Category When Is A Flow Compressible?

Introduction to Numerical. Techniques for Nonlinear. Supersonic Flow

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

С. K. Chu, 1978 Columbia University

13.1 Introduction: Philosophy of

Computational Fluid Dynamics

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

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

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

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

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

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

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

(a)

Figure 13.2 Grid points.

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

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

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

Interim Summary

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

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

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

(,b) Calculate the shear stress.

(c) Calculate the maximum temperature in the flow.

(d) Calculate the heat transfer to either wall.

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

Solution

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

(a) From Equation (16.6),

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

can be written as

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

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

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

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

Shock-Expansion Theory: Applications to Supersonic Airfoils

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

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

R’ = (рз – P2)c [9.46]

L’ — (p3 — p2)c cos a [9.47]

D’ = {рз — p2)c sin a [9.48]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Returning to Equation (9.47), we have

The lift coefficient is obtained from

From Equation (9.48),

D’ = (p} — pi )c sin a


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

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

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

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

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

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

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

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

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

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

9,10 Summary

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

Some of the more important results are summarized as follows:

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

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

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

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

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

Compare the results.

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

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

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

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

(a) A single normal shock wave

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

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

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

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

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

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

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

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

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

(a) a = 5° (b) a = 15° (c) a = 30°

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

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

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

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

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

Newtonian Theory

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

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

Figure 14.5 Streamlines in a hypersonic flow.

time rate of change of momentum is

Mass flow x change in normal component of velocity

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

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

IV = PocV^A sin2 в [14.1]

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

= PccV^ sin20 [14.2]

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

P – Poo = Poo sin2 в [ 14.3]

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

= 2 sin2#

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

Cp — 2 cos2 ф

which is an equally valid expression of newtonian theory.

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

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

Poo + PooV^ = P2 + P2V2 [14.6]

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

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

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

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

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

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

Laminar Boundary Layers

Lamina—A thin scale or sheet. A layer or coat lying over another.

18.1 Introduction

Within the panoply of boundary-layer analyses, the solution of laminar boundary layers is well in hand compared to the status for turbulent boundary layers. This chapter is exclusively devoted to laminar boundary layers; turbulent boundary layers is the subject of Chapter 19. The basic definitions of laminar and turbulent flows are discussed in Section 15.1, and some characteristics of these flows are illustrated in Figures 15.5 and 15.6; it is recommended that you review that material before progressing further.

The roadmap for this chapter is given in Figure 18.1. We will first deal with some well-established classical solutions that come under the heading of self-similar solutions, a term that is defined in Section 18.2. In this regard, we will deal with both incompressible and compressible flows over a flat plate, as noted on the left side of our roadmap in Figure 18.1. We will also discuss the boundary-layer solution in the region surrounding the stagnation point on a blunt-nosed body, because this solution gives us important information on aerodynamic heating at the stagnation point—vital information for high-speed flight vehicles. As part of the classical solution of com­pressible boundary layers, we will discuss the reference temperature method—a very useful engineering calculation that makes use of classical incompressible boundary – layer results to predict skin friction and aerodynamic heating for a compressible flow. Then we will move to the right side of the roadmap in Figure 18.1 and discuss some

more modern computational fluid dynamic solutions to laminar boundary layers. Un­like the classical self-similar solutions, which are limited to a few (albeit important) applications such as flat plates and the stagnation region, these CFD numerical solu­tions deal with the laminar boundary layer over bodies of arbitrary shapes.

Note: As we progress through this chapter, we will encounter ideas and results that are already familiar to us from our discussion of Couette flow in Chapter 16. Indeed, this is one of the primary reasons for Chapter 16—to introduce these concepts within the context of a relatively straightforward flow problem before dealing with the more intricate boundary-layer solutions.

Critical Mach Number

Return to the road map given in Figure 11.1. We have now finished our discussion of linearized flow and the associated compressibility corrections. Keep in mind that such linearized theory does not apply to the transonic flow regime, 0.8 < Мж <

1.2. Transonic flow is highly nonlinear, and theoretical transonic aerodynamics is a challenging and sophisticated subject. For the remainder of this chapter, we deal with several aspects of transonic flow from a qualitative point of view. The theory of transonic aerodynamics is beyond the scope of this book.

Consider an airfoil in a low-speed flow, say, with = 0.3, as sketched in Figure 11.5a. In the expansion over the top surface of the airfoil, the local flow Mach number M increases. Let point A represent the location on the airfoil surface

t Local MA

where the pressure is a minimum, hence where M is a maximum. In Figure 11.5a, let us say this maximum is MA = 0.435. Now assume that we gradually increase the freestream Mach number. As M^ increases, MA also increases. For example, if is increased to M = 0.5, the maximum local value of M will be 0.772, as shown in Figure 11.5b. Let us continue to increase Moo until we achieve just the right value such that the local Mach number at the minimum pressure point equals 1, that is, such that MA = 1.0, as shown in Figure 11.5c. When this happens, the freestream Mach number Mx is called the critical Mach number, denoted by Mcr. By definition, the critical Mach number is that freestream Mach number at which sonic flow is first achieved on the airfoil surface. In Figure 11.5c, Ma = 0.61.

One of the most important problems in high-speed aerodynamics is the determi­nation of the critical Mach number of a given airfoil, because at values of Мж slightly above Mcr, the airfoil experiences a dramatic increase in drag coefficient (discussed in Section 11.7). The purpose of the present section is to give a rather straightforward method for estimating Mcr.

Let poo and pA represent the static pressures in the freestream and at point A, respectively, in Figure 11.5. For isentropic flow, where the total pressure po is constant, these static pressures are related through Equation (8.42) as follows:

Pa_ = Pa/Po = /1 + [(у-1)/2]Л^у/(у~П [11 56]

Рос Poc/Po l + [(y -)/2]Mj) *

The pressure coefficient at point A is given by Equation (11.22) as

Combining Equations (11.56) and (11.57), we have

Equation (11.58) is useful in its own right; for a given freestream Mach number, it relates the local value of Cp to the local Mach number. [Note that Equation (11.58) is the compressible flow analogue of Bernoulli’s equation, Equation (3.13), which for incompressible flow with a given freestream velocity and pressure relates the local pressure at a point in the flow to the local velocity at that point.] However, for our purposes here, we ask the question, What is the value of the local Cp when the local Mach number is unity? By definition, this value of the pressure coefficient is called the critical pressure coefficient, denoted by Cp. cr. For a given freestream Mach number Moo, the value of Cp. cr can be obtained by inserting MA = 1 into Equation (11.58):

Equation (11.59) allows us to calculate the pressure coefficient at any point in the flow where the local Mach number is 1, for a given freestream Mach number M^. For example, if Mx is slightly greater than Mcr, say, Mx = 0.65 as shown in Figure

11.5d, then a finite region of supersonic flow will exist above the airfoil; Equation

(11.59) allows us to calculate the pressure coefficient at only those points where M = 1, that is, at only those points that fall on the sonic line in Figure 11,5d. Now, returning to Figure 11.5c, when the freestream Mach number is precisely equal to the critical Mach number, there is only one point where M = 1, namely, point A. The pressure coefficient at point A will be Cp<cr, which is obtained from Equation (11.59). In this case, Moo in Equation (11.59) is precisely Mcr. Hence,

[1 1.60]

Equation (11.60) shows that Cpa is a unique function of Mcr; this variation is plotted as curve C in Figure 11.6. Note that Equation (11.60) is simply an aerodynamic relation for isentropic flow—it has no connection with the shape of a given airfoil. In this sense, Equation (11.60), and hence curve C in Figure 11.6, is a type of “universal relation” which can be used for all airfoils.

Equation (11.60), in conjunction with any one of the compressibility corrections given by Equations (11.51), (11.54), or (11.55), allows us to estimate the critical Mach number for a given airfoil as follows:

1. By some means, either experimental or theoretical, obtain the low-speed incom­pressible value of the pressure coefficient Cpq at the minimum pressure point on the given airfoil.

2. Using any of the compressibility corrections, Equation (11.51), (11.54), or

(11.55) , plot the variation of Cp with Moo■ This is represented by curve В in Figure 11.6.

Eq. (11.51), (11.54), or (11.55)

cr

3. Somewhere on curve B, there will be a single point where the pressure coefficient corresponds to locally sonic flow. Indeed, this point must coincide with Equa­tion (11.60), represented by curve C in Figure 11.6. Hence, the intersection of curves В and C represents the point corresponding to sonic flow at the minimum pressure location on the airfoil. In turn, the value of M^ at this intersection is, by definition, the critical Mach number, as shown in Figure 11.6.

The graphical construction in Figure 11.6 is not an exact determination of MCT. Al­though curve C is exact, curve В is approximate because it represents the approximate compressibility correction. Hence, Figure 11.6 gives only an estimation of MCT. How­ever, such an estimation is quite useful for preliminary design, and the results from Figure 11.6 are accurate enough for most applications.

Consider two airfoils, one thin and the other thick, as sketched in Figure 11.7. First consider the low-speed incompressible flow over these airfoils. The flow over the thin airfoil is only slightly perturbed from the freestream. Hence, the expansion over the top surface is mild, and Cp о at the minimum pressure point is a negative number of only small absolute magnitude, as shown in Figure 11.7. [Recall from Equation

(11.32) that Cp ос й; hence, the smaller the perturbation, the smaller is the absolute magnitude of Cp.] In contrast, the flow over the thick airfoil experiences a large perturbation from the freestream. The expansion over the top surface is strong, and Cp (t at the minimum pressure point is a negative number of large magnitude, as shown in Figure 11.7. If we now perform the same construction for each airfoil as given in Figure 11.6, we see that the thick airfoil will have a lower critical Mach number than

the thin airfoil. This is clearly illustrated in Figure 11.7. For high-speed airplanes, it is desirable to have Mcr as high as possible. Hence, modern high-speed subsonic airplanes are usually designed with relatively thin airfoils. (The development of the supercritical airfoil has somewhat loosened this criterion, as discussed in Section

11.8. ) For example, the Gates Lear jet high-speed jet executive transport utilizes a 9 percent thick airfoil; contrast this with the low-speed Piper Aztec, a twin-engine propeller-driven general aviation aircraft designed with a 14 percent thick airfoil.

Example 1 1 .3 I In this example, we illustrate the estimation of the critical Mach number for an airfoil using (a) the graphical solution discussed in this section, and (b) an analytical solution using a closed – form equation obtained from a combination of Equations (11.51) and (11.60). Consider the NACA 0012 airfoil at zero angle of attack shown at the top of Figure 11.8. The pressure coefficient distribution over this airfoil, measured in a wind tunnel at low speed, is given at the

c

Figure 1 1.8 Low-speed pressure coefficient distribution

over the surface of an NACA 0012 airfoil at zero angle of attack. Re = 3.65 x 106. (Source; R. J. Freuler and G. M. Gregorek, "An Evaluation of Four Single Element Airfoil Analytical Methods," in Advanced Technology Airfoil Research, NASA CP 2045, 1978, pp. 133-162.)

Following step three, these values are plotted as Curve В in Figure 11.9. The intersection of curves В and C is at point D. The freestream Mach number associated with point D is the critical Mach number for the NACA 0012 airfoil. From Figure 11.9, we have

(.b) Analytical Solution. In Figure 11.9, curve В is given by Equation (11.61)

At the intersection point D, (Cp)mill in Equation (11.61) is the critical pressure coefficient and Moo is the critical Mach number

Cp a = 0,43 (at point D) [11.62]

Vі – Mcr

Also, at point D the value of Cp, cr is given by Equation (11.60). Hence, at point D we can equate the right-hand sides of Equations (11.62) and (11.60),

-0.43 _ _j?_ [( +[(y – 1)/2]МС2ГУ/У ‘

yi^Mj У Ml [ 1 + (y – l)/2 J

Equation (11.63) is one equation with one unknown, namely, Mcr. The solution of Equation (11.63) gives the value of Mcr associated with the intersection point D in Figure 11.9, that is, the solution of Equation (11.63) is the critical Mach number for the NACA 0012 airfoil. Since Mcr appears in a complicated fashion on both sides of Equation (11.63), we solve the equation by trial-and-error by assuming different values of Mcr, calculating the values of both sides of Equation (11.63), and iterating until we find a value of Ma that results in both the right and left sides being the same value.

м -0.43 2 ГП+[(у-])/2]МІу/г-‘ _ ~

СГ 1 + (к — l)/2 ) _

0.72

-0.6196

-0.6996

0.73

-0.6292

-0.6621

0.74

-0.6393

-0.6260

0.738

-0.6372

-0.6331

0.737

-0.6362

-0.6367

0.7371

-0.6363

-0.6363

To four-place accuracy, when Mcr = 0.7371, both the left and right sides of Equation (11.63) have the same value. Therefore, the analytical solution yields

Note: Within the two-place accuracy of the graphical solution in part (a), both the graphical and analytical solutions give the same value of Mcr.

Question: How accurate is the estimate of the critical Mach number in this example? To answer this question, we examine some experimental pressure coefficient distributions for the NACA 0012 airfoil obtained at higher freestream Mach numbers. Wind tunnel measurements of the surface pressure distributions for this airfoil at zero angle of attack in a high-speed flow are shown in Figure 11.10; for Figure 11.10a, Mx = 0.575, and for Figure ll. lOfc,

(a)

(h)

Figure 11.10 Wind tunnel measurements of surface pressure coefficient distribution for the NACA 0012 airfoil at zero angle of attack. Experimental data of Frueler and Gregorek, NASA CP 2045.

(a) Мх = 0.575, (b) Moo = 0.725.

Mx = 0.725. In Figure 11.10a, the value of Cpxr = -1.465 at = 0.575 is shown as the dashed horizontal line. From the definition of critical pressure coefficient, any local value of Cp above this horizontal line corresponds to locally supersonic flow, and any local value below the horizontal line corresponds to locally subsonic flow. Clearly from the measured surface pressure coefficient distribution at Mx = 0.575 shown in Figure 11.10a, the flow is

locally subsonic at every point on the surface. Hence, Mж = 0.575 is below the critical Mach number. In Figure 1 l. lOh, which is for a higher Mach number, the value of C pa = —0.681 at Moo = 0.725 is shown as the dashed horizontal line. Here, the local pressure coefficient on the airfoil is higher than Cpxr at every point on the surface except at the point of minimum pressure, where (Cp)mi„ is essentially equal to Cp, cl. This means that for = 0.725, the flow is locally subsonic at every point on the surface except the point of minimum pressure, where the flow is essentially sonic. Hence, these experimental measurements indicate that the critical Mach number of the NACA 0012 airfoil at zero angle of attack is approximately 0.73. Comparing this experimental result with the calculated value of Mcr = 0.74 in this example, we see that our calculations are amazingly accurate, to within about one percent.

Similarity Parameters

In Section 1.7, we introduced the concept of dimensional analysis, from which sprung the similarity parameters necessary to ensure the dynamic similarity between two or more different flows (see Section 1.8). In the present section, we revisit the governing similarity parameters, but cast them in a slightly different light.

Consider a steady two-dimensional, viscous, compressible flow. The x-momen­tum equation for such a flow is given by Equation (15.19a), which for the present case reduces to

In Equation (15.27), p, u, p, etc., are the actual dimensional variables, say, p = kg/m3, etc. Let us introduce the following dimensionless variables:

/ _ p

, u

, _ V

U = ——

Poo

Voo

V ~ Voo

/ p

, _ X

, У

/X = ——

У = –

P’00

c

с

where Poo, Vqo, poo, and рьж are reference values (say, e. g., freestream values) and c is a reference length (say, the chord of an airfoil). In terms of these dimensionless

variables, Equation (15.27) becomes

where M0о and Reoo are the freestream Mach and Reynolds numbers, respectively, Equation (15.28) becomes

Equation (15.29) tells us something important. Consider two different flows over two bodies of different shapes. In one flow, the ratio of specific heats, Mach number, and Reynolds number are у і, Мж, and Re^j, respectively; in the other flow, these parameters have different values, y2, M^, and Re^. Equation (15.29) is valid for both flows. It can, in principle, be solved to obtain u’ as a function of x’ and y’. However, since y, Moo, and Reoo are different for the two cases, the coefficients of the derivatives in Equation (15.29) will be different. This will ensure, if

f (x y’)

represents the solution for one flow and

и = h(x y’)

represents the solution for the other flow, that

However, consider now the case where the two different flows have the same values of y, Moo, and Reoo – Now the coefficients of the derivatives in Equation (15.29) will be the same for both flows; that is, Equation (15.29) is numerically identical for the two flows. In addition, assume the two bodies are geometrically similar, so that the boundary conditions in terms of the nondimensional variables are the same. Then, the solutions of Equation (15.29) for the two flows in terms of u’ — f(x’, v’) and и’ = /г(х’, у’) must be identical; that is,

f{(x’,y’) = f2(x, y) [15.30]

Recall the definition of dynamically similar flows given in Section 1.8. There, we stated in part that two flows are dynamically similar if the distributions of V/Vx, p! P<x>, etc., are the same throughout the flow field when plotted against common nondimensional coordinates. This is precisely what Equation (15.30) is saying—that

u’ as a function of x’ and v’ is the same for the two flows. That is, the variation of the nondimensional velocity as a function of the nondimensional coordinates is the same for the two flows. How did we obtain Equation (15.30)? Simply by saying that y, Mqo, and Reoo are the same for the two flows and that the two bodies are geometrically similar. These are precisely the criteria for two flows to be dynamically similar, as originally stated in Section 1.8.

What we have seen in the above derivation is a formal mechanism to identify governing similarity parameters for a flow. By couching the governing flow equations in terms of nondimensional variables, we find that the coefficients of the derivatives in these equations are dimensionless similarity parameters or combinations thereof.

To see this more clearly, and to extend our analysis further, consider the en­ergy equation for a steady, two-dimensional, viscous, compressible flow, which from Equation (15.26) can be written as (assuming no volumetric heating and neglecting the normal stresses)

Turbulence Modeling

The simple equations given in Section 19.2 for boundary-layer thickness and skin friction coefficient for a turbulent flow over a flat plate are simplified results that are heavily empirically based. Modern calculations of turbulent flows over arbitrarily – shaped bodies involve the solution of the continuity, momentum, and energy equations along with some model of the turbulence. The calculations are carried out by means of computational fluid dynamic techniques. Here we will discuss only one model of tubulence, the Baldwin-Lomax turbulence model, which has become popular over the past two decades. We emphasize that the following discussion is intended only to give you the flavor of what is meant by a turbulence model.

19.3.1 The Baldwin-Lomax Model

In order to include the effects of turbulence in any analysis or computation, it is first necessary to have a model for the turbulence itself. Turbulence modeling is a state – of-the-art subject, and a recent survey of such modeling as applied to computations is given in Reference 85. Again, it is beyond the scope of the present book to give a detailed presentation of various turbulence models; the reader is referred to the literature for such matters. Instead, we choose to discuss only one such model here, because: (a) it is a typical example of an engineering-oriented turbulence model, (b) it is the model used in the majority of modern applications in turbulent, sub­sonic, supersonic, and hypersonic flows, and (c) we will discuss in the next chapter several applications which use this model. The model is called the Baldwin-Lomax turbulence model, first proposed in Reference 86. It is in the class of what is called an “eddy viscosity” model, where the effects of turbulence in the governing viscous

flow equations (such as the boundary-layer equations or the Navier-Stokes equations) are included simply by adding an additional term to the transport coefficients. For example, in all our previous viscous flow equations, /л is replaced by (ц. + t±r) and к by (к + kj) where jij and kT are the eddy viscosity and eddy thermal conductivity, respectively—both due to turbulence. In these expressions, ji and к are denoted as the “molecular” viscosity and thermal conductivity, respectively. For example, the x momentum boundary-layer equation for turbulent flow is written as

Moreover, the Baldwin-Lomax model is also in the class of “algebraic,” or “zero – equation,” models meaning that the formulation of the turbulence model utilizes just algebraic relations involving the flow properties. This is in contrast to one – and two-equation models which involve partial differential equations for the convection, creation, and dissipation of the turbulent kinetic energy and (frequently) the local vorticity. (See Reference 87 for a concise description of such one – and two-equation turbulence models.)

The Baldwin-Lomax turbulence model is described below. We give just a “cook­book” prescription for the model; the motivation and justification for the model are described at length in Reference 86. This, like all other turbulence models, is highly empirical. The final justification for its use is that it yields reasonable results across a wide range of Mach numbers, from subsonic to hypersonic. The model assumes that the turbulent-boundary layer is split into two layers, an inner and an outer layer, with different expressions for jiT in each layer:

{

(Mr) inner У — ycrossover r 1

, , . [19.6]

‘ M ‘/’ Jouter У _ ycrossover

where у is the local normal distance from the wall, and the crossover point from the inner to the outer layer is denoted by ycrossover- By definition, ycrossover is that point in the turbulent boundary where (мг)outer becomes less than (mt)inner – F°r the inner region:

and к and А+ are two dimensionless constants, specified later. In Equation (19.7), m is the local vorticity, defined for a two dimensional flow as

du dv З у dx

For the outer region:

where К and Ccp are two additional constants, and Fwake and /’кіеь are related to the function

Equation (19.12) will have a maximum value along a given normal distance y; this maximum value and the location where it occurs are denoted by Fmax and ymax, respec­tively. ІП Equation (19.11), Fwake is taken to be either Углах Fmax ОҐ (2wk Углах Fj. j –

whichever is smaller, where Cwk is constant, and

t/dif = у/ U2 + V2

Also, in Equation (19.11), FKieb is the Klebanoff intermittency factor, given by

The six dimensionless constants which appear in the above equations are: A+ = 26.0, Ccp = 1.6, Скіеь = 0-3, Cwk = 0.25, к = 0.4, and К = 0.0168. These constants are taken directly from Reference 86 with the understanding that, while they are not precisely the correct constants for most flows in general, they have been used successfully for a number of different applications. Note that, unlike many algebraic eddy viscosity models which are based on a characteristic length, the Baldwin-Lomax model is based on the local vorticity a>. This is a distinct advantage for the analysis of flows without an obvious mixing length, such as separated flows. Note that, like all eddy-viscosity turbulent models, the value of iij obtained above is dependent on the flowfield properties themselves (for example ш and p); this is in contrast to the molecular viscosity ц. which is solely a property of the gas itself.

The molecular values of viscosity coefficient and thermal conductivity are related through the Prandtl number

[19.15]

In lieu of developing a detailed turbulence model for the turbulent thermal conductivity kT, the usual procedure is to define a “turbulent” Prandtl number as Pr7 = цтср/кт. Thus, analogous to Equation (19.15), we have

where the usual assumption is that Pr7 = 1. Therefore, /і 7 is obtained from a given eddy-viscosity model (such as the Baldwin-Lomax model), and the corresponding kT is obtained from Equation (19.16).

Turbulence itself is a flowfield; it is not a simple property of the gas. This is why, as mentioned above, in an algebraic eddy viscosity model the values of /x7 and kT depend on the solution of the flow field—they are not pure properties of the gas as are /г and k. This is clearly seen in the Baldwin-Lomax model via Equation (19.7),

where fi t is a function of the local vorticity in the flow, oj—a flow field variable which comes out as part of the solution for the particular case at hand.

When Is A Flow Compressible?

As a corollary to Section 8.4, we are now in a position to examine the question, When does a flow have to be considered compressible, that is, when do we have to use analyses based on Chapters 7 to 14 rather than the incompressible techniques discussed in Chapters 3 to 6? There is no specific answer to this question; for subsonic flows, it is a matter of the degree of accuracy desired whether we treat p as a constant or as a variable, whereas for supersonic flow the qualitative aspects of the flow are so different that the density must be treated as variable. We have stated several times in the preceding chapters the rule of thumb that a flow can be reasonably assumed to be incompressible when M < 0.3, whereas it should be considered compressible when M > 0.3. There is nothing magic about the value 0.3, but it is a convenient dividing line. We are now in a position to add substance to this rule of thumb.

Consider a fluid element initially at rest, say, an element of the air around you. The density of this gas at rest is po. Let us now accelerate this fluid element isentropically to some velocity V and Mach number M, say, by expanding the air through a nozzle. As the velocity of the fluid element increases, the other flow properties will change according to the basic governing equations derived in Chapter 7 and in this chapter. In particular, the density p of the fluid element will change according to Equation

(8.43) ;

When Is A Flow Compressible?[8.43]

For у = 1.4, this variation is illustrated in Figure 8.6, where p/po is plotted as a function of M from zero to sonic flow. Note that at low subsonic Mach numbers, the variation of p/po is relatively flat. Indeed, for M < 0.32, the value of p deviates from po by less than 5 percent, and for all practical purposes the flow can be treated as incompressible. However, for M > 0.32, the variation in p is larger than 5 percent, and its change becomes even more pronounced as M increases. As a result, many aerodynamicists have adopted the rule of thumb that the density variation should be accounted for at Mach numbers above 0.3; that is, the flow should be treated as compressible. Of course, keep in mind that all flows, even at the lowest Mach numbers, are, strictly speaking, compressible. Incompressible flow is really a myth. However, as shown in Figure 8.6, the assumption of incompressible flow is very reasonable at low Mach numbers. For this reason, the analyses in Chapters 3 to 6 and the vast bulk of existing literature for incompressible flow are quite practical for many aerodynamic applications.

To obtain additional insight into the significance of Figure 8.6, let us ask how the ratio p/po affects the change in pressure associated with a given change in velocity. The differential relation between pressure and velocity for a compressible flow is given by Euler’s equation, Equation (3.12) repeated below:

When Is A Flow Compressible?

Figure 8.6 Isentropic variation of density with Mach number.

This can be written as

ІЕ. = py2dV

P P V

This equation gives the fractional change in pressure for a given fractional change in velocity for a compressible flow with local density p. If we now assume that the density is constant, say, equal to po as denoted in Figure 8.6, then Equation (3.12) yields

{djA = P0y2dV V P Jo p y

where the subscript zero implies the assumption of constant density. Dividing the last two equations, and assuming the same dV/V and p, we have

dp/p = P_

(dp/p) о po

Hence, the degree by which p/po deviates from unity as shown in Figure 8.6 is re­lated to the same degree by which the fractional pressure change for a given dV/V is predicted. For example, if p/po = 0.95, which occurs at about M = 0.3 in Figure 8.5, then the fractional change in pressure for a compressible flow with local density p as compared to that for an incompressible flow with density p0 will be about 5 percent different. Keep in mind that the above comparison is for the local fractional

change in pressure, the actual integrated pressure change is less sensitive. For ex­ample, consider the flow of air through a nozzle starting in the reservoir at nearly zero velocity and standard sea level values of po = 2116 lb/ft2 and T0 = 510°R, and expanding to a velocity of 350 ft/s at the nozzle exit. The pressure at the nozzle exit will be calculated assuming first incompressible flow and then compressible flow.

Подпись: P — Po~
Подпись: pV2 = 2116—  (0.002377) (350)2 = Подпись: 1970 lb/ft2

Incompressible flow: From Bernoulli’s equation,

T = To – —

2cn

Подпись: 519
Подпись: (350)2 2(6006)
Подпись: 508.8°R

Compressible flow: From the energy equation, Equation (8.30), with cp = 6006[(ft) (lb)/slug°R] for air,

Подпись: P_ Po
Подпись: j  Y/(Y-i) To)
Подпись: 508.83'5 519 J
Подпись: 0.9329

From Equation (7.32),

Подпись: 1974 lb/ft2p = 0.9329po = 0.9329(2116)

Note that the two results are almost the same, with the compressible value of pressure only 0.2 percent higher than the incompressible value. Clearly, the assumption of incompressible flow (hence, the use of Bernoulli’s equation) is certainly justified in this case. Also, note that the Mach number at the exit is 0.317 (work this out for yourself). Hence, we have shown that for a flow wherein the Mach number ranges from zero to about 0.3, Bernoulli’s equation yields a reasonably accurate value for the pressure—another justification for the statement that flows wherein M < 0.3 are essentially incompressible flows. On the other hand, if this flow were to continue to expand to a velocity of 900 ft/s, a repeat of the above calculation yields the following results for the static pressure at the end of the expansion:

Incompressible (Bernoulli’s equation): p = 1153 lb/ft2

Compressible: p = 1300 lb/ft2

Here, the difference between the two sets of results is considerable—a 13 percent difference. In this case, the Mach number at the end of the expansion is 0.86. Clearly, for such values of Mach number, the flow must be treated as compressible.

In summary, although it may be somewhat conservative, this author suggests on the strength of all the above information, including Figure 8.6, that flows wherein the local Mach number exceeds 0.3 should be treated as compressible. Moreover, when M < 0.3, the assumption of incompressible flow is quite justified.

M < 1 (subsonic flow)

M = 1 (sonic flow)

M > 1 (supersonic flow)

In the definition of M, a is the local speed of sound, a = л/yRT. In the theory of supersonic flow, it is sometimes convenient to introduce a “characteristic” Mach number M* defined as

Подпись:и

a* where a* is the value of the speed of sound at sonic conditions, not the actual local value. This is the same a* introduced at the end of Section 7.5 and used in Equation (8.35). The value of a* is given by a* = *JyRT*. Let us now obtain a relation between the actual Mach number M and this defined characteristic Mach number M*. Dividing Equation (8.35) by u2, we have

(a/u)2 1 _ у + 1 / a*2

у – 1 2 “ 2(y – 1) V и )

(1/M)2 = y + 1 /J_2 _ 1

у – 1 2(y – 1) M*/ 2

Подпись:2 2

M2 = —————— Z————-

(у + 1 )/M*2 – (у – 1)

Подпись: _Jy±l)M2_ 2 + (y - 1)M2 Подпись: [8.48]

Equation (8.47) gives M as a function of M*. Solving Equation (8.47) for M*2, we have

which gives M* as a function of M. As can be shown by inserting numbers into Equation (8.48) (try some yourself),

M* = 1

if M = 1

M* < 1

if M < 1

M* > 1

if M > 1

Iy + 1

M* It———— r

if M —>■ oo

у – 1

Therefore, M* acts qualitatively in the same fashion as M except that M* approaches a finite value when the actual Mach number approaches infinity.

In summary, a number of equations have been derived in this section, all of which stem in one fashion or another from the basic energy equation for steady, inviscid, adiabatic flow. Make certain that you understand these equations and become very

Подпись: Example 8.1 When Is A Flow Compressible?

familiar with them before progressing further. These equations are pivotal in the analysis of shock waves and in the study of compressible flow in general.

Hence, M* = VA26 = 2.06, as obtained above.

Подпись: Example 8.2In Example 3.1, we illustrated for an incompressible flow, the calculation of the velocity at a point on an airfoil when we were given the pressure at that point and the freestream velocity and pressure. (It would be useful to review Example 3.1 before going further.) The solution involved the use of Bernoulli’s equation. Let us now examine the compressible flow analog of Example 3.1. Consider an airfoil in a freestream where M^ — 0.6 and poo = 1 atm, as sketched in Figure 8.5. At point 1 on the airfoil the pressure is p = 0.7545 atm. Calculate the local Mach number at point 1. Assume isentropic flow over the airfoil.

Solution

We cannot use Bernoulli’s equation because the freestream Mach number is high enough that the flow should be treated as compressible. The free stream total pressure for Mx = 0.6 is, from Appendix A

Po. oo = — Poo = (1.276) (1) = 1.276 atm

Poo

pі = 0.7545 atm

M, = ?

 

Подпись:M = 0.6

When Is A Flow Compressible?

Poo = 1 atm

This is the local Mach number at point 1 on the airfoil in Figure 8.5.

Подпись: Example 8.3Note that flow velocity did not enter the calculations in Example 8.2. For compressible flow, Mach number is a more fundamental variable than velocity; we will see this time-and-time again in the subsequent sections and chapters dealing with compressible flow. However, we can certainly calculate velocities for compressible flow problems, but in such cases we usually need to know something about the temperature level of the flow. For the conditions that prevail in Example 8.2, calculate the velocity at point 1 on the airfoil when the free stream temperature is 59°F.

Solution

We will need to deal with consistent units. Since 0°F is the same as 460°R,

Too = 460 + 59 = 519°R

Подпись: or Подпись: Ti =TX[ — When Is A Flow Compressible?

The flow is isentropic, hence, from Equation (7.32)

From Equation (8.25), the speed of sound at point 1 is

ai = JyRTi = У(1.4)(1716)(478.9) = 1072.6 ft/s

When Is A Flow Compressible?

Hence,

 

V, = Midi = (0.9) (1072.6) =

 

965.4 ft/s

 

Elements of the Method of Characteristics

In this section, we only introduce the basic elements of the method of characteristics. A full discussion is beyond the scope of this book; see References 21, 25, and 34 for more details.

Consider a two-dimensional, steady, inviscid, supersonic flow in xy space, as given in Figure 13.2a. The flow variables (p, и, T, etc.) are continuous throughout this space. However, there are certain lines in xy space along which the derivatives of the flow-field variables (др/дх, du/dy, etc.) are indeterminate and across which may even be discontinuous. Such lines are called characteristic lines. This may sound strange at first; however, let us prove that such lines exist, and let us find their precise directions in the xy plane.

In addition to the flow being supersonic, steady, inviscid, and two-dimensional, assume that it is also irrotational. The exact governing equation for such a flow is given by Equation (11.12):

ІІ 1.12]

[Keep in mind that we are dealing with the full velocity potential ф in Equation

(11.12) , not the perturbation potential.] Since дф/dx = и and дф/ду — v, Equation

(11.12) can be written as

The velocity potential and its derivatives are functions of x and y, for example,

3 ф

— = f{x, y) dx

Hence, from the relation for an exact differential,

Examine Equations (13.1) to (13.3) closely. Note that they contain the second deriva­tives д2ф/дх2, д2ф/ду2, and д2ф/дхду. If we imagine these derivatives as “un­knowns,” then Equations (13.1), (13.2), and (13.3) represent three equations with three unknowns. For example, to solve for д2ф/дх dy, use Cramer’s rule as follows:

where N and D represent the numerator and denominator determinants, respectively. The physical meaning of Equation (13.4) can be seen by considering point A and its surrounding neighborhood in the flow, as sketched in Figure 13.3. The derivative Ь2ф/Ьх dy has a specific value at point A. Equation (13.4) gives the solution for д2ф/дх dy for an arbitrary choice of dx and dy. The combination of dx and dy defines an arbitrary direction ds away from point A as shown in Figure 13.3. In general, this direction is different from the streamline direction going through point A. In Equation (13.4), the differentials du and dv represent the changes in velocity that take place over the increments dx and dy. Hence, although the choice of dx and dy is arbitrary, the values of du and dv in Equation (13.4) must correspond to this choice. No matter what values of dx and dy are arbitrarily chosen, the corresponding values of du and dv will always ensure obtaining the same value of d2ф/dx dy at point A from Equation (13.4).

The single exception to the above comments occurs when dx and dy are chosen so that D = 0 in Equation (13.4). In this case, d2ф/дxdy is not defined. This situation will occur for a specific direction ds away from point A in Figure 13.3, defined for that specific combination of dx and dy for which I) = 0. However, we know that d2ф/dx 3у has a specific defined value at point A. Therefore, the only

consistent result associated with D = 0 is that N = 0, also; that is,

д2ф _ N _ 0 Эх ду ~ Ъ ~ 0

Here, д2ф/дх ду is an indeterminate form, which is allowed to be a finite value, that is, that value of д2ф/дх ду which we know exists at point A. The important conclusion here is that there is some direction (or directions) through point A along which д2ф/Эх ду is indeterminate. Since д2ф/дх ду — ди/ду = dv/dx, this implies that the derivatives of the flow variables are indeterminate along these lines. Hence, we have proven that lines do exist in the flow field along which derivatives of the flow variables are indeterminate; earlier, we defined such lines as characteristic lines.

Consider again point A in Figure 13.3. From our previous discussion, there are one or more characteristic lines through point A. Question: How can we calculate the precise direction of these characteristic lines? The answer can be obtained by setting D = 0 in Equation (13.4). Expanding the denominator determinant in Equation

(13.4) , and setting it equal to zero, we have

In Equation (13.6), dy/dx is the slope of the characteristic lines; hence, the subscript “char” has been added to emphasize this fact. Solving Equation (13.6) for (dy /dx )ctm by means of the quadratic formula, we obtain

/dy —2uv/a2 ± y/(2uv/a2)2 — 4(1 — u2/a2)( 1 — v2/a2)

dx)c har 2(1 — u2/a2)

/dy —uv/a2 ± л/(и2 + v2)/a2 — 1

dx ) char 1-м2 /a2

From Figure 13.3, we see that и = V cos в and v = V sin в. Hence, Equation (13.7) becomes

/dy (—V2cos0 sin0)/a2 ± _

W/char 1 – [(V2/a2)cos20] l3’8

Recall that the local Mach angle p. is given by p. = sin^'(l/M), or sin p, = 1 /М. Thus, V2/a2 = M2 = 1/ sin2 p,, and Equation (13.8) becomes

/ dy (—cost? sin в)/sin2 p. ± vTcos^’+^in^X/sin^Ti^^

dx )char 1 – (COS2 0)/Sin2/Г

After considerable algebraic and trigonometric manipulation, Equation (13.9) reduces to

[13.10]

Equation (13.10) is an important result; it states that two characteristic lines run through point A in Figure 13.3, namely, one line with a slope equal to tan(6 — ;u) and the other with a slope equal to tan (б + /і). The physical significance of this result is illustrated in Figure 13.4. Here, a streamline through point A is inclined at the angle в with respect to the horizontal. The velocity at point A is V, which also makes the angle в with respect to the horizontal. Equation (13.10) states that one characteristic line at point A is inclined below the streamline direction by the angle /x this characteristic line is labeled as C in Figure 13.4. Equation (13.10) also states that the other characteristic line at point A is inclined above the streamline direction by the angle /x this characteristic line is labeled as C+ in Figure 13.4. Examining Figure 13.4, we see that the characteristic lines through point A are simply the left – and right-running Mach waves through point A. Hence, the characteristic lines are Mach lines. In Figure 13.4, the left-running Mach wave is denoted by C+, and the right-running Mach wave is denoted by C_. Hence, returning to Figure 13.2a, the characteristics mesh consists of left- and right-running Mach waves which crisscross the flow field. There are an infinite number of these waves; however, for practical calculations we deal with a finite number of waves, the intersections of which define the grid points shown in Figure 13.2a. Note that the characteristic lines are curved in space because (1) the local Mach angle depends on the local Mach number, which is

a function of x and y, and (2) the local streamline direction 9 varies throughout the flow.

The characteristic lines in Figure 13.2a are of no use to us by themselves. The practical consequence of these lines is that the governing partial differential equations which describe the flow reduce to ordinary differential equations along the charac­teristic lines. These equations are called the compatibility equations, which can be found by setting N = 0 in Equation (13.4), as follows. When N = 0, the numerator determinant yields

/ u2 ( v2

I 1—— — J du dy + I 1——- 1 dx dv = 0

dv —(1 — u2/a2) dy

or — = ———- —Ц—— [13.11]

du 1 — Vі/a2 dx

Keep in mind that N is set to zero only when D = 0 in order to keep the flow – field derivatives finite, albeit of the indeterminate form 0/0. When D = 0, we are restricted to considering directions only along the characteristic lines, as explained earlier. Hence, when N = 0, we are held to the same restriction. Therefore, Equation

(13.11) holds only along the characteristic lines. Therefore, in Equation (13.11),

<У = / dy

dx ~ у dx ) ch^

Substituting Equations (13.12) and (13.7) into (13.11), we obtain

dv 1 — и2 /a2 —uv/a2 ± (u2 + v2)/a2 — 1

du 1 — v2/a2 1 — u2/a2

dv uv/a2 =F у/(и2 + v2)/a2 — 1

du 1 — v2/a2

Recall from Figure 13.3 that и = V cos 9 and v = V sind. Also, (и2 + v2)/a2 = V2/а2 = M2. Hence, Equation (13.13) becomes

£?(Vsin0) M2 cos 9 sin в VM2 — 1 d(V cos9) 1-М2 sin2 9

which, after some algebraic manipulations, reduces to

Examine Equation (13.14). It is an ordinary differential equation obtained from the original governing partial differential equation, Equation (13.1). However, Equation (13.14) contains the restriction given by Equation (13.12); that is, Equation (13.14) holds only along the characteristic lines. Hence, Equation (13.14) gives the com­patibility relations along the characteristic lines. In particular, comparing Equation

(13.14) with Equation (13.10), we see that

(applies along the C characteristic) [13.15] (applies along the C+ characteristic) [13.16]

Examine Equation (13.14) further. It should look familiar; indeed, Equation (13.14) is identical to the expression obtained for Prandtl-Meyer flow in Section 9.6, namely, Equation (9.32). Hence, Equation (13.14) can be integrated to obtain a result in terms of the Prandtl-Meyer function, given by Equation (9.42). In particular, the integration of Equations (13.15) and (13.16) yields

в + v(M) = const = K_ (along the C_ characteristic) [13.17]

в — v(M) = const = K+ (along the C+ characteristic) [13.18]

In Equation (13.17), K – is a constant along a given C_ characteristic; it has different values for different C_ characteristics. In Equation (13.18), К t is a constant along a given C+ characteristic; it has different values for different C+ characteristics. Note that our compatibility relations are now given by Equations (13.17) and (13.18), which are algebraic equations which hold only along the characteristic lines. In a general inviscid, supersonic, steady flow, the compatibility equations are ordinary differential equations; only in the case of two-dimensional irrotational flow do they further reduce to algebraic equations.

What is the advantage of the characteristic lines and their associated compatibility equations discussed above? Simply this—to solve the nonlinear supersonic flow, we need deal only with ordinary differential equations (or in the present case, algebraic equations) instead of the original partial differential equations. Finding the solution of such ordinary differential equations is usually much simpler than dealing with partial differential equations.

How do we use the above results to solve a practical problem? The purpose of the next section is to give such an example, namely, the calculation of the supersonic flow inside a nozzle and the determination of a proper wall contour so that shock waves do not appear inside the nozzle. To carry out this calculation, we deal with two types of grid points: (1) internal points, away from the wall, and (2) wall points. Characteristics calculations at these two sets of points are carried out as follows.