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

Machine Learning for wall modeling in LES of separating flows

Large Eddy Simulations (LES) are of increasing interest for turbomachinery design since they provide a more reliable prediction of flow physics and component behavior than standard RANS simulations. However, they remain prohibitively expensive at high Reynolds numbers or realistic geometries. The cost of resolving the near-wall region has justified the development of wall-modeled LES (wmLES), which uses a wall model to account for the effect of the energetic near-wall eddies. The classical assumptions of algebraic wall models do not hold for more complex flow patterns that frequently occur in turbomachinery passages (i.e., misalignment, separation). This work focuses on the extension of wall models to the separation phenomenon. Among possibilities to solve the complex regression problem (i.e., predicting the wall-parallel components of the shear stress from instantaneous flow data and geometrical parameters), neural networks have been selected for their universal approximation capabilities. Since DNS and LES perform well on academic and several industrial configurations, they are used to produce databases to train various neural networks. In the present work, we investigate the possibility of using neural networks to improve wall-shear stress models for flows featuring severe pressure gradients and separation. The database is composed of three building-blocks flows: (1) a flow aligned turbulent boundary layer at equilibrium; (2) a turbulent boundary layer subjected to a moderate pressure gradient; and (3) a turbulent boundary layer that separates and reattaches from a curved wall. These building blocks are referred to as a channel flow at a friction Reynolds number of 950 and the two walls (i.e., the flat upper surface and the curved lower one) of the two-dimensional periodic hill at a bulk Reynolds number of $10{,}595$, respectively. This work is constructed around three main questions: which input points should be considered for the data-driven wall model, how should one normalize the in- and output data to obtain a unified and consistent database, and which neural networks are considered

Development of a Data-Driven Wall Model for Separated flows

Large Eddy Simulations (LES) are of increasing interest for turbomachinery design since they provide a more reliable prediction of flow physics and component behavior. However, they remain prohibitively expensive at high Reynolds numbers or actual complex geometries. Most of the cost is associated with the resolution of the boundary layer, and therefore, to save computational resources, wall-modeled LES (wmLES) has become a valued tool. However, wall models are not yet reliable in predicting the complex flow configurations occurring in turbomachinery passages. Most existing analytical wall models assume the flow to be fully turbulent, attached, flow aligned, and near-equilibrium. These assumptions no longer hold when different flow regimes and complex flow features coexist. Although significant progress has been made in recent years (e.g., non-equilibrium models using pressure gradients), they have not always brought a clear benefit for such realistic flows. This paper proposes an innovative data-driven wall model to treat separated flows. Among the many possibilities to solve this complex regression problem, deep neural networks have been selected for their universal approximation capabilities~\cite{hornik_approximation_1991}. In the present framework, the two-dimensional periodic hill problem is selected as a reference test case featuring the separation of a fully turbulent boundary layer. Gaussian Mixture Neural networks (GMN) and Convolutional Neural Networks (CNN) combined with a self-attention layer~\cite{Vaswani_sel_attention_2017} are trained to predict the wall-parallel components of the wall shear stress using instantaneous flow quantities and geometric parameters. The \textit{a priori} and \textit{a posteriori} validation of such data-driven wall models on the periodic hill problem will be presented.9. Industry, innovation and infrastructur

Coupling of a compressible vortex particle-mesh method with a near-body compressible discontinuous Galerkin solver

A hybrid approach, coupling a compressible vortex particle-mesh method (CVPM, also with ecient Poisson solver) and a high order compressible discontinuous Galerkin Eulerian solver, is being developed in order to eciently simulate flows past bodies; also in the transonic regime. The Eulerian solver is dedicated to capturing the anisotropic flow structures in the near-wall region whereas the CVPM solver is exploited away from the body and in the wake. An overlapping domain decomposition approach is used. The Eulerian solver, which captures the near-body region, also corrects the CVPM solution in that region at every time step. The CVPM solver, which captures the region away from the body and the wake, also provides the outer boundary conditions to the Eulerian solver. Because of the coupling, a bound- ary element method is also required for consistency. The approach is assessed on typical 2D benchmark cases

Topics in Vortex Methods for the Computation of Three- and Two-Dimensional Incompressible Unsteady Flows

Contributions to vortex methods for the computation of incompressible unsteady flows are presented. Three methods are investigated, both theoretically and numerically. The first method to be considered is the inviscid method of vortex filaments in three dimensions, and the following topics are presented: (a) review of the method of regularized vortex filaments and of convergence results for multiple-filament computations, (b) modeling of a vortex tube by a single filament convected with the regularized Biot-Savart velocity applied on the centerline: velocity of the thin filament vortex ring and dispersion relation of the rectilinear filament, and (c) development of a new regularization of the Biot-Savart law that reproduces the lowest mode dispersion relation of the rectilinear vortex tube in the range of large to medium wavelengths. Next the method of vortex particles in three dimensions is investigated, and the following contributions are discussed: (a) review of the method of singular vortex particles: investigation of different evolution equations for the particle strength vector and weak solutions of the vorticity equation, (b) review of the method of regularized vortex particles and of convergence results, and introduction of a new algebraic smoothing with convergence properties as good as those of Gaussian smoothing, (c) development of a new viscous method in which viscous diffusion is taken into account by a scheme that redistributes the particle strength vectors, and application of the method to the computation of the fusion of two vortex rings at Re = 400, and (d) investigation of the particle method with respect to the conservation laws and derivation of new expressions for the evaluation of the quadratic diagnostics: energy, helicity and enstrophy. The third method considered is the method of contour dynamics in two dimensions. The particular efforts presented are (a) review of the classical inviscid method and development of a new regularized version of the method, (b) development of a new vector particle version of the method, both singular and regularized: the method of particles of vorticity gradient, (c) development of a viscous version of the method of regularized particles and application of the method to computation of the reconnection of two vortex patches of same sign vorticity, and (d) investigation of the particle method with respect to the conservation laws and derivation of new expressions for the evaluation of linear and quadratic diagnostics.</p

Comments on a paper by Kiya et al. on the numerical simulation of pseudo-elliptical vortex rings using the vortex particle method

A paper by Kiya et al. (1992, Fluid Dyn. Res. 10, 117–131) on the numerical simulation of pseudo-elliptical vortex ring using the viscous version of the 3-D vortex particle method (Winckelmans 1989 (Ph.D. Thesis); Winckelmans and Leonard 1989 (Proc. Workshop SIAM, 1988), 1992a (to appear in J. Comput. Phys.)) was recently published in this journal. The blame for the poor quality of the numerical results was put on the “Winckelmans-Leonard viscous scheme” (actually, the credit for the viscous method by exchange of particle strengths should go to Degond and Mas-Gallic (1989, Math. Comput. 53, 485–526)). It is the author's hope that this short note will convince the reader that the method is perfectly sound as long as it is used with care

Contributions to vortex particle methods for the computation of three-dimensional incompressible unsteady flows

Recent contributions to vortex particle methods for the computation of three-dimensional incompressible unsteady flows are presented. Both singular and regularized vortex particle methods are reviewed, along with an investigation of different evolution equations for the particle strength vector. For the regularized method, a new algebraic smoothing is presented with convergence properties equal to those of Gaussian smoothing. A version of the regularized method which can account for viscous diffusion is developed using a scheme that redistributes the particle strength vectors. Finally, particle methods are investigated with respect to conservation laws, and new expressions for the quadratic diagnostics, energy, helicity, and enstrophy are derived

VLES of aircraft wake vortices in a turbulent atmosphere: A study of decay

Very Large-Eddy Simulations (VLES) of aircraft wake vortices in various atmospheric turbulence conditions are carried out. The turbulence level is characterized by the eddy dissipation rate and ranges from very strong to weak. The turbulence is assumed isotropic. Two LES models are considered. Most simulations were done using explicit gaussian filtering and a mixed model: the tensor-diffusivity model supplemented by a dynamic Smagorinsky term. As a point of comparison, some simulations were done using the classical dynamic Smagorinsky model alone (thus without explicit filtering). Initially, the wake vortices are assumed to follow the fairly "universal" circulation profile shortly after rollup. The vortex global circulation is investigated: its decay exhibits a similar behavior in all cases, leading to the exponential decay model based on the eddy dissipation rate. The present model differs from others by the presence of a time delay before the exponential decay. The constant defining the decay rate is also found to be quite different from that found in the literature. This lack of agreement is partially explained by the differences in the definitions of circulations. © 2002 Kluwer Academic Publishers

Investigation of eddy-viscosity models modified using discrete filters: A simplified "regularized variational multiscale model" and an "enhanced field model"

Subgrid-scale (SGS) models for large-eddy simulation (LES) having the formalism of an effective eddy-viscosity model, but that operates on a modified velocity field, are further evaluated and new ones are proposed. The modified field is obtained using regular filtering of the LES field carried out in physical space. This is actually done by using a discrete and compact operator (only using nearest neighbors values), eventually iterated; this ensures that the proper filtering behavior is preserved, even for near wall points. The first model investigated here is inspired by the variational multiscale approach originally proposed by T. J. Hughes [Phys. Fluids 13, 505 (2001)]. Here, the modelling is simplified, leading to a SGS viscosity effect operating on the "small-scale LES field" that is obtained by subtracting the LES field from its filtered counterpart. Such a model (here called RVMM for short) was already proposed and partially evaluated {e.g., see G. S. Winckelmans and H. Jeanmart [Direct and Large-Scale Eddy Simulation IV (Kluwer, Dordrecht, 2001)] and H. Jeanmart and G. S. Winckelmans (CTR Proceedings of the Summer Program, 2002), the "model M2" of A. W. Vreman [Phys. Fluids 15, L61 (2003)], the "high-pass filtered Smagorinsky model" of S. Stolz [Direct and Large-Eddy Simulation V (Kluwer, Dordrecht, 2004) and Phys. Fluids 17, 065103 (2005)]}. The other model investigated here is an "enhanced field model" (EFM). The SGS viscosity model then operates on a LES field that is artificially enhanced at the small scales; that obtained by adding to the LES field the small-scale field. The two model families are presented in a unified way; they have a behavior that combines viscous and hyperviscous effects, while remaining simple and practical. They however do not naturally have the proper y(3) near wall behavior for the SGS dissipation; hence, they need some near wall damping. To ensure the proper near-wall behavior, we use here the dynamic procedure (self-consistent for each model). The performance of both models is compared to that of other models (also dynamic): the Smagorinsky model, hyperviscosity models, and a hybrid model combining explicitly a Smagorinsky term and a hyperviscosity term. The cases here investigated are LES of decaying isotropic turbulence starting at Re-lambda=90 and LES of turbulent channel flow at Re-tau=395. A good behavior of the RVMM and EFM, as compared to the others, is observed in all cases. They constitute an easily implemented and better alternative than the dynamic Smagorinsky model. (C) 2007 American Institute of Physics