Simulation of Unsteady Turbomachinery Flows Using an Implicitly Coupled Nonlinear Harmonic Balance Method

2011 ◽  
Author(s):  
Jonathan M. Weiss ◽  
Venkataramanan Subramanian ◽  
Kenneth C. Hall
Keyword(s):  
Steady State ◽  
Harmonic Balance ◽  
Time Derivative ◽  
Computational Cost ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Computationally Efficient ◽  
State Equations ◽  
Approximate Factorization ◽  
Efficient Manner

A nonlinear harmonic balance method for the simulation of turbomachinery flows is presented. The method is based on representing an unsteady, time periodic flow by a Fourier series in time and then solving a set of mathematically steady-state equations to obtain the Fourier coefficients. The steady-state solutions are stored at discrete time levels distributed throughout one period of unsteadiness and are coupled via the physical time derivative and at periodic boundaries. Implicit coupling between time levels is achieved in a computationally efficient manner through approximate factorization of the linear system that results from the discretized equations. Unsteady, rotor-stator interactions are performed to validate the implementation. Results based on the harmonic balance method are compared against those obtained using a full unsteady, time-accurate calculation using moving meshes. The implicitly coupled nonlinear harmonic balance method is shown to produce a solution of reasonable accuracy compared to the full unsteady approach but with significantly less computational cost.

Unsteady Simulation of a 1.5 Stage Turbine Using an Implicitly Coupled Nonlinear Harmonic Balance Method

2012 ◽  
Author(s):  
Chad H. Custer ◽  
Jonathan M. Weiss ◽  
Venkataramanan Subramanian ◽  
William S. Clark ◽  
Kenneth C. Hall
Keyword(s):  
Unsteady Flow ◽  
Time Domain ◽  
Harmonic Balance ◽  
Fourier Coefficients ◽  
Fluid Dynamic ◽  
Time Derivative ◽  
Computational Cost ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Computational Domain

The harmonic balance method implemented within STAR-CCM+ is a mixed frequency/time domain computational fluid dynamic technique, which enables the efficient calculation of time-periodic flows. The unsteady solution is stored at a small number of fixed time levels over one temporal period of the unsteady flow in a single blade passage in each blade row; thus the solution is periodic by construction. The individual time levels are coupled to one another through a spectral operator representing the time derivative term in the Navier-Stokes equation, and at the boundaries of the computational domain through the application of periodic and nonreflecting boundary conditions. The blade rows are connected to one another via a small number of fluid dynamic spinning modes characterized by nodal diameter and frequency. This periodic solution is driven to the correct solution using conventional (steady) CFD acceleration techniques, and thus is computationally efficient. Upon convergence, the time level solutions are Fourier transformed to obtain spatially varying Fourier coefficients of the flow variables. We find that a small number of time levels (or, equivalently, Fourier coefficients) are adequate to model even strongly nonlinear flows. Consequently, the method provides an unsteady solution at a computational cost significantly lower than traditional unsteady time marching methods. The implementation of this nonlinear harmonic balance method within STAR-CCM+ allows for the simulation of multiple blade rows. This capability is demonstrated and validated using a 1.5 stage cold flow axial turbine developed by the University of Aachen. Results produced using the harmonic balance method are compared to conventional time domain simulations using STAR-CCM+, and are also compared to published experimental data. It is shown that the harmonic balance method is able to accurately model the unsteady flow structures at a computational cost significantly lower than unsteady time domain simulation.

Steady State Solutions to a Forced Nonlinear Oscillator: A Comparison of Results Using a Generalized Harmonic Balance Method and by Numerical Integrations

1993 ◽  
Author(s):  
Ben Noble ◽  
Julian J. Wu
Keyword(s):  
Steady State ◽  
Harmonic Balance ◽  
Nonlinear Oscillator ◽  
Initial Point ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Original Equation ◽  
Long Time ◽  
Steady State Solutions ◽  
Generalized Harmonic Balance Method

Abstract Steady state solutions for nonlinear dynamic problems are interesting because (1) the long time behaviors of many problems are of practical concern, and, (2) these behaviors are often difficult to predict. This paper first presents a brief description of a generalized harmonic balance method (GHB) for steady state solutions to nonlinear problems via a nonlinear oscillator problem with a quadratic nonlinearity. Using this approach, steady state solutions are obtained for problems with several parameters: damping, nonlinearity and frequency (subharmonic, superharmonic and primary resonance). These results, plotted in time evolution curves and phase diagrams are compared with those obtained by numerically integrating the original differential equations. The effect of initial conditions on long time solutions is discussed. This investigation indicates that (1) the GHB steady state is an excellent approximate solution to that of the original equation if such a solution is numerically stable, and (2) the GHB steady state simply indicates a region of instability when the numerical solution to the original equation, using a point in that region as the initial point, is unstable.

An Implicit Harmonic Balance Method in Graphics Processing Units for Vibrating Blades

2015 ◽  
Author(s):  
Javier Crespo ◽  
Roque Corral ◽  
Jesus Pueblas
Keyword(s):  
Finite Difference ◽  
Harmonic Balance ◽  
Graphics Processing Units ◽  
Stokes Equations ◽  
Computational Cost ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Transonic Compressor ◽  
Speed Up ◽  
Graphics Processing

An implicit harmonic balance method for modeling the unsteady non-linear periodic flow about vibrating airfoils in turbomachinery is presented. As departing point, an implicit edge-based three-dimensional Reynolds Averaged Navier-Stokes equations solver for unstructured grids that runs both on central processing units (CPUs) and graphics processing units (GPUs) is used. The harmonic balance method performs a spectral discretization of the time derivatives and marches in pseudo-time a new system of equations where the unknowns are the variables at different time samples. The application of the method to vibrating airfoils is discussed. It is shown that a time spectral scheme may achieve the same temporal accuracy at a much lower computational cost than a Backward Finite Difference method at the expense of using more memory. The performance of the implicit solver has been assessed with several application examples. A speed-up factor of 10 is obtained between the spectral and finite difference version of the code whereas and an additional speed-up factor of 10 is obtained when the code is ported to GPUs, totalizing a speed factor of 100. The performance of the solver in GPUs has been assessed using the 10th standard aeroelastic configuration and a transonic compressor.

scholarly journals Adjoint-Based Aeroelastic Design Optimization Using a Harmonic Balance Method

2020 ◽  
Author(s):  
Nitish Anand ◽  
Antonio Rubino ◽  
Piero Colonna ◽  
Matteo Pini
Keyword(s):  
Design Optimization ◽  
Harmonic Balance ◽  
Flow Analysis ◽  
Computational Cost ◽  
Harmonic Balance Method ◽  
Forced Response ◽  
Balance Method ◽  
Working Fluid ◽  
Compressor Cascade ◽  
Impulse Turbine

Abstract Turbomachinery blades characterized by highly-loaded, slender profiles and operating under unsteady flow may suffer from aeroelastic shortcomings, like forced response and flutter. One of the ways to mitigate these aeroelastic effects is to redesign the blade profiles, so as to increase aero-damping and decrease aero-forcing. Design optimization based on high-fidelity aeroelastic analysis methods is a formidable task due to the inherent computational cost. This work presents an adjoint-based aeroelastic shape-optimization framework based on reduced order methods for flow analysis and forced response computation. The flow analysis is carried out through a multi-frequency fully-turbulent harmonic balance method, while the forced response is computed by means of the energy method. The capability of the design framework is demonstrated by optimizing two candidate cascades, namely, i) a transonic compressor cascade and, ii) a supersonic impulse turbine rotor operating with toluene as working fluid, initially designed by means of the method of waves. The outcomes of the optimization show significant improvements in terms of forced-response in both cases as a consequence of aero-damping enhancement.

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