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At least 55 records · Page 3
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Characterization of the fractal dimension of helium plumes in laminar to turbulent transition

Abstract Schlieren imaging in conjunction with a high-speed camera was used to observe the behavior of metered helium plumes as they transition from laminar to turbulent flow in an air environment. The plumes were visualized at twelve jet Reynolds numbers ranging from 200 to 2980. The fractal dimension of the flows was obtained by applying a box counting algorithm to the recorded schlieren images. The results were analyzed to determine the correlation between the Reynolds number of the flow and the fractal dimension of the observed turbulence. A trend of increasing fractal dimension with increasing Reynolds number was observed for several different types of schlieren cutoffs including horizontal cutoff, vertical cutoff, circular cutoff, focused shadowgraphy and de-focused shadowgraphy. The vertical cutoff and focused shadowgraphy imaging methods showed the most consistent results for the fractal dimension characterization during the laminar to turbulent transition. For transitional plumes, it was observed that fractal dimension increased with distance from the jet outlet.

42 ENGINEERING↗
DOE OSTI2020

A viscoelastic model for seismic attenuation using fractal mechanical networks

SUMMARY Seismic attenuation (quantified by the quality factor Q) has a significant impact on the seismic waveforms, especially in the fluid-saturated rocks. This dissipative process can be phenomenologically represented by viscoelastic models. Previous seismological studies show that the Q value of Earth media exhibits a nearly frequency-independent behaviour (often referred to as constant-Q in literature) in the seismic frequency range. Such attenuation can be described by the mathematical Kjartansson constant-Q model, which lacks of a physical representation in the viscoelastic sense. Inspired by the fractal nature of the pore fluid distribution in patchy-saturated rocks, here we propose two fractal mechanical network (FMN) models, that is, a fractal tree model and a quasi-fractal ladder model, to phenomenologically represent the frequency-independent Q behaviour. As with the classic viscoelastic models, the FMN models are composed of mechanical elements (spring and dashpots) arranged in different hierarchical patterns. A particular parametrization of each model can produce the same complex modulus as in the Kjartansson model, which leads to the constant-Q. Applying the theory to several typical rock samples, we find that the seismic attenuation signature of these rocks can be accurately represented by either one of the FMN models. Besides, we demonstrate that the ladder model in particular exhibits the realistic multiscale fractal structure of the saturated rocks. Therefore, the FMN models as a proxy could provide a new way to estimate the microscopic rock structure property from macroscopic seismic attenuation observation.

Geochemistry & Geophysics↗
NASA NTRS1989

Laser light scattering as a probe of fractal colloid aggregates

The extensive use of laser light scattering is reviewed, both static and dynamic, in the study of colloid aggregation. Static light scattering enables the study of the fractal structure of the aggregates, while dynamic light scattering enables the study of aggregation kinetics. In addition, both techniques can be combined to demonstrate the universality of the aggregation process. Colloidal aggregates are now well understood and therefore represent an excellent experimental system to use in the study of the physical properties of fractal objects. However, the ultimate size of fractal aggregates is fundamentally limited by gravitational acceleration which will destroy the fractal structure as the size of the aggregates increases. This represents a great opportunity for spaceborne experimentation, where the reduced g will enable the growth of fractal structures of sufficient size for many interesting studies of their physical properties.

Weitz, David A.↗
NASA NTRS1989

A fractal approach to probabilistic seismic hazard assessment

The definition of a fractal distribution is that the number of objects (events) N with a characteristic size greater than r satisfies the relation N proportional to r exp - D is the fractal dimension. The applicability of a fractal relation implies that the underlying physical process is scale-invariant over the range of applicability of the relation. The empirical frequency-magnitude relation for earthquakes defining a b-value is a fractal relation with D = 2b. Accepting the fractal distribution, the level of regional seismicity can be related to the rate of regional strain and the magnitude of the largest characteristic earthquake. High levels of seismic activity indicate either a large regional strain or a low-magnitude maximum characteristic earthquake (or both). If the regional seismicity has a weak time dependence, the approach can be used to make probabilistic seismic hazard assessments.

Turcotte, D. L.↗
NASA NTRS1990

Implications of chaos, scale-invariance, and fractal statistics in geology

A set of three nonlinear total differential equations (Lorenz equations) exhibiting deterministic chaos is considered, and it is shown that these equations demonstrate that deterministic equations with deterministic initial conditions can yield stocastic solutions with fractal statistics. The logistic map, fractal distributions, and fragmentation are discussed. It is pointed out that well-defined fractal distributions of earthquakes are found both regionally and globally, and that the general applicability of the fractal relation for seismicity can provide the basis for a quantitative seismic hazard assessment. It is suggested that the governing physics of erosional topography is nonlinear and may be related to a fractal distribution of storms and floods that generate and renew erosional feature such as gullies and drainage systems.

Turcotte, D. L.↗
NASA NTRS1992

Lava flows are fractals

Results are presented of a preliminary investigation of the fractal nature of the plan-view shapes of lava flows in Hawaii (based on field measurements and aerial photographs), as well as in Idaho and the Galapagos Islands (using aerial photographs only). The shapes of the lava flow margins are found to be fractals: lava flow shape is scale-invariant. This observation suggests that nonlinear forces are operating in them because nonlinear systems frequently produce fractals. A'a and pahoehoe flows can be distinguished by their fractal dimensions (D). The majority of the a'a flows measured have D between 1.05 and 1.09, whereas the pahoehoe flows generally have higher D (1.14-1.23). The analysis is extended to other planetary bodies by measuring flows from orbital images of Venus, Mars, and the moon. All are fractal and have D consistent with the range of terrestrial a'a and have D consistent with the range of terrestrial a'a and pahoehoe values.

Bruno, B. C.↗
NASA NTRS1992

Fractals and cosmological large-scale structure

Observations of galaxy-galaxy and cluster-cluster correlations as well as other large-scale structure can be fit with a 'limited' fractal with dimension D of about 1.2. This is not a 'pure' fractal out to the horizon: the distribution shifts from power law to random behavior at some large scale. If the observed patterns and structures are formed through an aggregation growth process, the fractal dimension D can serve as an interesting constraint on the properties of the stochastic motion responsible for limiting the fractal structure. In particular, it is found that the observed fractal should have grown from two-dimensional sheetlike objects such as pancakes, domain walls, or string wakes. This result is generic and does not depend on the details of the growth process.

Luo, Xiaochun↗
NASA NTRS1999

Multi-Scale Fractal Analysis of Image Texture and Pattern

Analyses of the fractal dimension of Normalized Difference Vegetation Index (NDVI) images of homogeneous land covers near Huntsville, Alabama revealed that the fractal dimension of an image of an agricultural land cover indicates greater complexity as pixel size increases, a forested land cover gradually grows smoother, and an urban image remains roughly self-similar over the range of pixel sizes analyzed (10 to 80 meters). A similar analysis of Landsat Thematic Mapper images of the East Humboldt Range in Nevada taken four months apart show a more complex relation between pixel size and fractal dimension. The major visible difference between the spring and late summer NDVI images of the absence of high elevation snow cover in the summer image. This change significantly alters the relation between fractal dimension and pixel size. The slope of the fractal dimensional-resolution relation provides indications of how image classification or feature identification will be affected by changes in sensor spatial resolution.

Emerson, Charles W.↗
NASA NTRS1999

Multi-Scale Fractal Analysis of Image Texture and Pattern

Analyses of the fractal dimension of Normalized Difference Vegetation Index (NDVI) images of homogeneous land covers near Huntsville, Alabama revealed that the fractal dimension of an image of an agricultural land cover indicates greater complexity as pixel size increases, a forested land cover gradually grows smoother, and an urban image remains roughly self-similar over the range of pixel sizes analyzed (10 to 80 meters). A similar analysis of Landsat Thematic Mapper images of the East Humboldt Range in Nevada taken four months apart show a more complex relation between pixel size and fractal dimension. The major visible difference between the spring and late summer NDVI images is the absence of high elevation snow cover in the summer image. This change significantly alters the relation between fractal dimension and pixel size. The slope of the fractal dimension-resolution relation provides indications of how image classification or feature identification will be affected by changes in sensor spatial resolution.

Emerson, Charles W.↗
NASA NTRS1997

Characterizing Hyperspectral Imagery (AVIRIS) Using Fractal Technique

With the rapid increase in hyperspectral data acquired by various experimental hyperspectral imaging sensors, it is necessary to develop efficient and innovative tools to handle and analyze these data. The objective of this study is to seek effective spatial analytical tools for summarizing the spatial patterns of hyperspectral imaging data. In this paper, we (1) examine how fractal dimension D changes across spectral bands of hyperspectral imaging data and (2) determine the relationships between fractal dimension and image content. It has been documented that fractal dimension changes across spectral bands for the Landsat-TM data and its value [(D)] is largely a function of the complexity of the landscape under study. The newly available hyperspectral imaging data such as that from the Airborne Visible Infrared Imaging Spectrometer (AVIRIS) which has 224 bands, covers a wider spectral range with a much finer spectral resolution. Our preliminary result shows that fractal dimension values of AVIRIS scenes from the Santa Monica Mountains in California vary between 2.25 and 2.99. However, high fractal dimension values (D > 2.8) are found only from spectral bands with high noise level and bands with good image quality have a fairly stable dimension value (D = 2.5 - 2.6). This suggests that D can also be used as a summary statistics to represent the image quality or content of spectral bands.

Qiu, Hong-Lie↗
NASA NTRS1997

Multitemporal and Multiscaled Fractal Analysis of Landsat Satellite Data Using the Image Characterization and Modeling System (ICAMS)

The Image Characterization And Modeling System (ICAMS) is a public domain software package that is designed to provide scientists with innovative spatial analytical tools to visualize, measure, and characterize landscape patterns so that environmental conditions or processes can be assessed and monitored more effectively. In this study ICAMS has been used to evaluate how changes in fractal dimension, as a landscape characterization index, and resolution, are related to differences in Landsat images collected at different dates for the same area. Landsat Thematic Mapper (TM) data obtained in May and August 1993 over a portion of the Great Basin Desert in eastern Nevada were used for analysis. These data represent contrasting periods of peak "green-up" and "dry-down" for the study area. The TM data sets were converted into Normalized Difference Vegetation Index (NDVI) images to expedite analysis of differences in fractal dimension between the two dates. These NDVI images were also resampled to resolutions of 60, 120, 240, 480, and 960 meters from the original 30 meter pixel size, to permit an assessment of how fractal dimension varies with spatial resolution. Tests of fractal dimension for two dates at various pixel resolutions show that the D values in the August image become increasingly more complex as pixel size increases to 480 meters. The D values in the May image show an even more complex relationship to pixel size than that expressed in the August image. Fractal dimension for a difference image computed for the May and August dates increase with pixel size up to a resolution of 120 meters, and then decline with increasing pixel size. This means that the greatest complexity in the difference images occur around a resolution of 120 meters, which is analogous to the operational domain of changes in vegetation and snow cover that constitute differences between the two dates.

Quattrochi, Dale A.↗
NASA NTRS2000

Wavelet and Fractal Analysis of Remotely Sensed Surface Temperature with Applications to Estimation of Surface Sensible Heat Flux Density

Wavelet and fractal analyses have been used successfully to analyze one-dimensional data sets such as time series of financial, physical, and biological parameters. These techniques have been applied to two-dimensional problems in some instances, including the analysis of remote sensing imagery. In this respect, these techniques have not been widely used by the remote sensing community, and their overall capabilities as analytical tools for use on satellite and aircraft data sets is not well known. Wavelet and fractal analyses have the potential to provide fresh insight into the characterization of surface properties such as temperature and emissivity distributions, and surface processes such as the heat and water vapor exchange between the surface and the lower atmosphere. In particular, the variation of sensible heat flux density as a function of the change In scale of surface properties Is difficult to estimate, but - in general - wavelets and fractals have proved useful in determining the way a parameter varies with changes in scale. We present the results of a limited study on the relationship between spatial variations in surface temperature distribution and sensible heat flux distribution as determined by separate wavelet and fractal analyses. We analyzed aircraft imagery obtained in the thermal infrared (IR) bands from the multispectral TIMS and hyperspectral MASTER airborne sensors. The thermal IR data allows us to estimate the surface kinetic temperature distribution for a number of sites in the Midwestern and Southwestern United States (viz., San Pedro River Basin, Arizona; El Reno, Oklahoma; Jornada, New Mexico). The ground spatial resolution of the aircraft data varied from 5 to 15 meters. All sites were instrumented with meteorological and hydrological equipment including surface layer flux measuring stations such as Bowen Ratio systems and sonic anemometers. The ground and aircraft data sets provided the inputs for the wavelet and fractal analyses, and the validation of the results.

Schieldge, John↗
NASA NTRS2000

Roughness Perception of Haptically Displayed Fractal Surfaces

Surface profiles were generated by a fractal algorithm and haptically rendered on a force feedback joystick, Subjects were asked to use the joystick to explore pairs of surfaces and report to the experimenter which of the surfaces they felt was rougher. Surfaces were characterized by their root mean square (RMS) amplitude and their fractal dimension. The most important factor affecting the perceived roughness of the fractal surfaces was the RMS amplitude of the surface. When comparing surfaces of fractal dimension 1.2-1.35 it was found that the fractal dimension was negatively correlated with perceived roughness.

Costa, Michael A.↗
NASA NTRS2005

A Fractal Dimension Survey of Active Region Complexity

A new approach to quantifying the magnetic complexity of active regions using a fractal dimension measure is presented. This fully-automated approach uses full disc MDI magnetograms of active regions from a large data set (2742 days of the SoHO mission; 9342 active regions) to compare the calculated fractal dimension to both Mount Wilson classification and flare rate. The main Mount Wilson classes exhibit no distinct fractal dimension distribution, suggesting a self-similar nature of all active regions. Solar flare productivity exhibits an increase in both the frequency and GOES X-ray magnitude of flares from regions with higher fractal dimensions. Specifically a lower threshold fractal dimension of 1.2 and 1.25 exists as a necessary, but not sufficient, requirement for an active region to produce M- and X-class flares respectively .

McAteer, R. T. James↗
DOE OSTI2020

Scaling Analysis of Two–Phase Flow in Fractal Permeability Fields

Fluid mixing in permeable media is essential in many practical applications. The mixing process is a consequence of velocity fluctuations owing to geological heterogeneities and mobility contrast of fluids. Heterogeneities in natural rocks are often spatially correlated, and their properties, such as permeability, may be described using fractal distributions. This work models the fractal characteristics of such permeability fields in which the covariance function is expressed as a power-law function. A generalized scaling relation is derived relating various fractal permeability fields using the magnitude of their fluctuations. Here, this relation reveals the self-similar behavior of two-phase flow in such permeable media. To that end, a recently developed, high-resolution numerical simulator is employed to validate the analytically derived scaling relations. Two flow problems are considered in which flow is governed by 1) a linear, and 2) a nonlinear transport equation. Due to the probabilistic representation of the fractal permeability fields, a sensitivity study is conducted for each flow scenario to determine the number of realizations required for statistical convergence. Scaling analysis is performed using ensemble averages of simulated saturation profiles and their mixing lengths. Results support the validity of the developed scaling relation across the range of investigated flow conditions at intermediate times. The dynamics of linear flow in the asymptotic regime is affected by the correlation structure of heterogeneity. In nonlinear flow, scaling behavior appears to be dominated by the degree of nonlinearity.

58 GEOSCIENCES↗
DOE OSTI2022

Construction of Fractal Order and Phase Transition with Rydberg Atoms

We propose the construction of a many-body phase of matter with fractal structure using arrays of Rydberg atoms. The degenerate low energy excited states of this phase form a self-similar fractal structure. This phase is analogous to the so-called “type-II fracton topological states.” The main challenge in realizing fractonlike models in standard condensed matter platforms is the creation of multispin interactions, since realistic systems are typically dominated by two-body interactions. In this work, we demonstrate that the van der Waals interaction and experimental tunability of Rydberg-based platforms enable the simulation of exotic phases of matter with fractal structures, and the study of a quantum phase transition involving a fractal ordered phase.

74 ATOMIC AND MOLECULAR PHYSICS↗
NASA NTRS1995

Fractals in biology and medicine

Our purpose is to describe some recent progress in applying fractal concepts to systems of relevance to biology and medicine. We review several biological systems characterized by fractal geometry, with a particular focus on the long-range power-law correlations found recently in DNA sequences containing noncoding material. Furthermore, we discuss the finding that the exponent alpha quantifying these long-range correlations ("fractal complexity") is smaller for coding than for noncoding sequences. We also discuss the application of fractal scaling analysis to the dynamics of heartbeat regulation, and report the recent finding that the normal heart is characterized by long-range "anticorrelations" which are absent in the diseased heart.

Review↗
NASA NTRS1999

Fractal analysis of heart rate dynamics as a predictor of mortality in patients with depressed left ventricular function after acute myocardial infarction. TRACE Investigators. TRAndolapril Cardiac Evaluation

A number of new methods have been recently developed to quantify complex heart rate (HR) dynamics based on nonlinear and fractal analysis, but their value in risk stratification has not been evaluated. This study was designed to determine whether selected new dynamic analysis methods of HR variability predict mortality in patients with depressed left ventricular (LV) function after acute myocardial infarction (AMI). Traditional time- and frequency-domain HR variability indexes along with short-term fractal-like correlation properties of RR intervals (exponent alpha) and power-law scaling (exponent beta) were studied in 159 patients with depressed LV function (ejection fraction <35%) after an AMI. By the end of 4-year follow-up, 72 patients (45%) had died and 87 (55%) were still alive. Short-term scaling exponent alpha (1.07 +/- 0.26 vs 0.90 +/- 0.26, p <0.001) and power-law slope beta (-1.35 +/- 0.23 vs -1.44 +/- 0.25, p <0.05) differed between survivors and those who died, but none of the traditional HR variability measures differed between these groups. Among all analyzed variables, reduced scaling exponent alpha (<0.85) was the best univariable predictor of mortality (relative risk 3.17, 95% confidence interval 1.96 to 5.15, p <0.0001), with positive and negative predictive accuracies of 65% and 86%, respectively. In the multivariable Cox proportional hazards analysis, mortality was independently predicted by the reduced exponent alpha (p <0.001) after adjustment for several clinical variables and LV function. A short-term fractal-like scaling exponent was the most powerful HR variability index in predicting mortality in patients with depressed LV function. Reduction in fractal correlation properties implies more random short-term HR dynamics in patients with increased risk of death after AMI.

Non-NASA Center↗