Experimental investigation of intermediate-dissipation range energy spectra in shear turbulence

Abstract

The shape of the turbulent energy spectrum in the dissipation range, where viscous effects dominate, remains an open question despite decades of work. We report an experimental investigation of intermediate dissipation range energy spectra in turbulent shear layers at Taylor-scale Reynolds numbers, $Re_λ$, ranging from approximately 450 to 1500, which are among the highest achieved in shear flow experiments that resolved small scales. We generated turbulent shear layers in a wind tunnel and measured using nanoscale hot-wire probes with a sensing length $l_w \approx (0.2-0.5)η$ that was smaller than the Kolmogorov scale $η$ at all $Re_λ$. The measurements resolved wavenumbers up to $k_{max} η$ $\approx 17$ at the lowest $Re_λ$ and $k_{max} η$ $\approx 1$ at the highest $Re_λ$, where $k_{max}$ is the highest resolved wave number. In the range $0.1 \lesssim k η\lesssim 0.5$, the spectra collapse onto a universal stretched-exponential form, $E(kη) \sim $ exp$(-kη)^γ $, with $γ\approx 0.5$ independent of $Re_λ$. This value of stretching exponent, $γ$, is consistent with recent empirical and computational studies. The Reynolds-number invariance of $γ$ is strong evidence for universal scaling in the intermediate dissipation range of high-Reynolds-number shear turbulence.

Figures (4)

• Figure 1: Hot-wire probes used in the present measurements. (a) Nanoscale hot-wire with active sensing length $l_w \approx 60,\mu\mathrm{m}$. (b) Conventional hot-wire with $l_w \approx 1,\mathrm{mm}$.
• Figure 2: Compensated longitudinal energy spectra measured over a range of Taylor-scale Reynolds numbers, $Re_\lambda$. Spectra obtained using the nanoscale hot-wire probe are shown in color, while the spectrum measured with a conventional hot-wire at $Re_\lambda \approx 450$ is shown in gray. The agreement between the two probes extends up to $k\eta \approx 1$, indicating negligible spatial-resolution bias over the commonly resolved scales. Inset: Reynolds-number dependence of the Kolmogorov constant, $C_k$, estimated from the inertial-subrange plateau. Error bars represent the standard deviation computed over the identified plateau region for each dataset. Dark red circles denote estimates obtained from the nanoscale probe, while that from the conventional probe is in gray.
• Figure 3: (a) Logarithmic derivative of the longitudinal energy spectrum, $d\log E(k)/d\log k$, plotted as a function of the normalized wavenumber $k\eta$ for different $Re_\lambda$. The black line denotes the reference scaling $(k\eta)^{0.5}$. Inset: Enlarged view over the interval $0.1 \leq k\eta \leq 0.5$. Colors are consistent with Fig. \ref{['fig:Figure2']}. (b) Reynolds-number dependence of the intermediate-dissipation-range exponent, $\gamma$. Black circles correspond to data from Ref. buaria2020dissipation, and the gray dashed line indicates the theoretical prediction of canet2017spatiotemporal. Dark red circles denote estimates obtained using the nanoscale probe, while gray circles correspond to measurements from the conventional probe. The solid dark red line represents the mean value of the nanoscale probe measurements.
• Figure 4: (a) Compensated logarithmic derivative of the longitudinal energy spectrum plotted as a function of the normalized wavenumber $k\eta$ for different $Re_\lambda$. Inset: Enlarged view over the interval $0.1 \leq k\eta \leq 0.5$. Colors are consistent with Fig. \ref{['fig:Figure2']}. (b) Reynolds-number dependence of the exponent $\beta$. Dark red circles denote estimates obtained from the nanoscale hot-wire measurements, while gray circles correspond to measurements using the conventional probe.