A grid deformation technique for unsteady flow computations

A grid deformation technique is presented here based on a transfinite interpolation algorithm applied to the grid displacements. The method, tested using a two‐dimensional flow solver that uses an implicit dual‐time method for the solution of the unsteady Euler equations on deforming grids, is applicable to problems with time varying geometries arising from aeroelasticity and free surface marine problems. The present work is placed into a multi‐block framework and fits into the development of a generally applicable parallel multi‐block flow solver. The effect of grid deformation is examined and comparison with rigidly rotated grids is made for a series of pitching aerofoil test cases selected from the AGARD aeroelastic configurations for the NACA0012 aerofoil. The effect of using a geometric conservation law is also examined. Finally, a demonstration test case for the Williams aerofoil with an oscillating flap is presented, showing the capability of the grid deformation technique. Copyright © 2000 John Wiley & Sons, Ltd.     

Two‐dimensional implicit time‐dependent calculations on adaptive unstructured meshes with time evolving boundaries

An implicit multigrid‐driven algorithm for two‐dimensional incompressible laminar viscous flows has been coupled with a solution adaptation method and a mesh movement method for boundary movement. Time‐dependent calculations are performed implicitly by regarding each time step as a steady‐state problem in pseudo‐time. The method of artificial compressibility is used to solve the flow equations. The solution mesh adaptation method performs local mesh refinement using an incremental Delaunay algorithm and mesh coarsening by means of edge collapse. Mesh movement is achieved by modeling the computational domain as an elastic solid and solving the equilibrium equations for the stress field. The solution adaptation method has been validated by comparison with experimental results and other computational results for low Reynolds number flow over a shedding circular cylinder. Preliminary validation of the mesh movement method has been demonstrated by a comparison with experimental results of an oscillating airfoil and with computational results for an oscillating cylinder. Copyright © 2005 John Wiley & Sons, Ltd.     

Fractional step method for solution of incompressible Navier-Stokes equations on unstructured triangular meshes

In this paper an implicit fractional step method for the solution of the two-dimensional, time-dependent, incompressible Navier-Stokes equations is presented. The current method was developed for use on an unstructured grid made up of triangles. The basic principles of this method are that the evaluation of the time evolution is split into intermediate steps and that for the spatial discretization of the flow equations a finite volume discretization on an unstructured triangular mesh is used. The present approach has been used to simulate viscous, laminar flows for various Reynolds numbers in test cases such as a backward-facing step, a square cavity and a channel with wavy boundaries.     

二维对流扩散方程的高精度全隐式多重网格方法

提出了数值求解二维非定常变系数对流扩散方程的一种时间二阶、空间四阶精度的三层全隐紧致差分格式。为了加快迭代求解隐格式时在每一个时间步上的收敛速度，采用多重网格加速技术，建立了适用于本文高精度金隐紧致格式的多重网格算法。数值实验结果验证了本文方法的精确性、稳定性和对高网格雷诺数问题的强适应性。     

SIMULATION OF THE FLOW OVER ELLIPTIC AIRFOILS OSCILLATING AT LARGE ANGLES OF ATTACK

The incompressible, viscous flow over two-dimensional elliptic airfoils oscillating in pitch at large angles of attack, such that flow separation occurs, has been simulated numerically for a Reynolds number of 3000. A vortex method is used to solve the two-dimensional Navier–Stokes equations in vorticity/stream-function form using a time-marching approach. Using an operator-splitting method the convection and diffusion equations are solved sequentially at each time step. The convection equation is solved using a vortex-in-cell method, and the diffusion equation using a second-order ADI finite-difference scheme. Elliptic profiles are obtained by mapping a circle in a computational domain into the physical domain using a Joukowski transformation. The effects of several parameters on the flow field are considered, such as: frequency of oscillation, mean angle of attack, location of pitch-axis and the thickness ratio of the ellipse. The results obtained are shown to compare favourably with available experimental results.     

Prediction of oscillating thick cambered aerofoil aerodynamics by a locally analytic method

A complete first-order model and locally analytic solution method are developed to analyse the effects of mean flow incidence and aerofoil camber and thickness on the incompressible aerodynamics of an oscillating aerofoil. This method incorporates analytic solutions, with the discrete algebraic equations which represent the differential flow field equations obtained from analytic solutions in individual grid elements. The velocity potential is separated into steady and unsteady harmonic parts, with the unsteady potential further decomposed into circulatory and non-circulatory components. These velocity potentials are individually described by Laplace equations. The steady velocity potential is independent of the unsteady flow field. However, the unsteady flow is coupled to the steady flow field through the boundary conditions on the oscillating aerofoil. A body-fitted computational grid is then utilized. Solutions for both the steady and the coupled unsteady flow fields are obtained by a locally analytic numerical method in which locally analytic solutions in individual grid elements are determined. The complete flow field solution is obtained by assembling these locally analytic solutions. This model and solution method are shown to accurately predict the Theodorsen oscillating flat plate classical solution. Locally analytic solutions for a series of Joukowski aerofoils demonstrate the strong coupling between the aerofoil unsteady and steady flow fields, i.e. the strong dependence of the oscillating aerofoil aerodynamics on the steady flow effects of mean flow incidence angle and aerofoil camber and thickness.     

Projection‐based variational multiscale method for incompressible Navier–Stokes equations in time‐dependent domains

A variational multiscale method for computations of incompressible Navier–Stokes equations in time‐dependent domains is presented. The proposed scheme is a three‐scale variational multiscale method with a projection‐based scale separation that uses an additional tensor valued space for the large scales. The resolved large and small scales are computed in a coupled way with the effects of unresolved scales confined to the resolved small scales. In particular, the Smagorinsky eddy viscosity model is used to model the effects of unresolved scales. The deforming domain is handled by the arbitrary Lagrangian–Eulerian approach and by using an elastic mesh update technique with a mesh‐dependent stiffness. Further, the choice of orthogonal finite element basis function for the resolved large scale leads to a computationally efficient scheme. Simulations of flow around a static beam attached to a square base, around an oscillating beam and around a plunging aerofoil are presented. Copyright © 2016 John Wiley & Sons, Ltd.     

Unsteady supersonic flow over an oscillatory aerofoil using a second-order TVD scheme

The unsteady flow over an oscillatory NACA0012 aerofoil has been simulated by the calculation with Euler equations. The equations are discretized by an implicit Euler in time, and a second-order space-accurate TVD scheme based on flux vector splitting with van Leer's limiter. Modified eigenvalues are proposed to overcome the slope discontinuities of split eigenvalues at Mach = 0·0 and ± 1·0, and to generate a bow shock in front of the aerofoil. A moving grid system around the aerofoil is generated by Sorenson's boundary fitted co-ordinates for each time step. The calculations have been done for two angles of attack θ = 5·0° sin (ωt) and θ = 3·0° + 3·0° sin (ωt) for the free-stream Mach numbers 2·0 and 3·0. The results show that pressure and Mach cells flow along characteristic lines. To examine unsteady effects, the responses of wall pressure and normal force coefficients are analysed by a Fourier series expansion.     

Size reduction methods for the implicit time-dependent simulation of micro–macro viscoelastic flow problems

Traditionally, micro–macro simulations have been performed using simple explicit time-marching algorithms, which lack the desirable stability of implicit methods. In this study, a fully implicit time integration scheme is presented and implemented for the first time for the solution of time-dependent complex flows using the Brownian Configuration Fields approach. Special techniques need to be applied to deal with the very large size of the resulting linear systems. A novel size-reduction scheme is used, allowing an independent treatment for each molecular field and suited to parallel hardware architecture. To illustrate the method, a selected number of applications using linear spring chains are presented and the results are compared with their corresponding closed form constitutive equation. The excellent agreement between the results demonstrates the feasibility of the proposed approach.     

Is the steady viscous incompressible two-dimensional flow over a backward-facing step at Re = 800 stable?

A detailed case study is made of one particular solution of the 2D incompressible Navier–Stokes equations. Careful mesh refinement studies were made using four different methods (and computer codes): (1) a high-order finite-element method solving the unsteady equations by time-marching; (2) a high-order finite-element method solving both the steady equations and the associated linear-stability problem; (3) a second-order finite difference method solving the unsteady equations in streamfunction form by time-marching; and (4) a spectral-element method solving the unsteady equations by time-marching. The unanimous conclusion is that the correct solution for flow over the backward-facing step at Re = 800 is steady—and it is stable, to both small and large perturbations.     

Delay of finite-span excrescence-induced transition using plasma-based control

Direct numerical simulations are carried out to explore the use of flow control that delays transition generated by excrescence on a plate-like geometry in subsonic flow. Both forward-facing and rearward-facing steps of small roughness heights are considered in the investigation. These are representative of joints and other surface imperfections on wing sections that disrupt laminar flow, thereby increasing skin friction and configuration drag. Unlike previous studies, the steps have a finite lateral extent, such that sharp edges occur in both the spanwise and streamwise directions, and provide a more realistic characterisation of misaligned panels in aerodynamic configurations. The effect of spanwise corners upon transition is examined, and dielectric barrier discharge plasma-based flow control is applied to delay transition and increase the extent of the laminar flow region. Solutions are obtained to the Navier– Stokes equations that were augmented by source terms used to represent body forces imparted by plasma actuators on the fluid. A simple phenomenological model provided these forces resulting from the electric field generated by the plasma. The numerical method is based upon a high-fidelity scheme and an implicit time-marching approach, on an overset mesh system that is used to represent the finite-span steps. Very small-amplitude numerical forcing is employed to generate perturbations, which are amplified by the geometric disturbances and result in transition, similar to the physical situation. Both continuous and pulsed operations of actuators are considered, and the effectiveness of the control is quantified. Transition with the forward-facing step is considerably exacerbated by the presence of a spanwise edge. Plasma control is minimally effective, even with the use of multiple actuators and increased applied force. For the rearward-facing step, transition is substantially delayed by plasma control with small force application.     

Conservative unsteady aerodynamic simulation of arbitrary boundary motion using structured and unstructured meshes in time

Simulation of unsteady fluid behaviour with arbitrary boundary motion or topological change remains restricted owing to mesh deformation limitations, and usually requires the application of special techniques using overlapping meshes, sliding planes, remeshing or immersed boundaries. This work presents the application of a spacetime interpretation of the fluid conservation laws that unifies meshes in space and time. This effectively replaces the problem of mesh deformation with the problem of mesh generation and, because connectivity is no longer restricted to being constant in time, any motion or topological change may be simulated. Examples are given applying the method to a piston, a pitching NACA0012 aerofoil, an appearing/disappearing object, a two‐dimensional store separation and a rotation case to validate and then demonstrate the capabilities of the method. Results for the pitching aerofoil case are compared with a conventional moving mesh unsteady computation and shown to be consistent, whereas the demonstration cases show qualitatively correct behaviour and illustrate the general nature of the technique. Copyright © 2011 John Wiley & Sons, Ltd.     

A partially implicit method for simulating viscous aerofoil flows

A partially implicit method for the unsteady compressible Navier-Stokes equations is developed. The method is based on an explicit treatment of streamwise fluxes and an implicit treatment of normal fluxes. This leads to a linear system which is generated by an efficient finite difference procedure and which is block pentadiagonal. The method is tested on a shock-induced oscillatory flow over an aerofoil. Parallel implementations of an explicit, fully implicit and partially implicit method are investigated.     

Computational study of unsteady turbulent flows around oscillating and ramping aerofoils

The aim of this work is to computationally investigate subsonic and transonic turbulent flows around oscillating and ramping aerofoils under dynamic‐stall conditions. The investigation is based on a high‐resolution Godunov‐type method and several turbulence closures. The Navier–Stokes and turbulence transport equations are solved in a strongly coupled fashion via an implicit‐unfactored scheme. We present results from several computations of flows around oscillating and ramping aerofoils at various conditions in order to (i) assess the accuracy of different turbulence models and (ii) contribute towards a better understanding of dynamic‐stall flows. The results show that the employed non‐linear eddy‐viscosity model generally improves the accuracy of the computations compared to linear models, but at low incidence angles the Spalart–Allmaras one‐equation model was found to provide adequate results. Further, the computations reveal strong similarities between laminar and high‐Reynolds number dynamic‐stall flows as well as between ramping and oscillating aerofoil cases. Investigation of the Mach number effects on dynamic‐stall reveals a delay of the stall angle within a range of Mach numbers. Investigation of the reduced frequency effects suggests the existence of an (almost) linear variation between pitch rate and stall angle, with higher slope at lower pitch rates. The pitch rate affects both the onset of dynamic‐stall as well as the evolution of the associated vortical structures. Copyright © 2003 John Wiley & Sons, Ltd.     

基于特征分裂改进的WENO格式与Roe格式对比研究

双曲性守恒方程组采用高阶、高分辨率的WENO格式时有两类分裂方法,即逐点分裂和特征分裂。本文基于后者,对特征分裂重构时强间断和接触间断位置出现的振荡情况进行研究,对重构变量加以改进,发现改进后的WENO格式克服了间断处的振荡,然后以LU-SGS为子迭代的双时间步法求解Euler方程,选用一维Sod、二维前台阶和双马赫反射算例,并与Roe格式计算结果进行对比,发现WENO格式分辨率更高,耗散更小。     

Assessment of some iterative methods for non-symmetric linear systems arising in computational fluid dynamics

Various tests have been carried out in order to compare the performances of several methods used to solve the non-symmetric linear systems of equations arising from implicit discretizations of CFD problems, namely the scalar advection-diffusion equation and the compressible Euler equations. The iterative schemes under consideration belong to three families of algorithms: relaxation (Jacobi and Gauss-Seidel), gradient and Newton methods. Two gradient methods have been selected: a Krylov subspace iteration method (GMRES) and a non-symmetric extension of the conjugate gradient method (CGS). Finally, a quasi-Newton method has also been considered (Broyden). The aim of this paper is to provide indications of which appears to be the most adequate method according to the particular circumstances as well as to discuss the implementation aspects of each scheme.     

High‐fidelity aerodynamic shape optimization of modern transport wing using efficient hierarchical parameterization

Aerodynamic shape optimization technology is presented, using an efficient domain element parameterization approach. This provides a method that allows geometries to be parameterized at various levels, ranging from gross three‐dimensional planform alterations to detailed local surface changes. Design parameters control the domain element point locations and, through efficient global interpolation functions, deform both the surface geometry and corresponding computational fluid dynamics volume mesh, in a fast, high quality, and robust fashion. This results in total independence from the mesh type (structured or unstructured), and optimization independence from the flow‐solver is achieved by obtaining gradient information for an advanced gradient‐based optimizer by finite‐differences. Hence, the optimization tool can be used in conjunction with any flow‐solver and/or mesh generator. Results have been presented recently for two‐dimensional aerofoil cases, and shown impressive results; drag reductions of up to 45% were demonstrated using only 22 active design parameters. This paper presents the extension of these methods to three dimensions, with results for highly constrained optimization of a modern aircraft wing in transonic cruise. The optimization uses combined global and local parameters, giving 388 design variables, and produces a shock‐free geometry with an 18% reduction in drag, with the added advantage of significantly reduced root moments. Copyright © 2009 John Wiley & Sons, Ltd.     

A matrix‐free preconditioned Newton/GMRES method for unsteady Navier–Stokes solutions

The unsteady compressible Reynolds‐averaged Navier–Stokes equations are discretized using the Osher approximate Riemann solver with fully implicit time stepping. The resulting non‐linear system at each time step is solved iteratively using a Newton/GMRES method. In the solution process, the Jacobian matrix–vector products are replaced by directional derivatives so that the evaluation and storage of the Jacobian matrix is removed from the procedure. An effective matrix‐free preconditioner is proposed to fully avoid matrix storage. Convergence rates, computational costs and computer memory requirements of the present method are compared with those of a matrix Newton/GMRES method, a four stage Runge–Kutta explicit method, and an approximate factorization sub‐iteration method. Effects of convergence tolerances for the GMRES linear solver on the convergence and the efficiency of the Newton iteration for the non‐linear system at each time step are analysed for both matrix‐free and matrix methods. Differences in the performance of the matrix‐free method for laminar and turbulent flows are highlighted and analysed. Unsteady turbulent Navier–Stokes solutions of pitching and combined translation–pitching aerofoil oscillations are presented for unsteady shock‐induced separation problems associated with the rotor blade flows of forward flying helicopters. Copyright © 2000 John Wiley & Sons, Ltd.     

High-order ALE method for the Navier–Stokes equations on a moving hybrid unstructured mesh using flux reconstruction method

The flux reconstruction (FR) formulation can unify several popular discontinuous basis high-order methods for fluid dynamics, including the discontinuous Galerkin method, in a simple, efficient form. An arbitrary Lagrangian–Eulerian (ALE) extension to the high-order FR scheme is developed here for moving mesh fluid flow problems. The ALE Navier–Stokes equations are derived by introducing a grid velocity. The conservation law are spatially discretised on hybrid unstructured meshes using Huynh’s scheme (Huynh 2007) on anisotropic elements (quadrilaterals) and using Correction Procedure via Reconstruction scheme on isotropic elements (triangles). The temporal discretisation uses both explicit and implicit treatments. The mesh movement is described by node positions given as a time series, instead of an analytical formula. The geometric conservation law is tested using free stream preservation problem. An isentropic vortex propagation test case is performed to show the high-order accuracy of the developed method on both moving and fixed hybrid meshes. Flow around an oscillating cylinder shows the capability of the method to solve moving boundary viscous flow problems, with the numeric method further verified by comparison of the result on a smoothly deforming mesh and a rigid moving mesh.