Experimental test of the crossover between the inertial and the dissipative range in a turbulent swirling flow

The kinetic energy spectrum of high-Reynolds turbulent swirling flows is experimentally studied. This spectrum, obtained from direct measurements in space, exhibits nearly two decades of Kolmogorov $k^{-5/3}$ decay in the inertial range of scales. Beyond this regime, in the dissipative range of scales, a crossover to a stretched exponential decay on scale $k^{2/3}$ is observed, in full agreement with a recent theoretical prediction based on nonperturbative renormalization group theory.

Figure 1

Representation of the experimental setup and of the mean flow: in blue, the azimuthal motion, and in red, the pumping motion.

Figure 2

Velocity measurement areas for case A (blue) and cases B, C, D and E (orange). The green area corresponds to location were turbulence is less isotropic, and is used to test universality.

Figure 3

Fourier spectrum in universal representation, after nondimensionalization by $\epsilon$, the global energy dissipation, and $\eta$, the Kolmogorov scale. The symbols are the experimental points, with vertical lines marking out the Taylor scale $\lambda$. The continuous gray line is a numerical spectrum, obtained from a direct numerical simulation of Navier-Stokes equations in a fully periodic box of size ${1024}^3$. These numerical data come from the JHTDB [30].

Figure 4

Wavelet spectrum in universal representation, after nondimensionalization by $\epsilon$, the global energy dissipation, and $\eta$, the Kolmogorov scale.

Figure 5

Energy spectrum for each velocity component in case E, with vertical axis multiplied by $k^{5/3}$ and horizontal axis rescaled to $k^{2/3}$, plotted in log-lin scale. A stretched exponential is apparent in the dissipative range. The gray line is the spectrum of the JHU database. The black line is a fit of the spectrum according to Eq. (5)

Figure 6

Same energy spectrum as in Fig. 3, with vertical axis multiplied by $k^{5/3}$, and horizontal axis rescaled to $k^{2/3}$, plotted in log-lin scale. A stretched exponential is apparent in the dissipative range. The gray line is the spectrum of the JHU database. The black line is a fit of the spectrum according to Eq. (5).

Figure 7

Same energy spectrum as in Fig. 3, with vertical axis multiplied by $k^{5/3}$, and horizontal axis rescaled to $k^{2/3}$, plotted in log-log scale. A stretched exponential is apparent in the dissipative range. The gray line is the spectrum of the JHU database. The black line is a fit of the spectrum according to Eq. (5).

Figure 8

Same energy spectrum as in Fig. 4, with vertical axis multiplied by $k^{5/3}$, and horizontal axis rescaled to $k^{2/3}$, plotted in log-lin scale. A stretched exponential is apparent in the dissipative range. The black line is a fit of the spectrum according to Eq. (5).

Figure 9

Same energy spectrum as in Fig. 4, with vertical axis multiplied by $k^{5/3}$, and horizontal axis rescaled to $k^{2/3}$, plotted in log-log scale. A stretched exponential is apparent in the dissipative range. The black line is a fit of the spectrum according to Eq. (5).

Figure 10

Test of universality. The energy spectrum corresponding to different forcings, different Reynolds numbers, and different isotropy conditions are plotted on the same representation as in Fig. 6, with the vertical axis multiplied by $k^{5/3}$ and the horizontal axis rescaled to $k^{2/3}$, plotted in log-lin scale. A stretched exponential is apparent in the dissipative range. The gray line is the spectrum of the JHU database. The black line is a fit of the spectrum according to Eq. (5).

Figure 11

Fourier spectrum in standard log-log representation with spurious parts. The vertical dashed lines refer to the cutoff wave number due to the PIV method. For the spectrum at $\text{Re}=6.{10}^3$, we can see that the saturation due to PIV noise happens for $k<k_{max}$.

Figure 12

Total Fourier spectrum and Fourier spectra by components for case A.

Figure 13

Total Fourier spectrum and Fourier spectra by components for case B.

Figure 14

Total Fourier spectrum and Fourier spectra by components for case C.

Figure 15

Total Fourier spectrum and Fourier spectra by components for case D.