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fluidsCh11 - Chapter 11 Linearization This chapter combines...

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Chapter 11 Linearization This chapter combines wave motion and linear stability theory. Within the time-frame of the course, only a few simple cases can be treated, with the purpose of illustrating mathematical techniques and important physics. Be- cause of the ubiquitous importance of shear-layer instability, and the need for engineers to anticipate its presence and consequences, the Kelvin-Helmholtz problem is presented here. Also, the comparison of surface waves and inter- nal waves appears justified as an introduction to a very broad subfield with a common underlying technique: linearization. 11.1 Surface waves The interface between standing water and the surrouding air is familiar enough, as is the propagation of disturbances on the surface (Fig. 11.1) . This is a canonical situation, involving a reference flow (here, trivally static); a secondary flow of small amplitude that allows the linearization of the equations; and the (imaginary) exponential solution to these equations as propagating waves. A variant on this idea is found in linear stability theory, of which an example appears in the next section. The student is invited to compare the two cases carefully. Since the primary flow is static, it is certainly irrotational, and the veloc- ity potential φ is a constant (which obviously satisfies the Laplace equation). Let us take the body of water to be of finite depth H (a simpler solution is obtained for H → ∞ ). The free surface is described by y = η ( x, t ) , (11.1) 213
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214 CHAPTER 11. LINEARIZATION Figure 11.1: Definition sketch for surface waves with η = 0 for the primary ‘flow’. The disturbance associated with the motion of the interface is assumed to be irrotational also. The interface, as primary descriptor of the flow, is characterized by two features: 1. it is a material surface, for which the material derivative of the property ( y - η ) must be zero: t ( y - η ) + u∂ x ( y - η ) + v∂ y ( y - η ) = - t η - u∂ x η + v = 0; (11.2) 2. it is at ambient pressure p = 0, so that Bernoulli’s equation at the interface is t φ + 1 2 ( u 2 + v 2 ) + = 0 . (11.3) For small amplitude disturbances, we neglect quadratic terms in the per- turbations, and the interface equations become, respectively v = t η = y φ, (11.4) where we make use of the potential flow assumption , and t φ + = 0 . (11.5)
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11.1. SURFACE WAVES 215 In the bulk of the body of water, the Laplace equation for the velocity potential is already linear: 2 xx φ + 2 yy φ = 0 . (11.6) Note that the equation for the field is more easily derived than the boundary condition at the the interface. Exponential solutions are common solutions of linear equations (think about the solution of linear pde’s using Fourier or Laplace transforms). Here, we can look for propagating waves on the surface η = Ae i ( kx - ωt ) , (11.7) where A is the amplitude, and a real part can be extracted when needed. k is the wavenumber corresponding to a wavelength λ = 2 π/k , and ω/ 2 π is the frequency. One interface condition becomes v = - ωAη, (11.8) while the other suggests that φ should be exponential also: φ = f ( y ) e i ( kx - ωt ) . (11.9)
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