Marie-Pierre Chauve’s research while affiliated with Institut de Recherche sur les Phénomènes Hors Équilibre, French National Centre for Scientific Research and other places

The stability of a Batchelor flow enclosed between a stationary and a rotating disk with a central hub is investigated by extensive experimental visualizations and direct numerical simulations. The first instability appears as circular rolls (CR), which propagate along the stator towards the rotation axis. Above a sec-ond threshold, spiral rolls (SRI) appear at the periphery of the cavity due to the destabilization of the boundary layer along the external cylinder. These spirals coexist with the circular rolls in the experiment. For the first time, the presence of an inner rotating hub, which is of major industrial importance, measured by the influence of the curvature parameter has been studied experimentally. This parameter strongly modifies the wave number of the instabilities as well as the angle of the spirals. Finally, for a given geometry, a very good agreement on the characteristics of these instabilities is obtained between the experimental and nu-merical approaches.

Cette étude concerne les écoulements de type rotor–stator à couches limites séparées soumis à un flux axial. Suivant l’analyse faite par Schlichting [1] dans le cas d’un disque tournant de rayon infini, on détermine des lois analytiques simples permettant de prévoir le coefficient d’entraınement K du fluide à partir des paramètres de contrôle (taux de rotation, flux, espace interdisque, taux de prérotation) pour divers régimes d’écoulement (laminaire ou turbulent) et différentes configurations (géométrie, rotor lisse, rugueux ou aileté). Cette approche analytique est validée par d’extensives mesures de vitesse et de pression.

There have been numerous numerical simulations and experimental studies of flow between rotating and stationary discs with a stationary shroud and no throughflow (a “rotor-stator cavity”) (see references in Serre et al. 2001; Poncet et al. 2005; Randriamampianina & Poncet 2006). The flow has significant industrial applications, such as internal gas-turbine flows and computer hard disks, and the geometry is relatively simple. A characteristic feature of such flows is the coexistence of adjacent coupled flow regions that are radically different in terms of the flow properties (Serre et al. 2004). Moreover, the confinement, the flow curvature and the rotation effects create a strongly inhomogeneous and anisotropic turbulence. Consequently, these flows are very challenging for numerical modelling particularly in turbulent regimes (see a review in Crespo del Arco et al. 2005). Turbulent regimes are investigated here in an annular rotor-stator cavity, using experimental measurements as well as Large-Eddy Simulation (LES). At our knowledge, there has been no efficient investigation of turbulent rotor-stator flows within a closed interdisk cavity using LES. The mean flow is mainly governed by three control parameters: the aspect ratio of the cavity G(=(b-a)/h)=5, the rotational Reynolds number Re based on the outer radius b of the rotating disk and the radius ratio s(=a/b)=0.286. In this work, LES and experimental measurements have been used to characterize statistical properties of turbulent rotor-stator flows for Reynolds numbers up to one million. Till now, LES predictions have compared very favourably with experimental measurements for Reynolds number up to 0.7 million. In the oral presentation of this work it will be possible to show computations still in progress at the moment at Re equal to one million.

Cette étude expérimentale porte sur l’instabilité d’une couche de cisaillement au-dessus d’un disque en rotation avec surface libre. Cette instabilité est caractérisée par visualisations de l’écoulement pour une large gamme des paramètres de contrôle: le rapport d’aspect G de la cavité, le nombre de Reynolds global Re et le rapport s entre les rayons intérieur et extérieur du disque tournant. Cette instabilité spectaculaire se développe le long du cylindre extérieur sous la forme de polygones à m côtés. Ce nombre m dépend d’un nombre d’Ekman basé sur la hauteur d’eau au repos, ce qui confirme la nature « stewartsonienne » de la couche limite. Le seuil d’apparition du premier mode est constant si l’on considère le nombre de Reynolds mixte proposé par Niino and Misawa (1984). Pour des larges valeurs de s, une instabilité se développe le long du cylindre intérieur sous la forme de petites cellules.

Comparisons between Large Eddy Simulation (LES) and velocity measurements have been performed for the turbulent flow in a real shrouded rotor-stator configuration. To investigate turbulent flow regimes, Séverac et al.(2006) have developed a stabilization technique, called SVV (Spectral Vanishing Viscosity), which exhibits the properties of preserving the spectral accuracy of the approximation used in direct numerical simulation (DNS). Thus, numerical results and experimental data have been favourably compared for a large range of the rotational Reynolds number in an annular cavity of curvature parameter Rm=1.8 and of aspect ratio G=5. Coherent vortices have been identified numerically in both boundary layers using the Q-criterion of Hunt et al. (1988).

Turbulent flows are studied in an actual enclosed rotor-stator configuration with a rotating hub and a stationary shroud. Besides its fundamental importance - the disk boundary layer is one of the simplest platforms for investigating the underlying structure of three-dimensional boundary layers - this cavity models more complex configurations relevant to rotating machinery. Large Eddy Simulation (LES) is performed using a Spectral Vanishing Viscosity (SVV) technique which is shown leading to stable discretizations without sacrificing the formal accuracy of the spectral approximation. Numerical results and velocity measurements have been favorably compared for a large range of rotational Reynolds numbers up to one million in an annular cavity of curvature parameter Rm=(b+a)/(b-a)=1.8 and of aspect ratio G=(b-a)/h=5, where a and b are respectively the inner and outer radii of the rotating disk and h is the interdisk spacing. In the detailed picture of the flow structure that emerges, the turbulence is mainly confined in the boundary layers including in the Stewartson layer along the external cylinder. For Reynolds numbers larger than 0.1 million, the stator boundary layer is turbulent over most of the cavity. On the other hand, the rotor layer becomes progressively turbulent from the outer radial locations although the rotating hub is shown to destabilize the inner part of the boundary layers. The isosurface maps of the Q-criterion reveal that the three-dimensional spiral arms observed in the unstable laminar regime evolve to more axisymmetric structures when turbulence occurs. At Re equal to one million, the flow is fully turbulent and the anisotropy invariant map highlights turbulence structuring, which can be either a ``cigar-shaped'' structuring aligned on the tangential direction or a ``pancake-shaped'' structuring depending on the axial location. The reduction of the structural parameter a1 (the ratio of the magnitude of the shear stress vector to twice the turbulence kinetic energy) under the typical limit 0.15, as well as the misalignment between the shear stress vector and the mean velocity gradient vector, highlight the three-dimensional nature of both rotor and stator boundary layers with a degree of three-dimensionality much higher than in the idealized system studied by Lygren and Andersson (2001-2006).

The instability of the three-dimensional flow over a rotating disk is experimentally investigated by hot-wire measurements. The first part of this paper presents the experimental marginal stability curve and dispersion curves (linking frequencies to wave vector components). It is then observed that non-linear effects play a major role very early in the boundary layer. The second part is thus devoted to the study of a well controlled wave pattern excited by a roughness element placed under linear threshold. The system exhibits a weak non-linear behaviour which leads to the appearance of super harmonics modes and to a shift of the dispersion curves as compared to their linear values. Finally, we present an original method in order to infer the group velocity components from the dispersion curves associated with the propagation angle of the wave packet. In particular, the results demonstrates the convective nature of the instability.

Turbulent incompressible flows are studied both numerically and experimentally within an annular rotor-stator cavity of aspect ratio G=5 and radius ratio a/b=0.286. Besides its fundamental importance as a three-dimensional prototype flow, such flows are crude models of flows arising in many industrial devices, especially in turbomachinary applications. Our aim is to investigate turbulent regimes at three Reynolds numbers corresponding to different flow properties as the rotation of the rotor is increased. Experimental measurements have been obtained using a laser Doppler anemometer (LDA) technique. Numerical modelling is based on a Large Eddy Simulation (LES) using a spectral vanishing viscosity (SVV) technique implemented in a Chebyshev-collocation Fourier-Galerkin pseudo-spectral code. As far as the authors are aware, LES of fully turbulent flow in an actual shrouded rotor-stator cavity have not been performed before. Turbulent quantities are shown to compare very favourably with experimental measurements and are shown of interest in understanding the physics of turbulent rotor-stator flows from transitional to turbulent regimes. Moreover, averaged results may provide target data for workers employing RANS schemes.