APPH 4200 Physics of Fluids - Columbia...
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APPH 4200 Physics of Fluids Fluid Instabilities (Ch. 12) 1. Kelvin Helmholtz Instability 2. Viscous Effects on Parallel Flows 3. (Centrifugal Instability) 1
Transcript of APPH 4200 Physics of Fluids - Columbia...
APPH 4200 Physics of Fluids
Fluid Instabilities (Ch. 12)
1.!! Kelvin Helmholtz Instability2.! Viscous Effects on Parallel Flows3.! (Centrifugal Instability)
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Kelvin-Helmholtz Instability
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More Beauty
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More Beauty
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Also plasma!
Nature, vol. 430, pp 755-758 (2004)
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35
Sir G. I. Taylor’s Solution (1923)“The closeness of the agreement between his theoretical and experimental results was without precedent in the
history of fluid mechanics.” p. 476
Instability
3), neglecting
ain the pertur-
(12.37)
idmit solutionsnode solutions
nplies that the
es into (12.37)iûe. Under theLl (Ri +R2)/2,
)
(12.38)
(12.39)
'1 d/v)2(d/ R1)
(12.40)
il of () to zero.
pIe of exchange
5. Centrifugal Instability: Taylor Problem 475
of stabilties must be valid for this problem, and the marginal states are given by() = O. Ths was later proven to be tre for cylinders rotating in the same directions,but a general demonstration for all conditions is stil lacking.
Discussion of Taylor's Solution
A solution of the eigenvalue problem (12.38), subject to equation (12.40), wasobtained by Taylor. Figure 12.11 shows the results of his ca1culationsand his ownexperimental verification of the analysis. The vertical axis represents the angularveloGIty of the inner cylinder (taken positive), and the horizontal axis represents the
angular velocity of the outer cylinder. Cylinders rotating in opposite directions arerepresented by a negative Q2. Taylor's solution of the marginal state is indicated, withthe region above the curve corresponding to instability. Rayleigh's inviscid criterion isalso indicated by the straight dashed line. It is apparent that the presence of viscositycan stabilize a flow. Taylor's viscous solution indicates that the flow remains stableuntil a critical Taylor number of
1708Ta -er - (1/2) (1 + Qi/Q¡) , (12.41)
is attained. The nondimensional axial wavenumber at the onset of instabilty is foundto be ker = 3.12, which implies that the wavelength at onset is Åer = 2nd/ kef ~ 2d.The height of one cell is therefore nearly equal to d, so that the cross-section of a cellis nearly a square. In the limit Qi/ Qi -+ 1, the critical Taylor number is identical
to the critical Rayleigh number for thermal convection discussed in the precedingsection, for which the solution was given by Jeffreys five years later. The agreement
40 III/iiIIIIIiI
1""IIIIII/
D.iRi
v
D.¡ _Ri
D.i - Rr
40 D.iRiv
o 300
Figure 12.11 Taylor's observation and narow-gap calculation of marginal stabilty in rotating Couette
flow of water. The ratio of radii is Ri/ R¡ = 1.14. The region above the cure is unstable. The dashed linerepresents Rayleigh's inviscid criterion, with the region to the left of the line representing instabilty.
36
Centrifugal Convection
37
Taylor Vortex to Turbulent Vortex
From: http://wn.com/Couette_flow
38
Summary• Velocity shear is common place in natural
systems with stratified layers.
• When the velocity shear exceeds a threshold set by mass stratification, the Kelvin-Helmholtz Instability develops.
• Instability threshold can be calculated with amazing accuracy (like the centrifugal instability).
• Mixing at the interface, reduces the free energy available and drives unstable growth.
39
HW 5, Problem 1• A 1/25 scale model of a submarine is being tested in a wind tunnel
in which p = 200 kPa and T = 300 K. If the prototype speed is 30 km/hr, what should be the free-stream velocity in the wind tunnel? What is the drag ratio? Assume that the submarine would not operate near the free surface of the ocean.
APPH4200 Physics of Fluids: Homework 5
1. K&C, Chapter 8, problem 2. The correct answer for the velocity in the wind tunnel
may seem unrealistic, and strictly, would require us to consider an additional nondi-
mensional parameter not discussed in class, but that is mentioned in the chapter.
Extra credit for saying what this parameter is.
2. Consider the thermal energy equation for an incompressible flow satisfying Fourier’s
law, eq. 4.68 of K&C. Non-dimensionalize this equation, assuming a single length
scale and velocity scale. You should obtain one non-dimensional parameter. Inter-
pret this parameter physically, i.e., what does it mean if it is small or large?
3. Review the deep water equations derived in class. Then, describe the gravity surface
waves with viscosity. Assume wavelike solutions, e.g.
⌘ = Re{⌘0 ei(kx�!t)},
where, ⌘0 is a complex amplitude, and the frequency ! must be assumed complex.
Do the disturbances grow or decay in time? Is this physically reasonable? Verify
that the phase speed reduces to the expected value in the limit ⌫ ! 0.
1
40
HW 5, Problem 2APPH4200 Physics of Fluids: Homework 5
1. K&C, Chapter 8, problem 2. The correct answer for the velocity in the wind tunnel
may seem unrealistic, and strictly, would require us to consider an additional nondi-
mensional parameter not discussed in class, but that is mentioned in the chapter.
Extra credit for saying what this parameter is.
2. Consider the thermal energy equation for an incompressible flow satisfying Fourier’s
law, eq. 4.68 of K&C. Non-dimensionalize this equation, assuming a single length
scale and velocity scale. You should obtain one non-dimensional parameter. Inter-
pret this parameter physically, i.e., what does it mean if it is small or large?
3. Review the deep water equations derived in class. Then, describe the gravity surface
waves with viscosity. Assume wavelike solutions, e.g.
⌘ = Re{⌘0 ei(kx�!t)},
where, ⌘0 is a complex amplitude, and the frequency ! must be assumed complex.
Do the disturbances grow or decay in time? Is this physically reasonable? Verify
that the phase speed reduces to the expected value in the limit ⌫ ! 0.
1
41
HW 5, Problem 3
APPH4200 Physics of Fluids: Homework 5
1. K&C, Chapter 8, problem 2. The correct answer for the velocity in the wind tunnel
may seem unrealistic, and strictly, would require us to consider an additional nondi-
mensional parameter not discussed in class, but that is mentioned in the chapter.
Extra credit for saying what this parameter is.
2. Consider the thermal energy equation for an incompressible flow satisfying Fourier’s
law, eq. 4.68 of K&C. Non-dimensionalize this equation, assuming a single length
scale and velocity scale. You should obtain one non-dimensional parameter. Inter-
pret this parameter physically, i.e., what does it mean if it is small or large?
3. Review the deep water equations derived in class. Then, describe the gravity surface
waves with viscosity. Assume wavelike solutions, e.g.
⌘ = Re{⌘0 ei(kx�!t)},
where, ⌘0 is a complex amplitude, and the frequency ! must be assumed complex.
Do the disturbances grow or decay in time? Is this physically reasonable? Verify
that the phase speed reduces to the expected value in the limit ⌫ ! 0.
1
42
