Category When Is A Flow Compressible?

Application to Supersonic Airfoils

Equation (12.15) is very handy for estimating the lift and wave drag for thin supersonic airfoils, such as sketched in Figure 12.3. When applying Equation (12.15) to any surface, one can follow a formal sign convention for 9, which is different for regions of left-running waves (such as above the airfoil in Figure 12.3) than for regions of right-running waves (such as below the airfoil in Figure 12.3). This sign convention is

developed in detail in Reference 21. However, for our purpose here, there is no need to be concerned about the sign associated with в in Equation (12.15). Rather, keep in mind that when the surface is inclined into the freestream direction, linearized theory predicts a positive Cp. For example, points A and В in Figure 12.3 are on surfaces inclined into the freestream, and hence CPiA and CP. B are positive values given by

In contrast, when the surface is inclined away from the freestream direction, linearized theory predicts a negative Cp. For example, points C and D in Figure 12.3 are on surfaces inclined away from the freestream, and hence Cp c and C Ptp are negative values, given by

In the above expressions, в is always treated as a positive quantity, and the sign of Cp is determined simply by looking at the body and noting whether the surface is inclined into or away from the freestream.

With the distribution of Cp over the airfoil surface given by Equation (12.15), the lift and drag coefficients, c; and c(i, respectively, can be obtained from the integrals given by Equations (1.15) to (1.19).

Let us consider the simplest possible airfoil, namely, a flat plate at a small angle of attack a as shown in Figure 12.4. Looking at this picture, the lower surface of the

——————- c———————— «J

У Pu

………………. ………… Ї….. ……………………

Pi

Figure 1 2.4 A flat plate at angle of attack in a supersonic flow.

plate is a compression surface inclined at the angle a into the freestrearn. and from Equation (12.15),

Since the surface inclination angle is constant along the entire lower surface, Cpj is a constant value over the lower surface. Similarly, the top surface is an expansion surface inclined at the angle a away from the freestrearn, and from Equation (12.15),

Cp u is constant over the upper surface. The normal force coefficient for the flat plate can be obtained from Equation (1.15):

cn = – f (Cpj — Cp u) dx [12.18]

c Jo

Substituting Equations (12.16) and (12.17) into (12.18), we obtain

4a 1 Ґ 4a

Cn = . : – ax = , :

The axial force coefficient is given by Equation (1.16):

1

Q/ — ~ j (.Cp, u Epj’) dy c J LE

However, the flat plate has (theoretically) zero thickness. Hence, in Equation (12.20), dy = 0, and as a result, ca = 0. This is also clearly seen in Figure 12.4; the pressures act normal to the surface, and hence there is no component of the pressure force in the x direction. From Equations (1.18) and (1.19),

ci = cn cos a — ca sin a cd = c„ sin a + ca cos a

and, along with the assumption that a is small and hence cos a ~ 1 and sin a % a, we have

ci=cn—caa [12.21]

cd — c„a + ca [12.22]

Substituting Equation (12.19) and the fact that ca = 0 into Equations (12.21) and

(12.22) , we obtain

Equations (12.23) and (12.24) give the lift and wave-drag coefficients, respectively, for the supersonic flow over a flat plate. Keep in mind that they are results from linearized theory and therefore are valid only for small a.

For a thin airfoil of arbitrary shape at small angle of attack, linearized theory gives an expression for q identical to Equation (12.23); that is,

4a

Cl = 7^=i

Within the approximation of linearized theory, q depends only on a and is independent of the airfoil shape and thickness. However, the same linearized theory gives a wave – drag coefficient in the form of

Cd = ^r=i(a2 + 8c + 8f)

where gc and g, are functions of the airfoil camber and thickness, respectively. For more details, see References 25 and 26.

Example 12.1 | Using linearized theory, calculate the lift and drag coefficients for a flat plate at a 5° angle of attack in a Mach 3 flow. Compare with the exact results obtained in Example 9.10.

Solution

a = 5° = 0.087 rad

From Equation (12.23),

From Equation (12.24),

The results calculated in Example 9.10 for the same problem are exact results, utilizing the exact oblique shock theory and the exact Prandtl-Meyer expansion-wave analysis. These results were

(exact results from Example 9.10

Note that, for the relatively small angle of attack of 5°, the linearized theory results are quite accurate—to within 1.6 percent.

Figure 12.5 Transonic area ruling for the F-16. Variation af normal cross-sectional area as a function of location along the fuselage axis.

Problems

1. Using the results of linearized theory, calculate the lift and wave-drag coefficients for an infinitely thin flat plate in a Mach 2.6 freestream at angles of attack of

(a) a = 5° (b)a = 15° (c) a = 30°

Compare these approximate results with those from the exact shock-expansion theory obtained in Problem 9.13. What can you conclude about the accuracy of linearized theory in this case?

2. For the conditions of Problem 12.1, calculate the pressures (in the form of p/Poo) on the top and bottom surfaces of the flat plate, using linearized theory. Compare these approximate results with those obtained from exact shock-expansion theory in Problem 9.13. Make some appropriate conclusions regarding the accuracy of linearized theory for the calculation of pressures.

3. Consider a diamond-wedge airfoil such as shown in Figure 9.24, with a half-angle є = 10°. The airfoil is at an angle of attack a = 15° to a Mach 3 freestream. Using linear theory, calculate the lift and wave-drag coefficients for the airfoil. Compare these approximate results with those from the exact shock-expansion theory obtained in Problem 9.14.

Reynolds Analogy

Another useful engineering relation for the analysis of aerodynamic heating is Rey­nolds analogy, which can easily be introduced within the context of our discussion of Couette flow. Reynolds analogy is a relation between the skin friction coefficient and the heat transfer coefficient. The skin friction coefficient c/ was first introduced in Section 1.5. In our context here, we define the skin friction coefficient as

tPeUi

Let us define the Reynolds number for Couette flow as

PeUeD

Then, Equation (16.53) becomes

Equation (16.54) is interesting in its own right. It demonstrates that the skin friction coefficient is a function of just the Reynolds number—a result which applies in general for other incompressible viscous flows [although the function is not necessarily the same as given in Equation (16.54)].

Now let us define a heat transfer coefficient as

In Equation (16.55), Ся is called the Stanton number; it is one of several different types of heat transfer coefficient that is used in the analysis of aerodynamic heat­ing. It is a dimensionless quantity, in the same vein as the skin friction coefficient.

For Couette flow, from Equation (16.24), and dropping the absolute value signs for convenience, we have

. fi (he – hw + 5 Ргм^ 4w ~ Pr ( D

Inserting Equation (16.39) into (16.56), we have for Couette flow

/г (haw hu

D

Inserting Equation (16.57) into (16.55), we obtain

„ (tx/?mhaw-hw)/D] /r/Pr 1 ,.жи1

CH = ————————– = ——– = —— [16.58]

Pe^eiPaw ^w’) Pe^e^ Pi*

Equation (16.58) is interesting in its own right. It demonstrates that the Stanton number is a function of the Reynolds number and Prandtl number—a result that applies generally for other incompressible viscous flows [although the function is not necessarily the same as given in Equation (16.58)].

We now combine the results for Cf and CH obtained above. From Equa­tions (16.54) and (16.58), we have

Ch

( 1 ї

1 Re

cf ~

i Re Pr )

’ 2

Equation (16.59) is Reynolds analogy as applied to Couette flow. Reynolds analogy is, in general, a relation between the heat transfer coefficient and the skin friction coefficient. For Couette flow, this relation is given by Equation (16.59). Note that the ratio Сн/сf is simply a function of the Prandtl number—a result that applies usually for other incompressible viscous flows, although not necessarily the same function as given in Equation (16.59).

Prandtl-Meyer Expansion Waves

Oblique shock waves, as discussed in Sections 9.2 to 9.5, occur when a supersonic flow is turned into itself (see again Figure 9.1a). In contrast, when a supersonic flow is turned away from itself, an expansion wave is formed, as sketched in Figure 9.1 b. Examine this figure carefully, and review the surrounding discussion in Section 9.1 before progressing further. The purpose of the present section is to develop a theory which allows us to calculate the changes in flow properties across such expansion waves. To this stage in our discussion of oblique waves, we have completed the left – hand branch of the road map in Figure 9.5. In this section, we cover the right-hand branch.

The expansion fan in Figure 9.1 b is a continuous expansion region which can be visualized as an infinite number of Mach waves, each making the Mach angle /г [see Equation (9.1)] with the local flow direction. As sketched in Figure 9.23, the expansion fan is bounded upstream by a Mach wave which makes the angle i with respect to the upstream flow, where /xi = arcsin(l/M]). The expansion fan is bounded downstream by another Mach wave which makes the angle /x2 with respect to the downstream flow, where fi2 = arcsin(l/M2). Since the expansion through the

Figure 9.33 Prandtl-Meyer expansion.

wave takes place across a continuous succession of Mach waves, and since ds = 0 for each Mach wave, the expansion is isentropic. This is in direct contrast to flow across an oblique shock, which always experiences an entropy increase. The fact that the flow through an expansion wave is isentropic is a greatly simplifying aspect, as we will soon appreciate.

An expansion wave emanating from a sharp convex corner as sketched in Figures 9ЛЬ and 9.23 is called a centered expansion wave. Ludwig Prandtl and his student Theodor Meyer first worked out a theory for centered expansion waves in 1907-1908, and hence such waves are commonly denoted as Prandtl-Meyer expansion waves.

The problem of an expansion wave is as follows: Referring to Figure 9.23, given the upstream flow (region 1) and the deflection angle 6, calculate the downstream flow (region 2). Let us proceed.

Consider a very weak wave produced by an infinitesimally small flow deflection d6 as sketched in Figure 9.24. We consider the limit of this picture as d6 -> 0; hence, the wave is essentially a Mach wave at the angle p, to the upstream flow. The velocity ahead of the wave is V. As the flow is deflected downward through the angle d9, the velocity is increased by the infinitesimal amount dV, and hence the flow velocity behind the wave is V +dV inclined at the angle d0. Recall from the treatment of the momentum equation in Section 9.2 that any change in velocity across a wave takes place normal to the wave; the tangential component is unchanged across the wave. In Figure 9.24, the horizontal line segment A В with length V is drawn behind the wave. Also, the line segment AC is drawn to represent the new velocity V + dV behind the wave. Then line SC is normal to the wave because it represents the line along which the change in velocity occurs. Examining the geometry in Figure 9.24, from the law of sines applied to triangle ABC, we see that

However, from trigonometric identities,

From Equation (9.1), we know that ц, = arcsin(l/M). Hence, the right triangle in Figure 9.25 demonstrates that

1

Vm2 -1

Substituting Equation (9.31) into (9.30), we obtain

Equation (9.32) relates the infinitesimal change in velocity dV to the infinitesimal deflection dQ across a wave of vanishing strength. In the precise limit of a Mach wave, of course dV and hence dQ are zero. In this sense, Equation (9.32) is an approximate equation for a finite dQ, but it becomes a true equality as dQ —0. Since the expansion fan illustrated in Figures 9.1 b and 9.23 is a region of an infinite number of Mach waves, Equation (9.32) is a differential equation which precisely describes the flow inside the expansion wave.

Return to Figure 9.23. Let us integrate Equation (9.32) from region 1, where the deflection angle is zero and the Mach number is Mt, to region 2, where the deflection angle is в and the Mach number is My

[9.33]

To carry out the integral on the right-hand side of Equation (9.33), dV/V must be obtained in terms of M, as follows. From the definition of Mach number, M = V/a, we have V = Ma, or

In V — In M + lna

Differentiating Equation (9.34), we obtain

dV _ dM da

~V ~ ~M~ + ~a

From Equations (8.25) and (8.40), we have

[9.36]

Solving Equation (9.36) for a, we obtain

Differentiating Equation (9.37), we obtain

substituting Equation (9.38) into (9.35), we have

dV _ 1 dM

~V~ ~ 1 + [(y – 1)/2]M2 ~M

Equation (9.39) is a relation for dV/V strictly in terms of M—this is precisely what is desired for the integral in Equation (9.33). Hence, substituting Equation (9.39) into (9.33), we have

is called the Prandtl-Meyer function, denoted by v. Carrying out the integration, Equation (9.41) becomes

The constant of integration that would ordinarily appear in Equation (9.42) is not important, because it drops out when Equation (9.42) is used for the definite integral in Equation (9.40). For convenience, it is chosen as zero, such that v(M) = 0 when M = 1. Finally, we can now write Equation (9.40), combined with (9.41), as

where v(M) is given by Equation (9.42) for a calorically perfect gas. The Prandtl – Meyer function v is very important; it is the key to the calculation of changes across an expansion wave. Because of its importance, v is tabulated as a function of M in Appendix C. For convenience, values of д are also tabulated in Appendix C.

How do the above results solve the problem stated in Figure 9.23; that is how can we obtain the properties in region 2 from the known properties in region 1 and the known deflection angle 91 The answer is straightforward:

1. For the given M1, obtain v{M) from Appendix C.

2. Calculate v(M2) from Equation (9.43), using the known 9 and the value of v{M ) obtained in step 1.

3. Obtain M2 from Appendix C corresponding to the value of v(M2) from step 2.

4. The expansion wave is isentropic; hence, p0 and 7b are constant through the wave. That is, Tq 2 = To, і and /?0,2 = Po. i – From Equation (8.40), we have

T2 T2/Tq2 _ 1 + [(7 — 1)/2]M)2

T, T/Tq 1 + [(y – )/2]Mj

From Equation (8.42), we have

Pi = P2/P0 = /1+[(к – 1)/2]А79Гу/(у Pi Pi/Pa \+[(y – Y)/2Ml)

Since we know both M and M2, as well as T and p, Equations (9.44) and (9.45) allow the calculation of T2 and p2 downstream of the expansion wave.

(a) Scramjet powered air-to-surface-missile concept

(/?) Scramjet powered strike/reconnaissance vehicle concept

(c) Scramjet powered space access vehicle concept

Figure 9.26 Computer-generated images of possible future SCRAMjet-powered hypersonic vehicles. ICourtesy of James Weber, United States Air Force.)

Example 9.8 | In the preceding discussion on SCRAMjet engines, an isentropic compression wave was men­tioned as one of the possible compression mechanisms. Consider the isentropic compression surface sketched in Figure 9.32a. The Mach number and pressure upstream of the wave are M = 10 and pi = 1 atm, respectively. The flow is turned through a total angle of 15°. Calculate the Mach number and pressure in region 2 behind the compression wave.

Solution

From Appendix C, for M = 10, Ui = 102.3°. In Region 2,

u2 = У, – в = 102.3 – 15 = 87.3°

From Appendix C for v2 = 87.3°, we have (closest entry)

From Appendix A, for M = 10, рол/Рi = 0.4244 x 105 and for M2 = 6.4, po. i! Pi = 0.2355 x 104. Since the flow is isentropic, p0,2 = Рол, and hence

Consider the flow over a compression comer with the same upstream conditions of M{ = 10 and р = 1 atm as in Example 9.8, and the same turning angle of 15°, except in this case the comer is sharp and the compression takes place through an oblique shock wave as sketched in Figure 9.32b. Calculate the downstream Mach number, static pressure, and total pressure in region 2. Compare the results with those obtained in Example 9.8, and comment on the significance of the comparison.

Solution

From Figure 9.7 for M = 10 and в = 15°, the wave angle is /5 = 20°. The component of the upstream Mach number perpendicular to the wave is

M„,1 = Mi sin p = (10) sin 20° = 34.2

From Appendix В for M„,i = 3.42, we have (nearest entry), p2/P = 13.32, рол/Рол = 0.2322, and M„,2 = 0.4552. Hence

M„ 2 0.4552

M2 =——– —— =—————— = 5.22

sin(/3 — (9) sin(20 — 15) L _. J

Pi = (Pi/Pi)P = 13.32(1) =

The total pressure in region 1 can be obtained from Appendix A as follows. For M = 10, Poa/Pi = 0.4244 x 105. Hence, the total pressure in region 2 is

9.85 x 103 atm

Рол = (pi) = (0.2322X0.4244 x 105)(1) =

As a check, we can calculate p02 as follows. (This check also alerts us to the error incurred when we round to the nearest entry in the tables.) From Appendix A for M2 = 5.22, Рол! Рг = 0.6661 x 103 (nearest entry). Hence,

Po,2 = (P2) = (0.6661 x 103)(13.32) = 8.87 x 103 atm

Note this answer is 10 percent lower than that obtained above, which is simply due to our rounding to the nearest entry in the tables. The error incurred by taking the nearest entry is exacerbated by the very high Mach numbers in this example. Much better accuracy can be obtained by properly interpolating between table entries.

Comparing the results from this example and Example 9.8, we clearly see that the isentropic compression is a more efficient compression process, yielding a down­stream Mach number and pressure that are both considerably higher than in the case of the shock wave. The inefficiency of the shock wave is measured by the loss of total pressure across the shock; total pressure drops by about 77 percent across the shock. This emphasizes why designers of supersonic and hypersonic inlets would love to have the compression process carried out via isentropic compression waves. However, as noted in our discussion on SCRAMjets, it is very difficult to achieve such a compression in real life; the contour of the compression surface must be quite precise, and it is a point design for the given upsteam Mach number. At off-design Mach numbers, even the best-designed compression contour will result in shocks.

Qualitative Aspects of Hypersonic Flow

Consider a 15° half-angle wedge flying at Mx — 36. From Figure 9.7, we see that the wave angle of the oblique shock is only 18°; that is, the oblique shock wave is very close to the surface of the body. This situation is sketched in Figure 14.1. Clearly, the shock layer between the shock wave and the body is very thin. Such thin shock layers are one characteristic of hypersonic flow. A practical consequence of a thin shock layer is that a major interaction frequently occurs between the inviscid flow behind the shock and the viscous boundary layer on the surface. Indeed, hypersonic vehicles generally fly at high altitudes where the density, hence Reynolds number, is low, and therefore the boundary layers are thick. Moreover, at hypersonic speeds, the boundary-layer thickness on slender bodies is approximately proportional to hence, the high Mach numbers further contribute to a thickening of the boundary layer. In many cases, the boundary-layer thickness is of the same magnitude as the shock – layer thickness, such as sketched in the insert at the top of Figure 14.1. Here, the shock layer is fully viscous, and the shock-wave shape and surface pressure distribution are affected by such viscous effects. These phenomena are called viscous interaction phenomena—where the viscous flow greatly affects the external inviscid flow, and, of course, the external inviscid flow affects the boundary layer. A graphic example of such viscous interaction occurs on a flat plate at hypersonic speeds, as sketched in Figure 14.2. If the flow were completely inviscid, then we would have the case shown

Figure 14.1 For hypersonic flow, the shock layers are thin and viscous.

in Figure 14.2a, where a Mach wave trails downstream from the leading edge. Since there is no deflection of the flow, the pressure distribution over the surface of the plate is constant and equal to p^. In contrast, in real life there is a boundary layer over the flat plate, and at hypersonic conditions this boundary layer can be thick, as sketched in Figure 14.2b. The thick boundary layer deflects the external, inviscid flow, creating a comparably strong, curved shock wave which trails downstream from the leading edge. In turn, the surface pressure from the leading edge is considerably higher than poo, and only approaches px far downstream of the leading edge, as shown in Figure 14.2b. In addition to influencing the aerodynamic force, such high pressures increase the aerodynamic heating at the leading edge. Therefore, hypersonic viscous interaction can be important, and this has been one of the major areas of modem hypersonic aerodynamic research.

There is a second and frequently more dominant aspect of hypersonic flow, namely, high temperatures in the shock layer, along with large aerodynamic heat­ing of the vehicle. For example, consider a blunt body reentering the atmosphere at Mach 36, as sketched in Figure 14.3. Let us calculate the temperature in the shock layer immediately behind the normal portion of the bow shock wave. From Ap­pendix B, we find that the static temperature ratio across a normal shock wave with Moo = 36 is 252.9; this is denoted by Ts/Tx in Figure 14.3. Moreover, at a standard altitude of 59 km, Тж = 258 К. Hence, we obtain Ts = 65,248 К—an incredibly high temperature, which is more than six times hotter than the surface of the sun! This is, in reality, an incorrect value, because we have used Appendix В which is good only for a calorically perfect gas with у = 1.4. However, at high temperatures, the gas will become chemically reacting; у will no longer equal 1.4 and will no longer be constant. Nevertheless, we get the impression from this calculation that the temperature in the shock layer will be very high, albeit something less than 65,248 K. Indeed, if a proper calculation of Ts is made taking into account the chemically

reacting gas, we would find that Ts & 11,000 К— still a very high value. Clearly, high-temperature effects are very important in hypersonic flow.

Let us examine these high-temperature effects in more detail. If we consider air at p — 1 atm and T = 288 К (standard sea level), the chemical composition is essentially 20 percent O2 and 80 percent N2 by volume. The temperature is too low for any significant chemical reaction to take place. However, if we were to increase T to 2000 K, we would observe that the O2 begins to dissociate; that is,

2 -* 20 2000 К < T < 4000 К

If the temperature were increased to 4000 K, most of the O2 would be dissociated, and N2 dissociation would commence:

N2 -* 2N 4000 К < T < 9000 К

If the temperature were increased to 9000 K, most of the N2 would be dissociated, and ionization would commence:

N -* N+ + e_ O -* 0+ + e~

Hence, returning to Figure 14.3, the shock layer in the nose region of the body is a partially ionized plasma, consisting of the atoms N and O, the ions N+ and 0+, and electrons, e~. Indeed, the presence of these free electrons in the shock layer is responsible for the “communications blackout” experienced over portions of the trajectory of a reentry vehicle.

One consequence of these high-temperature effects is that all our equations and tables obtained in Chapters 7 to 13 which depended on a constant у = 1.4 are no longer valid. Indeed, the governing equations for the high-temperature, chemically reacting shock layer in Figure 14.3 must be solved numerically, taking into account the proper physics and chemistry of the gas itself. The analysis of aerodynamic flows with such real physical effects is discussed in detail in Chapters 16 and 17 of Reference 21; such matters are beyond the scope of this book.

Associated with the high-temperature shock layers is a large amount of heat trans­fer to the surface of a hypersonic vehicle. Indeed, for reentry velocities, aerodynamic

heating dominates the design of the vehicle, as explained at the end of Section 1.1. (Recall that the third historical example discussed in Section 1.1 was the evolution of the blunt-body concept to reduce aerodynamic heating; review this material be­fore progressing further.) The usual mode of aerodynamic heating is the transfer of energy from the hot shock layer to the surface by means of thermal conduction at the surface; that is, if 3T/3n represents the temperature gradient in the gas normal to the surface, then qc — —к(дТ/дп) is the heat transfer into the surface. Because ЗT/Эи is a flow-field property generated by the flow of the gas over the body, qc is called convective heating. For reentry velocities associated with ICBMs (about 28,000 ft/s), this is the only meaningful mode of heat transfer to the body. However, at higher velocities, the shock-layer temperature becomes even hotter. From expe­rience, we know that all bodies emit thermal radiation, and from physics you know that blackbody radiation varies as 7’4: hence, radiation becomes a dominant mode of heat transfer at high temperatures. (For example, the heat you feel by standing beside a fire in a fireplace is radiative heating from the flames and the hot walls.) When the shock layer reaches temperatures on the order of 11,000 K, as for the case given in Figure 14.3, thermal radiation from the hot gas becomes a substantial portion of the total heat transfer to the body surface. Denoting radiative heating by qr, we can express the total aerodynamic heating q as the sum of convective and radiative heating; q = qc + q, . For Apollo reentry, qr/q ~ 0.3, and hence radiative heating was an important consideration in the design of the Apollo heat shield. For the entry of a space probe into the atmosphere of Jupiter, the velocities will be so high and the shock-layer temperatures so large that the convective heating is negligible, and in this case q ~ qr. For such a vehicle, radiative heating becomes the dominant aspect in its design. Figure 14.4 illustrates the relative importance of qc and q, for a typical manned reentry vehicle in the earth’s atmosphere; note how rapidly qr dominates the aerodynamic heating of the body as velocities increase above 36,000 ft/s. The

Nose radius = 15 ft Altitude = 200,000 ft

heating rates of a blunt reentry vehicle as a function of flight velocity. (Source: Anderson, Reference 36.j

details of shock-layer radiative heating are interesting and important; however, they are beyond the scope of this book. For a thorough survey of the engineering aspects of shock-layer radiative heat transfer, see Reference 36.

In summary, the aspects of thin shock-layer viscous interaction and high – temperature, chemically reacting and radiative effects distinguish hypersonic flow from the more moderate supersonic regime. Hypersonic flow has been the subject of several complete books; see, for example, References 37 to 41. In particular, see Reference 55 for a modem textbook on the subject.

How Do We Solve the Boundary-Layer Equations?

Examine again the boundary-layer equations given by Equations (17.28) to (17.31). With these equations, are we still in the same “soup” as we are with the complete Navier-Stokes equations, in that the equations are a coupled system of nonlinear partial differential equations for which no general analytical solution has been obtained to date? The answer is partly yes, but with a difference. Because the boundary-layer equations are simpler than the Navier-Stokes equations, especially the boundary – layer у-momentum equation, Equation (17.30), which states that at any axial location along the surface the pressure is constant in the direction normal to the surface, there is more hope of obtaining meaningful solutions for the flow inside a boundary layer. For almost one hundred years, engineers and scientists have “nudged” the boundary layer equations in many different ways, and have come up with reasonable solutions for a number of practical applications. The most complete and authoritative book on such solutions is by Hermann Schlichting (Reference 42).

In Chapters 18 and 19 we will discuss some of these solutions—their technique and some practical results. Solutions of the boundary layer equations can be classi­fied into two groups: (1) classical solutions, some of which date back to 1908, and (2) numerical solutions obtained by modern computational fluid dynamic techniques. In the subsequent chapters, we will show examples from both groups. In addition to this subdivision based on the solution technique, boundary-layer solutions also subdivide on a physical basis into laminar boundary layers and turbulent boundary layers. This subdivision is natural for the reasons discussed in Section 15.2—the nature of turbulent flow is quite different than that of laminar flow. Indeed, for certain types of flow problems, exact solutions have been obtained for laminar boundary layers. To date, no exact solution has been obtained for turbulent boundary layers, because we still do not have a complete understanding of turbulence, and hence all turbulent boundary-layer solutions to date have depended on the use of some type of approximate model of turbulence. These contrasts will become more clear when you read Chapter 18, which deals with laminar boundary layers, and Chapter 19, which deals with turbulent boundary layers. You will see that laminar boundary-layer theory is well in hand, but turbulent boundary-layer theory is not. In some sense, what a

pity that nature always tends toward turbulent flow, and hence the vast majority of practical boundary-layer problems deal with turbulent boundary layers—if it were the other way around, the engineering calculation of skin friction and aerodynamic heating at the surface would be much simpler and more reliable.

Finally, we note what is meant by a “boundary-layer solution.” The solution of Equations (17.28) to (17.31) yields the velocity and temperature profiles throughout the boundary layer. However, the practical information we want is the solution for xw and qw, the surface shear stress and heat transfer, respectively. These are given by

where the subscript w denotes conditions at the wall. Question: Where do values of (du/dy)w and (dT/dy)w come from? Answer: From the velocity and temperature profiles obtained from a solution of the boundary layer equations, where the profiles evaluated at the wall provide both (du/dy)w and (dT/dy)w. Hence in order to obtain the values for xw and q„ along the wall, which are usually considered the engineering results of most importance, we first have to solve the boundary-layer equations for the velocity and temperature profiles throughout the boundary layer, which by themselves usually are of lesser practical interest.

17.5 Summary

Return to the road map given in Figure 17.2, and make certain that you feel at home with the material represented by each box. The highlights of our discussion of bound­ary layers are summarized as follows:

Improved Compressibility Corrections

The importance of accurate compressibility corrections reached new highs during the rapid increase in airplane speeds spurred by World War II. Efforts were made to improve upon the Prandtl-Glauert rule discussed in Section 11.4. Several of the more popular formulas are given below.

The Karman-Tsien rule states

This formula, derived in References 27 and 28, has been widely adopted by the aeronautical industry since World War II.

Laitone’s rule states

This formula is more recent than either the Prandtl-Glauert or the Karman-Tsien rule; it is derived in Reference 29.

These compressibility corrections are compared in Figure 11.4, which also shows experimental data for the Cp variation with Мж at the 0.3-chord location on an NACA 4412 airfoil. Note that the Prandtl-Glauert rule, although the simplest to apply, underpredicts the experimental data, whereas the improved compressibility corrections are clearly more accurate. Recall that the Prandtl-Glauert rule is based on linear theory. In contrast, both the Laitone and Karman-Tsien rules attempt to account for some of the nonlinear aspects of the flow.

The Viscous Flow Energy Equation

The energy equation was derived in Section 2.7, where the first law of thermodynamics was applied to a finite control volume fixed in space. The resulting integral form of the energy equation was given by Equation (2.95), and differential forms were obtained in Equations (2.96) and (2.114). In these equations, the influence of viscous effects was expressed generically by such terms as Gviscous anC^ Viscous’ ft ft recommended that you review Section 2.7 before progressing further.

In the present section, we derive the energy equation for a viscous flow using as our model an infinitesimal moving fluid element. This will be in keeping with

our derivation of the Navier-Stokes equation in Section 15.4, where the infinitesimal element was shown in Figure 15.11. In the process, we obtain explicit expressions for Qviscous and Viscous in terms of the flow-field variables. That is, we once again derive Equation (2.114), except the viscous terms are now displayed in detail.

Consider again the moving fluid element shown in Figure 15.11. To this element, apply the first law of thermodynamics, which states

Rate of change net flux of rate of work

of energy inside = heat into + done on element fluid element element due to pressure and

stress forces on surface

or A = В + C [15.20] where A, B, and C denote the respective terms above.

Let us first evaluate C; that is, let us obtain an expression for the rate of work done on the moving fluid element due to the pressure and stress forces on the surface of the element. (Note that we are neglecting body forces in this derivation.) These surface forces are illustrated in Figure 15.11, which for simplicity shows only the forces in the x direction. Recall from Section 2.7 that the rate of doing work by a force exerted on a moving body is equal to the product of the force and the component of velocity in the direction of the force. Hence, the rate of work done on the moving fluid element by the forces in the x direction shown in Figure 15.11 is simply the a component of velocity и multiplied by the forces; for example, on face abed the rate of work done by zyx dx dz is мг„ dx dz., with similar expressions for the other faces. To emphasize these energy considerations, the moving fluid element is redrawn in Figure 15.12, where the rate of work done on each face by forces in the a direction is shown explicitly. Study this figure carefully, referring frequently to its companion in Figure 15.11, until you feel comfortable with the work terms given in each face. To obtain the net rate of work done on the fluid element by the forces in the a direction, note that forces in the positive x direction do positive work and that forces in the negative x direction do negative work. Hence, comparing the pressure forces on faces adhe and begf in Figure 15.12, the net rate of work done by pressure in the a direction is

‘ / 9 (up) 1 3(up)

up — up H——— ax) ddz =—————- ax dy dz

9 a ) ‘ 9 a

Similarly, the net rate of work done by the shear stresses in the a direction on faces abed and efgh is

Considering all the forces shown in Figure 15.12, the net rate of work done on the moving fluid element is simply

3(up) 3{utxx) 9(m tvv) B(u tzx)

9a + 9a + By + Bz

The above expression considers only forces in the x direction. When the forces in the у and г directions are also included, similar expressions are obtained (draw some pictures and obtain these expressions yourself). In total, the net rate of work done on the moving fluid element is the sum of all contributions in the x, y, and г directions; this is denoted by C in Equation (15.20) and is given by

Note in Equation (15.21) that the term in large parentheses is simply V • p.

Let us turn our attention to В in Equation (15.20), that is, the net flux of heat into the element. This heat flux is due to (1) volumetric heating such as absorption or emission of radiation and (2) heat transfer across the surface due to temperature gradients (i. e., thermal conduction). Let us treat the volumetric heating the same as was done in Section 2.7; that is, define q as the rate of volumetric heat addition per unit mass. Noting that the mass of the moving fluid element in Figure 15.12 is p dx dy dz, we obtain

Volumetric heating of element = pq dx dy dz [1 5.22]

Thermal conduction was discussed in Section 15.3. In Figure 15.12, the heat trans­ferred by thermal conduction into the moving fluid element across face adhe is

qxdydz, and the heat transferred out of the element across face bcgf is [qx + (dqx/dx) dx I dy dz,. Thus, the net heat transferred in the x direction into the fluid element by thermal conduction is

Taking into account heat transfer in the у and z directions across the other faces in Figure 15.12, we obtain

The term В in Equation (15.20) is the sum of Equations (15.22) and (15.23). Also, re­calling that thermal conduction is proportional to temperature gradient, as exemplified by Equation (15.2), we have

Finally, the term A in Equation (15.20) denotes the time rate of change of energy of the fluid element. In Section 2.7, we stated that the energy of a moving fluid per unit mass is the sum of the internal and kinetic energies, for example, e + Vz/2. Since we are following a moving fluid element, the time rate of change of energy per unit mass is given by the substantial derivative (see Section 2.9). Since the mass of the fluid element is p dx dy dz, we have

D ( v.

A = p — I e + — ) dx dy dz

The final form of the energy equation for a viscous flow is obtained by substituting Equations (15.21), (15.24), and (15.25) into Equation (15.20), obtaining

Equation (15.26) is the general energy equation for unsteady, compressible, three­dimensional, viscous flow. Compare Equation (15.26) with Equation (2.105); the viscous terms are now explicitly spelled out in Equation (15.26). [Note that the body force term in Equation (15.26) has been neglected.] Moreover, the normal and shear

stresses that appear in Equation (15.26) can be expressed in terms of the velocity field via Equations (15.5) to (15.10). This substitution will not be made here because the resulting equation would simply occupy too much space.

Reflect on the viscous flow equations obtained in this chapter—the Navier-Stokes equations given by Equations (15.19a to c) and the energy equation given by Equa­tion (15.26). These equations are obviously more complex than the inviscid flow equations dealt with in previous chapters. This underscores the fact that viscous flows are inherently more difficult to analyze than inviscid flows. This is why, in the study of aerodynamics, the student is first introduced to the concepts associated with inviscid flow. Moreover, this is why we attempt to model a number of practical aerodynamic problems in real life as inviscid flows—simply to allow a reasonable analysis of such flows. However, there are many aerodynamic problems, especially those involving the prediction of drag and flow separation, which must take into ac­count viscous effects. For the analysis of such problems, the basic equations derived in this chapter form a starting point.

Question: What is the form of the continuity equation for a viscous flow? To answer this question, review the derivation of the continuity equation in Section 2.4. You will note that the consideration of the viscous or inviscid nature of the flow never enters the derivation—the continuity equation is simply a statement that mass is conserved, which is independent of whether the flow is viscous or inviscid. Hence, Equation (2.52) holds in general.

Reference Temperature Method for Turbulent Flow

The reference temperature method discussed in Section 18.4 for laminar boundary layers can be applied to turbulent boundary layers as well. With the reference tem­perature T* given by Equation (18.53), the incompressible turbulent flat plate result for C f given by Equation (19.2) can be modified for compressible turbulent flow as

0.074 (Re*)1/5

and we will assume that the reference temperature for this case is the same as given in Example

18.2. Hence, from Example 18.2, we have

Re* = 3.754 x 107 and p* = 0.574 kg/m3

From Equation (19.3) we have

_ 0.074 0.074

f ~ (Rer*)‘/5 (3.754 x 107)1’5 “ 226 X 10

From Equation (19.4),

Df = p*u)SC*f = |(0.574)( 1000)2(40)(2.26 x КГ3) = 25,945 N

Comparing this answer with that obtained in Example 19.1ft, we find a 27 percent discrepancy between the two methods of calculations. This is not surprising. It simply points out the great uncertainty in making calculations of turbulent skin friction.

Introduction to Numerical. Techniques for Nonlinear. Supersonic Flow

Regarding computing as a straightforward routine, some theoreticians still tend to underestimate its intellec­tual value and challenge, while practitioners often ignore its accuracy and overrate its validity.

С. K. Chu, 1978 Columbia University

13.1 Introduction: Philosophy of

Computational Fluid Dynamics

The above quotation underscores the phenomenally rapid increase in computer power available to engineers and scientists during the two decades between 1960 and 1980. This explosion in computer capability is still going on, with no specific limits in sight. As a result, an entirely new discipline in aerodynamics has evolved over the past three decades, namely, computational fluid dynamics (CFD). CFD is a new “third dimension” in aerodynamics, complementing the previous dimensions of both pure experiment and pure theory. It allows us to obtain answers to fluid dynamic problems which heretofore were intractable by classical analytical methods. Consequently, CFD is revolutionizing the airplane design process, and in many ways is modifying the way we conduct modem aeronautical research and development. For these reasons, every modem student of aerodynamics should be aware of the overall philosophy of

CFD, because you are bound to be affected by it to some greater or lesser degree in your education and professional life.

The philosophy of computational fluid dynamics was introduced in Section 2.17, where it was compared with the theoretical approach leading to closed-form analytical solutions. Please stop here, return to Section 2.17, and re-read the material presented there; now that you have progressed this far and have seen a number of analytical solutions for both incompressible and compressible flows in the proceeding chapters, the philosophy discussed in Section 2.17 will mean much more to you. Do this now, because the present chapter almost exclusively deals with numerical solutions with reference to Section 2.17.2 whereas Chapters 3-12 have dealt almost exclusively with analytical solutions with reference to Section 2.17.1.

In the present chapter we will experience the true essence of computational fluid dynamics for the first time in this book; we will actually see what is meant by the definition of CFD given in Section 2.17.2 as “the art of replacing the integrals or the partial derivatives (as the case may be) in the governing equations of fluid motion with discretized algebraic forms, which in turn are solved to obtain numbers for the flow field values at discrete points in time and/or space.” However, because modem CFD is such a sophisticated discipline that is usually the subject of graduate level studies, and which rests squarely on the foundations of applied mathematics, we can only hope to give you an elementary treatment in the present chapter, but a treatment significant enough to represent some of the essence of CFD. For your next step in learning CFD beyond the present book, you are recommended to read Anderson, Computational Fluid Dynamics: The Basics with Applications (Reference 64), which the author has written to help undergraduates understand the nature of CFD before going on to more advanced studies of the discipline.

The purpose of this chapter is to provide an introduction to some of the basic ideas of CFD as applied to inviscid supersonic flows. More details are given in Reference 21. Because CFD has developed so rapidly in recent years, we can only scratch the surface here. Indeed, the present chapter is intended to give you only some basic background as well as the incentive to pursue the subject further in the modem literature.

The road map for this chapter is given in Figure 13.1. We begin by introducing the classical method of characteristics—a numerical technique that has been available in aerodynamics since 1929, but which had to wait on the modern computer for practical, everyday implementation. For this reason, the author classifies the method of characteristics under the general heading of numerical techniques, although others may prefer to list it under a more classical heading. We also show how the method of characteristics is applied to design the divergent contour of a supersonic nozzle. Then we move to a discussion of the finite-difference approach, which we will use to illustrate the application of CFD to nozzle flows and the flow over a supersonic blunt body.

In contrast to the linearized solutions discussed in Chapters 11 and 12, CFD represents numerical solutions to the exact nonlinear governing equations, that is, the equations without simplifying assumptions such as small perturbations, and which apply to all speed regimes, transonic and hypersonic as well as subsonic and super-

(a)

Figure 13.2 Grid points.

sonic. Although numerical roundoff and truncation errors are always present in any numerical representation of the governing equations, we still think of CFD solutions as being “exact solutions.”

Both the method of characteristics and finite-difference methods have one thing in common: They represent a continuous flow field by a series of distinct grid points in space, as shown in Figure 13.2. The flow-field properties (и, v, p, T, etc.) are calculated at each one of these grid points. The mesh generated by these grid points is generally skewed for the method of characteristics, as shown in Figure 13.2a, but

is usually rectangular for finite-difference solutions, as shown in Figure 13.2b. We will soon appreciate why these different meshes occur.

Interim Summary

In this section, we have studied incompressible Couette flow. Although it is a some­what academic flow, it has all the trappings of many practical viscous flow problems, with the added advantage of lending itself to a simple, straightforward solution. We have taken this advantage, and have discussed incompressible Couette flow in great detail. Our major purpose in this discussion is to make the reader familiar with many concepts used in general in the analysis of viscous flows without clouding the picture with more fluid dynamic complexities. In the context of our study of Couette flow, we have one additional question to address, namely, What is the effect of compressibility? This question is addressed in the next section.

Consider the geometry sketched in Figure 16.2. The velocity of the upper plate is 200 ft/s, and the two plates are separated by a distance of 0.01 in. The fluid between the plates is air. Assume incompressible flow. The temperature of both plates is the standard sea level value of 519° R.

(a) Calculate the velocity in the middle of the flow.

(,b) Calculate the shear stress.

(c) Calculate the maximum temperature in the flow.

(d) Calculate the heat transfer to either wall.

(e) If the lower wall is suddenly made adiabatic, calculate its temperature.

Solution

Assume that д is constant throughout the flow, and that it is equal to its value of 3.7373 x 10~7 slug/ft/s at the standard sea level temperature of 519°R.

(a) From Equation (16.6),

difference between compressible and incompressible flows. With all this in mind, Equation (16.3), repeated below

can be written as

The temperature variation of ц is accurately given by Sutherland’s law, Equation

(15.3) , for the temperature range of interest in this book. Hence, from Equation (15.3) and recalling that it is written in the International System of Units, we have

The solution for compressible Couette flow requires a numerical solution of Equation (16.62). Note that, with p. and к as variables, Equation (16.62) is a nonlinear differential equation, and for the conditions stated, it does not have a neat, closed – form, analytic solution. Recognizing the need for a numerical solution, let us write Equation (16.62) in terms of the ordinary differential equation that it really is. (Recall that we have been using the partial differential notation only as a carry-over from the Navier-Stokes equations and to make the equations for our study of Couette flow look more familiar when treating the two-dimensional and three-dimensional viscous flows discussed in Chapters 17 to 20—just a pedagogical ploy on our part):

Equation (16.64) must be solved between у = 0, where T = 7„ , and у = D, where T = Te. Note that the boundary conditions must be satisfied at two different locations in the flow, namely, at у = 0 and у = D; this is called a two-point boundary value problem. We present two approaches to the numerical solution of this problem. Both approaches are used for the solutions of more complex viscous flows to be discussed in Chapters 17 through 20, and that is why they are presented here in the context of Couette flow—simply to “break the ice” for our subsequent discussions.