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This paper aims to present an improved finite element method used for achieving faster convergence in simulations of incompressible fluid flows. For stable computations of incompressible fluid flows, it is important to ensure that the flow field satisfies the equation of continuity in each element of a generally distorted mesh. The study aims to develop a numerical approach that satisfies this requirement based on the highly simplified marker-and-cell (HSMAC) method and increases computational speed by introducing a new algorithm into the simultaneous relaxation of velocity and pressure.

First, the paper shows that the classical HSMAC method is equivalent to a Jacobi-type method in terms of the simultaneous relaxation of velocity and pressure. Then, a Gauss–Seidel or successive over-relaxation (SOR)-type method is introduced in the Newton–Raphson iterations to take into account all the derivative terms in the first-order Taylor series expansion of a nodal-averaged error explicitly. Here, the nine-node quadrilateral (Q2–Q1) elements are used.

The new finite element approach based on the improved HSMAC algorithm is tested on fluid flow problems including the lid-driven square cavity flow and the flow past a circular cylinder. The results show significant improvement of the convergence property with the accuracy of the numerical solutions kept unchanged even on a highly distorted mesh.

To the best of the author’s knowledge, the idea of using the Gauss–Seidel or SOR method in the simultaneous relaxation procedure of the HSMAC method has not been proposed elsewhere.

The purpose of this paper is to study numerically and experimentally incompressible Newtonian flow in a three‐dimensional cylindrical branching channel.

The flow configuration studied in the present investigation is such that a fully developed laminar flow enters an abruptly expanded cylinder and the flow leaves this cylinder by two identical cylindrical outlet branch pipes. A numerical analysis was performed by developing a three‐dimensional numerical code using the highly simplified marker and cell method. Representative velocities in the flow field are recorded by Laser Doppler Velocimeter measurements and volume flow rate from each outlet branch pipe is measured. Flow visualization in representative symmetrical planes is also carried out. Comparisons of numerical predictions and experimental data are presented and the reasonable agreement between the numerical and experimental results is encouraging.

The flow field in the three‐dimensional cylindrical branching channel is clarified within the range of laminar flow. The characteristics of the branch flow rate are obtained and show that there exist two distinct domains of strong asymmetric flow distribution from the outlet branch pipes, depending on the Reynolds numbers. It is further observed that the flow became time periodic as the Reynolds number is increased. It becomes apparent that the swirl flow component plays a key role in the flow phenomena.

The present investigation sheds light on the three‐dimensionality in the prevailing flow field for various inlet Reynolds numbers in the laminar flow range. Flow rate deflection characteristics in a three‐dimensional cylindrical branching channel are also obtained.

An Unsteady Reynolds‐Averaged Navier‐Stokes (URANS) equation method has been applied to compute the flow over two‐dimensional smooth topography and compared with conventional RANS and large‐eddy simulation (LES) results. The URANS calculation with sufficient grid resolution near solid surface and an appropriate near‐wall model has been shown to simulate much of the large‐scale unsteadiness and some of the turbulent motion for flows with and without separation. Although the results with unadjusted model constants do not show an overwhelming improvement over a standard two‐equation model, it is demonstrated that it may be improved and, more importantly, can be generalized to a new simulation technique by refining the model, considering such factors as grid‐dependent length scales and by making a three‐dimensional calculation.

The influence of gravity on the Marangoni flow instability in half zone liquid bridges in the case of liquid metals is investigated by direct 3D and time‐dependent simulation of the problem. The computations are carried out for different heating conditions and environments (zero g conditions and on ground liquid zone heated from above or from below). The case of cylindrical shape (simplified model) and of melt/air interface deformed by the effect of gravity (real conditions) are considered. The comparison among these situations gives insight into the separate (gravity) effects of buoyancy forces and of the free surface deviation with respect to straight configuration. Body‐fitted curvilinear co‐ordinates are adopted to handle the non‐cylindrical problem. The liquid bridge exhibits different behaviours according to the allowed bridge shape. If the shape is forced to be cylindrical, the flow field is stabilized in the case of heating from above and destabilized if gravity is reversed. If the deformation is taken into account, gravity always stabilizes the Marangoni flow regardless of its direction (parallel or antiparallel to the axis) and the 3D flow structure is different according to the heating condition (from above or from below). In the latter case, the critical Marangoni number is larger and the critical wave number is smaller, compared with the opposite condition. In addition, for Pr=0.02 (Gallium), a surprising heretofore unseen behaviour arises. No steady bifurcation occurs and the flow becomes unstable directly to oscillatory disturbances. This phenomenon has never been reported before in the case of low Prandtl number liquids.