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.
In this paper, we propose a dimensional splitting method for the three dimensional (3D) rotating Navier-Stokes equations. Assume that the domain is a channel bounded by two surfaces and is decomposed by a series of surfaces ■i into several sub-domains, which are called the layers of the flow. Every interface i between two sub-domains shares the same geometry. After establishing a semi-geodesic coordinate (S-coordinate) system based on ■i , Navier-Stoke equations in this coordinate can be expressed as the sum of two operators, of which one is called the membrane operator defined on the tangent space on ■i , another one is called the bending operator taking value in the normal space on ■i . Then the derivatives of velocity with respect to the normal direction of the surface are approximated by the Euler central difference, and an approximate form of Navier-Stokes equations on the surface ■i is obtained, which is called the two-dimensional three-component (2D-3C) Navier-Stokes equations on a two dimensional manifold. Solving these equations by alternate iteration, an approximate solution to the original 3D Navier-Stokes equations is obtained. In addition, the proof of the existence of solutions to 2D-3C Navier-Stokes equations is provided, and some approximate methods for solving 2D-3C Navier-Stokes equations are presented.
