Length Scale Of Variation Research Articles

Stratified shear flow instabilities at large Richardson numbers

Numerical simulations of stratified shear flow instabilities are performed in two dimensions in the Boussinesq limit. The density variation length scale is chosen to be four times smaller than the velocity variation length scale so that Holmboe or Kelvin–Helmholtz unstable modes are present depending on the choice of the global Richardson number Ri. Three different values of Ri were examined Ri=0.2, 2, and 20. The flows for the three examined values are all unstable due to different modes, namely: the Kelvin–Helmholtz mode for Ri=0.2, the first Holmboe mode for Ri=2, and the second Holmboe mode for Ri=20 that has been discovered recently and this is the first time that it is examined in the nonlinear stage. It is found that the amplitude of the velocity perturbation of the second Holmboe mode at the nonlinear stage is smaller but comparable to first Holmboe mode. The increase in the potential energy, however, due to the second Holmboe modes is greater than that of the first mode. The Kelvin–Helmholtz mode is larger by two orders of magnitude in kinetic energy than the Holmboe modes and about ten times larger in potential energy than the Holmboe modes. The effect of increasing Prandtl number is also investigated, and a weak dependence on the Prandtl number is observed. The results in this paper suggest that although mixing is suppressed at large Richardson numbers it is not negligible, and turbulent mixing processes in strongly stratified environments cannot be excluded.

Solitonlike base pair opening in a helicoidal DNA: An analogy with a helimagnet and a cholesteric liquid crystal

We propose a model for DNA dynamics by introducing the helical structure through twist deformation in analogy with the structure of a helimagnet and a cholesteric liquid-crystal system. The dynamics in this case is found to be governed by the completely integrable sine Gordon equation, which admits kink-antikink solitons with increased width, representing a wide base-pair opening configuration in DNA. The results show that the helicity introduces a length-scale variation and thus provides a better representation of the base-pair opening in DNA.

Recent advances in multiscale simulations of heat transfer and fluid flow problems

In the thermal and fluid science, multiscale problems may be classified into two categories: multiscale process and multiscale system. In the multiscale process the overall behaviour is governed by the processes that occur on different length/time scales and are inherently related. Turbulent flow is an example. By multiscale system, we refer to a system that is characterised by large variation in length scales, for which the processes at different length scales are not so closely related. The cooling of an electronic system is an example. In this paper, recent advances in the numerical simulations of the first categories are presented.

Self-assembly of amorphous biophotonic nanostructures by phase separation

Some of the most vivid colors in the animal kingdom are created not by pigments, but by wavelength-selective scattering of light from nanostructures. Here we investigate quasi-ordered nanostructures of avian feather barbs which produce vivid non-iridescent colors. These β-keratin and air nanostructures are found in two basic morphologies: tortuous channels and amorphous packings of spheres. Each class of nanostructure is isotropic and has a pronounced characteristic length scale of variation in composition. These local structural correlations lead to strong backscattering over a narrow range of optical frequencies and little variation with angle of incidence. Such optical properties play important roles in social and sexual communication. To be effective, birds need to precisely control the development of these nanoscale structures, yet little is known about how they grow. We hypothesize that multiple lineages of birds have convergently evolved to exploit phase separation and kinetic arrest to self-assemble spongy color-producing nanostructures in feather barbs. Observed avian nanostructures are strikingly similar to those self-assembled during the phase separation of fluid mixtures; the channel and sphere morphologies are characteristic of phase separation by spinodal decomposition and nucleation and growth, respectively. These unstable structures are locked-in by the kinetic arrest of the β-keratin matrix, likely through the entanglement or cross-linking of supermolecular β-keratin fibers. Using the power of self-assembly, birds can robustly realize a diverse range of nanoscopic morphologies with relatively small physical and chemical changes during feather development.

Multi-scale characterization of orthotropic microstructures

Computer-generated 2D microstructures of varying second phase area fraction (5–30%), aspect ratio (1–16) and degree of alignment (where the reinforcement major-axis orientation is random, perfectly aligned or semi-aligned) are analyzed via the isotropic and directional forms of the computationally efficient multi-scale analysis of area fractions (MSAAF) technique. The impact of these microstructure parameters on the representative volume element (RVE) necessary to characterize a microstructure is ascertained with variations in isotropic and directional homogeneous length scales, derivative quantities of the MSAAF technique. Analysis of these results produces empirical expressions for the directional homogeneous length scale as a function of area fraction and aspect ratio for the limiting cases of random and ‘perfect’ second phase alignment. Generally, particle alignment is observed to increase the aspect ratio of a microstructure's RVE—a trend amplified by higher reinforcement aspect ratios and lower area fractions. Particle alignment also decreases the absolute size of such an element by reducing the directional homogeneous length scales transverse to the axis of alignment. Periodic boundary conditions on the perimeter of the synthetic microstructures are used to characterize the error in the MSAAF technique via multiple instantiations of the same microstructure, which further indicates that the statistical variation in the directional homogeneous length scale (measured by the directional MSAAF technique) can be an order of magnitude less than the variation in the isotropic homogeneous length scale (measured by the isotropic MSAAF technique).

On the relation between excess hydraulic head and flow in a conduit imbedded within a porous matrix

This article provides a partial answer to the question “What is the relation between excess hydraulic head and volume flux of water in a conduit within a porous matrix?”, focusing on the case that the forcing is steady. The conduit is modelled as a horizontal circular cylinder, imbedded within a porous matrix of rectangular cross section, having constant head prescribed on the sidewalls and being confined top and bottom. Laminar flow in the matrix is assumed to obey Darcy's law, while turbulent flow in the conduit is quantified using the Darcy–Weisbach equation. Analysis of the latter equation shows that the length scale of variations in the direction of the conduit is large compared with the scale of lateral and vertical variations. This permits separation of the full three-dimensional non-linear problem into a two-dimensional linear problem for head within the matrix and a one-dimensional non-linear problem for head within the conduit. Analytic solutions are obtained for the distribution of head in the matrix and in a conduit of either infinite or finite length. In both cases, the volume flux of water is proportional to the excess head to the 2/3 power, the conduit radius to the 5/3 power, the matrix permeability to the 1/3 power and gravity to the 1/3 power. The scale of variation of head along the conduit is proportional to the excess head to the −1/3 power, the conduit radius to the 5/3 power, the matrix permeability to the −2/3 power and gravity to the 1/3 power.

Rarefied gas flow over an in-line array of circular cylinders

A steady rarefied gas flow through periodic porous media kept at a uniform temperature is considered on the basis of the Bhatnagar–Gross–Krook equation and the diffuse reflection condition on the solid boundary. Under the assumption that the period is much smaller than the length scale of variation of the global pressure distribution, a macroscopic fluid model describing the pressure distribution and the mass flux of the gas in the medium is derived by the homogenization previously proposed by Charrier and Dubroca [Multiscale Model. Simul. 2, 124 (2003)]. The effective diffusion coefficient contained in the model is constructed numerically as a function of the Knudsen number, in the case of the medium consisting of an in-line array of circular cylinders, with the help of the numerical analysis of a rarefied gas flow in an infinite expanse of the cylinder array driven by a uniform small pressure gradient. An application of the model to an isothermal flow in a porous slab induced by a pressure difference is presented.

Analysis of dominant flow structures and their flow dynamics in chemical process equipment using snapshot proper orthogonal decomposition technique

Snapshot proper orthogonal decomposition (POD) technique has been applied to reveal the dominant flow structures, their dynamics and length scales in six widely used industrial equipments (stirred tank, bubble column, Taylor–Couette flow (annual contactor), ultrasonic reactor, jet reactor, and channel flow). The variation in length scale of structures within an equipment, with change in its operating conditions (Reynolds number and power input) or change in its geometric configuration (sparger and impeller designs), has been brought out in this work. The planar data set for POD analysis was obtained from particle image velocimetry (PIV) and large eddy simulation (LES) studies. The dominant spatial topology was analyzed by using the velocity and vorticity POD modes. The modes have revealed the following flow structures: the ascending streaks and bursts in channel flow, the vortex tube and leading edge vortices in jets, the irregular small chaotic vortices in Taylor–Couette flow, the variation in plume oscillation and flow structures in the vortical region of bubble column resulting from changes in sparger design, the high intensity vortices near the source of ultrasound in the ultrasonic reactor and the effect of impeller designs on dominant flow structures and near blade vortices in the stirred tank. The length scales of structures are obtained by applying image processing on the spatial modes. The dynamics of these flow structures in each of the items of equipment is captured by reconstructing the flow field using appropriate spatial and temporal modes that contribute to these structures. Further, a unique attempt has been made to correlate the length scale distribution with the mixing time.

Extension of the DSMC method to high pressure flows

A collision-limiter method, designated as equilibrium direct simulation Monte Carlo (eDSMC), is proposed to extend the DSMC technique to high pressure flows. The method is similar to collision-limiter schemes considered in the past with the important distinction that for inviscid flows, equilibrium is enforced in the entire flow by providing a sufficient number of collisions, based on pre-simulation testing. To test the method with standard DSMC and Navier–Stokes (NS) methods, axi-symmetric nozzle and embedded-channel flows are simulated and compared with experimental temperature data and pre-existing calculations, respectively. The method is shown to agree with third-order Eulerian nozzle flows and first-order channel flows. Chapman–Enskog theory is utilized to predict the range of initial conditions where eDSMC is potentially useful for modeling flows that contain viscous boundary layer regions. Comparison with supersonic nozzle data suggests that the eDSMC method is not adequate for capturing the large variation in flow length scales occurring in supersonic expansions into a vacuum. However, when eDSMC is used in combination with the baseline-DSMC method a near-exact solution is obtained with a considerable computational savings compared to the exact DSMC solution. Viscous flow channel calculations are found to agree well with an exact Navier–Stokes (NS) calculation for a small Knudsen number case as predicted by Chapman–Enskog theory.

Natural convection along a heated vertical plate immersed in a nonlinearly stratified medium: application to liquefied gas storage

We consider free convection driven by a heated vertical plate immersed in a nonlinearly stratified medium. The plate supplies a uniform horizontal heat flux to a fluid, the bulk of which has a stable stratification, characterized by a non-uniform vertical temperature gradient. This gradient is assumed to have a typical length scale of variation, denoted Z0, while Ψ0 is the order of magnitude of the related heat flux that crosses the medium vertically.We derive an analytic solution to the Boussinesq equations that extends the classical solution of Prandtl to the case of nonlinearly stratified media. This novel solution is asymptotically valid in the regime RaS ≫ 1, where RaS denotes the Rayleigh number of nonlinear stratification, based on Z0, Ψ0, and the physical properties of the medium.We then apply the new theory to the natural convection affecting the vapour phase in a liquefied pure gas tank (e.g. the cryogenic storage of hydrogen). It is assumed that the cylindrical storage tank is subject to a constant uniform heat flux on its lateral and top walls. We are interested in the vapour motion above a residual layer of liquid in equilibrium with the vapour. High-precision axisymmetric numerical computations show that the flow remains steady for a large range of parameters, and that a bulk stratification characterized by a quadratic temperature profile is undoubtedly present. The application of the theory permits a comparison of the numerical and analytic results, showing that the theory satisfactorily predicts the primary dynamical and thermal properties of the storage tank.

Within-urban variability in ambient air pollution: Comparison of estimation methods

An important component of air quality management and health risk assessment is improved understanding of spatial and temporal variability in pollutant concentrations. We compare, for Vancouver, Canada, three approaches for estimating within-urban spatiotemporal variability in ambient concentrations: spatial interpolation of monitoring data; an empirical/statistical model based on geographic analyses (“land-use regression”; LUR); and an Eulerian grid model (community multiscale air quality model, CMAQ). Four pollutants are considered—nitrogen oxide (NO), nitrogen dioxide (NO 2), carbon monoxide, and ozone—represent varying levels of spatiotemporal heterogeneity. Among the methods, differences in central tendencies (mean, median) and variability (standard deviation) are modest. LUR and CMAQ perform well in predicting concentrations at monitoring sites (average absolute bias: <50% for NO; <20% for NO 2). Monitors (LUR) offer the greatest (least) temporal resolution; LUR (monitors) offers the greatest (least) spatial resolution. Of note, the length scale of spatial variability is shorter for LUR (units: km; 0.3 for NO, 1 for NO 2) than for the other approaches (3–6 for NO, 4–6 for NO 2), indicating that the approaches offer different information about spatial attributes of air pollution. Results presented here suggest that for investigations incorporating spatiotemporal variability in ambient concentrations, the findings may depend on which estimation method is employed.

Marginally unstable Holmboe modes

Marginally unstable Holmboe modes for smooth density and velocity profiles are studied. For a large family of flows and stratification that exhibit Holmboe instability, it is shown that the modes with phase velocity equal to the maximum or the minimum velocity of the shear are marginally unstable. This allows us to determine the critical value of the control parameter R (expressing the ratio of the velocity variation length scale to the density variation length scale) above which Holmboe instability is present, Rcrit=2. Furthermore, systems for which the parameter R is very close to this critical value Rcrit are examined. For this case, an analytical expression for the dispersion relation of the complex phase speed c(k) in the unstable region is derived. The growth rate and the width of the region of unstable wavenumbers has a very strong (exponential) dependence on the deviation of R from the critical value. Two specific examples are examined and a physical interpretation of the results is described.

Can Wavelets Improve the Representation of Forecast Error Covariances in Variational Data Assimilation?

Abstract Two wavelet-based control variable transform schemes are described and are used to model some important features of forecast error statistics for use in variational data assimilation. The first is a conventional wavelet scheme and the other is an approximation of it. Their ability to capture the position and scale-dependent aspects of covariance structures is tested in a two-dimensional latitude–height context. This is done by comparing the covariance structures implied by the wavelet schemes with those found from the explicit forecast error covariance matrix, and with a non-wavelet-based covariance scheme used currently in an operational assimilation scheme. Qualitatively, the wavelet-based schemes show potential at modeling forecast error statistics well without giving preference to either position or scale-dependent aspects. The degree of spectral representation can be controlled by changing the number of spectral bands in the schemes, and the least number of bands that achieves adequate results is found for the model domain used. Evidence is found of a trade-off between the localization of features in positional and spectral spaces when the number of bands is changed. By examining implied covariance diagnostics, the wavelet-based schemes are found, on the whole, to give results that are closer to diagnostics found from the explicit matrix than from the nonwavelet scheme. Even though the nature of the covariances has the right qualities in spectral space, variances are found to be too low at some wavenumbers and vertical correlation length scales are found to be too long at most scales. The wavelet schemes are found to be good at resolving variations in position and scale-dependent horizontal length scales, although the length scales reproduced are usually too short. The second of the wavelet-based schemes is often found to be better than the first in some important respects, but, unlike the first, it has no exact inverse transform.

On the near-wall structure in reverse-flow and post-reattachment recovery regions of separated flow and its equivalence to the structure in wall and free-surface jets

The statistical and structural properties of two wall jets, one developing along a real wall and the other along a shear-free wall, are contrasted with the equivalent properties in the post-reattachment-recovery and reverse-flow regions of a separated backward-facing step flow, respectively. The study was motivated, principally, by the wish to isolate and understand the mechanisms responsible for the poor representation of the post-reattachment recovery returned by most RANS closures. It demonstrates a substantive commonality in the turbulent processes between, on the one hand, the real wall jet and post-reattachment recovery, and on the other hand, between the zero-wall-shear jet and the reattachment zone, as well as the reverse-flowing near-wall layer in the backward-facing-step flow. All regions under consideration are found to be characterised by an influential interaction between an outer shear layer and the wall, but an important distinction arises from the presence or absence of strong near-wall shear which introduces significant changes to the nature of this interaction: in the absence or near-absence of wall shear, the interaction is largely inviscid, associated with wall-blocking, while strong shear tends to shield the wall from the outer flow. The commonality between the two sets of flow conditions is explored by way of mean-flow properties, budgets, anisotropy maps, departures from local equilibrium, non-dimensional strain parameter, length-scale variations and structural properties. The study shows that the post-reattachment-recovery region, like the real wall jet, is far from the state of local equilibrium found in a standard boundary layer, that the recovery of the flow towards a state of equilibrium is slow, and that the near-wall region is strongly affected by turbulent diffusion associated with the migration of large-scale structures towards the wall. In statistical terms, this last process is represented by the importance of turbulent transport by third moments and pressure–velocity correlations. Within the reattachment and reverse–flow layer, near-wall shear is weak, and so is the shielding of the wall from the outer layer. In these regions, the flow shares many of the properties of the zero-wall-shear jet.

Fluctuations of the intergalactic UV background towards two lines of sight

Aims. We present a reanalysis of the He II Lyα absorption towards the quasars HS 1700+6416 and HE 2347-4342 using new high S/N, optical observations. Methods. An alternative analysis method is applied, which fits the high quality, optical HI data directly to the Hell spectrum. The results are compared to those inferred from standard line profile analyses. This new method enables us to derive redshift scales characterizing the fluctuations of the column density ratio η. Results. We find η changing smoothly with redshift on typical scales of Δ Z ∼ 0.01-0.03 corresponding to 8-24 h -1 Mpc comoving. The real length scales of variations of the column density ratio might be even larger, since part of the fluctuations may be caused by noise in the He II data and by systematic effects due to the applied method. However, variations on small scales cannot be ruled out completely. For η variations on scales of a few Mpc an amplitude of about ±1.5 dex cannot be excluded. The data shows an apparent correlation between low η regions and the presence of metal line absorbers, which corresponds to the more general correlation of low η and strong H absorption. Thermal line broadening is suggested as a probable explanation for this apparent correlation, since both fit methods would severely underestimate η for absorbers with N HI ≥ 10 13 cm -2 if the line width was dominated by thermal broadening. Indeed, lines located close to the cut-off of the b(N) distribution yield lower column density ratios compared to the whole sample, in particular if high density absorbers are considered. We argue that the apparent correlation of η with the strength of the H absorption is caused by insufficient consideration of thermal broadened lines by the standard analysis. As unbiased value of the column density ratio, we find η ∼ 80 in agreement with previous estimates.

Feedback between the Land Surface and Rainfall at Convective Length Scales

Abstract The surface fluxes of heat and moisture in semiarid regions are sensitive to spatial variability of soil moisture caused by convective rainfall. Under conditions typical of the Sahel, this variability may persist for several days after a storm, during which time it modifies the overlying boundary layer. A model of the land surface is used to quantify the dependence of surface fluxes of heat and moisture on antecedent rainfall amount, time since rainfall, and surface properties. Next, a coupled model of the land and atmosphere is used to characterize the boundary layer variability that results from this surface variability, and its dependence on factors including the length scale of the surface variability. Finally, two- and three-dimensional modeling of squall lines is used to examine the sensitivity of rainfall to boundary layer variability. Boundary layer variability tends to be greater for surface variability on long length scales, but squall-line rainfall shows the strongest response for anom...

Transition to hexagonal pattern under the variation of intrinsic length scales of a reaction diffusion system

The intrinsic length scales of a reaction diffusion system (Gierer-Meinhardt model) is varied by quasi-statically changing the diffusion constant of the activator and a transition from rolls to hexagon is detected. The transition is hysteretic or first order like. From stability analysis, we also analytically show the possibility of such transitions.

Scatter in Fatigue Crack Growth Rate in a Directionally Solidified Nickel-Base Superalloy

Abstract Directional solidification of nickel-base superalloy components for gas turbine rotor blades offers a significant improvement in creep strength while at the same time yielding an anisotropic, heterogeneous microstructure. In such a material, some of the continuum assumptions of linear elastic fracture mechanics begin to break down, and local (microscopic) conditions can lead to significant deviation in fatigue crack growth rate from the global (macroscopic) trend. Fatigue crack life prediction for a large population of fielded components requires a probabilistic treatment of the material fatigue crack growth behavior. A common approach to probabilistic fatigue crack life prediction involves sampling the Paris law coefficients from a large number of crack growth experiments, which can lead to effectively “smoothing” the local intraspecimen variability out of the model. The length scale of variability is discussed as it relates to material microstructure and crack life prediction. Results from fatigue crack growth experiments on a directionally solidified superalloy are presented and spatial variation in the fatigue crack growth rate is examined. Periodicity of the crack growth rate variation is compared with the scale of microstructural heterogeneity.

Atomic Scale Elastic Textures Coupled to Electrons in Superconductors

We present an atomic scale theory of lattice distortions using strain-related variables and their constraint equations. Our approach connects constrained atomic length scale variations to continuum elasticity and can describe elasticity at all length scales. We apply the general approach to a two-dimensional square lattice with a monatomic basis, and find the atomic scale elastic textures around a structural domain wall and a single defect, as exemplar textures. We clarify the microscopic origin of anisotropic gradient terms, some of which are included phenomenologically in Landau–Ginzburg theory. The obtained elastic textures are used to investigate the effects of elasticity-driven lattice deformation on the nanoscale electronic structure in superconductor by solving the Bogliubov–de Gennes equations with the electronic degrees of freedom coupled to the lattice ones. It is shown that the order parameter is depressed in the regions where the lattice deformation takes place. The calculated local density of states suggests the electronic structure is strongly modulated as a response to the lattice deformation—the elasticity propagates the electronic response over long distances. In particular, the trapping of low-lying quasiparticle states around the defects is possible. These predictions could be directly tested by STM experiments in superconducting materials.

Length Scale Analysis of Surface Energy Fluxes Derived from Remote Sensing

Wavelet multiresolution analysis was used to examine the variation in dominant length scales determined from remotely sensed airborne- and satellite-derived surface energy flux data. The wavelet cospectra are computed between surface radiometric temperature, fractional vegetation, and derived energy fluxes at airborne (12 m) and Advanced Very High Resolution Radiometer (AVHRR) (1000 m) resolutions. Length scale analysis of high-resolution data shows that small-scale variability in temperature dominates over other effects. Analysis of coarse-resolution data shows that small-scale variations in vegetation are important, although the large-scale variation in radiometric temperature dominates the derived fluxes. This is determined to be a result of the fact that, at smaller scales, the incoming solar radiation effect is muted by the small-scale variability in vegetation, temperature, and albedo, whereas at coarser scales, the large-scale effect of incoming radiation on temperature dominates over the smaller-scale features in surface variability.