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The purpose of this paper is the development of a new density-based (DB) semi-Lagrangian method to speed up the conventional pressure-based (PB) semi-Lagrangian methods.

The semi-Lagrangian-based solvers are typically PB, i.e. semi-Lagrangian pressure-based (SLPB) solvers, where a Poisson equation is solved for obtaining the pressure field and ensuring a divergence-free flow field. As an elliptic-type equation, the Poisson equation often relies on an iterative solution, so it can create a challenge of parallel computing and a bottleneck of computing speed. This study proposes a new DB semi-Lagrangian method, i.e. the semi-Lagrangian artificial compressibility (SLAC), which replaces the Poisson equation by a hyperbolic continuity equation with an added artificial compressibility (AC) term, so a time-marching solution is possible. Without the Poisson equation, the proposed SLAC solver is faster, particularly for the cases with more computational cells, and better suited for parallel computing.

The study compares the accuracy and the computing speeds of both SLPB and SLAC solvers for the lid-driven cavity flow and the step-flow problems. It shows that the proposed SLAC solver is able to achieve the same results as the SLPB, whereas with a 3.03 times speed up before using the OpenMP parallelization and a 3.35 times speed up for the large grid number case (512 × 512) after the parallelization. The speed up can be improved further for larger cases because of increasing the condition number of the coefficient matrixes of the Poisson equation.

This paper proposes a method of avoiding solving the Poisson equation, a typical computing bottleneck for semi-Lagrangian-based fluid solvers by converting the conventional PB solver (SLPB) to the DB solver (SLAC) through the addition of the AC term. The method simplifies and facilitates the parallelization process of semi-Lagrangian-based fluid solvers for modern HPC infrastructures, such as OpenMP and GPU computing.

The purpose of this paper is to develop a Vortex-In-Cell (VIC) method with the semi-Lagrangian scheme and apply it to the high-Re lid-driven cavity flow.

The VIC method is developed for simulating high Reynolds number incompressible flow. A semi-Lagrangian scheme is incorporated in the convection term to produce unconditional stability, which gets rid of the constraint of the convection Courant-Friedrichs-Lewy (CFL) condition; the adaptive time step is used to maintain the numerical stability of the diffusion term; and the velocity boundary condition is readily converted to the vorticity formulation to suit discontinuous boundary treatment. The VIC simulation results are compared with those produced by other gird methods reported in open literature studies.

The lid-driven cavity flow is simulated from Re = 100 to 100,000. Similar vortex birth mechanisms are exhibited though, but distinct flow characteristics are revealed. At Re = 100 to 7,500, the cavity flow is confirmed steady. At Re = 10,000, 15,000 and 20,000, the cavity flow is periodical with a primary vortex held spatially at the center. In particular, at Re = 100,000 highly turbulent characteristics is first revealed and an analogous primary vortex is formed but in motion rather than stationary, which is caused by the considerable flow separation at all the boundaries.

In the lid-driven cavity, the flow becomes extremely complex and highly turbulent at Re = 100,000, and the analogous primary vortex structure is observed. Boundary layer separation is observed at all walls, producing small vortices and causing the displacement of the analogous primary vortex. Such a finding original and has not yet been reported by other investigators. It may provide a basis for conducting in-depth studies of the lid-driven cavity flow.

The purpose of this paper is to develop a simple and efficient conservative semi-Lagrangian scheme (SL) for solving advection equation in fast fluid dynamics (FFD), so FFD can provide fast indoor airflow simulations while preserving conservation for energy and species transport.

This study thus proposed a mass-fixing type conservative SL that redistributes global surplus/deficit on the advected field after performing the standard semi-Lagrangian advection. The redistribution weights were designed to preserve the properties of conservatives and monotonicity.

The effectiveness of the conservative SL was validated with several test cases, and the results show that the proposed scheme is indeed conservative with negligible impact on the accuracy of the standard solutions. The numerical tests show that the proposed scheme was indeed conservative with negligible impact on the accuracy of the flow prediction.

The FFD with conservative SL can effectively enforce the energy and species conservation for indoor airflow and predict airflow distributions with reasonable accuracy.

The present study aims to develop a vortex method capable for solving the complex vortical flows past the moving/deforming bodies.

To achieve such a goal, some innovative work is conducted on the basis of vortex-in-cell (VIC) method that uses the improved semi-Lagrangian scheme. The penalization technique is incorporated with the VIC, which makes the complex boundaries of moving/deforming bodies readily treated. Iterative algorithm is further proposed for the penalization and used to solve the Poisson equation, which enhances the vorticity solution accuracy at the body boundary.

The developed method is used to simulate some distinct flows of different boundaries and features: the impulsively started circular cylinder flow represents the one-way coupling; the falling circular cylinder flow and ellipse leaf flow both represent the two-way coupling of moving boundary; the fish-like body flow represents the two-way fluid-structure interaction of deforming boundary. The vortical physics of the above flows are well revealed, and the developed method is proven capable in dealing with the complex fluid-structure interaction problems.

The penalization technique is incorporated with the semi-Lagrangian VIC method, which makes the complex boundaries of moving/deforming bodies readily treated. An iterative algorithm is further proposed for the penalization and used to solve the Poisson equation, which enhances the vorticity solution accuracy at the body boundary. The complex vortical physics of the moving/deforming body flows are well revealed, and the propulsive mechanism of fish-like swimmer is well illustrated with the present method.

A semi‐implicit semi‐Lagrangian mixed finite‐difference finite‐volume model for the shallow water equations on a rotating sphere is considered. The main features of the model are the finite‐volume approach for the continuity equation and the vectorial treatment of the momentum equation. Pressure and Coriolis terms in the momentum equation and velocity in the continuity equation are treated semi‐implicitly. Discretization of this model led to the introducion, in a previous paper, of a splitting technique which highly reduces the computational effort for the numerical solution. In this paper we solve the full set of equations, without splitting, introducing an ad hoc algorithm. A von Neumann stability analysis of this scheme is performed to establish the unconditional stability of the new proposed method. Finally, we compare the efficiency of the two approaches by numerical experiments on a standard test problem. Results show that, due to the devised algorithm, the solution of the full system of equations is much more accurate while slightly increasing the computational cost.

The primitive equations of a barotropic atmosphere in isobaric co‐ordinates are reformulated, in order to express the geopotential gradient as a function of the pressure at the Earth’s surface. Furthermore, the free surface equation is written in conservative form. A finite difference, semi‐implicit, semi‐Lagrangian scheme in isobaric co‐ordinates is developed. The numerical scheme is mass conservative, is proven to be stable and requires the solution of a single five‐diagonal system. Numerical simulations show that the model captures the main dynamical features of large scale atmospheric motion.

The NIRVANA project is concerned with the development of a nonoscillatory, integrally reconstructed, volume‐averaged numerical advection scheme. The conservative, flux‐based finite‐volume algorithm is built on an explicit, single‐step, forward‐in‐time update of the cell‐average variable, without restrictions on the size of the time‐step. There are similarities with semi‐Lagrangian schemes; a major difference is the introduction of a discrete integral variable, guaranteeing conservation. The crucial step is the interpolation of this variable, which is used in the calculation of the fluxes; the (analytic) derivative of the interpolant then gives sub‐cell behaviour of the advected variable. In this paper, basic principles are described, using the simplest possible conditions: pure one‐dimensional advection at constant velocity on a uniform grid. Piecewise Nth‐degree polynomial interpolation of the discrete integral variable leads to an Nth‐order advection scheme, in both space and time. Nonoscillatory results correspond to convexity preservation in the integrated variable, leading naturally to a large‐Δt generalisation of the universal limited. More restrictive TVD constraints are also extended to large Δt. Automatic compressive enhancement of step‐like profiles can be achieved without exciting “stair‐casing”. One‐dimensional simulations are shown for a number of different interpolations. In particular, convexity‐limited cubic‐spline and higher‐order polynomial schemes give very sharp, nonoscillatory results at any Courant number, without clipping of extrema. Some practical generalisations are briefly discussed.

The purpose of this study is to design numerical simulations of bubbly flow around a cylinder to better understand the characteristics of flow around a rigid obstacle.

The bubbly flow around a circular cylinder was numerically simulated using a semi-Lagrangian–Lagrangian method composed of a vortex-in-cell method for the liquid phase and a Lagrangian description of the gas phase. Additionally, a penalization method was applied to account for the cylinder inside the flow. The slip condition of the bubbles on the cylinder’s surface was enforced, and the outflow conditions were applied to the liquid flow at the far field.

The simulation clarified the characteristics of a bubbly flow around a circular cylinder. The bubbles were shown to move around and separate from both sides of the cylinder, because of entrainment by the liquid shear layers. Once the bubbly flow fully developed, the bubbles distributed into groups and were dispersed downstream of the cylinder. A three-dimensional vortex structure of various scales was also shown to form downstream, whereas a quasi-stable two-dimensional vortex structure was observed upstream. Overall, the proposed method captured the characteristics of a bubbly flow around a cylinder well.

A semi-Lagrangian–Lagrangian approach was applied to simulate a bubbly flow around a circular cylinder. The simulations provided the detail features of these flow phenomena.

A mixed implicit semi Lagrangian finite difference‐finite volume method for numerical simulation of 2D air motion inside cylinders is derived and discussed. A conformal mapping from a physical (moving) domain to a computational (fixed) one is resorted in order to deal with a grid independent of time, making the numerical code very efficient. The numerical method is mass and energy conservative, unconditionally stable and at each timestep requires the solution of two well structured five‐band systems of linear equations. Its accuracy is first order in time and second one in space where the solution is smooth, while due to FCT space accuracy drops to the first order where the solution is steep. Stability of the method is proved both by a classical Von Newmann analysis and analysis of the matrices involved in the systems of linear equations. All these elements make the numerical method particularly fast. Numerical experiments are performed that show the influence of the maximum Courant number (with respect to the fluid speed) on the performance of the numerical method; moreover, comparison of simulations with a major existing code for engines is worked out.

This paper aims to provide discussions of a numerical method for bubbly flows and the interaction between a vortex ring and a bubble plume.

Small bubbles are released into quiescent water from a cylinder tip. They rise under the buoyant force, forming a plume. A vortex ring is launched vertically upward into the bubble plume. The interactions between the vortex ring and the bubble plume are numerically simulated using a semi-Lagrangian–Lagrangian approach composed of a vortex-in-cell method for the fluid phase and a Lagrangian description of the gas phase.

A vortex ring can transport the bubbles surrounding it over a distance significantly depending on the correlative initial position between the bubbles and the core center. The motion of some bubbles is nearly periodic and gradually extinguishes with time. These bubble trajectories are similar to two-dimensional-helix shapes. The vortex is fragmented into multiple regions with high values of Q, the second invariant of velocity gradient tensor, settling at these regional centers. The entrained bubbles excite a growth rate of the vortex ring's azimuthal instability with a formation of the second- and third-harmonic oscillations of modes of 16 and 24, respectively.

A semi-Lagrangian–Lagrangian approach is applied to simulate the interactions between a vortex ring and a bubble plume. The simulations provide the detail features of the interactions.