Clear Scale Separation Research Articles

Hot-wire spatial resolution issues in adverse pressure gradient turbulent boundary layers

The effect of a finite length of hot-wire probe sensor length on the measured streamwise velocity fluctuations is well understood in canonical wall-bounded flow, where the small-scale energy has been found to be universal and invariant with Reynolds number. A straightforward application of that assumption to non-canonical flows such as strong adverse pressure gradient (APG) flows has, however, been hampered since the effect of Re and APG could not conclusively be studied separately due to the lack of data with a clear scale separation. The present experimental investigation at Reτ≈4000 in weak, moderate and strong APGs with different wire length shows that spatial averaging effects are not only limited to the inner layer. A note of caution is hence warranted for measurements that seemingly try to take the bias effect of spatial attenuation into account by performing measurements with albeit long but fixed viscous-scaled wire length.

Multiscale modeling of linear elastic heterogeneous structures via localized model order reduction

AbstractIn this article, a methodology for fine scale modeling of large scale linear elastic structures is proposed, which combines the variational multiscale method, domain decomposition and model order reduction. The influence of the fine scale on the coarse scale is modeled by the use of an additive split of the displacement field, addressing applications without a clear scale separation. Local reduced spaces are constructed by solving an oversampling problem with random boundary conditions. Herein, we inform the boundary conditions by a global reduced problem and compare our approach using physically meaningful correlated samples with existing approaches using uncorrelated samples. The local spaces are designed such that the local contribution of each subdomain can be coupled in a conforming way, which also preserves the sparsity pattern of standard finite element assembly procedures. Several numerical experiments show the accuracy and efficiency of the method, as well as its potential to reduce the size of the local spaces and the number of training samples compared to the uncorrelated sampling.

Bolgiano–Obukhov scaling in two-dimensional isotropic convection

The existence of Bolgiano–Obukhov (BO) scaling in Rayleigh–Bénard convection (RBC) has long been speculated. However, due to the inhomogeneity and anisotropy of the flow, and the lack of clear scale separation, no conclusive evidence has been found. To avoid these non-ideal factors, we construct an idealized isotropic convection system by introducing an additional horizontal buoyancy field to RBC in a doubly periodic domain. We focus on the two-dimensional case so that its upscale kinetic energy flux can enable a long inertial range for detecting the BO scaling. Through direct numerical simulations of this system, we justify the existence of BO scaling using second- and third-order structure functions, which are in good agreement with our theoretically obtained scaling relations from the Kármán–Howarth–Monin equations. These theoretical and numerical results provide direct support for the conjecture that the existence of the BO scaling in RBC is associated with the inverse kinetic energy cascade. For higher-order structure functions, we found strong intermittent effects in the buoyancy field, but not in the velocity. By comparing the present system with the canonical anisotropic RBC in a periodic domain, the effects of anisotropy on the scaling properties are elucidated.

Direct numerical simulation of forced thermal convection in square ducts up to

We carry out direct numerical simulations (DNS) of flow in a turbulent square duct by focusing on heat transfer effects, considering the case of unit Prandtl number. Reynolds numbers up to ${{Re}}_{\tau } \approx 2000$ are considered that are much higher than in previous studies, and that yield clear scale separation between inner- and outer-layer dynamics. Close similarity between the behaviour of the temperature and the streamwise velocity fields is confirmed as in previous studies related to plane channels and pipes. Just like the mean velocity, the mean temperature is found to exhibit logarithmic layers as a function of the nearest wall, but with a different slope. The most important practical implication is the validity of the traditional hydraulic diameter as the correct reference length for reporting heat transfer data, as we show rigorously here. Temperature and velocity fluctuations also have similar behaviour, but apparently logarithmic growth of their inner-scaled peak variances is not observed here, unlike in canonical wall-bounded flows. Analysis of the split contributions to the heat transfer coefficient shows that mean cross-stream convection associated with secondary motions is responsible for about $5\,\%$ of the total. Finally, we use the DNS database to highlight shortcomings of traditional linear closures for the turbulent heat flux, and show that substantial modelling improvement in principle may be obtained by retaining at least the three terms in the vector polynomial integrity basis expansion.

Direct FE[formula omitted] modeling of heterogeneous materials with a micromorphic computational homogenization framework

Direct FE2, recently proposed by Tan et al. (2020), is a concurrent multilevel modeling approach, which homogenizes a representative volume element (RVE) to a Cauchy continuum but calls for only a single finite element analysis. In this paper, the method is extended to homogenization towards a micromorphic continuum. It allows us to capture the deformation of composites with soft inclusions in the absence of scale separation and the size effects therein. Besides the displacement field, additional morphic fields are employed at the macroscale to characterize the micro-scale heterogeneity. Compared with the classical FE2 approach, the only effort in the development of the Direct FE2 is to formulate the macro–micro transition as kinematic constraints between the macro and micro DOFs, that are implemented through multi-point constraints (MPC) in Abaqus®. Automatically, the MPCs enable the micro–macro transition in an energetically consistent way and correlate the microscopic traction back to the macroscopic nodal internal force. The performance of the novel Direct FE2 modeling is benchmarked with three representative numerical simulations, where a better predictive capability than the classical one is demonstrated in situations without a clear separation of scales.

Waves in the gas centrifuge: asymptotic theory and similarities with the atmosphere

We study the stratified gas in a rapidly rotating centrifuge as a model for the Earth's atmosphere. Based on methods of perturbation theory, it is shown that in certain regimes, internal waves in the gas centrifuge have the same dispersion relation to leading order as their atmospheric siblings. Assuming an air filled centrifuge with a radius of around 50 cm, the optimal rotational frequency for realistic atmosphere-like waves is around 10 000 revolutions per minute. Using gases of lower heat capacities at constant pressure, such as xenon, the rotational frequencies can be even halved to obtain the same results. Similar to the atmosphere, it is feasible in the gas centrifuge to generate a clear scale separation of wave frequencies and therefore phase speeds between acoustic waves and internal waves. In addition to the centrifugal force, the Coriolis force acts in the same plane. However, its influence on axially homogeneous internal waves appears only as a higher-order correction. We conclude that the gas centrifuge provides an unprecedented opportunity to investigate atmospheric internal waves experimentally with a compressible working fluid.

Numerical homogenization beyond scale separation

Numerical homogenization is a methodology for the computational solution of multiscale partial differential equations. It aims at reducing complex large-scale problems to simplified numerical models valid on some target scale of interest, thereby accounting for the impact of features on smaller scales that are otherwise not resolved. While constructive approaches in the mathematical theory of homogenization are restricted to problems with a clear scale separation, modern numerical homogenization methods can accurately handle problems with a continuum of scales. This paper reviews such approaches embedded in a historical context and provides a unified variational framework for their design and numerical analysis. Apart from prototypical elliptic model problems, the class of partial differential equations covered here includes wave scattering in heterogeneous media and serves as a template for more general multi-physics problems.

Applications of matrix factorization methods to climate data

Abstract. An initial dimension reduction forms an integral part of many analyses in climate science. Different methods yield low-dimensional representations that are based on differing aspects of the data. Depending on the features of the data that are relevant for a given study, certain methods may be more suitable than others, for instance yielding bases that can be more easily identified with physically meaningful modes. To illustrate the distinction between particular methods and identify circumstances in which a given method might be preferred, in this paper we present a set of case studies comparing the results obtained using the traditional approaches of empirical orthogonal function analysis and k-means clustering with the more recently introduced methods such as archetypal analysis and convex coding. For data such as global sea surface temperature anomalies, in which there is a clear, dominant mode of variability, all of the methods considered yield rather similar bases with which to represent the data while differing in reconstruction accuracy for a given basis size. However, in the absence of such a clear scale separation, as in the case of daily geopotential height anomalies, the extracted bases differ much more significantly between the methods. We highlight the importance in such cases of carefully considering the relevant features of interest and of choosing the method that best targets precisely those features so as to obtain more easily interpretable results.

An improved multi-scale two phase method for bubbly flows

Multiple spatial scale is an important characteristic of two phase flow phenomena. The micro-scale and macro-scale flow structures are obviously different in flow state and have different effects on the mass, momentum and energy transfer between two phases. Different modeling approaches have been developed for each scale physical phenomenon in traditional numerical simulations. However, it is difficult to simulate two phase flow systems with multi-scale flow structures simultaneously. In order to address this problem, a multi-scale two phase method is developed based on the combination of Volume of Fluid (VOF) interface capture method and Euler–Lagrange particle tracking method. The fundamental assumption of the present method is that there is a clear scale separation between VOF interfaces and bubbles. Therefore, VOF method with artificial compressive algorithm is used to simulate the dynamic evolution of macro-scale air-water interface. While Euler–Lagrange method is used to track the micro-scale bubbles that cannot be captured in grids. Collision, coalescence and breakup of Lagrange bubbles and two-way coupling are fully considered to construct a comprehensive micro-bubble solving procedure. Transformation criteria and the corresponding algorithms between micro-scale and macro-scale flow structures are designed and discussed in detail. In addition, a new curvature-based algorithm for the transformation from VOF interface to Lagrange bubbles is proposed. Simulations of typical two phase flow problems involving multi-scale flow transformation are carried out to test the performance of the multi-scale solver. Results indicate that the multi-scale two phase method performs significantly better than the pure VOF method in capturing micro-scale phenomena. Besides, the curvature-based transformation algorithm proposed in this paper is proved to be more precise and efficient than the previous identify-based one. From the perspective of simulation accuracy and efficiency, the multi-scale two phase method is more promising for the simulation of actual complex two-phase flows.

A benchmark study of the multiscale and homogenization methods for fully implicit multiphase flow simulations

Accurate simulation of multiphase flow in subsurface formations is challenging, as the formations span large length scales (km) with high-resolution heterogeneous properties. To deal with this challenge, different multiscale methods have been developed. Such methods construct coarse-scale systems, based on a given high-resolution fine-scale system. Furthermore, they are amenable to parallel computing and allow for a-posteriori error control. The multiscale methods differ from each other in the way the transition between the different scales is made. Multiscale (finite element and finite volume) methods compute local basis functions to map the solutions (e.g. pressure) between coarse and fine scales. Instead, homogenization methods solve local periodic problems to determine effective models and parameters (e.g. permeability) at a coarser scale. It is yet unknown how these two methods compare with each other, especially when applied to complex geological formations, with no clear scale separation in the property fields. This paper develops the first comparison benchmark study of these two methods and extends their applicability to fully implicit simulations using the algebraic dynamic multilevel (ADM) method. At each time step, on the given fine-scale mesh and based on an error analysis, the fully implicit system is solved on a dynamic multilevel grid. The entries of this system are obtained by using multiscale local basis functions (ADM-MS), and, respectively, by homogenization over local domains (ADM-HO). Both sets of local basis functions (ADM-MS) and local effective parameters (ADM-HO) are computed at the beginning of the simulation, with no further updates during the multiphase flow simulation. The two methods are extended and implemented in the same open-source DARSim2 simulator (https://gitlab.com/darsim2simulator), to provide fair quality comparisons. The results reveal insightful understanding of the two approaches, and qualitatively benchmark their performance. It is re-emphasized that the test cases considered here include permeability fields with no clear scale separation. The development of this paper sheds new lights on advanced multiscale methods for simulation of coupled processes in porous media.

Energy transfers in magnetohydrodynamic shear turbulence

Astrophysical flows span a broad range of scales and thus require sub-grid modeling. We compute the energy fluxes in the Fourier and physical space for the turbulent magnetized shearing box simulations with a weak net vertical magnetic flux. The energy fluxes are dominated by the injection term from the mid-to-large scale range due to the Reynolds and Maxwell stresses in both cases while the small scales are dominated by dissipation. Since the nature of turbulent cascades requires a clear separation of scales, our work is unable to address these questions conclusively. But because our work is the first of its nature in trying to connect the Fourier and physical space energy flux behavior, we are able to make connections between the two profiles.

Investigating the Relative Contributions of Charge Deposition and Turbulence in Organizing Charge within a Thunderstorm

Abstract Large-eddy-resolving simulations using the Collaborative Model for Multiscale Atmospheric Simulation (COMMAS), which contains microphysical charging and branched-lightning parameterizations, produce much more complex net charge structures than conventionally visualized from previous observations, simulations, and conceptual diagrams. Many processes contribute to the hydrometeor charge budget within a thunderstorm, including advection, hydrometeor differential sedimentation, subgrid turbulent mixing and diffusion, ion drift, microphysical separation, and the attachment of ion charge deposited by the lightning channel. The lightning deposition, sedimentation, and noninductive charging tendencies contribute the most overall charge at relatively large scales, while the advection tendency, from resolved turbulence, provides the most “texture” at small scales to the net charge density near the updraft region of the storm. The scale separation increases for stronger storm simulations. In aggregate, lightning deposition and sedimentation resemble the smoother distribution of the electric potential, while evidence suggests individual flashes could be responding to the fine texture in the net charge. The clear scale separation between the advection and other net charge tendencies suggest the charge advection is most capable of providing net charge texture; however, a clear-cut causality is not obtained from this study.

Role of overturns in optimal mixing in stratified mixing layers

Turbulent mixing plays a major role in enabling the large-scale ocean circulation. The accuracy of mixing rates estimated from observations depends on our understanding of basic fluid mechanical processes underlying the nature of turbulence in a stratified fluid. Several of the key assumptions made in conventional mixing parameterizations have been increasingly scrutinized in recent years, primarily on the basis of adequately high resolution numerical simulations. We add to this evidence by compiling results from a suite of numerical simulations of the turbulence generated through stratified shear instability processes. We study the inherently intermittent and time-dependent nature of wave-induced turbulent life cycles and more specifically the tight coupling between inherently anisotropic scales upon which small-scale isotropic turbulence grows. The anisotropic scales stir and stretch fluid filaments enhancing irreversible diffusive mixing at smaller scales. We show that the characteristics of turbulent mixing depend on the relative time evolution of the Ozmidov length scale$L_{O}$compared to the so-called Thorpe overturning scale$L_{T}$which represents the scale containing available potential energy upon which turbulence feeds and grows. We find that when$L_{T}\sim L_{O}$, the mixing is most active and efficient since stirring by the largest overturns becomes ‘optimal’ in the sense that it is not suppressed by ambient stratification. We argue that the high mixing efficiency associated with this phase, along with observations of$L_{O}/L_{T}\sim 1$in oceanic turbulent patches, together point to the potential for systematically underestimating mixing in the ocean if the role of overturns is neglected. This neglect, arising through the assumption of a clear separation of scales between the background mean flow and small-scale quasi-isotropic turbulence, leads to the exclusion of an highly efficient mixing phase from conventional parameterizations of the vertical transport of density. Such an exclusion may well be significant if the mechanism of shear-induced turbulence is assumed to be representative of at least some turbulent events in the ocean. While our results are based upon simulations of shear instability, we show that they are potentially more generic by making direct comparisons with$L_{T}-L_{O}$data from ocean and lake observations which represent a much wider range of turbulence-inducing physical processes.

Simulation and validation of a spatially evolving turbulent boundary layer up to [formula omitted

Results of a finely resolved large-eddy simulation (LES) of a spatially developing zero-pressure-gradient turbulent boundary layer up to a Reynolds number of Reθ=8300 are presented. The very long computational domain provides substantial assessment for suggested high Reynolds number (Re) trends. Statistics, integral quantities and spectral data are validated using high quality direct numerical simulation (DNS) ranging up to Reθ=4300 and hot-wire measurements covering the remaining Re-range. The mean velocity, turbulent fluctuations, skin friction, and shape factor show excellent agreement with the reference data. Through utilisation of filtered DNS, subtle differences between the LES and DNS could to a large extent be explained by the reduced spanwise resolution of the LES. Spectra and correlations for the streamwise velocity and the wall-shear stress evidence a clear scale-separation and a footprint of large outer scales on the near-wall small scales. While the inner peak decreases in importance and reduces to 4% of the total energy at the end of the domain, the energy of the outer peak scales in outer units. In the near-wall region a clear k-1 region emerges. Consideration of the two-dimensional spectra in time and spanwise space reveals that an outer time scale λt≈10δ99/U∞, with the boundary layer thickness δ99 and free-stream velocity U∞, is the correct scale throughout the boundary layer rather than the transformed streamwise wavelength multiplied by a (scale independent) convection velocity. Maps for the covariance of small scale energy and large scale motions exhibit a stronger linear Re dependence for the amplitude of the off-diagonal peak compared to the diagonal one, thereby indicating that the strength of the amplitude modulation can only qualitatively be assessed through the diagonal peak. In addition, the magnitude of the wall-pressure fluctuations confirms mixed scaling, and pressure spectra at the highest Re give a first indication of a −7/3 wave number dependence.

The supernova-regulated ISM – II. The mean magnetic field

Abstract The origin and structure of the magnetic fields in the interstellar medium of spiral galaxies is investigated with 3D, non-ideal, compressible magnetohydrodynamic simulations, including stratification in the galactic gravity field, differential rotation and radiative cooling. A rectangular domain, 1 × 1 × 2 kpc3 in size, spans both sides of the galactic mid-plane. Supernova explosions drive transonic turbulence. A seed magnetic field grows exponentially to reach a statistically steady state within 1.6 Gyr. Following Germano (1992), we use volume averaging with a Gaussian kernel to separate magnetic field into a mean field and fluctuations. Such averaging does not satisfy all Reynolds rules, yet allows a formulation of mean-field theory. The mean field thus obtained varies in both space and time. Growth rates differ for the mean-field and fluctuating field and there is clear scale separation between the two elements, whose integral scales are about 0.7 and 0.3 kpc, respectively.

Flow Over Partially Forested Ridges

Numerical simulations of flow over hills that are partially covered with a forest canopy are performed. This represents a much more realistic situation than previous studies that have generally concentrated on hills that are fully-forested. The results show that the flow over the hill is sensitive to where on the hill the forest is positioned. In particular, for low slopes flow separation is predominantly located within the forest on the lee slope. This has implications for the transport of scalars in the forest canopy. For large hills the results show more variability in scalar concentrations within the canopy compared to either a fully-forested hill or a patch of forest over flat terrain. These results are likely to have implications for a range of applications including the siting and interpretation of flux measurements over forests in complex terrain, predicting wind damage to trees and wind-farm developments. Calculation of the hill-induced pressure drag and canopy-plus-surface stress shows a strong sensitivity to the position of the forest relative to the hill. Depending on the position of the forest the individual drag terms may be strongly enhanced or reduced and may even change sign. The net impact is generally to reduce the total drag compared to an equivalent fully-forested hill, but the amount of the reduction depends strongly on the position of the forest canopy on the hill. In many cases with large, wide hills there is a clear separation of scales between the adjustment of the canopy to a forest edge (of order 6 − 8L c, where L c is the canopy adjustment length scale) and the width of the hill. This separation means that the hill-induced pressure and flow fields and the forest-edge induced pressure and flow fields can in some sense be considered as acting separately. This provides a means of explaining the combined effects of partial forestation and terrain. It also offers a simple method for modelling the changes in drag over a hill due to partial forest cover by considering the impact of the hill and the partial canopy separately. Scaling arguments based on this idea successfully collapse the modelled drag over a range of different hill widths and heights and for different canopy parameters. This offers scope for a relatively simple parametrization of the effects of partial forest cover on the drag over a hill.

Multiscale Modelling Strategies for Heterogeneous Materials

This paper considers modelling strategies for the multiscale analysis of heterogeneous materials. Initial focus is on computational homogenization (e.g. [1-4]) which represents a powerful, albeit computationally intensive, technique for capturing the influence of an evolving microstrucuture on the overall macroscopic behaviour. First-order homogenization assumes that the representative volume element (RVE) is infinitely small compared to the macroscopic characteristic length scale and that there is clear separation of scales. Second-order schemes [2,4,5] are subsequently discussed as a possible solution. This approach introduces a material length scale into the macroscopic constitutive model and potentially enables geometric size effects to be captured. However, the size of the RVE now becomes an important parameter for the macroscopic response and there is no clear way of determining this length scale. Second-order computational homogenization is demonstrated in the context of an indentation problem. In situations where the RVE cannot be defined and there is no clear separation of scales, computational homogenization is not applicable and full resolution of the microstructure may be necessary. This paper summarises an efficient solution strategy for such situations following the work of Miehe and Bayreuther [6], with particular focus on fracturing heterogeneous materials, which utilises a geometric multi-grid preconditioner and the scale transition methodology from computational homogenization. This solution strategy is coupled with the use of the hybrid-Trefftz stress elements for modelling cohesive cracking [7]. The resulting modelling approach is demonstrated with the analysis of concrete dog-bone specimens where the different phases of matrix, aggregate and interfacial transition zone are explicitly modelled. Such an approach is still computationally intensive and so this paper also presents a novel hybrid multiscale approach that avoids the need for the finite element mesh to fully resolve all heterogeneities [8]. Classical micromechanics is utilised to enhance the approximation space in order to capture the effect of microstructural inclusions. In comparison to where the finite element mesh fully resolves the microstructure, this novel approach significantly reduces the number of degrees of freedom required while still accurately capturing the influence of the inclusions.

Small-Scale Properties of Turbulent Rayleigh-Bénard Convection

The properties of the structure functions and other small-scale quantities in turbulent Rayleigh-Bénard convection are reviewed, from an experimental, theoretical, and numerical point of view. In particular, we address the question of whether, and if so where in the flow, the so-called Bolgiano-Obukhov scaling exists, i.e., Sθ(r) ∼ r2/5 for the second-order temperature structure function and Su(r) ∼ r6/5 for the second-order velocity structure function. Apart from the anisotropy and inhomogeneity of the flow, insufficiently high Rayleigh numbers, and intermittency corrections (which all hinder the identification of such a potential regime), there are also reasons, as a matter of principle, why such a scaling regime may be limited to at most a decade, namely the lack of clear scale separation between the Bolgiano length scale LB and the height of the cell.

Mixing Measurements on an Equatorial Ocean Mooring

Abstract Vaned, internally recording instruments that measure temperature fluctuations using FP07 thermistors, including fluctuations in the turbulence wavenumber band, have been built, tested, and deployed on a Tropical Atmosphere Ocean (TAO) mooring at 0°, 140°W. These were supplemented with motion packages that measure linear accelerations, from which an assessment of cable displacement and speed was made. Motions due to vortex-induced vibrations caused by interaction of the mean flow with the cable are small (rms < 0.15 cable diameters) and unlikely to affect estimates of the temperature variance dissipation rate χT. Surface wave–induced cable motions are significant, commonly resulting in vertical displacements of ±1 m and vertical speeds of ±0.5 m s−1 on 2–10-s periods. These motions produce an enhancement to the measurement of temperature gradient in the surface wave band herein that is equal to the product of the vertical cable speed and the vertical temperature gradient (i.e., dT/dt ∼ wcdT/dz). However, the temperature gradient spectrum is largely unaltered at higher and lower frequencies; in particular, there exists a clear scale separation between frequencies contaminated by surface waves and the turbulence subrange. The effect of cable motions on spectral estimates of χT is evaluated and determined to result in acceptably small uncertainties (< a factor of two 95% of the time, based on 60-s averages). Time series of χT and the inferred turbulent kinetic energy dissipation rate ε are consistent with historical data from the same equatorial location.

Mixing‐induced precipitation: Experimental study and multiscale numerical analysis

Laboratory experiments, pore‐scale simulations, and continuum (Darcy) ‐scale simulations were used to study mixing‐induced precipitation in porous media. In the experimental investigation, solutions containing Na2CO3 and CaCl2 were each injected in different halves of a quasi two‐dimensional flow cell filled with quartz sand. As a result of the in situ mixing between the two solutions, a narrow calcite precipitate layer (less than 5 mm wide) of more or less uniform width was formed between the individual solutions. Pore‐scale simulations were conducted to help understand the mechanism of precipitation layer formation. The effect of the Peclet number, Pe, and the Damköhler number, Da, on mixing induced precipitation was also investigated. Pore‐scale simulations revealed the presence of large pore‐scale concentration gradients. This, and the presence of features, such as the precipitation layer, with characteristic lengths on the order of the average sand grain diameter, indicate the absence of a clear scale separation required for the strict derivation of Darcy‐scale advection‐dispersion equations. Nevertheless, we found that an adaptive high‐resolution model based on advection‐dispersion equations with grid sizes in the mixing zone smaller than the size of the sand grains can qualitatively reproduce the essential features of the experiment. As an alternative to computationally expensive high‐resolution simulations, we proposed new forms for the homogeneous and heterogeneous reaction terms in Darcy‐scale advection dispersion equations. These terms involve transport and mixing indices that account for highly nonuniform pore‐scale concentration distributions and highly localized reactions. The proposed model accurately estimates the changes in solute concentrations due to homogenous and heterogeneous reactions during precipitation of minerals, observed in the pore‐scale simulations, while conventional low‐resolution advective‐dispersion equations produced erroneous results.