A New Low-reynolds Version of An

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A NEW LOW-REYNOLDSVERSION OF AN EXPLICITALGEBRAIC STRESS MODEL

FOR TURBULENT CONVECTIVEHEAT TRANSFER IN DUCTSMasoud Rokn i

Availabl e onli ne: 29 Oct 2010

To cite this article: Masoud Rokni (2000): A NEW LOW-REYNOLDS VERSION OF ANEXPLICIT ALGEBRAIC STRESS MODEL FOR TURBULENT CONVECTIVE HEAT TRANSFEIN DUCTS, Num er ic al Heat Transf er, Part B: Fundament als: An Internat ional Journalof Computation and Methodology, 37:3, 331-363

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Numerical Heat Transfer, Part B, 37:331363, 2000Copyright Q 2000 Taylor & Francis10407790 r r r rr 00 $12.00 H .00

A NEW LOW-REYNOLDS VERSION OF ANEXPLICIT ALGEBRAIC STRESS MODEL FORTURBULENT CONVECTIVE HEAT TRANSFERIN DUCTS

Masoud Rokni Di v ision of Heat Transfer, Lund Institute of Technology, Box 118, 221 00 L und, S weden

This investigation concerns performance of a new low-Reynolds version of an explicit( ) algebraic stress model EASM for numerical calculation of turbulent forced-convective heat

transfer and fluid flow in straight ducts with fully developed conditions. The turbulent heat fluxes are modeled by a SED concept, the GGDH, and the WET methods. New versions of GGDH, WET, and EASM are presented for low Reynolds numbers. However, at high Reynolds numbers, two wall functions are used, one for velocity fields and one for the temperature field. All the models are computed in a general three-dimensional channel. Thelow-Reynoldsversion of the models presented is very stable and has been used for Reynolds numbers up to 70,000 with least demanded number of grid points, and without any convergence problem or stability problem.

INTRODUCTIONThe performance of a turbulence model in predicting the flow and tempera-

ture fields of relevant industrial problems has become increasingly importantduring the last few years. This is also valid for turbulent duct flow, which occursfrequently in many industrial applications such as compact heat exchangers, gasturbine cooling systems, recuperaters, cooling channels in combustion chambers,intercoolers, nuclear reactors, etc. The cross section of these ducts might be both

( ) ( )orthogonal square or rectangular and nonorthogonal such as trapezoidal , in

which the generated flow is extremely complex. Sometimes , the ducts are also wavyor corrugated in the main flow direction and might be manufactured with ribs inorder to achieve turbulence faster than usual. It is known that secondary motionstake place in the smooth corners of noncircular straight ducts in the planeperpendicular to the main flow direction. These motions are turbulence-inducedand commonly they are said to be of Prandtls second kind. Such motions are of importance since they redistribute the kinetic energy, influence the axial velocity,and thereby affect the wall shear stress and heat transfer. The effect of secondarymotions of Prandtls second kind on the wall shear stresses and heat fluxes

Received 16 June 1999; accepted 27 August 1999.Dr. Tom B. Gatski of NASA Langley Research Center contributed significantly to this article

and has been in extensive correspondence with the author from the beginning of this project. Hereby, Iwould like to thank him for very helpful comments and discussions during this period. I would also liketo thank Prof. Bengt Sunden of Lund University in Sweden, for valuable discussions and support during this project.

Address correspondence to Dr. Masoud Rokni, Department of Heat and Power Engineering,Lund Institute of Technology, Box 118, S-221 00 Lund, Sweden. E-mail: [email protected]

331

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NOMENCLATURE

a lower length T point temperature p A cross-section area T wall temperaturecross w

A area of the wall u u turbulent stressesw i jb anisotropy tensor u t turbulent heat fluxesi j j B , B constants in wall functions U velocityt jC specific heat at constant pressure U average ve locityP m

qC closure coefficients U dimensionless ve locityt C , C , C closure coefficients U , U * friction ve locitye 1 e 2 m t D hydraulic diameter W mean velocityh i j E constant in wall function x physical coordinates j

q ( ) f Fanning friction factor y normal distance, Eq. 42 f , f , f damping functions b cyclic pressure coefficient1 2 m

f t buoyancy-driven heat fluxG

numerical diffusivity jh duct height d Kronecker deltai jk kinetic energy e dissipation of kinetic energyNu Nusselt number e dissipation tensori j

( )Nu overall Nusse lt number h Eq. 23 invariant coefficientovNu local Nusse lt number at h distance normal to the wallxp

each point Q dimensionless temperatureNu Nusse lt number due to k , k von Karman constantsDB t

( )Dittus-Boelter l cyclic paramete r, Eq. 36P pressure m molecular viscosityP * cyclic pressure m turbulent viscosityt P , P , P production terms P pressure ] straink e t 2 i jPr Prandtl number r densityRe Reynolds number s turbulent Prandtl number

( )S mean strain rate j Eq. 23 invariant coefficienti jS source term for variable F j computational coordinateF iT temperature F variableT bulk temperature V cyclic temperature source termb

increases considerably when the ducts are corrugated. A linear k ] e model doesnot have the ability to predict secondary flows, but still it is one of the mostpopular models among engineers, thanks to its simplicity and decent overall

( )properties. An algebraic stress model ASM is able to predict secondary flows, butthe increased complexity of the model may cause stability problems together with asignificant increase in computational time and effort. An explicit algebraic stress

( )model EASM may therefore be a competitive choice model to predict reliableanisotropic normal stresses , and then secondary flows may be predicted withoutsolving additional equations.

For prediction of the flow and heat transfer characteristics for engineeringpurposes with available computer capacities, the need for turbulence models forflow and temperature fields see ms obvious. Limitation of computer capacitiesmakes it almost impossible to solve directly the details of the turbulent flow in

( )practical enginee ring duct flows by direct numerical simulation DNS } at present( )and for many years ahead if ever. Large-eddy simulations LES , on the other

hand, due to rapid increase in computer capacity and power , have found a place for

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ALGEBRAIC STRESS MODEL FOR HEAT TRANSFER IN DUCTS 333

some practical cases. However, for the time being, LES can be used for only a verylimited range of simple duct flows, which may not yet be of practical engineeringinterest. While most researchers agree that LES will be used in more complicatedduct flows in the future , how soon is not known: only development in computer

power can enable this.In this article the author develops a computational method to predict theturbulent Reynolds stresses in arbitrary ducts using an explicit algebraic stress

( )model EASM . The turbulent heat fluxes are modeled by the SED, GGDH, andWET methods. A new low-Reynolds version of the EASM, GGDH, and WET arepresented. The overall comparison between the models is presented in terms of thetwo most important hydraulic parameters, friction factor and Nusselt number. Thesecondary flow pattern is also of major concern.

Near the walls at high Reynolds numbers , the wall functions are used for

both the momentum equations and the temperature field, separately. JayatillekesP-function is thus abandoned . The authors experience shows that JayatillekesP-function has been suspected as one main source for numerically predicting the

(friction factor in ducts as much higher than experimental results if the wall)functions approach is used .

Several fundame ntal investigations concerning turbulent flow in square andw xrectangular ducts exist in the literature. Rokni and Sunden 1, 2 have employed a

nonlinear k ] e model for predicting the flow and heat flux in straight andcorrugated ducts with trapezoidal cross sections. Direct numerical simulations have

w xbee n carried out for a square duct by Gavrilakis 3 . Large-eddy simulations forw xsquare ducts are reported by Su and Friedrich 4 at a Reynolds number of 49,000

w xand by Madabhushi and Vanka 5 at a Reynolds number of 5 ,800.One difficulty associated with turbulent convective heat transfer and fluid

flow in ducts is obtaining satisfactory results for both friction factor and Nusseltnumber , if wall functions are used. Usually, either friction factor or Nusseltnumber can be predicted satisfactorily, not both of them. On the other hand,usually, the low-Reynolds version of the models cannot be extended to highReynolds numbers without additional considerations, such as significantly increas-ing the number of grid points near the walls. Comprehensive knowledge of flowstructure and reasonably accurate prediction of the flow field is a necessary firststep for analyzing and predicting the heat transfer phenomenon, so advance dturbulence models for Reynolds stresses will be discussed first.

PROBLEM STATEMENT

Straight ducts with square , rectangular , and trapezoidal cross sections areconsidered in this investigation. Only one-quarter of the ducts with square andrectangular cross sections and only half of the duct with trapezoidal cross sectionare considered by imposing symmetry conditions. A principle sketch of a duct isshown in Figure 1. A test calculation has also been carried out for a corrugatedtrapezoidal duct.

The calculation method has been focused on fully developed turbulent flowin an arbitrary three-dimensional duct. The results are presented in terms of thetwo most important hydraulic parameters, friction factor and Nusselt number. The

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Figure 1. Ducts under consideration.

secondary flow generation is also of major concern. Me an ve locity distributions ,friction factor, and Nusselt number are determined numerically for fully developedconditions.

GOVERNING EQUATIONS

The governing equations are the continuity, momentum, and energy equa-tions. Fully developed periodic turbulent flow and heat transfer are considered inthis investigation.

-r - ( ) ( )q r U s 0 1 j- t - x j

( )- r U - - P - - U - U - i i j( ) ( )q r U U s y q m q q y r u u 2( )i j i j( )- t - x - x - x - x - x - x j i j j i j

( )- r T - - m - T - ( ) ( )q r U T s q y r u t 3( ) j j( )- t - x - x Pr - x - x j j j j

There are basically three ways to solve these equations, namely, DNS, LES, andtraditional turbulence modeling. DNS has not yet been proved to be a generalmethod for solving the above equations due to limited computer capacity and

w xpower. For example , Speziale 6 postulated that a turbulent flow in a pipe atReynolds number of 500 ,000 would require a computer which is 10 million times

w xfaster than the Cray YMP. However , the DNS works of Kim et al. 7 , Gavrilakisw x w x3 , and Huser and Bringen 8 in a plane channel and a square duct withRe s 4,410 and Re f 10,000, respectively, have given valuable results for verysimple duct flows. In LES, the mean flow and the large energy-containing eddiesare computed directly with the time-dependent flow equations, but the small eddies

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} which are more universal in character } are modeled. Since the largest eddiesinteract strongly with the mean flow and contain most of the energies , thisapproach seems to yield a good approximation of the DNS. This turbulence modelhas re ached further than the DNS but still is limited to simple duct flows with

simple geometries. Nowadays researchers are attracted to the field of LES forsimpler geometries , and it seems that LES is very promising, with incredible fastw xincrease of computer power and capacity. Madabhushi and Vanka 5 , Su and

w x w xFriedrich 4 , and Meyer and Rehme 9 have performed LES calculations in squareand rectangular ducts with Re numbers up to 49,000.

The third way is to solve these equations by traditional turbulence modeling,also solving the flow equations by modeling both the large eddies and the small

( ) ( )eddies. The turbulent stresses y r u u and turbulent ` heat fluxes r C u t ,i j p jwhich are the unknown parameters in the governing equations , are modeled asdescribed in the following sections.

TWO-EQUATION MODELING FOR THE VELOCITY FIELD

A two-equation model is a computationally cheap approach to solve theNavier-Stokes equations by turbulence modeling. Obviously, the mixing-lengthapproach is much cheaper , but since the convection and diffusion terms areneglected in this approach and cannot be ignored in, e .g., recirculating flow, themixing-length approach is no longer feasible. The most widely used and validatedtwo-equation model, namely, the k ] e model, has also been used in this study. Itssuccessful application to a variety flows without case-by-case adjustment of themodel constants might be the main reason for its popularity in many industrialengineering applications.

( ) ( )The conventional k ] e model, given in Eqs. 4 and 5 , is based on our bestunderstanding of the relevant turbulence processes which cause changes to these

( w x)values Versteeg and Malalasekera 10 . This is why the two-equation model of

k ] e is still being used in commercial codes as well as in advanced researchprograms more than 20 years after its presentation. As can be seen from theseequations, expressing the turbulent Reynolds stresses, which appear in the produc-tion term, is still an important matter for the performance of these equations incomplex flows. A nonproper expression of the Reynolds stresses limits the use of this two-equation model in, e.g., duct flows. In other words, finding a properexpression for the Reynolds stresses may reveal the efficiency of this model incomplex flow applications.

( )- r k - - m - k t ( ) ( )q r U k s m q q P y re 4 j k ( )- t - x - x s - x j j k j2( )- re - - m -e e e t ( ) ( )q r U e s m q q f C P y f C r 5 j 1 e 1 k 2 e 2( )- t - x - x s - x k k j j e j

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where P is the production term, expressed ask

- U - U i i ( )P s t s y r u u 6k i j i j- x - x j j

At high Reynolds numbers, the constants C and C are set to 1.44 and 1.83,e 1 e 2respectively. The turbulent eddy viscosity m is calculated ast

K 2( )m s r f C where C f 0.09 7t m m me

The functions f , f , and f are the damping functions and are equal to unity in1 2 m(

the fully turbulent region remote from solid walls in other words, if wall functions)are used . However, in a region very close to a wall where the viscous effectsbecome very important , f , f , and f will differ considerably from unity. These1 2 mfunctions are given in the following sections.

At high Reynolds numbers, the constants s and s are set to 1.0 and 1.314,k e respectively. However, at low Reynolds numbers these constants may adopt othervalues, depending on the chosen model.

Damping Functions for the k and e equations

The high-Reynolds-number form of the k ] e model suggests that f shouldmapproximately be equal to unity in the fully turbulent region remote from solidwalls. However, in a region very close to a wall where the viscous effects becomevery important, f will differ considerably from unity. In this study, two differentm

w xdamping functions are used, the Lam-Bremhorst 11 damping functions and thew xAbe-Kondoh-Nagano 12 formulations of f , f , and f .1 2 m

( )In the Lam-Bremhorst LB method the f is determined asm

20.52y 0.0165 Re k ( ) ( ) f s 1 y e 1 q 8 am ( )Re t where

r k 0.5h r k 2Re s and Re sk t m me

The damping functions f and f are calculated as1 23

0.05 2y Re t ( ) f s 1 q and f s 1 y e 91 2( ) f mThe LB demands that the average yq value near a wall should be of the order of unity.

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( )The Abe-Kondoh-Nagan o AKN formulations of f , f , and f are given bym 1 2

5 22y y* r 14 y (Re r 200)t ( ) ( ) f s 1 y e 1 q e 10m 0.75( )Re t where

0.25 2 rh me rh r k y* s u s and Re se t ( )m r m me

f and f are calculated from1 2

2 2 y* r 3 .1 y (Re r 6.5)t ( ) ( ) ( ) f s 1 and f s 1 y e 1 y 0.3 e 111 2

( ) ( )Using this model, the constants in Eqs. 4 and 5 are set to s s s s 1.4 and thek e constants C and C are set to 1.5 and 1.9, respectively. In the above equations h e 1 e 2is the normal distance to a wall.

This damping approach demands a very small yq value near a solid wall. Theauthors experience shows that the average yq value near a wall should be of theorder of about 10 y 2 to yield the best results. This does not mean that additional

grid points are required; it is sufficient to change the stretching function. In otherwords, one needs finer grid points in the viscous sublaye r compare to the LBmodel.

EXPLICIT ALGEBRAIC STRESS MODEL

For the sake of simplicity, attention will be paid to incompressible turbulentflows that are statistically homogenous. The Reynolds stress tensor u u is thus ai jsolution of the transport equation as expressed below:

- u u - - U - U i j j i ( )q U u u s y u u y u u q P y e 12( )k i j i k j k i j i j- t - x - x - xk k k where

- u - u m - u - ui j i j ( )P s y p q and e s 2 13i j i j

( )- x

- x

r - x

- x

j i k k

are pressure ] strain correlation and dissipation rate tensors. Using the Kolmogorovassumption of local isotropy, the dissipation rate tensor can be expressed as

2( )e s ed 14i j i j3

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where e is the scalar turbulent dissipation rate. In equilibrium homogeneous flowsthe following expression may be used for Reynolds stresses:

- - u ui j( ) ( )u u q U u u s P y e 15( ) ( )i j k i j k - t - x k k

where P is the production term, expressed ask

- U i ( )P s y u u 16k i j - x j

( ) ( )and k is the kinetic energy. Using Eqs. 12 and 15 yields the linear system of algebraic equations

u u - U - U 2i j j i( ) ( )P y e s y u u y u u q P y ed 17k i k j k i j i jk - x - x 3k k

( ) ( )Equation 17 is the basic form of the traditional algebraic stress model ASM ,

which is an implicit system of equations for determination of the Reynolds stresses.w x ( )Gatski and Speziale 13 obtained an explicit solution of Eq. 17 by integrity basestechnique from linear algebra. The solution takes place as follows:

2 k U U U ( ) r u u s r k d y 2 m S y 2 m S W q S W i j i j t i j t ik k j jk k i3 e

k 1U U U ( )q 4 m S S y S S d 18t i k k j m n m n i j

( )e 3

where

k 2 k 2 k 2U U U U U U U U U ( )m s r C m s r C m s r C 19t m,1 t m,2 t m,3e e e

C U and C U arem,1 m, i

2( )3 1 q h UC s a m,1 1 2 2 2 23 q h q 6 j h q 6 j

2( )3 1 q h U ( )C s a a i s 2 ,3 20m, i 1 i 2 2 2 23 q h q 6 j h q 6 j

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and

4( ) ( ) ( )a s 0.5 y C g a s 0.5 2 y C g a s 0.5 2 y C g 211 1 2 2 3 3( )3

y 11 P k g s C q y 1 C s 0.36 C s 0.40 C s 1.25 C s 6.84 1 2 3 4( )2 e

( )22

The invariant coefficients are

1 a k a k 3 20.5 0 .5( ) ( ) ( )h s S S j s W W 23i j i j i j i j2 a e a e 1 1

where the mean strain rate and mean vorticity are

1 - U - U 1 - U - U i j i j ( )S s q W s y 24i j i j( ) ( )2 - x - x 2 - x - x j i j i( )Equation 20 reflects the regularization imposed by Gatski and Speziale in their

original formulation. A more robust regularization, also using a Pade approxima- w xtion, has been proposed by Speziale and Xu 14 , which has the correct asymptotic( )behavior for h 4 1 far from equilibrium ,

52 5 2( ) ( )1 q 2 j 1 q 6h q h 3UC s a m,1 1 2 2 2 6( ) ( )1 q 2 j 1 q 2 j q h q 6g h 1

22 4 2( ) ( )1 q 2 j 1 q h q h 3U ( )C s a a i s 2 ,3 25m, i 1 i2 2 6( ) ( )1 q 2 j 1 q 2 j q g h i

w xwhere the constants are g s 7.0, g s 6.3 , and g s 4.0, respectively. Gatski 151 2 3discussed anothe r form of Pade approximation which also diminishes the robust- ness of the original regularization,

2 6 6( ) ( )3 1 q h q 0.2 h q j UC s a m, 1 1 2 2 2 2 6 6( )3 q h q 6 j h q 6 j q h q j

2( )3 1 q h U ( )C s a a i s 2 ,3 26m, i 1 i 2 2 2 2 6 6( )3 q h q 6 j h q 6 j q h q j

( )While both the EASM in Eq. 18 and a nonlinear eddy viscosity model( w x)NLEVM, see e.g., 1 have similar functional forms, their development followsdifferent paths. In NLEVMs the functional form of the polynomial expansion is

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consistent with both the tensorial properties of a tensor to be represented and thevarious constraints imposed on the tensor expansion } such as coordinate invari-ance or material frame indifference. In ASMs the functional form is also based oncorrect tensorial properties , but expansion is also consistent with representation

theory in that a complete basis or some subset is chosen for the polynomial (expansion. In addition, the coefficients appearing in the tensorial expansion are or) ( )should be consistent with the Reynolds stress model RSM from which it is

derived. Recall that an ASM yields the same results as an RSM in the equilibriumlimit of homogeneous turbulence. The explicit form of the ASM is a consequenceof the fact that the expansion is in terms of mean flow variables. Thus, the EASMshould be a more consistent representation of the RSM than the NLEVM.

Low-Reynolds Version of EASM

The low-Reynolds-numbe r version of the EASM used in this study wasw xsuggested by Rokni 16 . It enables calculation of the Reynolds stresses at low

Reynolds numbers. The proposed model yielded very satisfactory results in straightducts with keeping its robust character,

k 2 k 2 k 2U U U U U U U U U ( )m s r f C m s r f C m s r f C 27t m m,1 t m m,2 t m m, 3e e e

However , these expressions were not sufficient, and he suggested that the invari-ance coefficients must also be damped. The damping functions used for theinvariant coefficients were the same as for the turbulent viscosity in order to keep

( )these coefficients consistent with the turbulent viscosity. Thus Eqs. 23 yield theforms

1 a k a k 3 20.5 0.5( ) ( ) ( )h s f S S j s f W W 28m i j i j m i j i j2 a e a e 1 1

TURBULENCE MODELS FOR HEAT FLUXES

Three models are used to express the turbulent heat flux.

( )1. Simple eddy diffusivity SED based on the Boussinesq viscosity model as

m - T t ( ) r u t s y 29 j s - xT j

At low Reynolds numbers the damping function f will appear in m . Them t turbulent Prandtl number for the temperature s is set to 0.89.T

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( )2. Generalized gradient diffusion hypothesis GGDH in its low-Reynoldsversion, defined as

k - T ( )u t s y f C u u 30 j m t j k

( )e - xk

( )3. The low-Reynolds version of wealth A earnings = time WET , defined as

k - T - U j ( )u t s y f C u u q u t q f t 31 j m t j k k j( )e - x - xk k where f t is the buoyancy-driven heat flux and is zero in this case. j

The constant C is set to 0.3 in both GGDH and WET.t

PERIODIC CONDITION

Periodic boundary conditions are imposed in the main flow direction fordecreasing the number of grid points in this direction. In practical applications,periodic conditions in the main flow direction are commonly justified , since in wavyor corrugated ducts such conditions occur naturally.

The pressure P decreases in the main flow direction and therefore thepressure should be handled in a special way, if periodic cases are considered. It isexpressed as

( ) ( ) ( )P x , y , z s y b x q P * x , y , z 32

where b is a constant representing the nonperiodic pressure gradient and P *behaves in a periodic manner from cycle to cycle in the flow direction.

The dimensionless temperature u is defined in the cyclic case as

( )T x , y , z y T w( ) ( )u x , y , z s 33( )T x y T b w

where T is the constant wall temperature and T is the fluid bulk temperature.w bUsing this expression and inserting it into the energy equation one obtains insteady state

- - m -u - ( ) ( ) r U u s q y r u u q V 34( ) j j( )- x - x Pr - x - x j j j j

where V is

-u - -l 2( ) ( )V s l G q Gu y r U u q Gu l q 35( )- x - x - x

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and

( ) r u t - T y T j b w( ) ( ) ( ) r u u s l s l x s r T y T 36 j b wT y T - xb w

The diffusion coefficient G s m r Pr and both l and V are periodic parameters.Using periodic conditions in the GGDH method, u t is then calculated from j

k -u -u -u ( ) ( )u t s y f C T y T u u lu q q u v q u w 37 j m t b w j j j( )e - x - y - z

In the WET method, u t is determined from j

k -u -u -u - U j( ) ( )u t s y f C T y T u u lu q q u v q u w q u t 38 j m t b w j j j k ( )e - x - y - z - xk An additional condition is needed to close the problem since the energy equationcontains two unknowns, u and l . This condition can be obtained from thedefinition of the bulk temperature. In dimensionless form one has

< < < < ( )H U u dA s H U dA 39cross cross

where A is the cross-sectional area in the main flow direction. In the fullycrossdeveloped periodic region, the shape of the nondimensional temperature profile

( )u x, y, z repeats itself.

BOUNDARY CONDITIONS

Periodicity conditions are imposed at the inlet and outlet for all variables:

( ) ( ) ( )F x , y , z s F x q L , y , z F s U , V , W , P * , k , e , u 40

In order to achieve numerical stability, u u , u u , l , and f are also handled asi j j m(periodic. Usually, at high Reynolds numbers more than about 10,000, based on the

)hydraulic diameter , the wall functions has bee n used and at low Reynolds numbersthe low-Reynolds versions of the models were applied.

WALL FUNCTIONS

Wall Functions for Momentum Equations

The law of the wall is assumed to be valid for both the flow and temperaturefields in the near-wall region at high Reynolds numbers. It is assumed that theregion near the wall consists of only two layers , the ` viscous sublayer, in whichthe turbulent viscosity is much smaller than the molecular viscosity, and the ` loglayer, in which the turbulent viscosity is much greater than the molecular viscosity.

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The ` buffer layer is ignored. The traditional law of the wall is defined as

U 1 1q q q( ) ( )U s ( ln y q B s ln Ey 41

U * k k

where

r U *h wallq ( ) y s 42m

and h is the distance normal to the wall. The von Karman constant is k s 0.41,wall(while the value for the roughness parameter E is set to 9.8 for a smooth wall or

) B s 5.5 . These values are supported from experimental results carried out byw x qHirota et al. 17 in a straight square duct. The point y s 11.53 is used to dispose

( )the intersection transition between the viscous sublayer and the log-law layer.(Below this point the flow is assumed to be purely viscous i.e., the turbulent

)stresses are negligible , and above this point the flow is assumed to be purelyw xturbulent. The details can be found in 1 .

Wall Functions for the Temperature Equation

Near the wall for the temperature field can be treated by either of twomethods: the so-called Pee-function method and the law of the wall.

Using the Pee-function method which is commonly used,

( )T y T r C U * Prw p pq q ( )T s s s U q P 43T ( )q s w T w xwhere the Pee-function may be expressed according to Jayatilleke 18 ,

0.75Pr Prwy 0.007*(Pr r s )xT w x ( )P s 9.24 y 1 1 q 0.28 e 44( ) ( )s s T T

This method is very popular, especially in commercial codes. However, the disad-vantages of this method is that the temperature field will be directly dependent on

( )the velocity field, and the von Karman constant k may play a significant roll indetermination of the Nusselt number in duct flows. Figure 2 shows clearly the

( )effect of the k value for the velocity field on the calculated friction factor and( )the Nusselt number in a square duct using EASM and SED models . Now a

question may arise, which value of the von Karman constant should be chosen; i.e.,k s 0.408 gives the best result for the friction factor , but k s 0.46 gives the bestresult for the Nusselt number. The average value k s 0.435 may also be a

w xreasonable choice and was chosen in 1 but not in the present study. The value of 0.89 for s was used in these calculations.t

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Figure 2. Calculated friction factor and Nu number compared withexperimental correlations in a square duct using EASM, SED, and twonear-wall treatments for the temperature field.

Using the law of the wall,

( )T y T r C U * 1w p pq q ( )T s s ln y q B 45t q k w t

w xwhere k s 0.46, as suggested by Launder 19 . This value is applied in this study.t w xHirota et al. 17 found k s 0.462 from experimental results in a straight squaret

duct. For a cold wall and a medium with Pr s 0.72, the value of B is B s 2.0; seet t w x qMohammadi and Pironneau 20 . The point y s 5.834 is used to dispose the

intersection between the sublayer and the log layer. The details of application of w xthis method can be found in 1 ,21 . It should be pointed out that the turbulent

( )Prandtl number for the temperature field s can be found by the ratio k r k ,t t which provides the value 0.89.

ADDITIONAL EQUATIONS

Some additional equations have been used to calculate Reynolds number ,mass flow, bulk ve locity, pressure drop, Fanning friction factor, and Nusselt

w xnumber. The details of the calculation procedures can be found in 1, 21 . Experi-w xments carried out by Lowdermilk et al. 22 showed that the Nusselt number in

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square ducts can be correlated with the Dittus-Boelter correlation and the frictionfactor can be correlated with the Prandtl-law friction factor.

w xThe calculated friction factor is thus compared with the Prandtl law 23 as

1 ( )s 2 log Re 4 f y 0.8 46( )4 f

The Reynolds number is based on the hydraulic diameter:

4 A cross3 walls D sh a q b q h r sin w

4 A cross ( )2 walls D s 47h a q h r sin w

where a , b , h , and w are base length, upper length, height, and base angle ,respectively.

( )The Nusselt number is calculated in two ways: 1 by calculating the localNusselt number at each point adjacent to a wall and then integrating over all the

( ) ( )walls NU ; 2 using the periodic conditions and calculating the overall Nusselt x

( )number by a heat balance equation Nu . These are called local and overallovNusselt numbers, respectively. The details of this procedure can also be found inw x1,21 . The calculated Nusselt number is compared with the Dittus-Boelter correla-

w xtion 23 as

0.8 0.3 ( )Nu s 0.023 Re Pr for Re R 8 ,000 48

The values calculated by these two Nusselt numbers should ideally be veryclose to each other. If not, something is not properly considered } for example, the yq value near the walls. The differences between these two calculated Nusseltnumbers are less than 1% for all c ases considered in this study.

NUMERICAL SOLUTION PROCEDURE

The partial differential equations are transformed to algebraic equations by ageneral finite-volume technique. The momentum equations are solved for thevelocity components on a nonstaggered grid arrangement. The Rhie-Chow interpo-lation method is used to interpolate the velocity components to the control-volumefaces from the grid points. The SIMPLEC algorithm is employed to handle the

( )pressure ] velocity coupling. A modified SIP strongly implicit procedure algorithmis used for solving the equations. The convective terms are treated by the QUICKscheme , while the diffusive terms are treated by the central difference scheme. Thehybrid scheme is used for solving the k and e equations.

The Prandtl number set to 0.72. The computations were terminated when thesum of the absolute residuals normalized by the inflow was less than 10 y 5 for all

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variables at high Reynolds numbers and less than 3 = 10 y 6 at low Reynoldsnumbers.

If the low-Reynolds versions of the models were used , it was important tomake sure that some grid points near the walls were in the viscous sublayer and

perhaps the mean values of yq

at the first grid points adjacent to each wall shouldbe of order less than unity. However, at high Reynolds numbers, when the wallfunctions are used, the average values of yq were kept between 41 and 43. Acomplete discussion about the yq value at the point adjacent to a wall can be

w xfound in 1 .At this point it is also important to remind the reader that all models

mentioned in the previous sections were presented in the Cartesian coordinate( )system. The transformation between the physical space x , y, z and the computa-

( )tional space j , , z can for a first-order derivative be written as

- F - F - xk ( )s 49-j - x -j i k i

As is obvious, any - r -j consists of three derivatives. The derivation of - r - x mayi( )also be written as generally speaking

- - ( )[ q extra terms 50

- x - x

in which the ` extra terms due to curvilinear transformation are very small instraight orthogonal ducts. This will be discussed later.

Numerical Stability

The general equation can be written as

( )- r F - - - F( ) ( )q r U F s G q S 51 j f F( )- t - x - x - x j j j

with G as the diffusion coefficient and S as the source term. All differentialF Fequations are compared with this equation and the diffusion coefficients and thesource terms are then identified. Numerical stability is secured by always settingthe diffusion coefficients as

G s m q m m s U , V , W m t , m

mt G s m q m s k , e m s m ( )52

m mt G s q m s T m Pr s T

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( )independent of whether the governing equation or model consists of such adiffusion coefficient or not. For example , if the temperature field is solved with thezero-equation WET model, the model does not have the term m r s , but this termt t is first added in the diffusion coefficient and then subtracted in the source term.

This treatment has been found to be very successful and is highly recommended bythe author.Another specific treatment implemented by the author to achieve a highly

numerical stable calculation method in periodic flows r heat transfer is the assump-tion that the matrices u u , u u , l , and f behave in a periodic fashion.i j j m

Wall-Surface Conditions

By using the low-Reynolds-numbe r version of the models , the wall-surface

conditions for k and e must be defined. These conditions are usually adopted fromthe proposed model. In the LB model, k s e s 0 at the walls. However , in theAKN model, k s 0 at the wall and the dissipation is set in the points adjacent to awall as source terms with following condition:

m k ( )e s 2 532 r h wall

where h is the normal distance from the point adjacent to a wall and thewallconsidered wall.

Symmetry Conditions

The derivatives of all the variables are set to zero in these planes.

RESULTS AND DISCUSSIONS

w x w xRokni and Sunden 24 and Rokni 16 , in previous studies , used the Lam- ( ) ( )Bremhorst 1981 damping functions LB in the k and e equations, but the study

showed that these damping functions were not proper functions for duct flows withperiodic conditions. It was found that these functions demanded initial values forthe k and e equations and these initial values were in a very narrow interval, whichwas difficult to find. The same problem was also noticed here. However , in thisstudy the wall damping functions proposed by the AKN model have also appliedfor duct flows and it has been found that these functions do not require anyspecific initial values for k and e , and the model is much more stable than the LBone in duct flows with periodic conditions. Significantly fewer grid points in thecross section can also be adopted with this model compared to the LB model. The

( )last two findings stability and fewer grid points may be explained by the fact thatthe LB damping functions have singularities in contrast to the AKN model. Thefriction velocity u* is used to account for the near-wall effect in the LB model,

( )0.25however, the Kolmogorov velocity scale u ve is used in the AKN model. Ite (is known that the friction velocity vanishes at separation at the corner of an

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)orthogonal duct and reattachment points, but the Kolmogorov velocity scale doesnot.

The calculations show that the LB model requires fewer grid points in theviscous sublaye r than the AKN model. However , this does not mean that the AKN

model requires more domain grid points than the LB model. Using the samedomain grid points and changing the stretching factor, one can insert more gridpoints in the viscous sublayer.

As discussed in previous sections, the GGDH and WET methods wereintroduced with the damping function f and these models are related to themtemperature field. It has been found that using the LB model in the GGDH andWET models gives better results than the AKN model. In short, the AKN model issuperior to the LB model for the flow field but the LB model gives better resultscompared to the AKN model in the GGDH and WET temperature models. The

difference between the AKN and LB models for the GGDH and WET tempera-ture methods are more obvious at higher Reynolds numbers, if the same number of grid points are used at the cross section. The results presented below are with the

(AKN damping functions for the flow field f in the turbulent viscosity, k ] e m)equations , and EASM e quations , while LB damping functions are used in the

GGDH and WET models. However , in order to extend the capability of the LB( )model to very high Reynolds numbers more than 10,000 , some modification has

been applied in this model, as shown below:

B2 A Re k ( ) ( ) f s 1 y e 1 q A s y 0.0125 B s 20.5 8 bm ( )Re t where

r k 0.5h r k 2Re s and Re sk t m me

Grid Investigation

A nonuniform grid distribution is employed in the plane perpendicular to themain flow direction. Close to each wall, the number of grid points or controlvolumes is increased to enhance the resolution and accuracy. From the duct centerto each wall the grid distance is multiplied by a factor less than unity, the stretching

( )factor ST . The lower this factor is chosen , the more grid points concentrate nearthe wall. Different numbers of grid points were used in the cross-sectional plane inorder to establish the accuracy of the calculations. Table 1 provides calculated Nunumber and friction factor in a square duct with different grid points and

(stretching factor , using the low-Reynolds version. The calculated Nu number by)SED and Fanning friction factor are compared with the correlations mentioned in

previous sections. In the table , Nu stands for Nu number calculated from theDBDittus-Boe lter correlation and f stands for friction factor calculated from thePrPrandtl-law correlation. As is evident from Table 1, decreasing the stretching

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Table 1. Calculated Nu number and Fanning friction factor for a square duct with different numbersof grid points and stretching factors using low-Re ynolds version

3 3ST Grid Re f = 10 f = 10 diff % SED Nu diff %Pr DB

0.9 21 = 21 5,448 10.245 9.123 y 12.3 21.4 20.3 y 5.40.9 31 = 31 5,685 9.411 9.014 y 4.4 20.6 21.0 1.90.9 35 = 35 5,697 9.372 9.008 y 4.0 20.6 21.1 2.40.9 41 = 41 5,590 9.733 9.057 y 7.5 20.2 20.7 2.40.9 51 = 51 5,611 9.659 9.047 y 6.8 20.2 20.8 2.90.85 21 = 21 5,640 9.562 9.034 y 5.8 20.9 20.9 00.83 41 = 41 5,614 9.648 9.046 y 6.7 20.3 20.8 2.4

( )diff % s 100 = correlated y calculated r correlated.

( )factor inserting more grid points in the viscous sublayers for a specific number of grid points increases the accuracy of calculations; see , e.g., 21 = 21 number of gridpoints.

(At high Reynolds numbers more than about 10,000 based on the hydraulic)diameter , 31 = 31 seemed to be enough to enhance reasonable accuracy for the

friction factor and Nu number in square ducts. However, at low Reynolds numbers31 = 31 or 41 = 41 grid points were used , depending on the Reynolds number.Only three grid points were applied in the main flow direction. The overall friction

factors and Nusselt numbers, which are the most interesting parameters from anengineering point of view, did not change significantly if the number of grid pointswas increased further. However, local values of the Reynolds stresses dependsomewhat on the number and distribution of the grid points, especially at lowReynolds numbers.

Secondary Flow Motion in a Square Duct

In Figure 3 b the secondary flow pattern in a square duct in the fullyw x (developed region obtained from the DNS work of Gavrilakis 3 is displayed the

reader should be warned about the quality of the figure since it was first scanned)and then enclosed . The corresponding secondary velocity vectors predicted by the

( ) ( )EASM with the low-Reynolds version at the same Re number s 4,800 areshown in Figure 3 a but rotated 180 8. As is evident from these figures, thequalitative agreement between the DNS data and the predicted results is quitesatisfactory.

Similar secondary flow is predicted at all the considered Reynolds numbers,at both high and low Reynolds numbers.

The driving forces for the secondary motion are concentrated to the regionclose to each corner. For laminar flow it can be shown that these secondary flowsdo not occur. In turbulent flow these motions are generated essentially by the

(gradient of the normal turbulent stresses. However , the LEVM linear k ] e model;w x)see , e.g., 25 does not predict these secondary motions because of its inability to

predict accurately the individual normal Reynolds stresses u u . The LEVM yieldsi i

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( )Figure 3. Predicted secondary motion in a square duct a( ) ( )compared with DNS work of Gavrilakis 1992 b . The Re

number is around 4,800. D o w n l o a

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( w x)erroneously, the physically incorrect expression see 6

uu f vv f ww

With predefined coordinates as in the section on ` Problem Statement, thecondition vv f ww is essentially sufficient for the LEVM to be unable to predictsecondary motions in straight square r orthogonal ducts. This problem is alleviatedin the EASMs.

It is worthwhile to point out that such secondary motions exist even withLEVM but they are extremely small, about 0.0001% to 0.001% of the main flow,which cannot normally be detected and cannot be regarded as secondary flowpattern. These very small values may also be in the limit of numerical r computer

(y 6 y 7

)accuracy absolute values of 10 ] 10 . One may then conclude that the turbu-lent shear stresses may also play an important role in generating secondarymotions. In fact , secondary motions originate from turbulent shear stresses butanisotropic turbulent normal stresses reflect them.

The predicted secondary velocity profile using the EASM model combinedwith both the wall functions and the low-Reynolds version of this model are shownin Figure 4. The velocity profile for the low Reynolds number is obviously muchsmaller than those for the high Reynolds number , but for the sake of clarity thevelocity profile for the low Reynolds number has been enlarged. A comprehensive

comparison between the predicted secondary motions and experimental results,w xdirect numerical simulation and large-eddy simulation, can be found in 1 . The

secondary motions consist of two counterrotating vortices which transport high-momentum fluid toward the duct corner along the bisector and then outward alongthe walls.

At low Reynolds numbers and close to the duct center the secondary flow isweak and i ts influence on the streamwise flow is expected to be small. However ,the secondary motions are concentrated near the corners of the duct and their

( )effect on the streamwise flow is expected to be large see Figure 5 . Very close tothe corners there exist no secondary flow. In general, it can be said that close toany corner there exists two counterrotating vortices. The sizes of these vortices aresimilar in a right-angle corner but essentially nonsimilar in a non-right-angl e

( )corner see below .( )Figure 5 shows the streamwise ve locity contours U r U predicted by thebulk

EASM using both the wall functions and its low-Reynolds version. More bulging is( )obtained at Re f 3,900 using the low-Reynolds version than at Re f 66,000

( )using the wall functions in the region close to the corner. However, the shapes of the contours are similar near the duct center.

Hydraulic Parameters and the Wall Functions in a Square Duct

The calculated Fanning friction factor and Nu number at high Reynolds( )numbers using wall functions are shown in Figure 6. The results obtained from

the EASM model agree very well with the Prandtl-law correlation for Fanning( )friction factor less than " 4% and the Dittus-Boelter correlation for Nu number.

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Figure 4. Secondary velocity profile predicted by the( )EASM model with different near-wall treatment: a using

( )wall functions, b using low-Reynolds version.

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Figure 5. Predicted streamwise contours using EASM model at twodifferent Reynolds numbers.

Both the GGDH and WET methods are in excellent agreement with the( )Dittus-Boelter correlation less than " 3% , but the SED deviates somewhat from

this correlation, as shown in Figure 6.One problem associated with using the wall functions is that the grid points

adjacent to a wall should be a certain distance away from the considered wall , in( )order to get the average y q value in an acceptable range more than about 35 .

The problem will be more evident when the ducts are corrugated and havetrapezoidal cross sections. This problem can be aviated by using a low-Reynoldsversion of the models. However, the damping functions demand the calculation of normal distance from any point to the nearest wall, which is not an easy task ingeneral complicated geometries.

Hydraulic Parameters and the Low-Reynolds Version in aSquare Duct

As mentioned above , the AKN damping functions are used for the flow field,while the LB model is used for GGDH and WET models. At low Reynoldsnumbers the calculated Fanning friction factor and calculated Nu number atdifferent Re numbers are shown in Figure 7. The figure shows that the calculatedFanning friction factors by the EASM agree excellently with the Prandtl-lawcorrelation. The figure shows also that Nu number obtained from SED modelsagrees well with the Dittus-Boelter correlation. However, the GGDH and WETmodels predict less Nu number than the correlation for Re numbers less than

w xabout 8 ,000. The experimental work of Lowdermilk et al. 22 shows also that the

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( ) ( )Figure 6. Calculated Fanning friction factor a and Nu number busing the EASM model with the wall functions.

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( ) ( )Figure 7. Calculated Fanning friction factor a and Nu number b(using the EASM model with the damping functions the low-

)Reynolds version .

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( )Nu number in square ducts deviates less values from the Dittus-Boelter equationfor Re numbers less than about 8,000.

The presented calculation procedure is highly stable and can be extended tomuch higher Re numbers than 10,000 with the least demanded number of grid

points. In Figure 8 all the calculations were performed with only 31 = 31 gridpoints for all Reynolds numbers. No convergence problems were faced even at thevery high Re number 71,000 using the low-Reynolds version of the modelspresented in previous sections. It should be mentioned that the LB model couldnot be used for the flow field up to such high Re numbers without significantlyincreasing the number of grid points. The LB model for the flow field at this highRe number also had convergence problems.

As mentioned in previous sections, the LB model is used for GGDH and( )WET models. Again the results both friction factor and Nu number are in

excellent agreement with the correlations, which in turn are shown to be inexcellent agreement with experiment data. The modified LB model is used inFigure 8.

Rectangular Ducts

Good agreement between the calculated results and the presented correla-tion for friction factor and Nu number are also obtained in the rectangular ducts

( )side ratios 2 , 3 , 5, and 10 are considered . The secondary flow are also predicted inall cases considered. For e xample , the secondary flow motion for two rectangularducts with aspect ratios 3 and 5 are presented in Figure 9. The secondary flowprofiles predicted by the EASM and the AKN damping functions are in goodagreement with what might be expected.

Table 2 provides calculated Fanning friction factor , Nu number , and the( )center-to-bulk velocity ratio U r U in a rectangular duct with different aspectc b

ratios. For a given cross section the U r U decreases slightly with increasing Rec bnumber , which is also evident from this table. The experimental value of U r U forc b

( w x)a rectangular duct with aspect ratio 10 at Re f 5,800 is 1.23 see Rokni et al. 26 ,( )which can be compared with the calculation result Table 2 1.22 at Re f 15,500.

Trapezoidal Ducts

Using wall functions and the LEVM predicts only one vortex, regardless of the cross section. Such a secondary flow pattern is actually not expected and doesnot exist in nonorthogona l ducts. The reason may be explained by the curvilinear

w ( )xtransformation e ffect. Recall from a previous section Eq. 50 that any gradientafter curvilinear transformation could be written as

- - [ q extra terms

- x - x

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( ) ( )Figure 8. Calculated Fanning friction factor a and Nu number busing the low-Reynolds version of the EASM model at high Renumbers.

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Figure 9. Predicted secondary flow profile in rectangular ducts with aspectratios 3 and 5.

Inserting this equation into the equation for LEVM yields

2 r u u s r k d y 2 m S q terms due to curvilinear transformationi j i j t i j3

The ` extra terms due to curvilinear transformation are obviously equal in the two(directions of cross sections of an orthogonal straight duct here , Y and Z direc-

)tions , while they are essentially not equal in the nonorthogonal straight ducts.

However , the presence of these ` extra terms together with the wall functions is(sufficient to provide the condition vv / ww and thereby generate a false not)correct secondary motion, see Fig. 10. In fact, these ` false secondary motions are

generated by the mathematical task and are geometry-driven rather than turbu-(lence-driven. Such calculation methods using the standard k ] e model with curvi-

)linear transformation are widely used and are adopted in most commercial codes.

Table 2. Calculated Nu number and Fanning friction factor for rectangular ducts with different aspect

ratio using low-Reynolds version of EASM and SED3Aspect ratio Re f = 10 Nu U r U c b

2 9,174 8.566 30.4 1.283 11,275 8.075 35.4 1.285 13,520 7.703 41.1 1.26

10 15,492 7.618 47.8 1.22

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(Figure 10. Secondary velocity ve ctors predicted by the LEVM linear)k ] e model in a straight trapezoidal duct using wall functions. These

secondary motions are called false secondary motions.

In wavy r corrugated ducts such false secondary patterns may change shape(and character but are still in the frame of false secondary motions or geometry-

)driven secondary flows rather than turbulence-driven flows . One may then con-clude that the LEVM or standard k ] e model cannot be used in duct flows.

The secondary velocity vectors in a trapezoidal duct predicted by the EASMare presented in Figure 11. Near-the-wall regions are treated using the wallfunctions and the damping functions of AKN. The trapezoidal duct chosen here

has a small upper side length compared with the lower side length and the duct( )height close to a triangular duct . No convergence problem was noticed inpredicting the secondary motion with the EASM and AKN damping functions ,even in such a trapezoidal duct. Only 51 = 31 grid points in the cross section wereused to perform the calculation with the AKN model. However, the LB dampingfunctions for the flow field had convergence and stability problems in trapezoidalducts, regardless of the number of grid points in the cross section.

In Figure 11 b , the Re number is about 13,250 and the calculated Fanning( ) y 3friction factor and the calculated Nu number SED model are 7.931 = 10 and

42.8, respectively. These values can be compared with the Prandtl-law and Dittus-w ( ) ( )x y 3Boelter correlation Eqs. 49 and 51 , which provide 7.188 = 10 and 41.4 ,

respectively. In Figure 11 a , the Re number is about 200,920 and the obtained( ) y 3Fanning friction factor as well as Nu number GGDH model are 4.010 = 10

and 385.6, respectively. The Prandtl-law and Dittus-Boelter correlations for this Renumber give 3.914 = 10 y 3 and 364.2, respectively.

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Figure 11. Secondary velocity profile predicted by

the EASM model with different near-wall treat-( ) ( )ment: a using wall functions; b using dampingfunctions.

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Wavy Ducts

A test calculation has been carried out on a wavy duct to evaluate theperformance of the low-Reynolds version of EASM presented in this study. Thewavy duct under consideration is shown in Figure 12. A symmetry plane is imposedat the cross section and the wavy amplitude is applied along the Y direction. Thenumber of grid points in the cross section is set to 51 = 31 for the y and zdirections, respectively. Also, the same cross section as in the previous section isused here. As mentioned before, a nonuniform grid distribution is employed in theplane perpendicular to the main flow direction. Close to each wall, the number of grid points or control volumes is increased to enhance the resolution and accuracy.Thirty grid points are set for the main flow direction, the x direction. However, auniform grid distribution is applied along the streamwise direction, as shown inFigure 12. The calculation is performed without any convergence problem orstability problem and the residuals reach the value 10 y 5 for the temperature fieldand 10 y 7 for the velocity filed and turbulence equations. The SED method wasused for the temperature equation.

Table 3 shows the calculated Nu number and Fanning friction factor for the( )wavy duct in comparison with the straight duct which has the same cross section .

In one column of the table , the calculated Fanning friction factor has beennormalized by the Prandtl law and the calculated Nu number has been normalizedby the Dittus-Boelter correlation.

As can be seen from Table 3, both the friction factor and the Nu number forthe corrugated duct are much higher than for the straight duct.

Figure 12. The wavy duct under consideration.

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Table 3. Comparison betwee n a wavy duct and a straight duct with similar cross sections

3 3Type Re f = 10 f = 10 f r f SED Nu NU r NUPr P r DB DB

Straight 13,258 7.931 7.188 1.10 42.8 41.4 1.03Wavy 9,431 15.470 7.845 1.97 46.1 31.5 1.46

CONCLUSION

A computational procedure for turbulent flow in 3D ducts has been pre-sented using an EASM model to predict the secondary velocity field, the stream-wise velocity, and other turbulent quantities. The turbulent heat fluxes are calcu-lated by a SED, the GGDH, and the WET methods. At high Reynolds numbers,the wall functions for the momentum equations and the temperature field are

employed , separately. A low-Reynolds number version of the EASM, the GGDH,and the WET are also presented. With the low-Reynolds version of the models, theAKN damping functions are used for flow field and the LB model is used for theGGDH and WET models. The presented calculation procedure is very stable, andsatisfactory results are obtained with least demanded number of grid points. Themodel was also tested for a trapezoidal straight duct with small upper length( )compared with the lower length and the duct height which is close to a triangularduct. Excellent results obtained without any convergence problem or stabilityproblem.

For ducts with square cross section, the calculated friction factors and Nunumbers agree very well with the Prandtl-law correlation when the low-Reynoldsversion of EASM presented here is used. Using the wall functions, the Nusseltnumbers obtained by the GGDH and the WET methods are in excellent agree-ment with the Dittus-Boelter correlation. However, there exist some discrepanciesbetween the calculated Nusselt numbers by the SED method and the Dittus-Boeltercorrelation using wall functions.

The model is used for ducts with different cross sections and satisfactory( )results in terms of friction factor and Nu number are obtained without any

convergence or stability problems.The model was also tested for a corrugated duct and the calculations were

performed without any convergence or stability problems. The model was tested forhigh Re numbers up to 70,000 without any convergence or stability problems.

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A New Low-reynolds Version of An

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