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NASA NTRS1988

Asymptotic modal analysis and statistical energy analysis of an acoustic cavity

A basic asymptotic theory for structural wall/acoustic cavity interaction is presented, and the analysis is illustrated with examples of the acoustic cavity response to a prescribed wall motion. Although, when spatially averaged, the classical modal analysis (CMA) response approaches the asymptotic modal analysis (AMA) response more rapidly as the number of modes increases, it is shown that information about local response intensification is lost in the averaging process. A larger bandwidth at a given center frequency is found to contain more excited modes than a smaller bandwidth; however, the AMA is slightly more accurate in the smaller bandwidth. All AMA asymptotes were shown to be approached from below by a CMA with fixed bandwidth and increasing center frequency.

Kubota, Y.↗

NASA NTRS1988

Asymptotic modal analysis and statistical energy analysis

Statistical Energy Analysis (SEA) is defined by considering the asymptotic limit of Classical Modal Analysis, an approach called Asymptotic Modal Analysis (AMA). The general approach is described for both structural and acoustical systems. The theoretical foundation is presented for structural systems, and experimental verification is presented for a structural plate responding to a random force. Work accomplished subsequent to the grant initiation focusses on the acoustic response of an interior cavity (i.e., an aircraft or spacecraft fuselage) with a portion of the wall vibrating in a large number of structural modes. First results were presented at the ASME Winter Annual Meeting in December, 1987, and accepted for publication in the Journal of Vibration, Acoustics, Stress and Reliability in Design. It is shown that asymptotically as the number of acoustic modes excited becomes large, the pressure level in the cavity becomes uniform except at the cavity boundaries. However, the mean square pressure at the cavity corner, edge and wall is, respectively, 8, 4, and 2 times the value in the cavity interior. Also it is shown that when the portion of the wall which is vibrating is near a cavity corner or edge, the response is significantly higher.

Dowell, Earl H.↗

NASA NTRS1989

Asymptotic modal analysis of a rectangular acoustic cavity and characterization of its intensification zones

Asymptotic modal analysis (AMA) is used to study a rectangular cavity with a flexible vibrating portion on one wall and five rigid walls. The agreement between mean square pressure levels of the cavity interior calculated from classical modal analysis and from the AMA method improved as the number of responding modes increased. It is shown that intensification effects were due to both the excitation location and the response location.

Peretti, Linda F.↗

NASA NTRS1990

Asymptotic modal analysis and statistical energy analysis

The sound field of a structural-acoustic enclosure was subject to experimental analysis and theoretical description in order to develop an efficient and accurate method for predicting sound pressure levels in enclosures such as aircraft fuselages. Asymptotic Modal Analysis (AMA) is the method under investigation. AMA is derived from classical modal analysis (CMA) by considering the asymptotic limit of the sound pressure level as the number of acoustic and/or structural modes approaches infinity. Using AMA, results identical to those of Statistical Energy Analysis (SEA) were obtained for the spatially-averaged sound pressure levels in the interior. AMA is systematically derived from CMA and therefore the degree of generality of the end result can be adjusted through the choice of appropriate simplifying assumptions. For example, AMA can be used to obtain local sound pressure levels at particular points inside the enclosure, or to include the effects of varying the size and/or location of the sound source. AMA theoretical results were compared with CMA theory and also with experiment for the case where the structural-acoustic enclosure is a rectangular cavity with part of one wall flexible and vibrating, while the rest of the cavity is rigid.

Dowell, Earl H.↗

NASA NTRS1972

Progress in nonlinear finite element analysis using asymptotic solution techniques.

The existence of standard forms of finite element equations for nonlinear analysis is observed. These are noted to provide a sufficiently special circumstance to admit the construction of asymptotic solution techniques. Although efforts in this direction have been made only recently, a number of distinct approaches to asymptotic solution techniques for finite element analysis have been put forward and are examined herein. Consideration is given to the differences and similarities of the approaches and the resulting methods of solution. Special characteristics regarding applicability, accuracy and efficiency are delineated.

Mallett, R. H.↗

NASA NTRSDate not supplied

Diffraction by Sharp Edges of Noncanonical Shape with Mean Flow and Surface Impedance

This paper presents a study on diffraction by sharp edges of noncanonical edge shape in the presence of mean flow and surface impedance, which are important features for aircraft noise but have not been accounted for in previous studies on sharp-edge diffraction. An analytical formula for the diffracted pressure is derived by applying asymptotic analysis to the integral representation of the total pressure scattered by a geometry that contains sharp edges. The asymptotic analysis is presented as a systematic process that can be carried out to any order, if needed. An illustration is given to use a higher-order result to eliminate the singularity in the first-order result to make the solution regular in the entire parameter domain. The formula for the diffracted pressure is validated by experimental data and is applied to parametric studies to reveal the effects of noncanonical edge shape, mean flow and surface impedance, as well as the impact of these effects on the total scattered pressure in the presence of other scattering components.

Yueping Guo↗

NASA NTRS1989

The analysis and modeling of dilatational terms in compressible turbulence

It is shown that the dilatational terms that need to be modeled in compressible turbulence include not only the pressure-dilatation term but also another term - the compressible dissipation. The nature of these dilatational terms in homogeneous turbulence is explored by asymptotic analysis of the compressible Navier-Stokes equations. A non-dimensional parameter which characterizes some compressible effects in moderate Mach number, homogeneous turbulence is identified. Direct numerical simulations (DNS) of isotropic, compressible turbulence are performed, and their results are found to be in agreement with the theoretical analysis. A model for the compressible dissipation is proposed; the model is based on the asymptotic analysis and the direct numerical simulations. This model is calibrated with reference to the DNS results regarding the influence of compressibility on the decay rate of isotropic turbulence. An application of the proposed model to the compressible mixing layer has shown that the model is able to predict the dramatically reduced growth rate of the compressible mixing layer.

Sarkar, S.↗

NASA NTRS1991

The analysis and modelling of dilatational terms in compressible turbulence

Sarkar, S.↗

NASA NTRS1999

Thin Interface Asymptotics for an Energy/Entropy Approach to Phase-Field Models with Unequal Conductivities

Karma and Rapped recently developed a new sharp interface asymptotic analysis of the phase-field equations that is especially appropriate for modeling dendritic growth at low undercoolings. Their approach relieves a stringent restriction on the interface thickness that applies in the conventional asymptotic analysis, and has the added advantage that interfacial kinetic effects can also be eliminated. However, their analysis focussed on the case of equal thermal conductivities in the solid and liquid phases; when applied to a standard phase-field model with unequal conductivities, anomalous terms arise in the limiting forms of the boundary conditions for the interfacial temperature that are not present in conventional sharp-interface solidification models, as discussed further by Almgren. In this paper we apply their asymptotic methodology to a generalized phase-field model which is derived using a thermodynamically consistent approach that is based on independent entropy and internal energy gradient functionals that include double wells in both the entropy and internal energy densities. The additional degrees of freedom associated with the generalized phased-field equations can be chosen to eliminate the anomalous terms that arise for unequal conductivities.

McFadden, G. B.↗

NASA NTRS1989

Numerical integration of asymptotic solutions of ordinary differential equations

Classical asymptotic analysis of ordinary differential equations derives approximate solutions that are numerically stable. However, the analysis also leads to tedious expansions in powers of the relevant parameter for a particular problem. The expansions are replaced with integrals that can be evaluated by numerical integration. The resulting numerical solutions retain the linear independence that is the main advantage of asymptotic solutions. Examples, including the Falkner-Skan equation from laminar boundary layer theory, illustrate the method of asymptotic analysis with numerical integration.

Thurston, Gaylen A.↗

NASA NTRS1986

Numerical studies of motion and decay of vortex filaments

A computational code is developed for the integro-differential equations governing the motion of the centerlines of vortex filaments submerged in a background potential flow. These equations, which are derived from the method of matched asymptotic analysis, include the effect of decaying large-magnitude circumferential and axial velocity components in the vortical cores. Numerical examples are presented to assess the effect of large axial velocity and of nonsimilar initial profiles in vortical cores. The initial configurations of the filaments are chosen so as to fulfill the basic assumption of asymptotic analysis, which is the effective vortical core size is much smaller than all other length scales in the flowfield, e.g., the radius of curvature and interfilament distance. The computations are continued until the basic assumption is no longer valid, that is, when the merging or intersection of filaments have begun. Various types of local or global merging or intersection of filaments are classified and demonstrated by numerical examples.

Liu, C. H.↗

NASA NTRS1992

An algebraic model for dissipation in supersonic boundary layers

With increasing mainstream Mach number, viscous dissipation becomes a progressively important influence in high-speed compressible turbulent boundary layers. An asymptotic analysis is carried out for high Reynolds numbers and Mach numbers of order 1, and it is shown that viscous dissipation gives rise to important terms in the solution of the total enthalpy equation. For simplicity, the case of supersonic flow over an adiabatic wall is considered. An expression for the adiabatic wall temperature is derived. It is shown that the asymptotic analysis constrains the types of turbulence models that can be used to represent the effects of viscous dissipation. A simple algebraic turbulence model is proposed and comparisons with measured total enthalpy profile data show good agreement.

He, J.↗

NASA NTRS1988

A hybrid asymptotic-modal analysis of the EM scattering by an open-ended S-shaped rectangular waveguide cavity

The electromagnetic fields (EM) backscatter from a 3-dimensional perfectly conducting S-shaped open-ended cavity with a planar interior termination is analyzed when it is illuminated by an external plane wave. The analysis is based on a self-consistent multiple scattering method which accounts for the multiple wave interactions between the open end and the interior termination. The scattering matrices which described the reflection and transmission coefficients of the waveguide modes reflected and transmitted at each junction between the different waveguide sections, as well at the scattering from the edges at the open end are found via asymptotic high frequency methods such as the geometrical and physical theories of diffraction used in conjunction with the equivalent current method. The numerical results for an S-shaped inlet cavity are compared with the backscatter from a straight inlet cavity; the backscattered patterns are different because the curvature of an S-shaped inlet cavity redistributes the energy reflected from the interior termination in a way that is different from a straight inlet cavity.

Law, P. H.↗

NASA NTRS1994

Triple flame structure and diffusion flame stabilization

The stabilization of diffusion flames is studied using asymptotic techniques and numerical tools. The configuration studied corresponds to parallel streams of cold oxidizer and fuel initially separated by a splitter plate. It is shown that stabilization of a diffusion flame may only occur in this situation by two processes. First, the flame may be stabilized behind the flame holder in the wake of the splitter plate. For this case, numerical simulations confirm scalings previously predicted by asymptotic analysis. Second, the flame may be lifted. In this case a triple flame is found at longer distances downstream of the flame holder. The structure and propagation speed of this flame are studied by using an actively controlled numerical technique in which the triple flame is tracked in its own reference frame. It is then possible to investigate the triple flame structure and velocity. It is shown, as suggested from asymptotic analysis, that heat release may induce displacement speeds of the triple flame larger than the laminar flame speed corresponding to the stoichiometric conditions prevailing in the mixture approaching the triple flame. In addition to studying the characteristics of triple flames in a uniform flow, their resistance to turbulence is investigated by subjecting triple flames to different vortical configurations.

Veynante, D.↗

NASA NTRS1982

The dependence of the shape and stability of captive rotating drops on multiple parameters

Asymptotic and numerical techniques in bifurcation theory are applied to the Young-Laplace equation governing meniscus shape in order to analyze the dependence of the shape and stability of rigidly rotating drops held captive between corotating solid faces on multiple parameters. Asymptotic analysis of the evolution of drop shape from the cylindrical as a function of distance between the solid faces, drop volume, rotational Bond number and gravitational Bond number shows that some shape bifurcations from cylinders to wavy, axisymmetric menisci are ruptured by small changes in drop volume or gravity. Computer calculations of axisymmetric drop shapes based on a finite element representation of the interface and numerical algorithms for tracking shape families and singular points are then used to map drop stability for the four-dimensional parameter space. The results of the asymptotic and numerical analyses are shown to agree well within the limited range of parameters where the asymptotic analysis is valid.

Ungar, L. H.↗

NASA NTRS1991

Ignition and structure of a laminar diffusion flame in the field of a vortex

The distortion of flames in flows with vortical motion is examined via asymptotic analysis and numerical simulation. The model consists of a constant density, one step, irreversible Arrhenius reaction between initially unmixed species occupying adjacent half-planes which are then allowed to mix and react in the presence of a vortex. The evolution in time of the temperature and mass fraction fields is followed. Emphasis is placed on the ignition time and location as a function of vortex Reynolds number and initial temperature differences of the reacting species. The study brings out the influence of the vortex on the chemical reaction. In all phases, good agreement is observed between asymptotic analysis and the full numerical solution of the model equations.

Macaraeg, Michele G.↗

NASA NTRS1989

Asymptotic/numerical analysis of supersonic propeller noise

An asymptotic analysis based on the Mach surface structure of the field of a supersonic helical source distribution is applied to predict thickness and loading noise radiated by high speed propeller blades. The theory utilizes an integral representation of the Ffowcs-Williams Hawkings equation in a fully linearized form. The asymptotic results are used for chordwise strips of the blade, while required spanwise integrations are performed numerically. The form of the analysis enables predicted waveforms to be interpreted in terms of Mach surface propagation. A computer code developed to implement the theory is described and found to yield results in close agreement with more exact computations.

Myers, M. K.↗

NASA NTRS1980

Transonic swept-wing analysis using asymptotic and other numerical methods

The paper presents asymptotic methods for high-aspect-ratio wings in transonic flow developed for straight unyawed wings and for oblique wings. They show that the three-dimensional mixed-flow calculations may be reduced to solving a set of two-dimensional problems at each span station; the development of this theory and the related computational studies are reviewed. Differences between the piloted (oblique) wing, the swept-back wing, and the swept-forward-wing in the induced upwash are discussed; examples of similarity solutions are demonstrated for high subcritical and slightly supercritical component flows, and comparisons made with relaxation solutions of a full potential equation. The examples include oblique and symmetric swept wings, and the adequacy of the existing full-potential computer code is examined.

Cheng, H. K.↗