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NASA NTRS1990

Absorbing boundary conditions for second-order hyperbolic equations

A uniform approach to construct absorbing artificial boundary conditions for second-order linear hyperbolic equations is proposed. The nonlocal boundary condition is given by a pseudodifferential operator that annihilates travelling waves. It is obtained through the dispersion relation of the differential equation by requiring that the initial-boundary value problem admits the wave solutions travelling in one direction only. Local approximation of this global boundary condition yields an nth-order differential operator. It is shown that the best approximations must be in the canonical forms which can be factorized into first-order operators. These boundary conditions are perfectly absorbing for wave packets propagating at certain group velocities. A hierarchy of absorbing boundary conditions is derived for transonic small perturbation equations of unsteady flows. These examples illustrate that the absorbing boundary conditions are easy to derive, and the effectiveness is demonstrated by the numerical experiments.

Jiang, Hong↗
NASA NTRS1990

Time-dependent boundary conditions for hyperbolic systems. II

A general boundary condition formalism is developed for all types of boundary conditions to which hyperbolic systems are subject; the formalism makes possible a 'cookbook' approach to boundary conditions, by means of which novel boundary 'recipes' may be derived and previously devised ones may be consulted as required. Numerous useful conditions are derived for such CFD problems as subsonic and supersonic inflows and outflows, nonreflecting boundaries, force-free boundaries, constant pressure boundaries, and constant mass flux. Attention is given to the computation and integration of time derivatives.

Thompson, Kevin W.↗
NASA NTRS1991

User's manual for the Langley boundary layer noise propagation program (MRS-BLP)

A computer program, McAninch-Rawls-Spence Boundary Layer Propagation (MRS-BLP), is described. This program models the refractive and scattering effects on acoustic pressure waves propagating through a boundary layer encompassing an aircraft's fuselage. The noise source is assumed known and generated by a propeller. The fuselage is represented by an infinitely long cylinder embedded in a longitudinal flow. By matching a numerical solution inside the boundary layer with an analytical solution outside the boundary layer, the program calculates the acoustic pressure at the surface of the cylinder given the incident field at the top of the boundary layer. The boundary layer flow velocity and sound speed profiles, as well as the boundary layer thickness may be specified by the user. A detailed description of the input parameters and how to execute the program is given. Example executions of MRS-BLP showing results are also included.

Spence, Peter L.↗
NASA NTRS1992

Interface of an uncoupled boundary layer algorithm with an inviscid core flow algorithm for unsteady supersonic engine inlets

An uncoupled boundary layer algorithm was combined with an inviscid core flow algorithm to model flows within supersonic engine inlets. The inviscid flow algorithm that was used was the LArge Perturbation INlet Code (LAPIN). The boundary layer and inviscid core flow algorithms were formulated in different manners. The boundary layer algorithm was two dimensional and solved in nonconservation form, while the core flow algorithm was one dimensional and solved in conservation form. In order to interface the two codes, the following modifications were important. The coordinate system was set up to maintain the parabolic nature of the boundary layer algorithm while approaching the one dimensional core flow solution far from a wall. The pressure gradient used in the boundary layer equation was calculated using the core flow values and the boundary layer equations, so the boundary layer solution smoothly approached the core flow values far from the wall. Flaring was used for the advection terms perpendicular to the core flow to maintain the stability of the algorithm. With these modifications, the combined viscous/inviscid algorithm matched well with experimental observations of pressure distributions with a supersonic inlet.

Darling, Douglas↗
NASA NTRS1992

Interface of an uncoupled boundary layer algorithm with an inviscid core flow algorithm for unsteady supersonic engine inlets

An uncoupled boundary layer algorithm was combined with an inviscid core flow algorithm to model flows within supersonic engine inlets. The inviscid flow algorithm that was used was the LArge Perturbation INlet Code (LAPIN). The boundary layer and inviscid core flow algorithms were formulated in different manners. The boundary layer algorithm was two dimensional and solved in nonconservation form, while the core flow algorithm was one dimensional and solved in conservation form. In order to interface the two codes, the following modifications were important. The coordinate system was set up to maintain the parabolic nature of the boundary layer algorithm while approaching the one dimensional core flow solution far from a wall. The pressure gradient used in the boundary layer equation was calculated using the core flow values and the boundary layer equations, so the boundary layer solution smoothly approached the core flow values far from the wall. Flaring was used for the advection terms perpendicular to the core flow to maintain the stability of the algorithm. With these modifications, the combined viscous/inviscid algorithm matched well experimental observations of pressure distributions with a supersonic inlet.

Darling, Douglas↗
NASA NTRS1993

Surface boundary conditions for the numerical solution of the Euler equations

We consider the implementation of boundary conditions at solid walls in inviscid Euler solutions by upwind, finite-volume methods. We review some current methods for the implementation of surface boundary conditions and examine their behavior for the problem of an oblique shock reflecting off a planar surface. We show the importance of characteristic boundary conditions for this problem and introduce a method of applying the classical flux-difference splitting of Roe as a characteristic boundary condition. Consideration of the equivalent problem of the intersection of two (equal and opposite) oblique shocks was very illuminating on the role of surface boundary conditions for an inviscid flow and led to the introduction of two new boundary-condition procedures, denoted as the symmetry technique and the curvature-corrected symmetry technique. Examples of the effects of the various surface boundary conditions considered are presented for the supersonic blunt body problem and the subcritical compressible flow over a circular cylinder. Dramatic advantages of the curvature-corrected symmetry technique over the other methods are shown, with regard to numerical entropy generation, total pressure loss, drag and grid convergence.

Dadone, A.↗
NASA NTRS1952

Generalization of Boundary-Layer Momentum-Integral Equations to Three-Dimensional Flows Including Those of Rotating System

The Navier-Stokes equations of motion and the equation of continuity are transformed so as to apply to an orthogonal curvilinear coordinate system rotating with a uniform angular velocity about an arbitrary axis in space. A usual simplification of these equations as consistent with the accepted boundary-layer theory and an integration of these equations through the boundary layer result in boundary-layer momentum-integral equations for three-dimensional flows that are applicable to either rotating or nonrotating fluid boundaries. These equations are simplified and an approximate solution in closed integral form is obtained for a generalized boundary-layer momentum-loss thickness and flow deflection at the wall in the turbulent case. A numerical evaluation of this solution carried out for data obtained in a curving nonrotating duct shows a fair quantitative agreement with the measures values. The form in which the equations are presented is readily adaptable to cases of steady, three-dimensional, incompressible boundary-layer flow like that over curved ducts or yawed wings; and it also may be used to describe the boundary-layer flow over various rotating surfaces, thus applying to turbomachinery, propellers, and helicopter blades.

Mager, Arthur↗
NASA NTRS1934

On the theory of laminar boundary layers involving separation

This paper presents a mathematical discussion of the laminar boundary layer, which was developed with a view of facilitating the investigation of those boundary layers in particular for which the phenomenon of separation occurs. The treatment starts with a slight modification of the form of the boundary layer equation first published by Von Mises. Two approximate solutions of this equation are found, one of which is exact at the outer edge of the boundary layer while the other is exact at the wall. The final solution is obtained by joining these two solutions at the inflection points of the velocity profiles. The final solution is given in terms of a series of universal functions for a fairly broad class of potential velocity distributions outside of the boundary layer. Detailed calculations of the boundary layer characteristics are worked out for the case in which the potential velocity is a linear function of the distance from the upstream stagnation point. Finally, the complete separation point characteristics are determined for the boundary layer associated with a potential velocity distribution made up of two linear functions of the distance from the stagnation point. It appears that extensions of the detailed calculations to more complex potential flows can be fairly easily carried out by using the explicit formulae given in the paper. (author)

Von Karman, TH↗
NASA NTRS1993

Geochemical evidences for two chondritic-like cometary or asteroidal impacts before and at the K/T boundary

A number of geological and palaeontological evidences support multiple impacts of cometary showers within a short time (approximately 1-3 Ma) and their connection with mass extinctions. Observations include clustered crater ages, stratigraphic horizons of impact ejecta closely spaced in time, and evidence for stepwise mass extinctions spanning intervals of 1-3 Ma. For the K/T boundary, three candidates, Popigai, Manson, and Yucatan, have been proposed as impact craters. Two distinct strata at the K/T boundary in western North America have been interpreted as evidence for two sequential impacts. If multiple impacts occurred within a time span of about 1 Ma then multiple Ir enrichments should be observed. DSDP Hole 577B on the Shatsky Plateau in the northern Pacific at K/T time is the first site. Samples contain approximately greater than 97 percent CaCO3, which exhibit clear chemical signals associated with asteroidal/cometary impact. Ir, Fe, and Cr data are presented. From the Th-normalized data, two satellite peaks below the major peak at 78 cm and 81 cm of 577B-1-4 are clearly shown. The major Ir peak (K/T boundary) is at 72 cm. Fe and Cr, from C1-like impactor ejecta fallout, also show two peaks at the same positions. For hole 738C on the southern Kerguelen Plateau, Ir values reach a peak concentration of 18 ppb in the clay layer at 96.0-96.2 cm in section 20R-5, and gradually tail off. In the sample 115 cm above the boundary, Ir concentrations have still not reached background levels. From the Ir peak downward to the lowermost sample analyzed at 102 cm, the Ir concentration is still as high as 1.7 ppb. From the Th-normalized data, we observe a small Ir/Th peak at 100-101 cm. Though this peak is within the error margin, the trend is clear. Fe and Cr exhibit the same pattern. The third case is Hole 690C on the Queen Maud Ridge. Again, the Ir/Th plot indicates the strong possibility of satellite peaks at approximately 52 cm. The main peak is at 39-40 cm. For the Stevns Klint K/T boundary layers, the stratification of trace elements appears threefold with peak concentrations in sublayers A1, A3, and B2 for different element groups, including Ir. C1 ratios for many siderophile elements found in combined layers III and IV, corresponding to layers A, B, C, and D, strongly support the impact hypothesis. Also, multiple Ir anomalies in the K/T section at Lattengebirge, Bavarian Alps are reported. Recent works on Ni-rich spinels and Ir at the K/T boundaries clearly establish cometary/asteroidal impacts at the K/T boundary. Lastly, cometary showers can explain the enhanced Ir contents over approximately a 1 Ma interval in Gubbio shales.

Liu, Y.-G.↗
NASA NTRS1995

Completed Beltrami-Michell formulation for analyzing mixed boundary value problems in elasticity

In elasticity, the method of forces, wherein stress parameters are considered as the primary unknowns, is known as the Beltrami-Michell formulation (BMF). The existing BMF can only solve stress boundary value problems; it cannot handle the more prevalent displacement of mixed boundary value problems of elasticity. Therefore, this formulation, which has restricted application, could not become a true alternative to the Navier's displacement method, which can solve all three types of boundary value problems. The restrictions in the BMF have been alleviated by augmenting the classical formulation with a novel set of conditions identified as the boundary compatibility conditions. This new method, which completes the classical force formulation, has been termed the completed Beltrami-Michell formulation (CBMF). The CBMF can solve general elasticity problems with stress, displacement, and mixed boundary conditions in terms of stresses as the primary unknowns. The CBMF is derived from the stationary condition of the variational functional of the integrated force method. In the CBMF, stresses for kinematically stable structures can be obtained without any reference to the displacements either in the field or on the boundary. This paper presents the CBMF and its derivation from the variational functional of the integrated force method. Several examples are presented to demonstrate the applicability of the completed formulation for analyzing mixed boundary value problems under thermomechanical loads. Selected example problems include a cylindrical shell wherein membrane and bending responses are coupled, and a composite circular plate.

Patnaik, Surya N.↗
NASA NTRS1994

Radiation boundary condition and anisotropy correction for finite difference solutions of the Helmholtz equation

In this paper finite-difference solutions of the Helmholtz equation in an open domain are considered. By using a second-order central difference scheme and the Bayliss-Turkel radiation boundary condition, reasonably accurate solutions can be obtained when the number of grid points per acoustic wavelength used is large. However, when a smaller number of grid points per wavelength is used excessive reflections occur which tend to overwhelm the computed solutions. Excessive reflections are due to the incompability between the governing finite difference equation and the Bayliss-Turkel radiation boundary condition. The Bayliss-Turkel radiation boundary condition was developed from the asymptotic solution of the partial differential equation. To obtain compatibility, the radiation boundary condition should be constructed from the asymptotic solution of the finite difference equation instead. Examples are provided using the improved radiation boundary condition based on the asymptotic solution of the governing finite difference equation. The computed results are free of reflections even when only five grid points per wavelength are used. The improved radiation boundary condition has also been tested for problems with complex acoustic sources and sources embedded in a uniform mean flow. The present method of developing a radiation boundary condition is also applicable to higher order finite difference schemes. In all these cases no reflected waves could be detected. The use of finite difference approximation inevita bly introduces anisotropy into the governing field equation. The effect of anisotropy is to distort the directional distribution of the amplitude and phase of the computed solution. It can be quite large when the number of grid points per wavelength used in the computation is small. A way to correct this effect is proposed. The correction factor developed from the asymptotic solutions is source independent and, hence, can be determined once and for all. The effectiveness of the correction factor in providing improvements to the computed solution is demonstrated in this paper.

Tam, Christopher K. W.↗
NASA NTRS1998

The Auroral Oval Boundaries on January 10, 1997: A Comparison of Global Magnetospheric Simulations with UVI Images

We present the results of a global magnetospheric simulation of the initial period of the January 10 - 11, 1997 magnetic cloud event. Distinct magnetospheric boundaries derived from the model are mapped down to the ionosphere and compared to UVI images of the auroral oval from 1:00 to 4:30 UT. The convection reversal boundary, which coincides with the maximum Region I currents, tends to generally match the UVI equatorward boundary. This boundary is almost always poleward of the boundary separating Region 1 and Region 2 currents. The separatrix between open and closed magnetic field lines as mapped in the model matches the poleward boundary of the UVI images very well during quiet periods. During dynamic periods, however, when the separatrix can move several degrees in latitude in some sectors, the poleward boundary of the Region 1 currents matches the UVI images better.

Elsen, R.↗
NASA NTRS1999

Sensitivity of High-Resolution Simulations of Hurricane Bob (1991) to Planetary Boundary Layer Parameterizations

The MM5 mesoscale model is used to simulate Hurricane Bob (1991) using grids nested to high resolution (4 km). Tests are conducted to determine the sensitivity of the simulation to the available planetary boundary layer parameterizations, including the bulk-aerodynamic, Blackadar, Medium-RanGe Forecast (MRF) model, and Burk-Thompson boundary-layer schemes. Significant sensitivity is seen, with minimum central pressures varying by up to 17 mb. The Burk-Thompson and bulk-aerodynamic boundary-layer schemes produced the strongest storms while the MRF scheme produced the weakest storm. Precipitation structure of the simulated hurricanes also varied substantially with the boundary layer parameterizations. Diagnostics of boundary-layer variables indicated that the intensity of the simulated hurricanes generally increased as the ratio of the surface exchange coefficients for heat and momentum, C(sub h)/C(sub M), although the manner in which the vertical mixing takes place was also important. Findings specific to the boundary-layer schemes include: 1) the MRF scheme produces mixing that is too deep and causes drying of the lower boundary layer in the inner-core region of the hurricane; 2) the bulk-aerodynamic scheme produces mixing that is probably too shallow, but results in a strong hurricane because of a large value of C(sub h)/C(sub M) (approximately 1.3); 3) the MRF and Blackadar schemes are weak partly because of smaller surface moisture fluxes that result in a reduced value of C(sub h)/C(sub M) (approximately 0.7); 4) the Burk-Thompson scheme produces a strong storm with C(sub h)/C(sub M) approximately 1; and 5) the formulation of the wind-speed dependence of the surface roughness parameter, z(sub 0), is important for getting appropriate values of the surface exchange coefficients in hurricanes based upon current estimates of these parameters.

Braun, Scott A.↗
NASA NTRS1986

A Conservation Treatment of Zonal Boundaries for Euler Equation Calculations

Finite-difference calculations require the generation of a grid for the region of interest. A zonal approach, wherein the given region is subdivided into zones and the grid for each zone is generated independently, makes the grid-generation process for complicated topologies and for regions requiring selective grid refinement a fairly simple task. This approach results in new boundaries within the given region, that is, zonal boundaries at the interfaces of the various zones. The zonal-boundary scheme (the integration scheme used to update the points on the zonal boundary) for the Euler equations must be conservative, accurate, stable, and applicable to general curvilinear coordinate systems. A zonal-boundary scheme with these desirable properties is developed in this study. The scheme is designed for explicit, first-order-accurate integration schemes but can be modified to accommodate second-order-accurate explicit and implicit integration schemes. Results for inviscid flow, including supersonic flow over a cylinder, blast-wave diffraction by a ramp, and one-dimensional shock-tube flow are obtained on zonal grids. The conservative nature of the zonal-boundary scheme permits the smooth transition of the discontinuities associated with these flows from one zone to another. The calculations also demonstrate the continuity of contour lines across zonal boundaries that can be achieved with the present zonal scheme.

Rai, Man Mohan↗
NASA NTRS2000

Abrupt Changes at the Permian/Triassic Boundary: Tempo of Events from High-Resolution Cyclostratigraphy

The Permian/Triassic (P/Tr) boundary (251.4 +/- 3 Myr) is marked by the most severe mass extinction in the geologic record. Recently, precise absolute dating has bracketed the marine extinctions and associated carbon-isotope anomaly within less than 1 Myr. We improve this resolution through high-resolution stratigraphy across the P/Tr boundary in the 331-m Gartnerkofel-1 core and nearby Reppwand outcrop section (Carnic Alps, Austria) utilizing FFT and wavelet timeseries analyses of cyclic components in down-hole core logs of density and natural gamma-ray intensity, and carbon-isotopic ratios of bulk samples. The wavelet analysis indicates continuity of deposition across the P/Tr boundary interval, and the timeseries analyses show evidence for persistent cycles in the ratio of approximately 40: 10: 4.7: 2.3 meters, correlated with Milankovitch-band orbital cycles of approximately 412: 100: 40: 20 kyr (eccentricity 1 and 2, obliquity, and precession), and giving a consistent average sedimentation rate of approximately 10 cm/1,000 yr. Milankovitch periods in delta C-13 and density in these shallow-water carbonates were most likely the result of climatically induced oscillations of sea level and climate, coupled with changes in ocean circulation and productivity, that affected sedimentation. Fluctuations in gamma radiation reflect varying input of clay minerals and the presence of shaly interbeds. Throughout the P/Tr boundary interval in the core, the 100,000-year eccentricity cycle seems to be dominant. Weaker obliquity and precession cycles are in line with the location of the Austrian section in the latest Permian, close to the Equator in the western bight of the Tethys, where obliquity and precessional effects on seasonal contrast might be subdued. Using the improved resolution provided by cycle analysis in the GK-1 core, we find that the dramatic change in the faunal record that marks the P/Tr boundary takes place over less than 6m, or less than 60,000 years. In the nearby Reppwand outcrop section, the same faunal changes occurs over only 0.8 m or about 8,000 years, close to the limit of time-resolution induced by bioturbation and reworking in these sediments. The sharp negative global carbon-isotope shift took place within less than or equal to 40,000 yr, and the isotope excursions persisted for approximately 480,000 yr into the Early Triassic. The results indicate that the severe marine faunal event that marks the P/Tr boundary was very sudden, perhaps less than the resolution window in the GK-1 core, and suggest a catastrophic cause. The wavelet-analysis approach to high-resolution cyclostratigraphy can be applied to other P/Tr boundary sections, and when combined with precise absolute dating and magnetostratigraphic methods promises a significant increase in resolution in determining the correlation and tempo of the end-Permian extinctions and related events worldwide.

Rampino, M. R.↗
NASA NTRS2003

Evaluation of Flush-Mounted, S-Duct Inlets With Large Amounts of Boundary Layer Ingestion

A new high Reynolds number test capability for boundary layer ingesting inlets has been developed for the NASA Langley Research Center 0.3-Meter Transonic Cryogenic Tunnel. Using this new capability, an experimental investigation of four S-duct inlet configurations with large amounts of boundary layer ingestion (nominal boundary layer thickness of about 40% of inlet height) was conducted at realistic operating conditions (high subsonic Mach numbers and full-scale Reynolds numbers). The objectives of this investigation were to 1) develop a new high Reynolds number, boundary-layer ingesting inlet test capability, 2) evaluate the performance of several boundary layer ingesting S-duct inlets, 3) provide a database for CFD tool validation, and 4) provide a baseline inlet for future inlet flow-control studies. Tests were conducted at Mach numbers from 0.25 to 0.83, Reynolds numbers (based on duct exit diameter) from 5.1 million to a fullscale value of 13.9 million, and inlet mass-flow ratios from 0.39 to 1.58 depending on Mach number. Results of this investigation indicate that inlet pressure recovery generally decreased and inlet distortion generally increased with increasing Mach number. Except at low Mach numbers, increasing inlet mass-flow increased pressure recovery and increased distortion. Increasing the amount of boundary layer ingestion (by decreasing inlet throat height and increasing inlet throat width) or ingesting a boundary layer with a distorted profile decreased pressure recovery and increased distortion. Finally, increasing Reynolds number had almost no effect on inlet distortion but increased inlet recovery by about one-half percent at a Mach number near cruise.

Berrier, Bobby L.↗
NASA NTRS2003

A Discrete Analysis of Non-reflecting Boundary Conditions for Discontinuous Galerkin Method

We present a discrete analysis of non-reflecting boundary conditions for the discontinuous Galerkin method. The boundary conditions considered in this paper include the recently proposed Perfectly Matched Layer absorbing boundary condition for the linearized Euler equation and two non-reflecting boundary conditions based on the characteristic decomposition of the flux on the boundary. The analyses for the three boundary conditions are carried out in a unifled way. In each case, eigensolutions of the discrete system are obtained and applied to compute the numerical reflection coefficients of a specified out-going wave. The dependencies of the reflections at the boundary on the out-going wave angle and frequency as well as the mesh sizes arc? studied. Comparisons with direct numerical simulation results are also presented.

Hu, Fang Q.↗
NASA NTRS2003

Dynamic Turbulence Modelling in Large-eddy Simulations of the Cloud-topped Atmospheric Boundary Layer

The use of large eddy simulation, or LES, to study the atmospheric boundary layer dates back to the early 1970s when Deardor (1972) used a three-dimensional simulation to determine velocity and temperature scales in the convective boundary layer. In 1974 he applied LES to the problem of mixing layer entrainment (Deardor 1974) and in 1980 to the cloud-topped boundary layer (Deardor 1980b). Since that time the LES approach has been applied to atmospheric boundary layer problems by numerous authors. While LES has been shown to be relatively robust for simple cases such as a clear, convective boundary layer (Mason 1989), simulation of the cloud-topped boundary layer has proved more of a challenge. The combination of small length scales and anisotropic turbulence coupled with cloud microphysics and radiation effects places a heavy burden on the turbulence model, especially in the cloud-top region. Consequently, over the past few decades considerable effort has been devoted to developing turbulence models that are better able to parameterize these processes. Much of this work has involved taking parameterizations developed for neutral boundary layers and deriving corrections to account for buoyancy effects associated with the background stratification and local buoyancy sources due to radiative and latent heat transfer within the cloud (see Lilly 1962; Deardor 1980a; Mason 1989; MacVean & Mason 1990, for example). In this paper we hope to contribute to this effort by presenting a number of turbulence models in which the model coefficients are calculated dynamically during the simulation rather than being prescribed a priori.

Kirkpatrick, M. P.↗