A finite element method for diffusion dominated unsteady viscous flows

A general conforming finite element scheme for computing viscous flows is presented which is of second-order accuracy in space and time. Viscous terms are treated implicitly and advection terms are treated explicitly in the time marching segment of the algorithm. A method for solving the algebraic equations at each time step is given. The method is demonstrated on two test problems, one of them being a plane vortex flow for which asymptotic methods are used to obtain suitable numerical boundary conditions at each time step.     

POD acceleration of fully implicit solver for unsteady nonlinear flows and its application on grid architecture

A method for the acceleration of a fully implicit solution of nonlinear unsteady boundary value problem is presented. The principle of acceleration is for provide to the inexact Newton backtracking method a better initial guess, for the current time step, than the conventional choice from the previous time step. This initial guess is built on the reduced model obtained by a proper orthogonal decomposition of solutions at the previous time steps. This approach is appealing to GRID computing: spare processors may help to improve the numerical efficiency and to manage the computing in a reliable way.     

An implicit high-order discontinuous Galerkin method for steady and unsteady incompressible flows

This paper presents the latest developments of a discontinuous Galerkin (DG) method for incompressible flows introduced in [Bassi F, Crivellini A, Di Pietro DA, Rebay S. An artificial compressibility flux for the discontinuous Galerkin solution of the incompressible Navier–Stokes equations. J Comput Phys 2006;218(2):794–815] for the steady Navier–Stokes equations and extended in [Bassi F, Crivellini A. A high-order discontinuous Galerkin method for natural convection problems. In: Wesseling P, Oñate E, Periaux J, editors. Electronic proceedings of the ECCOMAS CFD 2006 conference, Egmond aan Zee, The Netherlands, September 5–8; 2006. TU Delft] to the coupled Navier–Stokes and energy equations governing natural convection flows.

The method is fully implicit and applies to the governing equations in primitive variable form. Its distinguishing feature is the formulation of the inviscid interface flux, which is based on the solution of local Riemann problems associated with the artificial compressibility perturbation of the Euler equations. The tight coupling between pressure and velocity so introduced stabilizes the method and allows using equal-order approximation spaces for both pressure and velocity. Since, independently of the amount of artificial compressibility added, the interface flux reduces to the physical one for vanishing interface jumps, the resulting method is strongly consistent.

In this paper, we present a review of the method together with two recently developed issues: (i) the high-order DG discretization of the incompressible Euler equations; (ii) the high-order implicit time integration of unsteady flows. The accuracy and versatility of the method are demonstrated by a suite of computations of steady and unsteady, inviscid and viscous incompressible flows.     

Helicoidal vortex model for steady and unsteady flows

A helicoidal vortex model is used to predict the flow past the blades of a wind turbine. As the tip speed ratio (TSR) varies, the environment in which the blades operate varies, and for low enough TSR, the local angle of attack α will be larger than (α)_Clmax, the incidence of maximum lift. The problem becomes highly nonlinear and it is shown that adding an artificial viscosity term to the equation allows the iterative algorithm to converge toward a smooth solution that is physically acceptable.The introduction of unsteady effects is useful to understand the cyclic forces and moments due to yaw or tower interaction, both for the design of blades to account for fatigue and for power output prediction. The 2-D impulsively plunging plate problem is solved with a semi-implicit scheme and the integral and numerical solutions compared and shown to be in excellent agreement. A 2-D test case to study the convection of a periodic shedding of vorticity downstream of a blade element is analyzed using the same semi-implicit algorithm and a stretched mesh similar to that used to model a turbine vortex sheet. The scheme captures accurately the vorticity distribution in the wake. Finally, the scheme is applied to the simulation of the NREL two-bladed rotor in yaw to assess the validity of the approach.     

A second-order finite volume scheme for three dimensional truncated pyramidal quantum dot

Three dimensional truncated pyramidal quantum dots are simulated numerically to compute the energy states and the wave functions. The simulation of the hetero-structures is realized by using a novel finite volume scheme to solve the Schrödinger equation. The simulation benefits greatly from the finite volume scheme in threefold. Firstly, the BenDaniel-Duke hetero-junction interface condition is ingeniously embedded into the scheme. Secondly, the scheme uses uniform meshes in discretization and leads to simple computer implementation. Thirdly, the scheme is efficient as it achieves second-order convergence rates over varied mesh sizes. The scheme has successfully computed all the confined energy states and visualized the corresponding wave functions. The results further predict the relation of the energy states and wave functions versus the height of the truncated pyramidal quantum dots.     

High order finite difference methods for unsteady incompressible flows in multi-connected domains

Using the vorticity and stream function variables is an effective way to compute 2-D incompressible flow due to the facts that the incompressibility constraint for the velocity is automatically satisfied, the pressure variable is eliminated, and high order schemes can be efficiently implemented. However, a difficulty arises in a multi-connected computational domain in determining the constants for the stream function on the boundary of the “holes”. This is an especially challenging task for the calculation of unsteady flows, since these constants vary with time to reflect the total fluxes of the flow in each sub-channel. In this paper, we propose an efficient method in a finite difference setting to solve this problem and present some numerical experiments, including an accuracy check of a Taylor vortex-type flow, flow past a non-symmetric square, and flow in a heat exchanger.     

Topological active volumes: A topology-adaptive deformable model for volume segmentation

This paper proposes a generic methodology for segmentation and reconstruction of volumetric datasets based on a deformable model, the topological active volumes (TAV). This model, based on a polyhedral mesh, integrates features of region-based and boundary-based segmentation methods in order to fit the contours of the objects and model its inner topology. Moreover, it implements automatic procedures, the so-called topological changes, that alter the mesh structure and allow the segmentation of complex features such as pronounced curvatures or holes, as well as the detection of several objects in the scene. This work presents the TAV model and the segmentation methodology and explains how the changes in the TAV structure can improve the adjustment process. In particular, it is focused on the increase of the mesh density in complex image areas in order to improve the adjustment to object surfaces. The suitability of the mesh structure and the segmentation methodology is analyzed and the accuracy of the proposed model is proved with both synthetic and real images.     

A 3D finite element model for free-surface flows

The present work deals with the validation of 3D finite element model for free-surface flows. The model uses the non-hydrostatic pressure and the eddy viscosities from the conventional linear turbulence model are modified to account for the secondary effects generated by strong channel curvature in the natural rivers with meandering open channels. The unsteady Reynolds-averaged Navier–Stokes equations are solved on the unstructured grid using the Raviart–Thomas finite element for the horizontal velocity components, and the common P1 linear finite element in the vertical direction. To provide the accurate resolution at the bed and the free-surface, the governing equations are solved in the multi-layers system (the vertical plane of the domain is subdivided into fixed thickness layers). The up-to-date k–ε turbulence solver is implemented for computing eddy coefficients, the Eulerian–Lagrangian–Galerkin (ELG) temporal scheme is performed for enhancing numerical time integration to guarantee high degree of mass conservation while the CFL restriction is eliminated. The present paper reports on successful validation of the numerical model through available benchmark tests with increasing complexity, using the high quality and high spatial resolution three-dimensional data set collected from experiments.     

A 3rd order upwind finite volume method for generalised Newtonian fluid flows

The non-Newtonian effects in the flow of non-Newtonian fluids that can be modelled with generalised Newtonian constitutive equations are investigated using a numerical scheme based on the finite volume formulation. Collocated arrangement of variables is used and the pressure-correction method in conjunction with the SIMPLE scheme is applied. In order to avoid the diffusive effects of low order schemes, the approximation of the convection terms is carried out using the QUICK differencing scheme, which is of third order of accuracy. For modelling viscous non-Newtonian behaviour, the Power-Law, Quemada and the modified Bingham and Casson models are employed. Validation of the code is carried out upon comparison with numerical data available in the literature. Exploiting the lid-driven cavity flow a twofold investigation is carried out regarding the non-Newtonian effects first by using different models and secondly by using the shear-thinning and shear-thickening attributes of the fluid.     

A new second-order closure model for rough-wall turbulent flows using the Brinkman equation

A second-order turbulence closure is developed for the new rough-wall layer modeling approach using the Brinkman equation for turbulent flows over rough walls. In the proposed approach, we model the fluid dynamics of the volume averaged flow in the near-wall rough layer by using the Brinkman equation. The porosity can be calculated based on the volumetric characteristics of the roughness and the permeability is modeled. Interface stress jump conditions including the Reynolds stress components are also considered. The Reynolds-averaged Navier-Stokes equations are solved numerically above the near-wall rough layer, while a second-order turbulence closure is employed in all regions. The rough-wall second-order closure is developed by adopting an existing smooth-wall model. The computational results, including the skin friction coefficient, the log-law mean velocity, the roughness function, the Reynolds stresses, and the turbulent kinetic energy, are presented and compared with those obtained by using a previously developed two-equation turbulence closure. The results show that the new rough-wall layer modeling approach with the second-order turbulence closure model satisfactorily predicted the skin friction coefficient, the log-law mean velocity, the roughness function, and the Reynolds shear stress. However, the results for the Reynolds normal stresses are different from the measured data in the inner 20-60% of the boundary layer due to the interface stress jump conditions employed in the present rough-wall layer modeling approach.     

Benchmark computations based on lattice-Boltzmann,finite element and finite volume methods for laminar flows

The goal of this article is to contribute to the discussion of the efficiency of lattice-Boltzmann (LB) methods as CFD solvers. After a short review of the basic model and extensions, we compare the accuracy and computational efficiency of two research simulation codes based on the LB and the finite-element method (FEM) for two-dimensional incompressible laminar flow problems with complex geometries. We also study the influence of the Mach number on the solution, since LB methods are weakly compressible by nature, by comparing compressible and incompressible results obtained from the LB code and the commercial code CFX. Our results indicate, that for the quantities studied (lift, drag, pressure drop) our LB prototype is competitive for incompressible transient problems, but asymptotically slower for steady-state Stokes flow because the asymptotic algorithmic complexity of the classical LB-method is not optimal compared to the multigrid solvers incorporated in the FEM and CFX code. For the weakly compressible case, the LB approach has a significant wall clock time advantage as compared to CFX. In addition, we demonstrate that the influence of the finite Mach number in LB simulations of incompressible flow is easily underestimated.     

A finite volume method parallelization for the simulation of free surface shallow water flows

We construct a parallel algorithm, suitable for distributed memory architectures, of an explicit shock-capturing finite volume method for solving the two-dimensional shallow water equations. The finite volume method is based on the very popular approximate Riemann solver of Roe and is extended to second order spatial accuracy by an appropriate TVD technique. The parallel code is applied to distributed memory architectures using domain decomposition techniques and we investigate its performance on a grid computer and on a Distributed Shared Memory supercomputer. The effectiveness of the parallel algorithm is considered for specific benchmark test cases. The performance of the realization measured in terms of execution time and speedup factors reveals the efficiency of the implementation.     

Assessment of multiscale resolution for hybrid turbulence model in unsteady separated transonic flows

This paper presents results of a computational study conducted to assess the multi-scale resolution capabilities of a hybrid two-equation turbulence model in predicting unsteady separated high speed flows. Numerical solutions are obtained using a third order Roe scheme and the SST (shear-stress-transport) two-equation-based hybrid turbulence model for three-dimensional transonic flow over an open cavity. A detailed assessment of the effects of the computational grid and the hybrid turbulence model coefficient is presented for the unsteady flow field. Computed results are presented for both the resolved and the modeled turbulent kinetic energy (TKE) and for the predicted sound pressure level (SPL) spectra, which are compared to available experimental data and large Eddy simulation (LES) results. The comparison shows that the predicted SPL spectra agree well with the experimental results over a frequency range up to 2500 Hz, and that hybrid turbulence effectively models the shorter wavelengths. The results demonstrate improved agreement with experimental SPL spectra with increased grid resolution and a reduced hybrid turbulence model coefficient. In addition, they show that energy dissipation of the unresolved scales is over-predicted at low resolutions and that the hybrid coefficient influences the grid resolution requirements.     

A two-level computational graph method for the adjoint of a finite volume based compressible unsteady flow solver

The adjoint method is a useful tool for finding gradients of design objectives with respect to system parameters for fluid dynamics simulations. But the utility of this method is hampered by the difficulty in writing an efficient implementation for the adjoint flow solver, especially one that scales to thousands of cores. This paper demonstrates a Python library, called adFVM, that can be used to construct an explicit unsteady flow solver and derive the corresponding discrete adjoint flow solver using automatic differentiation (AD). The library uses a two-level computational graph method for representing the structure of both solvers. The library translates this structure into a sequence of optimized kernels, significantly reducing its execution time and memory footprint. Kernels can be generated for heterogeneous architectures including distributed memory, shared memory and accelerator based systems. The library is used to write a finite volume based compressible flow solver. A wall clock time comparison between different flow solvers and adjoint flow solvers built using this library and state of the art graph based AD libraries is presented on a turbomachinery flow problem. Performance analysis of the flow solvers is carried out for CPUs and GPUs. Results of strong and weak scaling of the flow solver and its adjoint are demonstrated on subsonic flow in a periodic box.     

The active geometric shape model: A new robust deformable shape model and its applications

We present a novel approach for fitting a geometric shape in images. Similar to active shape models and active contours, a force field is used in our approach. But the object to be detected is described with a geometric shape, represented by parametric equations. Our model associates each parameter of this geometric shape with a combination of integrals (summations in the discrete case) of the force field along the contour. By iteratively updating the shape parameters according to these integrals, we are able to find the optimal fit of the shape in the image. In this paper, we first explore simple cases such as fitting a line, circle, ellipse or cubic spline contour using this approach. Then we employ this technique to detect the cross-sections of subarachnoid spaces containing cerebrospinal fluid (CSF) in phase-contrast magnetic resonance (PC-MR) images, where the object of interest can be described by a distorted ellipse. The detection results can be further used by an s–t graph cut to generate a segmentation of the CSF structure. We demonstrate that, given a properly configured geometric shape model and force field, this approach is robust to noise and defects (disconnections and non-uniform contrast) in the image. By using a geometric shape model, this approach does not rely on large training datasets, and requires no manual labeling of the training images as is needed when using point distribution models.     

车辆检测中可变形部件模型的改进与应用

针对可变形部件模型的复杂性使其在检测车辆时速度慢的问题,对可变形部件模型进行了改进。一方面使用加权PCA对可变形部件模型的基础HOG特征进行降维来减少模型参数;另一方面将HOG特征层组合后,使用快速傅里叶变换（FFT）把滤波器与HOG特征层的卷积转换为频域乘积,来降低计算复杂度。仿真实验结果表明,改进的可变形部件模型在进行车辆检测时检测精度和召回率都与原始模型相当,但检测速度大幅提升,在UIUC和BIT两个数据集上的平均耗时分别仅占原始模型平均耗时的29.6%和26.3%。     

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.     

A finite difference method for 3D incompressible flows in cylindrical coordinates

In this work, a finite difference method to solve the incompressible Navier-Stokes equations in cylindrical geometries is presented. It is based upon the use of mimetic discrete first-order operators (divergence, gradient, curl), i.e. operators which satisfy in a discrete sense most of the usual properties of vector analysis in the continuum case. In particular the discrete divergence and gradient operators are negative adjoint with respect to suitable inner products. The axis r = 0 is dealt with within this framework and is therefore no longer considered as a singularity. Results concerning the stability with respect to 3D perturbations of steady axisymmetric flows in cylindrical cavities with one rotating lid, are presented.