A New Time-Averaging Procedure for Compressible, Unsteady Turbulent Flows

Abstract

The computations done at present for turbulent flow fields use either time-averaged (Reynolds) or time-and-mass-averaged (Favre) quantities, together with time-averaged fluctuating terms (FTs). Large numbers of these FTs, which are derived from the time-averaged balance equations, occur whenever all relevant time-averaged balance equations are being used and all quantities, including the transport quantities, must be considered as turbulent. The FTs require a great deal of experimental and theoretical work — for example, for closing time-averaged balance equations using familiar heuristic methods. For general simplification, then, a new averaging procedure is presented here which employs no arbitrary breakdown of the turbulent quantities, thus completely avoiding the FTs. This method is based on hypothetical reversible and irreversible equilibration processes on three control surfaces perpendicular to the major axes and in the control volume of a fluid element. The time-averaged balance equations take on the same manageable form as the instantaneous balance equations. In addition, the effort required for closing the time-averaged balance equations is reduced to a minimum.

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

“…deterministic, time dependency is given. See references [2] or [3] for the entire small-scale time averaging process. Similarly to this, the appropriate space avenging process of the remaining three-dimensional large-scale unsteady ow through control surfaces is described below, to deduce representative twoand one-dimensional large-scale unsteady quantities.…”

“…In references [2], [3] it has been demonstrated how the rst step, i.e. the reduction in turbulent uctuations by means of balance-based small-scale time averaging processes on a threefold in nitesimal volume element, is realized.…”

“…The method presented there is based on small-scale time-averaged conservative (total mass, total linear and angular momentum and total enthalpy) and, because of averaging process independence, non-conservative (enthalpy, kinetic energy and entropy) differential balance equations performing 'irreversible' and 'reversible' averaging processes. This 'balance-based' procedure avoids the arbitrarily de ned time-averaged eld quantities and therefore the timeaveraged uctuating terms of the 'statistics-based' procedures (Reynolds and Favre, see reference [2]).…”

“…A survey of alternative space averaging methods has also been given, including the 'correction factor method' [5][6][7], the 'mixing method' [8][9][10] and the 'perturbation or statistical method' [11][12][13]. Comparisons [2,4] with these well-known procedures have shown that they are formulated arbitrarily and create some problems of understanding.…”

On the choice of variables and matching criteria for the steady simulation of transonic axial turbine stages

“…The tangential mixing process is attractive for industrial applications where timeaveraged ow elds are of interest, rather than the detailed unsteady ow. The time averaging on the interface plane can be done by a simple normalized time integral of the conservation equation terms or, for example, by the balance-based approach proposed by Kreitmeier [17] who avoids time averaging of uctuating terms. Adamczyk [18] improves the simple mixing plane model by introducing the so-called deterministic stresses.…”