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Showing papers in "AIAA Journal in 1976"


Journal ArticleDOI
TL;DR: A simplified procedure is presented for the determination of the derivatives of eigenvectors of nth order algebraic eigensystems, applicable to symmetric or nonsymmetric systems, and requires knowledge of only one eigenvalue and its associated right and left eigenavectors.
Abstract: A simplified procedure is presented for the determination of the derivatives of eigenvectors of nth order algebraic eigensystems. The method is applicable to symmetric or nonsymmetric systems, and requires knowledge of only one eigenvalue and its associated right and left eigenvectors. In the procedure, the matrix of the original eigensystem of rank (/?-!) is modified to convert it to a matrix of rank /?, which then is solved directly for a vector which, together with the eigenvector, gives the eigenvector derivative to within an arbitrary constant. The norm of the eigenvector is used to determine this constant and complete the calculation. The method is simple, since the modified n rank matrix is formed without matrix multiplication or extensive manipulation. Since the matrix has the same bandedness as the original eigensystems, it can be treated efficiently using the same banded equation solution algorithms that are used to find the eigenvectors.

878 citations


Journal ArticleDOI
TL;DR: In this article, a theory of solid friction damping of mechanical vibrations is presented that is based on a solid friction mathematical model previously proposed by the author, and a summary and improved description of the general analytic features of the solid friction model are given as necessary background for the theory.
Abstract: A theory of solid friction damping of mechanical vibrations is presented that is based on a solid friction mathematical model previously proposed by the author. A summary and improved description of the general analytic features of the solid friction model are given as necessary background for the theory. The Coulomb friction damped oscillator is analyzed to establish an approach to the treatment of a simple friction damped oscillator. The approach then is generalized to treat a more general model of friction where the author's model is used to describe friction force primarily as a function of displacement. The solid friction damped oscillator studied is a wire pendulum where solid friction enters via inelastic flexing of the wire at the support. Theoretical results are generalized to be applicable to other types of oscillators and other sources of solid friction. An expression for the decay rate of the oscillation amplitude envelope of an unforced oscillator is derived. The decay rate and an equivalent linear damping ratio are determined for several values of an exponent parameter in the solid friction model.

592 citations


Journal ArticleDOI
TL;DR: The concept of large eddies has been explored in the context of turbulent shear flows and their properties have been discussed in detail in the literature as mentioned in this paper, with a focus on the role of large structures in the development of turbulent flows.
Abstract: The problem of turbulent now continues to be an outstanding one in technology and in physics. Of the nine Dryden research lectures so far, four have been on some aspect of the turbulence problem. At meetings such as this one the turbulence problem is always the subject of some sessions and lurks in the background of many others; for example, separated now, combustion, jet noise, chemical lasers, atmospheric problems, etc. It is continually the subject of conferences, workshops and reviews. In his time Hugh Dryden wrote several reviews of turbulent now. In reading some of them again, one statement particularly relevant to the present lecture caught my attention: "-it is necessary to separate the random processes from the nonrandom processes. It is not yet fully clear what the random elements are in turbulent now." Neither is it fully clear what the nonrandom, orderly elements are, but some of them are beginning to be recognized and described. Generally the picture one has had of turbulence is of chaos and disorder, implicit in the name. Although it was known that organized motion could exist, superimposed on the background of "turbulence," for example, vortex shedding from a circular cylinder up to Reynolds numbers of 10^7, such examples were regarded as special cases closely tied to their particular geometric origins and not characteristic of "well-developed" turbulence. It was known that large structures are important in the development of turbulent shear flows and that these ought to possess some definable features. But even when the concept of a characteristic "big eddy" was explored, it was usually in the context of a statistical quantity. The earliest and most decisive attempts to define the form of such large eddies were made by Townsend and his students. In recent years it has become increasingly evident that turbulent shear flows do contain structures or eddies whose description is more deterministic than had been thought, possessing identifiable characteristics, existing for significant lifetimes, and producing recognizable and important events. More accurate descriptions of their properties, how they fit into the complete description of a turbulent flow, to what extent are they central to its development, and how they can be reconciled with the apparent chaos and disorder, are problems which are becoming of interest to an increasing number of researchers. It is the purpose of this lecture to describe some of these new developments. The discussion will draw largely on experiences from our own laboratory; it is not intended to be a complete survey. Other discussions of these ideas can be found in various recent publications.

499 citations


Journal ArticleDOI
TL;DR: In this article, the leading edge geometry of an NACA 0012 airfoil has been studied in incompressible flow at moderately large Reynolds numbers and three different types of stall were produced.
Abstract: Dynamic stall and unsteady boundary layer separation have been studied in incompressible flow at moderately large Reynolds numbers. By varying the leading-edge geometry of an NACA 0012 airfoil, three different types of stall were produced. For most of the configurations studied, including the basic NACA 0012 profile, dynamic stall was found not to originate with the bursting of a leading-edge laminar separation bubble, as is commonly believed. Instead, the vortex shedding phenomenon, which is the predominant feature of dynamic stall, appears to be fed its vorticity by the breakdown of the turbulent boundary layer.

462 citations


Journal ArticleDOI
TL;DR: In this article, the effect on drag of a temperature difference between the sphere and the gas is incorporated, which simplifies in the limit to certain equations derived from theory, and which offers significantly improved agreement with the experimental data.
Abstract: A CCURATE representations of the drag coefficients of -^Vspheres over a wide range of flow conditions are a necessary prerequisite to the calculation of gas-particle flows. For greatest utility these representations should be in a form suitable for computer calculations. Two such correlations, both still used in computer programs, were published prior to the collection of a considerable body of experimental data, and, consequently are outdated. A third representation, published subsequently, requires the use of tabular data, and is inaccurate in some flow regimes of interest. This paper presents a new correlation, suitable for utilization in computer calculations, which simplifies in the limit to certain equations derived from theory, and which offers significantly improved agreement with the experimental data. The flow regimes of interest include continuum, slip, transition, and molecular flow at Mach numbers up to 6, and at Reynolds numbers up to the laminar-turbulent transition. The effect on drag of a temperature difference between the sphere and the gas is incorporated.

449 citations


Journal ArticleDOI
TL;DR: In this article, the authors consider the application of explicit Isecond-order, one-sided or "upwind," difference schemes for the numerical solution of hyperbolic systems in conservation-law form.
Abstract: Explicit second-order upwind difference schemes in combination with spatially symmetric schemes can produce larger stability bounds and better numerical resolution than symmetric schemes alone. However, if conservation form is essential, a special operator is required for transition between schemes. An operational approach has been devised for deriving transition operators so that strict conservation and local consistency are maintained. Various aspects of hybrid schemes are studied numerically for model linear and nonlinear equations. To demonstrate the utility of combining two different algorithms, MacCormack's explicit, noncentered, second-order method is combined with a completely upwind version, and numerical solutions of the Euler equations are obtained for two-dimensional, transonic flows with embedded supersonic regions and shock I. Introduction "1T4 this paper we consider the application of explicit Isecond-order, one-sided or "upwind," difference schemes for the numerical solution of hyperbolic systems in conservation-law form = 0

386 citations


Journal ArticleDOI
TL;DR: In this paper, a large parameter solution procedure was adapted to the task of calculating closed-form approximate solutions for the pressure and lift of a flat-plate, infinite-span airfoil.
Abstract: A large parameter solution procedure of Schwartzschild and Landahl is adapted to the task of calculating closed-form approximate solutions for the pressure and lift of a flat-plate, infinite-span airfoil. Two general cases are treated: 1) the two-dimensional subsonic flow problem, in which the large parameter is the upwash frequency, and 2) the three-dimensional incompressible flow problem, in which the large parameter is the spanwise wavenumber of the upwash. For the first case, the four problems of a gust drifting with the freestream, a gust moving at other than the freestream velocity, a plunging motion, and a linear upwash are treated. For the second case, the two problems of a gust drifting with the freestream and a generalized gust moving at other than the freestream velocity are considered. Comparison of the solutions with available numerical results generally shows good agreement when the appropriate parameter is large. The solutions for the gust convecting with the freestream for both the two-dimensional compressible and the three-dimensional incompressible cases were derived previously by Adamczyk using the Wiener-Hopf technique.

228 citations


Journal ArticleDOI
TL;DR: In this paper, an experimental and theoretical study of transonic flow over a thick airfoil, prompted by a need for adequately documented experiments that could provide rigorous verification of viscous flow simulation computer codes, is reported.
Abstract: An experimental and theoretical study of transonic flow over a thick airfoil, prompted by a need for adequately documented experiments that could provide rigorous verification of viscous flow simulation computer codes, is reported. Special attention is given to the shock-induced separation phenomenon in the turbulent regime. Measurements presented include surface pressures, streamline and flow separation patterns, and shadowgraphs. For a limited range of free-stream Mach numbers the airfoil flow field is found to be unsteady. Dynamic pressure measurements and high-speed shadowgraph movies were taken to investigate this phenomenon. Comparisons of experimentally determined and numerically simulated steady flows using a new viscous-turbulent code are also included. The comparisons show the importance of including an accurate turbulence model. When the shock-boundary layer interaction is weak the turbulence model employed appears adequate, but when the interaction is strong, and extensive regions of separation are present, the model is inadequate and needs further development.

189 citations


Journal ArticleDOI
TL;DR: In this paper, a substructure coupling method is proposed which employs free-interface substructure modes supplemented by reduced flexibility, and a numerical comparison with Hou's freeinterface method is given.
Abstract: Benfield, et al. (1972) showed that among fixed-interface, free-interface, and hybrid substructure coupling methods, the fixed-interface methods as the most accurate and the free-interface methods are the least accurate. In the present note, a substructure coupling method is proposed which employs free-interface substructure modes supplemented by 'reduced flexibility.' Substructure coupling based on the improved substructure model is discussed, and a numerical comparison with Hou's free-interface method is given. To simplify representation of the method proposed, the substructure equations are developed first for constrained substructures, and then the equations representing substructures with rigid-body modes are given. Finally, the equations for coupling of substructures are derived. Example calculations are included.

166 citations


Journal ArticleDOI
TL;DR: In this paper, the authors used a 24° compression corner mounted on the wall of the high Reynolds number wind tunnel at Mach 2.85 and Redo = 1.33 million.
Abstract: An experimental study is described in which detailed mean-flow measurements are made in a shock waveboundary-layer interaction. The interaction is produced by a 24° compression corner mounted on the wall of the Princeton University high Reynolds number wind tunnel. The experiments are performed at Mach 2.85 and Redo = 1.33 million. A detailed mapping of the flowfield is presented, including separated region shape and location and velocity profiles. Results indicate a relatively straight zero-velocity line, a persistent downstream normal pressure gradient, and reverse velocities up to 16% of u^.

159 citations


Journal ArticleDOI
Chul Park1
TL;DR: In this article, it was shown that a previously derived semi-empirical equation for describing observed ablation rates of isotropic graphites cannot be applied to low-density flows containing dissociated oxygen.
Abstract: It is shown that a previously derived semiempirical equation for describing observed ablation rates of isotropic graphites cannot be applied to low-density flows containing dissociated oxygen. Experimentally determined reaction probabilities of isotropic graphites to molecular and atomic oxygen are used to calculate heat-transfer rates and stagnation-point ablation rates for typical conditions. Integrated mass losses are computed for a group of flight trajectories which start from geosynchronous orbit and enter earth's atmosphere in a skipping motion following near-elliptic decaying orbits. A comparison of the results with those obtained by the equation under question shows excellent agreement for steep trajectories, but large discrepancies for shallow trajectories. The differences are attributed to surface oxidation by atomic oxygen.

Journal ArticleDOI
TL;DR: In this article, a 1-in. supersonic jet was investigated at nominal jet Mach numbers of 1.5, 2.0, and 2.5 using a directional microphone system.
Abstract: The noise generated by a 1-in. supersonic jet was investigated at nominal jet Mach numbers of 1.5, 2.0, and 2.5. In particular, a quantity W, referred to as the apparent source strength per unit length, was determined along the jet axis using a directional microphone system. The integrated value of W along the jet axis was found to agree with the sound intensity obtained by a conventional microphone. This result is consistent with the a priori assumption that the jet may be described in terms of independent, spatially compact acoustic sources. The main finding of the investigation is the discovery of two distinct intense noise-producing regions in a jet having supersonic source velocities: the upstream region associated with Mach wave radiation, and a zone, located downstream of the potential cone, exhibiting radiation similar in character to that of a subsonic jet. An estimate of the radiated intensity associated with the Mach waves also is made.

Journal ArticleDOI
TL;DR: In this paper, an efficient time-splitting, second-order accurate numerical scheme was used to solve the complete Navier-Stokes equations for supersonic and hypersonic laminar flow over a two-dimensional compression corner.
Abstract: An efficient time-splitting, second-order accurate, numerical scheme is used to solve the complete Navier-Stokes equations for supersonic and hypersonic laminar flow over a two-dimensional compression corner. A fine, exponentially stretched mesh spacing is used in the region near the wall for resolving the viscous layer. Good agreement is obtained between the present computed results and experimental measurement for a Mach number of 14.1 and a Reynolds number of 1.04 x 10(exp 5) with wedge angles of 15 deg, 18 deg, and 24 deg. The details of the pressure variation across the boundary layer are given, and a correlation between the leading edge shock and the peaks in surface pressure and heat transfer is observed.

Journal ArticleDOI
TL;DR: In this article, a simulative study is presented to clarify the characteristics of the flowfield in the grain port of a solid rocket motor under erosive burning conditions, where the experiments are performed by using channels with porous walls and air flow at ambient temperature.
Abstract: A simulative study is presented to clarify the characteristics of the flowfield in the grain port of a solid rocket motor under erosive burning conditions. The experiments are performed by using channels with porous walls and air flow at ambient temperature. It is shown that the boundary layer along the port develops rapidly similarly to pipe flow, which differs from Green's prediction. An empirical relation, similar to LenoirRobillard's semiempirical relation is obtained by correlating turbulence intensity with velocities and geometrical parameters of the channel. Thus, it is implied that an intimate relation exists between turbulence intensity and erosive burning.

Journal ArticleDOI
TL;DR: In this article, the singular perturbation technique was used to obtain an approximate solution to the aircraft minimum time-to-climb problem, which is the same as the one we consider in this paper.
Abstract: Application of singular perturbation techniques to trajectory optimization problems of flight mechanics is discussed. The method of matched asymptotic expansions is used to obtain an approximate solution to the aircraft minimum time-to-climb problem. Outer, boundary-layer, and composite solutions are obtained to zeroth and first orders. A stability criterion is derived for the zeroth-order boundary-layer solutions (the theory requires a form of boundary-layer stability). A numerical example is considered for which it is shown that the stability criterion is satisfied and a useful numerical solution is obtained. The zeroth-order solution proves to be a poor approximation, but the first-order solution gives a good approximation for both the trajectory and the minimum time-to-climb. The computational cost of the singular perturbation solution is considerably less than that of a steepest descent solution. Thus singular perturbation methods appear to be promising for the solution of optimal control problems.

Journal ArticleDOI
TL;DR: In this article, a cubic spline collocation procedure was developed for numerical solution of partial differential equations and the spline procedure was reformulated so that the accuracy of the second-derivative approximation is improved and parallels that previously obtained for lower derivative terms.
Abstract: A cubic spline collocation procedure was developed for the numerical solution of partial differential equations This spline procedure is reformulated so that the accuracy of the second-derivative approximation is improved and parallels that previously obtained for lower derivative terms The final result is a numerical procedure having overall third-order accuracy of a nonuniform mesh Solutions using both spline procedures, as well as three-point finite difference methods, are presented for several model problems

Journal ArticleDOI
TL;DR: The creation of an efficient automated capability for minimum weight design of structures is reported, and the efficiency of the ACCESS 1 program is demonstrated by giving results for several example problems.
Abstract: The creation of an efficient automated capability for minimum weight design of structures is reported. The ACCESS 1 computer program combines finite element analysis techniques and mathematical programming algorithms using an innovative collection of approximation concepts. Design variable linking, constraint deletion techniques and approximate analysis methods are used to generate a sequence of small explicit mathematical programming problems which retain the essential features of the design problem. Organization of the finite element analysis is carefully matched to the design optimization task. The efficiency of the ACCESS 1 program is demonstrated by giving results for several example problems.


Journal ArticleDOI
TL;DR: In this article, the floating shock fitting technique is examined for the calculation of internal two-dimensional flows with arbitrary number of shock waves and contact surfaces and a new procedure, based on the coalescence of characteristics, is developed to detect the formation of shock wave.
Abstract: The floating shock fitting technique is examined. Second-order difference formulas are developed for the computation of discontinuities. A procedure is developed to compute mesh points that are crossed by discontinuities. The technique is applied to the calculation of internal two-dimensional flows with arbitrary number of shock waves and contact surfaces. A new procedure, based on the coalescence of characteristics, is developed to detect the formation of shock waves. Results are presented to validate and demonstrate the versatility of the technique.

Journal ArticleDOI
TL;DR: In this article, a second-order variational analysis using a frequency-domain version of the classical Jacobi optimality condition is proposed to improve fuel efficiency of cycle cruise paths.
Abstract: For a fairly general aircraft model and a large class of drag models, steady-state cruise for a long time span is nonoptimal with respect to fuel economy. This is proved by a second-order variational analysis, using a frequency-domain version of the classical Jacobi (conjugate point) optimality condition. The variational analysis suggests a sinusoidal perturbation away from steady-state cruise which improves fuel economy (as confirmed numerically), but is still not optimal. The form of the optimal trajectory for long duration cruise is unknown. However, two intuitive reasons for improved fuel economy using cycle cruise paths are given.

Journal ArticleDOI
TL;DR: In this paper, a liquid finite element formulation which includes the potential energy due to compression but neglects the density change has been developed, where both kinetic and potential energy are expressed as functions of nodal displacements.
Abstract: A liquid finite element formulation which includes the potential energy due to compression but neglects the density change has been developed. Both kinetic and potential energy are expressed as functions of nodal displacements. Thus, the formulation is similar to that used for structural elements, with the only differences being that 1) the fluid can possess gravitational potential, and 2) the constitutive equations for fluid contain no Shear coefficients. Using this approach, structural and fluid elements can be used interchangeably in existing efficient sparse matrix structural computer programs such as SPAR. The theoretical development of the element formulations and the relationships of the local and global coordinates are shown. Solutions of fluid slosh, liquid compressibility, and coupled fluid-shell oscillation problems which were completed using a temporary digital computer program are shown. The frequency correlation of the solutions with classical theory is excellent.

Journal ArticleDOI
TL;DR: In this article, a method for treating certain troublesome boundary conditions in the numerical solution of time-dependent incompressible viscous flow problems is presented, based on an integral representation for the velocity vector which contains the entire kinematics of the problem, including the boundary conditions of concern.
Abstract: A method for treating certain troublesome boundary conditions in the numerical solution of time-dependent incompressible viscous flow problems is presented. This method is developed on the basis of an integral representation for the velocity vector which contains the entire kinematics of the problem, including the boundary conditions of concern. It is shown that for the exterior flow problem the freestream condition is satisfied at infinity exactly, and the need to treat a farfield condition is removed by the use of the integral representation. The distribution of a nonvelocity variable on the solid boundary, i.e., the "extraneous" boundary condition needed for both the exterior and the interior flows, are shown to be governed by the kinematics of the problem. The method is shown to accurately follow the local generation of vorticity on the solid boundary computationally.


Journal ArticleDOI
TL;DR: In this article, a two-dimensional compression corner and axisymmetric flare geometries were used in a study of shock wave interaction with a compressible turbulent boundary layer.
Abstract: Two-dimensional compression corner and axisymmetric flare geometries were used in this study of shock wave interaction with a compressible turbulent boundary layer. The study was carried out at a Mach number of 2.9 and over a Reynolds number range of 10 5

Journal ArticleDOI
TL;DR: In this paper, the complete set of Navier-Stokes equations for a compressible flow with an impinging shock was computed using a time-dependent finite-difference method.
Abstract: Two-dimensional viscous blunt body flows with an impinging shock have been computed using a time-dependent finite-difference method which solves the complete set of Navier-Stokes equations for a compressible flow. For low Reynolds number flows, the entire flow field, including the bow shock and impinging shock, has been captured in the computation. For higher Reynolds number flows, the bow shock is treated as a discontinuity across which the Rankine-Hugoniot equations are applied, while the boundary layer and interaction regions are captured as before. Using this latter shock-fitting approach, a Type III shock interaction flow field has been computed with flow conditions corresponding to the space shuttle orbiter freestream conditions at 61 km (200,000 ft).

Journal ArticleDOI
TL;DR: In this article, four different algebraic eddy viscoisity models are tested for viability to achieve turbulence closure for the class of flows considered, ranging from an unmodified boundary-layer mixing-length model to a relaxation model incorporating special considerations for the separation bubble region.
Abstract: The two-dimensional Reynolds averaged compressible Navier-Stokes equations are solved using MacCormack's second-order accurate explicit finite difference method to simulate the separated transonic tur- bulent flowfield over an airfoil. Four different algebraic eddy viscoisity models are tested for viability to achieve turbulence closure for the class of flows considered. These models range from an unmodified boundary-layer mixing-length model to a relaxation model incorporating special considerations for the separation bubble region. Results of this study indicate the necessity for special attention to the separated flow region and suggest limits of applicability of algebraic turbulence models to these separated flowfield. each of these studies the time-dependent Reynolds averaged Navier-Stokes equations for two-dimensional compressive flow are used and tur- bulence closure is achieved by means of model equations for the Reynolds stresses. Wilcox1'2 used a first-order accurate numerical scheme and the two equation differential tur- bulence model of Saffman 12 to simulate the supersonic shock boundary-layer interaction experiment of Reda and Mur- phy 13 and the compression corner flow of Law.14 Good quan- titative agreement with the Reda and Murphy data was ob- tained, but only the qualitative features of the compression corner flow were well simulated. Using a more sophisticated second-order accurate numerical scheme, Baldwin3'4 con- sidered both the two equation differential model of Saffman and a simpler algebraic mixing-length model to simulate the hypersonic shock boundary-layer interaction experiment of Holden.15 He found the more elaborate model of Saffman to yield somewhat better results than the algebraic model, but at the cost of considerably more computing time. Good quan- titative agreement with experiment was not obtained with either model. Following Baldwin's approach all subsequent investigations have been performed using the more rigorous second-order accurate numerical scheme of Mac- Cormack.17'18 Deiwert5'6'11 considered an algebraic mixing- length model to simulate the transonic airfoil experiment of McDevitt et al. 16 while Horstman et al. 8 used a similar ap- proach to simulate their hypersonic shock boundary-layer ex- periment on an axisymmetric cylinder. In each of these studies, while qualitative features of the flows were described well, good quantitative agreement with experiment in the in- teraction regions was not obtained. Using a relaxing turbulence model Shang and Hankey7 simulated the compression corner flow of Law, and Baldwin and Rose10 simulated the flat plate flow of Reda and Murphy. In each of these studies the relaxing model was found to per- form significantly better than the simpler algebraic model and, according to Shang and Hankey, provided significantly better comparisons with measurements than were obtained by Wilcox using the two equation differential model of Saffman. In each of these studies it was essential that the full Navier- Stokes equations be considered to describe the viscous- inviscid interaction and the elliptic nature of separating-

Journal ArticleDOI
TL;DR: In this article, the von Karman-Trefftz transformation was extended to multiply-connuffed regions, which can be used for the analysis of flow over multielement airfoils.
Abstract: With the advent of large-scale digital computers, numerical solution of the transonic flow equations in two dimensions has proven feasible. In such calculations it is convenient, and often crucial, to map the region of in- terest conformally onto a simple domain so that a simple finite difference system can be employed. This paper updates a well-known airfoil conformal mapping method (Theodorsen and Garrick) with modern techniques, greatly enhancing the mapping speed and accuracy while simplifying the analysis. Also, a powerful new class of conformal transformations is introduced, and applied to a two-element airfoil. N the calculation of transonic flow over airfoils it has proven convenient to map the infinite region of the flow exterior to an airfoil conformally onto a finite region, such as the interior of a circle, as in Sells,l Melnik and Ives,2 Jameson,3 and Bauer, Garabedian, and Korn.4 Such a mapping places a fine grid in the physical plane where needed, while retaining a uniform grid in the (con- formally mapped) computational plane. The gradients of flow properties in the computational plane are smaller than in the physical plane, so that finite difference approximations in the computational plane are more accurate than in the physical plane. This allows the use of a rather coarse grid. It is easier to apply boundary conditions in the mapped plane than it is in the physical plane, as the boundaries are coordinate lines in the mapped plane. The conformal mapping ensures that the nonlinear partial differential equations in the computational plane are only slightly more complicated than those in the physical plane.1'5 To take advantage of the above con- siderations, there is a need for conformal mapping techniques that are simple and rapid. This paper presents a conformal mapping technique which is at least an order of magnitude faster than conventional mapping techniques. In analogy to the fast Fourier transform techniques utilized here, the present work can be considered a "fast conformal mapping." This paper also introduces a powerful new class of con- formal transformations. These new transformations are an extension of the well-known von Karman-Trefftz trans- formation to multiply-conn ected regions, and can be used in the analysis of flow over multielement airfoils. They have the property that any number N of airfoil elements are simultaneously (with one application of the transformation) mapped to TV near circles. One member of this new class of transformations is used in mapping the region exterior to a two-element airfoil onto the region between two concentric circles. By combining the known solution for incompressible flow between two con- centric circles6 with this two-element mapping, we can very simply solve for the inviscid incompressible flow over a two- element air foil.

Journal ArticleDOI
TL;DR: The method is used to design several well-known truss-type structures and the results are shown to compare favorably with previous results obtained using mathematical programing algorithms and other optimality criteria methods.
Abstract: This paper presents a recursive design method for the minimum weight design of linear elastic redundant structures subject to multiple independent static loading conditions and with behavioral constraints on allowable element stresses and nodal displacements and constraints on design variables. This recursive method is based on the Kuhn-Tucker necessary conditions for a local optimum and gives, upon completion, a local optimum design. An iterative procedure is used to resize the structure until a design satisfying the Kuhn-Tucker necessary conditions is obtained. For resizing, it is necessary to identify the current near-active (critical) constraints and to use this data to construct the Kuhn-Tucker test. If the current design is not converged, then the information from the test is used to resize the design variables and improve the design. Each iteration or redesign requires only the solution of a set of linear algebraic equations equal in number to the number of currently active constraints. The method is used to design several well-known truss-type structures, and the results are shown to compare favorably with previous results obtained using mathematical programing algorithms and other optimality criteria methods.

Journal ArticleDOI
TL;DR: In this paper, the authors investigated the effect of the initial boundary layer on the performance of free jet mixing flows and found that the effects of boundary layers on mixing rates are often significant.
Abstract: Theme A MAJOR problem in many jet mixing flows is scaling, since research and development tests are usually done at much lower Reynolds numbers than the full scale application. It often occurs that if no special tripping is done, the initial boundary layer in the jet of a subscale test is laminar, while the full scale hardware will have a turbulent boundary layer. One of the primary goals of our investigation of initial boundary layer effects was to evaluate the importance of this difference on the accuracy of full scale predictions deduced from model tests. Secondary objectives included a study of the reasons for the differences in core lengths obtained by different investigators and possible isolation of particular boundary-layer characteristics that enhance turbulent mixing rates. Both axisymmetric and two-dimensional free jet flowfields exhibit a power law dependence for the decay of centerline velocity with distance from nozzle exit at points that are well downstream of the initial mixing region. However, variations in shear layer properties close to the jet exit can cause the centerline velocity profile to shift upstream or downstream under different operating conditions. Measurements made by different investigators in flowfields produced by similar nozzles frequently show significant variations in core length; the far field centerline velocity decay, however, follows a predictable variation with axial distance. Much of this earlier research on turbulent mixing in free jets has addressed cases where the effects of initial conditions were intentionally minimized. The few cases in which data regarding initial conditions are available indicate that the effects of nozzle boundary layers on mixing rates are often significant. While there are many aspects of turbulent mixing that warrant further investigation, it was the consensus of both the attendees and the review committee at the Langley Working Conference on Free Turbulent Shear Flows that the effect of initial conditions is a primary area requiring study. In this paper we describe some effects of nozzle wall boundary-layer conditions on incompressible air jets discharging into quiescent surroundings. This investigation included the influence of the boundary-layer state on the turbulent mixing layer surrounding axisymmetric and two-dimensional free jets and on time averaged measurements obtained in these flows.

Journal ArticleDOI
TL;DR: In this paper, a previously developed fatigue failure criterion for laminae in a state of plane stress is used to establish a failure criteria for first fatigue failure of balanced laminates, and an extensive tension-tension fatigue testing program for E-glass/epoxy angle plies with various reinforcement angles has been carried out to test the validity of the theory.
Abstract: A previously developed fatigue failure criterion for laminae in a state of plane stress is used to establish a failure criterion for first fatigue failure of balanced laminates An extensive tension-tension fatigue testing program for E-glass/epoxy angle plies with various reinforcement angles has been carried out in order to test the validity of the theory It has been found that for certain ranges of reinforcement angle there is good agreement between theory and experiment, whereas for other ranges of reinforcement angle this is not the case