Finite Difference Procedure Research Articles

Effect of temperature on the myoglobin-facilitated transport of oxygen in skeletal muscle

An analysis of thermal effects on the facilitative transport of oxygen in skeletal muscle fibers is presented. Steady-state mass and energy transport balances are written and solved analytically or numerically using a finite-difference procedure. It is shown that no significant spatial thermal gradients exist due to internal reactions or bulk conduction effects across a muscle fiber. At typical muscle conditions, it is predicted that increased global temperature reduces the fraction of oxygenated myoglobin, increases local oxygen concentrations, and increases the percentage of oxygen flux attributed to oxy-myoglobin. The maximum supportable oxygen consumption rate, mO2max, is defined as the highest consumption rate sustainable without developing anoxic regions at the center of the fiber. By considering only temperature sensitive effects within fibers, mO2max is found to increase slightly with temperature at low temperatures. This increase is due to thermal effects on the diffusion coefficients as opposed to effects associated with the kinetics of the myoglobin-oxygen reaction. If the simulations include the temperature effect associated with oxygen solubility in blood plasma, mO2max decreases with temperature. A sensitivity analysis was performed by varying the values of relevant parameters. The maximum consumption rate was least affected by parameters associated with the kinetic and equilibrium constants and most affected by the diffusion coefficients and the concentration of myoglobin.

A dual-porosity model for waterflooding in naturally fractured reservoirs

A dual-porosity model for saturated, two-phase, incompressible, immiscible flow in a naturally fractured petroleum reservoir is formulated and then approximated by a finite difference procedure. Calculations are presented to indicate the behavior of the model as a function of several parameters, including the size and shape of the matrix blocks, the form of the relative permeability curves in the blocks, the inclination of the reservoir, the absolute permeabilities and the fluid viscosities.

Dynamics of liquid membranes. II: Adaptive finite difference methods

Two domain-adaptive finite difference methods are presented and applied to study the dynamic response of incompressible, inviscid, axisymmetric liquid membranes subject to imposed sinusoidal pressure oscillations. Both finite difference methods map the time-dependent physical domain whose downstream boundary is unknown onto a fixed computational domain. The location of the unknown time-dependent downstream boundary of the physical domain is determined from the continuity equation and results in an integrodifferential equation which is non-linearly coupled with the partial differential equations which govern the conservation of mass and linear momentum and the radius of the liquid membrane. One of the finite difference methods solves the non-conservative form of the governing equations by means of a block implicit iterative method. This method possesses the property that the Jacobian matrix of the convection fluxes has an eigenvalue of algebraic multiplicity equal to four and of geometric multiplicity equal to one. The second finite difference procedure also uses a block implicit iterative method, but the governing equations are written in conservation law form and contain an axial velocity which is the difference between the physical axial velocity and the grid speed. It is shown that these methods yield almost identical results and are more accurate than the non-adaptive techniques presented in Part I. It is also shown that the actual value of the pressure coefficient determined from linear analyses can be exceeded without affecting the stability and convergence of liquid membranes if the liquid membranes are subjected to sinusoidal pressure variations of sufficiently high frequencies.

Turbulent Flow Around a Bluff Rectangular Plate. Part II: Numerical Predictions

This paper presents calculations of the time-averaged separated-reattaching flow around a bluff rectangular plate, using a finite difference procedure and the k-ε turbulence model. Two discretization methods are used: the hybrid differencing scheme, and the bounded skew hybrid differencing scheme. The latter, although superior to the former for all grid distributions, results in a reattachment length about 30 percent shorter than the measured value. When a modification which takes into account streamline curvature is incorporated into the k-ε model, a marked improvement in the predictions is obtained. A reattachment length of 4.3 plate thicknesses (D), compared to an experimental value of 4.7D, is obtained, and the predicted mean flow field, turbulent kinetic energy and pressure distributions within the separation bubble are found to be in good agreement with experiments.

Coupled space-marching method for the Navier-Stokes equations for subsonic flows

An implicit space-marching finite-difference procedure is described for solving the compressible form of the steady, two-dimensional Navier-Stokes equations in body-fitted curvilinear coordinates. The coupled system of equations is solved for the primitive variables of velocity and pressure by making multiple sweeps of the space-marching procedure. A new pressure correction method has been developed that significantly accelerates convergence of the iterative process. The scheme is used to compute incompressible flows by taking the low Mach number limit of the compressible formulation. Computed results are compared with other numerical predictions for low Reynolds number channel inlet flow, flow over a rearward-facing step in a channel, and external flow over a cylinder. NUMBER of different methods have been developed to solve the Navier-Stokes equations in primitive variables for steady, subsonic flows. Many of these algorithms are described in Anderson et al. 1 The present methods are not entirely adequate for all problems, and advanced techniques continue to be developed. New procedures and recent im- provements to existing methods are briefly reviewed in the following paragraphs. The most frequently used primitive variable algorithms for incompressible flows solve the momentum equations for the velocity components in an uncoupled (segregated) manner, holding pressure fixed. Although different in detail, the segre- gated solution schemes (e.g., Refs. 2-4) use the continuity equation indirectly in the formulation of a separate Poisson equation that is solved for the pressure. Since the velocity components and pressure are computed from different algo- rithms, rather than in a coupled manner, convergence is slowed. Furthermore, the solution of the elliptic Poisson equa- tion consumes a large part of the total computaion time for each global iteration. Van Doormaal et al.5 recently have eval- uated improved algorithms for solving the pressure Poisson equation, and Rhie4 has applied a multigrid method to acceler- ate the solution of the momentum equations and the pressure equation. Several strategies have been advanced for solving the cou- pled momentum and continuity equations for the primitive variables, including the pressure. The schemes all share the advantage that a separate procedure for imposing the continu- ity constraint and determining the pressure is not required. The choice of variables and the algebraic approach used to solve the system of nonlinear equations distinguish the direct, time- marching, and space-marching algorithms. Vanka and Leaf6 and recently Patankar et al. 7 have com- pared solutions for two-dimensional flows obtained by direct and segregated methods. A large sparse matrix solver is used to invert a system of equations spanning the entire flow domain.

Dynamics of planar flexible mechanisms by finite element method with truss-type elements

The kinematics and dynamics of planar mechanisms with revolute joints and flexible links are studied by using the finite element method. The mechanism finite element is based on a planar truss element. The equations of motion of a mechanism are formulated by applying Lagrange's equations to the kinetic and potential energy of the system. The analysis includes the higher order elastic acceleration terms. The presented method of solving equations of motion is based on a finite difference procedure and avoids the necessity of elimination of the singularity of the system stiffness matrix. A four-bar linkage is studied to illustrate the computational efficiency of the presented method. The results for flexible links are compared with those obtained for the rigid body.

A simple method for obtaining good bounds for solutions of reaction-diffusion equations with nonlinear kinetics

A kinetic model is proposed for a surface-supported biological film (bioparticle) employed in many biological processes. The equations that govern kinetic and diffusion-controlled substrate uptake by the attached organisms are invariably nonlinear and analytical solutions if any are impossible to find. It is therefore desirable to determine approximate analytical solutions or, failing this, bounds for the exact numerical solution. This article represents an attempt to provide increasingly better bounds to such equations by a linearization technique. An iterative scheme relying on the maximum principle is presented for obtaining upper and lower bounds to solutions to resulting nonlinear reaction-diffusion equations. In particular, a spherical bioparticle with Michaelis-Menten-type reaction kinetics is considered. Its application to more general equations is also discussed. These bounds are in good agreement with the numerical solutions obtained by shooting and finite-difference procedures. The method, which can easily be generalized to other geometries, is relatively simple to use and converges rapidly to a very good upper and lower bound.

Laminar Flow in a Pipe Rotating around a Perpendicular Axis.

A finite-difference procedure is applied to analyse the fully developed laminar flow characteristics in a pipe rotating about a perpendicular axis. We propose a new combination of two parameters of KL and number Ro as characteristic non-dimensional numbers. The four flow regimes are classified for large and srmall KL, Ro. We present a numerical formula for friction factor which is valid for wide range of KL and Ro. The formula agrees well with experimental data.

Passive solar heating of buildings using a fluidized bed plus Trombe wall system

The purpose of this study is to mathematically model the behaviour of an airfluidized bed solar collector. A new model for an air-fluidized solar collector is developed which is based on ‘Trombe wall’ structure and theoretical and experimental studies on fluidized beds. Computer solutions are obtained for the temperature and velocity fields and for the power extracted from the system. Results obtained for the fluidized bed Trombe-wall system are compared with those for a conventional type Trombe-wall system. A finite-difference procedure was used to solve the governing equations using air as the fluid.

Simulations of bifurcation processes of reflected-shock waves in ionizing argon in a shock tube

Abstract A finite-difference procedure, which numerically solves the thin-layer Navier-Stokes equations with ionization reactions, has been developed to predict ionizing nonequilibrium flows generated by strong shock waves. By using the procedure, simulations were carried out for bifurcated reflected-shock waves in ionizing argon in a shock tube. Results are shown for the reflection process of an ionizing shock moving at Mach No. 14 in argon at 400 Pa and 298 K. Computed flowfields are compared with snap-shot shadowgraphs for reflected-shock waves in ionizing argon. Qualitative agreement between the simulations and the experiments is excellent.

Investigations of mean and turbulent flow characteristics of a two dimensional plane diffuser

This paper reports the investigation of mean and turbulent flow characteristics of a two-dimensional plane diffuser. Both experimental and theoretical details are considered. The experimental investigation consists of the measurement of mean velocity profiles, wall static pressure and turbulence stresses. Theoretical study involves the prediction of downstream velocity profiles and the distribution of turbulence kinetic energy using a well tested finite difference procedure. Two models, viz., Prandtl's mixing length hypothesis and k-ɛ model of turbulence, have been used and compared. The nondimensional static pressure distribution, the longitudinal pressure gradient, the pressure recovery coefficient, percentage recovery of static pressure, the variation of Umax/Ubar along the length of the diffuser and the blockage factor have been valuated from the predicted results and compared with the experimental data. Further, the predicted and the measured value of kinetic energy of turbulence have also been compared. It is seen that for the prediction of mean flow characteristics and to evaluate the performance of the diffuser, a simple turbulence model like Prandtl's mixing length hypothesis is quite adequate.

Calculation of turbulent convection between corotating disks in axisymmetric enclosures

A numerical investigation is conducted for the case of turbulent flow in the unobstructed space between a pair of centrally clamped coaxial disks corotating in a fixed axisymmetric enclosure. The finite difference procedure of Chang et al. ( J. Heat Transfer 111, 625–632 (1989)) is extended to include a standard two-equation ( κ- ε) model of turbulence in the core of the flow. A van Driest relation, that accounts for the effects of streamline curvature and wall shear on the energy-containing length scales, is used in conjunction with Prandtl's mixing length hypothesis to model the near wall flow. The set of equations is solved assuming a constant property, circumferentially symmetric, statistically stationary flow. This approach predicts mean velocity and heat transfer results that are in good agreement with timeaveraged experimental data. The predictions reveal a flow that, in non-dimensional variables, tends to a limiting asymptotic state at high Reynolds numbers. In the absence of blowing a symmetrical pair of crossstream eddies appear near the enclosure wall the rotation of which increases with increasing disk rotational speed. High rotational speeds, in excess of 2400 rpm in the configuration studied, and small disk separations induce large values of shear and temperature (due to viscous dissipation) in the vicinity of the enclosure wall. The flow and its heat transfer characteristics can be drastically altered by the combined effects of radial and axial blowing. Specifically, it is shown that axial blowing significantly reduces the shear and heat transfer at the curved enclosure wall.

The Thermal Capacity Effect Upon Transient Natural Convection in a Rectangular Cavity

Transient natural convection in a rectangular enclosure is analyzed using a finite difference scheme. The enclosure is adiabatic and filled with water. The buoyancy induced flow is generated by a flat vertical uniform flux surface that has a finite thermal capacity. The full two-dimensional equations representing conservation of mass, momentum, and energy are solved in their time-dependent form. The solution technique used is a modified finite difference procedure very similar to Simple Arbitrary Lagrangian Eulerian (SALE) technique. Two values of surface thermal capacity are investigated, each resulting in a different flow regime during the transient. At short times a simple one-dimensional conduction regime is found to occur. As the leading edge effects arrive at any downstream location the conduction regime is terminated and true convection effects set in. At intermediate times a different flow regime is detected, namely a steady two-dimensional regime that approaches the steady state similarity solution for a similarly heated surface immersed in an infinite fluid medium. Excellent agreement is found with previous analyses and measurements during the early and intermediate transients.

Finite element simulation of nonlinear kinematic surface runoff

A deterministic model to simulate stormwater runoff from impervious surfaces is presented. The surface runoff model is based on a one-dimensional, variable width, kinematic wave approximation. Resistance to flow is described by a generalized flow resistance equation which includes laminar, turbulent and transitional flow representations or by a uniform flow resistance approximation. A Galerkin finite element formulation utilizing linear, quadratic and/or cubic one-dimensional Lagrangian elements is presented for the spatial discretization of the nonlinear kinematic runoff equations. Temporal discretization is based on a linear time variation which includes both explicit and implicit time marching algorithms, depending on the choice of the time weighting parameter. The implicit algorithm results in a system of nonlinear equations which are solved using successive substitutions. Initial estimates of the solution are generated with a linear finite difference procedure. A method to estimate a suitable computation time-step size based on the Courant condition is presented. Kinematic surface runoff simulation results are presented. The finite element simulations are shown to compare favorably with observed experimental data and previously published results.

Influence of far-field boundary on the calculated flowfield and performance of rotors

A experiment was conducted to demonstrate the influence of the physical location of the far-field boundary on the calculated performance and flowfield of a two-dimensional Darrieus rotor. A substantial sensitivity to the extent of the computational domain was noted and a series of calculations and comparisons with other approaches are presented to show the influence on pressure recovery and power estimates. Although the study was restricted to the case of a rotor, the sensitivity to far-field boundary will be of concern to the computation of other momentum devices. Introduction A two-dimensional Darrieus wind turbine, facing a uniform wind and spinning in the anticlockwise direction, is assumed to be placed at the center of a Cartesian domain. The computational domain is subdivided into control volumes by a series of orthogonal grid lines. The rotor-blades paths are circular in any horizontal plane. The velocity and presure field were obtained by solving steady, laminar, incompressible, NavierStokes equations using a primative variable finite-difference procedure known as SIMPLER. The influence of the spinning turbine blades was introduced as momentum source terms Sx and Sy (not known a priori) in the conservation equations. These terms are zero everywhere except at the circle described by the path of the blades. Rajagopalan and Fanucci proved this approach to be feasible in an application to inviscid flow around a two-dimensional vertical axis wind turbine in cylindrical coordinates, and further validation was obtained by Paraschivoiu et al. who compared the predictions with experiments after including dynamic stall effects. In functional notation, for a time-averaged calculation, the source terms can be written as Sx = Sx(Ci,Cd,a,Yabs,<*,R,e,c,p9B) (la) Sy = Sy(Cl9Cd9a9Yabs9u9R,0,c,p,B) (Ib) where Q and Cd are airfoil characteristics of the turbine blade, o: is the angle of attack of the blade to the relative velocity vector, Yabs is the absolute velocity of the fluid at the location (R ,6), co is the angular velocity of the rotor, c is the chord of the blade, and B is the number of blades. The dependence of Received July 6, 1987; presented as Paper 87-2284 at the AIAA 5th Applied Aerodynamics Conference, Monterey, CA, Aug. 17-19, 1987; synoptic received March 21, 1989. Copyright © 1989 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Full paper available at AIAA Library, 555 W. 57th St., New York, NY 10019. Price: microfiche, $4.00; hard copy, $9.00. Remittance must accompany order. *Assistant Professor, Aerospace Engineering Department. Member AIAA. tProfessor, Aerospace Engineering Department. Associate Fellow AIAA. Sx and Sy on Reynolds number is considered only implicitly through the airfoil sectional characteristics C/ and Cd. The source terms, evaluated using the procedure illustrated in Ref. 2, resolved for Cartesian directions are

Fourth-order accurate three-dimensional compressible boundary-layer calculations

A fourth-order accurate finite-difference procedure for solving compressible three-dimensional boundarylayer equations is presented and applied to wing and ellipsoid flows. The method is fourth-order accurate normal to the surface and second-order accurate in the streamwise and crossflow directions for arbitrary mesh increments. The procedure is validated for a number of standard test cases over a range of Mach numbers. An interface procedure is used to solve the surface Euler equations with the inviscid flow pressure field as the input to assure consistency at the boundary-layer edge. The importance of the interface procedure in ensuring accuracy in edge boundary conditions is discussed. Details of applying the method to solve laminar flow past wing- and fuselage-type bodies are presented.

Simulations of reflected-shock waves in shock tubes taking account of side-wall boundary-layer effects.

Numerical simulations are conducted for behaviors of reflected-shock waves interacting with side-wall boundary-layers in shock tubes. A finite-difference procedure is developed as a solver of the thin-layer Navier-Stokes equations. This procedure, which combines the Lax-Wendroff-FCT scheme with the Crank-Nicolson scheme, is applied to initial and boundary conditions in which effects of shock-tube side-wall boundary-layers. are taken account of by use of the Pohlhausen polynomials. Computations are carried out for reflected-shock waves in nitrogen and in argon gas, respectively. Bifurcated reflected-shock waves, observed experimentally in polyatomic gases in shock tubes, are reproduced in simulations for nitrogen gas and compared with semiempirical theory for bifurcated reflected-shock waves. Good agreements between them lead us to conclude that the present procedure can simulate essential parts of interactions between reflected-shock waves and side-wall boundary-layers.

The numerical simulation of an oldroyd liquid draining down a vertical surface

The problem of the drainage of an elastic liquid with a shear dependent viscosity from a vertical surface is solved numerically by using a time-marching finite difference procedure. Shear-thinning is found to have a damping effect on any elastic oscillations during start-up in all but very thin films, and is confirmed as the dominant non-Newtonian property at later times.

Draining non-Newtonian films: Part 1. The numerical simulation of a generalized Newtonian liquid draining down a vertical surface

A time-marching finite difference procedure is used to solve the problem of a generalized Newtonian liquid draining from rest down a vertical surface. It is confirmed that the unsteady start-up phase of the drainage can be ignored in most practical situations. It is suggested that the numerical procedure may be applicable to other similar, but more complicated, flows.

The unitarity of split-operator finite difference and finite-element methods: Applications to longitudinally varying semiconductor rib waveguides

Split-operator finite-difference and finite-element techniques are applied to the calculation of losses in semiconductor rib waveguide Y junctions. It is shown that, unlike the finite-difference procedure, which is unitary for both uniform and nonuniform grid-point spacings, considerable care must be exercised in formulating the split-operator finite-element method in order to preserve the power in the propagating electric field. The calculations are performed in the context of the Fresnel approximation to the scalar Helmholtz equation, which yields accurate solutions for transverse electric (TE)-polarized electric fields in rib waveguides far from cutoff.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>