Third-order structure function in the logarithmic layer of boundary-layer turbulence

It is well known that subgrid models such as Smagorinsky's cannot be used for the spatially growing simulation of the transition to turbulence of flat-plate boundary layers, unless large-amplitude perturbations are introduced at the upstream boundary: they are over-dissipative, and the flow simulated remains laminar. This is also the case for the structure-function model (SF) of Métais & Lesieur (1992). In the present paper we present a sequel to this model, the filtered-structure-function (FSF) model. It consists of removing the large-scale fluctuations of the field before computing its second-order structure function. Analytical arguments confirm the superiority of the FSF model over the SF model for large-eddy simulations of weakly unstable transitional flows. The FSF model is therefore used for the simulation of a quasi-incompressible (M∞ = 0.5) boundary layer developing spatially over an adiabatic flat plate, with a low level of upstream forcing. With the minimal resolution 650 × 32 × 20 grid points covering a range of streamwise Reynolds numbers Rex1 ε [3.4 × 105, 1.1 × 106], transition is obtained for 80 hours of time-processing on a CRAY 2 (whereas DNS of the whole transition takes about ten times longer). Statistics of the LES are found to be in acceptable agreement with experiments and empirical laws, in the laminar, transitional and turbulent parts of the domain. The dynamics of low-pressure and high-vorticity distributions is examined during transition, with particular emphasis on the neighbourhood of the critical layer (defined here as the height of the fluid travelling at a speed equal to the phase speed of the incoming Tollmien–Schlichting waves). Evidence is given that a subharmonic-type secondary instability grows, followed by a purely spanwise (i.e. time-independent) mode which yields peak-and-valley splitting and transition to turbulence. In the turbulent region, flow visualizations and local instantaneous profiles are provided. They confirm the presence of low- and high-speed streaks at the wall, weak hairpins stretched by the flow and bursting events. It is found that most of the vorticity is produced in the spanwise direction, at the wall, below the high-speed streaks. Isosurfaces of eddy viscosity confirm that the FSF model does not perturb transition much, and acts mostly in the vicinity of the hairpins.

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A note on Kolmogorov's third-order structure-function law, the local isotropy hypothesis and the pressure–velocity correlation

Journal of Fluid Mechanics ◽

10.1017/s0022112096008348 ◽

1996 ◽

Vol 326 ◽

pp. 343-356 ◽

Cited By ~ 28

Author(s):

Erik Lindborg

Keyword(s):

Structure Function ◽

Strain Tensor ◽

Inertial Range ◽

Order Structure ◽

Two Dimensional ◽

Mean Flow ◽

Velocity Correlation ◽

Third Order ◽

The Third ◽

Point Pressure

We show that Kolmogorov's (1941b) inertial-range law for the third-order structure function can be derived from a dynamical equation including pressure terms and mean flow gradient terms. A new inertial-range law, relating the two-point pressure–velocity correlation to the single-point pressure–strain tensor, is also derived. This law shows that the two-point pressure–velocity correlation, just like the third-order structure function, grows linearly with the separation distance in the inertial range. The physical meaning of both this law and Kolmogorov's law is illustrated by a Fourier analysis. An inertial-range law is also derived for the third-order velocity–enstrophy structure function of two-dimensional turbulence. It is suggested that the second-order vorticity structure function of two-dimensional turbulence is constant and scales with$\epsilon ^{2/3}_\omega$in the enstrophy inertial range, εωbeing the enstrophy dissipation. Owing to the constancy of this law, it does not imply a Fourier-space inertial-range law, and therefore it is not equivalent to thek−1law for the enstrophy spectrum, suggested by Kraichnan (1967) and Batchelor (1969).

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Some statistical properties of small scale turbulence in an atmospheric boundary layer

Journal of Fluid Mechanics ◽

10.1017/s002211207000054x ◽

1970 ◽

Vol 41 (1) ◽

pp. 141-152 ◽

Cited By ~ 48

Author(s):

R. W. Stewart ◽

J. R. Wilson ◽

R. W. Burling

Keyword(s):

Boundary Layer ◽

Lognormal Distribution ◽

A Priori ◽

Inertial Subrange ◽

Small Scale ◽

Order Structure ◽

Third Order ◽

The Difference ◽

Scale Turbulence ◽

Derivatives Of

Derivatives of velocity signals obtained in a turbulent boundary layer are examined for correspondence to the lognormal distribution. It is found that there is rough agreement but that unlikely events at high values are much less common in the observed fields than would be inferred from the lognormal distribution. The actual distributions correspond more to those obtained from a random walk with a limited number of steps, so the difference between these distributions and the lognormal may be related to the fact that the Reynolds number is finite.The third-order structure function is examined, and found to be roughly consistent with the existence of an inertial subrange of a Kolmogoroff equilibrium reacute;gime over a range of scale which is a priori reasonable but which is far less extensive than the $-\frac{5}{3}$ region of the spectrum.

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Structure functions of turbulence in the atmospheric boundary layer over the ocean

Journal of Fluid Mechanics ◽

10.1017/s002211207000174x ◽

1970 ◽

Vol 44 (1) ◽

pp. 145-159 ◽

Cited By ~ 133

Author(s):

C. W. Van Atta ◽

W. Y. Chen

Keyword(s):

Boundary Layer ◽

Atmospheric Boundary Layer ◽

Fourth Order ◽

Structure Functions ◽

Inertial Subrange ◽

Order Structure ◽

Third Order ◽

Wide Range ◽

Theoretical Predictions ◽

Order Quantities

Structure functions of turbulent velocity fluctuations up to fourth order have been measured at several heights in the atmospheric boundary layer over the open ocean, and the results are compared with theoretical predictions for separations in the inertial subrange. The behaviour of second- and third-order quantities shows substantial agreement with the predictions of Kolmogorov's original theory over a wide range of separations, but the results of a recent modification of the theory, attempting to account for intermittency in the local dissipation rate, are also consistent with the data over somewhat shorter separation intervals. The behaviour of the measured fourth-order structure function disagrees with that predicted from Kolmogorov's original work, but good agreement is found with the results of the modified theory.

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Shallow water wave turbulence

Journal of Fluid Mechanics ◽

10.1017/jfm.2019.375 ◽

2019 ◽

Vol 874 ◽

pp. 1169-1196 ◽

Cited By ~ 1

Author(s):

Pierre Augier ◽

Ashwin Vishnu Mohanan ◽

Erik Lindborg

Keyword(s):

Reynolds Number ◽

Structure Function ◽

Shallow Water ◽

Water Wave ◽

Structure Functions ◽

Wave Turbulence ◽

Order Structure ◽

Third Order ◽

Shallow Water Wave ◽

The Mean

The dynamics of irrotational shallow water wave turbulence forced at large scales and dissipated at small scales is investigated. First, we derive the shallow water analogue of the ‘four-fifths law’ of Kolmogorov turbulence for a third-order structure function involving velocity and displacement increments. Using this relation and assuming that the flow is dominated by shocks, we develop a simple model predicting that the shock amplitude scales as $(\unicode[STIX]{x1D716}d)^{1/3}$, where $\unicode[STIX]{x1D716}$ is the mean dissipation rate and $d$ the mean distance between the shocks, and that the $p$th-order displacement and velocity structure functions scale as $(\unicode[STIX]{x1D716}d)^{p/3}r/d$, where $r$ is the separation. Then we carry out a series of forced simulations with resolutions up to $7680^{2}$, varying the Froude number, $F_{f}=(\unicode[STIX]{x1D716}L_{f})^{1/3}/c$, where $L_{f}$ is the forcing length scale and $c$ is the wave speed. In all simulations a stationary state is reached in which there is a constant spectral energy flux and equipartition between kinetic and potential energy in the constant flux range. The third-order structure function relation is satisfied with a high degree of accuracy. Mean energy is found to scale approximately as $E\sim \sqrt{\unicode[STIX]{x1D716}L_{f}c}$, and is also dependent on resolution, indicating that shallow water wave turbulence does not fit into the paradigm of a Richardson–Kolmogorov cascade. In all simulations shocks develop, displayed as long thin bands of negative divergence in flow visualisations. The mean distance between the shocks is found to scale as $d\sim F_{f}^{1/2}L_{f}$. Structure functions of second and higher order are found to scale in good agreement with the model. We conclude that in the weak limit, $F_{f}\rightarrow 0$, shocks will become denser and weaker and finally disappear for a finite Reynolds number. On the other hand, for a given $F_{f}$, no matter how small, shocks will prevail if the Reynolds number is sufficiently large.

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Applicability of Kolmogorov's and Monin's equations of turbulence

Journal of Fluid Mechanics ◽

10.1017/s0022112097007362 ◽

1997 ◽

Vol 353 ◽

pp. 67-81 ◽

Cited By ~ 55

Author(s):

REGINALD J. HILL

Keyword(s):

Structure Function ◽

Structure Functions ◽

Inertial Range ◽

Homogeneous Turbulence ◽

Order Structure ◽

Third Order ◽

Local Isotropy ◽

The Third ◽

Moderate Reynolds Number ◽

Wide Range

The equation relating second- and third-order velocity structure functions was presented by Kolmogorov; Monin attempted to derive that equation on the basis of local isotropy. Recently, concerns have been raised to the effect that Kolmogorov's equation and an ancillary incompressibility condition governing the third-order structure function were proven only on the restrictive basis of isotropy and that the statistic involving pressure that appears in the derivation of Kolmogorov's equation might not vanish on the basis of local isotropy. These concerns are resolved. In so doing, results are obtained for the second- and third-order statistics on the basis of local homogeneity without use of local isotropy. These results are applicable to future studies of the approach toward local isotropy. Accuracy of Kolmogorov's equation is shown to be more sensitive to anisotropy of the third-order structure function than to anisotropy of the second-order structure function. Kolmogorov's 4/5 law for the inertial range of the third-order structure function is obtained without use of the incompressibility conditions on the second- and third-order structure functions. A generalization of Kolmogorov's 4/5 law, which applies to the inertial range of locally homogeneous turbulence at very large Reynolds numbers, is shown to also apply to the energy-containing range for the more restrictive case of stationary, homogeneous turbulence. The variety of derivations of Kolmogorov's and Monin's equations leads to a wide range of applicability to experimental conditions, including, in some cases, turbulence of moderate Reynolds number.

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Third-order structure function relations for quasi-geostrophic turbulence

Journal of Fluid Mechanics ◽

10.1017/s0022112006003697 ◽

2007 ◽

Vol 572 ◽

pp. 255-260 ◽

Cited By ~ 20

Author(s):

ERIK LINDBORG

Keyword(s):

Structure Function ◽

Three Dimensional ◽

Structure Functions ◽

Order Structure ◽

Third Order ◽

Longitudinal Structure ◽

The Third ◽

Inverse Cascade ◽

Geostrophic Turbulence ◽

Potential Enstrophy

We derive two third-order structure function relations for quasi-geostrophic turbulence, one for the forward cascade of potential enstrophy and one for the inverse cascade of energy. These relations are the counterparts of Kolmovorov's (1941) four-fifths law for the third-order longitudinal structure functions of three-dimensional turbulence.

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Exact second-order structure-function relationships

Journal of Fluid Mechanics ◽

10.1017/s0022112002001696 ◽

2002 ◽

Vol 468 ◽

pp. 317-326 ◽

Cited By ~ 78

Author(s):

REGINALD J. HILL

Keyword(s):

Structure Function ◽

Scaling Laws ◽

Velocity Structure ◽

Structure Functions ◽

Energy Dissipation Rate ◽

Second Order ◽

Order Structure ◽

Third Order ◽

Navier Stokes Equation ◽

Local Isotropy

Equations that follow from the Navier–Stokes equation and incompressibility but with no other approximations are ‘exact’. Exact equations relating second- and third- order structure functions are studied, as is an exact incompressibility condition on the second-order velocity structure function. Opportunities for investigations using these equations are discussed. Precisely defined averaging operations are required to obtain exact averaged equations. Ensemble, temporal and spatial averages are all considered because they produce different statistical equations and because they apply to theoretical purposes, experiment and numerical simulation of turbulence. Particularly simple exact equations are obtained for the following cases: (i) the trace of the structure functions, (ii) DNS that has periodic boundary conditions, and (iii) an average over a sphere in r-space. Case (iii) introduces the average over orientations of r into the structure-function equations. The energy dissipation rate ε appears in the exact trace equation without averaging, whereas in previous formulations ε appears after averaging and use of local isotropy. The trace mitigates the effect of anisotropy in the equations, thereby revealing that the trace of the third-order structure function is expected to be superior for quantifying asymptotic scaling laws. The orientation average has the same property.

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Impacts of Convergence on Structure Functions from Surface Drifters in the Gulf of Mexico

Journal of Physical Oceanography ◽

10.1175/jpo-d-18-0029.1 ◽

2019 ◽

Vol 49 (3) ◽

pp. 675-690 ◽

Cited By ~ 11

Author(s):

Jenna Pearson ◽

Baylor Fox-Kemper ◽

Roy Barkan ◽

Jun Choi ◽

Annalisa Bracco ◽

...

Keyword(s):

Gulf Of Mexico ◽

Structure Function ◽

Structure Functions ◽

Second Order ◽

Order Structure ◽

Third Order ◽

Second Order Statistics ◽

Surface Drifter ◽

Sign Change ◽

Surface Drifters

AbstractThere are limitations in approximating Eulerian statistics from surface drifters, due to biases from surface convergences. By contrasting second- and third-order Eulerian and surface drifter structure functions obtained from a model of the Gulf of Mexico, the consequences of the semi-Lagrangian nature of observations during the summer Grand Lagrangian Deployment (GLAD) and winter Lagrangian Submesoscale Experiment (LASER) are estimated. By varying launch pattern and location, the robustness and sensitivity of these statistics are evaluated. Over scales less than 10 km, second-order structure functions of surface drifters consistently have shallower slopes (~r2/3) than Eulerian statistics (~r), suggesting that surface drifter structure functions differ systematically and do not reproduce the scalings of the Eulerian fields. Medians of Eulerian and cluster release second-order statistics are also significantly different across all scales. Synthetic cluster release statistics depend on launch location and weakly on launch pattern. The observations suggest little seasonal difference in the second-order statistics, but the LASER third-order structure function shows a sign change around 1 km, while GLAD and the synthetic cluster releases show a third-order structure function sign change around 10 km. Further, synthetic surface drifter cluster releases (and therefore likely the GLAD observations) show robust biases in the negative third-order structure functions, which may lead to significant overestimation of the spectral energy flux and underestimation of the transition scale to a forward energy cascade. The Helmholtz decomposition, and curl and divergence statistics, of Eulerian and cluster releases differ, particularly on scales less than 10 km, in agreement with observations of drifters preferentially sampling convergences in coherent structures.

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