Ovsyannikov plane vortex: The equations of the submodel

A submodel of the equations of ideal magnetohydrodynamics is constructed that generalizes the classical motion of an ideal continuous medium with plane waves. It is shown that, in contrast to classical motion, in this submodel the velocity and magnetic-field vectors can change direction in a plane orthogonal to a distinguished spatial direction. The submodel is described by a system of equations with two independent variables and a finite relation specifying the orientation of the vector fields in space. The solutions of the submodel define substantially spatial processes and singularities in the motion of continuous media which cannot be studied in the classical one-dimensional formulation. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 27–40, September–October, 2008.     

Galilean-invariant axisymmetric self-similar submodel of gas dynamics without swirling

This paper deals with one insufficiently studied submodel of invariant solutions of rank 1 of the equations of gas dynamics. It is shown that, in cylindrical coordinates, the submodel without swirling reduces to a system of two ordinary differential equations. For the equation of state with additional invariance, a self-similar system is obtained. A pattern of phase trajectories is constructed, and particle motion is studied using asymptotic methods. The obtained solutions describe unsteady flows over axisymmetric bodies with possible strong discontinuities. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 46–52, March–April, 2009.     

Submodel of rotational motions in gas dynamics

An invariant submodel of the equations of gas dynamics constructed on a one-dimensional subalgebra consisting of the sum of operators of rotation and translation in time is studied within the framework of the SUBMODELS program. The system of equations of the submodel is brought to symmetric form. Hyperbolicity conditions for the system are derived. Group analysis is performed and an invariant solution is considered. Isobaric flows are listed. For the simplest of them, characteristics and strong discontinuities are considered. Necessary conditions for existence of solutions without singularities on the axis are derived. Institute of Mechanics, Ural Scientific Center, Russian Academy of Sciences, Ufa 450000. Translated from Prikladnya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 6, pp. 37–45, November–December, 1998.     

Convection mixte en régime transitoire de couche limite laminaire sur une plaque verticale

This work deals with transient laminar boundary layer along a vertical surface and system of equations is solved using finite difference implicite scheme. We show that the nature of the plate influence thermal and dynamical boundary layer thicknesses as well as the flow velocity. Moreover, we observe that a small perturbation of the velocity implies to flow laminar instabilities.     

The PROSPER General Circulation Model,an adapted form of the Sandia Ocean Modelling method: A numerical description

The PROSPER General Circulation Model (PGCM) is a three-dimensional model based on the incompressible Navier-Stokes equations, an equation of state and the heat equation. The hydrostatic approximation and the rigid lid approximation are used. The system of equations is converted into an equivalent form in which the surface pressure is more directly expressed in terms of a two-dimensional Poisson equation. The finite difference method is described and analysed. In particular, the iteration method within every time step to determine the new surface pressure and velocity components, and numerical diffusion aspects due to the use of the staggered Arakawa-C grid are looked at. Since part of the development of the PGCM code is a result of studying the Sandia Ocean Modelling System (SOMS), a comparison is made with respect to the concepts used in both models.     

Vortex Scattering of Monatomic Gas Along Plane Curves

An invariant submodel of gas dynamics equations constructed on a three-dimensional subalgebra with a projective operator for the case of monatomic gas is under consideration. The submodel is reduced to an Abel equation, with integral curves constructed for it. For a separatrix of a saddle, an approximate solution is studied. Such solutions describe the vortex scattering of gas along plane curves placed on the surface of revolution.     

Effects of thermal dispersion and stratification on non-darcy mixed convection from a vertical plate in a porous medium

A boundary layer analysis is presented for the mixed convection from a vertical plate embedded in a porous medium. The effects of thermal dispersion and stratification on the flow and temperature fields are investigated. The conservation equations that govern the problem are reduced to a system of nonlinear ordinary differential equations. The resulting equations were solved on the basis of the local similarity approach. Received on 12 February 1998     

Numerical Simulation of Pulverized Coal MILD Combustion Using a New Heterogeneous Combustion Submodel

The scope of this investigation is the application and analysis of a recently developed submodel (Schulze et al., Oil Gas Science Technol, 2013, doi:10.2516/ogst/2012069) for char particle combustion and gasification. The distinguishing feature of this model is a detailed representation of the diffusion and convection processes as well as the homogeneous reactions in the boundary layer around the char particle. These processes are fully coupled to the heterogeneous particle kinetics. The model was implemented into the CFD code ANSYS-Fluent. The coupled solver is used for simulating the IFRF full scale pulverized coal combustion MILD furnace, for which detailed experimental data are available for model evaluation (Orsino et al., IFRF Doc. No F46/y/3, 2000) The new model yields improved agreement with measured data as compared to the standard modeling approach. This can be directly related to the prediction of the char burnout rate. For further analysis, the mixing field in the IFRF furnace is investigated in detail by introducing four mixture fractions for pyrolysis products, char burn-off gases, primary and secondary air, respectively. The solutions of the respective transport equations are used to define the local stoichiometry both in the gas phase and on the particle surface in such a multi-stream system. The conditions in the particle surrounding gas phase as well as on the particle surface are used to define the regime of particle-gas interaction based on the simulations with the new submodel. It can be shown that for certain conditions the homogeneous reactions in the particle boundary must be accounted for.     

Calculation of vertical velocity in three-dimensional,shallow water equation,finite element models

Computation of vertical velocity within the confines of a three-dimensional, finite element model is a difficult but important task. This paper examines four approaches to the solution of the overdetermined system of equations arising when the first-order continuity equation is solved in conjunction with two boundary conditions. The traditional (TRAD) method neglects one boundary condition, solving the continuity equation with the remaining boundary condition. The vertical derivative of continuity (VDC) method involves solution of the second-order equation obtained by differentiation of the continuity equation with respect to the vertical co-ordinate. The least squares (LS) method minimizes the residuals of the continuity equation (in discrete form) and the two boundary conditions. The adjoint (ADJ) method minimizes the residuals of the continuity equation (in continuous form) and the two boundary conditions. Two domains are considered: a quarter-annular harbour and the southwest coast of Vancouver Island. Results indicate that the highest-quality solution is obtained with both LS and ADJ. Furthermore, ADJ requires less CPU and memory than LS. Therefore the optimal method for computation of vertical velocity in a three-dimensional finite element model is the adjoint (ADJ) method. © 1997 John Wiley & Sons, Ltd.     

Similarity analysis in magnetohydrodynamics: Hall effects on free convection flow and mass transfer past a semi-infinite vertical flat plate

Heat and mass transfer along a semi-infinite vertical flat plate under the combined buoyancy force effects of thermal and species diffusion is investigated in the presence of a strong non-uniform magnetic field and the Hall currents are taken into account. The induced magnetic field due to the motion of the electrically conducting fluid is negligible. This assumption is valid for a small magnetic Reynolds number. The similarity solutions are obtained using the scale group of transformations. These are the only symmetry transformations admitted by the field equations. The non-linear boundary layer equations with the boundary conditions are transferred to a system of non-linear ordinary differential equations with the appropriate boundary conditions. Furthermore, the similarity equations are solved numerically by using a fourth order Runge-Kutta scheme with the shooting method. Numerical results for the velocity profiles, the temperature profiles and the concentration profiles are presented graphically for various values of the magnetic parameter M in the range of 0-1 with the Hall parameter m taking the values 0.5, 1, 2, and 3.     

Boundary effect on free convection flow in a porous medium at given heat transfer from a vertical surface

The system of momentum and energy conservation equations governing free convection flow near a vertical surface in a semi-infinite porous medium subject to the boundary conditions of the third kind is considered in the boundary layer approximation. Asymptotic expansions in the powers of the Darcy parameter Da are constructed. The dependence of flow parameters on the parameters Da and xGr is studied. Analytical solutions applicable throughout the entire flow region are constructed.     

Effect of double dispersion on non-Darcy mixed convective flow over vertical surface embedded in porous medium

A numerical study of a non-Darcy mixed convective heat and mass transfer flow over a vertical surface embedded in a dispersion, melting, and thermal radiation is porous medium under the effects of double investigated. The set of governing boundary layer equations and the boundary conditions is transformed into a set of coupled nonlinear ordinary differential equations with the relevant boundary conditions. The transformed equations are solved numerically by using the Chebyshev pseudospectral method. Comparisons of the present results with the existing results in the literature are made, and good agreement is found. Numerical results for the velocity, temperature, concentration profiles, and local Nusselt and Sherwood numbers are discussed for various values of physical parameters.     

A qualitative analysis of the brachistochrone problem with dry friction and maximizing the horizontal range

The problem of maximizing the horizontal coordinate of a point moving in a vertical plane under the action of gravity and dry friction and the corresponding brachistochrone problem are considered. The optimal control problem is reduced to a boundary value problem for a system of two nonlinear differential equations. A qualitative analysis of the trajectories of this system is carried out, their typical features are found and illustrated by numerical solving of the boundary value problem. It is shown that the normal component of the support reaction should be positive when moving along the optimal curve. The optimality of the found extremals is discussed.     

Scaling Transformation for the Effect of Temperature-Dependent Fluid Viscosity with Thermophoresis Particle Deposition on MHD-Free Convective Heat and Mass Transfer Over a Porous Stretching Surface

This article concerns with a steady two-dimensional flow of an electrically conducting incompressible fluid over a vertical stretching sheet. The flow is permeated by a uniform transverse magnetic field. The fluid viscosity is assumed to vary as a linear function of temperature. A scaling group of transformations is applied to the governing equations. The system remains invariant due to some relations among the parameters of the transformations. After finding three absolute invariants, a third-order ordinary differential equation corresponding to the momentum equation, and two second-order ordinary differential equations corresponding to energy and diffusion equations are derived. The equations along with the boundary conditions are solved numerically. It is found that the decrease in the temperature-dependent fluid viscosity makes the velocity to decrease with the increasing distance of the stretching sheet. At a particular point of the sheet, the fluid velocity decreases with the decreasing viscosity but the temperature increases in this case. Impact of thermophoresis particle deposition in the presence of temperature-dependent fluid viscosity plays an important role on the concentration boundary layer. The results, thus, obtained are presented graphically and discussed.     

On the numerical solution of the turbulence energy equations for wave and tidal flows

This paper deals with the numerical solution, using finite difference methods, of the hydrodynamic and turbulence energy equations which describe wind wave and tidally induced flow. Calculations are performed using staggered and non-staggered finite difference grids in the vertical, with various time discretizations of the production and dissipation terms in the turbulence energy equations. It is shown that the time discretization of these terms can significantly influence the stability of the solution. The effect of time filtering on the numerical stability of the solution is also considered. The form of the mixing length is shown to significantly influence the bed stress in wind wave problems. A no-slip condition is applied at the sea bed, and the associated high-shear bottom boundary layer is resolved by transforming the equations onto a logarithmic or log-linear co-ordinate system before applying the finite difference scheme. A computationally economic method is developed which remains stable even when a very fine vertical grid (over 200 points) is used with a time step of up to 30 min.     

The Euler Limit of the Navier‐Stokes Equations, and Rotating Fluids with Boundary

In this paper we study the convergence of weak solutions of the Navier-Stokes equations in some particular domains, with different horizontal and vertical viscosities, when they go to zero with different speeds. The difficulty here comes from the Dirichlet boundary conditions. Precisely we show that if the ratio of the vertical viscosity to the horizontal viscosity also goes to zero, then the solutions converge to the solution of the Euler system. We study the same limit for rotating fluids with Rossby number also going to zero. (Accepted March 20, 1997)     

Corrigendum of “Similarity analysis in magnetohydrodynamics: Hall effects on free convection flow and mass transfer past a semi-infinite vertical flat plate” [International Journal of Non-Linear Mechanics 38 (2003) 513–520]

The problem of steady, laminar mixed convection heat and mass transfer past a semi-infinite vertical plate in the presence of Hall current has been studied. The governing partial differential equations describing the problem are transformed to a system of non-linear ordinary differential equations with appropriate boundary conditions using Lie's method of infinitesimal transformation groups. The non-linear ordinary differential equations are solved numerically using Chebyshev spectral method. The effects of various parameters on the velocity profiles, temperature and concentration profiles are presented and discussed. This work is an extension and correction for the paper by Megahed et al. [1], published in International Journal of Non-Linear Mechanics 38 (2003) 513–520.     

Mixed convection boundary layer flow near stagnation-point on vertical surface with slip

This paper considers the steady mixed convection boundary layer flow of a viscous and incompressible fluid near the stagnation-point on a vertical surface with the slip effect at the boundary. The temperature of the sheet and the velocity of the external flow are assumed to vary linearly with the distance from the stagnation-point. The governing partial differential equations are first transformed into a system of ordinary differential equations, which are then solved numerically by a shooting method. The features of the flow and heat transfer characteristics for different values of the governing parameters are analyzed and discussed. Both assisting and opposing flows are considered. The results indicate that for the opposing flow, the dual solutions exist in a certain range of the buoyancy parameter, while for the assisting flow, the solution is unique. In general, the velocity slip increases the heat transfer rate at the surface, while the thermal slip decreases it.     

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.     

Lie group analysis for the effect of temperature-dependent fluid viscosity with thermophoresis on magnetohydrodynamic free convective heat and mass transfer over a porous stretching surface

This article concerns with a steady two-dimensional flow of an electrically conducting incompressible fluid over a vertical stretching sheet. A scaling group of transformations is applied to the governing equations. The system remains invariant due to some relations among the parameters of the transformations. Impact of thermophoresis particle deposition in the presence of temperature-dependent fluid viscosity plays an important role on the concentration boundary layer. The results thus obtained are presented graphically and discussed.