Eulerian Spectra Research Articles

Turbulent diffusion from a quasi-kinematical point of view

An estimate for the coefficient of self-diffusion can be derived whenever the Eulerian velocity spectrum is known. The result is more general than those previously obtained. A comparison is made with computed test-particle diffusion in an inviscid two-dimensional Navier–Stokes fluid.

Turbulence correlations in a compressible wake

Space-time correlations have been measured in the turbulent supersonic wake behind a slender axisymmetric body. The data were obtained with two independently-positioned hot-wire anemometers, and interpreted with the aid of an extension of compressible anemometry theory. Comparison with earlier single-point Eulerian spectra showed a breakdown of Taylor's hypothesis near the wake edge, and thus also the expected differences in the scale lengths. The eddy convection speed appears equal to that of the mean flow, and the eddy shape is inclined to the wake axis, although it becomes nearly parallel to it at the wake edge. As a consequence, radial and longitudinal space correlation functions appear nearly equal. An upstream-downstream asymmetry was noted in the latter.

Eulerian and Lagrangian time microscales in isotropic turbulence

In isotropic ‘box’ turbulence without a mean flow, the Lagrangian frequency spectrum extends to frequencies of order$(\epsilon/\nu)^{\frac{1}{2}}$(ε is the rate of dissipation of kinetic energy per unit mass and ν is the kinematic viscosity of the fluid). This leads to an estimate that makes the r.m.s. value ofdu/dtof order$(\epsilon^3/\nu)^{\frac{1}{4}}$. The Eulerian frequency spectrum, however, extends to higher frequencies than its Lagrangian counterpart; this is caused by spectral broadening associated with large-scale advection of dissipative eddies. As a consequence, the r.m.s. value of ∂u/∂tat a fixed observation point is (apart from a numerical factor)$R_{\lambda}^{\frac{1}{2}}$times as large as the r.m.s. value ofdu/dt(RΛ is the turbulence Reynolds number based on the Taylor microscale). The results of a theoretical analysis based on these premises agree with data obtained by Comte-Bellot, Shlien and Corrsin. The analysis also suggests that the Eulerian frequency spectrum has a$\omega^{-\frac{5}{3}} $behaviour in the inertial subrange, and that it is not governed by Kolmogorov similarity.

Lagrangian-Eulerian time-scale ratios estimated from constant volume balloon flights past a tall tower

The ratio of Lagrangian and Eulerian time scales of motion in the planetary boundary layer is estimated from radar observations of 35 constant volume balloons (tetroons) flown past the 460 m high BREN tower at the Nevada Test Site. On the average, the peak energies of the Eulerian and tetroon vertical velocity spectra occur at periods near 6 and 18 minutes, yielding a mean Lagrangian-Eulerian scale ratio, β, of about three, β increases with increase in wind speed, decreases with increase in r.m.s. vertical velocity, and is inversely related to vertical turbulence intensity, with β ranging from 4.5 at a turbulence intensity of 0.05 to 2.5 at turbulence intensities exceeding 0.3.

Lagrangian‐Eulerian time‐scale ratios estimated from constant volume balloon flights past a tall tower

AbstractThe ratio of Lagrangian and Eulerian time scales of motion in the planetary boundary layer is estimated from radar observations of 35 constant volume balloons (tetroons) flown past the 460 m high BREN tower at the Nevada Test Site. On the average, the peak energies of the Eulerian and tetroon vertical velocity spectra occur at periods near 6 and 18 minutes, yielding a mean Lagrangian‐Eulerian scale ratio, β, of about three, β increases with increase in wind speed, decreases with increase in r.m.s. vertical velocity, and is inversely related to vertical turbulence intensity, with β ranging from 4.5 at a turbulence intensity of 0.05 to 2.5 at turbulence intensities exceeding 0.3.

Large-Scale Atmospheric Circulation Characteristics as Evident from Ghost Balloon Data

Abstract From GHOST balloon data obtained over the Southern Hemisphere at the 200-mb level, phase velocities of cyclone waves were found to vary between 6.3 and 9.2 degrees of longitude per day. Spectrum analysis of the relative velocities of balloon pairs, measured with respect to their common center of gravity, yielded a spectrum peak in the v component near frequencies of 1/52 hr, and a “−3” spectrum slope at higher frequencies. (The latter may be contaminated by a spline function smoothing technique.) Spectral densities in v were found to be slightly larger in summer than in winter, while densities in u were half as large in summer than in winter. Significantly stronger anisotropy of flow prevails at cyclone wavelengths in the Southern than the the Northern Hemisphere, with the v perturbations exceeding the zonal flow perturbations. GHOST balloon cospectra yielded similar results of momentum transport as did data from the Northern Hemisphere. Eulerian spectra of the v component over New Zealand agreed...

Lagrangian and Eulerian correlations and energy spectra of geostrophic velocities

Quarterly Journal of the Royal Meteorological SocietyVolume 93, Issue 397 p. 387-390 Discussion Lagrangian and Eulerian correlations and energy spectra of geostrophic velocities S.-K. Kao, S.-K. KaoSearch for more papers by this authorW. S. Bullock, W. S. BullockSearch for more papers by this author S.-K. Kao, S.-K. KaoSearch for more papers by this authorW. S. Bullock, W. S. BullockSearch for more papers by this author First published: July 1967 https://doi.org/10.1002/qj.49709339717Read the full textAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Volume93, Issue397July 1967Pages 387-390 RelatedInformation

Energy Spectra of Meso-Scale Turbulence Along and Across the Jet Stream

The wind velocities measured by an aircraft flying parallel and perpendicular to jet streams (Project Jet Stream, 1956–1957) have been analyzed; a smoothing technique has been used to separate the meso-scale turbulence from the mean flow. Eulerian auto-correlation coefficients and energy spectra are computed for the longitudinal and transversal components of the horizontal wind velocities. The distributions of the auto-correlation coefficients and the energy spectra appear to be similar for both the longitudinal and transversal components of the velocities, whereas the corrected meso-scale energy spectrum increases with decreasing wave number and is approximately proportional to k−2 in the range between 10−1 cycles km−1. An analysis is also made of the distribution of the Richardson number in a cross section perpendicular to the jet stream. A good relationship is found between the areas of turbulence and the regions of small Richardson number.

Lagrangian and eulerian correlations and energy spectra of geostrophic velocities

AbstractAn analysis is made of the dispersion of the end‐points of geostrophic trajectories initiated daily at 0000Z for the International Geophysical Year, 1958, at 500 mb over Seattle, Washington. Computations of the yearly mean of the Lagrangian autocorrelation coefficients and eddy diffusivities show that the Lagrangian integral time scale, ·, is equal to 5·0 × 104 sec. It is found that the zonal, x, and meridional, y, components of eddy diffusivity are respectively Kx = 5·90 × 1010 cm2 sec−1 and Ky = 2·30 × 1010 cm2 sec−1. Spectral analysis of the kinetic energy of the Lagrangian geostrophic velocities shows that the kinetic energy densities of both the zonal and meridional components of the geostrophic winds generally increase from high to low frequency with a peak of the meridional component occurring at the frequency of 0·01 cycles hr−1. Analyses are also made of the Eulerian autocorrelation coefficients and energy spectra of the geostrophic velocities. It is found that the Eulerian integral time‐scale, ϵ−1, is equal to 9·4 × 104 sec, making the ratio of the Lagrangian and Eulerian time‐scales near 0·5.

On the Range of Validity of Taylor's Hypothesis and the Kolmogoroff Spectral Law

Abstract Turbulence measurements obtained from concurrent tower and aircraft observations are analyzed to test G.I. Taylor's hypothesis (1938) of the equivalence between Eulerian space and Eulerian time spectra. The correlation and spectral data (along with data obtained from three other airplane turbulence programs) were also examined to ascertain the validity of Kolmogoroff's prediction (1940) for the behavior of the correlation coefficients and spectral densities in the inertial subrange. At heights of 300–400 ft above the ground, the tower and aircraft observations provide the same turbulence spectrum for wavelengths at least as large as 600 and 900 ft, for mean wind speeds of about 20 and 30 ft sec−1, respectively. The evidence is that the Kolmogoroff range is not approached for the wavelength range examined in this paper. The shortest wavelength sampled is on the order of 88 ft, at heights above the ground ranging from 200 to 2000 ft. It appears that at these heights, the Kolmogoroff spectrum, if va...

Estimates of the Relations between Eulerian and Lagrangian Scales in Large Reynolds Number Turbulence

With Eulerian and Lagrangian spectra approximated by their “inertial subrange” forms between appropriate wave number and frequency limits, it is found that the Lagrangian integral time scale is roughly equal to the Eulerian integral length scale divided by the root-mean-square velocity. The corresponding estimate for the “microscale” ratio shows fair agreement with the large Reynolds number form of Heisen-berg's result. A similar approach to Eulerian time scales gives values approximately equal to the Lagrangian scales.

Power spectra of temperature, humidity and refractive index from aircraft and tethered balloon measurements

Fifty-seven spectra of temperature, vapor pressure and refractive index were computed from captive balloon data taken at elevations up to 3000 feet MSL. Eight spectra of refractive index were obtained using an aircraft equipped with a microwave refractometer. It was found that atmospheric stability apparently has a pronounced effect on the variation of turbulent intensity with height. Although the spectra of all three parameters generally approach a -5/3 power law at high wave numbers, stability seems to have a controlling influence on spectral form at the low-frequency end of the wave number range studied. It is therefore concluded that methods of computing microwave scattered fields from the mean square dielectric perturbations and scale size obtained from the auto covariance are unreliable. The forms of the experimental auto covariances appear to be best represented by an exponential function or perhaps by a Modified Bessel Function of the second kind and 1/3 order. The temperature-humidity cospectrum may influence the shape of the refractive index spectrum, especially under unstable conditions. The equivalence of Eulerian space and time spectra is verified for refractive index by a series of aircraft-balloon fly-bys.

The relation between Eulerian time and space spectra

AbstractSimultaneous observations of space correlations and time correlations verify that Taylor's hypothesis x = Ut is valid for intervals up to at least 90 m, and for relative intensities of turbulence at least up to 0·26. It is also shown that, when the stratification is stable, the longitudinal scale of turbulence tends to be much larger along the flow than across it. In an unstable layer, the scale is of the same order of magnitude in all directions.