Lattice Boltzmann Method Simulation of Natural ConvectionHeat Transfer in Horizontal Annulus

Mo Yang,∗ Ziwen Ding,∗ Qin Lou,∗ and Zhiyun Wang∗

University of Shanghai for Science and Technology, 200093 Shanghai, China

andYuwen Zhang†

University of Missouri, Columbia, Missouri 65211

DOI: 10.2514/1.T4978

The lattice Boltzmann method is employed to simulate the stability and transitions of natural convection in a

horizontal annulus with the Prandtl number varying from 0.1 to 0.7 and the Rayleigh number ranging from 103 to5.0 × 105. Different flow patterns including steady upward flow, oscillatory upward flow, steady downward flow, and

oscillatorydownward floware found fordifferent values of the parameters. The flowpattern is very complexwhen the

Rayleigh number is large. The critical Rayleigh number increases with the increase of the Prandtl number and with

the decrease of the aspect ratio.

Nomenclature

A = aspect ratioCp = heat capacityc = fluid particle speedcs = speed of sound in lattice scalecα = discrete lattice velocityDi = inner cylinder diameterFy = force term along the vertical directionfα = density distribution functionGr = Grashof numberg = gravitational accelerationgα = temperature distribution functionj = Jacobiankeq = equivalent thermal conductivity

keqi = mean equivalent thermal conductivity of the innercylinder

keqo = mean equivalent thermal conductivity of the outercylinder

L = width gapN = grid quantityNu = Nusselt numberPr = Prandtl numberp = pressure, Pap 0 = lattice Boltzmann model macropressureQw = constant heat flux on the inner surfaceR = radius ratio; Ro∕Ri

Ra = Rayleigh numberRi = inner radiusRo = outer radiusr = radial coordinateTi = dimensionless temperature of inner cylinderTo = dimensionless temperature of outer cylinderT = temperaturet 0 = lattice Boltzmann model macrotemperatureu = dimensionless velocityu = dimensionless horizontal components of velocityu = velocityu 0 = lattice Boltzmann model macrovelocity

v = dimensionless vertical components of velocityw = weighting factorsβ = thermal expansion coefficientΔt = lattice time stepΘ = dimensionless temperatureθ = angleνk = kinematic viscosityρ = densityρ 0 = lattice Boltzmann model macrodensityτ = lattice relaxation time

I. Introduction

T HE importance of natural convection between horizontalconcentric cylinders has driven a great number of research in

engineering and industrial technology, including nuclear reactordesign, aircraft cabin insulation, and cooling systems in electroniccomponents. The annular shape enclosure is one of the applicablegeometries in engineering and industry; in particular, the horizontalcircular annulus shape is commonly found in solar collector–receiversystems, underground electric transmission cables, vapor condensersfor water distillation, heat exchangers, and food processingequipment. This topic has been widely investigated in the availableliterature [1,2], with studies focused on the large variety of flowstructures that arises from different aspect ratios, Rayleigh numbers,and Prandtl numbers [3,4].For the cases of lowRayleigh numbers, the basic flowfield consists

of two two-dimensional crescent-shaped cells, which aresymmetrical about the vertical plane containing the axes of thecylinders. In each of the two annular half-spaces, the fluid goesupward and downward along the hot inner and cold outer cylinders,respectively. Conduction is the major mode of heat transfer betweenthe differentially heated boundaries of the annulus. As the Rayleighnumber increases, the rotation of the main cells moves upward and athermal plume starts to form at the upper part of the annulus with animpingement region on the outer cylinder. The distributions of thethermal fluxes along the inner and outer cylinders show that thelargest part of the heat convection within the annulus is extractedfrom the lower part of the inner cylinder. The flow and temperaturefields of the crescent-shaped convection have been extensivelyinvestigated experimentally and numerically [5,6].The instability of the crescent-shaped convection with different

aspect ratios and Rayleigh numbers at moderate Prandtl numbers hasbeen investigated extensively. For example, Powe et al. [7] andBishop et al. [8] depicted flow regimes and spatial patterns for air-filled annuli as a function of the aspect ratio and Rayleigh number.Their work showed that, for wide gap annuli with aspect ratios lessthan 2.8, transitions happened from two-dimensional steady to

Received 9 April 2016; revision received 21 October 2016; accepted forpublication 17 November 2016; published online 3 March 2017. Copyright© 2016 by the American Institute of Aeronautics and Astronautics, Inc. Allrights reserved. All requests for copying and permission to reprint should besubmitted to CCC at www.copyright.com; employ the ISSN 0887-8722(print) or 1533-6808 (online) to initiate your request. See also AIAA Rightsand Permissions www.aiaa.org/randp.

*Department of Energy and Power Engineering.†Department of Mechanical and Aerospace Engineering.

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JOURNAL OF THERMOPHYSICS AND HEAT TRANSFERVol. 31, No. 3, July–September 2017

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oscillatory flows. For moderate-gap annuli (2.8 < A < 8.5), a three-dimensional spiral flowwas observed past the transition; whereas fornarrow-gap annuli (A > 8.5), the basic single-cellular flow changedto multicellular flows. Later, the experimental and numerical resultsof Rao et al. [9] were reported in a qualitative agreement with Poweet al.’s experimental map for moderate- and narrow-gap annuli [7].For narrow annuli (A > 10) in the case that three-dimensional flows,some discrepancies were observed. Therefore, the flow in the upperregion of the narrow-gap annulus was either transverse rolls or acombination of longitudinal and transverse rolls with respect to thecylinder axis [10]. If the radius ratio was sufficiently small, an oddnumber of transverse rolls could occur so that a longitudinal flowwaspresent at the midaxial plane [11].With the in-depth research, among the two-dimensional studies, a

large variety of research was focused on multicellular flow patternsand multiplicity of solutions for various sets of parameters [12–18].These results revealed the existence of an imperfect bifurcation. Thenumber of cells mainly depends on the aspect ratio and Rayleighnumber. For a fixed supercritical value of Rayleigh number, thenumber of cells ascends as the aspect ratio increases.When the aspectratio approaches a sufficiently large value, the number of cells tendstoward infinity and the classical Rayleigh–Benard problemcharacterized by a pitchfork bifurcation applies. The multicellularflows calculated in these studies ware shown to undergo an unsteadysecondary instability by increasing the Rayleigh number, providedthe aspect ratio is large enough (for example, A � 14.3). Theresulting periodic flow is composed of cellular patterns located in thebasic flow in the lateral regions. This second type of instability ishydrodynamic in its origin, as in air-filled vertical slots.There were also many studies about the stability in horizontal

concentric annulus. Choi and Moon-Uhn [19] studied the linearstability of the crescent-shaped convection by solving the linearequation for three-dimensional disturbances with a time-marchingmethod. It was shown that the transient behavior was valid for thecases of A ≥ 2.1. This implied that the resultant three-dimensionalspiral convectionwas a steady-state flow,whichwas in contrast to theprevious findings of Powe et al. [7] andMoukalled andAcharya [20].On the other hand, for the cases whereA ≤ 2.1, they could not find

any critical Rayleigh number. The linear stability theory for parallelflowswas applied to natural convection byWalton [21] andDyko andVafai [10] using an expansion in inverse power of the aspect ratio.Dyko and Vafai [10] evaluated critical Rayleigh numbers for three-dimensional instability by using a linear stability theory and energymethod, and they concluded that the instability was subcritical.Bifurcation diagrams were obtained for various radius ratios, and themain thresholds were tracked as a function of the radius ratio byPetrone et al. [22]. The vertical symmetry model was also consideredby Petrone et al. [23] to compute steady and oscillatory asymmetricalflows. A three-dimensional linear stability analysis using a spectralelementmethodwas conducted byAdachi and Imai [24] to evaluate acritical Rayleigh number with respect to three-dimensionaldisturbances over a wide range of aspect ratios. They proposednew transition lines of the critical Rayleigh number that consisted ofthree lines as a function of aspect ratio.In addition, many studies have also considered the influence of

Prandtl number [25–28]. Yoo [29] considered the natural convectionproblem in a narrowhorizontal annuluswithA � 12 and investigatedthe effects of the Prandtl number on the stability of the conductionregime. It was observed that the stability of the conduction regime ofnatural convection could be classified into two regimes. In otherresearch, Yoo [30] studied the natural convection in a narrowhorizontal concentric annulus for a fluid of Pr � 0.4. He found thatthe combined effects of thermal and hydrodynamic instabilityresulted in complex multicellular flow patterns. The influence of thePrandtl number to bifurcation and solutions in the horizontal annuluswere studied within the range of 0.3 ≤ Pr ≤ 1 [31].Many lattice Boltzmann model (LBMs) are designed to simulate

natural convection considering heat transfer. Chatterjee andChakraborty [32,33] and Jourabian et al. [34] used the latticeBoltzmannmethod instead of Navier–Stokes equations. Ren et al. [35]reported a LBM boundary condition-enforced immersed boundary

method to study the natural convection in a concentric annulus[31,32]. Liao and Lin adopted an LBM to simulate the naturalconvection flows with heat transfer [36]. Fattahi et al. [37] simulatednatural convection heat transfer in and eccentric annulus using thelattice Boltzmann model based on a double-population approach.They investigated mixed convection heat transfer in an eccentricannulus based on a multidistribution function, which is a double-population approach [38]. Osman and Sidik [39] investigated naturalconvection from a concentrically and eccentrically inner-heatedcylinder placed inside a cold outer cylinder.The flow patterns in a horizontal concentric annulus have been

studied by different methods, and their linear or nonlinear stabilityhas been investigated with different approaches. However, thestability and transitions of flowfields still require a deeper degree ofinvestigation in some ranges of Rayleigh and Prandtl numbers. Inaddition, none of the available research works simultaneouslyconsidered all the main parameters influencing the flow behavior inthis system. Therefore, the present study is motivated by the need ofrevealing the transitions and the flow structure while taking intoaccount all the main parameters by using the lattice Boltzmannmethod as a new solution technique. The flow and stability arediscussed in detail, and the effects of the aspect ratio and of thePrandtl number on the flow patterns, as well as the critical Rayleighnumber, are systematically investigated.

II. Problem Statement and Mathematical Model

A schematic diagram of the physical model of the investigatedsystem is shown in Fig. 1. The natural convection in a horizontalcylindrical annulus with an inner radius Ri and an outer radiusRo is simulated (The inner and outer diameters are denoted byDi and Do, respectively). The angle θ is measured clockwise fromthe upper vertical plane. The radial aspect ratio is denotedas A � Di∕�Ro − Ri�.In this paper, keq is the local equivalent thermal conductivity and

Qw is the constant heat flux on the inner surface. The velocity u andtemperature T are nondimensionalized as follows:

u � u

c(1)

T � T

Qw�Ro − Ri�∕keq(2)

The flow is assumed to be incompressible, laminar, Newtonian,and two-dimensional. The Boussinesq-approximated Navier–Stokesequations and the energy equationwith negligible viscous dissipationare used to determine the buoyancy-induced flowfield:

g

r

Ri

Ro

Ti

To

Ti > To

=0o

=180o

θ

θ

θ

Fig. 1 Physical model.

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θ � T − T0

Ti − T0

The governing equations written in dimensionless form can bewritten as follows:

∇ · u � 0 (3)

∂u∂t

� �u · ∇�u � −∇p� Pr · Δu� Ra · Pr · Θ · j (4)

∂Θ∂t

� �u · ∇�Θ � ΔΘ (5)

These equations show that this problemdepends on one geometricalparameter (the aspect ratio) and twodimensionless parameters (Prandtlnumber and Rayleigh number). They are defined as

Ra � cpρ2gβ�R0 − Ri�4Qw

k2eqvk; ρr �

Cpvkkeq

(6)

where ρ, Cp, and νk are the density, specific heat capacity, and kineticviscosity of the fluid, respectively.The Prandtl number is defined as the ratio of momentum

diffusivity (kinematic viscosity) to thermal diffusivity;andtheRayleigh number is defined as the product of the Grashof number,which describes the relationship between buoyancy and viscositywithin a fluid, and the Prandtl number, which describes therelationship between momentum diffusivity and thermal diffusivity.Hence, the Rayleigh number itself can also be viewed as the ratio ofbuoyancy to viscosity forces times the ratio of momentum to thermaldiffusivities. The boundary conditions at the inner and outer cylinderscan be written as follows:

r � Ri and 0 ≤ θ ≤ 2π∶ u � v � 0 and Θ � 1

r � Ro and 0 ≤ θ ≤ 2π∶ u � v � 0 and Θ � 0

The local equivalent thermal conductivity is given by

keq � −r ln R∂Θ∂r

(7)

The mean equivalent thermal conductivities on the inner and outercylinders are calculated by integrating the local equivalent thermalconductivity on the surface area of either cylinder:

keqi � −Ri ln R2π

Z2π

0

�∂Θ∂r

�r�Ri

dθ

keqo � −Ro ln R2π

Z2π

0

�∂Θ∂r

�r�Ro

dθ (8)

III. Lattice Boltzmann Methodand Interpolation Method

A. Lattice Boltzmann Method

The latticeBoltzmannmethod is a numerical scheme for the solutionof Boltzmann’s transport equation (BTE) for particle distributionfunction. The mass, momentum, and energy conservation equations offluid dynamics can be derived from BTE through a Chapman–Enskogexpansion [40–42]. The same scheme can be extended to show that theLattice Boltzmann Equations (LBE) method leads to the mass andmomentum conservation equations of fluid dynamics [43–52].

The discrete velocity set of two-dimensional nine velocity (D2Q9)lattice Boltzmann equation method in the Bhatnagar Gross Krook(BGK) approximation is used in this paper. The general form of theLBE in the presence of an external force can be written as follows:

fα�x� cαΔt; t� Δt� − fα�x; t� � Δtfeqα �x; t� − fα�x; t�

τ� Δt · F

(9)

where Δt is the lattice time step; cα �α � 0; 1 : : : ; 8� is the latticevelocity along the direction α in the discrete velocity set of two-dimensional nine velocity; F is the external force; fα is the particledistribution function along the direction α; and τ denotes the latticerelaxation time (the average time interval between particlescollision), which is related to the kinematic viscosity by the followingrelation:

τ ��

1

Δtc2s

�ν∂ � 0.5 (10)

where να is the kinematic viscosity, cα is the speed of sound, andfeq isthe equilibrium distribution function that depends on the type ofproblem. The equilibrium distribution functions for the fluid field arecalculated according to the following equation:

feqα � wα · ρ

�1� �cα · u�

c2s� �cα · u�2

2c4s−

u2

2c2s

�(11)

where wα is the set of weighting coefficients along each direction.The values ofw0 � 4∕9 for jc0j � 0,w1–4 � 1∕9 for jc1–4j � 1, andw5–8 � 1∕36 for jc5–8j �

���2

pare assigned in this model. Also, ρ and

u are the macroscopic fluid density and velocity, and they arecalculated as follows:

ρ �Xα

fα (12)

ρu �Xα

fαcα (13)

The thermal Lattice Boltzmann equation can bewritten as follows:

gα�x� cαΔt; t� Δt� − gα�x; t� � Δtgeqα �x; t� − gα�x; t�

τc(14)

where the components of the thermal equilibrium distributionfunction are given by the following:

geqα � wαT

�1� 2

�cα · u�c2

�(15)

where T is the fluid temperature, which can be calculated as follows:

T �Xα

geqα (16)

The temperature relaxation time is related to diffusivity by thefollowing equation:

τc ��

2

Δtc2

�α� 0.5 (17)

To incorporate the buoyancy force in the model, the Boussinesqapproximation was applied; therefore, the force term in Eq. (9) needsto be calculated as follows, in the vertical direction y:

Fy � 3wαgyβΔT (18)

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where gy is the acceleration of gravity acting in the y direction of thelattice links, β is the thermal expansion coefficient, and ΔT is thetemperature difference.

B. Interpolation Method Boundary Conditions

In the present study, the computational domain is set to be −L <� x and y <� L, where the cylinder’s center is taken to be thecoordinate origin and is covered by a square lattice. As shown inFig. 2. The evolution process at the lattice grids of the square ispreformed using the D2Q9 model. To obtain the variables in thehorizontal cylindrical annulus, it is divided into a series of concentriccircles as virtual grids. LBM grids variables are obtained by theinterpolation of the variables at the square grids.By keeping a fixed node distance at each virtual grid layer, the

amount of nodes in each virtual grid layer is accordingly not constant.When the physical boundaries (i.e., the inner and outer cylinders) arereached, the amount of nodes is defined to be the same as theneighbored virtual grid layer in order to use the nonequilibriumextrapolation method.In this paper, the Taylor-series expansion and the least-squares-

based interpolation methods are used. The following momentummatrix can be achieved:

�S� �

26666664

1 Δx0 Δy0�Δx0�2

2�Δy0�2

2Δx0Δy0

1 Δx1 Δy1�Δx1�2

2�Δy1�2

2Δx1Δy1

− − − − − −− − − − − −− − − − − −1 ΔxM ΔyM

�ΔxM�22

�ΔyM�22

ΔxMΔyM

37777775

�M�1�×6(19)

whereM is the number of neighboring points around P. The Δx andΔy values in the matrix �S� are given by

Δx0 � cαxΔt; Δy0 � cαyΔt (20)

Δxi � xi � cαxΔt − x0; Δyi � yi � cαyΔt − y0;

for i � 1; 2; · · · ;M(21)

When the coordinates of all mesh points are given and the particlevelocity and time-step size are specified, thematrix �S� is determined.To minimize the error used in the least-squares method, we obtain

fVg � ��S�T �S��−1�S�Tfhg � �B�fhg (22)

where

fhg � fh0; h1; · · · ; hMgT;hi � fα�xi; yi; t� � �feqα �xi; yi; t� − fα�xi; yi; t��∕τ (23)

Note that �B� in Eq. (22) is a 6 × �M� 1�-dimensional matrix.From Eq. (22), we find

fα�x0; y0; t� Δt� � V1 �XM�1

n�1

bα1;nhαn−1 (24)

where bα1;n are the elements of the first row of matrix [B], which areprecomputed before the LBM is applied.For the processing of boundary conditions, we choose the

nonequilibrium extrapolation boundary, which has a wideapplication range, and for which the convergence and numericalaccuracy can meet the requirement. The equilibrium distributionfunctions can be decomposed into two parts, corresponding toequilibrium and nonequilibrium:

fα � feqα � fneqα (25)

For the equilibrium part, all the physical parameters can becalculated by the adjacent internal nodes; whereas for thenonequilibrium part, the physical parameters can be calculated by asecond-order Taylor expansion:

fneqα �neighbor� � fneqα �O�δ2x� (26)

The equilibrium distribution functions can be written as follows:

fα � feqα � �fα − feqα ��neighbor� (27)

Fig. 2 Schematic diagram of the grid model.

Table 1 Averaged equivalent thermal conductivity of inner cylinder at A � 1.25 and Pr � 0.7

Rayleigh number 103 3 × 103 6 × 103 1 × 104 2 × 104 3 × 104 5 × 104

Kuehn and Goldstein [5] 1.082 1.403 1.735 2.007 2.400 2.652 2.998Present 1.079 1.392 1.708 1.968 2.352 2.588 2.891Relative error, % 0.277 0.784 1.556 1.943 2.000 2.413 3.569

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C. Numerical Solution Procedure

1) Initialize Macroparameters ρ, p, t, and u.2) Initialize equilibrium distribution functions fα and Tα.3) Initialize the equilibrium distribution functions in the

boundary.4) Calculate equilibrium distribution functions after the collision

and stream f 0α and T 0

α.5) Calculate macroparameters ρ 0, p 0, t 0, and u 0.6) Perform the convergence test.

7) Calculate macrocharacteristic parameters

max jΘn�1i;j − Θn

i;jj < 10−8

max���jujn�1

i;j − jujni;j��� < 10−8

If convergence is not achieved, the process resumes from step 3;otherwise, proceed to the last step.

IV. Grid Testing and Code Validation

To test and assess the grid independence of the solution method,numerical simulations are performed as shown in Table 1. The gridquantity is used as

N �Xn�ny

n�0

2π

�Ri � n ·

Ro − Ri

ny

�∕dry (28)

where ny indicates the number of grid layers in the radial direction,and dry denotes the distance between nodes on the same layer. Tomake the lattice Boltzmann method operate properly, the value ofdry � 1.5 is adopted.To evaluate the performance of the present method, the averaged

equivalent thermal conductivity keq is derived for natural convectionin a concentric horizontal annulus and the results are compared withthose of Kuehn and Goldstein [5] in Fig. 3 and Table 1, finding goodagreement between the isotherms and streamlines obtained from thepresent work and those taken from the literature. The relative error in

Angle

Equ

ival

ent t

herm

al c

ondu

ctiv

ity

0 50 100 150

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 keq of inner cylinder, presentkeq of outer cylinder, presentkeq of inner cylinder, Kuehn et al. [5]keq of outer cylinder, Kuehn et al. [5]

Fig. 3 Averaged equivalent thermal conductivity as a function of the angle.

Fig. 5 Isotherms at A � 2 and Pr � 0.3: a) Ra � 2.0 × 104; b) Ra � 3.0 × 104; and c) Ra � 1.0 × 105.

Fig. 4 Transient development of flow patterns for Pr � 0.3 and Ra 10000.

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terms of equivalent thermal conduction is less than 2% forRa � 103.As can be seen from Fig. 3, the mean equivalent thermalconductivities of both cylinders, at different Rayleigh numbers,closely match the values from [5].

V. Results and Discussions

In the present study, the effects of the Rayleigh number, the Prandtlnumber, and the aspect ratio on natural convection in a horizontalconcentric annulus are investigated using the LBE method asdiscussed previously. Calculations are performed in the parameterranges of 103 ≤ Ra ≤ 106, 0.1 ≤ Pr ≤ 0.7, and 0.4 ≤ A ≤ 10; andthe initial condition is assumed to be u � v � Θ � 0with the innercylinder suddenly heated to Θ � 1.

A. Effects of Rayleigh Number

We first investigate the effects of the Rayleigh number on thestability and the transitions of natural convection in a horizontalconcentric annulus. In the simulations discussed in this section, theaspect ratio and the Prandtl number are fixed at 2 and 0.3,respectively.The flow patterns are here labeled according to the classification

by Yoo [30] based on the flow direction atop the annulus. The flowpattern in which the fluid ascends along the center plane is called“upward” flow; the other, which descends, is called “downward”flow.Thevariation of flowpatterns atRa � 104 is displayed in Fig. 4,where the fluid is seen to form a crescent-shaped eddy at the earlystage (t � 0.0949). In particular, the fluid rises near the inner hotcylinder and sinks near the outer cold cylinder, forming a kind ofupward flow in which the fluid ascends in the top of the annulus(θ � 0). The flow patterns gradually change with time (Figs. 4b and4c). At t � 0.269, the downward flow pattern is established with thefluid descending in the top of the annulus (θ � 0) by forming twocounter-rotating eddies (see Fig. 4d). This kind of transition is inagreement with Yoo’s description. However, at a later stage, the flowpattern transforms to upward again, as illustrated in the figure fort � 12.65 (Fig. 4g) and eventuallymaintains this steady upward flow(see Fig. 4h). The detailed transition process from downward flow toupward flow is shown in Figs. 4e and 4f.To systematically study the effects of the Rayleigh number, a series

of Rayleigh numbers (in the range 103 ≤ Ra ≤ 106) are selected tostudy the transition phenomenon, and the final flowfields arepresented in Fig. 5. When the Rayleigh number is rather small(i.e.,Ra ≤ 6.0 × 103), the upward flow prevails at the beginning untilthe steady upward flow pattern is established. At higher values of theRayleigh number (i.e., Ra � 104), the downward flow occurs atthe beginning and then transforms to steady upward flow. AtRa � 2.0 × 104, a right side-skewed downward flow is observedafter an initial phase of symmetric downward flow. The trends ofdimensionless velocity and temperature over time, corresponding tothe flows of Fig. 5, are displayed in Fig. 6, which also reports thetransition of the flow pattern and the oscillation phenomenon. From

Dimensionless time

Dimensionless time

Dim

ensi

onle

ss v

alue

sD

imen

sion

less

val

ues

0a)

b)

c)

2 4 6 8

-30

-25

-20

-15

-10

-5

0

5

horizontal velocity-uvertical velocity-vtemperature

Enlargement of temperature0 2 4 6 80

0.2

0.4

0.6

0 2 4 6 8

-25

-20

-15

-10

-5

0

5

horizontal velocity-utemperature

Enlargement of temperature0 2 4 6 8

0.1

0.2

0.3

0.4

0.5

Dimensionless time

Dim

ensi

onle

ss v

alue

s

0 1 2 3 4 5

-120

-100

-80

-60

-40

-20

0

20

40

60

80

horizontal velocity-uvertical velocity-v

Fig. 6 Variation of dimensionless values (horizontal andvertical velocity,temperature) at Pr � 0.3: a) Ra � 2.0 × 104; b) Ra � 3.0 × 104; andc) Ra � 1.0 × 105.

104 1051.0

1.5

2.0

2.5

3.0

3.5

4.0

keq

re

dnilyc

re

nni

fo

Rayleigh number

steady upward flow

steady skewed downward flow

periodical upward flow

periodical downward flow

Fig. 7 Equivalent thermal conductivity of the inner cylinder atPr � 0.3when the inner cylinder is suddenly heated as an initial condition.

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the figure, we can see that, at Ra � 2.0 × 104, the dimensionlessvalues increase sharply to a certain level. The transition process isindicated by a gradually appearing steady decrease, and the steadyflow pattern is achieved with a similar right downward-skewedshape. At Ra � 3.0 × 104, the flow becomes oscillatory with aregular periodicity. At Ra � 1.0 × 105, the steady upward flowpattern is achieved again, after an initial stage of downward flow.To understand the effects of the Rayleigh number on flow pattern,

we performed simulations for different Rayleigh numbers; the resultsare shown in Fig. 7. When the inner cylinder is suddenly heated, thesteady upward flow can be established in the rangesRa ≤ 1.5 × 104,2.45 × 104 ≤ Ra ≤ 2.6 × 104, and 9.9 × 104 ≤ Ra ≤ 1.1 × 104.The steady right-skewed downward flow is maintained at1.55 × 104 ≤ Ra ≤ 2.4 × 104, whereas the oscillatory flow appearsat 2.65 × 104 ≤ Ra ≤ 9.7 × 104. The presence of these patterns canbe explained by noting that, for all the simulation conditions,momentumdiffusion is weak (i.e.,Pr � 0.3) and the flow itself is notvery stable. When the Rayleigh number is low, thermal diffusion is apredominant factor. The flow pattern evolves into an upward flow asthe Rayleigh number increases gradually. The flow of buoyancy liftgradually increases, and momentum diffusion becomes stronger,although it cannot balance the flow due to heat transfer. The flowpattern in these conditions is a periodical downward flow. If theRayleigh number is further increased, momentum diffusion becomesthe predominant factor and the flowpattern evolves into an oscillationupward flow.The influence of initial conditions is also studied, and the results

are shown in Fig. 8. The initial upward flow patterns are the same asdiscussed before, whereas the range of Rayleigh numbers leading tothe steady upward flow can be extended to Ra � 3.89 × 104;although, the flow symmetry at 3.5 × 104 ≤ Ra ≤ 3.89 × 104 is notcomplete, as can be seen in Fig. 5c. The range of values leading to thesteady right-skewed flow can be extended to Ra ≤ 2.85 × 104,anticipating the formation of the right-skewed flow whilemaintaining the zone of oscillatory flow in the same range. Because

the periodic oscillation can be regarded as a kind of dynamicbifurcation, it is in fact a combination of many solutions. We canconclude fromFig. 8 that, because of different initial conditions, withan increasing Rayleigh number, dual solutions exist for 1.55 × 104 ≤Ra ≤ 2.65 × 104 and multiple solutions are indeed valid for2.65 × 104 ≤ Ra ≤ 9.5 × 104. The transitions between flow patternregimes follow a clear route (i.e., the flow pattern evolves from steadyto oscillating, then back to steady, and then to oscillating again) whenthe Rayleigh number is sufficiently high.

0 20,000 40,000

1.5

2.0

2.5

initial condition of right-skewed flow

Rayleigh number

steady upward flow steady skewed downward flow periodical downward flow asymmetrical upward flow

Unique Dual Multiple

initial condition of oscillatory flow

initial condition of upward flowkeq

re

dnilyc

re

nni

fo

Fig. 8 Equivalent thermal conductivity of the inner cylinder at Pr � 0.3 when the inner cylinder is suddenly heated as an initial condition.

Fig. 9 Isotherms at Pr � 0.7 and Ra � 5.0 × 104 with different aspect ratios: a) A � 2, b)A � 3, and c) A � 4.

Dimensionless time

Dim

ensi

onle

ss te

mpe

ratu

re

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

A=2A=3A=4

Fig. 10 Time variation of dimensionless temperature atA � 2,A � 3,and A � 4.

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Asmentioned previously, themost important parameter governingfluid patterns transition is the Rayleigh number, for which theincrease can change the state of flow and heat transfer from steady tooscillatory.

B. Effects of Aspect Ratio

Wenow consider the effects of the aspect ratio on natural convectionin a horizontal concentric annular system. In the following simulations,the Prandtl number is kept constant at Pr � 0.7.

The isotherms corresponding to different aspect ratios (A � 2, 3,and 4) atRa � 5.0 × 104 are illustrated in Fig. 9,which clearly showsthat all the obtained isotherms are stable, although they are notcharacterized by the same flow conditions. For the case of an aspectratio of A � 2, the flow is initially upward and evolves into steady,with an according temperature stabilization (see Fig. 10). For thecases of aspect ratios of A � 3 and A � 4, the flow transforms fromdownward to steady upward.We also consider the case of larger aspect ratios, obtaining the

results shown in Fig. 11. When A � 6, the flow forms a stablecrescent-shaped eddy, whereas a pair of cells appear at the top part ofthe annulus for A � 8 and A � 10. This is mainly caused by the walleffect, which occurs in a limited space by interaction between theflow and thermal boundary layers. Moreover, the formation of flowpatterns at A � 6 undergoes a transition from a less symmetricdownward flow to a steady upward flow, whereas the upward flow ismaintained at all times when A � 8. The oscillatory upward flowappears at A � 10 (see Fig. 11). We also observe a temperatureincrease until a steady level is reached (see Fig. 12).From the aforementioned results, we can conclude that the

transition phenomenon does not occur when the aspect ratio is small.We now discuss the relationship between flow patterns and

Rayleigh numbers. A classification of the flow patterns at different

aspect ratios on a wide range of Rayleigh numbers is reported inFig. 13, where TIUF represents time-independent upward flow, TFrepresents transition flow, POF represents periodic oscillatory flow,and NPOF represents nonperiodic oscillatory flow. As can beseen in the figure, for A � 3, the transition occurs for Rayleighnumbers in the range 3.4 × 104 ≤ Ra ≤ 6.1 × 104, whereas for6.3 × 104 ≤ Ra ≤ 4.1 × 105, the upward flow is maintained until thesteady state is achieved. For Ra ≥ 4.3 × 105, a periodic oscillatoryupward flow occurs. At A � 4, interestingly, the oscillationphenomenon occurs for 2.5 × 103 ≤ Ra ≤ 3.2 × 103 because thehydrodynamic effect is not strong and the flow in this limited space isstrongly affected by the heat transfer process. After that, the transitionoccurs at 3.4 × 103 ≤ Ra ≤ 4.7 × 103 and the steady upward flow issustained in the range 4.9 × 103 ≤ Ra ≤ 3.5 × 104. Anothertransition is found when 3.8 × 104 ≤ Ra ≤ 2.6 × 105, leading to aperiodic oscillatory upward flow at Ra ≥ 2.8 × 105. When A � 6,the upward flow is present at Ra ≤ 2.3 × 103 and transition flowoccurs at 2.4 × 103 ≤ Ra ≤ 6.1 × 103. Periodic oscillatory flowhappens at 6.5 × 103 ≤ Ra ≤ 2.2 × 104. Time-independent upwardflow is formed again atRa � 2.5 × 104, and another transition in theflow pattern occurs at 2.8 × 104 ≤ Ra ≤ 5.5 × 104. Then, periodicoscillations are established at Ra � 6.0 × 104 and nonperiodicoscillations occur at 6.8 × 104 ≤ Ra ≤ 8.5 × 104. Another transitionin the flow pattern occurs at 9.8 × 104 ≤ Ra ≤ 2.5 × 105, andnonperiodic oscillatory flow is eventually established atRa ≥ 3.5 × 105. When A � 8, time-independent upward flow isfound at Ra ≤ 2.8 × 103 and the first transition in the flow patternoccurs at 2.9 × 103 ≤ Ra ≤ 3.3 × 103. Periodic oscillatory flow isthen established at 3.5 × 103 ≤ Ra ≤ 8.0 × 103, and nonperiodicoscillatory flow occurs at 8.5 × 103 ≤ Ra ≤ 4.7 × 104. If theRayleigh number is further increased, a transition in the flowbehavior occurs at 4.9 × 104 ≤ Ra ≤ 2.5 × 105 and nonperiodic

Fig. 11 Streamlines at Pr � 0.7 and Ra � 3.0 × 103 for a) A � 6, b) A � 8, and c)A � 10.

Dimensionless time

Dim

ensi

onle

ss te

mpe

ratu

re

0 20 40 60 800

0.2

0.4

0.6

0.8

A = 6A = 8A = 10

Fig. 12 Time variation of dimensionless temperature atA � 6,A � 8,and A � 10.

2 3 4 5 6 7 8 9 10 11103

104

105

106

TIUF TF POF NPOF

Ray

leig

h nu

mbe

r

A=D/LFig. 13 Categorization of flow patterns in a horizontal concentriccylindrical annulus; above the solid line, the flow becomes oscillatory.

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oscillatory flow is finally established at Ra ≥ 3.5 × 105. WhenA � 10, as the distance between the two cylinders is very small, thetransition phenomenon seldom occurs and the flow transformsdirectly from steady upward flow to periodic (at first) and tononperiodic oscillations with the increase of Rayleigh number.This analysis allows us to state that, in general, nonperiodic

oscillations are only found forA ≥ 6. The solid line in Fig. 13 dividesthe region of oscillatory flow patterns from the steady flow zone.Figure 14 shows the averaged equivalent thermal conductivity of

the inner cylinder for different aspect ratios on a wide range ofRayleigh numbers. As can be seen from the figure, no oscillations arefound for A ≤ 3; the oscillation zones for A � 4 and A � 6 are alsoshown, significantly pointing out the effects of variations in theaspect ratio.To summarize the results of this analysis, the increase of parameter

A corresponds to a narrower gapwidth, whichmeans that the thermalboundary layer and the flow boundary layer influence each other,making the flow behavior more chaotic. The critical Rayleighnumber for the transition of the flow and heat transfer from a steadystate to an oscillatory state becomes lower.

C. Effects of Prandtl Number

To investigate the effects of the Prandtl number on thecharacteristics of natural convection in a horizontal concentricannulus, the flow behavior and properties are studied as a function ofthe Prandtl number (0.1 ≤ Pr ≤ 0.7), wheeras the aspect ratio A isfixed at two.Yoo investigated [31] dual steady solutions for the specific case

of Pr � 0.7, obtaining the downward flow by introducing artificialnumerical disturbances. The upper part of an annulus with a heatedinner cylinder is thermally unstable; accordingly, the instability ofthe crescent-shaped upward flow that yields downward flow canpersist, if the unstable stratification in the top region is sufficientlystrong.Our results in terms of the equivalent thermal conductivity of the

inner cylinder aer presented with Pr � 0.2 for a wide range ofRayleigh numbers in Fig. 15, in which SU represents steady upwardflow,OU represents oscillatory upward flow, SLSD represents steadyleft-skewed downward flow, SRSD represents steady right-skeweddownward flow, OBLRSD represents oscillatory between left-and right-skewed downward flow, ORSD represents oscillatoryright-skewed downward flow, OLSD represents oscillatory left-skewed downward flow, OBUD represents oscillatory betweenupward and downward flow, and SSD represents steady symmetricdownward flow The figure shows that nine different types of flowpatterns appear for different Rayleigh numbers. Specifically, atRa ≤ 4.3 × 103, time-independent steady upward flow is found,whereas at 4.7 × 103 ≤ Ra ≤ 6.5 × 103, oscillatory upward flow isestablished. With further increase of the Rayleigh number, steadyleft- and right-skewed downward flows appear at 7.0 × 103 ≤ Ra ≤4.2 × 104 and 4.3 × 104 ≤ Ra ≤ 7.5 × 104, respectively. Afterward,the oscillatory downward flow swings from left to right as theRayleigh number ranges from 7.7 × 104 to 1.44 × 105. We remarkthat this kind of oscillation is different from the oscillation describedin the previous subsection with Pr � 0.3. In fact, this oscillation hasa fairly small amplitude and the flow mostly fluctuates from left- toright-skewed with a rather small deviation angle from the centersymmetric line, as opposed to the large deviation angle found atPr � 0.4. At higher Rayleigh numbers, oscillatory left- and right-skewed downward flows appear at 1.45 × 105 ≤ Ra ≤ 1.65 × 105

and 1.7 × 105 ≤ Ra ≤ 1.93 × 105, respectively. The oscillatory flowbetween upward and downward occurs when the Rayleigh number isapproximately 1.95 × 105. Then, a steady symmetrical downwardflow is established at 1.98 × 105 ≤ Ra ≤ 2.15 × 105. Beyond thisrange, the oscillatory right-skewed downward flow appears again.We may thus conclude that, for Pr � 0.2, the transitions in the flowpattern do not follow the same order as we indicated for the case ofPr � 0.3; but they follow a different, more complex route, which can

0 50,000 100,000 150,000 200,000 250,000

1.5

2.0

2.5

3.0

3.5

Rayleigh number

SU OU SLSD SRSD OBLRSD ORSD OLSD OBUD SSD

keq

re

dnilyc

re

nni

fo

Fig. 15 Equivalent thermal conductivity of the inner cylinder for different Rayleigh numbers at Pr � 0.2.

103 104 105

1.5

2

2.5

3

3.5

4

A=1.25A=2A=3A=4A=6

Oscillation

Oscillation

Rayleigh number

keq

re

dnilyc

re

nni

fo

Fig. 14 Equivalent thermal conductivity of the inner cylinder fordifferent aspect ratios as a function of the Rayleigh number (the Prandtlnumber is fixed at Pr � 0.7).

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be schematized as a “steady, oscillation, steady, oscillation, steady,oscillation” sequence.For Pr � 0.4, the same route situation can also be observed on

the whole range of Rayleigh numbers, as shown in Fig. 16. ForRa ≤ 2.4 × 104 or Ra ≥ 4.7 × 104, a steady upward flow pattern isobserved, whereas the oscillatory flow appears in the intermediaterange 2.5 × 104 ≤ Ra ≤ 4.5 × 104. The simpler transition route“steady, oscillation, steady, oscillation” is found in this case as well,which is also consistent with existing studies.Figure 17 shows the averaged equivalent thermal conductivity for

different Prandtl numbers, with a fixed aspect ratio of A � 2 and aRayleigh number ofRa � 104. As can be seen from the figure, in thisregion, keq exhibits a sharp decrease when the Prandtl numberincreases from 0.2 to 0.3 because the flow is downward at Pr � 0.2.However, further increasing the Prandtl number results in a gradualincrease of keq due to the upward flowfield in this region. Theseresults allowus to state that the transition does not appear atPr ≥ 0.4.The critical line beyond which the periodic oscillatory upward flowoccurs is displayed in Fig. 18. The critical Rayleigh number increaseswith the increase of Prandtl number for the case of A � 2.

As a final conclusion to this analysis, we may state that increasingthe Prandtl number reduces the amount of possible flow patternregimes.

VI. Conclusions

The lattice Boltzmann method is employed to simulate naturalconvective flow in a horizontal concentric annulus, focusing on itsstability and on the transitions between different flow patterns. Theeffects of the Rayleigh number, the annulus aspect ratio, and thePrandtl number on flow patterns and on the temperature distributionare systematically investigated. Different scenarios of transitions arestudied, revealing two important features of flows:1) At lowRayleigh numbers (Ra < 103) andwith an aspect ratio of

A � 2, the presence of crescent-shaped or kidney-shaped eddies isrevealed, regardless of the values of the Prandtl number and aspectratio. When Pr � 0.2, the evolution of flow patterns as the Rayleighnumber is increased can be schematized as a threefold switchingbetween steady and oscillation. The transition scheme is simpler (i.e.,steady, oscillation, steady, oscillation) at Pr � 0.3 and Pr � 0.4. AtPr ≥ 0.5, the transitional flow does not occur, leading to a single“steady, oscillation” switching sequence.2) For aspect ratios of A ≤ 2, the flow keeps upward until

stabilizing to steady upward flow and no transition occurs over thewhole investigated range of Rayleigh numbers of103 ≤ Ra ≤ 5.0 × 105. For A ≥ 3, the transition in flow patternoccurs at certain Rayleigh numbers. When A � 3, the flow patternsundergo a transitional route that can be schematized with thesequence “sustained stable flow, transitional flow, sustained stableflow, periodic oscillatory flow.” When A � 4, a more complextransition route is observedwith the increase of the Rayleigh number,namely, “sustained stable flow, periodic oscillatory flow, transitionalflow, sustained stable flow, transitional flow, periodic oscillatoryflow.” Finally, when A ≥ 6, the nonperiodic oscillatory flow occursafter the appearance of periodic oscillatory flow.3) The Rayleigh number is the most important parameter governing

flow pattern transitions. As the Rayleigh number increases, the state offlow and heat transfer evolves from steady to oscillatory.4)When the aspect ratio is increased, the critical Rayleigh number

for the transition of the flow and heat transfer from steady state tooscillatory state becomes lower.5) When the Prandtl is increased, fewer different flow patterns are

found for the fluid in the investigated horizontal annulus.

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (grant nos. 51476103 and 51406120), and it issupported by the Innovation Program of the Shanghai MunicipalEducation Commission (grant no. 12DZ1200100).

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0.3 0.4 0.5 0.6 0.70.0

2.0x105

4.0x105

6.0x105

8.0x105

1.0x106

Ray

leig

h nu

mbe

r

Prandtl number

oscillatory flow

steady flow

Fig. 18 Critical Rayleigh numbers for different Prandtl numbers at

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keq

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