Conditional Fourier spectral method for direct numerical simulation of incompressible flows

A new method of treating incompressible flows with nonslip boundaries is proposed as an extension of the Fourier spectral method. This is characteristic in using the function subspace that is a hyperplane in the Fourier-transformed velocity space, prescribed by the boundary condition, as well as in taking the solenoidal field representation in the Fourier space so that the pressure term need not be involved in the main dynamics and then time-integration can simply be made by the high-order Runge-Kutta scheme. The method can be applied in a more complicated case with an active scalar. As examples, the flow transitions to turbulence in a channel and in a rectangular duct heated from below are treated.     

Improved numerical simulations of incompressible flows based on viscous/inviscid interaction procedures

Viscous/inviscid interaction procedures consist usually of coupling potential flow and boundary layer calculations. In this study, the interaction is modeled using Helmholtz-type velocity decomposition where the gradient of the potential is augmented with a correction accounting for the vorticity effects in the viscous layers. Different ways to calculate the rotational components are discussed and methods to systematically improve the model are studied. Numerical results are compared with standard Navier–Stokes calculations to justify the present approach.     

自顶向下聚集型代数多重网格预条件的健壮性与参数敏感性研究

本文针对自顶向下聚集型代数多重网格预条件,进行了健壮性与参数敏感性研究。对从各向同性与各向异性偏微分方程边值问题离散所得的多种稀疏线性方程组,首先对问题规模敏感性进行了研究,并与基于强连接的经典聚集型算法进行了系统比较,发现虽然对沿不同坐标轴具有强各向异性的问题,基于坐标分割的自顶向下聚集型算法不如基于强连接的经典聚集算法,但对其它所有情形,自顶向下聚集型算法都具有明显优势,特别是在采用Jacobi光滑时,优势更加显著。之后,对最粗网格层的分割数与每次每个子图进行分割时的分割数这两个参数进行了敏感性分析,发现在采用Jacobi光滑求解五点差分离散所得的稀疏线性方程组时,自顶向下聚集型算法对这两个参数存在敏感性,虽然大部分情形下,迭代次数比较稳定,但在少量几种情形下,迭代次数明显增加。而对从九点差分离散得到的稀疏线性方程组,以及在采用Gauss-Seidel光滑的情况下,算法对这两个参数的选取不再具有敏感性,迭代次数都比较稳定。综合分析表明,自顶向下聚集型代数多重网格预条件具有较好的健壮性,特别是在采用Gauss-Seidel光滑,或采用九点差分离散时,健壮性表现更加充分。     

Comparisons of direct simulations and analytical predictions for multi-plate flows

Computational solutions are described for planar flow past a number of plates arranged in sequence or near-sequence, closely aligned with a uniform free stream. Comparisons are then made with recent analytical predictions. Fair agreement is found for Reynolds numbers in the low hundreds and above.     

Parallel simulation of three-dimensional complex flows: Application to two-phase compressible flows and turbulent wakes

In this paper, we present parallel simulations of three-dimensional complex flows obtained on an ORIGIN 3800 computer and on homogeneous and heterogeneous (processors of different speeds and RAM) computational grids. The solver under consideration, which is representative of modern numerics used in industrial computational fluid dynamics (CFD) software, is based on a mixed element-volume method on unstructured tedrahedrisations. The parallelisation strategy combines mesh partitioning techniques, a message-passing programming model and an additive Schwarz algorithm. The parallelisation performances are analysed on a two-phase compressible flow and a turbulent flow past a square cylinder.     

Three-dimensional numerical simulations of viscoelastic flows – predictability and accuracy

In this paper, fully three-dimensional (3-D) numerical simulations of viscoelastic flows using an implicit finite volume method are discussed with the focus on the predictability and accuracy of the method. The viscoelastic flow problems involving the stress singularity, including plane stick–slip flow, the flow past a junction in a channel, and the 3-D edge flow, are used to test the ability of the method to predict the singularity features with accuracy. The accuracy of the numerical predictions is judged by comparing with the known asymptotic behaviour for Newtonian fluids and some viscoelastic fluids, and the investigations are extended to the viscoelastic cases with unknown singular behaviour. The Phan-Thien–Tanner (PTT) model, and in some cases, the upper-convected Maxwell (UCM) model, are used to describe viscoelastic fluids. The numerical results with mesh refinement show that the accuracy is quite satisfactory, especially for Newtonian flows. For viscoelastic flows, the asymptotic results for the flow around a re-entrant corner for the UCM as well as the PTT fluid are reproduced numerically. In the stick–slip flow, a Newtonian-like asymptotic behaviour is predicted for the UCM fluid. In edge flow, it is verified numerically that the kinematics are Newtonian for viscoelastic fluids described by models with a constant viscosity and a zero second normal stress difference. For viscoelastic fluids described by the models with a shear-thinning viscosity and zero second normal stress difference, the fluid behaves like a power-law fluid, and the difference from its Newtonian kinematics is localized in the region near the singularity, and to capture the asymptotic behaviour, a parameter-dependent mesh has to be used. With the 3-D simulations, it is confirmed that in edge flow, the flow around the edge could not be rectilinear, and some secondary flows on the plane normal to the primary flow direction are expected for viscoelastic fluids described by the models with a shear-dependent second normal stress difference, such as the full PTT model. The strength of the secondary flows will depend on the level of the departure of the second normal stress difference from a fixed constant multiple of viscosity of the fluid.     

Adjoint-based error estimation and grid adaptation for functional outputs: Application to two-dimensional, inviscid, incompressible flows

Adjoint-based error estimation and grid adaptive procedures are investigated for their robustness and effectiveness in improving the accuracy of functional outputs such as lift and drag. The adjoint error estimates relate the global error in the output function to the local residual errors in the flow solution via adjoint variables as weight functions. These error estimates are used as a correction to produce improved functional estimates. Based on this error correction procedure, two output-based grid adaptive approaches are implemented and compared. While both approaches strive to improve the accuracy of the computed output, the means by which the adaptation parameters are formed differ. The first approach strives to improve the computable error estimates by forming adaptation parameters based on the level of error in the computable error estimates. The second approach uses the computable error estimates as adaptation parameters. Grid adaptation is performed with h-refinement and results are presented for two-dimensional, inviscid, incompressible flows.     

Particle Monte Carlo and lattice-Boltzmann methods for simulations of gas-particle flows

A numerical solution concept is presented for simulating the transport and deposition to surfaces of discrete, small (nano-)particles. The motion of single particles is calculated from the Langevin equation by Lagrangian integration under consideration of different forces such as drag force, van der Waals forces, electrical Coulomb forces and not negligible for small particles, under stochastic diffusion (Brownian diffusion). This so-called particle Monte Carlo method enables the computation of macroscopic filter properties as well the detailed resolution of the structure of the deposited particles. The flow force and the external forces depend on solutions of continuum equations, as the Navier-Stokes equations for viscous, incompressible flows or a Laplace equation of the electrical potential. Solutions of the flow and potential fields are computed here using lattice-Boltzmann methods. Essential advantage of these methods are the easy and efficient treatment of three-dimensional complex geometries, given by filter geometries or particle covered surfaces. A number of numerical improvements, as grid refinement or boundary fitting, were developed for lattice-Boltzmann methods in previous studies and applied to the present problem. The interaction between the deposited particle layer and the fluid field or the external forces is included by recomputing of these fields with changed boundaries. A number of simulation results show the influence of different effects on the particle motion and deposition.     

A comparative numerical study between dissipative particle dynamics and smoothed particle hydrodynamics when applied to simple unsteady flows in microfluidics

Laminar flows through channels, pipes and between two coaxial cylinders are of significant practical interest because they often appear in a wide range of industrial, environmental, and biological processes. Discrete particle modeling has increasingly been used in recent years and in this study we examined two of these methods: dissipative particle dynamics (DPD) and smoothed particle hydrodynamics (SPH) method when applied to (a) time-dependent, plane Poiseuille flow and (b) flow between two coaxial cylinders at low Reynolds numbers. The two examples presented in this paper give insight into different features of the two discrete particle methods. It was found that both methods give results with high accuracy, but CPU time is much larger (of order 10²–10³ in the second example) for DPD than for SPH model. This difference is due to the fact that the number of time steps for the DPD model is much greater than for the SPH model (since thermal fluctuations are taken into account in the DPD model).     

Algorithmic and theoretical results on computation of incompressible viscous flows by finite element methods

Primitive variable as well as streamfunction-vorticity and pure streamfunction formulations are discussed. For the primitive variable case alternative choices of the viscous stress term are shown to produce natural boundary conditions which are well suited for matching to various far field conditions. For the other cases recent analytical results, including error estimates are described, and an optical algorithm for pressure recovery as well as treatment for multiply connected domains are given.     

Simulation of incompressible two-phase flows with large density differences employing lattice Boltzmann and level set methods

A hybrid lattice Boltzmann and level set method (LBLSM) for two-phase immiscible fluids with large density differences is proposed. The lattice Boltzmann method is used for calculating the velocities, the interface is captured by the level set function and the surface tension force is replaced by an equivalent force field. The method can be applied to simulate two-phase fluid flows with the density ratio up to 1000. In case of zero or known pressure gradient the method is completely explicit. In order to validate the method, several examples are solved and the results are in agreement with analytical or experimental results.     

Manifold method coupled velocity and pressure for Navier-Stokes equations and direct numerical solution of unsteady incompressible viscous flow

Numerical manifold method (NMM) application to direct numerical solution for unsteady incompressible viscous flow Navier-Stokes (N-S) equations was discussed in this paper, and numerical manifold schemes for N-S equations were derived based on Galerkin weighted residuals method as well. Mixed covers with linear polynomial function for velocity and constant function for pressure was employed in finite element cover system. The patch test demonstrated that mixed covers manifold elements meet the stability conditions and can be applied to solve N-S equations coupled velocity and pressure variables directly. The numerical schemes with mixed covers have also been proved to be unconditionally stable. As applications, mixed cover 4-node rectangular manifold element has been used to simulate the unsteady incompressible viscous flow in typical driven cavity and flow around a square cylinder in a horizontal channel. High accurate results obtained from much less calculational variables and very large time steps are in very good agreement with the compact finite difference solutions from very fine element meshes and very less time steps in references. Numerical tests illustrate that NMM is an effective and high order accurate numerical method for unsteady incompressible viscous flow N-S equations.     

Scalability of parallel spatial direct numerical simulations on intel hypercube and IBM SP1 and SP2

The implementation and performance of a parallel spatial direct numerical simulation (PSDNS) approach on the Intel iPSC/860 hypercube and IBM SP1 and SP2 parallel computers is documented. Spatially evolving disturbances associated with laminar-to-turbulent transition in boundary-layer flows are computed with the PSDNS code. The feasibility of using the PSDNS to perform transition studies on these computers is examined. The results indicate that PSDNS approach can effectively be parallelized on a distributed-memory parallel machine by remapping the distributed data structure during the course of the calculation. Scalability information is provided to estimate computational costs to match the actual costs relative to changes in the number of grid points. By increasing the number of processors, slower than linear speedups are achieved with optimized (machine-dependent library) routines. This slower than linear speedup results because the computational cost is dominated by FFT routine, which yields less than ideal speedups. By using appropriate compile options and optimized library routines on the SP1, the serial code achieves 52–56 Mflops on a single node of the SP1 (45 percent of theoretical peak performance). The actual performance of the PSDNS code on the SP1 is evaluated with a real world simulation that consists of 1.7 million grid points. One time step of this simulation is calculated on eight nodes of the SP1 in the same time as required by a Cray Y/MP supercomputer. For the same simulation, 32-nodes of the SP1 and SP2 are required to reach the performance of a Cray C-90. A 32 node SP1 (SP2) configuration is 2.9 (4.6) times faster than a Cray Y/MP for this simulation, while the hypercube is roughly 2 times slower than the Y/MP for this application.     

Application of numerical optimization methods to molecular docking on graphics processing units

This paper is devoted to analyzing numerical optimization methods for solving the problem of molecular docking. Some additional requirements for optimization methods that take into account certain architectural features of graphics processing units (GPUs) have been formulated. A promising optimization method for use on graphics processors has been selected, its implementation is described, and its efficiency and accuracy have been estimated.     

Families of third and fourth algebraic order trigonometrically fitted symplectic methods for the numerical integration of Hamiltonian systems

The numerical integration of Hamiltonian systems by symplectic and trigonometrically fitted (TF) symplectic method is considered in this work. We construct new trigonometrically fitted symplectic methods of third and fourth order. We apply our new methods as well as other existing methods to the numerical integration of the harmonic oscillator, the 2D harmonic oscillator with an integer frequency ratio and an orbit problem studied by Stiefel and Bettis.     

The analysis of turbulent,free-shear,and channel flows by the finite element method

The finite element method is utilised to solve two-equation hydrodynamic models of turbulent flow subject to a prescribed pressure gradient. The method is used to analyse fully developed flow in smooth-walled channels and the plane mixing layer. The results are compared with experiment and with the results obtained by finite difference methods.     

Sensitivity of turbulent Schmidt number and turbulence model to simulations of jets in crossflow

Environmental discharges have been traditionally designed by means of cost-intensive and time-consuming experimental studies. Some extensively validated models based on an integral approach have been often employed for water quality problems, as recommended by USEPA (i.e.: CORMIX). In this study, FLOW-3D is employed for a full 3D RANS modelling of two turbulent jet-to-crossflow cases, including free surface jet impingement. Results are compared to both physical modelling and CORMIX to better assess model performance. Turbulence measurements have been collected for a better understanding of turbulent diffusion's parameter sensitivity. Although both studied models are generally able to reproduce jet trajectory, jet separation downstream of the impingement has been reproduced only by RANS modelling. Additionally, concentrations are better reproduced by FLOW-3D when the proper turbulent Schmidt number is used. This study provides a recommendation on the selection of the turbulence model and the turbulent Schmidt number for future outfall structures design studies.