A parallel fully coupled implicit domain decomposition method for numerical simulation of microfluidic mixing in 3D

A parallel fully coupled implicit fluid solver based on a Newton–Krylov–Schwarz algorithm is developed on top of the Portable, Extensible Toolkit for Scientific computation for the simulation of microfluidic mixing described by the three-dimensional unsteady incompressible Navier–Stokes equations. The popularly used fractional step method, originally designed for high Reynolds number flows, requires some modification of the inviscid-type pressure boundary condition in order to reduce the divergence error near the wall. On the other hand, the fully coupled approach works well without any special treatment of the boundary condition for low Reynolds number microchannel flows. A key component of the algorithm is an additive Schwarz preconditioner, which is used to accelerate the convergence of a linear Krylov-type solver for the saddle-point-type Jacobian systems. As a test case, we carefully study a three-dimensional passive serpentine micromixer and report the parallel performance of the algorithm obtained on a parallel machine with more than one hundred processors.     

An analysis of the time integration algorithms for the finite element solutions of incompressible Navier–Stokes equations based on a stabilised formulation

This work is concerned with the analysis of time integration procedures for the stabilised finite element formulation of unsteady incompressible fluid flows governed by the Navier–Stokes equations. The stabilisation technique is combined with several different implicit time integration procedures including both finite difference and finite element schemes. Particular attention is given to the generalised-α method and the linear discontinuous in time finite element scheme. The time integration schemes are first applied to two model problems, represented by a first order differential equation in time and the one dimensional advection–diffusion equation, and subjected to a detailed mathematical analysis based on the Fourier series expansion. In order to establish the accuracy and efficiency of the time integration schemes for the Navier–Stokes equations, a detailed computational study is performed of two standard numerical examples: unsteady flow around a cylinder and flow across a backward facing step. It is concluded that the semi-discrete generalised-α method provides a viable alternative to the more sophisticated and expensive space–time methods for simulations of unsteady flows of incompressible fluids governed by the Navier–Stokes equations.     

A Parallel Finite Element Method for 3D Two-Phase Moving Contact Line Problems in Complex Domains

Moving contact line problem plays an important role in fluid-fluid interface motion on solid surfaces. The problem can be described by a phase-field model consisting of the coupled Cahn–Hilliard and Navier–Stokes equations with the generalized Navier boundary condition (GNBC). Accurate simulation of the interface and contact line motion requires very fine meshes, and the computation in 3D is even more challenging. Thus, the use of high performance computers and scalable parallel algorithms are indispensable. In this paper, we generalize the GNBC to surfaces with complex geometry and introduce a finite element method on unstructured 3D meshes with a semi-implicit time integration scheme. A highly parallel solution strategy using different solvers for different components of the discretization is presented. More precisely, we apply a restricted additive Schwarz preconditioned GMRES method to solve the systems arising from implicit discretization of the Cahn–Hilliard equation and the velocity equation, and an algebraic multigrid preconditioned CG method to solve the pressure Poisson system. Numerical experiments show that the strategy is efficient and scalable for 3D problems with complex geometry and on a supercomputer with a large number of processors.     

Evaluation of Fully Implicit Runge Kutta Schemes for Unsteady Flow Calculations

This paper presents the formulation of a dual time stepping procedure to solve the equations of fully implicit Runge–Kutta schemes. In particular the method is applied to Gauss and Radau 2A schemes with either two or three stages. The schemes are tested for unsteady flows over a pitching airfoil modeled by both the Euler and the unsteady Reynolds averaged Navier Stokes equations. It is concluded that the Radau 2A schemes are more robust and less computationally expensive because they require a much smaller number of inner iterations. Moreover these schemes seem to be competitive with alternative implicit schemes.     

Numerical Analysis of Second Order,Fully Discrete Energy Stable Schemes for Phase Field Models of Two-Phase Incompressible Flows

In this paper, we propose several second order in time, fully discrete, linear and nonlinear numerical schemes for solving the phase field model of two-phase incompressible flows, in the framework of finite element method. The schemes are based on the second order Crank–Nicolson method for time discretization, projection method for Navier–Stokes equations, as well as several implicit–explicit treatments for phase field equations. The energy stability and unique solvability of the proposed schemes are proved. Ample numerical experiments are performed to validate the accuracy and efficiency of the proposed schemes.     

On a collocation B-spline method for the solution of the Navier–Stokes equations

A novel B-spline collocation method for the solution of the incompressible Navier–Stokes equations is presented. The discretization employs B-splines of maximum continuity, yielding schemes with high-resolution power. The Navier–Stokes equations are solved by using a fractional step method, where the projection step is considered as a Div–Grad problem, so that no pressure boundary conditions need to be prescribed. Pressure oscillations are prevented by introducing compatible B-spline bases for the velocity and pressure, yielding efficient schemes of arbitrary order of accuracy. The method is applied to two-dimensional benchmark flows, and mass lumping techniques for cost-effective computation of unsteady problems are discussed.     

A mesh adaptivity procedure for CFD and fluid-structure interactions

We present a procedure to adapt and repair meshes in the general solution of Navier–Stokes incompressible and compressible fluid flows, including structural interactions. For fluid-structure interactions, FSI, the fluid is described by an arbitrary-Lagrangian–Eulerian formulation fully coupled to general solids and structures described by Lagrangian formulations. The solids and structures can undergo highly nonlinear response due to large deformations, nonlinear material behavior, contact and temperature. We focus on the need to adapt the fluid mesh in pure CFD solutions when high gradients are present or boundary layer effects are important, and FSI solutions when large structural deformations take place. The procedure is a practical scheme to solve complex problems. We illustrate the proposed scheme in various example solutions.     

Parallel-multigrid computation of unsteady incompressible viscous flows using a matrix-free implicit method and high-resolution characteristics-based scheme

A three-dimensional parallel unstructured non-nested multigrid solver for solutions of unsteady incompressible viscous flow is developed and validated. The finite-volume Navier–Stokes solver is based on the artificial compressibility approach with a high-resolution method of characteristics-based scheme for handling convection terms. The unsteady flow is calculated with a matrix-free implicit dual time stepping scheme. The parallelization of the multigrid solver is achieved by multigrid domain decomposition approach (MG-DD), using single program multiple data (SPMD) and multiple instruction multiple data (MIMD) programming paradigm. There are two parallelization strategies proposed in this work, first strategy is a one-level parallelization strategy using geometric domain decomposition technique alone, second strategy is a two-level parallelization strategy that consists of a hybrid of both geometric domain decomposition and data decomposition techniques. Message-passing interface (MPI) and OpenMP standard are used to communicate data between processors and decompose loop iterations arrays, respectively. The parallel-multigrid code is used to simulate both steady and unsteady incompressible viscous flows over a circular cylinder and a lid-driven cavity flow. A maximum speedup of 22.5 could be achieved on 32 processors, for instance, the lid-driven cavity flow of Re = 1000. The results obtained agree well with numerical solutions obtained by other researchers as well as experimental measurements. A detailed study of the time step size and number of pseudo-sub-iterations per time step required for simulating unsteady flow are presented in this paper.     

Towards a three-dimensional moving body incompressible flow solver with a linear deformable model

In this study, a three-dimensional fluid–structured parallelized solver is extended from the previous work (Niu et al., 2009 [1]) for moving body simulations. Based on the unified Eulerian and Lagrangian coordinate transformations, the unsteady three-dimensional incompressible Navier–Stokes equations with artificial compressibility (Chorin, 1967 [2]) in a dual-time stepping approach are first derived. To implement unsteady flow calculations, the dual-time stepping strategy including the LU decomposition method is used in the pseudo-time iteration and the second-order accurate backward difference is adopted to discretize the unsteady flow terms. Also, a third-order Roe type flux limited splitting is derived to evaluate the spatial difference of the convective fluxes. The original FORTRAN code is converted to the MPI code and tested on a 64-CPU IBM SP2. The parallel strategy here is based on the partitions of all do-loops in the original FORTRAN code and transferring the calculations inside the do-loop into different CPUs. The partition of the do-loop can be applied on the innermost loop, only or the last two inner loops depending on two-dimensional or three-dimensional problems. This kind of the parallel data partition of the loops is independent of what kind of the explicit or implicit type numerical algorithm used. Therefore, the current parallel approach can take advantage of the MPI language fully to transfer data efficiently among CPUs even for solving the governing equation implicitly. The test results show that a significant reduction of computing time in running the model and a near-linear speed up rate is achieved up to 32 CPUs at IBM SP2. The speed up rate is as high as 31 for using 64 IBM SP2 processors The test shows efficient parallel processing to provide prompt simulation of 3D cavity, unsteady dropping airfoil and blood flows in an aortic tube with a linear elastic modeling of wall motion is included here.     

Toward advanced subgrid models for Lattice-Boltzmann-based Large-eddy simulation: Theoretical formulations

This paper addresses the issue of developing advanced subgrid model for large-eddy simulations (LES) of turbulent flows based on Lattice Boltzmann methods (LBM). Most of already existing subgrid closures used in LES-LBM are straightforward extensions of the most crude model developed within the Navier–Stokes equations, namely the Smagorinsky eddy-viscosity model. In a first part, it is shown how to obtain an improved eddy-viscosity subgrid model for LBM. The original implementation of the Inertial-Range Consistent Smagorinsky model proposed by Dong and Sagaut for the D3Q19 scheme is used as an illustration. In a second step, an original extension of the Approximate Deconvolution Method proposed by Adams and Stolz for Navier–Stokes simulation is proposed. This new LBM-LES approach does not rely on the eddy-viscosity concept and is written directly within the LBM framework. It is shown that it can be implemented thanks to a trivial modification of the existing LBM solvers for Direct Numerical Simulation.     

Energy Stability Analysis of Some Fully Discrete Numerical Schemes for Incompressible Navier–Stokes Equations on Staggered Grids

In this paper we consider the energy stability estimates for some fully discrete schemes which both consider time and spatial discretizations for the incompressible Navier–Stokes equations. We focus on three kinds of fully discrete schemes, i.e., the linear implicit scheme for time discretization with the finite difference method (FDM) on staggered grids for spatial discretization, pressure-correction schemes for time discretization with the FDM on staggered grids for the solutions of the decoupled velocity and pressure equations, and pressure-stabilization schemes for time discretization with the FDM on staggered grids for the solutions of the decoupled velocity and pressure equations. The energy stability estimates are obtained for the above each fully discrete scheme. The upwind scheme is used in the discretization of the convection term which plays an important role in the design of unconditionally stable discrete schemes. Numerical results are given to verify the theoretical analysis.     

Numerical predictions of backward-facing step flows in microchannels using extended Navier–Stokes equations

Flows in microchannels were successfully predicted, in the past, both analytically and numerically, employing the extended Navier–Stokes equations (ENSE). In ENSE, the self-diffusion transport of mass, together with the resulting momentum and heat transport, is taken into account properly and the same is omitted in the classical Navier–Stokes equations. The ENSE have been employed here to numerically predict backward-facing step flows in microchannels, and the predictions are summarized in this paper. The results obtained by employing ENSE are compared with the available literature data computed by both direct simulation Monte Carlo and slip-velocity-based simulations. The good agreement of the present results with those given in the literature evidently points out that the ENSE can be applied to gas flows through complex microchannel geometries.     

Numerical investigation of effectivity indices of space-time error indicators for Navier–Stokes equations

In this paper we propose two error indicators aimed at estimating the space discretization error and the time discretization error for the unsteady Navier–Stokes equations. We define a space error indicator for evaluating the quality of the mesh and a time error indicator for evaluating the time discretization error. Moreover, we verify the reliability of the estimations through numerical experiments and we propose an effective space-time adaptive strategy for the unsteady Navier–Stokes equations. Such technique is based on two residual-based error indicators that suitably drive the mesh and the timestep-length modifications. Adaptive simulations show that the presented strategy allows to obtain accurate solutions in efficient way.     

Topology optimization of steady and unsteady incompressible Navier–Stokes flows driven by body forces

This paper presents the topology optimization method for the steady and unsteady incompressible Navier–Stokes flows driven by body forces, which typically include the constant force (e.g. the gravity) and the centrifugal and Coriolis forces. In the topology optimization problem, the artificial friction force with design variable interpolated porosity is added into the Navier–Stokes equations as the conventional method, and the physical body forces in the Navier–Stokes equations are penalized using the power-law approach. The topology optimization problem is analyzed by the continuous adjoint method, and solved by the finite element method in conjunction with the gradient based approach. In the numerical examples, the topology optimization of the fluidic channel, mass distribution of the flow and local velocity control are presented for the flows driven by body forces. The numerical results demonstrate that the presented method achieves the topology optimization of the flows driven by body forces robustly.     

A parallel two-level method for simulating blood flows in branching arteries with the resistive boundary condition

Computer modeling of blood flows in the arteries is an important and very challenging problem. In order to understand, computationally, the sophisticated hemodynamics in the arteries, it is essential to couple the fluid flow and the elastic wall structure effectively and specify physiologically realistic boundary conditions. The computation is expensive and the parallel scalability of the solution algorithm is a key issue of the simulation. In this paper, we introduce and study a parallel two-level Newton–Krylov–Schwarz method for simulating blood flows in compliant branching arteries by using a fully coupled system of linear elasticity equation and incompressible Navier–Stokes equations with the resistive boundary condition. We first focus on the accuracy of the resistive boundary condition by comparing it with the standard pressure type boundary condition. We then show the parallel scalability results of the two-level approach obtained on a supercomputer with a large number of processors and on problems with millions of unknowns.     

A nonlinear POD-Galerkin reduced-order model for compressible flows taking into account rigid body motions

The construction of a nonlinear reduced-order model for fluid–structure interaction problems is investigated in this paper for unsteady compressible flows excited by the rigid body motion of a structure. The reduction is achieved by means of a Galerkin projection of the Navier–Stokes equations on the first POD modes resulting from the proper orthogonal decomposition. In the first part of the paper, the projection technique is carried out on a purely aerodynamic case in order (i) to validate an efficient iterative technique based on an updated QR decomposition to compute the POD modes, and (ii) to discuss the merits of different correction methods introduced to improve the long-term stability of the reduced-order model. The second and most original part of the paper deals with the construction of the reduced set of equations which arise from the projection of the compressible Navier–Stokes equations formulated in a suitable moving frame representing the rigid body motion. The expressions of the resulting non-autonomous terms appearing in the reduced-order model have also been optimized to reduce the computational costs.     

Kinetic-based numerical schemes for incompressible Navier–Stokes equations

Numerical schemes for incompressible Navier–Stokes equations based on low Mach number limits of kinetic equations are presented. Discretizations of the incompressible Navier–Stokes equations are derived based on discretizations of the Boltzmann equation and consideration for the low Mach number limit. In the incompressible Navier–Stokes limit the discretizations reduce to explicit high-order numerical schemes. Numerical results for several test cases and comparisons with other well-known approaches are also presented.     

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.     

Stabilization of the Spectral Element Method in Convection Dominated Flows by Recovery of Skew-Symmetry

We investigate stability properties of the spectral element method for advection dominated incompressible flows. In particular, properties of the widely used convective form of the nonlinear term are studied. We remark that problems which are usually associated with the nonlinearity of the governing Navier–Stokes equations also arise in linear scalar transport problems, which implicates advection rather than nonlinearity as a source of difficulty. Thus, errors arising from insufficient quadrature of the convective term, commonly referred to as ‘aliasing errors’, destroy the skew-symmetric properties of the convection operator. Recovery of skew-symmetry can be efficiently achieved by the use of over-integration. Moreover, we demonstrate that the stability problems are not simply connected to underresolution. We combine theory with analysis of the linear advection-diffusion equation in 2D and simulations of the incompressible Navier–Stokes equations in 2D of thin shear layers at a very high Reynolds number and in 3D of turbulent and transitional channel flow at moderate Reynolds number. For the Navier–Stokes equations, where the divergence-free constraint needs to be enforced iteratively to a certain accuracy, small divergence errors can be detrimental to the stability of the method and it is therefore advised to use additional stabilization (e.g. so-called filter-based stabilization, spectral vanishing viscosity or entropy viscosity) in order to assure a stable spectral element method.     

Computational simulation of model and full scale Class 8 trucks with drag reduction devices

Numerical solutions of the unsteady Reynolds-averaged Navier–Stokes equations using a parallel implicit flow solver are given to investigate unsteady aerodynamic flows affecting the fuel economy of Class 8 trucks. Both compressible and incompressible forms of the equations are solved using a finite-volume discretization for unstructured grids and using Riemann-based interfacial fluxes and characteristic-variable numerical boundary conditions. A preconditioned primitive-variable formulation is used for compressible solutions, and the incompressible solutions employ artificial compressibility. Detached eddy simulation (DES) versions of the one-equation Menter SAS and the two-equation k − ?/k − ω hybrid turbulence models are used. A fully nonlinear implicit backward-time approximation is solved using a parallel Newton-iterative algorithm with numerically computed flux Jacobians. Unsteady three-dimensional aerodynamic simulations with grids of 18–20 million points and 50,000 time steps are given for the Generic Conventional Model (GCM), a 1:8 scale tractor–trailer model that was tested in the NASA Ames 7 × 10 tunnel. Computed pressure coefficients and drag force are in good agreement with measurements for a zero-incidence case. Similar computations for a case with 10° yaw gave reasonable agreement for drag force, while the pressure distributions suggested the need for tighter grid resolution or possibly improved turbulence models. Unsteady incompressible flow simulations were performed for a modified full scale version of the GCM geometry to evaluate drag reduction devices. All of these simulations were performed with a moving ground plane and rotating rear wheels. A simulation with trailer base flaps is compared with drag reduction data from wind tunnels and track and road tests. A front spoiler and three mud-flap designs with modest drag reduction potential are also evaluated.