Patrick Bontoux

Annular and spiral patterns in flows between rotating and stationary discs

Different instabilities of the boundary layer flows that appear in the cavity between stationary and rotating discs are investigated using three-dimensional direct numerical simulations. The influence of curvature and confinement is studied using two geometrical configurations: (i) a cylindrical cavity including the rotation axis and (ii) an annular cavity radially confined by a shaft and a shroud. The numerical computations are based on a pseudo-spectral Chebyshev–Fourier method for solving the incompressible Navier–Stokes equations written in primitive variables. The high level accuracy of the spectral methods is imperative for the investigation of such instability structures. The basic flow is steady and of the Batchelor type. At a critical rotation rate, stationary axisymmetric and/or three-dimensional structures appear in the Bodewadt and Ekman layers while at higher rotation rates a second transition to unsteady flow is observed. All features of the transitions are documented. A comparison of the wavenumbers, frequencies, and phase velocities of the instabilities with available theoretical and experimental results shows that both type II (or A) and type I (or B) instabilities appear, depending on flow and geometric control parameters. Interesting patterns exhibiting the coexistence of circular and spiral waves are found under certain conditions.

A spectral projection method for the simulation of complex three-dimensional rotating flows

Abstract In this paper, we present an efficient projection method to solve the three-dimensional time-dependent incompressible Navier–Stokes equations in primitive variables formulation using spectral approximations. This method is based on a modification of the algorithm proposed by Goda [J. Comp. Phys. 30 (1979) 76]. It brings an improvement by introducing a preliminary step for the pressure in order to allow a temporal evolution of the normal pressure gradient at the boundaries. Its efficiency is brought to the fore by comparison with the Godas algorithm. The modified projection method is then applied to the simulation of complex three-dimensional flows in rotating cavities, involving either a throughflow or a differential rotation.

Vortex breakdown in a three-dimensional swirling flow

Time-dependent swirling flows inside an enclosed cylindrical rotor–stator cavity with aspect ratio H / R = 4, larger than the ones usually considered in the literature, are studied. Within a certain range of governing parameters, vortex breakdown phenomena can arise along the axis. Very recent papers exhibiting some particular three-dimensional effects have stimulated new interest in this topic. The study is carried out by a numerical resolution of the three-dimensional Navier–Stokes equations, based on high-order spectral approximations in order to ensure very high accuracy of the solutions. The first transition to an oscillatory regime occurs through an axisymmetric bifurcation (a supercritical Hopf bifurcation) at Re = 3500. The oscillatory regime is caused by an axisymmetric mode of centrifugal instability of the vertical boundary layer and the vortex breakdown is axisymmetric, being composed of two stationary bubbles. For Reynolds numbers up to Re = 3500, different three-dimensional solutions are identified. At Re = 4000, the flow supports the k = 5 mode of centrifugal instability. By increasing the rotation speed to Re = 4500, the vortex breakdown evolves to an S-shaped type after a long computational time. The structure is asymmetric and gyrates around the axis inducing a new time-dependent regime. At Re = 5500, the structure of the vortex breakdown is more complex: the upper part of the structure takes a spiral form. The maximum rotation speed is reached at Re = 10000 and the flow behaviour is now chaotic. The upper structure of the breakdown can be related to the spiral-type. Asymmetric flow separation on the container wall in the form of spiral arms of different angles is also prominent.

Coupled numerical and theoretical study of the flow transition between a rotating and a stationary disk

Both direct numerical simulation and theoretical stability analysis are performed together in order to study the transition process to turbulence in a flow between a rotating and a stationary disk. This linear stability analysis considers the complete rotor-stator flow and then extends the results of Lingwood [J. Fluid Mech. 299, 17 (1995); 314, 373 (1996)] obtained in a single disk case. The present linear analysis also extends the former two-disk computations of Itoh [ASME FED 114, 83 (1991)], only limited to a hydrodynamic spatial instability analysis. Moreover, in the present work, this approach is completed by discussing the effects of buoyancy-driven convection on the flow stability and by absolute/convective analysis of the flow. Coupled with accurate numerical computations based on an efficient pseudo-spectral Chebyshev-Fourier method, this study brings new insight on the spatio-temporal characteristics of this flow during the first stages of transition. For instance, an exchange of stability from a steady to a periodic flow with spiral structures is observed for the first time numerically in such cavity of large aspect ratio. The nature of the first bifurcation is discussed as well as the influence on it of disturbances coming from the end-wall boundary layer. Annular and spiral patterns are observed in the unstable stationary disk layer with characteristic parameters agreeing very well with the present theoretical results. Then, these structures are interpreted in terms of type I and type II generic instabilities. Moreover, the absolute instability regions which are supposed to be strongly connected with the turbulent breakdown process are also identified and the critical Reynolds numbers of the convective/absolute transition in both Ekman and Bodewadt layers are given.

Interaction between Ekman pumping and the centrifugal instability in Taylor–Couette flow

The endwalls in a Taylor–Couette cell introduce adjacent boundary layers that interact with the centrifugal instability. We investigate the interaction between the endwall Ekman layers and the Taylor vortices near transition from nonvortical to vortical flow via direct numerical simulation using a spectral method. We consider a radius ratio of η=0.75 in a short annulus having a length-to-gap ratio of Γ=6. To analyze the nature of the interaction between the vortices and the endwall layers, three endwall boundary conditions were considered: fixed endwalls, endwalls rotating with the inner cylinder, and stress-free endwalls. Below the critical Taylor number, endwall vortices for rotating endwalls are more than twice the strength of the vortices for fixed endwalls. This trend continues well above the transition to vortical flow, consistent with a simple force balance analysis near the endwalls. Stress-free endwalls result in endwall vortices that are similar in strength to those for rotating endwalls above t...

Convection in the vertical midplane of a horizontal cylinder. Comparison of two-dimensional approximations with three-dimensional results

Abstract Cylindrical, differentially heated and horizontal enclosures are commonly used in technological processes. A knowledge of the flow patterns in such a system is important for process optimization. In the past, flow predictions have often been made by using an asymptotic analytical approximation in the core, or by assuming a two-dimensional solution for the plane of symmetry. Laser Doppler anemometry studies, recently conducted by Schiroky and Rosenberger, have shown that in reality the free convection flows in the above configuration are highly three-dimensional. Here we compare the results for the vertical midplane obtained from the experiments and from 3-D numerical solutions with solutions of the aforementioned approximations. Both the core-driven and boundary layer-driven regimes are considered. In general the approximations give the correct Rayleigh-number-dependence of the velocities in the two regimes. However, the transition between the regimes and the magnitude and distribution of the velocity components were found to significantly depend on the 2-D approximation used.

Computational solution for fluid flow under solid/liquid phase change conditions

Abstract A fixed grid method based on an enthalpy–porosity formulation for liquid/solid phase transition is extended to compute the time-dependent solutal convection in the melt during directional solidification of alloys. A finite volumes approximation is used for uniform and refined grids with a second order Euler scheme. The ability of the method to describe accurately the flow transitions and regimes is considered with respect to the results of the linear theory of stability and of available spectral accurate calculations.

Numerical solution and analysis of asymmetric convention in a vertical cylinder: An effect of Prandtl number

Three‐dimensional steady flows are simulated with a finite‐difference technique in a cylinder of height‐to‐radius ratio A=4 heated from below. Basic axisymmetric (m=0) and asymmetric (m=1,2) modes are identified in the solutions. The relative contributions of these modes to the supercritical flow are assessed for Rayleigh numbers up to Ra≅10 Rac at Prandtl number Pr=6.7 and Ra≅2.5 Rac at Pr=0.02. At low Rayleigh number the core flow exhibits characteristic features of the m=1 dominant mode. At elevated Rayleigh numbers secondary vortices corresponding to the m=0 mode appear and develop differently in both size and magnitude for the two Prandtl numbers. The Ra dependence of important characteristic variables is analyzed in light of available nonlinear theories and laboratory measurements.

Turbulent Rayleigh–Bénard convection in a near-critical fluid by three-dimensional direct numerical simulation

This paper presents state of the art three-dimensional numerical simulations of the Rayleigh-Benard convection in a supercritical fluid. We consider a fluid slightly above its critical point in a cube-shaped cell heated from below with insulated sidewalls; the thermodynamic equilibrium of the fluid is described by the van der Waals equation of state. The acoustic filtering of the Navier-Stokes equations is revisited to account for the strong stratification of the fluid induced by its high compressibility under the effect of its own weight. The hydrodynamic stability of the fluid is briefly reviewed and we then focus on the convective regime and the transition to turbulence. Direct numerical simulations are carried out using a finite volume method for Rayleigh numbers varying from 10 6 up to 10 8 . A spatiotemporal description of the flow is presented from the convection onset until the attainment of a statistically steady state of heat transfer. This description concerns mainly the identification of the vortical structures in the flow, the distribution of the Nusselt numbers on the horizontal isothermal walls, the structure of the temperature field and the global thermal balance of the cavity. We focus on the influence of the strong stratification of the fluid on the penetrability of the convective structures in the core of the cavity and on its global thermal balance. Finally, a comparison with the case of a perfect gas, at the same Rayleigh number, is presented.

Axisymmetric and three-dimensional instabilities in an Ekman boundary layer flow

Abstract The instability of Ekman boundary layer flow is studied inside a rotating annular cavity with radial throughflow, which is a relevant geometry of the air cooling system in turbines. The flow is computed by direct numerical simulation using a time-dependent three-dimensional Navier–Stokes solver based on a pseudo-spectral method. The fluid entering the annulus at the inner section then develops into a rotating geostrophic core flanked above and below by two nonlinear Ekman boundary layers and exits at the outer section. In this study, the rotation rate of the cavity is fixed at a given high value, corresponding to an Ekman number E =2.24×10 −3 . When the throughflow is weak, the motion is steady and the boundary layer flow is well described by Ekmans analytical solution. On increasing the mass flow rate, the flow becomes unsteady and perturbations appear in the form of counter-rotating pairs of vortices adjacent to upper and lower surfaces of the cavity. Multiple stable solutions, involving circular and spiral waves with different numbers of arms, are obtained at fixed mass flow rate. The wavenumber and frequency of both circular and spiral waves are determined to be characteristic of the type II viscous Ekman layer instability.