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We are surrounded by examples of fluid dynamics phenomena; computational fluid dynamics (CFD) deals with the numerical analysis of these phenomena. Despite impressive progress in recent years it is evident that CFD remains an imperfect tool in the comparatively mature discipline of fluid dynamics, partly because electronic digital computers have been in widespread use for less than thirty years. History reminds us of ancient examples of fluid dynamics applications such as the Roman baths and aqueducts that fulfilled the requirements of the engineers who built them; of ships of various types with adequate hull designs, and of wind energy systems, built long before the subject of fluid mechanics was formalized by Reynolds, Newton, Euler, Navier, Stokes, Prandtl and others. The twentieth century has witnessed many more examples of applications of fluid dynamics for the use of humanity, all designed without the use of electronic computers. Examples include prime movers such as internal-combustion engines, gas and steam turbines, flight vehicles, and environmental systems for pollution control and ventilation.

The conservation equations for fluid flow are based on the principles of conservation of mass, momentum and energy and are known as the Navier-Stokes equations. They can be represented in both differential and integral forms. In this chapter we do not provide detailed derivations of these equations since they can be found in various textbooks such as References 1 to 5. In this book, we assume that the flow is incompressible and the temperature differences between the surface and freestream are small so that the fluid properties such as density ρ and dynamic viscosity μ in the conservation equations are not affected by temperature. This assumption allows us to direct our attention to the conservation equations for mass and momentum and ignore the conservation equation for energy.

Another simplification of the Navier Stokes equations occurs when the ratio of boundary-layer thickness δ to a reference length L, δ/L , is sufficiently small because terms that are smaller than the main terms by a factor of δ/L can be neglected. The resulting equations are known as the boundary-layer equations and include both laminar and turbulent flows as discussed here in this section for two-dimensional flows and axisymmetric flows in Section 3.2.

The boundary-layer equations of the previous chapter can be solved for a range of laminar and turbulent flows for the boundary and initial conditions discussed in Section 3.4. In this chapter we consider laminar flows and postpone the discussion on turbulent flows to Chapter 6. In both cases a solution can be obtained either by a “differential” method, solving the partial differential equations (3.1.2) and (3.1.10) or Eqs. (3.2.1) and (3.2.2) numerically; or by an “integral” method, solving ordinary differential equations already integrated in the y -direction, namely Eqs. (3.3.10) or (3.3.14).

At sufficiently high Reynolds numbers, most flows are turbulent rather than laminar, and this transition to turbulence has been the object of many studies utilizing several approaches. One approach discussed, for example, in [ 1 ], considers the solutions of the parabolized stability equations (PSE), and another considers the solutions of the unsteady Navier-Stokes equations (called direct numerical simulation, DNS) and another approach considers the solutions of the linear stability equations based on small-disturbance theory. Despite the progress being made with the PSE approach, this method currently is some time away from being used as an engineering tool; it is still under development. The direct numerical simulation (DNS) approach offers exciting possibilities; the excellent review of Kleiser and Zong [ 2 ] describes the rapid progress with this approach and shows that the prediction of transition can already be achieved in some simple flows with this method. The computer requirements of DNS, however, are large, and it is unlikely that this approach can be used for transition calculations on complex bodies in the near future. The only engineering calculation method for predicting transition at this time, aside from correlation methods, is the e ^n-method based on small disturbance theory to be discussed in this chapter for two-dimensional flows and in Chapter 8 for three-dimensional flows.

Unlike the solution of laminar flows discussed in Chapter 4, the solution of the turbulent boundary-layer equations can not be obtained without a model for the Reynolds shear stress term $$ - \rho \overline {u'v'} $$ , which introduces an additional unknown to the system given by the continuity and momentum equations. In addition, turbulent boundary-layers do not admit similarity solutions. This means that velocity profiles are not geometrically similar and do not reduce to a single curve if u / u _ e is plotted against a dimensionless y -coordinate, η. One velocity scale u * and one length scale l ( $$ u_e and\sqrt {vx/u_e } $$ , respectively, for laminar flows) are not sufficient to describe a turbulent boundary-layer velocity profile. Experiments indicate that the behavior of the flow close to the wall is different from the behavior of the flow away from the wall. This leads to a composite nature of a turbulent boundary-layer and requires different length scales in each region, as discussed in Section 6.2.

In this chapter and the following chapter we extend the discussion on two-dimensional flows of the previous chapters to three-dimensional flows. Unlike the separate discussion on laminar flows in Chapter 4 and turbulent flows in Chapter 6, this chapter considers both laminar and turbulent flows and begins with a discussion on the origin of crossflow in boundary-layers and presents three-dimensional boundary-layer equations in several coordinate systems. It is followed by Section 7.3 that discusses the initial conditions needed to solve the three-dimensional boundary-layer equations. Section 7.4 addresses the modeling of Reynolds stresses and presents an extension of Section 6.3 to three-dimensional flows. As in two-dimensional flows, it is computationally more convenient and economical to solve the three-dimensional boundary-layer equations in transformed coordinates. This is discussed in Section 7.5, including a discussion of the similarity solutions of the three-dimensional boundary-layer equations. Section 7.6 describes the numerical procedures used to solve the boundary-layer equations for a prescribed external velocity distribution. Unlike two-dimensional flows, the solution of the three-dimensional boundary-layer equations, even in the absence of flow separation, is rather involved due to the possible flow reversals in the spanwise velocity, w , which can cause numerical instabilities if appropriate numerical schemes are not used. For this reason, in addition to the box scheme discussed in the previous chapters for two-dimensional flows, another version of this scheme is discussed in Section 7.6.

As discussed in Chapter 5, transition in two-dimensional flows usually occurs in zero-pressure gradient or decelerating flows due to the growth of the Tollmien-Schlichting (TS) waves. Transition can also occur in accelerating flows if the Reynolds number is very high.

Throughout this book we have discussed the solution of the boundary-layer equations for external flows with prescribed external velocity distribution. This approach to boundary-layer theory, sometimes referred to as the standard problem or direct problem , allows viscous flow solutions provided that boundary-layer separation, which corresponds to vanishing wall shear in two-dimensional steady flows, does not occur. If the wall shear vanishes at some x -location, the solutions breakdown and convergence cannot be obtained. This is referred to as the singular behavior of the boundary-layer equations at separation. For laminar flows, the behavior of the wall shear τ _w close to the separation point x _s, has been shown to be of the form (9.1.1) $$ \left( {\frac{{\partial u}} {{\partial y}}} \right) \sim (x_s - x)^{1/2} $$ by Goldstein [ 1 ] who considered a uniformly retarded flow past a semi-infinite plate and showed that, with the relation given above, there is no real solution downstream of separation; the normal velocity component v becomes infinite at x _s (see Problem 9.6). Goldstein also pointed out that the pressure distribution around the separation point cannot be taken arbitrarily and must satisfy conditions associated with the existence of reverse flow downstream of separation.

In Chapters 2 and 3. we discussed the continuity and momentum equations for incompressible flows. Here, we extend the discussion to compressible flows. If the typical temperature difference in a gas flow is an appreciable fraction of the absolute temperature, the typical density difference will be an appreciable fraction of the absolute density, and the density appearing in the velocity field equations discussed in the previous chapters can no longer be taken as constant. Instead, the conservation equations for momentum and energy must be solved simultaneously since they are coupled, i.e., density appears in the momentum equations and is linked through an equation of state to the dependent variable of the energy equation.