A novel immersed boundary procedure for flow and heat simulations with moving boundary

This article describes a novel immersed boundary procedure for computing the flow and heat transfer problems with moving and complex boundary. Although the immersed boundary techniques have been successfully implemented to these flow and heat simulations, a frequently encountered drawback of this method is the relatively low accuracy proximate to the boundary due to the spreading of forcing function or the interpolation scheme. In this study, we propose a moving-grid process under the arbitrary Lagrangian-Eulerian framework to reduce the numerical diffusion near the immersed boundary. The incompressible Navier-Stokes equations are discretized spatially using unstructured finite element method, and advanced temporally by an operator-splitting scheme. The methodology is validated by the simulations of flow induced by an oscillating cylinder in a free stream. The capability of the proposed method is further demonstrated by good predictions of flow passing the rotating fan in a channel and also flow driven by two independent rotating fans in a circular cavity.     

圆柱绕流的二维数值模拟和尾迹分析

为指导机械设计中参数和布局的选择,研究固定在水流中的圆柱结构件的受力情况和流场分布．利用FLUENT中的三种湍流模型对雷诺数为3900的圆柱绕流进行二维数值模拟并进行对比,得到升力因数、阻力因数、分离角、斯特劳哈尔数和涡街尺寸等参数的模拟结果,与参考文献中的实验结果对比验证二维模拟的预测精度．RKE（Realizable k-ε）和雷诺应力模型（Reynolds Stress Model,RSM）均能在此雷诺数下得出接近实验结果的流场,RSM模型使用POWER LAW离散格式的结果优于QUICK格式．与三维模拟的对比表明二维模拟适合在设计初期的快速估算,能够快速得到合适精度的模拟结果．     

A Transformation-free HOC Scheme for Incompressible Viscous Flow Past a Rotating and Translating Circular Cylinder

In the present work, a numerical study is made using a recently developed Higher Order Compact (HOC) finite difference scheme to test its capacity in capturing the very complex flow phenomenon of unsteady flow past a rotating and translating circular cylinder. The streamfunction-vorticity formulation of the Navier-stokes equations in cylindrical polar coordinate are considered as the governing equations. In the present investigation, flow is computed for a fixed Reynolds number (Re) 200 and rotational parameter values 0.5, 1.0, 2.07 and 3.25 are considered. Firstly, the flow patterns for different α values and for long time range are computed and qualitative comparisons are made with existing experimental and numerical results. Then, as a further check on the consistency of the experimental and present numerical results, quantitative comparisons are made for the velocity profiles at several locations. All these qualitative and quantitative comparisons show excellent agreements with existing experimental and numerical results.     

A 3rd order upwind finite volume method for generalised Newtonian fluid flows

The non-Newtonian effects in the flow of non-Newtonian fluids that can be modelled with generalised Newtonian constitutive equations are investigated using a numerical scheme based on the finite volume formulation. Collocated arrangement of variables is used and the pressure-correction method in conjunction with the SIMPLE scheme is applied. In order to avoid the diffusive effects of low order schemes, the approximation of the convection terms is carried out using the QUICK differencing scheme, which is of third order of accuracy. For modelling viscous non-Newtonian behaviour, the Power-Law, Quemada and the modified Bingham and Casson models are employed. Validation of the code is carried out upon comparison with numerical data available in the literature. Exploiting the lid-driven cavity flow a twofold investigation is carried out regarding the non-Newtonian effects first by using different models and secondly by using the shear-thinning and shear-thickening attributes of the fluid.     

A three-dimensional numerical model for incompressible two-phase flow of a granular bed submitted to a laminar shearing flow

A numerical model for the simulation of incompressible two-phase flows of a flat granular bed submitted to a laminar shearing flow is presented, considering a two-fluid model and a mixed-fluid one. The governing equations are discretized by a finite element method and a penalisation method is introduced to cope with the incompressibility constraint. A regularisation technique is used to deal with the visco-plastic behaviour of the granular phase. Validations are carried out on three flow test cases: a Bingham fluid between two infinite parallel planes, a Bingham fluid in a square lid-driven cavity and a Newtonian fluid over a granular bed in a two-dimensional configuration, for which we compare our numerical results with existing analytical or numerical results. The accuracy and efficiency of the numerical models have been compared for the two formulations of the two-phase flow model. It turns out that the two-fluid model requires ten times more CPU time than the mixed-fluid one for a comparable accuracy, which can be achieved provided one takes a smaller regularisation parameter in the latter model. Finally, three-dimensional computations are presented for the flow of a Newtonian fluid over a granular bed in a square and circular cross-section ducts.     

A pressure-correction method and its applications on an unstructured Chimera grid

In this paper, an unstructured Chimera mesh method is used to compute incompressible flow around a rotating body. To implement the pressure correction algorithm on unstructured overlapping sub-grids, a novel interpolation scheme for pressure correction is proposed. This indirect interpolation scheme can ensure a tight coupling of pressure between sub-domains. A moving-mesh finite volume approach is used to treat the rotating sub-domain and the governing equations are formulated in an inertial reference frame. Since the mesh that surrounds the rotating body undergoes only solid body rotation and the background mesh remains stationary, no mesh deformation is encountered in the computation. As a benefit from the utilization of an inertial frame, tensorial transformation for velocity is not needed. Three numerical simulations are successfully performed. They include flow over a fixed circular cylinder, flow over a rotating circular cylinder and flow over a rotating elliptic cylinder. These numerical examples demonstrate the capability of the current scheme in handling moving boundaries. The numerical results are in good agreement with experimental and computational data in literature.     

Chebyshev pseudospectral solution of the incompressible Navier-Stokes equations in curvilinear domains

A pseudospectral method for the calculation of 2-D flows of a viscous incompressible fluid in curvilinear domains is presented. The incompressible Navier-Stokes equations, expressed in terms of the primitive variables velocity and pressure, are solved in a non-orthogonal coordinate system. All the variables are expanded in double truncated series of Chebyshev polynomials. Time integration is performed by an implicit finite differences scheme for both the advective and diffusive terms. The pressure is calculated by the use of a truncated influence matrix involving all the collocation points in the field. A preconditioned iterative method is used to solve the system of linear equations resulting from the pseudospectral Chebyshev approximation. The algorithm is applied to the classical problem of the Green-Taylor vortices in order to check its accuracy; then 2 examples of viscous flow calculation are given in the case of a driven polar cavity and of a 2-D channel.     

An upwind compact difference scheme for solving the streamfunction–velocity formulation of the unsteady incompressible Navier–Stokes equation

In this paper, an upwind compact difference method with second-order accuracy both in space and time is proposed for the streamfunction–velocity formulation of the unsteady incompressible Navier–Stokes equations. The first derivatives of streamfunction (velocities) are discretized by two type compact schemes, viz. the third-order upwind compact schemes suggested with the characteristic of low dispersion error are used for the advection terms and the fourth-order symmetric compact scheme is employed for the biharmonic term. As a result, a five point constant coefficient second-order compact scheme is established, in which the computational stencils for streamfunction only require grid values at five points at both $\left(n\right)$th and $(n+1)$th time levels. The new scheme can suppress non-physical oscillations. Moreover, the unconditional stability of the scheme for the linear model is proved by means of the discrete von Neumann analysis. Four numerical experiments involving a test problem with the analytic solution, doubly periodic double shear layer flow problem, lid driven square cavity flow problem and two-sided non-facing lid driven square cavity flow problem are solved numerically to demonstrate the accuracy and efficiency of the newly proposed scheme. The present scheme not only shows the good numerical performance for the problems with sharp gradients, but also proves more effective than the existing second-order compact scheme of the streamfunction–velocity formulation in the aspect of computational cost.     

Implementation of density-based solver for all speeds in the framework of OpenFOAM

In the framework of open source CFD code OpenFOAM, a density-based solver for all speeds flow field is developed. In this solver the preconditioned all speeds AUSM+(P) scheme is adopted and the dual time scheme is implemented to complete the unsteady process. Parallel computation could be implemented to accelerate the solving process. Different interface reconstruction algorithms are implemented, and their accuracy with respect to convection is compared. Three benchmark tests of lid-driven cavity flow, flow crossing over a bump, and flow over a forward-facing step are presented to show the accuracy of the AUSM+(P) solver for low-speed incompressible flow, transonic flow, and supersonic/hypersonic flow. Firstly, for the lid driven cavity flow, the computational results obtained by different interface reconstruction algorithms are compared. It is indicated that the one dimensional reconstruction scheme adopted in this solver possesses high accuracy and the solver developed in this paper can effectively catch the features of low incompressible flow. Then via the test cases regarding the flow crossing over bump and over forward step, the ability to capture characteristics of the transonic and supersonic/hypersonic flows are confirmed. The forward-facing step proves to be the most challenging for the preconditioned solvers with and without the dual time scheme. Nonetheless, the solvers described in this paper reproduce the main features of this flow, including the evolution of the initial transient.     

A segregated-implicit scheme for solving the incompressible Navier-Stokes equations

The proposed segregated-implicit (SI) scheme, which is based on the artificial compressibility method, is discretized by the finite difference numerical scheme and verified by simulating a shear-driven cavity flow. The current results demonstrate that the SI scheme is a simple algorithm capable of fast solving the incompressible Navier-Stokes equations.     

Solution domain decomposition method for viscous incompressible flow

A solution domain decomposition method is developed for steady state solution of the biharmonic-based Navier-Stokes equations. It consists of a domain decomposition in conjunction with Chebyshev collocation for spatial discretization. The interactions between subdomains are effectively decoupled by means of a superposition of auxiliary solutions to yield a set of independent elementary problems which can be solved concurrently on multiprocessor computers. Assessments are carried out to a number of test problems including the two-dimensional steady flow in a driven square cavity. Illustrative examples indicate a good performance of the proposed methodology which does not affect the convergence and stability of the discretization scheme. Spectral accuracy is retained with absolute error decaying in an exponential fashion. The numerical solutions for the driven cavity compare favorably against previously published numerical results except for a slight overprediction in the vertical velocity component at Reynolds number of 400. TheC ³ continuity is speculated to be its cause.     

不可压缩流两重稳定有限体积算法应用研究

通过将局部高斯积分稳定化方法和两重网格算法思想紧密结合,提出了粘性不可压缩流体的两重稳定有限体积算法。将该算法的三种迭代格式进行了效率的分析比较。理论分析和数值实验发现：当粗、细网格尺度比例选择适当时,两重算法与传统算法具有相同精度解的同时,效率大大提高;对不同格式的两重有限体积算法进行比较分析发现：Simple格式计算效率最高,Picard格式次之,Newton格式较低。     

On some approaches to solve CFD problems

This paper presents two efficient methods for spatial flows calculations. In order to simulate of incompressible viscous flows, a second-order accurate scheme with an incomplete LU decomposed implicit operator is developed. The scheme is based on the method of artificial compressibility and Roe flux-difference splitting technique for the convective terms. The numerical algorithm can be used to compute both steady-state and time-dependent flow problems. The second method is developed for modeling of stationary compressible inviscid flows. This numerical algorithm is based on a simple flux-difference splitting into physical processes method and combines a multi level grid technology with a convergence acceleration procedure for internal iterations. The capabilities of the methods are illustrated by computations of steady-state flow in a rotary pump, unsteady flow over a circular cylinder and stationary subsonic flow over an ellipsoid.     

Modeling complex boundaries using an external force field on fixed Cartesian grids in large-eddy simulations

In the present study a methodology to perform large-eddy simulations around complex boundaries on fixed Cartesian grids is presented. A novel interpolation scheme which is applicable to boundaries of arbitrary shape, does not involve special treatments, and allows the accurate imposition of the desired boundary conditions is introduced. A method to overcome the problems associated with the computation of the subgrid scale terms near solid boundaries is also discussed. A detailed study on the accuracy and efficiency of the method is carried out for the cases of Stokes flow around a cylinder in the vicinity of a moving plate, the three-dimensional flow around a circular cylinder, and fully developed turbulent flow in a plane channel with a wavy wall. It is demonstrated that the method is second-order accurate, and that the solid boundaries are mimicked “exactly” on the Cartesian grid within the overall accuracy of the scheme. For all cases under consideration the results obtained are in very good agreement with analytical and numerical data.     

基于预处理法非定常低速流动的数值模拟

本文基于预处理法,结合双时间步法,建立了应用高效隐式时间步进LU-SGS算法求解非定常低速流动问题的数值模拟方法．对典型的方腔顶盖瞬时启动驱动、周期振荡顶盖驱动等非定常低速流动问题进行了数值计算．结果表明,所建立的数值方法对非定常低速流动问题有较高的计算效率,并能有效的克服低速流动问题的系统刚性问题．     

An immersed boundary technique for simulating complex flows with rigid boundary

A new immersed boundary (IB) technique for the simulation of flow interacting with solid boundary is presented. The present formulation employs a mixture of Eulerian and Lagrangian variables, where the solid boundary is represented by discrete Lagrangian markers embedding in and exerting forces to the Eulerian fluid domain. The interactions between the Lagrangian markers and the fluid variables are linked by a simple discretized delta function. The numerical integration is based on a second-order fractional step method under the staggered grid spatial framework. Based on the direct momentum forcing on the Eulerian grids, a new force formulation on the Lagrangian marker is proposed, which ensures the satisfaction of the no-slip boundary condition on the immersed boundary in the intermediate time step. This forcing procedure involves solving a banded linear system of equations whose unknowns consist of the boundary forces on the Lagrangian markers; thus, the order of the unknowns is one-dimensional lower than the fluid variables. Numerical experiments show that the stability limit is not altered by the proposed force formulation, though the second-order accuracy of the adopted numerical scheme is degraded to 1.5 order. Four different test problems are simulated using the present technique (rotating ring flow, lid-driven cavity and flows over a stationary cylinder and an in-line oscillating cylinder), and the results are compared with previous experimental and numerical results. The numerical evidences show the accuracy and the capability of the proposed method for solving complex geometry flow problems both with stationary and moving boundaries.     

An algorithm for velocity calculation in close proximity to body surfaces in panel methods

To model incompressible flow over a body of arbitrary geometry when using vortex methods, it is necessary to construct an irrotational field to impose the impermeability condition at the surface of the object. In order to achieve this impermeability, this paper uses a boundary integral equation based on the single-layer representation for the velocity potential. Specifically, we formulate this exterior Neumann problem in terms of a source/sink boundary integral equation. The solution to this integral equation is then coupled with an interpolation procedure which smoothes the transition between near-wall and interior regimes. We describe the numerical scheme embedding this strategy and discuss its accuracy and efficiency. For validation purposes, we consider the potential and vortical flow over a circular cylinder, for which an analytical solution and the commonly used method of images are available.     

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.     

Boundary domain integral method for high Reynolds viscous fluid flows in complex planar geometries

The article presents new developments in boundary domain integral method (BDIM) for computation of viscous fluid flows, governed by the Navier–Stokes equations. The BDIM algorithm uses velocity–vorticity formulation and is based on Poisson velocity equation for flow kinematics. This results in accurate determination of boundary vorticity values, a crucial step in constructing an accurate numerical algorithm for computation of flows in complex geometries, i.e. geometries with sharp corners. The domain velocity computations are done by the segmentation technique using large segments. After solving the kinematics equation the vorticity transport equation is solved using macro-element approach. This enables the use of macro-element based diffusion–convection fundamental solution, a key factor in assuring accuracy of computations for high Reynolds value laminar flows. The versatility and accuracy of the proposed numerical algorithm is shown for several test problems, including the standard driven cavity together with the driven cavity flow in an L shaped cavity and flow in a Z shaped channel. The values of Reynolds number reach 10,000 for driven cavity and 7500 for L shaped driven cavity, whereas the Z shaped channel flow is computed up to Re = 400. The comparison of computational results shows that the developed algorithm is capable of accurate resolution of flow fields in complex geometries.     

Numerical simulation of the regularized driven cavity flows at high Reynolds numbers

We present a numerical simulation of the unsteady incompressible flows in the unit cavity as well as in the rectangular cavity of aspect ratio 2 by using a second order projection scheme in time and a Chebyshev-Tau approximation for the space variables. The time-accurate discretization and the high accuracy of the spectral methods enable us to observe dynamical features not apparent in previous studies.