INTRODUCTORY LECTURES ON FLUID DYNAMICS

1 INTRODUCTORY LECTURES ON FLUIDDYNAMICSR oger K. SmithVersion: June 13, 2008 Contents1 Descriptionoffluidflow .. Incompressibleflows .. Conservationofmass:thecontinuityequation .. 72 Equation of motion: some Rate-of-changemovingwiththefluid .. Internalforcesinafluid .. Fluidandsolids:pressure .. Pressure gradient forces in a FLUID in macroscopic equilibrium . Equilibrium of a horizontal element .. Equilibrium of a vertical element .. 183 Equations of motion for an inviscid Equationsofmotionforanincompressiblevisc ousfluid .. Dynamicpressure(orperturbationpressure). . Boundary conditions for FLUID An alternative boundary condition .. 294 Bernoulli s Application of Bernoulli s 325 The vorticity TheHelmholtzequationforvorticity .. Physical significance of the term ( ) Kelvin ResultsfollowingfromKelvinsTheorem.

2 Rotationalandirrotationalflow .. Vortexsheets .. Motionstartedfromrestimpulsively .. 496 Two dimensional flow of a homogeneous, incompressible, inviscidfluid507 Boundary layers in nonrotating Blasiussolution(U=constant).. Furtherreading .. 59 Chapter 1 IntroductionThese notes are intended to provide a survey of basic concepts in FLUID dynamicsas a preliminary to the study of dynamical meteorology. They are based on a moreextensive course of LECTURES prepared by Professor B. R. Morton of Monash University, Description of FLUID flowThe description of a FLUID flow requires a specification or determination of thevelocityfield, a specification of the FLUID velocity at every point in the region. In general,this will define avector fieldof position and time,u=u(x, t).Steady flowoccurs whenuis independent of time ( , u/ t 0). Otherwisethe flow lines which at a given instant are everywhere in the directionof the velocity (analogous to electric or magnetic field lines).

3 In steady flow thestreamlines are independent of time, but the velocity can vary in magnitude along astreamline (as in flow through a constriction in a pipe) - see Fig. : Schematic diagram of flow through a constriction in a 1. INTRODUCTION4 Particle pathsare lines traced out by marked particles as time evolves. Insteady flow particle paths are identical to streamlines; in unsteady flow they aredifferent, and sometimes very different. Particle paths are visualized in the laboratoryusing small floating particles of the same density as the FLUID . Sometimes they arereferred to lines or streaklines are tracedout over time by all particles passingthrough a given point; they may be visualized, for example, using a hypodermicneedle and releasing a slow stream of dye. In steady flow these are streamlines; inunsteady flow they are neither streamlines nor particle should be emphasized that streamlines represent the velocity field at a specificinstant of time, whereas particle paths and streaklines provide a representation ofthe velocity field over a finite period of time.

4 In the laboratory we can obtaina record of streamlines photographicallyby seeding the FLUID with small neutrallybuoyant particles that move with the flow and taking a short exposure ( sec),long enough for each particle to trace out a short segment of line; the eye readilylinks these segments into continuous streamlines. Particle paths and streaklines areobtained from a time exposure long enough for the particle or dye trace to traversethe region of Equations for streamlinesThe streamline through the pointP,say(x, y, z), has the direction ofu=(u, v, w).Figure : Velocity vector and streamlineLetQbe the neighbouring point (x+ x, y+ y, z+ z) on the streamline. Then x u t, y v t, z w tand as t 0, we obtain the differential relationshipdxu=dyv=dzw,( )between the displacementdxalong a streamline and the velocity components. Equa-tion ( ) gives two differential equations (why?). Alternatively, we can represent thestreamline parameterically(with time as parameter) asCHAPTER 1.

5 INTRODUCTION5 dxu= dt, dyv= dt, dzw= dt,( )Example 1 Find the streamlines for the velocity fieldu=( y, x,0), where is a ( ) gives dx y=dy x= first pair of ratios give (xdx+ydy)=0orx2+y2= (z),where is an arbitrary function ofz. The second pair give dz=0or z= the streamlines are circlesx2+y2=c2in planesz=constant(we havereplaced (z), a constant when z is constant, byc2).Note that the velocity at P with position vectorxcan be expressed asu= k xand corresponds with solid body rotation about thekaxis with angular velocity . Distinctive properties of fluidsAlthough fluids are molecular in nature , they can be treated ascontinuous mediafor most practical purposes, the exception being rarefied gases. Real fluids generallyshow somecompressibilitydefined as =1 d dp=change in density per unit change in pressuredensity,but at normal atmospheric flow speed, the compressibility of air is a relative bysmall effect and for liquids it is generally negligible.

6 Note that sound waves owe theirexistence to compressibility effects as do supersonic bangs produced by aircraftflying faster than sound. For many purposes it is accurate to assume that fluids areCHAPTER 1. INTRODUCTION6incompressible, they suffer no change in density with pressure. For the presentwe shall assume also that they arehomogeneous, , density = one solid body slides over another,frictional forcesact between them toreduce the relative motion. Friction acts also when layers of FLUID flow over oneanother. When two solid bodies are in contact (more precisely when there is anormal force acting between them) at rest, there is a threshold tangential forcebelowwhichrelative motion will not occur. It is called thelimiting a solid body resting on a flat surface under the action of gravity (see Fig. ).Figure : Forces acting on a rigid body at increased from zero,F=TuntilT= N,where is the so-calledcoefficient of limiting friction which depends on the degree of roughness between thesurface.

7 ForT> N, the body will overcome the frictional force and distinguishing characteristic of most fluids in their inability to support tangentialstresses between layers without motion occurring; there is no analogue of limitingfriction. Exceptions are certain types of so-calledvisco-elasticfluids such as friction is characterized byviscositywhich is a measure of the magnitudeof tangential frictional forces in flows with velocity forcesareimportant in many flows, but least important in flow past streamlined bodies. Weshall be concerned mainly withinviscidflows where friction is not important, but itis essential to acquire some idea of the sort of flow in which friction may be neglectedwithout completely misrepresenting the behaviour. The total neglect of friction isrisky!To begin with we shall be concerned mainly withhomogeneous, incompressibleinviscid Incompressible flowsConsider an element of FLUID bounded by a tube of streamlines , known as a streamtube.

8 In steady flow, no FLUID can cross the walls of the stream tube (as they areeverywhere in the direction of flow).Hence for incompressible fluids the mass flux ( = mass flow per unit time) acrosssection 1 (= v1S1) is equal to that across section 2 (= v2S2), as there can be noCHAPTER 1. INTRODUCTION7accumulation of FLUID between these sections. HencevS= constant and in the limit,for stream tubes of small cross-section,vS=constantalong an elementary constant along an elementary stream :It follows that, where streamlines contract the velocity increases, where they ex-pand it decreases. Clearly, the streamline pattern contains a great deal of informationabout the velocity vector fields with the property that(vector magnitude) (area of tube)remains constant along a tube are calledsolenoidal. The velocity field for an incom-pressible FLUID is Conservation of mass: the continuity equationApply the divergence theorem V udV= Su ndsto an arbitrarily chosen volumeVwith closed surfaceS(Fig.)

9 Letnbe a unitoutward normal to an element of the If the FLUID is incompressibleand there are no mass sources or sinks within S, then there can be neither continuingaccumulation of FLUID withinVnor continuing loss. It follows that the net flux offluid across the surfaceSmust be zero, , Su ndS=0,whereupon V udV= 0. This holds for an arbitrary volumeV, and therefore u= 0 throughout an incompressible flow without mass sources or sinks. This isthe continuity equation for ahomogeneous,incompressiblefluid. It corresponds withmass 1. INTRODUCTION8 Figure :Chapter 2 Equation of motion: somepreliminariesThe equation of motion is an expression of Newtons second law of motion:mass acceleration = apply this law we must focus our attention on a particular element of FLUID ,say the small rectangular element which at timethas vertex atP[= (x, y, z)] andedges of length x, y, z. The mass of this element is x y z,where is thefluiddensity(or mass per unit volume), which we shall assume to be : Configuration of a small rectangular element of velocity in the FLUID ,u=u(x, y, z, t) is a function both of position (x, y, z)and timet, and from this we must derive a formula for the acceleration of the elementof FLUID which is changing its position with time.

10 Consider, for example, steady flowthrough a constriction in a pipe (see Fig. ). Elements of FLUID must accelerateinto the constriction as the streamlines close in and decelerate beyond as they openout again. Thus, in general, the acceleration of an element ( , the rate-of-changeofuwith time for that element) includes arate-of-change at a fixed position u/ t9 CHAPTER 2. EQUATION OF MOTION: SOME PRELIMINARIES10 Figure :and in addition a change associated with its change of position with time. We derivean expression for the latter in section forces acting on the FLUID element consist of:(i)body forces, which are forces per unit mass acting throughout the FLUID becauseof external causes, such as the gravitationalweight,and(ii)contact forcesacting across the surface of the element from adjacent are discussed further in section Rate-of-change moving with the fluidWe consider first the rate-of-change of a scalar property, for example the temperatureof a FLUID , following a FLUID element.