Entropy generation of turbulent double-diffusive natural convection in a rectangle cavity

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Energy 36 (2011) 1721e1734

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Energy

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Entropy generation of turbulent double-diffusive natural convectionin a rectangle cavity

Sheng Chen a,b,*, Rui Du c

a State Key Laboratory of Coal Combustion, Huazhong University of Science and Technology, Wuhan 430074, PR ChinabR&D Center, WISCO, Wuhan 430083, PR ChinacDepartment of Mathematics, Southeast University, Nanjing 210003, PR China

a r t i c l e i n f o

Article history:Received 30 June 2010Received in revised form6 November 2010Accepted 23 December 2010Available online 3 February 2011

Keywords:Entropy generationDouble-diffusive convectionTurbulent

* Corresponding author. State Key Laboratory ofUniversity of Science and Technology, Wuhan 4300787542417.

E-mail address: [email protected] (S. Ch

0360-5442/$ e see front matter � 2011 Elsevier Ltd.doi:10.1016/j.energy.2010.12.056

a b s t r a c t

Turbulent double-diffusive natural convection is of fundamental interest and practical importance. In thepresent work we investigate systematically the effects of thermal Rayleigh number (Ra), ratio of buoy-ancy forces (N) and aspect ratio (A) on entropy generation of turbulent double-diffusive naturalconvection in a rectangle cavity. Several conclusions are obtained: (1) The total entropy generationnumber (Stotal) increases with Ra, and the relative total entropy generation rates are nearly insensitive toRa when Ra � 109; (2) Since N > 1, Stotal increases quickly and linearly with N and the relative totalentropy generation rate due to diffusive irreversibility becomes the dominant irreversibility; and (3) Stotalincreases nearly linearly with A. The relative total entropy generation rate due to diffusive and thermalirreversibilities both are monotonic decreasing functions against A while that due to viscous irrevers-ibility is a monotonic increasing function with A. More important, through the present work we observea new phenomenon named as “spatial self-copy” in such convectional flow. The “spatial self-copy”phenomenon implies that large-scale regular patterns may emerge through small-scale irregular andstochastic distributions. But it is still an open question required further investigation to reveal thephysical meanings hidden behind it.

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1. Introduction

Turbulent double-diffusive natural convection, i.e. flows gener-ated by buoyancy due to simultaneous temperature and concen-tration gradients are ubiquitous in natural as well as technicalsystems. In nature such flows are frequently encountered in oceans,lakes, solar pounds, shallow coastal waters and the atmosphere. Inindustry examples include chemical processes, crystal growth,energy storage, material processing such as solidification, foodprocessing, etc. But surprisingly, to date the open literature onturbulent double-diffusive natural convection is still sparse. VanDer Eyden et al. [1] numerically and experimentally investigatedturbulent double-diffusive natural convection of a mixture ina trapezoidal enclosure. They announced that the numerical resultsobtained by k� e model agreed well with the experimental data.Later, Papanicolaou and Belessiotis [2] reported the unsteady

Coal Combustion, Huazhong4, PR China. Tel./fax: þ86 27

en).

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behavior of double-diffusive natural convection in an asymmetrictrapezoidal enclosure with the thermal Rayleigh number Ra up to1010. They found that the ratio between the thermal and theconcentration (or solutal) buoyancy forces N is a key parameter todetermine the characteristics of convection patterns. A k� e modelfor treating turbulent double-diffusive flows in porous media wasproposed by de Lemos and Tofaneli but without any numericalvalidation [3]. Recently, they [4] validated their mathematicalframework by simulating double-diffusive turbulent naturalconvection in a porous square cavity with opposing temperatureand concentration gradients and the thermal Grashof number up to2.25 � 1010. In their work, the finite difference schemes were usedto discretize the governing equations.

The references mentioned above all are based on the first-law ofthermodynamics and avoid complicated analyses for optimumdesign. Recently the entropy generation analysis methodology [5]which based on the second-law of thermodynamics, is used tooptimize heat andmass transfer performance [6e8]. Although untilnow the entropy generation analysis has been extended widely forreactive flows [9e11], the available literature on entropy generationanalysis in double-diffusive convection is very few yet. To the bestknowledge of the present authors, there are only three publications

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Nomenclature

N ratio of buoyant forcesC Smagorinsky constantDe effective thermal diffusivityu! fluid velocity vectore!k discrete velocityFk source term in Eq. (7)g! gravitygk; fj;hj distribution function for velocity and scalar fields

geqk ; f eqj ;heqj equilibrium distribution function for velocity andscalar fields

H height of the cavityv! equilibrium velocity vectorp pressureNu Nusselt numberEc Eckert numberPr Prandtl numberBe Bejan numberRa thermal Rayleigh numberSh Sherwood numberStotal total entropy generation numberSv entropy generation numberT dimensionless temperatureW width of the cavity

x! phase space

Greek symbolsΔx, Δy grid spacing in x and y directionΔt time stepfab strain rate tensorne effective kinematic viscositym effective dynamic viscositysD relaxation time for concentrationsQ relaxation time for temperaturer densityae effective thermal diffusivityuk, ck the weights for equilibrium distribution functionD filter width4 irreversibility distribution ratio

Subscripts and superscriptsD thermalm viscousj,k discrete velocity direction0 initial index- filter operator or averaget turbulenta,b spatial indexU,T global, total

Fig. 1. Configuration of the computational domain and boundary conditions.

S. Chen, R. Du / Energy 36 (2011) 1721e17341722

on this topic. Magherbi et al. [12] investigated entropy generationof double diffusion in an inclined cavity. They found that the angleof inclination had a significant effect on entropy generation inconvective heat and mass transfer. The total entropy generationincreased with the thermal Grashof number and the buoyancy ratiofor moderate Lewis numbers. Later, they revealed the influence ofDufour effect on entropy generation in double-diffusive convection[13]. In their work, the total entropy generation was evaluated asa function of the buoyancy ratio, the Dufour parameter and thethermal Grashof number. Recently, Hidouri and Brahim [14]investigated the influence of Soret and Dufour effects on entropygeneration in transient double-diffusive convection of a binary gasmixture for the special case of opposing buoyancy forces with equalintensity. It was found that for moderate thermal Grashof number,Soret and Dufour parameters induced a slight increase of entropygeneration, but for relatively higher thermal Grashof number,oscillatory behavior of entropy generation was obtained.

As is seen, all of the studies in the above cited literature, there isno investigation that is conducted to analyze entropy generation inturbulent double-diffusive natural convection, which inspires thepresent work. The LES (large eddy simulationLES) based LB (latticeBoltzmannLB) method is employed to solve the turbulent convec-tional flow and part of the entropy generation equation, followingthe line proposed in our previous work [15,16]. The effects ofthermal Rayleigh number, ratio of buoyancy forces and aspect ratioon entropy generation of turbulent double-diffusive naturalconvection in a rectangle cavity are investigated systematically.More important, an interesting phenomenon, named as ”spatialself-copy”, is observed through the present study. It brings forwardan open question on turbulent natural convectional flow andrequires further investigation to explain it.

2. Governing equations

The configuration of the two-dimensional computationaldomain is illustrated in Fig. 1. The aspect ratio A ¼ H/W, where H is

the height of the cavity andW is the width. In the present study weset W ¼ 1.

With the aid of the normalizing characteristic quantities, i.e.length with H, velocity with (a/H)Ra0.5, pressure with r(a/H)2Ra,temperature with (T - T0)/DT, concentration with (Y � Y0)/DY andtime with H2/aRa0.5, the corresponding dimensionless governingequations read [3,17]

a b

c

Fig. 2. (a) Isotherms, (b) Isoconcentrations, (c) Stream lines for N ¼ 0.8.

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2a b

c

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

Fig. 3. (a) Isotherms, (b) Isoconcentrations, (c) Stream lines for N ¼ 1.3.

S. Chen, R. Du / Energy 36 (2011) 1721e1734 1723

Table 1Average Nusselt number Nu and average Sherwood number Sh.

Nu Sh

Ref. [38] Present Ref. [38] Present

N ¼ 0.8 3.67 3.6917 4.89 4.9038N ¼ 1.3 2.10 2.1135 3.15 3.1640

Table 2Comparison of average Nusselt number at the hot wall of convectional flow withprevious works.

Ra Ref. [39] Ref. [40] Present

107 16.79 e 16.7552108 30.506 32.045 30.4264109 57.350 e 51.25181010 103.663 156.85 99.9659


V$ u! ¼ 0 (1)

v u!vt

þ u!$V u! ¼ �Vpþ neD u!� PrðT � NYÞ g!j g!j (2)

vTvt

þ u!$VT ¼ aeDT (3)

vYvt

þ u!$VY ¼ DeDY (4)

where ne, ae and De are the effective viscosity, thermal and solutaldiffusivity, respectively. p is the pressure, T is the temperature and Yis the concentration. The gravity g! is downward. Pr is the Prandtlnumber and Ra is the thermal Rayleigh number. N is the ratio ofsolutal buoyancy force to thermal buoyancy force.

3. LB model

The LB method is a recently-developed promising alternative tosolve various challenging flow problems [19,20]. These brilliantachievements benefit from its inherent outstanding advantages:straightforward treatment of realistic boundary conditions in

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

a b

c d

Fig. 4. Isotherms for (a) Ra ¼ 107, (b) Ra

complicated geometries, the structural simplicity of the code, clearphysical pictures, the full parallelism of the method and its readyextension to 3D problems, which let the LB method be more suit-able for the current trend of massive computation and more easilysolve some problems that are difficult in conventional numericalmethods [19]. Moreover, the LB models have been comparedfavorably with spectral methods [21], artificial compressibilitymethods [22], finite volume methods [23], finite differencemethods [24e28], projection methods [29,30] and multigridmethod [31,32], all quantitative results further validate excellentperformance of the LB method not only in computational efficiencybut also in numerical accuracy. Especially, the LB method has beenextended on graphics processing unit architectures [33].

3.1. Flow field

The evolution equation for the flow field reads [15]

gkð x!þ c e!kDt; t þ DtÞ � gkð x!; tÞ ¼ UkðgÞ þ DtFk (5)

where gkð x!; tÞ is the distribution function associated with the fluidmolecules moving with the discrete velocity e!k. c ¼ Dx/DT is thefluid particle speed. Dx and DT are the lattice grid spacing and the

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

¼ 108, (c) Ra ¼ 109, (d) Ra ¼ 1010.

Fig. 5. (a) Entropy generation number, (b) relative entropy generation rate due to thermal irreversibility, (c) relative entropy generation rate due to diffusive irreversibility,(d) relative entropy generation rate due to viscous irreversibility for Ra ¼ 108.

Fig. 6. (a) Entropy generation number, (b) relative entropy generation rate due to thermal irreversibility, (c) relative entropy generation rate due to diffusive irreversibility,(d) relative entropy generation rate due to viscous irreversibility for Ra ¼ 1011.



time step, respectively. The term Fk is a forcing term accounting forthe body force experienced by the fluid particle, and Uk (g) is thediscrete collision operator. The most widely used collision operatoris the single relaxation time or BGK (Bhatnagar-Gross-Krook)model [34]

UkðgÞ ¼ �s�1hgkð x!; tÞ � gðeqÞk ð x!; tÞ

i(6)

where gðeqÞk is the discrete equilibrium distribution function.Models with (MRT) multiple-relaxation-times) were also proposed,where the collision operator is given by [35]

UkðgÞ ¼ �Xj

�M�1SM

�jk

�gj � gðeqÞj

�(7)

where S ¼ diag (s0,s1,.,sk-1)�1 is a non-negative diagonal matrix.For the D2Q9 (two-dimension-nine-velocity) lattice model [34]

e!k ¼8<:

ð0;0Þ : k ¼ 0ðcosðk� 1Þp=2; sinðk� 1Þp=2Þ : k ¼ 1;2;3;4ffiffiffi2

pðcosððk� 5Þp=2þ p=4Þ; sinððk� 5Þp=2þ p=4ÞÞ : k ¼ 5;6;7;8

S ¼ diag (1,0.2,0.1,1,1.2,1,1.2,1/s,1/s) and the transform matrix Mreads

M ¼

0BBBBBBBBBBBB@

1 1 1 1 1 1 1 1 1�4 �1 �1 �1 �1 2 2 2 24 �2 �2 �2 �2 1 1 1 10 1 0 �1 0 1 �1 �1 10 �2 0 2 0 1 �1 �1 10 0 1 0 �1 1 1 �1 �10 0 �2 0 2 1 1 �1 �10 1 �1 1 �1 0 0 0 00 0 0 0 0 1 �1 1 �1

1CCCCCCCCCCCCA

(8)

and

gðeqÞk ¼ ukr

"1þ c e!k$ v

!c2s

þ ðc e!k$ v!Þ2

2c4s� j v!j2

2c2s

#(9)

where u0 ¼ 4/9, u1,4 ¼ 1/9 and u5,8 ¼ 1/36. cs ¼ c=ffiffiffi3

pis the sound

speed. For consistency, the forcing terms Fk should be given by

Fk ¼ M�1�I� 1

2S�MjkFk (10)

where

Fk ¼ uk

�c e!k$ F

!c2s

þ ðc e!k$ v!Þðc e!k$ F

!Þc4s

� v!$ F!

c2s

�(11)

and I is the unity matrix. The velocity v! is defined as

v! ¼Xk�0

c e!kgk þ Dt2

F!

(12)

For the present work

F!hPrðT � NYÞ g!

j g!j (13)

The effective kinematic viscosity is determined by

ne ¼ ðs� 0:5Þc2sDt (14)

and ne can be split into two parts:

ne ¼ n0 þ nt (15)

where n0 ¼ PrRa�0.5 is the initial kinetic viscosity, and the turbulenteddy viscosity nt is obtained by

nt ¼ ðCDÞ2�jfj2 þ Pr

PrtVðT � NYÞ$ g!

j g!j

�1=2(16)

The first term in Eq. (16) represents stress forces while thesecond term represents buoyancy [15]. When N ¼ 0, Eq. (16) isreduced the model used in Refs. [15] in which only thermal buoy-ancy being considered. The constant C is called the Smagorinskyconstant and is adjustable [36]. In the present study, we take C¼ 0.1and the turbulent Prandtl number Prt is set to 0.4, which areidentical with that in Ref. [15]. And D is the filter width [15]. jfj isthe magnitude of the large-scale strain rate tensor

jfj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2fabfab

q(17)

where fab ¼ ðvaub þ vbuaÞ=2, and the over bar indicates filteredvalues. In LB model, fab can be computed directly from non-equi-librium moments, the detailed discussion can be found in Refs.[15,37]. After some tedious algebraic operations one can get:

s ¼ s0 þðCDÞ2c2sDt

�jfj2 þ Pr

PrtVðT � NYÞ$ g!

j g!j

�1=2(18)

where s0 ¼ n0=ðc2sDtÞ þ 0:5 .

3.2. Temperature field

The evolution equation for the temperature field reads [15]

fjx!þ c e!jDt; t þ Dt

� fjð x!; tÞ ¼ �s�1Q

hfjð x!; tÞ � f ðeqÞj ð x!; tÞ

i(19)

where sQ is the dimensionless relaxation time for temperature fieldand

f ðeqÞj ¼ Tb

�1þ b

e!j$ u!

2c

�(20)

where b is the number of discrete velocity directions for thetemperature field. The temperature T is obtained in terms of thedistribution function by

T ¼Xj

fj (21)

The effective thermal diffusivity ae is given by

ae ¼ 2c2ðsQ � 0:5ÞDt=b (22)

Similar as ne, the effective thermal diffusivity ae also can be splitinto two parts:

ae ¼ a0 þ at (23)

a0 ¼ Ra�0.5 is the initial thermal diffusivity. The turbulent thermaldiffusivity at ¼ nt/Prt. And


sQ ¼ sQ0þ bDt

2c2Dt(24)

where sQ0¼ ba0=ð2c2DtÞ þ 0:5 .

3.3. Concentration field

The evolution equation for the concentration field reads

hjx!þ c e!jDt; t þ Dt

� hjð x!; tÞ ¼ �s�1D

hhjð x!; tÞ � hðeqÞj ð x!; tÞ

i(25)

where sD is the dimensionless relaxation time for concentrationfield and

hðeqÞj ¼ Yb

�1þ b

e!j$ u!

2c

�(26)

a

b

Fig. 7. Variations of (a) total entropy generation number (b) relative total entropygeneration rates versus Ra.

where b is the number of discrete velocity directions for theconcentration field. The concentration Y is obtained in terms of thedistribution function by

Y ¼Xj

hj (27)

The effective solutal diffusivity De is given by

De ¼ 2c2ðsD � 0:5ÞDt=b (28)

According to Ref. [3], the effective solutal diffusivity De also canbe split into two parts:

De ¼ D0 þ Dt (29)

D0 ¼ LeRa�0.5 is the initial solutal diffusivity, where Le is the Lewisnumber. The turbulent solutal diffusivity Dt¼ nt/Sct, where Sct is theturbulent Schmidt number [3]. In the present simulation, we setSct ¼ 0.4. Consequently

sD ¼ sD0þ bDt

2c2Dt(30)

where sD0 ¼ bD0/(2c2Dt)þ0.5.In the present study, we employ the D2Q5 (two-dimension-five-

velocity) latticemodel [34] to solve the scalar fields. The advantagesof the present model, such as saving computational resources, havebeen presented in our previous study [18].

4. Entropy generation equation

The entropy generation number is given by Chen [11,12,15,16]:

Sv ¼ ðVTÞ2þ41jfj2 þ 42ðVCÞ2 (31)

where the irreversibility distribution ratios 41 ¼ BrT0/DT [11,15] and42 ¼ R2pT

20K

2T=ðLeC0Þ [12,16]. Br ¼ PrEc is the Brinkman number,

where Ec is the Eckert number [15]. KT¼ aT/ac is the expansion ratio[17]. jfj is the magnitude of the strain rate tensor. As shown in ourprevious work [15], jfj can be calculated straightforwardly in LBmethod.

Recognizing the first term in Eq. (31) as reflecting the entropygeneration due to thermal diffusion, the second due to viscousdissipation and the last due to concentration diffusion, the entropygeneration number can be expressed as

Sv ¼ Scond þ Svis þ Smix (32)

where the subscripts cond, vis and mix are used to indicate theeffect of thermal diffusion, viscous dissipation and concentrationdiffusion respectively. Usually Scond is referred to as HTI (HeatTransfer Irreversibility) and Svis as FFI (Fluid Friction Irreversibility)[5].

The total entropy generation number is defined as [5]

Stotal ¼ZU

SvvU (33)

where the subscript U means the global computational domain.

5. Numerical validation

Firstly we validated the present model by simulating double-diffusive convection in a rectangular cavity with A ¼ 2 [17,38]. Agrid resolution 100 � 200 is used for the simulation.

Figs. 2 and 3 illustrate the isothermal lines, the isoconcentrationlines and the stream lines for N ¼ 0.8 and N ¼ 1.3 respectively, with

Fig. 8. (a) Entropy generation number, (b) relative entropy generation rate due to thermal irreversibility, (c) relative entropy generation rate due to viscous irreversibility for N ¼ 0.1and Ra ¼ 1010.

Fig. 9. (a) Entropy generation number, (b) relative entropy generation rate due to thermal irreversibility, (c) relative entropy generation rate due to diffusive irreversibility,(d) relative entropy generation rate due to viscous irreversibility for N ¼ 2 and Ra ¼ 1010.


a

b

Fig. 10. Variations of (a) total entropy generation number (b) relative total entropygeneration rates versus N.


Pr ¼ 1.0, Le ¼ 2.0, Ra ¼ 105. When N < 1.0, the flow is primarilydominated by thermal buoyancy effects, and a large central clock-wise thermal recirculation is predicted with horizontally non-uniform isotherms in the core region within the enclosure.Furthermore, the concentration contours are distorted in the coreof the enclosure with a stable stratification in the vertical directionexcept near the insulated walls of the enclosure. A stagnant zone inthe corners of the enclosure is also observed. In contrast, for N> 1.0the flow is mainly dominated by compositional buoyancy effects.For N ¼ 1.3, a counterclockwise compositional recirculation existsin the core region of the enclosure along with two clockwisethermal recirculations occurring near the top-right and bottom-leftcorners of the enclosure. The contours for temperature andconcentration are almost parallel to each other within the center ofthe enclosure away from the walls. In both cases, the isothermallines, the isoconcentration lines and the stream lines are all point

symmetric with respect to the geometric center of the enclosure.The results obtained by the present model agree well with those inRefs. [17,38].

To further quantify the results, the average Nusselt number Nuand the average Sherwood number Sh at the left wall obtained bythe present model are listed in Table 1 together with referencevalues from Ref. [38]. The average Nusselt number Nu is calcu-lated by

Nu ¼ �1t

Zt

0

ZH0

�vTvx

�dydt (34)

and the average Sherwood number Sh is calculated by

Sh ¼ �1t

Zt

0

ZH0

�vCvx

�dydt (35)

The excellent agreement between them demonstrates thecapability of the present model again.

Then the natural convection in a square cavity with Pr ¼ 0.7 and107 � Ra � 1010 is used to validate the present model by settingN ¼ 0. When Ra � 109 the grid resolution 128 � 128 is employedwhile 256 � 256 for Ra ¼ 1010. It is well known that since Ra � 109

the convectional flows become fully turbulent. Fig. 4 illustrates thecorresponding isotherms. For the transitional flow 107 � Ra � 109,the isotherms at the center of the cavity are horizontal and becomevertical near the walls. When Ra¼ 109, the isothermal lines becomealmost straight at the center and very sharp inside the very thinboundary layers. However when Ra ¼ 1010, the isotherms exceptwhich very close to the mid-plane of the cavity are significantlydeformed by the turbulent flow and not straight any longer. Table 2reports the average Nusselt number at the hot wall, together withthat obtained in previous studies [39,40]. The present results areexcellent agreement with that in the publications [39,40] whenRa < 109. The slight deviations of Nu between different modelsresult from the effect of the eddy viscosity becoming significantsince Ra � 109.

The computer code for entropy generation analysis has beenvalidated in our previous works [11,15,16].

6. Results and discussions

In the present study, we simulate turbulent double-diffusivenatural convection in a rectangle cavity with 108 � Ra � 1011,1� A� 3 and 0.1�N� 2. The other parameters are Le¼ 1, Pr¼ 0.71.When Ra � 109 the grid resolution 128 � 128 is employed while256 � 256 for Ra � 1010 with A ¼ 1. The grid density along verticaldirection is changed according to different A.

6.1. Effect of Ra

We firstly investigate the effect of thermal Rayleigh number108� Ra� 1011 on entropy generation in turbulent double-diffusivenatural convection. The other parameters are A ¼ 1 and N ¼ 2.

Figs. 5 and 6 illustrate the distributions of entropy generationnumber, relative entropy generation rate due to thermal irrevers-ibility (gcond ¼ Scond/S), relative entropy generation rate due todiffusive irreversibility (gmix ¼ Smix/S) and relative entropy gener-ation rate due to viscous irreversibility (gvis¼ Svis/S) for different Ra.It is clear that maximum of entropy generation number increasessignificantly with Ra. Entropy generates intensively near the activewalls, especially in the vicinity of the corners. At higher Ra, forexample Ra ¼ 1011, the layers attached to the active walls in which

Fig. 12. Entropy generation number (a), entropy generation number due to diffusive irreversibility (b), entropy generation number due to thermal irreversibility (c), entropygeneration number due to viscous irreversibility (d) for A ¼ 2 and Ra ¼ 109.

Fig. 11. Entropy generation number (a), relative entropy generation rate due to thermal irreversibility (b), relative entropy generation rate due to diffusive irreversibility (c), relativeentropy generation rate due to viscous irreversibility (d) for A ¼ 1 and Ra ¼ 109.



intensive irreversibility occurs become thinner. At lower Ra such asRa¼ 108, the stratification of the layers is very obvious. However, asRa increasing, the stratification is disturbed and the layers break upinto a lot of plume-like fine structures, which is the characteristic offully turbulence. The same phenomena also can be found from themaps of gcond, gmix and gvis. Moreover, at low Ra the distributions ofentropy generation number and relative entropy generation ratesare nearly symmetric respect to the geometrical center of the cavitywhile not for large Ra.

Fig. 7 shows the variations of total entropy generation numberand relative total entropy generation rates versus Ra. It isstraightforward that Stotal is a monotonic increasing function of Raand this function can be split into two subsections aroundRa ¼ 1010. At each subsection Stotal increases nearly linearly withlog10 (Ra). From this figure, one can observe that when Ra� 109 therelative total entropy generation rates are nearly insensitive to Ra.Since Ra > 109, the relative total entropy generation rate due todiffusive irreversibility decreases against Ra while that due toviscous irreversibility increases. The variation of relative totalentropy generation rate due to thermal irreversibility is slightwithin the whole range.

6.2. Effect of N

To reveal the effect of ratio of buoyancy forces on entropygeneration in turbulent double-diffusive natural convection, in thissubsection we set Ra ¼ 1010 and A ¼ 1 while 0.1 � N � 2.

Figs. 8 and 9 plot the distributions of entropy generationnumber, relative entropy generation rate due to thermal irrevers-ibility, relative entropy generation rate due to diffusive irrevers-ibility and relative entropy generation rate due to viscous

Fig. 13. Entropy generation number (a), entropy generation number due to diffusive irregeneration number due to viscous irreversibility (d) for A ¼ 3 and Ra ¼ 109.

irreversibility for different N. What should be noted is that becausewhen N is very small, gmix nearly can be neglected, therefore themap of gmix for N ¼ 0.1 is not shown here. When N < 1, entropygenerates intensively near the bottom-right and top-left corners ofthe cavity while in the vicinity of the bottom-left and top-right onesfor N > 1. It is very interesting that as N < 1 the fine and irregularstructures of relative entropy generation rates also congregate nearthe bottom-right and top-left corners while that close to thebottom-left and top-right corners since N > 1. It may be explainedas: when N < 1, the convectional flow has the trend to moveclockwise, so the bottom-right and top-left corners become thebarrier of the recirculation, where vortices frequently break up intomore fine structures. Consequently, the gradients of velocity andscalar fields change significantly near these two corners. However,since N > 1, the convectional flow has the trend to move counter-clockwise. Therefore it is not doubtable the similar phenomenaappear near the bottom-left and top-right corners as they nowbecome the barrier for the fluid motion.

The variations of total entropy generation number and relativetotal entropy generation rates versus N are shown in Fig. 10. It isclear that Stotal increases monotonously with N. Especially, sinceN > 1, Stotal increases quickly and linearly with N becauseStotalfN2. The relative total entropy generation rate due to diffu-sive irreversibility approaches to zero when N is very small, but itbecomes the dominant irreversibility since N > 1. At low N, therelative total entropy generation rate due to thermal irreversibilityis the main contributor of entropy generation while it decreasesquickly against N. The relative total entropy generation rate due toviscous irreversibility is not a monotonic function of N. Itdecreases firstly and then increases. Its minimum is achievedabout N ¼ 1.

versibility (b), entropy generation number due to thermal irreversibility (c), entropy


6.3. Effect of A

To reveal the effect of aspect ratio on entropy generation inturbulent double-diffusive natural convection, in this subsectionwe set Ra ¼ 109 and N ¼ 2.

Figs. 11e13 show the distributions of entropy generationnumber, relative entropy generation rate due to thermal irrevers-ibility, relative entropy generation rate due to diffusive irrevers-ibility and relative entropy generation rate due to viscousirreversibility for different A. When A ¼ 1, one can see the stratifi-cation in the layers near the active walls clearly. As A ¼ 2, thestratification is disturbed significantly and it can hardly beobserved for A ¼ 3. The small structures become finer with Aincreasing, which denotes the turbulence intensity is enhancedaccordingly. Without surprise, the maximum of entropy generationnumber becomes larger in bigger A.

a

b

Fig. 14. Variations of (a) total entropy generation number (b) relative total entropygeneration rates versus A.

Fig. 14 illustrates the variations of total entropy generationnumber and relative total entropy generation rates versus A. It isobvious Stotal increases nearly linearly with A as

Stotal ¼ 2023:97Aþ 3924:35 (36)

From this figure, we can find that the relative total entropygeneration rate due to diffusive and thermal irreversibilities bothare monotonic decreasing functions against A while that due toviscous irreversibility is a monotonic increasing function with A,which implies that the motion of working fluid is enhanced as Aincreasing.

The similar results also can be found for Ra ¼ 1010. As illus-trated in Fig. 15, Stotal is also a linear monotonic increasing func-tion with A as

Stotal ¼ 3436:36Aþ 1642:35 (37)

6.4. Spatial self-copy of gcond

It is well-known that with the turbulence intensity beingenhanced, the maps of scalar fields will become more irregular dueto the stochastic motion, which has been demonstrated throughthe distributions of entropy generation number and relativeentropy generation rates presented above. However, through thepresent work, we observe a very interesting regular phenomenonin turbulent double-diffusive natural convection: for tall cavitywith A > 1, the irregular distributions of gcond will form regularperiodical maps along the vertical direction. In the present workwecall this phenomenon as “spatial self-copy”. It can be explainedwith the aid of Figs. 12(b) and 13(b). It is clear that in Fig. 12(b), thecontours within 0� y� 1 can fully overlap with that in 1� y� 2. InFig. 13(b), the contours within 0 � y � 1, 1 � y � 2 and 2 � y � 3 arecompletely same.

Furthermore, we find that the periodical length in space is unity,as shown in Fig. 16 with Ra ¼ 1010 and N ¼ 2. The aspect ratio Avaries from 1.25 to 2. The maps for different A are quite dissimilar.In Fig. 16(a) the contours within 1 � y � 1.25 fully overlap with thatin 0 � y � 0.25; in Fig. 16(b) the contours within 1 � y � 1.5 fullyoverlap with that in 0 � y � 0.5; in Fig. 16(c) the contours within

Fig. 15. Variation of total entropy generation number versus A for Ra ¼ 1010 and N ¼ 2.

Fig. 16. Relative total entropy generation rate due to thermal irreversibility at (a) A ¼ 1.25 (b) A ¼ 1.5 (c) A ¼ 1.75 (d) A ¼ 2 for Ra ¼ 1010.


1 � y � 1.75 fully overlap with that in 0 � y � 0.75 and in Fig. 16(d)the contours within 1 � y � 2 fully overlap with that in 0 � y � 1.

The “spatial self-copy” phenomenon implies that large-scaleregular patterns could emerge through small-scale irregular andstochastic distributions, which is a little similar with the coherentstructure [41] in shear turbulence but their mechanism is quitedifferent. As one of the referees pointed out: self-organized struc-tures usually show an integer number of patterns between theboundaries provided, but never fractional patterns. Moreover, thisphenomenon is observed only for gcond. Other scalar fields inves-tigated in the present work do not have such characteristic.However, through numerous numerical experiments with differentLB models, we have excluded the possibility that it is caused bynumerical technology. But, what is its physical meaning hiddenbehind this phenomenon? Or may it be a new indicator to describeturbulent double-diffusive natural convection? It is an open ques-tion and requires further investigation in future research.

7. Conclusion

In this work, entropy generation of turbulent double-diffusivenatural convection in a rectangle cavity is studied. The effects ofthermal Rayleigh number, ratio of buoyancy forces and aspect ratioon entropy generation are investigated systematically. Through thenumerical simulation, we find that:

1 The total entropy generation number increases with Ra, andthe relative total entropy generation rates are nearly insensi-tive to Ra when Ra � 109.

2 Since N > 1, Stotal increases quickly and linearly with N and therelative total entropy generation rate due to diffusive irre-versibility becomes the dominant irreversibility.

3 Stotal increases nearly linearly with A. The relative total entropygeneration rate due to diffusive and thermal irreversibilitiesboth is monotonic decreasing functions against A while thatdue to viscous irreversibility is a monotonic increasing functionwith A.

Furthermore, we observe a new phenomenon in turbulentdouble-diffusive natural convection, which named as “spatial self-copy” in this work. The “spatial self-copy” phenomenon showsthat regular patterns may emerge through irregular and stochasticcontours. It is a little similar with the coherent structure inturbulence. But the “spatial self-copy” phenomenon only can befound in the maps of gcond and with fractional patterns. Tounderstand it more clearly further detailed study is desired in thefuture.

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant Nos. 50936001 and 51006043) andthe State Key Development Programme for Basic Research ofChina (Grant Nos. 2011CB707301 and 2010CB227004). S. C.thanks the support from the Research Foundation forOutstanding Young Teachers, HUST (Grant No. 2010QN027). Thepresent authors would acknowledge the referees for their valu-able comments.


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Entropy generation of turbulent double-diffusive natural convection in a rectangle cavity

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