Solutions Manual for Fundamentals of Fluid Mechanics 7th edition by Munson Rothmayer Okiishi and Huebsch

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1 Solutions Manual for Fundamentals of Fluid Mecanics 7t edition by Munson Rotmayer Okiisi and Huebsc Link full download : ttps://digitalcontentmarket.org/download/solutions-manual-forfundamentals-of-fluid-mecanics-7t-edition-by-munson-rotmayer-okiisi-and-uebsc/ A.1 Introduction VA.1 Pouring a liquid Numerical metods using digital computers are, of course, commonly utilized to solve a wide variety of flow problems. As discussed in Capter 6, altoug te differential equations tat gov-ern te flow of Newtonian fluids [te Navier Stokes equations (Eq )] were derived many years ago, tere are few known analytical solutions to tem. However, wit te advent of ig-speed digital computers it as become possible to obtain approximate numerical solutions to tese (and oter fluid mecanics) equations for a wide variety of circumstances. Computational fluid dynamics (CFD) involves replacing te partial differential equations wit discretized algebraic equations tat approximate te partial differential equations. Tese equations are ten numerically solved to obtain flow field values at te discrete points in space and/or time. Since te Navier Stokes equations are valid everywere in te flow field of te fluid continuum, an analytical solution to tese equations provides te solution for an infinite num-ber of points in te flow. However, analytical solutions are available for only a limited number of simplified flow geometries. To overcome tis limitation, te governing equations can be discretized and put in algebraic form for te computer to solve. Te CFD simulation solves for te relevant flow variables only at te discrete points, wic make up te grid or mes of te solution (discussed in more detail below). Interpolation scemes are used to obtain values at non-grid point locations. CFD can be tougt of as a numerical experiment. In a typical fluids experiment, an exper-imental model is built, measurements of te flow interacting wit tat model are taken, and te results are analyzed. In CFD, te building of te model is replaced wit te formulation of te governing equations and te development of te numerical algoritm. Te process of obtaining measurements is replaced wit running an algoritm on te computer to simulate te flow inter-action. Of course, te analysis of results is common ground to bot tecniques. CFD can be classified as a subdiscipline to te study of fluid dynamics. However, it sould be pointed out tat a toroug coverage of CFD topics is well beyond te scope of tis textbook. Tis appendix igligts some of te more important topics in CFD, but is only intended as a brief introduction. Te topics include discretization of te governing equations, grid generation, bound-ary conditions, application of CFD, and some representative examples. A.2 Discretization Te process of discretization involves developing a set of algebraic equations (based on discrete points in te flow domain) to be used in place of te partial differential equations. Of te various discretization tecniques available for te numerical solution of te governing differential equations, te following tree types are most common: (1) te finite difference metod, (2) te finite element (or finite volume) metod, and (3) te boundary element metod. In eac of tese metods, te continuous flow field (i.e., velocity or pressure as a function of space and time) is described in terms of discrete (rater tan continuous) values at prescribed locations. Troug tis tecnique te dif-ferential equations are replaced by a set of algebraic equations tat can be solved on te computer. For te finite element (or finite volume) metod, te flow field is broken into a set of small fluid elements (usually triangular areas if te flow is two-dimensional, or small volume elements if te flow is tree-dimensional). Te conservation equations (i.e., conservation of mass, momentum, and energy) are written in an appropriate form for eac element, and te set of resulting 725

2 726 Appendix A Computational Fluid Dynamics i t panel U Γ i 1 Γ i Γ i+ 1 Γ i = strengt of vortex on i t panel Figure A.1 Panel metod for flow past an airfoil. algebraic equations for te flow field is solved numerically. Te number, size, and sape of elements are dictated in part by te particular flow geometry and flow conditions for te problem at and. As te number of elements increases (as is necessary for flows wit complex bound-aries), te number of simultaneous algebraic equations tat must be solved increases rapidly. Prob-lems involving one million to ten million (or more) grid cells are not uncommon in today s CFD community, particularly for complex tree-dimensional geometries. Furter information about tis metod can be found in Refs. 1 and 2. For te boundary element metod, te boundary of te flow field (not te entire flow field as in te finite element metod) is broken into discrete segments (Ref. 3) and appropriate singularities suc as sources, sinks, doublets, and vortices are distributed on tese boundary elements. Te strengts and type of te singularities are cosen so tat te appropriate boundary condi-tions of te flow are obtained on te boundary elements. For points in te flow field not on te boundary, te flow is calculated by adding te contributions from te various singularities on te boundary. Altoug te details of tis metod are rater matematically sopisticated, it may (depending on te particular problem) require less computational time and space tan te finite element metod. Typical boundary elements and teir associated singularities (vortices) for twodimensional flow past an airfoil are sown in Fig. A.1. Suc use of te boundary element metod in aerodynamics is often termed te panel metod in recognition of te fact tat eac element plays te role of a panel on te airfoil surface (Ref. 4). Te finite difference metod for computational fluid dynamics is peraps te most easily understood of te tree metods listed above. For tis metod te flow field is dissected into a set of grid points and te continuous functions (velocity, pressure, etc.) are approximated by dis-crete values of tese functions calculated at te grid points. Derivatives of te functions are approximated by using te differences between te function values at local grid points divided by te grid spacing. Te standard metod for converting te partial differential equations to algebraic equations is troug te use of Taylor series expansions. (See Ref. 5.) For example, assume a standard rectangular grid is applied to a flow domain as sown in Fig. A.2. Tis grid stencil sows five grid points in x y space wit te center point being labeled as i, j. Tis index notation is used as subscripts on variables to signify location. For example, u i1 1, j is te u component of velocity at te first point to te rigt of te center point i, j. Te grid spac-ing in te i and j directions is given as x and y, respectively. To find an algebraic approximation to a first derivative term suc as 0u/ 0x at te i, j grid point, consider a Taylor series expansion written for u at i 1 1 as u 5u 1 b i1 1, j i, j a 0u x 0 2 u 1 x u b 1 x23 0x i, j 1! 1 a 0x 2 i, j 2! 1 a 0x 3 3! b i, j 1 p (A.1) y i 1 i i + 1 j + 1 y j j 1 x x Figure A.2 Standard rectangular grid.

3 Solving for te underlined term in te above equation results in te following: u i1 1, j 0u b a 0x i, j 5 x 2u i, j A.3 Grids O1 x2 (A.2) were O1 x 2 contains iger order terms proportional to x, 1 x2 2, and so fort. Equation A.2 represents a forward difference equation to approximate te first derivative using values at i 1 1, j and i, j along wit te grid spacing in te x direction. Obviously in solving for te 0u/ 0x term we ave ignored iger order terms suc as te second and tird derivatives present in Eq. A.1. Tis process is termed truncation of te Taylor series expansion. Te lowest order term tat was truncated included 1 x2 2. Notice tat te first derivative term contains x. Wen solv-ing for te first derivative, all terms on te rigt-and side were divided by x. Terefore, te term O1 x2 signifies tat tis equation as error of order 1 x2, wic is due to te neglected terms in te Taylor series and is called truncation error. Hence, te forward difference is termed first-order accurate. Tus, we can transform a partial derivative into an algebraic expression involving values of te variable at neigboring grid points. Tis metod of using te Taylor series expansions to obtain discrete algebraic equations is called te finite difference metod. Similar procedures can be used to develop approximations termed backward difference and central difference representations of te first derivative. Te central difference makes use of bot te left and rigt points (i.e., i 2 1, j and i 1 1, j) and is second-order accurate. In addition, finite difference equations can be developed for te oter spatial directions (i.e., 0u/ 0y) as well as for second derivatives 10 2 u/ 0x 2 2, wic are also contained in te Navier Stokes equations (see Ref. 5 for details). Applying tis metod to all terms in te governing equations transfers te differential equations into a set of algebraic equations involving te pysical variables at te grid points (i.e., u i, j, p i, j for i 5 1, 2, 3, p and j 5 1, 2, 3, p, etc.). Tis set of equations is ten solved by appro-priate numerical tecniques. Te larger te number of grid points used, te larger te number of equations tat must be solved. A student of CFD sould realize tat te discretization of te continuum governing equations involves te use of algebraic equations tat are an approximation to te original partial differential equation. Along wit tis approximation comes some amount of error. Tis type of error is termed truncation error because te Taylor series expansion used to represent a deriv-ative is truncated at some reasonable point and te iger order terms are ignored. Te trun-cation errors tend to zero as te grid is refined by making x and y smaller, so grid refine-ment is one metod of reducing tis type of error. Anoter type of unavoidable numerical error is te so-called round-off error. Tis type of error is due to te limit of te computer on te number of digits it can retain in memory. Engineering students can run into round-off errors from teir calculators if tey plug values into te equations at an early stage of te solution process. Fortunately, for most CFD cases, if te algoritm is set up properly, round-off errors are usually negligible. A.3 Grids CFD computations using te finite difference metod provide te flow field at discrete points in te flow domain. Te arrangement of tese discrete points is termed te grid or te mes. Te type of grid developed for a given problem can ave a significant impact on te numer-ical simulation, including te accuracy of te solution. Te grid must represent te geometry correctly and accurately, since an error in tis representation can ave a significant effect on te solution. Te grid must also ave sufficient grid resolution to capture te relevant flow pysics, oterwise tey will be lost. Tis particular requirement is problem dependent. For example, if a flow field as small-scale structures, te grid resolution must be sufficient to capture tese structures. It is usually necessary to increase te number of grid points (i.e., use a finer mes) were large gradients are to be expected, suc as in te boundary layer near a solid surface. Te same can also be

4 728 Appendix A Computational Fluid Dynamics (a) (b) Figure A.3 Structured grids. (a) Rectangular grid. (b) Grid around a parabolic surface. VA.2 Dynamic grid said for te temporal resolution. Te time step, t, used for unsteady flows must be smaller tan te smallest time scale of te flow features being investigated. Generally, te types of grids fall into two categories: structured and unstructured, depending on weter or not tere exists a systematic pattern of connectivity of te grid points wit teir neigbors. As te name implies, a structured grid as some type of regular, coerent structure to te mes lay-out tat can be defined matematically. Te simplest structured grid is a uniform rectangular grid, as sown in Fig. A.3a. However, structured grids are not restricted to rectangular geometries. Figure A.3b sows a structured grid wrapped around a parabolic surface. Notice tat grid points are clustered near te surface (i.e., grid spacing in normal direction increases as one moves away from te surface) to elp capture te steep flow gradients found in te boundary layer region. Tis type of variable grid spacing is used werever tere is a need to increase grid resolution and is termed grid stretcing. For te unstructured grid, te grid cell arrangement is irregular and as no systematic pattern. Te grid cell geometry usually consists of various-sized triangles for two-dimensional prob-lems and tetraedrals for tree-dimensional grids. An example of an unstructured grid is sown in Fig. A.4. Unlike structured grids, for an unstructured grid eac grid cell and te connection information to neigboring cells is defined separately. Tis produces an increase in te computer code complexity as well as a significant computer storage requirement. Te advantage to an unstructured grid is tat it can be applied to complex geometries, were structured grids would ave severe difficulty. Te finite difference metod is usually restricted to structured grids wereas te finite volume (or finite element) metod can use eiter structured or unstructured grids. Oter grids include ybrid, moving, and adaptive grids. A grid tat uses a combination of grid elements (rectangles, triangles, etc.) is termed a ybrid grid. As te name implies, te moving grid Figure A.4 Anisotropic adaptive mes for flow induced by te rotor of a elicopter in over, wit ground effect. Left: flow; Rigt: grid. (From Newmerical Tecnologies International, Montreal, Canada. Used by permission.)

5 A.4 Boundary Conditions A.5 Basic Representative Examples 729 is elpful for flows involving a time-dependent geometry. If, for example, te problem involves simulating te flow witin a pumping eart or te flow around a flapping wing, a mes tat moves wit te geometry is desired. Te nature of te adaptive grid lies in its ability to literally adapt itself during te simulation. For tis type of grid, wile te CFD code is trying to reac a converged solution, te grid will adapt itself to place additional grid resources in regions of ig-flow gradients. Suc a grid is particularly useful wen a new problem arises and te user is not quite sure were to refine te grid due to ig-flow gradients. Te same governing equations, te Navier Stokes equations (Eq ), are valid for all incompressible Newtonian fluid flow problems. Tus, if te same equations are solved for all types of problems, ow is it possible to acieve different solutions for different types of flows involving different flow geometries? Te answer lies in te boundary conditions of te problem. Te boundary conditions are wat allow te governing equations to differentiate between different flow fields (for example, flow past an automobile and flow past a person running) and produce a solution unique to te given flow geometry. It is critical to specify te correct boundary conditions so tat te CFD simulation is a wellposed problem and is an accurate representation of te pysical problem. Poorly defined boundary conditions can ultimately affect te accuracy of te solution. One of te most common boundary conditions used for simulation of viscous flow is te no-slip condition, as discussed in Section 1.6. Tus, for example, for two-dimensional external or internal flows, te x and y components of velocity (u and v) are set to zero at te stationary wall to satisfy te no-slip condition. Oter boundary conditions tat must be appropriately specified involve inlets, outlets, far-field, wall gradients, etc. It is important to not only select te correct pysical boundary condition for te problem, but also to correctly implement tis boundary condition into te numerical simulation. A.5 Basic Representative Examples A very simple one-dimensional example of te finite difference tecnique is presented in te following example. E X A M P L E A. 1 Flow from a Tank A viscous oil flows from a large, open tank and troug a long, small-diameter pipe as sown in Fig. EA.1a. At time t 5 0 te fluid dept is H. Use a finite difference tecnique to determine te liquid dept as a function of time, 5 1t2. Compare tis result wit te exact solution of te governing equation. SOLUTION Altoug tis is an unsteady flow 1i.e., te deeper te oil, te faster it flows from te tank2 we assume tat te flow is quasisteady and apply steady flow equations as follows. As sown by Eq , te mean velocity, V, for steady lami-nar flow in a round pipe of diameter D is given by D 2 p V 5 32m/ (1) were p is te pressure drop over te lengt /. For tis problem te pressure at te bottom of te tank 1te inlet of te pipe2 is g and tat at te pipe exit is zero. Hence, p 5 g and Eq. 1 becomes D 2 g V 5 32m/ (2) Conservation of mass requires tat te flowrate from te tank, Q 5 pd 2 V/4, is related to te rate of cange of dept of oil in te tank, d/dt, by Q 5 2 p 2 d 4 D T dt were D T is te tank diameter. Tus, or p D 2 V 5 2 p DT 2 d 4 4 dt V 5 2a D T 2 d D b dt (3)

6 730 Appendix A Computational Fluid Dynamics H D T 2 3 i 1 i i 1 D i 0 t 2 t V i = i 1i t t (a) (b) H 0.8H t = H t = 0.1 Exact: = He -t 0.4H 0.2H 0 t Figure EA.1 (c) By combining Eqs. 2 and 3 we obtain or D 2 g D T 2 d 32m/ 5 2a D b dt d 5 2C dt were C 5 gd 4 /32m/D 2 T is a constant. For simplicity we assume te conditions are suc tat C 5 1. Tus, we must solve d 5 2wit 5 H at t 5 0 (4) dt Te exact solution to Eq. 4 is obtained by separating te variables and integrating to obtain 5 He 2t (5) However, assume tis solution was not known. Te following finite difference tecnique can be used to obtain an approximate solution. As sown in Fig. EA.1b, we select discrete points 1nodes or grid points2 in time and approximate te time derivative of by te expression d 2 i i2 1 dt ` t5 t i t were t is te time step between te different node points on te time axis and i and i2 1 are te approximate values of at nodes i and i 2 1. Equation 6 is called te backward-difference approxima-tion to d/dt. We are free to select watever value of t tat we wis. 1Altoug we do not need to space te nodes at equal distances, it is often convenient to do so.2 Since te governing equation 1Eq. 42 is an ordinary differential equation, te grid for te finite difference metod is a one-dimensional grid as sown in Fig. EA.1b rater tan a two-dimensional grid 1wic occurs for partial differential equa-tions2 as sown in Fig. EA.2b, or a tree-dimensional grid. Tus, for eac value of i 5 2, 3, 4,... we can approximate te governing equation, Eq. 4, as i 2 i i t (6)

7 A.5 Basic Representative Examples 731 or i 5 1 i t2 (7) We cannot use Eq. 7 for i 5 1 since it would involve te non-existing 0. Rater we use te initial condition 1Eq. 42, wic gives 1 5 H Te result is te following set of N algebraic equations for te N approximate values of at times t 1 5 0, t 2 5 t,..., t N 5 1N 2 12 t. 1 5 H / 11 1 t / 11 1 t2 or in general i 5 H/ 11 1 t2 i2 1 Te results for 0 6 t 6 1 are sown in Fig. EA.1c. Tabulated values of te dept for t 5 1 are listed in te table below. t i for t1 i for t H H H H Exact 1Eq H... N 5 N2 1/ 11 1 t2 For most problems te corresponding equations would be more complicated tan tose just given, and a computer would be used to solve for te i. For tis problem te solution is simply 2 5 H/ 11 1 t2 It is seen tat te approximate results compare quite favorably wit te exact solution given by Eq. 5. It is expected tat te finite difference results would more closely approximate te exact re-sults as t is decreased since in te limit of t S 0 te finite dif-ference approximation for te derivatives 1Eq. 62 approaces te actual definition of te derivative. 3 5 H/ 11 1 t For most CFD problems te governing equations to be solved are partial differential equations [rater tan an ordinary differential equation as in te above example (Eq. A.1)] and te finite difference metod becomes considerably more involved. Te following example illustrates some of te concepts involved. E X A M P L E A. 2 Flow Past a Cylinder Consider steady, incompressible flow of an inviscid fluid past a circular cylinder as sown in Fig. EA.2a. Te stream function, c, for tis flow is governed by te Laplace equation 1see Section c 0 2 c (1) 0x 0y Te exact analytical solution is given in Section Describe a simple finite difference tecnique tat can be used to solve tis problem. SOLUTION Te first step is to define a flow domain and set up an appropriate grid for te finite difference sceme. Since we expect te flow field to be symmetrical bot above and below and in front of and beind te cylinder, we consider only one-quarter of te entire flow domain as indicated in Fig. EA.2b. We locate te upper boundary and rigt-and boundary far enoug from te cylinder so tat we expect te flow to be essentially uniform at tese locations. It is not always clear ow far from te object tese boundaries must be located. If tey are not far enoug, te solution obtained will be incorrect because we ave imposed artificial, uniform flow conditions at a location were te actual flow is not uniform. If tese boundaries are farter tan necessary from te object, te flow domain will be larger tan necessary and excessive computer time and storage will be required. Experience in solving suc problems is invaluable! Once te flow domain as been selected, an appropriate grid is imposed on tis domain 1see Fig. EA.2b2. Various grid structures can be used. If te grid is too coarse, te numerical solution may not be capable of capturing te fine scale structure of te actual flow field. If te grid is too fine, excessive computer time and

8 U 732 Appendix A Computational Fluid Dynamics y r a θ j + y (a) + x x i (b) ψ i, j + 1 y ψ i 1, j x ψ i, j x ψ i + 1, j y (c) ψ i, j 1 Figure EA.2 storage may be required. Considerable work as gone into form-ing appropriate grids 1Ref. 62. We consider a grid tat is uniformly spaced in te x and y directions, as sown in Fig. EA.2b. As sown in Eq , te exact solution to Eq. 1 1in terms of polar coordinates r, u rater tan Cartesian coordinates x, y2 is c 5 Ur 11 2 a 2 /r 2 2 sin u. Te finite difference solution approximates tese stream function values at a discrete 1finite2 number of locations 1te grid points2 as c i, j, were te i and j in-dices refer to te corresponding x i and y j locations. Te derivatives of c can be approximated as follows: and 0c 1 0x x 1 c i1 1, j 2 c i, j 2 0c 1 0y y 1 c i, j1 1 2 c i, j2 Tis particular approximation is called a forward-difference approximation. Oter approximations are possible. By similar reasoning, it is possible to sow tat te second derivatives of c can be written as follows: and 0 2 c 1 0x 2 1 x c 1 0y 2 1 y22 c 2 2c 1c 1 i1 1, j i, j i2 1, j2 (2) c 2 2c 1c 1 i, j1 1 i, j i, j2 12 (3) Tus, by combining Eqs. 1, 2, and 3 we obtain 0 2 c 0 2 c 1 1 c i1 1, j 1 c i2 1, j x 2 1 0y 2 1 x22 1 y c 1 c i, j1 1 i, j a 1 x y22 b c i, j 5 0 (4) Equation 4 can be solved for te stream function at x i and y j to give c 5 i, j 2 31 x y y2 2 1 c 1 c i1 1, j i2 1, j x2 2 1c i, j1 1 1 c i, j (5) Note tat te value of c i, j depends on te values of te stream function at neigboring grid points on eiter side and above and below te point of interest 1see Eq. 5 and Fig. EA. 2c2. To solve te problem 1eiter exactly or by te finite difference tecnique2, it is necessary to specify boundary conditions for points located on te boundary of te flow domain 1see Section For example, we may specify tat c 5 0 on te lower boundary of te domain 1see Fig. EA.2b2 and c 5 C, a constant, on te upper boundary of te domain. Appropriate boundary con-ditions on te two vertical ends of te flow domain can also be specified. Tus, for points interior to te boundary Eq. 5 is valid; similar equations or specified values of c i, j are valid for boundary points. Te result is an equal number of equations and unknowns, c i, j, one for every grid point. For tis problem, tese equations represent a set of linear algebraic equations for c i, j, te solution

9 A.6 Metodology 733 of wic provides te finite difference approximation for te and stream function at discrete grid points in te flow field. Streamlines 1lines of constant c2 can be obtained by interpolating values v 5 2 0c 2 1 c 2 c 1 i 1 1, j i, j of c i, j between te grid points and connecting te dots of 0x x 2 c 5 constant. Te velocity field can be obtained from te deriva- Furter details of te finite difference tecnique can be found in tives of te stream function according to Eq Tat is, standard references on te topic 1Refs. 5, 7, 82. Also, see te com- 0c 1 pletely solved viscous flow CFD problem in Appendix I. u 5 0y y 1 c i, j c i, j 2 Te preceding two examples are rater simple because te governing equations are not too complex. A finite difference solution of te more complicated, nonlinear Navier Stokes equation (Eq ) requires considerably more effort and insigt and larger and faster computers. A typical finite difference grid for a more complex flow, te flow past a turbine blade, is sown in Fig. A.5. Note tat te mes is muc finer in regions were large gradients are to be expected (i.e., near te leading and trailing edges of te blade) and coarser away from te blade. Figure A.5 Finite difference grid for flow past a turbine blade. (From Ref. 9, used by permission.) A.6 Metodology In general, most applications of CFD take te same basic approac. Some of te differences include problem complexity, available computer resources, available expertise in CFD, and weter a commercially available CFD package is used, or a problem-specific CFD algoritm is developed. In today s market, tere are many commercial CFD codes available to solve a wide variety of problems. However, if te intent is to conduct a toroug investigation of a specific fluid flow problem suc as in a researc environment, it is possible tat taking te time to develop a problem-specific algoritm may be most efficient in te long run. Te features common to most CFD applications can be summarized in te flow cart sown in Fig. A.6. A complete, detailed CFD solution for a viscous flow obtained by using te steps summarized in te flow cart can be accessed from te book s web site at CFD Metodology Pysics Grid Discretize Solve Analyze Problem Geometry Discretization Algoritm Verification Meto d Development & Validation Steady Governing Structured or Accuracy / Postprocess Equations Unstructured Unsteady Values Model s Specia l Requirements Implicit or Explici t Run Simulation Visualize Flow Field Assumptions & Interpret Convergence Result Simplifications s Figure A.6 Flow cart of general CFD metodology.

10 734 Appendix A Computational Fluid Dynamics A.7 Application of CFD Te Algoritm Development box is grayed because tis step is required only wen developing your own CFD code. Wen using a commercial CFD code, tis step is not necessary. Tis cart represents a generalized metodology to CFD. Tere are oter more complex components tat are idden in te above steps, wic are beyond te scope of a brief introduction to CFD. In te early stages of CFD, researc and development was primarily driven by te aerospace industry. Today, CFD is still used as a researc tool, but it also as found a place in industry as a design tool. Tere is now a wide variety of industries tat make at least some use of CFD, including automotive, industrial, HVAC, naval, civil, cemical, biological, and oters. Industries are using CFD as an added engineering tool tat complements te experimental and teoretical work in fluid dynamics. VA.3 Tornado simulation A.7.1 Advantages of CFD Tere are many advantages to using CFD for simulation of fluid flow. One of te most important advantages is te realizable savings in time and cost for engineering design. In te past, coming up wit a new engineering design meant somewat of a trial-and-error metod of building and testing multiple prototypes prior to finalizing te design. Wit CFD, many of te issues dealing wit fluid flow can be flused out prior to building te actual prototype. Tis translates to a significant savings in time and cost. It sould be noted tat CFD is not meant to replace experimental testing, but rater to work in conjunction wit it. Experimental testing will always be a necessary component of engineering design. Oter advantages include te ability of CFD to: (1) obtain flow information in regions tat would be difficult to test experimentally, (2) simulate real flow conditions, (3) conduct large parametric tests on new designs in a sorter time, and (4) enance visualization of complex flow penomena. A good example of te advantages of CFD is sown in Figure A.7. Researcers use a type of CFD approac called large-eddy simulation or LES to simulate te fluid dynamics of a tornado as it encounters a debris field and begins to pick up sand-sized particles. A full animation of tis tor-nado simulation can be accessed by visiting te book web site. Te motivation for tis work is to investigate weter tere are significant differences in te fluid mecanics wen debris particles are present. Historically it as been difficult to get compreensive experimental data trougout a tor-nado, so CFD is elping to sine some ligt on te complex fluid dynamics involved in suc a flow. A.7.2 Difficulties in CFD One of te key points tat a beginning CFD student sould understand is tat one cannot treat te computer as a magic black box wen performing flow simulations. It is quite possible to obtain a fully converged solution for te CFD simulation, but tis is no guarantee tat te results are pysi-cally correct. Tis is wy it is important to ave a good understanding of te flow pysics and ow tey are modeled. Any numerical tecnique (including tose discussed above), no matter ow sim-ple in concept, contains many idden subtleties and potential problems. For example, it may seem reasonable tat a finer grid would ensure a more accurate numerical solution. Wile tis may be true (as Example A.1), it is not always so straigtforward; a variety of stability or convergence problems may occur. In suc cases te numerical solution obtained may exibit unreasonable oscillations or te numerical result may diverge to an unreasonable (and incorrect) result. Oter problems tat may arise include (but are not limited to): (1) difficulties in dealing wit te nonlinear terms of te Navier Stokes equations, (2) difficulties in modeling or capturing turbulent flows, (3) convergence issues, (4) difficulties in obtaining a quality grid for complex geometries, and (5) managing resources, bot time and computational, for complex problems suc as unsteady tree-dimensional flows. A.7.3 Verification and Validation Verification and validation of te simulation are critical steps in te CFD process. Tis is a neces-sary requirement for CFD, particularly since it is possible to ave a converged solution tat is non-pysical. Figure A.8 sows te streamlines for viscous flow past a circular cylinder at a given instant

11 A.7Application of CFD 735 Figure A.7 Results from a large-eddy simulation sowing te visual appearance of te debris and funnel cloud from a simulated medium swirl F3-F4 tornado. Te funnel cloud is translating at 15 m/s and is ingesting 1-mm-diameter sand from te surface as it encounters a debris field. Please visit te book web site to access a full animation of tis tornado simulation. (Potograps and animation courtesy of Dr. David Lewellen (Ref. 10) and Paul Lewellen, West Virginia University.) after it was impulsively started from rest. Te lower alf of te figure represents te results of a finite difference calculation; te upper alf of te figure represents te potograp from an experi-ment of te same flow situation. It is clear tat te numerical and experimental results agree quite well. For any CFD simulation, several levels of testing need to be accomplised before one can ave confidence in te solution. Te most important verification to be performed is grid conver-gence testing. In its simplest form, it consists of proving tat furter refinement of te grid (i.e., increasing te number of grid points) does not alter te final solution. Wen tis as been acieved, you ave a grid-independent solution. Oter verification factors tat need to be investigated include Figure A.8 Streamlines for flow past a circular cylinder at a sort time after te flow was impulsively started. Te upper alf is a potograp from a flow visualization experiment. Te lower alf is from a finite difference calculation. (See te potograp at te beginning of Capter 9.) (From Ref. 9, used by permission.)

12 736 Appendix A Computational Fluid Dynamics te suitability of te convergence criterion, weter te time step is adequate for te time scale of te problem, and comparison of CFD solutions to existing data, at least for baseline cases. Even wen using a commercial CFD code tat as been validated on many problems in te past, te CFD practitioner still needs to verify te results troug suc measures as grid-dependence testing. A.7.4 Summary In CFD, tere are many different numerical scemes, grid tecniques, etc. Tey all ave teir advantages and disadvantages. A great deal of care must be used in obtaining approximate numerical solutions to te governing equations of fluid motion. Te process is not as simple as te ofteneard just let te computer do it. Remember tat CFD is a tool and as suc needs to be used appropriately to produce meaningful results. Te general field of computational fluid dynam-ics, in wic computers and numerical analysis are combined to solve fluid flow problems, rep-resents an extremely important subject area in advanced fluid mecanics. Considerable progress as been made in te past relatively few years, but muc remains to be done. Te reader is encour-aged to consult some of te available literature. References 1. Baker, A. J., Finite Element Computational Fluid Mecanics, McGraw-Hill, New York, Carey, G. F., and Oden, J. T., Finite Elements: Fluid Mecanics, Prentice-Hall, Englewood Cliffs, N.J., Brebbia, C. A., and Dominguez, J., Boundary Elements: An Introductory Course, McGraw-Hill, New York, Moran, J., An Introduction to Teoretical and Computational Aerodynamics, Wiley, New York, Anderson, J. D., Computational Fluid Dynamics: Te Basics wit Applications, McGraw-Hill, New York, Tompson, J. F., Warsi, Z. U. A., and Mastin, C. W., Numerical Grid Generation: Foundations and Applications, Nort-Holland, New York, Peyret, R., and Taylor, T. D., Computational Metods for Fluid Flow, Springer-Verlag, New York, Tanneill, J. C., Anderson, D. A., and Pletcer, R. H., Computational Fluid Mecanics and Heat Transfer, 2nd Ed., Taylor and Francis, Wasington, D.C., Hall, E. J., and Pletcer, R. H., Simulation of Time Dependent, Compressible Viscous Flow Using Central and Upwind-Biased Finite-Difference Tecniques, Tecnical Report HTL-52, CFD-22, College of Engineering, lowa State University, Lewellen, D. C., Gong, B., and Lewellen, W. S., Effects of Debris on Near-Surface Tornado Dynamics, 22nd Conference on Severe Local Storms, Paper 15.5, American Meteorological Society, 2004.