scholarly journals Revisiting two-dimensional turbulence and mesoscale spectra

2021 ◽  
Author(s):  
Xiaoli Larsén ◽  
Søren Larsen ◽  
Erik Petersen ◽  
Torben Mikkelsen
Keyword(s):  
Boundary Layer ◽  
Stationary Time Series ◽  
Wave Number ◽  
Inertial Subrange ◽  
Spectral Structure ◽  
Two Dimensional ◽  
Spectral Slope ◽  
Lateral Velocity ◽  
2D Turbulence ◽  
Boundary Layer Meteorology

<p>Two-dimensional (2D) turbulence is not only a basic research topic that needs further investigations, it is also relevant for wind energy applications as today’s wind farm clusters can be as large as thousands of kilometers squared and individual turbines hundreds of meters tall. This challenges the use of classical turbulence models applicable for scales smaller than ~1 h, or as denoted in Högström et al. (2002) the Kolmogoroff inertial subrange, the shear production range, and for ranges the spectral gap region.</p><p>This study revisits some key characteristics of 2D turbulence and interpretation of the physics behind it, including general literatures as well as a series of our studies in recent years (Larsén et al. 2013, 2016, 2021). This includes</p><ul><li>The respective frequency/wave number range and the spectral behaviours for the wind speed: the synoptic scales where the spectral slope is -3, the mesoscale where the spectral slope is typically -5/3, and the gap region. We analyze at what scales the spectra meet and merge, and how the spectra are affected by weather types, seasons and stability.</li> <li>The 2D-isotropy characteristics. We analyze how the longitudinal and lateral velocities correlate across the scales.</li> <li>The application of stationarity. The validity of an assumption of stationary time series decides to how large scales we can perform the analysis of the longitudinal and lateral velocity components, the Taylor frozen hypothesis and 2D-isotropy.</li> </ul><p>The primary datasets are from several met stations over Denmark and the North Sea region, including both 10-min and sonic measurements from about 10 m up to 240 m.</p><p> </p><p>References:</p><p>Högström U, Hunt J, Smedman AS (2002) Theory and measurements for turbulence spectra and variances in the atmospheric neutral surface layer. Boundary-Layer Meteorol 103:101–124</p><p>Larsén, X. G., Larsen, S. E., Petersen, E. L., & Mikkelsen, T. K. (2021). A Model for the Spectrum of the Lateral Velocity Component from Mesoscale to Microscale and Its Application to Wind-Direction Variation. Boundary-Layer Meteorology, 178, 415-434. https://doi.org/10.1007/s10546-020-00575-0</p><p>Larsén X. Larsen S. and Petersen E. (2016): Full-scale spectrum of the boundary layer wind. Boundary-Layer Meteorology, Vol 159, p 349-371</p><p>Larsén X., Vincent C. and Larsen S.E. (2013): Spectral structure of mesoscale winds over the water, Q. J. R. Meteorol. Soc., DOI:10.1002/qj.2003, 139, 685-700.</p>

On the stability of three-dimensional boundary layers with application to the flow due to a rotating disk

1955 ◽  
Vol 248 (943) ◽  
pp. 155-199 ◽  
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Wave Velocity ◽  
Critical Points ◽  
Three Dimensional ◽  
Rotating Disk ◽  
Velocity Profiles ◽  
Wave Number ◽  
Two Dimensional ◽  
China Clay

A phenomenon of boundary-layer instability is discussed from the theoretical and experimental points of view. The china-clay evaporation technique shows streaks on the surface, denoting a vortex system generated in the region of flow upstream of transition. Experiments on a swept wing are described briefly, while experiments on the flow due to a rotating disk receive much greater attention. In the latter case, the axes of the disturbance vortices take the form of equi-angular spirals, bounded by radii of instability and of transition. A frequency analysis of the disturbances shows that there is a narrow band of disturbance components of high amplitude, some frequencies within this band corresponding to disturbances fixed relative to the surface and others corresponding to moving waves. Furthermore, the determination of velocity profiles for the rotating-disk flow is described, the agreement with the theoretical solution for laminar flow being quite satisfactory; for turbulent flow, however, the empirical theories are not very satisfactory. In order to explain the vortex phenomenon just discussed, the general equations of motion in orthogonal curvilinear co-ordinates are examined by superimposing an infinitesimal disturbance periodic in space and time on the main flow, and linearizing for small disturbances. An important result is that, within the range of certain approximations, the velocity component in the direction of propagation of the disturbance may be regarded as a two-dimensional flow for stability purposes; then the problem of stability formally resembles the well-known two dimensional problem. However, it is important to emphasize that this result—namely, that the flow curvature has little influence on stability—is applicable only to the possible modes of instability in a local region. The nature of three-dimensional flows is discussed, and the importance of co-ordinates along and normal to the stream-lines outside the boundary layer is examined. In accord with the formal two-dimensional nature of the instability, there is a whole class of velocity distributions, corresponding to different directions, which may exhibit instability. The question of stability at infinite Reynolds number is examined in detail for these profiles. As for ordinary two-dimensional flows, the wave velocity of the disturbance must lie somewhere between the maximum and minimum of the velocity profile considered. The points where the wave velocity equals the fluid velocity are called critical points, of which most of the profiles considered have two. Then Tollmien’s criterion that velocity profiles with a point of inflexion are unstable at infinite Reynolds number is extended to the case of profiles with two critical points. One particular profile—namely, that for which the point of inflexion lies at the point of zero velocity—may generate neutral disturbances of zero phase velocity, corresponding to the disturbances visualized by the china-clay technique. A variational method for the solution of certain of the eigenvalue problems associated with stability at infinite Reynolds number is derived, found by comparison with an exact solution to be very accurate, and applied to the rotating disk. The fixed vortices predicted by the theory have as their axes equi-angular spirals of angle 103°, in good agreement with experiment, but the agreement between theoretical and experimental wave number is not good, the discrepancy being attributed to viscosity. Finally, the correlation between the experimentally observed and theoretically possible disturbances is discussed and certain conclusions drawn therefrom. The streamlines of the disturbed boundary layer show the existence of a double row of vortices, one row of which produces the streaks in the china clay. Application of the theory to other physical phenomena is described.

On the two-dimensional mixing region

1976 ◽  
Vol 74 (2) ◽  
pp. 209-250 ◽  
Author(s):  
F. H. Champagne ◽  
Y. H. Pao ◽  
I. J. Wygnanski
Keyword(s):  
Initial Conditions ◽  
Longitudinal Velocity ◽  
Higher Order ◽  
Digital Data ◽  
Statistical Characteristics ◽  
Wave Number ◽  
Two Dimensional ◽  
Eighth Order ◽  
Lateral Velocity ◽  
Velocity Fluctuations

An experimental investigation of the two-dimensional incompressible mixing layer was carried out. The measurements provide new information on the development of the mean and turbulent fields towards a self-preserving state and on the higher-order statistical characteristics of the turbulent field. The relevance of initial conditions to the development of the flow is discussed in the light of both present and previous data. Measurements of spectra, probability densities and moments to eighth order of all three velocity-component fluctuations at various transverse positions across the flow were carried out using an on-line digital data acquisition system. The probability density distributions of the derivative and the squared derivative of the longitudinal and lateral velocity fluctuations were also determined. Direct measurements of moments to eighth order of the velocity derivatives were attempted and are discussed in the light of the simultaneously measured histograms. The problems in obtaining higher-order statistical data are considered in some detail. Estimates of the integral time scale of many of the higher-order statistics are presented. The high wave-number structure was found to be locally anisotropic according to both spectral and turbulent velocity-gradient moment requirements. Higher-order spectra to fourth order of the longitudinal velocity fluctuations were measured and are discussed. Finally the lognormality of the squared longitudinal and lateral velocity-derivative fluctuations was investigated and the universal lognormal constant μ was evaluated.

The Departure from Equilibrium of Turbulent Boundary Layers

1968 ◽  
Vol 19 (1) ◽  
pp. 1-19 ◽  
Author(s):  
H. McDonald
Keyword(s):  
Boundary Layer ◽  
Shear Stress ◽  
Kinetic Energy ◽  
Stress Distribution ◽  
Boundary Layers ◽  
Turbulent Boundary Layers ◽  
Two Dimensional ◽  
Pressure Distributions ◽  
Momentum Integral ◽  
State Examination

SummaryRecently two authors, Nash and Goldberg, have suggested, intuitively, that the rate at which the shear stress distribution in an incompressible, two-dimensional, turbulent boundary layer would return to its equilibrium value is directly proportional to the extent of the departure from the equilibrium state. Examination of the behaviour of the integral properties of the boundary layer supports this hypothesis. In the present paper a relationship similar to the suggestion of Nash and Goldberg is derived from the local balance of the kinetic energy of the turbulence. Coupling this simple derived relationship to the boundary layer momentum and moment-of-momentum integral equations results in quite accurate predictions of the behaviour of non-equilibrium turbulent boundary layers in arbitrary adverse (given) pressure distributions.

Acoustic receptivity and evolution of two-dimensional and oblique disturbances in a Blasius boundary layer

2001 ◽  
Vol 432 ◽  
pp. 69-90 ◽  
Author(s):  
RUDOLPH A. KING ◽  
KENNETH S. BREUER
Keyword(s):  
Boundary Layer ◽  
Surface Roughness ◽  
Acoustic Wave ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Surface Waviness ◽  
S Wave ◽  
Boundary Layer Instability ◽  
Blasius Boundary Layer ◽  
Tollmien Schlichting

An experimental investigation was conducted to examine acoustic receptivity and subsequent boundary-layer instability evolution for a Blasius boundary layer formed on a flat plate in the presence of two-dimensional and oblique (three-dimensional) surface waviness. The effect of the non-localized surface roughness geometry and acoustic wave amplitude on the receptivity process was explored. The surface roughness had a well-defined wavenumber spectrum with fundamental wavenumber kw. A planar downstream-travelling acoustic wave was created to temporally excite the flow near the resonance frequency of an unstable eigenmode corresponding to kts = kw. The range of acoustic forcing levels, ε, and roughness heights, Δh, examined resulted in a linear dependence of receptivity coefficients; however, the larger values of the forcing combination εΔh resulted in subsequent nonlinear development of the Tollmien–Schlichting (T–S) wave. This study provides the first experimental evidence of a marked increase in the receptivity coefficient with increasing obliqueness of the surface waviness in excellent agreement with theory. Detuning of the two-dimensional and oblique disturbances was investigated by varying the streamwise wall-roughness wavenumber αw and measuring the T–S response. For the configuration where laminar-to-turbulent breakdown occurred, the breakdown process was found to be dominated by energy at the fundamental and harmonic frequencies, indicative of K-type breakdown.

Observations of Small-Scale Turbulence and Energy Dissipation Rates in the Cloudy Boundary Layer

2006 ◽  
Vol 63 (5) ◽  
pp. 1451-1466 ◽  
Author(s):  
Holger Siebert ◽  
Katrin Lehmann ◽  
Manfred Wendisch
Keyword(s):  
Boundary Layer ◽  
Energy Dissipation ◽  
Wind Velocity ◽  
Turbulence Theory ◽  
Horizontal Wind ◽  
Inertial Subrange ◽  
Intermittent Flow ◽  
Small Scale ◽  
Dissipation Rates ◽  
Wide Range

Abstract Tethered balloon–borne measurements with a resolution in the order of 10 cm in a cloudy boundary layer are presented. Two examples sampled under different conditions concerning the clouds' stage of life are discussed. The hypothesis tested here is that basic ideas of classical turbulence theory in boundary layer clouds are valid even to the decimeter scale. Power spectral densities S( f ) of air temperature, liquid water content, and wind velocity components show an inertial subrange behavior down to ≈20 cm. The mean energy dissipation rates are ∼10−3 m2 s−3 for both datasets. Estimated Taylor Reynolds numbers (Reλ) are ∼104, which indicates the turbulence is fully developed. The ratios between longitudinal and transversal S( f ) converge to a value close to 4/3, which is predicted by classical turbulence theory for local isotropic conditions. Probability density functions (PDFs) of wind velocity increments Δu are derived. The PDFs show significant deviations from a Gaussian distribution with longer tails typical for an intermittent flow. Local energy dissipation rates ɛτ are derived from subsequences with a duration of τ = 1 s. With a mean horizontal wind velocity of 8 m s−1, τ corresponds to a spatial scale of 8 m. The PDFs of ɛτ can be well approximated with a lognormal distribution that agrees with classical theory. Maximum values of ɛτ ≈ 10−1 m2 s−3 are found in the analyzed clouds. The consequences of this wide range of ɛτ values for particle–turbulence interaction are discussed.

Direct simulation of transition in an oscillatory boundary layer

1998 ◽  
Vol 371 ◽  
pp. 207-232 ◽  
Author(s):  
G. VITTORI ◽  
R. VERZICCO
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Turbulent Regime ◽  
Two Dimensional ◽  
Temporal Development ◽  
Direct Simulation ◽  
Navier Stokes Equations ◽  
Oscillatory Boundary Layer

Numerical simulations of Navier–Stokes equations are performed to study the flow originated by an oscillating pressure gradient close to a wall characterized by small imperfections. The scenario of transition from the laminar to the turbulent regime is investigated and the results are interpreted in the light of existing analytical theories. The ‘disturbed-laminar’ and the ‘intermittently turbulent’ regimes detected experimentally are reproduced by the present simulations. Moreover it is found that imperfections of the wall are of fundamental importance in causing the growth of two-dimensional disturbances which in turn trigger turbulence in the Stokes boundary layer. Finally, in the intermittently turbulent regime, a description is given of the temporal development of turbulence characteristics.

Heat and Mass Transfer in an Incompressible Turbulent Boundary Layer

1972 ◽  
Vol 94 (1) ◽  
pp. 23-28 ◽  
Author(s):  
E. Brundrett ◽  
W. B. Nicoll ◽  
A. B. Strong
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Mass Transfer ◽  
Porous Plate ◽  
Mixing Length ◽  
Two Dimensional ◽  
Transfer Data ◽  
Incompressible Turbulent Boundary Layer ◽  
Wide Range ◽  
Incompressible Boundary

The van Driest damped mixing length has been extended to account for the effects of mass transfer through a porous plate into a turbulent, two-dimensional incompressible boundary layer. The present mixing length is continuous from the wall through to the inner-law region of the flow, and although empirical, has been shown to predict wall shear stress and heat transfer data for a wide range of blowing rates.

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