G. Bergeles Department of Mechanical Engineering, Nat. Technical University of Athens, Greece

Transcription

1 ELEMENTARY FLUID DYNAMICS G. Bergeles Department of Mechancal Engneerng, Nat. Techncal Unversty of Athens, Greece Keywords: Flud Mechancs, Statcs, Knematcs, Naver-Stokes, Mass Conservaton, Momentum balance, Bernoull equaton, Boundary layer, Moody, Compressble flow, pont sources, lft, drag, energy machne, pump, fan, wnd generator, propeller, Pelton turbne. Contents 1. Fundamental concepts of Fluds 2. Flud Statcs 3. Hydrodynamcs 4. Lamnar and Turbulent Flow 5. Aerodynamcs-gas dynamcs of the nvscd Flud 6. Flud Mechancs n energy machnes Glossary Bblography Summary Ths s an ntroductory chapter to the scence of Flud Mechancs; t s a self contaned text and t s wrtten wth the purpose of explanng defntons, concepts and laws of Flud Mechancs to such a level of practcalty so as the reader wll be able ether to solve problems of Flud Mechancs of engneerng nterest or to have the necessary background for a more n depth study of Flud Mechancs. The text covers all ntroductory aspects of Flud Mechancs, gvng emphass on Flud Mechancs of everyday lfe, such as flows n ppes or around bodes and less emphass n hgh speed flows, where compressblty effects or supersonc flow regons are present. For the latter cases however the basc background knowledge s presented and the reader has the necessary tools to proceed further. The text conssts of sx sectons, startng from the basc defntons and concepts of what s a flud and then by applyng known laws of mass and momentum conservaton, the laws of Flud Mechancs are establshed; whenever t s thought approprate examples are presented. Emphass s gven n the fourth secton on the vscous flows nsde ducts or around bodes and the dstncton between the two classes of flows, lamnar and turbulent, s clarfed. In secton fve, the theory of superposton of elementary flows s presented for nvscd fluds and complex flow felds are created by the applcaton of ths theory. In the last secton the laws of Flud Mechancs and the knowledge presented n the prevous sectons are appled n studyng energy flow machnes, such as fans, pumps, hydraulc turbnes, wnd turbnes and propellers, machnes whch form the bass of the energy cycle of the modern socety.

2 1. Fundamental Concepts of Fluds 1.1 What s Flud Mechancs? Flud Mechancs s the branch of scence whch, wth the use of Mathematcs and Experments studes the moton of fluds under the acton of external and nternal forces. The flud can normally be n two knetc states, ether n moton (flow n ppes, flow around cars or arplanes, flow n arteres or vanes, ar flow n the atmosphere) or no-moton, statc (flud n a reservor); the frst knetc state, flud n moton, s studed by the scence of Flud Dynamcs whlst the second, the flud beng statc, by the scence of Hydrostatcs. In Flud Mechancs the external forces act on the flud ether on the surfaces whch defne the volume of the flud body (the surface could be mpermeable or permeable to flud molecules) or remotely on the flud mass; for the latter force there s no need for any contact between the flud and the force orgn. Such forces are gravtatonal force, electrostatc or electromagnetc force or acoustc force; these forces can be- referred to as forces per unt flud mass or per unt flud volume. The forces actng on the (external) surface defnng the flud volume can have drecton normal and tangental to the surface and are, called normal forces (compresson and expanson) and tangental or shear forces What s a Flud? All materal substances consst of molecules whch can move n space and tme; the molecules attract each other (at dstances from a fracton of the mean dstance between molecules to some tens of the mean molecular dstance) and repel each other n the form of elastc collsons (at dstances smaller than a fracton of the mean molecular dstance). Dependng on the number of molecules packed n a gven volume, materal substances can be dvded n two categores, solds and fluds. In solds the molecules are closely packed, have a fxed poston n space, Fgure 1, and vbrate around ther equlbrum poston wth small ampltude compared to ther mean molecular dstance; the molecules n crystals are located at the corners of notonal geometrcal three dmensonal structures (cubes, pyramds). In solds the ntermolecular bonds are so strong that the sold has a fxed shape and volume. In fluds the molecules are not closely packed and they are separated by greater dstances than n solds; the attracton forces are not strong enough to keep these n fxed postons n space and hence move freely n space, Fgure 2; therefore the molecules cover dstances larger than the mean ntermolecular dstance before these collde wth each other and change path drecton.

3 Fgure 1. Locaton of molecules fxed n space (crystals) Fgure 2. Molecules movng n space (fluds) The fluds are further dvded nto lquds and gases dependng on the degree of the knetc state of ther molecules; to obtan a sense of magntude of the mean dstance between molecules, for ar at standard condtons (one atmosphere of pressure and at a temperature of 0 o degrees Celsus), the mean ntermolecular dstance s approxmately 3nm (3 nanometers= m) and the mean free path of the molecule (the average dstance a molecule travels n straght lne between two collsons) s approxmately 25 tmes the ntermolecular dstance. The number of molecules per mole of the gas s expressed by the Avogadro number N equal to ; therefore1m 3 of ar at standard condtons has molecules, a very large number beyond our comprehenson; even n a cube of space of 1mm sde there are molecules present and n a cube of even

4 1μm sde, there are molecules. It s also found that wthn approxmately 10 collsons wth other ar molecules, the ar molecule loses ts ntal knetc characterstcs and reaches a statstcally equlbrum condton. As the molecules move randomly n space colldng wth each other, the fluds do not posses ther own volume (.e. ther own shape ) by themselves but occupy the volume (maybe partly or whole) whch s avalable to them (e.g. a bottle, or a rver bed) and take the shape of ther contanment. The above descrpton of the nature of a flud s based on a mcroscopc vew of the materal structure whch has been developed out of the knetc theory of gases the last 100 years or so Contnuum Mathematcs s the unversal common language whch we have n expressng quanttatvely the laws of physcs. Mathematcs however uses the concept of contnuous functons wth contnuous dervatves and so on, n expressng these physcal laws, a concept whch requres that the values or the enttes expressng physcal quanttes as velocty, densty, etc, should be contnuous functons n space and tme. Wth the mcroscopc model of the flud presented above (&1.2) based on the dscrete nature of molecules, the knetc state and condton of each flud should be characterzed by the veloctes and the number of molecules at every tme and poston n space. Therefore a functon expressng a physcal quantty such as the velocty at a partcular pont n space and tme could not be contnuous because the pont n space cannot be occuped at all tmes by a molecule. Engneers and most scentsts are not nterested n length scales of the order of molecular dstances but dstances of the engneerng scale whch could nevertheless be small, for example the sze of a mcrometer. Then the statstcal approach of averagng over a regon of fnte volume ΔV s employed because wthn ths fnte volume of space, but nfntesmally small compared to engneerng length scales, there s an enormous number of molecules over whch the statstcal approach can be employed. Wthn ths volume ΔV there s a large number of molecules havng a specfed velocty; t s sad that the flud mass contaned wthn the volume has a (vector) velocty v whch s the average (vector) velocty of all the molecules v whch exst wthn the volume at the partcular tme, v = mv m (1) where m s the mass (or the number) of molecules wthn the volume havng velocty v and m s the total mass (or the number) of the molecules wthn the volume (.e. the mass of the flud contaned wthn the volume). A smlar defnton s used for the flud densty. The flud densty s the mass of flud contaned n the unt volume,

5 m ρ = ΔV (2) The value of the physcal quantty defned as above, such as the densty ρ, converges to a value representatve of the flud densty at a partcular pont as the sze of the volume ΔV over whch the property s averaged becomes smaller and smaller, but always contanng a large number of molecules for a statstcally sgnfcant estmate of the average. Therefore small whch for engneerng scence s effectvely nfntesmally small means a lnear dmenson of say 10 tmes the mean free path of a molecule. Even f the volume Δ V s taken smaller, say a lnear dmenson equal to the mean free path of the ar molecule at standard condtons the error whch could be made n the calculaton of densty would be around 0.8%. The contnuum hypothess s employed n the scence of Flud Mechancs and t allows the ntroducton of the concept of the flud element or flud partcle whch s a an nfntesmally small element of fxed mass wth propertes such as densty, temperature, representng the local propertes of the flud; ths hypothess s vald due to the large number of molecules wthn the length scales of scentfc nterest. However there are cases where the contnuum hypothess s not vald as n the case of very low pressure or temperature, arflow n the outer atmosphere, flow of gas across a shock wave and flow wthn devces of the nanometer length scale. In these cases the contnuum hypothess s not vald and dfferent approaches for the study of such phenomena are employed. A dmensonless number, known as Knudsen number (usually abbrevated to Kn ), s defned as the rato of the typcal lnear scale of the flow to the free molecular path. When the Knudsen number s of the order of unty, then the flow can not be regarded as contnuum, otherwse for Kn 1, whch s almost true n cases of engneerng nterest, the flow can be treated as a contnuum Physcal Propertes of a Flud Fluds have dstnctve characterstcs, some of whch are quantfed by numercal values called physcal propertes; such physcal propertes are, among others, the flud densty, temperature, compressblty, vscosty and surface tenson Densty Flud densty vares substantally from lquds to gases (due to the dfferent number of molecules packed n a gven volume) and has dmensons expressng mass per unt volume. In the nternatonal system of unts, the unt of densty s kg m -3.The densty of a lqud s a weak functon of pressure and temperature whlst n gases the densty depends strongly on pressure and temperature Temperature Temperature characterzes the random translatonal knetc energy of the flud molecules, so that the hgher the knetc energy of the molecules the hgher the temperature of the flud. The mean knetc energy of the molecule s

6 1 2 E = mvrel (3) 2 2 where vrel = v rel. v rel s the mean squared magntude of the relatve velocty of each molecule ( v ) to the bulk flud velocty ( v ), v rel = v - v and m the mass of the molecule. Accordng to the knetc theory of gases ths energy s equal to3/2kt, where k s the Boltzmann s constant (Boltzmann ), equal to k = JK. Therefore temperature s equal to 2k 1 2 T = mvrel (4) 3 2 In the nternatonal system of unts temperature s measured n Kelvn but usually use s made of the Celsus temperature scale whch has a lnear relaton to Kelvn scale T = θ (5) where θ s the temperature of the flud n degrees Celsus Pressure The molecules of a flud are n a contnuous moton; f the molecules ht a materal surface, let us say a sold boundary, then these rebound and retan ther knetc energy but change ther velocty normal to the surface. Ths change of the normal component of the momentum of all molecules strkng a unt surface n unt tme s due to a reactng force from the boundary to the flud whch s called pressure; the frequency of the mpact of the molecules on the surface s so hgh (order of Hz) that they can not be dstngushed as ndvdual mpacts; ths pressure s expressed for a gas by Eq. (6a), δ N P= 2mvn (6a) δt Where m s the mass of the molecule, v n s the mean velocty component normal to the surface of all molecules strkng the surface and δ N / δts the number of molecules strkng the unt surface per unt tme. Assumng that v n s equal for reasons of sotropy v to v n =, the knetc theory of gases gves the followng relaton for the pressure, 3 2 Nmv P = (6b) 3 2 where N s the number of molecules per unt volume and v= v rel. The pressure s then a characterstc property of the flud and s related to the knetc energy of molecules

7 (flud temperature) and the number of molecules per unt volume; these three quanttes are related by the so called equaton of state of the flud. For a gas the equaton of state s wrtten as P = ρ R MW T (6c) where P s the gas pressure, T the temperature, ρ the gas densty, MW the molecular weght of the gas and R a unversal gas constant equal to 8314 J.gmole -1.K -1. The pressure as defned above characterzes the force per unt surface whch the flud- n a compressve way-acts normally on not only any sold external surface surroundng the flud volume but on any notonal surface wthn the flud volume whch nstantaneously can be regarded as a materal surface Compressblty Compressblty s a flud property whch ndcates the way n whch the volume a flud occupes changes under the acton of compresson forces. A flud volume V under the acton of a change of a compresson force δ P undertakes a change of volume equal toδ V ; then the compressblty factor s defned as 1 δv K = (7a) V δp Smlarly the modulus of elastcty E s defned (to the nverse of the compressblty factor) as δp δp E = = ρ ( δv / V ) δρ The compressblty K vares for dfferent fluds; a value for water s K=510-5 atm -1, correspondng to a modulus of elastcty E equal to E = atm; ths compressblty factor for water ndcates that for a change of pressure of 1 atm the percentage change of water volume s only 0.005%. For gases the compressblty factor, usng the equaton of state for gases and assumng that the change of volume takes place under sothermal condtons, usng expresson (7a) leads to expresson (7c), or under sentropc condtons, to expresson (7d), K T (7b) 1 = (7c) P K S 1 = (7d) γ P Ths expresson ndcates that for ar at 1 atm, the compressblty factor s equal to K =1atm -1 T, ndcatng that ar, as for all gases, s tmes more compressble than

8 lquds and n fact n most cases lquds are treated as ncompressble fluds Vscosty The Dynamc vscosty ( μ ) s a flud property whch ndcates the resstance that the flud offers to rate of change of shape or to moton under the acton of shear forces. It arses n real fluds from molecular moton (manly n gases) and ntermolecular attracton (manly n lquds). When the flud s n moton, then at every pont wthn the flud there s a velocty gradent n the drecton normal to the flud velocty, Fgure 3; layers of flud partcles move n parallel drectons but wth dfferent veloctes; however the flud elements comprsng these partcles have nstantaneous veloctes n the drecton normal to the flud partcle s velocty and hence as these cross the notonal common nterface between any two adjacent layers, transfer the component of flud partcle momentum from a regon of hgher to a regon of lower velocty and vce versa. Fgure 3. Velocty gradent n a flow of flud partcles gves rse to shear stress τ xy Ths loss or gan of momentum whch the molecules experence n the drecton of the flud velocty averaged over the great number of molecules crossng the flud layer nterface under consderaton, manfests tself macroscopcally as a drag force of the lower velocty flud on the faster one and vce versa; ths force s called vscous force and accordng to Newton s prncple the so called vscous force per unt area, whch s called shear stress, actng between the two layers of the flud movng n parallel but wth dfferent veloctes s proportonal to the flud velocty gradent wth the constant of proportonalty beng the flud dynamc vscosty μ,

9 du τ xy = μ (8) dy Therefore a shear stress can develop wthn the flud only n the presence of velocty gradents. Ths s also an alternatve defnton of a flud, namely that a flud s a materal substance whch s set n moton under the acton of shear stresses. From the above consttutve law, dynamc vscosty has unts of [ μ ]=Pas. Frequent use s made of the knematc vscosty, whch s the rato of the dynamc vscosty to the flud densty wth unts [ v ]=m 2 s -1, μ ν = (9) ρ The symbol τ xy expresses the shear stress whch acts n x drecton and s located on the plane wth y constant. The dynamc vscosty of the flud s a property whch depends on temperature and pressure Surface Tenson Surface tenson s a flud property whch s pertnent n the nter-phase regon between two mmscble fluds A and B, Fgure 4 or between a lqud and a sold. On a molecule n flud A the surroundng molecules exert attracton forces wthn a sphere of nfluence around the molecule; ths sphere of nfluence has a radus of few molecular dstances and the attracton forces are neglgble beyond ths dstance. Snce there s sotropy n the flud the attracton forces result to a zero attracton force on the partcular molecule. Ths stuaton however does not hold f the partcular molecule s near the nter-phase between the two mmscble fluds, say flud A lqud and flud B ar, as more molecules of flud A (than flud B) exert attracton forces on the molecule. In ths case there s an attracton force actng on the molecule towards the nteror of flud A. Ths force becomes a maxmum when the molecule s on the nter-phase and gves rse to a force called surface tensonσ, wth dmensons of force per unt length. Ths force acts normally on any length on the surface of the nter-phase and performs work, stored as surface energy, durng any dsplacement of the lne on whch the surface tenson apples. In order for the nter-phase to be n stable equlbrum, the surface must take such a shape so as to mnmze ts surface energy,.e. an ntal flat surface wll be curved wth radus of curvature such that equlbrum of forces wll preval. In Fgure 5 a lqud sphercal droplet of radus R s shown n equlbrum n a surroundng of pressure P a. Equlbrum of forces on the surface of the droplet, results n the Eq. (10) P b 2σ = Pa + (10) R ndcatng that the pressure Pb nsde the droplet s hgher than the surroundng pressure P a.

10 Fgure 4. Sphere of nfluence of attracton forces on molecule A Fgure 5. Sphercal droplet n statc equlbrum wth the surroundng Table 1 gves values of varous physcal propertes for some fluds at 1 atm pressure and temperature of 20 o C. Flud Densty ρ (kg/m 3 ) Dynamc vscosty μ (Pas) Compressblty K (atm -1 ) T Surface tenson σ (Ν/m) (lqud to ar) Water x10-3 5x Normal heptane 879 4x x Ar x Hydrogen x Table 1. Physcal propertes of some typcal fluds

11 - - - TO ACCESS ALL THE 110 PAGES OF THIS CHAPTER, Vst: Bblography Anderson J. (1993), Modern compressble flow, 2 nd ed., McGraw Hll. [A well organzed textbook on compressble flows wth hstorcal nformaton on concept developments]. Batchelor G. (1979), An ntroducton to flud dynamcs, Cambrdge Unversty Press. [Advanced textbook on Theoretcal Flud Mechancs]. Brd R., Stewart W. and Lghtfoot E. (1960), Transport phenomena, John Wley. [Advanced textbook on Transport Phenomena ncludng mass transfer, puttng emphass on physcal prncples]. Blevns R.D. (1984), Appled Flud Dynamcs Handbook, Van Nostrand. [Reference book on Industral Flud Mechancs]. Fox R.W. and Mcdonald A. (1978), Introducton to flud mechancs, John Wley & Sons, 2 nd ed. [Well organzed textbook on Flud Mechancs for engneerng students wth a varety of solved examples and unsolved problems]. Houghton E.L. and Brock A.E. (1970), Aerodynamcs for engneerng students, Edward Arnold, 2 nd ed. [Excellent textbook on appled Aerodynamcs]. Massey B.S. (1975), Mechancs of Fluds, Van Nostrand, 3 rd ed. [Excellent textbook on Flud Mechancs wth engneerng applcatons and problems]. Panton R.L. (1984), Incompressble Flow, John Wley. [Excellent engneerng textbook wth mathematcal orentaton] Schlchtng H. (1979), Boundary layer theory, 7 th ed., McGraw Hll. [The reference book on Boundary layers and Flud Mechancs n general]. Whte F. (1974), Vscous flud flow, McGraw Hll. [Excellent textbook coverng all aspects of Flud Mechancs]. Bographcal Sketch Dr George Bergeles holds a dploma n Mechancal and Electrcal Engneerng (1968) from the Natonal Techncal Unversty of Athens, an M.Sc. (1973) and a Ph. D (1976) degree n Heat Transfer Engneerng from the Imperal College of Scence, Technology and Medcne, and a D.Sc. (Eng) (1994) degree from the Unversty of London. He has been an Esenhower Fellow for Greece (1983), and a Fellow of the UK Insttuton of Mechancal Engneers snce He s Professor of Flud Mechancs n the Department of Mechancal Engneerng of the Natonal Techncal Unversty of Athens and has been Vstng Professor n Kng s College London and s currently Honorary Vstng professor at Cty Unversty, London. Professor Bergeles has extensve experence n educaton, research and ndustry, has publshed more than 80 papers n academc journals, and has supervsed many Ph.D. researchers. Hs engneerng contrbutons nclude those to the desgn and manufacture of equpment for Wnd Energy, Industral Aerodynamcs, Ar Polluton and Combuston Engneerng. The desgn aspects have been facltated by the development of grd generaton technques and fnte-dfference and fnte volume methods for solvng the Naver-Stokes equatons n curvlnear coordnates wth automatc local grd refnement, and supported by experments n the largest subsonc wnd tunnel n Greece.

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