Coupled Navier–Stokes/molecular dynamics simulations in nonperiodic domains based on particle forcing

This paper focuses on coupling methods for hybrid Navier–Stokes/molecular dynamics (MD) simulations. The computational domain is split in a continuum flow region, where a finite‐volume discretisation of the Navier–Stokes equations is used, and one or more particle domains, where molecular level modelling of the flow is employed. The domains are defined with a partial overlap, in which the flow states are coupled through an exchange of the velocity components. For the steady flows considered, an under‐relaxed Newton iteration method is used to drive the coupled system to convergence. The main focus of the present work is on methods to impose nonperiodic boundary conditions on the particle domain(s). A particle forcing is applied in the direction normal to the particle domain boundary to impose the boundary normal velocity component. A novel aspect of the present work is the extension of this method to more general nonplanar particle domain boundaries. The main contribution of the paper is the development of a particle forcing method in the direction tangential to the domain boundary, which is based on the equivalent continuum‐flow boundary shear stresses along with an iterative forcing strength adjustment based on the extrapolated particle boundary velocity. Furthermore, an adaptation scheme is presented, which uses the finite‐volume flux residuals of the particle bin averaged velocity field as a truncation criterion for the iterative force‐update scheme. It is demonstrated that by comparing the residual reduction for the momentum equation in the nonhomogeneous directions during the molecular dynamics simulations with that for a homogeneous direction, the forcing iteration at which the statistical noise in the velocity field dominates the uncertainty in the forcing strength can be determined. At this point the iteration can be truncated. It is shown that with adaptive schemes of this type, the total number of MD evaluations required in a coupled Navier–Stokes/MD simulation can be reduced relative to a hybrid scheme with a fixed number of forcing‐strength updates. Copyright © 2011 John Wiley & Sons, Ltd.     

Parallel‐in‐time simulation of the unsteady Navier–Stokes equations for incompressible flow

In this paper, we present an application of a parallel‐in‐time algorithm for the solution of the unsteady Navier–Stokes model equations that are of parabolic–elliptic type. This method is based on the alternated use of a coarse global sequential solver and a fine local parallel one. A standard finite volume/finite differences first‐order approach is used for discretization of the unsteady two‐dimensional Navier–Stokes equations. The Taylor vortex decay problem and the confined flow around a square cylinder were selected as unsteady flow examples to illustrate and analyse the properties of the parallel‐in‐time method through numerical experiments. The influence of several parameters on the computing time required to perform a parallel‐in‐time calculation on a PC cluster was verified. Among them we have analysed the influence of the number of processors, the number of iterations for convergence, the resolution of the spatial domain and the influence of the time‐step sizes ratio between the coarse and fine grids. Significant computer time saving was achieved when compared with the single processor computing time, particularly when the spatial dimension of the problem is low and the temporal scale is large. Copyright © 2004 John Wiley & Sons, Ltd.     

Numerical simulation of free‐surface flow using the level‐set method with global mass correction

A new numerical method that couples the incompressible Navier–Stokes equations with the global mass correction level‐set method for simulating fluid problems with free surfaces and interfaces is presented in this paper. The finite volume method is used to discretize Navier–Stokes equations with the two‐step projection method on a staggered Cartesian grid. The free‐surface flow problem is solved on a fixed grid in which the free surface is captured by the zero level set. Mass conservation is improved significantly by applying a global mass correction scheme, in a novel combination with third‐order essentially non‐oscillatory schemes and a five stage Runge–Kutta method, to accomplish advection and re‐distancing of the level‐set function. The coupled solver is applied to simulate interface change and flow field in four benchmark test cases: (1) shear flow; (2) dam break; (3) travelling and reflection of solitary wave and (4) solitary wave over a submerged object. The computational results are in excellent agreement with theoretical predictions, experimental data and previous numerical simulations using a RANS‐VOF method. The simulations reveal some interesting free‐surface phenomena such as the free‐surface vortices, air entrapment and wave deformation over a submerged object. Copyright © 2009 John Wiley & Sons, Ltd.     

Incompressible Roe‐like scheme for turbulent flows

In the current study, numerical investigation of incompressible turbulent flow is presented. By the artificial compressibility method, momentum and continuity equations are coupled. Considering Reynolds averaged Navier–Stokes equations, the Spalart–Allmaras turbulence model, which has accurate results in two‐dimensional problems, is used to calculate Reynolds stresses. For convective fluxes a Roe‐like scheme is proposed for the steady Reynolds averaged Navier–Stokes equations. Also, Jameson averaging method was implemented. In comparison, the proposed characteristics‐based upwind incompressible turbulent Roe‐like scheme, demonstrated very accurate results, high stability, and fast convergence. The fifth‐order Runge–Kutta scheme is used for time discretization. The local time stepping and implicit residual smoothing were applied as the convergence acceleration techniques. Suitable boundary conditions have been implemented considering flow behavior. The problem has been studied at high Reynolds numbers for cross flow around the horizontal circular cylinder and NACA0012 hydrofoil. Results were compared with those of others and a good agreement has been observed. Copyright © 2012 John Wiley & Sons, Ltd.     

Flow simulation on moving boundary‐fitted grids and application to fluid–structure interaction problems

We present a method for the parallel numerical simulation of transient three‐dimensional fluid–structure interaction problems. Here, we consider the interaction of incompressible flow in the fluid domain and linear elastic deformation in the solid domain. The coupled problem is tackled by an approach based on the classical alternating Schwarz method with non‐overlapping subdomains, the subproblems are solved alternatingly and the coupling conditions are realized via the exchange of boundary conditions. The elasticity problem is solved by a standard linear finite element method. A main issue is that the flow solver has to be able to handle time‐dependent domains. To this end, we present a technique to solve the incompressible Navier–Stokes equation in three‐dimensional domains with moving boundaries. This numerical method is a generalization of a finite volume discretization using curvilinear coordinates to time‐dependent coordinate transformations. It corresponds to a discretization of the arbitrary Lagrangian–Eulerian formulation of the Navier–Stokes equations. Here the grid velocity is treated in such a way that the so‐called Geometric Conservation Law is implicitly satisfied. Altogether, our approach results in a scheme which is an extension of the well‐known MAC‐method to a staggered mesh in moving boundary‐fitted coordinates which uses grid‐dependent velocity components as the primary variables. To validate our method, we present some numerical results which show that second‐order convergence in space is obtained on moving grids. Finally, we give the results of a fully coupled fluid–structure interaction problem. It turns out that already a simple explicit coupling with one iteration of the Schwarz method, i.e. one solution of the fluid problem and one solution of the elasticity problem per time step, yields a convergent, simple, yet efficient overall method for fluid–structure interaction problems. Copyright © 2005 John Wiley & Sons, Ltd.     

An efficient and energy stable scheme for a phase‐field model for the moving contact line problem

In this paper, we propose for the first time a linearly coupled, energy stable scheme for the Navier–Stokes–Cahn–Hilliard system with generalized Navier boundary condition. We rigorously prove the unconditional energy stability for the proposed time discretization as well as for a fully discrete finite element scheme. Using numerical tests, we verify the accuracy, confirm the decreasing property of the discrete energy, and demonstrate the effectiveness of our method through numerical simulations in both 2‐D and 3‐D. Copyright © 2015 John Wiley & Sons, Ltd.     

Fourth‐order accurate compact‐difference discretization method for Helmholtz and incompressible Navier–Stokes equations

A fourth‐order accurate solution method for the three‐dimensional Helmholtz equations is described that is based on a compact finite‐difference stencil for the Laplace operator. Similar discretization methods for the Poisson equation have been presented by various researchers for Dirichlet boundary conditions. Here, the complicated issue of imposing Neumann boundary conditions is described in detail. The method is then applied to model Helmholtz problems to verify the accuracy of the discretization method. The implementation of the solution method is also described. The Helmholtz solver is used as the basis for a fourth‐order accurate solver for the incompressible Navier–Stokes equations. Numerical results obtained with this Navier–Stokes solver for the temporal evolution of a three‐dimensional instability in a counter‐rotating vortex pair are discussed. The time‐accurate Navier–Stokes simulations show the resolving properties of the developed discretization method and the correct prediction of the initial growth rate of the three‐dimensional instability in the vortex pair. Copyright © 2004 John Wiley & Sons, Ltd.     

A projection method for the spectral solution of non‐homogeneous and incompressible Navier–Stokes equations

This paper is devoted to the development of a parallel, spectral and second‐order time‐accurate method for solving the incompressible and variable density Navier–Stokes equations. The method is well suited for finite thickness density layers and is very efficient, especially for three‐dimensional computations. It is based on an exact projection technique. To enforce incompressibility, for a non‐homogeneous fluid, the pressure is computed using an iterative algorithm. A complete study of the convergence properties of this algorithm is done for different density variations. Numerical simulations showing, qualitatively, the capabilities of the developed Navier–Stokes solver for many realistic problems are presented. The numerical procedure is also validated quantitatively by reproducing growth rates from the linear instability theory in a three‐dimensional direct numerical simulation of an unstable, non‐homogeneous, flow configuration. It is also shown that, even in a turbulent flow, the spectral accuracy is recovered. Copyright © 2012 John Wiley & Sons, Ltd.     

Evaluation of Smagorinsky‐based subgrid‐scale models in a finite‐volume computation

Smagorinsky‐based models are assessed in a turbulent channel flow simulation at Re_b=2800 and Re_b=12500. The Navier–Stokes equations are solved with three different grid resolutions by using a co‐located finite‐volume method. Computations are repeated with Smagorinsky‐based subgrid‐scale models. A traditional Smagorinsky model is implemented with a van Driest damping function. A dynamic model assumes a similarity of the subgrid and the subtest Reynolds stresses and an explicit filtering operation is required. A top‐hat test filter is implemented with a trapezoidal and a Simpson rule. At the low Reynolds number computation none of the tested models improves the results at any grid level compared to the calculations with no model. The effect of the subgrid‐scale model is reduced as the grid is refined. The numerical implementation of the test filter influences on the result. At the higher Reynolds number the subgrid‐scale models stabilize the computation. An analysis of an accurately resolved flow field reveals that the discretization error overwhelms the subgrid term at Re_b=2800 in the most part of the computational domain. Copyright © 2002 John Wiley & Sons, Ltd.     

A coupled continuous and discontinuous finite element method for the incompressible flows

In this paper, we develop a coupled continuous Galerkin and discontinuous Galerkin finite element method based on a split scheme to solve the incompressible Navier–Stokes equations. In order to use the equal order interpolation functions for velocity and pressure, we decouple the original Navier–Stokes equations and obtain three distinct equations through the split method, which are nonlinear hyperbolic, elliptic, and Helmholtz equations, respectively. The hybrid method combines the merits of discontinuous Galerkin (DG) and finite element method (FEM). Therefore, DG is concerned to accomplish the spatial discretization of the nonlinear hyperbolic equation to avoid using the stabilization approaches that appeared in FEM. Moreover, FEM is utilized to deal with the Poisson and Helmholtz equations to reduce the computational cost compared with DG. As for the temporal discretization, a second‐order stiffly stable approach is employed. Several typical benchmarks, namely, the Poiseuille flow, the backward‐facing step flow, and the flow around the cylinder with a wide range of Reynolds numbers, are considered to demonstrate and validate the feasibility, accuracy, and efficiency of this coupled method. Copyright © 2016 John Wiley & Sons, Ltd.     

Non‐Newtonian blood flow study in a model cavopulmonary vascular system

A transient haemodynamic study in a model cavopulmonary vascular system has been carried out for a typical range of parameters using a finite element‐based Navier–Stokes solver. The focus of this study is to investigate the influence of non‐Newtonian behaviour of the blood on the haemodynamic quantities, such as wall shear stress (WSS) and flow pattern. The computational fluid dynamics (CFD) model is based on an artificial compressibility characteristic‐based split (AC‐CBS) scheme, which has been adopted to solve the Navier–Stokes equations in space–time domain. A power law model has been implemented to characterize the shear thinning nature of the blood depending on the local strain rate. Using the computational model, numerical investigations have been performed for Newtonian and non‐Newtonian flows for different frequencies and input pulse forms. The haemodynamic quantities observed in total cavopulmonary connection (TCPC) for the above conditions suggest that there are considerable differences in average (about 25–40%) and peak (about 50%) WSS distributions, when the non‐Newtonian behaviour of the blood is taken into account. The lower WSS levels observed for non‐Newtonian cases point to the higher risk of lesion formation, especially at higher pulsation frequencies. A realistic pulse form is relatively safer than a sinusoidal pulse as it has more energy distributed in the higher harmonics, which results in higher average WSS values. The present study highlights the importance of including non‐Newtonian shear thinning behaviour for modelling blood flow in the vicinity of repaired arterial connections. Copyright © 2010 John Wiley & Sons, Ltd.     

A variational multiscale Newton–Schur approach for the incompressible Navier–Stokes equations

In the following paper, we present a consistent Newton–Schur (NS) solution approach for variational multiscale formulations of the time‐dependent Navier–Stokes equations in three dimensions. The main contributions of this work are a systematic study of the variational multiscale method for three‐dimensional problems and an implementation of a consistent formulation suitable for large problems with high nonlinearity, unstructured meshes, and non‐symmetric matrices. In addition to the quadratic convergence characteristics of a Newton–Raphson‐based scheme, the NS approach increases computational efficiency and parallel scalability by implementing the tangent stiffness matrix in Schur complement form. As a result, more computations are performed at the element level. Using a variational multiscale framework, we construct a two‐level approach to stabilizing the incompressible Navier–Stokes equations based on a coarse and fine‐scale subproblem. We then derive the Schur complement form of the consistent tangent matrix. We demonstrate the performance of the method for a number of three‐dimensional problems for Reynolds number up to 1000 including steady and time‐dependent flows. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical simulation of bubble and droplet deformation by a level set approach with surface tension in three dimensions

In this paper we present a three‐dimensional Navier–Stokes solver for incompressible two‐phase flow problems with surface tension and apply the proposed scheme to the simulation of bubble and droplet deformation. One of the main concerns of this study is the impact of surface tension and its discretization on the overall convergence behavior and conservation properties. Our approach employs a standard finite difference/finite volume discretization on uniform Cartesian staggered grids and uses Chorin's projection approach. The free surface between the two fluid phases is tracked with a level set (LS) technique. Here, the interface conditions are implicitly incorporated into the momentum equations by the continuum surface force method. Surface tension is evaluated using a smoothed delta function and a third‐order interpolation. The problem of mass conservation for the two phases is treated by a reinitialization of the LS function employing a regularized signum function and a global fixed point iteration. All convective terms are discretized by a WENO scheme of fifth order. Altogether, our approach exhibits a second‐order convergence away from the free surface. The discretization of surface tension requires a smoothing scheme near the free surface, which leads to a first‐order convergence in the smoothing region. We discuss the details of the proposed numerical scheme and present the results of several numerical experiments concerning mass conservation, convergence of curvature, and the application of our solver to the simulation of two rising bubble problems, one with small and one with large jumps in material parameters, and the simulation of a droplet deformation due to a shear flow in three space dimensions. Furthermore, we compare our three‐dimensional results with those of quasi‐two‐dimensional and two‐dimensional simulations. This comparison clearly shows the need for full three‐dimensional simulations of droplet and bubble deformation to capture the correct physical behavior. Copyright © 2009 John Wiley & Sons, Ltd.     

A semi‐Lagrangian level set method for incompressible Navier–Stokes equations with free surface

In this paper, we formulate a level set method in the framework of finite elements‐semi‐Lagrangian methods to compute the solution of the incompressible Navier–Stokes equations with free surface. In our formulation, we use a quasi‐monotone semi‐Lagrangian scheme, which is both unconditionally stable and essentially non oscillatory, to compute the advective terms in the Navier–Stokes equations, the transport equation and the equation of the reinitialization stage for the level set function. The method we propose is quite robust and flexible with regard to the mesh and the geometry of the domain, as well as the magnitude of the Reynolds number. We illustrate the performance of the method in several examples, which range from a benchmark problem to test the volume conservation property of the method to the flow past a NACA0012 foil at high Reynolds number. Copyright © 2005 John Wiley & Sons, Ltd.     

Two‐level finite element method with a stabilizing subgrid for the incompressible Navier–Stokes equations

We consider the Galerkin finite element method for the incompressible Navier–Stokes equations in two dimensions. The domain is discretized into a set of regular triangular elements and the finite‐dimensional spaces employed consist of piecewise continuous linear interpolants enriched with the residual‐free bubble functions. To find the bubble part of the solution, a two‐level finite element method with a stabilizing subgrid of a single node is described, and its application to the Navier–Stokes equation is displayed. Numerical approximations employing the proposed algorithm are presented for three benchmark problems. The results show that the proper choice of the subgrid node is crucial in obtaining stable and accurate numerical approximations consistent with the physical configuration of the problem at a cheap computational cost. Copyright © 2008 John Wiley & Sons, Ltd.     

Parallel domain decomposition method for finite element approximation of 3D steady state non‐Newtonian fluids

We introduce a stabilized finite element method for the 3D non‐Newtonian Navier–Stokes equations and a parallel domain decomposition method for solving the sparse system of nonlinear equations arising from the discretization. Non‐Newtonian flow problems are, generally speaking, more challenging than Newtonian flows because the nonlinearities are not only in the convection term but also in the viscosity term, which depends on the shear rate. Many good iterative methods and preconditioning techniques that work well for the Newtonian flows do not work well for the non‐Newtonian flows. We employ a Galerkin/least squares finite element method, with stabilization parameters adjusted to count the non‐Newtonian effect, to discretize the equations, and the resulting highly nonlinear system of equations is solved by a Newton–Krylov–Schwarz algorithm. In this study, we apply the proposed method to some inelastic power‐law fluid flows through the eccentric annuli with inner cylinder rotation and investigate the robustness of the method with respect to some physical parameters, including the power‐law index and the Reynolds number ratios. We then report the superlinear speedup achieved by the domain decomposition algorithm on a computer with up to 512 processors. Copyright © 2015 John Wiley & Sons, Ltd.     

On coupling the Reynolds‐averaged Navier–Stokes equations with two‐equation turbulence model equations

Two methods for coupling the Reynolds‐averaged Navier–Stokes equations with the q–ω turbulence model equations on structured grid systems have been studied; namely a loosely coupled method and a strongly coupled method. The loosely coupled method first solves the Navier–Stokes equations with the turbulent viscosity fixed. In a subsequent step, the turbulence model equations are solved with all flow quantities fixed. On the other hand, the strongly coupled method solves the Reynolds‐averaged Navier–Stokes equations and the turbulence model equations simultaneously. In this paper, numerical stabilities of both methods in conjunction with the approximated factorization‐alternative direction implicit method are analysed. The effect of the turbulent kinetic energy terms in the governing equations on the convergence characteristics is also studied. The performance of the two methods is compared for several two‐ and three‐dimensional problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Coupled lubrication and Stokes flow finite elements

A method is developed for performing a local reduction of the governing physics for fluid problems with domains that contain a combination of narrow and non‐narrow regions, and the computational accuracy and performance of the method are measured. In the narrow regions of the domain, where the fluid is assumed to have no inertia and the domain height and curvature are assumed small, lubrication, or Reynolds, theory is used locally to reduce the two‐dimensional Navier–Stokes equations to the one‐dimensional Reynolds equation while retaining a high degree of accuracy in the overall solution. The Reynolds equation is coupled to the governing momentum and mass equations of the non‐narrow region with boundary conditions on the mass and momentum flux. The localized reduction technique, termed ‘stitching,’ is demonstrated on Stokes flow for various geometries of the hydrodynamic journal bearing—a non‐trivial test problem for which a known analytical solution is available. The computational advantage of the coupled Stokes–Reynolds method is illustrated on an industrially applicable fully‐flooded deformable‐roll coating example. The examples in this paper are limited to two‐dimensional Stokes flow, but extension to three‐dimensional and Navier–Stokes flow is possible. Copyright © 2003 John Wiley & Sons, Ltd.     

Weak imposition of boundary conditions for the Navier–Stokes equations by a penalty method

We prove convergence of the finite element method for the Navier–Stokes equations in which the no‐slip condition and no‐penetration condition on the flow boundary are imposed via a penalty method. This approach has been previously studied for the Stokes problem by Liakos (Weak imposition of boundary conditions in the Stokes problem. Ph.D. Thesis, University of Pittsburgh, 1999). Since, in most realistic applications, inertial effects dominate, it is crucial to extend the validity of the method to the nonlinear Navier–Stokes case. This report includes the analysis of this extension, as well as numerical results validating their analytical counterparts. Specifically, we show that optimal order of convergence can be achieved if the computational boundary follows the real flow boundary exactly. Copyright © 2008 John Wiley & Sons, Ltd.