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The use of Murman and Krupp's (1971) method for calculating steady, inviscid, transonic flows with imbedded shock waves is discussed in application to two-dimensional problems for transonic flows past sharp and moderately blunt nosed geometries. A relaxation algorithm is used to solve the equations.
The Cauchy-Riemann equations and the 1-D Euler equations are expressed in generalized coordinates and then cast in finite difference form by using central differencing throughout. The resulting matrix representation has an eigensystem that permits the development of an annihilation process using complex arithmetic in a block tridiagonal solver. Initial numerical experiments show that the process has potential for use as a relaxation procedure for the Euler equations.
The requirements for uniqueness of the calculated jump conditions across embedded shock waves are investigated for type-dependent difference systems used in transonic flow studies. A mathematical analysis shows that sufficient conditions are (1) the equations should be differenced in conservative form and (2) a special difference operator should be used when switching from a hyperbolic to an elliptic operator. The latter results in a consistency condition on the integral equations, rather than the differential, at these points. Calculated jump conditions for several embedded and detached shock waves are analyzed in the physical and hodograph planes. Comparisons are made with previous results, a time-dependent calculation, and data.
A mapping transformation is derived for the inverse solution of nonlinear and linear integral equations of the types encountered in remote sounding studies. The method is applied to the solution of specific problems for the determination of the thermal and composition structure of planetary atmospheres from a knowledge of their upwelling radiance.
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The relaxation method for inverse solution of the full radiative transfer equation is generalized to solve for all the atmospheric parameters that appear in the integrand as functions or functionals, without any a priori information about the expected solution. Illustrations are presented using the 7.5 micron CH4 band for determining temperature profiles in the Jovian atmosphere, and the 6.3 micron band for determining the water vapor mixing ratio in the earth's atmosphere.
The paper reviews the chief finite difference and finite element techniques used for numerical solution of nonlinear mixed elliptic-hyperbolic equations governing transonic flow. The forms of the governing equations for unsteady two-dimensional transonic flow considered are the Euler equation, the full potential equation in both conservative and nonconservative form, the transonic small-disturbance equation in both conservative and nonconservative form, and the hodograph equations for the small-disturbance case and the full-potential case. Finite difference methods considered include time-dependent methods, relaxation methods, semidirect methods, and hybrid methods. Finite element methods include finite element Lax-Wendroff schemes, implicit Galerkin method, mixed variational principles, dual iterative procedures, optimal control methods and least squares.
Relaxation schemes based on approximate and incomplete factorization technique (AF) are described. The AF schemes allow construction of a fast multigrid method for solving integral equations of the second and first kind. The smoothing factors for integral equations of the first kind, and comparison with similar results from the second kind of equations are a novel item. Application of the MD algorithm shows convergence to the level of truncation error of a second order accurate panel method.
A multi-level grid method was studied as a possible means of accelerating convergence in relaxation calculations for transonic flows. The method employs a hierarchy of grids, ranging from very coarse (e.g. 4 x 2 mesh cells) to fine (e.g. 64 x 32); the coarser grids are used to diminish the magnitude of the smooth part of the residuals, hopefully with far less total work than would be required with optimal iterations on the finest grid. To date the method was applied quite successfully to the solution of the transonic small-disturbance equation for the velocity potential in conservation form. Nonlifting transonic flow past a parabolic arc airfoil is the example studied, with meshes of both constant and variable step size.
The structure of slow magnetosonic shocks in the low beta regime is analyzed with attention given to ion heating and the effects of waves upstream of the electromagnetic ion/ion cyclotron (EMIIC) instability. Shock formation is assessed by means of three methods - a relaxation method and two based on dynamic flow interactions - to determine the effects of initialization and boundary conditions on the formation. Good solutions are found with the piston method and the similar flow-flow method in which the plasma is injected from two boundaries to form two slow shocks. Plasma parameters and shock normal angle are found to be the key variables dictating the structure of the magnetosonic shocks. Four unique classes of resultant shock structures are described in which classical, steady, or nonsteady behavior is found. The analysis also yields insight into the relationship between EMIIC instability and ion dissipation.
A multigrid relaxation method is applied to a pressure-based implicit procedure to solve unseady, incompressible Navier-Stokes equations. The present multigrid method is a Correction Scheme according to Brandt. This method is used to solve the scalar matrices resulting from the finite-volume formulation and uses flux averaging as the restriction operator. The accuracy and computational efficiency are demonstrated with a steady state driven cavity flow and an unsteady flow over a circular cylinder case. The results are compared with single grid results using the OrthoMin conjugate gradient method and experimental data.
This paper is intended as a review and summary of the advances made in a recently developed approach for rapid numerical solution of the equations of inviscid transonic aerodynamics. The investigation has been limited to two-dimensional, steady, inviscid flow over airfoils in a subsonic free stream, with emphasis on development of a rapid computational technique, rather than on generality of application. The approach uses finite-difference algorithms called "fast direct elliptic solvers" within an iteration scheme. "Direct" means that the entire computation field is solved at once, rather than in successive traverses over the field as in a point- or line-relaxation method. Such an iterative method is referred to as "semidirect." The iterative convergence can be faster than in other relaxation methods because changes are felt simultaneously at all points in each succeeding iteration. Direct elliptic solvers and semidirect methods have restrictions, but these are gradually being removed. Direct solvers were first developed for solving Poisson's equation on a rectangle without interior boundaries. A method to treat first-order systems, a direct Cauchy-Riemann solver has also been developed. Numerical treatment of part of a system of nonlinear equations by a Poisson solver has been reported. Also Poisson solvers in semidirect methods were used for nonseparable elliptic equations. The semidirect method was extended to the solution of a problem of mixed type, where the improved Murman-Cole transonic small-disturbance difference equations were solved. A slightly supercritical flow over a biconvex airfoil was treated successfully, but the iterations did not converge for more strongly supercritical conditions In another work the addition of terms ot both sides of the difference equations stabilized the iteration for supercritical conditions with large supersonic zones. For this, the Cauchy-Riemann solver was revised to incl,ude the needed terms. Most recently, the evaluation of parameters for rapid convergence and comparisons, with Murman's line-relaxation method was described. The method was extended to full second order accuracy in a fully conservative formulation in another work.
Use of an infrared chemiluminescence technique, called 'Method II,' or the 'method of arrested relaxation' to measure the distribution of energy among products of the Cl + HI and Cl + DI reactions. Preliminary results are also given for the Br + HI and Cl + HBr reactions. Instead of measuring vibrational relaxation, Method II attempts to arrest vibrational and rotational relaxation by the rapid removal of excited products at a cold surface.
A general method of design is developed for two-dimensional unbranched channels with prescribed velocities as a function of arc length along the channel walls. The method is developed for both compressible and incompressible, irrotational, nonviscous flow and applies to the design of elbows, diffusers, nozzles, and so forth. In part I solutions are obtained by relaxation methods; in part II solutions are obtained by a Green's function. Five numerical examples are given in part I including three elbow designs with the same prescribed velocity as a function of arc length along the channel walls but with incompressible, linearized compressible, and compressible flow. One numerical example is presented in part II for an accelerating elbow with linearized compressible flow, and the time required for the solution by a Green's function in part II was considerably less than the time required for the same solution by relaxation methods in part I.
Attention is given to a method which makes it possible to incorporate information from ancillary data sources into the results of an existing classification of remotely sensed data. The method is based upon probabilistic label relaxation procedures which are used for imbedding spatial context data in image-labeling problems. The source of ancillary information is utilized in the form of a set of probabilities. These are introduced into a modified relaxation method called supervised relaxation labeling which, on application, develops a labeling for remotely sensed data that strikes a balance in consistency between spectral, spatial, and ancillary data sources of information. A description is given of results of a forestry classification in which accuracy is improved from 68% to 81% by incorporating topographic elevation in the outlined manner.
The application of Chahine's (1970) inversion technique to remote sensing problems utilizing the limb viewing geometry is discussed. The problem considered here involves occultation-type measurements and limb radiance-type measurements from either spacecraft or balloon platforms. The kernel matrix of the inversion problem is either an upper or lower triangular matrix. It is demonstrated that the Chahine inversion technique always converges, provided the diagonal elements of the kernel matrix are nonzero.
The method of measured relaxation is described for the determination of initial vibrational energy distribution in the products of exothermic reaction. Hydrogen atoms coming from an orifice were diffused into flowing chlorine gas. Measurements were made of the resultant ir chemiluminescence at successive points along the line of flow. The concurrent processes of reaction, diffusion, flow, radiation, and deactivation were analyzed in some detail on a computer. A variety of relaxation models were used in an attempt to place limits on k(nu prime), the rate constant for reaction to form HCl in specified vibrational energy levels: H+Cl2 yields (sup K(nu prime) HCl(sub nu prime) + Cl. The set of k(?) obtained from this work is in satisfactory agreement with those obtained by another experimental method (the method of arrested relaxation described in Parts IV and V of the present series.