Interaction between two spherical bubbles rising in a viscous liquid

The three-dimensional flow around two spherical bubbles moving in a viscous fluid is studied numerically by solving the full Navier-Stokes equations. The study considers the interaction between two bubbles for moderate Reynolds numbers (50 ≤ Re ≤ 500, Re being based on the bubble diameter) and for positions described by the separation S (2.5 ≤ S ≤ 10, S being the distance between the bubble centres normalized by the bubble radius) and the angle θ (0o ≤ θ ≤ 90o ) formed between the line of centre and the direction perpendicular to the direction of the motion. We provide a general description of the interaction extending the results obtained for two bubbles moving side by side (θ = 0o ) by Legendre, Magnaudet & Mougin 2003 (J. Fluid Mech., 497,133-166) and for two bubbles moving in line (θ = 90o ) by Yuan & Prosperetti 1994 (J. Fluid Mech., 278, 325-349). Simple models based on physical arguments are given for the drag and lift forces experienced by each bubble. The interaction is the combination of three effects: a potential effect, a viscous correction (Moore correction) and a significant wake effect observed on both the drag and the transverse force of the second bubble when located in the wake of the first one

Functional codes arising from quadric intersections with Hermitian varieties

AbstractWe investigate the functional code Ch(X) introduced by G. Lachaud (1996) [10] in the special case where X is a non-singular Hermitian variety in PG(N,q2) and h=2. In [4], F.A.B. Edoukou (2007) solved the conjecture of Sørensen (1991) [11] on the minimum distance of this code for a Hermitian variety X in PG(3,q2). In this paper, we will answer the question about the minimum distance in general dimension N, with N<O(q2). We also prove that the small weight codewords correspond to the intersection of X with the union of 2 hyperplanes

Turbulence-induced secondary motion in a buoyancy-driven flow in a circular pipe.

We analyze the results of a direct numerical simulation of the turbulent buoyancy-driven flow that sets in after two miscible fluids of slightly different densities have been initially superimposed in an unstable configuration in an inclined circular pipe closed at both ends. In the central region located midway between the end walls, where the flow is fully developed, the resulting mean flow is found to exhibit nonzero secondary velocity components in the tube cross section. We present a detailed analysis of the generation mechanism of this secondary flow which turns out to be due to the combined effect of the lateral wall and the shear-induced anisotropy between the transverse components of the turbulent velocity fluctuations

A numerical investigation of horizontal viscous gravity currents

We study numerically the viscous phase of horizontal gravity currents of immiscible fluids in the lock-exchange configuration. A numerical technique capable of dealing with stiff density gradients is used, allowing us to mimic high-Schmidt-number situations similar to those encountered in most laboratory experiments. Plane two-dimensional computations with no-slip boundary conditions are run so as to compare numerical predictions with the ‘short reservoir’ solution of Huppert (J. Fluid Mech., vol. 121, 1982, pp. 43–58), which predicts the front position lf to evolve as t1/5, and the ‘long reservoir’ solution of Gratton & Minotti (J. Fluid Mech., vol. 210, 1990, pp. 155–182) which predicts a diffusive evolution of the distance travelled by the front xf ~ t1/2. In line with dimensional arguments, computations indicate that the self-similar power law governing the front position is selected by the flow Reynolds number and the initial volume of the released heavy fluid. We derive and validate a criterion predicting which type of viscous regime immediately succeeds the slumping phase. The computations also reveal that, under certain conditions, two different viscous regimes may appear successively during the life of a given current. Effects of sidewalls are examined through three-dimensional computations and are found to affect the transition time between the slumping phase and the viscous regime. In the various situations we consider, we make use of a force balance to estimate the transition time at which the viscous regime sets in and show that the corresponding prediction compares well with the computational results

Effects of channel geometry on buoyancy-driven mixing

The evolution of the concentration and flow fields resulting from the gravitational mixing of two interpenetrating miscible fluids placed in a tilted tube or channel is studied by using direct numerical simulation. Three-dimensional (3D) geometries, including a cylindrical tube and a square channel, are considered as well as a purely two-dimensional (2D) channel. Striking differences between the 2D and 3D geometries are observed during the long-time evolution of the flow. We show that these differences are due to those existing between the 2D and 3D dynamics of the vorticity field. More precisely, in two dimensions, the strong coherence and long persistence of vortices enable them to periodically cut the channels of pure fluid that feed the front. In contrast, in 3D geometries, the weaker coherence of the vortical motions makes the segregational effect due to the transverse component of buoyancy strong enough to preserve a fluid channel near the front of each current. This results in three different regimes for the front velocity (depending on the tilt angle), which is in agreement with the results of a recent experimental investigation. The evolution of the front topology and the relation between the front velocity and the concentration jump across the front are investigated in planar and cylindrical geometries and highlight the differences between 2D and 3D mixing dynamics

Influence of skull inhomogeneities on EEG source localization

We investigated the influence of using simplified models of the skull on electroencephalogram (EEG) source localization. An accurately segmented skull from computed tomography (CT) images, including spongy and compact bones as well as some air–filled cavities, was used as a reference model. The simplified models approximated the skull as a homogeneous compartment with: (1) isotropic, and (2) anisotropic conductivity. The results showed that these approximations could lead to errors of more than 2 cm in dipole estimation. We recommend the use of anisotropy but considering a different ratio for each region of the skull, according to the amount of spongy bone

On the relative impact of subgrid-scale modelling and conjugate heat transfer in LES of hot jets in cross-flow over cold plates

This work describes numerical simulations of a hot jet in cross-flow with applications to anti-ice systems of aircraft engine nacelles. Numerical results are compared with experimental measurements obtained at ONERA to evaluate the performances of LES in this industrial context. The combination of complex geometries requiring unstructured meshes and high Reynolds number does not allow the resolution of boundary layers so that wall models must be employed. In this framework, the relative influence of subgrid-scale modelling and conjugate heat transfer in LESs of aerothermal flows is evaluated. After a general overview of the transverse jet simulation results, a LES coupled with a heat transfer solver in the walls is used to show that thermal boundary conditions at the wall have more influence on the results than subgrid scale models. Coupling fluid flow and heat transfer in solids simulations is the only method to specify their respective thermal boundary conditions

On the small weight codewords of the functional codes C_2(Q), Q a non-singular quadric

We study the small weight codewords of the functional code C_2(Q), with Q a non-singular quadric of PG(N,q). We prove that the small weight codewords correspond to the intersections of Q with the singular quadrics of PG(N,q) consisting of two hyperplanes. We also calculate the number of codewords having these small weights

Sets of generators blocking all generators in finite classical polar spaces

We introduce generator blocking sets of finite classical polar spaces. These sets are a generalisation of maximal partial spreads. We prove a characterization of these minimal sets of the polar spaces Q(2n,q), Q-(2n+1,q) and H(2n,q^2), in terms of cones with vertex a subspace contained in the polar space and with base a generator blocking set in a polar space of rank 2.Comment: accepted for J. Comb. Theory

Incorporation of anisotropic conductivities in EEG source analysis

The electroencephalogram (EEG) is a measurement of brain activity over a period of time by placing electrodes at the scalp (surface EEG) or in the brain (depth EEG) and is used extensively in the clinical practice. In the past 20 years, EEG source analysis has been increasingly used as a tool in the diagnosis of neurological disorders (like epilepsy) and in the research of brain functionality. EEG source analysis estimates the origin of brain activity given the electrode potentials measured at the scalp. This involves solving an inverse problem where a forward solution, which depends on the source parameters, is fitted to the given set of electrode potentials. The forward solution are the electrode potentials caused by a source in a given head model. The head model is dependent on the geometry and the conductivity. Often an isotropic conductivity (i.e. the conductivity is equal in all directions) is used, although the skull and white matter have an anisotropic conductivity (i.e. the conductivity can differ depending on the direction the current flows). In this dissertation a way to incorporate the anisotropic conductivities is presented and the effect of not incorporating these anisotropic conductivities is investigated. Spherical head models are simple head models where an analytical solution to the forward problem exists. A small simulation study in a 5 shell spherical head model was performed to investigate the estimation error due to neglecting the anisotropic properties of skull and white matter. The results show that the errors in the dipole location can be larger than 15 mm, which is unacceptable for an accurate dipole estimation in the clinical practice. Therefore, anisotropic conductivities have to be included in the head model. However, these spherical head models are not representative for the human head. Realistic head models are usually made from magnetic resonance scans through segmentation and are a better approximation to the geometry of the human head. To solve the forward problem in these head models numerical methods are needed. In this dissertation we proposed a finite difference technique that can incorporate anisotropic conductivities. Moreover, by using the reciprocity theorem the forward calculation time during an dipole source estimation procedure can be significantly reduced. By comparing the analytical solution for the dipole estimation problem with the one using the numerical method, the anisotropic finite difference with reciprocity method (AFDRM) is validated. Therefore, a cubic grid is made on the 5 shell spherical head model. The electrode potentials are obtained in the spherical head model with anisotropic conductivities by solving the forward problem using the analytical solution. Using these electrode potentials the inverse problem was solved in the spherical head model using the AFDRM. In this way we can determine the location error due to using the numerical technique. We found that the incorporation of anisotropic conductivities results in a larger location error when the head models are fully isotropical conducting. Furthermore, the location error due to the numerical technique is smaller if the cubic grid is made finer. To minimize the errors due to the numerical technique, the cubic grid should be smaller than or equal to 1 mm. Once the numerical technique is validated, a realistic head model can now be constructed. As a cubic grid should be used of at most 1 mm, the use of segmented T1 magnetic resonance images is best suited the construction. The anisotropic conductivities of skull and white matter are added as follows: The anisotropic conductivity of the skull is derived by calculating the normal and tangential direction to the skull at each voxel. The conductivity in the tangential direction was set 10 times larger than the normal direction. The conductivity of the white matter was derived using diffusion weighted magnetic resonance imaging (DW-MRI), a technique that measures the diffusion of water in several directions. As diffusion is larger along the nerve fibers, it is assumed that the conductivity along the nerve fibers is larger than the perpendicular directions to the nerve bundle. From the diffusion along each direction, the conductivity can be derived using two approaches. A simplified approach takes the direction with the largest diffusion and sets the conductivity along that direction 9 times larger than the orthogonal direction. However, by calculating the fractional anisotropy, a well-known measure indicating the degree of anisotropy, we can appreciate that a fractional anisotropy of 0.8715 is an overestimation. In reality, the fractional anisotorpy is mostly smaller and variable throughout the white matter. A realistic approach was therefore presented, which states that the conductivity tensor is a scaling of the diffusion tensor. The volume constraint is used to determine the scaling factor. A comparison between the realistic approach and the simplified approach was made. The results showed that the location error was on average 4.0 mm with a maximum of 10 mm. The orientation error was found that the orientation could range up to 60 degrees. The large orientation error was located at regions where the anisotropic ratio was low using the realistic approach but was 9 using the simplified approach. Furthermore, as the DW-MRI can also be used to measure the anisotropic diffusion in a gray matter voxel, we can derive a conductivity tensor. After investigating the errors due to neglecting these anisotropic conductivities of the gray matter, we found that the location error was very small (average dipole location error: 2.8 mm). The orientation error was ranged up to 40 degrees, although the mean was 5.0 degrees. The large errors were mostly found at the regions that had a high anisotropic ratio in the anisotropic conducting gray matter. Mostly these effects were due to missegmentation or to partial volume effects near the boundary interfaces of the gray and white matter compartment. After the incorporation of the anisotropic conductivities in the realistic head model, simulation studies can be performed to investigate the dipole estimation errors when these anisotropic conductivities of the skull and brain tissues are not taken into account. This can be done by comparing the solution to the dipole estimation problem in a head model with anisotropic conductivities with the one in a head model, where all compartments are isotropic conducting. This way we determine the error when a simplified head model is used instead of a more realistic one. When the anisotropic conductivity of both the skull and white matter or the skull only was neglected, it was found that the location error between the original and the estimated dipole was on average, 10 mm (maximum: 25 mm). When the anisotropic conductivity of the brain tissue was neglected, the location error was much smaller (an average location error of 1.1 mm). It was found that the anisotropy of the skull acts as an extra shielding of the electrical activity as opposed to an isotropic skull. Moreover, we saw that if the dipole is close to a highly anisotropic region, the potential field is changed reasonable in the near vicinity of the location of the dipole. In reality EEG contains noise contributions. These noise contribution will interact with the systematical error by neglecting anisotropic conductivities. The question we wanted to solve was “Is it worthwhile to incorporate anisotropic conductivities, even if the EEG contains noise?” and “How much noise should the EEG contain so that incorporating anisotropic conductivities improves the accuracy of EEG source analysis?”. When considering the anisotropic conductivities of the skull and brain tissues and the skull only, the location error due to the noise and neglecting the anisotropic conductivities is larger then the location error due to noise only. When only neglecting the anisotropic conductivities of the brain tissues only, the location error due to noise is similar to the location error due to noise and neglecting the anisotropic conductivities. When more advanced MR techniques can be used a better model to construct the anisotropic conductivities of the soft brain tissues can be used, which could result in larger errors even in the presence of noise. However, this is subject to further investigation. This suggests that the anisotropic conductivities of the skull should be incorporated. The technique presented in the dissertation can be used to epileptic patients in the presurgical evaluation. In this procedure patients are evaluated by means of medical investigations to determine the cause of the epileptic seizures. Afterwards, a surgical procedure can be performed to render the patient seizure free. A data set from a patiënt was obtained from a database of the Reference Center of Refractory Epilepsy of the Department of Neurology and the Department of Radiology of the Ghent University Hospital (Ghent, Belgium). The patient was monitored with a video/EEG monitoring with scalp and with implanted depth electrodes. An MR image was taken from the patient with the implanted depth electrodes, therefore, we could pinpoint the hippocampus as the onset zone of the epileptic seizures. The patient underwent a resective surgery removing the hippocampus, which rendered the patient seizure free. As DW-MRI images were not available, the head model constructed in chapter 4 and 5 was used. A neuroradiologist aligned the hippocampus in the MR image from which the head model was constructed. A spike was picked from a dataset and was used to estimate the source in a head model where all compartments were isotropic conducting, on one hand, and where the skull and brain tissues were anisotropic conducting, on the other. It was found that using the anisotropic head model, the source was estimated closer to the segmented hippocampus than the isotropic head model. This example shows the possibilities of this technique and allows us to apply it in the clinical practice. Moreover, a thorough validation of the technique has yet to be performed. There is a lot of discussion in the clinical community whether the spikes and epileptical seizures originate from the same origin in the brain. This question can be solved by applying our technique in patient studies