Improving convergence of incremental harmonic balance method using homotopy analysis method

We have deduced incremental harmonic balance an iteration scheme in the （IHB） method using the harmonic balance plus the Newton-Raphson method. Since the convergence of the iteration is dependent upon the initial values in the iteration, the convergent region is greatly restricted for some cases. In this contribution, in order to enlarge the convergent region of the IHB method, we constructed the zeroth-order deformation equation using the homotopy analysis method, in which the IHB method is employed to solve the deformation equation with an embedding parameter as the active increment. Taking the Duffing and the van der Pol equations as examples, we obtained the highly accurate solutions. Importantly, the presented approach renders a convenient way to control and adjust the convergence.     

Implicit solution of harmonic balance equation system using the LU-SGS method and one-step Jacobi/Gauss-Seidel iteration

ABSTRACT

This paper presents efficient alternative numerical methods for an implicit solution of the harmonic balance equation system for analysing temporal periodic unsteady flows. The proposed method employs approximate factorisation to decouple the common residual term and the time spectral source term of a harmonic balance equation system when it is discretised implicitly. With this approximate factorisation, the complexity of implicit solution of the discrete system is greatly reduced. The common residual term can be dealt with using a lower-upper symmetric-Gauss-Seidel (LU-SGS) method and the time spectral source term is integrated using a Jacobi iteration (JI) or one step Gauss-Seidel (GS) iteration, leading to the LU-SGS/JI method or LU-SGS/GS method. The NASA stage 35 compressor and the 1.5 stage Aachen turbine were used to demonstrate the effectiveness of the proposed methods in stabilising solution and its advantages in comparison with the existing lower-upper symmetric-Gauss-Seidel/block Jacobi (LU-SGS/BJ) method. The LU-SGS/GS method and the LU-SGS/JI method are more robust than the LU-SGS/BJ method in stabilising solution. The LU-SGS/GS method also has faster and tighter convergence and lower memory consumption in comparison with the LU-SGS/BJ method.     

Transient flow around an impulsively started cylinder using a dynamic mesh method

The finite volume method (FVM) with a dynamic mesh method (DMM) to deal with the moving boundary was applied to the simulation of two-dimensional incompressible viscous flow past a circular cylinder that is impulsively started into rotation and translation. The non-dimensional rotating to translating speed ratio α is varied from 0.28 to 2.07, with the Reynolds number being 200 for the range of α. The computation covers a period, during which the cylinder translates seven times its diameter. The current scheme handles the impulsively moving boundary directly by DMM, which is implemented using both mesh deforming and local remeshing. The instantaneous asymmetrical flow configurations for various α are presented and compared with the experimental visualizations. Quantitatively, the velocity distributions with drag and lift coefficients are also compared with the experimental and numerical results. Results show that the flow is strongly influenced by the rotation. Comparisons are found to be satisfactory.