scholarly journals Statistics of Finite Scale Local Lyapunov Exponents in Fully Developed Homogeneous Isotropic Turbulence

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Nicola de Divitiis
Keyword(s):  
Lyapunov Exponents ◽  
Isotropic Turbulence ◽  
Longitudinal Velocity ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Homogeneous Isotropic Turbulence ◽  
Velocity Correlation ◽  
Uniform Function ◽  
Fully Developed Chaos ◽  
Effect Estimation

The present work analyzes the statistics of finite scale local Lyapunov exponents of pairs of fluid particles trajectories in fully developed incompressible homogeneous isotropic turbulence. According to the hypothesis of fully developed chaos, this statistics is here analyzed assuming that the entropy associated with the fluid kinematic state is maximum. The distribution of the local Lyapunov exponents results in an unsymmetrical uniform function in a proper interval of variation. From this PDF, we determine the relationship between average and maximum Lyapunov exponents and the longitudinal velocity correlation function. This link, which in turn leads to the closure of von Kármán–Howarth and Corrsin equations, agrees with results of previous works, supporting the proposed PDF calculation, at least for the purposes of the energy cascade main effect estimation. Furthermore, through the property that the Lyapunov vectors tend to align the direction of the maximum growth rate of trajectories distance, we obtain the link between maximum and average Lyapunov exponents in line with the previous results. To validate the proposed theoretical results, we present different numerical simulations whose results justify the hypotheses of the present analysis.

scholarly journals A Lagrangian direct-interaction approximation for homogeneous isotropic turbulence

1997 ◽  
Vol 345 ◽  
pp. 307-345 ◽  
Author(s):  
SHIGEO KIDA ◽  
SUSUMU GOTO
Keyword(s):  
Energy Spectrum ◽  
Differential Equations ◽  
Isotropic Turbulence ◽  
Longitudinal Velocity ◽  
Direct Interaction ◽  
Homogeneous Isotropic Turbulence ◽  
Velocity Correlation ◽  
Stationary Turbulence ◽  
Direct Interaction Approximation ◽  
Velocity Derivative

A set of integro-differential equations in the Lagrangian renormalized approximation (Kaneda 1981) is rederived by applying a perturbation method developed by Kraichnan (1959), which is based upon an extraction of direct interactions among Fourier modes of a velocity field and was applied to the Eulerian velocity correlation and response functions, to the Lagrangian ones for homogeneous isotropic turbulence. The resultant set of integro-differential equations for these functions has no adjustable free parameters. The shape of the energy spectrum function is determined numerically in the universal range for stationary turbulence, and in the whole wavenumber range in a similarly evolving form for the freely decaying case. The energy spectrum in the universal range takes the same shape in both cases, which also agrees excellently with many measurements of various kinds of real turbulence as well as numerical results obtained by Gotoh et al. (1988) for a decaying case as an initial value problem. The skewness factor of the longitudinal velocity derivative is calculated to be −0.66 for stationary turbulence. The wavenumber dependence of the eddy viscosity is also determined.

Turbulent energy density in scale space for inhomogeneous turbulence

2018 ◽  
Vol 842 ◽  
pp. 532-553 ◽  
Author(s):  
Fujihiro Hamba
Keyword(s):  
Energy Spectrum ◽  
Energy Density ◽  
Isotropic Turbulence ◽  
Scale Space ◽  
Scale Dependence ◽  
Energy Cascade ◽  
Homogeneous Isotropic Turbulence ◽  
Velocity Correlation ◽  
Inhomogeneous Turbulence ◽  
Point Velocity

The energy spectrum is commonly used to describe the scale dependence of the turbulent fluctuations in homogeneous isotropic turbulence. In contrast, one-point statistical quantities, such as the turbulent kinetic energy, are employed for inhomogeneous turbulence modelling. To obtain a better understanding of inhomogeneous turbulence, some attempts have been made to describe its scale dependence by using the second-order structure function and the two-point velocity correlation. However, previous expressions for the energy density in the scale space do not satisfy the requirement that it should be non-negative. In this work, a new expression for the energy density in the scale space is proposed on the basis of the two-point velocity correlation; the integral with a filter function is introduced to satisfy the non-negativity of the energy density. Direct numerical simulation (DNS) data of homogeneous isotropic turbulence were first used to assess the role of the energy density by comparing it with the energy spectrum. DNS data of a turbulent channel flow were then used to investigate the energy density and its transport equation in inhomogeneous turbulence. It was shown that the new energy density is positive in the scale space of the homogeneous direction. The energy transfer was successfully examined in the scale space both in the homogeneous and inhomogeneous directions. The energy cascade from large to small scales was clearly observed. Moreover, the inverse energy cascade from large to very large scales was observed in the scale space of the spanwise direction.

scholarly journals The theory of axisymmetric turbulence

1946 ◽  
Vol 186 (1007) ◽  
pp. 480-502 ◽  
Keyword(s):  
Viscous Dissipation ◽  
Invariant Theory ◽  
Isotropic Turbulence ◽  
Longitudinal Velocity ◽  
Practical Interest ◽  
Mean Flow ◽  
Velocity Correlation ◽  
Rates Of Change ◽  
The Mean ◽  
Velocity Pressure

This paper discusses a type of turbulence in a uniform stream which is next to isotropic turbulence in order of simplicity. Instead of spherical symmetry, or isotropy, axially symmetical turbulence possesses symmetry about an axis which in practice is usually the direction of mean flow. The analysis is developed with the aid of invariant theory, as suggested by a previous paper by Robertson. The form of the fundamental velocity correlation is obtained, and scales of axisymmetric turbulence are defined. The results of greatest practical interest concern the time rates of change of the mean squares of the lateral and longitudinal velocity components. The rates of change involve two terms, the first representing viscous dissipation, and the second representing a transfer of energy from one component to the other due to the finite correlation between the velocity and pressure at neighbouring points. The effect of the velocity-pressure correlation is to bring the two velocity components towards equality, while the effect of the viscous dissipation will only be towards equality if an inequality between the curvatures at the origin of two particular velocity correlation coefficient curves, both of which are measurable, is obeyed. The rates of change of the mean squares of the vorticity components are also obtained.

Scale invariance in finite Reynolds number homogeneous isotropic turbulence

2019 ◽  
Vol 864 ◽  
pp. 244-272 ◽  
Author(s):  
L. Djenidi ◽  
R. A. Antonia ◽  
S. L. Tang
Keyword(s):  
Reynolds Number ◽  
Scale Invariance ◽  
Isotropic Turbulence ◽  
Stokes Equations ◽  
Longitudinal Velocity ◽  
Navier Stokes ◽  
Invariance Property ◽  
Homogeneous Isotropic Turbulence ◽  
Navier Stokes Equations ◽  
Flatness Factor

The problem of homogeneous isotropic turbulence (HIT) is revisited within the analytical framework of the Navier–Stokes equations, with a view to assessing rigorously the consequences of the scale invariance (an exact property of the Navier–Stokes equations) for any Reynolds number. The analytical development, which is independent of the 1941 (K41) and 1962 (K62) theories of Kolmogorov for HIT for infinitely large Reynolds number, is applied to the transport equations for the second- and third-order moments of the longitudinal velocity increment, $(\unicode[STIX]{x1D6FF}u)$. Once the normalised equations and the constraints required for complying with the scale-invariance property of the equations are presented, results derived from these equations and constraints are discussed and compared with measurements. It is found that the fluid viscosity, $\unicode[STIX]{x1D708}$, and the mean kinetic energy dissipation rate, $\overline{\unicode[STIX]{x1D716}}$ (the overbar denotes spatial and/or temporal averaging), are the only scaling parameters that make the equations scale-invariant. The analysis further leads to expressions for the distributions of the skewness and the flatness factor of $(\unicode[STIX]{x1D6FF}u)$ and shows that these distributions must exhibit plateaus (of different magnitudes) in the dissipative and inertial ranges, as the Taylor microscale Reynolds number $Re_{\unicode[STIX]{x1D706}}$ increases indefinitely. Also, the skewness and flatness factor of the longitudinal velocity derivative become constant as $Re_{\unicode[STIX]{x1D706}}$ increases; this is supported by experimental data. Further, the analysis, backed up by experimental evidence, shows that, beyond the dissipative range, the behaviour of $\overline{(\unicode[STIX]{x1D6FF}u)^{n}}$ with $n=2$, 3 and 4 cannot be represented by a power law of the form $r^{\unicode[STIX]{x1D701}_{n}}$ when the Reynolds number is finite. It is shown that only when $Re_{\unicode[STIX]{x1D706}}\rightarrow \infty$ can an $n$-thirds law (i.e. $\overline{(\unicode[STIX]{x1D6FF}u)^{n}}\sim r^{\unicode[STIX]{x1D701}_{n}}$, with $\unicode[STIX]{x1D701}_{n}=n/3$) emerge, which is consistent with the onset of a scaling range.

A general self-preservation analysis for decaying homogeneous isotropic turbulence

2015 ◽  
Vol 773 ◽  
pp. 345-365 ◽  
Author(s):  
L. Djenidi ◽  
R. A. Antonia
Keyword(s):  
Velocity Structure ◽  
Isotropic Turbulence ◽  
Initial Conditions ◽  
Longitudinal Velocity ◽  
Length Scale ◽  
Grid Turbulence ◽  
Homogeneous Isotropic Turbulence ◽  
Kolmogorov Length Scale ◽  
Velocity Structure Function ◽  
Turbulence Data

A general framework of self-preservation (SP) is established, based on the transport equation of the second-order longitudinal velocity structure function in decaying homogeneous isotropic turbulence (HIT). The analysis introduces the skewness of the longitudinal velocity increment, $S(r,t)$ ($r$ and $t$ are space increment and time), as an SP controlling parameter. The present SP framework allows a critical appraisal of the specific assumptions that have been made in previous SP analyses. It is shown that SP is achieved when $S(r,t)$ varies in a self-similar manner, i.e. $S=c(t){\it\phi}(r/l)$ where $l$ is a scaling length, and $c(t)$ and ${\it\phi}(r/l)$ are dimensionless functions of time and $(r/l)$, respectively. When $c(t)$ is constant, $l$ can be identified with the Kolmogorov length scale ${\it\eta}$, even when the Reynolds number is relatively small. On the other hand, the Taylor microscale ${\it\lambda}$ is a relevant SP length scale only when certain conditions are met. The decay law for the turbulent kinetic energy ($k$) ensuing from the present SP is a generalization of the existing laws and can be expressed as $k\sim (t-t_{0})^{n}+B$, where $B$ is a constant representing the energy of the motions whose scales are excluded from the SP range of scales. When $B=0$, SP is achieved at all scales of motion and ${\it\lambda}$ becomes a relevant scaling length together with ${\it\eta}$. The analysis underlines the relation between the initial conditions and the power-law exponent $n$ and also provides a link between them. In particular, an expression relating $n$ to the initial values of the scaling length and velocity is developed. Finally, the present SP analysis is consistent with both experimental grid turbulence data and the eddy-damped quasi-normal Markovian numerical simulation of decaying HIT by Meldi & Sagaut (J. Turbul., vol. 14, 2013, pp. 24–53).

scholarly journals Closure model for homogeneous isotropic turbulence in the Lagrangian specification of the flow field

2018 ◽  
Vol 841 ◽  
pp. 521-551
Author(s):  
Makoto Okamura
Keyword(s):  
Flow Field ◽  
Isotropic Turbulence ◽  
Longitudinal Velocity ◽  
Navier Stokes ◽  
Homogeneous Isotropic Turbulence ◽  
Closure Model ◽  
Navier Stokes Equation ◽  
Closed Set ◽  
Proposed Model ◽  
Velocity Derivative

This paper proposes a new two-point closure model that is compatible with the Kolmogorov$-5/3$power law for homogeneous isotropic turbulence in an incompressible fluid using the Lagrangian specification of the flow field. A closed set of three equations was derived from the Navier–Stokes equation with no adjustable free parameters. The Kolmogorov constant and the skewness of the longitudinal velocity derivative were evaluated to be 1.779 and$-0.49$, respectively, using the proposed model. The bottleneck effect was also reproduced in the near-dissipation range.

Normalized dissipation rate in a moderate Taylor Reynolds number flow

2017 ◽  
Vol 818 ◽  
pp. 184-204 ◽  
Author(s):  
Alejandro J. Puga ◽  
John C. LaRue
Keyword(s):  
Reynolds Number ◽  
Dissipation Rate ◽  
Isotropic Turbulence ◽  
Longitudinal Velocity ◽  
Integral Length Scale ◽  
Length Scale ◽  
Velocity Correlation ◽  
Time Resolved ◽  
Reynolds Number Flow ◽  
Active Grid

Time-resolved velocity measurements are obtained using a hot-wire in a nearly homogeneous and isotropic flow downstream of an active grid for a range of Taylor Reynolds numbers from$191$to$659$. It is found that the dimensionless dissipation rate,$C_{\unicode[STIX]{x1D716}}$, is nearly a constant for sufficiently high values of Taylor Reynolds number,$R_{\unicode[STIX]{x1D706},u_{q}}$, and is approximately equal to$0.87$. This value is approximately$5\,\%$less than the value reported by Boset al.(Phys. Fluids, vol. 19 (4), 2007, 045101), which is obtained using DNS/LES (direct numerical simulation combined with large eddy simulation) for decaying homogeneous and isotropic turbulence, and is in excellent agreement with the active grid experiment of Thormann & Meneveau (Phys. Fluids, vol. 26 (2), 2014, 025112.). The results presented herein show that deviation from isotropy may cause inconsistencies in the computation of$C_{\unicode[STIX]{x1D716}}$. As a result, it is suggested that the velocity scale be the square root of the turbulence kinetic energy. The integral length scale measurements obtained from the longitudinal velocity correlation are in close agreement with the integral length scale measured from the peak of the energy spectrum,$\unicode[STIX]{x1D705}E_{11}(\unicode[STIX]{x1D705})$, where$\unicode[STIX]{x1D705}$is the wavenumber and$E_{11}(\unicode[STIX]{x1D705})$is the one-dimensional power spectrum of the downstream velocity.

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