APPH 4200 Physics of Fluids

Similarity (Ch. 8)October 15, 2009

1.!! Numerical solution to 2D potential flow2.! Dimensional analysis

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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

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Boat Models used by William Froude

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Boat Length = Speed2

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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

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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

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Drag Force on Sphere5. Nondimensional Parameters and Dynamic Similarity 271

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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,

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Drag Force on a SphereG)

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Cd for a flat plate(Fig. 10.12)

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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.

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