The main purpose of this chapter is to describe the non-local, multi-scale geometry of **Lagrangian** structures in the cascade process in **turbulence**. In Fourier space, the velocity field can be projected onto Fourier basis functions that represent, in some statistical sense, the hierarchy of eddy sizes. Although it is natural to present the energy spectrum in Fourier space, it is often difficult to at- tribute physical meaning to the amplitudes of the basis-function coefficients in terms, for example of structural elements such as vortical structures with different geometry like tubes and sheets. An at- tractive physically intuitive idea of energy cascade might be cast in terms of **vortex** dynamics where **vortex** stretching is a crucial agent. But the participating eddies may not be the often portrayed cartoon, blob-like structures of different sizes. While sheet and tubes are attractive alternative ge- ometries, there exist few relevant quantitative models with predictive or even postdictive capability. When the Reynolds number is infinite, **vortex** lines and surfaces can be considered as material lines and surfaces that are progressively stretched by chaotic motion in the inertial range to form highly convoluted shapes. This mechanism can occur at all scales, but the most efficient transfer of energy is perhaps caused by the interaction of vortices with similar sizes. There is, however, no explicit length scale in the vorticity equation in physical space. This suggests that a multi-scale method based on transforms with basis functions that are localized both in Fourier space and physical space is required (e.g., Meneveau, 1991; Farge, 1992). Furthermore, the knowledge of geometry of turbu- lent structures can inform a vorticity-based, small-scale description of **turbulence** (Lundgren, 1982; Pullin & Saffman, 1993) from which subgrid-scale models suitable for LES can be constructed (e.g., Misra & Pullin, 1997; Chung & Pullin, 2009).

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By means of direct numerical simulations of isotropic **turbulence** together with a scalar with a linear mean gradient, we have studied the micro-scales of the scalar fluctuation field in terms of the structures of the **turbulence** velocity field and of the Eulerian and **Lagrangian** statistics for a wide range of Schmidt numbers. First, we have reviewed in § 2 the well-established concepts about turbulent straining in combination with **vortex** stretching and roll-up and their effect on the scalar field. This leads to a picture of the generation of steep scalar gradients or cliffs by strong and short-lived straining deformations. The deformation tensor in this case has only one negative or compressive eigenvalue and two positive or expansive eigenvalues so that scalar cliffs are deformed into thin sheet-like structures which can be relatively flat or folded.

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We examine the probability distribution function (pdf) of energy injection rate (power) in nu- merical simulations of stationary two–dimensional (2D) **turbulence** in the **Lagrangian** frame. The simulation is designed to mimic an electromagnetically driven fluid layer, a well-documented system for generating two–dimensional **turbulence** in the laboratory. In our simulations, the forcing and velocity **fields** are close to Gaussian. On the other hand, the measured PDF of injected power is very sharply peaked at zero, suggestive of a singularity there, with tails which are exponential but asymmetric. Large positive fluctuations are more probable than large negative fluctuations. It is this asymmetry of the tails, which leads to a net positive mean value for the energy input despite the most probable value being zero. The main features of the power distribution are well described by Craig’s XY distribution for the PDF of the product of two correlated normal variables. We show that the power distribution should exhibit a logarithmic singularity at zero and decay exponentially for large absolute values of the power. We calculate the asymptotic behavior and express the asym- metry of the tails in terms of the correlation coefficient of the force and velocity. We compare the measured pdfs with the theoretical calculations and briefly discuss how the power pdf might change with other forcing mechanisms.

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be generated by the so-called Multi-scale Minimal **Lagrangian** Map (MMLM), and the Multi-scale Turnover **Lagrangian** Map (MTLM). Starting from a random field, the mappings allow the fluid particles to advect freely over short time scales while maintaining incompressibility and the energy spectrum. When the advections are applied over a set of nested grids with increasing resolution, it is shown that the synthetic **fields** not only reproduce accurately the multi-scaling properties of small scale **turbulence**, but also many properties related to small-scale geometrical structures as well as the pressure field. Using the synthetic **fields** as initial conditions for simulations, more realistic time evolution can be obtained for time evolving problems, and initial transient period can be significantly shortened for stationary problems. 25 It has been generalized to the synthesis of scalar **fields**. 29 The pressure field associated with the velocity field is also further investigated. 30

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When the cells are too distorted, the **Lagrangian** formalism is no longer reliable with respect to accuracy, consistency and stability and the use of a rezoning process to adjust the grid (Arbitrary-**Lagrangian**-Eulerian) is necessary. This step allows to obtain a better grid (smoothness, width) in order to improve the numerical accuracy of the first order (hyperbolic) operator and second order (anisotropic diffusion) operator of the coupled system.

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All applications discussed in the present study focus on the three-dimensional motion of an isolated **vortex** ring in the limit of high Reynolds number. Selection of this physical setting is motivated by the fact that the **vortex** ring is unstable to azimuthal perturbations 19] whose rapid growth leads to the formation of a complex \turbulent" vortical structure (see e.g. Fig. 6.56 in 15]). Thus, the setting proves ideal for our present purpose, since one of the primary objectives of **Lagrangian** LES is to be able to model the formation of complex vortical structures and to capture their large-scale features while using very coarse computational grids. Unfortunately, one of the disadvantages of the setting is that smallscale **turbulence** and the behavior of Helmholtz stresses are hard to characterize. This is the case because (1) the isotropy and/or homogeneity assumptions which form the basis of most analytical or semi-empirical predictions do not apply, and (2) experimental measurements of the 3D vector elds are scarce and interpretation of the data is dicult, in part due to the extreme sensitivity of the ow to perturbations and initial conditions. Briey, we have sacriced detailed quantitative analysis of the performance of subgrid- scale **turbulence** models in order to examine the behavior of the scheme in a complex, challenging environment.

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suggested that the BC recirculates in the platform (Stevenson and Souza, 1994; Stevenson, 1996). Following different paths along the SBCS, the period of recirculation computed from the LCDs' trajectories varied from 115 to 161 days. These studies suggested the BC return flow ends up becoming part of an MC extension inside the SBCS between 33S and 23S, but have not proposed any explanation for that. Our data suggest that in fact the BC does not recirculate inside the SBCS but exchanges mass and heat with BCC through **turbulence** along its western limit.

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The computational efforts to study annular **vortex** ﬂow are rather few as well. Xicheng and Zhengming [23] have used an algebraic **turbulence** model to study the three dimensional viscous ﬂow in an annular cascade with tip clearance. The objec- tive of their study was to determine the locations and structure of the leakage at the tip clearances. Garcia-Villalba and Frih- lich [24] have studied unconﬁned annular swirling ﬂow at different swirl numbers using large eddy simulation (LES). They were able to provide an analysis of the impact of swirl on the mean ﬂow and the precessing **vortex** structures. Similar tech- nique, assisted with PIV measurements, was used by Loureiro et al. [25] to investigate the characteristics of the ﬂow ﬁeld inside an annular space formed by two concentric cylinders with rotation of the inner cylinder. Such case is fundamentally different from the case presented in this paper since the **vortex** motion is imparted along the full length of the annulus, rather than through a single ﬁnite port such as the case in hand. There have been an interesting set of studies on the indus- trial application of **vortex** ﬂow in cylindrical passages in gas–liquid separators by Ahmed et al. in the past few years. They have proposed a modiﬁed Eulerian–**Lagrangian** approach that is able to enhance the numerical prediction of separation pro- cess in [26]. From their results, it is evident that the new approach captures the effect of the **vortex** ﬁeld on the separation efﬁciency more accurately than previous formulations. In some other recent studies from their group, the numerical mod- eling of high gas content gas–liquid **vortex** separators has been evaluated [27,28].

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Unlike the numerical techniques that apply an Eulerian system to solve the equations of turbulent ow, the **vortex** methods describe the ow in a **Lagrangian** fashion. **Vortex** methods have proven their validity in the numerical simulation of mixing layers and have been used extensively to simulate incompressible ows, see e.g., Ghoniem and Ng 8] and Inoue 10]. The objective of this study is to extend the prediction of mixing layers using the two dimensional **vortex** method to the uniformly sheared ows.

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兩 cos 共 4 j 兲 兩 + 兩 sin 共 4 j 兲 兩 , 共4兲 that is, harmonic plus a small perturbation which is aniso- tropic, where j gives the degree of rotational symmetry, is the polar angle, and ⑀ is a dimensionless parameter. Poten- tials of this form could in principle be created using multiple beam patterns of the desired symmetry 共see e.g., 关26兴兲. A harmonic potential can be achieved by a judicious choice of the external trapping potential to counteract the additional scalar potentials arising due to the light-atom interaction. Our motivation for considering potentials which are pre- dominantly harmonic form is twofold. First, this is of the same form as the rotating-frame potential of atoms in a ro- tating harmonic trap, allowing for a clear comparison. Sec- ond, because we are primarily interested in **vortex** nucle- ation, it is instructive to eliminate the scalar potential which can preclude **vortex** nucleation in inhomogeneous magnetic **fields** by preventing the necessary **surface** mode instabilities from occurring 关28兴.

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conclude, both the consistency in the inclination of the **vortex** packets and the collapse of all the main correlation cuts across different roughness topologies seem to offer a clear indication that the underlying spatial structures of the **turbulence** (i.e. **vortex** packets and their characteristics) are largely unaffected by changes in **surface** morphology - hence universal. This universality also extend to the smooth wall cases.

The South Pole region of the antarctic plateau provides an excellent site for the SBL research. The terrain has a gen- tle slope of ∼ 0.001 m/m, which virtually eliminates the influence of strong topographical forcings on the bound- ary layer evolutions. Furthermore, this region is devoid of several complicated atmospheric processes (e.g., hydraulic jump, barrier winds, flow splitting), which are omnipresent in some other parts of the antarctic continent. The predom- inant plateau “high” provides cold dry conditions through- out the year (King and Turner, 1997; Turner and Pendlebury, 2004). Some other notable site characteristics are: mostly clear skies, strong **surface** inversion, infrequent formation of precipitation, light (< 5 ms −1 ) northeast winds, and an aver- age annual temperature of − 49.4 ◦ C (King and Turner, 1997; Turner and Pendlebury, 2004).

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Abstract— The issue taken into consideration is sudden increase in drag over an aircraft wing due to three dimensional flow, tip vortices and flow separation. When flow separates its displacement thickness increases sharply this modifies the outside potential flow and pressure field. The pressure field modification results in an increase in pressure drag, and if severe enough will also result in loss of lift and stall. This study presents computational analysis results of a prototype wing with and without **vortex** generators of two different shapes located at leading and trailing edges of a linear wing. Here both wind tunnel testing and computational fluid dynamic analysis is carried out. The effect of the **vortex** generators are studied in four different cases. Nine sets of rectangular shaped **vortex** generators inclined at 15 degree were placed in the leading edge and trailing edge of the wing, nine sets of ogive shaped **vortex** generators inclined 15 degree were placed in the leading edge and trailing edge of the wing, are the cases analyzed. The studies also focus on prevention of downstream flow separation and improve overall performance by reducing drag. Both analytical and experimental results are compared where it shows that the pressure over the upper **surface** increases, so that the boundary layer is reenergized and attached with the body **surface** thus reducing the drag.

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The problem is that of forming an intensity image of a self-luminous object that emits monochromatic light which propagates through the atmosphere and is then collected by an ideal thin lens and recorded in the image plane. Fig. 1 shows the basic geometry of the problem. Without loss of generality, we will restrict our discussion to a linear system with **turbulence** using Taylor’s frozen-in hypothesis. r is at the object plane; v is at the lens plane; a is the thin lens radius; and p is at the image plane.

During hierarchical structure formation, clusters form from accretion of filaments, galaxies, galaxy groups, and cluster mergers. In clusters of galaxies, turbulent velocities can be inferred from the density perturbations, which, in turn, are obtained using the measured X-ray radiation intensities [23, 24, 25]. The **turbulence** in clusters is mainly sub-sonic at small scales (and near sonic at large scales), so density fluctuations (injected at large scales) behave like a passive scalar. Therefore the density and velocity spectra are expected to be the same [26]. The fact that **turbulence** is moderately supersonic in our experiment while sub-sonic in clusters, is likely to lead to only a a modest change in the power spectra (and at small enough scales, motions will in any event become sub-sonic). Indeed, spectroscopic observations of supersonic motions in molecular clouds [27] suggest a velocity power spectrum close to the classical Kolmogorov k − 5/3 law (where k is the wavenumber) that holds for incompressible fluids. Numerical simu-

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Figure 20. The ZRMS plot shows the contoured wing having a lower ZRMS in its core. However, the increased FSL **vortex**-core interaction may be clearly noted for the contoured wing case. At 4 there is lower ZRMS measured at the FSL-**vortex** interface which may indicate separation between the FSL and the wingtip **vortex**.

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One of the most powerful methods to implement the free **surface** is the Volume Of Fluid (VOF). In this study, an algorithm is developed, which includes an implicit pressure based method (SIMPLE) with a staggered grid and a **Lagrangian** propagation VOF method. Based on this algorithm, a computer code is generated and a cavity with a free **surface** and two test cases of dam-breaking problems are examined and, then, the eect of uid sloshing on a near wall is also analyzed and a time history of the normal force on the wall is presented. The results show good agreement with experimental and other computational results.

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The HF-radar data allows inclusion of the spatial current variability in the track computation with a high temporal resolution. This is a significant improvement on the Eulerian approach because tracking depends on spatial inhomogeneity of the **surface** current field. This is particularly important for the study region where the currents exhibit a large spatial variation imposed by tides, winds, large scale circulation and topography. One issue with HFR tracking at the moment is the need to fill gaps in the data sets, both in space and time. This can be solved by applying and validating current estimation- interpolation techniques to fill the gaps, or by assimilating the data into a model which produces **Lagrangian** tracks. The biggest issue to be addressed is the verisimilitude of the tracking when small scale fluctuations in **surface** velocities are ignored.

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Abstract. We use an elliptic differential equation of T ¸ it¸eica (or Toda) type to construct a minimal **Lagrangian** **surface** in CH 2 from the data of a com- pact hyperbolic Riemann **surface** and a cubic holomorphic differential. The minimal **Lagrangian** **surface** is equivariant for an SU (2, 1) representation of the fundamental group. We use this data to construct a diffeomorphism be- tween a neighbourhood of the zero section in a holomorphic vector bundle over Teichmuller space (whose fibres parameterise cubic holomorphic differentials) and a neighborhood of the R -Fuchsian representations in the SU (2, 1) repre- sentation space. We show that all the representations in this neighbourhood are complex-hyperbolic quasi-Fuchsian by constructing for each a fundamental domain using an SU (2, 1) frame for the minimal **Lagrangian** immersion: the Maurer-Cartan equation for this frame is the T ¸ it¸eica-type equation. A very similar equation to ours governs minimal surfaces in hyperbolic 3-space, and our paper can be interpreted as an analog of the theory of minimal surfaces in quasi-Fuchsian manifolds, as first studied by Uhlenbeck.

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Our density-functional 共 DFT-LDA 兲 calculations are based on a real-space multigrid implementation using nonlocal pseudopotentials. 31 The electron wave functions are mapped on a grid with a spacing corresponding to 4% of the GaAs bulk lattice constant. The **surface** is modeled by periodic super cells containing 12 atomic 共 001 兲 layers and a vacuum region 8 atomic layers thick. Further details of the DFT-LDA calculations are those in Refs. 3,32. The electronic structure obtained within DFT-LDA is used to calculate the **surface** optical anisotropy 33,34 in the independent-particle approxima- tion. In general, optical spectra are strongly modified by many-body effects such as self-energy corrections and electron-hole attraction. 15,35–37 However, RAS spectra are difference spectra, which are furthermore normalized to the bulk dielectric function. Due to the error cancellation, single- particle calculations within DFT-LDA are actually quite reli- able in predicting **surface** optical anisotropies. 3 Therefore, and because of the large number of optical spectra calculated in the present work, we simply use the scissors-operator approach 38 to take self-energy effects into account. Excitonic and local-field effects are neglected. A saw-tooth function added to the electrostatic potential entering the Kohn-Sham equations is used to mimic the effect of an electric field perpendicular to the crystal **surface**. From the self-consistent solution of the Kohn-Sham equations the influence of the electric field on both the wave functions and the eigenvalues is obtained. Additionally, the **surface** atomic geometry in the presence of an electric field is recalculated in case of the c(4 ⫻ 4) reconstruction. The sign convention used here is such that the field points in the direction of the **surface** normal.

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