Transactions on Engineering Sciences vol 5, 1994 WIT Press, ISSN

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1 Convergence of numerical solutions of free convection equations obtained by direct method T. Wlodarczyk" & S. Wyczolkowski& ^Institute of Mechanics and Machine Design Foundations ^Department of Industrial Furnaces, Technical University of Czestochowa, ul. Dabrowskiego 73, ASTRACT A new stabilization procedure for solutions of the thermoconvection equations has been presented. The procedure allow us to achieve, by using the direct method, the convergent solutions for the wide range of values for Ra and Pr numbers. On the examples of single-segmental region (and the problem formulated in the dimensionless form). The effect of the attenuation coefficient values of the stabilization procedure on the convergence of solutions has been described. Also, the influence of the node numbers and configuration of the differential grid nodes in the region, on the convergence and the stabilization rate of solution for various Ra and Pr numbers have been investigated. INTRODUCTION The paper refer to the new solution method of two-dimensional thermoconvection problems. The point of this new approach is that algebraic equation systems, corresponding to the particular-differential equation, are solved by the direct method (e.g. by the Gauss elimination method). The calculation cycle is as follows: first system of the equation for the stream function is solved. On the grounds of the calculated stream function values, the components of the velocity vector for the particular nodes of the differential grid are defined. Then, the system of equations for temperature, and finally the system of equation for stream function are solved. This calculation cycle is repeated so many times until the difference in the solution values between two successive cycles are less then a small, present value, e. In every consecutive cycle of calculations the vorticity values in the nodes of the differential grid, which are determined in the preceding cycle, are taken into consideration. The presented method of solution of the thermoconvection equation systems allows applying the division of the considered region into segments (semi-regions)

2 62 Heat Transfer [3-5]. Owing to that, the separation of equation systems, which correspond to the whole region, into semi-systems corresponding to the particular segment has been obtained. Thus defined semisystems of equations are solved successively in each calculation cycle. Division of the region into segments (separation of equation system on semi-systems) lowers significantly the load of the operational memory of the computer. Only the vorticity values for all the nodes have to be registered in the operational memory. Values of the stream functions and temperature are registered only for the nodes on the edge of segments. Correction of the particular function values in the consecutive computation cycles, for the nodes on the segment edges, is achieved by introducing partial overlapping of the adjacent segments. MATHEMATICAL FORMULATION OF THE PROLEM Natural convection in two-dimensional, rectangular, enclosed area (Fig. 1) filled with the Newtonian fluid (incompressible and of constant transport coefficients) is considered. C o o Figure 1. Geometry of the problem A o The laws of conservation of mass, momentum and energy, which constitute the base for description of the fluid motion, can be expressed by the following equations [6]: ax" (i) U + V (Ra d& ax (2)

3 Heat Transfer ax av \-i/2 ax* (3) The stream functions and vorticity can be defined as follows: av _ n = _ ' ax ax av (4) Dimensionless quantities which are present in the above equation, are formed from the physical quantities according to the relationship: * = H ' *-& (5) U = (a/h) (Ra Pr) 1/2 (a/h) (Ra Pr) 1/2 (6) T - (TH - Tc)/2 -, (7) oundary conditions for Eqns (1) and (3) are used: U = 0, V = 0, 8 = 0.5 for X = 0 (8) U = 0, V = 0, 8 = for X = g (9) U = 0, V = 0, = 0 for Y = 0 and f or Y = 1 (10) oundary conditions for the stream function and vorticity transport equations have been assumed in the form: = 0, Q = -, for X = 0 and for X = ^ (11) = 0, n = -, f or Y = 0 and for Y = 1 (12) The problem is solved by the finite difference method (upwind schema) [1] while applying a uniform, rectangular differential grid. Vorticity on the edge of the region is approximated by the formula proposed by Jensen and Person [2].

4 64 Heat Transfer STAILIZATION PROCEDURE Stabilization procedures [5] which allow us to obtain the convergent solutions of the convections equations (while applying the direct method) are based on the following generalized rule: the vorticity values in the nodes of the differential grid (for the given calculations cycle) are determined on the grounds of values obtained from solution of the equations system and vorticity defined in the previous calculation cycle (for the particular nodes) In this work the modified procedure of the following form has been applied: (13) where Aj = (m Aj (14) 1/2 1/2 (15) ut, if m < 0.2 [ig g + 4] + (16) (17) then m = 0.2 [ig g + 4J + (17a) where: j - is the node number, k - is the number of calculation cycle, Qj - is the function value resulting from the solution of the vorticity equations system. NUMERICAL CALCULATIONS AND RESULTS Numerical calculations have been performed for regions of the specified proportion: height to width (H/L) given in Table 1. In this Table also parameters of the differential grid applied to these regions have been specified. Calculations have been performed for the particular regions, assuming various coefficients and C values of the expression (16). As indicator of the solutions convergence the numbers of the obtained solution differ from the final solution less than 0.05% and 0.01%. These cycle are denoted as NQ 5 and NQ ^ respectively.

5 Heat Transfer 65 Table 1 Region parameters and differential grid parameters Region Number of step mesh Number of nodes Number of internal nodes Rate H/L N 1 13x N 2 16x N 3 9x N 4 9x I =2.4, C= = 1.0 a c r\o ' \ V. ~ V- i I c A O.C C Number of cycle : =2.8, C= = 1.2 b V V \ vllll V c ~~~.> V- i n nn Number of cycle Figure 2. Exemplary course of the stream function changes in nodes A, and C

6 66 Heat Transfer In Fig. 2 has been presented exemplary course of the stream function changes in nodes A,, i C (Fig. 1) of the region N 1 in the particular calculation cycles, which illustrates the effect of the coefficients and C on the stability and convergence of solutions. The stable solution has been obtained in this case for the coefficient value, &2.8. It means, that minimum value of the attenuation coefficient, m, which is necessary to obtain the convergent solution, is mg=5.4. It should be noted that for this value of attenuation coefficient (m=5.4, =2.8) the solution of the highest convergence rate is obtained. For attenuation coefficient, m which is m>5.4 (>2.8) also convergent solutions are obtained but their convergence rate are the lower the higher value of the coefficient m, is. For coefficient m<5.4 (<2.8) non-stable solutions are obtained (solution results oscillate around the exact solution). Amplitude of these oscillations decrease while value of m approaches to the values, m^. For too large difference in values m and m^ the solution becomes divergent. Coefficient C decides on the coefficient, m value only in the primary calculation cycles. Its influence on the convergence rate of solutions is insignificant. Data presented in Table 2 clearly confirm this statement. Table 2 Table 3 Effect of coefficient and C on the solution convergence for the region N 1 (Ra=lQ9, Pr=1.0) Values of the coefficients and C for region 1-4 (Ra=10?, Pr=1.0) C N N Region C N 0.5 "0.1 N l N N N Values of the coefficients and C, for which the solution of the highest convergence rate has been obtained, are presented in Table 3. Also, the cycle numbers Ng 5 and are given in that table. Values of coefficients and C, for which the solutions of the highest convergence in regions 1-4 have been obtained (for various Ra and Pr numbers) are presented in Table 4.

7 Heat Transfer 67 a) Figure 3: Isolines for stream function (a) and for temperature (b) for region N 1 and N 3 Values of the coefficients and C for region 1 and 4 (for highest convergence) Table 4 Region Ra Pr C N 0.5 "o.i N 1 10* icr N icr

8 68 Heat Transfer From the data contained in Table 4 results, that, the greatest influence on the convergence rate of the solutions has the proper choice of the coefficient, and C. Configuration of nodes and their number with the well-choosen coefficient,, are not significant importance. The final results of numerical calculations, for the choosen examples, are presented in Fig. 3 in the form of isoline diagrams for the temperature and stream function. CONCLUSIONS On the grounds of the performed calculations one can state that the condition for convergent solutions obtained by the direct methods is the proper selection of respective coefficient values in the stabilization procedure. The values of coefficient have decisive influence on the convergence rate of solution. Configuration of nodes and their number with the well-choosen coefficients, and C in stabilization procedure, have no bigger influence on the convergence rate of solution. However, because of the lower load of the computer operational memory and shorter computation time it is more convenient to divide the region into segments of bigger height to width ratio (e.g. region N 4). ACKNOWLEDGEMENTS This work was supported by State Commitee for Scientific Research under Grant N LITERATURE 1. Patankar, S.V., Numerical Heat Transfer and Fluid Flow, McGrow-Hill, New York, Roach, P.J., Computational Fluid Dynamics, Hermosa Publishers, Albuquerque, Wyczolkowski, S., Wlodarczyk, T., Methode der Numerischen Losung der Gleichungen der stationare Warmekonvektion, Z. Ang. Hath. Mech. Vol. 74, N 5, pp , Wyczolkowski, S., Wlodarczyk, T., Numerical Modelling of the Steady Natural Convection of Liquid Metal in Enclosed Space (Ed. Taylor, C.), pp , Proceedings of the 8th Int. Conf. on Numerical Methods in Laminar and Turbulent Flow, Pineridge Press, Swansea, Wyczolkowski, S., Wlodarczyk, T., Application of Direct Method for Numerical Solution of Natural Convection Equations, in prepare 6. Zhang, Z., ejan, A., Lage, J.L., Natural Convection in a Vertical Enclosure with Internal Permeable Screen, ASME Journal of Heat Transfer, Vol. 113, pp , 1991