Double Averaging Method and Turbulence Statistics for a Rough-Bed Flow

Natural streams can be classified as hydraulically rough-bed flows, and they represent one of the most complex issue of hydrodynamics. Although the dynamics of such flows has been extensively modelled in the past century, there are still many unsolved problems that need to be addressed with the support of theory, simulations and realistic experiments. In the past decades, new techniques and models have been proposed for the study of these turbulent systems, such as the well-known Reynolds time-Averaged Navier-Stokes (RANS) technique. More recently, the classical RANS equations have been combined with a spatial ensemble of the measurements. This technique, proposed for hydrodynamics in the pioneering work by Nikora et al. (2001), is known as the Double-Averaging Method (DAM). The DAM allows a general characterization of the macroscopic behaviour of rivers and channels, providing information about the mean flow profile as a function of the water depth, making use of the double-averaged turbulent stresses. In this preliminary work, we present an experiment on a turbulent flow on a rough bed, performed at the Laboratorio di Grandi Modelli Idraulici (Dipartimento di Ingegneria Civile, Università della Calabria, Italy). Results were analysed through the DAM, and the vertical distribution of the average Reynolds stresses, as well as the velocity profiles, were obtained, showing a good agreement with the existing literature. Moreover, we combined the DAM with an accurate estimate of the turbulent dissipation rate, ε, making use of the Kolmogorov 4/5 law. In the following paragraphs, the experimental facility is described, the double averaged profiles are computed and, finally, an estimate of ε is obtained. Experiments were conducted in a 1.00 m-wide, 0.80 m-deep and 16.00 m-long tilting flume with rectangular cross-section with longitudinal slope of 0.0025. A honeycomb was placed at the flume entrance, in order to regularize the flow. The test section was at 10.00 m from the inlet. A random spreading process was used to create the gravel bed. The sediments adopted were non-regular natural pebbles with median size d50 ≈ 70 mm spread in four layers. The bed surface framed in a square sampling box (0.5 m x 0.5 m) was acquired with a Laser Scanner by Minolta, Model Vivid 300. The scanned image is shown in Fig. 1(left), while Fig. 1(center) presents the roughness geometry function defined as φ(z)=A(z)/A0 where A(z) is the area occupied by fluid at z and A0 is the area of the square sampling box (0.25 m2). A Vectrino four-beam down-looking Acoustic Doppler Velocimeter (ADV) probe by Nortek was used to measure the 3D instantaneous velocity components, with 100 Hz resolution; the acquisition period of 300 s was considered adequate in order to measure statistically independent quantities as suggested by Dey and Das (2012). To resolve the spatial heterogeneity, the time averaging is conceptually supplemented by the area averaging in the layer parallel to the mean bed surface, computing the turbulence characteristic as velocity moments. Hereafter, we use overbar to indicate time averages of a generic variable ( ), prime to indicate fluctuations in time ( ), and brackets for spatial averages ( ). A logarithmic trend was clearly identified in the range z/ = 2 to 4.5 ( being the roughness height), as shown in Fig. 1(right), in which u* is the shear velocity. At lower levels, a linear trend was observed, and below the crests the mean velocity was negligible, indicating the presence of stationary vortexes through the dip. Second order moments were reported in Fig. 2, namely u’2 (top-left), v’2 (top-right), w’2 (bottom-left), and the stress u’w’ (bottom-right). Each point indicates a measurement at a given z and x (open red circles), while the double-average is reported with a full blue line. It is worth noting that the experimental data is in agreement with previous works. As expected, the peak of fluctuations was observed near the crest level (z/≈ 2). Below this level, a trend similar to that of the roughness geometry function can be recognized. We applied the DAM to the power spectrum of the streamwise velocity u,<Eu(k)>. The spectra were computed at z/= 4, where velocity shears are negligible, and where the Taylor frozen-in turbulence hypothesis can be applied. A clear inertial range was observed for k = 10 to 100 m-1, while at small scale other effects are present. At larger scales, k < 10, a trend similar to k-1 was observed. Finally we computed the exact third-order law of hydrodynamics (Kolmogorov’s 4/5 power law), where . Here u = u(t,z)-u(t+,z),  being the time increment which gives . Note that this is an exact law which can be used to measure the dissipation rate, , as a function of z. The compensated law is reported in Fig. 3(right), indicating that in the inertial range, 0.01 < r < 0.1. This value was used to reproduce the prediction of Kolmogorov -5/3 power law in Fig. 3(left) (with C = 0.52, from literature). As it can be observed, the value gives a very good estimate of the spectral distribution, and is in agreement with J/s. The above preliminary results confirm the validity of the DAM and suggest that the method can be used for higher order measurements of turbulence.