Carlos Härtel

Analysis and direct numerical simulation of the flow at a gravity-current head. Part 1. Flow topology and front speed for slip and no-slip boundaries

Direct numerical simulations are performed of gravity-current fronts in the lock-exchange configuration. The case of small density differences is considered, where the Boussinesq approximations can be adopted. The key objective of the investigation is a detailed analysis of the flow structure at the foremost part of the front, where no previous high-resolution data were available. For the simulations, high-order numerical methods are used, based on spectral and spectral-element discretizations and compact finite differences. A three-dimensional simulation is conducted of a front spreading along a no-slip boundary at a Reynolds number of about 750. The simulation exhibits all features typically observed in experimental flows near the gravity-current head, including the lobe-and-cleft structure at the leading edge. The results reveal that the flow topology at the head differs from what has been assumed previously, in that the foremost point is not a stagnation point in a translating system. Rather, the stagnation point is located below and slightly behind the foremost point in the vicinity of the wall. The relevance of this finding for the mechanism behind the lobe-and-cleft instability is discussed. In order to explore the high-Reynolds-number regime, and to assess potential Reynolds-number effects, two-dimensional simulations are conducted for Reynolds numbers up to about 30 000, for both no-slip and slip (i.e. shear-stress free) boundaries. It is shown that although quantitative Reynolds-number effects persist over the whole range examined, no qualitative changes in the flow structure at the head can be observed. A comparison of the two-dimensional results with laboratory data and the three-dimensional simulation provides evidence that a two-dimensional model is able to capture essential features of the flow at the head. The simulations also show that for the free-slip case the shape of the head agrees closely with the classical inviscid theory of Benjamin.

High-resolution simulations of particle-driven gravity currents

Abstract High-resolution simulations are presented of particle-driven gravity currents in the lock-exchange configuration. The study concentrates on dilute flows with small density differences between particle-laden and clear fluid. Moreover, particles are considered which have negligible inertia, and which are much smaller than the smallest length scales of the buoyancy-induced fluid motion. For the mathematical description of the particulate phase a Eulerian approach is employed with a transport equation for the local particle-number density. The governing equations are integrated numerically with a high-order mixed spectral/spectral-element technique. In the analysis of the results, special emphasis is placed on the sedimentation of particles and the influence of particle settling on the flow dynamics. Time-dependent sedimentation profiles at the channel floor are presented which agree closely with available experimental data. A detailed study is conducted of the balance between the various components of the energy budget of the flow, i.e. the potential and kinetic energy, and the dissipative losses. Furthermore, the simulation results, along with a modified Shields criterion, are used to show that resuspension of sediment back into the particle-driven current is unlikely to occur in the cases considered. Two-dimensional (2D) and three-dimensional (3D) computations are compared which reveal that, for the present configuration, a 2D model can predict reliably the flow development at early times. However, concerning the long-time evolution of the flow, more substantial differences exist between 2D and 3D simulations.

Mixing and dissipation in particle-driven gravity currents

Results are presented from a high-resolution computational study of particle-driven gravity currents in a plane channel. The investigation was conducted in order to obtain better insight into the energy budget and the mixing behaviour of such flows. Two- and three-dimensional simulations are discussed, and the effects of different factors influencing the flow are examined in detail. Among these are the aspect ratio of the initial suspension reservoir, the settling speed of the particles, and the initial level of turbulence in the suspension. While most of the study is concerned with the lock-exchange configuration, where the initial height of the suspension layer is equal to the height of the channel, part of the analysis is also done for a deeply submerged case. Here, the suspension layer is only one-tenth of the full channel height. Concerning the energy budget, a careful analysis is undertaken of dissipative losses in the flow. Dissipative losses arising from the macroscopic fluid motion are distinguished from those due to the microscopic flow around each sedimenting particle. It is found that over a large range of settling velocities and suspension reservoir aspect ratios, sedimentation accounts for roughly half of all dissipative losses. The analysis of the mixing behaviour of the flow concentrates on the mixing between interstitial and ambient fluid, which are taken to be of identical density. The former is assumed to carry a passive contaminant, whose dispersion with time is analysed qualitatively and quantitatively by means of Lagrangian markers. The simulations show the mixing between interstitial and ambient fluid to be more intense for larger values of the particle settling velocity. Finally, the question is addressed of whether or not initial turbulence in the suspension may exert a significant effect on the flow evolution. To this end, results from three simulations with widely different levels of initial kinetic energy are compared. While the initial turbulence level strongly affects the mixing within the current, it has only a small influence on the front velocity and the overall sedimentation rate.

Analysis and direct numerical simulation of the flow at a gravity-current head. Part 2. The lobe-and-cleft instability

Results are presented from a linear-stability analysis of the flow at the head of two-dimensional gravity-current fronts. The analysis was undertaken in order to clarify the instability mechanism that leads to the formation of the complex lobe-and-cleft pattern which is commonly observed at the leading edge of gravity currents propagating along solid boundaries. The stability analysis concentrates on the foremost part of the front, and is based on direct numerical simulation data of two-dimensional lock-exchange flows which are described in the companion paper, Hartel et al. (2000). High-order compact finite differences are employed to discretize the stability equations which results in an algebraic eigenvalue problem for the amplification rate, that is solved in an iterative fashion. The analysis reveals the existence of a vigorous linear instability that acts in a localized way at the leading edge of the front and originates in an unstable stratification in the flow region between the nose and stagnation point. It is shown that the amplification rate of this instability as well as its spanwise length scale depend strongly on Reynolds number. For validation, three-dimensional direct numerical simulations of the early stages of the frontal instability are performed, and close agreement with the results from the linear-stability analysis is demonstrated.

Energy transfer in numerically simulated wall‐bounded turbulent flows

Using velocity fields obtained in direct numerical simulations of turbulent convection and of turbulent channel flow, the energy transfer process among lateral scales of motion in these low Reynolds number flows is analyzed. In all cases the energy is transferred most effectively between scales of similar size. As a result, the subgrid‐scale energy transfer is caused almost exclusively by interactions between resolved scales and subgrid scales characterized by wave numbers not greater than twice the cutoff wave number. The scale dependence of forward and inverse energy transfers contributing to the total subgrid‐scale eddy viscosity is discussed. The local energy transfer between small scales is strongly affected by the nonlocal interactions characterized by a scale separation greater than a factor of 2 in wave number. However, the direct energy transfer between scales satisfying this condition is one order of magnitude less than the local energy transfer between scales of similar size.

Subgrid‐scale energy transfer in the near‐wall region of turbulent flows

Direct numerical simulation databases of turbulent channel and pipe flow have been used in order to assess the energy transfer between resolved and unresolved motions in large‐eddy simulations. To this end, the velocity fields are split into three parts: a statistically stationary mean flow, the resolved, and the unresolved turbulent fluctuations. The distinction between the resolved and unresolved motions is based on the application of a cutoff filter in spectral space. Within the buffer layer a backward transfer of averaged kinetic energy from subgrid to grid‐scale turbulent motions has been found to exist, which is primarily caused by subgrid‐scale stresses aligned with the mean rates of strain. Such reverse transfer generally cannot be described by the simple eddy‐viscosity‐type subgrid models usually applied in large‐eddy simulations. The use of a conditional averaging technique revealed that the reverse transfer of energy within the near‐wall flow is strongly enhanced by coherent motions, such as th...

Analysis and modelling of subgrid-scale motions in near-wall turbulence

A numerical study of turbulent channel flow at various Reynolds numbers ( Re τ =115, 210, 300) is conducted in order to examine the requirements for a reliable subgrid modelling in large-eddy simulations of wall-bounded flows. Using direct numerical simulation data, the interactions between large and small scales in the near-wall flow are analysed in detail which sheds light on the origin of the inverse cascade of turbulent kinetic energy observed in the buffer layer. It is shown that the correlation of the wall-normal subgrid stress and the wall-normal derivative of the streamwise grid-scale velocity plays the key role in the occurrence of the inverse cascade. A brief a priori test of several subgrid models shows that currently applied models are not capable of accounting properly for the complex interactions in the near-wall flow. A series of large-eddy simulations gives evidence that this deficiency may cause significant errors in important global quantities of the flow such as the mean wall shear stress. A study of the eddy-viscosity ansatz is conducted which reveals that the characteristic scales usually employed for the definition of the eddy viscosity are inappropriate in the vicinity of a wall. Therefore, a novel definition of the eddy viscosity is derived from the analysis of the near-wall energy budget. This new definition, which employs the wall-normal subgrid stress as a characteristic scale, is more consistent with the near-wall physics. No significant Reynolds-number effects are encountered in the present analysis which suggests that the findings may be generalized to flows at higher Reynolds numbers.

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 Hele-Shaw cell: linear stability analysis of the three-dimensional Stokes equations

We consider the situation of a heavier fluid placed above a lighter one in a vertically arranged Hele-Shaw cell. The two fluids are miscible in all proportions. For this configuration, experiments and nonlinear simulations recently reported by Fernandez et al. (2002) indicate the existence of a low-Rayleigh-number ( Ra ) ‘Hele-Shaw’ instability mode, along with a high- Ra ‘gap’ mode whose dominant wavelength is on the order of five times the gap width. These findings are in disagreement with linear stability results based on the gap-averaged Hele-Shaw approach, which predict much smaller wavelengths. Similar observations have been made for immiscible flows as well (Maxworthy 1989). In order to resolve the above discrepancy, we perform a linear stability analysis based on the full three-dimensional Stokes equations. A generalized eigenvalue problem is formulated, whose numerical solution yields both the growth rate and the two-dimensional eigenfunctions in the cross-gap plane as functions of the spanwise wavenumber, an ‘interface’ thickness parameter, and Ra . For large Ra , the dispersion relations confirm that the optimally amplified wavelength is about five times the gap width, with the exact value depending on the interface thickness. The corresponding growth rate is in very good agreement with the experimental data as well. The eigenfunctions indicate that the predominant fluid motion occurs within the plane of the Hele-Shaw cell. However, for large Ra purely two-dimensional modes are also amplified, for which there is no motion in the spanwise direction. Scaling laws are provided for the dependence of the maximum growth rate, the corresponding wavenumber, and the cutoff wavenumber on Ra and the interface thickness. Furthermore, the present results are compared both with experimental data, as well as with linear stability results obtained from the Hele-Shaw equations and a modified Brinkman equation.

A direct Numerical Simulation Approach to the Study of Intrusion Fronts

A direct numerical simulation approach for the study of gravity currents in a plane channel is described. The numerical method employed is based on a mixed spectral/spectral-element discretization in space together with finite differences in time. For the validation of the code, simulations of Rayleigh–Bénard convection are performed and the results are compared with theoretical predictions and reference data from the literature. The dynamics of gravity currents is then studied by simulations of two-dimensional lock-exchange flow. The results obtained in these simulations are in good agreement with recent experimental data. By a systematic variation of the Grashof number the influence of viscous diffusion on the characteristics of the propagating fronts is assessed.