Elements of Fluid Dynamics

Guido Buresti

• Publisher: ICP
• Year: 2012
• ISBN:
  9781908977045
• Topic: Tecnologia e ingegneria
• Subtopic: Ingegneria meccanica

Elements of Fluid Dynamics is intended to be a basic textbook, useful for undergraduate and graduate students in different fields of engineering, as well as in physics and applied mathematics. The main objective of the book is to provide an introduction to fluid dynamics in a simultaneously rigorous and accessible way, and its approach follows the idea that both the generation mechanisms and the main features of the fluid dynamic loads can be satisfactorily understood only after the equations of fluid motion and all their physical and mathematical implications have been thoroughly assimilated. Therefore, the complete equations of motion of a compressible viscous fluid are first derived and their physical and mathematical aspects are thoroughly discussed. Subsequently, the necessity of simplified treatments is highlighted, and a detailed analysis is made of the assumptions and range of applicability of the incompressible flow model, which is then adopted for most of the rest of the book. Furthermore, the role of the generation and dynamics of vorticity on the development of different flows is emphasized, as well as its influence on the characteristics, magnitude and predictability of the fluid dynamic loads acting on moving bodies.

The book is divided into two parts which differ in target and method of utilization. The first part contains the fundamentals of fluid dynamics that are essential for any student new to the subject. This part of the book is organized in a strictly sequential way, i.e. each chapter is assumed to be carefully read and studied before the next one is tackled, and its aim is to lead the reader in understanding the origin of the fluid dynamic forces on different types of bodies. The second part of the book is devoted to selected topics that may be of more specific interest to different students. In particular, some theoretical aspects of incompressible flows are first analysed and classical applications of fluid dynamics such as the aerodynamics of airfoils, wings and bluff bodies are then described. The one-dimensional treatment of compressible flows is finally considered, together with its application to the study of the motion in ducts.

Sample Chapter(s)
Chapter 1: Introduction (133 KB)

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Contents:

• Fundamentals:
  • Introduction
  • Properties of Fluids
  • Characterization of the Motion of Fluids
  • The Equations of Motion of Fluids
  • The Incompressible Flow Model
  • Vorticity Dynamics in Incompressible Flows
  • Incompressible Boundary Layers
  • Fluid Dynamic Loads on Bodies in Incompressible Flows
• Deeper Analyses and Classical Applications:
  • Exact Solutions of the Incompressible Flow Equations
  • The Role of the Energy Balance in Incompressible Flows
  • Supplementary Issues on Vorticity in Incompressible Flows
  • Airfoils in Incompressible Flows
  • Finite Wings in Incompressible Flows
  • An Outline of Bluff-Body Aerodynamics
  • One-Dimensional Compressible Flows

Readership: Graduates and undergraduates in the fields of fluid mechanics and engineering, specifically aerospace, mechanical and civil engineering.

Part II

DEEPER ANALYSES AND CLASSICAL APPLICATIONS

Chapter 9

EXACT SOLUTIONS OF THE
INCOMPRESSIBLE FLOW EQUATION

9.1. Introductory Remarks

As we have already pointed out in previous chapters, the equation of motion of a fluid are extremely complex, especially due to their non-linearity, and thus solutions cannot be derived easily for flow problems characterized by generic initial and boundary conditions. However, certain solutions of the Equation can be found, particularly for the case of incompressible flows. We shall restrict our attention to this case and briefly discuss some general features of the possible solutions of the relevant Equation . We recall that in the incompressible flow case the mass and momentum balance Equation are decoupled from the energy equation , and form a set of four scalar Equation in which the unknown functions are the three components of the velocity vector and the pressure. The solution of this set of Equation is indeed the mathematical problem to which one refers when dealing with incompressible flows. Thus, the energy equation is usually disregarded, and it is only used when it is rEquired to derive the temperature field, once the velocity and the pressure fields have been obtained.

Let us consider again the Equation governing an incompressible flow. The mass balance rEquires the volume of a material element of fluid to remain constant during the motion; thus, as the divergence of velocity represents the time variation of the volume of a fluid element, per unit volume, this rEquirement implies that the velocity field must be solenoidal, i.e. divergence-free, as expressed by Eq. (5.31).

Therefore, the momentum balance equation (5.35) becomes the basic equation describing the motion, and is now recast as follows:

(9.1)

It is often convenient to use relation (5.41) to derive the following alternative formulation of the equation :

(9.2)

which, when the body force is conservative (f = −grad Ψ), becomes

(9.3)

Finally, we recall that the viscous term in the above Equation may be also expressed in terms of the vorticity vector using relation (5.42), which gives

(9.4)

This expression is useful to recall that the viscous term in the momentum Equation disappears if the motion is such that curl ω =0, and in particular when ω is either zero or a constant.

It is now necessary to explain, in more detail, how we define an ‘exact solution’ of the equations of motion. In effect, there are several ways in which this definition may be given. In particular, a first possibility is to assert that two fields, V (x, t) and p(x, t), constitute an exact solution of the flow equations if, besides satisfying Eqs. (5.31) and (9.1) for all x and t and for given values of ρ, ν and f, they also fulfil initial and boundary conditions that are relevant to a real (or reasonably approximate)physical problem. This last condition restricts the number of the available exact solutions of the Navier–Stokes equations, as a direct consequence of the already mentioned intrinsic mathematical difficulties of the equations,mainly connected with their non-linearity.

Therefore, a second possibility might be to content ourselves with the rEquirement that the functions V(x,t) and p(x,t) be a solution of the Equation of motion, irrespective of whether or not they satisfy boundary conditions that may describe, in some way, a realistic physical problem. This is, for instance, the point of view adopted by Berker (1963), and it has several advantages. The first and most obvious one is that the number of known exact solutions increases significantly and a large number of them are listed by Berker (1963). More importantly, these solutions may give useful clues to the nature of the flows corresponding to certain real problems, or provide new ideas on how those problems might be described through simplified models that may be considered as first-order approximations of the real situation. A possible example might be the case in which two exact solutions, not satisfying a given boundary condition along a line or a surface, may be joined through a boundary layer, which is an approximate solution that might be representative of certain flow conditions. Furthermore, it might even happen that, by observing the flow corresponding to an exact solution, one may devise a new physical problem corresponding to that flow in a practical situation. For instance, it is possible that a streamline of a general solution might be replaced by a solid surface whose local shape and velocity coincide with those of that streamline. In this respect, it must be observed that the boundary conditions at solid surfaces may be suitably changed, in practice, by using suction or injection of fluid with different local velocities, and this is indeed a developing and widely studied technique for flow control.

In general, we then consider as valid this wider and more general definition of exact solution, even if the examples described in the following sections mainly refer to cases that may be representative of practical flow conditions. However, it will be immediately appreciated that this applicability to practical cases derives from the fact that the solid surfaces present in various situations are characterized by a shape and a velocity that coincide, in an appropriate reference system, with those of a streamline of a general solution.

Other possible distinctions may be introduced regarding the form of the velocity and pressure functions that may be accepted as exact solutions. In particular, one might accept only solutions that are expressed in closed form, by means of elementary functions or special well-known functions (such as the Bessel functions). Infinite series are also generally considered as acceptable, even if this opinion is not shared by all authors (see Wang, 1991). On the other hand, direct numerical simulations cannot be viewed as providing exact solutions because, apart fro...