Mathematical models and numerical simulations for the America’s Cup

This paper presents a review of the mathematical models which can be adopted to describe the different physical phenomena characterizing the flow around a sailing yacht. The complete model accounting for laminar–turbulent transition regime, free-surface dynamics and fluid–sails interaction is introduced as long as some simplified models that have been used to reduce the computational complexity. Drawing on the experience of the Ecole Polytechnique Fédérale de Lausanne (EPFL) as Official Scientific Advisor to the Alinghi Team, winner of the 2003 America’s Cup, we discuss the role of Computational Fluid Dynamics simulations based on Reynolds Averages Navier–Stokes (RANS) equations and their integration in standard yacht design process. Numerical results in different areas (appendages design, free-surface flows, aerodynamics of sails) are presented and discussed.     

Numerical simulation of turbulent combustion of subsonic gas jet flows

A physical and mathematical model of turbulent combustion of subsonic gas fuel jet flows flowing into an air space is proposed. The processes are described by averaged equations of the boundary layer with a turbulent viscosity model and a combustion diffusion model. As turbulent viscosity models, the well-known two-parameter k-? standard and k-?? models are taken. The results of the averaged and pulsating flow characteristics?? comparison of numerical calculations with the experimental data are presented.     

Particle mixing and reactive front motion in unsteady open shallow flow - Modelled using singular value decomposition

Passive and active tracers are used to examine particle mixing and reactive front dynamics in an open shallow flow of water past a circular cylinder. A quadtree grid based Godunov-type shallow water equation solver predicts the unsteady flow hydrodynamics of the wake behind the cylinder. The resulting periodic flow field consisting of a von Kármán vortex street is decomposed and stored over one oscillatory period using Singular Value Decomposition (SVD). Particles are advected according to the reconstructed flow field from the SVD modes, with continuous spatial velocity information obtained via bilinear interpolation. Passive particle dynamics driven by different SVD flow modes is investigated, and it is found that the flow field recovered from the mean flow and the first pair of time varying modes is adequate to represent the complicated dynamical properties induced by the original flow field. Active autocatalytic reaction, A + B → 2B, is incorporated into the particle advection model, assuming surface reaction. Active particles are found to trace out an expanded version of the unstable manifold of the chaotic saddle in the wake, in qualitative agreement with published analytical results. The numerical model is applicable to mixing and transport processes in more complicated shallow environmental flows.     

Computational modeling of flow over an ogee spillway

This paper presents an investigation into the hydraulics of regular ogee-profile spillways. The free-surfaces of the fluid for several flow heads as measured in the hydraulics laboratory are used as benchmarks. The finite element computational fluid dynamics software, ADINA, was used to predict the free surface over an ogee spillway and thus model the flow field. Since the actual flow is turbulent the k-ε flow model was used. For the cases considered in this paper, ADINA predicted reasonable free surface results that are consistent with general flow characteristics over spillways. The results are also in close agreement with measured free-surface profiles over the entire length of the spillway.     

Turbulence models and their applications to the prediction of internal flows: A review

The paper presents a brief account of various turbulence models employed in the computation of turbulent flows, and evaluates the application of these models to internal flows by examining the predictions of various turbulence models in selected important flow configurations. The main conclusions of this analysis are: (a) The κ-ε model is used in a majority of all the 2-D flow calculations reported in the literature. (b) Modified forms of the κ-ε model improve the performance for flows with streamline curvature and heat transfer. (c) For flows with swirl, the κ-ε model performs rather poorly; the algebraic stress model performs better in this case. (d) For flows with regions of secondary flow (noncircular duct flows), the algebraic stress model performs fairly well. Two important factors in the numerical solution of the model equations, namely false diffusion and inlet boundary conditions, are discussed. The existence of countergradient transport and its implications in turbulence modeling are examined. Finally, some recommendations for improving the model performance are made. The need for detailed experimental data in flows with strong curvature is emphasized.     

A vorticity based aeroacoustic prediction for the noise emission of a low-speed turbulent internal flow

Turbulent internal flows are known to generate intense noise as well as surface pressure fluctuations. Numerically predicting the noise emission near the prescribed boundaries requires that the sound-generating turbulent flow be adequately represented and described. The k-ε method provides a promising tool for obtaining the unsteady characteristics of a realistic turbulent flow interacting with a rectangular flat plate undergoing “ground effect”. The far-field acoustic calculation is facilitated by the Kambe model (from Lighthill’s theory) and an original post-processor has been developed to determine the far-field spectra and the source term characteristics. In pre-processed turbulent confined flows inside a rectangular enclosure, computations using the k-ε method, coupled with a convenient post-processor, is used to predict noise radiation over a wide range of frequencies and geometrical configurations. The portability and effectiveness of the present method, however, start to be less evident when the turbulent flow has very chaotic internal features; these suggest the need for a wider computational aeroacoustic domain and a new procedure suitable for the turbulent flow representation.     

Numerical study of the flow over shallow cavities

This paper reports on a series of numerical simulations of both laminar and turbulent flows over shallow cavities. For the turbulent case the influences of the following parameters were considered: (i) cavity aspect ratios, (ii) turbulence level of the oncoming flow, and (iii) Reynolds number. Several important results and conclusions are reported. We have found that for the turbulent case the external flow touches the floor of the cavity, and this depends on a specific value of each of these parameters. This condition has an important impact upon convective effects inside the cavity. The mathematical model corresponds to the incompressible, Reynolds-averaged, Navier-Stokes equations plus a high-Reynolds κ-ε model of turbulence, and the numerical computation is performed using the SIMPLER algorithm.     

An interface capturing method for free-surface hydrodynamic flows

An interface capturing method based on a numerically revisited procedure for velocity and pressure coupling is worked out. The new treatment of density discontinuity is formulated in the framework of the finite volume methodology for arbitrary unstructured grids. A simple analytical pressure-like model case is presented to illustrate the accuracy of the numerical implementation of the treatment of discontinuous variables. Then, the method is implemented in a viscous flow solver and applied to free-surface flows, including the two-dimensional Rayleigh-Taylor instability problem and three-dimensional hydrodynamic flows for the prediction of ship waves around the Series 60 model ship with and without drift angles. These latter simulations show excellent agreement with the experimental results illustrated by comparisons of free-surface elevations and also velocity field.     

Investigation of density flow in dam reservoirs using a three-dimensional mathematical model including Coriolis effect

This paper aims to simulate and discuss the propagation of density current and divergence flow in a dam reservoir. The density plunging flow is modeled in three-dimensions through a dam reservoir with diverging and sloping bottom channels, and the plunging phenomenon has been reproduced in the present model. Nonlinear and unsteady continuity, momentum, energy and turbulence model equations are formulated in the Cartesian coordinates both in a sloping and in a diverging channel. For the turbulence viscosity, a k-ε turbulence model including buoyancy effects is used to reproduce the main flow characteristics. To investigate the Coriolis force effect on the density flow in a dam reservoir, Coriolis force parameter is also included in the governing equations. The equations of the model are solved based on the initial and boundary conditions of the dam reservoir flow for a range of bottom slopes and divergence angles. In this paper the main interest is the formation of separated flows, such as wall-jet and free-jet flows. The model successfully simulates the formation of attached flow, wall jets, and free jets in a negatively buoyant environment. The simulation results obtained from this study are compared with previous experimental work, and the mathematical model studies data on density current generated by the plunging of cold water in ambient warm water in a diverging channel, and is found to be of the same magnitude as the experimental measurements and followed the expected basic trend.     

An analysis of the time integration algorithms for the finite element solutions of incompressible Navier–Stokes equations based on a stabilised formulation

This work is concerned with the analysis of time integration procedures for the stabilised finite element formulation of unsteady incompressible fluid flows governed by the Navier–Stokes equations. The stabilisation technique is combined with several different implicit time integration procedures including both finite difference and finite element schemes. Particular attention is given to the generalised-α method and the linear discontinuous in time finite element scheme. The time integration schemes are first applied to two model problems, represented by a first order differential equation in time and the one dimensional advection–diffusion equation, and subjected to a detailed mathematical analysis based on the Fourier series expansion. In order to establish the accuracy and efficiency of the time integration schemes for the Navier–Stokes equations, a detailed computational study is performed of two standard numerical examples: unsteady flow around a cylinder and flow across a backward facing step. It is concluded that the semi-discrete generalised-α method provides a viable alternative to the more sophisticated and expensive space–time methods for simulations of unsteady flows of incompressible fluids governed by the Navier–Stokes equations.     

Investigation of wall bounded flows using SPH and the unified semi-analytical wall boundary conditions

Semi-analytical wall boundary conditions present a mathematically rigorous framework to prescribe the influence of solid walls in smoothed particle hydrodynamics (SPH) for fluid flows. In this paper they are investigated with respect to the skew-adjoint property which implies exact energy conservation. It will be shown that this property holds only in the limit of the continuous SPH approximation, whereas in the discrete SPH formulation it is only approximately true, leading to numerical noise. This noise, interpreted as a form of “turbulence”, is treated using an additional volume diffusion term in the continuity equation which we show is equivalent to an approximate Riemann solver. Subsequently two extensions to the boundary conditions are presented. The first dealing with a variable driving force when imposing a volume flux in a periodic flow and the second showing a generalization of the wall boundary condition to Robin type and arbitrary-order interpolation. Two modifications for free-surface flows are presented for the volume diffusion term as well as the wall boundary condition. In order to validate the theoretical constructs numerical experiments are performed showing that the present volume flux term yields results with an error 5 orders of magnitude smaller then previous methods while the Robin boundary conditions are imposed correctly with an error depending on the order of the approximation. Furthermore, the proposed modifications for free-surface flows improve the behavior at the intersection of free surface and wall as well as prevent free-surface detachment when using the volume diffusion term. Finally, this paper is concluded by a simulation of a dam break over a wedge demonstrating the improvements proposed in this paper.     

A variational principle in terms of stream function for free-surface flows and its application to the finite element method

Steady free-surface flows under the influence of gravity are considered in this paper. The appropriate variational principle is derived, where the stream function within the flow and the free-surface elevation are independently subjected to variation. The formulation of the principle in terms of linear finite elements is presented, and the resulting set of non-linear equations is reduced to the iterative solution of a linear set. The computations are shown to be convergent regardless of whether the Froude number of the stream is greater or less than unity, and independent tests show that the accuracy of the results is well within the usual range expected for internal flows when linear elements are used.     

MHD flow past a circular cylinder using the immersed boundary method

The immersed boundary method (IB hereafter) is an efficient numerical methodology for treating purely hydrodynamic flows in geometrically complicated flow-domains. Recently Grigoriadis et als. [1] proposed an extension of the IB method that accounts for electromagnetic effects near non-conducting boundaries in magnetohydrodynamic (MHD) flows. The proposed extension (hereafter called MIB method) integrates naturally within the original IB concept and is suitable for magnetohydrodynamic (MHD) simulations of liquid metal flows. It is based on the proper definition of an externally applied current density field in order to satisfy the Maxwell equations in the presence of arbitrarily-shaped, non-conducting immersed boundaries. The efficiency of the proposed method is achieved by fast direct solutions of the two poisson equations for the hydrodynamic pressure and the electrostatic potential.The purpose of the present study is to establish the performance of the new MIB method in challenging configurations for which sufficient details are available in the literature. For this purpose, we have considered the classical MHD problem of a conducting fluid that is exposed to an external magnetic field while flowing across a circular cylinder with electrically insulated boundaries. Two- and three-dimensional, steady and unsteady, flow regimes were examined for Reynolds numbers Re_d ranging up to 200 based on the cylinder’s diameter. The intensity of the external magnetic field, as characterized by the magnetic interaction parameter N, varied from N=0 for the purely hydrodynamic cases up to N=5 for the MHD cases. For each simulation, a sufficiently fine Cartesian computational mesh was selected to ensure adequate resolution of the thin boundary layers developing due to the magnetic field, the so called Hartmann and sidewall layers. Results for a wide range of flow and magnetic field strength parameters show that the MIB method is capable of accurately reproducing integral parameters, such as the lift and drag coefficients, as well as the geometrical details of the recirculation zones. The results of the present study suggest that the proposed MIB methodology provides a powerful numerical tool for accurate MHD simulations, and that it can extend the applicability of existing Cartesian flow solvers as well as the range of computable MHD flows. Moreover, the new MIB method has been used to carrry out a series of accurate simulations allowing the determination of asymptotic laws for the lift and drag coefficients and the extent of the recirculation length as a function of the amplitude of the magnetic field. These results are reported herein.     

JOSEPHINE: A parallel SPH code for free-surface flows

JOSEPHINE is a parallel Smoothed Particle Hydrodynamics program, designed to solve unsteady free-surface flows. The adopted numerical scheme is efficient and has been validated on a first case, where a liquid drop is stretched over the time. Boundary conditions can also be modelled, as it is demonstrated in a second case: the collapse of a water column. Results show good agreement with both reference numerical solutions and experiments. The use of parallelism allows significant reduction of the computational time, even more with large number of particles. JOSEPHINE has been written so that any untrained developers can handle it easily and implement new features.Program summaryProgram title: JOSEPHINECatalogue identifier: AELV_v1_0Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AELV_v1_0.htmlProgram obtainable from: CPC Program Library, Queen?s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 5139No. of bytes in distributed program, including test data, etc.: 22 833Distribution format: tar.gzProgramming language: Fortran 90 and OpenMPIComputer: All shared or distributed memory parallel processors, tested on a Xeon W3520, 2.67 GHz.Operating system: Any system with a Fortran 90 compiler and MPI, tested on Debian Linux.Has the code been vectorised or parallelised?: The code has been parallelised but has not been explicitly vectorised.RAM: Dependent upon the number of particles.Classification: 4.12Nature of problem: JOSEPHINE is designed to solve unsteady incompressible flows with a free-surface and large deformations.Solution method: JOSEPHINE is an implementation of Smoothed Particle Hydrodynamics. SPH is a Lagrangian mesh free particle method, thus, no explicit tracking procedure is required to catch the free surface. Incompressibility is satisfied using a weakly compressible model. Boundary conditions at walls are enforced by means of the ghost particles technique. The free-surface dynamic and kinematic conditions are applied implicitly.Running time: 15 mn on 4 processors for the dam-break case with 5000 particles, dependent upon the real duration (2 s here).     

Neural networks based subgrid scale modeling in large eddy simulations

In this paper a multilayer feed-forward neural network (NN) is used as subgrid scale (SGS) model in a large eddy simulation (LES). The NN was previously off-line trained using numerical data generated by a LES of a channel flow at Re_τ=180 with Bardina's scale similar (BFR) SGS model. Results show the ability of NNs to identify and reproduce the highly nonlinear behavior of the turbulent flows, and therefore the possibility of using NN techniques in numerical simulations of turbulent flows.     

Multi-block lattice Boltzmann simulations of subcritical flow in open channel junctions

A two-dimensional lattice Boltzmann model (LBM) for subcritical flows in open channel junctions is developed. Shallow water equations coupled with the large eddy simulation model is numerically simulated by the lattice Boltzmann method, so that the turbulence, caused by the combination of the main channel and tributary flows, can be taken into account and modeled efficiently. In order to obtain more detailed and accurate results, a multi-block lattice scheme is designed and applied at the area of combining flows. The model is first verified by experimental data for a 90° junction flow, then is used to investigate the effect of the junction angle on flow characteristics, such as velocity field, water depth and separation zone. The objectives of this study are to validate the two-dimensional LBM in junction flow simulation and compare the results with available experimental data and classical analytical solutions in the separation zone.     

GPU-acceleration for Moving Particle Semi-Implicit method

The MPS (Moving Particle Semi-implicit) method has been proven useful in computation free-surface hydrodynamic flows. Despite its applicability, one of its drawbacks in practical application is the high computational load. On the other hand, Graphics Processing Unit (GPU), which was originally developed for acceleration of computer graphics, now provides unprecedented capability for scientific computations.The main objective of this study is to develop a GPU-accelerated MPS code using CUDA (Compute Unified Device Architecture) language. Several techniques have been shown to optimize calculations in CUDA. In order to promote the acceleration by GPU, particular attentions are given to both the search of neighboring particles and the iterative solution of simultaneous linear equations in the Poisson Pressure Equation.In this paper, 2-dimensional calculations of elliptical drop evolution and dam break flow have been carried out by the GPU-accelerated MPS method, and the accuracy and performance of GPU-based code are investigated by comparing the results with those by CPU. It is shown that results of GPU-based calculations can be obtained much faster with the same reliability as the CPU-based ones.     

Path Flow Estimator in an Entropy Model Using a Nonlinear L-Shaped Algorithm

The general problem of estimating origin-destination matrices in congested traffic networks is formulated as a mathematical program with equilibrium constraints. In this paper a path flow entropy model based on an equilibrium assignment approach is presented. It is assumed the flows on links are disaggregated due to some external reasons. Using the L-shaped algorithm the problem is transformed into a problem with less constraints. Then applying the lagrangian dual function the original problem is reduced to a nonlinear convex problem with only one linear constraint.     

改进GL-GIN的多意图识别和槽填充联合模型

在当前自然语言处理多意图识别模型研究中,存在建模方式均为从意图到插槽的单一方向的信息流建模,忽视了插槽到意图的信息流交互建模研究,意图识别任务易于混淆且错误捕获其他意图信息,上下文语义特征提取质量不佳,有待进一步提升等问题.本文以当前先进的典型代表GL-GIN模型为基础,进行优化改进,探索了插槽到意图的交互建模方法,运用槽到意图的单向注意力层,计算插槽到意图的注意力得分,纳入注意力机制,利用插槽到意图的注意力得分作为连接权重,使其可以传播和聚集与意图相关的插槽信息,使意图重点关注与其相关的插槽信息,从而实现多意图识别模型的双向信息流动;同时,引入BERT模型作为编码层,以提升了语义特征提取质量.实验表明,该交互建模方法效果提升明显,与原GL-GIN模型相比,在两个公共数据集(MixATIS和MixSNIPS)上,新模型的总准确率分别提高了5.2%和9%.