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# FLUID MECHANICS I MECH 371

Rice University

GPA 3.92

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This 48 page Class Notes was uploaded by Shaina Lowe Jr. on Monday October 19, 2015. The Class Notes belongs to MECH 371 at Rice University taught by Staff in Fall. Since its upload, it has received 44 views. For similar materials see /class/224975/mech-371-rice-university in Mechanical Engineering at Rice University.

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Date Created: 10/19/15

NECH 111 Boundary Layer Theory Exact Solution chHan 39 u 13 Boundary Layer Theory Inviscid Flow Past Wedges and Corners To start the derivation of the FalknerSkan and Blasius equations we must come up for an expression of the flow external to the boundary layer on a plane surface Let s say there s a possible potential ow solution in polar coordinates Vlr9 C r n1 sinm 19 where 11 satis es the Laplace Eq ViiI 0 VI is a valid potential ow solution The radial streamlines act as walls for a wedge ramp ow expansion comer or convergentdivergent wedge JefferyHamel ow 2m ml We39llde ne as 3 Then E NECH 111 Boundary Layer Theory S m S 0 2 S 3 S 0 ow around an expansion comer ofturning angle m0 30 a atplate r 0 S m S 00 0 S 3 S 2 owagainstawedgeofhalfangle 93 747x 1 l m l 3 l the plane stagnation point 9 180 J in quotECquot m Boundary Layer Theory m 2 3 4 This can be modeled by doublet ow near a plane wall 5 a m 3 3 5 Thls can be modeled by doublet ow near a 90 comer m l 3 oo This can be modeled by ow toward a line sink JelTeryHamel ow gt 7 Fl NECH 111 Boundary Layer Theory Exact Solution of the BL Equations for Laminar Flow by Similarity Recall the governing equations for steady incompressible ow ignoring effects of gravity C Bu BU azu all BK 3y BX ayl uUX at y6X In addition 139 0 at y 6 X NECH 111 Boundary Layer Theory Let s assume the velocity pro les are selfsimilar We would like 7 E L for ow along a surface 5X 6 6 C where I lt1 J1 laminar ow so 5 C1quot X Re De ne T suchthat T E y 2 where C2 willbe determined NECH 111 Boundary Layer Theory Assume the eXtemal velocity can be described by U KXm where K is just some proportionality constant Using selfsimilarity 1 i U T a Y We can write C V u dy BX y 2 so M u i JudyJUa UJr 8 11 X 0 X chHan a in 3554 Boundary Layer Theory 2 Using similarity E d U i 8 d f and BX dX dn BX d72 2 U 8 77 df 3y BX d72 8 Y V a Ju dy within the integral X is constant so x 0 an J71 877 J dn note that function of X ay ay quot5393quotquot i J Boundary Layer Theory 2 Assuming f0 at 710 then We U em z f Beef U Note that the B C on V is satis ed V 0 at y T 0 Bu P1 ug u ax d3f fd2f 2fd2f1 dnS g1 dnl g2 dnl g3 dnl and V into M 3 chHan Boundary Layer Theory 8277 3277 2 7 where g1 8y 2 g2 axay and W UPquot 3y 3y LU g3 ax 2 We will set C2 m 1 09 3y m 1 m 0 and g1 g2 2m g3 m1 3 2 2 s0 CM d f lf d f 2 1 i 0 the FalknerSkan Equation d73 2 d72 2 dn chH 111 Boundary Layer Theory 2m JL If 3 then UX K Kl m 1 This models B L over a wedge 1 mm tr r I Ll ll E77 For a at plate without an imposed pressure gradient we r o I 3 0 so m 0 and U K d3f l dzf gt C M f 0 the Blasius Equation d73 2 d72 chH 111 Boundary Layer Theory We re interested in the Blasius Equation To solve for f and hence u we only need to guess 2 the value of d f d72 at T 0 that satis es the BCs as we integrate the differential equation to 7100 That might take a while so engineering community de nes the actual boundary layer thickness as y 5 when u 099U 6 E a 5 i 99 boundary 47 layer thickness 77 3 2 l 7 Slope at i the wall 0 wwwm mlm 02 04 06 08 l f39uU NECH 111 u df We see that 099 corresponds to T E 5 U 77 TABLE 10 3 5010001101019 Blasius lammar quotat Hale boundary ayer 1n sImHaMy variables 17 rquot f39 r n r39 r39 f 00 033205 0 00000 100000 24 022809 072898 092229 OJ 133205 103321 100165 26 020645 077245 107250 02 033198 0 06641 000664 28 018401 081151 123098 03 033181 0 09960 001494 30 016136 084604 139681 04 033147 013276 002656 35 010777 091304 183770 05 033091 0 16589 004149 40 006423 035552 230574 06 033008 0 19894 005973 45 003398 037951 279013 03 032739 025471 010611 50 001591 099154 328327 10 032301 0 32978 016557 55 000658 099688 378057 12 031559 0 39378 023795 60 000240 099897 427962 14 030787 045626 032298 65 000077 039970 477932 16 029666 0 51676 042032 70 000022 039992 527923 18 028293 0 57476 052952 80 000001 100000 627921 20 026575 162977 055002 90 000000 100000 727921 22 024835 055131 018119 100 000000 100000 827921 k m lunchon 1 H lechmque N016 that rquot 15 pmpnmonal 10 me hea 739 15 pvopmtlunal Mammy in the boundavy Iayev r39 mu and I use Is 070007007131 02 the shear runcuon 739 s waited 1 055 r as a mnctlun 01 y m FIE 1099 In Eq 4 an to me 107000001110 NECH 111 Boundary Layer Theory If y 5 When 099 numerical integration of the Blasius equation gives Recall NECH 111 Boundary Layer Theory 2139 U dzf Cf WZ where 1w u uU 2 pU Byy0 0X 17 quot0 C 2 d2f 20332 0644 f ReX C1772 quot0 JReX ReX 2139 d9 d9 1 0644 7 c v x pU dX X 2 J v 0644 7 X 0644 9 X X J 0 ReX 1 chHan J Bounda La er Theo EXW ry y ry 15xL 1328 p UZbL VReL 6 H 6 X 259 9 0664 ECquot 371 N Questions ECquot 371 N Notes See you next time 10 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

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