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Unformatted text preview: ME 563  Intermediate Fluid Dynamics  Su Lecture 31  Instability: more on KelvinHelmholtz, and thermal convection Reading: Acheson, 9.29.3. To recap what we did with KelvinHelmholtz instability in the last lecture We defined our undisturbed, twodimensional flow system: a deep layer of inviscid fluid, density 2 , for y > 0, and a deep layer of inviscid fluid of density 1 for y < 0, with 1 > 2 . The interface is permitted to have surface tension. Initially, the upper fluid moves at uniform speed U in the xdirection and the lower fluid is at rest (Fig. 1). 2 1 U y= Figure 1: Undisturbed flow for the KelvinHelmholtz problem. We allowed the interface to be subject to sinusoidal, travelingwave disturbances with in finitesimal magnitude. We then wanted to see how the initially, spatiallydefined disturbances evolved with time; specifically, do they increase or decrease in magnitude with time? We call a flow system stable to a particular disturbance if that disturbance dies away in magnitude after its introduced. For this temporal stability analysis, we considered disturbances with real wavenumbers k but complex angular frequencies . We showed that disturbances grow exponentially with time, and thus destroy the undisturbed flow pattern (Fig. 1), when the imaginary part of the complex angular frequency is greater then zero. What we wanted, then, was to find the dispersion relation for the system, that is, = ( k ). This would tell us whether a disturbance of a given wavenumber either grows or decays with time. Linearizing the appropriate equations, then solving for ( k ), we were able to show that a given disturbance decays with time (i.e. doesnt grow and eventually destroy the initial flow pattern) when the following relation holds:...
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 Spring '11
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 Fluid Dynamics

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