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A parametric study of rigid body-viscous flow interaction Moorty, Shashi 1987

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A PARAMETRIC STUDY OF RIGID BODY-VISCOUS FLOW INTERACTION by SHASHI MOORTY B.Tech., Indian Institute of Technology, Kanpur, 1985 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Civil Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May 1987 © Shashi Moorty 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of C^iL. The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date MAV h r1 Abstract This thesis presents the numerical solution for two-dimensional incompressible viscous flow over a rigid bluff body which is elastically supported or alternately undergoing a specified harmonic oscillations. Solutions for the related associate flow in which the body is at rest in a two-dimensional incompressible time-dependent viscous flow have also been -obtained. This work is an extension of the work by Pattani [19] to include the effect of a steady far field flow on an oscillating body. The numerical model utilizes the finite element method based on a velocity-pressure primitive variable representation of the complete Navier-Stokes equations. Curved isopara metric elements with quadratic interpolation for velocities and bilinear interpolation for pressure are used. Nonlinear boundary conditions on the moving body are represented to the first order in the body amplitude parameter. The method of averaging is used to obtain the resulting periodic motion of the fluid. Three non-dimensional parameters are used to completely characterise the flow problem: the frequency Reynolds number Ru , the Reynolds number of steady flow Rei and the Reynolds number for time-dependent flow Re2 • Numerical results are obtained for a circular body, a square body and an equilateral triangular body. A parametric study is conducted for different values of the Reynolds numbers in the viscous flow regime. In all cases, results are obtained for streamlines, streaklines, added mass, added damping, added force and the drag coefficients. The lim iting cases of steady flow over a fixed body and an oscillating body in a stationary fluid are checked with known results. Results for the associated flow are also obtained. The transformations derived, between the two associated flows are checked. Good agreement is obtained between the present results and other known results. iii iv Contents Abstract iii List of Figures viList of Tables ix Acknowledgements x 1. Introduction and Literature Survey 1 1.1 General Remarks 1 1.2 Fluid-Structure Coupling 2 1.3 Comparison of the two Associated Flows 4 1.4 Present Investigations 4 2. Derivation of Governing Equations 6 2.1 General Remarks 6 2.2 Conservation Equations and Boundary Conditions 6 2.3 Comparison of the Two Associated Flows 9 2.4 Non-dimensional Form of Governing Equations 13 3. Finite Element Formulation of the Navier -Stokes Equation .... 17, 3.1 General Remarks 17 3.2 Restricted Variational Principle 13.3 The Discretised Form 8 3.4 Boundary Conditions 20 3.5 Steady State Solution 4 3.6 Finite Element Formulation for the Associated Flow 25 4. Characteristics of Fluid-Structure Interaction 29 4.1 General Remarks 24.2 Computation of the Body Boundary Velocity 29 4.3 Determination of Stream functions, Streamlines and Streaklines .... 30 V 4.3.1 General Remarks 30 4.3.2 Finite Element Representation of the Poisson Equation 32 4.4 Determination of Force Characteristics 33 4.4.1 Computation of Added Mass, Added Damping and Added Force . 33 4.4.2 Determination of Fluid Forces on the Bluff Body 35 4.4.3 Force Characteristics of the Associated Flow 8 4.5 The Morison Equation 40 5. Numerical Investigations 2 5.1 Introduction5.2 Result for a Circular Body 43 5.2.1 General Remarks5.2.2 Finite Element Grid and Boundary Conditions 44 5.2.3 Flow Results 46 5.2.4 Added Mass, 'Added Damping and Added Force 61 5.3 Results for a Square Body 67 5.3.1 General Remarks5.3.2 Finite Element Grid and Boundary Conditions 70 5.3.3 Flow Results 75.3.4 Added Mass, Added Damping and Added Force 80 5.4 Results for a Triangular Body 83 5.4.1 General Remarks5.4.2 Finite Element Grid and Boundary Conditions 86 5.4.3 Flow Results 87 5.4.4 Added Mass, Added Damping and Added Force 90 5.5 Case a Associated Flow 95.5.1 General Remarks 7 5.5.2 Finite Element Grid and Boundary Conditions 99 5.5.3 Flow Results 95.5.4 Added Mass, Added Damping and Added Force 102 5.5.5 Drag and Inertia Coefficients for the Morison Equation .... 106 vi 6. Conclusions 114 6.1 Concluding Remarks 116.2 Suggestions for Further Development 115 References . 116 Appendix A 9 Appendix B 120 Appendix C 2 Appendix D 3 Appendix E 126 vii Figures Figure 2.1 Problem Configuration 9 Figure 2.2 The Two Associated Flows 10 Figure 3.1 Isoparametric Element Used in Present Study 19 Figure 4.1 Stress Components 36 Figure 5.1 Finite element Grids for a Circular Body 4Figure 5.2 Limiting Case of an Oscillating Circular Body in Still Fluid R„ =250.0 R€l =0.0 Re2 =20.0 47 Figure 5.3 Limiting Case of Steady Flow over a Circular Body. Ru =0.0 Rei =20.0 Re, =0.0 48 Figure 5.4 Limiting Case of Steady Flow over a Circular Body. Ru =0.0 RCl =70.0 Re2 =0.0 48 Figure 5.5 Pressure Distribution along the Circular body Wall Ru =0.0 Rei =20.0 Re2 =0.0 49 Figure 5.6 Group l.a set of results for a Circular Body 53 Figure 5.7 Group l.b Set of Results for a Circular Body 5 Figure 5.8 Group 2 Set of Results for a Circular Body 56 Figure 5.9 Group 3 Set of Results for a Circular Body 8 Figure 5.10 Streaklines for a Circular Body Ru =250.0 RBi =20.0 Re2 =20.0 62 Figure 5.11 Streaklines for a Circular Body Ru =250.0 Rei =2.0 Re2 =20.0 3 Figure 5.12 Variation of Force Quantities with J?e,/i2ei in case l.a for a Circular Body 67 Figure 5.13 Variation of Force Quantities with Re2/Rei in Group l.b for a Circular Body 8 Figure 5.14 Variation of Force Quantities with /?2 in Group 2 69 Figure 5.15 Finite Element Grid for a Square Body 71 Figure 5.16 Group l.a Set of Results for a Square Body 4 Figure 5.17 Group l.b Set of Results for a Square Body 75 Figure 5.18 Group 2 Set of Results for a Square Body 7 viii Figure 5.19 Group 3 Set of Results for a Square Body 78 Figure 5.20 Streaklines for Ru =150.0 Rei =1.0 R£2 =20.0 for a Square Body 81 Figure 5.21 Variation of Force Quantities with Re^/Rei in Group l.b for a Square Body 83 Figure 5.22 Variation of Force Quantities with j32 in Group 2 for a Square Body . 84 Figure 5.23 Finite Element Grid for a Triangular Body 87 Figure 5.24 Limiting Case of an Oscillating Triangular Body in Still Fluid. R„ =156.0 Rei =0.0 R£2 =3.712 88 Figure 5.25 Group l.a Set of Results for a Triangular Body 92 Figure 5.26 Group l.b Set of Results for a Triangular Body 3 Figure 5.27 Group 2 Set of Results for a Triangular Body 95 Figure 5.28 Group 3 Set of Results for a Triangular BodyFigure 5.29 Streaklines for Ru =150.0 R€l =1.0 Re2 =20.0 for a Triangular Body 97 Figure 5.30 Finite Element Grids for Case a Flow problem 100 Figure 5.31 Streamline Plots for Case a and Case b for R„ =21.34 R£l =0.0 Re2 =0.6402 102 Figure 5.32 Streamline Plots for Case a and Case b for Ru =21.34 Res =0.0 Re2 =0.2134 10Figure 5.33 Streamline Plots for Case a and Case b for Rw =250.0 RCi =20.0 Re2 =20.0 103 Figure 5.34 Drag Coefficients vs Time for > •Ru, =800.0 Rei =40.0 Re2 =4.0 106' Figure 5.35 Streaklines for Ru =800.0 RCl =40.0 Ri2 =4.0 for case a and case b 107 Figure 5.36 Drag Coefficients vs time for Ru =30.0 R£l =40.0 Re2 =4.0 108 Figure 5.37 Streaklines for Ru =30.0 Rei =40.0 Rt2 =4.0 for case a and case b 109 Figure 5.38 Variation of Cd(l) and Cd(2) with Re2/Rei 112 Figure 5.39 Variation of C_(l) and Cd(2) with j32 3 ix Tables Table 5.1 Drag Coefficients for Steady flow over a Circular Body 50 Table 5.2 Parametric Study of the Flow Patterns for a Circular Body 51 Table 5.3 Stream Function Values of the Steady Component of the Velocity Field for a Circular Body 59 Table 5.4 Stream Function Values of the Total Velocity Field at Different Times t for a Circular Body 64 Table 5.5 Added Mass, Added Damping and Added Force for a Circular Body 65 Table 5.6 Parametric Study of the Flow Pattern for a Square Body 72 Table 5.7 Stream Function Values for the Steady Component of the velocity field for a square body 79 Table 5.8 Stream Function Values of the Total Velocity Field at Different Times t for a square body 80 Table 5.9 Added Mass, Added Damping and Added Force for a Square Body 85 Table 5.10 Parametric Study of the Flow Pattern for a Triangular Body .... 89 Table 5.11 Stream Function Values of the Steady Component of the Velocity Field for a Triangular Body 91 Table 5.12 Added Mass, Added Damping and Added Force for a Triangular Body 98>„• Table 5.13 i Stream Function Values for Case a and Case b 104 Table 5.14 Added mass-Inertia Force, Added Damping and Added Force for Case a and Case b 10Table 5.15 Inertia and Drag Coefficients for a Circular Body Ill X Acknowledgements My special thanks to my supervisor Dr.M.D. Olson, for his guidance and encouragement throughout the course of my research work and in the preparation of this thesis. Also my sincere thanks to Dr. Paresh Pattani for his support and many valuable suggestions. I finally wish to thank my sister Kamala and my brother-in-law Sudhakar for their encouragement and patience throughout my graduate studies. Financial support in the form of a University Graduate Fellowship from The University of British Columbia is gratefully acknowledged. CHAPTER 1 Introduction and Literature Survey 1.1 General Remarks The subject of bluff body flows has been receiving a great deal of attention recently. This is due in large part to its importance in design of offshore structures. In spite of the importance of bluff body flows, relatively little is known about them. The vortex shedding characteristics of even the simplest of bodies like circular and rectangular cylinders are not well understood. For most applications of practical interest, the forces acting on a structural section in a two-dimensional flow are given by the so-called Morison equation. This is an empirical equation but is extensively used in view of the fact that even in a steady, two-dimensional, separated flow past a smooth cylinder, one does not have a theoretical or numerical solution which explains all the characteristics of the flow as a function of Reynolds number. Many investigators have applied numerical methods to the Morison equation. Many fluid-structure interaction problems are of such complexity that the method of analysis must be numerical in nature. Although the theory of boundary layer permits us to use the influence of viscosity in many practically important cases, nevertheless some phenomena, important in practice, are not described by boundary layer theory. Among them must firstly be mentioned, flows with separation. This phenomenon occurs sometimes in flows around ships, planes or rockets, in spite of all possible means being used to avoid it. Solutions for incompressible real viscous fluid flow problems involve the complete Navier-Stokes equations. Two properties of the Navier-Stokes equations are the main 1 Chapter 1: Introduction 2 source of difficulty in their numerical solution: 1. The presence of nonlinear terms resulting in nonsymmetric convection operators. 2. The unbalanced domain of the solution, together with the elliptical character of the equations. A variety of efficient numerical methods are now available for the solution of 2D and 3D Navier-Stokes equations using finite difference or finite element methods [4]. These methods however are still only reliable at fairly modest Reynolds numbers. In fact, modern computers are not yet sufficiently effective for solving the Navier-Stokes equations in the fully developed turbulence domain. The finite element method is relatively recent as compared to the finite difference method but is gaining popularity due to its ability to easily model complex boundary geometry. Development of research in applying finite element and finite difference methods to the Navier-Stokes equations has progressed in three basic areas according to the variables used namely, 1 The velocity-pressure primitive variables u, v, p. 2 The stream function \J/ and the vorticity £. 3 Stream function \J/ alone. Few investigators have applied finite element methods to viscous fluid-structure inter action problems. Olson [17] has presented comparison between these methods in finite elements. Others like Liu [12] and Hughes [9] have adapted finite element methods for var ious problem configurations. Pattani [18, 19, 16] has conducted numerical investigations for the flow around an elastically supported rigid body. Excellent results have been ob tained by Davis and Moore [2, 3] for steady and unsteady flow around squares for Reynolds numbers between 100 and 2800 using finite difference techniques. Roache's book, [22] gives a detailed analysis of the various finite difference techniques. 1.2 Fluid-Structure Coupling Various numerical schemes exist for predicting the forces and the details of the flow around cylindrical bluff bodies in both uniform and oscillatory flow at high Reynolds nam-Chapter 1: Introduction 3 bers using the Morison equation [24]. The Euromech Colloquium, held in 1980 [8] aimed to bring together research work concerning bluff body flows in a unidirectional stream and those concerning the flow around bluff bodies in an oscillatory free stream. Investigations of the in-line oscillations of bluff bodies began in earnest following the troublesome and sometimes damaging vortex induced oscillations of pilings during the construction of an oil terminal on the Humber Estuary in England during 1960 [20]. Tanida and Okajima [26], conducted experiments for a circular cylinder oscillating in a uniform flow at low and high Reynolds numbers. They measured the lift and drag coefficients as well as determined the stability of the cylinder at different oscillating frequencies. Bertelsen [l], conducted exper iments to study the steady streaming induced by an oscillating cylinder in an otherwise still fluid at high values of Reynolds number associated with steady streaming. Considerable work has been done in recent years to develop numerical solutions for the 2D flow around bluff bodies using finite element and finite difference techniques. Re sults of acceptable accuracy have been obtained for flows and forces on fixed bodies for a finite range of Reynolds numbers. Goddard, [7] carried out a numerical analysis of the drag response of a cylinder to streamwise fluctuations. This work cannot be generalised to higher Reynolds numbers as it is based on the Navier-Stokes equations and since the diffusion of vorticity in the concentrated vortices for Reynolds numbers larger than about 200 is primarily turbulent. Olson and Pattani [16, 18, 19] addressed the problem of the flow around a rigid body which is elastically supported and obtained solutions using a finite element method based on a velocity-pressure primitive variables representation of; the complete Navier-Stokes equations. Various investigators, [21, 5] have applied the tech niques of higher order boundary layer theory to study the steady streaming induced by an oscillating cylinder in a still fluid. In order to solve the governing equations of the fluid-structure system simultaneously, suitable conditions must be prescribed along fluid-solid interfaces to match the velocities, tangential and normal to the solid body without gross distortion of the finite element grid. Hughes et al proposed a mixed Lagrangian-Eulerian approach to achieve this. In this method, each degree of freedom may be assigned to move at a fraction of the fluid particle velocity. Pattani and Olson successfully showed that it is possible to keep the finite element grid fixed, but allow the body to move past the grid. The relevant boundary terms Chapter 1: Introduction 4 were expanded by the Taylor series to approximate the velocities at the finite element grid points. The time-dependent character of the Navier-Stokes equation was taken care of by considering the steady state periodic solution. Steady state behaviour of forced oscillations of nonlinear systems can be determined by two basic kinds of techniques. 1 Method of Averaging and Multiple Scales. 2 Lindstedt-Poincare Techniques. In the present investigations, the method of averaging has been adapted as outlined by Pattani [18]. 1.3 Comparison of the two Associated Flows Wave flow past a bluff body is similar to a two-dimensional harmonic flow past a similar section. Thus waves can be modelled by harmonically oscillating fluid. Isaacson [10] has further indicated that the solution of harmonic flow past a bluff body is closely related to the solution of a harmonically oscillating body in an otherwise still fluid. This in turn is related to the solution of a rigid bluff body which is elastically supported and subjected to an external harmonic forcing function in an otherwise still fluid. Pattani [18] has modelled an elastically supported bluff body subjected to wave flow by considering a problem configuration of an oscillating body in an otherwise still fluid. He has given an excellent comparision between the two associated flows and derived a transformation from one flow to the other. In the present study, a similar transformation is used to model a combined flow of current and waves over bluff bodies. 1.4 Present Investigations This thesis extends the investigations conducted by Pattani [18] in the modelling of the interaction of viscous fluid flow over a moving body using the finite element method. He considered the case of a solid body oscillating in an otherwise still fluid. The present study considers the problem configuration of an elastically supported rigid body in a two-dimensional incompressible steady flow. Chapter 1: Introduction 5 In chapter 2, the fundamentals of fluid-structure interaction are discussed. The Navier-Stokes equations and continuity equation are derived and represented by the u, v, p primitive variables. A comparison of the two associated flows and the related transfor mations are derived. Suitable sets of non-dimensional parameters are used to effectively non-dimensionalise the governing equations. In chapter 3, the finite element formulation of the Navier-Stokes equations is discussed. The restricted variational principle for the governing equations is presented and the finite element matrix equations are obtained for a suitable choice of element. Boundary condi tions for the outer fluid boundary and the body-fluid interface are discussed and incorpo rated into the matrix equations. The method of slowly varying amplitudes as formulated by Pattani [18], is then used to obtain a steady state periodic solution. A Newton-Raphson iteration scheme for the solution of the resulting nonlinear algebraic equations, is outlined. Finite element representation of the other associated flow is also presented. The finite element flow results can be visualised by plotting the streamlines or streak lines. In chapter 4, a finite element scheme is outlined to obtain the stream functions from the flow results and thus obtain the streamlines and streaklines. The force on a bluff body oscillating in line with a steady fluid flow can be characterised by the drag coefficients or the added mass, added damping and added force. A numerical integration scheme to de termine these force characteristics is outlined. Force characteristics for the two associated flows and the transformation from one to the other is also presented. Finally, the Morison equation and the representation of the drag and inertia coefficients in terms of the added mass, added damping and the added force is presented. The numerical investigations are described in chapter 5. Numerical results are obtained for three different body shapes respectively. 1. A circular body oscillating in the direction of flow. 2. A square body oscillating parallel to one of its sides in the direction of flow. 3. An equilateral triangular shaped body oscillating parallel to one of its bisectors in the direction of flow. In all the cases, the results are presented in the form of streamlines streakline, added mass, added damping and added force. Results for the associated flow of a time-dependent flow over a stationary bluff body is also presented. The transformation between the two associated flows derived in chapters 3 and 4 are verified. Conclusions and suggestions for further development are presented in chapter 6. CHAPTER 2 Derivation of Governing Equations 2.1 General Remarks In this chapter, the equations governing two-dimensional incompressible fluid flow are derived and presented in terms of velocity-pressure primitive variables. A structural section in a two-dimensional flow can be represented as a spring mass system. The equation of motion of such an elastically supported single degree of freedom system is presented. Isaacson has indicated that there are strong similarities between the wave flow past a section of a structural member and a two-dimensional harmonic flow past a similar sec tion. Also, the problem configuration of a stationary body subjected to a time-dependent flow is interrelated to that of a body undergoing time-dependent motion in an otherwise still fluid. One configuration can be transformed to the other by simple reference frame transformations. Pattani has considered the wave flow past a body by considering an elastically sup ported rigid body of arbitrary shape undergoing specified harmonic oscillations in an other wise still fluid. In this study, a two-dimensional flow consisting of waves and currents past an elastically supported rigid body is considered. The governing equations are presented in their final non-dimensional form. 2.2 Conservation Equations and Boundary Conditions The equations governing two-dimensional, incompressible viscous fluid flow are the 6 Chapter 2: Governing Equations 7 Navier-Stokes equations, the continuity equations and the boundary and initial conditions. The solution of the problem will be sought within a plane domain Cl which is bounded by a contour T. T is composed of two distinct parts :- Tu and Ta respectively. Tu is the kinematic boundary where boundary conditions on the velocity components are specified. T8 is the natural boundary where tractions are specified. The equations of motion and boundary conditions are given in the (x,y) cartesian coordinate system. Equations of Motion:-(2.1) Constitutive Relations: -p + 2fx du dx dv a y P + 2p dy du (2.2) u,v are fluid velocities in the x and y directions respectively. p is the fluid pressure. a oy are normal stresses in the x and y directions respectively. r is the shear stress. p is the fluid density, p is the absolute viscosity. Dt d_ dt + u d_ dx + v dy d D j Dt is the total material derivative. Chapter 2: Governing Equations 8 Substituting equation (2.2) into (2.1), we get the well known Navier-Stokes equations. Dt p dx \ dx2 dy2 dxdy J Dv I dp (d2v nd2v d2u \ ~Dt I  d2v 2= + v\ V 2 1 ) p dy \dx2 dy2 dxdy J where u — p/p is the kinematic viscosity. Continuity Equation:-du dv dx dy Boundary Conditions:-(z,y)en, t>0 (2.3) (2.4) {x,y)eTs, t>0 (2.5a) (2.56) u = U, v = V (x,y)eTu, t > 0 oxn\ 4- Txyn2 = X Txyrt\ + oyni — Y ni, n2 are the direction cosines of the outward pointing normal to the boundary. AT, Y are the specified tractions on Ts. Substituting equations (2.2) into equations (2.5b), we get the boundary conditions in terms of the velocity components. Kinematic Boundary Conditions:-u = «V, v = V (x,y)£Tu, t>0 (2.6a) Natural Boundary Conditions du -p + 2p dx /du dv\ \dy dx) ni -f du dv dy dx) -p + 2p dv dy n2 = X n2 = Y {x,y)eTs, t>0 (2.66) The equations of motion of an elastically supported single degree of freedom rigid body as shown in figure 2.1 is given by ms + ks = f[t) (2.7) Chapter 2: Governing Equations 9 8 _ Fluid Domain Figure 2.1 Problem Configuration m is the mass of the body. k is the elastic spring constant 5 is the displacement of body f{t) = Fj(t) + Fe(t) =Total loading force. Fj(t) is the fluid force. Fe(t) is the external force on the body. Chapter 2: Governing Equations 10 2.3 Comparison of the Two Associated Flows It is sought to model the combined flow of current and waves over a structural section. This is similar to a two dimensional steady flow and a harmonic flow past a similar section [23]. The solution of this reference flow is closely related to the solution of steady flow over a harmonically oscillating structural member. Consider an associated pair of two-dimensional flows as defined in figure 2.2. -t> Xo Case a Case b Figure 2.2 The Two Associated Flows Case a: Fluid remote from the stationary body has two components of velocity in the x direction:-U3 is the steady component of velocity. Up(t) is the uniform time-dependent component of velocity. Inertial coordinate system (xa,ya) is fixed with respect to the body. All flow quantities are denoted by subscript a. Case b: Inertial coordinate system (xc,yc) is stationary. Fluid remote from the body has steady velocity U3. The body has a velocity —Up(t) with respect to the inertial Chapter 2: Governing Equations 11 reference frame. All flow quantities in the inertial reference frame are denoted by subscript c. Non-inertial coordinate system (ib,yb) is fixed with respect to the body. All flow quantities in the non-inertial reference frame are denoted by subscript 6. Case a: Equations of Motion and Continuity:-Dua ldpa fnd2ua d2ua d2va \ = \-1/ j 2 1 1 I Dt pdxa \ dx\ dyl dxadyaJ Dva \dpa (d2va d2va d2ua \ = (- v { (- 2 1 ) Dt pdya \dx2 dy2 dxadyaJ dua + dVa=Q (2-8) dxa dya where D_ _ d_ d d Dt dt adxa adya Boundary Conditions:-ua — 0, va = 0 On body-fluid interface ua — Ua + Up(t), va = 0 On fluid outer boundary Case b: Equations of motion and continuity in the (xc,yc) system:-1 Duc 1 dpc ( d2uc d2uc d2vc + v[ 2—^ + —r + Dt pdxc \ dx2 dy2 dxcdyc Dvc 1 dpc (d2vc d2vc d2uc — 1_ i j_ 2 1 Dt p dyc \ dx2 dy2 dxcdyc duc dvc (2.9) + —- = 0 dxc dyc where D d d d — = (- uc f- vc  Dt dt dxc dyc Boundary Conditions: uc = — Up(t), vc = 0 On body-fluid interface Chapter 2: Governing Equations 12 uc = Us, vc — 0 On fluid outer boundary Changing from reference frame (xc,yc) to reference frame (ib,yb)> we get the following transformations:-ub — uc + Up(t) Vb = vc Pb = Pc (2.10) xb-xc-+ / t/p(r) Jo Vb = Vc The absolute acceleration of a fluid particle, in the non-inertial coordinate system [xb,yb), moving with velocity — Up(t) relative to the inertial coordinate system (xc,yc) is given by Pattani [18] Duc Dub dUp(t) ~DT = ~DT~ (2.N) Dvc _ Dvb ~Dt ~ ~Dt Substituting equations (2.10) and (2.11) into equation (2.9), and rearranging terms, we get Equations of motion and continuity in (xb,yb) system:-Dub 1 (dpb dUp(t)\ fd2ub d2ub d2vh ~ P—7.— + v 2—^- + ——r + Dt p \dxb dt J \ dx2 dy2 dibdyb ;, Dvb _ Idpb | /d2v6 t 2d2vb | d2ub \ Dt p dyb \ dxi dy\ dxbdyb) dub dvb _ Q (2.12) dxb dyb where D d 8 d Dt at dxb dyb Boundary Conditions:-Ub = 0, Vb = 0 On body-fluid interface Ub = Us + Up(t), Vb — 0 On fluid outer boundary Chapter 2: Governing Equations 13 Comparing equations (2.8) with equations (2.12), we observe that the governing equations of motion, continuity and boundary conditions are the same. Thus assuming a unique solution for all time t, we get ua = ub dPa _ dp± _ dUp(t) (2.13) dxa dxi> dt dj>a _ dpb dya dyb Substitute equation (2.10) into equation (2.13) and get ua - uc + Up(t) Va-Vc dPa _ dPc _ dUp(t) (2.14) dxa dxc dt dpa _ dpc dya dyc Thus we see that the flow quantities and pressure in case a and case b are interrelated by simple reference transformations. In order to model a combination of steady and harmonic flow past a body as in case a, it is possible to model steady flow past a harmonically oscillating body as in case b and then use the appropriate transformations to get the solution for case a. 2.4 Non-dimensional Form of Governing Equations Consider a fluid with a velocity u3 + upcosut flowing over a body of arbitrary shape with a characteristic length b. Ua represents steady flow and up cos ut represents uniform harmonic flow with an oscillating frequency of u. The simple harmonic displacement of the fluid is given by s = s0 sin ut \Ue\ = us (2.15) WP\ = UP Chapter 2: Governing Equations 14 where s0 is the displacement amplitude such that up = us0 The problem can be transformed to that of case b where fluid with steady velocity us is flowing over a body of arbitrary shape with a characteristic length b undergoing uniform harmonic motion with a velocity of — upcoswt. u is the frequency of oscillations of the body. The simple harmonic displacement of the body is given by equation (2.15) Introduce the non-dimensional variables: i x i V i x — — y = T t = ut b 9 b , U , V , u = — v = — p u o where U0 — Ug + Up pu0jb (2.16) into equations (2.3) and (2.4), we get the non-dimensional form of the Navier-Stokes equa tions and the continuity equation. ^ du „ / du du\ d2u d2u d2v dp dt \ dx dy J dx2 dy2 dxdy dx R ^V + R (u^V + v^V\ — ^V + 2^V + ^U ^P (2 17) dt \ dx dy J dx2 dy2 dxdy dy du dv dx dy All quantities are their respective non-dimensional values and the primes have been omitted for convenience. Ru = ^~ ~ frequency Reynolds number Re — Rei + Ren i — — Reynolds number for steady flow Re? = = Reynolds number for oscillating flow. We thus have three non-dimensional parameters which completely characterise the flow problem under consideration. These are the three Reynolds numbers: R^, Rei and Ren. It should be noted that the pressure has been non-dimensionalised with respect to the characteristic shear stress rather than the dynamic pressure as it is anticipated that the flow regime would be a slow, shear dominated one. Chapter 2: Governing Equations 15 Introducing equations (2.16) and two additional non-dimensional variables X' Y' = Y pua/b ' pua/b into equation (2.6), we get the non-dimensional form of boundary conditions Kinematic Boundary Conditions:-u = U, v = V (x,y) € Tu, t > 0 (2.18a) Natural Boundary Conditions du -p + 2 dx ni + du dv dy dx (du dv \dy dx ni + -p + 2 dv dy n2 = X n2 = Y (x,y)eTs, t>0 (2.186) All quantities are their respective non-dimensional values. The primes have been omitted for convenience. Introducing equations (2.16) and additional non-dimensional variables 2 el 7"., = pu0jb y pua/b xy pu0/b into equations (2.2), we obtain the non-dimensional form of the constitutive relation. Constitutive Relations:--p + 2 du dx dv ay = -p + 2dy (dv du\ Txy =\d~x + dy) (2.19) All quantities are their respective non-dimensional values. The primes have been omitted for convenience. Introduce equations (2.16) and the non-dimensional variables m m pAbl k' u}2pAbl puDl f_ Ff Ft = pu0l Chapter 2: Governing Equations 16 into equations (2.7) Ab is the cross-sectional area of the cylinder body and / is the length of the cylinder. We obtain the non-dimensional form of the equation of motion for a single degree of freedom system. ms + ks = Jb(Fe + Ff) (2.20) where b ~ uHAb " RI \Ab) CHAPTER 3 Finite Element Formulation of the Navier-Stokes Equations 3.1 General Remarks In this chapter, the Navier-Stokes equations and the continuity equation (eqn 2.17) along with the boundary conditions (2.18) are discretised using the finite element method. The moving body boundary conditions are incorporated into the discretised equations which are then written in matrix form. The steady state solution is obtained using a modified method of averaging. Finally a Newton-Raphson iteration scheme is outlined for solving the resulting nonlinear equations. The whole procedure has been explained in detail for case b associated flow as described in section 2.3 in which the body is oscillating in a fluid domain moving with a steady velocity. A brief description of the finite element formulation of case a associated flow is presented later. 3.2 Restricted Variational Principle The starting point in the finite element discretisation is the derivation of the functional form of the governing equations (eqns 2.17, 2.18). The convective nonlinear terms in the complete Navier-Stokes equations are non-self adjoint terms, due to which no variational principle exists corresponding to these equations. Nevertheless, it is possible to construct 17 Chapter 3: Finite Element Formulation 18 a restricted variational principle of the form :-n = (3.1) The governing equations (eqn. 2.17 and 2.18) can be retrieved by taking the first variation of II as shown in appendix A. This is identical to the Galerkin method but allows one to think in the usual terms of seeking a stationary point to some functional. The velocities (u°,v°), are associated with the inertial terms and are held constant while taking the first variation of II. At the end of the process, u°, v° are equated to u, v and there by restoring the governing equations. 3.3 The Discretised Form The highest order of derivative in the functional II is unity. Hence an element with only C° continuity is required for the interpolation of each of the (u,v,p) variables. Olson [17], has shown that the finite element interpolation for pressure p should be at least one degree less than that for the velocity component u, v. This restriction comes about in order to avoid the spurious singularities for the [u,v,p) integrated formulation. Taking these conditions into consideration, curved isoparametric elements, shown in figure 3.1, with quadratic interpolation for velocities and bilinear interpolation for pressure are used to carry out the finite element discretisation of the Navier-Stokes equations. In isoparametric elements, the shape functions used are the same both for interpolat ing the variables u,v,p and for the transformation from s,t natural coordinates to x, y Chapter 3: Finite Element Formulation t Figure 3.1 Isoparametric element used in present study, (a.) Element in (s,t) space, (b.) Element in (x,y) space coordinates. The velocities and pressures are represented by u = NiUi{t) 1 = 1,2,...,8 v = NiVi{t) » = 1,2,...,8 p = MiVi(i) 1 = 1,...,4 i, Mi are the shape functions and are represented by Nx = 4 N2 = -]{l + s){l-t)[l-s + 4 N3 = -\{l + 8){l+t)(l-8-t) JV4 = --(l-s)(l + i)(l+5-N5 = \(l-s2)(l-t) N6 = \(l + a)(l-ti) N7 = 5(1-0(1+0 JV8 = i(l-5)(l-*2) Mi = \(l-s)(l-t) M2 = + M3 = ^(l+_)(l + t) M4 = + and ut, v,, p, are the time-dependent nodal variables. Chapter 3: Finite Element Formulation 20 Substituting equation 3.2 into the functional II for one element and carrying out the first variation of each variable, yields m 0 0 0 mVV 0 Uj Vj > + Kij Kij L.UV KVV Kji Klj -Pi V U-i Vj 6ijkujuk + &ijkVjUk ) ( 0 + Re <j SfjkUjVk + 6?jkVjVk > = < 0 (3.3) where TIe denotes the variational principle for one element and He denotes the domain of the element under consideration. m ij KVV Klj Kij 6h 2dNidN1 3Nj dNj dx dx dy dy jj R^NiNjdA = mV; IL JJnc oy dy dx dx JJUc dx dy JL.N'N>i£dA *=ILa-£M,iA * fL">^ ij,k = 1,2,...,8. i= 1,2,..., 8 y = i,...,4 The derivation of the finite element dicretised form is presented in appendix B. 3.4 Boundary Conditions For the elements not on the boundary, the line integral in the functional n cancels out between the adjacent elements. For elements on the boundary where u and v are not specified, the line integral becomes the consistant load vector. Chapter 3: Finite Element Formulation 21 At the fluid outer boundary, u velocity is specified to be the far field uniform flow and v velocity is specified to be zero. Along the fluid-body interface, u and v velocities are specified to be those of the moving body. The body is oscillating with an amplitude sQ, a frequency u and a velocity Up(t). s(t) = s0 sin ut Up(t) = Up cos ut (3.4) up- USa The moving body boundary conditions has been modelled by Pattani [18], where he con sidered the finite element grid to be fixed at the mean position of the body and the body moves past the grid. The relevant boundary terms are expanded by Taylor series to obtain the velocities at the mean position of the body at any time t > 0. We thus obtain u(0) = Up cos ut — s0 sin ut ^— ) «,(0) = -ao sin W*(g)o du\ dx)0 (3.5) The subscript 0 indicates that the derivatives are evaluated at 5 = 0, that is at the mean position of the body. Using the non-dimensional parameters as described in chapter 2 and additional parameters, _ Rei us Re2 sa Ru ws„ Rw b the non-dimensionalised form of equation (3.5) is obtained as u(0) = ubcost — /?2 ( ^- J s'mt "(o)=-ft(S)0sb' (3.6) All the quantities are in their respective non-dimensional form and the primes have been omitted for convenience. Uoo is the non-dimensional far field fluid velocity. Chapter 3: Finite Element Formulation 22 u0 is the non-dimensional body velocity. 02 is the body amplitude parameter which also governs the nonlinear convection terms. Equation (3.6) is then discretised by substituting equations (3.2) for nodal variables at the edge of the finite element grid interfacing with the mean position of the body. Thus obtain :-U{ = ub cos t — f32CijUj sin t — foubCik sin t cos t Vi = — 02CijVj s'mt u (3-7) where v ' 1. j is summed over the velocity degrees of freedom other than those on the edge inter facing with the mean position of the body. 2. k is summed over the velocity degrees of freedom on the edge interfacing with the mean position of the body. 3. The subscript i indicates that is evaluated at the location of Uj or Vj, which ever is appropriate. Suppose in a finite element domain, there are net u and v variables numbering n each and net p nodal variables numbering m each. Out of these, the u and v variables each numbering r are located on the fluid-body interface boundary whose values are known from equation (3.7). By suitable matrix manipulation, these variables are segregated and trasformed to the right hand side. This results in the matrix equation of form:-[M + /?2P sin t]d + [K + (/?2P + ReubQ) cos t + /?2R. sin t] d + Re\ sijkujvk + f>ijkvjvk (= ubFsint + ubGcost (3.8) I 0 J + 02ub [Hi cos2 t + J sin2 t] + ReulH2 cos2 t + 02ubL sin t cos t where d = {ui u2 ••• U„-„ ui v2 ••• un_r, Pl p2 ••• pm}T Chapter 3: Finite Element Formulation is the nodal vector of unknowns. M = m 0 0 0 0 vv Q 0 K = J.UU WD -p -Pi y 'ii E rn^dj 0 0 /=n-r+l R = 0 0 n /=n-r+l E k%>C,j l=n-r+l ~ t Pf&j l=n-r+l E mV^Qy 0 ;=n-r+l 0 0 E *„wc/y o l=n-r+l E *,7C,y 0 /=n-r+l ~ E pJJCy 0 /=n-r+l Q = , E (^, + ^y) E % o /=n-r+l Z=n-r+l ' 0 0 E ^ o l=n-r+l 0 0 E m F = < J="-»•+1 0 G = -E *$tt }=n-r+l . E *}? > E j=n-r+l Let q be the net number of u and v degrees of freedom, each at the edge of an element which is at the interface between the viscous fluid and the mean position of the body, then H2 , E m3«£cy l=n—r+l j—i 0 0 It-E E H2 = — ^ J=n-r+l fc=n-r+l 0 0 6?ik Chapter 3: Finite Element Formulation 24 t W ± Cy l-n-r+l j=l l=n-r+l j=l ~ t Pfi t Cy J=n-r+l j'=l 3.5 Steady State Solution Due to the quadratic nonlinear terms in the Navier-Stokes equations, the velocities have a steady component as well as a time-dependent component. Pattani has modified the method of averaging in order to obtain the steady streaming part of the solution from equation (3.8) which governs the fluid-structure problem under consideration. The same procedure is applied to obtain the steady state solution to equation (3.8). The starting point in the method of averaging is to assume a form of solution as d = A + B cos t + Cs'int (3.9) where B(<), C(t) are assumed to be slowly varying functions of non-dimensional time t. Equation (3.9) is substitued into equation (3.8) and method of averaging is applied. The resulting equation is presented in equation 3.10. Chapter 3: Finite Element Formulation 25 0 0 0 M 0 0 0 M K /?2P + ReUbQ &R f R M K -M K f 6fik{A)Al + BfBi/2 + C»C%/2) + 6?jk(A]Al + B]B%/2 + C;C£/2) ) 0 + Re < 6fik(AJBZ + A%B?) + 6fjk{AVBZ + A"kB?) 0 0 ^H2 UfcG (3.10) where A = B = The steady state solution corresponds to the singular points of the autonomous system of equations when B = C = 0. This results in a set of nonlinear algebraic equations for A,B,C which are solved using Newton-Raphson iteration procedure. The equations obtained for the increments to the solution vector, following the iterative scheme described by Pattani, are of the form [r]{Ax} = {-/} (3.11) [T\ is the tangent stiffness matrix. {Ax} is the incremental solution vector. { —/} is the unbalanced load vector. The details of these matrices are given in appendix C. Chapter 3: Finite Element Formulation 26 3.6 Finite Element Formulation for the Associated Flow So far in this chapter, the finite element representation of case b as described in section 2.3 has been derived. The finite element representation of case a is similar and in fact simpler than that of case b. In this section, the finite element formulation of case a is presented. The fluid remote from the stationary body has a velocity of U0 in the x direction and is given by U0 = Us + Up (3.12) \U0\ = ue + up Us is the steady component of the fluid velocity. Up = up cos ut is the periodic component of the fluid velocity. The transformation from reference frame (xa,ya) to reference frame (xc,yc), as given in equation 2.14 can be written as ua = uc -f up cos ut va = vc dpa dpc , . . (3 13) ~— = -z (- puup sin ut {o.xof dpa dPc dya dyc Using the non-dimensional variables introduced in sections 2.4 and 3.4, the non-dimensional form of equation 3.13 can be expressed as ua = Uc + U0 COS t dpa dPc i3U\ oxa oxc dpa dpc dya dyc All quantities are in their respective non-dimensional form and the primes have been omitted for convenience. Integrating the pressure terms in equation 3.14, the final trans-Chapter 3: Finite Element Formulation 27 formation for u, v, p is obtained. Ua = Uc + life cost ua = vc (3.15) Pa = Pc + Ru)XaUb sin /; + k k is the constant of integration. When pa at a point (xa„j J/o„) m tne finite element domain is assigned a specified value, say zero, then the value of A; = — R^x^Ubsmt. Thus the value of pa at a point (xa,ya) is Pa = Pc + Ru{Xa - Xa„)ub sin t The variational principle IT presented in equation 3.1 is the same in case a as in case b. The discretised form of II is identical to case b and is given by equation 3.3. In case a, no moving body boundary conditions need to be incorporated which makes the formulation simpler. Apply the method of averaging to equation 3.3 as described in section 3.5, and obtain the resulting equations in matrix form as "o 0 0 " K 0 0 " 0 M 0 0 K M 0 0 M 0 -M K + Re < f 6?ik[A?Al + B«B%/2 + C-CtJD + &>ik{A)A\ + B]B%/2 + CJC^/2) ) ^M^l + B?Bi/2 + CfCl/2) + 6?jk(A»Al + _?v_?«/2 + C/C^/2) 0 *i% + Al*") + + AiBj) *fik(AiBVk + AlBi) + 8UAVJBk + AtBj) 0 ^MlCt + AtC]) + ^jk{A)Ct + AtC]) ^k(Aich + Al^) + ^M^l + Alci) 0 = <_> I 0 ) (3.16) Chapter 3: Finite Element Formulation 28 This equation is similar to equation 3.10, except that the matrices P, Q, R, F, G, Hi, H2 and J which are associated with the moving body boundary conditions, are zero. The nodes at the outer boundary of the fluid domain have specified u and v velocities and are imposed as constraint equations in the formulation. The set of nonlinear equations are solved using Newton-Raphson iteration scheme just as described for case b. The tangent stiffness matrix and load vector is identical to that given in appendix C, except that P = Q = R = G = F = H2 = 0. CHAPTER 4 Characteristics of Fluid-Structure Interaction 4.1 General Remarks Once the flow quantitites such as the velocity and pressure are obtained for a fluid domain by the procedure outlined in chapter 3, various other characteristics of the flow can be determined. Flow characteristics like streamlines, streaklines and forces on the rigid bluff body are discussed and procedures to obtain them are outlined in this chapter. The first step in these procedures is to determine the velocities at the mean position of the body. The overall view of the flow pattern at any instant of time can be obtained by plotting the velocity vectors or the streaklines in the fluid domain. The basic nonlinear phenomenon of steady streaming is obtained by plotting the streamlines of the steady component of the velocity field. The forces on a rigid body is characterised by the concept of the drag coefficients or the added mass, added damping and added force. The force quantities can be obtained for one case of associated flow and the results can be transformed to the other case of associated flow using simple coordinate transformations. 4.2 Computation of the Body Boundary Velocity In obtaining the numerical solution of the complete Navier-Stokes equations, we made 29 Chapter 4: Characteristics of Flow 30 the assumption that the fluid flow is periodic and the flow results can be represented as Thus the velocity at any node in the fluid domain other than at the mean position of the body can be represented as For an element on the body boundary, substitute equation 4.1 into equation 3.7 to obtain the nodal velocities at the mean position of the body. The resulting equations are j is summed over the velocity degrees of freedom other than those on the edge inter facing with the mean position of the body. k is summed over the velocity degrees of freedom on the edge interfacing with the mean position of the body. At i = 0, when the body is at its mean position, the nodal velocities correspond to the body velocity. Neglecting higher harmonics, we can represent velocity at the mean position of the body as in equation 4.1. In this case d = A + B cos t + C sin t UJ = + Bj cos t + C;u sin t Vj = A) + B] cos t + Cvj sin t (4.1) (4.2) u -faCijAj V (4.3) Chapter 4: Characteristics of Flow 31 4.3 Determination of Stream functions, Streamlines and Streaklines 4.3.1 General Remarks A line in the fluid whose tangent is everywhere parallel to the fluid velocity U0 instan taneously, is a streamline. The family of streamlines at time t are solution of dx dy dz u{x,t) v(x,t) w(x,t) where u, v, w are the components of velocity U0 parallel to the rectilinear axes x, y, z. The path of a material element of fluid does not in general coincide with a streamline, although it does so when the motion is steady. Streakline is that on which lie all those fluid elements that at some earlier instant passed through a certain point of space. Thus, when a dye or some other marking material is discharged slowly at some fixed point in a moving fluid, the visible line produced in the fluid is a streakline. When the flow is steady, streaklines, streamlines and pathlines coincide. The stream function ^ is defined in terms of the velocity components u and v as dy dx The function \l? can also be regarded as the only nonzero component of vector potential for u and can be written as *) = dv_ ,4 4v u(x,t) dx Thus the lines of constant stream function are streamlines or streaklines as the case may be. The stream function can be obtained by solving the Poisson equation _o T du dv , 1. When u and v represent the total velocity field, (i.e the steady as well as the time-dependent components), lines of constant ^ represent streaklines. 2. When u and v represent only the steady flow component A, lines of constant \& represent streamlines. Chapter 4: Characteristics of Flow 32 3. C is the vorticity for the total velocity field or the steady flow conmponent as the case may be. 4.3.2 Finite Element Representation of the Poisson equation The Poisson equation 4.5 can be represented in its functional form as 2 / O.T.\ 2n 2 l — 1 •//0{i[S),+e)i^}"-i^" ^ fl is the domain of the problem. 9is) ~ H~ *s tne tangential velocity specified on the natural part of the boundary C8 The finite element discretisation is done using the same finite element interpolation of \I> as that for velocities described in section 3.3. The stream funtion \j? is represented by i = l,2,3,---,8 (4.7) Ni are the shape functions given in section 3.3. ^ are the nodal variables of stream function. Substituting equation 4.7 into the functional J for one element, the resulting equation is NitXNjiX + NiiyNJ!y -cATitf.J dA- j g{s)Ni*ids (4.8) Take the first variation of equation 4.8 with respect to \]/, and thus minimise the functional I ~i = IL { {Ni'xNj'x + Ni>vN>>»)~ <Ni) dA ~ jc 9^NidS = 0 (4-9) dx l'y dy Ie denotes the functional J for one element. fle denotes the domain of element under consideration. Chapter 4: Characteristics of Flow 33 All such elements are assembled to produce a set of global equations pertaining to the fluid domain. These equations can be written in matrix form as For an element not on the boundary, the line integral in the functional J cancels out between adjacent elements. For elements on the boundary where ^ is not specified, the line integral becomes the consistant load vector. 4.4 Determination of Force Characteristics 4.4.1 Computation of Added Mass, Added Damping and Added Force Consider the fluid-structure problem where a bluff body is oscillating in the direction of flow. This can be represented as a spring-mass system as shown in figure 2.1. 1. MD = Mass of the bluff body. 2. Kb = Stiffness of the bluff body. 3. F€(t) = External force applied to cause the motion 5 of the body. 4. s(t) = s0 sin art = Motion of the body. 5. Us = Steady velocity of fluid remote from the body. 6. Up — Up cosiot = Velocity of body where up = ws„. 7. Fj(t) = Fluid force on the body The fluid forces on a rigid body performing harmonic oscillations will in general consist of three components:- 1. a component in phase with the acceleration of the body, 2. a component in phase with the velocity of the body, 3. a constant component. The components are each associated with the added mass, added damping and added force, [««]{•} = {«} (4.10) Chapter 4: Characteristics of Flow 34 respectively. Added mass is defined as the quotient of the additional force required to produce acceleration throughout the fluid domain divided by the acceleration of the body. The equation of motion of the single degree of freedom is represented in its non-dimensional form as in equation 2.20 Mbs + Kbs = Jb{Fe + Ff) (4.11) where s = /?2 sin t _ usb V D - UPB He2 — V R -ub* V 01 Ru> Ru Rei tie Re* b is the characteristic length of the body. Ab is the cross-sectional area of the body. All variables are in their respective non-dimensional form. The non-dimensional vari ables introduced in section 2.4 and section 3.4 have been used. Substituting the expression for s into equation 4.11, obtain -Mbp2 sin t + Kbp2 sin t = ~ (^j (Fe + Ff) (4.12) We expect the fluid force to be of form Ff = P + Qcost + Rsint P represents the steady drag force on the body. Q and R are the periodic components of fluid force. Substitute equation 4.13 into 4.12 and rearrange terms to obtain an expression for the 4.13) Chapter 4: Characteristics of Flow 35 external force Fe. Fe = -P- (MbRuUbt^j + Rj sin* )A!\ (4-H) + KhRu,Ub ( -j-f J sin t - Q cos t Equation 4.12 can be represented in terms of the added mass Ma, the added damping Ca and the added force Fa. -{Mb + MaYRvuJ^Jsmt UH\ U,\ (4-15) + KbRwub (^J-in« + CaRuub ( -~ j cos t = Fe + Fa Comparing equations 4.14 and 4.15, the expressions for added mass, added damping and added force are obtained R fb2\ Ma = — ( —— 1 = added mass Ru,ub \AbJ Ca — —( — J = added damping (^-16) Rwub\Ab/ Fa = P = added force Appendix E presents the solution procedure to solve the equation of motion in case b flow configuration using Ma, Ca and Fa. 4.4.2 Determination of Fluid Forces on the Bluff Body The normal and shear stress at a point along a plane inclined at an angle a with the x-axis (figure 4.2), can be represented in terms of the stress components ax, oy and TXY as • 2 2 o = ax sin a + o~y cos a — 2rxy sin a cos a * x i ^ • {4'17) r = rxy [sm a — cos a) + (ax — oy) sin a cos a The resultant fluid force along the x direction on a length / of the cylinder due to a and T is given by Ff = —/ <j> (a sin a + r cos a) de (4-18) Chapter 4: Characteristics of Flow 36 7 a x Figure 4.1 Stress Components The integration around the cross-section is done in the counter- clockwise direction. The fluid force has two components: iv and FT. Fff is the normal force which is mainly due to pressure and hence is known as the pressure drag. FT is the tangential force which is the result of shearing stresses and hence is known as the friction drag. The drag force is made non-dimensional by dividing it by O.bpu2 where p is the density of the fluid and u3 is its steady velocity. We thus obtain three drag coefficients: 0.5pu = the pressure drag coefficient Cds T—z = the friction drag coefficient Cdv + Cdj = total drag coefficient Chapter 4: Characteristics of Flow 37 Since at low Reynolds number, flow over bluff bodies is symmetric along the axis normal to the direction of flow, the force along the y axis is zero. That is in our analysis, only the presence of drag force is considered and lift force is neglected. Using the non-dimensional variables z e , o , T e — T' = b pu0jb pua/b the non-dimensional form of equation 4.18 is obtained as 1 pu0l Ff = — j> (a sin a + r cos a) de All variables are in their respective non-dimensional form and primes have been omitted for convenience. F<r = — <j> o sin ade = fluid force due to a FT = — j> r cos ade — fluid force due to r With this non-dimensionalising, the drag coefficients are redefined as: (4.19) Cdf Rei ^oo FT Cd = Cdv + Cdj Rei is the Reynolds number for steady flow and Uoo is the far field fluid veloicty. All the variables are in their respective non-dimensional form and the primes have been omitted for convenience. The velocities and pressures are represented by the finite elements by u = Nim i = 1,2,-•• ,8 v = NiV{ i = 1,2,- --,8 p = MiPi i = !,-••, 4 Ni, Mi are the interpolating functions given in section 3.3. Ui, Vi, pi are the nodal variables. The stress components given by equation 2.19 can be represented in matrix form as (4.20) ~2NitX 0 -Mi' { Ui °y \ = 0 -Mi rxy ) MiiX 0 U Chapter 4: Characteristics of Flow 38 dNj dx dMi dNj dy dMi dx l,y dy Thus equation 4.19 can be written in matrix form as Cl- sin2 a cos2 a -2 sin a cos a • 2 2 sin a cos a — sin a cos a sin a — cos a (4.21) 'xy Equation 4.19 can be written as ?a\__l\ sin3a cos2asina —2sin2acosa 7T j J [sin a cos2 a —sin a cos2 a cos a (sin2 a — cos2 ct) 2NiiX 0 -Mi' ( Ui 0 2Ni<y -Mi de < Vi Mi>x 0 u (4.22) For an isoparametric element, the coordinates of a point in the element are also given by x = NiXi i = 1,2, • • • ,8 y = Niyi »=.l,2,---,8 where Xi and y, are the nodal variables. Thus the derivatives can be obtained as J is termed the Jacobian of the transformation and is given by (4.23) J = where dNj ds "J'L dt Substituting equation 4.23 into equation 4.22 and using two point Gauss quadrature for numerical integration, P, Q, R of Ff is obtained from which Ma, Ca and Fa and the drag coefficients can be calculated. Chapter 4: Characteristics of Flow 39 4.4.3 Force Characteristics of the Associated Flow The force characteristics for the case b where the body is oscillating in-line with a steady fluid flow has been considered. The force quantities for the case a where the body is at rest in an oscillating fluid flow can be easily determined by transforming the results obtained from case b. The transformation from reference frame a to c is obtained from equation 3.15. Sub scripts a and c denote case a and case b flow configurations respectively. The transfor mation for stresses is obtained as 0~xa - 0~xc — RuXaUbSint °ya - °yc - RuXaUbsint (4-24) Txya = Txyc The fluid forces in the two reference frames can be expressed as Ffa = — j'io'a sin a + ra cos a) de Ffe = — <j> (ac sin a + TC COS a) de The transformation for the fluid force in the two associated flows is written as Ffa = Ffc + R*ub (^j sin t (4.25) F/a — Pa + Qa cos t + Ra sin t Fje = Pc + Qc cos t + Rc sin t Thus the transformation for the fluid force components P, Q, R is expressed as Pa = Pc Qa = Qc ,4 26) Ra = Rc + RwUb f —2 The added mass is associated with R, the added damping with Q and the added force is associated with P. The added mass in case b is termed as the inertia force in case Chapter 4: Characteristics of Flow 40 a. Thus the transformation for added mass, added damping and added force for the two associated flows is expressed as Maa = Mac + 1 Ca. = Ca,- (4.27) F = F Maa is the inertia force in case a and Mac is the added mass in case b. From equation 4.27, it is noted that there is no change in the added damping and added force between the two associated flows. Maa is greater than Mac by unity. The transformation for Cap, Ca; and Cd for the two associated flows is expressed as: CdSa = Cdfc (4.28) Cda - Cdc + -~—RujUbl 4^ ) sin* •KeiUoo \° J From equation 4.28, it is evident that there is no change in the friction drag coefficient but there is change in the pressure and total drag coefficients in the sine component. 4.5 The Morison Equation In most applications of practical interest, the fluid forces acting on a body in a two-dimensional flow are determined by the Morison equation.The empirical coefficients used in the equation can be represented in terms of Ma, Ca and Fa. Consider a cylinder of length /, characteristic cross-sectional size b and cross-sectional area At,, in a viscous fluid. The fluid has a velocity U which consists of a steady component Us and an oscillating component Up and is given by U = U8 + UP \U\ = u0 = us + up where us = \US\ and Up = upcosu>t and u is the frequency of oscillations. The fluid force Chapter 4: Characteristics of Flow 41 Ff on the cylinder is given by the Morison equation as ^ = ;\pbCdU\V\ + PAbcJ^ 11 at = -pbCd(us + upcosut)\(u3 + upcosut)\ ~ pAbCmuupsmut (4.29) = FD + Fj is the drag coefficient associated with the drag force FD, and Cm is the inertia coefficient associated with the inertia force Fj. The drag force can be linearised by representing U\U\ in terms of a fourier series. Considering only the first two terms of the series, we obtain FT) 1 — = -pbCdV?o(a0 + ai cosut) (4.30) a0 and ai are constants determined by i?e, and Re2. Hence we can associate two drag coefficients with the drag force on a body. 1. Cd(l) = aDCd = Drag coefficient associated with the steady drag force. 2. Cd(2) = a\Cd = Drag coefficient associated with the oscillating drag force. The linearised form of the Morison equation is written as F 1 -j- = -pbu20\Cd{l) + Cd(2)coswr.] - pAbCmuup sin ut (4.31) From the present investigation, the fluid force is obtained as described in section 4.4 as Ff -j- = pu0[P + Q cos ut + R sin ut) The coefficients used in the Morison equation can be represented in terms of the added mass Ma, added damping Ca and the added force Fa. Crf(l) = J-Fa Ke C-(2) = \ (^-32) Cm = Ma + l where Rei , i?e, , Ru and Re are as defined in section 2.4. CHAPTER 5 Numerical Investigations 5.1 Introduction In this chapter, numerical results are presented for three different body shapes, namely 1. Circular body oscillating in the direction of the flow. 2. Square body oscillating parallel to one of its sides in the direction of the flow. 3. Equilateral triangular body oscillating parallel to one of its bisectors in the direction of the flow. The investigation is carried out in the viscous flow regime up to Reynolds number Rei = 100 with various values of Re2 and Rw, keeping the body amplitude parameter fl2 less than 0.2. The limiting cases of the body oscillating in an otherwise still fluid and that of a steady flow over a fixed body are verified with known results. No experimental or analytical results are available at such low values of Reynolds numbers for the combined problem of a body oscillating in line with a steady fluid flow. Hence a parametric study is conducted for this combined flow problem and is classified into four groups. Group l.a: Rei is kept constant and Re2 and Ru are changed such that 02 is kept constant. The effect of Re2/Rei on the flow pattern is observed. Group l.b: 02 is kept constant by keeping Ru and Re2 constant. Re2/Rei is changed by changing R6l Group 2: The effect of 02 on the flow pattern is observed. Rei and Re^ are kept constant and 02 is changed by changing Ru . 42 Chapter 5: Numerical Investigations 43 Group 3: Effect of Re2 on the flow pattern is observed. Ru and Rei are kept constant. Re2/Rei and 02 are changed due to the change in Re2 . The numerical results are also presented for the associated flow configuration of case a as described in section 2.3. These results are compared with those of case b and the transformations presented in equations 3.15 and 4.27 are checked. The flow results are presented in the form of streamlines and streaklines. Streamlines are the stream function contours of the steady part of the velocity field and streaklines are the stream function contours of the total velocity field at one instant in time. The contour plots are obtained by plotting 31 equally spaced contours between the maximum and minimum values of the stream function in the plotted domain. In order to plot stream function contours, the stream function at each node of the plotting grid is required. A procedure for obtaining these nodal values is outlined in appendix D. The forces on the rigid body are computed in terms of added mass, added damping, added force and the drag coefficients. All the computer programs are implemented on a 48-megabyte Amdahl 5840 system with double accelerator at the University of British Columbia. Double precision arithmetic is used throughout to reduce the effect of round-off errors. The solution of the linear algebraic equations and the matrix inversions are performed using a sparse matrix solving package called SPARSPAK. 5.2 Results for a Circular Body 5.2.1 General Remarks A considerable amount of work has been conducted on an oscillating circular body in a fluid. Bertelsen [l] conducted experiments to study the steady streaming phenomenon in the boundary layer on a cylinder performing simple harmonic motion in a viscous incom pressible fluid at rest. Riley [21] and Stuart tackled the same problem analytically using perturbation techniques to boundary layer theory. The experimental and analytical results are for very high values of R^ and i?e, , of the order of 100,000 and 5,000 respectively. Chapter 5: Numerical Investigations 44 The present numerical model is not accurate at such high values of Ru and Re2 . Pattani [18] has modelled the problem of a circular body oscillating in an otherwise still fluid and has obtained excellent agreement with Tatsuno's experimental results of the same problem configuration [28]. Tatsuno [27] and Tanida [26] have studied the problem of a circular cylinder oscillating in a steady viscous flow. But their experiments are conducted in the flow regime of a well developed vortex street behind the cylinder and at very high values of the body amplitude parameter in the range j32 >0.5. Hence a parametric study is conducted as described in section 5.1 for the combined problem. The limiting case of steady flow over a circular fixed body is verified with the results obtained from Olson [31]. The limiting case of a circular body oscillating in a stationary fluid is verified with the results obtained from Pattani [18]. 5.2.2 Finite Element Grid and Boundary Conditions The numerical results are obtained for a circular body with a diameter b of unity. This diameter is also the characteristic length of the body. The centroid of the body is located at the origin of the x-y axis system and the body is performing oscillations parallel to the x-axis. Using symmetry in the flow problem, only one half of the domain is modelled. Three finite element grids are developed as shown in figure 5.1. Grid 1 and Grid 2 have the same ratio of D/6=15.5, where D is the total length of the flow domain. Grid 3 has £)/6=30.0. Grid 2 has a rectangular outer boundry while Grid 1 and Grid 3 have a circular outer boundary. In finite element formulation, only the kinematic boundary conditions need to be spec ified and the homogenous natural boundary conditions come out as a part of the solution in the limit of grid refinement. Thus the symmetry condition is obtained by letting the velocity v = 0 and the shear stress rxy — 0 (which is implemented by letting the velocity u float, u —1) on the symmetry line Ts. Though there is no symmetry in the pressure distribution, the pressure is specified to be zero at two symmetric locations: points 1 and 2 shown in figure 5.1. The fluid outer boundary is assumed to be far enough so that the pressure at these two points is approximately zero. Negligible change in the numerical Chapter 5: Numerical Investigations 45 results is observed when the pressure is specified to be zero at only one point. Boundary Conditions for Grid 1 and Grid S Grid 1 and 3 have the same velocity and pressure boundary conditions. 1. u = UQO, v = 0 all along the fluid outer boundary. 2. u and v velocities at the body boundary are specified according to equation 3.7. 3. u=?, v=0 all along the symmetry line Ts. 4. p = 0 at points 1 and 2 shown in the figure 5.1a and c. Boundary Condition for Grid 2 1. u = UQO, v = 0 along T,, the inflow boundary. 2. u = UQO, U=? along Tu. 3. «=?, v — 0 along ro, the outflow boundary. 4. u=?, v = 0 along Ts, the symmetry line. 5. p = 0 at points 1 and 2 shown in figure 5.1b. Along ro, the velocity u is allowed to float This implies that the associated stress, ox = — p + p^ is zero. The pressure p — 0 at point 2 and hence is approximately zero all along ro. Thus ^ = 0 all along ro. For the flow problem of a body oscillating in a steady fluid flow, Grid 1 and Grid 3 have 368 net degrees of freedom of which 307 are for the velocities and 61 are for the pressure. Grid 2 has 370 net degrees of freedom of which 309 are for the velocities and 61 are for the pressure. There are three coeffiecients A, B, C for each degree of freedom. This results in 1104 variables in Grid 1 and Grid 3 and 1110 variables in Grid 2. er 5: Numerical Investigations 46 Chapter 5: Numerical Investigations 47 5.2.3 Flow Results Limiting Cases:-The limiting case of a circular bluff body oscillating in an otherwise still fluid is verified with [18] for =250.0, Rei =0.0 and Rt2 =20.0 using Grid 1 and Ru =278.2, R£l =0.0 and Re2 =10.572 using Grid 3. Identical numerical results are obtained in both cases. Streamlines are plotted for the first case in figure 5.2. Figure 5.2 Limiting Case of an Oscillating Circular Body in Still Fluid. =250.0 Rei =0.0 Re2 =20.0 The limiting case of a steady flow over a stationary body is verified with [31] for values of .Rei ranging from 2 to 70. The streamline plots for Rei =20 and 70 are presented in figures 5.3 and 5.4 respectively. The same contour values are plotted in the present study results as that of [31]. The kinks in the streamlines near the body in the present study are due to the coarseness of the grid. On refining the finite element grid, more stream function contours can be plotted and the kinks can be eliminated. Good agreement is obtained in the streamline plots. The pressure distribution around the body at RCi =20 is presented in figure 5.5. The drag coefficients for Rei =5, 20, 40, and 70 are presented in table 5.1. Good agreement is obtained with [31] for the pressure distribution and the drag coefficients. Chapter 5: Numerical Investigations 48 Chapter 5: Numerical Investigations Figure 5.5 Pressure Distribution along the Circular Body Wall for R„ =0.0 Rei =20.0 Rt2 =0.0 Chapter 5: Numerical Investigations 50 Table 5.1 Drag Coefficients for Different Values of Rei for a Circular Body R„ = 0.0 Re2 = 0.0 Rei Pressure Drag Friction Drag Total Drag 5.0 Present Study 2.382 2.020 4.402 Olson & Tuann 2.199 1.917 4.116 20.0 Present Study 1.211 0.791 2.002 Olson &; Tuann 1.233 0.812 2.045 40.0 Present Study 0.987 0.521 1.508 Olson &; Tuann 0.998 0.524 1.522 70.0 Present Study 0.870 0.372 1.242 Olson & Tuann 0.852 0.360 1.212 Parametric Study:-The parameters investigated and their groupings are given in table 5.2. Grid 1 is used in all the cases. Streamlines of the steady component of the velocity field are plotted for each of the groups. Group l.a set of results are presented in figures 5.6a to 5.6e. Rei and fi2 are kept constant at 20.0 and 0.08 respectively. Re2/Rei varies in the range 0.1 to 2. This set of results is at low values of Rt2jRtl , hence the flow pattern is dominated by the steady fluid flow. On introducing small oscillations (Re2/Rei =0.1) to the body, a small vortex appears at the front of the body but the wake behind the body remains intact (Fig. 5.6a). At Re2/Rei =0.75, the wake behind the body becomes very small and a small vortex at the top right of the body is formed (Fig 5.6c). At higher values of Re2/Rei , (Fig. 5.6d and e) the vortex behind the body dimnishes and that at the front and top right grow in size. All these vortices are at very localized regions near the body. Group l.b set of results are presented in figures 5.7a to 5.7c. Ru and Re2 are constant at 250.0 and 20.0 respectively. Re2/Rei is varied in the range 1.0 to 10.0 by changing Rei . On increasing Re2/Rei , the vortex at the front and top right of the body grow in size. It is interesting to compare this to the case when there is no steady flow around the body and two huge symmetric vortices appear on either side of the body as shown in figure 5.2. Chapter 5: Numerical Investigations 51 Table 5.2 Parametric Study of the Flow Pattern for a Circular Body Group Rw Re* Rt* R'i /?2 Figure 0.0* 20.0* 0.0* 5.3 25.0 20.0 2.0 0.1 0.08 5.6a la 62.5 20.0 5.0 0.25 0.08 5.6b 187.5 20.0 15.0 0.75 0.08 5.6c 250.0 20.0 20.0 1.0 0.08 5.6d 500.0 20.0 40.0 2.0 0.08 5.6e 250.0 20.0 20.0 1.0 0.08 5.6d 250.0 15.0 20.0 1.333 0.08 5.7a lb 250.0 5.0 20.0 4.0 0.08 5.7b 250.0 2.0 20.0 10.0 0.08 5.7c 250.0** 0.0** 20.0** 5.2 0.0* 2.0* 0.0* 5.8a 1000.0 2.0 20.0 10.0 0.02 5.8b 500.0 2.0 20.0 10.0 0.04 5.8c 2 333.33 2.0 20.0 10.0 0.06 5.8d 250.0 2.0 20.0 10.0 0.08 5.8e 200.0 2.0 20.0 10.0 0.1 5.8f 250.0 2.0 10.0 5.0 0.04 5.9a 250.0 2.0 15.0 7.5 0.06 5.9b 3 250.0 2.0 20.0 10.0 0.08 5.8e 250.0 2.0 25.0 12.5 0.1 5.9c * Limiting case of steady flow over a fixed body. ** Limiting case of an oscillating body in stationary fluid. Group 2 set of results are presented in figures 5.8a to 5.8f. Rei and Re^ are kept constant at 2.0 and 20.0 respectively. Ru is changed such that 02 varies in the range 0.02 to 0.1. This set of results is at a very high value of Re2/Rei (Re2/Rei =10.0). Thus the flow is dominated by the oscillations of the body. At RCl =2.0 and i?e, =R„ =0.0 (Fig. Chapter 5: Numerical Investigations 52 5.8a), no wake is formed behind the body. On introducing oscillations to the body (/?2 =0.02), a small vortex develops at the front of the body (Fig 5.8b). At 02 =0.04 (Fig. 5.8c), the localized vortex at the front of the body disappears and a huge vortex develops behind the body. At /32 =0.06, the vortex at the back moves to the front of the body (Fig. 5.8d). For higher values of /?2 =0.1 (Fig 5.8e), the vortex at the front of the body grows in size. Group 3 set of results are presented in figures 5.9a to 5.9c. i2w and RCi are kept constant at 250.0 and 2.0 respectively. Rt2 is changed such that (32 varies in the range 0.04 to 0.1. and Re2/Rei varies in the range 5 to 12.5. Just as in group 2, this set of results are at high Re2/Rei values. The phenomenon of the vortex moving from the back to the front of the body is observed to occur at /?2 =0.06, just as in Group 2 set of results. It is observed that irrespective of different values of Rw and Re2 , for the same /?2 values, the flow patterns in group 2 and group 3 set of results are quite similar. This indicates the significant effect of the /?2 parameter. The minimum and maximum values of the stream functions and their respective loca tions for different cases of Rw , Rei and R&2 are tabulated in table 5.3. In all the cases, a plotting domain from -2.5 to 3.7 along the x-axis and 0.0 to 3.1 along the y-axis is used. The dot (.) in the streamline plots represents the location of the minimum value of the stream function. The following observations are made from the foregoing flow results. 1. At low values of Re2/RSi (Re2/Rei <l), the flow pattern is dominated by the steady stream flow. Hence the body amplitude parameter 02 has little effect on the flow pattern except in a localized region near the body. 2. At high values of Re2/Rei [Re2/Rei >l), the flow pattern is dominated by the oscilla tions of the body. Thus R^ and Re2 and as a consequence (i2 influence the flow pattern significantly. 3. For a constant low value of Rei (Re2/Rei <l), the flow pattern for different cases of Rw and Re2 which result in the same value of 02 , are quite similar. This implies that the body amplitude parameter characterises the flow pattern to a greater extent than either Ru or Rej alone. Chapter 5: Numerical Investigati, Figure R, 'ei ~2°-° =5'° **JK =0.25 fi2 =0. 08 5-' NumericaJ Investigations 54 Chapter 5: Numerical Investigat ions Chapter 5: Numerical Investigations 56 Figure 5.8c Group 2 R„ =500.0 Rei =2.0 Re2 =20.0 RejRei =10.0 02 =0.04 Chapter 5: Numerical Investigations 57 Chapter 5: Numerical Investigations 58 Figure 5.9a Group 3 R„ =250.0 R6l =2.0 Re, =10.0 RejR6l =5.0 /32 =0.04 Figure 5.9b Group 3 Rw =250.0 RCi =2.0 Re2 =15.0 RejR6l =7.5 02 =0.06 Figure 5.9c Group 3 Ru =250.0 #6l =2.0 Re2 =25.0 i?e2/iZei =12.5 /?2 =0.1 Chapter 5: Numerical Investigations 59 Table 5.3 Maximum and Minimum Stream Function Values of the Steady Component of the Velocity Field Re2 Stream Function loca x tion y 0.0 2.0 0.0 min. val.=0.0000 max. val.=2.6750 0.0 -2.5 0.5 3.1 25.0 20.0 2.0 min. val.=-0.0121 max. val.=2.5040 0.65 -2.5 0.3 3.1 62.5 20.0 5.0 min. val.=-0.0086 max. val.=2.2000 0.65 -2.5 0.3 3.1 187.5 20.0 15.0 min. val.=-0.0183 max. val.=1.5510 0.2 -2.5 0.525 3.1 250.0 20.0 20.0 min. val.=-0.0189 max. val.=1.3560 0.2 -2.5 0.525 3.1 250.0 15.0 20.0 min. val.=-0.0219 max. val.=1.1610 0.2 -2.5 0.525 3.1 250.0 5.0 20.0 min. val.=-0.0309 max. val.=0.5410 0.2 -2.5 0.55 3.1 250.0 2.0 20.0 min. val.=-0.0350 max. val.=0.2326 0.2 -2.5 0.55 3.1 250.0 0.0 20.0 min. val.=-0.0519 max. val.=0.0519 -1.45 1.45 1.05 1.05 1000.0 2.0 20.0 min. val.=-0.0305 max. val.=0.2442 -0.55 -2.5 0.425 3.1 500.0 2.0 20.0 min. val.=-0.0394 max. val.=0.2547 0.8 -2.5 0.7 3.1 333.33 2.0 20.0 min. val.=-0.0330 max. val.=0.2087 -0.8 -2.5 0.55 3.1 200.0 2.0 20.0 min. val.=-0.0746 max. val.=0.2454 -0.6 3.25 0.80 3.1 250.0 2.0 10.0 min. val.=-0.0825 max. val.=0.4748 0.8 -2.5 0.7 3.1 250.0 2.0 25.0 min. val.=-0.0928 max. val.=0.2163 -0.6 3.3 0.8 3.1 Chapter 5: Numerical Investigations 60 4. Dramatic changes are observed as 02 changes at a constant low value of Rei . For instance, at low values of 02 (< 0.04), a vortex develops behind the body. At higher values of 02 (> 0.06), the vortex moves to the front of the body. 5. At a constant low value of Re2/Rei , the flow is dominated by the steady stream flow. On increasing 02 gradually by changing i?w , the following changes are observed in the flow pattern - The wake behind the body grows considerably in size. - The apparent separation point moves to the front of the body. - At Re2/Rei =0.25, there is a drastic decrease in the wake behind the body. At higher values of Re2/Rei the wake again increases in size. - At still higher values of Re2/Rei , a small vortex appears in a localized region at the front of the body. The streaklines of the total velocity field are plotted for t=0, ^, 7r, 4f, that is, equal time steps over one half cycle of the body motion. Streakline plots for Ru =250.0, RCl =20.0 and Re2 =20.0 are presented in figure 5.10. The streamline plot at these values of Reynolds numbers indicates two small vortices at the front and at the top right of the body (Fig. 5.6d). At t = IT/2, two small vortices appear at the front and back of the body. At t=?r, and 3TT/2, the body is enveloped by these vortices. At t=27r, the streaklines are the same as at t = 0. Streakline plots for R^ =250.0, Rei =2.0 and Re2 =20.0 are presented in figure 5.11. The streamline plot for the same values of Reynolds numbers (Fig. 5.7c) shows a large vortex at the front of the body. At t=0, the streakline plot shows small vortices on the top of the body and a big vortex at the front of the body. At t=7r and t=3?r/2, the body is enveloped by the vortices which grow in size. There is a dominant vortex at the front of the body. The minimum and maximum values of stream function and their respective locations at different times t for these two case are presented in table 5.4. The following observations are made from the foregoing streaklines: 1. At low values of Re2/Rei (Re2/Rei <l), the streaklines are quite similar to the stream-Chapter 5: Numerical Investigations 61 lines for the same values of Reynolds numbers except for small vortices near the body. This is expected as the flow pattern is dominated by the steady component of the velocity field. 2. At higher values of R£2/Rei , the streaklines show significant change in vortex at different time t. The overall flow pattern still resembles the streamline plots for the same values of Reynolds numbers. At different time t, the vortex moves to and fro from the front to the back and to the front again in one cycle. 5.2.4 Added Mass, Added Damping and Added Force The values of added mass, added damping and added force are tabulated for different cases of Rw , R6l and Re2 in table 5.5. The force quantities are observed to be intimately related to the flow pattern. A similar parametric study as described for the streamlines is carried out for the force quantities. Figures 5.12, 5.13 and 5.14 show the variation of the force quantities with change in 0i and Re2/RCi for Group 1 and Group 2, respectively. Figure 5.12 presents the variation of the force quantities with change in Re2/Rei for Group l.a. Rei is constant at 20.0. It is observed that with an increase in Re^/Rei , there is a decrease in added force while the added mass and added damping remain relatively constant. Figure 5.13 presents the variation of the force quantities with Re^/Rei for Group 1. b. Ru and Re2 are kept constant at 250.0 and 20.0 respectively. In both Group l.a and Group l.b, /?2 is contant at 0.08. From figure 5.13 it is evident that the added mass and added damping are constant while there is a decrease in added force with increase in Re* I Rei • Figure 5.14 presents the variation of the force quantities with change in 02 for Group 2. Rei and Re2 are kept constant at 2.0 and 20.0 respectively. At /?2 =0.06, there is a sharp decrease in the added mass and added damping and a sharp increase in the added force. At this value of 02 i the vortex moves from the back to the front of the body. Thus the flow pattern has a great influence on the forces on a body. The following observations are made from this parametric study:-1. The added mass and added damping are influenced by Rw and Re„ , while the added ChWer 5: JYIJm . 2 " 5i° S'-atJines fo at t=0.0, -» , "3r25°-0 =20.0 » Chapter 5: Numerical Investigations 63 Chapter 5: Numerical Investigations 64 Table 5.4 Maximum and Minimum Stream Function Values of the Total Velocity Field at Different Times t 7^=250.0 i?ei=20.0 iEB2=20.0 t Stream Function location X y 0 min. val.=-0.0020 -0.3 0.4 max. val.=1.3550 -2.5 3.1 jr 2 min. val.=-0.0119 -0.4 0.375 max. val.=1.3610 -2.5 3.1 7T min. val.=-0.0523 0.2 0.55 max. val.=1.3580 -2.5 3.1 3TT 2 min. val.=-0.0563 0.25 0.6 max. val.=1.3520 -2.5 3.1 #w=250.0 Rei=2.0 Re2=20.0 0 min. val.=-0.0161 -0.95 0.425 max. val.=0.2291 -2.5 3.1 7T 2 min. val.=-0.0138 -0.4 0.375 max. val.=0.2408 -2.5 3.1 7T min. val.=-0.0495 -0.4 0.4 max. val.=0.2405 -2.5 3.1 3JT 2 min. val.=-0.1133 0.25 0.6 max. val.=0.2239 -2.5 3.1 Chapter 5: Numerical Investigations 65 Table 5.5 Added Mass, Added Damping and Added Force for a Circular Body Ren Added Mass Added Damping Added Force 0.0* 2.0* 0.0* - - 7.1769 0.0* 20.0* 0.0* - - 20.0221 0.0* 40.0* 0.0* - - 30.1568 250.0** 0.0** 20.0** 1.2696 0.4260 0.0000 25.0 20.0 2.0 2.1312 1.6045 13.1090 62.5 20.0 5.0 1.5815 0.9020 11.7870 187.5 20.0 15.0 1.3003 0.4828 9.3309 250.0 20.0 20.0 1.2827 0.4014 7.9780 500.0 20.0 40.0 1.2161 0.2665 4.5587 250.0 15.0 20.0 1.2776 0.4107 5.8885 250.0 5.0 20.0 1.2706 0.4237 1.6123 250.0 2.0 20.0 1.2697 0.4259 0.4372 1000.0 2.0 20.0 1.1383 0.2093 0.7381 500.0 2.0 20.0 1.1723 0.3874 1.0247 333.33 2.0 20.0 1.3306 0.2161 0.3467 200.0 2.0 20.0 1.5158 0.3827 -0.0370 250.0 2.0 10.0 1.1873 0.7151 1.9281 250.0 2.0 15.0 1.3852 0.2793 0.6197 250.0 2.0 25.0 1.4734 0.3396 -0.3898 * Limiting case of steady flow over a fixed body. ** Limiting case of an oscillating body in stationary fluid. Chapter 5: Numerical Investigations 66 force is predominantly influenced by Rei . 2. At constant values of /32 , the added mass and added damping remain fairly constant indicating their relative insensitivity to the other parameters. On the other hand, they are affected significantly by changes in /32 itself. 3. At high values of Re2/Rei , the added force is also influenced considerably by /32 . 4. The force quantities are related to the flow pattern. Drastic changes in the flow pattern result in drastic changes in the added mass, added damping and added force. Chapter 5: Numerical Investigations 67 Force Quantities for a Circular Body Class la Re,=20.0 fi=0.08 o in-o <n o C3 o o o rP <o 0.00 0.23 0.50 LEGEND Q Added Mass A Added Damping + Added Force * -A —i 1 1 1 1 1 0.75 1.00 1.25 1.30 1.73 2.00 Figure 5.12 Variation of Force Quantities with Re2/Rei in Group l.a Chapter 5: Numerical Investigations Force Quantities for a Circular Bodv o CO-" Class lb Rea= 0=0.08 20.0 o «- + LEGEND o Added Mass * Added Damping + Added Force co CS •r» •r> 0 9 o Q) O c. \ O \ < 1 o o A 1.0 1 I 3.0 3.0 1 1 7.0 9.0 Figure 5.13 Variation of Force Quantities with RejRei in Group l.b Chapter 5: Numerical Investigations Force Quantities for a Circular Body Class 2 Re,=2.0 i?e8=20.0 Figure 5.14 Variation of Force Quantities with 02 in Group 2 Chapter 5: Numerical Investigations 70 5.3 Results for a Square Body 5.3.1 General Remarks A parametric study is conducted as described in section 5.1 for a square body. The limiting case of a square body oscillating in an otherwise still fluid is verified with the results obtained by Pattani [18]. No experimental or analytical results seem to be available for the limiting case of a steady flow around fixed square bodies at such low values of Rei . 5.3.2 Finite Element Grid and Boundary Conditions The numerical results are obtained for a square body having sides of unity with the centroid of its area located at the origin of the x-y coordinate system and the sides parallel to the axis. The body is performing harmonic oscillations parallel to the x-axis. Using symmetry, only one half of the domain is modelled. The finite element grid used is shown in figure 5.15. The grid has a circular outer boundary and a D/b ratio of 15.5. Similar boundary conditions are used as that of grid 1 and 3 for a circular body. Boundary Conditions:-1. u = UQO, v = 0 along the fluid outer boundary. 2. u and v velocities at the body boundary are specified according to equation 3.7. 3. u=?, v = 0 along ra, the symmetry line. 4. p = 0 at points 1 and 2 shown in figure 5.15. For the flow problem under consideration, there are 368 net degrees of freedom of which 307 are for the velocities and 61 are for the pressure. There are three coefficients A, B, C, for each degree of freedom resulting in 1104 variables in total. 5.3.3 Flow Results The limiting case of a square bluff body oscillating in an otherwise still fluid is verified Chapter 5: Numerical Investigations 71 fl/b=15.5 2 -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 Figure 5.15 Finite Element Grid for Square Body with [18] for Rw =150.0, Rei =0.0 and Re2 =10.0. Identical numerical results are obtained. Streamlines are plotted for this limiting case in Figure 5.17d. A listing of the parameters investigated and their groupings for a square body are shown in table 5.6. Streamlines of the steady component of the velocity field are plotted for each of the groups. Group l.a set of results are presented in figures 5.16a to 5.16e. Rei and /?2 are kept constant at 10.0 and 0.067 respectively. Re2/Rei is varied in the range 0.1 to 1.0. At Rei = 10.0 and Re2 =0.0 (Fig. 5.16a), a wake is present behind the body and the separation point is just after the front corner. When small oscillations are introduced on the body (Re2/Rei =0.1), the separation point moves to the front of the body (Fig. 5.16b). At Re2/Rei =0.75 (Fig. 5.16d), the wake behind the body becomes very small and a small vortex at the front corner of the body develops. At Re^/Rei =1.0 (Fig. 5.16e), the vortex behind the body is negligible while that at the front corner of the body grows in size. Group l.b set of results are presented in figures 5.17a to 5.17d. Ru and Re2 are kept constant at 150.0 and 10.0 respectively. Re2/Rei is varied in the range of 1.0 to 10.0. With increase in Re2/Rei values, the vortex at the left corner of the body increases and becomes well developed. It is no longer at a localized region near the body. Chapter 5: Numerical Investigations 72 Table 5.6 Parametric Study of the Flow Pattern For a Square Body Group Re, Re2 Re 2 Rc [ 02 Figure 0.0* 10.0* 0.0* 5.16a 15.0 10.0 1.0 0.1 0.0667 5.16b la 60.0 10.0 4.0 0.4 0.067 5.16c 112.5 10.0 7.5 0.75 0.067 5.16d 150.0 10.0 10.0 1.0 0.067 5.16e 150.0 10.0 10.0 1.0 0.067 5.16e 150.0 7.5 10.0 1.33 0.067 5.17a lb 150.0 4.0 10.0 2.5 0.067 5.17b 150.0 1.0 10.0 10.0 0.067 5.17c 150.0** 0.0** 10.0** 5.17d 1000.0 1.0 10.0 10.0 0.01 5.18a 250.0 1.0 10.0 10.0 0.04 5.18b 2 150.0 1.0 10.0 10.0 0.067 5.17c 100.0 1.0 10.0 10.0 0.1 5.18c 150.0 1.0 1.5 1.5 0.01 5.19a 150.0 1.0 6.0 6.0 0.04 5.19b 3 150.0 1.0 10.0 10.0 0.067 5.17c 150.0 1.0 15.0 15.0 0.1 5.19c * Limiting case of steady flow over a fixed body. ** Limiting case of an oscillating body in stationary fluid. Group 2 set of results are presented in figures 5.18a to 5.18c. Rei and Re2 are kept constant at 1.0 and 10.0 respectively. Ru is changed such that 02 varies in the range 0.01 to 0.1. At 02 =0.01, there are localized vortices at the front and the top of the body. At 02 =0.04 (Fig 5.18b), a huge vortex appears behind the body. This vortex moves to the front of the body at about 02 =0.067 (Fig 5.17c), just as in the case of a circular body. At 02 =0.1 (Fig. 5.18c), the vortex at the front of the body becomes well developed. Group 3 set of results are presented in figures 5.19a to 5.19c. Ru and Rei are kept Chapter 5: Numerical Investigations 73 constant at 150.0 and 1.0 respectively. Re7 is changed such that 02 varies in the range 0.01 to 0.1. For the same values of 02 as that in group 2, the flow patterns are very similar to those obtained in group 2. The phenomenon of the vortex moving from the back to the front of the body is also observed to occur at about 02 =0.067 (Fig. 5.17c). The minimum and maximum values of stream functions and their respective locations for different cases of , Rei and Re2 are tabulated in table 5.7. In all the cases a plotting domain from -2.5 to 3.7 along the x-axis and 0.0 to 3.1 along the y-axis is used. The dot (.) in the streamline plots represents the location of the minimum value of the stream function. Chapter 5: Numerical Investigations 74 Chapter 5: Numerical Investigations 75 Figure 5.17a Group l.b Rw =150.0 Rei =7.5 Re2 =10.0 RejRei =1.33 /?2 =0.067 Chapter 5: Numerical Investigations 76 Figure 5.17d Group l.b Rw =150.0 R6l =0.0 Re2 =10.0 /?2 =0.067 Chapter 5: Numerical Investigations 77 Group 2 Rw =100.0 Rei =1.0 Rt2 =10.0 RejRei =10.0 02 =0.1 Chapter 5: Numerical Investigations 78 Figure 5.19c Group 3 Ru =150.0 Rei =1.0 Re2 =15.0 RejRei =15.0 (32 =0.1 Chapter 5: Numerical Investigations 79 Table 5.7 Maximum and Minimum Stream Function Values of the Steady Component of the Velocity Field Ret Re2 Stream Function loca X tion y 0.0 10.0 0.0 min. val.=-0.0204 max. val.=2.6710 0.8 -2.5 0.425 3.1 15.0 10.0 1.0 min. val.=-0.0267 max. val.=2.4010 0.90 -2.5 0.475 3.1 60.0 10.0 4.0 min. val.=-0.0173 max. val.=1.8770 -0.55 -2.5 0.575 3.1 112.5 10.0 7.5 min. val.=-0.0293 max. val.=1.4930 -0.6 -2.5 0.575 3.1 150.0 10.0 10.0 min. val.=-0.0400 max. val.=1.3010 -0.6 -2.5 0.625 3.1 150.0 7.5 10.0 min. val. =-0.0462 max. val.=1.1150 -0.65 -2.5 0.65 3.1 150.0 4.0 10.0 min. val.=-0.0626 max. val.=0.7299 -0.7 -2.5 0.7 3.1 150.0 1.0 10.0 min. val.=-0.1213 max. val.=0.2837 -1.0 -2.5 0.975 3.1 150.0 0.0 10.0 min. val.=-0.1719 max. val.=0.1719 -1.7 1.7 1.45 1.45 1000.0 1.0 10.0 min. val.=-0.0016 max. val.=0.2316 -0.3 -2.5 0.625 3.1 250.0 1.0 10.0 min. val.=-0.0415 max. val.=0.2730 0.95 -2.5 0.75 3.1 100.0 1.0 10.0 min. val.=-0.0566 max. val.=0.2314 -0.95 3.35 0.95 3.1 150.0 1.0 1.5 min. val.=-0.0064 max. val.=1.0240 0.7 -2.5 0.4 3.1 150.0 1.0 6.0 min. val.=-0.0493 max. val.=0.4114 0.95 -2.5 0.675 3.1 150.0 1.0 15.0 min. val.=-0.0759 max. val.=0.1880 -0.95 3.15 0.95 3.1 Chapter 5: Numerical Investigations 80 On comparing the streamline plots of the square body with those of the circular body, it is obvious that the flow patterns in both the cases are quite similar. The parametric study conducted for both the body shapes indicate similar effects and influences of the parameters Rw , Rei , Re2 , Re2/RCl and 02 • Similar changes in the flow patterns are observed as in the case of a circular body in groups 1, 2 and 3. Thus it can be concluded that the vortex should be quite similar for any doubly-symmetric body for the same values of the governing parameters. The streaklines of the total velocity field are plotted for t = 0, f, n and Streakline plots for Ru =150.0, Rei =1.0 and Re2 =10.0 are presented in figure 5.20. The minimum and maximum values of the stream function and their respective locations at different time t are presented in table 5.8. Changes in vortex are observed at very localized regions near the body. At t=7r, the vortex moves closer to the body and finally moves away from the body at t=3|. The streamline plot for the same values of Reynolds numbers (Fig. 5.17c) shows a dominant vortex at the top front corner of the body. In all the streakline plots this dominant vortex remains, indicating that the basic flow pattern is maintained at all time. Table 5.8 Maximum and Minimum Stream Function Values of the Total Velocity Field at Different Times t ^ = 150.0 Rei =1.0 Re2 = 10.0 t Stream Function location X y 0 min. val.=-0.0825 -1.2 0.925 max. val.=0.2805 3.1 3.1 TT 2 min. val.=-0.0925 -1.15 0.95 max. val.=0.2947 3.15 3.1 7T min. val.=-0.1942 0.7 0.7 max. val.=0.2879 3.4 3.1 3ir 2 min. val.=-0.1593 -0.9 0.9 max. val.=0.2729 3.3 3.1 Chapter 5: Numerical Investigations 81 Chapter 5: Numerical Investigations 82 5.3.4 Added Mass, Added Damping and Added Force The added mass, added damping and added force are tabulated for different values of Ru , Rei and Rt2 in table 5.9. A parametric study described in section 5.1 is carried out for the force quantities. Figure 5.21 and 5.22 show the variation of the force quantities with change in Re2/Rei and 02 for group l.b and group 2 respectively. In group l.b, as observed in figure 5.21, the added mass and added damping remain relatively constant with change in Re2/Rei while there is a significant decrease in added force. In group 2, there are drastic changes in all the force quantities in the range of 0.04 <02 <0.07. In this range, there is an increase in added mass while the added damping and added force decrease. In this range of 02 > there are drastic changes in the flow patterns. The vortex moves from the back to the front of the body. From these results, we observe that the added mass and added damping are mainly governed by the 02 parameter while the added force is predominantly influenced by Rtl . Further, the force quantities appear to be intimately related to the flow pattern. Changes in the flow pattern, cause changes in the force quantities. The variations of the force quantities for the square body is very similar to those obtained for the circular body. Chapter 5: Numerical Investigations 83 Force Quantities for a Square Bodu Group lb Re =10.0 ft,=0.067 §9 o CD O q d. -A 1.0 I— 3.0 LEGEND ° Added Mass Added Damping + Added Force 5.0 Re/ Ret r "7.0" 8.0 Figure 5.21 Variation of Force Quantities with RejRei in Group l.b Chapter 5: Numerical Investigations Force Quantities for a Square Body o o oi" Croup 2 Re =1.0 i?c8=10.0 R. •A o «. / LEGEND / o Added Moss / A Added Damping + Added Force in 01 <0 -5" Force Quanta 0.75 1.00 * ** .* / •f" / / / N \ OC'O / / / / / / f / / i I $ t \ / ).00 0.29 i f / f / A // 1 J 0.00 0.02 0.04 0.08 0.08 0.10 Figure 5.22 Variation of Force Quantities with /?2 in Group 2 Chapter 5: Numerical Investigations 85 Table 5.9 Added Mass, Added Damping and Added Force for a Square Body Re2 Added Mass Added Damping Added Force 0.0* 10.0* 0.0* - - 15.3848 150.0** 0.0** 10.0** 1.8042 0.2876 0.0000 15.0 10.0 1.0 3.8392 -0.3190 14.5020 60.0 10.0 4.0 2.3470 0.3085 11.0010 112.5 10.0 7.5 2.0088 0.1996 8.1494 150.0 10.0 10.0 1.9035 0.1346 6.5288 150.0 7.5 10.0 1.8712 0.1835 4.8415 150.0 4.0 10.0 1.8323 0.2433 2.2049 150.0 1.0 10.0 1.8074 0.2826 0.1210 1000.0 1.0 10.0 1.2753 0.2028 0.7730 250.0 1.0 10.0 1.4066 0.7852 1.0243 100.0 1.0 10.0 1.8291 0.5704 0.4751 150.0 1.0 1.5 1,6181 0.5451 3.4605 150.0 1.0 6.0 1.4831 0.9657 1.4573 150.0 1.0 15.0 1.7006 0.4622 0.1227 * Limiting case of steady flow over a fixed body. ** Limiting case of an oscillating body in stationary fluid. Chapter 5: Numerical Investigations 86 5.4 Results for a Triangular Body 5.4.1 General Remarks So far we have investigated the flow pattern around doubly-symmetric bodies. In order to investigate the streaming around asymmetrical bodies, an equilateral triangular body is chosen. Tatsuno [30] has experimentally investigated the flow pattern of steady streaming in the vicinity of an equilateral triangular body oscillating sinisoidally in an otherwise still fluid. A similar body geometry is used for this numerical investigation. The limiting case of a triangular body oscillating in an otherwise still fluid is verified with Tatsuno [30]. No experimental or numerical results seem to be available for the limiting case of steady flow over a triangular body. Further, no results are known for the combined problem of a triangular body oscillating in steady fluid flow. Hence a parametric study is conducted as described in section 5.1. 5.4.2 Finite Element Grid and Boundary Conditions Numerical results are obtained for an equilateral triangular body having sides of unity with the centroid of its area located at the origin of the x-y coordinate system. One side is perpendicular to the x-axis and a bisector of the triangle is parallel to the x-axis. The body is performing harmonic oscillations parallel to the x-axis. Using symmetry, only one half of the domain is modelled. The finite element grid used is shown in figure 5.23. The grid has a rectangular outer boundary and a D/b ratio of 23.7. The boundary conditions are similar to that of grid 2 for a circular body. Boundary Conditions:-1. u = UQO, v = 0 along T,, the inflow boundary. 2. u = UQQ, f=? along Tu. 3. u=?, v = 0 along T0, the outflow boundary. 4. u=?, v = 0 along Ts, the symmetry line. Chapter 5: Numerical Investigations 87 5. Pressure p = 0 at points 1 and 2 shown in figure. For the flow problem under consideration, there are 368 net degrees of freedom, 307 for velocities and 61 for pressure. There are three coefficients A, B, C for each degree of freedom, resulting in 1104 variables in total. i n, 1 1 1 1 1 1 1 1 1 1 -12.3 -10.0 -75 -3.0 -25 0.0 2.3 3.0 75 10.0 125 Figure 5.23 Finite Element Grid for a Triangular Body 5.4.3 Flow Results The limiting case of a triangular bluff body oscillating in an otherwise still fluid is verified with [30]. Streamlines are plotted for Rw =156.0, Rei =0.0 and Ren =3.712 in figure 5.24. Tatsuno obtained photographs of the steady streaming around an oscillating triangular cylinder. The experimental result shows the formation of two vortices: one at the front and one on top of the triangular body. The present numerical results also present two such vortices. Few discrepancies are apparent between the experimental and numerical result. The vortex on the top of the body in the present study appears to be much closer to the body than the experimental one. This discrepency could be attributed to the coarseness of the finite element grid near the body. On refining the grid, the numerical results are expected to converge to those obtained experimentally. A listing of the parameters investigated and their groupings are shown in table 5.10. Streamlines of the steady component of the velocity field are plotted for each of the groups Chapter 5: Numerical Investigations 88 Figure 5.24 Limiting Case of an Oscillating Body in Still Fluid. RU =156.0 RCL =0.0 REO =3.712 Group l.a set of results are presented in figures 5.25a to 5.25e. REI and 02 are kept constant at 10.0 and 0.067 respectively. RE2/REI varies in the range 0.1 to 1.0. At REI = 10.0 (Fig. 5.25a), a small vortex is present on the top of the triangular body with the separation point slightly downstream of the corner. On introducing small oscillations to Chapter 5: Numerical Investigations 89 Table 5.10 Parametric Study of the Flow Pattern For a Triangular Body Group Re2 Rr2 Rc i 02 Figure 0.0* 10.0* 0.0* 5.24a 15.0 10.0 1.0 0.1 0.0667 5.24b la 60.0 10.0 4.0 0.4 0.067 5.24c 112.5 10.0 7.5 0.75 0.067 5.24d 150.0 10.0 10.0 1.0 0.067 5.24e 150.0 10.0 10.0 1.0 0.067 5.24e 150.0 7.5 10.0 1.33 0.067 5.25a lb 150.0 4.0 10.0 2.5 0.067 5.25b 150.0 1.0 10.0 10.0 0.067 5.25c 150.0** 0.0** 10.0** 5.26d 1000.0 1.0 10.0 10.0 0.01 5.26a 250.0 1.0 10.0 10.0 0.04 5.26b 2 150.0 1.0 10.0 10.0 0.067 5.25c 100.0 1.0 10.0 10.0 0.1 -150.0 1.0 1.5 1.5 0.01 5.27a 150.0 1.0 6.0 6.0 0.04 5.27b 3 150.0 1.0 10.0 10.0 0.067 5.25c 150.0 1.0 15.0 15.0 0.1 -* Limiting case of steady flow over a fixed body. ** Limiting case of an oscillating body in stationary fluid. the body (Rt2/Rei =0.1), this vortex moves to the front of the body (Fig. 5.25b). As Rer,/Rei increases, the vortex at the front of the body grows in size (Fig 5.25c to e). Group l.b set of results are presented in figures 5.26a to 5.26d. i2w.and Re2 are kept constant at 150.0 and 10.0 respectively. Re2/Rei varies in the range 1.0 to 10.0. As Ren/Rei is increased the vortex at the front of the body grows in size and moves further away from the body. At high values of Re2/Rei (Re2/R€l =10.0), the vortex encompasses a large area around the triangular body (Fig 5.26d). Chapter 5: Numerical Investigations 90 Group 2 set of results are presented in figures 5.27a to 5.27b. Rei and i2e, are kept constant at 1.0 and 10.0 respectively. -Rw is changed such that (32 varies in the range of 0.01 to 0.1. At /?2 =0.01, (Fig 5.27a) there is a small localized vortex at the front and top of the body. At fi2 =0.04 (Fig 5.27b), the vortex on the top of the body grows in size. At p2 =0.0667 (Fig. 5.26c), the small localized vortex at the front and top of the body disappears and a huge vortex to the top left of the body is formed. The solution does not converge for fi2 =0.1. Group 3 set of results are presented in figures 5.28a and 5.28b. Rw and Rei are kept constant at 150.0 and 1.0 respectively. Re2 is changed such that 02 varies in the range 0.01 to 0.1. For the same values of /32 as that in group 2, the flow patterns are very similar to those obtained in group 2. The same drastic changes occur in the flow pattern as described in group 2 set of results. This shows the significant effect the 02 parameter has on the flow pattern. The solution does not converge for fi2 =0.1, just as in group 2. The minimum and maximum values of the stream functions and their respective loca tions for different values of Rw , Rei and Rt2 are tabulated in table 5.11. Two kinds of plotting domains are used:-a. -2.5 to 3.7 along the x-axis and 0.0 to 3.1 along the y-axis. b. -7.75 to 7.75 along the x-axis and 0.0 to 7.75 along the y-axis. Plotting domain b. is used for the streamline plots where the vortex develops far from the body. This is indicated in table 5.11. The streaklines of the total velocity field (Fig. 5.29) are plotted for t=0, f, TT, ^ for Ru =150.0, Rei =1.0, and Re2 =10.0. It is observed that the streaklines at different time t are almost identical to the streamlines for the same values of Reynolds numbers, shown in figure 5.26c. Even the minimum and maximum values of the stream function occur at the same point. 5.4.4 Added Mass, Added Damping and Added Force The added mass, added damping and added force for a triangular body are presented Chapter 5: Numerical Investigations 91 Table 5.11 Maximum and Minimum Stream Function Values of the Steady Component of the Velocity Field Ru Re2 Stream Function local x Lion y 0.0 10.0 0.0 min. val.=-0.0030 max. val.=2.7690 -0.05 -2.5 0.475 3.1 15.0 10.0 1.0 min. val.=-0.0631 max. val.=2.5770 -0.55 -2.5 0.275 3.1 60.0 10.0 4.0 min. val.=-0.1002 max. val.=1.9240 -0.6 -2.5 0.325 3.1 112.5 10.0 7.5 min. val.=-0.1598 max. val.=1.4480 -0.65 -2.5 0.35 3.1 150.0 10.0 10.0 min. val.=-0.2081 max. val.=1.1760 -0.65 -2.5 0.35 3.1 150.0 7.5 10.0 min. val.=-0.5605 max. val.=0.2988 -2.15 3.7 1.15 3.1 150.0 * 4.0 10.0 min. val.=-1.0240 max. val.=1.8330 -3.75 7.75 2.187 7.75 150.0 * 1.0 10.0 min. val.=-1.6115 max. val.=-0.6112 -5.5 7.75 2.875 7.75 150.0 * 0.0 10.0 min. val.=-1.7160 max. val.=0.0000 -5.375 -3.25 3.125 0.0 1000.0 1.0 10.0 min. val.=-0.0017 max. val.=0.2320 -0.5 -2.5 0.275 3.1 250.0 1.0 10.0 min. val.=-0.0097 max. val.=0.2499 0.0 -2.5 0.625 3.1 100.0 * 1.0 10.0 min. val.= -max. val.= -- -150.0 1.0 1.5 min. val.=-0.0011 max. val.=1.0110 -0.05 -2.5 0.475 3.1 150.0 1.0 6.0 min. val.=-0.0131 max. val.=0.3811 -0.05 -2.5 0.625 3.1 150.0 * 1.0 15.0 min. val.= -max. val.= ---* Plotting domain b is used. Chapter 5: Numerical Investigations 93 Figure 5.26a Group l.b Ru =150.0 Rtl =7.5 Re9 =10.0 RejRei =1.33 fa =0.067 Chapter 5: Numerical Investigations 94 Figure 5.26d Group l.b R„ =150.0 Rei =0.0 Re2 =10.0 =0.067 Chapter 5: Numerical Investigations 96 Figure 5.28b Group 3 Ru =150.0 Re. =1.0 Re2 =6.0 RejREi =6.0 02 =0.04 in table 5.12. A similar parametric study, conducted on the circular and square bodies, is carrried out for the triangular body. The overall variation of the force quantities is similar to that observed for the circular and square body. The added mass and added damping are relatively constant for constant 02 values while the added force decreases with increasing i?e2/i?e, values. Chapter 5: Numerical Investigations 97 Chapter 5: Numerical Investigations 98 Table 5.12 Added Mass, Added Damping and Added Force for a Triangular Body Re2 Added Mass Added Damping Added Force 0.0* 10.0* 0.0* - - 16.2755 150.0** 0.0** 10.0** 2.6427 -0.3107 -0.9298 15.0 10.0 1.0 -6.0070 -4.7580 10.2330 60.0 10.0 4.0 2.8606 -0.8783 2.9475 112.5 10.0 7.5 2.7096 -0.3756 -1.8326 150.0 10.0 10.0 2.5700 -0.2752 -3.7253 150.0 7.5 10.0 2.6616 -0.3020 -2.7161 150.0 4.0 10.0 2.6473 -0.2899 -3.1777 150.0 1.0 10.0 2.6459 -0.3103 -1.4108 1000.0 1.0 10.0 1.6136 0.3380 -0.7303 250.0 1.0 10.0 1.9230 0.7759 -0.2386 100.0 1.0 10.0 - - -150.0 1.0 1.5 2.1854 1.0430 2.8431 150.0 1.0 6.0 2.1479 1.0283 0.4604 150.0 1.0 15.0 - - -* Limiting case of steady flow over a fixed body. ** Limiting case of an oscillating body in stationary fluid. Chapter 5: Numerical Investigations 99 5.5 Case a Associated Flow 5.5.1 General Remarks So far we have considered the flow problem of a body oscillating in the direction of steady flow which corresponds to case b in figure 2.2. In this section, the results for the associated flow problem (case a), of a stationary body in a fluid flow with a steady as well as an oscillating component are presented. The investigation is carried out only for a circular body. Goddard [7] has presented a numerical solution of the two-dimensional, time-dependent Navier-Stokes equations for the case of a fluctuating flow past a circular cylinder using a time-dependent explicit finite difference method. He obtained results for Rei =40 and 200 for different values of Ru and Re2 . The variation of the drag coefficients with time is determined and compared with [7]. Some comparisons are shown between the two associated flows. 5.5.2 Finite Element Grid and Boundary Conditions The finite element grids used in this section are shown in figure 5.30. Grid 1 has a D/b ratio of 15.5 while that of grid 2 is 30.0. D is the total length of the fluid domain and b is the characteristic length of the body. Both the grids have circular outer boundaries. Boundary Conditions for Grid 1 and Grid 2 Grid 1 and 2 have the same velocity and pressure boundary conditions. 1. u = Uoo, v = 0 all along the fluid outer boundary. 2. u=0 and v=0 at the body boundary. 3. u=?, v=0 all along the symmetry line Ts. 4. p = 0 at point 2 of the grid. For the flow problem of a stationary body in an oscillating fluid, both the finite element grids have 335 net degrees of freedom of which 273 are for the velocities and 62 are for the pressure. Chapter 5: Numerical Investigations 100 • 1 1 1 1 1 1 1 • -8.3 -85 -4.3 -2.3 -0.3 1.3 3.3 3.3 7.3 Figure 5.30a Finite Element Grid 1 for Case a Flow Problem o '-18.0 -12.0 -8.0 -4.0 0.0 4.0 8.0 12.0 16.0 Figure 5.30b Finite Element Grid 2 for Case a Flow Problem 5.5.3 Flow Results The limiting case of a stationary circular bluff body in an oscillating fluid is compared with the results obtained by Pattani, [18] in which he considered the same bluff body oscillating in an otherwise still fluid. The numerical solution obtained for the case b flow is transformed into that of case a using the transformation given by equation 3.15. These Chapter 5: Numerical Investigations 101 transformed results are presented in terms of streamline plots of the steady component of the velocity field. Figures 5.31a and 5.31b are the streamline plots for both the associated flows for Rw =21.34, Rei =0.0, Re2 =0.6402 and 02 =0.03. Figures 5.32a and 5.32b are the streamline plots for both the associated flows for R^ =21.34, Rei =0.0 ,Re2 =2.134 and /?2 =0.1. Grid 2 was used in both the cases. Figures 5.33a and 5.33b are the streamline plots for the two associated flows for R„ =250.0, R£i =20.0, i?e, =20.0. Grid 1 was used in this case. The general flow pattern in both case a and case b is similar. Table 5.13 presents the minimum and maximum values of the stream functions and their respective locations for case a and case b flows. In all the cases, the contour plots are obtained by plotting 31 equally spaced contours between the maximum and minimum values of the stream function in the plotted domain. The stream function values and their respective locations do not match well between the two associated flows. The transformation between the two associated flows is valid for flows oscillating at far field. The boundary conditions simulated by the finite element grid may be such that the far field boundary is not far enough. This implies that there would be effect of the outer boundary even near the body. It is expected that by refining the grid near the outer boundary, the results in the two associated flows would match better. The transformation appears to work better in the present study for low values of the body amplitude parameter, 02 . The steady component of the velocity field is a secondary phenomenon and is very small. Hence though the steady components of the velocity field do not match well, it is expected that the overall flow field would conform with the transformation. Chapter 5: Numerical Investigations Figure 5.31a Case a Rw =21.34, Rtl =0.0, Re2 =0.6402 Chapter 5: Numerical Investigations 103 Chapter 5: Numerical Investigations 104 Table 5.13 Maximum and Minimum Stream Function Values of the Steady Component of the Velocity Field Re* Case Stream Function location X y 21.34 0.0 0.640 a min. val.=-0.0045 -1.0 1.0 a max. val.=0.0045 1.0 1.0 21.34 0.0 0.640 b min. val.=-0.0088 -0.65 0.625 b max. val.=0.0088 0.65 0.625 21.34 0.0 2.134 a min. val.=-0.0150 -1.0 1.0 a max. val.=0.0150 1.0 1.0 21.34 0.0 2.134 b min. val.=-0.0798 -1.65 1.65 b max. val.=0.0798 1.65 1.65 250.0 20.0 20.0 a min. val.=-0.0145 0.75 0.4 a max. val.=1.3570 -2.5 3.1 250.0 20.0 20.0 b min. val.=-0.0189 0.2 0.525 b max. val.=1.3560 -2.5 3.1 Table 5.14 Added Mass/ Inertia Force, Added Damping and Added Force for the Two Associated Flows Ru Re* Case Added Mass/ Added Added Force Inertial Force Damping 21.34 0.0 0.6402 a 3.1827 1.5899 0.0000 b 2.1721 1.5899 0.0000 21.34 0.0 2.134 a 3.1803 1.5956 0.0000 b 2.0903 1.6232 0.0000 250.0 20.0 20.0 a 2.3081 0.4117 9.3566 b 1.2827 0.4014 7.7800 Chapter 5: Numerical Investigations 105 5.5.4 Inertia Force, Added Damping and Added Force The transformation of added mass-inertia force, added damping and added force be tween case a and case b, presented in equation 4.27 is verified in this section. Table 5.14 presents the force quantities for case a and case b. It is observed that the results conform to the transformation of equation 4.27 quite well. Hence it can be concluded that though the steady components of the velocity field do not match well between the two associated flows, the overall flow results do conform. Figures 5.34a to 5.34c presents the variation of the drag coefficients with time for Rw =800.0, Rei =40.0 and Re3 =4.0 for case a, case b and the results from [7] respectively. In this case Re2/Rei =0.1 and /?2 =0.005. The drag coefficients obtained for case b are transformed to those of case a using equation 4.28. The results of case b compare very well with [7] while those of case a show higher fluctuations in the pressure drag and total drag coefficients. Figure 5.35 presents the streaklines for case a and case b for Ru =800.0, Rei =40.0 and Re2 =4.0. The flow patterns in all three cases are quite similar. This is due to the high value of the steady Reynolds number (i?e, =40) and the low value of the body amplitude parameter (/?2 =0.005). Figures 5.36a to 5.36c presents the variation of the drag coefficients with time for R„ =30.0, Rei =40.0 and Re2 =4.0 for case a, case b and the results from [7] respectively. In this case, J?E,/J?£L =0.1 and /?2 =0.13. As in the previous case, the results of case b presented, are those after the transformation using equation 4.28. The results of case a compare quite well with those of [7], The results of case b show smaller fluctuations in the drag coefficients than [7] or case a. Figure 5.37 presents the streaklines for case a and case b for Rw =30.0, RCl =40.0 and RC2 =4.0. The flow patterns of case a and case b do not compare well as observed from figure 5.37. These results are at a high value of 02 and hence very accurate results are not expected by the present method. Goddard's results show that the variation of the drag coefficients at steady state is similar to a sine wave. This shows that the assumption of a periodic solution in the method of averaging is fairly accurate. The results seem to match better for low values of 02 • The discrepancy in the results can be attributed to the crudeness of the grid. Better agreement in results is expected with grid refinement. Chapter 5: Numerical Investigations 106 Figure 5.34 Drag Coefficients vs Time for Ru =800.0 Re Re2 =4.0 (a) case 1, (b) case b, (c) [7] =40.0 Chapter 5: Numerical Investigations 107 Chapter 5: Numerical Investigations 108 q «0 *i 8 qo-LEGEND Form Drag .YJ?9°.UA.Q[99.. Total Drag _ —i— 7.833 4.712 10.994 14.133 Time 20.417 23.338 a q ci" q LEGEND • Form Drop Viscous Drag_ Total Drag 4.712 7.833 10.994 14.133 Time 20.417 23.338 c Figure 5.36 Drag Coefficients vs Time for R„ =30.0 Rei =40.0 Re2 =4.0. (a) case a, (b) case b, (c) [7] Chapter 5: Numerical Investigations 109 Chapter 5: Numerical Investigations 110 5.5.5 Drag and Inertia Coefficients for the Morison Equation The coefficients Cd(l), Cd{2) and Cm are determined from the added mass, added damping and the added force results presented in table 5.5 using equation 4.32. These results are presented in table 5.15. Cd(l) for purely steady flow (i2e,=20.0, Ru=Re2=0.0) is 2.0 and Cm for purely oscillatory flow is also 2.0. It is observed that Cm decreases with increase in Re2/RCl and increases with increase in 02 • Overall, Cm lies in the range 2.2 to 2.5. Cd{\) decreases with increase in Re2/RCl and fo . Ca{2) increases with increase in Re2/Rei and decreases with increase in fo . Cd(l) and Cd(2) as obtained from the results of the present investigation using equation 4.32 and those obtained by linearising the Morison equation (equation 4.30), are presented in figures 5.38 and 5.39. Cd(l) and Cd{2) are obtained from equation 4.30 by considering Cd to be that for purely steady flow for the given value of Rei. Figure 5.38 presents the variation of Cd(l) and Cd(2) with respect to Re2/Rei . It is observed that at oscillations of low amplitude (Re2/Rei < 0.75), the results of the present investigation compare quite well with the results from the linearised Morison equation. However, for higher values of Re2/Rei , the two results deviate substantially. Figure 5.39 presents the variation of Cd{\) and Cd[2) with respect to fo . It is observed that the drag coefficients obtained from equation 4.30 are constant with changes in 02 • This implies that the drag coefficients (obtained from equation 4.30) are independent of Ru . Significant changes are observed in the variation of the drag coefficient Cd{2) (obtained from the present study using equation 4.32) with 02 . This variation is very similar to that observed in the variation of the added force given in figure 5.14. Chapter 5: Numerical Investigations 111 Table 5.15 Inertia and Drag Coefficients for a Circular Body Re2 Cm Cd(l) Cd(2) 0.0* 2.0* 0.0* 0.0 3.588 0.0 0.0* 20.0* 0.0* 0.0 2.000 0.0 0.0* 40.0* 0.0* 0.0 1.5 0.0 250.0** 0.0** 20.0** 2.2696 0.0000 8.3723 25.0 20.0 2.0 3.1312 . 1.1917 0.3150 62.5 20.0 5.0 2.5815 0.9427 1.1069 187.5 20.0 15.0 2.3003 0.5332 1.7411 250.0 20.0 20.0 2.2827 0.3989 1.9703 500.0 20.0 40.0 2.2161 0.1519 2.3256 250.0 15.0 20.0 2.2776 0.3365 2.6332 250.0 5.0 20.0 2.2706 0.1290 5.3244 250.0 2.0 20.0 2.2697 0.0397 6.9112 1000.0 2.0 20.0 2.1383 0.0671 13.5854 500.0 2.0 20.0 2.1723 0.0932 12.5729 333.33 2.0 20.0 2.3306 0.0315 4.6756 200.0 2.0 20.0 2.5158 -0.0034 4.9681 250.0 2.0 10.0 2.1873 0.3213 11.6041 250.0 1 2.0 15.0 2.3852 0.0729 5.6928 250.0 2.0 25.0 2.4734 -0.0288 4.5734 * Limiting case of steady flow over a fixed body. ** Limiting case of an oscillating body in stationary fluid. Chapter 5: Numerical Investigations 112 Draq Coefficients for a Circular Bodxi o eo" /?a=0.08 o W O o i i i i i i LEGEND Cc/f7) present study __pd(2)_Pfesent study Cd(l) Morison eqn Cd(2) Morison egn Drag C i i $ i i i f i i '• o Cvi" • • i i A \ . ^ V./"" ^- ' q 3.0 2.0 • i i 1 4.0 6.0 8.0 10.0 R*J -Re, Figure 5.38 Variation of Cd{l) and Cd(2) with RejRei Chapter 5: Numerical Investigations 113 q iri-q w-Draq Coefficients for a Circular Body ReJ Re =10 LEGEND \ Cap) present study C»" \ ..Cd(^prejsentjtudy__^ Cd(l) Morison eqn O v~j \ Cd(2) Morison eqn s K (O-«3 1 1 i 1 1 0.00 0.02 0.04 0.08 0.08 0.10 Figure 5.39 Variation of Cd{\) and Cd{2) with 02 CHAPTER 6. Conclusions 6.1 Concluding Remarks The present method of representing the interaction between a solid body and viscous flow seems to work very well. The modified method of averaging used is also observed to be quite accurate. The overall agreement between the present study and other known results is very good. Dramatic changes in the flow patterns are observed with changes in the three param eters Ru , Rei and Re2 . The body amplitude parameter fa is found to have the most significant effect on the flow patterns. At high values of Re2/Rei , the vortex behind the body moves to the front at a certain value of fa . Similar flow patterns are observed for the circular and square body for different values of Reynolds numbers. The limiting cases of steady flow over a fixed body and an oscillating body in a stationary fluid compare quite well with known results. The added mass, added damping and the added force are intimately related to the flow pattern. Drastic changes in the flow pattern result in drastic changes in the force quantities. The added mass and added damping are influenced by Rw and Re2 , while the added force is predominately influenced by Rei . The drag coefficients obtained for the limiting case of steady flow over a fixed body compare quite well with known results. The inertia and the drag coefficients obtained from the present study results compare quite will with those obtained from the linearised Morison equation for Re^/Rei < 0.75. At higher values of Re2/Rei , the two results deviate substantially indicating the inaccuracy of the Morison equation at high values of Re2/Rei . 114 Chapter 6.: Conclusions 115 The numerical results for the related associate flow in which the body is at rest in a two-dimensional, time-dependent, viscous flow, compare quite well with known results. The transformations derived between the two associated flows for the velocities, pressure and the force quantities agree with numerical results. The flow patterns for the two associated flows agree better at lower values of /?2 and high values of Rei . Better agreement in results is expected with grid refinement. 6.2 Suggestions for Further Development Some specific recommendations for further studies based on the work in this thesis are 1 Extend the analysis to include the unsymmetric vortex shedding by using the full fluid domain. 2 Incorporate turbulence modelling for the fluid to obtain results for large values of Reynolds numbers suitable for engineering applications 3 Conduct a grid refinement study to determine the effect of grid refinement on the numerical results. 116 References [l] Bertelsen, A. F. An Experimental Investigation of High Reynolds Number Steady Stream ing Generated by Osicllating Cylinders, J. Fluid Mech., 1974, vol 64, part 3, pp 589-597. [2] Davis, R.W. and Moore, E.F., The Numerical Solutions of Flow around Squares, Proc. 2nd Int. Conf. on Num. Meth. in Lam. and Turb. Flow, Venice,1981, Italy, pp 279-290. [3] Davis, R.W. and Moore, E.F., Numerical-Experimental Study of Confined Flow around Rectangular Cylinders, Physics of Fluids, vol 27 Nl, Jan 1984, pp 1-11. [4] Dorodnicyn, A.A., Review of Methods for Solving the Navier-Stokes Equations, 1981, Proc. of 3rd Int. Conf. on Num. Meth. in Fluid Mech., vol 1, pp 1-11. [5] Duck, P.W. and Smith, F.T., Steady Streaming Induced between Oscillating Cylinders, J. Fluid Mech., 1979, vol 91, part 1, pp 93-110. [6] Finlayson, B. A., The Method of Weighted Residuals and Variational Principles, 1972, Academemic Press, New York. [7] Goddard V.P., Numerical Solutions of the Drag Response of a Circular Cylinder to Streamwise Velocity Fluctuations, 1972, Ph.D. Thesis, Univ. of Notre Dame. [8] Graham, J.M.R, Bearman P.W., Vortex Shedding from Bluff Bodies in Oscillatory Flow, Report on Euromech 119, 1980, Journal of Fluid Mechanics, vol 9, pp 225-245. [9] Hughes, T.J.R., Liu, W.K., and Zimmerman, T.K., Langragian-Eulerian Finite El ement Formulation for Incompressible Viscous Flows, U.S.-Japan Conf. Interdisci plinary Finite Element Analysis. 10] Isaacson, M. de St. Q., The Force on Circular Cylinders in Waves, Ph.D. Thesis, 1974, University of Cambridge, England. 11] Leal, L.G., Lea, S.H., Low Reynolds Number Flow Past Cylindrical Bodies of Arbitrary Cross-Sectional Shape, J. Fluid Mech., (1986), vol 164, pp 401-427. 12] Liu, W.K., Development of Finite Element Procedures for Fluid-Structure Interaction, Ph.D. Thesis, 1980, California Instt. of Tech., Pasadena. 13] Nicolaides, R.A., Liu, C.H., Algorithmic and Theoretical Results on Computation of Incompressible Viscous Flows by Finite Element Method, Comput. Fluids,1985, vl3 N3, pp 361-373. 117 [14] Olson, M.D., and Irani, M., Finite Element Analysis of Viscous Flow-Solid Body Inter action, Proc. of 2nd Int. Conf. on Num. Meths. in Lam. and Turb. Flow, 1983,Seattle, U.S.A. [15] Okajima, A. Strouhal Numbers of Rectangular cylinders, J. Fluid Mech., 1982,vol 123, pp 379-398. [16] Olson, M.D. and Pattani, P.G., Nonlinear Analysis of Rigid Body-Viscous Flow Inter action, 5th Symp. on Finite Element Meths. in Flow Probs., 1984, Austin, Texas, also in Finite Elements in Flu ids,1985, vol 6, Eds. Gallagher ,R.H., Carey, G.F., Oden, J.T., Zienkiewicz, O.C.,Wiley. [17] Olson, M.D., Comparison of Various Finite Element Solution Methods for the Navier-Stokes Equations, Finite Elements in Water Resources,1977,Ed. Grey, W.G., at al, pp 4.185-4.203. [18] Pattani, P.G., Nonlinear Analysis of Rigid Body-Viscous Flow Interaction, Ph.D. The sis, 1986, The University of British Columbia, Vancouver, Canada. [19] Pattani, P.G., Forces on Oscillating Bodies in Viscous Fluids,to be published [20] Ramber, E.S., Griffin O.M., Vortex Shedding from a Cylinder Vibrating In Line with an Incident Uniform Flow, J. Fluid Mech., 1976, vol 75 part 2, pp 255-271. [21] Riley, N., The Steady Streaming Induced by a Vibrating Cylinder, J. Fluid Mech., 1975, vol 68, part 4, pp 801-812. [22] Roache, P.G., Computational Fluid Dynamics, 1972, Hermosa Publishers, Alburquerque, New Mexico. [23] Sarpkaya, T., and Isaacson, M., Mechanics of Wave Forces on Offshore Structures, 1981, Van Nostrand Reinhold Company, New York. [24] Sarpkaya, T., In Line Force on a Cylinder Translating in Oscillatory Flow, Storm M. Appl. Ocean Res., vol 7 N4, Oct 1985, pp 188-196. [25] Schlichting, H., Boundary Layer Theory, 1968, Mcgraw-Hill, 6th Ed. [26] Tanida, Y., Okajima, A., Watanabe, Y., Stability of a Circular Cylinder Oscillating in Uniform Flow or in a Wake, J. Fluid Mech., 1973,vol 61, pt. 4, pp 769-784. [27] Tatsuno M., Vortex Wakes Behind a Circular Cylinder Oscillating in the Flow Direc tion, Bulletin Research Instt. Applied Mech., 1972, Kyushu University, No. 36. [28] Tatsuno M., Circulatory Streaming around an Oscillating Circular Cylinder at Low Reynolds Numbers, 1973, J. Physical Society of Japan, Vol. 35, No. 3, pp 915-920. [29] Tatsuno M., Circulatory Streaming in the Viscinity of an Oscillating Square Cylinder, J. of Physical Society of Japan,1974, vol. 36, No. 4, pp 1185-1191. 118 [30] Tatsuno M., Circulatory Streaming in the Viscinity of an Oscillating Triangular Cylin der, J. Physical Soc. Japan,Jan 1975,vol. 38, No. 1, pp 257-264. [31] Tuann, S.Y., and Olson, M.D., Numerical Studies of the Flow around a Circular Cylin der by a Finite Element Method,1978, Computers in Fluids, vol 6, pp 219-240. [32] Zienkiewicz, O.C., The Finite Element Method, Ed. 3,Mcgraw Hill Book Company, U.K. 119 Appendix A Verification of the Restricted Variational Principle Take the first variation of fl (equation 3.1) keeping u°and v° constant. an Ru ~X~ + Re ( uP dt tduP dx .duP dy Su + dv Ru ~~Z. 1" Re 1 U® - „ dt \ dx tdvc + uL ,dv° dy Bv ^rdudSu ^dv dSv ,du dv.dSu ,du dv.dSv-. L dx dx dy dy dy dx' dy ^dy dx dx * r ,dSu dSv-. ,du dvK -,) f — — " WaT + ^ + fe + VyW} «" - J «« + - o Integrate the appropriate terms by parts //„{[ + Ru^ + Re[u — + V — n dv / 0dv° Qdv°\ 6u rd2v d2v d2u dp -[—+2— + dx2 dy2 dxdy dx dp-, r rdu dv-, „ ) , dx2 dy2 dxdy f du \ ,du dvs —•> f , ,du dv , dv s —. i%i and 712 are the direction cosines of the outward pointing normal to the boundary. But 6u and Sv are arbitrary variations. Hence by virtue of Lagrange's Lemma, we 120 get:-du 'dt dv Jdt du° ndu° + Re{u°^-+v°--dx dy „ . 0dv° 0dv° dx dy r d2u d2u d2v [2— + — + dx2 dp, dy2 ' dxdy dx dp rd2v d2v d2u - f 1- 2 1 dx2 dy2 dxdy dy ,du dvx ]=0 Boundary terms:-, du x .du dv. — ,du diK , dv x — Equating u° to u and v° to v, we obtain the governing equations (2.17) and (2.18). Appendix B Derivation of the Discretised Form of the Navier-Stokes Equations Substitute the interpolating functions (eqn 3.2) into the functional ITE (eqn 3.1) and take the first variation of ITE with respect each variable u, v and p. IT = JJ | [RMNj-utult + R^NiNjN^Uiuy, + N^N^u^)} + [RvNiNjVivlt + Re(NiNjNkiSViU°.v 0k + NiNj-N^ViV^)] + NiiXNitXUiUj + Ni,yNjtyViVj + ^{Ni,yNjtyUiUj + NilXNjtVviVj + 2Ni>xNj<yViUj) - (MjNi>xpjUi + MjN^yPjVi) ^jdA- J (XJVfU,- + F/V>t) dS 121 where N. N. -™i N. =8Ii N. -™L dx ">'v- dy Nk = ™± Nk (9u, dv., IX,- y = - V,' f = -ane |^ j^iV.iVyu^ + Re(NiNiNktXuyk + A^y/Vfc>yV°u°) + (2NiiXNjiX + W.yJVy.yjuy + NilXNityVj - Ni>xMjPj J <L4 XiV.dS = 0 arp |£ jjkWA^ + Re(NiNjNk,xuyk + NiNjNkiyVjVk) + (NitXNjtX + 2NiiyNjty)vj + NityNjtxUj - Ni>yMjPj j dA - f YNidS = 0 dYLe r f ~dp~ = ~ i'L (MjNi'xUi + M>Ni>vVi)dA = 0 Representing the above three equations in matrix format, equation (3.3) is obtained. For elements not on the boundary, the line integral vanishes, but for the elements on the boundary, the line integral becomes the consistant load vector. 122 Appendix C Details of Matrices used in Newton-Raphson Iteration Procedure Equation (3.11) can be written as A B C (AA) f2* D £ J AB > - < z2 9 I [AC where ^H2 K 02P + ReubQ /?2R K -M &R 2 M K -Re < f ^ki^Al + + C«C£/2) + 6V.k[A]Al + BfBi/2 + C/Cjf/2) ) ^iik^Al + BfBl/2 + C?CU2) + 6?ik{A)Al + BVB£/2 + C/C£/2) 0 ^(^u + AtBi) + *S*(^y*J. + ^y) ^k(AiBk + AkBj) + ,4»B£ + ^B;) 0 + Al°j) + 'S*MyCfcM + ^y) ^k(^CJi + AlC^ + S¥.k(AHC-k + AlC^ 0 >? = K + Re °i]mAm + "trnj^m tjm m 0 °ijmAm Sx .AU 4- Sy AV °tmj-^m ^ °imjAm + 6?. A" 0 0 0 B 2 W 2 SijmBm + 6imjBm %mBm 0 0 0 123 2 2 CI pu 0 0 P = /?2P + i?eU6Q + /2e + 6y Bv ij m m 0 tj rn m 0 0 0 J = M Q = /?2R + #e 4- &y Cv  + °tmy°m ci pu 0 + 6y. Cv ' xjm 0 # = -M J = X Appendix D Determination of the Nodal Values of the Stream Functions for the Plotting Grid from those of the Finite Element Grid Consider and eight noded isoparametric element as shown in the figure 3.1. The shape functions TV,- and Mi for the element are presented in section 3.4. The value of the stream function \I> and the coordinates (x,y) at a point can be obtained in terms of the nodal 124 values ty{ and nodal coordinates (xt-,y,). * = JV,*t x = NiXi i = 1,2, - •• ,8 y = JVt-yi The following procedure is adopted to obtain the value of ty accurately at each node of the the plotting grid which lie within the finite element. Step 1: Determination of the s,t interpolation within the finite element Assume a quadratic interpolation for s and t of the form s = Ai + A2x + A3y + AAx2 + A5xy + A&y2 + A7x2y + A8y2x t = Bx + B2x + Bzy + BAx2 + Bbxy + BGy2 + B7x2y + B8y2x The [x, y) and (s, t) values at each of the eight nodes on the element are known. Hence eight equations each in Ai,A2,---,A$ and B\,B2, - • • ,B$ are obtained. Thus for any point [xo,y0) in the finite element, it is possible to calculate the corresponding (s0,t0) coordinates using equation 1. Step 2: Determination of the approximate plotting grid area occupied by each element Determine the maximum and minimum value of x and y coordinates of the finite element. The approximate area occupied by this element is a rectangle of length xmax to xmin along the x—axis and height ymai to ymtn along y—axis. For an element which is not too skewed from a rectangular shape, this approximate method works reasonably well. Step 3: Determination of the (s,t) value of each point of the plotting grid with coordinates (xp,yp) within the finite element a. Take the initial guess value of (sp,tp) to be that obtained from equation 1. sp = Ai + A2xp + Azyp + AAx2 + (- A%y2pxp tp = Bi + B2xp + Bzyp + BAx\ + • • • + B%y\xp 125 b. Calculate the shape functions JV, for t = 1,2, • • •, 8 for these values of sp, tp using the equations given in section 3.4. c. Calculate the jacobian J. J = 3JV; ds X3 dt Xl dNj dNj as yj at yj d. Determine the (x*,y*) coordinates of the point using the calculated value of shape functions. x* = NjXj y' = Njyj e. Determine the error in the (x, y) coordinates y = i,2,.--,8 Ax = xp — x* Ay = yP - y* f. Determine the error in the (s,t) coordinates. g. The new value of (s, t) is given as s* = sp + As t* = tp + At-With these new values of {s,t), repeat steps b to f until convergence occurs. Step 4: Determination of at the nodal points of the plotting grid Once convergence has occured in step 3, the stream function value can be obtained using the calculated shape functions. * = Ni*i 126 Appendix E Solving the Equation of Motion in Case b Flow Configuration The case b flow configuration is presented in figures 2.1 and 2.2. The body is oscillating in the direction of steady fluid flow of velocity u3. Fe(t) is the external force and Ff is the fluid force acting on the body. The equation of motion is Ms + Ks = Fe + Ff (1) The fluid force Ff can be represented in terms of the added mass, added damping, and the added force given in equation 4.16. They are represented in their respective dimensional form as Ma = PAbM'a Ca = pAbuC'a Fa = pu0lF'a The primes represent the non-dimensional quantities. Thus the equation of motion be comes (M + Ma)s + Cas + Ks = Fe + Fa (2) Solution procedure to solve equation (2): 1. Let the external force Fe — F0s\nut, then we would expect the response of the body to be of form s = s0sin(u>< — 6). 2. Assume a displacement s0 of the body. A good initial value is the solution of the equation Ms + Ks = Fe. Hence, knowing the steady fluid velocity u3, the frequency of oscillations w, and the amplitude of oscillations s0, determine J?w, R6i and Re^. 3. The flow problem is completely defined by Rei and Rei. Obtain the added mass Ma, the added damping Ca and the added force Fa for these values of Re> , Re2 and Ru using the tables and graphs presented in chapter 5. 4. Solve the equation of motion (2) using the values of MQ, Ca and Fa obtained in step 3 and determine a new value of s0. 5. Repeat steps 2, 3 and 4 using the new value of s0 until convergence occurs. 

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