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A least squares formulation of the system divu = rho, curlu = zeta is surveyed from the viewpoint of both finite element and finite difference methods. Closely related arguments are shown to establish convergence estimates.
Conditions were determined in a continuous-flow, premixing-prevaporizing duct at which autoignition occurred. Test conditions were representative of an advanced, regenerative-cycle, automotive gas turbine. The test conditions inlet air temperatures from 600 to 1250 K (a vitiated preheater was used), pressures from 170 to 600 kPa, air velocities of 10 to 30 m/sec, equivalence ratios from 0.3 to 1.0, mixing lengths from 10 to 60 cm, and residence times of 2 to 100 ms. The fuel was diesel number 2. The duct was insulated and had an inside diameter of 12 cm. Three different fuel injection systems were used: One was a single simplex pressure atomizer, and the other two were multiple-source injectors. The data obtained with the simplex and one of the multiple-source injectors agreed satisfactorily with the references and correlated with an Arrenhius expression. The data obtained with the other multiple source injector, which used multiple cones to improve the fuel-air distribution, did not correlate well with residence time.
Finite difference, spectral and Galerkin methods for the approximate solution of time dependent problems are surveyed. A unified discussion on their accuracy, stability and convergence is given. In particular, the dilemma of high accuracy versus stability is studied in some detail.
A representative class of elliptic equations is treated by a dissipative compact finite difference scheme and a general solution technique by relaxation methods is discussed in detail for the Laplace equation.
Discrete approximation schemes for the solution of nonlinear hereditary control problems are constructed. The methods involve approximation by a sequence of optimal control problems in which the original infinite dimensional state equation has been approximated by a finite dimensional discrete difference equation. Convergence of the state approximations is argued using linear semigroup theory and is then used to demonstrate that solutions to the approximating optimal control problems in some sense approximate solutions to the original control problem. Two schemes, one based upon piecewise constant approximation, and the other involving spline functions are discussed. Numerical results are presented, analyzed and used to compare the schemes to other available approximation methods for the solution of hereditary control problems.
New convenient stability criteria are provided in this paper for a large class of finite difference approximations to initial-boundary value problems associated with the hyperbolic system u sub t = au sub x + Bu + f in the quarter plane x or = 0, t or = 0. Using the new criteria, stability is easily established for numerous combinations of well known basic schemes and boundary conditions, thus generalizing many special cases studied in recent literature.
An initial value problem for the autonomous system of ordinary differential equations dy/dt = f(y), where y is a vector, is considered. In a number of practical applications the interest lies in obtaining the curve traced by the solution y. These applications include the computation of trajectories in mechanical problems. The term 'trajectory problem' is employed to refer to these cases. Lambert and McLeod (1979) have introduced a method involving local rotation of the axes in the y-plane for the two-dimensional case. The present investigation continues the study of difference schemes specifically derived for trajectory problems. A simple geometrical way of constructing such methods is presented, and the local accuracy of the schemes is investigated. A circularly exact, fixed-step predictor-corrector algorithm is defined, and a variable-step version of a circularly exact algorithm is presented.
A method has been developed to solve the unsteady, compressible Navier-Stokes equation with the property of consistency and the ability of minimizing the equation stiffness. It relies on innovative extensions of the state-of-the-art finite-difference techniques and is composed of: (1) the upwind scheme for split-flux and the central scheme for conventional flux terms in the inviscid and viscous regions, respectively; (2) the characteristic treatment of both inviscid and viscous boundaries; (3) an ADI procedure compatible with interior and boundary points; and (4) a scalar matrix coefficient including viscous terms. The performance of this method is assessed with four sample problems; namely, a standing shock in the Laval duct, a shock reflected from the wall, the shock-induced boundary-layer separation, and a transient internal nozzle flow. The results from the present method, an existing hybrid block method, and a well-known two-step explicit method are compared and discussed. It is concluded that this method has an optimal trade-off between the solution accuracy and computational economy, and other desirable properties for analyzing transient viscous flow problems.
Spectrophotometry from 0.6-2.5 micron at 1.5% spectral resolution of Saturn's equator and visibly dark South Equatorial Belt near a 1980 ring-plane crossing shows substantially less absorption by methane over the equator than over the SEB. Model fits using Appleby's model atmosphere and a multiple scattering model lead to the conclusion that the pressure levels of the clouds in the two regions differ by 20%. The region of high clouds is coincident with the location of the equatorial fast jet observed by Voyager.
Implicit compact finite difference schemes for the Euler equations are described which furnish equivalent treatment of the conservation and nonconservation forms; a simple modification yields an entropy-producing scheme. An extension of the scheme also treats the compressible Navier-Stokes equations; when the viscosity and heat conduction coefficients are negligible only the boundary data appropriate to the Euler equation influence the solution to any significant extent, a result consistent with singular perturbation theory.
The class of one-dimensional stretching functions used in finite-difference calculations is studied. For solutions containing a highly localized region of rapid variation, simple criteria for a stretching function are derived using a truncation error analysis. These criteria are used to investigate two types of stretching functions. One an interior stretching function, for which the location and slope of an interior clustering region are specified. The simplest such function satisfying the criteria is found to be one based on the inverse hyperbolic sine. The other type of function is a two-sided stretching function, for which the arbitrary slopes at the two ends of the one-dimensional interval are specified. The simplest such general function is found to be one based on the inverse tangent. Previously announced in STAR as N80-25055
A new, fast, direct-inverse, finite-difference boundary-layer code has been developed and coupled with a full-potential transonic airfoil analysis code via new inviscid-viscous interaction algorithms. The resulting code has been used to calculate transonic separated flows. The results are in good agreement with Navier-Stokes calculations and experimental data. Solutions are obtained in considerably less computer time than Navier-Stokes solutions of equal resolution. Because efficient inviscid and viscous algorithms are used, it is expected this code will also compare favorably with other codes of its type as they become available.
An explicit finite difference scheme consisting of a predictor and a corrector has been developed and applied to solve some hyperbolic partial differential equations (PDEs). The corrector is a convex-type function which is applied at each time level and at each mesh point. It consists of a parameter which may be estimated such that for larger time steps the algorithm should remain stable and generate a fast speed of convergence to the steady-state solution. Some examples have been given.
This paper reports a study of the linear Vlasov electromagnetic dispersion equation for density drift instabilities in very low beta plasmas. A uniform magnetic field and a weak uniform density gradient are assumed. This paper differs from most other studies of this topic in three important ways: (1) no low frequency or long wavelength approximations are made, (2) no gauge condition is imposed and (3) a modification of the local approximation is used which is argued to be less arbitrary than the usual local approximation. Numerical solution of the resulting dispersion equation yields stabilization of the universal density drift instability at values of beta somewhat higher than those obtained from the local approximation, and near maximum growth shows no evidence of coupling between the universal and Alfven modes.
First and second order explicit difference schemes are derived for a three dimensional hyperbolic system of conservation laws, without recourse to dimensional factorization. All schemes are upwind (backward) biased and optimally stable.
Using a direct integration, transient response rotor dynamics computer code, the response of turbine engine rotors to two different blade tip - seal interference rub models was studied. The first model, an abradable seal rub model, is based on an energy-loss-per-unit-volume theory (applicable to a ceramic turbine blade tip seal). The second, a smearin model, is based on viscous hydrodynamic theory (applicable to a metallic blade tip seal). The results from these two models were compared with those from a previously studied model based on dry friction theory. The abradable model was very senitive to small changes in the energy per unit volume, and once a threshold was exceeded, the rotor went into a backward whirl. The amplitude seemed to grow without limit. This was similar to the dry friction model when the coefficient of friction exceeded a particular threshold. The smearing model was not as sensitive to small changes in the viscosity, but a threshold viscosity was found. When it was exceeded, the rotor went into backward whirl, but the amplitude seemed to grow to a finite limit.
A compact difference scheme is described for treating the first-order system of partial differential equations which describe the equilibrium equations of an elastic body. An algebraic simplification enables the solution to be obtained by standard direct or iterative techniques.
The stability of finite difference models of hyperbolic initial boundary value problems is connected with the propagation and reflection of parasitic waves. Wave propagation ideas are applied to models containing two boundaires or interfaces, where repeated reflection of trapped wave packets is a potential new source of instability. Various known instability phenomena are accounted for in a unified way. Results show: (1) dissipativity does not ensure stability when three or more formulas are concatenated at a boundary or internal interface; (2) algebraic GKS instabilities can be converted by a second boundary to exponential instabilities only when an infinite numerical reflection coefficient is present; and (3) GKS-stability and P-stability can be established in certain problems by showing that all numerical reflection coefficients have modulus less than 1.