International Journal of Pure and Applied Mathematics Volume 119 No. 12 2018, 37-41 ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu ijpam.eu Effects of Downstream Wall Heating on the Flow Characteristics of Inclined Backward Facing Step Sanju Santhosh and Ajith Kumar. S Department of Mechanical Engineering, Amrita Vishwa Vidyapeetham, Amritapuri, India, sanjusanthosh74@gmail.com. Abstract 2-D Incompressible flow over inclined backward facing step with hot downstream wall is analyzed numerically for the laminar mixed convection regime. A modified SIMPLE algorithm is used for solving the governing partial differential equations of the flow. The entire analysis is performed for a single expansion ratio of 2. The effect of Nusselt number on the downstream heated wall is investigated for wide range of Richardson numbers and at three different Reynold numbers with inclinations of 0 0, 30 0 and 60 0 holding the boussinesq assumption intact. T I. INTRODUCTION HE flow separation due to the sudden expansion in the geometry like in backward facing step has plenty of industrial applications even in the field of aeronautics and is of interest for great many scientists. When flow takes place over the step above a critical value of Reynolds number, (Re, defined based on the free stream velocity U and step height, s) flow separates and subsequently reattaches at a location downstream of the step, commonly referred as reattachment length (x r ). Immense number of researches have been done in this field focusing the effect of flow velocity, step height, expansion ratio (ratio between domain to step height) etc. on the reattachment length [1-4]. Studies were also carried out in the case of inclined backward facing step [5-6]. Flow over backward facing step with downstream hot wall in the forced convection regime (without considering buoyancy) is already been under investigation and the heat transfer characteristics are studied in previous researches [7-12]. Sanju et al. [13] conducted numerical investigations on the effects of heating on reattachment length for the vertical configuration of the backward facing step (α=90 0 ) and concluded that heating has strong influence on the reattachment length, formed behind the step. However, to the best of authors knowledge, there is very few researches carried out focusing of the mixed convection regime, considering the buoyancy, which added to the motivation towards this work. It is interesting to experience the changes which buoyancy would bring in flow over vertical backward facing step with inclination (α), when the downstream side wall is heated as shown in figure 1. The hot wall offers a spatial density stratification normal to the hot wall and buoyancy force is generated opposite to the direction of acceleration due to gravity, g. This buoyancy force may aid or oppose the mean flow, depending on the Fig. 1. A schematic representation of an inclined backward facing step with downstream heated wall. direction of g. The problem discussed in this paper finds wide variety of aeronautical applications as well as industrial applications such as in electronic cooling equipment, cooling passages of turbine blades, combustion chamber, heat exchangers etc. II. NUMERICAL METHOD The non-dimensional incompressible governing (continuity, Navier Stokes and energy) equations in the vector form are,.v = 0 (1) V/ t + (V. )V = - P + (1/Re) 2 V+ Ri.ϴ (2) ϴ/ t + (V. )ϴ = (1/Re.Pr) 2 ϴ (3) Where ϴ represents the non-dimensional temperature difference. A Finite volume method (FVM) is used for the discretization of these partial differential equations. A SIMPLE algorithm is employed to convert PDE to a system of algebraic equations and are solved simultaneously. We made use of the open source code OpenFOAM for this buoyantboussinesqsimplefoam. 37
International Journal of Pure and Applied Mathematics III. VALIDATION We have taken upstream=10s and downstream=20s in the domain and used a grid size 250x70, after grid independence test. The code is then validated for flow over backward facing step when Re = 100 at Ri = 0 and a comparison is made with the existing literature in table. 1. It is verified that the OpenFOAM code provides a reasonably good agreement with the existing literature. TABLE I COMPARISON OF THE REATTACHMENT LENGTH EVALUATED IN THE PRESENT STUDY WITH THE LITERATURE Sl No: Reference Recirculation length (Xr/s) 1 Armaly-Expt [1] 2.970 2 Armaly-Num [1] 2.840 3 Jabir et al. [2] 2.860 4 Ercan Erturk [3] 2.878 5 G Biswas e t al. [4] 2.810 6 Present 2.800. IV. RESULTS AND DISCUSSIONS Two-dimensional incompressible flow simulations have been performed to analyze numerically the effects of buoyancy (-0.4 < Ri < 0.4) on the reattachment length for three different Re (50, 100 and 150) and also for three different inclinations of 0 0, 30 0 and 60 0. A. Effect of Heating on the Recirculation Bubble The reattachment length is evaluated as that point closer to the downstream wall where streamwise velocity changes its direction. It is interesting to note that the reattachment length decreases with increase in Ri for all the Re and Ri considered in this analysis. Also, it is observed that the location of the maximum velocity shifts away from the hot wall for buoyancy opposed flow. It can be seen that the location of the maximum velocity shifts towards the hot wall for buoyancy aided flow which eventually accelerates the shear layer towards the hot wall pushing the recirculation bubble against the wall and is attributed to the reduction in the reattachment length. Whereas in the counterpart, buoyancy opposed flow, the shear layer accelerates and moves away from the hot wall resulting in the progression of the recirculation bubble and thereby increasing the reattachment length. The same trend is seen for all the different orientations (α) for the backward facing step as can be found in Figures 2(a) to 2(c). Also it can be seen that the reattachment length increases with α for any Re and Ri considered in this analysis. (a) α=0 0 (b) α=30 0 38
International Journal of Pure and Applied Mathematics (c) α=60 0 Fig. 2. The variation in the reattachment length with Ri for different Re. B. Effects of Heating on the Heat Transfer Behavior The Nusselt number is a measure of the convection heat transfer over the conduction heat transfer and is evaluated in this work using the relation, Fig. 4. The variation in Nusselt number with the length of downstream heated wall for α=60 0 at different Ri.. Nu= - ϴ/ y (4) The variation of local Nusselt number (Nu) is plotted on the downstream wall for different Re, Ri and α in figures 3, 4 and 5 respectively. In all the plots, it can be seen that the Nusselt number attains a peak value and then decreases. The peak value in Nu is attributed to the larger convection which takes place inside the recirculation bubble, irrespective of Re, Ri and α Fig. 5. The variation of Nusselt number with the length of downstream heated wall for different angles. Fig. 3. The variation in Nusselt number with the length of downstream heated wall for α=60 0 at different Re. As Re is increased, the convection heat transfer increases and is attributed to increase in Nu on the downstream hot wall as shown in figure 3. Similarly the recirculation bubble length increases with Re and is responsible for the shifting of the maximum Nu point towards the right. Also we observed there is not much difference in the local Nusselt number when Ri or α is changed, as seen in figures 4 and 5. 39
International Journal of Pure and Applied Mathematics Fig. 6. The variation in Nusselt number average with Re for Ri=0.4 at different angles. A broader idea about the heat transfer characteristics can be obtained from the space averaged Nusselt number (Nu avg ) plotted against Re for different α, shown in figure 6. Nu avg is found increasing with Re which substantiate the heat transfer theories that Re increases the convection heat transfer. [3] Ercan Erturk, Numerical solutions of 2-D steady incompressible flow over a backward-facing step, part I: High Reynolds number solutions. Computers & Fluids 37, 2007, pp. 633-655. [4] Biswas.G, Breuer.M, and Durst.F, Backward-facing step flows for various expansion ratios at low and moderate Reynolds numbers, 2004. [5] B. Hong, B.F. Armaly and T.S. Chen, Laminar mixed convection in a duct with a backward-facing step: the effects of inclination angle and prandtl number, 1992. [6] J.T. Lin, B.F. Armaly, and T.S. Chen, Mixed convection heat transfer in inclined backward-facing step flows, 1990. [7] Aung.W, An experimental study of laminar heat transfer downstream of backsteps, J. Heat Transfer 105, 1983, pp. 823-829. [8] Aung.W, Separated forced convection, Proc. ASMEIJSME Thermal Enana Joint Con/.. Vol. 2. DD. 499-515. ASME. New York, 1983. [9] Aung.W, Baron.A and Tsou.F.K, Wall independency and effect of initial shear-layer thickness in separated flow and heat transfer, Int. J. Hear Muss Transfer 28, 1757-1771, 1985. [10] Aung.W and Worku.G, Theory of fully developed. combined convection including flow reversal, J. Hear Transfer 108,485-488, 1986. [11] Sparrow.E.M, Chrysler.G.M and Azevedo.L.F, Observed flow reversals and measured-predicted Nusselt numbers for natural convection in a one-sided heated vertical channel, J. Heat Transfer 106.325-332, 1984. [12] Sparrow.E.M, Kang.S.S and Chuck.W, Relation between the points of flow reattachment and maximum heat transfer for regions of flow separation, Inr. J. Hear Mass Transfer 30, 1237-1246, 1987. [13] Sanju Santhosh and Ajith Kumar. S, Effects of buoyancy on the reattachment length in flow over heated vertical backward facing step, The Eleventh International conference on Thermal Engineering and Applications. V. CONCLUSION Laminar incompressible flow over inclined backward facing step with inclinations 0 0, 30 0 and 60 0, and with hot downstream wall is analyzed computationally for Re = 50, 100, and 150 and -0.4<Ri<0.4. It has been concluded from this study that, 1) The reattachment length decreases with increase in Ri for all inclination (α). 2) The reattachment length increases with increase in Re for all inclination (α). 3) The reattachment length increases with increase in α for all Re and Ri considered in this study. 4) The local Nusselt number, Nu increases along the downstream hot wall, attains a peak and then decreases. This trend is consistent for all Re and Ri considered in the present analysis. The peak value corresponds to the reattachment length, X r. 5) The local Nusselt number increases with the increase in Re and is almost unaffected by changes in Ri or α. 6) The average value of Nusselt number increases with increase in Re for all α considered in this study REFERENCES [1] Armaly.B.F, Durst.F, Pereira J.C.F, and B.Schonung, Experimental and theoretical investigation of backward-facing step flow, J. Fluid Mech, 1983, pp. 473-496.. [2] Anil Lal.S and Jabir.E, A hybrid finite element-finite volume method for incompressible flow through complex geometries using mixed grids, 2009. 40
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