scholarly journals Transfer equation for the strain rate tensor and description of an incompressible dispersed mixture (incompressible fluid) by a system of equations of dynamic type

2021 ◽  
Vol 264 ◽  
pp. 03025
Author(s):  
Askar Yuldashov ◽  
Otabek Abdisamatov ◽  
Behzod Abdullaev ◽  
Salima Dustova
Keyword(s):  
Angular Velocity ◽  
Incompressible Fluid ◽  
Strain Rate ◽  
Strain Rate Tensor ◽  
System Of Equations ◽  
Symmetric Part ◽  
Antisymmetric Part ◽  
Vector Operation ◽  
Dynamic Type ◽  
First Time

It is known that the application of the vector operation rot to the equations of hydrodynamics leads to the Helmholtz-Friedman equation for a vortex. A dispersed mixture, tensor transformations are used, in a certain sense generalizing the vector operation rot, which gives more than one, a couple of equations. One of them describes the transfer of vorticity is the well-known Helmholtz-Friedman equation. The second equation was obtained for the first time, and it describes the transfer of the strain rate tensor. Any tensor decomposes into symmetric and antisymmetric parts. By definition, the symmetric part of the tensor U is the strain rate tensor. The antisymmetric part of U is a tensor whose components are related in a known manner to the pseudovector angular velocity.

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 Regularities of Low-Temperature Deformation and Fracture of Polyimide Films of Kapton H Type of Different Thickness

2020 ◽  
Keyword(s):  
Strain Rate ◽  
Thick Film ◽  
Sharp Decrease ◽  
Rate Sensitivity ◽  
Temperature Deformation ◽  
Relative Deformation ◽  
Polyimide Films ◽  
Deformation Curves ◽  
Deformation And Fracture ◽  
First Time

The mechanical characteristics (limit of forced elasticity σforc, fracture stress σfr, relative deformation to failure εfr) of polyimide films of kapton H type under uniaxial tension conditions along the direction of drawing in the temperature range (4.2-293 K), deformation rates (10-5 - 10-3 s-1) and film thicknesses (25, 75 and 125 μm) were investigated. It is discovered, that the forced-elastic state remains for all films up to 4.2 K of all strain rates - σfors<σfr. In this case, the reserve of elasticity significantly depends on the thickness of the film with a decrease in temperature. A sharp decrease in εfr occurs in films: 125 μm thick - at 77 K, 75 μm thick - at 4.2 K. Two variants of deformation curves are possible in a 25 μm thick film at 4.2 K: with a short nonlinear stage or with a long one proceeding jumpily. The working surface of the samples that have undergone jump deformation is covered with a deformation relief, partially representing a delayed highly elastic deformation. The σfors limit is most sensitive to the strain rate. The nature of the strain rate sensitivity σfors(έ) depends on the temperature and film thickness. The change to the opposite in the character of σfors(έ) and σfr(έ) with a decrease in temperature to 4.2 K in 75 and 125 thick films was found for a first time. Change in the character of σfors(έ) is not observed in 25 μm thick film which retains the maximum reserve of elasticity at 4.2 K

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.

Stability of circular Couette flow with variable inner cylinder speed

1982 ◽  
Vol 123 ◽  
pp. 43-57 ◽  
Author(s):  
G. P. Neitzel
Keyword(s):  
Angular Velocity ◽  
Incompressible Fluid ◽  
Viscous Incompressible Fluid ◽  
Outer Cylinder ◽  
Finite Amplitude ◽  
Reverse Direction ◽  
Stability Bound ◽  
Concentric Cylinders ◽  
Theory Result ◽  
Stability Enhancement

Energy & ability theory is employed to study the finite-amplitude stability of a viscous incompressible fluid occupying the space between a pair of concentric cylinders when the inner-cylinder angular velocity varies linearly with time. For the case with a fixed outer cylinder and increasing inner-cylinder speed, we find an enhancement of stability, consistent with a linear-theory result due to Eagles. When the inner-cylinder speed decreases, we find an initially decreased stability bound, indicating the possibility of hysteresis, while, if the inner cylinder is allowed to reverse direction and linearly increase in speed, we find significant stability enhancement.

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.

Sign in / Sign up

Export Citation Format

Share Document