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.