Lecture-12_(10-19-10)

Lecture-12_(10-19-10) - APPH 4200 Physics of Fluids...

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Unformatted text preview: APPH 4200 Physics of Fluids Similarity (Ch. 8) October 19, 2010 1.! ! Dimensional analysis 1 “Dynamic Similarity” (or finding the key dimensionless parameters) • Wind tunnels (powerful method in experimental fluid mechanics!) • Physical insights (what governs dynamics and the solutions to equations) • Convenient; significantly helps validation of models; broadens impact; defines general properties; … 2 Q ~ .. r " t J ~ ~ IJ . \) l II CL A ib (" J~ ~ XL~ -t ~ l~ -l~ i f~ \l A I~ Th. l:J "- + l:j C1 Cb b\ ;. ii ~ ~ c. ~ \ ~ t ~ .. T Q.\ ~ CO C' ~ t \:' l i ~ ~ ( 0 v ~ \li ~ v. Dimensional Variables ~ ~ ~ -- i ~ i ~ '.,L v le- i "' ~ .. ~ ~ ~ ~ J a ~ LL ~.-1I ~ J V t V1 t" ~ l UJ -1 ~ .~ /J ~ ~ ~ V) l t( -- LL ~ ~ II ~ .l iJ ll" ~ F u .. v .. 1'- v ~ t ~ .r CJ i V' ~ V' VI .. \, v(ß ~ .. 0 ~ ~ :: v VJ ~ -i ~ io "- \/ l Ll. t - ~D ~ ~ i~ "u I~ ~ LJ ~-. " l~ Jl Ii '~ '"- '\ 3 (0 .. )-1- " l~ , '- .. " l~ j ~ I) i :: a ,,~ t I~ l:: i- J lJ V) v I~ .. l .l " VJ :¿ III ~ - 0 v lè t ") ') ~ ~ ~ '- i 'C lQ ~ I~ i. ß '0 ~ Q. )' ~ l' ~ Dimensionless Equations 'i N '- Q. ~ -\ , :: Q. J ~ '" II i :: lb I~ -i ". "- " ,~ti i. \ .: " '\ l:: \ \- rc n "- I:J ~ "v "- ~ ~ -)~ ~ "-L \I .. \A F ~ .J t t :r u t ~~ ~ '- \l \l Q Q. ~ ~ L~ \. VI ~ ~ ~ ~ ~ v - l. ¡ ~ Q i.. ( "t ~ "- ~ lW Q. ~ '- \/ Jv \L H c' \I \L -A "- G. sl N ~ Q 1\ I ~ .: \. t~ " f' \~ ~ ~ ~ S 'l ~ 2- ~. ~~ ~ ~\~ -1æ ~ ~ "-J ~~ h + '- " Q. I ~ l/ 1:: "- " l'\ l~ '- . L CQ C) l:: ).. .. lL c¿ 1m 1 i J "- \~ ::.ri ~ çi Ç) ~ !. ~ 4 Physical Similarity (J ~ ~ I YV i l- tA l l- if r' M lltZ ~Vè r lt c2 Kifi- Dl(\~S/o_\.JLçS' 12, 6L 0 iA L '1 E /V ( frí) S/~(LI~ F L- t. ! ~fZl( A-J~,ò.--r pO;A-/ A. Sit..'0 u£/( urFiFvL! .. - ( IUD7é ~ \ - FI2 òv Q. lJu~ (j £/ -- ~ .r l- -.-- ç,o £ ¿v; () F ,4 ( - l- A-r./ \ "" -, £" c.'I F /f c i: L. /t fi Lç , - -= FLl SfEërllì ( (.A vtZ ) c. \~. V (CHA-A-C.n=J/(Ç J~l~ ~ __ ~ .l""--- í\ :: c..4-- E LE vVq nl c¡ ilç, ~ fi £T 't c. iA1Z t A- Vi -! CCHi4vrC.7rJl t " n' (.~) \ -- tz!L( ~o LJ) S 1v¡A/j vi .- lJ! ~ - P l-: U)G M I lJ £a-( Æ-L. l- V2 ca.çil) V l ç (' "ù C; Ïr.J- (A rFZ/) 5 William Froude Chelston Cross Tank at Torquay Circa 1871 William Froude (1810-1879) 6 Boat Models used by William Froude 7 Boat Length = Speed2 8 Boat Speed/ Length Ratios • As speed (in knots) exceeds 1.34 × "(length, ft), then resistance increases exponentially. • 1 knot # 0.5 m/s, so critical Froude number is Fr # 0.4 9 Osborn Reynolds • Re ≡ ρLU/μ • Turbulence when Re > 3000 • Blood flow: Re # 100 • Swimmer: Re # 4,000,000 • HMS QE II: 5,000,000,000 (1842-1910) 10 Reynolds’s Experiment Reynolds apparatus for investigating the transition to turbulence in pipe flow, with photographs of near-laminar flow (left) and turbulent flow (right) in a clearpipe much like the one used by Reynolds 11 Drag Force on Sphere 5. Nondimensional Parameters and Dynamic Similarity 271 100 D CD= (12) pu2A 10 1 0.1 10-1 l(Y¡ 101 102 103 104 l()5 106 Re= pUd J. Figure 8.2 Drag coeffcient for a sphere, The characteristic area is taken as A = rrd2/4. The reason for the sudden drop of CD at Re '" 5 x 105 is the transition of the lamnar boundar layer to a turbulent one, as explained in Chapter 10. Prediction of Flow Behavior from Dimensional Considerations An interesting observation in Figure 8.2 is that CD ex 1 IRe at small Reynolds numbers. Ths can be justified solely on dimensional grounds as follows. At small values of 12 G) II ~~ J ~ \f ~ C) ... .. ¡. 'v' i .. -l 0 J ~ ~ 'IS ~ ~~ .j t .. i. ~ l. ~ ). "- -J ~ () '- t.. VI v \: \. '; ,'1 t ~ VI .. l. IU ,- ~ ~t l: , ~ "~ \j .. ~ ii ~~ (V~ ~ -l"" iii ~ ~ -~ I"- t, 0 \. .. V' ~ ,)- 1VI ~ i- \. - ~ ~ t- ~ \J " ~ ~ "- J ~ '- 'l ¿ ? t ~ \9 .~ "- ~ ¿ q: -' ~ ~ \l ~ .. :¿ i. ~ l. 't ~ u. li ~ ~ ~ t ( :: (V IS l.\ ~ '- ~\~ ~ -l ~ c+ \J \. ,. 7 ~ I. ~ i: L: l: ~\C "" ~ ~ t " is ~ D ,. ~ \1. 'l Ò 't ).. .. ~ l. ~ .. VI ~ ~ 'u ) ': \: t ~ .. a~ ~ ~0 ).. ~ ~ '- G) J :: ~ .0 ~ ~ ~ ~ ~ --\~ ~ C' çt C) ~ \f ~ ~~ II ~\~ ~ V\ t .. ¡. 'v' ... i .. -l 0 J ~ 'IS ~ ~ J .j .. i. ~ ). -J ~ () '- t.. VI v \: \. '; ,'1 t ~ .. VI l. IU ,- ~ ~t l: , "~ \j .. ~ ~ ii ~~ (V~ ~ -l"" iii ~ ~ -~ I"- t, 0 \. V' .. ~ ,- )- - VI 1- ~ i- \. - ~ ~ \J ~ t- ~ ~ "- " J ~ ''l ¿ ? t ~ \9 .~ "- ~ q: li ~ ~ .. :¿ i. ~ l. 't ( ~ \l ~ ~ ~ -' t ~ :: (V IS l.\ ¿ u. ~ '- ~\~ ~ -l ~ c+ \J \. ,. 7 ~ I. ~ i: L: l: ~\C "" ~ ~ t " is ~ D ,. ~ \1. 'l Ò 't .. ).. ~ ~ .. l. 'u ~ VI ~ ': ) \: t ~ .. a~ ).. ~ ~0 ~ ~ '- ~ V\ t çt :: C' ~ ~ ~ --\~ ~ ~ ~ ~ .0 14 ~~ l. ~ "- Ul 0- ¿ ). l" 0 tA '- .. ) ~ V- ~ , r~ V) q: ~ r- £: t ~s iì lJi lI 'l l '" " ll 13 ~ C\ -E -1 f' II c. Cd for a flat plate J Ul 0- ~s iì lJi lI 'l l '" " ll ¿ ). l" 0 '- tA V- .. ) ~ ~ r~ V) , q: r- ~ £: ~ C\ -E -1 f' II c. (Fig. 10.12) ~\~ Drag Force on a Sphere DIMENSIONLESS NUMBERS OF FLUID MECHANICS12 Dimensionless Numbers in Fluid Dynamics Name(s) Symbol Definition Significance Alfv´n, e Karman ´´ Al, Ka VA /V *(Magnetic force/ inertial force)1/2 Bond Bd (ρ − ρ )L 2 g / Σ Gravitational force/ surface tension (Inertial force/ gravitational force)1/2 1/ 2 Boussinesq B V /(2g R) Brinkman Br µV 2 /k∆T Viscous heat/conducted heat Capillary Cp µ V /Σ Viscous force/surface tension Carnot Ca (T2 − T1 )/T2 2 Cauchy, Hooke Chandrasekhar Clausius Cy, Hk ρ V /Γ = M Ch B 2 L2 /ρ ν η Cl LV 3 ρ/k∆T Cowling C (VA /V )2 = Al2 Crispation Cr µκ/ΣL Dean D D 3/ 2 [Drag coefficient] CD (ρ − ρ )L g / ρ V2 Eckert E V /ν (2r) 1/ 2 2 V /cp ∆T 2 1/ 2 Ekman Ek (ν /2ΩL ) = (Ro/Re)1/2 Euler Eu ∆p/ρV 2 Effect of diffusion/effect of surface tension Transverse flow due to curvature/longitudinal flow Drag force/inertial force Kinetic energy/change in thermal energy (Viscous force/Coriolis force)1/2 Pressure drop due to friction/ dynamic pressure †(Inertial force/gravitational or buoyancy force)1/2 1/ 2 Froude Fr V /(g L ) V /N L Gay–Lussac Ga 1/β ∆T Grashof From NRL’s Plasma Formulary Theoretical Carnot cycle efficiency Inertial force/ compressibility force Magnetic force/dissipative forces Kinetic energy flow rate/heat conduction rate Magnetic force/inertial force 2 Gr g L3 β ∆T /ν 2 Inverse of relative change in volume during heating Buoyancy force/viscous force [Hall CH λ/rL Gyrofrequency/ coefficient] collision frequency *(†) Also defined as the inverse (square) of the quantity shown. 15 23 Dimensionless Numbers in Fluid Dynamics Name(s) Symbol Definition Significance B L/(µη )1/2 = (Magnetic force/ (Rm Re C)1/2 dissipative force)1/2 Knudsen Kn λ/L Lewis Le κ/D Lorentz Lo V /c Lundquist Lu Mach M µ0 L VA /η = Al Rm V /CS Mm V /VA = Al−1 Rm µ0 L V /η Nt F /ρ L2 V 2 Nusselt N α L /k P´clet e Pe L V /κ Poisseuille Po D 2 ∆p/µLV Pressure force/viscous force Prandtl Pr ν /κ Rayleigh Ra g H 3 β ∆T /ν κ Momentum diffusion/ heat diffusion Buoyancy force/diffusion force Reynolds Re L V /ν Inertial force/viscous force Richardson Ri (N H/∆V )2 Rossby Ro V /2ΩL sin Λ Buoyancy effects/ vertical shear effects Inertial force/Coriolis force Schmidt Sc ν /D Stanton St α/ρcp V Stefan Sf σ L T 3 /k Stokes S ν /L2 f Strouhal Sr f L /V Taylor From NRL’s Plasma Formulary H Magnetic Mach Magnetic Reynolds Newton (Page 2) Hartmann Ta Thring, Boltzmann Weber Th, Bo (2ΩL2 /ν )2 R1/2 (∆R)3/2 ·(Ω/ν ) W Hydrodynamic time/ collision time *Thermal conduction/molecular diffusion Magnitude of relativistic effects J × B force/resistive magnetic diffusion force Magnitude of compressibility effects (Inertial force/magnetic force)1/2 Flow velocity/magnetic diffusion velocity Imposed force/inertial force Total heat transfer/thermal conduction Heat convection/heat conduction Momentum diffusion/ molecular diffusion Thermal conduction loss/ heat capacity Radiated heat/conducted heat Viscous damping rate/ vibration frequency Vibration speed/flow velocity ρcp V / σ T 3 2 ρ LV /Σ Centrifugal force/viscous force (Centrifugal force/ viscous force)1/2 Convective heat transport/ radiative heat transport Inertial force/surface tension 24 16 Example 8.1 17 Viscous Drag on Hull 18 Wave Drag 15,625 19 Summary • Dimensional analysis is a useful tool in many physical problems. Key scaling parameters can be identified and used to understand behaviors as size and velocity change. • When the Reynolds number is not too large, flow is laminar. Some relatively simple problems can be solved analytically to guide o ur understanding of viscosity. 20 ...
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