Numerical algorithms in computational fluid dynamics on vector computers

Many problems in Computational Electromagnetics (CEM) and Computational Fluid Dynamics (CFD) require millions of grid cells to properly represent the physics. Advances in parallel computing, both in hardware and software, make it feasible to solve such large scale simulations in a timely manner. The objective of this paper is to describe some of the recent work at Rockwell in performing parallel computing for providing quick-turnaround solutions to problems in CFD, CEM and other computational disciplines.

Wind profile prediction at different scales plays a crucial role for efficient operation of wind turbines and wind power prediction. This problem can be approached in two different ways: one is based on statistical signal processing techniques and both linear and nonlinear models can be employed either separately or combined together for profile prediction; on the other hand, wind/atmospheric flow analysis is a classical problem in computational fluid dynamics (CFD) in applied mathematics, which employs various numerical methods and algorithms, although it is an extremely time-consuming process with high computational complexity. In this work, a new method is proposed based on synergy's between the signal processing approach and the CFD approach, by alternating the operations of a quaternion-valued least mean square (QLMS) algorithm and the large eddy simulation (LES) in CFD. As demonstrated by simulation results, the proposed method has a much lower computational complexity while maintaining a comparable prediction result.

Multiphase flow simulations are essential methods for providing information such as the evolution of phase fraction (gas, liquid and solid), velocities, pressure, temperature and flow regimes at every time during a process. Dynamic flow simulations also help reservoir, drilling, and production engineers to develop a proper well design. DamBreak problem is one of the most well-known problems in computational fluid dynamics (CFD); it is a dynamic hydraulic phenomena and the numerical simulation requires sophisticated mathematical modeling. OpenFOAM, is used to run CFD simulations in this thesis.;One of the main issues in CFD is that the simulations are time-consuming. In this work, will use artificial intelligence (AI) to predict the behavior of the system at each time-step of the process at a lower run time. DamBreak problem is defined base on a two-dimensional rectangular tank with a barrier at the bottom, the liquid column (water in this study) at the left side of the tank behind the wall. As soon as the wall collapse, the water will pour down, resulting in complicated fluid dynamics. The main data-set, generated by OpenFOAM flow simulations, is used for building the smart proxy model (SPM), using the network toolbox in MATLAB. Neural network (NN) is applied with feed-forward back propagation method and the training algorithm is Levenberg Marquardt.;Results indicate that the smart proxy can run 3 seconds of the DamBreak process, which takes 8 hours of computational time with 4 processors when is done by using OpenFOAM, takes less than 2 minutes using the developed SPM on one processor. SPM is also capable of predicting the CFD results in non-cascading condition and up to around 40 time-steps in cascading condition with acceptable error (less than %10).

Two problems in computational fluid dynamics are studied in the context of transonic small-disturbance theory - namely, (1) how to speed up the convergence for currently available iterative procedures, and (2) how a shock-fitting method may be adapted to existing relaxation procedures with minimal alterations in computer programming and storage requirements. The paper contributes to a clarification of error analyses for sequence transformations based on the power method (including also the nonlinear transforms of Aitken, Shanks, and Wilkinson), and to developing a cyclic iterative procedure applying the transformations. Examples testing the procedure for a model Dirichlet problem and for a transonic airfoil problem show that savings in computer time by a factor of three to five are generally possible, depending on accuracy requirements and the particular iterative procedure used.-

In the past several years, domain decomposition has been a very popular topic, partly because of the poten tial of parallelization. Although numerous theories and algorithms have been developed for model elliptic problems, they are only recently starting to be tested on realistic applications. This paper investigates the applica tion of some of these methods to two model problems in computational fluid dynamics: two-dimensional con vection-diffusion problems and the incompressible driven cavity flow problem. Our approach is the con struction and analysis of efficient preconditioners for the interface operator to be used in the iterative solution of the interface solution. For the convection-diffusion problems, we discuss the effect of the convection term and its discretization on the performance of some of the preconditioners. For the driven cavity problem, we dis cuss the effectiveness of a class of boundary probe preconditioners.

Computational Fluid Dynamics (CFD) is widely used by polymer processing industries in order to evaluate polymeric fluid flows. A successful computational code must provide reliable predictions (modeling) in a fast and efficient way (simulation). In this work, a new approach to solve the governing equations of viscoelastic fluid flows is proposed. It is based on the finite-volume method with collocated arrangement of the variables, using high-order approximations for the linear and nonlinear average fluxes in the interfaces and for the nonlinear terms obtained from the discretization of the constitutive equations. The approximations are coupled to the Weighted Essentially Non-Oscillatory (WENO) scheme to avoid oscillations in the solution. The Oldroyd-B model is used to describe the rheological behavior of the viscoelastic fluid. The average values of the variables in the volumes are used during the resolution, and the point values are recovered in the post-processing step by deconvolution of the average values. The nonlinear system, resulting from the discretization of the equations, is solved simultaneously using a Newton-like method. The obtained solutions are oscillation-free and accurate, demonstrated by the application on a classic problem in computational fluid dynamics, the slip-stick flow.

The structure and memory organization of graphics processor units (GPUs) manufactured by NVIDIA and the use of CUDA programming technology to solve computational fluid dynamics (CFD) problems is reviewed and discussed. The potential of using a general-purpose GPU to solve fluid dynamics problems is examined. The code optimization with the utilization of various memory types is considered. Some CFD benchmark problems focused on simulation of viscous incompressible fluid flows are solved on GPUs. Consideration is given to the application of the finite volume method and projection method. Programming implementation of various components of the computational procedure, solution of Poisson equation for pressure and multigrid method to solve the system of algebraic equations, is provided. By using meshes of varying resolutions and different techniques for dividing up the input data into blocks, the speedup of the GPU solution is compared to the CPU approach.

SUMMARYThe solution of problems in computational fluid dynamics (CFD) represents a classical field for the application of advanced numerical methods. Many different approaches were developed over the years to address CFD applications. Good examples are finite volumes, finite differences (FD), and finite elements (FE) but also newer approaches such as the lattice‐Boltzmann (LB), smooth particle hydrodynamics or the particle finite element method. FD and LB methods on regular grids are known to be superior in terms of raw computing speed, but using such regular discretization represents an important limitation in dealing with complex geometries. Here, we concentrate on unstructured approaches which are less common in the GPU world. We employ a nonstandard FE approach which leverages an optimized edge‐based data structure allowing a highly parallel implementation. Such technique is applied to the ‘convection‐diffusion’ problem, which is often considered as a first step towards CFD because of similarities to the nonconservative form of the Navier–Stokes equations. In this regard, an existing highly optimized parallel OpenMP solver is ported to graphics hardware based on the OpenCL platform. The optimizations performed are discussed in detail. A number of benchmarks prove that the GPU‐accelerated OpenCL code consistently outperforms the OpenMP version. Copyright © 2011 John Wiley & Sons, Ltd.

The study of hydrodynamics and thermal characteristics inside a lid‐driven cavity has been one of the most captivating problems in computational fluid dynamics. In this numerical work, the mixed convection phenomenon inside a two‐dimensional, tall lid‐driven cavity with top and bottom lids moving in opposite directions, +x and –x, respectively, has been explored for non‐Newtonian power‐law fluids. The cavity contains a uniformly heated equilateral triangular obstacle at its geometric center. Numerical experimentation is performed for a range of flow governing parameters, such as aspect ratio (0.25, 0.5, and 0.75), Prandtl number (1, 50, and 100) Richardson number (0.1, 1, and 10), power‐law index (0.6–1.4) and Grashof number of 104. The physical perceptions of the cavity are explained by using streamline and isotherm contours. The fluid movement is limited adjacent to the moving wall concerning the Richardson number at the lower Prandtl number. With a rise in the aspect ratio of the cavity, the flow‐pattern becomes more dispersed inside the cavity. Heat transfer enhancement is observed at a lower aspect ratio equal to 0.25.

Lattice-Boltzmann methods are being increasingly used to solve problems in computational fluid dynamics. The combination of robustness and simplicity has made it the method of choice for problems involving fluid flow through geometrically complex structures. Progress in several areas is summarized in a recent review article [1]. In this paper I will focus on applications of the lattice-Boltzmann method to simulations of particle-fluid suspensions. Since the basic principles of the method have already been described [2, 3], the focus of this article will be a review of recent developments and a discussion of some of the fine points that arise in practical applications of the method. In particular I will contrast the accuracy and complexity of different solid-fluid boundary conditions, discuss the application of an external pressure gradient, and describe additional complications arising from particle motion.

Grid Adaption Based on Modified Anisotropic Diffusion Equations Formulated in the Parametric Domain

A parallel implicit hole-cutting method based on background mesh for unstructured Chimera grid

ABSTRACTA key problem in computational fluid dynamics (CFD) modelling of gravel‐bed rivers is the representation of multi‐scale roughness, which spans the range from grain size, through bedforms, to channel topography. These different elements of roughness do not clearly map onto a model mesh and use of simple grain‐scale roughness parameters may create numerical problems. This paper presents CFD simulations for three cases: a plane bed of fine gravel, a plane bed of fine gravel including large, widely‐spaced pebble clusters, and a plane gravel bed with smaller, more frequent, protruding elements. The plane bed of fine gravel is modelled using the conventional wall function approach. The plane bed of fine gravel including large, widely‐spaced pebble clusters is modelled using the wall function coupled with an explicit high‐resolution topographic representation of the pebble clusters. In these cases, the three‐dimensional Reynolds‐averaged continuity and Navier–Stokes equations are solved using the standard k − ε turbulence model, and model performance is assessed by comparing predicted results with experimental data. For gravel‐bed rivers in the field, it is generally impractical to map the bed topography in sufficient detail to enable the use of an explicit high‐resolution topography. Accordingly, an alternative model based on double‐averaging is developed. Here, the flow calculations are performed by solving the three‐dimensional double‐averaged continuity and Navier‐Stokes equations with the spatially‐averaged 〈k − ε〉 turbulence model. For the plane bed of fine gravel including large, widely‐spaced pebble clusters, the model performance is assessed by comparing the spatially‐averaged velocity with the experimental data. The case of a plane gravel bed with smaller, more frequent, protruding elements is represented by a series of idealized hypothetical cases. Here, the spatially‐averaged velocity and eddy viscosity are used to investigate the applicability of the model, compared with using the explicit high‐resolution topography. The results show the ability of the model to capture the spatially‐averaged flow field and, thus, illustrate its potential for representing flow processes in natural gravel‐bed rivers. Finally, practical data requirements for implementing such a model for a field example are given. Copyright © 2011 John Wiley & Sons, Ltd.

The use of the adjoint equations is now well established for design-optimization problems in computational fluid dynamics (CFD) for the Euler and NavierStokes equations. In the continuous adjoint method approach, the original continuous governing equations are first linearized, and then discretized, while in the discrete adjoint approach, these two steps are performed in reverse order. Because the continuous approach affords more flexibility at the discretization stage, formulations involving reduced memory and cpu overheads have generally been achieved with this approach [1]. On the other hand, the discrete approach reproduces the exact sensitivity derivatives of the original discretization of the governing equations, which provides a verifiable consistency check on the final gradients produced by the adjoint solution [2]. If f(w∗,D) represents a cost functional to be minimized in an optimization process, where D is a design variable, and w∗ represents flow-field values which result from the solution of the discretized flow equations, given in residual form as R(w,D) = 0, the sensitivity derivative of f with respect to D is given by:

Casting computational fluid mechanics into a convex quadratic optimization framework