Showing papers in "AIAA Journal in 1982"
TL;DR: In this paper, the Taylor series expansion technique was used to systematically investigate the proper behavior of the turbulent shear stress and the kinetic energy and its rate of dissipation near a solid wall.
Abstract: 2~5 However, the effects of the kinematic viscosity on the turbulence structure were ignored in many of these treatments. Consequently, the exact boundary conditions at the wall cannot be used when the turbulence Reynolds number is not high as, e.g., in flows with rapid expansions or near the transition/turbulence interface. The general goal of the present investigation was to develop a single transport model from the Navier-Stokes equation for accurate predictions of skin friction, heat transfer, and fluctuating kinetic energy distributions in transitional and turbulent flow regimes. As a first step toward this general goal, a new turbulence model valid down to the solid wall is formulated in this paper. Turbulence model equations which provide predictions of the flow within the viscous layer adjacent to the wall have been proposed by several investigators.3'4'6'7 Although the general approach of the present model is the same as that of Jones and Launder,3 the detailed proposals are substantially different. In the present study, the Taylor series expansion technique was used to systematically investigate the proper behavior of the turbulent shear stress and the kinetic energy and its rate of dissipation near a solid wall. The results were used in developing a new turbulence model which retains the proper physical behavior of the balance between the dissipation and the molecular diffusion of the turbulent kinetic energy at the solid wall. The model was applied to the problems of a fully developed turbulent channel flow and of a turbulent boundary-layer flow over a flat plate. Results on skin friction, the distribution of mean velocity, turbulent shear stress, and turbulent kinetic energy will be presented and compared with available experimental data and with the theory of Jones and Launder.
1,322 citations
686 citations
TL;DR: In this article, an efficient method for finite element modeling of three-layer laminates containing a viscoelastic layer is described, and modal damping ratios are estimated from undamped normal mode results by means of the modal strain energy method.
Abstract: An efficient method is described for finite element modelling of three-layer laminates containing a viscoelastic layer. Modal damping ratios are estimated from undamped normal mode results by means of the modal strain energy method. Comparisons are given between results obtained by the MSE method implemented in NASTRAN, by various exact solutions for approximate governing differential equations, and by experiment. Results are in terms of frequencies, modal damping ratios, and mechanical admittances for simple beams, plates, and rings. Application of the finite element -- MSE method in design of integrally damped structures is discussed.
542 citations
TL;DR: In this paper, a second-order accurate method for solving viscous flow equations has been proposed that preserves conservation form, requires no block or scalar tridiagonal inversions, is simple and straightforward to program (estimated 10% modification for the update of many existing programs), and should easily adapt to current and future computer architectures.
Abstract: Although much progress has already been made In solving problems in aerodynamic design, many new developments are still needed before the equations for unsteady compressible viscous flow can be solved routinely. This paper describes one such development. A new method for solving these equations has been devised that 1) is second-order accurate in space and time, 2) is unconditionally stable, 3) preserves conservation form, 4) requires no block or scalar tridiagonal inversions, 5) is simple and straightforward to program (estimated 10% modification for the update of many existing programs), 6) is more efficient than present methods, and 7) should easily adapt to current and future computer architectures. Computational results for laminar and turbulent flows at Reynolds numbers from 3 x 10(exp 5) to 3 x 10(exp 7) and at CFL numbers as high as 10(exp 3) are compared with theory and experiment.
326 citations
TL;DR: In this paper, the authors investigated the effect of modifications to the nozzle exit on the fundamental screech tone of a jet operating under underexpanded sonic nozzles, and found that a large reduction of the screech amplitude can be obtained from modifications, although the extent of this suppression is mode dependent.
Abstract: Jet screech from underexpanded sonic nozzles has been investigated experimentally. Multiple screech modes, or stages, are found to be present at some jet operating conditions. The fundamental screech tone of each mode attains a maximum amplitude at about 20 deg from the inlet axis, with higher harmonics exhibiting multiple lobes. The directivity of each harmonic is predicted quite well from a stationary array of acoustic monopoles, with phasing between consecutive monopoles determined by the shock cell spacing and eddy convection velocity. Large reduction of screech amplitude can be obtained from modifications to the nozzle exit, although the extent of this suppression is mode dependent.
246 citations
TL;DR: In this paper, the laminar separation, transition, and turbulent reattachment near the leading edge of a two-dimensional NACA 663 -018 airfoil were investigated using a low-speed, smoke visualization wind tunnel.
Abstract: The laminar separation, transition, and turbulent reattachment near the leading edge of a two-dimensional NACA 663 -018 airfoil were investigated using a low-speed, smoke visualization wind tunnel. Lift and drag force measurements were made using an external strain gage balance for a chord Reynolds number range of 40,GOO400,000. An extensive flow visualization study was performed and correlated with the force measurements. Experiments were also conducted with distributed surface roughness at the leading edge and external acoustic excitation to influence the development of the airfoil boundary layer. This study delineates the effects of angle of attack and chord Reynolds number on the separation characteristics and airfoil performance. Nomenclature c = model chord cd = section profile drag coefficient (uncorrected) cf = section lift coefficient (uncorrected) Cp = pressure coefficient / = acoustic frequency, Hz R = reattachment location Rc = Reynolds number based on chord length, U^ civ S = separation location T = location of approximate end of transition £/«, = freestream velocity x/c = nondimensional distance along chord a = angle of attack v - kinematic viscosity
240 citations
TL;DR: A computational procedure for generating three-dimensional nonorthogonal surface-fitted mesh systems around wing-fuselage configurations is presented, based on the concept of transfinite interpolation, which makes it possible to generate single-block mappings with geometry data specified only at the outer boundaries of the computational domain.
Abstract: A computational procedure for generating three-dimensional nonorthogonal surface-fitted mesh systems around wing-fuselage configurations is presented. The method is based on the concept of transfinite interpolation, which has been extended to handle very general mapping function specifications at the boundaries, thereby making it possible to generate single-block mappings with geometry data specified only at the outer boundaries of the computational domain. Since it is a direct algebraic mapping technique, the method is very inexpensive in terms of computer cost. Different types of possible mappings are compared with respect to resolution and economy of nodal points. A procedure for a novel type of mapping, designated type O-O, is described and several plots of generated grids demonstrate the capabilities of the method. The singular lines inherent in every three-dimensional mesh for this type of surface geometry are also discussed.
231 citations
TL;DR: In this article, the authors derived the sensitivity equations that yield the sensitivity derivatives directly, which avoids the costly and inaccurate "perturb-and-reoptimize" approach, and examined the solvability of the equations.
Abstract: Solution of the optimum sensitivity problem yields the values of derivatives of the optimal objective function and design variables with respect to those physical quantities which were kept constant as problem parameters during optimization. Examples of these sensitivity derivatives might include derivatives of cross-section al area and structural mass with respect to allowable stress and derivatives of fuel consumed and wing aspect ratio with respect to aircraft range. Derivation of the sensitivity equations that yield the sensitivity derivatives directly, which avoids the costly and inaccurate "perturb-and-reoptimize" approach, is discussed and solvability of the equations is examined. The equations apply to optimum solutions obtained by direct search methods as well as those generated by procedures of the sequential unconstrained minimization technique (SUMT) class. Applications are discussed for the use of the sensitivity derivatives in extrapolation of the optimal objective function and design variable values for incremented parameters, optimization with multiple objectives, and decomposition of large optimization problems. Several aspects of these applications and verification of the sensitivity equations are presented through numerical examples.
212 citations
TL;DR: In this article, the boundary-layer and disturbance equations are formulated in a general, orthogonal, curvilinear system of coordinates constructed from the inviscid flow over a curved surface.
Abstract: A formal analysis of Goertler-type instability is presented. The boundary-layer and disturbance equations are formulated in a general, orthogonal, curvilinear system of coordinates constructed from the inviscid flow over a curved surface. Effects of curvature on the boundary-layer flow are analyzed. The basic approximation for the disturbance equations is presented and solved numerically. Previous analyses are discussed and compared with our analysis. It is shown that the general system of coordinates developed in this analysis and the correct order-of-magnitude analysis of the disturbance velocities with two velocity scales leads to a rational foundation for future work in Goertler vortices.
210 citations
TL;DR: In this article, the equations of motion were numerically integrated utilizing a total of 36 elements to define the geometry of the corer and the maximum bending moments occurred at the location of the step taper and at the junction between the afterbody and the upper portion of the barrel.
Abstract: Discussion of Results The equations of motion were numerically integrated utilizing a total of 36 elements to define the geometry of the corer. The corer was allowed to free-fall for 3 s, at which time the global variables had the following values: u = 23.6 m/s, w = 2.3 m/s, 4 = 0.05 rad/s, 6 = 6.4 deg, ^=-0.8 m, Ze = 31.0 m. The corer assumed an elongated inverted-S shape. At ^ = 3 s, maximum bending moments occurred at the location of the step taper and at the junction between the afterbody and the upper portion of the barrel. The local element deflection, slope, and bending moment at these locations are given in Table 1 and are approximate due to the local variables being measured at discrete locations along the length of the corer. It is unlikely that the corer will actually experience such large bending moments during free-fall. However, it is intuitive that large bending moments will occur at the barrel/afterbody junction and may even become larger during penetration.
200 citations
TL;DR: In this paper, the authors report results from comprehensive pressure tests on an ogive cylinder in the low-turbulence 12-ft pressure wind tunnel at Ames Research Center, which consist of detailed pressure distributions over a wide range of Reynolds numbers and angles of attack (20 to 90 deg).
Abstract: This paper reports results From comprehensive pressure tests on an ogive cylinder in the low-turbulence 12-ft pressure wind tunnel at Ames Research Center. The results consist of detailed pressure distributions over a wide range of Reynolds numbers (0.2 x 10(exp 6) to 4.0 x 10(exp 6)) and angles of attack (20 to 90 deg). Most important, the tests encompassed a complete coverage of different roll orientations. This variation of roll orientation is shown to be essential in order to fully define all the possible flow conditions. When the various roll-angle results are combined, it is possible to interpret correctly the effects of changing angle of attack or Reynolds number. Two basic mechanisms for producing asymmetric flow are identified. One mechanism operates in both the laminar and the fully turbulent separation regimes; this mechanism Is the one qualitatively described by the impulsive flow analogy. The other mechanism occurs only in the transitional separation regime. This asymmetric flow has the same form as that found in the two-dimensional cross flow on a circular cylinder in the transitional flow regime. Finally, these results make it possible to draw up critical Reynolds number boundaries between the laminar, transitional, and fully turbulent separation regimes throughout the angle-of-attack range from 20 to 90 deg.
TL;DR: In this paper, the authors investigated the peak shock noise from unheated convergent nozzles and found that the relative importance of shock noise with respect to jet-mixing noise is maximum near the pressure ratio at which a Mach disk begins to form in the jet.
Abstract: Broadband shock noise from supersonic jets is investigated through acoustic measurements in both the near and far fields. The peak Helmholtz number of broadband shock noise from unheated convergent nozzles is found to be independent of nozzle pressure ratio when based on the length of the shock cells and the ambient speed of sound. Excellent agreement between power spectral densities measured at various far-field angles is obtained at and above the peak shock noise frequency when source convection and directivity effects are included. The directivity of broadband shock noise is found to be pointed in the upstream direction, with omnidirectionality being approached only at high pressure ratios. For both convergent and convergent-divergent nozzles, the relative importance of shock noise with respect to jet-mixing noise is found to be maximum near the pressure ratio at which a Mach disk begins to form in the jet. Near-field measurements point to a limited portion of the shock cell system as the region of dominant broadband noise emission from a highly underexpanded convergent nozzle.
TL;DR: In this article, high-speed schlieren cinematography combined with synchronized pressure transducer records was used to investigate the mechanism of combustion instabilities leading to flashback in a combustion chamber with a rearward facing step acting as a flameholder.
Abstract: High-speed schlieren cinematography, combined with synchronized pressure transducer records, was used to investigate the mechanism of combustion instabilities leading to flashback. The combustion chamber had an oblong rectangular cross-section to model the essential features of planar flow, and was provided with a rearward facing step acting as a flameholder. As the rich limit was approached, three instability modes were observed: (1) humming - a significant increase in the amplitude of the vortex pattern; (2) buzzing - a large-scale oscillation of the flame; and (3) chucking - a cyclic reformation of the flame, which results in flashback. The mechanism of these phenomena is ascribed to the action of vortices in the recirculation zone and their interactions with the trailing vortex pattern of the turbulent mixing layer behind the step.
TL;DR: In this article, the authors describe some currently available models for calculating turbulent stresses and heat or mass fluxes in incompressible flow which are more generally applicable than the Prandtl mixing-length hypothesis.
Abstract: The paper describes some currently available models for calculating turbulent stresses and heat or mass fluxes in incompressible flow which are more generally applicable than the Prandtl mixing-length hypothesis. These include models employing transport equations for the intensity and the length scale of the turbulent motion, notably the k-t model, as well as second-order closure schemes based on transport equations for the turbulent stresses and heat or mass fluxes themselves. The individual models are introduced briefly, their merits and demerits are discussed, and typical examples of calculations relevant to aerospace problems are presented.
TL;DR: In this paper, the effects of blade mistiming on the aeroelastic stability and response of a cascade in incompressible flow were investigated, and it was shown that the mistuning has a beneficial effect on the coupled bending-torsion and uncoupled torsion flutter.
Abstract: This paper presents an investigation of the effects of blade mistiming on the aeroelastic stability and response of a cascade in incompressible flow. The aerodynamic, inertial, and structural coupling between the bending and torsional motions of each blade and the aerodynamic coupling between the blades are included in the formulation. A digital computer program was developed to conduct parametric studies. Results indicate that the mistuning has a beneficial effect on the coupled bending-torsion and uncoupled torsion flutter. The effect of mistuning on forced response, however, may be either beneficial or adverse, depending on the engine order of the forcing function. Additionally, the results illustrate that it may be feasible to utilize mistuning as a passive control to increase flutter speed while maintaining forced response at an acceptable level. [A ] [Ar] {AD } { ADr } a b c [D], [Ds [E] E(s,r) [G],[G S
TL;DR: The use of numerical techniques in structural optimization is emphasized here because it provides insight into the design problem and because it often provides theoretical lower bounds against which more practical designs may be judged.
Abstract: ' Introduction T concept of optimization is intrinsically tied to natural phenomena as well as to the human desire to excel. Sir George Cayley (1773-1857) measured the shape of a trout and noted, without mathematical proof, that the trout was ideally proportioned to minimize flow resistance. Theodore von Kdrmdn observed that this is precisely the shape of a lowdrag airfoil. Oliver Wendell Holmes (1809-1894), in his classic verse, "The Deacon's Masterpiece; or, The Wonderful OneHoss Shay," recorded man's desire to produce a uniformly strong, durable product. In this case it was the structural design of a shay to last a hundred years. Perhaps the first analytical work in structural optimization was by Maxwell in 1869, followed by the better-known work of Michell in 1904. These works provided theoretical lower bounds on the weight of trusses, and, although highly idealized, offer considerable insight into the structural optimization problem and the design process. The 1940s and early 1950s saw development of component optimization in such works as Shanley's Weight-Strength Analysis of Aircraft Structures. Also during this period, availability of the digital computer led to application of linear programming techniques to plastic design of frames, for example, the work of Heyman. This early numerical work is particularly significant in that it used mathematical programming techniques developed in the operations research community to solve structural design problems. Schmit in 1960 was the first to offer a comprehensive statement of the use of mathematical programming techniques to solve the nonlinear-inequality-constrained problem of designing elastic structures under a multiplicity of loading conditions. This work is significant, not only in that it ushered in an era of structural optimization, but also because it offered a new philosophy of engineering design which is only now beginning to be broadly applied. In Ref. 9 Schmit provides an excellent historical review of the development of this concept. Although this discussion will emphasize numerical design techniques, it is important to note that there has been an extensive amount of research in analytical methods of design. That work, although sometimes lacking the practicality of being applied to realistic structures, is nonetheless of fundamental importance because it provides insight into the design problem and because it often provides theoretical lower bounds against which more practical designs may be judged. References 10 and 11 provide an extensive review of the state-of-the-art in analytical design techniques. It is the use of numerical techniques in structural optimization that is emphasized here. The purpose is not to offer a tutorial on optimization or a comprehensive literature survey, although such works are referenced. Rather, it is to look briefly at the short history of modern structural optimization and assess the state-of-the-art from a somewhat more philosophical viewpoint. In this way we may begin to understand the ramifications of this fascinating approach to design. By learning what is now possible and what is not now possible, we may encourage the use of these techniques by practicing designers as well as identify research and development needs of the future.
TL;DR: In this paper, a model combustor composed of two confined coaxial swirling jets under noncombusting conditions is presented, where mean flow results are obtained for five different flow conditions to determine the effect of outer swirl on the recirculation zone (which is used for flame stabilization under combustion conditions).
Abstract: Flow measurements in a model combustor composed of two confined coaxial swirling jets under noncombusting conditions are presented. Mean flow results are obtained for five different flow conditions to determine the effect of outer swirl on the recirculation zone (which is used for flame stabilization under combustion conditions). As the outer swirl magnitude is first decreased from maximum counter-swirl to zero and then increased to give co-swirl conditions, the size and the reverse flow velocity in the recirculation zone diminish. Detailed time mean and fluctuating flow measurements are obtained for a co-swirl and a counter-swirl condition with a directional pitot probe and hot-wire anenometry. For these two cases, recirculation zone occurs only with counter-swirl, near the exit of the inner jet. The recirculation zone is in the form of a one celled toroidal vortex having very low swirl velocity. Axial development of fluctuation levels, energy dissipation rates and turbulence length scales are described for the two flow conditions. Spectral analysis reveals periodic oscillations in both flows. The oscillations originate from the inner jet and up to 4 harmonics are observed. The fundamental frequencies are comparable to the rotational frequencies of the inner jets at the exit which are approximately under solid body rotation. The significance of the present results for the combustion process is discussed.
TL;DR: In this article, a generalization and numerical implementation of the Stein-Hedgepeth (1961) continuum theory was presented for the analysis of partly wrinkled membranes, such as the tensioned membrane surfaces which are found in spacecraft structural components.
Abstract: A generalization and numerical implementation is presented of the Stein-Hedgepeth (1961) continuum theory for the analysis of partly wrinkled membranes, such as the tensioned membrane surfaces which are found in spacecraft structural components. The approach has its basis in experimental observations showing that when wrinkles develop in a membrane parallel to the x direction, the associated overall contraction in the y direction exceeds that predicted by the Poisson's ratio effect.
TL;DR: In this article, a simple method for calculating the unsteady aerodynamic loadings on harmonically oscillating thin wings in subsonic flow has been developed, based on a concept of concentrated lift forces.
Abstract: A simple method for calculating the unsteady aerodynamic loadings on harmonically oscillating thin wings in subsonic flow has been developed. The method is based on a concept of concentrated lift forces. The wing is divided into the element surfaces on which lift distributions are represented by single concentrated lift forces. Since the procedure does not include any quadratures, it can be applied easily to calculate the unsteady aerodynamic loadings on complex planform wings even when they have partial span control surfaces. Numerical calculations are carried out for various wing geometries and compared with other analyses and experiments.
TL;DR: In this paper, the multilevel approach to minimum weight structural design is extended to wing box structures with fiber-composite stiffened-panel components, and a key feature of the method is selection of change in stiffness as the component level objective function to be minimized.
Abstract: The multilevel approach to minimum weight structural design is extended to wing box structures with fiber-composite stiffened-panel components. Strength, deflection, and panel buckling constraints are treated at the system level with equivalent-thickness-type design variables. Local buckling and panel buckling constraints are guarded against at the component level, employing detailed component dimensions as design variables. A key feature of the method is selection of change in stiffness as the component level objective function to be minimized. Numerical results are given for wing box structures with sandwich and hat-stiffened fiber-composite panels.
TL;DR: In this article, the shape design capability is demonstrated on practical problems which result in as much as 35% weight savings over a uniform thickness design with fixed boundaries, which can be achieved by including in the design process the capability for varying the shape of boundaries and the shape and location of cutouts.
Abstract: In the past, much of the work done in structural optimization consisted of resizing the members of fixed configuration models. There is, however, a broad class of plate and shell problems in which an additional 'reduction in mass can be attained by including in the design process the capability for varying the shape of boundaries and the shape and location of cutouts. This additional capability has made it necessary to address other problems such as how to maintain an adequate finite element model, how to define perfectly general shapes which satisfy a number of criteria, and how to impose the proper constraints so that a realistic design results. The shape design capability is demonstrated on practical problems which result in as much as 35% weight savings over a uniform thickness design with fixed boundaries.
TL;DR: In this paper, the authors present results from an experimental study of fin-induced shock wave/turbulent boundary layer interaction in semi-infinite fin models with hemicylindrical, unswept leading edges.
Abstract: This paper presents results from an experimental study of blunt fin-induced shock wave/turbulent boundarylayer interaction Semi-infinite fin models with hemicylindrical, unswept leading edges were tested in Mach 3, high Reynolds number, turbulent boundary layers All tests were made under approximately adiabatic wall conditions The program had two fundamental objectives The first was to examine the spanwise development of the disturbed flowfield and to determine its dependence on the configuration geometry and incoming flow conditions To achieve this, streamwise surface pressure distributions were measured in the region extending from the centerline to 110 fin diameters outboard The second objective was to determine the vertical extent of the interaction on the fin This was carried out using a fin model whose leading edge and side face were instrumented with pressure taps The results show that, on the test surface near the fin and on the fin itself, the leading-edge diameter plays a dominant role in determining the interaction's scale and characteristics
TL;DR: In this article, three primary turbulence amplifier-generator mechanisms are identified and shown, by linear analysis, to be responsible for turbulence amplification across a shock wave in excess of 100% of the incident turbulence intensity.
Abstract: Attention is directed to the acoustics research of the 1950s and 1960s for guidance in understanding and quantizing the turbulence amplification that can occur in regions of shock-wave boundary-layer interaction. Three primary turbulence amplifier-generator mechanisms are identified and shown, by linear analysis, to be responsible for turbulence amplification across a shock wave in excess of 100% of the incident turbulence intensity.
TL;DR: In this article, a numerical procedure is presented for predicting the static and dynamic aeroelastic characteristics of thin, clean swept wings in transonic flow, based upon the simultaneous time integration of the equations governing the coupled nonlinear fluid dynamic and structural aero-elastic system.
Abstract: A numerical procedure is presented for predicting the static and dynamic aeroelastic characteristics of thin, clean swept wings in transonic flow. The method is based upon the simultaneous time integration of the equations governing the coupled nonlinear fluid dynamic and structural aeroelastic system. Governing equations for the system are developed and the numerical algorithm, including the coupling procedure for their solution, is discussed. As a computational example, the flutter of a simple rectangular wing is considered. Solutions are presented for a range of Mach numbers and dynamic pressures and compared to other existing flutter analysis methods including doublet lattice, modified strip theory, and time linearization. Unlike other procedures, the method presented here is capable of predicting the nonlinear interaction between unsteady shock wave motions and the dynamic response of an elastic wing. Computed results indicate the existence of the "transonic bucket."
TL;DR: In this article, the results from an extensive oscillating-airfoil experiment are analyzed and reviewed, and four distinct regimes of viscous-inviscid interaction are identified, corresponding to varying degrees of unsteady flow separation.
Abstract: : Selected results from an extensive oscillating-airfoil experiment are analyzed and reviewed. Four distinct regimes of viscous-inviscid interaction are identified, corresponding to varying degrees of unsteady flow separation. The dominant fluid dynamic phenomena are described for each regime. Ten specific test cases, including the appropriate flow conditions and experimental results, are proposed for evaluating unsteady viscous theories and computational methods. (Author)
TL;DR: In this paper, it was shown that the transonic potential flow partial differential equation admits nonsymmetric solutions with large positive or negative lift, for symmetric airfoils at zero angle of attack.
Abstract: The two-dimensional transonic potential flow equation, when solved in discrete form for steady flow over an airfoil, has been found to yield more than one solution in certain bands of angle of attack and Mach number. The most striking ex- ample of this is the appearance of nonsymmetric solutions with large positive or negative lift, for symmetric airfoils at zero angle of attack. The behavior of these "anomalous" solutions is exam- ined as grid size is varied by large factors and found to be not qualitatively different from that of %ormalrf solutions (outside the nonuniqueness band). Thus it appears that the effect is not due to discretization error, and that the basic tran- sonic potential flow partial differential equation admits nonunique solutions for certain values of angle of attack and Mach number.
TL;DR: In this paper, the authors studied the evolution of the upstream boundary layers into the classical asymptotic wake of a smooth flat plate and showed that it takes about 350 wake momentum thicknesses.
Abstract: Detailed measurements of mean flow and turbulence in the developing symmetric wake of a smooth, flat plate are presented. The results are discussed in the light of previous data and theories for near and far wakes. It is shown that evolution of the upstream boundary layers into the classical asymptotic wake occurs in three quite distinct stages and takes about 350 wake momentum thicknesses.
TL;DR: In this paper, a thorough study of exact elasticity solutions reveals that there are two additional effects that are the same order as transverse shear in bending behavior, and a new theory accounting for them is presented, along with several applications.
Abstract: The classical engineering theory of bending due to Bernoulli and Euler serves as a cornerstone for structural analysis and design. Limitations of this theory, however, become apparent in flexural wave propagation studies; it predicts infinite phase velocity as the wavelength becomes shorter. This theoretical deficiency is corrected by Timoshenko theory which accounts for transverse shear deformation. A thorough study of exact elasticity solutions reveals that there are two additional effects that are the same order as transverse shear in bending behavior. A new theory accounting for them is presented, along with several applications. The new equations are no more complicated than those of Timoshenko-ty pe theory, yet they yield solutions which are exact or indistinguishable from exact in the examples studied.
TL;DR: In this paper, a computational procedure for solving the Euler equations for transonic flow around a wing and fuselage upon an O-O mesh generated by transfinite interpolation is presented.
Abstract: Inviscid transonic flows containing either strong shock waves or complex vortex structure call for the Euler equations as a realistic model. Presented here is a computational procedure for solving the Euler equations for transonic flow around a wing and fuselage upon an O-O mesh generated by transfinite interpolation. An explicit time-marching finite-volume procedure solves the flow equations and features a nonreflecting far field boundary condition, an internal mechanism for temporal damping, and use of the local time step, all of which improve the convergence of the computation. Converged after several hundred iterations, results computed on the CYBER 203 vector processor are compared with experimental data and potential-flow computations. The Euler-equation model is found to predict the existence of a tip vortex created by inviscid flow separation in the downstream region of the tip of the M6 wing where the radius of curvature approaches zero. AST year a workshop 1 was held in order to assess the currently used computational procedures, one against the other, in several carefully specified two-dimensional transonic flow problems. While reasonable agreement among the results given by the various full potential and Euler-equation methods was obtained in the subcritical cases, a disparity between the results of these two models was found to grow with cases of increasing shock strength. For the NACA 0012 airfoil and conditions M- 0.80 and a. = 1.25 deg, for example, the lift coefficients CL given by all the Euler methods ranged between 0.30 and 0.38, whereas the range of CL for the potential method was 0.28-1.1. This discrepancy has led to a reconsideration of the validity of the potential model when strong shocks are present in the flow, not only locally in terms of the isentropic shock jumps but also, and perhaps even more importantly, with regard to the correct modeling of vortex phenomena throughout the entire flowfield. In the potential representation the vorticity that is bound to the airfoil is accounted for by a jump in the potential across a line originating at the trailing edge and lying a priori along some chosen coordinate direction downstream, the so-called Kutta condition. The Euler equations, in contrast, admit vorticity in the solution and the equivalent to the Kutta condition evidently does not need to be enforced explicitly. This situation is currently under study by researchers using a variety of numerical methods. In three dimensions the flow past a finite wing is even more complex and less is known. Vortices, for example, are shed continuously from the wing tips and the entire trailing edge. Whether a Kutta condition or its equivalent is necessary in this case has not been in- vestigated, but because vorticity is so crucial to the realism achieved by inviscid flow models, interest has been aroused in questions like this and in numerical methods that solve the Euler equations. A number of methods 2'4 exist to solve the Euler equations for three-dimensional flow, but they have been developed exclusively for and applied only to internal flows. Apparently, the solution of the Euler equations for