LECTURES IN ELEMENTARY FLUID DYNAMICS

1 LECTURES IN ELEMENTARYFLUID DYNAMICS :Physics, Mathematics and ApplicationsJ. M. McDonoughDepartments of Mechanical Engineering and MathematicsUniversity of Kentucky, Lexington, KY 40506-0503c 1987, 1990, 2002, 2004, 2009 Contents1 Importance of Fluids .. Fluids in the pure sciences .. Fluids in technology .. The Study of Fluids .. The theoretical approach .. Experimental FLUID DYNAMICS .. Computational FLUID DYNAMICS .. Overview of Course .. 82 Some Background: Basic Physics of The Continuum Hypothesis .. Definition of a FLUID .. Shear stress induced deformations .. More on shear stress .. FLUID Properties .. Viscosity .. Thermal conductivity .. Mass diffusivity .. Other FLUID properties .. Classification of Flow Phenomena .. Steady and unsteady flows.

2 Flow dimensionality .. Uniform and non-uniform flows .. Rotational and irrotational flows .. Viscous and inviscid flows .. Incompressible and compressible flows .. Laminar and turbulent flows .. Separated and unseparated flows .. Flow Visualization .. Streamlines .. Pathlines .. Streaklines .. Summary .. 45iiiCONTENTS3 The Equations of FLUID Lagrangian & Eulerian Systems; the Substantial Derivative .. The Lagrangian viewpoint .. The Eulerian viewpoint .. The substantial derivative .. Review of Pertinent Vector Calculus .. Gauss s theorem .. Transport theorems .. Conservation of Mass the continuity equation .. Derivation of the continuity equation .. Other forms of the differential continuity equation .. Simple application of the continuity equation.

3 Control volume (integral) analysis of the continuityequation .. Momentum Balance the Navier Stokes Equations .. A basic force balance; Newton s second law of motion .. Treatment of surface forces .. The Navier Stokes equations .. Analysis of the Navier Stokes Equations .. Mathematical structure .. Physical interpretation .. Scaling and Dimensional Analysis .. Geometric and dynamic similarity .. Scaling the governing equations .. Dimensional analysis via the Buckingham theorem .. Physical description of important dimensionless parameters .. Summary .. 994 Applications of the Navier Stokes FLUID Statics .. Equations of FLUID statics .. Buoyancy in static fluids .. Bernoulli s Equation .. Derivation of Bernoulli s equation .. Example applications of Bernoulli s equation.

4 Control-Volume Momentum Equation .. Derivation of the control-volume momentum equation .. Application of control-volume momentum equation .. Classical Exact Solutions to N. S. Equations .. Couette flow .. Plane Poiseuille flow .. Pipe Flow .. Some terminology and basic physics of pipe flow .. The Hagen Poiseuille solution .. Practical Pipe Flow Analysis .. Summary .. 158 List of Mean Free Path and Requirements for Satisfaction of Continuum Hypothesis; (a)mean free path determined as average of distances between collisions; (b) a volumetoo small to permit averaging required for satisfaction of continuum hypothesis.. Comparison of deformation of solids and liquids under application of a shear stress;(a) solid, and (b) liquid.. Behavior of things that flow ; (a) granular sugar, and (b) coffee.

5 Flow between two horizontal, parallel plates with upperone moving at velocityU.. Physical situation giving rise to the no-slip condition.. Structure of water molecule and effect of heating on short-range order in liquids; (a)low temperature, (b) higher temperature.. Effects of temperature on molecular motion of gases; (a) low temperature, (b) highertemperature.. Diffusion of momentum initial transient of flow between parallel plates; (a) veryearly transient, (b) intermediate time showing significantdiffusion, (c) nearly steady-state profile.. Interaction of high-speed and low-speed FLUID parcels.. Pressure and shear stress.. Surface tension in spherical water droplet.. Capillarity for two different liquids.. Different types of time-dependent flows; (a) transient followed by steady state, (b)unsteady, but stationary, (c) unsteady.

6 Flow dimensionality; (a) 1-D flow between horizontal plates, (b) 2-D flow in a 3-Dbox, (c) 3-D flow in a 3-D box.. Uniform and non-uniform flows; (a) uniform flow, (b) non-uniform, but locallyuniform flow, (c) non-uniform flow.. 2-D vortex from flow over a step.. 3-D vortical flow of FLUID in a box.. Potential Vortex.. Laminar and turbulent flow of water from a faucet; (a) steady laminar, (b) periodic,wavy laminar, (c) turbulent.. da Vinci sketch depicting turbulent flow.. Reynolds experiment using water in a pipe to study transition to turbulence; (a)low-speed flow, (b) higher-speed flow.. Transition to turbulence in spatially-evolving flow.. (a) unseparated flow, (b) separated flow.. Geometry of streamlines.. Temporal development of a pathline.. 44iiiivLIST OF FLUID particles and trajectories in Lagrangian view of FLUID motion.

7 Eulerian view of FLUID motion.. Steady accelerating flow in a nozzle.. Integration of a vector field over a surface.. Contributions to a control surface: piston, cylinder and valves of internal combustionengine.. Simple control volume corresponding to flow through an expanding pipe.. Time-dependent control volume for simultaneously filling and draining a tank.. Calculation of fuel flow rate for jet aircraft engine.. Schematic of pressure and viscous stresses acting on a FLUID element.. Sources of angular deformation of face of FLUID .. Comparison of velocity profiles in duct flow for cases of (a) high viscosity, and (b)low viscosity.. Missile nose cone ogive (a) physical 3-D figure, and (b) cross section indicating linearlengths.. 2-D flow in a duct.. Prototype and model airfoils demonstrating dynamic similarity requirements.

8 Wind tunnel measurement of forces on sphere.. Dimensionless force on a sphere as function ofRe; plotted points are experimentaldata, lines are theory (laminar) and curve fit (turbulent).. Hydraulic jack used to lift automobile.. Schematic of a simple barometer.. Schematic of pressure measurement using a manometer.. Application of Archimedes principle to the case of a floating object.. Stagnation point and stagnation streamline.. Sketch of pitot tube.. Schematic of flow in a syphon.. Flow through a rapidly-expanding pipe.. Couette flow velocity profile.. Plane Poiseuille flow velocity profile.. Steady, fully-developed flow in a pipe of circular crosssection.. Steady, 2-D boundary-layer flow over a flat plate.. Steady, fully-developed pipe flow.. Turbulent flow near a solid boundary.. Graphical depiction of components of Reynolds decomposition.

9 Empirical turbulent pipe flow velocity profiles for different exponents in Eq. ( ).. Comparison of surface roughness height with viscous sublayer thickness for (a) lowRe, and (b) highRe.. Moody diagram: friction number for various dimensionless sur-face roughnesses.. Time series of velocity component undergoing transitional flow behavior.. Simple piping system containing a pump.. Flow through sharp-edged inlet.. Flow in contracting pipe.. Flow in expanding pipe.. Flow in pipe with 90 bend.. Liquid propellant rocket engine piping system.. 153 Chapter 1 IntroductionIt takes little more than a brief look around for us to recognize that FLUID DYNAMICS is one of themost important of all areas of physics life as we know it would not exist without fluids, andwithout the behavior that fluids exhibit. The air we breathe and the water we drink (and whichmakes up most of our body mass) are fluids.

10 Motion of air keeps us comfortable in a warm room,and air provides the oxygen we need to sustain life. Similarly, most of our (liquid) body fluidsare water based. And proper motion of these fluids within our bodies, even down to the cellularlevel, is essential to good health. It is clear that fluids arecompletely necessary for the support ofcarbon-based life the study of biological systems is only one (and a very recent one) possible applicationof a knowledge of FLUID DYNAMICS . Fluids occur, and often dominate physical phenomena, on allmacroscopic (non-quantum) length scales of the known universe from the megaparsecs of galacticstructure down to the micro and even nanoscales of biological cell activity. In a more practicalsetting, we easily see that fluids greatly influence our comfort (or lack thereof); they are involvedin our transportation systems in many ways; they have an effect on our recreation ( , basketballsand footballs are inflated with air) and entertainment (the sound from the speakers of a TV wouldnot reach our ears in the absence of air), and even on our sleep(water beds!)