19 ` eme Congr` es Franc ¸ais de M´ ecanique Marseille, 24-28 aoˆ ut 2009 Direct Numerical Simulation of rotor-stator flows in an annular cavity S. PONCET a , P. LE GAL b , E. SERRE a a. Laboratoire M2P2 (UMR 6181), Technopˆ ole Chˆ ateau-Gombert, 13451 MARSEILLE b. IRPHE (UMR 6594), Technopˆ ole Chˆ ateau-Gombert, 13384 MARSEILLE Abstract : The stability of the flow enclosed between a stationary and a rotating disk with a central hub is revisited by experimental visualizations and direct numerical simulations in the case of unmerged boundary layers. The first instability appears as circular rolls, denoted CR (type 2 instability), which propagate along the stator before vanishing in the vicinity of the hub. The calculations highlight the convective nature of these rolls, which is in agreement with previous experimental results [1]. It proves in particular that the CR instability observed in the experiment under permanent conditions is noise sustained. Above a second threshold, spiral rolls, denoted SR1 (type 1 instability), appear at the periphery and can coexist with the circular rolls. The DNS shows that they appear through a supercritical Hopf bifurcation. Keywords : rotor-stator, instability, DNS, flow visualization 1 Introduction The stability of the flow confined between a rotating and a stationary disk is mainly governed by two global parameters [2]: the aspect ratio G = h/b of the cavity and the rotational Reynolds number Re =Ωb 2 /ν , where h is the interdisk spacing, b the rotating disk radius and Ω the rotation rate. In such rotating disk cavity, two types of instability have been identified: type 1 instability results from an inviscid mechanism due to unstable inflection points in the boundary layer velocity profiles, whereas type 2 instability is viscous and associated with the Coriolis terms. The first instabilities occur in the stator boundary layer according to the linear stability analysis (LSA) of San’kov and Smirnov [3] and Serre et al. [4] and to the DNS of Serre et al. [5]. The transition to turbulence in a cylindrical rotor-stator cavity has been considered later by Schouveiler et al. [6, 2]. Their experiments reported that, for G> 0.071, the two first stages of the transition are due to the developments of instabilities in the stator boundary layer. Above a first threshold, they observed the formation of circular rolls (CR) centered on the rotation axis, which propagate towards the center of the cavity. The CR instability, which is a type 2 instability has been formerly observed by Savas [7] during the spin-down of a rotating disk and much later by Schouveiler et al. [6, 2] and Gauthier et al. [1] under permanent conditions. Above a second threshold, Schouveiler et al. [6, 2] observed spiral rolls (SR1), which appear at the periphery and coexist with the CR instability. The SR1 patterns are a type 1 instability with a band of stable modes limited by the Eckhaus secondary instability. Their radial wavelength strongly varies between DNS results (4.4 − 14.7 for G =0.2) and the results of a LSA (21.2 − 24.35) [4]. When the Reynolds number is increased further, a transition to a kind of wave turbulence occurs [8]. Gauthier et al. [1] showed, by flow visualizations, that the CR are very sensitive to an external (un)controlled forcing and then highlighted the convective nature of this instability. Numerous experimental [1, 6, 2], theoretical [3] or numerical [5, 4] works have already been dedicated to the stability of the flow with unmerged boundary layers corresponding to G> 0.071. Nevertheless, the nature of both the circular and spiral rolls remains unclear. The scenario first proposed by Schouveiler et al. [2] for the transition to turbulence in the enclosed rotor-stator cavity is revisited in the present paper essentially by numerical experiments using DNS but also by flow visualizations in an annular cavity with a central hub. 2 Experimental set-up The cavity consists of two smooth parallel disks enclosed by a rotating hub of radius a = 40 mm and a stationary shroud of radius b + j = 140.85 mm. One disk of outer radius b = 140 mm is rotating, while the other one is stationnary. The interdisk space h is equal to 16 mm. Thus, the values of the aspect ratio G = h/b and the curvature parameter R m =(b + a)/(b − a) are here fixed to G =0.114 and R m =1.8. The rotor and the hub rotate clockwise at the same uniform rotation rate Ω. The shroud is fixed. The cavity is filled up with water maintained at a working temperature of 20 ◦ C and seeded with reflective particles of kalliroscope (30 × 6 × 0.07μm) in order to visualize the hydrodynamic structures. Images (768 × 576 pixels) are taken at a video frequency of 25 images per second using a CCD camera (see [8] for more details). 1