Category When Is A Flow Compressible?

Some Special Cases; Couette and. Poiseuille Flows

The resistance arising from the want of lubricity in the parts of a fluid is, other things being equal, proportional to the velocity with which the parts of the fluid are separated from one another.

Isaac Newton, 1687, from Section IX of Book II of his Principia

16.1 Introduction

The general equations of viscous flow were derived and discussed in Chapter 15. In particular, the viscous flow momentum equations were treated in Section 15.4 and are given in partial differential equation form by Equations (15.19a to c)—the Navier-Stokes equations. These, along with the viscous flow energy equation, Equa­tion (15.26), derived in Section 15.5, are the theoretical tools for the study of viscous flows. However, examine these equations closely; as discussed in Section 15.7, they are a system of coupled, nonlinear partial differential equations—equations which contain more terms and which are inherently more elaborate than the inviscid flow equations treated in Parts 2 and 3 of this book. Three classes of solutions of these equations were itemized in Section 15.5. The first itemized class was that of “exact” solutions of the Navier-Stokes equations for a few specific physical problems which, by their physical and geometrical nature, allow many terms in the governing equations to be precisely zero, resulting in a system of equations simple enough to solve, either analytically or by simple numerical methods. Such exact problems are the subject of this chapter.

Reynolds analogy

The road map for this chapter is given in Figure 16.1. The types of flows consid­ered here are generally labeled as parallel flows because the streamlines are straight and parallel to each other. We will consider two of these flows, Couette and Poiseuille, which will be defined in due course. In addition to representing exact solutions of the Navier-Stokes equations, these flows illustrate some of the important practical facets of any viscous flow, as itemized on the right side of the road map. In a clear, uncom­plicated fashion, we will be able to calculate and study the surface skin friction and heat transfer. We will also use the results to define the recovery factor and Reynolds analogy—two practical engineering tools that are frequently used in the analysis of skin friction and heat transfer.

Examples

A numerical simulation of the flow over an airfoil using the Reynolds averaged Navier-Stokes equations can be conducted on today’s supercomputers in less than a half hour for less than $1000 cost in computer time. If just one such simulation had been attempted 20 years ago on computers of that time (e. g., the IBM 704 class) and with algorithms then known, the cost in computer time would have amounted to roughly $10 million, and the results for that single flow would not be available until 10 years from now, since the computation would have taken about 30 years to complete.

Dean R. Chapman, NASA, 1977

20.1 Introduction

This chapter is short. Its purpose is to discuss the third option for the solution of viscous flows as discussed in Section 15.7, namely, the exact numerical solution of the complete Navier-Stokes equations. This option is the purview of modern compu­tational fluid dynamics—it is a state-of-the-art research activity which is currently in a rapid state of development. This subject now occupies volumes of modern literature; for a basic treatment, see the definitive text on computational fluid dynamics listed as Reference 54. We will only list a few sample calculations here.

20.2 The Approach

Return to the complete Navier-Stokes equations, as derived in Chapter 15, and re­peated and renumbered below for convenience:

These equations have been written with the time derivatives on the left-hand side and all spatial derivatives on the right-hand side. This is the form suitable to a time – dependent solution of the equations, as discussed in Chapters 13 and 16. Indeed, Equations (20.1) to (20.5) are partial differential equations which have a mathemati­cally “elliptic” behavior; that is, on a physical basis they treat flow-field information and flow disturbances that can travel throughout the flow field, in both the upstream and downstream directions. The time-dependent technique is particularly suited to such a problem.

The time-dependent solution of Equations (20.1) to (20.5) can be carried out in direct parallel to the discussion in Section 16.4. It is important for you to return to that section and review our discussion of the time-dependent solution of compressible Couette flow using MacCormack’s technique. We suggest doing this before reading further. The approach to the solution of Equations (20.1) to (20.5) for other problems is exactly the same. Therefore, we will not elaborate further here.

Measurement of Velocity in a Compressible Flow

The use of a Pitot tube for measuring the velocity of a low-speed, incompressible flow was discussed in Section 3.4. Before progressing further, return to Section 3.4, and review the principal aspects of a Pitot tube, as well as the formulas used to obtain the flow velocity from the Pitot pressure, assuming incompressible flow.

For low-speed, incompressible flow, we saw in Section 3.4 that the velocity can be obtained from a knowledge of both the total pressure and the static pressure at

a point. The total pressure is measured by a Pitot tube, and the static pressure is obtained from a static pressure orifice or by some independent means. The important aspect of Section 3.4 is that the pressure sensed by a Pitot tube, along with the static pressure, is all that is necessary to extract the flow velocity for an incompressible flow. In the present section, we see that the same is true for a compressible flow, both subsonic and supersonic, if we consider the Mach number rather than the velocity. In both subsonic and supersonic compressible flows, a knowledge of the Pitot pressure and the static pressure is sufficient to calculate Mach number, although the formulas are different for each Mach-number regime. Let us examine this matter further.

13.2.2 Wall Points

In Figure 13.6, point 4 is an internal flow point near a wall. Assume that we know all the flow properties at point 4. The C_ characteristic through point 4 intersects the wall at point 5. At point 5, the slope of the wall 9$ is known. The flow properties at the wall point, point 5, can be obtained from the known properties at point 4 as follows. Along the C_ characteristic, К is constant. Hence, (К )4 = (K-)5. Moreover, the value of К_ is known from Equation (13.17) evaluated at point 4:

(tf-)4 = (K-)s = 6*4 + v4 [13.25]

Evaluating Equation (13.17) at point 5, we have

(K_)5 = 9S + v5 [13.26]

In Equation (13.26), (K-)5 and 65 are known; thus V5 follows directly. In turn, all other flow variables at point 5 can be obtained from V5 as explained earlier. The characteristic line between points 4 and 5 is assumed to be a straight-line segment with average slope given by |(64 + 05) — ^(/x4 + /x5).

From the above discussion of both internal and wall points, we see that properties at the grid points are calculated from known properties at other grid points. Hence, in order to start a calculation using the method of characteristics, we have to know the flow properties along some initial data line. Then we piece together the characteristics mesh and associated flow properties by “marching downstream” from the initial data line. This is illustrated in the next section.

We emphasize again that the method of characteristics is an exact solution of inviscid, nonlinear supersonic flow. However, in practice, there are numerical er­rors associated with the finite grid; the approximation of the characteristics mesh by straight-line segments between grid points is one such example. In principle, the method of characteristics is truly exact only in the limit of an infinite number of characteristic lines.

We have discussed the method of characteristics for two-dimensional, irrota – tional, steady flow. The method of characteristics can also be used for rotational and three-dimensional flows, as well as unsteady flows. See Reference 21 for more details.

Figure 1 3.6 Wall point.

Results for Compressible Couette Flow

Some typical results for compressible Couette flow are shown in Figure 16.9 for a cold wall case, and in Figure 16.10 for an adiabatic lower wall case. These results are

u_ A = (y – l )PrM;

ue

(a) (b)

A = (y- )PrM}

obtained from White (Reference 43); they assume a viscosity-temperature relation of ц/цтах = ("/’/ ‘/’ref)2/3> which is not quite as accurate for a gas as is Sutherland’s law [Equation (15.3)]. Recall from Section 15.6 that a compressible viscous flow is governed by the following similarity parameters: the Mach number, the Prandtl number, and the ratio of specific heats, y. Therefore, we expect the results for compressible Couette flow to be governed by the same parameters. Such is the case, as illustrated in Figures 16.9 and 16.10. Here we see the different flow-field profiles
for different values of the combined parameter A = (у – l)Pr М]. In particular, examining Figure 16.9 for the equal temperature, cold wall case, we note that;

1. From Figure 16.9a, the velocity profiles are not greatly affected by compress­ibility. The profile labeled A = 0 is the familiar linear incompressible case, and that labeled A = 30 corresponds to Me approximately 10. Clearly, the velocity profile (in terms of u/ue versus y/D) does not change greatly over such a large range of Mach number.

2. In contrast, from Figure 16.9b, there are huge temperature changes in the flow; these are due exclusively to viscous dissipation, which is a major effect at high Mach numbers. For example, for A = 30 (Me ~ 10), the temperature in the middle of the flow is almost five times the wall temperature. Contrast this with the very small temperature increase calculated in Example 16.1 for an incompressible flow. This is why, on the scale in Figure 16.9b, the incompressible case (A = 0) is seen as essentially a vertical line.

For the adiabatic wall case shown in Figure 16.10, we note the following:

1. From Figure 16.10a, the velocity profiles show a pronounced curvature due to compressibility.

2. From Figure 16.1 Ofo, the temperature increases are larger than for the cold wall case. Note that, for A = 30 <M,, ~ 10), the maximum temperature is over 15 times that of the upper wall. Also, note the results, familiar from our discussion in Section 16.3, that the temperature is the largest at the adiabatic wall; that is, Taw is the maximum temperature. As expected, Figure 16.1 Oh shows that Taw increases markedly as Me increases.

In summary, in a general comparison between the incompressible flow discussed in Section 16.3 and the compressible flow discussed here, there is no tremendous qual­itative change; that is, there is no discontinuous change in the flow-field behavior in going from subsonic to supersonic flow as is the case for an inviscid flow, such as discussed in Part 3. Qualitatively, a supersonic viscous flow is similar to a subsonic viscous flow. On the other hand, there are tremendous quantitative differences, es­pecially in regard to the large temperature changes that occur due to massive viscous dissipation in a high-speed compressible viscous flow. The physical reason for this difference in viscous versus inviscid flow is as follows. In an inviscid flow, informa­tion is propagated via the mechanism of pressure waves traveling throughout the flow. This mechanism changes radically when the flow goes from subsonic to supersonic. In contrast, for a viscous flow, information is propagated by the diffusive transport mechanisms of // and к (a molecular phenomenon), and these mechanisms are not basically changed when the flow goes from subsonic to supersonic. These statements hold in general for any viscous flow, not just for the Couette flow case treated here.

Nozzle Flows

In this section, we move to the left-hand branch of the road map given in Figure 10.3; that is, we study in detail the compressible flow through nozzles. To expedite this study, we first derive an important equation which relates Mach number to the ratio of duct area to sonic throat area.

Consider the duct shown in Figure 10.9. Assume that sonic flow exists at the throat, where the area is A*. The Mach number and the velocity at the throat are denoted by M* and u*, respectively. Since the flow is sonic at the throat, M* = 1 and u* = a*. (Note that the use of an asterisk to denote sonic conditions was introduced in Section 7.5; we continue this convention in our present discussion.) At any other section of this duct, the area, the Mach number, and the velocity are denoted by A, M, and m, respectively, as shown in Figure 10.9. Writing Equation (10.1) between A and A*, we have

p*u*A* = puA [10.26]

Since u* = a*, Equation (10.26) becomes

A p* a* p* po a* . _

A* p и po p и

where po is the stagnation density defined in Section 7.5 and is constant throughout

an isentropic flow. From Equation (8.46), we have

p*_ _ ( 2

Po VX + 1 /

Also, from Equation (8.43), we have

Also, recalling the definition of M* in Section 8.4, as well as Equation (8.48), we have

[10.30]

Squaring Equation (10.27) and substituting Equations (10.28) to (10.30), we obtain

AV = (_2_V(y~’) (l + ^_AM22,{У~П 1 + [(X – l)/2M2 A*) Vx + 1/ V 2 ) [(y+)/2]M2

[10.31]

Algebraically simplifying Equation (10.31), we have

Equation (10.32) is very important; it is called the area-Mach number relation, and it contains a striking result. “Turned inside out,” Equation (10.32) tells us that M = f (A / A*); that is, the Mach number at any location in the duct is a function of the ratio of the local duct area to the sonic throat area. Recall from our discussion of Equation (10.25) that A must be greater than or at least equal to A*; the case where A < A* is

physically not possible in an isentropic flow. Thus, in Equation (10.32), A/A* > 1. Also, Equation (10.32) yields two solutions for M at a given А/A*—a subsonic value and a supersonic value. Which value of M that actually holds in a given case depends on the pressures at the inlet and exit of the duct, as explained later. The results for А/A* as a function of M, obtained from Equation (10.32), are tabulated in Appendix A. Examining Appendix A, we note that for subsonic values of M, as M increases, AI A* decreases (i. e., the duct converges). At M = 1, A / A* = 1 in Appendix A. Finally, for supersonic values of M, as M increases, А/A* increases (i. e., the duct diverges). These trends in Appendix A are consistent with our physical discussion of convergent-divergent ducts at the end of Section 10.2. Moreover, Appendix A shows the double-valued nature of M as a function of A/A*. For example, for А/A* = 2, we have either M = 0.31 or M = 2.2.

Consider a given convergent-divergent nozzle, as sketched in Figure 10.10a. Assume that the area ratio at the inlet A,-/A* is very large and that the flow at the inlet is fed from a large gas reservoir where the gas is essentially stationary. The reservoir pressure and temperature are po and 7b, respectively. Since A,-/A* is very large, the subsonic Mach number at the inlet is very small, M ~ 0. Thus, the pressure and temperature at the inlet are essentially po and 7o, respectively. The area distribution of the nozzle A = A(x) is specified, so that A/ A* is known at every station along the nozzle. The area of the throat is denoted by A,, and the exit area is denoted by Ae. The Mach number and static pressure at the exit are denoted by Me and pe, respectively. Assume that we have an isentropic expansion of the gas through this nozzle to a supersonic Mach number Me = Me^ at the exit (the reason for the subscript 6 will be apparent later). The corresponding exit pressure is pe>6. For this expansion, the flow is sonic at the throat; hence, M = 1 and A, = A* at the throat. The flow properties through the nozzle are a function of the local area ratio A/A* and are obtained as follows:

1. The local Mach number as a function of x is obtained from Equation (10.32), or more directly from the tabulated values in Appendix A. For the specified A — A(x), we know the corresponding A/A* = fix). Then read the related subsonic Mach numbers in the convergent portion of the nozzle from the first part of Appendix A (for M < 1) and the related supersonic Mach numbers in the divergent portion of the nozzle from the second part of Appendix A (for M > 1). The Mach number distribution through the complete nozzle is thus obtained and is sketched in Figure 10.10Й.

2. Once the Mach number distribution is known, then the corresponding variation of temperature, pressure, and density can be found from Equations (8.40), (8.42), and (8.43), respectively, or more directly from Appendix A. The distributions of p/po and T/Tq are sketched in Figure 10.10c and d, respectively.

Examine the variations shown in Figure 10.10. For the isentropic expansion of a gas through a convergent-divergent nozzle, the Mach number monotonically increases from near 0 at the inlet to M = 1 at the throat, and to the supersonic value Mefi at the exit. The pressure monotonically decreases from p0 at the inlet to 0.528/?o at the throat and to the lower value рем at the exit. Similarly, the temperature monotonically

decreases from T0 at the inlet to 0.8337b at the throat and to the lower value 7′(,.f) at the exit. Again, for the isentropic flow shown in Figure 10.10, we emphasize that the distribution of M, and hence the resulting distributions of p and /’, through the nozzle depends only on the local area ratio A/A*. This is the key to the analysis of isentropic, supersonic, quasi-one-dimensional nozzle flows.

Imagine that you take a convergent-divergent nozzle, and simply place it on a table in front of you. What is going to happen? Is the air going to suddenly start flowing through the nozzle of its own accord? The answer is, of course not! Rather, by this stage in your study of aerodynamics, your intuition should tell you that we have to impose a force on the gas in order to produce any acceleration. Indeed, this is the essence of the momentum equation derived in Section 2.5. For the inviscid flows

considered here, the only mechanism to produce an accelerating force on a gas is a pressure gradient. Thus, returning to the nozzle on the table, a pressure difference must be created between the inlet and exit; only then will the gas start to flow through the nozzle. The exit pressure must be less than the inlet pressure; that is, pe < p0. Moreover, if we wish to produce the isentropic supersonic flow sketched in Figure

10.10, the pressure pjpo must be precisely the value stipulated by Appendix A for the known exit Mach number Me g; that is, pe/po = Ре, бІРо■ If the pressure ratio is different from the above isentropic value, the flow either inside or outside the nozzle will be different from that shown in Figure 10.10.

Let us examine the type of nozzle flows that occur when pe/po is not equal to the precise isentropic value for Me b, that is, when ре/Po ф Ре, б/Ро – To begin with, consider the convergent-divergent nozzle sketched in Figure 10.1 la. If pe = po, no pressure difference exists, and no flow occurs inside the nozzle. Now assume that pe is minutely reduced below pa, say, pe = 0.999p0. This small pressure difference will produce a very low-speed subsonic flow inside the nozzle—essentially a gentle wind. The local Mach number will increase slightly through the convergent portion, reaching a maximum value at the throat, as shown by curve 1 in Figure 10.1 li>. This Mach number at the throat will not be sonic; rather, it will be some small subsonic value. Downstream of the throat, the local Mach number will decrease in the divergent section, reaching a very small but finite value M,,, і at the exit. Correspondingly, the pressure in the convergent section will gradually decrease from p0 at the inlet to a minimum value at the throat, and then will gradually increase to the value pe, i at the exit. This variation is shown as curve 1 in Figure 10.1 lc. Please note that because the flow is not sonic at the throat in this case, At is not equal to A*. Recall that A*, which appears in Equation (10.32), is the sonic throat area. In the case of purely subsonic flow through a convergent-divergent nozzle, A* takes on the character of a reference area; it is not the same as the actual geometric area of the nozzle throat At. Rather, A* is the area the flow in Figure 10.11 would have if it were somehow accelerated to sonic velocity. If this did happen, the flow area would have to be decreased further than shown in Figure 10.1 la. Hence, for a purely subsonic flow A, > A*.

Assume that we further decrease the exit pressure in Figure 10.11, say, to the value pe = pe,2- The flow is now illustrated by the curves labeled 2 in Figure 10.11. The flow moves faster through the nozzle, and the maximum Mach number at the throat increases but remains less than 1. Now, let us reduce pe to the value pe = pe, з, such that the flow just reaches sonic conditions at the throat. This is shown by curve 3 in Figure 10.11. The throat Mach number is 1, and the throat pressure is 0.528p^. The flow downstream of the throat is subsonic.

Upon comparing Figures 10.10 and 10.11, we are struck by an important physical difference. For a given nozzle shape, there is only one allowable isentropic flow solution for the supersonic case shown in Figure 10.10. In contrast, there are an infinite number of possible isentropic subsonic solutions, each one corresponding to some value of pe, where po > pe > Ре. з – Only three solutions of this infinite set of solutions are sketched in Figure 10.11. Hence, the key factors for the analysis of purely subsonic flow in a convergent-divergent nozzle are both А/A* and pe/Po-

Consider the mass flow through the convergent-divergent nozzle in Figure 10.11. As the exit pressure is decreased, the flow velocity in the throat increases; hence, the

Figure 10.11 Isentropic subsonic nozzle flow.

mass flow increases. The mass flow can be calculated by evaluating Equation (10.1) at the throat; that is, m — p, utAt. As pe decreases, u, increases and p, decreases. However, the percentage increase in ut is much greater than the decrease in p,. As a result, m increases, as sketched in Figure 10.12. When pe — pe^, sonic flow is achieved at the throat, and m = p*u*A* = p*u*A,. Now, if pe is further reduced below pe>3, the conditions at the throat take on a new behavior; they remain unchanged. From our discussion in Section 10.2, the Mach number at the throat cannot exceed 1; hence, as pe is further reduced, M will remain equal to 1 at the throat. Consequently, the mass flow will remain constant as pe is reduced below pe 3, as shown in Figure 10.12. In a sense, the flow at the throat, as well as upstream of the throat, becomes “frozen.” Once the flow becomes sonic at the throat, disturbances cannot work their way upstream of the throat. Hence, the flow in the convergent section of the nozzle no longer communicates with the exit pressure and has no way of knowing that the

Figure 10.12 Variation of mass flow with exit

pressure; illustration of choked flow.

exit pressure is continuing to decrease. This situation—when the flow goes sonic at the throat, and the mass flow remains constant no matter how low pe is reduced—is called choked flow. It is a vital aspect of the compressible flow through ducts, and we consider it further in our subsequent discussions.

Return to the subsonic nozzle flows sketched in Figure 10.11. Question: What happens in the duct when pe is reduced below p,..fl In the convergent portion, as described above, nothing happens. The flow properties remain fixed at the conditions shown by curve 3 in the convergent section of the duct (the left side of Figure 10.11 b and c). However, a lot happens in the divergent section of the duct. As the exit pressure is reduced below pe 3, a region of supersonic flow appears downstream of the throat. However, the exit pressure is too high to allow an isentropic supersonic flow throughout the entire divergent section. Instead, for pe less than pe,3 but substantially higher than the fully isentropic value pe$ (see Figure 10.10c), a normal shock wave is formed downstream of the throat. This situation is sketched in Figure 10.13.

In Figure 10.13, the exit pressure has been reduced to pe^, where рсл < pe,3, but where pe 4 is also substantially higher than pe 6. Here we observe a normal shock wave standing inside the nozzle at a distance d downstream of the throat. Between the throat and the normal shock wave, the flow is given by the supersonic isentropic solution, as shown in Figure 10.13Z? and c. Behind the shock wave, the flow is subsonic. This subsonic flow sees the divergent duct and isentropically slows down further as it moves to the exit. Correspondingly, the pressure experiences a discontinuous increase across the shock wave and then is further increased as the flow slows down toward the exit. The flow on both the left and right sides of the shock wave is isentropic; however, the entropy increases across the shock wave. Hence, the flow on the left side of the shock wave is isentropic with one value of entropy, V|, and the flow on the right side of the shock wave is isentropic with another value of entropy S2, where 52 > ■Si – The location of the shock wave inside the nozzle, given by d in Figure 10.13a, is determined by the requirement that the increase in static pressure across the wave plus that in the divergent portion of the subsonic flow behind the shock be just right to achieve pe 4 at the exit. As pe is further reduced, the normal

Normal shock wave

Figure 10.13 Su personic nozzle flow with a normal shock inside the nozzle.

shock wave moves downstream, closer to the nozzle exit. At a certain value of exit pressure, pe — pe 5, the normal shock stands precisely at the exit. This is sketched in Figure 10.14a to c. At this stage, when pe — ре $, the flow through the entire nozzle, except precisely at the exit, is isentropic.

To this stage in our discussion, we have dealt with pe, which is the pressure right at the nozzle exit. In Figures 10.10, 10.11, 10.13, and 10.14a to c, we have not been concerned with the flow downstream of the nozzle exit. Now imagine that the nozzle in Figure 10.14a exhausts directly into a region of surrounding gas downstream of the exit. These surroundings could be, for example, the atmosphere. In any case, the pressure of the surroundings downstream of the exit is defined as the back pressure, denoted by pB. When the flow at the nozzle exit is subsonic, the exit pressure must equal the back pressure, pe = pB, because a pressure discontinuity cannot be maintained in a steady subsonic flow. That is, when the exit flow is subsonic, the surrounding back pressure is impressed on the exit flow. Hence, in Figure 10.II, Рв = Pe. і for curve 1, pB — Pe.2 for curve 2, and pB — pej for curve 3. For the same reason, pB = pe_4 in Figure 10.13, and pB = pe^ in Figure 10.14. Hence, in discussing these figures, instead of stating that we reduced the exit pressure pe and

observed the consequences, we could just as well have stated that we reduced the back pressure pB. It would have amounted to the same thing.

For the remainder of our discussion in this section, let us now imagine that we have control over pв and that we are going to continue to decrease pB. Consider the case when the back pressure is reduced below pe 5. Whence < pB < pe 5, the back pressure is still above the isentropic pressure at the nozzle exit. Hence, in flowing out to the surroundings, the jet of gas from the nozzle must somehow be compressed such that its pressure is compatible with pB. This compression takes place across oblique shock waves attached to the exit, as shown in Figure 10.14d. When pB is reduced to the value such that pB = pee, there is no mismatch of the exit pressure and the back pressure; the nozzle jet exhausts smoothly into the surroundings without passing through any waves. This is shown in Figure 10.14-е. Finally, as pB is reduced below pe (, the jet of gas from the nozzle must expand further in order to match the lower back pressure. This expansion takes place across centered expansion waves attached to the exit, as shown in Figure 10.14/.

When the situation in Figure 10. 14й? exists, the nozzle is said to be overexpanded, because the pressure at the exit has expanded below the back pressure, p,,^ < pB.

That is, the nozzle expansion has gone too far, and the jet must pass through oblique shocks in order to come back up to the higher back pressure. Conversely, when the situation in Figure 10.14/ exists, the nozzle is said to be underexpanded, because the exit pressure is higher than the back pressure, pe (, > pB, and hence the flow is capable of additional expansion after leaving the nozzle.

Surveying Figures 10.10 through 10.14, note that the purely isentropic supersonic flow originally illustrated in Figure 10.10 exists throughout the nozzle for all cases when pB < Pe, 5- For example, in Figure 10.14a, the isentropic supersonic flow solution holds throughout the nozzle except right at the exit, where a normal shock exists. In Figure 10.14d to /, the flow through the entire nozzle, including at the exit plane, is given by the isentropic supersonic flow solution.

Keep in mind that our entire discussion of nozzle flows in this section is predicated on having a duct of given shape. We assume that A = A(x) is prescribed. When this is the case, the quasi-one-dimensional theory of this chapter gives a reasonable prediction of the flow inside the duct, where the results are interpreted as mean properties averaged over each cross section. This theory does not tell us how to design the contour of the nozzle. In reality, if the walls of the nozzle are not curved just right, then oblique shocks occur inside the nozzle. To obtain the proper contour for a supersonic nozzle so that it produces isentropic shock-free flow inside the nozzle, we must account for the three-dimensionality of the actual flow. This is one purpose of the method of characteristics, a technique for analyzing two – and three-dimensional supersonic flow. A brief introduction to the method of characteristics is given in Chapter 13.

Consider the isentropic supersonic flow through a convergent-divergent nozzle with an exit- | Example 1 0.1 to-throat area ratio of 10.25. The reservoir pressure and temperature are 5 atm and 600°R, respectively. Calculate M, p, and T at the nozzle exit.

Solution

From the supersonic portion of Appendix A, for AJ A* = 10.25,

Also,

Thus,

Te = 0.24277o = 0.2427(600) = 145.6°R

Consider the isentropic flow through a convergent-divergent nozzle with an exit-to-throat area | Example 10.2 ratio of 2. The reservoir pressure and temperature are 1 atm and 288 K, respectively. Calculate the Mach number, pressure, and temperature at both the throat and the exit for the cases where

Te 1

Te = —To = ———— (288) =

To 1.968

(b) At the throat, the flow is still sonic. Hence, from above, M, = 1.0, p, = 0.528 atm, and T, = 240 K. However, at all other locations in the nozzle, the flow is subsonic. At the exit, where Ae/A* = 2, from the subsonic portion of Appendix A,

Te 1

Te = —T0 =————- (288) =

T0 1.018

From the subsonic portion of Appendix A, for p0/p,: = 1.028, we have

– = — — = 0.5(2.964) = 1.482 * AeA*

From the subsonic portion of Appendix A, for A,/A* = 1.482, we have

Hypersonic Shock-Wave Relations and Another Look at Newtonian Theory

The basic oblique shock relations are derived and discussed in Chapter 9. These are exact shock relations and hold for all Mach numbers greater than unity, supersonic or hypersonic (assuming a calorically perfect gas). However, some interesting approxi­mate and simplified forms of these shock relations are obtained in the limit of a high

Mach number. These limiting forms are called the hypersonic shock relations; they are obtained below.

Consider the flow through a straight oblique shock wave. (See, e. g., Figure 9.1.) Upstream and downstream conditions are denoted by subscripts 1 and 2, respectively. For a calorically perfect gas, the classical results for changes across the shock are given in Chapter 9. To begin with, the exact oblique shock relation for pressure ratio across the wave is given by Equation (9.16). Since Mn< = Mx sin/), this equation becomes

Exact: — = 1 + (M? sin2 /) – 1) [14.28]

P і У + 1

where /) is the wave angle. In the limit as M goes to infinity, the term M sin2 /) ;>> 1, and hence Equation (14.28) becomes

In a similar vein, the density and temperature ratios are given by Equations (9.15) and (9.17), respectively. These can be written as follows:

The relationship among Mach number M, shock angle and deflection angle в is expressed by the so-called в-fi-M relation given by Equation (9.23), repeated below:

This relation is plotted in Figure 9.7, which is a standard plot of the wave angle versus the deflection angle, with the Mach number as a parameter. Returning to Figure 9.7, we note that, in the hypersonic limit, where 9 is small, p is also small. Hence, in this limit, we can insert the usual small-angle approximation into Equation (9.23):

sin /J ~ p cos 2/3^1 tan 9 % sin 9 ~ 9

resulting in

2 Г Mjp2 – 1 ‘

P [Mf(y + l) + 2_

Applying the high Mach number limit to Equation (14.33), we have

2 Г M2p2 ‘

~P _M2(y + 1)_

In Equation (14.34), M cancels, and we finally obtain in both the small-angle and hypersonic limits,

Note that, for у = 1.4,

It is interesting to observe that, in the hypersonic limit for a slender wedge, the wave angle is only 20 percent larger than the wedge angle—a graphic demonstration of a thin shock layer in hypersonic flow.

In aerodynamics, pressure distributions are usually quoted in terms of the nondi­mensional pressure coefficient Cp, rather than the pressure itself. The pressure coef­ficient is defined as

where p and q are the upstream (freestream) static pressure and dynamic pressure, respectively. Recall from Section 11.3 that Equation (14.37) can also be written as Equation (11.22), repeated below:

Combining Equations (11.22) and (14.28), we obtain an exact relation for Cp behind an oblique shock wave as follows:

In the hypersonic limit,

Pause for a moment, and review our results. We have obtained limiting forms of the oblique shock equations, valid for the case when the upstream Mach number becomes very large. These limiting forms, called the hypersonic shock-wave rela­tions, are given by Equations (14.29), (14.31), and (14.32), which yield the pressure ratio, density ratio, and temperature ratio across the shock when Mx —> oo. Fur­thermore, in the limit of both M —> oo and small 9 (such as the hypersonic flow over a slender airfoil shape), the limiting relation for the wave angle as a function of the deflection angle is given by Equation (14.35). Finally, the form of the pressure coefficient behind an oblique shock is given in the limit of hypersonic Mach numbers by Equation (14.39). Note that the limiting forms of the equations are always simpler than their corresponding exact counterparts.

In terms of actual quantitative results, it is always recommended that the exact oblique shock equations be used, even for hypersonic flow. This is particularly conve­nient because the exact results are tabulated in Appendix B. The value of the relations obtained in the hypersonic limit (as described above) is more for theoretical analysis rather than for the calculation of actual numbers. For example, in this section, we use the hypersonic shock relations to shed additional understanding of the significance of newtonian theory. In the next section, we will examine the same hypersonic shock relations to demonstrate the principle of Mach number independence.

Newtonian theory was discussed at length in Sections 14.3 and 14.4. For our pur­poses here, temporarily discard any thoughts of newtonian theory, and simply recall the exact oblique shock relation for Cp as given by Equation (14.38), repeated below (with freestream conditions now denoted by a subscript oo rather than a subscript 1, as used earlier):

Equation (14.39) gave the limiting value of Cp as Mж -► oo, repeated below:

Now take the additional limit of у -► 1.0. From Equation (14.39), in both limits as Moo —»• oo and у —► 1.0, we have

Cp ^ 2 sin2 p [14.40]

Equation (14.40) is a result from exact oblique shock theory; it has nothing to do with newtonian theory (as yet). Keep in mind that p in Equation (14.40) is the wave angle, not the deflection angle.

Let us go further. Consider the exact oblique shock relation for the density ratio, P/Poc! given by Equation (14.30), repeated below (again with a subscript oo replacing the subscript 1):

Pi __ (y + l)Af^ sin2 p

Poo (y – l)Af^ sin2 p + 2

Equation (14.31) was obtained as the limit where Mx —► oo, namely,

Pi У + 1

Poo У 1

1, we find

that is, the density behind the shock is infinitely large. In turn, mass flow consider­ations then dictate that the shock wave is coincident with the body surface. This is further substantiated by Equation (14.35), which is good for Moo oo and small deflection angles:

[14.35]

In the additional limit as у —»■ 1, we have

that is, the shock wave lies on the body. In light of this result, Equation (14.40) is written as

[14.44]

Examine Equation (14.44). It is a result from exact oblique shock theory, taken in the combined limit of Мх —> oo and у —> 1. However, it is also precisely the newtonian results given by Equation (14.4). Therefore, we make the following conclusion. The closer the actual hypersonic flow problem is to the limits —> oo and у —> 1, the closer it should be physically described by newtonian flow. In this regard, we gain a better appreciation of the true significance of newtonian theory. We can also state that the application of newtonian theory to practical hypersonic flow problems, where у is always greater than unity, is theoretically not proper, and the agreement that is frequently obtained with experimental data has to be viewed as somewhat fortuitous. Nevertheless, the simplicity of newtonian theory along with its (sometimes) reasonable results (no matter how fortuitous) has made it a widely used and popular engineering method for the estimation of surface pressure distributions, hence lift – and wave-drag coefficients, for hypersonic bodies.

A Comment on Drag Variation with Velocity

Beginning with Chapter 1, indeed beginning with the most elementary studies of fluid dynamics, the point is usually made that the aerodynamic force on a body immersed in a flowing fluid is proportional to the square of the flow velocity. For example, from Section 1.5,

L = poaVl0SCL and D = poaVl0SCD

As long as Ci and CD are independent of velocity, then clearly L <x and D <x V^. This is the case for an inviscid, incompressible flow, where С/, and Co depend only on the shape and angle of attack of the body. However, from the dimensional analysis in Section 1.7, we also discovered that С/, and Co in general are functions of both Reynolds number and Mach number,

Ci = f (Re, Mgo) Co = /2(Re, MTO)

Of course, for an inviscid, incompressible flow, Re and are not players (indeed,

for inviscid flow, Re —>■ 00 and for incompressible flow, —»• 0). However for all

other types of flow, Re and are players, and the values of Ci and С о depend not only on the shape and angle of attack of the body, but also on Re and For this reason, in general the aerodynamic force is not exactly proportional to the square of the velocity. For example, examine the results from Example 18.1. In part (a), we calculated a value for drag to the 175.6 N when = 100 m/s. If the drag were proportional to V^, then in part (b) where = 1000 m/s, a factor of 10 larger, the drag would have been one hundred times larger, or 17,560 N. In contrast, our calculations in part (b) showed the drag to be considerably smaller, namely 5026 N. In other words, when V^_ was increased by a factor of 10, the drag increased by only a factor of 28.6, not by a factor of 100. The reason is obvious. The value of Cf decreases when the velocity is increased because: (1) the Reynolds number increases, which from Equation (18.22) causes Cf to decrease, and (2) the Mach number increases, which from Figure 18.8 causes Cf to decrease.

So be careful about thinking that aerodynamic force varies with the square of the velocity. For cases other than inviscid, incompressible flow, this is not true.

The Supercritical Airfoil

Let us return to a consideration of two-dimensional airfoils. A natural conclusion from the material in Section 11.6, and especially from Figure 11.11, is that an airfoil with a high critical Mach number is very desirable, indeed necessary, for high-speed subsonic aircraft. If we can increase Mcr, then we can increase T/drag-divergence, which follows closely after Mcr. This was the philosophy employed in aircraft design from 1945 to approximately 1965. Almost by accident, the NACA 64-series airfoils (see Section 4.2), although originally designed to encourage laminar flow, turned out to have relative high values of Mcr in comparison with other NACA shapes. Hence, the NACA 64 series has seen wide application on high-speed airplanes. Also, we know that thinner airfoils have higher values of Mcr (see Figure 11.7); hence, aircraft designers have used relatively thin airfoils on high-speed airplanes.

However, there is a limit to how thin a practical airfoil can be. For example, con­siderations other than aerodynamic influence the airfoil thickness; the airfoil requires a certain thickness for structural strength, and there must be room for the storage of fuel. This prompts the following question: For an airfoil of given thickness, how can we delay the large drag rise to higher Mach numbers? To increase Mcr is one obvious tack, as described above, but there is another approach. Rather than increasing Mcr, let us strive to increase the Mach number increment between Mcr and T/dr-ag-divcrgcncc ■ That is, referring to Figure 11.11, let us increase the distance between points e and c. This philosophy has been pursued since 1965, leading to the design of a new family of airfoils called supercritical airfoils, which are the subject of this section.

The purpose Of a Supercritical airfoil is tO increase the Value Of Mdrag-divergence> although Mcr may change very little. The shape of a supercritical airfoil is compared with an NACA 64-series airfoil in Figure 11.19. Here, an NACA 642-A215 airfoil is sketched in Figure 11.19a, and a 13-percent thick supercritical airfoil is shown in Figure 11.19c. (Note the similarity between the supercritical profile and the modem

low-speed airfoils discussed in Section 4.11.) The supercritical airfoil has a relatively flap top, thus encouraging a region of supersonic flow with lower local values of M than the NACA 64 series. In turn, the terminating shock is weaker, thus creating less drag. Similar trends can be seen by comparing the Cp distributions for the NACA 64 series (Figure 11.19A>) and the supercritical airfoil (Figure 11.19c/). Indeed, Figure 11.19a and b for the NACA 64-series airfoil pertain to a lower freestream Mach number, Мж = 0.69, than Figure 11.19c and d. which pertain to the supercritial airfoil at a higher freestream Mach number, M<*, = 0.79. In spite of the fact that the 64-series airfoil is at a lower M^, the extent of the supersonic flow reaches farther above the airfoil, the local supersonic Mach numbers are higher, and the terminating shock wave is stronger. Clearly, the supercritical airfoil shows more desirable flow – field characteristics; namely, the extent of the supersonic flow is closer to the surface, the local supersonic Mach numbers are lower, and the terminating shock wave is weaker. As a result, the value of T/drag-divergence will be higher for the supercritical airfoil. This is verified by the experimental data given in Figure 11.20, taken from Reference 32. Here, the value of Afdrag-divergence is 0.79 for the supercritical airfoil in comparison with 0.67 for the NACA 64 series.

Because the top of the supercritical airfoil is relatively flat, the forward 60 percent of the airfoil has negative camber, which lowers the lift. To compensate, the lift is increased by having extreme positive camber on the rearward 30 percent of the airfoil. This is the reason for the cusplike shape of the bottom surface near the trailing edge.

The supercritical airfoil was developed by Richard Whitcomb in 1965 at the NASA Langley Research Center. A detailed description of the rationale as well as some early experimental data for supercritial airfoils are given by Whitcomb in Reference 32, which should be consulted for more details. The supercritical airfoil, and many variations of such, are now used by the aircraft industry on modem high­speed airplane designs. Examples are the Boeing 757 and 767, and the latest model Lear jets. The supercritical airfoil is one of two major breakthroughs made in transonic airplane aerodynamics since 1945, the other being the area mle discussed in Section

11.8. It is a testimonial to the man that Richard Whitcomb was mainly responsible for both.

Couette Flow: General Discussion

Consider the flow model shown in Figure 16.2. Here we see a viscous fluid contained between two parallel plates separated by a distance D. The upper plate is moving to the right at velocity ue. Due to the no-slip condition, there can be no relative motion between the plate and the fluid; hence, at у = D the flow velocity is и = ue and is directed toward the right. Similarly, the flow velocity at у = 0, which is the surface of the stationary lower plate, is и = 0. In addition, the two plates may be at different temperatures; the upper plate is at temperature Te and the lower plate is at temperature Tw. Again, due to the no-slip condition as discussed in Section 15.7, the fluid temperature at у = D is T = Te and that at у = 0 is T = Tw.

Clearly, there is a flow field between the two plates; the driving force for this flow is the motion of the upper plate, dragging the flow along with it through the mechanism of friction. The upper plate is exerting a shear stress, re, acting toward the right on the fluid at у — D, thus causing the fluid to move toward the right. By an equal and opposite reaction, the fluid is exerting a shear stress ze on the upper plate acting toward the left, tending to retard its motion. We assume that the upper plate

is being driven by some external force which is sufficient to overcome the retarding shear stress and to allow the plate to move at the constant velocity ue. Similarly, the lower plate is exerting a shear stress rw acting toward the left on the fluid at у = 0. By an equal and opposite reaction, the fluid is exerting a shear stress tw acting toward the right on the lower plate. (In all subsequent diagrams dealing with viscous flow, the only shear stresses shown will be those due to the fluid acting on the surface, unless otherwise noted.)

In addition to the velocity field induced by the relative motion of the two plates, there will also be a temperature field induced by the following two mechanisms:

1. The plates in general will be at different temperatures, thus causing temperature gradients in the flow.

2. The kinetic energy of the flow will be partially dissipated by the influence of friction and will be transformed into internal energy within the fluid. These changes in internal energy will be reflected by changes in temperature. This phenomenon is called viscous dissipation.

Consequently, temperature gradients will exist within the flow; in turn, these temper­ature gradients result in the transfer of heat through the fluid. Of particular interest is the heat transfer at the upper and lower surfaces, denoted by qe and qw, respectively.

These heat transfers are shown in Figure 16.2; the directions for qe and qw show heat being transferred from the fluid to the wall in both cases. When heat flows from the fluid to the wall, this is called a cold wall case, such as sketched in Figure 16.2. When heat flows from the wall into the fluid, this is called a hot wall case. Keep in mind that the heat flux through the fluid at any point is given by the Fourier law expressed by Equation (15.2); that is, the heat flux in the у direction is expressed as

[15.2]

where the minus sign accounts for the fact that heat is transferred from a region of high temperature to a region of lower temperature; that is, qy is in the opposite direction of the temperature gradient.

Let us examine the geometry of Couette flow as illustrated in Figure 16.2. An x-y cartesian coordinate system is oriented with the x axis in the direction of the flow and the у axis perpendicular to the flow. Since the two plates are parallel, the only possible flow pattern consistent with this picture is that of straight, parallel streamlines. Moreover, since the plates are infinitely long (i. e., stretching to plus and minus infinity in the x direction), then the flow properties cannot change with x. (If the properties did change with x, then the flow-field properties would become infinitely large or infinitesimally small at large values of x—a physical inconsistency.) Thus, all partial derivatives with respect to x are zero. The only changes in the flow – field variables take place in the у direction. Moreover, the flow is steady, so that all time derivatives are zero. With this geometry in mind, return to the governing Navier-Stokes equations given by Equations (15.19a to c) and Equation (15.26). In these equations, for Couette flow,

ди ЗT dp

dx dx dx

Hence, from Equations (15.19a to c) and Equation (15.26), we have

„ 3 /, Э7Л 3 / du n

Energy equation: — 3y / + dy “ dy ) = °

Equations (16.1) to (16.3) are the governing equations for Couette flow. Note that these equations are exact forms of the Navier-Stokes equations applied to the geom­etry of Couette flow—no approximations have been made. Also, note from Equa­tion (16.2) that the variation of pressure in the у direction is zero; this in combination with the earlier result that dp/dx = 0 implies that the pressure is constant throughout the entire flow field. Couette flow is a constant pressure flow. It is interesting to note that all the previous flow problems discussed in Parts 2 and 3, being inviscid flows, were established and maintained by the existence of pressure gradients in the flow.

In these problems, the pressure gradient was nature’s mechanism of grabbing hold of the flow and making it move. However, in the problem we are discussing now—being a viscous flow—shear stress is another mechanism by which nature can exert a force on a flow. For Couette flow, the shear stress exerted by the moving plate on the fluid is the exclusive driving mechanism that maintains the flow; clearly, no pressure gradient is present, nor is it needed.

This section has presented the general nature of Couette flow. Note that we have made no distinction between incompressible and compressible flow; all aspects discussed here apply to both cases. Also, we note that, although Couette flow appears to be a rather academic problem, the following sections illustrate, in a simple fashion, many of the important characteristics of practical viscous flows in real engineering applications.

The next two sections will treat the separate cases of incompressible and com­pressible Couette flow. Incompressible flow will be discussed first because of its rel­ative simplicity; this is the subject of Section 16.3. Then compressible Couette flow, and how it differs from the incompressible case, will be examined in Section 16.4.

As a final note in this section, it is obvious from our general discussion of Couette flow that the flow-field properties vary only in the у direction; all derivatives in the x direction are zero. Therefore, as a matter of mathematical preciseness, all the partial derivatives in Equations (16.1) to (16.3) can be written as ordinary derivatives. For example, Equation (16.1) can be written as

However, our discussion of Couette flow is intended to serve as a straightforward example of a viscous flow problem, “breaking the ice” so-to-speak for the more practical but more complex problems to come—problems which involve changes in both the x and у directions, and which are described by partial differential equations. Therefore, on pedagogical grounds, we choose to continue the partial differential notation here, simply to make the reader feel more comfortable when we extend these concepts to the boundary layer and full Navier-Stokes solutions in Chapters 17 and 20, respectively.