Category When Is A Flow Compressible?

Linearized Supersonic Flow

lflfith the stabilizer setting at 2° the speed was allowed to increase to approximately 0.98 to 0.99 Mach number where elevator and rudder effectiveness were regained and the airplane seemed to smooth out to normal flying characteristics. This development lent added confidence and the airplane was allowed to continue until an indication of 1.02 on the cockpit Mach meter was obtained. At this indication the meter momentarily stopped and then jumped at 1.06, and this hesitation was assumed to be caused by the effect of shock waves on the static source. At this time the power units were cut and the airplane allowed to decelerate back to the subsonic flight condition.

Captain Charles Yeager, describing his flight on October 14, 1947—the first manned flight to exceed the speed of sound.

12.1 Introduction

The linearized perturbation velocity potential equation derived in Chapter 11, Equa­tion (11.18), is

and holds for both subsonic and supersonic flow. In Chapter 11, we treated the case of subsonic flow, where 1 — > 0 in Equation (11.18). However, for supersonic flow,

1 – Ml < 0. This seemingly innocent change in sign on the first term of Equation (11.18) is, in reality, a very dramatic change. Mathematically, when 1 — > 0 for

subsonic flow, Equation (11.18) is an elliptic partial differential equation, whereas when 1 — < 0 for supersonic flow, Equation (11.18) becomes a hyperbolic

differential equation. The details of this mathematical difference are beyond the scope of this book; however, the important point is that there is a difference. Moreover, this portends a fundamental difference in the physical aspects of subsonic and supersonic flow—something we have already demonstrated in previous chapters.

The purpose of this chapter is to obtain a solution of Equation (11.18) for super­sonic flow and to apply this solution to the calculation of supersonic airfoil properties. Since our purpose is straightforward, and since this chapter is relatively short, there is no need for a chapter road map to provide guidance on the flow of our ideas.

Adiabatic Wall Conditions (Adiabatic Wall Temperature)

Let us imagine the following situation. Assume that the flow illustrated in Figure 16.5 is established. We have the parabolic temperature profile established as shown, and we have heat transfer into the walls as just discussed. However, both wall temperatures are considered fixed, and both are equal to the same constant value. Question: How can the wall temperature remain fixed at the same time that heat is transferred into the wall? Answer: There must be some independent mechanism that conducts heat away from the wall at the same rate that the aerodynamic heating is pumping heat into the wall. This is the only way for the wall temperature to remain fixed at some cooler temperature than the adjacent fluid. For example, the wall can be some vast heat sink that can absorb heat without any appreciable change in temperature, or possibly there are cooling coils within the plate that can carry away the heat, much like the water coils that keep the engine of your automobile cool. In any event, to have the picture shown in Figure 16.5 with a constant wall temperature independent of time, some exterior mechanism must carry away the heat that is transferred from the fluid to the walls. Now imagine that, at the lower wall, this exterior mechanism is suddenly shut off. The lower wall will now begin to grow hotter in response to qw, and Tw will begin to increase with time. At any given instant during this transient process, the heat transfer to the lower wall is given by Equation (16.24), repeated below.

At time t = 0, when the exterior cooling mechanism is just shut off, hw = hc, and qw is given by Equation (16.35), namely,

However, as time now progresses, Tu, (and therefore hw) increases. From Equa­tion (16.24), as hy, increases, the numerator decreases in magnitude, and hence qw decreases. That is,

Hence, as time progresses from when the exterior cooling mechanism was first cut off at the lower wall, the wall temperature increases, and the aerodynamic heating to the wall decreases. This in turn slows the rate of increase of Tw as time progresses. The transient variations of both qw and Tw are sketched in Figure 16.6. In Figure 16.6a, we see that, as time increases to large values, the heat transfer to the wall approaches zero—this is defined as the equilibrium, or the adiabatic wall condition. For an adiabatic wall, the heat transfer is, by definition, equal to zero. Simultaneously, the wall temperature Tw approaches asymptotically a limiting value defined as the adiabatic wall temperature Taw, and the corresponding enthalpy is defined as the adiabatic wall enthalpy haw.

The purpose of this discussion is to define an adiabatic wall condition; the ex­ample involving a timewise approach to this condition was just for convenience and edification. Let us now assume that the lower wall in our Couette flow is an adiabatic wall. For this case, we already know the value of heat transfer to the wall—by defi­nition, it is zero. The question now becomes, What is the value of the adiabatic wall enthalpy haw, and in turn the adiabatic wall temperature Taw‘l The answer is given by Equation (16.23), where qw = 0 for an adiabatic wall.

(

In turn, the adiabatic wall temperature is given by

[16.40]

Clearly, the higher the value of ue, the higher is the adiabatic wall temperature.

The enthalpy profile across the flow for this case is given by a combination of Equations (16.16) and (16.40), as follows. Setting hw = haw in Equation (16.16), we

obtain

[16.43]

Equation (16.43) gives the enthalpy profile across the flow. The temperature profile follows from Equation (16.43) as

[16.44]

This variation of T is sketched in Figure 16.7. Note that Tuw is the maximum tem­perature in the flow. Moreover, the temperature curve is perpendicular at the plate for v = 0; that is, the temperature gradient at the lower plate is zero, as expected for an adiabatic wall. This result is also obtained by differentiating Equation (16.44):

which gives З T/9у = 0 at у = 0.

Flow over an Airfoil with a Protuberance

Here we show some very recent Navier-Stokes solutions carried out to study the aerodynamic effect of a small protuberance extending from the bottom surface of an airfoil. These calculations represent an example of the state-of-the-art of full Navier-Stokes solutions at the time of writing. The work was carried out by Beierle (Reference 89). The basic shape of the airfoil was an NACA 0015 section. The computational fluid dynamic solution of the Navier-Stokes equations was carried out using a time-marching finite volume code labeled OVERFLOW, developed by NASA (Reference 90). The flow was low speed, with a freestream Mach number of 0.15 and Reynolds number of 1.5 x 106. The fully turbulent flow field was simulated using a 1-equation turbulence model.

Using a proper grid is vital to the integrity of any Navier-Stokes CFD solution. For the present case, Figures 20.8-20.11 show the grid used, progressing from the

(a)

x~xo

So

(b)

Figure 20.7 Effects of shock-wave/boundary-layer interaction on (a) pressure distribution, and (b) shear stress for Mach 3 turbulent flow over a flat plate.

big picture of the whole grid (Figure 20.8) to the detail of the grid around the small protuberance on the bottom surface of the airfoil (Figure 20.11). The grid is an example of a chimera grid, a series of independent but overlapping grids that are generated about individual parts of the body and for specific flow regions.

Some results for the computed flow field are shown in Figures 20.12 and 20.13. In Figure 20.12, the local velocity vector field is shown; the flow separation and locally reversed flow can be seen downstream of the protuberance. In Figure 20.13, pressure contours are shown, illustrating how the small protuberance generates a substantially asymmetric flow over the otherwise symmetric airfoil.

Finally, results for a related flow are shown in Figure 20.14. Here, instead of a protuberance existing on the bottom surface, an array of small jets that are distributed

z

Figure 20.8 Individual grid boundary outlines used in the chimera grid scheme for calculating the flow over an airfoil with a protuberance.

over the bottom surface alternately blow and suck air into and out of the flow in such a manner that the net mass flow added is zero, so-called “zero-mass synthetic jets.” The resulting series of large-scale vortices is shown in Figure 20.14—another example of a flow field that can only be solved in detail by means of a full Navier-Stokes solution. (See Hassan and JanakiRam, Reference 91, for details.)

Shock Interactions and Reflections

Return to the oblique shock wave illustrated in Figure 9.1a. In this picture, we can imagine the shock wave extending unchanged above the comer to infinity. However, in real life this does not happen. In reality, the oblique shock in Figure 9.1a will impinge somewhere on another solid surface and/or will intersect other waves, either shock or expansion waves. Such wave intersections and interactions are important in the practical design and analysis of supersonic airplanes, missiles, wind tunnels, rocket engines, etc. A perfect historical example of this, as well as the consequences that can be caused by not paying suitable attention to wave interactions, is a ramjet flight-test program conducted in the early 1960s. During this period, a ramjet engine was mounted underneath the X-15 hypersonic airplane for a series of flight tests at high Mach numbers, in the range from 4 to 7. (The X-15, shown in Figure 9.16, was an experimental, rocket-powered airplane designed to probe the lower end of hypersonic manned flight.) During the first high-speed tests, the shock wave from the engine cowling impinged on the bottom surface of the X-15, and because of locally high aerodynamic heating in the impingement region, a hole was burned in the X-15 fuselage. Although this problem was later fixed, it is a graphic example of what shock-wave interactions can do to a practical configuration.

The purpose of this section is to present a mainly qualitative discussion of shock­wave interactions. For more details, see Chapter 4 of Reference 21.

First, consider an oblique shock wave generated by a concave corner, as shown in Figure 9.17. The deflection angle at the corner is в, thus generating an oblique shock at point A with a wave angle fi. Assume that a straight, horizontal wall is present above the comer, as also shown in Figure 9.17. The shock wave generated at point A, called the incident shock wave, impinges on the upper wall at point B. Question: Does the shock wave simply disappear at point B1 If not, what happens to it? To answer this question, we appeal to our knowledge of shock-wave properties. Examining Figure 9.17, we see that the flow in region 2 behind the incident shock is inclined upward at the deflection angle в. However, the flow must be tangent everywhere along the upper wall; if the flow in region 2 were to continue unchanged, it would mn into the wall and have no place to go. Hence, the flow in region 2 must eventually be bent downward through the angle в in order to maintain a flow tangent to the upper wall. Nature accomplishes this downward deflection via a second shock wave originating at the impingement point В in Figure 9.17. This second shock is called the reflected shock wave. The purpose of the reflected shock is to deflect the

Figure 9.1 6 The X-15 hypersonic research vehicle. Designed and built during the late 1950s, it served as a test vehicle for the U. S. Air Force and NASA, jCourtesy of Rockwell Inti, North America.)

flow in region 2 so that it is parallel to the upper wall in region 3, thus preserving the wall boundary condition.

The strength of the reflected shock wave is weaker than the incident shock. This is because М2 < Mi, and М2 represents the upstream Mach number for the reflected shock wave. Since the deflection angles are the same, whereas the reflected shock sees a lower upstream Mach number, we know from Section 9.2 that the reflected wave must be weaker. For this reason, the angle the reflected shock makes with the upper wall Ф is not equal to fi (i. e., the wave reflection is not specular). The properties of the reflected shock are uniquely defined by М2 and 0; since М2 is in turn uniquely defined by Mi and в, then the poperties in region 3 behind the reflected shock as well

as the angle Ф are easily determined from the given conditions of M and 9 by using the results of Section 9.2 as follows:

1. Calculate the properties in region 2 from the given M and 9. In particular, this

gives us M2.

2. Calculate the properties in region 3 from the value of M2 calculated above and

the known deflection angle 9.

An interesting situation can arise as follows. Assume that M is only slightly above the minimum Mach number necessary for a straight, attached shock wave at the given deflection angle 9. For this case, the oblique shock theory from Section 9.2 allows a solution for a straight, attached incident shock. However, we know that the Mach number decreases across a shock (i. e., M2 < M ). This decrease may be enough such that M2 is not above the minimum Mach number for the required deflection 9 through the reflected shock. In such a case, our oblique shock theory does not allow a solution for a straight reflected shock wave. The regular reflection as shown in Figure 9.17 is not possible. Nature handles this situation by creating the wave pattern shown in Figure 9.18. Here, the originally straight incident shock becomes curved as it nears the upper wall and becomes a normal shock wave at the upper wall. This allows the streamline at the wall to continue parallel to the wall behind the shock intersection. In addition, a curved reflected shock branches from the normal shock and propagates downstream. This wave pattern, shown in Figure 9.18, is called a Mach reflection. The calculation of the wave pattern and general properties for a Mach reflection requires numerical techniques such as those to be discussed in Chapter 13.

Another type of shock interaction is shown in Figure 9.19. Here, a shock wave is generated by the concave corner at point G and propagates upward. Denote this wave as shock A. Shock A is a left-running wave, so-called because if you stand on top of the wave and look downstream, you see the shock wave running in front of you

Figure 9.19 Intersection of right – and left-running shock waves.

toward the left. Another shock wave is generated by the concave comer at point H, and propagates downward. Denote this wave as shock B. Shock В is a right-running wave, so-called because if you stand on top of the wave and look downstream, you see the shock mnning in front of you toward the right. The picture shown in Figure 9.19 is the intersection of right – and left-running shock waves. The intersection occurs at point E. At the intersection, wave A is refracted and continues as wave D. Similarly, wave В is refracted and continues as wave C. The flow behind the refracted shock D is denoted by region 4; the flow behind the refracted shock C is denoted by region 4′. These two regions are divided by a slip line EF. Across the slip line, the pressures are constant (i. e„ p = p4<), and the direction (but not necessarily the magnitude) of velocity is the same, namely, parallel to the slip line. All other properties in regions 4 and 4′ are different, most notably the entropy (s4 / ,v4′). The conditions which must hold across the slip line, along with the known Mi, 6, and 6L, uniquely determine

the shock-wave interaction shown in Figure 9.19. (See Chapter 4 of Reference 21 for details concerning the calculation of this interaction.)

Figure 9.20 illustrates the intersection of two left-running shocks generated at comers A and B. The intersection occurs at point C, at which the two shocks merge and propagate as the stronger shock CD, usually along with a weak reflected wave CE. This reflected wave is necessary to adjust the flow so that the velocities in regions 4 and 5 are in the same direction. Again, a slip line CF trails downstream of the intersection point.

The above cases are by no means all the possible wave interactions in a supersonic flow. However, they represent some of the more common situations encountered frequently in practice.

Consider an oblique shock wave generated by a compression corner with a 10° deflection angle. The Mach number of the flow ahead of the corner is 3.6; the flow pressure and temperature are standard sea level conditions. The oblique shock wave subsequently impinges on a straight wall opposite the compression comer. The geometry for this flow is given in Figure 9.17. Calculate the angle of the reflected shock wave Ф relative to the straight wall. Also, obtain the pressure, temperature, and Mach number behind the reflected wave.

Solution

From the 6-fi-M diagram, Figure 9.7, for M = 3.6 and в = 10°, i = 24°. Hence,

Mn ] = M] sin /1] = 3.6 sin 24° = 1.464

From Appendix B,

Also,

These are the conditions behind the incident shock wave. They constitute the upstream flow properties for the reflected shock wave. We know that the flow must be deflected again by в = 10° in passing through the reflected shock. Thus, from the 6-fl-M diagram, for M2 = 2.96 and в = 10°, we have the wave angle for the reflected shock, fi2 = 27.3°. Note that fi2 is not the angle the reflected shock makes with respect to the upper wall; rather, by definition of the wave angle, f}2 is the angle between the reflected shock and the direction of the flow in region 2. The shock angle relative to the wall is, from the geometry shown in Figure 9.17,

Ф = p2 – в = 27.3 – 10

Also, the normal component of the upstream Mach number relative to the reflected shock is M2 sin /Т = (2.96) sin 27.3° = 1.358. From Appendix B,

— = 1.991 — = 1.229 M„,3 = 0.7572

Pi T2

M„ з 0.7572 ГТ7Т

kf3 =——– —– =———————- = 2.55

sin(f}2 — 0) sin(27.3 — 10) _____

For standard sea level conditions, p =2116 lb/ft3 and T, = 519°R. Thus,

Ръ = — — Px= (1 -991)(2.32)(2116) =

Pi Px

h =T^Ti = (1.229)(1.294)(519) ;

12 l

Note that the reflected shock is weaker than the incident shock, as indicated by the smaller pressure ratio for the reflected shock, p2/p2 = 1.991 as compared to p2/p = 2.32 for the incident shock.

Corrector Step

Inserting the flow variables obtained above into the governing equations, Equations

(13.59) to (13.62), using rearward differences for the spatial derivatives, predicted

values of the time derivatives at t + At are obtained, for example, [(3p/30,j](t+A,). In turn, these are averaged with the time derivatives from the predictor step to obtain; for example,

Finally, the average time derivative obtained from Equation (13.64) is inserted into Equation (13.63) to yield the corrected value of density at time t + At. The same procedure is used for all the dependent variables, u, v, etc.

Starting from the assumed initial conditions at t — 0, the repeated application of Equation (13.63) along with the above predictor-corrector algorithm at each time step allows the calculation of the flow-field variables and shock shape and location as a function of time. As stated above, after a large number of time steps, the calculated flow-field variables approach a steady state, where [(3p/3r)(j]ave —>■ 0 in Equation

(13.63) . Once again, we emphasize that we are interested in the steady-state answer, and the time-dependent technique is simply a means to that end.

Note that the applications of MacCormack’s technique to both the steady flow calculations described in Section 13.4 and the time-dependent calculations described in the present section are analogous; in the former, we march forward in the spatial coordinate x, starting with known values along with a constant у line, whereas, in the latter, we march forward in time starting with a known flow field at t = 0.

Why do we bother with a time-dependent solution? Is it not an added complica­tion to deal with an extra independent variable t in addition to the spatial variables x and _y? The answers to these questions are as follows. The governing unsteady flow equations given by Equations (13.59) to (13.62) are hyperbolic with respect to time, independent of whether the flow is locally subsonic or supersonic. In Figure 13.9a, some of the grid points are in the subsonic region and others are in the supersonic re­gion. However, the time-dependent solution progresses in the same manner at all these points, independent of the local Mach number. Hence, the time-dependent technique is the only approach known today which allows the uniform calculation of a mixed subsonic-supersonic flow field of arbitrary extent. For this reason, the application of the time-dependent technique, although it adds one additional independent variable, allows the straightforward solution of a flow field which is extremely difficult to solve by a purely steady-state approach.

A much more detailed description of the time-dependent technique is given in Chapter 12 of Reference 21, which you should study before attempting to apply this technique to a specific problem. The intent of our description here has been to give you simply a “feeling” for the philosophy and general approach of the technique.

Some typical results for supersonic blunt-body flow fields are given in Figures 13.12 to 13.15. These results were obtained with a time-dependent solution described in Reference 35. Figures 13.12 and 13.13 illustrate the behavior of a time-dependent solution during its approach to the steady state. In Figure 13.12, the time-dependent motion of the shock wave is shown for a parabolic cylinder in a Mach 4 freestream. The shock labeled 0 At is the initially assumed shock wave at t = 0. At early times, the shock wave rapidly moves away from the body; however, after about

300 time steps, it has slowed considerably, and between 300 and 500 time steps, the shock wave is virtually motionless—it has reached its steady-state shape and location. The time variation of the stagnation point pressure is given in Figure 13.13. Note that the pressure shows strong timewise oscillations at early times, but then it asymptotically approaches a steady value at large times. Again, it is this asymptotic steady state that we want, and the intermediate transient results are just a means to that

end. Concentrating on just the steady-state results, Figure 13.14 gives the pressure distribution (nondimensionalized by stagnation point pressure) over the body surface for the cases of both Мж = 4 and 8. The time-dependent numerical results are shown as the solid curves, whereas the open symbols are from newtonian theory, to be discussed in Chapter 14. Note that the pressure is a maximum at the stagnation point and decreases as a function of distance away from the stagnation point—a variation that we most certainly would expect based on our previous aerodynamic experience. The steady shock shapes and sonic lines are shown in Figure 13.15 for

Figure 13.1 5 Shock shapes and sonic lines, parabolic cylinder.

Problems

Note: The purpose of the following problem is to provide an exercise in carrying out a unit process for the method of characteristics. A more extensive application to a complete flow field is left to your specific desires. Also, an extensive practical problem utilizing the finite-difference method requires a large number of arithmetic operations and is practical only on a digital computer. You are encouraged to set up such a problem at your leisure. The main purpose of the present chapter is to present the essence of several numerical methods, not to burden the reader with a lot of calculations or the requirement to write an extensive computer program.

1. Consider two points in a supersonic flow. These points are located in a cartesian coordinate system at {x, yi) = (0,0.0684) and (xi, у2) = (0.0121,0), where the units are meters. At point {x, yi): и і = 639 m/s, г 1 = 232.6 m/s, p = 1 atm, T = 288 K. At point {x3, Уі)’- R2 = 680 m/s, i>2 = 0, P2 = 1 atm, T2 = 288 K. Consider point 3 downstream of points 1 and 2 located by the intersection of the C+ characteristic through point 2 and the C_ characteristic through point 1. At point 3, calculate: m3, i>3, p3, and T3. Also, calculate the location of point 3, assuming the characteristics between these points are straight lines.

Boundary-Layer Properties

Consider the viscous flow over a flat plate as sketched in Figure 17.3. The viscous effects are contained within a thin layer adjacent to the surface; the thickness is exaggerated in Figure 17.3 for clarity. Immediately at the surface, the flow velocity is zero; this is the “no-slip” condition discussed in Section 15.2. In addition, the temperature of the fluid immediately at the surface is equal to the temperature of the surface; this is called the wall temperature Tw, as shown in Figure 17.3. Above the surface, the flow velocity increases in the у direction until, for all practical purposes, it equals the freestream velocity. This will occur at a height above the wall equal to S, as shown in Figure 17.3. More precisely, S is defined as that distance above the wall

Figure I 7.3 Boundary-layer properties.

where и = 0.99ue; here, ue is the velocity at the outer edge of the boundary layer. In Figure 17.3, which illustrates the flow over a flat plate, the velocity at the edge of the boundary layer will be V, x; that is, ue = Уж. For a body of general shape, ue is the velocity obtained from an inviscid flow solution evaluated at the body surface (or at the “effective body” surface, as discussed later). The quantity S is called the velocity boundary-layer thickness. At any given x station, the variation of и between у = 0 and у = S, that is, и = и (у), is defined as the velocity profile within the boundary layer, as sketched in Figure 17.3. This profile is different for different x stations. Similarly, the flow temperature will change above the wall, ranging from T = Tw at у = 0 to T — 0.997), at у = 8T. Here, St is defined as the thermal boundary-layer thickness. At any given x station, the variation of T between у = 0 and у = ST, that is, Г = T(y), is called the temperature profile within the boundary layer, as sketched in Figure 17.3. (In the above, Te is the temperature at the edge of the thermal boundary layer. For the flow over a flat plate, as sketched in Figure 17.3, Te = Too. For a general body, Te is obtained from an inviscid flow solution evaluated at the body surface, or at the “effective body” surface, to be discussed later.) Hence, two boundary layers can be defined: a velocity boundary layer with thickness S and a temperature boundary layer with thickness St. In general, Sj ф S. The relative thicknesses depend on the Prandtl number: it can be shown that if Pr = 1, then S = 5^;ifPr > l, then<5r < <5;ifPr < l, then<5r > S. For air at standard conditions, Pr = 0.71; hence, the thermal boundary layer is thicker than the velocity boundary layer, as shown in Figure 17.3. Note that both boundary-layer thicknesses increase with distance from the leading edge; that is, 8 = 8(x) and 8T = ST(x).

The consequence of the velocity gradient at the wall is the generation of shear stress at the wall,

where (du/dy)w is the velocity gradient evaluated at у = 0 (i. e., at the wall). Simi­larly, the temperature gradient at the wall generates heat transfer at the wall,

[17.6]

Equation (17.6) is identical to the definition of S* given in Equation (17.3). Hence, clearly <5* is a height proportional to the missing mass flow. If this missing mass flow was crammed into a streamtube where the flow properties were constant at pe and ue, then Equation (17.5) says that <5* is the height of this hypothetical streamtube.

2. The second physical interpretation of 8* is more practical than the one discussed above. Consider the flow over a flat surface as sketched in Figure 17.5. At the left is a picture of the hypothetical inviscid flow over the surface; a streamline through point yi is straight and parallel to the surface. The actual viscous flow is shown at the right of Figure 17.5; here, the retarded flow inside the boundary layer acts as a partial obstruction to the freestream flow. As a result, the streamline external to the boundary layer passing through point yi is deflected upward through a distance 8*. We now prove that this 5* is precisely the displacement thickness defined by Equation (17.3). At station 1 in Figure 17.5, the mass flow (per unit depth perpendicular to the page) between the surface and the external streamline is

[17.7]

At station 2, the mass flow between the surface and the external streamline is

m= I pudy + peue8* [17.8]

Jo

Since the surface and the external streamline form the boundaries of a streamtube, the mass flows across stations 1 and 2 are equal. Hence, equating Equations (17.7) and (17.8), we have

Streamline

1

1

ue

1

1

1

L_____

Pe

”щшшіишшшшшшшшгшшишіт.

Hypothetical flow with no boundary layer (inviscid case)

Figure 1 7.5 Displacement thickness is the distance by which an external flow streamline is displaced by the presence of the boundary layer.

Л’1 AVI

I peue dy — I pu dy + peue <5*

Jo Jo

or s*= Г’ (l-—)dy [17.9]

Jo PeM-e )

Hence, the height by which the streamline in Figure 17.5 is displaced upward by the presence of the boundary layer, namely, &*, is given by Equation (17.9). However, Equation (17.9) is precisely the definition of the displacement thickness given by Equation (17.3). Thus, the displacement thickness, first defined by Equation (17.3), is physically the distance through which the external inviscid flow is displaced by the presence of the boundary layer.

This second interpretation of 8* gives rise to the concept of an effective body. Consider the aerodynamic shape sketched in Figure 17.6. The actual contour of the body is given by curve ab. However, due to the displacement effect of the boundary layer, the shape of the body effectively seen by the freestream is not given by curve ab; rather, the freestream sees an effective body given by curve ac. In order to obtain the conditions ue, Te, etc., at the outer edge of the boundary layer on the actual body ab, an inviscid flow solution should be carried out for the effective body, and pe, u,_, Te, etc., are obtained from this inviscid solution evaluated along curve ac.

Note that in order to solve for S* from Equation (17.3), we need the profiles of и and p from a solution of the boundary-layer flow. In turn, to solve the boundary-layer flow, we need pe, ue, 7,. etc. However, pe, ue, Tc, etc., depend on <5*. This leads to an iterative solution. To calculate accurately the boundary-layer properties as well as the pressure distribution over the surface of the body in Figure 17.6, we proceed as follows:

1. Carry out an inviscid solution for the given body shape ab. Evaluate p, . ue. Te, etc., along curve ab.

2. Using these values of pe, ue, Te, etc., solve the boundary-layer equations (dis­cussed in Sections 17.3 to 17.6) for и — и {у), p = p( у), etc., at various stations along the body.

3. Obtain 8* at these stations from Equation (17.3). This will not be an accurate 5* because pe, ue, Te, etc., were evaluated on curve ab, not the proper effective body. Using this intermediate 8*, calculate an effective body, given by a curve ad (not shown in Figure 17.6).

4. Carry out an inviscid solution for the flow over the intermediate effective body ad, and evaluate new values of pe, ue, Te, etc., along ad.

5. Repeat steps 2 to 4 above until the solution at one iteration essentially does not deviate from the solution at the previous iteration. At this stage, a converged solution will be obtained, and the final results will pertain to the flow over the proper effective body ac shown in Figure 17.6.

In some cases, the boundary layers are so thin that the effective body can be ignored, and a boundary-layer solution can proceed directly from pe, ue, Te, etc., obtained from an inviscid solution evaluated on the actual body surface (ab in Fig­ure 17.6). However, for highly accurate solutions, and for cases where the boundary – layer thickness is relatively large (such as for hypersonic flow as discussed in Chap­ter 14), the iterative procedure described above should be carried out. Also, we note parenthetically that 5* is usually smaller than 5; typically, 8* «a 0.3 8.

Another boundary-layer property of importance is the momentum thickness в, defined as

[17.10]

To understand the physical interpretation of в, return again to Figure 17.4. Consider the mass flow across a segment dy, given by dm — pu dy. Then

A = momentum flow across dy = dm и = pu2 dy

If this same elemental mass flow were associated with the freestream, where the velocity is ue, then

{

momentum flow at freestream velocity associated with mass dm = dm ue = (pu dy)ue

Hence, decrement in momentum flow

(missing momentum flow) associated = pu(ue — u) dy with mass dm

The total decrement in momentum flow across the vertical line from у = 0 to у = y in Figure 17.4 is the integral of Equation (17.11),

[i7.i a]

Assume that the missing momentum flow is the product of peu2e and a height в. Then, Missing momentum flow = peu2e6 [ 17.13]

Equating Equations (17.12) and (17.13), we obtain

[17.14]

Equation (17.14) is precisely the definition of the momentum thickness given by Equation (17.10). Therefore, в is an index that is proportional to the decrement in momentum flow due to the presence of the boundary layer. It is the height of a hypothetical streamtube which is carrying the missing momentum flow at freestream conditions.

Note that в =9(x). In more detailed discussions of boundary-layer theory, it can be shown that в evaluated at a given station x = X is proportional to the integrated friction drag coefficient from the leading edge to x;; that is,

0(xi) a — f x Jo

where Cf is the local skin friction coefficient defined in Section 1.5 and C/ is the total skin friction drag coefficient for the length of surface from x = 0 to x = x. Hence, the concept of momentum thickness is useful in the prediction of drag coefficient.

All the boundary-layer properties discussed above are general concepts; they apply to compressible as well as incompressible flows, and to turbulent as well as laminar flows. The differences between turbulent and laminar flows were introduced in Section 15.2. Here, we extend that discussion by noting that the increased momen­tum and energy exchange that occur within a turbulent flow cause a turbulent boundary layer to be thicker than a laminar boundary layer. That is, for the same edge condi­tions, /)(., Uc. Te, etC., We have ^turbulent ^ ^laminar ttnd ($7) turbulent ■> (^7′)laminar – A hСП a boundary layer changes from laminar to turbulent flow, as sketched in Figure 15.8, the boundary-layer thickness markedly increases. Similarly, 8* and в are larger for turbulent flows.

The Linearized Velocity Potential Equation

Consider the two-dimensional, irrotational, isentropic flow over the body shown in Figure 11.2. The body is immersed in a uniform flow with velocity Voo oriented in the x direction. At an arbitrary point P in the flow field, the velocity is V with the x and у components given by и and v, respectively. Let us now visualize the velocity V as the sum of the uniform flow velocity plus some extra increments in velocity. For example, the x component of velocity и in Figure 11.2 can be visualized as Voo plus an increment in velocity (positive or negative). Similarly, the у component of velocity v can be visualized as a simple increment itself, because the uniform flow has a zero component in the у direction. These increments are called perturbations, and

и — Voo + й v = v

where u and v are called the perturbation velocities. These perturbation velocities are not necessarily small; indeed, they can be quite large in the stagnation region in front of the blunt nose of the body shown in Figure 11.2. In the same vein, because V = Уф, we can define a perturbation velocity potential ф such that

ф = Voo* + ф

дф

dx

дф

dy

Hence,

З 2ф д2ф

дх ду дх ду

Substituting the above definitions into Equation (11.12), and multiplying by a2, we obtain

[1 1.14]

Equation (11.14) is called the perturbation velocity potential equation. It is precisely the same equation as Equation (11.12) except that it is expressed in terms of ф instead of ф. It is still a nonlinear equation.

To obtain better physical insight in some of our subsequent discussion, let us recast Equation (11.14) in terms of the perturbation velocities. From the definition of ф given earlier, Equation (11.14) can be written as

r9 /Т7 Л, 7. 9 „лЗе лл3л

[a ~ (Voo + и)2]— + (a2 – v2)—– 2(Тоо + u)v— = 0

dx dy dy

From the energy equation in the form of Equation (8.32), we have

Кэс _ I (Too + n)2 + P2

2 “ у – 1 +

Substituting Equation (11.15) into (11.14a), and algebraically rearranging, we obtain

й у + 1 й2 у — їй2

tr + ,)^ + — vi + ~K

‘ й у + 1 v2

2 V^ +

Equation (11.16) is still exact for irrotational, isentropic flow. Note that the left-hand side of Equation (11.16) is linear but the right-hand side is nonlinear. Also, keep in mind that the size of the perturbations й and v can be large or small; Equation (11.16) holds for both cases.

Let us now limit our considerations to small perturbations; that is, assume that the body in Figure 11.2 is a slender body at small angle of attack. In such a case, й and v will be small in comparison with V-^. Therefore, we have

V2 ’ V2

Keep in mind that products of й and v with their derivatives are also very small. With this in mind, examine Equation (11.16). Compare like terms (coefficients of like derivatives) on the left – and right-hand sides of Equation (11.16). We find

1. For 0 < Moo £ 0.8 or Moo > 1.2, the magnitude of

or in terms of the perturbation velocity potential,

Examine Equation (11.18). It is a linear partial differential equation, and therefore is inherently simpler to solve than its parent equation, Equation (11.16). However, we have paid a price for this simplicity. Equation (11.18) is no longer exact. It is only an approximation to the physics of the flow. Due to the assumptions made in obtaining Equation (11.18), it is reasonably valid (but not exact) for the following combined situations:

1. Small perturbation, that is, thin bodies at small angles of attack

2. Subsonic and supersonic Mach numbers

In contrast, Equation (11.18) is not valid for thick bodies and for large angles of attack. Moreover, it cannot be used for transonic flow, where 0.8 < Мж < 1.2, or for hypersonic flow, where Мж > 5.

We are interested in solving Equation (11.18) in order to obtain the pressure distribution along the surface of a slender body. Since we are now dealing with approximate equations, it is consistent to obtain a linearized expression for the pres­sure coefficient—an expression which is approximate to the same degree as Equation (11.18), but which is extremely simple and convenient to use. The linearized pressure coefficient can be derived as follows.

First, recall the definition of the pressure coefficient Cp given in Section 1.5:

C„ = -—— [11.19]

QoC

where qoo = 5Pco^tx; = dynamic pressure. The dynamic pressure can be expressed in terms of M0c as follows:

Substituting Equation (11.21) into (11.19), we have

[11.22]

Equation (11.22) is simply an alternate form of the pressure coefficient expressed in terms of Moo. It is still an exact representation of the definition of C

To obtain a linearized form of the pressure coefficient, recall that we are dealing with an adiabatic flow of a calorically perfect gas; hence, from Equation (8.39),

V2 V2

T+- = roo + -^ [11.23]

2,Cp 2cp

Recalling from Equation (7.9) that cp = yR/(y — 1), Equation (11.23) can be written as

T – Tso = —— 25———–

°° 2yR/(y – 1)

Also, recalling that = *Jy Equation (11.24) becomes T x у – 1 – У 2 ^ у _ ! y2 _ V2

Too 2 yRToo 2 a^

In terms of the perturbation velocities

V2 = (Voo + m)2 + v2

Equation (11.25) can be written as

Since the flow is isentropic, p/poo = (T/TooyRy 1 ’, and Equation (11.26) becomes

2 ‘ 00 V V2

[2]oo ‘oo

Equation (11.27) is still an exact expression. However, let us now make the as­sumption that the perturbations are small, that is, й/V.^ 1, u2/ <SY 1, and

v2/ <gc 1. In this case, Equation (11.27) is of the form

— = (1 – e)^"1) [11.28]

Poo

where e is small. From the binomial expansion, neglecting higher-order terms, Equa­tion (11.28) becomes

p v

— = 1——– -—£ + ••• [11.29]

Poo Y – 1

Comparing Equation (11.27) to (11.29), we can express Equation (11.27) as

7- = 1 –

Poo 2


Substituting Equation (11.30) into the expression for the pressure coefficient, Equa­tion (11.22), we obtain

cP =

2

_i – ^

Ґ 2й | й2 + v2 ^

■ – 1

умі

VEoo+ ) +

or

cP =

2 й Й2 + v2

[1 1.31]

Since й2/and v2/V^ <ж. 1, Equation (11.31) becomes

Equation (11.32) is the linearized form for the pressure coefficient; it is valid only for small perturbations. Equation (11.32) is consistent with the linearized perturbation velocity potential equation, Equation (11.18). Note the simplicity of Equation (11.32); it depends only on the x component of the velocity perturbation, namely, й.

To round out our discussion on the basics of the linearized equations, we note that any solution to Equation (11.18) must satisfy the usual boundary conditions at infinity and at the body surface. At infinity, clearly ф = constant; that is, й = v — 0. At the body, the flow-tangency condition holds. Let в be the angle between the tangent to the surface and the freestream. Then, at the surface, the boundary condition is obtained from Equation (3.48c):

v v

tan# = – = ———– – [11.33]

и Ех, + и

which is an exact expression for the flow-tangency condition at the body surface. A simpler, approximate expression for Equation (11.33), consistent with linearized theory, can be obtained by noting that for small perturbations, й « Hence, Equation (11.33) becomes v

Equation (11.34) is an approximate expression for the flow-tangency condition at the body surface, with accuracy of the same order as Equations (11.18) and (11.32).

Viscosity and Thermal Conduction

The basic physical phenomena of viscosity and thermal conduction in a fluid are due to the transport of momentum and energy via random molecular motion. Each molecule in a fluid has momentum and energy, which it carries with it when it moves from one location to another in space before colliding with another molecule. The transport of molecular momentum gives rise to the macroscopic effect we call viscosity, and the transport of molecular energy gives rise to the macroscopic effect we call thermal conduction. This is why viscosity and thermal conduction are labeled as transport phenomena. A study of these transport phenomena at the molecular level is part of kinetic theory, which is beyond the scope of this book. Instead, in this section we simply state the macroscopic results of such molecular motion.

Consider the flow sketched in Figure 15.9. For simplicity, we consider a one­dimensional shear flow, that is, a flow with horizontal streamlines in the x direction but with gradients in the у direction of velocity, du/dy, and temperature, dT/dy.

conduction to velocity and temperature gradients, respectively.

Consider a plane ab perpendicular to the у axis, as shown in Figure 15.9. The shear stress exerted on plane ab by the flow is denoted by xyx and is proportional to the velocity gradient in the у direction, xyx ос 3u/dy. The constant of proportionality is defined as the viscosity coefficient ц. Hence,

Эи

tyx = Iі T~

The subscripts on xyx denote that the shear stress is acting in the x direction and is being exerted on a plane perpendicular to the у axis. The velocity gradient du/dy is also taken perpendicular to this plane (i. e., in the у direction). The dimensions of /і are mass/length x time, as originally stated in Section 1.7 and as can be seen from Equation (15.1). In addition, the time rate of heat conducted per unit area across plane ab in Figure 15.9 is denoted by qY and is proportional to the temperature gradient in the у direction, qy ос ЗT/3y. The constant of proportionality is defined as the thermal conductivity k. Hence,

[15.2]

where the minus sign accounts for the fact that the heat is transferred from a region of high temperature to a region of lower temperature; that is, qy is in the opposite direction of the temperature gradient. The dimensions of к are mass x length/(s2 • K), which can be obtained from Equation (15.2) keeping in mind that qY is energy per second per unit area.

Both д and к are physical properties of the fluid and, for most normal situa­tions, are functions of temperature only. A conventional relation for the temperature variation of ц for air is given by Sutherland’s law,

[15.3]

where T is in kelvin and до is a reference viscosity at a reference temperature, 7q. For example, if we choose reference conditions to be standard sea level values, then до = 1 -7894 x 10-5kg/(m • s) and 7b = 288.16 K. The temperature variation of к is analogous to Equation (15.3) because the results of elementary kinetic theory show that к ос jiCp; for air at standard conditions,

k = l.45 fiCp

where cp = 1000 J/(kg • K).

Equations (15.3) and (15.4) are only approximate and do not hold at high tem­peratures. They are given here as representative expressions which are handy to use. For any detailed viscous flow calculation, you should consult the published literature for more precise values of д and k.

In order to simplify our introduction of the relation between shear stress and viscosity, we considered the case of a one-dimensional shear flow in Figure 15.9. In this picture, the у and z components of velocity, v and w, respectively, are zero. However, in a general three-dimensional flow, u, v, and w are finite, and this requires a generalization of our treatment of stress in the fluid. Consider the fluid element sketched in Figure 15.10. In a three-dimensional flow, each face of the fluid element experiences both tangential and normal stresses. For example, on face abed, xxy and xxz are the tangential stresses, and xxx is the normal stress. As before, the nomenclature rtj denotes a stress in the j direction exerted on a plane perpendicular to the і axis. Similarly, on face abfe, we have the tangential stresses xyx and xyz, and the normal stress xyy. On face adge, we have the tangential stresses x7X and xzy, and the normal stress xzz. Now recall the discussion in the last part of Section 2.12 concerning the strain of a fluid element, that is, the change in the angle к shown in Figure 2.31. What is the force which causes this deformation shown in Figure 2.31? Returning to Figure 15.10, we have to say that the strain is caused by the tangential shear stress.

z

Figure 1 5*10 Shear and normal stresses caused

by viscous action on a fluid element.

However, in contrast to solid mechanics where stress is proportional to strain, in fluid mechanics the stress is proportional to the time rate of strain. The time rate of strain in the xy plane was given in Section 2.12 as Equation (2.135a):

Examining Figure 15.10, the strain in the xy plane must be carried out by тху and ryx. Moreover, we assume that moments on the fluid element in Figure 15.10 are zero; hence, xxy = xyx. Finally, from the above, we know that xxy – xyx ос єху. The proportionality constant is the viscosity coefficient д. Hence, from Equation (2.135a), we have

f dv du

tXy — tyx — Д ( ~dy ) ^ ®-®]

which is a generalization of Equation (15.1), extended to the case of multidimensional flow. For the shear stresses in the other planes, Equations (2.135i> and c) yield

The normal stresses xxx, xyy, and xzz shown in Figure 15.10 may at first seem strange. In our previous treatments of inviscid flow, the only force normal to a surface in a fluid is the pressure force. However, if the gradients in velocity Эм/dx, dv/dy, and dw/dz are extremely large on the faces of the fluid element, there can be a meaningful viscous-induced normal force on each face which acts in addition to the pressure. These normal stresses act to compress or expand the fluid element, hence changing its volume. Recall from Section 2.12 that the derivatives du/dx, dv/dy, and dw/dz are related to the dilatation of a fluid element, that is, to V • V. Hence, the normal stresses should in turn be related to these derivatives. Indeed, it can be shown that

du

= A(V • V) + 2д — dx

[15.8]

dv

= A(V-V) + 2 pt-

dy

[15.9]

dw

= A(V • V) + 2д—— 3 z

[15.10]

In Equations (15.8) to (15.10), A. is called the bulk viscosity coefficient. In 1845, the Englishman George Stokes hypothesized that

A = -| д [15.11]

To this day, the correct expression for the bulk viscosity is still somewhat controversial, and so we continue to use the above expression given by Stokes. Once again, the

normal stresses are important only where the derivatives du/dx, dv/dy, and 3w/dz are very large. For most practical flow problems, xxx, xyy, and rzz are small, and hence the uncertainty regarding X is essentially an academic question. An example where the normal stress is important is inside the internal structure of a shock wave. Recall that, in real life, shock waves have a finite but small thickness. If we consider a normal shock wave across which large changes in velocity occur over a small distance (typically 10-5 cm), then clearly du/dx will be very large, and rxx becomes important inside the shock wave.

To this point in our discussion, the transport coefficients /x and к have been considered molecular phenomena, involving the transport of momentum and energy by random molecular motion. This molecular picture prevails in a laminar flow. The values of /x and к are physical properties of the fluid; that is, their values for different gases can be found in standard reference sources, such as the Handbook of Chemistry and Physics (The Chemical Rubber Co.). In contrast, for a turbulent flow the transport of momentum and energy can also take place by random motion of large turbulent eddies, or globs of fluid. This turbulent transport gives rise to effective values of viscosity and thermal conductivity defined as eddy viscosity є and eddy thermal conductivity k, respectively. (Please do not confuse this use of the symbols £ and к with the time rate of strain and strain itself, as used earlier.) These turbulent transport coefficients є and к can be much larger (typically 10 to 100 times larger) than the respective molecular values /x and k. Moreover, є and к predominantly depend on characteristics of the flow field, such as velocity gradients; they are not just a molecular property of the fluid such as ji and k. The proper calculation of є and к for a given flow has remained a state-of-the-art research question for the past 80 years; indeed, the attempt to model the complexities of turbulence by defining an eddy viscosity and thermal conductivity is even questionable. The details and basic understanding of turbulence remain one of the greatest unsolved problems in physics today. For our purpose here, we simply adopt the ideas of eddy viscosity and eddy thermal conductivity, and for the transport of momentum and energy in a turbulent flow, we replace /г and k in Equations (15.1) to (15.10) by the combination /x + £ and k + к; that is,

An example of the calculation of є and к is as follows. In 1925, Prandtl suggested

that

for a flow where the dominant velocity gradient is in the у direction. In Equa­tion (15.12),/ is called the mixing length, which is different for different applications; it is an empirical constant which must be obtained from experiment. Indeed, all turbulence models require the input of empirical data; no self-contained purely the-

oretical turbulence model exists today. Prandtl’s mixing length theory, embodied in Equation (15.12), is a simple relation which appears to be adequate for a number of engineering problems. For these reasons, the mixing length model for є has been used extensively since 1925. In regard to к, a relation similar to Equation (15.4) can be assumed (using 1.0 for the constant); that is,

к = єср [15.13]

The comments on eddy viscosity and thermal conductivity are purely introduc­tory. The modern aerodynamicist has a whole stable of turbulence models to choose from, and before tackling the analysis of a turbulent flow, you should be familiar with the modern approaches described in such books as References 42 to 45.

Turbulent Boundary Layers

The one uncontroversial fact about turbulence is that it is the most complicated kind of fluid motion.

Peter Bradshaw Imperial College of Science and Technology, London 1978

Turbulence was, and still is, one of the great unsolved mysteries of science, and it intrigued some of the best scientific minds of the day. Arnold Sommerfeld, the noted German theoretical physicist of the 1920s, once told me, for instance, that before he died he would like to understand two phenomena—quantum mechanics and turbulence. Sommerfeld died in 1924. I believe he was somewhat nearer to an understanding of the quantum, the discovery that led to modern physics, but no closer to the meaning of turbulence.

Theodore von Karman, 1967

19.1 Introduction

The subject of turbulent flow is deep, extensively studied, but at the time of writing still imprecise. The basic nature of turbulence, and therefore our ability to predict its characteristics, is still an unsolved problem in classical physics. Many books have been written on turbulent flows, and many people have spent their professional lives working on the subject. As a result, it is presumptuous for us to try to carry out a thorough discussion of turbulent boundary layers in this chapter. Instead, the purpose of this chapter is simply to provide a contrast with our study of laminar boundary layers in Chapter 18. Here, we will only be able to provide a flavor of turbulent

boundary layers, but this is all that is necessary in the present book. Turbulence is a subject that we leave for you to study more extensively as a subject on its own.

Before proceeding further, return to Section 15.2 and review the basic discussion of the nature of turbulence that is given there. In the present chapter, we will pick up where Section 15.2 leaves off.

Also, we note that no pure theory of turbulent flow exists. Every analysis of turbulent flows requires some type of empirical data in order to obtain a practical answer. As we examine the calculation of turbulent boundary layers in the following sections, the impact of this statement will become blatantly obvious. Finally, because this chapter is short, there is no need for a roadmap to act as a guide.

Derivation of the Linearized Supersonic Pressure Coefficient Formula

For the case of supersonic flow, let us write Equation (11.18) as

л^-4=о

Эх2 dy2

where Л = J— 1. A solution to this equation is the functional relation

Ф — f(x-ky) [12.2]

We can demonstrate this by substituting Equation (12.2) into Equation (12.1) as follows. The partial derivative of Equation (12.2) with respect to x can be written as

З ф. d(x — ky)

= / (x — ky)——- —

dx dx

In Equation (12.3), the prime denotes differentiation of / with respect to its argument, x — ky. Differentiating Equation (12.3) again with respect to x, we obtain

d-±= f" dx2 1

Similarly,

Substituting Equations (12.4) and (12.6) into (12.1), we obtain the identity

Xі f" – Xі f" = 0

Hence, Equation (12.2) is indeed a solution of Equation (12.1).

Examine Equation (12.2) closely. This solution is not very specific, because / can be any function of x — Xy. However, Equation (12.2) tells us something specific about the flow, namely, that ф is constant along lines of л – Xy = constant. The slope of these lines is obtained from

x — Xy — const

Hence, ± = ! =_______ ‘___

dx x !Mlc -1

From Equation (9.31) and the accompanying Figure 9.25, we know that

tan/r = , : [12.8]

where p. is the Mach angle. Therefore, comparing Equations (12.7) and (12.8), we see that a line along which ф is constant is a Mach line. This result is sketched in Figure 12.1, which shows supersonic flow over a surface with a small hump in the middle, where 9 is the angle of the surface relative to the horizontal. According to Equations (12.1) to (12.8), all disturbances created at the wall (represented by the perturbation potential ф) propagate unchanged away from the wall along Mach waves. All the Mach waves have the same slope, namely, dy/dx — (M^. — 1)~1/2. Note that the Mach waves slope downstream above the wall. Hence, any disturbance at the wall cannot propagate upstream; its effect is limited to the region of the flow downstream of the Mach wave emanating from the point of the disturbance. This is a further substantiation of the major difference between subsonic and supersonic flows mentioned in previous chapters, namely, disturbances propagate everywhere throughout a subsonic flow, whereas they cannot propagate upstream in a steady supersonic flow.

Keep in mind that the above results, as well as the picture in Figure 12.1, pertain to linearized supersonic flow [because Equation (12.1) is a linear equation]. Hence, these results assume small perturbations; that is, the hump in Figure 12.1 is small,

and thus в is small. Of course, we know from Chapter 9 that in reality a shock wave will be induced by the forward part of the hump, and an expansion wave will emanate from the rearward part of the hump. These are waves of finite strength and are not a part of linearized theory. Linearized theory is approximate; one of the consequences of this approximation is that waves of finite strength (shock and expansion waves) are not admitted.

The above results allow us to obtain a simple expression for the pressure coeffi­cient in supersonic flow, as follows. From Equation (12.3),

and from Equation (12.5),

~ 9<^ і f’

v = — = – A./
dy

Eliminating /’ from Equations (12.9) and (12.10), we obtain

Figure 1 2.2 Variation of the linearized

pressure coefficient with Mach number (schematic).

portion. This is denoted by the (+) and (—) signs in front of and behind the hump shown in Figure 12.1. This is also somewhat consistent with our discussions in Chapter 9; in the real flow over the hump, a shock wave forms above the front portion where the flow is being turned into itself, and hence p > whereas an expansion wave occurs over the remainder of the hump, and the pressure decreases. Think about the picture shown in Figure 12.1; the pressure is higher on the front section of the hump, and lower on the rear section. As a result, a drag force exists on the hump. This drag is called wave drag and is a characteristic of supersonic flows. Wave drag was discussed in Section 9.7 in conjunction with shock-expansion theory applied to supersonic airfoils. It is interesting that linearized supersonic theory also predicts a finite wave drag, although shock waves themselves are not treated in such linearized theory.

Examining Equation (12.15), we note that Cp oc (Af£, — l)-l/2; hence, for su­personic flow, Cp decreases as M0c increases. This is in direct contrast with subsonic flow, where Equation (11.51) shows that Cp cx (1 — M^,)^1/2; hence, for subsonic flow, Cp increases as M^ increases. These trends are illustrated in Figure 12.2. Note that both results predict Cp —>■ oo as M —> 1 from either side. However, keep in mind that neither Equation (12.15) nor (11.51) is valid in the transonic range around Mach 1.