Principal Directions Of Strain Rate Research Articles

On the physical nature of the turbulent/turbulent interface

The existence of a turbulent/turbulent interface (TTI) has recently been verified in the far wake of a circular cylinder exposed to free-stream turbulence (Kankanwadi & Buxton, J. Fluid Mech., vol. 905, 2020, p. A35). This study aims to understand the physics within the TTI. The wake boundary, approximately 40 diameters downstream of a circular cylinder subjected to grid-generated turbulence, was investigated through simultaneous cinematographic, stereoscopic particle image velocimetry and planar laser induced fluorescence experiments. With no grid placed upstream of the cylinder, the behaviour of the resultant interface, our closest approximation to a turbulent/non-turbulent interface, exactly matched what is observed in existing literature. When background turbulence is present, viscous action is no longer the only method by which enstrophy is transported to the background fluid, unlike for turbulent/non-turbulent interfaces. The presence of rotational fluid on both sides of the TTI allows the vorticity stretching term of the enstrophy budget equation to be the dominant actor in this process. The role of viscosity within a TTI is greatly diminished as the vorticity stretching term takes over responsibilities for enstrophy production. The turbulent strain rate normal to the TTI was found to be enhanced in the interfacial region. Decomposing the vorticity stretching term into components aligned with the three principal strain-rate directions, it was found that the term most aligned with the interface-normal direction contributed to the largest share of enstrophy production. This indicates that better ‘organised’ vorticity on the wake side of the interface yields the enstrophy amplification leading to the previously discovered enstrophy jump across the TTI by Kankanwadi & Buxton (J. Fluid Mech., vol. 905, 2020, p. A35).

Insights into the Bending Effect in Premixed Turbulent Combustion Using the Flame Surface Density Transport

ABSTRACTThe bending effect of turbulent flame speed variation (i.e., the deviation from the linear increase of flame speed with increasing root-mean-square turbulent velocity fluctuation) has been investigated based on a Direct Numerical Simulation database of statistically planar turbulent premixed flames propagating into forced unburned gas turbulence. The validity of Damköhler’s first hypothesis has been utilized to analyze the bending effect in terms of generalized Flame Surface Density (FSD) evolution. The volume-integrated value of the tangential strain rate term of the FSD transport equation remains positive, whereas the volume-integrated value of the curvature term assumes negative values. Under statistically stationary state, the positive value of the volume-integrated tangential strain rate term remains in equilibrium with the negative value of the volume-integrated curvature term. It has been found that the contribution of the normal strain rate to the flame surface area remains negative for small turbulence intensities, which eventually become positive for large turbulence intensities. This is a consequence of the change of collinear alignment of the reaction progress variable gradient from the most extensive principal strain rate direction to the direction of the eigenvector associated with the most compressive principal strain rate with increasing turbulence intensity. An increase in turbulence intensity increases the width of the probability density functions of flame curvature, and thereby increases the surface-averaged curvature squared values. This eventually makes the FSD curvature term due to the tangential diffusion component of displacement speed as the major contributor to the negative contribution of the volume-integrated curvature term in the FSD transport equation for large turbulence intensities. However, the negative contribution of the volume-integrated FSD curvature term does not increase indefinitely with increasing turbulence intensity and the inner cut-off scale, which also limits the maximum possible value of the volume-integrated FSD strain rate term under statistically stationary state, governs the maximum possible destruction of flame surface area. It has been argued that the upper limits of the flame surface area generation and destruction are responsible for the bending effects in the variations of turbulent flame speed and flame surface area.

Strain rate and flame orientation statistics in the near-wall region for turbulent flame-wall interaction

Flame–wall interaction (FWI) in premixed turbulent combustion has been analysed based on a counter-flow like configuration at the statistically stationary state. For the present configuration, the two FWI sub-zones, i.e the influence zone and the quenching zone, can be identified from the DNS results. Detailed analysis of the important quantities, such as the flame temperature, flame–wall distance, wall heat flux, flame curvature and dilatation (including the flame normal and tangential strain rates), and some orientation relations between the flame normal and the principal strain rate directions, have been reported, together with the physical explanations. All these statistical results are determined by the relative strengths of the wall heat flux, thermal expansion and the flame–turbulence interaction.

Laboratory and micromechanical investigation of soil anisotropy

This paper presents an experimental investigation revisiting the anisotropic behavior of geomaterials in drained monotonic shear using Hollow Cylinder Apparatus. The test program has been designed to cover the effect of material anisotropy, pre-shearing and material density on the behavior of Leighton Buzzard sand. Experiments have also been performed on glass beads to understand the effect of particle shape. The test results demonstrate that the stress-strain-strength behavior of the specimen shows strong dependence on the principal stress direction. The pre-shearing history, material density and particle shape are found to be influential. The deformation non-coaxiality, i.e., non-coincidence between the principal stress direction and the principal strain rate direction is observed. In particular, it was found that non-coaxiality is more significant in presheared specimens. This paper also explains the phenomenological observations based on the recently acquired understanding in micromechanics, with attention focused on strength and deformation non-coaxiality.

A laboratory study of anisotropic geomaterials incorporating recent micromechanical understanding

This paper presents an experimental investigation revisiting the anisotropic stress–strain–strength behaviour of geomaterials in drained monotonic shear using hollow cylinder apparatus. The test programme has been designed to cover the effect of material anisotropy, preshearing, material density and intermediate principal stress on the behaviour of Leighton Buzzard sand. Experiments have also been performed on glass beads to understand the effect of particle shape. This paper explains phenomenological observations based on recently acquired understanding in micromechanics, with attention focused on strength anisotropy and deformation non-coaxiality, i.e. non-coincidence between the principal stress direction and the principal strain rate direction. The test results demonstrate that the effects of initial anisotropy produced during sample preparation are significant. The stress–strain–strength behaviour of the specimen shows strong dependence on the principal stress direction. Preloading history, material density and particle shape are also found to be influential. In particular, it was found that non-coaxiality is more significant in presheared specimens. The observations on the strength anisotropy and deformation non-coaxiality were explained based on the stress–force–fabric relationship. It was observed that intermediate principal stress parameter b(b = (σ2 − σ3)/(σ1 − σ3)) has a significant effect on the non-coaxiality of sand. The lower the b-value, the higher the degree of non-coaxiality is induced. Visual inspection of shear band formed at the end of HCA testing has also been presented. The inclinations of the shear bands at different loading directions can be predicted well by taking account of the relative direction of the mobilized planes to the bedding plane.

Velocity gradient dynamics in compressible turbulence: Characterization of pressure-Hessian tensor

Pressure-Hessian tensor produces the most significant difference between incompressible and compressible velocity gradient dynamics in turbulent flows. Characterization of pressure-Hessian tensor as a function of the level of compressibility is therefore of much interest. Using direct numerical simulation results, we demonstrate that the pressure-Hessian tensor behavior can be most exclusively characterized in terms of the compressibility parameter δ which is defined to be the growth-rate of dilatation-rate. A key compressibility effect is the distinct change in the alignment between pressure-Hessian and velocity gradient tensors with increasing δ. In incompressible turbulence, the pressure-Hessian eigenvectors exhibit a mild tendency to align at 45° angle with the principal directions of strain rate. With increasing δ, the pressure-Hessian tensor shows progressively stronger tendency to align along principal directions of the local strain-rate tensor. We show that this change in pressure-Hessian orientation causes the compressible velocity gradient dynamics to significantly differ from its incompressible counterpart. In incompressible turbulence, pressure mildly moderates the inherent gradient-steepening tendencies of the nonlinear inertial term. On the other hand, in highly compressible turbulence (extreme values of δ), pressure effects can lead to intense gradient steepening or smoothing depending upon the growth-rate of dilatation rate, thereby profoundly altering the cascade process.

Fabric evolution and accessible geometrical states in granular materials

We analyze the geometrical states of granular materials by means of a fabric tensor involving the coordination number and fabric anisotropy as the lowest-order descriptors of the contact network. In particular, we show that the fabric states in this representation are constrained by steric exclusions and the condition of mechanical equilibrium required in the quasi-static limit. A simple model, supported by numerical data, allows us to characterize the range of accessible fabric states and the joint evolution of fabric parameters. The critical state in this framework appears as a jammed state in the sense of a saturation of contact gain and loss along the principal strain-rate directions.

Limit analysis with anisotropic fibre-reinforced soil

Fibre reinforcement is a technology to improve the strength of soils used as fill in geotechnical structures. The properties of the fibre-reinforced sand are anisotropic because of the distribution of orientation of the fibres, which is characterised by a preferred bedding plane. The development of an anisotropic yield condition for fibre-reinforced sand is presented. Application of the kinematic approach of limit analysis to anisotropic frictional materials is demonstrated. An interesting implication of anisotropy is a lack of uniqueness in the internal friction angle along a shear surface with given orientation. The anisotropic internal friction angle was found to be a function of the major principal strain rate direction, and not just a function of the orientation of the shear surface. While the inclination of principal strain rate directions can be inferred from the kinematics of the mechanism for a given boundary value problem, the internal friction angle cannot be determined a priori as a unique function of orientation in space. Two examples are presented to illustrate the use of the kinematic approach for anisotropic frictional materials: retaining wall loading, and the bearing capacity of a footing.

Limit analysis with anisotropic fibre-reinforced soil

Fibre reinforcement is a technology to improve the strength of soils used as fill in geotechnical structures. The properties of the fibre-reinforced sand are anisotropic because of the distribution of orientation of the fibres, which is characterised by a preferred bedding plane. The development of an anisotropic yield condition for fibre-reinforced sand is presented. Application of the kinematic approach of limit analysis to anisotropic frictional materials is demonstrated. An interesting implication of anisotropy is a lack of uniqueness in the internal friction angle along a shear surface with given orientation. The anisotropic internal friction angle was found to be a function of the major principal strain rate direction, and not just a function of the orientation of the shear surface. While the inclination of principal strain rate directions can be inferred from the kinematics of the mechanism for a given boundary value problem, the internal friction angle cannot be determined a priori as a unique function of orientation in space. Two examples are presented to illustrate the use of the kinematic approach for anisotropic frictional materials: retaining wall loading, and the bearing capacity of a footing.

Style and history of Andean deformation, Puna plateau, northwestern Argentina

Topographically, the Puna plateau of northwestern Argentina is the southern continuation of the Bolivian Altiplano. Its thickening and consecutive uplift result from the Andean orogeny. To better constrain the structural style and its progressive development, we have studied field data, topographic and satellite imagery, balanced cross sections, seismic reflection data, kinematic analysis of fault slip data, anisotropy of magnetic susceptibility (AMS), paleomagnetic data, and apatite fission track (AFT) data. Across the Puna plateau, Precambrian and Paleozoic basement ranges, bounded by high‐angle reverse faults (dips ≥ 60°), alternate with Cenozoic intermontane basins. Major thrusts trend NNE‐SSW and do not show a preferred vergence. Intermontane basins have various degrees of symmetry, depending on the geometries and attitudes of associated thrusts as well as on the magnitudes of their offsets. There is a close correlation between the surface expression of a basin and the amount of internal deformation. A line‐balanced cross section of the Puna at 25°S has yielded a Cenozoic shortening of 10–15%, in a direction subperpendicular to the orogen. By kinematic analysis of Cenozoic fault slip data we have obtained principal directions of strain rate across the Puna. Shortening axes are subhorizontal and trend on average WNW‐ESE (∼N110°), stretching axes are subvertical, and intermediate axes are subhorizontal and trend on average NNE‐SSW. Strain ellipsoids are dominantly of plane strain type, and they represent dip‐slip thrusting. From paleomagnetic and AMS data, shortening axes form a radial pattern around the eastern edge of the central Andes. The pattern is attributed to an inhomogeneous stress field, reflecting the eastward convex shape of the central Andean thrust front. From the history of burial and uplift, Andean shortening reached the northeastern part of the Puna in the late Eocene and the adjacent Eastern Cordillera in the late Eocene or early Oligocene. This shortening was presumably due to the Incaic phase of the Andean orogeny. In the eastern part of the orogen the onset of shortening was probably guided by preexisting Paleozoic and Mesozoic structures, so that Andean deformation propagated unevenly eastward.

Strain partitioning during oblique plate convergence in northern Sumatra: Geodetic and seismologic constraints and numerical modeling

Global Positioning System (GPS) measurements along the subduction zone of northern Sumatra (2°S to 3°N) reveal that the strain associated with the oblique convergence of the Australian plate with Eurasia is almost fully partitioned between trench‐normal contraction within the forearc and trench‐parallel shear strain within a few tens of kilometers of the Sumatran fault Kinematic analyses of interplate earthquake slip vectors provide slip rates on the Sumatran fault within a few millimeters per year of GPS and geologic rates, giving us more confidence in the use of slip vectors for inferring slip partitioning elsewhere. The inferred slip rate on the Sumatran fault is ∼1/3 less than the full margin parallel component of plate motion. An across‐forearc rotation in the slip vectors suggests that the missing arc‐parallel shear occurs seaward of the geodetic network, between the forearc islands and the trench. Simple finite element models are used to explore the conditions under which the change in the principal strain rate directions between the forearc and the arc region can occur. Modeling suggests that neither a preexisting strike‐slip fault nor a zone of thermally induced lithospheric weakness in the overriding plate is needed for strain partitioning to occur. In general, forearc slivers form over the region of interplate coupling and are driven along strike by the basal shear. A volcanic arc can help the partitioning process by localizing the margin‐parallel shear strain in the upper plate if its crust and mantle are weaker than its surroundings. Interplate slip vectors and geodetic results from Sumatra together suggest that the highest coupling on the plate boundary occurs beneath and seaward of the forearc islands, consistent with inferences about the rupture zones of great nineteenth century earthquakes there. The Sumatra example suggests that geodetic measurements of interseismic, margin‐parallel shear strain at oblique convergent margins can be used to map the landward extent of the relatively high basal stress beneath the overriding plate if one can correct for strain localization caused by weak upper plate strike‐slip faults.

Topology of fine-scale motions in turbulent channel flow

An investigation of topological features of the velocity gradient field of turbulent channel flow has been carried out using results from a direct numerical simulation for which the Reynolds number based on the channel half-width and the centreline velocity was 7860. Plots of the joint probability density functions of the invariants of the rate of strain and velocity gradient tensors indicated that away from the wall region, the fine-scale motions in the flow have many characteristics in common with a variety of other turbulent and transitional flows: the intermediate principal strain rate tended to be positive at sites of high viscous dissipation of kinetic energy, while the invariants of the velocity gradient tensor showed that a preference existed for stable focus/stretching and unstable node/saddle/saddle topologies. Visualization of regions in the flow with stable focus/stretching topologies revealed arrays of discrete downstream-leaning flow structures which originated near the wall and penetrated into the outer region of the flow. In all regions of the flow, there was a strong preference for the vorticity to be aligned with the intermediate principal strain rate direction, with the effect increasing near the walls in response to boundary conditions.

Structural evolution of intermittency and anisotropy at different scales analyzed using three-dimensional wavelet transforms

The three-dimensional Mexican hat wavelet transform is used as a Fourier-spectral space filter to study (1) the process by which the nonlinearities in the equations of motion create intermittent regions of concentrated vorticity and (2) the process by which isotropic turbulence becomes anisotropic by mean shear. In the first study, the three-dimensional wavelet transform is applied to direct numerical simulations of the transition from Gaussian initial conditions to isotropic turbulence. During this transition, the initially noncoherent vorticity field is transformed by the Navier–Stokes nonlinearities into coherent regions of concentrated vorticity. Analysis of the wavelet-transformed enstrophy field suggests (a) that small scales are always more intermittent than large scales, (b) that the level of intermittency at the small scales grows more rapidly than at the large scales, and most importantly (c) that the structural development of turbulence at different scales is correlated both with the evolution of global statistical measures that dominate at those scales (energy and dissipation rate) and with the evolution of the energy and dissipation-rate (or enstrophy) spectra. In the second study, the three-dimensional wavelet transform is applied to simulations of the transition from isotropic to homogeneous but anisotropic shear-dominated turbulence. It is found that the effect of mean shear on large scales is very different from the effect of shear on small scales. Enstrophy structures rotate continuously to the mean flow direction with time at all scales, while the structures are elongated by mean strain rate dominantly in the principal strain-rate direction. Small scales elongate more rapidly than large scales. However, scales both smaller and larger than the enstrophy peak had lower inclination angles relative to the mean flow direction. Furthermore, the anisotropic structure of the enstrophy field at scales smaller than the peak in the enstrophy spectrum is dominantly ‘‘rodlike,’’ whereas large-scale structure is more ‘‘disklike.’’

Cumulative deformation in the barnes ice cap and implications for the development of foliation

In recent years, it has become apparent that the development of foliation and crystallographic fabric in glacier ice is strongly dependent upon cumulative deformation. However, because glaciers lack suitable strain markers and past velocity fields are unknown, the nature of the cumulative deformation is not well understood. Part of the Barnes Ice Cap, Baffin Island N.W.T., Canada has undergone little overall change in position in the past 2000 years, so a long-term steady-state velocity field can be estimated from present-day measurements in a section in which flow is essentially two-dimensional. By means of a numerical model, particle paths, isochrons, strain rates, and cumulative strains are computed for this section. There is no correspondence between the principal directions of strain rate and cumulative strain in most of the ice, the greatest divergence of about 45° being near the base of the glacier where the maximum cumulative extension direction, X, is sub-horizontal. All ice particles initially undergo a restricted amount of pure shear, and then pass gradually into a long domain of nearly simple shear. They return to a second short stage of nearly pure shear near the margin. The effects of the early stage of pure shear are never eliminated, but the cumulative strain is increasingly dominated by simple shear down-glacier. Cumulative strain magnitude, magnitude of strain rate, and age of the ice all increase rapidly with depth, and change much more slowly with distance down-glacier. We argue that foliation is formed during flow by modification of primary inhomogeneities in ice. The three most important inhomogeneities are probably sedimentary layering, crevasse infillings, and irregularities such as ice glands or lenses. During deformation, the first will behave as isochrons, the second will rotate as material planes, initially perpendicular to the isochrons, and the third will be stretched parallel to the X-direction. The numerical model indicates that the isochrons, planes representing crevasse infillings, X-direction, and particle paths are within 1° of mutual parallelism in the most distal ice. Thus a single subhorizontal foliation, dipping slightly more steeply up-glacier than the flow lines, and curving upwards near the margin would be produced in this case. This corresponds with observations.

Finite and incremental deformations recorded by planar subfabrics in S-tectonites

Analysis of relative componental movements in foliated rocks is formulated in terms of space-continuous deformations assuming that a portion of the strain recorded by planar subfabrics results from differential movements on closely spaced shear surfaces (i.e. fiducial planes). Continuous and discontinuous velocity boundary conditions controlling deformation patterns within subdomains of folded layers are analyzed by combining the spatial velocity and finite deformation gradients. Within each subdomain internal rotations cause material elements oblique to the principal strain rate directions to undergo a series of complex strain transfers resulting in their compensatory lengthening and shortening during finite intervals. Equations are derived which continuously monitor successive variations in the logarithmic strain rates, ≐( N, t), for fabrics whose rotation axes are parallel to an intermediate principal axis. Values of ≐( N, t) at an angle N to the shear plane are numerically equal at time t, to the magnitude of the Hencky strain rate vector ( dh i dt ) referred to natural strain coordinates and used in conjunction with e ̈ (N,t) and the finite stretch, evaluate contemporary strain profiles for groups of planar fabrics replacing passive material planes. Applications to rectilinear shearing modes reveal that the most significant changes in local extensional rates are located between the maximum shearing and principal stretching directions. Assuming sectional continuity and constant material properties of the subfabrics and their matrix, these variations are correlated with systematic spacings between boudin structures suggesting that recognition of multiple orders of boudinage with respect to a potential shear surface in natural S-tectonites can be useful in deciphering local finite and incremental deformation coefficients as well as differentiate continuous ( dh i dt = ƒ(t) or constant) from pulsatory ( dh i dt is undefined at t) overprinting of the subdomain.