Simulation of a simple reacting gas flow with an upwind scheme

Good results have been obtained using the Random Choice Method (RCM) in the computation of reacting gas flow problems. The RCM is an unfamiliar method and difficult to program. The question arises as to whether a simpler difference approximation can obtain as effective results with less computational difficulty. Among all difference schemes upwind methods have been proven to have excellent properties. Thus, such methods serve as models for the effectiveness of all difference schemes.A standard upwind scheme modified to include a fractional heat conduction step is used to compute solutions of one dimensional compressible fluid flow equations with a finite heat conduction coefficient. The gas is assumed to be chemically reacting and thus to deposit energy in the field. Comparison is made to the known qualitative behavior of the solutions for different ratios of the reaction rate and the heat conduction coefficient. This difference scheme is seen to compare unfavorably with the RCM.     

Analytical solution for suction and injection flow of a viscoplastic Casson fluid past a stretching surface in the presence of viscous dissipation

In this study, steady two-dimensional flow of a viscoplastic Casson fluid past a stretching surface is considered under the effects of thermal radiation and viscous dissipation. Both suction and injection flows situations are considered. The partial differential governing equations are transformed into ordinary differential equations and solved analytical. Analytical solutions for velocity and temperature are obtained in terms of hypergeometric function and discussed graphically. Moreover, numerical results are also obtained by Runge–Kutta–Fehlberg fourth–fifth-order (RKF45) method and compared with the analytical results. The results showed that the injection and suction parameter can be used to control the direction and strength of flow. The effects of Casson parameter on the temperature and velocity are quite opposite. The effects of thermal radiation on the temperature are much more stronger in case of injection. The heat transfer coefficient shows higher value for Casson fluid while for Newtonian fluid is the lowest.

     

The λ-scheme

A scheme for integrating the Euler equations of compressible flow in any hyperbolic case is presented. The scheme relies on the concept of characteristics but is strictly a finite difference scheme. Improvements is accuracy and physical consistence due to the scheme are discussed and results of its application to complex flows are shown.     

Rotordynamic coefficients in stepped labyrinth seals

The continuity and circumferential momentum equations are given for compressible flow in a stepped labyrinth seal. Periodic and analytic solutions of these equations are obtained for the case of time dependent flow generated by a non-axisymmetric rotation of the rotor. Since it is necessary for the lateral stability analysis of the rotor, rotordynamic coefficients are then calculated.     

Thermal Investigation for Solid Rocket Motor Nozzle Inserts Material

A time-resolved numerical computational approach, involving the combustion of double-base propellant is performed on thermal protection material for SRM nozzle. An implicit Navier-Stokes Solver is selected to simulate two-dimensional axial-symmetric unsteady turbulent flow of compressible fluid. The governing equations are discredited by using the finite Volume method. S-A turbulence model is employed. CFD scheme is implemented to investigate the temperature distribution causes at nozzle throat inserts comp...     

Finite element analysis of viscous,incompressible fluid flow: Part 1 : Basic methodology

A numerical procedure is developed for the analysis of general two-dimensional flows of viscous, incompressible fluids using the finite element method. The partial differential equations describing the continuum motion of the fluid are discretized by using an integral energy balance approach in conjunction with the finite element approximation. The nonlinear algebraic equations resulting from the discretization process are solved using a Picard iteration technique.A number of computational procedures are developed that allow significant reductions to be made in the computational effort required for the analysis of many flow problems. These techniques include a coarse-to-fine-mesh rezone procedure for the detailed study of regions of particular interest in a flow field and a special finite element to model far-field regions in external flow problems.     

Behavior of micro-polar flow due to linear stretching of porous sheet with injection and suction

The boundary layer flow of a micro-polar fluid due to a linearly stretching sheet is investigated. The influence of various flow parameters like ‘suction and injection velocity through the porous surface’, ‘viscosity parameter causing the coupling of the micro-rotation field and the velocity field’ and ‘vortex viscosity parameter’ on ‘shear stress at the surface’, ‘fluid velocity’ and ‘micro-rotation’ are studied. The governing equations of the transformed boundary layer are solved analytically using homotopy analysis method (HAM). The convergence of the obtained series solutions is explicitly studied and a proper discussion is given for the obtained results. Comparison between the HAM and numerical solutions showed excellent agreement.     

A subgrid stabilization finite element method for incompressible magnetohydrodynamics

This paper studies a numerical scheme for approximating solutions of incompressible magnetohydrodynamic (MHD) equations that uses eddy viscosity stabilization only on the small scales of the fluid flow. This stabilization scheme for MHD equations uses a Galerkin finite element spatial discretization with Scott-Vogelius mixed finite elements and semi-implicit backward Euler temporal discretization. We prove its unconditional stability and prove how the coarse mesh can be chosen so that optimal convergence can be achieved. We also provide numerical experiments to confirm the theory and demonstrate the effectiveness of the scheme on a test problem for MHD channel flow.     

Application of a posteriori error estimation to finite element simulation of compressible Navier-Stokes flow

In this paper, we discuss how to improve the adaptive finite element simulation of compressible Navier-Stokes flow via a posteriori error estimate analysis. We use the moving space-time finite element method to globally discretize the time-dependent Navier-Stokes equations on a series of adapted meshes. The generalized compressible Stokes problem, which is the Stokes problem in its most generalized form, is presented and discussed. On the basis of the a posteriori error estimator for the generalized compressible Stokes problem, a numerical framework of a posteriori error estimation is established corresponding to the case of compressible Navier-Stokes equations. Guided by the a posteriori errors estimation, a combination of different mesh adaptive schemes involving simultaneous refinement/unrefinement and point-moving are applied to control the finite element mesh quality. Finally, a series of numerical experiments will be performed involving the compressible Stokes and Navier-Stokes flows around different aerodynamic shapes to prove the validity of our mesh adaptive algorithms.     

Uniformly Effective Numerical Methods for Hyperbolic Systems

A unified method to compute compressible and incompressible flows is presented. Accuracy and efficiency do not degrade as the Mach number tends to zero. A staggered scheme solved with a pressure correction method is used. The equation of state is arbitrary. A Riemann problem for the barotropic Euler equations with nonconvex equation of state is solved exactly and numericaly. A hydrodynamic flow with cavitation in which the Mach number varies between 10⁻³ and 20 is computed. Unified methods for compressible and incompressible flows are further discussed for the flow of a perfect gas. The staggered scheme with pressure correction is found to have Mach-uniform accuracy and efficiency, and for the fully compressible case the accuracy is comparable with that of established schemes for compressible flows. Received October 20, 1999; revised May 26, 2000     

Non-coaxial rotating flow of viscous fluid with heat and mass transfer

Heat and mass transfer in unsteady non-coaxial rotating flow of viscous fluid over an infinite vertical disk is investigated. The motion in the fluid is induced due to two sources. Firstly, due to the buoyancy force which is caused because of temperature and concentration gradients. Secondly, because of non-coaxial rotation of a disk such that the disk executes cosine or since oscillation in its plane and the fluid is at infinity. The problem is modeled in terms of coupled partial differential equations with some physical boundary and initial conditions. The dimensionless form of the problem is solved via Laplace transform method for exact solutions. Expressions for velocity field, temperature and concentration distributions are obtained, satisfying all the initial and boundary conditions. Skin friction, Nusselt number and Sherwood number are also evaluated. The physical significance of the mathematical results is shown in various plots and is discussed for several embedded parameters. It is found that magnitude of primary velocity is less than secondary velocity. In limiting sense, the present solutions are found identical with published results.

     

A mathematical model of laminar axisymmetrical natural gas flames

A computational procedure for determining the velocities, temperatures and species concentrations in a laminar Bunsen type flame is presented. The boundary layer form of the Navier-Stokes equations with coupled chemistry are solved for a compressible, viscous and axisymmetric flow. An implicit finite difference method is used in the solutions. Velocities, temperatures and stable species concentrations are compared with experimental data.     

A pressure-enthalpy finite element model for simulating hydrothermal reservoirs

A numerical model is presented for simulating single or two-phase flow and energy transport in hydrothermal reservoirs. The model is formulated via two non-linear equations for fluid pressure and enthalpy. Both equations are solved simultaneously using a new finite element technique which employs asymmetric weighting functions to overcome numerical oscillation. Non-linearity is treated by a modified Newton-Raphson scheme which takes into account derivative discontinuities in the non-linear coefficients. This scheme also treats unknown flux boundary conditions inplicitly, thus allowing larger time steps to be taken without inducing instability. The proposed model is applied to two test examples involving one-dimensional flow in both hot water and steam dominated reservoirs. Results indicate that the numerical technique presented is efficient and the model can be used to simulate both types of reservoirs.     

A finite element numerical solution of natural convection in enclosed cavities

Steady state free convective flow enclosed within a cavity and subjected to a temperature gradient is predicted using the finite element method. The matrix equations resulting from the finite element discretisation and formulation are solved using both an iterative and a modified Newton-Raphson scheme. An assessment of the variation in the characteristics of the flow regime is made in association with the dimensionless Prandtl and Rayleigh numbers. A further parameter of interest in such problems is the cavity aspect ratio. The upper limit for the Rayleigh number (based on cavity width) presented in the present paper is 10⁷. The flow patterns are obtained for Prandtl numbers in the range 10^?2 ? Pr ? 10³ and for aspect ratios 1, 10, 20. Where possible the results are compared with existing solutions obtained using the finite difference method. A satisfactory correlation exists where such comparisons can be made. The results complement and extend those obtained during previous theoretical and numerical investigations.     

A triangular finite difference scheme for large deflection problems

A lumped triangular element formulation is developed based on a finite difference approach for the large deflection analysis of plates and shallow shells. The presented formulation is independent of the boundary condition (unlike the finite difference formulation) and uses energy principles to derive a set of nonlinear algebraic equations which are solved by using an incremental Newton-Raphson iterative procedure. A study of the large deflection behaviour of thin plates is made for various edge conditions and aspect ratios, and the results obtained are compared with those using a finite element scheme. Representative nondimensional solutions for deflections and stresses are presented in the form of graphs.     

Adaptive Local Refinement with Octree Load Balancing for the Parallel Solution of Three-Dimensional Conservation Laws

Conservation laws are solved by a local Galerkin finite element procedure with adaptive space-time mesh refinement and explicit time integration. The Courant stability condition is used to select smaller time steps on smaller elements of the mesh, thereby greatly increasing efficiency relative to methods having a single global time step. Processor load imbalances, introduced at adaptive enrichment steps, are corrected by using traversals of an octree representing a spatial decomposition of the domain. To accommodate the variable time steps, octree partitioning is extended to use weights derived from element size. Partition boundary smoothing reduces the communications volume of partitioning procedures for a modest cost. Computational results comparing parallel octree and inertial partitioning procedures are presented for the three-dimensional Euler equations of compressible flow solved on an IBM SP2 computer.     

Numerical analysis of unsteady compressible laminar boundary layer flow

The unsteady compressible laminar boundary layer flow on an arbitrary cylinder due to an incident stream whose velocity varies arbitrarily with time is considered. The method presented is based on the separation between the convective and diffusive quantities. By defining some new variables, the splitting appears rather naturally, and the initialisation problem can be solved without difficulty. The transformed equations are solved with the help of a semi-implicit finite difference scheme which is unconditionally linearly stable. The computations have been applied to flows past a cylinder with constant and fluctuating free-stream velocities.     

Numerical integration of the time-dependent equations of motion for taylor vortex flow

A method is presented for the finite difference solution of the equations of fluid motion. The complete Navier-Stokes equations are expressed in terms of tangential velocity, vorticity and stream function. The transformed equations are solved using an alternating direction implicit scheme. The classical problem of hydrodynamic stability of the rotational Couette flow is solved in two dimensions. Comparison with other numerical and experimental works shows that the method reported here is computationally stable, even when used with coarse grids and relatively large time increments.     

Multiscale computational analysis in mechanics using finite calculus: an introduction

The paper introduces a general procedure for computational analysis of a wide class of multiscale problems in mechanics using a finite calculus (FIC) formulation. The FIC approach is based in expressing the governing equations in mechanics accepting that the domain where the standard balance laws are established has a finite size. This introduces naturally additional terms into the classical equations of infinitesimal theory in mechanics which are useful for the numerical solution of problems involving different scales in the physical parameters. The discrete nodal values obtained with the FIC formulation and the finite element method (FEM) can be effectively used as the starting point for obtaining a more refined solution in zones where high gradients of the relevant variables occur using hierarchical or enriched FEM. Typical multiscale problems in mechanics which can be solved with the FIC method include convection–diffusion-reaction problems with high localized gradients, incompressible problems in solid and fluid mechanics, localization problems such as prediction of shear bands in solids and shock waves in compressible fluids, turbulence, etc. The paper presents an introduction of the treatment of multiscale problems using the FIC approach in conjunction with the FEM. Examples of application of the FIC/FEM formulation to the solution of simple multiscale convection–diffusion problems are given.