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.     

Free convection of a dusty‐gas flow along a semi‐infinite vertical cylinder

An analysis is performed to study the free convection of a dusty‐gas flow along a semi‐infinite isothermal vertical cylinder. The governing equations of the flow problem are transformed into non‐dimensional form and the resulting nonlinear, coupled parabolic partial differential equations have been solved numerically using an implicit finite difference scheme of Crank–Nicholson type. The flow variables such as gas–velocity, dust‐particle velocity and temperature, shearing stress and heat transfer coefficients are calculated numerically for various parameters occurring in the problem. It is observed that due to the presence of dust particles, the gas velocity is found to decrease. Copyright © 2009 John Wiley & Sons, Ltd.     

Fully non‐linear free‐surface simulations by a 3D viscous numerical wave tank

A finite difference scheme using a modified marker‐and‐cell (MAC) method is applied to investigate the characteristics of non‐linear wave motions and their interactions with a stationary three‐dimensional body inside a numerical wave tank (NWT). The Navier–Stokes (NS) equation is solved for two fluid layers, and the boundary values are updated at each time step by a finite difference time marching scheme in the frame of a rectangular co‐ordinate system. The viscous stresses and surface tension are neglected in the dynamic free‐surface condition, and the fully non‐linear kinematic free‐surface condition is satisfied by the density function method developed for two fluid layers. The incident waves are generated from the inflow boundary by prescribing a velocity profile resembling flexible flap wavemaker motions, and the outgoing waves are numerically dissipated inside an artificial damping zone located at the end of the tank. The present NS–MAC NWT simulations for a vertical truncated circular cylinder inside a rectangular wave tank are compared with the experimental results of Mercier and Niedzwecki, an independently developed potential‐based fully non‐linear NWT, and the second‐order diffraction computation. Copyright © 1999 John Wiley & Sons, Ltd.     

Prediction of unobstructed flow for co‐rotating multi‐disk drive in an enclosure

The flow structure and isotherms of hard disk drives (HDD) are investigated using a finite element method (FEM). The governing equations are based on the three‐dimensional axisymmetric Navier–Stokes partial differential equations (PDEs) with Galerkin FE formulation. Co‐rotating models are selected that include the non‐ventilated configuration within an enclosure. With various operating conditions for the disk system, the following important parameters are considered: disk number (n), rotational speed (Ro), and wall temperature. The flow structure changes rapidly when the rotational Reynolds number (Re_ϕ) is increased. The flow has a greater tendency to flow radially outwards and the swirling velocity tends to be more vertically orientated, especially for high Re_ϕ values. The isotherms only have small varying regions near the rotating axis, forming outward arcs near the wall and inward arcs near the end gap of the disk. Different from the case without the enclosure, the vorticities exist along the outer disk ends. Both the swirling velocity and isotherms indicate nearly symmetrical characteristics, as expected. A higher temperature gradient occurs near the right outer disk ends, which implies the characteristic of higher heat flux. A commercial computational fluid dynamic (CFD) code, CFX‐5, was chosen to validate the results. Copyright © 2001 John Wiley & Sons, Ltd.     

A two‐dimensional vertical non‐hydrostatic σ model with an implicit method for free‐surface flows

An implicit finite difference model in the σ co‐ordinate system is developed for non‐hydrostatic, two‐dimensional vertical plane free‐surface flows. To accurately simulate interaction of free‐surface flows with uneven bottoms, the unsteady Navier–Stokes equations and the free‐surface boundary condition are solved simultaneously in a regular transformed σ domain using a fully implicit method in two steps. First, the vertical velocity and pressure are expressed as functions of horizontal velocity. Second, substituting these relationship into the horizontal momentum equation provides a block tri‐diagonal matrix system with the unknown of horizontal velocity, which can be solved by a direct matrix solver without iteration. A new treatment of non‐hydrostatic pressure condition at the top‐layer cell is developed and found to be important for resolving the phase of wave propagation. Additional terms introduced by the σ co‐ordinate transformation are discretized appropriately in order to obtain accurate and stable numerical results. The developed model has been validated by several tests involving free‐surface flows with strong vertical accelerations and non‐linear waves interacting with uneven bottoms. Comparisons among numerical results, analytical solutions and experimental data show the capability of the model to simulate free‐surface flow problems. Copyright © 2004 John Wiley & Sons, Ltd.     

A novel non‐upwind,interconnected, multi‐grid,overlapping numerical procedure for problems involving fluid flow

A novel numerical procedure for heat, mass and momentum transfer in fluid flow is presented. The new scheme is passed on a non‐upwind, interconnected, multi‐grid, overlapping (NIMO) finite‐difference algorithm. In 2D flows, the NIMO algorithm solves finite‐difference equations for each dependent variable on four overlapping grids. The finite‐difference equations are formulated using the control‐volume approach, such that no interpolations are needed for computing the convective fluxes. For a particular dependent variable, four fields of values are produced. The NIMO numerical procedure is tested against the exact solution of two test problems. The first test problem is an oblique laminar 2D flow with a double step abrupt change in a passive scalar variable for infinite Peclet number. The second test problem is a rotating radial flow in an annular sector with a single step abrupt change in a passive scalar variable for infinite Peclet number. The NIMO scheme produced essentially the exact solution using different uniform and non‐uniform square and rectangular grids for 45 and 30° angle of inclination. All other schemes were unable to capture the exact solution, especially for the rectangular and non‐uniform grids. The NIMO scheme was also successful in predicting the exact solution for the rotating radial flow, using a uniform cylindrical‐polar coordinate grid. Copyright © 2008 John Wiley & Sons, Ltd.     

A depth‐integrated non‐hydrostatic finite element model for wave propagation

This paper presents a two‐dimensional finite element model for simulating dynamic propagation of weakly dispersive waves. Shallow water equations including extra non‐hydrostatic pressure terms and a depth‐integrated vertical momentum equation are solved with linear distributions assumed in the vertical direction for the non‐hydrostatic pressure and the vertical velocity. The model is developed based on the platform of a finite element model, CCHE2D. A physically bounded upwind scheme for the advection term discretization is developed, and the quasi second‐order differential operators of this scheme result in no oscillation and little numerical diffusion. The depth‐integrated non‐hydrostatic wave model is solved semi‐implicitly: the provisional flow velocity is first implicitly solved using the shallow water equations; the non‐hydrostatic pressure, which is implicitly obtained by ensuring a divergence‐free velocity field, is used to correct the provisional velocity, and finally the depth‐integrated continuity equation is explicitly solved to satisfy global mass conservation. The developed wave model is verified by an analytical solution and validated by laboratory experiments, and the computed results show that the wave model can properly handle linear and nonlinear dispersive waves, wave shoaling, diffraction, refraction and focusing. Copyright © 2013 John Wiley & Sons, Ltd.     

Melting heat transfer in the stagnation‐point flow of an upper‐convected Maxwell (UCM) fluid past a stretching sheet

The steady laminar boundary layer flow and heat transfer past a stretching sheet arre considered. Upper‐convected Maxwell (UCM) fluid is treated as a rheological model. The resulting nonlinear differential system is solved by homotopy analysis method (HAM). The influence of melting parameter (M), Prandtl number (Pr), Deborah number (β) and stretching ratio (A = a/c) on the velocity and temperature profiles is thoroughly examined. It is noticed that fields are effected appreciably with the variation of parameters. Furthermore, it is seen that the local Nusselt number is a decreasing function of melting parameter. Copyright © 2011 John Wiley & Sons, Ltd.     

A σ‐coordinate non‐hydrostatic model with embedded Boussinesq‐type‐like equations for modeling deep‐water waves

A σ‐coordinate non‐hydrostatic model, combined with the embedded Boussinesq‐type‐like equations, a reference velocity, and an adapted top‐layer control, is developed to study the evolution of deep‐water waves. The advantage of using the Boussinesq‐type‐like equations with the reference velocity is to provide an analytical‐based non‐hydrostatic pressure distribution at the top‐layer and to optimize wave dispersion property. The σ‐based non‐hydrostatic model naturally tackles the so‐called overshooting issue in the case of non‐linear steep waves. Efficiency and accuracy of this non‐hydrostatic model in terms of wave dispersion and nonlinearity are critically examined. Overall results show that the newly developed model using a few layers is capable of resolving the evolution of non‐linear deep‐water wave groups. Copyright © 2009 John Wiley & Sons, Ltd.     

Viscous dissipative fluid flow past a semi‐infinite vertical plate with variable surface temperature

An analysis of the effect of viscous dissipative heat on two‐dimensional viscous incompressible fluid flow past a semi‐infinite vertical plate with variable surface temperature is carried out. The dimensionless governing equations are unsteady, two‐dimensional, coupled, and non‐linear governing equations. A most accurate, unconditionally stable and fast converging implicit finite‐difference scheme is used to solve the non‐dimensional governing equations. Velocity and temperature of the flow have been presented graphically for various parameters occurring in the problem. The local and average skin friction and Nusselt number are also shown graphically. It is observed that greater viscous dissipative heat causes a rise in the temperature. Copyright © 2007 John Wiley & Sons, Ltd.     

Explicit characteristic‐based finite volume element method for level set equation

Accurate modeling of interfacial flows requires a realistic representation of interface topology. To reduce the computational effort from the complexity of the interface topological changes, the level set method is widely used for solving two‐phase flow problems. This paper presents an explicit characteristic‐based finite volume element method for solving the two‐dimensional level set equation. The method is applicable for the case of non‐divergence‐free velocity field. Accuracy and performance of the proposed method are evaluated via test cases with prescribed velocity fields on structured grids. By given a velocity field, the motion of interface in the normal direction and the mean curvature, examples are presented to demonstrate the performance of the proposed method for calculating interface evolutions in time. Copyright © 2010 John Wiley & Sons, Ltd.     

Two‐dimensional prediction of time dependent,turbulent flow around a square cylinder confined in a channel

This paper presents two‐dimensional and unsteady RANS computations of time dependent, periodic, turbulent flow around a square block. Two turbulence models are used: the Launder–Sharma low‐Reynolds number k–ε model and a non‐linear extension sensitive to the anisotropy of turbulence. The Reynolds number based on the free stream velocity and obstacle side is Re=2.2×10⁴. The present numerical results have been obtained using a finite volume code that solves the governing equations in a vertical plane, located at the lateral mid‐point of the channel. The pressure field is obtained with the SIMPLE algorithm. A bounded version of the third‐order QUICK scheme is used for the convective terms. Comparisons of the numerical results with the experimental data indicate that a preliminary steady solution of the governing equations using the linear k–ε does not lead to correct flow field predictions in the wake region downstream of the square cylinder. Consequently, the time derivatives of dependent variables are included in the transport equations and are discretized using the second‐order Crank–Nicolson scheme. The unsteady computations using the linear and non‐linear k–ε models significantly improve the velocity field predictions. However, the linear k–ε shows a number of predictive deficiencies, even in unsteady flow computations, especially in the prediction of the turbulence field. The introduction of a non‐linear k–ε model brings the two‐dimensional unsteady predictions of the time‐averaged velocity and turbulence fields and also the predicted values of the global parameters such as the Strouhal number and the drag coefficient to close agreement with the data. Copyright © 2009 John Wiley & Sons, Ltd.     

A numerical method to solve the m‐terms of a submerged body with forward speed

To model mathematically the problem of a rigid body moving below the free surface, a control surface surrounding the body is introduced. The linear free surface condition of the steady waves created by the moving body is satisfied. To describe the fluid flow outside this surface a potential integral equation is constructed using the Kelvin wave Green function whereas inside the surface, a source integral equation is developed adopting a simple Green function. Source strengths are determined by matching the two integral equations through continuity conditions applied to velocity potential and its normal derivatives along the control surface. After solving for the induced fluid velocity on the body surface and the control surface, an integral equation is derived involving a mixed distribution of sources and dipoles using a simple Green function and one component of the fluid velocity. The normal derivatives of the fluid velocity on the body surface, namely the m‐terms, are then solved by this matching integral equation method (MIEM). Numerical results are presented for two elliptical sections moving at a prescribed Froude number and submerged depth and a sensitivity analysis undertaken to assess the influence of these parameters. Furthermore, comparisons are performed to analyse the impact of different assumptions adopted in the derivation of the m‐terms. It is found that the present method is easy to use in a panel method with satisfactory numerical precision. Copyright © 2002 John Wiley & Sons, Ltd.     

Slip effects on heat and mass transfer in MHD micropolar fluid flow over an inclined plate with thermal radiation and chemical reaction

The influence of partial slip, thermal radiation, chemical reaction and temperature‐dependent fluid properties on heat and mass transfer in hydro‐magnetic micropolar fluid flow over an inclined permeable plate with constant heat flux and non‐uniform heat source/sink is studied. The transverse magnetic field is assumed as a function of the distance from the origin. Also it is assumed that the fluid viscosity and the thermal conductivity vary as an inverse function and linear function of temperature, respectively. With the use of the similarity transformation, the governing system of non‐linear partial differential equations are transformed into non‐linear ordinary differential equations and are solved numerically using symbolic software MATHEMATICA 7.0 (Wolfram Research, Champaign, IL). The numerical values obtained for the velocity, microrotation, temperature, species concentration, skin friction coefficient and the Nusselt number are presented through graphs and tables for several sets of values of the parameters. The effects of various physical parameters on the flow and heat transfer characteristics are discussed.Copyright © 2011 John Wiley & Sons, Ltd.     

Computation of turbulent free‐surface flows around modern ships

This paper presents the calculated results for three classes of typical modern ships in modelling of ship‐generated waves. Simulations of turbulent free‐surface flows around ships are performed in a numerical water tank, based on the FINFLO‐RANS SHIP solver developed at Helsinki University of Technology. The Reynolds‐averaged Navier–Stokes (RANS) equations with the artificial compressibility and the non‐linear free‐surface boundary conditions are discretized by means of a cell‐centred finite‐volume scheme. The convergence performance is improved with the multigrid method. A free surface is tracked using a moving mesh technology, in which the non‐linear free‐surface boundary conditions are given on the actual location of the free surface. Test cases recommended are a container ship, a US Navy combatant and a tanker. The calculated results are compared with the experimental data available in the literature in terms of the wave profiles, wave pattern, and turbulent flow fields for two turbulence models, Chien's low Reynolds number k–εmodel and Baldwin–Lomax's model. Furthermore, the convergence performance, the grid refinement study and the effect of turbulence models on the waves have been investigated. Additionally, comparison of two types of the dynamic free‐surface boundary conditions is made. Copyright © 2003 John Wiley& Sons, Ltd.     

Unsteady hydromagnetic flow of a reactive variable viscosity third‐grade fluid in a channel with convective cooling

This paper examines the combined effects of a transverse magnetic field and variable viscosity on unsteady flow of a reactive third‐grade electrically conducting fluid and heat transfer in a channel with convective cooling at the surface. It is assumed that the fluid has small electrical conductivity and the electromagnetic force produced is very small. The coupled nonlinear partial differential equations governing the problem are derived and solved numerically using a semi‐implicit finite‐difference scheme. Both numerical and graphical results are presented and physical aspects of the problem are discussed with respect to various parameters embedded in the system. It is in general noted that those parameters that increase/decrase one flow quantity (velocity or temperature) also lead to the increase/decrease respectively of the other quantity. Copyright © 2011 John Wiley & Sons, Ltd.     

An implicit three‐dimensional fully non‐hydrostatic model for free‐surface flows

An implicit method is developed for solving the complete three‐dimensional (3D) Navier–Stokes equations. The algorithm is based upon a staggered finite difference Crank‐Nicholson scheme on a Cartesian grid. A new top‐layer pressure treatment and a partial cell bottom treatment are introduced so that the 3D model is fully non‐hydrostatic and is free of any hydrostatic assumption. A domain decomposition method is used to segregate the resulting 3D matrix system into a series of two‐dimensional vertical plane problems, for each of which a block tri‐diagonal system can be directly solved for the unknown horizontal velocity. Numerical tests including linear standing waves, nonlinear sloshing motions, and progressive wave interactions with uneven bottoms are performed. It is found that the model is capable to simulate accurately a range of free‐surface flow problems using a very small number of vertical layers (e.g. two–four layers). The developed model is second‐order accuracy in time and space and is unconditionally stable; and it can be effectively used to model 3D surface wave motions. Copyright © 2004 John Wiley & Sons, Ltd.     

Thermal radiation effects on the flow and heat transfer in a liquid film on an unsteady stretching sheet

The influence of thermal radiation on the flow and heat transfer within Newtonian liquid film over an unsteady stretching sheet with and without thermocapillarity is examined. The governing non‐linear partial differential equations describing the problem are reduced to a system of nonlinear ordinary differential equations using similarity transformation, which is solved numerically for different values of the thermal radiation parameter and the thermocapillarity parameter. The results show that the dimensionless velocity, the film thickness and the local Nusselt number increase as the thermocapillarity parameter increases, while the free surface temperature decreases with increasing the thermocapillarity parameter. Also, both the dimensionless temperature and the free surface temperature increase and the local Nusselt number decreases as the thermal radiation parameter increases. Copyright © 2010 John Wiley & Sons, Ltd.     

An implicit numerical algorithm for solving non‐hydrostatic free‐surface flow problems

Details are given of the development of a two‐dimensional vertical numerical model for simulating unsteady free‐surface flows, using a non‐hydrostatic pressure distribution. In this model, the Reynolds equations and the kinematic free‐surface boundary condition are solved simultaneously, so that the water surface elevation can be integrated into the solution and solved for, together with the velocity and pressure fields. An efficient numerical algorithm has been developed, deploying implicit parameters similar to those used in the Crank–Nicholson method, and generating a block tri‐diagonal algebraic system of equations. The model has been applied to simulate a range of unsteady flow problems involving relatively strong vertical accelerations. The results show that the numerical algorithm described is able to produce accurate predictions and is also easy to apply. Copyright © 2001 John Wiley & Sons, Ltd.     

Multi‐objective optimization of a rotating cooling channel with staggered pin‐fins for heat transfer augmentation

A rotating channel with staggered pin‐fins is formulated numerically and optimized for performance (heat transfer/required pumping power) using a Kriging meta‐model and hybrid multi‐objective evolutionary algorithm. Two design variables related to cooling channel height, pin diameter, and spacing between the pins are selected for optimization, and two‐objective functions related to the heat transfer and friction loss are employed. A design of experiment is performed, and 20 designs are generated by Latin hypercube sampling. The objective function values are evaluated using a Reynolds‐averaged Navier–Stokes solver, and a Kriging model is constructed to obtain a Pareto‐optimal front through a multi‐objective evolutionary algorithm. Rotation in a cooling channel with staggered pin‐fins induces Coriolis force that causes a heat transfer discrepancy between the trailing (pressure) and leading (suction) surfaces, with a higher Nusselt number on the trailing surface. The tradeoff between the two competing objective functions is determined, and the distribution of the Pareto‐optimal solutions in the design space is discussed through k‐means clustering. In the optimal designs, with a decrease in spacing between the pins, heat transfer is enhanced whereas friction loss is increased. Copyright © 2011 John Wiley & Sons, Ltd.