Handbook of Turbomachinery Second Edition Revised - Ventech!

significant difference is introduced to the flow decomposition:
Uðx; tÞ ¼UðxÞþU 0 ðx; tÞ ð34Þ
where UðxÞ is a time-averaged variable, instead of a steady one as defined in
Eq. (33). For a general case with Np disturbances, each disturbance at a
frequency ok can be expressed by a pair of complex conjugates, eUk and eU k,
so:
U 0 ðx; tÞ ¼ XNp
k¼1
h eUkðxÞe iokt þ eU kðxÞe
iokt i
ð35Þ
The flow decomposition as defined in Eq. (34) leads to two sets of equations,
one for the time-averaged flow, the other for the unsteady perturbations.
The set of Np unsteady perturbation equations are obtained by balancing
harmonic terms for each frequency.
Following the earlier discussion, we know that the time-averaged
equations will contain extra deterministic stress terms, which depend on the
unsteady solution. On the other hand, the unsteady perturbation equations
cannot be solved without a ‘‘known’’ time-averaged flow. This interdependence
between the two sets of equations reflects the physical interaction
between the two parts of flow and is modeled by a simultaneous coupling in
the solution process of the nonlinear harmonic method, as shown in Fig. 19.
The nonlinear harmonic method offers a significant gain in computational
efficiency, since neither a time-accurate integration, nor a multi-
Figure 19 Strongly coupled solution between two sets of decomposed flow
equations (n ¼ number of pseudotime iterations).
Copyright © 2003 Marcel Dekker, Inc.