scholarly journals Universality in Decaying Turbulence at High Reynolds Numbers

2021 ◽  
Author(s):  
Christian Kuechler ◽  
Gergory Bewley ◽  
Eberhard Bodenschatz
Keyword(s):  
Isotropic Turbulence ◽  
Power Laws ◽  
Reynolds Numbers ◽  
Variable Density ◽  
Scaling Exponent ◽  
Max Planck ◽  
Self Similarity ◽  
Extended Self ◽  
Variable Density Turbulence ◽  
Decaying Turbulence

Abstract In the limit of very large Reynolds numbers for homogeneous isotropic turbulence of an incompressible fluid, the statistics of the velocity differences between two points in space are expected to approach universal power laws at scales smaller than those at which energy is injected. Even at the highest Reynolds numbers available in laboratory and natural flows such universal power laws have remained elusive. On the other hand, power laws have been observed empirically in derived quantities, namely in the relative scaling in statistics of different orders according to the Extended Self Similarity hypothesis. Here we present experimental results from the Max Planck Variable Density Turbulence Tunnel over an unprecedented range of Reynolds numbers. We find that the velocity difference statistics take a universal functional form that is distinct from a power law. By applying a self-similar model derived for decaying turbulence to our data, an effective scaling exponent for the second moment can be derived that agrees well with that obtained from Extended Self Similarity.

Modified Self-Scaling Relation for the Inertial and Low Energy Containing Scales of Decaying Turbulence

1998 ◽  
Vol 12 (04) ◽  
pp. 405-431
Author(s):  
M. Hnatich ◽  
D. Horváth
Keyword(s):  
Reynolds Numbers ◽  
Structure Functions ◽  
Scaling Relation ◽  
Self Similarity ◽  
Extended Self ◽  
High Reynolds Numbers ◽  
The Mean ◽  
New Form ◽  
Decaying Turbulence ◽  
High Reynolds

The limits of a new form of scaling, named Extended Self Similarity (ESS) originally suggested [R. Benzi et al., Phys. Rev.E48 (1993), 29] for the inertial, dissipation and transition scales are discussed. A modification of the ESS concept is put forward using the model of decaying turbulence at high Reynolds numbers [L. Ts. Adzhemyan et al., Czech. J. Phys.45 (1995), 517]. In this model the statistical description is simplified by the hypotheses of homogeneity, isotropy, incompressibility and self-similarity, for the power law stage of decay the presence of a single scaling length — Karman scale — is assumed within the energy containing range. The second and third structure functions of the velocity field [S2(r) and S3(r)] have been calculated using the well-known connections between the mean energy spectrum and S2(r), and between mean spectral transfer and third structure function S3(r). Both structure functions have been investigated in the inertial and low enery containing ranges, then expressed in the form involving the leading Kolmogorov's K41 asymptotics [S2(r)∝ r2/3, S3(r)∝ r] and its asymptotical corrections. These corrections allow to determine corrections to the original ESS form [Formula: see text] (for K41) and to find out the modified variant of the ESS.

scholarly journals Numerical study of pressure and velocity fluctuations in nearly isotropic turbulence

1978 ◽  
Vol 88 (4) ◽  
pp. 685-709 ◽  
Author(s):  
U. Schumann ◽  
G. S. Patterson
Keyword(s):  
Isotropic Turbulence ◽  
Initial Conditions ◽  
Numerical Study ◽  
Pressure Fluctuations ◽  
Reynolds Numbers ◽  
Companion Paper ◽  
Real Space ◽  
Pressure Gradients ◽  
Velocity Statistics ◽  
Decaying Turbulence

The spectral method of Orszag & Patterson has been extended to calculate the static pressure fluctuations in incompressible homogeneous decaying turbulence at Reynolds numbers Reλ [lsim ] 35. In real space 323 points are treated. Several cases starting from different isotropic initial conditions have been studied. Some departure from isotropy exists owing to the small number of modes at small wavenumbers. Root-mean-square pressure fluctuations, pressure gradients and integral length scales have been evaluated. The results agree rather well with predictions based on velocity statistics and on the assumption of normality. The normality assumption has been tested extensively for the simulated fields and found to be approximately valid as far as fourth-order velocity correlations are concerned. In addition, a model for the dissipation tensor has been proposed. The application of the present method to the study of the return of axisymmetric turbulence to isotropy is described in the companion paper.

Extended Self-Similarity in Drag-Reduced Turbulent Flow of Viscoelastic Fluid

2010 ◽  
Author(s):  
Feng-Chen Li ◽  
Hong-Na Zhang ◽  
Wei-Hua Cai ◽  
Juan-Cheng Yang
Keyword(s):  
Channel Flow ◽  
Viscoelastic Fluid ◽  
Isotropic Turbulence ◽  
Scaling Law ◽  
Viscoelastic Fluids ◽  
Turbulent Channel Flow ◽  
Elastic Model ◽  
Homogeneous Isotropic Turbulence ◽  
Self Similarity ◽  
Extended Self

Direct numerical simulations (DNS) have been performed for drag-reduced turbulent channel flow with surfactant additives and forced homogeneous isotropic turbulence with polymer additives. Giesekus constitutive equation and finite extensible nonlinear elastic model with Peterlin closure were used to describe the elastic stress tensor for both cases, respectively. For comparison, DNS of water flows for both cases were also performed. Based on the DNS data, the extended self-similarity (ESS) of turbulence scaling law is investigated for water and viscoelastic fluids in turbulent channel flow and forced homogeneous isotropic turbulence. It is obtained that ESS still holds for drag-reduced turbulent flows of viscoelastic fluids. In viscoelastic fluid flows, the regions at which δu(r)∝r and Sp(r)∝S3(r)ζ(p) with ζ(p) = p/3, where r is the scale length, δu(r) is the longitudinal velocity difference along r and Sp(r) is the pth-order moment of velocity increments, in the K41 (Kolmogorov theory)-fashioned plots and ESS-fashioned plots, respectively, are all broadened to larger scale for all the investigated cases.

Velocity-derivative skewness and two-time velocity correlations of isotropic turbulence as predicted by the LET theory

1989 ◽  
Vol 208 ◽  
pp. 91-114 ◽  
Author(s):  
W. D. Mccomb ◽  
V. Shanmugasundaram ◽  
P. Hutchinson
Keyword(s):  
Isotropic Turbulence ◽  
Initial Conditions ◽  
Direct Interaction ◽  
Reynolds Numbers ◽  
Time Correlation ◽  
General Belief ◽  
Direct Interaction Approximation ◽  
The Difference ◽  
Decaying Turbulence ◽  
Interaction Approximation

The local-energy-transfer (LET) theory was used to calculate freely decaying turbulence for arbitrary initial conditions over a range of microscale-based Reynolds numbers 0.5 [les ] Rλ(tf) [les ] 1009, where tf is the final time of computation. The predicted skewness factor S(Rλ) agreed closely with the results of numerical simulations at low-to-moderate Reynolds numbers and followed the same general trend at larger values of Rλ. It was also found that, for Rλ(tf) [les ] 5, the LET calculation was almost indistinguishable from that of the direct-interaction approximation (DIA), with the difference between the two theories tending to zero as Rλ(tf)∞ 0.Two-time correlation and propagator (or response) functions were also obtained. Tests of their scaling behaviour suggest that, contrary to general belief, the convective sweeping of the energy-containing range is much less important than the Kolmogorov timescale in determining inertial-range behaviour. This result raises questions about the accepted explanation for the failure of the direct-interaction approximation, thus motivating a discussion about the relevance of random Galilean invariance (RGI). It is argued that, for a properly constructed ensemble of transformations to inertial frames, invariance in every realization necessarily implies RGI. It is suggested that the defects of the direct-interaction approximation can be understood in terms of a failure to renormalize the stirring forces.

scholarly journals Dynamics of helicity in homogeneous skew-isotropic turbulence

2017 ◽  
Vol 821 ◽  
pp. 539-581 ◽  
Author(s):  
Antoine Briard ◽  
Thomas Gomez
Keyword(s):  
Kinetic Energy ◽  
Isotropic Turbulence ◽  
Early Stage ◽  
Helical Structure ◽  
Reynolds Numbers ◽  
Inertial Range ◽  
Scalar Flux ◽  
Antisymmetric Part ◽  
Decaying Turbulence ◽  
High Reynolds

The dynamics of helicity in homogeneous skew-isotropic freely decaying turbulence is investigated, at very high Reynolds numbers, thanks to a classical eddy-damped quasi-normal Markovian (EDQNM) closure. In agreement with previous direct numerical simulations, a $k^{-5/3}$ inertial range is obtained for both the kinetic energy and helical spectra. In the early stage of the decay, when kinetic energy, initially only present at large scales cascades towards small scales, it is found that helicity slightly slows down the nonlinear transfers. Then, when the turbulence is fully developed, theoretical decay exponents are derived and assessed numerically for helicity. Furthermore, it is found that the presence of helicity does not modify the decay rate of the kinetic energy with respect to purely isotropic turbulence, except in Batchelor turbulence where the kinetic energy decays slightly more rapidly. In this case, non-local expansions are used to show analytically that the permanence of the large eddies hypothesis is verified for the helical spectrum, unlike the kinetic energy one. Moreover, the $4/3$ law for the two-point helical structure function is assessed numerically at very large Reynolds numbers. Afterwards, the evolution equation of the helicity dissipation rate is investigated analytically, which provides significant simplifications and leads notably to the definition of a helical derivative skewness and of a helical Taylor scale, which is numerically very close to the classical Taylor longitudinal scale at large Reynolds numbers. Finally, when both a mean scalar gradient and helicity are combined, the quadrature spectrum, linked to the antisymmetric part of the scalar flux, appears and scales like $k^{-7/3}$ and then like $k^{-5/3}$ in the inertial range.

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.

A semi-Lagrangian direct-interaction closure of the spectra of isotropic variable-density turbulence

2019 ◽  
Vol 876 ◽  
pp. 186-236 ◽  
Author(s):  
David J. Petty ◽  
C. Pantano
Keyword(s):  
Large Scale ◽  
Isotropic Turbulence ◽  
Density Ratio ◽  
Direct Interaction ◽  
Variable Density ◽  
Constant Density ◽  
Velocity Spectrum ◽  
Variable Density Flow ◽  
Density Flow ◽  
Variable Density Turbulence

A study of variable-density homogeneous stationary isotropic turbulence based on the sparse direct-interaction perturbation (SDIP) and supporting direct numerical simulations (DNS) is presented. The non-solenoidal flow considered here is an example of turbulent mixing of gases with different densities. The spectral statistics of this type of flow are substantially more difficult to understand theoretically than those of the similar solenoidal flows. In the approach described here, the nonlinearly coupled velocity and scalar (which determine the density of the fluid) equations are expanded in terms of a normalised density ratio parameter. A new set of coupled integro-differential SDIP equations are derived and then solved numerically for the first-order correction to the incompressible equations in the variable-density expansion parameter. By adopting a regular expansion approach, one obtains leading-order corrections that are universal and therefore interesting in their own right. The predictions are then compared with DNS of forced variable-density flow with different density contrasts. It is found that the velocity spectrum owing to variable density is indistinguishable from that of constant-density turbulence, as it is supported by a wealth of indirect experimental evidence, but the scalar spectra show significant deviations, and even loss of monotonicity, as a function of the type and strength of the large-scale source of the mixing. Furthermore, the analysis helps clarify what may be the proper approach to interpret the power spectrum of variable-density turbulence.

Small-scale isotropy of orificed, perforated plate turbulence

2010 ◽  
Vol 224 (4) ◽  
pp. 889-899
Author(s):  
A Singha
Keyword(s):  
Velocity Structure ◽  
Isotropic Turbulence ◽  
Perforated Plate ◽  
Reynolds Numbers ◽  
Streamwise Velocity ◽  
Inertial Subrange ◽  
Small Scale ◽  
Scaling Exponent ◽  
Velocity Spectra ◽  
Large Stretch

The small-scale statistics of orificed, perforated plate (OPP) turbulence was studied over a range of Reynolds numbers based on Taylor microscale, Reλ, from 23 to 108. The streamwise velocity skewness, flatness, and the velocity spectra, deduced from hot-wire measurements taken at 70 D (where D is the diameter of the holes of the OPP) downstream of the OPP confirmed the turbulence to be quasi-isotropic. Application of the extended self-similarity concept enabled the analysis of the velocity structure functions of the small-scale statistics up to the sixth order. The ESS analysis revealed a large stretch of scaling region, despite the absence of a perceptible Kolmogorov inertial subrange. The values of the scaling exponents seem to affirm the superiority of the OPP over conventional grid in generating quasi-isotropic turbulence. The effects of Reλ on the extent of the scaling range and the scaling exponent were also explored.

Numerical calculation of decaying isotropic turbulence using the LET theory

1984 ◽  
Vol 143 ◽  
pp. 95-123 ◽  
Author(s):  
W. D. Mccomb ◽  
V. Shanmugasundaram
Keyword(s):  
Reynolds Number ◽  
Energy Transfer ◽  
Energy Dissipation ◽  
Isotropic Turbulence ◽  
Direct Interaction ◽  
Reynolds Numbers ◽  
Energy Dissipation Rate ◽  
Skewness Factor ◽  
Decaying Turbulence ◽  
High Reynolds

The local-energy-transfer (LET) theory (McComb 1978) was used to calculate freely decaying turbulence for four different initial spectra at low-to-moderate values of microscale Reynolds numbers (Rλ up to about 40). The results for energy, dissipation and energy-transfer spectra and for skewness factor all agreed quite closely with the predictions of the well-known direct-interaction approximation (DIA: Kraichnan 1964). However, LET gave higher values of energy transfer and of evolved skewness factor than DIA. This may be related to the fact that LET yields the k−5/3 law for the energy spectrum at infinite Reynolds number.The LET equations were then integrated numerically for decaying isotropic turbulence at high Reynolds number. Values were obtained for the wavenumber spectra of energy, dissipation rate and inertial-transfer rate, along with the associated integral parameters, at an evolved microscale Reynolds number Rλ of 533. The predictions of LET agreed well with experimental results and with the Lagrangian-history theories (Herring & Kraichnan 1979). In particular, the purely Eulerian LET theory was found to agree rather closely with the strain-based Lagrangian-history approximation; and further comparisons suggested that this agreement extended to low Reynolds numbers as well.

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