Modal interactions and energy transfers in isotropic turbulence as predicted by local energy transfer theory

Energy transfer processes in decaying, three-dimensional, isotropic turbulence are investigated using numerical results from local energy transfer (LET) theory. The study covers a wide range of evolved, microscale Reynolds numbers (5 < R_λ < 250). It is found that the energy transfer is mainly local (between scales of similar size), but there are also some signs of nonlocal transfer at higher Reynolds number. The nature of the underlying triad-wavenumber interactions, on the other hand, seems to depend on both the Reynolds number and the wavenumber range of interest. In the energy containing and dissipation ranges, both local (all three scales of the triad interaction are of comparable size) and nonlocal (one scale is much larger than the remaining two) interactions are important, with the latter becoming more dominant as the Reynolds number increases. But our nonlocal interactions tend to be less severe than those observed by Domaradzki and Rogallo. More significantly, in the inertial range of high Reynolds number flows, the LET theory predicts dominance of local and near-local interactions. While this is contrary to the result from eddy damped quasi-normal Markovian theory that the important triad interactions are mainly nonlocal, it is closer to the Kolmogorov picture of turbulence. Another interesting result is that, despite their inherent differences, the LET theory and the full simulation of Ohkitani and Kida predict inertial-range values for the energy transfer locality function in fairly good agreement, not only with each other, but also with the analytical closure theory result for infinite Reynolds number, stationary turbulence by Kraichnan. The calculated values reveal that the contributions to the net energy transfer are predominantly from near-local interactions (scale ratios ≈ 4), which is indicative of cancellation of large numbers of highly nonlocal interactions.     

Numerical simulation of a developed shear turbulence

Results of two-dimensional numerical studies of turbulence that arises at the interface of two flows of poorly compressible gases are described. The results were obtained using a MAKh software system. The interrelation between spatial and time problems on the development of a turbulent zone induced by shear instability is analyzed. A constant that characterizes the degree of turbulent shear mixing is calculated. The effect of the density difference of the mixing fluids on the growth rate of the turbulence zone is studied. For all density differences considered, the coefficient of heterogeneity of the resultant mixture is evaluated. Institute for Technical Physics, Snezhinsk 456770. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 1, pp. 77–83, January–February, 2000.     

Spectrum of developed turbulence of incompressible viscoelastic fluids

It is shown that fine-scale turbulent motions of a viscoelastic fluid damp out as in a viscous fluid with some effective viscosity dependent on the scale of the motion. The elasticity of deformation results in a diminution in the dissipativity of the turbulence, and hence, to an elongation of the high-frequency tail of the spectrum for a given energy influx.Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 23–33, January–February, 1972.     

Nonlinear gravity-wave interactions in stratified turbulence

To investigate the dynamics of gravity waves in stratified Boussinesq flows, a model is derived that consists of all three-gravity-wave-mode interactions (the GGG model), excluding interactions involving the vortical mode. The GGG model is a natural extension of weak turbulence theory that accounts for exact three-gravity-wave resonances. The model is examined numerically by means of random, large-scale, high-frequency forcing. An immediate observation is a robust growth of the so-called vertically sheared horizontal flow (VSHF). In addition, there is a forward transfer of energy and equilibration of the nonzero-frequency (sometimes called “fast”) gravity-wave modes. These results show that gravity-wave-mode interactions by themselves are capable of systematic interscale energy transfer in a stratified fluid. Comparing numerical simulations of the GGG model and the full Boussinesq system, for the range of Froude numbers (Fr) considered (0.05 ≤ Fr ≤ 1), in both systems the VSHF is hardest to resolve. When adequately resolved, VSHF growth is more vigorous in the GGG model. Furthermore, a VSHF is observed to form in milder stratification scenarios in the GGG model than the full Boussinesq system. Finally, fully three-dimensional nonzero-frequency gravity-wave modes equilibrate in both systems and their scaling with vertical wavenumber follows similar power-laws. The slopes of the power-laws obtained depend on Fr and approach ?2 (from above) at Fr = 0.05, which is the strongest stratification that can be properly resolved with our computational resources.     

Flow-thermodynamics interactions in rapidly-sheared compressible turbulence

We investigate the behavior of flow variables, thermodynamic variables and their interaction in rapidly sheared (S) homogeneous compressible turbulence using rapid distortion theory (RDT). We subject an initially isotropic and incompressible flow field to homogeneous shear-rate of various strengths quantified by a gradient Mach number (M _g ) based on characteristic wavenumber. Our objective is to characterize the behavior of flow/thermodynamic fluctuations and their linear interactions during the course of turbulence evolution. Even though the mean shear-rate is held constant, the gradient Mach number progressively diminishes with time as the relevant wavenumber increases due to the mean deformation. The evolution exhibits three distinct phases which we categorize based on the character of pressure as: (i) Pressure-released (PR) stage which is observed when ${St < \sqrt{M_{g0}}}$ and pressure effects are negligible; (ii) Wave-character (WC) stage wherein ${\sqrt{M_{g0}} < St < M_{g0}}$ and the wave character of pressure is in evidence; and (iii) Low-Mach number (LM) stage when St > M _g0, where M _g0 is the initial gradient Mach number. In the PR regime we find that the thermodynamic fluctuations evolve from their initial state but velocity fluctuations grow unhindered by pressure fluctuations. In the WC regime, the pressure fluctuations become significant and flow-thermodynamic interaction commences. This interaction brings about equipartition of dilatational kinetic energy and thermodynamic potential energy. The interaction also results in stabilization of turbulence, and the total kinetic energy growth comes to a near standstill. Ultimately in the LM stage, kinetic energy starts increasing again with the growth rate being very similar to that in incompressible RDT. However, the thermodynamic fluctuations continue to grow despite the gradient Mach number being substantially smaller than unity. Overall, the study yields valuable insight into the linear processes in high Mach number shear flows and identifies important closure modeling issues.     

Self-similarity in fully developed homogeneous isotropic turbulence using the lyapunov analysis

In this work, we calculate the self-similar longitudinal velocity correlation function and the statistics of the velocity difference, using the results of the Lyapunov analysis of the fully developed isotropic homogeneous turbulence just presented by the author in a previous work (de Divitiis, Theor Comput Fluid Dyn, doi:10.1007/s00162-010-0211-9). There, a closure of the von Kármán-Howarth equation is proposed and the statistics of velocity difference is determined through a specific statistical analysis of the Fourier-transformed Navier-Stokes equations. The longitudinal correlation functions correspond to steady-state solutions of the von Kármán-Howarth equation under the self-similarity hypothesis introduced by von Kármán. These solutions and the corresponding statistics of the velocity difference are numerically determined for different Taylor-scale Reynolds numbers. The obtained results adequately describe the several properties of the fully developed isotropic turbulence.     

A new representation of energy spectra in fully developed turbulence

It is well known that the Kolmogorov 1941 theory is based on global invariance, in the limit of Reynolds number tending to infinity. Experimentally, it is well verified only for very high Reynolds numbers, namelyR 2000 (Monin and Yaglom 1975).We propose a new experimental representation for energy spectra. Using the Kolmogorov scales, a compilation of dimensionless spectra (E= (k)/(v ⁵)^1/4 andK=k(v ³/)^1/4) shows that log(0.154E)/log(R /R*) is a universal function of log(5.42K)/log(R /R*) withR*=75. This new representation is not compatible with neither local nor global scaling invariance. The constant 5.42 takes into account the small scale intermittency. Similar results have been obtained for velocity structure functions of order 2, 3 and 6. In particular the wavenumber constant 5.42 is independent on the order of the moments.     

Some remarks on nonlocal interactions and hysteresis in phase transitions

This paper examines models of shape memory alloys in which the usual multiwell, nonconvex elastic energy is modified by the addition of a nonlocal interaction term. It is shown that minimizing such an energy over uniform Young-measures yields a good prediction of hysteresis in a material subjected to biaxial loading. In addition, the relationship between the nonlocal interaction energy and the coherency energy is discussed.This work has been partially supported by the National Science Foundation under grants number DMS-9204304 and DMS-9403844     

Localisation issues in local and nonlocal continuum approaches to fracture

Continuum approaches to fracture regard crack initiation and growth as the ultimate consequences of a gradual, local loss of material integrity. The material models which are traditionally used to describe the degradation process, however, may predict premature crack initiation and instantaneous, perfectly brittle crack growth. This nonphysical response is caused by localisation instabilities due to loss of ellipticity of the governing equations and—more importantly—singularity of the damage rate at the crack tip. It is argued that this singularity results in instantaneous failure in a vanishing volume, even if ellipticity is not first lost. Adding strong nonlocality to the modelling is shown to preclude localisation instabilities and remove damage rate singularities. As a result, premature crack initiation is avoided and crack growth rates remain finite. Weak nonlocality, as provided by explicit gradient models, does not suffice for this purpose. In implementing the enhanced modelling, the crack must be excluded from the equilibrium problem and the nonlocal interactions in order to avoid unrealistic damage growth.     

On the consistency of two-phase local/nonlocal piezoelectric integral model

Applied Mathematics and Mechanics - In this paper, we propose general strain- and stress-driven two-phase local/nonlocal piezoelectric integral models, which can distinguish the difference of...     

Short-and resonant-range interactions between scales in turbulence and their applications

Interactions between different scales in turbulence were studied starting from the incompressible Navier-Stokes equations. The integral and differential formulae of the shortrange viscous stresses, which express the short-range interactions between contiguous scales in turbulence, were given. A concept of the resonant-range interactions between extreme contiguous scales was introduced and the differential formula of the resonant-range viscous stresses was obtained. The short- and resonant-range viscous stresses were applied to deduce the large-eddy simulation ( LES ) equations as well as the multiscale equations, which are approximately closed and do not contain any empirical constants or relations. The properties and advantages of using the multiscale equations to compute turbulent flows were discussed. The short-range character of the interactions between the scales in turbulence means that the multiscale simulation is a very valuable technique for the calculation of turbulent flows. A few numerical examples were also given.     

Nucleation and phase propagation in a multistable lattice with weak nonlocal interactions

We study the overdamped gradient flow dynamics of a chain of massless points connected by bistable nearest-neighbor (NN) interactions and harmonic next-nearest-neighbor (NNN) interactions under quasistatic loads of assigned displacements. The model reproduces experimental observations on the phase transition of shape-memory wires with the possibility of different microstructure evolution strategies: internal or boundary nucleations and one or two coherently propagating phase fronts. The presence or absence of a stress peak is also obtained by considering nonlocal interaction effects with the loading device. Similar results are also obtained under the hypothesis of global energy minimization. The system also retains the described properties in the continuum limit. Some rate effects are numerically analyzed.      

Resonant oscillations of a gas in an open-ended tube in a weak turbulence regime

An analytical theory of resonant oscillations of a gas in an open-ended tube is developed. The gas flow in the tube is assumed to be turbulent. A model of gas flow near the open end of the tube is constructed. This model allows a boundary condition that is free of empirical parameters to be obtained. Theoretical results are in reasonable agreement with experimental data obtained by other authors. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 3, pp. 92–99, May–June, 1998.     

Constitutive theories for nonlocal micropolar continua with implicity and with multiple interactions

In this paper the constitutive theory for nonlocal micropolar continua which was proposed by A. C. Eringen is extended to the cases for nonlocal micropolar continua with implicity and with multiple interactions. Here nonlocal micropolar thermoelastic solids with implicity and with multiple interactions are cited as instances to illustrate the procedure for the establishment of their constitutive theories as well as two relevant theorems concerning the constitutive theories for those solids are given.     

Using wavelet transform to study the Lipschitz local singular exponent in wall turbulence

I.IntroductionThenearwallcoherentstructureinaturbulentboundarylayer11aslongbeenthesubjectforalarge11umberofinvestigations.Thereisnolongeranydoubtthatcoherentstructuresareamajorcompollentinwall-boundedturbulentshearflows.Theyplayanimportantroleintheproduction,dissipationandtransportationofturbulentenergy.Oneofthenumerousdifficultiesofthesestudiesishowtoquantitativelymeasurecoherentstructurefromphysicalexperimentsorfi.omdirectnumericalsimulations.Thegoalistoisolateorcharacterizethecoherentstruc…     

Radiation interactions in transient energy transfer in fully developed laminar flows

Analysis and numerical procedures are presented to investigate the transient radiative interactions of nongray absorbing-emitting species in laminar fully-developed flows between two parallel black plates. The particular species considered are OH, CO, CO₂, and H₂O and different mixtures of these species. Transient and steady-state results are obtained for the temperature distribution and bulk temperature for different plate spacings, wall temperatures, and pressures. Results, in general, indicate that the rate of radiative heating can be quite high during earlier times. This information is useful in designing thermal protection systems for transient operations.Nomenclature A band absorptance = A(u, ), cm^–1 - A ₀ band width parameter, cm^–1 - C ₀ correlation parameter, atm^–1-cm^–1 - C _p specific heat at constant pressure, kJ/kg-K = erg/gm-K - e Planck's function, (W-cm^–2)/cm^–1 - e ₀ Planck's function evaluated at wave number ₀ - e ₁, e ₂ emissive power of surfaces with temperatures T ₁ and T ₂, W-cm^–2 - H _1i , H ₁ gas property for the large path length limit - k thermal conductivity, erg/cm-s-K - K ₁ gas property for the optically thin limit - M large path length parameter, nondimensional - N optically thin parameter, nondimensional - P pressure, atm - q _R total radiative heat flux, W/cm² - q _Rw spectral radiation heat flux, (W-cm^–2)/cm^–1 - q _w wall het flux, W/cm² - S integrated intensity of a wide band, atm^–1-cm² - t time, s - T temperature, K - T ₁ wall temperature, K; T ₁=T _w - T _b bulk temperature, K - u nondimensional coordinate = SPy/A ₀ - u ₀ nondimensional path length = SPL/A ₀ - y transverse coordinate, cm - line structures parameter - nondimensional temperature - _b dimensionless bulk temperature - spectral absorption coefficient, cm^–1 - nondimensional coordinate = y/L=u/u ₀ - density, kg/m³ - nondimensional time - wave number, cm^–1 - ₀ wave number at the band center, cm^–1     

Flow-thermodynamics interactions in decaying anisotropic compressible turbulence with imposed temperature fluctuations

The fundamental nature of the non-linear flow-thermodynamics interactions in a compressible turbulent flow with imposed temperature fluctuations is investigated. Direct numerical simulations (DNS) of decaying anisotropic compressible turbulence (turbulent Mach number 0.06–0.6) with imposed temperature fluctuations are performed to examine: (i) interactions between solenoidal and dilatational kinetic energy; (ii) partition between dilatational kinetic energy and thermodynamic potential energy; and (iii) redistribution of solenoidal and dilatational kinetic energy among the various Reynolds stress components. It is found that solenoidal kinetic energy levels and return-to-isotropy are weakly dependent on Mach number but independent of imposed temperature fluctuations in the parameter range studied. The dilatational kinetic energy generated is proportional to the square of the pressure fluctuations associated with the initial solenoidal and temperature fluctuations and thus a strong function of Mach number and heat release intensity. The energy exchange between dilatational kinetic and potential energy is driven by a strong proclivity toward equipartition. Consequently, the dynamics of pressure-dilatation ( ${\overline{pd}}$ ), which is the mechanism of this energy exchange between dilatational and potential energies, is dictated entirely by the requirement to impose energy equipartition. Based on the results, we provide a physical picture of the solenoidal–dilatational–potential energy interactions and the action of pressure-dilatation. The identification of the fundamental precepts underlying the various interactions is of great utility for turbulence closure model development.     

A priori evaluations and least-squares optimizations of turbulence models for fully developed rotating turbulent channel flow

The present study involves a priori tests of pressure-strain and dissipation rate tensor models using data from direct numerical simulations (DNS) of fully developed turbulent channel flow with and without spanwise system rotation. Three different pressure-strain rate models are tested ranging from a simple quasi-linear model to a realizable fourth order model. The evaluations demonstrate the difficulties of developing RANS-models that accurately describe the flow for a wide range of rotation numbers. Furthermore, least-squares based tensor representations of the exact pressure-strain and dissipation rate tensors are derived pointwise in space. The relation obtained for the rapid pressure-strain rate is exact for general 2D mean flows. Hence, the corresponding distribution of the optimized coefficients show the ideal behaviour. The corresponding representations for the slow pressure-strain and dissipation rate tensors are incomplete but still optimal in a least-squares sense. On basis of the least-squares analysis it is argued that the part of the representation that is tensorially linear in the Reynolds stress anisotropy is the most important for these parts.