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enme656report - Vorticity and Enstrophy Changes due to...

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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 Navier-Stokes 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
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A physical interpretation of the stretching/compression term can be under- stood as follows. If a vortex filament is aligned in the x -direction, 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 y -direction. 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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