Turbomachinery Linear Unsteady CFD

An important part of modern design of turbomachinery blading is the validation for aeromechanical soundness. By modelling the unsteady flow through turbomachinery, designers can ameliorate blade designs in order to avoid the occurrence of flutter and forced response. There exists a whole spectrum of methods in use for the simulation of turbomachinery unsteady gas flows. At one end of the spectrum, the classical semi-analytical aerodynamic methods based on linear inviscid flow theory are computationally extremely efficient but are limited to design operating conditions involving lightly loaded and thin airfoils. At the other end of the spectrum, time-accurate nonlinear methods which time-march the discrete 3D Reynolds-averaged Navier-Stokes equations to the final time-periodic solution are very general and include many of the complex effects occurring in turbomachinery: arbitrary blade geometries, blade vibration, unsteady multi-stage effects, complicated shock structures and turbulence. Yet, despite their impressive capabilities to predict the important unsteady flow phenomena, the cost and computer time usually associated with these methods pose a major limitation to their application in industrial design. Accordingly, there exist a strong interest in developing new classes of simpler methods that exploit the peculiar properties of the unsteady flows in turbomachinery and partly trade the versatility of nonlinear methods for computer efficiency.

During the last twenty years, the linear harmonic methods have emerged as an important such class of techniques, their modelling capabilities evolving in time from 2D linear potential equations to full 3D Navier-Stokes equations coupled with a turbulence model. At present, there exists a substantial body of practice to prove that the linearised viscous flow methods are satisfactory for a large range of aeroelastic applications. In terms of accuracy and capacity to analyse different flow features, the linear harmonic methods are much better than the classical methods and only slightly poorer than the full unsteady nonlinear methods. The pre-eminent advantage quoted by proponents of the linearised flow analysis is its computational efficiency over the traditional time-accurate time-marching algorithms. First, the unsteady response of the flow to time-periodic excitations is resolved in the frequency domain by treating different sources of unsteadiness or flow response in different frequencies independently. Calculations in the frequency domain are not restricted to a constant time-step, therefore the iterative solutions of the linearised methods are nearly as efficient as for the more common steady-state flow calculations. Also, local time-stepping and acceleration using multiple grids, techniques traditionally reserved for steady-state analysis, can be effectively employed in a linearised analysis. Additionally, any linear unsteady analysis is carried out on a single blade-to-blade passage through a frequency domain treatment of the phase-lagged periodicity conditions which eliminates the need to construct computational grids containing spatial periodicity.

These characteristics render the linear harmonic methods highly commendable in turbomachinery applications in which the important flow unsteadiness takes place at a set of isolated frequencies. Thus, using the linearised approach for unsteady flows associated with single frequency forced response and flutter, the essential physical features involved are obtained at a computational cost typically several orders of magnitude smaller than that required by time-domain nonlinear methods.