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)

1

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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Bernoulli’s Condition

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Setting up Normal Mode Solutions

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Applying Boundary Conditions

12

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Check: Condition for Gravitational Rayleigh-Taylor Instability

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14

Instability/Stability Condition

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16

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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