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Theorems are developed to support bifurcation and stability of nonlinear parabolic partial differential equations in the solution of the asymptotic behavior of functions with certain specified properties.
A viscous primary-secondary flow analysis designed for numerical solution by forward marching integration is applied to the tip vortex generation problem. Equations governing streamwise momentum, streamwise vorticity, and a gas law are solved in conjunction with equations governing scalar and vector surface potentials for the secondary velocities. The numerical method used combines a consistently split linearized block implicit (LBI) scheme for parabolic equations and a scalar iterative ADI scheme for elliptic equations. Computed results are presented for a wing of rectangular planform immersed in a high Reynolds number stream at 6 degrees incidence. The physical mechanism for generation of the tip vortex, as revealed by the analysis and computations, is discussed. Detailed flow results for the case of 6 degree incidence are included.
An asymptotic, large wave number approximation for the Helmholtz equation is derived. The theory is an extension of the geometric acoustic theory, and provides corrections to that theory in the form of multiplicative functions which satisfy parabolic equations. A simple example is used both to illustrate failure of the geometric theory for large propagation distances, and to show the improvement obtained by use of the new theory.
The formulation of Deal and Grove for the oxidation of silicon and the growth of a silica layer is revisited to include the effects of a protective coating layer. The linear-parabolic equation of the form x_o^2+Ax_o=Bt is still applicable, but the expressions for the constants A and B must be modified to account for the coating. The modified expression for A is a linear function of the coating thickness. The expression for B is modified by the relative solubility of the outer coating layer. The modified expressions are applied to assess recently published experimental results for the oxidation of a silicon bond coat in an environmental barrier coating (EBC). Conclusions are drawn regarding the diffusivity and solubility of oxidants in the EBC top coating.
Solution of the parabolic Navier-Stokes (PNS) equations for supersonic flows is discussed, and compatibility of the PNS method with the triple-deck theory of Stewartson (1974) is demonstrated. Characteristic and stability analyses show that use of an appropriate filter on the pressure term in the x-momentum equation can suppress the departure solutions, giving the usual desired weak interaction solution. An Alternating Direction Explicit procedure, with minimal computer storage requirements compared to the full Navier-Stokes solvers, is proposed to calculate strongly interacting flows using a global iteration procedure for the PNS equations. The PNS equations are used to solve the hypersonic viscous interaction problem, and good agreement is found with experimental results.
Several aspects of multigrid methods are briefly described. The main subjects include the development of very efficient multigrid algorithms for systems of elliptic equations (Cauchy-Riemann, Stokes, Navier-Stokes), as well as the development of control and prediction tools (based on local mode Fourier analysis), used to analyze, check and improve these algorithms. Preliminary research on multigrid algorithms for time dependent parabolic equations is also described. Improvements in existing multigrid processes and algorithms for elliptic equations were studied.
Efforts in transferring computational work from the LRC computer to the IRIS Graphics Workstation at MSU are reported and the computation of a conservative solution of a simple hyperbolic equation on an overlapping grid is discussed. Several conclusions concerning computations on overlapping grids are apparent. Problems only occur when there is a major difference in grid spacing on the individual component grids. In the case of hyperbolic equations, it is necessary that both interpolation and extrapolation be applied at the grid boundaries. When interpolated values are used at outflow boundary points, excessive oscillations in the numerical solution may be the result. The same conclusions would be valid for more complicated systems of hyperbolic equations such as the Euler equations for inviscid flow. Some of the solution values would be extrapolated at the overlap boundary, the exact number depending on the number of characteristics pointing out of the overlap region. It is also possible that similar boundary conditions may be needed for some parabolic equations such as high Reynolds number viscous flow equations. Efforts were also expended on the development of three-dimensional conservative interpolation procedures. Finally, the investigation of grid smoothing procedures were initiated during this reporting period. It was decided that the first grid smoothing algorithms will be based on the concepts of variational grid generation.
We investigate the influence of a fine-scale (FS) layered structure in the atmosphere on the propagation of infrasound signals generated by fragmenting meteoroids. Using a pseudo-differential parabolic equation (PPE) approach, we model broadband acoustic signals from point sources at altitudes of 35–100 km. The presence of FS fluctuations in the stratosphere (37–45 km) and the lower thermosphere (100–120 km) modifies ray trajectories, causing multiple arrivals and prolonged signal durations at ground stations. In particular, meteoroids fragmenting at 80–100 km can produce two distinct thermospheric arrivals beyond 150 km range, while meteoroids descending to 50 km or below yield weak, long-lived arrivals within the acoustic shadow zone via antiguiding propagation and diffraction. Comparison with observed infrasound data confirms that FS-layered inhomogeneities can account for multi-arrival “N-waves,” broadening potential interpretations of meteoroid signals. The results also apply to other atmospheric-entry objects, such as sample return capsules, emphasizing how FS structure impacts shock wave propagation. In conclusion, our findings advance understanding of wavefield evolution in a layered atmosphere and have broad relevance for global infrasound monitoring of diverse phenomena (e.g., re-entry capsules, rocket launches, and large-scale explosions).
Infrasonic signatures of meteoroid fragmentation are frequently ambiguous: do multiple arrivals signify a complex breakup or merely the distorting effects of a layered atmosphere? Resolving this ambiguity is critical for accurate energy estimates and source reconstruction. In this study, we address this challenge by analyzing a unique regional dataset of well-constrained meteoroid events observed by the Southern Ontario Meteor Network and the co-located Elginfield Infrasound Array. We employ pseudo-differential parabolic equation (PPE) simulations to quantify how fine-scale gravity-wave structures in the stratosphere and lower thermosphere modify acoustic waveforms at ranges <300 km. Our modeling reveals that while fine-scale layering can stretch signals and generate diffuse oscillatory tails, it does not produce discrete, high-amplitude pulse splitting at ranges below ∼140 km. By applying these results to the rare multi-arrival event 20060305, we demonstrate that its distinct double arrival at 100 km range is inconsistent with atmospheric multipathing and provides definitive evidence of separate fragmentation episodes. These findings establish new diagnostic criteria for separating source physics from propagation artifacts, improving the reliability of infrasound as a monitoring tool for natural bolides, space debris re-entries, and catastrophic launch failures.
This dataset contains acoustic propagation model outputs, computational subroutines, and analysis tools for underwater sound propagation in the Block Island region, including parabolic equation (PE) model results, visualization products, and comprehensive modeling software tools.
Numerical program solving partial differential equations parabolic system for internal turbulent gas flow extended to flows undergoing circular tube laminarization by heating
Room temperature thermal conductivity measurements were completed on 10 graphite samples selected from the lot of 120 to uniformly span the range of electrical resistivity values. From these data and other measurements a thermal conductivity-electrical conductivity correlation was established to predict room temperature thermal conductivity values for the graphite samples. The correlation, expressed by a parabolic equation, represents available thermal and electrical conductivity data to about + or - 8%.
A new technique for calculating the entire flow-field on spherically blunted circular cones at high angles of attack and high Reynolds numbers is described. The calculations are based on a single-layer system of three-dimensional parabolic equations which are approximations to the full steady Navier-Stokes equations. Initial conditions at the sphere-cone tangency plane are provided by using an inviscid time-dependent solution added to a viscous nonsimilar boundary layer solution. Calculated results are compared with experimental heat transfer and pressure data for a 15 deg half-angle cone with a 1-in. spherical nose at 15 deg angle of attack. The free-stream Mach number is 10.6, and the free-stream Reynolds number is 1,200,000 per foot. Excellent agreement between the calculated and experimental data for both pressure and heat transfer is obtained.
A new technique is reported for calculating the entire flow field on spherically blunted cones at high angles of attack and high laminar Reynolds numbers. An approximate system of parabolic equations obtained from the steady Navier-Stokes equations by assuming the viscous, streamwise derivative terms are small compared to the viscous normal and circumferential derivatives is the basis of the calculations. These equations are valid for both the inviscid and viscous regions, including the circumferential separation zone that develops on the leeward side at high angles of attack. Two different methods are used to obtain the initial conditions for these equations at the sphere cone tangency plane. For small nose Reynolds numbers, an axisymmetric merged layer solution around a sphere is rotated to provide a three-dimensional initial plane of data. For large nose Reynolds numbers, the nose region is solved using an inviscid, three dimensional time dependent solution combined with a boundary layer solution for the viscous flow. The computed flowfield including the leeward separation region is described and compared with data for a 7 deg half angle cone at 10 deg angle of attack, and a blunt 15 deg half angle cone at 15 deg angle of attack.
Spectral broadening of monochromatic radio waves by density fluctuations near the sun is analyzed using the parabolic equation method. Application of the analysis to the 1968 Pioneer-6 spectral broadening observations yields indirect measurements of the density spectrum within 14 solar radii of the sun. For spatial wavenumbers of at least 0.0084 inverse km, the three-dimensional density spectrum is found to be close to the Kolmogorov spectrum (power law with spectral index of 11/3) and is consistent with interplanetary-scintillation and in situ spacecraft measurements farther from the sun. The magnitude of the density spectrum increases, but its shape remains the same during 'solar events' marked by increases in bandwidth.
The dependence of spectral broadening on anisotropic electron-density irregularities and velocity fluctuations in the solar wind is investigated on the basis of results for a second moment mutual coherence function obtained by a parabolic-equation method. It is found that although anisotropic irregularities and wind-velocity fluctuations decrease the bandwidth of the spectrum relative to that for isotropic irregularities and no velocity fluctuations, the shape of the spectrum remains unchanged. Examination of the frequency dependence and shape of a computed angular spectrum indicates that the electron-density spectrum is consistent with a power law having a spectral index close to the value appropriate for the Kolmogorov spectrum. It is proposed that multiple-station observations of some total field be employed to measure the solar wind.
The generalized Galerkin method (or the method of integral relations) is applied to the type of problem described by quasilinear parabolic equations. As an example the problem of nonlinear transient slab diffusion with a general reservoir boundary condition is worked out. The integral relations are given for an arbitrary number of strips, and solutions using up to seven strips have been obtained in order to investigate the convergence of the method.
A fourth-order accurate numerical solution algorithm is derived using finite element interpolation theory for the non-linear parabolic equations governing turbulent boundary layer flow including a two-equation turbulence closure model. The results of carefully controlled numerical experiments firmly quantize for the first time performance differences between finite element and finite difference solution methodology for this type of equation. The developed algorithm takes advantage of the apparent semi-analytical formulational procedure, in establishment of a single, retarded-evaluation Jacobian matrix iterative solution algorithm. Numerical results document performance of solution economy features in terms of computer requirements and solution accuracy. The developed algorithm should find wide application in aerodynamics analysis.