Fractal Iteration Number Research Articles

Solid-liquid mixing characteristics in a fractal cut impeller stirred reactor with dense solid loading

The fractal cut impeller (FC impeller) based on fractal iteration of “H” pattern cuts on the impeller blades was introduced in the solid-liquid system with dense solid loading. The solid-liquid mixing characteristics in a stirred tank with pitched blade impeller (PB impeller) and FC impeller were comparatively investigated with experiment and numerical simulation. Results showed that FC impeller exhibited lower power consumption (P) and power number (Np) compared to PB impeller at the same impeller speed. It could be also observed that increasing the fractal iteration number of FC impeller led to a further decrease in P and Np. Meanwhile, FC impeller could reduce the size of impeller tailing vortex, enhance turbulent intensity and turbulent kinetic energy dissipation rate, and improve solid particle suspension quality compared to PB impeller under the constant power consumption, and an increase in fractal iteration number of FC impeller resulted in higher solid particle suspension efficiency. In addition, it was also found that the fractal arrangement of cuts on the impeller blades could increase turbulent intensity, enhance turbulent kinetic energy dissipation rate and facilitate an improvement in solid particle suspension quality as compared to a whole cut on the impeller blades with equal cut area.

Space-filling single square and square fractal grids induced turbulence: Reynolds stress model parameters-optimization

The employment of grid-induced turbulent flow structures as a passive means of augmenting solid-fluid heat transfer is receiving considerable attention, especially in HVAC applications. However, there presently exists a gap in simulation approaches, where industry-accessible CFD packages are unable to accurately express crucial second-order turbulent flow statistics generated by space-filling single square grids (SSGs) and square fractal grids (SFGs). The present study reports a successful application of a revised Reynolds Stress Model (RSM), which accurately replicates the streamwise distributions of centerline mean flow velocity and turbulence intensity in the lee of one SSG (operating under three different flow Reynolds number ReDh) and five geometrically different SFG test cases after undergoing Nelder-Mead downhill simplex optimization of key RSM kernel parameters. The optimized RSM presents a disagreement of, at worst, 4.30% and 9.98%, respectively against experimental hot-wire anemometry measurements of first-order and second-order statistics, and is the first known instance of the RSM being validated for turbulence intensity predictions of SSG- and SFG-induced turbulence. Examination of RSM parameters reveals that the pre-factors for the rates of turbulence dissipation production and destruction (C1,ε and C2,ε) hold greatest effect on simulation accuracy, with additional optimization of the turbulent viscosity pre-factor (Cμ) required for SFG cases. This is attributed to the effect of enhanced turbulent transport due to the cascading and multiscale nature of SFG turbulence, which insofar could not be replicated by Reynolds-Averaged Navier-Stokes (RANS) models. The values of optimized C1,ε, C2,ε, and Cμ range between 1.057 to 1.697, 2.226 to 2.556, and 0.17 to 2.27, respectively. This leads to the largest deviation of −26.6%, 33.1%, and 200% for C1,ε, C2,ε, and Cμ, respectively, when compared to their corresponding default values. With regards to the sensitivity of this parameter set on grid design, it is shown that the grid's fractal iteration number N and thickness ratio tr have greatest influence on the variation of C1,ε, C2,ε, and Cμ, while the effect of ReDh is insignificant. Overall, this study presents an alternative approach to capture the anisotropic and inhomogeneous nature of SSG- and SFG-induced turbulence for industrial heat-transfer applications via an accessible RANS package, which was previously constrained to expensive DNS and LES studies.

Sound-scattering properties of Sierpinski triangle fractal structures in the near field

This paper proposes a new design of an acoustic diffuser based on the construction rules of the Sierpinski triangle in order to broaden the effective diffusion frequency range. The diffuser is made of triangular blocks of different sizes attached to a plane surface. The effects of the number of fractal iterations, the height of triangular blocks, and arrangements of the blocks on the normal-incidence diffusion coefficients in the near field are examined through numerical simulations based on the boundary element method (BEM) in the frequency range of 100 Hz – 5 kHz. Furthermore, measurement results will be presented to validate the diffusion performance presented by the numerical simulations. The diffusion performance of a conventional quadratic residue diffuser (QRD) is compared to confirm the advantage of the designed diffuser for broadening the effective frequency range. It shows that the fractal patterns with various sizes of blocks improve diffusion performance compared to the conventional QRD of the same size, especially in the mid-low frequency range below 1 kHz.

Numerical investigation on mixing characteristics of floating and sinking particles in a stirred tank with fractal impellers

The hydrodynamics of floating and sinking particles mixing process in a stirred tank with four pitched-blade impellers and fractal impellers were investigated employing computational fluid dynamics (CFD) simulation. Axial solid concentration profile, solid concentration distribution, power consumption, trailing vortex, cloud height, water integrated velocity, just suspension speed (Njs) and just drawdown speed (Njd) in the stirred tank were predicted. Results showed that the mixing quality of floating and sinking particles increased with the increase of impeller speed. Smaller density difference between solid-liquid two phases and smaller particle diameter resulted in higher suspension degree. Fractal impeller can enhance water integrated velocity and improve suspension quality compared with four pitched-blade impeller under the constant power consumption, and increasingly so with fractal iteration number. Meanwhile, fractal impeller can reduce the size of trailing vortex and decrease the power consumption based on four pitched-blade impeller at the same impeller speed. In addition, the just suspension speed (Njs), just drawdown speed (Njd) and correlation equation for Njs and Njd in the three different impeller systems were obtained. Fractal impeller could decrease the Njs and Njd on the basis of four pitched-blade impeller, and decreasingly so with the fractal iteration number.

Small-Angle Scattering and Multifractal Analysis of DNA Sequences.

The arrangement of A, C, G and T nucleotides in large DNA sequences of many prokaryotic and eukaryotic cells exhibit long-range correlations with fractal properties. Chaos game representation (CGR) of such DNA sequences, followed by a multifractal analysis, is a useful way to analyze the corresponding scaling properties. This approach provides a powerful visualization method to characterize their spatial inhomogeneity, and allows discrimination between mono- and multifractal distributions. However, in some cases, two different arbitrary point distributions, may generate indistinguishable multifractal spectra. By using a new model based on multiplicative deterministic cascades, here it is shown that small-angle scattering (SAS) formalism can be used to address such issue, and to extract additional structural information. It is shown that the box-counting dimension given by multifractal spectra can be recovered from the scattering exponent of SAS intensity in the fractal region. This approach is illustrated for point distributions of CGR data corresponding to Escherichia coli, Phospholamban and Mouse mitochondrial DNA, and it is shown that for the latter two cases, SAS allows extraction of the fractal iteration number and the scaling factor corresponding to “ACGT” square, or to recover the number of bases. The results are compared with a model based on multiplicative deterministic cascades, and respectively with one which takes into account the existence of forbidden sequences in DNA. This allows a classification of the DNA sequences in terms of random and deterministic fractals structures emerging in CGR.

Numerical simulation of solid-liquid mixing characteristics in a stirred tank with fractal impellers

The hydrodynamics of solid-liquid mixing process in a stirred tank with four pitched-blade impellers, fractal 1 impellers, and fractal 2 impellers were investigated using computational fluid dynamics (CFD) simulation. An Eulerian-Eulerian approach, standard k-ε turbulence model, and multiple reference frames (MRF) technique were employed to simulate the solid-liquid two-phase flow, turbulent flow, and impeller rotation, respectively. The effects of impeller speed, impeller type, impeller spacing, impeller blade tilt angle, impeller blade shape, solid particle size and initial solid particle loading on the solid particle suspension quality were investigated. Results showed that the homogenous degree of solid-liquid system increased with the increase of impeller speed. The impeller spacing of T5/6 and T and impeller blade tilt angle of 60° and 45° were appropriate for the solid-liquid suspension process. Fractal shape impeller was more efficient than jagged shape impeller in solid-liquid mixing process. Larger particle diameter and higher initial solid particle loading resulted in less homogenous distribution of solid particles. It was found that fractal impeller could improve the solid particle suspension quality compared with four pitched-blade impeller under the same power consumption, increasingly so with the fractal iteration number of fractal impeller. Moreover, fractal impeller reduced the size of impeller trailing vortex and consumed less power consumption compared with four pitched-blade impeller at the same impeller speed, and the more the number of fractal iteration, the higher the impeller energy utilization rate of fractal impeller.

Plate-fin heat sink forced convective heat transfer augmentation with a fractal insert

The interaction of fractal grid-induced turbulence on plate-fin heat sink is numerically investigated at flow Reynolds number of ReDh = 7.3 × 104. Three fractal grids of different number of fractal iterations N, namely: The rectangular fractal grid of N = 2 (RFG2), square fractal grid of N = 3 (SFG3), and square fractal grid of N = 4 (SFG4) are employed to perturb the windward fluid flow. For each case, the effects of eight fractal grid first iterative bar thicknesses t0 at five different inter-fin distances δ are investigated. Results show that Nusselt number Nu and pressure drop ΔP increase with t0 for all cases. 57%, 51% and 43% of forced convective heat transfer augmentations are observed using SFG3, RFG2 and SFG4, respectively, than that of the control plate-fin heat sink. The thermal and fluid flow perturbation promoted by SFG3 outperforms the rest with Nu = 7.07 × 103 at δ = 10 and 25 mm, but at a cost of higher ΔP. RFG2's maximum Nu is 6.82 × 103 where wider δ of δ = 50 mm is preferred, and SFG4 is 6.42 × 103 at δ=10 mm. Interestingly, SFG4 enjoys a lower ΔP, which is highly energy sustainable. The strength of SFG3-induced turbulence intensity is able to infiltrate deeper into the fins at a higher flow rate, which may facilitate the continuous restructuring of inter-fin flow boundary layers, thus promoting thermal dissipation. In short, plate-fin heat sink forced convection is strongly dependent on the interaction between the insert configuration and the induced flow structures within fins, of which, the effects of t0 and δ are highly correlated.

Average view factors for extended surfaces with fractal perforations

Extended surfaces are often used for passive thermal management of electronic devices. By perforating extended surfaces in accordance with the Sierpinski carpet fractal pattern, an increase in surface area and a decrease in mass can be achieved. Intersurface thermal radiation, within the perforations, can account for a significant percentage of the total radiative heat transfer rate. As the perforations are of a non-uniform size, a correlation for the average fin view factor as a function of fractal iteration and width-to-thickness ratio was developed. For example, while a fin inspired by the fourth iteration of the Sierpinski carpet fractal pattern has 23.30% more surface area than a solid rectangular fin of equal width-to-thickness ratio the same fin only exchanges 67.37% of the radiation emitted with the surroundings due to intersurface thermal radiation. Regardless of width-to-thickness ratio, there was found to be a finite number of fractal iterations after which the average view factor of the extended surface approached zero. Similarly, the percentage of the total surface area that the perforations account for approaches an asymptote at approximately 100% by 10 iterations, regardless of width-to-thickness ratio. Finally, while there is an exponential increase in surface area with iteration, the increase in area did not offset the decrease in average view factor.

Scattering from surface fractals in terms of composing mass fractals

It is argued that a finite iteration of any surface fractal can be composed of mass-fractal iterations of the same fractal dimension. Within this assertion, the scattering amplitude of a surface fractal is shown to be a sum of the amplitudes of the composing mass fractals. Various approximations for the scattering intensity of surface fractals are considered. It is shown that small-angle scattering (SAS) from a surface fractal can be explained in terms of a power-law distribution of sizes of objects composing the fractal (internal polydispersity), provided the distance between objects is much larger than their size for each composing mass fractal. The power-law decay of the scattering intensity I(q) ∝ q^{D_{\rm s}-6}, where 2 < D s < 3 is the surface-fractal dimension of the system, is realized as a non-coherent sum of scattering amplitudes of three-dimensional objects composing the fractal and obeying a power-law distribution dN(r) ∝ r −τdr, with D s = τ − 1. The distribution is continuous for random fractals and discrete for deterministic fractals. A model of the surface deterministic fractal is suggested, the surface Cantor-like fractal, which is a sum of three-dimensional Cantor dusts at various iterations, and its scattering properties are studied. The present analysis allows one to extract additional information from SAS intensity for dilute aggregates of single-scaled surface fractals, such as the fractal iteration number and the scaling factor.

A MULTI-RESOLUTION RECONSTRUCTION METHOD FOR NATURAL LINEAR FEATURES BASED ON RESTRICTED FRACTAL INTERPOLATION

Multi-resolution representation has wide application demands in the field of Geographic Information System (GIS). The existing fractal interpolation methods can implement the reconstruction of natural linear features from low resolution to high resolution, but they are unable to quantify the relationship between the number of fractal iterations and the resolution, and the fractal interpolation results cannot maintain the geographical characteristics of natural linear features. This study presents a multi-resolution reconstruction method for natural linear features in GIS based on the restricted fractal interpolation. First, the geographical characteristics of natural linear features are extracted to restrict the parameters of fractal interpolation function and control the process of fractal interpolation. Second, a mathematical function is deduced for identifying the relationship between the number of fractal iterations and the resolution. The experiment results demonstrate that this study can reconstruct various preset high-resolution linear features from the low-resolution linear data while the geographical characteristics and random fractal characteristics of natural linear features are kept well.

Small-angle scattering from the Cantor surface fractal on the plane and the Koch snowflake.

The small-angle scattering (SAS) from the Cantor surface fractal on the plane and Koch snowflake is considered. We develop the construction algorithm for the Koch snowflake, which makes possible the recurrence relation for the scattering amplitude. The surface fractals can be decomposed into a sum of surface mass fractals for arbitrary fractal iteration, which enables various approximations for the scattering intensity. It is shown that for the Cantor fractal, one can neglect with good accuracy the correlations between the mass fractal amplitudes, while for the Koch snowflake, these correlations are important. It is shown that nevertheless, correlations can be built in the mass fractal amplitudes, which explains the decay of the scattering intensity I(q) ∼ qDs-4, with 1 < Ds < 2 being the fractal dimension of the perimeter. The curve I(q)q4-Ds is found to be log-periodic in the fractal region with a period equal to the scaling factor of the fractal. The log-periodicity arises from the self-similarity of the sizes of basic structural units rather than from correlations between their distances. A recurrence relation is obtained for the radius of gyration of the Koch snowflake, which is solved in the limit of infinite iterations. The present analysis allows us to obtain additional information from SAS data, such as the edges of the fractal regions, the fractal iteration number and the scaling factor.

Convex and concave successions of power-law decays in small-angle scattering

The small-angle scattering (SAS) structure factor from a new model of a 3D deterministic fractal in which the relative positions and the number of structural units vary with fractal iteration number is calculated. It is shown that, depending on the relative positions of scattering units inside the fractal, we can obtain various types of power-law successions, such as: convex/concave - when the absolute value of the scattering exponent of the first power-law decay is higher/smaller than that of the subsequent power- law decay, or any combination of them (i.e. convex-concave or concave-convex). The obtained results can explain experimental SAS (neutron or X-rays) data which are characterized by a succession of power-law decays of arbitrary length.

Verification of Multiband Characteristics in Iterative Fractal Antenna

This paper presents verification of multiband characteristics of a circularly shaped iterative fractal antenna with increasing number of fractal iterations. The four numbers of iterations are considered for this study. Also, the miniaturization characteristics of the fractal antenna with increasing number of fractal iterations have been studied. The proposed 4 th iterated fractal antenna is designed on FR4 substrate (h = 1.56 mm and er = 4.3). The antenna is coaxially fed using surface mount adapter. The proposed circularly shaped fractal antenna is found to resonate at four centre frequencies such as 0.683 GHz, 0.97 GHz, 1.29 GHz, and 1.68 GHz. The antenna finds applications in various compact multiband wireless communications due its smaller size , low cross polarization and multiband behaviour.

Power consumption and form drag of regular and fractal‐shaped turbines in a stirred tank

Previous wind‐tunnel measurements have shown that fractal‐shaped plates have increased drag compared to square plates of the same area. In this study, the power consumption and drag of turbines with fractal and rectangular blades in a stirred tank are measured. Power number decreases from rectangular to fractal impellers by over 10%, increasingly so with fractal iteration number. Our results suggest that this decrease is not caused by the wake interaction of the blades, nor solely by the wake interaction with the walls either. Pressure measurements on the blades’ surface show that fractal blades have lower drag than the rectangular ones, opposite to the wind tunnel experiment results. All tested blades’ center of pressure radius increases with Re, while their drag coefficient decreases, a possible effect of the solid body rotation expansion with Re. Spectral analysis of the pressure signal reveals two peaks possibly connected to the blades’ roll vortices. © 2016 American Institute of Chemical Engineers AIChE J, 63: 843–854, 2017

Verification of Multiband Characteristics in Iterative Fractal Antenna

This paper presents verification of multi-band characteristics of a circularly shaped iterative fractal antenna with increasing number of fractal iterations. The four numbers of iterations are considered for this study. Also, the miniaturization characteristics of the fractal antenna with increasing number of fractal iterations have been studied. The proposed 4 th iterated fractal antenna is designed on FR4 substrate (h = 1.56 mm and er = 4.3). The antenna is co-axially fed using surface mount adapter. The proposed circularly shaped fractal antenna is found to resonate at four centre frequencies such as 0.683 GHz, 0.97 GHz, 1.29 GHz, and 1.68 GHz. The antenna finds applications in various compact multi-band wireless communications due its smaller size, low cross polarization and multi-band behavior.

Bandwidth enhanced compact fractal antenna for UWB applications with 5-6 GHz band rejection

A compact microstrip fed novel U-shaped fractal monopole antenna with enhanced bandwidth and band notched characteristics is presented for ultrawideband applications. The self-similar fractal property is used in the U-shape monopole to obtain 2n surface current paths to achieve good impedance matching over a wider bandwidth. A parametric study is performed to optimize the crucial antenna parameters. The proposed fractal antenna exhibits wider impedance bandwidth from 2 to 13.7 GHz for VSWR ≤ 2. By increasing the number of fractal iterations to three (n = 3), wider bandwidth of about 149% (impedance bandwidth ratio 6.85:1) is achieved with 43% improvement in impedance bandwidth. By etching a novel complementary split ring resonator in the ground plane, the stop band for wireless local area network and HIPERLAN/2 (5.15–5.825 GHz) system is obtained. The total volume of the presented antenna is 20 × 20 × 1.0 mm3 and the simulated and measured antenna characteristics are in good agreement. © 2015 Wiley Periodicals, Inc. Microwave Opt Technol Lett 57:607–613, 2015

Multiband frequency selective surfaces with a modified multifractal cantor geometry

Self-affine property of a modified multifractal Cantor geometry is exploited to design frequency selective surfaces (FSS) with multiband response. The main advantage of the proposed structure is to design multiband FSS with multiple frequency ratios between the adjacent bands and easily-built structures. In addition, the proposed structure increases the degree of freedom in design of multiband FSS response according to the number of fractal iterations. The validation of the proposed structure was initially verified through simulations in Ansoft Designer 3.5 and then a prototype was built with a validation purpose.

Drag and near wake characteristics of flat plates normal to the flow with fractal edge geometries

Past results have suggested that the drag coefficient and the shedding frequencies of regular polygon plates all fall within a very narrow band of values. In this study, we introduce a variety of length scales into the perimeter of a square plate and study the effects this has on the wake characteristics and overall drag. The perimeter of the plate can be made as long as allowed by practical constraints with as many length scales as desired under these constraints without changing the area of the plate. A total of eight fractal-perimeter plates were developed, split into two families of different fractal dimensions all of which had the same frontal area. It is found that by increasing the number of fractal iterations and thus the perimeter, the drag coefficient increases by up to 7%. For the family of fractal plates with the higher dimension, it is also found that when the perimeter increases above a certain threshold the drag coefficient drops back again. Furthermore, the shedding frequency remains the same but the intensity of the shedding decreases with increasing fractal dimension. The size of the wake also decreases with increasing fractal dimension and has some dependence on iteration without changing the area of the plate.

Stable and compact multiband frequency selective surfaces with Peano pre‐fractal configurations

This work presents a fractal design methodology for frequency selective surfaces (FSSs) with Peano pre-fractal patch elements. The proposed FSS structures are composed of periodic arrays of metallic patches printed on a single-layer fibreglass dielectric. The shapes presented by pre-fractal patches allow one to design compact FSSs that behave like dual-polarised band-stop spatial filters. On the other side, the space-filling and self-similarity properties of Peano fractals became possible various configurations for patch elements. An FSS parametric analysis is performed in terms of the fractal iteration-number and cell-size of pre-fractal patches. To validate the used methodology four FSS prototypes are built and tested in the range from 1.0 to 13.5 GHz. Experimental characterisation of the FSS prototypes is accomplished through three different measurement setups with commercial horns and circular monopole microstrip antennas. Results show that the proposed FSS presents most of the desired features for spatial filters: compact design, multiband responses, dual-polarisation, excellent angular stability and facility for reconfiguration.

Prospects for investigating deterministic fractals: Extracting additional information from small angle scattering data

We examine a general problem of small angle scattering from deterministic fractals: whether one can extract some additional information from the small angle scattering data apart from the fractal dimension and the edges of the fractal region. It is shown that for single-scale mass fractals one can additionally obtain the fractal iteration number, the scaling factor, and the number of structural units composing the fractal, provided the quality of fractal samples is high enough.