A near-wall two-equation heat transfer model for wall turbulent flows

International Journal of Heat and Mass Transfer 44 (2001) 691±698

www.elsevier.com/locate/ijhmt

A near-wall two-equation heat transfer model for wall turbulent ¯ows Baoqing Deng a,*, Wenquan Wu a, Shitong Xi b a

Department of Power Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China b School of Power and Energy Engineering, Shanghai Jiaotong University, Shanghai 200030, China Received 4 November 1999; received in revised form 9 April 2000

Abstract A proposed near-wall t 2 ±et two-equation model for turbulent heat transport reproduces the correct near-wall behavior of temperature under various wall thermal boundary conditions. In this model, a mixing timescale is introduced to model the production term of et equation, and a more convenient boundary condition for et under the uniform wall heat ¯ux is suggested. The present model is tested through application to turbulent heat transfer for channel ¯ow. Predicted results are compared with direct numerical simulation (DNS) data. The near-wall t 2 ±et twoequation model predicts reasonably well the distributions of the time-mean temperature, normal turbulent heat ¯ux, temperature variance, dissipation rate and their near-wall budgets. 7 2001 Elsevier Science Ltd. All rights reserved. Keywords: Two-equation model; Turbulent ¯ow; Heat transfer

1. Introduction Numerical prediction of turbulent heat transfer phenomena has attracted substantial interest over the past few decades. A set of di€erential equations for the Reynolds stress …ÿui uj † and turbulent heat ¯ux …ÿui t† should be solved simultaneously for the time-mean velocity and temperature. The k±e turbulence model for velocity ®eld is widely used in engineering applications [1±4]. As for scalar turbulence, the conventional method is the zeroequation heat transfer model, in which the eddy di€usivity for heat at is obtained by at=nt/Prt with the turbulent Prandtl number Prt as a constant. However, no universal value of Prt was found. Nagano and Kim [5] developed a two-equation model for the thermal ®eld,

* Corresponding author.

in which eddy di€usivity for heat at was modeled using the temperature variance t 2 and its dissipation rate et, together with k and e. Youssef et al. [6] modi®ed the NK model to determine the wall-limiting behavior of turbulence quantities under various wall thermal conditions. However, it is not so convenient to calculate complex heat transfer problems with uniform wall heat ¯ux. Consequently, further improvement of their model would be needed. In the present study, we will develop a new nearwall t 2 ±et model. Using the near-wall behavior of turbulence quantities, we modify the modeling of the production term in et equation, and propose a new boundary condition for et under uniform wall heat ¯ux, which is more convenient for complex heat transfer, as compared with that of Youssef et al. [6]. Both the low-Reynolds-number k±e turbulence model [3] and our proposed two-equation heat transfer model

0017-9310/01/$ - see front matter 7 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 7 - 9 3 1 0 ( 0 0 ) 0 0 1 3 1 - 9

692

B. Deng et al. / Int. J. Heat Mass Transfer 44 (2001) 691±698

Nomenclature cp Cl, Cp1, Cp2, Cd1, Cd 2 Cm, Ce1, Ce2 fm, fe fl, fd1, fd 2 k p Pr, Prt qw R Rt Re tt t2 T, t T+ ut

speci®c heat at constant pressure turbulence model constants for the temperature ®eld turbulence model constants for the velocity ®eld turbulence model functions for the velocity ®eld turbulence model functions for the temperature ®eld turbulence kinetic energy mean pressure molecular and turbulent Prandtl number, respectively wall heat ¯ux timescale ratio, tt/tu turbulent Reynolds number, k 2/ne turbulent Reynolds number, y(ne )1/4/n friction temperature, qw/ rcput temperature variance mean and ¯uctuating temperature, respectively dimensionless temperature, (TwÿT )/tt friction velocity, (tw/r )1/2

were applied to study convective heat transfer for a two-dimensional turbulent channel ¯ow.

Ui, ui

mean and ¯uctuating velocity, respectively coordinates in streamwise and wall normal directions Reynolds stress tensor Reynolds heat ¯ux vector dimensionless distance from wall, uty/n

x, y ÿui uj ÿui t y+ Greek symbols a, at dij e et n, nt sk, se, sh, sf t tw tm tu, tt Subscripts w De @ ˆ Dt @xj

wall  n‡

ÿ Ce2 fe

The governing equations for an incompressible velocity ®eld with a low-Reynolds-number k±e model can be written as: @ Ui ˆ0 @xi

…1†

  DUi 1 @P @ @ Ui n ‡ ÿ ui uj ˆÿ r @xi @xj Dt @xj

…2†

Dk @ ˆ Dt @xj

" #  nt @ k @ Ui n‡ ÿ ui uj ÿe sk @ x j @xj

…3†

nt se



 @e e @ Ui ÿ Ce1 ui uj @xj k @xj

e2 k

…4†

k2 e

…5†

2 ÿui uj ˆ ÿ kdij ‡ 2nt Sij 3

…6†

nt ˆ Cm fm

2. Two-equation model for velocity ®eld

molecular and turbulent thermal di€usivities, respectively Kronecker delta dissipation rate of k dissipation rate of t 2 molecular and turbulent viscosities, respectively turbulence model constants for di€usion of k, e, t 2 , et time wall shear stress mixing timescale timescales of velocity and temperature

In the present study, the low-Reynolds-number k±e model by Abe et al. [3] is adopted, i.e. the model functions and constants are as follows: 2 3 (     2 2) Re 5 R t 41 ‡ 5 fm ˆ 1 ÿ exp ÿ exp ÿ 14 200 R3=4 t "   2 (  2 #)   Re Rt fe ˆ 1 ÿ exp ÿ 1 ÿ 0:3 exp ÿ 3:1 6:5

B. Deng et al. / Int. J. Heat Mass Transfer 44 (2001) 691±698

Cm ˆ 0:09,

sk ˆ 1:4,

Ce1 ˆ 1:5,

Ce2 ˆ 1:9

se ˆ 1:4 …7†

We follow Youssef et al. [6] to take simply the model constants sh and sf both equal to 1.0. The production term in Eq. (12) is given as Pet ˆ ÿ2a

3. Formulation of a two-equation model for the thermal ®eld 3.1. Modeling of temperature variance and its dissipation rate equations The governing equations for turbulent heat transport are expressed as:   DT @ @T ˆ a ÿ ui t …8† Dt @xi @xi

693

@ uj @ t@ t @ uj @ t@ T ÿ 2a @ x k@ x k@ x j @ x k@ x k@ x j

ÿ 2a

@ t@ t@ Uj @ t@ 2 T ÿ 2auj @ x k@ x j@ x k @ x j@ x k@ x k

…15†

It has more time and generation-rate scales to be determined [9,10]. Jones and Mansonge [11] and Nagano and Kim [4] try to take the following form Pet ˆ ÿCp2

e @T et @ Ui ui t ÿ Cp3 ui uj k @xj k @xi

…16†

and ÿui t ˆ at

@T @xi

…9†

!m  n k t2 at ˆ Cl fl k n‡mˆ1 e et

…10†

Here, the eddy di€usivity for heat at is associated with temperature variance, t 2 , and its dissipation rate, et. Cl denotes the model constant and fl is the model function including the near-wall e€ect in a thermal ®eld, to be described later. The exact governing equations of t 2 and et are given by [7] " # Dt 2 @ @t2 @T 2 ÿ uj t ÿ uj t ÿ 2et …11† a ˆ @xj @xj Dt @xj

@ uj @ t@ T @ t@ t@ Uj ÿ 2a @ x k@ x k@ x j @ x j@ x k@ x k

@ t@ 2 T @ 2t ÿ 2auj ÿ2 a @ x k@ x j@ x k @ x j@ x k

!2 …12†

In Eqs. (11) and (12), the turbulent di€usion terms uj t 2 and uj et0 are approximated, respectively, as [5,6] ÿuj t 2 ˆ

at @ t 2 sh @ x j

at @ et ÿuj et0 ˆ sf @ x j

et

ui t

t2

@T et @ Ui ÿ Cp3 ui uj , @xi k @xj

…17†

respectively. Then, both the production rates for velocity and temperature are used. Newman et al. [12] used only the thermal production rate and thermal timescale. Elgobashi and Launder [13] expressed it as Pet ˆ ÿCp1

et

ui t

t2

@T @xi

…18†

In the present study, instead of Eq. (15), a mixing timescale and only the thermal production rate are used to model the production term in Eq. (12) as Pet ˆ ÿCp1

  Det @ @ et @ uj @ t@ t a ÿ uj et0 ÿ 2a ˆ @xj @ x k@ x k@ x j Dt @xj ÿ 2a

Pet ˆ ÿCp1



eet

1=2

kt 2

ui t

@T @xi

…19†

The last term on the right hand side of Eq. (12) stands for destruction due to ®ne-scale turbulence interaction, which is dependent on both the velocity and thermal timescales. Launder [7] suggested using two terms proportional to eet/k and et2 =t 2 [7], which are adopted by many investigators [11±14] also. It reads eet ˆ ÿCd1 fd1

et2 t2

ÿ Cd2 fd2

et e k

…20†

…13†

The modeled governing equations can be ®nally described as

…14†

Dt 2 @ ˆ Dt @xj

"

at a‡ sh



@t2 @xj

# ÿ 2uj t

@T ÿ 2et @xj

…21†

694

B. Deng et al. / Int. J. Heat Mass Transfer 44 (2001) 691±698

Det @ ˆ @xj Dt

"

a‡

ÿ Cd1 fd1

at sf

et2 t2



@ et @xj

# ÿ Cp1

ÿ Cd2 fd2

r eet kt 2

uj t

t 2 ˆ d12 y 2 ‡




@T @xj

eet k

…22†

vt ˆ b2 d1 y3 ‡




…under uniform wall temperature†

t ˆ tw ‡ d2 y 2 ‡


t 2 ˆ tw2 ‡ 2d2 tw y 2 ‡




3.2. Near-wall modeling and model constants Using the thermal±mechanical timescale ratio R=tt/ tu, Eq. (10) may be rewritten as at ˆ Cl fl

…23†

nt =at AR1=2 :

…24†

Taking into account fm, we write fl as #  2 "   Re Bl …2R†0:5 1 ‡ 3=4 fl ˆ 1 ÿ exp ÿ Al …2R†m Rt

…25†

The near-wall behavior of turbulent quantities can be determined using a Taylor series expansion with respect to y: U ˆ A1 y ‡ A2 y 2 ‡ A3 y3 ‡


u ˆ a1 y ‡ a2 y 2 ‡ a3 y3 ‡


v ˆ b2 y 2 ‡ b3 y3 ‡





1ÿ 2 a ‡ c12 y 2 ‡


2 1


ÿ uv ˆ a1 b2 y ‡ a1 b3 ‡ a2 b2 y4 ‡


3

! ˆn

ÿ

a12

‡

c12

w

T ‡ ˆ Pry‡ ‡ e1 y‡2 ‡


t ˆ d1 y ‡ d2 y 2 ‡




etw

@ 2t 2 ˆa @y2




! ˆ w

  a 2 d1 ‡ 2d2 tw 2

…26†

From Eq. (26), in the vicinity of the wall, vtAy3 (under uniform wall temperature) vtAy 2 (under uniform wall heat ¯ux) must be satis®ed. Submitting Eqs. (23), (25) and (26) into Eq. (9), the turbulent heat ¯ux vt, obtained from Eq. (25), satis®es the near-wall limiting behavior under the thermal boundary conditions with and without wall temperature ¯uctuations, regardless of the mixing timescale tm. Hence, it is convenient to use di€erent tm in Eq. (25). For the sake of 0.5 simplicity, we choose tm=t0.5 u (2tt) , as used by Sommer et al. [9]. Thus, #  2 "   Re 3 fl ˆ 1 ÿ exp ÿ 1 ‡ 3=4 16 Rt

…27†

and at ˆ Cl fl

w ˆ c1 y ‡ c2 y 2 ‡ c3 y3 ‡




@ 2k ew ˆ n @y2

…under uniform wall heat

flux†

k2 …2R†m e

Considering that the ratio between the temperatureand velocity-timescales for dissipative motions is represented by (R/Pr )1/2 [15], the following relation holds in the vicinity of the wall [8]

kˆ

vt ˆ b2 tw y 2 ‡ b3 tw y3 ‡




k2 …2R†0:5 e

…28†

The direct numerical simulation (DNS) data [16,17] indicates that the molecular di€usion term balances with the dissipation term at the wall in et equation. From Eq. (11), we have at y = 0: a

@ 2 et e2 eet ˆ Cd1 fd1 t ‡ Cd2 fd2 2 @y k t2

…29†

Considering the limiting behavior of wall turbulence, fd2 A y 2 and fd1 A y 2 (under uniform wall temperature) of fd1 A y n with n > 0 (under uniform wall heat ¯ux) need to satisfy Eq. (29), the following equations are proposed to meet the requirements. 2#  Re ˆ 1 ÿ exp ÿ 1:7 "

fd1

…30†

B. Deng et al. / Int. J. Heat Mass Transfer 44 (2001) 691±698

 fd2 ˆ

"  2#  1 Re …Ce2 fe ÿ 1† 1 ÿ exp ÿ Cd2 5:8

…31†

with fe=1ÿ0.3 exp{ÿ(Rt/6.5)2. The model constants can be, therefore, determined in the following way. First, Cl, Cd1, Cd2, sh and sf are set to 0.1, 2.0, 0.9, 1.0, and 1.0, respectively, following the lead of the model by Youssef et al. [6]. The constant Cp1 is to be determined with the relation for the constant-stress and constant-heat-¯ux layer [5,6]: Cd1 …k 2 =Prt † p Cp1 ˆ p ‡ Cd2 ÿ sf Cm 2R

…32†

Then, Cp1=2.34 by substituting the standard value of Cm, Cd1, Cd2, Prt=0.9, sf and k=0.39±0.41. In summary, the model constants in the present model are given as follows: Cl ˆ 0:1, Cp1 ˆ 2:34,

Cd1 ˆ 2:0, sh ˆ 1:0,

Cd2 ˆ 0:9 sf ˆ 1:0

…33†

The governing equations are discretized by means of the ®nite volume. The QUICK scheme is used for the approximation of convection terms and central di€erence for di€usion terms, with SIMPLEC [18] algorithm to handle pressure±velocity coupling. The set of discretized linear algebraic equations is solved by ADI. The boundary conditions at the wall ( y = 0) for a 2 2 velocity ®eld are U=V=k p= 0 and ew ˆ n@ k=@ y or equivalently ew ˆ 2n…@ k=@ y† 2 : The wall thermal boundary conditions are taken as: t2 ˆ 0

t ˆ d1 y ‡ d2 y 2 ‡


t 2 ˆ d12 y 2 ‡


The actual et at the wall may be given by  2 @t ˆ ad12 etw ˆ a @y

…35†

…36†

w

That is, p ! 2 2 @ t ˆ ad12 a @y

p So, we can obtain etw ˆ a…@ t 2 =@ y†w2 for the case of uniform wall temperature. For uniform wall heat ¯ux and adiabatic wall, wall heat ¯ux would be a constant or 0. Hence, from Eq. (26), the near wall behavior of ¯uctuating temperature will be t ˆ tw ‡ d2 y 2 ‡




…37†

Substituting Eq. (37) into the de®nition of et :et ˆ a…@ t=@ x i † 2 , we obtain # "    2 @ tw @ tw @ d2 2 ‡ 4d2 y 2 ‡ . . . et ˆ a ‡a 2 …38† @x @ x@ x

4. Numerical scheme and boundary conditions

T ˆ Tw

695

In Eq. (38), no linear term exists. So, we can use the following relation as the boundary condition for et under a uniform wall heat ¯ux and an adiabatic wall:

…34a†

for uniform wall temperature, ÿ

@T ˆ qw @y

@t2 ˆ0 @y

…34b†

for uniform wall heat ¯ux, and ÿ

@T ˆ0 @y

@t2 ˆ0 @y

…34c†

for adiabatic wall. We derive the boundary condition for et according to di€erent wall thermal conditions. For uniform wall temperature, from Eq. (26), a ¯uctuating temperature near the wall can be expressed as

Fig. 1. Mean temperature for the case of constant wall heat ¯ux and constant wall temperature.

696

B. Deng et al. / Int. J. Heat Mass Transfer 44 (2001) 691±698

Fig. 4. Dissipation rate of temperature variance for the case of constant wall heat ¯ux and constant wall temperature.

Fig. 2. Temperature variance for the case of constant wall heat ¯ux and constant wall temperature.

@ et ˆ0 @y

…39†

q This, as compared with etw ˆ a…@ t 2 ÿ tw2 =@ y†w2 given by Youssef et al. [6], would be more convenient actually. 5. Results and discussion To validate the prediction of the proposed model, we applied it to a thermally and hydrodynamically fully developed turbulent ¯ow in a two-dimensional channel [16,17]. The predicted results with two typical thermal boundary conditions are presented in Figs. 1±7. The predicted mean-temperature in the turbulent boundary layer is plotted in Fig. 1. The predicted pro-

Fig. 3. Near-wall behavior of wall-normal turbulent heat ¯ux.

®les for cases of both uniform wall temperature and uniform wall heat ¯ux are in good agreement with the theoretical distribution, T +=Pry+, in the viscous sublayer, and agree with the law of the wall in the logarithmic region. The predicted temperature variance t 2 in the thermal ®eld is illustrated in log±log form in Fig. 2, using the friction temperature to normalize the root-mean-square temperature variance. The predicted t 2 undergoes a sharp rise nearby y +=20 in the near-wall region, which agrees well with the DNS data [16,17]. The predicted values are much similar to the DNS data in the turbulent core region. The predicted results in viscous sublayer for the case of uniform wall heat ¯ux is slightly larger than DNS data of Kasagi et al. [16], but seems similar to the result of Youssef et al. [6]. The predicted wall-normal turbulent heat ¯ux ÿvt is depicted in Fig. 3, which is in good agreement with the

Fig. 5. Comparison of predicted results and DNS data for dissipation timescales and their ratio for the case of constant wall heat ¯ux.

B. Deng et al. / Int. J. Heat Mass Transfer 44 (2001) 691±698

697

increase with increasing y+. These distributions accord qualitatively well with the DNS data [16]. The di€erence in the timescale ratio, R, between DNS data and predicted result is observed. The timescale ratio R keeps a ®nite value at the wall for DNS data [16]. While, the predicted timescale ratio R increases signi®cantly toward the wall and becomes in®nite at the wall. This may be consistent with the prediction based on the Taylor series expansion. The predicted budgets of temperature variance t 2 and its dissipation rate et are depicted in Figs. 6 and 7, respectively, which are compared with that of [9] and the DNS data [16]. In general, the present model gives a better prediction of the budgets of t 2 and et.

Fig. 6. The budget of temperature variance for the case of constant wall heat ¯ux, (a) present prediction; (b) prediction of Sommer et al. [9]; (c) DNS data [16].

DNS data [16,17] for both thermal boundary conditions except in very near wall region, but agrees well with the result of Youssef et al. [6] in the viscous sublayer. The prediction of the dissipation rate of the temperature variance et is depicted in Fig. 4. As shown, in the vicinity of the wall, the predicted results for both wall thermal conditions tw=constant and qw=constant, deviate from the DNS data [16], which is also shown by that of Youssef et al. [6], but predicted fair well in the region of y+ > 20. Fig. 5 presents the predicted results of the velocity dissipation timescale tu, the temperature dissipation timescale tt, and their ratio R=tt/tu, for the case of uniform wall heat ¯ux. tu would be larger than tt over the whole cross section of channel and both timescales

Fig. 7. The budget of the dissipation rate of temperature variance for the case of constant wall heat ¯ux, (a) present prediction; (b) prediction of Sommer et al. [9]; (c) DNS data [16].

698

B. Deng et al. / Int. J. Heat Mass Transfer 44 (2001) 691±698

6. Conclusions A new near-wall t 2 ±et two-equation heat transfer model has been derived in this study. It is based on the transport equations for temperature variance t 2 and its dissipation rate et. A modi®cation for modeling et equation is proposed, which use both velocity and thermal timescales and only the thermal production rate to describe the production term in et equation. Near-wall corrections are obtained by analyzing the near-wall limiting behavior of turbulent quantities. Eddy di€usivity for heat is used for relating turbulent heat ¯uxes to mean temperature ®eld. This involves both velocity and thermal timescales and reproduces the near-wall limiting behavior of normal heat ¯ux under di€erent thermal boundary conditions. Based on Taylor series expansion, the boundary condition for et under uniform wall heat ¯ux is proposed, which is more convenient than that of previous research for complex heat transfer [6]. References [1] B.E. Launder, On the computation of convective heat transfer in complex turbulent ¯ows, J. Heat Transfer 110 (1988) 1112±1128. [2] W.P. Jones, B.E. Launder, The calculation of lowReynolds-number phenomena with a two-equation model of turbulence, Int. J. Heat Mass Transfer 16 (1973) 1119±1130. [3] K. Abe, T. Kondoh, Y. Nagano, A new turbulence model for predicting ¯uid ¯ow and heat transfer in separating and ¯ows I. Flow ®eld calculations, Int. J. Heat Mass Transfer 37 (1994) 139±151. [4] C.K.G. Lam, K. Bremhorst, A modi®ed form of the k±e model for predicting wall turbulence, J. Fluids Eng. 103 (1981) 456±460. [5] Y. Nagano, C. Kim, A two-equation model for heat transport in wall turbulent shear ¯ows, J. Heat Transfer 110 (1988) 583±589.

[6] M.S. Youssef, Y. Nagano, M. Tagawa, A two-equation heat transfer model for predicting turbulent thermal ®elds under arbitrary wall thermal conditions, Int. J. Heat Mass Transfer 35 (1992) 3095±3104. [7] B.E. Launder, Heat and mass transport, in: P. Bradshaw (Ed.), Turbulence Ð Topics in Applied Physics, Springer, Berlin, 1976, pp. 232±287. [8] K. Abe, T. Kondoh, Y. Nagano, A new turbulence model for predicting ¯uid ¯ow and heat transfer in separating and reattaching ¯ows Ð II. Thermal ®eld calculations, Int. J. Heat Mass Transfer 38 (1995) 1467±1481. [9] T.P. Sommer, R.M.C. So, Y.G. Lai, A near-wall twoequation model for turbulent heat ¯uxes, Int. J. Heat Mass Transfer 35 (1992) 3375±3387. [10] S. Torri, W.J. Yang, New near-wall two-equation model for turbulent heat transport, Numer. Heat Transfer 29 (1996) 417±440. [11] W.P. Jones, P. Musonge, Closure of the Reynolds stress and scalar ¯ux equations, Phys. Fluids 31 (1988) 3589± 3604. [12] G.R. Newman, B.E. Launder, J.L. Lumely, Modeling the behavior of homogeneous scalar turbulence, J. Fluid Mech. 111 (1981) 217±232. [13] S.E. Elgobashi, B.E. Launder, Turbulence timescales and the dissipation rate of temperature variance in the thermal mixing layer, Phys. Fluids 26 (1983) 2415±2419. [14] T.S. Park, H.J. Sung, A nonlinear low-Reynolds-number k±e model for turbulent separated and reattaching ¯ows II. Thermal ®eld calculations, Int. J. Heat Mass Transfer 39 (1996) 3465±3474. [15] N. Shikazono, N. Kasagi, Modeling Prandtl number in¯uence on scalar transport in isotropic and sheared turbulence, Proceedings of Ninth Symposium on Turbulent Shear Flows, 1993, pp. 18.3.1±18.3.6. [16] N. Kasagi, Y. Tomita, M. Kuroda, Direct numerical simulation of passive scalar ®eld in a turbulent channel ¯ow, J. Heat Transfer 114 (1992) 598±606. [17] J. Kim, P. Moin, Transport of passive scalars in a turbulent channel ¯ow, in: Turbulent Shear Flows, 6, Springer, Berlin, 1989, pp. 85±96. [18] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington, DC, 1980.