comparison of various turbulence models in rotating...

International Journal of Rotating Machinery2000, Vol. 6, No. 5, pp. 375-382

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Comparison of Various Turbulence Models inRotating Machinery Blade-to-Blade Passages

E.Y.K. NGa’* and S.T. TANb

School of Mechanical and Production Engineering, Nanyang Technological University,Nanyang Avenue, Singapore 639798; bFerroTec Cooperation (S) Pte Ltd,

Kallang Basin Industry Estate, Kallang Ave., Singapore 639798

(Received 26 February 1999," In finalform 22 March 1999)

Numerical calculations on four blade passages are done using Q3D Navier-Stokes solverwith a simple mixing length turbulence model and two more advanced transport-equationapproaches. Mixing length is simple and cheap but crude, while more sophisticated transportapproaches are more physical but more expensive. Predicted results using different turbulentmodels are discussed and compared with the laminar flow and well documented experimentalresults. Studies show that the model with more transport-equation predicts improved result asit includes the effects of upstream history into the velocity scale.

Keywords." Turbulent, CFD, Turbomachinery

INTRODUCTION

Most flow fields in fluid machinery passage areturbulent flow and it is one of the most complexproblem in the area ofcomputational fluid dynamicsuch as stall and surge phenomena in compressorsystem [14]. It is believed that the solution of time-dependent three-dimensional full Navier-Stokesequations could describe turbulent flows com-

pletely. However, the computers such as work-stations are not large and fast enough to solve theequations directly, for the required range of lengthand time scales, even for simple flows. Hence, it ispractical of using some of the turbulent modelling

to describe the turbulence motion instead of solvingthe full Navier-Stokes equation. Many publica-tions recommended various types of turbulentmodels such as those by Baldwin-Lomax [1],Cebeci-Smith [2], Birch [3], Chien-Kim [4],Launder-Spalding [5], Myong-Kasagi [6], etc.

Depending on the number of transport-equationused, the turbulent models can be classified into

zero-, one- and two-equation and higher-ordermodels. Theoretically speaking, the more the num-ber of transport equations involved, the more accu-rate the prediction is, as less assumptions are used.The aim of this paper is to evaluate the differenttypes of turbulent models including zero-, one- and

Corresponding author. Tel." (01065)790-4455. Fax: (01065)791-1859. E-mail: [email protected].

375

376 E.Y.K. NG AND S.T. TAN

two-equation models using a Q3D Navier-Stokes[7] and full energy equation unless otherwise stated,in one host code, with experimental data for axialturbomachinery application.

TURBULENT MODELS

Baldwin-Lomax’s Zero-Equation Model [1]

In zero-equation model, the concept of mixinglength is used. Dimensional analysis of variablesshows that the turbulent viscosity, #T, divided bythe density p has the same dimensions as a lengthmultiplied by a velocity. Hence momentum argu-ments can be used to show that #T is a function ofthe flow density, a length scale in the flow and thelocal mean flow velocity. Typically, this relation-ship is given as

(T Dlm20U

For inner layer"

lm n YD,

for outer layer:

lm Yrnax FKLEB,

where n is the von-Karman constant (0.41), Y isthe distance normal to nearest wall, D is the vanDriest damping factor with D e-y+/A+, y+(Yv/PwWw)/#w, A+=26, Ymax is the value of Y atthe maximum value of the function Fw(Y) whichis Fw(Y)= YZlwl, [w[ is the magnitude of thevorticity given by Iwl =Ou/Oy-Ov/Ox, FKLEB isthe Klebanoff intermittency factor:

FKLEB 1+5.5Ymax

-1

The turbulent viscosity can be written as:

Inner layer

tT p(YD)2[Iwl]; (2)

Outer layer

[T pCcLCcpFwakeFKLEB (3)

with CCL 0.0168, CCp 1.6, CKLEB 0.3.

fwake minCwake Ymax 2Udir/Fmax,

where Ymax is the location of the maximum value,Fma of the function, Fw--Y[wlD, Udi is the dif-ference of the maximum and minimum of u: Udir

b/max- b/min, Cwake is the model constant, 0.25.

Birch’s One-Equation Model [3]

In one-equation model, the turbulent viscosity isrelated to the turbulent kinetic energy k and iswritten as

[T C#oGx/’-lm. (4)

The value of k can be found from the transportequation

O(pk) O(puk) O(pvk)Ot Ox Oy

OX [Ze-x +-O-fy [e + Pu Pe,

where e is the distribution of the dissipation rateof k,

I Ou’i Ou’ip Ox. OXj’

Ou Ou Ov Ov+ + + Oy’

T(j u u

Energy dissipation equation is

e L---/ZT + CD2/z (6)

with F-l-exp(-B,Rek), Rek-(pvL)/#,CD 0.164, CD2--0.336, Cu--0.548, Bu--0.029,

VARIOUS TURBULENCE MODELS 377

=0.41, lm is the mixing length scale which isexactly the same as that used in the Baldwin-Lomax’s model.

Standard Two-Equation Model [5]

A standard k-c model developed by Launder andSpalding [5] is also coded. This is the mostcommonly used model for CFD calculations. Here

(7)

The partial differential equations used to find thevalues of k and c are,

turbulence kinetic energy equation:

Ot Ox Oy0 0

and energy dissipation equation:

Ot Ox Oy0 0

-{- - C Gk (9)

where

Gk--#, 2 x + y + yy+x0.9, crc 1.22, cl 1.44, c2 1.92,0.09.

INITIAL AND BOUNDARY CONDITIONS

Inlet Conditions

Values of k and e are in general not known at theinlet but some reasonable assumptions can be

made. The kinetic energy of turbulence is estimatedas a percentage of the square of the average inletvelocity [8]:

kinlet Ti2, O)

where is the average inlet velocity, Ti is theturbulent intensity in percentage. Here, Ti is set tobe 3% for compressors and 1% for turbine. Thedissipation rate is calculated according to theequation

k312inlet Cl, 2aro

(11)

where ro is the outer radius of the machine annular,a 0.005, C, 0.09.

Outlet Conditions

For the fully developed flow:

Ok) O, (12)outlet

--0, (13)outlet

where n is the streamwise direction.

Wall Function

With the general k-e model and wall function,where the first computational grid point P close tothe wall is in the turbulent sublayer, the followingformulae are used:

y pu(14)

ln(y )+ B, (15)

(16)


where -w is the wall shear stress

y- vVtP U YP > 11.5, (17)

Vtp- 0, yp <_ 11.5, (18)

t 0.4-0.42, B- 5.0-5.5, Cu 0.09,

2

kp u (19)

t--,3/4/3/2Ep

"-’#"P (20)

(a)

0.10

0.05

0.00

NUMERICAL RESULTS

Four experiment test cases are used to validate thepredicted results. The first is a transonic compressorrotor, experimental work has been done at theDFVLR [9] in Gottingen. The computational H-mesh and measured IsoMach contour plot at 45%span are shown in Figs. l(a) and (b). This meshdensity has been determined to represent’ a goodcompromise between economy and grid indepen-dence. It is sufficient to resolve down to either linearsublayer (for y+<_ 11.25) or log-law layer(11.25 < y+ < 500) as the value of y+ for the gridnext to the solid boundary is between 0.7 and 36.The IsoMach contours computed in Figs. 2-4

show that, on the suction side, strong accelerationjust after the leading edge followed by a weakoblique shock. Different models produce slightlydifferent results. The shock predicted by zero-

equation model smears out into a wider numberof grid. However, both the one- and two-equationmodels predict a sharper shock which are closer tothe experimental result. All the models predict theshock at about 18-20% chord of the suctionsurface. For the same axial chord, the one-equationmodel produces the highest Mach number, the zero-

equation model gives the lowest value while thetwo-equation model predicts an in between valuethat is closest to the experimental results. In brief,

0.00 0.02 0.04

(b)

SPEED :100"/.nABLADE HEIGHT" 45%

FIGURE (a) Grid generated for transonic compressor rotor(mesh: 86x 45); (b) Measured IsoMach contour at 45%span [9].

VARIOUS TURBULENCE MODELS

0.05

0.00 0.05 0.10 0.15

FIGURE 2 IsoMach contour at rotor mid-span using zero-equation model.

0,20

0.15

0.10

0.05

379

0.05

FIGURE 3 IsoMach contour at rotor mid-span using one-equation model.

0.05

FIGURE 4 IsoMach contour at rotor mid-span using stan-dard k-c model.

0.00

0.00 0.05

FIGURE 5 Grid generated for UTRC turbine stator (mesh:42 x 35).

Experimtd1.00| Equo

.I.0.00 TEq

-2.00 .......-3.00 A

-4.00

x x-7.00

0.00 0.20 0.40 0.60 0.80 1.00

X/Cx

FIGURE 6 Blade pressure-coefficient prediction for UTRCturbine stator.

the predicted result by each model agrees qualita-tively well against the experimental data.The next test cases are the UTRC turbine blades.

Experimental results [10] have been widely pub-lished. The mesh are shown in Figs. 5 and 7 forstator and rotor respectively.The blade surface pressure-coefficient distribu-

tions (Cp- (P-P)/O.5p, U) based on the inlet


condition for UTRC turbine stator at the nominaloperating point are compared in Fig. 6. It showssimilarity between the predictions using eachturbulence model. On the suction surface at same

X/Cx chord position, the zero-equation modelpredicts a lower value as compared to predictionby one- and two-equation models. However, on thepressure surface, all the models predicted similar Cpvalues and are very close to the experimental result.As the boundary layer at pressure surface isrelatively thin and no significant separation occurstherefore all the turbulent models are able to predictmore accurate results. Included also is the Cpprediction by laminar flow having similar shape toturbulent assumption but with a much lower Cpvalue at suction side.

Similarity, Fig. 8 compares the Cp distributionsfor turbine rotor. On both the blade surfaces atsame X/Cx position, zero-equation model predicts alower value as compared to that by one- and two-equation models as well as the experimental result.The latter models are able to predict results whichare very close to experimental result on pressuresurface and second half of the suction surface.

0.10

0.05

0.000.10 0.15

FIGURE 7 Grid generated for UTRC turbine rotor (mesh:45 x 35).

At the first 50% chord on the suction surface, theexperimental result is lower than the prediction byone- or two-equation models. Same as in case ofstator the Cp prediction by laminar flow has similarshape with that predicted by turbulent models butwith a much lower Cp value at suction side. In themeasurement, the flow has passed through thestator, which results in wake forming, flow distor-tion and non-uniformity at the rotor inlet in con-trast to the computation. Large error at entranceof rotor was also found in the calculation by Leeet al. [11].The final case is on C4 compressor blade where

experimental data are available from [12] (Fig. 9).

Equa-3 TX OneEqua

/ Equat A

/’X_4= " .....................X,,

-ax [],.x

-7 ...............D.................................................x R x

[]-8 []

-9

0.00 0.20 0.40 0.60 0.80 1.00

X/Cx

FIGURE 8 Blade pressure-coefficient prediction for UTRCturbine rotor.

FIGURE 9 Grid generated for C4 compressor blade (mesh:98 x 45).

VARIOUS TURBULENCE MODELS 381

0.4 -Equa

-1.1 Equc

1.6

0.2 0.4 0.6 0.8X/Cx

0,9 Experimental Result

0,8 + [] Zero Equa. Model

X One Equa. Model0.7

/ Two Equa. Model

0.6 Laminar

0.4

0.3

X ............... []0.1

0.2 0.80.4 0.6U/LIe

FIGURE 10 Blade pressure-coefficient prediction for C4compressor blade.

FIGURE Non-dimensionalized Y-direction distance vs.velocity curve at 36% chord of suction side of C4 compressorblade.

Figure l0 compares blade surface pressure-coefficient distributions at the nominal operatingpoint, showing similarity between the predictionsusing each turbulence model. On suction surface, atsame position, it is shown that zero- and one-

equation models predict similar values which are

slightly lower than two-equation model as wellas the experiment values. However, on pressuresurface, zero- and one-equation models predic-tions agree well with measured data. Figure 10also shows that the two-equation model agreesvery well with the measurement at both bladesurfaces. The prediction of Cp for laminar flow issimilar to that predicted by turbulent models atmost portion of the blade surfaces, except theregion near trailing edge of the pressure side wherelaminar model predicts a small boundary separa-tion bubble and hence lower pressure recovery ascompared to turbulent models.

Local velocity profiles along Y-axis are plottedat 36% and 64% chords and compare withexperimental result [12] as shown in Figs. 11 and12 respectively. The Y/Ye term is the non-dimensionalized normal distance from the bladesurface Y with the boundary layer thickness Ye(defined as the grid point with a speed less than98% different compared to adjacent grid). Thespeed U is non-dimensionalized with speed Uewhich is the speed at Ye. At both chord-location,

0.9

0.8

0.7

0.6

0.4

0.3

0.2

0.

Experimental Result

[] Zero Equa. Model

One Equa. Model

Two Equa. Model

Laminar

0.2 0.4 0.6 0.8

FIGURE 12 Non-dimensionalized Y-direction distance vs.velocity curve at 64% chord of suction side of C4 compressorblade.

the velocity profiles predicted by two-equationmodel agree very well with the experiment. Atabove 20% of boundary layer thickness, zero- andone-equation models predictions show good agree-ment with the experimental result. However,below 20% of boundary layer thickness, zero-

equation model predicts lower value while theone-equation model gives a higher value. In brief,all the models are able to predict the growth ofboundary layer and boundary separation. Thelaminar flow .prediction is found to be quitesimilar to the zero-equation model. More detailscould be found in [13].


CONCLUDING REMARKS

The computed results were compared with measure-ment to validate the code and assess the quality ofthe numerical solution. The performance of theturbulence models to predict the flow through a

blade passage depends on the number of transportequations used and on the inlet flow conditions.Another observation from the models used is theirdifferent separation behavior within the boundarylayers.

It is shown that in most cases the two-equationmodel produces results which are closest to theexperimental results followed by one-equationmodel. As the less simplification is made the closerto physics it will be.

Finally, for accurate simulations of fluid machi-nery, extension to three-dimension with transitionand higher-order of turbulent model are needed.

References

[1] B. Baldwin and H. Lomax (1978). ’Thin layer approximationand algebraic model for separated turbulent flows’, AIAAPaper, no. 78-257.

[2] T. Cebeci and A.M.O. Smith (1974). ’Analysis of turbulentboundary layers’, Applied Mathematics and Mechanics,Vol. 15, Academic Press, New York.

[3] N.T. Birch (1987). ’Navier-Stokes predictions of transi-tion loss and heat transfer in a turbine cascade’, ASME87-GT-22.

[4] Y.S. Chien and S.W. Kim (1987). ’Computation of turbul-ent flows using an extended k-c turbulence model’, NASA-CR-179204.

[5] B.E. Launder and D.B. Spalding (1974). ’The numericalcomputation of turbulent flow’, Comp. Math. Appl. Mech.Eng., 3, 269-289.

[6] H.K. Myong and N. Kasagi (1990). ’A new approach to theimprovement of the k-c turbulence model for boundedshear flows’, JSME Int. J., Ser. B, 33, 63-72.

[7] E.Y.K. Ng and Y. Miao (1996). ’Viscous flow prediction ofthe single stage axial rotating machineries’, CFD Journal,403-420.

[8] H.K. Versteeg and W. Malalasekera (1995). An Introductionto Computational Fluid Dynamics, Longman Scientific &Technical, Loughborough.

[9] L. Fottner (1990). ’Test cases for computation of internalflows in aero-engine components’, NASA AGARDAdvisory Report No: 275 (VI.4 Test Case E/CO-4 byDunker, R.) pp. 245-285.

[10] R.P. Dring, H.D. Joslyn, L.W. Hardin and J.H. Wagner(1982). ’Turbine rotor and stator interaction’, ASMEJournal ofEngineeringfor Power, 104, 729-742.

[11] Y.T. Lee, T.W. Bein, J. Feng and C.L. Merkle (1993).’Unsteady rotor dynamics in cascade’, ASME Journal ofTurbomachinery, 115, 85-93.

[12] E.Y.K. Ng (1992). ’A high resolution coupled parabolic/elliptic Navier-Stokes solver for turbomachinery flows’,Ph.D. Thesis, University of Cambridge, UK.

[13] S.T. Tan (1998). Numerical investigation of viscous flow inrotating cavity with different turbulence models, M. Eng.Thesis, Nanyang Technological University, Singapore.

[14] E.Y.K. Ng, S.T. Tan and H.N. Lim (1997). ’Simulationof instability flow in compressor system’, Proceedingsof 8th Aerospace Technology Seminar, Singapore,pp. PR 4/1-4/7.

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