Category When Is A Flow Compressible?

Governing Equations for Quasi-One­Dimensional Flow

Recall the one-dimensional flow treated in Chapter 8. There, we considered the flow – field variables to be a function of a only, that is, p = p(x), и = u(x), etc. Strictly speaking, a streamtube for such a flow must be of constant area; that is, the one­dimensional flow discussed in Chapter 8 is constant-area flow, as sketched in Fig­ure 10.4a.

In contrast, assume that the area of the streamtube changes as a function of x, that is, A = A(x), as sketched in Figure 10.4b. Strictly speaking, this flow is three­dimensional; the flow-field variables are functions of x, y, and z, as can be seen simply by examining Figure 10.4b. In particular, the velocity at the boundary of the streamtube must be tangent to the boundary, and hence it has components in the у and z directions as well as the axial x direction. Flowever, if the area variation is moderate, the components in the у and z directions are small in comparison with the component in the x direction. In such a case, the flow-field variables can be assumed to vary with x only (i. e., the flow can be assumed to be uniform across any cross section at a given x station). Such a flow, where A = A (x), but p = p(x), p = p(x), и = u(x), etc., is defined as quasi-one-dimensionalflow, as sketched in Figure 10.4b. Such flow is the subject of this chapter. We have encountered quasi-one-dimensional flow earlier, in our discussion of incompressible flow through a duct in Section 3.3. Return to Section 3.3, and review the concepts presented there before progressing further.

Although the assumption of quasi-one-dimensional flow is an approximation to the actual flow in a variable-area duct, the integral forms of the conservation equations, namely, continuity [Equation (2.48)], momentum [Equation (2.64)], and energy [Equation (2.95)], can be used to obtain governing equations for quasi-one­dimensional flow which are physically consistent, as follows. Consider the control volume given in Figure 10.5. At station 1, the flow across area A i is assumed to be uniform with properties p, p, и , etc. Similarly, at station 2, the flow across area An

is assumed to be uniform with properties p2, p2, u2, etc. The application of the integral form of the continuity equation was made to such a variable-area control volume in Section 3.3. The resulting continuity equation for steady, quasi-one-dimensional flow was obtained as Equation (3.21), which in terms of the nomenclature in Figure 10.5 yields

Consider the integral form of the momentum equation, Equation (2.64). For a steady, inviscid flow with no body forces, this equation becomes

Since Equation (10.2) is a vector equation, let us examine its jc component, given below:

(pV • dS)n = -0> (pdS)

where (pdS)x denotes the x component of the pressure force. Since Equation (10.3) is a scalar equation, we must be careful about the sign of the x components when evaluating the surface integrals. All components pointing to the right in Figure 10.5 are positive, and those pointing to the left are negative. The upper and lower surfaces of the control volume in Figure 10.5 are streamlines; hence, V • dS = 0 along these surfaces. Also, recall that across A , V and dS are in opposite directions; hence, V • dS is negative. Therefore, the integral on the left of Equation (10.3) becomes ~Pi 4] А і + p2uA2. The pressure integral on the right of Equation (10.2), evaluated over the faces A) and A2 of the control volume, becomes — (—pj At + p2A2). (The negative sign in front of p, A is because dS over A! points to the left, which is the

©

negative direction for the x components.) Evaluated over the upper and lower surface of the control volume, the pressure integral can be expressed as

[10.4]

where dA is simply the x component of the vector dS, that is, the area dS projected on a plane perpendicular to the x axis. The negative sign inside the integral on the left of Equation (10.4) is due to the direction of dS along the upper and lower surfaces; note that dS points in the backward direction along these surfaces, as shown in Figure

10.5. Hence, the x component of p dS is to the left, and therefore appears in our equations as a negative component. [Recall from Section 2.5 that the negative sign outside the pressure integral, that is, outside the integral on the left of Equation (10.4), is always present to account for the physical fact that the pressure force p dS exerted on a control surface always acts in the opposite direction of dS. If you are unsure about this, review the derivation of the momentum equation in Section 2.5. Also, do not let the signs in the above results confuse you; they are all quite logical if you keep track of the direction of the x components.] With the above results, Equation (10.3) becomes

[10.5]

Equation (10.5) is the momentum equation for steady, quasi-one-dimensional flow.

Consider the energy equation given by Equation (2.95). For inviscid, adiabatic, steady flow with no body forces, this equation becomes

[10.6]

Applied to the control volume in Figure 10.5, Equation (10.6) yields

or

[10.7]

Dividing Equation (10.7) by Equation (10.1), we have

[10.8]

Recall that h = e + pv = e + р/р. Hence, Equation (10.8) becomes

©

volume at station 1, where the area is A, has properties p, u, and p. In traversing the length dx, where the area changes by dA, the flow properties change by the corresponding amounts dp, dp, and du. Hence, the flow leaving at station 2 has the properties p + dp, u+du, and p + dp, as shown in Figure 10.6. For this case, Equa­tion (10.5) becomes [recognizing that the integral in Equation (10.5) can be replaced by its integrand for the differential volume in Figure 10.6]

pA + pu2A + p dA = (p + dp)(A + dA) + (p + dp)(u + du)2(A + dA) [10.l 5]

In Equation (10.15), all products of differentials, such as dp dA, dp(du)2, are very small and can be ignored. Hence, Equation (10.15) becomes

A dp + Au2 dp + pu2 dA + 2puA du = 0 [10.16]

Expanding the continuity equation, Equation (10.14), and multiplying by u, we have

pu2 dA + puAdu + Au2 dp = 0 [10.17]

Subtracting Equation (10.17) from (10.16), we obtain

[10.18]

which is the differential form of the momentum equation for steady, inviscid, quasi- one-dimensional flow. Equation (10.18) is called Euler’s equation. We have seen it before—as Equation (3.12). In Section 3.2, it was derived from the differential form of the general momentum equation in three dimensions. (Make certain to review that derivation before progressing further.) In Section 3.2, we demonstrated that Equation

(3.12) holds along a streamline in a general three-dimensional flow. Now we see Euler’s equation again, in Equation (10.18), which was derived from the governing equations for quasi-one-dimensional flow.

A differential form of the energy equation follows directly from Equation (10.9), which states that

Differentiating this equation, we have

[10.19]

In summary, Equations (10.14), (10.18), and (10.19) are differential forms of the continuity, momentum, and energy equations, respectively, for a steady, inviscid, adiabatic, quasi-one-dimensional flow. We have obtained them from the algebraic forms of the equations derived earlier, applied essentially to the picture shown in Figure 10.6. Now you might ask the question, Since we spent some effort obtaining partial differential equations for continuity, momentum, and energy in Chapter 2, applicable to a general three-dimensional flow, why would we not simply set d/dy = 0 and 9/3z = 0 in those equations and obtain differential equations applicable to the one-dimensional flow treated in the present chapter? The answer is that we certainly could perform such a reduction, and we would obtain Equations (10.18) and (10.19) directly. [Return to the differential equations, Equations (2.113a) and (2.114), and prove this to yourself.] However, if we take the general continuity equation, Equation

(2.52) , and reduce it to one-dimensional flow, we obtain d(pu) = 0. Comparing this result with Equation (10.14) for quasi-one-dimensional flow, we see an inconsistency. This is another example of the physical inconsistency between the assumption of quasi-one-dimensional flow in a variable-area duct and the three-dimensional flow which actually occurs in such a duct. The result obtained from Equation (2.52), namely, d(pu) = 0, is a truly one-dimensional result, which applies to constant – area flows such as considered in Chapter 8. [Recall in Chapter 8 that the continuity equation was used in the form pu = constant, which is compatible with Equation

(2.52) .] However, once we make the quasi-one-dimensional assumption, that is, that uniform properties hold across a given cross section in a variable-area duct, then Equation (10.14) is the only differential form of the continuity equation which insures mass conservation for such an assumed flow.

Let us now use the differential forms of the governing equations, obtained above, to study some physical characteristics of quasi-one-dimensional flow. Such physical information can be obtained from a particular combination of these equations, as follows. From Equation (10.14),

[10.30]

We wish to obtain an equation which relates the change in velocity du to the change in area dA. Hence, to eliminate dp/p in Equation (10.20), consider Equation (10.18) written as

dp dp dp

p dp p

Keep in mind that we are dealing with inviscid, adiabatic flow. Moreover, for the time being, we are assuming no shock waves in the flow. Hence, the flow is isentropic. In particular, any change in density dp with respect to a change in pressure dp takes place isentropically; that is,

Substituting Equation (10.24) into (10.20), we have

Equation (10.25) is the desired equation which relates dA to du; it is called the area-velocity relation.

Equation (10.25) is very important; study it closely. In the process, recall the standard convention for differentials; for example, a positive value of du connotes an increase in velocity, a negative value of du connotes a decrease in velocity, etc. With this in mind, Equation (10.25) tells us the following information:

1. For 0 < M < 1 (subsonic flow), the quantity in parentheses in Equation (10.25) is negative. Hence, an increase in velocity (positive du) is associated with a decrease in area (negative dA). Likewise, a decrease in velocity (negative du) is associated with an increase in area (positive dA). Clearly, for a subsonic compressible flow, to increase the velocity, we must have a convergent duct, and to decrease the velocity, we must have a divergent duct. These results are illustrated at the top of Figure 10.7. Also, these results are similar to the familiar trends for incompressible flow studied in Section 3.3. Once again we see that subsonic compressible flow is qualitatively (but not quantitatively) similar to incompressible flow.

2. For M > 1 (supersonic flow), the quantity in parentheses in Equation (10.25) is positive. Hence, an increase in velocity (positive du) is associated with an increase in area (positive dA). Likewise, a decrease in velocity (negative du) is associated with a decrease in area (negative dA). For a supersonic flow, to

Figure 10*7 Compressible flow in converging and diverging ducts.

increase the velocity, we must have a divergent duct, and to decrease the velocity, we must have a convergent duct. These results are illustrated at the bottom of Figure 10.7; they are the direct opposite of the trends for subsonic flow.

3. ForM = 1 (sonic flow), Equation (10.25) shows thatrM = 0 even though a finite du exists. Mathematically, this corresponds to a local maximum or minimum in the area distribution. Physically, it corresponds to a minimum area, as discussed below.

Imagine that we want to take a gas at rest and isentropically expand it to supersonic speeds. The above results show that we must first accelerate the gas subsonically in a convergent duct. However, as soon as sonic conditions are achieved, we must further expand the gas to supersonic speeds by diverging the duct. Hence, a nozzle designed to achieve supersonic flow at its exit is a convergent-divergent duct, as sketched at the top of Figure 10.8. The minimum area of the duct is called the throat. Whenever an isentropic flow expands from subsonic to supersonic speeds, the flow must pass through a throat; moreover, in such a case, M = 1 at the throat. The converse is also true; if we wish to take a supersonic flow and slow it down isentropically to subsonic speeds, we must first decelerate the gas in a convergent duct, and then as soon as sonic flow is obtained, we must further decelerate it to subsonic speeds in a divergent duct. Here, the convergent-divergent duct at the bottom of Figure 10.8 is operating as a diffuser. Note that whenever an isentropic flow is slowed from supersonic to subsonic speeds, the flow must pass through a throat; moreover, in such a case, M = 1 at the throat.

As a final note on Equation (10.25), consider the case when M = 0. Then we have dA/A — —du/u. which integrates to Au = constant. This is the familiar continuity equation for incompressible flow in ducts as derived in Section 3.3 and as given by Equation (3.22).

Accuracy Considerations

How accurate is newtonian theory in the prediction of pressure distributions over hy­personic bodies? The comparison shown in Figure 14.9 indicates that Equation (14.7) leads to a reasonably accurate pressure distribution over the surface of a blunt body. Indeed, for “back-of-the-envelope” estimates of the pressure distributions over blunt bodies at hypersonic speeds, modified newtonian is quite satisfactory. However, what about relatively thin bodies at small angles of attack? We can provide an answer by using the newtonian flat-plate relations derived in the present section, and compare these results with exact shock-expansion theory (Section 9.7), for flat plates at small angles of attack. This is the purpose of the following worked example.

Consider an infinitely thin flat plate at an angle of attack of 15° in a Mach 8 flow. Calculate | Example 1 4.1 the pressure coefficients on the top and bottom surface, the lift and drag coefficients, and the lift-to-drag ratio using (a) exact shock-expansion theory, and (b) newtonian theory. Compare the results.

Solution

(a) Using the diagram in Figure 9.26 showing a flat plate at angle of attack, and following the shock-expansion technique given in Example 9.8, we have for the upper surface, for A?, =8 and V{ = 95.62°,

v2 = V, +0 = 95.62 + 15 = 110.62° From Appendix C, interpolating between entries,

M2 = 14.32

From Appendix A, for M = 8, poJp = 0.9763 x 104, and for M2 = 14.32, po-,/p2 = 0.4808 x 106. Since p0l = /+,,

The pressure coefficient is given by Equation (11.22), and the freestream static pressure in Figure 9.26 is denoted by p. Hence

To obtain the pressure coefficient on the bottom surface from the oblique shock theory, we have from the в-fi-M for Mt = 8 and в = 15°, /і = 21°:

Af„,i = M sin p = 8 sin 21° = 2.87

Interpolating from Appendix B, for Mn = 2.87, p3/pt = 9.443. Hence the pressure coeffi­cient on the bottom surface is

The lift coefficient can be obtained from the pressure coefficients via Equations (1.15), (1.16), and (1.18).

c„ = – f (СрЛ – Єр,,,) dx = Cm – CP2 = 0.1885 – (-0.0219) = 0.2104

c Jo

The axial force on the plate is zero, because the pressure acts only perpendicular to the plate. On a formal basis, dy/dx in Equation (1.16) is zero for a flat plate. Hence, from Equation (1.18),

C„, = 2 sin2 a = 2 sin2 15° =

From Equation (14.9), we have for the upper surface

r —

^ pi

Discussion. From the above worked example, we see that newtonian theory underpredicts the pressure coefficient on the bottom surface by 29 percent, and of course predicts a value of zero for the pressure coefficient on the upper surface in comparison to —0.0219 from exact theory—an error of 100 percent. Also, newtonian theory underpredicts q and c, j by 36.6 percent. However, the value of L/D from newtonian theory is exactly correct. This is no surprise, for two reasons. First, the

newtonian values of ct and q are both underpredicted by the same amount, hence their ratio is not affected. Second, the value of L/D for supersonic or hypersonic inviscid flow over a flat plate, no matter what theory is used to obtain the pressures on the top and bottom surfaces, is simply a matter of geometry. Because the pressure acts normal to the surface, the resultant aerodynamic force is perpendicular to the plate (i. e., the resultant force is the normal force N). Examining Figure 1.10, when this is the case, the vectors R and N are the same vectors, and L/D is geometrically given by

L

— = cot a

D

For the above worked example, where a = 15°, we have

L

— = cot 15° = 3.73 D

which agrees with the above calculations where q and c, i were first obtained, and L/D is found from the ratio, L/D = cy/cj. So, Equation (14.16), derived in our discussion of newtonian theory applied to a flat plate, is not unique to newtonian theory; it is a general result when the resultant aerodynamic force is perpendicular to the plate.

We induce from Example 14.1 the general fact that the newtonian sine-squared law, Equation (14.4), does not accurately predict the hypersonic pressure distribution on the surface of two-dimensional bodies with local tangent lines that are at small or moderate angles to the flow, such as the bi-convex airfoil shape shown in Figure 12.3. On the other hand, it generally turns out that the newtonian prediction of the lift-to-drag ratio for slender shapes at small to moderate angles of attack is reasonably accurate. These statements apply to a gas with the ratio of specific heats substantially greater than one, such as the case of air with у = 1.4 treated in Example 14.1. In the next section, we will see that newtonian theory becomes more accurate as Mx —» со and у —>■ 1. For more information on the accuracy of newtonian theory applied to two-dimensional slender shapes, see Reference 77 which is a study of this specific matter.

Finally, we note that newtonian theory does a better job of predicting the pressure on axisymmetric slender bodies, such as the 15° half-angle cone shown in Figure 14.13.

Compressible Flow over a Flat Plate

The properties of the incompressible, laminar, flat-plate boundary layer were devel­oped in Section 18.2. These results hold at low Mach numbers where the density is essentially constant through the boundary layer. However, what happens to these properties at high Mach numbers where the density becomes a variable; that is, what are the compressibility effects? The purpose of the present section is to outline briefly the effects of compressibility on both the derivations and the final results for laminar flow over a flat plate. We do not intend to present much detail; rather, we exam­ine some of the salient aspects which distinguish compressible from incompressible boundary layers.

The compressible boundary-layer equations were derived in Section 17.3, and were presented as Equations (17.28) to (17.31). For flow over a flat plate, where dpeldx = 0, these equations become

Compare these equations with those for the incompressible case given by Equations (18.1) to (18.3). Note that, for a compressible boundary layer, (1) the energy equation must be included, (2) the density is treated as a variable, and (3) in general, p and к are functions of temperature and hence also must be treated as variables. As a result, the system of equations for the compressible case, Equations (18.31) to (18.34), is more complex than for the incompressible case, Equations (18.1) to (18.3).

It is sometimes convenient to deal with total enthalpy, ho = h + V2/2, as the dependent variable in the energy equation, rather than the static enthalpy as given in Equation (18.34). Note that, consistent with the boundary-layer approximation, where v is small, ho — h + V2/2 = h + (n + v2)/2 ~ h + и2/2. To obtain the energy equation in terms of ho, multiply Equation (18.32) by u, and add to Equation (18.34), as follows. From Equation (18.32) multiplied by u,

Adding Equation (18.35) to (18.34), we obtain

Recall that for a calorically perfect gas, dh = cpdT hence,

Substituting Equations (18.39) and (18.40) into (18.38), we obtain

dh0 dh0 d /X dh0

pu——- h pv—- = —————–

dx dy dy Pr dy

which is an alternate form of the boundary-layer energy equation. In this equation, Pr is the local Prandtl number, which, in general, is a function of T and hence varies throughout the boundary layer.

For the laminar, compressible flow over a flat plate, the system of governing equations can now be considered to be Equations (18.31) to (18.33) and (18.41). These are nonlinear partial differential equations. As in the incompressible case, let us seek a self-similar solution; however, the transformed independent variables must be defined differently.

£ — PeP^e^eX

The dependent variables are transformed as follows:

и._

/ = — (which is consistent with defining stream function = v2|/)

ue

_ hp
8 ~ iho)e

The mechanics of the transformation using the chain rule are similar to that described in Section 18.2. Hence, without detailing the precise steps (which are left for your

solution to Equations (18.42) and (18.43) is the shooting technique described in Section 16.4. The approach here is directly analogous to that used for the solution of compressible Couette flow discussed in Section 16.4. Since Equation (18.42) is third order, we need three boundary conditions at r] = 0. We have only two, namely, / = /’ = 0. Therefore, assume a value for /"(0), and iterate until the boundary condition at the boundary-layer edge, /’ = 1, is matched. Similarly, Equation (18.43) is a second-order equation. It requires two boundary conditions at the wall in order to integrate numerically across the boundary layer; we have only one, namely, g(0) = gw. Thus, assume g'(0), and integrate Equation (18.43). Iterate until the outer boundary condition is satisfied; that is, g = 1. Since Equation (18.42) is coupled to Equation (18.43), that is, since pji in Equation (18.42) requires a knowledge of the enthalpy (or temperature) profile across the boundary layer, the entire process must be repeated again. This is directly analogous to the two minor iterations nested within the major iteration that was described in the discussion of the shooting method in Section 16.4. The approach here is virtually the same philosophy as described in Section 16.4, which should be reviewed at this stage. Therefore, no further details will be given here.

Typical solutions of Equations (18.42) and (18.43) for the velocity and temper­ature profiles through a compressible boundary layer on a flat plate are shown in Figures 18.4-18.7, obtained from van Driest (Reference 79). Figures 18.4 and 18.5 contain results for an insulated flat plate (zero-heat transfer) using Sutherland’s law for [jl, and assuming a constant Pr = 0.75. The velocity profiles are shown in Fig­ure 18.4 for different Mach numbers ranging from 0 (incompressible flow) to the large hypersonic value of 20. Note that at a given x station at a given Re*, the boundary – layer thickness increases markedly as Me is increased to hypersonic values. This clearly demonstrates one of the most important aspects of compressible boundary layers, namely, that the boundary-layer thickness becomes large at large Mach num­bers. Figure 18.5 illustrates the temperature profiles for the same case as Figure 18.4. Note the obvious physical trend that, as Me increases to large hypersonic values, the temperatures increase markedly. Also note in Figure 18.5 that at the wall (у = 0), (ЗT/3y)w = 0, as it should be for an insulated surface (qw = 0). Figures 18.6 and 18.7 also contain results by van Driest, but now for the case of heat transfer to the wall. Such a case is called a “cold wall” case, because Tw < Taw. (The opposite case would be a “hot wall,” where heat is transferred from the wall into the flow; in this case, Tw > Taw.) For the results shown in Figures 18.6 and 18.7, Tw/Te = 0.25 and Pr = 0.75 = constant. Figure 18.6 shows velocity profiles for various different values of Me, again demonstrating the rapid growth in boundary layer thickness with increasing Me. In addition, the effect of a cold wall on the boundary layer thickness can be seen by comparing Figures 18.4 and 18.6. For example, consider the case of Me = 20 in both figures. For the insulated wall at Mach 20 (Figure 18.4), the bound­ary layer thickness reaches out beyond a value of (y/x)^Rex = 60, whereas for the cold wall at Mach 20 (Figure 18.6), the boundary-layer thickness is slightly above (y/x)«/Re^ = 30. This illustrates the general fact that the effect of a cold wall is to reduce the boundary-layer thickness. This trend is easily explainable on a physical basis when we examine Figure 18.7, which illustrates the temperature profiles through

0 0.2 0.4 0.6 0.8 1.0

«/«

Figure 1 8.4 Velocity profiles in a compressible laminar boundary layer over an insulated flat plate (Source: van Driest, Reference 79.)

the boundary layer for the cold-wall case. Comparing Figures 18.5 and 18.7, we note that, as expected, the temperature levels in the cold-wall case are considerably lower than in the insulated case. In turn, because the pressure is the same in both cases, we have from the equation of state p = pRT, that the density in the cold-wall case is

T/Te

Figure 18.5 Temperature profiles in a compressible laminar boundary layer over an insulated flat plate. (Source: van Driest Reference 79.)

much higher. If the density is higher, the mass flow within the boundary layer can be accommodated within a smaller boundary-layer thickness; hence, the effect of a cold wall is to thin the boundary layer. Also note in Figure 18.7 that, starting at the outer edge of the boundary layer and going toward the wall, the temperature first increases, reaches a peak somewhere within the boundary layer, and then decreases to its pre­scribed cold-wall value of Tw. The peak temperature inside the boundary layer is an indication of the amount of viscous dissipation occurring within the boundary layer. Figure 18.7 clearly demonstrates the rapidly growing effect of this viscous dissipation as Me increases—yet another basic aspect of compressible boundary layers.

Carefully study the boundary-layer profiles shown in Figures 18.4-18.7. They are an example of the detailed results which emerge from a solution of Equations (18.42) and (18.43); indeed, these figures are graphical representations of Equations (18.43) and (18.42), with the results cast in the physical (x, y) space (rather than in terms of the transformed variable r). In turn, the surface values Cf and Сц can be obtained from the velocity and temperature gradients respectively at the wall as given by the velocity and temperature profiles evaluated at the wall. Recall from Equations (16.51) and (16.55) that Cf and Сц are defined as

rw

PeU2e

and where (3u/dy)w and (dT/dy)w are the values obtained from the velocity and temperature profiles, respectively, evaluated at the wall. In turn, the overall flat plate skin friction drag coefficient C/ can be obtained by integrating c/ over the plate via Equation (18.21).

Return to Equation (18.22) for the friction drag coefficient for incompressible flow. The analogous compressible result can be written as

In Equation (18.44), the function F is determined from the numerical solution. Sam­ple results are given in Figure 18.8, which shows that the product C/VRec decreases as Me increases. Moreover, the adiabatic wall is warmer than the wall in the case of Tw/Te = 1.0. Hence, Figure 18.8 demonstrates that a hot wall also reduces C/ VRec.

Return to Equation (18.23) for the thickness of the incompressible flat-plate boundary layer. The analogous result for compressible flow is

In Equation (18.45), the function G is obtained from the numerical solution. Sample results are given in Figure 18.9, which shows that the product (&^/ШГх/х) increases as Me increases. Everything else being equal, boundary layers are thicker at higher

Figure 1 8.8 Friction drag coefficient for laminar, compressible flow over a flat plate, illustrating the effect of Mach number and wall temperature. Pr = 0.75.

(Calculations by E. R. van Driest, N АСА Tech. Note 2597.)

Figure 1 8-9 Boundary-layer thickness for laminar, compressible flow over a flat plate, illustrating the effect of Mach number and wall temperature. Pr = 0.75.

(Calculations by E. R. van Driest, NАСА Tech. Note 2597.)

Mach numbers. This fact was stated earlier, as shown in Figures 18.4 and 18.6. Note also from Figure 18.9 that a hot wall thickens the boundary layer, as discussed earlier.

Recall our discussion of Couette flow in Chapter 16. There, we introduced the concept of the recovery factor r where

haw=he + r-^ [18.46]

This is a general concept, and can be applied to the boundary-layer solutions here. If we assume a constant Prandtl number for the compressible flat-plate flow, the numerical solution shows that

r = VPr

for the flat plate. Note that Equation (18.47) is analogous to the result given for Couette flow in that the recovery factor is a function of the Prandtl number only. However, for the flat plate, r = л/Рг, whereas for Couette flow, r = Pr.

Aerodynamic heating for the flat plate can be treated via Reynolds analogy. The Stanton number and skin friction coefficients are defined respectively as

(See our discussion of these coefficients in Chapter 16.) Our results for Couette flow proved that a relation existed between C# and c/—namely, Reynolds analogy, given by Equation (16.59) for Couette flow. Moreover, in this relation, the ratio Сн/cf was a function of the Prandtl number only. A directly analogous result holds for the compressible flat-plate flow. If we assume that the Prandtl number is constant, then for a flat plate, Reynolds analogy is, from the numerical solution,

In Equation (18.50), the local skin friction coefficient Cf which is given by Equa­tion (18.20) for the incompressible flat-plate case, becomes the following form for the compressible flat-plate flow:

In Equation (18.51), F is the same function as appears in Equation (18.44), and its variation with Me and Tw/Te is the same as shown in Figure 18.8.

The speed of sound is

«oo = yJyRTx = л/(1-4)(287)(288) = 340.2 m/s

The Mach number is Mx = 100/340.2 = 0.29. Hence, Mx is low enough to assume incompressible flow, and we can use Equation (18.22),

1.328

v’ReZ

Please note that for the flow over a flat plate at zero angle of attack, the freestream velocity and density, Vx and p0c, are the same as the velocity and density at the outer edge of the boundary layer, ue and pe. Hence, these quantities can be used interchangeably. Thus,

The total drag due to friction is generated by the shear stress acting on both the top and bottom of the plate. Since D f above is the friction drag on only one surface, we have

Total friction drag = D = 2D/ = 2(87.8) = (b) For Poo = 1000 m/s, we have

Clearly, the flow is compressible, and we have to use Equation (18.44), or more directly, Figure 18.8. From Figure 18.8, we have for Mx = Me = 2.94 and an adiabatic wall,

С/л/ReZ = 1.2

The friction drag on one surface is

Df = {pooVlSCf = і(1.22)(1 ООО)2(40)(1.03 x IQ-4) = 2513 N

Taking into account both the top and bottom surfaces,

Total friction drag = D = 2Df = 2(2513) =

Drag-Divergence Mach Number: The Sound Barrier

Imagine that we have a given airfoil at a fixed angle of attack in a wind tunnel, and we wish to measure its drag coefficient cd as a function of Mx. To begin with, we measure the drag coefficient at low subsonic speed to be cd 0, shown in Figure

11.11. Now, as we gradually increase the freestream Mach number, we observe that

cd remains relatively constant all the way to the critical Mach number, as illustrated in Figure 11.11. The flow fields associated with points a, b, and c in Figure 11.11 are represented by Figure 11.5a, b, and r, respectively. Asweverycarefullyincrea. se slightly above Mcr, say, to point d in Figure 11.11, a finite region of supersonic flow appears on the airfoil, as shown in Figure 11.5cf. The Mach number in this bubble of supersonic flow is only slightly above Mach 1, typically 1.02 to 1.05. Flowever, as we continue to nudge M00 higher, we encounter a point where the drag coefficient suddenly starts to increase. This is given as point e in Figure 11.11. The value of Мж at which this sudden increase in drag starts is defined as the drag-divergence Mach number. Beyond the drag-divergence Mach number, the drag coefficient can become very large, typically increasing by a factor of 10 or more. This large increase in drag is associated with an extensive region of supersonic flow over the airfoil, terminating in a shock wave, as sketched in the insert in Figure 11.11. Corresponding to point / on the drag curve, this insert shows that as Мж approaches unity, the flow on both the top and bottom surfaces can be supersonic, both terminated by shock waves. For example, consider the case of a reasonably thick airfoil, designed originally for low – speed applications, when Мж is beyond drag-divergence; in such a case, the local Mach number can reach 1.2 or higher. As a result, the terminating shock waves can be relatively strong. These shocks generally cause severe flow separation downstream of the shocks, with an attendant large increase in drag.

Now, put yourself in the place of an aeronautical engineer in 1936. You are familiar with the Prandtl-Glauert rule, given by Equation (11.51). You recognize that as Mх —>• 1, this equation shows the absolute magnitude of Cp approaching

Figure I I.1 I Sketch of the variation of profile drag coefficient with freestream Mach number, illustrating the critical and drag-divergence Mach numbers and showing the large drag rise near Mach 1.

infinity. This hints at some real problems near Mach 1. Furthermore, you know of some initial high-speed subsonic wind-tunnel tests that have generated drag curves which resemble the portion of Figure 11.11 from points a to /. How far will the drag coefficient increase as we get closer to MTO = 1? Will q go to infinity? At this stage, you might be pessimistic. You might visualize the drag increase to be so large that no airplane with the power plants existing in 1936, or even envisaged for the future, could ever overcome this “barrier.” It was this type of thought that led to the popular concept of a sound barrier and that prompted many people to claim that humans would never fly faster than the speed of sound.

Of course, today we know the sound barrier was a myth. We cannot use the Prandtl-Glauert rule to argue that q will become infinite at MTO = 1, because the Prandtl-Glauert rule is invalid at MTO = 1 (see Sections 11.3 and 11.4). Moreover, early transonic wind-tunnel tests carried out in the late 1940s clearly indicated that cj peaks at or around Mach 1 and then actually decreases as we enter the supersonic regime, as shown by points g and h in Figure 11.11. All we need is an aircraft with an engine powerful enough to overcome the large drag rise at Mach 1. The myth of the sound barrier was finally put to rest on October 14, 1947, when Captain Charles (Chuck) Yeager became the first human being to fly faster than sound in the sleek, bullet-shaped Bell XS-1. This rocket-propelled research aircraft is shown in Figure 11.12. Of course, today supersonic flight is a common reality; we have

Figure 11.12 The Bell XS-1 —the first manned airplane to fly faster than sound, October 14, 1947. (Courtesy of the National Air and Space Museum.)

(a)

Figure 11.14 By sweeping the wing, a streamline effectively sees a thinner airfoil.

Figure 11.15 A typical example of a swept-wing aircraft. The North American F-86 Sabre of Korean War fame.

sound. One of these—the area rule—is discussed in this section; the other—the supercritical airfoil—is the subject of Section 11.9.

For a moment, let us expand our discussion from two-dimensional airfoils to a consideration of a complete airplane. In this section, we introduce a design concept which has effectively reduced the drag rise near Mach 1 for a complete airplane.

As stated before, the first practical jet-powered aircraft appeared at the end of World War II in the form of the German Me 262. This was a subsonic fighter plane with a top speed near 550 mi/h. The next decade saw the design and production of many types of jet aircraft—all limited to subsonic flight by the large drag near Mach 1. Even the “century” series of fighter aircraft designed to give the U. S. Air Force supersonic capability in the early 1950s, such as the Convair F-102 delta-wing airplane, ran into difficulty and could not at first readily penetrate the sound barrier in level flight. The thrust of jet engines at that time simply could not overcome the large peak drag near Mach 1.

A planview, cross section, and area distribution (cross-sectional area versus dis­tance along the axis of the airplane) for a typical airplane of that decade are sketched in Figure 11.16. Let A denote the total cross-sectional area at any given station. Note that the cross-sectional area distribution experiences some abrupt changes along the axis, with discontinuities in both A and dA/dx in the regions of the wing.

In contrast, for almost a century, it was well known by ballisticians that the speed of a supersonic bullet or artillery shell with a smooth variation of cross-sectional area

Figure 11.16 A schematic of a non-area-ruled aircraft.

was higher than projectiles with abrupt or discontinuous area distributions. In the mid-1950s, an aeronautical engineer at the NACA Langley Aeronautical Laboratory, Richard T. Whitcomb, put this knowledge to work on the problem of transonic flight of airplanes. Whitcomb reasoned that the variation of cross-sectional area for an airplane should be smooth, with no discontinuities. This meant that, in the region of the wings and tail, the fuselage cross-sectional area should decrease to compensate for the addition of the wing and tail cross-sectional area. This led to a “coke bottle” fuselage shape, as shown in Figure 11.17. Here, the planview and area distribution are shown for an aircraft with a relatively smooth variation of A(x). This design philosophy is called the area rule, and it successfully reduced the peak drag near Mach 1 such that practical airplanes could fly supersonically by the mid-1950s. The variations of drag coefficient with for an area-ruled and non-area-ruled airplane are schematically compared in Figure 11.18; typically, the area rule leads to a factor – of-2 reduction in the peak drag near Mach 1.

The development of the area rule was a dramatic breakthrough in high-speed flight, and it earned a substantial reputation for Richard Whitcomb—a reputation which was to be later garnished by a similar breakthrough in transonic airfoil design, to be discussed in Section 11.9. The original work on the area rule was presented by Whitcomb in Reference 31, which should be consulted for more details.

Figure 11.17 A schematic of an area-ruled aircraft.

Figure 11.18 The drag-rise properties of

area-ruled and non-area-ruled aircraft (schematic only).

Some Special Cases; Couette and. Poiseuille Flows

The resistance arising from the want of lubricity in the parts of a fluid is, other things being equal, proportional to the velocity with which the parts of the fluid are separated from one another.

Isaac Newton, 1687, from Section IX of Book II of his Principia

16.1 Introduction

The general equations of viscous flow were derived and discussed in Chapter 15. In particular, the viscous flow momentum equations were treated in Section 15.4 and are given in partial differential equation form by Equations (15.19a to c)—the Navier-Stokes equations. These, along with the viscous flow energy equation, Equa­tion (15.26), derived in Section 15.5, are the theoretical tools for the study of viscous flows. However, examine these equations closely; as discussed in Section 15.7, they are a system of coupled, nonlinear partial differential equations—equations which contain more terms and which are inherently more elaborate than the inviscid flow equations treated in Parts 2 and 3 of this book. Three classes of solutions of these equations were itemized in Section 15.5. The first itemized class was that of “exact” solutions of the Navier-Stokes equations for a few specific physical problems which, by their physical and geometrical nature, allow many terms in the governing equations to be precisely zero, resulting in a system of equations simple enough to solve, either analytically or by simple numerical methods. Such exact problems are the subject of this chapter.

Reynolds analogy

The road map for this chapter is given in Figure 16.1. The types of flows consid­ered here are generally labeled as parallel flows because the streamlines are straight and parallel to each other. We will consider two of these flows, Couette and Poiseuille, which will be defined in due course. In addition to representing exact solutions of the Navier-Stokes equations, these flows illustrate some of the important practical facets of any viscous flow, as itemized on the right side of the road map. In a clear, uncom­plicated fashion, we will be able to calculate and study the surface skin friction and heat transfer. We will also use the results to define the recovery factor and Reynolds analogy—two practical engineering tools that are frequently used in the analysis of skin friction and heat transfer.

Examples

A numerical simulation of the flow over an airfoil using the Reynolds averaged Navier-Stokes equations can be conducted on today’s supercomputers in less than a half hour for less than $1000 cost in computer time. If just one such simulation had been attempted 20 years ago on computers of that time (e. g., the IBM 704 class) and with algorithms then known, the cost in computer time would have amounted to roughly $10 million, and the results for that single flow would not be available until 10 years from now, since the computation would have taken about 30 years to complete.

Dean R. Chapman, NASA, 1977

20.1 Introduction

This chapter is short. Its purpose is to discuss the third option for the solution of viscous flows as discussed in Section 15.7, namely, the exact numerical solution of the complete Navier-Stokes equations. This option is the purview of modern compu­tational fluid dynamics—it is a state-of-the-art research activity which is currently in a rapid state of development. This subject now occupies volumes of modern literature; for a basic treatment, see the definitive text on computational fluid dynamics listed as Reference 54. We will only list a few sample calculations here.

20.2 The Approach

Return to the complete Navier-Stokes equations, as derived in Chapter 15, and re­peated and renumbered below for convenience:

These equations have been written with the time derivatives on the left-hand side and all spatial derivatives on the right-hand side. This is the form suitable to a time – dependent solution of the equations, as discussed in Chapters 13 and 16. Indeed, Equations (20.1) to (20.5) are partial differential equations which have a mathemati­cally “elliptic” behavior; that is, on a physical basis they treat flow-field information and flow disturbances that can travel throughout the flow field, in both the upstream and downstream directions. The time-dependent technique is particularly suited to such a problem.

The time-dependent solution of Equations (20.1) to (20.5) can be carried out in direct parallel to the discussion in Section 16.4. It is important for you to return to that section and review our discussion of the time-dependent solution of compressible Couette flow using MacCormack’s technique. We suggest doing this before reading further. The approach to the solution of Equations (20.1) to (20.5) for other problems is exactly the same. Therefore, we will not elaborate further here.

Measurement of Velocity in a Compressible Flow

The use of a Pitot tube for measuring the velocity of a low-speed, incompressible flow was discussed in Section 3.4. Before progressing further, return to Section 3.4, and review the principal aspects of a Pitot tube, as well as the formulas used to obtain the flow velocity from the Pitot pressure, assuming incompressible flow.

For low-speed, incompressible flow, we saw in Section 3.4 that the velocity can be obtained from a knowledge of both the total pressure and the static pressure at

a point. The total pressure is measured by a Pitot tube, and the static pressure is obtained from a static pressure orifice or by some independent means. The important aspect of Section 3.4 is that the pressure sensed by a Pitot tube, along with the static pressure, is all that is necessary to extract the flow velocity for an incompressible flow. In the present section, we see that the same is true for a compressible flow, both subsonic and supersonic, if we consider the Mach number rather than the velocity. In both subsonic and supersonic compressible flows, a knowledge of the Pitot pressure and the static pressure is sufficient to calculate Mach number, although the formulas are different for each Mach-number regime. Let us examine this matter further.

13.2.2 Wall Points

In Figure 13.6, point 4 is an internal flow point near a wall. Assume that we know all the flow properties at point 4. The C_ characteristic through point 4 intersects the wall at point 5. At point 5, the slope of the wall 9$ is known. The flow properties at the wall point, point 5, can be obtained from the known properties at point 4 as follows. Along the C_ characteristic, К is constant. Hence, (К )4 = (K-)5. Moreover, the value of К_ is known from Equation (13.17) evaluated at point 4:

(tf-)4 = (K-)s = 6*4 + v4 [13.25]

Evaluating Equation (13.17) at point 5, we have

(K_)5 = 9S + v5 [13.26]

In Equation (13.26), (K-)5 and 65 are known; thus V5 follows directly. In turn, all other flow variables at point 5 can be obtained from V5 as explained earlier. The characteristic line between points 4 and 5 is assumed to be a straight-line segment with average slope given by |(64 + 05) — ^(/x4 + /x5).

From the above discussion of both internal and wall points, we see that properties at the grid points are calculated from known properties at other grid points. Hence, in order to start a calculation using the method of characteristics, we have to know the flow properties along some initial data line. Then we piece together the characteristics mesh and associated flow properties by “marching downstream” from the initial data line. This is illustrated in the next section.

We emphasize again that the method of characteristics is an exact solution of inviscid, nonlinear supersonic flow. However, in practice, there are numerical er­rors associated with the finite grid; the approximation of the characteristics mesh by straight-line segments between grid points is one such example. In principle, the method of characteristics is truly exact only in the limit of an infinite number of characteristic lines.

We have discussed the method of characteristics for two-dimensional, irrota – tional, steady flow. The method of characteristics can also be used for rotational and three-dimensional flows, as well as unsteady flows. See Reference 21 for more details.

Figure 1 3.6 Wall point.

Results for Compressible Couette Flow

Some typical results for compressible Couette flow are shown in Figure 16.9 for a cold wall case, and in Figure 16.10 for an adiabatic lower wall case. These results are

u_ A = (y – l )PrM;

ue

(a) (b)

A = (y- )PrM}

obtained from White (Reference 43); they assume a viscosity-temperature relation of ц/цтах = ("/’/ ‘/’ref)2/3> which is not quite as accurate for a gas as is Sutherland’s law [Equation (15.3)]. Recall from Section 15.6 that a compressible viscous flow is governed by the following similarity parameters: the Mach number, the Prandtl number, and the ratio of specific heats, y. Therefore, we expect the results for compressible Couette flow to be governed by the same parameters. Such is the case, as illustrated in Figures 16.9 and 16.10. Here we see the different flow-field profiles
for different values of the combined parameter A = (у – l)Pr М]. In particular, examining Figure 16.9 for the equal temperature, cold wall case, we note that;

1. From Figure 16.9a, the velocity profiles are not greatly affected by compress­ibility. The profile labeled A = 0 is the familiar linear incompressible case, and that labeled A = 30 corresponds to Me approximately 10. Clearly, the velocity profile (in terms of u/ue versus y/D) does not change greatly over such a large range of Mach number.

2. In contrast, from Figure 16.9b, there are huge temperature changes in the flow; these are due exclusively to viscous dissipation, which is a major effect at high Mach numbers. For example, for A = 30 (Me ~ 10), the temperature in the middle of the flow is almost five times the wall temperature. Contrast this with the very small temperature increase calculated in Example 16.1 for an incompressible flow. This is why, on the scale in Figure 16.9b, the incompressible case (A = 0) is seen as essentially a vertical line.

For the adiabatic wall case shown in Figure 16.10, we note the following:

1. From Figure 16.10a, the velocity profiles show a pronounced curvature due to compressibility.

2. From Figure 16.1 Ofo, the temperature increases are larger than for the cold wall case. Note that, for A = 30 <M,, ~ 10), the maximum temperature is over 15 times that of the upper wall. Also, note the results, familiar from our discussion in Section 16.3, that the temperature is the largest at the adiabatic wall; that is, Taw is the maximum temperature. As expected, Figure 16.1 Oh shows that Taw increases markedly as Me increases.

In summary, in a general comparison between the incompressible flow discussed in Section 16.3 and the compressible flow discussed here, there is no tremendous qual­itative change; that is, there is no discontinuous change in the flow-field behavior in going from subsonic to supersonic flow as is the case for an inviscid flow, such as discussed in Part 3. Qualitatively, a supersonic viscous flow is similar to a subsonic viscous flow. On the other hand, there are tremendous quantitative differences, es­pecially in regard to the large temperature changes that occur due to massive viscous dissipation in a high-speed compressible viscous flow. The physical reason for this difference in viscous versus inviscid flow is as follows. In an inviscid flow, informa­tion is propagated via the mechanism of pressure waves traveling throughout the flow. This mechanism changes radically when the flow goes from subsonic to supersonic. In contrast, for a viscous flow, information is propagated by the diffusive transport mechanisms of // and к (a molecular phenomenon), and these mechanisms are not basically changed when the flow goes from subsonic to supersonic. These statements hold in general for any viscous flow, not just for the Couette flow case treated here.

Nozzle Flows

In this section, we move to the left-hand branch of the road map given in Figure 10.3; that is, we study in detail the compressible flow through nozzles. To expedite this study, we first derive an important equation which relates Mach number to the ratio of duct area to sonic throat area.

Consider the duct shown in Figure 10.9. Assume that sonic flow exists at the throat, where the area is A*. The Mach number and the velocity at the throat are denoted by M* and u*, respectively. Since the flow is sonic at the throat, M* = 1 and u* = a*. (Note that the use of an asterisk to denote sonic conditions was introduced in Section 7.5; we continue this convention in our present discussion.) At any other section of this duct, the area, the Mach number, and the velocity are denoted by A, M, and m, respectively, as shown in Figure 10.9. Writing Equation (10.1) between A and A*, we have

p*u*A* = puA [10.26]

Since u* = a*, Equation (10.26) becomes

A p* a* p* po a* . _

A* p и po p и

where po is the stagnation density defined in Section 7.5 and is constant throughout

an isentropic flow. From Equation (8.46), we have

p*_ _ ( 2

Po VX + 1 /

Also, from Equation (8.43), we have

Also, recalling the definition of M* in Section 8.4, as well as Equation (8.48), we have

[10.30]

Squaring Equation (10.27) and substituting Equations (10.28) to (10.30), we obtain

AV = (_2_V(y~’) (l + ^_AM22,{У~П 1 + [(X – l)/2M2 A*) Vx + 1/ V 2 ) [(y+)/2]M2

[10.31]

Algebraically simplifying Equation (10.31), we have

Equation (10.32) is very important; it is called the area-Mach number relation, and it contains a striking result. “Turned inside out,” Equation (10.32) tells us that M = f (A / A*); that is, the Mach number at any location in the duct is a function of the ratio of the local duct area to the sonic throat area. Recall from our discussion of Equation (10.25) that A must be greater than or at least equal to A*; the case where A < A* is

physically not possible in an isentropic flow. Thus, in Equation (10.32), A/A* > 1. Also, Equation (10.32) yields two solutions for M at a given А/A*—a subsonic value and a supersonic value. Which value of M that actually holds in a given case depends on the pressures at the inlet and exit of the duct, as explained later. The results for А/A* as a function of M, obtained from Equation (10.32), are tabulated in Appendix A. Examining Appendix A, we note that for subsonic values of M, as M increases, AI A* decreases (i. e., the duct converges). At M = 1, A / A* = 1 in Appendix A. Finally, for supersonic values of M, as M increases, А/A* increases (i. e., the duct diverges). These trends in Appendix A are consistent with our physical discussion of convergent-divergent ducts at the end of Section 10.2. Moreover, Appendix A shows the double-valued nature of M as a function of A/A*. For example, for А/A* = 2, we have either M = 0.31 or M = 2.2.

Consider a given convergent-divergent nozzle, as sketched in Figure 10.10a. Assume that the area ratio at the inlet A,-/A* is very large and that the flow at the inlet is fed from a large gas reservoir where the gas is essentially stationary. The reservoir pressure and temperature are po and 7b, respectively. Since A,-/A* is very large, the subsonic Mach number at the inlet is very small, M ~ 0. Thus, the pressure and temperature at the inlet are essentially po and 7o, respectively. The area distribution of the nozzle A = A(x) is specified, so that A/ A* is known at every station along the nozzle. The area of the throat is denoted by A,, and the exit area is denoted by Ae. The Mach number and static pressure at the exit are denoted by Me and pe, respectively. Assume that we have an isentropic expansion of the gas through this nozzle to a supersonic Mach number Me = Me^ at the exit (the reason for the subscript 6 will be apparent later). The corresponding exit pressure is pe>6. For this expansion, the flow is sonic at the throat; hence, M = 1 and A, = A* at the throat. The flow properties through the nozzle are a function of the local area ratio A/A* and are obtained as follows:

1. The local Mach number as a function of x is obtained from Equation (10.32), or more directly from the tabulated values in Appendix A. For the specified A — A(x), we know the corresponding A/A* = fix). Then read the related subsonic Mach numbers in the convergent portion of the nozzle from the first part of Appendix A (for M < 1) and the related supersonic Mach numbers in the divergent portion of the nozzle from the second part of Appendix A (for M > 1). The Mach number distribution through the complete nozzle is thus obtained and is sketched in Figure 10.10Й.

2. Once the Mach number distribution is known, then the corresponding variation of temperature, pressure, and density can be found from Equations (8.40), (8.42), and (8.43), respectively, or more directly from Appendix A. The distributions of p/po and T/Tq are sketched in Figure 10.10c and d, respectively.

Examine the variations shown in Figure 10.10. For the isentropic expansion of a gas through a convergent-divergent nozzle, the Mach number monotonically increases from near 0 at the inlet to M = 1 at the throat, and to the supersonic value Mefi at the exit. The pressure monotonically decreases from p0 at the inlet to 0.528/?o at the throat and to the lower value рем at the exit. Similarly, the temperature monotonically

decreases from T0 at the inlet to 0.8337b at the throat and to the lower value 7′(,.f) at the exit. Again, for the isentropic flow shown in Figure 10.10, we emphasize that the distribution of M, and hence the resulting distributions of p and /’, through the nozzle depends only on the local area ratio A/A*. This is the key to the analysis of isentropic, supersonic, quasi-one-dimensional nozzle flows.

Imagine that you take a convergent-divergent nozzle, and simply place it on a table in front of you. What is going to happen? Is the air going to suddenly start flowing through the nozzle of its own accord? The answer is, of course not! Rather, by this stage in your study of aerodynamics, your intuition should tell you that we have to impose a force on the gas in order to produce any acceleration. Indeed, this is the essence of the momentum equation derived in Section 2.5. For the inviscid flows

considered here, the only mechanism to produce an accelerating force on a gas is a pressure gradient. Thus, returning to the nozzle on the table, a pressure difference must be created between the inlet and exit; only then will the gas start to flow through the nozzle. The exit pressure must be less than the inlet pressure; that is, pe < p0. Moreover, if we wish to produce the isentropic supersonic flow sketched in Figure

10.10, the pressure pjpo must be precisely the value stipulated by Appendix A for the known exit Mach number Me g; that is, pe/po = Ре, бІРо■ If the pressure ratio is different from the above isentropic value, the flow either inside or outside the nozzle will be different from that shown in Figure 10.10.

Let us examine the type of nozzle flows that occur when pe/po is not equal to the precise isentropic value for Me b, that is, when ре/Po ф Ре, б/Ро – To begin with, consider the convergent-divergent nozzle sketched in Figure 10.1 la. If pe = po, no pressure difference exists, and no flow occurs inside the nozzle. Now assume that pe is minutely reduced below pa, say, pe = 0.999p0. This small pressure difference will produce a very low-speed subsonic flow inside the nozzle—essentially a gentle wind. The local Mach number will increase slightly through the convergent portion, reaching a maximum value at the throat, as shown by curve 1 in Figure 10.1 li>. This Mach number at the throat will not be sonic; rather, it will be some small subsonic value. Downstream of the throat, the local Mach number will decrease in the divergent section, reaching a very small but finite value M,,, і at the exit. Correspondingly, the pressure in the convergent section will gradually decrease from p0 at the inlet to a minimum value at the throat, and then will gradually increase to the value pe, i at the exit. This variation is shown as curve 1 in Figure 10.1 lc. Please note that because the flow is not sonic at the throat in this case, At is not equal to A*. Recall that A*, which appears in Equation (10.32), is the sonic throat area. In the case of purely subsonic flow through a convergent-divergent nozzle, A* takes on the character of a reference area; it is not the same as the actual geometric area of the nozzle throat At. Rather, A* is the area the flow in Figure 10.11 would have if it were somehow accelerated to sonic velocity. If this did happen, the flow area would have to be decreased further than shown in Figure 10.1 la. Hence, for a purely subsonic flow A, > A*.

Assume that we further decrease the exit pressure in Figure 10.11, say, to the value pe = pe,2- The flow is now illustrated by the curves labeled 2 in Figure 10.11. The flow moves faster through the nozzle, and the maximum Mach number at the throat increases but remains less than 1. Now, let us reduce pe to the value pe = pe, з, such that the flow just reaches sonic conditions at the throat. This is shown by curve 3 in Figure 10.11. The throat Mach number is 1, and the throat pressure is 0.528p^. The flow downstream of the throat is subsonic.

Upon comparing Figures 10.10 and 10.11, we are struck by an important physical difference. For a given nozzle shape, there is only one allowable isentropic flow solution for the supersonic case shown in Figure 10.10. In contrast, there are an infinite number of possible isentropic subsonic solutions, each one corresponding to some value of pe, where po > pe > Ре. з – Only three solutions of this infinite set of solutions are sketched in Figure 10.11. Hence, the key factors for the analysis of purely subsonic flow in a convergent-divergent nozzle are both А/A* and pe/Po-

Consider the mass flow through the convergent-divergent nozzle in Figure 10.11. As the exit pressure is decreased, the flow velocity in the throat increases; hence, the

Figure 10.11 Isentropic subsonic nozzle flow.

mass flow increases. The mass flow can be calculated by evaluating Equation (10.1) at the throat; that is, m — p, utAt. As pe decreases, u, increases and p, decreases. However, the percentage increase in ut is much greater than the decrease in p,. As a result, m increases, as sketched in Figure 10.12. When pe — pe^, sonic flow is achieved at the throat, and m = p*u*A* = p*u*A,. Now, if pe is further reduced below pe>3, the conditions at the throat take on a new behavior; they remain unchanged. From our discussion in Section 10.2, the Mach number at the throat cannot exceed 1; hence, as pe is further reduced, M will remain equal to 1 at the throat. Consequently, the mass flow will remain constant as pe is reduced below pe 3, as shown in Figure 10.12. In a sense, the flow at the throat, as well as upstream of the throat, becomes “frozen.” Once the flow becomes sonic at the throat, disturbances cannot work their way upstream of the throat. Hence, the flow in the convergent section of the nozzle no longer communicates with the exit pressure and has no way of knowing that the

Figure 10.12 Variation of mass flow with exit

pressure; illustration of choked flow.

exit pressure is continuing to decrease. This situation—when the flow goes sonic at the throat, and the mass flow remains constant no matter how low pe is reduced—is called choked flow. It is a vital aspect of the compressible flow through ducts, and we consider it further in our subsequent discussions.

Return to the subsonic nozzle flows sketched in Figure 10.11. Question: What happens in the duct when pe is reduced below p,..fl In the convergent portion, as described above, nothing happens. The flow properties remain fixed at the conditions shown by curve 3 in the convergent section of the duct (the left side of Figure 10.11 b and c). However, a lot happens in the divergent section of the duct. As the exit pressure is reduced below pe 3, a region of supersonic flow appears downstream of the throat. However, the exit pressure is too high to allow an isentropic supersonic flow throughout the entire divergent section. Instead, for pe less than pe,3 but substantially higher than the fully isentropic value pe$ (see Figure 10.10c), a normal shock wave is formed downstream of the throat. This situation is sketched in Figure 10.13.

In Figure 10.13, the exit pressure has been reduced to pe^, where рсл < pe,3, but where pe 4 is also substantially higher than pe 6. Here we observe a normal shock wave standing inside the nozzle at a distance d downstream of the throat. Between the throat and the normal shock wave, the flow is given by the supersonic isentropic solution, as shown in Figure 10.13Z? and c. Behind the shock wave, the flow is subsonic. This subsonic flow sees the divergent duct and isentropically slows down further as it moves to the exit. Correspondingly, the pressure experiences a discontinuous increase across the shock wave and then is further increased as the flow slows down toward the exit. The flow on both the left and right sides of the shock wave is isentropic; however, the entropy increases across the shock wave. Hence, the flow on the left side of the shock wave is isentropic with one value of entropy, V|, and the flow on the right side of the shock wave is isentropic with another value of entropy S2, where 52 > ■Si – The location of the shock wave inside the nozzle, given by d in Figure 10.13a, is determined by the requirement that the increase in static pressure across the wave plus that in the divergent portion of the subsonic flow behind the shock be just right to achieve pe 4 at the exit. As pe is further reduced, the normal

Normal shock wave

Figure 10.13 Su personic nozzle flow with a normal shock inside the nozzle.

shock wave moves downstream, closer to the nozzle exit. At a certain value of exit pressure, pe — pe 5, the normal shock stands precisely at the exit. This is sketched in Figure 10.14a to c. At this stage, when pe — ре $, the flow through the entire nozzle, except precisely at the exit, is isentropic.

To this stage in our discussion, we have dealt with pe, which is the pressure right at the nozzle exit. In Figures 10.10, 10.11, 10.13, and 10.14a to c, we have not been concerned with the flow downstream of the nozzle exit. Now imagine that the nozzle in Figure 10.14a exhausts directly into a region of surrounding gas downstream of the exit. These surroundings could be, for example, the atmosphere. In any case, the pressure of the surroundings downstream of the exit is defined as the back pressure, denoted by pB. When the flow at the nozzle exit is subsonic, the exit pressure must equal the back pressure, pe = pB, because a pressure discontinuity cannot be maintained in a steady subsonic flow. That is, when the exit flow is subsonic, the surrounding back pressure is impressed on the exit flow. Hence, in Figure 10.II, Рв = Pe. і for curve 1, pB — Pe.2 for curve 2, and pB — pej for curve 3. For the same reason, pB = pe_4 in Figure 10.13, and pB = pe^ in Figure 10.14. Hence, in discussing these figures, instead of stating that we reduced the exit pressure pe and

observed the consequences, we could just as well have stated that we reduced the back pressure pB. It would have amounted to the same thing.

For the remainder of our discussion in this section, let us now imagine that we have control over pв and that we are going to continue to decrease pB. Consider the case when the back pressure is reduced below pe 5. Whence < pB < pe 5, the back pressure is still above the isentropic pressure at the nozzle exit. Hence, in flowing out to the surroundings, the jet of gas from the nozzle must somehow be compressed such that its pressure is compatible with pB. This compression takes place across oblique shock waves attached to the exit, as shown in Figure 10.14d. When pB is reduced to the value such that pB = pee, there is no mismatch of the exit pressure and the back pressure; the nozzle jet exhausts smoothly into the surroundings without passing through any waves. This is shown in Figure 10.14-е. Finally, as pB is reduced below pe (, the jet of gas from the nozzle must expand further in order to match the lower back pressure. This expansion takes place across centered expansion waves attached to the exit, as shown in Figure 10.14/.

When the situation in Figure 10. 14й? exists, the nozzle is said to be overexpanded, because the pressure at the exit has expanded below the back pressure, p,,^ < pB.

That is, the nozzle expansion has gone too far, and the jet must pass through oblique shocks in order to come back up to the higher back pressure. Conversely, when the situation in Figure 10.14/ exists, the nozzle is said to be underexpanded, because the exit pressure is higher than the back pressure, pe (, > pB, and hence the flow is capable of additional expansion after leaving the nozzle.

Surveying Figures 10.10 through 10.14, note that the purely isentropic supersonic flow originally illustrated in Figure 10.10 exists throughout the nozzle for all cases when pB < Pe, 5- For example, in Figure 10.14a, the isentropic supersonic flow solution holds throughout the nozzle except right at the exit, where a normal shock exists. In Figure 10.14d to /, the flow through the entire nozzle, including at the exit plane, is given by the isentropic supersonic flow solution.

Keep in mind that our entire discussion of nozzle flows in this section is predicated on having a duct of given shape. We assume that A = A(x) is prescribed. When this is the case, the quasi-one-dimensional theory of this chapter gives a reasonable prediction of the flow inside the duct, where the results are interpreted as mean properties averaged over each cross section. This theory does not tell us how to design the contour of the nozzle. In reality, if the walls of the nozzle are not curved just right, then oblique shocks occur inside the nozzle. To obtain the proper contour for a supersonic nozzle so that it produces isentropic shock-free flow inside the nozzle, we must account for the three-dimensionality of the actual flow. This is one purpose of the method of characteristics, a technique for analyzing two – and three-dimensional supersonic flow. A brief introduction to the method of characteristics is given in Chapter 13.

Consider the isentropic supersonic flow through a convergent-divergent nozzle with an exit- | Example 1 0.1 to-throat area ratio of 10.25. The reservoir pressure and temperature are 5 atm and 600°R, respectively. Calculate M, p, and T at the nozzle exit.

Solution

From the supersonic portion of Appendix A, for AJ A* = 10.25,

Also,

Thus,

Te = 0.24277o = 0.2427(600) = 145.6°R

Consider the isentropic flow through a convergent-divergent nozzle with an exit-to-throat area | Example 10.2 ratio of 2. The reservoir pressure and temperature are 1 atm and 288 K, respectively. Calculate the Mach number, pressure, and temperature at both the throat and the exit for the cases where

Te 1

Te = —To = ———— (288) =

To 1.968

(b) At the throat, the flow is still sonic. Hence, from above, M, = 1.0, p, = 0.528 atm, and T, = 240 K. However, at all other locations in the nozzle, the flow is subsonic. At the exit, where Ae/A* = 2, from the subsonic portion of Appendix A,

Te 1

Te = —T0 =————- (288) =

T0 1.018

From the subsonic portion of Appendix A, for p0/p,: = 1.028, we have

– = — — = 0.5(2.964) = 1.482 * AeA*

From the subsonic portion of Appendix A, for A,/A* = 1.482, we have