Strojniški vestnik - Journal of Mechanical Engineering 51(2005)7-8, 501-508 UDK-UDC 536.2 Izvirni znanstveni clanek - Original scientific paper (1.01) Effect of asymmetry on the steady thermal convection in a vertical torus filled with a porous medium Postelnicu Adrian1, Scurtu Nicoleta2 1Department of Thermal Engineering and Fluid Mechanics, Transilvania University, Bdul Eroilor, No 29, Brasov, Romania* e-mail: adip@unitbv.ro 2Numerical Mathematics and Scientific Computing, Weierstrass-Institute for Applied Analysis and Stochastic (WIAS), Mohrenstrasse 39, 10117 Berlin, Germany Abstract The paper deals with a study on the thermal convection in a fluid saturated porous medium confined in a vertical oriented toroidal loop heated from below and cooled from above, subjected to a vertically uniform temperature gradient. In fact, Sano, O., Journal of the Physical Society of Japan (1987 and 1988) studied such a configuration for clear fluids. We take the steady heat conduction state under a constant vertical temperature gradient (-k) as a fundamental solution. A cylindrical system of co-ordinates (r, <|>, s) is considered for the loop of 2nR length and cross sectional diameter 2a, where the ratio a/R is assumed to be very small. We expand all quantities in terms of double Fourier series in <|> and 6, where 6 = s/R. First-order perturbed fields from steady heat conduction state are examined and various plots are given: isotherms in ring and meridian plane, stream lines and velocities. Finally, an attempt is made in order to identify other fundamental solutions: S-type (cellular thermal convection) and A-types (coaxial flow, bidirectional flow and antisymmetric cellular flow). Introduction Thermal convection in a vertically oriented torus has various applications in engineering and nature, like cooling systems in nuclear engineering, solar heaters geothermal engineering, etc. Several papers considered this configuration, in Newtonian fluids, see for instance [1] and [2]. On the other hand, natural convection in porous media is of interest in many applications and recent books by Nield and Bejan [3] and Ingham and Pop [4-5] present a comprehensive account of the available information in the field. For example, natural convection in a horizontal porous annulus is well documented in the literature, a whole chapter in the reference [5] being dedicated to this theme. However, studies on thermal convection in toroidal configurations filled with fluid-saturated porous media seem to be scarce in the open literature. One example is the work [6] where it is investigated the natural convection and its stability in a toroidal thermosyphon filled with a porous medium. The onset of the thermal convection in that configuration was studied using a one-dimensional model. We remark at this point that the stability of flows in porous media differs considerably from that of Newtonian fluids, due to the very specific changes in the hydraulic and thermal properties. The objective of the present paper is to study the steady thermal convection in a vertical torus filled with a fluid-saturated porous medium. The fluid in the porous medium is considered incompressible and obeying the Boussinesq law. The loop is heated from below and cooled from above and it is subjected to a vertically uniform temperature gradient Mathematical formulation We consider a thermal convection in a vertical torus heated from below and cooled from above. We denote the cross sectional diameter of the torus by 2a and its loop length by 2nR, where the ratio a/R is assumed to be very small. The x and z axis are taken in the plane of the generator of the torus, with the z axis in the opposite direction of the gravity, see Fig. 1. The reference system (r, <|), s) is also introduced, where s is the distance measured counterclockwise from the bottom, along the generator, while r and <|> are polar co-ordinates of the cross section. The direction § = 0 is chosen so that it always coincides with the direction from the generator to the outer edge of the torus in the ring xz plane. 501 Strojniški vestnik - Journal of Mechanical Engineering 51(2005) 7-8, 501-508 Nomenclature a half cross-sectional diameter of the torus P thermal expansion coefficient k vertical temperature gradient R 90 Laplacean and 0 = s/r. Collecting the zeroth order of a/R (<<1), equations (15)-(19) become where du u 1 9w 1 9w -----1------1----------1---------= 0 dr r r 9(f> R 90 dp RaT u =--------- COs 0 cos dr v 1 dp ------- r =----------------h 9(p RaT cos 0 sin w = RaT sin 0 (u cos ([) - v sin (|>)cos 0 - w sin 0 = AT where d2 19 1 92 A =------- H------------1—------- + — t---------------,— dr2 r dr r (7(p 2 (20) (21) (22) (23) (24) o(r2), so that these expressions are the same as those given in terms of a straight circular cylindrical coordinate systems, except that directions of vz and ez change at different positions along the generator. We confine our attention to the steady thermal convection, which corresponds to neutrally stable states at some particular critical Rayleigh numbers. Some fundamental solutions for symmetric modes In this paper we are mainly interested in the analysis of symmetric modes which are caused by experimentally uncontrollable small disturbances under symmetric boundary conditions. The procedure is similar with that used by Sano in [1-2], by expanding all relevant quantities in terms of double Fourier series in <(> and 0 {T, p,u)= ^ \Tm,n(r), pmn[r, umn(r)) m,n = 0 • cos m§ cos n0 , (25) v Z_,vm n(r)sinm(|)cosn0, m-0,n-1 w = } \wm n(r)cosm(|)sinn0 m-0,n-1 Results for symmetric modes Due to the fact that temperature distributions with m = 0 and n = 0 do not lead to physically realizable symmetric convection in a vertically oriented torus, we focus on the simplest situation, represented by T1,1 , w1, 2 , p1,2 , u1, 2 and v1, 2 series. By truncating these series at the lowest order, we have u ' 1 ( \ 2 \U1 2 I H-----Iu 1 2 + v 1 2 I H------w 1 2 = 0 r R 1,2 \p 1,2 / 1 1,2 - p 1,2 r -w 1 1 I T '' 1 T ' 1 1,2 = Wl.lJ +-17UJ ~r2 T 1,1 Ra 2 Ra T w 1,2 = T 1,1 T1 1,1 1,2 1,2 1, 2 p 1 2 = 0 , at r = 1 (26) (27) (28) (29) (30) (31) The solution of the set of equations (26-30) is obtained, after some algebra, in the form u 1,2 = _c1 + 2 + AUJ0lknr ~ J2knr)\ r + B[N0(knr)- N2(knr)}}kn c 2 4 v1,2 = c1 + r 2 - — LAJ 1 knr J + BN 1 knr JJ w1,2 = 2kn2[AJ1knr) + BN1knr)] p1 1,2 c1r c2 4 2 — r R [AJ1(knr)+BN1(knr)] (32a) (32b) (32c) (32d) (32b) where Jn are Bessel functions of the first kind of nth order, Nn are Bessel functions of the second kind and nth order, c1, c2, A and B are constants and kn = Ra1/2 /4 . — u Effect of asymmetry on the steady thermal convection in a vertical torus filled with a porous medium 503 Strojniški vestnik - Journal of Mechanical Engineering 51(2005) 7-8, 501-508 Imposing the boundary conditions (31), the solutions (32) can be expressed in the form u - uB cos § cos 29, v - vB sin § cos 29, = = (33) w = wB cos (|) sin 29 , p - pB cos (|) cos 29, T -TB cos (|) cos 9 where TB = T1 1 = AJ 1 k 1 lr) u u1,2 "2l1 + VJ0k 1 lrJ 2k1l (J0k 1 lr)-J2k 1 lr v1,2 wB = w1 2k1 l AJ1yk1 lr) 1, 2 p B = p 1,2 =^\2\r--\J0 (k1 lr) ~ 4J1 (k1 lr) R L v r J (34a) (34b) (34c) (34d) (34e) The solution is valid for Ra = 4k12 l, where k1 l (l = 1, 2, 3, …) are the zeroes of J1, whilst A is a undetermined constant. The first zero of J1 is k1 l = 3.831706, so that the critical Rayleigh number is Ra1 = 58.72788. Fig. 2 shows the isotherms in the ring plane (xz-plane), T = TB(r)cos9 = const. Next, in Fig. 3 there are plotted the temperature distributions in the meridian plane (9 = 0, or 9 = n). The streamlines in the ring plane (xz-plane) are represented in Fig. 4. We notice that in the xz-plane v = (u, 0, w), so the flow is assumed as two-dimensional. Such a plot for clear fluids, based on the same assumption, can be found in Sano [1]. We can use here the same argument as there: such as a stream function gives good qualitative patterns for the flow in the loop with small a/R, when the w component exceeds the others in the most part of the porous medium. Finally, Fig. 5 shows the (u, v) velocity fields, in vector representation in the meridian plane (9 = 0 or 9 = n). All the previous results belong to the so-called S-type modes, specifically denoted by S12. Type S has also cellular thermal convection at Ra = 2k12 l, denoted by S11 and is defined by = 0.5ub cos (|) cos 9, 0.5vB sin <|) cos 9 , w B — wB cos (|) sin 9 , (35) p = 0.5pB cos (|) cos 9, T = TB cos Some fundamental solutions for antisymmetric modes There are several types of antisymmetric modes, as for clear fluids, see [2]. Let us begin with the A-type modes, characterized by 00 (36) {T, p,u)= Ł [Tmn (r), pmn{r, um,n{r)\ m, n-0 ¦ cos m§ sin n9 , v- / , vm n r) sin m<\> sin nd, m-0, n-1 w = / , wm n r) cos m§ cos n9 m-0, n-1 This type include a family of coaxial flow along the loop at Ra = 2k02 l , which is denoted as in [2] by A00 . The distribution of the relevant quantities in this case is u = v = 0 , w = Ak02 l J 0 k 0lr), T = AJ0\k0lr)sin 9 (37) where l = 1, 2, … Another A-type of solution corresponds to a bidirectional flow at Ra = 2k12 l , denoted by A10. In this case, u = v = 0, w = wB cos (|), T - TB cos (|) sin 9 (38) At Ra = 4k12 l, where l = 1, 2, …, we obtain an antisymmetric cellular flow, called A12. Now the distributions are u - uB cos (|) sin 29 , (39) v - vB sin <|) sin 29, w = -wB cos § cos 29, p = pB cos (|) sin 29, T - TB cos § sin 9 Other fundamental solutions By inverting the sine and cosine terms in <|> for S- and A-mode types, we can obtain other fundamental solutions, see also Sano [2], for clear fluids. The S -type is obtained from (25) as follows {T, p,u)= Y, Tm,n{r, pm,nir, um,n{r)j m, n=0 ¦ sin m(|) cos n0 , cc v = 2_j vm n V ) cos m§ cos n® , m^0, n = 1 (40) 504 Postelnicu A. - Scurtu N. B v - — Strojniški vestnik - Journal of Mechanical Engineering 51(2005) 7-8, 501-508 (41) w = y] wm nr)sinm<\> sinn9 m-0, n-1 The ^~ -type is obtained from (25) in the form {T, p,u)= m , n Tm,nr , pm , nr ,um,nr)} • sin m§ sin n0, v = 2_j vm n V ) cos ^ sin ^ , cc w = /.i wm n(r)sin m(l) cos n® m-0, n-1 The fundamental solutions are readily obtained from their counterparts in S- and A-modes, according to the basic rule stated above (inverting the sine and cosine terms in <(>). For example, the fundamental solution for A ~2 -type flow is obtained from (39) as u - uB sin § sin 29, v = vB cos <|) sin 29, w p - pB sin § sin 29, T = -wB sin <|) cos 29 , = TB sin § sin 9 . (42) Conclusion An analysis of steady thermal convection in a vertical torus filled with fluid-saturated porous medium was presented in this paper. Many similarities were found between this physical case and that of vertical torus filled with Newtonian fluid [1-2]. Taking the steady state as heat conduction under a constant vertical temperature gradient, all the perturbed quantities have been expanded in double Fourier series but only the lowest order terms have been retained in the analysis. The main reason in doing so was to keep close the procedure to that one used in clear fluids [1-2], in order to facilitate (qualitative) comparisons, and this line of study gave fruitful results. It is worth to check the influence of higher modes, given by increasing of the truncated terms in the Fourier expansions. A further step is the superposition of the antisymmetric modes, which can be performed similarly as in [2]. The results of these tasks will be presented elsewhere. References [1] O. Sano, Steady thermal convection in a vertical torus, Journal of the Physical Society of Japan, 56 (11) (1987) 3893-3898. [2] O. Sano, Effect of asymmetry thermal convection in a vertical torus, Journal of the Physical Society of Japan, 57 (5) (1988) 1662-1668. [3] D.A. Nield, A. Bejan, Convection in Porous Media (2nd edition), Springer, New York, 1999. [4] D. Ingham, I. Pop (eds), Transport Phenomena in Porous Media I, Pergamon, Oxford, 1998. [5] D. Ingham, I. Pop (eds), Transport Phenomena in Porous Media II, Pergamon, Oxford, 2002. [6] Y.Y. Jiang, M. Shoji, Thermal convection in a porous toroidal thermosyphon, International Journal of Heat and Mass Transfer, 45 (2002) 3459-3470. Effect of asymmetry on the steady thermal convection in a vertical torus filled with a porous medium 505 Strojniški vestnik - Journal of Mechanical Engineering 51(2005) 7-8, 501-508 Fig. 1. Sketch of the physical problem. Fig. 2 Isotherms in the ring plane (xz-plane). 506 Postelnicu A. - Scurtu N. Strojniški vestnik - Journal of Mechanical Engineering 51(2005) 7-8, 501-508 Fig. 3 Isotherms in the meridian plane (9 = 0 or 9 = n). Fig. 4 Streamlines in the ring plane (xz-plane). Effect of asymmetry on the steady thermal convection in a vertical torus filled with a porous medium 507 Strojniški vestnik - Journal of Mechanical Engineering 51(2005) 7-8, 501-508 Fig. 5. (u, v) velocity fields in the meridian plane (9 = 0 or 9 = n). 508 Postelnicu A. - Scurtu N.