Arthur Veldman

Symmetry-preserving discretization of turbulent flow

We propose to perform turbulent flow simulations in such manner that the difference operators do have the same symmetry properties as the underlying differential operators, i.e., the convective operator is represented by a skew-symmetric coefficient matrix and the diffusive operator is approximated by a symmetric, positive-definite matrix. Mimicking crucial properties of differential operators forms in itself a motivation for discretizing them in a certain manner. We give it a concrete form by noting that a symmetry-preserving discretization of the Navier-Stokes equations is stable on any grid, and conserves the total mass, momentum and kinetic energy (for the latter the physical dissipation is to be turned off, of coarse). Being stable on any grid, the choice of the grid may be based on the required accuracy solely, and the main question becomes: how accurate is a symmetry-preserving discretization? Its accuracy is tested for a turbulent flow in a channel by comparing the results to those of physical experiments and previous numerical studies. The comparison is carried out for a Reynolds number of 5600, which is based on the channel width and the mean bulk velocity (based on the channel half-width and wall shear velocity the Reynolds number becomes 180). The comparison shows that with a fourth-order, symmetry-preserving method a 64 × 64 × 32 grid suffices to perform an accurate numerical simulation.

Proper orthogonal decomposition and low-dimensional models for driven cavity flows

A proper orthogonal decomposition (POD) of the flow in a square lid-driven cavity at Re=22,000 is computed to educe the coherent structures in this flow and to construct a low-dimensional model for driven cavity flows. Among all linear decompositions, the POD is the most efficient in the sense that it captures the largest possible amount of kinetic energy (for any given number of modes). The first 80 POD modes of the driven cavity flow are computed from 700 snapshots that are taken from a direct numerical simulation (DNS). The first 80 spatial POD modes capture (on average) 95% of the fluctuating kinetic energy. From the snapshots a motion picture of the coherent structures is made by projecting the Navier–Stokes equation on a space spanned by the first 80 spatial POD modes. We have evaluated how well the dynamics of this 80-dimensional model mimics the dynamics given by the Navier–Stokes equations. The results can be summarized as follows. A closure model is needed to integrate the 80-dimensional system ...

New, quasi-simultaneous method to calculate interacting boundary layers

A quasi-simultaneous method is described to calculate laminar, incompressible boundary layers interacting with an inviscid outer flow. The essential feature of this method is an interactive boundary condition prescribing a linear combination of pressure and displacement thickness which models the behaviour of the outer flow. This way the quasi-simultaneous method avoids difficulties incurred when either direct or inverse methods are used, resulting in fast convergence of the iterative procedure involved. The method is consistent with asymptotic triple-deck theory. Results will be presented for two problems which exhibit strong interaction between the viscous and inviscid regions: i) a boundary layer with a separation bubble, and ii) the flow near the trailing edge of a flat plate.

The numerical simulation of liquid sloshing on board spacecraft

The subject of study is the influence of sloshing liquid on the dynamics of spacecraft. A combined theoretical and experimental approach has been followed. On the one hand, CFD simulations have been carried out to predict the combined liquid/solid body motion. Basically a volume-of-fluid (VOF) approach is followed, however with improvements in the treatment of the free liquid surface: these cover the surface reconstruction and displacement and the calculation of surface tension effects by means of a local height function. Also attention has been paid to the stability of the numerical coupling between solid-body dynamics and liquid dynamics. On the other hand, in-orbit experiments have been carried out with the Sloshsat FLEVO satellite. The paper describes a first comparison between theoretical predictions and experimental findings.

Dynamics of liquid-filled spacecraft

A method is presented for simulating coupled liquid-solid dynamics. An important example of a coupled liquid-solid system is a satellite carrying fuel. The dynamics of the satellite and the onboard fuel influence each other, which may lead to satellite motion that is uncontrollable. For better understanding of the complex dynamics of coupled systems, a numerical model is developed. The model consists of two parts. The first part that solves the liquid motion is only briefly discussed here. The focus in this paper is on the way in which the dynamics of the liquid and the solid body are coupled. For this, the governing equations are presented in which terms appear that represent the force and torque on the solid body due to the sloshing liquid. The governing equations are rewritten such that the discrete approximation of these equations can be integrated in a stable manner for arbitrary liquid/solid mass ratios. Results are presented demonstrating the stability of the present model. A grid-refinement study and a time-step analysis are performed. Finally, the flat-spin motion of a satellite, partially filled with liquid, that flew in 1992 as part of the Wet Satellite Model experiment is studied. Results from the simulation are compared with the actual flight data.

Spectro-consistent discretization of Navier-Stokes: a challenge to RANS and LES

In this paper, we discuss the results of a fourth-order, spectro-consistent discretization of the incompressible Navier-Stokes equations. In such an approach the discretization of a (skew-)symmetric operator is given by a (skew-)symmetric matrix. Numerical experiments with spectro-consistent discretizations and traditional methods are presented for a one-dimensional convection-diffusion equation. LES and RANS are challenged by giving a number of examples for which a fourth-order, spectro-consistent discretization of the Navier-Stokes equations without any turbulence model yields better (or at least equally good) results as large-eddy simulations or RANS computations, whereas the grids are comparable. The examples are taken from a number of recent workshops on complex turbulent flows.

Direct numerical simulation of turbulence at lower costs

Direct Numerical Simulation (DNS) is the most accurate, but also the most expensive, way of computing turbulent flow. To cut the costs of DNS we consider a family of second-order, explicit one-leg time-integration methods and look for the method with the best linear stability properties. It turns out that this method requires about two times less computational effort than Adams–Bashforth. Next, we discuss a fourth-order finite-volume method that is constructed as the Richardson extrapolate of a classical second-order method. We compare the results of this fourth-order method and the underlying second-order method for a DNS of the flow in a cubical driven cavity at Re= 104. Experimental results are available for comparison. For this example, the fourth-order results are clearly superior to the second-order results, whereas their computational effort is about twenty times less. With the improved simulation method, a DNS of a turbulent flow in a cubical lid-driven flow at Re = 50,000 and a DNS of a turbulent flow past a square cylinder at Re = 22,000 are performed.

Glottal flow through a two-mass model: Comparison of Navier–Stokes solutions with simplified models

A new numerical model of the vocal folds is presented based on the well-known two-mass models of the vocal folds. The two-mass model is coupled to a model of glottal airflow based on the incompressible Navier-Stokes equations. Glottal waves are produced using different initial glottal gaps and different subglottal pressures. Fundamental frequency, glottal peak flow, and closed phase of the glottal waves have been compared with values known from the literature. The phonation threshold pressure was determined for different initial glottal gaps. The phonation threshold pressure obtained using the flow model with Navier-Stokes equations corresponds better to values determined in normal phonation than the phonation threshold pressure obtained using the flow model based on the Bernoulli equation. Using the Navier-Stokes equations, an increase of the subglottal pressure causes the fundamental frequency and the glottal peak flow to increase, whereas the fundamental frequency in the Bernoulli-based model does not change with increasing pressure.

Axisymmetric liquid sloshing under low-gravity conditions☆

A numerical simulation program has been developed for the determination of dynamic liquid behaviour in partially filled cylindrical containers during weightlessness. The program, based on the unsteady Navier-Stokes equations, is capable to treat axisymmetric flow; arbitrary free-surface shapes are allowed. In the paper some examples are presented showing liquid response to rotation and axial vibration. The numerical simulations provide information which is used in the definition and evaluation of an experiment onboard Spacelab. The combined theoretical/experimental investigation is directed towards a more efficient deeign of attitude control systems.

Playing with nonuniform grids

Numerical experiments with discretization methods on nonuniform grids are presented for the convection-diffusion equation. These show that the accuracy of the discrete solution is not very well predicted by the local truncation error. The diagonal entries in the discrete coefficient matrix give a better clue: the convective term should not reduce the diagonal. Also, iterative solution of the discrete set of equations is discussed. The same criterion appears to be favourable.