Dirk Wilhelm

Computational analysis of the two-dimensional-three-dimensional transition in forward-facing step flow

Results are presented from a computational study of the flow over a forward-facing step in a plane channel. The aim of the study is to gain better insight into the three-dimensionality that is typically observed in the separation region of flows over steps and ribs, and in similar configurations. We consider laminar flow at a Reynolds number of 330, based on step height and bulk velocity of the oncoming flow, and the step is assumed to be infinitely extended in the spanwise direction. High-resolution simulations are undertaken using a mixed spectral/spectral-element code. Moreover, a linear stability study of the flow at the step is performed. The results show that, in the case considered, the three-dimensionality is not related to some absolute instability of the separation bubble in front of the step; rather, it is a sensitive reaction of the flow to three-dimensional perturbations present in the oncoming stream. It is demonstrated that disturbance amplitudes of less than 1% of the mean flow (at, say, 10 step heights ahead of the step) already suffice to produce a visibly three-dimensional structure of the separation zone. If the disturbance level is systematically decreased, the three-dimensional state evolves to an almost two-dimensional recirculation. Here, the key finding is that the intensity of the flow response is proportionate to the amplitude of the inflow disturbance, meaning that the breakup of the flow in the step region is a linear (i.e. small) perturbation of the two-dimensional base flow. A comparison of the present simulation results with experimental data shows close agreement concerning, for example, the flow topology in the step region, and the spanwise spacing of the characteristic streaks that form further downstream.

Density-driven instabilities of miscible fluids in a capillary tube: linear stability analysis

A linear stability analysis is presented for the miscible interface formed by placing ah eavier fluid above a lighter one in a vertically oriented capillary tube. The analysis is based on the three-dimensional Stokes equations, coupled to a convection– diffusion equation for the concentration field, in cylindrical coordinates. A generalized eigenvalue problem is formulated, whose numerical solution yields both the growth rate and the two-dimensional eigenmodes as functions of the governing parameters in the form of a Rayleigh number and a dimensionless interfacial thickness. The dispersion relations show that for all values of the governing parameters the threedimensional mode with an azimuthal wavenumber of 1 represents the most unstable disturbance. The stability results also indicate the existence of a critical Rayleigh number of about 920, below which all perturbations are stable. The growth rates are seen to reach a plateau for Rayleigh numbers in excess of 10 6 .I n order to analyse the experimental observations by Kuang et al. (2002), which show that a small amount of net flow can stabilize the azimuthal instability mode and maintain an axisymmetric evolution, a base flow of Poiseuille type is included in the linear stability analysis. Results show that a weak base flow leads to a slight reduction of the growth rates of both axisymmetric and azimuthal modes. However, within the velocity interval that could be analysed in the present investigation, there is no indication that the axisymmetric mode overtakes its azimuthal counterpart.

Three-dimensional spectral element simulations of variable density and viscosity, miscible displacements in a capillary tube

Abstract A high-accuracy numerical approach is introduced for three-dimensional, time-dependent simulations of variable density and viscosity, miscible flows in a circular tube. Towards this end, the conservation equations are treated in cylindrical coordinates. The spatial discretization is based on a mixed spectral element/Fourier spectral scheme, with careful treatment of the singularity at the axis. For the temporal discretization, an efficient semi-implicit method is applied to the variable viscosity momentum equation. This approach results in a constant coefficient Helmholtz equation, which is solved by a fast diagonalization method. Numerical validation data are presented, and simulations are conducted for the three-dimensionally evolving instability resulting from an unstable density stratification in a vertical tube. Some preliminary comparisons with corresponding experiments are undertaken.

Stable and unstable formulations of the convection operator in spectral element simulations

Abstract We show that for the PN–PN−2 spectral element method (SEM), in which the velocity and pressure are approximated by polynomials of order N and N−2, respectively, numerical instabilities may occur in Navier–Stokes simulations. These instabilities depend on the formulation of the convection operator. The numerical scheme is stable for the convective form and one version of the rotational form but unstable for the divergence form and the skew-symmetric form. Further numerical analysis indicates that this instability is not caused by nonlinear effects but occurs also for the linearized momentum equations. We demonstrate that the instability is a consequence of the staggered grid between velocity and pressure, as often used in SEM.

Application of a Spectral Element Method to Two-Dimensional Forward-Facing Step Flow

In the present study, we investigate the two-dimensional laminar flow through a one-sided constriction of a plane channel with a ratio of h:H=1:4 (where h is the step height and H is the channel height). The computational approach employed is based on a mixed implicit/explicit time discretization scheme together with a highly accurate spatial discretization using a PN−PN−2 spectral-element method. It is well known that this so-called forward-facing step (FFS) flow exhibits a singularity in the pressure and the velocity derivatives at the corner point. We account for this singularity by a geometric mesh refinement strategy that was proposed in a hp-FEM context. A detailed numerical study of the FFS flow reveals that length and height of the recirculation zone in front of the step are almost constant for creeping flow. In the limit of high Reynolds numbers the length and height of the recirculation zone increase proportional to Re0.6 and Re0.2, respectively.

High-resolution two-field nuclear magnetic resonance spectroscopy

Nuclear magnetic resonance (NMR) is a ubiquitous branch of spectroscopy that can explore matter at the scale of an atom. Significant improvements in sensitivity and resolution have been driven by a steady increase of static magnetic field strengths. However, some properties of nuclei may be more favourable at low magnetic fields. For example, transverse relaxation due to chemical shift anisotropy increases sharply at higher magnetic fields leading to line-broadening and inefficient coherence transfers. Here, we present a two-field NMR spectrometer that permits the application of rf-pulses and acquisition of NMR signals in two magnetic centres. Our prototype operates at 14.1 T and 0.33 T. The main features of this system are demonstrated by novel NMR experiments, in particular a proof-of-concept correlation between zero-quantum coherences at low magnetic field and single quantum coherences at high magnetic field, so that high resolution can be achieved in both dimensions, despite a ca. 10 ppm inhomogeneity of the low-field centre. Two-field NMR spectroscopy offers the possibility to circumvent the limits of high magnetic fields, while benefiting from their exceptional sensitivity and resolution. This approach opens new avenues for NMR above 1 GHz.

Fluid flow dynamics in MAS systems

The turbine system and the radial bearing of a high performance magic angle spinning (MAS) probe with 1.3mm-rotor diameter has been analyzed for spinning rates up to 67kHz. We focused mainly on the fluid flow properties of the MAS system. Therefore, computational fluid dynamics (CFD) simulations and fluid measurements of the turbine and the radial bearings have been performed. CFD simulation and measurement results of the 1.3mm-MAS rotor system show relatively low efficiency (about 25%) compared to standard turbo machines outside the realm of MAS. However, in particular, MAS turbines are mainly optimized for speed and stability instead of efficiency. We have compared MAS systems for rotor diameter of 1.3-7mm converted to dimensionless values with classical turbomachinery systems showing that the operation parameters (rotor diameter, inlet mass flow, spinning rate) are in the favorable range. This dimensionless analysis also supports radial turbines for low speed MAS probes and diagonal turbines for high speed MAS probes. Consequently, a change from Pelton type MAS turbines to diagonal turbines might be worth considering for high speed applications. CFD simulations of the radial bearings have been compared with basic theoretical values proposing considerably smaller frictional loss values. The discrepancies might be due to the simple linear flow profile employed for the theoretical model. Frictional losses generated inside the radial bearings result in undesired heat-up of the rotor. The rotor surface temperature distribution computed by CFD simulations show a large temperature gradient over the rotor.

On the Relation between Fronts and High-Shear Layers in Wall Turbulence

Direct numerical simulation results of turbulent channel flow are analyzed in order to examine the relation between two kinds of near-wall flow structures, namely the instantaneous shear layers and the fronts which are derived from two-point statistics of the streamwise velocity component. The near-wall shear layers are analyzed by flow visualizations and conditional sampling, while the fronts are examined by means of space-time correlations and spatial two-point correlation functions. The present study focuses on the analysis of the propagation speed and the spatial shape of the structures. Concerning the propagation speed it is shown that the results obtained from flow visualizations are in close agreement with the propagation velocities derived from space-time correlation functions. The comparison of VISA results for the instantaneous shear with spatial structures obtained from two-point correlations of the streamwise velocity and the shear gives evidence that the fronts are intimately related to the pronounced near-wall shear layers.

Density-Driven Instabilities of Variable-Viscosity Miscible Fluids in a Capillary Tube

Abstract: A linear stability analysis is presented for variable‐viscosity miscible fluids in an unstable configuration; that is, a heavier fluid placed above a lighter one in a vertically oriented capillary tube. The initial interface thickness is treated as a parameter to the problem. The analysis is based on the three‐dimensional Stokes equations, coupled to a convection‐diffusion equation for the concentration field, in cylindrical coordinates. When both fluids have identical viscosities, the dispersion relations show that for all values of the governing parameters the three‐dimensional mode with an azimuthal wave number of one represents the most unstable disturbance. The stability results also indicate the existence of a critical Rayleigh number of about 920, below which all perturbations are stable. For the variable viscosity case, the growth rate does not depend on which of the two fluids is more viscous. For every parameter combination the maximum of the eigenfunctions tends to shift toward the less viscous fluid. With increasing mobility ratio, the instability is damped uniformly. We observe a crossover of the most unstable mode from azimuthal to axisymmetric perturbations for Rayleigh numbers greater than 105 and high mobility ratios. Hence, the damping influence is much stronger on the three‐dimensional mode than the corresponding axisymmetric mode for large Rayleigh numbers. For a fixed mobility ratio, similar to the constant viscosity case, the growth rates are seen to reach a plateau for Rayleigh numbers in excess of 106. At higher mobility ratios, interestingly, the largest growth rates and unstable wave numbers are obtained for intermediate interface thicknesses. This demonstrates that, for variable viscosities, thicker interfaces can be more unstable than their thinner counterparts, which is in contrast to the constant viscosity result where growth rate was seen to decline monotonically with increasing interface thickness.

Domain Decomposition Method and Fast Diagonalization Solver for Spectral Element Simulations

To perform direct numerical simulations of flows in more complex geometries such as the flow over an obstacle or over backward- and forward facing steps, numerical methods are required that combine high accuracy with geometric flexibility. In the present work, we employ the P N — P N-2 spectral element method (SEM) of Maday and Patera [6] for the discretization of the Navier-Stokes equations. In this approach, velocity and pressure are approximated by polynomials of order N and N — 2, respectively. The efficiency of this method depends strongly on the solution algorithm for the algebraic systems associated with the Helmholtz equation for the velocity u and the pseudo-Laplacian equation for the pressure q. Since the pseudo-Laplacian equation can become ill-conditioned [4], the solution for q tends to be the computational bottleneck when iterative solvers are employed. Therefore, a considerable effort has been made on the development of efficient pressure preconditioners in the past [2–4]. On the other hand, the costly iterative procedure can by avoided if direct solvers are applied. Very efficient direct solvers exist that are based on fast diagonalization methods, which, however, can only be used for rectangular Cartesian domains [1, 5]. For geometries that are more complex, but may still be decomposed into Cartesian subdomains, suitable domain decomposition methods are needed in order to apply fast diagonalization solvers. In the context of SEM, Couzy and Deville [3] employed a Schur complement method for the decomposition of the Helmholtz matrix in the momentum equation, while still solving the pressure equation it- eratively. In the present work, we will extend the approach of Couzy and Deville [3] to obtain a direct solution of the pressure equation also, employing a suitable domain decomposition along with the influence-matrix technique of Schumann and Benner [9].