Development of LBGK and incompressible LBGK‐based lattice Boltzmann flux solvers for simulation of incompressible flows

This paper presents lattice Boltzmann Bhatnagar–Gross–Krook (LBGK) model and incompressible LBGK model‐based lattice Boltzmann flux solvers (LBFS) for simulation of incompressible flows. LBFS applies the finite volume method to directly discretize the governing differential equations recovered by lattice Boltzmann equations. The fluxes of LBFS at each cell interface are evaluated by local reconstruction of lattice Boltzmann solution. Because LBFS is applied locally at each cell interface independently, it removes the major drawbacks of conventional lattice Boltzmann method such as lattice uniformity, coupling between mesh spacing, and time interval. With LBGK and incompressible LBGK models, LBFS are examined by simulating decaying vortex flow, polar cavity flow, plane Poiseuille flow, Womersley flow, and double shear flows. The obtained numerical results show that both the LBGK and incompressible LBGK‐based LBFS have the second order of accuracy and high computational efficiency on nonuniform grids. Furthermore, LBFS with both LBGK models are also stable for the double shear flows at a high Reynolds number of 10⁵. However, for the pressure‐driven plane Poiseuille flow, when the pressure gradient is increased, the relative error associated with LBGK model grows faster than that associated with incompressible LBGK model. It seems that the incompressible LBGK‐based LBFS is more suitable for simulating incompressible flows with large pressure gradients. Copyright © 2014 John Wiley & Sons, Ltd.     

Non-conservative pressure-based compressible formulation for multiphase flows with heat and mass transfer

A pressure-based compressible multiphase flow solver has been developed based on non-conservative discretization of the mixture continuity equation. The formulation is an extension of the single phase incompressible pressure-correction approach, such that it can be applied to both two-phase flows using interface resolving methods and general n-phase ensemble-averaged mixture flows. The formulation is currently presented with the single pressure and single temperature assumption, but extension to multiple temperatures is straightforward. A robust treatment of phase change allows the method to model conditions with rapid phase change such as expansion through nozzles and valves. The method has been validated thoroughly using canonical single phase problems such as the shock tube, tank filling and sudden valve closure problems. Multiphase flow validation has been carried out for sound propagation in mixtures using the ensemble-averaged model and pressure wave transmission and reflection across an air-water interface, using the level set interface tracking method. The method has been used to study sound propagation in saturated steam-water systems under thermodynamic non-equilibrium, where the expected drastic reduction in the speed of sound is reproduced. Finally the method is applied to the problem of critical (choked) flow in a nozzle for a saturated steam-water system.     

Unsteady flows of an inhomogeneous incompressible viscous fluid

This paper deals with a theoretical analysis of the transfer of reactive impurities by open and filtration flows of an incompressible viscous fluid. The first section of the paper studies the model of an inhomogeneous incompressible viscous fluid, which is widely used in meteorology and oceanology, with additional allowance for the drag of the magnetic field or porous medium. Another object of research in this paper is the model of filtration of an inhomogeneous incompressible fluid in porous media proposed by V. N. Monakhov (1977) (Section 2). In both models, hydrodynamic flows determine the motion of the mixture as a whole and the temperature and concentration distributions of the components of an inhomogeneous fluid are described by a common nonlinear system of equations of diffusive heat and mass transfer.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 2, pp. 44–51, March–April, 2005.     

A matrix-free high-order discontinuous Galerkin compressible Navier-Stokes solver: A performance comparison of compressible and incompressible formulations for turbulent incompressible flows

Both compressible and incompressible Navier-Stokes solvers can be used and are used to solve incompressible turbulent flow problems. In the compressible case, the Mach number is then considered as a solver parameter that is set to a small value, M ≈0.1, in order to mimic incompressible flows. This strategy is widely used for high-order discontinuous Galerkin (DG) discretizations of the compressible Navier-Stokes equations. The present work raises the question regarding the computational efficiency of compressible DG solvers as compared to an incompressible formulation. Our contributions to the state of the art are twofold: Firstly, we present a high-performance DG solver for the compressible Navier-Stokes equations based on a highly efficient matrix-free implementation that targets modern cache-based multicore architectures with Flop/Byte ratios significantly larger than 1. The performance results presented in this work focus on the node-level performance, and our results suggest that there is great potential for further performance improvements for current state-of-the-art DG implementations of the compressible Navier-Stokes equations. Secondly, this compressible Navier-Stokes solver is put into perspective by comparing it to an incompressible DG solver that uses the same matrix-free implementation. We discuss algorithmic differences between both solution strategies and present an in-depth numerical investigation of the performance. The considered benchmark test cases are the three-dimensional Taylor-Green vortex problem as a representative of transitional flows and the turbulent channel flow problem as a representative of wall-bounded turbulent flows. The results indicate a clear performance advantage of the incompressible formulation over the compressible one.     

Partial-average-based equations of incompressible turbulent flow

This paper presents a detailed study of incompressible turbulent flow based on a newly developed statistical partial average scheme. As the ensemble average is taken on two groups of turbulent fluctuations separately, the partial average scheme is able to capture the first-order statistical moment of the fluctuation field, providing valuable information in addition to what have been known in the past from the conventional Reynolds average. The first-order statistical moment serves as the foundation in formulating theories of orthotropic turbulence and a momentum transfer chain in the modeling of second-order correlation terms, and eventually leads to a complete set of equations of incompressible turbulent flow. Without any empirical coefficients, the same set of the equations is used to simulate statistical mean behaviors and coherent structures of various benchmark turbulent flows. The simulated results are in good agreement with experimental data.     

Modeling low Reynolds number incompressible flows with curved boundaries using SPH

In this paper, a method that can be used to model low Reynolds number incompressible flows with curved boundaries using SPH was presented. In contrast to that usually used for the flows with flat and straight boundaries, the hydrostatic pressure gradient is treated as a variable body force in this method, and thus, it can be applied to simulate the flows with curved boundaries. Three numerical examples of low Reynolds number incompressible flows, including Poiseuille flow, flow in a section of blood vessel with a local expansion, and flow between inclined plates were calculated to test the method. The results obtained with the proposed method were in good agreement with the analytical solutions. It implies that the method presented in this paper can be successfully used to simulate low Reynolds number incompressible flows not only with flat and straight boundaries but also with curved boundaries. Copyright © 2011 John Wiley & Sons, Ltd.     

Buoyancy modelling with incompressible SPH for laminar and turbulent flows

This work aims to model buoyant, laminar or turbulent flows, using a two‐dimensional incompressible smoothed particle hydrodynamics model with accurate wall boundary conditions. The buoyancy effects are modelled through the Boussinesq approximation coupled to a heat equation, which makes it possible to apply an incompressible algorithm to compute the pressure field from a Poisson equation. Based on our previous work [1], we extend the unified semi‐analytical wall boundary conditions to the present model. The latter is also combined to a Reynolds‐averaged Navier–Stokes approach to treat turbulent flows. The k ? ? turbulence model is used, where buoyancy is modelled through an additional term in the k ? ? equations like in mesh‐based methods. We propose a unified framework to prescribe isothermal (Dirichlet) or to impose heat flux (Neumann) wall boundary conditions in incompressible smoothed particle hydrodynamics. To illustrate this, a theoretical case is presented (laminar heated Poiseuille flow), where excellent agreement with the theoretical solution is obtained. Several benchmark cases are then proposed: a lock‐exchange flow, two laminar and one turbulent flow in differentially heated cavities, and finally a turbulent heated Poiseuille flow. Comparisons are provided with a finite volume approach using an open‐source industrial code. Copyright © 2015 John Wiley & Sons, Ltd.     

A segregated method for compressible flow computation Part I: isothermal compressible flows

Traditionally, coupled methods have been employed for the computation of compressible flows, whereas segregated methods have been preferred for the computation of incompressible flows. Compared to coupled methods, segregated solvers present the advantage of reduced computer memory and CPU time requirements, although at the cost of an inferior robustness. Therefore, in a series of papers we present unified computational techniques to compute compressible and incompressible flows with segregated stabilized methods. The proposed algorithms have an increased robustness compared to existing techniques, while possessing additional benefits such as employing standard pressure boundary conditions. In this first part, the thermodynamics of isothermal, thermally perfect compressible flows is set up in the framework of symmetric systems and the corresponding segregated algorithms are introduced. Copyright © 2005 John Wiley & Sons, Ltd.     

Gauge finite element method for incompressible flows

A finite element method for computing viscous incompressible flows based on the gauge formulation introduced in [Weinan E, Liu J‐G. Gauge method for viscous incompressible flows. Journal of Computational Physics (submitted)] is presented. This formulation replaces the pressure by a gauge variable. This new gauge variable is a numerical tool and differs from the standard gauge variable that arises from decomposing a compressible velocity field. It has the advantage that an additional boundary condition can be assigned to the gauge variable, thus eliminating the issue of a pressure boundary condition associated with the original primitive variable formulation. The computational task is then reduced to solving standard heat and Poisson equations, which are approximated by straightforward, piecewise linear (or higher‐order) finite elements. This method can achieve high‐order accuracy at a cost comparable with that of solving standard heat and Poisson equations. It is naturally adapted to complex geometry and it is much simpler than traditional finite element methods for incompressible flows. Several numerical examples on both structured and unstructured grids are presented. Copyright © 2000 John Wiley & Sons, Ltd.     

Efficient computation of the eigenspectrum of viscoelastic flows using submatrix-based transformation of the linearized equations

Linear stability analysis of incompressible viscoelastic flows based on normal mode expansions of the eigenfunctions requires the numerical solution of a generalized eigenvalue problem (GEVP). The complex boundary layer structure of the leading eigenfunctions and the singular character of the continuous set of eigenvalues, necessitate the use of fine mesh sizes, leading to large algebraic GEVPs. In this paper, we present a submatrix-based transformation of the linearized equations (SubTLE) that converts the GEVP into a simple eigenvalue problem (EVP) of half the original dimension for the purely elastic isothermal and non-isothermal flows of an Oldroyd-B liquid. This leads to significant (up to an order of magnitude) reduction in the CPU time and memory required for the solution of the EVP. This is illustrated in the context of isothermal and non-isothermal shear flows.     

Development of new finite volume schemes on unstructured triangular grid for simulating the gas–liquid two‐phase flow

This paper presents a coupled finite volume inner doubly iterative efficient algorithm for linked equations (IDEAL) with level set method to simulate the incompressible gas–liquid two‐phase flows with moving interfaces on unstructured triangular grid. The finite volume IDEAL method on a collocated grid is employed to solve the incompressible two‐phase Navier–Stokes equations, and the level set method is used to capture the moving interfaces. For the sake of mass conservation, an effective second‐order accurate finite volume scheme is developed to solve the level set equation on triangular grid, which can be implemented much easier than the classical high‐order level set solvers. In this scheme, the value of level set function on the boundary of control volume is approximated using a linear combination of a high‐order Larangian interpolation and a second‐order upwind interpolation. By the rotating slotted disk and stretching and shrinking of a circular fluid element benchmark cases, the mass conservation and accuracy of the new scheme is verified. Then the coupled method is applied to two‐phase flows, including a 2D bubble rising problem and a 2D dam breaking problem. The computational results agree well with those reported in literatures and experimental data. Copyright © 2015 John Wiley & Sons, Ltd.     

Simulation of the Navier–Stokes equations in three dimensions with a spectral collocation method

This paper describes a nonlinear, three‐dimensional spectral collocation method for the simulation of the incompressible Navier–Stokes equations under the Boussinesq approximation, motivated by geophysical and environmental flows. These flows are driven by the interaction of stratified fluid with topography, which this model accurately accounts for by using a mapped coordinate system. The spectral collocation method is implemented with both a Fourier trigonometric expansion and the Chebyshev polynomials, as appropriate for the domain boundary conditions. The coordinate mapping prohibits the use of existing, fast solution methods that rely on the separation of variables, so a preconditioner based on the approximate solution of a corresponding finite‐difference problem with geometric multigrid is used. The model is parallelized with the Message Passing Interface library, and it runs effectively on shared and distributed‐memory systems. Copyright © 2013 John Wiley & Sons, Ltd.     

NUMERICAL SIMULATION OF INCOMPRESSIBLE FLOWS WITH MOVING INTERFACES

A numerical model has been developed for the 2D simulation of free surface flows or, more generally speaking, moving interface ones. The bulk fluids on both sides of the interface are taken into account in simulating the incompressible laminar flow state. In the case of heat transfer the whole system, i.e. walls as well as possible obstacles, is considered. This model is based on finite element analysis with an Eulerian approach and an unstructured fixed mesh. A special technique to localize the interface allows its temporal evolution through this mesh. Several numerical examples are presented to demonstrate the capabilities of the model. © 1997 by John Wiley & Sons, Ltd.     

An elastoplasticity model in porous media theories

In this article, porous media theories are referred to as mixture theories extended by the well-known concept of volume fractions. This approach implies the diverse field functions of both the porous solid matrix and the pore fluid to be represented by average functions of the macroscale.The present investigations are based on a binary model of incompressible constituents, solid skeleton, and pore liquid, where, in the constitutive range, use is made of the second-grade character of general heterogeneous media. Within the framework of geometrically finite theories, the paper offers a set of constitutive equations for the solid matrix, the viscous pore liquid and the different interactions between the constituents. The constitutive model applies to saturated as well as to empty solid materials, taking into account the physical nonlinearities based on elasto-plastic solid deformations. In particular, the constitutive model concentrates on granular materials like soil or concrete, where the elastic deformations are usually small and the plastic range is governed by kinematically hardening properties.     

Chemically Reacting Fluid Flows: Strong Solutions and Global Attractors

In this paper we consider the existence and properties of strong solutions for a model of incompressible chemically reacting flows where reactants enter the domain, react, and then leave the domain. We show results which exactly parallel those of the Navier–Stokes equations, i.e., in two dimensions strong solutions exist for all time, and in three dimensions we show existence only for small times. In two dimensions, we also show the existence of global attractors which are compact in L ². Rather than considering a specific set of boundary conditions, we instead state our results based on a series of assumptions, which would be proved using the boundary conditions. This allows our results to be applied directly to the two sets of boundary conditions which appear in the literature.     

Pressure‐based unified solver for gas‐liquid two‐phase flows where compressible and incompressible flows coexist

We propose a pressure‐based unified solver for gas‐liquid two‐phase flows where compressible and incompressible flows coexist. Unlike the original thermo–Cubic Interpolated Propagation Combined Unified Procedure (CIP‐CUP) method proposed by Himeno et al (Transactions of the Japan Society of Mechanical Engineers, Series B, 2003), we split the advection term of the governing equations into a conservation part and into the rest. The splitting of advection term has two advantages. One is the high degree of freedom in choosing discretization schemes such as central‐difference schemes, upwind schemes, and Total Variation Diminishing (TVD) schemes. The other is the ease of implementation on unstructured grids. The advantages enable the analyses of various flows such as turbulent and supersonic ones in actual complicated boundaries. Therefore, the solver is useful for practical analyses. The solver was validated on the following test cases: subsonic single‐phase flows, incompressible single‐phase turbulent flows, and incompressible gas‐liquid two‐phase flows. With unstructured grids, we obtained the equivalent results as the ones with structured grids. After the validations, subsonic jet impinging on a water pool was calculated and compared with experimental results. It was confirmed that the calculated results were consistent with the experimental ones.     

On the immersed boundary‐lattice Boltzmann simulations of incompressible flows with freely moving objects

For simulating freely moving problems, conventional immersed boundary‐lattice Boltzmann methods encounter two major difficulties of an extremely large flow domain and the incompressible limit. To remove these two difficulties, this work proposes an immersed boundary‐lattice Boltzmann flux solver (IB‐LBFS) in the arbitrary Lagragian–Eulerian (ALE) coordinates and establishes a dynamic similarity theory. In the ALE‐based IB‐LBFS, the flow filed is obtained by using the LBFS on a moving Cartesian mesh, and the no‐slip boundary condition is implemented by using the boundary condition‐enforced immersed boundary method. The velocity of the Cartesian mesh is set the same as the translational velocity of the freely moving object so that there is no relative motion between the plate center and the mesh. This enables the ALE‐based IB‐LBFS to study flows with a freely moving object in a large open flow domain. By normalizing the governing equations for the flow domain and the motion of rigid body, six non‐dimensional parameters are derived and maintained to be the same in both physical systems and the lattice Boltzmann framework. This similarity algorithm enables the lattice Boltzmann equation‐based solver to study a general freely moving problem within the incompressible limit. The proposed solver and dynamic similarity theory have been successfully validated by simulating the flow around an in‐line oscillating cylinder, single particle sedimentation, and flows with a freely falling plate. The obtained results agree well with both numerical and experimental data. Copyright © 2016 John Wiley & Sons, Ltd.     

Stabilized finite element solution to handle complex heat and fluid flows in industrial furnaces using the immersed volume method

We consider the numerical simulation of conjugate heat transfer, incompressible turbulent flows for multicomponents systems using a stabilized finite element method. We present an immersed volume approach for thermal coupling between fluids and solids for heating high‐alloy steel inside industrial furnaces. It consists in considering a single 3D grid of the furnace and solving one set of equations with different thermal properties. A distance function enables to define precisely the position and the interface of any objects inside the volume and to provide homogeneous physical and thermodynamic properties for each subdomain. An anisotropic mesh adaptation algorithm based on the variations of the distance function is then applied to ensure an accurate capture of the discontinuities that characterize the highly heterogeneous domain. The proposed method demonstrates the capability of the model to simulate an unsteady three‐dimensional heat transfers and turbulent flows in an industrial furnace with the presence of three conducting solid bodies. Copyright © 2010 John Wiley & Sons, Ltd.     

Simulation of gas–solid particle flows over a wide range of concentration

A two-fluid model of gas–solid particle flows that is valid for a wide range of the solid-phase volume concentration (dilute to dense) is presented. The governing equations of the fluid phase are obtained by volume averaging the Navier–Stokes equations for an incompressible fluid. The solid-phase macroscopic equations are derived using an approach that is based on the kinetic theory of dense gases. This approach accounts for particle–particle collisions. The model is implemented in a control-volume finite element method for simulations of the flows of interest in two-dimensional, planar or axisymmetric, domains. The chosen mathematical model and the proposed numerical method are applied to three test problems and one demonstration problem. © 1998 John Wiley & Sons, Ltd.     

The complementary RANS equations for the simulation of viscous flows

A complementary set of Reynolds‐averaged Navier–Stokes (RANS) equations has been developed for steady incompressible, turbulent flows. The method is based on the Helmholtz decomposition of the velocity vector field into a viscous and a potential components. In the complementary RANS solver a potential solution coexists with a viscous solution with the purpose of contributing to a fastest decay of the viscous solution in the far field. The proposed complementary RANS equations have been validated for steady laminar and turbulent flows. The computational results show that the complementary RANS solver is able to produce less grid‐dependent solutions than a conventional RANS solver. Copyright © 2005 John Wiley & Sons, Ltd.