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Unformatted text preview: Vorticity and Enstrophy Changes due to Stretching/Compression in a Turbulent Mixing Layer Patrick Rabenold ENME 656 Midterm Project April 14, 2005 Abstract A joint probability density function (JPDF) and covariance integrand analysis is performed on experimental mixing layer data in order to better understand the term in the transport equations for instantaneous vortic ity and enstrophy that represents stretching and compression of vortic ity filaments due to reorientation. The data is found to contain several anomalies which must be corrected, at least approximately. Using this data, strong correlations are found between Ω j and ∂U i /∂x j , which help in understanding correlations between Ω i and Ω j ∂U i /∂x j , which result in a net increase in the rate of change of enstrophy due to vorticity filament stretching or compression. It is found that vorticity filament stretching dominates compression. 1 Transport Equations The vorticity field, Ω ≡ ∇× u , can be very useful in understanding many aspects of turbulent flows. Therefore, it is helpful to derive its own transport equation by taking the curl of the incompressible NavierStokes equation, which results in ∂ Ω i ∂t + U j ∂ Ω i ∂x j = Ω j ∂U i ∂x j + ν ∂ 2 Ω i ∂x j ∂x j . (1) The enstrophy, 1 2 Ω i Ω i , is a scalar measure of the magnitude of vorticity, similar to kinetic energy for velocity. The enstrophy transport equation is derived from equation (1) as ∂ ( 1 2 Ω i Ω i ) ∂t + U j ∂ ( 1 2 Ω i Ω i ) ∂x j = Ω i Ω j ∂U i ∂x j + ν ∂ 2 ( 1 2 Ω i Ω i ) ∂x j ∂x j − ν ∂ Ω i ∂x j ∂ Ω i ∂x j . (2) The terms in both of these equations represent, from left to right, the Eulerian time rate of change of vorticity or enstrophy, advection, stretching and com pression due to reorientation, viscous diffusion, and viscous dissipation for the enstrophy. 1 A physical interpretation of the stretching/compression term can be under stood as follows. If a vortex filament is aligned in the xdirection, then the term Ω x ∂U/∂x represents a gain or loss of Ω x due to pure stretching or com pression, depending on the signs of Ω x and ∂U/∂x . However, the Ω y equation contains the term Ω x ∂V/∂x , which represents pure rotation of the filament into the ydirection. An arbitrarily oriented vortex filament can be thought of as undergoing simultaneous stretching and compression due to reorientation by the velocity gradients. This physical description can be extended to the stretching/compression term in the enstrophy equation, by considering a vortex filament projected onto a coordinate plane. All combinations of the signs of Ω i , Ω j , and ∂U i /∂x j determine whether there is an increase due to stretching or a decrease due to compression in the contribution to the change in enstrophy....
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This note was uploaded on 10/09/2011 for the course FLUID MECH 4020 taught by Professor Chandra during the Spring '11 term at Université de Liège.
 Spring '11
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