Category When Is A Flow Compressible?

Hypersonic Shock-Wave Relations and Another Look at Newtonian Theory

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

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

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

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

P і У + 1

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

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

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

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

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

resulting in

2 Г Mjp2 – 1 ‘

P [Mf(y + l) + 2_

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

2 Г M2p2 ‘

~P _M2(y + 1)_

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

Note that, for у = 1.4,

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

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

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

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

In the hypersonic limit,

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

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

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

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

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

Cp ^ 2 sin2 p [14.40]

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

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

Pi __ (y + l)Af^ sin2 p

Poo (y – l)Af^ sin2 p + 2

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

Pi У + 1

Poo У 1

1, we find

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

[14.35]

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

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

[14.44]

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

A Comment on Drag Variation with Velocity

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

L = poaVl0SCL and D = poaVl0SCD

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

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

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

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

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

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

The Supercritical Airfoil

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

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

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

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

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

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

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

Couette Flow: General Discussion

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

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

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

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

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

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

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

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

[15.2]

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

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

ди ЗT dp

dx dx dx

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

„ 3 /, Э7Л 3 / du n

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

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

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

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

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

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

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

Examples of Some Solutions

In this section, we present samples of a few numerical solutions of the complete

Navier-Stokes equations. Most of these solutions have the following in common:

1. They were obtained by means of a time-dependent solution using MacCormack’s technique as described in Section 16.4.

2. They utilize the Baldwin-Lomax turbulence model (see Section 19.3.1 for a discussion of this model). Hence, turbulent flow is modeled in these calculations.

3. They require anywhere from thousands to close to a million grid points for their solution. Therefore, these are problems that must be solved on large-scale digital computers.

20.3.1 Flow over a Rearward-Facing Step

The supersonic viscous flow over a rearward-facing step was examined in Refer­ence 46. Some results are shown in Figures 20.1 and 20.2. The flow is moving from left to right. In the velocity vector diagram in Figure 20.1, note the separated, recirculating flow region just downstream of the step. The calculation of such sep­arated flows is the forte of solutions of the complete Navier-Stokes equations. In contrast, the boundary-layer equations discussed in Chapter 17 are not suited for the analysis of separated flows; boundary-layer calculations usually “blow up” in regions of separated flow. Figure 20.2 shows the temperature contours (lines of constant temperature) for the same flow in Figure

Figure 20.2 Temperature contours for the flow shown in Figure 20.1. The separated region just downstream of the step is a reasonably constant pressure, constant temperature region.

Subsonic Compressible Flow

Подпись: Po.i Pi Подпись: 1 + Подпись: -Mi Подпись: y/(y-О Подпись: [8.42]

Consider a Pitot tube in a subsonic, compressible flow, as sketched in Figure 8.10a. As usual, the mouth of the Pitot tube (point b) is a stagnation region. Hence, a fluid element moving along streamline ab is brought to rest isentropically at point b. In turn, the pressure sensed at point b is the total pressure of the freestream, p„ i. This is the Pitot pressure read at the end of the tube. If, in addition, we know the freestream static pressure p, then the Mach number in region 1 can be obtained from Equation (8.42),

Подпись: Mi Подпись: 2 Подпись: (y-U/y P 0.1  Pi ) Подпись: [8.74]

or solving for Mf,

Clearly, from Equation (8.74), the Pitot pressure p,,., and the static pressure p, allow the direct calculation of Mach number.

Subsonic Compressible Flow Подпись: 2 a 7^1 Подпись:  (y — l)/y Po,i  Pi / Подпись: [8.75]

The flow velocity can be obtained from Equation (8.74) by recalling that M = u/a. Hence,

From Equation (8.75), we see that, unlike incompressible flow, a knowledge of pop and p is not sufficient to obtain и,; we also need the freestream speed of sound, a.

8.7.8 Supersonic Flow

Consider a Pitot tube in a supersonic freestream, as sketched in Figure 8.106. As usual, the mouth of the Pitot tube (point e) is a stagnation region. Hence, a fluid element moving along streamline cde is brought to rest at point e. However, because the freestream is supersonic and the Pitot tube presents an obstruction to the flow,

Subsonic Compressible Flow

Subsonic flow

Subsonic Compressible Flow

 

Supersonic flow

Subsonic Compressible Flow

Figure 8.10 A Pitot tube in (a) subsonic flow and (b) supersonic flow.

 

there is a strong bow shock wave in front of the tube, much like the picture shown at the left of Figure 8.1 for supersonic flow over a blunt body. Hence, streamline cde crosses the normal portion of the bow shock. A fluid element moving along streamline cde will first be decelerated nonisentropically to a subsonic velocity at point d just behind the shock. Then it is isentropically compressed to zero velocity at point e. As a result, the pressure at point e is not the total pressure of the freestream but rather the total pressure behind a normal shock wave, po,2- This is the Pitot pressure read at the end of the tube. Keep in mind that because of the entropy increase across the shock, there is a loss in total pressure across the shock, po,2 < Po, і ■ However, knowing po,2 and the freestream static pressure p is still sufficient to calculate the freestream Mach number Mb as follows:

Подпись:P0,2 __ P0,2 Pi Pi Pi Pi

Here, p^ijpi is the ratio of total pressure to static pressure in region 2 immediately behind the normal shock, and рг/Pi is the static pressure ratio across the shock. From

Subsonic Compressible Flow Subsonic Compressible Flow Подпись: 1) Подпись: [8.79]
Subsonic Compressible Flow
Подпись: [8.77]
Подпись: [8.78І

Also, from Equation (8.65),

Подпись: ^2 = / (у + 1 )2M2 K/(K 0 1 -y+2yM2 pi AyM - 2(y - 1)У у + 1

Substituting Equation (8.78) into (8.77), and substituting the result as well as Equa­tion (8.79) into Equation (8.76), we obtain, after some algebraic simplification (see Problem 8.14),

Equation (8.80) is called the Rayleigh Pitot tube formula. It relates the Pitot pressure po,2 and the freestream static pressure p to the freestream Mach number M. Equa­tion (8.80) gives Mx as an implicit function of рол/Pi and allows the calculation of Mx fromaknown рол/Рі – For convenience in making calculations, the ratio Рол/Рі is tabulated versus Mi in Appendix B.

Подпись: Example 8.7A Pitot tube is inserted into an airflow where the static pressure is 1 atm. Calculate the flow Mach number when the Pitot tube measures (a) 1.276 atm, (b) 2.714 atm, (c) 12.06 atm.

Solution

First, we must assess whether the flow is subsonic or supersonic. At Mach 1, the Pitot tube would measure p0 = p/0.528 = 1.893/?. Hence, when p0 < 1.893 atm, the flow is subsonic, and when p0 > 1.893 atm, the flow is supersonic.

(a)

Подпись: M = 0.6

Pitot tube measurement = 1.276 atm. The flow is subsonic. Hence, the Pitot tube is directly sensing the total pressure of the flow. From Appendix A, for р0/p = 1.276,

(b) Pitot tube measurement = 2.714 atm. The flow is supersonic. Hence, the Pitot tube is sensing the total pressure behind a normal shock wave. From Appendix B, for po,2/Pi = 2.714,

Subsonic Compressible Flow

Mi = 1.3

 

(c)

Подпись: M, = 3.0

Pitot tube measurement = 12.06 atm. The flow is supersonic. From Appendix B, for Ро. г/Рі = 12.06,

Example 8.8 I Consider a hypersonic missile flying at Mach 8 at an altitude of 20,000 ft, where the pressure is 973.3 lb/ft2. The nose of the missle is blunt and is shaped like that shown at the left of Figure 8.1. Calculate the pressure at the stagnation point on the nose.

Solution

Examining the blunt body shown in Figure 8.1, the streamline that impinges at the stagnation point has traversed the normal portion of the bow shock wave. By definition, V = 0 at the stagnation point. Since the flow is isentropic between the shock and the body, the pressure at the stagnation point on the body is the total pressure behind a normal shock with an upstream Mach number of 8. Let us denote the pressure at the stagnation point by ps. Since p0 2 is the total pressure behind the normal shock, then ps = p02. From Appendix B, for Mach 8, Ро. г/Рі = 82.87. Hence,

Подпись:Ps = Po,2 = (Pi) = 82.87(973.3) =

Since 1 atm = 2116 lb/ft2,

Подпись: 38.1 atm_ 8.07 x 104

Ps ~ 2П6 _

Note that the pressure at the nose of the missile is quite high—38.1 atm. This is typical of hypersonic flight at low altitude.

Check on the calculation This problem can also be solved by first calculating the upstream total pressure from Appendix A, and then using the total pressure ratio across the normal shock from Appendix B. From Appendix A for Mach 8, Po, i/Pi = 0.9763 x 10-4. Hence,

p0,i = (y1) Pt = (0.9763 x 104)973.3 = 9.502 x 106

From Appendix В for Mach 8, р0,г/Po. i = 8.8488 x 10 2. Hence,

Подпись:p, = рог = ( — ) Po і = (0.8488 x 10~z)(9.502 x 106) =

’ Po. i/ ’

This is the same result as obtained earlier.

Supersonic Nozzle Design

In Chapter 10, we demonstrated that a nozzle designed to expand a gas from rest to supersonic speeds must have a convergent-divergent shape. Moreover, the quasi- one-dimensional analysis of Chapter 10 led to the prediction of flow properties as a function of x through a nozzle of specified shape (see, e. g., Figure 10.10). The flow properties at any x station obtained from the quasi-one-dimensional analysis represent an average of the flow over the given nozzle cross section. The beauty of the quasi – one-dimensional approach is its simplicity. On the other hand, its disadvantages are (1) it cannot predict the details of the actual three-dimensional flow in a convergent – divergent nozzle and (2) it gives no information on the proper wall contour of such nozzles.

The purpose of the present section is to describe how the method of characteristics can supply the above information which is missing from a quasi-one-dimensional analysis. For simplicity, we treat a two-dimensional flow, as sketched in Figure 13.7. Here, the flow properties are a function of x and y. Such a two-dimensional flow is applicable to supersonic nozzles of rectangular cross section, such as sketched in the insert at the top of Figure 13.7. Two-dimensional (rectangular) nozzles are used in many supersonic wind tunnels. They are also the heart of gas-dynamic lasers (see Reference 1). In addition, there is current discussion of employing rectangular exhaust nozzles on advanced military jet airplanes envisaged for the future.

Consider the following problem. We wish to design a convergent-divergent nozzle to expand a gas from rest to a given supersonic Mach number at the exit Me. How do we design the proper contour so that we have shock-free, isentropic flow in the nozzle? The answer to this question is discussed in the remainder of this section.

For the convergent, subsonic section, there is no specific contour which is better than any other. There are rules of thumb based on experience and guided by subsonic flow theory; however, we are not concerned with the details here. We simply assume that we have a reasonable contour for the subsonic section.

Due to the two-dimensional nature of the flow in the throat region, the sonic line is generally curved, as sketched in Figure 13.7. A line called the limiting characteristic is sketched just downstream of the sonic line. The limiting characteristic is defined such that any characteristic line originating downstream of the limiting characteristic does not intersect the sonic line; in contrast, a characteristic line originating in the small region between the sonic line and the limiting characteristic can intersect the sonic line (for more details on the limiting characteristic, see Reference 21). To begin a method of characteristics solution, we must use an initial data line which is downstream of the limiting characteristic.

Let us assume that by independent calculation of the subsonic-transonic flow in the throat region, we know the flow properties at all points on the limiting character­istic. That is, we use the limiting characteristic as our initial data line. For example, we know the flow properties at points 1 and 2 on the limiting characteristic in Figure 13.7. Moreover, consider the nozzle contour just downstream of the throat. Letting в denote the angle between a tangent to the wall and the horizontal, the section of the divergent nozzle where в is increasing is called the expansion section, as shown in Figure 13.7. The end of the expansion section occurs where в — втм (point 8 in Figure 13.7). Downstream of this point, в decreases until it equals zero at the nozzle exit. The portion of the contour where в decreases is called the straightening section. The shape of the expansion section is somewhat arbitrary; typically, a circular arc of large radius is used for the expansion section of many wind-tunnel nozzles. Conse­quently, in addition to knowing the flow properties along the limiting characteristic, we also have an expansion section of specified shape; that is, we know 6, 65, and (9X in Figure 13.7. The purpose of our application of the method of characteristics now becomes the proper design of the contour of the straightening section (from points 8 to 13 in Figure 13.7).

The characteristics mesh sketched in Figure 13.7 is very coarse—this is done intentionally to keep our discussion simple. In an actual calculation, the mesh should be much finer. The characteristics mesh and the flow properties at the associated grid points are calculated as follows:

1. Draw a C_ characteristic from point 2, intersecting the centerline at point 3.

Evaluating Equation (13.17) at point 3, we have

$3 + U3 = (А"_)з

In the above equation, 03 = 0 (the flow is horizontal along the centerline). Also,

(А"_)з is known because (A"_)3 = (K – )3. Hence, the above equation can be

solved for v3.

2. Point 4 is located by the intersection of the C_ characteristic from point 1 and the C+ characteristic from point 3. In turn, the flow properties at the internal point 4 are determined as discussed in the last part of Section 13.2.

3. Point 5 is located by the intersection of the C+ characteristic from point 4 with the wall. Since 05 is known, the flow properties at point 5 are determined as discussed in Section 13.2 for wall points.

4. Points 6 through 11 are located in a manner similar to the above, and the flow properties at these points are determined as discussed before, using the internal point or wall point method as appropriate.

5. Point 12 is a wall point on the straightening section of the contour. The purpose of the straightening section is to cancel the expansion waves generated by the expansion section. Hence, there are no waves which are reflected from the straightening section. In turn, no right-running waves cross the characteristic line between points 9 and 12. Asa result, the characteristic line between points 9 and 12 is a straight line, along which в is constant, that is, 612 = O9. The section of the wall contour between points 8 and 12 is approximated by a straight line with an average slope of ^(в% + 612).

6. Along the centerline, the Mach number continuously increases. Let us assume that at point 11, the design exit Mach number Me is reached. The characteristic line from points 11 to 13 is the last line of the calculation. Again, вц = 9ц, and the contour from point 12 to point 13 is approximated by a straight-line segment with an average slope of 5(^12 + віз).

The above description is intended to give you a “feel” for the application of the method of characteristics. If you wish to carry out an actual nozzle design, and/or if you are interested in more details, read the more complete treatments in References 21 and 34.

Note in Figure 13.7 that the nozzle flow is symmetrical about the centerline. Hence, the points below the centerline (T, 2′, 3′, etc.) are simply mirror images of the corresponding points above the centerline. In making a calculation of the flow through the nozzle, we need to concern ourselves only with those points in the upper half of Figure 13.7, above and on the centerline.

Some Analytical Considerations

For air temperatures up to 1000 K, the specific heats are essentially constant, thus justifying the assumption of a calorically perfect gas for this range. Moreover, the

temperature variations of /і and к over this range are virtually identical. As a result, the Prandtl number, ficp/ k, is essentially constant up to temperatures on the order of 1000 K. This is shown in Figure 16.11, obtained from Schetz (Reference 53). Note that Pr ~ 0.71 for air; this is the value that was used in Example 16.1.

Question: How high a Mach number can exist before we would expect to en­counter temperatures in the flow above 1000 K? Answer: An approximate answer is to calculate that Mach number at which the total temperature is 1000 K. Assuming a static temperature T = 288 K, from Equation (8.40),

Hence, for most aeronautical applications involving flight at a Mach number of 3.5 or less, the temperature within the viscous portions of the flow field will not exceed 1000 K. A Mach number of 3.5 or less encompasses virtually all operational aircraft today, with the exception of a few hypersonic test vehicles.

In light of the above, many viscous flow solutions are carried out making the justi­fiable assumption of a constant Prandtl number. For the case of compressible Couette flow, the assumption of Pr = constant allows the following analysis. Consider the energy equation, Equation (16.3), repeated below:

Inserting b and c into Equation (16.79) and simplifying, we obtain

и2 и Pr

h – b Pr — =hw—– (he – hw) + —(uue)

2 ue 2

Assume the lower wall is adiabatic; that is, (Bh/Sy)w = 0. Differentiating Equa­tion (16.80) with respect to y, we have

Recall that the condition for an adiabatic wall is that (dh/dy)w = 0. Applying Equation (16.81) at у = 0 for an adiabatic wall, where и = 0 and by definition hw = haw, we have

Since (du/dy)w is finite, then

This is identical to Equation (16.39) obtained for incompressible flow. Hence, we have just proven that the recovery factor for compressible Couette flow, assuming constant Prandtl number, is also

[16.83]

Since the recovery factors for the incompressible and compressible cases are the same (as long as Pr = constant), what can we say about Reynolds analogy? Does Equation (16.59) hold for the compressible case? Let us examine this question. Return to Equation (16.3), repeated below:

Recalling that, from the definitions,

then Equation (16.3) can be written as

Integrating Equation (16.86) with respect to y, we have

where a is a constant of integration. Evaluating Equation (16.87) at у = 0, where и = 0 and q = qw, we find that

Hence, Equation (16.87) is

q + xu = qw

Inserting Equations (16.84) and (16.85) into (16.88), we have

. dT du 4w=k~ + їли — dy dy

[16.89]

or

qw к dT du x x dy dy

[16.90]

Recall that the shear stress

is constant throughout the flow; hence,

du

r = ~ r>"

dy

or

?W

a — ——–

du/dy

[16.91]

Also,

к _ Cp

fl Pr

[16.92]

Inserting Equation (16.91) into the left-hand side of Equation (16.90), and Equa­tion (16.92) into the right-hand side of Equation (16.90), we have

qw du cp dT d{u2/2)

rw dy Pr 3у dy

Integrate Equation (16.93) between the two plates, keeping in mind that qw, ru

and Pr are all fixed values:

or which yields

CtLUe^%Te~Tw)+1^ Tw Pr 2

Rearranging Equation (16.94), and recalling that h = cpT, we have

Equation (16.98) is Reynolds analogy—a relation between heat transfer and skin friction coefficients. Moreover, it is precisely the same result as obtained in Equa­tion (16.59) for incompressible flow. Hence, for a constant Prandtl number, we have shown that Reynolds analogy is precisely the same form for incompressible and com­pressible flow.

Consider the geometry given in Figure 16.2. The two plates are separated by a distance of 0.01 in (the same as in Example 16.1). The temperature of the two plates is equal, at a value of 288 К (standard sea level temperature). The air pressure is constant throughout the flow and equal to 1 atm. The upper plate is moving at Mach 3. The shear stress at the lower wall is 72 N/m2. (This is about 1.5 lb/ft2—a much larger value than that associated with the low-speed case treated in Example 16.1.) Calculate the heat transfer to either plate. (Since the shear stress is constant throughout the flow, and the plates are at equal temperature, the heat transfer to the upper and lower plates is the same.)

Solution

The velocity of the upper plate is

ue = Meae = Mey/yRTe = 3v7(1 -4)(288)(287) = 1020 m/s The air density at both plates is (noting that 1 atm =1.01 x 105 N/m2)

1.22 kg/m3

Hence, the skin friction coefficient is

(1020)2 ,

haw = (1004.5)(288) + (0.71)- – – – = 6.59 x 105 J/kg

[Note: This gives Taw = huw/cp = (6.59 x 105)/1004.5 = 656 K. In the adiabatic case, the wall would be quite warm.] Hence, from the definition of the Stanton number [Equa­tion (16.55)], and noting that hw = cpT„, = (1004.6)(288) = 2.89 x 105 J/kg,

qw = peue(hau,-hw)CH = ( 1.22)(1020)[(6.59- 2.89) x 105](8 x КГ5)

Diffusers

The role of a diffuser was first introduced in Section 3.3 in the context of a low-speed subsonic wind tunnel. There, a diffuser was a divergent duct downstream of the test section whose role was to slow the higher-velocity air from the test section down to a very low velocity at the diffuser exit (see Figure 3.8). Indeed, in general, we can define a diffuser as any duct designed to slow an incoming gas flow to lower velocity at the exit of the diffuser. The incoming flow can be subsonic, as discussed in Figure

3.8, or it can be supersonic, as discussed in the present section. However, the shape of the diffuser is drastically different, depending on whether the incoming flow is subsonic or supersonic.

Before pursuing this matter further, let us elaborate on the concept of total pres­sure p0 as discussed in Section 7.5. In a semiqualitative sense, the total pressure of a flowing gas is a measure of the capacity of the flow to perform useful work. Let us consider two examples:

1. A pressure vessel containing stagnant air at 10 atm

2. A supersonic flow at M = 2.16 and p = 1 atm

In case 1, the air velocity is zero; hence, p0 = p = 10 atm. Now, imagine that we want to use air to drive a piston in a piston-cylinder arrangement, where useful work is performed by the piston being displaced through a distance. The air is ducted into the cylinder from a large manifold, in the same vein as the reciprocating internal combustion engine in our automobile. In case, 1, the pressure vessel can act as the manifold; hence, the pressure on the piston is 10 atm, and a certain amount of useful work is performed, say, Wi. However, in case 2, the supersonic flow must be slowed to a low velocity before we can readily feed it into the manifold. If this slowing process can be achieved without loss of total pressure, then the pressure in the manifold in this case is also 10 atm (assuming V ~ 0), and the same amount of useful work Wi is performed. On the other hand, assume that in slowing down the supersonic stream, a loss of 3 atm takes place in the total pressure. Then the pressure in the manifold is only 7 atm, with the consequent generation of useful work №2, which is less than in the first case; that is, ИА < W. The purpose of this simple example is to indicate that the total pressure of a flowing gas is indeed a measure of its capability to perform useful work. On this basis, a loss of total pressure is always an inefficiency—a loss of the capability to do a certain amount of useful work.

In light of the above, let us expand our definition of a diffuser. A diffuser is a duct designed to slow an incoming gas flow to lower velocity at the exit of the diffuser with as small a loss in total pressure as possible. Consequently, an ideal diffuser would be characterized by an isentropic compression to lower velocities; this is sketched in Figure 10.15a, where a supersonic flow enters the diffuser at M, is isentropically

j2> J,

compressed in a convergent duct to Mach 1 at the throat, where the area is A*, and then is further isentropically compressed in a divergent duct to a low subsonic Mach number at the exit. Because the flow is isentropic, s2 = si, and from Equation (8.73), pQ 2 = pop – Indeed, po is constant throughout the entire diffuser—a characteristic of isentropic flow. However, common sense should tell you that the ideal diffuser in Figure 10.15a can never be achieved. It is extremely difficult to slow a supersonic flow without generating shock waves in the process. For example, examine the convergent portion of the diffuser in Figure 10.15a. Note that the supersonic flow is turned into itself; hence, the converging flow will inherently generate oblique shock waves, which will destroy the isentropic nature of the flow. Moreover, in real life, the flow is viscous; there will be an entropy increase within the boundary layers on the walls of the diffuser. For these reasons, an ideal isentropic diffuser can never be constructed; an ideal diffuser is of the nature of a “perpetual motion machine”—only a utopian wish in the minds of engineers.

An actual supersonic diffuser is sketched in Figure 10.15/7. Here, the incoming flow is slowed by a series of reflected oblique shocks, first in a convergent section usually consisting of straight walls, and then in a constant-area throat. Due to the interaction of the shock waves with the viscous flow near the wall, the reflected shock pattern eventually weakens and becomes quite diffuse, sometimes ending in a weak normal shock wave at the end of the constant-area throat. Finally, the subsonic flow downstream of the constant-area throat is further slowed by moving through a divergent section. At the exit, clearly s2 hence pop < Pop – The art of diffuser design is to obtain as small a total pressure loss as possible, that is, to design the

convergent, divergent, and constant-area throat sections so that ро. г/Рол is as close to unity as possible. Unfortunately, in most cases, we fall far short of that goal. For more details on supersonic diffusers, see Chapter 5 of Reference 21 and Chapter 12 of Reference 1.

Please note that due to the entropy increase across the shock waves and in the boundary layers, the real diffuser throat area A, is larger than,4*. that is, in Figure 10.15, A, > AU

Mach Number Independence

Examine again the hypersonic shock-wave relation for pressure ratio as given by Equation (14.29); note that, as the freestream Mach number approaches infinity, the pressure ratio itself also becomes infinitely large. On the other hand, the pressure coefficient behind the shock, given in the hypersonic limit by Equation (14.39), is a constant value at high values of the Mach number. This hints strongly of a situation where certain aspects of a hypersonic flow do not depend on Mach number, as long as the Mach number is sufficiently high. This is a type of “independence” from the Mach number, formally called the hypersonic Mach number independence principle. From the above argument, Cp clearly demonstrates Mach number independence. In turn, recall that the lift- and wave-drag coefficients for a body shape are obtained by integrating the local Cp, as shown by Equations (1.15), (1.16), (1.18), and (1.19). These equations demonstrate that, since Cp is independent of the Mach number at high values of Mx, the lift and drag coefficients are also Mach number independent. Keep in mind that these conclusions are theoretical, based on the limiting form of the hypersonic shock relations.

Let us examine an example that clearly illustrates the Mach number independence principle. In Figure 14.13, the pressure coefficients for a 15° half-angle wedge and a 15° half-angle cone are plotted versus freestream Mach number for у = 1.4. The exact wedge results are obtained from Equation (14.38), and the exact cone results are obtained from the solution of the classical Taylor-Maccoll equation. (See Reference 21 for a detailed discussion of the solution of the supersonic flow over a cone. There, you will find that the governing continuity, momentum, and energy equations for a conical flow cascade into a single differential equation called the

Taylor-Maccoll equation. In turn, this equation allows the exact solution of this conical flow field.) Both sets of results are compared with newtonian theory, Cp = 2 sin2 0, shown as the dashed line in Figure 14.13. This comparison demonstrates two general aspects of newtonian results:

1. The accuracy of the newtonian results improves as Мж increases. This is to be expected from our discussion in Section 14.5. Note from Figure 14.13 that below Moo = 5 the newtonian results are not even close, but the comparison becomes much closer as M^ increases above 5.

2. Newtonian theory is usually more accurate for three-dimensional bodies (e. g., the cone) than for two-dimensional bodies (e. g., the wedge). This is clearly evident in Figure 14.13 where the newtonian result is much closer to the cone results than to the wedge results.

However, more to the point of Mach number independence, Figure 14.13 also shows the following trends. For both the wedge and the cone, the exact results show that, at low supersonic Mach numbers, Cp decreases rapidly as Мж is increased. However, at hypersonic speeds, the rate of decrease diminishes considerably, and Cp appears to reach a plateau as M<*, becomes large; that is, Cp becomes relatively independent of Moo at high values of the Mach number. This is the essence of the Mach number independence principle; at high Mach numbers, certain aerodynamic quantities such as pressure coefficient, lift – and wave-drag coefficients, and flow-field structure (such as shock-wave shapes and Mach wave patterns) become essentially independent of the Mach number. Indeed, newtonian theory gives results that are totally independent of the Mach number, as clearly demonstrated by Equation (14.4).

Another example of Mach number independence is shown in Figure 14.14. Here, the measured drag coefficients for spheres and for a large-angle cone cylinder are plot­ted versus the Mach number, cutting across the subsonic, supersonic, and hypersonic regimes. Note the large drag rise in the subsonic regime associated with the drag – divergence phenomenon near Mach 1 and the decrease in Сд in the supersonic regime beyond Mach 1. Both of these variations are expected and well understood. For our purposes in the present section, note, in particular, the variation of Co in the hyper­sonic regime; for both the sphere and cone cylinder, CD approaches a plateau and becomes relatively independent of the Mach number as M^ becomes large. Note also that the sphere data appear to achieve “Mach number independence” at lower Mach numbers than the cone cylinder.

Keep in mind from the above analysis that it is the nondimensional variables that become Mach number independent. Some of the dimensional variables, such as p, are not Mach number independent; indeed, p —>• oc and Mcc —»• oo.

Finally, the Mach number independence principle is well grounded mathemati­cally. The governing inviscid flow equations (the Euler equations) expressed in terms of suitable nondimensional quantities, along with the boundary conditions for the limiting hypersonic case, do not have the Mach number appearing in them—hence, by definition, the solution to these equations is independent of the Mach number. See References 21 and 55 for more details.

Figure 14.14 Drag coefficient for a sphere and a cone

cylinder from ballistic range measurements; an example of Mach number independence at hypersonic speeds. (Source: Cox and Crabtree, Reference 61 .j