Turbulence statistics from three different nacelle lidars

Atmospheric turbulence can be characterized by the Reynolds stress tensor, which consists of the second-order moments of the wind field components. Most of the commercial nacelle lidars cannot estimate all components of the Reynolds stress tensor due to their limited number of beams; most can estimate the along-wind velocity variance relatively well. Other components are however also important to understand the behavior of, e.g., the vertical wind profile and meandering of wakes. The SpinnerLidar, a research lidar with multiple beams and a very high sampling frequency, was deployed together with two commercial lidars in a forward-looking mode on the nacelle of a Vestas V52 turbine to scan the inflow. Here, we compare the lidar-derived turbulence estimates with those from a sonic anemometer using both numerical simulations and measurements from a nearby mast. We show that from these lidars, the SpinnerLidar is the only one able to retrieve all Reynolds stress components. For the two- and four-beam lidars, we study different methods to compute the along-wind velocity variance. By using the SpinnerLidar's Doppler spectra of the radial velocity, we can partly compensate for the lidar's probe volume averaging effect and thus reduce the systematic error of turbulence estimates. We find that the variances of the radial velocities estimated from the maximum of the Doppler spectrum are less affected by the lidar probe volume compared to those estimated from the median or the centroid of the Doppler spectrum.

Reynolds' Turbulence Solution

This study revisits the Reynolds-averaged Navier--Stokes equations (RANS) and finds that the existing literature is erroneous regarding the primary unknowns and the number of independent unknowns in the RANS. The literature claims that the Reynolds stress tensor has six independent unknowns, but in fact the six unknowns can be reduced to three that are functions of the three velocity fluctuation components, because the Reynolds stress tensor is simply an integration of a second-order dyadic tensor of flow velocity fluctuations rather than a general symmetric tensor. This difficult situation is resolved by returning to the time of Reynolds in 1895 and revisiting Reynolds' averaging formulation of turbulence. The study of turbulence modeling could focus on the velocity fluctuations instead of on the Reynolds stress. An advantage of modeling the velocity fluctuations is, from both physical and experimental perspectives, that the velocity fluctuation components are observable whereas the Reynolds stress tensor is not.

Deriving Six Components of Reynolds Stress Tensor from Single-ADCP Data

Acoustic Doppler current profilers (ADCP) are widely used in geophysical studies for mean velocity profiling and calculation of energy dissipation rate. On the other hand, the estimation of turbulent stresses from ADCP data still remains challenging. With the four-beam version of the device, only two shear stresses are derivable; and even for the five-beam version (Janus+), the calculation of the full Reynolds stress tensor is problematic currently. The known attempts to overcome the problem are based on the “coupled ADCP” experimental setup and include some hard restrictions, not to mention the essential complexity of performing experiments. In this paper, a new method is presented which allows to derive the stresses from single-ADCP data. Its essence is that interbeam correlations are taken into account as producing the missing equations for stresses. This method is applicable only for the depth range, for which the distance between the beams is comparable to the scales, where the turbulence is locally isotropic and homogeneous. The validation of this method was carried out for convectively-mixed layer in a boreal ice-covered lake. The results of computations turned out to be physically sustainable in the sense that realizability conditions were basically fulfilled. The additional verification was carried out by comparing the results, obtained by the new method and “coupled ADCPs” one.

Reynolds stress tensor measurements using magnetic resonance velocimetry: expansion of the dynamic measurement range and analysis of systematic measurement errors

This study presents magnetic resonance velocimetry (MRV) Reynolds Stress measurements in a periodic hill channel with a hill Reynolds number of Re = 29,500. The velocity encoding scheme is based on the ICOSA6 method with six icosahedral encoding directions and multiple encoding values are measured to increase the dynamic range. The full Reynolds stress tensor is obtained from a voxel-wise three-dimensional Gaussian fit using the magnitude data of all acquisitions. The MRV results are compared to a wall-resolved large eddy simulation and laser Doppler velocimetry measurements conducted in the same channel. It is shown that the MRV Reynolds stress data have excellent precision and agree qualitatively with the reference data. However, there are apparent systematic deviations. One of the most prominent error contributions is the signal attenuation caused by higher orders of motion, which leads to an overestimation of the turbulence level. Another fundamental error is identified in the assumption that the turbulence is Gaussian distributed. With the presented reconstruction technique, the MRV data are fitted to a statistical model, and depending on the examined flow setup, the Gaussian model can lead to considerable errors. Possible ways of how to reduce all identified errors are presented. In summary, this technique enables Reynolds stress tensor measurements in complex internal flows with high dynamic range and excellent precision. However, several issues need to be resolved to make the turbulence quantification more accurate. Graphic abstract

Revisiting the Reynolds-averaged Navier–Stokes equations

This study revisits the Reynolds-averaged Navier–Stokes (RANS) equations and finds that the existing literature is erroneous regarding the primary unknowns and the number of independent unknowns in the RANS. The literature claims that the Reynolds stress tensor has six independent unknowns, but in fact the six unknowns can be reduced to three that are functions of the three velocity fluctuation components, because the Reynolds stress tensor is simply an integration of a second-order dyadic tensor of flow velocity fluctuations rather than a general symmetric tensor. This difficult situation is resolved by returning to the time of Reynolds in 1895 and revisiting Reynolds’ averaging formulation of turbulence. The study of turbulence modeling could focus on the velocity fluctuations instead of the Reynolds stress. An advantage of modeling the velocity fluctuations is, from both physical and experimental perspectives, that the velocity fluctuation components are observable whereas the Reynolds stress tensor is not.