Mixing Losses in Steady and Unsteady Simulations of Turbomachinery Flows

Abstract

The aim of the present work is to facilitate insight into the modeling errors in the context of blade row coupling approaches which capture unsteady flow phenomena at different levels of detail. The focus is on RANS-based steady mixing plane computations as well as time domain and frequency domain unsteady computations. The concept of mixing loss is revisited to quantify the amount of unsteadiness in a flow field. Following an idea by Fritsch and Giles, we compute a second order approximation of the mixing losses which are generated at blade row interfaces. The resulting formula decomposes the entropy jump into contributions of circumferential and temporal fluctuations. The mathematical derivation, however, is based upon simpler arguments. It is shown that Fritsch and Giles’ main result can be extended to non-ideal gases. Moreover, the second order mixing loss formula is applied to time and frequency domain unsteady simulations. It is shown that an additional term has to be computed which accounts for the interaction of evanescent acoustic modes if the method is applied to unsteady flows. The methodology decomposes the overall mixing entropy into contributions of single perturbation types and harmonics. This may be used to assess whether unsteady flow phenomena of interest are adequately resolved and, in particular, to quantify the unsteadiness contained in the unresolved harmonics. A detailed investigation of the transonic IGV-rotor configuration of DLR’s Rig 250 compressor demonstrates the approach.

Analyzing Unsteady Turbomachinery Flow Simulations With Mixing Entropy

“…Moreover, for each frequency and mode order, the contribution can be further decomposed into contributions from different wave types. Schlüß and Frey [15] gave a more insightful interpretation of the resulting formula by relating it to the waves' group velocities and a certain norm, whose square is defined by the Hessian of the entropy density. This way, Fritsch and Giles' main result could be shown to hold in much more general contexts, e.g., for imperfect gas or multicomponent flow.…”

“…In this section, we summarize the derivation of the asymptotic expansion of the mixing entropy presented in [15] and relate the result for acoustic modes to sound intensity. In the following, DF(q) denotes the Jacobian of a function F in several variables q 1 , .…”

“…where the scalar product is generalized to complex vectors such that it is anti-linear in the first component. If a general disturbance q with angular frequency ω and tangential wave number η is given as a sum of fundamental waves of the form ( 16), q = j Re r (j) e i(ωt+ξ j x+ηy) , then (15) and (17) imply…”

“…Expressing DF x r 1 , r 2 ϕ in a modal basis would thus yield zero diagonal terms for cut-off acoustic modes. On the other hand, the axial wavenumbers as well as the righteigenvectors of the up-and downstream running cut-off modes are complex conjugates of each other, so the off-diagonal terms contain expressions of the form DF x r , r ϕ with cut-off acoustic eigenvectors r and r. Note that these crosscoupling terms are non-zero and may represent a significant contribution to the overall mixing entropy at blade row interfaces where both up-and downstream disturbances exist [15].…”

“…see [15]. Here, and in the following, we simply write ρ, p,... for the averaged variables ρ a , p a etc.…”