Rotating incompressible flow with a pressure Neumann condition

This work considers the internal flow of an incompressible viscous fluid contained in a rectangular duct subject to a rotation. A direct velocity–pressure algorithm in primitive variables with a Neumann condition for the pressure is employed. The spatial discretization is made with finite central differences on a staggered grid. The pressure and velocity fields are directly updated without any iteration. Numerical simulations with several Reynolds numbers and rotation rates were performed for ducts of aspect ratios 2:1 and 8:1. Copyright © 2005 John Wiley & Sons, Ltd.     

Simulation in primitive variables of incompressible flow with pressure Neumann condition

A velocity–pressure algorithm, in primitive variables and finite differences, is developed for incompressible viscous flow with a Neumann pressure boundary condition. The pressure field is initialized by least‐squares and updated from the Poisson equation in a direct weighted manner. Simulations with the cavity problem were made for several Reynolds numbers. The expected displacement of the central vortex was obtained, as well as the development of secondary and tertiary eddies. Copyright © 1999 John Wiley & Sons, Ltd.     

Numerical simulation for rotating internal weakly viscoelastic flows in rectangular ducts

The present work develops a numerical method for the solution of rotating internal weakly viscoelastic flows in rectangular ducts for dimensionless parameters such as the Reynolds, Rossby and Weissenberg numbers, taken respectively in the intervals between 171 and 12000, 0.047 and 1/12 and up to 1/10000. It is shown that the usual counter‐rotating double‐vortex configuration of secondary flow breaks down with the increase of the Reynolds number (over the threshold of 171). For higher Reynolds numbers such as 7500 and 12000 the secondary flow diffuses to the interior of the duct where it assumes a fully developed configuration and the transition to the turbulence structure is observed. The Sobolev norms increase almost proportionally to the increase of the Reynolds number, and play an essential role for more complex problems involving transition to turbulence modelling. Copyright © 2002 John Wiley & Sons, Ltd.     

On a boundary condition for pressure‐driven laminar flow of incompressible fluids

We prove in Theorem 1 a new relationship between the stress, pressure, velocity, and mean curvature for embedded surfaces in incompressible viscous flows. This is then used to define a corresponding modified pressure boundary condition for flow of Newtonian and generalized Newtonian fluids. These results agree with an intuitive notion of the flow physics but apparently have not previously been shown rigorously. We describe some of the implementation issues for inflow and outflow boundaries in this context and give details for a penalty treatment of the associated tangential velocity constraint. This is then implemented and applied in high‐resolution 3D benchmark calculations for a representative generalized viscosity model. Copyright © 2007 John Wiley & Sons, Ltd.     

Magnetohydrodynamic flow of a Sisko fluid in annular pipe: A numerical study

This paper presents a numerical study for the unsteady flow of a magnetohydrodynamic (MHD) Sisko fluid in annular pipe. The fluid is assumed to be electrically conducting in the presence of a uniform magnetic field. Based on the constitutive relationship of a Sisko fluid, the non‐linear equation governing the flow is first modelled and then numerically solved. The effects of the various parameters especially the power index n, the material parameter of the non‐Newtonian fluid b and the magnetic parameter B on the flow characteristics are explored numerically and presented through several graphs. Moreover, the shear‐thinning and shear‐thickening characteristics of the non‐Newtonian Sisko fluid are investigated and a comparison is also made with the Newtonian fluid. Copyright © 2009 John Wiley & Sons, Ltd.     

A multigrid finite difference approach to steady flow between eccentric rotating cylinders

A simple finite difference scheme over a non‐uniform grid is proposed to solve the two‐dimensional, steady Navier–Stokes equations. Instead of the Newton method, a more straightforward line search algorithm is used to solve the resultant system of non‐linear equations. By adopting the multigrid methodology, a fast convergence is achieved, at least for low‐Reynolds number flow. This scheme is applied, in particular, to flow between eccentric rotating cylinders. The computed results are shown in good agreement with some analytic findings. Copyright © 2000 John Wiley & Sons, Ltd.     

Numerical investigation of two-phase rotating flow

Finite difference solutions of the two-fluid equations of motion for a particle (droplet)-fluid mixture in a rotating finite axisymmetric cylinder are presented. The numerical method, which can be regarded as an extension of the Harlow & Amsden approach, employs forward time and centred space discretization and treats implicitly the pressure, Coriolis and volume flux terms. The computed flow fields are examined via a detailed comparison to previous analytic approximations, which illuminates both the physical and numerical aspects and the validity of these approximations.     

A computational solution of hydromagnetic‐free convective flow past a vertical plate in a rotating heat‐generating fluid with Hall and ion‐slip currents

Taking Hall and ion‐slip current into account, the unsteady magnetohydrodynamic heat‐generating free convective flow of a partially ionized gas past an infinite vertical plate in a rotating frame of reference is investigated theoretically. A computer program using finite elements is employed to solve the coupled non‐linear differential equations for velocity and temperature fields. The effects of Hall and ion‐slip currents as well as the other parameters entering into the problem are discussed extensively and shown graphically. Copyright © 2006 John Wiley & Sons, Ltd.     

A study on numerical methods for transient rotating flow induced by starting blades

Based on the finite volume method, three methods for rotational region treatment were presented and validated by simulating two-dimensional accelerating rotational flows. Separate transient incompressible flows induced by cross-shaped blades during starting process were simulated using the dynamic mesh, sliding mesh and dynamic reference frame methods. The computing performance and stability of the three methods were evaluated by comparing numerical results, and the transient characteristics of the accelerating rotational flow were analysed numerically. Results showed that the starting process affected the flow structure and transient characteristics of the accelerating rotational flows. The sliding mesh method showed higher computational efficiency and accuracy compared with other methods, and could easily be extended to solve three-dimensional transient flows in hydraulic machineries under transient operations, such as start-up and shutdown.     

Axisymmetric Powell-Eyring fluid flow with convective boundary condition: optimal analysis

The effects of axisymmetric flow of a Powell-Eyring fluid over an impermeable radially stretching surface are presented. Characteristics of the heat transfer process are analyzed with a more realistic condition named the convective boundary condition. Governing equations for the flow problem are derived by the boundary layer approximations. The modeled highly coupled partial differential system is converted into a system of ordinary differential equations with acceptable similarity transformations. The convergent series solutions for the resulting system are constructed and analyzed. Optimal values are obtained and presented in a numerical form using an optimal homotopy analysis method (OHAM). The rheological characteristics of different parameters of the velocity and temperature profiles are presented graphically. Tabular variations of the skin friction coefficient and the Nusselt number are also calculated. It is observed that the temperature distribution shows opposite behavior for Prandtl and Biot numbers. Furthermore, the rate of heating/cooling is higher for both the Prandtl and Biot numbers.     

A high‐order scheme for the incompressible Navier–Stokes equations with open boundary condition

We present a numerical scheme to solve the incompressible Navier–Stokes equations with open boundary condition. After replacing the incompressibility constraint by the pressure Poisson equation, the key is how to give an appropriate boundary condition for the pressure Poisson equation. We propose a new boundary condition for the pressure on the open boundary. Some numerical experiments are presented to verify the accuracy and stability of scheme. Copyright © 2013 John Wiley & Sons, Ltd.     

Numerical study of vortex shedding from a rotating cylinder immersed in a uniform flow field

A numerical study is made of the unsteady two‐dimensional, incompressible flow past an impulsively started translating and rotating circular cylinder. The Reynolds number (Re) and the rotating‐to‐translating speed ratio (α) are two controlled parameters, and the influence of their different combinations on vortex shedding from the cylinder is investigated by the numerical scheme sketched below. Associated with the streamfunction (ψ)–vorticity (ω) formulation of the Navier–Stokes equations, the Poisson equation for ψ is solved by a Fourier/finite‐analytic, separation of variable approach. This approach allows one to attenuate the artificial far‐field boundary, and also yields a global conditioning on the wall vorticity in response to the no‐slip condition. As for the vorticity transport equation, spatial discretization is done by means of finite difference in which the convection terms are handled with the aid of an ENO (essentially non‐oscillatory)‐like data reconstruction process. Finally, the interior vorticity is updated by an explicit, second‐order Runge–Kutta method. Present computations fall into two categories. One with Re=10³ and α≤3; the other with Re=10⁴ and α≤2. Comparisons with other numerical or physical experiments are included. Copyright © 2000 John Wiley & Sons, Ltd.     

Monolithic Newton‐multigrid solution techniques for incompressible nonlinear flow models

We present special Newton‐multigrid techniques for stationary incompressible nonlinear flow models discretized by the high order LBB‐stable Q₂P₁ element pair. We treat the resulting nonlinear and the corresponding linear discrete systems by a fully coupled monolithic approach to maintain high accuracy and robustness, particularly with respect to different rheological behaviors and also regarding different problem sizes and types of nonlinearity. Here, local pressure Schur complement techniques are presented as a generalization of the classical Vanka smoother. The discussed methodology is implemented for the well‐known flow around cylinder benchmark configuration for generalized Newtonian as well as non‐Newtonian flows including non‐isothermal, shear/pressure dependent and viscoelastic effects.Copyright © 2012 John Wiley & Sons, Ltd.     

Improving Eulerian two‐phase flow finite element approximation with discontinuous gradient pressure shape functions

In this paper we present a problem we have encountered using a stabilized finite element method on fixed grids for flows with interfaces modelled with the level set approach. We propose a solution based on enriching the pressure shape functions on the elements cut by the interface. The enrichment is used to enable the pressure gradient to be discontinuous at the interface, thus improving the ability to simulate the behaviour of fluids with different density under a gravitational force. The additional shape function used is local to each element and the corresponding degree of freedom can therefore be condensed prior to assembly, making the implementation quite simple on any existing finite element code. Copyright © 2005 John Wiley & Sons, Ltd.     

Modelling of two‐dimensional laminar flow using finite element method

In this paper, a Galerkin weighted residual finite element numerical solution method, with velocity material time derivative discretisation, is applied to solve for a classical fluid mechanics system of partial differential equations modelling two‐dimensional stationary incompressible Newtonian fluid flow. Classical examples of driven cavity laminar flow and laminar flow past a cylinder are presented. Numerical results are compared with data found in the literature. Copyright © 1999 John Wiley & Sons, Ltd.     

Fourier analysis of an equal‐order incompressible flow solver stabilized by pressure gradient projection

Fourier analysis techniques are applied to the stabilized finite element method (FEM) recently proposed by Codina and Blasco for the approximation of the incompressible Navier–Stokes equations, here denoted by pressure gradient projection (SPGP) method. The analysis is motivated by spurious waves that pollute the computed pressure in start‐up flow simulation. An example of this spurious phenomenon is reported. It is shown that Fourier techniques can predict the numerical behaviour of stabilized methods with remarkable accuracy, even though the original Navier–Stokes setting must be significantly simplified to apply them. In the steady state case, good estimates for the stabilization parameters are obtained. In the transient case, spurious long waves are shown to be persistent when the element Reynolds number is large and the Courant number is small. This can be avoided by treating the pressure gradient projection implicitly, though this implies additional computing effort. Standard extrapolation variants are unfortunately unstable. Comparisons with Galerkin–least‐squares (GLS) method and Chorin's projection method are also addressed. Copyright © 2000 John Wiley & Sons, Ltd.     

Numerical simulation of liquid crystalline polymer flow into two‐dimensional thin cavity moulds

In this paper, flows of liquid crystalline polymers into two‐dimensional thin cavity moulds are simulated. The flows are modelled by Ericksen–Leslie equations of motion in the high viscosity limit. An elliptic pressure equation is derived under Hele–Shaw approximations, and the non‐isothermal natures of the flow are modelled. The equations are solved using the finite‐difference technique. A new boundary‐mapping technique is developed in this study to solve the difficulty in the finite‐difference treatment of arbitrarily shaped boundaries, which possess no natural coordinate system. This new method avoids the difficult mesh control in the body‐fitted mapping process and makes the mapping process easy to implement. It can also solve the problems caused by the uneven distribution of grid nodes in the traditional body‐fitted mapping technique. Copyright © 2008 John Wiley & Sons, Ltd.     

A novel nonreflecting boundary condition for unsteady flow

A novel nonreflecting boundary condition, which converges to the specified time‐dependent boundary condition within any degree of accuracy, is introduced for the numerical simulation of hyperbolic systems and validated against the solution of two fundamental boundary value problems in fluids. First, transonic nozzle flow with backward acoustic disturbance is considered. Using high‐order aeroacoustic numerical schemes, the proposed nonreflecting boundary condition yields results that are in excellent agreement with those obtained using conventional nonreflecting boundary conditions based on the method of characteristics as well as with the results of the exact solution. The novel nonreflecting boundary condition, implemented into a semi‐analytical solution algorithm of unsteady bubbly cavitating nozzle flows, is also validated against results obtained using a Lagrangian finite volume scheme. Copyright © 2013 John Wiley & Sons, Ltd.     

Incompressible fluid flow simulations with flow rate as the sole information at synthetic inflow and outflow boundaries

In numerically simulating heat and mass transport processes in an unconfined domain involving synthetic open (inflow and/or outflow) boundaries, how to properly specify flow conditions at these boundaries can become a challenging issue. In this work, within the context of a pressure‐based finite volume method under an unstructured grid, a solution procedure without the need for explicit specification of flow profiles at any of these boundaries when simulating incompressible fluid flow is proposed and numerically examined. Within this methodology, the flow at any open boundary is not necessarily assumed to be unidirectional or fully developed; indeed, the sole information required is the mass flow rate crossing the boundary. As a result, one can select the specific region of interest to perform simulations, rather than having to artificially increase the flow domain so as to invoke fully developed flow at all open boundaries. This not only greatly reduces computational costs (both in terms of memory requirements and simulation run‐time) but provides the means to engage with flow problems, which otherwise cannot be solved with currently available methods for handling the flow conditions at open boundaries. The proposed methodology is demonstrated by simulating laminar flow of an incompressible fluid in a two‐dimensional planar channel with a 90° T‐branch, a known inflow rate, and flow splits for the two outflow channels. The results obtained by placing the entrance and the two exits at different locations show that the flow behavior predicted is completely unaffected by using a highly truncated domain. Copyright © 2015 John Wiley & Sons, Ltd.     

Free surface tracking with polynomial reconstruction and error correction

A non‐linear method, PREC, for computation of the movement of a free surface is proposed here. The method is composed of three steps: identifying the free surface by using a non‐linear function from the volume fraction matrix, updating the volume fraction matrix using a volume projection method with error correction, and treatment of the results using overshooting or undershooting. Identification of the free surface includes using a polynomial function with 2, 4, or 8 coefficients for one‐, two‐, or three‐dimensional problems, respectively. The polynomial reconstruction involves non‐negligible numerical error. The second advection step includes a linear projection method in space and time. Advection of the volume fraction matrix is computed from the occupying volume of the mesh at the previous time step. At the new time step, the error at each grid point is assumed to be similar to the error at the previous time step and is used for correction. Overshooting or undershooting develops around the free surface mesh points due to the solution's finite time increment. The third step includes truncating the numerical overshooting or undershooting volumes, i.e. isotropic spreading of the excess fluid volumes. The PREC method is evaluated for a one‐dimensional flow case and several two‐dimensional simple flow cases with circular sections (cases include transition parallel to a coordinate, transition with an intersection angle to a coordinate, and rotation). The results from the present method are compared with analytical solutions and results from a donor‐cell VOF method. As a result of these comparisons, the PREC method is validated. Copyright © 2008 John Wiley & Sons, Ltd.