Computing Streamfunction and Velocity Potential in a Limited Domain of Arbitrary Shape. Part I: Theory and Integral Formulae |
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Authors: | Qin XU CAO Jie and GAO Shouting |
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Institution: | NOAA/National Severe Storms Laboratory, Norman, Oklahoma, USA,Cooperative Institute for Mesoscale Meteorological Studies, University of Oklahoma, USA, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029,Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029 |
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Abstract: | The non-uniqueness of solution and compatibility between the coupled
boundary conditions in computing velocity potential and streamfunction from
horizontal velocity in a limited domain of arbitrary shape are revisited
theoretically with rigorous mathematic treatments. Classic integral formulas
and their variants are used to formulate solutions for the coupled problems.
In the absence of data holes, the total solution is the sum of two integral
solutions. One is the internally induced solution produced purely and
uniquely by the domain internal divergence and vorticity, and its two
components (velocity potential and streamfunction) can be constructed by
applying Green's function for Poisson equation in unbounded domain to the
divergence and vorticity inside the domain. The other is the externally
induced solution produced purely but non-uniquely by the domain external
divergence and vorticity, and the non-uniqueness is caused by the harmonic
nature of the solution and the unknown divergence and vorticity
distributions outside the domain. By setting either the velocity potential
(or streamfunction) component to zero, the other component of the externally
induced solution can be expressed by the imaginary (or real) part of the
Cauchy integral constructed using the coupled boundary conditions and
solvability conditions that exclude the internally induced solution. The
streamfunction (or velocity potential) for the externally induced solution
can also be expressed by the boundary integral of a double-layer (or
single-layer) density function. In the presence of data holes, the total
solution includes a data-hole--induced solution in addition to the above
internally and externally induced solutions. |
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Keywords: | integral formulae streamfunction velocity potential domain of arbitrary shape |
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