15TH EUROPEAN TURBULENCE CONFERENCE, 25-28 AUGUST, 2015, DELFT, THE NETHERLANDS LIFETIME OF TURBULENT PATCH IN TAYLOR COUETTE SETUP

In linearly stable shear flows like pipe and plane Couette flows, the transition from the laminar to the turbulent regime occurs abruptly. To better understand this transition, the time evolution of turbulent patches, created by controlled finite amplitude perturbations, have been studied in the literature. These studies mostly focused on pipe flows for which a finite lifetime of the patch was proven. The same conclusion was drawn in the only available study performed in a Taylor Couette setup. Here, we measured the lifetime in a different size TC setup. We show that the lifetime is indeed finite and also very sensitive to the boundary condition, but not much to the perturbation mechanism. We suggest that in addition to the Reynolds number, the lifetime depends on the aspect ratio to the radius ratio of the setup. INTRODUCTION Linearly stable shear flows such as pipe and plane Couette flows show an abrupt transition from the laminar to the turbulent regime. The time evolution of a turbulent patch, created by sufficiently large finite amplitude perturbations, have been investigated to better understand this transition [7, 6, 8, 3, 2]. All these studies have agreed on the exponential probability distribution of the lifetime and the increase of the mean lifetime with Reynolds number. However, there was a dispute over the divergence of the mean lifetime at a critical Reynolds number. This dispute was finally settled by Avila et al. [2] which proved a super-exponential scaling of the mean lifetimes in pipe flows that indicates that no threshold of lifetime divergence exists. Consistent with the obtained results in pipe flows, a finite lifetime was reported in the only available study performed in a Taylor Couette setup [5]. But there, the influence of geometrical properties, boundary condition and perturbation type (global or local) were not examined. In the current work, we aim to examine the effect of these parameters on the patch lifetime in a TC setup. EXPERIMENTAL MEASUREMENT SETUP The setup consisted of two concentric cylinders with the geometrical properties listed in Table 1. The end plates were attached to the outer cylinder with a gap of 1.5 mm between the end surfaces of the inner cylinder and the plates. Two geometrical non-dimensional numbers, the aspect ratio Γ = L/d and the radius ratio η = ri/ro, are defined. The Reynolds number is described as Re = roΩod ν where Ωo and ν are the angular velocity of the outer cylinder and the kinematic viscosity of fluid. Two mechanisms were deployed to create the patch: first, Global Perturbation (GP hereafter) by counter-rotating of the inner cylinder (at a frequency of 0.65 Hz) for 3 sec, and second, Local Perturbation (LP hereafter) by simultaneous injection and suction of water from two holes in the middle of the inner cylinder. It has to be noted that except during the global perturbation, the inner cylinder was always stationary. For flow visualisation, Iriodin pigments (Iriodin 121) were added to water. The particles were illuminated by exposing the cylinders to white (halogen) light from a projector. To capture the flow a CCD camera with a frequency of 10 Hz and resolution of 500 × 2800 pixels (corresponding to 37.5 mm width × 200 mm height in physical space) was used. For determining the lifetime, each image matrix was subtracted and normalized by the average of 100 images taken from the laminar flow. Standard deviation of the normalized image displayed a clear distinction between the images with and without the patch. Parameter ri(mm) ro(mm) d(mm) η Γ Recr Current study 110 120 10.0 0.917 22 ≈6000 Borrero-Echeverry et al.[6] 66.38 76.20 9.82 0.871 36 ≈16000 Table 1. Geometrical properties of setup. ri and ro are the inner and outer cylinder radius and d = ro − ri is the gap size. Recr is the Reynolds number where sudden transition from laminar to turbulent flow takes place. RESULTS A common feature was observed at all Reynolds number (Figure 1); the distribution had an exponentially decaying tail after an initial patch formation time. The probability of survival was described as an exponential function of the time and Reynolds number, P (t, Re) ∼ exp(−(t)/τ(Re)), where τ is the mean characteristic lifetime determined by fitting a line to the tail of each distribution. The mean characteristic lifetime grew by increasing the Reynolds number, with the super-exponential function suggested by Hof et al. [7, 6], τ(Re)−1 = exp(−exp(aRe+b)), as the best fit for this growth. This indicates the boundedness of the lifetime in TC setup. 0 100 200 300 400 500 600 700 800 90