scholarly journals Non-local dispersion and the reassessment of Richardson's t3-scaling law

2021 ◽  
Vol 932 ◽  
Author(s):  
G.E. Elsinga ◽  
T. Ishihara ◽  
J.C.R. Hunt
Keyword(s):  
Large Scale ◽  
Scaling Law ◽  
Taylor Dispersion ◽  
Small Time ◽  
Inertial Range ◽  
Small Scale ◽  
Mean Square ◽  
Dispersion Regime ◽  
Non Local ◽  
Scale Properties

The Richardson-scaling law states that the mean square separation of a fluid particle pair grows according to t3 within the inertial range and at intermediate times. The theories predicting this scaling regime assume that the pair separation is within the inertial range and that the dispersion is local, which means that only eddies at the scale of the separation contribute. These assumptions ignore the structural organization of the turbulent flow into large-scale shear layers, where the intense small-scale motions are bounded by the large-scale energetic motions. Therefore, the large scales contribute to the velocity difference across the small-scale structures. It is shown that, indeed, the pair dispersion inside these layers is highly non-local and approaches Taylor dispersion in a way that is fundamentally different from the Richardson-scaling law. Also, the layer's contribution to the overall mean square separation remains significant as the Reynolds number increases. This calls into question the validity of the theoretical assumptions. Moreover, a literature survey reveals that, so far, t3 scaling is not observed for initial separations within the inertial range. We propose that the intermediate pair dispersion regime is a transition region that connects the initial Batchelor- with the final Taylor-dispersion regime. Such a simple interpretation is shown to be consistent with observations and is able to explain why t3 scaling is found only for one specific initial separation outside the inertial range. Moreover, the model incorporates the observed non-local contribution to the dispersion, because it requires only small-time-scale properties and large-scale properties.

Kolmogorov’s contributions to the physical and geometrical understanding of small-scale turbulence and recent developments

1991 ◽  
Vol 434 (1890) ◽  
pp. 183-210 ◽  
Keyword(s):  
Turbulent Flows ◽  
Large Scale ◽  
Inertial Range ◽  
Small Scale ◽  
Small Scales ◽  
Recent Developments ◽  
Self Similar ◽  
Scale Turbulence ◽  
High Reynolds ◽  
Non Gaussian

This paper reviews how Kolmogorov postulated for the first time the existence of a steady statistical state for small-scale turbulence, and its defining parameters of dissipation rate and kinematic viscosity. Thence he made quantitative predictions of the statistics by extending previous methods of dimensional scaling to multiscale random processes. We present theoretical arguments and experimental evidence to indicate when the small-scale motions might tend to a universal form (paradoxically not necessarily in uniform flows when the large scales are gaussian and isotropic), and discuss the implications for the kinematics and dynamics of the fact that there must be singularities in the velocity field associated with the - 5/3 inertial range spectrum. These may be particular forms of eddy or ‘eigenstructure’ such as spiral vortices, which may not be unique to turbulent flows. Also, they tend to lead to the notable spiral contours of scalars in turbulence, whose self-similar structure enables the ‘box-counting’ technique to be used to measure the ‘capacity’ D K of the contours themselves or of their intersections with lines, D' K . Although the capacity, a term invented by Kolmogorov (and studied thoroughly by Kolmogorov & Tikhomirov), is like the exponent 2 p of a spectrum in being a measure of the distribution of length scales ( D' K being related to 2 p in the limit of very high Reynolds numbers), the capacity is also different in that experimentally it can be evaluated at local regions within a flow and at lower values of the Reynolds number. Thus Kolmogorov & Tikhomirov provide the basis for a more widely applicable measure of the self-similar structure of turbulence. Finally, we also review how Kolmogorov’s concept of the universal spatial structure of the small scales, together with appropriate additional physical hypotheses, enables other aspects of turbulence to be understood at these scales; in particular the general forms of the temporal statistics such as the high-frequency (inertial range) spectra in eulerian and lagrangian frames of reference, and the perturbations to the small scales caused by non-isotropic, non-gaussian and inhomogeneous large-scale motions.

Scaling Effect on Dynamic Impact Response of Structures in Fluid Fields

2013 ◽  
Author(s):  
Zhongheng Guo ◽  
Lingyu Sun ◽  
Taikun Wang ◽  
Junmin Du ◽  
Han Li ◽  
...  
Keyword(s):  
Numerical Simulation ◽  
Dimensional Analysis ◽  
Large Scale ◽  
Scaling Laws ◽  
Scaling Law ◽  
Fluid Pressure ◽  
Structural Safety ◽  
Scale Model ◽  
Small Scale ◽  
Water Tank

At the conceptual design phase of a large-scale underwater structure, a small-scale model in a water tank is often used for the experimental verification of kinematic principles and structural safety. However, a general scaling law for structure-fluid interaction (FSI) problems has not been established. In the present paper, the scaling laws for three typical FSI problems under the water, rigid body moves at a given kinematic equation or is driven by time-dependent fluids with given initial condition, as well as elastic-plastic body moves and then deforms subject to underwater impact loads, are investigated, respectively. First, the power laws for these three types of FSI problems were derived by dimensional analysis method. Then, the laws for the first two types were verified by numerical simulation. In addition, a multipurpose small-scale water sink test device was developed for numerical model updating. For the third type of problem, the dimensional analysis is no longer suitable due to its limitation on identifying the fluid pressure and structural stress, a simulation-based procedure for dynamics evaluation of large-scale structure was provided. The results show that, for some complex FSI problems, if small-scale prototype is tested safely, it doesn’t mean the full-scale product is also safe if both their pressure and stress are the main concerns, it needs further demonstration, at least by numerical simulation.

Strong MHD helical turbulence and the nonlinear dynamo effect

1976 ◽  
Vol 77 (2) ◽  
pp. 321-354 ◽  
Author(s):  
A. Pouquet ◽  
U. Frisch ◽  
J. Léorat
Keyword(s):  
Large Scale ◽  
Magnetic Energy ◽  
Three Dimensional ◽  
Power Laws ◽  
Inertial Range ◽  
Small Scale ◽  
Mhd Turbulence ◽  
Inverse Cascade ◽  
Small Scales ◽  
Dynamo Effect

To understand the turbulent generation of large-scale magnetic fields and to advance beyond purely kinematic approaches to the dynamo effect like that introduced by Steenbeck, Krause & Radler (1966)’ a new nonlinear theory is developed for three-dimensional, homogeneous, isotropic, incompressible MHD turbulence with helicity, i.e. not statistically invariant under plane reflexions. For this, techniques introduced for ordinary turbulence in recent years by Kraichnan (1971 a)’ Orszag (1970, 1976) and others are generalized to MHD; in particular we make use of the eddy-damped quasi-normal Markovian approximation. The resulting closed equations for the evolution of the kinetic and magnetic energy and helicity spectra are studied both theoretically and numerically in situations with high Reynolds number and unit magnetic Prandtl number.Interactions between widely separated scales are much more important than for non-magnetic turbulence. Large-scale magnetic energy brings to equipartition small-scale kinetic and magnetic excitation (energy or helicity) by the ‘Alfvén effect’; the small-scale ‘residual’ helicity, which is the difference between a purely kinetic and a purely magnetic helical term, induces growth of large-scale magnetic energy and helicity by the ‘helicity effect’. In the absence of helicity an inertial range occurs with a cascade of energy to small scales; to lowest order it is a −3/2 power law with equipartition of kinetic and magnetic energy spectra as in Kraichnan (1965) but there are −2 corrections (and possibly higher ones) leading to a slight excess of magnetic energy. When kinetic energy is continuously injected, an initial seed of magnetic field will grow to approximate equipartition, at least in the small scales. If in addition kinetic helicity is injected, an inverse cascade of magnetic helicity is obtained leading to the appearance of magnetic energy and helicity in ever-increasing scales (in fact, limited by the size of the system). This inverse cascade, predicted by Frischet al.(1975), results from a competition between the helicity and Alféh effects and yields an inertial range with approximately — 1 and — 2 power laws for magnetic energy and helicity. When kinetic helicity is injected at the scale linjand the rate$\tilde{\epsilon}^V$(per unit mass), the time of build-up of magnetic energy with scaleL[Gt ] linjis$t \approx L(|\tilde{\epsilon}^V|l^2_{\rm inj})^{-1/3}.$

scholarly journals Micromechanical study of potential scale effects in small-scale modelling of sinker tree roots

2021 ◽  
Author(s):  
Xingyu Zhang ◽  
◽  
Matteo Ciantia ◽  
Jonathan Knappett ◽  
Anthony Leung ◽  
...  
Keyword(s):  
Particle Size ◽  
Large Scale ◽  
Scale Effects ◽  
Distinct Element ◽  
Scaling Law ◽  
Scale Model ◽  
Small Scale ◽  
Tree Roots ◽  
Scale Modelling ◽  
Element Method

When testing an 1:N geotechnical structure in the centrifuge, it is desirable to choose a large scale factor (N) that can fit the small-scale model in a model container and avoid unwanted boundary effects, however, this in turn may cause scale effects when the structure is overscaled. This is more significant when it comes to small-scale modelling of sinker root-soil interaction, where root-particle size ratio is much lower. In this study the Distinct Element Method (DEM) is used to investigate this problem. The sinker root of a model root system under axial loading was analysed, with both upward and downward behaviour compared with the Finite Element Method (FEM), where the soil is modelled as a continuum in which case particle-size effects are not taken into consideration. Based on the scaling law, with the same prototype scale and particle size distribution, different scale factors/g-levels were applied to quantify effects of the ratio of root diameter (𝑑𝑟) to mean particle size (𝐷50) on the root rootsoil interaction.

Buoyancy- to inertial-range transition in forced stratified turbulence

2001 ◽  
Vol 427 ◽  
pp. 205-239 ◽  
Author(s):  
GEORGE F. CARNEVALE ◽  
M. BRISCOLINI ◽  
P. ORLANDI
Keyword(s):  
Wave Breaking ◽  
Large Scale ◽  
High Strain ◽  
Inertial Range ◽  
Large Eddy Simulations ◽  
Small Scale ◽  
Stratified Turbulence ◽  
Large Eddy ◽  
Scale Turbulence ◽  
Range Density

The buoyancy range, which represents a transition from large-scale wave-dominated motions to small-scale turbulence in the oceans and the atmosphere, is investigated through large-eddy simulations. The model presented here uses a continual forcing based on large-scale standing internal waves and has a spectral truncation in the isotropic inertial range. Evidence is presented for a break in the energy spectra from the anisotropic k−3 buoyancy range to the small-scale k−5/3 isotropic inertial range. Density structures that form during wave breaking and periods of high strain rate are analysed. Elongated vertical structures produced during periods of strong straining motion are found to collapse in the subsequent vertically compressional phase of the strain resulting in a zone or patch of mixed fluid.

scholarly journals A stochastic model for the polygonal tundra based on Poisson-Voronoi Diagrams

2012 ◽  
Vol 3 (1) ◽  
pp. 453-483
Author(s):  
F. Cresto Aleina ◽  
V. Brovkin ◽  
S. Muster ◽  
J. Boike ◽  
L. Kutzbach ◽  
...  
Keyword(s):  
Stochastic Model ◽  
Large Scale ◽  
Climate Models ◽  
Voronoi Diagrams ◽  
Field Studies ◽  
Statistical Characteristics ◽  
Small Scale ◽  
Landscape Response ◽  
Soil Heterogeneities ◽  
Scale Properties

Abstract. Sub-grid processes occur in various ecosystems and landscapes but, because of their small scale, they are not represented or poorly parameterized in climate models. These local heterogeneities are often important or even fundamental for energy and carbon balances. This is especially true for northern peatlands and in particular for the polygonal tundra where methane emissions are strongly influenced by spatial soil heterogeneities. We present a stochastic model for the surface topography of polygonal tundra using Poisson-Voronoi Diagrams and we compare the results with available recent field studies. We analyze seasonal dynamics of water table variations and the landscape response under different scenarios of precipitation income. We upscale methane fluxes by using a simple idealized model for methane emission. Hydraulic interconnectivities and large-scale drainage may also be investigated through percolation properties and thresholds in the Voronoi graph. The model captures the main statistical characteristics of the landscape topography, such as polygon area and surface properties as well as the water balance. This approach enables us to statistically relate large-scale properties of the system taking into account the main small-scale processes within the single polygons.

scholarly journals Approach and Development of Effective Models for Simulation of Thermal Stratification and Mixing Induced by Steam Injection into a Large Pool of Water

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Hua Li ◽  
Walter Villanueva ◽  
Pavel Kudinov
Keyword(s):  
Thermal Stratification ◽  
Large Scale ◽  
Steam Injection ◽  
Small Time ◽  
Small Scale ◽  
Computationally Efficient ◽  
Thermal Mixing ◽  
Large Pool ◽  
Effective Models ◽  
Contact Condensation

Steam venting and condensation in a large pool of water can lead to either thermal stratification or thermal mixing. In a pressure suppression pool (PSP) of a boiling water reactor (BWR), consistent thermal mixing maximizes the capacity of the pool while the development of thermal stratification can reduce the steam condensation capacity of the pool which in turn can lead to pressure increase in the containment and thereafter the consequences can be severe. Advanced modeling and simulation of direct contact condensation in large systems remain a challenge as evident in commercial and research codes mainly due to small time-steps necessary to resolve contact condensation in long transients. In this work, effective models, namely, the effective heat source (EHS) and effective momentum source (EMS) models, are proposed to model and simulate thermal stratification and mixing during a steam injection into a large pool of water. Specifically, the EHS/EMS models are developed for steam injection through a single vertical pipe submerged in a pool under two condensation regimes: complete condensation inside the pipe and chugging. These models are computationally efficient since small scale behaviors are not resolved but their integral effect on the large scale flow structure in the pool is taken into account.

scholarly journals Wave turbulence in the two-layer ocean model

2014 ◽  
Vol 756 ◽  
pp. 309-327 ◽  
Author(s):  
Katie L. Harper ◽  
Sergey V. Nazarenko ◽  
Sergey B. Medvedev ◽  
Colm Connaughton
Keyword(s):  
Large Scale ◽  
Resonance Condition ◽  
Ocean Model ◽  
Wave Turbulence ◽  
Small Scale ◽  
Coupled Equations ◽  
Canonical Variables ◽  
Beta Effect ◽  
Non Local ◽  
Baroclinic Modes

AbstractThis paper looks at the two-layer ocean model from a wave-turbulence (WT) perspective. A symmetric form of the two-layer kinetic equation for Rossby waves is derived using canonical variables, allowing the turbulent cascade of energy between the barotropic and baroclinic modes to be studied. It is already well known that in two-layers, energy is transferred via triad interactions from the large-scale baroclinic modes to the baroclinic and barotropic modes at the Rossby deformation scale, where barotropization takes place, and from there to the large-scale barotropic modes via an inverse transfer. However, by applying WT theory, we find that energy is transferred via dominant $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\{+--\}$ triads with one barotropic component and two baroclinic components, and that the direct transfer of energy is local and the inverse energy transfer is non-local. We study this non-locality using scale separation and obtain a system of coupled equations for the small-scale baroclinic component and the large-scale barotropic component. Since the total energy of the small-scale component is not conserved, but the total barotropic plus baroclinic energy is conserved, the baroclinic energy loss at small scales will be compensated by the growth of the barotropic energy at large scales. Using the frequency resonance condition, we show that in the presence of the beta-effect this transfer is mostly anisotropic and mostly to the zonal component.

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