Introduction To Fluid Dynamics* - Scientia Marina
SCIENTIA MARINASCI. MAR., 61 (Supl. 1): 7-241997LECTURES ON PLANKTON AND TURBULENCE. C. MARRASÉ, E. SAIZ and J.M. REDONDO (eds.)Introduction to Fluid Dynamics*T.J. PEDLEYDepartment of Applied Mathematics and Theoretical Physics, University of Cambridge,Silver St., Cambridge CB3 9EW, U.K.SUMMARY: The basic equations of fluid mechanics are stated, with enough derivation to make them plausible but without rigour. The physical meanings of the terms in the equations are explained. Again, the behaviour of fluids in real situations is made plausible, in the light of the fundamental equations, and explained in physical terms. Some applications relevant to life in the ocean are given.Key words: Kinematics, fluid dynamics, mass conservation, Navier-Stokes equations, hydrostatics, Reynolds number, drag,lift, added mass, boundary layers, vorticity, water waves, internal waves, geostrophic flow, hydrodynamic instability.FUNDAMENTAL LAWS AND EQUATIONSKinematicsWhat is a fluid? Specification of motionA fluid is anything that flows, usually a liquid ora gas, the latter being distinguished by its great relative compressibility.Fluids are treated as continuous media, and theirmotion and state can be specified in terms of thevelocity u, pressure p, density ρ, etc evaluated atevery point in space x and time t. To define the density at a point, for example, suppose the point to besurrounded by a very small element (small compared with length scales of interest in experiments)which nevertheless contains a very large number ofmolecules. The density is then the total mass of all*Received February 20, 1996. Accepted March 25, 1996.the molecules in the element divided by the volumeof the element.Considering the velocity, pressure, etc as functions of time and position in space is consistent withmeasurement techniques using fixed instruments inmoving fluids. It is called the Eulerian specification.However, Newton’s laws of motion (see below) areexpressed in terms of individual particles, or fluidelements, which move about. Specifying a fluidmotion in terms of the position X(t) of an individualparticle (identified by its initial position, say) iscalled the Lagrangian specification. The two arelinked by the fact that the velocity of such an element is equal to the velocity of the fluid evaluated atthe position occupied by the element:dX u[ X(t),t ] .dt(1)The path followed by a fluid element is called aparticle path, while a curve which, at any instant, iseverywhere parallel to the local fluid velocity vectorINTRODUCTION TO FLUID DYNAMICS 7
is called a streamline. Particle paths are coincidentwith streamlines in steady flows, for which thevelocity u at any fixed point x does not vary withtime t.Material derivative; acceleration.Newton’s Laws refer to the acceleration of a particle. A fluid element may have acceleration bothbecause the velocity at its location in space is changing (local acceleration) and because it is moving toa location where the velocity is different (convectiveacceleration). The latter exists even in a steady flow.How to evaluate the rate of change of a quantityat a moving fluid element, in the Eulerian specification? Consider a scalar such as density ρ (x ,t). Letthe particle be at position x at time t, and move to x δx at time t δt, where (in the limit of small δt)δ x u(x , t)δ t .(2)FIG. 1. – Mass flow into and out of a small rectangular region ofspace.etc. Combining all three components in vector shorthand we writeDu u (u. )u,Dt tThen the rate of change of ρ following the fluid,or material derivative, isbut care is needed because the quantity u is notdefined in standard vector notation. Note that u/ tis the local acceleration, (u. )u the convectiveacceleration. Note too that the convective acceleration is nonlinear in u, which is the source of thegreat complexity of the mathematics and physics offluid motion.Dρρ (x δ x , t δ t) ρ (x,t) limδ t 0Dtδt ρ δ x ρ δ y ρ δ z ρ x δt y δt z δt t(by the chain rule for partial differentiation) ρ ρ ρ ρ u v w t x y zConservation of mass(3a)(using (2)) ρ u. ρ t(3b)in vector notation, where the vector ρ is the gradient of the scalar field ρ :A similar exercise can be performed for eachcomponent of velocity, and we can write the x-component of acceleration as8 T.J. PEDLEYThis is a fundamental principle, stating that forany closed volume fixed in space, the rate ofincrease of mass within the volume is equal to thenet rate at which fluid enters across the surface ofthe volume. When applied to the arbitrary small rectangular volume depicted in fig. 1, this principlegives: x y z ρ ρ ρ ρ , , . x y z Du u u u u u v w , x y zDt t(4b)(4a) ρ y z [ρ u ]x [ρ u ]x x t(()) z x [ρ v ]y [ρ v ]y y () x y [ρ w ]z [ρ w ]z z .Dividing by x y z and taking the limit as thevolume becomes very small we get
ρ (ρ u ) (ρ v ) (ρ w ) t x y z(5a)or (in shorthand) ρ div(ρ u ) t(5b)where we have introduced the divergence of a vector. Differentiating the products in (5a) and using(3), we obtainDρ ρ divu.(6)DtThis says that the rate of change of density of a fluidelement is positive if the divergence of the velocityfield is negative, i.e. if there is a tendency for theflow to converge on that element.If a fluid is incompressible (as liquids often are,effectively) then even if its density is not uniformeverywhere (e.g. in a stratified ocean) the density ofeach fluid element cannot change, soDρ 0Dt(7)everywhere, and the velocity field must satisfyFIG. 2. – An arbitrary region of fluid divided up into small rectangular elements (depicted only in two dimensions).that, if two elements A and B exert forces on eachother, the force exerted by A on B is the negative ofthe force exerted by B on A.To apply these laws to a region of continuousfluid, the region must be thought of as split up intoa large number of small fluid elements (fig. 2), oneof which, at point x and time t, has volume V , say.Then the mass of the element is ρ (x,t) V , and itsacceleration is Du/Dt evaluated at (x,t). What is theforce?Body force and stressdivu 0(8a) u v w 0 . x y z(8b)orThis is an important constraint on the flow of anincompressible fluid.The Navier-Stokes equationsNewton’s Laws of MotionNewton’s first two laws state that if a particle (orfluid element) has an acceleration then it must beexperiencing a force (vector) equal to the product ofthe acceleration and the mass of the particle:The force on an element consists in general oftwo parts, a body force such as gravity exerted onthe element independently of its neighbours, andsurface forces exerted on the element by all the otherelements (or boundaries) with which it is in contact.The gravitational body force on the element V isgρ (x, t) V , where g is the gravitational acceleration. The surface force acting on a small planar surface, part of the surface of the element of interest,can be shown to be proportional to the area of thesurface, A say, and simply related to its orientation,as represented by the perpendicular (normal) unitvector n (fig. 3). The force per unit area, or stress, isthen given byforce mass acceleration.For any collection of particles this becomesnet force rate of change of momentumwhere the momentum of a particle is the product ofits mass and its velocity. Newton’s third law statesFIG. 3. – Surface force on an arbitrary small surface element embedded in the fluid, with area A and normal n. F is the force exertedby the fluid on side 1, on the fluid on side 2.INTRODUCTION TO FLUID DYNAMICS 9
F x σ xx n x σ xy n y σ xz nzF y σ yx n x σ yy n y σ yz nz(9a)Fz σ zx n x σ zy n y σ zz nzor, in shorthand,F σn (9b)where σ is a matrix quantity, or tensor, depending on x and t but not n or A. σ is called the stress ten sor, and can be shown to be symmetric (i.e. σyx σxy,etc) so it has just 6 independent components.It is an experimental observation that the stress ina fluid at rest has a magnitude independent of n andis always parallel to n and negative, i.e. compressive. This means that σxy σyz σzx 0, σxx σyy σzz p, say, where p is the positive pressure (hydrostatic pressure); alternatively,σ –p I (10)where I is the identity matrix. The relation between stress and deformation rateIn a moving fluid, the motion of a general fluidelement can be thought of as being broken up intothree parts: translation as a rigid body, rotation as arigid body, and deformation (see fig. 4).Quantitatively, the translation is represented by thevelocity field u, the rigid rotation is represented bythe curl of the velocity field, or vorticity,ω curlu ,(11)and the deformation is represented by the rate ofdeformation (or rate of strain) e which, like stress, is a symmetric tensor quantity made up of the symmetric part of the velocity gradient tensor. Formally,1e ( u u T ) 2or, in full component form,(12) u1 u v 1 u w 2 y x 2 z x x 1 v u v1 v w e 2 z y y 2 x y w 1 w u 1 w v 2 x z 2 y z z (13)Note that the sum of the diagonal elements of e is equal to div u.It is a further matter of experimental observationthat, whenever there is motion in which deformationis taking place, a stress is set up in the fluid whichtends to resist that deformation, analogous to friction. The property of the fluid that causes this stressis its viscosity. Leaving aside pathological (‘nonNewtonian’) fluids the resisting stress is generallyproportional to the deformation rate. Combining thisstress with pressure, we obtain the constitutive equation for a Newtonian fluid:σ –p I 2µ e – 2/3µ div uI (14)The last term is zero in an incompressible fluid, andwe shall ignore it henceforth. The quantity µ is thedynamic viscosity of the fluid.To illustrate the concept of viscosity, consider theunidirectional shear flow depicted in fig. 4 wherethe plane y 0 is taken to be a rigid boundary. Thenormal vector n is in the y-direction, so equations(9) show that the stress on the boundary is()F σ xy , σ yy , σ zy .From (14) this becomesFIG. 4. – A unidirectional shear flow in which the velocity is in thex- direction and varies linearly with the perpendicular componenty : u αy. In time t a small rectangular fluid element at level y0 istranslated a distance αy0 t, rotated through an angle α/2, anddeformed so that the horizontal surfaces remain horizontal, and thevertical surfaces are rotated through an angle α.10 T.J. PEDLEY()F 2 µ exy , p µ eyy , µ ezy ,but because the velocity is in the x-direction onlyand varies with y only, the only non-zero component
(σof e is e xy 1 u . Hence 2 yxx x x σ xxx) y z.If x is small enough, this is u F µ , p,0 y In other words, the boundary experiences a perpendicular stress, downwards, of magnitude p, the pressure, and a tangential stress, in the x-direction, equalto µ times the velocity gradient u/ y. (It can beseen from (9) and (14) that tangential stresses arealways of viscous origin.)The Navier-Stokes equationsThe easiest way to apply Newton’s Laws to amoving fluid is to consider the rectangular blockelement in fig. 5. Newton’s Law says that the massof the element multiplied by its acceleration is equalto the total force acting on it, i.e. the sum of the bodyforce and the surface forces over all six faces. Theresulting equation is a vector equation; we will consider just the x-component in detail. The x-component of the stress forces on the faces perpendicularto the x-axis is the difference between the perpendicular stress σxx evaluated at the right-hand face(x x) and that evaluated at the left-hand face (x)multiplied by the area of those faces, y z, i.e. σ xx x y z. xThe x-component of the forces on the faces perpendicular to the y-axis is σ xyy y σ xyσ xy z x x y z,y yand similarly for the faces perpendicular to the zaxis. Hence the x-component of Newton’s LawgivesDu ρ g x x y z Dt σ xy σ σxz x y z xx x y z (ρ x y z )( )or, dividing by the element volume,ρ σ xx σ xy σ xzDu ρ gx .Dt x y z(15a)Similar equations arise for the y- and z-components,and they can be combined in vector form to giveFIG. 5. – Normal and tangential surface forces per unit area (stress) on a small rectangular fluid element in motion.INTRODUCTION TO FLUID DYNAMICS 11
Duρ ρ g div σ Dt(15b)The equations can be further transformed, usingthe constitutive equation (14) (with div u 0) and(13) to express e in terms of u, to give for (15a) ρ 2u 2u 2u Du p ρgx µ 2 2 2 . x x y z Dt(16a)permitted are discussed below. When it is allowed,however, we can put µ 0 in equations (16) andthese are greatly simplified.For quantitative purposes we should note the values of density and viscosity for fresh water and airat 1 atmosphere pressure and at different temperatures:Tempρ(kgm-3)Waterµ(kgm-1s-1)Air (dry)ρ(kgm-3) µ(kgm-1s-1)Similarly in the y- and z-directions:222ρ v v v Dv p ρ gy µ 2 2 2 y y z Dt xρ 2w 2w 2w Dw p ρ gz µ 2 2 2 . (16c) x z y z Dt(16b)In these equations, it should not be forgotten thatDu/Dt etc are given by equations (4).Finally, the above three equations can be compressed into a single vector equation as follows:ρDu ρ g p µ 2 uDt1.0000 x 1030.9997 x 1030.9982 x 1031.787 x 10-31.304 x 10-31.002 x 10-31.2931.2471.2051.71 x 10-51.76 x 10-51.81 x 10-5Boundary conditionsWhether the fluid is viscous or not, it cannotcross the interface between itself and another medium (fluid or solid), so the normal component ofvelocity of the fluid at the interface must equal thenormal component of the velocity of the interfaceitself:(16d)2where the symbol is shorthand for 2 2 .22 x y z 2Equations (16a-c), or (16d), are the Navier-Stokesequations for the motion of a Newtonian viscousfluid. Recall that the left side of (16d) represents themass-acceleration, or inertia terms in the equation,while the three terms on the right side are respectively the body force, the pressure gradient, and theviscous term.The four equations (16a-c) and (8b) are four nonlinear partial differential equations governing fourunknowns, the three velocity components u,v,w, andthe pressure p, each of which is in general a functionof four variables, x, y, z and t. Note that if the density ρ is variable, that is a fifth unknown, and the corresponding fifth equation is (7). Not surprisingly,such equations cannot be solved in general, but theycan be used as a framework to understand thephysics of fluid motion in a variety of circumstances.A particular simplification that can sometimes bemade is to neglect viscosity altogether (to assumethat the fluid is inviscid). Conditions in which this is12 T.J. PEDLEY0 C10 C20 Cun Un or n.u n.U(17a)where U is the interface velocity. In particular, on asolid boundary at rest,n.u 0(17b)In a viscous fluid it is another empirical fact thatthe velocity is continuous everywhere, and in particular that the tangential component of the velocity ofthe fluid at the interface is equal to that of the interface - the no-slip condition. Henceu U(18)at the interface (u 0 on a solid boundary at rest).There are boundary conditions on stress aswell as on velocity. In general they can be summarised by the statement that the stress F (eq.9)must be continuous across every surface (not thestress tensor, note, just σ .n), a condition that fol lows from Newton’s third law. At a solid boundary this condition tells you what the force per unitarea is and the total stress force on the boundaryas a whole is obtained by integrating the stressover the boundary (thus the total force exerted bythe fluid on an immersed solid body can be calculated).
When the fluid of interest is water, and theboundary is its interface with the air, the dynamicsof the air can often be neglected and the atmospherecan be thought of as just exerting a pressure on theliquid. Then the boundary conditions on the liquid’smotion are that its pressure (modified by a small viscous normal stress) is equal to atmospheric pressureand that the viscous shear stress is zero.CONSEQUENCES: PHYSICAL PHENOMENAFIG. 6. – Flow of a uniform stream with velocity U in the x-direction past a body with boundary S which has a typical length scale L.HydrostaticsWe consider a fluid at rest in the gravitationalfield, with a free upper surface at which the pressureis atmospheric. We choose a coordinate system x, y,z such that z is measured vertically upwards, so gx gy 0 and gz -g, and we choose z 0 as the levelof the free surface. The density ρ may vary withheight, z. Thus all components of u are zero, andpressure p patm at z 0. The Navier-Stokes equations (16) reduce simply to p p p 0, ρ g. x y zHenceNote that, for constant density problems in whichthe pressure does not arise explicitly in the boundaryconditions (e.g. at a free surface), the gravity termcan be removed from the equations by including it inan effective pressure, pe. Putpe p gρ z(21)in equations (16) (with gx gy 0, gz -g) and seethat g disappears from the equations, as long as pereplaces p.Flow past bodies0p patm g z ρ dz(19)or, for a fluid of constant density,p patm gρ z :the pressure increases with depth below the free surface (z increasingly negative).The above results are independent of whetherthere is a body at rest submerged in the fluid. If thereis, one can calculate the total force exerted by thefluid by integrating the pressure, multiplied by theappropriate component of the normal vector n, overthe body surface. The result is that, whatever theshape of the body, the net force is an upthrust andequal to g times the mass of fluid displaced by thebody. This is Archimedes’ principle. If the fluid density is uniform, and the body has uniform density ρb,then the net force on the body, gravitational andupthrust, corresponds to a downwards force equal to(ρ b ρ )VgThe flow of a homogeneous incompressiblefluid of density ρ and viscosity µ past bodies hasalways been of interest to fluid dynamicists ingeneral and to oceanographers or ocean engineersin particular. We are concerned both with fixedbodies, past which the flow is driven at a givenspeed (or, equivalently, bodies impelled by anexternal force through a fluid otherwise at rest)and with self-propelled bodies such as marineorganisms.Non-dimensionalisation: the Reynolds numberConsider a fixed rigid body, with a typicallength scale L, in a fluid which far away has constant, uniform velocity U in the x-direction (fig.6). Whenever we want to consider a particularbody, we choose a sphere of radius a, diameter L 2a. The governing equations are (8) and (16),and the boundary conditions on the velocity fieldare(20)where V is the volume of the body. The quantity(ρb – ρ) is called the reduced density of the body.u v w 0on the body surface, S(22)u U , v 0, w 0 at infinity.(23)INTRODUCTION TO FLUID DYNAMICS 13
Usually the flow will be taken to be steady, ie 0 , but we shall also wish to think about devel topment of the flow from rest.For a body of given shape, the details of the flow(i.e. the velocity and pressure at all points in thefluid, the force on the body, etc) will depend on U ,L , µ and ρ as well as on the shape of the body.However, we can show that the flow in fact dependsonly on one dimensionless parameter, the Reynoldsnumberρ LU Re ,(24)µand not on all four quantities separately, so onlyone range of experiments (or computations) wouldbe required to investigate the flow, not four. Theproof arises when we express the equations indimensionless form by making the following transformations:x′ x / L, y′ y / L, z′ z / L, t′ U t / L,u′ u / U , v′ v / U , w′ w / U , p′ p / ρ U 2 .Then the equations become: (8b): u′ v′ w′ 0; x′ y′ z′(25)Du(16a), with Dt replaced by (4a): u′ u′ u′
INTRODUCTION TO FLUID DYNAMICS9 FIG. 2. – An arbitrary region of fluid divided up into small rectan-gular elements (depicted only in two dimensions). FIG. 3. – Surface force on an arbitrary small surface element embed-ded in the fluid, with area A and normal n. F is the force exerted by the fluid on side 1, on the fluid on side 2.
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