Sur l'évolution du taux moyen de dissipation de l'énergie cinétique de la turbulence dans une turbulence de grilleTransport equation for the homogeneous mean energy dissipation rate in decaying grid turbulence

The transport equation for the mean turbulent energy dissipation rate $\overline{{\it\epsilon}}$ along the centreline of a fully developed channel flow is derived by applying the limit at small separations to the two-point budget equation. Since the ratio of the isotropic energy dissipation rate to the mean turbulent energy dissipation rate $\overline{{\it\epsilon}}_{iso}/\overline{{\it\epsilon}}$ is sufficiently close to 1 on the centreline, our main focus is on the isotropic form of the transport equation. It is found that the imbalance between the production of $\overline{{\it\epsilon}}$ due to vortex stretching and the destruction of $\overline{{\it\epsilon}}$ caused by the action of viscosity is governed by the diffusion of $\overline{{\it\epsilon}}$ by the wall-normal velocity fluctuation. This imbalance is intrinsically different from the advection-driven imbalance in decaying-type flows, such as grid turbulence, jets and wakes. In effect, the different types of imbalance represent different constraints on the relation between the skewness of the longitudinal velocity derivative $S_{1,1}$ and the destruction coefficient $G$ of enstrophy in different flows, thus resulting in non-universal approaches of $S_{1,1}$ towards a constant value as the Taylor microscale Reynolds number, $R_{{\it\lambda}}$, increases. For example, the approach is slower for the measured values of $S_{1,1}$ along either the channel or pipe centreline than along the axis in the self-preserving region of a round jet. The data for $S_{1,1}$ collected in different flows strongly suggest that, in each flow, the magnitude of $S_{1,1}$ is bounded, the value being slightly larger than 0.5.

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Three-component vorticity measurements in a turbulent grid flow

Journal of Fluid Mechanics ◽

10.1017/s0022112098002547 ◽

1998 ◽

Vol 374 ◽

pp. 29-57 ◽

Cited By ~ 49

Author(s):

R. A. ANTONIA ◽

T. ZHOU ◽

Y. ZHU

Keyword(s):

Energy Dissipation ◽

Dissipation Rate ◽

Velocity Structure ◽

Transverse Velocity ◽

Longitudinal Velocity ◽

Energy Dissipation Rate ◽

Grid Turbulence ◽

Local Isotropy ◽

Velocity Increments ◽

The Mean

All components of the fluctuating vorticity vector have been measured in decaying grid turbulence using a vorticity probe of relatively simple geometry (four X-probes, i.e. a total of eight hot wires). The data indicate that local isotropy is more closely satisfied than global isotropy, the r.m.s. vorticities being more nearly equal than the r.m.s. velocities. Two checks indicate that the performance of the probe is satisfactory. Firstly, the fully measured mean energy dissipation rate 〈ε〉 is in good agreement with the value inferred from the rate of decay of the mean turbulent energy 〈q2〉 in the quasi-homogeneous region; the isotropic mean energy dissipation rate 〈εiso〉 agrees closely with this value even though individual elements of 〈ε〉 indicate departures from isotropy. Secondly, the measured decay rate of the mean-square vorticity 〈ω2〉 is consistent with that of 〈q2〉 and in reasonable agreement with the isotropic form of the transport equation for 〈ω2〉. Although 〈ε〉≃〈εiso〉, there are discernible differences between the statistics of ε and εiso; in particular, εiso is poorly correlated with either ε or ω2. The behaviour of velocity increments has been examined over a narrow range of separations for which the third-order longitudinal velocity structure function is approximately linear. In this range, transverse velocity increments show larger departures than longitudinal increments from predictions of Kolmogorov (1941). The data indicate that this discrepancy is only partly associated with differences between statistics of locally averaged ε and ω2, the latter remaining more intermittent than the former across this range. It is more likely caused by a departure from isotropy due to the small value of Rλ, the Taylor microscale Reynolds number, in this experiment.

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Transport equation for the isotropic turbulent energy dissipation rate in the far-wake of a circular cylinder

Journal of Fluid Mechanics ◽

10.1017/jfm.2015.597 ◽

2015 ◽

Vol 784 ◽

pp. 109-129 ◽

Cited By ~ 18

Author(s):

S. L. Tang ◽

R. A. Antonia ◽

L. Djenidi ◽

Y. Zhou

Keyword(s):

Energy Dissipation ◽

Circular Cylinder ◽

Transport Equation ◽

Turbulent Diffusion ◽

Dissipation Rate ◽

Turbulent Energy ◽

Energy Dissipation Rate ◽

Grid Turbulence ◽

Turbulent Energy Dissipation Rate ◽

Far Wake

The transport equation for the isotropic turbulent energy dissipation rate $\overline{{\it\epsilon}}_{iso}$ along the centreline in the far-wake of a circular cylinder is derived by applying the limit at small separations to the two-point energy budget equation. It is found that the imbalance between the production and the destruction of $\overline{{\it\epsilon}}_{iso}$, respectively due to vortex stretching and viscosity, is governed by both the streamwise advection and the lateral turbulent diffusion (the former contributes more to the budget than the latter). This imbalance differs intrinsically from that in other flows, e.g. grid turbulence and the flow along the centreline of a fully developed channel, where either the streamwise advection or the lateral turbulent diffusion of $\overline{{\it\epsilon}}_{iso}$ governs the imbalance. More importantly, the different types of imbalance represent different constraints on the relation between the skewness of the longitudinal velocity derivative $S$ and the destruction coefficient of enstrophy $G$. This results in a non-universal approach of $S$ towards a constant value as the Taylor microscale Reynolds number $R_{{\it\lambda}}$ increases. For the present flow, the magnitude of $S$ decreases initially ($R_{{\it\lambda}}\leqslant 40$) before increasing ($R_{{\it\lambda}}>40$) towards this constant value. The constancy of $S$ at large $R_{{\it\lambda}}$ violates the modified similarity hypothesis introduced by Kolmogorov (J. Fluid Mech., vol. 13, 1962, pp. 82–85) but is consistent with the original similarity hypotheses (Kolmogorov, Dokl. Akad. Nauk SSSR, vol. 30, 1941b, pp. 299–303 (see also 1991 Proc. R. Soc. Lond. A, vol. 434, pp. 9–13)) ($K41$), and, more importantly, with the almost completely self-preserving nature of the plane far-wake.

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

Journal of Fluid Mechanics ◽

10.1017/jfm.2014.157 ◽

2014 ◽

Vol 747 ◽

pp. 288-315 ◽

Cited By ~ 7

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).

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On the normalized dissipation parameter in decaying turbulence

Journal of Fluid Mechanics ◽

10.1017/jfm.2017.110 ◽

2017 ◽

Vol 817 ◽

pp. 61-79 ◽

Cited By ~ 16

Author(s):

L. Djenidi ◽

N. Lefeuvre ◽

M. Kamruzzaman ◽

R. A. Antonia

Keyword(s):

Reynolds Number ◽

Energy Dissipation ◽

Kinetic Energy ◽

Turbulent Kinetic Energy ◽

Dissipation Rate ◽

Energy Dissipation Rate ◽

Grid Turbulence ◽

Far Field ◽

Round Jet ◽

Decaying Turbulence

The Reynolds number dependence of the non-dimensional mean turbulent kinetic energy dissipation rate$C_{\unicode[STIX]{x1D716}}=\overline{\unicode[STIX]{x1D716}}L/u^{\prime 3}$(where$\unicode[STIX]{x1D716}$is the mean turbulent kinetic energy dissipation rate,$L$is an integral length scale and$u^{\prime }$is the velocity root-mean-square) is investigated in decaying turbulence. Expressions for$C_{\unicode[STIX]{x1D716}}$in homogeneous isotropic turbulent (HIT), as approximated by grid turbulence, and in local HIT, as on the axis of the far field of a turbulent round jet, are developed from the Navier–Stokes equations within the framework of a scale-by-scale energy budget. The analysis shows that when turbulence decays/evolves in compliance with self-preservation (SP),$C_{\unicode[STIX]{x1D716}}$remains constant for a given flow condition, e.g. a given initial Reynolds number. Measurements in grid turbulence, which does not satisfy SP, and on the axis in the far field of a round jet, which does comply with SP, show that$C_{\unicode[STIX]{x1D716}}$decreases in the former case and remains constant in the latter, thus supporting the theoretical results. Further, while$C_{\unicode[STIX]{x1D716}}$can remain constant during the decay for a given initial Reynolds number, both the theory and measurements show that it decreases towards a constant,$C_{\unicode[STIX]{x1D716},\infty }$, as$Re_{\unicode[STIX]{x1D706}}$increases. This trend, in agreement with existing data, is not inconsistent with the possibility that$C_{\unicode[STIX]{x1D716}}$tends to a universal constant.

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