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Approximate Bayesian computation (ABC) is used to estimate unknown parameter values, as well as their uncertainties, in Reynolds-averaged Navier–Stokes (RANS) simulations of turbulent flows. The AB...

In the present work, premixed combustion in a turbulent boundary layer under auto-ignitive conditions is investigated using direct numerical simulation (DNS). The turbulent inflow of the reactive DNS is obtained by temporal sampling of a corresponding inert DNS of a turbulent boundary layer at a location with R e τ = 360, where R e τ is the friction Reynolds number. The reactants of the DNS are determined by mixing the products of lean natural gas combustion and a H 2 /N 2 fuel jet, resulting in a lean mixture of high temperature with a short ignition delay time. In the free stream the reaction front is stabilized at a streamwise location which can be predicted using the free stream velocity U ∞ and the ignition delay time τ i g . Inside the boundary layer, combustion modifies the near-wall coherent turbulent structures considerably and turbulence results in reaction front wrinkling. The combustion modes in various regions were examined based on the results of displacement velocity, species budget and chemical explosive mode analysis (CEMA). It was indicated that flame propagation prevails in the near-wall region and auto-ignition becomes increasingly important as the wall-normal distance increases. The interactions of turbulence and combustion were studied through statistics of reaction front normal vector and strain rate tensor. It was found that the reaction front normal preferentially aligns with the most compressive strain rate in regions where the effects of heat release on the strain rate are minor and with the most extensive strain rate where its effects are significant. Negative correlations between the wall heat flux and flame quenching distance were observed. A new quenching mode, back-on quenching, was identified. It was found that the heat release rate at the wall is the highest when head-on quenching occurs and lowest when back-on quenching occurs.

The primary parameter of a standard k-ϵ model, Cμ, was calculated from stereoscopic particle image velocimetry (PIV) data for a supersonic jet exhausting into a transonic crossflow. This required t...

In the paper, a pragmatic approach to finding the dual formulation for isotropic perfectly plastic materials given a dissipation potential dependent on three cylindrical invariants and involving the Ottosen shape function is proposed and illustrated by examples. The main goal is to provide instructions on how to perform the Legendre transformation used when passing from a dissipation potential to its conjugate yield condition and offer some suggestions regarding calibration for particular potentials dependent on the trace of the strain rate tensor and the product of the norm of its deviator and the Ottosen shape function, which covers a wide class of engineering materials. The classic framework for constitutive modelling of thermodynamically consistent materials within the small deformation theory is used. First, general formulae connecting a dissipation potential dependent on three invariants of the strain rate tensor to the coupled yield condition are derived. Then, they are narrowed down for the aforementioned case of dissipation functions dependent on the Lode angle in a way proposed by Ottosen. Finally, three examples are given involving classical potentials: Beltrami’s, Drucker–Prager’s and Mises–Schleicher’s generalised potential using the shape function. Detailed calculations exposing the introduced technique are performed. Also, a method of the calibration of such potentials leading to explicit mathematical formulae is demonstrated, based on the typical tests located on the tension and compression meridians.

The Global Navigation Satellite System (GNSS) over the years has become an essential instrument in modelling geodynamic processes, because of its high spatio-temporal characteristics. Accurate velocity modelling of GNSS stations and strain rate analysis can provide an accurate mechanism for studying the deformation of the earth crust. Therefore, this study presents the strain rate tensor of the Nigeria crust from GNSS network, but first, the GAMIT/GLOBK was used to process and obtain accurate velocity solution of the Nigerian Permanent GNSS Network (NigNET). The surface deformation of Nigeria deduced from strain analysis was computed using splines in tension. The results revealed that the highest extensional and compressional strain rates of up to 2.67 to 2.89μstrain/year were observed around parts of the north-west (Jigawa, Kaduna and Kano) and north central region (parts Abuja and Jos) and some parts of the north-east (parts of Bauchi and Gombe), while the lowest minimal or low extensional and compressional strain rates of 0.09 to 0.534μstrain/year were observed around some parts of the north central (parts of Kwara and Niger) and parts of north-west (parts of Kebbi, Zamfara and Sokoto). The highest maximum shear strain rate of about 0.4μstrain/year which has its peak around some parts of Kaduna, Abuja, Plateau and Bauchi was observed. The highest areal dilatation rate observed in Nigeria is about 1.02μstrain/year. The study serves as a benchmark for further studies.

A new subgrid-scale (SGS) stress model is proposed for rotating turbulent flows, and the new model is based on the traceless symmetric part of the square of the velocity gradient tensor and the symmetric part of the vorticity gradient tensor (or the so-called vorticity strain rate tensor). The new subgrid-scale stress model is taken into account the effect of the vortex motions in turbulence, which is reflected on the anti-symmetric part of the velocity gradient tensor. In addition, the eddy viscosity of the new model reproduces the proper scaling as O(y3) near the wall. Then, the new SGS model is applied in large-eddy simulation of the spanwise rotating turbulent channel flow. Different simulating cases are selected to test the new model. The results demonstrate that the present model can well predict the mean velocity profiles, the turbulence intensities, and the rotating turbulence structures. In addition, it needs no a second filter, and is convenient to be used in the engineering rotational flows.

The present east–west crustal extension of the Tibetan Plateau has been demonstrated through field investigations, satellite imagery, geodetic deformation, and earthquake focal mechanisms. Normal faulting earthquakes in the interior Tibetan Plateau are almost entirely confined to regions at elevations over 4000 m. However, our knowledge of the eastward extent of normal faulting in the plateau is still uncertain due to the limited occurrence of well-documented earthquakes. Based on a retrospective analysis of the 2016 Mw 5.9 Zaduo earthquake in the Tibetan Plateau, we consider the NE trending Zaduo-Shanglaxiu fault as the most likely rupture fault through a comprehensive analysis of relocated aftershock sequences, mapped active faults, and newly acquired strain rate tensor. We further determine seismogenic fault geometry using a Bayesian approach and sample with a Markov Chain Monte Carlo method. We interpret the Zaduo earthquake to reflect the release of slowly accumulated elastic strain accumulated mainly by gravitational forces rather than a delay triggering event from the 2010 Yushu earthquake. The viscoelastic calculations to estimate Coulomb stress changes over time indicate that long-term viscous flow in a weak mid-crust can load adjacent faults far more than static stress changes alone. Our results show that the Zaduo earthquake was a Mw 5.9 oblique normal faulting event that occurred in the easternmost part of the Tibetan Plateau, suggesting that the Qiangtang block at longitude ~ 95° E accommodates east–west extensional crustal deformation by small-scale oblique normal faults, which may act as the boundary of micro-blocks. This may also mean that the normal faulting in the Qiangtang block is expanding outwards, and a new rifting system may be formed, which requires more geological evidence.

The article considers the topic of mechanics of a deformable solid. Mathematical modeling of the process of deformation of shell structures by deforming a flat sample (part of the shell) under conditions of a plane stress state with a discontinuous field of displacement velocities is considered. Analytical solutions were obtained for the fields of strain tensors observed in shells of various materials during their deformation. A comparative analysis is carried out under various deformation states. Such decisions are due to the need to obtain deformation fields at various points of the shells used at objects of various directions (military, industrial, etc.) and there are significant difficulties in determining the strain fields by numerical methods (for example, the finite element method).

Physical conservation laws are inherently Lagrangian. However, analyses in fluid mechanics using the Lagrangian framework are often forgone in favor of those using the Eulerian framework. This is perhaps due to a lack of experimental techniques with high temporal and spatial resolution that track the movement of fluid tracers in a flow domain. The development of time-resolved Particle Tracking Velocimetry/Accelerometry (TR-PTV/A) that measures flows with high seeding density has made the use of the Lagrangian framework more accessible. A challenge facing PTV/A is the need for robust mesh-free numerical schemes that handle random particle locations. Such a scheme can be created with high-order accuracy using Radial Basis Functions (RBFs). RBFs allow direct evaluation of derivatives of vector and scalar fields at random locations with infinite-order smoothness. The current work uses RBF-based differential schemes to develop a post-processing tool for PTV/A data, which can accurately evaluate spatial derivatives directly from Lagrangian particle tracks. This RBF-based strain/rotation-rate tensor evaluation tool is validated with two and three-dimensional flows from analytical solutions and is then tested with experimental data measured by a multi-pulse PTV/A system.

Nonuniformity of texture component and microstructure parameter distribution in the cross section of an alloy 8011 billet during hot rolling in a continuous group is studied. The study is performed by optical microscopy and x-ray diffraction analysis of microsections taken from hot-rolled strip in all stages of rolling in a hot-rolling mill continuous group. It is established that at first the texture is inhomogeneous over the thickness of a specimen, but it levels out in the hot rolling finishing passes. The results obtained are in good agreement with the theoretical concept of O. Engler in which the ratio of the components of the strain rate tensor I13/I11 is used to estimate the probability of appearance of a friction texture. Finite modeling using DEFORM 2D/3D software is used for theoretical analysis of the strain tensor components.

Folds in rocks are commonly used in geology for strain analysis. They contain information about rock rheology, and the kinematics and mechanics of deformation. The systematic analysis of the evolution of stress and strain during rock folding can provide key information on the mechanical behaviour of rocks and the efficiency of the folding process. In order to investigate the evolution of rock rheology during fold development, a series of two-dimensional simulations of single-layer folding are presented here. They are run using the finite element method BASIL, integrated within the software platform ELLE, to simulate linear and non-linear viscous deformation. The kinematics of deformation, the competence ratio between the folding layer and surrounding matrix as well as the stress exponent of the power-law viscous material are systematically varied. The results allow comparing the stages of folding under different deformation kinematic conditions. For all simulations the folding amplification process starts when the second invariant of the strain rate tensor in the competent layer deviates from the theoretical strain rate curve for a homogeneous material, which corresponds to a viscosity ratio between layer and matrix of m = 1. The relative time when the fold amplification starts is determined by the viscosity ratio between the competent layer and its surrounding matrix, the initial layer orientation with respect to the shear plane, the kinematics of deformation and the stress exponent. The folding process is more effective in cases with high viscosity ratio, non-linear rheology and layers initially oriented at a low angle with respect to the shear plane, because the second invariant of the strain rate tensor at the layer deviates earlier from the theoretical curve. The results also show differences depending on the boundary conditions, where (1) folding a competent layer requires less work in simple shear than in pure shear, and (2) the geometrical softening experienced by the competent layer due to fold development is followed by a hardening stage in pure shear and by a major softening phase in simple shear. The simulation results suggest that the decrease of stress of a competent layer without decreasing the mechanical strength has a direct influence on the behaviour of a lithospheric layer around the crust-mantle boundary, which may experience geometrical softening depending on the tectonic settings rather than material softening due metamorphic reactions or grain size reduction.

The theory of diffusive stresses based on the time-fractional diffusion equation with the Liouville–Caputo derivative is discussed. The diffusion process is characterized by the chemical potential tensor and the concentration tensor. Eliminating the chemical potential tensor and the concentration tensor from the constitutive equations for the stress tensor, the space–time-nonlocal equations for the mean stress and the deviatoric stress are obtained with the kernel being the Wright function.

Fiber reinforced polymer matrix composites are increasingly being used to fabricate energy-absorbing aerospace structural components susceptible to impact loading , such as jet engine containment structures subjected to blade-out events. The development and certification of such structures necessitates a detailed understanding of the dynamic behavior of the constituent materials, particularly the strain rate, temperature, and pressure dependent polymer matrix , and their complex interaction. As the rate of deformation increases, the thermodynamic condition transitions from isothermal to adiabatic. The conversion of plastic work to heat causes local adiabatic heating in the polymer matrix, but the rapid nature of impact events does not allow sufficient time for heat to dissipate; ballistic impact events can therefore be regarded as fully adiabatic. Local adiabatic heating can significantly affect the high rate deformation response if thermal softening effects outweigh strain and strain rate hardening effects and therefore, must be explicitly modeled at the appropriate length scale. In this paper, an existing isothermal viscoplastic strain rate and pressure-dependent polymer constitutive formulation is extended to nonisothermal conditions by modifying the inelastic strain rate tensor components to explicitly depend on temperature based on the Arrhenius equation for nonisothermal processes. A methodology based on shifting neat resin dynamic mechanical analysis data is utilized to determine the polymer elastic properties over a range of strain rates and temperatures. Temperature rises due to the conversion of plastic work to heat are computed, assuming adiabatic conditions. It is demonstrated that the modified polymer constitutive model is capable of capturing strain rate and temperature dependent yield as well as thermal softening associated with the conversion of plastic work to heat at high rates of strain. The nonisothermal polymer constitutive model is then embedded within the Generalized Method of Cells micromechanics framework to investigate the manifestation of matrix thermal softening, due to the conversion of plastic work to heat, on the high strain rate response of a T700/Epon 862 unidirectional composite. Adiabatic model predictions for high strain rate composite longitudinal tensile, transverse tensile, and in-plane shear loading are presented. Results show a substantial deviation from isothermal conditions ; significant thermal softening is observed for matrix dominated deformation modes (transverse tension and in-plane shear), highlighting the importance of accounting for local adiabatic heating in the polymer matrix in the high strain rate analysis of polymer matrix composite structures.

Numerical investigations of the supersonic axisymmetric base flow at Mach number 2.46 and Reynolds number 2.858 × 10 6 are carried out using the scale-adaptive simulation (SAS) approach. To achieve SAS computation, three mathematical forms of von Karman length scale are utilized. The first one is calculated from the strain rate tensor and second derivative of the velocity field (this SAS can be marked as “SAS-1”). Computation of the second one is employed using the vorticity and its first derivative (this SAS can be denoted as “SAS-2”). The third one is computed from the vorticity and the second derivative of the velocity field (this SAS can be named as “SAS-3”). Their effects on the quantitative predictions are assessed. Two inflow conditions are also investigated, i.e., inflow condition with/without turbulent-boundary-layer profiles. Results show that the inflow conditions have the significant effects on the base-pressure and streamwise velocity distributions, and the predicted results at the inflow condition with profiles show better agreement with experimental data than that at the inflow condition without profiles. The inflow conditions and von Karman length scales only have the relatively slight effects on the radial velocity distributions. Similarly, at the inflow condition with profiles, the base-pressure distributions are also not sensitive to mathematical forms of von Karman length scale. More comparisons reveal that the best predictions can be obtained by SAS-2 and SAS-3. It may be associated with the vortex stretching effects explicitly considered in the von Karman length scales of SAS-2 and SAS-3. Although the mean flow field predicted by SAS-2 is slightly better than that obtained by SAS-3. However, SAS-3 can give the slightly better predictions for the Reynolds stresses than SAS-2.

In this paper, we obtained analytical solutions of the fields of strain tensors under uniaxial tension of a rigidplasticstrip underthe conditions of a plane stress state.The topicalityof the construction of these solutions is connected with significant difficulties in determining the strain fields by numerical methods (for example, the finite element method).In the construction of these solutions, the change in the geometric characteristics of the strip (thickness, width) was taken into account, which led to the solution of the nonlinear problem of the continuum mechanics.

This study examines the flow of dense granular materials under external shear stress and pressure using discrete element method simulations. In this method, the material is allowed to strain along all periodic directions and adapt its solid volume fraction in response to an imbalance between the internal state of stress and the external applied stress. By systematically varying the external shear stress and pressure, the steady rheological response is simulated for: (1) rate-independent quasi-static flow; and (2) rate-dependent inertial flow. The simulated flow is viscometric with non-negligible first and second normal stress differences. While both normal stress differences are negative in inertial flows, the first normal stress difference switches from negative to slightly positive, and second normal stress difference tends to zero in quasi-static flows. The first normal stress difference emerges from a lack of coaxiality between a second-rank contact fabric tensor and strain rate tensor in the flow plane, while the second normal stress difference is linked to an excess of contacts in the shear plane compared with the vorticity direction. A general rheological model of second order (in terms of strain rate tensor) is proposed to describe the two types of flow, and the model is calibrated for various values of interparticle friction from simulations on nearly monodisperse spheres. The model incorporates normal stress differences in both regimes of flow and provides a complete viscometric description of steady dense granular flows., Accepted Version, 28 pages, 8 figures

The anomaly of the energy reverse transmission process makes the turbulent kinetic energy production term in the turbulent kinetic energy transport equation negative. The negative production of turbulent kinetic energy (NPTKE) will affect the redistribution of energy in the flow field, and the conventional gradient assumption is not applicable in many flow situations. In this paper, the Reynolds stress turbulence model is used to solve the two-dimensional compressible turbulent kinetic energy transport equation. The commercial software Fluent v19.1 is utilized to numerically simulate the supersonic channel flow with the effect of shock waves. The results indicate that the Reynolds stress model considering flow anisotropy can characterize the NPTKE. The inherent properties of the mean strain rate tensor influence the turbulent kinetic energy production, and the NPTKE is dominated by the stretching factors. The compression caused by the shock wave leads to a constant positive turbulent kinetic energy production at the position, and the local maximum value is approximately obtained.

We derive the macroscopic stress tensor for plug-free dense granular flow, using a first-principles coarse-graining of the intergranular forces. The derivation is based on the assumption, which defines the da Vinci fluid model, that the intergranular interactions are dominated by normal contact forces and solid friction. An explicit form for the stress---strain rate relation in the slow flow regime is obtained, providing, together with previously derived equations for the formation and growth dynamics of plug regions, a full closure for the rheology of dense granular fluids, in terms of well-defined material parameters. This relation allows us to quantify the strain rate, at which the flow crosses over from solid-friction-dominated to viscosity-dominated flow.

The stress in the quasi-static particle flow is often modeled through the Mohr-Coulomb failure criterion. In the extension to complex three-dimensional flows, a granular viscosity is introduced through a tensorial rheology and the deviatoric frictional stress tensor is assumed aligned with the strain rate tensor. This granular viscosity is singular as the shear rate approaches zero, regardless of the local rheology. We discuss the influence of regularizing such a frictional viscosity on the particle circulation rate and other measured characteristics in a laboratory scale draft tube spouted bed. The friction between particles is modeled either with a constant Coulomb rheology or using a local particle pressure and strain-rate based friction known as μ I -rheology. The predictions appear very dependent on the regularization parameter introduced by the method. The mean properties of the flow (e.g. circulation and pressure drop) monotonically converge towards the measurements when the regularization parameter tends to zero. In other respects, the two regularization models regarded in this study induced similar hydrodynamics within the spouted bed of interest. But the analysis of the conditional averages of the inertial number and the fraction of the solids in the quasi-static regime shows that the extent and staticity of the quasi-static region is sensitive to changes to the regularization parameter or regularization function.

This work criticizes the entrenched views according to which the orientation of the principal tectonic stress axes can be determined from local (in time and space) observations of the kinematic indicators which jointly allow estimating the strain rate of a crustal block under study. The criticized approach ignores or replaces by subjective assumptions the following factors: (1) The block’s interaction with a hosting medium expressed in terms of the equilibrium conditions of the block; (2) The stress rate which (in addition to stresses) can affect the strain rate; (3)The specific macroscopic mechanical properties of the block’s material under the unknown sought stresses, including the ratio of the stress relaxation time to the period of observations. This approach, which is developed in some Solid Earth sciences and mainly in tectonophysics, is referred to in our paper as the method of local kinematic reconstruction (MLKR) of stresses. After briefly surveying the concept of forces and stresses and discussing the importance of studying the tectonic stresses, this paper refutes the MLKR notions based on general arguments and by the example of certain thought experiments. It is shown that the use of the MLKR for the conditions of the Earth’s interior does not guarantee against obtaining the results that fundamentally and drastically differ from the true tectonic stresses. In the studied rock block, depending on the factors ignored in the MLKR, the principal axes of the strain rate tensor, on one hand, and the principal stress axes, on the other hand, can be oriented discordantly in any arbitrary given fashion. In particular, in the processes accompanied by the release of elastic energy, the maximal rate of elongation can be oriented along the axis of maximal compression, whereas the maximal rate of shortening can be aligned with the axis of the maximal tension. In this paper, the deformation processes that are most detrimental to the results of stress reconstruction by the MLKR are revealed. We introduce the notion of the inherited stress-state regime in which the orientation of the axes of principal stresses during the observation period does not depend on the deformation process and, hence, cannot be in principle determined by the MLKR. An attempt to directly locally recover the stress axes from the kinematic data is a false objective because neither the physical meaning of the stress tensor nor the way it is introduced has anything to do with strains. It is concluded that the MLKR is physically inadequate and that the tectonophysical concept of locality should be abandoned in favor of returning to the notions of classical physics, namely, to using the conservation laws. By the example of several guides on tectonophysics, this paper exposes typical errors in understanding the stress reconstruction problem.

Mechanical forces and interactions participate in ontogenesis at all scale levels: intracellular, cellular, and supra-cellular, the latter including tissue level. This concept, now almost trivial, was finding its way with difficulties, and the works of L.V. Beloussov have played a decisive role in its establishment. The continuum approach presented in this study makes it possible to take at the tissue level into account both relative motion of cells and forces that control this motion. The characteristics which allow us to take into account general active properties of the cell medium are described, possible mechanisms represented by these characteristics are discussed, and a concise review of our results obtained to date is presented. In the strain rate tensor, two separate components are distinguished, one of them being related to deformation of individual cells and the other to cell rearrangement. A separate phase (submedium) that corresponds to active subcellular elements associated with rearrangement-controlling active stresses is also introduced. Within this general approach two specific models are considered. The first made it possible to establish general mechanisms whose account enabled us to satisfactorily describe the experimental results of L.V. Beloussov and collaborators, concerning mechano-dependent reactions of embryonic epithelium explants. On the assumptions that the active stress responds to cell shape deviations and the rearrangement strain rate component depends on the active stresses developed by pseudopodia, the cell shape and tissue stress evolution observed experimentally in stretched explants, as well as their post-release deformation, are reproduced. The second particular model considers self-organization in a conglomerate of loosely connected cells in the presence of a fluid phase. In this case, the active stress was assumed to nonlocally depend on the density of cells and the rearrangement strain rate on the active and passive stresses. Due to loss of stability of the spatially homogeneous state, various structures similar to those observed in embryogenesis develop. In particular, within the conglomerate, a cavity can be formed, a certain level of the fluid pressure being necessary for this.

Based on the published GPS data for the Upper Amur region, the parameters of the tensor of strain rates are calculated. In particular, we obtain such parameters of the main rates and main directions of strains, maximum shear strain rates and its directions, dilatancy rate, and second invariant of the strain rate tensor. The obtained results indicate a high tectonic activity in the zone of interaction between the Eurasian and Amurian plates and Stanovoy geoblock. The zones of high-rate aseismic displacements are revealed. It is supposed that NE-trending faults will be activated at the contemporary stage at the boundary between the Eurasian and Amurian plates.

Clustering of coherent fine-scale eddies in a turbulent mixing layer has been analyzed by using direct numerical simulation (DNS) data at Reλ ≃ 250. The coherent fine-scale eddies are defined based on the second invariant of the velocity gradient tensor and the vorticity vector. The clustering is evaluated by the number density of coherent fine-scale eddies, and the large-scale structures are extracted by low-pass filtered velocity fields. Conditional averaging shows that the large-scale enstrophy increases with the number density, whereas the large-scale strain rate stays around the average in the high number density region. The alignments of the vorticity vector and the eigenvectors of the large-scale strain rate tensor are conditioned by the number density or the strain rate magnitude. The eigenvectors and the vorticity vector indicate strong preferential alignments under the intense large-scale strain rate condition. On the other hand, those alignments become weak in the high number density regions. The inter-scale energy transfer between grid and subgrid scales is significantly correlated with the magnitude of the large-scale strain rate while there is no apparent correlation with the number density.

It is known that turbulence is characterized by intermittence which is closely related to the development of unsteady nonisotropic intense small-scale vortex structures. In this study, small fluid particles from the inertial range of isotropic turbulence are considered. It is shown that the phenomenon of rotation intensification and stretching of the particles can be analyzed theoretically. In recent experimental and numerical studies, where this phenomenon was called “the pirouette effect”, its significance in the mechanism of the intense small-scale structures generation was discussed. In this study, a linear stochastic Lagrangian model for the effect is developed. In this model, the kinetic equation for the distribution function of the squared cosine of the angle between the vorticity and the eigenvector of the strain rate tensor of a fluid particle is derived and time history asymptotics of this quantity are analytically calculated at large and small times. The results are in good agreement with the recent experiments and numerical calculations. An analysis made in this study shows that the linear processes probably play the crucial role in certain processes in the isotropic turbulence, which is known to be a principally nonlinear phenomenon. The model developed makes it possible to analyze the statistics of the Lagrangian dynamics of small fluid particles in the inertial range which can be useful in some computational approaches to turbulence.

A new optimality criterion algorithm is presented for producing modified shape designs for fluid flow inside channels. To compute the fluid motion in a channel, the lattice Boltzmann method (LBM) was used based on D2Q9 and D3Q15 lattice spaces associated with the Bhatnagar-Gross-Krook (BGK) collision term. An experiential optimality method to design channels with the lowest pressure drop along the passage is introduced. The positions of solid cells and fluid cells are exchanged based on the strain rate tensor at the solid-fluid interface. To obtain the optimized shape, the cells are changed until the optimality condition is obtained with the restriction of constant fluid volume. Examples are presented to validate the algorithm, including an elbow tube as well as symmetrical and nonsymmetrical Tjunction channels. The validation exercises demonstrate that the algorithm is suitable for optimal channel design.

In the non-classical continuum theories for solid continua the presence of internal rotations and their gradients arising due to Jacobian of deformation and/or consideration of Cosserat rotations as additional unknown degrees of freedom at a material point necessitate existence of moment tensor. For small deformation, small strains theories, in Lagrangian description the Cauchy moment tensor and the rates of rotation gradients are rate of work conjugate pair in addition to the rate of work conjugate Cauchy stress tensor and the strain rate tensor. It is well established that in such non-classical theories the Cauchy stress tensor is non-symmetric and the antisymmetric components of the Cauchy stress tensor are balanced by gradients of the Cauchy moment tensor, the balance of angular momenta balance law. In the non-classical continuum theories incorporating internal rotations and conjugate moment tensor that are absent in the classical continuum theories, the fundamental question is “are the conservation and balance laws used in classical continuum mechanics sufficient to ensure dynamic equilibrium of the deforming volume of matter”. At this stage the Cauchy moment tensor remains non-symmetric if we only consider standard balance laws that are used in classical continuum theories. Thus, requiring constitutive theories for the symmetric as well as anti-symmetric Cauchy moment tensors. The work presented in this paper shows that when the thermodynamically consistent constitutive theories are used for symmetric as well as antisymmetric Cauchy moment tensor non physical and spurious solutions result even in simple model problems. This suggests that perhaps the additional conjugate tensors resulting due to presence of internal rotations, namely the Cauchy moment tensor and the antisymmetric part of the Cauchy stress stress tensor must obey some additional law or restriction so that the spurious behavior is precluded. This paper demonstrates that in the non-classical theory with internal rotations considered here the law of balance of moment of moments and the consideration of the equilibrium of moment of moments are in fact identical. When this balance law is considered the Cauchy moment tensor becomes symmetric, hence eliminating the constitutive theory for the antisymmetric Cauchy moment tensor and thereby eliminating spurious and non physical solutions. The necessity of this balance law is established theoretically and is also demonstrated through model problems using thermoelastic solids with small strain small deformation as an example. The findings reported in this paper hold for thermoviscoelastic solids with and without memory as well as when deformation and strains are small. Extensions of the concepts presented here for finite deformation and finite strain will be presented in a follow up paper.

The average seismic strain rate is estimated for the seismotectonic zone of the northern/central parts of the Gulf of Suez. The principal strain rate tensor and velocity tensor were derived from a combination of earthquake focal mechanisms data and seismic moment of small-sized earthquakes covering a time span of 13 years (1992–2004). A total of 17 focal mechanism solutions have been used in the calculation of the moment tensor summation. The local magnitudes (MLs) of these events range from 2.8 to 4.7. The analysis indicates that the dominant mode of deformation in the central and northern parts of the Gulf of Suez is extension at a rate of 0.008 mm/year in N28°E direction and a small crustal thinning of 0.0034 mm/year. This low level of strain means that this zone experienced a little seismic deformation. There is also a right lateral shear motion along the ESE–WNW direction. This strain pattern is consistent with the predominant NW–SE normal faulting and ESE–WNW dextral transtensive faults in this zone. Comparing the results obtained from both stress and strain tensors, we find that the orientations of the principal axes of both tensors have the same direction with a small difference between them. Both tensors show a predominantly extensional domain. The nearly good correspondence between principal stress and strain orientations in the area suggests that the tectonic strength is relatively uniform for this crustal volume.

The tip leakage vortex (TLV) cavitation is investigated by a commercial Reynolds averaged Navier-Stokes (RANS) solver. A referenced test on a NACA0009 hydrofoil is used to validate the numerical simulation. Considering the local rotation characteristics of the vortical flow, a rotation-curvature corrected Shear-Stress-Transport model (SST-CC model) is applied to simulate the time-averaged turbulent flow. Compared to the original SST model, the SST-CC model improves the prediction of the velocity in TLV on the measured sections in downstream, and the vorticity and pressure features along the TLV trajectory are analysed numerically. In order to increase the prediction accuracy for the TLV cavitation, the empirical condensation coefficient (Fc) in Zwart's cavitation model is calibrated based on the referenced experiment. By introducing a vortex identification parameter (f∗) related to the strain rate tensor and the vorticity tensor, a relationship between the Fc and f∗ is built, and the effects of the rotational motion of the vortex on the cavity are embodied in a modified Zwart's cavitation model. Compared to the conventional Zwart's cavitation model, the modified cavitation model significantly improves the prediction of the TLV cavitation and gets a better agreement with the referenced test on different conditions with various gap widths.

In this paper, we present a fractional dashpot that is capable of continuously adjusting the degree of dissipation between the whole dashpot and the Maxwell model (spring and dashpot in serial connection). It is related to the springpot by a frequency shift of the relaxation modulus in the complex domain. In contrast to the springpot, which is a viscoelastic solid-like model, the fractional dashpot is an intrinsically viscoelastic fluid model. We demonstrate that the fractional dashpot allows one to better model the polydispersity effect of typical viscoelastic fluids, overcoming undesired stationary predictions and reducing or even eliminating multiple modes in data fitting. The linear version of the fractional dashpot model is also scaled up to large deformation by incorporation of corotational tensor derivatives and a projected strain rate tensor for constructing an objective fractional constitutive equation. The resulting model having five model parameters (one for fractionality, two for linear viscoe...

This article is focused on the derivation of constitutive equations for magnetic liquids. The results can be used for both ferromagnetic and magnetorheological fluids after the introduced simplifications. The formulation of constitutive equations is based on two approaches. The intuitive approach is based on experimental experience of non-Newtonian fluids, which exhibit a generally non-linear dependence of mechanical stress on shear rate; this is consistent with experimental experience with magnetic liquids. In these general equations, it is necessary to determine the viscosity of a liquid as a function of magnetic induction; however, these equations only apply to the symmetric stress tensor and can only be used for an incompressible fluid. As a result of this limitation, in the next part of the work, this approach is extended by the asymmetry of the stress tensor, depending on the angular velocity tensor. All constitutive equations are formulated in Cartesian coordinates in 3D space. The second approach to determining constitutive equations is more general: it takes the basis of non-equilibrium thermodynamics and is based on the physical approach, using the definition of density of the entropy production. The production of entropy is expressed by irreversible thermodynamic flows, which are caused by the effect of generalized thermodynamic forces after disturbance of the thermodynamic equilibrium. The dependence between fluxes and forces determines the constitutive equations between stress tensors, depending on the strain rate tensor and the magnetization vector, which depends on the intensity of the magnetic field. Their interdependencies are described in this article on the basis of the Curie principle and on the Onsager conditions of symmetry.

The basis of many turbulence models in computational fluid dynamics is the linear Boussinesq hypothesis that assumes an alignment between the mean strain rate tensor and the Reynolds stress tensor. The validity of this main assumption is analyzed for the test case of an infinite tube bundle with periodic boundary conditions at Re b ≈ 34 , 800 . This work focuses on the application of five methods based on the Reynolds-averaged Navier–Stokes equations and two scale resolving methods and their ability to accurately reproduce the mean velocity field and the Reynolds stresses. In addition, their capability to predict the anisotropic behavior of the turbulent flow is analyzed. The results indicate that only scale resolving methods are able to predict the turbulent flow field properly for the case under investigation. The results of the large eddy simulation are used to further analyze the distribution of the validity parameter ρ RS and the anisotropic turbulence field. Only small areas between the tubes are identified in which alignment occurs and the linear Boussinesq hypothesis is therefore fully valid.

In this study, direct numerical simulations are performed to investigate the spatial evolution of a dual-plane jet flow with L d / L 0 = 6, where L d and L 0 are the separation distance between the two jets and the jet width, respectively. The formation mechanism of the turbulent kinetic energy along the centerline can be quite different. In the reversal flow region, the transport equation of turbulent kinetic energy can be seen as the combination of the transport, convection, production, pressure, and viscous dissipation terms, whereas in the downstream combined region, it can be further simplified as the combination of convection and viscous dissipation terms. The gain of the centerline TKE mainly occurs in the vertical direction through the turbulence transport effects. Concerning the strain product process, in all three regions the strain self-amplification term is the dominant source term, whereas the enstrophy production and viscous dissipation terms are main sink terms. Two important aspects of small-scale motions are considered, one being the correlation between the vorticity vector and the corresponding eigenvectors of strain rate tensor and the other being the characteristics of invariant of the velocity gradient. The joint PDFs of the strain self-amplification and enstrophy production are highly squeezed in the vertical direction in the converging region. In contrast, in the reversal flow region, where the flow is still highly intermittent, the contour lines already take similar albeit slightly squeezed shapes as those in HIT. The joint PDFs of the second and third invariants in the reversal and highly intermittent flow region and the subsequent fully turbulent regions acquire the well-known ‘teardrop’ shape. It is demonstrated that even in the highly intermittent region, the characteristics of small-scale motions are already close to the fully developed flow. These above findings contribute to a better understanding of the dual-plane jet flows.

The deformation of fluid particles is usually directly related to various Lagrangian phenomena in fluids. We show typical analytical expressions under external strain rate fields by defining the deformation limit for small fluid particles, aiming at providing analytical tools for supporting future investigations on the Lagrangian phenomena in fluids. In particular, when particles are driven by a constant strain rate which is greater or equal to the rotation rate, the orientation of fluid elements aligns with strain tensor in short times and then decouples with them in long times, while the fluid particles are flattened in most cases, which are similar to previous observations in turbulent flows. This supports the simplifications in previous Lagrangian models (e.g., Fang et al., 2015 [20] ), where the strain rates were assumed to be locally homogeneous.

We present an estimation method of fully-resolved/coarse-grained gradients from randomly distributed points in turbulence. The method is based on a linear approximation of spatial gradients expressed with the volumetric tensor, which is a 3 × 3 matrix determined by a geometric distribution of the points. The coarse grained gradient can be considered as a low pass filtered gradient, whose cutoff is estimated with the eigenvalues of the volumetric tensor. The present method, the volumetric tensor approximation, is tested for velocity and passive scalar gradients in incompressible planar jet and mixing layer. Comparison with a finite difference approximation on a Cartesian grid shows that the volumetric tensor approximation computes the coarse grained gradients fairly well at a moderate computational cost under various conditions of spatial distributions of points. We also show that imposing the solenoidal condition improves the accuracy of the present method for solenoidal vectors, such as a velocity vector in incompressible flows, especially when the number of the points is not large. The volumetric tensor approximation with 4 points poorly estimates the gradient because of anisotropic distribution of the points. Increasing the number of points from 4 significantly improves the accuracy. Although the coarse grained gradient changes with the cutoff length, the volumetric tensor approximation yields the coarse grained gradient whose magnitude is close to the one obtained by the finite difference. We also show that the velocity gradient estimated with the present method well captures the turbulence characteristics such as local flow topology, amplification of enstrophy and strain, and energy transfer across scales.

In this work, a data-driven reduced-order model is presented to predict the microscale spatial distribution of the plastic strain rate tensor in an isotropic two-phase composite subjected to an arbitrary macroscopically imposed strain rate tensor. This model was built using the framework of localization linkages called Material Knowledge Systems (MKS), which has been demonstrated to exhibit a remarkable combination of accuracy and low computational cost. In prior work, the MKS framework was successfully used to predict the local strain rate fields in multiphase composites subjected to a selected macroscale strain rate tensor. In this work, the MKS framework is extended to include the complete set of all macroscale strain rate tensors that could be applied. This is accomplished by developing novel representations that allow a parametrization of the localization kernel over the complete space of unit symmetric traceless second-rank tensors and implementing them with the required fast computational strategies. The MKS localization linkage produced in this work was calibrated and validated to results from microscale finite element models.

Aim at avoiding the issue of energy anomaly near the stagnation point in bluff body flows, a new reformulated production limiter considering the efforts of strain rate tensor and rotation tensor is proposed for the Scale-Adaptive Simulation method (SAS). Meanwhile, two other limiters only considering strain or vorticity are also conducted for comparison. The concrete mechanism and capabilities of different limiters in adjusting the turbulent energy are carefully tested through the typical circular cylinder flow at the subcritical Reynolds number 3 900 and 1.4 × 10 5 . As for the validation, detailed numerical comparisons with the relevant experimental data are presented. The numerical results show that, with the recommended production limiter, the issue of spurious energy generation is effectively remedied and the turbulent viscosity is reduced to an appropriate level in the wake region, the concerned flow qualities such as recirculation length, velocity profiles and integral parameters are in excellent agreement with the measurements benefiting from the attractive representation of the shear layers. Moreover, the prediction of high order Reynolds stress is also comparable with that of some DES and LES outputs. In general, the well established production limiter is available for the development of SAS and other hybrid SST formulations.

The practice of periodically reinitializing the level set function is well established in two-phase flow applications as a way of controlling the growth of anomalies and/or numerical errors. In the present work, the underlying roots of this anomalous growth are studied, where it is established that the augmentation of the magnitude of the level set gradient (||) is directly connected to the nature of the flow field; hence, it is not necessarily the result of some type of numerical error. More specifically, for a general flow field advecting the level set function, it is shown that the eigenpairs of the strain rate tensor are responsible for the rate of change of || along a fluid particle trajectory. This straining action not only affects the magnitude of ||, but the general character of , and consequently contributes to the growth in numerical error. These numerical consequences are examined by adopting the Gradient Augmented Level Set method. Specifically, it is shown that the local error for is directly connected to the size of || and to the magnitude of the second and fourth order derivatives of . These analytical findings are subsequently supported by various examples. The role of reinitialization is discussed, where it is shown that in cases where the zero level set contour has a local radius of curvature that is below the local grid resolution, reinitialization exacerbates rather than diminishes the degree of error. For other cases, where the interface is well resolved, reinitialization helps stabilize the error as intended.

The Eshelby tensor field is evaluated analytically for polygonal inclusions of arbitrary shape. The formula contributed in the paper is directly expressed as function of the coordinates defining the vertices of the polygon, thus avoiding the use of complex variables and anomalies exploited in previous contributions on the subject. It has been obtained by evaluating analytically the displacement field induced by a uniform eigenstrain within the inclusion and differentiating its expression in order to derive the infinitesimal strain tensor and, hence, the Eshelby tensor. This allows us to deal only with the first derivative of the Green tensor, appearing in the expression of the displacement field, rather than addressing its second derivative as commonly done in the literature. The proposed formulation has been implemented in a Matlab code and we report the numerical results concerning inclusions whose boundary is defined by Laurent polynomials or by quasi-circular shapes.

This article reports an efficient method to characterize constitutive responses based on multiscale modeling for fluid flow in heterogeneous media based on the concept of representative volume element (RVE). Between different scales, it is considered as the basic principles for down-scaling information the conservation of velocity and of the strain rate tensor. Within this context, we formulate (i) the problem to be solved at the micro-scale, (ii) the up-scaling procedure which involves homogenization rules, and (iii) the generalized principle of multiscale virtual power. The complete theory for constitutive modeling is revisited and shown that when employing multiscale analysis among the suitable variational arguments we are able to obtain, in a straightforward manner, new constitutive behavior between kinematic motions and actions. Some examples of application of fluid flow in heterogeneous media with obstacles are presented to show the consequences of the proposed approach.

A mathematical model of the hydroelasticity of a flow of a non-Newtonian fluid in a wedge-shaped clearance with an elastic wall has been constructed. This wall represents a thin cylindrical elastic shell inextensible along its central axis, for which the Kirchhoff–Love hypotheses are true. A fluid flow in the clearance was simulated in the approximation of the Reynolds lubrication theory with the use of an asymptotic approximation for the second invariant of the strain rate tensor. Results of numerical investigations with the mathematical model proposed are presented.

In this paper, to simulate free surface flows of granular materials in a dense regime as a continuum media, a 2D SPH model is developed. The dense flow is characterized as a pressure dependent visco-plastic material based on a local constitutive law to calculate effective viscosity related to local pressure and the norm of strain rate tensor in the numerical method. A simple regularization technique is proposed to reproduce stopping condition and the free surface of a granular flow where the pressure vanishes. Pressure fluctuation as the main drawback of the weakly compressible SPH method leads to an inaccurate pressure distribution. This numerical instability increases at the free surface due to errors associated with the truncated kernels. In this work, a new algorithm is proposed to remove the nonphysical fluctuations by relating divergence of velocity to the Laplacian of pressure. The algorithm is validated for reproducing the dynamics and deposits of collapsing granular columns. The excellent agreement with experimental data is obtained. The maximum thickness of a granular flow on a rough inclined plane is obtained based on the local rheology model. The run-out distances and slopes of the deposits in the simulations also show good agreement with the values found in the experiments.

In this article, we consider the problem of finding the effective moment tensor for shear slip along a bimaterial interface. When the slip occurs at the jump of the values of elastic parameters, it is not clear how the elasticity tensor of each side should be combined in a unique way to obtain the effective elasticity tensor that then can be used in the moment tensor expression. Moreover, the Green’s function of the medium is not differentiable at the bimaterial interface, which prevents writing down the moment tensor form of the representation theorem, therefore leading to an ambiguity. In this article, we first show that the derivative of the Green’s function can be obtained once the Green’s function is considered as a hyperfunction, which is an equivalent concept to generalized functions or distributions. The resulting Green’s function at the bimaterial interface is the average of the Green’s function derivatives evaluated on either side. Finally, to obtain the effective elastic parameters, we used the equivalence of the unambiguous potency form and the moment tensor form of the representation theorems constrained with the boundary conditions, which are the continuity of the traction acting on the fault and the tangential strain across the fault. A unique solution is obtained for the isotropic case as well as transversely isotropic and orthotropic media if the symmetry axes of the tensors are aligned with the fault orientation. However, for general anisotropy or arbitrarily oriented structures, the solution is nonunique, for which certain physical constraints such as positive definiteness of the elasticity tensor can be used to confine the solution.

A stress tensor eigenvector projection space is created to describe reaction pathways on the (H 2 O) 5 MP2 potential energy surface. Evidence for the stabilizing role of the O---O bonding interactions is found from the length of the recently introduced stress tensor trajectory in the stress tensor eigenvector projection space. The stress tensor trajectories demonstrate coupling behavior of the adjoining covalent (σ) O-H and hydrogen bonds due to sharing of covalent character. Additionally, the stress tensor trajectories can show dynamic coupling effects of pairs of σ bonds and of pairs of hydrogen bonds.

The effects of combustion on the strain rate field are investigated in turbulent premixed CH4/air Bunsen flames using simultaneous tomographic PIV and OH LIF measurements. Tomographic PIV provides three-dimensional velocity measurements, from which the complete strain rate tensor is determined. The OH LIF measurements are used to determine the position of the flame surface and the flame-normal orientation within the imaging plane. This combination of diagnostic techniques enables quantification of divergence as well as flame-normal and tangential strain rates, which are otherwise biased using only planar measurements. Measurements are compared in three lean-to-stoichiometric flames that have different amounts of heat release and Damkohler numbers greater than unity. The effects of heat release on the principal strain rates and their alignment relative to the local flame normal are analyzed. The extensive strain rate preferentially aligns with the flame normal in the reaction zone, which has been indicated by previous studies. The strength of this alignment increases with increasing heat release and, as a result, the flame-normal strain rate becomes highly extensive. These effects are associated with the gas expansion normal to the flame surface, which is largest for the stoichiometric flame. In the preheat zone, the compressive strain rate has a tendency to align with the flame normal. Away from the flame front, the flame – strain rate alignment is arbitrary in both the reactants and products. The flame-tangential strain rate is on average positive across the flame front, and therefore the turbulent strain rate field contributes to the enhancement of scalar gradients as in passive scalar turbulence. Although increases in heat release result in larger positive values of the divergence as well as flame-normal and tangential strain rates, the tangential strain rate has a weaker dependence on heat release than the flame-normal strain rate and the divergence.

This paper concerns finite deformation in the strain-gradient continuum. In order to take account of the geometric nonlinearity, the original strain-gradient theory which is based on the infinitesimal strain tensor is rewritten given the Green–Lagrange strain tensor. Following introducing the generalized isotropic Saint Venant–Kirchhoff material model for the strain-gradient elasticity, the boundary value problem is investigated in not only the material configuration but also the spatial configuration building upon the principle of virtual work for a three-dimensional solid. By presenting one example, the convergence of the strain-gradient and classical theories is studied.

International audience; Kinematics of divergent boundaries and Rift-Rift-Rift junctions are classically studied using long-term geodetic observations. Since significant magma-related displacements are expected, short-term deformation provides important constraints on the crustal mechanisms involved both in active rifting and in transfer of extensional deformation between spreading axes. Using InSAR and GPS data, we analyse the surface deformation in the whole Central Afar region in detail, focusing on both the extensional deformation across the Quaternary magmato-tectonic rift segments, and on the zones of deformation transfer between active segments and spreading axes. The largest deformation occurs across the two recently activated Asal-Ghoubbet (AG) and Manda Hararo-Dabbahu (MH-D) magmato-tectonic segments with very high strain rates, whereas the other Quaternary active segments do not concentrate any large strain, suggesting that these rifts are either sealed during interdyking periods or not mature enough to remain a plate boundary. Outside of these segments, the GPS horizontal velocity field shows a regular gradient following a clockwise rotation of the displacements from the Southeast to the East of Afar, with respect to Nubia. Very few shallow creeping structures can be identified as well in the InSAR data. However, using these data together with the strain rate tensor and the rotations rates deduced from GPS baselines, the present-day strain field over Central Afar is consistent with the main tectonic structures, and therefore with the long-term deformation. We investigate the current kinematics of the triple junction included in our GPS data set by building simple block models. The deformation in Central Afar can be described by adding a central microblock evolving separately from the three surrounding plates. In this model, the northern block boundary corresponds to a deep EW-trending trans-tensional dislocation, locked from the surface to 10–13 km and joining at depth the active spreading axes of the Red Sea and the Aden Ridge, from AG to MH-D rift segments. Over the long-term, this plate configuration could explain the presence of the en-´ echelon magmatic basins and subrifts. However, the transient behaviour of the spreading axes implies that the deformation in Central Afar evolves depending on the availability of magma supply within the well-established segments.

Different theoretical and laboratory studies on the propagation of elastic waves in layered hydrocarbon reservoir have shown characteristic velocity dispersion and attenuation of seismic waves. The wave-induced fluid flow between mesoscopic-scale heterogeneities (larger than the pore size but smaller than the predominant wavelengths) is the most important cause of attenuation for frequencies below 1 kHz. Most studies on mesoscopic wave-induced fluid flow in the seismic frequency band are based on the representative elementary volume, which does not consider interaction of fluid flow due to the symmetrical structure of representative elementary volume. However, in strongly heterogeneous media with unsymmetrical structures, different courses of wave-induced fluid flow may lead to the interaction of the fluid flux in the seismic band; this has not yet been explored. This paper analyses the interaction of different courses of wave-induced fluid flow in layered porous media. We apply a one-dimensional finite-element numerical creep test based on Biot's theory of consolidation to obtain the fluid flux in the frequency domain. The characteristic frequency of the fluid flux and the strain rate tensor are introduced to characterise the interaction of different courses of fluid flux. We also compare the behaviours of characteristic frequencies and the strain rate tensor on two scales: the local scale and the global scale. It is shown that, at the local scale, the interaction between different courses of fluid flux is a dynamic process, and the weak fluid flux and corresponding characteristic frequencies contain detailed information about the interaction of the fluid flux. At the global scale, the averaged strain rate tensor can facilitate the identification of the interaction degree of the fluid flux for the porous medium with a random distribution of mesoscopic heterogeneities, and the characteristic frequency of the fluid flux is potentially related to that of the peak attenuation. The results are helpful for the prediction of the distribution of oil–gas patches based on the statistical properties of phase velocities and attenuation in layered porous media with random disorder.

In this work we study the effect of the nuclear tensor force on properties related with deformation. We focus on isotopes in the Mg, Si, S, Ar, Sr and Zr chains within the Hartree–Fock–Bogoliubov theory using the D1ST2a Gogny interaction. Contributions to the tensor energy in terms of saturated and unsaturated subshells are analyzed. Like–particle and proton–neutron parts of the tensor term are independently examinated. We found that the tensor term may considerably modify the potential energy landscapes and change the ground state shape. We analyze too how the pairing characteristics of the ground state change when the tensor force is included.