Christoph Egbers

A comparative study of different CFD-codes for numerical simulation of gas–solid fluidized bed hydrodynamics

Abstract The hydrodynamics of a gas–solid fluidized bed reactor were studied numerically. Computational two-dimensional results from open source software packages MFIX and OpenFOAM , and those obtained from the commercial software package Fluent were discussed and compared with numerical and experimental data existing in the literature. The gas–solid flow was simulated applying the multifluid Eulerian–Eulerian model, where the solid phase is treated as a continuum. The solid-phase properties were calculated by using the kinetic theory of granular flow. Momentum exchange coefficients were calculated using the Gidaspow and Syamlal–O’Brien drag functions. Pressure drop and bed expansion ratio predicted by the simulations were in relatively close agreement with benchmark numerical and experimental data sets in the bubbling regime. Contrary to the OpenFOAM predictions, computations with MFIX and Fluent predicted instantaneous and time-average local voidage and velocity profiles which are comparable with results from the literature.

Routes to chaos in wide-gap spherical Couette flow

The dynamical behavior of wide-gap instabilities in spherical Couette flow is investigated experimentally with chaos analyzing techniques applied on time series from Laser-Doppler-Velocimetry (LDV) measurements. With an increasing Reynolds number of the rotating inner sphere, the flow undergoes two Hopf bifurcations and several mode changes. The transition scenarios for two different gap widths investigated can be described primarily as the Ruelle–Takens–Newhouse type and show a strong dependence on the meridional coordinate. The transition processes are accompanied by phenomena such as a locked torus in reconstructed phase space and periodic long wave modulations of chaotic flows. Although the investigated gap widths are all classified as wide, a comparison exhibits significant differences in the development of chaotic motion.

Inertial waves in a spherical shell induced by librations of the inner sphere: experimental and numerical results

We begin with an experimental investigation of the flow induced in a rotating spherical shell. The shell globally rotates with angular velocity Ω. A further periodic oscillation with angular velocity 0 ≤ ω ≤ 2Ω, a so-called longitudinal libration, is added on the inner spheres rotation. The primary response is inertial waves spawned at the critical latitudes on the inner sphere, and propagating throughout the shell along inclined characteristics. For sufficiently large libration amplitudes, the higher harmonics also become important. Those harmonics whose frequencies are still less than 2Ω behave as inertial waves themselves, propagating along their own characteristics. The steady component of the flow consists of a prograde zonal jet on the cylinder tangent to the inner sphere and parallel to the axis of rotation, and increases with decreasing Ekman number. The jet becomes unstable for larger forcing amplitudes as can be deduced from the preliminary particle image velocimetry observations. Finally, a wave attractor is experimentally detected in the spherical shell as the pattern of largest variance. These findings are reproduced in a two-dimensional numerical investigation of the flow, and certain aspects can be studied numerically in greater detail. One aspect is the scaling of the width of the inertial shear layers and the width of the steady jet. Another is the partitioning of the kinetic energy between the forced wave, its harmonics and the mean flow. Finally, the numerical simulations allow for an investigation of instabilities, too local to be found experimentally. For strong libration amplitudes, the boundary layer on the inner sphere becomes unstable, triggering localized Gortler vortices during the prograde phase of the forcing. This instability is important for the transition to turbulence of the spherical shell flow.

The geoflow-experiment on ISS (Part II): Numerical simulation

Convection in spherical shells under the influence of a radial force field is an important problem in classical convection theory. It is difficult to reproduce in a terrestrial laboratory though, because there gravity is everywhere downwards rather than radially inwards. It turns out that one can produce a radial force field, by applying a high voltage difference between the inner and outer spheres. The combination of the electric field and the temperature-dependence of the fluids dielectric coefficient then produces an r−5 central force field. Of course, in a terrestrial laboratory one still has the external gravity as well. In order to eliminate this effect and produce a purely radial force field, an experiment is planned on the International Space Station. In this paper we will present the results of numerical simulations intended to assist in the design of this experiment, for example in choosing the optimal radius ratios. We solve for the onset of convection in such an r−5 force field, as a function of the radius ratio (varying between 0.3 and 0.6), the Prandtl number (varying between 1 and 100), and the Taylor number (measuring an overall rotation of the whole shell, which will also be possible in the experiment).

THE GEOFLOW-EXPERIMENT ON ISS (PART I): EXPERIMENTAL PREPARATION AND DESIGN OF LABORATORY TESTING HARDWARE

Thermal convection in a spherical shell represents an important model in fluid dynamics and geophysics. Investigations on thermal convective instabilities occuring in the spherical gap flow under terrestrial conditions are of basic importance especially for the understanding of symmetry-breaking bifurcations during the transition to chaos. Microgravity ,experiments on thermal convection with a simulated central force field are important for the understanding of large scale geophysical motions as the convective transport phenomena in the Earth’s liquid outer core. This report summari zes the concurrent experimental (part I), numerical (part II) and theoretical (part ICI) studies for the preparation of an International Space Station (ISS) experiment inside the Fluid Science Laboratory (FSL). This special experimental device with respect to geophysical simulations is called GEOFLOW. A central symmetric force field similar to the gravity field acting on planets can be produced using the effect of the dielectrophoretic force field by applying a high voltage potential difference to the inner and outer sphere. How visualization, Wollaston Shearing Interferometry and Laser Doppler Velocimetry will be used to determ@e the expected flow pattern during the space experiments. 0 2003 COSPAR. Published by Elsevier Ltd. All rights reserved.

Estimates on diagnostic methods for investigations of thermal convection between spherical shells in space

Thermal convection in a spherical shell represents an important model in fluid dynamics and geophysics. Investigations on thermal convective instabilities occurring in the spherical gap flow under terrestrial conditions are of basic importance, especially for the understanding of symmetry-breaking bifurcations during the transition to chaos. Microgravity experiments on thermal convection with a simulated central force field are important for the understanding of large-scale geophysical motions such as the convective transport phenomena in the Earths liquid outer core. More than one diagnostic tool is needed to examine and characterize the different flow types. Flow visualization, Wollaston shearing interferometry and laser Doppler velocimetry should be available for space experiments. This report summarizes concurrent theoretical, numerical and experimental studies for the preparation of a Get Away Special (Shuttle) experiment as well as a Space Station experiment inside the Fluid Science Laboratory. The special experimental device for investigations of supercritical and turbulent thermal convection in spherical shells under a central force field with respect to geophysical simulations is called an electrohydrodynamic container. A central symmetric force field similar to the gravity field acting on planets can be produced using the effect of the dielectrophoretic force field by applying a high-voltage potential difference to the inner and outer sphere.

A Study on Flow Transition and Development in Circular and Rectangular Ducts

The present paper reports observations on some aspects regarding the dependence of the transition Reynolds number and flow development on the inlet flow conditions and the entrance length in circular and rectangular ducts for Re m ≤ 106 × 10 3 , where Re m is the Reynolds number based on the bulk fiow velocity (U b ) and the duct integral length scale (D). The hot-wire anemometer was used to carry out measurements close to the circular duct exit; however, the laser-Doppler anemometry was utilized for the rectangular duct measurements. Particular considerations were given to the bulk flow velocity, the mean-velocity profile, the centerline-average-velocity, and the centerline turbulence statistics to the fourth order. Transition criteria in both ducts were discussed, reflecting effects of flow geometry, entrance flow conditions, and entrance length on the transition Reynolds number. A laminar behavior was maintained up to Re m ≈ 15.4 × 10 3 and Re m ≈2 × 10 3 in the circular and rectangular ducts, respectively, and the transition was observed to take place at different downstream positions as the inlet flow velocity varied.

Stability of natural convection between spherical shells: energy theory

Abstract The energy stability problem with respect to axisymmetric disturbances of the natural convection in the narrow gap between two spherical shells under the earth gravity is discussed. The results are compared with the results of the linear stability analysis for the same problem. The problem is solved for different fluids with Pr=0–100 and different radius ratios η=0.9,0.925,0.95. With the aid of the variational principle Euler–Lagrange equations are received, which have the form of an eigenvalue problem, that is solved by means of Galerkin–Chebyshev spectral method. The convergence problem and the dependence of the critical stability parameter on Prandtl number are discussed. The calculations show that there is a big difference between critical numbers for energy and linear stability theories for the small Prandtl numbers. For large Prandtl numbers this difference is very small.

Higher order dynamics of baroclinic waves

Instabilities in the form of baroclinic waves occur in a rotating cylindrical annulus cooled from within. Flow visualisation studies and LDV-measurements of the radial velocity component were carried out in an annulus with an aspect ratio of 4.4. The flow undergoes transitions from the laminar stable state through baroclinic waves, both stable and time-varying, to an irregular state. Based on the time series of the radial velocity at fixed point in the rotating annulus, the attractors of the flow match previous results based on temperature measurements. The bifurcation diagram of extrema in the radial velocity shows the existence of low dimensional chaos at the transition from the axisymmetric flow to periodic baroclinic waves. This bifurcation scenario at low rotation rates is substantially different from the nonlinear behaviour of Taylor-Couette flow.

The GEOFLOW-experiment on ISS (part III): Bifurcation analysis

Codimension 2 bifurcation of convective patterns in the nonrotating spherical Benard Problem of Boussinesq fluid is studied using center manifold reduction. The aim is to determine, in the GEOFLOW-experiment framework, the physical parameters in order to obtain intermittent-like behaviour dynamics. We compute the critical aspect ratio and Rayleigh number corresponding to the (l, l+1) mode interaction. In the case of the (2,3) interaction, the existence of heteroclinic cycles are examined versus the Prandtl number.