Turbulent flow between a rotating and a stationary disk

Abstract

Turbulent flow between a rotating and a stationary disk is studied. Besides its fundamental importance as a three-dimensional prototype flow, such flow fields are frequently encountered in rotor–stator configurations in turbomachinery applications. A direct numerical simulation is therefore performed by integrating the time-dependent Navier–Stokes equations until a statistically steady state is reached and with the aim of providing both long-time statistics and an exposition of coherent structures obtained by conditional sampling. The simulated flow has local Reynolds number r2ω/v = 4 × 105 and local gap ratio s/r = 0.02, where ω is the angular velocity of the rotating disk, r the radial distance from the axis of rotation, v the kinematic viscosity of the fluid, and s the gap width.The three components of the mean velocity vector and the six independent Reynolds stresses are compared with experimental measurements in a rotor–stator flow configuration. In the numerically generated flow field, the structural parameter a1 (i.e. the ratio of the magnitude of the shear stress vector to twice the mean turbulent kinetic energy) is lower near the two disks than in two-dimensional boundary layers. This characteristic feature is typical for three-dimensional boundary layers, and so are the misalignment between the shear stress vector and the mean velocity gradient vector, although the degree of misalignment turns out to be smaller in the present flow than in unsteady three-dimensional boundary layer flow. It is also observed that the wall friction at the rotating disk is substantially higher than at the stationary disk.Coherent structures near the disks are identified by means of the λ2 vortex criterion in order to provide sufficient information to resolve a controversy regarding the roles played by sweeps and ejections in shear stress production. An ensemble average of the detected structures reveals that the coherent structures in the rotor–stator flow are similar to the ones found in two-dimensional flows. It is shown, however, that the three-dimensionality of the mean flow reduces the inter-vortical alignment and the tendency of structures of opposite sense of rotation to overlap. The coherent structures near the disks generate weaker sweeps (i.e. quadrant 4 events) than structures in conventional two-dimensional boundary layers. This reduction in the quadrant 4 contribution from the coherent structures is believed to explain the reduced efficiency of the mean flow in producing Reynolds shear stress.

Large eddy simulation and measurements of turbulent enclosed rotor-stator flows

“…We have numerically verified that the axial velocity component V + z is fairly close to zero along the disks. The velocities have been normalized by the tangential friction velocity v τ θ = (ν∂V θ /∂z) 1/2 to enable direct comparisons with the work of Lygren and Andersson [21]. The magnitude of the velocity vector (Fig.5c) follows rather closely the law of the wall as obtained by Lygren and Andersson [21] in an open rotor-stator cavity.…”

“…The velocities have been normalized by the tangential friction velocity v τ θ = (ν∂V θ /∂z) 1/2 to enable direct comparisons with the work of Lygren and Andersson [21]. The magnitude of the velocity vector (Fig.5c) follows rather closely the law of the wall as obtained by Lygren and Andersson [21] in an open rotor-stator cavity. The profiles along both disks are compared to the profile of a turbulent boundary layer on a flat plate obtained by Rotta [30].…”

“…At Re = 10 6 , 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 a 1 (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 [1,21,22]. …”

“…The rotor-stator problem has also proved to be a fruitful means of studying turbulence properties with wall confinement and rotation as this specific configuration is a relatively simple case where rotation brings significant modifications to the turbulent field. Finally, rotating disk flows are also among the simplest flows where the boundary layers are three-dimensional from their inception and they are therefore well suited for studying the effects of mean-flow three-dimensionality on the turbulence and its structure [14,19,21]. Until now, numerical studies have been dedicated to simpler flows: single disk flows [44], axisymmetric flows using statistical approaches (Reynolds Averaged Navier-Stokes) [27], idealized rotor-stator cavities [21], or enclosed rotor-stator cavities but at much lower rotation rates using Direct Numerical Simulation (DNS) [28,35].…”

“…At present, computer performances only permit DNS of transitionally turbulent cavity flows (Re ≈ 10 5 ) [27,35]. In a simpler flow model, where the flow is restricted to an angular section of the cavity and assumed homogeneous also in the radial direction (assuming slow variation of the turbulence statistics in the radial direction far from the transition), Lygren and Andersson [21] performed DNS at higher Reynolds number (Re = 4 × 10 5 ) using a second-order finite-difference scheme. They provided a detailed set of data to analyze the coherent structures near the two disks.…”