Allen Plotkin

Unsteady Incompressible Potential Flow

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: 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). 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). 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.

Effect of airfoil (trailing-edge) thickness on the numerical solution of panel methods based on the Dirichlet boundary condition

The practical limit of airfoil thickness ratio for which acceptable engineering results are obtainable with the Dirichlet boundary-condition-based numerical methods is investigated. This is done by studying the effect of thickness on the calculated pressure distribution near the trailing edge and by comparing the aerodynamic coefficients with available exact solutions. The first objective of this study, owing to the wide use of such computational methods, is to demonstrate the numerical symptoms that occur when the body or wing thickness approaches zero and to increase the awareness of potential users of these methods. Additionally, an effort is made to obtain the practical limits of the trailing-edge thickness where such problems will appear in the flow solution, and to propose some possible cures for very thin airfoils or those with cusped trailing edges.

Laminar Incompressible Flow Past a Circulation-Controlled Circular Lifting Rotor

Prandtl first order boundary layer equations for two dimensional laminar incompressible flow past circulation controlled circular lifting rotor

Low-Speed Aerodynamics: Sample Computer Programs

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 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 2. Discrete vortex, thin wing method, based on Section 11.1.1. 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. 4. Constant strength doublet method, based on Section 11.2.2. 5. Constant strength vortex method, based on Section 11.2.3. 6. Linear strength source method, based on Section 11.4.1. 7. Linear strength vortex method, based on Section 11.4.2.

Low-Speed Aerodynamics: Fundamentals of Inviscid, Incompressible Flow

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 = t 0 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 Δ x and Δ y , 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 )Δ y ), where higher order terms in the small quantities Δ x and Δ y are neglected.

Low-Speed Aerodynamics: Small-Disturbance Flow over Three-Dimensional Wings: Formulation of the Problem

One of the first important applications of potential flow theory was the study of lifting surfaces (wings). Since the boundary conditions on a complex surface can considerably complicate the attempt to solve the problem by analytical means, some simplifying assumptions need to be introduced. In this chapter assumptions will be applied to the formulation of the 3D thin wing problem and the scene for the singularity solution technique will be set.

Low-Speed Aerodynamics: Singularity Elements and Influence Coefficients

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 Laplaces 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 “elements 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 ( x P , y P , z P ) 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.