comparison of low reynolds number k-e turbulence models in fully developed pipe flow

Pergamon Chemical Engineering Science, Vol. 50, No. 12, pp. 1923-1941, 1995 Elsevier Science Ltd

Printed in Great Britain.

0009-2509(95)00035-6

C O M P A R I S O N OF LOW R E Y N O L D S N U M B E R k-e T U R B U L E N C E M O D E L S IN P R E D I C T I N G F U L L Y

D E V E L O P E D PIPE F L O W

C. M. HRENYA, E. J. BOLIO, D. CHAKRABARTI* and J. L. SINCLAIR Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A.

(Received 20 December 1993; accepted in revised form 26 December 1994)

Abstraet--A comparative study is presented of ten different versions of the low Reynolds number k-e turbulence model. The individual models are briefly outlined and evaluated by application to fully developed pipe flow. Numerical simulations were performed at Reynolds numbers of 7000, 21,800, 50,000, and 500,000. Predictions of the mean axial velocity, turbulent kinetic energy, dissipation rate of turbulent kinetic energy, Reynolds shear stress, eddy viscosity, and skin friction coefficients are compared to both experimental and direct numerical simulation data. The relative performance of the models is assessed. It has been found that the predictions between the models can vary considerably, particularly for the turbulent kinetic energy, its dissipation rate, and the eddy viscosity. The results indicate that the model proposed by Myong and Kasagi (1990, JSME Int. J. 33, 63-72) has the best overall performance in predicting fully developed, turbulent pipe flow.

INTRODUCTION

For wall-bounded flows, the high Reynolds number version of the k-~ model cannot be applied in the immediate vicinity of the wall since this model neglects the effects of viscosity. In order to avoid modeling these viscous effects, empirical wall functions are often employed to bridge the gap between the solid boundary and the turbulent core. The uni- versality of such functions breaks down, however, for complex flows. Hence, near-wall k-e turbulence models or low Reynolds number k-e models, which attempt to model the direct influence of viscosity, have been developed. Although computationally more intensive, these low Reynolds number models allow integration of the transport equations for the turbulent kinetic energy and its dissipation rate to the wall. For flows where anisotropic effects are impor- tant, such as rotating flows and flows in square ducts, the algebraic stress and Reynolds stress models are better able to predict turbulent quantities than the closures based on the eddy viscosity concept (Speziale, 1991). However, for simpler flows, such as shear flow over a flat plate, the isotropic eddy viscosity assumption inherent in the k-e model does not break down.

This paper presents a systematic evaluation of existing low Reynolds number k-e turbulence models. The study is restricted to the test case of fully developed turbulent pipe flow, a relatively simple flow yet one of utmost importance to the chemical engineering community. (This particular investigation was motivated by the present authors' research interest in

tPresent address: Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA, U.S.A.

modeling circulating fluidized bed reactors and was an integral part of that continuing program of work.) This selection of low Reynolds number models was guided by consideration of the results of a two-part investigation carried out by Martinuzzi and Pollard (1989) and Pollard and Martinuzzi (1989). Their study compared the ability of the four turbulence models mentioned above (high Reynolds number k-e model, low Reynolds number k-e model, algebraic stress model, and Reynolds stress model) to predict develop- ing, turbulent pipe flow at Re ranging from 10,000 to 380,000. It was found that the predictions from a low Reynolds number k-e model, the Lam and Bremhorst (1981) model, were in better overall agreement with the experimental data than the predictions from the Reynolds stress and algebraic stress models. In addition, the results from the low Reynolds number k-e model were superior to the high Reynolds number form for lower Reynolds number flows. This last result is not surprising, however, as the empirical wall functions required by the high Reynolds number closure are based on experimental data obtained at high Reynolds numbers.

Several versions of the low Reynolds number k-e turbulence model were reviewed and evaluated by Patel et al. (1985). The purpose of the current investigation is twofold:

(1) To extend Patel et al.'s (1985) study to include the more recent versions of the low Reynolds number k-e models documented in the literature. Ten different existing models are summarized. The origin of the different functions, constants, and assumptions inherent in each of the models is outlined.

(2) To test the predictive capabilities of the models for the case of fully developed pipe flow.

1923

1924

In genera L analyses of turbulence modeling for pipe flow applications are notably lacking in the literature. Patel et al.'s (1985) study focused on comparisons with experimental data obtained from external boundary layer flows. In fact, only five of the ten original papers outlining the models showed any comparisons with pipe flow data. It has been shown, however, that differences do exist between the flow in a pipe and a plane channel. For example, data obtained from experiments (Patel and Head, 1969) and direct numerical simulations (Kim et al., 1987; Eggels et al., 1994) indicate that fully developed flow in a pipe does not conform to the universal logarithmic velocity distribution until Re ~ 10,000, whereas plane channel flows agree with the law of the wall at Re > 3000. Hence, it is not clear which of the many proposed models can be used with the most confidence for the turbulent pipe flow application.

THE LOW REYNOLDS NUMBER MODEL

For steady, isothermal, incompressible, fully developed turbulent pipe flow, the axial component of the time-averaged Navier-Stokes equation is given in cylindrical coordinates as

O = r ~ r r v--~- r - - p dzz c (1)

where (dp/dz)c is the constant pressure gradient. The

Reynolds stress u'v' is described using the eddy viscosity assumption, which relates the turbulent transport of momentum to the eddy viscosity Vr and the mean velocity gradient:

- - dV~ u'v' = - VT d~-' (2)

The momentum balance then becomes

1 d [ d V . ] _ _ l ( d p ~ . 0 = r d r r ( v + v r ) d r j p \ d z ] c (3)

The turbulent kinematic viscosity in the low Reynolds number turbulence models is determined from the relation

k 2 Vr = c u fu - - (4)

C. M. HRENYA et al.

where k is the turbulent kinetic energy, e is the dissipation rate of this energy, defined as (using the standard Einstein notation)

1 t2 k = : v~ (5)

e v \ ? x f f (6)

andf , is a damping function to account for near-wall effects. The general form of the transport equations which determine k and e in the low Reynolds number models, simplified for the flows considered here, is given as

O = + vr~--~r ) - - e - - D ( 7 )

1

C2/2 ~2 - - - + E ( 8 )

k

where cu, c~, c2, ak, and a, are the same empirical constants found in the standard high Reynolds number model. The functions fa,f2 and, in some cases, D and E are included in the low Reynolds closure models in order to render the models valid to the wall.

Ten different low Reynolds number k-e models are evaluated. The models selected for comparison are the models of Jones and Launder (1972, 1973), Launder and Sharma (1974), Lam and Bremhorst (1981), Chien (1982), Lai and So (1990), Myong and Kasagi (1990), So et al. (1991), Yang and Shih (1993), Fan et al. (1993), and a modified form of the model proposed by Rodi and Mansour (1993) as given by Michelassi et al. (1993). Although the models proposed by Hoffman (1975), Hassid and Poreh (1978), and Dutoya and Michard (1981) were also examined, the predictions obtained from these models will not be shown since the other pre-1985 models performed considerably better, as was also found in Patel et al.'s (1985) study. The models vary within the general framework outlined above in terms of the assigned model constants, the functions f~,./'l, and f2, the extra terms D and E, and the boundary condition for e at the wall. This information is summarized in Tables 1, 2, and 3 for

Table 1. Numerical values for the constants cu, cl, c2, irk, and tr,

Researchers Model G, cl c2 trk or

Jones and Launder (1972, 1973) JL 0.09 1.45 2.0 1.0 1.3 Launder and Sharma (1974) LS 0.09 1.44 1.92 1.0 1.3 Lam and Bremhorst (1981) LB 0.09 1.44 1.92 1.0 1.3 Chien (1982) CH 0.09 1.35 1.8 1.0 1.3 Lai and So (1990) LSO 0.09 1.35 1.8 1.0 1.3 Myong and Kasagi (1990) MK 0.09 1.4 1.8 1.4 1.3 So et al. (1991) SZS 0.096 1.5 1.83 0.75 1.45 Yang and Shih (1993) YS 0.09 1.44 1.92 1.0 1.3 Fan et al. (1993) FLB 0.09 1.4 1.8 1.0 1.3 Rodi and Mansour (1993) and Michelassi et al. (1993) RMM 0.09 1.44 1.92 1.3 1.3

Comparison of low Reynolds number k-e Table 2. Summary of the functionsf, ,f l and f2

1925

Model f , A A

JL exp[ - 2.5/(1 + Rr/50)] 1 1 - 0.3 exp( - g 2)

LS exp[ - 3.4/(1 + RT/50) 2] 1 1 -- 0.3 exp( -- R 2)

LB [1 - exp( - 0.0165R,)]2(1 + 20.5/Rr) 1 + (0.05/f,) 3 1 - exp( - R~)

C H 1 - exp( - 0.0115 y+) 1 1 - {exp[ - (Rr/6) 2]

[ ( Re)] 1 - { e x p [ - ( R r / 6 , 2] LSO 1 - e x p ( - 0 . 0 1 1 5 y +) 1 + 1 - 0 . 6 e x p - ] ~

MK

SZS

YS

FLB

RMM

1-1 - exp( - y+/70)] (1 + 3 . 4 5 / ~ r ) 1

(1 + 3.45/x/~r)tanh(y + /l15) 1

(1 + l i a r ) [ 1 --exp ~ r

( - 1 . 5 × 1 0 - 4 R r

-- 5.0 x 10- 7Rya

- 1.0 x 1 0 - l °R~)] 1/2

0.4f./x//~r + (1 -- 0.4 f . / x / ~ r ) 1

x I t -- exp( -- R/42.63)] 3

where

f , = 1 - exp{ - ~f~/2 .3

+ (x/~r/2.3 - R/8.89)

[1 -- exp( -- Rr/20)] 3}

f~l[1 - exp( - 0.095Rr) ] where 1

f ~ = l f o r y + /> 100

f.' = [1 - exp( - 2 x 10-4y +

- 6 x 10-4y +2 + 2.5 x 10-7y+3)]

for y+ ~< 100

1+ d- 7

1

1 + ,,/-~7

1 - ~ e x p l / Rr'~2-]) 2 -tT) }s

[1 - 0.22 exp( - 0.3357x/~r) ]

x [1 - exp( - 0.095 Rr) ]

+ exp(1.8 Rp a) - 1 where = v, (t3 ~'~ 2

Rp k x / ~ \ &,]

each of the ten models. The models are listed in chronological order with an abbrevia ted des ignat ion [e.g. Launder and S h a r m a (1974), designated LS] which will be employed t h r o u g h o u t this paper.

DESCRIPTION OF MODEL CONSTANTS, FUNCTIONS,

AND BOUNDARY CONDITIONS

Model constants c,, Cl, c2, a~, and tr~ Table 1 summarizes the numerical values assigned

to the cons tan ts of the ten different low Reynolds n u m b e r models. The value of cu used by mos t models is 0.09. This value is fixed by the requi rement tha t for local equi l ibr ium shear layers, the p roduc t ion of

k equals diss ipat ion of k, and thus c~ = (u--7/k) 2.

Measurement of the Reynolds shear stress and turbulent kinetic energy in such flows indica te tha t c, is abou t 0.09.

The value of cl is est imated from a cons t ra in t relat ion result ing from consistency with the law of the wall region, where the molecular viscosity effects are negligible. For local equi l ibr ium flows with zero pressure gradient and logar i thmic velocity profile, the e equa t ion reduces to

/£2 cl = c2 a~x/~. (9)

where x is the Von K h r m a n constant . The set of cons tants in all of the ten models satisfy the above relat ion such tha t the Yon Kfirman cons tan t takes on

1926 C. M. HRENYA et al.

Table 3. Summary of D and E terms, and wall boundary conditions for k and e

Model D E Wall boundary condition

JL 2v (dx/k~2 [d2 G'X2

0 0

LS

LB

CH

LSO

k = e = 0

2vk/y 2 ( - 2ve/y2)exp( - 0.5 y+)

e /dx/k \ 2 expI_ RT 2

xe-(e-2v(dx~')2~ 1 (e 2vk~2q k \ dr / / - 2 - k - y2 )

MK* 0 0 d2k k=0, e = v - - dr 2

s z s 0 -

3 x l - 2-~ (e - 2v (d x/k ]2 dr ] / + 2"k ( e - 2vk']2 ]

vs o )

FLB 0 0

X'2vv~-~r2 ) + 0"0075v e \ d r / \ dr ] \ dr 2 ] RMM 0

k=e=0 dk d2k d r = 0 ' e=Vdr ~

k = e = 0

k = O, e \dr-r ]

= 2 v ( d , , / Z ~ 2 k = O , e \ d r /

= 2 v ( d v / k ' ] 2 k = O , e \ dr J

de = 0 k =0, dS

k = o, e \ - - ~ - /

*A numerically equivalent e boundary condition is given is given by ewaH = (4vk,e~t/ynext)2 _ e .... (Myong and Kasagi). This form was used for the solution of the MK model.

the value x = 0.41 +_ 0.02 with the exception of the JL model in which the set of constants imply x = 0.463.

The value for the model constant c2 in all ten models lies between 1.8 and 2.0. This range of values is based on experiments of decaying homogeneous grid turbulence. In grid turbulence, the diffusion and production terms in the k and e equations are zero, rendering an analytic solution for k in which c2 is the only constant appearing; hence, c2 was determined directly from measurements of the decay of k behind a grid at high Reynolds numbers.

The constants trk and a, are obtained by computer optimization. Most models use values of 1.0 and 1.3 for ak and a~, respectively. Exceptions to these values are found in the models of MK, SZS and RMM. MK increased the value of irk since most other models overpredicted the turbulent kinetic energy, as well as the eddy diffusivity, in the center of internal flows, which indicated that the value of ak was too low relative to a,. Similarly, RMM set the values of both ak and a, equal to 1.3. The different set of model constants employed by SZS was chosen based on calculations of fiat-plate boundary-layer flows.

Model functions f~, fl and 1"2 Table 2 outlines the different functions f , , fl, and

f2 for the ten models. The purpose of these functions is to modify the constants c~, c1, c2 to account for low Reynolds number effects. At high Reynolds number flows remote from the wall, all of the functions asym- ptote (Rr ~o0, y+ ~ o0) to the value one in accord- ance with the high Reynolds form of the model. Table 4 presents a summary of the motivation for the choi- ces of thefu,fl, and f2 descriptions and, in some cases, includes comments on how these descriptions affect the prediction of e.

The boundary conditions and the D and E terms At the tube centerline, the symmetry condition is

enforced for all of the variables. Assuming there is no slip at the solid boundary, both the mean axial velocity and the turbulent kinetic energy are equal to zero at the wall. Unlike these other variables, the boundary condition for e at the wall is not intuitive. It can be derived, however, by first considering the expansions of the fluctuating velocity components near the wall:

u' = aly + a2y 2 W a3y 3 + O(y 4) (10)

Comparison of low Reynolds number k-e

Table 4. Origin off. , fl , and f2

1927

Basis for choice/comments

Model f~ ft f2

JL, LS From predictions of constant Found no advantage in making stress Couette flow; vr was ft a function of RT obtained using a mixing length model and Van Driest's damping function and not eq. (4)

LB Derived by combining the eddy viscosity and dissipation rate equations of a one-equation model by Hassid and Porch (1978)

CH Direct wall effect through y+

Augmented to match experimental data as no E term was included in the model which solves for the true dissipation rate

Followed JL, LS

Determined from measurements of isotropic grid turbulence at high and low turbulence intensities

Tends to zero as R T tends to zero; does not correctly satisfy the asymptotic behavior of the e equation

Followed Hanjalic and Launder (1976); based on the lowest Re decay pattern data of Batchelor and Townsend (1948)

LSO Followed CH Enhanced generation ofe near the Followed CH wall; form suggested by Shima (1988)

Folowed JL, LS

Followed JL, LS

Form obtained from the equation which was developed

based on characteristic time scale

Did not modify e production since near-wall effects result in a net destruction of e

MK Rigorous derivation; developed a characteristic length scale expression for e valid over the entire turbulent Re range

SZS Modified the MK function to better fit the near-wall data given in Patel et al. (1985)

Based on characteristic time scale which approaches the Kolmogorov time scale in the near-wall region

FLB Determined using data of Patel Folowed JL, LS et al. (1985); function of RT and Ry only (not y+) in order to accommodate unsteady flow situations

Obtained by fitting DNS data of Kim et al. (1987) for flow between flat plates

YS

RMM

Proportional to y2 near the wall; correctly satisfies the asymptotic behavior of the e equation

All low Re effects subsumed into the E term

Form obtained from the e equation which was developed based on characteristic time scale

Modified Hanjalic and Launder (1976) to ensure asymptotic consistency of the e equation near the wall

Derived from evaluation of the terms in the exact e equation using DNS data of Kim et al. (1987) for flow betwen flat plates

v' = b2y 2 + bay 3 + O ( y 4) (11)

w' = c l y + c2y 2 W cay a + O ( y 4) (12)

where the coefficients a t, a2, etc. are constants . Substi- tu t ing these expansions into eq. (5) gives the near-wall behavior of k:

k = ay 2 + by a + cy 4 + O(yS) . (13)

Eva lua t ion of the k t r anspor t equa t ion at the wall using the above expression leads to the bounda ry condi t ion for e at the wall:

a2k swan = v - - (14) c3y 2"

Since k is p ropor t iona l to y2 in the near-wall region, an asymptot ical ly equivalent bounda ry condi t ion is

given as

(15) = 2 v

The more recent models (LB, LSO, MK, SZS, YS, and R M M ) implement one of the two bounda ry condi- t ions given above. In order to avoid the numerical stiffness tha t can be associated with these bounda ry condit ions, some of the earlier models (JL, LS, and CH) formulate the governing equat ions in terms of e*, which is equal to the true diss ipat ion rate minus its value at the wall D,

e* = e -- D (16)

ra ther than e itself. In these models e* is set equal to zero at the wall, a numerical ly convenient bounda ry

1928

condition. Hence, when this approach is taken, the additional term D is needed to balance the molecular diffusion of k in order to satisfy the k transport equation in the near-wall region. This term is negligible far from the wall. In the three models, D is either of the form given in eq. (15) or in an asymptotically equivalent form.

In several of the models (JL, LS, and YS) the E term had no physical justification. It was included to in- crease the predicted dissipation rate in order to obtain

C. M. HRENYA et al.

a realistic k profile in the near-wall region. LB, MK, and FLB altered other functions to achieve the same result, omitting the E term. The E term in CH was included to achieve a balance between wall dissipation and molecular diffusion of e at the wall. LSO adopted a form for E proposed by Shima (1988) who analyzed the near-wall behavior of the e equation in one-dimensional shear flow. SZS optimized the same form as LSO using the boundary layer flow data summarized in Patel et al. (1985). The E term used by

Vg

Ux

V Z

Ux

20

15

10

(a)

V z / U, r = 2.5 In Eggels et al. (1994)

. . . . . . . . . . . . . . . . . . . L S

Ctt

. . . . . . . . . . LSO • . . • , , , | • . . . . . • I

10 100

20

15

10

5

y +

r

(b) .

V z/u -- 2 5 In y+ + S5

Eggels etal . (1994)

. . . . . . . . . . YS

. . . . . FLB

i e . . . . a i I I . . . . l I I I

lO 100

y +

Fig. 1. Comparison of the mean axial velocity predictions at Re = 7000.


RMM was added in order to account for the near- wall e-production term involving the second velocity derivative.

SELECTION OF DATA

Model predictions were compared with the experimental data of Laufer (1954), Schildknecht et al. (1979), Patel and Head (1969) and the direct numerical simulation (DNS) data of Eggels et al. (1994). These

1929

data sets were chosen because they include complete, detailed measurements of fully developed, turbulent pipe flow at a large range of Re. More specific- ally, the data of Laufer (Re = 50,000 and 500,000), Schildknecht et al. (Re = 21,800) and Eggels et at. (Re = 7000) contain measurements of the mean axial velocity, the fluctuating velocity components, and the Reynolds shear stress, while the data of Patel and Head contain measurements of the coefficient of friction at Re < I0,000. In addition, the DNS data set generated by Eggels et al. (Re = 7000) also includes

V z

U~

V Z

Ux

30

25

20

15

10

25

(a)

0

v & " t , l r j s ' J f J f z , f ~ ~.1 .--~ f ~ i r~ r . ~ z

• ~l~hildknccht (1979)

LS

" ' " " ' " " LB

CH

. . . . . . . . . . LSO

a ! I . . . . . I . J |

0.2 0.4 0.6 0.8

r i g

20

15

10

. . . . " - " - "

YS and RMM " ' e ( ' l ' ~ ~

• Schildknecht (1979) " ~

MK

. . . . . . . . . . . . SZS

. . . . . . . . . . YS

FIB

. . . . . . . . . . RMM

0 0.2 0.4 0.6 0.8

r / R

Fig. 2. Comparison of the mean axial velocity predictions at Re = 21,800.

1930

the profile for dissipation rate of turbulent kinetic energy.

NUMERICAL PROCEDURE

A modified form of the finite-difference procedure proposed by Patankar (1980) was used for the solution of the three coupled ordinary differential equations [eqs (3), (7), and (8)]. For each case, a non- uniform grid was used. The number and concentra-

C. M. HRENYA et al.

tion of points used depended on the Reynolds number of that case, where denser grids were used for flows at high Re in order to resolve the steeper gradients present near the wall. Grid densities ranged from 50 to 100 points where at least half of the points were located in the range r/R > 0.9. The solutions were found to be insensitive to the grids since doubling the number of grid points changed the solution profiles by less than 1%. It should also be noted that for each experimental system, the axial pressure gradient was

V Z

Ux

V Z

Ux

30

25

20

15

10

,It. - LS andOt (a)

• Laufer (1954)

JL

. . . . . . . . . LS

. . . . . . . . . . L ~

CH

. . . . . . . . . . LSO

0 , I , I , I ~ I ,

0 0.2 0.4 0.6 0.8 1

r / R

Fig. 3. Compar ison of the mean axial velocity predictions at Re = 500,000.

r/R

30 (b)

• Wa fa fa B41 ird pi~a ~d "~ °°

25

15 • Laufer (1954)

MK 10 . . . . . . . . . . . SZS

. . . . . . . . . . YS

FLB 5

. . . . . . . . . . RMM

0 , I , l , l , l ,

0 0.2 0.4 0.6 0.8


determined based on the reported value of the friction velocity.

PRESENTATION AND DISCUSSION OF RESULTS Model predictions of the mean axial velocity, tur-

bulent kinetic energy, eddy viscosity, Reynolds shear stress, and dissipation rate of turbulent kinetic energy were obtained at Re = 7000, 21,800, 50,000 and 500,000. Since the ability of each model to predict a given variable are often similar for a range of Re,

1931

only profiles at a "representative" Re are shown. For the sake of clarity, the predictions of the ten models are divided into two plots, "a" and "b" in each case. Figures labeled "a" show the results from the first five models listed in the tables; figures labeled "b" show the results from the five most recent models.

The predicted mean axial velocity profiles and the DNS data at Re = 7000 are shown in Fig. 1 using semi-logarithmic coordinates. As mentioned earlier, at this low Re, pipe flow data do not conform to the

k

Ux 2

k

Ux 2

6

0

(a) Eggels et al. (1994)

JL

.................... L S

. . . . . . . . . . LB s , , ° ¢ ~

..... CH .'" . ~f-~:% 0o* • ,*

. . . . . . . . . . , s o . , . ; , : > ; 7 ~,.s • B*

s • S o"

"S;"" " "

. . a k . s . m,.. aw.. e,... m..- a - " ~ ' ' P " "d]~

0 0.2 0.4 0.6 0.8 1

r / R

6 L MKE e'seta ' 19 5 ................... SZS

YS / '%

L . . . . . . . . . . RMM

3 .

01 ~ " "' ; ' ; " " " '

0 0.2 0.4 0.6 0.8 1

r / R

Fig. 4. Comparison of the turbulent kinetic energy predictions at Re = 7000.

1932

universal velocity distribution for y+ > 30. The models which provide the best comparison with the DNS pipe flow data include LS, CH, MK, and FLB, while the other models show considerable differences from the data in both the slope and the value of the velocity distribution. Similar plots at Re = 21,800 and 50,000 are not shown since the data agree well with the universal velocity distribution over the entire pipe radius.

In Figs 2 and 3, the mean axial velocity profiles are plotted with the experimental data at Re = 21,800

C. M. HRENYA et al.

and 500,000, respectively. In general, these figures reveal an underprediction of the experimental velocity near the pipe core. A significant variation also exists between the results of the different models. Over the entire Reynolds number range, the model of JL yields the poorest predictions of the mean axial velocity profile. The centerline velocity predictions from this model differ from the experimental data by more than 10%. Again, the best reproduction of the mean axial velocity is generated from the models of LS, CH, MK, and FLB. At the lower Re, the profiles from all of the

6 I 5

k

Ux 2

0

(a)

Schildknecht (1979)

JL :" . . . . . . . . . . . . . .

. . . . . . . . . . ,;'!'; I

. . . . . CH , .{, .

. . . . . . . . . . LSO ~ , ~ ~ ~ .~"

. . . . . __. ~ ~ ' " # "

, I , I , I , I ,

0.2 0.4 0.6 0.8

r / R

3 k

U,c 2 2

(b)

• Schildknecht (1979) I I

MK ; .............. SZS /~-¢~;1 . . . . . . . . . . YS z,"

. . . . . FLB . :

. . . . . . . . . . RMM z~/"

0.2 0.4 0.6 0.8

r / R

Fig. 5. Comparison of the turbulent kinetic energy predictions at Re = 21,800.

Comparison of low

models are flatter than indicated by the experimental data. The agreement between the slope of the data and the predictions is improved at the highest Reynolds number investigated, Re = 500,000.

The turbulent kinetic energy profiles are shown for increasing Reynolds numbers in Figs 4-6. For both low and high Re flow, the radial position at which the maximum value of the turbulent kinetic energy occurs does not vary significantly between models. The results at the lowest Reynolds number, shown in Fig. 4,

Reynolds number k-e t933

indicate that only the model of MK is able to predict both the centerline and peak turbulent kinetic energy within 15% of the DNS values. The models of LB, CH, LSO, SZS, YS, FLB, and RMM, while showing relatively good agreement with the magnitude of the energy maximum, overpredict the centerline intensity. The remaining models (JL and LS) show a consistent underprediction of the peak intensity. The results shown in Fig. 5 for Re = 21,800 are similar to those at the lower Re, except that both the MK and RMM

6

5

4

3 k

Ux 2 2

0

(a)

• Laufer (1954)

- JL

. . . . . . . . . . LS

. . . . . . . . . . I.,B • • i

..... CH

. . . . . . . . . . LSO

,lul,v Lv ii,,v lUl,9 n * ~ ' ~

0

, I ~ I , I , I ,

0.2 0.4 0.6 0.8

rlR

5

4

3 k

Ux 2 2

1

• Laufer (1954)

MK

.. . . . . . . . . . . . SZS

. . . . . . . . . . YS •

..... PLB •

RMM • ~ . #

• "~

............. ---: -.-'~-'-'~'i;~,g.

0 , I ~ I , I , I ,

0 0.2 0.4 0.6 0.8

r/R

Fig. 6. Comparison of the turbulent kinetic energy predictions at R e = 500,000.

1934

models are able to predict the centerline turbulent kinetic energy within 10% of the experimental measurements. The increased ratio of trk/a~ in the MK and RMM models is responsible for the ability of these models to resolve this weakness inherent in the other models. At the highest Re, shown in Fig. 6, the MK and RMM models give the best agreement at the pipe centerline, while none of the models yield good predictions of k near the wall.

The distributions of the eddy viscosity are depicted in Figs 7-9. While the eddy viscosity cannot be dir-

C. M. HRENYA et al.

ectly measured, its value can be extracted from the experimentally obtained mean axial velocity and Reynolds stress profiles via eq. (2). The slight scatter in the data at Re = 500,000 is due to the inability to resolve the small velocity gradients which were present near the tube centerline. Although the velocity gradient was determined from three- and five-point finite-difference approximations, as well as cubic spline approximations, no significant improvement in the scatter was obtained. Considerable qualitative and quantitative differences exist between the model

0.2

0.15

0.1 v t

U x R

0.05

0.2

0.15

0.1 vt

U x R

0.05

(a) • Schildknecht (1979)

JL

................. LS

.......... lob

CH

.......... LSO

: : -_: ) )

~'..~. , . ) ) )

0.2 0.4 0.6 0.8 1

r / R

Co) * Schildknecht (1979)

MK

......................... , ................... SZS

- "" ""'~,. " ........ " YS

" "-.,, - .... FLB

".,...,.. - ......... RMM i

"°'°~.°

2:'-'-':" .............. "\,

, I , I , I , ~ I . -i

0 0.2 0.4 0.6 0.8 1

r / R

Fig. 7. Comparison of the eddy viscosity predictions at Re = 21,800.

Comparison of low Reynolds number k - e 1935

0.2

0.15

0.1 v t

Ux R

0.05

(a) • Laufer (1954)

JL

LS

• . . . . . . " • = L ~

CH

. . . . . . . . . . LSO . ; ' ; ' ; ' ; . : . . ' : - . . . . . . . . . . . . .

• • • • -'~;"~"~O-"'~ %,~__~,~

0 0.2 0.4 0.6 0.8 1

r/R

0.2

0.15

0.1 v t

U x R

0.05

0

Co) • Laufer (1954)

MK

.............. SZS

: ........ ... . . . . . . . . . . YS

" ~ ' " " ............. .. . . . . . F L B

....... • ..,,. ~ . . . . . . . . . . RMM

, . . . . . . . ' . 2 . = . . . . . ~." ~.° . . . . " ' . .

0 0.2 0.4 0.6 0.8 I

r/R

Fig. 8. Comparison of the eddy viscosity predictions at R e = 50,000.

predictions at all levels of R e . With the exception of the MK and RMM models, all of the models predict that the eddy viscosity increases monotonically from the wall and reaches a maximum at the pipe centerline. This behavior is in contrast to the experimental results in which the maximum in the eddy viscosity occurs away from the pipe center. At R e = 50,000 and 500,000 the eddy viscosity predictions of MK are in good quantitative agreement with the experimental data, while the predictions of the remaining models are significantly larger than experi-

mental values near the pipe center (15-95% error). MK attribute the improved predictions to the larger ak/a, ratio. At R e = 21,800, while the relative performance of the models does not change, the absolute deviation of the predictions from the measurements increases. Even though significant differences exist between the model predictions of eddy viscosity for both low and high R e flows, these differences do not seem to greatly influence prediction of the mean axial velocity profile. Such differences will become impor- tant, however, for heat and mass transfer applications.

1936 C.M. HRENYA et al.

0.2

0.15

0.I vt

U x R

0.05

(a) • Laufer (1954)

JL

................ LS CH and I.SO

. . . . . . . . . . L ~

LS and LB CH

. . . . . . . . . . LSO

~ ° o . . ~ . I L .

0.2 0.4 0.6 0.8

rlR

0.2

0.15

0.I V t m

U, t R

0.05

(b) • Laufer (1954)

MK

. . . . . . . . . SZS

. . . . . . . . . . YS

! . . . . . ..,,..~. FLB

• ~""'~..,.~ . . . . . . . . . . RMM

.-.:..--..- • - . . . . . . . . . . . . . . . . . . . . . . . . a--_.-.-~. ~r,,

0 , I , I , I , I ,

0 0.2 0.4 0.6 0.8

r/R

Fig. 9. Comparison of the eddy viscosity predictions at Re = 500,000.

m

The radial distributions of the u'v' field over the entire pipe cross-section are shown in Figs 10 and 11. On this macroscopic scale for which experimental data are available, the results from the different models are barely indistinguishable at Re = 21,800 and Re = 500,000 and compare well with the data.

Figure 12 shows predictions of the dissipation rate of turbulent kinetic energy at Re = 7000 along with the corresponding DNS data. Comparisons with data at higher Re were not possible since neither of the experimental data sets contained all of the turbulent correlations necessary to determine e [eq. (6)]. It

Comparison of low Reynolds number k-e 1937

0.8

l l * V i

0.6

Ux2 0.4

0.2

0 w

0

• Schildknecht (1979)

JL

................. LS

. . . . . . . . . . L B

CH

. . . . . . . . . . LSO

/ / , I , I , I a I

0.2 0.4 0.6 0.8 1

r / R

0.75

0.5

Ux 2

0.25

0

• Schildknecht (1979) 0a)

.................... SZS

. . . . . . . . . . y s

. . . . . FLB

. . . . . . . . . . RMM •

w

0 0.2 0.4 0.6 0.8 1

r/R

Fig. 10. Comparison of the Reynolds stress predictions at Re = 21,800.

should be noted, however, that the qualitative features of the model predictions were the same at all Re. In the case of models which solve for e*, the true dissipation rate minus its value at the wall, the D term is added in order to generate results which reflect the true dissipation rate and a corresponding finite value of e at the wall. A significant qualitative difference in

the shape of the e profile is observed with the SZS model; namely, e reaches its maximum value at the wall and plateaux where the other models show a maximum. This qualitative behavior displayed by the SZS model matches the DNS behavior, al though the predicted values are considerably larger than the simulation data. SZS were able to obtain this near-

CES 50-12-G

1938

l l ' V '

v?

0.8

0.6

0.4

0.2

C. M. HRENYA et al.

• Laufer (1954)

JL

LS

CH

. . . . . . . . . . LSO

/ (a) 0 , I , I , l , I i

0 0.2 0.4 0.6 0.8 1

r / R

0 . 8

0.6

Ut¥ ~

Ux2 0 . 4

0.2

• Laufer (1954)

MK

................. SZS

. . . . . . . . . . YS

FLB

. . . . . . . . . . RMM

I , I , I i I

0 0.2 0.4 0.6 0.8 l

r / R

Fig. 11. Comparison of the Reynolds stress predictions at Re = 500,000.

wall behavior by optimizing the E term proposed by Shima (1988). Although the YS and RMM models also predict that ~ reaches a maximum at the wall, the local minimum and maximum value of e displayed away from the wall is considerably more pronounced than that shown by the DNS data.

The computed friction factors for pipe flow from the JL model are shown together with the data of

Patel and Head (1969) and the Blausius formula in Fig. 13. The prediction from only the JL model is shown as this is the only model which revealed any slight discrepancy with the data over the range of Re

numbers investigated, In addition, this is the same model which showed the poorest axial velocity predictions. It should be noted that there is a slight deviation with the data, similar in magnitude to the JL

Comparison of low Reynolds number k-e 1939

0.3

0.2

£ v

Ux 4

0.1

0 0.8

(a) Eggels (1994)

JL

. . . . . . . . . . LS

. . . . . . . . . . L B

CH

. . . . . . . . . . LSO

0.85 0.9 0.95 1

r / R

0.3

0.2

~ v

l.Jx4

0.1

0 0.8

(b) Eggels (1994)

MK

.................. SZS

. . . . . . . . . . YS

FLB

. . . . . . . . . . RMM Ill

0.85 0.9 0.95 1

r / R

Fig. 12. Predictions of dissipation rate of turbulent kinetic energy at Re = 7000.

deviation, with the LSO and SZS models at R e numbers less than 5000.

CONCLUSIONS

In this paper, ten different versions of the low Reynolds number k - e models are examined for application to fully developed pipe flow. It is shown that significant qualitative and quantitative differences

exist between the model predictions for the range of R e investigated. These relative differences are most apparent in the predictions of the turbulent kinetic energy, eddy viscosity, and dissipation rate. Overall, the MK model offers the best performance for the pipe flow application. The model is able to reproduce the experimental measurements of the turbulent kinetic energy throughout the domain and to predict the correct qualitative behavior of the eddy viscosity pro-

1940 C.M. HRENYA et al.

Cf

0.1

0.01

0.001 1000

Blausius

• Patel & Head (1969)

• JL

• . • • . • • | • • • • • • •

10000 1 ~

Re

Fig. 13. Comparison of the JL model prediction of friction factor.

file. In general, the M K model yields predictions for u,v ~ the axial velocity, Reynolds stress, and friction factor U~ that are comparable to the best predictions from the VcL other models over the range of Re. However, a clear V~ deficiency in the M K model is in the predicted e distri-

Y bution. Only the SZS model captures the same shape y+ of the e distribution as is found in current DNS data.

Acknowledgements--The authors gratefully acknowledge funding support from the DOE University Coal Research Program Grant No. DE-FG22-92PC92540, the National Science Foundation Presidential Young Investigator Awards Program Grant No. CTS-9157185 with matching funds from the Amoco Oil Company, and the Alcoa Foun- dation. The authors also appreciate the cooperation of Pro- fessor Nieuwstadt who provided the complete DNS data set.

NOTATION

G,c~,c2 constants in k-e model C: coefficient of friction (dp/dz)c constant axial pressure gradient D term contained in the k equation E term contained in the e equation f . , f l , f2 functions in k-e model k turbulent kinetic energy P mean pressure r radial coordinate R pipe radius Re Reynolds number based on the centerline

velocity ( = 2R VcL/V) Rr turbulent Reynolds number [ = k2/(w)] Ry turbulent Reynolds number based on

Y ( = ykl/2/v) u', v', w' fluctuating velocity component in the z-, r-,

and 0-directions, respectively

Reynolds shear stress

friction velocity [ = x / - R(dp/dz)c/(2p)] mean axial velocity at pipe centerline mean axial velocity normal distance from the wall ( = R - r) dimensionless distance from the wall ( = yU~/v) axial coordinate

Greek letters

g*

K 2

V

~T 0

P O " k

O " e

Subscripts next

wall

dissipation rate of turbulent kinetic energy dissipation variable ( = t - D) Von K~irman constant

Taylor microscale ( = ~ ) kinematic viscosity eddy viscosity ( = c~fvk2/e) azimuthal coordinate density turbulent Prandtl number for k turbulent Prandtl number for e

indicates property value at the grid point next to the wall indicates property value at the wall

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