EFFECT OF VARYING THE HEATED LOWER REGION ON FLOW WITHIN A HORIZONTAL CYLINDER

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1 ISTP-1, 5, PRAGUE 1 TH INTERNATIONAL SYMPOSIUM ON TRANSPORT PHENOMENA EFFECT OF VARYING THE HEATED LOWER REGION ON FLOW WITHIN A HORIZONTAL CYLINDER S. S. Leong School of Mechanical and Manufacturing Engineering The University of New South Wales, Sydney, Australia Fax: s.leong@unsw.edu.au Keywords: CFD, cylinder, transient, instability, convection Abstract The time dependent natural convection in a horizontal circular cylinder, heated from below and cooled from above, is computed using the stream function-vorticity formulation for the two-dimensional equations of motion and energy. The extents of the heated lower and upper cooled regions are varied. The fluid studied has a Prandtl number of 7. The values of Rayleigh number (Ra) presented in this paper are 1, and 1,. When the extent of the heated region is small, the flow consists of two cells and steady state solutions are obtained. The flow becomes oscillatory when the extent of the heated is increased. The temperature, streamlines, relative total kinetic energy and sselt number are presented as a function of time. 1 Introduction There are very few published papers on natural convection in a horizontal circular cylinder because of the singularity at the axis. When the lower region is heated and the rest of the cylindrical wall is cooled, natural convection exists even at very low Rayleigh numbers because of the curved cylindrical boundary. When the lower half is heated and the upper half cooled, Leong and de Vahl Davis [1] obtained steady state solutions for Ra<1, using finite difference for the governing equations in cylindrical coordinates. The singularity at the centre was avoided by not placing a mesh point at the center []. Heinrich and Yu [3] used these results to compare with their numerical results using finite element method based on a penalty function and a Petrov-Galerkin formulation. Nagai and Emery [4] used finite volume with a staggered grid (which also avoid placing a mesh point at the center) in their studies and found that it was impossible to achieve grid independence for any characteristic of the flow. They also found that the use of different timesteps give different results. Huang and Heinrich [5], using the same method as Heinrich and Yu found that the results using two different meshes are qualitatively the same but are quantitatively different. Leong [] showed that grid and time independent solutions can be obtained using the same thermal boundary conditions as Huang and Heinrich. In this paper the transient flow within a horizontal circular cylinder when the extents of the heated lower and upper cooled regions are varied is presented. The fluid has a Prandtl number of 7. and the values of Rayleigh numbers used are 1, and 1,. Governing Equations The physical model is shown in Figure 1. The extent of the heated lower and cooled upper regions are δ and ( π δ ) respectively. φ is the azimuthal angle measured from the gravity vector g. The energy and momentum equations are non-dimensionalized using R (radius of the cylinder), R /κ and κ/r as scale factors for length, time and velocity respectively. κ is the 1

2 coefficient and υ is the kinematic viscosity. A ψ stream function ψ, defined by: u = and r φ ψ v = is introduced and is related to the r vorticity by ψ ψ ψ ζ = r rr r φ (3) Fig. 1. Physical model. fluid thermal diffusivity. Applying the Boussinesq approximation, the energy and vorticity transport equations are given by ( ruθ) ( vθ) θ = + θ + θ + θ (1) t r r r φ r r r r φ ( ruζ ) ( vζ ) ζ = + RaPrsinφ θ t r r r φ r RaPrcos θ + φ + Pr + + () r φ ζ ζ ζ r r r r φ The dimensionless temperature θ is given by θ=(t-t m )/T d where T m =(T h +T c )/, T d =(T h -T c )/, T h is the temperature of the heated lower half and T c is the temperature of the cold upper half of the cylinder. u and v are the radial and azimuthal velocities respectively. φ is the angle from the gravitational vector. The vorticity and the two non-dimensional parameters Ra and Pr are given by 3 ( rv) ( u) β gtd R ζ =, Ra = and Pr = υ κ r r r φ υκ respectively. β is the volumetric expansion The thermal boundary conditions are θ=½ for -δ/<φ<δ/, θ= ½ for δ/<φ<(π-δ/) and θ= when φ=δ/ and φ=-δ/. The temperatures at all the interior mesh-points are set to zeros. The initial velocities, vorticities and vectorpotentials in the solution domain and at the rigid walls were all set to zeros. 3 merical Solution A uniform mesh consisting of LxM discrete points in the radial (r) and azimuthal (φ) directions respectively is superimposed on the solution domain so that the radial mesh points are given by r i =(i ½) r for i=1,,3,...l and the azimuthal mesh points are given by φ j =(j 1) φ for j=1,,3,...m, where r=1/(l ½) and φ=π/m. Using this mesh, centered finite difference approximations are used for equations (1) and (), except at r= r/ where second order forward differences are used for the radial derivatives. The resulting finite difference equations are then solved using a modified Samarskii-Andreyev [7] alternating direction implicit (ADI) scheme at each timestep. The elliptic equation (3) is solved directly using fast Fourier transform. 4 Results A mesh and time-step sensitivity was presented by Leong [] when the lower half of the cylinder is heated and the upper half is cooled. For Ra=1, and Pr=7., Leong showed that a 31x4 mesh and a time-step of 1 - are adequate for presenting results for transient flows. This 31x4 mesh and time-step of 1 - are used for presenting the following results.

3 EFFECT OF VARYING THE HEATED LOWER REGION ON FLOW WITHIN A HORIZONTAL CYLINDER Figure shows the total relative kinetic energy (KE) defined as 1 1 A c v rdφdr KE = (4) for Ra=1 3 for various values of δ 1.75º. For these values of δ, steady state solutions are obtained. When δ=33.75º, KE increases sharply to a maximum value of KE max =15.17 at t=.175 before decreasing to KE steady =7.53 at steady state. Table 1 shows the values of KE max and KE steady for various values of δ. As δ is increased to 11.5º, KE max increases to 9.5, after which KE max decreases to a when as δ=1.75º. ψ [-.7,.7,15] (a) t=.4 θ [-1.,1.,13] 3 KE º 5.5º 7.75º 11.5º 13.75º 14.5º 1.75º δ ψ [-4.,4.,15] (b) t=. θ [-1.,1.,13] Fig. 3. Plots of ψ and θ for Ra=1 3 and δ=11.5º. 1 3 Fig.. KE as a function of time for Ra=1 3 and various values of δ. Table 1. KE max and KE steady for Ra=1 3 and various values of δ. δ KE max KE steady 33.75º º º º º º º Figure 3 shows the typical flow patterns and isotherms for Ra=1 3 and δ Figure 3(a) shows that when Ra=1 3 and δ=11.5º four weak cells are initially formed. The two lower cells are very weak and disappear at t=.4 as the two upper cells get stronger. When KE is maximum, the flow consists of two cells as shown in Figure 3(b). This two cells structure persists to steady state. Figure 4 shows the sselt number () defined as α θ = φ (5) r α 1 at the cooled wall and α 1 and α are points at which θ=. Figure 5 shows the transient sselt number after the initial sharp drop in. For δ 7.75º, decreased to a constant value at steady state. For 11.5º δ 1.75º, initially decreased sharply to a minimum, then increased to a maximum value before decreasing to a constant value at steady state. When δ is increased to 1º, plots of KE and are shown in Figures 5 and. This boundary condition is different from Leong []. Leong presented results with θ=½ for 7.75º φ 7.75º, θ= ½ for 11.5º φ 5.75º and the temperature is linear for 7.75º φ 11.5º and 5.75º φ 1.5º. In this paper θ=½ for 3

4 4.375º φ 4.375º, θ= ½ for 95.5º φ 4.375º and θ= when φ=9º and 7º. The extents of the heated and cooled regions are slightly greater than that used by Leong []. The initial flow consists of four equal cells as shown in Figure 7(a). The maximum KE during the four equal cells pattern is.75 at time t=.1 before it drops to 1.97 for a period of time. At time t>3.5, the cells in the second and third quadrants start to merge as shown in 9 δ 33.75º 5.5º 7.75º 11. 5º 13.75º 14.5º 1.75º Figure 7(b). KE and reached values of 39.3 and 7.95 respectively. The two merged cells form a large single clockwise rotating cell with two very weak counter-clockwise rotating cells clinging to the boundary shown in Figure 7(c). The values of KE and at steady state are 3.13 and 7.5 respectively. When Ra=1 4, Figure shows that the transient KE for δ=33.75º, 7.75º and 13.75º oscillates a bit. There is a maximum in KE for all three values of δ, before it settles to a constant value at steady state. Stream function contours and isotherms for Ra=1 4 and δ=13.75º are shown in Figure 9. It shows that two small weak cells are form in the third and fourth quadrants near the wall Fig. 4. as a function of time for Ra=1 3 and various values of δ. 4 KE 3 1 ψ [-.9,.9,15] (a) t=.1 θ [-1.,1.,13] Fig. 5. KE as a function of time for Ra=1 3 and δ=1º ψ [-1.,.5,13] (b) t=3. θ [-1.,1.,13] Fig.. as a function of time for Ra=1 3 and δ=1º. ψ [-.4,.,13] (c) t=. θ [-1.,1.,13] Fig. 7. Plots of ψ and θ for Ra=1 3 and δ=1º. 4

5 EFFECT OF VARYING THE HEATED LOWER REGION ON FLOW WITHIN A HORIZONTAL CYLINDER When δ is increased to 135º, KE becomes cyclic, oscillating between 7.5 and 47.5 as shown in Figure 1. Figure 11 shows the transient plot of against time. oscillates between and Figure 1 shows the isotherms and flow patterns during one cycle. Figure 1(a) shows the streamlines and isotherms when KE is maximum at time t= At this time, the left counterclockwise rotating cell is decreasing in size and the cell on the right is increasing in size. At time t=4.553, KE is a minimum and the streamlines and isotherms are shown in Figure 1(b). The KE (x1 ) Fig.. KE as a function of time for Ra=1 4 and δ=33.75º, 7.75º and 13.75º. δ 33.75º 7.75º 13.75º left cell continue to increase in size and KE is again a maximum at time t=4.3. the isotherms shown in Figure 1(c) is a mirror image of that shown in Figure 1(a). The flow at this time has gone through approximately half a cycle. At t=4.73, KE is a minimum again and at time t=4.14, the flow has gone through one complete cycle. The period of each cycle is.39. When δ is increased to 1.75º, there are two distinct cyclic modes as shown in Figure 13 and 14. In the first cyclic mode KE oscillates between and and oscillates between 1.3 and Figure 15 shows the streamlines and isotherms during the first cyclic mode. Figure 15(a) shows the upper two cells are stronger and larger while the two lower cells are weaker. The upper two cells decrease in size while the lower two cells increase in size and Figure 15(b) shows the lower two cells are stronger than the upper two cells. The two lower cells then decrease in size while the upper two KE (x1 ) 4 ψ [-.,.,15] (a) t=.4 θ [-1.,1.,13] Fig. 1. KE as a function of time for Ra=1 4 and δ=135º ψ [-1.,1.,15] (b) t=3.7 θ [-1.,1.,13] Fig. 9. Plots of ψ and θ for Ra=1 4 and δ=13.75º Fig. 11. as a function of time for Ra=1 4 and δ=135º. 5

6 cells increase in size and strength and at time t=1.1 is almost the same size as that in Figure 15(a). This cyclic mode shows the flow oscillating in the vertical direction. The period during this oscillation is.174. (a) t=4.445 (b) t=4.553 In the second cyclic mode KE oscillates between.43 and with a period of.. Figure 1(a) shows the streamlines and isotherms at time t=4.4 when KE is 7.. The main counter-clockwise cell is at an angle of 45º to the vertical with two smaller cells in the first and third quadrants. This main cell decreases in size while the other two cells enlarge and merged together at time t=4.9, dividing the main cell into two as shown in Figure 1(b). Figure 1(c) shows a large clockwise cell with two smaller cells in the second and third quadrants. This large cell then decreases in size while the two smaller cells increase in size and at time t=4. split the clockwise cell into two as shown in Figure (c) t=4.3 (d) t=4.73 KE (x1 ) Fig. 13. KE as a function of time for Ra=1 4 and δ=1.75º ψ (e) t=4.14 θ Fig. 1. Plots of ψ[-,,15] and θ [-1,1,13] for Ra=1 4 and δ=135º Fig. 14. as a function of time for Ra=1 4 and δ=1.75º.

7 EFFECT OF VARYING THE HEATED LOWER REGION ON FLOW WITHIN A HORIZONTAL CYLINDER 1(d). At time t=4.5, Figure 1(e) shows the flow pattern has return to that shown in Figure 1(a). When δ is increased to 1º, plots of KE and are shown in Figures 17 and 1. Figure 17 shows KE increased rapidly to 1.7 and remains in a quasi steady-state for sometime before it starts to unsteady and finally oscillates between 75.9 and 3.7. The period of the oscillation is.. Figure 1 shows a rapid drop in to a seemingly steady state value of 9.37 before it finally oscillates between and 1.. Figure 19 shows the streamlines and isotherms at time t=1. during the quasi steady state. The flow consists of four equal cells. The maximum absolute value of ψ (a) t=4.4 (b) t=4.9 (a) t=1. (c) t=4.15 (b) t=1.1 (d) t=4. ψ (c) t=1.1 θ Fig. 15. Plots of ψ[-15,15,15] and θ [-1,1,13] for Ra=1 4 and δ=135º. ψ (e) t=4.5 θ Fig. 1. Plots of ψ[-,,15] and θ [-1,1,13] for Ra=1 4 and δ=1.75º. 7

8 is 9.3. The streamlines are symmetrical about a diameter inclined at 45º and the isotherms are symmetrical about the vertical diameter At time t>., the flow becomes unsteady and enters into a steady oscillatory mode at time t>3.5. Figure and 1 show the transient values of KE and over one cycle. Figure (a) shows streamlines and isotherms at time KE (x1 ) Fig. 17. KE as a function of time for Ra=1 4 and δ=1º. 5 t=1.4 when KE has a maximum value of The flow consists of large counterclockwise (CCW) rotating cell with two clockwise (CW) cells in the first and third quadrants. At this time the large cell decreases in size while the two smaller cells increase in size. At time t=1.79 when has a maximum value of 14.55, the streamlines and isotherms are shown in Figure (b). The main CCW cell is now smaller and the two CW cells have grown bigger. Figure (c) shows the splitting of the CCW cell into two and the flow consists of two CCW cells and two CW cells. After splitting the CCW cell into two, the two CW cells start to merge and Figure (d) shows the merging of the two cells at t=1.1 when KE has a minimum value of 3.7. The merged cells grow to become the main CW cell and the two CCW cells get smaller in the second and fourth quadrants as shown in Figure (e) at t= This is the time at which Fig. 1. as a function of time for Ra=1 4 and δ=1º. 5 KE (x1 ) Fig.. KE as a function of time for Ra=1 4 and δ=1º for a cycle ψ[-9,9,15] θ [-1,1,13] Fig. 19. Plots of ψ and θ at time t=1. for Ra=1 4 and δ=1º Fig. 1. as a function of time for Ra=1 4 and δ=1º for a cycle.

9 EFFECT OF VARYING THE HEATED LOWER REGION ON FLOW WITHIN A HORIZONTAL CYLINDER has a minimum value of 11. and KE is increasing. At t=1.15, the CW cell is the main cell and KE is again at the maximum value of The main cell is decreasing in size while the two CCW cells are getting bigger. Figure (h) shows the splitting of the CW cell into two cells and Figure (i) shows the merging of the two CCW cells. After merging to form the main CCW cell, the flow pattern shown in Figure (k) at t=1. is the same as that in Figure (a). The flow has gone through one complete cycle. 5 Conclusions The results showed that the KE, and the flow are affected by the extents of the heated and cooled regions. When Ra=1 3 and δ 1.75º, (a) t=1.4 (e) t=1.14 (b) t=1.79 (f) t=1.15 (c) t=1.19 (g) t=1.1 ψ (d) t=1.1 θ Fig.. Plots of ψ[-19,19,15] and θ [-1,1,13] for Ra=1 4 and δ=1º. ψ (h) t=1.1 θ Fig. (cont.). Plots of ψ[-19,19,15] and θ [-1,1,13] for Ra=1 4 and δ=1º. 9

10 the flow quickly changed from a four cell pattern consisting of two large upper cells and two small lower cells to a two cell pattern which persists to steady state. When δ=1º, the four equal cell pattern changes into a single large cell with two very small cells clinging to the wall. When Ra=1 4, the two cell pattern is only formed for δ 13.75º. When δ=1.75º, the flow has two cyclic modes with different periods. The amplitude of KE during the first cyclic mode is smaller than that for the second mode. The amplitude of during the first cyclic mode is greater than that for the second mode. When δ=1º, the four equal cell pattern persists for some time before changing into a (i) t=1.1 cyclic mode. References [1] Leong, S.S. and De Vahl Davis, G. Natural convection in a horizontal cylinder, merical Methods in Thermal Problems, eds Lewis, R.W. and Morgan, K., Pineridge Press, Swansea, pp 7-9, [] de Vahl Davis, G. A Note on a Mesh for use with Polar Coordinates, merical Heat Transfer, vol., pp 1-, [3] Heinrich, J.C. and Yu, C.C. Finite Element Simulation of Buoyancy-driven Flows with Emphasis on Natural Convection in a Horizontal Circular Cylinder, Computer Methods in Applied Mechanics and Engineering, Vol. 9, pp 1-7, 19. [4] Nagai, H. and Emery, A.F. The Computation of the time dependent Natural Convection Flow in a Horizontal Cylinder Heated from Below, Proc. of the ASME Heat Transfer Division, Vol. 4, pp 1-, 199. [5] Huang, H.W. and Heinrich, J.C. Benchmark Problem: Natural Convection within a Horizontal Circular Cylinder, Proc. of the ASME Heat Transfer Division, Vol. 4, pp , 199. [] Leong, S.S. Instability within a Horizontal Cylinder Heated from the Below and Cooled from the Top. To be presented at Proceedings of 4 th ICCHMT, Paris- Cachan, FRANCE, May 17, 5. [7] Samarskii, A.A. and Andreyev, V.B. On a highaccuracy difference scheme for an elliptic equation with several space variables, USSR Comp. Math. and Math. Phys., Vol. 3, pp , 193. (j) t=1.4 ψ (k) t=1. θ Fig. (cont.). Plots of ψ[-19,19,15] and θ [-1,1,13] for Ra=1 4 and δ=1º. 1