Handbook of Turbomachinery Second Edition Revised - Ventech!

It should be noted that the effects of the extra terms depend on the spatial
gradients of the unsteady disturbances (not just their absolute magnitudes).
Also note that the pressure term remains unchanged since it is linear. Thus,
an unsteady flow with large velocity fluctuations (e.g., unsteady vortices) is
likely to be have more significant effect on the time-averaged flow than one
with mainly pressure fluctuation (e.g., pressure waves). Clearly, the timeaveraged
results of an unsteady flow cannot be obtained by simply solving
the steady flow equations.
The most common and probably most complicated type of unsteady
flows is ‘‘steady’’ turbulent flows. It is somehow ironic that we hear much
more about how turbulence affects our ‘‘steady’’ flow results than we do
about a periodic unsteadiness. In a case of a turbulent flow, the extra terms
(e.g., ru 0 v 0 in a 2D case) are the turbulence (Reynolds) stress terms, which
need to be closed by turbulence modeling in one form or another. The basic
mechanism to generate turbulence stress terms is exactly the same as what is
illustrated above using the 1D flow equations, i.e., by the nonlinearity of the
inviscid part of the equations. The only difference is that the Reynolds
stresses are generated by essentially random unsteadiness, while for the cases
we consider here, the stresses terms are generated by unsteadiness with
distinctive frequencies, and for this reason they are called deterministic
stresses as cast in the framework for blade-row interactions by Adamczyk
[20].
Likewise, an appreciation of the effects of turbulence is helpful in
identifying those of a periodic unsteadiness on time-averaged flows. For
instance, we know that magnitudes of turbulence fluctuations are generally
much smaller than an averaged flow velocity. It is therefore not surprising
that a periodic unsteadiness with a small magnitude could have a significant
effect on a time-averaged flow. In general, as far as a time-averaged flow is
concerned, periodic unsteadiness might be regarded as ‘‘regular turbulence.’’
The main difference is that the extra terms due to periodic disturbances can
be directly evaluated by unsteady flow calculations or modeling with less
uncertainties than conventional turbulence modeling for the Reynolds stress
terms.
Here are some simple examples concerning the loss associated with a
periodic unsteady flow. Firstly, consider a shock wave in a quasi 1D duct
flow (Fig. 15). In asteady flow situation with fixed upstream conditions, the
solution of the shock wave (position, strength, and entropy rise, etc.) is
determined by the downstream static pressure P2. Now the shock wave is
made to oscillate by the downstream static pressure changing in the
following form:
P ¼ P2 þ Am sinðot þ fÞ ð16Þ
Copyright © 2003 Marcel Dekker, Inc.