This thesis presents the development and application of a method for the solution of steady and unsteady constant or variable density subsonic fluid flow through ...
A FINITE VOLUME METHOD FOR COMPUTATION OF FLUID FLOW IN COMPLEX GEOMETRIES
BY
ISMET AHMED DEMIRDZIC B.SC.(MECH.ENG.); B.SC.(MATH.)
THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY OF THE UNIVERSITY OF LONDON AND FOR THE DIPLOMA OF IMPERIAL COLLEGE
IMPERIAL COLLEGE OCTOBER 1982
MECH. ENG. DEPT. LONDON SW7
ABSTRACT
This thesis presents the development and application of a method for the solution of steady and unsteady constant or variable density subsonic fluid flow through a domain of arbitrary shape. A mathematical description of the problem is developed in terms of general non-orthogonal curvilinear coordinates using general tensor calculus. To facilitate the physical interpretation of the resulting equations of motion a new definition of physical tensor components is given and a novel method of deriving transport equations directly in terms of these components is developed. The many forms of the transport equations that can be employed to describe fluid flows in arbitrary geometries are reviewed and evaluated with respect to their suitability for numerical solution by the finite volume approach. Arising from this a semi-strong conservation form is selected which employs contravariant physical tensor components. The differential equations are discretized in two dimensions by the finite volume method. The discretization incorporates a fully-implicit time differencing scheme with hybrid central/upwind spatial differencing. The resulting set of algebraic equations is characterised by nine-diagonal coefficient matrices, as compared with five-diagonal ones in the case of an orthogonal coordinate system. Moreover some of the coefficients in the non-orthogonal case are not unconditionally positive. These equations are solved iteratively in a sequential manner by a modification of an existing method that incorporates a procedure for coupling the continuity and momentum equations through an approximate pressure correction equation. In the present application this is done in a way which avoids the adverse effects of negative coefficients and retains five-diagonal coefficients matrix of the original pressure correction equation.
-ii
The boundary fitted computational grid allowed by the procedure consists of arbitrary contigous quadrilateral-shaped control volumes defined by the coordinates of their vertices. The necessary geometrical information characterising the grid is provided by applying a nine-point quadratic interpolation function to each control volume. The method is tested by applying it to a number of cases for which an exact analytical solution, results from other computational methods or reliable experimental data exists, including inviscid, laminar and turbulent flow in straight and constant-curvature ducts, irrotational stagnation flow and Jeffery-Hamel flow. With a few exceptions agreement to within a few percent is obtained, even with relatively large degrees of grid non-orthogonality. Calculations are also presented of some complex turbulent flows, including variations of the backward-facing step problem, a curved free shear layer and the flow through an inlet port/ valve assembly of a reciprocating internal combustion engine. The results are compared with experimental data and/or with predictions obtained from other computational methods, when available. Reasonable agreement with experimental results is obtained when considered in the light of the turbulence model employed.
iii
ACKNOWLEDGEMENTS
I would like to express my deep gratitude to my supervisor Dr. A.D.Gosman for his advice and guidance throughout this project and also for his always friendly and patient introduction into the fascinating world of computational fluid mechanics. I would like also to thank Dr. R.I.Issa whose keen interest, sharp criticism and above all helpful comments have greatly contributed to the completion of this work. I have also benefited from occasional_discussions with other members of Fluid Section who are too numerous to list here, but are no less appreciated. I am grateful to my wife Senka and our children who made me forget all my 'numerical stability' problems during the short hours I could spare from my work. Mrs Helen Bastin has typed this difficult manuscript with the rare mastery. The financial support of the Central Electricity Generating Board is gratefully acknowledged.
iv
CONTENTS page
ABSTRACT
ACKNOWLEDGEMENTS
CONTENTS
LIST OF TABLES
LIST OF FIGURES
i
iii
iv
viii
ix
CHAPTER 1 INTRODUCTION 1.1 Background
1
1.2 Present Contributions
4
1.3 Contents of the Thesis
6
CHAPTER 2 PRELIMINARIES FROM GENERAL TENSOR CALCULUS 2.1 Introduction
10
2.2 Coordinate Transformation
11
2.3 Contravariant, Covariant and Mixed Components
14
2.4 The Metric
15
2.5 Christoffel Symbols
17
2.6 Covariant Derivatives
19
2.7 Scalar Multiplication and Addition
20
2.8 Transposition, Symmetrization and Alternation
21
2.9 Dot and Tensor Products
22
2.10 Gradient and Divergence
23
2.11 Closure
25
page CHAPTER 3 TENSOR CALCULUS IN TERMS OF PHYSICAL COMPONENTS 3.1 Introduction
26
3.2 Definition of Physical Components
29
3.3 Physical Metric Tensor Components
31
3.4 Physical Christoffel Symbols
33
3.5 Covariant Derivatives of Physical Components
34
3.6 Algebraic Operations
36
3.7 Gradient and Divergence
36
3.8 Dual Physical Components
38
3.9 Closure
39
CHAPTER 4 ALTERNATIVE FORMULATIONS OF TRANSPORT EQUATIONS IN GENERAL COORDINATES 4.1 Introduction
4.2 Governing Equations
41
4.3 Weak and Strong Conservation Forms of Governing Equations
42
44
4.4 Relative Merits of Contravariant and Covariant Forms of Governing Equations
51
4.5 Choice of Direction of Resolution of Momentum Equations
53
4.6 Summary of Different Forms of Governing Equations
54
4.7 Previous Work on Numerical Solutions of Transport Equations of Fluid Flow in General Coordinates 4.8 Summary and Selection
55 60
CHAPTER 5 DIFFERENTIAL EQUATIONS OF LAMINAR AND TURBULENT FLUID FLOW IN GENERAL NON-STEADY COORDINATES 5.1 Introduction
65
5.2 Governing Differential Equations
65
5.3 Governing Equations for Moving Coordinate Frame
68
vi
page 5.4 Equations for Turbulent Flow
78
5.5 General Form of Transport Equations
83
5.6 Boundary Conditions
83
5.7 Closure
90
CHAPTER 6 THE NUMERICAL SOLUTION METHOD 6.1 Introduction
91
6.2 Expanded Form of Differential Equations
92
6.3 Discretization Procedure
97
6.4 Choice of Differencing Scheme
108
6.5 Treatment of the Cross-Derivative Diffusion Terms
116
6.6 The form of Discretized Equations Used in this Study
118
6.7 Differencing of the Source Terms
119
6.8 Interpolation Practices
123
6.9 Solution Procedure
124
6.10 Solution of the Discretized Equations
132
6.11 Treatment of the Boundary Conditions
135
6.12 Closure
139
CHAPTER 7 GRID AND ASSOCIATED GEOMETRICAL PROPERTIES 7.1 Introduction
141
7.2 Properties of the Computational Grid
142
7.3 Grid Generation
143
7.4 Local Coordinate Transformation
147
7.5 Geometrical Properties of the Coordinate System
150
7.6 Calculation of the Basic Geometrical Quantities
156
7.7 Cell-Related Quantities
158
7.8 Closure
159
vii
page CHAPTER 8 ASSESSMENT OF THE METHOD 8.1 Introduction
160
8.2 Inviscid Flows
161
8.3 Laminar Flows
166
8.4 Turbulent Flows
182
8.5 Closure
183
CHAPTER 9 APPLICATION OF THE METHOD 9.1 Introduction
185
9.2 Backward-Facing Step Problems
185
9.3 Highly Curved Mixing Layer
201
9.4 Flow Across Tube Banks
208
9.5 Flow in an Inlet Port/Valve Assembly
212
9.6 Closure
213
CHAPTER 10 SUMMARY AND CONCLUSIONS 10.1 Achievements and Findings
216
10.2 Improvements and Extensions
218
REFERENCES
224
NOMENCLATURE
237
APPENDICES 1
Transformation of Equations from Cartesian into General Coordinates
244
2
Cross-Derivative Diffusion Flux Terms
248
3
An Alternative Discretization Practice for Diffusion Fluxes
249
4
Parameters of Jeffery-Hamel Flow
FIGURES
253
255
viii
LIST OF TABLES page
TABLE 4.1
Definitions of Quantities in Equation (4.4)
43
TABLE 4.2 An Estimate of the Execution Time and Storage Requirements for Some Forms of Transport Equations for Steady Two-Dimensional Flow
63
TABLE 5.1
Values of the k-E Turbulence Model Constants
82
TABLE 5.2
Definitions of Exchange Coefficients and Source Terms in General Transport Equation (5.52) for Particular Variables
TABLE 8.1
84
Results of Calculations for Inviscid Constant 163
Curvature Duct Flow TABLE 8.2
Results of Calculations for Inviscid Stagnation Flow
166
TABLE 8.3
Jeffery-Hamel Flow Test Cases
TABLE 8.4
The Main Characteristics of the Methods and
174
Results of Calculations for the Curved Diffuser Flow Obtained with a 20x20 Mesh (or FiniteElement Equivalent)
181
TABLE 9.1
Reattachment Length for BFS1
192
TABLE 9.2
Reattachment Length for BFS2
196
TABLE 9.3
Reattachment Length for BFS3
198
ix
LIST OF FIGURES
Fig. 1.1
Irregular boundary nodes in a Cartesian grid
Fig. 2.1
Curvilinear coordinates, natural and dual bases at a point in two dimensions
Fig. 3.1
Physical and dual physical vector components in two dimensions
Fig. 4.1
Vector conservation in the case of resolution into spatially variable directions
Fig. 4.2
Expression for the base vectors in terms of locally constant basis in two dimensions
Fig. 4.3
Possible forms of the transport equations
Fig. 4.4
Two-dimensional ICED-ALE grid
Fig. 4.5
TURF grid
Fig. 4.6
Definitions of geometrical quantities in equations (4.12) and (4.24)
Fig. 4.7
Usual staggered variable arrangement and Cartesian velocity components
Fig. 4.8
Grid and variable arrangement employed by Liu (1976) and Demird/i6 et al (1980)
Fig. 5.1
Moving and fixed coordinate frame in two dimensions
Fig. 5.2
Wall boundary
Fig. 6.1
Computational grid arrangement
Fig. 6.2
Typical control volume and labelling scheme
Fig. 6.3
Notation for skew-upwind differencing scheme
Fig. 6.4
Scalar (main or continuity) control volume
x
Fig. 6.5
(1) v velocity computational cell
Fig. 7.1
An example of the computational grid
Fig. 7.2
Notation related to the calculation of the points in the middle of the scalar cell faces
Fig. 7.3
Definition of the local coordinate system
Fig. 7.4
Local geometrical characteristics of the coordinate system
Fig. 8.1
Channel flow: geometry and grid
Fig. 8.2
Constant curvature duct flow: geometry and (non-uniform) grid
Fig. 8.3
Stagnation-flow: geometry and grid
Fig. 8.4
Comparison of numerical diffusion for flow-oriented and Cartesian grid
Fig. 8.5
Geometry of Jeffery-Hamel flow
Fig. 8.6
Grid arrangement for Jeffery-Hamel flow calculations
Fig. 8.7
Geometry and boundary conditions for curved diffuser flow
Fig. 8.8
Typical grid (20 x 20 CV) used for curved diffuser flow calculations
Fig. 8.9
Convergence test for curved diffuser flow
Fig. 8.10
Streamlines and pressure field for Case 1
Fig. 8.11
Streamlines for Case 3
Fig. 8.12
Wall vorticity for curved diffuser flow
Fig. 8.13
Fully-developed channel flow predictions
Fig. 8.14
Fully developed pipe flow predictions compared with experiment
Fig. 9.1
Geometry of backward-facing step flows
Fig. 9.2
Modified wall functions
Fig. 9.3
Grid refinement test (BFS1)
Fig. 9.4
Computational grids for backward-facing step flow calculations
xi
Fig. 9.5
Wall pressure distribution (BFS1)
Fig. 9.6
Profiles of streamwise velocity (BFS1)
Fig. 9.7
Streamwise shear stress profiles (BFS1)
Fig. 9.8
Axial variation of maximum streamwise shear stress (BFS1)
Fig. 9.9
Variation of reattachment length with wall angle for different turbulence models and wall functions (BFS2)
Fig. 9.10
Inclined-wall backward-facing step flow predictions
Fig. 9.11
Pressure distribution along step-side wall (BFS2)
Fig. 9.12
Pressure distribution along step-side wall for different wall angles (BFS2)
a
6° (BFS2)
Fig. 9.13
Streamwise velocity profiles for
Fig. 9.14
Streamwise shear stress profiles for 8 = 6° (BFS2)
Fig. 9.15
Variation at reattachment length with turning angle 8 (BFS3)
Fig. 9.16
Reattachment length as a function at expansion ratio
Fig. 9.17
Turned-passage backward-facing step flow predictions
Fig. 9.18
Pressure distribution along step-side wall (BFS3)
Fig. 9.19
Pressure distribution along opposite-side wall (BFS3)
Fig. 9.20
Comparison of calculated and measured pressure distribution along step- and opposite-side walls for different turning angles 8 (BFS3)
a
Fig. 9.21
Streamwise velocity profiles for
10° (BFS3)
Fig. 9.22
Flow geometry of curved mixing layer of Castro and Bradshaw (1976)
Fig. 9.23
Grid arrangements and calculated velocity field for curved shear layer
Fig. 9.24
Grid refinement test for curved shear layer
Fig. 9.25
Curved shear layer flow predictions
Fig. 9.26
Velocity profiles for curved shear layer
Fig. 9.27
Shear stress profiles for curved shear layer
xii
Fig. 9.28
Turbulence intensity for curved shear layer
Fig. 9.29
Streamwise variation of the maximum velocity for curved shear layer
Fig. 9.30
Streamwise variation of maximum turbulence intensity for curved shear layer
Fig. 9.31
Streamwise variation of maximum shear stress for curved shear layer
Fig. 9.32
Generation and dissipation rate of the turbulent kinetic 3 energy on the centre-line (non-dimensionalised by pUref/s 3 /s respectively) for curved shear layer re f 3/2 Variation of the length scale Z = k /c along the centerand U
Fig. 9.33
line for curved shear layer Fig. 9.34
Entrainment for the curved shear layer flow
Fig. 9.35
A cross sectional view of a staggered tube bank
Fig. 9.36
Solution Domain and a typical grid arrangement for the flow across tube bank
Fig. 9.37
Grid dependence test for the flow across tube bank
Fig. 9.38
Predicted flow fields for the flow across tube bank
Fig. 9.39
Predictions of the fluid temperature variation around a tube in a tube bank using two different interpolation practices
Fig. 9.40
Velocity variation around a tube in a bank of tubes in cross flow
Fig. 9.41
Local turbulence intensity variation around a tube in a bank of tubes in cross flow
Fig. 9.42
Fluid temperature variation around a tube in a bank of tubes in cross flow
Fig. 9.43
Geometry and results of calculation for an inlet port/valve assembly (maximim valve lift)
Fig. 9.44
Geometry and results of calculation for an inlet port/valve assembly (small valve lift)
Diffusion flux vector components
Fig. A1.1 Fig. A3.1
Model problem used to assess approximations of diffusion terms
CHAPTER 1 INTRODUCTION
1.1 Background The rapid development of the digital computers in the last 30 years has enabled solution of fluid flow problems that none could have seriously considered before this period. As a consequence computational fluid dynamics has emerged as a new separate discipline. It started with crude calculations of simple flow fields of little practical interest. In that early stage integral and finite-difference methods dominated the scene, the latter being much more general. Indeed, numerous solutions have been obtained for a variety of simple compressible and incompressible flow problems using finite-difference methods. When, however, attention was turned to practical flow situations, many problems emerged, two main ones being the increasing degree of physical complexity of the flow, notably the phenomenon of turbulence, and the geometrical complexity of the flow domains in which most practical flows occur. Numerous 'turbulence models' have been proposed and used with varying degrees of success to overcome the difficulties imposed by the small scale motions of high Reynolds number flows (see Reynolds and Cebeci (1978), for example, for a brief review). These are not, however, the focus of this study. The aspect of practical application of numerical methods in fluid dynamics of interest here is the treatment of irregularly shaped flow domains. In this connection attempts to fit curved boundaries into simple rectilinear grids have proved at the very least to be cumbersome and aesthetically unpleasing, if not costly and/or unsuccessful. The reasons for this are numerous:
(1 )
By placing 'irregular nodes' at the boundary (see fig. 1.1), in order to enable implementation of the boundary conditions, the distances between near boundary nodes (Sy i may become very small. This affects both the accuracy and cost of the calculations (especially for explicit methods) since the formal accuracy deteriorates (see Blottner and Roache, 1971) and the stability requirement might be determined by this locally small spacing.
(ii)
The introduction of fine grid in regions of high gradients (near boundaries, for example) usually results in the grid being fine in regions where high resolution is not required; thus additional computer storage and time are required.
(iii) Substantial programming is usually necessary in order to incorporate the boundary conditions.
These are the reasons why the view has often been held that finiteelement methods are superior to finite-difference and finite-volume techniques in handling non-trivial geometries. The finite-element methods, which have been successfully applied in solid mechanics and heat conduction, have an inherent advantage in the relative ease of handling complicated solution domains by discretizing them into elements of (nearly) arbitrary shape. However, these methods usually require more complex matrix operations to solve the resulting discretized equations than do the finitedifference and finite-volume approaches and there are still some problems to be overcome, especially in the case of turbulent flows. An obvious way of overcoming the forementioned deficiencies of the finite-volume/finite-difference methods is the use of boundary-fitted coordinates. Here there are two main options:
(i)
the use of curvilinear orthogonal coordinates (Hung and Brown, 1978; Antonopoulos et al, 1978; Pope, 1978).
(ii)
the use of general non-orthogonal coordinates (Steger, 1978; Demird2id et al, 1980).
There is also a class of methods in which the computing mesh is defined independently of the coordinates in which the governing equations are written. Typically quadrilateral or triangular meshes are employed as in, for example, the methods of Hirt et al (1974) and Winslow (1966). The main argument in favour of curvilinear orthogonal coordinates is that the transport equations are little more complicated than their Cartesian counterparts and moreover their discretized versions retain nearly the same general form as for the simple coordinate systems. By contrast, the equations in general non-orthogonal coordinates and their discretized counterparts are considerably more complicated. Further, these equations usually contain cross-derivatives in pressure and other variables which give rise to major computational problems: these may, however, be surmounted, as is demonstrated in the present study. Although the curvilinear orthogonal coordinates approach is the more straightforward of the two, it too possesses , spveral drawbacks. Firstly, the coordinate mesh must be generated by some means, such as conformal mapping (useful only in a very limited number of cases) or by (usually numerical) solution of differential equations expressing the transformation relationships between the coordinates of the computing mesh in the physical and transformed planes. This process can itself be , costly notably in the case of problems with moving boundaries and in three-dimensional applications.
Secondly, the orthogonal coordinate lines so generated often distribute themselves in an inconvenient and inefficient manner; for example, they may bunch together in a region of the flow where high resolution is not required or be too widely separated where resolution is needed. This leads to loss of accuracy and/or economy. Unfortunately, only limited control can be exercised on the distribution of orthogonal coordinate lines for a given shape of domain. Thirdly, for the regions with non-orthogonally intersecting boundaries orthogonal coordinates strictly cannot be used although in practice it is usually possible to develop special treatments for the non-orthogonal regions. In the case of general non-orthogonal coordinates, however, the coordinate mesh is considerably easier to generate (often simple algebraic specifications suffice) and the mesh distribution can be controlled and optimized (points can be essentially distributed at will). Furthermore, the possibility exists of aligning the mesh with the flow streamlines thereby alleviating the effects of 'false diffusion' which plagues most if not all numerical methods. Thus, judicious arrangement of the coordinate lines can lead to great improvement in the accuracy of the solution for a given number of points, so that the additional computations caused by the grid non-orthogonality may be more than compensated for. Consequently, although in specific cases an orthogonal mesh can be equally effective, general non-orthogonal coordinates are to be preferred.
1.2
Present Contributions The work presented here is concerned with development and application
of a numerical method for solving the general-coordinate versions of the partial differential equations of fluid flow governing the balance of mass, momentum and arbitrary scalar quantities such as enthalpy or a
chemical species concentration. Specifically this work makes contributions in the following areas:
(1)
Development of the tensor calculus in terms of physical tensor components, which enables one to work with physical quantities and still retain the simplicity of mathematical operations possessed by contravariant or covariant non-physical components.
(2)
Review of the existing formulations of the transport equations in general coordinates and proposal of some new alternatives, with assessment of their suitability for solution by a numerical procedure.
(3)
Derivation of a semi-strong conservation form of transport equations for laminar and turbulent flow in terms of contravariant physical vector and tensor components in general non-steady coordinates that can accommodate situations requiring moving non-Eulerian coordinate frames, such as flows with moving boundaries.
(4)
Discretization of the above equations (for the case of a general two-dimensional Eulerian coordinate frame) in a form that allows alternative spatial and temporal differencing schemes to be examined. Special attention has been paid to the treatment of the crossderivative diffusion and pressure gradient terms arising from the coordinate system non-orthogonality. A procedure is also developed for the calculation of the grid and associated geometrical parameters required by the method.
(5)
Assessment of the accuracy and stability of the method by comparison with existing exact or numerical solutions or experimental data for
a range of flows.
(6)
Further assessment by comparison with data for various complex turbulent flow problems, comprising ducts with stepped and sloping walls, impinging jets, and tube bundles.
1.3
Contents of the Thesis This study employs the tools of general tensor calculus which are,
it is supposed, unfamiliar to most readers. Therefore, in Chapter 2 concepts and definitions are reviewed necessary for understanding of the subsequent material of the thesis. The reader familiar with the above subject may skip to Chapter 3, where results of novel developments from tensor calculus are presented concerning the physical tensor components. Several types of physical vector and tensor components are defined and a mathematical apparatus is developed in terms of these components which is completely analogous to the existing one which deals with contravariant and covariant non-physical components. The main advantageous feature of this development is that it allows one to work with components that, unlike non-physical ones, always bear the physical significance of vectors and tensors themselves, while retaining at the same time the full compactness of expression and versatility of mathematical operations of tensor calculus. Within the framework of general coordinates the equations of fluid flow can be expressed in many different ways arising from the various options for expressing both the vector and tensor components and of manipulating the resulting differential equations. In Chapter 4 these options are reviewed and their merits are assessed with respect to their potential impact on the accuracy, stability and economy of numerical
7
solution procedures. Methods developed by previous authors are also reviewed. In the light of these considerations the final choice is made of the vector and tensor components and the form of transport equations to be used in the present study. In Chapter 5 the selected versions of the differential equations which are in 'semi-strong' conservation form and employ contravariant physical vector and tensor components are derived for both laminar and turbulent flow in an arbitrary moving coordinate frame. It has also been shown that an additional equation expressing the 'space conservation law' has to be solved simultaneously with mass, momentum and scalar transport equations if non-steady coordinate systems are to be employed. For the closure of time averaged equations a two-equation k-c turbulence model is selected for use in this study and the appropriate differential equations are presented in terms of general non-steady coordinates and physical components. The imposition of boundary conditions is also discussed with the focus on the particularities arising from the nonorthogonality of the computing mesh. In Chapter 6 the governing differential equations for the case of two-dimensional plane and axi-symmetrical flow are expressed in expanded form and discretized by employing the finite-volume method. The discretized equations are presented in a general form that allows a range of differencing schemes to be incorporated. Several spatial and temporlal differencing practices are discussed. The treatment of the terms that arise from the use of general non-orthogonal coordinates, and pose problems to stability and accuracy, is especially emphasized. An algorithm is then described for solving the discretized equations, which uses a semi-implicit momentum-pressure coupling method for the pressure calculation and an alternating direction line iteration • procedure for the solution of the discretized and linearized equations.
The semi-strong form of the equations employed requires a smooth computational grid since the second derivatives of the computational coordinates with respect to Cartesian coordinates are necessary for evaluation of 'curvature terms'. A procedure that generates such a grid is developed in Chapter 7. Also, a method is presented for the calculation of all geometrical parameters required by the solution procedure. Chapter 8 is concerned with testing the accuracy and stability of the overall solution method. A range of (relatively) simple test cases is selected for independent testing of different aspects of the method such as (i) the influence of the cross-derivative pressure term on the stability of the method; (ii) effects of the accuracy of calculation of the geometrical parameters of the grid on the overall accuracy of the solution; (iii) the consequences of the possibly negative coefficients arising from the crossderivative diffusion terms, etc. The results of the calculations are compared with exact analytical solutions, previous numerical solutions, or experimental data, where available.. In Chapter 9 the full capabilities of the method are demonstrated by applying it to the calculation of physically and geometrically complex turbulent flows. These include several variations of the backward-facing step flow problem, a free shear layer with the strong streamline curvature, and flows across tube bundles and through the inlet port/valve assembly of a reciprocating internal combustion engine. For the first two cases reliable detailed measurements of both mean and turbulent quantities exist and are used for comparison with the computations. The last two cases are chosen to demonstrate the ability of the present method to calculate geometrically complex flows which are practically important. Unfortunately, little or no experimental data exists for these cases.
9
Additional details of the theoretical work are presented in appendices. Appendix 1 presents an alternative way of deriving the differential transport equations in general coordinates by a simple transformation of their Cartesian tensor counterparts. Appendix 2 explains the origin and nature of the cross-derivative diffusion terms arising from the use of non-orthogonal coordinates and Appendix 3 deals with an alternative discretization of the diffusion flux terms which reduces the possibility of negative coefficients while maintaining conservation of fluxes.
10
CHAPTER 2 PRELIMINARIES FROM GENERAL TENSOR CALCULUS
2.1 Introduction The purpose of this chapter is to review the concepts and definitions from general tensor analysis necessary for the understanding of the subsequent chapters on the fluid dynamics equations. The reader familiar with this subject may skip to the next chapter. Most of the material presented here is given without formal proofs, which can be found in the following references: Sokolnikoff (1964), Aris (1962), Sedov (1971) and Truesdell (1977). As a matter of terminology, the term 'tensor' will be used for the second order tensors only, although vectors and scalars can be considered as tensors of the order one and zero respectively. In such a way some generality will be lost, but on the other hand the different physical meaning of scalars, vectors and tensors will be emphasized. Although most of the analysis will be valid for rather general spaces, it will be generally restricted to a three-dimensional Euclidean space E3. The Einstein summation convention is assumed throughout the thesis, i.e. matching upper and lower indices are to be summed over the range of their values (from 1 to 3 in three-dimensional space), unless otherwise stated. Now the contents of this chapter will be outlined. In Section 2.2 a general non-orthogonal coordinate system is established by means of a non-singular coordinate transformation. Different ways of presenting vectors and tensors by their contravariant, covariant or mixed components are introduced in Section 2.3. In Section 2.4 the metric tensor is defined by which all essential metric properties of the Euclidean space are determined and in Section 2.5 some combinations of partial derivatives of metric tensor components, so-called Christoffel symbols, are introduced,
11
which have proved useful in the tensor calculus development. The derivatives of vectors and tensors in general coordinates expressed by so-called covariant derivatives are given in Section 2.6. Finally, some useful rules and expressions from tensor algebra and analysis are presented in Sections 2.7 to 2.10, that will be used in the later manipulations of the equations.
2.2 Coordinate Transformation Let R
3
(the solution domain considered) be a subspace of E
3
. Let y
denote the coordinates of a fixed Cartesian frame with base vectors
i
-01 -
i'
and let x i be a system of arbitrary curvilinear coordinates relative to the Cartesian coordinates such that:
y i = y i (x j ) = y i (x l , x 2 , x3)
(2.1)
If the Jacobian of the transformation (2.1), given by:
J = det ( F1-) . ax3
(2.2)
is non-zero everywhere on R 3 , the transformation (2.1) is called nonsingular*. If this is true, then there exists an inverse transformation:
xj .
*
x3(y1) .
x j (y l , y 2 , y3)
If J = 0 at some points in R 3 , then these have to be dealt with separately.
(2.3)
-12
and associated tangent vectors:
m
-0e. - -- 1 1 l ill ax
(2.4)
which are defined at each point in R 3 . Since the latter are linearly independent, they can be taken as the base vectors in R 3 . Also the vectors:
41
e -
axi 4m m1
(2.5)
aym
that are orthogonal to the coordinate surface x i = constant and reciprocal* 4.
to the vectors e. can form a basis in R3. 1
As a matter of terminology the basis -4. i is called the natural basis and basis ;1 is called the dual (natural) basis. Orthogonality and reciprocity of the natural and dual base vectors follow from their definitions (equations (2.4) and (2.5)) thus:
• v . aym i • axi in _ aym ax j ic n 3x j ay n ax / 11/ - DX / ay n -fi/ - Dx i
;i
j
(2.6)
= 1.i
where 64 is Kronecker delta (which is unity if i = j, and zero otherwise). 1
It is not difficult to show also that:
e 41 e -
4.
e
÷
4.
2
x e
; 1 • (;.
2
3
, e -
x ; )
3
;
x e 4•
1
4. 0. e x e -03 l l 2 , e - 4. 4. ÷ 4. x e ) e • (e x e )
4.
3
4-2
• (e
2
3
1
2
(2.7)
3
and:
*
Two base vectors l i and 1J are said to be reciprocal to each other if:
li • li = Si i
13
4-2
e
4-3
e
e
l
e
• (e
4-3
ej)
x
e
= 4
e
•
l
+1
e
(e 4
4-1
e
_
x
e"
3
-
•
4-2
x
142
e
(2
. 8)
x e"'")
where the symbols • and x denote the ordinary 'dot' and 'cross' products, respectively. These relationships enable one to calculate the dual base vectors if the natural basis is given and vice versa. A two-dimensional illustration of the curvilinear coordinates, natural and dual basis at a point in R 3 is shown in fig. 2.1. If two arbitrary coordinate systems x
i
and E
i
with respective natural
41 base vectors e. and e. and dual base vectors e and e are defined then the relations between these base vectors are, according to equations (2.4) and (2.5):
, a = 1
41
6 =
3E
aE
bla
t , a
ta
3y a @x m Dx m la 3E 3x 3E
3E
i
=
m
e
(2.9)
m'
i
3x 3E 7.tm
3xm
-
ay a
-
-
(2.10)
3x
Quantities which transform like the natural base vectors as in equation (2.9) are called covariant. Quantities which transform like the dual base vectors as in equation (2.10) are said to be contravariant. Consequently, the vectors e. and e are sometimes called covariant and contravariant base vectors, respectively. Conventionally superscripts are associated with contravariant quantities and subscripts are related to covariant quantities. However, since in a +i Cartesiancoordinatesysteme=e.
0
4.111
=
m
), there is no disti nction
between the covariant and contravariant quantities and the use of subscripts and superscripts has no significance. The bases e 1 or
el
are sufficient for the representation of vectors.
To be able to represent tensors, one has to introduce new base objects,
14
namely the tensor products of the base vectors, or 'dyadics':
;.
e. • el
j '
0
ej ; el
ej • ;.
'
0 ;.
(2.11)
By definition dyadics are assumed to be linearly independent and the tensor product has the linearity property. It is also associative, but it is not commutative. Transformation formulae for the dyadics of equations (2.11) are readily obtained, use the transformation formulae (2.9) and (2.10) and the property of linearity of the tensor product. For example:
4.
®
m n
9. E . =
ax ax
e
0 e n
m
(2.12)
or:
9.
4J_ axm a j E---F e m
411
e
(2.13)
3x
2.3 Contravariant, Covariant and Mixed Components Each vector of a vector field ; has unique components relative to any basis, and in particular relative to the natural and dual bases:
i
a = a e.
a. e
(2.14)
Since, according to '(2.9) and (2.10):
m a=a e
=a m
and:
m g ax
EM1
(2.15)
15
m.
-K11
a=ae m
=a
Dx 41 M1
(2.16)
the transformation formulae for a i and a are:
ai -
n
i a
m
(2.17)
Dx
ax al -
Dx
(2.18)
am
+1 e
+ where a = a e. =a1
Therefore the scalar fields a
i
and a. are called the contravariant and
covariant components of the vector
r
Similarly, if
r. A
ii ;
where the A
ij
0
is a tensor field, then in each point in R 3 : -
-4* j
and A
respectively.
A
ij
ij
el 0
= A i . -e. 0ej = A i j el 0 J 1
(2.19)
are called contravariant and covariant tensor
cornponents,respectively,andA i j and A1 3 are said to be mixed tensor mwonents.TheA l
and A 1 3 are sometimes also called 'once covariant'
and 'once contravariant' tensor components, respectively and, accordingly, the A 13 and A l. j are called 'twice contravariant' and 'twice covariant' tensor components, respectively.
2.4 The Metric The topology of the space R 3 is defined in the standard way by the metric:
ds
2
=dx
i
j dx ( 2.20)
16
where ds is the arc length differential and the g ij are given by the dot products of the natural base vectors:
3 m m 4. -4. r 3y ay g ij = e i • e i = L 1 J J m=1 3x 3x
(2.21)
The g ij are called 'covariant metric tensor components'. Similarly the dot products of the dual base vectors define 'contravariant metric tensor components':
ij _ 4-i - e g
;)(
4j = • e
m=1 3
i m
ax
j
cof(gij) (2.22)
3 m
9
where cof(g ij ) are the co-factors of the elements of g ij in the determinant:
(2.23)
g = det(g ij ) = J 2*
From the definition of the base vectors it is clear that the mixed metric tensor components reduce to the Kronecker delta (see equation (2.6)):
. . i --g = e • e = S
(2.24)
i
It also follows that gij, g ij and g j are symmetric in the indices i and j. When the metric of R3 is defined, the magnitudes of the base vectors as well as the angles between them can be calculated, as follows:
*
g
0 everywhere in R 3 if transformation (2.1) is non-singular.
17
4-1
II
= vg
cos 3( -e i
(2.25)
, (no summation)
,e.j )
ij
= cos aij -
6,7; iFiT The metric tensor components enable one to relate natural and dual base vectors:
= g im ;ffi ;
=
;111 ;
e1
= gTem
(2.26)
as well as contravariant and covariant vector and tensor components:
a
i
= g
im
am ; ai = gim a
m
A ij = g im A j = g mj A i = g im gin A m mn m
(2.27)
(2.28)
From another point of view, it can be seen that the metric tensor components can be used for raising and lowering indices.
2.5 Christoffel Symbols Since the base vectors are not constant in R 3 (except in the case of the Cartesian coordinate system) their derivatives, by definition, also form vectors, characterizing some properties of the curvilinear coordinate system (see Section 7.5). These derivatives are compactly represented by:
18
ae,
m (2.29)
{ 1J 1 em
and:
De _ —S ax
(2.30)
= tjmj
where the quantities
qd are functions of x i and are called 'Christoffel
symbols'. They can be calculated as follows from the metric tensor components:
ag km m 1 im ag ik-+,_ i 1 - 7 C jk 9 (-Dx 3xJ
ag-jk)
(2.31)
Dxm
In Euclidean space the Christoffel symbols are symmetric with respect to their subscripts:
(2.32)
}
One can also prove that for i
j, equation (2.31) reduces to the
following relation:
)
= I
(2.33)
Another relationship that relates derivatives of the metric tensor components to the Christoffel symbols is:
19
NAJTT _ 1 qi
il gim
ax i
(2.34)
v§Ti
2.6 Covariant Derivatives On the basis of (2.29) one can write the derivative of a (continuously differentiable in R 3 ) vector field 1 in the following form:
i 3-e . i • 31 3 i 4 - 3a -+ i -> —-3 - - ---10 e.) = --r e. + a --- - v. a e. 1 1 J J J 1 3xJ 3x 3xJ ax
(2.35)
where:
3a i m { i.} i + a Vj - a -E Mj J ax
(2.36)
is called the covariant derivative of the contravariant component of the vector
1.
Similarly, using relation (2.30):
= 3xj
41 3a. 41 1 ÷ 1. ae +1 . e + a. ----r = V. a.e .(a.e ) = j 1 J 1 J 1 ax 3xJ Dx
3
(2.37)
where:
3ai
j1
3xj
a
rm,
m
1..r lj
is the covariant derivative of the covariant vector component. For the tensor field rone can derive, for example:
(2.38)
20
ar
a ij
-
ax"
4-
ei 0 ej ) =
ax
ij aA
--IT ax
4- 4- e.0e
Bx"
ij
a4e.
ax
0e j
+A e. Bx
4.
_ii4
_ Vk A e
+A
(2.39)
i e j
where:
v
Aim {41
Amj
Aij -
k
(2.40)
axk
is called the covariant derivative of the contravariant component of the tensor
r
The covariant derivatives of the covariant and mixed tensor components are defined in a similar manner. A very important property of the metric tensor is that the covariant derivative of its components is equal to zero, i.e.:
V
V g ijo = k
g.. k ij
(2.41)
2.7 Scalar Multiplication and Addition In addition to the familiar definitions of scalar multiplication and addition of vectors the same operations will now be defined for tensors. If
r.
A ij -e".i
sA is a tensor P
-e). is a tensor and s is a scalar, then the product P
ij
-4-
e.
The sum of two tensors tensor C
ij 1:
e.
'
r.
whose components are P A ij
ij
sA
-e% and r= B ij
e. 0 e whose components are C
ij•
A
ij
+ B
ij
.
-e% is the ij
.
It is obvious that one can sum only two tensors of the same type and
21
that the same rules apply if tensors are represented by their covariant or mixed components.
2.8 Transposition, Symmetrization and Alternation The tensor
(AT ) ii
A
rT
ji
is called the 'transpose' of the tensor
T , (A ). =
Ti
9 kM )
r
Further, tensors
and
r
A'
A i j
if:
(2.42)
= Mj
defined such that:
r-r e = _
(2.43)
are called 'symmetric' and 'skew' respectively. Any tensor
r
has a unique
representation as a sum of a symmetric and a skew tensor:
=,› ==>
A = B + C
(2.44)
where:
=> 1 => —>T ) B = -2-( A +
and:
(2.45)
e = l(r
rT
)
are called symmetric and skew part of the tensor operations of generating tensors . and 'alternation', respectively.
r
and
r
r,
respectively. The
are called 'symmetrization'
22
2.9 Dot and Tensor Products The dot product of two vectors
I
and
11-
(
known as the 'inner' or
'scalar' product) is a scalar defined as:
;•
= g ij a i b j = a i b i (2.46)
r and a vector
The dot product of a tensor
1:,
is a vector defined
as follows:
ij k
=
b e. = A
A
ij
r and
The dot product of two tensors
r
(2.47)
b. e. J
r
is a tensor defined by:
=
g
jk
A ij B kL
;
;
(2.48)
= A ij B jk e 1 ® ;1(
The double-dot product of two tensors
r and
r
(also known as the
'inner' or 'scalar' product) is a scalar defined as:
=ij kt i gjk g iz A B =A
:
k
B
k
(2.49)
i
The tensor product of two vectors -a). and
5
(sometimes called the
'outer' product) is a tensor defined as:
®5
=
;. j
a i b j ;.
a.b
j e."1
0;j
(2.50)
The following tensor product of the base vectors:
4.
I
=
41
e.0e
4.j
=e0e.
(2.51)
23
is called the identity or unit tensor. It has the property of leaving other vectors and tensors unchanged when a dot product on either side is taken. The tensor (outer) product of a vector and a tensor or the tensor product of two tensors is a third or fourth order tensor, respectively, defined similar to equation (2.5G). Using rules (2.27) and (2.28) for raising and lowering indices one can get some alternative expressions for products (2.46) to (2.51), which employ some other vector and tensor components.
2.10 Gradient and Divergence The gradient of a scalar field f is given by the expression:
grad f =
0-4 = g jm 2F-ir;m ax3 axJ
(2.52)
The gradient of a vector field I is defined as the following tensor:
grad
= -ej 0
V. a. ej 0 -4'1 =V 1 DxJ J
j a i I 0-4'. m
(2.53)
The gradient of a tensor field r is defined in an analogous manner:
grad e = -44(
''K>
g
km V Aij em ® e i k
ej
(2.54)
By replacing the tensor product with the dot product one gets the following expressions for the divergence of a vector field I:
div
_ ij
= -4j . 3x j
g
7 i ai =V
3 ai
(2.55)
24
and a tensor field r:
44(
div
= e •
g'
=
Ali
(2.56)
ax"
Using the relation (2.33) one can obtain an expression for the divergence of a vector field in the so-called strong conservation form:
-
div
a'
v.
a
j s J .
aj)
V1 ax
.3 — v97 ar]
gjm am)
(2.57)
If one introduces the following notation:
A
j
(...)
E
a i
1
(..)]
(2.58)
Lx ax-
-).
then div a can be written in the simple form:
j J
4m _ A div a = Aa am) Le Ax ) (gj
(2.59)
In the same way the divergence of a tensor field r becomes:
div r= v.
A
ii
AAii m 1–f1. . 1 )
e i = (s --AxJ
+ A *
mj
46.
(2.60)
or:
div
(2.61)
25
2.11 Closure In this chapter a mathematical foundation and terminology necessary for the analysis of fluid flow in general curvilinear coordinates have been established. The material of this chapter will be used in the next chapter for developing of an analogous mathematical apparatus for vectors and tensors expressed in terms of so-called 'physical' components which, unlike ones presented here, always bear the physical significance of vectors and tensors themselves.
26
CHAPTER 3 TENSOR CALCULUS IN TERMS OF PHYSICAL COMPONENTS
3.1 Introduction Tensor calculus, whose main features have been presented in the previous chapter, is an extremely valuable and powerful tool in the analysis of physical phenomena in general coordinates. However, it has some pitfalls arising from the fact that neither contravariant nor covariant vector or tensor components have the same dimensions as vectors or tensors themselves. Moreover, the dimensions of different contravariant or covariant components of the same vector or tensor are often not the same. For example, in plane polar coordinates:
x l
r , x
2
(3.1)
=
with the metric tensor components:
g
n
=
1
2 ' g 22 = r
' g 12 = g 21 =
(3.2)
the contravariant velocity components:
“ 1
2 dx d e
2
dx l dr
V =
,
V
=
-
(3.3)
do not have the same dimensions, since:
dim vl _ length. time
'
dim v2 -
time
The corresponding covariant velocities suffer in a similar way,
(3.4)
27
since (according to (2.27)):
1 dr . j v l = g lj v = v = a. ,
2d0 j v 2 = g 2j v = r .a.T
(3.5)
and:
dim v
length 1 - time
Thus both v
2
,
and v
dim v
2
2 -
(3.6)
time me
do not have the physical significance of what
is usually understood by a 'velocity component'. Another drawback of the contravariant and covariant components is that they change in general from point to point according to the geometrical properties of the coordinate system employed, irrespective of the flow behaviour. For example, if a uniform velocity parallel flow is analysed in the plane polar coordinates (3.1), then radial and circumferential velocity components are:
v _ dr de r - -a-f = v1 = const , v0 . r TE
where v
1
and v
2
_ -
2 r • v = const
(3.7)
are defined by equations (3.3). One can see that the
contravariant velocity component:
v
2
2 constant = v (r) r
and moreover when r -* 0, v
(3.8)
2
-)-
00,
which can introduce serious errors in
numerical calculations. There have been several attempts to overcome these problems and to define vectors and tensors in terms of so-called 'physical components'. As a matter of fact, Ricci and Levi-Civita (1901) in their historic
28
memoir that set out the foundations of tensor calculus introduced the following four sets of quantities as vector representations:
VT77 v
i
v. ;
5-7 v. ;
vi
4
, (no summation)
(3.9)
,/g77 471-1-
which all have the same dimension (the dimension oft), but different geometrical interpretation (see fig. 3.1). According to this interpretation they called the first two, 1/qTT v i and 47 v i , components along the coordinate lines and along the normals to the coordinate surfaces, respectively and the last two, v i / i
and vigT //i vg ii , projections
on the coordinate lines and on the normals to the coordinate surfaces, respectively. It should be noted that all of these quantities reduce to a single set in the case of an orthogonal system. Unfortunately, as noted by Truesdell (1953), "in books intended for mathematicians these components are no longer mentioned, and what Ricci and Levi-Civita called covariant and contravariant systems are now called the covariant and contravariant components" and are almost exclusively used as a general tensor representations, primarily because they transform more simply under coordinate transformations. In contemporary mathematical physics the first set of quantities (3.9)
/67T
v i is referred to as the 'physical vector components'.
Several authors (Synge and Schild, 1949; Green and Zerna, 1950; 011endorf, 1950; Truesdell, 1953) have defined physical components of second order tensors. Their definitions differ except in the case of orthogonal coordinates when they all reduce to the McConnell's components (McConnell, 1931), that are considered entirely satisfactory for orthogonal coordinates. Neither of these definitions is found to be satisfactory for applications in fluid mechanics. Finally, Ericksen (1960) introduced physical vector and tensor components as a special case of his method of
29
anholonomic components. Although his approach is quite general and very flexible, its importance and potential have not been recognised, possibly because he gives little detail of its potential applications. In Section 3.2 a different approach is used to define physical vector and tensor components which enables the subsequent development of tensor calculus in terms of these components. In Sections 3.3, 3.4 and 3.5 the metric tensor, Christoffel symbols and covariant derivatives are defined in terms of physical components. Further in Sections 3.6 and 3.7, algebraic and differential relationships between physical 'vector and tensor components are defined, equivalent to the relationships between contravariant and covariant vector and tensor components presented in Sections 2.7 to 2.10 of the previous chapter. Finally, in Section 3.8 the dual physical components are defined.
3.2 Definition of Physical Components If one takes normalized natural base vectors:
(no summation)
(3.10)
vgii
rather than natural base vectors themselves as a basis in R 3 , then the vectors:
=
e
( no summation)
(3.11)
form the reciprocal basis. It is important to note that the vectors are non-dimensional unit vectors (colinear with the tangent vectors while their reciprocal vectors -41 are also non-dimensional but not unit vectors (colinear with the normal vectors el )• Vectors and tensors can now be expressed in terms of their components relative to the base
30
vectors
n)
(i)÷ a = a e d) (3.12)
A(J)+ =
4.
eci) ® ed)
where, according to equations (2.14), (2.19) and (3.10):
aci) =ai 11
(no summation) (3.13)
Ali ). ATT
Aij
(no summation)
Since eti) are non-dimensional unit vectors, quantities a
d)
all have the
same dimension as vector ; itself. Moreover they conform with the usual representation of vector components as directed line segments that add by the parallelogram rule to form vector ;. The quantities Pcoincide with Ricci and Levi-Civita's components along the coordinate lines (see fig. 3.1). As was mentioned above, they are usually called 'physical vector components' and are denoted by a(i), i.e. with an index that is neither a superscript nor a subscript (see Aris, 1962; Truesdell, 1953, for example). In this study this practice is not employed (for the reasons that will become clear very shortly) and the Pare called contravariant physical vector components. Analogously quantities gij) are called 'contravariant physical tensor components'. Since they too arerelatedtonon-dimensionalbasevectorsivtiley all have the same dimensions, those of the tensor r. If the reciprocal basis
is employed, one gets:
31
4d)
a = a(1)
(1)
(3.14)
rA..
,ch 4d)
e
0j)
e
where:
a.
(3.15)
(no summation) Ai. = j
167 i
Quantities act all have the same dimensions as vector
I,
but since
vectors i'Sh are not unit vectors, they cannot be regarded as vector components in the usual sense. In fact, they coincide with Ricci and Levi-Civita's projections on the coordinate lines and consequently they do not obey the parallelogram rule. Here they will be called covariant physical vector components. Following this convention the quantities
Ali)
will be called 'covariant physical tensor components'. The geometrical interpretation of contravariant and covariant physical vector components is shown in fig. 3.1.
3.3 Physical Metric Tensor Components The metric of R
3
can be defined in terms of the physical metric
tensor components in a manner analogous to the definition of metric tensor components (see equations (2.21), (2.22) and (2.24)) as follows:
-1- /1j) = `(1) • e(j) -
(no summation)
(3.16)
32
t gij) = -)(i) • ij e•e - _Ig ii gji g
(no summation)
) =
(j) =
(3.17)
(3.18)
grb
It is clear that these physical components are all symmetric. The arc length differential can also be defined by an expression which is analogous to the equation (2.20), thus:
ds 2 g = -(ij)
(3.19)
dx6)
Moreover, the physical tensor components can be used for raising and lowering indices of physical vector and tensor components in exactly the same way as in the case of covariant and contravariant components (of equations (2.26) to (2.28)). For example:
grim)
e
(3.20) hi j) =
m) xj) =
rijn) mn)
Here the A
ti)
are mixed physical tensor components defined by the
following relation:
CJ) -)th A = A. e clo e co
(3.21)
They coincide with Truesdell's physical tensor components (Truesdell, 1953).
33
3.4 Physical Christoffel Symbols Using equations (2.29), (2.33), (2.34) and (3.10) the derivative of 4theunitvectore.can be written in the form: (1)
ail)
-
3
1
D ei —71 /-=) rg.. 3x g.. JJ (3.22) 11 ( iinj ) 477-7 { i1j } gin 91i ' 9 11 gjj 91 1 9 jj
where the following (covariant) operator has been defined (according to equation (3.15)):
a
(...)
(j)
a
=
3x
!TT
JJ
(...)
(3.23)
3xg
Further, using the last expression in (2.26) equation (3.22) becomes:
÷
a 1)
g =
Dx
i
({ P
.1 ji k iv
4. , ,, m g in r ino) a i 77 t UM u l/
J
(3.24)
After introducing the notation:
m g in cn.}) —
j ({m}
(3.25)
gi gii i
an expression analogous to the derivative of the natural base vector (2.29) is obtained:
(j)
3x
• =
1 j (•1
(111)
(3.26)
34
Similarly one can get an expression analogous to the derivative of the dual base vector (2.30):
3e 3x(J)
(i mj
-
-411)
It is obvious that the symbol
(3.27)
(Z)
can be regarded as the physical
Christoffel symbol analogous to the Christoffel symbol {.1}. It should m be noted, however, that unlike {. .}, the physical Christoffel symbols ij are not in general symmetric* in the indices i and j:
(ml
(mi
(3.28)
Another important property of quantities
is that, when they
are written in expanded form, Christoffel symbols with all equal indices never appear.
3.5 Covariant Derivatives of Physical Components The definition of the physical Christoffel symbols enables one to make a complete analogy between expressions in terms of physical and contravariant and covariant components. The derivative of a vector field a can be now expressed as:
(i) { (i) 3a () a e. = — e. + ai (i) (j) n ax 3x 3X 3
4.
ch d) - V a e (j) C)) d)
The antisymmetric part of the physical Christoffel symbol m. g in n {/..} is called the 'object of anholonomity' g ii gjj g/ gii J (Schouten, 1951). gmm
(3.29)
35 where:
d) aa Vo ) a -
rb
+ dm(ii
(3.30)
Mj
f,j) aX
is the covariant derivative of the contravariant physical vector component. It is completely analogous to the covariant derivative of the contravariant vector component (2.36) and it is related to it as follows:
V.
)
d) gii a = — V. ai gJJ
(3.31)
The above relation can easily be obtained by comparing (2.35) and (3.29) and using (3.10) and (3.12). Following the same path, one can get an expression for the covariant derivative of the covariant vector component analogous to the expression (2.38):
aach
d)
v
ad)
= T xu )
Irm
(3.32) (illiji
where:
V. aft) = ) 0
)41 /g1
1
7j i
(3.33) Vd)
ar• )
as well as expressions for the covariant derivatives of the different second order tensor physical components, for example:
(j
ck)
V
where:
fijj)
-
3Adj)
a) )
+
Amj)
(
ink) +
si
Ad
ni)
mk
1
(3.34)
36
gij)
v(
gii
k)
Aij k
V
gkk
(no summation) mm
n
(3.35)
It is apparent that the expressions (3.34) and (2.40) have exactly the same form. The covariant derivatives of covariant and mixed physical tensor components can be obtained in a similar manner. From equation (3.35) it follows that the physical metric tensor components also have the property defined by equation (2.41):
- 7
7(k1
-
(3.36)
- 0 -
-
3.6 Algebraic Operations It is clear that the operations of scalar multiplication, summation, transposition, symmetrization, alternation, dot and tensor product defined in Sections 2.7, 2.8 and 2.9 retain the same form if physical rather than contravariant or covariant components are used. For example:
I • ;
=
=
(3.37)
n)
or:
•
ID'
0 0 )
n)
0
u)
= a .
n)
'ff) 0 4. - ed)
e
3.7 Gradient and Divergence The gradient and divergence operators, when expressed in terms of physical components, also retain forms analogous to the corresponding expressions in terms of contravariant or covariant components, defined in Section 2.10, thus:
(3.38)
37
(
grad f=
af = im)
CIO
grad
= +e(j)
= • a • -4d)
31
_
V 0) (1)
3x°)
—
grad
(3.39)
u) 1m)
9x
a)/J)
®
*
k) r =V-0a p
dm
=g
V
do
4d) e
dm) = g
di) 4- A-e
mril
a(i) -4- as.
4ee ee d)
+0
11) (M)
(J)
d)
(3.40)
(3.41)
and:
div .
div
. 31 v. ad) @xo d)
=
. are ( k) •70 r = ,e a
(3.42)
0)
(3.43)
It follows from (2.57) and using (3.13) that the strong conservation form of the divergence operator becomes:
di v
= -1— —a, 3x3
(3.44)
By defining a differential operator:
ig
A
a
\c_a
g
(3.45) 3x°)
which is analogous to the operator (2.58), the analogy with the expressions (2.59), (2.60) and (2.61) can be retained to give:
d)
di v a =
Aa
Ax(i)
(
dm) a
Axd) g
11111
(3.46)
38
di v
r = [6d - '.) Axd )
Ami, ( i ii mj
(3.47)
(i)
or:
dot
T->
m =
A -r
ri j) +
A Om) (A e.) --r(g
Ax(1)
A
-ehi))
m
Ax
(3.48)
/
m)
0)
(1)'
3.8 Dual Physical Components By defining contravariant and covariant physical components in Section 3.2, not all possibilities have been explored for expressing vectors and tensors in terms of physical quantities. If instead of normalized natural base vectors, as in Section 3.2, the normalized dual basis:
(no summation)
(3.49)
(no summation)
(3.50)
and its reciprocal basis:
ett
/TT 4g ei
are used, the following dual physical vector and tensor components can be defined:
' it
'
(3.51)
it
=
t
®
It can be noted that vectors
=
ett
it
are non-dimensional unit vectors,
(3.52)
39
while %are non-dimensional but not unit vectors, as was the case with -4ti)
vectors e. and e , respectively. One can also easily see that:
_ /71
in
ia.. - g
(3.53)
a1
and:
(3.54)
coincide with Ricci and Levi-Civita's components and projections along/on the normals to the coordinate surfaces (see fig. 3.1). It is clear that an analysis completely analogous to the one performed in Sections 3.3 to 3.7 in terms of physical components could now be conducted in terms of dual physical components. However, in order to avoid repetition, further details will not be presented here.
3.9 Closure All physical and dual physical components defined here have appropriate physical dimensions and all reduce to McConnell's physical components in the case of an orthogonal coordinate system. These are considered as necessary conditions for any proper definition of physical components (Ericksen, 1960). However, the fact that the base vectors e and i t are not unit, but variable intensity vectors, should be emphasized. One of the consequences is that the corresponding vector components l b and
113 do
not obey the parallelogram rule for vector components. Clearly the physical interpretation of vector, and especially tensor, components related to a basis of non-dimensional unit vectors is easier than in the case of base vectors whose length varies from point to point.
40
This may be a reason why one should prefer to use contravariant physical components
(0, Acii)
or covariant dual physical components (it ,
rather than covariant physical (ad) , ht ) or mixed physical (0 )
1)
or contravariant dual physical
At .) components, although one might '
[33
find reasons for the opposite decision. Indeed, Parameswaran (1982) has used covariant physical components for some fluid flow calculations and Issa (1976) has expressed the fluid flow equations in terms of contravariant dual physical components. Truesdell (1977, p. 248) recommends the use of physical components and suggests: "The easiest way to get expressions in terms of physical components is to derive them first in terms of contravariant or covariant components, which is a simple routine matter, and then convert the results". An application of this procedure, however, usually leads to cumbersome, inconvenient expressions in terms of physical components. According to the results in this section the derivation of the expressions directly in terms of physical components has also become "a simple routine matter", since it is identical to the derivation in terms of contravariant or covariant components. Thus, the present analysis developed for physical components enables one to work with physical quantities and still retain the simplicity of mathematical operations possessed by 'non-physical' contravariant and covariant components.
41
CHAPTER 4 ALTERNATIVE FORMULATIONS OF TRANSPORT EQUATIONS IN GENERAL COORDINATES
4.1 Introduction When a simple coordinate system is employed, (Cartesian, for example) the choice of the vector and tensor components and the form of the conservation equations is obvious and is usually not discussed. In the case of general coordinates, however, there are many options with respect to both the vector and tensor components and also the form of the equations. When the merits of these options are assessed it is important to recognize that they will have an important bearing on the viability, accuracy and cost of numerical solution procedures. In this chapter the more obvious options are discussed with respect to all above properties and the choice for the present study is made. In Section 4.2 the equations governing fluid flow are given in general coordinate free notation. In Section 4.3 different conservation-law forms of the transport equations are presented and possible ways of achieving strong conservation forms of momentum equations are discussed. The differences between equations, written in terms of contravariant or covariant-type components, are discussed in Section 4.4. Section 4.5 discusses the different possibilities of resolving the vector momentum equation into the three scalar equations for the components. In Section 4.6 a summary of the different forms of the previously obtained transport equations is presented. The previous work on numerical solution of the fluid flow equations in general coordinates is analysed in Section 4.7 with emphasis on subsonic flow calculations.
42
Finally, in Section 4.8 the results of the previous sections are summarized and a form of the equations is selected for subsequent use in this study.
4.2. Governing Equations The conservation of mass, momentum and scalars such as specific energy, concentration, etc. in a fluid flow can be expressed in the following coordinate-free form:
3P + div(p) = Sm at
div(pnie
pni) +
a
(p0
(4.1)
=
div(p4 -) = S
+
where density
-
p,
velocity -. v> and scalar (I) are the basic dependent variables.
To close the system of equations (4.1) one has to supply constitutive relations which define the stress tensor
r
and the flux vector
in terms
of the basic dependent variables. For a Newtonian fluid the stress tensor is given by the Navier-Stokes law:
= -(p +
4
div n;)
(4.2)
+
where p is the pressure, 3i is the viscosity,
_
r
is the unit tensor and
D> is the deformation (or rate of strain) tensor, which is defined to be the symmetric part of the velocity gradient. The flux vector 4. is usually given by a Fourier type law:
43
(4.3)
-4 . r grad (/) (I)
where
r
is the 'conductivity' or 'diffusivity' according to the variable
(0
in question. The terms on the right hand side of the equations (4.1) represent sources or sinks of the corresponding basic variable. It is well known (see Truesdell, 1977, for example) that the equations (4.1) can be written in a general form:
a
-Ft- (
pip) +
where V and
E
div(pip ® -nsi — fl) = E
(4.4)
are tensor fields of the same order and 0 is a tensor field
of one order greater than that of V and
E.
The definitions of these
quantities for the case of the mass, momentum and scalar equations are given in Table 4.1.
TABLE 4.1 DEFINITIONS OF QUANTITIES IN EQUATION (4.4)
Transport Equation
IV
Q
Mass
1
0
Momentum
-,;
-1.-->
Scalar
(I)
7:1
E S
m v
S (0
However, it has been found in the present study that the following alternative expression of the general field equation (4.4) is very useful as a starting point for extraction of the different possible forms of the conservation equations in terms of particular coordinates and variables:
44
A
(0) +
-0-
4-j
(pipv • e -
--01,
(4.5)
• e-) = E
AxJ
The above has been derived from (4.4) by making use of the definitions of the divergences of vectors (2.59) and tensors (2.61).
4.3 Weak and Strong Conservation Forms of Governing Equations All the discussion in this section will be conducted in terms of the contravariant-component equations, but the conclusions will also apply when the conservation equations are written in terms of covariant, physical or dual physical components. When: (i) the contravariant velocity vector, stress tensor and flux vector components are used, (ii) the divergence operator is calculated from (2.55) and (2.56) and (iii) the definition of the covariant derivatives (2.36) and (2.40) are taken into account, the conservation equations (4.1) can be written in the following form:.
2.2/ v3 ) o at 727
(Pvi )
+
-7; 3x
fj!),) pv m sm
i i m . m j (pv v- - T' 3 ) + { I .} (pv v - T J ) + mj (4.6) fj 1 (pvivm _Tim) im
m (povi - qj) + f1 (p(pvm - q ) = S
3 7 (0) + @xJ
Si
(1)
If instead relation (2.33) is applied to the above system, or
4-5
equivalently if the alternative expressions (2.59) and (2.61) for the divergence are used, the conservation equations (4.1) become:
ap + A (pv i ) = S
at Axi
a ,
e i ) +
Ax
A -Fta - (0) +-,-
i ij 4-1 i j [(pv v - T ) e.1 = S e.
A
v
J
( 100
-
q 3 )
(4.7)
S4) 0
By comparing the sets of equations (4.6) and (4.7) one can see that each equation in (4.6) has extra terms related to the Christoffel symbols and arising from the representation of the divergence operator. But much more significant is the fact that the application of the divergence operator has produced undifferentiated terms in the case of equations (4.6), while in equations (4.7) all terms arising from the divergence operator appear within a differential operator
A/Ax i .
The significant factor is
that if the latter set is integrated with respect to x j there results integrals of exact differentials which are known to depend on the boundary values only. In the terms of discretized equations this implies that, provided appropriate care-fs taken in the discretizing,the flux terms cancel in pairs at the cell faces in the interior of the solution domain so that only boundary fluxes remain. The result is that the discretization is conservative (Roache, 1976). In the case of equations (4.6), however, there is no obvious way of discretizing the undifferentiated terms which would assure their cancellation from cell to cell and consequently produce overall conservation of the c6rresponding quantity. For these reasons equations (4.6) are said to be in a weak conservation form, while set (4.7) is said to have a strong conservation form.
46
While the above definitions are sufficient in the case of mass and scalar conservation equations, the fact that the momentum equation has to be resolved into specific directions before it can be solved makes the issue of the type of conservation form of the momentum equations more complicated. Since in this case a vector, (the momentum vector in the absence of all forces), and not its components, is the quantity that is physically conserved, if the directions of resolution are not spatially constant, the resolved equations must have the terms which redistribute this vector from one direction to the other, as illustrated in fig. 4.1. For the given coordinate system x i it is natural to resolve the momentum equation into the coordinate directions
In this case the
momentum equation of the set (4.7) assumes the following form:
9t
(pv i ) +
(pvivj - T ij ) + { i .1 (pv mvj - Tmj ) AxJ
MJ
(4.8)
Si
From the foregoing discussion it is clear that the centrifugal and mj m j , Coriolis forces {.} pv v and the stress-tensor counterparts1 T MJ
MJ
which redistribute the conserved vector quantity between coordinate directions and do not appear under the A/Axi operator, generally do not cancel from cell to cell. 'Appropriate discretization' will, however, ensure that this transfer of the vector quantity from one direction to the other is conservative and will lead to the conservation of this vector quantity. Since the 'appropriate discretization' in this case is not so obvious as in the case of equations without undifferentiated terms*, equations (4.7) with the momentum equation resolved in the form (4.8) are
Since in this case it is necessary to consider all momentum equations simultaneously.
47
said to be in a semi-strong conservation form. As mentioned above, the strong conservation form of the momentum equation can be obtained if resolved into spatially invariant directions because in this case the conservation of a vector implies also conservation of its components. This can be done in essentially two different ways: either by expressing the base vectors constant base vectors
1.
with respect to the arbitrary
= const by the following linear combination:
(4..9)
=
or by expressing the vector -v). and tensor
r
in terms of their components
with respect to a constant bases:
-0-
i*
V=UT
.
>
T
,
1
= T
i l
1.
(4.10)
From the first way the momentum equation (4.7) becomes:
i
3 (Pv
m my) -A [(pv j v m - T jm ) y i
71
AX
m]
1-
Sm
v Ym
(4.11)
while in the second approach one can get the following strong conservation form:
(pu i ) +
[(puium - Tlim Ax°
)01} mi
= Si
1.
(4.12)
where:
= 1m • v
(4:13)
.
48
Now some relative merits of the equations (4.8), (4.11) and (4.12) will be discussed. The set represented by (4.11) has a major drawback from the computational point of view in that the individual equations do not contain a dominant velocity component. This means that if the equations are solved numerically, instead of solving three weakly coupled linearized first order systems by iterating for each velocity component in turn, as would be possible for sets (4.8) and (4.12), one would have to solve set (4.11) simultaneously, which would require much more computer storage and time. A further advantageous feature of equations (4.12) is the fact that
ij ij the expressions for stress tensor components T i are simpler than for T because they are referred to a fixed basis and, therefore, do not contain any 'curvature terms'. On the other hand, while the mass and scalar equations that correspond to equations (4.8) and (4.11) remain the same as in set (4.7), those that correspond to equations (4.12) are more complicated (i.e. there is additional summation on m):
L. °+
3t
A, ( pu m d) = s 6,x j m m
(4.14) a
j
m. A m (PO + --- [(O u - q 1 ) 13, m ] = S cp Ax J
Finally, it should be noted that the strong-conservation forms (4.11) and (4.12) generally have more terms than the semi-strong formulation (4.8) due to summation associated with the introduction of the quantities yi and 3. This increase is partially compensated, in the case of equations (4.12), by the simplicity of the stress tensor Tlj.
In previous analyses of the general-coordinate forms of the equations several approaches have been devised for obtaining strong conservation
49
forms of the momentum equations (e.g. the method of integrating factors, variational principles, etc.), as discussed by Eiseman and Stone (1980), but they all reduce to the two presented above. In a further alternative approach Vinokur (1974) considered the discrete analogue of equations (4.7) as used in finite-volume numerical -* solutions.Hetakesasafixedbasisthenaturalbasevectorse . (P) at
1
the 'central' point P of each discrete subdomain or 'cell' in the overall domain and expands the base vectors -4.., at the other points in each cell (points on the cell faces, for example) with respect to the locallydefined -4. 1 (P) (fig. 4.2). In this case the momentum equations have the form (4.11) with the coefficient yi defined as:
,4= -4-. • V(P) 1
1
1
-
°Jr UX Dx 1 3ym
(PI\ % /
(4.15)
Although formally of the same type as the equations (4.11) Vinokur's formulation has some properties that make it superior. Firstly, the coefficients yi in Vinokur's equations reduce to the Kronecker delta at the point P:
(4.16)
Y47 ( P ) = 641
and, accordingly, the time derivative term (which is calculated at the point P) contains only one velocity component v i . Secondly, Vinokur's coefficients yi have the following properties (see fig. 4.2):
i _ - 1 Ii
(no summation) (4.17)
50
Both of the above properties promote the velocity component v i as a dominant one in the equations (4.11). Thus, the main drawback of these equations identified earlier in relation to equation (4.11) has been avoided. One should, however, note that the coefficients yi, and hence the fluxes, are now discontinuous at the cell faces. It seems no longer possible to unconditionally ensure overall momentum balance (so in this sense the apparent strong conservation form is illusory). This also implies more storage and/or more computations. Extending Vinokur's idea of a locally constant basis there is a further possibility of achieving a strong conservation form of the momentum equations, namely, expression of vector and tensor components in terms of the locally constant base vectors -e i ( P). The resulting momentum equations have the same form as (4.12) with coefficients Bi now given by:
j+ = -e'^ i (P) •
V-
3Y11 3x1
(p) VX j m 3y
(4.18)
This formulation retains the simplicity of the stress tensor possessed by all the approaches that express vectors and tensors in terms of fixed bases, as discussed earlier, which makes it superior to the Vinokur formulation. Moreover, it is preferable to the approach that uses spatiallyinvariant-direction components, e.g. Cartesian ones, because here velocity components change their directions consistently with the change of coordinate system directions. The importance of this will be illustrated in Section 4.7. The remark about the discontinuity of coefficients yi and its consequences apply also to coefficients Bi. From the foregoing considerations it follows that the strong conservation form of the equations can be obtained only if an appeal to a
51
spatially constant basis (Cartesian, for example) is made. The use of locally fixed bases gives an apparently strong conservation which, however, suffers from the drawbacks discussed above.
4.4 Relative Merits of Contravariant and Covariant Form of Governing Equations All the discussion in the previous section has been centred around the contravariant-component equations. There are, however, two essentially different forms of the conservation equations emerging from two different types of vector and tensor components employed, namely, contravariant and covariant (see Chapters 2 and 3). In this section these two forms will be presented and examined from the point of view of a solution method. If the contravariant components are employed, the stress tensor (4.2) and the flux vector (4.3) can be written in the following way:
2 =
{
(p
7
Av
m I
11
g
Ax
J
20
ij
e.
(2:0
e. J
(4.19)
The governing equations then assume their (semi-strong) contravariant form:
+ at
(03) = S 6,xj
(4.20)
m
....
Continued
52
a (pvi) Tf
(pvivj _ 20 ii ) Ax
+ {
i
, kp
g i j
2 1.1
Av
m
Axm
3x
.1 (Pv mvj - 2.1Dmi ) = Si
(4.20)
MJ
a ,
o ci).) + --T ( P
r
v j
(i)
Ax-
s
g jm j± )
3x
m
If the covariant components are used the stress tensor and the flux vector become:
(171 (gmnv n )]
2 p . A 7 z--
= {-[p
+ 21.1D ij l e ® e
(4.21)
and the (semi-strong) governing equations are:
' P + —L. (gl y ) m at AxJ
-3-aj (pv i ) +
Ax
sm
[ g inkv iVM -
m
rp +
)1 +
3x
2
A
1-1 - -7.1 Ax
,
mn v
(4.22) -
a
[givnvm - 20nm )] = Svi
A r (PCP) -----r Lig 3111 (OV m Axv
r IL)] = sq) (/) )(111
Since one of the main difficulties in devising a computational algorithm is the treatment of the (non-linear) convective terms and the
53
pressure gradient terms, it is useful to examine the properties of the above equations sets in respect of these terms. One can see that the contravariant form has three convection terms in each equation (summation on j only) and three pressure gradients in the momentum equations, while the covariant form contains nine convection terms in each equation (summation on j and m), but only one pressure gradient term. The presence of more than one pressure gradient term in the momentum equation can affect stability of the solution algorithm, as it will be discussed later (see Section 6.9), while the presence of the additional convective terms in the covariant form implies more computations will be involved in solving them. From the results of Chapter 3 it follows that the form of equations (4.20) and (4.22) will remain the same if physical or dual physical components are used. In addition if the strong rather than semi-strong conservation form of equations (4.20) and (4.22) is used, the number of convective and pressure gradient terms increases but the relative numbers remain the same. Thus, whatever the form of equations one chooses, it is never possible to simultaneously achieve all desirable features, i.e. three convection terms and, in the momentum equations, one pressure gradient term. Some compromise is always necessary.
4.5 Choice of Direction of Resolution of Momentum Equation In the previous sections it was assumed that, with the exception of the strong conservation form (4.11), the momentum equation is resolved in the directions normal to the velocity components, i.e. equations (4.20) are resolved in the product with
V
directions (equivalent to stating that the dot
is taken) and equations (4.22) are resolved in the
directions, for example. The other possibilities are to resolve in the
54
directions aligned with the velocity components, or in any other direction. For example, resolving the momentum equation (4.7) in the directions gives equations (4.8) and resolving the same equation in the
e
- i direction yields:
j
ij
A a i i ( qv ) + -- (pv v - T ) + - [-g ik at AxJ
r
(4.23) i m im i + { • }(pv v - T )] = g ik Sy
This equation is actually a linear combination of equations (4.8) and, therefore, has three times as many terms as equation (4.8) and has no dominant velocity component. Therefore, resolution in the directions normal to the velocity components is generally preferable.
4.6 Summary of Different Forms of Governing Equations It can be seen from the foregoing sections of this chapter that there are many possible ways of expressing the transport equations of fluid flow. Starting from the equations in a coordinate free form (4.1) the following options are possible:
1.
One can choose vector and tensor representation to be of either contravariant or covariant types, leading to the contravariant or covariant form of the equations, respectively. Each of these forms can be expressed in terms of non-physical, physical or dual physical components.
2.
For each of the above one can write the conservation equations in the weak, semi-strong or strong conservation form.
55
3.
As far as the strong conservation form is concerned, there are two main possibilities: expansion of the base vectors or vectors and tensors with respect to a basis of constant vectors, which can be a Cartesian-like basis or a locally constant basis.
4.
Finally, there is the choice of the direction of resolution of the momentum equation: resolution in the directions normal to or colinear with the velocity components, for example.
The various permutations of all these possibilities sum up to 72 different forms of the conservation equations, as illustrated in the diagram in fig. 4.3.
4.7 Previous Work on Numerical Solutions of Transport Equations of Fluid -Flow in General Coordinates Although all options mentioned in the previous section are formally equivalent to each other, they possess substantial differences from a numerical analysis point of view. The most extensively used of all options is the strong conservation form with vectors and tensors resolved into their Cartesian components. The vast majority of the calculation procedures which solve these equations have been devised for high Mach number (transonic, supersonic, hypersonic) flow problems using differencing schemes of MacCormac (1969, 1975, 1976) or Beam and Warming (1976, 1977) type, derived by either the finite-volume approach (Deiwart, 1975) or by the finite-difference method (Tannehill, 1976; Shang and Hankey, 1977; Steger, 1978). Since the present study is not concerned with high Mach number flows and since an extensive review of these methods is presented by Hollanders and Viviand (1980), they will not be further discussed here.
56
The rather less numerous methods devised for the solution of subsonic (constant and/or variable density) flows in arbitrary domains will now be reviewed. A finite-volume method developed at the Los Alamos Scientific Laboratory was first presented by Hirt et al (1974) for two-dimensional flow and later extended for three-dimensional calculations by Pracht (1975). The method named 'ICED-ALE' is an extension of the Arbitrary-LagrangianEulerian (ALE) procedure of Hirt (1970) for irregular geometries and/or moving boundaries and the Implicit Continuous-Fluid Eulerian (ICE) algorithm of Harlow and Amsden (1971) for all flow speeds. The method uses Cartesian velocity components which implies the strong conservation form of the equations. The ICED-ALE grid consists of trapezoidal cells with velocity components located at the cell vertices and the scalar variables (p, p, k, ....) assigned to the cell 'centres', as indicated in fig. 4.4. It should be noted that the only geometrical information needed by this method is the Cartesian coordinates of the cell vertices, since the governing equations contain no curvature terms. The solution procedure consists of three phases: (i) an explicit Lagrangian calculation, in which the mesh vertices are caused to move with fluid; (ii) iteration to adjust the pressures and (iii) a rezone or convective phase, in which the mesh is moved in any desired fashion relative to the fluid. It could be shown (see Hirt et al, 1974) that the ICED-ALE finitedifference equations are independent of the position of the vertex within the control volume (see fig. 4.4), which may cause decoupling between adjacent vertices, as reported by Hirt et al (1974). In order to prevent this decoupling an artificial restoring force is introduced which is nonphysical and diffusive. Moreover, since the pressure iterations in the second phase of the solution procedure are based on an equation of state, the calculations
57
become excessively sensitive to the volume changes in the case of subsonic flow and difficulties with pressure iterations may occur, as discussed by Amsden et al (1980). This method has been used most extensively for high Mach number flow calculations, but some applications have also been made to subsonic flows (e.g. Harlow and Amsden, 1975; Daly, 1976; Boni et al, 1976). Another procedure which uses Cartesian velocity components and consequently strong conservation form of equations is TURF method of Wachspress (1979). It uses an arbitrary curvilinear grid with geometrical parameters (like, arc lengths, cell volumes, etc.) calculated by employing a local coordinate transformation defined in terms of the pressure node and four neighbouring nodes (see fig. 4.5). The arrangement of variables is shown in fig. 4.5. Both velocity components are located at each cell face, with four pressure nodes surrounding each velocity node (which is also a characteristic of the ICED-ALE variable arrangement). The solution procedure is based on SIMPLE algorithm of Patankar and Spalding (1972). As with ICED-ALE, the TURF method also suffers from possibility of decoupling between adjacent nodes. In fact, according to the author of the method, the coupling of the finite-difference equations between odd and even columns/lines is relatively weak and arises only from the cross-derivatives associated with the grid non-orthogonality. It would appear that no calculations performed by the TURF method have been published. Now some reasons will be offered for the use in ICED-ALE and TURF of variable arrangements that differ from the one that is usually employed by finite-volume methods based on Cartesian components and Cartesian grid
(Patankar, 1980, for example). Also analysed will be possible reasons for the decoupling phenomena observed in both methods. The discussion will be conducted with reference to the simple situation of steady two-dimensional inviscid flow, for which, in the notation shown in fig. 4.6, the momentum
58
equations (4.12) for i = 1 and for physical components reduce to:
A
7)
Ax
r,JDO
sin(a + ex)
L P
(1)(2) pu u
sina
cos(a + 0 ) x sina
r__JDJD sine cos°x (1)(2) x u 4- -TO Pu sinaj sina + pu Ax
sin(a + e x ) sina
a
sine
+
415 ax
(4,24)
x ap
sina
ax
Here a is the angle between coordinate lines and
ex is the angle
between x l and y l directions, as shown in fig. 4.6. For the sake of simplicity the case of a = 90 As
e
x
0
will be considered.
varies, the following Cartesian-like equation (in the sense that it
contains one pressure gradient which 'drives' the dominant Pvelocity component) obtained for
A A (1)(1) -71-) ( pu u) + Ax Ax
transforms for
A
e
x
ex = 0:
(1)(21) ( pu u = -
(4.25)
= 90 0 into:
121
A (1)(1). ,1-) ( pu(1) u ) - 772) (pu u ) -
Ax'
Apr () ax
Ax
ap2
(4.26)
(2 ) 3x
where the roles of the derivatives in the
x
1
and x 2 directions are inter-
changed. The significance of this behaviour with varying
ex can be seen
from fig. 4.7 where the usual variable arrangement (in which velocities are defined normal to the continuity control volume boundaries and lie between two principal pressure nodes, see Section 6.3) transforms, as ex is increased to 90 0 , into an anomalous situation where the velocity
59
components all lie parallel to the associated cell faces (and hence the convective fluxes are all equal to zero (fig. 4.7b)). Also, the pressure force, which is approximated by:
f•dX IL 1)d 2 : (p - p ) 6)0 ) = n s j (2 ) Dx V
(4.27)
l xJ
= [WS(P NE' P NW' P E' PW ) - WS(P E' PW' P SE' SW / ]J '"" D
where WS stands for weighting sums of the corresponding pressures, exhibits anomalous behaviour. Specifically, the coefficients of p E and
pw
are
generally weak and for a uniform grid equal to zero, which causes a complete decoupling between the adjacent pressure nodes. In order to avoid the anomaly concerning the convective fluxes through the cell faces the ICED-ALE and TURF methods define both velocity components at the same locations. It seems, however, that neither of these methods have managed to completely overcome difficulties with the pressure gradients treatment and, as mentioned above, some decoupling has occurred in both methods. It should be mentioned that neither of the above problems arises if vector and tensor components in terms of locally constant base vectors rather than Cartesian components are employed, since their direction depends on the local orientation of the coordinate system. This approach has, however, its own drawbacks, as discussed in Section 4.3. Few attempts appear to have been made to use non-Cartesian velocity formulations in subsonic flow calculations. Liu (1976) employs the governing equations in a semi-strong conservation form and contravariant velocity components, with an arbitrary curvilinear grid and variable arrangement as shown in fig. 4.8. The method is designed to solve unsteady three-dimensional incompressible
60
flows by approximating time derivatives by forward and spatial derivatives by central differences. After establishing the finite-difference equations for contravariant components, the corresponding equations in terms of contravariant physical components are obtained and solved iteratively using the method of Viecelly (1971). The method of Liu has been applied to a few simple flow configurations with variable success (Liu, 1976). DemirdliC et al (1980) also use the semi-strong conservation form of the equations, non-physical contravariant velocity components and the grid arrangement as shown in fig. 4.8. Their solution method, however, is based on the SIMPLE procedure of Patankar and Spalding (1972) and is very similar to the one used in this study, details of which will be given later in Chapter 6. The older method has been successfully applied to several flow situations described in the forementioned reference. Subsequently, however, it was found that in some cases numerical errors occurred, solely from the use of the non-physical contravariant velocity components. The nature of these errors was discussed earlier in Section 3.1.
4.8 Summary and Selection In this chapter various different forms of the transport equations that govern the fluid flow in general coordinates have been reviewed and their relative merits have been discussed from the point of view of a finite-volume solution method. It has been found that:
(a) The strong conservation form with base vectors expressed in terms of spatially fixed base vectors has no dominant velocity component in the momentum equations and, therefore, requires simultaneous solution
61
for all three components, which can be much less efficient than solving them in turn.
(b)
The strong conservation form with vector and tensor components expressed in terms of constant base vectors (Cartesian, for example) requires arrangements of variables different from the one usually employed in the simple circumstances of Cartesian velocities and rectilinear coordinates and even then some decoupling between adjacent nodes usually occurs.
(c)
The use of locally fixed bases eliminates deficiencies of both formulations (a) and (b), but at the expense of discontinuity of coefficients yi and
ai
which implies more computer storage and
introduces additional computations since the reciprocity of the finite-difference coefficients is lost. In addition, this discontinuity of coefficients also affects the overall balance of momentum.
(d)
In the case of the semi-strong conservation forms (4.8) the overall balance of momentum may be more difficult to achieve than in the case of the strong conservation formulation, but consistent discretization of non-differentiated 'curvature terms' in all equations should lead to the exact conservation of the physically conserved quantities.
(e)
The weak conservation form of the equations, has many more terms than the other formulations and, therefore, requires more computer time to solve. In addition, the presence of non-differentiated terms in all transport equations makes overall conservation even more difficult to ensure.
62
(f) The choice between contravariant and covariant form of the equations involves a trade-off between more pressure terms in the contravariant case and more convection terms in the covariant one. The former may cause some difficulties with the convergence of the solution procedure and latter requires a little more computation.
(g) The resolution of momentum equations in the directions normal to the velocity components is superior to the resolution in any other direction.
(h) For completeness, it should be recalled that in Chapter 3 the superiority of physical vector and tensor components when compared with non-physical ones has been emphasised. It has also been shown that the contravarient physical and covariant dual physical components have the advantage of being related to non-dimensional unit base vectors as compared with covariant physical or contravariant dual physical components which are related to non-dimensional but spatially varying base vectors.
In the light of the above observations the following alternatives are now rejected: (i) the strong conservation form with base vectors expressed in terms of spatially fixed basis; (ii) the weak conservation form of the equations; (iii) momentum equations resolved in the directions other than normal to the velocity components; (iv) the use of non-physical vector and tensor components and (v) the covariant form of the equations. In order to facilitate evaluation of the remaining options estimates have been made of the computer time per iteration, and the storage requirements (additional to those of the version based on Cartesian velocities and coordinates) for their solution. These estimates, which are given in Table 4.2 are based on what was considered as a reasonable
63
balance between quantities which could be computed once for all and stored and quantities that require recalculation at each iteration. It should be emphasised that the data in Table 4.2 are very approximate, since they strongly depend on the choice of variables to be stored, on the discretisation procedure employed as well as on the programing itself.
TABLE 4.2 AN ESTIMATE OF THE EXECUTION TIME AND STORAGE REQUIREMENTS FOR SOME FORMS OF TRANSPORT EQUATIONS FOR STEADY TWO-DIMENSIONAL FLOW (VALUES ARE NORMALIZED BY THE VALUES OF OPTION A)
A
B
C
D
Execution time
1.
0.93
1.04
1.58
Storage requirement
1.
1.00
1.50
2.50
Option
Option A: Semi-strong conservation form with contravariant physical components Option B: Strong conservation form with Cartesian components Option C: Strong conservation form with physical contravariant components expanded with respect to locally constant bases Option D: Strong conservation form and contravariant physical components with base vectors expanded with respect to locally constant bases
64
Considering (i) results from Table 4.2; (ii) uncertainty about handling pressure gradients in the case of Cartesian components and (iii) the consequences of discontinuity of fluxes if locally fixed bases are used the equations in semi-strong conservation form with contravariant physical components have been chosen for the subsequent use in this study. This decision has also been influenced by the fact that it draws an experience gained in the development and application of solution algorithms for the simple circumstances of Cartesian velocities and rectilinear coordinates (Patankar, 1980, for example) as well as for general orthogonal coordinates that use semi-strong form of the equations and physical components (Anthonopoulos et al, 1978; Pope, 1978). However, the equations in terms of (i) covariant dual physical components; (ii) Cartesian components and (iii) components in terms of locally fixed bases deserve further attention.
65
CHAPTER 5 DIFFERENTIAL EQUATIONS OF LAMINAR AND TURBULENT FLUID FLOW IN GENERAL NON-STEADY COORDINATES
5.1 Introduction In the previous chapter the semi-strong conservation form of the transport equations with contravariant physical vector and tensor components has been chosen for the subsequent use in the present study. In this chapter the differential equations governing both laminar and turbulent fluid flow in an arbitrary moving coordinate frame are presented in this framework. In Section 5.2 the governing differential equations are assembled. In order to accommodate situations requiring a moving non-Eulerian coordinate frame, such as flows with moving boundaries, the equations are recast in Section 5.3 in terms of such a frame. Section 5.4 outlines the derivation of the time-averaged version of the governing differential equations appropriate for turbulent flow calculations. Also given in the chosen framework are the transport equations of the 'k-E' model of turbulence. Finally, in Section 5.5 the general form of the governing equations is presented and in Section 5.6 the treatment of the boundary conditions is outlined.
5.2 Governing Differential Equations If the vectors and tensors are expressed in terms of their contravariant physical components, as defined in Chapter 3, then the velocity vector, stress tensor and flux vector can be respectively written as:
66
r
e.
= Td j) -
4. _ q -
n)
0 -e".
(5.1)
q)
(i) 4-
q ed)
BY employing the definition of the divergence of vectors (3.46) and tensors (3.47) the transport equations (4.1) take the following form:
al
+A
(D/) .
sm
Ax
a
Tf•
(P
4i) + ) + e d)
S(i) ii)
A
a , A TE 0(0) + --AO
— [(P vci\fj) - Tdi ) )0)
(00-
(5.2)
it} =
cid ) = S qb
where differential operator A/APis defined by equation (3.45). If the stress tensor
r
and the flux vector
CI"
obey relations (4.2)
and (4.3) respectively, then their contravariant physical components are:
Tri j) = -
(P +
2
Mb
Av
-5' 1.1 ---.7.:)
ti j) 41 g + 23-1 u
j)
anw (5.3)
cid)
= r 4)
gdm)
ail) ax(m)
when the expressions (3.40) for the gradient and (3.46) for the divergence of a vector are employed. The deformation tensor is defined as the symmetric part of the velocity gradient and, according to the definitions
67
in Section 2.8 (equations (2.42) and (2.45)) and the relation (3.40), its components can be expressed by:
dij) = 1 „On
ji) + dim) v Irn)
7
vd))
(5.4)
n)
If the following notation:
P
2 AP) = P
7 u ,Jm) (5.5)
pj) . 2p ji,j) p(gdm)7(m)
vo)
dimv( M)
is used to express the isotropic and anisotropic part of the stress tensor, respectively, then:
Tdi) = - P grii)
(5.6)
+
Using (5.3) and (5.6) and with the aid of (3.47), the transport equations (5.2) can be written in the following semi-strong conservation form:
ap at
A (J)) ) = S 71 6,
O
3 ( P N)
(0)
+ A
m
‘p) ‘ij) _
j)
—7AxJ)
A — AA
_ 4i j) 3P ax(j)
(pp°
0) - q = S
i] , Am) (j) mi kco v V
_ imi) )
T,
sci) (5.7)
V
68
where use of equation (3.36) is made to express the divergence of the isotropic part of the stress tensor: from (3.36), (3.43) and (3.48) it follows that:
A CD AX
(pdii)-4-.) . (1)
dj) 4V. (Pg ) ed ) _(3)
. p v. gtii) + 4ii) v.0)p ) CD
.
;;.. . 0)
(5.8)
--i `ri)
a)P
Equations (5.7) are, of course, completely analogous to the equations (4.20) expressed in terms of the contravariant (non-physical) components. An alternative way of deriving the governing equations (5.7) by a transformation of their Cartesian counterparts is presented in Appendix 1.
5.3 Governirq Equations for Moving Coordinate Frame In some cases of fluid flow the use of a non-Eulerian coordinate system is often desirable and sometimes essential, as in the case of flow with moving boundaries. In order to enable the solution of such problems the conservation equations in general non-steady coordinate frame will now be defined. Previous work on this subject has been done by McVittie (1949), Viviand (1974), Vinocur (1974) and Warsi (1981). The last-named has independently used an approach similar to the one presented here.
Coordinate Transformation If
0
and
T
are independent variables of the moving coordinate
69
frame, they can be related to a fixed coordinate system by (see fig. 5.1):
= el ( xj , t) (5.9) T =
C
T(t) = t
(C = const)
For any dependent variable * defined on the moving coordinate frame by * =
T) the following relationships hold:
aI). .
aP
ae a
en) ax(3)
+
ae
aT _
t
3
a)(( )
a
en) a)0) (5.10)
a* . a* ae + a* aT _ a* ae + a* .11170 at aT BT at at ae, at
If the Jacobian of the transformation (5.9):
J
det(
(5.11)
ae is non-zero everywhere in the solution domain, then the inverse transformation exists:
xj = x j (el , T) (5.12) t = t(T) =
T -
C
(C = const)
from which it follows for any variable * = *( xi, t):
70
at
aa
_
a0
axd)
1
at
at
ay.
at a en) a ,p) aam)_
aP)
aen) (5.13)
=
at
311) 30 , at _ 311) 30 „ 31) DT DT ' at at 3;(5 aT ' at
Metric of Moving Coordinate Frame If the metric of the el coordinate system is defined by:
Y(2,m)
EQ.)
5M)
(5.14) y (2.111) = E • E
then:
= det(ytm ) = J 2 det(g ii ) = J2g
(5.15)
where J is given by (5.11). The base vectors 5m) and coordinate system
5
are related to the base vectors of the xj
and -40l by the relations (see equations (2.9) and
(2.10)):
4- 5m) -
(j) Dx
aeb-(J) (5.16)
-31111) E
4Q) (j)
3x
Vector and Tensor Components in Moving Coordinate Frame If the vector and tensor components in the el coordinate system are given by their contravariant physical components:
71
= a
e (m)
(5.17) ikm)4-
A
=
9v
5m)
then the following relationships hold (see (2.17)):
d)
3
Ini)
a
a
3x (5.18) "Altm)
-
DeDem)Adj) " (1) ) 3x 3x(J
where overbar ^ indicates the components related to the moving coordinate system
el.
Divergence and Gradient in Moving Coordinate Frame From equation (3.42) and (5.16) it follows that the divergence of an arbitrary vector ; in the moving coordinate system
el
can be expressed
by:
di v
-H3) 3a 3a
3 . -4d)
a)P)
;
3
,arn —73;
at. ax
-vm)
(5.19)
em
or with the aid of (3.46):
div a
Aa
d)
1m) Aa
(5.20)
Aem
Analogously from (3.43) and (5.16) it follows that for an arbitrary tensor
r:
72
.
r
div
_
g'
4A)
g> aem) -e•-
•
aem
axq )
A>
(5.21)
aem
or, with respect to (3.48):
A
div r
- .) 00 j ) -4CD
° Cemq-u)
(5.22)
Le)
Similarly for the gradient of an arbitrary scalar f and an arbitrary vector
I.
one gets from (3.39), (3.40) and (5.16):
ae_ 1m 3f grad f = e20) -2f, - af 3XJ) a4 3r, 3x
(5.23)
and:
-00A
grad
_ -4d) 0
al' ae 4m) --T5
3x
11
aen)
(5.24)
or:
4-
grad a
where V
CM
)
=
nil) y
is
Vo
') a
etm 0 eal
stands for the covariant derivative defined by (3.30) in the
(5.25)
m
coordinate system. From the expressions just derived one can see that the divergence and the gradient operators in the moving coordinate system el have exactly the same form as in the fixed coordinate system x j . This similarity of the divergence operators will be used in the derivation of the transport equations in the moving coordinate frame, while the similarity of the gradient operators implies that the expressions for the stress tensor and the flux vector in the moving coordinate system
73
retain the form given by (5.3), (5.5) and (5.6).
Absolute, Relative and Grid Velocity The fluid velocity relative to the moving coordinate frame Em (according to the relation (5.13)) is given by:
11) r
@ern _ aen) ax(j) @em DT ---71) 3T at 3x
(5.26)
or:
11) -
(5.27)
4Th)
where, according to (5.18);
Dc (j) a en) • ax . v oi DT ) 3x 3xl]
(5.28)
is the absolute fluid velocity and:
(5.29)
is the velocity of the coordinate frame
el
, i.e. the so-called grid
velocity.
Space Conservation Law In the fixed coordinate frame x j the elementary control volume does not change with time and, therefore:
34: - 0 at
(5.30)
74
Using (5.10) one gets:
221 — at
3T
aem
_ 0
(5.31)
km) 3t
or:
a)q
1
-3T)
3T MM
ar
=
vYMM "
3
(5.32)
1.7 ri— lim
If the following relationship is used (whose proof can be found in, for example, Aris (1962)):
a
,en
1
a
(-5F) =
r-
fft1M -71M)
=
3
(5.33) 1 aJ _
_
1 f aJ 4.
It
k
DJ Del)\
DT
at)
then equation (5.31) becomes:
r n 1 ( aJ aT
3T
aJ a
@en; +
IM)
3t ' • (5.34)
After multiplying by J and after some manipulation the above becomes:
a Tr-
r,;„ ( WOo
,em
3t
=0
(5.35)
75
Employing (5.15) and (5.29) one finally gets:
1
-. v);
a
_ ziem (
, v Y )
=
0
(5.36)
A
This equation relates the rate of change of the elementary control volume in the moving coordinate system to the coordinate frame velocity and it is called, according to Trulio and Trigger (1961), the 'space conservation law'. They seem to be the first to include this equation together with mass, momentum and energy equations in their 'fundamental equations of motion' for numerical analysis applications involving moving meshes and to use it for some one-dimensional flow calculations. However, it seems that the fact that this equation has to be solved simultaneously with the other conservation equations has not subsequently been recognised until it was rediscovered by Thomas and Lombard (1979) and, independently, by the present author.
Transport Equations in Moving Coordinate Frame If lp stands for an arbitrary scalar or vector, using equations (3.45), (5.10) and (5.29) and employing the space conservation law (5.36) the following expression for the time derivative term is obtained:
a
13
(0) = ,A7 -
1 =
at
(67p0 - 10*
, r7 ,
at
a
1
aT 0'14)
-
-
/77 \ ae OrYPIP/ 3t
(5.37)
rY a
_7 aV at
0
1 DV7
e)
aell) at -
....
Continued
.
76
Vini
a
=
(470) -
A
g
(PliwAC,„) -
0/7in
pTh
p
--g-- A
WE-) =
(5.37)
Aen m
env
'T r ((rip) a V— 0 -1—) (N)
=
g
Using this equation and the expressions for the divergence of vectors (5.20) and tensors (5.22) the conservation equations (5.7) can be written as follows:
A
a ( )67p)
Aen
vi
1
^soa 757 T (VC7PV 5k) ) - 77
/7 aT
r (
LPt
( ,/7.p
m) -
4111g 5]
= Sm
417) 1[ 13#%.1129171)- 4M)
9)
Ae—
) + A [p(150m)- 1Pp - 21
If (i) the overbars
A
1
(5.38)
= st'uiz V
so
are dropped; (ii) the independent variables of
the moving coordinate system are denoted by x j and t; (iii) the following new differential operator is introduced:
)
= -
_I pg. ( . )
(5.39)
at
and (iv) the relative velocities defined by (5.27) are used; the governing equations for the fluid flow in an arbitrary moving coordinate frame can be written in their final form:
77
( )q)
+ At
A
--r) (v(3)) = Ax° g
( vd) - S P r m
AA ( p v ) Ax3)
-
MJ
d) - -e ) r
(ofm vth - 4ini))
sch
j)
v
-
(5.40)
3x3)
-
(i ) pe)
A TEE (4)
A. (#0() q Ax9 ) r
S I)
These equations differ from their counterparts for a fixed coordinate frame (5.7) by the following:
(a)
an additional, space conservation equation arisen from the fact that the elementary control volume dimensions change in time;
(b)
the different time derivative operator, now defined by (5.39);
(c)
the convection terms, which are now expressed in terms of the relative velocities v
(d)
r'
rather than by the absolute velocities A
the inertial terms in the momentum equation, which are now proportional to the product of the absolute and relative velocities r'
(e)
an additional term in the momentum equation ( i ) pPI whicn arises
78
frcathefactthatthebasevectorse.change not only in space, (1)
but also in time. The Christoffel symbol-like quantities ( ] are i defined by the following relation:
(1) _
imi t
(5.41)
i
at
which is equivalent to the definition of the physical Christoffel symbols of equation (3.26).
5.4 Equations for Turbulent Flow It is a well-known fact that the small computational mesh size required for the resolution of the small time and length scale turbulence processes cannot be accommodated in present computers. Therefore, the instantaneous equations (5.40) must be time averaged to remove explicit reference to the small scale turbulent motions. The first step in establishing time-averaged turbulent flow equations is to expand each variable into the sum of two terms:
=
+
(5.42)
where the 'mean' value varies slowly (or, in the case of stationary flows, is constant) and 'turbulent' component
fluctuates at the high frequencies.
With unsteady compressible flow there are several ways of accomplishing this expansion, one being the classical Reynolds time averaging, another being mass averaging first suggested by Favre (1965) and a third is a combination of these proposed by Rubesin and Rose (1973). (In any case the time averaging interval must be sufficiently long compared to the time scale of the turbulence, but small in comparison with the time scale of the changes within the flow field as a whole).
79
If the fluctuations of laminar viscosity and density are ignored, which is usually justifiable in non-reacting flows, then substitution of equation (5.42) into the transport equations (5.40) and then time averaging gives the following:
a
A (Vi)
9
y
.(vd) 0
LXT
A
A
g
_4
(pv r
1/4P )
A
A --d)
(pv )
Lpv
Ax'
— (
•
A
r--N
=
m
—d)
vr
p- v
_ crei _ -
mi Lpv v—d) r —
(q
-
P (0 1
v )
1(x ]
..10) IQ) 1
_
p v
-d) -
A
a 04 ) A x J r
in)
v
1
d), )j
-sch
(5.43)
mi
)1 — ( ) pv
s
These equations are similar to their instantaneous counterparts (5.40), except for the replacement of instantaneous quantities with the averaged ones (denoted by overbar
and for the appearance of additional
unknowns, namely the turbulent (scalar) fluxes:
IT
_ —
(5.44)
P (I) v
q t
and the Reynolds stresses:
pj) t —
_ _ —
p v
v
3
)
(5.45)
80
A variety of 'turbulence models' exists for linking the Reynolds stresses (and turbulent scalar fluxes) with the main time-averaged properties and thus closing the system of equations (5.43). The most widely used alternative approaches to the 'closure problem' may be identified as follows: (i) models based on the Boussinesq's suggestion (Boussinesq, 1877), often referred to as 'turbulent' or 'eddy' viscosity models; (ii) Reynolds Stress Models (RSM); and (iii) the most recent Large Eddy Simulation (LES) approach. The methods of the first group range from Prandtl's mixing length hypothesis to different variations of two-equation differential transport models and have been widely used for the calculation of many turbulent flows. The models of the second, and especially third, groups have been relatively less used, despite their superiority. The reason for this is a much higher complexity of the last two groups of models: the RSM requires solution of seven or more additional differential equations (in 3D case), while the LES requires always 3D time-dependent calculations of the large scale turbulence and can only be accommodated on the biggest computers. More details about above turbulence models can be found in Launder and Spalding (1972) and Bradshaw (1978), for example. For the calculations presented in this study, the two-equation k-e model of Jones and Launder (1972) is used. It defines the turbulent viscosity and the turbulent scalar flux by:
— k2 t = P — e (5.46) r
_
qt
Pt qt
where C and a
the turbulent Prandtl/Schmidt number are both empirical
81
coefficients. The turbulent kinetic energy k = 0.5
v qi
dissipation rate e
d
in)
v°11)V.
V
v.(i)*
ri)
(11)
v" v"and its
are obtained by solving
the following differential equations:
) + -Lr At ( A k
(T,k0
AO
A /7 k pE)
At
( 7,6 -0 ) _
+
r
Ap
dim) a
-
r
- G - -
aer -
ak
dm)
a g
PE
ae
(5.47)
ax
2
= C 6 G + 1 k*
Here a
k'
a
e'
C
C
2
c 2
P
6
and C
+ r
-3 Pc —
AP)
3
are further empirical coefficients and
the effective viscosity and diffusivity are defined as:
(5.48)
where
m
and
r
are respectively the molecular viscosity and diffusivity. Om
The quantity G is the generation rate of turbulent kinetic energy, defined by (see equation (2.49)):
G = 7-> t . gra d d v - a
T6i)
t Vd)
(5.49)
V I°
where (see equations (5.5) and (5.6)):
1
The exact definition is C = p- q. T ' dm)
v
,
which reduces to the
expression given in the main text for isotropic turbulence (Corrsin, 1953).
82
-(m)
g
[ em)
T t= t
Lg
-ch cm) v
v 1 + dim) Vain -d
rpk + -
6' \1 1 (RI Ax
di j)
(5.50)
It can be shown that in this case:
(D 2 - 13 3 )2
P t [ (D 1 - D2 )2
G =
(D 3 -
0
1 )2] (5.51)
TA (D i + D 2 + D3)
-
where D
D
1 ,
2
D3
are the (real) eigenvalues of the deformation tensor
rdefined by equation (5.4), and div n,/). = D
1
+ D
2
+
3'
From (5.51) it
follows that in the case of constant density flow G is a non-negative quantity. The values of empirical coefficients employed in the present applications of this turbulence model were taken from Launder (1981) and are given in Table 5.1.
TABLE 5.1 VALUES OF THE k-E TURBULENCE MODEL CONSTANTS
Constant
C
Value
0.09
il
ak 1.0
a
E
1.22
C
1
1.44
C
2
1.92
C
3
1.0
K
E
0.4187
9.0
83
5.5 General Form of Transport Equations From the previous section, it follows that if k-c turbulence model is employed, the governing equations (5.40) remain valid in the case of turbulent flow if the dependent variables are considered to be timeaveraged and if p and
1' 4)
are taken as the effective exchange coefficients.
The set of equations (5.40) and (5.47) with (5.46) is then a closed system that can be solved by a numerical solution procedure. All these equations (except for the space conservation equation) can be written in a general form:
aA (
A
(0J'r - r 4im) a'P,
(5.52)
afri
where 11) stands for
0,
k, e, any scalar quantity (I) (stagnation enthalpy,
concentration, etc.) and in the case of mass conservation, unity. The definitions of the exchange coefficients
r
and source terms
S pertaining to particular variables are given in Table 5.2.
5.6 Boundary Conditions In this section the different types of boundary conditions which may pertain in particular problems are considered and an approach used for the wall boundary conditions in simple circumstances of Cartesian components and coordinates is extended to the case of general coordinates and contravariant physical components. Apart from the geometry of the boundaries, which can be quite arbitrary, some classification of the boundary conditions can be made. One can distinguish the following main types:
(a)
walls;
(b)
planes and axes of symmetry;
84
TABLE 5.2 DEFINITIONS OF EXCHANGE COEFFICIENTS AND SOURCE TERMS IN GENERAL TRANSPORT EQUATION (5.52) FOR PARTICULAR VARIABLES
lp
r
s
IP
IP
o
1
s
(ij) d m) -g (T
A
ci)
v
P
(j) Ax
(I)
D4 1)
ci j)
P
3)1
1
_ (
m
mj
i min vc1j) _ ,
) _
rn,j)
(m ii
aP
p4m)
‘r,
s
r 4)
k
-ILa k
E
-41-
vE
4)
G - pe
2 Ar) G - C 2 + p k_ C 3 pe Ap)
Ci t
where:
P
dj)
=
T =
2 k P 7 (P
d m)
blip)
(i)
p (g Valo
dj) 2 G = gdk) [ T t -
dimv 0) (M)
4i j)
( pk
+ Pam
\Pm)]
.
--6 t 3x
V .) \it'd (J
g i) v
_
85
(c)
inflow and outflow surfaces;
(d)
'infinity' or 'free-stream' boundaries.
Details of the treatment of these will now be given:
Wall Boundaries In the case of inviscid or laminar flows near walls the respective slip and no-slip boundary conditions are sufficient. The situation is, however, much more complicated if turbulent flow is calculated. The reasons for this are very steep gradients of the flow properties and a significant role of the molecular viscosity in the near-wall regions. These effects can be accounted for by employing a 'low Reynolds number' version of the k-E turbulence model (see Jones and Launder (1972), for example). However, such an approach requires a very fine grid in near wall zones and special turbulence models; it is used only if the flow details near the wall are of special interest or if the alternative described below is unsuccessful. An alternative approach is, therefore, very often employed that uses 'wall functions' (described by Launder and Spalding (1972), and others) to link the solution in the interior of the solution domain with the near-wall region. The main assumption of the wall function concept is that the flow next to the wall exhibits certain regularities that allow simplified representation: in the present case it is assumed to behave as a steady one-dimensional Couette flow. Starting from this assumption a simple analysis results in the following expressions for the wall shear stress:
T
H r1/4 61/2 P B u B ""11 N B
w -
in(EY/I)
K
(5.53)
86
and the local dissipation and production rate of the kinetic ener gY of turbulence:
C B
-
G =
B
3/4 3/2 kB
u
KY
(5.54)
B
U
(5.55)
w 3Y
where: 1/4
p _Bp C
y+
k
1/2
BBY
(5.56)
m
and K and E are logarithmic velocity profile parameters whose values are (for smooth walls) given in Table 5.1. The quantities Y B and U B denote the distance normal to the wall and velocity component parallel to the wall at the matching point B respectively, which must be situated in a zone where Y
B
> 30. For purposes of illustration of the application of
the above calculations in general coordinates, the calculation of y v UB and 3U/3Y will be given for a two-dimensional case. Assuming the first interior grid line in the x l direction, for example, to be fairly parallel to the wall (see fig. 5.2), one gets:
(2) Y Z sin a 6x B B
(5.57)
(2) 2 where 6xi s the distance from point B to the wall along the x coB ordinate line, and:
U
B
•
- ‘P )
->
eci ) • en )
. ( 1) = v +
cosa v(2)
(5.58)
87
The derivative of the velocity vector in the direction normal to the wall is given by (see equations (3.40) and (7.9)):
-42] e • g dm)
e • grad
4. . ji) eum 0 ed) (j)
v
(5.59) =
1 ( sin
(i) v a 12)
cos a
sin
r,
si) \ ;.
a "(1) "
o)
For the case where the normal flux at the wall of a scalar quantity (I) is prescribed:
(5.60)
the derivative normal to the wall @WY is given by (see equations (3.39) and (7.9)):
dm) acp • 4.4123 • grad (I) = e •• g e = (J) mm Dx
sin
(2) a ax
cos
(5.61)
a
sin a 3]K"
Attention should be paid here to the fact that expressions (5.59) and (5.61) contain derivatives along both coordinate lines in contrast to the orthogonal case, where only the normal gradient appears. Expressions (5.58), (5.59) and (5.61) take into account the angle between the grid lines and also the wall curvature and are, therefore, appropriate for use in the case of general coordinates. However, in the near wall region the following relationships hold:
88
2) n! , 0 e e
(6.30)
1 42),,(111 1 e Ye I \ ( 2) (1) 1 v e Ve
/T
f D-S- and: For the situation in fig. 6.3 ae = 1, ye = 1, O e = --5
4'e = 9's + (1 - Se ) *p
(6.31)
(For comparison, the upwind differencing scheme would in the same circumstances give IP e =
11)p)-
Some other nine-point schemes that also suffer less from false diffusion have been put forward. Examples are: the 'quadratic upstream' differencing (QUICK) scheme of Leonard (1979) and the 'finite analitic' scheme independently proposed by Chen and Li (1979) and Stubley et al (1980). As originally proposed all these nine-point schemes suffered from the possibility of negative coefficients, even when applied in orthogonal coordinates. This may cause oscillatory (unbounded) solutions and make these schemes intrinsically less stable than the simple five-point molecule scheme (see Leschziner and Rodi (1981) and Han et al (1981), for example). Moreover, the imposition of the boundary conditions is more difficult. For the above reasons and since (i) these schemes are considerably more complicated than the simple five-point ones and (ii) the flexibility of non-orthogonal grid usually allows fairly good alignment of the grid lines with the main flow direction, the upwind and hybrid schemes are used in this study. However, proposals are made in the final chapter for the inclusion of a newly-developed variant of SUD by Gosman and Lai (1982), which has unconditionally positive coefficients.
114
Temporal Differencing The value of the temporal weighting factor K determines the contribution of the 'old' and 'new' time-level quantities in the discretized equations and, therefore, defines the time differencing scheme. There are three well-known schemes that correspond to:
K= 0
:
explicit scheme, which assumes: 11, = e for t° < t< t° + St
K= 1
:
fully-implicit scheme which assumes: =n for t° < t < t o + St
1
K = 2-
:
time-centred or 'Crank-Nicholson' scheme which assumes that lp varies linearly between e and e.
The explicit scheme (K
=
0) relies only on the 'old' time-level
values at neighbouring nodes to P and, therefore, obtains variables at the 'new' time-level by simple direct substitution. However, as is wellknown, this scheme suffers from the Courant time step restriction (Courant et al, 1967) for the reasons of numerical stability. Thus, its application for many flows (especially at high Reynolds numbers) becomes prohibitively expensive. The linear variation assumed by the Crank-Nicolson scheme (K
=
0.5)
seems the most reasonable one and, indeed for sufficiently small time steps it produces very accurate results. However, for larger time steps oscillatory, physically unrealistic solutions may occur (Patankar and Baliga, 1978; Johns, 1980). The fully-implicit method (K = 1) is unconditionally stable, although less accurate for small time steps than Crank-Nicolson scheme. Patankar and Baliga (1978) have described a further alternative, called
115
the 'exponential' scheme, that is also unconditionally stable and is as accurate as Crank-Nicolson scheme for the case of small time steps. However, it requires recalculation of temporal weighting factors K (K
constant) at each time step. Some other temporal practices have also been used. For example,
the 'hybrid' methods of Harlow and Amsden (1971), MacCormack (1976), Li (1976) or Shang (1978) split the equations into several terms and use implicit treatments for those that would severely restrict the time step and explicit treatments for all others. The stability of particular schemes can be explained by examining the sign of the coefficients of equations (6.21). One can see that the only constant value of K that ensures positivity of all the main parts of coefficients is K = 1. For K < 1 the coefficient a
o
may become
negative. This may cause oscillatory unbounded solutions. It also reduces the diagonal dominance of the coefficient matrix which is detrimental for the convergence of iterative techniques (Varga, 1962). In addition to stability and accuracy there is another important factor influencing the choice of temporal differencing schemes, namely, computational efficiency. This can be characterized by examining the number and nature of the coefficients of equations (6.21) which arise in each scheme. It can be seen that in the case of an explicit (K = 0) or a fully-implicit (K = 1) scheme the 'new' or 'old' time-level coefficients (except for a; or a (l;) are equal to zero, respectively. In any other case (0 < K < 1) both sets of coefficients are required in order to advance the calculation in time. This means that the 'old' time-level coefficients have to be either stored or recalculated which implies an increase in either computer storage or time. In addition to unconditional stability and reasonable computer requirements there is one more property that favours the fully-implicit scheme. It is a fact that the steady state solution is often costly to
116
obtain by other time-differencing practices because of their time step restrictions, while in the case of fully-implicit scheme the time step may be taken as infinity. On the grounds of the foregoing considerations the fully-implicit time differencing scheme is selected for use in this study.
6.5 Treatment of the Cross-Derivative Diffusion Terms From the earlier observations about the part III contribution to the finite-difference coefficients (see equations (6.23) and (6.24)) it follows that regardless of the differencing scheme used, the presence ofthecross-derivativediffusioncontributionsE.mkes the coefficient matrices nine-diagonal and admits the possibility of negative coefficients, in the presence of grid non-orthogonality. Indeed, if fully-implicit time differencing and upwind or hybrid spatial differencing are used, the ninediagonal coefficient matrices and negative
coefficients arise solely due
to the cross-derivative part a o f the coefficient a K
a
III
f
1P
(1 - f
2P
) (E
One can see that the a
e
+ En ) -
III
f 1P f2S (Ee
Es)
(6.32)
are proportional to the product of two
spatial interpolation-factors f and the cross-derivative diffusion coefficients IE 1 which are relatively small and vanish when the grid is orthogonal, for example:
E
e
- (cosa)
(1) dx ) e ‘-"Tae dx
(6.33)
Therefore, it has been decided to take them into account 'explicitly',
117
i.e. to absorb them into the 'source' term sip. This approach reduces the coefficient matrices to a five-diagonal one with unconditionally positive coefficients, thus allowing the use of a simpler solution procedure for the set of algebraic equations (6.20). However, the solution procedure must be iterative even if the parent differential equation is linear due to the fact that the terms in s
IP
must
necessarily be calculated from previous iterates. However, convergence will now depend in part on the nature of the components that are relegated into the source term. Alternative interpolation practices for evaluation of crossderivative diffusion terms have also been evaluated in an effort to find one which produces positive coefficients while maintaining conservation. One such procedure has been derived and is described in detail in Appendix 3. It is based on a two-point rather than four-point (see equation (6.16)) interpolation formula for
*ne' il)nw' 11)se and
Ipsw .
The scheme is con-
servative and reduces, but does not eliminate, the possibility of negative coefficients. In fact its contributions to the corner coefficients are always positive, while the contributions to the principal coefficients are positive if a criterion of the following form is obeyed:
0) 0 6x 6x lcosal max ( -75 , --1) 6x 6x
where 6X
D
(6.34)
1
(2) and 6x are the cell dimensions and a is the angle between the
grid lines. It should be noted that for a square grid (6 X
D
= 6X
2)
the
above condition is always satisfied irrespective of the angle a. Attempts to avoid it completely have been unsuccessful.
118
• 6.6
The Form of Discretized Equations Used in this Study In the previous two sections, the decision was made to use a fully-
implicit time differencing scheme
(K
= 1) with upwind (equation (6.28)) or
hybrid (equation (6.29)) spatial differencing (8 i 0) and with the crossderivative part III of the coefficients relegated to the 'source' term. If, further, the continuity equation is used, which can be obtained from the general equation (6.20) by setting
tp
1 and
-a- [(pv); - (pv)] + Ce - Cw + C n - C s 0
r= S = 0*, thus:
(6.35)
the finite-difference equation (6.11) may be written in the following form:-
a pip p, m
amIpm + b " "
(6.36)
where the summation on M is now confined to the four principal neighbours of P (i.e. M
E, W, N, S).
The non-zero finite-difference coefficients of equations (6.21) reduce, for the above version, to the following:
a
a
E D e
W
= D
w
- (1 - a ) C e e
+ aC ww (6.37)
a
a
N D n
- (1 - a ) C n n
S D s
+ a C s s ...,Continw2d
The superscript n is dropped henceforth and all variables refer to the 'new' time-level, unless stated otherwise.
119
a
=la
P
o a p =
1
b=
M
o +a -s P 2
(pv)p
r III + L av K
(6.37)
(11) K - 1)13)
where the summation on
K
is, it should be recalled, over all eight
neighbours of P. The spatial weighing factors a i are calculated from (6.28) or (6.29) and the source term s (equation (6.19)) has been linearized in the following manner:
s lp =
s210,p
(s 2 0)
(6.38)
Details of the finite-difference form of the source terms for each variable are given further below.
6.7 Differencing of the Source Terms The differencing of the source terms, i.e. those terms appearing on the right hand side of the equations of Section 6.2 is dealt with here with the exception of the pressure gradient terms in the momentum equations which are treated separately in Section 6.9. Many of these terms are similar in form in which case only a representative example is considered. As far as the evaluation of geometrical quantities entering the expressions for the source terms are concerned, they will be dealt with in Chapter 7. Where reference is made to ip values at locations other than nodal
120
points, interpolation is implied. The practices employed for these are described later in Section 6.8.
Momentum Equations 1)
Since the momentum equations have similar structure the only v( momentum equation will be considered here (see fig. 6.5). Considering first the divergence of the stress tensor, this is approximated in the following manner, e.g.:
( A
01) -
01)
)
T
a (T
-- i
V
dV -
A
Ax
-
01)
A
T w
e
e
w
where, e.g.:
To e 1)
,
. 2pel
yo)
7 ol
di u { a
+
(1
I a) + ( 11 j
v -
j2)i
4.
2-i i - J -
+ go z [ av(1) + f 1 1 va) + ( 212 ] ,;21}e
0
7
2j
(6.39)
and:
@ v
(1) v - E
a75)e
-
0)
(
MI
t
av
k--(72i f
ax
-
(1) P
- v )
6x(
e (6.40)
(1) (1) vne vse
e -
0
6x
e
The stress tensor source terms are approximated as below:
I V
( 1 11
1 '
To
p
dv
z
i i ii
Tirbvp
' J P
r
(6.41)
121
where:
.p) . 211 { 411) [ 3‘i (1) v(1) p p —7) on
3x
‘52 fl (21) (6.42)
g(124
f1 112 ) vcD ( 212 )
2
ax`
and: (1)
av( (—OP 3x
v
e
- v
(1)
w
() 6x
P
(6.43)
0) s n ( 3v ` 70P —725— aX (5xP 0)
v ( - v
All above terms are incorporated into the s l part of the linearized source term of equation (6.38). The contributions to the s 2 part are obtained only from the approximations of centrifugal and Coriolis inertia terms containing the 41) velocity component. For example:
J/DPP dV Z v(1)
(6.44)
V
The term in the bracket is taken into s
2
if it is negative, other-
wise the whole term on the right hand side of equation (6.44) is absorbed into the s
I
part.
Turbulence Model Equations Special attention has to be paid to the linearization of the source terms in the k and E equations because: (i) strong coupling exists
122
between these equations through their source terms and (ii) the source terms usually make an important contribution to the local turbulence energy balance, reducing the dependence on the surrounding cells and sometimes causing numerical instability. It has been found in the present study that the following linearisation gives good results:
s
k =
(G - pc) dV
2(GV)p - [(G + pe)
= s
lk
+ s
C M
V] p kp=
(6.45)
k
2k P
where use of equation (5.46) is made, and:
2
S
f
- C 2 p
[C i G
+ C,pe
V
(G I G
V) p - ( C 2 pt
+ er AP 3P A
r?P E P
=
AP)
Ar
V)p Ep
le
dV
(6.46)
S 2E EP
where the term containing brackets is incorporated into s 2E if it is negative and into s le if it is positive.
123
6.8
Interpolation Practices In order to provide information about dependent variables at all the
required locations some interpolation is often necessary. For this purpose the spatial interpolation factors defined by equation (6.17) are used. It should be recalled that the interpolation used to evaluate the dependent variables at the corners of the corresponding cell faces was defined earlier, by equation (6.16). It should also be noted that the velocity components are immediately available at all the scalar cell faces and so are the scalar quantities at the two velocity component cell faces (e.g. east and west in the case of Because of the staggered grid arrangement the scalar and momentum component cells will be treated separately. Since many variables are treated in the same way, only examples are given.
Scalar Cells The values of the scalar variables (like density, pressure, viscosity, etc.) at the scalar cell faces are calculated in the following way (see fig. 6.4):
Pe
=
(1 - f lP ) P P
+ flPPE
(6.47)
The velocity components at the centre of the cell and at the cell faces are approximated by:
(1)
vp =
1 2-
(
(1)
v
(1)
+ vw)
e
(6.48) (1)(1) 1 — l 2 p) Vp n c
V
= (1
_s_
4
(1)
T i2pVN
124
Velocity Cells The values of the v
(1)
0 and v velocity components at locations
0) associated with the v cell of fig. 6.5 are approximated by:
(1) ve =
0) n
V
1
7
(1) (1) (v p + vE)
J.
c 1 c 0) = (1 ''' 12p) VI) 1
0)
,2pvN
(2) (2) (2) 1 + v ) v = 7 (v se e ne
(6.49)
(2) 1(2) (2) + v ) v = 7 (v nw ne n
0 v P
(2) (2) 0 1 0 +v ) + v (v +v sw se T ne nw
=-
The scalar variables are approximated by:
1 PP =
7
(Pe + Pw) (6.50)
p
(1 - f
2P
) p
P
+ f
p 2P N
6.9 Solution Procedure
Choice of Dependent Variables Before the solution of discretized equations (6.36) is attempted the appropriate dependent variables have to be selected. Ideally, each equation should have a dominant dependent variable. In the case of the compressible high Mach number flow, the density
125
may be regarded as one of the main dependent variables and the pressure may be obtained when required from an equation of state. Then the continuity, momentum and energy equations form a closed set of five (1) 0 0 equations of the form (6.36) with the five unknowns: p, v , v , v and T. Such an approach is not, however, useful for constant density flow simply because the density no longer features in the continuity equation. Indeed, even in the case of variable-density subsonic flow the local variations of pressure which are very important in the momentum balance do not appreciably affect the density. Therefore, the continuity equation can no longer be regarded as a density equation and some other means for obtaining the pressure field is needed.
Pressure-Correction Equation Researchers who were confronted with the above problem recognized that it is natural to attempt to extract the pressure, which does not have 'its own equation' from the continuity equation that does not have 'its own main dependent variable'. This was done in various ways by Harlow and Amsden (1971), Patankar and Spalding (1972), Patankar (1980) and Issa (1982), among others. The SIMPLE procedure (Semi-Implicit Method for Pressure-Linked Equations) of Patankar and Spalding (1972) is taken as the basis for the present method. In order to apply this procedure the discretized momentum equation for each of the four velocity components normal to the surfaces of a scalar cell is written in the form of the following example, in the notation of fig. 6.4:
0)
r
(1)
a v = L a v + b - (g e e m m
f sina mAA )e k p E - pp) -
M
(6.51) - (g02 ) sina)e
APE (P ne
Pse
126
in which the pressure gradient terms have been extracted from the source term. Here the summation on m is over the four neighbours of e. The above equations are solved for a guessed pressure field p* to yield a provisional velocity field v
v421 -that by definition satisfies
the momentum equations at least in a linearised sense (the coefficients are usually held constant during part of this procedure, as is explained later). If the 'correct' pressure and velocity fields (i.e. those which simultaneously satisfy momentum and continuity) are now expressed as the sum of the provisional (starred) and correction (primed) values:
,(1) (l) 41) V = V +V
(6.52)
\J D = v 42) + v, (2)
and if they are inserted in place of their counterparts in equation (6.51), there result equations of the same form in terms of primed variables, for example:
,(1) a e v e =
, (1) a mv m m
(
01) . g sina A) e -
pi) ) -
m
(6.53) -
12) sina) e A pE (p y'le - pie)
where p rile and p e are interpolated from surrounding nodal values by equation (6.16). Omission of the first term on the right hand side of equation (6.53) (the justification for which will be discussed later) and combination with (6.52) gives relationships of the following form:
127
41) (1) - d (p - p') - e (p' v = v eeeEPene
(6.54)
se
;there:
d
e
OD . = — 1 (g sum A)e a e (6.55)
e
(12) l sina) = — (9 e APE e a e
(1) 0
(2)
Substitution of expression (6.54) and similar ones for v , v n and v s w into the continuity equation (6.35) yields, after some manipulation, the following equation for p':
(6.56)
ap t') = i aok + b
where the summation on K is over all eight locations surrounding point P (see fig. 6.4) and:
a E = (p* sina A) e [d e + f 1p (1 - f2 p - f25 ) e el +
+ (p* sina A) n f 1p (1 - f2p ) e n - (p* sina A)s f1Pf25e5
aw = (p* sina A) w [dw - (1 - f lw ) ( 1 - f2 p - f2S ) e w] -
- (p* sina A) n (1 - f i w) (1 - f2p ) e n +
+ (p* sina A)s
(1 - f lW ) f25e5 .... Continued
(6.57)
128
a N = (p* sina A) n [ d n + f 2p (1 - f lp - flw ) e n] +
+ (p* sina A) e
1
2p (1 - f / p) ee
(p* sina A) w f2pf/wew
-
a s = (p* sina A) s [ d s - (
1 - f2S ) (1 - f lP - flW ) e s] -
(p* sina A) e (1 - f2S ) (1 - f lP ) e e +
-
+ (p* sina A) w (1 12S ) flWew
a NE =
[(p* sina A) n e + (p* sina A) e e e l f1pf2p n
a N w = - [(p* sina A) n e n + (p* sina A) w e w] (1 - f lw ) f2p
a SE
a
= - [(p* sina A) s es + (p* sina A)e eel f 1P (1 - f2S)
SW --
[(p* sina A) s e + ( p* sina A) w ew] s (1 - f lW ) (1 - f2S)
a p = 1 aK K
b
=
if,*v n )p _ (poe )p] .1. CI, _ c* I. c* _ ,,,,, 'St '" us/ e w n
11
where Ct are calculated from (6.22)
i
by using 'starred' velocities.
(6.57)
129
Equation (6.56) is called the pressure correction equation. It should be noted that just as in the case of the cross-derivative diffusiontermsthecoefficientse .arising from the cross-pressure 1 gradient terms may be of either sign and vanish when the grid is orthogonal. Consequently, the finite-difference coefficients of the pressure correction equation are not unconditionally positive: hence the coefficient matrix is not diagonally dominant. This may affect the stability of the solution procedure, which is an essential property. It should also be noted that the 'source' term b, usually called the 'mass source' in the pressure correction equation (6.56) is actually (the negative of) the left hand side of the continuity equation (6.35) expressed in terms of starred velocity field. If b is everywhere zero, the primed quantities are also equal to zero and a converged solution is obtained. This means that these quantities only serve in a temporary way to procure a satisfactory linkage between the pressure and velocity fields so that they ultimately satisfy both the momentum and continuity equations. Thus any approximation made in the process of the derivation of the pressure correction equation is justified as long as the procedure converges. The neglected terms like a
1(1) which represent one such approximation m vm
would,if retained,introduce an implicit influence of the local pressure corrections on more distant velocities, which would, therefore, lead to much more complicated discretized equations, with non-sparce matrices. In practice, their omission usually does not prevent the convergence of the solution procedure, although under-relaxation is often necessary (Wachspress, 1979). In order to avoid possibility of negative coefficients in the pressure correction equation a further approximation is here made: the contributions of the cross-pressure gradients e i to the coefficients (6.57) are neglected, leading to the pressure correction equation of the same form as the general
130
equation (6.36):
a p g r
aoL + b
(6.58)
M " "
where the summation is over four principal nodes (M = E, W, N, S) and coefficients a are unconditionally positive:
a
E
= (p* sina A d)
a = (p* sina A d)
e
w
a N = (p* sina A d)n (6.59) a = (p* sina A d)
aP =
s
a
b = -
[(b*Vn)p - (p°V°)p]+ C: - C: + C /4 - CV
It should be noted that velocity corrections now reduce to, for example:
v
ia)
e
= de (p' - p') P E
(6.60)
The relative magnitude of the neglected cross-derivative terms depends
131
on the local grid skewness and for an orthogonal grid is equal to zero, thus (see equations (6.55) and (7.9)):
0) = - (cosa). (7254 Ti ` 6x e e
2.
6x
(6.61)
Moreover, the coefficients e i in the expressions for a K of equation (6.57) are always multiplied by the product of two spatial interpolation factors f i , and, therefore, it follows that the contribution of the crossderivative pressure terms is usually small compared with the contribution of the 'normal' pressure gradients. However, in the case of a significant departure from the orthogonality and unfavourable large 'aspect ratio' 1) 2) dx /dx the neglected terms may become important and adversely affect the convergence rate of the procedure (see Section 6.10 for the treatment in this case). It should also be mentioned that one further assumption was implicitly made in the above derivation, namely, the effect of the pressure on density was neglected. Mathematically this is equivalent to assuming that:
p = p* + p '
p
= p* + ( 2- )
p' ::- p*
(6.62)
aP T
This practice is consistent with the focus —Of the present study on subsonic flows. However, for highly compressible (high Mach number) flows the influence of pressure on density is significant and a 'compressible form' of the pressure correction equation should be used, along with other modifications. For a derivation of such an equation see, for example, Watkins (1977).
132
Overall Solution Algorithm The main steps in this iterative solution algorithm can now be summarized as follows:
1.
Guess a pressure field p*.
2.
Assemble the discretized momentum equations of the form (6.36) using the prevailing velocities to calculate the coefficients and solve to obtain the starred velocities v*CI) and v421 and the quantities d i and b used to calculate coefficients of the pressure correction equation.
3.
Solve the pressure correction equations (6.58) and hence obtain p'.
4.
Correct the velocities and pressures by using equations (6.52).
5.
Solve discretized equations (6.36) for other ip's (such as energy, turbulence quantities, etc.).
6.
Take the corrected pressures as the new guessed pressure field and return to step 2 unless a converged solution is obtained.
7.
Start the next time step using the final
tp
values from the previous
time step as guessed fields.
6.10 Solution of the Discretized Equations It has been shown that all discretized equations, including the pressure correction equation (6.58), have the form of (6.36). They, therefore, consist of a set of non-linear coupled equations, comprising subsets for each of the unknowns. These equations are solved by the
133
following iterative procedure. First the equations are linearized and the subsets are effectively temporarily decoupled by assuming that all quantities entering the coefficients are temporarily known. In such a manner subsets of linear algebraic equations for each unknown are obtained with five-diagonal coefficient matrices and unconditionally positive coefficients. Each of these subsets is then solved by an alternating direction line iteration procedure that employs a tri-diagonal matrix solution algorithm (Ames, 1977). These are called 'inner' iterations. After the subsets for all the unknowns are solved in this manner it is said that one 'outer' iteration is completed and the degree of convergence of the procedure is then examined. If the convergence criteria are not satisfied, the finite-difference coefficients are updated according to the new values of dependent variables and the whole procedure is repeated. The solution is assumed to be converged if (i) the sum over the whole solution domain of the normalized absolute residuals, defined with respect to the general discretized equation (6.36) as:
amipm + b - apipp
R
(6.63)
M
have fallen below a specified level, i.e.:
y IR,I < A N4) V
where X is usually of the order 10
(6.64)
-3
and N is a normalization factor, and 1.1)
(ii) if the dependent variables satisfy the following relationship:
(6.65)
134
where p is usually of the order 10
-3
and the superscripts n and n-1 refer
to the values of two subsequent outer iterations. The normalization factors for the mass and momentum residuals are usually taken as the total mass and momentum inflow, respectively, although other choices may be appropriate, depending on the problem. The normalization factors for other variables are usually also related to the total inflow: for the thermal energy it is the total inflow of thermal energy, and for the kinetic energy of turbulence it is the total inflow of the kinetic energy of the main flow. It is important to note that there is no necessity to procure full convergence of the inner iterations because of the need to continually update the non-linear and/or coupled coefficients. Indeed, better convergence of the inner iterations does not necessarily produce faster overall convergence. Therefore, the number of inner iterations ('sweeps') is specified in advance (usually 1 to 4) and held fixed for the duration of calculations for all unknowns except for the pressure correction equation, whose convergence is of the special importance for the success of the whole solution procedure. It has been found (Issa, 1981) satisfactory to ensure that the absolute sum of the residuals of this equation over the whole field satisfy:
1bl
/ I R II < X V
P
(6.66)
PV
where A p , = 0.3 to 0.6 and
1
lbl, it should be noted, is the absolute sum V of the local residual mass imbalances. There are several aspects of the solution procedure that may impair, or even prevent, convergence; the main ones being the possibility of negative coefficients and the strong non-linearity of the equations. Avoiding unnecessarily large departures from orthogonality of the corn-
135
putational grid will reduce the possibility of negative coefficients. Appropriate under-relaxation of the variables and linearization of the source terms will damp changes caused by non-linearity. The under-relaxation factors
wi
are specified separately for each
variable (although equal values are employed for the velocities) and generally held fixed during the solution procedure. The values employed in the present study are:
1)
Variable lp
Ni
wilj
0.5
' \i`)
0.5
pi
k
e
1 - 0.5
0.7
0.7
although other values may be appropriate, depending on the problem. The choice of w p , depends on the grid non-orthogonality and 'aspect-ratio', varying from w p , = 1 for an orthogonal grid to
wp , =
0.5 for highly non-
orthogonal one. Details of some explorations into this will be given later in Chapter 8. An alternative approach is to incorporate a criterion for dynamic control of stability and convergence through an algorithm that updates the under-relaxation factors during the solution procedure, as was done by Johns (1980) for the calculation of the flow inside Diesel engine cylinders. However, this was not attempted here.
6.11 Treatment of the Boundary Conditions In this section the way of incorporating the boundary conditions, introduced earlier in Section 5.6, is described, for the case when the approximate relationships (5.65) and (5.66) are valid, which is the case in most practical applications. If, however, this is not the case and
136
more than one gradient is involved in the boundary fluxes expressions, as in equations (5.59) and (5.61), the source terms containing the additional derivatives arising from non-orthogonality have to be modified appropriately. These changes will be additional to those described below for the simpler situation.
Wall Boundaries The boundary condition for all variables are invoked by setting to zero the coefficient
am
that would normally link the value at the bOundary
to the control volume next to the boundary and inserting the boundary conditions described in Section 5.6 through a linearized form of the 'source' term:
.
5 1B
+ s
2B B ip
(s 2B
0)
(6.67)
as described by Gosman and Pun (1974). Further details are given below.
Momentum Equation In the case of laminar flow a linear profile is assumed to apply to the tangential velocity and the wall shear stress is given by:
U B Tw = p m y-
(6.68)
B
The above is included through the linearized source term in the following manner:
S .
where:
existing terms + s 2B UB
(6.69)
137
s
U
2B B
= - f Tw A
and A
w
um dA : - ( v- Aw ) UB '13
(6.70)
w
is the area of the wall on which
T
w
apply.
In the case of turbulent flow a distinction is first made between the boundary nodes B which lie in the 'viscous sublayer' and those which lie in the 'logarithmic law' region, where the 'buffer layer' (3 < y+ < 40 roughly) has been ignored. The limiting value is
= 11.63 (see equation
(5.56)) which is determined by the intersection of linear and logarithmic law functions. Thus, for Yil- < 11.63 the approximation (6.68) applies. For Y 1E; > 11.63 T
w
is calculated from (5.53) and linearized in the way given by (6.69) where:
s2BUB = -
f A
dA -
OBC1 414 412
-
TW
tn(EY4B-)
K A U w B
(6.71)
w
It should be mentioned here that since pressure values at the wall are not usually known they are obtained by extrapolation in order to evaluate the cross-pressure derivatives at the near wall cells.
k Equations
The source term s
k
given by (6.45) is modified for the near wall grid
nodes in the following manner. The expression for the generation rate G of the turbulent kinetic energy is replaced by:
=f
GB
IT!
( 1j ) 2 dV
'
2.: IT
v
I tiY_N 2 w' '31°B B
(6.72)
V
where V the Y
+
B
B
is the volume of the near wall cell,
T
w
is evaluated according to
value, and 3U/3Y is defined by (5.59) or (5.66).
138
The dissipation rate c of the turbulent kinetic energy at the near wall grid node is replaced by:
C Y 1 E B =
B
r
j
0
314
k
Y+ 312 B B Y
,
Y
11.63
B
8
CB a =
(6.73) 241(EY-1-) 314 3/2 B C k KY B B
Y
B
> 11 63 '
e Equation The value of c at the near wall nodes is fixed at the value by equation (5.54) (irrespective of Y
=
lb
s E -2B- B = L
B
E
B
given
value) in the following manner:
LB
(6.74)
where L is a large number, typically 10 30 .
p' Equation
Since the velocity component intersecting the wall is prescribed (equal to zero in the case of fixed walls), it follows from (6.54) that the gradient ap i /a0= 0. This is implemented by setting the appropriate coefficient a in the pressure correction equation to zero.
Symmetry Axis
The boundary conditions at the axis of symmetry is easy to implement. The velocity component intersecting the axis is set to zero there and the appropriate coefficients am for the other unknowns are set to zero.
139
Inflow and Outflow Boundaries The distribution of any variable * on the inflow or outflow boundary is, when known, specified by assigning the appropriate values at all grid nodes lying on that boundary. In the case when pressure boundary conditions at the outlet are avoided by forcing an overall mass balance via the outflow velocities, d) the velocity component intersecting the outflow boundary v can be B obtained by some form of extrapolation from the interior velocities in a way that satisfies the mass balance. For example:
ci)
vB
ci)
d)
+ (Sv
(6.75)
where subscripts B and B-1 denote boundary and next to the boundary velocity nodes, respectively and the velocity increment
60
is calculated
from:
!kin -
f pB A
(50 -
1
sinaB dAB
B
(6.76)
B sinaB dAB Ip A
where
tin
B
is total mass inflow.
6.12 Closure In this chapter the discretized counterparts of the transport equations of Chapter 5 have been assembled and a solution procedure has been described for the case of two-dimensional plane and axisymmetric flow. However, much of the solution method developed here is valid for three-dimensional flows. The governing equations have been discretized in a general way that
140
allows a range of different differencing schemes to be incorporated. Some alternative spatial and temporal differencing schemes have been outlined and the fully-implicit upwind/hybrid scheme has been selected, for reasons of economy and stability, for use in the present study. However, the method is sufficiently flexible to enable other practices such as skew differencing to be employed. Special attention has been paid to the treatment of the crossderivative diffusion and pressure gradient terms which arise from the coordinate system non-orthogonality. A partial solution to the problem of negative coefficients arising from the cross-derivative diffusion terms has been found, while a stable solution procedure has been obtained by suppressing the possibly-negative contributions of the cross-derivative
pressure terms in the pressure correction equation. An alternating direction line iteration procedure has been selected for the solution of linearized discretized equations. The criteria for the convergence and the stability of the solution procedure have also been discussed, as has the manner of implementing the boundary conditions. The following chapter is concerned with geometrical aspects of the solution procedure and in the Chapters 8 and 9, some applications of this method will be described. Chapter 8 will present tests of accuracy and stability for which in most instances analytical or previous numerical solutions are available. Chapter 9 contains calculations of some complex turbulent flows.
141
CHAPTER 7 GRID AND ASSOCIATED GEOMETRICAL PROPERTIES
7.1 Introduction It follows from the previous chapters that a computational algorithm for the solution of fluid flow in arbitrary regions on a curvilinear grid requires a considerable amount of geometrical information such as cell volumes, curvature of the coordinate lines, etc. due to the complexities of the coordinate system. The accuracy with which this information is supplied has, consequently, a decisive influence on the overall accuracy of the results. In this chapter the method is presented for the calculation of the geometrical parameters required by the present solution procedure. Also, problems are discussed in relation to the semi-strong form of the equations and the solution method employed. In Section 7.2 the properties sought in the computational grid are discussed together with the constraints on achieving these. The grid generation procedure employed in this study is presented in Section 7.3. The Cartesian coordinates of the grid nodes obtained by this procedure are used for the local coordinate transformation of Section 7.4 which is necessary for the calculation of geometrical quantities. In Section 7.5 all the geometrical parameters entering the finitedifference equations of the previous chapter are expressed in terms of the so-called 'basic geometrical quantities' for the case of a two-dimensional geometry. In Section 7.6 these basic geometrical quantities are related to the Cartesian coordinates of the mesh points defined in Section 7.3 by employing the coordinate transformation of Section 7.4. In Section 7.7 some particularities of the calculation of the geometrical quantities with respect to the scalar and velocity control volumes are presented.
142
7.2 Properties of the Computational Grid The computational grid is a very important factor in the achievement of an economical numerical solution of fluid flow problems. The properties sought in the grid and geometrical parameters required by the solution method are governed by several factors: (i) the form of the differential equations (e.g. strong, semi-strong, etc.); (ii) the method of solution (e.g. finite-volume, finite-difference, etc.); (iii) the properties of the flow itself (flow direction and curvature, steepness of gradients of the dependent variables, etc.), etc. The consequential desirable properties of the computational grid are:
1.
High concentration of the grid lines only in the regions with steeply varying derivatives of the variables (e.g. wall bounding layers, embodied shear layers, etc.) where high resolution is required.
2.
Alignment of the grid lines with the flow streamlines in order to alleviate numerical diffusion.
3.
Smoothness of the grid lines to allow accurate evaluation of curvature terms (i.e. physical Christoffel symbols).
4.
Control over the important geometrical properties of the computational cells, which are (see fig. 6.2): (1) 6x P spect ratio E -75 , a P
and
(1) 6x E expansion ratio = —75 or 6xp
These affect both the stability and accuracy of the solution procedure
143
as implied earlier (see also Roache, 1976). The ideal value for both of them is unity and experience suggests that their maximum values should not exceed 8.0 for the aspect ratio and 1.5 for the expansion ratio.
5.
Control over departure from orthogonality.
As usually happens, simultaneous achievement of all these requirements, which will hereafter be referred to as property 1, property 2, etc., is difficult, if not impossible. As for the geometrical parameters required by the present solution method, these are the following: cell volumes; cell-face areas; angles between the grid lines; curvature of the grid lines and, in the case of an axisymmetric geometry, distance from the axis of symmetry.
7.3 Grid Generation It follows from the discussion in the previous section that a grid generation procedure has to compromise between the requirements for an ideal grid. The use of non-orthogonal coordinates greatly facilitates grid generation and allows considerable control over the grid properties. In essence, the grid can be specified in any desired way including drawing grid lines by hand. However, this is a tedious process, so some means of 'automatic' grid generation is desirable. Since some of the grid properties depend on the flow itself, which is not known in advance, an ideal grid generation procedure would be of an adaptive nature. This, however, brings in additional problems of precise specification of criteria for grid generation, which have only recently come under investigation. Many authors have proposed different grid generation procedures, to
144
mention only:(i) the iterative method of Amsden and Hirt (1973) which is equivalent to the procedure of Barfield (1970) who solved a quasi-elliptic system for the mesh coordinates (modified Laplace's equations) in the physical plane; (ii) the procedure of Thompson et al (1977) that relies on the solution of a quasi-linear elliptic system of equations with Dirichlet boundary conditions for the physical coordinates in the transformed plane, and (iii) the algebraic grid generation procedure of Eiseman (1980). All these methods usually need some experimentation and tuning of adjustable parameters before a suitable grid is obtained. The term 'grid generation procedure' usually means generation of the main control volume vertices. However, the accurate evaluation of the curvature terms (i.e. physical Christoffel symbols) requires more information. Therefore, the grid generation procedure developed and employed in this study consists of two steps:
(i)
Generation of the main (scalar) control volume vertices, and,
(ii) A procedure that provides Cartesian coordinates of the scalar and velocity grid nodes taking into account the coordinate line curvature.
Generation of the Main Control Volume Vertices In the present study a simple procedure for generating the physical coordinates of the main control volume vertices has been devised for nontrivial domains. It consists of:
(i) specification of the physical locations of pairs of points on two opposite boundaries (e.g. Bl and B2 in fig. 7.1), and,
145
(ii) specification of a distribution function that distributes the mesh points along straight lines joining the forementioned pairs.
Thus, if the Cartesian coordinates of the points on the boundaries Bl and B2 are
41
and 4 2 (i . 1, 2) and if the distribution function is
f, then the coordinates of the main control volume vertices are given by:
i y= Y B1
+ f(Y
B2 - Y B1 )
(i . 1, 2)
(7.1)
Each of the above two steps determine in advance the desired grid line distribution in one of the coordinate directions, which means that this procedure enables the user to specify to some extent properties 1 and 4 of the previous section. Further, specifying a different distribution function for certain grid lines (e.g. AB in fig. 7.1) a smooth grid can be obtained even in the case of boundaries with discontinuous slope, which means that the property 3 is also controllable. In addition, the present procedure is very suitable for the implementation of an adaptive grid generation scheme, which can ensure, for example, the alignment of the grid lines with the flow streamlines (property 2). A disadvantage of this procedure is that one set of grid lines is constrained to be straight. This can sometimes cause significant departures from orthogonality in some regions (property 5). Some alternative algebraic procedures have also been used in this study and will be described in Chapters 8 and 9. Indeed, the overall solution method and the remaining geometrical considerations in this section are quite independent of the way in which the main control volume vertices are specified.
•• •
• 146
Specification of Grid Nodes Locations As defined in Chapter 6 the velocity components are situated in the middle of the main control volume cell faces (e, w, n and s in fig. 6.4) and all other variables are located in the 'middle' of the scalar cells: however, the latter locations have not yet been precisely defined. In order to obtain the coordinates of the points M in the middle of the scalar cell faces, taking into account at the same time the curvature of the coordinate lines, an isoparametric parabola is passed through three ordered vertices on a coordinate line, two of them, V1 and V2, being the vertices of the scalar cell and the third one, V3R or V3L, being the vertex of the adjacent left or right cell respectively (see fig. 7.2). The resulting coordinates of the point M are:
„2 \ „1 \ ,_ C 1 ,2 „1 B ,,1 M = A ` .1 1/2 - J Vl i ' A. °'112 - JVli
J
(7.2)
Y
2 B M = A
1
„2 ,2 V2 - J Vl
‘J
C ( „1 ,1 \ - A ` J V2 - JVli
1 i
where:
1 V2
1 2. V1 )
2 22 V2 - YV1)
A
=
B
= 2-1- [(Y 1/ 2) 2 - (Y 11/ 1) 2
(3/
1 "Ft = $ r
- Y
I-, L\Y
2 V2
-
4- (Y
4'
(.4 2 ) 2 - (4)2]
„2 1 (1v1 .7 1" -J V1 +
-4
\ 1 „1 V2 - J V3R i
-
(7.3)
-
1 (Y V2 -
1 rf„ 2 C L = $ L "V2 "1..
1 Y V1
k-7
4- 6
42 -
4
3R)]
1 “1 _,_ a 1 ,,2 ) tivl - J 1" -J V2 ' " Y V1 - '1131.. 1 -
1 1 fl
( „1 -
)
2 (3Y V1
112 -
Y
v1)
k ".Y
„ 2 0 1 2 .,_ V2 ' 6YV1 - J V3LII
147
where C . C
R
if V3 E V3R and C . C
L
if V3 E V3L.
The possibility to choose between V3R and V3L enables easy handling of the cells next to the boundaries where either V3R or V3L does not exist and of points on boundaries whose profile exhibits a discontinuity of slope. The coordinates of the central point P are defined in such a manner that it coincides with the intersection of the middle lines (i.e. segments joining the middles of the two opposite sides) for the case of rectilinear control volume and takes into account the departure of the straight lines in the case of curvilinear control volumes, thus:
i i
Yp -
Ye
i jr
Yvi
i + Y
n
i 4- Y
i
i
s Y ne + Y nw
2
+
Y si e
+
Yi SW
4
(7.4)
Thus each scalar control volume is characterized in terms of the physical coordinates of the nine points in fig. 7.3. These will be used in the next section to obtain a local coordinate transformation function.
7.4 Local Coordinate Transformation In order to calculate the metric tensor components of Section 3.3 and other geometrical parameters entering the discretized equations of the previous chapter, relationships of the form of (2.1) are necessary between the general coordinates x i and the Cartesian coordinates yi. In some cases these relations can be specified analytically. In general, however, only discrete information about the coordinate system
148
x
i
is available, usually in the form of the Cartesian coordinates of
the points defining the computational mesh. In this case, the coordinate transformation can be done on a local basis, which is the approach adopted in this study. Different interpolation functions can be used according to how many points are used to describe the control volume. Wachspress (1979), for example, used the five points P, e, w, n, and s (see fig. 7.3) to define the local coordinate system. This is, however, insufficient when values of the coordinate curvature are required (since these involve the calculation of second derivatives). In the present study all nine points shown in fig. 7.3 are included in the definition of the local coordinate system. For each (scalar) control volume an isoparametric quadratic interpolation function is used of the form:
i
il2
i
il
i2
il2
i2 2
il 2
y = y (x ,x ) = C l + C 2 x + C 3x + C 4 (x ) + C 5 (x ) + Co x +
(7.5) i 1 2 2 i 1 2 2 i 1 2 2 + C 7 (x ) x + C 8 x (x ) + C9 (x ) (x2 )
(i = 1, 2)
If the values x i - ± 1 (i = 1, 2) are associated with the sides of the scalar control volume and the values x i = 0 with the sides of the velocity control volumes, as shown in fig. 7.3, then the constants C i.(i = 1, 2; j = 1, 9; note that C i. is not a tensor) can be determined J J and the transformation function (7.5) becomes:
149
„i “ i “i „i ,, i „. J e - j w 1 Jn - Ys 2 2 J = J P -F x 4- --2---- x + v i
y
i
+ y
i
- 2y
ew + 2
i P1 2 () x
y
i
+
+ y
i
ns 2
- 2y
2 2 P( x ) +
i --i i -A AY e AY w 1 2 s Y n Y s Y n Y 1) 2 + 4 x x + (---2--- --2--) (x ) x +
(7.6)
_
-A i i -A , Y e - Yw Y e - Yw, 1, 2,2 + k---2--- —2--) x kx )
(i . 1, 2)
where:
i
i
-4 Y ne I- Ynw Y n - 2
_A Y
Y
s
-
-i
Y
e
-
-i
se
+ Y sw 2
Yne i + Yise 2
Y
(i . 1, 2)
(7.7)
i
nw + Y sw Y w 2
AY
e
. Y
ne
-
' se
i Y w = Y ni w - Ysw
(It is interesting to note that these transformation formulae (7.6) can also be obtained by applying the concept of shape functions to the 'quadratic element with the central node' shown in fig. 7.3, as proposed by Ergatoudis et al (1968) for use in conjunction with finite element methods).
150
In this study the coordinate transformation formulae (7.6) are used for the calculation of the geometrical characteristics of the coordinate system. This matter will be dealt with in further detail in the next section.
7.5 Geometrical Properties of the Coordinate System In order to devise a procedure for obtaining all the geometrical parameters entering the finite-difference equations of Chapter 6 such as the cell volumes, cell-face areas, physical Christoffel symbols, etc., it will be first shown that they all can be expressed in terms of the following eight 'basic' geometrical quantities: g 11 , g 22 , cosa, sina, o) cn ae /ax , ae /a>: , 30 /3X, 30 / Dx for planar geometries (see fig. 7.4), x x Y Y with the addition of one more parameter, namely the distance from the axis of symmetry r, for the case of axisymmetrical geometries. Then expressions for these basic quantities will be derived for each cell from the coordinate transformation formulae (7.6).
Planar Geometry For the case of planar geometries it follows from equations (3.16) and (3.17) that:
gn = 1 , 922) = 1 , ga = cosa
(7.8)
and:
(1 1) g =
1 _7_z_ sin a
(22) g
1
sin a
g (1 2)
_
cos a
sin a
(7.9)
151
Cell Volumes and Surface Areas Since (see equation (2.23)):
g ll
g12
g = det() =
= g 11 g 22 sin2a
(7.10)
g 21 g22
the elementary volume is given by:
(1) CD . 1 2 dx dx = dx dx sina
dV =
(7.11)
where, according to equations (3.13), the differential arc length is:
dV
(no summation)
dxi
=
(7.12)
The volume of the computational cell is then given by: 1 Vp
= f dV
1
1
= I f
1
9 22 sin a dx1dx2 dx 1 dx 2 v---
(7.13)
-1 -1
which is difficult to calculate analytically and may be approximated by the area of a trapezoidal cell:
(1) (2) . - V - (dx Sx sina)p
(7.14)
where, for example:
(1) Sx =
f
1 (1) dx =
f ()qii) -1
2 dx1 : 21(g11) 1 2 x =0 x =x =0
(7.15)
152
The accuracy of the above approximations is consistent with the overall accuracy of the solution procedure. The cell-face areas are then, for example:
(2)
dA e = (Sx
e
=
2
() -1
x=1 dx
(7.16)
1
dA
n
a)
= 6x
n
= f ( -1
2 4TT ) x= 1
dx1
Physical Christoffel Symbols By definition (see equation (3.26)), the physical Christoffel symbols are related to the rate of change of the base vectors ; :t along the coordinate lines. It is obvious that they are identically equal to zero in the case of constant base vector fields (e.g. the Cartesian basis) and that they are related to the local curvature of the coordinate lines in the case of curvilinear coordinates x
2
(In plane polar coordinates x / = r,
= 6, for example, the only non-vanishing physical Christoffel symbols
are related to x i = r, which is the local radius of curvature of coordinate lines: thus
(L) = -
3- and ( 122 1
= ;-).
x2
= 6
They can be calculated
in the following way. The derivative of the unit vector
iv along
the coordinate line x l is
given by:
-1-
Dem
4-
_(1)
36
x _
_42)
--(15'(11e
3x
(7.17)
x 3x
where:
;ex K 11 = - cr)
(7.18)
153
is the curvature of the coordinate line x
1
and:
1 R = 1 1 1 IK111
(7.19)
is the corresponding radius of the curvature. On the other hand, the derivative of the unit vector ea) is also given by the expression (3.26):
---(1)
_ m + 11 cm)
(7.20)
By taking the dot product of the expression that follows from (7.17) and (7.20) with el23:
K
421
11 e
• e
_ -
11
-e)- • 123 mv
(7.21)
one gets:
K 11 = ( 121 ] sina
(7.22)
In a related manner, the following relationships between physical Christoffel symbols and geometrical properties of the coordinate system are obtained:
I
2) _ 1 .11J sina
De
x
K
11 0) = sina Dx (7.23)
. (111 22J
30
y _
70
sina -3
K
22
sinct
154
ae
K
aey
K
x _ 12 (7t --r2") sina 3x
1 ( 121 = 2
21 ( 211] = -s-1171) - sina
= - cosa
G il)
ae cos a x cosa :11111a (12 1] = - sina ---(15 - 3x (7.23)
(22]
= -
ae K 22 = - cosa y a cosa sin 7/ = - cos sina (212] @x
(112]
= -
cosa
( L) = _
(221 ) = - co s a (211 ) = -
ae cosa X K12 cosa sina ---(23 = sina 3x
ae cosa y sina --(15 3x
_
K
cosa
21 sina
Now one can see, for example, that the first two physical Christoffel symbols are equal to zero if the corresponding coordinate lines are straight, the second two physical Christoffel symbols are equal to zero if the corresponding coordinate lines are parallel to each other and the last four physical Christoffel symbols are equal to zero if the coordinate lines are orthogonal.
Axisymmetrical Geometry In the case of axisymmetrical coordinate frame (in which g
13
= g
31
= 0,
2 g 33 = r ) the additional physical metric tensor components are:
5 (1
g
3) =
(13)
= 0
'
5(31)
,
(31) g = 0
=
'
,
=
(33) g
(7.24)
1
=1
(7.25)
'I" 155
Since the determinant of the metric tensor is now:
g = deg) =
g11 gll
g 12
°
g21
g 22
°
0
0
r
= gl1g22
r2
Sin2a
(7.26)
2
the elementary control volume is given by:
dxl dx 2 dx 3 = dOdX2)r sina
dV =
for dx
3
(7.27)
= de = 1, where 0 is the polar angle of the cylindrical polar
coordinate system. Thus the cell volume is given by: 1 V P =
dV
1
= f
1 dx1dx2
1
= f f 1/4TT )/6-
r sin a dx i dx2 (7.28)
or approximately:
_ 0) 0 V - (Sx d x r sina)p
where Sx
P'
(7.29)
for example, is given by equation (7.15). The cell-face areas
are now, for example:
dA
e
= (Sx r)
e (7.30)
dA
= (&r)
n
where Sx
e
and
1) (5)
are given by equations of the form (7.16).
156 The additional (n"..ro) Physical Christoffel symbols for an axisymmetrical system are:
1 (313) = j
•3r
3)15 cosa 9r r sin 2 a (-
cosa
3r OY 3x (7.31)
ar = lr--M ( 13) 3 ax
3 ( 23] =
—725 9r
Dx
7.6 Calculation of the Basic Geometrical Quantities From the definition of the metric tensor components (2.21) it follows that:
a‘,1 2
a 2 2 (7.32)
g ll
(-LT) ax
ax
2 2 1 2 4. Dy \ ( ay g 22 =x-a72-1 ax2
cosa =
g12
(7.33)
2 2 1 33/ ay 9y 4. ay 75 —72-) - -75 -70 ' — ax ax ax ax
(7.34)
Further from equation (7.11): 1 ay2 sina =
g_
-- 0) ax aX
ay l ay2 3x ax
(7.35)
157
The rates of change of the angles x
2
e
x
and 6 with respect to x
1
and
Y
are obtained in the manner described below, taking as an example:
2,,(1) tan@ -
33/
X
1
" x(I5
(7.36)
ay /3x
Taking the derivative of this expression with respect to x
cos@ x
1
and using:
-40) 3x
(7.37)
--
one gets:
2 2 1 x _ 9 y 3y —115(1) 2
2 1
ae
3x
(30)
30
a y
3y
2
2
(7.38)
(ax ) 2 3x
and analogously:
3e
2 2 ay 1 2 2 1 xa y 3y a y a 2) = lT a a72 1) —a7Ti 3 03x(2) 31)
36
2 1 2 Y aY _ (1) (2) —(2)
(7.39)
2 1 Y aY (1) (2) —TO
(7.40)
ay' 32y2 —125 ---A-2 i Oc —6-7 —125
(7.41)
Y _
-75 Dx
ae y
3x
a
3x 3x 3x
_ a 2y 1
2
a
ax ax 3x
a 2
(3x ) ax
(ax ) ax
All derivatives on the right hand sides of the equations (7.32) to (7.41) can easily be calculated from the coordinate transformation formulae (7.6). As a result, the expressions for all the geometrical quantities featuring in the finite-difference equations of Chapter 6 can be obtained in terms of the Cartesian coordinates of the nine points shown
158
in fig. 7.3 that define a control volume.
7.7 Cell-Related Quantities In this section some specific aspects of the calculation of geometrical quantities for the scalar and velocity cells will be presented. All basic geometrical quantities, as well as the spatial interpolation factors f
1P
and f
2P
defined by equations (6.17), are calculated for the
scalar cell (and, therefore, denoted here by the subscript P) by using the relations (7.32) to (7.41) and (7.6) and setting x l = x2 = 0 (see fig. 7.3). The geometrical quantities relating to the locations where they are not defined,such as the scalar cell faces, are linearly interpolated using the interpolation factors (6.17), for example:
(2) (2) (2) Ae = 6xe = (1 - f lp ) dx p + f lp dxE
except for the volumes of the velocity cells and the arc lengths dx
(7.42)
(1) (1) , (Sx , w e
0 (2) which may be obtained exactly, thus: (Sx and a s' n
1 V = — (V + V E ,1 2 • P e
(7.43)
(I) 1 (1) (1). dx e = A P E = 2- (dx p + (5)( E )
(7.44)
It is important to mention that the interpolations of the form (7.42), when applied in regions where the interpolated quantity actually exhibits maxima or minima, may 'smear' these extrema. This is most noticeable in the case of coordinate line curvature, which can sometimes have very sharp peaks whose smearing may introduce serious errors.
159
7.8 Closure In this chapter the geometrical aspects of the present solution method have been discussed. A simple grid generation procedure has been described which produces a smooth grid with a prespecified distribution, thus allowing high concentration of the grid lines only in the regions where it is required. However, the departure from orthogonality and the alignement of the grid lines with -flow streamlines are sometimes difficult to control. A local coordinate transformation has been used for calculation of all the required geometrical quantities for the case of both plane and axisymmetrical geometries. The particularities arising from the staggered variable arrangement have also been outlined.
160
CHAPTER 8 ASSESSMENT OF THE METHOD
8.1
Introduction The purpose of this chapter is to examine the capabilities of the
solution method described in the previous chapters in respect of accuracy, cost and reliability. In order to fullfil this l a set of simple steady flow test cases is assembled. Each test case is designed to examine a particular feature of the method, as independently as possible from all other features. In Section 7.2 inviscid test cases are examined; then Section 7.3 deals with laminar flows; and finally in Section 7.4 the additional equations for turbulent flow calculations are tested. The calculation results are compared with exact solutions where available, and the errors are defined and evaluated by the following formulae:
4-i) exact
v(i) Eu -
U
ref (i = 1, 2)
EP -
where U
ref
P - Pexact 1 2 7 Puref
is a reference fluid velocity.
Where an exact solution does not exist, the computed results are compared either with experimental results or with previous numerical calculations obtained by other methods.
(8.1)
161
8.2
Inviscid Flows If inviscid flows are considered, then the governing equations
reduce to a balance between inertia and pressure forces. The former are expressed, in general coordinates, via the terms representing convection, centrifugal and Coriolis forces U lk ]
(where, it should
be recalled, physical Christoffel symbols represent curvature of the coordinate lines); while the latter consist of normal and cross pressure gradient terms. Thus, although very simple, these test cases allow assessment of the following features of the method: the treatment of the cross-derivative pressure terms that arise from grid non-orthogonality, and of the 'curvature' terms which appear in the semi-strong formulation of governing equations used here.
Uniform Flow in Parallel Straight Channel This test case is designed to assess the treatment of cross pressure gradient terms. The flow geometry and the grid employed (an 8 x 6 CV*, deliberately arranged to be non-uniformly spaced) are shown in fig. 8.1. Since the grid lines are straight, all curvature terms are equal to zero and only convective and pressure terms remain. By decreasing the angle a between grid lines from the orthogonal (a = 90 0 ) value the magnitude of the cross pressure gradient terms increases. Starting from the same initial 'guessed' fields, the angle between grid lines is changed from 90° to 45
0
• The calculations converge
in all cases. However, in order to achieve a converged solution the under-relaxation factor for pressure-correction w p ,
had to be decreased
from 1 to 0.5, to account for the increased importance of the neglected cross-derivative pressure gradient terms. As a consequence the convergence
*
This notation denotes the numbers of main control volumes (equivalent to the number of interior grid nodes) in the x l and x' directions respectively: in this instance they are 8 and 6.
162
rate decreases and the execution time increases by about 60%. The accuracy of all calculations was better than 0.1%.
Fully Developed Flow in Constant Curvature Parallel Duct The momentum equations in this case reduce to balances between the centrifugal and pressure forces. The purpose of this test case is to investigate the discretization of the curvature terms (i.e. those containing the Christoffel symbols). For this purpose different grid arrangements are used: 8 x 6 CV grid non-uniform in the radial x 2 direction and both non-uniform and uniform in the circumferential x l direction. Calculations are done for different lengths of duct expressed in terms of the included angle 8 between inlet and outlet, but with the same number of control volumes and relative grid spacing. An example of the grid employed is shown in fig. 8.2. The geometrical properties of the grid are calculated by the method described in Chapter 7, which essentially means that the circular arcs are approximated by isoparametric second order parabolic arcs. For comparison, calculations are also done with the exact geometrical parameters which are simply those of a polar coordinate system. The results are presented in Table 8.1. (The errors E vA and
E
Kll
are defined analogously to E U , by
equation (8.1), with exact values taken as reference values). Apart from the curvature of
x
1
=
e coordinate lines
(K 11
= DOx/ax*CD
sina (]) the procedure of Chapter 7 calculates all other geometrical parameters quite accurately, especially in the case of 8 = 45
0
. Nonetheless,
inaccurate evaluation of curvature terms causes unacceptable errors. It can be noted that it also increases computing time, which can be explained to be a consequence of the more complicated, non-uniform velocity field which is generated. Also, noticeable is the larger number of iterations required for 8 = 180 0 , as compared with the corresponding ones for 8 = 45 larger grid aspect ratio.
0
, due to the
163
4-• VI c 0 0 S.- •L". a) T.0-
XIs...
,— c..1
•cr .---
,
CM
•:2-
ce)
f....
c...)
ta
0
CO
rn
al
0
• co
• 0.
al
rn
01
.—
E aj Z 4..) = ,.._,
X
,., e .......
co ,..-
C\ I
, ...Q
C
cr,
f
ra a. E 1.11
x ra
= E
,....•
Cr
0 ,--
LL.1
• • X
..--.
C)
S —..•
ta
r-
sz L.L.1
,•-n
•— ‘
0
'Cr
ch 1•0
01
C . 0• C.0
0
V
V
co
c)
cr)
0
•cr co
0 • co
co . co
a". 0
V
V
V
c..1
LA •
C)
CD
1".•
I-C) C)
C)
CD
• 0
0
0
CD
C co
LC) d.•
c) co ,--
1-r)
,--
cr.) C • CD
1--
„,.
•-•
.........
s>4 11:$ 40 S
.......•
t.0 • r--
0 C)
Co
CD
0
CD
al
co.
I--.
c)
• 0
V
>1 4-) ..- ----
a)
= •
s _
,-- tel CO
a)
4.4 0 = C CU 5.- S- 0 S'' rCS
4_
a. a CD
)
a) c) c) cl) = 4_) ...... so
c., 1—
0
co
Cr)
•::1'
..--. (11
co.
a) a.) sa) a) ci .......
C Co ,—
Lc) •cl-
co Lc) o cocl f•
,—
*1-
LI1 .S-
a) op g cu s
4-)
0 ,„ cu ms s_ c.D
-0 ., s_ (.0
-av xodd
432)(3
W.10j.!_un-uoN
• xcuddv
q.oex3
uoi.40a-IP Tx U. w-ooj.!.un
164
When 'exact' geometrical information is supplied very good results are obtained (q1ax < 0.01%, Egax 0.4%) for all grid arrangements and duct lengths despite the coarse (8 x 6 CV) grid employed . The results obtained with the approximate geometrical parameters are slightly better for
a
45° than for e.
180° because of the improved
approximation of the arc lengths in the former case. It can be concluded that the inaccurate evaluation of geometrical quantities may introduce significant errors. In this connection it is noteworthy that the calculation of the curvature of the coordinate lines (i.e. physical Christoffel symbols) tends to be much more sensitive and prone to errors than is the case with the other geometrical parameters (areas, volumes, angles), since for the evaluation of the physical Christoffel symbols the second derivatives of the computational c000rdinates with respect to Cartesian coordinates are involved. However, a grid arrangement that takes into account not only the properties of the flow but also the geometrical properties of the grid employed can greatly reduce this source of error, as is illustrated here by the better results obtained with the grid which is uniform in the xl direction. Another way of reducing this error is grid refinement. It is expected that in most cases the grid required for fine resolution of the flow gradients will also be fine enough to eliminate the errors in the curvature evaluation.
Irrotational Plane Stagnation Flow The aim of this test case, in which a uniform irrotational flow impinges normally onto a plane surface, is to examine simultaneously the effects of curvature and cross-pressure derivatives on the solution method. In order to do this the flow streamlines, known from the exact solution to
165
1 2 this problem to have locii given by y y = constant, are taken as one set of grid lines; while the other set consists of straight lines passing through a fixed point A shown in fig. 8.3. In this manner the flow is calculated in a coordinate system in which the angle between the grid lines varies from 52.7 0 to 127.3° and one set of grid lines has variable curvature. Calculations are done for the 10 x 9 CV grid shown in fig. 8.3 and also for a 20 x 18 CV grid. The geometrical quantities are calculated both in the approximate manner described in Chapter 7 and also using the exact coordinate transformation:
x1= 1 x
y
2 ,
x2 . yly2
(8.2)
However, in both cases these parameters are calculated and stored at the scalar grid node locations. Their evaluation at the other locations required by the method is performed by interpolation in the usual way. Special care has to be taken here with the geometrical quantities associated with the x
1
= constant coordinate line passing through the
stagnation point. These quantities (e.g. cell face area and x
2
= constant
line curvature) have their maximum values along this line, but since they are determined by interpolation from the neighbouring scalar nodes their values may be substantially underestimated. The results are presented in Table 8.2, where 'Approximate' and 'Exact' refer to the different geometrical calculations described above. The results show that there are substantial maximum errors E
max
in
both the velocity and pressure calculations, although the average errors are considerably lower and diminish in the expected way with grid refinement. It is believed that the appearance of the large local maximum errors can be attributed to one or more of the following: inaccuracies in the coordinate curvature evaluation; the extrapolation procedure used to obtain
166
TABLE 8.2 RESULTS OF CALCULATIONS FOR INVISCID STAGNATION FLOW
Geometrical Parameters
U av (%)
EP max (%)
No. of Iterations
(CV)
EU max (%)
Approximate
10 x 9 20 x 18
10.6 11.1
1.7 1.1
28.3 6.6
15 23
Exact
10 x 9 20 x 18
10.2 9.6
1.9 0.98
10.0 4.7
15 21
Grid
E
the boundary pressures; and errors in overall momentum conservation. Further research will be required to identify and remove these errors. The fact that one set of grid lines here nearly coincides with the flow streamlines enables the demonstration of the advantage of floworiented meshes with respect to reduction of numerical diffusion. In order to examine this feature, a step profile of a scalar property (0 is imposed at the inlet of the solution domain and the 45 transport equation is solved with diffusion suppressed. Results are obtained with a floworiented mesh defined by transformation (8.2) and with a uniform Cartesian mesh and are presented in fig. 8.4. One can see that the profile is very much diffused in the Cartesian grid case, while with the flow-aligned grid the numerical diffusion is comparatively very small.
8.3
Laminar Flows In this section some additional properties of the solution method are
tested related to the treatment of the viscous terms, particularly the crossderivative and curvature components.
167
Fully Developed Parallel Channel Flow The aim of this test case is to examine the influence of the crossderivative diffusion terms on the accuracy and stability of the numerical solution. The flow geometry and the grid employed is the same as for inviscid channel flow (fig. 8.1). Starting from the exact velocity profile at the inlet and zero initial fields the following accuracy is obtained for angles between grid lines of 90° and 45°:
E
(Degrees)
U max (%)
EP max (%)
90 45
1.7 2.0
2.8 3.4
a
These errors are evaluated using the exact solution as given below:
v(1) .
(2) V
a Tr
sina _2 „
uu -IF-
A kl -
sina X) 2
---8--
=0
(8.3)
121111[(x' P = P ref - --E70
_ X) ref
+ cosa(x2 - x2 )] ref
where U is the average velocity (U = t in /pB). Calculations are done for Reynolds number:
Re =
riBp 11
- 50
(8.4)
168
As was the case with inviscid channel flow, in order to obtain convergence the under-relaxation factor for pressure correction had to be 0
reduced from= 1 for a = 90 0 to w = 0.5 for a = 45 , causing a 30% P I Ps reduction in the convergence rate. Only slightly lower accuracy was obtained in the a = 45 the large non-orthogonality and the aspect ratio
(60/dX21)
0
max
case despite = 8. This
suggests that the cross-derivative diffusion terms do not significantly influence the accuracy of the method, at least in these simple circumstances.
Fully Developed Constant Curvature Duct Flow The feature of interest in calculations of this flow is the non-zero anisotropic part of the stress tensor with non-zero curvature terms. The geometry and uniform grid employed for the calculation is the same as that used for the f3 = 45° inviscid curved duct flow (fig. 8.2). The results of calculations are compared with the exact solution:
(1)
V
1 4 -,
2 A2 ,\ X = ;2- + Bx 2 + p ax (21nx - 1 )
(8.5)
42)
where:
,
0
169
A -
1 2 2 - R. Rout in
1 + 41.1
2 2 2 2 1nR. - R ut in Rout) + (Rout - Rin)] in in o
3
PT 3x
—
[2(R.
2 2 R R B . 1 3p out in 1 out in ri . 3x1 R 2 2 2 - R. i in out in R
(8.6)
21X(Rout - Rin) IPT 3x
2 R2 R out in in 2 2 R - R. out in
2 2 - R. in out 42Rout
R
Rin
The following errors are obtained when the geometrical quantities are calculated using procedure described in Chapter 7:
EUax = 4.5% • E P = 7.3% m ' max
If, however, the exact geometrical information is supplied, these errors are reduced to:
E U = 2.8% ; E P = 4.5% max max
The increase in solution error in comparison with the inviscid curved duct flow may be attributed to the more rapid variations in pressure and velocity in this case, which the coarse grid employed (8 x 6 CV) was unable to resolve accurately.
Jeffery-Hamel Flow The flow between two infinite flat diverging or converging plates, known as 'Jeffery-Hamel' flow, possesses one of the few known exact solutions of the full Navier-Stokes equations (Schlichting, 1968). In particular, as opposed to the previous test cases relating to channel flows, the governing
170
equations for Jeffery-Hamel flow have non-trivial (non-vanishing) convective terms, thus allowing examination of the simultaneous effects of inertial, pressure and viscous forces. With respect to the notation given in fig. 8.5 the exact solution, first obtained by Jeffery (1915) and, independently, by Hamel (1916) may be expressed in the following form:
e . ( 3) 1/2 4 .
dF
(8.7)
Re, - F)(e 2 - F)(e 3 - 9]-1/2
where:
F = F(e) -
(1) v r
(8.8)
Here v is the kinematic viscosity and the roots of the polynomial e
e
2
and e
3
are to be determined from the boundary conditions. The
velocity distribution may then be written in the following (self-similar) form for:
(a)
Converging Flow
j
2 dn (me,k) sn me,k 2 sn (me.,k) dn (me,k)
,
u o
(8.9)
where operators 'sn' and 'dn' stand for Jacobian elliptic functions (see, for example, Korn and Korn, 1961), U 0 = U 0 (r) is the velocity along the
o:
axial streamline F 1 +
2
o (8.10)
=
171
and the elliptic modulus k is a solution of the transcendental equation: 2 F0 (1 - 2k )
2
sn (me,k) -
Fn 9 2k 2 [3k 2 - 3 + u (k' -l)]
(8.11)
where:
F
(b)
0
. F(0)
(8.12)
Diverging Flow
(1) v _ i IT- o
2 sn
(me,k)
(8.13)
2
sn (me,k)
where: Fo 2 m
1 + 7
(8.14) - -7-121- (
and k is a solution of:
2 sn (me,k) -
2 21
+ k
2 3k (1 + -.-) lo
(8.15)
In both converging and diverging flow cases the pressure is calculated from:
1) 2( 0 P r
(8.16)
172
where the wall pressure:
2 / PV (e i e 2 + e 2 e 3 + e 3e 1 ) pw = --2-
(8.17)
and:
2 2 e 2 [m (1 + k ) 1
lj i
2 e 2 [m (1 - 2k 2 ) - lj 2
(8.18)
2 2 e 2 [m (k - 2) - 1] 3
Although this solution has been known and studied since 1915 few numerically calculated velocity profiles appear in literature that can be used for testing numerical methods. The most probable reasons are difficulties in numerical evaluation of the exact solution, which involves solving transcendental equations (8.11) or/and (8.15) which contain the Jacobian elliptic functions. The velocity profiles presented by Millsaps and Polhausen (1953), obtained for B
0 5 , have almost exclusively been
used to illustrate typical solutions (Schlichting, 1968; Batchelor, 1967) or to test numerical methods (Liu, 1976). More recently the calculations for other angles B have appeared (Marshall, 1979) and have been used for testing performances of finite-element numerical methods (Gartling and Nickell, 1977; Hughset al, 1978). For the purpose of the present study the transcendental equations (8.11) and (8.15) are solved by a modified regule-falsi method (Conte and de Boor, 1972) and the Jacobian elliptic functions are evaluated by employing the subroutine of Umstatter (1975) to yield results for a wider range of angles (
0 5 - 45 ) between the plates'and Reynolds numbers
173
(Re = 10 - 3000). An extract from these calculations is given in Appendix 4. The predictions of the computational procedure are compared with the analytical solution for different Reynolds numbers defined as:-
8Uor Re
= i3F
0 =
(8.19)
Various grid arrangements are also employed, as shown in fig. 8.6. For the symmetric grids the calculations were limited to the half flow domain and a uniform 10 x 10 CV grid was used (fig. 8.6a and b), while for asymmetric grids (y
0) a 10 x 20 CV grid was employed (fig. 8.6c).
For the calculation of geometrical properties of the grids the following (exact) coordinate transformations (rather than the local coordinate transformation described in Chapter 7) were used:
(a)
Orthogonal grid (polar coordinates, fig. 8.6a)
y y
(b)
1 2
= x
1
cosx
2 (8.20)
1 . 2 = x sinx
Non-orthogonal grids (figs. 8.6b and c)
y
1
= X
1 cosy cosx
2
cos(y - x )
y
2
2 1 cosy sinx = X 2 cos(y - x )
(8.21)
The flow and grid characteristics, accuracy and required number of iterations for the calculations performed are summarized in Table 8.3. It should be noted that reasonably good accuracy for relatively coarse (10x1OCV)
174
cn a 4- 0
0 ..--1
X ra 0- E w
.
01 .--
-1-
S.0• ms Z a) 4-) .-1
N.
t.0 .-
01
•zr
N.
al al
c%)
.—
a*
8-•
CV
h. co
a
."44
c.r)
,
0 x --, -a•Z
r—
= E ...--..• Li.)
7 L.) C)
-0
c0
r-0
GI0
MS
7 C..) 0
C) II
s_. C.0
CV
0)
0
7 C.--) 0
0
f.'
•••
CT)
7 L..) 0
0
II
X
X
X
CD
CD
CD
CD
.
1..
.•nnn
f
0
I-
II
II
•
X
I
1
4o cu o_ >1 I--
li
.Y
cu =
1
•
iii
1
•
CV I•••n
CV CT)
(7) 01
01
01
at
C)
C)
0
0 co t
r... , ri
co Cr) Ct• 1
..--cn
ea
a) cu san a) CI .....
14
co
I
0 C:) 0 n-•
c) L.0
175
(.11 C
4- 0 0 ••Pr-• • co 0 1Z CL1 4-)
r C%.1
d• C\I
r
0
1.0 1-.. 0
0
r
0
r
r
Cr)
Cy, r.
CO
r
r
r
Cn
r c..1
ksa
.4-
r r
CV
e-8
Cr)
X cl. (CS .,,e E LAJ
rs.
X ••-•• co 2rE.
=La-IE n--.
1:5 ••SCD
0 rX 0 r
S-
rcf 1--
0
0 0
X 0 r
CO.
C\ I
II
II
X 0 ,—
r
7 C...)
l—
X 0 r-
0
0
4-
X if) CnJ
V
0 a) Ci. >1 1--
I •
-1C
CU CC
?
I
k.0 0 r....
C \J 0'1
o
0
f•••• r r-
iC) Cr) •Ch
cc)
..--. 0 a) a.) scn cu cm —...
Lo
0
Cs..1
II
!
03 r
co.
C.) 0
• •
0
7
C.) 0
I—
r.
•
7
7
7
LC)
col
II
176
o 0 4— 0 0
r•-• CV
4-)
•
(6
Osa.) C) .P In1
c..) CV
X La.i
X---••••
=
al
L.LI
0 ._
nck 03
Lc)
,—
0
Cr)
7 C..) 0 0 .--- II X >0
7 L.) 0 ,— X 0
,••
l'•
C)
Cn
7 C.) 0 ,-X 0
S.as r• 0 0.
7 C...)
0
cu cc
ca
CU CU
scm cs) co ,-....
co c:,
c) ,--
c) Lc)
c) c•-)
II )••• 0
c)
VI
ca
X ›-
CV
----.
0 CV
ii
1.
a— 1---
-0 CI) 0 C •t4.) C 0 C-)
c), CV
0")
Y.,
-le
c•-)
C\ I
,...
>)
(.0
01
1.0
E ...—..•
-0 •rC.CD
CV CO
vr
03
in
to_ 63 E Z.ca
CV CV
177
0 C 4- 0 0 4-)
X CL 4:5 E ?A LLI
X—. =
rCS '. .R
t.0
C\ I
01
P..
Cr)
k.0
C\I
Cr)
Cr)
CO
en
Lc)
r,-)
r---
c.,
LO CO
E n-•
L.L.I
CD II
X
CO C\ I
T••n
›. T.-)
0 r
SCC
in
in
C) CO
C...)
-0 • 1-
cr)
CV
C\ I
• TO 0 SZ Q) .1.-) r
›. C-)
L.) 0
0 II
0 r-X
X
>-
0 r
0
0 II
0 X
....
0 C\ I
3
0 r 11.
i
All
40
if
II
CL) CL
>) I-
Yll
-14
01 0
ul CO
it 1 i
ct
C\i
C\I fn 0
LO CO 0 0 0 CV
a) CC
..--. CA 0) CD Cil SCT) a) cl ........
0 CI"
1I i i .1
0 II ?..-
178
grids is obtained: qiax = 0.18 to 3.73% and Eg ax = 0.31 to 9.4%. The J exceptions are the cases of 8 = 5°, Re = - 436 (E rinax = 9.39%) and
a.
30°,
Re = 200 (E ax = 7.57 to 8.55%) where the 10 x 10 CV grid was insufficiently fine to resolve the particularly high velocity gradients in those cases. This is shown by the fact that finer 20 x 20 CV grid in the last case reduces the
from 3.6 to 3.2%. ax from 7.57 to 2.82% and EP m max
EU
The tests with
y = 13,
where for 8 = 30 0 the departure from
orthogonality of the grid lines varies from 0 0 to 60°, show that these calculations are completely insensitive to the variation of the angle between the grid lines, implying that in this case the treatment of the cross-derivative terms is adequate.
Flow in a Curved Diffuser* The feature which distinguishes this flow from the ones previously calculated is the occurrence of flow separation and recirculation. The flow geometry and the prescribed boundary conditions are presented in fig. 8.7. The shape of the upper channel wall is defined by a hyperbolic tangent function given, in Cartesian coordinates y
2 y
w
1
= 1 +
2-
tanh 2 -
1 2-
30 1 tanh (2 - r-- y ) "ref
1 ,
y
2
by:
(8.22)
The calculations are performed for the following three combinations of geometrical and flow parameters:
Case 1
L
= 10 ref 1 , 1 1 'ref ' max -
Re = 10
* This flow was taken as a test case for the 5th Meeting of the IAHR Working Group on Refined Modelling of Flows, held in Rome in June, 1982, the proceedings of which are proposed to be published in the near future.
179
L
Case 2
ref
= 100
1 1 1 'max = 7 -ref Re = 100
L
Case 3
ref
y
Max
= 10 = 2.3 L
ref
Re = 100
where, for the notation in fig. 8.7:
a
(8.23)
Re = — v
and
U
is the bulk velocity (U = itin/pB). The grid arrangement employed is shown in fig. 8.8: it is uniform
in the x l direction and non-uniform, with concentration near the wall in the x 2 direction. The geometrical parameters of the grid are obtained using the exact coordinate transformation:
1 1 x =y
2
2 y ; x =-2-
(8.24)
Yw
where y
2 is given by equation (8.22). w
Grid refinement tests were performed for Case 1 by comparing the wall vorticities for grids from 10 x 10 to 40 x 40 CV. Apart from the 10 x 10 CV grid all others produce identical results to within plotting accuracy. Therefore, the 20 x 20 CV grid was considered adequate for this case (fig. 8.8). It was also used for the Cases 2 and 3 due to insufficient time to perform grid tests for these. The convergence characteristics of the calculation is examined by
180
plotting the minimum and outlet values of the wall vorticity versus the cumulative number of iterations performed. Plots for the Case 1 (fig. 8.9) show that the value of the convergence criterion X = 10
-3
(1) (21 (* = v , v
p1)
IP
employed, with the total momentum and mass inflow as the normalization factors, is satisfactory. In fact, the outlet value is still slightly changing but it does not have any significant influence on the rest of the field since the flow is parabolic there. Plots of streamlines and pressure field for the Case 1 (Re = 10) are presented in fig. 8.10, while fig. 8.11 shows the streamline pattern for the Case 3 (Re = 100) with its considerably extended recirculation region. The present results are compared in Table 8.4 with those obtained by some other numerical methods, whose main characteristics are also summarized. Plots of the wall vorticity obtained by the present method and some others for which data were available are presented in fig. 8.12. All these results are obtained with a 20 x 20 mesh except for the calculations of Cliffe and Jackson which are done on a 60 x 30 finite-element grid (equivalent to 120 x 60 CV). The poorer agreement between the results for Case 1 near inlet is due to a discontinuity of the wall vorticity at the inlet, caused by an inappropriate specification of the inlet velocity profile by the proposers of the test case. This discontinuity is a function of the angle between the wall and the horizontal and is much less influential in Case 2 where this angle is smaller (0.6° as compared with 6° in Case 1). The slightly larger discrepancies in the results for Case 3 are probably due to the inadequacies of the grids employed by Porter and the present author (20 x 20 CV) which are not fine enough in this case.
▪
181
-c Nll N ••n ....
C.,
•n• 4.)
.n
p.a..
I =
co
04 .-•
*I 0 0
L 0 L0
r. LO Ill
,..t.3
0
0
0
a
E
CV .
CV ••n .
CV CV t.
S. O. 0 CU
I
0 CS. LCI ro• CV
rn CO
CV n-•• CV .
CV V0 .cr c.., %.0 %CS • • a o
CO , • a
U1 CV L
LO
o I
•
V0 LO •••• CV mo •n• • • 0 0 I I
san • a I
Lc, .
Ld 01
CV CV
LO CV LO
= CV 4.1 VI tO C..)
LaJ
J
•2 gi S .1.• .1 o. 4..) IV = E 0 Li =
LLI LLI . r. LU LLJ U.I
LI . CD
I. W c In
4=,
7 0
to C..)
j
C
nn E
S.J3 .
4.1 VI
C0 CU •-• '10 .0 C tO 0 •• C. S. 8 g
LLI
S. 0
..= 4
0 CC
cm .
•
o
ras. 0
a I
0
01
0 F...
0 I...
I
CV t.0
01 I...
LC" ••n
CV LO Col
V0 Co) Co)
01 1.0 scar
0
111 CV ••n •
CV X
0
CV CV XI" .
0
0
0
rn
1.0
VI
g 1.
0
I
Zt E •r 0 U tn 0Vcal) •L CU C ..1:1..
_ 0. 00 I.3 CO • 4-1 ..0 ..-4.) C S. .1.-. 0 IL ••-••
4-1 a.,
S.. CU LI S. 0 Ca.
0
p.•
a
h
I.
I.
II
0 I
0 I
.4-
>,4.,
LI S..
••••• 0 a..) an 0 1/1 CU el) S›, 0.
.-•
8
.
4.)
0
.
CS
I
0
I
›.)
4) 0 C./
Q
•
C ..
u.. -.-
E 4S.-.) RI 0 CU L. 44 VI
>, 9 4.1 S.. .0-• = UV) 0 an t.-CI OI 1. ›. (2.
01 4./ 4.I Sn •••n = u an 0 tli ro. C U CU S. 0. O.
1
4-4 CIJ C ..)
4
_ N CC) L s••n U.. LLI
-0 C 63
4, I1) C 4.) .42 °I.
C 43
.1 .• I..n
IL U.I
'0 C ID II -0 r.m. C
. to
•
0
C ...0
I Ill
a I
a I
a7 -.. S.
-0 0 .4.1 .0 2
CV LO .
a
al/0 ‘In 111 4- -X '-U 1-n to C..) '7
CU (0 C..1-1
to'4..) •-n 1- 0 RS 0. 0 RI
az
43
4„
C
La L. C C) ..n ..... LL. U.!
0 C4
a
VD 01
• 0
t-• (",) 0
a
.er
1. . 4. . 00 I I
0 I
•er 'Cr •
CO
C 0 .... 4..) >.
0 .
U4. C ••n• 7U LL. •In 4.1 e sc0 0 CU S.. LI _ VI
>,V 4.) S.. 4•• 7 U V) (1 01 o-n CU IV S,-
. ,..,
= S., _. 00 0 ›. C...)
......
_. . 00 >•• U • 4 .1 ..0 .r• J CS. .... CD Li. s.... 4./
CU C .)
4
9,
..n S. = - 4.1 . ----.... . 4IL 4.• •... 0
CU 4 -1 • ”-• c C CU .r-• CZ U. .......
'0 C
to
•t)
C 10
>1
an _Id = N
C4CU CU 44.. to 0 .=
=
0
.4.01 0 4.) VI ,13
cc
= to
4.) 61. 1... 8."0.1 tO IS
zx
mlig
S. 0
3
4-1 C CU
VI
MI
o.S..
182
8.4
Turbulent Flows
Fully Developed Flow in a Parallel Channel This case concerns the turbulent flow through a straight channel._ The purpose is to assess solutions when the additional equations for turbulent flow, i.e. those for the kinetic energy of turbulence k and its dissipation rate e are solved. Of particular interest is the influence of the cross-derivative diffusion terms in these equations on the stability and accuracy of the method. The length L of the solution domain, which is confined between one wall and the symmetry plane was taken to be two hundred times the distance B between the walls. A uniform 10 x 10 CV grid was employed, and uniform inlet profiles were prescribed for all variables. The Reynolds number based on the bulk velocity U and the channel width Re =
a/v was set at
4 2.10 . The boundary conditions for the channel wall and plane of symmetry were specified as described in Section 5.6. The results are compared in fig. 8.13 with the fine-grid finitedifference solution of Young (1975) and with the finite-volume solution of Antonopoulos et al (1976) obtained with coarse 8 x 8 CV grid. In these plots Y is the distance from the wall and u = )/'r /p is the friction velocity. w The results obtained with an orthogonal Cartesian grid (i.e. a = 90°) are in very good agreement with the calculations of Antonopoulos et al (1976) and are reasonably close to Young's (1975) fine-grid solution. There are, however, some differences between the results for a = 90 0 and
a
= 60 0 which
indicates that the cross-derivative diffusion terms may affect the accuracy of the method in the case of strong departure from orthogonality and/or (I)
(2)
large grid aspect ratio (in this case dx /dx = 400).
183
Fully Developed Pipe Flow This test case allows further testing of the solution of the turbulent flow equations in circumstances where the additional terms for an axi-symmetrical flow situation are active. The length of the solution domain is taken as two hundred pipe diameters (L
200 D) and the grid and boundary conditions are identical
to those for the previous case. The results are compared with the experimental data of Laufer (1954) in fig. 8.14. The agreement is satisfactory, being comparable to that obtained by other methods employing the same turbulence model (e.g. Jones and Launder, 1973).
8.5
Closure The following conclusions may be drawn from the results of this
chapter about performance of the present method:
1.
The treatment of the cross-derivative pressure gradients described in Section 6.9 gives very satisfactory results. The procedure is stable for the test range of angles between the grid lines
(a =
45
to 150 0 ). However, pressure under-relaxation is necessary to promote convergence if the departure from orthogonality is large (typically Au > 15°).
2.
The presence of the cross-derivative diffusion terms in the momentum equations does not have a significant effect on the accuracy of the calculations performed even for the large departure from orthogonality (Act =
60 0 in the case of Jeffery-Hamel flow). However, their combined
presence in the momentum and scalar transport equations (e.g. turbulent
184
kinetic energy) may affect the accuracy of the method. No influence on the stability of the method is noted of these terms and the possibly negative coefficients arising from them.
3.
The accuracy of the method depends significantly on the accuracy of the geometrical parameters supplied. The procedure for their calculation described in Chapter 7 gives always good results when arc lengths, volumes and angles are calculated, but it may be less accurate in respect of the curvature terms, since they represent the rate of change of angles and are, therefore, much more sensitive and prone to errors. This feature should be considered as a drawback of semi-strong forms of the equations, as compared with the strong formulations which do not contain any curvature terms. The interpolations necessary for the evaluation of geometrical quantities at all required locations can also introduce serious errors, especially in regions of non-monotonic variation of these parameters. In such cases special treatment is required. Also the alternative approach of storing geometrical quantities at the scalar cell faces rather than at the scalar cell centres should be considered.
4.
The overall accuracy and stability of the method may be considered as satisfactory.
Additional insight into the above aspects will be gained from the calculations of the more complex flows presented in the next chapter.
185
CHAPTER 9 APPLICATION OF THE METHOD
9.1
Introduction This chapter describes the application of the present method to more
complex flow situations, both geometrically and physically. Unfortunately, it appears that data for flows in really complex geometries either do not exist or are incomplete or unreliable. Therefore, a compromise is necessary, in this case by focussing on complex turbulent flows with well documented experimental results occurring in relatively simple geometrical configurations. These include several variants of the backward-facing step problem and a highly curved mixing layer formed by an impinging jet. These are dealt with in Sections 9.2 and 9.3 respectively. In addition, in Section 9.4 the method is applied to the practicallyimportant case of flow through tube banks, for which only a limited amount of experimental data is available. Finally, in Section 9.5 some calculations are presented of flow in the geometrically complex inlet port/valve assembly of a reciprocating internal combustion engine, for which no data for comparison have been found.
9.2
Backward-Facing Step Problems
Preliminary Remarks Internal flows with sudden changes in cross section are very interesting both from the theoretical and practical points of view. Within this class, two-dimensional (plane and axi-symmetrical) backwardfacing step flows have been investigated by many authors because of their
186
geometrical simplicity and wide engineering application (e.g. Abbott and Kline, 1962; Durst and Withlaw, 1971; Kim et al, 1980). Eaton and Johnston (1980) have evaluated the available data for steady subsonic turbulent flows over two-dimensional backward-facing steps in straight channels for their suitability as test cases for numerical methods. The data they selected have been taken as the experimental basis for one of the test cases (Case 0421, in the Conference terminology) at the 1980-81 AFOSR-HTTM Stanford Conference on Complex Turbulent Flows. In addition, two slightly more complex cases involving a variable-angled wall (Case 0422) and a turned flow passage (Case 0423) (fig. 9.1) were proposed as 'predictive' test cases for the same conference, for which the experimental data were not provided until after the predictions had been made. These three flows have been selected as test cases for the present method application and will be referred to as BFS1, BFS2 and BFS3, respectively*. It can be seen from fig. 9.1 that BFS1 can be easily calculated by a . simple Cartesian code, while the other two require more general boundary fitted coordinates. For the present predictions the turbulence model and wall functions described in Chapter 5 were used along with several modifications which attempt to take better into account effects of the streamline curvature and non-equilibrium flow in the near-wall region. These modifications are described below.
Modifications to Turbulence Model and Wall Functions Launder (1981) has proposed the following modifications to the 'standard' k-e turbulence model described in Sections 5.4 and 6.2 for the
The results of calculation have been submitted in part to the forementioned Conference and will appear in the proceedings.
187
use in the recirculating flow calculations:
1.
Replace the generation term G in the c equation by the following modified version:
dm, GM =
(i)
cm) Th) v
%in) g
(2/2 = li t {(Vm v ) + (Vo
6,e) Axsn)
2
V
7 "11‘
) 2
(1) + 2V(1) v
)
n)
+ 31
,o)
(2) v + (9.1)
r (v +
L
sin a
03
2 Av - -3- [Pk + 11(-75 Ax
2.
„ 03, 1 2 1 _ 4 23) + cos a, ( vo , v ( 2) 12) v -I
_
(1)
(23 Av
(1)
Av
(2) Av
-75)] (-7D + 125) Ax
Ax
A
Obtain the Prandtl number cr in equation (6.6) from the following expression:
C a eM
aE C
2
2
- C - C
1
(9.2)
1M
where:
C
1M
and c
min(C1,C1
c'
C
1
and C
(9.3)
2
are given in Table 5.1.
Modification 1 is intended to improve the predictions of turbulent viscocity in cases of streamline curvature, while 2 is aimed at producing more realistic length-scale behaviour in the near-wall region.
188
In the discussion below the 'standard' k-e model will be referred to as the 'STM' and the one with the above modifications as 'MTM'. Apart from the 'one-layer' 'standard' wall functions (SWF) described in Section 5.6, several new 'two-layer' models were used for the present calculations. They all assume the existence of 'viscous' and 'inertial' sublayers and one or both of the following variations of the shear stress T and turbulent kinetic energy k in the near wall region:
T=T
w
(9.4)
SY
0 4Y < Y
)
v
k =
(9.5) b + mY
Y
v
< Y < Y
e
where Y is the normal distance from the wall, s, b and in are constants and the subscripts v and e refer to the edge of the viscous sublayer and the outer edge of the inertial sublayer, respectively (see fig. 9.2). Various combinations of these approximations have been made by different authors, thus leading to the following new wall functions (NWF):
NWF1 (Chieng and Launder, 1980) * " 1/2
K puBkv T
wl 9,n
1/2 E*Y k B v
T w (U - U )
G
Bl
-
T (T - T
ev wef, w) Y 1/2 e pek v Ye 3/2
Yvl
k 1 2 /2 v , 1/2 1 4. x] (k3/2 - k 3/2 ) + 2b(k e - K 1 6 81 = Y e Re v ye L3 e v '
(9.6)
189
where:
b 3/2 £n X
k 1/2 - b 1/2 e k 1/2 + b 1/2 e k 1/2 - b 1/2 V k 1/2 + b 1/2 V
, if b > 0
k 1/2 k 1/2 3/2 (tan -1 e v -1 2(- b) - tan ) (- b) 1/2 (- b) 112
(9.7)
,
if b < 0
NWF2 (Johnson, 1981)
v 2 * YB " k 1/2 1/2,iVK + 2 Yv \ - sb K * (Due k—N— 1/2' 1/2 k + kV B Tw2 - E**Y k B v 2,n v
G
B2 . G Bl k 3/2
) + b'" 2 ) Yve (k 1/2 , 1 v 8 /n EB2 - -5e 7— k-C— + R---) + b3/2 112 4- b"2) t ev Y v (k e )+
+
4- [i (0e12 - q/12) + 2b(kle /2 _ klv/2)]
(9.8)
190
NWF3 (Launder, 1981)
T
w3
=
T
=
T
W1
U G
33
w
e
- U
v
(9.9)
--Ter--
3/2 Y. e 1 8 1 933 = --T-- ( r- 4- TRT- + r n f") e ZvZv k
The values of the constants and parameters in the equations (9.5) to (9.91 are specified or evaluated as follows:
K— * - K C 114 = 0.23 P
K*Re
*
v
E = exp _Ri;__ - 5.88
C
Z
Re
-3/4 = KC = 2.55 p
21.
1/2 1/2 k + b v 1 E ** - exp( 1/2 1/2 Re k + b v 8
(9.10) 1/2 1/9 K*Rev) k''' v
b
k - k B E ‘ m = 7137—:—yi
b = k B - mYB
Here k
v
is the solution of the following cubic equation:
k 3/2 -bk 1/2 - Re vm = 0 v 3 v
(9.11)
191
and:
(9.12)
It should be noted that in the most general approach of Johnson (NWF2, equations (9.8)) the position where T w = 0 does not generally coincide with the position U s = 0. This is a consequence of allowing in the analysis for the effects of pressure gradient, which are ignored in all the other wall functions. It should also be noted that NWF3 is obtained by simply setting s
0 in NWF2, which is equivalent to assuming zero
pressure gradient.
Standard Backward-Facing Step Flow (BFS1) This flow (fig. 9.1) has been selected for initial tests for grid dependence and assessment of the performance of the various combinations of turbulence model and wall function employed. The measured inlet conditions were not supplied at a single plane, so the streamwise velocity profiles are linearly interpolated onto the plane X/H = -1.333, at which the turbulence information is given. The kin distribution is estimated from the latter with the assumption that:
(V/(3)
while the e
E .
71
in -
in
2 If -, -(15)in
(v i( 1)2 ii 'n
LAY
(9.13)
distribution is obtained from:
2
(9.14)
= C k. p in v
AD
v
in
192
using the measured velocity and (v
,(1)
v
x profiles. Zero streamwise -a)in
gradients were imposed on all variables at the outlet plane. Grid dependence tests were run with the STM turbulence model and the SWF wall functions. These tests focussed primarily on the reattachment length, X R the results for which are presented in fig. 9.3. Although XR still shows some variation between the two finest grids, it is small and, therefore, in view of the demands on computing resources* the 27 x 28 CV result was taken as 'sensibly grid-independent' and all subsequent calculations were done with this grid arrangement (shown in fig. 9.4). For reasons of lack of time such systematic grid refinement tests have not been performed for the other two backward facing step flows: instead it was assumed that the form of grid selected above would be suitable for these as well. The calculated reattachment length for the different combinations of turbulence model and wall function are presented, together with the experimental value in Table 9.1. The two different X R /H values reported
TABLE 9.1 REATTACHMENT LENGTH FOR BFS1
Test No.
Turbulence Model
Wall Functions
1 2 3 4 5
STM STM MTM MTM MTM
SWF NWF2 SWF NWF2 NWF3
X /H R T
w
= 0
U
B
= 0
,
Measurements
*
5.6 6.3 5.8 6.7 6.1
5.6 5.8 5.8 6.1 6.1 7.0
The cost of the solution obtained with the 37 x 36 CV grid is approximately five times the cost of the 27 x 28 CV grid solution.
193
for the case NWF2 correspond to two alternative definitions, one being the usual definition of the position where the wall shear stress T w = 0 and the other corresponds to the position where the streamwise velocity component at the near-wall node U B = O. The following conclusions can be drawn from these results:
(i)
The standard model and wall functions underpredict X R by 20% (Test No. 1).
(ii)
Introduction of the MTM alone produces a slight improvement, giving X /H = 5.8 (Test No. 3). R
(iii) Introduction of the NWF2 alone produces a slightly better result than (ii), giving X R /H = 6.3 (Test No. 2).
(iv)
The use of MTM in combination with NWF2 and NWF3 increases XR/H by 19.6% and 8.9% respectively (Tests No. 4 and 5).
(v)
The highest X R/H are produced when NWF2 is used: however, the data for the U
B
= 0 locations show that this prolongation is
(unrealistically) confined to the region between the wall and the nearby node.
Overall, therefore, all combinations are seen to underpredict the reattachment length. The further details of the results presented below were obtained from the calculations using the MTM-NWF3 combination. The static pressure distributions along the step wall and opposite wall are presented in fig. 9.5. It can be seen that the general trend
194
is predicted well and that discrepancies are mainly due to the underprediction of the reattachment length, which causes the pressure recovery to begin too soon. The axial velocity profiles (fig. 9.6) show the kinds of discrepancies that would be expected from the discussion above, as do the axial Reynolds shear stresses in fig. 9.7. The maximum shear stress is replotted as a function of the distance from the reattachment in fig. 9.8. The severe overprediction in the region before reattachment is typical of the k-e model predictions and is chiefly responsible for the early reattachment. However, this sharp increase is followed by a similarly sharp decay, resulting in quite good predictions in the recovery region downstream of reattachment. This is also the reason for fairly good predictions of the shear stress profiles in the recovery region, as shown in fig. 9.7. Previous experiences with the k-c turbulence models as well as the evidence from the 1980-81 Stanford Conference confirms that the errors just described are almost certainly due to turbulence modelling. The comparison of the different near-wall treatments suggests that this part of the modelling is not the major source of error, so the blame must be attached to the k-e model itself. Indeed, other predictions submitted to the same Conference showed that when the Algebraic Stress Model (ASM) variant of , the-k-c model is employed, much more satisfactory agreement is obtained (see contribution of Launder et alb Finally, it should be noted that in the light of the above results similar errors can also be expected to a lesser or greater extent in the predictions of the other backward-facing step flow cases to be presented below.
195
Backward-Facing Step: Variable Opposite-Wall Angle (BFS2) This is one of the modifications of the backward-facing step flow in which the wall opposite the step is set at various angles, as shown in fig. 9.1. The inlet experimental data were specified at the plane X/H = -4. The conditions there were specified using the measured values of the boundary layer thickness (S, displacement thickness Sp drag coefficient c
f'
and U
o
there (the values of the first three being different for the
two inlet walls) and the following relations derived from the flat-plate boundary layer theory (Schlichting, 1968; Launder and Spalding, 1972) were used:
Y (1) (—). } = min{U v o' Uo (5 in in
k in
=
-1/2 2 dv(p 2 cp tin(7Hin
(9.15)
1) 2 dv( -1 E . = C k. 0. ---). II in dY in in
where:
(9.16) . = min(KY, 0.09 (5) tin
The outlet boundary conditions were the same as for BFS1. All predictions for this flow were obtained with the modified k-E model of turbulence (MTM) but all four wall functions were explored. The results of the reattachment length calculations together with the measured values are given in Table 9.2 and shown in fig. 9.9.
196
TABLE 9.2 REATTACHMENT LENGTH FOR BFS2
Test No.
Turbulence Model
Wall Functions
1
MTM
2
X /H R
o
-2°
0°
6
NWF2
4.9 (4.3)
5.3 (4.6)
6.9 (5.7)
MTM
NWF3
4.3
4.6
6.3
10.3
3
MTM
NWF1
4.3
4.7
6.5
10.9
4
MTM
SWF
4.2
4.5
5.8
8.6
5.8
6.3
8.2
10.2
Measurements
10°
- * (10.6)**
* Flow did not reattach according to this criterion. ** Values in brackets correspond to the position of U B = O.
An interesting observation can be made about these results. For angles up to 6° the reattachment length is seriously underpredicted but the trend is reasonably good. However, after 6 0 a steep increase in the predicted reattachment length is obtained. Moreover, when NWF2 is employed the wall shear stress does not change its sign inside the calculation domain (X/H < 17.58), although near wall velocity U B does, at X/H = 10.62. Close inspection of the implied flow pattern revealed an unrealistic, thin prolongation of the separation zone, which suggests that the NWF2 are inadequate in the presence of strong pressure gradients at least in their present form. It is clear from the foregoing considerations that more testing is required before the reasons can be found for the failure of the predictions to perform consistently for the angles longer than 6 0 . The results obtained with the MTM and NWF3 are examined further below. The plots of calculated streamlines, isobars and turbulence intensity and length scale profiles for the wall inclination angle B = 6°
197
are shown in fig. 9.10. The streamlines exhibit the expected behaviour downstream of the step and also show that, due to the high B/H ratio the bulk of the flow is relatively undisturbed by the step. The isobars plot shows that, apart from the low pressure region in the recirculation zone, an initially constant and then slightly diminishing adverse axial pressure gradient exists. There is also a cross-stream gradient with the higher pressure levels at the opposite-wall in the recirculation region and at the step-wall further downstream. The turbulence intensity and length scale profiles show the build-up of the turbulence activity in the shear and wall layers and a large potential core with negligible turbulence activity. Also noticeable is the gradual merging of the shear and wall layers in the initial region of the recirculation zone. For the intermediate angle B = 6°, which is the only case for which data were provided, comparisons with predictions are shown of the static pressure distributions (fig. 9.11); the mean velocity profiles (fig. 9.13); and the shear stress profiles (fig. 9.14). The agreement is similar to that for BFS1. The influence of the angle of inclination B on the pressure coefficient for the step-wall is illustrated in fig. 9.12, where there is seen to be qualitative agreement between the experimental and numerical results. The pressure at any location tends to increase as (3 is increased. When B < 0 the initial pressure rise is followed by decay due to wall shear and flow acceleration, whereas for B. > 0 there is a rise due to the dominant effect of deceleration. The small difference between the results for 6 0 and 10 0 may be due to the larger recirculation region in the latter case. It can be seen that the relative slope of the calculated curves in the pressure recovery region for two highest angles is comparatively small
198
so that they intersect with the curves for
-2 and 0 degrees. Inter-
section also occurs between the lines for $
6 and 10 degrees. Neither
of these features is exhibited by the data.
Backward-Facing Step: Turned Flow Passage (BFS3) In this case the duct has parallel walls, but they are aligned at an angle
a to the axis of the upstream duct as shown in fig. 9.1.
The inflow condition treatment here follows that of the previous case apart from being imposed at X/H
-3. The outflow boundary conditions
are the same as for BFS1 and BFS2. Predictions are done with the MTM and both NWF2 and NWF3. Again, in the case of NWF2 the reattachment length defined by
Tw = 0 and the location
where U B 0 differ and the latter agree approximately with the X
R /H values
for NWF3 (see Table 9.3). Also the field values for the two wall function treatments do not show appreciable difference.
TABLE 9.3 REATTACHMENT LENGTH FOR BFS3
X /H
R
Test No.
Turbulence Model
Wall Functions
00
5
1
MTM
NWF2
7.4 (6.7)
7.8 (6.9)
8.2 (7.4)
2
MTM
NWF3
6.7
7.1
7.5
7.8
8.7
9.1
9.5
9.7
Measurements
o
10 0
* Values in brackets correspond to the position of U BO .
15 o 8.5 (7.8)*
199
The comparison of the predicted (using MTM and NWF3) and measured reattachment lengths shown in fig.9.15 reveals that the trend is predicted quite well but X R is severely underpredicted by about two step-heights. This is to be compared with the underprediction of about one step height in the case of BFS1 and with two step heights in the case of BFS2 (for < 6°) which have different B/H ratios. Thus the agreement between predicted and measured reattachment lengths exhibits a non-uniform variation with B/H ratio. More light can be shed on this matter if fig. 9.16 is analysed on which the data of the forementioned Conference cases, the curve fitted by Durst and Tropea (1981) through a collection of data from numerous sources and the measurements of Durst and Tropea (1981) are presented, along with the present predictions. There is considerable disagreement amongst the experimental data, which complicates assessment of the predictions. In searching for an explanation several possibilities arise:
(i)
If Durst and Tropea data are correct, the k-E model underpredicts X
R
by about 2H for all values of the aspect ratio (H + B)/B
considered.
(ii)
If, however, the curve fitted to the collected data is correct, then the k-E model perform better for larger aspect ratios.
(iii) There is also possibility of some unknown factors in the particular experiments (e.g. three-dimensionality, outflow conditions, etc.) which are not accounted for in this correlation.
200
It is not possible on the basis of the available information to judge which, if any, of these explanations is correct. The predicted streamlines, streamwise velocity profiles, isobars and the turbulence intensity and length scale profiles for largest angle (B = 15°) are presented in fig. 9.17. The first two show that the flow away from the recirculation zone is relatively undisturbed and that the fully-developed condition has not been established for the length of the calculation domain (X/H < 25.5). The isobar plots show the build-up of the pressure at entry where the flow impinges on the outer wall of the bend and a low pressure region in the recirculation zone. Also noticeable is the decrease in the adverse pressure gradient as the flow moves towards the fully developed state. The turbulence intensity and length scale profiles show behaviour similar to the ones obtained in the case of BFS2 (fig. 9.10), apart from the fact that a potential core practically does not exist here, because of the smaller B/H ratio. The predictions of the static pressure distribution shown in figs. 9.18 and 9.19 are again consistent with the reattachment length underprediction. The trends as well as the influence of the turning angle 6. are, however, reproduced very well, as shown in fig. 9.20. The mean velocity profiles at the streamwise locations of the computed and measured reattachment points agree well with experiments (fig. 9.21), as was the case with BFS2 (fig. 9.13). In the downstream region the predicted profiles are not recovering as rapidly as the measurements, as occurred also with both BFS1 and BFS2 (figs 9.6 and 9.13).
201
9.3
Highly Curved Mixing Layer
Description of the Problem Curved shear layers are considered 'complex' turbulent flows since their turbulence structure exhibits special features resulting from streamline curvature. These are described by Bradshaw (1973) who also gives a review of the work on the subject. Apart from the less detailed data of Wyngaard et al (1968) the measurements of Castro (1973), reported also by Castro and Bradshaw (1976) (hereafter referred to as 'CB'),
the
only experimental data for free curved shear layers. In addition to their turbulence complexities these flows provide other tests for a calculation method notably the ability to follow the highly curved mixing layer without smearing it through numerical diffusion. They, therefore, provide a useful testing ground for the present calculation method. The configuration of the CB flow is shown in fig. 9.22. The flow can be thought of as half of a two-dimensional impinging jet with a potential core and with a wall or 'floor' replacing the plane of symmetry. Boundary layer separation in the corner between the 'floor' and the 'backplate l was suppressed by allowing a certain volume of air to escape through a slot in the corner. Observation showed that initially self-preserving shear layer eminating from the jet nozzle is perturbed by the strong curvature and subsequently returns to the self-preserving state of a classical plane mixing layer. The results of CB measurements have indicated, according to them, that many of the principles on which current turbulence models are based, including the gradient diffusion hypothesis, are not valid in the case of such complex turbulent flows. The measurements reported by CB were already used by Gibson and Rodi
202
(1981) (hereafter referred to as 'GR') who evaluated the performance of a calculation method that combines a Reynolds-stress closure model with a parabolic solution procedure. They adapted the Reynolds-stress model (RSM) of Launder et al (1975) so as to represent the effects of extra strain due to streamline curvature and succeeded in reproducing many of the curvature effects reported by CB. However, since they used a parabolic, 'marching', solution algorithm, the curvature of the mixing layer had to be supplied as an input to the calculations rather than be predicted. The same measurements have been taken as one of the test cases for the 1980-81 AFOSR-HTTM-Stanford Conference on Complex Turbulent Flows (Case 0331, in the Conference terminology).
Boundary Conditions For the purposes of the calculations, the uniform velocity profile U
ref
= 33 m/s and constant tunnel turbulence level of 0.1% were prescribed
at the nozzle outlet plane, as given by CB. The wall functions described in Section 5.6 are applied at the floor and backplate boundaries. At the free boundary defined by the outer edge of the shear layer the entrainment velocity was calculated from the assumption of constant total pressure, while the zero gradient condition was applied to the other velocity component. The kinetic energy of turbulence and its dissipation rate were set to zero there. For example, &t-the 'north' boundary:
(2) v B =
PB, (1),2 tot - PB - 7 kvB)
(9.17)
k
B
= e
B
= 0
203
The first of the above equations was also used to derive the corresponding expression for the velocity correction:
0
42) vB vB
(2) ay.
42) U Ap Tfii; B = IsfE
1
42) pvB
A
n
"Pli
(9.18)
At the outlet plane zero gradient conditions were applied along the intersecting grid lines. The bleed through the slot in the corner was taken to be 0.04 U
ref
3 m /s per unit span, as reported by CB.
Computational Grid An example of the mesh employed is shown in fig. 9.23b. It coincides with polar coordinate lines in the curved region and is nearly Cartesian upstream and further downstream. In the potential core of the curved portion the polar mesh is smoothly distorted in order to match the solid boundaries. Such a mesh is fairly well aligned with the flow streamlines because the origin of the polar coordinates is specified to be close to the centre of curvature of the streamlines. It also conveniently coincides with the way in which the CB data are presented. In fig. 9.23 the geometrical flexibility of the method is further demonstrated. Initial calculations were done with the grid shown in fig. 9.23a. However, the resulting predicted velocity field plot shown in the same figure reveals that in a large region next to the free boundary the calculations are unnecessary since the air there is practically undisturbed. Therefore, in further calculations this region is excluded from the calculations by redefining the grid as shown in fig. 9.23b.
Grid Refinement Before the comparisons were made between the calculations and experiments efforts were made to obtain results which are grid independent.
204
Extracts from the grid refinement tests are presented in fig. 9.24. Although the increments in the numbers of mesh lines between the various grids might look small, most of the added lines were inserted in the shear layer, so that the number of grid lines inside the shear layer (for s > 0.388 m ) was increased by about 9 in each case. The velocity profiles of fig. 9.24 coincide closely for the meshes 37 x 40 CV and 37 x 49 CV, but there is still a small difference between the shear stress profiles for
e
< 60° (s
0.388 m ) caused by inadequate
resolution of the initial portion of the mixing layer. This difference, however, becomes negligible further downstream. The results from the 37 x 49 CV grid are, therefore, considered sufficiently grid insensitive. For the 37 x 49 CV mesh 419 iterations were needed to obtain a converged solution
= 10
-3
;
= v 0) , v0, p'), requiring 1466 CP sec on
(A11)
the CDC6600 computer.
Presentation of Overall Characteristics In fig. 9.25 the calculated streamlines, velocity profiles, isobars and turbulence intensity
(A7 grref )
and length scale contours and profiles
(normalized by the nozzle height B; C lI k 3/2 /(EB)) are presented. Comparing fig. 9.23b and the streamlines of fig. 9.25 one can see fairly close correspondence between the streamlines and mesh-lines, as was mentioned earlier. The streamlines also show very well the direction of the entrainment from the ambient air. The pressure field shows the rapid build-up of pressure as the jet approaches the backplate, which causes the change of the shear layer direction. One can also see the straightening of the isobars and decrease of gradients further downstream as the flow approaches the self-preserving state of a plane mixing layer. The turbulence intensity predictions show the expected reduction of the turbulence intensity in the curved portion of the shear layer and its
205
recovery to the plane shear layer value after the removal of the extra 1) rate of strain %i /R
1
due to the curvature. As will be shown below, this
behaviour is in qualitative accord with the measurements. CB observed a decrease of the length scale in the region of maximum curvature. Such a trend is not, however, reproduced by the calculations, as indicated by the length scale calculations in fig. 9.25 and the centreline plot of fig. 9.33.
Comparison with Experiments The results of the calculations are now compared with the measurements of CB and with the predictions obtained by Rodi (to appear in the Proceedings of the forementioned Conference) with the same k-e model of turbulence as employed in the present study, but with a parabolic calculation procedure. In fig. 9.26 the velocity profiles are presented. The better agreement with the data for the maximum velocity in the calculations of Rodi is not surprising because the parabolic procedure calculates the edge velocity on the high velocity side by integrating a simplified cross-stream momentum equation across the layer, taking as an input the measured curvature of the shear layer. Otherwise, both calculation methods produce very similar velocity profiles. The profiles of the shear stress both in the curved region and in the recovery region further downstream are compared with experiments of CB and with the calculations of Rodi in fig. 9.27. The overprediction of the maximum shear stress and of the shear stress at the high velocity edge of the layer are due to inability of the k-e model (STM version) to take fully into account the stabilizing effect of curvature. The differences between the present predictions and those of Rodi can be attributed to the one or more of the following reasons. Although the turbulence model employed is the same in both cases the solution
206
methods are different and employ different input and boundary conditions. In particular, parabolic method requires specification of the curvature of the shear layer and the velocity at the inner edge of the shear layer from the experiments, whereas the present method predicts these properties. In addition numerical errors in one or both sets of calculations are not excluded. The turbulence intensity profiles are presented in fig. 9.28 and compared with experimental data of CB. It is surprising that predictions follow the measurements as closely as they do in the curved region. The model, however, fails badly (as expected) to predict the sharp increase in the turbulence intensity in the recovery region further downstream, as is also evident in fig. 9.30. Although the forementioned profiles give a good impression on the accuracy of the calculations, further insight into this and the flow phenomena related to the reduction of the turbulent activity in the curved region followed by the non-monotonic recovery can be gained by analysing the streamwise variation of the centreline or maximum values of the turbulent quantities. In fig. 9.29, the streamwise variation of the maximum velocity is presented. As indicated by the velocity profile plots the maximum velocity in the region of maximum curvature is severely underpredicted. This may be a turbulence model effect, but there is also the possibility of interpolation errors in the calculation of the geometrical quantities at the cell faces laying at the x l = constant coordinate line passing through the corner formed by the backplate and the floor, where the cell face area, sina and the curvature of the x
2
= constant coordinate line exhibit their
maximum values. (The latter possibility was recognized too late for further investigation in the present study). In fig. 9.30 the variation of maximum turbulence intensity is plotted and compared with both the experiments of CB and the calculations of GR,
207
obtained in the latter case with both RSM and k-E turbulence models. While the fall in the turbulence intensity is reproduced reasonably well by all predictions, they simulate the recovery much less satisfactorily, especially in the case of the present calculations. By contrast the predicted behaviour of the maximum shear stress (fig. 9.31) is inadequate in the region before and within the zone of maximum curvature, but it follows the measurements better in the recovery region. The GR calculations obtained with the k-E model show a similar behaviour, while their results for the RSM model (not shown*) agree well with the experiments in both the high curvature and recovery regions. In the fig. 9.32 the generation and the dissipation rates of the turbulent kinetic energy on the centreline are presented. Again the k-E model fails to predict the observed increase and overshoot in the recovery region. The RSM calculations of GR, also shown in fig. 9.32, do predict an overshoot, but an excessive one. • The variation of the length scale Z e = k 3/2 /E is plotted in fig. 9.33 where it is evident that the agreement with measurement is poor. One reason is that the only effect of curvature on the predicted values of the length scale is to slightly diminish its rate of increase with s whereas the data show a large reduction. It is interesting that GR obtained the same type of behaviour as the present predictions with their RSM turbulence model. Fig. 9.34 shows the predicted rate of entrainment. The reduction in turbulent activity in the curved region results in a slightly diminished rate of the entrainment there. The only available experimental data point is also presented in fig. 9.34 and it indicates that the quantitative agreement is also good.
These results are not shown in fig. 9.31 because they were presented in a different coordinate system from the one employed in this study.
208
9.4
Flow Across Tube Banks
Description of the Problem This section is concerned with the application of the present method to the prediction of cross flow around tube banks as occurs, for example, in heat exchangers. More details about this flow, a review of the theoretical and experimental work and calculations of laminar and turbulent flow for different geometrical characteristics of the tube bank can be found in Antonopoulos (1979). In this study the flow across the staggered tube bank illustrated in fig. 9.35 is calculated and comparisons were made with the experimental data of Neal and Hitchcock (1967) for which both fluid dynamics and heat transfer data are available. The calculations were done for the typical 'subchannel' or 'unit of symmetry' indicated in fig. 9.35, for which fully developed conditions are assumed. In this context the expresion 'fully developed' denotes a flow structure that repeats itself from one unit of symmetry to another in a symmetrical or anti-symmetrical (as in fig. 9.35) fashion. Such situation occurs at locations sufficiently remote from the influence of boundaries.
Boundary Conditions For the walls and the planes of symmetry of the solution domain (shown in fig. 9.36) the boundary conditions described in Section 5.6 were applied. In the fully developed case considered the flow repeats itself at the inflow and outflow boundaries in an anti-symmetrical manner. The 'explicit periodic' procedure of Antonopoulos (1979) was used here to link the variables at the inlet and outlet of the subchannel. This
209
procedure requires an additional row of cells at the outlet and an antisymmetric grid arrangement at the inlet and outlet of the solution domain, as illustrated in fig. 9.36. However, since in the Neal and Hitchcock (1967) experiments only one tube was heated, periodic boundary conditions do not apply to the energy equation. Therefore, a two step procedure was used to obtain the complete temperature field, after the flow field was calculated. (i) Firstly it was assumed that only the trailing tube is heated and the temperature solution was obtained, with the inlet temperature set at 0 °C. (ii) Then the outflow temperature distribution from the calculation of (i) was antisymmetrically mapped onto the inflow cross section and the calculations were done again with a heated leading tube and a non-heated trailing one, thus giving the temperature distribution for the rest of the field. In both cases zero gradient conditions were applied at the outlet and at the non-heated tube wall.
Grid Refinement The results of grid dependence calculations are presented in fig. 9.37 where the computed wall shear stress at the tube surface (which is a particularly sensitive quantity) is plotted for grids with different number of controle volumes. Although some differences still exist between the results for the two finest grids (mainly in the impingement region), the calculations for the 39 x 20 CV grid were considered sufficiently accurate and are presented in further detail below. 5 These are obtained for the Reynolds number of 1.4 . 10 , based on the average velocity If through the minimum flow passage area in the bank and the tube diameter D, which corresponds to the Reynolds number of the Neal and Hitchcock (1967) experiments.
210
Presentation of Results The results of the calculations are presented in fig. 9.38 in the form of field plots. The velocity field (fig. 9.38a)shows flow deceleration in the impingement region as well as in the wake of the tube, where the velocity is nearly zero. However, no separation was predicted, although a weak recirculation was reported in the latter region by Neal and Hitchcock (1967) with the separation point at about 150 0 . The reason for the near-suppression of recirculation is the close pitch of the tubes. The most probable cause of the failure to predict recirculation is inadequacy of the wall functions in the presence of steep pressure gradients and high turbulence levels. The isobar plot (fig. 9.38b) shows a build-up of pressure as the flow approaches the impingement zone on the front side of the tube and a pressure decrease as the flow accelerates through the minimum gap. The wake region is one of relatively uniform pressure. The turbulence intensity
(A70) contours (fig. 9.38c) show that the
most of the turbulence is generated in the impingement region and to a lesser extent where the flow enters the minimum gap. The density of the contours near the tube walls indicates very sharp gradients there. Also shown is the length scale (C u k 3/2 /(eD))distribution (fig.9.38d) which displays the expected regular increase from the wall, with maximum levels forward of the impingement location and near the centre of the minimum gap. The temperature calculations revealed that in the present application negative coefficients did arise in the energy equation (the possibility of this occuring was discussed in Section 6.5 and in Appendix 3). When the standard interpolation formula of the form (6.16) were used to determine the cell corner temperature values T
ne
T
se'
T
nw
and T negative temperasw
211
ture was generated at the impingement cell and then convected downstream. However, when the alternative interpolation scheme described in Appendix 3 was employed, this unrealistic behaviour did not occur. The fluid temperature variation around the heated tube calculated by these two approaches, shown in fig. 9.39, demonstrates the advantage of the second approach in this case.
Comparison with Experiment Comparisons of the predictions with the data of Neal and Hitchcock (1967) are shown in figs. 9.40 to 9.42. The velocity variation adjacent to the tube wall (fig. 9.40) is predicted relatively well (the maximum error is about 14.5%). The local turbulence intensity (i.e. Vkill, where U is the local mean velocity) is, however, less well predicted (fig. 9.41), the tendency being for the calculated level to be too high everywhere but near the rear of the tube. The possible reasons for this are numerous and could lie in the k-E model, the wall functions or both. The temperature predictions, shown in fig. 9.42, are also less accurate than those of velocity, indicating that perhaps the temperature wall functions are also somewhat in error, although the gross trends are reproduced.
Computational Details It should be mentioned that some numerical instabilities have been n experienced when a grid was employed with large aspect ratio 41)/(SZ2)i the region of substantial departure from orthogonality near the impingement zone, i.e. when condition (6.34) was strongly violated. However, a small reduction of the aspect ratio stabilized the calculations. Convergence in this case was relatively slow due, it is believed, to the explicit procedure used to apply the periodic boundary conditions. In the
212
case of the 39 x 28 CV grid the program required 1566 iterations to converge (A = 10 -3 ;
9.5
=
V2),
p') and 605 CP sec on a CDC 7600 computer.
Flow in an Inlet Port/Valve Assembly
It is well known that the fluid dynamics aspects of the design of inlet port/valve assemblies are of great importance to the performance of reciprocating internal combustion engines. In this section calculations are presented of steady turbulent flow through an idealized axi-symmetric inlet port/valve assembly. The aim of these calculations is to further demonstrate the ability of the present method to predict geometrically complicated and practically important flows. A series of calculations were done for different valve lifts with a relatively coarse grid (23 x 13 CV) which is probably not fine enough to resolve all details of the flow. However, since no experimental data are available for comparison, no efforts were mada to refine the calculations. Uniform profiles were prescribed for all dependent variables at the inlet cross section. At the wall and at the outlet the boundary conditions described in Section 5.6 were imposed. The results for two valve lifts are presented in figs. 9.43 and 9.44. Fig. 9.43 shows the grid, velocity field and turbulence intensity and length scale profiles for the maximum valve lift, while fig. 9.44 provides similar information for a smaller valve lift. The results are qualitatively as expected. A recirculation zone exists in the wake of the valve guide for all valve lifts, while the other recirculation region near the base of the valve, noticable in the case of the maximum valve lift, is suppressed at the smaller lifts by the acceleration of the flow caused by the narrowing of the passage. The pressure field presented in fig. 9.44 shows also the qualitatively
213
correct behaviour: high pressure occurs in the impingement regions at the upper wall bend and at the valve base; and nearly one-dimensional variations occur in the channel-like passage at the outlet. Also expected is the build-up of the turbulence near the walls, especially in the region of high shear around the upper protruding corner and impingement region near the baseof the value (fig. 9.43). Generally the same pattern of the turbulence intensity is obtained for both valve lifts, as can be seen by comparing the contour plots in fig. 9.44 with the corresponding profiles in fig. 9.43. On the basis of the above results it may be speculated that the present method could be a useful tool for optimization of the shape of an inlet port/valve assembly from the hydrodynamics point of view. However, a three-dimensional version of this method would be necessary for the generally three-dimensional geometries exhibited by practical arrangements.
9.6
Closure In this chapter applications have been described of the present method
to complex turbulent flows. The calculations have been compared with experimental data and with predictions obtained by other methods, where available. The results suggest that the method is capable of predicting complex turbulent flows within the accuracy of the turbulence model employed, although there is scope for some further improvement in the accuracy of the method itself. The detailed findings may be summarized as follows:
1. The modifications to the 'standard' k-E model as well as the new wallfunction treatments have produced only slight improvement to the predictions of the backward-facing step cases. It has also been found
214
that one of the new wall functions (NWF2) produces anomalous results when applied to flows with strong adverse pressure gradients.
2.
The predictions of the highly curved mixing layer have demonstrated the advantage of a method which allows the mesh to be adjusted to suit the complex flow field, thereby emphasizing the importance of mesh flexibility even when the solution domain is of simple form.
3.
Although some previous calculations of the curved mixing layer have been done with parabolic solution procedures, the present elliptic calculations should be considered as more representative of predictive methods since, unlike the parabolic ones, the only input information the elliptic method requires is the boundary conditions.
4.
The present results for the curved layer show that once again the k-e model was unable to reproduce accurately the effects of streamline curvature.
5.
The tube-bank case has illustrated the capability of the method to treat problems in which both the flow and the geometry are complex.
6.
The predictions for the tube bank were qualitatively correct, but exhibited appreciable errors. Again, it is belived that the turbulence modelling is at fault, but it is not possible to determine whether the errors lie in the k-e model, the wall functions (which must in this instance cope with very thin boundary layer subject to strong pressure gradients) or both.
215
7.
Application of the present method to a practical example of a flow through an inlet port/valve assembly has produced physically plausible results.
8.
There is an urgent need for experimental data for flows in complex geometries.
216
CHAPTER 10 SUMMARY AND CONCLUSIONS
10.1 Achievements and Findings
Methodology A numerical method has been developed for calculation of subsonic separated or elliptic flows in regions of arbitrary complexity. The novelties of the research are as follows:
(a)
Analytical The differential equations governing such flows have been formulated
in a general non-orthogonal non-Eulerian coordinate frame which can accommodate many different situations including problems with moving boundaries and/or an interactive adaptive grid solution procedure. The semi-strong form of these equations and contravariant physical vector and tensor components have been selected for the use in this study. This has been done after a detailed examination of many forms of the equations and vector and tensor components. A mathematical apparatus has been developed for these components that enables compact tensor notation and easy manipulation of the equations to be achieved.
(b)
Numerical The finite volume method has been used to discretize the governing
differential equations in a general way. The stability problems associated with the presence of negative coefficients arising from cross derivatives have been overcome by (justifiably) neglecting these terms in the pressure correction equation and relegating their contributions to the 'source' term in the case of all other equations. This has resulted in five-diagonal coefficient matrices with unconditionally
217
positive coefficients thus allowing use of an existing solution procedure originally developed for the equations in orthogonal coordinates. However, the possibility of negative coefficients arising from the cross-derivative diffusion terms can still, in the case of an extremely unfavourable combination of the grid and flow parameters, impair or even prevent convergence of the procedure and/or produce physically unrealistic solutions. The latter problems of possibly improperly bounded solution have been partially solved by introducing a scheme for differencing cross-derivative diffusion terms, which reduces the possibility of negative coefficients. Another possible source of numerical errors is the presence of the 'curvature terms' in the momentum equations which require evaluation of the second derivatives of the computational coordinates with respect to Cartesian coordinates. In order to minimize these errors a simple smooth grid generation and geometrical quantities calculation procedure have been developed.
Testing The important features of the method have been tested by running a number of simple test cases and comparing the results of calculations with exact analytical solutions, where available, or with the numerical results obtained with other methods or experimental data. These calculations have shown that the present method is stable, accurate and economical in most cases, although some improvements with respect to all above properties can be made. These will be discussed in the next section.
218
Applications The method has also been applied to the calculation of some complex turbulent flows for which reliable experimental data exists. The grid independent (or nearly so) results of calculations have been compared with the data and the agreement can generally be considered as good, although there are certain cases for which some problems still remain. The discrepancies obtained can be attributed mainly to well-known deficiencies of the form of k-c turbulence model employed. Finally, predictions have been made of some practical flow situations. For these very little or no experimental data has been found. However, the predictions obtained have shown the expected qualitative behaviour.
10.2 Improvements and Extensions
Differencing Scheme The non-orthogonality of the coordinate system employed leads to nine-diagonal coefficient matrices with the possibility of negative coefficients. This should be compared with the five-diagonal unconditionally-positive coefficient matrix in the case of an orthogonal coordinate frame. In Appendix 3 an attempt is presented to construct a differencing scheme that would produce unconditionally positive coefficients while maintaining conservation of fluxes. Only partial success has been achieved and further work is required. Further scope for improvement lies in the elimination of numerical diffusion. Alignment of the grid lines with the flow can greatly reduce false diffusion, as shown in Section 8.2. However, this alignment is often difficult to achieve since it generally requires an adaptive grid
219
generation procedure which itself involves many additional problems as discussed further below. An alternative way of alleviating false diffusion is incorporation of a more accurate scheme such as the 'bounded skew upwind' (BSU) scheme of Gosman and Lai (1982), which is the only nine-point computational molecule scheme that has unconditionally positive coefficients (for the case of an orthogonal grid).
Solution Method Recently several methods for handling the coupling between momentum and continuity equations have appeared, which are more efficient than the SIMPLE method employed in this study. Two of them will be shortly described further below. The SIMPLER (SIMPLE Revised) method of Patankar (1980) is an improved version of the SIMPLE algorithm. It retains the pressure correction equation but only for correcting velocities and solves an additional 'pressure equation' to obtain the pressure field. Although it requires more time per iteration than SIMPLE, it has been found to give faster overall convergence (Patankar, 1980). The PISO method of Issa (1982a) is based on the splitting of the process of solution into a series of predictor and corrector steps, resulting in simplified equations whose solution can be obtained by standard techniques (direct or iterative such as the line iteration procedure used in this study). Although designed for unsteady flow calculations, where its efficiency is fully realized (it has been found to be up to 8 times faster than the SIMPLE algorithm), the PISO method is also applicable to steady-state flows (with appropriate modifications, Issa, 1982b) where the time steps replace (outer) iterations. Since it is capable of coping with very large time steps reductions of CPU time of 60 to 80% have been obtained. This method is still undergoing testing and development, but the
220
results obtained so far suggest that its modification for the use in the present method would considerably reduce the execution time. However, some difficulties arising from the cross-derivative pressure gradients may be expected. Another means for reducing the CPU time would be a faster solver for discretized (linearized) equations, e.g. fully-implicit method of Stone (1968).
Grid Generation'
(a)
Non-orthogonality Although satisfactory results have been obtained in most applications
by employing the simple grid generation procedure described in Chapter 7,
the strong departure from orthogonality that sometimes cannot be avoided may produce physically unrealistic results and instability of the solution procedure, as was the case with flow across rod bundles. Therefore, a grid generation procedure that has more control over the grid non-orthogonality would improve both accuracy and stability of the method. Amongst the many such methods which now exist (see Proceedings of the Symposium on the Numerical Generation of Curvilinear Coordinate Systems and use in the Numerical Solution of Partial Differential Equations, Elsevier, 1982) the method of Smith and Wiegel modified by Kowalski (1980) seems to be the most suitable one. It is in fact a direct extension of the procedure used in this study, the only difference being that the straight lines joining the points on the two opposite boundaries used here are replaced by the third order parabolas. This allows the orthogonality of the grid lines at the boundaries to be controlled. It requires, however, some measures to be undertaken (the 'ramping-functions' used by Kowalski, 1980, for example) for prevention from the possible grid lines cross-over.
221
(b)
Smoothness The smooth grid is a necessary condition for an accurate calculation
of the curvature of coordinate lines (i.e. physical Christoffel symbols) required by the present method. However, since these are calculated at the centres of the main control volumes only, the interpolation employed for providing them at the required locations may introduce significant errors especially in the case of non-monotonic variation of curvature. In this case the grid should be refined in the regions of the extremum values of curvature in order to reduce the interpolation error, or the curvature (and sometimes some other geometrical parameters) should be calculated at all required locations so that interpolation is not required. One should be reminded here that the strong conservation form of the equations, obtained by the reference to a fixed basis, requires only angles between x i and y i grid lines and not their rate of change (i.e. physical Christoffel symbols). Thus, if such a form of equations is employed, a smooth grid is not required and the problems related to this grid property do not exist.
(c)
Adaptive grids A further extension that would improve the accuracy of the method
would be the use of an adaptive grid generation procedure in which the grid would be dynamically adjusted during the fluid dynamics calculation towards an 'optimum' distribution. Ideally, this distribution should be governed by such properties as: inclination of mesh to the local flow direction, steepness of the gradients of the dependent variables, rate of change of these gradients, etc. There are, however, some difficulties in specifying the precise criteria for an adaptive grid procedure. For example: (i) the alignment of the grid lines with the flow streamlines in the regions of recirculation, (ii) it needs to be established whether
222
it is best to concentrate grid lines in the regions of high first or high second derivatives of the dependent variable, (iii) it must also be decided which dependent variable gradients should be used as a criterion for grid positioning, since the locations of high gradients of different dependent variables will not generally coincide, etc. Ideally the grid should be changed according to the above criteria during the iteration procedure. This brings in the problems related to a moving coordinate frame. However, an adaptive grid generation procedure does not have to be dynamic. Instead a fixed grid could be used to obtain partially or fully converged solutions and then be adjusted according to the results, in an intermittent fashion. In such a manner the calculations could be done on a steady Eulerian frame and the problems related to the moving grids would be avoided.
Non-Eulerian (Moving) Coordinate Frame In Chapter 5 the differential equations have been derived that govern fluid flow in an arbitrary non-Eulerian coordinate system. Some preliminary numerical tests performed by the present author have indicated that some problems related to the space conservation law and spatial and temporal differencing have to be resolved before a reliable computer code with an arbitrary moving grid is obtained.
Three-Dimensional Flows All the equations of Chapter 5 are valid for three-dimensional flows and the application of the solution method of Chapter 6 to such problems is formally straight forward. In practice, however, substantial increases in computer time and storage will arise mainly from the considerable geometrical complexity of the semi-strong form of equations. One way around this difficulty would be to make use of particular
223
geometrical features of the problem which allow a reduction in the number of geometrical parameters involved as in for example the studies of DemirdtiC (1982) and Raithby and Elliott (1982). Another alternative is to use a strong conservation form of the equations (e.g. with vectors and tensors expressed in terms of locally fixed or Cartesian basis) which do not require any curvature information. One must, however, bear in mind that in this case problems related to the discontinuity of fluxes will have to be overcome.
Turbulence Modelling As has been shown in Chapter 9, the k-E turbulence model is incapable of predicting to high accuracy some of the important features of complex turbulent flows (e.g. reattachment length in the case of backward facing step flows, effects of curvature in the case of curved shear layer, etc.). There are a number of more refined turbulence models, e.g. the algebraic stress model of Rodi (1976), the Reynolds stress model of Launder et al (1975) or the large eddy simulation model of Kwak et al (1975). They are all based on less restricted assumptions than the k-e model and are, therefore, capable of predicting better some flow phenomena. However, their employment involves considerable increase in computer resources and a balance should be made between the requirements on the results of calculation and the availability_of computer resources and costs of calculations. It seems that at present the k-E model is still an optimum option for most of the engineering calculations.
224
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237
NOMENCLATURE
Arbitrary tensor
A ij , A.., A
Contravariant, covariant and mixed tensor components j
1, 2, 3)
Contravariant, covariant and mixed physical tensor
A.., )
components (i, j = 1, 2, 3) A.
DV
VI
Contravariant, covariant and mixed dual physical tensor components (i, j
1, 2, 3)
Cell—face area (i = e, w, n, s)
Ai
Area or constant
A
Arbitrary vector
a i , a. , ad)
Contravariant and covariant vector components (i
1,2,3)
Contravariant and covariant physical vector components (i = 1, 2, 3)
a , iContravariant and covariant dual physical vector components (i = 1, 2, 3) a
Coefficients of discretized equations (K = P, E, W, N, S,
K
NE, NW, SE, SW) Arbitrary tensor (components analogous to those of
r)
Channel width or constant Finite-difference coefficient or constant of the modified wall functions Arbitrary tensor (componerit analogous to those of Convection coefficient (i
ci Cl
, C 2' C 3
k-E turbulence model constants
C , C
u
Constant
cf
e, w, n, s)
Drag coefficient
r)
238
Deformation (rate of strain) tensor (components analogous to those of r) Diffusion coefficient (i . e, w, n, s)
Di D
03 1, 02,
Eigenvalues of deformation tensor r
di
Coefficients in pressure correction equation (i = e,w,n, ․ )
ds
Arc length differential
Ei
Cross-derivative diffusion coefficient (i . e, w, n, s) Turbulence model constant or error Natural and dual natural base vectors (i = 1, 2, 3) Normalized natural and its reciprocal base vector (i = 1, 2, 3)
4ti] 4e , eti3
Normalized dual natural and its reciprocal base vector (i = 1, 2, 3) Coefficient in pressure-correction equation (i . e,w,n, ․ )
e.
1
e
l'
e
2'
e
Parameters of Jeffery-Hamel flow
3
Integral flux or dimensionless velocity of Jeffery-Hamel
F
flow Covariant and contravariant fixed base vectors (i . 1,2,3) Spatial interpolation factor (i . 1, 2; M = P, E, W, N, S)
fiM f
Scalar field or distribution function
G
Generation rate of turbulent energy
g 1. j ., g
ij
,
i
• g,)
Covariant, contravariant and mixed metric tensor components (i, j = 1, 2, 3)
gd;) , 4,1))
Covariant, contravariant and mixed physical metric tensor components (i, j = 1, 2, 3)
,
nri i]
11
Covariant, contravariant and mixed dual physical metric
.3
tensor components (i, j = 1, 2, 3)
g
Metric tensor determinant
H
Step height
239
r -).
Identity or unit tensor -)1 1
r 1 1 1 )k i ijiki
Cartesian base vectors (i = 1, 2, 3) Christoffel symbols 0, j, k = 1, 2, 3) Physical Christoffel symbols (i, j, k = 1, 2, 3)
J
Jacobian
k
Turbulent kinetic energy or elliptic modulus of Jacobian elliptic functions
L
Length Length scale
t
Mass flow rate
m
Constant of modified wall functions or parameter of Jacobian elliptic functions
N IP
Normalization factor Op = p', 0, k, ...)
N
Number of iterations
P
Isotropic part of stress tensor
Pe
Peclet number
p
Pressure
il
Flux vector (components analogous to those of I)
R
Residual of discretized equation Op = p', vd), k, ... )
4) Re
Reynolds number
R
Radius
r
Radial coordinate of cylindrical-polar coordinate system
S IP s 11) s
Source term in differential equation Op = m, V1), k, ...) Source term in discretized equation Op = m, 0, k, ...) Constant of modified wall functions or distance along center-line
..._T
Stress tensor (components analogous to those of r)
t
Time
U
Reference or near wall velocity
u i , i i)
Cartesian velocity components (i = 1, 2, 3)
240
V
Volume Velocity vector (components analogous to those of -a)
X x
Longitudinal coordinate i
General coordinates (i
1, 2, 3)
Distance from the boundary y
i
Cartesian coordinates (i
1, 2, 3)
Greek Symbols c. ,• al
Contravariant and covariant components of vector coordinate system (i
in
1, 2, 3)
ai
Spatial weighting factor (i = e, w, n, s)
a
Angle between x l and x 2 coordinate lines Spatial weighting factor ( i = e,w,n,s ) Coefficients in strong conservation form of momentum equations Angle Exchange coefficient ( 0
k,E,T,... )
Covariant and contravariant metric tensor components in 0 coordinate system ( i,j = 1,2,3 ) Coefficients in strong conservation form of momentum equations Determinant of metric tensor in
0
coordinate
system or angle A
Difference
7j , Vth
Covariant differentiation operators ( j
A
A
Differential operators ( j = 1,2,3 ) Ax j ' AXJ)
1,2,3 )
241
(s i •,
6i i
d (5
Kronecker's delta ( i,j = 1,2,3 ) Increment or boundary layer thickness Displacement thickness
1
3 —7(j) Dx
Differential operator ( j = 1,2,3 )
41 E i , E
Natural und dual natural base vectors in Ei coordinate syatem ( i = 1,2,3 )
4d) ccif c
Nonnalized natural and its reciprocal base vectors in E i coordinate system ( i = 1,2,3 ) Dissipation rate of turbulent energy Vorticity
o
Circumferential coordinate of cylindrical-polar coordinate system
ex , e y .
2 1 2 1 Angles between x , y and x , y coordinate lines Curvature (i, j
1, 2)
Temporal weighting factor or turbulence model constant Small number used for convergence tests Op
X 11) 1.1
Small number used for convergence tests Op
1-1
Dynamic viscosity Kinematic viscosity General coordinates (i = 1, 2, 3) Density
E
Summation
a
Prandtl/Schmidt number
di) ij
Anisotropic part of stress tensor (i, j = 1, 2, 3) Stress tensor components in Cartesian coordinates (i, j
1, 2, 3)
242
Time Arbitrary scalar property of fluid flow A dependent variable or stream function Under-relaxation factor
1)
Op = p', n1 ,
k,
Subscripts a
Atmospheric
av
Average Boundary Outer edge of inertial sublayer
e, w, n, s
Cell faces
in
Inlet or inner Modified
in
Mass or molecular
max
Maximum
min
Minimum
ne, nw, se, sw
Cell corners
out
Outlet or outer
P, E, W, N, S
1
Compass notation for grid points NE, NW, SE, SW j Reattachment ref
Reference Constant temperature Turbulent
tot
Total
V1, V2, V3
Ordered vertices of the main control volumes on x i = constant line Velocity or edge of viscous sublayer Wall
243
4)
Scalar property
IP
Variable lp
0
Centreline
1, 2, 3
Coordinate directions
Superscripts n
New time level
o
Old time level
p
Pressure
T
Transposed
U
Velocity
+
Dimensionless
*
Prevailing or modified
**
Modified Fluctuating or correction
_ „ 1, 2, 3
(Overbar) time or space averaged (Overbar) referring to moving coordinate frame Coordinate directions
244
APPENDIX 1 TRANSFORMATION OF EQUATIONS FROM CARTESIAN INTO GENERAL COORDINATES
A novel procedure for transforming the equations of fluid flow written in Cartesian tensor notation, into general coordinates is presented. This is similar to the procedure of Pope (1978), for curvilinear orthogonal coordinates. The familiar Cartesian tensor notation form of the equations which represent the transport of mass, momentum and an arbitrary scalar is:
ap
, j =s 4. a ) m 3t —73- lP v 3y
3
Tt-
i
(pv ) +
ij i j (pv v - T ) . - a P. + si v 3y 3 3313 3
a a it- (0) + —-, (ov j ay]
( Al .1)
r 21,-) . s 4) ay J
(0
where the stress tensor has been divided into its isotropic and anisotropic parts:
3 m P = p + -?- (pk +p ---z) 3 Dym
(A1.2) T
ij
=
,3vi pk--r aY j
30
4 ---..)
Vi
These equations can now be transformed into general coordinates by the following two-step procedure.
245
Step 1 The components of the metric tensor which reduce to the Kronecker delta in Cartesian coordinates (g . = d .. g lj ' i j
ij
= d
ij
) are used for
raising and lowering indices (see equations (2.27) and (2.28)) in the above equations, in order to bring them in accord with the Einstein summation convention:
ap at
vj
a
—P T(
) = sm
ay
i
9
_
( P vivi - T ij )
a- ( Pv )
+S Ot
a
„ k po
a ,
- -T
cov - r
(A1.3)
V
S
.5jm
ay°
i
6 i i aP
axm
where:
P = p +
2
(pk +
a ym @x (A1.4)
ijlm T
=i-(
av j „jm avi
—)
3 m 9x
Step 2 Introduce the following transformations:
246
Scalars
±tax(J)
3Yj
Vectors d) a ==> a ci)
i
3a Da ci) => V . a (J)
(m)
+a
3y 3 3x•-"
Mj (A1.5)
ay J
(J) 70° - A° AX
Tensors Ali
dii)
6-
g4 )
Ani) ( i Am j) AxmjJ (j)
—> V(j).A(ii)
under which equations (A1.3) and (A1.4) assume their general coordinates form (see equation (5.7)):
+ L
( p (j) at —75-) v
, 41) Tt-
A )
s
d) d) (Pv v -
dj)
T ) + (
I
(po
A
_ 4ii)
3P .1. sch
d)
3x
Axx
a
(0)9,d) _
(0‘)3)_ridjm)4
s q)
V
(A1.6)
247
where (see equation (5.5)):
)
2 P = p + 7 (pk + p Ax
(A1.7) u nCi
j)=
rfj97 ni
4j)
(n)
i) \
The above prOcedure can be used for the transformation of any expression from Cartesians into general coordinates. For example, the generation rate of turbulent kinetic energy, defined in Cartesian notation by:
G
, ij 't
2 t,k
w Dv
, Dv
7 `"
Dy
n
m
Dv
m'
Dy
n
(A1.8)
after the first step becomes:
G = 6. Ti m Dv i .1 lm t 3y-
2
7
(pk +
Dv n Dym ayn Dvm
(A1.9)
and after the second step assumes its final form in general coordinates:
G
=
L)
I
T rrn
(1) V.V (j)
2
(pk +
AP) AP ""(n) Ax
In the special case of curvilinear orthogonal coordinates (g ii = g ii = 1/h1 and g ij = g ij = 0 for i
j) the transformations (A1.5) and
the equations (A1.6) reduce to those of Pope (1978).
(A1.10)
248
APPENDIX 2 CROSS-DERIVATIVE DIFFUSION FLUX TERMS
Let
4.
be the diffusion flux vector at a cell face (east, for example)
of the two-dimensional control volume shown in fig. A2.1. According to can be resolved into its covariant physical
equations (4.3) and (3.39) components:
+
q = Ckte = 1'4)
aq).
4 4d) -
r
.
4)
4e1)
+ r(I)
42)
(A2.1)
e
or its contravariant physical components:
r adm)
-6-4 1 1-
=
=
r (gal)
4 ar)
acp +
4
Wr)
(1)
a)
,(22) a(p
))
goaat,) @)Fr
42)
(A2.2)
r(2 CI)
(i)
ax
Thus, the integral diffusion flux through the east cell face of the control volume of fig. A2.1 is given by:
=
A
1 ( . • IA =
f -4 • e 7
)
dA
r
(
P
( ,
4. a2) 34) g11) ax —7) ' g
si na A
(A2.3)
A
where A is the cell face area, 01 e
4
a is the angle between the grid lines and
. 41) sina e . One can see in the case of non-orthogonal coordinates that two
gradients are always involved in the expression for the diffusion flux, while in the case of an orthogonal coordinate frame (where g and cf
t
= qth =
t 340X
),
(12).
cosa/sin 2 a = 0
only gradient, normal to the cell face appears.
249
APPENDIX 3 AN ALTERNATIVE DISCRETIZATION PRACTICE FOR DIFFUSION FLUXES
Starting from the Laplace operator in general coordinates which expresses the diffusion fluxes for variable V in two dimensions:
A L(*) = --r[) Ax
(11)
A (221. -72') (22r Ax
DV
(g r th
9x
+ A ( ,,a2,
api
73
3V
I (21), , kg It1) )4
(A3.1)
and employing the control-volume approach of Section 6.3, one gets (see equations (6.36) and (6.37)):
L(p)
De0Pp
1PE )
pw(Vw - VI))
DrI(Vp
4i N)
ps(Vs - VI)) (A3.2)
Ew (11)nw V sw )
1Pse )
Ee (Vne
E n (lPne 4)nw )
E s (lP se
1Psw)
whereD.andE.are given by equations (6.22) and they, according to equation (7.9), may be written in the following form, for example:
,J11) De =
t
since A \ =
6x
1
,
6x --Oe 6x (A3.3)
Ee
(g02)1,
sina
A \ =
tCOSa r ` -sTriTt `vie
If instead of equation (6.16) for V ne the following interpolation formula (Bramble and Hubbard, 1964; O'Carroll, 1976)'is used (see fig. 6.2),
250
which involves only two 'nearest' neighbours of point ne:
I. (41,1 4- lit) Iljne
, for cosa > 0 (A3.4)
= 1
t.7
(4) 4) ) ' 1)
for cosa < 0
NE
ipnw ,
and similar expressions for
L(4))
ip
se'
*
sw'
the Laplace operator (A3.2) becomes:
alAp
(A3.5)
aOK
where the summation is now over the six nodes surrounding point P (K = E, W, N, S, NW, SE, for cosa > 0; there is no contribution from the 'far' nodes NE and SW, see fig. 6.2) and the finite difference coefficients are for cosa > 0 given by:
a
1
E =
a
W
-
N
aS a
ei-Cric
NW
,
-
r Sx _ 1 r ( COSa r Sx( Die L'sina
1 sina
r
1 slna
r
I
1 Sx 7711/w - Sx
(13
6x 1
(1)
1
r ( cosa
r
1-T1r —Fc-t
'VW '
1 [( cosa sina
dx
L`TiliCi (Sx
2 _ "SE -
1 r ( COSa L sina z
1
r ( cosa
7L
`TiTi3
)
(cosa
r n
_ 1 r ( cosa r )
r 6x
=
=
(cosa
n
r
cosa sina
r
L
ipis
)
w
(Cosa r s
1/4T1flTi IpJei
r
'
r 411)1s-1
cosa ) sina * e
( cosa ‘TiTCci
r
(A3.6)
251
Analogous equations can be obtained for cosa < O. One can see from (A3.6) that (since cosa > 0) the corner coefficients a mw and as E are always positive, while the principal coefficients a E , aw, a
N
and a are positive if the criterion of the following form is satisfied:
(1)
'cased max(
6x
2-) ,
dx
CD dx
(A3.7)
1 dx
It should be noted that this condition is approximate in the sense that
r ,p , sina and cosa are generally not the same at the east and north
cell faces, for example. To illustrate the advantage of this approach as compared with the nine-point formula, the solution of:
L(T) = 0
(A3.8)
will be considered, where T is the temperature for a single cell in a rectilinear non-orthogonal coordinate system with the uniform mesh and uniform conductivity (i.e. dx
(1)
CD = d x , cosa
const, sina = const,
rT
const) for the situation shown in fig. A3.1. In the case of the nine-point approximation (i.e. when ip ne is approximated by (6.16)) the solution of the equation (A3.8) is:
T p =
1
(T E + Tw + TN + T s ) -
Cosa
( T
NE
- T
NW
- T
SE
+ T ) SW'
(A3.9)
or for the given data:
T p = 100(1 + cos a ) > 100
for cosa > 0
If instead, the seven-point approximation (A3.5) is used, then:
(A3.10)
252
T
1
P . 4 - 2cosa [(1 - cosa) (T
E + Tw + TN + Ts ) +
+ cosa(Tmw + TsE)]
(A3.11)
or for the situation in fig. A3.1:
T . 100 P
(A3.12)
It can be seen that the temperature of the point P in the case (A3.10) is higher than the maximum surrounding temperature which is unrealistic since there are no sources of internal energy, while the second approach (equation (A3.12)) produces a realistic solution irrespective of the angle a, as indicated in fig. A3.1 by possible isotherms. The reason for this is that the former approach takes into account the 'remote' nodes NE and SW.
253
APPENDIX 4 PARAMETERS OF JEFFERY-HAMEL FLOW
f3.
k
m
p* 1) w
100
0.712822
19.50845
-4.34557 . 105
500
0.911295
39.56781
-8.42411
1500
0.776534 0.919498 0.991702
73.22592 68.24612 65.82928
-8.74744 . 10' -7.54342 . 107 -7.38957 . 10'
50
0.715261
7.98968
-1.22229 . 10'
500
0.776621 0.919804 0.991714
24.41897 22.75582 21.95316
-1.08180 . 106 -9.32366 . 105 -9.13987 . 105
1500
0.817412 0.903284 0.968698
41.44793 39.72587 38.45060
-9.18769 . 106 -8.46591 . 106 -8.23764 . 106
3000
0.833164 0.894392 0.950120 0.987228 0.995923
58.15963 56.42487 54.87959 53.87131 53.63762
-3.60501 -3.40566 -3.30970 -3.28558 -3.28412
Re
. 106
50
15°
1) 1:)
1
(e1e2 + e 2 e 3 + eel)
. . . . .
107 10' 107 107 10'
254
13
30 0
k
m
P:
10
0.719383
2.63661
-1.41038 . 102
50
0.826859
5.38071
-2.62381 . 103
200
0.722342 0.854599 0.987148
11.23205 10.53341 9.87148
-4.77744 . 10" -3.95401 . 10" -3.68799
1000
0.712866 0.748866 0.886272 0.954542 0.985358
25.17599 24.74796 23.13853 22.36483 22.02304
-1.20533 -1.13087 -9.53371 -9.19723 -9.14394
20
0.727137
2.99715
1500
0.811993 0.861139 0.920909 0.962346 0.982305
24.00194 23.42846 22.74329 22.27780 22.05668
-1.02935 -9.74167 -9.32324 -9.17832 -9.14671
. . . . .
106 105 105 105 105
3000
0.721507 0.805675 0.839676 0.890076 0.924108 0.951593 0.975638 0.986197
35.44955 34.03998 33.47689 32.65253 32.10423 31.66698 31.28881 31.12407
-4.74033 -4.14730 -3.97916 -3.79859 -3.71933 -3.67830 -3.65804 -3.65349
. . . . . . . .
10 6 106 106 106 106 106 106 106
Re
. . . . .
106 106 105 105 105
-2.38343 . 102
45°
255
FIGURES
256
0
o
0 Regular nodes • Irregular nodes
Fig. 1.1 Irregular boundary nodes in a Cartesian grid
)12
—..
12
1;
Y1
Fig. 2.1 Curvilinear coordinates, natural and dual bases at a point in two dimensions
y2
12 Y1
=
)/T 1- al
= .4
1)
; 17[T =
,/.y
(2) 2 a =a Ricci and Levi-Civita's components
PC = ign
a/ = an]
• Pu =
a
a[23
2
Ricci and Levi-Civita's projections 42]
Fig. 3.1 Physical and dual physical vector components in two dimensions
258
, 1 ,2 "A - 'A '1A "A '2A A
2 1+ = a e + a e B 2B B B 1B
-> a
Fig. 4.1 Vector conservation in the case of resolution into spatially variable directions
'T X
; IW
;
1Y1 .
1 2w
= W1 2
v:f4
;203)
I
2 -> e l (P) 4- W 2 e2(P)
A/tit-Pi
Fig. 4.2 Expression for the base vectors in terms of locally constant basis in two dimensions
259
COORDINATE -FREE EQUATIONS
CONTRAVARIANT
ON-PHYSICAL OMPONENTS
FORM
PHYSICAL COMPONENTS
COVARIANT
DUAL PHYS CAL COMPONENTS
NON- PHYSYCAL COMPONENTS
FORM
PHYSICAL COMPONENTS
DUAL PHYS CAL COMPONENTS
Immn=11•••••n•
SEMI
WEEK STRONG CONSERV CONSERV. FORM FORM
STRONG CONSERV FORM
(2)
VECTORS AND TENSORS expressed in terms of:
BASE VECTORS expressed in terms of:
(3) LOCALLY FIXED BASIS
CARTE- LOCALLY SIAN FIXED BASS BASIS
CARTESIAN BASS nn••
•••••1 CICZ L.LJ
-
2
0
-0 002 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0 014 - VI(/)02/U2
Fig. 9.14 Streamwise shear stress profiles for
13
= 6° (BFS2)
287
12
10
6
EXPERIMENTS -0- PREDICTIONS
4 -5
0
5
10
20
15
BETA (deg)
Fig. 9.15 Variation of reattachment length with turning angle 13 (BFS3)
9.0
• ao
ao•
do.
7.0
n
•
•
6.0
Shmfford cod data Present predictions Expaurst and Tropea,1981)
5.0
Exp.(Fit to collected data)
0 4.0 1.0
12
Fig. 9.16
1.4
1.6
1.8 (13+H)/B
2D
22
2.4
Reattachment length as a function of expansion ratio
2.6
288
5TRERMLINES A: 8: C: D: E: F: G:
0.9 0.8 0.7 0.6 0.5 0.4 0.8
H:.0.2 I: 0.1 J: 0.00 K:-0.01 L:-0.02 M:-0.04 8:-0.06
VELOCITY PROFILES
A: 0.13 8: 0.095 C: 0.063 0: 0.031 E: 0.00 F:-0.033 G:-0.064 H:-0.096
R 9LLENCE INTENSITY
-
0.221
_ c= NG T H SCP,E
0.003
Fig. 9.17 Turned-passage backward-facing step flow predictions
289
Fig. 9.18 Pressure distribution along step-side wall (BFS3)
290
Fig. 9.19 Pressure distribution along opposite-side wall (BFS3)
291
0.4
0.4
/3,0°
Predictions
Experiments 0.3
0.3 ,
/3-0°
5°
10•
0.2
0.2
15°
-0.1
- 0.1
-0.2
- 0.2
- 0.3 -4
0
4
8
12
16
- 0.
20
4
0
4
X /H
8
12
16
20
X /H
0.4
0.4
Experiments 0.3
0.3
/3-0° 5° 10°
0.2
_
0.2
15° 0.1
L.) 0.0
- 0.1
-0.2
- 0.2
- 0. 3
-4
O
4
8
X /H
12
160
-0.
.?4
0
4
8
12
16
20
X /H
Fig. 9.20 Comparison of calculated and measured pressure distribution along step- and opposite-side walls for different turning angles f3 (BFS3)'
292
3.0
2.5 2.0
= n ›-
1 .5
1.0 0.5 0.0
-0 4
0.0
0.4 /0/ U0
0.8
12
3.0 2.5 2.0
0.5 0.0
-0 4
0.0
0.4
0.8
12
v")/U
Fig. 9.21 Streamwise velocity profiles for 13. 100 (BFS3)
293
1 I shear layer I
entrainment from the atmosphere
I I
polar coordinate origin
I ackplate
I
/ / /
center-line
/
./
reference streamline floor
Fig. 9.22 Flow geometry of curved mixing layer of Castro and Bradshow (1976)
—4:-
bleed
294
111111110 11110111111111 111111111111111111 1111111111111111111
„III! 011111111111011: .0..
Raw MIIMML‘ MM U MS EL
ktenillillinilli Bs 0441: Haan
l OV Sial II
12=0 S i wimmunil • immumull**0
mm=11,,gp .. •
n •• n•••;1 0 =--=4%, I -assa•-* n ml.m...os..... .. .1..m........ =.1.1..........
i
00 ti
....„„if
b) m
ufti!!
...,muuffit!!! I
_•muffin!!
...„„mtffit!!!!
'Unit **** •••t ritt
•.
tfr
•—•••
Fig. 9.23 Grid arrangements and calculated velocity field for curved shear layer flow: a) 23x25 CV grid, b) 37x31 CV grid
I
295
0•12
I
•
0.12
I
I
THETR=60°
0•10
.''. \\,, \'\ n\ \s#. 0.
1'1\
0.08
0.08
\\/k‘\
I
THETR=90°
•
0.10
\
•
\
\\\
0.06
\ \ 7:\* \n \ • \:\ • ‘ *
c; 0.06 cc
\\\* \\\
\
\\\\,s,,t
0.04
0.04
0.02
0.02
.
. 28 2
0.0
I
0.2
.
.
0.4
I
0.6
"J.-of
0.8
1.0
0O8 2
1 2
0.24
\
\\
0.0
0.2 0.40 0u .: 0.8 1-0 1 2
0.24
X=0.352
0.20
'
I
X=0.556
\I\
rn
0.20
0.16
.
0•16 \
0.12
ssx%
0.12
ss'k,
0.08
0.08
0.04
0.04
I
.
0.98 2 0.0 0.2 0.4 0.6 0.8
1.0
0
1 2
. 08 2
0.0
0.2
Uref
---23x25 CV;
37x31 CV; ----37x40 CV;
0.4 0.6 V" /) Uref
0.8
1.0
---37x49 CV
Fig. 9.24 Grid refinement test for curved shear layer 0.2.8 m) (velocity profiles) ( RO
....
Continued
12
296
0.12
1
1
"
0.12
1
1
1
• \N,‘
THETA:60° 0.10
0.10
THETR=90°
• \‘‘
0.08
•
0.08
0.06 cc
0.04
0.04
°,1
/
• fr/ 0.02
° . 98 2
0.02
0.0 0.2 0.4 0.6 0.8 1.0 2 2 V1( 0 V1(2)/, Uref '10
1 2
. . ° • -°8 2 O002 0.4 0.6 0.8
1.0
1 2
-77\7/ U 2ref • 101
0.24
0.24
X:0.352 m 0.20
0.20
0.16
0.16
0.12
0.12
0.08
0.08
X=.0 .556 rn
•
• 0.04
0.04
1
0.98 2 0.0 0.2
0.4 -
23x25 CV;
0.6
0.8
1.0
1 2
° . °8 2
0.0 0.2
0.6
0 ..8
.
1
1.0
- 7/7,/U 2ref " 10 1
023/Ur2ef.- 102
37x31 CV;
0.4
----37x40 CV;
Fig. 9.24 Grid refinement test for curved shear layer (shear stress profiles)
----37x49 CV
12
297
116.40 916.20 C.5.90 0.5.40 ts5.30 F.5.05 5.4.60 Ms4.00 113.50 J13.00 9.2.50 1.12.00 1.1.50 9.1.00
910.16 810.31 C•0.26 O10.21 (10.15 P.0.11 80.06 910.01 1.0.0e J.-0.0015 t o-0.0030 .. —0.0045 68-0.0060 Iv-0.0075
\ \NN-,\ NN-V.0.\ \ ‘,.\\\ \\ NAN\N \ N\N-",\ \ \\\NON\
STRERMLINES
ISOBFIRS
410.14 810.12 C10.091 00.065 810.039 F:0.013
A:0.091 8.0.074 00.058 010.041 8:0.025 Ft0.0082
\ \ \
\ \ \
N's \
TURBULENCE INTENSITY
VC\
LENGTH SCALE
Fig. 9.25 Curved shear layer flow predictions .... Continued
298
\\nnn\\\‘‘
n\.nnnnnnnnnnnn‘‘ v.nnnnnnnnnn N‘ nnnnnnnnnn nnnnnnnnnn ‘nnn
\AA.,
Cr) CI 0
o
II
\\\\nn\\\\
‘,‘,‘‘\ .s.,,)‘‘‘‘\\‘‘‘‘),,,,,,II, \NAN\ \\\ 1.01.4":::n\\‘. ‘VIA
//
299
0.12
0•10
0.08
cc
0.06
0.04
0-02
° .
98.2 0.0 0.2 0.4 0.6 0.8 1.0 1 2 v")/Uref
0.98 2 0.0 0.2 0.4 0.6 0.8 • 1.0 V(1)/Uref
0.12
0•12
0•10
0•10
0-08
0.08
• 0.06
▪ 0.06
0.04
0..04
0.02
0.02
° • 28 2
0.0
0.2
0.4
0.6 0.8 V")/Uref
1.0
1 2
13•98 2
0.0
0.2 • 0.4
0.6 0.8 V(1)/Urcf
1.0
Fig. 9.26 Velocity profiles for curved shear layer (Ro.0.23 rn) .... Continued
12
12
300 N -
•••••n
o 0
CV
w
a
co
cm
o
o
cv o
(w)) /1.
CO 0
v 0
00 0 i
0
0
0
301
0.12
0.12
0.10
0.10
THETR=30°
.
o EXP. -PRE0---PRED.(RODI)
0.08 -
0.08
E
E-
ce o'
cc
0.06
cx
0.06
cc
0.04
0.04 ..
0.02
0.98
0.02
2
0 0 ' 0.2 ' 0.4 - 0.6 0.8 - \eV) af • 102
1.0 . 1 2
0.98 2 0.0 0.2 0.4 0.6 0.8 /PULf • 101 -‘7)71
0.12
0.12
0.10
0.10
0.08
0.08
i
12
1.0
1
er
cc
a' 0.06
.:1) 0.06 cc
ce
0.04
0.04
0.02
0.02
0.98 2 0.0 0.2 0.4 0.6 0.8 -vi rl vi,u f • lot
1.0
1.0
1 2
0
.93
2
0.0
0.2 -
0.4 v'u-7
0.6 0.8 ULf • 101
Fig. 9.27 Shear stress profiles for curved shear layer ( R0 .- 0.28 rn) .... Continued
2
302
.
1
on
ao
E CD LD
•
in
6
cc . ---•. 0 0
o-Luu, xerct
C:/
/
4.1 6. O.
x11
0 1
/
/
/
X
—
e
\
• \\
/
C .n
\
\
4
Cl?
/
/
/
/
•
• •
•
0
1
\
/
1
1
1 1
ir
o cv o
? 1 1
0 n.. .. 7 5
0
-
N I 0 o o
0
N .... 0 (us ) A
co 0 0
s .>
1 * \
1.0 ....
c.0 ...—
0
•
N 0
./.....
.
I I /
0.1
I \
,
L.
/
/
/
/
/
. .'
cv
.r 0 0
00 0 i
0
•0 .. 4.) C 0 0 sm./
303
0.12
THETR=0°
0.10
—PREO• 0.08
,
0.04
0.02
0.0D U
00
0.01
2 K/1.64
0.01
0.02
0.02 K/U%f
0 03
0 03
0.01
0.01
2 K/Uref
2 it /Uref
0.02
0 03
0.02
0 03
Fig. 9.28 Turbulence intensity for curved shear layer (R0=0.2B rn) .... Continued
304
0.24
X=0.352 0.20
•
EXP. -PRO.
0.16
0.12
• •
•
0.08
•
•
0.04
0.0
8
-0-6-
0.01
•
2 K/Ure
,•n•••n•
0.01
0 03
0.02
2. K/Uref
0.02
Fig. 9.28 Turbulence intensity for curved shear layer (cont'd)
1 .0
0.9
0 -
0.8
0.7 00
0.2
0.4
EXP. PRED. PRED.(Gibson&Rodi,1981)
0.6
0.8
1.0
S(m)
Fig. 9.29 Streamwise variation of the maximum velocity for curved shear layer
12
305
0.05
EXP. PRED. PRED.(GR, k-E model) PRED.(GR, ASM)
0.04
,._ 0.03
w C4, = =
0.02
0.01
0.00 00
0.2
0.4
0.6
0.8
1.0
12
S(m)
Fig. 9.30 Streamwise variation of maximum turbulence intensity for curved shear layer
Fig. 9.31 Streamwise variation of maximum shear stress for curved shear layer
306
0.12 n•n MEW
0.10
0 —
0.08
EXPERIMENTS PREDICTIONS PRED.(Gibseiw&Rodi,1981,RSM)//
0/ tcy, 0.06
/
\ 0
\
0.04
/
•
0- 0• 0.02
0/1
460.00 0.0
0.2
0.4
0.6 S(M)
0.8
1.0
1-2
0.08 •
0.06
0 EXPERIMENTS — PREDICTIONS PRED.(Gibson&Rodi,1981,RSM)
g 0.04
0.02
0.00 0 0
0.2
0.4
0.6
0.8
1.0
12
5(m)
Fig. 9.32 Generation and dissipation rate of the turbulent kinetic energy3 on the center-line (non-dimensionalised by 03refis and Uref/s respectively) for curved shear layer
----
307
0.20
0 EXPERIMENTS PREDICTIONS
-
0.15
o o
Z
o
:0.10
o
o
o
-
o
0.05
o 0•00
o.o
0.2
o
0.4
0.6 S(m)
0.8
3/2 Fig. 9.33 Variation of the length scale 2, = k center-line for curved shear layer
1.0
/e along the
Fig. 9.34 Entrainment for the curved shear layer flow
12
308
I Heated rube (isotherrrial)
PL = 17.78 cm PT = 20.32 cm D = 15.24 cm Fig. 9.35 A cross sectional view of a staggered tube tube bank
Inflow
Trailing tube /---
Additional row of cells ,
Leading tube
Outflow
Fig. 9.36 Solution Domain and a typical grid arrangement for the flow across tube bank
309
310
mi 11111 io
1
11111171M N i i iii r l\ \ \\\\,
//iiiiitit;i\\\\\\\\\\\
1 \\ \\\ \_ \ \
111141
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7
4-)
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s.
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0 CU
—1L
as G.)
4—)
V)
V lq 4 7i;41.
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i
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311
312
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c
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(1)
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-a = 4-)
v) tn o s0
ca
3 o 47 a3
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o 4tn -o , cu , 4-
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313
CC CC — V) N COM n Ish, COr• yr MI m CV.—.0 0 0000000000 . . . . . 2 0000000000 e7 la.1 ,J CL CD 1.2
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dl.— L&
314
20
Modified interpolation (A3.4) 10 ----------------
Standard interpolation (6.16) ••nnnn•
r"
-20
0
90
60
30 , THETA (Deg)
Fig. 9.39 Predictions of the fluid temperature variation around a tube in a tube bank using two different interpolation practices
15
12 s,s
,— 9
0
6 EXPEIMEITS 1
PREDICTIONS
3
H Y = 5.8
30
50
90
120
mm
ISO
I8C
THETA (Deg) Fig. 9.40 Velocity variation around a tube in a bank of tubes in cross flow
315
100
80
Sas*.
•
60
• CC
40
20
0
o 30
90
60
120
150
180
THETA (Deg)
Fig. 9.41 Local turbulence intensity (1'V7U) variation around a tube in a bank of tubes in cross flow
20
15
----- EXPERIMENTS PREDICTIONS
Y = 2.4 mm 30
60
SO
120
150
180
THETA (Deg)
Fig. 9.42 Fluid temperature variation around a tube in a bank of tubes in cross flow
316
I
317
0
0 . 0 8. 6 — 0. 7 • •
0
CD
4-)
6—
0 4– c 0 4-) (E)
0
cu
s4 -)
0 cu
318
Fig. A2.1 Diffusion flux vector components
\ \
\
Tt4w=1th\
\ \
\ \ TN =100
\\\ \
Tsw=8 0 \\
------ Possible
T -100 \s-
\
;i.--100
\
isotherms
Fig. A3.1 Model pi-oblem used to assess approximations of diffusion terms
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