Abstract
For some special flows, especially the potential flow in a plane, using the hodograph method has obvious advantages. Realistic flows have a stream surface, namely, a two-dimensional manifold, on which the velocity vector of the flow lies on its tangent space. By introducing a stream function and a potential function, we establish the hodograph method for potential flows on a surface using the tensor analysis. For the derived hodograph equation, we obtain a characteristic equation and its characteristic roots, from which we can classify the type of the second-order hodograph equation. Moreover, we give some examples for special surfaces.
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References
Cherry, T. M. Flow of a compressible fluid about a cylinder. Proceedings of the Royal Society of London, Series A: Mathematical and Physical Sciences 192(1028), 45–79 (1947)
von Kármán, T. Compressibility effects in aerodynamics. Journal of the Aeronautical Sciences 8(9), 337–356 (1941)
Tsien, H. S. Two-dimensional subsonic flow of compressible fluids. Journal of the Aeronautical Sciences 6(10), 399–407 (1939)
Li, K. T. and Huang, A. X. Mathematical aspect of the stream-function equations of compressible turbomachinery flows and their finite element approximation using optimal control. Computer Methods in Applied Mechanics and Engineering 41(2), 175–194 (1983)
Li, K. T., and Huang, A. X. Tensor Analysis and Its Applications (in Chinese), Chinese Scientific Press, Beijing (2000)
Li, K. T., Huang, A. X., and Zhang, W. L. A dimension split method for the 3-D compressible Navier-Stokes equations in turbomachine. Communications in Numerical Methods in Engineering 18(1), 1–14 (2002)
Li, K. T., Su, J., and Huang, A. X. Geometrical design of blade’s surface and boundary control of Navier-Stokes equations. Academic Journal of Xi’an Jiaotong University 19(1), 1–6 (2007)
Il’in, A. A. The Navier-Stokes and Euler equations on two-dimensional closed manifolds. Mathematics of USSR Sbornik 69(2), 559–579 (1991)
Wu, C. H. A general theory of three-dimensional flow in subsonic and supersonic turbomachines of axial, radial and mixed-flow types. NACA TN2604 (1952)
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Communicated by Xing-ming GUO
Project supported by the National Natural Science Foundation of China (Nos. 10971165, 10771167, and 10926080)
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Li, Kt., Shi, F. Hodograph method of flow on two-dimensional manifold. Appl. Math. Mech.-Engl. Ed. 31, 363–376 (2010). https://doi.org/10.1007/s10483-010-0309-x
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DOI: https://doi.org/10.1007/s10483-010-0309-x