Study of a Freely Falling Ellipse with a Variety of Aspect Ratios and Initial Angles

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1 Study of a Freely Falling Ellipe with a Variety of Apect Ratio and Initial Angle Dedy Zulhidayat Noor*, Ming-Jyh Chern*, Tzyy-Leng Horng** *Department of Mechanical Engineering, National Taiwan Univerity of Science and Technology, Taipei 0607, Taiwan, ROC. ( d @mail.ntut.edu.tw) ** Department of Applied Mathematic, Feng Chia Univerity, Taichung 40724, Taiwan, ROC. Abtract:The dynamic behavior of an ellipe falling freely inide a Newtonian fluid i invetigated. The immered boundary method i ued to imulate the cae at different apect ratio and initial falling angle. Within the conidered apect ratio and falling angle, the trajectorie fall into ome mode including fluttering, tumbling, fluttering-tumbling, turning and teady falling motion. The cae are very enitive to perturbation. Neverthele, when the ellipe ha the cloer form to a circular cylinder, it can perit it poition a initial due to a high moment of inertia. Furthermore, the initial falling angle doe not affect the flow pattern ignificantly. Keyword: Falling ellipe, immered boundary method, fluid-tructure interaction, nonlinear dynamic.. INTRODUCTION It i a common fact that falling object move preciely under the influence of gravity except when their motion are retarded through fluid interaction. For example, leave, tree eed, and paper card all follow complicated downward trajectorie a they fall under the gravity. They flutter and tumble down in a eemingly unpredictable and complex manner. Some experimental and numerical work have been conducted to tudy the rich dynamical behavior of freely falling object [-6]. Belmonte et al. [] and Mahadevan et al. [2] have done the experimental tudie for the thin flat trip falling through a fluid in a vertical cell and the tumbling card, repectively. Wang and Peavento [3] tudied the aerodynamic of freely falling plate for a quai two-dimenional flow, which i a typical tate for a leaf of buine card falling in air. They performed the experimental and the direct numerical imulation of the two-dimenional incompreible Navier-Stoke equation for the falling object. Anderen et al. [4, 5] have invetigated the unteady aerodynamic of fluttering and tumbling plate. The thickne-to-width ratio and the dimenionle moment of inertia were varied to oberve the tranition between fluttering, tumbling, and teady decent of falling object. Recently, Jin and Xu [6] have carried out the experimental and numerical imulation to tudy the unteady aerodynamic of freely falling plate. The elliptical and rectangular plate were conidered in their work. The plate falling velocity and trajectory were almot the ame for the elliptical and rectangular plate under the experimental condition. The main dynamic difference due to the geometrical hape were the angular velocity at which the rectangular plate rotated much lower than the elliptical plate. From the previou tudie, it hould be noted that the cae of freely falling object are very enitive to perturbation or initial condition. Therefore, we invetigate the effect of apect ratio and initial angle of a freely falling ellipe immered in a Newtonian fluid in the preent tudy. According to the obtained reult, we oberve ome intereting trajectorie of a freely falling ellipe and categorize them into a mode diagram. 2. NUMERICAL METHODS The non-dimenional continuity and momentum equation are written a follow u = 0 ()

2 u = t Re 2 ( uu) = p + u + f (2) We ued the immered boundary method [7] to olve the problem in thi tudy. The three-tep time-plit i ued to advance the flow field. Firt the velocity i tepped from the n th time level to the firt intermediate level * by olving the convective-diffuive term without the preure and virtual force for the momentum Equation (2). Subequently, thi tep can be tated in the following form * n u u = S n, (3) where S i the convective and diffuive term of the momentum equation. The intermediate velocity in Equation (3), in general, doe not atify the divergence-free condition (). At the econd tep we advance the firt intermediate velocity by including the preure term u ** u* n+ = p. (4) By applying the divergence on both ide, Equation (4) become u ** u* 2 n+ = p. (5) Due to conervation of ma we have u ** = 0. (6) Then ubtitution of Equation (6) to (5) give the Poion equation 2 + p n = u*. (7) Once Equation (7) i olved, we can advance the intermediate velocity in Equation (4). Furthermore, we update the velocity to the (n+) th time level by impoing the virtual force, f, a follow n+ u u* * n+. (8) = f The virtual force term, f, in Equation (8) repreent the action of a olid upon a fluid. The force reveal the exitence of a force to hold or drive a olid body when it i tationary or moving. To atify the no-lip condition for a olid motion, the force acting on the olid hould enure that the fluid velocity (u) equal the olid velocity ( u ) at the (n+) th + time tep i.e. u n+ = u n. Hence, the virtual force i defined a the rate of momentum change of olid body and proportional to the difference between the olid velocity at the (n+) th time tep and the fluid velocity at the n th time tep. The force exit on the olid body and zero elewhere. Furthermore, it can be imply written a f n+ n n+ n u u u = η = η, (9) n+ u where u i the velocity of olid. η denote the fraction of olid within a cell where η equal to and 0 for olid and fluid cell, repectively. There i no different computational time between tationary and moving body cae due to a fixed grid arrangement. Subequently it reduce computational cot and memory requirement, epecially for moving object cae. The motion of a olid body/particle, u, will be tracked in the Lagrangian reference by the equation of linear momentum and tranportation of a rigid body a follow

3 n+ n v v n m = f V + ( m m f )g ηρ d. (0) Ω Eq. (0) can be rewritten in the differential form a ( m v ) d = F + G, () dt while the torque acting upon the olid, T, and the tranlation of the olid can be tated a d( I. ω ) = T (2) dt and dx = v. (3) dt When conidering the rotation, the particle velocity inide the olid body moving with the tranlational velocity, v, at the center of rotation and the angular velocity, ω, can be expreed a: u = v + ω r, (4) where, F i the hydrodynamic force, G i the external force including gravitational and buoyancy force, T i the torque acting upon the olid particle, I i the inertia tenor, ω i the angular velocity of the olid particle, m i the ma of the particle, v i the tranlational velocity of the center of ma, x i the center poition of the olid particle. 3. RESULTS AND DISCUSSION We perform the imulation to tudy flow behavior of a freely falling ellipe in a fluid domain. Since the flow i very enitive to perturbation, the three value, 0 0, 45 0, and 90 0, of the initial angle of attack of the falling ellipe with repect to the horizontal axi, are choen in the imulation. The apect ratio i changed alo to invetigate the effect of the thickne. The denity of the ellipe and the fluid are 2.7 g/cm 3 and.0 g/cm 3, repectively. The kinematic vicoity of the fluid i cm 2 /. The chord of the ellipe i cm while the thickne i varied. The vortex hedding and the wake tructure of the freely falling ellipe are hown in Fig. and 2 for the fluttering and tumbling mode, repectively. The vortex hed when the ellipe glide at a mall initial angle (a). The ellipe turn (b) and vortice are hed from the leading and trailing edge (c). Subequently, the ellipe reume gliding and the old leading edge become the new trailing edge by hedding the vortex (d). For the tumbling motion, the ellipe glide by hedding vortex at the trailing edge (2a). The ellipe tart turning (2b) and hed vortex from the old leading edge (2c). Turning i completed and the ellipe repeat to glide (2d). When the ellipe glide with an angle of attack, it lead to two high preure region, one below the ellipe cloe to the leading edge and the other above the ellipe cloe to the trailing edge. Thee region create a torque. The reulting torque rotate the ellipe a it begin to move broadide-on and low down. If the rotation angle i larger than 90 0, then the ellipe tumble. Otherwie, it periodically ocillate from ide to ide. Plot of the trajectory of the falling ellipe i preented in Fig 3. Within the conidered range of the apect ratio and α, the flow variation due to the ellipe falling motion can be categorized to be four mode, i.e. fluttering, tumbling, fluttering-tumbling, turning, and traight/teady falling mode. The motion of the falling ellipe tend to be fluttering when the ellipe i initially placed in horizontal poition, α = 0 o The ellipe ocillate from ide to ide

4 a it decend with alternating gliding at a low initial angle and a rotational motion at the turning point. The rotational angle at the turning point become larger for a decreaing apect ratio due to a maller moment of inertia. The ellipe can not perit the fluttering motion and it tumble finally (ee the fluttering-tumbling mode). In term of the effect of the initial poition, our reult how a good agreement with the previou tudy [8] A horizontally held ellipe hould have fluttered to the ground a it fall. On the other hand, a vertically held ellipe hould have fallen for a moment before tumbling and landing far from the point directly below the releae point. The fluttering and tumbling motion can be explained by following the obervation of Maxwell [9]. When the ellipe glide with an angle of attack, α 0, it lead to two high preure region, one below the ellipe cloe to the leading edge, and the other above the ellipe cloe to the trailing edge. Thee region create a torque. The reulting torque rotate the ellipe a it begin to move broadide-on and low down. If the rotation angle i larger than 90 0, then the ellipe tumble. Otherwie, it periodically ocillate from ide to ide. An initial angle, α = 45 o lead the ellipe tumble immediately after being releaed except for the low apect ratio cae. Intead of tumbling, the ellipe travel form ide to ide. It hould be noted that, when the ellipe fall in the fluid field, it can hold it motion a the initial at horizontal, α = 0 o and vertical, α = 90 o for a bigger apect ratio becaue of a higher moment of inertia. The initial angle doe not influence the motion ignificantly o long a the ellipe ha a cloe form to a circular cylinder. Fig. : The vorticity contour at the fluttering motion

5 Fig. 2: The vorticity contour at the tumbling motion Fig. 3: Trajectory mode of a freely falling ellipe at different apect ratio and initial angle

6 4. CONCLUSIONS A freely falling ellipe with a variety of apect ratio and initial angle ha been invetigated uing an immered boundary method. A horizontally held ellipe hould have fluttered to the ground a it fall. On the other hand, a vertically held ellipe hould have fallen for a moment before tumbling and landing far from the point directly below the releae point. When the ellipe with a higher apect ratio fall in the fluid field, it can hold it motion a initial due to a higher moment of inertia. REFERENCES. Belmonte, A., Eienberg, H., and Moe, E., (998), From flutter to tumble: Inertial drag and Froude imilarity in falling paper. Phyical Review Letter 8, 2 pp Mahadevan, L., Ryu, W.S., and Samuel, A.D., (999), Tumbling chard, Phyic of Fluid, pp Peavento, U. and Wang, Z. J., 2004, Falling paper: Navier-Stoke olution, model of fluid force and center of ma elevation, Phyical Review Letter 93, pp Anderen, A., Peavento, U., and Wang, Z.J, (2005), Unteady aerodynamic of fluttering and tumbling plate, Journal of Fluid Mechanic 54, pp Anderen, A., Peavento, U., and Wang, Z.J, (2005), Analyi of tranition between fluttering, tumbling and teady decent of falling card, Journal of Fluid Mechanic 54, pp Jin, C. and Xu, K., (2008), Falling paper: Numerical tudy of the unteady aerodynamic of freely falling plate, Communication in Computational Phyic 3, pp Noor, D.Z, Chern, M.J., and Horng, T.Z, (2009), An immered boundary method to olve fluid-olid interaction problem, Computational Mechanic 44, pp Finn, D.L., 2007, Falling paper and flying buine card, SIAM New 40, pp Maxwell, J.C., 940, Scientific paper of Maxwell J.C., (Dover, New York), pp. 5.