Akihiro Hori

Probability density function of turbulent velocity fluctuations.

The probability density function (PDF) of velocity fluctuations is studied experimentally for grid turbulence in a systematical manner. At small distances from the grid, where the turbulence is still developing, the PDF is sub-Gaussian. At intermediate distances, where the turbulence is fully developed, the PDF is Gaussian. At large distances, where the turbulence has decayed, the PDF is hyper-Gaussian. The Fourier transforms of the velocity fluctuations always have Gaussian PDFs. At intermediate distances from the grid, the Fourier transforms are statistically independent of each other. This is the necessary and sufficient condition for Gaussianity of the velocity fluctuations. At small and large distances, the Fourier transforms are dependent.

Laboratory experiments for intense vortical structures in turbulence velocity fields

Vortical structures of turbulence, i.e., vortex tubes and sheets, are studied using one-dimensional velocity data obtained in laboratory experiments for duct flows and boundary layers at microscale Reynolds numbers from 332 to 1934. We study the mean velocity profile of intense vortical structures. The contribution from vortex tubes is dominant. The radius scales with the Kolmogorov length. The circulation velocity scales with the rms velocity fluctuation. We also study the spatial distribution of intense vortical structures. The distribution is self-similar over small scales and is random over large scales. Since these features are independent of the microscale Reynolds number and of the configuration for turbulence production, they appear to be universal.

Vortex tubes in velocity fields of laboratory isotropic turbulence: dependence on the Reynolds number.

: The streamwise and transverse velocities are measured simultaneously in isotropic grid turbulence at relatively high Reynolds numbers Re(lambda) approximately 110-330. Using a conditional averaging technique, we extract typical intermittency patterns that are consistent with velocity profiles of a model for a vortex tube, i.e., Burgers vortex. The radii of the vortex tubes are several of the Kolmogorov length, regardless of the Reynolds number. Using the distribution of an interval between successive enhancements of a small-scale velocity increment, we study the spatial distribution of vortex tubes. The vortex tubes tend to cluster together. This tendency is increasingly significant with the Reynolds number. Using statistics of velocity increments, we also study the energetical importance of vortex tubes as a function of the scale. The vortex tubes are important over the background flow at small scales especially below the Taylor microscale. At a fixed scale, the importance is increasingly significant with the Reynolds number.

On Landau’s prediction for large-scale fluctuation of turbulence energy dissipation

Kolmogorov’s theory for turbulence, proposed in 1941, is based on a hypothesis that small-scale statistics are uniquely determined by the kinematic viscosity and the mean rate of energy dissipation. Landau remarked that the local rate of energy dissipation should fluctuate in space over large scales and hence should affect small-scale statistics. Experimentally, we confirm the significance of this large-scale fluctuation, which is comparable to the mean rate of energy dissipation at the typical scale for energy-containing eddies. The significance is independent of the Reynolds number and the configuration for turbulence production. With an increase of scale r above the scale of largest energy-containing eddies, the fluctuation comes to have the scaling r−1∕2 and becomes close to Gaussian. We also confirm that the large-scale fluctuation affects small-scale statistics.

Vortex tubes in turbulence velocity fields at Reynolds numbers Re lambda approximately equal to 300-1300.

The most elementary structures of turbulence, i.e., vortex tubes, are studied using velocity data obtained in a laboratory experiment for boundary layers with microscale Reynolds numbers 295-1258. We conduct conditional averaging for enhancements of a small-scale velocity increment and obtain the typical velocity profile for vortex tubes. Their radii are of the order of the Kolmogorov length. Their circulation velocities are of the order of the root-mean-square velocity fluctuation. We also obtain the distribution of the interval between successive enhancements of the velocity increment as the measure of the spatial distribution of vortex tubes. They tend to cluster together below about the integral length and more significantly below about the Taylor microscale. These properties are independent of the Reynolds number and are hence expected to be universal.

Two-point velocity average of turbulence: Statistics and their implications

For turbulence, although the two-point velocity difference u(x+r)−u(x) at each scale r has been studied in detail, the velocity average [u(x+r)+u(x)]/2 has not thus far. Theoretically or experimentally, we find interesting features of the velocity average. It satisfies an exact scale-by-scale energy budget equation. The flatness factor varies with the scale r in a universal manner. These features are not consistent with the existing assumption that the velocity average is independent of r and represents energy-containing large-scale motions alone. We accordingly propose that it represents motions over scales ≥r as long as the velocity difference represents motions at the scale r.

Large-scale lognormal fluctuations in turbulence velocity fields

For several flows of laboratory turbulence, we obtain long records of velocity data. These records are divided into numerous segments. In each segment, we calculate the mean rate of energy dissipation, the mean energy at each scale, and the mean total energy. Their values fluctuate significantly among the segments. The fluctuations are lognormal, if the segment length lies within the range of large scales where the velocity correlations are weak but not yet absent. Since the lognormality is observed regardless of the Reynolds number and the configuration for turbulence production, it is expected to be universal. The likely origin is some multiplicative stochastic process related to interactions among scales through the energy transfer.

Large-scale length that determines the mean rate of energy dissipation in turbulence.

The mean rate of energy dissipation [ε] per unit mass of turbulence is often written in the form of [ε]=C(u)[u(2)](3/2)/L(u), where the root-mean-square velocity fluctuation [u(2)](1/2) and the velocity correlation length L(u) are parameters of the energy-containing large scales. However, the dimensionless coefficient C(u) is known to depend on the flow configuration that is to induce the turbulence. We define the correlation length L(u(2)) of the local energy u(2), study C(u(2))=[ε]L(u(2))/[u(2)](3/2) with experimental data of several flows, and find that C(u(2)) does not depend on the flow configuration. Not L(u) but L(u(2)) could serve universally as the typical size of the energy-containing eddies, so that [u(2)](3/2)/L(u(2)) is proportional to the rate at which the kinetic energy is removed from those eddies and is eventually dissipated into heat. The independence from the flow configuration is also found for the two-point correlations and so on when L(u(2)) is used to normalize the scale.

Probability density function of turbulent velocity fluctuations in a rough-wall boundary layer.

The probability density function of single-point velocity fluctuations in turbulence is studied systematically using Fourier coefficients in the energy-containing range. In ideal turbulence where energy-containing motions are random and independent, the Fourier coefficients tend to Gaussian and independent of each other. Velocity fluctuations accordingly tend to Gaussian. However, if energy-containing motions are intermittent or contaminated with bounded-amplitude motions such as wavy wakes, the Fourier coefficients tend to non-Gaussian and dependent of each other. Velocity fluctuations accordingly tend to non-Gaussian. These situations are found in our experiment of a rough-wall boundary layer.

Fluctuations of statistics among subregions of a turbulence velocity field

To study subregions of a turbulence velocity field, a long record of velocity data of grid turbulence is divided into smaller segments. For each segment, we calculate statistics such as the mean rate of energy dissipation and the mean energy at each scale. Their values significantly fluctuate, in lognormal distributions at least as a good approximation. Each segment is not under equilibrium between the mean rate of energy dissipation and the mean rate of energy transfer that determines the mean energy. These two rates still correlate among segments when their length exceeds the correlation length. Also between the mean rate of energy dissipation and the mean total energy, there is a correlation characterized by the Reynolds number for the whole record, implying that the large-scale flow affects each of the segments.