scholarly journals Three-Dimensional Central Moment Lattice Boltzmann Method on a Cuboid Lattice for Anisotropic and Inhomogeneous Flows

Fluids ◽  
2021 ◽  
Vol 6 (9) ◽  
pp. 326
Author(s):  
Eman Yahia ◽  
William Schupbach ◽  
Kannan N. Premnath
Keyword(s):  
Stress Tensor ◽  
Shear Viscosity ◽  
Lattice Boltzmann ◽  
Central Moment ◽  
Second Order ◽  
Memory Storage ◽  
Viscous Stress ◽  
Viscous Stress Tensor ◽  
Cubic Lattice ◽  
Aspect Ratios

Lattice Boltzmann (LB) methods are usually developed on cubic lattices that discretize the configuration space using uniform grids. For efficient computations of anisotropic and inhomogeneous flows, it would be beneficial to develop LB algorithms involving the collision-and-stream steps based on orthorhombic cuboid lattices. We present a new 3D central moment LB scheme based on a cuboid D3Q27 lattice. This scheme involves two free parameters representing the ratios of the characteristic particle speeds along the two directions with respect to those in the remaining direction, and these parameters are referred to as the grid aspect ratios. Unlike the existing LB schemes for cuboid lattices, which are based on orthogonalized raw moments, we construct the collision step based on the relaxation of central moments and avoid the orthogonalization of moment basis, which leads to a more robust formulation. Moreover, prior cuboid LB algorithms prescribe the mappings between the distribution functions and raw moments before and after collision by using a moment basis designed to separate the trace of the second order moments (related to bulk viscosity) from its other components (related to shear viscosity), which lead to cumbersome relations for the transformations. By contrast, in our approach, the bulk and shear viscosity effects associated with the viscous stress tensor are naturally segregated only within the collision step and not for such mappings, while the grid aspect ratios are introduced via simpler pre- and post-collision diagonal scaling matrices in the above mappings. These lead to a compact approach, which can be interpreted based on special matrices. It also results in a modular 3D LB scheme on the cuboid lattice, which allows the existing cubic lattice implementations to be readily extended to those based on the more general cuboid lattices. To maintain the isotropy of the viscous stress tensor of the 3D Navier–Stokes equations using the cuboid lattice, corrections for eliminating the truncation errors resulting from the grid anisotropy as well as those from the aliasing effects are derived using a Chapman–Enskog analysis. Such local corrections, which involve the diagonal components of the velocity gradient tensor and are parameterized by two grid aspect ratios, augment the second order moment equilibria in the collision step. We present a numerical study validating the accuracy of our approach for various benchmark problems at different grid aspect ratios. In addition, we show that our 3D cuboid central moment LB method is numerically more robust than its corresponding raw moment formulation. Finally, we demonstrate the effectiveness of the 3D cuboid central moment LB scheme for the simulations of anisotropic and inhomogeneous flows and show significant savings in memory storage and computational cost when used in lieu of that based on the cubic lattice.

Neoclassical viscous stress tensor for non-linear MHD simulations with XTOR-2F

2013 ◽  
Vol 53 (4) ◽  
pp. 043022 ◽  
Author(s):  
N. Mellet ◽  
P. Maget ◽  
H. Lütjens ◽  
D. Meshcheriakov ◽  
Keyword(s):  
Stress Tensor ◽  
Viscous Stress ◽  
Viscous Stress Tensor ◽  
Mhd Simulations ◽  
Non Linear

scholarly journals Semiflow Selection to Models of General Compressible Viscous Fluids

2020 ◽  
Vol 23 (1) ◽  
Author(s):  
Danica Basarić
Keyword(s):  
Hilbert Space ◽  
Viscous Fluid ◽  
Stress Tensor ◽  
Velocity Gradient ◽  
Nonlinear Function ◽  
Viscous Fluids ◽  
Viscous Stress ◽  
Compressible Viscous Fluid ◽  
Abstract Result ◽  
Viscous Stress Tensor

AbstractWe prove the existence of a semiflow selection with range the space of càglàd, i.e. left-continuous and having right-hand limits functions defined on $$[0,\infty )$$ [ 0 , ∞ ) and taking values in a Hilbert space. Afterwards, we apply this abstract result to the system arising from a compressible viscous fluid with a barotropic pressure of the type $$a\varrho ^{\gamma }, \gamma \ge 1$$ a ϱ γ , γ ≥ 1 , with a viscous stress tensor being a nonlinear function of the symmetric velocity gradient.

scholarly journals Viscous stress tensor and stability of laminar contravortical flows

Vestnik MGSU ◽  
2019 ◽  
pp. 870-884
Author(s):  
Andrey L. Zuikov ◽  
Elena V. Bazhina
Keyword(s):  
Stress Tensor ◽  
Vortex Core ◽  
Local Stability ◽  
Vortex Breakdown ◽  
Hydrodynamic Instability ◽  
Laminar Flows ◽  
Viscous Stress ◽  
Unstable Flow ◽  
Hydraulic Unit ◽  
Viscous Stress Tensor

Introduction: coaxial layers in contravortical flows rotate in the opposite directions. This determines their complicated spatial structure. The relevance of the subject is in the uniquely effective mixing of the moving medium. This property has a great potential of application from microbiology and missile building for obtaining highly dispersed mixtures to heat engineering for increasing the intensity of heat transfer. However, contravortical flows have a high degree of hydrodynamic instability. This hinders effective development of these technologies. Contravortical flows are observed behind Francis hydroturbines, whose derated operation causes modes with a significant increase of hydraulic unit vibrations up to destruction of the units. The purpose of the study is to identify physical laws of the contravortical flow hydrodynamics, common for both laminar and turbulent fluid flow modes. Materials and methods: theoretical analysis of the viscous stress tensor and local stability zones of contravortical laminar flows. Results: the article provides a mathematical description of the tensor of viscous tangential (τij) and normal (σii) stresses as well as local stability zones of the flow according to Rayleigh (Ra) and Richardson (Ri) criteria. The graphs of the radial-axial distributions of the viscous stress components are given, local stability zones are shown and the point of “vortex breakdown” is indicated. The solutions are obtained in the form of Fourier – Bessel series. The hydrodynamic structure of the flow is analysed. Conclusions: it is established that the most significant viscous stresses are observed at the beginning of the interaction zone of contrarotating layers. It is established that the areas with the most unstable flow are localized in the flow vortex core. Three zones can be distinguished in the vortex core: a zone of weak instability with local Richardson numbers to Ri = –1, passing into a zone of flow destabilization with high negative values of Richardson numbers Ri = –10 to –100, in turn, transforming into a zone with rapidly increasing instability up to Ri = –1000. This is a zone of loss of flow stability, culminating in the “ortex breakdown”.

S-wave propagation across material discontinuities in poroelasticity

Geophysics ◽  
2021 ◽  
pp. 1-39
Author(s):  
Josué G. Gonzalez ◽  
Pratap N. Sahay ◽  
Tobias M. Mueller
Keyword(s):  
Porous Medium ◽  
Wave Propagation ◽  
Stress Tensor ◽  
Elastic Solid ◽  
Slip Condition ◽  
Biot Theory ◽  
Viscous Stress ◽  
S Wave ◽  
Viscous Stress Tensor ◽  
Frequency Regime

The shear motion in Newtonian fluids, i.e., the fluid vorticity, represents an intrinsic loss mechanism governed by a diffusion equation. Its description involves the trace-free part of the fluid viscous stress tensor. This part is missing in the Biot theory of poroelasticity. As a result, the fluid vorticity is not captured, and only one S-wave is predicted. The missing fluid vorticity has implications for the propagation of S-waves across discontinuities. This becomes most apparent in the problem of S-wave propagation across the welded contact of an elastic solid with a porous medium. At such a contact, the no-slip condition between the elastic solid and the constituent parts of the porous medium, the solid-frame, and the pore-fluid, must hold. This requirement translates into a vanishing relative motion of the fluid with respect to the solid-frame, i.e., filtration field, at the contact. Nevertheless, our analysis shows that for the Biot theory, in the low-frequency regime, a non-zero, although insignificantly small filtration field exists at the contact. But, more importantly, the filtration field is noticeable when the transition to the high-frequency regime occurs. This constitutes a disagreement with the requirement of a no-slip boundary condition and renders the prediction unphysical. This shortcoming is circumvented by including the fluid viscous stress tensor into the poroelastic constitutive relations, as stipulated by the de la Cruz-Spanos poroelasticity theory. Then, a second S-wave is predicted which manifests as the fluid vorticity at macroscale. This process is distinct from the fast S-wave, the other predicted S-wave akin to the Biot S-wave. We find that the generation of this process at the contact induces a filtration field equal and opposite to that associated with the fast S-wave. Therefore, the no-slip condition is satisfied, and the S-wave reflection/transmission across a discontinuity becomes physically meaningful.

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