Solution-Manual-Fundamentals-Of-Fluid-Mechanics-7th-Edition-Munson

Numerical methods using digital computers are, of course, commonly utilized to solve a wide variety of flow problems. As discussed in Chapter 6, although the differential equations that govern the flow of Newtonian fluids [the Navier–Stokes equations (Eq. 6.127)] were derived many years ago, there are few known analytical solutions to them. However, with the advent of highspeed digital computers it has become possible to obtain approximate numerical solutions to these (and other fluid mechanics) equations for a wide variety of circumstances. Computational fluid dynamics (CFD) involves replacing the partial differential equations with discretized algebraic equations that approximate the partial differential equations. These equations are then numerically solved to obtain flow field values at the discrete points in space and/or time. Since the Navier–Stokes equations are valid everywhere in the flow field of the fluid continuum, an analytical solution to these equations provides the solution for an infinite number of points in the flow. However, analytical solutions are available for only a limited number of simplified flow geometries. To overcome this limitation, the governing equations can be discretized and put in algebraic form for the computer to solve. The CFD simulation solves for the relevant flow variables only at the discrete points, which make up the grid or mesh of the solution (discussed in more detail below). Interpolation schemes are used to obtain values at non-grid point locations. VA.1 Pouring a liquid CFD can be thought of as a numerical experiment. In a typical fluids experiment, an experimental model is built, measurements of the flow interacting with that model are taken, and the results are analyzed. In CFD, the building of the model is replaced with the formulation of the governing equations and the development of the numerical algorithm. The process of obtaining measurements is replaced with running an algorithm on the computer to simulate the flow interaction. Of course, the analysis of results is common ground to both techniques. CFD can be classified as a subdiscipline to the study of fluid dynamics. However, it should be pointed out that a thorough coverage of CFD topics is well beyond the scope of this textbook. This appendix highlights some of the more important topics in CFD, but is only intended as a brief introduction. The topics include discretization of the governing equations, grid generation, boundary conditions, application of CFD, and some representative examples. A.2 Discretization The process of discretization involves developing a set of algebraic equations (based on discrete points in the flow domain) to be used in place of the partial differential equations. Of the various discretization techniques available for the numerical solution of the governing differential equations, the following three types are most common: (1) the finite difference method, (2) the finite element (or finite volume) method, and (3) the boundary element method. In each of these methods, the continuous flow field (i.e., velocity or pressure as a function of space and time) is described in terms of discrete (rather than continuous) values at prescribed locations. Through this technique the differential equations are replaced by a set of algebraic equations that can be solved on the computer. For the finite element (or finite volume) method, the flow field is broken into a set of small fluid elements (usually triangular areas if the flow is two-dimensional, or small volume elements if the flow is three-dimensional). The conservation equations (i.e., conservation of mass, momentum, and energy) are written in an appropriate form for each element, and the set of resulting 725 Γi = strength of vortex on ■ Figure A.1 Panel method for ith panel flow past an airfoil. algebraic equations for the flow field is solved numerically. The number, size, and shape of elements are dictated in part by the particular flow geometry and flow conditions for the problem at hand. As the number of elements increases (as is necessary for flows with complex boundaries), the number of simultaneous algebraic equations that must be solved increases rapidly. Problems involving one million to ten million (or more) grid cells are not uncommon in today’s CFD community, particularly for complex three-dimensional geometries. Further information about this method can be found in Refs. 1 and 2. For the boundary element method, the boundary of the flow field (not the entire flow field as in the finite element method) is broken into discrete segments (Ref. 3) and appropriate singularities such as sources, sinks, doublets, and vortices are distributed on these boundary elements. The strengths and type of the singularities are chosen so that the appropriate boundary conditions of the flow are obtained on the boundary elements. For points in the flow field not on the boundary, the flow is calculated by adding the contributions from the various singularities on the boundary. Although the details of this method are rather mathematically sophisticated, it may (depending on the particular problem) require less computational time and space than the finite element method. Typical boundary elements and their associated singularities (vortices) for twodimensional flow past an airfoil are shown in Fig. A.1. Such use of the boundary element method in aerodynamics is often termed the panel method in recognition of the fact that each element plays the role of a panel on the airfoil surface (Ref. 4). The finite difference method for computational fluid dynamics is perhaps the most easily understood of the three methods listed above. For this method the flow field is dissected into a set of grid points and the continuous functions (velocity, pressure, etc.) are approximated by discrete values of these functions calculated at the grid points. Derivatives of the functions are approximated by using the differences between the function values at local grid points divided by the grid spacing. The standard method for converting the partial differential equations to algebraic equations is through the use of Taylor series expansions. (See Ref. 5.) For example, assume a standard rectangular grid is applied to a flow domain as shown in Fig. A.2. This grid stencil shows five grid points in x–y space with the center point being labeled as i, j. This index notation is used as subscripts on variables to signify location. For example, ui1, j is the u component of velocity at the first point to the right of the center point i, j. The grid spacing in the i and j directions is given as ¢x and ¢y, respectively. To find an algebraic approximation to a first derivative term such as 0u0x at the i, j grid point, consider a Taylor series expansion written for u at i 1 as u b ¢x a 2 b 1¢x22 a 03ub 1¢x23 p ui1, ji, j 2! 0x3 i, j 3! (A.1) x ■ Figure A.2 Standard rectangular grid. A.3 Grids Solving for the underlined term in the above equation results in the following: a 0u ui1, j ui, j b ¢ O1¢x2 (A.2) 0x i, j x where O1¢x2 contains higher order terms proportional to ¢x, 1¢x22, and so forth. Equation A.2 represents a forward difference equation to approximate the first derivative using values at i 1, j and i, j along with the grid spacing in the x direction. Obviously in solving for the 0u0x term we have ignored higher order terms such as the second and third derivatives present in Eq. A.1. This process is termed truncation of the Taylor series expansion. The lowest order term that was truncated included 1¢x22. Notice that the first derivative term contains ¢x. When solving for the first derivative, all terms on the right-hand side were divided by ¢x. Therefore, the term O1¢x2 signifies that this equation has error of “order 1¢x2,” which is due to the neglected terms in the Taylor series and is called truncation error. Hence, the forward difference is termed first-order accurate. Thus, we can transform a partial derivative into an algebraic expression involving values of the variable at neighboring grid points. This method of using the Taylor series expansions to obtain discrete algebraic equations is called the finite difference method. Similar procedures can be used to develop approximations termed backward difference and central difference representations of the first derivative. The central difference makes use of both the left and right points (i.e., i 1, j and i 1, j) and is second-order accurate. In addition, finite difference equations can be developed for the other spatial directions (i.e., 0u0y) as well as for second derivatives 102u0x22, which are also contained in the Navier–Stokes equations (see Ref. 5 for details). Applying this method to all terms in the governing equations transfers the differential equations into a set of algebraic equations involving the physical variables at the grid points (i.e., ui, j , pi, j for i 1, 2, 3, p and j 1, 2, 3, p , etc.). This set of equations is then solved by appropriate numerical techniques. The larger the number of grid points used, the larger the number of equations that must be solved. A student of CFD should realize that the discretization of the continuum governing equations involves the use of algebraic equations that are an approximation to the original partial differential equation. Along with this approximation comes some amount of error. This type of error is termed truncation error because the Taylor series expansion used to represent a derivative is “truncated” at some reasonable point and the higher order terms are ignored. The truncation errors tend to zero as the grid is refined by making ¢x and ¢y smaller, so grid refinement is one method of reducing this type of error. Another type of unavoidable numerical error is the so-called round-off error. This type of error is due to the limit of the computer on the number of digits it can retain in memory. Engineering students can run into round-off errors from their calculators if they plug values into the equations at an early stage of the solution process. Fortunately, for most CFD cases, if the algorithm is set up properly, round-off errors are usually negligible. A.3 Grids CFD computations using the finite difference method provide the flow field at discrete points in the flow domain. The arrangement of these discrete points is termed the grid or the mesh. The type of grid developed for a given problem can have a significant impact on the numerical simulation, including the accuracy of the solution. The grid must represent the geometry correctly and accurately, since an error in this representation can have a significant effect on the solution. The grid must also have sufficient grid resolution to capture the relevant flow physics, otherwise they will be lost. This particular requirement is problem dependent. For example, if a flow field has small-scale structures, the grid resolution must be sufficient to capture these structures. It is usually necessary to increase the number of grid points (i.e., use a finer mesh) where large gradients are to be expected, such as in the boundary layer near a solid surface. The same can also be ■ Figure A.3 Structured grids. (a) Rectangular grid. (b) Grid around a parabolic surface. said for the temporal resolution. The time step, ¢t, used for unsteady flows must be smaller than the smallest time scale of the flow features being investigated. VA.2 Dynamic grid Generally, the types of grids fall into two categories: structured and unstructured, depending on whether or not there exists a systematic pattern of connectivity of the grid points with their neighbors. As the name implies, a structured grid has some type of regular, cohere