Spatiotemporal statistics of the dissipation rate at the boundary of a turbulent flow using Diffusing-Wave Spectroscopy

Abstract

We use Diffusing Wave Spectroscopy (DWS) to perform the first direct space- and time-resolved measurement of the dissipation rate~$ε$ at the boundary of a turbulent flow. We have shown in a previous publication that this technique provides maps of the dissipation rate of Newtonian fluids~\cite{Francisco}. Here, we apply the technique at the boundary of a turbulent flow generated in a square box by an impeller stirring the fluids. Although the measurement is made on a small region near the boundary, we show that the dissipation remains proportional to the injected power and follows the turbulent scaling $ε\propto \mathrm{Re}^3$, with Re being the Reynolds number ranging from $1.5 \times 10^4$ to $6 \times 10^5$. With this flow, there is no need for logarithmic corrections to reproduce the dissipation near the flat boundary. In addition, our setup allows us to measure the spatio-temporal fluctuations of the dissipation near the boundary. These fluctuations are quite large (the relative fluctuations are about 50\%) and are well described by a log-normal distribution, as expected for the dissipation rate in the bulk of homogeneous and isotropic turbulence (HIT) but Power Density Spectra (PDS) do not correspond to those expected for HIT \cite{Li07,Graham16,K62}

Figures (5)

• Figure 1: Top sketch of the experimental setup viewed from the top. The DWS involves a 2-Watt Neodymium-YAG laser illuminating a large area of the cell through a microscope lens. The diffused light is collected either in the far field by a Photo Multiplier Tube (PMT) through a cross-polarizer or by an ultra-fast camera. We continuously control the value of $l^*$ with the spatially resolved reflectance technique involving another Neodymium-YAG laser and a CCD camera. Bottom Side view of the experimental cell. The gray rectangle shows the impeller position with the disk (light gray) and the four blades (dark gray). The blue square represents the area measured with the high-speed camera. The orange circle delimits the light collected by the monomodal fiber.
• Figure 2: Scaling of the mean injected power per unit mass $\overline{P}$ (blue crosses) and the mean power dissipated per unit mass near the boundary. $\overline{\langle \epsilon \rangle}$ measured with the PMT (red circles) is averaged in time and over a disk of 12 cm diameter. For the fast camera (green asterisks), the measurement is averaged over all the pixels (corresponding to a square of 5.1 cm). $\overline{P}$ is multiplied by a factor 2.4 to align with the dissipated power, highlighting an excess of dissipation at our measurement spot. The inset compares the dimensionless dissipation $\tilde{\epsilon}=\overline{\langle \epsilon \rangle}(L/\nu)$ compensated by $Re^3$ (red circles) with the theoretical logarithmic correction (black line) or without (dot-dashed line).
• Figure 3: Snapshots of dissipation maps in a window of $5.1\times 5.1$ cm$^2$ as shown by the blue square in figure \ref{['Setup']}. a and b show two successive maps from the experiment at $Re=3.2\times10^5$. The impeller rotates around the x-axis, generating a mean flow in the y-direction. c shows a map from the experiment at $Re=1\times10^5$ and d shows a map from DNS at $Re=3\times10^4$ (horizontal streamwise direction). The black line represents 1 cm.
• Figure 4: Probability Density Function of the centered and normalized logarithm of dissipation fluctuations, $\left(\log(\epsilon)-\langle\log(\epsilon)\rangle\right)/\sigma_{\log(\epsilon)}$. Green dots correspond to the long measurement at $Re=3.2\times10^5$, blue dots correspond to DNS data from a channel flow. The dashed line represents the centered and normalized log-normal distribution.
• Figure 5: Power Density Spectra (PDS) of spatial fluctuations of $\epsilon$, averaged over time, as a function of wavenumber $k$, for $Re$ ranging from $2.4\times10^4$ to $1.68\times10^5$ (arrow indicates increasing $Re$). The inset shows the PDS of $\epsilon$ extracted from DNS at $Re=4\times 10^4$ in the spanwise (dotted line) and streamwise (dot-dashed line) directions.The dashed line in the main panel and the inset represents the power $k^{-(1-\mu)}$ predicted for the dissipation in HIT, with $\mu=0.2$.