Computational Fluid Dynamics

Transcription

1 Advanced Wind Energy Bernhard Stoevesandt, Gerald Steinfeld, Leo Höning Computational Fluid Dynamics Lehrbrief PUBLIKATION DER BILD UNGS ALLI ANZ MINT.ONL INE: UNIVERSITÄT OLDENBUR G, UNIVERSITÄT K ASSE L, UNIVERSITÄT STUTTG AR T, FERNUNIVERSITÄT IN H AGEN, FRAUNHOFER-GESELLSCH AF T, DLR-INSTITUT FÜR VERNETZTE ENERGIESYS TEME (EHEM ALS NEXT ENERGY)

2 Das diesem Bericht zugrundeliegende Vorhaben wurde mit Mitteln des Bundesministeriums für Bildung, und Forschung unter dem Förderkennzeichen 16OH12044 gefördert. Die Verantwortung für den Inhalt dieser Veröffentlichung liegt beim Autor/bei der Autorin.

3 Renewable Energy Online Computational Fluid Dynamics by Bernhard Stoevesandt Gerald Steinfeld Leo Höning (C) CARL VON OSSIETZKY UNIVERSITY OF OLDENBURG (2017)

4 This document has been typeset using the L A TEX2e bundle on TEX. compilation date: September 26, 2017 Imprint: Author: Bernhard Stoevesandt, Gerald Steinfeld, Leo Höning Publisher: Carl von Ossietzky University of Oldenburg Edition: First edition (2017) Editor: Leo Höning Layout: Robin Knecht Copyright: c 2017 Carl von Ossietzky University of Oldenburg. Any unauthorized reprint or use of this material is prohibited. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system without express written permission from the author/publisher. Oldenburg, September 2017

5 Contents I Theoretical Part Introduction to this Course Introduction to CFD Introduction Navier-Stokes equations Various notations Literature 6 3 Governing equations Elements of CFD Criteria for a good CFD simulation Methodology for a good CFD simulation Literature 8

6 4 Finite Differences Method Introduction Basic Concepts Definition Forward, backward, and central differences Example: Error and approximation Achieving a better resolution Polynomial fitting Compact schemes Non-uniform grids Approximation of mixed derivatives and other terms The algebraic equation system Heat transfer equation Heat transfer equation with backward finite differences scheme Crank-Nicolson Scheme Exercise Literature 25 5 Finite Volume Method Introduction to the Finite Volume Method Different grids used in FVMF Integral approximations Interpolation and Differentiation Boundary conditions The algebraic equation system 12

7 5.7 Practical examples Exercise Literature 25 6 Linear Algebraic Equation Systems Introduction Solution methods for particular systems Solution methods for arbitrary systems: Direct methods Solution methods for arbitrary systems: Iterative methods Iterative methods: Jacobi s Method Exercise Literature 18 7 Non-Linear Algebraic Equation Systems Introduction Solution methods for one equation Algorithm for fixed-point iterations Solution methods for a system of non-linear equations Exercise Literature 25 8 Ordinary Differential Equations: Initial Value Problems Introduction and example Initial value problems Solutions using numerical approximations Explicit and implicit methods for non-linear systems 10

8 8.5 Exercise Literature 13 9 Classification of partial differential equations Hyperbolic equations Parabolic equations Elliptic equations Literature 4 10 Navier-Stokes-Solver The solution of the pressure equation A simple explicit solution The simple implicit Euler method Further implicit methods Literature 8 11 Turbulent flows RANS Modeling Boussinesq Assumption Large Eddy Simulations Detached Eddy Simulation Literature 0

9 II Practical Part Introduction to CFD Introduction to Ubuntu Installation of OpenFOAM Literature 5 13 Getting started with OpenFOAM OpenFOAM overview Pre-processing Solving Post-processing OpenFOAM tutorials Lessons learned Literature 8 14 The blockmesh tool Meshing tools Lessons learned Literature 7 15 Advanced blockmesh Cell grading in blockmesh Boundary conditions - patch creation Curved geometry meshing Task Lessons learned 5

10 15.6 Literature 6 16 Boundary conditions The boundary file The zero folder Task Lessons learned Literature The final setup Turbulence modeling Fluid properties Solver settings Task Lessons learned Literature 9 18 Post-processing Restarting a case Force calculation Task Lessons learned Literature 6 19 Convergence FAQ s about convergence Mesh independence study 3

11 19.3 Task Lessons learned Literature 7 20 Airfoil simulation Airfoil aerodynamics Task Lessons learned Literature 8 21 Project task Simulation of a car Lessons learned Literature 6

12 Part I Theoretical Part CFD 0-1

13 3 Governing equations Learning objectives After processing the theoretical part of this chapter you will CFD understand the basics of computational fluid dynamics know about the Navier-Stokes equations know various kinds of notation understand the motivations of CFD 3-1

14 Governing equations 3.1 Elements of CFD In CFD, the differential equations, which represent the governing equations are discretized with the aid of quotients of differences. Various tools are required for discretization, which are associated with approximations. Detailed treatment would require a separate lecture on this topic, so only a brief discussion about discretization is presented here. References for this topic are added at the end of this chapter Mathematical Models A mathematical model adapted to the problem should be derived. This involves decisions on whether to select a 2D or 3D model, turbulence models, compressibility effects, and to consider or neglect certain processes. The derivation of a mathematical model involves a lot of experience of the user about the given flow problem Discretization Methods Discretization describes the process in which we transfer the temporal and spatial structure of our real world into purely numerical form for computers. For this purpose, a method has to be chosen that is capable of solving differential equations. There are various methods, such as the Finite Element Method (FEM), Finite Differences Methods (FDM), Finite Volume Methods (FVM) and Spectral Methods (SM). Some methods are better designed for certain problems than others. Available solvers generally use one specific discretization method, ex: OpenFOAM and Star CCM+ uses FVM Coordinate System The selection of a specific coordinate system may be crucial depending on the problem at hand. Spherical Couette flow, for example, can be readily calculated in spherical coordinates. The most common coordinate systems are: Cartesian coordinates Polar coordinates Spherical coordinates The comparison of two cases using two different coordinate systems is based on a coordinate transformation Grids In most methods, discrete calculation of the flow in space and time. The discretization of the physical domain results in a discrete grid. The discretization is simply done by splitting the physical domain into sub-elements called cells, where in each cell a grid point is present. On the cells, calculations will take place and the variables are represented in these cells. The accuracy of the calculation depends on the quality of the grid. Information (or variable values) which are not captured by the grid points are also omitted from 3-2 CFD

15 3.1 Elements of CFD the flow calculation. A good grid resolution is therefore desirable - but comes along with a considerable strain on computational time (and memory). Spectral methods represent an exception, which enable the calculation by transformation of the space. However, geometric elements in the flow are in some cases difficult to consider using Spectral Methods. There are various types of grids, the examples provided here generally only deal with the simplest - structured grids. However, creating good grids is a key factor in the CFD calculation. Structured grid A structured grid is the simplest grid structure, and it is equivalent to a Cartesian grid. The grid consist of groups of grid lines that do not cross each other. This enables that any grid point within the domain is uniquely identified by its indices. Neighbor points differ only by ±1 in the index, this properties simplifies the automatic generation of a structured grid (using an algorithm). Fig. 3.1 shows a structured grid. Figure 3.1: 3D multiblock structured grid (H-type). Source: [1] Structured grids may be of different types, we distinguish between H-, O- and C-type. A grid of O-type is shown in Fig Unstructured grid Unstructured grids are more complex than structured ones. They are usually used with Finite Element Methods (FEM) and Finite Volume Methods (FVM). The elements of an unstructured grid may have any shape and there is no fixed value of neighbor points for the grid elements, like it is the case for structured grids. Common shapes used in unstructured grids are triangles or quadrilaterals for a 2D geometry, and tetrahedrals or hexahedrals for a 3D geometry. One advantage of unstructured grids is that the grid generation is more simple and flexible and that complex geometries can be easily meshed. Fig. 3.3 shows an unstructured grid around an airfoil. CFD 3-3

16 Governing equations Figure 3.2: A structured grid of O-type. Figure 3.3: An unstructured grid around a NACA 0012 airfoil. Source: [2] 3-4 CFD

17 3.2 Criteria for a good CFD simulation Approximation Methods Besides making the right choice for the grid, is it also very important to choose the right approximation method to be used for the equations discretization process. The selected method differs when approximating the derivative on surfaces or volumes. For FVM for example, the derivatives at each grid point has to be approximated. In most cases, more than one approximation is available. The choice of the approximation influences the accuracy of the solution. More detailed approximation method give better resolution, but will result in a heavy usage of memory and computational time. In every discretization method approximations are involved and an appropriate discretization has to be used and adapted to the accuracy of the grid. Nevertheless, every approximation involves an approximation error. The difference between the exact equation and the discretized one is called the truncation error. In CFD, it is very important to deal properly with truncation errors Solution Methods Generally, in CFD iterative solution processes are used, which can vary according to whether the problem treated is steady or unsteady in time. In addition, it is also necessary to specify convergence criteria for the solution methods used. 3.2 Criteria for a good CFD simulation 1. Consistency: Results should not vary even for smaller t, x, where t is defined as the time increment and x is defined as the spacing between two points of a grid (or mesh). These increments are defined as follows: x := x i x i 1, (3.1) t := t i t i 1, (3.2) i = 1,..., N. (3.3) 2. Stability: Errors must not increase during the iterative process. 3. Convergence: The calculation should converge to one result with a certain accuracy after a certain simulation time. 4. Conservativeness: Retention of the conserved quantities should be guaranteed through the solution method used or it has to be enforced. 5. Boundedness: Physical quantities have to comply with physical conditions - kinetic energy, density, for example cannot become negative. 6. Accuracy: As approximation calculations are concerned, it is important to have a measure of its accuracy. In the first step, this can only be determined through measurements. At a later stage, comparisons are possible using different simulations with known accuracy. Various errors may occur as a matter of principle: Model errors which result because of an approximated mathematical model (e.g. RANS calculations). Discretization errors which constitute a difference between the exact solution through the algebraic equations used and the results derived through the finite grid and the CFD 3-5

18 Governing equations finite step size. Iteration errors that occur during the iteration process in the calculation. The solution will almost always contain errors because it is always necessary to make approximations. The aim is therefore to minimize the respective errors. One criterion for a good transient CFD simulation is the Courant-Friedrich-Levy number (CFL) criteria, which is calculated as: CF L = Re t x, (3.4) where, x is the smallest cell length in the direction of the main flow, and Re represents the local Reynolds number. The CFL number should be significantly less than 1 so that events in the flow can be recorded with the chosen grid. Consequently, if one part of the grid is very fine but the flow is fast, the time step has to be made smaller according to Eq. 3.4 so to satisfy the CF L < 1 criterion. 3.3 Methodology for a good CFD simulation To obtain a good simulation using CFD methods, it is advisable to follow a certain sequence: A. Representation of the geometry: The given geometry is generally created in an approximate form to the real one by means of a CAD model. In this process, the geometry features determine to what degree the details can be included in the simulation. B. Generation of a grid: The space in which the flow is to be calculated, must (except in the case of Lagrange and Spectral Methods) be discretized (split up) into sub-elements and represented with a discrete grid. Here, it is necessary to consider a higher resolution in regions with a major change. This means that the grid has to be adapted to the flow phenomena. For example, in the case of a wing, a higher resolution should be given around the wing and the near wake regions (cf. Fig 3.3). C. Determination of the modeling parameters: Many of the equations are ultimately dependent upon approximation calculations. Empirical parameters governing the problem at hand have to be set. For example, for a flow over an airfoil, the flow velocity, the airfoil chord length (size) and the turbulence intensity are important parameters. 3-6 CFD

19 3.3 Methodology for a good CFD simulation D. Calculation of the variables: With discrete calculations, computation of the flow problem at hand is iterative. Starting from an assumption at the origin, the calculation is repeatedly refined until the residual values become so small that it is possible to refer to a converged solution. E. Determination of the convergence: It is necessary to determine when the solution has sufficiently converged. For this purpose it is possible to examine whether the values to be calculated fluctuate during the iteration process. F. Evaluation of the flow field: Depending on what sort of parameters are of interest, it is possible to depict the field graphically in different forms. (a) Velocity distribution (b) Pressure distribution Figure 3.4: Paraview[3] visualization of the flow field around an airfoil. G. Verification and validation: As many approximations are made in the calculation process, various simulation settings have to be used to verify that this simulation can be reproduced. This is called verification of the simulation results. On the other hand, the result should be examined using measured data in order to show that the calculations are in agreement with the experiments conducted. This process is a very important part of a CFD simulation and is called validation. CFD 3-7

20 Governing equations 3.4 Literature References [1] Ansys Fluent. Examples of Acceptable Grid Topologies URL: sharcnet.ca/software/fluent6/html/ug/node150.htm. [2] Jens-Dominik Müller. Grid Generation Tools URL: edu.tr/tuncer/ae546/prj/delaundo/files/small0012.gif. [3] Paraview. Large Data Visualization Made Easier URL: org/. 3-8 CFD

21 4 Finite Differences Method Learning objectives After processing the theoretical part of this chapter you will have learned the basics of finite differences method understand the necessity of spatial discretization CFD 4-1

22 Finite Differences Method 4.1 Introduction In general, Finite Differences Methods (FDM) represent a class of numerical methods for solving ordinary and partial differential equations. It is the oldest method, dating back to Euler, who developed the Finite Differences Method during the 18th century. This method is the simplest discretization method, when applied on simple geometries. The geometrical domain must be first discretized. A numerical grid needs to be defined equivalent to the domain of the calculation. Fig. 4.1 shows the comparison between the geometric domain and the grid. Figure 4.1: Schematic representation of a geometric 2D domain and an equivalent numerical grid. In FDM it is possible to use a structured grid and at each grid point, the discretized governing equations have to be solved. In the discretization step, the differential equations are replaced by differences equations: y t = y t Continuum = Discretized It follows, that an algebraic equation system will be created, for which one equation will exist for each grid point, in which the variable value at that and a certain number of neighboring nodes appear as unknown, which has to be solved for. In general, for linear equation systems, the number of equations has to be equal to the number of unknowns to be solved. It should be noted, that not all finite differences approximations lead to precise numerical schemes. Hence stability and convergence issues are important. 4.2 Basic Concepts As mentioned above, it is necessary to replace the continuum (or fluid domain) by N discrete mesh (or grid) points: 4-2 CFD

23 4.2 Basic Concepts x x i {x 1,... x Ni }, t t j {t 1,..., t Nj }, i { } 1,..., j Nx. (4.1) 1,..., N t Fig. 4.2 shows examples of a one-dimensional (1D) and a two-dimensional (2D) grid. Figure 4.2: Examples of a 1D (upper part) and a 2D (lower part) grid. Source: [1]. At each grid point, the differential equation is approximated by replacing the partial derivatives by approximations in terms of the nodal values of the functions. Each point of the grid is uniquely identified by its indices. The main idea of finite differences approximations is the transition from continuum to discrete space and the usage of the following definition of a derivative: ( ) u = lim x x 0 x i FDM consists of three different schemes to solve a problem: Central Finite Difference Forward Finite Difference Backward Finite Difference u(x i x) u(x i ). (4.2) x All three schemes will be presented in the following section. Fig. 4.3 shows the geometrical interpretation of all three schemes cited above. CFD 4-3

24 Finite Differences Method Figure 4.3: Slope of a function at x i and the geometrical interpretation of Central FD, Forward FD and Backward FD. Source: [2] As it can be seen in Fig. 4.3, some approximations of the real slope at the point x i are better than others. In this example the forward and backward scheme are not a good approximation for the slope at point x i. A better choice is the central finite difference scheme. But the decision which of these three methods should be used differs from one problem to another. 4.3 Definition Definition 4.1 FDM stems from the definition of the derivative in Eq. 4.2, if we remove the limit in this equation, we obtain the finite difference approximation. This approximation will be improved by reducing the spacing x, but for any finite value of x an error is always introduced by this approximation. The basic concepts of finite differences approximations are based on the properties of Taylor expansions. Expanding u(x + x) around u(x) we obtain: u(x + x) = u(x) + x u x + x2 2 u 2 x + x3 3 u 2 3! x xn n u + H, (4.3) 3 n! xn where H stands for higher-order terms, of order four and greater. Eq. 4.3 can be re-written as follows: (considering the notation: u x := u x(x)) u(x + x) u(x) x = u x (x) + x 2 u xx(x) + x2 6 u xxx(x) +..., (4.4) }{{} Truncation error The right hand side of Eq. 4.2 is an approximation of the first derivative at the point x. This is 4-4 CFD

25 4.4 Forward, backward, and central differences the so called truncation error. The power of x with which the truncation error tends to zero is called the order of accuracy of the finite differences approximation. The remaining terms of Eq. 4.4 represent the truncation error associated with this formula. When higher-order terms are neglected, the Eq. 4.2 can be re-written as: u(x + x) u(x) x = ux (x) + x 2 u xx(x) = u x (x) + O( x). (4.5) 4.4 Forward, backward, and central differences In the 1D case (as shown in Fig. 4.2), we obtain at the point i, a first order forward difference scheme, with Eq. 4.3 and Eq. 4.4, as follows: (u x ) i = ( ) u = u i+1 u i x x i x 2 (u xx) i x2 6 (u xxx) i +... }{{} Truncation error = u i+1 u i x + O( x). (4.6) With the same ansatz of Eq. 4.3 and Eq. 4.4 at the point i, one can also obtain the first order backward difference formula as follows: (u x ) i = ( ) u = u i u i 1 + x x i x 2 (u xx) i x2 6 (u xxx) i +... }{{} Truncation error = u i u i 1 x + O( x). (4.7) The Eqs. 4.6 and 4.7 are called one-sided differences formulas. This is very intuitive, as from Fig. 4.2, is it shown that, to solve for the value of the unknown at point i, another point at one side of i is used. The approximations of the first derivative can be written as the first term at point i with neglecting truncation error as follows: ( ) u u i+1 u i = u i+1 u i, (4.8) x i x i+1 x i x ( ) u u i u i 1 = u i u i 1, (4.9) x i x 1 x i 1 x ( ) u u i+1 u i 1 = u i+1 u i 1. (4.10) x x i+1 x i 1 2 x i CFD 4-5

26 Finite Differences Method These formulas are forward-, backward-, and central difference schemes, respectively, without including the error terms. The truncation error part of the approximation gives a criterion of the accuracy of the approximation and of the rate at which the error grows. Eq is obtained by adding Eq. 4.7 and Eq. 4.8 at the point i. It becomes the central differences formula and is a second-order approximation, as follows: (u x ) i = u i+1 u i 1 2 x = u i+1 u i 1 2 x x2 6 (u xx) i O( x 2 ). (4.11) To minimize the approximation error, the grid has to be refined ( x has to be smaller), as a smaller x leads to smaller errors. 4.5 Example: Error and approximation Some short examples will be given here to show the importance of errors in approximations formulas. Example 4.1 The first function to approximate, will be be a sine curve at the point 1. We set: f(x) = sin(x), (4.12) and the first derivative gives at x = 1 the value: f (1) = cos(1) = (4.13) As shown earlier, we can approximate the value at x = 1 by the following: cos(1) sin(1 + x) sin(1). (4.14) x The following table shows the values of the approximation and the truncation error with different values of x. x Approximation Error CFD

27 4.6 Achieving a better resolution Table 3.1: Error of a first-order finite-differences approximation for different x. The error gets smaller with smaller x. It is an approximation of first-order. If the step size gets reduced by a factor of 10, the error is reduced by the same factor. Some remarks: The truncation error of a first-order finite differences approximation is proportional to the second derivative: A first order finite difference formula gives the exact result for a linear function. The truncation error of a second-order finite differences approximation is proportional to the third derivative: A second order finite difference formula gives the exact result for a quadratic function. 4.6 Achieving a better resolution To achieve a better resolution, a first-order forward difference approximation of u(x) i can be considered as a central difference approximation with respect to the mid point x i+1/2 = (x i+x i+1 ). 2 The usage of the midpoint leads to a second-order approximation for the derivative (u x ) i+1/2 at this point. So a second-order approximation at i + 1/2 and i 1/2 is obtained as follows: ( ) u (u x ) i+1/2 = x ( ) u (u x ) i 1/2 = x i+1/2 i 1/2 = u i+1 u i x = u i u i 1 x + O( x 2 ), (4.15) + O( x 2 ). (4.16) Another way to obtain a second-order approximation of the derivative, is to use more points to eliminate more truncation error terms. In general, approximating a derivative of order n, requires n + 1 grid points. For an approximation of the second derivative, three grid points are used. The following equations use the points x, x + x and x x. Adding the two Taylor expansions (around x, x + x and x x) results in a second-order approximation, as follows: f(x + x) = f(x) + f (x) x f (x)( x) f (x)( x) 3 + O ( ( x) 4), (4.17) f(x x) = f(x) f (x) x f (x)( x) f (x)( x) 3 + O ( ( x) 4). (4.18) Adding Eq and Eq results in the following equation: f(x + x) + f(x x) = 2f(x) + f (x)( x) 2 + O ( ( x) 4). (4.19) CFD 4-7

28 Finite Differences Method For the second derivative we then obtain the following equation: f (x) = f(x + x) 2f(x) + f(x x) ( x) 2 + O ( ( x) 2). (4.20) Example 4.2 As shown for the sine curve example, we investigate the error for a second order approximation of the second derivative. Therefore we use the expression: f(x) = exp(x 2 ) f (x) = ( 4x ) exp(x 2 ). (4.21) The exact value of the second derivative at point x = 1 is: f (1) = 6 exp(1) = (4.22) Using Eq we obtain: f (1) = 6 exp(1) exp((1 + x)2 ) 2 exp(1) + exp((1 x) 2 ) ( x) 2. (4.23) The values of the approximation and the error for different grid space values are shown in the following table: x Approximation Error Table 3.2: Values of the error and the approximation for different values of x. When x is divided by 10, the error is divided by 100, therefore the approximation is of second-order. In finite differences approximation, there is a division over ( x) 2 present (see Eq. 4.20), which can lead to numerical problems, like round-off error which in turn leads to instabilities. Overall, to achieve a better resolution, it is common to include more terms in the approximation. Example 4.3 In the next example, an approximation of second-order of the first derivative will be investigated. Three grid points are needed for this approximation. First, we use Taylor expansion around the points x + x and the point x x. 4-8 CFD

29 4.7 Polynomial fitting f(x + x) = f(x) + f (x) x f (x)( x) f (x)( x) 3 + O(( x) 4 ) (4.24) f(x x) = f(x) f (x) x f (x)( x) f (x)( x) 3 + O(( x) 4 ) (4.25) We subtract Eq from Eq and we obtain: f(x + x) f(x x) = 2f (x) x f (x)( x) 3 O(( x) 5 ). (4.26) Using Eq. 4.26, a Central Finite Difference approximation of the first derivative of second order is then stated by: f (x) = f(x + x) f(x x) 2 x + O(( x) 2 ). (4.27) To further increase the resolution, more grid points must be used. With three grid points (i), (i + 1) and (i + 2) a forward finite differences approximation is obtained as follows: (u x ) i = 3u i + 4u i+1 u i+2 2 x + O( x 2 ). (4.28) A backward finite differences approximation with three points (i, i 1 and i )) is obtained as follows: (u x ) i = 3u i 4u i 1 + u i 2 2 x + O( x 2 ). (4.29) 4.7 Polynomial fitting So far, Taylor series expansions were used to approximate the derivatives. Another way of obtaining approximations for the derivative is to use polynomial fitting. The basic idea behind polynomial fitting is to fit the function to an interpolation curve, and differentiate the curve at the point of interest. Some interpolations are related to the finite differences approximations presented earlier in this document. A piece-wise linear interpolation leads to forward or a backward differentiation schemes. Fitting a parabola to the data at points x i 1, x i, and x i+1, the derivative at x i is: CFD 4-9

30 Finite Differences Method ( ) f = f i+1( x i ) 2 ) f i 1 ( x i+1 ) 2 + f (( x i+1 ) 2 ( x i ) 2 ), (4.30) x i x i+1 x i ( x i + x i+1 ) using x i := x i x i 1. Higher-order polynomials can also be used for polynomial fitting. Usually, the order of approximation is equal to the degree of the polynomial used to approximate the unknown variable or function. Example 4.4 Here, an example of a polynomial fitting is given: the following equations are the result of fitting a cubic polynomial using four points: ( ) u x ( ) u x i i = 2u i+1 + 3u i 6u i 1 + u i 2 6 x = u i+2 + 6u i+1 3u i 2u i 1 6 x + O(( x) 3 ), (4.31) + O(( x) 3 ). (4.32) 4.8 Compact schemes Compact schemes are used for uniformly spaced grids. They can be derived by the use of polynomial fitting, but the variable and the derivative values are needed to obtain the coefficients of the polynomials. With this basic idea, a forth order Padé scheme is derived. The goal is to use only the information from near-neighboring points. This makes the solution of the resulting equation simpler and reduces the difficulty of finding approximations near the domain boundaries. The Padé approximation is the best approximation of a function by a rational function of a given order. Using this technique, the approximant s power-series are equivalent to the power-series of the function it is approximating. This method was developed by Henri Padé and this approximation often gives better approximations of a function than truncating its Taylor series. The Padé schemes are extensively used in numerical analysis and computer calculations. Here we use three variable values at the nodes x i, x i+1 and x i 1 and the derivatives at nodes x i+1 and x i 1 to obtain an approximation for the first derivative at the node (i). A polynomial of forth-order around x i is written as: f(x) = a 0 + a 1 (x x i ) + a 2 (x x i ) 2 + a 3 (x x i ) 3 + a 4 (x x i ) 4. (4.33) The values of the coefficients a i can be found by fitting Eq to the three variables and two derivative values. For the first derivative we obtain the expression: 4-10 CFD

31 4.8 Compact schemes f (x) = a 1 + 2a 2 (x x i ) + 3a 3 (x x i ) 2 + 4a 4 (x x i ) 3. (4.34) This leads to f (x i ) = a 1. In the next steps we obtain by writing Eq for x = x i, x = x i+1 and x = x i 1 and Eq for x = x i 1 and x = x i+1 the following equation after rearrangement of the equation: f i = 1 4 f i f i (f i+1 f i 1 ). (4.35) x The complete set of equations obtained for each point leads to a tridiagonal system for the derivatives at the grid points. To compute the derivatives, this system of equations has to be solved. To get a family of compact centred approximations up to the sixth order the following equation is derived: α ( ) u + x i+1 ( ) u + α x i ( ) u = β u i+1 u i 1 x i 1 2 x + γ u i+2 u i 2. (4.36) 4 x The derivation of the second- and fourth-order central difference schemes (CDS), and the fourthand sixth-order Padé schemes depends on the choice of the parameters α, β and γ in Eq Tab. 4.1 shows choices for the parameters α, β and γ, as well as the associated truncation errors. Scheme Truncation Error α β γ CDS-2 CDS-4 Padé-4 Padé-6 ( x) 2 3! 13( x) 4 3 3! ( x) 4 5! 4( x) 6 7! 3 u x 3 5 u 0 4/3-1/3 x 5 5 u 1/4 3/2 0 x 5 7 u 1/3 14/9 1/9 x 7 Table 4.1: Truncation Error of central differences schemes (CDS) and Padé-schemes associated with different values of the parameters α, β and γ Comparing for the same order of approximation, Padé schemes use less points (computational nodes) than central differences approximations and have more compact computational molecules (the number of nodes or points needed for the computation at one node). It is also possible to derive schemes for non-uniform grids but this task is impractical for different reasons. Note that: CFD 4-11

32 Finite Differences Method The schemes presented so far in this document are only a few of the existing possibilities of CDS and Padé-schemes. It is possible to extend the approximations to a higher order and to multi-dimensional configurations. It is possible to derive schemes for non-uniform grids, but this can be impractical using finite differences methods, so different methods has to be used in this case. Computational molecules Definition 4.2 Computational molecules are defined as a fixed grid structure with fixed neighboring points. It is comparable to a chemical crystal structure. Fig. 4.4 shows an example of a 3D computational molecule. The neighboring points of the point P are named equal to geographic points, the left point in the P plane is the western point, the right one is the eastern point and so on (see Fig. 4.4). The arrangement of the points in the molecule is also important to the algebraic equation system. Figure 4.4: A 3D computational molecule with its center at point P. N stands for North, S for South, E for East, W for West, T for Top and B for Bottom. Source: [1]. 4.9 Non-uniform grids The truncation error is not only dependent on the grid spacing, it depends as well on the derivatives of the variable. The reason is that a uniform distribution of the discretization cannot be achieved on a uniform grid. The basic idea behind using non-uniform grids is to use large x values in smooth regions of the function and small x values where the derivatives of the function can be large. So, the error is uniformly distributed over the solution domain and we obtain a better solution for a given grid number of points. The following is given without a proof: A systematic grid refinement of a non-uniform grid 4-12 CFD

33 4.10 Approximation of mixed derivatives and other terms produces the same error decrease as for a uniform grid. Also the order of the approximation is the same for both grids. As user it is possible to define where to refine the grid, and an automatic grid refinement can be implemented to achieve better results Approximation of mixed derivatives and other terms The following section is a short overview about approximations for mixed derivatives, other terms (like sources) and boundary conditions. For detailed analysis have a look to relevant literature like [1]. The problem of mixed derivatives can be approximated by combining the one-dimensional approach presented earlier. In general mixed derivatives can be expressed by: 2 u x y = x ( ) u = y y ( ) u. (4.37) x Example 4.5 Here, an example is shown for a central difference scheme used to approximate mixed derivatives. For a 2D grid, the mixed derivative is expressed as follows: ( ) 2 u = x y i,j ( u y ) i+1,j ( 2 x u y ) i 1,j + O( x) 2. (4.38) The first derivative with respect to y at the points (x i+1, y j ) and (x i 1, y j ) is evaluated first. The resulting approximation is then evaluated with respect to x. Up to now, all terms already approximated in this document included some derivatives, but some terms which do not involve differentiation, i.e. scalars, sources, may also be present and need to be approximated. For these terms only the values at the nodes are needed and have to be evaluated. In some cases it is possible that sources involve the dependent variable and they may also be expressed in terms of nodal values. If this dependence is not linear, special care is to be taken in the approximation process (advanced FD). (add a reference) Boundary conditions are needed in partial difference equations to solve given problems. Finite differences methods are the relation between continuum and discrete space, so boundary conditions are needed as well. There are common methods to provide the values of the variables at the boundaries: Dirichlet boundary condition provides the value of the unknown variable at the desired boundary condition, Neumann boundary condition provides the gradient of the unknown variable at the desired boundary condition, CFD 4-13

34 Finite Differences Method Robin boundary condition provides a linear combination of both the value and the gradient of the unknown at the boundary. Since the values of the unknown are provided at the boundaries, no equations are to be solved for the boundary conditions. A problem can arise when higher orders of the derivatives are needed since they require data at more than one grid point. The solution for this problem is to use different approximations for the derivatives at points close to the boundary. This is called low-order approximations at the boundaries, or one-sided approximations The algebraic equation system The finite differences discretization provides an algebraic equation at each grid point. Each of these equations contains the variable values at the particular node and at the neighboring points. If the differential equation to be solved is non-linear, the approximation will contain non-linear terms. In this case the numerical solution process will require linearization to deal with the non-linear part. In the linear case, the system of the linear algebraic equations is the result of discretization and is of the following form: A P Φ P + l A l Φ l = Q P. (4.39) P is the node of interest at which the partial differential equation is approximated, l is an index and it runs over the neighbouring nodes which are involved in the finite differences approximation (the computational molecule). The values of the coefficients A l depend on geometrical quantities and fluid properties. The matrix Q P contains the source terms, i.e., all terms which do not contain unknown variable values. One equation is needed for each unknown, thus a large set of algebraic equations will be generated from the process of discretization. These equations have to be solved. Eq can be re-written as follows: AΦ = Q, (4.40) where A is a coefficient matrix and Φ is a vector which contains the values at the grid points. The vector Q contains all terms on the right-hand side of Eq How to solve and work with such systems we be introduced later in this document Heat transfer equation An example will be shown for the algebraic system of equations obtained after discretizing a governing equation. The equation which will be presented is the heat transfer equation, where the temperature T (t, x) needs to be computed in a beam of length l for t > 0. The temporal change in the beam temperature is described by the heat equation, which can be written as follows: 4-14 CFD

35 4.12 Heat transfer equation T t (t, x) = γ 2 T (t, x), for 0 < x < l, t 0. (4.41) x2 The boundary conditions at the upper and lower end of the beam are: T (t, 0) = α(t), T (t, 1) = β(t). (4.42) The initial condition at time t = 0 is: T (0, x) = f(x). (4.43) The first step in solving the Eq is to discretize the physical domain (here the beam) and thus create a uniform grid. We set for (t i, x j ): 0 = t 0 < t 1 < t 2 <..., 0 = x 0 < x 1 <... < x n = l. (4.44) In this simple case the grid is uniform in time and space, where: t = t i+1 t i and x = x j+1 x j = l n. (4.45) The resulting 1D grid is shown in Fig Figure 4.5: The discretized beam with initial and boundary conditions. After discretizing the physical domain, the original partial differential equation needs to be also discretized into a finite difference form as follows: CFD 4-15

36 Finite Differences Method T t (t, x) = γ 2 T (t, x), 0 < x < l, t 0. (4.46) x2 For the left side of Eq. 4.46, we use a first-order forward finite differences scheme, we then obtain: T t (t i, x j ) T (t i+1, x j ) T (t i, x j ) + O( t) t T i+1,j T i,j t + O( t). (4.47) A second-order central differences scheme is used for the right side of Eq. 4.46, which results in the following: ( ) γ 2 T T x (t (ti, x j ) 2T (t i, x j ) + T (t i, x j 1 ) i, x 2 j ) γ + O(( x) 2 ) ( x) ( 2 ) Ti,j+1 2T i,j + T i,j 1 γ + O(( x) 2 ). (4.48) ( x) 2 The finite differences form of the partial differential equation (Eq. 4.46) is then written as: T i+1,j T i,j t = γ T i,j+1 2T i,j + T i,j 1 ( x) 2. (4.49) After rearrangement, the linear system of equations representing the heat equation is expressed as: T i+1,j = γ t ( x) 2 (T i,j+1 2T i,j + T i,j 1 ) + T i,j = µ (T i,j+1 2T i,j + T i,j 1 ) + T i,j = µt i,j+1 + (1 2µ)T i,j + µt i,j 1 for i = 0, 1, 2,..., j = 1,..., n 1 and γ t := µ. (4.50) ( x) 2 This is an iterative linear system, where the temperature values at time t i+1 are calculated using the temperature values at time t i. The initial and boundary conditions mentioned in Eq and Eq are expressed as follows: 4-16 CFD

37 4.12 Heat transfer equation T 0,j = f j = f(x j ), j = 1,..., n 1, T i,0 = α i = α(t i ), i = 0, 1, 2,..., T i,n = β i = β(t i ), i = 0, 1, 2,.... (4.51) The linear system obtained is the re-written in matrix form: T i+1 = AT i + b i, (4.52) where T i, A and b i are defined as: T i = T i,1 T i,2. T i,n 1, (4.53) 1 2µ µ µ 1 2µ µ µ 1 2µ µ.. 0 A = , (4.54) µ 1 2µ µ µ 1 2µ µα i 0 b i = (4.55). µβ i. The matrix A is a symmetric tri-diagonal matrix. An explicit scheme is used to calculate the state of the system at a later time from the state of the system at the current time. Hence the time index i ends at n 1 in the definition of T i Numerical solution of the heat transfer equation To solve the example presented above, values for initial and boundary conditions are needed. We set: CFD 4-17

38 Finite Differences Method Figure 4.6: Distributions of T (t, x) for different t at various time steps t. γ = 1, (4.56) l = 1, (4.57) x = 0.1. (4.58) x, 0 x 1 5 T (0, x) = f(x) = x 2 5, 1 x 7, x, 7 x 1 10 T (t, 0) = 0, T (t, l) = 0. (4.59) Now a numerical solution can be obtained for the heat transfer equation. Fig. 4.6 shows the distributions for T (t, x) after solving the system of equations obtained above. In Fig. 4.6 the solution becomes unstable for t = 0.01 and converges to the exact solution for t = When t = 0.04 and t = 0.01 the solutions is not converging and an oscillating solution is obtained. This means that the solution of the heat equations is diverging for this value of t. We address this issue in the next section Stability analysis The question we want to answer in this section is: why is the solution of the heat transfer converging in some cases and in some cases not? (see Fig. 4.6) It is important to understand, that the general solution of the heat transfer equation can, for each time step, be represented by a sum of Fourier modes. With this in mind, our question about the solution can be transferred into: how will the solution of a complex exponential function 4-18 CFD

39 4.12 Heat transfer equation depend on the numerical schemes chosen? We assume at t = t i, that the solution can be expressed by: T (t i, x) = exp(ikx) T (t i, x j ) = T i,j = exp(ikx j ). (4.60) Advancing one time step forward, we obtain with Eq. 4.50: T i+1,j = µt i,j+1 + (1 2µ)T i,j + µt i,j 1 = µ exp(ikx j+1 ) + (1 2µ) exp(ikx j ) + µ exp(ikx j 1 ) = µ exp(ik(x j + x)) + (1 2µ) exp(ikx j ) + µ exp(ik(x j x)) = (1 4µ sin 2 (0.5k x)) exp(ikx j ) = λ exp(ikx j ), with λ = (1 4µ sin 2 (0.5k x)). (4.61) The effect of advancing the time step, is the multiplication of the temperature value of the previous time step with the factor λ. In a mathematical sense, the exponential function is like an eigenfunction and λ is the corresponding eigenvalue for the linear operator of the numerical scheme. The solution T will be defined in the point x j after n time steps as: T i+n,j = λ n T i,j = λ n exp(ikx j ). (4.62) Definition 4.3 Hence, the stability of the numerical scheme will be defined by the factor λ. Two cases are possible: λ > 1 λ n for n and t. Unstable system. λ 1 λ n 1 for n and t. Stable system (Stability criteria, analysis of von Neumann, cf. [3]) With this information, a new choice of the time step t is needed. To obtain a stable solution we start with λ 1: λ 1 1 4µ sin 2 (0.5k x) 1 1 4µ sin 2 (0.5k x) 1 and 1 4µ sin 2 (0.5k x) 1, because 1 sin 2 (0.5k x) 0. (4.63) Which results in: CFD 4-19

40 Finite Differences Method µ 0 µ 1 2 γ t γ t 0 and ( x) 2 ( x) for x > 0, t > 0, γ > 0. (4.64) After rearrangement we obtain: t 1 2γ ( x)2, with γ = 1, x = 0.1 t (4.65) In this case, the solution with the time step t = 0.01 is unstable. Not all combinations of time steps and grid size will be guarantee stability of the numerical scheme. This behavior is known as: conditionally stable schemes Heat transfer equation with backward finite differences scheme The heat transfer equation will be solved again, but with the aim of obtaining an unconditionally stable scheme. This is obtained when the left side of the heat transfer equation (see Eq. 4.41) is approximated using backward finite differences scheme. We then obtain: T t (t i, x j ) T (t i, x j ) T (t i 1, x j ) + O( t) T i,j T i 1,j t t + O( t) (4.66) Combined with the right hand side of Eq the approximation of the heat transfer equation is expressed as follows: T i,j T i 1,j t = γ T i,j+1 2T i,j + T i,j 1 ( x) 2 (4.67) Re-writing Eq leads to: T i 1,j = T i,j γ t ( x) 2 (T i,j+1 2T i,j + T i,j 1 ) = T i,j µ(t i,j+1 2T i,j + T i,j 1 ) = µt i,j+1 + (1 + 2µ)T i,j µt i,j 1 (4.68) Due to the normalization of the indices, we shift i 1 to i and i to i + 1, we then obtain: 4-20 CFD

41 4.13 Heat transfer equation with backward finite differences scheme T i,j = µt i+1,j+1 + (1 + 2µ)T i+1.j µt i+1,j 1. (4.69) The linear algebraic equation system written in matrix form for the heat transfer equation is then stated as follows: AT i+1 = T i + b i+1, with (4.70) 1 + 2µ µ T i,1 µ 1 + 2µ µ T T i = i,2 0 µ 1 + 2µ µ.. 0, A = ,. 0. T i,n µ 1 + 2µ µ µ 1 + 2µ b i+1 = µα i+1 0. µβ i+1. (4.71) In this case we use an implicit scheme to obtain the numerical solution Stability analysis As for Section we replace the temperature T by the following expressions: By replacing this equation in Eq we obtain: T i,j = exp(ikx j ), (4.72) T i+1,j = λt i,j. (4.73) T i,j = µt i+1,j+1 + (1 + 2µ)T i+1,j µt i+1,j 1 (4.74) and after re-writting Eq. 4.74, the stability factor λ is given as: λ = µ(1 cos(k x)) = µ sin 2 (0.5k x). (4.75) In case of µ > 0 and λ < 1 all combinations of time and grid steps are stable. This scheme is called an unconditionally stable scheme. CFD 4-21

42 Finite Differences Method Numerical solution of the heat transfer equation using backward finite differences scheme When the used scheme is unconditionally stable, various combinations of time steps and grid sizes should lead to a stable solution. In comparison to the conditionally stable scheme we use the same x and t values. Fig. 4.7 shows the numerical solution of the backward finite differences scheme used. Figure 4.7: Different choices of time and grid spacing have no influence to the stability of the solution. Both choices of time steps and grid size leads to stable solutions Crank-Nicolson Scheme Another popular scheme is the so-called Crank-Nicolson scheme. It is constructed from the implicit and the explicit schemes and is used to solve the heat transfer equation or similar equations. We subtract Eq from Eq and obtain: T i+1,j T i,j = = µt i,j+1 + (1 2µ)T i,j + µt i,j 1 + µt i+1,j+1 (1 + 2µ)T i+1,j + µt i+1,j 1 T i+1,j T i,j = 1 2 µ(t i+1,j+1 2T i+1,j + T i+1,j 1 + T i, j + 1 2T i,j + T i,j 1 ). (4.76) This equation system is written in matrix form as: 4-22 CFD

43 4.14 Crank-Nicolson Scheme BT i+1 = CT i (b i + b i+1 ), (4.77) 1 + µ 1µ 0 2 1µ 1 + µ 1µ B = , µ µ 1 + µ µ µ µ 1 µ 1µ C = (4.78) µ µ 1 µ 2 The analysis of the stability leads to the stability factor λ: λ = 1 µ sin2 (0.5k x) 1 + 2µ sin 2 (0.5k x), (4.79) with µ > 0 the stability factor will be λ 1 for any combination of time and grid step and will lead to an unconditionally stable solution. The truncation error of this approximation are of the order ( x) 2 and ( t) 2. The numerical solution of the heat transfer example using the Crank-Nicolson scheme is shown in Fig CFD 4-23

44 Finite Differences Method Figure 4.8: Numerical solution of the heat equation using the Crank-Nicolson scheme Exercise Self-reflection 4.1 Stability analysis: 1. Make a table of the error for a second order approximation of the first derivative for the function f(x) = sin(x), x = 1. (4.80) Use the central finite difference formula (Eq. 4.27). 2. Derive formula (4.28) and (4.29) using the Taylor series and the finite difference formulas presented so far. 3. What kind of finite difference formulas are Eqs and 4.32? FDS, BDS or CDS and why? 4. Start with Eq and proof that Eq is right (analogue to ) CFD

45 4.16 Literature 4.16 Literature References [1] J. H. Ferziger & M. Peric. Computational Methods for Fluid Dynamics. Springer Berlin Heidelberg, [2] Institut für Mikroelektronik an der TU Wien. Numerical Discretization Schemes URL: [3] C. Hirsch. Numerical Computation of Internal and External Flows: The Fundamentals of Computational Fluid Dynamics. Butterworth-Heinemann, CFD 4-25

46 Part II Practical Part CFD 11-1

47 14 The blockmesh tool Learning objectives After processing the practical part of this chapter you will be able to use blockmesh as a meshing tool mesh a block by a given geometry CFD 14-1

48 The blockmesh tool Figure 14.1: (Source: [1], modified) In Ch. 13 we already learned how to use OpenFOAM as a simulation toolbox and we ran some tutorials. In the following chapters we will go step by step through the different processes of a simulation. As shown in Fig. 14.1, we will now focus on the pre-processing step, especially the meshing tool blockmesh Meshing tools In OpenFOAM version two main meshing tools are present, i.e. snappyhexmesh and blockmesh. Both are illustrated in Fig The snappyhexmesh tool (a) needs an STL surface (here a 2D car) and adjusts the mesh shape and resolution according to it. Normally snappyhexmesh creates a three dimensional mesh but for clarification we use a 2D example. The mesh properties can be adjusted in a dictionary, so that snappyhexmesh refines the mesh closer to the surface. With the help of this algorithm, an unstructured mesh can be created around very complex geometries. In this tutorial we will focus on the blockmesh tool (Fig: 14.2 (b)). It gives the basics of creating a structured numerical mesh. Contrary to the snappyhexmesh tool, the cell distribution has to be completely user defined around any geometry. Working with blockmesh requires the steps shown below. Have a look at the provided video (4_blockMesh) alongside Ex Example 14.1 How to work with blockmesh. It is dictionary based Dictionary and mesh are located under constant/polymesh (this may differ for newer versions of OpenFOAM) The blockmesh command has to be executed from the case directory After the blockmeshcase was copied to the $FOAM_RUN directory, we go to the case folder by typing: 14-2 CFD

49 14.1 Meshing tools (a) A snappyhexmesh[2] (b) A blockmesh Figure 14.2: The two meshing options in OpenFOAM 1. cd blockmeshcase We open the dictionary file with our preferable texteditor (vi, gedit, emacs...) and configure it to our needs: 2. cd constant/polymesh 3. gedit blockmeshdict Example 14.2 How to edit the blockmeshdict. A nice description on how to use the blockmesh utility is given in [2]. 1. Create all the vertices (the first vertex has the number 0, the second vertex number 1 etc.) 2. Create the blocks with the given vertices, while sticking to the rule of a right handed coordinate system: Each block has a local coordinate system (x 1, x 2, x 3 ) that must be right-handed. A right-handed set of axes is defined such that to an observer looking down the O z axis, with O nearest them, the arc from a point on the O x axis to a point on the O y axis is in a clockwise sense. [2] After saving all the changes, we can run the mesher from the case directory: 4. cd../../ 5.a. blockmesh The command output is then written to the screen. CFD 14-3

50 You can store the command output into a file, so that later on you can still lookup what happened while creating a mesh. (Good for locating mistakes and errors in your dictionary) The blockmesh tool So better write: 5.b. blockmesh > log.blockmesh & We can now open the log.blockmesh file anytime to see the output. In case the execution of the command takes longer than a few seconds, the tail command is very helpful. Type tail -f log.blockmesh and the output of the blockmesh command will be printed to the screen and is updated at runtime. You can end this command by pressing Ctrl+c. The mesh generation is finished now. It can be found in the constant/polymesh folder. After creating a mesh, always run the checkmesh command. Its an Open- FOAM tool to check the quality of a mesh and will give errors in case something went wrong. 6. checkmesh > log.checkmesh & Now check if the mesh was created without any errors ( mesh ok in the log.checkmesh file). Use parafoam as your post-processing tool. 7.a. parafoam For checking block and vertice structures, use the following command: (After that command, blockmesh has to be rerun). 7.b. parafoam -block Exercise 14.1 Creating your own blockmesh. In the example 14.1 it is shown how to create a blockmesh. Now it is your turn. 1. Copy the blockmeshcase from this chapter to your $FOAM_RUN folder CFD

51 14.1 Meshing tools 2. Create your own blockmesh by editing the blockmeshdict file. The geometry is shown in Fig Use the given order to define your blocks. Be aware of the fact that OpenFOAM starts counting from 0, so that you have to create five blocks (0 to 4). 3. Hand in your blockmeshcase folder with your results. 1. The coordinate of the bottom left corner should be (0 0 0). 2. One cell into the third dimension. 3. Use example case as base file. 4. You can first run the provided blockmesh to see what is going on. Then change vertices and blocks according to your needs. 5. The final result should look like Fig (except for the number of cells) Figure 14.3: Geometry of the blockmesh Self-reflection 14.1 How does OpenFOAM know how to create a block? And how is the local coordinate system defined? CFD 14-5

52 The blockmesh tool Figure 14.4: Final result of the blockmesh 14.2 Lessons learned The blockmesh utility gave a first impression on creating structured grids for numerical purposes. In the dictionary one has to define all vertices and blocks to describe the mesh geometry. To visualize and investigate the created grid, we learned how to use parafoam. By choosing the Surface With Edges property in the drop-down list, all vertices on the outer surfaces can be seen. This makes it easy to understand the cell distribution inside the domain. The checkmesh utility is a nice tool to understand the quality of the mesh. In case no errors are detected by this tool, the mesh is defined correctly. Be aware of the fact that this does not mean that your simulation is creating the correct results. It just means that OpenFOAM is able to perform numerical calculations for your spatial discretization. If your mesh contains errors, a simulation will crash instantly CFD

53 14.3 Literature 14.3 Literature References [1] OpenFOAM. The Open Source Computational Fluid Dynamics (CFD) Toolbox URL: [2] CFD Direct. OpenFOAM v5 User Guide: 5.3 Mesh generation with blockmesh URL: CFD 14-7

54

55 15 Advanced blockmesh Learning objectives After processing the practical part of this chapter you will have created a curved geometry by using blockmesh be able to apply cell grading towards surfaces be able to create boundary patches inside blockmesh CFD 15-1

56 Advanced blockmesh Figure 15.1: (Source: [1], modified) This chapter will focus on further parts of the blockmesh utility. After giving a general overview in Ch. 14, the attention will now be set towards cell grading at round obstacles with different boundary conditions Cell grading in blockmesh Areas with high flow gradients like near wall regions need higher cell resolutions than the freestream region. Simply applying this high clustering of cells in the whole domain is possible but computationally unfeasible. Therefor, cell grading has to be applied to the mesh to get a good compromise between accuracy and numerical costs. For the blockmesh in OpenFOAM, the cell grading properties are again specified in the blockmeshdict. The grading is defined in the blocks setup as follows: Example 15.1 Cell grading in the blockmeshdict. vertices (... ); blocks (... simplegrading (3 1 1) ); Here, a grading of 1 results in a uniform distribution of the cells (axis x 2 and x 3 ). A grading of 3 means that the cell at the end of that block coordinate axis is three times larger than at the beginning of that block (axis x 1 ). A distinct illustration is shown in Fig CFD