Nonuniform catalyst distributions in porous supports are obtained primarily by multicomponcnt impregnation techniques. In general, an intermediate level of interaction between catalyst precursor and support is required, so that the precursor can attach to the support, but can also desorb if another competing adsorbing species is present. Depending upon the interplay between competitive adsorption and diffusion of the various species in the porous support, a variety of nonuniform catalyst distributions can be obtained. The above physicochemical processes are also encountered in chromatographic separations (Ruthven, 1984). This chapter is divided in two parts. The first deals with adsorption on powders, while the second is focused on simultaneous diffusion and adsorption phenomena.

Although diffusion–adsorption methods are dominant for the preparation of nonuniform catalyst pellets, other procedures have also been employed. One such technique is deposition precipitation in preshaped carriers (l)e Jong, 1991). It involves deposition inside pellets of insoluble compounds, such as hydroxides which are formed by a precipitation reaction. The latter can be induced by a change of solution pH. Immediately after imbibition, a pH profile develops inside the pellets, which depends on the initial solution pH and the isoelectric point of the carrier. Since precipitation reactions depend on pH, the insoluble compound distribution reflects the pH gradient. Hence, by appropriate choice of the impregnation conditions, precipitation can occur in either the inner (egg-yolk distribution) or the outer (eggshell distribution) region of the pellet. However, preparation of eggshell catalysts leads to the problem of precipitation outside the pellets.

The principles of singular element based numerical solutions were introduced in Chapter 9 and the first examples 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 time for investigating improvements in larger codes and are also suitable for homework assignments and class demonstrations.

Based on the level of approximation of the singularity distribution, surface geometry, and type of boundary conditions, numerous computational methods can be constructed, some of which are presented in Table 11.1. We will not attempt to demonstrate all the possible combinations but will try to cover some of the most frequently 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 will be exercised and mainly the outer Neumann normal velocity and the internal Dirichlet boundary conditions will be used (and there are additional options, 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 areas of high surface curvature the solution can be improved by using more panels.

In this chapter and in the following ones the primary concern is the simplicity of the explanation and the ease of constructing the numerical technique, while numerical efficiency considerations are secondary.

We have seen in the previous chapters that in an incompressible, irrotational fluid the velocity field can be obtained by solving the continuity equation. However, the incompressible continuity equation does not directly include time-dependent terms, and the time dependency is introduced through the boundary conditions. Therefore, the first objective is to demonstrate that the methods of solution that were developed for steady flows can be used with only small modifications. These modifications will include the treatment of the “zero normal flow on a solid surface” boundary conditions and the use of the unsteady Bernoulli equation. Furthermore, as a result of the nonuniform motion, the wake becomes more complex than in the corresponding steady flow case and it should be properly accounted for. Consequently, this chapter is divided into three parts, as follows:

1. a. Formulation of the problem and of the proposed modifications for converting steady-state flow methods to treat unsteady flows (Sections 13.1–13.6).

2. b. Examples of converting analytical models to treat time-dependent flows (e.g., thin lifting airfoil and slender wing in Sections 13.8–13.9).

3. c. Examples of converting numerical models to treat time-dependent flows (Sections 13.10–13.13).

For the numerical examples only the simplest models are presented; however, application of the approach to any of the other methods of Chapter 11 is strongly recommended (e.g., can be given as a student project).

In the general case of the arbitrary motion of a solid body submerged in a fluid (e.g., a maneuvering wing or aircraft) the motion path is determined by the combined dynamic and fluid dynamic equations.

This appendix lists several computer programs that are based on the methods presented in the previous chapters. These FORTRAN programs were prepared mainly by students during regular class work and their algorithms were not optimized for clear programming and computational efficiency. Also, an effort was made to list only the simplest versions without interactive and graphic input/output sections owing to the rapid changes and improvements in computer operation systems. In spite of this brevity these computer programs can help the readers to construct their baseline algorithms upon which their customized computer programs may be developed.

Two-Dimensional Panel Methods

1. Grid generator for van de Vooren airfoil shapes, based on the formulas of Section 6.7. The program also calculates the exact chordwise velocity components and pressure coefficient for the purpose of comparison. All the two-dimensional codes (Programs 3–11) use the input generated by this subroutine.

Two-Dimensional Panel Methods Based on the Neumann Boundary Condition

1. 2. Discrete vortex, thin wing method, based on Section 11.1.1.

2. 3. Constant strength source method (based on Section 11.2.1). Note that the matrix solver (SUBROUTINE MATRX) is attached to this program only and is not listed with Programs 4–11, for brevity.

3. 4. Constant strength doublet method, based on Section 11.2.2.

4. 5. Constant strength vortex method, based on Section 11.2.3.

5. 6. Linear strength source method, based on Section 11.4.1.

6. 7. Linear strength vortex method, based on Section 11.4.2.

Three dimensional numerical solutions based on surface singularity distributions are similar, in principle, to methods presented for the two-dimensional case. From the theoretical aspect, only the wake and the trailing-edge conditions (three-dimensional Kutta condition) will require some additional attention. The most difficult aspect in three dimensions, though, is the modeling of the geometry, especially when arbitrary geometry capability is sought.

In the first part of this chapter the geometry (of the wing) is kept relatively simple and the aerodynamics of a thin lifting surface is modeled. In principle, this simple method has all the elements of the more complex panel methods and is capable of modeling the effect of wing planform shape on the fluid dynamic loads. In addition, the examples that are being presented require only limited programming effort and, therefore, are suitable for classroom instruction. Furthermore, the introduction in class of the numerical lifting-line model (Section 12.1), next to Prandtl's lifting-line model of Section 8.1, provides additional insight and a clear explanation of the spanwise integral equation.

In the second part of this chapter the principles of panel codes capable of solving the flow over bodies with arbitrary three-dimensional geometry will be presented. Over the years many such methods were developed and improved, but recent trends indicate an increased use of the approach that is based on the combination of surface source and doublet distributions with the inner potential boundary condition (for closed bodies).

For the small-disturbance solution techniques that are treated in this book, approximations to the exact mathematical problem formulation are made to facilitate the determination of a solution. Since for incompressible and irrotational flow the governing partial differential equation is linear, the approximations are made to the body boundary condition. For example, for the three-dimensional wing in Chapter 4, only terms linear in thickness, camber, and angle of attack are kept and the boundary condition is transferred to the x–y plane. The solution technique is therefore a “first-order” thin wing theory.

The small-disturbance methods developed here can be thought of as providing the first term in a perturbation series expansion of the solution to the exact mathematical problem and terms that were neglected in determining the first term will come into play in the solution for the following terms. In this book we will follow the lead of Van Dyke and use the thin-airfoil problem as the vehicle for the presentation of the ideas and some of the details of perturbation methods and their applicability to aerodynamics. First, the thin-airfoil solution will be introduced as the first term in a small-disturbance expansion and the mathematical problem for the next term will be derived. An example of a second-order solution will be presented and the failure of the expansion in the leading-edge region will be noted. A local solution applicable in the leading-edge region will be obtained and the method of matched asymptotic expansions will be used to provide a solution valid for the complete airfoil.

In Chapter 1 it was established that for flows at high Reynolds number the effects of viscosity are effectively confined to thin boundary layers and thin wakes. For this reason our study of low-speed aerodynamics will be limited to flows outside these limited regions where the flow is assumed to be inviscid and incompressible. To develop the mathematical equations that govern these flows and the tools that we will need to solve the equations it is necessary to study rotation in the fluid and to demonstrate its relationship to the effects of viscosity.

It is the goal of this chapter to define the mathematical problem (differential equation and boundary conditions) of low-speed aerodynamics whose solution will occupy us for the remainder of the book.

Angular Velocity, Vorticity, and Circulation

The arbitrary motion of a fluid element consists of translation, rotation, and deformation. To illustrate the rotation of a moving fluid element, consider at t = t0 the control volume shown in Fig. 2.1. Here, for simplicity, we select an infinitesimal rectangular element that is being translated in the z = 0 plane by a velocity (u, v) of its corner no. 1. The lengths of the sides, parallel to the x and y directions, are Δx and Δy, respectively. Because of the velocity variations within the fluid the element may deform and rotate, and, for example, the x component of the velocity at the upper corner (no. 4) of the element will be (u + (∂u/∂y)Δy), where higher order terms in the small quantities Δx and Δy are neglected.

It was demonstrated in the previous chapters that the solution of potential flow problems over bodies and wings can be obtained by the distribution of elementary solutions. The strengths of these elementary solutions of Laplace's equation are obtained by enforcing the zero normal flow condition on the solid boundaries. The steps toward a numerical solution of this boundary value problem are described schematically in Section 9.7. In general, as the complexity of the method is increased, the “element's influence” calculation becomes more elaborate. Therefore, in this chapter, emphasis is placed on presenting some of the typical numerical elements upon which some numerical solutions are based (the list is not complete and an infinite number of elements can be developed). A generic element is shown schematically in Fig. 10.1. To calculate the induced potential and velocity increments at an arbitrary point P(xP, yP, zP) requires information on the element geometry and strength of singularity.

For simplicity, the symbol Δ is dropped in the following description of the singularity elements. However, it must be clear that the values of the velocity potential and velocity components are incremental values and can be added up according to the principle of superposition.

In the following sections some two-dimensional elements will be presented, whose derivation is rather simple. Three-dimensional elements will be presented later and their complexity increases with the order of the polynomial approximation of the singularity strength.

The differential equations that are generally used in the solution of problems relevant to low-speed aerodynamics are a simplified version of the governing equations of fluid dynamics. Also, most engineers when faced with finding a solution to a practical aerodynamic problem, find themselves operating large computer codes rather than developing simple analytical models to guide them in their analysis. For this reason, it is important to start with a brief development of the principles upon which the general fluid dynamic equations are based. Then we will be in a position to consider the physical reasoning behind the assumptions introduced to generate simplified versions of the equations that still correctly model the aerodynamic phenomena being studied. It is hoped that this approach will give the engineer the ability to appreciate both the power and the limitations of the techniques that will be presented in this text. In this chapter we will derive the conservation of mass and momentum balance equations and show how they are reduced to obtain the equations that will be used in the rest of the text to model flows of interest to the low-speed aerodynamicist.

Description of Fluid Motion

The fluid being studied here is modeled as a continuum, and infinitesimally small regions of the fluid (with a fixed mass) are called fluid elements or fluid particles. The motion of the fluid can be described by two different methods. One adopts the particle point of view and follows the motion of the individual particles.

The discussion so far has focused mainly on the potential flow model whose solution provides a useful but restricted description of the flow. For practical problems such as the flow over an airfoil, however, effects of the viscous flow near the solid surface must be included. The objective of this chapter, therefore, is to explain how a viscous boundary layer model can be combined with the inviscid flow model to provide a more complete representation of the flowfield. These principles can be demonstrated by using the laminar boundary layer model, which provides all the necessary elements for combining the viscous and inviscid flow models. We must remember, though, that the Reynolds number of the flow over actual airplanes or other vehicles is such that large portions of the flow are turbulent, and the solely laminar flow model must be augmented to reflect this. However, the principles of the matching process remain similar. Extensions of this laminar boundary layer based approach to flows with transition, to turbulent boundary layers, or to cases with flow separation, and other aspects of airfoil design, will be discussed briefly in Chapter 15. (Although in these cases the viscous flow model may change substantially from the laminar model, the viscous–inviscid coupling strategy remains unchanged.)

Boundary layer theory is a very wide topic and there are several textbooks that focus solely on this subject (e.g., see Ref. 1.6).

Our goal in writing this book is to present a comprehensive and up-to-date treatment of the subject of inviscid, incompressible, and irrotational aerodynamics. Over the last several years there has been a widespread use of computational (surface singularity) methods for the solution of problems of concern to the low-speed aerodynamicist and a need has developed for a text to provide the theoretical basis for these methods as well as to provide a smooth transition from the classical small-disturbance methods of the past to the computational methods of the present. This book was written in response to this need. A unique feature of this book is that the computational approach (from a single vortex element to a three-dimensional panel formulation) is interwoven throughout so that it serves as a teaching tool in the understanding of the classical methods as well as a vehicle for the reader to obtain solutions to complex problems that previously could not be dealt with in the context of a textbook. The reader will be introduced to different levels of complexity in the numerical modeling of an aerodynamic problem and will be able to assemble codes to implement a solution.

We have purposely limited our scope to inviscid, incompressible, and irrotational aerodynamics so that we can present a truly comprehensive coverage of the material. The book brings together topics currently scattered throughout the literature. It provides a detailed presentation of computational techniques for three-dimensional and unsteady flows.