Comparative Numerical Simulation of natural Convection in a Porous Horizontal Cylindrical Annulus

Jabrane BELABID1, a and Abdelkhalek CHEDDADI1, b

1Systèmes Thermiques et Ecoulements Réels, Ecole Mohammadia d’Ingénieurs, Université Mohammed V-Agdal, B.P. 765, Rabat, Morocco

abelabide@gmail.com, bcheddadi@emi.ac.ma

Keywords: Natural convection, porous medium, annular space, heat transfer.

Abstract. This work presents a numerical study of the natural convection in a saturated porous

medium bounded by two horizontal concentric cylinders. The governing equations (in the stream

function and temperature formulation) were solved using the ADI (Alternating Direction Implicit)

method and the Samarskii-Andreev scheme. A comparison between the two methods is conducted. In

both cases, the results obtained for the heat transfer rate given by the Nusselt number are in a good

agreement with the available published data.

Introduction

Natural convection in horizontal annular porous media has become a subject receiving increasing

attention due to its relevance in a wide variety of practical applications such as the insulation of

aircraft cabin or horizontal pipes, thermal storage system, underground cable systems, compact

cryogenic devices, gain storage in silos and nuclear waste repositories. The problem of natural

convection in a saturated porous medium bounded by two concentric, horizontal and isothermal

cylinders was investigated firstly by [1]. The regimes considered in this paper have been investigated

also by other authors in [2], [3], [4], [5, 6], [7], [8], [9] and [10].

This paper aims to present a numerical simulation of natural convection in a space confined between

two horizontal concentric cylinders, filled with a saturated porous medium and compare two

methods, namely the ADI method and the Samarskii-Andreev scheme.

Equations

Sketched in Fig. 1 is the physical system consisting in an annular geometry bounded by two long

concentric impermeable cylinders placed in a horizontal position. The inner cylinder of radius ri and

the outer cylinder of radius ro are both kept at uniform and constant temperatures Ti and To,

respectively, with Ti > To.

Fig. 1. Schematic of the problem

The porous medium is saturated by an incompressible Newtonian fluid. The polar coordinates are

used. Thus the basic equations for steady state natural convection with Oberbeck-Boussinesq

approximation are given as follows:

Applied Mechanics and Materials Vols. 670-671 (2014) pp 613-616 Submitted: 01.08.2014Online available since 2014/Oct/08 at www.scientific.net Accepted: 06.08.2014© (2014) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/AMM.670-671.613

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∇. ∗ = 0. (1) ∗ = − ∇ ∗ − . (2) ∗∗ + ∗. ∇ ∗ = ∇ ∗. (3)

The asterisk (*) is used to designate the dimensional variables. We introduce the non-dimensional

variables as follows: =∗; =

∗; =

∗ ∗∗ ∗ ; =

∗ ; =

∗. The governing equations

reduce to the following:

∇. = 0. (4)

= −∇ − . (5)

+ . ∇ = ∇ . (6)

Where V is the adimensional velocity, K is the permeability, µ is the dynamic viscosity, ρ is the

density, a is the thermal diffusivity and T is the adimensional temperature. The equations Eq. 4, Eq. 5

and Eq. 6 in stream function formulation write:

∇ = − sin + . (7)

+ − = ∇ . (8)

ψ is the stream function defined by: = and = − . Where u and v are respectively the

radial and tangential adimensional velocities and r and ϕ are the polar coordinates. Ra is the

dimensionless Rayleigh number given by: = ∆. Where ν is the kinematic viscosity, βT is

the coefficient of thermal expansion and ∆T is the temperature difference. Another dimensionless

parameter of the problem is the aspect ratio: R = ro/ri.

The boundary conditions are handled as follows: r = 1: T = 1 and = 0; r = R: T = 0 and = 0.

The geometric symmetry of the problem studied leads to the addition of two new boundary

conditions: φ = 0, π: = 0 and = 0. The steady state average Nusselt number characterizes the

rate of heat transfer through the whole area. It is given by: = − ln .

Model validation

The governing equations are discretized using the Finite Difference Method with the Alternating

Direction Implicit (ADI) Scheme and the Samarski-Andreev algorithm. The two methods lead to

three diagonal systems of simultaneous equations. The algebraic systems were solved using the

Thomas Tridiagonal Matrix Algorithm. As the study is dedicated to steady-state regimes, the iteration

process is terminated when the following criterion is satisfied in each node of the

grid: ≤ 10 . Where χ refers to T or ψ and n denotes the iteration number performed.

Various flow regimes may develop depending on the initial conditions introduced in the

computations (Fig. 2). To qualify the method of solution, the program was validated by solving the

present convection problem in cases for which solutions are available in the literature. Table 1 shows

the average Nusselt number obtained by previous studies for monocellular flow regimes.

614 Applied Mechanics, Materials and Manufacturing IV

Fig. 2. Monocellular and bicellular flows for R=2 and Ra=120

Table 1. Comparison with literature. Nusselt number for R=2

Ra [4] [3] [11] [12] [5, 6] This study

ADI

This study

Samarskii-Andreev

50 1.341 1.335 1.342 1.362 1.344 1.3439 1.3440

100 1.861 1.844 1.835 1.902 --- 1.8686 1.8687

A good agreement is found between the present work and the literature. In order to guarantee grid

independent solutions, runs were performed in a high refined mesh.

Results and discussion

The comparison between both methods is conducted for monocellular flow regimes. The calculation

is done in a refined mesh of 61×901, respectively in r-ϕ directions. Table 2 shows the comparison for

aspect ratio R = 2 and 21/2 and different values of Rayleigh number, using the same adimensional time

step ∆t for both methods in the calculation. This table clarifies that Samarskii-Andreev method, in

this case, converges faster than the ADI one.

Table 2. Comparison between the ADI and Samarskii-Andreev methods

using the same ∆t=0.0001 for both methods

ADI Samarskii-Andreev

Ra R Nu ψmax CPU time Nu ψmax CPU time

30 2 1.1430 3.4618 174.0938 1.1434 3.4634 31.484

100 1.8686 9.9713 1024.20 1.8687 9.9716 194.125

30 21/2 1.0075 1.5289 248.0469 1.0075 1.5290 42.203

80 1.0512 4.0381 572.2031 1.0513 4.0383 94.312

Using the suitable adimensional time step for each method, Table 3 shows the values of Nu and ψmax

obtained by the codes developed in this work.

Table 3. Comparison between the ADI and Samarskii-Andreev methods

using suitable ∆t for each method

ADI Samarskii-Andreev

Ra R Nu ψmax CPU time Nu ψmax CPU time

20 2 1.0667 2.3384 24.375 1.0671 2.3395 26.437

200 2.6910 16.3145 844.234 2.6910 16.3145 1096.14

20 21/2 1.0033 1.0202 24.796 1.0034 1.0207 34.843

100 1.0782 5.0178 114.812 1.0783 5.0201 159.906

In this case, the results indicate that the ADI method converges faster than Samarskii-Andreev one.

Concerning the bifurcation point, no change is observed in the results given by the two methods. The

bifurcation point is represented by the critical value of the Rayleigh number Rac, under which only

the unicellular flow is obtained (Fig. 3).

Applied Mechanics and Materials Vols. 670-671 615

Fig. 3. Bifurcation point for R=2

Table 4 presents the bifurcation point obtained for different values of R. A good agreement is

observed between the results found and the already published data.

Table 4. Comparison with literature of critical Rayleigh number

This study

ADI

This study

Samarskii-Andreev

[5,6] [4] [10]

R =21/4 225.5 225.5 --- --- ---

R =21/2 112.4 112.4 111.5< Rac<112 ---- 112

R =22/3 86 86 --- --- ---

R =2 62.4 62.4 60.5< Rac <61.5 65.5±0.5 62

Conclusion

Natural convection heat transfer in a porous medium bounded by two horizontal, isothermal, and

concentric cylinders is studied numerically. This work based on a simulation using a finite difference

model has allowed us to compare the ADI method and the Samarskii-Andreev one. Computations for

different initial conditions show that the ADI method is faster if we use the suitable time step ∆t.

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616 Applied Mechanics, Materials and Manufacturing IV

Applied Mechanics, Materials and Manufacturing IV 10.4028/www.scientific.net/AMM.670-671 Comparative Numerical Simulation of Natural Convection in a Porous Horizontal Cylindrical Annulus 10.4028/www.scientific.net/AMM.670-671.613

DOI References

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http://dx.doi.org/10.1115/1.3245638