Category When Is A Flow Compressible?

Shooting Method

This method is a classic method for the solution of the boundary-layer equations to be discussed in Chapter 17. For the solution of compressible Couette flow, the same philosophy follows as that to be applied to boundary-layer solutions, and that is why we discuss it now. The method involves a double iteration, that is, two minor iterations nested within a major iteration. The scheme is as follows:

1. Assume a value for r in Equation (16.64). A reasonable assumption to start with is the incompressible value, r = p(ue/D). Also, assume that the variation of u(y) is given by the incompressible result from Equation (16.6).

2. Starting at у = 0 with the known boundary condition T = Tw, integrate Equa­tion (16.64) across the flow until у = D. Use any standard numerical technique for ordinary differential equations, such as the well-known Runge-Kutta method (see, e. g., Reference 52). However, to start this numerical integration, because Equation (16.64) is second order, two boundary conditions must be specified at у 0. We only have one physical condition, namely, T = Tw. Therefore, we have to assume a second condition; let us assume a value for the temperature gradient at the wall, i. e., assume a value for (dT/dy)w. A value based on the incompressible flow solution discussed in Section 16.3 would be a reasonable assumption. With the assumed (dT/dy)w and the known T„ at _y = 0, then Equation (16.64) is integrated numerically away from the wall, starting at у = 0 and moving in small increments, Ay in the direction of increasing y. Values of T at each increment in у are produced by the numerical algorithm.

3. Stop the numerical integration when у = D is reached. Check to see if the numerical value of T at у = D equals the specified boundary condition, T = 7). Most likely, it will not because we have had to assume a value for (dT/dy)w in step 2. Hence, return to step 2, assume another value of (dT/dy)w, and repeat the integration. Continue to repeat steps 2 and 3 until convergence is obtained, that is, until a value of (dT/dy)w is found such that, after the numerical integration, T = Tt. at у = D. From the converged temperature profile obtained by repetition of steps 2 and 3, we now have numerical values for Г as a function of у that satisfy both boundary conditions; that is, T = Tw at the lower wall and T = Te at the upper wall. However, do not forget that this converged solution was obtained for the assumed value of r and the assumed velocity profile u(y) in step 1. Therefore, the converged profile for T is not necessarily the correct profile. We must continue further; this time to find the correct value for r.

4. From the converged temperature profile obtained by the repetitive iteration in steps 2 and 3, we can obtain ft. = ц.(у) from Equation (15.3).

5. From the definition of shear stress,

du

we have — = — [16.65]

dy fi

Recall from the solution of the momentum equation, Equation (16.60), that г is a constant. Using the assumed value of r from step 1, and the values of ft = ft(y) from step 4, numerically integrate Equation (16.65) starting at у = 0 and using the known boundary condition и = 0 at у = 0. Since Equation (16.65) is first order, this single boundary condition is sufficient to initiate the numerical integration. Values of и at each increment in y, Ay, are produced by the numerical algorithm.

6. Stop the numerical integration when у = D is reached. Check to see if the numerical value of и at у = D equals the specified boundary condition, и = ue. Most likely, it will not, because we have had to assume a value of г and и (у) all the way back in step 1, which has carried through to this point in our iterative

solution. Hence, return to step 5, assume another value for r, and repeat the integration of Equation (16.65). Continue to repeat steps 5 and 6 [using the same values of д = д(у) from step 4] until convergence is obtained, that is, until a value of r is found that, after the numerical integration of Equation (16.65), и = ue at у = D. From the converged velocity profile obtained by repetition of steps 5 and 6, we now have numerical values for и as a function of у that satisfy both boundary conditions; that is, и = Oat у = 0 and и = и,, at v = D. However, do not forget that this converged solution was obtained using /j. — fi(y) from step 4, which was obtained using the initially assumed r and u(y) from step 1. Therefore, the converged profile for и obtained here is not necessarily the correct profile. We must continue one big step further.

7. Return to step 2, using the new value of r and the new и (у) obtained from step 6. Repeat steps 2 through 7 until total convergence is obtained. When this double iteration is completed, then the profile for Г = T (y) obtained at the last cycle of step 3, the profile for m = u(y) obtained at the last cycle of step 6, and the value of r obtained at the last cycle of step 7 are all the correct values for the given boundary conditions. The problem is solved!

Looking over the shooting method as described above, we see two minor iterations nested within a major iteration. Steps 2 and 3 constitute the first minor iteration and provide ultimately the temperature profile. Steps 5 and 6 are the second minor iteration and provide ultimately the velocity profile. Steps 2 to 7 constitute the major iteration and ultimately result in the proper value of r.

The shooting method described above for the solution of compressible Couette flow is carried over almost directly for the solution of the boundary-layer equations to be described in Chapter 18. In the same vein, there is another completely different approach to the solution of compressible Couette flow which carries over directly for the solution of the Navier-Stokes equations to be described in Chapter 20. This is the time-dependent, finite-difference method, first discussed in Chapter 13 and applied to the inviscid flow over a supersonic blunt body in Section 13.5. In order to prepare ourselves for Chapter 20, we briefly discuss the application of this method to the solution of compressible Couette flow.

When Is A Flow Compressible?

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

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

(8.43) ;

When Is A Flow Compressible?[8.43]

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

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

When Is A Flow Compressible?

Figure 8.6 Isentropic variation of density with Mach number.

This can be written as

ІЕ. = py2dV

P P V

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

{djA = P0y2dV V P Jo p y

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

dp/p = P_

(dp/p) о po

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

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

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

Incompressible flow: From Bernoulli’s equation,

T = To – —

2cn

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

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

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

From Equation (7.32),

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

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

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

Compressible: p = 1300 lb/ft2

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

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

M < 1 (subsonic flow)

M = 1 (sonic flow)

M > 1 (supersonic flow)

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

Подпись:и

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

(a/u)2 1 _ у + 1 / a*2

у – 1 2 “ 2(y – 1) V и )

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

у – 1 2(y – 1) M*/ 2

Подпись:2 2

M2 = —————— Z————-

(у + 1 )/M*2 – (у – 1)

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

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

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

M* = 1

if M = 1

M* < 1

if M < 1

M* > 1

if M > 1

Iy + 1

M* It———— r

if M —>■ oo

у – 1

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

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

Подпись: Example 8.1 When Is A Flow Compressible?

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

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

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

Solution

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

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

Poo

pі = 0.7545 atm

M, = ?

 

Подпись:M = 0.6

When Is A Flow Compressible?

Poo = 1 atm

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

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

Solution

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

Too = 460 + 59 = 519°R

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

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

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

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

When Is A Flow Compressible?

Hence,

 

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

 

965.4 ft/s

 

Elements of the Method of Characteristics

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

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

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

ІІ 1.12]

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

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

(11.12) can be written as

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

3 ф

— = f{x, y) dx

Hence, from the relation for an exact differential,

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

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

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

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

д2ф _ N _ 0 Эх ду ~ Ъ ~ 0

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

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

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

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

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

dx)c har 2(1 — u2/a2)

/dy —uv/a2 ± л/(и2 + v2)/a2 — 1

dx ) char 1-м2 /a2

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

/dy (—V2cos0 sin0)/a2 ± _

W/char 1 – [(V2/a2)cos20] l3’8

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

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

dx )char 1 – (COS2 0)/Sin2/Г

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

[13.10]

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

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

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

/ u2 ( v2

I 1—— — J du dy + I 1——- 1 dx dv = 0

dv —(1 — u2/a2) dy

or — = ———- —Ц—— [13.11]

du 1 — Vі/a2 dx

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

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

<У = / dy

dx ~ у dx ) ch^

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

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

du 1 — v2/a2 1 — u2/a2

dv uv/a2 =F у/(и2 + v2)/a2 — 1

du 1 — v2/a2

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

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

which, after some algebraic manipulations, reduces to

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

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

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

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

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

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

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

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

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

Incompressible Flow over a Flat Plate: The Blasius Solution

Consider the incompressible, two-dimensional flow over a flat plate at 0° angle of attack, such as sketched in Figure 17.7. For such a flow, p = constant, p = constant, and dpe/dx = 0 (because the inviscid flow over a flat plate at a = 0 yields a constant pressure over the surface). Moreover, recall that the energy equation is not needed to calculate the velocity field for an incompressible flow. Hence, the boundary-layer equations, Equations (17.28) to (17.31), reduce to

3 и dv

— + — = 0

дх 3 у

ди ди 32u

Ud^ + V^ = Vd?

d-?=o

dy

where v is the kinematic viscosity, defined as v = р/р.

Equation (18.12) is of particular note. The function /(p) defined in Equation (18.11) has the property that its derivative /’ gives the a component of velocity as

/'(>?) =

Substitute Equations (18.8) to (18.10), (18.12), and (18.13) into the momentum equa­tion, Equation (18.2). Writing each term explicitly so that you can see what is hap­pening, we have

I V VoО r / 91? „Д.. I Voo „ Voo n

—— f + VvxVoo — f Voo J——- / =vV0o——- /

V x ox І V vx vx

[18.15]

Equation (18.15) is important; it is called Blasius’ equation, after H. Blasius, who obtained it in his Ph. D. dissertation in 1908. Blasius was a student of Prandtl, and his flat-plate solution using Equation (18.15) was the first practical application of Prandtl’s boundary-layer hypothesis since its announcement in 1904. Examine Equa­tion (18.15) closely. Amazingly enough it is an ordinary differential equation. Look what has happened! Starting with the partial differential equations for a flat-plate boundary layer given by Equations (18.1) to (18.3), and transforming both the inde­pendent and dependent variables through Equations (18.4) and (18.11), we obtain an ordinary differential equation for f(tj). In the same breath, we can say that Equa­tion (18.15) is also an equation for the velocity и because и = V*, fit]). Because Equation (18.15) is a single ordinary differential equation, it is simpler to solve than the original boundary-layer equations. However, it is still a nonlinear equation and must be solved numerically, subject to the transformed boundary conditions,

Atrj = 0: / = 0, /’ = 0

Atrj^-oo: f = 1

[Note that at the wall where rj — 0, /’ = 0 because и = 0, and therefore / = 0 from Equation (18.13) evaluated at the wall.]

Equation (18.15) is a third-order, nonlinear, ordinary differential equation; it can be solved numerically by means of standard techniques, such as the Runge-Kutta method (such as that described in Reference 52). The integration begins at the wall and is carried out in small increments Ay in the direction of increasing у away from the wall. However, since Equation (18.15) is third order, three boundary conditions must be known at rj = 0; from the above, only two are specified. A third boundary condition, namely, some value for /"(0), must be assumed-, Equation (18.15) is then integrated across the boundary layer to a large value of rj. The value of /’ at large eta is then examined. Does it match the boundary condition at the edge of the boundary layer, namely, is /’ = 1 satisfied at the edge of the boundary layer? If not, assume a different value of /"(0) and integrate again. Repeat this process until convergence

is obtained. This numerical approach is called the ‘‘shooting technique”; it is a clas­sical approach, and its basic philosophy and details are discussed at great length in Section 16.4. Its application to Equation (18.15) is more straightforward than the dis­cussion in Section 16.4, because here we are dealing with an incompressible flow and only one equation, namely, the momentum equation as embodied in Equation (18.15).

The solution of Equation (18.15) is plotted in Figure 18.2 in the form of f'(r) — m/Voc as a function of r. Note that this curve is the velocity profile and that it is a function of г] only. Think about this for a moment. Consider two different x stations along the plate, as shown in Figure 18.3. In general, и = u(x, у), and the velocity profiles in terms of и = и (у ) at given x stations will be different. Clearly, the variation of и normal to the wall will change as the flow progresses downstream. However, when plotted versus r), we see that the profile, и = и (if), is the same for all x stations, as illustrated in Figure 18.3. This result is an example of a self­similar solution—solutions where the boundary-layer profiles, when plotted versus a similarity variable r) are the same for all x stations. For such self-similar solutions, the governing boundary-layer equations reduce to one or more ordinary differential equations in terms of a transformed independent variable. Self-similar solutions occur only for certain special types of flows—the flow over a flat plate is one such example. In general, for the flow over an arbitrary body, the boundary-layer solutions are nonsimilar; the governing partial differential equations cannot be reduced to ordinary differential equations.

Numerical values of /, /’, and f" tabulated versus r can be found in Ref­erence 42. Of particular interest is the value of /" at the wall; /"(0) = 0.332. Consider the local skin friction coefficient defined as С/ = zw/ . From

/'(r?) = u/Voe

Figure 1 8.2 Incompressible velocity profile for a flat plate; solution of the Blasius equation.

du df

__ — у J— — у

~ — [3] 00 r, — Kcx

ay ay

which is a classic expression for the local skin friction coefficient for the incompress­ible laminar flow over a flat plate—a result that stems directly from boundary-layer theory. Its validity has been amply verified by experiment. Note that Cf oc Re“l/2 oc x-1/2; that is, cf decreases inversely proportional to the square root of distance from the leading edge. Examining the flat plate sketched in Figure 17.7, the total drag on the top surface of the entire plate is the integrated contribution of rw{x) from x = 0 to x = c. Letting Cf denote the skin friction drag coefficient, we obtain from Equation (1.16)

Cf = – f cf dx [18.21]

c Jo

Substituting Equation (18.20) into (18.21), we obtain

where Rec is the Reynolds number based on the total plate length c.

An examination of Figure 18.2 shows that f = 0.99 at approximately rj — 5.0. Hence, the boundary-layer thickness, which was defined earlier as that distance above the surface where и = 0.99ue, is V

Note that the boundary-layer thickness is inversely proportional to the square root of the Reynolds number (based on the local distance x). Also, 8 oc. rl/2; the laminar boundary layer over a flat plate grows parabolically with distance from the leading edge.

The displacement thickness 5*, defined by Equation (17.3), becomes for an in­compressible flow

8* = [ (l ——’j dy [18.24]

Jo ue)

In terms of the transformed variables f and rj given by Equations (18.4) and (18.12), the integral in Equation (18.24) can be written as

s* = f [1 – ҐШ dr] = УттЧт – /(hi)] [1 8.25]

у voo Jo V boo

where T]X is an arbitrary point above the boundary layer. The numerical solution for f(rj) obtained from Equation (18.15) shows that, amazingly enough, r/, – f(t]) =

1.72 for all values of r] above 5.0. Therefore, from Equation (18.25), we have

8* = 1.72.

Note that, as in the case of the boundary-layer thickness itself, 8* varies inversely with the square root of the Reynolds number, and <5* a x,/2. Also, comparing Equations (18.23) and (18.26), we see that 8* = 0.34<5; the displacement thickness is smaller than the boundary-layer thickness, confirming our earlier statement in Section 17.2.

The momentum thickness for an incompressible flow is, from Equation (17.10),

or in terms of our transformed variables,

Гш~ Ґh

9 = ПГ [18.27]

V Гоо Jo

Equation (18.27) can be integrated numerically from r) = 0 to any arbitrary point Г)і > 5.0. The result gives

Note that, as in the case of our previous thicknesses, в varies inversely with the square root of the Reynolds number and that 9 ос xl/2. Also, 9 = 0.39<5*, and 9 = 0.135. Another property of momentum thickness can be demonstrated by evaluating в at the trailing edge of the flat plate sketched in Figure 17.7. In this case, x = c, and from Equation (18.28), we obtain

0.664c

9X=C = —=

VRi;

Comparing Equations (18.22) and (18.29), we have

„ 2 0X=C

Equation (18.30) demonstrates that the integrated skin friction coefficient for the flat plate is directly proportional to the value of в evaluated at the trailing edge.

Compressible Flow. Through Nozzles,. Diffusers, and Wind Tunnels

Having wondered from what source there is so much difficulty in successfully applying the principles of dynamics to fluids than to solids, finally, turning the matter over more carefully in my mind, I found the true origin of the difficulty; I discovered it to consist of the fact that a certain part of the pressing forces important in forming the throat (so called by me, not considered by others) was neglected, and moreover regarded as if of no importance, for no other reason than the throat is composed of a very small, or even an infinitely small, quantity of fluid, such as occurs whenever fluid passes from a wider place to a narrower, or vice versa, from a narrower to a wider.

Johann Bernoulli; from his Hydraulics, 1743

10.1 Introduction

Chapters 8 and 9 treated normal and oblique waves in supersonic flow. These waves are present on any aerodynamic vehicle in supersonic flight. Aeronautical engineers are concerned with observing the characteristics of such vehicles, especially the gen­eration of lift and drag at supersonic speeds, as well as details of the flow field, including the shock – and expansion-wave patterns. To make such observations, we usually have two standard choices: (1) conduct flight tests using the actual vehicle, and (2) run wind-tunnel tests on a small-scale model of the vehicle. Flight tests, al­though providing the final answers in the full-scale environment, are costly and, not to

say the least, dangerous if the vehicle is unproven. Hence, the vast bulk of supersonic aerodynamic data have been obtained in wind tunnels on the ground. What do such supersonic wind tunnels look like? How do we produce a uniform flow of supersonic gas in a laboratory environment? What are the characteristics of supersonic wind tunnels? The answers to these and other questions are addressed in this chapter.

The first practical supersonic wind tunnel was built and operated by Adolf Buse – mann in Germany in the mid – 1930s, although Prandtl had a small supersonic facility operating as early as 1905 for the study of shock waves. A photograph of Busemann’s tunnel is shown in Figure 10.1. Such facilities proliferated quickly during and after World War II. Today, all modern aerodynamic laboratories have one or more super­sonic wind tunnels, and many are equipped with hypersonic tunnels as well. Such machines come in all sizes; an example of a moderately large hypersonic tunnel is shown in Figure 10.2.

In this chapter, we discuss the aerodynamic fundamentals of compressible flow through ducts. Such fundamentals are vital to the proper design of high-speed wind tunnels, rocket engines, high-energy gas-dynamic and chemical lasers, and jet en­gines, to list just a few. Indeed, the material developed in this chapter is used almost daily by practicing aerodynamicists and is indispensable toward a full understanding of compressible flow.

The road map for this chapter is given in Figure 10.3. After deriving the governing equations, we treat the cases of a nozzle and diffuser separately. Then we merge this information to examine the case of supersonic wind tunnels.

Figure 1 0.3 Road map for Chapter 1 0.

The Lift and Drag of Wings at Hypersonic Speeds: Newtonian Results for a Flat Plate at Angle of Attack

Question: At subsonic speeds, how do the lift coefficient Cl and drag coefficient Co for a wing vary with angle of attack al

Answer: As shown in Chapter 5, we know that:

1. The lift coefficient varies linearly with angle of attack, at least up to the stall; see, for example, Figure 5.22.

2. The drag coefficient is given by the drag polar, as expressed in Equation (5.63), repeated below:

[5.63]

Since Cl is proportional to a, then Ci> varies as the square of a.

Question: At supersonic speeds, how do Cl and Сд for a wing vary with a? Answer: In Chapter 12, we demonstrated for an airfoil at supersonic speeds that:

1. Lift coefficient varies linearly with a, as seen from Equation (12.23), repeated

[12.23]

2. Drag coefficient varies as the square of a, as seen from Equation (12.24) for the flat plate, repeated below:

[12.24]

The characteristics of a finite wing at supersonic speeds follow essentially the same functional variation with the angle of attack, namely, Cl is proportional to a and CD is proportional to a2.

Question: At hypersonic speeds, how do Cl and CD for a wing vary with al We have shown that Cl is proportional to a for both subsonic and supersonic speeds— does the same proportionality hold for hypersonic speeds? We have shown that Cq is proportional to a2 for both subsonic and supersonic speeds—does the same proportionality hold for hypersonic speeds? The purpose of the present section is to address these questions.

In an approximate fashion, the lift and drag characteristics of a wing in hypersonic flow can be modeled by a flat plate at an angle of attack, as sketched in Figure 14.10. The exact flow field over the flat plate involves a series of expansion and shock waves as shown in Figure 14.10; the exact lift- and wave-drag coefficients can be obtained from the shock-expansion method as described in Section 9.7. However, for hypersonic speeds, the lift – and wave-drag coefficients can be further approximated by the use of newtonian theory, as described in this equation.

Consider Figure 14.11. Here, a two-dimensional flat plate with chord length c is at an angle of attack a to the freestream. Since we are not including friction, and because surface pressure always acts normal to the surface, the resultant aerodynamic force is perpendicular to the plate; that is, in this case, the normal force N is the resultant aerodynamic force. (For an infinitely thin flat plate, this is a general result which is not limited to newtonian theory, or even to hypersonic flow.) In turn, N is resolved into

lift and drag, denoted by L and D, respectively, as shown in Figure 14.11. According to newtonian theory, the pressure coefficient on the lower surface is

Cpj = 2 sin2 a [ 14.8]

The upper surface of the flat plate shown in Figure 14.11, in the spirit of newtonian theory, receives no direct “impact” of the freestream particles; the upper surface is said to be in the “shadow” of the flow. Hence, consistent with the basic model of newtonian flow, only freestream pressure acts on the upper surface, and we have

Returning to the discussion of aerodynamic force coefficients in Section 1.5, we note that the normal force coefficient is given by Equation (1.15). Neglecting friction, this becomes

1 fc

cn = – (CPti-CPtll)dx [14.10]

c Jo

where x is the distance along the chord from the leading edge. (Please note: In this section, we treat a flat plate as an airfoil section; hence, we will use lowercase letters to denote the force coefficients, as first described in Chapter 1.) Substituting Equations (14.8) and (14.9) into (14.10), we obtain

1 2

cn = “(2 sin a)c c

= 2 sin2 a

From the geometry of Figure 14.11, we see that the lift and drag coefficients, defined as ci = L/q^S and q = D/q^S, respectively, where S = (c)(1), are given by

Substituting Equation (14.11) into Equations (14.12) and (14.13), we obtain

ci — 2 sin2 a cos a [14.14]

Cd — 2 sin3 a [14.15]

Finally, from the geometry of Figure 14.11, the lift-to-drag ratio is given by

L

— = cot a [14.16]

[Note: Equation (14.16) is a general result for inviscid supersonic or hypersonic flow over a flat plate. For such flows, the resultant aerodynamic force is the normal force N. From the geometry shown in Figure 14.11, the resultant aerodynamic force makes the angle a with respect to lift, and clearly, from the right triangle between L, D, and N, we have L/D = cot a. Hence, Equation (14.16) is not limited to newtonian theory.]

The aerodynamic characteristics of a flat plate based on newtonian theory are shown in Figure 14.12. Although an infinitely thin flat plate, by itself, is not a practical aerodynamic configuration, its aerodynamic behavior at hypersonic speeds is consistent with some of the basic characteristics of other hypersonic shapes. For example, consider the variation of a shown in Figure 14.12. First, note that, at a small angle of attack, say, in the range of a from 0 to 15°, с/ varies in a nonlinear fashion; that is, the slope of the lift curve is not constant. This is in direct contrast to the subsonic case we studied in Chapters 4 and 5, where the lift coefficient for an airfoil or a finite wing was shown to vary linearly with a at small angles of attack, up to the stalling angle. This is also in contrast with the results from linearized supersonic theory as itemized in Section 12.3, leading to Equation (12.23) where a linear variation of q with a for a flat plate is indicated. However, the nonlinear lift curve shown in Figure 14.12 is totally consistent with the results discussed in Section 11.3, where hypersonic flow was shown to be governed by the nonlinear velocity potential equation, not by the linear equation expressed by Equation (11.18). In that section, we noted that both transonic and hypersonic flow cannot be described by a linear theory—both these flows are inherently nonlinear regimes, even for low angles of attack. Once again, the flat-plate lift curve shown in Figure 14.12 certainly demonstrates the nonlinearity of hypersonic flow.

Also, note from the lift curve in Figure 14.12 that q first increases as a increases, reaches a maximum value at an angle of attack of about 55° (54.7° to be exact), and then decreases, reaching zero at a = 90°. However, the attainment of q max (point A) in Figure 14.12 is not due to any viscous, separated flow phenomenon analogous to that which occurs in subsonic flow. Rather, in Figure 14.12, the attainment of a maximum ci is purely a geometric effect. To understand this better, return to Figure 14.11. Note that, as a increases, Cp continues to increase via the newtonian expression

Cp = 2 sin2 a

That is, Cp reaches a maximum value at a = 90°. In turn, the normal force N shown in Figure 14.11 continues to increase as a increases, also reaching a maximum value at a = 90°. However, recall from Equation (14.12) that the vertical component of

101—

the aerodynamic force, namely, the lift, is given by

L = N cos a [14.17]

Hence, as a increases to 90°, although N continues to increase monotonically, the value of L reaches a maximum value around a = 55°, and then begins to decrease at higher a due to the effect of the cosine variation shown in Equation (14.17)— strictly a geometric effect. In other words, in Figure 14.11, although N is increasing with a, it eventually becomes inclined enough relative to the vertical that its vertical component (lift) begins to decrease gradually. It is interesting to note that a large number of practical hypersonic configurations achieve a maximum CL at an angle of attack in the neighborhood of that shown in Figure 14.12, namely, around 55°.

The maximum lift coefficient for a hypersonic flat plate, and the angle at which it occurs, is easily quantified using newtonian theory. Differentiating Equation (14.14) with respect to a, and setting the derivative equal to zero (for the condition of maxi­mum ci), we have

This is the angle of attack at which q is a maximum. The maximum value of q is obtained by substituting the above result for a into Equation (14.14):

Q. max = 2sin2(54.7°) cos(54.7°) = 0.77

Note, although С/ increases over a wide latitude in the angle of attack (q increases in the range from a = 0 to a = 54.7°), its rate of increase is small (that is, the effective lift slope is small). In turn, the resulting value for the maximum lift coefficient is relatively small—at least in comparison to the much higher q. max values associated with low-speed flows (see Figures 4.20 and 4.22). Returning to Figure 14.12, we now note the precise values associated with the peak of the lift curve (point Л), namely, the peak value of q is 0.77, and it occurs at an angle of attack of 54.7°.

Examining the variation of drag coefficient q/ in Figure 14.12, we note that it monotonically increases from zero at a = 0 to a maximum of 2 at a = 90°. The newtonian result for drag is essentially wave drag at hypersonic speeds because we are dealing with an inviscid flow, hence no friction drag. The variation of cj with a for the low angle of attack in Figure 14.12 is essentially a cubic variation, in contrast to the result from linearized supersonic flow, namely, Equation (12.24), which shows that c, i varies as the square angle of attack. The hypersonic result that q varies as a3 is easily obtained from Equation (14.15), which for small a becomes

[14.18]

The variation of the lift-to-drag ratio as predicted by newtonian theory is also shown in Figure 14.12. The solid curve is the pure newtonian result; it shows that L/D is infinitely large at a = 0 and monotonically decreases to zero at a = 90°. The infinite value of L/ D at a = 0 is purely fictional—it is due to the neglect of skin friction. When skin friction is added to the picture, denoted by the dashed curve in Figure 14.12, L/D reaches a maximum value at a small angle of attack (point В in Figure 14.12) and is equal to zero at a = 0. (At a — 0, no lift is produced, but there is a finite drag due to friction; hence, L/D = 0 at a — 0.)

Let us examine the conditions associated with (L/D)nm more closely. The value of (L/D)mM and the angle of attack at which it occurs (i. e., the coordinates of point В in Figure 14.12) are strictly a function of the zero-lift drag coefficient, denoted by Q o – The zero-lift drag coefficient is simply due to the integrated effect of skin friction over the plate surface at zero angle of attack. At small angles of attack, the skin friction exerted on the plate should be essentially that at zero angle of attack; hence, we can write the total drag coefficient [referring to Eq. (14.15)] as

Furthermore, when a is small, we can write Equations (14.14) and (14.19) as

Q = 2a2 [14.20]

and cd — 2a3 + Cd, o 114.21 ]

Dividing Equation (14.20) by (14.21), we have

ci 2a2

Cd 2 a3 + Cd, о

The conditions associated with maximum lift-to-drag ratio can be found by differen­tiating Equation (14.22) and setting the result equal to zero:

d(ci/cd) _ (:2a3 + Cd, o)4a – 2q;2(6q!2) _ da (2a3 + cd, o)

or 8a4 + 4ac^o — 12a4 = 0

Substituting Equation (14.23) into Equation (14.21), we obtain

Q = 2(c^0)2/3 = 2/3

Q/max 2cd, o + cd, o (Q, o)1/3

Equations (14.23) and (14.24) are important results. They clearly state that the coor­dinates of the maximum L/D point in Figure 14.12, when friction is included (point В in Figure 14.12), are strictly a function of cd, o- In particular, note the expected trend that (L/D)max decreases as cd, о increases—the higher the friction drag, the lower is L/D. Also, the angle of attack at which maximum L/D occurs increases as Cd, o increases. There is yet another interesting aerodynamic condition that holds at (L/D)max, derived as follows. Substituting Equation (14.23) into (14.21), we have

Cd — 2с^о + Cd, о = 3q о

Since the total drag coefficient is the sum of the wave-drag coefficient cd, w and the friction drag coefficient cd, o we can write

Cd ~cd, w+ cd, о [14.26]

However, at the point of maximum L/D (point В in Figure 14.12), we know from Equation (14.25) that cd = 3q. o. Substituting this result into Equation (14.26), we obtain

3Cd,0 — Cd, w З – са, о

This clearly shows that, for the hypersonic flat plate using newtonian theory, at the flight condition associated with maximum lift-to-drag ratio, wave drag is twice the friction drag.

This brings to an end our short discussion of the lift and drag of wings at hyper­sonic speeds as modeled by the newtonian flat-plate problem. The quantitative and qualitative results presented here are reasonable representations of the hypersonic aerodynamic characteristics of a number of practical hypersonic vehicles; the flat – plate problem is simply a straightforward way of demonstrating these characteristics.

Final Comments

This chapter and the previous two have dealt with boundary layers, especially those on a flat plate. We end with the presentation of a photograph in Figure 19.2 showing the development of velocity profiles in the boundary layer over a flat plate. The fluid is water, which flows from left to right. The profiles are made visible by the hydrogen bubble technique, the same used for Figure 16.13. The Reynolds number is low (the freestream velocity is only 0.6 m/s); hence, the boundary-layer thickness is large. However, the thickness of the plate is only 0.5 mm, which means that the boundary layer shown here is on the order of 1 mm thick—still small on an absolute scale. In any event, if you need any further proof of the existence of boundary layers, Figure 19.2 is it.

19.5 Summary

Approximations for the turbulent, incompressible flow over a flat plate are

0.37 x

[19.1]

Re[./5

0.074

[19.2]

Cf = —– Г77

Re,1/5

To account for compressibility effects, the data shown in Figure 19.1 can be used, temperature method can be employed.

or alternatively the reference

Problems

Note: The standard sea level value of viscosity coefficient for air is д = 1.7894 x 10-5 kg/(m • s) = 3.7373 x 1СГ7 slug/(ft • s).

1. The wing on a Piper Cherokee general aviation aircraft is rectangular, with a span of 9.75 m and a chord of 1.6 m. The aircraft is flying at cruising speed (141 mi/h) at sea level. Assume that the skin friction drag on the wing can be approximated by the drag on a flat plate of the same dimensions. Calculate the skin friction drag:

(a) If the flow were completely laminar (which is not the case in real life)

(b) If the flow were completely turbulent (which is more realistic)

Compare the two results.

2. For the case in Problem 19.1, calculate the boundary-layer thickness at the trailing edge for

(a) Completely laminar flow

(b) Completely turbulent flow

3. For the case in Problem 19.1, calculate the skin friction drag accounting for transition. Assume the transition Reynolds number = 5 x 10s.

4. Consider Mach 4 flow at standard sea level conditions over a flat plate of chord 5 in. Assuming all laminar flow and adiabatic wall conditions, calculate the skin friction drag on the plate per unit span.

5. Repeat Problem 19.4 for the case of all turbulent flow.

6. Consider a compressible, laminar boundary layer over a flat plate. Assuming Pr = 1 and a calorically perfect gas, show that the profile of total temperature through the boundary layer is a function of the velocity profile via

To = Tw + (7oe — Tw) — ue

where Tw = wall temperature and To,,, and ue are the total temperature and velocity, respectively, at the outer edge of the boundary layer. [Hint: Compare Equations (18.32) and (18.41).]

7. Consider a high-speed vehicle flying at a standard altitude of 35 km, where the ambient pressure and temperature are 583.59 N/m2 and 246.1 K, respectively. The radius of the spherical nose of the vehicle is 2.54 cm. Assume the Prandtl number for air at these conditions is 0.72, that cp is 1008 joules/(kg K), and that the viscosity coefficient is given by Sutherland’s law. The wall temperature at the nose is 400 K. Assume the recovery factor at the nose is 1.0. Calculate the aero­dynamic heat transfer to the stagnation point for flight velocities of (a) 1500 m/s, and (b) 4500 m/s. From these results, make a comment about how the heat transfer varies with flight velocity.

A Comment on the Location of Minimum Pressure (Maximum Velocity)

Examining the shape of the NACA 0012 airfoil shown at the top of Figure 11.8, note that the maximum thickness occurs at x/c = 0.3. However, examining the pressure coefficient distribution shown at the bottom of Figure 11.8, note that the point of minimum pressure occurs on the surface at xjc = 0.11, considerably ahead of the point of maximum thickness. This is a graphic illustration of the general fact that the point of minimum pressure (hence maximum velocity) does not correspond to the location of maximum thickness of the airfoil. Intuition might at first suggest that, at least for a symmetric airfoil at zero degrees angle of attack, the location of minimum pressure (maximum velocity) on the surface might be at the maximum thickness of the airfoil, but our intuition would be completely wrong. Nature places the maximum velocity at a point which satisfies the physics of the whole flow field, not just what is happening in a local region of the flow. The point of maximum velocity is dictated by the complete shape of the airfoil, not just by the shape in a local region.

We also note that it is implicit in the approximate compressibility corrections discussed in Sections 11.4 and 11.5, and their use for the estimation of the critical Mach number as discussed in Section 11.6, that the point of minimum pressure remains at a fixed location on the body surface as Mx is increased from a very low to a high subsonic value. This is indeed approximately the case. Examine the experimental pressure distributions in Figures 11.8 and 11.10, which are for three different Mach numbers ranging from a low, incompressible value (Figure 11.8) to Мж = 0.725 (Figure 11.10&). Note that in each case the minimum pressure point is at the same approximate location, that is, at x/c = 0.11.

Solutions of Viscous Flows: A. Preliminary Discussion

The governing continuity, momentum, and energy equations for a general unsteady, compressible, viscous, three-dimensional flow are given by Equations (2.52), (15.19a to c), and (15.26), respectively. Examine these equations closely. They are nonlin­ear, coupled, partial differential equations. Moreover, they have additional terms— namely, the viscous terms—in comparison to the analogous equations for an inviscid flow treated in Part 3. Since we have already seen that the nonlinear inviscid flow equations do not lend themselves to a general analytical solution, we can certainly expect the viscous flow equations also not to have any general solutions (at least, at the time of this writing, no general analytical solutions have been found). This leads to the following question: How, then, can we make use of the viscous flow equations in order to obtain some practical results? The answer is much like our approach to the solution of inviscid flows. We have the following options:

1. There are a few viscous flow problems which, by their physical and geometrical nature, allow many terms in the Navier-Stokes solutions to be precisely zero, with the resulting equations being simple enough to solve, either analytically or by simple numerical methods. Sometimes this class of solutions is called “exact solutions” of the Navier-Stokes equations, because no simplifying approxima­tions are made to reduce the equations—just precise conditions are applied to reduce the equations. Chapter 16 is devoted to this class of solutions; examples are Couette flow and Poiseuille flow (to be defined later).

2. We can simplify the equations by treating certain classes of physical problems for which some terms in the viscous flow equations are small and can be neglected. This is an approximation, not a precise condition. The boundary-layer equations developed and discussed in Chapter 17 are a case in point. However, as we will see, the boundary-layer equations may be simpler than the full viscous flow equations, but they are still nonlinear.

3. We can tackle the solution of the full viscous flow equations by modem numerical techniques. For example, some of the computational fluid dynamic algorithms discussed in Chapter 13 in conjunction with “exact” solutions for the inviscid flow equations carry over to exact solutions for the viscous flow equations. These matters will be discussed in Chapter 20.

There are some inherent very important differences between the analysis of vis­cous flows and the study of inviscid flows that were presented in Parts 2 and 3. The remainder of this section highlights these differences.

First, we have already demonstrated in Example 2.4 that viscous flows are ro­tational flows. Therefore, a velocity potential cannot be defined for a viscous flow, thus losing the attendant advantages that were discussed in Sections 2.15 and 11.2. On the other hand, a stream function can be defined, because the stream function satisfies the continuity equation and has nothing to do with the flow being rotational or irrotational (see Section 2.14).

Second, the boundary condition at a solid surface for a viscous flow is the no­slip condition. Due to the presence of friction between the surface material and the adjacent layer of fluid, the fluid velocity right at the surface is zero. This no-slip condition was discussed in Section 15.2. For example, if the surface is located at у = 0 in a cartesian coordinate system, then the no-slip boundary condition on velocity is

At у = 0; и = 0 u = 0 w = 0

This is in contrast to the analogous boundary condition for an inviscid flow, namely, the flow-tangency condition at a surface as discussed in Section 3.7, where only the component of the velocity normal to the surface is zero. Also, recall that for an inviscid flow, there is no boundary condition on the temperature; the temperature of the gas adjacent to a solid surface in an inviscid flow is governed by the physics of the flow field and has no connection whatsoever with the actual wall temperature. However, for a viscous flow, the mechanism of thermal conduction ensures that the temperature of the fluid immediately adjacent to the surface is the same as the temperature of

the material surface. In this respect, the no-slip condition is more general than that applied to the velocity; in addition to и = v = 0 at the wall, we also have T = Tw at the wall, where T is the gas temperature immediately adjacent to the wall and Tw is the temperature of the surface material. Thus,

Aty= 0: T = TW [15.34]

Tn many problems, Tw is specified and held constant; this boundary condition is easily applied. However, consider the following, more general case. Imagine a viscous flow over a surface where heat is being transferred from the gas to the surface, or vice versa. Also, assume that the surface is at a certain temperature, Tw, when the flow first starts, but that Tw changes as a function of time as the surface is either heated or cooled by the flow [i. e., Tw = Tw(t)]. Because this timewise variation is dictated in part by the flow which is being calculated, Tw becomes an unknown in the problem and must be calculated along with the solution of the viscous flow. For this general case, the boundary condition at the surface is obtained from Equation (15.2) applied at the wall; that is,

Aty= 0: qw = -(kd-^j [15.35]

Here, the surface material is responding to the heat transfer to the wall qw, hence changing Tw, which in turn affects qw. This general, unsteady heat transfer problem must be solved by treating the viscous flow and the thermal response of the material simultaneously. This problem is beyond the scope of the present book.

Finally, let us imagine the above, unsteady case carried out to the limit of large times. That is, imagine a wind-tunnel model which is at room temperature suddenly inserted in a supersonic or hypersonic stream. At early times, say, for the first few sec­onds, the surface temperature remains relatively cool, and the assumption of constant wall temperature Tw is reasonable [Equation (15.34)]. However, due to the heat trans­fer to the model [Equation (15.35)], the surface temperature soon starts to increase and becomes a function of time, as discussed in the previous paragraph. However, as Tw increases, the heating rate decreases. Finally, at large times, Tw increases to a high enough value that the net heat transfer rate to the surface becomes zero, that is, from Equation (15.35),

q<u

or (— =0 [15.36]

Ь К

When the situation of zero heat transfer is achieved, a state of equilibrium exists; the wall temperature at which this occurs is, by definition, the equilibrium wall tempera­ture, or, as it is more commonly denoted, the adiabatic wall temperature, Taw. Hence, for the case of an adiabatic wall (no heat transfer), the wall boundary condition is given by Equation (15.36).

In summary, for the wall boundary condition associated with the solution of the energy equation [Equation (15.26)], we have three possible cases:

1. Constant temperature wall, where Tw is a specified constant [Equation (15.34)]. For this given wall temperature, the temperature gradient at the wall (dT/dy)w is obtained as part of the flow-field solution and allows the direct calculation of the aerodynamic heating to the wall via Equation (15.35).

2. The general, unsteady case, where the heat transfer to the wall qw causes the wall temperature Tw to change, which in turn causes qw to change. Here, both Tw and (ЗT/3 v )„ change as a function of time, and the problem must be solved by treating jointly the viscous flow as well as the thermal response of the wall mate­rial (which usually implies a separate thermal conduction heat transfer numerical analysis).

3. The adiabatic wall case (zero heat transfer), where (дТ/dy)w = 0 [Equa­tion)^.36)]. Here, the boundary condition is applied to the temperature gradient at the wall, not to the wall temperature itself. Indeed, the wall temperature for this case is defined as the adiabatic wall temperature Taw and is obtained as part of the flow-field solution.

Finally, we emphasize again that, from the point of view of applied aerodynamics, the practical results obtained from a viscous flow analysis are the skin friction and heat transfer at the surface. However, to obtain these quantities, we usually need a complete solution of the viscous flow field; among the data obtained from such a solution are the velocity and temperature gradients at the wall. These, in turn, allow the direct calculation of rw and qw from

and

Another practical result provided by a viscous flow analysis is the prediction and calculation of flow separation; we have discussed numerous cases in the preceding chapters where the pressure field around an aerodynamic body can be greatly changed by flow separation; the flows over cylinders and spheres (see Sections 3.18 and 6.6) are cases in point.

Clearly, the study of viscous flow is important within the entire scope of aero­dynamics. The purpose of the following chapters is to provide an introduction to such flows. We will organize our study following the three options itemized at the beginning of this section; that is, we will treat, in turn, certain specialized “exact” solutions of the Navier-Stokes equations, boundary-layer solutions, and then “exact” numerical solutions of Navier-Stokes equations. In so doing, we hope that the reader will gain an overall, introductory picture of the whole area of viscous flow. Entire books have been written on this subject, see, for example, References 42 and 43. We cannot possibly present such detail here; rather, our objective is simply to provide a “feel” for and a basic understanding of the material. Let us proceed.

Problems

1. Consider the incompressible viscous flow of air between two infinitely long parallel plates separated by a distance h. The bottom plate is stationary, and the top plate is moving at the constant velocity ue in the direction of the plate. Assume that no pressure gradient exists in the flow direction.

(a) Obtain an expression for the variation of velocity between the plates.

(b) If T = constant = 320 K, ue = 30 m/s, and h = 0.01 m, calculate the shear stress on the top and bottom plates.

2. Assume that the two parallel plates in Problem 15.1 are both stationary but that a constant pressure gradient exists in the flow direction (i. e., dp/dx = constant).

(a) Obtain an expression for the variation of velocity between the plates.

(b) Obtain an expression for the shear stress on the plates in terms of dp/dx.

Time-Dependent Finite-Difference Method

Return to the picture of Couette flow in Figure 16.2. Imagine, for a moment, that the space between the upper and lower plates is filled with a flow field which is not a Couette flow; for example, imagine some arbitrary flow field with gradients in both the x and у directions, including gradients in pressure. We can imagine such a flow existing at some instant during the start-up process just after the upper plate is set into motion. This would be a transient flow field, where и, T, p, etc., would be functions of time t as well as of r and y. Finally, after enough time elapses, the flow will approach a steady state, and this steady state will be the Couette flow solution discussed above. Let us track this picture numerically. That is, starting from an assumed initial flow field at time t = 0, let us solve the unsteady Navier-Stokes equations in steps of

time until a steady flow is obtained at large times. As discussed in Section 13.5, the time-asymptotic steady flow is the desired result; the time-dependent approach is just a means to that end. At this stage in our discussion, it would be well for you to review the philosophy (not the details) presented in Section 13.5 before progressing further.

The Navier-Stokes equations are given by Equations (15.18a to c) and (15.26). For an unsteady, two-dimensional flow, they are Continuity:

dp d(pu) d(pv)

dt dx dy

x momentum:

du

~dt

у momentum:

dv

~dt

d{uTxx) d(uryx) d(vrxy) d(vzyy) |

dx dy dx dy J

Note that Equations (16.66) to (16.69) are written with the time derivatives on the left-hand side and spatial derivatives on the right-hand side. These are analogous to the form of the Euler equations given by Equations (13.59) to (13.62). In Equa­tions (16.67) to (16.69), rxy, rxx, and xyy are given by Equations (15.5), (15.8), and (15.9), respectively.

The above equations can be solved by means of MacCormack’s method as de­scribed in Chapter 13. This is a predictor-corrector approach, and its arrangement for the time-dependent method is described in Section 13.5. The application to com­pressible Couette flow is outlined as follows:

1. Divide the space between the two plates into a finite-difference grid, as sketched in Figure 16.8a. The length L of the grid is somewhat arbitrary, but it must be longer than a certain minimum, to be described shortly.

2. At x — 0 (the inflow boundary), specify some inflow conditions for u, v, p, and T (hence, e, since e = cvT). The incompressible solution for Couette flow makes reasonable inflow boundary conditions.

3. At all the remaining grid points, arbitrarily assign values for all the flow-field variables, u, v, p, and T. This arbitrary flow field, which constitutes the initial conditions at t = 0, can have finite values of v, and can include pressure gradients.

4. Starting with the initial flow field established in step 3, solve Equations (16.66) to

(16.69) in steps of time. For example, consider the x-momentum equation in the form of Equation (16.67). MacCormack’s predictor-corrector method, applied to this equation, is as follows.

Predictor: Assume that we know the complete flow field at time t, and we wish to advance the flow-field variables to time / + Д/. Replace the spatial derivatives with forward differences:

(Ljx)/,_/+1 (j-yx)i. j

Ду

All the quantities on the right-hand side are known at time t; we want to advance the flow-field values to the next time, t + At. That is, the right-hand side of Equation (16.70) is a known number at time t. Form the predicted value of u, j at time t + At, denoted by m; j from the first two terms of a Taylor’s series as

Calculate predicted values for p, v, and e, namely, /),. /, v,,;, and e, ,, by the same approach applied to Equations (16.66), (16.68), and (16.69), respectively. Do this for all the grid points in Figure 16.8a.

Corrector: Return to Equation (16.67), and replace the spatial derivatives with rearward differences using the predicted (barred) quantities obtained from the predictor step:

(tyx)i, j (tyx)i. j— 1

~Ay

Finally, calculate the corrected value of m,,/ at time t + At, denoted by from the first two terms of a Taylor’s series using an average time derivative

obtained from Equations (16.70) and (16.72). That is,

[16.73]

Carry out the same process using Equations (16.66), (16.68), and (16.69) to obtain p^At, и’+л’, and е’+л’. The complete flow field at time t + At is now obtained.

5. Repeat step 4, except starting with the newly calculated flow-field variables at the previous time. The flow-field variables will change from one time step to the next. This transient flow field will not even have parallel streamlines; i. e., there will be finite values of v throughout the flow. This is sketched in Figure 16.86. Make the calculations for a large number of time steps; as we go out to large times, the changes in the flow-field variables from one time step to another will become smaller. Finally, if we go out to a large enough time (hundreds, sometimes even thousands, of time steps in some problems), the flow-field variables will not change anymore—a steady flow will be achieved, as sketched in Figure 16.8c. Moving from left to right in Figure 16.8c, we see a developing flow near the entrance, influenced by the assumed inflow profile. However, at the right of Figure 16.8c, the history of the inflow has died out, and the flow-field profiles become independent of distance. Indeed, we have chosen L to be a sufficient length for this to occur. The flow field near the exit is the desired solution to the compressible Couette flow problem.

The value of At in Equations (16.71) and (16.73) is not arbitrary. The steps outlined above constitute an explicit finite-difference method, and hence there is a stability bound on At. The value of At must be less than some prescribed maximum, or else the numerical solution will become unstable and “blow up” on the computer. A useful expression for At is the Courant-Friedrichs-Lewy (CFL) criterion, which states that At should be the minimum of Atx and Aty, where

In Equation (16.74), a is the local speed of sound. Equation (16.74) is evaluated at every grid point, and the minimum value is used to advance the whole flow field.

The time-dependent technique described above is a common approach to the solution of the compressible Navier-Stokes equations, and for that reason, it has been outlined here. Our purpose has been not so much to outline the solution of Couette flow by means of this technique, but rather to present the technique as a precursor to our later discussions on Navier-Stokes solutions.

(b) Transient flow

Illustration of the finite-difference grid, and characteristics of the flow during its transient approach to the steady state.