Category When Is A Flow Compressible?

The Boundary-Layer Equations

For the remainder of this chapter, we consider two-dimensional, steady flow. The nondimensionalized form of the x-momentum equation (one of the Navier-Stokes equations) was developed in Section 15.6 and was given by Equation (15.29);

Let us now reduce Equation (15.29) to an approximate form which holds reasonably well within a boundary layer.

Consider the boundary layer along a flat plate of length c as sketched in Fig­ure 17.7. The basic assumption of boundary-layer theory is that a boundary layer is

very thin in comparison with the scale of the body; that is,

Consider the continuity equation for a steady, two-dimensional flow,

d(pu) d(pv) __ dx dy

In terms of the nondimensional variables defined in Section 15.6, Equation (17.16) becomes

d(p’u’) Э(рУ) dx1 dy’

Because u’ varies from 0 at the wall to 1 at the edge of the boundary layer, let us say that u’ is of the order of magnitude equal to 1, symbolized by 0(1). Similarly, p’ = 0(1). Also, since x varies from 0 to c, x’ = 0(1). However, since у varies from 0 to <5, where 8 <£ c, then y’ is of the smaller order of magnitude, denoted by у’ = О(8/с). Without loss of generality, we can assume that c is a unit length. Therefore, y’ = 0(8). Putting these orders of magnitude in Equation (17.17), we have

!в + Ш=0 [17.18]

0(1) 0(8)

Hence, from Equation (17.18), clearly v’ must be of an order of magnitude equal to 8; that is, v’ = 0(8). Now examine the order of magnitude of the terms in Equation (15.29). We have

Let us now introduce another assumption of boundary-layer theory, namely, the Reynolds number is large, indeed, large enough such that

Then, Equation (17.19) becomes

From Equation (17.26), we see that dp’/dy’ = 0(8) or smaller, assuming that yM^ = 0(1). Since 8 is very small, this implies that dp’/dy’ is very small. There­fore, from the у-momentum equation specialized to a boundary layer, we have

Equation (17.26a) is important; it states that at a given x station, the pressure is constant through the boundary layer in a direction normal to the surface. This implies that the pressure distribution at the outer edge of the boundary layer is impressed directly to the surface without change. Hence, throughout the boundary layer, p — P(x) = pe (x).

It is interesting to note that if is very large, as in the case of large hypersonic Mach numbers, then from Equation (17.26) dp’/dy’ does not have to be small. For example, if Mqo were large enough such that 1/yM^ = 0(8), then dp’/dy’ could be as large as 0(1), and Equation (17.26) would still be satisfied. Thus, for very large hypersonic Mach numbers, the assumption that p is constant in the normal direction through a boundary layer is not always valid.

Consider the general energy equation given by Equation (15.26). The nondi­mensional form of this equation for two-dimensional, steady flow is given in Equa­tion (15.33). Inserting e = h — p/p into this equation, subtracting the momentum equation multiplied by velocity, and performing an order-of-magnitude analysis sim­ilar to those above, we can obtain the boundary-layer energy equation as

The details are left to you.

In summary, by making the combined assumptions of 8 <SC c and Re > 1/82, the complete Navier-Stokes equations derived in Chapter 15 can be reduced to simpler forms which apply to a boundary layer. These boundary-layer equations are

[17.30]

[17.31]

Note that, as in the case of the Navier-Stokes equations, the boundary-layer equations are nonlinear. However, the boundary-layer equations are simpler, and therefore are more readily solved. Also, since p = p,. (x), the pressure gradient expressed as dp/dx in Equations (17.23) and (17.27) is reexpressed as dpe/dx in Equations (17.29) and

(17.31) . In the above equations, the unknowns are u, v, p, and h p is known from p = pe (x), and д and к are properties of the fluid which vary with temperature. To complete the system, we have

and h=cpT [17.33]

Hence, Equations (17.28), (17.29), and (17.31) to (17.33) are five equations for the five unknowns, и, n, p, T, and h.

The boundary conditions for the above equations are as follows:

At the wall: у = 0 и = 0 v = О T — Tw

At the boundary-layer edge: у —»• oo и —*■ ue T —*■ Te

Note that since the boundary-layer thickness is not known a priori, the boundary con­dition at the edge of the boundary layer is given at large y, essentially у approaching infinity.

Prandtl-Glauert Compressibility orrection

The aerodynamic theory for incompressible flow over thin airfoils at small angles of attack was presented in Chapter 4. For aircraft of the period 1903-1940, such theory

was adequate for predicting airfoil properties. However, with the rapid evolution of high-power reciprocating engines spurred by World war II, the velocities of military fighter planes began to push close to 450 mi/h. Then, with the advent of the first operational jet-propelled airplanes in 1944 (the German Me 262), flight velocities took a sudden spurt into the 550 mi/h range and faster. As a result, the incompressible flow theory of Chapter 4 was no longer applicable to such aircraft; rather, high-speed airfoil theory had to deal with compressible flow. Because a vast bulk of data and experience had been collected over the years in low-speed aerodynamics, and because there was no desire to totally discard such data, the natural approach to high-speed subsonic aerodynamics was to search for methods that would allow relatively simple corrections to existing incompressible flow results which would approximately take into account the effects of compressibility. Such methods are called compressibility corrections. The first, and most widely known of these corrections is the Prandtl – Glauert compressibility correction, to be derived in this section. The Prandtl-Glauert method is based on the linearized perturbation velocity potential equation given by Equation (11.18). Therefore, it is limited to thin airfoils at small angles of attack. Moreover, it is purely a subsonic theory and begins to give inappropriate results at values of Moo = 0.7 and above.

Consider the subsonic, compressible, inviscid flow over the airfoil sketched in Figure 11.3. The shape of the airfoil is given by у = f(x). Assume that the airfoil is thin and that the angle of attack is small; in such a case, the flow is reasonably approximated by Equation (11.18). Define

P2 – і – мі

so that Equation (11.18) can be written as

2д2ф д2ф

Let us transform the independent variables x and у into a new space, £ and ij, such that

у

Ко

%=x [11.36a]

П = РУ [11.36b]

Moreover, in this transformed space, consider a new velocity potential ф such that

</>(£, rj) = Рф(х, у) [11.36c]

To recast Equation (11.35) in terms of the transformed variables, recall the chain rule of partial differentiation; that is,

Зф 3ф 3§ 3ф dt]

дх 3£ dx dr] dx

dip d0 3§ 30 dr]

dy Э§ 3у dr] dy

From Equations (11.36a and b), we have

Differentiating Equation (11.41) with respect to x (again using the chain rule), we obtain

З20 1 d2ф

3×2 “ W

Differentiating Equation (11.42) with respect to y, we hnd that the result is

Substitute Equations (11.43) and (11.44) into (11.35):

Examine Equation (11.45)—it should look familiar. Indeed, Equation (11.45) is Laplace’s equation. Recall from Chapter 3 that Laplace’s equation is the governing relation for incompressible flow. Hence, starting with a subsonic compressible flow in physical (x, у) space where the flow is represented by ф(х, у) obtained from Equation (11.35), we have related this flow to an incompressible flow in transformed (£, і)) space, where the flow is represented by ф(%, і) ‘) obtained from Equation (11.45). The relation between ф and ф is given by Equation (11.36c).

Consider again the shape of the airfoil given in physical space by у = f(x). The shape of the airfoil in the transformed space is expressed as r) — q(^). Let us compare the two shapes. First, apply the approximate boundary condition, Equation

(11.34) , in physical space, noting that df/dx = tan в. We obtain

df _дф _ 1 дф _ дф 00 dx ду p ду 3 г)

Similarly, apply the flow-tangency condition in transformed space, which from Equa­tion (11.34) is

dq_ _ дф

°°d$ ~ dr)

Examine Equations (11.46) and (11.47) closely. Note that the right-hand sides of these two equations are identical. Thus, from the left-hand sides, we obtain

df_ _dq_

dx </£

Equation (11.48) implies that the shape of the airfoil in the transformed space is the same as in the physical space. Hence, the above transformation relates the compress­ible flow over an airfoil in (x, y) space to the incompressible flow in (£, >)) space over the same airfoil.

The above theory leads to an immensely practical result, as follows. Recall Equa­tion (11.32) for the linearized pressure coefficient. Inserting the above transformation into Equation (11.32), we obtain

2Й 2 3ф 2 1 3ф 2 1 3ф ^ ^ 49]

Question: What is the significance of 3$/3£ in Equation (11.49)? Recall that ф is the perturbation velocity potential for an incompressible flow in transformed space. Hence, from the definition of velocity potential, дф/ді; = й, where й is a perturbation

velocity for the incompressible flow. Hence, Equation (11.49) can be written as

From Equation (11.32), the expression f—2u/V00) is simply the linearized pressure coefficient for the incompressible flow. Denote this incompressible pressure coeffi­cient by Орд. Hence, Equation (11.50) gives

or recalling that f = ,/F— M^, we have

[11.51]

Equation (11.51) is called the Prandtl-Glauert rule; it states that, if we know the incompressible pressure distribution over an airfoil, then the compressible pressure distribution over the same airfoil can be obtained from Equation (11.51). Therefore, Equation (11.51) is truly a compressibility correction to incompressible data.

Consider the lift and moment coefficients for the airfoil. For an inviscid flow, the aerodynamic lift and moment on a body are simply integrals of the pressure distribution over the body, as described in Section 1.5. (If this is somewhat foggy in your mind, review Section 1.5 before progressing further.) In turn, the lift and moment coefficients are obtained from the integral of the pressure coefficient via Equations (1.15) to (1.19). Since Equation (11.51) relates the compressible and incompressible pressure coefficients, the same relation must therefore hold for lift and moment coefficients:

The Prandtl-Glauert rule, embodied in Equations (11.51) to (11.53), was histor­ically the first compressibility correction to be obtained. As early as 1922, Prandtl was using this result in his lectures at Gottingen, although without written proof. The derivation of Equations (11.51) to (11.53) was first formally published by the British aerodynamicist, Hermann Glauert, in 1928. Hence, the rule is named after both men. The Prandtl-Glauert rule was used exclusively until 1939, when an improved com­pressibility correction was developed. Because of their simplicity, Equations (11.51) to (11.53) are still used today for initial estimates of compressibility effects.

Recall that the results of Chapters 3 and 4 proved that inviscid, incompressible flow over a closed, two-dimensional body theoretically produces zero drag—the well – known d’Alembert’s paradox. Does the same paradox hold for inviscid, subsonic,

compressible flow? The answer can be obtained by again noting that the only source of drag is the integral of the pressure distribution. If this integral is zero for an incompressible flow, and since the compressible pressure coefficient differs from the incompressible pressure coefficient by only a constant scale factor, p, then the integral must also be zero for a compressible flow. Hence, d’Alembert’s paradox also prevails for inviscid, subsonic, compressible flow. However, as soon as the freestream Mach number is high enough to produce locally supersonic flow on the body surface with attendant shock waves, as shown in Figure 137b, then a positive wave drag is produced, and d’Alembert’s paradox no longer prevails.

Example 11.1 | At a given point on the surface of an airfoil, the pressure coefficient is —0.3 at very low speeds. If the freestream Mach number is 0.6, calculate Cp at this point.

Solution

From Equation (11.51),

^ _ Cp, o _ ~~0-3

p ~~ Vl – M2 ~ y/ – (0.6)2

Example 1 1.2 | From Chapter 4, the theoretical lift coefficient for a thin, symmetric airfoil in an incompressible

flow is с/ = 2ла. Calculate the lift coefficient for Mx = 0.7.

Solution

From Equation (11.52),

C/,o 2na

C‘ ~ У1 – Ml ~ Vl – (0.7)2

Note: The effect of compressibility at Mach 0.7 is to increase the lift slope by the ratio 8.8/27Г = 1.4, or by 40 percent.

The Navier-Stokes Equations

In Chapter 2, Newton’s second law was applied to obtain the fluid-flow momentum equation in both integral and differential forms. In particular, recall Equations (2.13a to c), where the influence of viscous forces was expressed simply by the generic terms (•Tudviscous, (dd'(viscousi and (d%)viSCous – The purpose of this section is to obtain the analogous forms of Equations (2.13a to c) where the viscous forces are expressed explicitly in terms of the appropriate flow-field variables. The resulting equations are called the Navier-Stokes equations—probably the most pivotal equations in all of theoretical fluid dynamics.

In Section 2.3, we discussed the philosophy behind the derivation of the governing equations, namely, certain physical principles are applied to a suitable model of the fluid flow. Moreover, we saw that such a model could be either a finite control volume (moving or fixed in space) or an infinitesimally small element (moving or fixed in space). In Chapter 2, we chose the fixed, finite control volume for our model and obtained integral forms of the continuity, momentum, and energy equations directly from this model. Then, indirectly, we went on to extract partial differential equations from the integral forms. Before progressing further, it would be wise for you to review these matters from Chapter 2.

For the sake of variety, let us not use the fixed, finite control volume employed in Chapter 2; rather, in this section, let us adopt an infinitesimally small moving fluid element of fixed mass as our model of the flow, as sketched in Figure 15.11. To this model let us apply Newton’s second law in the form F = ma. Moreover, for the time being consider only the jc component of Newton’s second law:

Fx = max [15.14]

In Equation (15.14), Fx is the sum of all the body and surface forces acting on the fluid element in the x direction. Let us ignore body forces; hence, the net force acting on the element in Figure 15.11 is simply due to the pressure and viscous stress distributions over the surface of the element. For example, on face abed, the only force in the jc direction is that due to shear stress, rvv dx dz. Face efgh is a

Figure 1 5.1 1 Infinitesimally small, moving fluid element. Only the forces in the x direction are shown.

distance dy above face abed; hence, the shear force in the x direction on face efgh is [xyx + (дтух/ду) dy] dx dz. Note the directions of the shear stress on faces abed and efgh; on the bottom face, zyx is to the left (the negative x direction), whereas on the top face, zyx + (dzyx/dy) dy is to the right (the positive x direction). These directions are due to the convention that positive increases in all three components of velocity, u, v, and w, occur in the positive directions of the axes. For example, in Figure 15.11, и increases in the positive у direction. Therefore, concentrating on face efgh, и is higher just above the face than on the face; this causes a “tugging” action which tries to pull the fluid element in the positive x direction (to the right) as shown in Figure 15.11. In turn, concentrating on face abed, и is lower just beneath the face than on the face; this causes a retarding or dragging action on the fluid element, which acts in the negative x direction (to the left), as shown in Figure 15.11. The directions of all the other viscous stresses shown in Figure 15.11, including zxx, can be justified in a like fashion. Specifically, on face degh, zzx acts in the negative x direction, whereas on face abfe, zzx + (dzzx/8z)dz acts in the positive x direction. On face adhe, which is perpendicular to the x axis, the only forces in the x direction are the pressure force p dy dz, which always acts in the direction into the fluid element, and zxx dy dz, which is in the negative x direction. In Figure 15.11, the reason why zxx on face adhe is to the left hinges on the convention mentioned earlier for the direction of increasing velocity. Here, by convention, a positive increase in и takes place in the positive x direction. Hence, the value of и just to the left of face adhe is smaller than the value of и on the face itself. As a result, the viscous action of the normal stress

acts as a “suction” on face adhe: that is, there is a dragging action toward the left that wants to retard the motion of the fluid element. In contrast, on face bcgf, the pressure force [p + (dp/dx) dx] dy dz presses inward on the fluid element (in the negative x direction), and because the value of и just to the right of face bcgf is larger than the value of и on the face, there is a “suction” due to the viscous normal stress which tries to pull the element to the right (in the positive x direction) with a force equal to [txx + (dzxx/dx)dx]dydz.

Return to Equation (15.14). Examining Figure 15.11 in light of our previous discussion, we can write for the net force in the л direction acting on the fluid element:

і dP
p-p+-dx

Equation (15.15) represents the left-hand side of Equation (15.14). Considering the right-hand side of Equation (15.14), recall that the mass of the fluid element is fixed and is equal to

m = p dx dy dz

Also, recall that the acceleration of the fluid element is the time rate of change of its velocity. Hence, the component of acceleration in the x direction, denoted by ax, is simply the time rate of change of n; since we are following a moving fluid element, this time rate of change is given by the substantial derivative (see Section 2.9 for a review of the meaning of substantial derivative). Thus,

Du _ .

ax = — [15.17]

Dt

Combining Equations (15.14) to (15.17), we obtain

[15.18a]

which is the x component of the momentum equation for a viscous flow. In a similar fashion, the у and z components can be obtained as

Dw dp drxz dzyz drzz

Dt dz dx dy dz

Equations (15.18a to c) are the momentum equations in the x, y, and z directions, respectively. They are scalar equations and are called the Navier-Stokes equations in

honor of two men—the Frenchman M. Navier and the Englishman G. Stokes—who independently obtained the equations in the first half of the nineteenth century.

With the expressions for rxy = xyx, xyi = xzy, rzx = rxz, rxx, xyy, and xzz from Equations (15.5) to (15.10), the Navier-Stokes equations, Equations (15.18a to c), can be written as

dи Эи Эи Зи dp Э / Эи

РЧ7 + риТ + pv^~ + pwV~ = UV • V + 2/х —

dt dx dy dz dx dx dx J

Equations (15.19a to c) represent the complete Navier-Stokes equations for an un­steady, compressible, three-dimensional viscous flow. To analyze incompressible vis­cous flow, Equations (15.19a to c) and the continuity equation [say, Equation (2.52)] are sufficient. However, for a compressible flow, we need an additional equation, namely, the energy equation to be discussed in the next section.

In the above form, the Navier-Stokes equations are suitable for the analysis of laminar flow. For a turbulent flow, the flow variables in Equations (15.19a to c) can be assumed to be time-mean values over the turbulent fluctuations, and p can be replaced by p + є, as discussed in Section 15.3. For more details, see References 42 and 43.

Results for Turbulent Boundary Layers on a Flat Plate

In this section, we discuss a few results for the turbulent boundary layer on a flat plate, both incompressible and compressible, simply to provide a basis of comparison with the laminar results described in the previous section. For considerably more detail on the subject of turbulent boundary layers, consult References 42 to 44.

For incompressible flow over a flat plate, the boundary-layer thickness is given approximately by

Note from Equation (19.1) that the turbulent boundary-layer thickness varies approx­imately as RejTl/5 in contrast to Re“I/2 for a laminar boundary layer. Also, turbulent values of <5 grow more rapidly with distance along the surface; 8 <x x4/5 for a turbulent flow in contrast to 8 oc x1 /2 for a laminar flow. With regard to skin friction drag, for incompressible turbulent flow over a flat plate, we have

Note that for turbulent flow, C f varies as Re~1//5 in comparison with the Re’1/2 vari­ation for laminar flow. Hence, Equation (19.2) yields larger friction drag coefficients for turbulent flow in comparison with Equation (18.22) for laminar flow.

The effects of compressibility on Equation (19.2) are shown in Figure 19.1, where C f is plotted versus ReTO with Мто as a parameter. The turbulent flow results are shown toward the right of Figure 19.1, at the higher values of Reynolds numbers where turbulent conditions are expected to occur, and laminar flow results are shown toward the left of the figure, at lower values of Reynolds numbers. This type of figure—friction drag coefficient for both laminar and turbulent flow as a function of Re on a log-log plot—is a classic picture, and it allows a ready contrast of the two types of flow. From this figure, we can see that, for the same Re^, turbulent skin

friction is higher than laminar; also, the slopes of the turbulent curves are smaller than the slopes of the laminar curves—a graphic comparison of the Re,/? variation in contrast to the laminar Re 1/2 variation. Note that the effect of increasing Мж is to reduce Cf at constant Re and that this effect is stronger on the turbulent flow results. Indeed, C/ for the turbulent results decreases by almost an order of magnitude (at the higher values of Re,*,) when M0c is increased from 0 to 10. For the laminar flow, the decrease in Су as Мж is increased though the same Mach number range is far less pronounced.

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.