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

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.

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The intermediate wake of a body of revolution at high Reynolds numbers

Journal of Fluid Mechanics ◽

10.1017/s0022112010002715 ◽

2010 ◽

Vol 659 ◽

pp. 516-539 ◽

Cited By ~ 33

Author(s):

JUAN M. JIMÉNEZ ◽

M. HULTMARK ◽

A. J. SMITS

Keyword(s):

Reynolds Number ◽

Reynolds Numbers ◽

Stress Distributions ◽

Body Of Revolution ◽

Self Similarity ◽

Mean Velocity ◽

High Reynolds Numbers ◽

Order Of Magnitude ◽

The Mean ◽

High Reynolds

Results are presented on the flow field downstream of a body of revolution for Reynolds numbers based on a model length ranging from 1.1 × 106 to 67 × 106. The maximum Reynolds number is more than an order of magnitude larger than that obtained in previous laboratory wake studies. Measurements are taken in the intermediate wake at locations 3, 6, 9, 12 and 15 diameters downstream from the stern in the midline plane. The model is based on an idealized submarine shape (DARPA SUBOFF), and it is mounted in a wind tunnel on a support shaped like a semi-infinite sail. The mean velocity distributions on the side opposite the support demonstrate self-similarity at all locations and Reynolds numbers, whereas the mean velocity distribution on the side of the support displays significant effects of the support wake. None of the Reynolds stress distributions of the flow attain self-similarity, and for all except the lowest Reynolds number, the support introduces a significant asymmetry into the wake which results in a decrease in the radial and streamwise turbulence intensities on the support side. The distributions continue to evolve with downstream position and Reynolds number, although a slow approach to the expected asymptotic behaviour is observed with increasing distance downstream.

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Universality of the energy-containing structures in wall-bounded turbulence

Journal of Fluid Mechanics ◽

10.1017/jfm.2017.315 ◽

2017 ◽

Vol 823 ◽

pp. 498-510 ◽

Cited By ~ 6

Author(s):

Charitha M. de Silva ◽

Dominik Krug ◽

Detlef Lohse ◽

Ivan Marusic

Keyword(s):

Turbulent Flows ◽

Friction Velocity ◽

Velocity Structure ◽

Longitudinal Velocity ◽

Reynolds Numbers ◽

Structure Functions ◽

Self Similarity ◽

Logarithmic Region ◽

High Reynolds ◽

Wall Bounded Turbulence

The scaling behaviour of the longitudinal velocity structure functions $\langle (\unicode[STIX]{x1D6E5}_{r}u)^{2p}\rangle ^{1/p}$ (where $2p$ represents the order) is studied for various wall-bounded turbulent flows. It has been known that for very large Reynolds numbers within the logarithmic region, the structure functions can be described by $\langle (\unicode[STIX]{x1D6E5}_{r}u)^{2p}\rangle ^{1/p}/U_{\unicode[STIX]{x1D70F}}^{2}\approx D_{p}\ln (r/z)+E_{p}$ (where $r$ is the longitudinal distance, $z$ the distance from the wall, $U_{\unicode[STIX]{x1D70F}}$ the friction velocity and $D_{p}$, $E_{p}$ are constants) in accordance with Townsend’s attached eddy hypothesis. Here we show that the ratios $D_{p}/D_{1}$ extracted from plots between structure functions – in the spirit of the extended self-similarity hypothesis – have further reaching universality for the energy containing range of scales. Specifically, we confirm that this description is universal across wall-bounded flows with different flow geometries, and also for both the longitudinal and transversal structure functions, where previously the scaling has been either difficult to discern or differences have been reported when examining the direct representation of $\langle (\unicode[STIX]{x1D6E5}_{r}u)^{2p}\rangle ^{1/p}$. In addition, we present evidence of this universality at much lower Reynolds numbers, which opens up avenues to examine structure functions that are not readily available from high Reynolds number databases.

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The vortex-induced forces on a multi-column platform in current at high Reynolds numbers

Modern Physics Letters B ◽

10.1142/s0217984918503566 ◽

2018 ◽

Vol 32 (29) ◽

pp. 1850356 ◽

Cited By ~ 1

Author(s):

Xiaofeng Hu ◽

Xinshu Zhang ◽

Yunxiang You

Keyword(s):

Numerical Simulations ◽

Three Dimensional ◽

Reynolds Numbers ◽

Transverse Force ◽

Single Cylinder ◽

Force Coefficients ◽

High Reynolds Numbers ◽

The Mean ◽

High Reynolds ◽

Current Incidence

The unsteady vortex-induced forces on a multi-column platform in current from subcritical up to supercritical Reynolds numbers have been investigated using three-dimensional numerical simulations. Two different current incidences, 0[Formula: see text] and 45[Formula: see text], are considered. The results show that for 0[Formula: see text] current incidence, the mean streamwise force coefficients [Formula: see text] increase with the rise of [Formula: see text] when [Formula: see text], while they grow slightly when [Formula: see text]. For 45[Formula: see text] current incidence, a decrease of streamwise force with increasing [Formula: see text] is observed. Similar to a single cylinder, the fluctuating transverse force coefficients [Formula: see text] of the multi-column platform drop at [Formula: see text] for 0[Formula: see text] and 45[Formula: see text] current incidences. In addition, it is found that for 0[Formula: see text] current incidence, the [Formula: see text] values of the downstream columns are much larger than those of the upstream columns, while for 45[Formula: see text] current incidence, the [Formula: see text] values of each column are similar. Furthermore, the results of correlations between the forces on each column and total forces indicates that for 0[Formula: see text] current incidence, the fluctuating transverse forces on the downstream columns are mainly responsible for the total fluctuating transverse force on the multi-column platform.

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Fluid motion between parallel planes. Dynamical stability

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1946.0051 ◽

1946 ◽

Vol 186 (1006) ◽

pp. 391-409 ◽

Cited By ~ 3

Keyword(s):

Reynolds Number ◽

Critical Reynolds Number ◽

Fluid Motion ◽

Reynolds Numbers ◽

Small Disturbance ◽

Mean Velocity ◽

High Reynolds Numbers ◽

The Mean ◽

Parabolic Flow ◽

High Reynolds

If U is the velocity of the mean motion the following main results are obtained: 1. The region where U = c , c being the wave velocity, is the source where vibrations are generated; i.e. the slowly varying vibrations give rise to large rapidly varying vibrations in passing through the critical point. 2. Curved profiles admit a periodic motion at sufficiently high Reynolds numbers. 3. Parabolic flow is unstable at high Reynolds numbers; i.e. an infinitely small disturbance is sufficient to break up such flow. The critical Reynolds number is equal to R = U 0 h/v =6700, and the corresponding wavelength is about three times the width of the channel ( U 0 is the mean velocity at the axis, and h is the half-width of the channel).

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Universality in Decaying Turbulence at High Reynolds Numbers

10.21203/rs.3.rs-587483/v1 ◽

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.

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Revisiting Batchelor's theory of two-dimensional turbulence

Journal of Fluid Mechanics ◽

10.1017/s0022112007008427 ◽

2007 ◽

Vol 591 ◽

pp. 379-391 ◽

Cited By ~ 30

Author(s):

DAVID G. DRITSCHEL ◽

CHUONG V. TRAN ◽

RICHARD K. SCOTT

Keyword(s):

Reynolds Number ◽

Mathematical Analysis ◽

High Reynolds Number ◽

Reynolds Numbers ◽

Inertial Range ◽

Two Dimensional ◽

Simple Modification ◽

High Reynolds Numbers ◽

Decaying Turbulence ◽

High Reynolds

Recent mathematical results have shown that a central assumption in the theory of two-dimensional turbulence proposed by Batchelor (Phys. Fluids, vol. 12, 1969, p. 233) is false. That theory, which predicts a χ2/3k−1 enstrophy spectrum in the inertial range of freely-decaying turbulence, and which has evidently been successful in describing certain aspects of numerical simulations at high Reynolds numbers Re, assumes that there is a finite, non-zero enstrophy dissipation χ in the limit of infinite Re. This, however, is not true for flows having finite vorticity. The enstrophy dissipation in fact vanishes.We revisit Batchelor's theory and propose a simple modification of it to ensure vanishing χ in the limit Re → ∞. Our proposal is supported by high Reynolds number simulations which confirm that χ decays like 1/ln Re, and which, following the time of peak enstrophy dissipation, exhibit enstrophy spectra containing an increasing proportion of the total enstrophy 〈ω2〉/2 in the inertial range as Re increases. Together with the mathematical analysis of vanishing χ, these observations motivate a straightforward and, indeed, alarmingly simple modification of Batchelor's theory: just replace Batchelor's enstrophy spectrum χ2/3k−1 with 〈ω2〉 k−1 (ln Re)−1).

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Extended self-similarity works for the Burgers equation and why

Journal of Fluid Mechanics ◽

10.1017/s0022112010000595 ◽

2010 ◽

Vol 649 ◽

pp. 275-285 ◽

Cited By ~ 30

Author(s):

SAGAR CHAKRABORTY ◽

URIEL FRISCH ◽

SAMRIDDHI SANKAR RAY

Keyword(s):

Three Dimensional ◽

Reynolds Numbers ◽

Theoretical Explanation ◽

Navier Stokes ◽

Self Similarity ◽

Extended Self ◽

Burgers Turbulence ◽

High Reynolds ◽

Incompressible Turbulence

Extended self-similarity (ESS), a procedure that remarkably extends the range of scaling for structure functions in Navier–Stokes turbulence and thus allows improved determination of intermittency exponents, has never been fully explained. We show that ESS applies to Burgers turbulence at high Reynolds numbers and we give the theoretical explanation of the numerically observed improved scaling at both the IR and UV end, in total a gain of about three quarters of a decade: there is a reduction of subdominant contributions to scaling when going from the standard structure function representation to the ESS representation. We conjecture that a similar situation holds for three-dimensional incompressible turbulence and suggest ways of capturing subdominant contributions to scaling.

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