Proceedings of the ASME 3st International Conference on Ocean, Offshore and Arctic Engineering OMAE July -6,, Rio de Janeiro, Brazil OMAE-84 Stability Analysis Hinged Vertical Flat Plate Rotation in a Uniform Flow Antonio Carlos Fernandes LOC COPPE UFRJ Rio de Janeiro, Brazil acfernandes@peno.coppe.ufrj.br Sina Mirzaei Sefat LOC COPPE UFRJ Rio de Janeiro, Brazil smsefat@peno.coppe.ufrj.br ). Though, the study on flow induced rotation of a vertical hinged flat plate under the influence of uniform current is a DOF problem, but it aims at the understanding of the fluttering and autorotation problems of falling objects in air or water. The results could be generalized for some other phenomena which are 3-DOF, like as PIM method, falling objects, windborne debris, etc. ABSTRACT The free falling of objects is a phenomenon that has been observed in the nature. The Pendulous Installation Method (PIM) of heavy devices is an example of free falling which occurs in the installation of heavy offshore devices on sea bed. Hence, the motivation of the present research is to study the fluttering and tumbling (autorotation) motions that may occur during the free fall of object. The fluttering is a periodic or chaotic oscillation of body about a vertical axis. On the other hand, the tumbling is end-over-end rotation of body. In order to access the main physical aspects, the present work decided to attack a more fundamental problem and describes the investigations on fluttering and autorotation motions of the interaction of uniform current and freely rotating plate about a fixed vertical axis. A quasi-steady model is suggested to model the trajectories of flow induced rotation phenomenon and a stability analysis performed to gain insight into the nature of the bifurcation from fluttering to autorotation. At first, the fixed points for different models of motion is obtained and each point analyzed by using the linearized equation. Secondly, the phase diagrams as a function of angular velocity and angle of rotation have been presented for different dynamic models. Fig. Schematic of flow induced rotation of a flat plate hinged at the center and is free to rotate in uniform current. INTRODUCTION In this study, the authors are under the impression of fluttering and tumbling phenomenon which happens in the falling objects problems in air or water. In fluttering, the body oscillates either periodically or chaotically from side to side as it descends alternating gliding at low angle of attack and fast rotational motion [] and tumbling motion is characterized by the end-over-end rotational motion of body. In the present case, controlling the oscillatory behavior of manifolds during the pendulous installation method of a manifold [4,5,6,7,8,9] was the first motivation to study the flow induced rotation phenomenon. Therefore, the authors chose to study the behavior of a hinged flat plate allowed to rotate about its vertical axis, under the influence of uniform current (see Fig. The fluttering and autorotation are two different phenomena which may occur in flow induced rotation of a plate about a fixed axis, which is free to rotate in the current. In the autorotation, the plate rotates continuously around a vertical axis and it never damps out. On the other hand, the fluttering motion is another unexpected periodic oscillation of the plate around a stable position which the plate is normal to the flow. Maxwell [6] was the first who tried to explain tumbling motion of a flat-plate. Later, several experimental researches under both free-fall and fixed-axis conditions were made in order to classify and quantify the types of these rotational motions. The theoretical and experimental results before 979, Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 9//3 Terms of Use: http://asme.org/terms Copyright by ASME
such as Dupleich [], and etc, have been evaluated and summarized by Iversen [3]. According to the Iversen [3], the motion of a freely falling and also fixed axis rectangular plate is governed by the Reynolds number, the aspect ratio (ratio of width to span of plate), thickness ratio (ratio of width to thickness of plate) and dimensionless moment of inertia of the flat plate. Actually the motion in free fall, and, in particular, the transition from fluttering to tumbling, should be completely governed by these parameters. Tachikawa [9] firstly, presented a two-dimensional equation of motion to explain the trajectories of free falling of square and rectangular plates in a uniform flow. Secondly, he obtained by experiments, the aerodynamic coefficients of autorotation in different square and rectangular flat plates, which are required for solving the equations of motion. The results show that the trajectories of a flat plate released into a flow from rest is closely related to the initial mode of motion and is distributed over a wide range. Tanabe et al. [] gave a phenomenological model for free fall of a paper, assuming zero thickness and incompressible ideal fluid. They discovered five different patterns of: periodic rotation, chaotic rotation, chaotic fluttering, periodic fluttering and simple perpendicular fall. Mittal et al. [8] show a computational study on flow induced motion of a hinged plate pinned at its center. Their focus is on the effect of Reynolds number and plate thickness ratio and non-dimensional moment of inertia on vortex induced rotation of plates. Based on their numerical results, they suggest that the flutter and tumble frequencies of large aspect-ratio plates are governed by the Von-Kármán vortex shedding process. Anderen et al. [] investigated the dynamics of freely falling plates experimentally, numerically and by quasi-steady modeling, at Reynolds number of 3, which describes the motion of freely falling rigid plates based on detailed measurement of the plate trajectories and from this to assess the instantaneous fluid forces. Lugt [5] argued that, for a viscous fluid, at least qualitatively, a fifth-order polynomial for the damping terms in the pendulum equation is necessary to simulate the self-excited oscillation (fluttering) and autorotation for bodies with axis fixed in a parallel flow. It should be mentioned that most of the previous researches on flow induced rotation, conducted experimentally and rarely numerically. In the present study, a quasi-steady model is suggested to model the trajectory of autorotation motion via analytical modeling. The comparison of analytical modeling with experiments at water flume of LOC was made. Also, a stability analysis is performed on flow induced rotation phenomenon to gain insight into the nature of transition from fluttering to autorotation. HYDRODYNAMIC FORCES AND MOMENTS OF FLAT PLATE A flat plate has a simple shape and actually is the simplest airfoil which having no thickness or shape. For small incidence angles of attack, several results were recovered for inviscid and irrotational flow such as the classical results of vortex sheet approach to calculate loads acting in flat plate. But for large incidence angles because of presence of wake behind the flat plate, it is not possible to assume that the flow is inviscid and viscous term plays an important role on the loads and moments acting on the flat plate. One of the methods used to attack the problem was Free-Streamline theory. The main objective of this theory is to find the free streamlines to define the wake, the location of these free streamlines are initially unknown and must be found as a part of solution, outside the wake the flow is potential and to compute the resulting pressure drag. According to the experiments, the pressure on the free streamline does tend to remain constant for some distance downstream of the separation point. The shear layers do not continue far downstream as assumed and roll up to form vortices, alternately on each side. This vortex formation occurs behind all bluff bodies, at a frequency which is a characteristic of each body shape. The flow past a bluff body is considered in two parts (see Fig. ). Near the body it may be described by the free-streamline theory. As mentioned before the experiments show that, the pressure in a certain distance downstream of the separation points is approximately constant; this part of the wake will be called the near-wake and the pressure in this region. Downstream regime of wake after the near-wake region, the vortices mix and diffuse rapidly and are eventually dissipated in the wake, in such a range, the shape of the free streamline cannot be determined definitely. This part of the wake will be called the far-wake, or the mixing range. Along the far-wake the mean pressure increases gradually from the wake underpressure and finally recovers the main stream pressure far downstream. Fig. Schematic of free streamline theory, for flow past a flat plate. The subject of free streamline methods has an interesting history; Kirchhoff was the first who proposed the idea of a wake bounded by free streamlines as a model for the flow behind a bluff body. He used the mathematical methods of Helmholtz to find the irrotational solution for a flat plate set normal to an oncoming stream. According to his assumption the velocity on the free streamline at separation point is equal to the free-stream velocity U. In this case, the pressure at the separation points and behind the separation points is equal to the free-stream pressure, which it is considerable loss of reality. Because this is not in agreement with experience, which shows Copyright by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 9//3 Terms of Use: http://asme.org/terms
that the pressure is actually always lower than the free-stream value. In this paper we will focus on the results of a modern view of free streamline theory by Wu [4,5,6] to calculate the loads and moment acting on a flat plate. Wu assumed that the pressure in the near-wake region (p c ), is uniform and constant, which it leads to a dimensionless equivalent parameter of p c, usually called the wake under-pressure coefficient (σ). This parameter characterizes the pressure of wake flow, based on free stream pressure. In fact, the different flow regimes of the fully and partially developed flows can also be indicated by different ranges of σ. P pc ().5 U where P denotes the pressure of the undisturbed free stream, U is its relative velocity, is the fluid density. The wake flow will be called fully developed, if the region of the constantpressure near-wake extends beyond the trailing edge of the plate and will be called partially developed, if the near-wake region terminates in front of the trailing edge. Fig. 3 shows the flow in the physical space (z-plan, where z = x+ iy). Fig. 3 The flow in the physical space (Hint: q is the velocity in streamline). Hereby, the velocity at separation is normalized to q=and remains at this value along the free streamlines until the latter reach points D and D. Downstream of these two points, the free streamlines keep parallel to the free stream on DE and D E along which q decreases from unity back to the free stream value U. In order that under-pressure coefficient of the flow be σ with q= on AD and BD, U takes the value (+σ) /. According to Kirchhoff s classical result, by considering that, the normal coefficient of flow past a flat plate ( ) is rewritten as below cos C N () 4 cos In the fully wake flow past a flat plate, the local pressure is everywhere normal to the plate, and there is no singular force at leading edge, therefore C L and C D should satisfy the condition C D C L tan ; For all and (3) In the special cases of (i) or (ii) close to zero and, the Eq. (3) condition is obviously satisfied. Each angle of attack must be assumed a different bluff body, which has a different pressure on the downstream of separation points. But in the general case, because of the complicated manner in which the dependence on and appears, the results of streamline theory show that for small angles of, less than 45 degree, the values of C L and C D approach respectively the asymptotes [6] C (, ) ( ) C (, ) (4) L D L C (, ) ( ) C (, ) (5) D cos C N (, ) ( ) (6) 4 cos This argument is supported by experimental and numerical evidences. The loads and moment coefficients were measured for a fixed flat plate, as a function of different angles of attack by experiments in a flume (m x.4m x.5m) at the LOC- COPPE-UFRJ (Laboratory of Waves and Current of COPPE, Federal University of Rio de Janeiro). Similar measurements were carried out by the present authors with a CFD simulation using the ANSYS CFX code (Fernandes et al. [4,6,7]). Fig. 4 shows the comparison of normal coefficient of streamline theory with σ.5 and experimental results of Hoerner [], and also Fernandes et al. [4,6,7] numerical results. C N,5,75,5,5,75,5,5 Streamline Theory - σ =.5 CFX - CB=.6 LOC Horner 3 4 5 6 7 8 9 θ [deg.] Fig. 4 Comparison of normal coefficient of streamline theory with.5, and experimental results at LOC and Hoerner [], and also Fernandes et al. [4,6,7] numerical results. Fig. 4 shows that streamline theory results agree very well numerical and experimental results, especially when θ is less than 55 degrees. This is about the range of interest for modeling the fluttering motion. As we expected, because of constant pressure distribution on the downstream side of separated side of the plate, the location of center of pressure is independent of all practical range of σ. The constant pressure in downstream side of plate cause to the center of pressure of this side of plate always is in center of plate. Hence the overall center of pressure of plate is only depending on pressure distribution of upstream side of plate. The Kirchhoff s classical formulation for the center of pressure in term of the width is 3 Copyright by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 9//3 Terms of Use: http://asme.org/terms
3 cos c b (7) 4 4 sin Based on above discussion, the Kirchhoff s classical formulation for the center of pressure (Eq. (7)) shows a good agreement with numerical results of Fernandes et al. [4,6,7] and two experimental results of Flachsbart and Tachikawa [] when θ is less than 55 degrees (see Fig. 5). Center of pressure (c/b),35,3,5,,5,,5 Fernanses et al TACHIKAWA FLACHSBART Kirchhoff s Classical Model 3 4 5 6 7 8 9 θ[deg.] Fig. 5 Comparison of position of the center of pressure with streamline theory (Eq. (7)) and Fernandes et al. [4,6,7] and two experimental results of Flachsbart and Tachikawa (apud []). The moment coefficient could be obtained by considering of Eq. (7) for center of pressure and Eq. (6) for normal coefficient. Hence, the moment coefficient formulation is: 3 sin C M ( ) (8) 4 4 cos As noted before the center of pressure position is independent of σ, and also the normal coefficient formulation of streamline theory has a good accuracy with numerical and experimental results with σ.5. Fig. 6 shows the comparison of moment coefficient of streamline theory (Eq. (8)) with σ. and also σ.5 and experimental results from LOC and numerical modeling of Fernandes et al. [4,6,7]. According to the Fig. 6 the linear piecewise of streamline theory is in a good agreement with numerical and experimental results, for angles of attack smaller than 55 degree. This means that the streamline theory works very well in this range; perhaps more important point of Fig. 6 is that the linear behavior that observed by experiments in moment coefficient curve has a technical support in this large range. C M -,5 -, -,5 -, -,5 -,3 -,35 -,4 θ [ang.] 3 4 5 6 7 8 9 LOC CFX - CB=.6 Streamline Theory - σ=. Streamline Theory - σ=.5 Fig. 6 Comparison of moment coefficient of streamline theory (Eq. (8)) with σ. and also σ.5 and numerical results of Fernandes et al. [4,6,7] and experiments at LOC. Based on the D'alembert's paradox the net force on a body which is subjected on an inviscid flow is zero, but not necessarily a zero moment and any shape other than a sphere generates a moment. This moment may be called Munk moment in the elongated body context. The Munk moment arises from the asymmetric location of the stagnation points, where the pressure is highest. In the front of the body there is a decelerating flow and on the back there is an accelerating flow. Letting represent the angle of attack, the Munk moment is given by (9) [4]. M m m yy mxx U sin (9) where m xx is the added mass along the body x-axis (longitudinal), and m yy is along the body lateral y-axis (transversal). This moment may be approximated for small angles of incidence. According to the above statements, the Munk moment coefficient for a flat plate can be express as C M ( ) sin () 4 The comparison of the moment coefficient calculated by numerical modeling (Fernandes et al. [4,6,7]) and experiments at LOC and also the Munk moment coefficient given by () are presented in Fig. 7. This comparison shows that the range of validity of Eq. () is just for 75 9, again as expected, since this is an asymptotic result. 4 Copyright by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 9//3 Terms of Use: http://asme.org/terms
C M -, -, -,3 -,4 -,5 -,6 -,7 -,8 -,9 Fig. 7 Moment coefficient obtained by experiments at LOC, numerical modeling (Fernandes et al. [4,6,7]) and also the Munk moment coefficient given by (), and moment coefficient segments, proposed in Table. Table List of moment coefficient segments Moment Coefficient Angle 3 sin C M 4 cos 75 4sin θ[deg.] 3 4 5 6 7 8 9 C M = -(3/)πsinθ/(4+cosθ) -.45θ C M = -(π/4)sinθ LOC Fernandes et al. Munk C M 75 9 DIMENSIONAL ANALYSIS In order to improve the understanding of flow induced rotation of flat plate, a dimensional analysis is performed. The flow induced rotation motion of a flat plate may be characterized by eight dimensional parameters: b the width of the plate, t the thickness of the plate, the density of the plate, the density of the fluid, moment of inertia of plate, A 66 added moment of inertia, the kinematic viscosity of the fluid, and the uniform flow velocity, as summarized in () R f b, t, s, f, I, A66,, U () The dimensional analysis leads to (): R f,, Re, Iˆ () Where =t/b is the thickness ratio, / the specific density of the plate, Re=Ub/v is Reynolds number and I/A 66 the dimensionless added moment of inertia (It should be mentioned that for the comparison purpose, instead of, in this research, we will use I =(8/π)., as used by Iversen [3] and Field et al. []). It should be noted that the dimensionless moment of inertia can be rewritten as a function of dimensionless density and thickness ratio. Therefore, the motion in flow induced rotation motion and in particular, the transition from fluttering to autorotation, should be completely governed by dimensionless moment of inertia and Reynolds number. EQUATION OF MOTION OF AUTOROTATION A simple linear model for the flow induced rotation of a flat plate in a uniform current which has only one degree of freedom is suggested by applying the angular moment theorem ( I A66) f CM SbU (3) By incorporating dimensionless variables Iˆ I A66 and t t ( U b), the equation of motion become d Iˆ 64 CM (4) dt This equation can be solved numerically using Runge-Kutta fourth order method. The normal force coefficient and centre of pressure positions obtained for static plates at various angles of attack can be used to obtain moment coefficients in the above equation (Table ). The numerical calculations were made to compare with the experimental trajectories for autorotation motion recorded at LOC. With the moment coefficient proposed from the experimental results of LOC, the obtained equations of motion are shown at Table : Table Equations of motion based on moment coefficient proposed by experimental results of LOC. Equation of Motion Angle ˆ d 64 3 sin I 75 dt 4 cos ˆ d 64 I sin dt 4 75 9 Figs. 8 and 9 show the comparison of the autorotation trajectories recorded by experiments at LOC and the numerical simulation based on the equations of motion of Table. The Reynolds Number was kept around 9.95 x 4. The initial angular position in all Figures is θ =9 (= α = ). Angle of Rotatin [rad.] 4 3,5 3,5,5,5 LOC Experiments - I=.58 Quasi-Steady Modeling - I=.58,4,8,,6 Time [sec.] Fig. 8 The autorotation motion of plate with I =.58 and Re=9.95 x 4. 5 Copyright by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 9//3 Terms of Use: http://asme.org/terms
Angle of Rotation [rad.] 4 3,5 3,5,5,5 LOC Experiments - I=.5 Quasi-Steady Modeling - I=.5,4,8,,6 Time [sec.] Fig. 9 The autorotation motion of plate with I =.5 and Re=9.95 x 4. As we expected the numerical modeling simulation of autorotation motion based on LOC moment coefficient model has a good agreement with experiments. STABILITY ANALYSIS OF FLOW INDUCED ROTATION To gain future insight into the nature of transition between fluttering and autorotation, we considered a phenomenological model of the flow induced rotation of a flat plate hinged at center. The aim of this section is to do a simple stability analysis on the dynamic model of flow induced rotation phenomenon. The fixed points for differential equation models of motion are presented in Table 3: Table 3 Fixed points for dynamic equation of motion. Hence, for a given solution x(t), the total energy is constant as a function of time: E mx V( x) (7) The energy is often called a conversed quantity and these systems called conservative systems. The conservative energy system for differential equation model of flow induced rotation motion is presented in Table 5. Table 4 Stability analysis of the linearized equations in the neighborhood of the fixed pints. Linearized Equations of Motion Fixed Points Munk (or Lift) theory moment For fixed points:,, n=,, 4,.. : coefficient model 6/. Hence, we have: Where =, det 6/ and 4q 64/, so we can express that, the critical points nπ,, n=,, 4,...are all Centers. y y y k y For fixed points:,, n=,,.. : 6/. Hence, we have: Where =, det 6/ and 4q 64/, so we can express that, the critical points nπ,, n=, 3,...are all Saddle points. Table 5 The conservative energy system for differential equation model. Munk (Lift) theory moment coefficient model ( A ) 4 M sin 66 d M dt 8( A 66 cos ) For linearization of equations of motion of Table 3, to obtain a system of ODEs, we set, and also considering the Maclaurin series. The linearized equations in the neighborhood of the fixed pints can be obtained as Table 4. The Newton s second low ( ) is the source of many important second-order systems. We can show that energy is conversed, as follow. Let denote the potential energy, defined by. Then dv m x (5) dx By multiplying both sides by, we have dv d m x x x mx V( x) dx dt (6) M 8( A 66 cos E ) Fig. shows trajectories for various values of E and based on the equations of motion of table. These graphs continue periodically with period to the left and right. We can see some of the trajectories are ellipse-like and closed (fluttering motion) and some other are wavy (autorotation motion). By considering of Table 5, at point: y, we get (8): M cos C (8) 8( A ) 66 6 Copyright by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 9//3 Terms of Use: http://asme.org/terms
If y, then E C, where E ( 8) ( M /( A66)). Then if C E, the autorotation occurs and if C E, we have the fluttering motion. 7 Fluttering E>C Autorotation E=C 3,5 5 5-3,5 E<C -7 Fig. The phase diagram for various values of E and based on equations of motion of table. According to the Fig., the value of the separation energy level from fluttering to autorotation is C.95. CONCLUSIONS The free streamline theory is applied to approximate the fully developed wake far downstream. The normal and moment coefficient formulations have been suggested by considering streamline theory for different angles of attack, which these formulations have a good accuracy with experimental and numerical results. The dimensional analysis proves that the motion in flow induced rotation and in particular, the transition from fluttering to autorotation, should be completely governed by dimensionless moment of inertia (I ) and Reynolds number (Re). The stability analysis was performed on flow induced rotation phenomenon to gain more insight into the nature of transition between fluttering and autorotation. The system looks conservative. The amount of energy treasure that separates the fluttering from autorotation is identified. ACKNOWLEDGEMENTS The authors express their thanks to ANP (Brazilian National Petroleum Agency), CNPq (Brazilian National Research Council) and LOC/COPPE/UFRJ (Laboratory of Waves and Current of COPPE, Federal University of Rio de Janeiro). REFERENCES [] Andersen, V., Pesavento, U., Wang, Z. J., 5. Analysis of transitions between fluttering, tumbling and steady descent of falling cards. Journal of Fluid Mechanics 54, 9 4. [] Dupleich, P., 94. Rotation in Free Fall of Rectangular Wings of Elongated Shape. NACA Technical Memo., - 99. [3] Fernandes, A. C., Mirzaei Sefat, S.,. Fluttering and Autorotation of a vertical Flat Plate about a Fixed Axis Submitted to a Uniform Horizontal Flow. IUTAM Symposium on Bluff Body Flows, December -6,, IIT Kanpur, India. [4] Fernandes, A. C., Mirzaei Sefat, S.,. The Hydrodynamic Torsional Spring in the Flow Induced Fluttering of a Hinged Vertical Flat Plate. Submitted for Journal of Ocean Engineering,. [5] Fernandes, A. C., Mirzaei Sefat, S., Coelho, F. M., Albuquerque, A. S.,. Experimental Investigation of Flow Induced Rotation of Hinged Plates with Different Bluffness in Uniform Flow. 3th International Conference on Ocean, Offshore and Artic Engineering - OMAE, June 9-4,, Rotterdam, Netherland. [6] Fernandes, A. C., Mirzaei Sefat, S., Coelho, F. M., Ribeiro, M.,. Investigation of the Flow Induced Small Amplitude Rotation Triggering Flat Plate Fluttering. 6th IUTAM Symposium on Bluff Body Wakes and Vortex-Induced Vibrations, Capri Island, Italy. [7] Fernandes, A. C., Mirzaei Sefat, S., Coelho, F. M., Ribeiro, M.,. Towards the understanding of Manifold Fluttering during Pendulous Installation: Induced Rotation of Flat Plates in Uniform Flow. 9th International Conference on Ocean, Offshore and Arctic Engineering, OMAE-95, China. [8] Fernandes, A. C., Sales JR, J.S., Neves, C. R., 7. Development of the Pendulous Installation Method for Deploying Heavy Devices in Deep Water by Using Model Testing Concomitantly with Numeric Code. Submitted for publication in Journal of Offshore Mechanic and Arctic Engineering, OMAE7-33. [9] Fernandes, A. C., Sales JR, J.S., NeveS, C. R., 7. The Concomitant Model Testing Approach for the Development of the Pendulous Installation Method of Heavy Devices in Deep Water. 6th Sympocium on Offshore Mechanics and Arctic Engineering (OMAE). OMAE7-945, USA. [] Field, S. B., Klaus, M., Moore, M. G., Nori, F., 997. Chaotic dynamics of falling disks. Nature. 388, 5-54. [] Hoerner, S. F., 965. Fluid-Dynamic Drag. Published by Author. New York, US. [] Holmes, JD, Letchford, CW, Ning Lin.; Investigations of plate-type windborne debris Part II: Computed trajectories, 7 Copyright by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 9//3 Terms of Use: http://asme.org/terms
Journal of Wind Engineering and Industrial Aerodynamics; Vol. 94, pp 39, 6. [3] Iversen, J. D., 979. Autorotating flat-plate wings: the effect of the moment of inertia, geometry and Reynolds number. Journal of Fluid Mechanics. 9, 37-48. [4] Lamb, S. H., 93. Hydrodynamics, 6th ed. Cambridge University Press. [5] Lugt, H. J., 983. Autorotation. Annual Review of Fluid Mechanics 5, 3-47. [6] Maxwell, J. C., 854. On a particular case of the descent of a heavy body in a resisting medium. J. Cambridge and Dublin Math 9, 45-48. [7] Mirzaei Sefat. S., Fernandes, A. C.,. Investigation of the Flow Induced Small Amplitude Rotation. 3th International Conference on Ocean, Offshore and Artic Engineering - OMAE, June 9-4,, Rotterdam, Netherland. [8] Mittal, R., Seshadri, V., Udaykumar, H. S., 4. Flutter, Tumble and Vortex Induced Autorotation. Journal of Theoretical and Computational Fluid Dynamics 7(3), 65-7. [9] Tachikawa, M., 983. Trajectories of flat plates in uniform flow with application to wind-generated missiles. Journal of Wind Engineering and Industrial Aerodynamics 4(-3) 443-453. [] Tanabe, Y., Kaneko, K., 994. Behavior of a falling paper. Journal of Physical Review Letters 73(), 37-375. [] Roshko, A., 953. On the development of turbulent wakes from vortex streets. NACA, Tecnical Note:93. [] Roshko, A., 954. Hodograph for Free-Streamline Theory. NACA, Tecnical Note:368. [3] Roshko, A., 954. On the drag and shedding frequency of two-dimensional bluff bodies. NACA, Tecnical Note:369. [4] Wu, T., 956. A free streamline theory for two-dimensional fully cavitated hydrofoils. Journal of Mathematical Physics 35, 36-65. [5] Wu, T., Yao-TSU., 96. A Wake Model for Free- Streamline Flow Theory Part I. Fully and Partially Developed Wake Flows and Cavity Flows past an Oblique Flat Plate. Journal of Fluid Mechanics 3(), 6-8. [6] Wu, T., Yao-TSU., Wang, D. P., 964. A wake model for free-streamline flow theory. Part. Cavity flows past obstacles of arbitrary profile. Journal of Fluid Mechanics 8(), 65-93. 8 Copyright by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 9//3 Terms of Use: http://asme.org/terms