A dual-time method for two-dimensional unsteady incompressible flow calculations

A method for computing unsteady incompressible viscous flows on moving or deforming meshes is described. It uses a well-established time-marching finite-volume flow solver, developed for steady compressible flows past rigid bodies. Time-marching methods cannot be applied directly to incompressible flows because the governing equations are not hyperbolic. Such methods can be extended to steady incompressible flows using an artificial compressibility scheme. A time-accurate scheme for unsteady incompressible flows is achieved by using an implicit real-time discretization and a dual-time approach, which uses a technique similar to the artificial compressibility scheme. Results are presented for test cases on both fixed and deforming meshes. Experimental, numerical and theoretical data have been included for comparison where available and reasonable agreement has been achieved. © 1998 John Wiley & Sons, Ltd.     

An efficient parallel computation of unsteady incompressible viscous flow with elastic moving and compliant boundaries on unstructured grids

This paper presents the development and validation of a parallel unstructured‐grid fluid–structure interaction (FSI) solver for the simulation of unsteady incompressible viscous flow with long elastic moving and compliant boundaries. The Navier–Stokes solver on unstructured moving grid using the arbitrary Lagrangian Eulerian formulation is based on the artificial compressibility approach and a high‐order characteristics‐based finite‐volume scheme. Both unsteady flow and FSI are calculated with a matrix‐free implicit dual time‐stepping scheme. A membrane model has been formulated to study fluid flow in a channel with an elastic membrane wall and their interactions. This model can be employed to calculate arbitrary wall movement and variable tension along the membrane, together with a dynamic mesh method for large deformation of the flow field. The parallelization of the fluid–structure solver is achieved using the single program multiple data programming paradigm and message passing interface for communication of data. The parallel solver is used to simulate fluid flow in a two‐dimensional channel with and without moving membrane for validation and performance evaluation purposes. The speedups and parallel efficiencies obtained by this method are excellent, using up to 16 processors on a SGI Origin 2000 parallel computer. A maximum speedup of 23.14 could be achieved on 16 processors taking advantage of an improved handling of the membrane solver. The parallel results obtained are compared with those using serial code and they are found to be identical. Copyright © 2005 John Wiley & Sons, Ltd.     

Co-located pressure-correction formulations on unstructured 2-D grids

 By extending a Navier–Stokes solution method for structured grids (SGs), a pressure-correction, finite-volume formulation for the numerical solution of laminar, incompressible, 2-D flows on unstructured grids (UGs) with triangular elements has been deviced. Since a co-located storage arrangement for all of the flow variables is used, the velocity and pressure fields should be artificially coupled. This is achieved through the careful extension of the Pressure-Weighted Interpolation Method (PWIM), successfully used for SGs in the past. In the first part of the paper, the method formulation for UGs is analyzed. Then, the PWIM for UGs and the boundary conditions' implementation along solid walls are investigated, on the basis of two flow problems.     

The effect of the horizontal vibrations on natural heat transfer from an isothermal array of cylinders

A numerical investigation is carried out to study an unsteady laminar natural convection heat transfer caused by an array of isothermal oscillating circular cylinders. Under oscillating conditions, flow and thermal fields are categorized into a class of moving boundary problems. In this study, the moving interfaces between the fluid and cylinders have been considered. The numerical model used in the present paper, is based on a 2D Navier–Stokes momentum and energy equations for an incompressible flow solver on an unstructured grid. Discretization of the governing equations including continuity, momentum and energy equations is achieved through a finite element scheme based on characteristic based split algorithm using the arbitrary Lagrangian–Eulerian approach to satisfy boundary movement. Besides a dual time stepping method is employed to capture unsteady flow and thermal characteristics. The working fluid is designated a Prandtl number of 0.71(air) and assumed to be incompressible with constant physical properties. The radiation, viscous dissipation and pressure work are also assumed to be negligible throughout this investigation. Fluid flow and heat transfer characteristics are examined in the domain of the Rayleigh number, cylinders spacing, amplitude, and frequency of oscillations such that: 10³ ≤ Ra ≤ 10⁵, 2 ≤ s/d ≤ 4, 0.5 ≤ l ≤ 2, and 0.1 ≤ f ≤ 0.4. The obtained results reveal that increment of Rayleigh number and cylinders’ spacing augment the average Nusselt of each cylinder as well as higher oscillation amplitude and frequency. Moreover, it was found that horizontal vibration makes vortices appear in the left and right area of the cylinders. These vortices reduce heat transfer from two upper cylinders.     

Numerical simulation of the gas flow in a circuit-breaker

A computational methodology for the solution of unsteady two-dimensional/axisymmetric Euler equations within geometries with moving boundaries is presented. The flow simulation is carried out by applying a finite-volume method which makes use of a Lagrangian-Eulerian version of Roe's approximate Riemann solver. The domain discretization is handled via unstructured triangular grids. Grid adaptation is applied on the basis of geometric and physical requirements. The importance of the implicit treatment of the space conservation laws, based on geometric analysis, is evoked. The procedure for reconstructing Roe's method for moving meshes is described and validated. Finally, the ability of the method for the prediction of the transient flow in a circuit-breaker during its opening phase is illustrated.     

振荡压气机叶栅叶片表面非定常响应以及气弹稳定性分析

发展并验证了一种适用于叶轮机内部非定常跨音流动诱导的叶片气弹问题的高效、准确的数值模拟方法。采用有限体积的多块结构化网格形式,多重网格方法加速收敛,隐式的双时间步时间推进,Spalart-Allmaras（S-A）湍流模型求解非定常雷诺平均Navier-Stokes方程。通过气动弹性标准算例10,叶片在高亚音和跨音流动下做弯曲振动,分析了流动状态、折合频率以及叶片间相位角对叶片表面非定常气动力响应以及叶栅气弹稳定性的影响。分析结果表明激波在此跨音振荡压气机叶栅中起失稳作用,叶片间相位角对气弹稳定性的影响在高折合频率下被加强。     

An implicit–explicit solution method for electro‐osmotic flow through three‐dimensional micro‐channels

This paper presents a comprehensive finite‐element modelling approach to electro‐osmotic flows on unstructured meshes. The non‐linear equation governing the electric potential is solved using an iterative algorithm. The employed algorithm is based on a preconditioned GMRES scheme. The linear Laplace equation governing the external electric potential is solved using a standard pre‐conditioned conjugate gradient solver. The coupled fluid dynamics equations are solved using a fractional step‐based, fully explicit, artificial compressibility scheme. This combination of an implicit approach to the electric potential equations and an explicit discretization to the Navier–Stokes equations is one of the best ways of solving the coupled equations in a memory‐efficient manner. The local time‐stepping approach used in the solution of the fluid flow equations accelerates the solution to a steady state faster than by using a global time‐stepping approach. The fully explicit form and the fractional stages of the fluid dynamics equations make the system memory efficient and free of pressure instability. In addition to these advantages, the proposed method is suitable for use on both structured and unstructured meshes with a highly non‐uniform distribution of element sizes. The accuracy of the proposed procedure is demonstrated by solving a basic micro‐channel flow problem and comparing the results against an analytical solution. The comparisons show excellent agreement between the numerical and analytical data. In addition to the benchmark solution, we have also presented results for flow through a fully three‐dimensional rectangular channel to further demonstrate the application of the presented method. Copyright © 2007 John Wiley & Sons, Ltd.     

Unstructured mesh methods for the solution of the unsteady compressible flow equations with moving boundary components

A review of a procedure for the simulation of time-dependent, inviscid and turbulent viscous, compressible flows involving geometries that change in time is presented. The adopted discretization technique employs unstructured meshes and both explicit and implicit time-stepping schemes. A dual time-stepping procedure and an ALE formulation enable flows involving moving boundary components to be included. Techniques that have been developed to maintain the validity of the unstructured mesh and to allow for the capture of moving flow features are also reviewed. Using the in-house developed techniques, some examples are included to demonstrate the use of the approach for the simulation of a number of flows of practical industrial interest.     

Adaptive embedded unstructured grid methods

A simple embedded domain method for node‐based unstructured grid solvers is presented. The key modification of the original, edge‐based solver is to remove all geometry‐parameters (essentially the normals) belonging to edges cut by embedded surface faces. Several techniques to improve the treatment of boundary points close to the immersed surfaces are explored. Alternatively, higher‐order boundary conditions are achieved by duplicating crossed edges and their endpoints. Adaptive mesh refinement based on proximity to or the curvature of the embedded CSD surfaces is used to enhance the accuracy of the solution. User‐defined or automatic deactivation for the regions inside immersed solid bodies is employed to avoid unnecessary work. Several examples are included that show the viability of this approach for inviscid and viscous, compressible and incompressible, steady and unsteady flows, as well as coupled fluid–structure problems. Copyright © 2004 John Wiley & Sons, Ltd.     

An efficient numerical method for highly loaded transonic cascade flow

In this paper, an efficient numerical method for transonic viscous flow in a highly loaded turbine vane cascade, where the interaction of a shock wave and boundary layer often leads to very complicated flow phenomena, is developed. The numerical code, a modified implicit flux-vector-splitting solver of the Navier–Stokes equations (MIFVS), is extended to simulate such transonic cascade flow. A compressible low-Reynolds-number k–ɛ model, together with a transition-modified damping function, has been implemented into the MIFVS code. With this extended MIFVS solver, the main feature of transonic flow and shock and boundary-layer interactions in the highly loaded transonic turbine vane are efficiently predicted with satisfactory accuracy. The convergence rate is found to be three times faster than that of flux-vector-splitting (FVS) methods.     

Numerical analysis of the influence of the physical viscosity on the vortex heat transfer in laminar and turbulent flows around a heated plate with a shallow spherical hole

The influence of the Reynolds number on the patterns of separation laminar and turbulent flows around a plane plate with a shallow hole as well as on the thermal-hydraulic characteristics of this plate was analyzed on the basis of the solution of the differential mass- and energy-conservation equations, the Navier–Stokes equations, and the Reynolds equations closed with the use of the shear-stress transfer model by the factorized finite-volume method.     

Effects of Mesh Motion on the Stability and Convergence of ALE Based Formulations for Moving Boundary Flows

This paper investigates the effects of mesh motion on the stability of fluid-flow equations when written in an Arbitrary Lagrangian–Eulerian frame for solving moving boundary flow problems. Employing the advection-diffusion equation as a model problem we present a mathematical proof of the destabilizing effects induced by an arbitrary mesh motion on the stability and convergence of an otherwise stable scheme. We show that the satisfaction of the so-called geometric conservation laws is essential to the development of an identity that plays a crucial role in establishing stability. We explicitly show that the advection dominated case is susceptible to growth in error because of the motion of the computational grid. To retain the bound on the growth in error, the mesh motion techniques need to account for a domain based constraint that minimizes the relative mesh velocity. Analysis presented in this work can also be extended to the Navier–Stokes equations when written in an ALE frame for FSI problems.     

A fast integral approach for drag force calculation due to oscillatory slip stokes flows

In this paper, we describe an efficient numerical method for modelling oscillatory incompressible slip Stokes flows in three dimensions. The efficiency is achieved by employing an integral approach combined with an accelerated boundary‐element‐method (BEM) solver. First the integral representations for slip flows with two different slip models are formulated. The resulting integral equations are then solved using the BEM combined with the precorrected‐FFT accelerated technique. 3D numerical codes have been developed based on the method described above. These codes are then used to calculate the drag forces on oscillating objects immersed in an unbounded slip flow. Three objects are considered, namely a sphere, a pair of plates and a comb structure. The simulated drag forces on these objects obtained from the two slip models are compared. In the sphere case, the simulated results are also compared with the analytical solutions for both the steady‐state case and the no‐slip oscillatory case and are found to be in good agreement. In addition, qualitative comparison of the simulation results with the experimental results in the plate problem is also presented in this paper. Copyright © 2004 John Wiley & Sons, Ltd.     

Parallel edge-based solution of viscoplastic flows with the SUPG/PSPG formulation

A parallel edge-based solution of three dimensional viscoplastic flows governed by the steady Navier–Stokes equations is presented. The governing partial differential equations are discretized using the SUPG/PSPG stabilized finite element method on unstructured grids. The highly nonlinear algebraic system arising from the convective and material effects is solved by an inexact Newton-Krylov method. The locally linear Newton equations are solved by GMRES with nodal block diagonal preconditioner. Matrix-vector products within GMRES are computed edge-by-edge (EDE), diminishing flop counts and memory requirements. A comparison between EDE and element-by-element data structures is presented. The parallel computations were based in a message passing interface standard. Performance tests were carried out in representative three dimensional problems, the sudden expansion for power-law fluids and the flow of Bingham fluids in a lid-driven cavity. Results have shown that edge based schemes requires less CPU time and memory than element-based solutions.     

An implicit finite-element method for high-speed flows

A fast algorithm is presented for constructing continuous lines, made up of element sides, which pass once through each node of a general unstructured triangular mesh and which are generally aligned in prescribed directions. The lines are used as the basis of an adaptive fully implicit unstructured grid procedure for the solution of two-dimensional problems of steady compressible inviscid and laminar viscous high-speed flows, where the equation system is solved by line relaxation using a block tridiagonal equation solver. For three-dimensional laminar viscous simulations it is proposed to utilize an implicit/explicit finite-element formulation. In the vicinity of solid walls a grid exhibiting structure in the normal direction is employed while, away from this region, the grid will be totally unstructured. In the structured region, lines in the normal direction to the wall are readily identified, while lines in the surfaces parallel to the solid wall are constructed using the proposed two-dimensional procedure. The implicit algorithm is then used in the structured region and the equation solution is achieved via line relaxation. An explicit form of the solution algorithm is used elsewhere. To illustrate the performance of the proposed method, solutions are obtained for both transonic inviscid and transonic and hypersonic laminar viscous problems in two dimensions. The application of the proposed procedure to the solution of three-dimensional hypersonic laminar viscous flow over a double ellipsoid configuration is also described.     

A multiscale stabilized ALE formulation for incompressible flows with moving boundaries

This paper presents a variational multiscale stabilized finite element method for the incompressible Navier–Stokes equations. The formulation is written in an Arbitrary Lagrangian–Eulerian (ALE) frame to model problems with moving boundaries. The structure of the stabilization parameter is derived via the solution of the fine-scale problem that is furnished by the variational multiscale framework. The projection of the fine-scale solution onto the coarse-scale space leads to the new stabilized method. The formulation is integrated with a mesh moving scheme that adapts the computational grid to the evolving fluid boundaries and fluid-solid interfaces. Several test problems are presented to show the accuracy and stability of the new formulation.     

Cancellation of the Heating Piston Effect by Convective Enhancement of a Cooling Piston Effect

This work brings new insight to the question of the piston effect, which has been found to be the main cause of temperature equilibration in the vicinity of the liquid–vapor critical point under weightlessness conditions. The thermalization process of a near-critical fluid confined in a cavity and submitted to local heating is modeled with special emphasis on the role of gravity and boundary conditions. The solution of the unsteady Navier–Stokes equations written for a hypercom-pressible low-heat-diffusing van der Waals gas is obtained in a 2-D configuration by means of a finite-volume numerical code. Under Earth gravity conditions, the results show that the thermal plume rising from a heat source strongly decreases and rapidly cancels bulk fluid heating when it strikes the top thermo-stated wall. It is proved that convection does not prevent heat transfer by the piston effect but that it causes a sudden enhancement of the cooling piston effect generated at the thermostated top boundary, which leads to an early equilibrium between the cooling and heating piston effects.     

Effect of flapping kinematics on aerodynamic force of a flapping two-dimensional flat plate

Potential applications of flapping-wing micro-aerial vehicles (MAVs) have prompted enthusiasm among the engineers and researchers to understand the flow physics associated with flapping flight. An incompressible Navier–Stokes solver that is capable of handling flapping flight kind of moving boundary problem is developed. Arbitrary Lagrangian–Eulerian (ALE) method is used to handle the moving boundaries of the problem. The solver is validated with the results of problems like inline oscillation of a circular cylinder in still fluid and a flat plate rapidly accelerating at constant angle of attack. Numerical simulations of flapping flat plate mimicking the kinematics of those like insect wings are simulated, and the unsteady fluid dynamic phenomena that enhance the aerodynamic force are studied. The solution methodology provides the velocity field and pressure field details, which are used to derive the force coefficients and the vorticity field. Time history of force coefficients and vortical structures gives insight into the unsteady mechanism associated with the unsteady aerodynamic force production. The scope of the work is to develop a computational fluid dynamic (CFD) solver with the ALE method that is capable of handling moving boundary problems, and to understand the flow physics associated with the flapping-wing aerofoil kinematics and flow parameters on aerodynamic forces. Results show that delayed stall, wing–wake interaction and rotational effect are the important unsteady mechanisms that enhance the aerodynamic forces. Major contribution to the lift force is due to the presence of leading edge vortex in delayed stall mechanism.     

叶栅内非定常不可压流动的数值模拟

基于SMAC(SimplifiedMarkerandCell)方法推导出直接求解二维非定常、不可压N-S方程的隐式数值方法。求解的基本方程是任意曲线坐标系中以逆变速度为变量的N-S方程和椭圆型的压力Poisson方程。采用该方法,对二维叶栅非定常分离流场进行了数值模拟,叶栅表面压力的计算结果与试验结果相比比较吻合,从而验证了这种方法的可靠性。同时对叶栅非定常流场的流场结构和流动机理做了初步的探讨。在均匀来流和定常边界条件下,叶栅内部流动表现出强烈的非定常性;在小冲角和高雷诺数时,叶栅尾部产生类似卡门涡街的周期性流动。     

Comparison of methods for wall boundary conditions as applied to the calculation of turbulent heat exchange characteristics

Methods for setting and realizing wall boundary conditions numerically in calculating turbulent flows is considered. A method for realizing weak boundary conditions on the wall with discretization of Reynolds-averaged Navier–Stokes equations by the control volume approach is discussed. The results of calculations for a number of model problems obtained within the framework of different approaches to the near-wall modeling are compared to the data of the physical experiment and the available correlation dependences. The grid dependence of the solution in using the method of near-wall functions is compared to that in using weak boundary conditions.