Engineering turbulence modelling and experiments 5 [i.e. measurements 5] : proceedings of the 5th International Symposium on Engineering Turbulence Modelling and Measurements, Mallorca, Spain, 16-18 September 2002

Turbulence is one of the key issues in tackling engineering flow problems. As powerful computers and accurate numerical methods are now available for solving the flow equations, and since engineering applications nearly always involve turbulence effects, the reliability of CFD analysis depends increasingly on the performance of the turbulence models. This series of symposia provides a forum for presenting and discussing new developments in the area of turbulence modelling and measurements, with particular emphasis on engineering-related problems. The papers in this set of proceedings were presented at the 5th International Symposium on Engineering Turbulence Modelling and Measurements in September 2002. They look at a variety of areas, including: Turbulence modelling; Direct and large-eddy simulations; Applications of turbulence models; Experimental studies; Transition; Turbulence control; Aerodynamic flow; Aero-acoustics; Turbomachinery flows; Heat transfer; Combustion systems; Two-phase flows. These papers are preceded by a section containing 6 invited papers covering various aspects of turbulence modelling and simulation as well as their practical application, combustion modelling and particle-image velocimetry.

Chapter One

Invited Lectures

A PERSPECTIVE ON RECENT DEVELOPMENTS IN RANS MODELING

P. A. Durbin

Mechanical Engineering Department Stanford University Stanford, CA 94305-3030, U.S.A.

ABSTRACT

Some recent developments in RANS modeling are reviewed under the organizing theme 'connections between theory and practice'. Several cases where theory has proved useful are described: the nature of unsteady, Reynolds averaged computation is discussed; models that might be considered as spawned from realizability, or from rapid distortion theory are cited; the role of bifurcation theory in explicating aspects of second moment closure is noted; constraints devised from edge solutions are mentioned. Practical connections of such ideas range from devising the analytical form of the closure, to fixing faults that occur in widely used models. Faults include the likes of spurious energy production, insensitivity to external forces, and anomalous free-stream dependencies. In connection with unsteady RANS, remarks are made on using bridled RANS closures for detached eddy simulation.

KEYWORDS

turbulence, closure modelling, computational fluid mechanics, RANS, DES

INTRODUCTION

My title begins with a disclaimer: this is a perspective on some aspects of Reynolds averaged (RANS) closure modeling that I think are interesting. This article is not meant as a thorough review: the prospect of reviewing developments in RANS modeling, even without the page constraints of this proceedings, would be quite daunting. It is inevitable that many recent developments in the field will not be mentioned here — my apologies to those who feel slighted. In fact, I have decided that my theme should be connections between basic, analytical ideas about models and operational modeling, for use in computational fluid dynamics (CFD). There are several practical aspects of single point closure that have benefited from applied mathematical approaches. Of course, the practical level usually is reached after a good deal of massaging of this more abstruse material.

I regard the single point moment closure model to be among the most remarkable accomplishments of research into fluid turbulence. That opinion is certainly influenced by its practical value. Closure models for turbulence transport arefindingan increasing number of applications, in increasingly complex flows. The coming of age of commercial CFD software, and advances in computing speeds, are fueling that growth. But, from an intellectual standpoint, the remarkable accomplishment is that the statistics underlying a complex, irregular flow field are predicted, without having to simulate that complexity and then extract its statistics by processing data. Indeed, one reason why simulation methods (DNS or LES) have made little ingress into the realm of practical flow prediction is the computationally expensive requirement of a lengthy averaging process. Figure 1 illustrates the idea: were a large ensemble of flow fields like that in the lower part averaged, a streamline pattern quite similar to that shown at the top would emerge; but the top figure is obtained directly with the help of a RANS closure. I need not belabor the point to the audience of these proceedings, but I do want to counter the perception that modeling is about fitting data.

That said, the predictive capability of closure schemes is the end product. Efforts to assess accuracy and computational tractability, such as the workshops organized by ERCOFTAC, provide a valuable service. They also spur research into new models, or revisions to existing formulations, which might correct systematic defects identified by these efforts. I would like to look at some situations where model development has involved interplay between theoretical concepts and practicality.

UNSTEADY RANS

This first topic is not so much one to which theory has contributed, as it is one over which I have encountered confusion, perhaps of a philosophical, rather than operational, order. The subject is the application of RANS to statistically unsteady flow.

Unsteadiness can be imposed externally, as in rotor-stator interactions; or it can be spontaneous, internal unsteadiness, as in vortex shedding. In both cases, one simply includes time derivatives into the governing transport equations and integrates in time; there is no operational problem. The confusion is as follows.

One point of view I have encountered is that unsteady RANS is needed in flows like figure 1 because low frequency eddies are observed in experiments. When I ask experimentalists whether they have seen periodic shedding in flow over a backstep, all agree that they have not. For instance, temporal spectra show no spike (ideally, a 8 function; experimentally, something like figure 2) at a shedding frequency. The absence of a spike means that the ensemble, or Reynolds, average is not a function of time, and a suitably formulated model should produce a steady solution, even though it might be solved by time accurate integration.

An example of this is flow past a sphere. Unlikeflowpast a cylinder, there is no coherent frequency. The wake contains a low frequency, helical mode, which consists of vortex loops arranged along a helix. In turbulent flow, statistical symmetry is respected by fluctuations between right and left spiraling helices; unsteadiness does not survive ensemble averaging. So when Constantinescu, et al (2001) computed this flow with time accurate RANS they obtained, essentially, steady flow (see the top part of figure 5 later in the present paper, where S-A is the RANS computation). This means that the models were producing the qualitatively correct statistical behavior. Aside from the 2-layer k — ε solution, RANS gave good predictions of statistics (lower part of figure 5). The 2-layer model here is out of line simply because it uses a turbulent length scale near the wall, even in laminar flow — such as the boundary layer on a sphere at this subcritical Reynolds number.

The otiier point of view I have encountered is that RANS cannot be applied to unsteady flow, unless there is a spectral gap between the unsteadiness and the turbulence. As far as I can tell, this is based on an insistence that Reynolds averaging equals temporal averaging — which it does not. A more correct criterion for statistical periodicity is that the spectrum should contain a very narrow spike, representing the mean unsteadiness, as illustrated by figure 2. That spike can occur right in the midst of the broadband, turbulence spectrum; there is no need for a spectral gap. This is usually the case, and a number of studies (an early one being by Rodi and Franke) have shown how unsteady RANS can be applied correctly to flows in which the coherent unsteadiness lies right in the range of highest turbulent energy. Essentially, it is no more irrational to apply RANS to flow that is not statistically stationary than it is to apply RANS to flow that is not statistically homogeneous.

In other words, the velocity can be written as the mean plus a fluctuation

U = [??] {x, t) + u'(x, t) (1)

as usual; no further separation of the mean flow into periodic and time-averaged components is required. If unsteadiness is due to vortex shedding, then u' contains a contribution caused by jitter in the position and strength of the vortices, so it inevitably will contain significant energy at the mean shedding frequency. A spectral gap is unlikely.

While [??] can be time-averaged, this is a post-processing step. We can evaluate

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

but this time average is not a separate component in the modeling. Of course, some assumptions behind the closure will be violated if there is strong disequilibrium; but that restriction exists also in steady mean flows. For instance, think of the schematic idea of relaxation to eddy viscosity equilibrium.

[partial derivative]tbij = - [bij/T] - [2vT/kT]Sij + ... (3)

where bij = [??]i[??]j/k - 2/3δij. If the mean shedding frequency is ω, then the condition ωT