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ARTICLE

Expression for the Gradient of the First Normal Derivative of the Velocity Potential

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1 Department of Aerodynamics, Nanjing University of Aeronautics and Astronautics, Yu Dao Street 29, City of Nan Jing, Jiang Su Province, P. R.China, 210016. Email: jutsjtu@yahoo.com.cn

Computer Modeling in Engineering & Sciences 2009, 46(1), 1-20. https://doi.org/10.3970/cmes.2009.046.001

Abstract

It is well-known that the velocity potential and its first normal derivative on the structure surface can be easily found in the boundary element method for problems of potential flow. Based on an investigation in progress, the gradient of the normal derivative of the velocity potential will be very helpful in the treatment of the so-called hypersingular integral. Through a coordinate transformation, such gradient can be expressed by the combination of the first and the second normal derivatives of the velocity potential. Then one interesting problem is how to find the second normal derivative of the velocity potential through the combination of the velocity potential and its first normal derivative on the structure surface. Here a detailed derivation of the second normal derivative of the velocity potential and the gradient of the first normal derivative of the velocity potential are presented for isoparametric curvilinear boundary elements. To validate these expressions, some numerical results for the second normal derivative of the velocity potential are compared to the corresponding analytical solutions.

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