INTRODUCTORY LECTURES ON FLUID DYNAMICS - LMU

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.

2 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 .. Rotationalandirrotationalflow .. Vortexsheets .. Motionstartedfromrestimpulsively .. 496 Two dimensional flow of a homogeneous , incompressible, inviscidfluid507 Boundary layers in nonrotating Blasiussolution(U=constant).

3 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).

4 Otherwisethe flow lines which at a given instant are everywhere in the directionof the velocity (analogous to electric or magnetic field lines). 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.

5 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.

6 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).

7 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. 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.

8 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.

9 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.

10 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. 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.