APPH 4200 Physics of Fluids

Similarity (Ch. 8)

1. Dimensional analysis

“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

William Froude

William Froude (1810-1879)Chelston Cross Tank at Torquay Circa 1871

Boat Models used by William Froude

Boat Length = Speed2

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

1.34 V/√L (knot/√ft)

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

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

Drag Force on Sphere5. Nondimensional Parameters and Dynamic Similarity 271

100

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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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Cd for a flat plate (Fig. 10.12)

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 (ρ′ − ρ)L2g/Σ 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/Σ Viscous force/surface tension

Carnot Ca (T2 − T1)/T2 Theoretical Carnot cycleefficiency

Cauchy, Cy, Hk ρV 2/Γ = M2 Inertial force/Hooke compressibility force

Chandra- Ch B2L2/ρνη Magnetic force/dissipativesekhar forces

Clausius Cl LV 3ρ/k∆T Kinetic energy flow rate/heatconduction rate

Cowling C (VA/V )2 = Al2 Magnetic force/inertial force

Crispation Cr µκ/ΣL Effect of diffusion/effect ofsurface tension

Dean D D3/2V/ν(2r)1/2 Transverse flow due tocurvature/longitudinal flow

[Drag CD (ρ′ − ρ)Lg/ Drag force/inertial forcecoefficient] ρ′V 2

Eckert E V 2/cp∆T Kinetic energy/change inthermal energy

Ekman Ek (ν/2ΩL2)1/2 = (Viscous force/Coriolis force)1/2

(Ro/Re)1/2

Euler Eu ∆p/ρV 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/β∆T Inverse of relative change involume during heating

Grashof Gr gL3β∆T/ν2 Buoyancy force/viscous force

[Hall CH λ/rL Gyrofrequency/coefficient] collision frequency

*(†) Also defined as the inverse (square) of the quantity shown.

23

From NRL’s Plasma Formulary

Dimensionless Numbers in Fluid

Dynamics (Page 2)

From NRL’s Plasma Formulary

Name(s) Symbol Definition Significance

Hartmann H BL/(µη)1/2 = (Magnetic force/(Rm ReC)1/2 dissipative force)1/2

Knudsen Kn λ/L Hydrodynamic time/collision time

Lewis Le κ/D *Thermal conduction/moleculardiffusion

Lorentz Lo V/c Magnitude of relativistic effects

Lundquist Lu µ0LVA/η = J × B force/resistive magneticAl Rm diffusion force

Mach M V/CS Magnitude of compressibilityeffects

Magnetic Mm V/VA = Al−1 (Inertial force/magnetic force)1/2

MachMagnetic Rm µ0LV/η Flow velocity/magnetic diffusionReynolds velocity

Newton Nt F/ρL2V 2 Imposed force/inertial force

Nusselt N αL/k Total heat transfer/thermalconduction

Peclet Pe LV/κ Heat convection/heat conduction

Poisseuille Po D2∆p/µLV Pressure force/viscous force

Prandtl Pr ν/κ Momentum diffusion/heat diffusion

Rayleigh Ra gH3β∆T/νκ Buoyancy force/diffusion force

Reynolds Re LV/ν Inertial force/viscous force

Richardson Ri (NH/∆V )2 Buoyancy effects/vertical shear effects

Rossby Ro V/2ΩL sinΛ Inertial force/Coriolis force

Schmidt Sc ν/D Momentum diffusion/molecular diffusion

Stanton St α/ρcpV Thermal conduction loss/heat capacity

Stefan Sf σLT 3/k Radiated heat/conducted heat

Stokes S ν/L2f Viscous damping rate/vibration frequency

Strouhal Sr fL/V Vibration speed/flow velocity

Taylor Ta (2ΩL2/ν)2 Centrifugal force/viscous forceR1/2(∆R)3/2 (Centrifugal force/

·(Ω/ν) viscous force)1/2

Thring, Th, Bo ρcpV/ϵσT 3 Convective heat transport/Boltzmann radiative heat transport

Weber W ρLV 2/Σ Inertial force/surface tension

24

Example 8.1

Viscous Drag on Hull

Wave Drag

15,625

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.