Numerical investigation of non-Newtonian fluid flow through a pipe sudden contraction

Sudden contraction represents a geometrical heterogeneity leading to the significant flow structure changes and intensifying energy losses. The fluid flowing through a sudden contraction is of great scientific interest as it is found to be an intermediate stage of the processes taking place in a technical equipment such as pumps, engines, reactors, etc.
In this paper, the problem of a laminar stationary flow of non-Newtonian fluid in a pipe with sudden contraction was numerically solved. The fluid rheological properties were described by the Ostwald–de Waele power law. The constitutive equations were written using the stream function and vorticity variables in a cylindrical coordinate system. The asymptotic time solution to the unsteady flow equations was obtained in order to derive a steady-state solution to the initial problem. The main equations were discretized using the finite-difference method based on the alternative directions scheme and solved using the sweep method. To verify numerical algorithm developed, the approximating convergence was tested on the sequence of square grids.
According to the flow patterns, pseudoplastic and dilatant fluid flow structures both consist of one-dimensional zones next to the inlet and outlet sections and two-dimensional zones in the vicinity of contraction plane. To evaluate the impact of the Reynolds number, pipe contraction ratio, and power-law index on the length of two-dimensional flow regions, the dependency diagrams were plotted in a wide range of the parameters. Two different methods were used to calculate the local resistance coefficient. The obtained values were found to be in a good agreement. A parametric study was performed to reveal the influence of the governing parameters on the local energy losses.