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APPH 4200 Physics of Fluids
Similarity (Ch. 8)October 15, 2009
1.!! Numerical solution to 2D potential flow2.! Dimensional analysis
1
Numerical Solution to Potential Flow in 2D
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Mathematica is available at all Columbia computer labs. See http://www.columbia.edu/acis/facilities/software.html
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“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; …
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Physical Similarity
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William Froude
William Froude(1810-1879)Chelston Cross Tank at Torquay Circa 1871
9
Boat Models used by William Froude
10
Boat Length = Speed2
11
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
12
Osborn Reynolds
(1842-1910)
• Re ≡ ρLU/μ
• Turbulence when Re > 3000
• Blood flow: Re # 100
• Swimmer: Re # 4,000,000
• HMS QE II: 5,000,000,000
13
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
14
Drag Force on Sphere5. Nondimensional Parameters and Dynamic Similarity 271
100
DCD= (12) pu2A
10
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10-1 101 104 106103 l()5l(Y¡ 102
Re= pUdJ.
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.
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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
Reynolds numbers we expect that the inertia forces in the equations of motion must
become negligible. Then p drops out of equation (8.15), requirng
D = fed, U, /L).
The only dimensionless product that can be formed from the preceding is D I /LU d.
Because there is no other nondimensional parameter on which D I f. U d can depend,
15
Drag Force on a SphereG)
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Cd for a flat plate(Fig. 10.12)
17
Dimensionless Numbers in Fluid
Dynamics
DIMENSIONLESS NUMBERS OF FLUID MECHANICS12
Name(s) Symbol Definition Significance
Alfven, Al, Ka VA/V *(Magnetic force/Karman inertial force)1/2
Bond Bd (!! ! !)L2g/! Gravitational force/surface tension
Boussinesq B V/(2gR)1/2 (Inertial force/gravitational force)1/2
Brinkman Br µV 2/k"T Viscous heat/conducted heat
Capillary Cp µV/! Viscous force/surface tension
Carnot Ca (T2 ! T1)/T2 Theoretical Carnot cyclee#ciency
Cauchy, Cy, Hk !V 2/$ = M2 Inertial force/Hooke compressibility force
Chandra- Ch B2L2/!"# Magnetic force/dissipativesekhar forces
Clausius Cl LV 3!/k"T Kinetic energy flow rate/heatconduction rate
Cowling C (VA/V )2 = Al2 Magnetic force/inertial force
Crispation Cr µ$/!L E%ect of di%usion/e%ect ofsurface tension
Dean D D3/2V/"(2r)1/2 Transverse flow due tocurvature/longitudinal flow
[Drag CD (!! ! !)Lg/ Drag force/inertial forcecoe#cient] !!V 2
Eckert E V 2/cp"T Kinetic energy/change inthermal energy
Ekman Ek ("/2&L2)1/2 = (Viscous force/Coriolis force)1/2
(Ro/Re)1/2
Euler Eu "p/!V 2 Pressure drop due to friction/dynamic pressure
Froude Fr V/(gL)1/2 †(Inertial force/gravitational orV/NL buoyancy force)1/2
Gay–Lussac Ga 1/%"T Inverse of relative change involume during heating
Grashof Gr gL3%"T/"2 Buoyancy force/viscous force
[Hall CH &/rL Gyrofrequency/coe#cient] collision frequency
*(†) Also defined as the inverse (square) of the quantity shown.
23
From NRL’s Plasma Formulary
18
Dimensionless Numbers in Fluid
Dynamics(Page 2)
From NRL’s Plasma Formulary
Name(s) Symbol Definition Significance
Hartmann H BL/(µ!)1/2 = (Magnetic force/(Rm ReC)1/2 dissipative force)1/2
Knudsen Kn "/L Hydrodynamic time/collision time
Lewis Le #/D *Thermal conduction/moleculardi!usion
Lorentz Lo V/c Magnitude of relativistic e!ects
Lundquist Lu µ0LVA/! = J ! B force/resistive magneticAl Rm di!usion force
Mach M V/CS Magnitude of compressibilitye!ects
Magnetic Mm V/VA = Al!1 (Inertial force/magnetic force)1/2
MachMagnetic Rm µ0LV/! Flow velocity/magnetic di!usionReynolds velocity
Newton Nt F/$L2V 2 Imposed force/inertial force
Nusselt N %L/k Total heat transfer/thermalconduction
Peclet Pe LV/# Heat convection/heat conduction
Poisseuille Po D2"p/µLV Pressure force/viscous force
Prandtl Pr &/# Momentum di!usion/heat di!usion
Rayleigh Ra gH3'"T/&# Buoyancy force/di!usion force
Reynolds Re LV/& Inertial force/viscous force
Richardson Ri (NH/"V )2 Buoyancy e!ects/vertical shear e!ects
Rossby Ro V/2#L sin $ Inertial force/Coriolis force
Schmidt Sc &/D Momentum di!usion/molecular di!usion
Stanton St %/$cpV Thermal conduction loss/heat capacity
Stefan Sf (LT 3/k Radiated heat/conducted heat
Stokes S &/L2f Viscous damping rate/vibration frequency
Strouhal Sr fL/V Vibration speed/flow velocity
Taylor Ta (2#L2/&)2 Centrifugal force/viscous forceR1/2("R)3/2 (Centrifugal force/
·(#/&) viscous force)1/2
Thring, Th, Bo $cpV/)(T 3 Convective heat transport/Boltzmann radiative heat transport
Weber W $LV 2/% Inertial force/surface tension
2419
Example 8.1
20
Viscous Drag on Hull
21
Wave Drag
15,625
22
Laminar Flow (Ch. 9)
• Two simple examples:
• Couette Flow (moving top plate)
• Poiseuille Flow (pressure-driven)
• With viscosity
23
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 our understanding of viscosity.
24
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