In the absence of large-scale coherent structures, a widely used statistical theory of two-dimensional turbulence developed by Kraichnan, Leith, and Batchelor (KLB) predicts a power-law scaling for the energy, $E(k)\propto k^\alpha$ with an integral exponent $\alpha ={-3}$, in the inertial range associated with the direct cascade. A power-law scaling is also observed in the presence of coherent structures, but the scaling exponent becomes fractal and often differs substantially from the value predicted by the KLB theory. Here we present a dynamical theory that sheds new light on the relationship between the spatial and temporal structure of the large-scale flow and the scaling of small-scale structures representing filamentary vorticity. Specifically, we find hyperbolic regions of the large-scale flow to play a key role in the flux of enstrophy between scales. Small-scale vorticity in these regions can be described by dynamically self-similar solutions of the Euler equation, which explains the power-law scaling. Furthermore, we find that correlations between different hyperbolic regions are responsible for the emergence of fractal scaling exponents.

The nature and behaviour of the drag coefficient $C_D$ of irregularly shaped grains within a wide range of Reynolds numbers $Re$ is discussed. The morphology of the grains is controlled by their fractal description, and they differ in shape. Using computational fluid dynamics tools, the characteristics of the boundary layer at high $Re$ has been determined by applying the Reynolds-averaged Navier–Stokes turbulence model. Both grid resolution and mesh size dependence are validated with well-reported previous experimental results applied in flow around isolated smooth spheres. The drag coefficient for irregularly shaped grains is shown to be higher than that for spherical shapes, also showing a strong drop in its value at high $Re$. This drag crisis is reported at lower $Re$ compared to the smooth sphere, but higher critical $C_D$, demonstrating that the morphology of the particle accelerates this crisis. Furthermore, the dependence of $C_D$ on $Re$ in this type of geometry can be represented qualitatively by four defined zones: subcritical, critical, supercritical and transcritical. The orientational dependence for both particles with respect to the fluid flow is analysed, where our findings show an interesting oscillatory behaviour of $C_D$ as a function of the angle of incidence, fitting the results to a sine-squared interpolation, predicted for particles within the Stokes laminar regime ($Re\ll 1$) and for elongated/flattened spheroids up to $Re=2000$. A statistical analysis shows that this system satisfies a Weibullian behaviour of the drag coefficient when random azimuthal and polar rotation angles are considered.

Particle-size distribution (PSD) is a fundamental soil property usually reported as discrete clay, silt, and sand percentages. Models and methods to effectively generate a continuous PSD from such poor descriptions using another property would be extremely useful to predict and understand in fragmented distributions, which are ubiquitous in nature. Power laws for soil PSDs imply scale invariance (or selfsimilarity), a property which has proven useful in PSD description. This work is based on two novel ideas in modeling PSDs: (1) the concept of selfsimilarity in PSDs; and (2) mathematical tools to calculate fractal distributions for specific soil PSDs using few actual texture data. Based on these ideas, a random, multiplicative cascade model was developed that relies on a regularity of scale invariance called ‘log-selfsimilarity.’ The model allows the estimation of intermediate particle size values from common texture data. Using equivalent inputs, this new modeling approach was checked using soil data and shown to provide greatly improved results in comparison to the selfsimilar model for soil PSD data. The Kolmogorov-Smirnov D-statistic for the log-selfsimilar model was smaller than the selfsimilar model in 92.94% of cases. The average error was 0.74 times that of the selfsimilar model. The proposed method allows measurement of a heterogeneity index, H, defined using Hölder exponents, which facilitates quantitative characterization of soil textural classes. The average H value ranged from 0.381 for silt texture to 0.838 for sandy loam texture, with a variance of <0.034 for all textural classes. The index can also be used to distinguish textures within the same textural class. These results strongly suggest that the model and its parameters might be useful in estimating other soil physical properties and in developing new soil PSD pedotransfer functions. This modeling approach, along with its potential applications, might be extended to fine-grained mineral and material studies.

The presence of non-local interactions and intermittent signals in the homogeneous isotropic turbulence grant multi-point statistical functions a key role in formulating a new generation of large-eddy simulation (LES) models of higher fidelity. We establish a tempered fractional-order modelling framework for developing non-local LES subgrid-scale models, starting from the kinetic transport. We employ a tempered Lévy-stable distribution to represent the source of turbulent effects at the kinetic level, and we rigorously show that the corresponding turbulence closure term emerges as the tempered fractional Laplacian, $(\varDelta +\lambda )^{\alpha } (\cdot )$, for $\alpha \in (0,1)$, $\alpha \neq \frac {1}{2}$ and $\lambda >0$ in the filtered Navier–Stokes equations. Moreover, we prove the frame invariant properties of the proposed model, complying with the subgrid-scale stresses. To characterize the optimum values of model parameters and infer the enhanced efficiency of the tempered fractional subgrid-scale model, we develop a robust algorithm, involving two-point structure functions and conventional correlation coefficients. In an a priori statistical study, we evaluate the capabilities of the developed model in fulfilling the closed essential requirements, obtained for a weaker sense of the ideal LES model (Meneveau, Phys. Fluids, vol. 6, issue 2, 1994, pp. 815–833). Finally, the model undergoes the a posteriori analysis to ensure the numerical stability and pragmatic efficiency of the model.

Energy transfer in turbulent fluids is non-Gaussian. We quantify non-Gaussian energy transfer between the atmosphere and bodies of water using a turbulent diffusion operator coupled with temporally self-affine velocity distributions and a recursive integration method that produce multifractal measures. The measures serve as input to a system of moment field equations (derived from Navier–Stokes) that generate and track high-frequency gravity waves that propagate through the water surface (as a result of the air–water interactions). The dimension of the support of the air–water turbulence produced by our methods falls within the range of theory and observation, and correspondingly, hindcast statistical measures of the water-wave surface such as significant water-wave height and wave period are well correlated to observational buoy data. Further, our recursive integration method can be used by spectral resolving phase-averaged models to interpolate temporal wind data to smaller scales to capture the non-Gaussian behaviour of the air–water interaction.

We have conducted an extensive study of the scaling properties of small scale turbulence using both numerical and experimental data of a flow in the same von Kármán geometry. We have computed the wavelet structure functions, and the structure functions of the vortical part of the flow and of the local energy transfers. We find that the latter obey a generalized extended scaling, similar to that already observed for the wavelet structure functions. We compute the multi-fractal spectra of all the structure functions and show that they all coincide with each other, providing a local refined hypothesis. We find that both areas of strong vorticity and strong local energy transfer are highly intermittent and are correlated. For most cases, the location of local maximum of the energy transfer is shifted with respect to the location of local maximum of the vorticity. We, however, observe a much stronger correlation between vorticity and local energy transfer in the shear layer, that may be an indication of a self-similar quasi-singular structure that may dominate the scaling properties of large order structure functions.

We use well resolved numerical simulations with the lattice Boltzmann method to study Rayleigh–Bénard convection in cells with a fractal boundary in two dimensions for $Pr = 1$ and $Ra \in \left [10^7, 10^{10}\right ]$, where Pr and Ra are the Prandtl and Rayleigh numbers. The fractal boundaries are functions characterized by power spectral densities $S(k)$ that decay with wavenumber, $k$, as $S(k) \sim k^{p}$ ($p < 0$). The degree of roughness is quantified by the exponent $p$ with $p < -3$ for smooth (differentiable) surfaces and $-3 \le p < -1$ for rough surfaces with Hausdorff dimension $D_f=\frac {1}{2}(p+5)$. By computing the exponent $\beta$ using power law fits of $Nu \sim Ra^{\beta }$, where $Nu$ is the Nusselt number, we find that the heat transport scaling increases with roughness through the top two decades of $Ra \in \left [10^8, 10^{10}\right ]$. For $p$$= -3.0$, $-2.0$ and $-1.5$ we find $\beta = 0.288 \pm 0.005, 0.329 \pm 0.006$ and $0.352 \pm 0.011$, respectively. We also find that the Reynolds number, $Re$, scales as $Re \sim Ra^{\xi }$, where $\xi \approx 0.57$ over $Ra \in \left [10^7, 10^{10}\right ]$, for all $p$ used in the study. For a given value of $p$, the averaged $Nu$ and $Re$ are insensitive to the specific realization of the roughness.

Ramp–cliff patterns visible in scalar turbulent time series have long been suspected to enhance the fine-scale intermittency of scalar fluctuations compared to longitudinal velocity fluctuations. Here, we use the wavelet transform modulus maxima method to perform a multifractal analysis of air temperature time series collected at a pine forest canopy top for different atmospheric stability regimes. We show that the multifractal spectra exhibit a phase transition as the signature of the presence of strong singularities corresponding to sharp temperature drops (respectively jumps) bordering the so-called ramp (respectively inverted ramp) cliff patterns commonly observed in unstable (respectively stable) atmospheric conditions and previously suspected to contaminate and possibly enhance the internal intermittency of (scalar) temperature fluctuations. Under unstable (respectively stable) atmospheric conditions, these ‘cliff’ singularities are indeed found to be hierarchically distributed on a ‘Cantor-like’ set surrounded by singularities of weaker strength typical of intermittent temperature fluctuations observed in homogeneous and isotropic turbulence. Under near-neutral conditions, no such a phase transition is observed in the temperature multifractal spectra, which is a strong indication that the statistical contribution of the ‘cliffs’ is not important enough to account for the stronger intermittency of temperature fluctuations when compared to corresponding longitudinal velocity fluctuations.

Gas turbine combustors are susceptible to thermoacoustic instability, which manifests as large amplitude periodic oscillations in acoustic pressure and heat release rate. The transition from a stable operation characterized by combustion noise to thermoacoustic instability in turbulent combustors has been described as an emergence of order (periodicity) from chaos in the temporal dynamics. This emergence of order in the acoustic pressure oscillations corresponds to a loss of multifractality in the pressure signal. In this study, we investigate the spatiotemporal dynamics of a turbulent flame in a bluff-body stabilized combustor during the transition from combustion noise to thermoacoustic instability. During the occurrence of combustion noise, the flame wrinkles due to the presence of small-scale vortices in the turbulent flow. On the other hand, during thermoacoustic instability, large-scale coherent structures emerge periodically. These large-scale coherent structures roll up the wrinkled flame surface further and introduce additional complexity in the flame topology. We perform multifractal analysis on the flame contours detected from high-speed planar Mie scattering images of the reactive flow seeded with non-reactive tracer particles. We find that multifractality exists in the flame topology for all the dynamical states during the transition to thermoacoustic instability. We discuss the variation of multifractal parameters for the different states. We find that the multifractal spectrum oscillates periodically during the occurrence of thermoacoustic instability at the time scale of the acoustic pressure oscillations. The loss of multifractality in the temporal dynamics and the oscillation of the multifractal spectrum of the spatial dynamics go hand in hand.

Fractal features of the turbulent/non-turbulent interface (TNTI) in shock wave/turbulent boundary-layer interaction (SWBLI) flows are essential in understanding the physics of the SWBLI and the supersonic turbulent boundary layer, yet have received almost no attention previously. Accordingly, this study utilises a high spatiotemporal resolution visualisation technique, ice-cluster-based planar laser scattering (IC-PLS), to acquire the TNTI downstream of the reattachment in a SWBLI flow. Evolution of the fractal features of the TNTI in this SWBLI flow is analysed by comparing the parameters of the TNTI acquired in this study with those from a previous result (Zhuang et al.J. Fluid Mech., vol. 843, 2018a).

A compact coplanar waveguide (CPW)-fed circular polarization (CP)-antenna for new generation applications with dual bands filtering performance along with CP feature based on unit-cell semi-fractal is proposed in this paper. The CP-antenna privileges from semi-fractal radiator causes to have a miniaturized size. The stopped bands are designed to suppress the interference with present WLAN and ITU-R satellite systems. These properties are obtained by embedding semi-fractal unit-cell patterns stubs at the radiator and applying two rectangular-shaped slits inside CPW ground plane and a pair of grounded L-shaped strips. By introducing the first step of semi-fractal strips, and the mirrored defected ground surface structures, dual-band rejection functionality at WLAN (5–6 GHz) and ITU-R (7.725–8.5 GHz) are practically obtained. Besides that, semi-fractal strips results to two orthogonal modes stimulation on the radiator and CP attribute are obtained at WiMAX (3.1–3.7 GHz). CP-antenna presents omni-directional radiation H-plane patterns over the applicational frequency band. The CP-antennas size is 25 mm × 25 mm and fabricated on commercially available FR4-epoxy substrate with 1 mm thickness. Measured results illustrate that the proposed ultimate CP-antenna with miniaturized structure, efficient impedance tuning characteristics, and adequate radiation performances is the best choice for new generation of wireless communications.

The turbulent–non-turbulent interface (TNTI) of supersonic turbulent boundary layers is a fundamental but relatively unexplored physics problem. In this study, we present experimental results from fractal analysis on the TNTI of supersonic turbulent boundary layers, and test the applicability of the additive law for these flows. By applying the nanoparticle-tracer planar laser scattering (NPLS) technique in a supersonic wind tunnel, we obtain data covering nearly three decades in scale. The box-counting results indicate that the TNTI of supersonic turbulent boundary layers is a self-similar fractal with a fractal dimension of 2.31. By comparing data sets acquired from two orthogonal planes, we find that the scaling exponent does not depend on direction, consistent with the validity of the additive law for the TNTI of turbulent boundary layers in a scale range with the large-scale limit not exceeding approximately $0.05\unicode[STIX]{x1D6FF}$.

A characteristic feature of axisymmetric jets, and turbulent shear flows in general, is the entrainment of mass across the turbulent/non-turbulent interface (TNTI). The multi-scale nature of the TNTI surface area was recently observed to exhibit power-law scaling with a fractal dimension, $D_{f}$, between $D_{f}=2.3{-}2.4$, inferred from two-dimensional data, in both high Reynolds number boundary layers and the far field of axisymmetric jets. In this paper, we show that the fractal scaling previously observed in the far field of an axisymmetric jet is established at the end of the potential core. Simultaneous measurements of the velocity and scalar fields were obtained and coarse grain filtering was applied over two decades of scale separation, showing that $D_{f}$ evolves to ${\approx}2.35$ at $x/d=4.6$, which is similar to $D_{f}$ found in the far field between $x/d=40{-}60$. This is evidence that scale separation becomes sufficiently developed to achieve scale invariance of the TNTI surface area in the near field of the jet well before self-similarity is established. We also observe that the onset of this geometric scale invariance coincides with the onset of radial homogeneity shown by two-point velocity correlations. Finally, we present a simple theoretical basis for these results using an exact fractal construction based on the Koch curve and applying a coarse-grain filtering analysis.

In classical literature, blowout is described as loss of static stability of the combustion system whereas thermoacoustic instability is seen as loss of dynamic stability of the system. At blowout, the system transitions from a stable reacting state to a non-reacting state, indicating loss of static stability of the reaction. However, this simple description of stability margin is inadequate since recent studies have shown that combustors exhibit complex nonlinear behaviour prior to blowout. Recently, it was shown that combustion noise that characterizes the regime of stable operation is itself dynamically complex and exhibits multifractal characteristics. Researchers have already described the transition from combustion noise to combustion instability as a loss of multifractality. In this work, we provide a multifractal description for lean blowout in combustors with turbulent flow and thus introduce a unified framework within which both thermoacoustic instability and blowout can be described. Further, we introduce a method for predicting blowout based on the multifractal description of blowout.