Category When Is A Flow Compressible?

Subsonic Compressible Flow over. Airfoils: Linear Theory

During the war a British engineer named Frank Whittle invented the jet engine, and deHavilland built the first production-type model. He produced a jet plane named Vampire, the first to exceed 500 mph. Then he built the experimental DH 108, and released it to young Geoffrey for test. In the first cautious trials the new plane behaved beautifully; but as Geoffrey stepped up the speed he unsuspectingly drew closer to an invisible wall in the sky then unknown to anyone, later named the sound barrier, which can destroy a plane not designed to pierce it. One evening he hit the speed of sound, and the plane disintegrated. Young Geoffrey’s body was not found for ten days.

From the Royal Air Force Flying Review, as digested in Reader’s Digest, 1959

11.1 Introduction

The above quotation refers to an accident which took place on September 27, 1946, when Geoffrey deHavilland, son of the famed British airplane designer Sir Geoffrey deHavilland, took the D. H. 108 Swallow up for an attack on the world’s speed record. At that time, no airplane had flown at or beyond the speed of sound. The Swallow was an experimental jet-propelled aircraft with swept wings and no tail. During its first high-speed, low-level run, the Swallow encountered major compressibility problems and broke up in the air. deHavilland was killed instantly. This accident strengthened the opinion of many that Mach 1 stood as a barrier to manned flight and that no airplane would ever fly faster than the speed of sound. This myth of the “sound barrier” originated in the early 1930s. It was in full force by the time of the

Volta Conference in 1935 (see Section 7.1). In light of the opening quotation, the idea of a sound barrier was still being discussed in the popular literature as late as 1959, 12 years after the first successful supersonic flight by Captain Charles Yeager on October 14, 1947.

Of course, we know today that the sound barrier is indeed a myth; the supersonic transport Concorde flies at Mach 2, and some military aircraft are capable of Mach 3 and slightly beyond. The X-15 hypersonic research airplane has flown at Mach 7, and the Apollo lunar return capsule successfully reentered the earth’s atmosphere at Mach 36. Supersonic flight is now an everyday occurrence. So, what caused the early concern about a sound barrier? In the present chapter, we develop a theory applicable to high-speed subsonic flight, and we see how the theory predicts a monotonically increasing drag going to infinity as Mx —> 1. It was this type of result that led some people in the early 1930s to believe that flight beyond the speed of sound was impossible. However, we also show in this chapter that the approximations made in the theory break down near Mach 1 and that in reality, although the drag coefficient at Mach 1 is large, it is still a manageable finite number.

Specifically, the purpose of this chapter is to examine the properties of two­dimensional airfoils at Mach numbers above 0.3, where we can no longer assume incompressible flow, but below Mach 1. That is, this chapter is an extension of the airfoil discussions in Chapter 4 (which applied to incompressible flow) to the high-speed subsonic regime.

In the process, we climb to a new tier in our study of compressible flow. If you survey our discussions so far of compressible flow, you will observe that they treat one-dimensional cases such as normal shock waves and flows in ducts. Even oblique shock waves, which are two – and three-dimensional in nature, depend only on the component of Mach number normal to the wave. Therefore, we have not been explicitly concerned with a multidimensional flow. As a consequence, note that the types of equations which allow an analysis of these flows are algebraic equations, and hence are relatively easy to solve in comparison with partial differential equations. In Chapters 8 to 10, we have dealt primarily with such algebraic equations. These algebraic equations were obtained by applying the integral forms of the conservation equations [Equations (2.48), (2.64), and (2.95)] to appropriate control volumes where the flow properties were uniform over the inflow and outflow faces of the control volume. However, for general two – and three-dimensional flows, we are usually not afforded such a luxury. Instead, we must deal directly with the governing equations in their partial differential equation form (see Chapter 2). Such is the nature of the present chapter. Indeed, for the remainder of our aerodynamic discussions in this book, we appeal mainly to the differential forms of the continuity, momentum, and energy equations [such as Equations (2.52), (2.113a to c), and (2.114)].

The road map for this chapter is given in Figure 11.1. We are going to return to the concept of a velocity potential, first introduced in Section 2.15. We are going to combine our governing equations so as to obtain a single equation simply in terms of the velocity potential; that is, we are going to obtain for compressible flow an equation analogous to Laplace’s equation derived for incompressible flow in Section 3.7 [see Equation (3.40)]. However, unlike Laplace’s equation, which is linear, the exact

velocity potential equation for compressible flow is nonlinear. By making suitable approximations, we are able to linearize this equation and apply it to thin airfoils at small angles of attack. The results enable us to correct incompressible airfoil data for the effects of compressibility—so-called compressibility corrections. Finally, we conclude this chapter by discussing several practical aspects of airfoil and general wing-body aerodynamics at speeds near Mach 1.

Introduction то the Fundamental Principles and Equations of Viscous Flow

I do not see then, I admit, how one can explain the resistance of fluids by the theory in a satisfactory manner. It seems to me on the contrary that this theory, dealt with and studied with profound attention gives, at least in most cases, resistance absolutely zero: a singular paradox which I leave to geometricians to explain.

Jean LeRond d’Alembert, 1768

15.1 Introduction

In the above quotation, the “theory” referred to by d’Alembert is inviscid, incom­pressible flow theory; we have seen in Chapter 3 that such theory leads to a prediction of zero drag on a closed two-dimensional body—this is d’Alembert’s paradox. In reality, there is always a finite drag on any body immersed in a moving fluid. Our earlier predictions of zero drag are a result of the inadequacy of the theory rather than some fluke of nature. With the exception of induced drag and supersonic wave drag, which can be obtained from inviscid theory, the calculation of all other forms of drag must explicitly take into account the presence of viscosity, which has not been included in our previous inviscid analyses. The purpose of the remaining chapters in this book is to discuss the basic aspects of viscous flows, thus “rounding out” our overall presentation of the fundamentals of aerodynamics. In so doing, we ad­dress the predictions of aerodynamic drag and aerodynamic heating. To help put our

current discussion in perspective, return to the block diagram of flow categories given in Figure 1.31. All of our previous discussions have focused on blocks D, E, and F—inviscid, incompressible and compressible flows. Now, for the remaining six chapters, we move to the left branch in Figure 1.38, and deal with blocks С, E, and F—viscous, incompressible and compressible flows.

Our treatment of viscous flows will be intentionally brief—our purpose is to present enough of the fundamental concepts and equations to give you the flavor of viscous flows. A thorough presentation of viscous flow theory would double the size of this book (at the very least) and is clearly beyond our scope. A study of viscous flow is an essential part of any serious study of aerodynamics. Many books have been exclusively devoted to the presentation of viscous flows; References 42 and 43 are two good examples. You are encouraged to examine these references closely.

The road map for the present chapter is given in Figure 15.1. Our course is to first examine some qualitative aspects of viscous flows as shown on the left branch of Figure 15.1. Then we quantify some of these aspects as given on the right branch. In the process, we obtain the governing equations for a general viscous flow—in particular, the Navier-Stokes equations (the momentum equations) and the viscous flow energy equation. Finally, we examine a numerical solution to these equations.

Boundary Layers over Arbitrary Bodies: Finite-Difference Solution

“Exact” solutions of the boundary layer equations. Equations (17.28)—(17.31), for the flow over bodies of arbitrary shape did not occur until the advent of the high-speed digital computer and ultimately not until the beginnings of computational fluid dy­namics. In this section we discuss a finite-difference technique for solving the general boundary layer equations; such finite-difference solutions represent the current state of the art in the analysis of boundary layers.

Let us set the perspective for our discussion. Equations (17.28)—(17.31) are the general boundary layer equations. For the special case of the flat plate, these equa­tions reduced to Equations (18.42) and (18.43), and for the stagnation region they reduced to Equations (18.63) and (18.64). In both special cases, these equations in terms of the transformed dependent and independent variables led to self-similar solu­tions (flow variations only in the transformed i] direction). For the general case of an arbitrary body, it is still useful to transform the full boundary-layer equations, Equa­tions (17.28)—(17.31), via the transformation given by Equations (18.59)-( 18.62). For a detailed derivation of these transformed equations, see Chapter 6 of Reference 55. The resulting form of the equation is:

x momentum:


where as before C = ррь/pe/ze, /’ = и/ие, and g = hj he. In Equations (18.84)—

(18.86) , the prime denotes the partial derivative with respect to q, that is, f = df/dq. Equations (18.84)—(18.86) are simply the transformed versions of Equations (17.28)-

(17.31) , with no loss of authority.

Examine Equations (18.84)—(18.86); they are the transformed compressible boundary layer equations. They are still partial differential equations, where both / and g are functions of § and r]. They contain no further approximations or assump­tions beyond those associated with the original boundary-layer equations. However, they are certainly in a less recognizable, somewhat more complicated-looking form than the original equations. However, do not be disturbed by this; in reality Equa­tions (18.84)—(18.86) are in a form that proves to be practical and useful.

The above transformed boundary-layer equations must be solved subject to the following boundary conditions. The physical boundary conditions were given imme­diately following Equations (17.28)—(17.31); the corresponding transformed bound­ary conditions are:

At the wall: q = 0 / = /’ = 0 g = gw (fixed wall temperature)

or g’ = 0 (adiabatic wall)

At the boundary-layer edge: q —>• oo /’ = 1 g=l

In general, solutions of Equations (18.84), (18.85), and (18.86) along with the appropriate boundary conditions yield variations of velocity and enthalpy throughout the boundary layer, via и = uef(%, q) and h = heg(£, q). The pressure throughout the boundary layer is known, because the known pressure distribution (or equivalently the known velocity distribution) at the edge of the boundary is given by pe = pe(^ ), and this pressure is impressed without change through the boundary layer in the lo­cally normal direction via Equation (18.85), which says that p = constant in the normal direction at any § location. Finally, knowing h and p throughout the bound­ary layer, equilibrium thermodynamics provides the remaining variables through the appropriate equations of state, for example, T = T{h, p), p = p(h, p), etc.

Whitcomb—Architect of the Area Rule and the Supercritical Wing

The developments of the area rule (Section 11.8) and the supercritical airfoil (Section 11.9) are two of the most important advancements in aerodynamics since 1950. That both developments were made by the same man—Richard T. Whitcomb—is remark­able. Who is this man? What qualities lead to such accomplishments? Let us pursue these matters further.

Richard Whitcomb was born on February 21, 1921, in Evanston, Illinois. At an early age, he was influenced by his grandfather, who had known Thomas A. Edison. In an interview with The Washington Post on August 31, 1969, Whitcomb is quoted as saying: “I used to sit around and hear stories about Edison. He sort of developed into my idol.” Whitcomb entered the Worcester Polytechnic Institute in 1939. (This is the same school from which the rocket pioneer, Robert H. Goddard, had graduated 31 years earlier.) Whitcomb distinguished himself in college and graduated with a mechanical engineering degree with honors in 1943. Informed by a Fortune magazine article on the research facilities at the NACA Langley Memorial Laboratory,

Whitcomb immediately joined the NACA. He became a wind-tunnel engineer, and as an early assignment he worked on design problems associated with the Boeing B-29 Superfortress. He remained with the NACA and later its successor, NASA, until his retirement in 1980—spending his entire career with the wind tunnels at the Langley Research Center. In the process, he rose to become head of the Eight-foot Tunnel Branch at Langley.

Whitcomb conceived the idea of the area rule as early as 1951. He tested his idea in the transonic wind tunnel at Langley. The results were so promising that the aeronautical industry changed designs in midstream. For example, the Convair F – 102 delta-wing fighter had been designed for supersonic flight, but was having major difficulty even exceeding the speed of sound—the increase in drag near Mach 1 was simply too large. The F-102 was redesigned to incorporate Whitcomb’s area rule and afterward was able to achieve its originally intended supersonic Mach number. The area rule was such an important aerodynamic breakthrough that it was classified “secret” from 1952 to 1954, when airplanes incorporating the area rule began to roll off the production line. In 1954, Whitcomb was given the Collier Trophy—an annual award for the “greatest achievement in aviation in America.”

In the early 1960s, Whitcomb turned his attention to airfoil design, with the objective again of decreasing the large drag rise near Mach 1. Using the existing knowledge about airfoil properties, a great deal of wind-tunnel testing, and intuition honed by years of experience, Whitcomb produced the supercritical airfoil. Again, this development had a major impact on the aeronautical industry, and today virtu­ally all new commercial transport and executive aircraft designs are incorporating a supercritical wing. Because of his development of the supercritical airfoil, in 1974 NASA gave Whitcomb a cash award of $25,000—the largest cash award ever given by NASA to a single individual.

There are certain parallels between the personalities of the Wright brothers and Richard Whitcomb: (1) they all had powerful intuitive abilities which they brought to bear on the problem of flight, (2) they were totally dedicated to their work (none of them ever married), and (3) they did a great deal of their work themselves, trusting only their own results. For example, here is a quote from Whitcomb which appears in the same Washington Post interview mentioned above. Concerning the detailed work on the development of the supercritical airfoil, Whitcomb says:

I modified the shape of the wing myself as we tested it. It’s just plain easier this way.

In fact my reputation for filing the wing’s shape has become so notorious that the people at North American have threatened to provide me with a 10-foot file to work on the real airplane, also.

Perhaps the real ingredient for Whitcomb’s success is his personal philosophy, as well as his long hours at work daily. In his own words:

There’s been a continual drive in me ever since I was a teenager to find a better way to do everything. A lot of very intelligent people are willing to adapt, but only to a certain extent. If a human mind can figure out a better way to do something, let’s do it. I can’t just sit around. I have to think.

Students take note!

Equal Wall Temperatures

Here we assume that Te = T„; that is, he = h„. The enthalpy profile for this case, from Equation (16.16), is

. 2


hw + – Pr u]

In terms of temperature, this becomes

Note that the temperature varies parabolically with у, as sketched in Figure 16.5. The maximum value of temperature occurs at the midpoint, у = D/2. This maximum value is obtained by evaluating Equation (16.33) at у = D/2.

The heat transfer at the walls is obtained from Equations (16.24) and (16.25) as

At у = 0: qw = r-‘-^ [16.35]

At у = D:

Equations (16.35) and (16.36) are identical; the heat transfers at the upper and lower walls are equal. In this case, as can be seen by inspecting the temperature distribution shown in Figure 16.5, the upper and lower walls are both cooler than the adjacent fluid. Hence, at both the upper and lower walls, heat is transferred from the fluid to the wall.

Question: Since the walls are at equal temperature, where is the heat transfer coming from? Answer: Viscous dissipation. The local temperature increase in the flow as sketched in Figure 16.5 is due solely to viscous dissipation within the fluid. In turn, both walls experience an aerodynamic heating effect due to this viscous dis­sipation. This is clearly evident in Equations (16.35) and (16.36), where qw depends on the velocity ue. Indeed, qw is directly proportional to the square of ue. In light of Equation (16.9), Equations (16.35) and (16.36) can be written as


which further emphasizes that qw is due entirely to the action of shear stress in the flow. From Equations (16.35) to (16.37), we can make the following conclusions that reflect general properties of most viscous flows:

1. Everything else being equal, aerodynamic heating increases as the flow velocity increases. This is why aerodynamic heating becomes an important design factor in high-speed aerodynamics. Indeed, for most hypersonic vehicles, you can begin to appreciate that viscous dissipation generates extreme temperatures within the boundary layer adjacent to the vehicle surface and frequently makes aerodynamic heating the dominant design factor. In the Couette flow case shown here—a far cry from hypersonic flow—we see that qw varies directly as u2e.

2. Everything else being equal, aerodynamic heating decreases as the thickness of the viscous layer increases. For the case considered here, qw is inversely proportional to D. This conclusion is the same as that made for the above case of negligible viscous dissipation but with unequal wall temperature.

Shock-Wave/Boundary-Layer Interaction

The flow field that results when a shock wave impinges on a boundary layer can only be calculated in detail by means of a numerical solution of the complete Navier – Stokes equations. The qualitative physical aspects of a two-dimensional shock – wave/boundary-layer interaction are sketched in Figure 20.6. Here we see a boundary layer growing along a flat plate, where at some downstream location an incident shock wave impinges on the boundary layer. The large pressure rise across the shock wave acts as a severe adverse pressure gradient imposed on the boundary layer, thus causing the boundary layer to locally separate from the surface. Because the high pressure be­hind the shock feeds upstream through the subsonic portion of the boundary layer, the separation takes place ahead of the impingement point of the incident shock wave. In turn, the separated boundary layer induces a shock wave, identified here as the induced separation shock. The separated boundary layer subsequently turns back toward the plate, reattaching to the surface at the reattachment shock. Between the separation and reattachment shocks, expansion waves are generated where the boundary layer is turning back toward the surface. At the point of reattachment, the boundary layer has become relatively thin, the pressure is high, and consequently this becomes a region of high local aerodynamic heating. Further away from the plate, the separation and reattachment shocks merge to form the conventional “reflected shock wave” which is

Figure 20.4 Streamlines for the low Reynolds flow

over a Wortmann airfoil. Re = 100,000. (a) Laminar flow, (b) Turbulent flow.

expected from the classical inviscid picture (see, for example, Figure 9.17). The scale and severity of the interaction picture shown in Figure 20.6 depends on whether the boundary layer is laminar or turbulent. Since laminar boundary layers separate more readily than turbulent boundary layers, the laminar interaction usually takes place more readily with more severe attendant consequences than the turbulent interaction. However, the general qualitative aspects of the interaction as sketched in Figure 20.6 are the same.

The fluid dynamic and mathematical details of the interaction region sketched in Figure 20.6 are complex, and the full prediction of this flow is still a state-of-the-art research problem. However, great strides have been made in recent years with the application of computational fluid dynamics to this problem, and solutions of the full

Navier-Stokes equations for the flow sketched in Figure 20.6 have been obtained. For example, experimental and computational data for the two-dimensional interaction of a shock wave impinging on a turbulent flat plate boundary layer are given in Fig­ure 20.7, obtained from Reference 86. In Figure 20.7a, the ratio of surface pressure to freestream total pressure is plotted versus distance along the surface (nondimension – alized by <$o, the boundary-layer thickness ahead of the interaction). Here, л’о is taken as the theoretical inviscid flow impingement point for the incident shock wave. The freestream Mach number is 3. The Reynolds number based on <50 is about 106. Note in Figure 20.7a that the surface pressure first increases at the front of the interaction region (ahead of the theoretical incident shock impingement point), reaches a plateau through the center of the separated region, and then increases again as the reattachment

point is approached. The pressure variation shown in Figure 20.7a is typical of that for a two-dimensional shock-wave/boundary-layer interaction. The open circles cor­respond to experimental measurements of Reda and Murphy (Reference 88). The curve is obtained from a numerical solution of the Navier-Stokes equations as re­ported in Reference 86 and using the Baldwin-Lomax turbulence model discussed in Section 19.3.1. In Figure 20.1b the variation of surface shear stress plummets to zero, reverses its direction (negative values) in a rather complex variation, and then recovers to a positive value in the vicinity of the reattachment point. The two circles on the horizontal axis denote measured separation and reattachment points, and the curve is obtained from the calculations of Reference 86.

Supersonic Flow Over Wedges and Cones

For the supersonic flow over wedges, as shown in Figures 9.10 and 9.11, the oblique shock theory developed in Section 9.2 is an exact solution of the flow field; no simpli­fying assumptions have been made. Supersonic flow over a wedge is characterized by an attached, straight oblique shock wave from the nose, a uniform flow downstream of the shock with streamlines parallel to the wedge surface, and a surface pressure equal to the static pressure behind the oblique shock p2. These properties are sum­marized in Figure 9.14a. Note that the wedge is a two-dimensional profile; in Figure 9.14a, it is a section of a body that stretches to plus or minus infinity in the direction perpendicular to the page. Hence, wedge flow is, by definition, two-dimensional flow, and our two-dimensional oblique shock theory fits this case nicely.

In contrast, consider the supersonic flow over a cone, as sketched in Figure 9.14b. There is a straight oblique shock which emanates from the tip, just as in the case of a wedge, but the similarity stops there. Recall from Chapter 6 that flow over a three-dimensional body experiences a “three-dimensional relieving effect.” That is, in comparing the wedge and cone in Figure 9.14, both with the same 20° angle, the flow over the cone has an extra dimension in which to move, and hence it more easily adjusts to the presence of the conical body in comparison to the two-dimensional wedge. One consequence of this three-dimensional relieving effect is that the shock wave on the cone is weaker than on the wedge; that is, it has a smaller wave angle, as compared in Figure 9.14. Specifically, the wave angles for the wedge and cone are 53.3 and 37°, respectively, for the same body angle of 20° and the same upstream Mach number of 2.0. In the case of the wedge (Figure 9.14a), the streamlines are deflected by exactly 20° through the shock wave, and hence downstream of the shock the flow is exactly parallel to the wedge surface. In contrast, because of the weaker shock on the cone, the streamlines are deflected by only 8° through the shock, as shown in Figure 9.14b. Therefore, between the shock wave and the cone surface, the streamlines must gradually curve upward in order to accommodate the 20° cone. Also, as a consequence of the three-dimensional relieving effect, the pressure on the surface of the cone, pc, is less than the wedge surface pressure P2, and the cone surface Mach number Mc is greater than that on the wedge surface М2. In short, the main differences between the supersonic flow over a cone and wedge, both with the same body angle, are that (1) the shock wave on the cone is weaker, (2) the cone surface pressure is less, and (3) the streamlines above the cone surface are curved rather than straight.

The analysis of the supersonic flow over a cone is more sophisticated than the oblique shock theory given in this chapter and is beyond the scope of this book. For details concerning supersonic conical flow analysis, see Chapter 10 of Reference 21. However, it is important for you to recognize that conical flows are inherently different from wedge flows and to recognize in what manner they differ. This has been the purpose of the present section.

Consider a wedge with a 15° half angle in a Mach 5 flow, as sketched in Figure 9.15. Calculate | Example 9.5 the drag coefficient for this wedge. (Assume that the pressure over the base is equal to freestream static pressure, as shown in Figure 9.15.)


Consider the drag on a unit span of the wedge D’. Hence,

D’ D’

From Figure 9.15,

D’ = 2p2l sin6? — 2pl sinfl = (21 sin$)(p2 — P)

(Note: The drag is finite for this case. In a supersonic or hypersonic inviscid flow over a two-dimensional body, the drag is always finite. D’Alembert’s paradox does not hold for freestream Mach numbers such that shock waves appear in the flow. The fundamental reason for the generation of drag here is the presence of shock waves. Shocks are always a dissipative, drag-producing mechanism. For this reason, the drag in this case is called wave drag, and cd is the wave-drag coefficient, more properly denoted as cdtW.)

The Time-Dependent Technique: Application to Supersonic Blunt Bodies

The method of characteristics described in Section 13.2 is applicable only to super­sonic flows; the characteristic lines are not defined in a practical fashion for steady, subsonic flow. Also, the particular finite-difference method outlined in Section 13.4 applies only to supersonic flows; if it were to be used in a locally subsonic region, the calculation would blow up. The reason for both of the above comments is that the method of characteristics and the steady flow, forward-marching finite-difference technique depend on the governing equations being mathematically “hyperbolic.” In contrast, the equations for steady subsonic flow are “elliptic.” (See Reference 21 for a description of these mathematical classifications.) The fact that the governing equa­tions change their mathematical nature in going from locally supersonic to locally subsonic flow has historically caused theoretical aerodynamicists much grief. One problem in particular, namely, the mixed subsonic-supersonic flow over a supersonic blunt body as described in Section 9.5, was a major research area until a breakthrough was made in the late 1960s for its proper numerical solution. The purpose of this section is to describe a numerical finite-difference solution which readily allows the calculation of mixed subsonic-supersonic flows—the time-dependent method—and to show how it is used to solve supersonic blunt-body flows. Time-dependent tech­niques are very common in modem computational fluid dynamics, and as a student of aerodynamics, you should be familiar with their philosophy.

Consider a blunt body in a supersonic stream, as sketched in Figure 13.9a. The shape of the body is known and is given by b = b(y). For a given freestream Mach number Moo, we wish to calculate the shape and location of the detached shock wave, as well as the flow-field properties between the shock and the body. The physical aspects of this flow field were described in Section 9.5, which you should review before progressing further.

The flow around a blunt body in a supersonic stream is rotational. Why? Examine Figure 13.10, which illustrates several streamlines around the blunt body. The flow is inviscid and adiabatic. In the uniform freestream ahead of the shock wave, the entropy is the same for each streamline. However, in crossing the shock wave, each streamline traverses a different part of the wave, and hence experiences a different increase in entropy. That is, the streamline at point a in Figure 13.10 crosses a normal shock, and hence experiences a large increase in entropy, whereas the streamline at point b crosses a weaker, oblique shock, and therefore experiences a smaller increase in entropy, Sb < sa. The streamline at point c experiences an even weaker portion



(b) Computational plane

Figure 13.9 Blunt-body flow field in both the

physical and computational planes.

of the shock, and hence sc < sh < sa. The net result is that in the flow between the shock and the body, the entropy along a given streamline is constant, whereas the entropy changes from one streamline to the next; that is, an entropy gradient exists normal to the streamlines. It can readily be shown (see chapter 6 of Reference 21) that an adiabatic flow with entropy gradients is rotational. Hence, the flow field over a supersonic blunt body is rotational.

In light of the above, we cannot use the velocity potential equation to analyze the blunt-body flow. Rather, the basic continuity, momentum, and energy equations must be employed in their fundamental form, given by Equations (7.40), (7.42a and b), and (7.44). With no body forces, these equations are

Notice the form of the above equations; the time derivatives are on the left, and all spatial derivatives are on the right. These equations are in the form necessary for a time-dependent finite-difference solution, as described below.

Return to Figure 13.9a. Recall that the body shape and freestream conditions are given, and we wish to calculate the shape and location of the shock wave as well as the flow field between the shock and body. We are interested in the steady flow over the blunt body; however, we use a time-dependent method to obtain the steady flow. The basic philosophy of this method is as follows. First, assume a shock-wave shape and location. Also, cover the flow field between the shock and body with a series of grid points, as sketched in Figure 13.9a. At each of these grid points, assume values of all the flow variables, p, u, v, etc. These assumed values are identified as initial conditions at time t = 0. With these assumed values, the spatial derivatives on the right sides of Equations (13.59) to (13.62) are known values (obtained from finite differences). Hence, Equations (13.59) to (13.62) allow the calculation of the time derivatives dp/dt, du/dt, etc. In turn, these time derivatives allow us to calculate the flow properties at each grid point at a later instant in time, say, At. The flow properties at time t = At are different from at t = 0. A repetition of this cycle gives the flow-field variables at all grid points at time t — 2At. As this cycle is repeated many hundreds of times, the flow-field properties at each grid point are calculated as a function of time. For example, the time variation of ut J is sketched in Figure

13.11. At each time step, the value of u, j is different; however, at large times the changes in Uij from one time step to another become small, and Uij approaches a steady-state value, as shown in Figure 13.11. It is this steady-state value that we want; the time-dependent approach is simply a means to that end. Moreover, the shock-wave shape and location will change with time; the new shock location and shape at each time step are calculated so as to satisfy the shock relations across the wave at each of the grid points immediately behind the wave. At large times, as the flow-field variables approach a steady state, the shock shape and location also approach a steady state. Because of the time-dependent motion of the shock wave, the wave shape is a function of both t and у as shown in Figure 13.9a, s = s(y, t).

Given this philosophy, let us examine a few details of the method. First, note that the finite-difference grid in Figure 13.9a is curved. We would like to apply our finite differences in a rectangular grid; hence, in Equations (13.59) to (13.62) the independent variables can be transformed as

£ =———- and г/ — у

where b = b(y) gives the abscissa of the body and і = s(y, t) gives the abscissa of the shock. The above transformation produces a rectangular grid in the computational plane, shown in Figure 13.9b, where the body corresponds to £ = 0 and the shock corresponds to $ — 1. All calculations are made in this transformed, computational plane.

The finite-difference calculations themselves can be carried out using MacCor – mack’s method (see Section 13.4) applied as follows. The flow-field variables can be advanced in time using a Taylor series in time; for example,


In Equation (13.63), we know the density at grid point (i, j) at time t; that is, we know pij(t). Then Equation (13.63) allows us to calculate the density at the same grid point at time t + At, that is, Pij(t + At), if we know a value of the average time derivative [(dp/df), j]ave. This time derivative is an average between times t and t + At and is obtained from a predictor-corrector process as follows.

Introduction to Boundary Layers

A very satisfactory explanation of the physical process in the boundary layer between a fluid and a solid body could be obtained by the hypothesis of an adhesion of the fluid to the walls, that is, by the hypothesis of a zero relative velocity between fluid and wall. If the viscosity was very small and the fluid path along the wall not too long, the fluid velocity ought to resume its normal value at a very short distance from the wall. In the thin transition layer however, the sharp changes of velocity, even with small coefficient of friction, produce marked results.

Ludwig Prandtl, 1904

1 7.1 Introduction

The above quotation is taken from an historic paper given by Ludwig Prandtl at the third Congress of Mathematicians at Heidelberg, Germany, in 1904. In this paper, the concept of the boundary layer was first introduced—a concept which eventually revolutionized the analysis of viscous flows in the twentieth century and which allowed the practical calculation of drag and flow separation over aerodynamic bodies. Before Prandtl’s 1904 paper, the Navier-Stokes equations discussed in Chapter 15 were well known, but fluid dynamicists were frustrated in their attempts to solve these equations for practical engineering problems. After 1904, the picture changed completely. Using Prandtl’s concept of a boundary layer adjacent to an aerodynamic surface, the Navier-Stokes equations can be reduced to a more tractable form called the boundary – layer equations. In turn, these boundary-layer equations can be solved to obtain the distributions of shear stress and aerodynamic heat transfer to the surface. Prandtl’s boundary-layer concept was an advancement in the science of fluid mechanics of the caliber of a Nobel prize, although he never received that honor. The purpose of this chapter is to present the general concept of the boundary layer and to give a

few representative samples of its application. Our purpose here is to provide only an introduction to boundary-layer theory; consult Reference 42 for a rigorous and thorough discussion of boundary-layer analysis and applications.

What is a boundary layer? We have used this term in several places in our previous chapters, first introducing the idea in Section 1.10 and illustrating the concept in Figure 1.35. The boundary layer is the thin region of flow adjacent to a surface, where the flow is retarded by the influence of friction between a solid surface and the fluid. For example, a photograph of the flow over a supersonic body is shown in Figure 17.1, where the boundary layer (along with shock and expansion waves and the wake) is made visible by a special optical technique called a shadowgraph (see References 25 and 26 for discussions of the shadowgraph method). Note how thin the boundary layer is in comparison with the size of the body; however, although the boundary layer occupies geometrically only a small portion of the flow field, its influence on the drag and heat transfer to the body is immense—in Prandtl’s own words as quoted above, it produces “marked results.”

The purpose of the remaining chapters is to examine these “marked results.” The road map for the present chapter is given in Figure 17.2. In the next section, we discuss some fundamental properties of boundary layers. This is followed by a development

Figure I 7.1 The boundary layer on an aerodynamic body. (Courtesy of the U. S. Army Ballistics Laboratory, Aberdeen, Maryland.}

Figure I 7.2 Road map for Chapter 17.

of the boundary-layer equations, which are the continuity, momentum, and energy equations written in a special form applicable to the flow in the thin viscous region adjacent to a surface. The boundary layer equations are partial differential equations that apply inside the boundary layer.

Finally, we note that this chapter represents the second of the three options discussed in Section 15.7 for the solution of the viscous flow equations, namely, the simplification of the Navier-Stokes equations by neglecting certain terms that are smaller than other terms. This is an approximation, not a precise condition as in the case of Couette and Poiseuille flows in Chapter 16. In this chapter, we will see that the Navier-Stokes equations, when applied to the thin viscous boundary layer adjacent to a surface, can be reduced to simpler forms, albeit approximate, which lend themselves to simpler solutions. These simpler forms of the equations are called the boundary-layer equations—they are the subject of the present chapter.

The Velocity Potential Equation

The inviscid, compressible, subsonic flow over a body immersed in a uniform stream is irrotational; there is no mechanism in such a flow to start rotating the fluid elements (see Section 2.12). Thus, a velocity potential (see Section 2.15) can be defined. Since we are dealing with irrotational flow and the velocity potential, review Sections 2.12 and 2.15 before progressing further.

Consider two-dimensional, steady, irrotational, isentropic flow. A velocity po­tential, ф = <p(x, y), can be defined such that [from Equation (2.154)]

V = V0 [11.1]

or in terms of the cartesian velocity components,

Эф r,

u = —— [11.2a]


_ 9 ф

V dy

Let us proceed to obtain an equation for ф which represents a combination of the continuity, momentum, and energy equations. Such an equation would be very useful, because it would be simply one governing equation in terms of one unknown, namely the velocity potential ф.

The continuity equation for steady, two-dimensional flow is obtained from Equa­tion (2.52) as

We are attempting to obtain an equation completely in terms of </>; hence, we need to eliminate p from Equation (11.5). To do this, consider the momentum equation in terms of Euler’s equation:

dp = —pV dV

This equation holds for a steady, compressible, inviscid flow and relates p and V along a streamline. It can readily be shown that Equation (3.12) holds in any direction throughout an irrotational flow, not just along a streamline (try it yourself). Therefore, from Equations (3.12) and (11.2a and b), we have

dp = – pVdV = ~^d(V2) = ~^d(u2 + v2)

Recall that we are also considering the flow to be isentropic. Hence, any change in pressure dp in the flow is automatically accompanied by a corresponding isentropic change in density dp. Thus, by definition

dp dp)s

The right-hand side of Equation (11.7) is simply the square of the speed of sound. Thus, Equation (11.7) yields

[1 1.8]

Substituting Equation (11.8) for the left side of Equation (11.6), we have

Considering changes in the x direction, Equation (11.9) directly yields

Similarly, for changes in the у direction, Equation (11.9) gives

Эр = p /дф д2ф дф д2ф пі ці

Эу а2 Эт Эх ду ду ду2 )

Substituting Equations (11.10) and (11.11) into (11.5), canceling the p which appears in each term, and factoring out the second derivatives of ф, we obtain

[1 1.12]

which is called the velocity potential equation. It is almost completely in terms of ф only the speed of sound appears in addition to ф. However, a can be readily expressed in terms of ф as follows. From Equation (8.33), we have

Since ao is a known constant of the flow, Equation (11.13) gives the speed of sound a as a function of ф. Hence, substitution of Equation (11.13) into (11.12) yields a single partial differential equation in terms of the unknown ф. This equation represents a combination of the continuity, momentum, and energy equations. In principle, it can be solved to obtain ф for the flow field around any two-dimensional shape, subject of course to the usual boundary conditions at infinity and along the body surface. These boundary conditions on ф are detailed in Section 3.7, and are given by Equations (3.47a and b) and (3.48Z?).

Because Equation (11.12) Lalong with Equation (11.13)] is a single equation in terms of one dependent variable ф, the analysis of isentropic, irrotational, steady, compressible flow is greatly simplified—we only have to solve one equation instead of three or more. Once ф is known, all the other flow variables are directly obtained as follows:

1. Calculate и and v from Equations (11.2a and b).

2. Calculate a from Equation (11.13).

3. Calculate M = V/a — u2 + v2/a.

4. Calculate T, p, and p from Equations (8.40), (8.42), and (8.43), respectively. In these equations, the total conditions Tq, po, and po are known quantities; they are constant throughout the flow field and hence are obtained from the given freestream conditions.

Although Equation (11.12) has the advantage of being one equation with one unknown, it also has the distinct disadvantage of being a nonlinear partial differential equation. Such nonlinear equations are very difficult to solve analytically, and in modem aerodynamics, solutions of Equation (11.12) are usually sought by means of sophisticated finite-difference numerical techniques. Indeed, no general analytical solution of Equation (11.12) has been found to this day. Contrast this situation with that for incompressible flow, which is governed by Laplace’s equation—a linear partial differential equation for which numerous analytical solutions are well known.

Given this situation, aerodynamicists over the years have made assumptions regarding the physical nature of the flow field which are designed to simplify Equation

(11.12) . These assumptions limit our considerations to the flow over slender bodies at small angles of attack. For subsonic and supersonic flows, these assumptions lead to an approximate form of Equation (11.12) which is linear, and hence can be solved analytically. These matters are the subject of the next section.

Keep in mind that, within the framework of steady, irrotational, isentropic flow, Equation (11.12) is exact and holds for all Mach numbers, from subsonic to hyper­sonic, and for all two-dimensional body shapes, thin and thick.