scholarly journals Universal Scaling Laws of Hydraulic Fracturing

2020 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Hydraulic Fracturing ◽  
Response Time ◽  
Scaling Laws ◽  
Scaling Law ◽  
Control Variable ◽  
Universal Scaling ◽  
Geometric Similarity ◽  
Damage Number ◽  
Similarity Law ◽  
Total Damage

Hydraulic fracturing has been studied by using dimensional analysis. The universal scaling law of the problem has been formulated. From this investigation, it has found that the key control variable in the process is total damage number $J=\frac{p+\frac{1}{2}\rho U^2}{\sigma_Y}$. The crack length is satisfied with the geometric similarity law if no response time is concerned, otherwise, it would not satisfy the similarity.

scholarly journals Universal Scaling Laws of Hydraulic Fracturing

2020 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Hydraulic Fracturing ◽  
Dimensional Analysis ◽  
Scaling Laws ◽  
Scaling Law ◽  
Simple Relation ◽  
Universal Scaling

The hydraulic fracturing is studied by using dimensional analysis. A universal scaling law of the hydraulic fracturing is obtained. This simple relation has not been seen in the literature.

scholarly journals Asymmetric Unimodal Maps with Non-universal Period-Doubling Scaling Laws

2020 ◽  
Vol 379 (1) ◽  
pp. 103-143
Author(s):  
Oleg Kozlovski ◽  
Sebastian van Strien
Keyword(s):  
Scaling Laws ◽  
Scaling Law ◽  
Period Doubling ◽  
Cantor Sets ◽  
Unimodal Maps ◽  
Renormalization Theory

Abstract We consider a family of strongly-asymmetric unimodal maps $$\{f_t\}_{t\in [0,1]}$$ { f t } t ∈ [ 0 , 1 ] of the form $$f_t=t\cdot f$$ f t = t · f where $$f:[0,1]\rightarrow [0,1]$$ f : [ 0 , 1 ] → [ 0 , 1 ] is unimodal, $$f(0)=f(1)=0$$ f ( 0 ) = f ( 1 ) = 0 , $$f(c)=1$$ f ( c ) = 1 is of the form and $$\begin{aligned} f(x)=\left\{ \begin{array}{ll} 1-K_-|x-c|+o(|x-c|)&{} \text{ for } x<c, \\ 1-K_+|x-c|^\beta + o(|x-c|^\beta ) &{} \text{ for } x>c, \end{array}\right. \end{aligned}$$ f ( x ) = 1 - K - | x - c | + o ( | x - c | ) for x < c , 1 - K + | x - c | β + o ( | x - c | β ) for x > c , where we assume that $$\beta >1$$ β > 1 . We show that such a family contains a Feigenbaum–Coullet–Tresser $$2^\infty $$ 2 ∞ map, and develop a renormalization theory for these maps. The scalings of the renormalization intervals of the $$2^\infty $$ 2 ∞ map turn out to be super-exponential and non-universal (i.e. to depend on the map) and the scaling-law is different for odd and even steps of the renormalization. The conjugacy between the attracting Cantor sets of two such maps is smooth if and only if some invariant is satisfied. We also show that the Feigenbaum–Coullet–Tresser map does not have wandering intervals, but surprisingly we were only able to prove this using our rather detailed scaling results.

Universal scaling laws and the QCD scale anomaly

1994 ◽  
Vol 242 (4-6) ◽  
pp. 355-361
Author(s):  
Georges Ripka ◽  
Martine Jaminon
Keyword(s):  
Scaling Laws ◽  
Universal Scaling

Universal scaling laws of transients

2002 ◽  
Vol 47 (3) ◽  
pp. 181-183 ◽  
Author(s):  
A. A. Koronovskii ◽  
D. I. Trubetskov ◽  
A. E. Khramov ◽  
A. E. Khramova
Keyword(s):  
Scaling Laws ◽  
Universal Scaling

scholarly journals Cope’s Rule and the Universal Scaling Law of Ornament Complexity

2015 ◽  
Vol 186 (2) ◽  
pp. 165-175 ◽  
Author(s):  
Pasquale Raia ◽  
Federico Passaro ◽  
Francesco Carotenuto ◽  
Leonardo Maiorino ◽  
Paolo Piras ◽  
...  
Keyword(s):  
Scaling Law ◽  
Universal Scaling ◽  
Cope’S Rule

scholarly journals A universal scaling law for the evolution of granular gases

2016 ◽  
Vol 114 (1) ◽  
pp. 10002 ◽  
Author(s):  
Mathias Hummel ◽  
James P. D. Clewett ◽  
Marco G. Mazza
Keyword(s):  
Scaling Law ◽  
Granular Gases ◽  
Universal Scaling

scholarly journals On the Scaling of Bubble Cluster Collapse in Cloud Cavitating Flow Around a Slender Projectile

2015 ◽  
Author(s):  
Yiwei Wang ◽  
Chenguang Huang ◽  
Xiaocui Wu
Keyword(s):  
Phase Change ◽  
Numerical Simulations ◽  
Scaling Law ◽  
Cavitating Flow ◽  
Cavitation Model ◽  
Bubble Cluster ◽  
Collapse Pressure ◽  
Variation Range ◽  
Main Factors ◽  
Similarity Law

The scaling law of bubble cluster collapse in cloud cavitating flow around a slender projectile is investigated in the present paper. The influence of compressibility is mainly discussed. Firstly the governing parameters are obtained by dimensional analysis, and the numerical method is established in order to verify the similarity law and obtain the influence of parameters based on a mixture approach with Singhal cavitation model. Moreover, the similarity law is validated by numerical simulations. Two main factors of compressibility of mixture fluid, including compressibility of non-condensable gas and phase change, are studied, respectively. Results indicated that the phase change has little influence on both flowing and collapse pressure. In the condition that the variation range of the mixture compressibility is small, the compressibility of non-condensable gas has notable impact the local collapse pressure peaks, however the macroscopic flow pattern does not change.

scholarly journals Scaling and Similarity of the Anisotropic Coherent Eddies in Near-Surface Atmospheric Turbulence

2018 ◽  
Vol 75 (3) ◽  
pp. 943-964 ◽  
Author(s):  
Khaled Ghannam ◽  
Gabriel G. Katul ◽  
Elie Bou-Zeid ◽  
Tobias Gerken ◽  
Marcelo Chamecki
Keyword(s):  
Scaling Laws ◽  
Velocity Structure ◽  
Scaling Law ◽  
Longitudinal Velocity ◽  
Structure Functions ◽  
Length Scale ◽  
Near Surface ◽  
Vegetation Canopies ◽  
Velocity Spectra ◽  
Attached Eddies

Abstract The low-wavenumber regime of the spectrum of turbulence commensurate with Townsend’s “attached” eddies is investigated here for the near-neutral atmospheric surface layer (ASL) and the roughness sublayer (RSL) above vegetation canopies. The central thesis corroborates the significance of the imbalance between local production and dissipation of turbulence kinetic energy (TKE) and canopy shear in challenging the classical distance-from-the-wall scaling of canonical turbulent boundary layers. Using five experimental datasets (two vegetation canopy RSL flows, two ASL flows, and one open-channel experiment), this paper explores (i) the existence of a low-wavenumber k−1 scaling law in the (wind) velocity spectra or, equivalently, a logarithmic scaling ln(r) in the velocity structure functions; (ii) phenomenological aspects of these anisotropic scales as a departure from homogeneous and isotropic scales; and (iii) the collapse of experimental data when plotted with different similarity coordinates. The results show that the extent of the k−1 and/or ln(r) scaling for the longitudinal velocity is shorter in the RSL above canopies than in the ASL because of smaller scale separation in the former. Conversely, these scaling laws are absent in the vertical velocity spectra except at large distances from the wall. The analysis reveals that the statistics of the velocity differences Δu and Δw approach a Gaussian-like behavior at large scales and that these eddies are responsible for momentum/energy production corroborated by large positive (negative) excursions in Δu accompanied by negative (positive) ones in Δw. A length scale based on TKE dissipation collapses the velocity structure functions at different heights better than the inertial length scale.

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