Numerical Solution of the Equations Governing the Steady State of a Thin Cylindrical Web Supported by an Air Cushion

1999 ◽  
Author(s):  
Sinan Müftü
Keyword(s):  
Steady State ◽  
Nonlinear Partial Differential Equations ◽  
Coupled System ◽  
Steady State Solution ◽  
Iteration Scheme ◽  
Two Dimensions ◽  
Navier Stokes ◽  
Air Cushion ◽  
Raphson Method ◽  
The Web

Abstract The numerical method used to solve the coupled nonlinear, partial differential equations (PDEs) representing the interaction between a thin, flexible, cylindrical web and an air cushion at steady state is analyzed. The web deflections are modeled by a cylindrical shell theory that allows moderately large deflections. The airflow is modeled in two-dimensions with a modified form of the Navier-Stokes and mass balance equations that have non-linear source terms. The coupled fluid/structure system is solved numerically in a stacked iteration scheme: The fluid equations are solved using pseudo-compressibility method with artificial viscosity; and the web equations are solved with a modified Newton-Raphson method. The convergence characteristics of the coupled system and the effects of the numerical parameters on the steady state solution are studied by numerical experiments.

The Mechanics of a Helically Wrapped Thin-Shell Supported by an Externally Pressurized Air Cushion

2002 ◽  
Author(s):  
Sinan Mu¨ftu¨
Keyword(s):  
Steady State ◽  
Thin Shell ◽  
Previous Model ◽  
Fluid Structure Interactions ◽  
Oblique Angle ◽  
Equilibrium Equations ◽  
State Equilibrium ◽  
Air Cushion ◽  
The Web

The mechanics of the fluid structure interactions between a flexible web and an externally pressurized air cushion is modeled. The web is wrapped around the porous cylindrical air-reverser at an oblique angle. The air reverser supplies pressurized air into the web/air-reverser clearance. This model is an extension of a previous model and allows the web to be wrapped around the cylinder in a helical fashion. The geometric relations are based on Rongen’s work (1994) and steady state equilibrium equations are developed based on the work of Mu¨ftu¨ and Cole (1999). This paper describes the theory. A case study is presented.

Numerical Analysis of Unstable Flow in Last-Stage Blades of Steam Turbines

2006 ◽  
Author(s):  
J. C. Garci´a ◽  
J. Kubiak ◽  
F. Sierra ◽  
G. Gonza´lez ◽  
G. Urquiza
Keyword(s):  
Steady State ◽  
Pressure Field ◽  
Stokes Equations ◽  
Two Dimensions ◽  
Navier Stokes ◽  
Steam Turbines ◽  
Unstable Flow ◽  
Geometry Model ◽  
Last Stage ◽  
Exit Edge

As well known steam turbines are strongly affected because of vibrations. Unstable vibrations can appear together with steady-state vibrations. We present the results of numerical computations about unstable flow and its interaction on blades of steam turbines, which can lead to unstable modes of vibration. Unstable phenomena appear as a result of interaction of blades with the stream of steam flow where the pressure field provides the force. The analysis centers particularly in the last stage or L-0 of a 110 MW turbine. Navier-Stokes equations are resolved in two dimensions using a commercial program called Fluent based on finite-volume method. A 2-D geometry model was built in order to represent the dimensional aspects of the diaphragm as well as the rotor located in the last stage of the turbine. Periodic boundary conditions were applied to both sides of the blade with the purpose of simplifying the computation avoiding resolve for the whole wheel. The computations were conducted in both modes, steady state and time dependent. The results show the distribution of pressure fields as a function of the distance to the exit edge of the diaphragm blades. Also, the pressure and velocity fields are shown through contours along the flow channel between the diaphragm blades. The paper includes the time-dependence behavior of pressure field. A Fourier analysis is used to determine the characteristic frequencies of the system, based on numerical results.

Analytical solutions for the unsteady MHD rotating flow over a rotating sphere near the equator

Open Physics ◽  
2011 ◽  
Vol 9 (1) ◽  
Author(s):  
Erik Sweet ◽  
Kuppalapalle Vajravelu ◽  
Robert Gorder
Keyword(s):  
Steady State ◽  
Differential Equations ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Nonlinear Partial Differential Equations ◽  
Coupled System ◽  
Polar Coordinates ◽  
Rotating Flow ◽  
Rotating Sphere

AbstractIn this paper we investigate the three-dimensional magnetohydrodynamic (MHD) rotating flow of a viscous fluid over a rotating sphere near the equator. The Navier-Stokes equations in spherical polar coordinates are reduced to a coupled system of nonlinear partial differential equations. Self-similar solutions are obtained for the steady state system, resulting from a coupled system of nonlinear ordinary differential equations. Analytical solutions are obtained and are used to study the effects of the magnetic field and the suction/injection parameter on the flow characteristics. The analytical solutions agree well with the numerical solutions of Chamkha et al. [31]. Moreover, the obtained analytical solutions for the steady state are used to obtain the unsteady state results. Furthermore, for various values of the temporal variable, we obtain analytical solutions for the flow field and present through figures.

Comparison of Transient and Steady-State Behaviors in Unwinding Mechanism

2011 ◽  
Author(s):  
Wan-Suk Yoo ◽  
Kun-Woo Kim ◽  
Deuk-Man An ◽  
Jae-Wook Lee
Keyword(s):  
Steady State ◽  
Tensile Force ◽  
Equations Of Motion ◽  
Finite Difference Methods ◽  
Nonlinear Partial Differential Equations ◽  
Inertial Force ◽  
Steady State Solution ◽  
State Solution ◽  
Resistance Force ◽  
Transient Solution

In this study, the transient analysis of a cable unwinding from a cylindrical spool package is first studied and compared to experiment. Then, a steady-state solution is also compared to transient solution. Cables are assumed to be withdrawn with a constant velocity through a fixed point which is located along the axis of the package. When the cable is flown out of the package, several dynamic forces, such as inertial force, Coriolis force, centrifugal force, tensile force, and fluid-resistance force are acting on the cable. Consequently, the cable becomes to undergo very nonlinear and complex unwinding behavior which is called unwinding balloon. In this paper, to prevent the problems during unwinding such as tangling or cutting, unwinding behaviors of cables in transient state were derived and analyzed. First of all, the governing equations of motion of cables unwinding from a cylindrical spool package were systematically derived using the extended Hamilton’s principles of an open system in which mass is transported at each boundary. And the modified finite difference methods are suggested to solve the derived nonlinear partial differential equations. Time responses of unwinding cables are calculated using Newmark time integration methods. The transient solution is compared to physical experiment, and then the steady-state solution is compared to transient solution.

scholarly journals Conservation Laws for Some Systems of Nonlinear Partial Differential Equations via Multiplier Approach

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Rehana Naz
Keyword(s):  
Conservation Laws ◽  
Stokes Equations ◽  
Nonlinear Partial Differential Equations ◽  
Type System ◽  
Coupled System ◽  
Magnetized Plasma ◽  
Navier Stokes ◽  
Local Conservation ◽  
Navier Stokes Equations ◽  
Dependent Variables

The conservation laws for the integrable coupled KDV type system, complexly coupled kdv system, coupled system arising from complex-valued KDV in magnetized plasma, Ito integrable system, and Navier stokes equations of gas dynamics are computed by multipliers approach. First of all, we calculate the multipliers depending on dependent variables, independent variables, and derivatives of dependent variables up to some fixed order. The conservation laws fluxes are computed corresponding to each conserved vector. For all understudying systems, the local conservation laws are established by utilizing the multiplier approach.

Acceleration Methods for Coarse-Grained Numerical Solution of the Boltzmann Equation

2006 ◽  
Vol 129 (7) ◽  
pp. 908-912 ◽  
Author(s):  
Husain A. Al-Mohssen ◽  
Nicolas G. Hadjiconstantinou ◽  
Ioannis G. Kevrekidis
Keyword(s):  
Steady State ◽  
Steady State Solution ◽  
Boltzmann Kinetic Equation ◽  
Coarse Grained ◽  
Navier Stokes ◽  
Explicit Integration ◽  
Simulation Tools ◽  
Efficient Simulation ◽  
Acceleration Methods ◽  
O 10

We present a coarse-grained steady-state solution framework for the Boltzmann kinetic equation based on a Newton-Broyden iteration. This approach is an extension of the equation-free framework proposed by Kevrekidis and coworkers, whose objective is the use of fine-scale simulation tools to directly extract coarse-grained, macroscopic information. Our current objective is the development of efficient simulation tools for modeling complex micro- and nanoscale flows. The iterative method proposed and used here consists of a short Boltzmann transient evolution step and a Newton-Broyden contraction mapping step based on the Boltzmann solution; the latter step only solves for the macroscopic field of interest (e.g., flow velocity). The predicted macroscopic field is then used as an initial condition for the Boltzmann solver for the next iteration. We have validated this approach for isothermal, one-dimensional flows in the low Knudsen number regime. We find that the Newton-Broyden iteration converges in O(10) iterations, starting from arbitrary guess solutions and a Navier-Stokes based initial Jacobian. This results in computational savings compared to time-explicit integration to steady states when the time to steady state is longer than O(40) mean collision times.

Computational Models in Nano-Bioelectronics: Simulation of Ionic Transport in Voltage Operated Channels

2008 ◽  
Vol 8 (7) ◽  
pp. 3686-3694
Author(s):  
Massimo Longaretti ◽  
Giovambattista Marino ◽  
Bice Chini ◽  
Joseph W. Jerome ◽  
Riccardo Sacco
Keyword(s):  
Computational Models ◽  
Stokes System ◽  
Nonlinear Partial Differential Equations ◽  
Coupled System ◽  
Ionic Transport ◽  
Ionic Channels ◽  
Measured Data ◽  
Navier Stokes ◽  
Electrical Parameters ◽  
Functional Iteration

In this article, a novel mathematical and computational model is proposed for the numerical simulation of Voltage Operated ionic Channels (VOC) in Nano-bioelectronics applications. This is a first step towards a multi-physics description of hybrid bio-electronical devices such as bio-chips. The model consists of a coupled system of nonlinear partial differential equations, comprising a Poisson-Nernst-Planck system to account for electro-chemical phenomena, and a Navier-Stokes system to account for fluid-mechanical phenomena. Suitable functional iteration techniques for problem decoupling and finite element methods for discretization are proposed and discussed. Numerical results on realistic VOCs illustrate the validity of the model and its accuracy by comparison with relevant computed channel equivalent electrical parameters with measured data.

scholarly journals An Error Analysis for the Finite Element Approximation to the Steady-State Poisson-Nernst-Planck Equations

2013 ◽  
Vol 5 (1) ◽  
pp. 113-130 ◽  
Author(s):  
Ying Yang ◽  
Benzhuo Lu
Keyword(s):  
Finite Element ◽  
Steady State ◽  
Nonlinear Partial Differential Equations ◽  
Planck Equation ◽  
Coupled System ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Numerical Examples ◽  
Delta Distribution ◽  
Biomolecular System

AbstractPoisson-Nernst-Planck equations are a coupled system of nonlinear partial differential equations consisting of the Nernst-Planck equation and the electrostatic Poisson equation with delta distribution sources, which describe the electrodiffusion of ions in a solvated biomolecular system. In this paper, some error bounds for a piecewise finite element approximation to this problem are derived. Several numerical examples including biomolecular problems are shown to support our analysis.

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