Category When Is A Flow Compressible?

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.

Shock-Expansion Theory: Applications to Supersonic Airfoils

Consider a flat plate of length c at an angle of attack a in a supersonic flow, as sketched in Figure 9.33. On the top surface, the flow is turned away from itself; hence, an expansion wave occurs at the leading edge, and the pressure on the top surface p2 is less than the freestream pressure p2 < p. At the trailing edge, the flow must return to approximately (but not precisely) the freestream direction. Here, the flow is turned back into itself, and consequently a shock wave occurs at the trailing edge. On the bottom surface, the flow is turned into itself; an oblique shock wave occurs at the leading edge, and the pressure on the bottom surface p2 is greater than the freestream pressure p2 > p. At the trailing edge, the flow is turned into approximately (but not precisely) the freestream direction by means of an expansion wave. Examining Figure 9.33, note that the top and bottom surfaces of the flat plate experience uniform pressure distribution of p2 and p2, respectively, and that p2 > p2. This creates a

net pressure imbalance which generates the resultant aerodynamic force R, shown in Figure 9.33. Indeed, for a unit span, the resultant force and its components, lift and drag, per unit span are

R’ = (рз – P2)c [9.46]

L’ — (p3 — p2)c cos a [9.47]

D’ = {рз — p2)c sin a [9.48]

In Equations (9.47) and (9.48), рз is calculated from oblique shock properties (Section 9.2), and p2 is calculated from expansion-wave properties (Section 9.6). Moreover, these are exact calculations; no approximations have been made. The inviscid, su­personic flow over a flat plate at angle of attack is exactly given by the combination of shock and expansion waves sketched in Figure 9.33.

The flat-plate case given above is the simplest example of a general technique called shock-expansion theory. Whenever we have a body made up of straight-line segments and the deflection angles are small enough so that no detached shock waves occur, the flow over the body goes through a series of distinct oblique shock and expansion waves, and the pressure distribution on the surface (hence the lift and drag) can be obtained exactly from both the shock – and expansion-wave theories discussed in this chapter.

As another example of the application of shock-expansion theory, consider the diamond-shape airfoil in Figure 9.34. Assume the airfoil is at 0° angle of attack. The supersonic flow over the airfoil is first compressed and deflected through the angle є by the oblique shock wave at the leading edge. At midchord, the flow is expanded through an angle 2e, creating an expansion wave. At the trailing edge, the flow is turned back to the freestream direction through another oblique shock. The pressure distributions on the front and back faces of the airfoil are sketched in Figure 9.34;

note that the pressures on faces a and c are uniform and equal to pi and that the pressures on faces b and d are also uniform but equal to pi, where pi < pi – In the lift direction, perpendicular to the freestream, the pressure distributions on the top and bottom faces exactly cancel (i. e., L’ = 0). In contrast, in the drag direction, parallel to the freestream, the pressure on the front faces a and c is larger than on the back faces b and d, and this results in a finite drag. To calculate this drag (per unit span), consider the geometry of the diamond airfoil in Figure 9.34, where l is the length of each face and t is the airfoil thickness. Then,

In Equation (9.49), P2 is calculated from oblique shock theory, and pi is obtained from expansion-wave theory. Moreover, these pressures are the exact values for supersonic, inviscid flow over the diamond airfoil.

At this stage, it is worthwhile to recall our discussion in Section 1.5 concerning the source of aerodynamic force on a body. In particular, examine Equations (1.1),

(1.2) , (1.7), and (1.8). These equations give the means to calculate L’ and D’ from the pressure and shear stress distributions over the surface of a body of general shape. The results of the present section, namely, Equations (9.47) and (9.48) for a flat plate and Equation (9.49) for the diamond airfoil, are simply specialized results from the more general formulas given in Section 1.5. However, rather than formally going through the integration indicated in Equations (1.7) and (1.8), we obtained our results for the simple bodies in Figures 9.33 and 9.34 in a more direct fashion.

The results of this section illustrate a very important aspect of inviscid, supersonic flow. Note that Equation (9.48) for the flat plate and Equation (9.49) for the diamond airfoil predict a finite drag for these two-dimensional profiles. This is in direct contrast

to our results for two-dimensional bodies in a low-speed, incompressible flow, as discussed in Chapters 3 and 4, where the drag was theoretically zero. That is, in supersonic flow, d’Alembert’s paradox does not occur. In a supersonic, inviscid flow, the drag per unit span on a two-dimensional body is finite. This new source of drag is called wave drag, and it represents a serious consideration in the design of all supersonic airfoils. The existence of wave drag is inherently related to the increase in entropy and consequently to the loss of total pressure across the oblique shock waves created by the airfoil.

Finally, the results of this section represent a merger of both the left – and right- hand branches of our road map shown in Figure 9.5. As such, it brings us to a logical conclusion of our discussion of oblique waves in supersonic flows.

Calculate the lift and drag coefficiens for a flat plate at a 5° angle of attack in a Mach 3 flow. | Example 9.1 О Solution

Refer to Figure 9.33. First, calculate p2/pi on the upper surface. From Equation (9.43),

£i=Toi/£oi = 36ЛЗ= 0 668 P і P і / P2 55

Next, calculate рз/рі on the bottom surface. From the в-ji-M diagram (Figure 9.7), for Mi = 3 and в = 5°, p = 23.1 °. Hence,

M„,| = Mt sin fi = 3 sin 23.1° = 1.177

From Appendix B, for M„ і = 1.177, ръ/р = 1.458 (nearest entry).

Returning to Equation (9.47), we have

The lift coefficient is obtained from

From Equation (9.48),

D’ = (p} — pi )c sin a


seclusion, working on various improved methods of farming, including silkworm cultivation. Ernst’s mother, on the other hand, came from a family of lawyers and doctors and brought with her a love of poetry and music. Ernst seemed to thrive in this family atmosphere. Until the age of 14, his education came exclusively from instruction by his father, who read extensively in the Greek and Latin classics. In 1853, Mach entered public school, where he became interested in the world of science. He went on to obtain a Ph. D. in physics in 1860 at the University of Vienna, writing his dissertation on electrical discharge and induction. In 1864, he became a full professor of mathematics at the University of Graz and was given the title of Professor of Physics in 1866. Mach’s work during this period centered on optics—a subject which was to interest him for the rest of his life. The year 1867 was important for Mach—during that year he married, and he also became a professor of experimental physics at the University of Prague, a position he held for the next 28 years. While at Prague, Mach published over 100 technical papers—work which was to constitute the bulk of his technical contributions.

Mach’s contribution to supersonic aerodynamics involves a series of experiments covering the period from 1873 to 1893. In collaboration with his son, Ludwig, Mach studied the flow over supersonic projectiles, as well as the propagation of sound waves and shock waves. His work included the flow fields associated with meteorites, explosions, and gas jets. The main experimental data were photographic results. Mach combined his interest in optics and supersonic motion by designing several photographic techniques for making shock waves in air visible. He was the first to use the schlieren system in aerodynamics; this system senses density gradients and allows shock waves to appear on screens or photographic negatives. He also devised an interferometric technique which senses directly the change in density in a flow. A pattern of alternate dark and light bands are set up on a screen by the superposition of light rays passing through regions of different density. Shock waves are visible as a shift in this pattern along the shock. Mach’s optical device still perpetuates today in the form of the Mach-Zehnder interferometer, an instrument present in many aerodynamic laboratories. Mach’s major contributions in supersonic aerodynamics are contained in a paper given to the Academy of Sciences in Vienna in 1887. Here, for the first time in history, Mach shows a photograph of a weak wave on a slender cone moving at supersonic speed, and he demonstrates that the angle ц between this wave and the direction of flight is given by sin д = a/V. This angle was later denoted as the Mach angle by Prandtl and his colleagues after their work on shock and expansion waves in 1907 and 1908. Also, Mach was the first person to point out the discontinuous and marked changes in a flow field as the ratio Vla changes from below 1 to above 1.

It is interesting to note that the ratio V/a was not denoted as Mach number by Mach himself. Rather, the term “Mach number” was coined by the Swiss engineer Jacob Ackeret in his inaugural lecture in 1929 as Privatdozent at the Eidgenossiche Technische Hochschule in Zurich. Hence, the term “Mach number” is of fairly recent usage, not being introduced into the English literature until the mid-1930s.

In 1895, the University of Vienna established the Ernst Mach chair in the philos­ophy of inductive sciences. Mach moved to Vienna to occupy this chair. In 1897 he suffered a stroke which paralyzed the right side of his body. Although he eventually

partially recovered, he officially retired in 1901. From that time until his death on February 19,1916 near Munich, Mach continued to be an active thinker, lecturer, and writer.

In our time, Mach is most remembered for his early experiments on supersonic flow and, of course, through the Mach number itself. However, Mach’s contempo­raries, as well as Mach himself, viewed him more as a philosopher and historian of science. Coming at the end of the nineteenth century, when most physicists felt com­fortable with newtonian mechanics, and many believed that virtually all was known about physics, Mach’s outlook on science is summarized by the following passage from his book Die Mechanik:

The most important result of our reflections is that precisely the apparently simplest mechanical theorems are of a very complicated nature; that they are founded on incomplete experiences, even on experiences that never can be fully completed; that in view of the tolerable stability of our environment they are, in fact, practically safeguarded to serve as the foundation of mathematical deduction; but that they by no means themselves can be regarded as mathematically established truths, but only as theorems that not only admit of constant control by experience but actually require it.

In other words, Mach was a staunch experimentalist who believed that the established laws of nature were simply theories and that only observations that are apparent to the senses are the fundamental truth. In particular, Mach could not accept the elementary ideas of atomic theory or the basis of relativity, both of which were beginning to surface during Mach’s later years and, of course, were to form the basis of twentieth – century modern physics. As a result, Mach’s philosophy did not earn him favor with most of the important physicists of his day. Indeed, at the time of his death, Mach was planning to write a book pointing out the flaws of Einstein’s theory of relativity.

Although Mach’s philosophy was controversial, he was respected for being a thinker. In fact, in spite of Mach’s critical outlook on the theory of relativity, Albert Einstein had the following to say in the year of Mach’s death: “I even believe that those who consider themselves to be adversaries of Mach scarcely know how much of Mach’s outlook they have, so to speak, adsorbed with their mother’s milk.”

Hopefully, this section has given you a new dimension to think about whenever you encounter the term “Mach number.” Maybe you will pause now and then to reflect on the man himself and to appreciate that the term “Mach number” is in honor of a man who devoted his life to experimental physics, but who at the same time was bold enough to view the physical world through the eyes of a self-styled philosopher.

9,10 Summary

The road map given in Figure 9.5 illustrates the flow of our discussion on oblique waves in supersonic flow. Review this road map, and make certain that you are familiar with all the ideas and results that are represented in Figure 9.5.

Some of the more important results are summarized as follows:

2. Consider an oblique shock wave with a wave angle of 30° in a Mach 4 flow. The upstream pressure and temperature are 2.65 x 104 N/m2 and 223.3 K, respec­tively (corresponding to a standard altitude of 10,000 m). Calculate the pressure, temperature, Mach number, total pressure, and total temperature behind the wave and the entropy increase across the wave.

3. Equation (8.80) does not hold for an oblique shock wave, and hence the column in Appendix В labeled pop/p cannot be used, in conjunction with the normal component of the upstream Mach number, to obtain the total pressure behind an oblique shock wave. On the other hand, the column labeled pop/po. i can be used for an oblique shock wave, using МП:. Explain why all this is so.

4. Consider an oblique shock wave with a wave angle of 36.87°. The upstream flow is given by Mi =3 and p — 1 atm. Calculate the total pressure behind the shock using

(a) ро г /Pd. і from Appendix В (the correct way)

(b) po i/pi from Appendix В (the incorrect way)

Compare the results.

5. Consider the flow over a 22.2° half-angle wedge. If Mi = 2.5, p — 1 atm, and 7) = 300 K, calculate the wave angle and p2, T2, and M2.

6. Consider a flat plate at an angle of attack a to a Mach 2.4 airflow at 1 atm pressure. What is the maximum pressure that can occur on the plate surface and still have an attached shock wave at the leading edge? At what value of a does this occur?

7. A 30.2° half-angle wedge is inserted into a freestream with Mx — 3.5 and Poo = 0.5 atm. A Pitot tube is located above the wedge surface and behind the shock wave. Calculate the magnitude of the pressure sensed by the Pitot tube.

8. Consider a Mach 4 airflow at a pressure of 1 atm. We wish to slow this flow to subsonic speed through a system of shock waves with as small a loss in total pressure as possible. Compare the loss in total pressure for the following three shock systems:

(a) A single normal shock wave

(b) An oblique shock with a deflection angle of 25.3°, followed by a normal shock

(c) An oblique shock with a deflection angle of 25.3°, followed by a second oblique shock of deflection angle of 20°, followed by a normal shock

From the results of (a), (b), and (c), what can you induce about the efficiency of the various shock systems?

9. Consider an oblique shock generated at a compression corner with a deflection angle 9 = 18.2°. A straight horizontal wall is present above the comer, as shown in Figure 9.17. If the upstream flow has the properties Mi = 3.2, p = 1 atm and T = 520°R, calculate М3, p3, and T2 behind the reflected shock from the upper wall. Also, obtain the angle Ф which the reflected shock makes with the upper wall.

10. Consider the supersonic flow over an expansion comer, such as given in Figure 9.23. The deflection angle 9 = 23.38°. If the flow upstream of the corner is

given by Mi =2, p = 0.7 atm, 7j = 630°R, calculate M2, p2, T2, p2, P0.2, and 7І) 2 downstream of the comer. Also, obtain the angles the forward and rearward Mach lines make with respect to the upstream direction.

11. A supersonic flow at M] = 1.58 and p — 1 atm expands around a sharp corner. If the pressure downstream of the corner is 0.1306 atm, calculate the deflection angle of the corner.

12. A supersonic flow at M1 = 3, 7j = 285 K, and p = 1 atm is deflected upward through a compression corner with 9 = 30.6° and then is subsequently expanded around a corner of the same angle such that the flow direction is the same as its original direction. Calculate М3, p2, and 7) downstream of the expansion comer. Since the resulting flow is in the same direction as the original flow, would you expect М3 = Mi, p = p, and 7з = 7У? Explain.

13. Consider an infinitely thin flat plate at an angle of attack a in a Mach 2.6 flow. Calculate the lift and wave-drag coefficients for

(a) a = 5° (b) a = 15° (c) a = 30°

{Note: Save the results of this problem for use in Chapter 12.)

14. Consider a diamond-wedge airfoil such as shown in Figure 9.34, with a half-angle є = 10°. The airfoil is at an angle of attack a = 15° to a Mach 3 freestream. Calculate the lift and wave-drag coefficients for the airfoil.

15. Consider sonic flow. Calculate the maximum deflection angle through which this flow can be expanded via a centered expansion wave.

16. Consider a circular cylinder (oriented with its axis perpendicular to the flow) and a symmetric diamond-wedge airfoil with a half-angle of 5° at zero angle of attack; both bodies are in the same Mach 5 freestream. The thickness of the airfoil and the diameter of the cylinder are the same. The drag coefficient (based on projected frontal area) of the cylinder is 4/3. Calculate the ratio of the cylinder drag to the diamond airfoil drag. What does this say about the aerodynamic performance of a blunt body compared to a sharp-nosed slender body in supersonic flow?

17. 9.17 Consider the supersonic flow over a flat plate at angle of attack, as sketched in Figure 9.33. As stated in Section 9.7, the flow direction downstream of the trailing edge of the plate, behind the trailing edge shock and expansion waves, is not precisely in the free stream direction. Why? Outline a method to calculate the strengths of the trailing edge shock and expansion waves, and the direction of the flow downstream of the trailing edge.

Newtonian Theory

Return to Figure 14.1; note how close the shock wave lies to the body surface. This figure is redrawn in Figure 14.5 with the streamlines added to the sketch. When viewed from afar, the straight, horizontal streamlines in the freestream appear to almost impact the body, and then move tangentially along the body. Return to Figure 1.1, which illustrates Isaac Newton’s model for fluid flow, and compare it with the hypersonic flow field shown in Figure 14.5; they have certain distinct similarities. (Also, review the discussion surrounding Figure 1.1 before progressing further.) Indeed, the thin shock layers around hypersonic bodies are the closest example in fluid mechanics to Newton’s model. Therefore, we might expect that results based on Newton’s model would have some applicability in hypersonic flows. This is indeed the case; newtonian theory is used frequently to estimate the pressure distribution over the surface of a hypersonic body. The purpose of this section is to derive the famous newtonian sine – squared law first mentioned in Section 1.1 and to show how it is applied to hypersonic flows.

Consider a surface inclined at the angle в to the freestream, as sketched in Fig­ure 14.6. According to the newtonian model, the flow consists of a large number of individual particles which impact the surface and then move tangentially to the surface. During collision with the surface, the particles lose their component of mo­mentum normal to the surface, but the tangential component is preserved. The time rate of change of the normal component of momentum equals the force exerted on the surface by the particle impacts. To quantify this model, examine Figure 14.6. The component of the freestream velocity normal to the surface is Too sin в. If the area of the surface is A, the mass flow incident on the surface is p(A sin 9)V0O. Hence, the

Figure 14.5 Streamlines in a hypersonic flow.

time rate of change of momentum is

Mass flow x change in normal component of velocity

or (PccVccA sin0)(Voosin(9) = PocV^A sin2#

In turn, from Newton’s second law, the force on the surface is

IV = PocV^A sin2 в [14.1]

This force acts along the same line as the time rate of change of momentum (i. e., normal to the surface), as sketched in Figure 14.6. From Equation (14.1), the normal force per unit area is

= PccV^ sin20 [14.2]

Let us now interpret the physical meaning of the normal force per unit area in Equa­tion (14.2), N/A, in terms of our modem knowledge of aerodynamics. Newton’s model assumes a stream of individual particles all moving in straight, parallel paths toward the surface; that is, the particles have a completely directed, rectilinear motion. There is no random motion of the particles—it is simply a stream of particles such as pellets from a shotgun. In terms of our modern concepts, we know that a moving gas has molecular motion that is a composite of random motion of the molecules as well as a directed motion. Moreover, we know that the freestream static pressure рж is simply a measure of the purely random motion of the molecules. Therefore, when the purely directed motion of the particles in Newton’s model results in the normal force per unit area, N/A in Equation (14.2), this normal force per unit area must be construed as the pressure difference above px, namely, p — p^ on the surface. Hence, Equation (14.2) becomes

P – Poo = Poo sin2 в [ 14.3]

Equation (14.3) can be written in terms of the pressure coefficient Cp — (p — Poe)/ poeV£>’ as follows

= 2 sin2#

Equation (14.4) is Newton’s sine-squared law; it states that the pressure coefficient is proportional to the sine square of the angle between a tangent to the surface and the direction of the freestream. This angle в is illustrated in Figure 14.7. Frequently, the results of newtonian theory are expressed in terms of the angle between a normal to the surface and the freestream direction, denoted by ф as shown in Figure 14.7. In terms of ф, Equation (14.4) becomes

Cp — 2 cos2 ф

which is an equally valid expression of newtonian theory.

Consider the blunt body sketched in Figure 14.7. Clearly, the maximum pressure, hence the maximum value of Cp, occurs at the stagnation point, where в = тг/2 and ф — 0. Equation (14.4) predicts Cp = 2 at the stagnation point. Contrast this hypersonic result with the result obtained for incompressible flow theory in Chapter 3, where Cp = 1 at a stagnation point. Indeed, the stagnation pressure coefficient increases continuously from 1.0 at M= 0 to 1.28 at Мж = 1.0 to 1.86 for у = 1.4 as Mqo —► °o. (Prove this to yourself.)

The result that the maximum pressure coefficient approaches 2 at Moo —► oo can be obtained independently from the one-dimensional momentum equation, namely, Equation (8.6). Consider a normal shock wave at hypersonic speeds, as sketched in Figure 14.8. For this flow, Equation (8.6) gives

Poo + PooV^ = P2 + P2V2 [14.6]

Recall that across a normal shock wave the flow velocity decreases, V2 < Fool indeed, the flow behind the normal shock is subsonic. This change becomes more severe as Mco increases. Hence, at hypersonic speeds, we can assume that (pж V^) УР (p2

As stated above, the result that Cp = 2 at a stagnation point is a limiting value as Moo —*■ oo. For large but finite Mach numbers, the value of Cp at a stagnation point is less than 2. Return again to the blunt body shown in Figure 14.7. Considering the distribution of Cp as a function of distance 5 along the surface, the largest value of Cp will occur at the stagnation point. Denote the stagnation point value of Cp by Cp, max, as shown in Figure 14.7. Cp, max for a given Mx can be readily calculated from normal shock-wave theory. [If у = 1.4, then Cp, max can be obtained from Рол/Pi — Рол/Poo, tabulated in Appendix B. Recall from Equation (11.22) that Cp, max = (2/уМ^0)(рол/Poo ~ 1) ] Downstream of the stagnation point, Cp can be assumed to follow the sine-squared variation predicted by newtonian theory; that is,

Equation (14.7) is called the modified newtonian law. For the calculation of the Cp dis­tribution around blunt bodies, Equation (14.7) is more accurate than Equation (14.4).

Return to Figure 13.14, which gives the numerical results for the pressure distri­butions around a blunt, parabolic cylinder at = 4 and 8. The open symbols in this figure represent the results of modified newtonian theory, namely, Equation (14.7). For this two-dimensional body, modified newtonian theory is reasonably accurate only in the nose region, although the comparison improves at the higher Mach num­bers. It is generally true that newtonian theory is more accurate at larger values of both Moo and в. The case for an axisymmetric body, a paraboloid at Moo = 4, is given in Figure 14.9. Here, although Moo is relatively low, the agreement between the time- dependent numerical solution (see Chapter 13) and newtonian theory is much better.

paraboloid, = 4. Comparison of modified newtonian theory and time-dependent finite-difference calculations.

It is generally true that newtonian theory works better for three-dimensional bodies. In general, the modified newtonian law, Equation (14.7), is sufficiently accurate that it is used very frequently in the preliminary design of hypersonic vehicles. Indeed, extensive computer codes have been developed to apply Equation (14.7) to three­dimensional hypersonic bodies of general shape. Therefore, we can be thankful to Isaac Newton for supplying us with a law which holds reasonably well at hypersonic speeds, although such an application most likely never crossed his mind. Neverthe­less, it is fitting that three centuries later, Newton’s fluid mechanics has finally found a reasonable application.