Handbook of Turbomachinery Second Edition Revised - Ventech!

For most practical situations of interest, first few harmonics (typically less
than 5) have been shown to be sufficient.
A major advantage of expressing unsteadiness in a Fourier series is its
ability to include multiple disturbances with unrelated frequencies. If the
total number of unsteady disturbances of interest is Nd, a flow variable at
periodic boundaries is expressed in the following general form:
Uðx; tÞ ¼UðxÞþ XNd
i¼1
Uiðx; tÞ ð32Þ
Each disturbance Ui is identified by its fundamental and higher harmonics
in a Fourier series as in Eq. (31). The capability of including multiple
disturbances has been demonstrated for a transonic fan rotor subject to inlet
distortion and blade oscillation [19].
Frequency-Domain Modeling
Frequency domain methods have been well established for unsteady flow
calculations, mainly for blade aeroelasticity applications, e.g., Verdon and
Caspar [32], Hall et al. [33]. The methodology of this kind starts with
decomposing a flow variable U into a steady part and an unsteady part:
Uðx; tÞ ¼U0ðxÞþU 0 ðx; tÞ ð33Þ
And the unsteady part is assumed to be very small so that nonlinear effects
(product terms of the unsteady perturbations) can be neglected. It follows
then that the time-averaged flow is the same as the steady one. On the basis
of a steady flow, a linear equation can be formed with respect to the
unsteady perturbation. The linear unsteady perturbation is normally cast in
a complex harmonic form, to remove the time dependence, resulting in the
perturbation equation with respect to the complex amplitude. The real and
imaginary parts of the complex amplitude determine the phase and
amplitude of unsteadiness. A solution to the complex harmonic amplitude
is obtained at a given frequency, thus is in ‘‘frequency domain.’’ Compared
to unsteady time-domain (time-marching) methods, frequency-domain
methods are much more efficient. A frequency-domain unsteady calculation
can be regarded as being equivalent to solving two ‘‘steady’’ flow problems
since the amplitude is not time-dependent, and can be conveniently solved in
a single-passage domain. The key limitation of the conventional frequencydomain
methods is the linear assumption about unsteadiness.
To include nonlinear effects, a new methodology, called nonlinear
harmonic method, has been proposed [2] and developed for analysis of blade
aeroelasticity [34], [35] and blade-row interactions [6]. Here, a simple but
Copyright © 2003 Marcel Dekker, Inc.