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。     

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     

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.     

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.     

Transonic viscous-inviscid interaction by a finite element method

A method is outlined for solving two-dimensional transonic viscous flow problems, in which the velocity vector is split into the gradient of a potential and a rotational component. The approach takes advantage of the fact that for high-Reynolds-number flows the viscous terms of the Navier-Stokes equations are important only in a thin shear layer and therefore solution of the full equations may not be needed everywhere. Most of the flow can be considered inviscid and, neglecting the entropy and vorticity effects, a potential model is a good approximation in the flow core. The rotational part of the flow can then be calculated by solution of the potential, streamfunction and vorticity transport equations. Implementation of the no-slip and no-penetration boundary conditions at the walls provides a simple mechanism for the interaction between the viscous and inviscid solutions and no extra coupling procedures are needed. Results are presented for turbulent transonic internal choked flows.     

Boundary linear integral method for compressible potential flows

A boundary linear integral method based on Green function theory has been developed to solve the full potential equation for subsonic and transonic flows. In this integral method, potential values in the flow region are determined by potential values represented by boundary integrals and a volume integral. The boundary potential values are obtained by implementing the boundary integrals along boundary segments where a linear potential relation is assumed. The volume integral is evaluated in a grid generated by finite element discretization. The volume integral is evaluated only outside the body. Therefore there is no extra boundary treatment required for evaluation of the volume integral. The source term is assumed to be constant in an element integral volume. The volume integral needs to be evaluated only once and can be stored in computer memory for further usage.     

Non-linear heat transfer of solids in orthogonal coordinate systems

Solutions to the non-linear partial differential equation of heat conduction, (Poisson type), are obtained in which the conductivity is temperature dependent, by solving a linear partial differential equation and transforming it to the non-linear form using the Kirchhoff transformation. The method applies to any orthogonal coordinate system.

Transformations for handling boundary conditions of the Dirichlet, Neumann, convection and non-zero type are developed. The method is extended to solve a special class of non-linear unsteady-state conduction problems.

Two non-linear examples are solved to illustrate the method.     

On the Riemann problem for liquid or gas-liquid media

Self-similar solutions to the Riemann problem for water with the modified Tait equation of state are presented. The methods of Smoller for gas dynamics are employed to reduce the problem to the solution of a single non-linear equation. The same methods are used for solving the Riemann problem at a gas-water interface. In both cases the method of interval bisections affords a solution technique free of problems with convergence.     

A comparative analysis of methods for solving equations in the nonlinear elasticity theory

An approach to solving a variational equation used in geometrically and physically nonlinear problems of deformable body mechanics is considered. This approach is based on the continuation of a solution with respect to the loading parameter. Large systems of nonlinear ordinary differential equations arise in such problems. Usually, these systems are solved by the Euler methods. It is proposed to use the Runge-Kutta and multistep methods and to estimate the total computational cost. A dependence of numerical errors on the number of integration steps is obtained. An optimal method for solving nonlinear problems is chosen on the basis of this dependence.     

ANALYSIS/DESIGN CALCULATIONS OF TRANSONIC FLOWS

SUMMARY

Analysis/design calculations of transonic flow are discussed and several improvements are made. The nonisentropic potential method is used to calculate the inviscid transonic flow analysis problem instead of the traditional potential method. An inverse integral 3D boundary layer method is used to calculate the boundary layer in the viscous transonic flow analysis problem. The viscous/inviscid interaction calculations are carried out by a semi-inverse coupling scheme. In design problem calculations, an improved residual-correction method is used. Three individual methods are combined in a global algorithm and computing code. The improvements speed up the convergence, increase applicability and computational efficiency. Some numerical results are given to illustrate that the present method provides an effective engineering tool of high accuracy and efficiency in three dimensional transonic analysis and design situations.     

Numerical solution of single and multi‐phase internal transonic flow problems

This paper presents numerical methods for solving turbulent and two‐phase transonic flow problems. The Navier–Stokes equations are solved using cell‐vertex Lax–Wendroff method with artificial dissipation or cell‐centred upwind method with Roe's Riemann solver and linear reconstruction. Due to a big difference of time scales in two‐phase flow of condensing steam a fractional step method is used. Test cases including 2D condensing flow in a nozzle and one‐phase transonic flow in a turbine cascade with transition to turbulence are presented. Copyright © 2005 John Wiley & Sons, Ltd.     

Global and bifurcation analysis of a structure with cyclic symmetry

This article combines the application of a global analysis approach and the more classical continuation, bifurcation and stability analysis approach of a cyclic symmetric system. A solid disc with four blades, linearly coupled, but with an intrinsic non-linear cubic stiffness is at stake. Dynamic equations are turned into a set of non-linear algebraic equations using the harmonic balance method. Then periodic solutions are sought using a recursive application of a global analysis method for various pulsation values. This exhibits disconnected branches in both the free undamped case (non-linear normal modes, NNMs) and in a forced case which shows the link between NNMs and forced response. For each case, a full bifurcation diagram is provided and commented using tools devoted to continuation, bifurcation and stability analysis.     

THE PARALLEL BLOCK ADAPTIVE MULTIGRID METHOD FOR THE IMPLICIT SOLUTION OF THE EULER EQUATIONS

A method capable of solving very fast and robust complex non-linear systems of equations is presented. The block adaptive multigrid (BAM) method combines mesh adaptive techniques with multigrid and domain decomposition methods. The overall method is based on the FAS multigrid, but instead of using global grids, locally enriched subgrids are also employed in regions where excessive solution errors are encountered. The final mesh is a composite grid with uniform rectangular subgrids of various mesh densities. The regions where finer grid resolution is necessary are detected using an estimation of the solution error by comparing solutions between grid levels. Furthermore, an alternative domain decomposition strategy has been developed to take advantage of parallel computing machines. The proposed method has been applied to an implicit upwind Euler code (EuFlex) for the solution of complex transonic flows around aerofoils. The efficiency and robustness of the BAM method are demonstrated for two popular inviscid test cases. Up to 19-fold acceleration with respect to the single-grid solution has been achieved, but a further twofold speed-up is possible on four-processor parallel computers.     

Non-linear vibration of a single link viscoelastic Cartesian manipulator

In this present work, the non-linear behavior of a single-link flexible visco-elastic Cartesian manipulator is studied. The temporal equation of motion with complex coefficients of the system is obtained by using D’Alembert's principle and generalized Galarkin method. The temporal equation of motion contains non-linear geometric and inertia terms with forced and non-linear parametric excitations. It may also be found that linear and non-linear damping terms originated from the geometry of the large deformation of the system exist in this equation of motion. Method of multiple scales is used to determine the approximate solution of the complex temporal equation of motion and to study the stability and bifurcation of the system. The response obtained using method of multiple scales are compared with those obtained by numerically solving the temporal equation of motion and are found to be in good agreement. The response curves obtained using viscoelastic beams are compared with those obtained from a linear Kelvin-Voigt model and also with an equivalent elastic beam. The effect of the material loss factor, amplitude of base excitation, and mass ratio on the steady state responses for both simple and subharmonic resonance conditions are investigated.     

Isospectral Euler-Bernoulli beams via factorization and the Lie method

We obtain isospectral Euler-Bernoulli beams by using factorization and Lie symmetry techniques. The canonical Euler-Bernoulli beam operator is factorized as the product of a second-order linear differential operator and its adjoint. The factors are then reversed to obtain isospectral beams. The factorization is possible provided the coefficients of the factors satisfy a system of non-linear ordinary differential equations. The uncoupling of this system yields a single non-linear third-order ordinary differential equation. This ordinary differential equation, called the principal equation, is analyzed, reduced or solved using Lie group methods. We show that the principal equation may admit a one-dimensional or three-dimensional symmetry Lie algebra. When the principal system admits a unique symmetry, the best we can do is to depress its order by one. We obtain a one-parameter family of invariant solutions in this case. The maximally symmetric case is shown to be isomorphic to a Chazy equation which is solved in closed form to derive the general solution of the principal equation. The transformations connecting isospectral pairs are obtained by numerically solving systems of ordinary differential equations using the fourth-order Runge-Kutta method.     

平面跨音速叶栅正、反混合问题的一种高效率解法

本文在文献[1]的基础上提出了平面跨音速叶栅正、反混合问题的一种高效率解法,它保持了文献[1]中方法的优点,避免了打靶,从而达到了扩大适用范围,减少了计算时间。 文中导出了以Von Mises坐标为自变量的流线控制方程,简单而有效地处理了速度的双值问题,作为考核,计算了两个跨音速平面叶栅和一个拉伐尔喷管的流场,结果令人满意,本方法具有精度高、收敛快的特点,即使对于624所跨音涡轮动叶中部叶栅中比较复杂的流场,也只需迭代50次,费时不到1分钟(DPS-8机),即可获得满意的结果。     

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.     

Computation of unsteady transonic flow over a fighter wing using a zonal Navier–Stokes/full‐potential method

An improved hybrid method for computing unsteady compressible viscous flows is presented. This method divides the computational domain into two zones. In the inner zone, the Navier–Stokes equations are solved using a diagonal form of an alternating‐direction implicit (ADI) approximate factorisation procedure. In the outer zone, the unsteady full‐potential equation (FPE) is solved. The two zones are tightly coupled so that steady and unsteady flows may be efficiently solved. Characteristic‐based viscous/inviscid interface boundary conditions are employed to avoid spurious reflections at that interface. The resulting CPU times are about 60% of the full Navier–Stokes CPU times for unsteady flows in non‐vector processing machines. Applications of the method are presented for a F‐5 wing in steady and unsteady transonic flows. Steady surface pressures are in very good agreement with experimental data and are essentially identical to the full Navier–Stokes predictions. Density contours show that shocks cross the viscous/inviscid interface smoothly, so that the accuracy of full Navier–Stokes equations can be retained with significant savings in computational time. Copyright © 1999 John Wiley & Sons, Ltd.     

Numerical solutions of coupled Burgers' equation

Two new algorithms based on cubic spline function technique are proposed for solving Burgers' equation in one space variable and coupled Burgers' equation in two space variables. The algorithms have been analysed for their stability and convergence. Two test examples have been solved for illustrating the merits of the proposed numerical method. The method can be extended for solving non-linear problems arising in mechanics and other areas.