A near-wall strategy for buoyancy-affected turbulent flows using stabilized FEM with applications to indoor air flow simulation

We consider the numerical simulation of buoyancy-affected, incompressible turbulent flows using a stabilized finite-element method. We present an approach which combines two domain decomposition methods (DDM). Firstly, we apply a DDM with full overlap for near-wall modelling, which can be interpreted as an improved wall-function concept. Secondly, a non-overlapping DDM of iteration-by-subdomains-type for the parallel solution of the linearized problems is employed. For this scheme, we demonstrate both the accuracy for a benchmark problem and the applicability to realistic indoor-air flow problems.     

Investigation of multiscale fluid flow characteristics based on a hybrid atomistic–continuum method

A computer program based on a molecular dynamics–continuum hybrid method has been developed in which the Navier–Stokes equations are solved in the continuum region and the molecular dynamics in the atomistic region. The coupling between the atomistic and continuum is constructed through constrained dynamics within an overlap region where both molecular and continuum equations are solved simultaneously. The simulation geometries are solved in three dimensions and an overlap region is introduced in two directions to improve the choice of using the molecular region in smaller areas. The proposed method is used to simulate steady and start-up Couette flow showing quantitative agreement with results from analytical solutions and full molecular dynamics simulations. The prepared algorithm and the computer code are capable of modeling fluid flows in micro and nano-scale geometries.     

Hybrid molecular dynamics-continuum simulation for nano/mesoscale channel flows

The present study deals with multiscale simulation of the fluid flows in nano/mesoscale channels. A hybrid molecular dynamics (MD)-continuum simulation with the principle of crude constrained Lagrangian dynamics for data exchange between continuum and MD regions is performed to resolve the Couette and Poiseuille flows. Unlike the smaller channel heights, H < 50σ (σ is the molecular length scale, σ ≈ 0.34 nm for liquid Ar), considered in the previous works, this study deals with nano/mesoscale channels with height falling into the range of 44σ ≤ H ≤ 400σ, i.e., O(10)–O(10²) nm. The major concerns are: (1) to alleviate statistic fluctuations so as to improve convergence characteristics of the hybrid simulation—a novel treatment for evaluation of force exerted on individual particle is proposed and its effectiveness is demonstrated; (2) to explore the appropriate sizes of the pure MD region and the overlap region for hybrid MD-continuum simulations—the results disclosed that, the pure MD region of at least 12σ and the overlap region of the height 10σ have to be used in this class of hybrid MD-continuum simulations; and (3) to investigate the influences of channel height on the predictions of the flow field and the slip length—a slip length correlation is formulated and the effects of channel size on the flow field and the slip length are discussed. An erratum to this article can be found at     

On Boundary Condition Capturing for Multiphase Interfaces

This review paper begins with an overview of the boundary condition capturing approach to solving problems with interfaces. Although the authors’ original motivation was to extend the ghost fluid method from compressible to incompressible flow, the elliptic nature of incompressible flow quickly quenched the idea that ghost cells could be defined and used in the usual manner. Instead the boundary conditions had to be implicitly captured by the matrix formulation itself, leading to the novel approach. We first review the work on the variable coefficient Poisson equation, noting that the simplicity of the method allowed for an elegant convergence proof. Simplicity and robustness also allowed for a quick extension to three-dimensional two-phase incompressible flows including the effects of viscosity and surface tension, which is discussed subsequently. The method has enjoyed popularity in both computational physics and computer graphics, and we show some comparisons with the traditional delta function approach for the visual simulation of bubbles. Finally, we discuss extensions to problems where the velocity is discontinuous as well, as is the case for premixed flames, and show an example of multiple interacting liquids that includes all of the aforementioned phenomena.     

Hybrid atomistic�Ccontinuum simulations of fluid flows involving interfaces

In this work, we use an hybrid atomistic–continuum (HAC) simulation method to study transient and steady isothermal flows of Lennard-Jones fluids near interfaces. Our hybrid method is based on a domain decomposition algorithm. The flow domain is composed of two overlapping regions: an atomistic region described by molecular dynamics, and a continuum region described by a finite volume discretization of the incompressible Navier–Stokes equations. To show the interest of such an hybrid method to compute flows near fluid/solid interface, we first applied our hybrid scheme to the classical Couette flow, where the moving wall is modelled at the atomistic scale. In addition, we also studied an oscillatory shear flow. Then, to compute flows near fluid/fluid interface, we applied our method to a two-phase Couette flow (liquid/gas), where the interface is modelled at the molecular scale. We show that hybrid results can sometimes differ from those provided by analytical solutions deduced from continuum mechanics equations combined with usual boundary/interface relations. For the Couette and oscillatory shear flows, a good agreement is found between hybrid simulations and macroscopic analytical solutions, however, we noticed that the fluid in contact with the wall can be more entailed than what expected. For the liquid/gas Couette flow, the hybrid simulation exhibits an unexpected jump of the velocity in the interfacial region, corresponding to a partial slip between the two fluid phases. Those interesting results highlight the interest of using an HAC method to deal with systems for which surfaces/interfaces effects are important.     

Time domain computation of flow induced sound

We present a recently developed numerical scheme for computational aeroacoustics (CAA). Therewith, we solve the flow field by a large eddy simulation (LES) and the generation as well as propagation of acoustic noise by Lighthill’s analogy applying the finite element method. The developed scheme allows a direct coupling in time domain as well as a sequential coupling in frequency domain and provides the acoustic sound field not only in the far field but also in the region of the flow. Furthermore, we can directly investigate the acoustic source terms in the flow region. The scheme is well suited for interior aeroacoustic problems with complex geometries as well as for fluid-structure interaction problems. Implementation is validated and a two-dimensional simple application example is used to investigate the acoustic sources and to evaluate the acoustic pressure field from both transient and harmonic analyses.     

Computational haemodynamics: geometry and non-newtonian modelling using spectral/hp element methods

The increased availability of large scale computational facilities is enabling the use of computational modelling for the treatment of complex problems of industrial interest. Computational haemodynamics is an area where improved computational power and algorithmic development have permitted the application in biomechanics of techniques originally devised for direct numerical simulations in others areas of scientific interest. In this paper we present a highly accurate algorithm for the numerical solution of the incompressible Navier–Stokes equations which employs a spectral/hp element spatial discretisation and a high-order time splitting. Two important developments discussed in this paper are the coupling of these techniques with unstructured mesh generation, used in many everyday engineering applications, and the introduction of a non-Newtonian viscosity model to account for the shear thinning properties of blood. The proposed method will be applied to the analysis of shear stress distributions in flows of interest in haemodynamics. Received: 5 May 1999 / Accepted: 21 september 1999     

The variational multiscale method for laminar and turbulent flow

Summary  The present article reviews the variational multiscale method as a framework for the development of computational methods for the simulation of laminar and turbulent flows, with the emphasis placed on incompressible flows. Starting with a variational formulation of the Navier-Stokes equations, a separation of the scales of the flow problem into two and three different scale groups, respectively, is shown. The approaches resulting from these two different separations are interpreted against the background of two traditional concepts for the numerical simulation of turbulent flows, namely direct numerical simulation (DNS) and large eddy simulation (LES). It is then focused on a three-scale separation, which explicitly distinguishes large resolved scales, small resolved scales, and unresolved scales. In view of turbulent flow simulations as a LES, the variational multiscale method with three separated scale groups is refered to as a “variational multiscale LES”. The two distinguishing features of the variational multiscale LES in comparison to the traditional LES are the replacement of the traditional filter by a variational projection and the restriction of the effect of the unresolved scales to the smaller of the resolved scales. Existing solution strategies for the variational multiscale LES are presented and categorized for various numerical methods. The main focus is on the finite element method (FEM) and the finite volume method (FVM). The inclusion of the effect of the unresolved scales within the multiscale environment via constant-coefficient and dynamic subgrid-scale modeling based on the subgrid viscosity concept is also addressed. Selected numerical examples, a laminar and two turbulent flow situations, illustrate the suitability of the variational multiscale method for the numerical simulation of both states of flow. This article concludes with a view on potential future research directions for the variational multiscale method with respect to problems of fluid mechanics.     

A finite element based level set method for two-phase incompressible flows

We present a method that has been developed for the efficient numerical simulation of two-phase incompressible flows. For capturing the interface between the phases the level set technique is applied. The continuous model consists of the incompressible Navier–Stokes equations coupled with an advection equation for the level set function. The effect of surface tension is modeled by a localized force term at the interface (so-called continuum surface force approach). For spatial discretization of velocity, pressure and the level set function conforming finite elements on a hierarchy of nested tetrahedral grids are used. In the finite element setting we can apply a special technique to the localized force term, which is based on a partial integration rule for the Laplace–Beltrami operator. Due to this approach the second order derivatives coming from the curvature can be eliminated. For the time discretization we apply a variant of the fractional step θ-scheme. The discrete saddle point problems that occur in each time step are solved using an inexact Uzawa method combined with multigrid techniques. For reparametrization of the level set function a new variant of the fast marching method is introduced. A special feature of the solver is that it combines the level set method with finite element discretization, Laplace–Beltrami partial integration, multilevel local refinement and multigrid solution techniques. All these components of the solver are described. Results of numerical experiments are presented.     

Adaptive particles for incompressible fluid simulation

We propose a particle-based technique for simulating incompressible fluid that includes adaptive refinement of particle sampling. Each particle represents a mass of fluid in its local region. Particles are split into several particles for finer sampling in regions of complex flow. In regions of smooth flow, neighboring particles can be merged. Depth below the surface and Reynolds number are exploited as our criteria for determining whether splitting or merging should take place. For the fluid dynamics calculations, we use the hybrid FLIP method, which is computationally simple and efficient. Since the fluid is incompressible, each particle has a volume proportional to its mass. A kernel function, whose effective range is based on this volume, is used for transferring and updating the particle’s physical properties such as mass and velocity. Our adaptive particle-based simulation is demonstrated in several scenarios that show its effectiveness in capturing fine detail of the flow, where needed, while efficiently sampling regions where less detail is required.     

E(FG)<Superscript>2</Superscript>: a new fixed-grid shape optimization method based on the element-free galerkin mesh-free analysis

We propose a shape optimization method over a fixed grid. Nodes at the intersection with the fixed grid lines track the domain’s boundary. These “floating” boundary nodes are the only ones that can move/appear/disappear in the optimization process. The element-free Galerkin (EFG) method, used for the analysis problem, provides a simple way to create these nodes. The fixed grid (FG) defines integration cells for EFG method. We project the physical domain onto the FG and numerical integration is performed over partially cut cells. The integration procedure converges quadratically. The performance of the method is shown with examples from shape optimization of thermal systems involving large shape changes between iterations. The method is applicable, without change, to shape optimization problems in elasticity, etc. and appears to eliminate non-differentiability of the objective noticed in finite element method (FEM)-based fictitious domain shape optimization methods. We give arguments to support this statement. A mathematical proof is needed.     

On the application of the minimum polynomial extrapolation method to incompressible flows with heat transfer

In this paper, the Minimum Polynomial Extrapolation method (MPE) is used to accelerate the convergence of the Characteristic–Based–Split (CBS) scheme for the numerical solution of steady state incompressible flows with heat transfer. The CBS scheme is a fractional step method for the solution of the Navier–Stokes equations while the MPE method is a vector extrapolation method which transforms the original sequence into another sequence converging to the same limit faster then the original one without the explicit knowledge of the sequence generator. The developed algorithm is tested on a two-dimensional benchmark problem (buoyancy–driven convection problem) where the Navier–Stokes equations are coupled with the temperature equation. The obtained results show the feature of the extrapolation procedure to the CBS scheme and the reduction of the computational time of the simulation.     

Evaluation of a bounded high order upwind scheme for 3D incompressible free surface flow computations

In the context of normalized variable formulation (NVF) of Leonard and total variation diminishing (TVD) constraints of Harten, this paper presents an extension of a previous work by the authors for solving unsteady incompressible flow problems. The main contributions of the paper are threefold. First, it presents the results of the development and implementation of a bounded high order upwind adaptative QUICKEST scheme in the 3D robust code (Freeflow), for the numerical solution of the full incompressible Navier–Stokes equations. Second, it reports numerical simulation results for 1D shock tube problem, 2D impinging jet and 2D/3D broken dam flows. Furthermore, these results are compared with existing analytical and experimental data. And third, it presents the application of the numerical method for solving 3D free surface flow problems.     

A stabilized finite element method for stochastic incompressible Navier–Stokes equations

The major emphasis of this work is the development of a stabilized finite element method for solving incompressible Navier–Stokes equations with stochastic input data. The polynomial chaos expansion is used to represent stochastic processes in the variational problem, resulting in a set of deterministic variational problems to be solved for each Wiener polynomial chaos. To obtain the chaos coefficients in the corresponding deterministic incompressible Navier–Stokes equations, we combine the modified method of characteristics with the finite element discretization. The obtained Stokes problem is solved using a robust conjugate-gradient algorithm. This algorithm avoids projection procedures and any special correction for the pressure. These numerical techniques associate the geometrical flexibility of the finite element method with the ability offered by the modified method of characteristics to solve convection-dominated problems using time steps larger than its Eulerian counterpart. Numerical results are shown for the benchmark problems of driven cavity flow and backward-facing step flow. We also present numerical results for a problem of stochastic natural convection. It is found that the proposed stabilized finite element method offers a robust and accurate approach for solving the stochastic incompressible Navier–Stokes equations, even when high Reynolds and Rayleigh numbers are used in the simulations.     

Linear stability analysis in fluid-structure interaction with transpiration. Part II: Numerical analysis and applications

This paper constitutes the numerical counterpart of the mathematical framework introduced in Part I. We address the problem of flutter analysis of a coupled fluid-structure system involving an incompressible Newtonian fluid and a reduced structure. We use the Linearization Principle approach developed in Part I, particularly suited for fluid-structure problems involving moving boundaries. Thus, the stability analysis is reduced to the computation of the leftmost eigenvalues of a coupled eigenproblem of minimal complexity. This eigenproblem involves the linearized incompressible Navier-Stokes equations and those of a reduced linear structure. The coupling is realized through specific transpiration interface conditions. The eigenproblem is discretized using a finite element approximation and its smallest real part eigenvalues are computed by combining a generalized Cayley transform and an implicit restarted Arnoldi method. Finally, we report three numerical experiments: a structure immersed in a fluid at rest, a cantilever pipe conveying a fluid flow and a rectangular bridge deck profile under wind effects. The numerical results are compared to former approaches and experimental data. The quality of these numerical results is very satisfactory and promising.     

基于国产十亿亿次超算系统的近连续过渡流区N-S/DSMC耦合算法并行优化研究

过渡流区气动问题的数值模拟一直是空气动力学领域的难点。首先介绍了在已有 N-S解算器和 DSMC方法研究基础上,采用 MPC耦合技术建立N-S/DSMC 耦合算法,把 DSMC 方法和 N-S 方法的应用范围拓展到近连续过渡流区。然后详述了基于国家超级计算无锡中心的国产十亿亿次超级计算机开展的耦合算法多级并行优化技术,并首次实现了耦合算法的众核并行。测试表明,本文的进程级优化技术取得了超线性加速比;众核级优化受制于原算法特点和系统特点没有取得预期效果,但进行了探讨和分析,为 N-S/DSMC 耦合算法的众核并行提供了研究和分析依据,为过渡流区高超声速气动特性数值模拟研究提供了有效的途径。     

Molecular dynamics–continuum hybrid simulation for condensation of gas flow in a microchannel

A molecular dynamics-continuum coupling method combining fluid flow and heat transfer is developed to study the condensation process of gas flow in a microchannel. The computational domain is decomposed into particle (P), continuum (C) and overlap (O) regions with solving approaches of molecular dynamics simulation, finite volume method and the developed coupling method, respectively. Continuities of momentum and energy in O region are ensured by constraint dynamics and the Langevin method. The validity of the developed method is confirmed by a good agreement between hybrid results and analytical solutions from two cases including the unsteady dynamical and thermal problems. For the condensation process of gas flow, the hybrid transient velocity and temperature fields indicate that the process does not progress smoothly but wavily with noticeable fluctuation, leading to oscillation in temperature field and recirculation flow in velocity field. Analysis based on heat and mass transfer is carried out in P region, and the Kapitza resistance and the thermal conductivity in liquid are obtained with the satisfying agreement with experimental data, which shows the availability of the developed model for the investigation on the thermal boundary resistance. The good performance had demonstrated that the developed coupling method and computational model are available to provide a multiscale overview in dynamical and thermal problems including phase-transition from nanoscale to microscale, which will show significantly potential in micro fluidics and thermal engineering.     

Numerical simulation of the motion of red blood cells and vesicles in microfluidic flows

We study the mathematical modeling and numerical simulation of the motion of red blood cells (RBC) and vesicles subject to an external incompressible flow in a microchannel. RBC and vesicles are viscoelastic bodies consisting of a deformable elastic membrane enclosing an incompressible fluid. We provide an extension of the finite element immersed boundary method by Boffi and Gastaldi (Comput Struct 81:491–501, 2003), Boffi et al. (Math Mod Meth Appl Sci 17:1479–1505, 2007), Boffi et al. (Comput Struct 85:775–783, 2007) based on a model for the membrane that additionally accounts for bending energy and also consider inflow/outflow conditions for the external fluid flow. The stability analysis requires both the approximation of the membrane by cubic splines (instead of linear splines without bending energy) and an upper bound on the inflow velocity. In the fully discrete case, the resulting CFL-type condition on the time step size is also more restrictive. We perform numerical simulations for various scenarios including the tank treading motion of vesicles in microchannels, the behavior of ‘healthy’ and ‘sick’ RBC which differ by their stiffness, and the motion of RBC through thin capillaries. The simulation results are in very good agreement with experimentally available data.     

Simulation of low speed 3D nanochannel flow

This paper presents a new technique of simulating low speed nanochannel flows using molecular dynamics simulation (MDS) method. When using the molecular dynamics simulation method to study low speed nanoscale flow problems, a major difficulty is the extraction of the true flow velocity because of the highly nonlinear coupling of the low bulk flow velocity and the high velocity of molecules’ thermal motion. In all published papers the reported flow velocity is the average value of the sum of these two velocities over time. For high speed flow problems the conventional MDS method can give satisfactory result. However, when the flow velocity is much smaller than the thermal velocity, the conventional molecular dynamics simulation method cannot predict the true flow velocity. To overcome this difficulty, in this study, a new linearized algorithm is developed. The new algorithm separates the flow velocity increment caused by external forces from the thermal motion velocity at each time step. The detailed process of the new algorithm is derived in this paper and several cases of 3D nanochannel flows of liquid argon are simulated by using this method. The numerical results show that the new algorithm is valid for nanoscale flows.