Second-order Structure Function Research Articles

Effects of free-stream turbulence on rough surface turbulent boundary layers

Several effects of nearly isotropic free-stream turbulence in transitionally rough turbulent boundary layers are studied using data obtained from laser Doppler anemometry measurements. The free-stream turbulence is generated with the use of an active grid, resulting in free-stream turbulence levels of up to 6.2%. The rough surface is characterized by a roughness parameterk+≈ 53, and measurements are performed at Reynolds numbers of up toReθ= 11300. It is confirmed that the free-stream turbulence significantly alters the mean velocity deficit profiles in the outer region of the boundary layer. Consequently, the previously observed ability of the Zagarola & Smits (J. Fluid Mech., vol. 373, 1998, p. 33) velocity scaleU∞δ*/δ to collapse results from both smooth and rough surface boundary layers, no longer applies in this boundary layer subjected to high free-stream turbulence. In inner variables, the wake region is significantly reduced with increasing free-stream turbulence, leading to decreased mean velocity gradient and production of Reynolds stress components. The effects of free-stream turbulence are clearly identifiable and significant augmentation of the streamwise Reynolds stress profiles throughout the entire boundary layer are observed, all the way down to the inner region. In contrast, the Reynolds wall-normal and shear stress profiles increase due to free-stream turbulence only in the outer part of the boundary layer due to the blocking effect of the wall. As a consequence, there is a significant portion of the boundary layer in which the addition of nearly isotropic turbulence in the free-stream, results in significant increases in anisotropy of the turbulence. To quantify which turbulence length scales contribute to this trend, second-order structure functions are examined at various distances from the wall. Results show that the anisotropy created by adding nearly isotropic turbulence in the free-stream resides mostly in the larger scales of the flow. Furthermore, by analysing the streamwise Reynolds stress equation, it can be predicted that it is the wall-normal gradient of 〈u2v〉 term that is responsible for the increase in 〈u2〉 profiles throughout the boundary layer (i.e. an efficient turbulent transport of turbulence away from the wall). Furthermore, a noticeable difference between the triple correlations for smooth and rough surfaces exists in the inner region, but no significant differences are seen due to free-stream turbulence. In addition, the boundary layer parameters δ*/δ95,Handcfare also evaluated from the experimental data. The flow parameters δ*/δ95andHare found to increase due to roughness, but decrease due to free-stream turbulence, which has significance for flow control, particularly in delaying separation. Increases incfdue to high free-stream turbulence are also observed, associated with increased momentum flux towards the wall.

Effect of the Scalar Injection Mechanism on Passive Scalar Structure Functions in a Turbulent Flow

Higher-order passive scalar (temperature) structure functions are measured in the turbulent wake of a circular cylinder at a Taylor-microscale Reynolds number (Rlambda) of 370. The scalar is injected by two different means: (i) heating of the cylinder and (ii) use of a mandoline. Even though the second-order statistics (e.g., power spectra, second-order structure functions) of the scalar field are experimentally indistinguishable in the inertial and dissipative ranges, we observe notable differences in the inertial-range scaling exponents (xin) of the scalar structure functions at higher orders. The implication is therefore that the variations in previous estimates of xin may be attributable to differences in the scalar field initial conditions (and may not be deemed characteristic of a universal nature of the small-scale statistics of turbulent passive scalars).

A simple model for the streamwise fluctuations in the log-law region of a boundary layer

We build on the work of Davidson et al. [Phys. Fluids 18, 065112 (2006)] and propose an elementary model for the log-law region of a boundary layer. The model is remarkably simple, contains only one free parameter (which we set equal to unity) and, unlike Davidson et al., assumes very little about the shape of the boundary-layer eddies. The physical content of the model is simple: we assume that the two-point statistics of the streamwise velocity fluctuations know about the presence of the wall only to the extent that, over a range of eddy sizes, it imposes a kinetic energy scale proportional to the square of the shear velocity. Little else is assumed, other than classic Kolmogorov phenomenology for the small scales. Despite its naivety, and the minimal number of free parameters, the model is a good fit to experimental data for the k−1 law of the one-dimensional, longitudinal spectrum, Φuu(k), and also to Φuu(k) in the inertial range. It is also an excellent fit to experimental data for the real-space analog of the k−1 law; that is, the logarithmic law for the longitudinal structure function. In addition, the model predicts the cross-stream variation of the variance of the streamwise velocity fluctuations, ⟨ux2⟩, and, to within an additive constant, it too is a reasonable fit to the data. Our model differs from the classic analyses of Townsend [J. Fluid Mech. 165, 163 (1986)] and Perry et al. [J. Fluid Mech. 165, 163 (1986)] in three respects. First, we do not invoke the attached eddy hypothesis, which assumes that the vortical structures in the log-layer extend to the wall. Second, we predict the precise shape of Φuu(k) for all values of k and not just the k−1 region. Third, there are three adjustable parameters in the theory of Perry et al., whereas we have only one. It is instructive that a model which contains very little information about the morphology of the vorticity field, and has only one free parameter, is able to reproduce both the structural form for, and magnitude of, many features of the one-dimensional spectrum and the second-order structure function. This suggests that these two measures of the scale-by-scale energy distribution are insensitive to the detailed structure of the vorticity field. We argue that this is indeed the case.

SPECTRAL INDICES FOR MULTI-DIMENSIONAL INTERPLANETARY TURBULENCE AT 1 AU

We examine Advanced Composition Explorer and Helios 1 data in search of evidence for an anisotropic spectrum of interplanetary magnetic and velocity field fluctuations. Specifically, we focus on the power-law indices of the fluctuation spectra and associated second-order structure functions and ask whether the index varies systematically with the angle between the mean magnetic field and the wind velocity. We extend previous results to show convincingly that it does not. Several popular theories for magnetohydrodynamic turbulence predict a significant variation as part of the turbulent cascade dynamic. We offer some observations on why the predicted anisotropy is not present.

A Geometric Interpretation of the Second-Order Structure Function Arising in Turbulence

We primarily deal with homogeneous isotropic turbulence and use a closure model for the von Karman-Howarth equation to study several geometric properties of turbulent fluid dynamics. We focus our attention on the application of Riemannian geometry methods in turbulence. Some advantage of this approach consists in exploring the specific form of a closure model for the von Karman-Howarth equation that enables to equip a model manifold (a cylindrical domain in the correlation space) by a family of inner metrics (length scales of turbulent motion) which depends on time. We show that for large Reynolds numbers (in the limit of large Reynolds numbers) the radius of this manifold can be evaluated in terms of the second-order structure function and the correlation distance. This model manifold presents a shrinking cylindrical domain as time evolves. This result is derived by using a selfsimilar solution of the closure model for the von Karman-Howarth equation under consideration. We demonstrate that in the new variables the selfsimilar solution obtained coincides with the element of Beltrami surface (or pseudo-sphere): a canonical surface of the constant sectional curvature equals − 1.

Interaction of Upper-Tropospheric Turbulence and Gravity Waves as Obtained from Spectral and Structure Function Analyses

Abstract Spectral and structure function analyses of horizontal velocity fields observed in the upper troposphere and lower stratosphere during the Severe Clear Air Turbulence Collides with Air Traffic (SCATCAT) field program, conducted over the Pacific, were carried out in an effort to identify the scale interactions of turbulence and small-scale gravity waves. Because of the intermittent nature of turbulence, these analyses were conducted by clearly separating out the cases when turbulence did or did not occur in the data. In the presence of turbulence, transitional power spectra from k−2 to k−5/3 were found to be associated with gravity waves and turbulence, respectively. The second-order structure function analysis was able to translate these spectral slopes into r and r 2/3 scaling, consistent with the Monin and Yaglom conversion law, in physical space, which presented clearer pictures of scale interactions between turbulence and gravity waves. The third-order structure function analysis indicated the existence of a narrow region of inverse energy cascade from the scales of turbulence up to the gravity waves scales. This inverse energy cascade region was linked to the occurrence of Kelvin–Helmholtz instability and other wave-amplifying mechanisms, which were conjectured to lead to the breaking of small-scale gravity waves and the ensuing generation of turbulence. The multifractal analyses revealed further scale breaks between gravity waves and turbulence. The roughness and intermittent properties were also calculated for turbulence and gravity waves, respectively. Based on these properties, turbulence and gravity waves in a bifractal parameter space were mapped. In this way, their physical and statistical attributes were clearly manifested and understood.

Studies of Regular and Random Magnetic Fields in the ISM: Statistics of Polarization Vectors and the Chandrasekhar‐Fermi Technique

Polarimetry is extensively used as a tool to trace the interstellar magnetic field projected on the plane of sky. Moreover, it is also possible to estimate the magnetic field intensity from polarimetric maps based on the Chandrasekhar-Fermi method. In this work, we present results for turbulent, isothermal, three-dimensional simulations of sub/supersonic and sub/super-Alfvenic cases. With the cubes, assuming perfect grain alignment, we created synthetic polarimetric maps for different orientations of the mean magnetic field with respect to the line of sight (LOS). We show that the dispersion of the polarization angle depends on the angle of the mean magnetic field regarding the LOS and on the Alfvenic Mach number. However, the second-order structure function of the polarization angle follows the relation SF ∝ lα, α being dependent exclusively on the Alfvenic Mach number. The results show an anticorrelation between the polarization degree and the column density, with exponent γ ~ − 0.5, in agreement with observations, which is explained by the increase in the dispersion of the polarization angle along the LOS within denser regions. However, this effect was observed exclusively on supersonic, but sub-Alfvenic, simulations. For the super-Alfvenic, and the subsonic model, the polarization degree showed to be independent of the column density. Our major quantitative result is a generalized equation for the CF method, which allowed us to determine the magnetic field strength from the polarization maps with errors <20%. We also account for the role of observational resolution on the CF method.

On the deficiency of even-order structure functions as inertial-range diagnostics

In the limit of vanishing viscosity, ν→0, Kolmogorov's two-thirds, 〈(Δυ)2〉~ε2/3r2/3, and five-thirds, E~ε2/3k−5/3, laws are formally equivalent. (Here 〈(Δυ)2〉 is the second-order structure function, ε the dissipation rate, r the separation in physical space, E the three-dimensional energy spectrum, and k the wavenumber.) However, for the Reynolds numbers encountered in terrestrial experiments, or numerical simulations, it is invariably easier to observe the five-thirds law. We ask why this should be. To this end, we create artificial fields of isotropic turbulence composed of a random sea of Gaussian eddies whose size and energy distribution can be controlled. We choose the energy of eddies of scale, s, to vary as s2/3, in accordance with Kolmogorov's 1941 law, and vary the range of scales, γ=smax/smin, in any one realization from γ=25 to γ=800. This is equivalent to varying the Reynolds number in an experiment from Rλ=60 to Rλ=600. We find that, while there is some evidence of a five-thirds law for γ&gt;50; (Rλ&gt;100), the two-thirds law only starts to become apparent when γ approaches 200 (Rλ~240). The reason for this discrepancy is that the second-order structure function is a poor filter, mixing information about energy and enstrophy, and from scales larger and smaller than r. In particular, in the inertial range, 〈(Δυ)2〉 takes the form of a mixed power law, a1 + a2r2 + a3r2/3, where a2r2 tracks the variation in enstrophy and a3r2/3 the variation in energy. These findings are shown to be consistent with experimental data where the ‘pollution’ of the r2/3 law by the enstrophy contribution, a2r2, is clearly evident. We show that higher-order structure functions (of even order) suffer from a similar deficiency.

Calculation of scalar structure functions from a vortex model of turbulent passive scalar transport

A Saffman and Pullin [Phys. Fluids 8, 3072 (1996)] type vortex model for passive scalar structure functions is formulated. The intermittent turbulent fine-scale dynamics in the model is represented by numerical solutions of the advection-diffusion and Navier–Stokes equations in the form of axially strained vortex-scalar structures. The use of these structures is motivated by Pullin and Lundgren’s [Phys. Fluids 13, 2553 (2001)] asymptotic strained spiral vortex model of turbulent passive scalar transport. Ensemble-averaged scalar structure functions, of even orders 2–10, are calculated from a range of vortex-scalar structures using Monte Carlo integration. For axisymmetric strained scalar fields, acceptable agreement of the second-order structure function with experimental data reported by Antonia and Van Atta [J. Fluid Mech. 84, 561 (1978)] is obtained. Structure functions are also calculated for a range of passive scalar spiral structures. These are generated by the winding of single and double scalar patches in single strained vortex patches and in merging strained vortices. Power-law scaling of the second- and higher-order structure functions is obtained from cases involving the winding of single scalar patches in an axisymmetric strained vortex patch. The second-order scaling exponents from these cases are in reasonable agreement with Kolmogorov–Oboukhov–Corrsin scaling and the experimental results of Antonia et al. [Phys. Rev. A 30, 2704 (1984)] and Gylfason and Warhaft [Phys. Fluids 16, 4012 (2004)]. However, the higher-order scaling exponents from these cases fall below theoretical predictions and experimental results. Higher-order moments are sensitive to the composition of the vortex-scalar structures, and various improvements are suggested that could enhance the performance of the model. The present approach is promising, and it is the first demonstration that a vortex model using simplified Navier–Stokes dynamics can produce some scalar structure functions that compare favorably with experimental observations.

Should structure functions be used to estimate power laws in turbulence? A comparative study

Second-order structure functions are widely used to characterize turbulence in the inertial range because they are simple to estimate, particularly in comparison to spectral density functions and wavelet variances. Structure function estimators, however, are highly autocorrelated and, as a result, no suitable theory has been established to provide confidence intervals for turbulence parameters when determined via regression fits in log/log space. Monte Carlo simulations were performed to compare the performance of structure function estimators of turbulence parameters with corresponding multitaper spectral and wavelet variance estimators. The simulations indicate that these latter estimators have smaller variances than estimators based upon the structure function. In contrast to structure function estimators, the statistical properties of the multitaper spectral and wavelet variance estimators allow for the construction of confidence intervals for turbulence parameters. The Monte Carlo simulations also confirm the validity of the statistical theory behind the multitaper spectral and wavelet variance estimators. The strengths and weaknesses of the various estimators are further illustrated by analyzing an atmospheric temperature time series.

Response of mean turbulent energy dissipation rate and spectra to concentrated wall suction

The response of mean turbulent energy dissipation rate and spectra to concentrated suction applied through a porous wall strip has been quantified. Both suction and no suction data of the spectra collapsed reasonably well for Kolmogorov normalised wavenumber k1* > 0.2. Similar results were also observed for second-order structure functions (not shown) for Kolmogorov normalised radius r* 20, respectively. The departure at the larger scales with collapse at the small scales suggests that suction induce a change in the small-scale motion. This is also reflected in the alteration of mean turbulent energy dissipation rate and Taylor microscale Reynolds number. This change is a result of the weakening of the large-scale structures. The effect is increased as the suction rate is increased.

A conditionally cubic-Gaussian stochastic Lagrangian model for acceleration in isotropic turbulence

The modelling of fluid-particle acceleration in homogeneous isotropic turbulence in terms of stochastic models for the Lagrangian velocity, acceleration and a dissipation rate variable is considered. The basis for the Reynolds model (A. M. Reynolds, Phys. Rev. Lett. vol. 91, 2003, 084503) is reviewed and examined by reference to direct numerical simulations (DNS) of isotropic turbulence at Taylor-scale Reynolds number (Rλ) up to about 650. In particular, we show DNS data that support stochastic modelling of the logarithm of pseudo-dissipation as an Ornstein–Uhlenbeck process and reveal non-Gaussianity of the acceleration conditioned on fluctuations of the pseudo-dissipation rate. The DNS data are used to construct a new stochastic model that is exactly consistent with Gaussian velocity and conditionally cubic-Gaussian acceleration statistics. This model captures the effects of small-scale intermittency on acceleration and the conditional dependence of acceleration on pseudo-dissipation (which differs from that predicted by the refined Kolmogorov hypotheses). Non-Gaussianity of the conditionally standardized acceleration probability density function (PDF) is accounted for in terms of model nonlinearity. The large-time behaviour of the new model is that of a velocity-dissipation model that can be matched with DNS data for conditional second-order Lagrangian velocity structure functions. As a result, the diffusion coefficient for the new model incorporates two-time information and its Reynolds-number dependence as observed in DNS. The resulting model predictions for conditional and unconditional velocity autocorrelations and time scales are shown to be in very good agreement with DNS.

Spatial resolution of PIV for the measurement of turbulence

Recent technological advancements have made the use of particle image velocimetry (PIV) more widespread for studying turbulent flows over a wide range of scales. Although PIV does not threaten to make obsolete more mature techniques, such as hot-wire anemometry (HWA), it is justifiably becoming an increasingly important tool for turbulence research. This paper assesses the ability of PIV to resolve all relevant scales in a classical turbulent flow, namely grid turbulence, via a comparison with theoretical predictions as well as HWA measurements. Particular attention is given to the statistical convergence of mean turbulent quantities and the spatial resolution of PIV. An analytical method is developed to quantify and correct for the effect of the finite spatial resolution of PIV measurements. While the present uncorrected PIV results largely underestimate the mean turbulent kinetic energy and energy dissipation rate, the corrected measurements agree to a close approximation with the HWA data. The transport equation for the second-order structure function in grid turbulence is used to establish the range of scales affected by the limited resolution. The results show that PIV, due to the geometry of its sensing domain, must meet slightly more stringent requirements in terms of resolution, compared with HWA, in order to provide reliable measurements in turbulence.

Examination of anisotropy of the small-scale motion in a perturbed low Reynolds number turbulent boundary layer

Hot wire measurements have been made in a turbulent boundary layer subjected to concentrated suction applied through a porous wall strip in order, to examine the influence of suction on the anisotropy of the small-scale motion. Compensated energy spectra and normalised turbulent energy structure function for Kolmogorov normalised wavenumber k 1 ∗ < 0.2 and Kolmogorov normalised radius r ∗ > 20 depart from those for suction rate σ = 0, reflecting a modification in the energy level. This change is a result of the alteration in the large-scale motion. The effect is increased as the suction rate is increased. The degree of anisotropy as measured by the ratio of calculated second-order structure function to the measured second-order structure function (〈( δv c) 2〉/〈( δv m) 2〉) is increased as σ is increased albeit marginally as distance from the wall normalised by boundary layer thickness ( y/ δ) is increased. The overall result suggests that suction alter the pseudo-equilibrium and the anisotropy of the small-scale motion.

Self-similar two-particle separation model

We present a new stochastic model for relative two-particle separation in turbulence. Inspired by material line stretching, we suggest that a similar process also occurs beyond the viscous range, with time scaling according to the longitudinal second-order structure function S2(r), e.g.; in the inertial range as ε−1∕3r2∕3. Particle separation is modeled as a Gaussian process without invoking information of Eulerian acceleration statistics or of precise shapes of Eulerian velocity distribution functions. The time scale is a function of S2(r) and thus of the Lagrangian evolving separation. The model predictions agree with numerical and experimental results for various initial particle separations. We present model results for fixed time and fixed scale statistics. We find that for the Richardson-Obukhov law, i.e., ⟨r(t)2⟩=gεt3, to hold and to also be observed in experiments, high Reynolds numbers are necessary, i.e., Reλ&amp;gt;O(1000), and the integral scale needs to be large compared to initial separation, i.e., L∕r0&amp;gt;30 and d∕L&amp;gt;3 need to be fulfilled, where d is the size of the field of view. Removing the constraint of finite inertial range, the model is used to explore separation dynamics in the asymptotic regime. As Reλ→∞, the distance neighbor function takes on a constant shape, almost as predicted by the Richardson diffusion equation. For the Richardson constant we obtain that g→0.95 as Reλ→∞. This asymptotic limit is reached at Reλ&amp;gt;1000. For the Richardson constant g, the model predicts a ratio of gb∕gf≈1.9 between backwards and forwards dispersion.

A refined interpretation of the logarithmic structure function law in wall layer turbulence

It has been shown recently that the real-space equivalent of the k−1 law for near-wall turbulence is a logarithmic law for the second-order longitudinal structure function, ⟨(Δu)2⟩(r)=u*2(A+Bln(r∕y)). Here y is the distance from the wall, u* is the shear velocity, A and B are coefficients of order unity and r is measured in the streamwise direction. In this paper we provide theoretical arguments to suggest that, in the limit of large Reynolds number, B is a universal constant while A is of the form A=A′−Bln(P∕ϵ), where A′ is a universal constant and P and ϵ are the rates of production and dissipation of energy, respectively. Hence A is a weak, universal, function of y. Two independent sets of data are examined and it is shown that, to within experimental error, our predictions are consistent with the data.

Small-scale anisotropy in Lagrangian turbulence

We report measurements of the second-order Lagrangian structure function and the Lagrangian velocity spectrum in an intensely turbulent laboratory flow. We find that the asymmetries of the large-scale flow are reflected in the small-scale statistics. In addition, we present new measurements of the Lagrangian structure function scaling constant C0, which is of central importance to stochastic turbulence models as well as to the understanding of turbulent pair dispersion and scalar mixing. The scaling of C0 with the turbulence level is also investigated, and found to be in agreement with an existing model.

Rough-pipe flows and the existence of fully developed turbulence

It is widely believed that at high Reynolds number (Re) all turbulent flows approach a limiting state of “fully developed turbulence" in which the statistics of the velocity fluctuations are independent of Re. Nevertheless, direct measurements of the velocity fluctuations have failed to yield firm empirical evidence that even the second-order structure function becomes independent of Re at high Re, let alone structure functions of higher order. Here we relate the friction coefficient (f) of rough-pipe flows to the second-order structure function. Then we show that in light of experimental measurements of f our results yield unequivocal evidence that the second-order structure function becomes independent of Re at high Re, compatible with the existence of fully developed turbulence.

Approach to the 4/5 law in homogeneous isotropic turbulence

Kolmogorov's similarity hypotheses and his 4/5 law are valid at very large Reynolds numbers. For flows encountered in the laboratory, the effect of a finite Reynolds number and of the non-stationarity or inhomogeneity associated with the large scales can affect the behaviour of the scales in the inertial range significantly. This paper focuses on the source of inhomogeneity in two types of flows, those dominated mainly by a decay of energy in the streamwise direction and those which are forced, through a continuous injection of energy at large scales. Results based on a parameterization of the second-order velocity structure function indicate that the normalized third-order structure function approaches 4/5 much more rapidly for forced than for decaying turbulence. This trend is supported by grid turbulence measurements and numerical data in a periodic box.

Identifying Turbulent Energy Distributions in Real, Rather than Fourier, Space

It has been suggested that the equilibrium-range properties of high-Reynolds number turbulence are more readily observed in spectral space, using E(k) or T(k), than in real space, using second- or third-order structure functions. For example, the -5/3 law is usually easier to see in experimental data than the equivalent 2/3 law. We argue that this is not an implicit feature of a real-space description of turbulence. Rather, it is because the second-order structure function mixes small and large-scale information. To remedy this problem we adopt a real-space function, the signature function, which plays the role of an energy density, somewhat analogous to E(k). In this Letter we determine the form of the signature function in a variety of turbulent flows. We find that dissipation-range phenomena, such as the so-called bottleneck effect, are evident in the signature function, while absent in the structure function.