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Notes on Fluid Dynamics

Notes on Fluid Dynamics
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Notes on Fluid Dynamics Rodolfo Repetto Department of Civil, Chemical and Environmental Engineering University of Genoa, Italy rodolfo.repettounige.it phone number: +39 010 3532471 http://www.dicca.unige.it/rrepetto/ skype contact: rodolfo-repetto January 13, 2016 Rodolfo Repetto (University of Genoa) Fluid dynamics January 13, 2016 1 / 161Table of contents I 1 Acknowledgements 2 Stress in uids The continuum approach Forces on a continuum The stress tensor Tension in a uid at rest 3 Statics of uids The equation of statics Implications of the equation of statics Statics of incompressible uids in the gravitational eld Equilibrium conditions at interfaces Hydrostatic forces on at surfaces Hydrostatic forces of curved surfaces 4 Kinematics of uids Spatial and material coordinates The material derivative De nition of some kinematic quantities Reynolds transport theorem Principle of conservation of mass The streamfunction The velocity gradient tensor Physical interpretation of the rate of deformation tensor D Physical interpretation of the rate of rotation tensor Rodolfo Repetto (University of Genoa) Fluid dynamics January 13, 2016 2 / 161Table of contents II 5 Dynamics of uids Momentum equation in integral form Momentum equation in di erential form Principle of conservation of the moment of momentum Equation for the mechanical energy 6 The equations of motion for Newtonian incompressible uids De nition of pressure in a moving uid Constitutive relationship for Newtonian uids The Navier-Stokes equations The dynamic pressure 7 Initial and boundary conditions Initial and boundary conditions for the Navier-Stokes equations Kinematic boundary condition Continuity of the tangential component of the velocity Dynamic boundary conditions Two relevant cases 8 Scaling and dimensional analysis Units of measurement and systems of units Dimension of a physical quantity Quantities with independent dimensions Buckingham's  theorem Dimensionless Navier-Stokes equations Rodolfo Repetto (University of Genoa) Fluid dynamics January 13, 2016 3 / 161Table of contents III 9 Unidirectional ows Introduction to unidirectional ows Some examples of unidirectional ows Unsteady unidirectional ows Axisymmetric ow with circular streamlines 10 Low Reynolds number ows Introduction to low Reynolds number ows Slow ow past a sphere Lubrication Theory 11 High Reynolds number ows The Bernoulli theorem Vorticity equation and vorticity production Irrotational ows Bernoulli equation for irrotational ows 12 Appendix A: material derivative of the Jacobian Determinants Derivative of the Jacobian 13 Appendix B: the equations of motion in di erent coordinates systems Cylindrical coordinates Spherical polar coordinates 14 References Rodolfo Repetto (University of Genoa) Fluid dynamics January 13, 2016 4 / 161Acknowledgements Acknowledgements These lecture notes were originally written for the course in \Fluid Dynamics", taught in L'Aquila within the MathMods, Erasmus Mundus MSc Course. A large body of the material presented here is based on notes written by Prof. Giovanni Seminara from the University of Genoa to whom I am deeply indebted. Further sources of material have been the following textbooks: Acheson (1990), Aris (1962), Barenblatt (2003), Batchelor (1967), Ockendon and Ockendon (1995), Pozrikidis (2010). I wish to thank Julia Meskauskas (University of L'Aquila) and Andrea Bon glio (University of Genoa) for carefully checking these notes. Very instructive lms about uid motion have been released by the National Committee for Fluid Mechanics and are available at the following link: http://web.mit.edu/hml/ncfmf.html Rodolfo Repetto (University of Genoa) Fluid dynamics January 13, 2016 5 / 161Stress in uids The state of stress in uids Rodolfo Repetto (University of Genoa) Fluid dynamics January 13, 2016 6 / 161Stress in uids The continuum approach The continuum approach I De nition of a simple uid The characteristic property of uids (both liquids and gases) consists in the ease with which they can be deformed. A proper de nition of a uid is not easy to state as, in many circumstances, it is not obvious to distinguish a uid from a solid. In this course we will deal with \simple uids", which Batchelor (1967) de nes as follows. \A simple uid is a material such that the relative positions of elements of the material change by an amount which is not small when suitable chosen forces, however small in magnitude, are applied to the material. . . . In particular a simple uid cannot withstand any tendency by applied forces to deform it in a way which leaves the volume unchanged." Note: the above de nition does not imply that there will not be resistance to deformation. Rather, it implies that this resistance goes to zero as the rate of deformation vanishes. Microscopic structure of uids The macroscopic properties of solids and uids are related repulsion to their molecular nature and to the forces acting between molecules. In the gure a qualitative diagram of the force between two molecules as a function of their distance d is d 0 shown. d d repulsion; d d attraction, where 0 0 attraction 10 d  10 m. 0 distance d Rodolfo Repetto (University of Genoa) Fluid dynamics January 13, 2016 7 / 161 forceStress in uids The continuum approach The continuum approach II Let d be the average distance between molecules. We have gases d d ; 0 solids and liquids d d . 0 In solids the relative position of particles is xed, in uids (liquids and gases) it can be freely rearranged. Continuum assumption Molecules are separated by voids and the percentage of volume occupied by molecules is very small compared to the total volume. In most applications of uid mechanics the typical spatial scale L under consideration is much larger than the spacing between molecules d. We can then suppose that the behaviour of the uid is the same as if the uid was perfectly continuous in structure. This means that any physical property of the uid, say f , can be regarded as a continuous function of space x (and possibly time t) f = f (x; t):  In order for the continuum approach to be valid it has to be possible to nd a length scale L which is much smaller than the smallest spatial scale at which macroscopic changes take place and much larger than the microscopic (molecular) scale.  5 For instance in uid mechanics normally a length scale L = 10 m is much smaller than the  scale of macroscopic changes but still we have L  d. Rodolfo Repetto (University of Genoa) Fluid dynamics January 13, 2016 8 / 161Stress in uids Forces on a continuum Forces on a continuum I Two kind of forces can act on a continuum body: long distance forces; short distance forces. Long distance forces Such forces are slowly varying in space. This means that if we consider a small volume V the force is approximately constant over it. Therefore, we may write F = fV: As long distance forces are proportional to the volume of uid they act on, they are referred to as volume or body forces. In most cases of interest for this course F will be proportional to the mass of the element F =fV; 3 where  denotes density, i.e. mass per unit volume. The dimensions of  are  = ML (with M 3 mass and L length), and in the International System (SI) it is measured in kg m . The vector eld f is denominated body force eld. f has the dimension of an acceleration, or 2 2 force per unit mass f = LT (with T time), and in the SI it is measured in m/s . In general f depends on space and time: f = f(x; t). If we want to compute the force F on a nite volume V we need to integrate f over V ZZZ F = fdV: V Rodolfo Repetto (University of Genoa) Fluid dynamics January 13, 2016 9 / 161Stress in uids Forces on a continuum Forces on a continuum II Short distance forces Such forces are extremely rapidly variable in space and they act on very short distances. This means that short distance forces are only felt on the surface of contact between adjacent portions of uid. Therefore, we may write  = tS: As short distance forces are proportional to the surface they act on, they are referred to as surface forces. The vector t is denominated tension. The tension t has the dimension of a force 2 1 2 2 per unit surface t = FL = ML T , and in the SI it is measured in Pa=N m . The vector t depends on space x, time t and on the unit vector n normal to the surface on which the stress acts: t = t(x; t; n). Convention: we assume that t is the force per unit surface that the uid on the side of the surface towards which n points exerts on the uid on the other side. Important note: t(n) =t(n). If we want to compute the force  on a nite surface S we need integrating t over S: ZZ  = tdS: S Note that, if S is a closed surface,  represents the force that the uid outside of S exerts on the uid inside. Rodolfo Repetto (University of Genoa) Fluid dynamics January 13, 2016 10 / 161Stress in uids The stress tensor The stress tensor I Cauchy's stress principle We now wish to characterise the state of stress at a point P of a continuum. To this end we consider a small tetrahedron of volume V centred in P. In the gure on the right e i denotes the unit vector in the direction of the axis x i (i = 1; 2; 3). The total surface force acting on the tetrahedron is t(n)S + t(e )S + t(e )S + t(e )S = 0: 1 1 2 2 3 3 In the above expression we have not displayed the dependence of t on x, as the value of x is approximately constant over the small tetrahedron. Moreover, t is xed. Note that if we wrote the momentum balance for the tetrahedron, volume forces would vanish more rapidly than surface forces as the volume tends to zero. Therefore, at leading order, only surface forces contribute to the balance. We note that S = e  nS: i i Therefore S t(n) t(e )e  n t(e )e  n t(e )e  n = 0; 1 1 2 2 3 3 Rodolfo Repetto (University of Genoa) Fluid dynamics January 13, 2016 11 / 161Stress in uids The stress tensor The stress tensor II or, in index notation,   S t (n) t (e )e n t (e )e n t (e )e n = 0: i i 1 1j j i 2 2j j i 3 3j j Note that, throughout the course we will adopt Einstein notation or Einstein summation convention. According to this convention, when an index variable appears twice in a single term of a mathematical expression, it implies that we are summing over all possible values of the index (typically 1, 2, 3). Thus, for instance f f f f i i i i f g = f g + f g + f g or f = f + f + f : j j 1 1 2 2 3 3 j 1 2 3 x x x x j 1 2 3 We can now write   t (n) = t (e )e + t (e )e + t (e )e n: i i 1 1j i 2 2j i 3 3j j Since neither the vector t nor n depend on the coordinate system, the term in square brackets in the above equation is also independent of it. Thus it represents a second order tensor, say (or in index notation  ). ij We can thus write t (n) = n; or, in vector notation, t(n) =n: (1) i ij j  is named the Cauchy stress tensor, or simply stress tensor.  represents the i component of ij ij the stress on the plane orthogonal to the unit vector e . j Rodolfo Repetto (University of Genoa) Fluid dynamics January 13, 2016 12 / 161Stress in uids The stress tensor The stress tensor III Equation (1) implies that to characterise the stress in a point of a continuum we need a second order tensor, i.e. (given a coordinate system) 9 scalar quantities. We will show in the following (section 5) that  is symmetric ( = ), and therefore such scalar quantities reduce to 6. ij ij ji The terms appearing in the principal diagonal of the matrix  represent the so called normal ij stresses, those out of the principal diagonal are named tangential or shear stresses. It is always possible to choose Cartesian coordinates such that takes a diagonal form 0 1  0 0 I A 0  0 ; II 0 0  III and  ,  ,  are named principal stresses and they are the eigenvalues of the matrix I II III representing  . The corresponding directions are called principal directions. ij Obviously, the components of  depend on the coordinate system but the stress tensor does not ij as it is a quantity with a precise physical meaning. For any second order tensor it is possible to de ne 3 invariants, i.e. 3 quantities that do not depend on the choice of the coordinate system. A commonly used set of invariants is given by I = + + = tr = ; I =  +  +  ; I =   = det: 1 I II III jj 2 I II II III III I 3 I II III Rodolfo Repetto (University of Genoa) Fluid dynamics January 13, 2016 13 / 161Stress in uids Tension in a uid at rest Tension in a uid at rest I The structure of in a uid at rest is a consequence of the de nition of simple uid put forward. We consider a small spherical domain in a uid at rest. Since the sphere is very small must be approximately constant at all points within the sphere. We locally choose the principal axes so that we can write as 0 1  0 0 I A  = 0  0 : II 0 0  III We can now write = + , where 1 2 0 1 0 1 1=3 0 0  1=3 0 0 jj I jj A A  = 0 1=3 0 ;  = 0  1=3 0 : 1 jj 2 II jj 0 0 1=3 0 0  1=3 jj III jj The tensor is spherical. It represents a normal compression on the sphere (see gure (a) 1 below). In fact on any portion S of normal n the force is given by S n = 1=3S n. 1 jj The second tensor is diagonal and the sum of the terms on the diagonal is zero. This means 2 that, excluding the trivial case in which all terms are zero, at least one term is positive and one is negative. Referring to the gure on the right this implies that this state of stress necessarily tend to change the shape of the small volume we are considering. This is not compatible with the de nition of simple uid given before, according to which such uid is not able to withstand a system of forces that tends to change its shape. Rodolfo Repetto (University of Genoa) Fluid dynamics January 13, 2016 14 / 161Stress in uids Tension in a uid at rest Tension in a uid at rest II Therefore, must be equal to zero in a uid at rest. Since uids are normally in a state of 2 compression we set  =p ; or, in vector form,  =pI; (2) ij ij where the scalar quantity p is called pressure, and I is the identity matrix. Note that, due to the minus sign in the above equation, p 0 implies compression. In general the pressure is a function of space and time p(x; t). p has the dimension of a force per unit area 2 1 2 (p = FL = ML T ) and in the SI is measured in Pa. Equation (2) implies that at a given point P of a uid at rest the force acting on a small surface passing from P is equal topn, i.e. it is always normal to the surface and its magnitude does not depend on the orientation of the surface. Note: in some textbooks (2) is assumed as an indirect de nition of a simple uid (Euler assumption). Rodolfo Repetto (University of Genoa) Fluid dynamics January 13, 2016 15 / 161Statics of uids Statics of uids Rodolfo Repetto (University of Genoa) Fluid dynamics January 13, 2016 16 / 161Statics of uids The equation of statics The equation of statics I Equation of statics in integral form Let V be a volume of uid within a body of uid at rest and let S be its bounding surface. We wish to write the equilibrium equation for this volume. From the equilibrium of forces we have ZZZ ZZ fdV + tdS = 0: (3) V S Equation (2) allows to rewrite the above expression as ZZZ ZZ fdV + pndS = 0; (4) V S which represents the integral form of the equation of statics. The above equation is often conveniently written in compact form as F +  = 0; (5) with F resultant of all body forces acting on V and  resultant of surface forces acting on S. Rodolfo Repetto (University of Genoa) Fluid dynamics January 13, 2016 17 / 161Statics of uids The equation of statics The equation of statics II Equation of statics in di erential form Using Gauss theorem equation (4) can be written as ZZZ frpdV = 0: V Since V is arbitrary the following di erential equation must hold p frp = 0; or, in index notation, f = 0; (6) i x i which is the equation of statics in di erential form. Equilibrium to rotation In principle, the above equation alone is not sucient to ensure equilibrium as we also have to impose an equilibrium balance to rotation. This can be written as ZZZ ZZ x fdV + px ndS = 0; (7) V S Rodolfo Repetto (University of Genoa) Fluid dynamics January 13, 2016 18 / 161Statics of uids The equation of statics The equation of statics III or, in index notation, ZZZ ZZ  x f dV + p x n dS = 0: (8) ijk j k ijk j k V S Note:  is the alternating tensor. Its terms are all equal to zero unless when i, j and k are ijk di erent from each other, in which case  takes the values 1 or -1 depending if i, j and k are or ijk not in cyclic order. Thus, we have i j k  ijk 1 2 3 1 3 1 2 1 2 3 1 1 2 1 3 -1 1 3 2 -1 3 2 1 -1 Applying Gauss theorem to equation (8) we have: ZZZ    x f (px ) dV = 0: ijk j k j x k V Rodolfo Repetto (University of Genoa) Fluid dynamics January 13, 2016 19 / 161Statics of uids The equation of statics The equation of statics IV Carrying on the calculations: ZZZ    x f (px ) dV = ijk j k j x k V ZZZ   p x j  x f x p dV = ijk j k j x x k k V   (9) ZZZ p  x f x p dV = ijk j k j jk x k V   ZZZ ZZZ p  x f x dV p  dV = 0; ijk j k j ijk jk x k V V where  is the Kronecker delta ( = 0 if i =6 j and  = 1 if i = j). The above equation is ij ij ij automatically satis ed as the rst integral vanishes due to equation (6) and   = 0 by ijk ij de nition. Rodolfo Repetto (University of Genoa) Fluid dynamics January 13, 2016 20 / 161