Results for “variable geometry” — Page 48

NASA NTRS2021

A Summary of Normal Incidence Tube Tests of NASA'S Bio-inspired Broadband Acoustic Absorber

Motivated by the need to reduce noise pollution from aircraft engines, NASA has continued to design, manufacture and test recently patented structures that mimic the geometry and the broadband and low frequency acoustic absorption of assemblies of natural reeds. This report documents a screening test, which is a precursor to more controlled modelling experiments. The purpose of this experiment was to identify variables and prototype design features that might play a dominant role in the acoustic performance of these structures. To date, twenty-four prototypes, each with an overall shape of a cube (nominally 50 mm x 50 mm x 50 mm), were designed and additively manufactured from thermoplastics using a Fused Filament Fabrication technique. The prototypes were tested in the NASA Glenn and Langley Normal Incidence Tubes to experimentally determine the acoustic absorption as a function of frequency from 400 Hz to 3000 Hz. Results indicate that a variety of structures exhibit substantial acoustic absorption in that frequency range, with an average absorption coefficient greater than 0.6, especially in the frequency range of 400 Hz to 1000 Hz. Six basic prototype designs were chosen to inform future modeling research. Results of these experiments can be used to evaluate existing physics-based models of the interaction of sound waves with these structures or to develop new models. Validated physics-based models of these structures can help engineers optimize the designs for different commercial noise control applications.

Acoustics↗

NASA NTRS2018

Methane on Mars: New Insights into the Sensitivity of CH4 with the NOMAD/ExoMars Spectrometer Through Its First In-Flight Calibration

The Nadir and Occultation for MArs Discovery instrument (NOMAD), onboard the ExoMars Trace Gas Orbiter (TGO) spacecraft was conceived to observe Mars in solar occultation, nadir, and limb geometries, and will be able to produce an outstanding amount of diverse data, mostly focused on properties of the atmosphere. The infrared channels of the instrument operate by combining an echelle grating spectrometer with an Acousto- Optical Tunable Filter (AOTF). Using in-flight data, we characterized the instrument performance and parameterized its calibration. In particular: an accurate frequency calibration was achieved, together with its variability due to thermal effects on the grating. The AOTF properties and transfer function were also quantified, and we developed and tested a realistic method to compute the spectral continuum transmitted through the coupled grating and AOTF system. The calibration results enabled unprecedented insights into the important problem of the sensitivity of NOMAD to methane abundances in the atmosphere. We also deeply characterized its performance under realistic conditions of varying aerosol abundances, diverse albedos and changing illumination conditions as foreseen over the nominal mission. The results show that, in low aerosol conditions, NOMAD single spectrum, 1σ sensitivity to CH4 is around 0.33 ppbv at 20 km of altitude when performing solar occultation's, and better than 1 ppbv below 30 km. In dusty conditions, we show that the sensitivity drops to 0 below 10 km. In Nadir geometry, results demonstrate that NOMAD will be able to produce seasonal maps of CH4 with a sensitivity around 5 ppbv over most of planet's surface with spatial integration over 5×5° bins. Results show also that such numbers can be improved by a factor of ~10 to ~30 by data binning. Overall, our results quantify NOMAD's capability to address the variable aspects of Martian climate.

Liuzzi, Giuliano↗

NASA NTRS1991

Thermal modeling of a Ni-H2 battery cell

The nickel-hydrogen secondary battery has many desirable features which make it attractive for satellite power systems. It can provide a significant improvement over the energy density of present spacecraft nickel-cadnium batteries, combined with longer life, tolerance to overcharge and possibility of state-of-charge indication. However, to realize these advantages, accurate thermal modeling of nickel-hydrogen cells is required in order to properly design the battery pack so that it operates within a specified temperature range during the operation. Maintenance of a low operating temperature and a uniform temperature profile within the cell will yield better reliability, improved cycle life and better charge/discharge efficiencies. This research has the objective of developing and testing a thermal model which can be used to characterize battery operation. Primarily, temperature distribution with the heat generation rates as a function of position and time will be evaluated for a Ni-H2 cell in the three operating modes: (1) charge cycle, (2) discharge cycle, and (3) overcharge condition, if applicable. Variables to be examined include charging current, discharge rates, state of charge, pressure and temperature. Once the thermal model has been developed, this resulting model will predict the actual operating temperature and temperature gradient for the specific cell geometry to be used.

Ryu, Si-Ok↗

NASA NTRS1994

Prediction of Undsteady Flows in Turbomachinery Using the Linearized Euler Equations on Deforming Grids

A linearized Euler solver for calculating unsteady flows in turbomachinery blade rows due to both incident gusts and blade motion is presented. The model accounts for blade loading, blade geometry, shock motion, and wake motion. Assuming that the unsteadiness in the flow is small relative to the nonlinear mean solution, the unsteady Euler equations can be linearized about the mean flow. This yields a set of linear variable coefficient equations that describe the small amplitude harmonic motion of the fluid. These linear equations are then discretized on a computational grid and solved using standard numerical techniques. For transonic flows, however, one must use a linear discretization which is a conservative linearization of the non-linear discretized Euler equations to ensure that shock impulse loads are accurately captured. Other important features of this analysis include a continuously deforming grid which eliminates extrapolation errors and hence, increases accuracy, and a new numerically exact, nonreflecting far-field boundary condition treatment based on an eigenanalysis of the discretized equations. Computational results are presented which demonstrate the computational accuracy and efficiency of the method and demonstrate the effectiveness of the deforming grid, far-field nonreflecting boundary conditions, and shock capturing techniques. A comparison of the present unsteady flow predictions to other numerical, semi-analytical, and experimental methods shows excellent agreement. In addition, the linearized Euler method presented requires one or two orders-of-magnitude less computational time than traditional time marching techniques making the present method a viable design tool for aeroelastic analyses.

Clark, William S.↗

NASA NTRS1994

Spectral study of GX 339-4 in the low-intensity state observed with Ginga

The energy spectra of the black-hole candidate GX 339-4 in the low-intensity state were observed on four occasions through 1989 to 1991 with the Large Area Counter on board the Ginga satellite. The spectra showed significant deviations from a power-law, with an iron K(sub alpha) emission line at approximetaly 6.4 keV and a broad iron K-edge structure above approximately 7 keV. The enrgy spectra above approximately 4 keV were successfully explained with a reflection model, in which part of the incident X-rays with a power-law spectrum is Compton reflected by optically thick matter, resulting in a harder continuum component with iron K-edge absorption and an iron flourescent line. The line equivalent width with respect to the reflection component decreases as the source flux increases. This is consistent with an increase in the ionization state of the material, so that resonant absorption followed by Auger ionization depletes the line. The photon-index of the power-law component was clearly variable, and it correlated with the relative amount of the reflection component. Such a correlation may be explained in the context of the anisotropic Comptonization models of Haardt et al. (1993), or by a variation of the relative geometry of the source and disk.

Ueda, Yoshihiro↗

NASA NTRS1997

Radio Occultation Investigation of the Rings of Saturn and Uranus

The proposed work addresses two main objectives: (1) to pursue the development of the random diffraction screen model for analytical/computational characterization of the extinction and near-forward scattering by ring models that include particle crowding, uniform clustering, and clustering along preferred orientations (anisotropy). The characterization is crucial for proper interpretation of past (Voyager) and future (Cassini) ring, occultation observations in terms of physical ring properties, and is needed to address outstanding puzzles in the interpretation of the Voyager radio occultation data sets; (2) to continue the development of spectral analysis techniques to identify and characterize the power scattered by all features of Saturn's rings that can be resolved in the Voyager radio occultation observations, and to use the results to constrain the maximum particle size and its abundance. Characterization of the variability of surface mass density among the main ring, features and within individual features is important for constraining the ring mass and is relevant to investigations of ring dynamics and origin. We completed the developed of the stochastic geometry (random screen) model for the interaction of electromagnetic waves with of planetary ring models; used the model to relate the oblique optical depth and the angular spectrum of the near forward scattered signal to statistical averages of the stochastic geometry of the randomly blocked area. WE developed analytical results based on the assumption of Poisson statistics for particle positions, and investigated the dependence of the oblique optical depth and angular spectrum on the fractional area blocked, vertical ring profile, and incidence angle when the volume fraction is small. Demonstrated agreement with the classical radiative transfer predictions for oblique incidence. Also developed simulation procedures to generate statistical realizations of random screens corresponding to uniformly packed ring models, and used the results to characterize dependence of the extinction and near-forward scattering on ring thickness, packing fraction, and the ring opening angle.

Marouf, Essam A.↗

NASA NTRS2005

Disk-averaged synthetic spectra of Mars

The principal goal of the NASA Terrestrial Planet Finder (TPF) and European Space Agency's Darwin mission concepts is to directly detect and characterize extrasolar terrestrial (Earthsized) planets. This first generation of instruments is expected to provide disk-averaged spectra with modest spectral resolution and signal-to-noise. Here we use a spatially and spectrally resolved model of a Mars-like planet to study the detectability of a planet's surface and atmospheric properties from disk-averaged spectra. We explore the detectability as a function of spectral resolution and wavelength range, for both the proposed visible coronograph (TPFC) and mid-infrared interferometer (TPF-I/Darwin) architectures. At the core of our model is a spectrum-resolving (line-by-line) atmospheric/surface radiative transfer model. This model uses observational data as input to generate a database of spatially resolved synthetic spectra for a range of illumination conditions and viewing geometries. The model was validated against spectra recorded by the Mars Global Surveyor-Thermal Emission Spectrometer and the Mariner 9-Infrared Interferometer Spectrometer. Results presented here include disk-averaged synthetic spectra, light curves, and the spectral variability at visible and mid-infrared wavelengths for Mars as a function of viewing angle, illumination, and season. We also considered the differences in the spectral appearance of an increasingly ice-covered Mars, as a function of spectral resolution, signal-to-noise and integration time for both TPF-C and TPFI/ Darwin.

Mars↗

NASA NTRS2014

Statistical Mechanics of Turbulent Dynamos

Incompressible magnetohydrodynamic (MHD) turbulence and magnetic dynamos, which occur in magnetofluids with large fluid and magnetic Reynolds numbers, will be discussed. When Reynolds numbers are large and energy decays slowly, the distribution of energy with respect to length scale becomes quasi-stationary and MHD turbulence can be described statistically. In the limit of infinite Reynolds numbers, viscosity and resistivity become zero and if these values are used in the MHD equations ab initio, a model system called ideal MHD turbulence results. This model system is typically confined in simple geometries with some form of homogeneous boundary conditions, allowing for velocity and magnetic field to be represented by orthogonal function expansions. One advantage to this is that the coefficients of the expansions form a set of nonlinearly interacting variables whose behavior can be described by equilibrium statistical mechanics, i.e., by a canonical ensemble theory based on the global invariants (energy, cross helicity and magnetic helicity) of ideal MHD turbulence. Another advantage is that truncated expansions provide a finite dynamical system whose time evolution can be numerically simulated to test the predictions of the associated statistical mechanics. If ensemble predictions are the same as time averages, then the system is said to be ergodic; if not, the system is nonergodic. Although it had been implicitly assumed in the early days of ideal MHD statistical theory development that these finite dynamical systems were ergodic, numerical simulations provided sufficient evidence that they were, in fact, nonergodic. Specifically, while canonical ensemble theory predicted that expansion coefficients would be (i) zero-mean random variables with (ii) energy that decreased with length scale, it was found that although (ii) was correct, (i) was not and the expected ergodicity was broken. The exact cause of this broken ergodicity was explained, after much investigation, by greatly extending the statistical theory of ideal MHD turbulence. The mathematical details of broken ergodicity, in fact, give a quantitative explanation of how coherent structure, dynamic alignment and force-free states appear in turbulent magnetofluids. The relevance of these ideal results to real MHD turbulence occurs because broken ergodicity is most manifest in the ideal case at the largest length scales and it is in these largest scales that a real magnetofluid has the least dissipation, i.e., most closely approaches the behavior of an ideal magnetofluid. Furthermore, the effects grow stronger when cross and magnetic helicities grow large with respect to energy, and this is exactly what occurs with time in a real magnetofluid, where it is called selective decay. The relevance of these results found in ideal MHD turbulence theory to the real world is that they provide at least a qualitative explanation of why confined turbulent magnetofluids, such as the liquid iron that fills the Earth's outer core, produce stationary, large-scale magnetic fields, i.e., the geomagnetic field. These results should also apply to other planets as well as to plasma confinement devices on Earth and in space, and the effects should be manifest if Reynolds numbers are high enough and there is enough time for stationarity to occur, at least approximately. In the presentation, details will be given for both theoretical and numerical results, and references will be provided.

Shebalin, John V.↗

NASA NTRS2014

MHD Turbulence and Magnetic Dynamos

Incompressible magnetohydrodynamic (MHD) turbulence and magnetic dynamos, which occur in magnetofluids with large fluid and magnetic Reynolds numbers, will be discussed. When Reynolds numbers are large and energy decays slowly, the distribution of energy with respect to length scale becomes quasi-stationary and MHD turbulence can be described statistically. In the limit of infinite Reynolds numbers, viscosity and resistivity become zero and if these values are used in the MHD equations ab initio, a model system called ideal MHD turbulence results. This model system is typically confined in simple geometries with some form of homogeneous boundary conditions, allowing for velocity and magnetic field to be represented by orthogonal function expansions. One advantage to this is that the coefficients of the expansions form a set of nonlinearly interacting variables whose behavior can be described by equilibrium statistical mechanics, i.e., by a canonical ensemble theory based on the global invariants (energy, cross helicity and magnetic helicity) of ideal MHD turbulence. Another advantage is that truncated expansions provide a finite dynamical system whose time evolution can be numerically simulated to test the predictions of the associated statistical mechanics. If ensemble predictions are the same as time averages, then the system is said to be ergodic; if not, the system is nonergodic. Although it had been implicitly assumed in the early days of ideal MHD statistical theory development that these finite dynamical systems were ergodic, numerical simulations provided sufficient evidence that they were, in fact, nonergodic. Specifically, while canonical ensemble theory predicted that expansion coefficients would be (i) zero-mean random variables with (ii) energy that decreased with length scale, it was found that although (ii) was correct, (i) was not and the expected ergodicity was broken. The exact cause of this broken ergodicity was explained, after much investigation, by greatly extending the statistical theory of ideal MHD turbulence. The mathematical details of broken ergodicity, in fact, give a quantitative explanation of how coherent structure, dynamic alignment and force-free states appear in turbulent magnetofluids. The relevance of these ideal results to real MHD turbulence occurs because broken ergodicity is most manifest in the ideal case at the largest length scales and it is in these largest scales that a real magnetofluid has the least dissipation, i.e., most closely approaches the behavior of an ideal magnetofluid. Furthermore, the effects grow stronger when cross and magnetic helicities grow large with respect to energy, and this is exactly what occurs with time in a real magnetofluid, where it is called selective decay. The relevance of these results found in ideal MHD turbulence theory to the real world is that they provide at least a qualitative explanation of why confined turbulent magnetofluids, such as the liquid iron that fills the Earth's outer core, produce stationary, large-scale magnetic fields, i.e., the geomagnetic field. These results should also apply to other planets as well as to plasma confinement devices on Earth and in space, and the effects should be manifest if Reynolds numbers are high enough and there is enough time for stationarity to occur, at least approximately. In the presentation, details will be given for both theoretical and numerical results, and references will be provided.

Shebalin, John V↗

NASA NTRS2023

Shape from spectra

We introduce a new unified atmospheric–topographic correction approach that estimates surface geometry directly from the radiance measurement. Surface topography influences the at-sensor radiance measurement, making precise topography modeling critical in applications like vegetation or snow studies in mountainous terrain. Currently, elevation maps are used to derive topographic variables such as the slope and sky-view factor. This process is error-prone since static global digital elevation models do not generally achieve the accuracy required, and even minor mismatches in spatial resolution can introduce significant artifacts in downstream processing. Here we demonstrate that it is possible to estimate topographic parameters directly from spectral data, ensuring perfect physical consistency, temporal coincidence, and spatial alignment. We present experiments estimating topographic slope in two scenes in Southern California, with data from NASA’s Next Generation Airborne Visible/Near Infrared Imaging Spectrometer (AVIRIS-NG). We compared our radiance-based estimates against high-resolution lidar datasets. Our initial validation result showed a correlation of R 2 = 0.864 (n = 160) over the homogeneous surface of Beckman Auditorium’s cone-shaped roof on the Caltech campus in Pasadena, California. We then validate the model over a larger study site near Santa Clarita, California, finding R 2 = 0.923 (n = 40,000) in a 350 x 350m area. The accuracy of our model estimates, combined with its systematic advantages over the alternative, show the potential of the approach for use in both airborne campaigns and orbital missions.

Nimrod Carmon↗

NASA NTRS1991

The orbital ephemeris and eclipse transitions of the low-mass X-ray binary EXO 0748 - 676

Using the eclipses as fiducial markers, an updated ephemeris for EXO 0748 - 676 is derived and evidence is found that between February 1985 and March 1989 the 3.82-h orbital period of EXO 0748 - 676 decreased with a time scale of -5 x 10 to the 6th yr. The sense of this change is the same as that predicted by simple models for the evolution of low-mass X-ray binaries containing main-sequence companions, but is a factor about 100 faster than expected. This rapid change in orbital period could result from the expansion of the companion due to the effects of X-ray heating. The eclipse transition durations are variable, with the shortest observed taking 1.5 s and the longest 40 s. This latter figure is about an order of magnitude too large to be due to absorption effects in the atmosphere of the secondary assuming a Roche geometry and likely stellar temperature. Either flaring activity or the presence of an X-ray heated evaporative wind or a corona may enhance the scale height of the companion's atmosphere producing the extended eclipse transitions.

Parmar, A. N.↗

NASA NTRS1992

Influence of airfoil geometry on delta wing leading-edge vortices and vortex-induced aerodynamics at supersonic speeds

An assessment of the influence of airfoil geometry on delta wing leading edge vortex flow and vortex induced aerodynamics at supersonic speeds is discussed. A series of delta wing wind tunnel models were tested over a Mach number range from 1.7 to 2.0. The model geometric variables included leading edge sweep and airfoil shape. Surface pressure data, vapor screen, and oil flow photograph data were taken to evaluate the complex structure of the vortices and shocks on the family of wings tested. The data show that airfoil shape has a significant impact on the wing upper surface flow structure and pressure distribution, but has a minimal impact on the integrated upper surface pressure increments.

Wood, Richard M.↗

NASA NTRS1992

Dynamic interactions between hypersonic vehicle aerodynamics and propulsion system performance

Described here is the development of a flexible simulation model for scramjet hypersonic propulsion systems. The primary goal is determination of sensitivity of the thrust vector and other system parameters to angle of attack changes of the vehicle. Such information is crucial in design and analysis of control system performance for hypersonic vehicles. The code is also intended to be a key element in carrying out dynamic interaction studies involving the influence of vehicle vibrations on propulsion system/control system coupling and flight stability. Simple models are employed to represent the various processes comprising the propulsion system. A method of characteristics (MOC) approach is used to solve the forebody and external nozzle flow fields. This results in a very fast computational algorithm capable of carrying out the vast number of simulation computations needed in guidance, stability, and control studies. The three-dimensional fore- and aft body (nozzle) geometry is characterized by the centerline profiles as represented by a series of coordinate points and body cross-section curvature. The engine module geometry is represented by an adjustable vertical grid to accommodate variations of the field parameters throughout the inlet and combustor. The scramjet inlet is modeled as a two-dimensional supersonic flow containing adjustable sidewall wedges and multiple fuel injection struts. The inlet geometry including the sidewall wedge angles, the number of injection struts, their sweepback relative to the vehicle reference line, and strut cross-section are user selectable. Combustion is currently represented by a Rayleigh line calculation including corrections for variable gas properties; improved models are being developed for this important element of the propulsion flow field. The program generates (1) variation of thrust magnitude and direction with angle of attack, (2) pitching moment and line of action of the thrust vector, (3) pressure and temperature distributions throughout the system, and (4) performance parameters such as thrust coefficient, specific impulse, mass flow rates, and equivalence ratio. Preliminary results are in good agreement with available performance data for systems resembling the NASP vehicle configuration.

Flandro, G. A.↗

NASA NTRSDate not supplied

Detection of Drizzle and Light Snowfall Over the Southern Ocean Using the GPM Combined Algorithm

Following the 2014 launch of the Global Precipitation Measurement Mission (GPM), an unprecedented combination of coincident active and passive microwave observations are available for state of the art precipitation retrieval. The GPM Combined Algorithm forms the backbone of this effort, optimizing geophysical variables for agreement with the multi-spectral information content. Combined retrievals are then utilized, along with a radiative transfer model, as a database applied for retrievals across a constellation of passive microwave radiometers of varying frequencies and viewing geometries. By keeping such retrievals related through the transfer standard of the combined algorithm, level 3 products such as the Integrated Multi-satellitE Retrievals for GPM (IMERG) are able to provide consistent global products for users at the higher temporal resolution required for hydrological applications on the global scale. In the current version of the combined product, precipitation retrievals are carried out only in the presence of a signal from the active radar. As a result, light precipitation and drizzle below the threshold of radar sensitivity are not included in any of the products down the chain from the constellation to IMERG. In this work, the effects of enhancing the retrievals with an optimal estimation-type (OE) water vapor and cloud retrieval using the passive observations are explored over the Southern Ocean on a regional scale. Non-convergence of the OE in areas with no detectable radar signal is used to identify areas with high probability of light precipitation and drizzle. Microphysical scale characterization of the light precipitation will be explored using information derived from the higher sensitivity CloudSat mission along with model information to identify and associate the related atmospheric state and dynamics. This is a physically-based approach requiring radiometric consistency with all available multi-spectral observations. Retrieved drizzle can then be included in the constellation databases continuing across all scales through to the level 3 products. The technique, successfully demonstrated for the Southern Ocean, can be easily adapted for light precipitation retrieval over other areas and surfaces leading to a global climatology of light precipitation.

Sarah Rinegerud↗

NASA NTRS1974

An analytical study of reduced-gravity propellant settling

Full-scale propellant reorientation flow dynamics for the Centaur D-1T fuel tank were analyzed. A computer code using the simplified marker and cell technique was modified to include the capability for a variable-grid mesh configuration. Use of smaller cells near the boundary, near baffles, and in corners provides improved flow resolution. Two drop tower model cases were simulated to verify program validity: one case without baffles, the other with baffles and geometry identical to Centaur D-1T. Flow phenomena using the new code successfully modeled drop tower data. Baffles are a positive factor in the settling flow. Two full-scale Centaur D-1T cases were simulated using parameters based on the Titan/Centaur proof flight. These flow simulations indicated the time to clear the vent area and an indication of time to orient and collect the propellant. The results further indicated the complexity of the reorientation flow and the long time period required for settling.

Bradshaw, R. D.↗

NASA NTRS1976

Design of structures for optimum geometry

A general method is proposed for geometric optimization of finite element structures on the assumption that a reasonable initial geometry is specified. The number of finite elements, number of joints, and the element-joint relationships are specified and are not changed during the optimization process. The structure may be statically indeterminate and may support multiple loading conditions. Design variables include geometric and member sizing parameters. Design objectives may be minimum weight or cost, and constraints include strength and stiffness limitations. The optimization procedure is a general mathematical programming approach. An iterative design algorithm is outlined, and the method is specialized for truss structures. Design examples are given.

Vanderplaats, G. N.↗

NASA NTRS1987

A linearized Euler analysis of unsteady flows in turbomachinery

A method for calculating unsteady flows in cascades is presented. The model, which is based on the linearized unsteady Euler equations, accounts for blade loading shock motion, wake motion, and blade geometry. The mean flow through the cascade is determined by solving the full nonlinear Euler equations. Assuming the unsteadiness in the flow is small, then the Euler equations are linearized about the mean flow to obtain a set of linear variable coefficient equations which describe the small amplitude, harmonic motion of the flow. These equations are discretized on a computational grid via a finite volume operator and solved directly subject to an appropriate set of linearized boundary conditions. The steady flow, which is calculated prior to the unsteady flow, is found via a Newton iteration procedure. An important feature of the analysis is the use of shock fitting to model steady and unsteady shocks. Use of the Euler equations with the unsteady Rankine-Hugoniot shock jump conditions correctly models the generation of steady and unsteady entropy and vorticity at shocks. In particular, the low frequency shock displacement is correctly predicted. Results of this method are presented for a variety of test cases. Predicted unsteady transonic flows in channels are compared to full nonlinear Euler solutions obtained using time-accurate, time-marching methods. The agreement between the two methods is excellent for small to moderate levels of flow unsteadiness. The method is also used to predict unsteady flows in cascades due to blade motion (flutter problem) and incoming disturbances (gust response problem).

Hall, Kenneth C.↗

NASA NTRS1989

Calculation of unsteady flows in turbomachinery using the linearized Euler equations

A method for calculating unsteady flows in cascades is presented. The model, which is based on the linearized unsteady Euler equations, accounts for blade loading shock motion, wake motion, and blade geometry. The mean flow through the cascade is determined by solving the full nonlinear Euler equations. Assuming the unsteadiness in the flow is small, then the Euler equations are linearized about the mean flow to obtain a set of linear variable coefficient equations which describe the small amplitude, harmonic motion of the flow. These equations are discretized on a computational grid via a finite volume operator and solved directly subject to an appropriate set of linearized boundary conditions. The steady flow, which is calculated prior to the unsteady flow, is found via a Newton iteration procedure. An important feature of the analysis is the use of shock fitting to model steady and unsteady shocks. Use of the Euler equations with the unsteady Rankine-Hugoniot shock jump conditions correctly models the generation of steady and unsteady entropy and vorticity at shocks. In particular, the low frequency shock displacement is correctly predicted. Results of this method are presented for a variety of test cases. Predicted unsteady transonic flows in channels are compared to full nonlinear Euler solutions obtained using time-accurate, time-marching methods. The agreement between the two methods is excellent for small to moderate levels of flow unsteadiness. The method is also used to predict unsteady flows in cascades due to blade motion (flutter problem) and incoming disturbances (gust response problem).

Hall, Kenneth C.↗