

ABSTRACT

Measurements and computations of a highly swirling flow are presented. The

isothermal flow expands into a highly confined test channel which represents

the geometry of typical industrial furnaces.

With the technique of Laser Doppler Velocimetry (LDV), measurements of

mean and fluctuation velocities were made at several positions in the test

channel. These data are used for a detailed evaluation of various turbulence

models: the usually used standard k-e-model and two higher-order turbulence

models, the Algebraic Stress Model (ASM) and the Effective Viscosity

Hypothesis (EVH). These models are incorporated into a finite element

program. In addition to the LDV-measurements, the finite element results are

compared to predictions obtained with a finite volume code. Attention was

paid to inlet conditions for the turbulence energy k and the dissipation rate e.

INTRODUCTION

In combustion applications, a swirl component is usually imparted to the axial

flow in order to achieve a stably burning flame or to influence the flame

Transactions on Modelling and Simulation vol 4, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X

338 Computational Methods and Experimental Measurements

shape. In such cases, an adequate modelling of the turbulent swirling flow is

of utmost importance for obtaining reliable results especially in predicting the

near-burner zone, which is an important region for pollutant formation. In

practical applications, the turbulent flow is normally treated by means of a

turbulence model, which serves to close the Favre or Reynolds averaged

governing equations by introducing modelled expressions for the Reynolds

stresses in terms of the known averaged quantities.

The existing turbulence models vary in a wide range in degree of

sophistication. The most frequently employed turbulence model in finite

element and finite volume codes is still the k-e model [1], since it provides an

optimal choice between accuracy and economy in a wide range of engineering

applications. However, the performance of the k-e model is often not sufficient

in swirling flows (and in other kinds of flows exhibiting strong streamline

curvature and rotation) [2], since the strong non-isotropic turbulence structures

arising in such flows can not adequately be described by the k-e model or any

other two-equation turbulence model based on a standard scalar turbulent

viscosity hypothesis.

Due to the importance of the swirling flow in combustion applications, and

due to the above mentioned inadequacies of the k-e model in swirling flows,

we have initiated an analysis of higher-order turbulence models within the

framework of our finite element modelling. A first step towards this aim is to

investigate the finite element application of the Algebraic Stress Model (ASM)

[3] for isothermal swirling flows. The ASM-model provides a principally more

accurate turbulence model compared to the k-e-model, because the local values

of Reynolds stresses are not assumed to be proportional to the mean velocity

gradients. They are obtained directly by solving algebraic equations, which are

derived from Reynolds stress transport equations under certain assumptions.

In the present study, we are investigating one further higher-order turbulence

model, namely the Effective Viscosity Hypothesis (EVH) proposed by Pope

[4]. This hypothesis is comparable to the ASM-model, as far as the accuracy

is concerned, since similar modelling assumptions are employed in modelling

the Reynolds stresses. The advantage of the EVH in opposite to the ASM is

given by the fact that the simultaneous calculation of the unknown Reynolds

stresses which are strongly interlinked is avoided. Thus the EVH-model seems

to possess better stability properties.


Computational Methods and Experimental Measurements 339

As numerical test case for a semi-industrial coal combustion facility an

isothermal expanding swirl burner flow is considered. The diameter and length

of the horizontal combustion chamber are 0.22 m and 4 m, respectively.

Velocity measurements are carried out by the technique of Laser-Doppler-

Velocimetry (LDV).

The finite element predictions obtained with different turbulence models are

evaluated by detailed comparisons with these measurements, and with finite

volume predictions.

Fluid Flow and Turbulence Modelling

In case of turbulent flows, the velocity components must be divided into mean

values ui and fluctuation quantities u/. With this decomposition the Favre

averaged Navier-Stokes equations for a steady, incompressible and

axisymmetrical Newtonian fluid are given in equation (1).

with

-

The task of a turbulence model is to find suitable approximations for the

Reynolds stresses Q uju/, which are included in equation (1) as second term

on the right side. Predominantly, a Boussinesq hypotheses is used that relates

the Reynolds stresses with the local velocity gradients

by introducing a scalar constant called turbulent viscosity /v



— with C = 0.09 (4)

In eq. (3) and (4) two additional quantities are introduced: the kinetic energy

of turbulence k, which extracts energy from the mean motion and transfers it

to the turbulent motion, and the dissipation rate e, which describes the

annihilation of the turbulent eddies. For high Reynolds number flows, the

transport equations of k and e can be expressed in the general form:

-A(p^-0.)=D* +3* (5)a%. ' "

In the standard k-e-model D denotes the diffusive transport

D. - -I •=—2 I (6)

where the sum of the laminar and turbulent viscosity is divided by the Prandtl

number of k or e. The source terms S* for k and e are given by

e) (7)

with Ci = 1.43 and C^ = 1.92.

The term P represents the production of k and is defined by

The restriction of the k-e model on only one scalar quantity & implies an

isotropic and homogeneous structure of turbulence, which does not apply for

strongly swirling flows.

An improved approach was proposed by Rodi [3] assuming that the turbulent

shear stresses are not calculated from the same eddy viscosity and thus the

individual Reynolds stresses in the flow field are obtained directly by solving

algebraic equations (eq. (10)-(12)).



where

with Ci = 2.5 , C% = 0.55 and

I dli- dll- I (\ 9\D _ _ ,,',, ' L j. u ' 11 ' L \i^/

This assumption is valid if the variation of the Reynolds stresses along the

flow is small and if there are only small deviations from homogenity.

Replacing equation (3), each algebraic Reynolds stress equation is a function

of other Reynolds stresses and of mean velocity gradients. This turbulence

model called ASM is comparable with an extended Effective-Viscosity

Hypothesis (EVH), which relates the components of the Reynolds stress tensor

with the strain rates and local scalar quantities in a definite way [4].

(13)^ . \JJ±- I t, I vy v • v v • r/I. . f7A.

With the use of the following relation for g

L - iV (14)

the modified constant CD in equation (4)

\-i(15)

and with the provision of the invariants of the symmetric tensor {s}

i k (du. d



and the antisymmetric tensor {w}

W = w,w,. mth w, = -- _N (17)'* * *

the Reynolds stresses can be calculated according to equation (13).

The influence of the boundary conditions on the turbulent stresses and the time

and length-scales that are used in the deduction of the hypothesis is embodied

in the following set of constants.

bi = 8/15 02 = (5-9Q)/ll bg = (7Q4-1)/11

Cg = 1.5 €4 = 0.4

The advantage of the EVH compared to the ASM-model is given by the fact

that the simultaneous calculation of the unknown Reynolds stresses which are

strongly interlinked is avoided. The numerical effort is thus significantly

reduced.

For both turbulence models the structure of an existing finite element or finite

volume algorithm as well as the transport equations for k and e can be

retained. Only the diffusive term D , in eq. (5) has to replaced by

with Q = 0.22 and C, = 0.15.

Nevertheless both theories demand that the transport of the turbulent stresses

brought about by the triple correlations appearing in the full Reynolds stress

transport equation is negligible. Thus homogenity of the turbulent stresses and

consequently of the rates of strain is a necessary demand, as it applies to high

Reynolds number flows. Because there is the same critical assumption in the

deduction of both turbulence models, the computer simulations should render

the same values for the Reynolds stresses.

A complete formulation of the ASM-model in cylindrical coordinates was

recently presented [5] as well as a formulation of the EVH-model [6].



FLOW CHANNEL AND LDV-MEASUREMENTS

To validate the different turbulence models a test case of the International

Flame Research Foundation (IFRF) [7] is considered. The geometry of the

horizontal IFRF test channel with a length of 4 m and a diameter of 0.22 m is

shown in Figure 1.

z 0.11 0,44 0,61

Figure 1 Geometry of the IFRF test-channel

The burner quarl has an angle of about 20°, which represents a typical

example of a highly confined flow.

After the swirl generator which is located upstream of the inlet, the isothermal

flow has a swirl number (SJ of 0.7, thus the flow is supercritical (S<,<0.96).

In the expanding quarl section it undergoes transition to a subcritical state with

a significant loss of total energy and an increase of turbulent kinetic energy.

At several positions LDV-measurements of the mean velocities and their

fluctuations are available. The LDV-system used to study velocity distributions

in the discussed IFRF test channel was a dual beam backscatter instrument. By

this optical assembly only the velocity component of one spatial direction can

be measured at the same time.

Measurements inside the test channel faced restrictions by the optical set-up,

which result in a loss of accuracy. It was necessary to use a front lens with a

focal length of 600 mm and a beam separation of less than 40 mm. A large



focal length of the front lens results in large dimensions of the optical probe

volume, which reduces the spatial resolution of measurements. Some

characteristic values of the optical probe volume are summarized in Table 1.

Table 1 Characteristic values of the sampling volume

Focal length of

the front lens

[mm]

600

600

600

Beam

separation

[mm]

13

26

39

Spatial dimension of the

probe volume [mm]

dx dy dz

0.16 0.16 7.6

0.16 0.16 3.8

0.16 0.16 2.5

Number of

fringes

H

13

26

40

A prerequisite for LDV-measurements is the seeding of light scattering

particles in the air stream. At the IFRF, the air stream was seeded with MgO-

particles with diameters in the range of 1/zm. Therefore, it might be expected

that the air flow is correctly represented by measuring particle velocity,

assuming a no-mean-slip condition.

The IFRF-measurement program comprises measurements of velocity

components in axial and tangential direction. Assuming an axisymmetric air

flow, measurements were taken at different axial distances downstream of the

swirl generator at several radial positions between the axis and the wall of the

test-channel. At each location the mean velocity and its averaged standard

deviation (RMS-value), representing turbulence intensity, was calculated. Each

mean value is represented by 2000 bursts.

The statistical accuracy was estimated with a simple test. At several locations

measurements were repeated nine times, acquiring 2000 bursts each. The

averaged values of this test were considered "true" mean values. It was

reported that the mean velocity calculated from one data unit deviated from

the "true" mean value by +/- 0.3%. The variation in turbulence intensity

(RMS-value) was within +/- 4% of the "true" mean turbulence intensity.

To consider the accuracy of the measurements, the resulting fluxes were

calculated for several traverses along the test channel. They are compiled in

Table 2.



Table 2 Integrated Fluxes

Axial distance

from swirl

generator [mm]

Integrated fluxes

[irrVh]

-35

479

50

552

200

524

310

540

440

419

610

523

1060

458

As published by the IFRF the inlet flux totalled 480 mVh. As shown in

Table 2, the integrated fluxes give a maximum deviation of +/- 15% of the

correct value, thus indicating that the measurements are relatively accurate.

NUMERICAL COMPUTATIONS

The inlet data for the computer simulation (axial and tangential velocity and

the fluctuation velocities, see Table 3) were measured by LDV 35 mm

upstream of the quarl.

Table 3 Measured inlet data at different radial distances

r [mm]

0

12

24

36

48

60

72

84

90

93

u [m/s]

4.35

4.71

4.63

4.69

4.85

4.96

5.00

4.92

4.15

3.29

u' [m/s]

0.109

0.218

0.244

0.257

0.272

0.268

0.294

0.349

0.508

0.843

w [m/s]

0.210

1.094

1.890

2.730

3.650

4.660

5.730

6.890

6.860

6.010

w' [m/s]

0.256

0.246

0.237

0.184

0.147

0.129

0.163

0.291

0.525

1.380



The mean axial velocity is 4.7 m/s and the radial velocity is specified as zero.

The values of the turbulent kinetic energy k are derived from the measured u'

and w' profiles,

k = — (uu + vv + (19)

assuming locally at the inlet

v v =u u + w w (20)

The inlet conditions for the turbulence dissipation rate are obtained from

&%2 (21)£ —

0.03 r.

Figure 2 shows the finite element mesh (0. <z<0.5m), with totally 909 nodes

and 848 elements. In the burner quarl the elements are adapted to the wall. No

stepwise discretization is necessary, as it is usually done in finite volume

codes without body fitted coordinates.

Figure 2 Finite element mesh

The results of the finite element predictions are compared with results of the

commerically available FLOW3D code. With this finite volume code the

number of grid lines was increased from 40x22 to 70x45 in order to check

the independence of the chosen numerical mesh. Since the fine grid solution

showed only marginal differences of predicted axial and tangential velocities,

the computations on the above described finite elment mesh are assumed to be

grid-independent.



The computed axial velocities are compared with measurements at four axial

positions (z=0.11m, 0.31m, 0.44m and 0.61m marked in Figure 1). At each

position the solutions of both k-e computations (FE and FV) are plotted. The

results of the ASM-model and the Effective Viscosity hypothesis (EVH) show

only negligibly small differences, which would not be perceptible in any

figure. Thus only the ASM-results are shown in Figure 3 and 4.

8.0

-2.0000 0.04 0.08 0.12 0.16 0.20rodiol distance [m]

6.0-

2- 4.0-

2.0-

0.0

-2.0

ASM- - k-e (FE)& A k-« (FV)

RSMO measurement*

0.00 0.04 0.08 0.12 0.16 0.20radial distance [m]

4.0

E 2.0-

1.0-

-1.0-

-2.00.00 0.04 0.08 0.12 0.16 0.20

rodiol distance [m]

-2.0

.00

z - 0.61 m


Figure 3 Mean axial velocities at different axial distances

a) z=0.11m b) z=0.31m

c) z=0.44m d) z=0.61m



The last curve plotted in both figures are results of a Reynolds Stress Model

(RSM, see Ref. [8]) included in FLOW3D. In opposite to the ASM where the

Reynolds stresses are estimated by means of algebraic equations, differential

equations (totally six) are solved for these quantities. The RSM-model

accounts for convective and diffusive transport of the Reynolds stresses, in

some cases leading to a more accurate and more universal method. Thus the

RSM represents a more expensive but also a more accurate modelling

compared to the ASM.5.0

m_ 4.0-|

'I 3.0-

12.0-

1.0-

o.o 4**

Z - 0.11 m

0.00 0.04 0.08 0.12 0,16 0.20rodiol distance [m]

2.5

_ 2.0I

1.5

.21.0-

0.5-

0.00.00 0.04 0-08 0.12 0.16 0.20

radial distance [m]


2.5

1.5-

1.0-

0.5-

0.00-00 0.04 0.08 0.12 0.16 0.20

radial distance [m]

Figure 4 Mean tangential velocities at different axial distances

a) z=0.11m b) z=0.31m

c) z=0.44m d) z=0.61m



Inspection of the various numerical results shows at the first axial position

(z = O.llm, Fig. 3a and 4a) good agreement with the experimental data.

However, at positions further downstream the predictions of both k-e-models

(FE and FV) differ more significantly from the measurements than results

obtained with the ASM and RSM-model. This applies primarily to the profiles

at z = 0.31m where the k-e-model overestimates the tangential velocity (Fig.

4b). The radial slope of both ASM and RSM predictions agrees quite

accurately with the measured data. Near the symmetry axis only the RSM-

model corresponds with the experimental axial velocity (Fig. 3a and 3b).

In Figure 5 the axial and tangential fluctuation velocities are compared at two

axial positions.

1,4-

0.8-

o 0.6-

lo.4

0.2

o.o

z - 0.11 m


0.00.00 0.04 0.08 0.12 0.16 0.20

radial distance [m]

1.4-

1.0-

0.8-

0.6-

0.4-

0.2-

0.0-

z - 0.11 m

0

\ :

Qtf?o\o-

77 1.4-

0•§ 1-0-

o 0.8-

o 0.6-

o 0.4-

§» 0,2-o

n n —

Z - 0.44 m

_— -%'-••"'••""

' ' A U« . '. _ ' '0.00 0.04 0.08 0.12 0.16 0.20radial distance [m] radial distance [m]

Figure 5 Mean fluctuation velocities; above: u\ below: w\



The k-e-model (only FE-results are plotted) overestimates the turbulence level

in the quarl (z = O.llm), whereas the RSM-model agrees very well with the

measured data. At the axial distance of 0.44m the ASM-results agree much

better with measured fluctuation velocities.

A further investigation covered the influence of the inlet data for k and e.

Instead of using the measured fluctuation velocities, the turbulence energy at

the inlet was estimated by

— \2 (22)

It depends on the degree of turbulence (Tu% = 5%) and the mean axial

velocity U;.

In equation (21) for the dissipation rate, the factor 0.03 was changed to 0.41

and the inlet radius % was replaced by the local distance to the wall.

3.0

mm 2.0-

xZ

1.0-

-1.0

— k-fk-s (k voriotion)k-e (e voriotion)

O m«osur«ments00

0.00 0.04 0.08 0.12 0.16 0.20

rodiol distance [m]

3.0

^ 2.0-

-1.0

ASM- - ASM (k voriotion)

ASM (t voriotion)O meosurements

O 0

0.00 0.04 0.08 0.12 0.16 0.20

rodiol distonce [m]

Figure 6 Mean axial velocities for different inlet conditions at z=0.61 m



Figure 6 shows the influence of this modification on the axial velocities for the

k-e-model and the ASM-model. The changed inlet data for k has little impact

on the results, the ASM-model being a little more sensitive. The plots of the

tangential velocities are changed in a similar way. Modifying the inlet data for

the dissipation rate e has a much greater influence (see Fig. 6) for both

turbulence models.

CONCLUSIONS

Numerical computations and LDV-measurements of a highly confined swirl

flow were reported. Two higher-order turbulence models (ASM and EVH)

which are incorporated in a finite element program, were investigated. The

numerical results were compared with the standard k-e-model and the

Reynolds Stress Model (RSM), which is included in a finite volume code

(FLOW3D). It has been shown that improved results can be obtained using the

ASM instead of the k-e model. The EVH-model predicted almost the same

results as the ASM. Nevertheless, some features of the flow (especially near

the symmetry axis) are even better predicted using a RSM-model. However,

the prediction quality of the finite element program is expected to be further

improved by introducing the RSM, which will be investigated in a future

work.

ACKNOWLEDGEMENTS

This work was carried out within the TECFLAM project on Mathematical

Modelling and Laser Diagnostics of Combustion Processes. Funding by the

German Ministry of Research and Technology and the Federal Government of

Baden-Wiirttemberg is gratefully acknowledged.

The FVM results were obtained using the software FLOW3D as being

developed and owned by the United Kingdom Atomic Energy Authority.



REFERENCES

[1] Launder, B.E. and Spalding, D.B. 'The Numerical Computation of

Turbulent Flows', Comput. Appl. Mech. Engrg. (1974) pp. 269-289.

[2] Srinivasan, R. and Mongia, H.C. 'Numerical Computation of Swirling

Recirculating Flows: Final Report', NASA CR-165196 (1980).

[3] Rodi, W. 'A New Algebraic Relation for Calculating the Reynolds

Stresses', Zeits. angew. Math. Mech. (ZAMM), 56 (1976), pp. T219-

T221.

[4] Pope, S.B. 'A More General Effective-Viscosity Hypothesis', J. Fluid

Mech. 72 (1975) pp. 331-340.

[5] Benim, A.C. 'Finite Element Analysis of Confined Turbulent Swirling

Flows', Int. J. Num. Methods Eng. 11 (1990) pp. 697-717.

[6] Schnell, U. 'New Developments in Modelling Near Field Swirl Burner

Flows', in: C. Taylor, P. Gresho, R.L. Sani and J. Mauser, (Eds.),

Numerical Methods in Laminar and Turbulent Flow, Volume 6, Part 1

(Pineridge Press, Swansea, 1989) pp. 307-317.

[7] Hagiwara, A.; Borz, S. and Weber, R. 'Theoretical and Experimental

Studies on Isothermal Expanding Swirling Flows with Application to

Swirl Burner Design - Results of the NFA 2-1 Investigations', IFRF

Report, Doc. F 259/a/3, 1986.

[8] D.S. Sloan, PJ. Smith and L.D. Smoot, "Modelling of Swirl in

Turbulent Flow Systems", Prog. Energy. Combust. Sci. 12 (1986) pp.

163-250.


Documents

shape. In such cases, an adequate modelling of the turbulent … · 2014. 5. 14. · The finite element predictions obtained with different turbulence models are evaluated by detailed