Category AERODYNAMICS, AERONAUTICS, AND FLIGHT MECHANICS

Wings designed to operate at high speeds are generally thin and employ sweepback in order to increase the critical Mach number. In some instances the sweep is variable to accommodate operation at both low and high speeds. Many airplanes for which the primary mission involves supersonic flight employ delta planforms. Generally, the trend is to increase the sweepback with increasing Mach number until a point is reached where the delta planform becomes more advantageous from a structural as well as aerody­namic viewpoint.

Typical operational aircraft with differing planform shapes are illus­trated in Figure 5.25. The Boeing 727 has a midchord sweep angle of approximately 27° with a cruise Mach number of around 0.84 at 6705 m (22,000 ft). The aspect ratio of its wing equals 7.2. The Vought A-7D has about the same midchord sweep angle with a maximum Mach number in level flight at 1525 m (5000 ft) of around 0.88. Its aspect ratio of 4.0 is much lower than that of the 727. The Grumman F-14A and Aerospatiale/BAC Concorde have supersonic capability and employ planforms of a low aspect ratio that are approximately delta shaped. The F-14A employs variable sweep, sometimes

referred to as a “swing wing.” Fully forward its aspect ratio equals ap­proximately 7.6. When swung to the rear, the wing and horizontal tail combine to produce a delta wing having an aspect ratio of approximately 1.0. The maximum design Mach number of the F-14A, according to Reference 5.11, is 2.4. The Concorde’s planform shape is referred to as an ogee. It has an aspect ratio of 1.7 and is designed to cruise at a Mach number of 2.02 at 15,635 m (51,300 ft).

If M is greater than unity, Equation 5.56 changes to a hyperbolic partial differential equation, specifically, to the following wave equation.

where / and g represent arbitrary functions of their arguments. As an exercise, verify that Equation 5.69 satisfies Equation 5.68. ф is seen to be constant along families of straight lines defined by

x — By = constant x + By = constant

The slope of the lines represented by Equation 5.70a is

dy _ 1 dx~ В

= (M2-l)~m

But this is the tangent of the Mach wave angle as defined by Equation 5.45. Thus ф is constant along a Mach wave. In the case of Equation 5.706,

dy _ _І_ dx~ В

On the upper surface of a body, this would correspond to a disturbance being propagated forward in the flow, which is physically impossible in a supersonic flow. Thus Equation 5.706 is ruled out for the upper surface. However, on the lower surface of a body, g(x + By) is a physically valid flow and represents a disturbance being propagated rearward along a Mach wave. Similarly, f(x – By) is not allowed as a solution on the lower surface of a body. The net result is pictured in Figure 5.23, where it is seen that f(x – By) and g(x + By) are solutions to ф on the upper and lower surfaces, respec-

Flgure 5.23 Mach waves emanating from upper and lower surfaces of a body.

tively. Since ф is constant along Mach waves emanating from disturbances from the upper and lower surfaces, it follows that the properties of the flow (velocity and state) are also constant along these waves.

Along the surface of the body the flow must be tangent to the body. Thus, if Y(jc) represents the body surface, it follows, to a first order in the perturbation velocities, that

Consider the upper surface and let x – By = z. Then

/дф =d£d£

dy)u dz dy

= – Bf (5.72)

where /’ denotes dfldz. In addition, from Equation 5.64,

r = 2 дФ

p“ V„.dx

Thus, according to Ackeret’s linearized theory for supersonic flow around a slender body, the pressure locally on the body is determined by the slope of the surface at the particular location in question.

This simple result leads quickly to some interesting conclusions regarding the characteristics of thin, supersonic airfoils at low angles of attack. Since, for a unit chord,

C, = [ (CPI – CPu) dx

JO

you can quickly verify that

4a

C, = – g (5.76)

Thus, within the limitations of the linearized theory, the section lift coefficient of a supersonic airfoil depends only on its angle of attack. Camber is predicted to have no effect on Ct.

The wave drag coefficient is obtained by integrating the component of C„ in the drag direction around the airfoil.

(5.77)

If dyldx is expressed in the form

— a + e

where a is the angle of attack of the chord line and є is the slope of the surface relative to the chord line, Cdw becomes

Cd*=Ч~+1 і (£“2+e,2) dx (5 -78)

The wave drag coefficient can thus be viewed as the sum of two terms; the first results from lift and the second results from thickness and camber.

Cloi+sI (eu + €,) dx

The first term, CLa, is simply the streamwise component of the normal pressures integrated over the airfoil. In the case of a subsonic airfoil, this term is canceled by the leading edge suction force.

The pitching moment coefficient about the leading edge of a thin, super­sonic airfoil can be written

= j CPlx dx – j Cpx dx 2 2 f’

= ~ga + ffj0 + e^x dx

Table 5.1 compares the results of the linearized theory with the more exact predictions made earlier for the symmetrical wedge airfoil pictured in Figure 5.22. In this particular case, the linearized theory is seen to be somewhat optimistic with regard to lift and drag and predicts the center of pressure to be further aft than the position obtained from the more exact calculations. Nevertheless, the Ackeret theory is valuable for predicting trends. For example, for symmetrical airfoils, the expressions of Cdw and Cm reduce to

4

The center of pressure for a symmetrical airfoil in supersonic flow is thus predicted to be at the midchord point.

Figure 5.24 (taken from Ref. 5.6) provides a comparison between the linear theory and experiment for a 10% thick biconvex airfoil. This figure shows fairly good agreement of the theory with experiment with the differences being of the same signs as those in Table 5.1.

The foregoing treatment based on Prandtl-Meyer and oblique shock relationships is somewhat tedious to apply. Also, the general behavior of supersonic airfoils is not disclosed by this approach. Therefore we will now consider a linearized solution that holds for slender profiles and for Mach numbers that are not too close to unity or not too high.

Assuming that the free-stream velocity is only perturbed by the presence of a slender body at a small angle of attack, the x and у components of the local velocity can be written as

Vx=V~+u Vy = v

д2фс _ 1 д2фі дХ2 ~P дХ2

Substituting this into the left side of Equation 5.56 gives

Since the terms within the parentheses are equal to zero, it follows that Equation 5.58 is a solution of Equation 5.56.

Now consider a body contour У(х). At any point along the contour, the following boundary condition must hold.

dY v dx Vo, + и

_ v

Vac

Equation 5.59 holds to the first order in the perturbation velocities.

Relating v to the incompressible perturbation velocity potential leads to

(5.60)

In the compressible case,

(5.61)

дф^ду can be expanded in a Maclaurin series to give

дфі(х, у) дфі(х, 0) /f £2)

dy ~ ду 1 ‘ ’

Thus, by comparing Equations 5.60 and 5.61, it follows that the body contour for which фс holds is the same (to a first order) as that for </>,.

We are now in a position to determine the pressure distribution for a given slender body shape as a function of Mach number. Along a streamline the resultant velocity, U, in terms of the perturbation velocities, can be written as

U = [(V„+uf+v2im = V» + и (to a first order)

Euler’s equation along a streamline was derived earlier in differential form. Expressed in finite difference form, it can be written as

UAU + ^- = 0

Using Equation 5.63, this becomes

Finally,

p (ll2)pVj

(t/2)PVj

(5.64)

Since и = дфідх, it follows from Equations 5.64 and 5.58 that the pressure distribution over a slender body at a finite subsonic Mach number is related to the pressure distribution over the same body at M = 0 by

(5.65)

This was assumed earlier in this chapter as Equation 5.2.

Thus, to predict the lift and moment on a two-dimensional shape such as an airfoil, one simply calculates these quantities in coefficient form for the incompressible case and then multiplies the results by the factor 1//3.

The three-dimensional case is somewhat more complicated, but not much. Here,

<t>c = – jp4>i(x, f3y, f3z) (5.66)

Hence, to find the compressible flow past a three-dimensional body with coordinates of x, y, and z, one solves for the incompressible flow around a body having the coordinates x, fiy, and jQz. The pressure coefficients are then related by

(5.67)

When the free-stream Mach number exceeds unity, the flow around an airfoil will appear as shown in Figure 5.21a or 5.21b. If the nose of the airfoil is blunt, a detached bow shock will occur, causing a small region of subsonic flow over the nose of the airfoil. After the flow is deflected subsonically around the nose, it expands again through Mach waves fanning out from the convex surfaces to supersonic conditions. As it leaves the trailing edge, the flows along the upper and lower surfaces are deflected by oblique shock waves and become parallel to each other and to the free stream.

In the case of a sharp leading edge, which is the case for an airfoil designed to operate supersonically, the flow is deflected at the leading edge by oblique shock waves attached to the leading edge.

The diamond-shaped supersonic airfoil illustrated in Figure 5.21b is rela­tively easy to analyze, given the oblique shock and Prandtl-Meyer flow relationships. To begin, since pressure distributions cannot be propagated ahead, the flow will be uniform until it is deflected by the oblique shock waves above and below the leading edge. The streamlines, after passing through the oblique shocks, will remain parallel and straight until they are turned through the expansion fan, after which they are again straight and parallel until they are deflected to approximately the free-stream direction by the oblique shock waves from the trailing edge. This flow is illustrated in detail in Figure 5.22.

The flow from the trailing edge does not necessarily have to satisfy the Kutta conditions, as in the subsonic case. Instead, the final deflection, and hence the strength of the trailing oblique shock waves, is fixed by stipulating that the pressure and flow directions be the same for the flows from the upper and lower surfaces as they meet behind the trailing edge.

As an example, consider a supersonic airfoil having a symmetrical wedge configuration. as shown in Figure 5.22. We are given

M„ = 2.0

a =2°

For this case, the angle e in Figure 5.22 becomes 11.3°. Thus the required deflection angles are as follows.

region 1 to region 2 8 = 3.65°

region 2 to region 3 8 = 11.3°

region 3 to region 4 5 = 7.65° + у

region 1 to region 5 8 = 7.65°

region 5 to region 6 5 = 11.3°

region 6 to region 7 8 = 3.65° – у

У is the unknown angle that must satisfy 8^ = 8^. Figures 5.14 and 5.17 cannot be read with sufficient accuracy, so it is necessary to work with the oblique shock and Prandtl-Meyer relationships on which these figures are based. These relationships are easily programmed on a programmable cal­culator. In this way, the following numbers were readily obtained. For M, = 2.000 and 8^2 = 3.65°,

в = 33.08°

1.224

Pi

= 1.155 Pi

M2= 1.911

Pm = ggg Pm

For M2 = 1.911 and S2_3 = 11.3°, the Prandtl-Меуег relationships are as fol­lows.

— = 0.1467 Po

& = 0.2539 Po

M3 = 2.337

& = 0.07548

Po

& = 0.1579

Po

& = 0.2973 Pi

fe = 0.6219 Pi

For M, = 2.000 and 8 M = 7.65°,

в = 36.86°

— = 1.513 Pi

— = 1.341 Pi

M5= 1.808

For the preceding with S5-6 = 11.3°, the Prandtl-Meyer relationships applied to the lower surface are as follows.

— = 0.1719

Po

— = 0.2843

Po

Mb = 2.220

— = 0.09057 Po

— = 0.1799

Po

— = 0.3186 Pi [2]

With this information, we can determine that

— = 0.3639 —

Pi Рг

Also,

— = 0.4820 —

Pi Рб

Thus, if p4 and p? are to be equal, it must follow that

^ = 1.325 —

Pi Pe

The problem now is finding the value of у that results in oblique shocks trailing from the upper and lower surfaces having compression ratios that satisfy the above.

Through the process of trial and error this was accomplished with the following results.

The ratio of the actual velocities can be found from

V4 _ M4 a4 _ M5 p4 p7 V7 M7 a7 M7 p7 p4 = 0.982

Thus a shear layer, or vorticity, is generated downstream of the airfoil, since the loss in total temperature is slightly different between the upper and lower surfaces.

The lift coefficient can be expressed as

c, = f'(&–&) dx (5.53)

yMJJo p« pJ

where x is the dimensionless distance along the chord and the subscripts l and и refer to lower and upper surfaces. In this example, the pressures are

constant, but with different values, over each half of the chord. Thus

Unlike the two-dimensional, inviscid, subsonic flow, a drag known as wave drag exists for the supersonic case. This drag can be obtained by resolving the integral of the normal pressure forces over the body in the drag direction.

For the symmetrical wedge pictured in Figure 5.22, the wave drag is; therefore

D = [(p2-Ре) sin 3.65° + (ps ~ Рз) sin 7.65°] |

In dimensionless form, this becomes

The moment coefficient about the leading edge will be given by

From Cm and Ct, the center of pressure for this symmetrical airfoil is seen to be 0.395 chord lengths behind the leading edge.

An airfoil operating at high, but subsonic, Mach numbers is pictured in Figure 5.18. If the free-stream Mach number is sufficiently high, the local flow as it progresses back along the upper surface will reach a point where the local Mach is equal to, or greater than, unity. As the flow continues along the concave surface, a region of supersonic flow develops. However, as the flow

m. < i

approaches the trailing edge, it must eventually become subsonic again. As we saw in the preceding section, this can occur only through a shock wave.

If the compression to subsonic flow occurs before the trailing edge, as shown in Figure 5.18a, there is no deflection of the flow as it passes through the wave, so that the shock wave is normal to the flow. As the free-stream Mach number is increased, a similar transonic region will develop on the lower surface, as shown in Figure 5.18b. Immediately behind both normal shocks, the boundary layer will separate because of the strong positive pressure gradients. This results in a loss in lift and a sharp increase in the drag. Increasing still further will result in the shock waves on both the upper and lower surfaces moving to the trailing edge. Here they become oblique shocks in order to turn the flow from both surfaces in the free-stream direction. Because of the developing boundary layer, as the shocks move toward the trailing edge, they assume a bifurcated or A form, as shown. Here, within the boundary layer, compression begins initially through an oblique shock and continues through to a normal shock.

Increasing the Mach number also causes the sonic line, defining the forward extent of the supersonic flow, to move forward. This line, shown dashed in Figure 5.18, is a constant pressure surface along which M = 1.

A detailed treatment of analytical methods for predicting airfoil charac­teristics is beyond the scope of this text. However, an interesting aspect of transonic airfoil behavior, discussed in Reference 5.8, is the limiting Mach number concept, which also leads to a limit on pressure coefficients. This particular reference presents semiempirical methods for estimating two – dimensional and three-dimensional values of CL and CD through the transonic regime.

Combining Equations 5.35 and 5.17b, the pressure, p2, immediately downstream of the normal shock can be written in terms of upstream reservoir pressure, po, and the local Mach number just ahead of the normal shock.

(5.49)

This ratio reaches a maximum value at a Mach number denoted as the limiting Mach number, М, шь and given by,

= 1.483 (y = 1.4)

Laitone argues in Reference 5.9 that the normal shock will be positioned on the surface of a transonic airfoil at the location where the local Mach number equals the limiting Mach number, thus assuring the maximum positive pressure downstream of the shock wave.

This limit on the local Mach number leads to a minimum pressure coefficient that can be attained on an airfoil surface ahead of the shock wave. Cp is defined as

, _ P~P«.

p (1/2 )p„Vj

which can be written as

The ratio of the local pressure to the reservoir pressure, p0, is a function of the local M, according to Equation 5.17b, and decreases monotonically with

M. When the local, M reaches Мц, this ratio attains a minimum value of 0.279. Using Equation 5.17b also to relate the free-stream static pressure, Po„, to po, a limiting value for Cp is obtained as a function of the free-stream Mach number.

This relationship is presented graphically in Figure 5.19. The limiting value of Cp is seen to decrease rapidly in magnitude as М» increases. ^Pcr is also presented on this same figure and is a value of Cp necessary to achieve local sonic flow. The value of M« corresponding to G* is equal to Mcr, the critical Mach number. G, is obtained from Equation 5.51 by setting the local M equal to unity to obtain p/po – The result is identical to Equation 5.52 except for replacing the constant 0.279 by 0.528.

Before discussing the significance of these relationships, let us return to Equation 5.2, which allows us to predict Cp at subsonic Mach numbers based on predictions for incompressible flow. К CPc is the pressure coefficient at a given Mach number, the Prandtl-Glauert correction states that, for the same geometry, Cp at M = 0 will equal /3CPc. Using this scaling relationship, the critical value for the incompressible Cp can be calculated from the com­pressible CPa. This result is also presented in Figure 5.19.

As an example of the use of Figure 5.19, consider the Liebeck airfoil in Figure 3.43. The minimum Cp at M„ = 0 for this airfoil is approximately -2.8. Hence its critical Mach number is estimated from the lower curve of Figure 5.19 to be approximately 0.43. Its limiting Mach number, based on Cp = -2.8, would be 0.57. However, at this Mach number, the Prandtl-Glauert factor, /3, equals 0.82, so that the minimum Cp at this Mach number is estimated to be -3.4. A second iteration on Мім then gives a value of 0.52. Continuing this iterative procedure, a value of Мім = 0.53 is finally obtained.

Next, consider the chordwise pressure distributions presented in Figure

5.20. Here, Cp as a function of chordwise position is presented for the NACA 64A010 airfoil for free-stream Mach numbers of 0.31, 0.71, and 0.85, all at a constant angle of attack of 6.2°. Only the pressure distributions over the upper

surface are shown. The critical Mach number for this airfoil, corresponding to the minimum Cp of approximately -3.0, is approximately Mcr= 0.43. Thus, at М» = 0.31, this airfoil is operating in the subsonic regime. At M„ = 0.71, the flow is transonic and theoretically limited to a Cp of -1.7. Near the nose this value is exceeded slightly. However, the experimental values of Cp are indeed nearly constant and equal to Cp,^, over the leading 30% of the chord. At between the 30 and 40% chord locations, a normal shock compresses the flow,

and the pressure rises over the after 60% to equal approximately the subsonic distribution of Cp over this region. Using the preceding relationships for isentropic flow, the pressure rise across a normal shock, and the limiting Mach number of 1.483, one would expect an increase in Cp of 1.56. The experiment shows a value of around 1.2. This smaller value may be the result of flow separation downstream of the shock.

The results at М» = 0.85 are somewhat similar. Over the forward 35% of the chord, Cp is nearly constant and approximately equal to Cp, imit. Behind the normal shock, at around the 40% chord location, the increase in Cp is only approximately 0.25 as compared to an expected increment of approximately

1.24. Here the separation after the shock is probably more pronounced, as evidenced by the negative Cp values all the way to the trailing edge.

Notice the appreciable reduction in the area under the Cp curve for Moo = 0.85 as compared to M„ = 0.71 because of the difference in the C^, values of the two Mach numbers. This limiting effect on Cp is certainly a contribution to the decrease in Q at the very high subsonic Mach numbers, the other major contributor being shock stall.

Considering the mixed flow in the transonic regime, results such as those shown in Figures 5.3, 5.5, and 5.20, and nonlinear effects such as shock stall and limiting Mach number, the prediction of wing and airfoil characteristics is a difficult task of questionable accuracy. Although the foregoing material may help to provide an understanding of transonic airfoil behavior, one will normally resort to experimental data to determine Q, Cd, and Cm accurately in this operating regime.

Let us consider the two supersonic flows pictured in Figure 5.15a and 5.15b. When the flow is turned by a surface concave to the flow, as in Figure 5.15a, we have seen that an oblique shock originating from the bend in the surface will compress the flow and turn it though the angle, 8. The question then posed is, how is the flow turned around a bend convex to the flow, as shown in Figure 15b. As suggested by the figure, this is accomplished through a continuous ensemble of weak expansion waves, known as an expansion fan.

In order to examine the flow relationships in this case, we take an approach similar to that for oblique shock waves. Consider supersonic flow through a single, weak wave, known as a Mach wave, as illustrated in Figure 5.16. The wave represents a limiting case of zero entropy gain across the wave. Hence the turning and velocity changes are shown as differentials instead of as finite changes. Since the wave is a weak wave, it propagates

Figure 5.15b Deflection of a supersonic flow by a series of Mach waves (expan­sion).

normal to itself at the acoustic velocity, a, which added vectorially to the free-stream velocity, V, defines the angle of the wave p,.

■ – i a M = sm —

= sin-1 – гг (5.45)

Applying momentum principles across the wave, as done previously for the oblique shock wave, results in

dVt = 0

and

-dp = pa[(V + dV) sinGu. + dS)~ a]

This reduces to

-dp = pa2|^pr +VM2- 1dS j

Since the tangential velocity component is unchanged across the wave, it follows that

—^— = (V + d V) cos (p + d8) tan ju.

Expanding this and substituting Equation 5.45 results in

dV V dS л/м2-1

dp __ ypM2

dS ~ VM!- 1

Thus, this weak wave, deflecting the flow in the direction shown in Figure

5.16, results in an expansion of the flow, since dp/dS is negative. It is also possible for small deflections in the opposite direction to produce a com­pression with a Mach wave. This fepresents a limiting case of an oblique shock wave.

The expansion fan shown in Figure 5.15b represents a continuous dis­tribution of Mach waves. Each wave deflects the flow a small amount, so that the integrated effect produces the total deflection, 8. The changes in the flow can be related to the total deflection by integrating Equation 5.46. The energy equation is used to relate the local sonic velocity to V. It is convenient in so doing to let 8 = 0 at M = 1.0. This corresponds to V = a* for a given set of reservoir conditions. Therefore,

The details of performing this integration will not be presented here. They can be found in several texts and in Reference 5.7. The final expression for 8 becomes

This relationship is presented graphically in Figure 5.17 and is referred to as Prandtl-Meyer flow. To use this graph one relates a given flow state back to the M = 1 condition. For example, suppose the local Mach is equal to 3.0. This means that, relative to M = 1, the flow has already been deflected

Figure 5.17

through an angle of approximately 50°. Suppose the flow is turned an additional 50°. Relative to M = 1, this gives a total deflection of 100°. Thus, one enters Figure 5.17 with this value of 8 to determine a final Mach number slightly in excess of 9.0. Since the Prandtl-Meyer flow is isentropic, the flow state is determined completely by the reservoir conditions and the local Mach number (Equation 5.17).

Generally, a shock wave is not normal to the flow. In Figure 5.11, for example, the wave becomes oblique to the flow as one moves away from the nose. Let us therefore examine this more general case in the same manner as we did the normal shock wave.

Figure 5.12 pictures a flow passing through an oblique shock wave that is

In view of Equation 5.41, the energy equation becomes

V? V?

CpT, + ^=CpT2 + -^

Equations 5.39, 5.40, and 5.42 are identical to Equations 5.25, 5.26, and 5.28 if Vin is replaced by Vi and V2„ by V2. Thus, all of the relationships previously derived for a normal shock wave apply to an oblique shock wave if the Mach numbers normal to the wave are used. These relationships, together with the fact that the tangential velocity remains unchanged through the wave, allow us to determine the flow conditions downstream of the wave as well as the angles в and S.

2.0.

1.92

M2„ = 0.69

Now we must be careful, because the tangential velocity is constant across the wave, not the tangential Mach number. To obtain M2f, we write

or, in this case,

Notice for this example that

M2 = VMi + Ml, = 1.31

Thus the flow is still supersonic after it has passed through the wave, unlike the flow through a normal shock wave.

As an exercise, repeat the foregoing example, but with а 6 of 76.5°.

Surprisingly, the same turning angle is obtained for the same upstream Mach number but different wave angle, 0. For this steeper wave, which is more like a normal shock, the flow becomes subsonic behind the wave, M2 being equal to 0.69.

The deflection angle, 3, as a function of 6, for a constant Mach will appear as shown qualitatively in Figure 5.13. For a given Mi, a maximum deflection angle exists with a corresponding shock wave angle. For deflections less than the maximum, two different в values can accomplish the same deflection. The oblique shock waves corresponding to the higher 6 values are referred to as strong waves, while the shock waves having the lower в values are known as weak waves. There appears to be no analytical reason for rejecting either possible family of waves but, experimentally, one finds only the weak oblique shock waves. Thus, the flow tends to remain supersonic through the wave unless it has no other choice. If, for a given Mb the boundary of the airfoil requires a turning greater than Smax, the wave will become detached, as illustrated in Figure 5.11. The flow then becomes subsonic just behind the normal part of the wave and navigates around the blunt nose under the influence of pressure gradients propagated ahead of the

airfoil in the subsonic flow region. It then accelerates downstream, again attaining supersonic speeds.

An explicit relationship for 8 as a function of Mi and 6 can be obtained by applying the equations for the normal shock wave to the Mach number normal components of the oblique shock wave, Mt sin в and M2 sin (6 ~ 8). After a considerable amount of algebraic reduction, one obtains the result

Mi2 sin 20-2 cot в 6 2 + Mi2(y + cos 26)

_ _ Mi2 sin 26-2 cot 0 ~ iO + Mi2(7 + 5 cos 26) (for у = 7/5)

This equation is presented graphically in Figure 5.14 (taken from Ref. 5.7) for a range of Mach numbers and shock wave angles from 0 to 90°. 0 values

lying below the broken line correspond to weak oblique shock waves. This dividing line is close to but slightly below the solid line through the maximum deflection angles.

Before proceeding further into the question of airfoil characteristics at Mach numbers higher than Mcr, it is necessary to develop some basic relationships relating to compressible subsonic and supersonic flows.

One-DImenslonal Isentroplc Flow

We will begin by considering briefly a reversible, adiabatic flow where the state of the flow is a function only of the position along the flow direction as, for example, a uniform flow through a duct. This simple case illustrates some of the pronounced differences between subsonic and supersonic flows. Ap­plying the momentum theorem to a differential fluid element, Euler’s equation of motion along a streamline was derived in Chapter Two and is again stated here.

VdV + —= 0 P

Also, the continuity equation was derived earlier. For one-dimensional flow through a pipe having a variable cross-sectional area of A,

pAV = constant (5.10)

For an inviscid fluid, the density and pressure are related through the isentropic process

■—r = constant Py

Finally, the properties of the gas are related through the equation of state.

p = pRT (5.12)

Differentiating by parts, Equation 5. JO can be written as

* + £+£-0

p V A

From Chapter Two, the local acoustic velocity is given by

dp

Defining the local Mach number, M, as Via and substituting Equations 5.14 and 5.13 into Equation 5.9 leads to a relationship between и and A.

dV -1 VdA ds 1 — M2 A ds

Since V and A are both positive, we arrive at the surprising result (at least to those who have never seen it) that, for supersonic flow through a duct, an increase in cross-sectional area in the direction of flow will cause the flow to accelerate. Also, Equation 5.15 shows that a Mach number of unity can only occur if dAlds =0 since, for M = 1, dV/ds will be finite only if the cross – sectional area does not change with distance along the duct. This does not mean that M must equal unity when dAlds equals zero but, instead, that dAlds equal to zero is a necessary condition for M = 1.

Consider flow from a reservoir through a converging-diverging nozzle, as pictured in Figure 5.6. Such a nozzle is referred to as a Laval nozzle. If the

reservoir pressure, po, is sufficiently high relative to the exit pressure, pE, the flow will accelerate to Mach 1 at the throat. Beyond the throat, with dA/ds positive, the flow will continue to accelerate, thereby producing a supersonic flow. Downstream of the throat, pressure and density decrease as the velocity increases with the increasing area.

The compressible Bernoulli equation governing one-dimensional isen – tropic flow was derived earlier. In terms of the local acoustic velocity,

V2 a2

— +——- = constant

2 у -1

For this case of flow from a reservoir,

V2. a2 _ ao2 2 у – 1 у — 1

Dividing this equation through by a2 and using the isentropic relationships among p, p, and T leads to these three quantities as a function of the local Mach number.

These relationships, presented graphically in Figure 5.7, are valid for Mach numbers greater than unity if the flow is shockless. The subject of shock waves will be treated later.

At the throat the local velocity and the local acoustic velocity are equal. Designating this velocity by a*, Equation 5.16 can be written as

(5.18)

Using Equations 5.11 and 5.14, it follows that

2 "p-u

(*y + l)J

= 0.528

(5.20)

In these equations, the superscript * refers to the throat.

Equation 5.19 shows that the airflow from a reservoir will reach Mach 1 if the reservoir pressure exceeds the exit pressure by a factor of at least 1.894. The mass flow rate, m, through the nozzle will be

m = p*A*a* (5.21)

where A* is the throat area.

Using Equations 5.18 and 5.20, this becomes

m = 0.579p0a0A* (5.22)

Observe that this is the maximum mass flow rate that can be obtained from a given reservoir independent of the exit pressure. For example, consider two tanks connected together through a nozzle having a throat area of 1 m2, as pictured in Figure 5.6. Assume that the air in both tanks is at standard sea level conditions. We will now begin to lower the pressure in one tank, causing air to flow from the other tank into the one with the vacuum. As the pressure in the vacuum tank is gradually reduced, the mass flow through the pipe will increase continuously, assuming the volume in the other tank, or reservoir, is sufficiently large so that its pressure and density does not change

significantly. The pressure drop along the pipe resulting from skin friction will exactly equal the pressure difference between the two tanks. However, when the pressure in the vacuum tank is reduced to 53.5 kN/m2 (from Equation 5.19), a value of M = 1 occurs at the throat. The nozzle, or flow, is then said to be “choked,” since a further reduction in the pressure downstream of the throat will not result in any further increase in the mass flow. From Equation 5.22, this critical mass flow will equal 242.3 kg/s.

If we assume that the flow beyond the throat is still isentropic, Equations 5.17, 5.18, 5.20, and 5.21 can be combined to give

(A2 y- (2Іу+1)(у+ту~1)

UV 2 [l-(plpo)(y~"ly](plpo)2ly

This is known as St. Venant’s equation.

Substituting for the local pressure ratio in terms of Mach number, this can also be written as

Since p, p, and T are related through the adiabatic process and the equation of state, it follows from the foregoing that p, p, and T are all uniquely related to their corresponding reservoir values by the ratio of A to A*. This obviously raises some problems since, in the example of Figure 5.6, the pressure at the exit into the vacuum tank does not necessarily have to match the pressure from Equation 5.23 corresponding to the area of the duct at its connection to the vacuum tank. Some nonisentropic mechanism must exist that will allow the pressure to adjust to exit conditions. This leads us to the concept of a shock wave.

Normal Shock Waves

across which the flow properties p, p, T, and V change discontinuously. We will now examine the equations governing the flow to see if such a standing wave is possible and to determine the relationships between the upstream and downstream fluid properties. To begin, the equations governing the con­servation of mass and momentum must hold.

In addition, the equation of state,(Equation 2.1, must also hold. It is repeated here for convenience. ‘ 1

equation of state p = pRT

A fourth relationship, which has not been used as yet, is the energy equation.

energy

V,2 V->2

CpTt + Ц – = CPT2 + Ц – = CpTo

This equation, derived in Reference 5.6, applies to adiabatic flows where no heat is added to the flow. Cp is the specific heat at constant pressure. The product CPT is the enthalpy of the flow per unit mass. Thus Equation 5.28 states that the sum of the enthalpy and kinetic energy per unit mass of an adiabatic flow remains constant. Cp, R, y, and the specific heat at constant volume, C„, are all interrelated.

R = Cp — Cv y = ~C

V-‘t)

Ry

(y — l)

If Equation 5.29 is substituted into the energy equation (Equation 5.28), it is interesting to note that one obtains the compressible Bernoulli equation (Equation 5.16). Thus Equation 5.16 and the energy equation are equivalent for isentropic flow. However, across the shock wave the flow is not a reversible, adiabatic process, so the changes in state are not related by Equation 5.11.

In order to see how p, p, T, and M change across a normal shock wave, we begin by substituting Equation 5.25 into Equation 5.26 so that,

Pi + PiVi2 = p2 + pV22

Since a2 = yplp, it follows that

Pi + ypiM2 = p2 + УР2М22

P2^i + yM2 p2 1 + yM-2

Manipulating Equation 5.28 in a similar manner, it follows that

/02V 1 + [(7 ~ 1)/2]M^ _ P2P1 T2 W 1 + [(-y – l)/2]M2i pip2 Ti

Thus,

P2 p2l + l(y-W]Mi2 Pi p, l + [(y-l)/2]M25

Next, Equation 5.25 is written as

£lMi£i = t л Р2М2С12 u

Substituting Equations 5.30, 5.31, and 5.32 into the preceding equations leads to an implicit relationship for M2 as a function of Mt.

МЛ/І + К7-І)/2JM,1 M2Vl + [(y-m]M2i

——– гтда———————— ГТтМ?——– – f(M)

Obviously, one solution of the above is M2 = M,, in which case p2 = pi, p2 = pu and T2 = T, so that there is no discontinuity in the flow and the solution is trivial. The other solution is apparent from Figure 5.9, where f(M)

0 0.4 0.8 1.2 1.6 2.0

Mach number, M

Figure 5.9 Function to determine conditions across a normal shock wave.

is presented as a. function of M. It is seen that the same value for /(M) is obtained from two different values of M, one greater and one less than unity. For example, if Mi is equal to 2.0, Equation 5.33 would be satisfied by an M2 value of approximately 0.57. One might, of course, say that a value of Mi equal to 0.57 with M2 equal to 2.0 would also satisfy Equation 5.33 which, indeed, would be the case. However, it can be argued on the basis of the second law of thermodynamics that the flow ahead of the shock wave must be supersonic (see Ref. 5.6, p. 234). An entropy loss, in violation of the second law, will occur if Mi is less than unity. Therefore, in Figure 5.9, M, is greater than unity and M2 is the value of M less than unity for which

, , ; /(М2) = f(Mi)

l’*

Notice from Equation 5.33 that /(M) approaches a value of (у ~ 1 )mly, or approximately 0.319, as M,-»°°. Thus, behind a normal shock wave, the Mach number has a lower limit of approximately 0.38.

Since there is an entropy gain across the normal shock wave, a loss occurs in the total, or reservoir, pressure as the flow passes through the wave. With some algebraic manipulation of the energy equation and application of the isentropic relationships before and after the wave (but not across it), the following equation can be obtained.

Pb = P2 Г1 + [(у – l)/2]МЛ»-‘1 Po, Pi ІЛ + [(y – 1)/2]M,2J

Poj and po, are the reservoir pressures behind and ahead of the wave, respectively.

Equations 5.30 to 5.34 are unwieldy to use because of their implicit nature. However, after some algebraic manipulation, they can all be reduced to explicit functions of Mx. These can be found in Reference 5.7 and are repeated here. Following the lead of Reference 5.7, the second form of each equation is for у = 7/5.

Figure 5.10 presents these relationships graphically and can be used for approximate calculations.

Let us now return to the problem of flow in the duct illustrated by Figure 5.8. Suppose the pressure in the vacuum tank is lowered to a value of 80 kN/m2. Furthermore, let us assume that the area of the duct entering the second tank is large so that, as the air enters the tank, its velocity, and hence dynamic pressure, is low. Thus, 80 kN/m2 would represent approximately the reservoir pressure downstream of a normal shock in the duct. The upstream reservoir pressure is equal to the standard sea level value of 101.3 kN/m2. Thus,

— = 0.790

P o,

From Figure 5.10, the Mach number, Mb just upstream of the normal shock wave equals 1.85. In addition,

— = 3.85 Pi

^ = 2.43 Pi

M2 = 0.605

Thus, from Equation 5.24

A.,.50

Therefore, we predict for the pressure in the vacuum tank of 80 kN/m2 that a normal shock wave must be positioned in the duct at a location where the duct area is one and a half times greater than the throat area.

This supersonic flow through a duct that must ultimately come to rest in the vacuum is directly comparable to a blunt-nosed body, or airfoil, traveling at supersonic speeds. Figure 5.11 depicts a supersonic airfoil with a rounded leading edge traveling at a Mach number of 1.85. Since the flow must come to rest at the stagnation point on the nose, it obviously must be subsonic for some extent ahead of the nose. The result is a shock wave that is normal to the flow in the vicinity of the nose. As in the case of the duct flow, immediately behind this wave, the flow is subsonic with a Mach number of 0.605. The shock wave, positioned away from the nose some small distance, is referred to as a “detached” shock wave, since it is detached from the surface.

Since weak pressure disturbances propagate at the speed of sound, the time that a fluid particle ahead of a moving body is influenced by the pressure field around the body is proportional to the difference between the acoustic velocity, a, and the speed of the body, V. As V increases to a (i. e., as М» approaches unity), the fluid is displaced less and less ahead of the body. Thus the streamline pattern around an airfoil and hence its pressure distribution can be expected to change with M„, even though the flow is subsonic everywhere.

As long as the flow remains entirely subsonic, the effect of M* on airfoil characteristics can be estimated by the use of a factor, p, where /3 is defined as

p = Vl-Mo»2 (5.1)

P is known as the Prandtl-Glauert compressibility correction factor.

In a later, more complete treatment of p, it will be noted that the local pressure coefficient at a given point on an airfoil in subsonic compressible flow, CPc, is related to the pressure coefficient in incompressible flow, CPi, by

where x is the dimensionless distance along the airfoil chord and the sub­scripts l and и refer to lower and upper surfaces, respectively. Thus, it follows from Equations 5.2, 5.3, and 5.4 that the lift and moment coefficients for compressible flow are related to those for incompressible flow in a manner similar to Equation 5.2.

, =9l

ІС p

Cm,

P

Notice from the use of Equations 3.11 and 3.12 that neither the center of pressure, Xcp, nor the location of the aerodynamic center, Xac, varies with Mach number in the purely subsonic regime.

Obviously, the lift curve slope, Cio, also obeys Equation 5.5.

(5.7)

This relationship is presented graphically in Figure 5.1 together with the corresponding supersonic relationship, which will be discussed later. However, it must be used with caution. First, the theoretical basis on which it rests is valid only f<?r Mach numbers less than critical. Second, by comparison with experiment, the ratio QJG, is overestimated by Equation 5.5 in some cases and underestimated for others, depending on the airfoil geometry.

Reference 5.4 presents data on nine different airfoils at Mach numbers up to 1.0. These airfoils vary in thickness, design lift coefficient, and thickness distribution; they are illustrated in Figure 5.2. Pressure distribution*’ measurements were made in order to determine lift and pitching moment, and wake surveys were taken for determination of drag. Unfortunately, it is difficult to generalize on the data, and they are too voluminous to present here. A sample of the data is presented in Figure 5.3 for the 64A009 airfoil (taken from Ref. 5.4). The normal force coefficient, C„, is defined as the force normal to the chord line (obtained by integrating the normal pressure around the airfoil contour) divided by the product of the free-stream dynamic

d

c

Figure 5.2 Airfoil profiles.

pressure and the airfoil chord. Cd is the usual drag coefficient and is com­posed of the skin friction drag and the component of C„ in the drag direction. The lift coefficient is slightly less than C„ and can be obtained from

Q = C„ cos a (5.8)

Estimated critical Mach numbers are indicated by arrows in Figure 5.3 and were obtained from calculated graphs found in Reference 3.13. An example of such a graph is presented in Figure 5.4a and 5.4b. The results of Figure 5.4a apply approximately to the airfoils of Figure 5.3 and were used to obtain the Mcr values shown there. The 64Axxx airfoils are similar to the 64-xxx airfoils except that the rear portion of the 64Axxx airfoils are less curved than the corresponding surfaces of the 64-xxx airfoils.

Observe that the thinner symmetrical airfoils, as one might guess, have the higher critical Mach numbers at а С/ of zero. However, the rate at which M„ decreases with С/ is greater for the thinner airfoils. Thus, the thicker airfoils become relatively more favorable as C increases. As shown in Figure 5.4b, camber results in shifting the peak Mcr to the right. As a function of

thickness, the curves for the cambered airfoils are similar in appearance to those for the symmetrical airfoils.

It can be seen from Figure 5.3 that Equation 5.5 holds in a qualitative sense. At a given angle of attack, the lift coefficient increases with Mach number; however, the increase is not as great as Equation 5.5 predicts. For example, at an angle of attack of 6° and a Mach number of 0.3, C, is equal to 0.51. Therefore, at this same angle of attack, one would predict a C( of 0.61 at a Mach number of 0.6. Experimeiitally, however, Q equals only 0.57 at the higher M value.

Figure 5.5, also based on the data of Reference 5.4, presents the variation with Mach number of the slope of the normal force coefficient curve for 4, 6, 9, and 12% thick airfoils. The theoretical variation of C„a with M, matched to the experiment at an M of 0.3, is also included. Again, the Glauert correction is seen to be too high by comparison to the experimental results. Contrary to these observations, Reference 5.2 states that Equation 5.2 underestimates the effect of Mach number and presents a comparison between theory and experiment for a 4412 airfoil to substantiate the statement.

Reference 5.6 presents a graph similar to Figure 5.5 for symmetrical airfoils varying in thickness from 6 to 18%. The results are somewhat similar except that, at the lower Mach numbers, below approximately 0.8, the trend of C/a with thickness is reversed. Both graphs show Qa continuing to increase with a Mach number above the critical Mach number. Unlike Figure 5.5, the

results presented in Reference 5.6 show a closer agreement with the Prandtl – Glauert factor for the lower thickness ratios.

Reference 5.5 is a voluminous collection of data pertaining to aircraft and missiles. Subsonic and supersonic data are given for airfoil sections, wings, bodies, and wing-body combinations. Any practicing aeronautical engineer should be aware of its existence and have access to the wealth of material contained therein. In Section 4 of this reference, the Prandtl-Glauert factor is used up to the critical Mach number. Isolated examples given in this reference using j8 show reasonably good agreement with test results. Thus, in the absence of pliable data, it is recommended that the Prandtl-Glauert com­pressibility correction be used, but with caution, keeping in mind dis­crepancies such as those shown in Figure 5.5.

From Figures 5.3 and 5.5 it is interesting to note that nothing drastic happens to the lift or drag when the critical Mach number is attained. Indeed, the lift appears to increase at a faster rate with Mach number for M values higher than M„. Only when M„ is exceeded by as much as 0.2 to 0.4 does the normal force coefficient drop suddenly with increasing M„. The same general behavior is observed for Cd, except that the increments in above Mcr where the Cd curves suddenly bend upward are somewhat less than those for the breaks in the G, curves.

The value of М» above which Q increases rapidly with Mach number is known as the drag-divergence Mach number. A reliable determination of this number is of obvious importance in estimating the performance of an airplane such as a jet transport, designed to operate at high subsonic Mach numbers.

The preceding material on lift and drag was limited primarily to incompressible flows, that is, to Mach numbers less than approximately 0.4. Compressibility becomes more and more important as the Mach number increases. In the vicinity of a Mach number of unity, airfoils and wings undergo a radical change in their behavior. It is not too surprising, therefore, to find that the equations covering the flow of air undergo a similar change at around М» = 1.0.

Many textbooks are devoted entirely to the subject of compressible flows. References 5.1 and 5.2 are two such examples. Here, the several equations and techniques for the study of gas dynamics are developed in considerable detail. An excellent, lucid, qualitative explanation of com­pressibility effects on wings and airfoils is found in Reference 5.3.

QUALITATIVE BEHAVIOR OF AIRFOILS AS A FUNCTION OF MACH NUMBER

We will consider three regimes of flow around an airfoil. In the first, the flow is everywhere subsonic with a relatively high Mach number. The second regime is referred to as transonic flow. Here the free-stream Mach number is less than unity, but sufficiently high so that the flow locally, as it accelerates over the airfoil, exceeds the local speed of sound; that is, locally the flow becomes supersonic. The lowest free-stream Mach number at which the local flow at some point on the airfoil becomes supersonic is known as the critical Mach number. The third regime is the supersonic flow regime, in which the free-stream Mach number exceeds unity. Even here, a small region of subsonic flow may exist near the leading edge of the airfoil immediately behind a shock wave depending on the bluntness of the leading edge. The sharper the leading edge, the smaller is the extent of the subsonic flow region.