Category When Is A Flow Compressible?

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.

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

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

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.

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.

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.

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

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

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.

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.

lflfith the stabilizer setting at 2° the speed was allowed to increase to approximately 0.98 to 0.99 Mach number where elevator and rudder effectiveness were regained and the airplane seemed to smooth out to normal flying characteristics. This development lent added confidence and the airplane was allowed to continue until an indication of 1.02 on the cockpit Mach meter was obtained. At this indication the meter momentarily stopped and then jumped at 1.06, and this hesitation was assumed to be caused by the effect of shock waves on the static source. At this time the power units were cut and the airplane allowed to decelerate back to the subsonic flight condition.

Captain Charles Yeager, describing his flight on October 14, 1947—the first manned flight to exceed the speed of sound.

12.1 Introduction

The linearized perturbation velocity potential equation derived in Chapter 11, Equa­tion (11.18), is

and holds for both subsonic and supersonic flow. In Chapter 11, we treated the case of subsonic flow, where 1 — > 0 in Equation (11.18). However, for supersonic flow,

1 – Ml < 0. This seemingly innocent change in sign on the first term of Equation (11.18) is, in reality, a very dramatic change. Mathematically, when 1 — > 0 for

subsonic flow, Equation (11.18) is an elliptic partial differential equation, whereas when 1 — < 0 for supersonic flow, Equation (11.18) becomes a hyperbolic

differential equation. The details of this mathematical difference are beyond the scope of this book; however, the important point is that there is a difference. Moreover, this portends a fundamental difference in the physical aspects of subsonic and supersonic flow—something we have already demonstrated in previous chapters.

The purpose of this chapter is to obtain a solution of Equation (11.18) for super­sonic flow and to apply this solution to the calculation of supersonic airfoil properties. Since our purpose is straightforward, and since this chapter is relatively short, there is no need for a chapter road map to provide guidance on the flow of our ideas.

Let us imagine the following situation. Assume that the flow illustrated in Figure 16.5 is established. We have the parabolic temperature profile established as shown, and we have heat transfer into the walls as just discussed. However, both wall temperatures are considered fixed, and both are equal to the same constant value. Question: How can the wall temperature remain fixed at the same time that heat is transferred into the wall? Answer: There must be some independent mechanism that conducts heat away from the wall at the same rate that the aerodynamic heating is pumping heat into the wall. This is the only way for the wall temperature to remain fixed at some cooler temperature than the adjacent fluid. For example, the wall can be some vast heat sink that can absorb heat without any appreciable change in temperature, or possibly there are cooling coils within the plate that can carry away the heat, much like the water coils that keep the engine of your automobile cool. In any event, to have the picture shown in Figure 16.5 with a constant wall temperature independent of time, some exterior mechanism must carry away the heat that is transferred from the fluid to the walls. Now imagine that, at the lower wall, this exterior mechanism is suddenly shut off. The lower wall will now begin to grow hotter in response to qw, and Tw will begin to increase with time. At any given instant during this transient process, the heat transfer to the lower wall is given by Equation (16.24), repeated below.

At time t = 0, when the exterior cooling mechanism is just shut off, hw = hc, and qw is given by Equation (16.35), namely,

However, as time now progresses, Tu, (and therefore hw) increases. From Equa­tion (16.24), as hy, increases, the numerator decreases in magnitude, and hence qw decreases. That is,

Hence, as time progresses from when the exterior cooling mechanism was first cut off at the lower wall, the wall temperature increases, and the aerodynamic heating to the wall decreases. This in turn slows the rate of increase of Tw as time progresses. The transient variations of both qw and Tw are sketched in Figure 16.6. In Figure 16.6a, we see that, as time increases to large values, the heat transfer to the wall approaches zero—this is defined as the equilibrium, or the adiabatic wall condition. For an adiabatic wall, the heat transfer is, by definition, equal to zero. Simultaneously, the wall temperature Tw approaches asymptotically a limiting value defined as the adiabatic wall temperature Taw, and the corresponding enthalpy is defined as the adiabatic wall enthalpy haw.

The purpose of this discussion is to define an adiabatic wall condition; the ex­ample involving a timewise approach to this condition was just for convenience and edification. Let us now assume that the lower wall in our Couette flow is an adiabatic wall. For this case, we already know the value of heat transfer to the wall—by defi­nition, it is zero. The question now becomes, What is the value of the adiabatic wall enthalpy haw, and in turn the adiabatic wall temperature Taw‘l The answer is given by Equation (16.23), where qw = 0 for an adiabatic wall.

In turn, the adiabatic wall temperature is given by

[16.40]

Clearly, the higher the value of ue, the higher is the adiabatic wall temperature.

The enthalpy profile across the flow for this case is given by a combination of Equations (16.16) and (16.40), as follows. Setting hw = haw in Equation (16.16), we

obtain

[16.43]

Equation (16.43) gives the enthalpy profile across the flow. The temperature profile follows from Equation (16.43) as

[16.44]

This variation of T is sketched in Figure 16.7. Note that Tuw is the maximum tem­perature in the flow. Moreover, the temperature curve is perpendicular at the plate for v = 0; that is, the temperature gradient at the lower plate is zero, as expected for an adiabatic wall. This result is also obtained by differentiating Equation (16.44):

which gives З T/9у = 0 at у = 0.

Here we show some very recent Navier-Stokes solutions carried out to study the aerodynamic effect of a small protuberance extending from the bottom surface of an airfoil. These calculations represent an example of the state-of-the-art of full Navier-Stokes solutions at the time of writing. The work was carried out by Beierle (Reference 89). The basic shape of the airfoil was an NACA 0015 section. The computational fluid dynamic solution of the Navier-Stokes equations was carried out using a time-marching finite volume code labeled OVERFLOW, developed by NASA (Reference 90). The flow was low speed, with a freestream Mach number of 0.15 and Reynolds number of 1.5 x 106. The fully turbulent flow field was simulated using a 1-equation turbulence model.

Using a proper grid is vital to the integrity of any Navier-Stokes CFD solution. For the present case, Figures 20.8-20.11 show the grid used, progressing from the

Figure 20.7 Effects of shock-wave/boundary-layer interaction on (a) pressure distribution, and (b) shear stress for Mach 3 turbulent flow over a flat plate.

big picture of the whole grid (Figure 20.8) to the detail of the grid around the small protuberance on the bottom surface of the airfoil (Figure 20.11). The grid is an example of a chimera grid, a series of independent but overlapping grids that are generated about individual parts of the body and for specific flow regions.

Some results for the computed flow field are shown in Figures 20.12 and 20.13. In Figure 20.12, the local velocity vector field is shown; the flow separation and locally reversed flow can be seen downstream of the protuberance. In Figure 20.13, pressure contours are shown, illustrating how the small protuberance generates a substantially asymmetric flow over the otherwise symmetric airfoil.

Finally, results for a related flow are shown in Figure 20.14. Here, instead of a protuberance existing on the bottom surface, an array of small jets that are distributed

over the bottom surface alternately blow and suck air into and out of the flow in such a manner that the net mass flow added is zero, so-called “zero-mass synthetic jets.” The resulting series of large-scale vortices is shown in Figure 20.14—another example of a flow field that can only be solved in detail by means of a full Navier-Stokes solution. (See Hassan and JanakiRam, Reference 91, for details.)

Return to the oblique shock wave illustrated in Figure 9.1a. In this picture, we can imagine the shock wave extending unchanged above the comer to infinity. However, in real life this does not happen. In reality, the oblique shock in Figure 9.1a will impinge somewhere on another solid surface and/or will intersect other waves, either shock or expansion waves. Such wave intersections and interactions are important in the practical design and analysis of supersonic airplanes, missiles, wind tunnels, rocket engines, etc. A perfect historical example of this, as well as the consequences that can be caused by not paying suitable attention to wave interactions, is a ramjet flight-test program conducted in the early 1960s. During this period, a ramjet engine was mounted underneath the X-15 hypersonic airplane for a series of flight tests at high Mach numbers, in the range from 4 to 7. (The X-15, shown in Figure 9.16, was an experimental, rocket-powered airplane designed to probe the lower end of hypersonic manned flight.) During the first high-speed tests, the shock wave from the engine cowling impinged on the bottom surface of the X-15, and because of locally high aerodynamic heating in the impingement region, a hole was burned in the X-15 fuselage. Although this problem was later fixed, it is a graphic example of what shock-wave interactions can do to a practical configuration.

The purpose of this section is to present a mainly qualitative discussion of shock­wave interactions. For more details, see Chapter 4 of Reference 21.

First, consider an oblique shock wave generated by a concave corner, as shown in Figure 9.17. The deflection angle at the corner is в, thus generating an oblique shock at point A with a wave angle fi. Assume that a straight, horizontal wall is present above the comer, as also shown in Figure 9.17. The shock wave generated at point A, called the incident shock wave, impinges on the upper wall at point B. Question: Does the shock wave simply disappear at point B1 If not, what happens to it? To answer this question, we appeal to our knowledge of shock-wave properties. Examining Figure 9.17, we see that the flow in region 2 behind the incident shock is inclined upward at the deflection angle в. However, the flow must be tangent everywhere along the upper wall; if the flow in region 2 were to continue unchanged, it would mn into the wall and have no place to go. Hence, the flow in region 2 must eventually be bent downward through the angle в in order to maintain a flow tangent to the upper wall. Nature accomplishes this downward deflection via a second shock wave originating at the impingement point В in Figure 9.17. This second shock is called the reflected shock wave. The purpose of the reflected shock is to deflect the

flow in region 2 so that it is parallel to the upper wall in region 3, thus preserving the wall boundary condition.

The strength of the reflected shock wave is weaker than the incident shock. This is because М2 < Mi, and М2 represents the upstream Mach number for the reflected shock wave. Since the deflection angles are the same, whereas the reflected shock sees a lower upstream Mach number, we know from Section 9.2 that the reflected wave must be weaker. For this reason, the angle the reflected shock makes with the upper wall Ф is not equal to fi (i. e., the wave reflection is not specular). The properties of the reflected shock are uniquely defined by М2 and 0; since М2 is in turn uniquely defined by Mi and в, then the poperties in region 3 behind the reflected shock as well

as the angle Ф are easily determined from the given conditions of M and 9 by using the results of Section 9.2 as follows:

1. Calculate the properties in region 2 from the given M and 9. In particular, this

gives us M2.

2. Calculate the properties in region 3 from the value of M2 calculated above and

the known deflection angle 9.

An interesting situation can arise as follows. Assume that M is only slightly above the minimum Mach number necessary for a straight, attached shock wave at the given deflection angle 9. For this case, the oblique shock theory from Section 9.2 allows a solution for a straight, attached incident shock. However, we know that the Mach number decreases across a shock (i. e., M2 < M ). This decrease may be enough such that M2 is not above the minimum Mach number for the required deflection 9 through the reflected shock. In such a case, our oblique shock theory does not allow a solution for a straight reflected shock wave. The regular reflection as shown in Figure 9.17 is not possible. Nature handles this situation by creating the wave pattern shown in Figure 9.18. Here, the originally straight incident shock becomes curved as it nears the upper wall and becomes a normal shock wave at the upper wall. This allows the streamline at the wall to continue parallel to the wall behind the shock intersection. In addition, a curved reflected shock branches from the normal shock and propagates downstream. This wave pattern, shown in Figure 9.18, is called a Mach reflection. The calculation of the wave pattern and general properties for a Mach reflection requires numerical techniques such as those to be discussed in Chapter 13.

Another type of shock interaction is shown in Figure 9.19. Here, a shock wave is generated by the concave corner at point G and propagates upward. Denote this wave as shock A. Shock A is a left-running wave, so-called because if you stand on top of the wave and look downstream, you see the shock wave running in front of you

toward the left. Another shock wave is generated by the concave comer at point H, and propagates downward. Denote this wave as shock B. Shock В is a right-running wave, so-called because if you stand on top of the wave and look downstream, you see the shock mnning in front of you toward the right. The picture shown in Figure 9.19 is the intersection of right – and left-running shock waves. The intersection occurs at point E. At the intersection, wave A is refracted and continues as wave D. Similarly, wave В is refracted and continues as wave C. The flow behind the refracted shock D is denoted by region 4; the flow behind the refracted shock C is denoted by region 4′. These two regions are divided by a slip line EF. Across the slip line, the pressures are constant (i. e„ p = p4<), and the direction (but not necessarily the magnitude) of velocity is the same, namely, parallel to the slip line. All other properties in regions 4 and 4′ are different, most notably the entropy (s4 / ,v4′). The conditions which must hold across the slip line, along with the known Mi, 6, and 6L, uniquely determine

the shock-wave interaction shown in Figure 9.19. (See Chapter 4 of Reference 21 for details concerning the calculation of this interaction.)

Figure 9.20 illustrates the intersection of two left-running shocks generated at comers A and B. The intersection occurs at point C, at which the two shocks merge and propagate as the stronger shock CD, usually along with a weak reflected wave CE. This reflected wave is necessary to adjust the flow so that the velocities in regions 4 and 5 are in the same direction. Again, a slip line CF trails downstream of the intersection point.

The above cases are by no means all the possible wave interactions in a supersonic flow. However, they represent some of the more common situations encountered frequently in practice.

Consider an oblique shock wave generated by a compression corner with a 10° deflection angle. The Mach number of the flow ahead of the corner is 3.6; the flow pressure and temperature are standard sea level conditions. The oblique shock wave subsequently impinges on a straight wall opposite the compression comer. The geometry for this flow is given in Figure 9.17. Calculate the angle of the reflected shock wave Ф relative to the straight wall. Also, obtain the pressure, temperature, and Mach number behind the reflected wave.

Solution

From the 6-fi-M diagram, Figure 9.7, for M = 3.6 and в = 10°, i = 24°. Hence,

Mn ] = M] sin /1] = 3.6 sin 24° = 1.464

From Appendix B,

Also,

These are the conditions behind the incident shock wave. They constitute the upstream flow properties for the reflected shock wave. We know that the flow must be deflected again by в = 10° in passing through the reflected shock. Thus, from the 6-fl-M diagram, for M2 = 2.96 and в = 10°, we have the wave angle for the reflected shock, fi2 = 27.3°. Note that fi2 is not the angle the reflected shock makes with respect to the upper wall; rather, by definition of the wave angle, f}2 is the angle between the reflected shock and the direction of the flow in region 2. The shock angle relative to the wall is, from the geometry shown in Figure 9.17,

Ф = p2 – в = 27.3 – 10

Also, the normal component of the upstream Mach number relative to the reflected shock is M2 sin /Т = (2.96) sin 27.3° = 1.358. From Appendix B,

— = 1.991 — = 1.229 M„,3 = 0.7572

Pi T2

M„ з 0.7572 ГТ7Т

kf3 =——– —– =———————- = 2.55

sin(f}2 — 0) sin(27.3 — 10) _____

For standard sea level conditions, p =2116 lb/ft3 and T, = 519°R. Thus,

Ръ = — — Px= (1 -991)(2.32)(2116) =

Pi Px

h =T^Ti = (1.229)(1.294)(519) ;

12 l

Note that the reflected shock is weaker than the incident shock, as indicated by the smaller pressure ratio for the reflected shock, p2/p2 = 1.991 as compared to p2/p = 2.32 for the incident shock.

Inserting the flow variables obtained above into the governing equations, Equations

(13.59) to (13.62), using rearward differences for the spatial derivatives, predicted

values of the time derivatives at t + At are obtained, for example, [(3p/30,j](t+A,). In turn, these are averaged with the time derivatives from the predictor step to obtain; for example,

Finally, the average time derivative obtained from Equation (13.64) is inserted into Equation (13.63) to yield the corrected value of density at time t + At. The same procedure is used for all the dependent variables, u, v, etc.

Starting from the assumed initial conditions at t — 0, the repeated application of Equation (13.63) along with the above predictor-corrector algorithm at each time step allows the calculation of the flow-field variables and shock shape and location as a function of time. As stated above, after a large number of time steps, the calculated flow-field variables approach a steady state, where [(3p/3r)(j]ave —>■ 0 in Equation

(13.63) . Once again, we emphasize that we are interested in the steady-state answer, and the time-dependent technique is simply a means to that end.

Note that the applications of MacCormack’s technique to both the steady flow calculations described in Section 13.4 and the time-dependent calculations described in the present section are analogous; in the former, we march forward in the spatial coordinate x, starting with known values along with a constant у line, whereas, in the latter, we march forward in time starting with a known flow field at t = 0.

Why do we bother with a time-dependent solution? Is it not an added complica­tion to deal with an extra independent variable t in addition to the spatial variables x and _y? The answers to these questions are as follows. The governing unsteady flow equations given by Equations (13.59) to (13.62) are hyperbolic with respect to time, independent of whether the flow is locally subsonic or supersonic. In Figure 13.9a, some of the grid points are in the subsonic region and others are in the supersonic re­gion. However, the time-dependent solution progresses in the same manner at all these points, independent of the local Mach number. Hence, the time-dependent technique is the only approach known today which allows the uniform calculation of a mixed subsonic-supersonic flow field of arbitrary extent. For this reason, the application of the time-dependent technique, although it adds one additional independent variable, allows the straightforward solution of a flow field which is extremely difficult to solve by a purely steady-state approach.

A much more detailed description of the time-dependent technique is given in Chapter 12 of Reference 21, which you should study before attempting to apply this technique to a specific problem. The intent of our description here has been to give you simply a “feeling” for the philosophy and general approach of the technique.

Some typical results for supersonic blunt-body flow fields are given in Figures 13.12 to 13.15. These results were obtained with a time-dependent solution described in Reference 35. Figures 13.12 and 13.13 illustrate the behavior of a time-dependent solution during its approach to the steady state. In Figure 13.12, the time-dependent motion of the shock wave is shown for a parabolic cylinder in a Mach 4 freestream. The shock labeled 0 At is the initially assumed shock wave at t = 0. At early times, the shock wave rapidly moves away from the body; however, after about

300 time steps, it has slowed considerably, and between 300 and 500 time steps, the shock wave is virtually motionless—it has reached its steady-state shape and location. The time variation of the stagnation point pressure is given in Figure 13.13. Note that the pressure shows strong timewise oscillations at early times, but then it asymptotically approaches a steady value at large times. Again, it is this asymptotic steady state that we want, and the intermediate transient results are just a means to that

end. Concentrating on just the steady-state results, Figure 13.14 gives the pressure distribution (nondimensionalized by stagnation point pressure) over the body surface for the cases of both Мж = 4 and 8. The time-dependent numerical results are shown as the solid curves, whereas the open symbols are from newtonian theory, to be discussed in Chapter 14. Note that the pressure is a maximum at the stagnation point and decreases as a function of distance away from the stagnation point—a variation that we most certainly would expect based on our previous aerodynamic experience. The steady shock shapes and sonic lines are shown in Figure 13.15 for

Problems

Note: The purpose of the following problem is to provide an exercise in carrying out a unit process for the method of characteristics. A more extensive application to a complete flow field is left to your specific desires. Also, an extensive practical problem utilizing the finite-difference method requires a large number of arithmetic operations and is practical only on a digital computer. You are encouraged to set up such a problem at your leisure. The main purpose of the present chapter is to present the essence of several numerical methods, not to burden the reader with a lot of calculations or the requirement to write an extensive computer program.

1. Consider two points in a supersonic flow. These points are located in a cartesian coordinate system at {x, yi) = (0,0.0684) and (xi, у2) = (0.0121,0), where the units are meters. At point {x, yi): и і = 639 m/s, г 1 = 232.6 m/s, p = 1 atm, T = 288 K. At point {x3, Уі)’- R2 = 680 m/s, i>2 = 0, P2 = 1 atm, T2 = 288 K. Consider point 3 downstream of points 1 and 2 located by the intersection of the C+ characteristic through point 2 and the C_ characteristic through point 1. At point 3, calculate: m3, i>3, p3, and T3. Also, calculate the location of point 3, assuming the characteristics between these points are straight lines.

Consider the viscous flow over a flat plate as sketched in Figure 17.3. The viscous effects are contained within a thin layer adjacent to the surface; the thickness is exaggerated in Figure 17.3 for clarity. Immediately at the surface, the flow velocity is zero; this is the “no-slip” condition discussed in Section 15.2. In addition, the temperature of the fluid immediately at the surface is equal to the temperature of the surface; this is called the wall temperature Tw, as shown in Figure 17.3. Above the surface, the flow velocity increases in the у direction until, for all practical purposes, it equals the freestream velocity. This will occur at a height above the wall equal to S, as shown in Figure 17.3. More precisely, S is defined as that distance above the wall

Figure I 7.3 Boundary-layer properties.

where и = 0.99ue; here, ue is the velocity at the outer edge of the boundary layer. In Figure 17.3, which illustrates the flow over a flat plate, the velocity at the edge of the boundary layer will be V, x; that is, ue = Уж. For a body of general shape, ue is the velocity obtained from an inviscid flow solution evaluated at the body surface (or at the “effective body” surface, as discussed later). The quantity S is called the velocity boundary-layer thickness. At any given x station, the variation of и between у = 0 and у = S, that is, и = и (у), is defined as the velocity profile within the boundary layer, as sketched in Figure 17.3. This profile is different for different x stations. Similarly, the flow temperature will change above the wall, ranging from T = Tw at у = 0 to T — 0.997), at у = 8T. Here, St is defined as the thermal boundary-layer thickness. At any given x station, the variation of T between у = 0 and у = ST, that is, Г = T(y), is called the temperature profile within the boundary layer, as sketched in Figure 17.3. (In the above, Te is the temperature at the edge of the thermal boundary layer. For the flow over a flat plate, as sketched in Figure 17.3, Te = Too. For a general body, Te is obtained from an inviscid flow solution evaluated at the body surface, or at the “effective body” surface, to be discussed later.) Hence, two boundary layers can be defined: a velocity boundary layer with thickness S and a temperature boundary layer with thickness St. In general, Sj ф S. The relative thicknesses depend on the Prandtl number: it can be shown that if Pr = 1, then S = 5^;ifPr > l, then<5r < <5;ifPr < l, then<5r > S. For air at standard conditions, Pr = 0.71; hence, the thermal boundary layer is thicker than the velocity boundary layer, as shown in Figure 17.3. Note that both boundary-layer thicknesses increase with distance from the leading edge; that is, 8 = 8(x) and 8T = ST(x).

The consequence of the velocity gradient at the wall is the generation of shear stress at the wall,

where (du/dy)w is the velocity gradient evaluated at у = 0 (i. e., at the wall). Simi­larly, the temperature gradient at the wall generates heat transfer at the wall,

[17.6]

Equation (17.6) is identical to the definition of S* given in Equation (17.3). Hence, clearly <5* is a height proportional to the missing mass flow. If this missing mass flow was crammed into a streamtube where the flow properties were constant at pe and ue, then Equation (17.5) says that <5* is the height of this hypothetical streamtube.

2. The second physical interpretation of 8* is more practical than the one discussed above. Consider the flow over a flat surface as sketched in Figure 17.5. At the left is a picture of the hypothetical inviscid flow over the surface; a streamline through point yi is straight and parallel to the surface. The actual viscous flow is shown at the right of Figure 17.5; here, the retarded flow inside the boundary layer acts as a partial obstruction to the freestream flow. As a result, the streamline external to the boundary layer passing through point yi is deflected upward through a distance 8*. We now prove that this 5* is precisely the displacement thickness defined by Equation (17.3). At station 1 in Figure 17.5, the mass flow (per unit depth perpendicular to the page) between the surface and the external streamline is

[17.7]

At station 2, the mass flow between the surface and the external streamline is

m= I pudy + peue8* [17.8]

Jo

Since the surface and the external streamline form the boundaries of a streamtube, the mass flows across stations 1 and 2 are equal. Hence, equating Equations (17.7) and (17.8), we have

Figure 1 7.5 Displacement thickness is the distance by which an external flow streamline is displaced by the presence of the boundary layer.

Л’1 AVI

I peue dy — I pu dy + peue <5*

Jo Jo

or s*= Г’ (l-—)dy [17.9]

Jo PeM-e )

Hence, the height by which the streamline in Figure 17.5 is displaced upward by the presence of the boundary layer, namely, &*, is given by Equation (17.9). However, Equation (17.9) is precisely the definition of the displacement thickness given by Equation (17.3). Thus, the displacement thickness, first defined by Equation (17.3), is physically the distance through which the external inviscid flow is displaced by the presence of the boundary layer.

This second interpretation of 8* gives rise to the concept of an effective body. Consider the aerodynamic shape sketched in Figure 17.6. The actual contour of the body is given by curve ab. However, due to the displacement effect of the boundary layer, the shape of the body effectively seen by the freestream is not given by curve ab; rather, the freestream sees an effective body given by curve ac. In order to obtain the conditions ue, Te, etc., at the outer edge of the boundary layer on the actual body ab, an inviscid flow solution should be carried out for the effective body, and pe, u,_, Te, etc., are obtained from this inviscid solution evaluated along curve ac.

Note that in order to solve for S* from Equation (17.3), we need the profiles of и and p from a solution of the boundary-layer flow. In turn, to solve the boundary-layer flow, we need pe, ue, 7,. etc. However, pe, ue, Tc, etc., depend on <5*. This leads to an iterative solution. To calculate accurately the boundary-layer properties as well as the pressure distribution over the surface of the body in Figure 17.6, we proceed as follows:

1. Carry out an inviscid solution for the given body shape ab. Evaluate p, . ue. Te, etc., along curve ab.

2. Using these values of pe, ue, Te, etc., solve the boundary-layer equations (dis­cussed in Sections 17.3 to 17.6) for и — и {у), p = p( у), etc., at various stations along the body.

3. Obtain 8* at these stations from Equation (17.3). This will not be an accurate 5* because pe, ue, Te, etc., were evaluated on curve ab, not the proper effective body. Using this intermediate 8*, calculate an effective body, given by a curve ad (not shown in Figure 17.6).

4. Carry out an inviscid solution for the flow over the intermediate effective body ad, and evaluate new values of pe, ue, Te, etc., along ad.

5. Repeat steps 2 to 4 above until the solution at one iteration essentially does not deviate from the solution at the previous iteration. At this stage, a converged solution will be obtained, and the final results will pertain to the flow over the proper effective body ac shown in Figure 17.6.

In some cases, the boundary layers are so thin that the effective body can be ignored, and a boundary-layer solution can proceed directly from pe, ue, Te, etc., obtained from an inviscid solution evaluated on the actual body surface (ab in Fig­ure 17.6). However, for highly accurate solutions, and for cases where the boundary – layer thickness is relatively large (such as for hypersonic flow as discussed in Chap­ter 14), the iterative procedure described above should be carried out. Also, we note parenthetically that 5* is usually smaller than 5; typically, 8* «a 0.3 8.

Another boundary-layer property of importance is the momentum thickness в, defined as

[17.10]

To understand the physical interpretation of в, return again to Figure 17.4. Consider the mass flow across a segment dy, given by dm — pu dy. Then

A = momentum flow across dy = dm и = pu2 dy

If this same elemental mass flow were associated with the freestream, where the velocity is ue, then

momentum flow at freestream velocity associated with mass dm = dm ue = (pu dy)ue

Hence, decrement in momentum flow

(missing momentum flow) associated = pu(ue — u) dy with mass dm

The total decrement in momentum flow across the vertical line from у = 0 to у = y in Figure 17.4 is the integral of Equation (17.11),

[i7.i a]

Assume that the missing momentum flow is the product of peu2e and a height в. Then, Missing momentum flow = peu2e6 [ 17.13]

Equating Equations (17.12) and (17.13), we obtain

[17.14]

Equation (17.14) is precisely the definition of the momentum thickness given by Equation (17.10). Therefore, в is an index that is proportional to the decrement in momentum flow due to the presence of the boundary layer. It is the height of a hypothetical streamtube which is carrying the missing momentum flow at freestream conditions.

Note that в =9(x). In more detailed discussions of boundary-layer theory, it can be shown that в evaluated at a given station x = X is proportional to the integrated friction drag coefficient from the leading edge to x;; that is,

0(xi) a — f x Jo

where Cf is the local skin friction coefficient defined in Section 1.5 and C/ is the total skin friction drag coefficient for the length of surface from x = 0 to x = x. Hence, the concept of momentum thickness is useful in the prediction of drag coefficient.

All the boundary-layer properties discussed above are general concepts; they apply to compressible as well as incompressible flows, and to turbulent as well as laminar flows. The differences between turbulent and laminar flows were introduced in Section 15.2. Here, we extend that discussion by noting that the increased momen­tum and energy exchange that occur within a turbulent flow cause a turbulent boundary layer to be thicker than a laminar boundary layer. That is, for the same edge condi­tions, /)(., Uc. Te, etC., We have ^turbulent ^ ^laminar ttnd ($7) turbulent ■> (^7′)laminar – A hСП a boundary layer changes from laminar to turbulent flow, as sketched in Figure 15.8, the boundary-layer thickness markedly increases. Similarly, 8* and в are larger for turbulent flows.

Consider the two-dimensional, irrotational, isentropic flow over the body shown in Figure 11.2. The body is immersed in a uniform flow with velocity Voo oriented in the x direction. At an arbitrary point P in the flow field, the velocity is V with the x and у components given by и and v, respectively. Let us now visualize the velocity V as the sum of the uniform flow velocity plus some extra increments in velocity. For example, the x component of velocity и in Figure 11.2 can be visualized as Voo plus an increment in velocity (positive or negative). Similarly, the у component of velocity v can be visualized as a simple increment itself, because the uniform flow has a zero component in the у direction. These increments are called perturbations, and

и — Voo + й v = v

where u and v are called the perturbation velocities. These perturbation velocities are not necessarily small; indeed, they can be quite large in the stagnation region in front of the blunt nose of the body shown in Figure 11.2. In the same vein, because V = Уф, we can define a perturbation velocity potential ф such that

ф = Voo* + ф

дф

dx

дф

dy

Hence,

З 2ф д2ф

дх ду дх ду

Substituting the above definitions into Equation (11.12), and multiplying by a2, we obtain

[1 1.14]

Equation (11.14) is called the perturbation velocity potential equation. It is precisely the same equation as Equation (11.12) except that it is expressed in terms of ф instead of ф. It is still a nonlinear equation.

To obtain better physical insight in some of our subsequent discussion, let us recast Equation (11.14) in terms of the perturbation velocities. From the definition of ф given earlier, Equation (11.14) can be written as

r9 /Т7 Л, 7. 9 „лЗе лл3л

[a ~ (Voo + и)2]— + (a2 – v2)—– 2(Тоо + u)v— = 0

dx dy dy

From the energy equation in the form of Equation (8.32), we have

Кэс _ I (Too + n)2 + P2

2 “ у – 1 +

Substituting Equation (11.15) into (11.14a), and algebraically rearranging, we obtain

й у + 1 й2 у — їй2

tr + ,)^ + — vi + ~K

‘ й у + 1 v2

2 V^ +

Equation (11.16) is still exact for irrotational, isentropic flow. Note that the left-hand side of Equation (11.16) is linear but the right-hand side is nonlinear. Also, keep in mind that the size of the perturbations й and v can be large or small; Equation (11.16) holds for both cases.

Let us now limit our considerations to small perturbations; that is, assume that the body in Figure 11.2 is a slender body at small angle of attack. In such a case, й and v will be small in comparison with V-^. Therefore, we have

V2 ’ V2

Keep in mind that products of й and v with their derivatives are also very small. With this in mind, examine Equation (11.16). Compare like terms (coefficients of like derivatives) on the left – and right-hand sides of Equation (11.16). We find

1. For 0 < Moo £ 0.8 or Moo > 1.2, the magnitude of

or in terms of the perturbation velocity potential,

Examine Equation (11.18). It is a linear partial differential equation, and therefore is inherently simpler to solve than its parent equation, Equation (11.16). However, we have paid a price for this simplicity. Equation (11.18) is no longer exact. It is only an approximation to the physics of the flow. Due to the assumptions made in obtaining Equation (11.18), it is reasonably valid (but not exact) for the following combined situations:

1. Small perturbation, that is, thin bodies at small angles of attack

2. Subsonic and supersonic Mach numbers

In contrast, Equation (11.18) is not valid for thick bodies and for large angles of attack. Moreover, it cannot be used for transonic flow, where 0.8 < Мж < 1.2, or for hypersonic flow, where Мж > 5.

We are interested in solving Equation (11.18) in order to obtain the pressure distribution along the surface of a slender body. Since we are now dealing with approximate equations, it is consistent to obtain a linearized expression for the pres­sure coefficient—an expression which is approximate to the same degree as Equation (11.18), but which is extremely simple and convenient to use. The linearized pressure coefficient can be derived as follows.

First, recall the definition of the pressure coefficient Cp given in Section 1.5:

C„ = -—— [11.19]

QoC

where qoo = 5Pco^tx; = dynamic pressure. The dynamic pressure can be expressed in terms of M0c as follows:

Substituting Equation (11.21) into (11.19), we have

[11.22]

Equation (11.22) is simply an alternate form of the pressure coefficient expressed in terms of Moo. It is still an exact representation of the definition of C

To obtain a linearized form of the pressure coefficient, recall that we are dealing with an adiabatic flow of a calorically perfect gas; hence, from Equation (8.39),

V2 V2

T+- = roo + -^ [11.23]

2,Cp 2cp

Recalling from Equation (7.9) that cp = yR/(y — 1), Equation (11.23) can be written as

T – Tso = —— 25———–

°° 2yR/(y – 1)

Also, recalling that = *Jy Equation (11.24) becomes T x у – 1 – У 2 ^ у _ ! y2 _ V2

Too 2 yRToo 2 a^

In terms of the perturbation velocities

V2 = (Voo + m)2 + v2

Equation (11.25) can be written as

Since the flow is isentropic, p/poo = (T/TooyRy 1 ’, and Equation (11.26) becomes

Equation (11.27) is still an exact expression. However, let us now make the as­sumption that the perturbations are small, that is, й/V.^ 1, u2/ <SY 1, and

v2/ <gc 1. In this case, Equation (11.27) is of the form

— = (1 – e)^"1) [11.28]

Poo

where e is small. From the binomial expansion, neglecting higher-order terms, Equa­tion (11.28) becomes

p v

— = 1——– -—£ + ••• [11.29]

Poo Y – 1

Comparing Equation (11.27) to (11.29), we can express Equation (11.27) as

7- = 1 –

Poo 2

Substituting Equation (11.30) into the expression for the pressure coefficient, Equa­tion (11.22), we obtain

Equation (11.32) is the linearized form for the pressure coefficient; it is valid only for small perturbations. Equation (11.32) is consistent with the linearized perturbation velocity potential equation, Equation (11.18). Note the simplicity of Equation (11.32); it depends only on the x component of the velocity perturbation, namely, й.

To round out our discussion on the basics of the linearized equations, we note that any solution to Equation (11.18) must satisfy the usual boundary conditions at infinity and at the body surface. At infinity, clearly ф = constant; that is, й = v — 0. At the body, the flow-tangency condition holds. Let в be the angle between the tangent to the surface and the freestream. Then, at the surface, the boundary condition is obtained from Equation (3.48c):

v v

tan# = – = ———– – [11.33]

и Ех, + и

which is an exact expression for the flow-tangency condition at the body surface. A simpler, approximate expression for Equation (11.33), consistent with linearized theory, can be obtained by noting that for small perturbations, й « Hence, Equation (11.33) becomes v

Equation (11.34) is an approximate expression for the flow-tangency condition at the body surface, with accuracy of the same order as Equations (11.18) and (11.32).