Category When Is A Flow Compressible?

Introduction to Boundary Layers

A very satisfactory explanation of the physical process in the boundary layer between a fluid and a solid body could be obtained by the hypothesis of an adhesion of the fluid to the walls, that is, by the hypothesis of a zero relative velocity between fluid and wall. If the viscosity was very small and the fluid path along the wall not too long, the fluid velocity ought to resume its normal value at a very short distance from the wall. In the thin transition layer however, the sharp changes of velocity, even with small coefficient of friction, produce marked results.

Ludwig Prandtl, 1904

1 7.1 Introduction

The above quotation is taken from an historic paper given by Ludwig Prandtl at the third Congress of Mathematicians at Heidelberg, Germany, in 1904. In this paper, the concept of the boundary layer was first introduced—a concept which eventually revolutionized the analysis of viscous flows in the twentieth century and which allowed the practical calculation of drag and flow separation over aerodynamic bodies. Before Prandtl’s 1904 paper, the Navier-Stokes equations discussed in Chapter 15 were well known, but fluid dynamicists were frustrated in their attempts to solve these equations for practical engineering problems. After 1904, the picture changed completely. Using Prandtl’s concept of a boundary layer adjacent to an aerodynamic surface, the Navier-Stokes equations can be reduced to a more tractable form called the boundary – layer equations. In turn, these boundary-layer equations can be solved to obtain the distributions of shear stress and aerodynamic heat transfer to the surface. Prandtl’s boundary-layer concept was an advancement in the science of fluid mechanics of the caliber of a Nobel prize, although he never received that honor. The purpose of this chapter is to present the general concept of the boundary layer and to give a

few representative samples of its application. Our purpose here is to provide only an introduction to boundary-layer theory; consult Reference 42 for a rigorous and thorough discussion of boundary-layer analysis and applications.

What is a boundary layer? We have used this term in several places in our previous chapters, first introducing the idea in Section 1.10 and illustrating the concept in Figure 1.35. The boundary layer is the thin region of flow adjacent to a surface, where the flow is retarded by the influence of friction between a solid surface and the fluid. For example, a photograph of the flow over a supersonic body is shown in Figure 17.1, where the boundary layer (along with shock and expansion waves and the wake) is made visible by a special optical technique called a shadowgraph (see References 25 and 26 for discussions of the shadowgraph method). Note how thin the boundary layer is in comparison with the size of the body; however, although the boundary layer occupies geometrically only a small portion of the flow field, its influence on the drag and heat transfer to the body is immense—in Prandtl’s own words as quoted above, it produces “marked results.”

The purpose of the remaining chapters is to examine these “marked results.” The road map for the present chapter is given in Figure 17.2. In the next section, we discuss some fundamental properties of boundary layers. This is followed by a development

Figure I 7.1 The boundary layer on an aerodynamic body. (Courtesy of the U. S. Army Ballistics Laboratory, Aberdeen, Maryland.}

Figure I 7.2 Road map for Chapter 17.

of the boundary-layer equations, which are the continuity, momentum, and energy equations written in a special form applicable to the flow in the thin viscous region adjacent to a surface. The boundary layer equations are partial differential equations that apply inside the boundary layer.

Finally, we note that this chapter represents the second of the three options discussed in Section 15.7 for the solution of the viscous flow equations, namely, the simplification of the Navier-Stokes equations by neglecting certain terms that are smaller than other terms. This is an approximation, not a precise condition as in the case of Couette and Poiseuille flows in Chapter 16. In this chapter, we will see that the Navier-Stokes equations, when applied to the thin viscous boundary layer adjacent to a surface, can be reduced to simpler forms, albeit approximate, which lend themselves to simpler solutions. These simpler forms of the equations are called the boundary-layer equations—they are the subject of the present chapter.

The Velocity Potential Equation

The inviscid, compressible, subsonic flow over a body immersed in a uniform stream is irrotational; there is no mechanism in such a flow to start rotating the fluid elements (see Section 2.12). Thus, a velocity potential (see Section 2.15) can be defined. Since we are dealing with irrotational flow and the velocity potential, review Sections 2.12 and 2.15 before progressing further.

Consider two-dimensional, steady, irrotational, isentropic flow. A velocity po­tential, ф = <p(x, y), can be defined such that [from Equation (2.154)]

V = V0 [11.1]

or in terms of the cartesian velocity components,

Эф r,

u = —— [11.2a]

dx

_ 9 ф

V dy

Let us proceed to obtain an equation for ф which represents a combination of the continuity, momentum, and energy equations. Such an equation would be very useful, because it would be simply one governing equation in terms of one unknown, namely the velocity potential ф.

The continuity equation for steady, two-dimensional flow is obtained from Equa­tion (2.52) as

We are attempting to obtain an equation completely in terms of </>; hence, we need to eliminate p from Equation (11.5). To do this, consider the momentum equation in terms of Euler’s equation:

dp = —pV dV

This equation holds for a steady, compressible, inviscid flow and relates p and V along a streamline. It can readily be shown that Equation (3.12) holds in any direction throughout an irrotational flow, not just along a streamline (try it yourself). Therefore, from Equations (3.12) and (11.2a and b), we have

dp = – pVdV = ~^d(V2) = ~^d(u2 + v2)

Recall that we are also considering the flow to be isentropic. Hence, any change in pressure dp in the flow is automatically accompanied by a corresponding isentropic change in density dp. Thus, by definition

dp dp)s

The right-hand side of Equation (11.7) is simply the square of the speed of sound. Thus, Equation (11.7) yields

[1 1.8]

Substituting Equation (11.8) for the left side of Equation (11.6), we have

Considering changes in the x direction, Equation (11.9) directly yields

Similarly, for changes in the у direction, Equation (11.9) gives

Эр = p /дф д2ф дф д2ф пі ці

Эу а2 Эт Эх ду ду ду2 )

Substituting Equations (11.10) and (11.11) into (11.5), canceling the p which appears in each term, and factoring out the second derivatives of ф, we obtain

[1 1.12]

which is called the velocity potential equation. It is almost completely in terms of ф only the speed of sound appears in addition to ф. However, a can be readily expressed in terms of ф as follows. From Equation (8.33), we have

Since ao is a known constant of the flow, Equation (11.13) gives the speed of sound a as a function of ф. Hence, substitution of Equation (11.13) into (11.12) yields a single partial differential equation in terms of the unknown ф. This equation represents a combination of the continuity, momentum, and energy equations. In principle, it can be solved to obtain ф for the flow field around any two-dimensional shape, subject of course to the usual boundary conditions at infinity and along the body surface. These boundary conditions on ф are detailed in Section 3.7, and are given by Equations (3.47a and b) and (3.48Z?).

Because Equation (11.12) Lalong with Equation (11.13)] is a single equation in terms of one dependent variable ф, the analysis of isentropic, irrotational, steady, compressible flow is greatly simplified—we only have to solve one equation instead of three or more. Once ф is known, all the other flow variables are directly obtained as follows:

1. Calculate и and v from Equations (11.2a and b).

2. Calculate a from Equation (11.13).

3. Calculate M = V/a — u2 + v2/a.

4. Calculate T, p, and p from Equations (8.40), (8.42), and (8.43), respectively. In these equations, the total conditions Tq, po, and po are known quantities; they are constant throughout the flow field and hence are obtained from the given freestream conditions.

Although Equation (11.12) has the advantage of being one equation with one unknown, it also has the distinct disadvantage of being a nonlinear partial differential equation. Such nonlinear equations are very difficult to solve analytically, and in modem aerodynamics, solutions of Equation (11.12) are usually sought by means of sophisticated finite-difference numerical techniques. Indeed, no general analytical solution of Equation (11.12) has been found to this day. Contrast this situation with that for incompressible flow, which is governed by Laplace’s equation—a linear partial differential equation for which numerous analytical solutions are well known.

Given this situation, aerodynamicists over the years have made assumptions regarding the physical nature of the flow field which are designed to simplify Equation

(11.12) . These assumptions limit our considerations to the flow over slender bodies at small angles of attack. For subsonic and supersonic flows, these assumptions lead to an approximate form of Equation (11.12) which is linear, and hence can be solved analytically. These matters are the subject of the next section.

Keep in mind that, within the framework of steady, irrotational, isentropic flow, Equation (11.12) is exact and holds for all Mach numbers, from subsonic to hyper­sonic, and for all two-dimensional body shapes, thin and thick.

Qualitative Aspects of Viscous Flow

What is a viscous flow? Answer: A flow where the effects of viscosity, thermal conduction, and mass diffusion are important. The phenomenon of mass diffusion is important in a gas with gradients in its chemical species, for example, the flow of air over a surface through which helium is being injected or the chemically reacting flow through a jet engine or over a high-speed reentry body. In this book, we are not concerned with the effects of diffusion, and therefore we treat a viscous flow as one where only viscosity and thermal conduction are important.

First, consider the influence of viscosity. Imagine two solid surfaces slipping over each other, such as this book being pushed across a table. Clearly, there will be a frictional force between these objects which will retard their relative motion. The same is true for the flow of a fluid over a solid surface; the influence of friction between the surface and the fluid adjacent to the surface acts to create a frictional force which retards the relative motion. This has an effect on both the surface and the fluid. The surface feels a “tugging” force in the direction of the flow, tangential to the surface. This tangential force per unit area is defined as the shear stress r, first introduced in Section 1.5 and illustrated in Figure 15.2. As an equal and opposite reaction, the fluid adjacent to the surface feels a retarding force which decreases its local flow velocity, as shown in insert a of Figure 15.2. Indeed, the influence of friction is to create V = 0 right at the body surface—this is called the по-slip condition which dominates viscous flow. In any real continuum fluid flow over a solid surface, the flow velocity is zero at the surface. Just above the surface, the flow velocity is finite, but retarded, as shown in insert a. If и represents the coordinate normal to the surface, then in

Figure 1 5.2 Effect of viscosity on a body in a moving fluid: shear stress and separated flow.

the region near the surface, V = V(n), where V = 0 at n = 0, and V increases as n increases. The plot of V versus n as shown in insert a is called a velocity profile. Clearly, the region of flow near the surface has velocity gradients, 9 V/дп, which are due to the frictional force between the surface and the fliud.

In addition to the generation of shear stress, friction also plays another (but related) role in dictating the flow over the body in Figure 15.2. Consider a fluid element moving in the viscous flow near a surface, as sketched in Figure 15.3. Assume that the flow is in its earliest moments of being started. At the station si, the velocity of the fluid element is Vi. Assume that the flow over the surface produces an increasing pressure distribution in the flow direction (i. e., assume p3 > рг > Pi). Such a region of increasing pressure is called an adverse pressure gradient. Now follow the fluid element as it moves downstream. The motion of the element is already retarded by the effect of friction; in addition, it must work its way along the flow against an increasing pressure, which tends to further reduce its velocity. Consequently, at station 2 along the surface, its velocity V2 is less than Vi. As the fluid element continues to move downstream, it may completely “run out of steam,” come to a stop, and then, under the action of the adverse pressure gradient, actually reverse its direction and start moving back upstream. This “reversed flow” is illustrated at station S3 in Figure 15.3, where the fluid element is now moving upstream at the velocity V3. The picture shown in Figure 15.3 is meant to show the flow details very near the surface at the very initiation of the flow. In the bigger picture of this flow at later times shown in Figure 15.2, the consequence of such reversed-flow phenomena is to cause the flow to separate from

Figure 1 5.3 Separated flow induced by an adverse pressure gradient. This picture corresponds to the early evolution of the flow; once the flow separates from the surface between points 2 and 3, the fluid element shown at S3 is in reality different from that shown at S] and S2 because the primary flow moves away from the surface, as shown in Figure 15.2.

the surface and create a large wake of recirculating flow downstream of the surface. The point of separation on the surface in Figure 15.2 occurs where dV/dn = 0 at the surface, as sketched in insert b of Figure 15.2. Beyond this point, reversed flow occurs. Therefore, in addition to the generation of shear stress, the influence of friction can cause the flow over a body to separate from the surface. When such separated flow occurs, the pressure distribution over the surface is greatly altered. The primary flow over the body in Figure 15.2 no longer sees the complete body shape; rather, it sees the body shape upstream of the separation point, but downstream of the separation point it sees a greatly deformed “effective body” due to the large separated region. The net effect is to create a pressure distribution over the actual body surface which results in an integrated force in the flow direction, that is, a drag. To see this more clearly, consider the pressure distribution over the upper surface of the body as sketched in Figure 15.4. If the flow were attached, the pressure over the downstream portion of the body would be given by the dashed curve. Flowever, for separated flow, the pressure over the downstream portion of the body is smaller, given by the solid curve in Figure 15.4. Now return to Figure 15.2. Note that the pressure over the upper rearward surface contributes a force in the negative drag direction; that is, p acting over the element of surface ds shown in Figure 15.2 has a horizontal component in the upstream direction. If the flow were inviscid, subsonic, and attached and the body were two-dimensional, the forward-acting components of the pressure distribution shown in Figure 15.2 would exactly cancel the rearward-acting components due to the pressure distribution over other parts of the body such that the net, integrated pressure distribution would give zero drag. This would be d’Alembert’s paradox discussed in Chapter 3. Flowever, for the viscous, separated flow, we see that p is reduced in the separated region; hence, it can no longer fully cancel the pressure distribution over the remainder of the body. The net result is the production of drag; this is called the pressure drag due to flow separation and is denoted by Dp.

Figure 1 5.4 Schematic of the pressure

distributions for attached and separated flow over the upper surface of the body illustrated in Figure 15.2.

In summary, we see that the effects of viscosity are to produce two types of drag as follows:

Df is the skin friction drag, that is, the component in the drag direction of the integral of the shear stress r over the body.

Dp is the pressure drag due to separation, that is, the component in the drag direction of the integral of the pressure distribution over the body.

Dp is sometimes called form drag. The sum Df + Dp is called the profile drag of a two-dimensional body. For a three-dimensional body such as a complete airplane, the sum Df + Dp is frequently called parasite drag. (See Reference 2 for a more extensive discussion of the classification of different drag contributions.)

The occurrence of separated flow over an aerodynamic body not only increases the drag but also results in a substantial loss of lift. Such separated flow is the cause of airfoil stall as discussed in Section 4.3. For these reasons, the study, understanding, and prediction of separated flow is an important aspect of viscous flow.

Let us turn our attention to the influence of thermal conduction—another overall physical characteristic of viscous flow in addition to friction. Again, let us draw an analogy from two solid bodies slipping over each other, such as the motion of this book over the top of a table. If we would press hard on the book, and vigorously rub it back and forth over the table, the cover of the book as well as the table top would soon become warm. Some of the energy we expend in pushing the book over the table will be dissipated by friction, and this shows up as a form of heating of the bodies. The same phenomenon occurs in the flow of a fluid over a body. The moving fluid has a certain amount of kinetic energy; in the process of flowing over a surface, the flow velocity is decreased by the influence of friction, as discussed earlier, and hence the kinetic energy is decreased. This lost kinetic energy reappears in the form of internal energy of the fluid, hence causing the temperature to rise. This phenomenon is called viscous dissipation within the fluid. In turn, when the fluid temperature increases, there is an overall temperature difference between the warmer fluid and the cooler body. We know from experience that heat is transferred from a warmer body to a cooler body; therefore, heat will be transferred from the warmer fluid to the cooler surface. This is the mechanism of aerodynamic heating of a body. Aerodynamic heating becomes more severe as the flow velocity increases, because more kinetic energy is dissipated by friction, and hence the overall temperature difference between the warm fluid and the cool surface increases. As discussed in Chapter 14, at hypersonic speeds, aerodynamic heating becomes a dominant aspect of the flow.

All the aspects discussed above—shear stress, flow separation, aerodynamic heating, etc.—are dominated by a single major question in viscous flow, namely, Is the flow laminar or turbulent? Consider the viscous flow over a surface as sketched in Figure 15.5. If the path lines of various fluid elements are smooth and regular, as shown in Figure 15.5a, the flow is called laminar flow. In contrast, if the motion of a fluid element is very irregular and tortuous, as shown in Figure 15.5b, the flow is called turbulent flow. Because of the agitated motion in a turbulent flow, the higher-energy fluid elements from the outer regions of the flow are pumped close to the surface. Hence, the average flow velocity near a solid surface is larger for a turbulent flow

(b) Turbulent flow

Figure 1 5.5 Path lines for laminar and turbulent flows.

in comparison with laminar flow. This comparison is shown in Figure 15.6, which gives velocity profiles for laminar and turbulent flow. Note that immediately above the surface, the turbulent flow velocities are much larger than the laminar values. If (3 V/3n)„=0 denotes the velocity gradient at the surface, we have

Because of this difference, the frictional effects are more severe for a turbulent flow; both the shear stress and aerodynamic heating are larger for the turbulent flow in comparison with laminar flow. However, turbulent flow has a major redeeming value; because the energy of the fluid elements close to the surface is larger in a turbulent flow, a turbulent flow does not separate from the surface as readily as a laminar flow. If the flow over a body is turbulent, it is less likely to separate from the body surface, and if flow separation does occur, the separated region will be smaller. As a result, the pressure drag due to flow separation Dp will be smaller for turbulent flow.

This discussion points out one of the great compromises in aerodynamics. For the flow over a body, is laminar or turbulent flow preferable? There is no pat answer; it depends on the shape of the body. In general, if the body is slender, as sketched in Figure 15.7a, the friction drag Df is much greater than Dp. For this case, because Df is smaller for laminar than for turbulent flow, laminar flow is desirable for slender bodies. In contrast, if the body is blunt, as sketched in Figure 15.7b, Dp is much greater than Df. For this case, because Dp is smaller for turbulent than for laminar flow, turbulent flow is desirable for blunt bodies. The above comments are not all­inclusive; they simply state general trends, and for any given body, the aerodynamic virtues of laminar versus turbulent flow must always be assessed.

Although, from the above discussion, laminar flow is preferable for some cases, and turbulent flow for other cases, in reality we have little control over what actually happens. Nature makes the ultimate decision as to whether a flow will be laminar or turbulent. There is a general principle in nature that a system, when left to itself, will always move toward its state of maximum disorder. To bring order to the system, we generally have to exert some work on the system or expend energy in some manner. (This analogy can be carried over to daily life; a room will soon become cluttered and disordered unless we exert some effort to keep it clean.) Since turbulent flow is much more “disordered” than laminar flow, nature will always favor the occurrence of turbulent flow. Indeed, in the vast majority of practical aerodynamic problems, turbulent flow is usually present.

Let us examine this phenomenon in more detail. Consider the viscous flow over a flat plate, as sketched in Figure 15.8. The flow immediately upstream of the leading edge is uniform at the freestream velocity. However, downstream of the leading edge, the influence of friction will begin to retard the flow adjacent to the surface, and the extent of this retarded flow will grow higher above the plate as we move downstream, as shown in Figure 15.8. To begin with, the flow just downstream of the leading edge will be laminar. However, after a certain distance, instabilities will appear in the laminar flow; these instabilities rapidly grow, causing transition to turbulent flow. The transition from laminar to turbulent flow takes place over a finite region, as sketched in Figure 15.8. However, for purposes of analysis, we frequently model the

Figure 1 5.7 Drag on slender and blunt bodies.

Figure 15.8 Transition from laminar to turbulent flow.

transition region as a single point, called the transition point, upstream of which the flow is laminar and downstream of which the flow is turbulent. The distance from the leading edge to the transition point is denoted by xCT. The value of xcr depends on a whole host of phenomena. For example, some characteristics which encourage transition from laminar to turbulent flow, and hence reduce xa, are:

1. Increased surface roughness. Indeed, to promote turbulent flow over a body, rough grit can be placed on the surface near the leading edge to “trip” the laminar flow into turbulent flow. This is a frequently used technique in wind-tunnel testing. Also, the dimples on the surface of a golf ball are designed to encourage turbulent flow, thus reducing Dp. In contrast, in situations where we desire large regions of laminar flow, such as the flow over the NACA six-series laminar-flow airfoils, the surface should be as smooth as possible. The main reason why such airfoils do not produce in actual flight the large regions of laminar flow observed in the laboratory is that manufacturing irregularities and bug spots (believe it or not) roughen the surface and promote early transition to turbulent flow.

2. Increased turbulence in the free stream. This is particularly a problem in wind – tunnel testing; if two wind tunnels have different levels of freestream turbulence, then data generated in one tunnel are not repeatable in the other.

3. Adverse pressure gradients. In addition to causing flow-field separation as dis­cussed earlier, an adverse pressure gradient strongly favors transition to turbulent flow. In contrast, strong favorable pressure gradients (where p decreases in the downstream direction) tend to preserve initially laminar flow.

4. Heating of the fluid by the surface. If the surface temperature is warmer than the adjacent fluid, such that heat is transferred to the fluid from the surface, the instabilities in the laminar flow will be amplified, thus favoring early transition. In contrast, a cold wall will tend to encourage laminar flow.

There are many other parameters which influence transition; see Reference 42 for a more extensive discussion. Among these are the similarity parameters of the flow, principally Mach number and Reynolds number. High values of Мж and low values of Re tend to encourage laminar flow; hence, for high-altitude hypersonic flight, laminar flow can be quite extensive. The Reynolds number itself is a dominant factor

in transition to turbulent flow. Referring to Figure 15.8, we define a critical Reynolds number, Recr, as

D ______ Poo F-x-l’cT

КЄСГ =

P’00

The value of Recr for a given body under specified conditions is difficult to predict; indeed, the analysis of transition is still a very active area of modem aerodynamic research. As a rule of thumb in practical applications, we frequently take Recr ~ 500,000; if the flow at a given x station is such that Re = рж V^x/poo is considerably below 500,000, then the flow at that station is ihost likely laminar, and if the value of Re is much larger than 500,000, then the flow is most likely turbulent.

To obtain a better feeling for Recr, let us imagine that the flat plate in Figure 15.8 is a wind-tunnel model. Assume that we carry out an experiment under standard sea level conditions [рж — 1.23 kg/m3 and = 1.79 x 10-5 kg/(m ■ s)] and measure xcr for a certain freestream velocity; for example, say that xCI = 0.05 m when Voo = 120 m/s. In turn, this measured value of xcr determines the measured Recr as

Hence, for the given flow conditions and the surface characteristics of the flat plate, transition will occur whenever the local Re exceeds 412,000. For example, if we double Voo, that is, = 240 m/s, then we will observe transition to occur at xcr =

0. 05/2 = 0.025 m, such that Recr remains the same value of 412,000.

This brings to an end our introductory qualitative discussion of viscous flow. The physical principles and trends discussed in this section are very important, and you should study them carefully and feel comfortable with them before progressing further.

Finite-Difference Method

Return for a moment to Section 2.17.2 where we introduced some ideas from compu­tation fluid dynamics, and especially review the finite-difference expressions derived there. Recall that we can simulate the partial derivatives with forward, rearward, or central differences. We will use these concepts in the following discussion.

Also consider Figure 18.13, which shows a schematic of a finite-difference grid inside the boundary layer. The grid is shown in the physical x-y space, where it is curvilinear and unequally spaced. However, in the £-77 space, where the calculations are made, the grid takes the form of a rectangular grid with uniform spacing A£ and A rj. In Figure 18.13, the portion of the grid at four different £ (or x ) stations is shown, namely, at (7 — 2), (7 — 1), 7, and (7 + 1).

Consider again the general, transformed boundary-layer equations given by Equations (18.84) and (18.86). Assume that we wish to calculate the boundary layer at station (7 + 1) in Figure 18.13. As discussed in Section 2.17.2, the general philos­ophy of finite-difference approaches is to evaluate the governing partial differential equations at a given grid point by replacing the derivatives by finite-difference quo­tients at that point. Consider, for example, the grid point (7, j) in Figure 18.13. At this point, replace the derivatives in Equations (18.84) and (18.86) by finite-difference expressions of the form:

[18.87]

[18.88]

where в is a parameter which adjusts Equations (18.87)—(18.90) to various finite – difference approaches (to be discussed below). Similar relations for the derivatives of g are employed. When Equations (18.87)-( 18.90) are inserted into Equations (18.84) and (18.86), along with the analogous expressions for g, two algebraic equations are obtained. If в = 0, the only unknowns that appear are fi+ij and gi+ij, which can be obtained directly from the two algebraic equations. This is an explicit approach. Using this approach, the boundary layer properties at grid point (; + 1, j) are solved explicitly in terms of the known properties at points (i, j + 1), (i, j) and O’, j — 1). The boundary-layer solution is a downstream marching procedure; we are calculating the boundary layer profiles at station (; + 1) only after the flow at the previous station (і) has been obtained.

When 0 < в < 1, then fi+ij+i, fi+,j-, g;+u+i, gi+ij, and gi+ij-i

appear as unknowns in Equations (18.84) and (18.86). We have six unknowns and only two equations. Therefore, the finite-difference forms of Equations (18.84) and (18.86) must be evaluated at all the grid points through the boundary layer at station (i + 1) simultaneously, leading to an implicit formulation of the unknowns. In particular, if в = the scheme becomes the well-known Crank-Nicolson implicit procedure, and if в — 1, the scheme is called “fully implicit.” These implicit schemes result in large systems of simultaneous algebraic equations, the coefficients of which constitute block tridiagonal matrices.

Already the reader can sense that implicit solutions are more elaborate than ex­plicit solutions. Indeed, we remind ourselves that the subject of this book is the fundamentals of aerodynamics, and it is beyond our scope to go into great computa­tional fluid dynamic detail. Therefore, we will not elaborate any further. Our purpose here is only to give the flavor of the finite-difference approach to boundary-layer solutions. For more information on explicit and implicit finite-difference methods, see the author’s book Computational Fluid Dynamics: The Basics with Applications (Reference 64).

In summary, a finite-difference solution of a general, nonsimilar boundary-layer proceeds as follows:

1. The solution must be started from a given solution at the leading edge, or at a stagnation point (say station 1 in Figure 18.13). This can be obtained from appropriate self-similar solutions.

2. At station 2, the next downstream station, the finite-difference procedure reflected by Equations (18.87)—(18.90) yields a solution of the flowfield variables across the boundary layer.

3. Once the boundary-layer profiles of и and T are obtained, the skin friction and heat transfer at the wall are determined from

„ = (»£)_

Here, the velocity gradients can be obtained from the known profiles of и and 7 by using one-sided differences (see References 64), such as

‘dus

—3u]

-f – 4w 2 — и з

[18.91]

1 –

W

2 Ay

dT2

-37,

+ 47) – 7,

[18.92]

9>’y

) –

W

2 Ay

In Equations (18.91) and (18.92), the subscripts 1,2, and 3 denote the wall point and the next two adjacent grid points above the wall. Of course, due to the specified boundary conditions of no velocity slip and a fixed wall temperature, и і = 0 and T = Tw in Equations (18.91) and (18.92).

4. The above steps are repeated for the next downstream location, say station 3 in Figure 18.13. In this fashion, by repeating applications of these steps, the complete boundary layer is computed, marching downstream from a given initial solution.

An example of results obtained from such finite-difference boundary-layer solu­tions is given in Figures 18.14 and 18.15 obtained by Blottner (Reference 84). These are calculated for flow over an axisymmetric hyperboloid flying at 20,000 ft/s at an altitude of 100,000 ft, with a wall temperature of 1000 K. At these conditions, the boundary layer will involve dissociation, and such chemical reactions were included in the calculations of Reference 84. Chemically reacting boundary layers are not the purview of this book; however, some results of Reference 84 are presented here just to illustrate the finite-difference method. For example, Figure 18.14 gives the calculated velocity and temperature profiles as a station located at x/RN = 50, where Rn is the nose radius. The local values of velocity and temperature at the boundary layer edge are also quoted in Figure 18.14. Considering the surface properties, the variations of Ся and ty as functions of distance from the stagnation point are shown in Figure 18.15. Note the following physical trends illustrated in Figure 18.15.

1. The shear stress is zero at the stagnation point (as is always the case), then it in­creases around the nose, reaches a maximum, and decreases further downstream.

2. The values of Ся are relatively constant near the nose, and then decrease further downstream.

3. Reynolds analogy can be written as

Сц = [18.93]

2s

where 5 is called the “Reynolds analogy factor.” For the flat plate case, we see from Equation (18.50) that л = Pr1. However, clearly from the results of Figure 18.15 we see that s is a variable in the nose region because Ся is relatively constant while су is rapidly increasing. In contrast, for the downstream region, Cf and Ся are essentially equal, and we can state that Reynolds analogy becomes

u/ue or T/Te

Figure 18.14 Velocity and temperature profiles across the boundary layer at x/Rn = 50 on an axisymmetric hyperbloid. (Source: Blottner, Reference 84.}

approximately C#/c/ = 1. The point here is that Reynolds analogy is greatly affected by strong pressure gradients in the flow, and hence loses its usefulness as an engineering tool in such cases, at least when Ся and с/ are based on freestream quantities as shown in Figure 18.15.

Figure 18.15 Stanton number and skin friction coefficient (based on freestream properties) along a hyperbloid. (Source: Blottner, Reference 84.]

18.7 Summary

This brings to an end our discussion of laminar boundary layers. Return to the roadmap in Figure 18.1 and remind yourself of the territory we have covered. Some of the important results are summarized below.

For incompressible laminar flow over a flat plate, the boundary-layer equations reduce to the Blasius equation

2/"’ + //" = 0

[18.15]

where /’ = м/м,,. This produces a self-similar solution where /’ = independent of any particular x station along the surface. A numerical solution of Equation (17.48) yields numbers which lead to the following results.

tw 0.664

Local skin friction coefficient: cf = i——————– г = ,_____

2 Poo V Л-®*

[18.30]

1-328

Integrated friction drag coefficient: Cf= ________

vRec

[18.33]

5.О*

Boundary-layer thickness: S = -……………………….. -■

VRe^

[18.33]

(continued)