BL_FlatPlate_Theory.pdf - 2 Boundary Lager on a Flat Plate...

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Unformatted text preview: 2) Boundary Lager on a Flat Plate The simplest example of the application of the boundary layer equations is that of flow along a thin flat plate. From the historical point of View this was the first example illustrating the application of Prandtl's boundary layer equations. In the sketch above, the presence of the plate hardly disturbs the inviscid flow at all. Even though there isra boundary layer, the thickness is sufficiently small (providing Re is high enough) and to a good first approximation; the external flow is unperturbed. Bernoulli's equation tells us that in the external flow, \ 1 _ P+ ecu—if —- Combat 2 “1.0:? : V911)— : o (3 (ix doc. so that fOr the flat plate leflix (the streamwise pressure gradient) is zero. We thus have to solve U§E+V§szV§§ 53¢ a“\ 6:5 Ion/91 Note that we have assumed we are dealing with a steady flow, so that bu/bt = 0, subject to boundary conditions u v = O at y = O u = U at y 2 CD Let's introduce the stream function 9/, which, since we may write, LAz; Eli? Ckynfi \f :: -}5 9” ,.....——r- 5.3 ' ox satisfies the continuity equation‘ Le. 322+?! : 5 Whats. ii” M4 2);: EMS bar. 515/} Db E31 1: 0 With the stream function_q’, we have satisfied continuity and are left with one unknown and one equation. We will now attempt to do a similar thing to that which we did in the Rayleigh problem ~——-convert a p.d.e. to an o.d.e., through a transformation' of variables. In other words, we are looking for a similarity solution; Let's look in a little more detail at this phrase ”similarity solution.” Consider the boundary layer along a flat plate in air at constant velocity Ul' Due to viscous diffusion and streamwise conyection of vorticity, We expect that the boundary layer will increase in thickness with increasing distance downstream. The velocity profiles at different X will be different. 39).};3 ,“23‘32250‘21 If these profiles can be collapsed onto a single curve through a suitable adjustment of the y ordinate, then the profiles at various values of x are said to be SIMILAR and independent of x. An obvious choice for scaling y would be S (x). Now, if we increased the velocity to U2 and fixed attention at a given x the velocity profiles will be different, with E) depending on the velo- city. -» Same :6 - Same Huu‘oi —thwstTT DC Now, if the local velocity in the boundary layer is u, then scaling u by U, and y by g (x,U), may again collapse the profiles so that they are similar. If we now changed fluids (say air to water) {2 would be different and at a fixed velocity and location u and 8 would again be different. Again, if u/U is a function of y/d (X,U,V) alone and is independent of x, U and Y , then the profiles are similar. Huai Same be Some U chfiwstiimdi imaz CDC- Referring back to the Rayleigh problem, we found that the shear layer thickness was approximately LI ‘1th . E these results are roughly appli— cable to the boundary layer along a flat plate, We may say that t E x/U and thus S = 1+ .19 x/U. Now; in the similarity relationship outlined above, we have :1: ‘3 U wag—.— 1)‘ g(xav3 v) If we included an arbitrary constant c it will make no difference to the similarity argument. . 3— : Q lil— 1.e. ’0“ C. {)(UCJU‘) 13) For convenience (you will see why later) we will choose ng, = DC M» »> [2; Now, introducing a dimensionless coordinate Q : b / 8 we have {Z '1 E5 hlffi 2N 36— Thus the similarity means that the velocity profile 5‘— “ HQ) again for convenience (again you will soonsee why) we take u/U to be the first derivative of f . i.e. I where ¥ 7- rig/c711 We will now substitute U: U‘F ('1) Omci 'Q: 3 £3: in the boundary layer equation and see if We can obtain an o.d.e. whiCh does not explicitly contain x, U, or \3 Note: (i) @311 =1 ”.1. 3 l: 7- #:2— by. 2x ZVJC 2:“: Taking the terms one by one, we have a) 3‘3: 2333 : Vtuég : 5015,32 b7"; bi 53C 03‘;— ’21 mag: 2245.2: U951 fl 3: U15" b3) 5? 62> 5:3) 1V3: ‘5 Q i Now 9/: Udks "5 v #dqolgfir o 0 d7, : 2 3c 1 ‘U % (l + Wot >> If we choose that V] 2 f = 0 at y =I’L : O (at the wall) then WOW) I 0 and then we have the stream function tr: W4 (Now you see why we needed u/U 2 f'.) NOW V: HM : fl}?— CZY'U-x ' '9) 1! __.-«( 72. 2%:6-4- + \lQY—U-J; EAL—BYE We now have U: V; ‘u! 3 EU! CLmoK bl“ ___.-_ has SE5 bu; Substituting all these into the boundary layer equation We have (on simpli— fiCation —n-try this yourself) This equation was first discussed by H. Blasius in 1908 in his Ph.D thesis and it is commonly referred to as the BLASIUS equation. Note that (as we'd hoped) it is an o.d.e. and does not contain x, U, or 1) explicitly. The solution concerns only f and 1’ and the relationship between f‘($u/U) and V1; will be universal and the velocity profiles are similar. Blasius obtained a solution in the form of a series expansion around VL: 0 and an asymptotic expansion for large fiL , with matching occurring at a suitable value of V1,. This is a very tedious method! Others have tried better methods since then. Nowadays, solutions of such equations can be obtained numerically on computers in a matter of seconds. The tabulated solution is shown on the next page. Ill u @412ka *F ; O 0-00.30 000. . o-nooo no 1 .. 034-666 60 '0on 45° '0469 59 ' ” was 63 ‘0093 9143 - ' '0939 05 '4693 06 1:2” 275 H408 06 16M) 09 '0375 492- 1376 as ' 1672 54 125% 427 '2342 27 - 1650 30 '0843 856 '2R05 75 '46:? 34 ””47 447 '3265 32 '45?! 77 '1496 745 , 37:9 63_ '45” 90 'I89: I48 '- ‘4167 13 1436 23 0-2329 900 . 01606 32 ‘ 0-4343 79 '28:: 075 '5035 as '4233 68 '3336 572 ‘54s: 46 ' '4Ios 65. '3902 In ‘ '5355 88 ' '3959 3+ '4507 234 - ' '6243 86 - '3796 92 ‘slso 3:2 , 66:4 73 ~36|8 04 '53:“) 56° - '6966 99 ‘ 33434 37- '5543 045 '7299 30 '3219 so 7288 7-8 * 7610.57 - V '3004 .45 '8064 429 7 "1399 96 ‘2732 51 0-8867 962 0-8166 94 . 0-2556369; 1-05.49 463 £633 03 ’za'bgso l~33|5 :67 -9mo 65 , 1675 6!. {-4148 13: ' ~9306 ox 1286 :3 16032 823 «9528 75 1595: :4 :'7955 666 ‘ . '9690 54 «>677 n r9905 796 - '9803 65 ' '0463 70 “874 6.58 '9379 70 mos 35 2-3855 886 '9928 38 ' mo: 29 2'5344 972 '9559 44 '0”? so 2-7838 848 0-997; 70 0-0068 7.;- z-gflgs 535- ~9988 :8 13038 61 1-1833 808‘ ' ~9993 96 ‘0020 84 233832 94: ‘ '9997 on mm B: 3‘5832 52° '9993 59 was 39 37332 324- 1399936 '0002 58 3-9832 135. '9999 7! ‘ we! :9 4:183: :97 . @999 88 - 'oooo 52. $3831 :81. A '9999 95 '0000 2: $5832 I73 - ‘9999 93 * 'oooo o9 4:283; 170. ‘ 0'9999 99 0,0909 9_3 00 ME” (‘5 SCI-{Ddtzi-Zl67-8 (a) 90-95% % o~ Lrecrso The boundary conditions needed for solving this equation are The velocity distributions obtained from the Blasius solution are shown below. Note that at the outer edge of the boundary layer (i.e.azrtrd3) the v component differs from zero. This means that at the outer edge, there is a flow outward due to the fact that the increasing boundary layer thickness causes the fluid to be displaced away from the wall as it flows along it. i) Shear stress on the Elate At the wall ‘{ :— flCbgfl ~53!» ‘22 v no :2: /u FUMX From the tabular solutionj f"(0) : 0.H695 C ?. g3) ”C =~ 6% 'UV 1: 0.337. rt“ m L? W 313.1: flvj’gc .The shear stress at the plate is called the SKIN FRICTION. The skin friction coefficient C+ ~— f‘w/i vi : «Mfg .. Us: where Rex : Reynolds number based on the "wetted length” x. Q: Calculate the total friction resistance for one side of a flat plate of length (, and width b . (ii) Boundary Layer ThiCKnes's 8 We see from the velocity profile that the influence of viscosity decreases asymptotically outwards (i.e., u tends asymptotically to Ujfflm.vebcfii>o£ the potential flow). Generally, the thickness 23 of a boundary layer is defined as that distance for which u/U : 0.99. ow -; :9: h J» = j; N 132W: aa ngmx 25mm w e We : The tabular solution shows that f' = 0.99 at Q: = 3.5 (iii) Boundary Layer Displacement Thickness We can see from the above that S is rather an arbitrary quantity since we could have picked the edge at u/U = 0.98, or 0.995 etc. A more physically meaningful measure_of the thickness is the DISPLACEMENT thickness (5*. Inside of the boundary layer, the fluid is slowed up and because of this, the outer potential flow is pushed away from the plate. The displacement outward, denoted by g* (see sketch below) GE ‘3 We dense-g5 at “me baudcanb EMS-24' gilcbfl‘ where Yl is outside of the boundary layer. We can rewrite this as 553, 23¢ _ (“M3 ”* 65—0—0135 *" fevgi‘: rI‘he upper limit of the integral yl, can be taken as g or 00 since the integrand tends to zero towards the boundary layer edge. Now 1%: ~;, Skit?) II In )o -<.’. B 01" Note that 3* grows at about 1/3 the rate of S . Q: Show that the inclination of the streamlines in the potential flow 3ust outSide the boundary layer 18 dB/dx. Note also that dgK/dx is less than dS/dx so that streamlines near the boundary layer enter it and then the fluid particles lose momentum through the action of viscosity. (iv) Momentum—Deficit Thickness Just as $31K provides an indication of the mass flow reduction in the boundary layer, 9 , the MOMENTUM DEFICIT THICKNESS, defines the momentum loss (as compared to the potential flow). boundanb huge: edge (verb-cg; Stale hx‘g‘nkfi Quoflmfid E) In the sketch on the previous page, the mass flow per unit time at section 2 is eudy, so that the momentum flow per unit time is c.) u2dy. Before entering the boundary layer the momentum flow was 6 U 2 dyl. Now continuity says that Pu dob «1 EEVG‘EBL hm”, (31)}de :- Eevadtti The momentum deficit is E u (VF H) (11% OD The total def1c1t 15 f) tLVA(U"U~> {*6 0 If we define 9 by the expression 00 Gay 8 «'-‘- GU (—01,“de weget e: e“ ‘_ i or if E = constant (incompressible) Form factor . _ 3* . . We can define a form factor H h / 9 which 18 useful as a measure of the shape of the velocity profile. For the Blasius 'flow considered here, H 1' 2.5. (Q. cm H 5% LESS {’Lm. :L_ Experimental Verification of Blasius's results W a) A comparison of Liepmann's velocity profile measurements with the IBQlasiqsmgelgcity profile is shown‘below. } 1.0 J E_ 0.8 f; 0.5 . . Symbol Ba, x 10*6 ‘ 0.85 ' 0.86 0.4 0.93 0.32 ,. , 0.93 0.2 — 1,03- 1.11 O 1.24 0 1 2 3 4 5 5 7 5 Jam, f1 PX b) A comparison (again Liepmann‘s measurement) of measured and predicted skin friction coefficients (at the wall) is shown below. 0.0305 ' * 0 Merry my mum measwsmmt 1119003 — firm velar/Y; profile 0030.? 0/?ch skirt mic-rim 11123.91”an X -28.é‘£m o I . . I 1 ,stfi‘m H mm L L JILH - J l b“ 870 6 19 This good agreement of theory and experiment for the velocity distri- bution and shear stress in a laminar boundary layer on a flat plate at zero angle of attack clearly demonstrates the validity of the boundary layer approximations from the physical point of View. Q: Will the boundary layer equations hold near the leading edge of the plate? ...
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