ROBUST NUMERICAL METHODS FOR TRANSONIC FLOWS

In this paper, numerical methods for solving the transonic full potential equation are developed. The governing equation is discretized by a flux-biasing finite volume method. The resulting non-linear algebraic system is solved by using a continuation method with full Newton iteration. The continuation method is based on solving a highly ‘upstream-weighted’ discretization and then gradually reducing the upstream weighting. A general PCG-like sparse matrix iterative solver is used to solve the Jacobians at each non-linear step. Various types of incomplete LU (ILU) preconditioners and ordering techniques are compared. Numerical results are presented to demonstrate that these methods are efficient and robust for solving the transonic potential equation in the workstation computing environment. © 1997 by John Wiley & Sons, Ltd.     

Flow-induced vibrations and the Landau equation

The initial and general transient temporal behaviours of flow induced vibrations were studied particularly with regard to the extent with which self-excited, flow-induced vibrations can be described by the Landau equation.Three different cases are studied experimentally, using bodies with generic shape. The first case represents a bluff body: resembling a simplified section of the Tacoma bridge, which has a single torsional degree of freedom. The second case is a two-dimensional airfoil in transonic flow having a heave and a torsional degree of freedom. The third case is an elastic half-wing model, also investigated in transonic flow.It is shown that in all three cases, beyond the critical point and at small initial amplitude the temporal development of the oscillations up to the limit cycle i.e. the envelope of the measured time functions, agrees with the corresponding curve, given by the solution of the Landau equation. For the Tacoma section and the airfoil the same agreement was demonstrated for initial amplitudes much larger than that of the limit cycle. In addition for both last cases the bifurcation behaviour was investigated and the Landau constants were determined. Finally an elementary physical explanation for the instability phenomenon was given.     

Global convergence of inexact Newton methods for transonic flow

In computational fluid dynamics, non-linear differential equations are essential to represent important effects such as shock waves in transonic flow. Discretized versions of these non-linear equations are solved using iterative methods. In this paper an inexact Newton method using the GMRES algorithm of Saad and Schultz is examined in the context of the full potential equation of aerodynamics. In this setting, reliable and efficient convergence of Newton methods is difficult to achieve. A poor initial solution guess often leads to divergence or very slow convergence. This paper examines several possible solutions to these problems, including a standard local damping strategy for Newton's method and two continuation methods, one of which utilizes interpolation from a coarse grid solution to obtain the initial guess on a finer grid. It is shown that the continuation methods can be used to augment the local damping strategy to achieve convergence for difficult transonic flow problems. These include simple wings with shock waves as well as problems involving engine power effects. These latter cases are modelled using the assumption that each exhaust plume is isentropic but has a different total pressure and/or temperature than the freestream.     

基于自适应直角网格的二维全速势方程有限体积解法A finite volume method for 2D Full-Potential equation on adaptive Cartesian grids

为满足亚声速和跨声速飞机概念设计中快速气动计算的需求,研究和发展一种基于自适应直角网格的非线性全速势方程有限体积解法。要点如下。（1）在几何自适应直角网格的基础上,使用结合单元融合的网格切割算法处理物面边界,提出一种修正非贴体切割网格的方法。（2）采用隐式格式结合GM RES算法求解该非线性位流方程,针对流场的自适应来捕捉激波。（3）采用镜像法处理物面边界处的无穿透条件,并提出解析的方法来修正镜像单元的值。（4）针对直角网格的特点,提出在库塔线上插入库塔单元的方法施加库塔条件。NACA0012翼型绕流的算例结果表明,该方法用于亚声速和跨声速气动计算能得到令人满意的结果,且自动化程度高、收敛速度快。     

Far nonlinear field of a nonlifting airfoil for a generalised equation of transonic flow

The transonic flow equation [1] for plane unsteady irrotational idealgas flows is extended to the case of subsonic, transonic or supersonic flows in a region with an almost constant value of the velocity using orthogonal flow coordinates (family of equipotential lines and streamlines). A solution for the nonlinear far field of steady transonic flow past an airfoil has been obtained for the transonic equation [2]. In this paper it is obtained for a generalized transonic equation and its asymptotic expansion is given. In using difference methods of calculating the flow past an airfoil in the transonic regime a knowledge of the nonlinear field makes it possible to reduce the dimensions of the calculation region (near field) as compared with the region determined by the far field of the linear theory.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 87–91, January–February, 1986.     

Adaptive grids in numerical fluid dynamics

A solution adaptive grid (SAG) method which redistributes the nodal points of a function according to its curvature is presented. A single, user-selected step parameter, P, is available for controlling the maximum step size, allowing the application of the technique to a wide variety of problems. Three test cases are cited: (1) the 1-dimensional inviscid Burgers equation, (2) the Falkner-Skan equation and (3) the finite-volume form of the Navier-Stokes equations for transonic aerofoil flows. In all three cases, significant solution improvement in terms of accuracy and convergence acceleration were achieved.     

Invariance groups and reduction of the unsteady transonic small disturbance equation

A geometric approach is undertaken towards the solution of the unsteady transonic small disturbance equation describing the low frequency flow field about a thin airfoil. From group properties given in Anderson and Ibragimov, Lie-Bäcklund Transformations in Applications (1979) we derive three invariance groups for the equation. Based on these groups a reduction of the equation is performed. The reduction leads to a steady equation and to an ordinary differential equation from which group invariant solutions of the unsteady transonic small disturbance equation can be obtained.     

Existence and Stability of Multidimensional Transonic Flows through an Infinite Nozzle of Arbitrary Cross-Sections

We establish the existence and stability of multidimensional steady transonic flows with transonic shocks through an infinite nozzle of arbitrary cross-sections, including a slowly varying de Laval nozzle. The transonic flow is governed by the inviscid potential flow equation with supersonic upstream flow at the entrance, uniform subsonic downstream flow at the exit at infinity, and the slip boundary condition on the nozzle boundary. Our results indicate that, if the supersonic upstream flow at the entrance is sufficiently close to a uniform flow, there exists a solution that consists of a C ^1,α subsonic flow in the unbounded downstream region, converging to a uniform velocity state at infinity, and a C ^1,α multidimensional transonic shock separating the subsonic flow from the supersonic upstream flow; the uniform velocity state at the exit at infinity in the downstream direction is uniquely determined by the supersonic upstream flow; and the shock is orthogonal to the nozzle boundary at every point of their intersection. In order to construct such a transonic flow, we reformulate the multidimensional transonic nozzle problem into a free boundary problem for the subsonic phase, in which the equation is elliptic and the free boundary is a transonic shock. The free boundary conditions are determined by the Rankine–Hugoniot conditions along the shock. We further develop a nonlinear iteration approach and employ its advantages to deal with such a free boundary problem in the unbounded domain. We also prove that the transonic flow with a transonic shock is unique and stable with respect to the nozzle boundary and the smooth supersonic upstream flow at the entrance.     

Laval nozzle flow calculation

The inverse problem of the theory of the Laval nozzle is considered, which leads to the Cauchy problem for the gasdynamic equations; the streamlines and the flow parameters are found from the known velocity distribution on the axis of symmetry.The inverse problem of Laval nozzle theory was considered in 1908 by Meyer [1], who expanded the velocity potential into a series in powers of the Cartesian coordinates and constructed the subsonic and supersonic solutions in the vicinity of the center of the nozzle. Taylor [2] used a similar method to construct a flowfield which is subsonic but has local supersonic zones in the vicinity of the minimal section. Frankl [3] and Fal'kovich [4] studied the flow in the vicinity of the nozzle center in the hodograph plane. Their solution, just as the Meyer solution, made it possible to obtain an idea of the structure of the transonic flow in the vicinity of the center of the nozzle.A large number of studies on transonic flow in the vicinity of the center of the nozzle have been made using the method of small perturbations. The approximate equation for the transonic velocity potential in the physical plane, obtained in [3–6], has been studied in detail for the plane and axisymmetric cases. In [7] Ryzhov used this equation to study the question of the formation of shock waves in the vicinity of the center of the nozzle, and conditions were formulated for the plane and axisymmetric cases under which the flow will not contain shock waves. However, none of the solutions listed above for the inverse problem of Laval nozzle theory makes it possible to calculate the flow in the subsonic and transonic parts of the nozzles with large gradients of the gasdynamic parameters along the normal to the axis of symmetry.Among the studies devoted to the numerical calculation of the flow in the subsonic portion of the Laval nozzle we should note the study of Alikhashkin et al., and the work of Favorskii [9], in which the method of integral relations was used to solve the direct problem for the plane and axisymmetric cases.The present paper provides a numerical solution of the inverse problem of Laval nozzle theory. A stable difference scheme is presented which permits analysis with a high degree of accuracy of the subsonic, transonic, and supersonic flow regions. The result of the calculations is a series of nozzles with rectilinear and curvilinear transition surfaces in which the flow is significantly different from the one-dimensional flow. The flowfield in the subsonic and transonic portions of the nozzles is studied. Several asymptotic solutions are obtained and a comparison is made of these solutions with the numerical solution.The author wishes to thank G. D. Vladimirov for compiling the large number of programs and carrying out the calculations on the M-20 computer.     

Implicit schemes for unsteady Euler equations on triangular meshes

An implicit finite element method is presented for the solution of steady and unsteady inviscid compressible flows on triangular meshes under transonic conditions. The method involves a first-order time-stepping scheme with a finite element discretization that reduces to central differencing on a rectangular mesh. On a solid wall the slip condition is prescribed and the pressure is obtained from an approximation of the normal momentum equation. With this solver no artificial viscosity is added to ensure the success of the calculation. Numerical examples are given for steady and unsteady cases.     

Finite volume method for three-dimensional transonic potential flow through turbomachinery blade rows

A finite-volume method has been developed for the calculation of transonic, potential flows through 3-D turbomachinery blades with complex geometries. The exact transonic potential flow equation is solved on a mesh constructed from small volume elements. A transformation is introduced through which cuboids of the physical plane are mapped into computational cubes. Two sets of overlapping volumes are used. While the thermodynamic properties are calculated at the primary volume centres, the flux balance is established on the secondary volumes. For transonic flows an artificial compressibility term (upwind density gradient) is added to density to produce the necessary directional bias in the hyperbolic region. The successive point over-relaxation Gauss-Seidel method has been used to solve the non-linear partial differential equations. Comparisons with experiments and/or other numerical solutions for various turbomachinery configurations show that the 3-D finite-volume approach is a relatively accurate, reliable and fast method for inviscid, transonic flow predictions through turbomachinery blade rows     

Multiple solutions and stability of the steady transonic small-disturbance equation

Numerical solutions of the steady transonic small-disturbance(TSD) potential equation are computed using the conservative Murman-Cole scheme. Multiple solutions are discovered and mapped out for the Mach number range at zero angle of attack and the angle of attack range at Mach number 0.85 for the NACA 0012 airfoil. We present a linear stability analysis method by directly assembling and evaluating the Jacobian matrix of the nonlinear finite-difference equation of the TSD equation. The stability of all the discovered multiple solutions are then determined by the proposed eigen analysis. The relation of stability to convergence of the iterative method for solving the TSD equation is discussed. Computations and the stability analysis demonstrate the possibility of eliminating the multiple solutions and stabilizing the remaining unique solution by adding a sufficiently long splitter plate downstream the airfoil trailing edge. Finally, instability of the solution of the TSD equation is shown to be closely connected to the onset of transonic buffet by comparing with experimental data.     

Study of modeling unsteady blade row interaction in a transonic compressor stage part 1: code development and deterministic correlation analysis

The average-passage equation system (APES) provides a rigorous mathematical framework for accounting for the unsteady blade row interaction through multistage compressors in steady state environment by introducing deterministic correlations (DC) that need to be modeled to close the equation system.The primary purpose of this study is to provide insight into the DC characteristics and the influence of DC on the time-averaged flow field of the APES.In Part 1 of this two-part paper,firstly a 3D viscous unsteady and time-averaging flow CFD solver is developed to investigate the APES technique.Then steady and unsteady simulations are conducted in a transonic compressor stage.The results from both simulations are compared to highlight the significance of the unsteady interactions.Furthermore,the distribution characteristics of DC are studied and the DC at the rotor/stator interface are compared with their spatial correlations (SC).Lastly,steady and time-averaging (employing APES with DC) simulations for the downstream stator alone are conducted employing DC derived from the unsteady results.The results from steady and time-averaging simulations are compared with the time-averaged unsteady results.The comparisons demonstrate that the simulation employing APES with DC can reproduce the time-averaged field and the 3D viscous time-averaging flow solver is validated.     

A hybrid pressure‐based solver for nonideal single‐phase fluid flows at all speeds

This paper describes the implementation of a numerical solver that is capable of simulating compressible flows of nonideal single‐phase fluids. The proposed method can be applied to arbitrary equations of state and is suitable for all Mach numbers. The pressure‐based solver uses the operator‐splitting technique and is based on the PISO/SIMPLE algorithm: the density, velocity, and temperature fields are predicted by solving the linearized versions of the balance equations using the convective fluxes from the previous iteration or time step. The overall mass continuity is ensured by solving the pressure equation derived from the continuity equation, the momentum equation, and the equation of state. Nonphysical oscillations of the numerical solution near discontinuities are damped using the Kurganov‐Tadmor/Kurganov‐Noelle‐Petrova (KT/KNP) scheme for convective fluxes. The solver was validated using different test cases, where analytical and/or numerical solutions are present or can be derived: (1) A convergent‐divergent nozzle with three different operating conditions; (2) the Riemann problem for the Peng‐Robinson equation of state; (3) the Riemann problem for the covolume equation of state; (4) the development of a laminar velocity profile in a circular pipe (also known as Poiseuille flow); (5) a laminar flow over a circular cylinder; (6) a subsonic flow over a backward‐facing step at low Reynolds numbers; (7) a transonic flow over the RAE 2822 airfoil; and (8) a supersonic flow around a blunt cylinder‐flare model. The spatial approximation order of the scheme is second order. The mesh convergence of the numerical solution was achieved for all cases. The accuracy order for highly compressible flows with discontinuities is close to first order and, for incompressible viscous flows, it is close to second order. The proposed solver is named rhoPimpleCentralFoam and is implemented in the open‐source CFD library OpenFOAM^®. For high speed flows, it shows a similar behavior as the KT/KNP schemes (implemented as rhoCentralFoam‐solver, Int. J. Numer. Meth. Fluids 2010), and for flows with small Mach numbers, it behaves like solvers that are based on the PISO/SIMPLE algorithm.     

An improved nonlinear reduced-order modeling for transonic aeroelastic systems

In this study, an improved nonlinear reduced-order model composed of a linear part and a nonlinear part is explored for transonic aeroelastic systems. The linear part is identified via the eigensystem realization algorithm and the nonlinear part is obtained via the Levenberg–Marquardt algorithm. The impulsive signal is chosen as the training signal for the linear part and the sinusoidal signal is used to determine the order of the linear part. The training signal for the nonlinear part is selected as the filtered white Gaussian noise with the maximal amplitude and frequency range to be designed via the aeroelastic responses. An NACA64A010 airfoil and an NACA0012 airfoil are taken as illustrative examples to demonstrate the performance of the presented reduced-order model in modeling transonic aerodynamic forces. The aeroelastic behaviors of the two airfoils are obtained via computational fluid dynamics to solve the Euler equation and the Navier–Stokes equation, respectively. The numerical results demonstrate that the presented reduced-order model can successfully predict the nonlinear aerodynamic forces with and without viscous flows. Moreover, the presented reduced-order model is capable of capturing the flutter velocity and modeling complex aeroelastic behaviors, including limit-cycle oscillations, beat phenomena and nodal-shaped oscillations at the transonic Mach numbers with high accuracy.     

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。     

A finite volume method for two-dimensional transonic potential flow through turbomachinery blade rows

To predict inviscid transonic flow through turbomachinery blade rows, the exact transonic potential flow equation is solved on a mesh constructed from small area elements. A transformation is introduced through which distorted squares of the physical plane are mapped into computational squares. Two sets of overlapping elements are used; while the thermodynamic properties are calculated at the primary element centres, the flux balance is established on the secondary elements. For transonic flows an artificial compressibility term (upwind density gradient) is added to density in order to produce the desired directional bias in the hyperbolic region. while the entropy does not increase across mass conservative shock jump regions. Comparisons withexperiments and with other numerical and analytical solutions for various turbomachinery configurations show that this approach is comparatively accurate, reliable, and fast.     

HyperFLOW软件非结构网格亚跨声速湍流模拟的验证与确认

计算流体力学(computational fluid dynamics,CFD)数值模拟在航空航天等领域发挥越来越重要的作用,然而CFD数值模拟结果的可信度仍然需要通过不断地验证与确认来提高.本文给出了从制造解精度测试、简单到复杂外形湍流模拟网格收敛性研究等三个方面开展CFD软件验证与确认的方法,并对自主研发的CFD软件平台HyperFLOW在非结构网格上模拟亚跨声速湍流问题的能力进行了验证与确认.首先通过基于Euler方程和标量扩散方程的制造解精度测试,分别验证了HyperFLOW在非结构网格上对Euler方程和黏性项的求解精度,结果表明其能够在任意非结构网格上达到设计的二阶精度. 其次,通过NASATurbulence Modeling Resource中的湍流平板、二维翼型近尾迹流动、二维Bump等几个典型的亚声速湍流算例的网格收敛性研究,量化考察了数值结果的观测精度阶和网格收敛性指数,并与国外知名CFD解算器CFL3D,FUN3D的计算结果进行了对比,验证了HyperFLOW对简单湍流问题的模拟能力,且具有良好的网格收敛性和计算精度(阶). 最后,通过NASA CommonResearchModel标模定升力系数的网格收敛性研究和升阻极曲线预测,验证了软件在复杂外形亚跨声速湍流流动数值模拟中也具有良好的可信度.      

Conservative calculations of non-isentropic transonic flows

The classical potential formulation of inviscid transonic flows is modified to account for non-isentropic effects. The density is determined in terms of the speed as well as the pressure, which in turn is calculated from a second-order mixed-type equation derived via differentiating the momentum equations. The present model differs in general from the exact inviscid Euler equations since the flow is assumed irrotational. On the other hand, since the shocks are not isentropic, they are weaker and are placed further upstream compared to the classical potential solution. Furthermore, the streamline leaving the aerofoil does not necessarily bisect the trailing edge. Results for the present conservative calculations are presented for non-lifting and lifting aerofoils at subsonic and transonic speeds and compared to potential and Euler solutions.     

REDUCED-ORDER MODELS OF UNSTEADY TRANSONIC VISCOUS FLOWS IN TURBOMACHINERY

The proper orthogonal decomposition (POD) technique is applied in the frequency domain to obtain a reduced-order model of the unsteady flow in a transonic turbomachinery cascade of oscillating blades. The flow is described by a inviscid—viscous model, i.e. a full potential equation outer flow model and an integral equation boundary layer model. The nonlinear transonic steady flow is computed first and then the unsteady flow is determined by a small perturbation linearization about the nonlinear steady solution. Solutions are determined for a full range of frequencies and validated. The full model results and the POD method are used to construct a reduced-order model in the frequency domain. A cascade of airfoils forming the Tenth Standard Configuration is investigated to show that the reduced-order model with only 15–75 degrees of freedom accurately predicts the unsteady response of the full system with approximately 15 000 degrees of freedom.