^{1}, Pierre-Philippe Cortet

^{1,a)}, Frédéric Moisy

^{1}and Leo R. M. Maas

^{2}

### Abstract

We report an experimental study of the decay of grid-generated turbulence in a confined geometry submitted to a global rotation. Turbulence is generated by rapidly towing a grid in a parallelepipedic water tank. The velocity fields of a large number of independent decays are measured in a vertical plane parallel to the rotation axis using a corotating particle imagevelocimetry system. We first show that, when a “simple” grid is used, a significant amount of the kinetic energy (typically 50%) is stored in a reproducible flow composed of resonant inertial modes. The spatial structure of those inertial modes, extracted by band-pass filtering, is found compatible with the numerical results of L. R. M. Maas [Fluid Dyn. Res.33, 373 (2003)]. The possible coupling between these modes and turbulence suggests that turbulence cannot be considered as freely decaying in this configuration. We demonstrate however that these inertial modes may be significantly reduced (down to 15% of the total energy) by adding a set of inner tanks attached to the grid. These results suggest that it is possible to produce an effectively freely decaying rotating turbulence in a confined geometry.

We acknowledge C. Morize and M. Rabaud for discussions about the manuscript, and A. Aubertin, L. Auffray, C. Borget, G.-J. Michon, and R. Pidoux for experimental help. The rotating platform “Gyroflow” was funded by the ANR (Grant No. 06-BLAN-0363-01 “HiSpeedPIV”) and the “Triangle de la Physique.”

I. INTRODUCTION

II. EXPERIMENTAL SETUP

A. Tank and rotating platform

B. Particle imagevelocimetry(PIV)measurements

C. Reynolds decomposition

III. INERTIAL MODES PRODUCED BY THE SIMPLE GRID CONFIGURATION

A. Kinetic energy decay

B. Fourier analysis of the inertial modes

C. Spatial structure of the inertial modes

IV. MODIFIED CONFIGURATION WITH INNER WALLS

A. Comparison with a previous configuration

B. Modified experimental setup

C. Turbulence generated with the tanks configuration

V. DISCUSSION AND CONCLUSION

### Key Topics

- Turbulent flows
- 87.0
- Inertial waves
- 17.0
- Velocimetry
- 12.0
- Rotating flows
- 11.0
- Energy transfer
- 10.0

## Figures

Schematic view of the experimental setup, in the “simple grid” configuration. The tank is filled with 52 cm of water and is rotating at an angular velocity of . A square grid of 40 mm mesh is towed by four shafts from the bottom to the top at constant velocity . PIV measurements in a vertical plane are achieved in the rotating frame, based on a laser sheet illuminating the vertical plane and a camera aiming normally at it.

Schematic view of the experimental setup, in the “simple grid” configuration. The tank is filled with 52 cm of water and is rotating at an angular velocity of . A square grid of 40 mm mesh is towed by four shafts from the bottom to the top at constant velocity . PIV measurements in a vertical plane are achieved in the rotating frame, based on a laser sheet illuminating the vertical plane and a camera aiming normally at it.

Times series of the vertical velocity measured at the center of the flow , for 20 realizations performed at (in various colors). The black thick line shows the ensemble average of these time series.

Times series of the vertical velocity measured at the center of the flow , for 20 realizations performed at (in various colors). The black thick line shows the ensemble average of these time series.

Total (dashed), mean (continuous), and turbulent (dashed-dotted) kinetic energies as a function of the number of tank rotations from 40 realizations performed at . Inset: ratio of turbulent to total kinetic energy (4), measured at times of maximum mean energy.

Total (dashed), mean (continuous), and turbulent (dashed-dotted) kinetic energies as a function of the number of tank rotations from 40 realizations performed at . Inset: ratio of turbulent to total kinetic energy (4), measured at times of maximum mean energy.

Temporal energy spectrum of the total (dashed), mean (continuous), and turbulent (dashed-dotted) component of the flow as a function of computed from 40 decay realizations performed at , with (a) linear and (b) logarithmic -axis. Inertial modes can develop for angular frequencies . The modes corresponding to the peak frequencies are given in Table I.

Temporal energy spectrum of the total (dashed), mean (continuous), and turbulent (dashed-dotted) component of the flow as a function of computed from 40 decay realizations performed at , with (a) linear and (b) logarithmic -axis. Inertial modes can develop for angular frequencies . The modes corresponding to the peak frequencies are given in Table I.

Spatial structure of the five dominating inertial modes listed in Table I, extracted by band-pass filtering of the ensemble-averaged fields. The ellipses show the velocity orbit, and the arrows illustrate the velocity field at a given arbitrary phase of the oscillation. The color of the ellipses traces the ellipticity (see the text for details). (a) ; (b) ; (c) ; (d) ; (e) . Resolution of the fields has been reduced by a factor 6 for a better visibility.

Spatial structure of the five dominating inertial modes listed in Table I, extracted by band-pass filtering of the ensemble-averaged fields. The ellipses show the velocity orbit, and the arrows illustrate the velocity field at a given arbitrary phase of the oscillation. The color of the ellipses traces the ellipticity (see the text for details). (a) ; (b) ; (c) ; (d) ; (e) . Resolution of the fields has been reduced by a factor 6 for a better visibility.

Vertical velocity field in the wake of the grid. (a) Simple grid configuration. (b) Modified configuration with the inner walls attached to the grid, represented by the six vertical thick lines. The grid is towed from the bottom, and is at the height in these snapshots. (c) Horizontal profile of the vertical velocity, at a distance below the grid (indicated by the horizontal lines in a and b). Continuous line: simple grid configuration. Dashed line: modified configuration with inner walls. The vertical black lines show the locations of the vertical inner walls.

Vertical velocity field in the wake of the grid. (a) Simple grid configuration. (b) Modified configuration with the inner walls attached to the grid, represented by the six vertical thick lines. The grid is towed from the bottom, and is at the height in these snapshots. (c) Horizontal profile of the vertical velocity, at a distance below the grid (indicated by the horizontal lines in a and b). Continuous line: simple grid configuration. Dashed line: modified configuration with inner walls. The vertical black lines show the locations of the vertical inner walls.

Schematic view of the modified grid configuration (the outer water tank is not shown; see Fig. 1). Three inner tanks are mounted on the grid, and turbulence is generated by raising the set . Each inner tank consists in four vertical sidewalls, without top and bottom walls.

Schematic view of the modified grid configuration (the outer water tank is not shown; see Fig. 1). Three inner tanks are mounted on the grid, and turbulence is generated by raising the set . Each inner tank consists in four vertical sidewalls, without top and bottom walls.

Total (dashed), mean (continuous), and turbulent (dashed-dotted) kinetic energies as a function of reduced time from 40 decay realizations performed at with the modified configuration with inner tanks. Inset: ratio of turbulent to total kinetic energy as a function of reduced time with (points) and without (triangles) inner tanks.

Total (dashed), mean (continuous), and turbulent (dashed-dotted) kinetic energies as a function of reduced time from 40 decay realizations performed at with the modified configuration with inner tanks. Inset: ratio of turbulent to total kinetic energy as a function of reduced time with (points) and without (triangles) inner tanks.

Temporal energy spectrum of the total (dashed), mean (continuous), and turbulent (dashed-dotted) component of the flow as a function of computed from 40 decay realizations performed at with inner tanks, with (a) linear and (b) logarithmic -axis. The dotted line reproduces spectrum of the mean flow for comparison (Fig. 4), obtained with the simple grid configuration.

Temporal energy spectrum of the total (dashed), mean (continuous), and turbulent (dashed-dotted) component of the flow as a function of computed from 40 decay realizations performed at with inner tanks, with (a) linear and (b) logarithmic -axis. The dotted line reproduces spectrum of the mean flow for comparison (Fig. 4), obtained with the simple grid configuration.

## Tables

Numerical values of normalized frequencies for different modes of order (where is the vertical wavenumber, characterizes the horizontal structure, and is the symmetry of the mode) compared to the experimental peaks in Fig. 4. The uncertainty of the experimental values is . The numerical values are computed for an aspect ratio identical to the experimental one .

Numerical values of normalized frequencies for different modes of order (where is the vertical wavenumber, characterizes the horizontal structure, and is the symmetry of the mode) compared to the experimental peaks in Fig. 4. The uncertainty of the experimental values is . The numerical values are computed for an aspect ratio identical to the experimental one .

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