Fractal Grid Research Articles

Direct numerical simulation of turbulent mixing in regular and fractal grid turbulence

Turbulent mixing in regular and fractal grid turbulence is investigated in this work by using direct numerical simulation (DNS). Two types of turbulence-generating grids are used: a biplane square grid (regular grid) and a square fractal grid. The thickness ratios tr of the fractal grids are set at 5.0 and 8.5. The grid solidity is maintained at σ=0.36 for all the grids. The mesh Reynolds number, ReM=U0Meff/ν, is set at 2500 for all cases, where U0 is the cross-sectionally averaged mean velocity; Meff, the effective mesh size; and ν, the kinematic viscosity. The grids are numerically generated using the immersed boundary method at 4Meff downstream of the entrance to the computational domain. The computational domain size normalized by Meff is 64×8×8 in the streamwise, vertical and spanwise directions for the regular grid and 64×16×16 for the fractal grids. Scalar mixing layers that initially have a step profile develop downstream of the grids. The Prandtl number is set at Pr=0.71 considering the heat transfer in air flow. Instantaneous temperature fields, instantaneous fluctuating temperature fields and fundamental turbulent statistics are presented. The results show that turbulent mixing is more strongly enhanced in fractal grid turbulence than in regular grid turbulence for the same ReM. In fractal grid turbulence, turbulent mixing is more strongly enhanced at tr=8.5 than at tr=5.0.

High-Schmidt-number scalar transfer in regular and fractal grid turbulence

Turbulent mixing of high-Schmidt-number passive scalars in regular and fractal grid turbulence is experimentally investigated using a water channel. A turbulence-generating grid is installed at the entrance of the test section, which is 1.5 m in length and 0.1 m×0.1 m in cross section. Two types of grids are used: one is a regular grid consisting of square-mesh and biplane constructions, and the other is a square-type fractal grid, which was first investigated by Hurst and Vassilicos (2007 Phys. Fluids 19 035103) and Seoud and Vassilicos (2007 Phys. Fluids 19 105108). The two grids have the same solidity of 0.36. The Reynolds number based on the mesh size, ReM=U0Meff/ν, is 2500 in both flows, where U0 is the cross-sectionally averaged mean velocity, Meff is the effective mesh size and ν is the kinematic viscosity. A fluorescent dye (rhodamine B) is homogeneously premixed only in the lower stream and therefore the scalar mixing layers with an initial step profile develop downstream of the grids. The Schmidt number of the dye is O(103). The time-resolved particle image velocimetry and the planar laser-induced fluorescence techniques are used to measure the velocity and concentration fields. The results show that the turbulent mixing in fractal grid turbulence is more strongly enhanced than that in regular grid turbulence for the same mesh Reynolds number ReM. The profile of instantaneous scalar dissipation shows that scalar dissipation takes place locally even in the far downstream region at x/Meff=120 in fractal grid turbulence.

Fractal-Generated Turbulence in Opposed Jet Flows

The opposed jet configuration presents a canonical geometry suitable for the evaluation of calculation methods seeking to reproduce the impact of strain and re-distribution on turbulent transport in reacting and non-reacting flows. The geometry has the advantage of good optical access and, in principle, an absence of complex boundary conditions. Disadvantages include low frequency flow motion at high nozzle separations and comparatively low turbulence levels causing bulk strain to exceed the turbulent contribution at small nozzle separations. In the current work, fractal generated turbulence has been used to increase the turbulent strain and velocity measurements for isothermal flows are reported with an emphasis on the axis, stagnation plane and the distribution of mean and instantaneous strain rates. Energy spectra were also determined. The instrumentation comprised hot-wire anemometry and particle image velocimetry with the flows to both nozzles seeded with 1 \(\upmu\)m silicon oil droplets providing a relaxation time of ≃ 3 \(\upmu\)s. It is shown that fractal grids increase the turbulent Reynolds number range from 48–125 to 109–220 for bulk velocities from 4 to 8 m/s as compared to conventional perforated plate turbulence generators. Low frequency motion of the order 10 Hz could not be completely eliminated and probability density functions were determined for the location of the stagnation plane. Results show that the fluctuation in the position of the stagnation plane is of the order of the integral length scale, which was determined to be 3.1±0.1 mm at the nozzle exits through the use of hot-wire anemometry. Flow statistics close to the fractal plate located upstream of the nozzle exit were also determined using a transparent glass nozzle.

Turbulence without Richardson–Kolmogorov cascade

We investigate experimentally wind tunnel turbulence generated by multiscale/fractal grids pertaining to the same class of low-blockage space-filling fractal square grids. These grids are not active but nevertheless produce very much higher turbulence intensities u′/U and Reynolds numbers Reλ than higher blockage regular grids. Our hot wire anemometry confirms the existence of a protracted production region where turbulence intensity grows followed by a decay region where it decreases, as first reported by Hurst and Vassilicos [“Scalings and decay of fractal-generated turbulence,” Phys. Fluids 19, 035103 (2007)]. We introduce the wake-interaction length scale x⋆ and show that the peak of turbulence intensity demarcating these two regions along the centerline is positioned at about 0.5x⋆. The streamwise evolutions on the centerline of the streamwise mean flow and of various statistics of the streamwise fluctuating velocity all scale with x⋆. Mean flow and turbulence intensity profiles are inhomogeneous at streamwise distances from the fractal grid smaller than 0.5x⋆, but appear quite homogeneous beyond 0.5x⋆. The velocity fluctuations are highly non-Gaussian in the production region but approximately Gaussian in the decay region. Our results confirm the finding of Seoud and Vassilicos [“Dissipation and decay of fractal-generated turbulence,” Phys. Fluids 19, 105108 (2007)] that the ratio of the integral length-scale Lu to the Taylor microscale λ remains constant even though the Reynolds number Reλ decreases during turbulence decay in the region beyond 0.5x⋆. As a result, the scaling Lu/λ∼Reλ, which follows from the u′3/Lu scaling of the dissipation rate in boundary-free shear flows and in usual grid-generated turbulence, does not hold here. This extraordinary decoupling is consistent with a noncascading and instead self-preserving single-length scale type of decaying homogeneous turbulence proposed by George and Wang [“The exponential decay of homogeneous turbulence,” Phys. Fluids 21, 025108 (2009)], but we also show that Lu/λ is nevertheless an increasing function of the inlet Reynolds number Re0. Finally, we offer a detailed comparison of the main assumption and consequences of the George and Wang theory against our fractal-generated turbulence data.

S0502-7-3 高精度PLIFによる高Sc数スカラー乱流混合の実験解析([S0502-7]噴流,後流および剥離流れ現象の解明と制御(7))

High-Schmidt-number scalar mixing layers in the grid turbulence are experimentally investigated in a water channel by means of the planar laser induced fluorescence (PLIF) technique and the time-resolved particle image velocimetry (PIV). A biplane square grid and a fractal square grid are used to generate turbulence. The Reynolds number based on the effective mesh size and mean velocity is 2,500 in both flows. Fluorescent dye (Rhodamine B; Schmidt number is about 2,100) is homogeneously premixed only in the upper stream, and therefore, the scalar mixing layer with an initial step profile develops downstream of the grid. Various turbulence quantities are evaluated and comparisons are made between turbulence and scalar fields generated by the regular and fractal grids.

New statistical symmetries of the multi-point equations and its importance for turbulent scaling laws

We presently show that the infinite set of multi-point correlation equations, which are direct statistical consequences of the Navier-Stokes equations, admit a rather large set of Lie symmetry groups. This set is considerable extended compared to the set of groups which are implied from the original set of equations of fluid mechanics. Specifically a new scaling group and translational groups of the correlation vectors and all independent variables have been discovered. These new statistical groups have important consequences on our understanding of turbulent scaling laws to be exemplarily revealed by two examples. Firstly, one of the key foundations of statistical turbulence theory is the universal law of the wall with its essential ingredient is the logarithmic law. We demonstrate that the log-law fundamentally relies on one of the new translational groups. Second, we demonstrate that the recently discovered exponential decay law of isotropic turbulence generated by fractal grids is only admissible with the new statistical scaling symmetry. It may not be borne from the two classical scaling groups implied by the fundamental equations of fluid motion and which has dictated our understanding of turbulence decay since the early thirties implicated by the von-Kármán-Howarth equation.

フラクタル格子により生成されるマルチスケール誘起乱流の構造とスカラー拡散機構 : 第2報,DNSによる一様等方性と間欠性に及ぼす格子パラメータの影響の検討(流体工学,流体機械)

フラクタル格子により生成されるマルチスケール誘起乱流の構造とスカラー拡散機構 : 第2報,DNSによる一様等方性と間欠性に及ぼす格子パラメータの影響の検討(流体工学,流体機械)

A numerical strategy to combine high-order schemes, complex geometry and parallel computing for high resolution DNS of fractal generated turbulence

Impressive advances in parallel platform architectures over the past decade have made Direct Numerical Simulation (DNS) a powerful tool which can provide full access to the spatial structure of turbulent flows with complex geometries. An innovative approach which combines high-order schemes and a dual domain decomposition method is presented in this paper and is applied to DNS of multiscale-generated turbulent flows by a fractal grid. These DNS illustrate the applicability of our approach to the simulation of complex turbulent flows and provide results which are compared with recent laboratory experiments thus providing new insights for the interpretation of the experimental measurements.

The exponential decay of homogeneous turbulence

Similarity equations of the spectral equations for decaying homogeneous turbulence are considered for which the similarity length scale is not allowed to grow. Two types of solutions are found: an inviscid solution and one involving viscosity. For the former, the energy decays asymptotically as t−2, while for the latter the energy decays exponentially and the ratio of integral scale to Taylor microscale is constant. For both the spectra for fixed initial conditions collapse during decay with simply the energy and a single length scale. The exponentially decaying solution appears to provide an excellent description of the turbulence generated in recent space-filling fractal grid experiments.

MULTISCALE GENERATION OF TURBULENCE

This paper presents a brief but general introduction to the physics and engineering of fractals, followed by a brief introduction to fluid turbulence generated by multiscale flow actuation. Numerical computations of such turbulent flows are now beginning to be possible because of the immersed boundary method (IBM) and terascale parallel high performance computing capabilities. The first-ever direct numerical simulation (DNS) results of turbulence generated by fractal grids are detailed and compared with recent wind tunnel measurements.

フラクタル格子により生成されるマルチスケール誘起乱流の構造とスカラー拡散機構 : 第1報,DNSによるフラクタル基本形状の影響に関する検討(流体工学,流体機械)

フラクタル格子により生成されるマルチスケール誘起乱流の構造とスカラー拡散機構 : 第1報,DNSによるフラクタル基本形状の影響に関する検討(流体工学,流体機械)

Direct numerical simulation of turbulent mixing in grid-generated turbulence

Turbulent mixing of passive scalar (heat) in grid-generated turbulence (GGT) is simulated by means of direct numerical simulation (DNS). A turbulence-generating grid, on which the velocity components are set to zero, is located downstream of the channel entrance, and it is numerically constructed on the staggered mesh arrangement using the immersed boundary method. The grid types constructed are: (a) square-mesh biplane grid, (b) square-mesh single-plane grid, (c) composite grid consisting of parallel square-bars and (d) fractal grid. Two fluids with different temperatures are provided separately in the upper and lower streams upstream of the turbulence-generating grids, generating the thermal mixing layer behind the grids. For the grid (a), simulations for two different Prandtl numbers of 0.71 and 7.1, corresponding to air and water flows, are conducted to investigate the effect of the Prandtl number. The results show that the typical grid turbulence and shearless mixing layer are generated downstream of the grids. The results of the scalar field show that a typical thermal mixing layer is generated as well, and the effects of the Prandtl numbers on turbulent heat transfer are observed.

029 Novel Parameters for Computer-Assisted Image Analysis to Study Angiogenesis

Angiogenesis is a dynamic process that requires interaction of proangiogenic and antiangiogenic factors. Several methodologies have been used to evaluate this process. The quail chorioallantoic membrane (CAM) is a suitable model to explore angiogenesis in vivo. Recently, the morphometric parameters of fractal dimension (Df) and grid intersection (ρ) are used to quantify angiogenesis in CAM. However, more information is necessary to understand angiogenic mechanisms. New methods to measure additional parameters such as the average blood vessel thickness, tortuousity, total branch length and complexity (junction node number), are described herein. For the analysis, CAM images were obtained from ex ovo culture of E8-E9 quail embryos. MatLab®(The MathWorks, Natick, MA) software was used for the analyses of branches nodes, thickness and length of the forming branches. Conventional morphometry was done using IPLab (Scanalytics, Fairfax, VA). To validate the new parameters, leptin, VEGF and neutralizing antibodies were evaluated. Leptin and VEGF treated CAM had increased complexity and vascular densities (Df = 1.5 ± 0.005; ρ = 12.3 ± 0.689) and (Df = 1.6 ± 0.006; ρ = 14.3 ± 0.465), respectively, compared to PBS control (Df = 1.3 ± 0.035; ρ = 8.6 ± 0.439). There was a corresponding decrease antileptin and anti-VEGF antibodies. The node-structural map applied to CAM images reveled 53% increase in the number of one-branch nodes and 50% increase in three-branch nodes with leptin compared to controls. Additionally, the total branch length was 30% greater with leptin along with a decrease in vessel thickness, The analyses allowed quantification of branching, tortuousity, thickness, and length of the new blood vessels. Together with the conventional morphometric parameters, these new analyses will help understand the mechanisms exerted by angiogenesis-modulating molecule.

Superconductivity in Cu-Nb with extremely fine structure

Electrodynamic properties of ‘three-dimensional’ composites CuNb with extremely fine structure and different concentration of components are investigated. The samples have been fabricated from the melt using slow and extremely fast cooling rate. The slowly cooled samples (SCS) are characterized by a fractal niobium grid with a characteristic size of the micron scale immersed into a copper matrix. The fast melt cooling results in the nano-scale composite. It has a rather pure copper matrix in which almost pure isolated nano-inclusions of niobium are distributed. We have found the electromagnetic properties of these two (micron and nano scale) CuNb composites have some qualitative difference. The SCS shows features of superconducting transition, typical for granular superconductors. The superconducting transition in quenched samples (QS) carries the feature of an effective superconducting media which a critical temperature Tc are renormalized to a rather lowered value for the small size of superconducting regions. On the basis of the experimental data we have estimated a coherence length of the effective superconducting media of the QS that occurs close to the Nb droplet diameter. Voltage-current characteristics (VCC) of the QS in magnetic fields parallel and perpendicular to the current show abnormally low slope in some intermediate current region while at lower and higher currents the VCC features are characteristic for a hard superconductor. The opportunity of a relation of the abnormal dissipation in the QS with the Josephson generation is discussed.

Dissipation and decay of fractal-generated turbulence

Space-filling fractal square grids fitted at the entrance of a wind tunnel’s test section generate unusually high Reynolds number homogeneous isotropic turbulence which decays locked into a single length-scale l. Specifically, during turbulence decay along the streamwise coordinate x, E11(k1,x)=u′2lf(k1l) over the entire range of wavenumbers, where l and the function f are about the same for all the grids tried here. As a result, this fractal-generated turbulence has the following properties which we have also observed in the decaying region: L∕λ is constant, independent of the x grid and Reλ; ϵ∼Reλ−1u′3∕Lu; and E11(k1)∼(u′3∕Lu)2∕3k1−5∕3 instead of E11(k1)∼ϵ2∕3k1−5∕3 in the observed range of wavenumbers where f(k1l)∼(k1l)−5∕3.

Scalings and decay of fractal-generated turbulence

A total of 21 planar fractal grids pertaining to three different fractal families have been used in two different wind tunnels to generate turbulence. The resulting turbulent flows have been studied using hot wire anemometry. Irrespective of fractal family, the fractal-generated turbulent flows and their homogeneity, isotropy, and decay properties are strongly dependent on the fractal dimension Df≤2 of the grid, its effective mesh size Meff (which we introduce and define) and its ratio tr of largest to smallest bar thicknesses, tr=tmax∕tmin. With relatively small blockage ratios, as low as σ=25%, the fractal grids generate turbulent flows with higher turbulence intensities and Reynolds numbers than can be achieved with higher blockage ratio classical grids in similar wind tunnels and wind speeds U. The scalings and decay of the turbulence intensity u′∕U in the x direction along the tunnel’s center line are as follows (in terms of the normalized pressure drop CΔP and with similar results for v′∕U and w′∕U): (i) for fractal cross grids (Df=2), (u′∕U)2=tr2CΔPfct(x∕Meff); (ii) for fractal I grids, (u′∕U)2=tr(T∕Lmax)2CΔPfct(x∕Meff), where T is the tunnel width and Lmax is the maximum bar length on the grid; (iii) for space-filling (Df=2) fractal square grids, the turbulence intensity builds up as the turbulence is convected downstream until a distance xpeak from the grid is reached where the turbulence intensity peaks and then decays exponentially, u′2=upeak′2exp[−(x−xpeak)∕lturb], where upeak′2 increases linearly with tr, xpeak∝tminT∕Lmin (Lmin being the minimum bar length on the grid), and lturb∝λ2U∕ν (ν being the kinematic viscosity of the air and λ being the Taylor microscale); λ remains approximately constant during decay at x≫xpeak. The longitudinal and lateral integral length scales also remain approximately constant during decay at x≫xpeak.

Moiré by fractal structures

The theoretical and experimental results of the Moiré effect observed by superposing two grids containing fractal Cantor structures are presented in this paper. It is also analysed the equivalence between the information obtained by the Fraunhofer diffraction through those fractal grids and that obtained through Moiré. In a recently published paper [1] it was verified that the diffraction pattern is highly sensitive to variations in dimension, order of growth and lacunarity of the Cantor fractal, becoming a powerful tool to analyse and determine these parameters. So, in this paper it is intended to determine the effect that the order of growth, the dimension, and the lacunarity of one-dimensional fractal Cantor structures have over the Moiré patterns that result from superposing these structures over replicas of themselves which have been rotated through a small angle α. The main goal is to verify if there exists an intimate relation between the resulting Moiré pattern and the parameters that describe this fractal structure, as occurs in the case of the diffraction pattern.

Generational Analysis Reveals that TGF-β1 Inhibits the Rate of Angiogenesis in Vivo by Selective Decrease in the Number of New Vessels

Quantitative analysis of vascular generational branching demonstrated that transforming growth factor-β1 (TGF-β1), a multifunctional cytokine and angiogenic regulator, strongly inhibited angiogenesis in the arterial tree of the developing quail chorioallantoic membrane (CAM) by inhibition of the normal increase in the number of new, small vessels. The cytokine was applied uniformly in solution at embryonic day 7 (E7) to the CAMs of quail embryos cultured in petri dishes. After 24 h the rate of arterial growth was inhibited by as much as 105% as a function of increasing TGF-β1 concentration. Inhibition of the rate of angiogenesis in the arterial tree by TGF-β1 relative to controls was measured in digital images by three well-correlated, computerized methods. The first computerized method, direct measurement by the computer code VESGEN of vascular morphological parameters according to branching generations G1 through G≥5, revealed that TGF-β1 selectively inhibited the increase in the number density of small vessels, Nv≥5 (382 ± 85 cm−2 for specimens treated with 1 μg TGF-β1/CAM for 24 h, compared to 583 ± 99 cm−2 for controls), but did not significantly affect other parameters such as average vessel length or vessel diameter. The second and third methods, the fractal dimension (Df) and grid intersection (ρv), are statistical descriptors of spatial pattern and density. According to Df and ρv, arterial density increased in control specimens from 1.382 ± 0.007 and 662 ± 52 cm−2 at E7 (0 h) to 1.439 ± 0.013 and 884 ± 55 cm−2 at E8 (24 h), compared to 1.379 ± 0.039 and 650 ± 111 cm−2 for specimens treated with 1 μg TGF-β1/CAM for 24 h. TGF-β1 therefore regulates vascular pattern and the rate of angiogenesis in a unique “fingerprint” manner, as do other major angiogenic regulators that include VEGF, FGF-2 (bFGF), and angiostatin. TGF-β1 did not stimulate angiogenesis significantly at low cytokine concentrations, which suggests that this quail CAM model of angiogenesis is not associated with an inflammatory response.

A Novel Assay of Angiogenesis in the Quail Chorioallantoic Membrane: Stimulation by bFGF and Inhibition by Angiostatin According to Fractal Dimension and Grid Intersection

In a novel assay of angiogenesis in the quail chorioallantoic membrane (CAM), we measured vascular pattern and angiogenic rate after homogeneous exposure of the entire vascular tree to recognized modulators of vessel growth. In comparison to phosphate-buffered saline (PBS)-treated controls, the vascular stimulator, basic fibroblast growth factor (bFGF or FGF-2), increased the rate of angiogenesis by a maximum of 72%, whereas a recently discovered angiogenic inhibitor, angiostatin, decreased the rate of vascular growth by a maximum of 68%. The perturbants were applied in PBS to the CAM of 7-day-old embryos (E7) cultured in petri dishes, and the embryos were cultured further until fixation at E8 or E9. For morphometry of the quasi-two-dimensional CAM vasculature, digital images of arterial endpoints from the middle region of the CAM were acquired in grayscale at a magnification of 10×, binarized to black/white, and skeletonized. The pattern of vessel branching was assessed by measurement of the fractal dimension (Df), and vessel density (ρv), with the method of grid intersection. Correlations between these two statistical techniques were linear (r2ranged from 0.967 to 0.985). For skeletonized images at E9,Dfand ρvof bFGF-treated samples were 1.55 ± 0.01 and 782 ± 26/cm2, respectively (relative to 1.49 ± 0.02 and 583 ± 60/cm2for controls), and of angiostatin-treated samples, 1.43 ± 0.02 and 424 ± 74/cm2(relative to 1.50 ± 0.02 and 616 ± 59/cm2for controls). To establish normalization values for rates of angiogenesis, we analyzed untreated CAMs of E6 to E12. From E7 to E10 in skeletonized images,Dfincreased linearly from 1.37 ± 0.01 to 1.54 ± 0.01 and ρvfrom 311 ± 67 to 746 ± 124/cm2(in both cases,r2= 1.000). Thus, the rates of normal angiogenic growth as measured byDfand ρvwere 0.06/day and 138/cm2-day, respectively. From E10 to E12,Dfand ρvdeclined slightly. Differences between the vasculature of untreated and PBS-treated CAMs were statistically insignificant. In conclusion, vascular branching pattern and density in the quail CAM were stimulated by bFGF and inhibited by angiostatin. We quantified these changes with statistical significance byDfand ρv, which are expressed relative to the rates of normal developmental angiogenesis measured for the two parameters in untreated quail embryos.

Control of volume holograms

The mismatch in the number of degrees of freedom supported by volume holograms and the boundary fields that control them limits the dynamic range of recorded holograms. For holograms controlled by using fractal sampling grids, the maximum dynamic range falls inversely with the minimum number of exposures needed to record the hologram, the rank of the hologram. In adaptive holography, feedback between coupled holograms prevents the dynamic range from decreasing faster than the fundamental limit. If the control problem is overcome, the maximum dynamic range that a hologram can support falls inversely with the square root of the rank. In principle, holograms in which the dynamic range falls inversely with the square root of the rank can be recorded by using cross-spectrally coherent polychromatic pulses.