Intermediate Mechanics of Fluids
Intermediate Mechanics of Fluids 058 160
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0580160 Chapter 1 Professor Fred Stern Fall 2007 l 058 160 Intermediate Mechanics of Fluids Class Notes Fall 2007 Prepared by Professor Fred Stern Typed by Derek Schnabel Fall 2004 Nobuaki Sakamoto Fall 2006 Hamid SadatHosseini Fall 2006 Maysam Mousaviraad Fall 2006 Corrected by Jun Shao Fall 2004 Mani Kandasamy Fall 2005 Tao Xing Hyun Se Yoon Fall 2006 Hamid SadatHosseini Fall 2007 0580160 Chapter 1 Professor Fred Stem Fall 2007 2 Chapter 1 Introduction Definition of a uid A uid cannot resist an applied shear stress and remain at rest whereas a non uid ie solid can Solids resist shear by static deformation up to an elastic limit of the material after which they undergo fracture 39 y Elastic solid T 0c y strain E T G Y 1 G shear modulus 10 PM Fluids deform continuously undergo motion when subjected to shear stress Consider a uid between two parallel plates with the lower one fixed and the upper moving at speed U which is an example of Couette ow ie wall shear driven ows y X 11Y i uU 1D ow velocity 11 uy pro le 0580160 Chapter 1 Professor Fred Stern Fall 2007 3 No slip condition Length scale of molecular mean free path 7 ltlt length scale of uid motion E therefore macroscopically there is no relative motion or temperature between the solid and uid in contact Knudsen number Kn NE ltlt 1 Exceptions are rare ed gases and gasliquid contact line Newtonian uids 6 0c 9 rate of strain 1 ue u coefficient of Viscosity Rate of Strain my dygtdt uuydy Fluid element with sides parallel to the Fluid element deformation at 39 time t dt coordinate axes at time t0 0580160 Chapter 1 Professor Fred Stern Fall 2007 4 dd 39 tand6uy yt 6u dy dt y du T 6 dy rate of strain velocity gradient For 3D ow rate of strain is a second order symmetric l Bui auj 3 Bx Bxi zaji tensor 8 Diagonal terms are elongationcontraction in Xyz and off diagonal terms are shear in Xy Xz and yz Liquids vs Gases Liquids Gases Closely spaced With large Widely spaced With small intermolecular cohesive intermolecular cohesive forces forces Retain volume but take Take volume and shape of shape of container container 3 ltlt 1 3 gtgt 1 p constant p ppT Where 3 coef cient of compressibility change in volumedensity With external pressure 0580160 Chapter 1 Professor Fred Stem Fall 2007 5 3p 3p1 Bulk modulus K VE 10 Recall pvT diagram from thermodynamics Single phase two phase triple point point at which solid liquid and vapor are all in equilibrium critical point maximum pressure at which liquid and vapor are both in equilibrium Liquid gases and twophase liquidvapor behave as uids 0580160 Chapter 1 Professor Fred Stern Fall 2007 6 Continuum Hypothesis Fluids are composed of molecules in constant motion and collosion however in most cases molecular motion can be disregarded and the assumption is made that the uid behaves as a continuum ie the number of molecules Within the smallest region of interest a point are suf cient that all uid properties are point functions single valued at a point For example Consider de nition of density p of a uid hm 5M g position vector xiyjzk Xt 2 p 8V gt8V 8v t me 5V limiting volume below Which molecular variations may be important and above Which macroscopic variations may be important 5V z 10399 mm3 for all liquids and for gases at atmospheric pressure 10399 mm3 air at standard conditions 200C and 1 atm contains 3X107 molecules such that 5M5V constant p 0580160 Chapter 1 Professor Fred Stern Fall 2007 7 II M WI h V W l quotr 31 ll MSW M Mr ml thin 7m mt 27136 1 lv il im m Exception rare ed gas ow Note that typical smallest measurement volumes are about 10393 100 mm3 gtgt 8V and that the scale of macroscopic variations are very problem dependent A point in a uid is equivalently used to de ne a uid particle or in nitesimal material element used in de ning the governing differential equations of uid dynamics At a more advanced level the Knudsen number is used to quantify the separation of molecular and uid motion length scales Kn l molecular length scale I uid motion length scale 0580160 Chapter 1 Professor Fred Stern Fall 2007 8 Molecular scales Air atmosphere conditions l6gtlt108m mean ee path tx 1010 s time between collisions Smallest uid motion scales Olmm10394m Vmax 100 ms incompressible ow Mg 03 133 Z 106 S Thus Kn10393 ltlt 1 and i scales larger than 3 order of magnitude 1 scales An intermediate scale is used to de ne a uid particle t ltlt 3 ltlt Z And continuum uid properties are an average over Vquot f3 vquot 3 10399 mm 3016 5 10396m Previously given smallest uid motion scales are rough estimates for incompressible ow Estimates are VERY conservative for laminar ow since for laminar ow I is usually taken as smallest characteristic length of the ow domain and Vrmx can not exceed Re restriction imposed by transition om laminar to turbulent ow 0580160 Chapter 1 Professor Fred Stern Fall 2007 9 For turbulent ow the smallest uid motion scales are estimated by the Kolmogorov scales which de ne the length velocity and time scales at Which viscous dissipation takes place ie at Which turbulent kinetic energy is destroyed nV3gl4 Tvg12 upper4 V kinematic viscosity 5 dissipation rate Which can also be written nzloRe 34 0 zL uolo UL u z uO Re 4 R60 2 z 7 V V 12 Q QRe Which even for large Re of interest given 77 gtgt 1 For example 100 watt mixer in 1 kg water 9 100 wattkg 100 mZs3 V210 6m2Sf0rwater 7710 2mmgtl 0580160 Professor Fred Stern Fall 2007 Chapter 1 10 The smallest uid motion scales for ship and airplane Ums Lm Re 77 M 277 Ship 40 150 6E09 7E6 014 5E5 Airplane 260 20 5E08 6E6 174 34E6 Ma08 Fluid Properties 1 Kinematic linear X angular 92 velocity rate of strain sij vorticity Q and acceleration a 2 T ransgort viscosity u thermal conductivity k and mass diffusivity D 3 Thermodynamic pressure p density p temperature T internal energy f1 enthalpy h u pp entropy s speci c heat CV Cp 7 Cp CV etc 4 Miscellaneous surface tension 6 vapor pressure pv etc 0580160 Chapter 1 Professor Fred Stern Fall 2007 ll 1 Kinematic Properties Kinematics refers to the description of the ow pattern without consideration of forces and moments whereas dynamics refers to descriptions of E and M Lagrangian vs Eulerian description of velocity and acceleration a Lagrangian approach focuses on tracking individual fixed particles J k5 P VJ Air 5t Yalhll oox p All All E 1 quotE b Eulerian approach focuses on fixed points in space 1 f virm A A A NJ Yum L XL 31 EA 0580160 Chapter 1 Professor Fred Stern Fall 2007 12 uvW f t are velocity components in Xyz directions MEL LiHLKimLZiZ d a axar ayaz azaz However are not arbitrary but assumed to g 8 follow a uid particle ie u DZ 8K 8K 8K 8K u v w Dt at 8x 8y 82 DV 8V a a a VVV V radzent z Dt at 7 g 8x 8y J 82 D a E E JFK V substantialmaterial derivative DV 3 Lagrangian time rate of change of velocity 3K 5K VK local amp convective acceleration in terms of Eulerian derivatives DI derivative following motion of particle 0580160 Chapter 1 Professor Fred Stern Fall 2007 13 aaxiayjazk 8u 8u 8u 8u ax u v w 81 8x 8y 82 8v 8v 8v 8v u v w y 81 8x 8y 82 8w 8w 8w 8w v w u Z 81 8x 8y 82 The Eulerian approach is more convenient since we are seldom interested in simultaneous time history of individual uid particles but rather time history of uid motion and E M in xed regions in space control volumes However three fundamental laws of uid mechanics ie conservation of mass momentum and energy are formulated for systems ie particles and not control volumes ie regions and therefore must be converted om system to CV Reynolds Transport Theorem X t is a vector eld Vector operators divergence and curl lead to other kinematics properties VK2divergence Kzalav aw ax 8y 82 V volume of uid particle 0580160 Chapter 1 Professor Fred Stern Fall 2007 14 1 DV V 1 Dp g K EE 9 Cont1nu1ty equatIOIl rate of change V per unit V rate of change p per unit p For incompressible uids p constant V Z O ie uid particles have constantv but not necessarily shape VxZ curl Z 2Q wxfwyjwzl vorticity 2 angular velocity of uid particle if aaa W753 LIV 8W 8V 5 av 81 quot z 1 k 8y 82 8x 82 8x 8y For irrotational ow VXK 0 ie K W9 and for p constant VK V V V2 0 9 Potential FlowTheory 0580160 Chapter 1 Profs or Fred Stem Fall 2007 15 Other useful kinematic properties include volume and mass owrate Q1 average velocity V and circulation F L I 5 y J As S g V 2130 J 39j wax AA 343 where Q volume of uid per unit time through A ux of Vn through A bounded by S uX generally used to mean surface integral of variable 5141 where 72 mass of uid per unit time A through A V QA where V average velocity through A A jdA where A surface area 0580160 Chapter 1 Professor Fred Stern Fall 2007 16 F lK lVXK39 Stokes theorem relates line S A and area integrals line integral for tangential velocity component i Q E 0114 ux surface integral of normal vorticity component KuttaJoukowski Theorem lift L per unit span for an aribitrary 2D cylinder in uniform stream U With density p is L pUF with direction of L is perpendicular to U 2 Transport Properties There is a close analogy between momentum heat and mass transport therefore coef cient of viscosity u thermal conductivity k and mass diffusivity D are referred to as transport properties Heat Flux J Fourier s Law 1 kVT mzs 0580160 Chapter 1 Professor Fred Stern Fall 2007 17 rate of heat ux is proportional to the temperature gradient per unit area W k fXyz 9 solid constant 9 liquid isotropic Mass Flux k Fick s Law DVC m S rate of mas ux is proportional to concentration c gradient per unit area m2 D H S du i Newtonian Fluid 139 2 0 m2 1D ow y rate of momentum uX shear stress is proportional to the velocity gradient per unit area Ns kg m2 MUS Momentum Flux 0580160 Chapter 1 Professor Fred Stern Fall 2007 18 For 3D ow the shearrate of strain relationship is more complex as will be shown later in the derivation of the momentum equation T 5 au 51VV U p 17 39u ax 8x1 7 Where ul uvwj x1 xyz 9 2nd coef cient of viscosity For heat and mass transported quantities are scalars and ux is a vector whereas for momentum transported quantity is a vector and ux is a tensor Also all three laws are phenomenological ie based on empirical evidence experience and experiments Non Newtonian uids follow nonlinear shearrate of strain relationships I or 8in n lt l pseudoplastic n 1 Newtonian n gt 1 dilatant u and k are also thermodynamic properties u ugas or liquid T p Fig A1 0580160 Chapter 1 Professor Fred Stern Fall 2007 19 For both gases and liquids u increases With p but A u is small and usually neglected For gases u increases With T Whereas for liquids u decreases With T Differences are due to more molecular activity and decreased cohesive forces for gases Kinematic viscosity 2 m U u 0 arises in equations as diffusion coef cient Reynolds Number Re 2 g U velocity scale 1 U L length scale The Reynolds number is an important nondimensional parameter ratio inertiaviscous forces Which characterizes uid ow 0580160 Chapter 1 Professor Fred Stern Fall 2007 20 3 Thermodynamic Properties Classical Thermodynamics the study of equilibrium states of matter in which properties are assumed uniform in space and time Thermodynamic system xed mass separated from surroundings by boundary through which heat and work are exchanged but not mass Properties are state functions whereas heat and work are path functions A classical thermodynamic system is assumed static whereas uids are often in motion however if the relaxation time time it takes material to adjust to a new state is small compared to the time scale of uid motion an assumption is made that thermodynamic properties are point functions and that laws and state relations of static equilibrium thermodynamics are valid In gases and liquids at normal pressure relaxation time is very small hence only a few molecular collisions are needed for adjustment Exceptions are rare ed gases chemically reacting ows sudden changes such as shock waves etc For singlephase pure substances only two properties are independent and all others follow through equations of state which are determined experimentally or theoretically Some mixtures such as air can also be 0580160 Chapter 1 Professor Fred Stern Fall 2007 21 considered a pure substance Whereas others such as salt water cannot and require additional numbers of independent properties eg sea water requires three salinity T and p Pressure p Nmz Density p kgm3 Temperature T K Internal Energy a Nmkg Jkg Enthalpy h a p p Nmkg Jkg Entropy s Jkg K p ppT 13p7T h hpT s spT Speci c weight 7 pg Nm3 pair 1205 M 118 Nm3 pwam 1000 Me 9790 Nm3 pmercury 39Ymercury NII3 SG pgas pgas g p 1205 kgm3 01139 uid 01139 uid SG 4 4 quot mer4c 1000 kgm3 SGHg 136 0580160 Chapter 1 Professor Fred Stern Fall 2007 22 Total stored energy per unit mass 16 2 e l2V2gz a energy due to molecular activity and bonding forces l2V2 work required to change speed of mass from 0 to V per unit mass gz work required to move mass from 0 to L xf yj zkA against g gl per unit mass mg z m 4 Miscellaneous Properties Surface Tension Two nonmixing liquids or liquids and gases form an interface across which there is a discontinuity in density The interface behaves like a stretched membrane under tension The tension originates due to strong intermolecular cohesive forces in the liquid that are unbalanced at the interface due to loss of neighbors resulting in surface tension per unit length 6 coef cient of surface tension Nm Line force FCS 6L where L length of cut 0580160 Chapter 1 Professor Fred Stern Fall 2007 23 Fluidl through interface L F o f two uids T Fluid 2 Direction of F0 is normal to cut Effects of surface tension 1 Pressure jump across curved interfaces in An mil 3 A M I 13 7 x 39 it xx W 4m m Fig l Irmme clungu M J curved Interface due to mm mnmm ml mlenm at 4 MW Qnmm my vmenor m a spherical mp1ch n 1 general fumed lnledacl a Cylindrical interface Force Balance 20L 2pip0RL Ap oR pigt p0 ie pressure is larger on concave vs convex side of interface b Spherical interface 0580160 Chapter 1 Professor Fred Stem Fall 2007 24 27tR6 nRZ Ap 9 Ap ZoR c General interface AP 7Rf1 R24 R1 principle radii of curvature 2 Contact Angle Li When the surface of a solid intersects 5am QM interface contact angle either be in e wetting 9 lt 90 or non wetting 9 gt I 90 9 depends on both two uids and 1 solid surface properties a Capillary tube patm 1N a Q F r Jump across 31w l a boundi gt due to 6 1 H1 102 0580160 Chapter 1 Professor Fred Stern Fall 2007 25 Surface Tension Force Weight of uid 27tRo cos 9 pghatR2 2039 cos 9 h h 0L R 1 74 h gt 0 wetting h lt 0 nonwetting Consider Glt900 patrn patm patrn h Pressure jump V due to o 3 Transformation liquid jet into droplets 4 Binding of wetted granular material such as sand 5 Capillary waves 0580160 Chapter 1 Professor Fred Stern Fall 2007 26 Similar stretched membrane string waves surface tension acts as restoring force resulting in interfacial waves called capillary waves Cavitation When the pressure in a liquid falls below the vapor pressure it will evaporate ie become a gas If due to temperature changes alone the process is called boiling whereas if due to liquid velocity the process is called cavitation Ca pa pv l 2 0U 2 Ca Cavitation pV vapor pressure pa ambient pressure U characteristic velocity p pa W falls below the cavitation number Ca the liquid will cavitate If the local pressure coef cient Cp Cp Ca f liquidproperties T Effects of cavitation l erosion 058 0160 Chapter 1 Dr F n d y m Fall 2007 27 2 vibration 3 noise b Computation Cavitation comparison with experiment 0580160 Chapter 1 Professor Fred Stern Fall 2007 28 Flow Classi cation 1 Spatial dimensions 1D 2D 3D 2 Steady or unsteady g 0 or 0 3 Compressible p constant or incompressible p constant 4 Inviscid or Viscous u 0 or u 72 0 5 Rotational or Irrotational Q 72 0 or Q 0 6 InviscidIrrotational potential ow 7 Viscous laminar or turbulent Remus 8 Viscous low Re Stokes ow 9 Viscous high Re external ow boundary layer 10 Etc Depending on ow classi cation different approximations can be made to exact governing differential equations resulting in different forms of approximate equations and analysis techniques 0580160 Chapter 1 Professor Fred Stern Fall 2007 29 Flow Analysis Techniques I I I Fluids Eng Systems Components Idealized I I I I I I EFD Mathematical Physics Problem Formulation UD JBZ P2 UM l I AFD CFD UM Us 2 V USZM USZN