Low speed aerodynamics (second edition) part 2
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Low speed aerodynamics (second edition) part 2
CHAPTER 11Two-Dimensional Numerical SolutionsThe principles of singular element based numerical solutions were introduced in Chapter 9 and the first e Low speed aerodynamics (second edition) part 2examples are provided in this chapter. The following two-dimensional examples will have all the elements of more refined three-dimensional methods, but because of the simple two-dimensional geometry, the programming effort is substantially less. Consequently, such methods can be developed in a short Low speed aerodynamics (second edition) part 2 time for investigating improvements in larger codes and are also suitable for homework assignments and class demonstrations.Based on the level of appLow speed aerodynamics (second edition) part 2
roximation of the singularity distribution, surface geometry, and type of boundary conditions, numerous computational methods can be constructed, someCHAPTER 11Two-Dimensional Numerical SolutionsThe principles of singular element based numerical solutions were introduced in Chapter 9 and the first e Low speed aerodynamics (second edition) part 2tly used methods (denoted by the word “example" in Table 11.1), including discrete singular elements and constant-strength, linear, and quadratic elements (as an example for higher order singularity distributions). The different approaches in specifying the zero normal velocity boundary condition wi Low speed aerodynamics (second edition) part 2ll be exercised and mainly the outer Neumann normal velocity and the internal Dirichlet boundary conditions will be used (and there are additional optLow speed aerodynamics (second edition) part 2
ions, e.g., an internal Neumann condition). In terms of the surface geometry, for simplicity, only the flat panel element will be used here and in areCHAPTER 11Two-Dimensional Numerical SolutionsThe principles of singular element based numerical solutions were introduced in Chapter 9 and the first e Low speed aerodynamics (second edition) part 2mplicity of the explanation and the ease of constructing the numerical technique, while numerical efficiency considerations are secondary. Consequently, the numerical economy of the methods presented can be improved (with some compromise in regard to the ease of code readability). Also, the methods Low speed aerodynamics (second edition) part 2are presented in their simplest form and each can be further developed to match the requirements of a particular problem. Such improvement can be obtaLow speed aerodynamics (second edition) part 2
ined by changing grid spacing and density, location of collocation points, or wake model, or altering the method of enforcing the boundary conditions CHAPTER 11Two-Dimensional Numerical SolutionsThe principles of singular element based numerical solutions were introduced in Chapter 9 and the first e Low speed aerodynamics (second edition) part 2re details. As the chapter evolves, some redundant details are omitted and the description may appear inadequate without reading the previous sections.11.1 Point Singularity SolutionsThe basic idea behind point singularity solutions is presented schematically in Fig. 11.1. If an exact solution in a Low speed aerodynamics (second edition) part 2form of a continuous singularity distribution (e.g., a vortex distribution y(.v)) exists, then it can be divided into several finite segments (e.g.. tLow speed aerodynamics (second edition) part 2
he segment between X| and A2). The local average strength of the element is then To = /*"’ y(A‘ k/.v and it can be placed at a point A'o within the inCHAPTER 11Two-Dimensional Numerical SolutionsThe principles of singular element based numerical solutions were introduced in Chapter 9 and the first e Low speed aerodynamics (second edition) part 2s for their solution. This can be done by specifying the boundary conditions at /V262li. I Point Singularity SolutionsTable 11.1. List of possible two-dimensional panel methods and of those tested in this chapterSingularity distributionBoundary conditionsSurface paneling flaL'high-orderNeumann (exte Low speed aerodynamics (second edition) part 2rnal)Dirichlet (internal)PointsourceexampleilaldoubletvortexexampleflatConstant strengthsourceexampleexampleflatdoubletexampleexampleHatvortexexampleHLow speed aerodynamics (second edition) part 2
at1 .inear strengthsourceexampleexampleHatdoubletexampleexampleflatvortexQuadratic strengthsourcedoubletexampleflatvortexcollocation points along the CHAPTER 11Two-Dimensional Numerical SolutionsThe principles of singular element based numerical solutions were introduced in Chapter 9 and the first e Low speed aerodynamics (second edition) part 2 the wake) must be addressed.As a first example for this very simple approach the lifting and thickness problems of thin airfoils are solved based on models (such as the lumped-vortex element) generated during examination of the analytical solutions in Chapter 5.11.1.1 Discrete Vortex MethodThe disc Low speed aerodynamics (second edition) part 2rete vortex method presented here for solving the thin lifting airfoil problem is based on the lumped-vortex clement and serves for solving numericallLow speed aerodynamics (second edition) part 2
y the integral equation (F.q. (5.39)) presented in Chapter 5. The advantage of the numerical approach is that the boundary conditions can be specifiedCHAPTER 11Two-Dimensional Numerical SolutionsThe principles of singular element based numerical solutions were introduced in Chapter 9 and the first e Low speed aerodynamics (second edition) part 2nd effect or multielement airfoils, can be studied with great case./fl J Yịxìdx•toX| 4- k(x2 - X|)Figure 11.1 Discretization of a continuous singularity distribution.264Ị Ị / Two-Dimensional. Low speed aerodynamics (second edition) part 2CHAPTER 11Two-Dimensional Numerical SolutionsThe principles of singular element based numerical solutions were introduced in Chapter 9 and the first eGọi ngay
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