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This paper aims to design and analyze implicit/explicit/semi-implicit schemes and a universal error estimator within the Generalized Single-step Single-Solve computational framework for First-order transient systems (GS4-I), which also fosters the adaptive time-stepping procedure to improve stability, accuracy and efficiency applied for fluid dynamics.

The newly proposed child-explicit and semi-implicit schemes emanate from the parent implicit GS4-I framework, providing numerous options with flexible and controllable numerical properties to the analyst. A universal error estimator is developed based on the consistent algorithmic variables and it works for all the developed methods. Applications are demonstrated by merging the developed algorithms into the iterated pressure-projection method for incompressible Navier–Stokes equations.

The child-explicit GS4-I has improved solution accuracy and stability properties, and the most stable option is the child explicit GS4-I(0,0)/second-order backward differentiation formula/Gear’s methods, which is new and novel. Numerical tests validate that the universal error estimator emanating from implicit designs works well for the newly proposed explicit/semi-implicit algorithms. The iterative pressure-correction projection algorithm is efficiently fostered by the error estimator-based adaptive time-stepping.

The implicit/explicit/semi-implicit methods within a unified computational framework are easy to implement and have flexible options in practical applications. In contrast to traditional error estimators, which work only on an algorithm-by-algorithm basis, the proposed error estimator is universal. They work for the entire class of implicit/explicit/semi-implicit linear multi-step methods that are second-order time accurate. Based on the accurately estimated local error, balance amongst stability, accuracy and efficiency can be well achieved in the dynamic simulation.

The purpose of this paper is to simulate two macrosegregation benchmarks with a newly developed stabilized finite element algorithm based on a semi-implicit pressure correction scheme.

A streamline-upwind/Petrov–Galerkin (SUPG) stabilized finite element algorithm is developed for the coupled conservation equations of mass, momentum, energy and species. A semi-implicit pressure correction method combined with SUPG stabilization technique is proposed to solve the convection flow during solidification. An analytically derived enthalpy method is adopted to solve the energy conservation equation. The nonlinearities of the energy and species equations are tackled by Newton–Raphson method. Two macrosegregation benchmarks considering the solidification of an Al-4.5 per cent Cu alloy and a Sn-10 per cent Pb alloy are simulated.

A very good agreement is achieved by comparison with the classical finite volume method and a novel meshless method for the Al-4.5 per cent Cu alloy solidification benchmark. Moreover, a unique reference numerical solution has been obtained. Besides, it is demonstrated that the stabilized finite element algorithm can capture the flow instability and channel segregation during solidification of the Sn-10 per cent Pb alloy.

A semi-implicit pressure correction method combined with SUPG stabilization technique is adopted to develop robust stabilized finite element algorithm for the macrosegregation model. A new enthalpy formulation for heat transfer problems with phase change is derived analytically.

We introduce a second‐order accurate time‐marching pressure‐correction algorithm to accommodate weakly‐compressible highly‐viscous liquid flows at low Mach number. As the incompressible limit is approached (Ma ≈ 0), the consistency of the compressible scheme is highlighted in recovering equivalent incompressible solutions. In the viscous‐dominated regime of low Reynolds number (zone of interest), the algorithm treats the viscous part of the equations in a semi‐implicit form. Two discrete representations are proposed to interpolate density: a piecewise‐constant form with gradient recovery and a linear interpolation form, akin to that on pressure. Numerical performance is considered on a number of classical benchmark problems for highly viscous liquid flows to highlight consistency, accuracy and stability properties. Validation bears out the high quality of performance of both compressible flow implementations, at low to vanishing Mach number. Neither linear nor constant density interpolations schemes degrade the second‐order accuracy of the original incompressible fractional‐staged pressure‐correction scheme. The piecewise‐constant interpolation scheme is advocated as a viable method of choice, with its advantages of order retention, yet efficiency in implementation.

Simple methods for the steady‐state analysis of a flow network are readily available, but the dynamic behavior of a large‐scale flow network is difficult to study due to the complex differential‐algebraic equation system resulting from its modeling. It is the aim of this paper to present two simple methods for the dynamic analysis of large‐scale flow networks and to demonstrate their use by examining the dynamics of a self‐similar branching tree network.

Two numerical projection methods are proposed for one‐dimensional dynamic analysis of large piping networks. Both are extensions of that suggested by Chorin for the nonlinear differential‐algebraic system resulting from the Navier‐Stokes equations. Each numerical algorithm is discussed and verified for turbulent flow in a nonlinear, self‐similar, branching tree network with constant friction factor for which an exact solution is available.

The dynamics of this network are calculated for more realistic friction factors and described as system parameters are varied. Self‐excited oscillations due to laminar‐turbulent transition are found for some parameter values and dynamic component behavior is observed in the network which is not observable in components apart from it.

It is shown that the dynamics of a flow network can exhibit unexpected behavior, reinforcing the need for simple methods to perform dynamic analysis.

This paper presents two numerical projection schemes for dynamic analysis of large‐scale flow networks to aid in their study and design.

Three‐dimensional flows over backward facing s.tif are analysed by means of a finite element procedure, which shares many features with the SIMPLER method. In fact, given an initial or guessed velocity field, the pseudovelocities, i.e. the velocities that would prevail in the absence of the pressure field, are found first. Then, by enforcing continuity on the pseudovelocity field, the tentative pressure is estimated, and the momentum equations are solved in sequence for velocity components. Afterwards, continuity is enforced again to find corrections that are used to modify the velocity field and the estimated pressure field. Finally, whenever necessary, the energy equation is solved before moving to the next step.

A splitting technique for solutions of the Navier—Stokes and the energy equations, in Boussinesq approximately, is presented. The equations are first integrated in time using a splitting procedure and then discretized spatially by means of a high‐order spectral element method. The whole technique is validated on the flow in a differentially‐heated cavity at intermediate and transitional Rayliegh numbers. The results are in a very good agreement with other available numerical solutions.

A numerical study on the stretching of a Newtonian fluid filament is analysed. Stretching is performed between two retracting plates, moving under constant extension rate. A semi‐implicit Taylor‐Galerkin/pressure‐correction finite element formulation is employed on variable‐structure triangular meshes. Stability and accuracy of the scheme is maintained up to large Hencky‐strain levels. A non‐uniform radius profile, minimum at the filament mid‐plane, is observed along the filament‐length at all times. We have found maintenance of a suitable mesh aspect‐ratio around the mid‐plane region (maximum stretch zone) to restrict early filament break‐up and consequently solution divergence. As such, true transient flow evolution is traced and the numerical results bear close agreement with the literature.

Three‐dimensional laminar forced convective heat transfer in ribbed square channels is investigated. In these channels, transverse and angled ribs are placed on one or two of the walls to form a repetitive geometry. After a short distance from the entrance, also the flow and the dimensionless thermal fields repeat themselves from module to module allowing the assumption of periodic, or anti‐periodic, conditions at the inlet/outlet sections of the calculation cell. Prescribed temperature boundary conditions are assumed at all solid walls, including the ribs. Pressure drop and heat transfer characteristics are compared for rib angles ranging from 90° (transverse ribs) to 45°, and different values of the Reynolds number. The influence of rib geometries is investigated below and above the onset of the self‐sustained flow oscillations that precede the transition to turbulence. Numerical simulations are carried out employing an equal order finite‐element procedure based on a projection algorithm.

This paper aims to propose two parallel two-step finite element algorithms based on fully overlapping domain decomposition for solving the 2D/3D time-dependent natural convection problem.

The first-order implicit Euler formula and second-order Crank–Nicolson formula are used to time discretization respectively. Each processor of the algorithms computes a stabilized solution in its own global composite mesh in parallel. These algorithms compute a nonlinear system for the velocity, pressure and temperature based on a lower-order element pair (P₁b-P₁-P₁) and solve a linear approximation based on a higher-order element pair (P₂-P₁-P₂) on the same mesh, which shows that the new algorithms have the same convergence rate as the two-step finite element methods. What is more, the stability analysis of the proposed algorithms is derived. Finally, numerical experiments are presented to demonstrate the efficacy and accuracy of the proposed algorithms.

Finally, numerical experiments are presented to demonstrate the efficacy and accuracy of the proposed algorithms.

The novel parallel two-step algorithms for incompressible natural convection problem are proposed. The rigorous analysis of the stability is given for the proposed parallel two-step algorithms. Extensive 2D/3D numerical tests demonstrate that the parallel two-step algorithms can deal with the incompressible natural convection problem for high Rayleigh number well.

The purpose of the paper is to propose some modifications to the SIMPLE (semi-implicit method for pressure-linked equations) algorithm. These modifications can ensure the numerical robustness and optimize computational efficiency. They remarkably promote the ability of the SIMPLE algorithm for incompressible DNS (direct numerical simulation) of multiscale problems, such as transitional flows and turbulent flows, by improving the properties of dispersion and dissipation.

The MDCD (minimized dispersion and controllable dissipation) scheme and MMIM (modified momentum interpolation method) are introduced. Six typical test cases are used to validate the modified algorithm, including the linear convective flow, lid-driven cavity flow, laminar boundary layer, Taylor vortex and DHIT (decaying homogenous isotropic turbulence). Particularly, a highly unsteady DNS of separated-flow transition in turbomachinery is precisely predicted by the modified algorithm.

The numerical examples show the distinct superiority of the modified algorithm in both internal flows and external flows. The advantages of the MDCD scheme and MMIM make the SIMPLE algorithm a promising method for DNS.

Some effective modifications to the SIMPLE algorithm are addressed. It is the first attempt to introduce the MDCD approach into the SIMPLE-type algorithms. The new algorithm is especially suitable for the incompressible DNS of convection-dominated flows.