Abstract:
New methods are presented for generating 3-D stream lines and stream surfaces based on the evaluation of two 3-D stream functions. Two approaches for solving the dual stream functions are considered, firstly a global solution and secondly a local solution. The global approach enables stream surfaces to be visualised by constructing surfaces of constant value (isosurfaces) although it can only be applied to certain classes of flows. Modifications to the 'marching cubes' isosurface algorithm are described which correct the problem of ambiguous surface elements. An algorithm is presented which generates a look-up table containing all the surface element geometries. A simple gradient method is also presented which selects the correct surface element geometry when ambiguous cases are encountered. Two algorithms are presented for constructing stream lines based on a local solution of two trilinear stream functions. Direct analytical integration is used to calculate the stream functions from the discretely defined mass flow within a cell. A mapping technique is used to convert the 3-D stream functions into a 2-D graphical form. Stream lines are calculated by tracking constant values of each stream function, a process which corresponds to finding the intersection of two stream surfaces. The tracking process is mass conservative, which ensures consistent stream lines, and does not use a time-stepping method for integration. The method can be applied generally to any 3-D compressible or incompressible steady flow. The accuracy of the new methods was evaluated in five mathematical flows ranging from a simple vortex flow to a complex chaotic one. Stream lines constructed from these flows were compared with exact solutions and to those calculated with time-stepping algorithms. The stream function method produced very accurate stream lines in all the test flows, graphical analysis revealing the errors to be lower than 1%. The accuracy was on par with the best time stepping scheme tested but computation times for generating the stream lines were significantly reduced. Results show that the stream function method is up to an order of magnitude faster than conventional schemes.