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Two-dimensional incompressible fluid flows around a rectangular shape placed over a larger rectangular shape at low Reynolds numbers (Re) have been numerically analyzed in the present work. The vortex shedding is investigated at different arrangements of the two shapes allowing the investigation of three possible configurations. The calculations are carried out for several values of Re ranging from 1 to 200. The effect of the obstacle geometry on the vortex shedding is analyzed for crawling, steady and unsteady regimes. The analysis of the flow evolution shows that with increasing Re beyond a certain critical value, the flow becomes unstable and undergoes a bifurcation. This paper aims to observe that the transition of the unsteady regime is performed by a Hopf bifurcation. The critical Re beyond which the flow becomes unsteady is determined for each configuration. A special attention is paid to compute the drag and lift forces acting on the rectangular shapes, which allowed determining; the best configuration in terms of both drag and lift. The unsteady periodic wake is characterized by the Strouhal number, which varies with the Re and the obstacle geometry. Hence, the values of vortex shedding frequencies are calculated in this work.

The dimensionless Navier–Stokes equations were numerically solved using the following numerical technique based on the finite volume method. The temporal discretization of the time derivative is performed by an Euler backward second-order implicit scheme. Non-linear terms are evaluated explicitly; while, viscous terms are treated implicitly. The strong velocity–pressure coupling present in the continuity and the momentum equations are handled by implementing the projection method.

The present paper aims to numerically study the effect of the obstacle geometry on the vortex shedding and on the drag and lift forces to analyze the flow structure around three configurations at crawling, steady and unsteady regimes.

A special attention is paid to compute the drag and lift forces acting on the rectangular shapes, which allowed determining; the best shapes configuration in terms of both drag and lift.

To accurately predict transient flow in homogeneous gas‐liquid mixtures in rigid and quasi‐rigid pipes, two mathematical models based on the gas‐fluid mass ratio are presented. The fluid pressure and velocity are considered as two principal dependent variables and the gas‐fluid mass ratio is assumed to be constant. By application of the conservation of mass and momentum laws, non‐linear hyperbolic systems of two differential equations are obtained and integrated numerically by a finite difference conservative scheme. The fluid density is defined by an expression averaging the two‐component densities where a polytropic process of the gaseous phase is admitted. The rigid model is deduced by neglecting the liquid compressibility and the pipe wall elasticity against the gas deformability. The quasi‐rigid model takes into account these two parameters. The effect of fluid compressibility on transient pressure behaviour is then analysed and confronted to the pipe wall elasticity. Numerical solutions are compared with numerical results available in literature and experiment developed in the laboratory. The results show that the pressure wave propagation is significantly influenced by the gas‐fluid mass ratio and the elasticity of the pipe wall. They indicate that the pipe elasticity and liquid compressibility may be neglected for great values of gas‐fluid mass ratio but not for the smaller ones.