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<HTML><HEAD><title>Boussinesq Bubbles</title><BODY bgcolor="#ffffff"><H2><UL>Investigation of singularities in Boussinesq convection<HR><Center> Michael Minion </Center><HR></UL></H2><H4><!WA0><IMG SRC="http://math.nyu.edu/postdocs/minion/res/color_bubbles.gif" ALT="COLOR BUBBLE"><P> The figure above was produced using a Godunov/Projection methodapplied to the two-dimensional equations for Boussinesq convection.The initial conditions for this computation were taken froma paper by E and Shu and consist of a smooth "bubble" of densityin a zero flow field. As the bubble begins to rise, strongfronts in the density field form (shown by the contour lines in thefigure) and vorticity is baroclinically produced along this front(shown by the color shading).As the numerical method begins to under-resolve the flow, thedensity front destabilizes and rolls up under the influenceof the vorticity. Because of the experience gained fromstudying the appearance of <!WA1><a href="http://math.nyu.edu/postdocs/minion/res/under.html">spurious vortices</a> in under-resolved flowsthe correctness of the front roll-up is questionable despitethe fact that it is consistent with the evolution of the bubble.In order to investigate this problem further I have developedadaptive mesh refinement techniques for the Boussinesq equations.Below is an example of a simulation using initial conditions fromthe paper of Siggia and Pumir.</P><!WA2><IMG SRC="http://math.nyu.edu/postdocs/minion/res/s3ac.gif" ALT="REFINED BUBBLE"><P><P><H3> Impact: </H3>Besides providing an excellent test problem for the development,improvement and evaluation of adaptive mesh refinement methodsfor incompressible flow, I hope to also shed some light onthe open and sometimes controversal question as to whether ornot the two-dimensional Boussinesq equations admit finitetime singularities.</P><H2> References </H2><UL><LI> Weinan E and Chi-Wang Shu, <B>Small-scale structures in Boussinesq convection</B>, <CITE> Physics of Fluids, </CITE> January 1994.<LI> Alain Pumir and Eric D, Siggia, <B>Development of singular solutions to the axisymmetric Euler equations</B> <CITE> Physics of Fluids, </CITE>July 1992.</li><LI>A paper containing results from my study is in preparation.</LI><P><!WA3><A HREF="http://math.nyu.edu/postdocs/minion/index.html"> To Michael Minion's Homepage</A></P>⌨️ 快捷键说明
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