Demonstration of balance-based space averaging and modelling in view of blading—diffuser interaction

Abstract

Advanced experimental and numerical methods in the eld of uid dynamics and turbomachinery are increasingly successful in describing real ow elds, i.e. elds that are generally three-dimensional and unsteady. For many purposes, e.g. ow characterization, it is necessary to reduce these ow elds step by step to three-, two-or one-dimensional large-scale unsteady ow elds. This procedure permits a lower-level simulation of the ow elds. However, many averaging approaches are arbitrary or succeed in balancing the ow elds in only a few physical aspects. The rst author has already shown the steps of a balance-based procedure that avoids this limitation. Small-scale time averaging of (probabilistically) turbulent inhomogeneities by means of irreversible and reversible small-scale time averaging processes on a threefold in nitesimal control volume element has already been demonstrated. The present paper demonstrates the balance-based procedure of space averaging. It is carried out by averaging generally three-dimensional small-scale time-averaged (deterministic) inhomogeneities using irreversible and reversible space averaging processes on onefold in nitesimal and nite control surfaces. The procedure is, similarly to small-scale time averaging, based on conservative and independent non-conservative small-scale time-averaged integral balance equations. The general concept is to represent all the relevant uxes through the control surface by appropriate average quantities or numbers. The full use of the vector equations for the linear and angular momentum is important. One of the consequences in space averaging is the introduction of a wrench (parallel linear and angular momentum vectors), which is generally used only in mechanics for the reduction of force systems in space. The ow eld inhomogeneity is described on all dimensional levels via the diffusion intensity of the irreversible averaging process, and, only for space averaging, via the distance vector and the parameter of the wrench. A numerical example on different dimensional levels is presented in detail. The procedure also illustrates the basis of a new and more complete twoand one-dimensional large-scale unsteady theory generally in uid dynamics and especially in turbomachinery.

Optimization of blade—diffuser interaction for improved turbine performance

“…Typically, this kind of stage design exhibits a downstream total pressure distribution increasing from hub to tip in the control surface CS2. This total pressure distribution is unfavourable for the diffuser, causing a ow separation at the hub, as has been demonstrated by a three-dimensional Navier-Stokes calculation in reference [2]. The resulting poor diffuser performance parameters are w 23ˆ1 6.7 per cent and x 23ˆ7 9.0 per cent (x ¤ 23ˆ3 2:9 per cent).…”

“…For guidance, the minimum total pressure coef cient is a measure of the maximum diffuser recovery achievable and for the diffuser length required. Characterization of the ow inhomogeneity affected by the contour corners and their development within the interaction zone are described in reference [2]. The resulting slightly bell-shaped optimized diffuser with the corner angles according to Fig.…”

“…Unlike classical diffuser analysis, where the parameters of recovery factor, ideal recovery factor and loss [1], based on geometry and on mass-averaged total pressures and velocities across the control surfaces, are used, in the following the diffuser performance will be characterized by parameters based on the mechanical energy equation and on reversible averaged one-dimensional eld quantities(superscript ¹) [2]:…”

“…Here, the minimal kinetic energy K ¤ 3 is calculated using the reversible pressurep p 3 and the reversible total enthalpyh h 02ˆh h 03 at the diffuser exit, assuming a homogeneous swirl-free ow (for details, see reference [2]).…”

New space-averaging procedure for inhomogeneous flow fields using balance equation formulations

“…Dzung's averaging is simple, but the flow incidence angles calculated from the mean radial and tangential velocities may be misleading in flows with a strongly non-uniform velocity profile [4]. In an attempt to merge both previous theories, a hybrid procedure is proposed by Kreitmeier [2,5]. His technique avoids the need for any correction factor, as in Dzung's procedure, and he still maintains the advantages of Traupel's idea as well.…”