Introduction

In this chapter, solutions of the conservation equations in partial-differential equation form are sought for a simple case—namely, steady, incompressible, inviscid two-dimensional flow. Each of these crucial assumptions is discussed in detail and their applicability as models of real flow-field situations are justified. Body forces such as gravity effects are neglected because they are negligible in most aerodynamics problems. Simple geometries are considered first. The analysis is then extended so that finally it is possible to represent the complex flow field around realistic airfoil shapes, such as those needed to efficiently produce lift forces for flight vehicles. Chapter 5 is a detailed treatment of two-dimensional airfoil flows.

The intention here is to obtain solutions valid throughout the entire flow field; hence, the differential-conservation equations are integrated so as to work from the small (i.e., the differential element) to the large (i.e., the flow field). In this regard, the integral form of the conservation equations is not a useful starting point because in steady flow, the integral equations describe events over the surface of only some fixed control volume. We are seeking detailed information regarding the pressure and velocity fields at any point in the flow. What are the implications of each assumption listed previously?

1. Steady flow. The assumption of steady flow enables the definition of a streamline as the path traced by a fluid particle moving in the flow field, from which it follows that a streamline is a line in the flow that is everywhere tangent to the local velocity vector. Also, all time-derivative terms in the governing equations can be dropped; this results in a much simpler formulation.

2. Incompressible flow. The assumption of incompressible flow means that the density is assumed to be constant. As shown herein, and as the conservation equations in Chapter 3 indicate, the assumption of incompressibility in a problem leads to enormous simplifications. The obvious one is that terms in the equations containing derivatives of density are zero. The other major simplification is that the number of equations to be solved is reduced. If the density is constant, then there cannot be large variations in temperature, and the temperature may be assumed to be constant as well. With density and temperature no longer variables, the equation of state and the energy equation may be set to one side and the continuity and momentum equations solved for the remaining variables—namely, velocity and pressure.In other words, for incompressible flows, the equation of state and the energy equation may be uncoupled from the continuity and momentum equations. It is true that no fluid (liquid or gas) is absolutely incompressible; however, at low speeds, the variation in density of an airflow is small and can be considered essentially incompressible. For example, considerations of compressible flow show that at a Mach number of 0.3 (a velocity of 335 ft/s, or 228 mph, at sea level), the maximum possible change in density in a flow field is about 6 percent and the maximum change in temperature of the flow is less than 2 percent. For flows of this velocity or less, the incompressible assumption is good. However, at Mach number 0.5 (558 ft/s, or 380 mph, at sea level), the maximum change in density in a flow field is almost 19 percent. An incompressible-flow assumption for such a case leads to prohibitive errors.Results from an assumed incompressible flow around thin airfoils or wings and around slender bodies provide a foundation for the prediction of the flow around these bodies at higher, compressible-flow Mach numbers (i.e., less than unity). It turns out that the effects of compressibility on pressure distribution, lift, and moment at flow Mach numbers less than 1 can be expressed as a correction factor times a related incompressible flow value. Thus, results using the incompressible model are useful not only for low-speed flight, they also provide a database for the accurate prediction of vehicle operation at much higher (but subsonic) speeds.

3. Inviscid flow. The inviscid-flow assumption means physically that viscous-shear and normal stresses are negligible. Thus, all of the viscous shear-stress terms on the force side of the momentum equations drop out, as well as the normal stresses due to viscosity. As a result, the only stresses acting on the body surface are the normal stresses due to pressure. Recall from Chapter 3 that when considering incompressible viscous-flow theory (see Chapter 8), the viscous-shear stresses are assumed to be proportional to the rate of strain of a fluid particle, with the constant of proportionality as the coefficient of viscosity. Thus, an assumption equivalent to that of negligible viscous stresses is the assumption that the coefficient of viscosity is essentially zero. Such a flow is termed inviscid (i.e., of zero viscosity). In effect, the boundary layer on the surface of the body is deleted by this assumption. This implies that the boundary layer must be very thin compared to a dimension of the body and that the presence or absence of the boundary layer has a negligible effect relative to modifications to the body geometry as “seen” by the flow.The inviscid, incompressible-fluid model is often termed a perfect fluid (not to be confused with a perfect or ideal gas as defined in Chapter 1). The boundary layer in many practical situations is extremely thin compared to a typical dimension of the body under study such that the body shape that a viscous flow “sees” is essentially the geometric shape. The exception is where the flow separates and the boundary layer leaves the body, resulting in a major change in the effective geometry of the body. Such separated regions occur on wings, for example, at large angles of attack. However, the wing angle of attack of a vehicle at a cruise condition is only a few degrees so that the effects of separation are minimal. Thus, the inviscid-flow assumption provides useful results that match closely the experiment for conditions corresponding to cruise, and the inviscid-flow model breaks down when large regions of separated flow occur.Because the presence of the boundary layer is neglected in perfect-fluid theory, the theory does not predict the frictional drag of a body; that must be left to viscous-flow theory. However, within the framework of incompressible inviscid flow, predictions for low-speed pressure distribution, lift, and pitching moment are valid and useful.

4. Two-dimensional flow. The assumption of two-dimensional flow is a simplifying assumption in that it reduces the vector-component momentum equations from three to two. Two-dimensional simply means that the flow (and the body shape) is identical in all planes parallel to, say, a page of this book; there are no variations in any quantity in a direction normal to the plane. Consider a cylinder or wing extending into and out of this page, with each cross section of the body exactly the same as any other. The flow around the body in all planes parallel to the page are then identical. It follows that the cylinder in two-dimensional flow has an infinite axis length and the wing has an infinite span. Any cross section of this wing of infinite span is termed an airfoil section. Theoretical predictions for such an airfoil may be validated by experiments in a wind tunnel in which the wing model extends from one wall to the opposite wall. If the wing/wall interfaces are properly sealed, the model then behaves as if it were a wing of infinite span—that is, as if it has no wing tips around which there would be a flow due to the difference in pressure between the top and bottom surfaces of the wing. Theoretical results for an airfoil (i.e., a two-dimensional problem) form the basis for predicting the behavior of wings of finite span (i.e., a three-dimensional problem) because each cross section (i.e., airfoil section) of the finite wing is assumed to behave as if the flow around it were locally two-dimensional (see Chapters 5 and 6). Thus, two-dimensional results have considerable value. Most (but not all) of the concepts discussed in this chapter may be extended to three dimensions and/or to compressible flow. Such extensions are introduced at appropriate points.

Introduction

To solve the fundamental problems of aerodynamics defined in Chapter 1, it is necessary to formulate a mathematical representation of the underlying fluid dynamics. The appropriate mathematical expressions or sets of equations may be algebraic, integral, or differential in character but will always represent basic physical laws or principles. In this chapter, the fundamental equations necessary for the solution of aerodynamics problems are derived directly from the basic laws of nature. The resulting mathematical formulations represent a large class of fluid mechanics problems within which aerodynamics is an important subclass.

Some problems in aerodynamics require solutions for all of the variables needed to describe a moving stream of gas—namely, velocity, pressure, temperature, and density. Because velocity is a vector quantity (i.e., with magnitude and direction), in a general case there are three scalar velocity components. Thus, in many cases of interest, there is a total of six unknowns: three velocity components and the scalar thermodynamic quantities of pressure, temperature, and density.* This requires six independent equations to be written to solve for the six unknowns. The physical laws of conservation of mass, momentum, and energy supply five such equations (i.e., the momentum equation is a vector equation; therefore, conservation of momentum leads in general to three component equations). For all of the subject matter in this book, the assumption of an ideal gas is physically realistic. Thus, the perfect gas law (i.e., equation of state) p = ρRT, which relates pressure, density, and temperature, supplies the final equation needed to solve for the six unknowns.

Introduction

This chapter considers steady, inviscid, incompressible flow about a lifting wing of arbitrary section and planform. Because the flow around a wing is not identical at all stations between the two ends of the wing, the lifting finite wing constitutes a three-dimensional flow problem. The two wing tips are located at distance ±b/2, where b is the wing span.

Certain terms must be defined before a study of finite wings can be begun (Fig. 6.1). The coordinate axis system used is shown in Fig. 6.1a. A wing section is defined as any cross section of a wing as viewed in any vertical plane parallel to the x-z plane. It also is called an airfoil section. The wing may be of constant section or variable section. If a wing is of constant section, wing sections at any spanwise station have the same shape (e.g., NACA 2312). If a wing is of variable section, the wing-section shape varies at different spanwise locations. For example, a wing of variable section might have a NACA 0012 at the root section (i.e., the section in the plane of symmetry at y = 0), then smoothly change in the spanwise direction until the wing had, a NACA 2312 section at the tip.

Rationale and organization

Models for the turbulent stresses and scalar fluxes have been in widespread use since the 1960s, incorporated within CFD codes of a wide range of types and capabilities. Over this period the vast majority of computations have been made using turbulence models simpler than second-moment closure. Quite clearly, such simpler models must deliver satisfactory predictions of some of the flows of interest – for otherwise they would be discarded. This chapter is devoted to such reduced models. The position adopted is that, of course, such simplification makes sense, provided it is made with an appreciation of what has been lost in the process.

This truism applies as much to the numerical solver as to the physical model of turbulence employed, for one would surely never use a three-dimensional, elliptic, compressible-flow solver if one's interests were simply in computing a range of axisymmetric, unseparating boundary layers in liquids. But, if we proceed in the reverse direction, while it is not usually possible to apply a simple numerical solver to flows well beyond the solver's capability, it is all too easy to assume that a turbulence model that functioned very satisfactorily in computing simple shear flows, will perform equally as well in computing complex strains or in the presence of strong external force fields. That is why it is seen as important that simple (or simpler) turbulence models should be arrived at by a rational simplification of the full second-moment closure (having regard for the particular features of the flow to be computed) rather than by adopting some constitutive equation as an article of faith.

Early proposals

The label wall functions was first applied by Patankar and Spalding (1967) as the collective name for the set of algebraic relations linking the values of the effective wall-normal gradients of dependent variables between the wall and the wall-adjacent node (in a numerical solver) to the shear stress, heat or mass flux at the wall.

The underlying purpose of wall functions, as originally proposed, was to allow computations to escape the need to model the very complex flow dynamics associated with the low-Re region that formed the subject of Chapter 6. It may seem absurd that in the region which, from a physical point of view, contains the most complex viscous and turbulent interactions, one resorts to algebraic rather than differential relations to resolve the flow. We note, however, that in Chapter 7 the power of using very simple eddy-viscosity models of turbulence to handle the sublayer has been demonstrated. Wall functions may just be seen as an extrapolation of that simplification strategy; that is, an even cheaper approach to capturing the essentials of the viscosity-affected layer, by exploiting the fact that gradients of dependent variables normal to the wall are dominant and that transport effects are relatively uninfluential. The present chapter first summarizes conventional wall functions and then introduces four more powerful approaches that the authors and their colleagues have developed more recently.

The fact of turbulent flow

Man has evolved within a world where air and water are, by far, the most common fluids encountered. The scales of the environment around him and of the machines and artefacts his ingenuity has created mean that, given their relatively low kinematic viscosities, the relevant global Reynolds number, Re, associated with the motion of both fluids is, in most cases, sufficiently high that the resultant flow is of the continually time-varying, spatially irregular kind we call turbulent.

If, however, our Reynolds number is chosen not by the overall physical dimension of the body of interest – an aircraft wing, say – and the fluid velocity past it but by the smallest distance over which the velocity found within a turbulent eddy changes appreciably and the time over which such a velocity change will occur, its value then turns out to be of order unity. Indeed, one might observe that if this last Reynolds number, traditionally called the micro-scale Reynolds number, Reη, were significantly greater than unity, the rate at which the turbulent kinetic energy is destroyed by viscous dissipation could not balance the overall rate at which turbulence ‘captures’ kinetic energy from the mean flow.

The nature of viscous and wall effects: options for modelling

The turbulence models considered in earlier chapters were based on the assumption that the turbulent Reynolds numbers were high enough everywhere to permit the neglect of viscous effects. Thus, they are not applicable to flows with a low bulk Reynolds number (where the effects of viscosity may permeate the whole flow) or to the viscosity-affected regions adjacent to solid walls (commonly referred to as the viscous sublayer and buffer regions but which we shall normally collectively refer to as the viscous region) which always exist on a smooth wall irrespective of how high the bulk Reynolds number may be. In other words, while at high Reynolds number, viscous effects on the energy-containing turbulent motions are indeed negligible throughout most of the flow, the condition of no-slip at solid interfaces always ensures that, in the immediate vicinity of a wall, viscous contributions will be influential, perhaps dominant. Figure 6.1 shows the typical ‘layered’ composition for a near-wall turbulent flow (though with an expanded scale for the near-wall region) as found in a constant-pressure boundary layer, channel or pipe flow. Although the thickness of this viscosity-affected zone is usually two or more orders of magnitude less than the overall width of the flow (and decreases as the Reynolds number increases), its effects extend over the whole flow field since, typically, half of the velocity change from the wall to the free stream occurs in this region.

Because viscosity dampens velocity fluctuations equally in all directions, one may argue that viscosity has a ‘scalar’ effect. However, turbulence in the proximity of a solid wall or a phase interface is also subjected to non-viscous damping arising from the impermeability of the wall and the consequent reflection of pressure fluctuations. This ‘wall-blocking’ effect, which is also felt outside the viscous layer well into the fully turbulent wall region, directly dampens the velocity fluctuations in the wall-normal direction and thus it has a ‘vector’ character. A good illustration of this effect is the reduction of the surface-normal velocity fluctuations that has been observed in flow regions close to a phase interface, where there are no viscous effects, for example the DNS of Perot and Moin (1995).

Scientific papers on how to represent in mathematical form the types of fluid motion we call turbulent flow have been appearing for over a century while, for the last sixty years or so, a sufficient body of knowledge has been accumulated to tempt a succession of authors to collect, systematize and distil a proportion of that knowledge into textbooks. From the start a bewildering variety of approaches has been advocated: thus, even in the 1970s, the algebraic mixing-length models presented in the book by Cebeci and Smith jostled on the book-shelves with Leslie's manful attempt to make comprehensible to a less specialized readership the direct-interaction approach developed by Kraichnan and colleagues. As the progressive advance in computing power made it possible to apply the emerging strategy of computational fluid dynamics to an ever-widening array of industrially important flows, however, eddy-viscosity models (EVMs) based on the solution of two transport equations for scalar properties of turbulence (essentially, length and time scales of the energy-containing eddies) emerged as the modelling strategy of choice and, correspondingly, have been the principal focus in several textbooks on the modelling of turbulent flows (for example, Launder and Spalding, Wilcox and Piquet).

Today, two-equation EVMs remain the work-horse of industrial CFD and are applied through commercially marketed software to flows of a quite bewildering complexity, though often with uncertain accuracy. However, there has been a major shift among the modelling research community to abandon approaches based on the Reynolds-averaged Navier–Stokes (RANS) equations in favour of large-eddy simulation (LES) where the numerical solution for any flow adopts a three-dimensional, time-dependent discretization of the Navier–Stokes equations using a model to account simply for the effects of turbulent motions too fine in scale to be resolved with the mesh adopted – that is, a sub-grid-scale (or sgs) model. While acknowledging that LES offers the prospects of tackling turbulence problems beyond the scope of RANS, a further major driver for this changeover has been the manifold inadequacies of the stress-strain hypothesis adopted by linear eddy-viscosity models. While such a simple linkage between mean strain rate and turbulent stress seemed adequate for a large proportion of two-dimensional, nearly parallel flows, its weaknesses became abundantly clear as attention shifted to recirculating, impinging and three-dimensional shear flows. Although an LES approach will, most probably, also adopt an sgs model of eddy-viscosity type, the consequences are less serious for two reasons. First, the majority of the transport caused by the turbulent motion will be directly resolved by the large eddies and secondly, the finer scale eddies that must still be resolved by the sub-grid-scale model of turbulence will arguably be a good deal closer to isotropy. Thus, adopting an isotropic eddy viscosity as the sgs model may not significantly impair the accuracy of the solution.

Overview

Thermodynamics is unquestionably the most powerful and most elegant of the engineering sciences. Its power arises from the fact that it can be applied to any discipline, technology, application, or process. The origins of thermodynamics can be traced to the development of the steam engine in the 1700's, and thermodynamic principles do govern the performance of these types of machines. However, the power of thermodynamics lies in its generality. Thermodynamics is used to understand the energy exchanges accompanying a wide range of mechanical, chemical, and biological processes that bear little resemblance to the engines that gave birth to the discipline. Thermodynamics has even been used to study the energy exchanges that are involved in nuclear phenomena and it has been helpful in identifying sub-atomic particles. The elegance of thermodynamics is the simplicity of its basic postulates. There are two primary ‘laws’ of thermodynamics, the First Law and the Second Law, and they always apply with no exceptions. No other engineering science achieves such a broad range of applicability based on such a simple set of postulates.

So, what is thermodynamics? We can begin to answer this question by dissecting the word into its roots: ‘thermo’ and ‘dynamics’. The term ‘thermo’ originates from a Greek word meaning warm or hot, which is related to temperature. This suggests a concept that is related to temperature and referred to as heat. The concept of heat will receive much attention in this text. ‘Dynamics’ suggests motion or movement. Thus the term ‘thermodynamics’ may be loosely interpreted as ‘heat motion’. This interpretation of the word reflects the origins of the science. Thermodynamics was developed in order to explain how heat, usually generated from combusting a fuel, can be provided to a machine in order to generate mechanical power or ‘motion’. However, as noted above, thermodynamics has since matured into a more general science that can be applied to a wide range of situations, including those for which heat is not involved at all. The term thermodynamics is sometimes criticized because the science of thermodynamics is ordinarily limited to systems that are in equilibrium. Systems in equilibrium are not ‘dynamic’. This fact has prompted some to suggest that the science would be better named ‘thermostatics’ (Tribus, 1961).