THREE-DIMENSIONAL FLOW CHARACTERISTICS FOR FLOW ALONG A SQUARE PRISM HAVING BUILT-IN PLANE SIDE FLAPS

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1 Proceedings of the Forty Second National Conference on Fluid Mechanics and Fluid Power December 14-16, 2015, NITK Surathkal, Karnataka, India FMFP THREE-DIMENSIONAL FLOW CHARACTERISTICS FOR FLOW ALONG A SQUARE PRISM HAVING BUILT-IN PLANE SIDE FLAPS Amit Soni * Department of Mechanical Engineering Indian Institute of Technology Madras Chennai , India me14s046@smail.iitm.ac.in Hemant Naik Department of Mechanical Engineering Indian Institute of Technology Madras Chennai , India me14d014@smail.iitm.ac.in Shaligram Tiwari Department of Mechanical Engineering Indian Institute of Technology Madras Chennai , India shaligt@iitm.ac.in ABSTRACT Three-dimensional numerical simulations have been carried out using commercial software ANSYS Fluent 14.0 to study the flow along an elongated square prism having coplanar side flaps. Three different arrangements for position of the two coplanar side flaps are considered, viz. both at midplane, both at the ends and mutually staggered. Flow characteristics have been investigated for different lengths of the prism and flaps by varying the flow Reynolds number. The width of each flap is kept fixed while considering the three proposed arrangements. The aim is to identify the value of Re which is associated with minimum drag and high lift with an aim to mimic the situation of bird flight. Investigation has been carried out to find the effect of prism length and flap width on characteristics of leading edge vortex, trailing edge vortex and wing tip vortices. Results are presented in the form of streamlines, vorticity contour and pressure contour to identify the arrangements as well as Re value that is associated with minimum drag and maximum lift. For each arrangement, the effect of length of the prism and width of flaps has been studied for different Reynolds numbers on the characteristics of leading edge vortex and the bound vortices. Keywords: Elongated square prism, coplanar side flaps, leading edge and bound vortices, lift and drag. 1. INTRODUCTION Flow past a bird in flight is difficult to mimic without considering actual shape of the bird. However, taking advantage of rich literature for 1

2 flow past various body shapes, a simple model for a bird with stationary wings could be proposed. The most basic configuration may be assumed to consist of an elongated square prism with built-in side flaps. For actual flight situation, an oscillatory motion may be imparted to the flaps. In order to understand the effect of wings on mechanism of flow past a bird, the side flaps may be assumed to adapt various shapes. This helps in appreciating the optimized evolution of actual wing shape. Three different types of arrangements have been assumed for the flaps with respect to the square prism. The two side flaps can be placed at midplane, top-plane and one at mid and other at top (staggered) as shown in Fig.1. In study of bird and bat flight, it is necessary to consider effect of various parameters, such as wing area and wing span. This would help in finding a relation between the weight of the bird, wing loading and cruising. Aerodynamics of low Reynolds number flyers has been described by Shyy et al. [1]. Based on dimensional analysis, Lighthill [2], Norberg [3], Pennycuick [4] and Schmidt-Nielsen [5] have studied the effect of above parameters. Tennekes [6] presented correlation between the weight of the bird, wing loading and cruising and established "The Great Flight Diagram". The diagram shows amount of force required (weight in N) along with wing loading (per meter square) with cruising speed. In a recent study, Pesavento and Wamg [7] have reported using 2-D numerical model that the flapping wing is more effective than steady wing due to wing-wake interaction. However, Lissaman [8] pointed out that the prediction and control of wing-wake interaction is difficult because on a wing laminar boundary layer can easily separate and reattach after it has become turbulent. Spedding et al. [9] suggested that many features of bird flight can be understood by simple aerodynamic models, like fixed wing model. Moreover, they reported that behaviour of wing-wake interaction is sensitive to small changes in geometry and flexibility. Spedding et al. [10] observed the family of vortex wakes during the fluid interaction with bird wakes. According to flight speed they proposed two vortex models, viz. discrete vortex model at low flight speeds and continuous vortex model at moderate to high speeds. In the present work, simulations have been carried out to investigate the flow characteristics during the interaction of flow with the prism. Also the effect of Reynolds number on drag coefficient has been examined for all the three arrangements of flapper. 2. PROBLEM STATEMENT Figure 1 shows the arrangement of the body in the computational domain where all the dimensions have been considered in terms of the side of crosssection of the square prism (d). The height of domain, H=5d, length of the domain, L=25d and width of the domain, B=12d. The front face of the body is placed at a distance of 5d from inlet and its centerline lies in the horizontal mid-plane of the domain. The built-in flappers all along length of prism are parallel to the horizontal mid-plane. Each flapper is assumed to have a thickness of 0.1d. Fig. 1: Flappers in (a) top and mid-plane, (b) midplane and (c) top-plane 2

3 The computation has been carried out for various arrangements of prism and flapper, which are shown in Table-1. Table-1: Different arrangements by changing the dimensions of prism and flappers Arrangement Mid-plane Top-plane Staggered arrangement Length of prism and flapper 5d Width of flapper (left flapper, right flapper) (1d, 1d), (1d, 2d), (2d, 2d) 3d (2d, 2d) 1d (2d, 2d) 5d (1d, 1d), (1d, 2d), (2d, 2d) 5d (1d, 1d), (1d, 2d), (2d, 1d), (2d, 2d) 3. GOVERNING EQUATIONS AND BOUNDARY CONDITIONS 3.1 Governing equations The governing equations for three-dimensional, incompressible fluid flow along an elongated square prism having coplanar side flaps in Cartesian coordinates are those of mass and momentum conservation. For an incompressible flow of air these equations are written in tensor form as ui 0 (1) x i ( ) 2 u uu i i j p 1 u i t x j xi Re x j x j (2) where, u i is the Cartesian velocity component along xi - coordinate direction and p is the nondimensional pressure. For a three-dimensional Cartesian coordinate system, Eq. (2) collectively represents the x, y and z-components of the momentum equation for the corresponding velocity components u, v and w respectively. The nondimensionalization of all length scales is based on a characteristic length scale which is the side of the square prism (d). The coordinate axes are as shown in Fig. 1 with x = 0, y = 0 and z = 0 being at the right bottom corner at the leading face of the prism. 3.2 Boundary conditions The boundary conditions employed for the computations are as follows Inlet: uniform velocity ( u U, v w 0) Outlet: pressure outlet ( p ) Side walls: free-slip and impermeable u w p boundaries ( v 0, 0) y y y Top and bottom walls: free-slip and impermeable boundaries ( w 0, u v p 0) z z z Prism and flapper surfaces: no-slip and impermeable boundaries ( u v w 0) 4. GRID MESH AND NUMERICAL METHODOLOGY 4.1 Grid mesh In the present work, structured grid of hexahedral type mesh is used. Fine mesh is used near the solid surfaces and coarse mesh is used in the fluid region away from the body surface. This is mainly to capture the steep gradients near the body surface. Nodes have uniform grid spacing throughout the domain. A schematic of mesh employed for computations has been shown in Fig. 2. Four different grid sizes have been taken for same domain of mid-plane prism arrangement and the grid independence test is carried out for same value of Re which as shown in Table-2. Finally, Grid 3 has been chosen for further computations because coefficient of drag does not vary when Grid 3 is refined to Grid 4. p 3

4 coupling and least square cell based technique used for spatial discretization. Second order upwind scheme has been used for discretizing the convective terms. 5. RESULTS AND DISCUSSION Computations are carried out to study the flow features for different arrangements. Results are presented in the form of vorticity contours, bound vortex, streamlines and force coefficients. 5.1 Vorticity Contours Vorticity magnitudes are shown in the different planes for all the three arrangements and for various prism length and flapper width at Re = 100. Fig.2: Mesh used for simulation Sl. No. Table-2: Grid independence study Total number of nodes Nodes along the length of square prism Mean Cd Effect of arrangements on vorticity Figure 3 presents the vorticity contour for the case when flappers are mounted in the mid-plane. Similarly, Fig.4 and Fig.5 show the corresponding vorticity contours for flappers in top plane and in staggered arrangement respectively. In all the cases, symmetric vortex pattern is observed. Vorticity contours consist of the vorticity of different strength and the high vorticity zone near the flapper is clearly visible. Formation of such high vorticity zone is caused mainly due to sudden expansion in the flow forming the body of vortex and due to the induced pressure gradient in the fluid. 4.2 Numerical technique ANSYS Fluent 14.0 has been used for the present computations which is based on finite volume algorithm. It uses semi-implicit pressure linked equation (SIMPLE) scheme for pressure-velocity 3(a) plane at x = 5 mm 4

5 3(b) plane at y = 5 mm 4(c) plane at z = 5 mm Fig.4: Vorticity contours in top-plane arrangement 3(c) plane at z = 5 mm Fig.3: Vorticity contours in mid-plane arrangement 5(a) plane at x = 5 mm 4(a) plane at x = 5 mm 5(b) plane at y = 5 mm 4(b) plane at y = 5 mm 5(c) plane at z = 10 mm 5

6 5(d) plane at z = 5 mm 6(b) Flapper width (1d, 1d) at z = 5 mm 6(c) Flapper width (1d, 2d) at z = 5 mm 5(e) plane at z = 0 mm Fig.5: Vorticity contours in staggered arrangement Effect of width of flapper on vorticity Figure 6 presents vorticity contour for different width of the flapper. It is observed that increase in flapper width decreases the strength of vorticity. In Fig. 6(c), the vorticity contour for unsymmetrical flapper arrangement shows that unbalanced forces act on the body which tend to move or turn the body towards high vorticity region. Fig.6: Vorticity contours in mid-plane arrangement for flapper width (2d, 2d), (1d, 1d), (1d, 2d) Effect of length of prism and flapper on vorticity Figure 7 presents vorticity contours for different length of prism and flapper. It is observed that increase in length of prism and flapper decreases vorticity magnitude. 7(a) Prism length 1d at z = 5 mm 6(a) Flapper width (2d,2d) at z = 5 mm 7(b) Prism length 3d at z = 5 mm 6

7 7(c) Prism length 5d at z = 5 mm Fig.7: Vorticity contours in mid-plane arrangement for prism and flapper length 1d, 3d and 5d 5.2 Pressure contours Figure 8 describes pressure contours of flow along a square prism having built-in plane side flaps for different length of prism and flapper. It is observed that increase in length decreases the region of adverse pressure gradient. 5.3 Bound Vortex Formation along the Body Bound vortex is a vortex which is tightly associated with body when any fluid flow across it. It is equivalent to circulation caused by boundary layer formation. This phenomenon express the viscous effect of fluid that balances the fluid speed when it leaves the trailing edge. The shape of these vortices change continuously across the body. 9(a) 8(a) 9(b) 8(b) 8(c) Fig.8: Pressure contours in mid-plane arrangement for prism and flapper length 1d, 3d and 5d Fig.9: (a) Front-view, (b) 3-D view of bound vortex in mid-plane arrangement Bound vortices in various cross-stream planes with flapper in mid-plane are shown in Figs.9 (a) and (b). It is seen that the orientation of the vortex axis changes by a right angle after the body. The moment it comes nearer the body it starts changing the shape due to the geometry of the body and envelope the whole body. 7

8 10(a) 11(b) Fig.11: (a) Front-view, (b) 3-D view of bound vortex in top-plane arrangement Figures 11(a) and (b) show the bound vortices for top-plane arrangement. The bound vortex shifts upward in comparison to Fig.9 in order to cover the geometry. 10(b) Fig.10: (a) Front-view, (b) 3-D view of bound vortex in staggered arrangement The behavior of bound vortices shown in Figs.10 (a) and (b) can be explained similar to that of Fig.9. However, the strength of vortex core is greater in this case because of geometric arrangement of flappers. 5.4 Streamline Plot Streamlines are presented for all the arrangements in Figs.12, 13 and 14. Near the surface of the body, the low velocity zones are clearly visible, which are also high vorticity zones as already shown previously. 11(a) Fig.12: 3-D view of streamline behind the square prism with flappers in mid -plane arrangement 8

9 Figure 12 shows the streamlines behind the body for which symmetric pattern is found. The four cores are formed which show the path of fluid particle and state that it will remain at that place for longer time and region of low velocity will develop. Figure 14 shows the streamline in prism with flappers in top-plane arrangement. 5.5 Force Coefficient The drag coefficient is calculated for different values of Re ranging from 40 to100 for all three arrangements with prism and flapper length of 5d. Figure 15 shows mean value of Cd for different values of Re for all the arrangements. Here one can see that the Cd value for staggered flapper lies between the other two arrangements in certain ranges of Re. Except this range the drag coefficient for staggered-plane becomes dominant. The drag coefficient for different lengths of prism and flapper have also been computed. With increase in length of prism and flapper coefficient of drag also increases. Fig.13: 3-D view of streamline behind the square prism with flappers in staggered arrangement Figure 13 shows the streamlines corresponding to the case where one flapper is at mid-plane and another is at top. Because of asymmetry in the arrangement itself, asymmetric pattern of streamlines is observed behind the body. Fig.15: Variation of mean drag coefficient for Re Fig.14: 3-D view of streamline behind the square prism with flappers in top-plane arrangement 6. CONCLUSIONS Numerical investigations are carried out to study characteristics of flow along a square prism with flappers on both sides. The vortex formation in midplane arrangement is more stable than all the other cases. Accordingly, mid-plane arrangement is 9

10 preferred for the fixed wing flyers as well as flexible wing flyers. As shown in vorticity contours of top-plane arrangement in z-plane, the arrangement helps in generating lift along z-axis. The staggered arrangement illustrates how roll could develop during flight about x-axis. Thus topplane position explains the sweeping motion and staggered plane position explains the rolling motion. By changing the length of one wing, yawing motion is induced in the body due to which centre of drag comes nearer to the body. It helps in changing the direction which is observed from the contours of vorticity for different widths of the flappers. REFERENCES 1. Wei Shyy, Yongsheng L, Dragos Viier, Haou Liu, Aerodynamics of low Reynolds number flyers, Cambridge University Press, M. J. Lighthill, Introduction to the scaling of aerial locomotion, in T. J. Pedley (Ed.), Scale Effects in Animal Locomotion (New York Academic), (1977) U. M. Norberg, Vertebrate Flight: Mechanics, Physiology, Morphology, Ecology and Evolution (Berlin, Springer-Verlag), C. J. Pennycuick, Newton Rules Biology: A Physical Approach to Biological Problem, Oxford University Press, K. Schmidt-Nielsen, Scaling: Why Is Animal Size So Important?, Cambridge University Press, H. Tennekes, The Simplicity Science of Flight (From Insects to Jumbo Jets), MIT Press, U. Pesavento and Z. J. Wang, Flapping Wing Flight Can Save Aerodynamic Power Compared to Steady Flight, Physical Review Letters, (2009)103: P. Lissaman, Low-Reynolds-Number Airfoils, Annual Review of Fluid Mechanics, 15(1983) G. R. Spedding, A. H. Hedenstrom, J. McArthur and M. Rosen, The implications of low speed fixed-wing aerofoil measurements on the analysis and performance of flapping bird wings, Journal of Experimental Biology, 211(2008) G. R. Spedding, M. Rosen, A. Hedenstorm, A family of vortex wakes generated by a thrush nightingale in free flight in a wind tunnel over its entire natural range of flight speeds, Journal of Experimental Biology, 206 (2003) Nomenclature B Cd d H L p p Re U Breadth of domain Coefficient of drag Side of the square prism Height of domain Length of domain Free stream pressure Pressure Reynolds number Free stream velocity u, v, w Velocity components along x, y and z-directions respectively x, y, z Cartesian coordinates Subscripts Free stream 10