Experimental measurements of spectral subgrid-scale Prandtl number in a heated turbulent wake flow, EDQNM predictions and 4/3-law

Abstract

For turbulent flows including scalar transport, the spectral diffusivity as well as the spectral Prandtl number characterizes the scalar-variance transfer rate from any wavenumber κ to all scales smaller than a specified filter scale. Classical theories predict a plateau behavior at inertial-range wavenumbers with deviations of the diffusivity at scales close to the cutoff wavenumber. However, to date these quantities have not been measured in high Reynolds number flows. To examine realistic turbulence at a moderately high Reynolds number, in this study an experimental measurement technique previously applied to obtain spectral eddy viscosities (Cerutti et al. 2000 Journal of Fluid Mechanics 421 307–338) is extended and applied to the case of scalar eddy diffusivity. We show that the subgrid-scale (SGS) dissipation spectrum of scalar variance can be expressed as the co-spectrum of negative SGS scalar flux and filtered scalar gradient. Using local isotropy, the result is expressed in terms of the streamwise components of the SGS scalar flux and filtered scalar gradient that can be measured in experiments. Using an array of four X-type hot-wire and four I-type cold-wire probes, two-dimensional box-filtered velocities and temperatures in the streamwise (invoking Taylor's hypothesis) and cross-stream directions are obtained at the centerline of a heated wake flow, at a Taylor-scale Reynolds number of 350. From the radial dissipation spectra, the spectral eddy viscosity and Prandtl number are evaluated. The measured spectral eddy viscosity, diffusivity and Prandtl number (the ratio of the spectral eddy viscosity to diffusivity) decrease near the filter scale. They also decrease with decreasing wavenumber near the integral scale of turbulence. Measurement results are compared with predictions from a classical two-point closure (eddy-damped quasi-normal Markovian, EDQNM) applied with a Gaussian filter and finite integral scale. Measurement and predictions are in qualitative agreement (decreasing trends observed near the filter wavenumber), but the details differ especially in the low wavenumber limit. This could be due to the assumption of isotropy which does not hold for the experiment at large scales. We also find significant sensitivity of the EDQNM predictions to free parameters of the eddy-damping term. Analysis in physical space shows that the two-point correlation between the SGS scalar flux and filtered scalar plays an important role to predict a mixed third-order structure function in the von Kármán–Howarth–Kolmogorov equation for the resolved scalar field (the LES equivalent of the 4/3-law).