Dissipation Tensor Research Articles

The dissipation tensor in wall turbulence

The paper investigates the dissipation tensor$\unicode[STIX]{x1D700}_{ij}$in wall turbulence. Available direct numerical simulation (DNS) data are examined to illustrate the differences in the anisotropy of the dissipation tensor$\unicode[STIX]{x1D700}_{ij}$with respect to the anisotropy of the Reynolds stresses$\unicode[STIX]{x1D633}_{ij}$. The budgets of the transport equations of the dissipation tensor$\unicode[STIX]{x1D700}_{ij}$are studied using novel DNS data of low Reynolds number turbulent plane channel flow with spatial resolution sufficiently fine to accurately determine the correlations of products of two-derivatives of fluctuating velocities$u_{i}^{\prime }$and pressure$p^{\prime }$which appear in various terms. Finally, the influence of the Reynolds number on the diagonal components of$\unicode[STIX]{x1D700}_{ij}$($\unicode[STIX]{x1D700}_{xx}$,$\unicode[STIX]{x1D700}_{yy}$,$\unicode[STIX]{x1D700}_{zz}$) and on the various terms in their transport equations is studied using available DNS data of Vreman and Kuerten (Phys. Fluids, vol. 26, 2014b, 085103).

Four-dimensional equation of motion for viscous compressible and charged fluid with regard to the acceleration field, pressure field and dissipation field

From the principle of least action the equation of motion for viscous compressible and charged fluid is derived. The viscosity effect is described by the 4-potential of the energy dissipation field, dissipation tensor and dissipation stress-energy tensor. In the weak field limit it is shown that the obtained equation is equivalent to the Navier-Stokes equation. The equation for the power of the kinetic energy loss is provided, the equation of motion is integrated, and the dependence of the velocity magnitude is determined. A complete set of equations is presented, which suffices to solve the problem of motion of viscous compressible and charged fluid in the gravitational and electromagnetic fields.

Anisotropic Characteristics of Turbulence Dissipation in Swirling Flow: A Direct Numerical Simulation Study

This study investigates the anisotropic characteristics of turbulent energy dissipation rate in a rotating jet flow via direct numerical simulation. The turbulent energy dissipation tensor, including its eigenvalues in the swirling flows with different rotating velocities, is analyzed to investigate the anisotropic characteristics of turbulence and dissipation. In addition, the probability density function of the eigenvalues of turbulence dissipation tensor is presented. The isotropic subrange of PDF always exists in swirling flows relevant to small-scale vortex structure. Thus, with remarkable large-scale vortex breakdown, the isotropic subrange of PDF is reduced in strongly swirling flows, and anisotropic energy dissipation is proven to exist in the core region of the vortex breakdown. More specifically, strong anisotropic turbulence dissipation occurs concentratively in the vortex breakdown region, whereas nearly isotropic turbulence dissipation occurs dispersively in the peripheral region of the strong swirling flows.

Relaxation of nonequilibrium states of a uniaxial ferromagnet

Relaxation of the nonequilibrium state of a uniaxial ferrimagnetic is studied. It is shown that the character of evolution of the magnetization vector is determined by anisotropy of the dissipation tensor.

An Improved Model for the Return to Isotropy of Homogeneous Turbulence

In CFD modeling, the most widely used Reynolds stress models is the Speziale, Sarkar, Gatski (SSG) model. The present formulation, though similar in structure to the SSG model, is a mathematical variation assuming homogeneity of turbulence and is an improved model for the slow pressure strain of turbulence. The basic thrust is that anisotropy of dissipation tensor is not negligible when compared to the anisotropy of turbulent kinetic energy and affects the slow pressure strain rate. After an exhaustive survey of the available experimental results on return to isotropy, graphical plots reveal that the model performs as good as the SSG model.

Excitation Energy Transfer in a Classical Analogue of Photosynthetic Antennae

We formulate a classical pure dephasing system-bath interaction model in a full correspondence to the well-studied quantum model of natural light-harvesting antennae. The equations of motion of our classical model not only represent the correct classical analogy to the quantum description of excitonic systems, but they also have exactly the same functional form. We demonstrate derivation of classical dissipation and relaxation tensor in second order perturbation theory. We find that the only difference between the classical and quantum descriptions is in the interpretation of the state and in certain limitations imposed on the parameters of the model by classical physics. The effects of delocalization, transfer pathway interference, and the transition from coherent to diffusive transfer can be found already in the classical realm. The only qualitatively new effect occurring in quantum systems is the preference for a downhill energy transfer and the resulting possibility of trapping the energy in the lowest energy state.

Budget of Equations for Reynolds Stresses in a Turbulent Patch Arising from Internal Wave Breaking

Results of the direct numerical simulations are used to obtain statistical moments in a quasi-steady turbulent patch arising in a stably stratified flow above an obstacle after internal wave breaking. Temporal evolution and spatial behavior of the Reynolds-stress tensor components and budgets of the Reynolds-stress transport equations have been studied. Such an analysis is helpful to explore the turbulent patch and its energetic characteristics by means of statistical moments, to examine closure hypotheses in turbulence models, and to evaluate geophysically important quantites. In particular, the calculated global value of the mixing efficiency is about 0.2 as in oceanic applications. Moreover, simple algebraic relations are proposed between the turbulent kinetic energy and its dissipation rate. In the Reynolds stress tensor equation, the pressure-strain correlation term provides the redistribution of the normal stress components according to the linear returnto-isotropy approximation, whereas the dissipation tensor roughly follows the local-isotropy assumption

Excitation energy transfer in chromophore aggregates within a dissipative medium

The Redfield theory-based model of excitation energy transfer in chromophore ensembles within dissipative environment is proposed. Application of the multipole expansion to an operator of interaction between the chromophore molecules and the environment together with some assumptions about the latter led to the closed-form expressions for the elements of the dissipation tensor. These expressions relate the rates of transition between eigenstates of a chromophore ensemble with the spectral and electronic properties of an environment and the chromophore molecules. For several model cases the exact solution of the Redfield equations was obtained.

Near-wall extension of a non-equilibrium, omega-based Reynolds stress model

In this paper, the development of a new ω-based Reynolds stress model that is consistent with asymptotic analysis in the near wall region and with rapid distortion theory in homogeneous turbulence is reported. The model is based on the SSG/LRR-ω model developed by Eisfeld (2006) with three main modifications. Firstly, the near wall behaviors of the redistribution, dissipation and diffusion terms are modified according to the asymptotic analysis and a new blending function based on low Reynolds number is proposed. Secondly, an anisotropic dissipation tensor based on the Reynolds stress inhomogeneity (Jakirlic et al., 2007) is used instead of the original isotropic model. Lastly, the SSG redistribution term, which is activated far from the wall, is replaced by Speziale's non-equilibrium model (Speziale, 1998).

Fission-fragment mass distributions from strongly damped shape evolution

Random walks on five-dimensional potential-energy surfaces were recently found to yield fission-fragment mass distributions that are in remarkable agreement with experimental data. Within the framework of the Smoluchowski equation of motion, which is appropriate for highly dissipative evolutions, we discuss the physical justification for that treatment and investigate the sensitivity of the resulting mass yields to a variety of model ingredients, including in particular the dimensionality and discretization of the shape space and the structure of the dissipation tensor. The mass yields are found to be relatively robust, suggesting that the simple random walk presents a useful calculational tool. Quantitatively refined results can be obtained by including physically plausible forms of the dissipation, which amounts to simulating the Brownian shape motion in an anisotropic medium.

Acoustic spectroscopy of fluid saturation effects in carbonate rock

The results of acoustic spectroscopy of a carbonate sedimentary rock in the case of saturation degree variations are given. The data on the elasticity and dissipation tensors within a two-octave range are obtained for one and the same sample using one and the same measurement technique. High accuracy of the measurements allowed distinguishing all three saturation stages, namely, condensation, meniscus creation, and pore filling with a fluid. The observed variations are in good agreement with the results of a standard granulometric analysis. The character of variations in the frequency dependence of attenuation coefficients make it possible to make reasonable assumption concerning the mechanisms responsible for acoustic energy absorption in porous materials.

Resonant acoustic spectroscopy of the fluid saturation effects in a carbonate rock.

We present the results of the acoustic spectroscopy of a carbonate sedimentary rock under various saturation degrees. The data for the same sample were obtained both for elasticity and dissipation tensors within the 2‐octaves frequency range. High accuracy of acoustic measurements allowed distinguishing all three stages in the saturation process: condensation, meniscus creation, and filling the pore space with fluid. Both dry rock parameters and the variations observed are in a good agreement with standard granulometry and chemical analysis. Simple estimations of the relative changes in elastic moduli enable clear explanations of these changes observed. The alteration in the frequency dependence of attenuation coefficient allowed proposing reasonable dissipation mechanisms in porous media that can also be useful in the acoustics of ocean sediments.

Effect of Particle Clusters on Carrier Flow Turbulence: A Direct Numerical Simulation Study

Experiments indicate that particle clusters that form in fluidized–bed risers can enhance gas-phase velocity fluctuations. Direct numerical simulations (DNS) of turbulent flow past uniform and clustered configurations of fixed particle assemblies at the same solid volume fraction are performed to gain insight into particle clustering effects on gas-phase turbulence, and to guide model development. The DNS approach is based on a discrete-time, direct-forcing immersed boundary method (IBM) that imposes no-slip and no-penetration boundary conditions on each particle’s surface. Results are reported for mean flow Reynolds number Rep = 50 and the ratio of the particle diameter dp to Kolmogorov scale is 5.5. The DNS confirm experimental observations that the clustered configurations enhance the level of fluid-phase turbulent kinetic energy (TKE) more than the uniform configurations, and this increase is found to arise from a lower dissipation rate in the clustered particle configuration. The simulations also reveal that the particle-fluid interaction results in significantly anisotropic fluid-phase turbulence, the source of which is traced to the anisotropic nature of the interphase TKE transfer and dissipation tensors. This study indicates that when particles are larger than the Kolmogorov scale (dp > η), modeling the fluid-phase TKE alone may not be adequate to capture the underlying physics in multiphase turbulence because the Reynolds stress is anisotropic. It also shows that multiphase turbulence models should consider the effect of particle clustering in the dissipation model.

Prediction of stably stratified homogeneous shear flows with second-order turbulence models

The present study investigated the role of pressure-correlation second-order turbulence modelling schemes on the predicted behaviour of stably stratified homogeneous vertical-sheared turbulence. The pressure-correlation terms were modelled with a nonlinear formulation (Craft 1991), which was compared with a linear pressure–strain model and the ‘isotropization of production’ model for the pressure–scalar correlation. Two additional modelling issues were investigated: the influence of the buoyancy term in the kinetic energy dissipation rate equation and the time scale in the thermal production term in the scalar variance dissipation equation. The predicted effects of increasing the Richardson number on turbulence characteristics were compared against a comprehensive set of direct numerical simulation databases. The linear models provide a broadly satisfactory description of the major effects of the Richardson number on stratified shear flow. The buoyancy term in the dissipation equation of the turbulent kinetic energy generates excessively low levels of dissipation. For moderate and large Richardson numbers, the term yields unrealistic linear oscillations in the shear and buoyancy production terms, and therefore should be dropped in this flow (or at least their coefficient cε3 should be substantially reduced from its standard value). The mechanical dissipation time scale provides marginal improvements in comparison to the scalar time scale in the production. The observed inaccuracy of the linear model in predicting the magnitude of the effects on the velocity anisotropy was demonstrated to be attributed mainly to the defective behaviour of the pressure-correlation model, especially for stronger stratification. The turbulence closure embodying a nonlinear formulation for the pressure-correlations and specific versions of the dissipation equations failed to predict the tendency of the flow to anisotropy with increasing stratification. By isolating the effects of the dissipation rate equations, it was demonstrated that nonlinear closure provides an improved account of the normal components of the anisotropy tensor over the linear model; however, the vertical velocity correlation and the shear stress still present too large a discrepancy. The observed shortcomings may be intrinsic to the current structure of the pressure-correlation terms. None of the models were able to predict the ultimate collapse of the turbulent fluxes and counter-gradient fluxes. However, this weakness may be beyond the modelling of the redistribution processes, but instead in fundamental shortcomings in the energy dissipation rate equations and the dissipation tensor to adequately represent the phenomenology of the transition from shear- to buoyancy-dominated flows.

Measurement and modeling of homogeneous axisymmetric turbulence

A new method for modeling the unknown correlations in Reynolds stress transport equations is developed taking the kinematic relationship of turbulence tensors in homogeneous axisymmetric turbulence. Both the rapid-straining and the return-to-isotropy process of homogeneous axisymmetric turbulence are studied with the aim of improving Reynolds stress closures. The partition of the stress dissipation is also studied to assess the possible existence of local isotropy for turbulence at small scales. Homogeneous, axisymmetric turbulence is a simple flow, where the axes of anisotropy of the Reynolds stresses and dissipation tensor are found to be aligned. Using the theory of barycentric coordinates, the relationship between the Reynolds stress and dissipation tensors is derived, satisfying restrictions for the limiting states of turbulence and its assumed behavior for large Reynolds number and arbitrary anisotropy. The role of the anisotropy in constraining models for the turbulent dissipation rate and the pressure–strain correlations is discussed. Comparisons of the resulting closure based on barycentric coordinates with the experimental data for axisymmetric flow measurements of Ertunç and Durst (On the high contraction ratio anomaly of axisymmetric contraction of grid-generated turbulence, Phys. Fluids. 20, February 2008) are good within the limitations of the data.

Simulation of shear orientation effects on stably stratified homogeneous turbulence with RANS second-order modelling

The paper investigates the ability of a differential second-order turbulence closure to predict the effects of shear directionality on stably stratified homogeneous turbulence. The study does not aim to propose new turbulence models but rather aims to provide a detailed examination of the behaviour of the turbulence closure from an extensive set of available direct numerical simulations (DNS) of homogeneous and stably stratified flows that have been subjected to non-vertical shear. The study disclosed several previously unknown features about the predicted behaviour of turbulence subjected to a mean shear gradient that makes an arbitrary angle with the plane of the temperature gradient (which is aligned with the gravitational acceleration vector) for a wide range of Ri numbers. The turbulence closure comprises transport equations for all of the components of the Reynolds stress tensor, the heat flux vector and for the dissipation rates of kinetic energy and temperature fluctuations. Further, an anisotropic dissipation tensor is considered. The results show that the model captures the impact on the turbulence growth rate of increasing the horizontal shear. The model yields more energetic turbulence levels with increasing orientation angle, and it predicts accurately the “critical” inclination angle. One of the most important findings is that the model is capable of estimating the critical Ri for both the vertical and the horizontal shear cases when a properly calibrated buoyancy term is included in the ϵ-equation. For both cases, the model predicts the same turbulent Froude number for critical Richardson number, in agreement with the DNS data. Moreover, the present model is able to predict the strong increase in the mixing efficiency parameter from the vertical shear to the horizontal shear cases, which was not achieved in previous studies. The model is sensitive to buoyancy damping of vertical velocity fluctuations, which ultimately lead to the overall decay of all of the velocity components. The comparison of the present predictions with rapid distortion theory (RDT) analysis attests the importance of non-linear processes in the approach to “sudden suppression” of turbulence. However, some concerns remain in the behaviour of the model. A major source of uncertainty relates to the anisotropy of Reynolds stresses. It was found that the observed departure in the prediction of anisotropies is not simply related to the accuracy of the pressure–strain redistribution, but the anisotropic dissipation modelling has a role to play in stratified flows.

Characterization of the Reynolds‐stress and dissipation‐rate decay and anisotropy from DNS of grid‐generated turbulence

AbstractGrid–generated turbulence is a classical but still controversial topic, one open issue being the spatial decay rate of turbulent energy. We study the influence of grid geometry on the Reynolds–stress and dissipation–rate tensors, including the range and exponent of their self–similar spatial decay. DNS using a validated lattice Boltzmann code at mean–flow Reynolds numbers up to 1400 are performed, comparing square grids with blockage ratios from 0.05 to 0.49. A clear picture of spatial distribution and self–similarity emerges for the statistics of interest: Axisymmetry is excellently confirmed. A consistent power law decay is found in the self–similar decay region beyond 10 grid stride lengths downstream. Its exponent of –5/3 can be obtained, for weak turbulence, from a spatial flux balance reminiscent of the constant transport through the inertial range of isotropic turbulence. In the near–grid region, on the other hand, differences in Reynolds stress components are pronounced while those between dissipation tensor components are only recognizable very close to the grid, where a strong dependence on grid porosity is found. A normalization with respect to porosity is proposed that collapses the data from all runs. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Anisotropy of a turbulent boundary layer

A two-dimensional turbulent boundary layer with a Reynolds number 2000 based on the displacement thickness was calculated by using direct numerical simulation. The goal of this study was to establish turbulence models for the closure of the Reynolds stress tensor and the anisotropy tensor, distinguish the global and local anisotropy and calculate the orientation of turbulent structures occurring in the turbulent boundary layers. This paper proposes an anisotropic model, which approximates the anisotropy tensor bij using a normalized turbulent dissipation tensor dij . The multiscale behavior of anisotropy occurring in turbulent boundary layers is investigated and analyzed. The results reveal that the anisotropy does not decay as the scale decreases but persists in the local scales. Additionally, this paper analyzes the orientation of the turbulent energy transportation and the distribution of the positive and negative vorticities by using the skewness of the velocity fluctuations and turbulent vorticities.

The residual anisotropy at small scales in high shear turbulence

It has always been believed that turbulence in fluids can achieve a universal state at small scales with fluctuations that, becoming statistically isotropic, are characterized by universal scaling laws. In fact, in different branches of physics it is common to find conditions such that statistical isotropy is never recovered and the anisotropy induced by large scale shear contaminates the entire range of scales up to velocity gradients. We address this issue here, of particular significance, for wall bounded flows. The systematic decomposition in spherical harmonics of the correlation functions of velocity fluctuations enables us to extract the different anisotropic contributions. They vanish at small scale at a relatively fast rate under weak shear. Under strong shear instead they keep a significant amplitude up to viscous scales, thus leaving a persistent signature on the gradients which can be detected even in the statistics of low order, e.g., in the energy dissipation tensor.

The Dependency on the Dissipation Tensor of Multi-modal Nuclear Fission

In the study of the fission of actinide nuclei at low excitation energies including the spontaneous fission, it was found that the fragment mass distribution and the total kinetic energy (TKE) distribution consist of more than one component, in contrast to the simple single peak structure that is found in the fission at high excitation energies. 1– 6 This phenomenon is attributed to the existence of more than one fission path and is called the multi-modal fission. The mass and TKE distributions depend sensitively on the excitation energy and the position of the peaks of the mass distribution suggests the influence of the closed shell structure of the fragments. Therefore, it is supposed that the microscopic energy plays an important role for the manifestation of this phenomenon. It is a great challenge for us to understand this phenomenon in terms of nuclear many-body dynamics. Several authors studied the potential energy surface (PES) including the microscopic energy in a multi-dimensional parameter space that describes various nuclear shapes; one can deduce the possible fission paths by studying the location of the saddle points and the fission valleys in multi-dimensional parameter space. 7 With this method, they could explain the general trend of the position of the peaks of the mass distribution. The dynamical point of view is necessary to progress the study of the fission mode. We have applied the Langevin approach to the study of the fission modes in uranium nuclei and in fermium nuclei. 8–10 We studied the mass and TKE distributions and demonstrated that we can decompose the fission events into several components by tracing the Langevin trajectories. We also studied the isotope dependence and the excitation energy dependence of the fission mode. 8–10 In the previous studies, we adopted the wall-and-window type one-body friction as the dissipation mechanism of the nuclear fission dynamics. The validity of this dissipation mechanism has been demonstrated by one of the authors (T.W.) who studied the dissipation tensor dependence of the pre-scission neutron multiplicity and the mean TKE. 11–14 From the comparison of the results of the dynamical calculation with experimental data, they excluded the possibility of the two-body type dissipation to be the dominant mechanism by showing that it cannot reproduce the pre-scission neutron data and the TKE data simultaneously. On the other hand, they showed that the wall-and-window type one-body friction can reproduce both data reasonably well and concluded that it is a reasonable model for the dissipation mechanism of nuclear fission. There are other models that are of one-body nature, e.g. surface-plus-window formula, modified wall-and-window formula and chaos weighted wall formula. 15–17 Though there were no free parameters in the original derivation of the one-body friction, 18, 19 the strength has been modified frequently in order to reproduce some experimental data. For example, in the study of the light particle evaporation and the mass distribution, Schmitt et al. used the strength as a free parameter. 17 The modification itself should be acceptable when we take account of the simplicity of the model; it is a macroscopic model without any microscopic effect and it has no dependence on the temperature. However, when one modifies the strength of the nuclear dissipation just to reproduce only one physical quantity, it might be inappropriate to conclude that the deduced strength has definite physical meanings. It may reflect the other effects completely different from the dissipation, like the insufficiency of the model space. It is very important to compare many (at least more than one) physical quantities at the same time. Among the physical quantities that are measured in nuclear fission, the TKE and mass distributions are well investigated experimentally in many cases. In this study, we use these quantities to discuss the dissipation dependence of the fission modes. It is shown that the TKE distribution is, as was expected, directly connected to the strength of the dissipation and we can put some constraints on the strength of the dissipation. Furthermore, it is shown that the mass distribution changes rather drastically when one uses different models for the dissipation mechanism. These results demonstrate the importance and the usefulness of the dynamical approach to the study of the fission mode. Section 2 gives a concise description of our framework. Results are shown in Sec. 3 concerning the fission of 264 Fm nucleus at Ex = 20 MeV. Summary is given in Sec. 4. 2. Methods