Fourier modelling for nonlinear steady and unsteady aerodynamics

Fourier Model of Nonlinear Flows

The starting point of a Fourier modelling is the recognition that a nonlinear periodic flow distribution (the periodicity can be in time and/or in space) can be expressed by a temporal and/or spatial Fourier series. In many practical engineering situations, very good approximations can be obtained by truncated Fourier series with a very small number of low order harmonics.

Fourier 'Shape Correction' for Single–Passage Time-Marching Solution

The Fourier modelling approach to nonlinear flows was proposed in 1990 for time-marching solutions of unsteady turbomachinery flows (He 1990).  This was the first Fourier method for turbomachinery. The objective at the time was to enable an unsteady flow solution to be carried out in a single blade passage domain but without requiring a large amount of computer meomory, as in the Erdos's Direct Store method.  The main ingredient is to carry out the temporal Fourier transform at the ‘periodic boundaries of the single blade passage domain. Then the Fourier harmonics (temporal shape) are used to correct the corresponding boundaries according to the phase shift periodicity. The method was then called ‘Shape Correction’. he validity of the single passage Shape-Correction method can be examined by comparing with the direct multi-passage solution.   It was shown that the Fourier modelling as implemented in the Shape-Correction can capture flow disturbances and responses with large nonlinearity (e.g. a large scale shock oscillation in fan blade passage under an inlet distortion of long circumferential wave length, Fig.1 (Li and He 2001).

Given only 3-5 harmonics were required for capturing sufficiently accurately the temporal variation, the computer memory requirement is very low compared to the Erdos’s Direct Store approach. A key advantage of splitting flow components represented by Fourier harmonics is the ability in dealing with multiple disturbances with distinctive frequencies (He 1992). The generalised shape correction has been applied to unsteady flows in multi-rows (IGT-rotor-stator) with vibrating rotor blades for optimization of intra-row gap effects on both aerothermal performance and flutter stability, Fig.2 (Li and He 2005).

NASA rotor67 under inlet distortion

Fig.1 NASA rotor67 under inlet distortion (stagnation pressure contours)

 

Pressure contours of IGV-Rotor-Stator Interaction

Pressure contours of IGV-Rotor-Stator Interaction

 

 Time History of Force on Vibrating Rotor blade in IGV-rotor-stator configuration

Time History of Force on Vibrating Rotor blade in IGV-rotor-stator configuration
(SP-Single passage, MP-Multi-passage)

References:

L. He, "An Euler Solution for Unsteady Flows Around Oscillating Blades", ASME,  Journal of Turbomachinery, Vol.112, No.4, pp.714-722, 1990.


L. He, "Method of Simulating Unsteady Turbomachinery Flows with Multiple Perturbations", AIAA Journal, Vol.30, No.11, pp2730-2735, 1992.


H.D. Li and L. He, “Single-Passage Analysis of Unsteady Flows around Vibrating Blades of a Transonic Fan under Inlet Distortion”, ASME Journal of Turbomachinery, Vol.124, No.2, pp285-292, April, 2002.


H.D. Li and L. He, “Towards Intra-row Gap Optimization for 1&1/2 Stage Transonic Compressor”, ASME Journal of Turbomachinery, Vol.127, No.3, July 2005.

 

Nonlinear Harmonic Approach (frequency domain Fourier model)

A conventional frequency-domain time-linearised solution method offers a significant advantage in solution efficiency compared to the nonlinear time-domain method. The restriction is of course that the unsteady disturbances need to be small perturbations to a steady state with negligible effects on the base steady flow. Consequently with a given steady state, we only need to solve a time-independent harmonic equation for one harmonic unsteady disturbance.  The solution to the complex number amplitude field for the amplitude and phase angle of the unsteadiness is equivalent to solving two steady flow fields.  

To relax the fundamental linear assumption while taking advantage of the high solution efficiency, a nonlinear harmonic method was proposed (Ning and He, 1998). Similarly to the time-domain Fourier model, the unsteadiness is represented by the Fourier series. But now each harmonic will be balanced (‘harmonic balancing’) respectively in the nonlinear flow equations. Consequently, for a Fourier series retaining N harmonics, we will have 2N equations for the complex harmonics. In addition, the time-averaged flow will now be different from the steady flow due to the added deterministic stresses. So in total we have 2N+1 steady-like flow equations, which are solved simultaneously to reflect the interactions between the unsteady harmonics and the time mean flows. The interactions among the harmonics are included in a more complete nonlinear harmonic formulation by Hall’s harmonic balance formulations.

The nonlinear harmonic approach have been extended to effectively solve rotor-rotor/stator-stator interactions in multistage turbomachines (He et al 2002).

Recently efforts have been made to harness nonlinearity to stabilize the harmonic solution at highly loaded conditions with large scale flow separations (He, 2008).

References

L. He and W. Ning, "An Efficient Approach for Analysis  of  Unsteady Viscous Flows in Turbomachinery”,  AIAA Journal, Vol.36, No.11, 1998.

W. Ning and L. He, "Computation of Unsteady Flows around Oscillating Blades using Linear and Nonlinear Harmonic Euler Methods”, ASME, Journal of Turbomachinery, Vol.120, No.3, pp.508-514, 1998.

L. He, T. Chen, R.G. Wells, Y.S. Li and W. Ning, ‘Analysis of Rotor-Rotor and Stator-Stator Interferences in Multi-stage Turbomachines’, ASME Journal of Turbomachinery, Vol.124, No.4,  pp. 564-571, Oct, 2002.

L. He , "Harmonic Solution of Unsteady Flows around Blades with Separation”,  AIAA Journal. Vol.46, No6, 2008.

L. He, “Fourier Methods for Turbomachinery Applications”, Progress in Aerospace Sciences, Vol.46, Issue 8, Nov, 2010.

Spatial Fourier Spectral Model for Non-Axisymmetric Flows

In many cases of practical interest, large scale steady and/or unsteady flow disturbances develop in large cylindrical domains (e.g. intake, exhaust ducts, rotor disk cavities). The circumferential domain truncation similar to the single-passage method is difficult without a known circumferential wave length.  Thus, a computational domain to cover the whole 360° circumference would be necessary.

In these cases with a large circumferential domain, a circumferential Fourier spectrum can be introduced to achieve an efficient solution.  The model can be quite simply implemented in an existing time-marching solver, and the validity has been demonstrated for an intake duct case subject to a cross-wind, and a case of acoustic pressure wave corresponding to fan tone noise propagated in a distorted duct (He, 2005, 2006)  

Recently the spatial Fourier model has been applied to analysis of self-excited coherent unsteady flows and convective heat-transfer in rotating disk cavities.

Circumferential flow Angle
(exit plane from an intake duct, subject to 10 cross wind)

Circumferential flow Angle

References

L. He, “On Circumferential Phase-Shift Periodicity for Turbomachinery Aerodynamics and Aeromechanics Applications”,  AIAA Paper 2005-0016, Invited Paper on Aeroelasticity, the 43rd AIAA Aerospace Sciences Meeting, Reno, U.S.A., Jan 2005.

L. He, “Fourier Modelling of Nonaxisymmetrical Steady and Unsteady Flows”, AIAA Journal of Propulsion and Power, Vol.22, No.1, pp197-201 Jan-Feb, 2006.

L. He, “Efficient Computational Model for Non-axisymmetric Flow and Heat Transfer in Rotating Cavity”,  ASME Journal of Turbomachinery, Vol.133. No.2, 2011.