Additionally, for solution procedures which require higher order derivatives or high order of accuracy, this method can be better suited than other methods despite the limitation of mesh regularity.
We will start with this to build an understanding of the accuracy of the various formulations. The expression in Equation 5. Equation 5. A geometrical interpretation of the three difference schemes is shown in Figure 5. It can be seen that the central difference approximation provides a better representation of the slope of the curve at the point of interest which is the first derivative.
Figure Geometric interpretation of difference formulae. In this example, we will revisit the fin example presented in Section 3. The objective is to show how the finite difference gradient boundary condition can be applied. The objective is to solve the equation:. It indicates that the heat flux at the tip of the fin is zero, indicating that temperature gradient is zero.
At point 1, the temperature is specified by the base temperature and thus we do not need to elaborate further as we have:. However, for point 6, there is a problem if we attempt to use the central difference scheme as there is no grid point to the right of point 6. Remember that we need to apply the gradient boundary condition at this point. We could be tempted to use a backward difference formula Equation 5. At the first instance, this might seem a good idea. This can be used with the other set of equations to solve the problem, however, an error is found near that point which propagates to the rest of the domain.
This error results from the fact that the discretisation at point 6 is first order accurate which is not consistent with the second order accurate scheme for the internal nodes. Figure 5. Figure Solution with first order discretisation of boundary conditions. To enable this, we add an imaginary point after nodes 6, lets say node 7, and we work out the value of the field variable at this node in terms of the values at nodes 5 and 6. This then allows us to formulate a central difference equation.
Thus, with reference to Figure 5. Which is clearly different from that obtained using the first order backward formula above. Using this equation together with the other equations for the rest of the nodes leads to the solution shown in Figure 5.
Figure Solutions with second order discretisation of boundary conditions. However, finite difference formulae for the first derivative can be formulated using any number of adjacent points with the order of approximation increasing with the number of points. In a particular numerical scheme, a balance need to be found between the order of accuracy and the number of grid points involved in the computation. This will dictate the computational memory and effort both need to be kept to a minimum for a given overall solution accuracy.
It is usually difficult to devise rules that govern the optimum combination of discretisation accuracy and number of grid points as these vary from problem to problem and only experience with a large number of cases can produce guidance in this respect. Consider the stencil shown in Figure 5. Figure Stencil for second order backward difference formula. If we are required to formulate a second order backward difference formula, we use the following procedure:. Comparing this to Equation 5.
This gives the following three Equations:. The above procedure with undetermined coefficients can by systematically used to obtain finite difference formulae for all derivatives at any required degree of accuracy. As an exercise for the student, try to derive a second order forward formula using a three point stencil to give the following:. The way that partial derivatives of a function of several variables can be discretised using the same methods of the previous sections for each variable and for each coordinate direction.
Our objective is to discretise this equation for a two dimensional domain. The index m is incremented in the x-direction, while the index n is incremented in the y direction. If we apply the second order difference formula of Equation 3. A similar equation can be obtained for each internal grid node. For boundary nodes, the discretisation will depend on the type of boundary conditions applied. If a given temperature boundary conditions is to be applies, then the equation for that node is deleted and replaced by the given value for the temperature.
If a flux boundary condition is to be applied, then a procedure similar to that explained in Section 5. In this case a fictious node is added and the value of the temperature at that node is computed as a function of the internal nodes. The same central difference scheme then can be used at boundary nodes. The resulting discretisation leads to a number of equations, which equals the number of unknown temperatures. These equations can then be solved simultaneously for the unknown nodal temperatures.
We will discuss in more detail solution methods to systems of algebraic equations in Chapter 8. The case described here illustrates the simplicity of extension of the finite difference method to two dimensions.
Extension to three-dimensions is equally simple and follows the same logic. In this case, there are two possible approaches.
The first is to generate a regular grid as shown in Figure 5. All internal nodes can be treated in the usual way as described in the previous section. For nodes on the boundary, some grid cells will not be complete. In this case, a special procedure is required for the nodes surrounding those cells.
There are several ways to approximate the discretised equations at those nodes. These might include shifted stencils that use the inner nodes or fictious nodes outside the domain. In either case, the fact that cell is not complete need to be taken into account. The other alternative is to generate a body fitted grid see Chapter In this case, the grid is what is termed the physical plane is mapped onto a regular grid in a computational plane See Figure 5.
The differential equations are then transformed onto the computational plane using the same mapping. Once this is done, all the standard difference formulae discussed before can be applied in the computational domain. Once the discrete equations are constructed, they can be solved and the results mapped back to the physical domain, or the discrete equations are mapped back to the physical domain and solved.
We then outlined the history of the method, its advantages. We also outlined the theoretical background and the framework by which the Galerkin weighted residual method is used to discretise differential equations. In this Chapter, we will present the method in more detail. There is a wide volume of literature discussing the theoretical background of the Finite Element Method and its application to engineering problems.
If you are interested in more detail, refer to one of the reference listed in the back of this Chapter. The structure is then re-assembled after each element has been analysed.
The technique was developed further in what is now known as the Finite Element Method between and , mainly in the field of structural dynamics. The technique was then expanded to solve field problems in the s See Zeinkiewicz Nowadays, the Finite Element Method has been put in an engineering rigorous framework with precise mathematical conditions for existence, convergence and error bounds.
In this book, we will not concentrate on the mathematical derivation of the method, but rather on its application for the discretisation of differential equations. We could choose a model of the full Navier-Stokes equations or any of the approximation levels. We then require solving the mathematical model over a given physical domain with some boundary conditions. The first step is to discretise the spatial domain into non-overlapping elements or sub-regions.
The Finite Element Method allows a variety of element shapes, for example, triangles, quadrilaterals in two dimensions and tetrahedral, hexahedral, pentahedral, and prisms in three dimensions. Each element is formed by the connection of a certain number of nodes, with the number of nodes in an element depending on the type of the element Figure 6.
The number of nodes in each element does not depend only on the number of corner points in the element, but also on the type of the element interpolation function as we will explain in the next section. Once a mesh is generated, we choose the type of interpolation function that represents the variation of the field variable over the element. A clear distinction can be seen here from the Finite Difference Method. In the Finite Difference Method, we were only interested in the values of the field variable at grid nodes, and no information was required for the behaviour between the nodes.
We may have implicitly assumed that it is linear, but we did not have to do that. The next stage is to determine the matrix equations that express the properties of the individual element by forming a left hand side matrix and a load vector. A typical left hand side matrix and a load vector for a one dimensional element may look like:. The next stage is to assemble the element equations to obtain a system of simultaneous equations that can be solved for the unknown field variables at the mesh nodes.
The final system of equations will be represented in matrix notation as:. In the next section, we will illustrate those steps using a simple example. But before we do that, we will present more information about the element shape functions and the Finite Element discretisation process of differential equations. In this process, the variables are represented in a piece- wise manner over the domain.
By dividing the solution domain into elements, and approximating the solution over these elements using a suitable known function, a relationship between the elements and the differential equation is established. The functions used to represent the variation of the solution within each element are called shape functions, or interpolation functions or basis functions. Typically, polynomial functions are used because they can be easily integrated or differentiated.
The accuracy of the results can be improved by increasing the order of the polynomial used. These are the simplest elements and their discussion will help illustrate the basic principles.
The simplest element has a piece-wise linear interpolation function and contains two nodes. If the x-coordinate is discretised to a set of elements each containing two nodal points, and the function is approximated using linear variation between each two nodes, then the variation for a typical element is shown in Figure 6. The linear variation over the element of Figure 6. Each shape function will have a maximum of unity and the corresponding point and varies linearly to zero at the other point as shown in Figure 6.
This is essentially the same as the linear variation in Figure 6. We can use the expression in Equation 6. Noting that xj — xi is the length of the element. Then Equation 6. This indicates that the first derivative of the function over the domain will be stepwise constant over the entire domain indicating that it is not a continuous function. Higher order shape functions can be obtained by using more nodes within the element. For example a quadratic shape function can be obtained by using three nodes in the element as shown in Figure 6.
Exercise: dT e As an exercise, the student is required to derive a matrix expression for using the quadratic element dx shape function similar to that derived for the linear element in Equation 6. The most popular element for arbitrary two dimensional geometries is the triangular element. This is mainly because triangular meshes are relatively easier to generate and to control their quality. We will discuss methods if generating triangular grids in Chapter 9.
In this section, we will present the shape functions for these elements. A two dimensional linear element is shown in Figure 6. We can represent the variation of the field function on this element using a linear polynomial follows:. Discover the truth 69 at www. Equations 6. Substituting the values of the three coefficients from Equations 6.
Additionally, at any point in the triangle:. It should be noted, as in the linear one dimensional element, that the first derivative of the field function is constant within the triangular two dimensional element.
Higher order triangular elements can be obtained by placing more nodes within the element and using higher order polynomials to obtain the shape functions. As most CFD methods use the linear elements, we will not analyse higher order elements in detail and the interested reader can refer to one of the books in the reference list of this book.
It is sufficient in this book to show some of these high order elements. Figure 6. Bilinear quadrilateral elements have four nodes located at the vertices as shown in Figure 6.
A quadrilateral Finite Element grid may look like a Finite Difference grid. But in the Finite Difference mesh, the grid need to be orthogonal, that is all grid lines intersect at right angles, whereas in the Finite Element mesh, this restriction is removed and each element can have a unique shape.
In addition, In the Finite Difference mesh, each grid point has to have the same number of neighbours. For example, in two dimensional grids, each grid point should be surrounded by four points, while in Finite Element Method; a point can have an arbitrary number of neighbours.
Unstructured quadrilateral grids with arbitrary number of neighbours for each node can be generated using paving techniques that will be discussed in Chapter 9. Then the shape functions are defined on the master element, or isoperimetric element. The inverse transformation function can be used to transform the discretised system to the actual physical space.
Figure Isoparametric element mapping for a quadrilateral element. We will not go into the derivation of these shape functions and the interested reader is referred to Zeinkeiwichz However, we will state she shape functions in terms of the local coordinates. These take the form:. To obtain a transformation between the physical coordinates and the local or isoparametric coordinates, we can use the expression in Equation 6. The result in Equation 6. Higher order quadrilateral elements can be obtained in a similar manner to higher order triangular elements by placing more nodes in the element and using higher order polynomials to represent the shape functions.
The additional spatial coordinate allows for the possibility of more type of elements. The three most popular three dimensional linear elements are tetrahedral, hexahedral and prismatic elements shown in Figure 6. Because of the larger number of nodes in three dimensional elements, the amount of data to required to establish the shape functions become significantly greater than two-dimensional elements.
Consequently, the amount of computational time and memory required become significantly greater. This means that care should be exercised when generating three-dimensional grids to get the most appropriate element for a particular application which produces the least possible computational effort. Detailed analysis of the shape functions will not be presented in this introductory book and the interested reader is referred to Zeinkeiwickz It was also mentioned that the method of weighted residuals, particularly the Galerkin method is the most popular one.
We are not interested in the theoretical derivation of this method. Our interest here is mainly on how to apply this method to transform a given differential equation into its equivalent discrete form over a Finite Element grid. We will illustrate this by way of an example for a one-dimensional problem that we addressed in Chapters 3 and 5.
That is the fin problem of Section 3. The choice of this example serves at least two purposes. The first is the illustration of the Finite Element discretisation using the Galerkin method. The second, if we use the same regular grid that was used for the Finite Difference Discretisation, we can compare the two discretisations. We can start the discretisation using the shape functions of the linear one-dimensional element of Equations 6.
However to follow a more general procedure that can be used in two and three-dimensional isoprametric elements, we will express the one-dimensional shape functions using isoparametric one-dimensional element. If we pick any of the elements in Figure 6. The next stage is to use the Galerkin formulation outlined in Section 3. The subscript k indicates the nodes in the domain.
This means that by using the shape functions at all nodes one at a time, we obtain a set of equations which equals the number of nodes.
Since the first derivative of the linear shape function Equation 6. To be able to represent the second derivative, we need to obtain the so called weak formulation, by which the first term of Equation 6. The end result when the equations are assembled together is that this term cancels out for all internal nodes and will be either 1 or -1 at the boundary nodes.
Substituting into Equation 6. If we weight the equation with Nj and repeat the integration, we obtain the following equation for the second node of element 1, where the outward normal here is -1 because it is in the opposite direction:. No assumption was made so far that this should be equal for all elements. Thus this formulation is general for arbitrary distribution of grid nodes.
Once we have chosen the grid, we can substitute the length of each element in Equation 6. Those matrices can then be assembled in the global system matrix as we will see next. The contributions from all elements can be assembled by placing the matrix for each element in the global matrix. This means that for internal nodes where the node is shared between two elements, the contributions need to be overlapped, or in other words added together.
The process is explained schematically in Figure 6. Each colour represents a column of the assembled matrix. Notice that the two middle columns and rows each containing contributions from two neighbouring elements. Following the same procedure, the system matrix can be assembled for the six nodes from the contribution matrices of Equation 6.
Comparing the system of Equations in Equation 3. This can be assured by multiplying all terms on the left and right of Equation 6.
We obtain Equation 3. This means that the solution for this system is the same as Equation 3. This leads to the verification of Equations 6. It also leads us to the conclusion that the discretisation using linear one-dimensional elements is equivalent to the second order accurate central difference scheme of the Finite Difference formulation if we used equal grid spacing.
However, the important point is that the Finite Element scheme can be used in a straight forward manner using irregular grids. In additions, the incorporation of higher order schemes by using higher order elements is straightforward. You might think that the derivation of the Finite Element formulation is cumbersome compared to the Finite Difference formulation, and at the end, we obtained the same results as the simple finite difference scheme. That is correct. However, the advantages of the Finite Element formulations become more apparent in two and three dimensional problems when arbitrary geometries are handled.
It was mentioned that it can be viewed as a special case of the weighted residual method, were the weighting function is 1. We also presented an alternative way of obtaining a Finite Volume discretisation using the integral form of the differential equation. The solution domain needs to be divided into non-overlapping cells surrounded by boundary edges in two-dimensions or boundary faces in three-dimensions. Integrating by parts leads the volume integral to equal a flux through the volume boundary.
Working out these fluxes in terms of the unknown field variables at grid points either at the cell corners or centres leads to a system of algebraic equations which can be solved for the unknown field variables. In this Chapter, we will explain the principles of the Finite Difference Method through worked examples that start by one-dimensional simple models. We will build these up gradually to more complex models and multiple dimensions.
We will also compare the discretisation resulting from the Finite Volume formulation to that of the Finite Difference and Finite Element formulations to the fin problem. This equation is known as the Stokes equation, which contains the pressure and viscous terms in the Navier-Stokes equations. In one-dimensional steady state formulation, it takes the form:.
To simplify matters further, we will ignore the pressure term for the moment. Once this is done, the equation resembles the heat conduction equation which also includes a source term. Here, k is the thermal conductivity, S is the source or internal heat generation per unit volume and T is the temperature. To derive a Finite Volume discretisation, we will use a three point stencil as shown in Figure 7. We are seeking to derive the discretisation for the middle point P. We have also used the conventional Finite Volume notation for the surrounding point of E and W for east and west.
The next stage is to to integrate Equation 7. Figure Finite Volume stencil for the 1D conduction equation. To enable the calculation of the temperature gradient at the east and west boundaries of the cell, we need to make an assumption of the temperature profiles within the grid. A reasonable assumption would be that the temperature is varying linearly between grid points as shown in Figure 7.
This allows a straight forward evaluation of the first two terms of Equation 7. For the source term, we will assume that the average value S prevails over the control volume.
Example 7. Equation 7. Thus the finite volume discretisation in Equation 7. We also found in Chapter 6, for the same example that this is also equivalent to the Finite Element discretisation over the same grid size. In fact, the use of non-uniform grid spacing is often desirable as it enables the effective use of computing power. In general, an accurate solution will be obtained when the grid is sufficiently fine.
However, there is no need to use fine grids in regions where the field variable changes slowly with the space coordinate. On the other hand, fine grids are required when the variation is steep. Similar equations can be formulated for all internal cells as discussed above. For boundary points, the last cell is not complete as shown in Figure 7. Boundary conditions need to be applied. This is done depending on the type of boundary conditions. If the value of the field variable is given, then there is no need to formulate a flux calculation there as the equation for that cell will be replaced by the given temperature for the end node.
For a give gradient of the flow field, a treatment similar to that performed with the Finite Difference methods in Chapter 5 is performed. The approach proposed here, which ensures second order accuracy of the boundary conditions discretisation is to extend the domain by an additional virtual node which completes the cell as shown in Figure 7.
The value of the field variable at the virtual node is computed in terms of the two neighbouring internal nodes using the inner cell scheme. Once this is done, an equation for the boundary node can be established as in Chapter 5. These are similar to the diffusion terms in the flow equations. Flow equations however contain convection terms in addition to pressure and source terms.
We will keep the simple approach to explain the basic principles here by presenting the treatment of a simplification to the flow equations by including the convection and diffusion terms only. Thus the equation we will consider is:.
In Equation 7. The term u on the left hand side indicates that we are considering convection by this component of the velocity field.
Interfacing fundamental concepts and practical methods of scienti? In one approach, theory and implementation are kept complementary and presented in a sequential fashion.
In another approach, the coupling involves deriving computational methods and simulation algorithms, and translating equations into computer code - structions immediately following problem formulations. Seamlessly interjecting methods of scienti?
Instead of following an approach that focuses on mathematics first, this book allows you to develop an intuitive physical understanding of various fluid flows, including internal compressible flows with simultaneous area change, friction, heat transfer, and rotation. Drawing on over 40 years of industry and teaching experience, the author emphasizes physics-based analyses and quantitative predictions needed in the state-of-the-art thermofluids research and industrial design applications.
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Turbulent flow modeling has many applications in industry. The relevant numerical methods have advanced to the level that could be used by industry professionals to model many natural turbulent flows with acceptable accuracy. Save to Library Save. Create Alert Alert. Share This Paper. Background Citations. Methods Citations. Results Citations. Citation Type. Has PDF. Publication Type. More Filters. This book provides an accessible introduction to the basic theory of fluid mechanics and computational fluid dynamics CFD from a modern perspective that unifies theory and numerical computation.
This book is an accessible introduction to theoretical and computational fluid dynamics CFD , written from a modern perspective that unifies theory and numerical practice. For the third edition of this textbook , the authors have replaced the previous editions ' BASIC programs with MathCad , updated problems , a nd made some MR e : Warsi , Z. Theoretical and computational approaches. Third edition. ISBN ; From Theoretical frameworks and models require validation in order to determine their value.
Further details on the theoretical aspects of the non-Boussinesq equations can be found in [22] and [23]. Skip to content Many introductions to fluid dynamics offer an illustrative approach that demonstrates some aspects of fluid behavior, but often leave you without the tools necessary to confront new problems.
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