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This paper aims to analyze the effect of aspect ratio A and aspect velocity ratio a on the bifurcation occurrence phenomena in lid-driven cavity by using finite volume method (FVM) and multigrid acceleration. This study has been performed for certain pertinent parameters; a wide range of the Reynolds number values has been adopted, and aspect ratios ranging from 0.25 to 1 and various velocity ratios from 0.25 to 0.825 have been considered in this investigation. Results show that the transition to the unsteady regime follows the classical scheme of Hopf bifurcation, giving rise to a perfectly periodic state. Flow periodicity has been verified through time history plots for the velocity component and phase-space trajectories as a function of Reynolds number. Velocity profile for special case of a square cavity (A = 1) was found to be in good agreement between current numerical results and published ones. Flow characteristics inside the cavity have been presented and discussed in terms of streamlines and vorticity contours at a fixed Reynolds number (Re = 5,000) for various aspect ratios (a = 0).

The numerical method is based on the FVM and multigrid acceleration.

Computations have been investigated for several Reynolds numbers and aspect ratios A (0.25, 0.5, 0.75, 0.825 and 1). Besides, various velocity ratios (a = 0.25, 0.5, 0.75 and 0.825) at fixed aspect ratios (A = 0.25, 0.5 and 0.75) were considered. It is observed that the transition to the unsteady regime follows the classical scheme of Hopf bifurcation, giving rise to a perfectly periodic state. Flow periodicity is verified through time history plots for velocity components and phase-space trajectories.

The bifurcations between steady and unsteady states are investigated.

The fractional order HIV model has an important role in biological science. To study the HIV model in a better way, the model is presented with the help of Atangana- Baleanu operator which is in Caputo sense. Also, the characteristics of the solutions are described briefly with the help of the advance numerical techniques for the different values of fractional order derivatives. This paper aims to discuss the aforementioned objectives.

In this work, Adams-Bashforth method and Euler method are used to get the solution of the HIV model. These are the important numerical methods. The comparison results also are described with the physical meaning of the solutions of the model.

HIV model is analyzed under the view of fractional and AB derivative in Atangana-Baleanu-Caputo sense. The uniqueness of the solution is proved by using Banach Fixed point. The solution is derived with the help of Sumudu transform. Further, the authors employed fractional Adam-Bashforth method and Euler method to enumerate numerical results. The authors have used several values of fractional orders to present the outcomes graphically. The above calculations have been done with the help of MATLAB (R2016a). The numerical scheme used in the proposed study is valid and fruitful, and the same can be used to explore other real issues.

This investigation can be done for the real data sets.

This paper aims to express the solution of the HIV model in a better way with the effect of non-locality, this work is very useful.

In this work, HIV model is developed with the help of Atangana- Baleanu operator in Caputo sense. By using Banach Fixed point, the authors proved that the solution is unique. Also, the solution is presented with the help of Sumudu transform. The behaviors of the solutions are checked for different values of fractional order derivatives with the physical meaning with help of the Adam-Bashforth method and the Euler method.

In this contribution, the solution of Mass-Spring-Damper Systems in the fractional context by using Caputo derivative and Orthogonal Collocation Method is investigated. For this purpose, different case studies considering constant and periodic sources are evaluated. The dimensional consistency of the model is guaranteed by introducing an auxiliary parameter. The obtained results are compared with those found by using both the analytical solution and the predictor-corrector method of Adams–Bashforth–Moulton type. The influence of the fractional order on the mechanical system is evaluated.

In the present contribution, an extension of the Orthogonal Collocation Method to solve fractional differential equations is proposed.

In general, the proposed methodology was able to solve a classical mechanical engineering problem with different characteristics.

The development of a new numerical method to solve fractional differential equations is the major contribution.

The research is directed at development of an efficient and accurate technique for modelling incompressible free surface flows in which viscous effects may not be neglected. The paper describes the methodology, and gives illustrative results for simple geometries.

The pseudospectral matrix element method of discretisation is selected as the basis for the CFD technique adopted, because of its high spectral accuracy. It is implemented as a means of solving the Navier‐Stokes equations coupled with the modified compressibility method.

The viscous solver has been validated for the benchmark cases of uniform flow past a cylinder at a Reynolds number of 40, and 2D cavity flows. Results for sloshing of a viscous fluid in a tank have been successfully compared with those from a linearised analytical solution. Application of the method is illustrated by the results for the interaction of an impulsive wave with a surface piercing circular cylinder in a cylindrical tank.

The paper demonstrates the viability of the approach adopted. The limitation of small amplitude waves should be tackled in future work.

The results will have particular significance in the context of validating computations from more complex schemes applicable to arbitrary geometries.

The new methodology and results are of interest to the community of those developing numerical models of flow past marine structures.

Variable density flows play an important role in many technological devices and natural phenomena. The purpose of this paper is to develop a robust and accurate method for low Mach number flows with large density and temperature variations.

Low Mach number approximation approach is used in the paper combined with a predictor-corrector method and accurate compact scheme of fourth and sixth order. A novel algorithm is formulated for the projection method in which the boundary conditions for the pressure are implemented in such a way that the continuity equation is fulfilled everywhere in the computational domain, including the boundary nodes.

It is shown that proposed implementation of the boundary conditions considerably improves a solution accuracy. Assessment of the accuracy was performed based on the constant density Burggraf flow and for two benchmark cases for the natural convection problems: steady flow in a square cavity and unsteady flow in a tall cavity. In all the cases the results agree very well with exemplary solutions.

A staggered or half-staggered grid arrangement is usually used for the projection method for both constant and low Mach number flows. The staggered approach ensures stability and strong pressure-velocity coupling. In the paper a high-order compact method has been implemented in the framework of low Mach number approximation on collocated meshes. The resulting algorithm is accurate, robust for large density variations and is almost free from the pressure oscillations.

This study is concerned with the direct numerical simulation (DNS) of a turbulent channel flow by an improved vortex in cell (VIC) method. The paper aims to discuss these issues.

First, two improvements for VIC method are proposed to heighten the numerical accuracy and efficiency. A discretization method employing a staggered grid is presented to ensure the consistency among the discretized equations as well as to prevent the numerical oscillation of the solution. A correction method for vorticity is also proposed to compute the vorticity field satisfying the solenoidal condition. Second, the DNS for a turbulent channel flow is conducted by the improved VIC method. The Reynolds number based on the friction velocity and the channel half width is 180.

It is highlighted that the simulated turbulence statistics, such as the mean velocity, the Reynolds shear stress and the budget of the mean enstrophy, agree well with the existing DNS results. It is also shown that the organized flow structures in the near-wall region, such as the streaks and the streamwise vortices, are favourably captured. These demonstrate the high applicability of the improved VIC method to the DNS for wall turbulent flows.

This study enables the VIC method to perform the DNS for wall turbulent flows.

The purpose of this paper is to describe and analyse a class of finite element fractional step methods for solving the incompressible Navier-Stokes equations. The objective is not to reproduce the extensive contributions on the subject, but to report on long-term experience with and provide a unified overview of a particular approach: the characteristic-based split method. Three procedures, the semi-implicit, quasi-implicit and fully explicit, are studied and compared.

This work provides a thorough assessment of the accuracy and efficiency of these schemes, both for a first and second order pressure split.

In transient problems, the quasi-implicit form significantly outperforms the fully explicit approach. The second order (pressure) fractional step method displays significant convergence and accuracy benefits when the quasi-implicit projection method is employed. The fully explicit method, utilising artificial compressibility and a pseudo time stepping procedure, requires no second order fractional split to achieve second order or higher accuracy. While the fully explicit form is efficient for steady state problems, due to its ability to handle local time stepping, the quasi-implicit is the best choice for transient flow calculations with time independent boundary conditions. The semi-implicit form, with its stability restrictions, is the least favoured of all the three forms for incompressible flow calculations.

A comprehensive comparison between three versions of the CBS method is provided for the first time.

Detailed results of numerical calculations of transient, 2D incompressible flow around and in the wake of a square prism at Re = 100, 200 and 500 are presented. An implicit finite‐difference operator‐splitting method, a version of the known SIMPLEC‐like method on a staggered grid, is described. Appropriate theoretical results are presented. The method has second‐order accuracy in space, conserving mass, momentum and kinetic energy. A new modification of the multigrid method is employed to solve the elliptic pressure problem. Calculations are performed on a sequence of spatial grids with up to 401 × 321 grid points, at sequentially halved time steps to ensure grid‐independent results. Three types of flow are shown to exist at Re = 500: a steady‐state unstable flow and two which are transient, fully periodic and asymmetric about the centre line but mirror symmetric to each other. Discrete frequency spectra of drag and lift coefficients are presented.

The quality of crystals grown from melt depends on the flow field in the melt. To simulate melt conditions, a finite element analysis is performed on flow in a heated cavity under the driving forces of natural convection, thermocapillary effects and rotation. In addition, the gravity field is modulated to simulate a microgravitational environment. The purpose for conducting this research is to determine whether the use of baffles can effectively reduce convection and suppress temperature oscillations. The results show that the baffle is able to suppress convection and reduce the amplitude of the temperature oscillations when placed perpendicular to the modulation direction. Under crystal and crucible rotation, the results with and without baffles are similar. In all cases, baffles did not induce temperature oscillations. From this study, it can be concluded that the effects of baffles on the flow behaviour depends greatly on the direction of gravity modulation and frequency.

To investigate the effect of transient mixed convective flow interaction between circular cylinders and channel walls on heat transfer with three circular cylinders arranged in an isosceles right‐angled triangle within a horizontal channel.

This paper uses a semi‐implicit finite element method to solve the incompressible Navier‐Stokes equation, energy equation and continuity equation in primitive‐variable form by assuming the flow to be two‐dimensional and laminar.

Provides information indicating that the transient streamlines, isotherms, drag coefficient and time‐mean Nusselt number around the surfaces of three cylinders are affected by various gap‐to‐diameter ratio, Reynolds numbers and Grashof numbers. The results show that the maximum value of surface‐ and time‐mean Nusselt number along cylinders exists at S=0.75.

It is limited to two‐dimensional laminar flow for the transient mixed convective flow interaction between circular cylinders and channel walls in a horizontal channel.

A very useful source of information and favorable advice for people is applied to heat exchangers, space heating, power generators and other thermal apparatus.

The results of this study may be of interest to engineers attempting to develop thermal control of thermal apparatus and to researchers interested in the flow‐modification aspects of mixed convection between circular cylinders and channel walls in a horizontal channel.