lecture31 - ME 563 - Intermediate Fluid Dynamics - Su...

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Unformatted text preview: ME 563 - Intermediate Fluid Dynamics - Su Lecture 31 - Instability: more on Kelvin-Helmholtz, and thermal convection Reading: Acheson, 9.2-9.3. To recap what we did with Kelvin-Helmholtz instability in the last lecture We defined our undisturbed, two-dimensional 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 x-direction and the lower fluid is at rest (Fig. 1). 2 1 U y= Figure 1: Undisturbed flow for the Kelvin-Helmholtz problem. We allowed the interface to be subject to sinusoidal, traveling-wave disturbances with in- finitesimal magnitude. We then wanted to see how the initially, spatially-defined 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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lecture31 - ME 563 - Intermediate Fluid Dynamics - Su...

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