Power-law exponent in the transition period of decay in grid turbulence

2015 ◽  
Vol 779 ◽  
pp. 544-555 ◽  
Author(s):  
L. Djenidi ◽  
Md. Kamruzzaman ◽  
R. A. Antonia
Keyword(s):  
Transition Region ◽  
Power Law ◽  
Isotropic Turbulence ◽  
Transition Period ◽  
Initial Period ◽  
Power Laws ◽  
Grid Turbulence ◽  
Power Law Decay ◽  
Initial Stage ◽  
Reported Data

Hot-wire measurements are carried out in grid-generated turbulence at moderate to low Taylor microscale Reynolds number $Re_{{\it\lambda}}$ to assess the appropriateness of the commonly used power-law decay for the mean turbulent kinetic energy (e.g. $k\sim x^{n}$, with $n\leqslant -1$). It is found that in the region outside the initial and final periods of decay, which we designate a transition region, a power law with a constant exponent $n$ cannot describe adequately the decay of turbulence from its initial to final stages. One is forced to use a family of power laws of the form $x^{n_{i}}$, where $n_{i}$ is a different constant over a portion $i$ of the decay time during the decay period. Accordingly, it is currently not possible to determine whether any grid-generated turbulence reported in the literature decays according to Saffman or Batchelor because the reported data fall in the transition period where $n$ differs from its initial and final values. It is suggested that a power law of the form $k\sim x^{n_{init}+m(x)}$, where $m(x)$ is a continuous function of $x$, could be used to describe the decay from the initial period to the final stage. The present results, which corroborate the numerical simulations of decaying homogeneous isotropic turbulence of Orlandi & Antonia (J. Turbul., vol. 5, 2004, doi:10.1088/1468-5248/5/1/009) and Meldi & Sagaut (J. Turbul., vol. 14, 2013, pp. 24–53), show that the values of $n$ reported in the literature, and which fall in the transition region, have been mistakenly assigned to the initial stage of decay.

The use of dual heat injection to infer scalar covariance decay in grid turbulence

1981 ◽  
Vol 104 ◽  
pp. 93-109 ◽  
Author(s):  
Z. Warhaft
Keyword(s):  
Cross Correlation ◽  
Isotropic Turbulence ◽  
Thermal Fluctuations ◽  
Grid Turbulence ◽  
Mixing Rate ◽  
Quantitative Measurements ◽  
Power Law Decay ◽  
Covariance Term ◽  
Heat Injection ◽  
Grid Mesh

Thermal fluctuations introduced into decaying grid turbulence at two different downstream locations are shown to be initially correlated and this correlation decays with distance from the grid. The fluctuations are introduced by placing two mandolines (Warhaft & Lumley 1978) at different distances downstream from the grid. The sum of the thermal variances produced by each mandoline operating separately, $\overline{\theta^2_1}+\overline{\theta^2_2}$, is significantly less than the total variance produced by both mandolines operating simultaneously, $\overline{(\theta_1+\theta_2)^2} = \overline{\theta^2_1} + \overline{\theta^2_2} + \overline{2\theta_1\theta_2} $, i.e. the deficit is due to the covariance term $\overline{2\overline{\theta_1\theta_2}}$. This covariance is responsible for a cross-correlation, $\rho = \overline{\theta_1\theta_2}/(\overline{\theta_1^2}\overline{\theta^2_2})^{\frac{1}{2}}$, as great as 0·6. The decay of $\overline{\theta_1\theta_2} $ and ρ is studied for various initial input thermal scale sizes and for various input locations. It is shown that the covariance follows a power-law decay, the exponent varying from - 5·5 if the thermal fluctuations are introduced close to the grid where the turbulence dissipation rate is large and the flow is inhomogeneous to - 4 if they are introduced further downstream (x/M ≥ 10, where x is the distance from the grid and M is the grid mesh length) in the region where the approximately isotropic turbulence is beginning to develop. The decay rate of $\overline{\theta_1\theta_2} $ and ρ was insensitive to the intensity of the thermal fluctuations. In all these experiments the cross-correlation between velocity and temperature fluctuations was very small (∼ − 0·05) and temperature was a passive additive. The results, which appear to be the first quantitative measurements of the rate of destruction of scalar covariance and hence of the mixing rate between two scalars, are shown to provide good confirmation of recent predictions of the decay of ρ by the second-order closure techniques of Lumley (1978 a, b).

scholarly journals Effects of initial conditions in decaying turbulence generated by passive grids

2007 ◽  
Vol 585 ◽  
pp. 395-420 ◽  
Author(s):  
P. LAVOIE ◽  
L. DJENIDI ◽  
R. A. ANTONIA
Keyword(s):  
Power Law ◽  
Large Scale ◽  
Velocity Structure ◽  
Isotropic Turbulence ◽  
Initial Conditions ◽  
Reynolds Numbers ◽  
Recent Analysis ◽  
Grid Turbulence ◽  
Power Law Exponent ◽  
Decaying Turbulence

The effects of initial conditions on grid turbulence are investigated for low to moderate Reynolds numbers. Four grid geometries are used to yield variations in initial conditions and a secondary contraction is introduced to improve the isotropy of the turbulence. The hot-wire measurements, believed to be the most detailed to date for this flow, indicate that initial conditions have a persistent impact on the large-scale organization of the flow over the length of the tunnel. The power-law coefficients, determined via an improved method, also depend on the initial conditions. For example, the power-law exponent m is affected by the various levels of large-scale organization and anisotropy generated by the different grids and the shape of the energy spectrum at low wavenumbers. However, the results show that these effects are primarily related to deviations between the turbulence produced in the wind tunnel and true decaying homogenous isotropic turbulence (HIT). Indeed, when isotropy is improved and the intensity of the large-scale periodicity, which is primarily associated with round-rod grids, is decreased, the importance of initial conditions on both the character of the turbulence and m is diminished. However, even in the case where the turbulence is nearly perfectly isotropic, m is not equal to −1, nor does it show an asymptotic trend in x towards this value, as suggested by recent analysis. Furthermore, the evolution of the second- and third-order velocity structure functions satisfies equilibrium similarity only approximately.

scholarly journals The energy decay in self-preserving isotropic turbulence revisited

1992 ◽  
Vol 241 ◽  
pp. 645-667 ◽  
Author(s):  
Charles G. Speziale ◽  
Peter S. Bernard
Keyword(s):  
Reynolds Number ◽  
Power Law ◽  
Isotropic Turbulence ◽  
Initial Conditions ◽  
Energy Decay ◽  
Analytical Determination ◽  
Asymptotic State ◽  
Power Law Decay ◽  
Point Analysis ◽  
High Reynolds

The assumption of self-preservation permits an analytical determination of the energy decay in isotropic turbulence. Batchelor (1948), who was the first to carry out a detailed study of this problem, based his analysis on the assumption that the Loitsianskii integral is a dynamic invariant – a widely accepted hypothesis that was later discovered to be invalid. Nonetheless, it appears that the self-preserving isotropic decay problem has never been reinvestigated in depth subsequent to this earlier work. In the present paper such an analysis is carried out, yielding a much more complete picture of self-preserving isotropic turbulence. It is proven rigorously that complete self-preserving isotropic turbulence admits two general types of asymptotic solutions: one where the turbulent kinetic energy K ∼ t−1 and one where K ∼ t−α with an exponent α > 1 that is determined explicitly by the initial conditions. By a fixed-point analysis and numerical integration of the exact one-point equations, it is demonstrated that the K ∼ t−1 power law decay is the asymptotically consistent high-Reynolds-number solution; the K ∼ t−α decay law is only achieved in the limit as t → ∞ and the turbulence Reynolds number Rt vanishes. Arguments are provided which indicate that a t−1 power law decay is the asymptotic state toward which a complete self-preserving isotropic turbulence is driven at high Reynolds numbers in order to resolve an O(R1½) imbalance between vortex stretching and viscous diffusion. Unlike in previous studies, the asymptotic approach to a complete self-preserving state is investigated which uncovers some surprising results.

Unifying Laws in Multidisciplinary Power-Law Phenomena: Fixed-Point Universality and Nonextensive Entropy

2004 ◽  
Author(s):  
Alberto Robledo
Keyword(s):  
Random Walks ◽  
Fixed Points ◽  
Power Law ◽  
Nonlinear Dynamical Systems ◽  
Power Laws ◽  
Basins Of Attraction ◽  
Market Prices ◽  
Nonlinear Dynamical ◽  
Power Law Decay ◽  
Non Gaussian

Critical, power-law behavior in space and/or time manifests in a large variety of complex systems [12] within physics and, nowadays, more conspicuously in other fields, such as biology, ecology, geophysics, and economics. Universality, the same power law holding for completely different systems, is a consequence of the characteristic self-similar, scale-invariant property of criticality, and can be understood in terms of basins of attraction of the renormalization-group (RG) fixed points. However, the guiding quality of a variatkmal approach has been seemingly lacking in the theoretical studies of critical phenomena. Here we give an account of entropy extrema associated with fixed points of RG transformations. As illustrations, we consider simple one-dimensional models of random walks and nonlinear dynamical systems. In describing these systems we consider distribution and/or time relaxation functions with power-law decay that may have infinite first- or second- and higher-order moments. When these moments diverge, we observe the emergence of nonexponential or non-Gaussian fractal properties that can be measured by the nonextensive Tsallis entropy index q. We note that the presence of nonextensive properties may signal situations of hindered movement among the system's possible configurations. Some representative applications within physics, but with suggested or recognized connections to other fields, are critical behavior in fluids and magnets, anomalous diffusion processes, transitions to chaos in nonlinear systems, and relaxation properties of supercooled liquids near the glass formation. Two prototypical model systems serve to illustrate the development of critical states characterized by power laws from generic states described by exponential behavior. These are random walks and nonlinear iterated maps that we discuss below in some detail. Random walks [18] are suitable, for example, for representing Brownian motion (molecular thermal motion under the microscope), but also for many types of data originating from diverse disciplines. One type is that which comes in the form of a "time series," a temporal sequence of measured values, for instance, stock market prices in economics or electroencephalographic potentials in medicine.

Time-dependent scaling relations and a cascade model of turbulence

1978 ◽  
Vol 88 (2) ◽  
pp. 369-391 ◽  
Author(s):  
Thomas L. Bell ◽  
Mark Nelkin
Keyword(s):  
Energy Spectrum ◽  
Power Law ◽  
Isotropic Turbulence ◽  
Cascade Model ◽  
Time Dependent ◽  
Wave Number ◽  
Grid Turbulence ◽  
A Value ◽  
Model Equations ◽  
Spatial Dimensionality

We study the time-dependent solutions of a nonlinear cascade model for homogeneous isotropic turbulence first introduced by Novikov & Desnyansky. The dynamical variables of the model are the turbulent kinetic energies in discrete wave-number shells of thickness one octave. The model equations contain a parameter C whose size governs the amount of energy cascaded to small wavenumbers relative to the amount cascaded to large wavenumbers. We show that the equations permit scale-similar evolution of the energy spectrum. For 0 ≤ C ≤ 1 and no external force, the freely evolving energy spectrum displays the Kolmogorov k power law, and the total energy decreases in time as a power t−w, where the exponent w depends on the value of C. Grid-turbulence experiments seem to favour a value of C in the range 0·3-0·6. In the presence of an external stirring force acting near a wavenumber k0, the model predicts, in addition to the Kolmogorov k spectrum for k > k0, a scale-similar flow of energy to wavenumbers k < k0. This backward energy flow falls off as a power law in time, and establishes a stationary energy spectrum for k < k0 which is a power law in k less steep than k. We discuss the similarity of the behaviour of the model for C > 1 to the behaviour of turbulent fluid for a spatial dimensionality near 2. The model is shown to approach the Kovasznay and the Leith diffusion approximation equations in the limit in which the thickness of the wavenumber shells approaches zero. However, the cascade model with finite shell thicknesses appears to behave in a more physically reasonable way than the limiting differential equations.

Transport equation for the mean turbulent energy dissipation rate in low- grid turbulence

2014 ◽  
Vol 747 ◽  
pp. 288-315 ◽  
Author(s):  
L. Djenidi ◽  
R. A. Antonia
Keyword(s):  
Reynolds Number ◽  
Energy Dissipation ◽  
Transport Equation ◽  
Power Law ◽  
Dissipation Rate ◽  
Low Reynolds Number ◽  
Energy Dissipation Rate ◽  
Grid Turbulence ◽  
Power Law Decay ◽  
The Mean

AbstractA direct numerical simulation (DNS) based on the lattice Boltzmann method (LBM) is carried out in low-Reynolds-number grid turbulence to analyse the mean turbulent kinetic energy dissipation rate, $\overline{\epsilon }$, and its transport equation during decay. All the components of $\overline{\epsilon }$ and its transport equation terms are computed, providing for the first time the opportunity to assess the contribution of each term to the decay. The results indicate that although small departures from isotropy are observed in the components of $\overline{\epsilon }$ and its destruction term, there is sufficient compensation among the components for these two quantities to satisfy isotropy to a close approximation. A short distance downstream of the grid, the transport equation of $\overline{\epsilon }$ simplifies to its high-Reynolds-number homogeneous and isotropic form. The decay rate of $\overline{\epsilon }$ is governed by the imbalance between the production due to vortex stretching and the destruction caused by the action of viscosity, the latter becoming larger than the former as the distance from the grid increases. This imbalance, which is not constant during the decay as argued by Batchelor & Townsend (Proc. R. Soc. Lond. A, vol. 190, 1947, pp. 534–550), varies according to a power law of $x$, the distance downstream of the grid. The non-constancy implies a lack of dynamical similarity in the mechanisms controlling the transport of $\overline{\epsilon }$. This is consistent with the fact that the power-law-decay ($\overline{q^2} \sim x^n$) exponent $n$ is not equal to $-$1. It is actually close to $-$1.6, a value in keeping with the relatively low Reynolds number of the simulation. These results highlight the importance of the imbalance in establishing the value of $n$. The $\overline{\epsilon }$-transport equation is also analysed in relation to the power-law decay. The results show that the power-law exponent $n$ is controlled by the imbalance between production and destruction. Further, a relatively straightforward analysis provides information on the behaviour of $n$ during the entire decay process and an interesting theoretical result, which is yet to be confirmed, when $R_{\lambda } \rightarrow 0 $, namely, the destruction coefficient $G$ is constant and its value must lie between $15/7$ and $30/7$. These two limits encompass the predictions for the final period of decay by Batchelor & Townsend (1947) and Saffman (J. Fluid Mech., vol. 27, 1967, pp. 581–593).

Energy spectra power laws and structures

2009 ◽  
Vol 623 ◽  
pp. 353-374 ◽  
Author(s):  
P. ORLANDI
Keyword(s):  
Strain Rate ◽  
Power Law ◽  
Isotropic Turbulence ◽  
Power Laws ◽  
Radius Of Curvature ◽  
Strain Rate Tensor ◽  
Inviscid Flows ◽  
Principal Axes ◽  
Velocity Derivative ◽  
Finite Time Singularity

Direct numerical simulations (DNS) of two inviscid flows, the Taylor–Green flow and two orthogonal interacting Lamb dipoles, together with the DNS of forced isotropic turbulence, were performed to generate data for a comparative study. The isotropic turbulent field was considered after the transient and, in particular, when the velocity derivative skewness oscillates around −0.5. At this time, Rλ ≈ 257 and a one decade wide k−5/3 range was present in the energy spectrum. For the inviscid flows the fields were considered when a wide k−3 range was achieved. This power law spectral decay corresponds to infinite enstrophy and is considered one of the requirements to demonstrate that the Euler equations lead to a finite time singularity (FTS). Flow visualizations and statistics of the strain rate tensor and vorticity components in the principal axes of the strain rate tensor (λ, λ) were used to classify structures. The key role of the intermediate component 2 is demonstrated by its good correlation with enstrophy production. Filtering of the fields shows that the slope of the power law is directly connected to self-similar structures, whose radius of curvature is smaller the steeper the spectrum.

scholarly journals Decay of turbulence generated by a square-fractal-element grid

2014 ◽  
Vol 741 ◽  
pp. 567-584 ◽  
Author(s):  
R. J. Hearst ◽  
P. Lavoie
Keyword(s):  
Kinetic Energy ◽  
Power Law ◽  
Measurement Range ◽  
Grid Turbulence ◽  
Space Filling ◽  
Apparent Contrast ◽  
Turbulence Measurements ◽  
Power Law Decay ◽  
Background Mesh ◽  
Significant Extension

AbstractA novel square-fractal-element grid was designed in order to increase the downstream measurement range of fractal grid experiments relative to the largest element of the grid. The grid consists of a series of square fractal elements mounted to a background mesh with spacing$L_0 = 100\, {\rm mm}$. Measurements were performed in the region$3.5 \le x/L_0 \le 48.5$, which represents a significant extension to the$x/L_0 < 20$of previously reported square fractal grid measurements. For the region$x/L_0 \gtrsim 24$it was found that a power-law decay region following$\langle {q}^2 \rangle \sim (x - x_0)^m$exists with decay exponents of$m = -1.39$and$-1.37$at$\mathit{Re}_{L_0} = 57\, 000$and$65\, 000$, respectively. This agrees with decay values previously measured for regular grids ($-1 \gtrsim m \gtrsim -1.4$). The turbulence in the near-grid region,$x/L_0 < 20$, is shown to be inhomogeneous and anisotropic, in apparent contrast with previous fractal grid measurements. Nonetheless, power-law fits to the decay of turbulent kinetic energy in this region result in$m = -2.79$, similar to$m \approx -2.5$recently reported by Valente & Vassilicos (J. Fluid Mech., vol. 687, 2011, pp. 300–340) for space-filling square fractals. It was also found that$C_\epsilon $is approximately constant for$x/L_0 \ge 25$, while it grows rapidly for$x/L_0 < 20$. These results reconcile previous fractal-generated turbulence measurements with classical grid turbulence measurements.

Invariants for slightly heated decaying grid turbulence

2013 ◽  
Vol 727 ◽  
pp. 379-406 ◽  
Author(s):  
R. A. Antonia ◽  
S. K. Lee ◽  
L. Djenidi ◽  
P. Lavoie ◽  
L. Danaila
Keyword(s):  
Reynolds Number ◽  
Isotropic Turbulence ◽  
Longitudinal Velocity ◽  
Power Laws ◽  
Published Data ◽  
Grid Turbulence ◽  
Point Energy ◽  
Dimensionless Energy ◽  
Scalar Variance ◽  
Small Reynolds Numbers

AbstractThe paper examines the validity of velocity and scalar invariants in slightly heated and approximately isotropic turbulence generated by passive conventional grids. By assuming that the variances $\langle {u}^{2} \rangle $ and $\langle {\theta }^{2} \rangle $ ($u$ and $\theta $ represent the longitudinal velocity and temperature fluctuations) decay along the streamwise direction $x$ according to power laws $\langle {u}^{2} \rangle \sim {(x- {x}_{0} )}^{{n}_{u} } $ and $\langle {\theta }^{2} \rangle \sim {(x- {x}_{0} )}^{{n}_{\theta } } $ (${x}_{0} $ is the virtual origin of the flow) and with the further assumption that the one-point energy and scalar variance budgets are represented closely by a balance between the rates of change of $\langle {u}^{2} \rangle $ and $\langle {\theta }^{2} \rangle $ and the corresponding mean energy dissipation rates, the products $\langle {u}^{2} \rangle { \lambda }_{u}^{- 2{n}_{u} } $ and $\langle {\theta }^{2} \rangle { \lambda }_{\theta }^{- 2{n}_{\theta } } $ must remain constant with respect to $x$. Here ${\lambda }_{u} $ and ${\lambda }_{\theta } $ are the Taylor and Corrsin microscales. This is unambiguously supported by previously available data, as well as new measurements of $u$ and $\theta $ made at small Reynolds numbers downstream of three different biplane grids. Implications for invariants based on measured integral length scales of $u$ and $\theta $ are also tested after confirming that the dimensionless energy and scalar variance dissipation rate parameters are approximately constant with $x$. Since the magnitudes of ${n}_{u} $ and ${n}_{\theta } $ vary from grid to grid and may also depend on the Reynolds number, the Saffman and Corrsin invariants which correspond to a value of $- 1. 2$ for ${n}_{u} $ and ${n}_{\theta } $ are unlikely to apply in general. The effect of the Reynolds number on ${n}_{u} $ is discussed in the context of published data for both passive and active grids.

Double Power Law in the Japanese Financial Market

2019 ◽  
Vol 7 (1) ◽  
pp. 28-35
Author(s):  
Sudhir Jain ◽  
Takuya Yamano
Keyword(s):  
Time Dependence ◽  
Financial Market ◽  
Power Law ◽  
Power Laws ◽  
Recent Analysis ◽  
Short Term ◽  
Power Law Decay ◽  
Long Time ◽  
Ising Spins

The authors study the persistence phenomenon in the Japanese stock market by using a novel mapping of the time evolution of the values of shares quoted on the Nikkei Index onto Ising spins. The method is applied to historical end of day data from the Japanese financial market. By studying the time dependence of the spins, they find clear evidence for a double-power law decay of the proportion of shares that remain either above or below ‘starting' values chosen at random. The results are consistent with a recent analysis of the data from the London FTSE100 market. The slopes of the power-laws are also in agreement. The authors estimate a long time persistence exponent for the underlying Japanese financial market to be 0.5. Furthermore, they argue that the presence of a double power law in the decay of the persistence probability could be the signature of the presence of both speculative (short-term) and long-term traders in the market.

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