scholarly journals Extended power-law scaling of air permeabilities measured on a block of tuff

2011 ◽  
Vol 8 (4) ◽  
pp. 7805-7843 ◽  
Author(s):  
M. Siena ◽  
A. Guadagnini ◽  
M. Riva ◽  
S. P. Neuman
Keyword(s):  
Power Law ◽  
Spatial Data ◽  
Structure Functions ◽  
Air Permeability ◽  
Self Similarity ◽  
Scaling Exponents ◽  
Theoretical Support ◽  
Sample Structure ◽  
Extended Power ◽  
Permeability Data

Abstract. We use three methods to identify power law scaling of (natural) log air permeability data collected by Tidwell and Wilson (1999) on the faces of a laboratory-scale block of Topopah Spring tuff: method of moments (M), extended power-law scaling also known as Extended Self-Similarity (ESS) and a generalized version thereof (G-ESS). All three methods focus on qth-order sample structure functions of absolute increments. Most such functions exhibit power-law scaling at best over a limited midrange of experimental separation scales, or lags, which are sometimes difficult to identify unambiguously by means of M. ESS and G-ESS extend this range in a way that renders power-law scaling easier to characterize. Most analyses of this type published to date concern time series or one-dimensional transects of spatial data associated with a unique measurement (support) scale. We consider log air permeability data having diverse support scales on the faces of a cube. Our analysis confirms the superiority of ESS and G-ESS over M in identifying the scaling exponents ξ(q) of corresponding structure functions of orders q, suggesting further that ESS is more reliable than G-ESS. The exponents vary in a nonlinear fashion with q as is typical of real or apparent (Guadagnini and Neuman, 2011; Guadagnini et al., 2011) multifractals. Our estimates of the Hurst scaling coefficient increase with support scale, implying a reduction in roughness (anti-persistence) of the log permeability field with measurement volume. ESS and G-ESS ratios between scaling exponents ξ(q) associated with various orders q show no distinct dependence on support volume or on two out of three Cartesian directions (there being no distinct power law scaling in the third direction). The finding by Tidwell and Wilson (1999) that log permeabilities associated with all tip sizes can be characterized by stationary variogram models, coupled with our findings that log permeability increments associated with the smallest tip size are approximately Gaussian and those associated with all tip size scales show nonlinear (multifractal) variations in ξ(q) with q, are consistent with a view of these data as a sample from a truncated version (tfBm) of self-affine fractional Brownian motion (fBm). Since in theory the scaling exponents, ξ(q), of tfBm vary linearly with q we conclude, in accord with Neuman (2010a, b, 2011), that nonlinear scaling in our case is not an indication of multifractality but an artifact of sampling from tfBm. This allows us to explain theoretically how power law scaling is extended by ESS. It further allows us to identify the functional form and estimate all parameters of the corresponding tfBm based on sample structure functions of first and second orders. Our estimate of lower cutoff is consistent with a theoretical support scale of the data.

scholarly journals Extended power-law scaling of air permeabilities measured on a block of tuff

2012 ◽  
Vol 16 (1) ◽  
pp. 29-42 ◽  
Author(s):  
M. Siena ◽  
A. Guadagnini ◽  
M. Riva ◽  
S. P. Neuman
Keyword(s):  
Power Law ◽  
Structure Functions ◽  
Self Similarity ◽  
Scaling Exponents ◽  
Multi Scale ◽  
Sample Structure ◽  
Heavy Tailed ◽  
Non Gaussian ◽  
Nonlinear Fashion ◽  
Extended Power

Abstract. We use three methods to identify power-law scaling of multi-scale log air permeability data collected by Tidwell and Wilson on the faces of a laboratory-scale block of Topopah Spring tuff: method of moments (M), Extended Self-Similarity (ESS) and a generalized version thereof (G-ESS). All three methods focus on q-th-order sample structure functions of absolute increments. Most such functions exhibit power-law scaling at best over a limited midrange of experimental separation scales, or lags, which are sometimes difficult to identify unambiguously by means of M. ESS and G-ESS extend this range in a way that renders power-law scaling easier to characterize. Our analysis confirms the superiority of ESS and G-ESS over M in identifying the scaling exponents, ξ(q), of corresponding structure functions of orders q, suggesting further that ESS is more reliable than G-ESS. The exponents vary in a nonlinear fashion with q as is typical of real or apparent multifractals. Our estimates of the Hurst scaling coefficient increase with support scale, implying a reduction in roughness (anti-persistence) of the log permeability field with measurement volume. The finding by Tidwell and Wilson that log permeabilities associated with all tip sizes can be characterized by stationary variogram models, coupled with our findings that log permeability increments associated with the smallest tip size are approximately Gaussian and those associated with all tip sizes scale show nonlinear variations in ξ(q) with q, are consistent with a view of these data as a sample from a truncated version (tfBm) of self-affine fractional Brownian motion (fBm). Since in theory the scaling exponents, ξ(q), of tfBm vary linearly with q we conclude that nonlinear scaling in our case is not an indication of multifractality but an artifact of sampling from tfBm. This allows us to explain theoretically how power-law scaling of our data, as well as of non-Gaussian heavy-tailed signals subordinated to tfBm, are extended by ESS. It further allows us to identify the functional form and estimate all parameters of the corresponding tfBm based on sample structure functions of first and second orders.

scholarly journals Solar wind low-frequency magnetohydrodynamic turbulence: extended self-similarity and scaling laws

1996 ◽  
Vol 3 (4) ◽  
pp. 247-261 ◽  
Author(s):  
V. Carbone ◽  
P. Veltri ◽  
R. Bruno
Keyword(s):  
Solar Wind ◽  
Scaling Laws ◽  
Velocity Structure ◽  
Fluid Flows ◽  
Structure Functions ◽  
Point Of View ◽  
Self Similarity ◽  
Extended Self ◽  
Scaling Exponents ◽  
Magnetohydrodynamic Turbulence

Abstract. In this paper we review some of the work done in investigating the scaling properties of Magnetohydrodynamic turbulence, by using velocity fluctuations measurements performed in the interplanetary space plasma by the Helios spacecraft. The set of scaling exponents ξq for the q-th order velocity structure functions, have been determined by using the Extended Self-Similarity hypothesis. We have found that the q-th order velocity structure function, when plotted vs. the 4-th order structure function, displays a range of self-similarity which extends over all the lengths covered by measurements, thus allowing for a very good determination of ξq. Moreover the results seem to show that the scaling exponents are the same regardless the various observation periods considered. The obtained scaling exponents have been compared with the results of some intermittency models for Kraichnan's turbulence, derived in the framework of infinitely divisible fragmentation processes, showing the good agreement between these models and our observations. Finally, on the basis of the actually available data sets, we show that scaling laws in Solar Wind turbulence seem to be different from turbulent scaling laws in the ordinary fluid flows. This is true for high-order velocity structure functions, while low-order velocity structure functions show the same scaling laws. Since our measurements involve length scales which extend over many order of magnitude where dissipation is practically absent, our results show that Solar Wind turbulence can be regarded as a testing bench for the investigation of general scaling behaviour in turbulent flows. In particular our results strongly support the point of view which attributes a key role to the inertial range dynamics in determining the intermittency characteristics in fluid flows, in contrast with the point of view which attributes intermittency to a finite Reynolds number effect.

scholarly journals First-Order Structure Function Analysis of Statistical Scale Invariance in the AIRS-Observed Water Vapor Field

2012 ◽  
Vol 25 (16) ◽  
pp. 5538-5555 ◽  
Author(s):  
Kyle G. Pressel ◽  
William D. Collins
Keyword(s):  
Water Vapor ◽  
Structure Function ◽  
Power Law ◽  
Scale Invariance ◽  
Structure Functions ◽  
Scale Dependence ◽  
Global Maximum ◽  
Order Structure ◽  
Scaling Exponents ◽  
First Order

Abstract The power-law scale dependence, or scaling, of first-order structure functions of the tropospheric water vapor field between 58°S and 58°N is investigated using observations from the Atmospheric Infrared Sounder (AIRS). Power-law scale dependence of the first-order structure function would indicate that the water vapor field exhibits statistical scale invariance. Directional and directionally independent first-order structure functions are computed to assess the directional dependence of derived first-order structure function scaling exponents (H) for a range of scales from 50 to 500 km. In comparison to other methods of assessing statistical scale invariance, the methodology used here requires minimal assumptions regarding the homogeneity of the spatial distribution of data within regions of analysis. Additionally, the methodology facilitates the evaluation of anisotropy and quantifies the extent to which the structure functions exhibit scale invariance. The spatial and seasonal dependence of the computed scaling exponents are explored. Minimum scaling exponents at all levels are shown to occur proximate to the equator, while the global maximum is shown to occur in the middle troposphere near the tropical–subtropical margin of the winter hemisphere. From a detailed analysis of AIRS maritime scaling exponents, it is concluded that the AIRS observations suggest the existence of two scaling regimes in the extratropics. One of these regimes characterizes the statistical scale invariance the free troposphere with H approximately = 0.55 and a second that characterizes the statistical scale invariance of the boundary layer with H approximately = ⅓.

Self‐Similarity, the Power Law Form of the Species‐Area Relationship, and a Probability Rule: A Reply to Maddux

2004 ◽  
Vol 163 (4) ◽  
pp. 627-633 ◽  
Author(s):  
Annette Ostling ◽  
John Harte ◽  
Jessica L. Green ◽  
Ann P. Kinzig
Keyword(s):  
Power Law ◽  
Self Similarity ◽  
Species Area ◽  
Species Area Relationship

scholarly journals Eulerian and Lagrangian Structure Function’s Scaling Exponents in Turbulent Channel Flow

2006 ◽  
Vol 61 (12) ◽  
pp. 624-628
Author(s):  
Jian-Ping Luo ◽  
Tatsuo Ushijima ◽  
Osami Kitoh ◽  
Zhi-Ming Lu ◽  
Yu-Lu Liu ◽  
...  
Keyword(s):  
Channel Flow ◽  
Scaling Laws ◽  
Turbulent Channel Flow ◽  
Self Similarity ◽  
Extended Self ◽  
Scaling Exponents ◽  
Turbulent Channel Flows ◽  
Frame Work ◽  
Simulation Results ◽  
Similarity Method

The relation between Eulerian structure function’s scaling exponents and Lagrangian ones in turbulent channel flows is explored both theoretically and numerically. A nonlinear parametric transformation between Eulerian structure function’s scaling exponents and Lagrangian ones is derived, following Landau and Novikov’s frame work. This relation is then compared to some known experimental and numerical results, but mainly to our DNS (direct numerical simulation) results of a fully developed channel flow with Reτ = 100. The scaling exponents are evaluated in terms of the ESS (extended self-similarity) method, since the Reynolds number is too low to make the standard scaling laws applicable. The agreement between theory and simulation is satisfactory

scholarly journals Froissart bound and self-similarity based models of proton structure functions

2018 ◽  
Vol 33 (08) ◽  
pp. 1850046 ◽  
Author(s):  
D. K. Choudhury ◽  
Baishali Saikia
Keyword(s):  
Structure Function ◽  
Cross Section ◽  
Structure Functions ◽  
Proton Structure Function ◽  
Self Similarity ◽  
Proton Proton ◽  
Proton Structure

Froissart bound implies that the total proton–proton cross-section (or equivalently proton structure function) cannot rise faster than [Formula: see text]. Compatibility of such behavior with the notion of self-similarity in proton structure function was suggested by us sometime back. In the present work, we generalize and improve it further by considering more recent self-similarity based models of proton structure functions and compare with recent data as well as with the model of Block, Durand, Ha and McKay.

Finite-Péclet-number effects on the scaling exponents of high-order passive scalar structure functions

2012 ◽  
Vol 713 ◽  
pp. 453-481 ◽  
Author(s):  
J. Lepore ◽  
L. Mydlarski
Keyword(s):  
Scalar Field ◽  
Boundary Conditions ◽  
Passive Scalar ◽  
Structure Functions ◽  
High Order ◽  
Field Boundary ◽  
Scaling Exponents ◽  
Heated Cylinder ◽  
Scalar Structure ◽  
The Difference

AbstractThe effect of scalar-field (temperature) boundary conditions on the inertial-convective-range scaling exponents of the high-order passive scalar structure functions is studied in the turbulent, heated wake downstream of a circular cylinder. The temperature field is generated two ways: using (i) a heating element embedded within the cylinder that generates the hydrodynamic wake (thus creating a heated cylinder) and (ii) a mandoline (an array of fine, heated wires) installed downstream of the cylinder. The hydrodynamic field is independent of the scalar-field boundary conditions/injection methods, and the same in both flows. Using the two heat injection mechanisms outlined above, the inertial-convective-range scaling exponents of the high-order passive scalar structure functions were measured. It is observed that the different scalar-field boundary conditions yield significantly different scaling exponents (with the magnitude of the difference increasing with structure function order). Moreover, the exponents obtained from the mandoline experiment are smaller than the analogous exponents from the heated cylinder experiment (both of which exhibit a significant departure from the Kolmogorov prediction). Since the observed deviation from the Kolmogorov $n/ 3$ prediction arises due to the effects of internal intermittency, the typical interpretation of this result would be that the scalar field downstream of the mandoline is more internally intermittent than that generated by the heated cylinder. However, additional measures of internal intermittency (namely the inertial-convective-range scaling exponents of the mixed, sixth-order, velocity–temperature structure functions and the non-centred autocorrelations of the dissipation rate of scalar variance) suggest that both scalar fields possess similar levels of internal intermittency – a distinctly different conclusion. Examination of the normalized high-order moments reveals that the smaller scaling exponents (of the high-order passive scalar structure functions) obtained for the mandoline experiment arise due to the smaller thermal integral length scale of the flow (i.e. the narrower inertial-convective subrange) and are not solely the result of a more intermittent scalar field. The difference in the passive scalar structure function scaling exponents can therefore be interpreted as an artifact of the different, finite Péclet numbers of the flows under consideration – an effect that is notably less prominent in the other measures of internal intermittency.

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