MAIN TOPICS iensional i l Analysis Bukingha Pi Theore eterination of Pi Ters Coents about iensional Analysis Coon iensionless Groups in Fluid Mehanis

Transcription

1 FUNAMENTALS OF FLUI MECHANICS Chapter 7 iensional Analysis Modeling, and Siilitude

2 MAIN TOPICS iensional i l Analysis Bukingha Pi Theore eterination of Pi Ters Coents about iensional Analysis Coon iensionless Groups in Fluid Mehanis Correlation of Experiental ata Modeling and Siilitude Typial Model Studies Siilitude Based on Governing ifferential Equation

3 iensional Analysis /4 A typial fluid ehanis proble in whih experientation is required onsider the steady flow of an inopressible Newtonian fluid through a long, sooth- walled, horizontal, irular pipe. An iportant harateristi of this syste, whih would be interest to an engineer designing a pipeline, is the pressure drop per unit length that develops along the pipe as a result of frition. 3

4 iensional Analysis /4 The first step in the planning of an experient to study this proble would be to deide the fators, or variables, that will have an effet on the pressure drop. Pressure drop per unit length p f (,,, ) Pressure drop per unit length depends on FOUR variables: sphere size (); speed (); fluid density (ρ); fluid visosity () 4

5 iensional Analysis 3/4 To perfor the experients in a eaningful and systeati anner, it would be neessary to hange the variable, suh as the veloity, whih holding all other onstant, and easure the orresponding pressure drop. iffiulty to deterine the funtional relationship between the pressure drop and the various fats that influene it. 5

6 Series of Tests 6

7 iensional Analysis 4/4 Fortunately, there is a uh sipler approah to the proble that will eliinate the diffiulties desribed above. Colleting these variables into two non-diensional obinations of the variables (alled diensionless produt or diensionless groups) Only one dependent and one p independent variable Easy to set up experients to deterine dependeny Easy to present results (one graph) 7

8 Plot of Pressure rop ata Using p (FL L(F / L 4 T 3 ) )(FT ) F L T (FL 4 T (FL )(LT T) )(L) F L T diensionless produt or diensionless groups 8

9 Bukingha Pi Theore /5 A fundaental question we ust answer is how any diensionless produts are required to replae the original list of variables? The answer to this question is supplied by the basi theore of diensional analysis that states If an equation involving k variables is diensionally hoogeneous, it an be redued to a relationship aong k-r independent diensionless produts, where r is the iniu nuber of referene diensions required to desribe the variables. Bukingha Pi Theore Pi ters 9

10 Bukingha Pi Theore /5 Given a physial proble in whih the dependent variable is a funtion of k- independent variables. u f (u, u 3,..., u k ) Matheatially, we an express the funtional relationship in the equivalent for g(u, u, u 3,...,, u k ) where g is an unspeified funtion, different fro f.

11 Bukingha Pi Theore 3/5 The Bukingha Pi theore states that: Given a relation aong k variables of the for g(u, u, u 3,..., u k ) The k variables ay be grouped di into k-r independent d diensionless produts, or Π ters, expressible in funtional lf for by,,,,, ) ( 3 kr or (,, 3,,,, kr ) r?? Π??

12 Bukingha Pi Theore 4/5 The nuber of r is usually, but not always, equal to the iniu nuber of independent diensions required to speify the diensions of all the paraeters. Usually the referene diensions required to desribe the variables will be the basi diensions M, L, and T or F, L, and T. This theore does not predit the partiular funtional for of or. The funtional relation aong the independent diensionless produts Π ust be deterined experientally. The k-r diensionless Π ters obtained fro the proedure are independent.

13 Bukingha Pi Theore 5/5 A A Π ter is not independent if it an be obtained fro a produt or quotient of the other diensionless produts of the proble. For exaple, if 3 / or then neither Π 5 nor Π 6 is independent of the other diensionless produts or diensionless groups. 3

14 eterination of Pi Ters / Several ethods an be used to for the diensionless produts, or pi ter, that arise in a diensional analysis. Regardless of the ethod to be used to deterine the diensionless produts, one begins by listing all (diensional) variables that are known (or believed) to affet the given flow phenoenon. Eight steps listed below outline a reoended proedure for deterining the Π ters. 4

15 eterination of Pi Ters / Step List all the variables. List all the diensional variables involved. Keep the nuber of variables to a iniu, so that we an iniize the aount of laboratory work. All variables ust be independent. For exaple, if the ross-setional setional area of a pipe is an iportant variable, either the area or the pipe diaeter ould be used, but not both, sine they are obviously not independent. Ex. γ=ρ g, that is, γ,ρ,, and g are not independent. 5

16 eterination of Pi Ters 3/ Step List all the variables. Let k be the nuber of variables. Exaple: For pressure drop per unit length, k=5. (All variables are p,,,,, and ) p f (,,,) 6

17 eterination of Pi Ters 4/ Step Express eah of the variables in ters of basi diensions and find the nuber of referene diensions. Selet a set of fundaental (priary) diensions. For exaple: MLT, or FLT. Exaple: For pressure drop per unit length, we hoose FLT. p FL 3 FL T L LT FL 4 T r=3 7

18 eterination of Pi Ters 5/ Step 3 eterine the required nuber of pi ters. Let k be the nuber of variables in the proble. Let r be the nuber of referene diensions (priary diensions) required to desribe these variables. The nuber of pi ters is k-r Exaple: For pressure drop per unit length k=5, r = 3, the nuber of pi ters is k-r=5-3=. 8

19 eterination of Pi Ters 6/ Step 4 Selet a nuber of repeating variables, where the nuber required is equal to the nuber of referene diensions. Selet a set of r diensional variables that inludes all the priary diensions (repeating variables). These repeating variables will all be obined with eah of the reaining paraeters. Exaple: For pressure drop per unit length ( r = 3) selet ρ,,. p FL FL 3 T L LT FL 4 T 9

20 eterination of Pi Ters 7/ Step 5 For a pi ter by ultiplying one of the nonrepeating variables by the produt of the repeating variables, eah raised to an exponent that will ake the obination diensionless. Set up diensional equations, obining the variables seleted in Step 4 with eah of the other variables (nonrepeating variables) in turn, to for diensionless groups or diensionless produt. There will be k r equations. Exaple: For pressure drop per unit length

21 eterination of Pi Ters 8/ Step 5 (Continued) a b p (FL 3 )(L) a (LT ) b (FL 4 T ) F L T F : L : 3 a T : b a b 4,b, p

22 eterination of Pi Ters 9/ Step 6 Repeat Step 5 for eah of the reaining non-repeating variables. a b (FL T)(L) a (LT ) b (FL 4 T ) F L T F : L : a b 4 T : b a,b,

23 eterination of Pi Ters / Step 7 Chek all the resulting pi ters to ake sure they are diensionless. Chek to see that eah group obtained is diensionless. Exaple: For pressure drop per unit length. p F L T M L T F L T M L T 3

24 eterination of Pi Ters / Step 8 Express the final for as a relationship aong the pi ters, and think about what is eans. Express the result of the diensional analysis. (, 3,,,, kr Exaple: p For pressure drop pp per unit length. ) p iensional analysis will not provide the for of the funtion. The funtion an only be obtained fro a suitable set of experients. 4

25 eterination of Pi Ters / The pi ters an be rearranged. For exaple, Π, ould be expressed as p 5

26 Exaple 7. Method of Repeating ariables A thin retangular plate having a width w and a height h is loated so that it is noral to a oving strea of fluid. Assue that the drag,, that the fluid exerts on the plate is a funtion of w and h, the fluid visosity, µ,and ρ, respetively, and the veloity,, of the fluid approahing the plate. eterine a suitable set of pi ters to study this proble experientally. 6

27 Exaple 7. Solution /5 rag fore on a PLATE f (w, h,,, ) Step :List all the diensional variables involved.,w,h, ρ,μ, k=6 6di diensional i paraeters. Step :Selet priary diensions M,L, and T. Express eah of the variables in ters of basi diensions MLT w L h L ML T ML 3 LT 7

28 Exaple 7. Solution /5 Step 3: eterine the required nuber of pi ters. k-r=6-3=3 Step 4:Selet repeating variables w,,. Step 5~6:obining the repeating variables with eah of the other variables in turn, to for diensionless groups or diensionless produts. MLT w L h L ML T ML 3 LT 8

29 Exaple 7 Exaple 7 Solution Solution 3/5 3/5 Exaple 7. Exaple 7. Solution Solution 3/5 3/5 3 b a b a M : T L M ) (ML ) (LT )( L) (MLT w 3 b a b a b b T : 3 b a L : w, b, a w T L M ) (ML ) (LT L(L) hw 3 b a b a 3 b a L : M : ) ( ) ( ) ( h b a b T : 3 b a L : w h 9, b, a

30 Exaple 7. Solution 4/5 3 w a M : L : T : b (ML a b 3 b a,b, T )(L) a (LT ) b (ML 3 3 ) M w L T 3

31 Exaple 7. Solution 5/5 Step 7: Chek all the resulting pi ters to ake sure they are diensionless. Step 8: Express the final for as a relationship aong the pi ters. The funtional relationship is w (, 3 ), or h, w w 3

32 Seletion of ariables /4 One of the ost iportant, and diffiult, steps in applying diensional analysis to any given proble is the seletion of the variables that are involved. There is no siple proedure whereby the variable an be easily identified. Generally, one ust rely on a good understanding of the phenoenon involved and the governing physial laws. If extraneous variables are inluded, then too any pi ters appear in the final solution, and it ay be diffiult, tie onsuing, and expensive to eliinate these experientally. 3

33 Seletion of ariables /4 If iportant variables are oitted, then an inorret result will be obtained; and again, this ay prove to be ostly and diffiult to asertain. Most engineering probles involve ertain siplifying assuptions that have an influene on the variables to be onsidered. Usually we wish to keep the probles as siple as possible, perhaps even if soe auray is sarified. 33

34 Seletion of ariables 3/4 A suitable balane between sipliity and auray is an desirable goal.~~~~~ Enrio Feri ariables an be lassified into three general group: Geoetry: : lengths and angles. Material Properties: relate the external effets and the responses. External Effets (Input & Output): produe, or tend to produe, a hange in the syste. Suh as fore, pressure, gravity,(< (<- Input) or veloity (<- Output). 34

35 Seletion of ariables 4-/4 Points should be onsidered in the seletion of variables: Clearly define the proble. What s the ain variable of interest? Consider the basi laws that govern the phenoenon. Start the variable seletion proess by grouping the variables into three broad lasses. Continued 35

36 Seletion of ariables 4-/4 Points should be onsidered in the seletion of variables: Consider other variables that ay not fall into one the three ategories. For exaple, tie and tie dependent variables. Be sure to inlude all quantities that ay be held onstant (e.g., g). Make sure that all variables are independent. Look for relationships aong subsets of the variables. 36

37 eterination of Referene iension /3 When to deterine the nuber of pi ters, it is iportant to know how any referene diensions are required to desribe the variables. In fluid ehanis, the required nuber of referene diensions is three, but in soe probles only one or two are required. In soe probles, we oasionally find the nuber of referene diensions needed to desribe all variables is saller than the nuber of basi diensions. Illustrated in Exaple e

38 Exaple 7. eterination of Pi Ters An open, ylindrial tank having a diaeter is supported around its botto iruferene and is filled to a depth h with a liquid having a speifi weight.. The vertial defletion,, of the enter of the botto is a funtion of, h, d,,, and E, where d is the thikness of the botto and E is the odulus of elastiity of the botto aterial. Perfor a diensional analysis of this proble. 38

39 Exaple 7. Solution /3 The vertial defletion f,f,d,,e L For FL F,L,T. Pit ters=6-=4 h L For M,L,T Pi ters=6-3=3 d L E L FL FL 3,,E ML ML T T 39

40 Exaple 7. Solution /3 For F,L,T syste, Pi ters=6-=4 andγ γ are seleted as repeating variables a h a b3 3 d 4 b a b 3 a4 b4 E, h, 3 d, 4 E h, d, E 4

41 Exaple 7. Solution 3/3 For M,L,T syste, Pi ters=6-3=3? A loser look at the diensions of the variables listed reveal that only two referene diensions, L and MT - are required. 4

42 eterination of Referene iension /3 EXAMPLE h f,, h L MLT SYSTEM L L M T M T FLT SYSTEM h L L F L 3 F L Pi ter=4-3= Pi ter=4-= 4

43 eterination of Referene iension eterination of Referene iension 3/3 3/3 (option) (option) Set iensional Matrix Set iensional Matrix MLT SYSTEM MLT SYSTEM FLT SYSTEM FLT SYSTEM M h F h T L M 3 L F T T h h Rank= Pi ter=4 Rank= Pi ter=4-= = 43

44 Uniqueness of Pi Ters /4 The Pi ters obtained depend on the soewhat arbitrary seletion of repeating variables. For exaple, in the proble of studying the pressure drop in a pipe. p f (,,,) Seleting,, and as repeating variables: p Seleting,, and as repeating variables: p 44

45 Uniqueness of Pi Ters /4 p p Both are orret, and both would lead to the sae final equation for the pressure drop. There is not a unique set of pi ters whih arises fro a diensional analysis. The funtions Φ and Φ are will be different beause the dependent pi ters are different for the two relationships.` 45

46 Uniqueness of Pi Ters 3/4 EXAMPLE For a new pi ter,3 ' a b 3 ' ', 3, All are orret 46

47 Uniqueness of Pi Ters Uniqueness of Pi Ters 4/4 Uniqueness of Pi Ters 4/4 Uniqueness of Pi Ters 4/4 4/4 p Seleting,, and as repeating variables: p p repeating variables: p p p p p p 47

48 Coon iensionless Groups / A list of variables that oonly arise in fluid ehanial probles. Possible to provide a physial interpretation to the diensionless groups whih h an be helpful l in assessing their influene in a partiular appliation. 48

49 Reynolds Nuber / Re In honor of Osborne Reynolds (84~9), 9) the British engineer who first deonstrated that this obination of variables ould be used as a riterion to distinguish between lainar and turbulent flow. The Reynolds nuber is a easure of the ration of the inertia fores to visous fores. If the Reynolds nuber is sall (Re<<), this is an indiation that the visous fores are doinant in the proble, and it ay be possible to neglet the inertial effets; that is, the density of the fluid will no be an iportant variable. 49

50 Reynolds Nuber / Flows with very sall Reynolds nubers are oonly referred to as reeping flows. For large Reynolds nuber flow, the visous effets are sall relative to inertial effets and for these ases it ay be possible to neglet the effet of visosity and onsider the proble as one involving a nonvisous fluid. Flows with large Reynolds nuber generally are turbulent. Flows in whih the inertia fores are sall opared with the visous fores are harateristially lainar flows. 5

51 Correlation of Experiental ata iensional analysis only provides the diensionless groups desribing the phenoenon, and not the speifi relationship between the groups. To deterine this relationship, suitable experiental data ust be obtained. The degree of diffiulty depends on the nuber of pi ters. 5

52 Probles with One Pi Ter The funtional relationship for one Pi ter. C where C is a onstant. The value of the onstant ust still be deterined by experient. 5

53 Exaple 7.3 Flow with Only One Pi Ter Assue that the drag,, ating on a spherial partile that falls very slowly through a visous fluid is a funtion of the partile diaeter, d, the partile veloity,, and the fluid visosity, μ. eterine, with the aid the diensional analysis, how the drag depends on the partile veloity. 53

54 Exaple 7.3 Solution The drag f (d,, ) 3 F ML C Cd d d L LT FL For a given partile and fluids, the drag varies diretly with the veloity T 54

55 Probles with Two or More Pi Ter / Probles with two pi ters ( ) the funtional relationship aong the variables an the be deterined by varying Π and easuring the orresponding value of Π. The epirial equation relating Π and Π by using a standard urve-fitting tehnique. An epirial relationship is valid over the range of Π. angerous to extrapolate beyond valid range 55

56 Probles with Two or More Pi Ter / Probles with three pi ters. To deterine a suitable epirial equation, 3 relating the three pi ters. To show data orrelations on siple graphs. Failies urve of urves 56

57 Modeling and Siilitude To develop the proedures for designing odels so that the odel and prototype will behave in a siilar fashion. 57

58 Model vs. Prototype / Model l?a odel dli is a representation tti of a physial syste that t ay be used to predit the behavior of the syste in soe desired respet. Matheatial or oputer odels ay also onfor to this definition, iti our interest t will be in physial odel. Prototype? The physial syste for whih the predition are to be ade. Models that reseble the prototype but are generally of a different size, ay involve different fluid, and often operate under different onditions. Usually a odel is saller than the prototype. Oasionally, if the prototype is very sall, it ay be advantageous to have a odel that is larger than the prototype so that it an be ore easily studied. For exaple, large odels have been used to study the otion of red blood ells. 58

59 Model vs. Prototype / With the suessful developent of a valid odel, it is possible to predit the behavior of the prototype under a ertain set of onditions. There is an inherent danger in the use of odels in that preditions an be ade that are in error and the error not deteted until the prototype is found not to perfor as predited. It is iperative that the odel be properly designed and tested and that the results be interpreted orretly. 59

60 Siilarity of Model and Prototype What onditions i ust be et to ensure the siilarity il i of odel and prototype? Geoetri Siilarity Model and prototype have sae shape. Linear diensions on odel and prototype orrespond within onstant sale fator. Kineati Siilarity eloities at orresponding points on odel and prototype differ only by a onstant sale fator. ynai Siilarity il it Fores on odel and prototype differ only by a onstant sale fator. 6

61 Theory of Models /5 The theory of odels an be readily developed by using the priniples of diensional analysis. For given proble whih an be desribed in ters of a set of pi ters as (, 3,,, n ) This relationship an be forulated with a knowledge of the general nature of the physial phenoenon and the variables involved. This equation applies to any syste that is governed by the sae variables. 6

62 Theory of Models /5 A A siilar relationship an be written for a odel of this prototype; that is, (, 3,,,, n ) where the for of the funtion will be the sae as long as the sae phenoenon is involved in both the prototype and the odel. The prototype and the odel ust have the sae phenoenon. 6

63 Theory of Models 3/5 Model design (the odel is designed and operated) onditions, also alled siilarity requireents or odeling laws n n The for of Φ is the sae for odel and prototype, it follows that This is the desired predition equation and indiates that the easured of Π obtained with the odel will be equal to the orresponding Π for the prototype as long as the other Π paraeters are equal. 63

64 Theory of Models 4/5 Suary The prototype and the odel ust have the sae phenoenon. For prototype,,,, ) ( 3 n For odel,,,,, ) ( 3 n 64

65 Theory of Models 5/5 Suary The odel is designed and operated under the following onditions (alled design onditions, also alled siilarity requireents or odeling laws) n n The easured of Π obtained with the odel will be equal lt to the orresponding Π for the prototype. t Called p q Called predition equation 65

66 Theory of Models EXAMPLE Exaple: Considering i the drag fore on a sphere. F F f (,,, ) f The prototype and the odel ust have the sae phenoenon. F F f f esign onditions. odel prototype prototype Then F F odel prototype 66

67 Theory of Models Theory of Models EXAMPLE Theory of Models EXAMPLE Theory of Models EXAMPLE EXAMPLE E l i i h d f hi l l ( E l i i h d f hi l l ( Exaple: eterining the drag fore on a thin retangular plate ( Exaple: eterining the drag fore on a thin retangular plate (w h in size) in size) h f w w The prototype and the odel ust have the sae phenoenon. The prototype and the odel ust have the sae phenoenon.,, h, w, f, h w The prototype and the odel ust have the sae phenoenon. The prototype and the odel ust have the sae phenoenon. prototype w, h w w w, h w w esign onditions. esign onditions. prototype w w w, h w h w Then Then h h w 67 w w w

68 Exaple 7.5 Predition of Prototype Perforane fro Model ata / A long strutural oponent of a bridge has the ross setion shown in Figure E7.5. It is known that when a steady wind blows past this type of bluff body, vorties ay develop on the downwind side that are shed in a regular fashion at soe definite frequeny. Sine these vorties an reate harful periodi fores ating on the struture, it is iportant to deterine the shedding frequeny. For the speifi struture of interest, =., H=.4, and a representative wind veloity 5k/hr. Standard air an be assued. The shedding frequeny is to be deterined through the use of a sall-sale sale odel that is to be tested in a water tunnel. For the odel = and the water teperature is. 68

69 Exaple 7.5 Predition of Prototype Perforane fro Model ata / eterine the odel diension, H, and the veloity at whih the test should be perfored. If the shedding frequeny ω for the odel is found to be 49.9Hz, 9Hz what is the orresponding frequeny for the prototype? For air at standard ondition For water at, water 5.79 kg / s,.3kg / 3 kg / s, water 998kg /

70 Exaple 7.5 Solution /4 Step :List all the diensional variables involved. ω,h,,ρ,μ k=6 diensional variables. Step :Selet priary diensions F,L and T. List the diensions of all variables in ters of priary diensions. r=3 priary diensions T L H L LT FL 4 T ML T 7

71 Exaple 7.5 Solution /4 Step 3: eterine the required nuber of pi ters. k-r=6-3=3 Step 4:Selet repeating variables,, μ. Step 5~6:obining the repeating variables with eah of the other variables in turn, to for diensionless groups. 3 a b a H b a 3 b 3 3 H 7

72 Exaple 7.5 Solution 3/4 The funtional relationship is H, Strouhal nuber The prototype and the odel ust have the sae phenoenon. H, 7

73 Exaple 7.5 Solution 4/4 The esign onditions. H H Then. H H / s... 9.Hz 73

74 Model Sales The ratio of a odel variable to the orresponding prototype variable is alled the sale for that variable. Length Sale eloity Sale ensity Sale isosity Sale 74

75 istorted Models In any odel studies, to ahieve dynai siilarity requires dupliation of several diensionless groups. In soe ases, oplete dynai siilarity between odel and prototype ay not be attainable. If one or ore of the siilarity requireents are not et, for exaple, if, then it follows that the predition equation is not true; that is, MOELS for whih one or ore of the siilarity requireents are not satisfied are alled ISTORTE MOELS. 75

76 EXAMPLE- /3 /3 istorted Models EXAMPLE eterine the drag fore on a surfae ship, oplete dynai siilarity requires that both Reynolds and Froude nubers be dupliated between odel and prototype. Fr Re (g / ) Fr p Re (g p / p ) p To ath hf Froude nubers between odel and prototype p p p Froude nubers Reynolds nubers p p / 76

77 EXAMPLE- /3 istorted Models EXAMPLE To ath Reynolds nubers between odel and prototype p p p p p / p p 3 / / p p If / p equals /(a typial length sale for ship odel tests), then υ /υ p ust be /. >>> The kineati visosity ratio required to dupliate Reynolds nubers annot be attained. 77

78 EXAMPLE- 3/3 istorted Models EXAMPLE It is ipossible in pratie for this odel/prototype sale of / to satisfy both the Froude nuber and Reynolds nuber riteria; at best we will be able to satisfy only one of the. If water is the only pratial liquid for ost odel test of free-surfae flows, a full-sale test is required to obtain oplete dynai siilarity. 78

79 Typial Model Studies Flow through losed onduits. Flow around iersed bodies. Flow with a free surfae. 79

80 Flow Through Closed Conduits /5 This type of flow inludes flow through pipes, valves, fittings, and etering devies. The onduits are often irular, they ould have other shapes as well and ay ontain expansions or ontrations. Sine there are no fluid interfaes or free surfae, the doinant fores are inertial and visous fores so that the Reynolds nuber is an iportant siilarity paraeter. 8

81 Flow Through Closed Conduits /5 For low Mah nubers (Ma<.3), opressibility effets are usually negligible for both the flow of liquids or gases. For flow in losed onduits at low Mah nubers, and dependent pi ter, suh as pressure drop, an be expressed as ependent pi ter= i,, Where is soe partiular length of the syste and i represents a series of length ters, ε/ is the relative roughness of the surfae, and ρ/μ is the Reynolds nuber. 8

82 Flow Through Closed Conduits Flow Through Closed Conduits 3/5 Flow Through Closed Conduits 3/5 Flow Through Closed Conduits 3/5 3/5 To eet the requireent of geoetri siilarity To eet the requireent of geoetri siilarity i i i i To eet the requireent of Reynolds nuber To eet the requireent of Reynolds nuber If the sae fluid is used, then If the sae fluid is used, then / 8

83 Flow Through Closed Conduits 4/5 The fluid veloity in the odel will be larger than that in the prototype for any length sale less than. Sine length sales are typially uh less than unity. Reynolds nuber siilarity ay be diffiult to ahieve beause of the large odel veloities required. / 83

84 Flow Through Closed Conduits 5/5 With these siilarity requireents satisfied, it follows that the dependent pi ter will be equal in odel and prototype. For exaple, ependent pi it ter The prototype pressure drop p p pp In general p p 84

85 Flow Around Iersed Bodies /7 This type of flow inludes flow around airraft, autoobiles, golf balls, and building. For these probles, geoetri and Reynolds nuber siilarity is required. Sine there are no fluid interfaes, surfae tension is not iportant. Also, gravity will not affet the flow pattern, so the Froude nuber need not to be onsidered. For inopressible flow, the Mah nuber an be oitted. 85

86 Flow Around Iersed Bodies /7 A A general forulation for these probles is ependent pi ter= i,, Where is soe harateristi length of the syste and i represents other pertinent lengths, ε/ is the relative roughness of the surfae, and ρ/μ is the Reynolds nuber. Model of the National Bank of Coere, San Antonio, Texas, for easureent of peak, rs, and ean pressure distributions. The odel is loated in a long-test-setion, eteorologial wind tunnel. 86

87 Flow Around Iersed Bodies 3/7 Frequently, the dependent variable of interest for this type of proble is the drag,, developed on the body. The dependent pi ters would usually be expressed in the for of a drag oeffiient C To eet the requireent of geoetri siilarity i i i i 87

88 Flow Around Iersed Bodies Flow Around Iersed Bodies 4/7 Flow Around Iersed Bodies 4/7 Flow Around Iersed Bodies 4/7 4/7 To eet the requireent of Reynolds nuber siilarity To eet the requireent of Reynolds nuber siilarity The sae fluid is used, then The sae fluid is used, then / 88

89 Flow Around Iersed Bodies 5/7 The fluid veloity in the odel will be larger than that in the prototype for any length sale less than. Sine length sales are typially uh less than unity. Reynolds nuber siilarity ay be diffiult to ahieve beause of the large odel veloities required. 89

90 Flow Around Iersed Bodies 6/7 How to redue the fluid veloity in the odel? A different fluid is used in the odel suh that / For exaple, the ratio of the kineati visosity of water to that of air is approxiately /, so that if the prototype fluid were air, test ight be run on the odel using water. This would redue the required odel veloity, but it still ay be diffiult to ahieve the neessary veloity in a suitable test faility, suh as a water tunnel. 9

91 Flow Around Iersed Bodies 7/7 How to redue the fluid veloity in the odel? Sae fluid with different density.. > An alternative way to redue is to inrease the air pressure in the tunnel so that >. The pressurized tunnels are obviously opliated and expensive. 9

92 Siilitude Based on Governing ifferential Siilitude Based on Governing ifferential Equations Equations /5 /5 For a For a steady inopressible two steady inopressible two-diensional diensional flow of a flow of a Newtonian fluid with onstant visosity. Newtonian fluid with onstant visosity. The ass onservation equation is The ass onservation equation is Has diensions of /tie Has diensions of /tie v u The Navier The Navier-Stokes equations are Stokes equations are Has diensions of /tie. Has diensions of /tie. y x y u x u x p y u v x u u Has diensions of Has diensions of y v x v g y p y v v x v u y fore/volue fore/volue 9 y x y y x

93 Siilitude Based on Governing ifferential Siilitude Based on Governing ifferential Equations Equations /5 /5 iensionli e ith harateristi (standard) q antities iensionli e ith harateristi (standard) q antities iensionlize with harateristi (standard) quantities iensionlize with harateristi (standard) quantities suh that suh that the quantities of diensionless paraeters are the quantities of diensionless paraeters are O () () t p u l x (). (). How to non How to non-diensionalize these equations? diensionalize these equations? t t t p p p u u l l x x * * * * * t t t p p p v v u u y y x x * * * * * * * * * * x u x u x x u x u x u x x x x x x v u The ass onservation equation The ass onservation equation 93 x y

94 Siilitude Based on Governing ifferential Siilitude Based on Governing ifferential Equations Equations 3/5 3/5 R ld b R ld b The Navier The Navier-Stokes equations Stokes equations Reynolds nuber Reynolds nuber y u x u x p p y u v x u u t u t y v x v g y p p y v v x v u t v t y Strouhal nuber Strouhal nuber Euler nuber Euler nuber Reiproal of the sq are Reiproal of the sq are Reiproal of the square Reiproal of the square of the Froude nuber of the Froude nuber 94

95 Siilitude Based on Governing ifferential Equations 4/5 Fro these equations it follows that if two systes are governed by these equations, then the solutions (in ters of u*,v*,p*,x*,y*, and t*) will be the sae if the four paraeters are equal for the two systes. The two systes will be dynaially siilar. Of ourse, boundary and initial onditions expressed in diensionless for ust also be equal for the two systes, and this will require oplete geoetri siilarity. 95

96 Siilitude Based on Governing ifferential Equations 5/5 These are the sae siilarity requireents that would be deterined by a diensional analysis if the sae variables were onsidered. These variables appear naturally in the equations. All the oon diensionless groups that we previously developed by using diensional analysis appear in the governing equations that desribe fluid otion when these equations are expressed in ter of diensionless variables. The use of governing equations to obtain siilarity laws provides an alternative to onventional diensional analysis. 96