Member AIAA Research Articles

Reply by Authors to C. D. Mikkelsen

Submitted June 1, 1987; received May 12, 1989. Copyright © 1989 American Institute of Aeronautics and Astronautics, Inc. All rights reserved. *Associate Professor of Engineering and Graduate Program Director, Department of Mechanical Engineering. Member AIAA. tGraduate Student, Department of Mechanical Engineering. lected without priori knowledge of fully developed entrance length Lf. The solution thus is constrained. We did not mean to suggest that extent of computational be constrained, without prior knowledge, to some [necessarily] finite entrance length. The transformation cited by C. D. Mikkelsen, namely z = x( + jc), is convenient for scaling flow problems. Indeed, it has been used by Barbee and Mikkelsen in the case of steady, laminar, Newtonian tube flow developing from an initial flow of zero Reynolds number. In cited paper, authors selected an axial computational 0 oo) to allow space for an tube of [fixed radius] to run from zero to Extending infinitely also can be employed to advantage in other situations, such as where boundary conditions at finite distances are unknown or difficult to resolve. We consider that cases with the streamwise extent of computational domain presumed to be have constraint imposed and so a priori knowledge of fully developed flow length Lf is not needed. In our problems of interest, neither extent of flow field nor fully developed flow length are known or defined in advance. However, they are presumed to be finite. Therefore, in our opinion, transformation z = x/( + x) does not offer us advantages presented when is presumed infinite. Furthermore, for formulations based on case of infinite extent, our computer resource limitations (memory and speed) may encumber arriving at acceptable solutions to problems over reasonable periods of time. Therefore, an alternate route for numerical solution of partial differential equations involved had to be found. Our alternate route, as reported in our submission, involves applying Green's formula transformation to Galerkinweighted residual formulation of stream function and vorticity expression of Navier-Stokes equations. This technique causes downstream boundary integrals to vanish and thereby precludes either requirement for priori definition of spacial extent or extension to infinity. Consequently, our formulation does not require advantages that transformation z = x/( + x) has to offer when is presumed infinite.

The Effect of Nonlinear Heat Conduction on the Pressure-Coupled Response of Solid Propellants

A ZELDOVICH-NOVOZHILOV model for the pressurecoupled response of solid propellants was developed with heat conduction in the condensed phase governed by a relaxation time, non-Fourier relation. Results indicate that, although the relaxation time appears in a frequency-squared relaxation time product combination, the non-Fourier model has negligible effect on the response in the frequency domain of interest to combustion instability for reasonable values of relaxation time. For large values of this parameter, the resonance is shifted to lower frequencies, reduced in magnitude, and narrowed. Presented as Paper 82-1102 at the AIAA/SAE/ASME 18th Joint Propulsion Conference, Cleveland, Ohio, June 21-23, 1982; submitted June 25, 1982; synoptic received July 5, 1983. Copyright © American Institute of Aeronautics and Astronautics, Inc., 1982. All rights reserved. Full paper available from AIAA Library, 555 W. 57th Street, New York, N.Y. 10019. Price: Microfiche, $4.00; hard copy, $8.00. Remittance must accompany order. * Graduate Student, School of Aeronautics and Astronautics (currently, Engineer, Morton Thiokol, Inc., Huntsville, Ala.). Student Member AIAA. tSenior Researcher, School of Aeronautics and Astronautics. Associate Fellow AIAA.

Comment on "Generalized Inverse of a Matrix: The Minimization Approach "

Received March 29, 1977. Index categories: Vibration, Structural Dynamics. *Professor. Member AIAA. S and Decarli have given a useful summary of the application of the generalized inverse of a matrix to the solution of a set of linear equations. Although they have given the solution for the extended vector norm, there are some additional points concerning the case of overdetermined sets of equations that may be mentioned usefully. We consider the algebraic matrix equation:

Viscous Flowfield Calculations on Pointed Bodies at Angle of Attack in Nonuniform Freestreams

determined from Eq. (1) and the condition of constant static pressure and total enthalpy. The inviscid flowfield is then calculated from a modified form of Rakich's three-dimensional method of characteristics (NASA TN D-5341). Typical results for the non-dimensional pressure (pb — Pb/p^v^) and inviscid surface Mach Number (Mb) are shown in Fig. 2 for a 20° half-angle cone at a = 10° and Mx = 3. For a velocity defect of A = 0.2 the surface Mach number goes to unity at x = 0.95 (based on B = 10) in the windward plane, and the three-dimensional method of characteristics cannot be continued beyond this point. Member AIAA.

Comment on "Numerical Solution of the Boundary-Layer Equations"

Received August 25, 1967. * Research Scientist. Associate Member AIAA. ample, is demonstrated in Ref. 1 for the implicit finite-difference solution of Fliigge-Lotz and Blottner. The remark in Ref. 1 regarding Smith and Clutter's method is motivated by Eq. (9) of Ref. 2. That equation expresses the ^-derivatives by means of a three-point difference formula, presumably to achieve high accuracy. When the three-point formula is used to start the integration, it is necessary to specify the initial data at two previous stations instead of one. This is equivalent to specifying the function and its x-derivative at one station. The suggestion of the comment to use a two-point difference formula for the ^-derivative does, in principle, eliminate the difficulties in starting the integration.