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

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.

S-wave propagation across material discontinuities in poroelasticity

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.

Semiflow Selection to Models of General Compressible Viscous Fluids

We 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.

Dynamics of dusty vortices – I. Extensions and limitations of the terminal velocity approximation

ABSTRACT Motivated by the stability of dust laden vortices, in this paper we study the terminal velocity approximation equations for a gas coupled to a pressureless dust fluid and present a numerical solver for the equations embedded in the FARGO3D hydrodynamics code. We show that for protoplanetary discs it is possible to use the barycentre velocity in the viscous stress tensor, making it trivial to simulate viscous dusty protoplanetary discs with this model. We also show that the terminal velocity model breaks down around shocks, becoming incompatible with the two-fluid model it is derived from. Finally we produce a set of test cases for numerical schemes and demonstrate the performance of our code on these tests. Our implementation embedded in FARGO3D using an unconditionally stable explicit integrator is fast, and exhibits the desired second-order spatial convergence for smooth problems.

Steady Flow of a Cement Slurry

Understanidng the rheological behavior of cement slurries is important in cement and petroleum industries. In this paper, we study the fully developed flow of a cement slurry inside a wellbore. The slurry is modeled as a non-linear fluid, where a constitutive relation for the viscous stress tensor based on a modified form of the second grade (Rivlin–Ericksen) fluid is used;we also propose a diffusion flux vector for the concentration of particles. The one-dimensional forms of the governing equations and the boundary conditions are made dimensionless and solved numerically. A parametric study is performed to present the effect of various dimensionless numbers on the velocity and the volume fraction profiles.

Viscous stress tensor and stability of laminar contravortical flows

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”.

On the energy dissipative spatial discretization of the barotropic quasi-gasdynamic and compressible Navier–Stokes equations in polar coordinates

The barotropic quasi-gasdynamic system of equations in polar coordinates is treated. It can be considered a kinetically motivated parabolic regularization of the compressible Navier–Stokes system involving additional 2nd order terms with a regularizing parameter τ > 0. A potential body force is taken into account. The energy equality is proved ensuring that the total energy is non-increasing in time. This is the crucial physical property. The main result is the construction of symmetric spatial discretization on a non-uniform mesh in a ring such that the property is preserved. The unknown density and velocity are defined on the same mesh whereas the mass flux and the viscous stress tensor are defined on the staggered meshes. Additional difficulties in comparison with the Cartesian coordinates are overcome, and a number of novel elements are implemented to this end, in particular, a self-adjoint and positive definite discretization for the Navier–Stokes viscous stress, special discretizations of the pressure gradient and regularizing terms using enthalpy, non-standard mesh averages for various products of functions, etc. The discretization is also well-balanced. The main results are valid for τ = 0 as well, i.e., for the barotropic compressible Navier–Stokes system.