Jump conditions for filtered quantities at an under-resolved discontinuous interface. Part 2: A priori tests

In order to study and validate the jump conditions established in part 1, we realize a priori tests thanks to the data of a 3D Direct Numerical Simulation (DNS) of a strongly deformable bubble in a spatially decaying turbulence. The complex interactions between interface and turbulence are fully resolved. An explicit filtering of the DNS has been employed to evaluate the filtered quantities and to check the potential of the models for two-phase flows in the Interface and Subgrid Scales (ISS) modeling case proposed in part 1. The ISS concept is our proposal of a two-phase equivalent for the one-phase Large Eddy Simulation (LES) modeling case with sharp-interfaces. In this concept, bubbles remain bigger than the mesh size. Due to the impossibility to define a filter equivalent to the matched asymptotic expansions, we only test the modeling of the equivalent interface transport (the momentum jump conditions are not tested in this article, but will deserve additional results in a posteriori tests). Because the closure of the transport equation of the under-resolved discontinuous interface requires more modeling assumptions than the closure of the momentum equation, we think that the most relevant test has been done. The a priori tests realized show excellent agreement between the ISS models and the real contributions.     

Localised knife waves in a structured interface

We consider a Mode III lattice with an interface layer where the dynamic crack growth is caused by a localised sinusoidal wave. In the wave–fracture scenario, the ‘feeding wave’ (here also called the knife wave) delivers energy to the moving crack front, while the dissipative waves carry a part of this energy away from the front. The questions addressed here are:

• What are the conditions of existence of the localised knife wave?

• What is the lower bound of the amplitude of the feeding wave, which supports the crack propagation, for a given deformational fracture criterion?

• How does the crack speed depend on the amplitude of the feeding wave?

• What are the dissipative waves? How much energy is irradiated by these waves and what is the total dissipation?

• What are the conditions of existence of the steady-state regime for the propagating crack?

We consider analytically two established regimes: the steady-state regime, where the motion of neighbouring masses (along the interface) differs only by a constant shift in time, and an alternating-strain regime, where the corresponding amplitudes differ by sign. We also present the numerical simulation results for a model of a high-contrast interface structure. Along with the energy of the feeding and dissipative waves, an energy radiated to the bulk of the lattice is identified.

Simulating air entrainment and vortex dynamics in a hydraulic jump

The air entrainment characteristics of three separate Froude number hydraulic jumps are investigated numerically using an unsteady RANS, realizable k–ε turbulence model, with a Volume of Fluid treatment for the free surface. Mean velocity profiles, average void fraction, and Sauter mean diameter compare favorably with experimental data reported in literature. In all simulations, time-averaged void fraction profiles show good agreement with experimental values in the turbulent shear layer and an accurate representation of interfacial aeration at the free surface. Sauter mean diameter is well represented in the shear layer, and free surface entrainment results indicate bubble size remains relatively unchanged throughout the depth of the jump. Several different grid resolutions are tested in the simulations. Significant improvements in void fraction and bubble size comparison are seen when the diameter to grid size ratio of the largest bubbles in the shear layer surpasses eight. A three-dimensional simulation is carried out for one Froude number jump, showing an improvement in the prediction of entrained air and bubble size compared with two-dimensional results at a substantial increase in computation time. An analysis of three-dimensional vorticity shows a complex interaction between spanwise and streamwise vortical structures and entrained air bubbles. The jump is similar to a turbulent mixing layer, constrained by the free surface, with vortex pairing and subsequent fluctuations in free surface elevation. Downstream fluctuations of the toe are associated with a roll up of the primary spanwise vortex, fluctuations of the free surface, and counter-rotating streamwise vortex pairs. The action of these flow structures is likely responsible for the improvement in three-dimensional results.     

Flow property and self-similarity in steady hydraulic jumps

The flow structure in a steady hydraulic jump in both the non-aerated and aerated regions was measured using the image-based particle image velocimetry and bubble image velocimetry techniques, respectively. Three highly aerated steady jumps with Froude numbers varying from 4.51 to 5.35 were tested, and a weak jump with a Froude number of 2.43 was generated for comparison. Mean velocities and turbulence statistics were obtained by ensemble averaging the repeated velocity measurements. Based on the mean velocities, the flow structure in the steady jumps was classified into four regions to distinguish their distinct flow behaviors; they are the potential core region, the boundary layer region, the mixing layer region, and the recirculation region. The flow structure in the weak jump features only three regions without the recirculation region. In addition, spatial variations of mean velocities, turbulence intensity, and Reynolds stresses were also presented. It was observed that the maximum horizontal bubble velocity and maximum horizontal water velocity occur at the same location in the overlapping regions of potential core and mixing layer. The ratio between the maximum horizontal bubble velocity and maximum horizontal water velocity is between 0.6 and 0.8, depending on the Froude number. Examining the mean horizontal bubble velocities in the mixing layer, a similarity profile was revealed with representative mixing layer thickness as the characteristic length scale and the difference between the maximum positive and maximum negative velocities as the characteristic velocity scale. It was also found that the mean horizontal water velocities in the near-wall region are self-similar and behave like a wall jet. Further analyzing autocorrelation functions and energy spectra of the water and bubble velocity fluctuations found that the energy spectra in the water region follow the ?5/3 slope, whereas the spectra in the bubble region follow a ?2/5 slope. In addition, the integral length scale of bubbles is one order of magnitude shorter than that of water.     

A numerical strategy for the direct 3D simulation of the expansion of bubbles into a molten polymer during a foaming process

In the framework of the foam process modelling, this paper presents a numerical strategy for the direct 3D simulation of the expansion of gas bubbles into a molten polymer. This expansion is due to a gas overpressure. The polymer is assumed to be incompressible and to behave as a pseudo‐plastic fluid. Each bubble is governed by a simple ideal gas law. The velocity and the pressure fields, defined in the liquid by a Stokes system, are subsequently extended to each bubble in a way of not perturbing the interface velocity. Hence, a global velocity–pressure‐mixed system is solved over the whole computational domain, thanks to a discretization based on an unstructured first‐order finite element. Since dealing with an Eulerian approach, an interface capturing method is used to follow the bubble evolution. For each bubble, a pure advection equation is solved by using a space–time discontinuous‐Galerkin method, coupled with an r‐adaptation technique. Finally, the numerical strategy is achieved by considering a global mesh expansion motion, which conserves the amount of liquid into the computational domain during the expansion. The expansion of one bubble is firstly considered, and the simulations are compared with an analytical model. The formation of a cellular structure is then investigated by considering the expansion of 64 bubbles in 2D and the expansion of 400 bubbles in 3D. Copyright © 2007 John Wiley & Sons, Ltd.     

On the application of LDA to bubbly flow in the wobbling regime

 Considerations for applying LDA to bubbly flows with bubbles about 3 to 4 mm in diameter were investigated by means of detailed experiments in the model geometry of a train of bubbles. Both forward scatter and backscatter LDA were studied. The validity of phase discrimination via burst amplitude was tested and special attention was paid to the impact of bubble interface response to the laser beams. Forward and backscatter measurements can be compared well. In both configurations, predominantly the liquid phase is “seen” by LDA. A bubble itself only leads to a velocity realization in special conditions. In those cases the Doppler shift is determined by the motion of the bubble interface which consists of the motion of the center of gravity of the bubble as well as shape oscillations. In backscatter bubbles only give velocity realizations when their “cheeks” pass through the measuring volume virtually perpendicularly. It is shown that the bubble-caused velocity realization frequency is very low for bubbles of the size used. Phase discrimination on burst amplitude does not hold. In ambient cases such as bubble columns one can assume that only the liquid phase is being studied. Received: 4 May 1998/Accepted: 30 September 1998     

Interaction of vehicle and terrain results from 10 years research at IKK

The IKK research program comprises:

• experimental evaluation of terrain data and formulation of analytical models for soil properties

• development of models for the different soil-wheel interactions

• development of a highly sophisticated model, ORIS, for off-road interactive simulation of the driver-vehicle-terrain system and the utilization for vehicle development and mission optimization

• development of finite element methods, e.g. VENUS, and their utilization for multiple terramechanical problems.

With these achievements, the team of scientists of the Institute of Automotive Engineering of the University of the Federal Armed Forces, Hamburg (IKK) could contribute essentially to the state of the art.It is the nature of scientific development that problems are recognized to be important and treated by several institutions in the world often at the same time. Also, in the case of the IKK research program, some problems may have been investigated in parallel by others. This indicates the actuality of the research program, and gives the chance for discussion and cooperation between the different scientific groups in the world, to ensure the dissemination of new pieces of knowledge.     

An advanced LIF-PLV system for analysing the hydrodynamics in a laboratory bubble column at higher void fractions

Bubble columns are widely used in the chemical industry and biotechnology. Flow and turbulence in such an apparatus are induced by the bubble rise, and the bubble behaviour is strongly affected by swarm effects (i.e. the interaction between bubbles). For analysing the bubble swarm behaviour and simultaneously evaluating the flow structure and bubble-induced turbulence, a bubble column of 140 mm diameter and a height of 650 mm or 1,400 mm (initial water level) were considered. The bubble column was aerated with relatively fine bubbles having a mean size between about 0.5 and 4.0 mm. The gas hold-up was varied in the range between 0.5 and 19%. A two-phase pulsed-light velocimetry (PLV) system was developed to evaluate instantaneous flow fields of both rising bubbles and the continuous phase. The measurement of the liquid velocities in the bubble swarm was achieved by adding fluorescing seed particles. Images of bubbles and fluorescing tracer particles were acquired by two CCD cameras. Hence, the images from tracers and bubbles were easily separated by optical interference filters with a bandwidth corresponding to the emitting wavelength of the fluorescing tracer particles and the wavelength of the applied Nd-YAG pulsed laser, respectively. To improve the phase separation of the system, the CCD cameras were additionally placed in a non-perpendicular arrangement with respect to the light sheet. The acquired images were evaluated with the minimum-quadratic-difference algorithm. The potential of this technique for the analysis of bubbly flows with higher void fraction was explored. In order to obtain averaged velocity maps of bubble and fluid within the entire column, about 1,000 image pairs were recorded and evaluated for each phase. In addition, turbulence intensities of the fluid were deduced from the measurements. The turbulence properties were used to characterise bubble-induced turbulence for various bubble mean diameters and gas hold-ups. Moreover, the determination of the average bubble slip velocity within the bubble swarm was possible.     

Modeling air entrainment and transport in a hydraulic jump using two-fluid RANS and DES turbulence models

Both RaNS (Reynolds-averaged Navier–Stokes) and DES (Detached Eddy Simulation) type turbulence models were used in conjunction with a two-fluid model of bubbly flow and a new subgrid air entrainment model to predict air entrainment and transport in a hydraulic jump. It was found that the void fraction profiles predicted by both methods are in agreement with the experimental data in the lower shear layer region, which contains the air bubbles entrained at the so-called toe of the hydraulic jump. In contrast, in the upper roller region behind the toe, the averaged results of the DES turbulence model gives accurate predictions while a RaNS turbulence model does not. This is because the DES turbulence model successfully captures the strong fluctuations on the free surface which allows it to entrain air near the top of the roller region. In contrast, RaNS type turbulence model results in a steady, smooth interface which fails to capture the wave-induced bubble sources in that region. To our knowledge, this study is the first successful quantitative numerical simulation of the overall void fraction profiles in a hydraulic jump.     

Influence of the spatial resolution on fine-scale features in DNS of turbulence generated by a single square grid

We focus in this paper on the effect of the resolution of direct numerical simulations (DNS) on the spatio-temporal development of the turbulence downstream of a single square grid. The aims of this study are to validate our numerical approach by comparing experimental and numerical one-point statistics downstream of a single square grid and then investigate how the resolution is impacting the dynamics of the flow. In particular, using the Q–R diagram, we focus on the interaction between the strain-rate and rotation tensors, the symmetric and skew-symmetric parts of the velocity gradient tensor, respectively. We first show good agreement between our simulations and hot-wire experiment for one-point statistics on the centreline of the single square grid. Then, by analysing the shape of the Q–R diagram for various streamwise locations, we evaluate the ability of under-resolved DNS to capture the main features of the turbulence downstream of the single square grid.     

Effect of weak compressibility of a fluid on bubble-bubble interaction in strong acoustic fields

The effect of weak compressibility of a fluid on the interaction between spherical bubbles in a strong acoustic field is considered. A small parameter ɛ which represents the ratio of the characteristic velocity of radial oscillations of the bubbles to the speed of sound in the fluid is used as a parameter characterizing the fluid compressibility. The equations governing the interaction between two bubbles are derived with an accuracy O(ɛ) in the case in which the ratio of the characteristic velocities of their translational and radial motions is of the order of ɛ. It is shown that neglecting the fluid compressibility effect due to the bubble interaction can lead to either enhancement or attenuation of their radial oscillations following the main compression stage, variation in the oscillation frequency, the bubble approach velocity, and the velocity of the spatial motion of the coupled pair, and the bubble approach and collision rather than their moving away from one another with the formation of a coupled pair.     

Structure of turbulent bubbly jets—I. Methods and centerline properties

The structure of dilute bubbly turbulent round jets, injected vertically upward in still water, was studied both theoretically and experimentally. All measurements were nonintrusive, including mean and fluctuating phase velocities, bubble number intensities, bubble-size distributions and calibration of the motion of individual bubbles. Predictions from three analyses were compared with measurements: (1) locally homogeneous flow analysis, where velocity differences between the phases were neglected; (2) deterministic separated flow analysis, where relative velocity was considered but bubble/turbulence interactions were ignored; and (3) stochasic separated flow analysis, where both relative velocity and bubble/turbulence interactions were considered using random-walk methods. This paper describes theoretical and experimental methods, flow structure near the source and mean properties along the jet axis. Effects of relative velocity were important almost everywhere in the flow; therefore, only the separated flow models yielded satisfactory predictions of bubble velocities along the axis. A companion paper treats mean and fluctuating properties in other regions of the flow.     

Effects of sub-millimeter-bubble injection on transition to turbulence in natural convection boundary layer along a vertical plate in water

Temperature and velocity measurements are performed to clarify the effects of sub-millimeter-bubble injection on the transition to turbulence in the natural convection boundary layer along a vertical plate in water. In particular, we focus on the relationship between the bubble injection position L and the transition to turbulence in the natural convection boundary layer. The bubble injection positions used in our experiments are L = 1.6 and 3.6 mm. Bubble injection at L = 1.6 mm delays the transition to turbulence in the natural convection boundary layer, while that at L = 3.6 mm accelerates the transition to turbulence in the boundary layer. In the case of L = 1.6 mm, the appearance region of the liquid velocity fluctuation in the bubble-induced upward flow in the upstream unheated section is restricted to near the wall, although the peak of the liquid velocity fluctuation is high. In contrast, in the case of L = 3.6 mm, the relatively large liquid velocity fluctuation is distributed widely over the laminar boundary layer width. These results suggest that the effect of the liquid velocity fluctuation on the laminar boundary layer is quite different between L = 1.6 and 3.6 mm. It is therefore expected that the transition to turbulence in the natural convection boundary layer for the case with bubble injection is dependent on the magnitude and appearance region of the liquid velocity fluctuation in the bubble-induced upward flow in the upstream unheated section.     

The effect of the VOF–CSF static contact angle boundary condition on the dynamics of sliding and bouncing ellipsoidal bubbles

The static contact angle is the only empiricism introduced in a Volume of Fluid–Continuum Surface Force (VOF–CSF) model of bubbly flow. Although it has previously been shown to have a relatively limited effect on the accuracy of velocity and shape predictions in the case of large gas bubbles sliding under inclined walls (e.g. Cook and Behnia, 2001), it may have a more determining influence on the numerical prediction of the dynamics of smaller ellipsoidal bubbles which were shown by Tsao and Koch (1997) to bounce repeatedly when sliding under inclined walls at certain wall inclinations. The present paper reports on the influence of surface tension and the static contact angle on the dynamics of an ellipsoidal air bubble of equivalent diameter D_e = 3.4 mm. The bubble Eötvös and Morton numbers are Eo = 1.56 and Mo = 2 × 10⁻¹¹ respectively. The computational results are achieved with a Piecewise Linear Construction (PLIC) of the interface and are reviewed with reference to experimental measurements of bubble velocity and interface shape oscillations recorded using a high speed digital camera. Tests are performed at plate inclination angles θ ∈ {10°, 20°, 30°, 45°} to the horizontal and computational models consider three static contact angles θ_c ∈ {10°, 20°, 30°}. The static contact angle has been found to have a significant effect on the bubble dynamics but to varying degree depending on the plate inclination. It is shown to promote lift off and bouncing when the plate inclination angle reaches 30° in a way that does not necessarily reflect experimental observations.     

Bubbly flow measurements in hydraulic jumps with small inflow Froude numbers

The transition from supercritical to subcritical open channel flow is characterised by a strong dissipative mechanism called a hydraulic jump. A hydraulic jump is turbulent and associated with the development of large-scale turbulence and air entrainment. In the present study, some new physical experiments were conducted to characterise the bubbly flow region of hydraulic jumps with relatively small Froude numbers (2.4 < Fr₁ < 5.1) and relatively large Reynolds numbers (6.6 × 10⁴ < Re < 1.3 × 10⁵). The shape of the time-averaged free-surface profiles was well defined and the longitudinal profiles were in agreement with visual observations. The turbulent free-surface fluctuation profiles exhibited a peak of maximum intensity in the first half of the hydraulic jump roller, and the fluctuations exhibited some characteristic frequencies typically below 3 Hz. The air–water flow properties showed two characteristic regions: the shear layer region in the lower part of the flow and an upper free-surface region above. The air–water shear layer region was characterised by local maxima in terms of void fraction and bubble count rate. Other air–water flow characteristics were documented including the distributions of interfacial velocity and turbulence intensity. The probability distribution functions (PDF) of bubble chord time showed that the bubble chord times exhibited a broad spectrum, with a majority of bubble chord times between 0.5 and 2 ms. An analysis of the longitudinal air–water structure highlighted a significant proportion of bubbles travelling within a cluster structure.     

The dynamics and dissolution of gas bubbles in a viscoelastic fluid

This paper reports an experimental study of the motion of dissolving and non-dissolving gas bubbles in a quiescent viscoelastic fluid. The objective of the investigation was to determine the influence of the abrupt transition in bubble velocity, which had been observed at a critical radius of approx. on the rate of mass transfer. Thus, a range of bubble sizes from an equivalent (spherical) radius of 0.2–0.4 cm was employed using CO₂ gas, and five different fluids, including one Newtonion glycerine/water solution and four viscoelastic solutions of Separan AP30 in water (0.1, 0.5, 1% by weight) and in a water/glycerine mixture.The experimental data on bubble velocity shows that the discontinuous increase with bubble volume observed previously for air bubbles in viscoelastic fluids, does not occur for dissolving CO₂ bubbles—presumably due to the continuous decrease in bubble volume. Instead, a very steep but definitely continuous transition is found. Mass transfer rates are found to be significantly enhanced by viscoelasticity, and comparison with available theoretical results shows that the increase is greater than expected for purely viscous, power-law fluids. We conclude that a fully viscoelastic constitutive model would be necessary for a successful analysis of the dissolution of a gas bubble which is translating through a (high molecular weight) polymer solution.     

Residual‐based turbulence models for moving boundary flows: hierarchical application of variational multiscale method and three‐level scale separation

This paper presents residual‐based turbulence models for problems with moving boundaries and interfaces. The method is developed via a hierarchical application of variational multiscale ideas and the models are cast in an arbitrary Lagrangian–Eulerian (ALE) frame to accommodate the deformation of domain boundaries. An overlapping additive decomposition of velocity and pressure fields into coarse and fine scale components leads to coarse and fine scale mixed‐field problems. The problem governing fine scales is subjected to a further decomposition of the fine scale velocity into overlapping components termed as fine scales level I and level II. In turn, in the bottom‐up integration of scales, the model for level II fine scales serves to stabilize the problem governing level I fine scales, and model for level I fields yields the turbulence models. From the computational perspective, the coarse scales are represented in terms of the standard Lagrange shape functions, whereas level I and level II scales are represented via quadratic and fourth order polynomial bubbles, respectively. Because of the bubble functions approach employed in the consistently derived fine scale models, the resulting method is free of any embedded or tunable parameters. The proposed turbulence models share a common feature with the LES models in that the largest scales in the flow are numerically resolved, whereas the subgrid scales are modeled. The method is applied to flow around a plunging airfoil at Re = 40,000, and results are compared with experimental and numerical data published in the literature. Also presented are the results for the plunging airfoil at Re = 60,000 to show the robustness and range of applicability of the method. Copyright © 2013 John Wiley & Sons, Ltd.     

Air transport in chute flows

The air bubble rise velocity in still water depends mainly on the bubble size and is basically influenced by buoyancy, viscosity and surface tension. In high-speed flows the number of forces acting on air bubbles increases with turbulence, non-hydrostatic pressure gradient, shear forces, bubble clouds and free-surface entrainment. Air bubbles in these flows are used for cavitation protection of hydraulic structures such as chutes, spillways and bottom outlets. Here, air is normally added by means of aerators upstream of regions where the cavitation number falls below a critical value mainly to reduce the sonic velocity of the fluid and cushion the cavitation bubble collapse process. The distance between successive aerators depends basically on the bubble rise velocity. Until today, the bubble rise velocity in high-speed flows was not thoroughly investigated because of limited laboratory instrumentation. The present project focused on the streamwise development of air concentrations in high-speed flows along a 14 m long model chute. The bubble rise velocity was indirectly derived from the air detrainment gradient of the air concentration contour lines downstream of an aeration device. It accounts for the main hydraulic parameters chute slope, Froude number and air concentration. It is demonstrated that the bubble rise velocity in high-speed flow and stagnant water differ significantly due to fracturing processes, turbulence, and the ambient air concentration.     

Experimental investigation of horizontal air–water bubbly-to-plug and bubbly-to-slug transition flows in a 3.81 cm ID pipe

The present study seeks to investigate horizontal bubbly-to-plug and bubbly-to-slug transition flows. The two-phase flow structures and transition mechanisms in these transition flows are studied based on experimental database established using the local four-sensor conductivity probe in a 3.81 cm inner diameter pipe. While slug flow needs to be distinguished from plug flow due to the presence of large number of small bubbles (and thus, large interfacial area concentration), both differences and similarities are observed in the evolution of interfacial structures in bubbly-to-plug and bubbly-to-slug transitions. The bubbly-to-plug transition is studied by decreasing the liquid flow rate at a fixed gas flow rate. It is found that as the liquid flow rate is lowered, bubbles pack near the top wall of the pipe due to the diminished role of turbulent mixing. As the flow rate is lowered further, bubbles begin to coalesce and form the large bubbles characteristic of plug flow. Bubble size increases while bubble velocity decreases as liquid flow rate decreases, and the profile of the bubble velocity changes its shape due to the changing interfacial structure. The bubbly-to-slug transition is investigated by increasing the gas flow rate at a fixed liquid flow rate. In this transition, gas phase becomes more uniformly distributed throughout the cross-section due to the formation of large bubbles and the increasing bubble-induced turbulence. The size of small bubbles decreases while bubble velocity increases as gas flow rate increases. The distributions of bubble size and bubble velocity become more symmetric in this transition. While differences are observed in these two transitions, similarities are also noticed. As bubbly-to-plug or bubbly-to-slug transition occurs, the formation of large elongated bubbles is observed not in the uppermost region of bubble layer, but in a lower region. At the beginning of transitions, relative differences in phase velocities near the top of the pipe cross-section to those near the pipe center become larger for both gas and liquid phases, because more densely packed bubbles introduce more resistance to both phases.