INTRODUCTION TO AERODYNAMICS
INTRODUCTION TO AERODYNAMICS MECH 594
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Propulsion Characteristics MECH 594 Fowler Flap r rm quotWm rm C mer H mm mm 1 m no la Fowler nap W up 535mm Ckuu d Mm Dye a Cumplex slotted ap 0 Beam 737 A mm Han mg in u Nvtgcd 3 mm H mm mm um and Mum Hm mm 5 mm mm nmaya suucmmw nnwnmmcrmsc Hmumbcr Thcmmpmxsmmd apshawuh mm m usus a hadngngu 5an m 2 mph 5mm rrmhng cdgm w Recip Prop Pitch Angle Recip Prop AlE Skyraider Recip Prop 4str0ke cycle Recip Prop Eta VS J Recip Prop annual 9 a 1 a o um or DIETiRCE rteML gimt mrImwmu u 29mm 3i rjpiwr39 as 4 rw39ns an7315ui Eir 1c3rr v Lunar Q5 355 Comparison of xed variable amp constant speed propellers Turbojet Axial Flow Turbojet Sketch SNng a nll uakaen w w ull UH indulgingIraquot J Illi mu Eamon S Turbojets B 5 8 Hustler Turbofans Turbofan Sketch Turbofans Boeing 777 Turbofans cum1 EHHL J 1m ING LIN JLECTELTI FAN Emmi3 COUNTER HQIMINE WEB N ES 39quotLJHE INE CC39MBLJETOH Ultra High Bypass Engine Turboprops Turboprop Sketch Turboprops C13 O Hercules Engine Flight Limits Operational Mach No5 and Altitudes Speci c Thmst Speci c Fuel Consumption Summary Symbols Conservation Equations in Derivative Form Nomenclature Explanation uid property of the nite sized blob and nite sized control volume dBdt as we travel along with the uid blob Lagrangian frame of reference uid property of an in nitesimal sized blob and differential control volume differential mass dB dm differential control volume CV volume of the in nitesimal sized blob uid velocity vector with components u v and w in Cartesian space the gradient a vector operator the divergence of the uid velocity vector divl7 force vector acting on a differential mass heat added to a differential mass work done by a differential mass The Material Derivative Recall that we derived a form of the Reynolds Transport Theorem RTT for a differential control volume ie7 a point in space7 1 D dB dV Dt Vbpx7 Alternater7 we could have started with the original RTT7 D 8 a dB dB 1 V a dA Dt blob at cv cs 0 lt n and if we assume that B7 p and l7 are continuous throughout the blob and the control volume then we can use the divergence theorem from vector calculus D Dt AladeCVdBCVVltbpl7gt dV D 8 a de deVdeV Dtblobp b at cvp cs If we consider the CV to be rigid then D i 8bp a allobbpdvbVl at W GM W and if we are only interested in the differential blob and differential control volume This is the differential form of the RTT which is also the de nition of the material derivative evaluated at a point in space Please note that Eq 1 is not a conservation equation but the translation between the change in some property of a uid particle differential uid blob as we travel along with it Lagrangian frame of reference and how that same property changes at some point in space differential control volume that we are observing Eulerian frame of reference We have to develop this translation between the Lagrangian and Eulerian frames of references because the conservations of mass mo mentum and energy were originally derived for a Lagrangian frame of reference PHY 101 MECH 211 and ENGI 200 The Conservation of Mass As previously mentioned the conservation equations were originally derived in a Lagrangian frame of reference so the conservation of mass for our uid particle is written as D dm 0 Dt We can now use our translator to rewrite the conservation equation in a form we can used in uid mechanics D dm 8p dV Dt at Vltpl7dVO But since dV is rigid then the equation can be written as 1 Ddm 7 8p dv D25 7 8t Vltpl70 2 The speci c forms of Eq 4 were mentioned in class 0 Steady ow 0 fmyz V pi 0 3 o lncompressible ow 0 constant a VV0 4 Material Derivative for b We know from our work with the integral forms of the conservation equations that we re going to have to deal with the material derivative of dB Since we know mass is conserved we might as well investigate the form of the material derivative of b ie D dB D b dm Db D dm Db d b D25 D25 mm D25 D25 From Eq 1 we can write D dB E 7 8b dm D25 7 mm 7 8t V b di 5 The application of the chain rule to the partial with respect to time is straightforward but we better look at the divergence term a bit closer In a Cartesian coordinate system V M vj wk For the scalar E and the vector V we have 7 85 85a 85a 814 81 8w k d V V5 8mz ayj 82 an V 8m 8y 82 So with a second scalar c we can write 7 NEW 8560 WSW Vi Wib yiT 8W 81 8W 856 856 856 EC 8m Ect3y 56 82 u 8m 1 8y w 82 7 5CVX7 i7vgc Returning to our problem v b di bdil7bltl7Vgtdmdmltl7Vgtb and we can write D dB 7 Db 7 8dm 8b a a a Dt 7 mW bTMirnabdmvvbltvvdmdmltvvb or bringing back the material derivative of dm dm Db bib 7 WWW 7 7b Ddm 7 lt8ltdm dmvv vwdm E 5 D25 8t ibiwwww From the conservation of mass equation we know that the right hand side of the equation above is zero Therefore VVb 6 Note that because we made use of the conservation of mass Eq 6 applies only to b and not dB So Eq 5 can be written as Db 8b a p papVVb 7 Testing Eq 7 for the conservation of mass we have b 1 and so the equation is satis ed identically Conservation of Linear Momentum The conservation of linear momentum is a statement of the fact that the time rate of change of the momentum of a mass that we re following is equivalent to the sum of the forces acting on that mass Written in our uid mechanics nomenclature for the differential mass it would appear as D Vdm a D9 D9 Dt ZdF dm p dv or D c113 8 7 a a p ZWplE V39Vle a since I V Conservation of Energy The conservation of energy is a statement of the fact that the time rate of change of the total energy of a mass that we re following is equivalent to the sum of the rate of heat added to and rate of work done by the mass For the differential mass we can write De dm 7 dVDe 7 DdQ DdW Dt 7p D25 7 D25 D25 0r Def 1DdQ7 1DdW th dv Dt dV Dt sincebe Limiting Forms of the NavierStokes Equations Low Reynolds Flows Recall for incompressible Newtonian uid gt C V 0 V 0 D 7 gt M p pg VpuV2V Dt In 3D three equations are involved from M which are extremely dif cult impossible to solve analytically It is possible to simplify N S for special conditions 37lliml Derivation of Stokes Eq Low Reynolds Number Flows Obtained from nondimensionalization of N S eqs and setting Re lt lt 1 Low Reynolds ows also called creeping ows highly Viscous ows lubrication theory Consider ow over a sphere SS neglect graVity gt p7 V Vp W23 37llirn2 Nondimensionalized with respect to U R a and p0 53 sz 953 yz 5 p0 U R R a a a V1 k gtV RV ax Jay az M ampV 0 V V p Vp gV2V R R R Vquot Vquot LL 13 371lim3 Physical experiments have shown that the in uence of pressure is never negligible therefore as Q gt O p 0 gt 1 H R Q Re ltlt 1 so for V p V 2V 0 2 or Vp LLV V Stokes Equation UR valid for Re 2 p7 ltlt 1 37llim4 Stokes Equation is in a relatively simple form but can still be quite dif cult to solve generally There are some very special cases where simple solutions are possible Simple cases incompressible Newtonian neglect gravity SS 1 Couette ow with pressure gradient 2 Couette ow without pressure gradient 3 Plane parabolic ow 4 HagenPoiseuille ow circular pipe Chap 8 37llim5 Couette Flow ss39 39 39 39 zoutofthepage w 0 7 7 0 371mm Governing equations gt C V 0 V 0 M p7o V Vp W2 7 Governing equations in Cartesian coordinates c a ua V 0 3X 3y Bu Bu 3p azu azu M ltxgt phast 2 2 3V 3V 3p 3V 3V M u V n v y 3X 3X2 3711im7 Nondimensionalize with L hO U A pO We have chosen L hO U A pO so that maximum values of dimensionless variables are of the order 1 37llim8 Substitute into C gt E 811 A at 0 L 8X h0 By C must be satis ed for all conditions therefore Df AEU 61116 L 371lim9 Nondimensionalize M Mx pUh0h 0J H L 3X 3y 371lim10 371lim11 From the way we have nondimensionalized velocity and spatial variables the derivatives are of the same order eg 311 311 av av Ll N V N U N V 3X 3y 3X 3y d 32 32 32v 32v an 2 N 2 N 2 N 2 3X 3y 3X 3 2 2 2 2 2 2 h h Therefore 2 312 ltlt 312 and 0 3 112 ltlt 3 112 L 3X By L 3X 3y Uh h and if p 1 LL L 372cou3 pUh Note Actually Therefore 372cou4 0 can be moderate 3103 E 3i L 3X L 3p h0 3y II II Recall Then My 372cou5 If the pressure force balances the Viscous force and the length ratio f0 then p0 N1 E L ho a 32 So ap 3132 O X y 2 2 M ap 11 3 31220 y By L 3y 372cou6 h 2 2 but 2 a 2 ltlt 312 L 3y 3y 3y Neglecting My we have 2 1 a u a d MX 2 p p 3y 3X dX the dimensional form of the governing equations become Bu 3V C 3X 3y 0 dP dX 372cou7 Boundary conditions y O V 0 u 0 y 2 ho V O u U Since the plates are in nite and parallel uid particles move parallel to the plates and there is no velocity in the y direction V y O Bu 3V Bu C 0 d o 3X By an ax So u uy 372cou8 2 Governing equations become LL 1 121 2 d p dy dX B constant since uuy and p px c12u z i dp dy2 H dX B 2 Integrat1ngtw1ceg1ves u y C1 372cou9 Elm Using the boundary conditions uyO 2 Lly2h0 U 2 110 C1110 So we can write Cl De ne 372cou10 L 2 L ZHhojho 2 Lh L 1 ho 25UltBgtho1 ho 2 h d p ZHIOJdX So the velocity pro le depends on P l i P i 1 1 CouetteFlow U h0 h0 h0 372cou11 The simplest type of Couette ow is zero pressure gradient 3 Q O and P 0 dX 2 2 hi the velocity varies linearly between the plates 0 The uid is being dragged along by the upper plate 372cou12 h 0 Volume ow rate per unit width q I udy 0 h0 B 2 y q U y hydy d ho 2 0 3 qzho Bh0 2 12H 2 Ph0B ZHU 372cou13 Mean velocity q u mean U P umean 314 Maximum velocity id u i1P 2 3 P U dy hO ho 1 du 1 P 0 gt for u u U dy y 2P 0 max Bu U UP 2y Shearstress TH H H 1 3y h0 h0 h0 372cou14 Creeping Flow About Immersed Bodies Creeping ow is the only Viscous ow about an immersed body you can analyze by hand 3D Problem Stoke39s Solution for an Immersed Sphere Assumptions steadystate incompressible aXisymmetric gravity free creeping ow Re ltlt 1 Governing eqs lt n O C V M Vp uV2 7 372crel V 0 M gives 2 2 gt 2 gt VonzV pquoV Vqu VoVO So from C Vzp O The pressure eld in creeping motion satis es the potential equation and the pressure p is a potential function Various solutions can be found through superpositioning We also want to nd a linear equation for the velocity 372CI82 Let39s de ne some vector F with the property gt VXFV and VOFO Recall the vector identities VXVIIEO and V0VgtltFEO So VgtltM gives VgtltV2VV2VgtltVV2VgtltVgtltF But from another vector identity and de nition of F VxVxFVVoF V2F V2F and so VXVp0MVXV2V MV4F Therefore V4F O 372cre3 We will use spherical polar coordinates r 9 with no 1 variation 0 gt ra u r a A gt gt 372cre4 De ne the vector F so that A 11 FFrF 61 1 r 9 M M rsin6 Where the only nonzero component of Fis a function of r and 6 and we have the stream function 1 1r6 We write modi ed M as V4 1 O which is a linear partial differential equation 372creS The velocity components ur and u9 can be described by a stream function 11 1 8w 1 Bu 2 d u r2 Sin 9 36 an ue r sin 6 3r Plug into C 13 2 1 a r 2rur rsi116u9s1n6 O r2 sin 6 Brae r2 sin 6 Brae Which was expected from the vector identity V 0 V x F E O 372CI86 BCs LIV LIV 0 Br 36 22 r gtoo wz Ur s1n Gconst Since the governing equation is linear we can use the separation of variables method III f rg9 Applying to the biharrnonic equation V4 11 0 372cre7 Solution for a sphere is 1 2 11 Ua2 s1n2 6E 2L2 4 r a a 3 ur Ucos612a 3 a r r 3 L19 2 Usin6 1 Ail 3 3 21 r 372cre8 Note the following properties 1 The streamlines and velocities are independent of LL 2 2 The stream11nes possess perfect symmetry sm 6 there 1s m wake Inert1al terms cause wakes Typical of higher Re 3 The local velocity is everywhere retarded from its freestream value Different from potential ow 4 The effect of the sphere extends to enormous distances 372cre9 We integrate the M to nd p p pm Wcose 2 r pm 2 uniform freestream pressure The pressure difference p poo is antisymmetric positive in front and negative in the rear of the sphere This creates pressure drag on the sphere 372CI810 Shearstress T V9 r86 8r LLUsin6 3a 5a3 r 4r 4r Total drag force F F jrr9 sinGdA jp cosGdA 0 0 ra ra dA2139Ia2 sinGdG so F 41tLLUa ZNLLUa 61tLLUa Stokes Law 372crell Valid for Re ltlt 1 Experiments give 0 lt Re lt 1 The drag coef cient uses the projected area Ap area of a circle C 2 say Rezm D 1U2A 5 0 P where Ap 7ra2 CD E Sphere O ltRe lt1 Re 372CI812 Solutions for other bodies Disk normal to freestream F 16 HUa Disk parallel quot quot F LLUa Stokes sphere law is accurate for roughly spherical bodies grains of sand dust pollen etc 372crel3 MAPPINGS A mapping is a function that is used to transform one set of coordinates into another eg z transforms zxz39y to z n Applying a mapping to the complex potential F for example transfers the value that F has at z to the new point 9quot So this generates a new potential Fz We may then calculate the new ow velocity by differentiation N dF dF dz dz W d Ed WZd This new potential describes a valid new ow as long as 147 is analytic ie 511351 in nity Since we require that d dz d 720 dz Where if 2 0 the mapping is called CONFORMAL Z Points where d dz 0 are called critical points Angles of intersection are preserved under conformal mapping since for small changes 6 5147512 5z if dfdz 2 0 difdz will be in general a complex number Multipication of the line element dz by this number will thus rotate it by an angle of argd 7dz and increase its magnitude by a factor dfdz These effects will be the same regardless of the initial direction of dz Angles of intersection are not in general preserved at a critical point Here the connection between 5 and amp depends on higher derivatives It can be expressed as a function of the form 5 Aamp f where b is a real but not necessarily integer power It is clear therefore that angles of intersection at a critical point in the 2 plane will be multiplied by the factor b during transformation The circulation around and out ow from a closed loop are unaltered by mapping rag jmzmzjW dzjw7dgfHQ It is normal to apply mappings to the complex potential Any mapping however may alternatively be applied to the complex velocity In general though this will not produce the same result ie 1sz Dimensional Analysis and Similaritv Overview The control volume and differential approach are the analytic tools used by engineers to solve ow problems Unfortunately purely analytic methods are limited 0 lack of complete information turbulence dif culty of math and computations required 100 years of research has yet to yield a complete theory for turbulent ow in a pipe the variables are known but not their relationship 37ldiml Engineering uid mechanics is a combination of theory and experiment more so than most other elds Analytical Models Theory V Physical Phenomena Computational Model CFD PhysicalEXperiments 371dim15 Experiments Turbulent ow in a pipe Investigate me pressure amp e prpz sh P Wecanwrite ghffLDVpug where g isameasure ofthe wall roughness Finding me function r is me objective ofthe experiment rneed not be an analytical function Tables chm and curve ts are perfectly acceptable 106 Ifwe 39439 r39 experiments wim uid wim different p and p Also anyone using your data would have to make a a parameter interpolation To reduce me number ofparameters we take advantage of 37mm Dimensional Analysis a packaging or compacting technique used to reduce the complexity of experimental programs and increase the generality of the results h L VD 8 Turbulent p1pe ow gt 2 F p l 2 D LL D V 2 helps our way of thinking and planning for an experiment g theory It suggests ways of writing equations gives a great deal of insight into the form of physical relationships provides scaling laws which can convert data from models to prototypes 37ldim3 To use dimensional analysis we must make use of the principle of dimensional homogeneity PDH which we spoke of in the beginning of this course PDH Any equation that completely describes a physical phenomenon must be valid regardless of the units of measurements 0 Dimensions of all additive terms must be the same Not affected by derivatives and integrals Example 2 2 p1 V1 p2 V2 Bernoulli39s equation p 2 ng g 7 gZZ is dimensionally homogeneous 37ldim4 Example Manning39s formula for open channel ow English units 2 1 v R5 s4 R radius ft 11 n n S slope o is not dimensionally homogeneous n number V velocity ftsec 0 Eq is valid only for English units Motivation any dimensionally homogeneous equation or process can be written in nond1mens1ona1 form 371dim5 Nondimensionalization of experimental data or an equation requires the Buckingham Pi H Theorem of dimensionless parameters H groups of important parameters involved of dimensions involved For DA there are in general four quotdimensionsquot used MLTe or FLTe where 6 temperature We usually neglect heat transfer except for gas dynamics problems So we only need to use three dimensions in this course MLT or FLT 37ldim6 Buckingham Pi Theorem Pipe Flow Example more detail vclmzw Wall pram raLgham n i where ghf represents mechanical energy loss 1 Obtain important parameters of the problem 0 ow properties nghf 0 geometry L D 8 o uid properties 314 SoN7 ghf 7D 7V7 p 9 H 98 2 Determine the number of dimensionless parameters H groups you need to construct gt H groups N K K the maximum number of dimensional parameters repeating variables that cannot form a dimensionless H group among themselves We39ll need to use these repeating variables later in the method K is usually equal to the number of fundamental dimensions involved in the dimensional parameters 372bu02 To nd these repeating variables it39s usual to nd the parameters whose dimensions are as close to quotpurequot as possible This is an art that can involve knowledge of the experiment and trial and error Rules ranked in order of importance Select as repeating variables K of the dimensional parameters with all of the fundamental dimensions Do not use the dependent parameter ghf Do not use a parameter that may be important only part of the time u 8 Use parameters whose dimensions are as close to quotpurequot as possible Parameters whose dimensions are as close to quotpurequot as possible L L pure L D L pure L V LT 1 closest to pure T p ML 3 closest to pure M n ML 1T4 8 L pure L Using the previously listed rules K 3 and we chose D V and p gtN K7 34H groups 37lbuc3 3 Nondimensionalization of the remaining dimensional parameters using the repeating variables Take the remaining parameters and combine them with repeating ones ghf 7L My 3 to form the 4 H groups Start with H1ghfpavbD Find ab and c so H1 is dimensionless 13911L2T2ML3aLT1bLc MaL2 3abcT 2 b 2 a0 2b0 a0b 2c0 2 3abc0 372buc 3 cont39d ghf H1 V2 H2 LpaVch a 1 3abc b a0 b0 13abc0 a0b0c 1 HL 372buc7 abc H3upVD 1a0 13abc0 1b0 a1b1c1 H 39H3pVD a b c 8 H4 epVD 5 So we can write H1 FH2H3H4 F M g V2 D pVD D 372bu08 4 Rearrange the H groups to correspond with customary usage Use traditional types of dimensionless parameters Rearrangement It39s apparent that BPT will not give a unique set of H terms Therefore we are allowed to V2 2 multiply Hs by a constant raise the Hs to any positive or negative power 0 multiply any power of a H with any power of another H ghf 2 ghf pVD LL 1 1V2 V2 u pVD 2 ghf GL pVD a 1 B M 75 As long as the required number of H terms is xed 372buc9 quotStandardquot Dimensionless Parameters Two generic problems Internal ow Luu ar pal mm ul lr mumm was I J Timurv umiw uI us va drf am I gt 39L a41539m u brdr E39lrcu 14 kmIn m want u N h w 39 s Mquot can a m hard law Ir Hm mi 1w lulu rd m n 1511 External ow The general functional relation for these problems Dimensional parameter of interest f size shape uid velocity uid properties dimensional constants In general the type of parameters on the right hand side changes very little from problem to problem 371sta1 On the left hand side we are interested usually in External ow force on the body by the uid total force lift force or drag or oscillation 0 Internal ow wall force shear stress pressure drop mechanical energy loss or oscillations 0 For our example we will use drag and mechanical energy loss ghf respectively f fLVpua6cpcVg86w and ghf fLVpua6cpcvg8ew a speed of sound o E surface tension 6 W wall temperature 37 l sta2 Using the drag equation we put it in fundamental dimensional form llflmltll lllll lTMl lizzel lizzel L e The drag equation has N 12 dimensional parameters and K 4 fundamental dimensions M L T 6 Since p V L and cp cannot form a H K 4 and we use them as the repeating variables ofPigroups NK1248 37lsta3 Perform dimensional analysis on the original equation to get i 2 F X V2 p E 9w lpszz u a o gL c L GO 2 Each of the groups on the right hand side is a quotstandardquot dimensionless parameter in uid mechanics The standard dimensionless parameter are used as tools to classify ow as laminar or turbulent compressible or incompressible etc Dimensionless parameters and their signi cance are given on page 415 of the text 37lsta4 mu 7 Mam Mum Mummvnm quotma mm mm An trmmulunvhyg quotmy mm 1 madman1mmbumuo mmquot Nahum Mm My M4 Mm WWW mm Wm mmmm mmmmmmm mm m um vae um mm mnmm u mmmwmnm m m wwwwmmw 7 Au mm mm mm WWW m mm mm mm g WW W mum m Mm hm mm H mmpr m W MWWWN W WW quotMyc u quot m H m mmquot mm mm mm m m m Wm 7 g m Wm W m quotM Ht rm W mmwmh M V Vquot Wm M mm mm 5lt My W mm M MM 7 mm man lwmnu H mm m w m 7 W m M Note for the pipe ow 2 1 ghf F pVL V pVZL V2 Cp 8 9w ivz u 7a o gL cVLG0 2 Important parameters that appear on the right hand side of our equations are Re through 6 W These act as the dependent variables 0 On the left hand side what we are usually interested in are drag force 13 0 lift force pressure difference Ap p poo wall shear stress Tw mechanical energy loss ghL frequency of oscillation 0 37lsta7 These appear in the following dimensionless groups CD CL Cp in book Also C f a ITWZ loss coef cient important for internal ows 5 9L St E 7 Strouhal number osc1llat1ng ows These act as dependent variables in our experiments For example gtCD F ReMaWeFry e W Z FHIIH V2 29 37lsta8 Ph sical Si nificance of Dimensionless Parameters Standard H groups can be interpreted as a ratio of typical values of two physical entities force or energy Imagine a moving uid particle under the in uence of an inertia force ma Estimate the inertia force 3 ma pL X where 1s an order of magnitude estimation since V L T are characteristic velocity length and time we can write is V V2 T L L V 32 22 mapLT pVL Estimate the Viscous shear force Viscous shear force shear stress gtlt area 3u 3y shear stress 2 c 2 LL 372phy2 Assuming the velocity changes from O to the characteristic value V over the length L then the Viscous force is u J L LLVL Note then that the ratio of 2 2 1nert1a force N pV L Viscous force LLVL amp Re the Reynolds number u 372phy3 2 Pressure force pressure dlfference gtlt area ApL 2 Press force lApL 1A1 Cp 2 coef cient of pressure 1nert1a force EszLz E pvz 3 graV1ty force pL g 39 rt39 f 2L2 2 me 121 orce pV 3 V Froude number squared grav1ty force pL g gL Note that not all US are force ratios L Strouhal number m L Char39 ow tlme N 371phy Model Testing and S tude ACJLA We plan to test models to study the drag Fv7hrmw39v39muzmwn WW awmmwwuw has wquot quot 39 quot 1 mm snm Wm rramhu y mm 0 First requirement of testing the model should look exactly like the prototype geometric similarity All model lengths scale gtlt all prototype lengths L1 11L L2 A1112 117 7 Therefore model areas iLl gtlt prototype areas 37lmod2 From dimensional analysis we can write for the prototype CDp FpRep Map yp up 1 where OL angle of attack When the model is tested it should obey the same form CDm FmRemMamjym0tm 2 Note by using H groups F p and F mdepend only on the shape of the prototype and model 37lmod3 So from geometric similarity F F Fand0t 20 p m p m 1 gt CD p FRepMapypoc CD FRemMamymoc Ul Kinematic similarity Model and prototype have the same lengthscale ratio and the same time ratio velocityscale ratio is the same for both 37lhnod4 If we were interested in the Froude number Fr having Frm a l3 N H t M H w H Dynamic similarity Model and prototype have the same lengthscale timescale and forcescale massscale ratio Dynamic similarity exists along with kinematic similarity if force and pressure coef cients for model and prototype are equal 371mod5 For our model compressible ow CDm CDp only if For incompressible ow a No free surface Rem 2 Rep b Free surface Rem 2 Rep Frm 2 Fr Wem We Ca 2 Ca 37lmod7 Measurements of Static pressure taps head loss in a pipe Vinturi meter I 4 Test pipe Flow control Flow 4 Motor Tank 3 ft gt TABLE 43 Results of Pressure Drop in 2 Pipe Experiment Test Q39gpm Mun vats pVDp39 LD AllL AhLpvop1 Group I I50 250 206 9097 Ho 004I7 504 X IO 1 2 I 85 369 254 I I 220 006I5 489 3 200 438 275 2 I 30 0073 497 4 245 625 337 I4 859 0 04 47 5 280 888 385 I6 982 0 I48 5 I3 6 330 I 75 454 20 OH 0 I96 489 7 350 I33 48I 2 227 02I9 486 8 390 I6 I 3 536 23 653 0269 48I 9 4 I0 I78I 564 24866 0297 480 Group II I l I I50 I50 206 9097 66 004I6 504 X IOquoto I2 I85 220 254 I I 220 006I I 489 I3 200 263 275 I2 I 30 0073 I 497 I4 245 380 337 I4 859 0 I06 47I 5 280 533 385 I6 982 0I49 5l3 I6 330 700 454 200I4 0 I94 489 I7 350 788 48I 2 227 02I9 486 I8 390 970 536 23 653 0269 48I I9 4 IO I070 564 24 866 0297 48I Group In 2 I50 I00 206 9097 440 004I6 504 x I0 0 22 LBS I48 254 I I 220 006I7 489 23 200 I 75 275 I2 I 30 00729 497 24 245 250 337 I4859 0 I04 47 25 280 356 385 I6982 0 I48 5 I3 26 330 470 454 200I4 0 I95 489 27 350 525 48I 2 227 02I9 486 28 390 645 536 23 653 0269 48I 29 4 IO 7 I 3 564 24866 0297 48I average 490 x I0 Note Group I data for 5ft length group II data for 3ft length group III data for 2ft length T 70 F v I03 X IO ft s D 0545 in A 000l62 ft Raw datiaiafand reduced data b of the pipe friction experiment E 5 035 3 5 0 30 f f A E 0 25 Z c d 2 020 E 1 393 71 g 015 010 A 005 0 I i l l I I l 7 8 910 15 2 25 3 4 Flow rate Q gpml pVDp a b Problems with Model Testing Suppose we want to determine the drag on a large ship hull by experiments on a scale model CD F Re Fr The model is 125 the size of the prototype 14 31le i geometric similarity Lp 25 37lprol For kinematic and dynamic similarity 371pro2 FrIn Frp Fr SO and Rem 2 Rep V2 V2 m P gm 2 gp by today39 S technology gmLm gPLP L 1 vii7 P P pmeLm pPVPLP VPLP Vm LmliL v Vpr 5 25 125 lt l m practical test uid for this case is water v the Viscosity of water There is no such uid since mercury is 19 The only we cannot match both Re and Fr For our previous aircraft experiment CD F Re Ma y Ot Try to match Re and Ma for the model and prototype Ifwe assume the uid is an ideal gas V V V or m M a 4 lymRmem zprpep Vp 37lpro3 Re pmeLm pPVPL Hm Hp V L V1scos1ty ratlo H m pm m H p ppr p mePeP YmRmem Lm pm YmRPeP Lm ppRmGm prpGp Lp R Prototype gas is air and so is the model test gas ym 2 VP Rp 371pro4 The Viscosity obeys 075 1 25 HmEem OLmxlgampem l LP ep LP pm ep Say the prototype will y at 3km in standard atmosphere pp 2 70kPa 9p 2 270K Say Gm 2270K tunnel temp pm El750kPa a good deal of compressor power We have to satisfy parameters in order of importance is we can39t satisfy them simultaneously 37lpr05 Typical importance of H groups in model tests Tvne of test Primarv Second Low speed ow no free Re 06 eL surface pipes valves ttings low speed aerodynamics Surface vessels Fr Re eL Dams rivers Fr Re eL High speed ow Ma y 0L Re eL Unsteady ows St Others 37lpr06 Nondimensionalization of the Governing Equations An alternative to nding the H parameters for various experiments is to nondimensionalize the governing uid ow equations that are applicable to speci c conditions Recall from Chapter 6 Continuity Momentum 0 Energy State 37lnonl a pVopVO at DV v V P Dt Pg P T 39O CD39O pVOVVOkVGCD phone a h 110979 2 temperature h enthalpy Looking at incompressible ow of a Newtonian uid const cV const cp const p constu C VoVO 2 M Ppg Vpuvv D6 2 E kVGCIgt ltgt pcVDt k coef cient of thermal conductivity Note that energy is uncoupled from C and M 37lnon2 Typical boundary conditions for the equations are Fixed solid surfaceV O Inlet or outlet V p are known DT assumin z isu Dt g p Free surface z n w 1 1 p pa 6RX Ry where pa reference pressure surface tension 6 R radii of curvature 37lnon3 Using characteristic velocity length time and dynamic pressure U L T pU2 we nondimensionalize dependent and independent variables gtxlt V gtxlt Z V p p 9 pU U Assuming all spatial variables can be related to L 1 aY L7Z X 1 L For time t but we can relate T to U and L TN Clb b p U L are constants 37lnon4 Using the X component as an example 371non5 au 3Uu E ax 8LX L 8X 8p 3PU2 13 3PgL Z pU2 ap ng azquot 3X 8LX 8LX L 8X L 8X Applied to C C 0V 0 or V 0V 0 Applied to M 2 2 DU J rwkwwm w 2 371non6 For the boundary conditions Fixed solid surface V O Inlet or outlet known V p Free surface z n j W D n loquotwe lmW 37lnon7 Note that the various quantities Within the brackets can be seen as various forces per unit volume pU2 inertia force L unit volume MU Viscous force L2 unit volume g Eressure force L unit volume pg gravity force unit volume 2 surface tension force L2 J unit volume 37lnon8 inertia force Dividing the bracketed parameters by the unit volume we have for C and M C V V 0 2 M Vp iiV V Dt Re and for the free surface boundary condition p EuFrz LW6JR R 371non9 Re Reynolds number Eu Euler number Fr Froude number We Weber number The Euler number is usually written in terms of pressure differences pa Ap A Eu pz pU lf Ap involves vapor pressure pV we have the cavitation number Capr v pV Where pr is some reference pressure Within the flow field 37lnon lO For the energy equation we Will nondimensionalize the temperature using a characteristic value 60 reference temperature 6 i 80 SO Dt pcVUL pcVE0 L Hrrwl h FrRe Re 371non11 Dividing the bracketed parameters by the m we have for C and M un1t volume C V V 0 M D i Vp LV2V Dt Re and for the free surface boundary condition p Eu kr l 111 371non9 Re Reynolds number Eu Euler number Fr Froude number We Weber number The Euler number is usually written in terms of pressure differences pa Ap Ap Eu pUZ If Ap involves vapor pressure pV we have the cavitation number Ca IDr V pV where pr is some reference pressure within the ow field 37lnon10 For the energy equation we will nondimensionalize the temperature using a characteristic value 60 reference temperature So Dt pcVUL pcVG0 L y V26 Fr Re Re 371non11 371non12 Dimensional Analysis and Similarity My The integral and differential methods previously described are the analytical methods used in the eld of uid ow analysis Experimentation supplemented by dimensional analysis is the second analysis method Numerical analysis is the third Experiments are a Vital part of uid mechanics and are needed to validate theory and numerical methods 37lno n13 We have 0 recast familiar physical relations in dimensionless form to satisfy the principle of dimensional homogeneity presented a systematic method to uncover dimensionless parameters the Buckingham H theorem found H groups that affect particular physical processes Re Fr Pr etc 0 found difficulties in achieving true similarity between models and prototypes scaling difficulties Compromises must be made using more experiments andor theory showed that even though the governing differential equations may be near impossible to solve analytically we can use them to uncover important dimensionless parameters We will use dimensionless groups with the differential equations to investigate speci c ow phenomena in MECH 372