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A computer model based on the fractional step method is presented for modelling density coupled mass transport in groundwater. Although several models utilising the fractional step method had been developed previously, all were based on the Eulerian solution approach. The model developed by the authors uses the Langrangian approach which has some inherent advantages and disadvantages. The problems associated with the implementation of the fractional step method and techniques by which they were overcome are discussed. The performance of the model is examined and results obtained for standard problems are compared with those from other computer packages.

In the present work, a novel dual time stepping approach is applied to a quasi-implicit (QI) fractional step method and its performance is assessed against the classical versions of the QI procedure for the solution of incompressible Navier-Stokes equations. The paper aims to discuss these issues.

In the proposed method, a local time stepping algorithm is utilised to accelerate the solution to steady state, while the transient solution is recovered through the use of a dual time step. It is demonstrated that, unlike the classical fractional step method, the temporal convergence rate of the proposed method depends solely upon the choice of the time discretisation.

While additional stabilisation is the prerequisite for obtaining higher order accuracy in the standard QI methods, the proposed dual time stepping approach completely eliminates this requirement. In addition, the dual time stepping approach proposed achieves the correct formal accuracy in time for both velocity and pressure. It is also demonstrated that a time accuracy beyond second order for both pressure and velocity is possible. In summary, the proposed dual time approach to QI methods simplifies the algorithm, accelerates solution and achieves a higher order time accuracy.

The dual time stepping removed first order pressure error.

The purpose of this paper is to apply the fractional sub-equation method to research on coupled fractional variant Boussinesq equation and fractional approximate long water wave equation.

The algorithm is implemented with the aid of fractional Ricatti equation and the symbol computational system Mathematica.

New travelling wave solutions, which include generalized hyperbolic function solutions, generalized trigonometric function solutions and rational solutions, for these two equations are obtained.

The algorithm is demonstrated to be direct and precise, and can be used for many other nonlinear fractional partial differential equations. The fractional derivatives described in this paper are in the Jumarie's modified Riemann-Liouville sense.

This paper aims to present briefly a unified fractional step method for fluid dynamics, incompressible solid mechanics and heat transfer calculations. The proposed method is demonstrated by solving compressible and incompressible flows, solid mechanics and conjugate heat transfer problems.

The finite element method is used for the spatial discretization of the equations. The fluid dynamics algorithm used is often referred to as the characteristic‐based split scheme.

The proposed method can be employed as a unified approach to fluid dynamics, heat transfer and solid mechanics problems.

The idea of using a unified approach to fluid dynamics and incompressible solid mechanics problems is proposed. The proposed approach will be valuable in complicated engineering problems such as fluid‐structure interaction and problems involving conjugate heat transfer and thermal stresses.

The purpose of this paper is to analyse algorithms for fluid‐structure interaction (FSI) from a purely algorithmic point of view.

First of all a 1D model problem is selected, for which both the fluid and structural behavior are represented through a minimum number of parameters. Different coupling algorithm and time integration schemes are then applied to the simplified model problem and their properties are discussed depending on the values assumed by the parameters. Both exact and approximate time integration schemes are considered in the same framework so to allow an assessment of the different sources of error.

The properties of staggered coupling schemes are confirmed. An insight on the convergence behavior of iterative coupling schemes is provided. A technique to improve such convergence is then discussed.

All the results are proved for a given family of time integration schemes. The technique proposed can be applied to other families of time integration techniques, but some of the analytical results need to be reworked under this assumption.

The problems that are commonly encountered in FSI can be justified by simple arguments. It can also be shown that the limit at which trivial iterative schemes experience convergence difficulties is very close to that at which staggered schemes become unstable.

All the results shown are based on simple mathematics. The problems are presented so to be independent of the particular choice for the solution of the fluid flow.

The purpose of this paper is to develop and test an implicit scheme, accurate to the second order, for solving full Navier‐Stokes equations for three dimensional problems, using parallel algorithm.

Parallel solution to the 3‐D incompressible full Navier‐Stokes equations is presented, based on two fractional steps in time and finite element in space. The accuracy of the scheme is second order in both time and space domains. Large time‐step sizes, with Courant‐Friedrichs‐Lewy (CFL) numbers much larger than unity, are taken since the momentum equation is solved implicitly. A fourth order artificial viscosity term is added. In order to stabilize the numerical solution, fourth order artificial viscosity term is used for high Reynolds number flows. The domain decomposition technique is implemented for parallel solution to the problem with matching and non‐overlapping sub‐domains. It is aimed to study both a 3D free and mixed convection problems using the developed scheme. The segregate solution for temperature field is calibrated by a 3‐D free convection problem. Then the flow case where the forced convection is one order of magnitude higher than the free convection is studied.

It is observed that the long time solution to the flow field shows oscillatory behaviour as the Reynolds number of the flow doubled while keeping the ratio of the forced to free convection fixed. The solution using a parallel algorithm gives satisfactory results, in terms of computation time and accuracy, for the natural convection problem in cubic cavity, and, the forced cooling of a room with chilled ceiling having a parabolic geometry as presented at the end. It is observed that doubling the Reynolds number, while keeping all the parameters unchanged, varies the flow behaviour completely.

A code previously developed and published by the author only solved momentum equation and studied the velocity field. In this study, full Navier Stokes equation is solved and the code is calibrated with a well‐known 3D free‐convection for two different Rayleigh number cases and then 3D mixed convection problem is studied for two cases. Re=2000 case results, solved both by the scheme in this study and by commercial code, presented an interesting physics of the problem. For Re=2000 case, continuous cooling of the room is not possible. Doubling the Reynolds number, raising it from 1000 to 2000, while keeping all the parameters unchanged, varies the flow behaviour completely.

This study proposes to develop and investigate different iterative solvers for non‐Newtonian flow equations.

Existing approaches for the time discretization of the flow equation and for an iterative solution of the discrete systems are discussed. Ideas for further development of existing preconditioners are proposed, implemented and investigated numerically.

A two‐level preconditioning, consisting of a transformation of the original system in the first step and subsequent preconditioning of the transformed system is suggested. The GMRES iterative method, which usually performs well when applied to academic problems, showed dissatisfactory performance for the type of industrial flow simulations investigated in this work. It was found that the BiCGStab method performed best in the tests presented here.

The iterative solvers considered here were investigated only for a certain class of polymer flows. More detailed studies for other non‐Newtonian flows should be carried out.

The work presented in this paper fills a gap related to the usage of efficient iterative methods for non‐Newtonian flow simulations.

In part I uses an iterative point successive over‐relaxation (PSOR) finite difference scheme to solve the coupled unsteady Navier‐Stokes and energy equations for incompressible, viscous and laminar flows in their primitive variable form. Presents the details concerning the derivation of the solution scheme, as well as details on its computer implementation. For validation purposes, includes the results of the two‐dimensional and three‐dimensional benchmark problem of natural convection in a cavity with differentially heated vertical walls. Benchmark computations have been performed for a Prandtl number of 0.71, and different values of the Rayleigh number ranging between 10³ and 10⁶ depending on the problem. By comparison with other approaches in the literature, the scheme has been found to be accurate even for large Rayleigh numbers.

The numerical simulation of two‐dimensional incompressible complex flows of viscoelastic fluids is presented. The context is one, relevant to the food industry (dough kneading), of stirring within a cylindrical vessel, where stirrers are attached to the lid of the vessel. The motion is driven by the rotation of the outer vessel wall, with various stirrer locations. With a single stirrer, both a concentric and an eccentric configuration are considered. A double‐stirrer eccentric case, with two symmetrically arranged stirrers, is also contrasted against the above. A parallel numerical method is adopted, based on a finite element semi‐implicit time‐stepping Taylor‐Galerkin/pressure‐correction scheme. For viscoelastic fluids, constant viscosity Oldroyd‐B and two shear‐thinning Phan‐Thien/Tanner constitutive models are employed. Both linear and exponential models at two different material parameters are considered. This permits a comparison of various stress, shear and extensional properties and their respective influences upon the flow fields generated. Variation with increasing speed of vessel and change in mixer geometry are analysed with respect to the flow kinematics and stress fields produced. Optimal kneading scenarios are commended with asymmetrical stirrer positioning, one‐stirrer proving better than two. Then, models with enhanced strain‐hardening, amplify levels of localised maxima in rate‐of‐work done per unit power consumed. Simulations are conducted via distributed parallel processing, performed on work‐station clusters, employing a conventional message passing protocol (PVM). Parallel results are compared against those obtained on a single processor (sequential computation). Ideal linear speed‐up with the number of processors has been observed.

Turbomachinery, including pumps, are mainly designed to extract/produce energy from/to the flow. A major challenge in the numerical simulation of turbomachinery is the inlet flow rate, which is routinely treated as a known boundary condition for simulation purposes but is properly a dependent output of the solution. As a consequence, the results from numerical simulations may be erroneous due to the incorrect specification of the discharge flow rate. Moreover, the transient behavior of the pumps in their initial states of startup and final states of shutoff phases has not been studied numerically. This paper aims to develop a coupled procedure for calculating the transient inlet flow rate as a part of the solution via application of the control volume method for linear momentum. Large eddy simulation of a four-blade axial hydraulic pump is carried out to calculate the forces at every time step. The sharp interface immersed boundary method is used to resolve the flow around the complex geometry of the propeller, stator and the pipe casing. The effect of the spurious pressure fluctuations, inherent in the sharp interface immersed boundary method, is damped by local time-averaging of the forces. The developed code is validated by comparing the steady-state volumetric flow rate with the experimental data provided by the pump manufacturer. The instantaneous and time-averaged flow fields are also studied to reveal the flow pattern and turbulence characteristics in the pump flow field.

The authors use control volume analysis for linear momentum to simulate the discharge rate as part of the solution in a large eddy simulation of an axial hydraulic pump. The linear momentum balance equation is used to update the inlet flow rate. The sharp interface immersed boundary method with dynamic Smagorinsky sub-grid stress model and a proper wall model is used.

The steady-state volumetric flow rate has been computed and validated by comparing to the flow rate specified by the manufacturer at the simulation conditions, which shows a promising result. The instantaneous and time averaged flow fields are also studied to reveal the flow pattern and turbulence characteristics in the pump flow field.

An approach is proposed for computing the volumetric flow rate as a coupled part of the flow solution, enabling the simulation of turbomachinery at all phases, including the startup/shutdown phase. To the best of the authors’ knowledge, this is the first large eddy simulation of a hydraulic pump to calculate the transient inlet flow rate as a part of the solution rather than specifying it as a fixed boundary condition. The method serves as a numerical framework for simulating problems incorporating complex shapes with moving/stationary parts at all regimes including the transient start-up and shut-down phases.