A space-time lower–upper symmetric Gauss–Seidel scheme for the time-spectral method

The time-spectral method (TSM) offers the advantage of increased order of accuracy compared to methods using finite-difference in time for periodic unsteady flow problems. Explicit Runge–Kutta pseudo-time marching and implicit schemes have been developed to solve iteratively the space-time coupled nonlinear equations resulting from TSM. Convergence of the explicit schemes is slow because of the stringent time-step limit. Many implicit methods have been developed for TSM. Their computational efficiency is, however, still limited in practice because of delayed implicit temporal coupling, multiple iterative loops, costly matrix operations, or lack of strong diagonal dominance of the implicit operator matrix. To overcome these shortcomings, an efficient space-time lower–upper symmetric Gauss–Seidel (ST-LU-SGS) implicit scheme with multigrid acceleration is presented. In this scheme, the implicit temporal coupling term is split as one additional dimension of space in the LU-SGS sweeps. To improve numerical stability for periodic flows with high frequency, a modification to the ST-LU-SGS scheme is proposed. Numerical results show that fast convergence is achieved using large or even infinite Courant–Friedrichs–Lewy (CFL) numbers for unsteady flow problems with moderately high frequency and with the use of moderately high numbers of time intervals. The ST-LU-SGS implicit scheme is also found to work well in calculating periodic flow problems where the frequency is not known a priori and needed to be determined by using a combined Fourier analysis and gradient-based search algorithm.     

Computation of three-dimensional flow using the Euler equations and a multiple-grid scheme

The unsteady Euler equations are numerically solved using the finite volume one-step scheme recently developed by Ron-Ho Ni. The multiple-grid procedure of Ni is also implemented. The flows are assumed to be homo-enthalpic; the energy equation is eliminated and the static pressure is determined by the steady Bernoulli equation; a local time-step technique is used. Inflow and outflow boundaries are treated with the compatibility relations method of ONERA. The efficiency of the multiple-grid scheme is demonstrated by a two-dimensional calculation (transonic flow past the NACA 12 aerofoil) and also by a three-dimensional one (transonic lifting flow past the M6 wing). The third application presented shows the ability of the method to compute the vortical flow around a delta wing with leading-edge separation. No condition is applied at the leading-edge; the vortex sheets are captured in the same sense as shock waves. Results indicate that the Euler equations method is well suited for the prediction of flows with shock waves and contact discontinuities, the multiple-grid procedure allowing a substantial reduction of the computational time.     

A study of thermal nonequilibrium effects in low-pressure wet-steam turbines using a blade-to-blade time-marching technique

A theoretical analysis of the wet-steam flow in the low pressure cylinder of a 500 MW steam turbine using a blade-to-blade time-marching computer program is described. The calculating method can compute most types of wet-steam flow found in LP turbines, including those involving both primary and secondary nucleations in transonic and supersonic blading with shock waves. In particular, condensing flows in highly staggered rotor tip sections can be computed without difficulty. Extensive results are presented showing the effect of departures from thermal equilibrium on the blade surface pressure and velocity distributions, the blade outlet relative flow angle, the mass flow coefficient and the thermodynamic loss coefficient. On the basis of the analysis, recommendations are made concerning the application of nonequilibrium wet-steam theory to steam turbine design.     

Analysis of High-order Explicit LES Dynamic Modeling Applied to Airfoil Flows

A high-order low dissipative numerical framework is discussed to tackle simultaneously the modeling of unresolved sub-grid scale flow turbulence and the capturing of shock waves. The flows around two different airfoil profiles are simulated using a Spectral Difference discretisation scheme. First, a transitional, almost incompressible, low Reynolds number flow over a Selig-Donovan 7003 airfoil. Second, a high Reynolds number flow over a RAE2822 airfoil under transonic conditions. These flows feature both laminar and turbulent flow physics and are thus particularly challenging for turbulence sub-grid scale modeling. The accuracy of the recently developed Spectral Element Dynamic Model, specifically capable of detecting spatial under-resolution in high-order flow simulations, is evaluated. Concerning the test in transonic conditions, the additional complexity due to the presence of shock waves has been handled using an artificial viscosity shock-capturing technique based on bulk viscosity. To mitigate the impact of the shock-capturing on turbulence dissipation, it was necessary to combine the high-order modal-type shock detection with a usual sensor measuring the local flow compressibility.

     

格子Boltzmann方法模拟多孔介质惯性流的边界条件改进

格子Boltzmann方法可以有效地模拟水动力学问题,边界处理方法的选择对于可靠的模拟计算至关重要.本文基于多松弛时间格子Boltzmann模型开展了不同边界条件下,周期对称性结构和不规则结构中流体流动模拟,阐述了不同边界条件的精度和适用范围. 此外,引入一种混合式边界处理方法来模拟多孔介质惯性流, 结果表明:对于周期性对称结构流动模拟,体力格式边界条件和压力边界处理方法是等效的,两者都能精确地捕捉流体流动特点; 而对于非周期性不规则结构,两种边界处理方法并不等价,体力格式边界条件只适用于周期性结构;由于广义化周期性边界条件忽略了垂直主流方向上流体与固体格点的碰撞作用,同样不适合处理不规则模型;体力-压力混合式边界格式能够用来模拟周期性或非周期性结构流体流动,在模拟多孔介质流体惯性流时,比压力边界条件有更大的应用优势,可以获得更大的雷诺数且能保证计算的准确性.      

An efficient approach for calculation of pitching moment in nonlinear reduced frequency method at low Mach number transonic flows

In this research, an efficient methodology for calculation of pitching moment coefficient at low Mach number transonic flows by using the perturbed nonlinear reduced frequency approach is presented. The proposed approach uses the perturbation technique in the nonlinear frequency domain (NLFD) method to estimate the solution at high harmonics. In this approach, the density and velocity fields at high harmonics are perturbed about those at low harmonics. Perturbing the density and velocity fields, the semi‐linear form of the governing equations is obtained. The resulting solution vector and spatial operator are then approximated by discrete form of Fourier transformation and governing equations are solved by using the pseudo‐spectral approach. Numerical results show that the proposed approach predicts good pitching moment coefficient at low Mach number transonic flows with up to 50% savings in computational time. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical solution of subsonic and transonic cascade flows

In this work a study of the application of the finite element method to transonic flows in axial turbomachines is undertaken. Solution techniques capable of accurately predicting flows from the incompressible regime up to the establishment of shocks in the transonic regime are presented. In the subsonic and shockless transonic regimes a local linearization method capable of very rapid convergence is used. In the full transonic regime the artificial compressibility method is employed to exclude downstream influences in the supersonic regions. The two approaches can be combined in a unified package and appropriate switches introduced to select the relevant method in any flow regime.     

跨音速翼型上的激波／边界层干扰自适应控制计算

在激波区使用自适应壁对跨音速翼型的激波／边界层的相互作用（干扰）进行控制，可改变机翼的气动性能，这种被动控制可通过在翼型的激波区开一凹腔，其上覆盖一弹性橡胶膜柔壁来，本文给出用Ｎａｖｉｅｒ－Ｓｔｏｋｅｒ方程数值模拟这一自适应控制翼型的跨音速粘性绕流，提出了一个适应于本特殊情况（物面边界局部地区在求解过程中有变化）的处理办法。并探讨了自适应柔壁对当代跨音速翼绕流的影响。     

High-order strand grid methods for shock turbulence interaction

ABSTRACT

In this work, we examine the flux correction method for three-dimensional transonic turbulent flows on strand grids. Building upon previous work, we treat flux derivatives along strands with high-order summation-by-parts operators and penalty-based boundary conditions. A finite-volume like limiting strategy is implemented in the flux correction algorithm in order to sharply capture shocks. To achieve turbulence closure in the Reynolds-Averaged Navier–Stokes equations, a robust version of the Spalart–Allmaras turbulence model is employed that accommodates negative values of the turbulence working variable. Validation studies are considered which demonstrate the flux correction method achieves a high degree of accuracy for turbulent shock interaction flows.     

A high‐order finite difference method for incompressible fluid turbulence simulations

A Hermitian–Fourier numerical method for solving the Navier–Stokes equations with one non‐homogeneous direction had been presented by Schiestel and Viazzo (Internat. J. Comput. Fluids 1995; 24 (6):739). In the present paper, an extension of the method is devised for solving problems with two non‐homogeneous directions. This extension is indeed not trivial since new algorithms will be necessary, in particular for pressure calculation. The method uses Hermitian finite differences in the non‐periodic directions whereas Fourier pseudo‐spectral developments are used in the remaining periodic direction. Pressure–velocity coupling is solved by a simplified Poisson equation for the pressure correction using direct method of solution that preserves Hermitian accuracy for pressure. The turbulent flow after a backward facing step has been used as a test case to show the capabilities of the method. The applications in view are mainly concerning the numerical simulation of turbulent and transitional flows. Copyright © 2003 John Wiley & Sons, Ltd.     

Performance Improvement of a Supercritical Airfoil by a Multi-point Optimized Shock Control Channel

A shock control channel (SCC) is a flow control method introduced here to control the shock wave/boundarylayer interaction (SWBLI) in order to reduce the resulting wave drag in transonic flows. An SCC transfers an appropriate amount of mass and momentum from downstream of the shock wave location to its upstream to decrease the pressure gradient across the shock wave and as a result the shock-wave strength is reduced. Here, a multi-point optimization method under a constant-lift-coefficient constraint is used to find the optimum design of the SCC. This flow control method is implemented on a RAE-2822 supercritical airfoil for a wide range of off-design transonic Mach numbers. The RANS flow equations are solved using Roe’s averages scheme and a gradient-based adjoint algorithm is used to find the optimum location and shape of the SCC. The solver is validated against experimental works studying effect of cavities in transonic airfoils. It is shown that the application of an SCC improves the average aerodynamic efficiency in a range of off-design conditions by 13.2% in comparison with the original airfoil. The SCC is shown to be an effective tool also for higher angle of attack at transonic flows. We have also studied the SWBLI and how the optimization algorithm makes the flow wave structure and interactions of the shock wave with the boundary layer favorable.     

The effect of an unsteady drag force on the structure of a non-equilibrium region behind a shock wave in a gas-particle mixture

For numerical analysis of shock wave propagation in gas-particle mixtures, drag coefficients of a sphere in steady flows are generally used. However, it is shown both experimentally and numerically that a shock loaded solid sphere experiences unsteady drag forces. The paper describes a model of unsteady drag force and its effect on the structure of the non-equilibrium region behind a shock front traveling in a dusty gas. The results are compared with those obtained by using a steady drag coefficient and are discussed. It is demonstrated that the large drag force at the early stage of the interaction between shock-wave induced flow and a solid particle affects the flow structure that is obtained with a steady drag force.      

A parallel pressure implicit splitting of operators algorithm applied to flows at all speeds

A parallel implementation of the pressure‐based implicit splitting of operators (PISO) method is described and applied to both compressible and incompressible flows. The treatment of variables at the interfaces between adjacent blocks is highlighted, and, for compressible flow, a straightforward method for the implicit handling of density is described. Steady state and oscillatory flow through a sudden expansion are considered at low speeds for both two‐ and three‐dimensional geometries. Extension of the incompressible method to compressible flow is assessed for subsonic, transonic and supersonic flow through a two‐dimensional bump. Although good accuracy is achieved in these high‐speed flows, including the automatic capturing of shock waves, the method is deemed unsuitable for simulating steady state high‐speed flows on fine grids due to the requirement of very small time steps. Copyright © 2001 John Wiley & Sons, Ltd.     

Algorithm for analysis of flows in ribbed annuli

Spectral methods for analyses of steady flows in annuli bounded by walls with either axi‐symmetric or longitudinal ribs are developed. The physical boundary conditions are enforced using the immersed boundary conditions concept. In the former case, the Stokes stream function is used to eliminate pressure and to reduce system of field equations to a single fourth‐order partial differential equation. The ribs are assumed to be periodic in the axial direction and this permits representation of the solution in terms of the Fourier expansion. In the latter case, the problem is reduced to the Laplace equations for the flow modifications that can be expressed in terms of the Fourier expansions. The modal functions, which are functions of the radial coordinate, are discretized using Chebyshev polynomials. The problem formulations are closed using either the fixed volume flow rate constraint or the fixed pressure gradient constraint. Various tests have been carried out in order to demonstrate the spectral accuracy of the discretizations, as well as the spectral accuracy of the enforcement of the flow boundary conditions at the ribbed walls using the immersed boundary conditions concept. Special linear solver that takes advantage of the matrix structure has been implemented in order to reduce computational time and memory requirements. It is shown that the algorithm has superior performance when one is interested in the analysis of a large number of geometries, as part of the coefficient matrix that corresponds to the field equation is always the same and one needs to change only the part of the matrix that corresponds to the boundary relations when changing geometry of the flow domain. Copyright © 2011 John Wiley & Sons, Ltd.     

SUBGRID VOLUMETRIC QUADRATURE ACCURACY FOR TRANSIENT COMPRESSIBLE FLOW PREDICTIONS

A finite volume–element formulation of the Navier–Stokes equations for compressible flows is applied to the transient shock tube problem. A second-order spatial quadrature for volumetric integration is studied because of its effects on the shock wave resolution and positioning. Low quadrature order is shown to produce solution anomalies in regions with a transonic character as well as poor predictions of shock wave propagation. The second-order volumetric quadrature includes the proper upstream and downstream solution behaviour and eliminates both the transonic and shock speed errors in the transient shock tube problem. © 1997 John Wiley & Sons, Ltd.     

基于滑移速度壁模型的复杂边界湍流大涡模拟

采用滑移速度壁模型实现了浸入边界方法与壁模型相结合的大涡模拟.本文首先分别采用平衡层模型和非平衡壁模型对周期山状流进行数值模拟,以考查在壁模型中考虑切向压力梯度的作用.数值结果表明,流场的压力对本文所采用的壁模型形式并不敏感,但是考虑切向压力梯度可以显著改进壁面摩擦力的计算结果,并且能够准确的预测强压力梯度区以及分离区内的流动平均统计特性.不考虑压力梯度效应的平衡层模型显著低估了壁面摩擦力的分布,同时无法准确预测分离区内的平均速度剖面.非平衡模型的修正项正比于切向压力梯度和壁面法向距离,因此在强压力梯度区或者网格较粗时,计算得到的平均压力和摩擦力分布以及流动的低阶统计量均与参考的实验和计算结果吻合.在此基础上,通过回转体绕流的大涡模拟考查了该方法用于模拟高雷诺数壁湍流的适用性,非平衡壁模型可以准确地捕捉流动的物理结构并较准确地预测其水动力学特性.结果表明,将浸入边界方法与非平衡滑移速度壁模型相结合的大涡模拟,有望成为数值模拟复杂边界高雷诺数壁湍流的工具.      

AN EXPLICIT FINITE VOLUME SPATIAL MARCHING METHOD FOR REDUCED NAVIER–STOKES EQUATIONS

This paper develops a spatial marching method for high-speed flows based on a finite volume approach. The method employs the reduced Navier– Stokes equations and a pressure splitting in the streamwise direction based on the Vigneron strategy. For marching from an upstream station to one downstream the modified five-level Runge–Kutta integration scheme due to Jameson and Schmidt is used. In addition, for shock handling and for good convergence properties the method employs a matrix form of the artificial dissipation terms, which has been shown to improve the accuracy of predictions. To achieve a fast rate of convergence, a local time-stepping concept is used. The method retains the time derivative in the governing equations and the solution at every spatial station is obtained in an iterative manner. The developed method is validated against two test cases: (a) supersonic flow past a flat plate; and (b) hypersonic flow past a compression corner involving a strong viscous–inviscid interaction. The computed wall pressure and wall heat transfer coefficients exhibit good general agreement with previous computations by other investigators and with experiments.     

Role of the momentum interpolation mechanism of the Roe scheme in shock instability

The shock instability phenomenon is a well‐known problem for hypersonic flow computation by the shock‐capturing Roe scheme. The pressure checkerboard is another well‐known problem for low‐Mach‐number flow computation. The momentum interpolation method (MIM) is necessary for low‐Mach‐number flows to suppress the pressure checkerboard problem, and the pressure‐difference‐driven modification for cell face velocity can be regarded as a version of the MIM by subdividing the numerical dissipation of the Roe scheme. In this paper, MIM has been discovered through analysis and numerical tests to have the most important function in shock instability. MIM should be completely removed for nonlinear flows. However, the unexpected MIM is activated on the cell face nearly parallel to the flow for the high‐Mach‐number flows or low‐Mach‐number cells in numerical shock. Therefore, MIM should be retained for low‐Mach‐number flows and be completely removed for high‐Mach‐number flows and low‐Mach‐number cells in numerical shock. For such conditions, two coefficients are designed on the basis of the local Mach number and a shock detector. Thereafter, the improved Roe scheme is proposed. This scheme considers the requirement of MIM for incompressible and compressible flows, and is validated for good performance of numerical tests. An acceptable result can also be obtained with only the Mach number coefficient for general practical computation. Therefore, the objective of decreasing rather than increasing numerical dissipation to cure shock instability can be achieved with simple modification. Moreover, the mechanism of shock instability has been profoundly understood, in which MIM plays the most important role, although it is not the only factor. Copyright © 2016 John Wiley & Sons, Ltd.     

An implicit high-order preconditioned flux reconstruction method for low-Mach-number flow simulation with dynamic meshes

A fully implicit high-order preconditioned flux reconstruction/correction procedure via reconstruction (FR/CPR) method is developed to solve the compressible Navier-Stokes equations at low Mach numbers. A dual-time stepping approach with the second-order backward differentiation formula (BDF2) is employed to ensure temporal accuracy for unsteady flow simulation. When dynamic meshes are used to handle moving/deforming domains, the geometric conservation law is implicitly enforced to eliminate errors due to the resolution discrepancy between BDF2 and the spatial FR/CPR discretization. The large linear system resulted from the spatial and temporal discretizations is tackled with the restarted generalized minimal residual solver in the PETSc (portable, extensible toolkit for scientific computation) library. Through several benchmark steady and unsteady numerical tests, the preconditioned FR/CPR methods have demonstrated good convergence and accuracy for simulating flows at low Mach numbers. The new flow solver is then used to study the effects of Mach number on unsteady force generation over a plunging airfoil when operating in low-Mach-number flows. It is observed that weak compressibility has a significant impact on thrust generation but has a negligible effect on lift generation of an oscillating airfoil.     

A hybrid pressure–density‐based algorithm for the Euler equations at all Mach number regimes

In the present work, we propose a reformulation of the fluxes and interpolation calculations in the PISO method, a well‐known pressure‐correction solver. This new reformulation introduces the AUSM⁺ ? up flux definition as a replacement for the standard Rhie and Chow method of obtaining fluxes and central interpolation of pressure at the control volume faces. This algorithm tries to compatibilize the good efficiency of a pressure based method for low Mach number applications with the advantages of AUSM⁺ ? up at high Mach number flows. The algorithm is carefully validated using exact solutions. Results for subsonic, transonic and supersonic axisymmetric flows in a nozzle are presented and compared with exact analytical solutions. Further, we also present and discuss subsonic, transonic and supersonic results for the well known bump test‐case. The code is also benchmarked against a very tough test‐case for the supersonic and hypersonic flow over a cylinder. Copyright © 2011 John Wiley & Sons, Ltd.