Olivier Daube

Resolution of the 2D Navier-Stokes equations in velocity-vorticity form by means of an influence matrix technique

Abstract An influence matrix technique is proposed to enforce both the continuity equation and the definition of the vorticity in the treatment of the 2D incompressible Navier-Stokes equations. It is shown and supported by numerical experiments that at each time step the divergence is actually equal to zero within machine accuracy. The same result is obtained for the definition of the vorticity.

Further experiments on vortex formation around an oscillating and translating airfoil at large incidences

The starting flows past a two-dimensional NACA 0012 airfoil translating and oscillating at large incidences are investigated by visualization experiments and numerical calculations. The airfoil model is set in motion impulsively and subjected simultaneously to a constant translation and harmonic oscillation in pitch. The evolution of the vortex wake is followed in a sequence of streamline visualizations and the wake pattern generated is analysed. The parameters varied in the visualization experiment are the Reynolds number ranging from 1500 to 10000, the reduced frequency from 0.1 to 1.0, the mean incidence 30° or 15° and the angular amplitude 15° or 7°. There are also two additional parameters of special interest: the airfoil cross-section and the pitching axis. The effects of these parameters are discussed in relation to the resultant wake patterns. Some comparison is made with the results of earlier experiments.

The 1[ratio]2 mode interaction in exactly counter-rotating von Kármán swirling flow

The flow produced in an enclosed cylinder of height-to-radius ratio of two by the counter-rotation of the top and bottom disks is numerically investigated. When the Reynolds number based on cylinder radius and disk rotation is increased, the axisymmetric basic state loses stability and different complex flows appear successively: steady states with an azimuthal wavenumber of 1; travelling waves; near-heteroclinic cycles; and steady states with an azimuthal wavenumber of 2. This scenario is understood in a dynamical system context as being due to a nearly codimension-two bifurcation in the presence of

Experimental and numerical study of the shear layer instability between two counter-rotating disks

O(2)

Survey of instability thresholds of flow between exactly counter-rotating disks

symmetry. A bifurcation diagram is determined, together with the most dangerous eigenvalues as functions of the Reynolds number. Two distinct types of near-heteroclinic cycles are observed, with either two or four bursts per cycle. The physical mechanism for the primary instability could be the Kelvin–Helmholtz instability of the equatorial azimuthal shear layer of the basic state.

Natural convection in a differentially heated horizontal cylinder: Effects of Prandtl number on flow structure and instability

The shear layer instability in the flow between two counter-rotating disks enclosed by a cylinder is investigated experimentally and numerically, for radius-to-height ratio Γ=R/h between 2 and 21. For sufficiently large rotation ratio, the internal shear layer that separates two regions of opposite azimuthal velocities is prone to an azimuthal symmetry breaking, which is investigated experimentally by means of visualization and particle image velocimetry. The associated pattern is a combination of a sharp-cornered polygonal pattern, as observed by Lopez et al. (2002) for low aspect ratio, surrounded by a set of spiral arms, first described by Gauthier et al. (2002) for high aspect ratio. The spiral arms result from the interaction of the shear layer instability with the Ekman boundary layer over the faster rotating disk. Stability curves and critical modes are experimentally measured for the whole range of aspect ratios, and are found to compare well with numerical simulations of the three-dimensional time-dependent Navier–Stokes equations over an extensive range of parameters. Measurements of a local Reynolds number based on the shear layer thickness confirm that a shear layer instability, with only weak curvature effect, is responsible for the observed patterns. This scenario is supported by the observed onset modes, which scale as the shear layer radius, and by the measured phase velocities.

Axisymmetric numerical simulations of turbulent flow in rotor stator enclosures

The three-dimensional linear instability of axisymmetric flow between exactly counter-rotating disks is studied numerically. The dynamics are governed by two parameters, the Reynolds number Re based on cylinder radius and disk rotation speed and the height-to-radius ratio Γ. The stability analysis performed for 0.5 ≤ Γ ≤ 3 shows that non-axisymmetric modes are dominant and stationary and that the critical azimuthal wavenumber is a decreasing function of Γ. The patterns of the dominant perturbations are analysed and a physical mechanism related to a shear layer instability is discussed. No evidence of complex dynamical behaviour is seen in the neighbourhood of the 1: 2 codimension-two point when the m = 2 threshold precedes that of m = 1. Axisymmetric instabilities are also calculated; these may be stationary or Hopf bifurcations. Their thresholds are always higher than those of non-axisymmetric modes

Numerical investigation of the first bifurcation for the flow in a rotor-stator cavity of radial aspect ratio 10

Natural convection in a differentially heated horizontal cylinder is investigated numerically and analytically. Particular attention is paid to the structure of steady convection, the nature of the transients and the onset of unsteadiness for a range of Prandtl numbers extending from 0.7 to infinity. The numerical algorithm integrates the 2-D Navier-Stokes equations in velocity-pressure formulation with a Chebyshev-Fourier spatial approximation. A gradual shift from the conduction to the boundary layer regime is observed for increasing Rayleigh number and the steady flow structure becomes rapidly independent of Pr. Whereas classical scalings are obtained for the azimuthal velocity and the thermal boundary layer thickness, the dynamic boundary layer thickness is found to be independent of the Prandtl number. A simplified semi-analytical model derived from projecting the governing equations on the lowest Fourier modes is proposed, which explains this property. Its solutions are in good quantitative agreemen...

Competition between axisymmetric and three-dimensional patterns between exactly counter-rotating disks

We present axisymmetric numerical simulations of transitional and chaotic flow regimes in rotor–stator cavities of radial aspect ratio approximately 8. These simulations are carried out using a second order time and space accurate algorithm which integrates the axisymmetric unsteady Navier–Stokes equations in stream function-azimuthal vorticity-azimuthal velocity form. Detailed flow analysis has been carried out for selected values of the rotational Reynolds number (Reθ) up to 106. At the largest value considered, computations have been performed using 4096 mesh points in the radial direction, which has required using a multi-domain decomposition algorithm implemented on a parallel machine. The limitations and consequences of the axisymmetry assumption are first discussed and checked against available experimental results. The evolutions of the instantaneous flow structure and of its first and second order statistic moments as the Reynolds number increases are discussed. It is shown that the dynamics of the flow mainly consists of travelling waves propagating in the stator and rotor boundary layers and of inertial waves in the core region, and that for moderate Reynolds numbers (Reθ≃3×105), the rotor boundary layer is almost completely steady while large amplitude fluctuations are found in the stator boundary layer. The evolution of second order moments confirms the fundamentally asymmetrical role of the boundary layers along the rotor and along the stator. A turbulent kinetic energy budget is shown which exhibits some specific features attributed to the rotation effects and to a lesser extent to the axisymmetry assumption.

Numerical and Experimental Study of the Unsteady Viscous Flow Generated by an Impulsively Started Elliptic Cylinder

Abstract The nature of transition to unsteadiness of rotor–stator disk flows of large radial aspect ratio is investigated by means of several numerical tools which consist in computing the base flow even when unstable, performing linearized or non-linear time integrations starting from initial conditions of different amplitudes and computing the spectrum of the Jacobian using the ARPACK library. From these numerical experiments we conclude that, in a cavity of radial aspect ratio 10, the transition to unsteadiness occurs through a subcritical Hopf bifurcation. In addition these calculations show the existence of a large amplitude chaotic branch for values of the Reynolds number far below the linear stability threshold, and onto which the solutions are attracted for large subcritical values due to the strong non-normality of the Jacobian of the evolution operator.