Incompressible Flow Model Research Articles

Incompressible lattice Boltzmann model for axisymmetric flows through porous media

In this paper, an axisymmetric LBE model for incompressible flows through porous media is proposed. In this model, the influence of density change caused by large pressure difference can be overcome by replacing density distribution function with pressure distribution function. A more simple processing format for external force is introduced so as to make the involved method in this paper more perfect. The coupling between flow velocity and pressure also can be significantly reduced when calculating the macroscopic quantities. Good agreement between the analytical solution and numerical results is also obtained based on this model and it also can provide guidance for other problem with such complicated force forms.

Energy analysis and improved regularity estimates for multiscale deconvolution models of incompressible flows

This paper presents new analytical results and the first numerical results for a recently proposed multiscale deconvolution model (MDM) recently proposed. The model involves a large‐eddy simulation closure that uses a novel deconvolution approach based on the introduction of two distinct filtering length scales. We establish connections between the MDM and two other models, and, on the basis of one of these connections, we establish an improved regularity estimate for MDM solutions. We also prove that the MDM preserves Taylor‐eddy solutions of the Navier–Stokes equations and therefore does not distort this particular vortex structure. Simulations of the MDM are performed to examine the accuracy of the MDM and the effect of the filtering length scales on energy spectra for three‐dimensional homogeneous and isotropic flows. Numerical evidence for all tests clearly indicates that the MDM gives very accurate coarse‐mesh solutions and that this multiscale approach to deconvolution is effective. Copyright © 2015 John Wiley & Sons, Ltd.

Multiple-relaxation-time lattice Boltzmann modeling of incompressible flows in porous media

In this paper, a two-dimensional eight-velocity multiple-relaxation-time (MRT) lattice Boltzmann (LB) model is proposed for incompressible porous flows at the representative elementary volume scale based on the Brinkman–Forchheimer-extended Darcy model. In the model, the porosity is included into the pressure-based equilibrium moments, and the linear and nonlinear drag forces of the porous matrix are incorporated into the model by adding a forcing term to the MRT-LB equation in the moment space. Through the Chapman–Enskog analysis, the incompressible generalized Navier–Stokes equations can be recovered. Numerical simulations of several typical porous flows are carried out to validate the present MRT-LB model. It is found that the present numerical results agree well with the analytical solutions and/or other numerical results reported in the literature.

An efficient numerical method for simulating multiphase flows using a diffuse interface model

This paper presents a new diffuse interface model for multiphase incompressible immiscible fluid flows with surface tension and buoyancy effects. In the new model, we employ a new chemical potential that can eliminate spurious phases at binary interfaces, and consider a phase-dependent variable mobility to investigate the effect of the mobility on the fluid dynamics. We also significantly improve the computational efficiency of the numerical algorithm by adapting the recently developed scheme for the multiphase-field equation. To illustrate the robustness and accuracy of the diffuse interface model for surface tension- and buoyancy-dominant multi-component fluid flows, we perform numerical experiments, such as equilibrium phase-field profiles, the deformation of drops in shear flow, a pressure field distribution, the efficiency of the proposed scheme, a buoyancy-driven bubble in ambient fluids, and the mixing of a six-component mixture in a gravitational field. The numerical result obtained by the present model and solution algorithm is in good agreement with the analytical solution and, furthermore, we not only remove the spurious phase-field profiles, but also improve the computational efficiency of the numerical solver.

Decoupled, Energy Stable Schemes for Phase-Field Models of Two-Phase Incompressible Flows

In this paper we construct two classes, based on stabilization and convex splitting, of decoupled, unconditionally energy stable schemes for Cahn--Hilliard phase-field models of two-phase incompressible flows. At each time step, these schemes require solving only a sequence of elliptic equations, including a pressure Poisson equation. Furthermore, all of these elliptic equations are linear for the schemes based on stabilization, making them the first, to the best of the authors' knowledge, totally decoupled, linear, unconditionally energy stable schemes for phase-field models of two-phase incompressible flows. Thus, the schemes constructed in this paper are very efficient and easy to implement.

On local $L_p$-$L_q$ well-posedness of incompressible two-phase flows with phase transitions: the case of non equal densities

The basic model for incompressible two-phase flows with phase transitions where the interface is nearly flat in the case of non-equal densities is considered. The local well-posedness of the model in $L_p$-setting was proved in Prüss and Shimizu [14]. In this paper, we prove local well-posedness of the model $L_p$ in time $L_q$ in space settting.

Homogenization in algebras with mean value

In several works, the theory of strongly continuous groups is used to build a framework for solving random homogenization problems. Following this idea, we present a detailed and comprehensive framework enabling one to solve homogenization problems in algebras with mean value, regardless of whether they are ergodic or not. We also state and prove a compactness result for Young measures in these algebras. As an important achievement we study the homogenization problem associated with a stochastic Ladyzhenskaya model for incompressible viscous flow, and we present and solve a few examples of homogenization problems related to nonergodic algebras.

Regularized family of models for incompressible Cahn–Hilliard two-phase flows

We consider a general family of regularized models for incompressible two-phase flows based on the Cahn–Hilliard formulation in n-dimensional compact Riemannian manifolds (with or without boundary) for n=2,3. The system we consider consists of a regularized family of Navier–Stokes equations (including the Navier–Stokes-α-like model, the Leray-α model, the Modified Leray-α model, the Simplified Bardina model, the Navier–Stokes–Voight model, the Navier–Stokes model, and many others) for the fluid velocity u suitably coupled with a convective Cahn–Hilliard equation for the order (phase) parameter ϕ. We give a unified analysis of the entire three-parameter family of two-phase models. We first establish existence, stability and regularity results. Then, we show the existence of a global attractor and exponential attractor for our general model, and then establish precise conditions under which each trajectory (u,ϕ) converges to a single equilibrium by means of a Lojasiewicz–Simon inequality.

Analysis of some projection method based preconditioners for models of incompressible flow

In this paper, several projection method based preconditioners for various incompressible flow models are studied. In the derivations of these projection method based preconditioners, we use three different types of the approximations of the inverse of the Schur complement, i.e., the exact inverse, the Cahouet–Chabard type approximation and the BFBt type approximation. We illuminate the connections and the distinctions between these projection method based preconditioners and those related preconditioners. For the preconditioners using the Cahouet–Chabard type approximation, we show that the eigenvalues of the preconditioned systems have uniform bounds independent of the parameters and most of them are equal to 1. The analysis is based on a detailed discussion of the commutator difference operator. Moreover, these results demonstrate the stability of the staggered grid discretization and reveal the effects of the boundary treatment. To further illustrate the effectiveness of these projection method based preconditioners, numerical experiments are given to compare their performances with those of the related preconditioners. Generalizations of the projection method based preconditioners to other saddle point problems are also discussed.

A hyperbolic model for viscous Newtonian flows

We discuss a pure hyperbolic alternative to the Navier–Stokes equations, which are of parabolic type. As a result of the substitution of the concept of the viscosity coefficient by a microphysics-based temporal characteristic, particle settled life (PSL) time, it becomes possible to formulate a model for viscous fluids in a form of first-order hyperbolic partial differential equations. Moreover, the concept of PSL time allows the use of the same model for flows of viscous fluids (Newtonian or non-Newtonian) as well as irreversible deformation of solids. In the theory presented, a continuum is interpreted as a system of material particles connected by bonds; the internal resistance to flow is interpreted as elastic stretching of the particle bonds; and a flow is a result of bond destructions and rearrangements of particles. Finally, we examine the model for simple shear flows, arbitrary incompressible and compressible flows of Newtonian fluids and demonstrate that Newton’s viscous law can be obtained in the framework of the developed hyperbolic theory as a steady-state limit. A basic relation between the viscosity coefficient, PSL time, and the shear sound velocity is also obtained.

Dynamics of a parallel, high-speed, lubricated thrust bearing with Navier slip boundary conditions

An incompressible fluid flow model for a thin-film thrust bearing with slip flow is derived, leading to a modified Reynolds equation for a highly rotating rotor that incorporates a slip length shear condition on the bearing faces, extending previous bearing studies for new bearing applications associated with decreasing film thickness. Mathematical and numerical modelling is applied to the coupled process of the pressurized fluid flow through the bearing, with a Navier slip condition replacing a no-slip condition, and the axial motion of the rotor and stator. The derived modified Reynolds equation is coupled with the dynamic motion of the stator through the pressure exerted by the fluid film, with explicit analytical expressions for the pressure and force determined and the equation for the bearing gap reduced to a non-linear second-order non-autonomous ordinary differential equation. A mapping solver is used to investigate the time-dependent bearing gap for prescribed periodic motion of the rotor. A parametric study focuses on bearing operation under close contact motion to examine the minimum film thickness and possibility of bearing face contact.

Traveling wave solutions for (3 + 1) dimensional equations arising in fluid mechanics

Abstract In this note, traveling wave solutions for (3 + 1) dimensional fluid models of incompressible flow are considered. The governing partial differential equations of two models are reduced to ordinary differential equation by employing wave parameter and exact solutions are obtained. It is shown that these fluid models allow 3 + 1 dimensional solutions amongst each other. The methodology used in this work is independent of symmetric consideration and other restrictive assumption. Finally, a set of example of boundary condition is discussed for the couple stress fluid. It is observed that velocity profile strongly depends upon couple stress parameter.

On a Regularized Family of Models for Homogeneous Incompressible Two-Phase Flows

We consider a general family of regularized models for incompressible two-phase flows based on the Allen–Cahn formulation in $$n$$ -dimensional compact Riemannian manifolds for $$n=2,3$$ . The system we consider consists of a regularized family of Navier–Stokes equations (including the Navier–Stokes- $$\alpha $$ -like model, the Leray- $$\alpha $$ model, the modified Leray- $$\alpha $$ model, the simplified Bardina model, the Navier–Stokes–Voight model, and the Navier–Stokes model) for the fluid velocity $$u$$ suitably coupled with a convective Allen–Cahn equation for the order (phase) parameter $$\phi $$ . We give a unified analysis of the entire three-parameter family of two-phase models using only abstract mapping properties of the principal dissipation and smoothing operators and then use assumptions about the specific form of the parameterizations, leading to specific models, only when necessary to obtain the sharpest results. We establish existence, stability, and regularity results and some results for singular perturbations, which as special cases include the inviscid limit of viscous models and the $$\alpha \rightarrow 0$$ limit in $$\alpha $$ models. Then we show the existence of a global attractor and exponential attractor for our general model and establish precise conditions under which each trajectory $$\left( u,\phi \right) $$ converges to a single equilibrium by means of a Lojasiewicz–Simon inequality. We also derive new results on the existence of global and exponential attractors for the regularized family of Navier–Stokes equations and magnetohydrodynamics models that improve and complement the results of Holst et al. (J Nonlinear Sci 20(5):523–567, 2010). Finally, our analysis is applied to certain regularized Ericksen–Leslie models for the hydrodynamics of liquid crystals in $$n$$ -dimensional compact Riemannian manifolds.

A note on helicity conservation in Leray models of incompressible flow

We prove conservation of a regularized helicity HL:=∫Ωu⋅wdx for the Leray model (and its variants) of turbulent flow, where w is the solution of a Leray-regularized vorticity equation. The usual definition of helicity is H=∫Ωu⋅(∇×u)dx, which is considered by Navier–Stokes flows, but is not a conserved quantity of the Leray model. However, if u is a Leray solution, then the difference between H and HL is that HL uses a regularized vorticity and H uses the curl of a regularized velocity. The results are extended to show that the standard Crank–Nicolson finite element method for Leray models conserves both discrete energy and discrete regularized helicity.

An alternative lattice Boltzmann model for three-dimensional incompressible flow

In this work, an alternative lattice Boltzmann (LB) model for three-dimensional (3D) incompressible flow is proposed. The equilibrium distribution function (EDF) of the present model is directly derived in accordance with the incompressibility conditions by applying the Hermite expansion. Moreover, an alternative formula for pressure computation is designed from the second order moment of the distribution function. The present 3D LB model inherits the advantageous features of Guo’s LB model: the density is a constant, the fluid pressure is independent of density and the Navier–Stokes (N–S) equations for incompressible flow can be derived. Two benchmark tests, flow over a backward-facing step and the lid-driven cavity flow, are applied to validate the present model. Accurate results for these tests are obtained with the present model, and further comparisons with the previous LB models (the standard LB model, the He–Luo model and Guo’s LB model) demonstrate that the present model provides better accuracy in the region of high deviatoric stress and such advantage is further enhanced by using the D3Q27 lattice.

On Crank–Nicolson Adams–Bashforth timestepping for approximate deconvolution models in two dimensions

We consider a Crank–Nicolson–Adams–Bashforth temporal discretization, together with a finite element spatial discretization, for efficiently computing solutions to approximate deconvolution models of incompressible flow in two dimensions. We prove a restriction on the timestep that will guarantee stability, and provide several numerical experiments that show the proposed method is very effective at finding accurate coarse mesh approximations for benchmark flow problems.

Applications of CFD Techniques in the Design of Fabric Filters

Analyzing the gas flow inside a bag filter experimentally is a difficult challenge to overcome, since the complexity of the highly three-dimensional flow hinders any form of directly measuring the flow. In recent years, the increasing availability of computational resources together with the increasing accuracy of numerical methods built up techniques that describe the flow of fluids in industrial equipments through mathematical modelling and numerical simulations. Among these techniques, Computational Fluid Dynamics (CFD) is noteworthy regarding the design of new equipment, since it uses experimental data in their simulations. In this work, the CFD techniques are applied to investigate the best position for feeding the dirty gas in a filter bag, in order to support the search for a more efficient and durable equipment design. To evaluate the filter bag performance, two different inlet positions were simulated. The first, called “conventional”, with gas jet reaching the bags in the centre and the second with concentric triple feed (called “triple”), in which the same gas flow rate is fed by in three regions roughly in the top, center, and bottom of the filter medium. The aim was to determine which design has improved uniformity of mass flow distribution between the bags and lower the equipment pressure drop, in order to increase the filter lifetime. For this purpose, it was used the isothermal and incompressible single phase flow model with turbulence being treated by realizable k-e model. It was concluded that the “triple” showed a better gas distribution to be filtered among the bags system, thus representing a more advantageous layout option when compared to “conventional”.

Numerical simulation of multiply connected axisymmetric discontinuous incompressible potential flows

The ascend and evolution of an axisymmetric gas bubble are studied numerically using an inviscid incompressible potential flow model. The volume of the gas bubble varies adiabatically. The transition from a simply connected bubble to a doubly connected toroidal one and its interaction with the free surface are simulated. The change in connectedness is accompanied by a nonzero velocity circulation and a discontinuous velocity potential occurring over an arbitrary toroidal liquid surface enclosing the bubble.

Phase-field-based multiple-relaxation-time lattice Boltzmann model for incompressible multiphase flows.

In this paper, a phase-field-based multiple-relaxation-time lattice Boltzmann (LB) model is proposed for incompressible multiphase flow systems. In this model, one distribution function is used to solve the Chan-Hilliard equation and the other is adopted to solve the Navier-Stokes equations. Unlike previous phase-field-based LB models, a proper source term is incorporated in the interfacial evolution equation such that the Chan-Hilliard equation can be derived exactly and also a pressure distribution is designed to recover the correct hydrodynamic equations. Furthermore, the pressure and velocity fields can be calculated explicitly. A series of numerical tests, including Zalesak's disk rotation, a single vortex, a deformation field, and a static droplet, have been performed to test the accuracy and stability of the present model. The results show that, compared with the previous models, the present model is more stable and achieves an overall improvement in the accuracy of the capturing interface. In addition, compared to the single-relaxation-time LB model, the present model can effectively reduce the spurious velocity and fluctuation of the kinetic energy. Finally, as an application, the Rayleigh-Taylor instability at high Reynolds numbers is investigated.

Decoupled Energy Stable Schemes for a Phase-Field Model of Two-Phase Incompressible Flows with Variable Density

We consider in this paper numerical approximations of two-phase incompressible flows with different densities and viscosities. We present a variational derivation for a thermodynamically consistent phase-field model that admits an energy law. Two decoupled time discretization schemes for the coupled nonlinear phase-field model are constructed and shown to be energy stable. Numerical experiments are carried out to validate the model and the schemes for problems with large density and viscosity ratios.