Optimal transient growth in an incompressible flow past a backward-slanted step

With the aim of providing a first step in the quest for a reduction of the aerodynamic drag on the rear-end of a car, we study the phenomena of separation and reattachment of an incompressible flow focusing on a specific aerodynamic geometry, namely a backward-slanted step at 25 degrees of inclination. The ensuing recirculation bubble provides the basis for an analytical and numerical investigation of streamwise-streak generation, lift-up effect, and turbulent-wake and Kelvin-Helmholtz instabilities. A linear stability analysis is performed, and an optimal control problem with a steady volumic forcing is tackled by means of variational formulation, adjoint method, penalization scheme and orthogonalization algorithm. Dealing with the transient growth of spanwise-periodic perturbations and inspired by the need of physically-realizable disturbances, we finally provide a procedure attaining a kinetic-energy maximal gain of the order of one million with respect to the power introduced by the external forcing.Comment: 17 figure

Large Eddy Simulation of Non-Isothermal Turbulent Rotor-Stator Flows

Non-isothermal turbulent flows in an enclosed rotorstator cavity are here investigated using large eddy simulation (LES). Besides their fundamental importance as three-dimensional prototype flows, such flows arise in many industrial applications and especially in turbomachineries. The LES is performed using a Spectral Vanishing Viscosity technique, which is shown leading to stable discretizations without sacrificing the formal accuracy of the spectral approximation. The LES results have been favorably compared to velocity measurements in the isothermal case. The Boussinesq approximation is then used to take into account the centrifugal-buoyancy effects. The thermal effects have been examined for Re equal to 1 million in a rotor-stator cavity of aspect ratio G=(b-a)/h=5 and curvature parameter Rm=(b-a)/(b+a)=1.8 (a, b the inner and outer radii of the rotor and h the interdisk spacing) and for Rayleigh numbers up to Ra=108. These LES results provide accurate, instantaneous quantities which are of interest in understanding the physics of turbulent flows and heat transfers in an interdisk cavity. The averaged results show small effects of density variation on the mean and turbulent fields

High-order LES of turbulent heat transfer in a rotor–stator cavity

International audienceThe present work examines the turbulent flow in an enclosed rotor–stator system subjected to heat transfer effects. Besides their fundamental importance as three-dimensional prototype flows, such flows arise in many industrial applications but also in many geophysical and astrophysical settings. Large eddy simulations(LES) are here performed using a spectral vanishing viscosity technique. The LES results have already been favorably compared to velocity measurements in the isothermal case (Séverac et al. 2007) for a large range of Reynolds numbers in an annular cavity of large aspect ratio and weak curvature parameter. The purpose of this paper is to extend these previous results in the non-isothermal case using the Boussinesq approximation to take into account the buoyancy effects. Thus, the effects of thermal convection have been examined for a turbulent flow of air in the same rotor–stator system for Rayleigh numbers up to 100 millions. These LES results provide accurate, instantaneous quantities which are of interest in understanding the physics of turbulent flows and heat transfers in an interdisk cavity. Even at high Rayleigh numbers, the structure of the iso-values of the instantaneous normal temperature gradient at the disk surfaces resembles the one of the iso-values of the tangential velocity with large spiral arms along the rotor and more thin structures along the stator. The averaged results show small effects of density variation on the mean and turbulent fields. The turbulent Prandtl number is a decreasing function of the distance to the wall with 1.4 close to the disks and about 0.3 in the outer layers. The local Nusselt number is found to be proportional to the local Reynolds number to the power 0.7. The evolution of the averaged Bolgiano length scale with the Rayleigh number indicates that temperature fluctuations may have a large influence on the dynamics only at the largest scales of the system for Ra larger than 10 millions, since the averaged Bolgiano length scale remains lower than the thermal boundary layer thicknesses

A 3D pseudospectral algorithm to simulate rotating flows in cylindrical cavities

International audienceWhen simulating flows in cylindrical rotating cavities a difficulty arises from the singularities appearing on the axis. Its singularities are due to the presence of terms 1/r^n (n = 1, 2) in the Navier-Stokes equations, where r is the radial dis-tance. To avoid evaluating differential equation coefficients which are infinite at that point, the grid must exclude the origin or specific pole conditions must be imposed. An efficient and accurate pseudo-spectral method has been here developed using collocation Chebyshev polynomials in the radial and axial directions and Fourier approximation in the azimuthal direction. To avoid the difficulty on the axis without prescribing any pole conditions, a new approach based on the work of Heinrichs [W. Heinrichs J. Comp. Phys. 199 (2004) 66-86] has been developed. The calculation domain is defined as (r, θ , z) ∈ [−1, 1] × [0, 2π] × [−1, 1] using an even number N of collocation points in the radial direction. Thus, r = 0 is not a collocation point. The clustering of collocation points around the rotation axis is also avoided due to the utilization of a Gauss-Lobatto distribution. The flow being indeed laminar close to the axis in most of the rotating flows. In the azimuthal direction, the overlap in the discretization is avoided by introducing a shift equal to π/2K (K the number of mesh points in θ -direction) for θ > π in the Fourier transform. The accuracy of the method was checked on the exact steady and unsteady analytical solutions and the capability of the method to simulate complex flows is illustrated considering the well documented case of the vortex breakdown phenomenon

Large eddy simulation and measurements in a turbulent rotor-stator flow

International audienceThere 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

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

International audienceTurbulent 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)

NUMERICAL TREATMENT OF CYLINDRICAL COORDINATE SINGULARITIES

The present work proposes direct numerical simulations of some rotating disk flows using a pseudo-spectral method with collocation Chebyshev polynomials in the radial and axial directions and Fourier approximation in the periodic azimuthal direction. When using cylindrical coordinates to calculate the Navier-Stokes equations the singularity that appears on the axis (r = 0), because of the terms 1/r and 1/r2 is only apparent. To avoid evaluating differential equation coefficients which are infinite the spectral grid must exclude the origin. The interesting issue is how does one impose boundary conditions at the origin? With spectral methods, there are various ways to avoid this difficulty without prescribing any pole conditions. In this work, we have developed a method which consists in discretizing the whole diameter −R ≤ r ≤ R with an even number of radial Gauss-Lobatto nodes. In the azimuthal direction, the overlap in the discretization is avoided by introducing a shift equal to Pi /2K (K the number of mesh points in that direction) for Theta>Pi in the Fourier transform. Spectral convergence of the method is illustrated on an analytical solution. The ability of our numerical method to investigate complex unsteady flows is illustrated for three rotating flows where other reliable experimental and numerical results are available

Transition mechanisms to turbulence in a cylindrical rotor-stator cavity by pseudo-spectral simulations of Navier-Stokes equations

The flow above an infinite rotating disk is an example of 3D boundary-layers where crossflow instability can develop as over swept wings. With a second rotating disk parallel to the first, the configuration schematizes the cavity between the disks holding the blades of a turbine or compressor. Centrifugal and Coriolis forces produce a secondary flow in the meridian plane composed of two thin boundary-layers along the disks separated by a non-viscous geostrophic core. That produces adjacent coupled flow regions that are radically different in terms of flow stability and thickness scales involving very challenging simulations. Identify and characterize the transition mechanism is a necessity for developing future efficient control strategies of turbulent rotating boundary layers. The matter of the transition scenario is currently much debated (see in Viaud, Serre & Chomaz JFM 07, APS 2010) around the idea that a global instability might take place and lead to transition to turbulence. This work addresses the study of coherent structures related to the transition mechanisms in a rotor-stator cavity. An accurate pseudo-spectral algorithm has been developed dealing with the singularity at the centreline of the cylindrical coordinate system. The very unstable stator boundary is characterized at moderate Reynolds number by travelling axisymmetric structures that eventually damped close to the axis and localized rotating spiral at the periphery. The much more stable rotor layer seems governed by the same mechanisms than recently identified by Viaud et al 2010 of a cascade of global modes leading eventually to turbulence

Sensitivity of 2-D turbulent flow past a D-shaped cylinder using global stability

International audienceWe use adjoint-based gradients to analyze the sensitivity of turbulent wake past a D-shaped cylinder at Re = 13000. We assess the ability of a much smaller control cylinder in altering the shedding frequency, as predicted by the eigenfrequency of the most unstable global mode to the mean flow. This allows performing beforehand identification of the sensitive regions, i.e., without computing the actually controlled states. Our results obtained in the frame of 2-D, unsteady Reynolds-averaged Navier-Stokes compare favorably with experimental data reported by Parezanović and Cadot [J. Fluid Mech.693, 115 (2012)] and suggest that the control cylinder acts primarily through a local modification of the mean flow profiles