Solution Of Fluid Flow Problems Research Articles

A numerical analysis of the performance of linear interpolation schemes coupled with finite volume method in determining velocity distribution for confined convection-diffusion turbulent flow field

Numerical methods are widely used to obtain solutions of fluid flow problems because they well compliment experimental methods. The numerical results obtained are however never exact due to errors emanating from the scheme used in discretizing the governing equations and the flow domain. For convection-diffusion flow, the magnitudes of these errors vary depending on the scheme used to interpolate the nodal values of the flow quantities to the interfaces. The precision level of an interpolation scheme is determined by its ability to minimize these errors hence generating results that are consistent with experimental results. This paper documents the performance of three linear interpolation schemes; upwind differencing, central differencing scheme and the hybrid scheme in obtaining velocity profiles for a convection-diffusion turbulent flow field. The field variables present in the governing equations are decomposed into a mean and a fluctuating component and averaged so as to reduce the enormous scales inherent in a turbulent flow regime. The closure problem was solved using the turbulence model. The turbulence equations have been converted into discrete form using the robust finite volume discretization technique. The discretized equations are solved using a segregated pressure-based algorithm. The numerical results were validated using the benchmark results of Ampofo and Karayiannis, (2003). The results revealed that linear interpolation schemes are not appropriate in analyzing velocity distribution for confined convection-diffusion turbulent flows because the results obtained using all the three linear schemes were inconsistent with experimental results.

On the solutions of fluid flow problems with a Chebyshev collocation method using Mathematica

AbstractFluid flow problems are important in engineering applications like heat exchangers, renewable energy sources, and electromagnetic processing of materials, among other applications. Some examples related to a computational fluid dynamics (CFD) course are explored in this study. A CFD course provides tools for the numerical solution of fluid flows where an analytical cannot be found. Flows can be described by numerically solving the mass, momentum, and energy conservation equations for incompressible fluids. A Chebyshev collocation spectral method has been used to solve the governing equations. A high‐level functional programming language strategy was implemented in Mathematica including readability and adaptability. The programs are built by applying and composing functions, mappings, iterators, and pattern matching. It was noted that students can achieve a better comprehension of the algorithms using this programming approach.

Intelligent computing for the dynamics of fluidic system of electrically conducting Ag/Cu nanoparticles with mixed convection for hydrogen possessions

The present study aims to provide an innovative stochastic numerical solver's application by the use of neural networks with Levenberg-Marquardt backpropagation to examine the dynamics of hydrogen possessions and variable viscosity in the fluidic system of electrically conducting copper and silver nanoparticles with mixed convection. The system of PDEs obtained by mathematical modeling of the physical phenomena are reduced into non-linear ODEs by utilizing suitable transformations. The ODEs dataset is constructed through Adams numerical solver and target parameters for input and output parameter of neural networks. The testing, validation and training processes are exploited in neural network models with learning based on backpropagation of LM method to calculate the solution for different scenarios created on variation of physical parameters of the proposed flow of Reynolds and Vogel models. Validation and verification of neural network model to find the solution of fluid flow problem is endorsed on the assessment of achieved accuracy through mean squared error, error histograms and regression studies.

A phase field based discrete fracture model (PFDFM) for fluid flow in fractured porous media

In this study we propose a novel phase field based discrete fracture model (PFDFM) to simulate fluid flow in fractured porous media. The common discrete fracture models represent fractures by sharp topology in an explicit way, regardless of using conforming or non-conforming mesh. Inspired by the definition of crack phase field, the sharp fracture topology is treated as a diffusive one in the solution of fluid flow problems, and the integration of fluid flow equation over fractures can be transformed to the one over the matrix. The algorithm to determine the fracture phase field and finite element discretization have been described in detail. The performance of the proposed method is validated against the classic discrete fracture model on several numerical cases in both two and three dimensions. We further investigate the convergency behavior of the proposed method through sensitivity analysis to mesh resolution and fracture parameters. Numerical results are obtained, which demonstrates that the proposed method is accurate, convergent and quite promising for simulating fluid flow in fractured porous media.

Meshless velocity – vorticity local boundary integral equation (LBIE) method for two dimensional incompressible Navier-Stokes equations

PurposeThis paper aims to describe the 2D meshless local boundary integral equation (LBIE) method for solving the Navier–Stokes equations.Design/methodology/approachThe velocity–vorticity formulation is selected to eliminate the pressure gradient of the equations. The local integral representations of flow kinematics and transport kinetics are derived. The integral equations are discretized using the local RBF interpolation of velocities and vorticities, while the unknown fluxes are kept as independent variables. The resulting volume integrals are computed using the general radial transformation algorithm.FindingsThe efficiency and accuracy of the method are illustrated with several examples chosen from reference problems in computational fluid dynamics.Originality/valueThe meshless LBIE method is applied to the 2D Navier–Stokes equations. No derivatives of interpolation functions are used in the formulation, rendering the present method a robust numerical scheme for the solution of fluid flow problems.

Some useful optimisations for unstructured computational fluid dynamics codes on multicore and manycore architectures

This paper presents a number of optimisations for improving the performance of unstructured computational fluid dynamics codes on multicore and manycore architectures such as the Intel Sandy Bridge, Broadwell and Skylake CPUs and the Intel Xeon Phi Knights Corner and Knights Landing manycore processors. We discuss and demonstrate their implementation in two distinct classes of computational kernels: face-based loops represented by the computation of fluxes and cell-based loops representing updates to state vectors. We present the importance of making efficient use of the underlying vector units in both classes of computational kernels with special emphasis on the changes required for vectorising face-based loops and their intrinsic indirect and irregular access patterns. We demonstrate the advantage of different data layouts for cell-centred as well as face data structures and architectural specific optimisations for improving the performance of gather and scatter operations which are prevalent in unstructured mesh applications. The implementation of a software prefetching strategy based on auto-tuning is also shown along with an empirical evaluation on the importance of multithreading for in-order architectures such as Knights Corner. We explore the various memory modes available on the Intel Xeon Phi Knights Landing architecture and present an approach whereby both traditional DRAM as well as MCDRAM interfaces are exploited for maximum performance. We obtain significant full application speed-ups between 2.8 and 3X across the multicore CPUs in two-socket node configurations, 8.6X on the Intel Xeon Phi Knights Corner coprocessor and 5.6X on the Intel Xeon Phi Knights Landing processor in an unstructured finite volume CFD code representative in size and complexity to an industrial application. Program summaryProgram Title: some_opt_for_unstructured_cfdProgram Files doi:http://dx.doi.org/10.17632/zyh2zkf3jw.1Licensing provisions: GNU General Public License 3 (GPL)Programming language: C/C++Nature of problem: The solution of fluid flow problems in the vicinity of complex geometries mandates the utilisation of unstructured grids. However, this flexibility of unstructured mesh methods in dealing with complicated geometries comes at a cost of increased difficulty in extracting high performance out of modern processors. We provide implementations for a number of optimisations useful for improving the performance of unstructured CFD codes on modern multicore and manycore architectures.Solution method: grid renumbering via Reverse Cuthill–Mckee, code transformations necessary for enabling vectorisation, face colouring/reordering for removing dependencies at the face end-points when accumulating residuals, data layout transformations for reducing cache misses, hand-tuned gather and scatter primitives for in-register transpositions, software prefetching via auto-tuning and multithreading for exploiting SMT features of modern processors.

Comparison of additional second-order terms in finite-difference Euler equations and regularized fluid dynamics equations

In recent years, an area of research in computational mathematics has emerged that is associated with the numerical solution of fluid flow problems based on regularized fluid dynamics equations involving additional terms with velocity, pressure, and body force. The inclusion of these functions in the additional terms has been physically substantiated only for pressure and body force. In this paper, the continuity equation obtained geometrically by Euler is shown to involve second-order terms in time that contain Jacobians of the velocity field and are consistent with some of the additional terms in the regularized fluid dynamics equations. The same Jacobians are contained in the inhomogeneous right-hand side of the wave equation and generate waves of pressure, density, and sound. Physical interpretations of the additional terms used in the regularized fluid dynamics equations are given.

A Generic Computer Tool for the Solution of Fluid Flow Problems: Case Study of a Thermal Radiative Flow of an Optically Thick Gray Gas in the Presence of Indirect Natural Convection

A Generic Computer Tool for the Solution of Fluid Flow Problems: Case Study of a Thermal Radiative Flow of an Optically Thick Gray Gas in the Presence of Indirect Natural Convection

A Generic Computer Tool for the Solution of Fluid Flow Problems: Case Study of a Thermal Radiative Flow of an Optically Thick Gray Gas in the Presence of Indirect Natural Convection

A Generic Computer Tool for the Solution of Fluid Flow Problems: Case Study of a Thermal Radiative Flow of an Optically Thick Gray Gas in the Presence of Indirect Natural Convection

Parallel control-volume method based on compact local integrated RBFs for the solution of fluid flow problems

In this paper, a high performance computing method based on the Integrated Radial Basis Function (IRBF), Control Volume (CV) and Domain Decomposition technique for solving Partial Differential Equations is presented. The goal is to develop an efficient parallel algorithm based on the Compact Local IRBF method using the CV approach, especially for problems with non-rectangular domain. The results showed that the goal is achieved as the computational efficiency is quite significant. For the case of square lid driven cavity problem with Renoylds number 100, super-linear speed-up is also achieved. The parallel algorithm is implemented in the Matlab environment using Parallel Computing Toolbox based on Distributed Computing Engine.

The Convergence Rate of Iterative Procedures for Elliptic Problems in Heterogeneous Media

We investigate the convergence rate of two iterative procedures that approximate the solutionof fluid flow problems in heterogeneous porous media. Porous media flows at large scales arecomplex problems, which require fine grid solutions to provide accurate results. Pressures and velocitiesassociated to these problems are governed by second order elliptic equations. We discretizesuch equations by a mixed and hybrid finite element method, combined with domain-decompositioniterative procedures. In order to minimize the computational effort involved in the numerical approximation,we have presented an iterative procedure to accelerate the convergence rate in the approximation.In this paper, we perform numerical experiments to compare iterative procedures in order tocheck which one provides the best convergence rate. We believe that steady-state diffusion problemscan be solved efficient and accurately by the most robust procedure.

A GPU application for high-order compact finite difference scheme

In this study, a high-order compact finite difference scheme for the solution of fluid flow problems is implemented to run on a Graphical Processing Unit (GPU) using Compute Unified Device Architecture (CUDA). Besides the compact scheme, a high-order low pass filter is also employed. For time integration, the classical fourth-order Runge–Kutta method is used. Advection of a vortical disturbance and a temporal mixing layer, two basic flows, are chosen for the application of this numerical method on a Tesla C1060, one of NVIDIA’s scientific computing GPUs. Obtained results are compared with those obtained on a single core CPU (AMD Phenom 2.5 GHz) in terms of calculation time. The CPU code exploits LAPACK/BLAS library to solve cyclic tridiagonal systems generated by the compact solution and filtering schemes, whereas the GPU code uses the inverse of the coefficient matrix to solve the same linear systems by utilizing the CUBLAS library. Moreover, the shared memory feature of the GPU is also employed to ease coalescing issues on some parts of the GPU code. Speedups between 9x–16.5x are achieved for different mesh sizes in comparison to CPU computations.

Why the Fluid Friction Factor should be Abandoned, and the Moody Chart Transformed

The friction factor (f) should be abandoned because it is a mathematically undesirable parameter that complicates the solution of fluid flow problems. f is the dimensionless group  2 g PD 5 /8LW 2 . This group is mathematically undesirable because it includes  P, W, and D. Therefore if f is used in the solution of a problem, the problem must be solved with  P, W, and D in the same term, even though fluid flow problems are generally much easier to solve if  P, W, and D are in separate terms. (Just as it is generally much easier to solve equations if x and y are in separate terms). The mathematical complication introduced by f is illustrated by the Moody chart (Fig. 1). Because the chart is based on f, it must be read iteratively (or by trial-and-error) to determine W or D. But if the Moody chart is transformed in order to eliminate f, the transformed chart (Fig. 2) is read directly to determine  P, W, or D. The fluid flow methodology described herein altogether abandons f, and allows fluid flow problems to be solved in the simplest possible manner.

Parallel Meshless EFG Solution for Fluid Flow Problems

ABSTRACT This article deals with the parallel computing solution of fluid flow problems using a meshless element-free Galerkin (EFG) method. The EFG method utilizes moving least-square (MLS) approximants to approximate the unknown field variables. The MLS approximant consists of three components: a weight function, a basis function, and a set of nonconstant coefficients. A new parallel algorithm is proposed for the EFG method. The code has been written in FORTRAN language using MPI message passing library and implemented on a MIMD (multiple-instruction multiple-data)-type PARAM 10000 supercomputer. The code has been validated by solving three model fluid flow problems. A comparison is made among the results (velocities values) obtained by the EFG method with those obtained by the finite-element method (FEM) at a few typical locations. For 8 processors, speedup and efficiency have been obtained as 2.73 and 34.22% for N = 1,552 in 1-D (Example I), 7.20 and 90.00% for N = 1,462 in 2-D (Example II), and 7.30 and 91.27% for N = 1,346 in 2-D (Example III), respectively.

Finite volume scheme for the solution of fluid flow problems on unstructured non‐staggered grids

AbstractA recently developed non‐staggered methodology which uses the principle of applying fourth‐order dissipation to the governing pressure‐correction equation is developed so it can be applied to unstructured grids. A finite volume methodology is used for discretization. The fourth‐order dissipation term is found using second‐order gradient operators. This makes it straightforward to incorporate the dissipation term on unstructured grids. The new methodology is compared with solutions from a standard finite volume second‐order flow solver and is also tested for a standard laminar driven‐lid flow problem with grids systems that do not have a uniform structure. Finally, we demonstrate how the new methodology can be used to predict flow over a wavy boundary. Copyright © 2002 John Wiley & Sons, Ltd.

Computing singular solutions of the Navier–Stokes equations with the Chebyshev‐collocation method

AbstractThe solution of fluid flow problems exhibits a singular behaviour when the conditions imposed on the boundary display some discontinuities or change in type. A treatment of these singularities has to be considered in order to preserve the accuracy of high‐order methods, such as spectral methods. The present work concerns the computation of a singular solution of the Navier–Stokes equations using the Chebyshev‐collocation method. A singularity subtraction technique is employed, which amounts to computing a smooth solution thanks to the subtraction of the leading part of the singular solution. The latter is determined from an asymptotic expansion of the solution near the singular points. In the case of non‐homogeneous boundary conditions, where the leading terms of the expansion are completely determined by the local analysis, the high accuracy of the method is assessed on both steady and unsteady lid‐driven cavity flows. An extension of this technique suitable for homogenous boundary conditions is developed for the injection of fluid into a channel. The ability of the method to compute high‐Reynolds number flows is demonstrated on a piston‐driven two‐dimensional flow. Copyright © 2001 John Wiley & Sons, Ltd.

Finite difference scheme for the solution of fluid flow problems on non‐staggered grids

A new finite difference methodology is developed for the solution of computational fluid dynamics problems that do not require the use of staggered grid systems. Previous successful and robust non-staggered methods, which used primitive variables and mass conservation in order to solve the pressure field, either interpolate cell-face velocities or interpolate the pressure gradients in a special way, usually with an upwind-bias to avoid the problem of odd–even coupling between the velocity and pressure fields. The new methodology presented does not detail a ‘special interpolation procedure for a primitive variable’, however, it manages to avoid the problem of odd–even coupling. The odd–even coupling is avoided by applying fourth-order dissipation to the pressure field. It is shown that this approach can be regarded as a modified Rhie and Chow scheme. The method is implemented using a SIMPLE-type algorithm and is applied to two test problems: laminar flow over a backward-facing step and laminar flow in a square cavity with a driven lid. Good agreement is obtained between the numerical solutions and the corresponding benchmark solutions. The pressure dissipation term was found to successfully suppress wiggles in the pressure field. Copyright © 2000 John Wiley & Sons, Ltd.

Finite difference scheme for the solution of fluid flow problems on non‐staggered grids

A new finite difference methodology is developed for the solution of computational fluid dynamics problems that do not require the use of staggered grid systems. Previous successful and robust non-staggered methods, which used primitive variables and mass conservation in order to solve the pressure field, either interpolate cell-face velocities or interpolate the pressure gradients in a special way, usually with an upwind-bias to avoid the problem of odd-even coupling between the velocity and pressure fields. The new methodology presented does not detail a special interpolation procedure for a primitive variable', however, it manages to avoid the problem of odd-even coupling. The odd-even coupling is avoided by applying fourth-order dissipation to the pressure field. It is shown that this approach can be regarded as a modified Rhie and Chow scheme. The method is implemented using a SIMPLE-type algorithm and is applied to two test problems: laminar flow over a backward-facing step and laminar flow in a square cavity with a driven lid

Finite difference scheme for the solution of fluid flow problems on non-staggered grids

Finite difference scheme for the solution of fluid flow problems on non-staggered grids

SUCCA—an alternative scheme to reduce the effects of multidimensional false diffusion

Multidimensional false diffusion effects within numerical solutions of fluid flow problems are a direct result of the use of the conventional upwind differencing scheme. The problem lies within the inaccurate modelling of convective transport within the flow field when the upwind scheme is used, particularly when the local flow direction is inclined to the solution cell. An alternative scheme, SUCCA (skew upwind corner convection algorithm), is presented that attempts to reduce false diffusion errors. The scheme takes into account the local flow direction by introducing the influence of upwind corner cells into the discretized conservation equation for momentum. A comparison is made between SUCCA and the similar flow-oriented scheme CUPID. The scheme is tested on a two-dimensional benchmark flow problem, which demonstrates its ability to accurately represent convective transport and hence help reduce the effects of false diffusion in the numerical solution of the Navier—Stokes equation applied to laminar flow.