Weak solutions of the three-dimensional vorticity equation with vortex singularities

The use of a modified scheme for the dynamics of vortex singularities is shown to lead to a weak solution of the three-dimensional inviscid incompressible vorticity equation

Non-singular Green's functions for the unbounded Poisson equation in one, two and three dimensions

This paper is a revised version of the original paper of same title--published in Applied Mathematics Letters 89--containing some corrections and clarifications to the original text. We derive non-singular Green's functions for the unbounded Poisson equation in one, two and three dimensions, using a cut-off function in the Fourier domain to impose a smallest length scale when deriving the Green's function. The resulting non-singular Green's functions are relevant to applications which are restricted to a minimum resolved length scale (e.g. a mesh size h) and thus cannot handle the singular Green's function of the continuous Poisson equation. We furthermore derive the gradient vector of the non-singular Green's function, as this is useful in applications where the Poisson equation represents potential functions of a vector field

Some Progress in Large-Eddy Simulation using the 3-D Vortex Particle Method

This two-month visit at CTR was devoted to investigating possibilities in LES modeling in the context of the 3-D vortex particle method (=vortex element method, VEM) for unbounded flows. A dedicated code was developed for that purpose. Although O(N(sup 2)) and thus slow, it offers the advantage that it can easily be modified to try out many ideas on problems involving up to N approx. 10(exp 4) particles. Energy spectrums (which require O(N(sup 2)) operations per wavenumber) are also computed. Progress was realized in the following areas: particle redistribution schemes, relaxation schemes to maintain the solenoidal condition on the particle vorticity field, simple LES models and their VEM extension, possible new avenues in LES. Model problems that involve strong interaction between vortex tubes were computed, together with diagnostics: total vorticity, linear and angular impulse, energy and energy spectrum, enstrophy. More work is needed, however, especially regarding relaxation schemes and further validation and development of LES models for VEM. Finally, what works well will eventually have to be incorporated into the fast parallel tree code

Vortex Particle-Mesh with Immersed Lifting Lines for Aerospace and Wind Engineering

AbstractWe present the treatment of lifting lines with a Vortex Particle-Mesh (VPM) methodology. The VPM method relies on the Lagrangian discretization of the Navier-Stokes equations in vorticity-velocity formulation. The use of this hybrid discretization offers several advantages. The particles are used solely for the advection, thereby waiving classical time stability constraints. They also exploit the compactness of vorticity support, leading to high computational gains for external flow simulations. The mesh, on the other hand, handles all the other computationally intensive tasks, such as the evaluation of the differential operators and the use of fast Fourier-based Poisson solvers, which allow the combination of unbounded directions and inlet/outlet boundaries. Both discretizations communicate through high order interpolation. The mesh and the interpolation also allow for additional advances; they are used to handle Lagrangian distortion by reinitializing the particle positions onto a regular grid. This crucial step, referred to as remeshing, guarantees the accuracy of the method. In addition, the resulting methodology provides computational efficiency and scalability to massively parallel architectures.Sources of vorticity are accounted for through a lifting line approach. This line handles the attached and shed vorticity contributions in a Lagrangian manner. Its immersed treatment efficiently captures the development of vorticity from thin sheets into a three-dimensional field. We apply this approach to the simulation of wake flows encountered in aeronautical and wind energy applications. An important aspect in these fields is the handling of turbulent inflows. We have developed a technique for the introduction of pre-computed or synthetic turbulent flow fields in vorticity form. Our treatment is based on particles as well and consistent with the Lagrangian character of the method. We apply here our method to the investigation of wind turbine wakes over very large distances, reaching cluster or wind farm sizes

Large Eddy Simulation of Turbulent Channel Flows by the Rational LES Model

The rational large eddy simulation (RLES) model is applied to turbulent channel flows. This approximate deconvolution model is based on a rational (subdiagonal Pade') approximation of the Fourier transform of the Gaussian filter and is proposed as an alternative to the gradient (also known as the nonlinear or tensor-diffusivity) model. We used a spectral element code to perform large eddy simulations of incompressible channel flows at Reynolds numbers based on the friction velocity and the channel half-width Re{sub tau} = 180 and Re{sub tau} = 395. We compared the RLES model with the gradient model. The RLES results showed a clear improvement over those corresponding to the gradient model, comparing well with the fine direct numerical simulation. For comparison, we also present results corresponding to a classical subgrid-scale eddy-viscosity model such as the standard Smagorinsky model.Comment: 31 pages including 15 figure

Discontinuous Galerkin methodology for Large-Eddy Simulations of wind turbine airfoils

peer reviewedThis paper aims at evaluating the potential of the Discontinuous Galerkin (DG) methodology for Large-Eddy Simulation (LES) of wind turbine airfoils. The DG method has shown high accuracy, excellent scalability and capacity to handle unstructured meshes. It is however not used in the wind energy sector yet. The present study aims at evaluating this methodology on an application which is relevant for that sector and focuses on blade section aerodynamics characterization. To be pertinent for large wind turbines, the simulations would need to be at low Mach numbers (M ≤ 0.3) where compressible approaches are often limited and at large Reynolds numbers (Re ≥ 106) where wall-resolved LES is still unaffordable. At these high Re, a wall-modeled LES (WMLES) approach is thus required. In order to first validate the LES methodology, before the WMLES approach, this study presents airfoil flow simulations at low and high Reynolds numbers and compares the results to state-of-the-art models used in industry, namely the panel method (XFOIL with boundary layer modeling) and Reynolds Averaged Navier-Stokes (RANS). At low Reynolds number (Re = 6 x 104), involving laminar boundary layer separation and transition in the detached shear layer, the Eppler 387 airfoil is studied at two angles of attack. The LES results agree slightly better with the experimental chordwise pressure distribution than both XFOIL and RANS results. At high Reynolds number (Re = 1.64 x 106), the NACA4412 airfoil is studied close to stall condition. In this case, although the wall model approach used for the WMLES is very basic and not supposed to handle separation nor adverse pressure gradients, all three methods provide equivalent accuracy on averaged quantities. The present work is hence considered as a strong step forward in the use of LES at high Reynolds numbers

A priori testing of subgrid-scale models for the velocity-pressure and vorticity-velocity formulations

Subgrid-scale models for Large Eddy Simulation (LES) in both the velocity-pressure and the vorticity-velocity formulations were evaluated and compared in a priori tests using spectral Direct Numerical Simulation (DNS) databases of isotropic turbulence: 128(exp 3) DNS of forced turbulence (Re(sub(lambda))=95.8) filtered, using the sharp cutoff filter, to both 32(exp 3) and 16(exp 3) synthetic LES fields; 512(exp 3) DNS of decaying turbulence (Re(sub(Lambda))=63.5) filtered to both 64(exp 3) and 32(exp 3) LES fields. Gaussian and top-hat filters were also used with the 128(exp 3) database. Different LES models were evaluated for each formulation: eddy-viscosity models, hyper eddy-viscosity models, mixed models, and scale-similarity models. Correlations between exact versus modeled subgrid-scale quantities were measured at three levels: tensor (traceless), vector (solenoidal 'force'), and scalar (dissipation) levels, and for both cases of uniform and variable coefficient(s). Different choices for the 1/T scaling appearing in the eddy-viscosity were also evaluated. It was found that the models for the vorticity-velocity formulation produce higher correlations with the filtered DNS data than their counterpart in the velocity-pressure formulation. It was also found that the hyper eddy-viscosity model performs better than the eddy viscosity model, in both formulations

Efficient FMM accelerated vortex methods in three dimensions via the Lamb-Helmholtz decomposition

Vortex element methods are often used to efficiently simulate incompressible flows using Lagrangian techniques. Use of the FMM (Fast Multipole Method) allows considerable speed up of both velocity evaluation and vorticity evolution terms in these methods. Both equations require field evaluation of constrained (divergence free) vector valued quantities (velocity, vorticity) and cross terms from these. These are usually evaluated by performing several FMM accelerated sums of scalar harmonic functions. We present a formulation of the vortex methods based on the Lamb-Helmholtz decomposition of the velocity in terms of two scalar potentials. In its original form, this decomposition is not invariant with respect to translation, violating a key requirement for the FMM. One of the key contributions of this paper is a theory for translation for this representation. The translation theory is developed by introducing "conversion" operators, which enable the representation to be restored in an arbitrary reference frame. Using this form, extremely efficient vortex element computations can be made, which need evaluation of just two scalar harmonic FMM sums for evaluating the velocity and vorticity evolution terms. Details of the decomposition, translation and conversion formulae, and sample numerical results are presented

Peripheral blood mitochondrial DNA content in relation to circulating metabolites and inflammatory markers: a population study

Mitochondrial DNA (mtDNA) content might undergo significant changes caused by metabolic derangements, oxidative stress and inflammation that lead to development and progression of cardiovascular diseases. We, therefore, investigated in a general population the association of peripheral blood mtDNA content with circulating metabolites and inflammatory markers. We examined 310 subjects (50.6% women; mean age, 53.3 years) randomly selected from a Flemish population. Relative mtDNA content was measured by quantitative real-time PCR in peripheral blood cells. Peak circulating metabolites were quantified using nuclear magnetic resonance spectroscopy. The level of inflammation was assessed via established inflammatory markers. Using Partial Least Squares analysis, we constructed 3 latent factors from the 44 measured metabolites that explained 62.5% and 8.5% of the variance in the contributing metabolites and the mtDNA content, respectively. With adjustments applied, mtDNA content was positively associated with the first latent factor (P = 0.002). We identified 6 metabolites with a major impact on the construction of this latent factor including HDL3 apolipoproteins, tyrosine, fatty acid with αCH2, creatinine, β-glucose and valine. We summarized them into a single composite metabolite score. We observed a negative association between the composite metabolic score and mtDNA content (P = 0.001). We also found that mtDNA content was inversely associated with inflammatory markers including hs-CRP, hs-IL6, white blood cell and neutrophil counts as well as neutrophil-to-lymphocyte ratio (P≤0.0024). We demonstrated that in a general population relative peripheral blood mtDNA content was associated with circulating metabolites indicative of perturbed lipid metabolism and with inflammatory biomarkers