ME 305 Fluid Mechanics I. Part 3 Introduction to Fluid Flow. Field Representation. Different Viewpoints for Fluid and Solid Mechanics (cont d)

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1 ME 305 Fluid Mechanics I Pat 3 Intoduction to Fluid Flow Field Repesentation As a fluid moves, its popeties in geneal change fom point to point in space and fom time to time. In field epesentation of a flow, fluid and flow popeties ae given as functions of space coodinates and time. p = p,,, t, =,,, t, etc. If thee is no time dependenc in a flow field, it is said to be stead, othewise it is unstead. These pesentations ae pepaed b D. Cünet Set Depatment of Mechanical Engineeing Middle East Technical Univesit Stead and unstead flows von Kaman vote steet Wing tip votices Flow aound a ca Ankaa, Tuke cset@metu.edu.t You can get the most ecent vesion of this document fom D. Set s web site. Please ask fo pemission befoe using them to teach. You ae NOT allowed to modif them Diffeent iewpoints fo Fluid and Solid Mechanics In solid mechanics we ae usuall inteested in how mateial moves o defoms. We focus ou attention on mateial and follow its motion/defomation. Diffeent iewpoints fo Fluid and Solid Mechanics (cont d) Howeve, in fluid mechanics we ae geneall inteested in how things behave/change at a point, on a suface o inside a volume. We focus ou attention not on mateial, but on space (location) We locate a solid paticle (o goup of paticles) at an initial time and stud thei motion in time to detemine whee the go. We ae inteested in paticles tajectoies and thei final positions, such as golf ball s point of hitting o maimum deflection of the beam s cente point. 3-3 Fo a lift foce geneating wing, we need to know the pessue distibution ove the wing. We ae not eall inteested in the oiginal locations of fluid paticles that cause the lift o whee the go afte the passed ove the wing. To measue the amount of liquid flowing in a pipe, we need to make calculations at the eit coss section of it. We do not need to follow the fluid paticles that pass though that eit section. 3-4

2 Lagangian (Mateial) Desciption Lagangian (Mateial) Desciption (cont d) Identified fluid paticles ae followed in the couse of time as the move in a flow field. NOT pefeed in fluid mechanics, moe suitable to solid mechanics. Conside the following epeiment whee a fluid flows in a conveging duct. We located a paticle P at time t 0 at the entance of the duct and follow it in time and measue its speed. t 0 t t t 2 3 t 4 1 P P P P P Time Paticle P s speed [m/s] t 0 5 t 1 8 t 2 10 t 3 15 t 4 20 In following a paticle, the onl independent vaiable is time. Space coodinates (,, ) of paticle P ae NOT independent vaiables. When we select a paticle b identifing it at its initial location at an initial time, its location at a futue time, sa t 3, depends on which paticle we ae following and the value of t 3. Popeties of paticle P ae in geneal epessed as position of P : velocit of P : pessue of P : etc. P t P t p P t Lagangian desciption t 0 t t t 2 3 t 4 1 P P P P P P (t 3 ) Euleian (Spatial) Desciption Attention is focused at fied points (o aea o volume) in the flow field and the vaiation of popeties at these points ae detemined as fluid paticles pass though these points. This is the pefeed viewpoint fo fluid mechanics. Conside the same flow in the conveging duct, but now concentating at two points, A and B (o two sections, inlet and eit). A B Time Speed at A Speed at B t t t t t Euleian (Spatial) Desciption (cont d) Now both time and space coodinates ae independent vaiables. Location of point A (o B) does NOT depend on the flow field o time. Fluid and flow popeties at a point (e.g. point A) ae epessed as position : A A, A, A velocit : A A, A, A, t pessue : p A A, A, A, t, etc. Euleian desciption The duct flow descibed in the pevious slides is said to be stead if the flow popeties (such as velocit, pessue, etc.) do not change with time. Fo stead flows time is NOT a vaiable in the Euleian desciption. But time is alwas an independent vaiable in the Lagangian desciption, even fo stead flows. Without time, a fluid paticle simpl can not move. A A B 3-8

3 Lagangian vs. Euleian Desciption Eecise : Ae the following desciptions Lagangian o Euleian? A docto using X-a opaque de to tace blood flow in ateies. A civil enginee studing the taffic load of a highwa b focusing at a cetain section of the oad and counting the numbe of cas passing in font of him duing a cetain peiod of time. A student pefoming a wind tunnel epeiment and measuing the velocit at diffeent points of a flow field b manuall moving a velocit measuing pobe. Fluid dnamic measuements pefomed in the lab ae suited to the Euleian desciption. A velocit o pessue pobe inseted in a flow field do NOT move with the flow, but povide data at the locations we point it to. Eecise : isit Stom Chase s web site to see the use of a Lagangian tpe pobe fo gatheing data inside a twiste. Also ou can watch the movie Twiste to see such pobes in action. 3-9 Use of Euleian Desciption fo Solid Mechanics Euleian desciption is pefeed in studing ve high defomation solid mechanics poblems, in which solids show fluid-like behavio. Eecise : Do a eseach on the woking pinciple of shaped chage used fo amo penetation. Watch the movie Eecise : Watch the following movies in which solids undego ve ecessive (fluidlike) defomation. Aluminum etusion : Deep dawing : Cash test : Shaped chage and its penetation Use of Lagangian Desciption fo Fluid Mechanics A docto using an X-a opaque de to tace blood flow in ateies pefoms a Lagangian stud. In some Computational Fluid Dnamics (CFD) studies motion of fluid paticles ae modeled in a Lagangian wa. Eecise : Watch this paticle simulation of flow aound a ca Eecise : RealFlow is a paticle based fluid simulation softwae used in film-making and television indust. isit to see its capabilities. SPH Simulation of a tanke in wave RealFlow Demo Reel Lagangian - Euleian Relation Conside a popet N (can be velocit, densit, pessue, etc.) in a flow field. At time t fluid paticle P passes though a point A in space. P (t) Eecise: Wok on the details of the following impotant elation A At time t, paticle P passes though point A Path of paticle P Rate of change of popet N of paticle P at time t fom a Lagangian point of view = Rate of change of popet N at point A fom an Euleian point of view

4 Lagangian - Euleian Relation (cont d) Undestanding the convective deivative N dn = N + N Let the popet N epesents the tempeatue T. Conside the following simple tempeatue field (shown with constant T lines). Mateial deivative (Substantial deivative) (Total deivative) Local deivative (Patial deivative) Convective deivative Mateial deivative : Rate of change of popet N in the mateial desciption (following a paticle) Local deivative : Rate of change of popet N (at a fied point) with time onl. Fo a stead flow this tem is eo fo an popet. Convective deivative : Change of popet N (at a fied point) with space onl, i.e. at a fied time. If thee is no flow this tem is eo. In this 2D tempeatue field T changes onl in the vetical diection. Gadient of tempeatue ( T) is in the vetical diection. If ou move in this diection ou ll feel the most apid change of T. T Undestanding the convective deivative N Undestanding the convective deivative (cont d) Conside moving in this T field in diffeent diections. T If we move paallel to T we feel the maimum tempeatue change T = T = maimum T T If we move pependicula to T we feel no tempeatue change T = T = If we move at an angle to T we feel a noneo tempeatue change T = T 0 Cold T in dn Hot T out > T in = N Stead state opeation of a wate heate. T/ at an point is eo, but dt/ of a moving fluid paticle is not eo. Patial deivative of T is eo, but convective deivative is not. + N 1 u 1 u2 > u 1 u 3 < u 2 2 Stead, unifom flow in a convegingdiveging nole. Fluid paticles fist acceleate and than deceleate. u/ at an point is eo, but du/ of a moving fluid paticle is not. Patial deivative of u is eo, but convective deivative is not

5 Acceleation of a Fluid Paticle Acceleation of a Fluid Paticle (cont d) Selecting N as in equation dn = N + N, acceleation of a fluid paticle can be obtained as Acceleation ( a) d = + Local acceleation Convective acceleation Eecise : Using the following opeato in the clindical coodinate sstem = i + 1 θ i θ + i and the fact that in clindical coodinate sstem unit vectos have the following non-eo deivatives i θ = i θ and i θ θ = i Components of the acceleation vecto in Catesian coodinate sstem ae deive the following acceleation components a = du = u + u = u + u u u u + v + w a = + + θ θ + 2 θ a = dv = v + v = v + u v v v + v + w a θ = θ + θ + θ θ θ + θ + θ a = dw = w + w = w + u w w w + v + w 3-17 a = + + θ θ Acceleation of a Fluid Paticle (cont d) Flow Classification as 1D, 2D and 3D Eecise : Conside one-dimensional, stead, incompessible flow though a conveging channel. Following Munson s book (diffeent in Aksel s book) Thee-dimensional flow : all 3 velocit components ae noneo. L elocit field is given b u = U o 1 + L v = 0 w = 0 a) Detemine the acceleation field, a(), b using the Euleian method. b) Using the Lagangian method, detemine the equations fo the position and acceleation of the fluid paticle, which is located at = 0 at time t = 0. c) Show that both epessions fo the acceleation give identical esults, as the fluid paticle eits the channel at = L Two-dimensional flow : onl 2 velocit components ae noneo. One-dimensional flow : onl 1 velocit component is noneo. Couette flow 1D flow u 0 v = 0 w = 0 U 0 Aismmetic nole flow 2D flow 0 0 θ =

6 Unifom Flow The flow is said to be unifom at a coss section if the onl noneo velocit component is the one pependicula to the coss section, and the velocit is not changing acoss the section. Unifom flow simplification as used above disegads the no slip condition. Instead of the actual velocit pofile it uses the aveage speed at a coss section. Actual velocit pofile Simplified as unifom 3-21 Flow Classification as Stead, Unstead Stead flow : Local deivatives ( /) ae eo in a flow field. Popeties at a fied point do not change in time. See slide 3-16 fo two eamples. A centifugal pump woking constantl at the same speed between the same input and output conditions is said to be woking steadil, although thee is a otating blade inside it. Ai flow aound a ca moving at constant speed is consideed to be stead, although thee ae fluctuations in the wake egion behind the ca. Unstead flow : Local deivatives (at least fo 1 popet) ae noneo. Popeties at a fied point change in time. If the inlet wate tempeatue of the heate shown in slide 3-16 changes with time, it will be an unstead flow. Pulsatile blood flow in ou veins is unstead. But it is a special kind of unstead flow, it is time peiodic. It epeats itself afte a cetain peiod. von Kaman vote steet of slide 3-2 is also unstead and time peiodic. A gust wind blowing ove a house is unstead Flow Classification as Stead, Unstead (cont d) Flow Classification as Lamina, Tubulence Sometimes an unstead flow can be studied as stead b a pope choice of efeence fame. Conside the following wing moving at a constant speed in still ai. Fo an obseve fied at the gound this flow is unstead. At an upsteam point A, initiall ai speed is eo. But as the wing appoaches point A, it will push the ai thee. Obseve fied at the gound will obseve diffeent things at point A at diffeent times. A The same flow becomes stead with espect to an obseve moving with the wing. This obseve will alwas see the same ai motion aound him/he. Nothing will change in time. Lamina flow is a well-odeed state of flow in which adjacent fluid laes move smoothl with espect to each othe. Lamina flow is usuall associated with low speeds and high viscosities. Tubulent flow has andom, unstead, 3D fluctuations. Thee is intense miing and otation. Tubulent flow is usuall associated with high speeds. Tubulent flows ae, b fa, moe common than lamina ones. Although a tubulent flow alwas have unsteadiness in it, it ma be stead in the mean (in a time aveaged sense). Similal it can be 2D o 1D in the mean. Lamina flow ove clindes and aifoils Lamina to tubulent tansition in a pipe Wake behind a clinde Simila simplifications ae obseved when tubomachine flows ae studied using a otating efeence fame

7 Pathlines, Steaklines and Steamlines These ae the common was used to visualie a flow field. Pathline is a line taced out b a fluid paticle as it flows in a flow field. Pathline is a Lagangian concept. In laboato it can be geneated b making (ding) a small fluid element and taking time eposue photogaph of its motion. Steakline is a line that joins the paticles in a flow that have peviousl passed though a common point. In laboato it can be geneated b continuousl injecting de (o bubbles) at a point and obseving the collection of ded paticles as the move in the flow. Steamline is a line that is evewhee tangent to the velocit field. It is a mathematical tool, athe than an epeimental technique. Fo a stead flow all these thee ae the same. Fo an unstead flow the ae all diffeent. NCFMF Pathline, steakline and steamline compaison fo Equation of a Steamline Conside a steamline in a 2D flow field. At an point velocit vecto is tangent to it. Slope of the line at an point (d/d) should be equal to the velocit component atio (v/u) d d = v u which can also be witten as d u = d v This can be genealied to a 3D flow as d u = d v = d w If the velocit field is known as a function of, and (and t if the flow is unstead), the above equation can be integated to give the equation of steamlines. Eecise : Fo the velocit field given b = 2 i 2 j, detemine the equation of the steamline that passes though point P(2,2,0). unstead oscillating plate flow ds ds d d u v Diffeential vs. Integal Fomulation Diffeential fomulation povides a ve detailed solution of a flow field. When used with Euleian point of view, it povides infomation at all points in the poblem egion at all times of inteest. It equies the solution of diffeential equations fo consevation laws (mass, linea momentum and eneg). Analtical solution of consevation equations ae available onl fo a few ve simple poblems. Computational Fluid Dnamics (CFD) povides an altenative. Detailed solution of the flow field inside a pump Diffeential vs. Integal Fomulation (cont d) Integal fomulation used with Euleian viewpoint focuses at a fied egion of space (contol volume). It studies the inteaction of this contol volume with its suoundings. It is used to detemine goss flow effects (not details), such as the lift foce geneated b a wing, thust geneated b a jet engine o the shaft wok equied to un a pump. It povides less infomation compaed to diffeential appoach. But it has much simple mathematics. No diffeential equation is solved. 2, A 2, p 2 1, A 1, p 1 Shaft wok Pats of a centifugal pump Integal analsis fo a contol volume aound a pump 3-28

8 Closed Sstem vs. Contol olume A closed sstem (o just sstem) is a fied, identifiable quantit of mass. It can change its position and shape, but it alwas contains the same fluid paticles. It is sepaated fom the suoundings b the sstem boundaies, which is closed to mass tansfe. Fluid paticles can not pass though it. It is closel linked to the Lagangian desciption. It has the advantage that basic laws (consevation of mass, momentum, eneg) can be witten fo it in a ve natual and simple wa. Closed Sstem vs. Contol olume (cont d) A contol volume (C) is a fied egion of a flow field. It can NOT change its position o shape, but it contains diffeent fluid paticles at diffeent times (Note: Moving/defoming Cs can also be defined). It is sepaated fom the suoundings b the contol suface (CS), which is open to mass tansfe. Fluid paticles can pass though the CS. It is closel linked to the Euleian viewpoint. Renolds Tanspot Theoem (RTT) is used to convet basic consevation laws witten fo a closed sstem to equations that can be used fo a C. Closed sstem fo the pefume is initiall inside the spa can Afte using it, the closed sstem is patl inside and patl outside. It follows pefume s motion Initial shape of the C. C does not change shape afte using the pefume. It does not follow pefume's motion. It has an eit, though which pefume leaves Basic Laws Witten fo a Sstem Basic Laws Witten fo a Sstem (cont d) Consevation of Mass : Mass of a closed sstem does not change, i.e. time ate of change of a closed sstem s mass is eo. Consevation of Eneg (1 st Law of Themodnamics) : Eneg of a closed sstem changes b heat and wok inteaction with its suoundings as follows dm ss = 0 whee m ss = ss ρ d Consevation of Linea Momentum (Newton s 2 nd Law) : Sum of all etenal foces acting on a sstem is equal to the time ate of change of its linea momentum. Dot means time ate of change of. Same as d Q + W = de ss e = u whee E ss = ss ρ e d + g F = dp ss whee P ss = ss ρ d Total eneg pe unit mass Intenal eneg pe unit mass Kinetic eneg pe unit mass Potential eneg pe unit mass Consevation of Angula Momentum : Sum of all etenal toques acting on a sstem is equal to the time ate of change of its angula momentum. Q : ate of heat tansfe (heat coming into the sstem is positive) T = dh ss whee H ss = ss ρ d W : ate of wok done (wok done on the sstem is positive)

9 Renolds Tanspot Theoem (RTT) All basic laws ae natuall witten in ve simple foms fo a closed sstem. But we want to use Cs to stud fluid mechanics poblems. RTT is a geneal elation between the ate of change of a fluid popet in a closed sstem and the coesponding C. C Moved sstem Region C This fluid was inside the C and left it (outflow) Renolds Tanspot Theoem (cont d) We ae inteested in the change of an etensive popet N such as m, P, H o E. The coesponding intensive (pe mass) popet is η, such as 1,, o e. N ss = Fom time t to t + t, N ss ma change dn ss = d 1 ρηd = lim ss t 0 t ρ η d ss ρηd t+ t ρηd t ss ss Eqn ( ) Region A This fluid was outside the C and enteed into it (inflow) Region B This fluid was inside the C and it is still in it. ρηd t+ t + ρηd t+ t ρηd t+ t C C A ρηd t C At time t, C and the sstem coincide At time t + t, C stas at its oiginal place but the sstem moves/defoms with the flow 3-33 N leaving the C though the eit N enteing the C though the inlet 3-34 dn ss n 1 = lim t 0 t CS C Renolds Tanspot Theoem (cont d) ρηd t+ t ρηd t + ρηd t+ t ρηd t+ t C C C A n C ρηd A C ρη( n)da is the velocit of the fluid at the contol suface CS Eecise : Stud the detailed deivation of RTT fom a tetbook. ρη( n)da A A ρη n da CS n is the unit outwad nomal of the CS RTT: Rate of change of popet N within the sstem. Renolds Tanspot Theoem (cont d) Lagangian pat dn ss = d ρηd = ρηd + ss C = Rate of change of popet N in the coesponding C (eo fo stead flows) + Euleian pat Impotant: Left hand side is NOT calculated diectl, instead we use Zeo F T Q + W : fo mass consevation : fo linea momentum consevation : fo angula momentum consevation : fo eneg consevation ρη( n)da CS Net flowate of popet N acoss the CS (positive fo outflow, negative fo inflow) 3-36

10 Renolds Tanspot Theoem (cont d) Eample : Let s use RTT to stud mass consevation in the spa can. Renolds Tanspot Theoem (cont d) Eample : Let s use RTT to stud the thust geneated b a jet engine. Fo mass consevation, N = m and η = 1 dm ss = ρ d + ρ n da C CS Fo linea momentum consevation, N = P and η = dp ss = ρd + ρ n da C CS A in, in p in, ρ in Engine on test stand A out, out p out, ρ out Zeo = Time ate of change of pefume's mass inside the spa can. A negative value. + Amount of mass that leaves the can in unit time. A positive value (Noneo onl at the little opening that the pefume can escape fom) F Sum of etenal foces acting on the C = Zeo fo the stead + opeation of the engine. Net ate of momentum outflow (Noneo onl at the inlet and eit of the engine) Foce to hold the engine in place = Thust foce Renolds Tanspot Theoem (cont d) Eample : Let s use RTT to stud the powe necessa to un a pump. Fo eneg consevation, N = E and η = e de ss = ρed + ρe n da C CS Outlet 2, A 2, p 2 Inlet 1, A 1, p 1 Shaft wok Q + W Rate of heat tansfe and wok done on the C = Zeo fo the stead opeation of the pump. + Net ate of eneg flow though the CS (Noneo onl at the inlet and outlet of the pump) 3-39