有限体积法计算流体力学的误差分析

有限体积法计算流体力学的误差分析（Error An alysis in finite Volume CFD）

摘要：

本文针对二阶精度有限体积法离散误差量化与网格优化展开研究，提出全新单元面残差误差估计器（FREE），依托单元两侧外推场量差值计算面离散误差，适配对流、扩散主导多类算例。基于该估计器构建全自动各向异性自适应加密流程，沿高误差面拆分六面体单元，在保证精度前提下大幅削减网格量；同时基于 Delaunay 算法开发多面体网格生成与自适应重构工具。通过解析解基准算例与空腔、弯管等工程湍流算例验证方法，对比四边形、三角形、多面体网格精度，证实四边形网格计算误差最低。整套误差评估与自适应网格体系可为 CFD 仿真可控高精度求解提供完整支撑。

Error An alysis in Finite Volume CFD

Franjo Juretic

Thesis submitted for the

Degree of Doctor of Philosophy of the University of London  and   Diploma of Imperial College

Department of Mechanical Engineering

Imperial College London

December, 2004

Abstract

This study was aimed towards improving the accuracy of Computational Fluid Dy- namics  (CFD) by developing methods for reliable estimation of the discretisation error and its reduction.

A new method for error estimation of the discretisation error for the second- order accurate Finite Volume Method is presented, called the Face Residual Error Estimator (FREE), which estimates the discretisation error on the cell faces. The es- timator is tested on a set of cases with an alytical solutions, ranging from convection- dominated to diffusion-dominated dominated ones.  Testing is also performed on a set of cases of engineering interest and on polygonal meshes.

In order to automatically produce a solution of pre-determined accuracy an au- tomatic error-controlled adaptive mesh refinement procedure is set up. It uses local mesh refinement to control the local error magnitude by refining hexahedral cells parallel to the face with large discretisation error.  The procedure is tested on four cases with a nalytical solutions and on several laminar and turbulent flow cases of engineering interest.  It was found able to produce accurate solutions with savings in computational resources.

In order to explore the possibilities of different mesh structures, a mesh generator producing polyhedral meshes based on the Delaunay technique is developed.   An adaptive mesh generation technique for polyhedral meshes is also developed and is based on remeshing parts of the mesh which are selected for refinement.  The mesh adaptation technique is tested on a case with an an alytical solution. A comparison of accuracy achieved on quadrilateral, triangular and polygonal meshes is also given, where quadrilateral meshes perform best followed by polygonal meshes.

Acknowledgements

I would like to express my gratitude to my supervisor Prof A. D. Gosman for his interest and continuous guidance during this study.

I would like to use the opportunity to thank my colleagues from Prof Gosman’s CFD group, especially Dr. Hrvoje Jasak, Mr.  Henry Weller and Mr Mattijs Janssens for developing the FOAM C++ simulation code which made the implementation of the ideas easier. Their support and suggestions were invaluable.

This study and the text of this thesis has benefited a lot from the numerous suggestions and comments by Dr. Hrvoje Jasak.

It would be unfair not to thank Mrs Nicky Scott-Knight and Mrs Susan Clegg for arranging administrative matters.

Finally, the financial support provided by the Computational Dynamics Ltd.  is gratefully acknowledged.

Nomenclature

Latin Characters

a   – general vector property

aN     – matrix coefficient corresponding to the neighbour N

aP     – central coefficient

C   – constant dependent on the scheme for temporal discretisation CF     – force coefficient

Cp     – pressure coefficient

Co   – Courant number

d   – vector between P and N

dn     – vector between the cell centre and the boundary face

E   – exact error, required error tolerance

e   – total specific energy, solution error, truncation error

ef     – error on the face

F   – mass flux through the face

Fconv     – convection transport coefficient

Fdiff     – diffusion transport coefficient

Fnorm     – normalisation factor for the residual

f   – face, point in the centre of the face

fi     – point of interpolation on the face fx     – interpolation factor

gb     – boundary condition on the fixed gradient boundary g   – acceleration of the gravity force

G   – matrix for Least-Squares Fit

H     – transport part, Hessian matrix

h   – mesh size

I   – unit tensor

k   – non-orthogonal part of the face area vector

k   – turbulent kinetic energy

L   – functional, set of edges

m   – skewness correction vector, second moment

M   – geometric moment of inertia, momentum

N   – point in the centre of the neighbouring cell, number of cells P   – pressure, point in the centre of the cell, set of points

P   – atmospheric pressure

xdist     – position difference vector

p   – kinematic pressure, order of accuracy q   – q in the q _ ζ turbulence model

QP     – source for the system of linear equations

QV     – body forces

QS     – surface forces

Re   – Reynolds number

r   – ratio

Resf     – face residual

Res     – face residual from the cell owner

Res     – face residual from the cell neighbour ResP     – cell residual

S   – outward-pointing face area vector

Sf     – face area vector s   – parametric curve Sφ     – source term

Se     – error source term

Sp   – linear part of the source term

Su   – constant part of the source term SCV     – area of a control volume

T   – temperature, time-scale

t   – time

U   – velocity

Ub     – velocity of the arbitrary volume’s face V   – volume

VM     – material volume VCV     – control volume Vi     – Voronoi Polygon VP     – volume of the cell x   – x component

x   – position vector y   – y component  z   – z component

Greek Characters

α  – under-relaxation factor

αN     – non-orthogonality angle Γφ     – diffusivity

γ  – blending factor

Δ   – orthogonal part of the face area vector

∈  – dissipation rate of turbulent kinetic energy

ζ  – effectivity index, ζ-field in the q _ ζ turbulence model

λ  – heat conduction coefficient µ  – dynamic viscosity

ν  – kinematic viscosity

νT     – turbulent kinematic viscosity ρ  – density

σ  – turbulent Prandtl number σ  – stress tensor

Φ   – exact solution

φ  – general scalar property

ψ  – measure of mesh skewness

Superscripts 

qT     – transpose

q   – mean

q,     – fluctuation around the mean value, shadow points 

qn     – new time-level

qnew     – new value 

qo     – old time-level 

qold     – old value

q    – unit vector

q   – normalised, recovered value

Subscripts

qf     – value on the face

qb     – value on the boundary face

qP     – value in the cell owner

qN     – value in the cell neighbour, value in new point

qt     – turbulent

qreal     – real flow

qgovEqn     – exact solution of governing equations

qFV     – solution obtained by using the Finite Volume Method 

qiteration    – solution obtained by using an iterative procedure 

q   – interpolated value, approximated value

Abbreviations

ADT   – Alternating Digital Tree

AFT   – Advancing Front Method

Bi-CGSTAB   – Bi-Conjugate Gradient Stabilised 

CAD   – Computer-Aided Design

CD   – Central Differencing

CFD   – Computational Fluid Dynamics

CG   – Conjugate Gradient

CV   – Control Volume

DNS   – Direct Numerical Simulation

FD   – Finite Difference Method

FEM   – Finite Element Method

FV   – Finite Volume

FVM   – Finite Volume Method

FREE   – Face Residual Error Estimator

ICCG   – Incomplete Cholesky Conjugate Gradient LES   – Large Eddy Simulation

LSF   – Least Squares Fit

LU   – Lower-Upper

NVA   – Normalised Variable Approach

NS   – Navier-Stokes equations

PISO   – Pressure-Implicit with Splitting of Operators

RANS   – Reynolds Averaged Navier-Stokes

RE   – Richardson Extrapolation

SIMPLE   – Semi-Implicit Method for Pressure-Linked Equations 

TDMA   – Thomas algorithm

UD   – Upwind Differencing

2D   – Two-dimensional space

3D   – Three-dimensional space

Chapter 1  Introduction

1.1    Background

Computational Fluid Dynamics (CFD) provides solutions to fluid flow problems by solving the governing equations on a computer.  CFD has undergone rapid develop- ment in the last two decades and problems which can be solved with it range from simple laminar flows to very complicated multi-phase flows, including heat exchange. Many of the existing CFD codes are coupled with the CAD systems to make the design process easier and less expensive.  As CFD is becoming an engineering tool its accuracy is gaining more and more importance, introducing the need for a reliable method for assessing and controlling accuracy.

The governing equations are of partial differential form, coupled in most cases. Closed form solutions cannot be found except for some simple problems, which are not of much practical interest.  The numerical methods used for CFD provide solu- tions by dividing the domain into smaller domains and assuming a certain variation of the dependent fields over each subdomain.   This, together with the conditions specified at the boundary of the original domain, generates a system of N algebraic equations with N unknowns for each dependent variable, N representing the number of subdomains, which can be solved using a computer.  The process of converting a differential equation into a system of algebraic equations is called discretisation. This process may introduce errors which can have a great influence on the quality of the results obtained.

There are many different discretisation practices.   The most widely used ones are [47]: Finite Difference Method (FD), Finite Element Method (FEM) and Finite  Volume Method (FVM). The most popular discretisation practice in the CFD com- munity is the Finite Volume (FV) Method and it is also adopted in this study.  Finite Volume discretisation is performed using the integral formulation of the conserva- tion laws which are then discretised in physical space by assuming linear variation of the dependent variables over each discrete volume, called a control volume (CV). The nature of the FV discretisation practice allows the use of an arbitrary mesh, where control volumes can be of arbitrary topology consisting of general polyhedral volumes. The FV discretisation practice is conservative in its nature and the quan- tities like momentum, mass, energy,  etc.  remain conserved during the calculation process.

CFD solutions may contain errors which can be divided into four main groups [47]:

• Modelling errors   are defined as the difference between the real flow Φreal and the exact solution of the governing equations ΦgovEqn , thus:

Complex problems like turbulence, combustion and multi-phase flows are very difficult to describe exactly and require approximations in the process of de- riving  the  governing  equations.   Initial  and  boundary  conditions  may  also introduce errors if they are not known exactly. The geometry description can also introduce errors if it is not exact.

• Discretisation errors  are defined as the difference between the exact solu- tion of the differential equations ΦgovEqn  and the exact solution of their discrete FV approximations φFV:

Discretisation errors originate from the discretisation approximations, the mesh density and quality, and the size of the time-step in transient calculations.  A discretisation procedure, which can be partly characterised by its approxima- tion order, can be exact in the parts of the flow where the variable distribution corresponds exactly to the discretisation assumptions and very inaccurate in the regions where the variables vary in different ways.

1.1 Background                                                                                                                    27

• Iteration errors  are a group of errors which arise if the governing equations are solved using an iterative procedure.   They are defined as the difference between the exact solution of the FV equations and the solution obtained by using an iterative procedure φiteration , thus:

These errors can be reduced to the level of computer truncation error for any given problem at the expense of time needed to complete the calculation.

• Programming and User errors are a result of the incorrect implementation or use of the CFD methodology in a computer code.

Error estimators are tools for estimation of the discretisation error in the nu- merical solution by using properties of the discretisation practice and the governing equations. They provide information about the discretisation error distribution and its magnitude in some norm and therefore measure the quality of the results in this respect.

The required discretisation accuracy is known before the an alysis is performed. It depends on the objective of the calculations and on the accuracy of the differential equations which are used to describe the physics.  Error estimators can be used as indicators of where and how to modify the mesh to achieve solutions of the required discretisation accuracy.   This can be achieved by locally refining the mesh where the error is large and coarsening the mesh in the regions where the error is small, in order to maintain the error at the required level and equidistribute it over the computational  domain.   An  adaptive  procedure,  used  for  achieving  the  required accuracy, should be composed of a number of cycles, each cycle consisting of solving equations using a current mesh and the discretisation practice,  followed by error estimation and finally modification of the mesh.

The quality of a computational mesh is an important factor in minimising dis- cretisation error [47].  The quality is influenced by spatial resolution, skewness and non-orthogonality of the mesh and also by the type of cells used.

The aim of the present study is to develop:

1.  An accurate method for estimation of the discretisation error in the FV so- lution which is applicable to different types of differential equations and for problems ranging from convection to diffusion dominated ones.

2.  A method capable of reducing the discretisation error below the required level without any user intervention and with the smallest computational load by producing the optimum mesh for the given problem.

1.2    Present Contributions

This study has contributed to the field of Computational Fluid Dynamics in the following respects:

• A Face Residual Error Estimator  (FREE) which estimates the error on the faces of the computational cells is developed.  It is a result of an an alysis of the discretisation error and its transport through cell faces.   The estimator measures the error in the field under consideration by comparing the values extrapolated onto the face from the neighbouring nodes.

• A fully automatic mesh adaptation procedure is set up and applied to several steady-state flows, both laminar and turbulent.  The mesh is adapted by re- fining the cells sharing a face with large error by splitting them parallel to the face.

• A mesh generator for polyhedral meshes for the Finite Volume Method based on the Delaunay method is developed.

• A mesh adaptation procedure for polyhedral meshes is developed.  It is per- formed by enriching the Delaunay structure with vertices where more resolu- tion is needed resulting in local refinement.

• An  an alysis  of discretisation  errors  on  meshes  consisting  of squares,  trian- gles, hexagons and split-hexahedra is presented. A comparison of the relative accuracy on quadrilateral, polygonal and triangular meshes is performed on laminar flow cases.

1.3    Thesis Outline

In Chapter 2, a summary of the governing equations of continuum mechanics can be found along with the Newtonian constitutive relations.  A transport equation as a model equation is introduced. A brief description of turbulence modelling is given.

Chapter 3 presents the Finite Volume Method used in this study.  It is a second- order accurate method for arbitrary unstructured meshes.  Discretisation of spatial terms in the transport equation is described term by term along with the errors which may arise.   Temporal discretisation and the errors which may results from temporal discretisation are briefly discussed. An an alysis of the discretisation error on different shapes of computational cells is performed.  A solution algorithm for Navier-Stokes equations is presented at the end of the chapter.

Developments in the field of a _ posteriori error estimation made during this study are presented in Chapter 4.  A literature survey of the existing methods is given first.  A new method for error estimation is proposed.   The performance of the proposed error estimator is tested on a set of cases with an alytical solutions, including convection and diffusion-dominated ones.

In Chapter 5 a mesh refinement procedure is proposed.  A literature survey of mesh adaptation methods is presented first, followed by the proposed mesh refine- ment procedure based on directional cell-by-cell refinement of hexahedral cells.  The performance  of the  refinement  procedure  is  examined  on  a  set  of test  for which an alytical solutions are available.

In Chapter 6, the mesh refinement procedure proposed in Chapter 5 is further tested on four cases of engineering interest, involving laminar and turbulent flows.

Chapter 7 presents an algorithm for polyhedral mesh generation developed during this study.  A survey of mesh generation methods is given at the beginning of the chapter. It is followed by an algorithm assembled for calculating polyhedral meshes from the Delaunay Triangulation and the Voronoi Polygons which is described step by step.  A mesh adaptation technique for polyhedral meshes is also presented.  A comparison of relative accuracy which can be achieved on triangular, quadrilateral and  polygonal  meshes  is  performed  on  cases  introduced  in  earlier  chapters.   An example of the adaptive mesh generation is also given.

Finally,  a  summary  of the  Thesis  with  some  conclusions  and  suggestions  for future work are given in Chapter 8.

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