Calculation of the local resistance coefficient of viscous incombressible fluid flow in a pipe with sudden contraction

Most technical applications for fluid transportation include such a structural particularity as sudden contraction. This geometrical feature has an effect on the flow characteristics that should be taken into account in order to provide the appropriate mode and conditions of the flow.
This work was intended to investigate the kinematic characteristics and pressure losses of Newtonian fluid flow through an axisymmetric sudden contraction. The mathematical statement of the problem of laminar flow was formulated using stream function and vorticity variables. The stationary solution was obtained using the relaxation method with the following realization of numerical algorithm based on the finite difference alternative directions scheme. The obtained flow pattern showed that the flow structure apart from the one-dimensional zones included twodimensional regions in the vicinity of contracted part ($l_1$ and $l_2$ are the lengths of two-dimensional regions upstream and downstream of the sudden contraction, respectively; $L$ is the length of recirculation zone in the inner corner). The effect of the main parameters on these characteristics was studied and represented as dependency diagrams.
For the calculations of local resistance coefficient ($C_{\mathcal{M}}$), two different formulas were used. Implemented parametrical investigations allowed plotting the local resistance coefficient as a function of the Reynolds number and pipe contraction ratio: $C_{\mathcal{M}}$ decreased with an increase in the Reynolds number; an increase in contraction ratio caused an increase in $C_{\mathcal{M}}$. The obtained results were verified and compared with available data.