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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.

As the penalized vortex-in-cell (pVIC) method is based on the vorticity-velocity form of the Navier–Stokes equation, the pressure variable is not incorporated in its solution procedure. This is one of the advantages of vorticity-based methods such as pVIC. However, dynamic pressure is an essential flow property in engineering problems. In pVIC, the pressure field can be explicitly evaluated by a pressure Poisson equation (PPE) from the velocity and vorticity solutions. How to specify far-field boundary conditions is then an important numerical issue. Therefore, this paper aims to robustly and accurately determine the boundary conditions for solving the PPE.

This paper introduces a novel non-iterative method for specifying Dirichlet far-field boundary conditions to solve the PPE in a bounded domain. The pressure field is computed using the velocity and vorticity fields obtained from pVIC, and the solid boundary conditions for pressure are also imposed by a penalization term within the framework of pVIC. The basic idea of our approach is that the pressure at any position can be evaluated from its gradient field in a closed contour because the contour integration for conservative vector fields is path-independent. The proposed approach is validated and assessed by a comparative study.

This non-iterative method is successfully implemented to the pressure calculation of the benchmark problems in both 2D and 3D. The method is much faster than all the other methods tested without compromising accuracy and enables one to obtain reasonable pressure field even for small computation domains that are used regardless of a source distribution (the right-hand side in the Poisson equation).

The strategy introduced in this paper provides an effective means of specifying Dirichlet boundary conditions at the exterior domain boundaries for the pressure Poisson problems. It is very efficient and robust compared with the conventional methods. The proposed idea can also be adopted in other fields dealing with infinite-domain Poisson problems.

The purpose of this paper is to explore the application of the mesh‐free local radial basis function collocation method (RBFCM) in solution of coupled heat transfer and fluid‐flow problems.

The involved temperature, velocity and pressure fields are represented on overlapping five nodded sub‐domains through collocation by using multiquadrics radial basis functions (RBF). The involved first and second derivatives of the fields are calculated from the respective derivatives of the RBFs. The energy and momentum equations are solved through explicit time stepping.

The performance of the method is assessed on the classical two dimensional de Vahl Davis steady natural convection benchmark for Rayleigh numbers from 10³ to 10⁸ and Prandtl number 0.71. The results show good agreement with other methods at a given range.

The pressure‐velocity coupling is calculated iteratively, with pressure correction, predicted from the local mass continuity equation violation. This formulation does not require solution of pressure Poisson or pressure correction Poisson equations and thus much simplifies the previous attempts in the field.

A new fractional step method in conjunction with the finite element method is proposed for the analysis of the thermal convection and conduction in a fluid region expressed by the momentum equations, the equation of continuity and the energy equation. This paper focuses on the features of the present finite element method which gives a simple way of treating the Neumann boundary condition for the pressure Poisson equation. The applicability and effectiveness of the proposed scheme are illustrated through the numerical examples of the two‐dimensional natural convection flow in enclosures with several Rayleigh numbers.

The jet impingement usually accompanying large interface movement is studied by the in-house solver MLParticle-SJTU based on the modified moving particle semi-implicit (MPS) method, which can provide more accurate pressure fields and deformed interface shape. The comparisons of the pressure distribution and the shape of free surface between the presented numerical results and the analytical solution are investigated. The paper aims to discuss these issues.

To avoid the instability in traditional MPS, a modified MPS method is employed, which include mixed source term for Poisson pressure equation (PPE), kernel function without singularity, momentum conservative gradient model and highly precise free surface detection approach. Detailed analysis on improved schemes in the modified MPS is carried out. In particular, three kinds of source term in PPE are considered, including: particle number density (PND) method, mixed source term method and divergence-free method. Two typical kernel functions containing original kernel function with singularity and modified kernel function without singularity are analyzed. Three kinds of pressure gradient are considered: original pressure gradient (OPG), conservative pressure gradient (CPG) and modified pressure gradient (MPG). In addition, particle convergence is performed by running the simulation with various spatial resolutions. Finally, the comparison of the pressure fields by the modified MPS and by SPH is presented.

The modified MPS method can provide a reliable pressure distribution and the shape of the free surface compared to the analytical solution in a steady state after the water jet impinging on the wall. Specifically, mixed source term in PPE can give a reasonable profile of the shape of free surface and pressure distribution, while PND method adopted in the traditional MPS is not stable in simulation, and divergence-free method cannot produce rational pressure field near the wall. Two kernel functions show similar pressure field, however, the kernel function without singularity is preferred in this case to predict the profile of free surface and pressure on the wall. The shape of free surface by CPG and MPG is agreement with the analytical solution, while a great discrepancy can be observed by OPG. The pressure peak by MPG is closer to the analytical solution than that by CPG, while the pressure distribution on the right hand side of the pressure peak by latter is better match with the analytical solution than that by former. Besides, fine spatial resolution is necessary to achieve a good agreement with analytical results. In addition, the pressure field by the modified MPS is also quite similar to that by SPH, and this can further validate the reliable of current modified MPS.

The present modified MPS appears to be a stable and reliable tool to deal with the impinging jet flow problems involving large interface movement. Mixed source term in PPE is superior to PND adopted in the traditional MPS and divergence-free method. The kernel function without singularity is preferred to improve the computational accuracy in this case. CPG is a good choice to obtain the shape of free surface and the pressure distribution by jet impingement.

The purpose of this paper is to show how a surface tension model and an eddy viscosity based on the Smagorinsky sub‐grid scale model, which belongs to the Large‐Eddy Simulation (LES) theory for turbulent flow, have been introduced into ISPH (Incompressible smoothed particle hydrodynamics) method. In addition, a small modification in the source term of pressure Poisson equation has been introduced as a stabilizer for robust simulations. This stabilization generates a smoothed pressure distribution and keeps the total volume of fluid, and it is analogous to the recent modification in MPS.

The surface tension force in free surface flow is evaluated without a direct modeling of surrounding air for decreasing computational costs. The proposed model was validated by calculating the surface tension force in the free surface interface for a cubic‐droplet under null‐gravity and the milk crown problem with different resolution models. Finally, effects of the eddy viscosity have been discussed with a fluid‐fluid interaction simulation.

From the numerical tests, the surface tension model can handle free surface tension problems including high curvature without special treatments. The eddy viscosity has clear effects in adjusting the splashes and reduces the deformation of free surface in the interaction. Finally, the proposed stabilization appeared in the source term of pressure Poisson equation has an important role in the simulation to keep the total volume of fluid.

An incompressible smoothed particle hydrodynamics is developed to simulate milk crown problem using a surface tension model and the eddy viscosity.

Modeling of multi-phase flows for Rayleigh-Taylor instability and natural convection in a square cavity has been investigated using an incompressible smoothed particle hydrodynamics (ISPH) technique. In this technique, incompressibility is enforced by using SPH projection method and a stabilized incompressible SPH method by relaxing the density invariance condition is applied. The paper aims to discuss these issues.

The Rayleigh-Taylor instability is introduced in two and three phases by using ISPH method. The author simulated natural convection in a square/cubic cavity using ISPH method in two and three dimensions. The solutions represented in temperature, vertical velocity and horizontal velocity have been studied with different values of Rayleigh number Ra parameter (103=Ra=105). In addition, characteristic based scheme in Finite Element Method is introduced for modeling the natural convection in a square cavity.

The results for Rayleigh-Taylor instability and natural convection flow had been compared with the previous researches.

Modeling of multi-phase flows for Rayleigh-Taylor instability and natural convection in a square cavity has been investigated using an ISPH technique. In ISPH method, incompressibility is enforced by using SPH projection method and a stabilized incompressible SPH method by relaxing the density invariance condition is introduced. The Rayleigh-Taylor instability is introduced in two and three phases by using ISPH method. The author simulated natural convection in a square/cubic cavity using ISPH method in two and three dimensions.

To study the steady viscous incompressible electrically conducting fluid flow past a circular cylinder under the influence of an external magnetic field at high Reynolds numbers (Re).

The finite difference method is applied to solve the governing non‐linear Navier‐Stokes equations. First order upwind difference scheme is applied to the convective terms. The multigrid method with coarse grid correction is used to enhance the convergence rate. The defect correction technique is employed to achieve the second order accuracy.

A non‐monotonic behavior in separation angle when N≥5 and separation length when N≥3 is found with the increase of external magnetic field. The drag coefficient is found to increase with increase of N. The pressure drag coefficient, total drag coefficient and rear pressure are found to exhibit a linear dependence with N^0.5. The pressure Poisson equation is solved to find pressure fields in the flow region. It is found that the upstream base pressure increases with increase of external magnetic field while the downstream base pressure decreases with the increase of the external magnetic field.

The non‐monotonic behaviors in the separation angle and separation length at high Re are explained through pressure fields which are found first time for this problem. The linear dependence of the pressure drag coefficient, total drag coefficient and the pressure at rear stagnation point with N^0.5 is in agreement with experimental findings.

A convection‐diffusion‐reaction scheme is proposed in this study to simulate the high gradient electroosmotic flow behavior in microchannels. The equations governing the total electric field include the Laplace equation for the effective electrical potential and the Poisson‐Boltzmann equation for the electrical potential in the electric double layer.

Mixed electroosmotic/pressure‐driven flow in a straight microchannel is studied with the emphasis on the Joule heat in the equations of motion. The nonlinear behaviors resulting from the hydrodynamic, thermal and electrical three‐field coupling and the temperature‐dependent fluid viscosity, thermal conductivity, electrical permittivity, and conductivity of the investigated buffer solution are analyzed.

The solutions computed from the employed flux discretization scheme for the hydrodynamic, thermal and electric field equations have been verified to have good agreement with the analytical solution. Parametric studies have been carried out by varying the electrical conductivity at the fixed zeta potential and varying the zeta potential at the fixed electrical conductivity.

Investigation is also addressed on the predicted velocity boundary layer and the electric double layer near the negatively charged channel wall.

The purpose of this paper is to propose an efficient/robust numerical algorithm for solving the two‐dimensional laminar mixed‐convection in a lid‐driven cavity using the mixed finite element (FE) technique.

A numerical algorithm was based on the so‐called consistent splitting scheme, which improved the numerical accuracy of the primitive variables. In order to obtain a stable solution, two choices of mixed FEs, the Taylor‐Hood and Crouzeix‐Raviart types, were used. Two mesh layouts were considered; uniform and non‐uniform.

To verify that the proposed scheme had a second‐order accuracy, some numerical results are presented and compared with the known solution. The answer was confirmative. Numerically accurate solutions were obtained for a fixed Prandtl number, Pr=0.71, for a range of the Reynolds number, Re from 100 to 3,000, and for a range of the Richardson number, Ri from 0.001 to 100. The results from these calculations, using the mixed FE consistent splitting scheme, agreed with the existing ones.

Further extensions of this work could include the influence of various choices of Reynolds numbers, Prandtl numbers and Richardson numbers, and the effect of aspect ratio.

The present work was the first to apply a mixed FE in association with the consistent splitting scheme to the mixed convection problem.