"Anti-squeeze" for Mantle Convection Simulations in Two-Dimensional Spherical Geometry

2020 ◽  
Author(s):  
Paul Tackley
Keyword(s):  
Strain Rate ◽  
Mantle Convection ◽  
Spherical Geometry ◽  
High Viscosity ◽  
Three Dimensions ◽  
Perpendicular Direction ◽  
Spherical Approximation ◽  
Strain Rate Tensor ◽  
Normal Strain ◽  
Scaling Relationships

<p>It is common to perform 2-dimensional simulations of mantle convection in spherical geometry, either with (r, theta) axisymmetry or the (r, phi) spherical annulus geometry (Hernlund and Tackley, PEPI 2008). </p><p>A problem with both of these is that the geometrical restriction forces deformation that is not present in 3 dimensions. Specifically, in a 2-D spherical approximation, a downwelling is forced to contract in the plane-perpendicular direction, requiring it to extend in the 2 in-plane directions. In other words, it is "squeezed" in the plane-perpendicular direction.  If the downwelling has a high viscosity, as a cold slab does, then it resists this forced deformation, sinking much more slowly than in three dimensions, in which it could sink with no deformation. This can cause unrealistic behaviour and scaling relationships for high viscosity contrasts. </p><p>This problem can be solved by subtracting the geometrically-forced deformation ("squeezing") from the strain-rate tensor when calculating the stress tensor. Specifically, components of in-plane and plane-normal strain rate that are required by and proportional to the vertical (radial) velocity are subtracted, a procedure that is here termed "anti-squeeze". It is demonstrated here that this "anti-squeeze" correction results in sinking rates and scaling relationships that are similar to those in 3-D geometry whereas without it, abnormal and physically unrealistic results can be obtained for high viscosity contrasts. This correction has been used for 2-D geometries in the code StagYY (Tackley, PEPI 2008; Hernlund and Tackley, PEPI 2008) since 2010.</p>

LES Investigation of Boussinesq Constitutive Relation Validity in a Corner Separation Flow

2018 ◽  
Author(s):  
Jean-François Monier ◽  
Nicolas Poujol ◽  
Mathieu Laurent ◽  
Feng Gao ◽  
Jérôme Boudet ◽  
...  
Keyword(s):  
Strain Rate ◽  
Stress Tensor ◽  
Reynolds Stress ◽  
Constitutive Relation ◽  
Strain Rate Tensor ◽  
Separation Flow ◽  
Compressor Cascade ◽  
Corner Separation ◽  
Reynolds Stress Tensor ◽  
Mean Strain

The present study aims at analysing the Boussinesq constitutive relation validity in a corner separation flow of a compressor cascade. The Boussinesq constitutive relation is commonly used in Reynolds-averaged Navier-Stokes (RANS) simulations for turbomachinery design. It assumes an alignment between the Reynolds stress tensor and the zero-trace mean strain-rate tensor. An indicator that measures the alignment between these tensors is used to test the validity of this assumption in a high fidelity large-eddy simulation. Eddy-viscosities are also computed using the LES database and compared. A large-eddy simulation (LES) of a LMFA-NACA65 compressor cascade, in which a corner separation is present, is considered as reference. With LES, both the Reynolds stress tensor and the mean strain-rate tensor are known, which allows the construction of the indicator and the eddy-viscosities. Two constitutive relations are evaluated. The first one is the Boussinesq constitutive relation, while the second one is the quadratic constitutive relation (QCR), expected to render more anisotropy, thus to present a better alignment between the tensors. The Boussinesq constitutive relation is rarely valid, but the QCR tends to improve the alignment. The improvement is mainly present at the inlet, upstream of the corner separation. At the outlet, the correction is milder. The eddy-viscosity built with the LES results are of the same order of magnitude as those built as the ratio of the turbulent kinetic energy k and the turbulence specific dissipation rate ω. They also show that the main impact of the QCR is to rotate the mean strain-rate tensor in order to realign it with the Reynolds stress tensor, without dilating it.

scholarly journals Seismic deformation at the Alban Hills volcano during the 1989-1990 seismic sequence

1998 ◽  
Vol 41 (2) ◽  
Author(s):  
G. Selvaggi ◽  
F. D'Ajello Caracciolo
Keyword(s):  
Strain Rate ◽  
Fluid Pressure ◽  
Strain Rate Tensor ◽  
Seismic Structure ◽  
Fault Plane Solutions ◽  
Alban Hills ◽  
Shear Failures ◽  
One Year ◽  
The One ◽  
Seismic Moment Release

We analysed the one-year-long seismic swarm at the Alban Hills volcano which occurred during 1989-1990. We portray spatial distribution of seismic moment release, better delineating the activated volume during the swarm. The seismic structure is imaged as a 7-km long, 3-km wide, and 3-km thick volume, located between 2 and 5 km depth, and NW-SE striking. Fault plane solutions and scalar seismic moments for the largest earthquakes provide the description of the average strain rate tensor. The principal strain rate axes show a dominant extension in NE-SW direction, a SE-NW direction of compression and a negligible thickening rate. P and T axes direction of the smaller earthquakes suggests that the same mode of deformation is distributed all over the activated volume. These results are discussed in terms of seismic deforming processes active at the Alban Hills volcano, in the frame of magmatic inflation recently invoked to explain the rapid vertical uplift affecting part of the volcano. The observed average deformation is consistent with shear failures occurring on faults connecting stress-oriented dykes in response to an increasing fluid pressure.

scholarly journals Strain Rates On Rutford Ice Stream, Antarctica (Abstract)

1986 ◽  
Vol 8 ◽  
pp. 207
Author(s):  
N. Stephenson ◽  
C.S.M. Doake
Keyword(s):  
Strain Rate ◽  
Velocity Gradient ◽  
Surface Strain ◽  
Lateral Strain ◽  
Strain Rate Tensor ◽  
Strain Rates ◽  
Average Value ◽  
Divergent Flow ◽  
Ice Stream ◽  
Thickness Variations

In a study of the Rutford Ice Stream, strain rates were measured on a transverse section. Magnitudes ranged up to 40 × 10−3 a−1 but were typically in the order of 3 × 10−3 a−1 with an error of 0.1 χ 10−3 a−1. Variations in the strain rate between adjacent stakes of 0.2 χ 10−3 a−1 to 2 × 10−3 a−1 were matched to the thickness variations on the glacier. For each set of three adjacent stakes, the velocity gradient components of the surface strain rate tensor were calculated by assuming that the gradients were linear over the distance between adjacent stakes. When plotted against distance across the ice stream, each strain rate component revealed different aspects of the flow field. The longitudinal strain rate was compressive, with an almost constant magnitude of 10−3 a−1. The lateral strain rate is extensive, with an average value of 1.1 × 10−3 a−1 which agreed with the angle between the divergent flow lines observed on a Landsat image. Peaks in the lateral strain rate, corresponding to longitudinal bands of thicker ice, showed that these thicker bands were spreading more rapidly at the expense of thinner areas. The two velocity gradient components of the shear rate tensor also reflected differences in ice thickness.

scholarly journals How to Realize Volume Conservation During Finite Plastic Deformation

2017 ◽  
Vol 84 (11) ◽  
Author(s):  
Heling Wang ◽  
Dong-Jie Jiang ◽  
Li-Yuan Zhang ◽  
Bin Liu
Keyword(s):  
Plastic Deformation ◽  
Plastic Strain ◽  
Strain Rate ◽  
Plastic Strain Rate ◽  
Strain Rate Tensor ◽  
Cauchy Stress ◽  
Volume Conservation ◽  
Tangential Stiffness ◽  
Manufacture Process ◽  
Universal Approach

Volume conservation during plastic deformation is the most important feature and should be realized in elastoplastic theories. However, it is found in this paper that an elastoplastic theory is not volume conserved if it improperly sets an arbitrary plastic strain rate tensor to be deviatoric. We discuss how to rigorously realize volume conservation in finite strain regime, especially when the unloading stress free configuration is not adopted in the elastoplastic theories. An accurate condition of volume conservation is first clarified and used in this paper that the density of a volume element after the applied loads are completely removed should be identical to that of the initial stress free states. For the elastoplastic theories that adopt the unloading stress free configuration (i.e., the intermediate configuration), the accurate condition of volume conservation is satisfied only if specific definitions of the plastic strain rate are used among many other different definitions. For the elastoplastic theories that do not adopt the unloading stress free configuration, it is even more difficult to realize volume conservation as the information of the stress free configuration lacks. To find a universal approach of realizing volume conservation for elastoplastic theories whether or not adopt the unloading stress free configuration, we propose a single assumption that the density of material only depends on the trace of the Cauchy stress by using their objectivities. Two strategies are further discussed to satisfy the accurate condition of volume conservation: directly and slightly revising the tangential stiffness tensor or using a properly chosen stress/strain measure and elastic compliance tensor. They are implemented into existing elastoplastic theories, and the volume conservation is demonstrated by both theoretical proof and numerical examples. The potential application of the proposed theories is a better simulation of manufacture process such as metal forming.

A Corotational Elastoplastic Formulation With Subloading Surface Model for Hardening Materials

Volume 1 ◽  
2004 ◽  
Author(s):  
Ali Reza Saidi ◽  
Koichi Hashiguchi
Keyword(s):  
Constitutive Model ◽  
Strain Rate ◽  
Simple Shear ◽  
Deformation Theory ◽  
Elastoplastic Deformation ◽  
Strain Tensor ◽  
Surface Model ◽  
Strain Rate Tensor ◽  
Hencky Strain ◽  
Stress Components

In this paper a corotational constitutive model for the large elastoplastic deformation of hardening materials using subloading surface model is formulated. This formulation is obtained by refining the large deformation theory of Naghdabadi and Saidi (2002) adopting the corotational logarithmic (Hencky) strain rate tensor and incorporating it into the subloading surface model of Hashiguchi (1980, 2003) falling within the framework of the unconventional plasticity. As an application of the proposed constitutive model, the large Elastoplastic deformation of simple shear example has been solved and the results have been compared with classical elasto-plastic model using the Hencky strain tensor. Also the effect of the choice of corotational rates on stress components has been studied.

scholarly journals An anisotropic flow law for ice-sheet ice and its implications

1996 ◽  
Vol 23 ◽  
pp. 202-208 ◽  
Author(s):  
Nobuhiko Azuma ◽  
Kumiko Goto-Azuma
Keyword(s):  
Strain Rate ◽  
Stress Condition ◽  
Ice Sheet ◽  
Geometrical Factor ◽  
Strain Rate Tensor ◽  
Horizontal Shear ◽  
Anisotropic Flow ◽  
Single Maximum ◽  
Polycrystalline Ice ◽  
Flow Law

A new flow law for anisotropic polycrystalline ice is presented. The strain-rate tensor is related by a geometrical factor tensor (G) to the stress tensor. The G factor tensor can be obtained front the c-axis fabric data and stress condition. This new flow law describes well the direction-dependent mechanical properties of anisotropic ice which cannot be demonstrated by Glen’s flow law. For example, the new flow law can explain the fact that a strong single-maximum fabric ice, such as Dye 3 Wisconsin ice, can deform several times faster than isotropic ice under horizontal shear but can hardly deform under vertical or horizontal normal stress. We also show that at a deeper part of an ice sheet, where a single-maximum fabric develops, a positive vertical strain rate can be produced with only a horizontal shear stress.

Subgrid-scale structure and fluxes of turbulence underneath a surface wave

2019 ◽  
Vol 878 ◽  
pp. 768-795
Author(s):  
Kuanyu Chen ◽  
Minping Wan ◽  
Lian-Ping Wang ◽  
Shiyi Chen
Keyword(s):  
Strain Rate ◽  
Eddy Viscosity ◽  
Isotropic Turbulence ◽  
Wave Theory ◽  
Viscosity Model ◽  
Strain Rate Tensor ◽  
Linear Wave ◽  
Subgrid Scale ◽  
Eddy Viscosity Model ◽  
Wave Induced

In this study, the behaviours of subgrid-scale (SGS) turbulence are investigated with direct numerical simulations when an isotropic turbulence is brought to interact with imposed rapid waves. A partition of the velocity field is used to decompose the SGS stress into three parts, namely, the turbulent part $\unicode[STIX]{x1D749}^{T}$, the wave-induced part $\unicode[STIX]{x1D749}^{W}$ and the cross-interaction part $\unicode[STIX]{x1D749}^{C}$. Under strong wave straining, $\unicode[STIX]{x1D749}^{T}$ is found to follow the Kolmogorov scaling $\unicode[STIX]{x1D6E5}_{c}^{2/3}$, where $\unicode[STIX]{x1D6E5}_{c}$ is the filter width. Based on the linear Airy wave theory, $\unicode[STIX]{x1D749}^{W}$ and the filtered strain-rate tensor due to the wave motion, $\tilde{\unicode[STIX]{x1D64E}}^{W}$, are found to have different phases, posing a difficulty in applying the usual eddy-viscosity model. On the other hand, $\unicode[STIX]{x1D749}^{T}$ and the filtered strain-rate tensor due to the turbulent motion, $\tilde{\unicode[STIX]{x1D64E}}^{T}$, are only weakly wave-phase-dependent and could be well related by an eddy-viscosity model. The linear wave theory is also used to describe the vertical distributions of SGS statistics driven by the wave-induced motion. The predictions are in good agreement with the direct numerical simulation results. The budget equation for the turbulent SGS kinetic energy shows that the transport terms related to turbulence are important near the free surface and they compensate the imbalance between the energy flux and the SGS energy dissipation.

scholarly journals Measurement of the Deformation of Ice in a Tunnel at the Foot of an Ice Fall

1956 ◽  
Vol 2 (20) ◽  
pp. 735-746 ◽  
Author(s):  
J. W. Glen
Keyword(s):  
Strain Rate ◽  
Compressive Stress ◽  
Compressive Strain ◽  
Longitudinal Direction ◽  
Strain Rate Tensor ◽  
Surface Slope ◽  
Local Surface ◽  
Maximum Compressive Stress

AbstractDuring the Cambridge Austerdalsbre Expedition 1955, a tunnel was dug horizontally into the ice at the foot of a large ice fall, and its subsequent movements were surveyed. The axis of the tunnel was rotating very fast, as might be expected from the change in surface slope under the ice fall, but as well as this the tunnel was being bent, i.e. the tunnel axis was an active anticline. As the local surface of the glacier above the tunnel was convex, this bending was tending to increase the convexity, and as it is in this region that waves form across the glacier under the ice fall this observation is interpreted as showing that the wave ogives are actually being formed in the ice surrounding the tunnel. The distances between pegs in the tunnel walls were also measured. There was a compressive strain occurring along the tunnel axis (i.e. the tunnel was decreasing in length) and the tunnel was also closing far more rapidly than would be predicted from the weight of overlying ice alone. These two observations suggest that a large compressive stress is acting in the glacier in a longitudinal direction. Measurements on surface stakes, though not of the same accuracy as the internal measurements, confirmed these predictions, and also allow the strain rate tensor to be estimated. From this the magnitude of the stresses can be computed, and the longitudinal compressive stress is estimated to be about 3 bars. Observations of the banding on the walls of the tunnel showed that the fine bands were within 10° of being perpendicular to the maximum compressive stress.

scholarly journals Influence of the Ringwoodite-Perovskite transition on mantle convection in spherical geometry as a function of Clapeyron slope and Rayleigh number

2011 ◽  
Vol 3 (2) ◽  
pp. 713-741 ◽  
Author(s):  
M. Wolstencroft ◽  
J. H. Davies
Keyword(s):  
Phase Transition ◽  
Rayleigh Number ◽  
Phase Change ◽  
Mantle Convection ◽  
Spherical Geometry ◽  
Earth History ◽  
Buoyancy Parameter ◽  
Seismic Discontinuity ◽  
Clapeyron Slope

Abstract. We investigate the influence on mantle convection of the negative Clapeyron slope ringwoodite to perovskite and ferro-periclase mantle phase transition, which is correlated with the seismic discontinuity at 660 km depth. In particular, we focus on understanding the influence of the magnitude of the Clapeyron slope (as measured by the Phase Buoyancy parameter, P) and the vigour of convection (as measured by the Rayleigh number, Ra) on mantle convection. We have undertaken 76 simulations of isoviscous mantle convection in spherical geometry varying Ra and P. Three domains of behaviour were found: layered convection for high Ra and more negative P, whole mantle convection for low Ra and less negative P and transitional behaviour in an intervening domain. The boundary between the layered and transitional domain was fit by a curve P = αRaβ where α = −1.05, and β = −0.1, and the fit for the boundary between the transitional and whole mantle convection domain was α = −4.8, and β = −0.25. These two curves converge at Ra≈2.5×104 and P≈−0.38. Extrapolating to high Ra, which is likely earlier in Earth history, this work suggests a large transitional domain. It is therefore likely that convection in the Archean would have been influenced by this phase change, with Earth being at least in the transitional domain, if not the layered domain.

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