A METHOD TO SOLVE A PROBLEM OF FLOW IN A CHANNEL WITH RECTANGULAR EXPANSION IN THE VARIABLES STREAM FUNCTION-VORTICITY - Борисюк, Андрій Олександрович (Science-based technologies, 2019)
A numerical method in the variables stream function-vorticity is developed to solve a problem of fluid motion in an infinite straight flat rigid-walled channel with a local rigid axisymmetric expansion of rectangular shape. The method has the first order of accuracy in a temporal and the second order of accuracy in spatial coordinates. In the developed method, the formulated problem is solved by means of (a) introducing the stream function and the vorticity, and the corresponding transition from the variables velocity-pressure to the variables stream function-vorticity-pressure, (b) subsequent non-dimensionalizing the relationships obtained on the basis of that transition, (c) choice of both the computational domain and the corresponding spatial and temporal integration mesh, having small constant steps in the temporal and spatial coordinates, (d) discretization of the noted non-dimensional relationships in the appropriate nodes of the chosen mesh and subsequent solving the algebraic equations obtained with the use of the indicated discretization. In making the discretization, its temporal part is carried out on the basis of the two-point upward differencing scheme, whereas the spatial one is based on the two-point adverse flow differencing schemes (for the convective term of the non-linear vorticity equation) and the fifths-point differencing schemes (for the diffusive term of the noted equation and the Poisson’s equations for the stream function and the pressure) for the corresponding co-ordinates. The iterative method of successive over relaxation is applied to solve the linear algebraic equations for the stream function and the pressure (the only difference between these equations is in their right parts, which are known). As for the algebraic relationship for the vorticity (that was obtained after performing the noted discretization), it does not require application of any method for its solution, because actually it is a computational scheme to find immediately the vorticity on the basis of the known values of the corresponding magnitudes found at the previous time step (at the initial time, the values of all the magnitudes are prescribed).